210Pb-based dating models for recent sediments. A review

Review 210 Pb-based dating models for recent sediments. A review Jos´ e M. Abril-Hern´ andez Departamento de Física Aplicada I, ETSI Agron´ omica, Universidad de Sevilla, 41004, Sevilla, Spain ARTICLE INFO Handling Editor: Sheldon Landsberger Keywords: 210 Pb-dating 210 Pb-based models Sedimentation rates Polyphasic porous media Aquatic sediments ABSTRACT The 210 Pb-based dating method provides absolute ages determination in recent aquatic sediments at centennial scales. It is widely used to support a large variety of environmental studies. However, any empirical data set is compatible with an infinite number of chronologies that need to be constrained by a series of assumptions (models) on the particular sedimentary conditions of the studied environment, and validated with independent chronostratigraphic markers. During five decades, about thirty models have been developed to cope with the wide diversity of natural conditions, a good number of them appearing in recent years, along with new concepts such as model errors, attractors for χ -mapping, or kinetic reactive transport, which have changed common views and practices. This paper aims to present a comprehensive review of this dating method to provide to final users updated tools and a renewed understanding to improve the reliability of their applications. Models are classified in terms of their assumptions on the sedimentary systems, which are better understood from a revisited theory of early compaction and the description of the microcosms of saturated porous media, where composite fluxes of tracers undergo different deposition pathways in terms of physical and kinetic reactive transport. The article reviews empirical evidence on the natural variability in mass flows and initial activity concentrations. Some models allow analytical solutions, while others require numerical techniques. The review is illustrated with examples from real case studies. 1. Introduction The radiometric dating has represented a revolutionary contribution to the study of sedimentary processes (Carroll and Lerche, 2003). The method based on the natural radionuclide 210 Pb is widely used to determine ages and sedimentation rates in the past 100–150 years, with about 4000 scientific documents in Scopus (search criteria ‘ 210 Pb and sediment’). 210 Pb-based chronologies can support a wide diversity of studies, including, among others, decoding past natural and anthropogenic impacts in sedimentary conditions (e.g., Lu and Matsumoto, 2005; Trabelsi et al., 2012; Begy et al., 2018; Chen et al., 2019), the history of anthropogenic pollutants (e.g., Kirwan and Megonigal, 2013; Ontiveros-Cuadras et al., 2024), quantification of carbon sequestration capacity of vegetated coastal sediments (e.g., Kristensen et al., 2008; Marchand, 2017), or studying the interplay of global sea level rise and accretion rate of saltmarshes sediments (e.g., Madsen et al., 2007; Gore et al., 2024). During the past three decades, a gap has emerged between method developers and end users. Most studies use only three of the approximately 30 models developed to address the wide diversity of sedimentary systems, and often apply them beyond their intended use. Far of being a closed problem, there has been an intense activity during recent years around the fundamentals of the technique, with the rise of new models such as TERESA, PLUM, RUS2023 and the family of χ -mapping models, as reviewed latter in this work. Lessons learnt have shown some shocking results. Thus, any empirical data set (a mass depth profile of 210 Pb and 226 Ra) is compatible with infinite exact solutions for the chronology (e.g., Robbins, 1978; Abril, 2015). The chronology must be constrained by a set of assumptions on the sedimentary conditions, called a 210 Pb-based model. It could be though that different assumptions should produce different chronologies, reducing the problem to find the right assumptions for each specific scenario. However, different models can often produce very close chronologies (eg, Abril, 2020). This leads to the concept of model errors (deviations from model output and the true solution due to a partial or null accomplishment of the model assumptions). Their quantification in varved and synthetic cores has revealed that the model records of sedimentation rates, profusely used in literature for tracking past environmental changes, are undermined by large model errors, which makes futile their use for this goal, while the proper magnitude, the initial activity concentrations, never captured our attention. It is also surprising that the model solution that best fits the empirical profile does not E-mail address: [email protected]. Contents lists available at ScienceDirect Journal of Environmental Radioactivity journal homepage: www.elsevier.com/locate/jenvrad https://doi.org/10.1016/j.jenvrad.2025.107749 Received 14 June 2025; Accepted 29 June 2025 Journal of Environmental Radioactivity 289 (2025) 107749 0265-931X/© 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). necessarily lead to the most likely chronology, as has been shown with the study of the attractors (Abril, 2023a). These new concepts will be presented in detail in this review. The 210 Pb dating method needs the support of independent chronostratigraphic markers, and bomb-fallout radionuclides such as 137 Cs, 241 Am and 239+240 Pu have been profusely used for this goal. A large set of additional data can support the holistic analysis of a