Analysis of Distance Measures for Detection of Manoeuvres for Satellites in Low Orbits with Radar Data

Proyecto Fin de Carrera Ingeniería de Telecomunicación Formato de Publicación de la Escuela Técnica Superior de Ingeniería Autor: F. Javier Payán Somet Tutor: Juan José Murillo Fuentes Dep. Teoría de la Señal y Comunicaciones Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2013 Ingeniería Aeroespacial ESI - Universidad de Sevilla Trabajo Fin de Máster Ingeniería Aeroespacial Analysis of Distance Measures for Detection of Manoeuvres for Satellites in Low Orbits with Radar Data Autora: Ana Sánchez Rivero Tutor: Rafael Vázquez Valenzuela Dpto. Ingeniería Aeroespacial y Mecánica de Fluidos Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2021 Trabajo Fin de Máster Ingeniería Aeroespacial Analysis of Distance Measures for Detection of Manoeuvres for Satellites in Low Orbits with Radar Data Autora: Ana Sánchez Rivero Tutor: Rafael Vázquez Valenzuela Profesor Titular Dpto. Ingeniería Aeroespacial y Mecánica de Fluidos Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2021 Trabajo Fin de Máster: Analysis of Distance Measures for Detection of Manoeuvres for Satellites in Low Orbits with Radar Data Autora: Ana Sánchez Rivero Tutor: Rafael Vázquez Valenzuela El tribunal nombrado para juzgar el trabajo arriba indicado, compuesto por los siguientes profesores: Presidente: Vocal/es: Secretario: acuerdan otorgarle la calificación de: El Secretario del Tribunal Fecha: Resumen Título: Análisis de Medidas de Distancia para la Detección de Maniobras para Satélites en Órbitas Bajas con Datos de Radar. Resumen del trabajo: Uno de los principales problemas del sector espacial actualmente es la gran población de objetos y basura espacial que se encuentra orbitando la Tierra, y que se prevee que continue creciendo en el futuro. A causa de ello, la detección de maniobras se ha convertido en un tema de actualidad en el sector; ya que permite mejorar la conciencia situacional reduciendo el número de objetos no correlacionados. Este trabajo se centra en un algoritmo de detección de maniobras en órbita baja usando datos de radar. La idea principal del mismo es la comparación, mediante una distancia estadística, de los datos esperados según los parámetros iniciales de la órbita y los datos obtenidos por el radar en la siguiente pasada. El objetivo del trabajo es caracterizar la dependencia de los resultados en función de los diferentes parámetros de los que depende el algoritmo, destacando entre ellos, la distancia estadística empleada. Palabras clave: Detección de Maniobras, Divergencia de Kullback-Leibler y Distancia de Bhattacharyya. Conclusiones: Las tres distancias empleadas en el proyecto obtienen buenos resultados en la detección de maniobras con datos sintéticos. Con todas ellas se obtiene un elevado valor del F-score, especialmente con la distancia de Bhattacharyya, que resulta ser también la que ofrece mejores resultados en el caso de escenarios con datos reales. El desempeño del algoritmo se reduce considerablemente en estos últimos casos. En concreto, utilizando la divergencia de Kulback-Leibler no se logra detectar prácticamente ninguna maniobra. Por otra parte, se ha comprobado que, así como la órbita afecta de forma considerable en los resultados, los parámetros de propagación parecen no impactar en la detección de maniobras. El otro aspecto importante tratado a lo largo del proyecto es la propagación de la incertidumbre. La elección de este método repercute tanto en los resultados como en el tiempo de computación. Se ha detectado que el orden de Taylor aumenta considerablemente este tiempo, sin embargo, no incrementa el número de maniobras detectadas ni disminuye el número de falsos positivos. Concluyendo así, que establecer el segundo orden de Taylor es la mejor opción. Por otra parte, el método de unscented transform consigue reducir el tiempo de computación en un 35% con respecto al álgebra diferencial de Taylor y Montecarlo, sin detrimento de los resultados. I Abstract Nowadays , one of the main problems in the space sector is the amount of space objects and debris in orbit about Earth, which will increase continually in the future. This is the reason why manoeuvres detection has become a topic of broad and current interest, because it can enable to improve the space situational awareness by reducing the number of uncorrelated targets detected. This project focuses on an algorithm for manoeuvres detection for satellites in Low Earth Orbit from radar data. The main idea of this algorithm is the comparison, using a statistical distance, between the predicted orbit from the initial data and those obtain from the radar in the next track. The aim of the project is to characterize the impact of different parameters of the algorithm on the results, highlighting among them, the statistical distance used. III 2Chapter 1. Introduction on the algorithm 1 presented in [ 3 ], a method for the detection of manoeuvres in Low Earth Orbit from radar data. This algorithm is applicable to objects whose orbit is known. The main idea is to compare the predicted position of the satellite, which is computed by the propagation of its orbit, with the next track obtained from the radar. This comparison is carried out with the Mahalanobis distance. However, there are some other metrics used to determine whether an object is correlated or not with a previously tracked object. The Kullbak-Leibler divergence and the Bhattacharyya distance are some of them [4]. This algorithm is applied to synthetic data, but also to real data. Those are provided from S3TSR, the Spanish surveillance radar developed by Indra with the funding of the Spanish Government and tecnically managed by ESA. It is a ground-based radar capable of providing positional information of orbital object. It provides surveillance and tracking of space objects in Low Earth Orbit, from 200 km to 2000 km of orbit height above Earth [5]. 1.1 Scope of the project The final aim of this project is to use radar data to detect possible manoeuvres of known objects in Low Earth Orbits; the idea is to compute a probability of manoeuvre occurrence. The project is based on the algorithm developed in [ 3 ]. One of the main pieces of this algorithm is the measurement of the distance between the propagation of the orbit and radar data. It is here where the objective of this project lies. This work seeks to determine which statistical distance (the original one, the Mahalanobis distance, or the other two distances aforementioned, the Kullback-Leibler divergence and the Bhattacharyya distance) offers better results, i.e., detects a greater number of manoeuvres and avoids a higher number of false positives. The algorithm depends on other parameters, both satellite scenario data and specific aspects of the algorithm. Some of the parameters that define the scenario are the orbit or the physical characteristics of the satellite, its mass or its ballistic coefficient. On the other hand, the computation of the covariance matrix is an aspect of the algorithm that needs to be considered. This project also pretends to characterize the impact of these parameters on the detection of manoeuvres. 1.2 Document structure The organization of the document and a brief description of the content of each chapter are presented below: • Chapter 1: Introduction. It is the current chapter. It contains an explanation of the context in which this project is located. The problem of the increasing population of space objects and orbital debris is introduced. The chapter also establishes the scope of the project. • Chapter 2: Preliminary concepts. It is contains a review of the theoretical background of the project, including orbital mechanics and basic concepts of statistics needed to work with the statistical distances. Some tools needed on the project are also introduced in this chapter. • Chapter 3: Algorithm for manoeuvre detection. Here, the description of the algorithm for manoeuvre detection is laid out. In addition, the Theory of attributables and the Theory of distances, which are employed by the algorithm, are also presented, as well as the concept of F-score. • Chapter 4: Results. In this chapter, the results of applying the algorithms to a number of synthetic and real scenarios are presented, as well as the analyses of the distances and the impact of the parameters . • Chapter 5: Conclusions and future work. This final chapter closes the document with the conclusions obtained from the project. In addition, possible future considerations are mentioned in order to improve the study. 1.2 Document structure 3 • Appendix A: Distances and probabilities table. This appendix contains the values of the distances and the probabilities of manoeuvre occurence of all the analyses performed. •Bibliography. It contains the reference documents used through the report. 2 Preliminary concepts In this chapter, an introduction of the theoretical background of the project is posed, together with the explanation of some tools used for the development. It is structured as follows: Section 2.1 contains a brief explanation of orbital mechanics. Basic statistics concepts are introduced in Section 2.2. Sections 2.3 and 2.4 contains a brief introduccion of OREKIT and Distribution Fitter MATLAB App, respectively. In Section 2.5 Taylor differential algebra is explained and the unscented transform is described in Section 2.6. 2.1 Orbital mechanics Thi section contains the introduction to orbital mechanics, in order to define the orbits and positions of the satellites detected by the radar, taking also into account the considered orbital perturbations. 2.1.1 Reference Frames First of all, different reference systems used in the project will be defined. Depending on the situation, one reference frame or another may be better. For instance, radar data is always given in its corresponding topocentric frame, however, the propagation algorithms usually need to be expressed in an inertial frame. Earth-centred inertial frame The Earth-centred inertial frame (ECI) has its origin at the Earth’s centre of mass. XY-plane contains the Equator, X-axis points in the direction of the Vernal equinox (  point) and Z-axis is orthogonal to the former ones, so the system is directly oriented. Moreover, Z-axis points towards the celestial north pole. Figure 2.1 Earth-centred inertial frame [6]. 5 6Chapter 2. Preliminary concepts The modulus of the position vector and two angles are necessary to describe the position of a certain point on this frame: • Right Ascension ( α ): it is the angle between X-axis and the projection of the position vector on the equatorial plane. •Declination (δ): it is the angle between the vector position and the equatorial plane, XY-plane. Earth-centred Earth-fixed frame The Earth-centred Earth-fixed frame (ECEF), known as well as Geographic equatorial frame, also has its origin at the Earth’s centre of mass and its XY-plane contains the Equator. However, the X-axis passes through the intersection between the Equator and the Greenwich meridian, thus, it rotates with the Earth. The following variables are used to define the position of a satellite on this frame: •The modulus of the position vector. • Longitude ( λ ): angle between X-axis and the normal plane to the Equator which contains the satellite. •Latitude (φ): angle between the vector position and the equatorial plane, XY-plane. Figure 2.2 Earth-centred Earth-fixed frame (g) and topocentric frame (w) [7]. In the project, conversion between this frame and the Inertial ECI frame is provided by OREKIT (a tool explained in Section 2.3). It takes into account the effects of precession and nutation, as well as the poles’ wandering displacement with precision, and the ellipsoidel shape of the WGS84 Earth. Topocentric frame As it was said at the beginning of this section, the radar data are always expressed on the topocentric frame corresponding to it. The origin of the frame is placed on the position of the radar. The XY-plane is tangent to the Earth’s surface, taking into account that, in this project, it has been approximated by the shape of the WGS84 Earth. X-axis points to the East and Y-axis to the North. Z-axis is orthogonal to the tangent plane and points up, to the zenith. This frame is also represents on Figure 2.2, with the suffix w. The three coordinates that define the position on this frame are the following: •The modulus of the position vector. • Azimuth (Az): angle between North and the projection of the position vector on the XY-plane, positive clockwise. •Elevation (El): angle between XY-plane and the position vector. Other important concepts are the variation of these coordinates over time: •Range-rate ( ˙ ρ): is the radial component of the velocity. •Azimuth-rate ( ˙ Az): is the time derivate function of the azimuth angle. •Elevation-rate ( ˙ El): is the time derivate function of the elevation angle. 2.1 Orbital mechanics 7 Local-vertical local-horizontal frame Local-vertical local-horizontal frame (LVLH) is placed on the target point, in this project, on the satelite. Z-axis points down, to the Earth centre, nadir. Y-axis is orthogonal to the orbit plane and X-axis completes the reference frame. This frame is usually employed to define manoeuvres. Figure 2.3 Local-vertical local-horizontal frame [8]. 2.1.2 Two-body problem The aim of the two-body problem is to determine the motion of two massive objects in the space. It is assumed that the system is isolated from the rest of the Universe, only their own mutual gravitational attraction force is taken into account. The masses are considered punctual and located in their centers of mass or equivalently sphere with density depending only on the radius. Equation of motion As it has been said, the gravitational force is the main element of the orbital motion of bodies. This force describes the attraction between two bodies by the simple fact of having mass. It is directly proportional to each of their masses, m1and m2, and inversely proportional to the square of the distance between them, r. ~ F1=G·m1·m2 r2·~r r;~ F2=−G·m1·m2 r2·~r r(2.1) Where G = 6.67 ·10−11 Nm 2 /kg 2 is the gravitational constant and ~r is the vector going from the first body to the second one,~r=~ R2−~ R1, as it is shown in Figure 2.4. Thus, the equation of motion of each body with respect to an inertial reference frame defined by Newton’s second law is the Equation 2.2. m1¨ ~ R1=G·m1·m2 r2·~r r,m2¨ ~ R2=−G·m1·m2 r2·~r r(2.2) 8Chapter 2. Preliminary concepts Figure 2.4 Two masses located in an inertial frame [6]. The coordinate origin of the reference frame can be translated to the center of mass of the system, and the frame will remain inertial. Equations 2.3 are obtained by redefining the position vectors of both bodies. ~r1=~ R1−~ RG=−~r·m2 m1+m2 ~r2=~ R2−~ RG=~r·m1 m1+m2 (2.3) If the mass of one of the bodies is much greater than that of the other, m 1 » m 2 , it can be assumed that the center of mass of the system coincides with that of the most massive body. Approximating the problem to that of the relative motion of one body around a massive one. ¨ ~r=¨ ~ R2−¨ ~ R1=−Gm1 r2 ~r r−Gm2 r2 ~r r=−G(m1+m2)~r r3(2.4) Whether the gravitational parameter, µ, is defined by Equation 2.5, µ=G(m1+m2)(2.5) the equation of motion can be written as Equation 2.6. ¨ ~r=−µ~r r3(2.6) As with the center of mass of the system, if one body is more massive than another, the gravitational parameter is practically function of the greater mass only, µ≃µ1 . Taking into account that the two bodies involved in this project are the Earth and the satellites, due to the difference of masses, M ⊕>> m sat , the gravitational parameter is defined by the Earth mass (as expressed in Equation 2.7). µ≃GM⊕=3.986004 ×1014 m3 s2(2.7) Angular momentum and orbit equation Defining the angular momentum of the orbit with Equation 2.8, which is constant (see [ 6 ] for the demonstration), and taking the cross product of the equation of motion with ~ h , the orbit equation (which is the Equation 2.9) can be determined . It defines the path of the orbiting body, the satellite, around the massive body, planet Earth, relative to the second one. ~ h=~r×˙ ~r(2.8) r=h2 ν 1 1+ecosθ(2.9) 2.1 Orbital mechanics 9 The orbit equation represents the parametric equation of a conic curve, where eis the eccentricity, related to the type of conic curve and θ is the true anomaly, which is the angle between vector ~r , position of the object, and position vector of the periapse (perigee on Earth case). Position as a function of time The definition of the variation of the position in time depends on the conic curve type. The satellite orbits around Earth are for the most part elliptical, so this is the only case explained in this project (see [ 6 ] for parabolic and hyperbolic orbits). First, some previous concepts are established. • Mean motion: average angular velocity in elliptical orbit. This velocity is equivalent to real angular velocity in circular orbits, n=2π T=rµ a3(2.10) where Tis the period and athe semimajor axis • Mean anomaly: angle between the position of a fictitious body moving around the ellipse at the constant angular speed, n. For a circular orbit, the mean anomaly, M, and the true anomaly, θ , are identical, for elliptical orbit, this is true solely in periapsis and apoapsis. M=n∆t(2.11) • Eccentric anomaly: it is the angle Eshown in Figure 2.5. Drawing the circumscribed circumference to the ellipse and a perpendicular to the mayor axis through point S (the location of the object defined by the true anomaly) is determined point Q. The angle between the apse line and the radius of the circle that passes through Q is the eccentric anomaly, E. Figure 2.5 Eccentric anomaly definition [6]. The eccentric anomaly and the true anomaly are related by Equation 2.12. tan θ 2=r1+e 1−etan E 2(2.12) 10 Chapter 2. Preliminary concepts Now, by means of Kepler’s second law, the relationship between time increment and the eccentric anomaly can be established, as expressed in Equation 2.13. ∆t=E−esinE n(2.13) Finally, combining Equation 2.12 and 2.13 and the definition of mean anomaly results in Equation 2.14. called Kepler equation relates the true anomaly, θ, with the mean anomaly, M: M=E−esinE(2.14) 2.1.3 Orbital elements Orbital elements is the minimum data set, together with the epoch, that allows to define the position of a body in an orbit at a certain time. Two parameters are necessary to define the orbit in the plane, the semimajor axis, a, and the eccentricity, e. Sometimes acan be replaced by the angular momentum, h, the period, T, or the average angular velocity, n (the last two can only be used if the orbit is elliptical). Then, a third parameter is required to define a specific point on the orbit. This parameter is the true anomaly, which is the angle between the perigee and the body position. It can be replaced by the mean anomaly, M, the eccentric anomaly, Eor the increment of time, ∆t . Three additional parameters are requiered in order to describe the orientation of a plane in three dimensions, known as Euler angles. The intersection between the orbit plane and the reference plane is known as the node line; for terrestrial orbits, the reference is the equatorial plane. The intersection between the orbit and this line are two points. The one where the orbit crossed the equatorial plane from below to above is called the ascending node,  , and the other one the descending node,  . The node line vector, ~n , extends outwards from the focus through the ascending node. Thus, the angle between the positive X-axis and the node line vector is the right ascension of the ascending node,Ω, positive counterclockwise. The eccentricity vector, ~e , intersects the orbit at perigee. Then, the argument of perigee, ω , is required to determine the orientation of the orbit. This is the angle between the node line vector, ~n , and the eccentricity vector,~e, and it is positive in the movement direction. Figure 2.6 Keplerian orbital elements [6]. 2.1 Orbital mechanics 11 Finally, the angle between the reference plane, equatorial plane, and the orbit plane is the inclination, i, positive in the direction indicated by~n. In summary, six orbital elements are required to define the position of a body in a certain orbit (Figure 2.6 shows all of them). The Keplerian elements are: •a: semimajor axis. •e: eccentricity. •θ: true anomaly. •Ω: right ascension of the ascending node. •ω: argument of perigee. •i: inclination. There are special orbits where some Keplerian elements are not well-defined. • Elliptical equatorial orbit: inclination is equal to 0 so the orbit is contained in the equatorial plane, i.e. there is no node line. Thus, the argument of perigee and the right ascension of the ascending node are not well-defined. • Circular non-equatorial orbit: there is no perigee, consequently, the argument of perigee and the true anomaly are not well-defined. • Circular equatorial orbit: there is neither perigee nor line of nodes. Therefore, neither right ascension of the ascending node, nor argumen of perigee, nor true anomaly are well-defined. There are other representations of orbital elements which removes these singularities. In particular, in this project the Equinoctial element set is used, [ 9 ], which removes the mathematical singularities at e = 0 and i = 0 and 90 degrees. Therefore, it is especially well adapted to orbits with small eccentricity and small inclination. •a: semimajor axis •ey =e·sin(Ω+ω) •ex =e·cos(Ω+ω) •hy =tani 2·sin(Ω) •hx =tani 2·cos(Ω) •lv =θ+ω+Ω 2.1.4 Orbital perturbations The two-body problem is not realistic. One of their main assumptions is to consider the system isolated from the rest of the Universe, which allows the forces to be reduced only to the gravitational force between the two bodies. Nevertheless, there are many other forces acting on the system, for instance, gravitational force exerted by other bodies, such as the Sun or the Moon, atmospheric drag, quite significant in Low Earth Orbits, or solar radiation pressure. The other essential assumption is to consider the masses punctual and located in the center of mass. This statement is valid just if the bodies are homogeneous solid spheres. Typically, neither planets nor satellites are perfect spheres. All these factors are called perturbations and it is necessary to analyze them in order to get a precise position of the satellites. 18 Chapter 2. Preliminary concepts Different distributions are used in the project; both to characterize the errors and to compute the probabilities from the statistical distances. These distributions are defined below. Definition 2.4 The normal distribution is a type of continuous probability distribution for a real-value random variable. Its pdf and its cdf are given by Equations 2.32. Where µ is the mean value, σ is the standard deviation and er f is the related error function, which is defined by Equation 2.33. f(x) = 1 σ√2πe−1 2(x−µ σ)2 F(x) = 1 21+er f x−µ σ√2 (2.32) er f (x) = 2 √πZx 0 e−t2dt (2.33) This distribution plays an important role in classic statistics due to the Central Limit Theorem, which is explained in Theorem 2.1. Furthermore, an important property of this distribution is that the sum of normal distributions is also a normal distribution. That means, whether X∼N(µx,σ2 x) , Y∼N(µy,σ2 y) and they are independents, then Z=X+Yis distributed as a normal; Z∼N(µx+µy,σ2 x+σ2 y). Theorem 2.1 The central limit theorem establishes that the sum of random variables, with any kind of distribution, tends towards a normal distribution. Definition 2.5 The gamma distribution is a two-parameter family of continuous probability distributions. Its pdf and its cdf are given by Equations 2.34. Where α is the shape parameter, β is the scale parameter, Γ is the gamma function and γ is the lower incomplete gamma function. These two functions are defined by Equations 2.35. f(x) = xα−1 βαΓ(α)e−x βF(x) = γ(α,βx) Γ(α)(2.34) Γ(α) = Z∞ 0 tα−1e−tdt γ(α,βx) = Zβx 0 tα−1e−tdt (2.35) Definition 2.6 The chi-squared distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. It is a special case of the gamma distribution with α=k 2and β=2. Definition 2.7 The exponential distribution is a particular case of the gamma distribution, with α=1 and β=1 λ . λ is the parameter of the distribution, called rate parameter. Its cdf is defined by Equation 2.36. F(x) = 1−e−λx(2.36) In addition, the variable of this project is not one-dimentional. Thus, multidimention random variables are also introduced. Moreover, since the large-scale errors come from the sum and accumulation of many small-scale errors, the central limit theorem justifies the use of the normal distribution as error model. Hence, the multivariate normal distribution is also explained. Definition 2.8 Let ~ X∈ℜn be a multivariate continium random variable, each component of it follows an unidimensional distribution. As in one-dimensional case, a joint distribution function is defined, which is computed by the density function. The two important parameters are, the mean, which is defined by Equation 2.37, and the covariance matrix, that is characterized by Equation 2.38. The covariance matrix is always symmetric and positive semi-definite. The diagonal values represent the variance of each component of ~ X . The rest of the matrix elements are the correlations between two components of ~ X ; the component ij of the covariance matrix is the correlation 2.3 OREKIT 19 between the components Xiand Xj. ~ µ(~ X) = Eh~ Xi=Zℜn~y f (~y)dy (2.37) Σ(~ X) = E~ X−µ(~ X)~ X−µ(~ X)T(2.38) Definition 2.9 The multivariate normal distribution is expressed as ~ X∼Nn(~ µ,Σ) and its density function is defined by Equation 2.39. f(~x) = 1 Det(Σ)(2π)n/2e(−1 2(~x−~ µ)TΣ−1(~x−~ µ))(2.39) Finally, some additional important concepts for the error model are introduced [14], [15]. Definition 2.10 Astochastic or random process is a time-dependant random variable, ~ X(t) . Thus, the mean and the covariance also depend on time, ~ µ(t),Σ(t). The autocorrelation for a process is defined by Equation 2.40. It shows how much the past history affects its current value. Whether a stochastic process is distributed following a multivariate normal, it is called gaussian process. R(t,τ) = Eh~ X(t)~ X(τ)Ti(2.40) Definition 2.11 A process,~ ν(t)is called white noise if it verifies the following characteristics: •E[~ ν(t)] =~ 0. •E~ ν(t)~ ν(t)T=σ2Id. •R(t,τ) = E~ ν(t)~ ν(τ)T=δ(t−τ)σ2Id, where δ(x) = 1if x = 0 and δ(x) = 0otherwise. The last condition means that a white noise at a certain instant is independet of its value at any previous instant. Finally, a gaussian white noise is a process which satisfies the previous conditions and is also a gaussian process. 