Polarimetric formulation of the visibility function equation including cross-polar antenna patterns

292 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 2, NO. 3, JULY 2005 Polarimetric Formulation of the Visibility Function Equation Including Cross-Polar Antenna Patterns Adriano Camps, Senior Member, IEEE, Ignasi Corbella, Member, IEEE, Francesc Torres, Member, IEEE, Mercè Vall-llossera, Member, IEEE, and Nuria Duffo, Member, IEEE Abstract—The European Space Agency’s Soil Moisture and Ocean Salinity (SMOS) mission will be the first one using two-di- mensional aperture synthesis radiometry for earth observation. This study presents the formulation that relates instrument observables and brightness temperature maps including cross-polar antenna voltage patterns, which may be also different from element to element. Finally, the radiometric accuracy degradation if cross-polar patterns are neglected in the image reconstruction is studied. Index Terms—Antenna arrays, interferometry, radiometry. I. INTRODUCTION THE Microwave Imaging Radiometer by Aperture Synthesis (MIRAS) instrument [1], [2] is the single payload of the European Space Agency’s Soil Moisture and Ocean Salinity (SMOS) mission, and it will be the first two-dimen- sional aperture synthesis radiometer for earth observation. The brightness temperature image reconstruction is performed through a Fourier synthesis process of the cross-correlations measured between the band pass signals and -centered at , - collected by every pair of elements in the array located in the plane. According to [3] (1) where is the Boltzmann’s constant, and are the receivers’ noise bandwidth and power gain, is the solid angle of the antennas, is the brightness temperature of the scene [see (4)], is the physical temperature of the receivers [3], if and 0 if , Manuscript received October 10, 2003; revised February 4, 2004. This work was supported in part by the Spanish Comisión Interministerial de Ciencia y Tecnología under Grant MCYT TIC2002-04451-C02-01 and in part by EADSCASA Space Division (Madrid, Spain), prime contractor of SMOS PLM during Phase B Activities. The authors are with the Universitat Politècnica de Catalunya, E-08034 Barcelona, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/LGRS.2005.846885 are the normalized antenna copolar voltage patterns at and polarizations, is the fringe-washing function, is the spatial frequency (baseline) that depends on the antenna position difference: , and the director cosines are defined with respect to the and axes. II. BRIGHTNESS TEMPERATURE MEASURED BY THE INTERFEROMETRIC RADIOMETER:IDEAL CASE WITH ISOTROPIC ANTENNA VOLTAGE PATTERNS AND ZERO CROSS-POLAR PATTERNS Due to the following: • the relative orientation [ ] between the pixel’s reference frame over the earth’s surface ( or : vertical and horizontal polarizations) and the antenna reference frame [4] ( is perpendicular to the orbital plane; is 32 upward titled from the velocity vector); • Faraday rotation effects [ ] the electric fields incident in the antennas are rotated an angle [4] (2) where , . Therefore, since is proportional to , the brightness temperatures in the antenna reference frame at a given polarization ( or ) can be expressed as a linear combination of the brightness temperatures in the earth’s pixel reference frame ( or ) [5, eq. (57)]. (3) where , and and are the third and fourth Stokes parameters in the earth’s reference frame. Compared to and , and have negligible values, and even more at L-band. Therefore, and , and . Equation (3) reduces to (4) 1545-598X/$20.00 © 2005 IEEE CAMPS et al.: POLARIMETRIC FORMULATION 293 Fig. 1. Image errors maps in a high-contrast scene if antenna cross-polarization patterns are neglected: T = A T + B T and T = AB ( T 0 T ) . Root mean squared errors in the alias-free field of view:  =0 : 18 K,  =5 : 510 K,  =0 : 15 K,  = 3 : 610 K,  =0 : 30 K and  =3 : 5 K. III. BRIGHTNESS TEMPERATURE MEASURED BY THE INTERFEROMETRIC RADIOMETER:R EAL CASE INCLUDING ANTENNA COPOLAR AND NONZERO CROSS-POLAR PATTERNS If nonzero cross-polar antenna patterns are taken into account, a similar process to the one followed to derive (2)–(4), can be used to derive the cross-brightness temperatures to be used in (1) (5) where the symbol denotes the received electric field including the effect of the antenna voltage patterns, is the copolar pattern of antenna 1 or 2, at or polarization, and is the cross-polar antenna pattern, both normalized to the maximum of , and defined according to Ludwig’s third definition [6].1Expanding (5), the relationship between 1In (5) the measured co- and cross-polar patterns at X - and Y -polarizations correspond to the two outputs of a dual-polarization antenna as normally measured. If the Y -pattern were obtained by a rotation of the X -pattern, then the cross-polar pattern C must be replaced by 0 C . , and and , and can be readily obtained (6) (7) (8) The term “ ” in (1) accounts for the thermal noise radiated by the receivers at both polarizations (but not at the cross-polar one) that is collected by other elements because of the nonzero antenna coupling. This noise is collected together with the noise coming from the scene being imaged. Therefore, following a similar procedure to the one described in previous paragraphs for , (1) should be modified for the term accordingly (9) Substituting (6)–(9) in (1), (1) can be rewritten making the following substitutions, as in (10), shown at the bottom of the page, where it has been assumed that . (10) 294 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 2, NO. 3, JULY 2005 IV. ERRORS ASSOCIATED TO NONZERO CROSS-POLAR ANTENNA PATTERNS IN THE IMAGE RECONSTRUCTION ALGORITHMS IN THE ANTENNA REFERENCE FRAME In order to study the impact of neglecting antenna cross-polar pattern in the image reconstruction algorithms, synthetic brightness temperature images have been generated using the SMOS End-to-end Performance Simulator (SEPS) [7]. The co- and cross-polar antenna radiation voltage patterns used correspond to the first nine receiver units (LICEF-2) of MIRAS and exhibit a directivity of 9 dB approximately, with cross-polar level at boresight of dB approximately, and dB worst case at the border of the alias-free field of view, which is determined by the periodic repetition of six replicas of the earth’s image [7]. Fig. 1 (top) shows the difference between when the cross-polar component of the antenna voltage patterns is assumed to be zero, and (7) with nonzero cross-polar component, for a high-contrast scenario, including land and sea. Fig. 1 (bottom) presents a similar result for . Error images at polarization are very similar to the ones at polarization, and are not shown. The rms errors computed in the alias-free field-of-view at different polarizations are: K, K, K, K, K and K where the superscript “”refers to the if the cross-polar patterns were zero. It would be expected that the smaller the difference between and , the smaller the effect of nonzero cross-polar pattern errors. However, a low-contrast only-ocean scene has similar errors at - and - polarizations, and just slightly smaller at -polarization: K, and K. Errors in the imaginary parts of and come from neglecting cross-polar antenna patterns in the image reconstruction. At the present time there is not an explanation for the fact that the error on is found with the same sign throughout the field of view, but it is believed that it may be due to a phase offset between the and antenna patterns, due to different path lengths from the antenna to the receivers’input. V. ERROR AMPLIFICATION IN THE TRANSFORMATION FROM ANTENNA REFERENCE FRAME TO EARTH REFERENCE FRAME MIRAS has two operation modes. •Full-polarimetric mode, in which , and are measured by a sophisticated combination of the antenna polarization switches at and polarizations [5], and •Dual-polarization mode, in which and are sequentially measured, by selecting all the antenna polarization switches at and polarizations simultaneously. In the full-polarimetric mode, the brightness temperatures in the earth reference frame ( , , , and ) are computed from the measured brightness temperatures in the antenna reference frame ( , , and ) by inverting (3) (11) and errors in the antenna reference frame translate into errors in the earth reference frame (12) Note that the inverse of the matrix in (3) is equal to its transpose. In the dual-polarization mode, the brightness temperatures in the earth reference frame ( and ) are computed from (4), assuming (13) and errors in the antenna reference frame translate into errors in the earth reference frame (14) Note that the errors tend to infinite wherever ,as studied in [8]. Errors in [see (8)] are specially important because of the phase difference between the co- and cross-polar antenna voltage patterns, and may ultimately limit the capabilities of using the measurement of the third Stokes parameter to correct for Faraday rotation effects2[see (8)] [9], necessary to derive and accurately.3Additionally, even though the measurement of allows to obtain and maps in the whole alias-free field of view without singularities [see (11); matrix in (3) is never singular], errors in directly translate into and through (12). The singularity problem can be avoided and cross-polar pattern errors minimized if the geophysical parameter retrieval algorithms are designed to operate with brightness temperature maps in the antenna reference frame—although Faraday rotation correction will still be necessary—or to operate on , which is invariant to rotations (see [11] for ocean salinity retrieval study). can be computed as , where and are measured in the dual polarization mode. VI. CONCLUSION This work has presented the extension of the formulation of the visibility function to include the cross-polar voltage patterns, which may be different from element to element. Its impact on the radiometric accuracy has been analyzed using SEPS and antenna patterns measured by EADS-CASA espacio for the first nine MIRAS receivers, showing a degradation smaller than 0.18 K root mean square within the alias-free field of view. 2The average daytime Faraday rotation at L-band is 17 =f ( f in gigahertz), which is 8.7 approximately [10]. However, it can exhibit very large variations, depending on the local time, season, solar activity, and geographic location. 3The brightness temperature sensitivity to sea surface salinity is 0.5 K/psu at 25 C, and the salinity retrieval accuracy is 0.1 psu after spatio-temporal averaging in 200-km boxes and 30 days. CAMPS et al.: POLARIMETRIC FORMULATION 295 REFERENCES [1] M. Martín-Neira and J. M. Goutoule, “MIRAS—A two-dimensional aperture-synthesis radiometer for soil-moisture and ocean salinity observations,”ESA Bull., no. 92, pp. 95–104, Nov. 1997. [2] P. Sivestrin, M. Berger, Y. Kerr, and J. 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