sediment core. Thus, sediment dating becomes only a part of a more general and interesting problem consisting of understanding the particular sedimentary conditions and processes that govern the environmental behaviour of radionuclides and anthropogenic pollutants. The application of reactive kinetic transport models for the upper regions of aquatic sediments has changed our view on these processes (Abril and Barros, 2022; Abril, 2023b). This work aims to present a comprehensive review of about 30 different 210 Pb dating models developed during the last five decades, with a classification based on the specific sedimentary conditions they represent. For this end, it is necessary to review the main features of the porous medium of sediments and on the kinetic reactive transport of different radiotracers to understand under which circumstances it can be considered as a continuous medium or a polyphasic one. Radiotracers may reach the sediment-water interface (SWI) in dissolved form or attached to a wide diversity of carriers with diverse provenances and path histories. Depending on the scenario, different contributions may govern this composite nature of the fluxes. Due to the large number of models considered in this work, they cannot be presented in full mathematical detail. It would be advisable for the reader to be already familiarised with at least some of the simpler classical models. There are some reviews that can be particularly helpful for beginners (see, e.g., Sanchez-Cabeza and Ruiz-Fern´ andez, 2012) and some essential references such as Robbins (1978) and Appleby (1998). Although not necessarily at a fundamental level, it would be helpful to get a general idea of other 210 Pb-based models used in the scientific literature. Some helpful review papers can be found, among others, in Mabit et al. (2014) and Arias-Ortiz et al. (2018). In this work, the reader will also find references from the scientific literature for each specific model mentioned. Not treated here are the tasks related to the sampling of sediment cores, the sample treatment, and the nuclear techniques for measuring the radionuclides of interest. For the first topics, the reader is addressed to the IAEA TEC-DOC 1360 (IAEA, 2003). A study of methods for 210 Pb determination in environmental samples can be found in Zaborska et al. (2007) and Cuesta et al. (2022), among others. 2. 210 Pb cycling in nature and the rise of dating models Aquatic sediments accrete with mass flows from a diversity of provenances that contribute with varying intensities over time. 210 Pb (T 1/2 =22.3 yr) is generated in the radioactive decay chain from 226 Ra (T 1/2 =1600 yr), 222 Rn being the radioisotope in between with the longest half-live (T1/2 =3.82 days). Measuring 226 Ra and 210 Pb mass activity concentrations in the surficial layers of sediments uses to reveal a lack of secular equilibrium, with 210 Pb being in excess with respect to 226 Ra. The amount in excess is denoted hereafter as 210 Pb exc (following Krishnaswamy et al., 1971), while the amount that equates the mass activity concentration of 226 Ra is referred to as supported 210 Pb, and denoted as 210 Pb s . 210 Pb exc ultimately comes from the 222 Rn exhaled by the continental crust and dispersed in the atmosphere where it decays to 210 Pb, which is attached to aerosols and dust particles. It is scavenged by wet and dry fallout and accumulates in the surface layers of soils and sediments. The 210 Pb fallout rate is locally governed by wet deposition and it largely varies over time in the range from hours to years, with annual fallout fluxes that at a global scale ranging from 0.1 to 360 Bq m −2 y −1 (Mabit et al., 2014). In aquatic sediments, the meteoric 210 Pb that reaches the SWI has met the particulate matter of the mass flows at different stages of their paths from their provenances. Thus, the 210 Pb exc flux at the SWI is conceptually different from the atmospheric deposition in the area, taking different values and with a temporal variability driven by different factors. The discrete slicing of the sediment cores in environmental studies implies a temporal discretization that, depending on the sedimentation rates, may range from a few months to a few years. Thus, the physically significant values of the 210 Pb exc flux at the SWI must refer to these time intervals. Independently of its path history, the mass activity concentration of 210 Pb found in excess in the material accreting on the sediment is referred to as the initial activity concentration, and denoted as A0. Aquatic sediments are porous media that remain permanently saturated with water, so the diffusion and exhalation of 222 Rn is generally negligible. Thus, once meteoric 210 Pb has been incorporated into the sediments, the amount of 210 Pb exc decreases with time by radioactive decay, and in some scenarios it can also undergo some redistribution processes. These features have inspired the 210 Pb dating method, initially proposed by Goldberg (1963) for dating glacier ice. It was first applied to lacustrine sediments by Krishnaswamy et al. (1971) and to marine sediments by Koide et al. (1972). The first models used very simple assumptions on the sedimentary conditions. Thus, Krishnaswamy et al. (1971) presented a linear fit for the logarithmic plot of 210 Pb exc versus depth in two sediment cores, which they interpreted with the assumptions of constant 210 Pb exc flux, constant sedimentation rate (CFCS model), and uniform bulk density. A similar analysis was used by