2.3 OREKIT The language chosen for the development of the project has been MATLAB. Moreover, an important tool for this has been OREKIT (ORbit Extrapolation KIT) [ 16 ], a low level space dynamics library written in Java which can interfaced with MATLAB. It aims at providing accurate and efficient low level components for the development of flight dynamics applications. OREKIT provides basic elements, like orbits, dates or frames; and various algorithms to handle them, such as conversions between different frames or analytical and numerical propagation. OREKIT has been developed by the CS GROUP. Some of the elements of OREKIT which are used in the project are the definition of the different frames, that are explained in Section 2.1.1 (Earth-centred inertial, Earth-centred Earth-fixed, topocentric and localvertical local-horizontal frame); or the conversions among them. For instance, it provides the transformation between the ECEF frame and the inertial frame, taking into account the effects of precession and nutation, as well as the poles’ wandering displacement with precision. OREKIT also includes absolute propagators in cartesian coordinates or orbital elements and a Taylor differential algebra framework (an other important tool 20 Chapter 2. Preliminary concepts of the project which is explained following in Section 2.5). Furthermore, OREKIT allows to include all basic perturbations models when propagating orbits. For the altitudes in the scenarios cosidered in this project, LEO propagation, the most relevant perturbations are the first tesseral Earth harmonics and the atmospheric drag. For the atmospheric model, it has been used the standar Harris-Priester model and the Naval Research Laboratory Mass Spectrometer and Incoherent Scatter Radar Exosphere (NRLMSISE) model, in particular the Marshall Solar Activity Future Estimation (MSAFE) for the empirical solar data. Both models are included in the OREKIT library. 2.4 Distribution Fitter MATLAB App The Distribution Fitter App is a tool from MATLAB that allows one to fit distributions to the data. This app has been employed to compute the distribution of one of the statistical distances used in the project. Firstly, you have to introduce the target sample data in the app. There are two options for this purpose; a vector of the data can be created directly in the app or it can be imported from the MATLAB workspace. It is necessary to choose a distribution to create the fit. Moreover, the results, both the data and the fit, can be displayed on the app. You can choose to display, for instance, the density (pdf) or the cumulative probability function (cdf). The fit can be improved by excluding certain data. For example, bounds for the data to exclude can be specified or data can be excluded graphically. Additional fit can be created and certain parameters of the fit can be changed manually (See [17] for more information about this app). 2.5 Taylor differential algebra One of the basic problem in dynamics, in particular, in satellite dynamics, relies on the fact that, even if the physical laws of motion are well-known with reasonable accuracy and the algorithm is precise enough, the set of initial data at a given epoch always contains a certain amount of uncertainty. The aim of this section is to explain a method to propagate the data with its uncertainty along time to a future epoch. The set of initial data are expressed with their nominal value, and the uncertainty with respect of this value is define by a covariance matrix (symmetric and positive semi-definite matrix, whose main diagonal contains the variance of each component and the other elements represent the correlations between two components). There are several methods to achive this goal. A typical one is the Montecarlo-like procedure, which consists in sampling the domain of initial data and propagating along time each point independetly (as seen in Figure 2.12). Nevertheless, it is easy to notice that this Montecarlo-like procedure is an expensive computational process. In this project, an alternative method that can reduce this cost is used. This method is known as jet transport, which is a flexible, accuracy and efficient method, and it is based on high-order Taylor expansions. It is known that, for instance, computations of Lyapunov exponents or solution of boundary value problems by shooting requiere the use of first order variation equation. The jet propagation is a generalization of this idea to higher order, so it is not just based on the linear approximation around the nominal solution. Let x0 be the initial data and ε the initial uncertainty on a given domain. The main idea of the jet transport is to propagate this initial data expressed like x0+ε along time and to obtain its evolution. The Montecarlo-like procedure would sample the domain and propagates for many ε values (this procedure is represented by Figure 2.12). On the other hand, jet transport uses arbitrary high-order variational equations to express the solution at any time as a Taylor series expansion in ε around the nominal solution, truncated at the desired order [ 18 ] (Figure 2.13 represents this method). Besides, the coefficients of the Taylor series are computed along the nominal trajectory as extended states using a suitable numerical scheme. 2.6 Unscented transform 21 Figure 2.12 Montecarlo-like procedure [19]. Figure 2.13 Taylor based jet transport [19]. Once the coefficients are computed, the series is defined. At that point, the computational effort involved in propagating a small sampled initial domain is reduced to the direct polynomial evaluation. With respect to accuracy, it can be controlled by the step of the numeric scheme or by the order of Taylor expansion. A further advantage is that the effect of parameters can be included with the same methodology, in addition of the inicial condition. This means that more in-depth and detailed analyses can be performed. It is important to take into account that Taylor’s method is not always a better option than the typical Montecarlo procedure. Montecarlo procedure is faster than jet transport for low orders and low number of variables. However, for samples of moderate to high size the Taylor-based method becomes competitive. Due to project conditions jet transport technique using Taylor differential algebra will be one of the method employed. 2.6 Unscented transform The purpose of this section is to explain another method to propagate the uncertainty along time to a future epoch, called unscented transform. This project pretends to compare the results obtained with both methods and their computation times. The unscented transform (UT) was originally prosed by Jeffrey Uhlmann as part of his PhD thesis (Uhlmann, 1995). The basic premise of the UT is that it easier to approximate a Gaussian distribution than it is to approximate an arbitrary density after a nonlinear transformation [20]. The problem that solved the UT is the following: having some n-dimensional Gaussian distributed variable, as expressed in Equation 2.41 (where µX is the mean and ΣX is the covariance marix), a nonlinear transformation is applied to it, as shown in Equation 2.42. The aim of the problem is to estimate the density distribution of this new variable. xk∼Nn(µX,ΣX)(2.41) yk=f(xk)(2.42) The main idea of the UT is to compute a subset points, called sigma points S={si}2D+1 i=1 , and propagate them through the nonlinear map f(·) . Then, it is necessary to estimate a Gaussian distribution, as expressed in Equation 2.43 (where µYis the sample mean and ΣYis the sample covariance matrix). yi=f(si),∀i∈1,...,2D+1, yi∼N(˜ µY,˜ ΣY)(2.43) 22 Chapter 2. Preliminary concepts There are different proposals to choose sigma points. In this project, a weighted unscented transform is used, so it is also necessary to compute the weights associated ( wi ) to the sigma points ( Xi ). Equation 2.44 shows how to compute them: X0=¯x w0=k n+k Xi=¯x+p(n+k)ΣXiwi=1 2(n+k) Xi+n=¯x−p(n+k)ΣXiwi=1 2(n+k) (2.44) where p(n+k)ΣXi is the ith column of p(n+k)ΣX . k is a free parameter, which has been set k=n−3 to minimize the fourth-order moment mismatch [21]. Next step is to propagate the sigma points through the nonlinear map, as expressed in Equation 2.45. yi=f(Xi)(2.45) Finally, the sample mean of the transformed sigma points and the sample covariance can be computed by Equations 2.46 and 2.47, respectively. ¯y= 2n ∑ i=0 wi·yi(2.46) Σy= 2n ∑ i=0 wi(yi−¯y)(yi−¯y)T(2.47) In this project, the nonlinear map is a combination of the propagation along time of position and velocity, and the trasformation from these variables to range-range rate. The dimension of the variable is 6, hence, 13 propagations are needed. 3 Algorithm for manoeuvres detection This chapter is about the algorithm for manoeuvres detection employed in this project. The Theory of attributables is posed in Section 3.1. In section 3.2, the algorithm itself is explained. In section 3.3, the different statistical distances, which are going to be analyzed, are explained; as well as the computation of the probability that a manouevre has been performed. Finally, the concepts of precision, recall and F-score are defined in Section 3.4. 3.1 Theory of Attributables The aim of the attributable is to condense the information of all measurements and obtain a singe, virtual one [ 22 ]. The radar provides range, ρ , range-rate, ˙ ρ , elevation, El , and azimuth, Az , and together with the chosen reference epoch, t, the attributable can be computed (as expressed in Equation 3.1). A={t,ρ,˙ ρ,El,Az}(3.1) The fit to an observable’s measurement vector is computed with an unweighted linear least squares approach for a polynomial of degree nbased on a Taylor series (as expressed in Equations 3.2). Moreover, if the definition of the range-rate is included into the modelling, it is possible to improve the uncertainty of the resulting virtual measurement; in this case, the observables range and range-rate share the parameters, because range-rate is the time derivative of range. ρ(t) = ρ0+ρ1·t+ρ2·t2 2! +...+ρn·tn n! ˙ ρ(t) = ρ1+ρ2·2t 2! +...+ρn·ntn−1 n! El(t) = El0+El1·t+El2·t2 2! +...+Eln·tn n! Az(t) = Az0+Az1·t+Az2·t2 2! +...+Azn·tn n! (3.2) For the fitting process, the origin of time is at the middle time of the tracklet. The classical least squares algorithm is employed to compute the parameters, using the Equation 3.3. ~m=    ~ ρ∗ ~ El∗ ~ Az∗ ~ ˙ ρ∗     =ASYS ~p+~v=    A0 0 0A0 0 0 A A˙ ρ0 0       ~ ˆ ρ ~ ˆ El ~ ˆ Az   +~v(3.3) 23 24 Chapter 3. Algorithm for manoeuvres detection In Equation 3.3, ~m contains the measurements of all observables, ~p are the parameters which are going to be computed (named ρ0 , ρ1 , El0 , Az0 ,... in Equations 3.2) and the matrices Aand A ˙ ρ have coefficients found from the attributable time-varying formula for ( ρ ,El,Az) and ˙ ρ , as expressed in Equation 3.4 (where m is the number of observables). Finally, ~v contains the error, which follows the distribution of the measurements. A=    1t1 t2 1 2! ··· tn 1 n! . . .. . .. . ..... . . 1tm t2 m 2! ··· tn m n!     A˙ ρ=    0 1 2t1 2! ··· ntn−1 1 n! . . .. . .. . ..... . . 0 1 2tm 2! ··· ntn−1 m n!     (3.4) As the measurement errors are uncorrelated (its covariace matrix, Σv , has only elements on the main diagonal); the problem is posed (as seen in Equation 3.5) as weighted least-squares, whose solution is well-know (as expressed in Equation 3.6). min p~vTΣ−1 v~v=min p(~m−ASY S ~p)TΣ−1 v(~m−ASY S ~p)(3.5) ~p= (AT SYSΣ−1 vASYS)−1AT SYSΣ−1 v~m(3.6) Finally, the covariance matrix of the parameters, Σp , can be computed directly from the covariance matrix of the measurements, with the expression shown in Equation 3.7. Σp= (AT SYSΣ−1 vASYS)−1(3.7) 3.1.1 Position and velocity from attributables In this section is explained how to compute the position and the velocity of the satellites from their attributables. As it has been mentioned the range rate value can be derived from the parameters of the range attributable. Thus, in order to obtain approximately a complete representation of the satellite state, the same computation can be done with elevation and azimuth. Then, the definition of the attributable is expressed as Equation 3.8. A=t,ρ,˙ ρ,El,˙ El,Az,˙ Az(3.8) Radar data are given in its topocentric frame. Thus, it is necessary to go from this frame to the Earth-fixed and then to the inertial. The satellite coordinates (~ sg) in topocentric frame are shown in Equation 3.9. ~ sg=ρ·  cos(El)·sin(Az) cos(El)·cos(Az) sin(El) (3.9) The satellite position in the Earth-fixed frame is expressed as Equation 3.10. Where Ce g is the rotation matrix from the topocentric to the Earth-fixed frame, and ~xe radar is the vector from the radar to the satellite, both are defined with the coordinates of the radar in the Earth-fixed frame. ~xe=Ce g·~ sg+~xe radar (3.10) The translation from the Earth-fixed frame to the inertial frame depends on the epoch. In the project, OREKIT has been used to compute the rotation matrix required, Ci e . The satellite coordinates in the inertial 3.2 Algorithm for manoeuvre detection 25 frame are computed by Equation 3.11. ~xi(A) = Ci e·~xe=Ci eCe g·~ sg+Ci e~xe radar (3.11) Now that the position is computed, the velocity can be calculated by differentiating it, as expressed in Equation 3.12; where d dt (~ sg)g= [~ sg ρ,~ sg El ,~ sg Az]·[˙ ρ,˙ El,˙ Az]T. ~vi(A) = d dt (~xi)i=Ci e·d dt (~xe)e+~wi e/i×~xi=Ci eCe g·d dt (~ sg)g+~wi e/i×~xi(3.12) Also the uncertainty information of the attributable can be passed on to the inertial frame. Let ~ Xi= [~xi,~v]T be the complete satellite state in cartesian coordinates, the covariance matrix can be computed with Equation 3.13, where ∂~ Xi ∂Asis the jacobian of the complete transformation. Σ~ Xi= ∂~ Xi ∂As!ΣAs ∂~ Xi ∂As!T (3.13) 3.2 Algorithm for manoeuvre detection In this section the employed algorithm for manoeuvre detection is explained. The algorithm employed in this project is based on comparing the attributable obtained from the real measurements taken by the radar and the one obtained from the predicted states at the time of the measurements. It has been applied computing the attributable only with range and range rate, only with azimuth and elevation, and using all the variables. At the end of the project we will see that the most important one (with the best results) is the range-range rate attributable. The inputs requiered by the algorithm are the following: • The satellite’s orbit before the measurements with its uncertainty information. As it has been mentioned above, the data are expressed in TLE format in case of synthetic data, and the uncertainty in the form of a covariance matrix. • The satellite’s measurements, together with its associated uncertainty in the form of a covariance matrix. The output of the algorithm is the probability that a manoeuvre has been performed by the object. Moreover, the following assumptions are taken into account: • Measurements come from radar data. It is considered that the measurements contain quite accurate range and range rate measurements and rather imprecise elevation/azimuth angle measurementes. •Only one manoeuvre at most happens in between measurements. •Manoeuvre are approximated as impulses. Using the idea of the Theory of Attributables, from the measurements at certain time, the attributable of range, range-rate, azimuth and elevation can be obtain at the middle of a tracklet, as well as the associated uncertainty, which is expressed in the form of a covariance matrix, Σ . The scope of this algorithm is to obtain a predicted measurement from the initial value of the reference orbit, which is known with quite some accuracy. The initial condition is sampled, obtaining m sampled points. These point will be denoted as ~x0j for j = 1, ..., m. Thus, the set of initial conditions is approximated by these points. 26 Chapter 3. Algorithm for manoeuvres detection The sampled points are propagated using an OREKIT propagator up to time t f . This computation can be made using different methods. The original algorithm employed Taylor differential algebra. In this project, the unscented transform is also used, in order to compare both procedures. Predicted values at measurement times are obtained as a cloud of points ~xj(ti) . The density of points gives an approximate measure of the probability associated to the real trajectory. Once it has the sampled orbit, the measurements at the attributable time can be computed. A “cloud” of measurements has been obtained, so its mean can be computed ( ˆ ρ0,ˆ Az0,ˆ El0,ˆ ρ1 ), as well as the covariance matrix which expresses the uncertainty. This value would be the expected one whether the satellite had not performed any maneuver, given the starting point of the orbit. Finally, the attributables and the predicted measurements can be compared. If there is no manoeuvre, both values are expected to agree. This can be defined by Equation 3.14. (∆ρ0,∆Az0,∆El0,∆ρ1) = (ρ0,Az0,El0,ρ1)−(ˆ ρ0,ˆ Az0,ˆ El0,ˆ ρ1),∆Σ =Σ+ˆ Σ(3.14) Thus, under assumption of normality and whether the satellite has not performed any manoeuvre, (∆ρ0,∆Az0, ∆El0,∆ρ1) is expected to belong to a normal distribution of zero mean and covariance matrix ∆Σ . This can be checked computing a statistical measurement. It is precisely here where the aim of this project lies. Together with the Mahalanobis distance, the Bathacharyya distance and the Kullback Leibler divergence are the statistical distances used in the project. All of them are explained in the following section. 3.3 Theory of distances In Statistics, the study of distances is of fundamental importance in solving practical problems. Statistical distances are measurements that quatify how close two statistical objects are, which can be, for instance, two random variables or two probability distributions. From the point of view of Geometry, a metric is define as follows: Let Mbe a non-empty set, a distance function or metric over Mis an application δ:M×M−→Rif the following conditions are satisfied for all x, y, z ∈M. 1. δ>0if x6=y, and δ(x,x) = 0; 2. δ(x,y) = δ(y,x), (symetry); 3. δ(x,y)≤δ(x,z)+δ(z,y), (triangle inequality); Then, statistical distances are mostly not metrics because they lack one or more properties of proper metrics. A metric is distinguished from a semimetric when the application does not satisfy the triangle inequality and from a quasimetrics when it does not achieve the symmetry condition. Statistical distances that just fulfill the first property are referred to as divergences [23]. In this project, three statistical distances are used. The Mahalanobis distance (MD), which is the original one used by the algorithm; and the Kullback-Leibler divergece (KLD) and the Bhattacharyya distance (BD), which are those that will be studied in the project. As it has been explained before, the scope of the algorithm is to compute the distance between the attributables and projected measurements and make an estimation of the probability of manoeuvres occurrence. 3.3.1 Mahalanobis Distance The Mahalanobis distance is the original one employed in the algorithm. The MD is a measure of the distance between a point xand a distribution. If the distribution has mean µ and covariance matrix Σ , the MD is 3.3 Theory of distances 27 define as expressed in Equation 3.15. MD =q(~x−~ µ)TΣ−1(~x−~ µ)(3.15) In the project, the radar measurements have also error which is defined by a covariance matrix. Thus, the matrix Σ of the Equation 3.15 is define, under a Gaussian hypothesis, as Σ=Σ1+Σ2 , [ 24 ]. Where Σ1 and Σ2 are the covariance matrices of the radar measurements and the predicted ones. Probability associated with the MD In particular, if the ditribution is a multivariate normal, as the distribution of the project data, the square of the MD has a χ2 distribution with ndegrees of freedom, being nthe number of variables. Then, the probability of manoeuvre is computed by Equation 3.16 PR =max0,(χ2(MD2,n)−0.5)·2·100 (3.16) Finally, to reduce false positive, if the MD has a probability of 50% or less of occurring, it is assumed that there is no manoeuvre. Conversely, if the probability is more than 50%, then one substracts 50 from the probability and multiplies it by two. 3.3.2 Kullback-Leibler divergence The Kullback-Leibler divergence, also called relative entropy, is a measure of the similarity or difference between two probability distributions, [ 25 ]. It is a divergence, since it does not satisfy the condition of symmetry, δ(x,y) = δ(y,x), nor the triangular inequality, δ(x,y)≤δ(x,z)+ δ(z,y). For two continuous random variables, P and Q, whose probability density funtions are p(x) and q(x), the Kullback-Leibler divergence is computed by the Equation 3.17. KLD(P||Q) = Zp(x)log p(x) q(x)dx (3.17) Its value is always greater than or equal to zero and the equality will be fulfilled only when the two distributions are identical. As the working distributions are multivariate normal, the divergence between these two distributions, with means µ1and µ2, and covariance matrices Σ1and Σ2, not singular, is defined by Equation 3.18, [26]. KLD(P||Q) = 1 2logdet(Σ2) det(Σ1)+tr(Σ−1 2Σ1)+(µ1−µ2)TΣ−1 2(µ1−µ2)−k(3.18) Probability associated with the KLD The distribution of the KLD is unknown. Thus, the determination of the probability distribution of this distance has been done experimentally. The Distribution Filter App from MATLAB, which is explained in Section 2.4, has been used. The KLD has been computed for 5 satellites; it has been used 27 different scenarios with manoeuvre for each satellite 4 Results This chapter presents the different results obtained. It is structured as follows: In Section 4.1, the synthetic scenarios considered for the analyses are described. Section 4.2, which is the most important, contains the comparison of the distances. Section 4.3 contains analyses on different parameters and their impact the results. Finally, in Section 4.4, the results from real scenarios and the discussion between the three distances are shown. 4.1 Synthetic simulation scenarios To analyze and compare the three distances, five scenarios are considered, based on Sentinel-1A, Sentinel-1B, Sentinel-2A, Sentinel-2B and Swarm-C, whose orbital elements at the initial epoch of the propagations are shown in Table 4.1. It can be noticed that the orbits have a very small eccentricity, i.e. they are almost circular. Moreover, all of them are polar orbit because they have an inclination of about 90 degrees. Orbital element Sentinel-1A Sentinel-1B Sentinel-2A Sentinel-2B Swarm-C n [deg/s] 0.0608 0.0608 0.0596 0.0596 0.0644 e [-] 1.3340E-04 1.3540E-04 1.1260E-04 1.1300E-04 2.2750E-04 M [deg] 279.8708 273.7215 267.1320 265.7235 277.3101 Ω[deg] 236.9954 236.8547 305.1974 115.8571 122.0350 ω[deg] 80.2640 86.4133 92.9988 94.4088 82.8415 i [deg] 98.1828 98.1828 98.5660 98.5725 87.3481 Table 4.1 Orbital element of the considered orbits at initial epoch. On this study, the parameters of the propagator have been kept constant for all cases (varying only the orbit from one satellite to another), which take the values presented in Table 4.2. Real values are employed to compute radar data and estimated values are used to calculate the attributables. Different real and estimated values are used to take into account that in real cases, the parameters are not perfectly known. In subsequent analyses, these values are varied to study their impact the results. Magnitud Real Value Estimated value Mass 1000 Kg 1000 Kg CD2.2 2 Surface 10 m29.5 m2 Atmospheric model Harris Priester Harris Priester Table 4.2 Data for the propagation. 35 36 Chapter 4. Results Then, the synthetic manoeuvre set maintains a constant acceleration of 10 −3 m/s 2 and the main parameters that are varied are the following: •Manoeuvre type: the type of impulse can be: –Tangential: the impulse is tangent to the orbit. –Out-of-plane: the impulse is orthogonal to the orbital plane. – Hybrid: the impulse contains significant components in both directions (in particular, it has been modeled with the tangetial component twice as high as the other). • Manoeuvre intensity: this parameter is regulated through the duration of the impulse. It can be high (as the largest in the data set), medium (a typical intense manoeuvre in the data set) or low (small manoeuvres that will be challenging to detect). Depending on the manoeuvre type, the intensity takes different values: –Tangential: Low Medium High t [s] 5 30 120 ∆V[m/s] 5×10−33×10−21.2×10−1 –Out-of-plane: Low Medium High t [s] 500 1000 2000 ∆V[m/s] 0.5 1 2 –Hybrid: Low Medium High t [s] 50 250 1000 ∆V[m/s] 5×10−20.25 1 •Manoeuvre location: distance between the manoeuvre and the next radar observation: 2h, 6h or 12h. Combining these parameters, 27 scenarios with different manoeuvres are computed. On the other hand, for scenarios without manoeuvres, from 18 to 27 scenarios of this type are considered for each satellite (this number depends on the satellite). These scenarios are computed by propagating the initial orbits for 15 or 20 days, obtaining a scenario every time the satellite cross the radar mask. Sentinel-1A and Sentinel-2B are propagated for 20 days, because they are the satellites employed in all the analyses, in order to consider more scenarios without manoeuvres in those analyses with only two satellites. Swarm-C are also propagated for 20 days because this satellite cross fewer times the mask of the radar. Finally, the other two satellites, Sentinel-1B and Sentinel-2A, are propagated for 15 days. The number of the scenarios depends on how many times the satellites cross the radar mask in the propagation time. Altogether, these are all the scenarios for the satellites. 4.2 Analysis of the distances The results of each satellite are presented in four different tables 1 . The first one, Table 4.5, contains the percentages of true negatives, false positives, true positives and false negatives. The fact of having a small number of radar plots in a segment introduces a lot of false positives, so segments with fewer than 4 radar 1 For the sake of a better understanding, the references to the tables correspond only to the first satellite, Sentinel-1A. The tables of the rest of the satellites follows the same structure. 4.2 Analysis of the distances 37 plots data have not been taken into account. Moreover, with the aim of reducing false positives, in the case of the MD, if the probability of manoeuvre is less than 50%, it is assumed that there is no manoeuvre. In the case of the KLD, it has been considered that there is no manoeuvre if the probability is less than 40% and, using the BD, if it is less than 5%. The rest of tables, which are presented in Appendix A, show the results of the different distances and the probability of manoeuvre. The second table, Table A.1, contains the results of the MD. It has been computed considering only range-range rate (column 1), elevation-azimuth (column 3) and all data (column 5) (with its respective probability value in the adjacent column on the right). The third table, Table A.2, contains the results of the KLD; it has been computed considering only range-range rate (column 1) and elevation-azimuth (column 3). The computation of the KLD with all data has not been possible because, as explained in Section 3.1, the KLD is not symmetric, therefore, it does not work equally in both directions. Considering range-range rate and elevation-azimuth, the distance makes sense using the attributable as the first factor but, with all data, inconsistencies arise in both cases, using the attributable as the first factor and using it as the second one. The fourth table, Table A.3, contains the results of the BD and its coefficient. They have been computed, as the MD, considering range-range rate (the distance appeares in column 1 and the coefficient in column 2), elevation-azimuth (columns 4 and 5) and all data (columns 7 and 8). Five different satellites are employed to see whether the orbit impacts the results. Therefore, individual results have been studied for each satellite, as well as the total percentage of manoeuvres detected and false positives, in order to obtain generic conclusions. Table 4.3 contains the summary of the results for manoeuvre detection. They are presented for the three distances and for the diverse ways of computing them. Analyzing the obtained results, several conclusions can be drawn: •Mahalanobis distance: –Using only range-range rate is more sensitive. –Employing only elevation-azimuth, 42% fewer manoeuvre are detected. – Computing the distance with all data, approximately the same pencentage of manoeuvres as using only range-range rate is obtained, but it introduces false positives. –All high-intensity manoeuvres are detected. –Medium-intensity manoeuvres are mostly detected. – Low-intensity manoeuvres are in general detected in the out-of-plane and the hybrid cases. Some of them are detected in the tangential case, although not the majority of them. •Kullback-Leibler divergence: – Using only elevation-azimuth it always results positive, so it is invalid. On the other hand, employing only range-range rate, good results are obtained, especially avoiding false positives. –All high-intensity manoeuvre are detected. –Medium-intensity manoeuvre are detected, except for the tangential case. – Low-intensity manoeuvres are in general not detected in the tangential case. In the out-of-plane and hybrid case, the manoeuvres are detected in 40% of the cases. •Bhattacharyya distance: –Computing the BD with range-range rate presents excellent results. – Using only elevation-azimuth and all data almost always detects manoeuvre, thus, both options are useless. –All high-intensity manoeuvre are detected. –Medium-intensity manoeuvre are in general detected. –Low-intensity manoeuvre are detected, except for the tangential case. 38 Chapter 4. Results Common conclusion of the three distances is that the time between the manoeuvre and the next radar measurement does not seem to affect these results. This is general for all the scenarios, as well as the fact that the distances computing using only range-range rate are more sensitive, and those calculated using elevation and azimuth introduce a great number of false positives. Comparing the results of the three distances, some conclusions can be drawn. The distance which detects more manoeuvres is the MD, with 87% of true positives. Nevertheless, the BD also detects a great number of them, 83%. Regarding false positives, the KLD and the BD are the distances which detect more scenarios without manoeuvres, 94% and 92%, respectively. Hence, the MD is the best option for detection of manoeuvres. On the other hand, the KLD offers the greatest results avoiding false positives. However, the results of the BD are very close to these maxima in both targets. Thus, we could conclude that the best choice for the detection of manoeuvres is the BD. MD(R, Rr) KLD (R,Rr) BD (R, Rr) MD (el, az) KLD (el, az) BD (el, az) MD (All) BD (All) % True negatives 81% 94% 92% 79% 0% 18% 76% 0% % False positives 19% 6% 8% 21% 100% 82% 24% 100% % True positives 87% 66% 83% 45% 100% 70% 83% 100% % False negatives 13% 34% 17% 55% 0% 30% 17% 0% Table 4.3 Results of all the satellites for 247 scenarios (102 without manoeuvre and 135 with manoeuvre). Finally, the F-score of each distance computed using range-range rate has been calculated to support this conclusion. Table 4.4 presents the values of precision, recall, F 1 and F β , with β taking the value 0.5. The F 1 is the harmonic mean of the precision and recall, and F β with the value 0.5 weighs recall lower than precision. In view of these results, the hypothesis raised before is verified. The BD is the best distance for the detection of manoeuvres. Furthermore, after this one, the MD offers the best results whether precision and recall are equally important. However, if precision is considered more important than recall, the KLD is the best option after the BD. MD(R, Rr) KLD (R,Rr) BD (R, Rr) Precision 0.824 0.918 0.914 Recall 0.874 0.659 0.830 F10.848 0.767 0.870 Fβ(β=0.5) 0.834 0.851 0.895 Table 4.4 Precision, recall, F1and Fβ(β=0.5). Sentinel-1A Scenario The Sentinel-1A scenario with manoeuvre spans from 00:00:00 18/08/2020 to 00:00:00 22/08/2020, and the manoeuvre is placed with respect to 18:25:00 20/08/2020. On the other hand, the Sentinel-1A scenario without manoeuvre spans from 00:00:00 18/08/2020 to 00:00:00 07/09/2020, in order to simulate a greater number of segments without manoeuvre. Table 4.5 presents the percentages of the results, Table A.1 contains the results from the MD, Table A.2 those from the KLD and Table A.3 shows the BD values and their probabilities. Analyzing the obtained results for the satellite Sentinel-1A, several conclusions can be drawn: •The three distances detect a great number of manoeuvres and hardly any false positives are obtained. • All high-intensity manoeuvres are detected and medium-intensity manoeuvres are also in general detected. • Low-intensity manoeuvres are detected in the out-of-plane and hybrid cases; in the tangential cases are in general not detected. 4.2 Analysis of the distances 39 •The distance to the radar measurement does not seem to affect these results. MD(R, Rr) KLD (R,Rr) BD (R, Rr) MD (el, az) KLD (el, az) BD (el, az) MD (All) BD (All) % True negatives 95% 91% 100% 82% 0% 23% 82% 0% % False positives 5% 9% 0% 18% 100% 77% 18% 100% % True positives 93% 85% 89% 48% 100% 63% 93% 100% % False negatives 7% 15% 11% 52% 0% 37% 7% 0% Table 4.5 Results of satellite Sentinel-1A for 49 scenarios (22 without manoeuvre and 27 with manoeuvre). Sentinel-1B Scenario The Sentinel-1B scenario with manoeuvre spans from 00:00:00 18/08/2020 to 00:00:00 22/08/2020. The manoeuvre is placed with respect to 17:35:00 20/08/2020. The scenario without manoeuvre spans from 00:00:00 18/08/2020 to 00:00:00 02/09/2020. Table 4.6 presents the percentages of the results, Table A.4 contains the results from the MD, those from the KLD are shown on Table A.5 and those from the BD on Table A.6. In view of the obtained results for the satellite Sentinel-1A, several conclusions can be drawn: •The MD and the BD detect a great number of manoeuvres, 25% more than the KLD. •The MD and the KLD introduce more false positive than the BD. •All high-intensity manoeuvres are detected. • Medium-intensity manoeuvres are in general detected with all the distances, except in the tangential case with the KLD. • Low-intensity manoeuvres are in general detected in the out-of-plane and the hybrid cases; in the tangential case some of them are detected with the MD and the BD . MD(R, Rr) KLD (R,Rr) BD (R, Rr) MD (el, az) KLD (el, az) BD (el, az) MD (All) BD (All) % True negatives 82% 82% 94% 82% 0% 12% 65% 0% % False positives 18% 18% 6% 18% 100% 88% 35% 100% % True positives 93% 70% 96% 44% 100% 74% 81% 100% % False negatives 7% 30% 4% 56% 0% 26% 19% 0% Table 4.6 Results of satellite Sentinel-1B for 44 scenarios (17 without manoeuvre and 27 with manoeuvre). Sentinel-2A Scenario The Sentinel-2A scenario with manoeuvre spans from 00:00:00 19/08/2020 to 00:00:00 23/08/2020. The manoeuvre is placed with respect to 11:10:00 21/08/2020. The scenario without manoeuvre spans from 00:00:00 19/08/2020 to 00:00:00 04/09/2020. Table 4.7 presents the percentages of the results, Table A.7 contains the results from the MD, Table A.8 presents those from the KLD and Table A.9 shows the BD values and their probabilities. Analyzing the obtained results, several conclusions can be extracted: • The BD and the MD detect a great percentage of manoeuvres. Nevertheless, the KLD just detects 67% of them. •Both the KLD and the BD detect correctly all the scenarios without manoeuvre. •All high-intensity manoeuvres are detected. • Medium-intensity manoeuvres are all detected in the out-of-plane and the hybrid cases; but in the tangential case with the KLD they are not detected. • Low-intensity manoeuvres in the tangential case are not detected with any of the distances, and the KLD in general does not detect any low manoeuvre. 40 Chapter 4. Results MD(R, Rr) KLD (R,Rr) BD (R, Rr) MD (el, az) KLD (el, az) BD (el, az) MD (All) BD (All) % True negatives 89% 100% 100% 89% 0% 5% 84% 0% % False positives 11% 0% 0% 11% 100% 95% 16% 100% % True positives 89% 67% 89% 41% 100% 89% 89% 100% % False negatives 11% 33% 11% 59% 0% 11% 11% 0% Table 4.7 Results of satellite Sentinel-2A for 46 scenarios (19 without manoeuvre and 27 with manoeuvre). Sentinel-2B Scenario The Sentinel-2B scenario with manoeuvre spans from 00:00:00 09/02/2020 to 00:00:00 13/02/2020. The manoeuvre is placed with respect to 11:20:00 11/02/2020. The Sentinel-2B scenario without manoeuvre extents from 00:00:00 09/02/2020 to 00:00:00 29/02/2020. Table 4.8 presents the percentages of the results, Table A.10 contains the results from the MD, Table A.11 those from the KLD and Table A.12 shows the BD values and their probabilities. Analyzing the obtained results, several conclusions can be drawn: • In the Sentinel-2B scenario, a lower percentage of manoeuvres is obtained, and also a larger number of false positives is avoided. •The KLD detects far fewer manoeuvres. •In this case, the distance which offers the best results is the MD. •High-intensity manoeuvres are in general detected. • Medium-intensity manoeuvres in the hybrid case are detected with the three distances. In the tangential and the out-of-plane cases, they are detected with the MD and with the BD but with the KLD are in general not detected. • Low-intensity manoeuvres are in general not detected, except for the hybrid case with the MD and the BD. MD(R, Rr) KLD (R,Rr) BD (R, Rr) MD (el, az) KLD (el, az) BD (el, az) MD (All) BD (All) % True negatives 92% 96% 96% 84% 0% 16% 92% 0% % False positives 8% 4% 4% 16% 100% 84% 8% 100% % True positives 74% 44% 63% 44% 100% 100% 70% 100% % False negatives 26% 56% 37% 56% 0% 0% 30% 0% Table 4.8 Results of satellite Sentinel-2B for 52 scenarios (25 without manoeuvre and 27 with manoeuvre). Swarm-C Scenario The Swarm-C scenario with manoeuvre spans from 00:00:00 14/07/2020 to 00:00:00 20/07/2020. The manoeuvre is placed with respect to 12:30:00 17/07/2020. The scenario without manoeuvre extents from 00:00:00 14/07/2020 to 00:00:00 03/08/2020. Table 4.9 presents the percentages of the results, Table A.13 contains the results from the MD, Table A.14 shows those from the KLD and the results from the BD are presented in Table A.15. Analyzing the obtained results, several conclusions can be drawn: • In Swarm-C scenario, the MD introduces a high percentage of false positives. The BD also introduces a considerable number of false positives. This is because these scenarios present a number of plots between 5 and 7, while those of the rest of satellites have around 10 plots. •All high-intensity manoeuvre are detected. • Medium-intensity manoeuvre are detected in the hybrid and the out-of-plane cases. In the tangential case they are detected just with the MD. • Low–intensity manoeuvres detection success varies, with all of them being detected in the out-of-plane case and none in the tangential case. 4.2 Analysis of the distances 41 MD(R, Rr) KLD (R,Rr) BD (R, Rr) MD (el, az) KLD (el, az) BD (el, az) MD (All) BD (All) % True negatives 42% 100% 68% 58% 0% 32% 53% 0% % False positives 58% 0% 32% 42% 100% 68% 47% 100% % True positives 89% 63% 78% 48% 100% 26% 81% 100% % False negatives 11% 37% 22% 52% 0% 74% 19% 0% Table 4.9 Results of satellite Swarm-C for 46 scenarios (19 without manoeuvre and 27 with manoeuvre). 4.2.1 Small manoeuvres After the previous analysis, the manoeuvre intensity has been reduced to check how small these manoeuvres can be and remain detectable for each distance. As it has been shown by the first results, the detection depends on the orbit, so it has been decided to carry out the five scenarios based on Sentinel-1A, Sentinel-1B, Sentinel-2A, Sentinel-2B and Swarm-C. Since the intensity depends on manoeuvre type, it has taken the following values for each type: •Tangential: Low Medium High t [s] 0.5 1 2.5 ∆V[m/s] 5×10−410−32.5×10−3 •Out-of-plane: Low Medium High t [s] 10 50 250 ∆V[m/s] 10−25×10−20.25 •Hybrid: Low Medium High t [s] 1 5 10 ∆V[m/s] 10−35×10−310−2 Combining these values with the locations aforementioned, 27 scenarios have been computed with different manoeuvres, all of them smaller than the manoeuvres used in the first analysis. Tables 4.10, 4.11 and 4.12 show the percentages of detecting manoeuvres depending on the intensity and the distance (it is shown the results for the intensities mentioned above and for the low intensity of the previous analysis). Each table presents the results of a manoeuvre type: tangential, out-of-plane and hybrid, respectively. As it has been found that the distances computed with elevation and azimuth introduce a large number of false positives, the study is going to be focused on the results of the distances computed using only range-range rate. Analyzing the obtained results, it can be claimed that the distance capable of detecting smaller manoeuvres is the MD, followed by the BD. In the tangential case, the manoeuvres of 5 s, studied in Section 4.2, are undetectable with the KLD and the BD; and the MD detects only 20% of them. The results show that the MD detects also 20% of tangential manoeuvres of 1 s of duration. However, the manoeuvres detected are the three of the Swarm-C scenario, in 42 Chapter 4. Results which the MD presents 58% of false positives. Noticing also that it only detects 7% of the manoeuvres of 2.5 s, it cannot be concluded that the MD is able to detects any manoeuvre smaller than 5 s. In the out-of-plane case, the manoeuvres of 500 s are generally detected with the three distances, with better results using the MD or the BD. The MD detects out-of-plane manoeuvers up to a duration of 250 s. Using the KLD or the BD approximately 50% of them are detected. For lower intensity this type of manoeuvre is practically undetectable. Finally, in the hybrid case, the manoeuvres of 50 s of duration are almost all detected with the MD and the BD. If the intensity is reduced, they are no longer detectable with the BD, and the MD detects just 50% of them when its duration is 10 s. In conclusion, it can be claimed that the tangential manoeuvres are detected when its duration is higher than 30 s with the MD and the BD. In the out-of-plane case, manoeuvres larger than 250 s are detected using the MD. In the hybrid case, all the manoeuvres higher than 50 s are detected employing the MD and the BD. In addition, 50% of the manoeuvres of 10 s are detected using the MD. Tangent MD(R, Rr) KLD (R,Rr) BD (R, Rr) MD (el, az) KLD (el, az) BD (el, az) MD (All) BD (All) 5 s 20% 0% 0% 13% 100% 73% 7% 100% 2.5 s 7% 0% 0% 27% 100% 87% 20% 100% 1 s 20% 0% 0% 27% 100% 80% 20% 100% 0.5 s 0% 0% 0% 27% 100% 80% 7% 100% Table 4.10 Results for tangent manoeuvre. Out-of-plane MD(R, Rr) KLD (R,Rr) BD (R, Rr) MD (el, az) KLD (el, az) BD (el, az) MD (All) BD (All) 500 s 80% 73% 100% 20% 100% 80% 80% 100% 250 s 87% 47% 60% 33% 100% 87% 73% 100% 50 s 20% 0% 20% 27% 100% 87% 13% 100% 10 s 13% 0% 0% 27% 100% 80% 20% 100% Table 4.11 Results for out-of-plane manoeuvre. Hybrid MD(R, Rr) KLD (R,Rr) BD (R, Rr) MD (el, az) KLD (el, az) BD (el, az) MD (All) BD (All) 50 s 100% 40% 93% 33% 100% 80% 87% 100% 10 s 47% 0% 20% 27% 100% 87% 40% 100% 5 s 20% 0% 7% 27% 100% 80% 27% 100% 1 s 7% 0% 0% 13% 100% 80% 7% 100% Table 4.12 Results for hybrid manoeuvre). 4.3 Analyses of the impact of different parameters In this section, several results are presented: the analyses of the influence of propagation parameters (mass, ballistic coefficient and atmospheric model), Taylor order and the computation of the covariance matrix on manouevres detection. To study these impacts, two scenarios are considered, based on Sentinel-1A and Sentinel-2B (the orbit impacts the results). These scenarios have been computed as explained in Section 4.1. Therefore, 27 scenarios with different manoeuvres have been calculated for each satellite. 4.3 Analyses of the impact of different parameters 43 For scenarios without manoeuvre, in the case of the Sentinel-1A, the scenario spans from 00:00:00 18/08/2020 to 00:00:00 07/09/2020, so 25 different segments without manoeuvres are computed. In the case of Sentinel-2B, the scenario spans from 00:00:00 09/02/2020 to 00:00:00 29/09/2020, obtaining 27 segments. As the scenarios with fewer plots than 4 have been eliminated, the analysis has been conducted on 101 scenarios, 47 without manoeuvre and 54 with manoeuvre. In these analyses, only the distances computed using range-range rate have been employed, given the poor results of including elevation and azimuth. In the first three subsections, the influence of the propagation parameters is studied. In the fourth subsection, the Taylor order, and in the last one a different method for the computation of the covariance matrix. 4.3.1 Mass To study the impact of the satellite mass on the results, it has been considered that this parameter is wellknown, so the real and the estimated values are identical. Five mass values have been taken between 1000 Kg and 3000 Kg, in intervals of 500 Kg. The distances and their probabilities for the satellite mass equal to 1000 Kg have been presented on Section 4.2 and the results for the other mass values are shown below on this section. The tables below show the percentages of true negatives, false positives, true positives and false negatives for each value of the satellite mass. Table 4.13 presents the results using the MD, Table 4.14 the results obtained with the KLD and Table 4.15 with BD. In view of the obtained results, it is concluded that no direct relationship can be drawn between the mass of the satellite and the performance of the algorithm. The percentages vary slightly from one mass value to another but the biggest different between two values is 9%. Taking into account that the algorithm works with statistical variables and in the generation of scenarios a random error is introduced, these differences are not relevant. Furthermore, these variations do not follow any pattern. MD (R,Rr) % True negatives % False positives % True positives % False negatives 1000 Kg 94% 6% 83% 17% 1500 Kg 94% 6% 83% 17% 2000 Kg 85% 15% 91% 9% 2500 Kg 89% 11% 83% 17% 3000 Kg 91% 9% 85% 15% Table 4.13 Results for the MD computed using only range-range rate. KLD (R,Rr) % True negatives % False positives % True positives % False negatives 1000 Kg 94% 6% 65% 35% 1500 Kg 98% 2% 69% 31% 2000 Kg 96% 4% 69% 31% 2500 Kg 94% 6% 65% 35% 3000 Kg 94% 6% 65% 35% Table 4.14 Results for the KLD computed using only range-range rate. 5 Conclusions and future works As mentioned in Chapter 1, the interest of this project lies in the increasing population of space objects in Earth orbit. A great amount of this population is space debris, specially dangerous in Low Earth Orbits, where they travel at high speed. All the operating satellites may perform manoeuvres occasionally for diverse purposes, or change their orbits due to perturbances. This makes the satellite operation activities and the space situational awareness more difficult. These are the main reasons why methods for the detection of manoeuvres are being developed. The project focus on the detection of manoeuvres of known objects in Low Earth Orbits using radar data. For this purpose, the project is based on the algorithm developed in [ 3 ], whose main idea is to study the correlation between the propagation of the satellite orbit and radar data of the next track. The concepts developed along the project have been those necessary to understand two things; first, the way that the algorithm operates and second, the changes that haven been implemented with the aim of trying to improve the detection of manoeuvres. The first analyses (done to compare the distances and study the impact of the parameters on the results) have been performed with synthetic data. Then, the algorithm has been applied to real scenarios. Here, the sensitivity of the algorithm with the different distances in real cases has been analyzed. Moreover, all the three distances behave well in synthetic scenarios. All of them result in a high value of F-score, especially the BD, which turns out to have also the highest value with real scenarios. The algorithm performance is greatly reduced in real cases. The KLD is useless in those ones because the algorithm cannot detect any manoeuvre in real cases when this metric is used. In addition, while the orbit seems to have a great effect on the results, the propagation parameters do not seem to impact the results. The algorithm detects more or less the same percentage of manoeuvres regardless the mass or the ballistic coefficient of the satellite. Another important aspect of the algorithm, that has been explained throughout the project, is the propagation of uncertainty. This do not impact only the results, but also the computation time of the algorithm. The Taylor order increase considerably the computation time, nevertheless, it does not increase the number of manoeuvres detected, so that choosing second order is the best option. On the other hand, to use the unscented transform instead of Taylor differential algebra (with second Taylor order) and montecarlo (with 50000 samples) reduces the computation time by 35%, without detriment on the results. 51 52 Chapter 5. Conclusions and future works 5.1 Future work In this section, some suggestions for future works which would be interesting to continue with are presented. The performance and effectiveness of the algorithm is not only assessed by the manoeuvres detected, but also by the number of false positives; being the last one even more important. In the project, some false positives have been eliminated by filtering with the number of radar plots, because the fact of having too few radar plots in a segment introduces a lot of false positives. Some other filters could be studied. It was mentioned in the analysis of the atmospheric model that false positives appear in different scenarios comparing one model to another. Therefore, the combination of results with both models could be studied with the aim of avoiding false positives. As well as the combination of the atmospheric models, it would be interesting to study the combination of the results obtained with two different distances. It was seen in the analysis of real scenarios that the results of the MD and the BD depend differently on the number of plots and the duration of the propagation. Thus, the study of the results of these two distances together could lead to a reduction of false positives or increase the number of manoeuvres detected. The distribution of the KLD has been computed experimentally because it is unknown in the literature. The fit has been done with synthetic data, but this does not work well for real data. Thus, this fit could be improved by studying the data sample and introducing more filters to exclude some data, or even using real data. In that way, the results of the algorithm with the KLD on real cases could be improved. Finally, it is important to mention that the algorithm studied in this project just detects manoeuvres, but it does not characterize them. Thus, the satellite orbit after the manoeuvre is unknown. It would be interesting to combine this algorithm with some manoeuvre reconstruction method, in order to know the satellite orbital elements after the manoeuvre. Appendix A Distances and probabilities tables This appendix shows the tables that contain the distances and their probabilities computed in the different analyses. A.1 Analysis of the distances In this section, as it is explained before, the different distances and the probabilities of manoeuvre of each satellite are presented in three tables 1 . The first table, Table A.1, contains the results of the MD. The second table, Table A.2, contains the results of the KLD. The third table, Table A.3, contains the results of the BD and its coefficient. The tables show the distances (and the coefficient in the BD case) computed with the different attributables and its probability, which is in the adjacent column on the right. In addition, they also present the number of plots of each segment. A.1.1 Sentinel-1A Scenario Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR plots Sentinel-1A No manoeuvre 1.26E-01 0% 2.35E+00 87% 2.35E+00 53% 9 Sentinel-1A No manoeuvre 1.28E+00 12% 3.79E-01 0% 1.34E+00 0% 10 Sentinel-1A No manoeuvre 6.73E-01 0% 6.57E-01 0% 9.46E-01 0% 9 Sentinel-1A No manoeuvre 1.43E+00 28% 8.35E-01 0% 1.66E+00 0% 9 Sentinel-1A No manoeuvre 1.39E+00 23% 1.05E+00 0% 1.74E+00 0% 10 Sentinel-1A No manoeuvre 1.26E-01 0% 2.85E+00 97% 2.86E+00 83% 8 Sentinel-1A No manoeuvre 1.60E+01 100% 1.59E+00 44% 1.61E+01 100% 2 Sentinel-1A No manoeuvre 4.73E-01 0% 1.21E+00 4% 1.32E+00 0% 9 Sentinel-1A No manoeuvre 8.56E-01 0% 1.90E+00 67% 2.10E+00 29% 8 Sentinel-1A No manoeuvre 1.62E+01 100% 8.54E-01 0% 1.62E+01 100% 2 Sentinel-1A No manoeuvre 8.91E-01 0% 1.29E+00 13% 1.57E+00 0% 9 Sentinel-1A No manoeuvre 1.62E+00 46% 2.09E+00 77% 2.65E+00 73% 10 Sentinel-1A No manoeuvre 1.06E+00 0% 1.18E+00 0% 1.76E+00 0% 9 Sentinel-1A No manoeuvre 1.41E-01 0% 6.60E-01 0% 6.76E-01 0% 9 Sentinel-1A No manoeuvre 8.24E-01 0% 1.51E+00 36% 1.78E+00 0% 9 Sentinel-1A No manoeuvre 7.43E-01 0% 9.02E-01 0% 1.18E+00 0% 9 Sentinel-1A No manoeuvre 1.22E+00 5% 1.52E+00 37% 2.18E+00 37% 9 Sentinel-1A No manoeuvre 1.44E-01 0% 1.06E+00 0% 1.07E+00 0% 10 Sentinel-1A No manoeuvre 2.15E-01 0% 9.92E-01 0% 1.14E+00 0% 10 Sentinel-1A No manoeuvre 4.28E-01 0% 9.90E-01 0% 1.08E+00 0% 10 Sentinel-1A No manoeuvre 8.90E-01 0% 1.68E+00 51% 1.90E+00 8% 11 Sentinel-1A No manoeuvre 6.00E-01 0% 1.43E+00 28% 1.56E+00 0% 9 Sentinel-1A No manoeuvre 1.42E+01 100% 9.66E-01 0% 1.42E+01 100% 2 Sentinel-1A No manoeuvre 6.47E-01 0% 2.11E+00 79% 2.21E+00 40% 9 1 For the sake of a better understanding, the references to the tables correspond only to the first satellite, Sentinel-1A. The tables of the rest of the satellites follows the same order. 53 54 Chapter A. Distances and probabilities tables Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR plots Sentinel-1A No manoeuvre 2.13E+00 79% 2.20E+00 82% 3.06E+00 90% 9 Sentinel-1A Tangential Low -2 1.71E+00 54% 2.44E+00 90% 2.98E+00 87% 9 Sentinel-1A Tangential Low -6 2.60E-01 0% 4.40E-01 0% 7.17E-01 0% 9 Sentinel-1A Tangential Low -12 9.05E-01 0% 1.44E-01 0% 9.68E-01 0% 9 Sentinel-1A Tangential Medium -2 2.94E+00 97% 1.48E+00 33% 3.57E+00 97% 9 Sentinel-1A Tangential Medium -6 7.54E+00 100% 1.57E+00 42% 7.62E+00 100% 9 Sentinel-1A Tangential Medium -12 6.85E+00 100% 2.62E+00 94% 7.11E+00 100% 9 Sentinel-1A Tangential High -2 1.50E+01 100% 2.71E-01 0% 1.50E+01 100% 9 Sentinel-1A Tangential High -6 3.21E+01 100% 2.69E+00 95% 3.21E+01 100% 9 Sentinel-1A Tangential High -12 3.53E+01 100% 4.44E+00 100% 3.53E+01 100% 9 Sentinel-1A Out-of-plane Low -2 3.05E+01 100% 1.47E+00 32% 3.05E+01 100% 9 Sentinel-1A Out-of-plane Low -6 2.61E+01 100% 8.63E-01 0% 2.61E+01 100% 9 Sentinel-1A Out-of-plane Low -12 2.96E+01 100% 1.12E+00 0% 2.97E+01 100% 9 Sentinel-1A Out-of-plane Medium -2 6.46E+01 100% 1.93E+00 69% 6.46E+01 100% 9 Sentinel-1A Out-of-plane Medium -6 5.13E+01 100% 1.08E+00 0% 5.14E+01 100% 9 Sentinel-1A Out-of-plane Medium -12 5.91E+01 100% 2.57E+00 93% 5.91E+01 100% 9 Sentinel-1A Out-of-plane High -2 1.25E+02 100% 1.04E+00 0% 1.25E+02 100% 9 Sentinel-1A Out-of-plane High -6 1.01E+02 100% 1.26E+00 10% 1.01E+02 100% 9 Sentinel-1A Out-of-plane High -12 1.22E+02 100% 3.16E-01 0% 1.22E+02 100% 9 Sentinel-1A Hybrid Low -2 7.02E+00 100% 1.61E+00 45% 7.39E+00 100% 9 Sentinel-1A Hybrid Low -6 9.54E+00 100% 2.12E+00 79% 9.74E+00 100% 9 Sentinel-1A Hybrid Low -12 1.04E+01 100% 1.75E+00 57% 1.05E+01 100% 9 Sentinel-1A Hybrid Medium -2 3.75E+01 100% 1.23E+00 7% 3.75E+01 100% 9 Sentinel-1A Hybrid Medium -6 5.46E+01 100% 4.62E+00 100% 5.47E+01 100% 9 Sentinel-1A Hybrid Medium -12 8.83E+01 100% 7.27E+00 100% 8.83E+01 100% 10 Sentinel-1A Hybrid High -2 1.57E+02 100% 4.56E+00 100% 1.57E+02 100% 9 Sentinel-1A Hybrid High -6 4.35E+02 100% 1.65E+01 100% 4.35E+02 100% 9 Sentinel-1A Hybrid High -12 1.16E+03 100% 3.39E+01 100% 1.16E+03 100% 9 Table A.1 Sentinel-1A Reachability analysis and probability from MD. Scenario KLD(R,Rr) PR KLD(el,az) PR plots Sentinel-1A No manoeuvre 6.72E+00 0% 8.05E+04 100% 9 Sentinel-1A No manoeuvre 7.80E+01 11% 1.02E+06 100% 10 Sentinel-1A No manoeuvre 8.85E+00 0% 2.98E+06 100% 9 Sentinel-1A No manoeuvre 1.75E+01 0% 2.74E+05 100% 9 Sentinel-1A No manoeuvre 1.78E+02 56% 3.26E+06 100% 10 Sentinel-1A No manoeuvre 2.53E+01 0% 7.48E+05 100% 8 Sentinel-1A No manoeuvre 1.28E+04 100% 1.16E+06 100% 2 Sentinel-1A No manoeuvre 5.25E+01 0% 6.52E+05 100% 9 Sentinel-1A No manoeuvre 6.09E+01 0% 3.62E+05 100% 8 Sentinel-1A No manoeuvre 5.49E+03 100% 1.57E+06 100% 2 Sentinel-1A No manoeuvre 4.40E+01 0% 4.26E+05 100% 9 Sentinel-1A No manoeuvre 6.59E+01 2% 1.38E+06 100% 10 Sentinel-1A No manoeuvre 7.54E+00 0% 2.62E+05 100% 9 Sentinel-1A No manoeuvre 2.54E+01 0% 1.10E+06 100% 9 Sentinel-1A No manoeuvre 3.88E+01 0% 4.41E+05 100% 9 Sentinel-1A No manoeuvre 3.45E+01 0% 2.42E+05 100% 9 Sentinel-1A No manoeuvre 6.53E+00 0% 1.55E+05 100% 9 Sentinel-1A No manoeuvre 1.12E+02 31% 9.76E+05 100% 10 Sentinel-1A No manoeuvre 1.04E+01 0% 8.55E+05 100% 10 Sentinel-1A No manoeuvre 1.72E+01 0% 6.12E+05 100% 10 Sentinel-1A No manoeuvre 1.15E+02 32% 2.69E+06 100% 11 Sentinel-1A No manoeuvre 2.38E+01 0% 1.27E+06 100% 9 Sentinel-1A No manoeuvre 1.90E+04 100% 7.33E+05 100% 2 Sentinel-1A No manoeuvre 2.71E+01 0% 1.41E+06 100% 9 A.1 Analysis of the distances 55 Scenario KLD(R,Rr) PR KLD(el,az) PR plots Sentinel-1A No manoeuvre 2.33E+02 70% 1.41E+06 100% 9 Sentinel-1A Tangential Low -2 2.13E+01 0% 3.12E+06 100% 9 Sentinel-1A Tangential Low -6 7.58E+00 0% 4.46E+05 100% 9 Sentinel-1A Tangential Low -12 1.10E+01 0% 4.62E+05 100% 9 Sentinel-1A Tangential Medium -2 4.74E+01 0% 4.79E+05 100% 9 Sentinel-1A Tangential Medium -6 2.68E+02 76% 4.47E+05 100% 9 Sentinel-1A Tangential Medium -12 2.14E+02 66% 1.37E+06 100% 9 Sentinel-1A Tangential High -2 1.07E+03 100% 4.73E+05 100% 9 Sentinel-1A Tangential High -6 4.80E+03 100% 4.54E+05 100% 9 Sentinel-1A Tangential High -12 5.58E+03 100% 9.60E+05 100% 9 Sentinel-1A Out-of-plane Low -2 4.44E+03 100% 1.06E+06 100% 9 Sentinel-1A Out-of-plane Low -6 3.23E+03 100% 7.49E+05 100% 9 Sentinel-1A Out-of-plane Low -12 4.25E+03 100% 4.70E+05 100% 9 Sentinel-1A Out-of-plane Medium -2 1.99E+04 100% 2.11E+06 100% 9 Sentinel-1A Out-of-plane Medium -6 1.26E+04 100% 6.23E+05 100% 9 Sentinel-1A Out-of-plane Medium -12 1.65E+04 100% 3.31E+06 100% 9 Sentinel-1A Out-of-plane High -2 7.55E+04 100% 6.12E+05 100% 9 Sentinel-1A Out-of-plane High -6 4.87E+04 100% 7.68E+05 100% 9 Sentinel-1A Out-of-plane High -12 7.15E+04 100% 4.67E+05 100% 9 Sentinel-1A Hybrid Low -2 2.40E+02 71% 4.90E+05 100% 9 Sentinel-1A Hybrid Low -6 4.35E+02 91% 2.17E+06 100% 9 Sentinel-1A Hybrid Low -12 4.89E+02 94% 6.93E+05 100% 9 Sentinel-1A Hybrid Medium -2 6.69E+03 100% 6.88E+05 100% 9 Sentinel-1A Hybrid Medium -6 1.39E+04 100% 2.71E+06 100% 9 Sentinel-1A Hybrid Medium -12 4.03E+04 100% 1.30E+06 100% 10 Sentinel-1A Hybrid High -2 1.16E+05 100% 5.53E+05 100% 9 Sentinel-1A Hybrid High -6 6.55E+05 100% 4.55E+05 100% 9 Sentinel-1A Hybrid High -12 3.42E+06 100% 7.26E+05 100% 9 Table A.2 Sentinel-1A Reachability analysis and probability from KLD. Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR plots NM 2.76E+00 6.30E-02 0% 2.62E+00 7.30E-02 0% 9.78E+00 5.65E-05 23% 9 NM 2.38E+00 9.28E-02 0% 4.28E+00 1.38E-02 16% 1.02E+01 3.61E-05 25% 10 NM 2.57E+00 7.69E-02 0% 3.99E+00 1.85E-02 13% 1.14E+01 1.17E-05 30% 9 NM 2.46E+00 8.58E-02 0% 3.14E+00 4.32E-02 3% 9.75E+00 5.83E-05 23% 9 NM 2.96E+00 5.18E-02 0% 4.18E+00 1.54E-02 15% 1.10E+01 1.65E-05 29% 10 NM 2.07E+00 1.27E-01 0% 3.98E+00 1.86E-02 13% 9.91E+00 4.95E-05 23% 8 NM 2.15E+01 4.70E-10 91% 3.73E+00 2.39E-02 10% 2.92E+01 1.99E-13 79% 2 NM 2.68E+00 6.85E-02 0% 3.72E+00 2.42E-02 10% 1.05E+01 2.89E-05 26% 9 NM 2.35E+00 9.49E-02 0% 4.12E+00 1.63E-02 14% 1.02E+01 3.76E-05 25% 8 NM 2.95E+01 1.53E-13 97% 3.66E+00 2.58E-02 9% 3.78E+01 3.85E-17 88% 2 NM 2.81E+00 6.00E-02 0% 3.20E+00 4.07E-02 4% 1.03E+01 3.49E-05 25% 9 NM 2.43E+00 8.83E-02 0% 4.60E+00 1.01E-02 20% 1.07E+01 2.32E-05 27% 10 NM 2.59E+00 7.52E-02 0% 2.97E+00 5.11E-02 1% 9.97E+00 4.70E-05 24% 9 NM 2.54E+00 7.91E-02 0% 3.61E+00 2.70E-02 9% 1.03E+01 3.26E-05 25% 9 NM 2.68E+00 6.86E-02 0% 3.53E+00 2.93E-02 8% 1.02E+01 3.56E-05 25% 9 NM 2.24E+00 1.06E-01 0% 3.70E+00 2.47E-02 10% 9.80E+00 5.56E-05 23% 9 NM 2.76E+00 6.36E-02 0% 2.99E+00 5.04E-02 1% 1.03E+01 3.30E-05 25% 9 NM 2.58E+00 7.57E-02 0% 4.30E+00 1.36E-02 16% 1.06E+01 2.39E-05 27% 10 NM 2.62E+00 7.29E-02 0% 3.34E+00 3.53E-02 5% 1.06E+01 2.54E-05 27% 10 NM 2.40E+00 9.10E-02 0% 3.35E+00 3.51E-02 5% 1.00E+01 4.44E-05 24% 10 NM 2.96E+00 5.18E-02 0% 4.02E+00 1.80E-02 13% 1.10E+01 1.70E-05 29% 11 NM 2.11E+00 1.22E-01 0% 3.91E+00 2.01E-02 12% 9.94E+00 4.81E-05 24% 9 NM 2.79E+01 7.73E-13 96% 3.71E+00 2.44E-02 10% 3.57E+01 3.04E-16 86% 2 NM 2.63E+00 7.19E-02 0% 3.96E+00 1.91E-02 13% 1.07E+01 2.28E-05 27% 9 NM 3.00E+00 5.00E-02 1% 4.27E+00 1.40E-02 16% 1.11E+01 1.51E-05 29% 9 TL -2 2.89E+00 5.55E-02 0% 3.85E+00 2.12E-02 11% 1.14E+01 1.08E-05 31% 9 TL -6 2.53E+00 7.94E-02 0% 3.13E+00 4.38E-02 3% 1.04E+01 3.08E-05 26% 9 TL -12 2.63E+00 7.20E-02 0% 3.11E+00 4.45E-02 2% 1.04E+01 2.91E-05 26% 9 56 Chapter A. Distances and probabilities tables Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR plots TM -2 3.61E+00 2.72E-02 8% 3.38E+00 3.41E-02 6% 1.19E+01 6.65E-06 33% 9 TM -6 9.63E+00 6.60E-05 59% 3.41E+00 3.30E-02 6% 1.76E+01 2.32E-08 54% 9 TM -12 8.40E+00 2.25E-04 51% 3.97E+00 1.89E-02 13% 1.67E+01 5.86E-08 51% 9 TH -2 3.06E+01 5.02E-14 97% 3.12E+00 4.43E-02 2% 3.85E+01 1.90E-17 88% 9 TH -6 1.31E+02 1.36E-57 100% 4.01E+00 1.82E-02 13% 1.39E+02 5.57E-61 100% 9 TH -12 1.58E+02 1.93E-69 100% 5.57E+00 3.80E-03 29% 1.66E+02 6.56E-73 100% 9 OL -2 1.19E+02 2.38E-52 100% 3.38E+00 3.41E-02 6% 1.27E+02 7.59E-56 100% 9 OL -6 8.75E+01 9.66E-39 100% 3.20E+00 4.08E-02 3% 9.55E+01 3.50E-42 100% 9 OL -12 1.12E+02 1.91E-49 100% 3.27E+00 3.81E-02 4% 1.20E+02 5.64E-53 100% 9 OM -2 5.24E+02 3.50E-228 100% 3.57E+00 2.81E-02 8% 5.32E+02 8.78E-232 100% 9 OM -6 3.32E+02 6.78E-145 100% 3.25E+00 3.87E-02 4% 3.40E+02 1.93E-148 100% 9 OM -12 4.39E+02 3.50E-191 100% 3.93E+00 1.96E-02 12% 4.47E+02 6.59E-195 100% 9 OH -2 1.95E+03 0.00E+00 100% 3.25E+00 3.89E-02 4% 1.96E+03 0.00E+00 100% 9 OH -6 1.29E+03 0.00E+00 100% 3.30E+00 3.68E-02 5% 1.30E+03 0.00E+00 100% 9 OH -12 1.87E+03 0.00E+00 100% 3.12E+00 4.41E-02 2% 1.88E+03 0.00E+00 100% 9 HL -2 8.69E+00 1.69E-04 53% 3.43E+00 3.24E-02 6% 1.72E+01 3.56E-08 52% 9 HL -6 1.39E+01 9.11E-07 76% 3.67E+00 2.54E-02 9% 2.22E+01 2.35E-10 66% 9 HL -12 1.62E+01 9.43E-08 82% 3.49E+00 3.06E-02 7% 2.40E+01 3.62E-11 70% 9 HM -2 1.78E+02 4.97E-78 100% 3.30E+00 3.69E-02 5% 1.86E+02 1.83E-81 100% 9 HM -6 3.76E+02 6.96E-164 100% 5.77E+00 3.11E-03 31% 3.84E+02 1.24E-167 100% 9 HM -12 9.76E+02 0.00E+00 100% 9.78E+00 5.63E-05 59% 9.85E+02 0.00E+00 100% 10 HH -2 3.09E+03 0.00E+00 100% 5.71E+00 3.33E-03 31% 3.10E+03 0.00E+00 100% 9 HH -6 2.36E+04 0.00E+00 100% 3.69E+01 9.09E-17 99% 2.36E+04 0.00E+00 100% 9 HH -12 1.68E+05 0.00E+00 100% 1.47E+02 1.78E-64 100% 1.68E+05 0.00E+00 100% 9 Table A.3 Sentinel-1A Reachability analysis and probability from BD 2. A.1.2 Sentinel-1B Scenario Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR plots Sentinel-1B No manoeuvre 2.52E-01 0% 2.98E+00 98% 3.02E+00 88% 9 Sentinel-1B No manoeuvre 1.47E-01 0% 1.29E+00 13% 1.29E+00 0% 9 Sentinel-1B No manoeuvre 1.38E+01 100% 4.54E-01 0% 1.39E+01 100% 2 Sentinel-1B No manoeuvre 2.00E+00 73% 1.78E+00 59% 2.73E+00 77% 9 Sentinel-1B No manoeuvre 1.49E+00 34% 3.10E+00 98% 3.44E+00 96% 11 Sentinel-1B No manoeuvre 4.55E-02 0% 1.69E+00 52% 1.75E+00 0% 9 Sentinel-1B No manoeuvre 4.60E-01 0% 6.11E-01 0% 7.86E-01 0% 9 Sentinel-1B No manoeuvre 3.67E-01 0% 1.44E+00 29% 1.49E+00 0% 10 Sentinel-1B No manoeuvre 2.64E+00 94% 1.69E+00 52% 3.13E+00 91% 8 Sentinel-1B No manoeuvre 4.80E-01 0% 5.45E-01 0% 8.44E-01 0% 9 Sentinel-1B No manoeuvre 6.32E-01 0% 9.60E-01 0% 1.15E+00 0% 10 Sentinel-1B No manoeuvre 9.84E-01 0% 1.16E+00 0% 1.52E+00 0% 10 Sentinel-1B No manoeuvre 3.36E-01 0% 1.03E+00 0% 1.14E+00 0% 10 Sentinel-1B No manoeuvre 2.09E+00 77% 1.59E+00 43% 2.62E+00 72% 10 Sentinel-1B No manoeuvre 1.17E+00 0% 2.92E-01 0% 1.20E+00 0% 8 Sentinel-1B No manoeuvre 1.27E+01 100% 1.65E+00 49% 1.28E+01 100% 2 Sentinel-1B No manoeuvre 5.38E-01 0% 1.19E+00 1% 1.33E+00 0% 9 Sentinel-1B No manoeuvre 1.25E-01 0% 1.52E-01 0% 1.92E-01 0% 8 Sentinel-1B No manoeuvre 1.55E+01 100% 7.07E-01 0% 1.55E+01 100% 2 Sentinel-1B No manoeuvre 6.93E-01 0% 2.22E+00 83% 2.51E+00 64% 9 Sentinel-1B Tangential Low -2 1.66E+00 50% 9.30E-01 0% 1.97E+00 15% 11 Sentinel-1B Tangential Low -6 1.86E+00 65% 4.02E-01 0% 1.93E+00 11% 11 Sentinel-1B Tangential Low -12 1.01E+00 0% 4.07E-01 0% 1.05E+00 0% 11 Sentinel-1B Tangential Medium -2 2.46E+00 90% 9.36E-01 0% 2.62E+00 71% 11 Sentinel-1B Tangential Medium -6 1.45E+00 30% 1.74E+00 56% 2.10E+00 29% 11 Sentinel-1B Tangential Medium -12 2.47E+00 90% 1.40E+00 25% 3.00E+00 88% 10 Sentinel-1B Tangential High -2 1.20E+01 100% 2.63E+00 94% 1.23E+01 100% 11 Sentinel-1B Tangential High -6 6.68E+00 100% 2.64E+00 94% 6.76E+00 100% 10 2 NM: No manoeuvre, TL: Tangential Low, TM: Tangential Medium, TH: Tangential High, OL: Out-of-plane Low, OM: Out-of-plane Medium, OH: Out-of-plane High, HL: Hybrid Low, HM: Hybrid Medium, HH: Hybrid High A.1 Analysis of the distances 57 Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR plots Sentinel-1B Tangential High -12 1.47E+01 100% 3.77E+00 100% 1.47E+01 100% 10 Sentinel-1B Out-of-plane Low -2 1.30E+01 100% 1.38E+00 23% 1.31E+01 100% 11 Sentinel-1B Out-of-plane Low -6 1.10E+01 100% 7.31E-01 0% 1.10E+01 100% 11 Sentinel-1B Out-of-plane Low -12 1.21E+01 100% 5.17E-01 0% 1.21E+01 100% 11 Sentinel-1B Out-of-plane Medium -2 2.72E+01 100% 1.16E+00 0% 2.72E+01 100% 11 Sentinel-1B Out-of-plane Medium -6 2.22E+01 100% 9.43E-01 0% 2.22E+01 100% 11 Sentinel-1B Out-of-plane Medium -12 2.67E+01 100% 2.35E+00 87% 2.69E+01 100% 11 Sentinel-1B Out-of-plane High -2 5.32E+01 100% 7.52E-01 0% 5.32E+01 100% 11 Sentinel-1B Out-of-plane High -6 4.21E+01 100% 8.04E-01 0% 4.21E+01 100% 11 Sentinel-1B Out-of-plane High -12 5.16E+01 100% 6.18E-01 0% 5.16E+01 100% 11 Sentinel-1B Hybrid Low -2 5.17E+00 100% 1.76E+00 58% 5.45E+00 100% 11 Sentinel-1B Hybrid Low -6 1.75E+00 57% 1.47E+00 32% 2.11E+00 30% 11 Sentinel-1B Hybrid Low -12 3.89E+00 100% 1.58E+00 42% 3.95E+00 99% 10 Sentinel-1B Hybrid Medium -2 2.42E+01 100% 1.73E+00 55% 2.42E+01 100% 10 Sentinel-1B Hybrid Medium -6 1.20E+01 100% 2.94E+00 97% 1.21E+01 100% 10 Sentinel-1B Hybrid Medium -12 4.65E+01 100% 6.95E+00 100% 4.66E+01 100% 11 Sentinel-1B Hybrid High -2 1.09E+02 100% 2.96E+00 98% 1.09E+02 100% 10 Sentinel-1B Hybrid High -6 1.10E+02 100% 1.24E+01 100% 1.10E+02 100% 11 Sentinel-1B Hybrid High -12 5.76E+02 100% 2.98E+01 100% 5.76E+02 100% 10 Table A.4 Sentinel-1B Reachability analysis and probability from MD. Scenario KLD(R,Rr) PR KLD(el,az) PR plots Sentinel-1B No manoeuvre 1.59E+01 0% 9.80E+05 100% 9 Sentinel-1B No manoeuvre 3.58E+01 0% 6.51E+05 100% 9 Sentinel-1B No manoeuvre 4.67E+03 100% 1.46E+06 100% 2 Sentinel-1B No manoeuvre 6.11E+01 0% 5.16E+05 100% 9 Sentinel-1B No manoeuvre 1.66E+02 53% 1.13E+07 100% 11 Sentinel-1B No manoeuvre 7.30E+00 0% 7.14E+05 100% 9 Sentinel-1B No manoeuvre 2.39E+01 0% 4.12E+05 100% 9 Sentinel-1B No manoeuvre 6.09E+01 0% 2.17E+06 100% 10 Sentinel-1B No manoeuvre 2.77E+02 77% 3.91E+05 100% 8 Sentinel-1B No manoeuvre 6.08E+00 0% 3.00E+04 100% 9 Sentinel-1B No manoeuvre 8.16E+01 13% 1.20E+06 100% 10 Sentinel-1B No manoeuvre 1.67E+01 0% 1.33E+06 100% 10 Sentinel-1B No manoeuvre 1.63E+01 0% 3.27E+05 100% 10 Sentinel-1B No manoeuvre 3.93E+02 89% 3.09E+06 100% 10 Sentinel-1B No manoeuvre 5.86E+01 0% 6.06E+05 100% 8 Sentinel-1B No manoeuvre 1.51E+04 100% 1.71E+06 100% 2 Sentinel-1B No manoeuvre 2.81E+01 0% 4.29E+05 100% 9 Sentinel-1B No manoeuvre 4.86E+01 0% 2.90E+05 100% 8 Sentinel-1B No manoeuvre 6.57E+03 100% 1.70E+06 100% 2 Sentinel-1B No manoeuvre 1.59E+01 0% 1.66E+05 100% 9 Sentinel-1B Tangential Low -2 6.65E+01 2% 6.84E+05 100% 11 Sentinel-1B Tangential Low -6 7.71E+01 10% 6.83E+05 100% 11 Sentinel-1B Tangential Low -12 3.50E+01 0% 6.77E+05 100% 11 Sentinel-1B Tangential Medium -2 1.18E+02 34% 6.64E+05 100% 11 Sentinel-1B Tangential Medium -6 4.14E+01 0% 1.17E+06 100% 11 Sentinel-1B Tangential Medium -12 5.00E+01 0% 2.10E+06 100% 10 Sentinel-1B Tangential High -2 2.48E+03 100% 8.86E+05 100% 11 Sentinel-1B Tangential High -6 5.82E+02 96% 7.49E+05 100% 10 Sentinel-1B Tangential High -12 2.56E+03 100% 9.10E+05 100% 10 Sentinel-1B Out-of-plane Low -2 2.90E+03 100% 8.59E+05 100% 11 Sentinel-1B Out-of-plane Low -6 2.03E+03 100% 8.88E+05 100% 11 Sentinel-1B Out-of-plane Low -12 2.56E+03 100% 6.89E+05 100% 11 Sentinel-1B Out-of-plane Medium -2 1.28E+04 100% 7.36E+05 100% 11 58 Chapter A. Distances and probabilities tables Scenario KLD(R,Rr) PR KLD(el,az) PR plots Sentinel-1B Out-of-plane Medium -6 8.45E+03 100% 1.15E+06 100% 11 Sentinel-1B Out-of-plane Medium -12 1.20E+04 100% 8.59E+05 100% 11 Sentinel-1B Out-of-plane High -2 4.86E+04 100% 1.06E+06 100% 11 Sentinel-1B Out-of-plane High -6 3.01E+04 100% 7.07E+05 100% 11 Sentinel-1B Out-of-plane High -12 4.53E+04 100% 9.10E+05 100% 11 Sentinel-1B Hybrid Low -2 4.69E+02 93% 7.19E+05 100% 11 Sentinel-1B Hybrid Low -6 4.27E+01 0% 1.54E+06 100% 11 Sentinel-1B Hybrid Low -12 1.11E+02 30% 1.06E+06 100% 10 Sentinel-1B Hybrid Medium -2 1.03E+04 100% 2.02E+06 100% 10 Sentinel-1B Hybrid Medium -6 1.77E+03 100% 1.43E+06 100% 10 Sentinel-1B Hybrid Medium -12 2.56E+04 100% 1.00E+06 100% 11 Sentinel-1B Hybrid High -2 2.06E+05 100% 7.31E+05 100% 10 Sentinel-1B Hybrid High -6 1.50E+05 100% 1.61E+06 100% 11 Sentinel-1B Hybrid High -12 3.44E+06 100% 1.77E+06 100% 10 Table A.5 Sentinel-1B Reachability analysis and probability from KLD. Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR plots NM 2.40E+00 9.10E-02 0% 4.51E+00 1.10E-02 19% 1.11E+01 1.58E-05 29% 9 NM 2.20E+00 1.11E-01 0% 4.04E+00 1.76E-02 14% 9.89E+00 5.05E-05 23% 9 NM 2.68E+01 2.32E-12 96% 3.48E+00 3.07E-02 7% 3.48E+01 7.37E-16 85% 2 NM 2.97E+00 5.14E-02 0% 3.37E+00 3.43E-02 6% 1.05E+01 2.81E-05 26% 9 NM 2.60E+00 7.45E-02 0% 5.56E+00 3.85E-03 29% 1.17E+01 8.28E-06 32% 11 NM 2.58E+00 7.60E-02 0% 3.24E+00 3.91E-02 4% 1.03E+01 3.38E-05 25% 9 NM 2.54E+00 7.86E-02 0% 3.35E+00 3.52E-02 5% 1.00E+01 4.44E-05 24% 9 NM 2.75E+00 6.37E-02 0% 3.97E+00 1.89E-02 13% 1.07E+01 2.32E-05 27% 10 NM 3.08E+00 4.59E-02 2% 4.08E+00 1.70E-02 14% 1.10E+01 1.68E-05 29% 8 NM 2.75E+00 6.41E-02 0% 2.43E+00 8.78E-02 0% 9.69E+00 6.19E-05 22% 9 NM 2.45E+00 8.60E-02 0% 4.23E+00 1.45E-02 16% 1.04E+01 3.17E-05 26% 10 NM 2.76E+00 6.35E-02 0% 3.37E+00 3.44E-02 6% 1.08E+01 2.02E-05 28% 10 NM 2.39E+00 9.16E-02 0% 3.34E+00 3.55E-02 5% 9.95E+00 4.78E-05 24% 10 NM 3.40E+00 3.34E-02 9% 4.12E+00 1.62E-02 15% 1.15E+01 1.05E-05 31% 10 NM 2.27E+00 1.04E-01 0% 3.84E+00 2.14E-02 11% 1.00E+01 4.54E-05 24% 8 NM 2.30E+01 1.05E-10 93% 3.94E+00 1.94E-02 12% 3.09E+01 3.64E-14 81% 2 NM 2.60E+00 7.44E-02 0% 3.59E+00 2.75E-02 8% 1.03E+01 3.43E-05 25% 9 NM 2.39E+00 9.17E-02 0% 3.79E+00 2.27E-02 11% 9.97E+00 4.66E-05 24% 8 NM 3.29E+01 5.18E-15 98% 3.53E+00 2.93E-02 8% 4.10E+01 1.61E-18 90% 2 NM 2.47E+00 8.44E-02 0% 3.62E+00 2.68E-02 9% 1.05E+01 2.85E-05 26% 9 TL -2 3.38E+00 3.40E-02 6% 3.38E+00 3.39E-02 6% 1.10E+01 1.63E-05 29% 11 TL -6 3.47E+00 3.12E-02 7% 3.29E+00 3.72E-02 5% 1.10E+01 1.66E-05 29% 11 TL -12 3.16E+00 4.22E-02 3% 3.30E+00 3.71E-02 5% 1.07E+01 2.30E-05 27% 11 TM -2 3.79E+00 2.26E-02 11% 3.38E+00 3.41E-02 6% 1.14E+01 1.12E-05 31% 11 TM -6 3.30E+00 3.70E-02 5% 3.65E+00 2.60E-02 9% 1.11E+01 1.53E-05 29% 11 TM -12 3.78E+00 2.28E-02 11% 3.54E+00 2.90E-02 8% 1.17E+01 8.26E-06 32% 10 TH -2 2.11E+01 6.55E-10 91% 4.14E+00 1.59E-02 15% 2.93E+01 1.88E-13 79% 11 TH -6 8.59E+00 1.86E-04 53% 4.17E+00 1.55E-02 15% 1.63E+01 8.41E-08 50% 10 TH -12 3.01E+01 8.61E-14 97% 5.07E+00 6.27E-03 25% 3.77E+01 4.30E-17 88% 10 OL -2 2.42E+01 2.97E-11 94% 3.51E+00 2.98E-02 7% 3.21E+01 1.16E-14 82% 11 OL -6 1.81E+01 1.43E-08 86% 3.34E+00 3.55E-02 5% 2.56E+01 7.45E-12 73% 11 OL -12 2.13E+01 5.38E-10 91% 3.31E+00 3.64E-02 5% 2.89E+01 2.78E-13 78% 11 OM -2 9.55E+01 3.22E-42 100% 3.45E+00 3.19E-02 7% 1.03E+02 1.54E-45 100% 11 OM -6 6.46E+01 9.16E-29 100% 3.39E+00 3.38E-02 6% 7.22E+01 4.59E-32 99% 11 OM -12 9.24E+01 7.25E-41 100% 3.97E+00 1.90E-02 13% 1.01E+02 1.40E-44 100% 11 OH -2 3.56E+02 2.06E-155 100% 3.35E+00 3.51E-02 5% 3.64E+02 1.01E-158 100% 11 OH -6 2.25E+02 2.56E-98 100% 3.35E+00 3.49E-02 5% 2.32E+02 1.13E-101 100% 11 OH -12 3.36E+02 1.50E-146 100% 3.32E+00 3.61E-02 5% 3.43E+02 7.47E-150 100% 11 HL -2 6.38E+00 1.70E-03 36% 3.66E+00 2.57E-02 9% 1.43E+01 6.42E-07 42% 11 HL -6 3.42E+00 3.26E-02 6% 3.55E+00 2.88E-02 8% 1.11E+01 1.51E-05 29% 11 HL -12 4.91E+00 7.40E-03 23% 3.60E+00 2.72E-02 8% 1.25E+01 3.64E-06 35% 10 HM -2 7.61E+01 8.77E-34 100% 3.67E+00 2.55E-02 9% 8.39E+01 3.62E-37 99% 10 HM -6 2.11E+01 6.93E-10 91% 4.37E+00 1.26E-02 17% 2.88E+01 3.10E-13 78% 10 HM -12 2.74E+02 1.40E-119 100% 9.25E+00 9.61E-05 56% 2.82E+02 5.28E-123 100% 11 HH -2 1.50E+03 0.00E+00 100% 4.38E+00 1.25E-02 17% 1.50E+03 0.00E+00 100% 10 HH -6 1.53E+03 0.00E+00 100% 2.25E+01 1.75E-10 92% 1.54E+03 0.00E+00 100% 11 A.1 Analysis of the distances 59 Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR plots HH -12 4.15E+04 0.00E+00 100% 1.14E+02 3.08E-50 100% 4.15E+04 0.00E+00 100% 10 Table A.6 Sentinel-1B Reachability analysis and probability from BD. A.1.3 Sentinel-2A Scenario Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR plots Sentinel-2A No manoeuvre 1.10E+00 0% 1.34E+00 18% 1.80E+00 0% 12 Sentinel-2A No manoeuvre 7.06E-01 0% 1.02E+00 0% 1.25E+00 0% 13 Sentinel-2A No manoeuvre 4.80E-01 0% 5.01E-01 0% 7.06E-01 0% 10 Sentinel-2A No manoeuvre 1.05E+00 0% 4.75E-01 0% 1.15E+00 0% 10 Sentinel-2A No manoeuvre 8.78E-01 0% 1.56E+00 41% 1.79E+00 0% 5 Sentinel-2A No manoeuvre 1.33E+01 100% 1.03E+00 0% 1.33E+01 100% 3 Sentinel-2A No manoeuvre 1.49E+00 34% 1.37E+00 22% 2.03E+00 22% 10 Sentinel-2A No manoeuvre 1.59E+00 44% 2.36E+00 88% 2.86E+00 83% 10 Sentinel-2A No manoeuvre 1.97E-01 0% 1.78E+00 59% 1.82E+00 0% 12 Sentinel-2A No manoeuvre 5.83E-01 0% 1.37E+00 22% 1.49E+00 0% 12 Sentinel-2A No manoeuvre 1.24E+00 8% 5.47E-01 0% 1.36E+00 0% 11 Sentinel-2A No manoeuvre 1.64E+00 48% 1.27E+00 10% 2.08E+00 27% 11 Sentinel-2A No manoeuvre 1.61E+00 45% 1.11E+00 0% 1.95E+00 14% 10 Sentinel-2A No manoeuvre 6.44E-01 0% 1.95E+00 70% 2.06E+00 25% 10 Sentinel-2A No manoeuvre 6.75E-01 0% 1.40E+00 25% 1.72E+00 0% 12 Sentinel-2A No manoeuvre 4.18E-02 0% 1.60E+00 44% 1.60E+00 0% 12 Sentinel-2A No manoeuvre 7.93E-01 0% 2.52E-01 0% 8.32E-01 0% 10 Sentinel-2A No manoeuvre 6.49E-01 0% 2.69E+00 95% 2.79E+00 80% 10 Sentinel-2A No manoeuvre 1.68E+00 51% 1.51E+00 36% 2.26E+00 45% 5 Sentinel-2A No manoeuvre 1.49E+01 100% 5.71E-01 0% 1.49E+01 100% 3 Sentinel-2A No manoeuvre 2.43E+00 90% 1.00E+00 0% 2.63E+00 72% 10 Sentinel-2A Tangential Low -2 1.03E+00 0% 1.03E+00 0% 1.45E+00 0% 10 Sentinel-2A Tangential Low -6 5.72E-01 0% 9.34E-01 0% 1.15E+00 0% 10 Sentinel-2A Tangential Low -12 9.93E-01 0% 1.10E+00 0% 1.37E+00 0% 10 Sentinel-2A Tangential Medium -2 3.93E+00 100% 2.29E+00 86% 4.55E+00 100% 10 Sentinel-2A Tangential Medium -6 3.65E+00 100% 2.12E+00 79% 3.94E+00 99% 10 Sentinel-2A Tangential Medium -12 6.08E+00 100% 8.31E-01 0% 6.13E+00 100% 9 Sentinel-2A Tangential High -2 1.81E+01 100% 1.12E+00 0% 1.81E+01 100% 10 Sentinel-2A Tangential High -6 1.83E+01 100% 4.40E+00 100% 1.84E+01 100% 9 Sentinel-2A Tangential High -12 2.43E+01 100% 3.70E+00 100% 2.43E+01 100% 10 Sentinel-2A Out-of-plane Low -2 6.38E+00 100% 1.29E+00 13% 6.52E+00 100% 10 Sentinel-2A Out-of-plane Low -6 8.59E+00 100% 1.44E+00 29% 8.72E+00 100% 10 Sentinel-2A Out-of-plane Low -12 5.70E+00 100% 8.42E-01 0% 5.77E+00 100% 10 Sentinel-2A Out-of-plane Medium -2 1.22E+01 100% 1.40E+00 25% 1.23E+01 100% 10 Sentinel-2A Out-of-plane Medium -6 1.23E+01 100% 9.33E-01 0% 1.24E+01 100% 10 Sentinel-2A Out-of-plane Medium -12 1.30E+01 100% 1.22E+00 5% 1.31E+01 100% 10 Sentinel-2A Out-of-plane High -2 2.82E+01 100% 1.27E+00 11% 2.82E+01 100% 10 Sentinel-2A Out-of-plane High -6 2.39E+01 100% 1.61E+00 45% 2.39E+01 100% 10 Sentinel-2A Out-of-plane High -12 2.85E+01 100% 1.05E+00 0% 2.86E+01 100% 10 Sentinel-2A Hybrid Low -2 7.16E+00 100% 1.33E+00 17% 7.30E+00 100% 10 Sentinel-2A Hybrid Low -6 6.04E+00 100% 1.04E+00 0% 6.04E+00 100% 9 Sentinel-2A Hybrid Low -12 8.99E+00 100% 1.89E+00 66% 8.99E+00 100% 9 Sentinel-2A Hybrid Medium -2 3.71E+01 100% 2.05E+00 75% 3.71E+01 100% 9 Sentinel-2A Hybrid Medium -6 3.42E+01 100% 4.40E+00 100% 3.42E+01 100% 10 Sentinel-2A Hybrid Medium -12 5.02E+01 100% 9.37E+00 100% 5.02E+01 100% 10 Sentinel-2A Hybrid High -2 1.29E+02 100% 3.40E+00 99% 1.29E+02 100% 10 Sentinel-2A Hybrid High -6 2.10E+02 100% 1.83E+01 100% 2.10E+02 100% 10 Sentinel-2A Hybrid High -12 4.01E+02 100% 3.19E+01 100% 4.01E+02 100% 10 Table A.7 Sentinel-2A Reachability analysis and probability from MD. 66 Chapter A. Distances and probabilities tables Scenario KLD(R,Rr) PR KLD(el,az) PR plots Swarm-C Tangential Medium -2 2.76E+01 0% 2.97E+05 100% 7 Swarm-C Tangential Medium -6 1.33E+01 0% 4.43E+05 100% 7 Swarm-C Tangential Medium -12 1.40E+01 0% 2.81E+05 100% 7 Swarm-C Tangential High -2 8.17E+02 99% 3.32E+05 100% 7 Swarm-C Tangential High -6 8.49E+01 15% 5.23E+05 100% 7 Swarm-C Tangential High -12 2.22E+03 100% 3.87E+05 100% 7 Swarm-C Out-of-plane Low -2 1.80E+03 100% 2.24E+05 100% 7 Swarm-C Out-of-plane Low -6 8.58E+02 99% 6.36E+05 100% 7 Swarm-C Out-of-plane Low -12 1.70E+03 100% 4.11E+05 100% 7 Swarm-C Out-of-plane Medium -2 7.27E+03 100% 3.50E+05 100% 7 Swarm-C Out-of-plane Medium -6 3.84E+03 100% 5.21E+05 100% 7 Swarm-C Out-of-plane Medium -12 7.15E+03 100% 2.51E+05 100% 7 Swarm-C Out-of-plane High -2 2.92E+04 100% 3.79E+05 100% 7 Swarm-C Out-of-plane High -6 1.68E+04 100% 4.15E+05 100% 7 Swarm-C Out-of-plane High -12 2.62E+04 100% 3.15E+05 100% 6 Swarm-C Hybrid Low -2 7.62E+01 10% 2.45E+05 100% 7 Swarm-C Hybrid Low -6 1.12E+01 0% 2.78E+05 100% 7 Swarm-C Hybrid Low -12 1.01E+02 25% 2.37E+05 100% 7 Swarm-C Hybrid Medium -2 2.31E+03 100% 2.49E+05 100% 7 Swarm-C Hybrid Medium -6 1.11E+03 100% 4.44E+05 100% 7 Swarm-C Hybrid Medium -12 2.06E+04 100% 4.90E+05 100% 6 Swarm-C Hybrid High -2 4.74E+04 100% 2.24E+05 100% 7 Swarm-C Hybrid High -6 2.40E+05 100% 1.85E+06 100% 7 Swarm-C Hybrid High -12 3.54E+06 100% 3.19E+06 100% 7 Table A.14 Swarm-C Reachability analysis and probability from KLD. Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR plots NM 8.64E-01 4.21E-01 0% 4.29E+00 1.37E-02 16% 8.78E+00 1.53E-04 17% 6 NM 3.68E+00 2.52E-02 9% 3.94E+00 1.95E-02 12% 1.19E+01 7.03E-06 33% 5 NM 3.60E+00 2.73E-02 8% 2.98E+00 5.10E-02 1% 1.08E+01 1.98E-05 28% 7 NM 2.30E+00 1.00E-01 0% 3.10E+00 4.49E-02 2% 9.57E+00 6.96E-05 22% 6 NM 2.25E+00 1.06E-01 0% 3.84E+00 2.15E-02 11% 1.08E+01 2.12E-05 28% 7 NM 2.91E+00 5.47E-02 0% 3.03E+00 4.81E-02 1% 9.97E+00 4.69E-05 24% 4 NM 2.64E+00 7.11E-02 0% 3.89E+00 2.04E-02 12% 1.07E+01 2.27E-05 27% 5 NM 2.43E+00 8.85E-02 0% 3.65E+00 2.59E-02 9% 1.05E+01 2.86E-05 26% 7 NM 9.59E+00 6.84E-05 58% 3.18E+00 4.18E-02 3% 1.78E+01 1.85E-08 54% 5 NM 2.36E+00 9.40E-02 0% 3.75E+00 2.35E-02 10% 1.05E+01 2.82E-05 26% 5 NM 3.66E+00 2.57E-02 9% 2.81E+00 6.04E-02 0% 1.08E+01 2.09E-05 28% 6 NM 2.21E+00 1.09E-01 0% 3.26E+00 3.84E-02 4% 9.54E+00 7.22E-05 21% 6 NM 6.49E+00 1.52E-03 37% 5.12E+00 5.98E-03 25% 1.41E+01 7.29E-07 42% 6 NM 2.47E+00 8.47E-02 0% 3.42E+00 3.27E-02 6% 1.01E+01 4.13E-05 24% 6 NM 3.65E+00 2.60E-02 9% 3.30E+00 3.71E-02 5% 1.11E+01 1.47E-05 29% 4 NM 2.51E+00 8.09E-02 0% 3.43E+00 3.23E-02 6% 1.01E+01 4.12E-05 24% 5 NM 2.50E+00 8.20E-02 0% 4.06E+00 1.73E-02 14% 1.12E+01 1.30E-05 30% 6 NM 3.09E+00 4.54E-02 2% 3.36E+00 3.47E-02 6% 1.05E+01 2.76E-05 26% 7 NM 2.45E+00 8.63E-02 0% 3.59E+00 2.75E-02 8% 1.02E+01 3.64E-05 25% 6 TL -2 2.76E+00 6.32E-02 0% 3.21E+00 4.03E-02 4% 1.04E+01 3.15E-05 26% 7 TL -6 2.60E+00 7.43E-02 0% 3.29E+00 3.73E-02 5% 1.03E+01 3.41E-05 25% 7 TL -12 2.61E+00 7.37E-02 0% 3.06E+00 4.70E-02 2% 1.02E+01 3.56E-05 25% 7 TM -2 3.73E+00 2.39E-02 10% 3.31E+00 3.65E-02 5% 1.15E+01 1.03E-05 31% 7 TM -6 2.88E+00 5.59E-02 0% 3.24E+00 3.90E-02 4% 1.08E+01 2.08E-05 28% 7 TM -12 3.02E+00 4.88E-02 1% 2.95E+00 5.22E-02 0% 1.05E+01 2.85E-05 26% 7 TH -2 4.88E+01 6.47E-22 100% 2.97E+00 5.11E-02 1% 5.62E+01 3.81E-25 96% 7 TH -6 8.05E+00 3.19E-04 49% 3.28E+00 3.77E-02 4% 1.58E+01 1.39E-07 48% 7 TH -12 1.35E+02 3.78E-59 100% 5.98E+00 2.53E-03 33% 1.42E+02 2.14E-62 100% 7 OL -2 1.06E+02 8.03E-47 100% 3.14E+00 4.33E-02 3% 1.14E+02 4.81E-50 100% 7 OL -6 5.23E+01 1.97E-23 100% 3.43E+00 3.25E-02 6% 6.01E+01 7.98E-27 97% 7 OL -12 9.69E+01 8.04E-43 100% 3.33E+00 3.57E-02 5% 1.05E+02 3.52E-46 100% 7 OM -2 4.17E+02 9.56E-182 100% 3.07E+00 4.63E-02 2% 4.24E+02 5.47E-185 100% 7 OM -6 2.22E+02 5.69E-97 100% 3.34E+00 3.53E-02 5% 2.29E+02 2.36E-100 100% 7 A.2 Small manoeuvres 67 Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR plots OM -12 4.15E+02 5.39E-181 100% 3.18E+00 4.15E-02 3% 4.23E+02 2.50E-184 100% 7 OH -2 1.69E+03 0.00E+00 100% 3.01E+00 4.93E-02 1% 1.69E+03 0.00E+00 100% 7 OH -6 9.63E+02 0.00E+00 100% 3.26E+00 3.82E-02 4% 9.71E+02 0.00E+00 100% 7 OH -12 1.03E+03 0.00E+00 100% 3.13E+00 4.39E-02 3% 1.04E+03 0.00E+00 100% 6 HL -2 6.57E+00 1.40E-03 38% 2.92E+00 5.38E-02 0% 1.40E+01 8.00E-07 42% 7 HL -6 2.79E+00 6.16E-02 0% 2.94E+00 5.28E-02 0% 1.03E+01 3.51E-05 25% 7 HL -12 8.34E+00 2.40E-04 51% 2.97E+00 5.11E-02 1% 1.59E+01 1.31E-07 48% 7 HM -2 1.34E+02 7.26E-59 100% 2.93E+00 5.32E-02 0% 1.41E+02 4.04E-62 100% 7 HM -6 7.06E+01 2.19E-31 100% 6.96E+00 9.52E-04 41% 7.83E+01 1.02E-34 99% 7 HM -12 1.19E+03 0.00E+00 100% 1.28E+01 2.65E-06 73% 1.19E+03 0.00E+00 100% 6 HH -2 2.76E+03 0.00E+00 100% 6.11E+00 2.23E-03 34% 2.77E+03 0.00E+00 100% 7 HH -6 1.95E+04 0.00E+00 100% 5.46E+01 1.89E-24 100% 1.95E+04 0.00E+00 100% 7 HH -12 4.03E+05 0.00E+00 100% 2.22E+02 3.01E-97 100% 4.03E+05 0.00E+00 100% 7 Table A.15 Swarm-C Reachability analysis and probability from BD. A.2 Small manoeuvres In this section, the distances and the probabilities are presented in tables with the same structure as those of the previous section, except for the last column. Scenarios without manoeuvres are not necessary in this analysis, so the tables does not present the number of plots of each segment. Sentinel-1A Scenario Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR Sentinel-1A Tangential Low -2 9.22E-02 0% 1.83E+00 62% 1.84E+00 1% Sentinel-1A Tangential Low -6 1.65E-01 0% 1.19E+00 1% 1.22E+00 0% Sentinel-1A Tangential Low -12 1.24E+00 7% 1.87E+00 65% 2.24E+00 43% Sentinel-1A Tangential Medium -2 2.07E-01 0% 6.95E-01 0% 7.10E-01 0% Sentinel-1A Tangential Medium -6 1.08E+00 0% 1.69E+00 52% 2.00E+00 18% Sentinel-1A Tangential Medium -12 8.21E-01 0% 1.84E+00 63% 2.01E+00 20% Sentinel-1A Tangential High -2 1.80E+00 60% 3.31E+00 99% 3.72E+00 98% Sentinel-1A Tangential High -6 1.32E+00 17% 8.42E-01 0% 1.50E+00 0% Sentinel-1A Tangential High -12 1.22E+00 4% 5.24E-01 0% 1.36E+00 0% Sentinel-1A Out-of-plane Low -2 1.37E-01 0% 6.22E-01 0% 6.61E-01 0% Sentinel-1A Out-of-plane Low -6 1.03E+00 0% 8.50E-01 0% 1.34E+00 0% Sentinel-1A Out-of-plane Low -12 1.00E+00 0% 5.65E-01 0% 1.15E+00 0% Sentinel-1A Out-of-plane Medium -2 2.12E-01 0% 1.74E+00 56% 1.74E+00 0% Sentinel-1A Out-of-plane Medium -6 6.13E-01 0% 1.42E+00 27% 1.54E+00 0% Sentinel-1A Out-of-plane Medium -12 1.85E-01 0% 2.28E+00 85% 2.30E+00 48% Sentinel-1A Out-of-plane High -2 1.66E+00 50% 6.23E-01 0% 1.79E+00 0% Sentinel-1A Out-of-plane High -6 2.47E+00 91% 6.17E-01 0% 2.55E+00 67% Sentinel-1A Out-of-plane High -12 2.00E+00 73% 2.11E+00 78% 2.91E+00 85% Sentinel-1A Hybrid Low -2 1.19E+00 2% 3.58E-01 0% 1.24E+00 0% Sentinel-1A Hybrid Low -6 1.25E+00 9% 3.81E-01 0% 1.30E+00 0% Sentinel-1A Hybrid Low -12 4.83E-01 0% 1.07E+00 0% 1.20E+00 0% Sentinel-1A Hybrid Medium -2 1.61E+00 46% 1.07E+00 0% 1.96E+00 14% Sentinel-1A Hybrid Medium -6 1.47E+00 32% 1.63E+00 47% 2.03E+00 22% Sentinel-1A Hybrid Medium -12 2.57E+00 93% 1.48E+00 33% 2.80E+00 81% Sentinel-1A Hybrid High -2 2.55E+00 92% 9.29E-01 0% 2.84E+00 82% Sentinel-1A Hybrid High -6 3.17E+00 99% 3.21E+00 99% 4.69E+00 100% Sentinel-1A Hybrid High -12 4.31E+00 100% 1.45E+00 30% 4.43E+00 100% Table A.16 Sentinel-1A Reachability analysis and probability from MD for small manoeuvres. 68 Chapter A. Distances and probabilities tables Scenario KLD(R,Rr) PR KLD(el,az) PR Sentinel-1A Tangential Low -2 2.51E+01 0% 8.98E+05 100% Sentinel-1A Tangential Low -6 2.55E+01 0% 4.00E+05 100% Sentinel-1A Tangential Low -12 6.14E+01 0% 1.13E+06 100% Sentinel-1A Tangential Medium -2 2.54E+01 0% 2.61E+05 100% Sentinel-1A Tangential Medium -6 5.12E+01 0% 2.60E+05 100% Sentinel-1A Tangential Medium -12 3.79E+01 0% 1.11E+06 100% Sentinel-1A Tangential High -2 9.35E+01 21% 9.78E+05 100% Sentinel-1A Tangential High -6 5.29E+01 0% 2.57E+05 100% Sentinel-1A Tangential High -12 3.26E+01 0% 3.24E+05 100% Sentinel-1A Out-of-plane Low -2 2.49E+01 0% 3.00E+05 100% Sentinel-1A Out-of-plane Low -6 5.02E+01 0% 4.39E+05 100% Sentinel-1A Out-of-plane Low -12 4.91E+01 0% 3.29E+05 100% Sentinel-1A Out-of-plane Medium -2 2.56E+01 0% 9.76E+05 100% Sentinel-1A Out-of-plane Medium -6 3.35E+01 0% 7.45E+05 100% Sentinel-1A Out-of-plane Medium -12 2.50E+01 0% 1.48E+06 100% Sentinel-1A Out-of-plane High -2 8.83E+01 17% 2.59E+05 100% Sentinel-1A Out-of-plane High -6 1.69E+02 54% 3.19E+05 100% Sentinel-1A Out-of-plane High -12 1.19E+02 34% 1.30E+06 100% Sentinel-1A Hybrid Low -2 5.80E+01 0% 2.58E+05 100% Sentinel-1A Hybrid Low -6 6.16E+01 0% 2.61E+05 100% Sentinel-1A Hybrid Low -12 2.85E+01 0% 5.01E+05 100% Sentinel-1A Hybrid Medium -2 5.49E+01 0% 5.51E+05 100% Sentinel-1A Hybrid Medium -6 2.71E+01 0% 2.84E+05 100% Sentinel-1A Hybrid Medium -12 8.45E+01 15% 2.67E+05 100% Sentinel-1A Hybrid High -2 3.73E+01 0% 4.10E+05 100% Sentinel-1A Hybrid High -6 4.89E+01 0% 2.61E+06 100% Sentinel-1A Hybrid High -12 4.67E+01 0% 4.97E+05 100% Table A.17 Sentinel-1A Reachability analysis and probability from KLD for small manoeuvres. Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR TL -2 2.53E+00 7.94E-02 0% 3.87E+00 2.09E-02 12% 1.04E+01 2.93E-05 26% TL -6 2.53E+00 7.93E-02 0% 3.63E+00 2.66E-02 9% 1.02E+01 3.71E-05 25% TL -12 2.73E+00 6.55E-02 0% 3.89E+00 2.05E-02 12% 1.06E+01 2.38E-05 27% TM -2 2.54E+00 7.89E-02 0% 3.51E+00 2.99E-02 7% 1.01E+01 4.19E-05 24% TM -6 2.68E+00 6.89E-02 0% 3.81E+00 2.22E-02 11% 1.05E+01 2.72E-05 26% TM -12 2.61E+00 7.32E-02 0% 3.87E+00 2.08E-02 12% 1.05E+01 2.70E-05 26% TH -2 2.94E+00 5.30E-02 0% 4.82E+00 8.05E-03 22% 1.17E+01 7.91E-06 32% TH -6 2.75E+00 6.38E-02 0% 3.53E+00 2.92E-02 8% 1.03E+01 3.37E-05 25% TH -12 2.72E+00 6.59E-02 0% 3.48E+00 3.07E-02 7% 1.03E+01 3.54E-05 25% OL -2 2.54E+00 7.92E-02 0% 3.50E+00 3.03E-02 7% 1.01E+01 4.23E-05 24% OL -6 2.67E+00 6.94E-02 0% 3.54E+00 2.91E-02 8% 1.02E+01 3.56E-05 25% OL -12 2.66E+00 6.99E-02 0% 3.49E+00 3.04E-02 7% 1.02E+01 3.77E-05 25% OM -2 2.54E+00 7.89E-02 0% 3.83E+00 2.18E-02 11% 1.04E+01 3.06E-05 26% OM -6 2.58E+00 7.59E-02 0% 3.70E+00 2.47E-02 10% 1.03E+01 3.33E-05 25% OM -12 2.53E+00 7.93E-02 0% 4.10E+00 1.66E-02 14% 1.07E+01 2.31E-05 27% OH -2 2.87E+00 5.64E-02 0% 3.49E+00 3.04E-02 7% 1.04E+01 3.00E-05 26% OH -6 3.30E+00 3.71E-02 5% 3.49E+00 3.04E-02 7% 1.08E+01 1.99E-05 28% OH -12 3.03E+00 4.82E-02 1% 4.00E+00 1.83E-02 13% 1.11E+01 1.55E-05 29% HL -2 2.71E+00 6.66E-02 0% 3.47E+00 3.12E-02 7% 1.02E+01 3.69E-05 25% HL -6 2.73E+00 6.53E-02 0% 3.47E+00 3.12E-02 7% 1.02E+01 3.62E-05 25% HL -12 2.57E+00 7.69E-02 0% 3.59E+00 2.76E-02 8% 1.02E+01 3.73E-05 25% HM -2 2.86E+00 5.73E-02 0% 3.59E+00 2.76E-02 8% 1.05E+01 2.76E-05 26% HM -6 2.81E+00 6.04E-02 0% 3.78E+00 2.27E-02 11% 1.05E+01 2.67E-05 26% HM -12 3.36E+00 3.48E-02 5% 3.72E+00 2.41E-02 10% 1.10E+01 1.67E-05 29% HH -2 3.34E+00 3.54E-02 5% 3.55E+00 2.86E-02 8% 1.10E+01 1.65E-05 29% A.2 Small manoeuvres 69 Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR HH -6 3.79E+00 2.26E-02 11% 4.73E+00 8.79E-03 21% 1.28E+01 2.86E-06 36% HH -12 4.86E+00 7.77E-03 22% 3.71E+00 2.44E-02 10% 1.25E+01 3.86E-06 35% Table A.18 Sentinel-1A Reachability analysis and probability from BD for small manoeuvres. Sentinel-1B Scenario Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR Sentinel-1B Tangential Low -2 7.90E-01 0% 5.48E-01 0% 9.83E-01 0% Sentinel-1B Tangential Low -6 7.44E-01 0% 1.11E+00 0% 1.36E+00 0% Sentinel-1B Tangential Low -12 4.72E-01 0% 1.90E+00 67% 1.97E+00 15% Sentinel-1B Tangential Medium -2 5.34E-01 0% 7.63E-01 0% 1.03E+00 0% Sentinel-1B Tangential Medium -6 1.20E+00 2% 1.85E+00 64% 2.41E+00 58% Sentinel-1B Tangential Medium -12 6.69E-01 0% 4.23E-01 0% 8.10E-01 0% Sentinel-1B Tangential High -2 6.59E-02 0% 8.28E-01 0% 9.04E-01 0% Sentinel-1B Tangential High -6 2.64E-01 0% 1.82E+00 61% 1.93E+00 11% Sentinel-1B Tangential High -12 6.20E-01 0% 8.49E-01 0% 1.05E+00 0% Sentinel-1B Out-of-plane Low -2 5.10E-01 0% 6.43E-01 0% 8.84E-01 0% Sentinel-1B Out-of-plane Low -6 8.59E-02 0% 1.45E+00 30% 1.62E+00 0% Sentinel-1B Out-of-plane Low -12 7.20E-01 0% 2.24E+00 84% 2.50E+00 64% Sentinel-1B Out-of-plane Medium -2 2.13E+00 79% 3.64E-01 0% 2.17E+00 37% Sentinel-1B Out-of-plane Medium -6 2.24E+00 84% 5.17E-01 0% 2.30E+00 48% Sentinel-1B Out-of-plane Medium -12 9.86E-01 0% 6.64E-01 0% 1.23E+00 0% Sentinel-1B Out-of-plane High -2 6.40E+00 100% 1.96E+00 71% 6.75E+00 100% Sentinel-1B Out-of-plane High -6 5.40E+00 100% 9.54E-01 0% 5.48E+00 100% Sentinel-1B Out-of-plane High -12 6.43E+00 100% 1.14E+00 0% 6.53E+00 100% Sentinel-1B Hybrid Low -2 4.55E-01 0% 2.56E+00 92% 2.61E+00 71% Sentinel-1B Hybrid Low -6 7.66E-01 0% 1.23E+00 6% 1.54E+00 0% Sentinel-1B Hybrid Low -12 9.92E-02 0% 7.75E-01 0% 7.87E-01 0% Sentinel-1B Hybrid Medium -2 3.12E-02 0% 1.03E+00 0% 1.16E+00 0% Sentinel-1B Hybrid Medium -6 1.84E-01 0% 2.15E+00 80% 2.36E+00 54% Sentinel-1B Hybrid Medium -12 5.21E-01 0% 1.97E+00 71% 2.05E+00 24% Sentinel-1B Hybrid High -2 1.11E+00 0% 1.33E+00 17% 1.76E+00 0% Sentinel-1B Hybrid High -6 4.38E-01 0% 2.51E+00 91% 2.56E+00 68% Sentinel-1B Hybrid High -12 6.07E-01 0% 2.22E+00 83% 2.25E+00 44% Table A.19 Sentinel-1B Reachability analysis and probability from MD for small manoeuvres. Scenario KLD(R,Rr) PR KLD(el,az) PR Sentinel-1B Tangential Low -2 2.97E+01 0% 7.24E+05 100% Sentinel-1B Tangential Low -6 2.86E+01 0% 1.38E+06 100% Sentinel-1B Tangential Low -12 2.32E+01 0% 3.01E+06 100% Sentinel-1B Tangential Medium -2 2.44E+01 0% 7.00E+05 100% Sentinel-1B Tangential Medium -6 4.37E+01 0% 7.32E+05 100% Sentinel-1B Tangential Medium -12 2.74E+01 0% 6.93E+05 100% Sentinel-1B Tangential High -2 1.93E+01 0% 8.47E+05 100% Sentinel-1B Tangential High -6 2.06E+01 0% 2.14E+06 100% Sentinel-1B Tangential High -12 2.61E+01 0% 1.01E+06 100% Sentinel-1B Out-of-plane Low -2 2.37E+01 0% 7.46E+05 100% Sentinel-1B Out-of-plane Low -6 1.93E+01 0% 6.78E+05 100% Sentinel-1B Out-of-plane Low -12 2.85E+01 0% 2.56E+06 100% Sentinel-1B Out-of-plane Medium -2 9.53E+01 22% 6.75E+05 100% 70 Chapter A. Distances and probabilities tables Scenario KLD(R,Rr) PR KLD(el,az) PR Sentinel-1B Out-of-plane Medium -6 1.05E+02 27% 7.98E+05 100% Sentinel-1B Out-of-plane Medium -12 3.63E+01 0% 6.90E+05 100% Sentinel-1B Out-of-plane High -2 7.07E+02 98% 1.94E+06 100% Sentinel-1B Out-of-plane High -6 5.14E+02 95% 1.13E+06 100% Sentinel-1B Out-of-plane High -12 7.16E+02 98% 1.39E+06 100% Sentinel-1B Hybrid Low -2 2.26E+01 0% 4.90E+06 100% Sentinel-1B Hybrid Low -6 2.97E+01 0% 1.03E+06 100% Sentinel-1B Hybrid Low -12 1.96E+01 0% 1.07E+06 100% Sentinel-1B Hybrid Medium -2 1.94E+01 0% 6.73E+05 100% Sentinel-1B Hybrid Medium -6 1.93E+01 0% 9.43E+05 100% Sentinel-1B Hybrid Medium -12 2.24E+01 0% 2.25E+06 100% Sentinel-1B Hybrid High -2 3.97E+01 0% 1.72E+06 100% Sentinel-1B Hybrid High -6 2.19E+01 0% 4.05E+06 100% Sentinel-1B Hybrid High -12 2.00E+01 0% 1.58E+06 100% Table A.20 Sentinel-1B Reachability analysis and probability from KLD for small manoeuvres. Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR TL -2 3.11E+00 4.45E-02 2% 3.31E+00 3.66E-02 5% 1.07E+01 2.35E-05 27% TL -6 3.10E+00 4.49E-02 2% 3.43E+00 3.25E-02 6% 1.08E+01 2.10E-05 28% TL -12 3.06E+00 4.68E-02 2% 3.73E+00 2.41E-02 10% 1.10E+01 1.63E-05 29% TM -2 3.07E+00 4.63E-02 2% 3.35E+00 3.51E-02 5% 1.07E+01 2.31E-05 27% TM -6 3.21E+00 4.02E-02 4% 3.70E+00 2.47E-02 10% 1.13E+01 1.28E-05 30% TM -12 3.10E+00 4.52E-02 2% 3.30E+00 3.69E-02 5% 1.06E+01 2.43E-05 27% TH -2 3.04E+00 4.81E-02 1% 3.36E+00 3.49E-02 5% 1.06E+01 2.40E-05 27% TH -6 3.05E+00 4.76E-02 1% 3.69E+00 2.51E-02 9% 1.10E+01 1.66E-05 29% TH -12 3.09E+00 4.56E-02 2% 3.37E+00 3.44E-02 6% 1.07E+01 2.29E-05 27% OL -2 3.07E+00 4.66E-02 2% 3.32E+00 3.60E-02 5% 1.06E+01 2.40E-05 27% OL -6 3.04E+00 4.80E-02 1% 3.53E+00 2.92E-02 8% 1.09E+01 1.91E-05 28% OL -12 3.10E+00 4.50E-02 2% 3.91E+00 2.01E-02 12% 1.13E+01 1.21E-05 30% OM -2 3.60E+00 2.73E-02 8% 3.29E+00 3.73E-02 5% 1.11E+01 1.47E-05 29% OM -6 3.66E+00 2.57E-02 9% 3.31E+00 3.66E-02 5% 1.12E+01 1.37E-05 30% OM -12 3.16E+00 4.24E-02 3% 3.33E+00 3.57E-02 5% 1.07E+01 2.18E-05 27% OH -2 8.16E+00 2.85E-04 50% 3.75E+00 2.35E-02 10% 1.62E+01 8.90E-08 49% OH -6 6.68E+00 1.26E-03 39% 3.39E+00 3.38E-02 6% 1.43E+01 6.17E-07 43% OH -12 8.20E+00 2.75E-04 50% 3.43E+00 3.23E-02 6% 1.59E+01 1.29E-07 48% HL -2 3.06E+00 4.69E-02 2% 4.09E+00 1.67E-02 14% 1.14E+01 1.13E-05 30% HL -6 3.11E+00 4.45E-02 2% 3.46E+00 3.13E-02 7% 1.08E+01 1.96E-05 28% HL -12 3.04E+00 4.79E-02 1% 3.35E+00 3.51E-02 5% 1.06E+01 2.44E-05 27% HM -2 3.03E+00 4.81E-02 1% 3.41E+00 3.31E-02 6% 1.07E+01 2.24E-05 27% HM -6 3.04E+00 4.81E-02 1% 3.84E+00 2.14E-02 11% 1.12E+01 1.32E-05 30% HM -12 3.07E+00 4.66E-02 2% 3.76E+00 2.33E-02 10% 1.11E+01 1.57E-05 29% HH -2 3.19E+00 4.13E-02 3% 3.49E+00 3.04E-02 7% 1.09E+01 1.80E-05 28% HH -6 3.06E+00 4.68E-02 2% 4.06E+00 1.72E-02 14% 1.14E+01 1.16E-05 30% HH -12 3.09E+00 4.57E-02 2% 3.89E+00 2.04E-02 12% 1.12E+01 1.40E-05 30% Table A.21 Sentinel-1B Reachability analysis and probability from BD for small manoeuvres. Sentinel-2A Scenario Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR Sentinel-2A Tangential Low -2 1.26E+00 0% 2.62E+00 100% 2.95E+00 100% Sentinel-2A Tangential Low -6 9.33E-01 0% 4.32E-01 0% 1.03E+00 0% Sentinel-2A Tangential Low -12 9.36E-01 0% 6.94E-01 0% 1.19E+00 0% A.2 Small manoeuvres 71 Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR Sentinel-2A Tangential Medium -2 7.10E-01 0% 5.79E-01 0% 9.16E-01 0% Sentinel-2A Tangential Medium -6 1.02E+00 0% 1.26E+00 0% 1.63E+00 0% Sentinel-2A Tangential Medium -12 7.27E-01 0% 1.12E+00 0% 1.38E+00 0% Sentinel-2A Tangential High -2 1.22E+00 0% 6.69E-01 0% 1.40E+00 0% Sentinel-2A Tangential High -6 6.62E-01 0% 1.62E+00 0% 1.73E+00 0% Sentinel-2A Tangential High -12 5.79E-01 0% 1.33E-01 0% 5.89E-01 0% Sentinel-2A Out-of-plane Low -2 8.77E-02 0% 2.69E+00 100% 2.70E+00 100% Sentinel-2A Out-of-plane Low -6 7.83E-01 0% 6.98E-01 0% 1.05E+00 0% Sentinel-2A Out-of-plane Low -12 1.15E+00 0% 1.27E+00 0% 1.71E+00 0% Sentinel-2A Out-of-plane Medium -2 2.38E-01 0% 7.14E-01 0% 7.71E-01 0% Sentinel-2A Out-of-plane Medium -6 6.50E-01 0% 5.33E-01 0% 8.41E-01 0% Sentinel-2A Out-of-plane Medium -12 4.17E-01 0% 1.34E+00 0% 1.43E+00 0% Sentinel-2A Out-of-plane High -2 3.83E+00 100% 2.71E+00 100% 4.70E+00 100% Sentinel-2A Out-of-plane High -6 2.67E+00 100% 1.20E+00 0% 2.93E+00 100% Sentinel-2A Out-of-plane High -12 4.34E+00 100% 1.72E+00 100% 4.67E+00 100% Sentinel-2A Hybrid Low -2 1.15E+00 0% 1.63E+00 0% 2.01E+00 0% Sentinel-2A Hybrid Low -6 6.49E-01 0% 1.39E+00 0% 1.53E+00 0% Sentinel-2A Hybrid Low -12 1.73E+00 100% 1.27E+00 0% 2.14E+00 0% Sentinel-2A Hybrid Medium -2 1.43E+00 0% 8.31E-01 0% 1.65E+00 0% Sentinel-2A Hybrid Medium -6 1.69E+00 100% 5.83E-01 0% 1.78E+00 0% Sentinel-2A Hybrid Medium -12 1.58E+00 0% 3.41E-01 0% 1.60E+00 0% Sentinel-2A Hybrid High -2 1.58E+00 0% 1.50E+00 0% 2.17E+00 0% Sentinel-2A Hybrid High -6 1.28E+00 0% 4.40E-01 0% 1.44E+00 0% Sentinel-2A Hybrid High -12 1.98E+00 100% 6.32E-01 0% 2.00E+00 0% Table A.22 Sentinel-2A Reachability analysis and probability from MD for small manoeuvres. Scenario KLD(R,Rr) PR KLD(el,az) PR Sentinel-2A Tangential Low -2 9.98E+00 0% 3.48E+05 100% Sentinel-2A Tangential Low -6 7.89E+00 0% 1.85E+05 100% Sentinel-2A Tangential Low -12 7.94E+00 0% 1.69E+05 100% Sentinel-2A Tangential Medium -2 6.79E+00 0% 2.09E+05 100% Sentinel-2A Tangential Medium -6 8.39E+00 0% 4.07E+05 100% Sentinel-2A Tangential Medium -12 6.74E+00 0% 2.38E+05 100% Sentinel-2A Tangential High -2 9.82E+00 0% 1.70E+05 100% Sentinel-2A Tangential High -6 6.39E+00 0% 1.74E+05 100% Sentinel-2A Tangential High -12 5.59E+00 0% 1.71E+05 100% Sentinel-2A Out-of-plane Low -2 5.22E+00 0% 1.23E+06 100% Sentinel-2A Out-of-plane Low -6 7.11E+00 0% 2.20E+05 100% Sentinel-2A Out-of-plane Low -12 9.32E+00 0% 4.10E+05 100% Sentinel-2A Out-of-plane Medium -2 5.36E+00 0% 1.93E+05 100% Sentinel-2A Out-of-plane Medium -6 6.48E+00 0% 1.97E+05 100% Sentinel-2A Out-of-plane Medium -12 5.75E+00 0% 2.83E+05 100% Sentinel-2A Out-of-plane High -2 5.06E+01 0% 7.40E+05 100% Sentinel-2A Out-of-plane High -6 2.72E+01 0% 3.91E+05 100% Sentinel-2A Out-of-plane High -12 6.38E+01 0% 6.05E+05 100% Sentinel-2A Hybrid Low -2 9.27E+00 0% 2.99E+05 100% Sentinel-2A Hybrid Low -6 6.45E+00 0% 4.61E+05 100% Sentinel-2A Hybrid Low -12 1.44E+01 0% 3.62E+05 100% Sentinel-2A Hybrid Medium -2 1.15E+01 0% 2.82E+05 100% Sentinel-2A Hybrid Medium -6 1.35E+01 0% 2.11E+05 100% Sentinel-2A Hybrid Medium -12 1.10E+01 0% 1.78E+05 100% Sentinel-2A Hybrid High -2 1.28E+01 0% 1.75E+05 100% Sentinel-2A Hybrid High -6 7.93E+00 0% 1.71E+05 100% 72 Chapter A. Distances and probabilities tables Scenario KLD(R,Rr) PR KLD(el,az) PR Sentinel-2A Hybrid High -12 9.12E+00 0% 1.68E+05 100% Table A.23 Sentinel-2A Reachability analysis and probability from KLD for small manoeuvres. Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR TL -2 2.21E+00 1.10E-01 0% 4.13E+00 1.61E-02 17% 1.07E+01 2.16E-05 30% TL -6 2.13E+00 1.19E-01 0% 3.29E+00 3.71E-02 7% 9.80E+00 5.56E-05 25% TL -12 2.13E+00 1.19E-01 0% 3.33E+00 3.57E-02 8% 9.83E+00 5.37E-05 25% TM -2 2.08E+00 1.25E-01 0% 3.32E+00 3.62E-02 8% 9.76E+00 5.76E-05 25% TM -6 2.15E+00 1.17E-01 0% 3.47E+00 3.12E-02 9% 9.99E+00 4.58E-05 26% TM -12 2.08E+00 1.25E-01 0% 3.43E+00 3.25E-02 9% 9.89E+00 5.05E-05 25% TH -2 2.21E+00 1.10E-01 0% 3.33E+00 3.59E-02 8% 9.91E+00 4.99E-05 26% TH -6 2.07E+00 1.26E-01 0% 3.60E+00 2.74E-02 11% 1.00E+01 4.39E-05 26% TH -12 2.06E+00 1.27E-01 0% 3.27E+00 3.78E-02 7% 9.70E+00 6.10E-05 25% OL -2 2.02E+00 1.33E-01 0% 4.17E+00 1.54E-02 17% 1.06E+01 2.57E-05 29% OL -6 2.10E+00 1.23E-01 0% 3.33E+00 3.58E-02 8% 9.80E+00 5.53E-05 25% OL -12 2.18E+00 1.13E-01 0% 3.47E+00 3.10E-02 10% 1.00E+01 4.44E-05 26% OM -2 2.03E+00 1.32E-01 0% 3.33E+00 3.57E-02 8% 9.74E+00 5.91E-05 25% OM -6 2.07E+00 1.26E-01 0% 3.31E+00 3.66E-02 8% 9.74E+00 5.86E-05 25% OM -12 2.04E+00 1.30E-01 0% 3.49E+00 3.04E-02 10% 9.92E+00 4.94E-05 26% OH -2 3.86E+00 2.12E-02 14% 4.19E+00 1.52E-02 18% 1.24E+01 4.03E-06 37% OH -6 2.91E+00 5.46E-02 3% 3.45E+00 3.16E-02 9% 1.07E+01 2.20E-05 29% OH -12 4.37E+00 1.26E-02 20% 3.64E+00 2.62E-02 12% 1.24E+01 4.20E-06 37% HL -2 2.18E+00 1.13E-01 0% 3.60E+00 2.73E-02 11% 1.02E+01 3.86E-05 27% HL -6 2.07E+00 1.26E-01 0% 3.51E+00 2.98E-02 10% 9.95E+00 4.76E-05 26% HL -12 2.39E+00 9.13E-02 0% 3.47E+00 3.12E-02 9% 1.02E+01 3.59E-05 27% HM -2 2.27E+00 1.03E-01 0% 3.36E+00 3.48E-02 8% 1.00E+01 4.54E-05 26% HM -6 2.38E+00 9.29E-02 0% 3.31E+00 3.64E-02 8% 1.01E+01 4.30E-05 26% HM -12 2.33E+00 9.71E-02 0% 3.29E+00 3.74E-02 7% 9.98E+00 4.62E-05 26% HH -2 2.33E+00 9.75E-02 0% 3.55E+00 2.87E-02 10% 1.02E+01 3.54E-05 27% HH -6 2.22E+00 1.08E-01 0% 3.29E+00 3.72E-02 7% 9.92E+00 4.94E-05 26% HH -12 2.51E+00 8.14E-02 0% 3.32E+00 3.61E-02 8% 1.02E+01 3.87E-05 27% Table A.24 Sentinel-2A Reachability analysis and probability from BD for small manoeuvres. Sentinel-2B Scenario Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR Sentinel-2B Tangential Low -2 9.38E-01 0% 7.95E-01 0% 1.23E+00 0% Sentinel-2B Tangential Low -6 8.67E-02 0% 1.37E+00 22% 1.41E+00 0% Sentinel-2B Tangential Low -12 2.06E-01 0% 9.26E-01 0% 9.59E-01 0% Sentinel-2B Tangential Medium -2 1.34E+00 18% 9.12E-01 0% 1.62E+00 0% Sentinel-2B Tangential Medium -6 1.17E+00 0% 2.20E+00 82% 2.49E+00 63% Sentinel-2B Tangential Medium -12 2.26E-01 0% 1.48E+00 33% 1.49E+00 0% Sentinel-2B Tangential High -2 1.26E-01 0% 4.66E-01 0% 4.82E-01 0% Sentinel-2B Tangential High -6 2.10E-01 0% 5.68E-01 0% 6.41E-01 0% Sentinel-2B Tangential High -12 6.80E-01 0% 1.47E+00 32% 1.58E+00 0% Sentinel-2B Out-of-plane Low -2 2.38E-01 0% 2.06E+00 76% 2.12E+00 31% Sentinel-2B Out-of-plane Low -6 1.03E+00 0% 1.86E+00 64% 2.13E+00 33% Sentinel-2B Out-of-plane Low -12 2.08E-01 0% 3.82E-01 0% 4.41E-01 0% Sentinel-2B Out-of-plane Medium -2 8.87E-01 0% 1.70E+00 53% 1.92E+00 10% Sentinel-2B Out-of-plane Medium -6 1.12E+00 0% 1.07E+00 0% 1.56E+00 0% Sentinel-2B Out-of-plane Medium -12 1.15E-01 0% 7.58E-01 0% 7.89E-01 0% Sentinel-2B Out-of-plane High -2 1.61E+00 45% 1.18E+00 1% 2.00E+00 19% A.2 Small manoeuvres 73 Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR Sentinel-2B Out-of-plane High -6 6.73E-01 0% 1.36E+00 21% 1.54E+00 0% Sentinel-2B Out-of-plane High -12 1.66E+00 50% 7.82E-01 0% 1.84E+00 1% Sentinel-2B Hybrid Low -2 1.33E+00 17% 1.07E+00 0% 1.71E+00 0% Sentinel-2B Hybrid Low -6 1.13E+00 0% 1.31E+00 15% 1.74E+00 0% Sentinel-2B Hybrid Low -12 9.27E-01 0% 2.05E+00 76% 2.25E+00 44% Sentinel-2B Hybrid Medium -2 8.79E-01 0% 2.58E+00 93% 2.82E+00 81% Sentinel-2B Hybrid Medium -6 6.69E-01 0% 8.16E-01 0% 1.13E+00 0% Sentinel-2B Hybrid Medium -12 1.94E+00 70% 2.20E+00 82% 2.88E+00 84% Sentinel-2B Hybrid High -2 6.15E-01 0% 1.22E+00 5% 1.36E+00 0% Sentinel-2B Hybrid High -6 1.34E+00 18% 1.10E+00 0% 1.69E+00 0% Sentinel-2B Hybrid High -12 1.95E+00 70% 7.24E-01 0% 2.00E+00 19% Table A.25 Sentinel-2B Reachability analysis and probability from MD for small manoeuvres. Scenario KLD(R,Rr) PR KLD(el,az) PR Sentinel-2B Tangential Low -2 1.49E+01 0% 5.62E+05 100% Sentinel-2B Tangential Low -6 8.94E+00 0% 3.89E+05 100% Sentinel-2B Tangential Low -12 9.21E+00 0% 6.56E+05 100% Sentinel-2B Tangential Medium -2 2.09E+01 0% 6.44E+05 100% Sentinel-2B Tangential Medium -6 1.84E+01 0% 2.20E+06 100% Sentinel-2B Tangential Medium -12 9.15E+00 0% 1.15E+06 100% Sentinel-2B Tangential High -2 9.06E+00 0% 4.35E+05 100% Sentinel-2B Tangential High -6 9.05E+00 0% 4.59E+05 100% Sentinel-2B Tangential High -12 1.15E+01 0% 3.88E+05 100% Sentinel-2B Out-of-plane Low -2 9.34E+00 0% 1.06E+06 100% Sentinel-2B Out-of-plane Low -6 1.63E+01 0% 1.51E+06 100% Sentinel-2B Out-of-plane Low -12 9.19E+00 0% 4.07E+05 100% Sentinel-2B Out-of-plane Medium -2 1.43E+01 0% 1.41E+06 100% Sentinel-2B Out-of-plane Medium -6 1.73E+01 0% 5.10E+05 100% Sentinel-2B Out-of-plane Medium -12 9.10E+00 0% 3.92E+05 100% Sentinel-2B Out-of-plane High -2 2.64E+01 0% 9.08E+05 100% Sentinel-2B Out-of-plane High -6 1.20E+01 0% 7.11E+05 100% Sentinel-2B Out-of-plane High -12 2.74E+01 0% 4.96E+05 100% Sentinel-2B Hybrid Low -2 2.10E+01 0% 7.80E+05 100% Sentinel-2B Hybrid Low -6 1.76E+01 0% 5.32E+05 100% Sentinel-2B Hybrid Low -12 1.46E+01 0% 1.76E+06 100% Sentinel-2B Hybrid Medium -2 1.41E+01 0% 5.38E+05 100% Sentinel-2B Hybrid Medium -6 1.14E+01 0% 4.18E+05 100% Sentinel-2B Hybrid Medium -12 3.25E+01 0% 4.01E+05 100% Sentinel-2B Hybrid High -2 1.14E+01 0% 6.61E+05 100% Sentinel-2B Hybrid High -6 1.88E+01 0% 6.96E+05 100% Sentinel-2B Hybrid High -12 2.57E+01 0% 3.96E+05 100% Table A.26 Sentinel-2B Reachability analysis and probability from KLD for small manoeuvres. Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR TL -2 2.57E+00 7.64E-02 0% 3.47E+00 3.10E-02 7% 1.01E+01 4.00E-05 24% TL -6 2.47E+00 8.49E-02 0% 3.62E+00 2.67E-02 9% 1.02E+01 3.75E-05 25% TL -12 2.47E+00 8.48E-02 0% 3.50E+00 3.02E-02 7% 1.01E+01 4.28E-05 24% TM -2 2.69E+00 6.82E-02 0% 3.50E+00 3.03E-02 7% 1.03E+01 3.46E-05 25% TM -6 2.64E+00 7.13E-02 0% 4.00E+00 1.84E-02 13% 1.07E+01 2.21E-05 27% TM -12 2.47E+00 8.45E-02 0% 3.66E+00 2.57E-02 9% 1.02E+01 3.66E-05 25% 74 Chapter A. Distances and probabilities tables Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR TM -2 2.47E+00 8.48E-02 0% 3.42E+00 3.27E-02 6% 9.98E+00 4.64E-05 24% TH -6 2.47E+00 8.46E-02 0% 3.43E+00 3.23E-02 6% 1.00E+01 4.56E-05 24% TH -12 2.53E+00 7.98E-02 0% 3.66E+00 2.57E-02 9% 1.03E+01 3.52E-05 25% OL -2 2.47E+00 8.45E-02 0% 3.92E+00 1.97E-02 12% 1.05E+01 2.74E-05 26% OL -6 2.60E+00 7.42E-02 0% 3.82E+00 2.18E-02 11% 1.05E+01 2.70E-05 26% OL -12 2.47E+00 8.44E-02 0% 3.41E+00 3.31E-02 6% 9.97E+00 4.70E-05 24% OM -2 2.56E+00 7.70E-02 0% 3.75E+00 2.35E-02 10% 1.04E+01 3.04E-05 26% OM -6 2.62E+00 7.29E-02 0% 3.54E+00 2.91E-02 8% 1.02E+01 3.56E-05 25% OM -12 2.47E+00 8.47E-02 0% 3.47E+00 3.13E-02 7% 1.00E+01 4.44E-05 24% OH -2 2.79E+00 6.15E-02 0% 3.57E+00 2.82E-02 8% 1.04E+01 2.91E-05 26% OH -6 2.52E+00 8.02E-02 0% 3.62E+00 2.68E-02 9% 1.02E+01 3.60E-05 25% OH -12 2.81E+00 6.02E-02 0% 3.46E+00 3.13E-02 7% 1.04E+01 3.15E-05 26% HL -2 2.69E+00 6.82E-02 0% 3.54E+00 2.91E-02 8% 1.03E+01 3.34E-05 25% HL -6 2.62E+00 7.25E-02 0% 3.61E+00 2.71E-02 9% 1.03E+01 3.28E-05 25% HL -12 2.57E+00 7.63E-02 0% 3.91E+00 2.00E-02 12% 1.06E+01 2.55E-05 27% HM -2 2.56E+00 7.73E-02 0% 4.22E+00 1.46E-02 16% 1.09E+01 1.78E-05 28% HM -6 2.52E+00 8.06E-02 0% 3.47E+00 3.11E-02 7% 1.01E+01 4.13E-05 24% HM -12 2.94E+00 5.30E-02 0% 4.00E+00 1.83E-02 13% 1.10E+01 1.70E-05 29% HH -2 2.51E+00 8.11E-02 0% 3.57E+00 2.81E-02 8% 1.02E+01 3.84E-05 25% HH -6 2.69E+00 6.80E-02 0% 3.55E+00 2.89E-02 8% 1.03E+01 3.36E-05 25% HH -12 2.94E+00 5.28E-02 0% 3.46E+00 3.16E-02 7% 1.04E+01 2.92E-05 26% Table A.27 Sentinel-2B Reachability analysis and probability from BD for small manoeuvres. Swarm-C Scenario Scenario MD(R,Rr) PR MD(el,az) PR MD(All) PR Swarm-C Tangential Low -2 1.15E+00 0% 8.62E-01 0% 1.17E+00 0% Swarm-C Tangential Low -6 1.12E+00 0% 8.79E-01 0% 1.36E+00 0% Swarm-C Tangential Low -12 1.05E+00 0% 1.51E+00 36% 1.88E+00 6% Swarm-C Tangential Medium -2 1.78E+00 59% 1.06E-01 0% 1.96E+00 14% Swarm-C Tangential Medium -6 2.10E+00 78% 1.58E+00 43% 2.48E+00 62% Swarm-C Tangential Medium -12 1.83E+00 63% 1.60E+00 45% 2.23E+00 42% Swarm-C Tangential High -2 1.07E+00 0% 2.31E+00 86% 2.33E+00 51% Swarm-C Tangential High -6 1.00E+00 0% 1.88E+00 66% 1.91E+00 9% Swarm-C Tangential High -12 1.38E+00 22% 1.50E+00 35% 2.61E+00 71% Swarm-C Out-of-plane Low -2 1.90E+00 67% 1.24E+00 8% 2.22E+00 41% Swarm-C Out-of-plane Low -6 1.43E+00 28% 1.09E+00 0% 1.84E+00 0% Swarm-C Out-of-plane Low -12 1.88E+00 66% 1.09E+00 0% 3.02E+00 88% Swarm-C Out-of-plane Medium -2 3.12E+00 98% 2.19E+00 82% 3.66E+00 98% Swarm-C Out-of-plane Medium -6 1.40E+00 25% 1.55E+00 39% 1.81E+00 0% Swarm-C Out-of-plane Medium -12 1.33E+00 18% 8.97E-01 0% 2.43E+00 59% Swarm-C Out-of-plane High -2 1.50E+01 100% 1.84E+00 63% 1.50E+01 100% Swarm-C Out-of-plane High -6 9.90E+00 100% 1.09E+00 0% 1.01E+01 100% Swarm-C Out-of-plane High -12 1.23E+01 100% 1.43E+00 28% 1.24E+01 100% Swarm-C Hybrid Low -2 1.05E+00 0% 8.56E-01 0% 1.17E+00 0% Swarm-C Hybrid Low -6 1.25E+00 8% 5.82E-01 0% 1.35E+00 0% Swarm-C Hybrid Low -12 1.63E+00 47% 1.27E+00 10% 1.83E+00 0% Swarm-C Hybrid Medium -2 1.02E+00 0% 9.13E-01 0% 1.09E+00 0% Swarm-C Hybrid Medium -6 1.01E+00 0% 8.16E-01 0% 1.09E+00 0% Swarm-C Hybrid Medium -12 1.10E+00 0% 6.69E-01 0% 1.51E+00 0% Swarm-C Hybrid High -2 1.85E+00 64% 2.04E+00 75% 2.59E+00 70% Swarm-C Hybrid High -6 1.78E+00 59% 1.28E+00 12% 2.65E+00 73% Swarm-C Hybrid High -12 4.76E-01 0% 6.53E-01 0% 1.34E+00 0% Table A.28 Swarm-C Reachability analysis and probability from MD for small manoeuvres. A.2 Small manoeuvres 75 Scenario KLD(R,Rr) PR KLD(el,az) PR Swarm-C Tangential Low -2 6.44E+00 0% 2.29E+05 100% Swarm-C Tangential Low -6 6.37E+00 0% 3.37E+05 100% Swarm-C Tangential Low -12 6.07E+00 0% 7.17E+05 100% Swarm-C Tangential Medium -2 1.06E+01 0% 2.23E+05 100% Swarm-C Tangential Medium -6 1.33E+01 0% 5.59E+05 100% Swarm-C Tangential Medium -12 1.12E+01 0% 3.79E+05 100% Swarm-C Tangential High -2 6.15E+00 0% 9.77E+05 100% Swarm-C Tangential High -6 6.05E+00 0% 4.04E+05 100% Swarm-C Tangential High -12 8.17E+00 0% 6.04E+05 100% Swarm-C Out-of-plane Low -2 1.14E+01 0% 5.10E+05 100% Swarm-C Out-of-plane Low -6 8.09E+00 0% 4.74E+05 100% Swarm-C Out-of-plane Low -12 1.14E+01 0% 2.69E+05 100% Swarm-C Out-of-plane Medium -2 2.47E+01 0% 5.01E+05 100% Swarm-C Out-of-plane Medium -6 7.91E+00 0% 2.22E+05 100% Swarm-C Out-of-plane Medium -12 7.53E+00 0% 2.75E+05 100% Swarm-C Out-of-plane High -2 4.90E+02 94% 5.78E+05 100% Swarm-C Out-of-plane High -6 2.16E+02 66% 4.44E+05 100% Swarm-C Out-of-plane High -12 3.33E+02 84% 2.25E+05 100% Swarm-C Hybrid Low -2 6.05E+00 0% 2.87E+05 100% Swarm-C Hybrid Low -6 7.07E+00 0% 2.52E+05 100% Swarm-C Hybrid Low -12 9.63E+00 0% 2.98E+05 100% Swarm-C Hybrid Medium -2 6.00E+00 0% 2.49E+05 100% Swarm-C Hybrid Medium -6 6.29E+00 0% 2.52E+05 100% Swarm-C Hybrid Medium -12 7.16E+00 0% 3.18E+05 100% Swarm-C Hybrid High -2 1.14E+01 0% 5.34E+05 100% Swarm-C Hybrid High -6 1.14E+01 0% 4.69E+05 100% Swarm-C Hybrid High -12 5.64E+00 0% 2.33E+05 100% Table A.29 Swarm-C Reachability analysis and probability from KLD for small manoeuvres. Scenario BD(R,Rr) ρ(R,Rr) PR BD(el,az) ρ(el,az) PR BD(All) ρ(All) PR TL -2 2.59E+00 7.47E-02 0% 3.01E+00 4.95E-02 1% 9.98E+00 4.65E-05 24% TL -6 2.58E+00 7.54E-02 0% 3.00E+00 4.97E-02 1% 1.00E+01 4.38E-05 24% TL -12 2.57E+00 7.69E-02 0% 3.20E+00 4.09E-02 3% 1.02E+01 3.55E-05 25% TM -2 2.83E+00 5.93E-02 0% 2.91E+00 5.46E-02 0% 1.03E+01 3.42E-05 25% TM -6 2.98E+00 5.07E-02 1% 3.22E+00 4.01E-02 4% 1.06E+01 2.57E-05 27% TM -12 2.85E+00 5.79E-02 0% 3.23E+00 3.95E-02 4% 1.04E+01 2.98E-05 26% TH -2 2.57E+00 7.65E-02 0% 3.57E+00 2.81E-02 8% 1.05E+01 2.81E-05 26% TH -6 2.55E+00 7.79E-02 0% 3.35E+00 3.49E-02 5% 1.03E+01 3.51E-05 25% TH -12 2.66E+00 6.98E-02 0% 3.18E+00 4.15E-02 3% 1.07E+01 2.36E-05 27% OL -2 2.88E+00 5.62E-02 0% 3.10E+00 4.49E-02 2% 1.04E+01 2.98E-05 26% OL -6 2.68E+00 6.83E-02 0% 3.06E+00 4.68E-02 2% 1.02E+01 3.62E-05 25% OL -12 2.87E+00 5.65E-02 0% 3.06E+00 4.71E-02 2% 1.09E+01 1.77E-05 28% OM -2 3.65E+00 2.61E-02 9% 3.51E+00 2.99E-02 7% 1.15E+01 1.03E-05 31% OM -6 2.67E+00 6.91E-02 0% 3.20E+00 4.07E-02 4% 1.02E+01 3.66E-05 25% OM -12 2.65E+00 7.07E-02 0% 3.01E+00 4.95E-02 1% 1.05E+01 2.64E-05 26% OH -2 3.05E+01 5.84E-14 97% 3.33E+00 3.58E-02 5% 3.81E+01 2.82E-17 88% OH -6 1.47E+01 4.22E-07 79% 3.05E+00 4.72E-02 2% 2.24E+01 1.80E-10 66% OH -12 2.14E+01 4.90E-10 91% 3.16E+00 4.23E-02 3% 2.90E+01 2.65E-13 78% HL -2 2.57E+00 7.68E-02 0% 3.00E+00 4.97E-02 1% 9.97E+00 4.67E-05 24% HL -6 2.63E+00 7.24E-02 0% 2.95E+00 5.24E-02 0% 1.00E+01 4.40E-05 24% HL -12 2.76E+00 6.33E-02 0% 3.11E+00 4.44E-02 2% 1.02E+01 3.61E-05 25% HM -2 2.56E+00 7.74E-02 0% 3.01E+00 4.91E-02 1% 9.95E+00 4.75E-05 24% HM -6 2.56E+00 7.75E-02 0% 2.99E+00 5.04E-02 1% 9.95E+00 4.75E-05 24% HM -12 2.58E+00 7.58E-02 0% 2.96E+00 5.18E-02 0% 1.01E+01 4.18E-05 24% HH -2 2.86E+00 5.75E-02 0% 3.44E+00 3.22E-02 6% 1.06E+01 2.38E-05 27% 82 Chapter A. Distances and probabilities tables Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-1A No manoeuvre 2.26E-01 0% 7.60E+00 0% 2.54E+00 7.91E-02 0% 9 Sentinel-1A No manoeuvre 4.37E-01 0% 1.71E+01 0% 2.45E+00 8.65E-02 0% 9 Sentinel-1A No manoeuvre 1.34E+00 19% 1.98E+02 62% 3.02E+00 4.86E-02 1% 10 Sentinel-1A No manoeuvre 1.45E+00 30% 5.59E+01 0% 2.25E+00 1.06E-01 0% 9 Sentinel-1A No manoeuvre 1.54E+01 100% 1.95E+04 100% 3.24E+01 8.87E-15 98% 2 Sentinel-1A No manoeuvre 7.99E-01 0% 3.22E+01 0% 2.59E+00 7.52E-02 0% 9 Sentinel-1A No manoeuvre 3.55E-01 0% 3.98E+01 0% 2.28E+00 1.02E-01 0% 9 Sentinel-1A No manoeuvre 1.56E+01 100% 6.86E+03 100% 3.34E+01 3.10E-15 98% 2 Sentinel-1A No manoeuvre 2.56E-01 0% 1.24E+01 0% 2.43E+00 8.79E-02 0% 9 Sentinel-1A No manoeuvre 8.35E-01 0% 8.03E+01 12% 2.46E+00 8.51E-02 0% 11 Sentinel-1A No manoeuvre 1.48E+00 33% 2.31E+01 0% 2.93E+00 5.35E-02 0% 10 Sentinel-1A No manoeuvre 3.96E-01 0% 2.71E+01 0% 2.53E+00 7.97E-02 0% 8 Sentinel-1A No manoeuvre 8.40E-01 0% 5.47E+01 0% 2.77E+00 6.25E-02 0% 9 Sentinel-1A No manoeuvre 6.08E-01 0% 3.67E+01 0% 2.25E+00 1.05E-01 0% 8 Sentinel-1A No manoeuvre 2.46E-01 0% 5.68E+00 0% 2.69E+00 6.78E-02 0% 9 Sentinel-1A No manoeuvre 6.29E-01 0% 5.70E+01 0% 2.43E+00 8.76E-02 0% 10 Sentinel-1A No manoeuvre 2.74E-01 0% 9.21E+00 0% 2.54E+00 7.90E-02 0% 9 Sentinel-1A No manoeuvre 1.42E-01 0% 1.72E+01 0% 2.45E+00 8.67E-02 0% 10 Sentinel-1A No manoeuvre 1.60E+00 44% 2.26E+02 68% 3.20E+00 4.06E-02 4% 10 Sentinel-1A No manoeuvre 1.29E+00 12% 6.04E+01 0% 2.35E+00 9.56E-02 0% 9 Sentinel-1A No manoeuvre 1.51E+01 100% 2.06E+04 100% 3.12E+01 2.84E-14 98% 2 Sentinel-1A No manoeuvre 6.86E-01 0% 2.32E+01 0% 2.59E+00 7.48E-02 0% 9 Sentinel-1A No manoeuvre 8.60E-01 0% 8.42E+01 15% 2.54E+00 7.91E-02 0% 9 Sentinel-1A Tangential Low -2 1.10E+00 0% 1.33E+01 0% 2.69E+00 6.81E-02 0% 9 Sentinel-1A Tangential Low -6 1.17E+00 0% 1.38E+01 0% 2.71E+00 6.67E-02 0% 9 Sentinel-1A Tangential Low -12 8.51E-01 0% 9.92E+00 0% 2.62E+00 7.25E-02 0% 9 Sentinel-1A Tangential Medium -2 3.42E+00 99% 6.38E+01 0% 4.00E+00 1.84E-02 13% 9 Sentinel-1A Tangential Medium -6 6.24E+00 100% 1.88E+02 59% 7.41E+00 6.06E-04 45% 9 Sentinel-1A Tangential Medium -12 5.58E+00 100% 1.35E+02 41% 6.42E+00 1.62E-03 37% 9 Sentinel-1A Tangential High -2 1.64E+01 100% 1.31E+03 100% 3.60E+01 2.36E-16 99% 9 Sentinel-1A Tangential High -6 3.10E+01 100% 4.54E+03 100% 1.23E+02 4.03E-54 100% 9 Sentinel-1A Tangential High -12 3.65E+01 100% 6.08E+03 100% 1.69E+02 4.06E-74 100% 9 Sentinel-1A Out-of-plane Low -2 3.10E+01 100% 4.69E+03 100% 1.23E+02 4.38E-54 100% 9 Sentinel-1A Out-of-plane Low -6 2.72E+01 100% 3.58E+03 100% 9.47E+01 7.49E-42 100% 9 Sentinel-1A Out-of-plane Low -12 3.03E+01 100% 4.42E+03 100% 1.18E+02 9.03E-52 100% 9 Sentinel-1A Out-of-plane Medium -2 6.39E+01 100% 2.00E+04 100% 5.14E+02 8.33E-224 100% 9 Sentinel-1A Out-of-plane Medium -6 5.10E+01 100% 1.25E+04 100% 3.28E+02 3.46E-143 100% 9 Sentinel-1A Out-of-plane Medium -12 6.01E+01 100% 1.76E+04 100% 4.55E+02 3.27E-198 100% 9 Sentinel-1A Out-of-plane High -2 1.24E+02 100% 7.36E+04 100% 1.92E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -6 9.97E+01 100% 4.81E+04 100% 1.25E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -12 1.20E+02 100% 7.07E+04 100% 1.80E+03 0.00E+00 100% 9 Sentinel-1A Hybrid Low -2 7.46E+00 100% 2.76E+02 77% 9.49E+00 7.54E-05 58% 9 Sentinel-1A Hybrid Low -6 1.14E+01 100% 6.11E+02 97% 1.87E+01 7.88E-09 87% 9 Sentinel-1A Hybrid Low -12 1.09E+01 100% 5.34E+02 95% 1.74E+01 2.77E-08 85% 9 Sentinel-1A Hybrid Medium -2 3.77E+01 100% 6.77E+03 100% 1.80E+02 5.43E-79 100% 9 Sentinel-1A Hybrid Medium -6 5.70E+01 100% 1.54E+04 100% 4.09E+02 2.47E-178 100% 9 Sentinel-1A Hybrid Medium -12 8.79E+01 100% 4.07E+04 100% 9.69E+02 0.00E+00 100% 10 Sentinel-1A Hybrid High -2 1.56E+02 100% 1.17E+05 100% 3.06E+03 0.00E+00 100% 9 Sentinel-1A Hybrid High -6 4.35E+02 100% 6.74E+05 100% 2.36E+04 0.00E+00 100% 9 Sentinel-1A Hybrid High -12 1.16E+03 100% 3.47E+06 100% 1.68E+05 0.00E+00 100% 9 Table A.37 Sentinel-1A Reachability analysis and probability from the three distances computed using rangerange rate. Sentinel-2B Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B No manoeuvre 1.91E-01 0% 1.18E+01 0% 2.22E+00 1.08E-01 0% 5 Sentinel-2B No manoeuvre 1.38E+01 100% 5.46E+03 100% 2.67E+01 2.47E-12 96% 3 Sentinel-2B No manoeuvre 4.10E-01 0% 9.95E+00 0% 2.48E+00 8.36E-02 0% 10 Sentinel-2B No manoeuvre 3.87E-01 0% 2.08E+01 0% 2.28E+00 1.02E-01 0% 10 Sentinel-2B No manoeuvre 5.06E-01 0% 5.85E+00 0% 2.11E+00 1.22E-01 0% 12 Sentinel-2B No manoeuvre 1.09E-01 0% 4.35E+01 0% 2.58E+00 7.55E-02 0% 11 Sentinel-2B No manoeuvre 9.63E-02 0% 6.82E+00 0% 2.88E+00 5.59E-02 0% 10 Sentinel-2B No manoeuvre 1.28E+00 12% 4.07E+01 0% 2.34E+00 9.67E-02 0% 11 Sentinel-2B No manoeuvre 1.84E+00 63% 1.63E+01 0% 2.59E+00 7.48E-02 0% 11 Sentinel-2B No manoeuvre 7.89E-01 0% 4.70E+01 0% 2.55E+00 7.79E-02 0% 11 Sentinel-2B No manoeuvre 7.34E-01 0% 8.81E+00 0% 2.99E+00 5.01E-02 1% 12 A.3 Analyses of the impact of different parameters 83 Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B No manoeuvre 8.67E-01 0% 1.98E+01 0% 2.05E+00 1.29E-01 0% 12 Sentinel-2B No manoeuvre 5.85E-01 0% 9.54E+00 0% 2.44E+00 8.71E-02 0% 10 Sentinel-2B No manoeuvre 1.51E+00 36% 5.97E+01 0% 2.61E+00 7.35E-02 0% 9 Sentinel-2B No manoeuvre 7.69E-01 0% 1.55E+01 0% 2.42E+00 8.88E-02 0% 5 Sentinel-2B No manoeuvre 1.25E+01 100% 5.53E+03 100% 2.24E+01 1.80E-10 92% 3 Sentinel-2B No manoeuvre 5.19E-01 0% 1.24E+01 0% 2.58E+00 7.61E-02 0% 10 Sentinel-2B No manoeuvre 2.06E+00 76% 1.14E+02 32% 2.85E+00 5.81E-02 0% 10 Sentinel-2B No manoeuvre 7.47E-01 0% 6.24E+00 0% 2.10E+00 1.22E-01 0% 12 Sentinel-2B No manoeuvre 1.41E+00 25% 1.05E+02 27% 2.76E+00 6.34E-02 0% 12 Sentinel-2B No manoeuvre 1.10E+00 0% 9.16E+00 0% 3.02E+00 4.90E-02 1% 11 Sentinel-2B No manoeuvre 5.59E-01 0% 1.95E+01 0% 2.13E+00 1.18E-01 0% 11 Sentinel-2B No manoeuvre 6.94E-01 0% 8.04E+00 0% 2.32E+00 9.85E-02 0% 11 Sentinel-2B No manoeuvre 8.49E-01 0% 5.69E+01 0% 2.55E+00 7.77E-02 0% 10 Sentinel-2B No manoeuvre 1.93E+00 69% 1.91E+01 0% 3.43E+00 3.24E-02 6% 13 Sentinel-2B No manoeuvre 8.05E-01 0% 2.31E+01 0% 2.08E+00 1.25E-01 0% 12 Sentinel-2B No manoeuvre 1.81E+00 61% 2.46E+01 0% 2.79E+00 6.14E-02 0% 10 Sentinel-2B Tangential Low -2 6.74E-01 0% 1.18E+01 0% 2.52E+00 8.06E-02 0% 10 Sentinel-2B Tangential Low -6 1.17E+00 0% 1.73E+01 0% 2.63E+00 7.18E-02 0% 10 Sentinel-2B Tangential Low -12 1.72E+00 55% 2.59E+01 0% 2.84E+00 5.86E-02 0% 10 Sentinel-2B Tangential Medium -2 1.33E+00 17% 1.91E+01 0% 2.69E+00 6.81E-02 0% 10 Sentinel-2B Tangential Medium -6 5.29E+00 100% 1.72E+02 54% 5.96E+00 2.57E-03 33% 10 Sentinel-2B Tangential Medium -12 4.74E+00 100% 5.83E+01 0% 5.28E+00 5.11E-03 27% 10 Sentinel-2B Tangential High -2 5.89E+00 100% 2.17E+02 66% 6.80E+00 1.11E-03 40% 10 Sentinel-2B Tangential High -6 2.22E+01 100% 2.95E+03 100% 6.43E+01 1.18E-28 100% 10 Sentinel-2B Tangential High -12 1.64E+01 100% 2.56E+02 74% 3.60E+01 2.21E-16 99% 10 Sentinel-2B Out-of-plane Low -2 2.46E+00 90% 4.95E+01 0% 3.22E+00 3.98E-02 4% 10 Sentinel-2B Out-of-plane Low -6 9.26E-01 0% 1.45E+01 0% 2.57E+00 7.68E-02 0% 10 Sentinel-2B Out-of-plane Low -12 1.92E+00 68% 3.38E+01 0% 2.93E+00 5.36E-02 0% 10 Sentinel-2B Out-of-plane Medium -2 3.35E+00 99% 8.46E+01 15% 3.87E+00 2.08E-02 12% 10 Sentinel-2B Out-of-plane Medium -6 2.85E+00 97% 6.33E+01 0% 3.48E+00 3.09E-02 7% 10 Sentinel-2B Out-of-plane Medium -12 3.20E+00 99% 7.76E+01 10% 3.75E+00 2.36E-02 10% 10 Sentinel-2B Out-of-plane High -2 6.63E+00 100% 3.02E+02 80% 7.95E+00 3.51E-04 48% 10 Sentinel-2B Out-of-plane High -6 3.50E+00 100% 9.12E+01 19% 4.00E+00 1.84E-02 13% 10 Sentinel-2B Out-of-plane High -12 4.11E+00 100% 1.22E+02 36% 4.58E+00 1.02E-02 20% 10 Sentinel-2B Hybrid Low -2 1.01E+00 0% 1.21E+01 0% 2.59E+00 7.49E-02 0% 10 Sentinel-2B Hybrid Low -6 8.33E+00 100% 4.20E+02 91% 1.11E+01 1.47E-05 66% 10 Sentinel-2B Hybrid Low -12 6.29E+00 100% 4.95E+01 0% 7.41E+00 6.07E-04 45% 10 Sentinel-2B Hybrid Medium -2 1.45E+01 100% 1.33E+03 100% 2.86E+01 3.81E-13 97% 10 Sentinel-2B Hybrid Medium -6 4.35E+01 100% 1.15E+04 100% 2.39E+02 2.54E-104 100% 10 Sentinel-2B Hybrid Medium -12 2.96E+01 100% 6.73E+02 98% 1.12E+02 1.83E-49 100% 10 Sentinel-2B Hybrid High -2 5.85E+01 100% 2.18E+04 100% 4.31E+02 9.03E-188 100% 10 Sentinel-2B Hybrid High -6 2.37E+02 100% 4.58E+05 100% 7.02E+03 0.00E+00 100% 10 Sentinel-2B Hybrid High -12 3.21E+02 100% 9.77E+05 100% 1.29E+04 0.00E+00 100% 10 Table A.38 Sentinel-2B Reachability analysis and probability from the three distances computed using rangerange rate. A.3.2 Ballistic coefficient CD= 1 Sentinel-1A Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-1A No manoeuvre 7.12E-01 0% 6.76E+00 0% 2.77E+00 6.28E-02 0% 9 Sentinel-1A No manoeuvre 1.13E+00 0% 1.36E+02 42% 2.50E+00 8.21E-02 0% 10 Sentinel-1A No manoeuvre 1.70E-01 0% 7.68E+00 0% 2.54E+00 7.89E-02 0% 9 Sentinel-1A No manoeuvre 3.14E-01 0% 1.58E+01 0% 2.43E+00 8.77E-02 0% 9 Sentinel-1A No manoeuvre 2.31E-01 0% 8.43E+01 15% 2.81E+00 6.01E-02 0% 10 Sentinel-1A No manoeuvre 1.20E-01 0% 1.85E+01 0% 1.98E+00 1.38E-01 0% 9 Sentinel-1A No manoeuvre 1.60E+01 100% 2.13E+04 100% 3.49E+01 6.99E-16 99% 2 Sentinel-1A No manoeuvre 9.55E-01 0% 4.39E+01 0% 2.63E+00 7.24E-02 0% 9 Sentinel-1A No manoeuvre 1.65E-01 0% 3.69E+01 0% 2.26E+00 1.04E-01 0% 9 Sentinel-1A No manoeuvre 1.54E+01 100% 6.50E+03 100% 3.24E+01 8.15E-15 98% 2 Sentinel-1A No manoeuvre 4.54E-01 0% 1.45E+01 0% 2.47E+00 8.47E-02 0% 9 Sentinel-1A No manoeuvre 1.59E+00 44% 1.67E+02 53% 2.68E+00 6.83E-02 0% 11 Sentinel-1A No manoeuvre 7.93E-01 0% 1.26E+01 0% 2.71E+00 6.67E-02 0% 10 Sentinel-1A No manoeuvre 1.95E-01 0% 2.61E+01 0% 2.54E+00 7.88E-02 0% 8 84 Chapter A. Distances and probabilities tables Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-1A No manoeuvre 1.02E+00 0% 9.16E+01 20% 2.85E+00 5.81E-02 0% 9 Sentinel-1A No manoeuvre 6.12E-01 0% 5.04E+01 0% 2.32E+00 9.86E-02 0% 8 Sentinel-1A No manoeuvre 7.78E-01 0% 6.64E+00 0% 2.78E+00 6.20E-02 0% 9 Sentinel-1A No manoeuvre 7.35E-02 0% 5.91E+01 0% 2.47E+00 8.45E-02 0% 10 Sentinel-1A No manoeuvre 1.14E+00 0% 2.07E+01 0% 2.79E+00 6.16E-02 0% 9 Sentinel-1A No manoeuvre 5.81E-02 0% 1.41E+01 0% 2.35E+00 9.56E-02 0% 10 Sentinel-1A No manoeuvre 8.92E-01 0% 1.39E+02 43% 3.01E+00 4.94E-02 1% 10 Sentinel-1A No manoeuvre 1.13E-01 0% 1.99E+01 0% 2.12E+00 1.21E-01 0% 9 Sentinel-1A No manoeuvre 1.44E+01 100% 2.08E+04 100% 2.87E+01 3.51E-13 97% 2 Sentinel-1A No manoeuvre 1.78E+00 59% 6.57E+01 2% 2.91E+00 5.45E-02 0% 9 Sentinel-1A No manoeuvre 5.91E-01 0% 5.34E+01 0% 2.45E+00 8.62E-02 0% 9 Sentinel-1A Tangential Low -2 6.52E-01 0% 9.53E+00 0% 2.59E+00 7.53E-02 0% 9 Sentinel-1A Tangential Low -6 3.70E-01 0% 8.01E+00 0% 2.55E+00 7.82E-02 0% 9 Sentinel-1A Tangential Low -12 2.19E+00 82% 2.98E+01 0% 3.13E+00 4.38E-02 3% 9 Sentinel-1A Tangential Medium -2 4.78E+00 100% 1.16E+02 33% 5.39E+00 4.58E-03 28% 9 Sentinel-1A Tangential Medium -6 4.36E+00 100% 9.25E+01 20% 4.91E+00 7.37E-03 23% 9 Sentinel-1A Tangential Medium -12 6.85E+00 100% 2.17E+02 66% 8.40E+00 2.24E-04 51% 9 Sentinel-1A Tangential High -2 1.59E+01 100% 1.21E+03 100% 3.42E+01 1.46E-15 98% 9 Sentinel-1A Tangential High -6 2.88E+01 100% 3.89E+03 100% 1.06E+02 7.03E-47 100% 9 Sentinel-1A Tangential High -12 3.47E+01 100% 5.44E+03 100% 1.53E+02 2.66E-67 100% 9 Sentinel-1A Out-of-plane Low -2 3.17E+01 100% 4.84E+03 100% 1.28E+02 1.85E-56 100% 9 Sentinel-1A Out-of-plane Low -6 2.57E+01 100% 3.16E+03 100% 8.51E+01 1.06E-37 100% 9 Sentinel-1A Out-of-plane Low -12 2.70E+01 100% 3.59E+03 100% 9.37E+01 1.95E-41 100% 9 Sentinel-1A Out-of-plane Medium -2 6.41E+01 100% 2.03E+04 100% 5.16E+02 9.53E-225 100% 9 Sentinel-1A Out-of-plane Medium -6 5.21E+01 100% 1.31E+04 100% 3.42E+02 2.75E-149 100% 9 Sentinel-1A Out-of-plane Medium -12 5.89E+01 100% 1.68E+04 100% 4.36E+02 5.68E-190 100% 9 Sentinel-1A Out-of-plane High -2 1.23E+02 100% 7.25E+04 100% 1.90E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -6 1.00E+02 100% 4.83E+04 100% 1.26E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -12 1.21E+02 100% 6.98E+04 100% 1.85E+03 0.00E+00 100% 9 Sentinel-1A Hybrid Low -2 6.83E+00 100% 2.34E+02 70% 8.36E+00 2.34E-04 51% 9 Sentinel-1A Hybrid Low -6 1.02E+01 100% 4.96E+02 94% 1.55E+01 1.91E-07 81% 9 Sentinel-1A Hybrid Low -12 1.22E+01 100% 6.74E+02 98% 2.10E+01 7.36E-10 91% 9 Sentinel-1A Hybrid Medium -2 3.74E+01 100% 6.88E+03 100% 1.78E+02 6.92E-78 100% 9 Sentinel-1A Hybrid Medium -6 5.49E+01 100% 1.42E+04 100% 3.79E+02 1.68E-165 100% 9 Sentinel-1A Hybrid Medium -12 8.79E+01 100% 4.00E+04 100% 9.67E+02 0.00E+00 100% 10 Sentinel-1A Hybrid High -2 1.56E+02 100% 1.17E+05 100% 3.04E+03 0.00E+00 100% 9 Sentinel-1A Hybrid High -6 4.32E+02 100% 6.69E+05 100% 2.33E+04 0.00E+00 100% 9 Sentinel-1A Hybrid High -12 1.16E+03 100% 3.49E+06 100% 1.67E+05 0.00E+00 100% 9 Table A.39 Sentinel-1A Reachability analysis and probability from the three distances computed using rangerange rate. Sentinel-2B Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B No manoeuvre 1.31E+00 16% 2.90E+01 0% 2.44E+00 8.75E-02 0% 5 Sentinel-2B No manoeuvre 1.39E+01 100% 5.30E+03 100% 2.69E+01 2.00E-12 96% 3 Sentinel-2B No manoeuvre 1.46E+00 31% 2.33E+01 0% 2.73E+00 6.51E-02 0% 10 Sentinel-2B No manoeuvre 1.60E+00 45% 6.31E+01 0% 2.58E+00 7.55E-02 0% 10 Sentinel-2B No manoeuvre 1.01E+00 0% 8.05E+00 0% 2.20E+00 1.10E-01 0% 12 Sentinel-2B No manoeuvre 4.67E-02 0% 4.26E+01 0% 2.58E+00 7.60E-02 0% 11 Sentinel-2B No manoeuvre 1.07E+00 0% 1.03E+01 0% 3.03E+00 4.83E-02 1% 10 Sentinel-2B No manoeuvre 1.53E+00 38% 5.05E+01 0% 2.46E+00 8.58E-02 0% 11 Sentinel-2B No manoeuvre 1.42E+00 27% 1.22E+01 0% 2.43E+00 8.79E-02 0% 11 Sentinel-2B No manoeuvre 2.33E-01 0% 3.11E+01 0% 2.49E+00 8.33E-02 0% 11 Sentinel-2B No manoeuvre 4.05E-01 0% 7.62E+00 0% 2.95E+00 5.23E-02 0% 12 Sentinel-2B No manoeuvre 6.86E-01 0% 1.64E+01 0% 2.00E+00 1.35E-01 0% 12 Sentinel-2B No manoeuvre 6.47E-01 0% 1.00E+01 0% 2.45E+00 8.61E-02 0% 10 Sentinel-2B No manoeuvre 1.25E+00 8% 6.95E+01 5% 2.63E+00 7.24E-02 0% 9 Sentinel-2B No manoeuvre 3.73E-01 0% 1.18E+01 0% 2.37E+00 9.34E-02 0% 5 Sentinel-2B No manoeuvre 1.41E+01 100% 6.96E+03 100% 2.76E+01 9.84E-13 96% 3 Sentinel-2B No manoeuvre 5.61E-01 0% 1.30E+01 0% 2.59E+00 7.50E-02 0% 10 Sentinel-2B No manoeuvre 5.56E-01 0% 2.85E+01 0% 2.34E+00 9.61E-02 0% 10 Sentinel-2B No manoeuvre 2.49E+00 91% 2.09E+01 0% 2.82E+00 5.96E-02 0% 12 Sentinel-2B No manoeuvre 1.02E-01 0% 3.52E+01 0% 2.51E+00 8.12E-02 0% 12 Sentinel-2B No manoeuvre 9.67E-02 0% 6.46E+00 0% 2.88E+00 5.63E-02 0% 11 Sentinel-2B No manoeuvre 1.40E+00 25% 5.59E+01 0% 2.41E+00 8.96E-02 0% 11 Sentinel-2B No manoeuvre 1.54E+00 39% 1.58E+01 0% 2.53E+00 7.95E-02 0% 11 A.3 Analyses of the impact of different parameters 85 Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B No manoeuvre 8.91E-01 0% 5.75E+01 0% 2.55E+00 7.80E-02 0% 10 Sentinel-2B No manoeuvre 6.34E-01 0% 7.88E+00 0% 2.96E+00 5.19E-02 0% 13 Sentinel-2B No manoeuvre 5.83E-01 0% 1.81E+01 0% 2.02E+00 1.33E-01 0% 12 Sentinel-2B No manoeuvre 2.31E-01 0% 8.06E+00 0% 2.41E+00 9.01E-02 0% 10 Sentinel-2B Tangential Low -2 3.18E-01 0% 9.56E+00 0% 2.48E+00 8.41E-02 0% 10 Sentinel-2B Tangential Low -6 1.68E+00 51% 2.73E+01 0% 2.82E+00 5.98E-02 0% 10 Sentinel-2B Tangential Low -12 2.07E+00 76% 3.46E+01 0% 3.00E+00 5.00E-02 1% 10 Sentinel-2B Tangential Medium -2 2.31E+00 86% 4.36E+01 0% 3.13E+00 4.37E-02 3% 10 Sentinel-2B Tangential Medium -6 6.35E+00 100% 2.60E+02 74% 7.51E+00 5.45E-04 45% 10 Sentinel-2B Tangential Medium -12 6.65E+00 100% 2.07E+02 64% 7.99E+00 3.40E-04 49% 10 Sentinel-2B Tangential High -2 7.17E+00 100% 3.28E+02 83% 8.89E+00 1.38E-04 54% 10 Sentinel-2B Tangential High -6 2.02E+01 100% 2.42E+03 100% 5.37E+01 4.84E-24 100% 10 Sentinel-2B Tangential High -12 1.62E+01 100% 1.94E+02 61% 3.52E+01 4.94E-16 99% 10 Sentinel-2B Out-of-plane Low -2 1.93E+00 69% 3.42E+01 0% 2.93E+00 5.33E-02 0% 10 Sentinel-2B Out-of-plane Low -6 9.58E-01 0% 1.50E+01 0% 2.58E+00 7.60E-02 0% 10 Sentinel-2B Out-of-plane Low -12 1.76E+00 57% 2.96E+01 0% 2.85E+00 5.79E-02 0% 10 Sentinel-2B Out-of-plane Medium -2 2.21E+00 82% 4.17E+01 0% 3.08E+00 4.62E-02 2% 10 Sentinel-2B Out-of-plane Medium -6 1.92E+00 69% 3.37E+01 0% 2.93E+00 5.35E-02 0% 10 Sentinel-2B Out-of-plane Medium -12 1.27E+00 11% 1.96E+01 0% 2.66E+00 6.97E-02 0% 10 Sentinel-2B Out-of-plane High -2 6.74E+00 100% 3.14E+02 82% 8.14E+00 2.91E-04 50% 10 Sentinel-2B Out-of-plane High -6 1.48E+00 33% 2.33E+01 0% 2.74E+00 6.48E-02 0% 10 Sentinel-2B Out-of-plane High -12 5.45E+00 100% 2.07E+02 64% 6.17E+00 2.09E-03 35% 10 Sentinel-2B Hybrid Low -2 3.14E+00 99% 7.14E+01 6% 3.70E+00 2.48E-02 10% 10 Sentinel-2B Hybrid Low -6 7.40E+00 100% 3.27E+02 83% 9.31E+00 9.08E-05 57% 10 Sentinel-2B Hybrid Low -12 6.67E+00 100% 8.82E+01 17% 8.02E+00 3.28E-04 49% 10 Sentinel-2B Hybrid Medium -2 1.29E+01 100% 1.03E+03 100% 2.32E+01 8.75E-11 93% 10 Sentinel-2B Hybrid Medium -6 4.16E+01 100% 1.05E+04 100% 2.19E+02 6.57E-96 100% 10 Sentinel-2B Hybrid Medium -12 2.93E+01 100% 5.61E+02 96% 1.10E+02 1.39E-48 100% 10 Sentinel-2B Hybrid High -2 5.61E+01 100% 1.99E+04 100% 3.96E+02 8.50E-173 100% 10 Sentinel-2B Hybrid High -6 2.36E+02 100% 4.47E+05 100% 6.99E+03 0.00E+00 100% 10 Sentinel-2B Hybrid High -12 3.21E+02 100% 9.63E+05 100% 1.29E+04 0.00E+00 100% 10 Table A.40 Sentinel-2B Reachability analysis and probability from the three distances computed using rangerange rate. CD= 1.5 Sentinel-1A Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-1A No manoeuvre 1.25E+00 8% 8.70E+00 0% 2.90E+00 5.50E-02 0% 9 Sentinel-1A No manoeuvre 2.99E-01 0% 6.45E+01 1% 2.35E+00 9.55E-02 0% 10 Sentinel-1A No manoeuvre 2.69E-01 0% 7.90E+00 0% 2.54E+00 7.89E-02 0% 9 Sentinel-1A No manoeuvre 6.07E-01 0% 1.98E+01 0% 2.47E+00 8.42E-02 0% 9 Sentinel-1A No manoeuvre 1.20E-01 0% 8.14E+01 13% 2.81E+00 6.04E-02 0% 10 Sentinel-1A No manoeuvre 5.30E-01 0% 2.27E+01 0% 2.00E+00 1.35E-01 0% 9 Sentinel-1A No manoeuvre 1.49E+01 100% 1.86E+04 100% 3.04E+01 6.08E-14 97% 2 Sentinel-1A No manoeuvre 4.30E-01 0% 2.84E+01 0% 2.55E+00 7.83E-02 0% 9 Sentinel-1A No manoeuvre 8.20E-01 0% 5.87E+01 0% 2.33E+00 9.76E-02 0% 9 Sentinel-1A No manoeuvre 1.52E+01 100% 6.19E+03 100% 3.18E+01 1.54E-14 98% 2 Sentinel-1A No manoeuvre 1.22E+00 5% 2.91E+01 0% 2.65E+00 7.08E-02 0% 9 Sentinel-1A No manoeuvre 3.93E-01 0% 5.18E+01 0% 2.37E+00 9.37E-02 0% 11 Sentinel-1A No manoeuvre 9.91E-01 0% 1.41E+01 0% 2.72E+00 6.57E-02 0% 10 Sentinel-1A No manoeuvre 3.13E-01 0% 2.13E+01 0% 2.49E+00 8.32E-02 0% 8 Sentinel-1A No manoeuvre 5.66E-01 0% 6.51E+01 1% 2.79E+00 6.12E-02 0% 9 Sentinel-1A No manoeuvre 6.49E-01 0% 4.48E+01 0% 2.31E+00 9.90E-02 0% 8 Sentinel-1A No manoeuvre 7.65E-01 0% 6.96E+00 0% 2.77E+00 6.24E-02 0% 9 Sentinel-1A No manoeuvre 2.39E+00 89% 2.94E+02 79% 3.12E+00 4.41E-02 2% 10 Sentinel-1A No manoeuvre 8.24E-01 0% 1.35E+01 0% 2.63E+00 7.20E-02 0% 9 Sentinel-1A No manoeuvre 1.74E+00 56% 6.37E+01 0% 2.82E+00 5.97E-02 0% 10 Sentinel-1A No manoeuvre 7.02E-01 0% 1.08E+02 29% 2.93E+00 5.31E-02 0% 10 Sentinel-1A No manoeuvre 2.15E-01 0% 2.25E+01 0% 2.16E+00 1.16E-01 0% 9 Sentinel-1A No manoeuvre 1.56E+01 100% 2.28E+04 100% 3.31E+01 4.30E-15 98% 2 Sentinel-1A No manoeuvre 4.37E-01 0% 2.59E+01 0% 2.61E+00 7.38E-02 0% 9 Sentinel-1A No manoeuvre 2.37E-01 0% 4.26E+01 0% 2.42E+00 8.91E-02 0% 9 Sentinel-1A Tangential Low -2 6.29E-01 0% 9.36E+00 0% 2.58E+00 7.57E-02 0% 9 Sentinel-1A Tangential Low -6 2.72E+00 95% 4.27E+01 0% 3.45E+00 3.17E-02 7% 9 Sentinel-1A Tangential Low -12 4.20E-01 0% 7.94E+00 0% 2.55E+00 7.79E-02 0% 9 86 Chapter A. Distances and probabilities tables Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-1A Tangential Medium -2 5.21E+00 100% 1.35E+02 41% 5.91E+00 2.70E-03 32% 9 Sentinel-1A Tangential Medium -6 7.74E+00 100% 2.90E+02 79% 1.00E+01 4.47E-05 61% 9 Sentinel-1A Tangential Medium -12 6.53E+00 100% 1.91E+02 60% 7.85E+00 3.89E-04 48% 9 Sentinel-1A Tangential High -2 1.54E+01 100% 1.15E+03 100% 3.23E+01 9.14E-15 98% 9 Sentinel-1A Tangential High -6 2.96E+01 100% 4.03E+03 100% 1.12E+02 2.08E-49 100% 9 Sentinel-1A Tangential High -12 3.64E+01 100% 6.04E+03 100% 1.68E+02 1.26E-73 100% 9 Sentinel-1A Out-of-plane Low -2 3.18E+01 100% 4.81E+03 100% 1.29E+02 8.95E-57 100% 9 Sentinel-1A Out-of-plane Low -6 2.70E+01 100% 3.47E+03 100% 9.40E+01 1.52E-41 100% 9 Sentinel-1A Out-of-plane Low -12 2.96E+01 100% 4.15E+03 100% 1.12E+02 1.71E-49 100% 9 Sentinel-1A Out-of-plane Medium -2 6.35E+01 100% 1.93E+04 100% 5.07E+02 9.94E-221 100% 9 Sentinel-1A Out-of-plane Medium -6 5.25E+01 100% 1.30E+04 100% 3.47E+02 2.77E-151 100% 9 Sentinel-1A Out-of-plane Medium -12 5.80E+01 100% 1.61E+04 100% 4.23E+02 2.10E-184 100% 9 Sentinel-1A Out-of-plane High -2 1.24E+02 100% 7.43E+04 100% 1.94E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -6 1.00E+02 100% 4.82E+04 100% 1.26E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -12 1.22E+02 100% 6.98E+04 100% 1.85E+03 0.00E+00 100% 9 Sentinel-1A Hybrid Low -2 6.06E+00 100% 1.84E+02 58% 7.12E+00 8.06E-04 42% 9 Sentinel-1A Hybrid Low -6 8.88E+00 100% 3.67E+02 87% 1.24E+01 4.16E-06 71% 9 Sentinel-1A Hybrid Low -12 1.09E+01 100% 5.43E+02 96% 1.74E+01 2.66E-08 85% 9 Sentinel-1A Hybrid Medium -2 3.85E+01 100% 6.96E+03 100% 1.88E+02 3.01E-82 100% 9 Sentinel-1A Hybrid Medium -6 5.54E+01 100% 1.42E+04 100% 3.86E+02 1.58E-168 100% 9 Sentinel-1A Hybrid Medium -12 8.91E+01 100% 4.10E+04 100% 9.95E+02 0.00E+00 100% 10 Sentinel-1A Hybrid High -2 1.56E+02 100% 1.15E+05 100% 3.06E+03 0.00E+00 100% 9 Sentinel-1A Hybrid High -6 4.31E+02 100% 6.64E+05 100% 2.32E+04 0.00E+00 100% 9 Sentinel-1A Hybrid High -12 1.16E+03 100% 3.49E+06 100% 1.68E+05 0.00E+00 100% 9 Table A.41 Sentinel-1A Reachability analysis and probability from the three distances computed using rangerange rate. Sentinel-2B Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B No manoeuvre 3.60E-01 0% 1.28E+01 0% 2.24E+00 1.06E-01 0% 5 Sentinel-2B No manoeuvre 1.29E+01 100% 4.74E+03 100% 2.37E+01 5.05E-11 94% 3 Sentinel-2B No manoeuvre 1.49E+00 34% 2.37E+01 0% 2.74E+00 6.47E-02 0% 10 Sentinel-2B No manoeuvre 1.90E+00 67% 8.09E+01 13% 2.71E+00 6.65E-02 0% 10 Sentinel-2B No manoeuvre 5.87E-01 0% 6.15E+00 0% 2.12E+00 1.20E-01 0% 12 Sentinel-2B No manoeuvre 4.34E-01 0% 5.06E+01 0% 2.60E+00 7.42E-02 0% 11 Sentinel-2B No manoeuvre 1.91E-02 0% 6.84E+00 0% 2.89E+00 5.57E-02 0% 10 Sentinel-2B No manoeuvre 2.29E-01 0% 1.65E+01 0% 2.17E+00 1.14E-01 0% 11 Sentinel-2B No manoeuvre 7.10E-01 0% 7.33E+00 0% 2.25E+00 1.06E-01 0% 11 Sentinel-2B No manoeuvre 1.18E+00 0% 6.79E+01 3% 2.64E+00 7.13E-02 0% 11 Sentinel-2B No manoeuvre 8.23E-01 0% 9.46E+00 0% 3.02E+00 4.87E-02 1% 12 Sentinel-2B No manoeuvre 1.06E+00 0% 2.28E+01 0% 2.07E+00 1.26E-01 0% 12 Sentinel-2B No manoeuvre 1.99E-01 0% 8.06E+00 0% 2.41E+00 8.96E-02 0% 10 Sentinel-2B No manoeuvre 1.02E+00 0% 5.38E+01 0% 2.55E+00 7.80E-02 0% 9 Sentinel-2B No manoeuvre 7.88E-01 0% 1.65E+01 0% 2.45E+00 8.67E-02 0% 5 Sentinel-2B No manoeuvre 1.38E+01 100% 6.45E+03 100% 2.65E+01 3.22E-12 95% 3 Sentinel-2B No manoeuvre 5.80E-01 0% 1.10E+01 0% 2.54E+00 7.92E-02 0% 10 Sentinel-2B No manoeuvre 3.57E-01 0% 2.36E+01 0% 2.30E+00 9.99E-02 0% 10 Sentinel-2B No manoeuvre 2.25E+00 84% 1.86E+01 0% 2.70E+00 6.74E-02 0% 12 Sentinel-2B No manoeuvre 4.08E-01 0% 3.88E+01 0% 2.51E+00 8.12E-02 0% 12 Sentinel-2B No manoeuvre 2.79E+00 96% 2.58E+01 0% 3.87E+00 2.09E-02 12% 11 Sentinel-2B No manoeuvre 2.11E-01 0% 1.90E+01 0% 2.15E+00 1.17E-01 0% 11 Sentinel-2B No manoeuvre 6.55E-01 0% 8.45E+00 0% 2.32E+00 9.82E-02 0% 11 Sentinel-2B No manoeuvre 9.68E-01 0% 5.52E+01 0% 2.56E+00 7.76E-02 0% 10 Sentinel-2B No manoeuvre 1.38E+00 22% 1.25E+01 0% 3.17E+00 4.21E-02 3% 13 Sentinel-2B No manoeuvre 2.07E+00 77% 6.33E+01 0% 2.47E+00 8.49E-02 0% 12 Sentinel-2B No manoeuvre 1.08E+00 0% 1.53E+01 0% 2.59E+00 7.51E-02 0% 10 Sentinel-2B Tangential Low -2 8.43E-01 0% 1.37E+01 0% 2.55E+00 7.79E-02 0% 10 Sentinel-2B Tangential Low -6 8.20E-01 0% 1.28E+01 0% 2.55E+00 7.82E-02 0% 10 Sentinel-2B Tangential Low -12 1.36E+00 20% 1.85E+01 0% 2.70E+00 6.75E-02 0% 10 Sentinel-2B Tangential Medium -2 2.02E+00 74% 3.52E+01 0% 2.98E+00 5.09E-02 1% 10 Sentinel-2B Tangential Medium -6 4.32E+00 100% 1.12E+02 31% 4.80E+00 8.26E-03 22% 10 Sentinel-2B Tangential Medium -12 4.54E+00 100% 4.77E+01 0% 5.04E+00 6.50E-03 24% 10 Sentinel-2B Tangential High -2 6.55E+00 100% 2.68E+02 76% 7.83E+00 3.97E-04 48% 10 Sentinel-2B Tangential High -6 2.18E+01 100% 2.86E+03 100% 6.17E+01 1.64E-27 100% 10 Sentinel-2B Tangential High -12 1.63E+01 100% 2.49E+02 73% 3.57E+01 3.16E-16 99% 10 Sentinel-2B Out-of-plane Low -2 1.67E+00 50% 2.75E+01 0% 2.81E+00 6.00E-02 0% 10 A.3 Analyses of the impact of different parameters 87 Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B Out-of-plane Low -6 9.26E-01 0% 1.47E+01 0% 2.57E+00 7.63E-02 0% 10 Sentinel-2B Out-of-plane Low -12 3.37E+00 99% 8.51E+01 16% 3.89E+00 2.05E-02 12% 10 Sentinel-2B Out-of-plane Medium -2 2.93E+00 97% 6.60E+01 2% 3.53E+00 2.92E-02 8% 10 Sentinel-2B Out-of-plane Medium -6 2.14E+00 80% 3.98E+01 0% 3.04E+00 4.81E-02 1% 10 Sentinel-2B Out-of-plane Medium -12 1.88E+00 66% 3.25E+01 0% 2.91E+00 5.46E-02 0% 10 Sentinel-2B Out-of-plane High -2 8.01E+00 100% 4.44E+02 92% 1.05E+01 2.79E-05 63% 10 Sentinel-2B Out-of-plane High -6 3.05E+00 98% 7.09E+01 6% 3.62E+00 2.68E-02 9% 10 Sentinel-2B Out-of-plane High -12 7.77E+00 100% 4.18E+02 90% 1.00E+01 4.51E-05 61% 10 Sentinel-2B Hybrid Low -2 2.47E+00 91% 4.65E+01 0% 3.23E+00 3.96E-02 4% 10 Sentinel-2B Hybrid Low -6 6.40E+00 100% 2.33E+02 69% 7.58E+00 5.09E-04 46% 10 Sentinel-2B Hybrid Low -12 8.03E+00 100% 2.22E+02 67% 1.05E+01 2.67E-05 63% 10 Sentinel-2B Hybrid Medium -2 1.26E+01 100% 9.81E+02 100% 2.22E+01 2.32E-10 92% 10 Sentinel-2B Hybrid Medium -6 4.18E+01 100% 1.04E+04 100% 2.21E+02 1.30E-96 100% 10 Sentinel-2B Hybrid Medium -12 2.93E+01 100% 4.70E+02 93% 1.10E+02 2.67E-48 100% 10 Sentinel-2B Hybrid High -2 5.72E+01 100% 2.06E+04 100% 4.12E+02 1.09E-179 100% 10 Sentinel-2B Hybrid High -6 2.36E+02 100% 4.55E+05 100% 6.98E+03 0.00E+00 100% 10 Sentinel-2B Hybrid High -12 3.21E+02 100% 9.72E+05 100% 1.29E+04 0.00E+00 100% 10 Table A.42 Sentinel-2B Reachability analysis and probability from the three distances computed using rangerange rate. CD= 2.5 Sentinel-1A Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-1A No manoeuvre 1.45E+00 30% 9.72E+00 0% 2.97E+00 5.14E-02 0% 9 Sentinel-1A No manoeuvre 1.42E+00 27% 1.78E+02 56% 2.59E+00 7.49E-02 0% 10 Sentinel-1A No manoeuvre 1.71E+00 54% 2.11E+01 0% 2.89E+00 5.56E-02 0% 9 Sentinel-1A No manoeuvre 6.27E-01 0% 2.22E+01 0% 2.46E+00 8.58E-02 0% 9 Sentinel-1A No manoeuvre 4.40E-01 0% 9.78E+01 23% 2.84E+00 5.81E-02 0% 10 Sentinel-1A No manoeuvre 5.18E-01 0% 3.14E+01 0% 2.09E+00 1.24E-01 0% 9 Sentinel-1A No manoeuvre 1.45E+01 100% 1.81E+04 100% 2.92E+01 2.09E-13 97% 2 Sentinel-1A No manoeuvre 6.17E-01 0% 3.44E+01 0% 2.59E+00 7.47E-02 0% 9 Sentinel-1A No manoeuvre 9.07E-02 0% 4.64E+01 0% 2.31E+00 9.93E-02 0% 9 Sentinel-1A No manoeuvre 1.89E+01 100% 6.28E+03 100% 4.77E+01 1.99E-21 100% 2 Sentinel-1A No manoeuvre 1.29E+00 13% 3.88E+01 0% 2.70E+00 6.69E-02 0% 9 Sentinel-1A No manoeuvre 1.10E+00 0% 1.23E+02 36% 2.54E+00 7.86E-02 0% 11 Sentinel-1A No manoeuvre 1.56E+00 41% 6.41E+01 1% 2.29E+00 1.02E-01 0% 10 Sentinel-1A No manoeuvre 5.29E-01 0% 2.97E+01 0% 2.54E+00 7.88E-02 0% 8 Sentinel-1A No manoeuvre 4.13E-01 0% 4.44E+01 0% 2.73E+00 6.55E-02 0% 9 Sentinel-1A No manoeuvre 7.55E-01 0% 5.74E+01 0% 2.34E+00 9.63E-02 0% 8 Sentinel-1A No manoeuvre 6.73E-01 0% 6.59E+00 0% 2.75E+00 6.42E-02 0% 9 Sentinel-1A No manoeuvre 1.68E+00 52% 1.71E+02 54% 2.76E+00 6.33E-02 0% 10 Sentinel-1A No manoeuvre 6.25E-01 0% 1.13E+01 0% 2.58E+00 7.56E-02 0% 9 Sentinel-1A No manoeuvre 9.14E-01 0% 3.13E+01 0% 2.57E+00 7.67E-02 0% 10 Sentinel-1A No manoeuvre 1.72E+00 54% 3.12E+02 82% 3.28E+00 3.77E-02 4% 10 Sentinel-1A No manoeuvre 3.19E-01 0% 2.14E+01 0% 2.12E+00 1.21E-01 0% 9 Sentinel-1A No manoeuvre 1.26E+00 10% 4.60E+01 0% 2.76E+00 6.34E-02 0% 2 Sentinel-1A No manoeuvre 2.50E-01 0% 4.59E+01 0% 2.44E+00 8.68E-02 0% 9 Sentinel-1A No manoeuvre 3.57E-02 0% 7.43E+00 0% 2.52E+00 8.02E-02 0% 9 Sentinel-1A Tangential Low -2 4.05E-01 0% 8.05E+00 0% 2.54E+00 7.87E-02 0% 9 Sentinel-1A Tangential Low -6 2.35E+00 87% 3.32E+01 0% 3.22E+00 4.01E-02 4% 9 Sentinel-1A Tangential Low -12 4.13E+00 100% 8.79E+01 17% 4.66E+00 9.50E-03 20% 9 Sentinel-1A Tangential Medium -2 8.10E+00 100% 3.09E+02 81% 1.07E+01 2.20E-05 64% 9 Sentinel-1A Tangential Medium -6 7.08E+00 100% 2.29E+02 69% 8.80E+00 1.51E-04 54% 9 Sentinel-1A Tangential Medium -12 1.40E+01 100% 9.28E+02 100% 2.69E+01 2.04E-12 96% 9 Sentinel-1A Tangential High -2 3.06E+01 100% 4.34E+03 100% 1.20E+02 1.24E-52 100% 9 Sentinel-1A Tangential High -6 3.47E+01 100% 5.36E+03 100% 1.53E+02 2.41E-67 100% 9 Sentinel-1A Tangential High -12 3.13E+01 100% 4.76E+03 100% 1.25E+02 3.72E-55 100% 9 Sentinel-1A Out-of-plane Low -2 2.58E+01 100% 3.17E+03 100% 8.56E+01 6.80E-38 100% 9 Sentinel-1A Out-of-plane Low -6 2.81E+01 100% 3.69E+03 100% 1.01E+02 1.16E-44 100% 9 Sentinel-1A Out-of-plane Low -12 6.36E+01 100% 1.93E+04 100% 5.08E+02 1.72E-221 100% 9 Sentinel-1A Out-of-plane Medium -2 5.28E+01 100% 1.33E+04 100% 3.51E+02 3.84E-153 100% 9 Sentinel-1A Out-of-plane Medium -6 5.82E+01 100% 1.61E+04 100% 4.25E+02 2.16E-185 100% 9 Sentinel-1A Out-of-plane Medium -12 1.25E+02 100% 7.33E+04 100% 1.95E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -2 1.01E+02 100% 4.89E+04 100% 1.29E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -6 1.21E+02 100% 6.78E+04 100% 1.83E+03 0.00E+00 100% 9 88 Chapter A. Distances and probabilities tables Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-1A Out-of-plane High -12 6.94E+00 100% 2.33E+02 69% 8.55E+00 1.94E-04 52% 9 Sentinel-1A Hybrid Low -2 9.22E+00 100% 4.00E+02 89% 1.31E+01 1.95E-06 74% 9 Sentinel-1A Hybrid Low -6 1.18E+01 100% 6.27E+02 97% 2.01E+01 1.95E-09 89% 9 Sentinel-1A Hybrid Low -12 3.94E+01 100% 7.35E+03 100% 1.96E+02 5.68E-86 100% 9 Sentinel-1A Hybrid Medium -2 5.77E+01 100% 1.55E+04 100% 4.18E+02 2.95E-182 100% 9 Sentinel-1A Hybrid Medium -6 8.69E+01 100% 3.86E+04 100% 9.46E+02 0.00E+00 100% 9 Sentinel-1A Hybrid Medium -12 1.58E+02 100% 1.20E+05 100% 3.12E+03 0.00E+00 100% 10 Sentinel-1A Hybrid High -2 4.34E+02 100% 6.68E+05 100% 2.36E+04 0.00E+00 100% 9 Sentinel-1A Hybrid High -6 1.16E+03 100% 3.41E+06 100% 1.69E+05 0.00E+00 100% 9 Sentinel-1A Hybrid High -12 1.16E+03 100% 3.49E+06 100% 1.68E+05 0.00E+00 100% 9 Table A.43 Sentinel-1A Reachability analysis and probability from the three distances computed using rangerange rate. Sentinel-2B Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B No manoeuvre 2.03E+00 75% 5.29E+01 0% 2.74E+00 6.48E-02 0% 5 Sentinel-2B No manoeuvre 1.34E+01 100% 5.02E+03 100% 2.54E+01 9.45E-12 95% 3 Sentinel-2B No manoeuvre 1.50E+00 35% 2.42E+01 0% 2.75E+00 6.41E-02 0% 10 Sentinel-2B No manoeuvre 6.11E-01 0% 2.51E+01 0% 2.31E+00 9.92E-02 0% 10 Sentinel-2B No manoeuvre 7.05E-01 0% 6.57E+00 0% 2.14E+00 1.17E-01 0% 12 Sentinel-2B No manoeuvre 2.50E+00 91% 3.07E+02 81% 3.36E+00 3.48E-02 5% 11 Sentinel-2B No manoeuvre 2.51E-01 0% 7.05E+00 0% 2.90E+00 5.51E-02 0% 10 Sentinel-2B No manoeuvre 1.85E-01 0% 1.60E+01 0% 2.16E+00 1.16E-01 0% 11 Sentinel-2B No manoeuvre 3.95E-01 0% 6.24E+00 0% 2.21E+00 1.10E-01 0% 11 Sentinel-2B No manoeuvre 1.57E-01 0% 2.88E+01 0% 2.46E+00 8.54E-02 0% 11 Sentinel-2B No manoeuvre 1.01E-01 0% 7.27E+00 0% 2.95E+00 5.25E-02 0% 12 Sentinel-2B No manoeuvre 2.38E-01 0% 1.13E+01 0% 1.92E+00 1.46E-01 0% 12 Sentinel-2B No manoeuvre 4.23E-01 0% 7.07E+00 0% 2.35E+00 9.55E-02 0% 10 Sentinel-2B No manoeuvre 1.36E+00 21% 7.23E+01 7% 2.64E+00 7.16E-02 0% 9 Sentinel-2B No manoeuvre 5.10E-01 0% 1.35E+01 0% 2.41E+00 8.94E-02 0% 5 Sentinel-2B No manoeuvre 1.27E+01 100% 6.36E+03 100% 2.32E+01 8.73E-11 93% 3 Sentinel-2B No manoeuvre 2.34E-01 0% 9.78E+00 0% 2.50E+00 8.18E-02 0% 10 Sentinel-2B No manoeuvre 5.72E-01 0% 2.56E+01 0% 2.30E+00 1.01E-01 0% 10 Sentinel-2B No manoeuvre 2.94E-01 0% 5.53E+00 0% 2.12E+00 1.20E-01 0% 12 Sentinel-2B No manoeuvre 3.95E-01 0% 4.59E+01 0% 2.55E+00 7.78E-02 0% 12 Sentinel-2B No manoeuvre 5.94E-01 0% 7.08E+00 0% 2.90E+00 5.52E-02 0% 11 Sentinel-2B No manoeuvre 1.53E-01 0% 1.62E+01 0% 2.12E+00 1.20E-01 0% 11 Sentinel-2B No manoeuvre 4.26E-01 0% 6.79E+00 0% 2.27E+00 1.03E-01 0% 11 Sentinel-2B No manoeuvre 1.74E+00 56% 1.33E+02 40% 2.85E+00 5.78E-02 0% 10 Sentinel-2B No manoeuvre 5.09E-01 0% 7.46E+00 0% 2.94E+00 5.30E-02 0% 13 Sentinel-2B No manoeuvre 1.71E-01 0% 1.41E+01 0% 1.98E+00 1.38E-01 0% 12 Sentinel-2B No manoeuvre 7.42E-01 0% 1.09E+01 0% 2.48E+00 8.39E-02 0% 10 Sentinel-2B Tangential Low -2 7.99E-01 0% 1.32E+01 0% 2.54E+00 7.86E-02 0% 10 Sentinel-2B Tangential Low -6 1.46E+00 32% 2.28E+01 0% 2.73E+00 6.50E-02 0% 10 Sentinel-2B Tangential Low -12 9.08E-01 0% 1.18E+01 0% 2.57E+00 7.67E-02 0% 10 Sentinel-2B Tangential Medium -2 2.20E+00 82% 4.00E+01 0% 3.07E+00 4.65E-02 2% 10 Sentinel-2B Tangential Medium -6 4.86E+00 100% 1.46E+02 46% 5.42E+00 4.45E-03 28% 10 Sentinel-2B Tangential Medium -12 5.76E+00 100% 1.33E+02 40% 6.61E+00 1.34E-03 38% 10 Sentinel-2B Tangential High -2 4.71E+00 100% 1.35E+02 41% 5.24E+00 5.31E-03 26% 10 Sentinel-2B Tangential High -6 2.07E+01 100% 2.51E+03 100% 5.58E+01 6.01E-25 100% 10 Sentinel-2B Tangential High -12 1.66E+01 100% 3.08E+02 81% 3.67E+01 1.13E-16 99% 10 Sentinel-2B Out-of-plane Low -2 2.44E+00 90% 4.96E+01 0% 3.21E+00 4.02E-02 4% 10 Sentinel-2B Out-of-plane Low -6 4.12E-01 0% 1.01E+01 0% 2.49E+00 8.30E-02 0% 10 Sentinel-2B Out-of-plane Low -12 2.50E+00 91% 5.15E+01 0% 3.25E+00 3.88E-02 4% 10 Sentinel-2B Out-of-plane Medium -2 1.46E+00 31% 2.33E+01 0% 2.73E+00 6.49E-02 0% 10 Sentinel-2B Out-of-plane Medium -6 1.50E+00 35% 2.38E+01 0% 2.74E+00 6.43E-02 0% 10 Sentinel-2B Out-of-plane Medium -12 1.73E+00 55% 2.89E+01 0% 2.84E+00 5.86E-02 0% 10 Sentinel-2B Out-of-plane High -2 9.74E+00 100% 6.46E+02 98% 1.43E+01 6.08E-07 78% 10 Sentinel-2B Out-of-plane High -6 2.84E+00 96% 6.31E+01 0% 3.47E+00 3.10E-02 7% 10 Sentinel-2B Out-of-plane High -12 6.48E+00 100% 2.90E+02 79% 7.71E+00 4.49E-04 47% 10 Sentinel-2B Hybrid Low -2 3.05E+00 98% 6.80E+01 4% 3.63E+00 2.66E-02 9% 10 Sentinel-2B Hybrid Low -6 7.18E+00 100% 3.04E+02 81% 8.90E+00 1.36E-04 54% 10 Sentinel-2B Hybrid Low -12 7.68E+00 100% 1.86E+02 59% 9.84E+00 5.34E-05 60% 10 Sentinel-2B Hybrid Medium -2 1.20E+01 100% 8.95E+02 99% 2.04E+01 1.32E-09 90% 10 Sentinel-2B Hybrid Medium -6 4.13E+01 100% 1.03E+04 100% 2.15E+02 3.41E-94 100% 10 Sentinel-2B Hybrid Medium -12 2.98E+01 100% 6.78E+02 98% 1.14E+02 3.63E-50 100% 10 A.3 Analyses of the impact of different parameters 89 Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B Hybrid High -2 5.78E+01 100% 2.13E+04 100% 4.20E+02 5.19E-183 100% 10 Sentinel-2B Hybrid High -6 2.36E+02 100% 4.53E+05 100% 6.97E+03 0.00E+00 100% 10 Sentinel-2B Hybrid High -12 3.20E+02 100% 9.77E+05 100% 1.28E+04 0.00E+00 100% 10 Table A.44 Sentinel-2B Reachability analysis and probability from the three distances computed using rangerange rate. CD= 3 Sentinel-1A Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-1A No manoeuvre 3.35E-01 0% 6.03E+00 0% 2.72E+00 6.59E-02 0% 9 Sentinel-1A No manoeuvre 1.04E+00 0% 1.23E+02 36% 2.47E+00 8.45E-02 0% 10 Sentinel-1A No manoeuvre 7.12E-02 0% 9.63E+00 0% 2.64E+00 7.16E-02 0% 9 Sentinel-1A No manoeuvre 3.38E-01 0% 1.82E+01 0% 2.42E+00 8.85E-02 0% 9 Sentinel-1A No manoeuvre 6.18E-01 0% 1.12E+02 31% 2.87E+00 5.67E-02 0% 10 Sentinel-1A No manoeuvre 4.94E-01 0% 2.85E+01 0% 2.11E+00 1.21E-01 0% 9 Sentinel-1A No manoeuvre 1.26E+01 100% 1.37E+04 100% 2.25E+01 1.73E-10 92% 2 Sentinel-1A No manoeuvre 4.09E-01 0% 3.11E+01 0% 2.55E+00 7.78E-02 0% 9 Sentinel-1A No manoeuvre 1.11E+00 0% 9.33E+01 21% 2.47E+00 8.45E-02 0% 9 Sentinel-1A No manoeuvre 1.62E+01 100% 4.51E+03 100% 3.56E+01 3.47E-16 99% 2 Sentinel-1A No manoeuvre 3.46E-01 0% 1.24E+01 0% 2.41E+00 8.96E-02 0% 9 Sentinel-1A No manoeuvre 9.92E-01 0% 9.99E+01 24% 2.51E+00 8.10E-02 0% 11 Sentinel-1A No manoeuvre 1.99E-01 0% 1.80E+01 0% 1.95E+00 1.42E-01 0% 10 Sentinel-1A No manoeuvre 3.70E-01 0% 2.84E+01 0% 2.56E+00 7.73E-02 0% 8 Sentinel-1A No manoeuvre 1.12E+00 0% 1.09E+02 29% 2.90E+00 5.49E-02 0% 9 Sentinel-1A No manoeuvre 3.90E-01 0% 3.59E+01 0% 2.27E+00 1.03E-01 0% 8 Sentinel-1A No manoeuvre 8.76E-01 0% 6.85E+00 0% 2.78E+00 6.22E-02 0% 9 Sentinel-1A No manoeuvre 8.61E-01 0% 1.06E+02 27% 2.56E+00 7.70E-02 0% 10 Sentinel-1A No manoeuvre 1.03E-01 0% 7.97E+00 0% 2.51E+00 8.16E-02 0% 9 Sentinel-1A No manoeuvre 1.47E-01 0% 1.59E+01 0% 2.40E+00 9.03E-02 0% 10 Sentinel-1A No manoeuvre 1.10E+00 0% 1.63E+02 52% 3.03E+00 4.84E-02 1% 10 Sentinel-1A No manoeuvre 2.02E-01 0% 2.22E+01 0% 2.15E+00 1.16E-01 0% 9 Sentinel-1A No manoeuvre 1.57E+01 100% 2.51E+04 100% 3.37E+01 2.26E-15 98% 2 Sentinel-1A No manoeuvre 7.90E-01 0% 2.79E+01 0% 2.62E+00 7.30E-02 0% 9 Sentinel-1A No manoeuvre 2.36E-01 0% 5.04E+01 0% 2.43E+00 8.77E-02 0% 9 Sentinel-1A Tangential Low -2 2.19E+00 82% 4.33E+01 0% 3.24E+00 3.93E-02 4% 9 Sentinel-1A Tangential Low -6 8.30E-01 0% 1.42E+01 0% 2.72E+00 6.59E-02 0% 9 Sentinel-1A Tangential Low -12 3.09E-01 0% 9.69E+00 0% 2.65E+00 7.07E-02 0% 9 Sentinel-1A Tangential Medium -2 4.54E+00 100% 1.54E+02 48% 5.21E+00 5.48E-03 26% 9 Sentinel-1A Tangential Medium -6 8.27E+00 100% 3.27E+02 83% 1.11E+01 1.56E-05 66% 9 Sentinel-1A Tangential Medium -12 6.66E+00 100% 2.02E+02 63% 8.08E+00 3.11E-04 49% 9 Sentinel-1A Tangential High -2 1.69E+01 100% 1.32E+03 100% 3.81E+01 2.78E-17 99% 9 Sentinel-1A Tangential High -6 3.18E+01 100% 4.63E+03 100% 1.29E+02 8.25E-57 100% 9 Sentinel-1A Tangential High -12 3.39E+01 100% 5.20E+03 100% 1.46E+02 2.89E-64 100% 9 Sentinel-1A Out-of-plane Low -2 2.52E+01 100% 4.55E+03 100% 8.23E+01 1.78E-36 100% 9 Sentinel-1A Out-of-plane Low -6 2.13E+01 100% 3.22E+03 100% 5.96E+01 1.34E-26 100% 9 Sentinel-1A Out-of-plane Low -12 2.36E+01 100% 3.97E+03 100% 7.21E+01 4.79E-32 100% 9 Sentinel-1A Out-of-plane Medium -2 5.32E+01 100% 2.00E+04 100% 3.56E+02 2.17E-155 100% 9 Sentinel-1A Out-of-plane Medium -6 4.14E+01 100% 1.18E+04 100% 2.17E+02 8.76E-95 100% 9 Sentinel-1A Out-of-plane Medium -12 4.76E+01 100% 1.64E+04 100% 2.86E+02 5.29E-125 100% 9 Sentinel-1A Out-of-plane High -2 1.02E+02 100% 7.32E+04 100% 1.30E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -6 8.19E+01 100% 4.67E+04 100% 8.40E+02 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -12 9.90E+01 100% 6.75E+04 100% 1.23E+03 0.00E+00 100% 9 Sentinel-1A Hybrid Low -2 4.86E+00 100% 1.70E+02 54% 5.59E+00 3.74E-03 30% 9 Sentinel-1A Hybrid Low -6 9.26E+00 100% 3.97E+02 89% 1.32E+01 1.79E-06 74% 9 Sentinel-1A Hybrid Low -12 1.11E+01 100% 5.43E+02 96% 1.79E+01 1.68E-08 86% 9 Sentinel-1A Hybrid Medium -2 3.70E+01 100% 6.29E+03 100% 1.73E+02 5.25E-76 100% 9 Sentinel-1A Hybrid Medium -6 5.56E+01 100% 1.41E+04 100% 3.89E+02 1.22E-169 100% 9 Sentinel-1A Hybrid Medium -12 9.25E+01 100% 3.87E+04 100% 1.07E+03 0.00E+00 100% 10 Sentinel-1A Hybrid High -2 1.60E+02 100% 1.20E+05 100% 3.21E+03 0.00E+00 100% 9 Sentinel-1A Hybrid High -6 4.35E+02 100% 6.54E+05 100% 2.36E+04 0.00E+00 100% 9 Sentinel-1A Hybrid High -12 1.17E+03 100% 3.44E+06 100% 1.70E+05 0.00E+00 100% 9 Table A.45 Sentinel-1A Reachability analysis and probability from the three distances computed using rangerange rate. 90 Chapter A. Distances and probabilities tables Sentinel-2B Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B No manoeuvre 1.37E-01 0% 1.16E+01 0% 2.23E+00 1.08E-01 0% 5 Sentinel-2B No manoeuvre 1.25E+01 100% 4.30E+03 100% 2.25E+01 1.72E-10 92% 3 Sentinel-2B No manoeuvre 1.69E-01 0% 9.15E+00 0% 2.47E+00 8.48E-02 0% 10 Sentinel-2B No manoeuvre 1.79E+00 60% 7.45E+01 8% 2.66E+00 6.97E-02 0% 10 Sentinel-2B No manoeuvre 5.83E-01 0% 6.05E+00 0% 2.08E+00 1.25E-01 0% 12 Sentinel-2B No manoeuvre 8.03E-01 0% 6.91E+01 4% 2.65E+00 7.04E-02 0% 11 Sentinel-2B No manoeuvre 2.91E-01 0% 7.09E+00 0% 2.90E+00 5.49E-02 0% 10 Sentinel-2B No manoeuvre 4.16E-01 0% 1.79E+01 0% 2.17E+00 1.14E-01 0% 11 Sentinel-2B No manoeuvre 1.39E+00 24% 1.23E+01 0% 2.44E+00 8.74E-02 0% 11 Sentinel-2B No manoeuvre 1.38E+00 23% 7.97E+01 12% 2.69E+00 6.79E-02 0% 11 Sentinel-2B No manoeuvre 5.41E-01 0% 8.28E+00 0% 2.99E+00 5.05E-02 1% 12 Sentinel-2B No manoeuvre 1.65E-01 0% 1.11E+01 0% 1.96E+00 1.41E-01 0% 12 Sentinel-2B No manoeuvre 7.50E-01 0% 9.50E+00 0% 2.40E+00 9.08E-02 0% 10 Sentinel-2B No manoeuvre 2.23E+00 83% 1.45E+02 45% 3.01E+00 4.92E-02 1% 9 Sentinel-2B No manoeuvre 2.50E-01 0% 1.19E+01 0% 2.40E+00 9.07E-02 0% 5 Sentinel-2B No manoeuvre 1.59E+01 100% 9.49E+03 100% 3.44E+01 1.12E-15 98% 3 Sentinel-2B No manoeuvre 4.44E-01 0% 1.11E+01 0% 2.54E+00 7.93E-02 0% 10 Sentinel-2B No manoeuvre 4.83E-01 0% 2.29E+01 0% 2.27E+00 1.04E-01 0% 10 Sentinel-2B No manoeuvre 3.69E-01 0% 5.66E+00 0% 2.09E+00 1.24E-01 0% 12 Sentinel-2B No manoeuvre 3.48E-01 0% 4.12E+01 0% 2.55E+00 7.82E-02 0% 12 Sentinel-2B No manoeuvre 7.19E-01 0% 7.58E+00 0% 2.93E+00 5.34E-02 0% 11 Sentinel-2B No manoeuvre 6.52E-01 0% 2.10E+01 0% 2.14E+00 1.17E-01 0% 11 Sentinel-2B No manoeuvre 7.84E-01 0% 8.73E+00 0% 2.31E+00 9.90E-02 0% 11 Sentinel-2B No manoeuvre 1.54E+00 39% 9.85E+01 24% 2.76E+00 6.36E-02 0% 10 Sentinel-2B No manoeuvre 3.34E-02 0% 6.87E+00 0% 2.92E+00 5.37E-02 0% 13 Sentinel-2B No manoeuvre 6.90E-01 0% 1.82E+01 0% 2.00E+00 1.36E-01 0% 12 Sentinel-2B No manoeuvre 8.73E-02 0% 8.36E+00 0% 2.45E+00 8.64E-02 0% 10 Sentinel-2B Tangential Low -2 1.51E+00 36% 2.43E+01 0% 2.75E+00 6.40E-02 0% 10 Sentinel-2B Tangential Low -6 6.62E-01 0% 1.12E+01 0% 2.52E+00 8.05E-02 0% 10 Sentinel-2B Tangential Low -12 2.11E+00 78% 3.62E+01 0% 3.02E+00 4.88E-02 1% 10 Sentinel-2B Tangential Medium -2 1.51E+00 36% 2.29E+01 0% 2.75E+00 6.37E-02 0% 10 Sentinel-2B Tangential Medium -6 3.95E+00 100% 9.10E+01 19% 4.41E+00 1.21E-02 18% 10 Sentinel-2B Tangential Medium -12 4.52E+00 100% 4.82E+01 0% 5.02E+00 6.60E-03 24% 10 Sentinel-2B Tangential High -2 8.27E+00 100% 4.50E+02 92% 1.10E+01 1.64E-05 66% 10 Sentinel-2B Tangential High -6 2.09E+01 100% 2.59E+03 100% 5.68E+01 2.06E-25 100% 10 Sentinel-2B Tangential High -12 1.65E+01 100% 2.95E+02 79% 3.66E+01 1.22E-16 99% 10 Sentinel-2B Out-of-plane Low -2 2.28E+00 85% 4.43E+01 0% 3.12E+00 4.43E-02 2% 10 Sentinel-2B Out-of-plane Low -6 1.69E+00 52% 2.84E+01 0% 2.82E+00 5.94E-02 0% 10 Sentinel-2B Out-of-plane Low -12 8.65E-01 0% 1.40E+01 0% 2.56E+00 7.74E-02 0% 10 Sentinel-2B Out-of-plane Medium -2 3.03E+00 98% 7.14E+01 6% 3.62E+00 2.69E-02 9% 10 Sentinel-2B Out-of-plane Medium -6 2.01E+00 74% 3.63E+01 0% 2.97E+00 5.11E-02 1% 10 Sentinel-2B Out-of-plane Medium -12 4.70E+00 100% 1.58E+02 50% 5.23E+00 5.37E-03 26% 10 Sentinel-2B Out-of-plane High -2 6.94E+00 100% 3.35E+02 84% 8.48E+00 2.07E-04 52% 10 Sentinel-2B Out-of-plane High -6 3.20E+00 99% 7.85E+01 11% 3.75E+00 2.35E-02 10% 10 Sentinel-2B Out-of-plane High -12 5.62E+00 100% 2.23E+02 67% 6.42E+00 1.63E-03 37% 10 Sentinel-2B Hybrid Low -2 3.04E+00 98% 6.82E+01 4% 3.62E+00 2.67E-02 9% 10 Sentinel-2B Hybrid Low -6 6.95E+00 100% 2.85E+02 78% 8.51E+00 2.01E-04 52% 10 Sentinel-2B Hybrid Low -12 7.19E+00 100% 1.39E+02 43% 8.92E+00 1.33E-04 55% 10 Sentinel-2B Hybrid Medium -2 1.40E+01 100% 1.25E+03 100% 2.70E+01 1.93E-12 96% 10 Sentinel-2B Hybrid Medium -6 4.23E+01 100% 1.09E+04 100% 2.26E+02 7.23E-99 100% 10 Sentinel-2B Hybrid Medium -12 2.95E+01 100% 5.16E+02 95% 1.11E+02 3.93E-49 100% 10 Sentinel-2B Hybrid High -2 5.78E+01 100% 2.13E+04 100% 4.21E+02 2.07E-183 100% 10 Sentinel-2B Hybrid High -6 2.35E+02 100% 4.48E+05 100% 6.90E+03 0.00E+00 100% 10 Sentinel-2B Hybrid High -12 3.18E+02 100% 9.54E+05 100% 1.26E+04 0.00E+00 100% 10 Table A.46 Sentinel-2B Reachability analysis and probability from the three distances computed using rangerange rate. A.3 Analyses of the impact of different parameters 91 A.3.3 Atmospheric model Marshall Sentinel-1A Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-1A No manoeuvre 4.27E-01 0% 6.16E+00 0% 2.73E+00 6.55E-02 0% 9 Sentinel-1A No manoeuvre 7.77E-01 0% 9.59E+01 22% 2.42E+00 8.92E-02 0% 10 Sentinel-1A No manoeuvre 7.39E-01 0% 1.03E+01 0% 2.61E+00 7.37E-02 0% 9 Sentinel-1A No manoeuvre 3.94E-01 0% 1.63E+01 0% 2.43E+00 8.77E-02 0% 9 Sentinel-1A No manoeuvre 5.68E-01 0% 9.95E+01 24% 2.85E+00 5.76E-02 0% 10 Sentinel-1A No manoeuvre 8.24E-01 0% 3.14E+01 0% 2.08E+00 1.25E-01 0% 9 Sentinel-1A No manoeuvre 1.51E+01 100% 1.84E+04 100% 3.13E+01 2.62E-14 98% 2 Sentinel-1A No manoeuvre 3.03E-01 0% 2.45E+01 0% 2.51E+00 8.17E-02 0% 9 Sentinel-1A No manoeuvre 1.60E-01 0% 3.89E+01 0% 2.28E+00 1.02E-01 0% 9 Sentinel-1A No manoeuvre 1.66E+01 100% 8.00E+03 100% 3.73E+01 6.15E-17 99% 2 Sentinel-1A No manoeuvre 1.90E-01 0% 1.18E+01 0% 2.41E+00 9.02E-02 0% 9 Sentinel-1A No manoeuvre 7.48E-01 0% 7.95E+01 12% 2.47E+00 8.47E-02 0% 11 Sentinel-1A No manoeuvre 1.42E+00 27% 1.67E+01 0% 2.82E+00 5.94E-02 0% 10 Sentinel-1A No manoeuvre 9.72E-01 0% 4.03E+01 0% 2.63E+00 7.20E-02 0% 8 Sentinel-1A No manoeuvre 1.34E+00 18% 1.41E+02 44% 2.99E+00 5.03E-02 1% 9 Sentinel-1A No manoeuvre 1.28E+00 12% 8.14E+01 13% 2.46E+00 8.57E-02 0% 8 Sentinel-1A No manoeuvre 2.66E-01 0% 5.84E+00 0% 2.69E+00 6.76E-02 0% 9 Sentinel-1A No manoeuvre 9.54E-01 0% 1.07E+02 28% 2.57E+00 7.67E-02 0% 10 Sentinel-1A No manoeuvre 4.96E-01 0% 1.23E+01 0% 2.65E+00 7.08E-02 0% 9 Sentinel-1A No manoeuvre 1.21E+00 4% 3.25E+01 0% 2.53E+00 7.99E-02 0% 10 Sentinel-1A No manoeuvre 1.12E+00 0% 1.67E+02 53% 3.05E+00 4.74E-02 2% 10 Sentinel-1A No manoeuvre 3.74E-01 0% 2.34E+01 0% 2.16E+00 1.15E-01 0% 9 Sentinel-1A No manoeuvre 1.49E+01 100% 2.10E+04 100% 3.04E+01 6.25E-14 97% 2 Sentinel-1A No manoeuvre 7.52E-02 0% 1.89E+01 0% 2.55E+00 7.82E-02 0% 9 Sentinel-1A No manoeuvre 3.27E-01 0% 5.31E+01 0% 2.43E+00 8.79E-02 0% 9 Sentinel-1A Tangential Low -2 1.75E+00 57% 2.24E+01 0% 2.92E+00 5.39E-02 0% 9 Sentinel-1A Tangential Low -6 2.98E-01 0% 7.78E+00 0% 2.55E+00 7.83E-02 0% 9 Sentinel-1A Tangential Low -12 9.24E-01 0% 1.11E+01 0% 2.64E+00 7.13E-02 0% 9 Sentinel-1A Tangential Medium -2 4.07E+00 100% 8.68E+01 17% 4.60E+00 1.00E-02 20% 9 Sentinel-1A Tangential Medium -6 7.00E+00 100% 2.37E+02 70% 8.65E+00 1.75E-04 53% 9 Sentinel-1A Tangential Medium -12 6.85E+00 100% 2.19E+02 67% 8.40E+00 2.24E-04 51% 9 Sentinel-1A Tangential High -2 1.49E+01 100% 1.07E+03 100% 3.03E+01 6.90E-14 97% 9 Sentinel-1A Tangential High -6 3.18E+01 100% 4.82E+03 100% 1.29E+02 6.84E-57 100% 9 Sentinel-1A Tangential High -12 3.51E+01 100% 5.61E+03 100% 1.57E+02 1.00E-68 100% 9 Sentinel-1A Out-of-plane Low -2 3.02E+01 100% 4.42E+03 100% 1.16E+02 2.76E-51 100% 9 Sentinel-1A Out-of-plane Low -6 2.58E+01 100% 3.21E+03 100% 8.55E+01 7.66E-38 100% 9 Sentinel-1A Out-of-plane Low -12 2.94E+01 100% 4.28E+03 100% 1.10E+02 1.32E-48 100% 9 Sentinel-1A Out-of-plane Medium -2 6.40E+01 100% 2.04E+04 100% 5.15E+02 1.96E-224 100% 9 Sentinel-1A Out-of-plane Medium -6 5.15E+01 100% 1.29E+04 100% 3.34E+02 1.15E-145 100% 9 Sentinel-1A Out-of-plane Medium -12 5.88E+01 100% 1.69E+04 100% 4.34E+02 2.13E-189 100% 9 Sentinel-1A Out-of-plane High -2 1.24E+02 100% 7.36E+04 100% 1.91E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -6 1.01E+02 100% 4.90E+04 100% 1.27E+03 0.00E+00 100% 9 Sentinel-1A Out-of-plane High -12 1.20E+02 100% 6.91E+04 100% 1.81E+03 0.00E+00 100% 9 Sentinel-1A Hybrid Low -2 6.38E+00 100% 2.04E+02 63% 7.62E+00 4.90E-04 46% 9 Sentinel-1A Hybrid Low -6 9.49E+00 100% 4.34E+02 91% 1.38E+01 1.02E-06 76% 9 Sentinel-1A Hybrid Low -12 1.04E+01 100% 4.98E+02 94% 1.62E+01 9.68E-08 82% 9 Sentinel-1A Hybrid Medium -2 3.83E+01 100% 7.14E+03 100% 1.86E+02 1.94E-81 100% 9 Sentinel-1A Hybrid Medium -6 5.49E+01 100% 1.45E+04 100% 3.80E+02 1.11E-165 100% 9 Sentinel-1A Hybrid Medium -12 8.59E+01 100% 3.89E+04 100% 9.24E+02 0.00E+00 100% 10 Sentinel-1A Hybrid High -2 1.56E+02 100% 1.17E+05 100% 3.05E+03 0.00E+00 100% 9 Sentinel-1A Hybrid High -6 4.30E+02 100% 6.63E+05 100% 2.31E+04 0.00E+00 100% 9 Sentinel-1A Hybrid High -12 1.15E+03 100% 3.47E+06 100% 1.65E+05 0.00E+00 100% 9 Table A.47 Sentinel-1A Reachability analysis and probability from the three distances computed using rangerange rate. Sentinel-2B Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B No manoeuvre 1.40E+00 25% 3.11E+01 0% 2.47E+00 8.49E-02 0% 5 Sentinel-2B No manoeuvre 1.27E+01 100% 4.55E+03 100% 2.33E+01 7.82E-11 93% 3 98 Chapter A. Distances and probabilities tables Scenario MD(R,Rr) PR KLD(R,Rr) PR BD(R,Rr) ρ(R,Rr) PR plots Sentinel-2B Out-of-plane Medium -6 6.51E-01 0% 2.07E+01 0% 2.32E+00 9.85E-02 0% 10 Sentinel-2B Out-of-plane Medium -12 3.11E+00 98% 1.53E+02 48% 3.48E+00 3.09E-02 7% 10 Sentinel-2B Out-of-plane High -2 7.55E+00 100% 8.24E+02 99% 9.40E+00 8.29E-05 57% 10 Sentinel-2B Out-of-plane High -6 2.15E+00 80% 7.92E+01 12% 2.84E+00 5.82E-02 0% 10 Sentinel-2B Out-of-plane High -12 6.54E+00 100% 6.21E+02 97% 7.61E+00 4.95E-04 46% 10 Sentinel-2B Hybrid Low -2 3.56E+00 100% 1.63E+02 52% 3.85E+00 2.13E-02 11% 10 Sentinel-2B Hybrid Low -6 8.58E+00 100% 5.45E+02 96% 1.15E+01 1.05E-05 67% 10 Sentinel-2B Hybrid Low -12 1.27E+01 100% 1.19E+02 34% 2.25E+01 1.71E-10 92% 10 Sentinel-2B Hybrid Medium -2 1.67E+01 100% 3.16E+03 100% 3.72E+01 6.67E-17 99% 10 Sentinel-2B Hybrid Medium -6 5.50E+01 100% 3.05E+04 100% 3.80E+02 9.71E-166 100% 10 Sentinel-2B Hybrid Medium -12 6.40E+01 100% 5.84E+03 100% 5.15E+02 2.16E-224 100% 10 Sentinel-2B Hybrid High -2 6.95E+01 100% 5.54E+04 100% 6.05E+02 1.41E-263 100% 10 Sentinel-2B Hybrid High -6 2.79E+02 100% 1.17E+06 100% 9.75E+03 0.00E+00 100% 10 Sentinel-2B Hybrid High -12 4.14E+02 100% 2.67E+06 100% 2.14E+04 0.00E+00 100% 10 Table A.54 Sentinel-2B Reachability analysis and probability from the three distances computed using rangerange rate. A.4 Analysis on real scenarios The last analysis of the project is the one performed on real scenarios. Fifteen different scenarios have been used. The distances and the probabilities are presented in one table for each scenario. The tables show the number of manoeuvres in each segment (column 2), the statistical distances computed using only range-range rate and the probability of manoeuvre occurence. The MD is in column 3, the KLD is in column 5 and the BD and its coefficient are in columns 7 and 8 respectively. The probability computed from each distance is presented in the adjacent column of each one, on the right. Scenario 1 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 8.49E-01 0% 2.26E+00 0% 8.52E-01 4.26E-01 0% 2 0 3.70E-01 0% 4.66E+00 0% 2.13E+00 1.19E-01 0% 3 2 2.58E-01 0% 7.03E+00 0% 3.33E+00 3.57E-02 5% 4 0 8.99E-01 0% 6.05E+00 0% 2.17E+00 1.14E-01 0% 5 0 4.65E-01 0% 6.08E+00 0% 2.87E+00 5.69E-02 0% 6 0 2.69E-01 0% 7.09E+00 0% 3.37E+00 3.44E-02 6% 7 0 3.88E-01 0% 4.44E+00 0% 2.04E+00 1.30E-01 0% 8 0 4.78E-01 0% 7.18E+00 0% 3.39E+00 3.37E-02 6% 9 0 4.65E-01 0% 4.80E+00 0% 2.21E+00 1.10E-01 0% 10 0 2.94E-01 0% 5.48E+00 0% 2.53E+00 7.96E-02 0% 11 2 3.76E+00 100% 2.58E+01 0% 4.21E+00 1.49E-02 15% 12 0 2.99E-02 0% 2.24E+01 0% 3.13E+00 4.35E-02 3% 13 0 2.96E-01 0% 4.21E+00 0% 1.98E+00 1.38E-01 0% Table A.55 Scenario 1 probability from the three distances computed using range-range rate. Scenario 2 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 2.05E-01 0% 1.51E+00 0% 6.12E-01 5.42E-01 0% 2 0 1.17E+00 0% 7.57E+00 0% 3.41E+00 3.30E-02 6% 3 0 4.60E-01 0% 5.88E+00 0% 2.77E+00 6.28E-02 0% 4 0 1.74E+00 56% 1.68E+01 0% 3.04E+00 4.78E-02 1% 5 0 2.85E-01 0% 5.63E+00 0% 2.69E+00 6.80E-02 0% 6 0 1.61E+00 45% 1.86E+01 0% 3.21E+00 4.04E-02 4% A.4 Analysis on real scenarios 99 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 7 0 5.30E+00 100% 2.26E+02 68% 5.93E+00 2.67E-03 33% 8 2 5.01E+00 100% 2.25E+01 0% 5.17E+00 5.68E-03 26% 9 0 7.50E-02 0% 6.17E+00 0% 2.99E+00 5.05E-02 1% Table A.56 Scenario 2 probability from the three distances computed using range-range rate. Scenario 3 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 3.28E-01 0% 2.63E+00 0% 8.64E-01 4.21E-01 0% 2 0 2.10E-01 0% 2.70E+01 0% 3.18E+00 4.15E-02 3% 3 0 6.90E-01 0% 5.21E+00 0% 2.29E+00 1.01E-01 0% 4 0 2.46E-01 0% 6.93E+00 0% 3.30E+00 3.67E-02 5% 5 0 7.87E-01 0% 5.55E+00 0% 2.27E+00 1.03E-01 0% 6 2 2.95E-01 0% 7.17E+00 0% 3.40E+00 3.35E-02 6% 7 0 2.25E+00 84% 1.24E+01 0% 2.75E+00 6.40E-02 0% 8 0 7.13E-01 0% 6.03E+00 0% 2.78E+00 6.23E-02 0% 9 0 1.19E-01 0% 4.57E+00 0% 2.18E+00 1.13E-01 0% 10 0 1.18E+00 1% 1.72E+01 0% 3.14E+00 4.33E-02 3% 11 0 5.89E-01 0% 4.97E+00 0% 2.24E+00 1.07E-01 0% Table A.57 Scenario 3 probability from the three distances computed using range-range rate. Scenario 4 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 1.31E+00 16% 8.05E+00 0% 1.87E+00 1.53E-01 0% 2 2 3.62E+00 100% 1.68E+01 0% 4.09E+00 1.67E-02 14% 3 0 3.42E+00 99% 2.44E+01 0% 3.87E+00 2.10E-02 12% 4 0 1.08E+00 0% 8.21E+01 14% 3.29E+00 3.73E-02 5% 5 0 3.49E-01 0% 4.38E+00 0% 2.01E+00 1.34E-01 0% 6 0 3.73E-01 0% 6.41E+00 0% 3.05E+00 4.76E-02 1% 7 0 5.17E-01 0% 5.13E+00 0% 2.17E+00 1.15E-01 0% 8 0 5.99E-01 0% 6.95E+00 0% 3.26E+00 3.86E-02 4% 9 0 5.90E-02 0% 4.63E+00 0% 2.11E+00 1.21E-01 0% 10 1 1.19E-01 0% 5.72E+00 0% 2.74E+00 6.44E-02 0% Table A.58 Scenario 4 probability from the three distances computed using range-range rate. Scenario 5 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 2.82E-02 0% 2.89E+00 0% 1.27E+00 2.80E-01 0% 2 0 5.01E-02 0% 4.33E+00 0% 2.07E+00 1.27E-01 0% 3 0 4.36E-01 0% 7.12E+00 0% 3.36E+00 3.46E-02 6% 4 0 6.70E-01 0% 5.07E+00 0% 2.23E+00 1.08E-01 0% 5 2 1.09E+00 0% 6.61E+00 0% 2.78E+00 6.19E-02 0% 6 0 2.81E-01 0% 6.94E+00 0% 3.30E+00 3.68E-02 5% 7 0 8.19E-01 0% 4.94E+00 0% 2.08E+00 1.24E-01 0% 8 0 9.81E-01 0% 7.20E+00 0% 3.28E+00 3.78E-02 4% 9 0 2.57E-02 0% 4.37E+00 0% 2.05E+00 1.29E-01 0% 10 0 8.11E-01 0% 7.55E+00 0% 3.50E+00 3.01E-02 7% 11 0 1.42E+00 27% 8.59E+00 0% 2.13E+00 1.19E-01 0% 100 Chapter A. Distances and probabilities tables Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 12 2 2.85E+00 97% 1.13E+01 0% 3.89E+00 2.05E-02 12% 13 0 3.64E-01 0% 5.19E+00 0% 2.30E+00 1.01E-01 0% 14 0 1.28E+00 12% 3.36E+01 0% 3.23E+00 3.97E-02 4% Table A.59 Scenario 5 probability from the three distances computed using range-range rate. Scenario 6 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 8.43E-01 0% 3.85E+00 0% 1.33E+00 2.65E-01 0% 2 0 5.36E-01 0% 6.64E+00 0% 3.12E+00 4.41E-02 2% 3 0 8.77E-01 0% 5.96E+00 0% 2.46E+00 8.56E-02 0% 4 0 4.71E-01 0% 6.27E+00 0% 2.94E+00 5.31E-02 0% 5 0 6.82E-01 0% 1.59E+01 0% 2.94E+00 5.28E-02 0% 6 0 1.51E-01 0% 4.27E+00 0% 2.01E+00 1.34E-01 0% 7 0 1.01E+00 0% 6.99E+00 0% 3.13E+00 4.37E-02 3% 8 0 6.41E-01 0% 5.73E+00 0% 2.15E+00 1.16E-01 0% 9 3 8.17E-01 0% 6.98E+00 0% 3.22E+00 3.99E-02 4% 10 0 7.61E-01 0% 4.57E+00 0% 1.95E+00 1.42E-01 0% 11 0 6.76E-01 0% 5.96E+00 0% 2.69E+00 6.80E-02 0% 12 0 5.57E-01 0% 5.43E+00 0% 2.18E+00 1.13E-01 0% 13 0 1.02E+00 0% 1.61E+01 0% 3.09E+00 4.55E-02 2% Table A.60 Scenario 6 probability from the three distances computed using range-range rate. Scenario 7 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 1.03E-01 0% 2.71E+00 0% 1.26E+00 2.84E-01 0% 2 0 7.36E-01 0% 7.05E+00 0% 3.28E+00 3.78E-02 4% 3 0 2.72E+00 95% 2.08E+01 0% 3.19E+00 4.12E-02 3% 4 0 9.76E-01 0% 7.63E+00 0% 3.50E+00 3.02E-02 7% 5 0 1.31E+00 15% 6.59E+00 0% 2.24E+00 1.06E-01 0% 6 0 4.44E-01 0% 6.08E+00 0% 2.87E+00 5.69E-02 0% 7 0 5.06E-02 0% 4.43E+00 0% 2.11E+00 1.22E-01 0% 8 0 1.06E-01 0% 7.36E+00 0% 2.95E+00 5.25E-02 0% 9 0 9.04E-01 0% 5.76E+00 0% 2.29E+00 1.02E-01 0% 10 2 2.34E-01 0% 6.89E+00 0% 3.27E+00 3.82E-02 4% 11 0 4.89E-03 0% 4.93E+00 0% 2.37E+00 9.38E-02 0% 12 0 1.27E-02 0% 6.62E+00 0% 3.13E+00 4.36E-02 3% 13 0 1.76E+00 58% 3.61E+01 0% 3.22E+00 3.99E-02 4% 14 0 2.93E-01 0% 4.56E+00 0% 2.08E+00 1.25E-01 0% 15 0 1.62E+00 46% 8.44E+00 0% 3.63E+00 2.66E-02 9% 16 0 7.18E-01 0% 5.90E+00 0% 2.30E+00 1.00E-01 0% 17 0 4.59E-01 0% 7.39E+00 0% 3.49E+00 3.06E-02 7% 18 0 7.83E-02 0% 5.21E+00 0% 2.17E+00 1.14E-01 0% Table A.61 Scenario 7 probability from the three distances computed using range-range rate. A.4 Analysis on real scenarios 101 Scenario 8 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 1.12E+00 0% 3.80E+01 0% 1.91E+00 2.20E-05 0% 2 0 1.15E-01 0% 6.21E+00 0% 2.96E+00 4.27E-05 0% 3 0 8.25E-01 0% 5.06E+00 0% 2.20E+00 4.54E-05 0% 4 0 7.48E-01 0% 2.46E+01 0% 3.09E+00 2.93E-05 2% 5 0 8.95E-01 0% 5.64E+00 0% 2.15E+00 5.02E-05 0% 6 0 8.78E-01 0% 7.57E+00 0% 3.50E+00 2.68E-05 7% 7 0 4.99E-02 0% 4.76E+00 0% 2.25E+00 6.55E-05 0% 8 0 1.81E+00 61% 7.72E+00 0% 2.98E+00 1.27E-05 1% 9 0 1.95E-01 0% 4.72E+00 0% 2.21E+00 9.36E-06 0% 10 0 2.75E-01 0% 2.34E+01 0% 2.99E+00 1.25E-05 1% 11 0 1.64E+00 48% 7.76E+00 0% 2.32E+00 6.52E-05 0% 12 2 5.88E-01 0% 7.37E+00 0% 3.47E+00 5.17E-05 7% 13 0 2.69E-01 0% 4.52E+00 0% 2.03E+00 3.88E-05 0% 14 0 8.87E-01 0% 7.81E+00 0% 3.62E+00 5.49E-05 9% 15 0 2.55E+00 92% 2.04E+01 0% 2.82E+00 6.87E-06 0% 16 0 7.50E-01 0% 6.43E+00 0% 2.97E+00 3.05E-05 0% 17 0 2.04E+00 75% 8.13E+00 0% 2.71E+00 2.81E-05 0% 18 0 1.53E+00 37% 5.40E+01 0% 3.28E+00 3.10E-05 5% 19 0 1.25E+00 8% 7.28E+00 0% 2.32E+00 4.93E-05 0% Table A.62 Scenario 8 probability from the three distances computed using range-range rate. Scenario 9 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 8.48E-01 0% 3.15E+00 0% 9.99E-01 3.68E-01 0% 2 2 1.04E+00 0% 8.54E+00 0% 3.94E+00 1.94E-02 12% 3 0 2.37E+00 88% 1.03E+01 0% 2.83E+00 5.90E-02 0% 4 0 2.30E+00 86% 2.66E+01 0% 3.59E+00 2.76E-02 8% 5 0 5.69E-01 0% 7.67E+00 0% 3.61E+00 2.71E-02 9% 6 0 1.95E-01 0% 5.41E+00 0% 2.39E+00 9.21E-02 0% 7 0 2.87E+00 97% 1.40E+01 0% 4.08E+00 1.68E-02 14% 8 0 7.67E-01 0% 7.23E+00 0% 2.80E+00 6.10E-02 0% 9 0 7.63E-01 0% 6.03E+00 0% 2.58E+00 7.59E-02 0% Table A.63 Scenario 9 probability from the three distances computed using range-range rate. Scenario 10 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 4.14E-01 0% 1.06E+00 0% 3.65E-01 6.94E-01 0% 2 0 5.59E+00 100% 1.21E+02 35% 6.25E+00 1.93E-03 35% 3 0 7.04E-01 0% 4.28E+00 0% 1.79E+00 1.67E-01 0% 4 0 6.17E-01 0% 7.24E+00 0% 3.40E+00 3.33E-02 6% 5 0 4.69E-01 0% 5.85E+00 0% 2.71E+00 6.66E-02 0% 6 0 6.69E-01 0% 7.23E+00 0% 3.39E+00 3.38E-02 6% 7 2 1.47E-01 0% 6.29E+00 0% 3.03E+00 4.83E-02 1% Table A.64 Scenario 10 probability from the three distances computed using range-range rate. 102 Chapter A. Distances and probabilities tables Scenario 11 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 7.03E-01 0% 2.75E+00 0% 1.14E+00 3.20E-01 0% 2 0 4.99E-01 0% 6.81E+00 0% 2.69E+00 6.76E-02 0% 3 0 4.07E-01 0% 6.43E+00 0% 3.05E+00 4.73E-02 2% 4 0 7.80E-01 0% 9.37E+00 0% 4.42E+00 1.21E-02 18% 5 2 1.19E+01 100% 1.07E+02 28% 2.07E+01 9.86E-10 90% 6 0 1.23E+00 6% 9.38E+00 0% 2.96E+00 5.21E-02 0% Table A.65 Scenario 11 probability from the three distances computed using range-range rate. Scenario 12 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 8.16E-01 0% 1.61E+00 0% 4.38E-01 6.46E-01 0% 2 0 1.19E+00 2% 3.47E+01 0% 2.89E+00 5.58E-02 0% 3 0 3.54E-01 0% 6.54E+00 0% 3.11E+00 4.44E-02 2% 4 0 3.13E-01 0% 5.90E+00 0% 2.81E+00 6.02E-02 0% 5 0 1.06E+00 0% 9.40E+00 0% 2.69E+00 6.79E-02 0% 6 0 9.09E-02 0% 6.35E+00 0% 3.04E+00 4.78E-02 1% 7 0 5.48E-01 0% 7.81E+00 0% 2.77E+00 6.28E-02 0% 8 2 1.69E+00 52% 8.28E+00 0% 3.37E+00 3.45E-02 6% 9 0 2.71E+00 95% 6.82E+01 4% 3.51E+00 2.99E-02 7% 10 0 3.95E-01 0% 3.59E+00 0% 1.65E+00 1.93E-01 0% Table A.66 Scenario 12 probability from the three distances computed using range-range rate. Scenario 13 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 8.86E-01 0% 5.51E+00 0% 1.40E+00 2.46E-01 0% 2 0 1.13E+00 0% 6.09E+00 0% 2.57E+00 7.65E-02 0% 3 0 6.89E-01 0% 8.80E+00 0% 4.15E+00 1.58E-02 15% 4 0 1.42E+00 27% 6.49E+00 0% 2.54E+00 7.87E-02 0% 5 0 6.62E-01 0% 8.31E+00 0% 3.91E+00 2.00E-02 12% 6 0 7.99E-01 0% 5.61E+00 0% 2.50E+00 8.21E-02 0% 7 0 7.52E-01 0% 7.59E+00 0% 3.54E+00 2.91E-02 8% 8 0 4.47E-01 0% 5.26E+00 0% 2.37E+00 9.33E-02 0% 9 1 4.04E-01 0% 7.58E+00 0% 3.58E+00 2.78E-02 8% 10 0 5.42E-01 0% 5.05E+00 0% 2.30E+00 1.00E-01 0% 11 0 1.67E+00 51% 7.87E+00 0% 2.45E+00 8.65E-02 0% 12 0 5.28E-01 0% 8.26E+00 0% 3.91E+00 2.01E-02 12% 13 0 1.01E+00 0% 5.85E+00 0% 2.55E+00 7.79E-02 0% Table A.67 Scenario 13 probability from the three distances computed using range-range rate. Scenario 14 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 6.41E+00 100% 1.27E+02 38% 5.99E+00 2.51E-03 33% 2 1 7.53E-01 0% 6.83E+00 0% 2.50E+00 8.24E-02 0% 3 4 4.16E+00 100% 1.58E+01 0% 5.30E+00 5.01E-03 27% 4 2 9.99E-01 0% 6.75E+00 0% 2.98E+00 5.06E-02 1% A.4 Analysis on real scenarios 103 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 5 2 4.09E-01 0% 6.95E+00 0% 3.28E+00 3.78E-02 4% Table A.68 Scenario 14 probability from the three distances computed using range-range rate. Scenario 15 Segment Manoeuvre MD(R, Rr) PR KLD (R,Rr) PR BD (R, Rr)ρBD (R, Rr) PR 1 0 9.46E+00 100% 1.60E+03 100% 1.18E+01 7.55E-06 69% 2 1 6.28E+00 100% 3.17E+01 0% 7.95E+00 3.53E-04 48% 3 1 6.62E+00 100% 5.12E+01 0% 7.90E+00 3.71E-04 48% 4 3 4.57E-01 0% 7.51E+00 0% 3.54E+00 2.90E-02 8% 5 2 3.72E+00 100% 1.58E+01 0% 4.74E+00 8.74E-03 21% Table A.69 Scenario 15 probability from the three distances computed using range-range rate. List of Figures 1.1 Objects in orbit around Earth [1] 1 2.1 Earth-centred inertial frame [6] 5 2.2 Earth-centred Earth-fixed frame (g) and topocentric frame (w) [7] 6 2.3 Local-vertical local-horizontal frame [8] 7 2.4 Two masses located in an inertial frame [6] 8 2.5 Eccentric anomaly definition [6] 9 2.6 Keplerian orbital elements [6] 10 2.7 US Standard Atmosphere 1976 [6] 12 2.8 Perturbation effects in Low earth Orbit [10] 15 2.9 Two-line element set format [11] 15 2.10 Satellite visibility from Earth station (Re=R⊕) [12] 16 2.11 Elevation-azimuth mask 17 2.12 Montecarlo-like procedure [19] 21 2.13 Taylor based jet transport [19] 21 3.1 Histogram of the KLD 28 3.2 Probability density function of the gamma distribution 28 3.3 Cumulative density function of the KLD 29 3.4 Probability density function of the KLD 29 3.5 Cumulative density function of the BD 31 3.6 Probability density function of the BD 31 105 List of Tables 2.1 Earth zonal harmonics 13 3.1 Contingency table 32 4.1 Orbital element of the considered orbits at initial epoch 35 4.2 Data for the propagation 35 4.3 Results of all the satellites for 247 scenarios (102 without manoeuvre and 135 with manoeuvre) 38 4.4 Precision, recall, F1and Fβ(β=0.5) 38 4.5 Results of satellite Sentinel-1A for 49 scenarios (22 without manoeuvre and 27 with manoeuvre) 39 4.6 Results of satellite Sentinel-1B for 44 scenarios (17 without manoeuvre and 27 with manoeuvre) 39 4.7 Results of satellite Sentinel-2A for 46 scenarios (19 without manoeuvre and 27 with manoeuvre) 40 4.8 Results of satellite Sentinel-2B for 52 scenarios (25 without manoeuvre and 27 with manoeuvre) 40 4.9 Results of satellite Swarm-C for 46 scenarios (19 without manoeuvre and 27 with manoeuvre) 41 4.10 Results for tangent manoeuvre 42 4.11 Results for out-of-plane manoeuvre 42 4.12 Results for hybrid manoeuvre) 42 4.13 Results for the MD computed using only range-range rate 43 4.14 Results for the KLD computed using only range-range rate 43 4.15 Results for the BD computed using only range-range rate 44 4.16 Ballistic coefficient values 44 4.17 Results for the MD computed using only range-range rate 44 4.18 Results for the KLD computed using only range-range rate 45 4.19 Results for the BD computed using only range-range rate 45 4.20 Results for the MD computed using only range-range rate 45 4.21 Results for the KLD computed using only range-range rate 45 4.22 Results for the BD computed using only range-range rate 46 4.23 Results for the MD computed using only range-range rate 46 4.24 Results for the KLD computed using only range-range rate 47 4.25 Results for the BD computed using only range-range rate 47 4.26 Results for the MD computed using only range-range rate 47 4.27 Results for the KLD computed using only range-range rate 47 4.28 Results for the BD computed using only range-range rate 48 4.29 Results of 162 real segments (138 without manoeuvre and 24 with manoeuvre) 48 4.30 Precision, recall, F1and Fβ(β=0.5) 49 A.1 Sentinel-1A Reachability analysis and probability from MD 54 A.2 Sentinel-1A Reachability analysis and probability from KLD 55 A.3 Caption for LOF 56 A.4 Sentinel-1B Reachability analysis and probability from MD 57 A.5 Sentinel-1B Reachability analysis and probability from KLD 58 A.6 Sentinel-1B Reachability analysis and probability from BD 59 A.7 Sentinel-2A Reachability analysis and probability from MD 59 107