Koide et al. (1972) for a varved marine sediment core, demonstrating the use of the 210 Pb method with the independent chronology from varves. As the application cases to other lacustrine and marine environments increased, certain limitations were found in the above approach, leading to the rise of new models considering temporal variability in fluxes, mixing, and diffusion. Only six years after the paper by Koide et al. (1972), Robbins (1978) presented a comprehensive review of the state of the art and stated the pillars of the 210 Pb dating method. It is worth summarising the following points: i) The mass depth scale should be used in the models instead of the metric depth to account for compaction and shortening. ii) There are infinite chronologies being the solution of any given empirical 210 Pb exc profile, so assumptions on the fluxes of matter and 210 Pb exc are necessary to constrain a solution. iii) The author presented analytical solutions for models assuming a constant flux (CF), a constant flux with a constant sedimentation rate (CFCS), including its piecewise version, a constant initial concentration (CIC), and a differential equation for a model with the 210 Pb exc flux being a linear function of the sedimentation rate. iv) Solutions were provided for a family of models considering diffusion, raising as particular solutions of an advection-diffusion equation written in terms of metric depths (previously used by Guinasso and Schink, 1975). The work included applications to real cores and identified special cases with 210 Pb exc profiles showing disruptions due to slump events. It is worth noting the limitation in the above work of using the diagenetic equations by Berner (1971, 1980), which were latter revisited by Abril (2003a). In the same year, Appleby and Oldfield (1978) published their version of the CF model, named as the constant rate of supply (CRS) model. Appleby et al. (1979) applied this model to three varved sediments from Finland, with a reasonable agreement between the model and varve chronologies, while the CIC model failed in two cases. It is worth noting that the CRS model was used here with the addition of a reference date (the varve date of the deepest sediment slice), so that the CRS chronology necessarily fits the varve chronology at this point and at the SWI, and captures the main value of the sedimentation rate. This is J.M. Abril-Hern´ andez Journal of Environmental Radioactivity 289 (2025) 107749 2 known as the reference-date method (Appleby, 2001; Abril, 2019), which is different from the basic and widely used version of the model when the age of the deepest slice is unknown. The CRS model has a simple analytical formulation and apparently powerful outputs: the chronology and the history of the sedimentation rates. This explains its popularity. By the use, the model assumptions seem to have acquired the category of physical principles for some authors, being applied without adaptation to sedimentary scenarios where the assumptions do not hold. Some authors, such as Smith (2001), with outstanding contributions to the development of the 210 Pb dating technique, claimed that it should be mandatory that any 210 Pb-based chronology be validated by independent chronostratigraphic markers. For this end, the artificial 137 Cs, 241 Am and 239+240 Pu, with characteristic peaks in their history of atmospheric deposition, are being profusely used. Very soon, it was noticed that mixing could translocate the position of the peaks of 137 Cs (e.g., Robbins and Edgington, 1975), while diffusion and mixing affect differently to high-particle-reactive radionuclides, such as 239+240 Pu and 210 Pb. To distinguish post-depositional mixing from pre-depositional processes it was necessary the development of specific models for these man-made radionuclides (e.g., Robbins et al., 2000). A more recent view explains the effects attributed in some cases to diffusion, as result of the kinetic reactive transport of radionuclides through the pore fluid (Abril and Barros, 2022). 3. The porous media of recent sediments 3.1. Bulk density and early compaction Due to natural compaction, and the shortening during coring, storage, and extrusion, the metric scale does not represent the undisturbed conditions. However, the mass-depth scale remains invariant, and the bulk density profiles still preserve information on early compaction. The dry-bulk density is defined as the mass of dry solids divided by the total volume of the wet sample. There are two basic methodologies for estimating bulk density: i) Method I involves direct measurements of dry mass and bulk volume; ii) method II involves other index properties and a conceptual model of the porous media. Method I uses a sediment sample of well-known unperturbed bulk volume, V, (usually from a known cross-sectional area and thickness), and measures (by weight) its mass (wet sediment), mws. Alternatively, the wet volume can be estimated from the volume of water displaced by the wet sediment. The dry sample weight can be determined by drying the same sample in a freeze-dryer or in an oven at 110◦±0.5 ◦C, for 24 h (Dadey et al., 1992), although in the scientific literature one can find a large diversity of drying protocols. The remaining mass, ms, corresponds to the dry solids (along with the precipitated salts from the interstitial solution). Then, the bulk density, ρ b , is ρ b=ms V.(1) Equation (1) provides a good estimate for freshwater environments, but for saline pore fluids, ms must be corrected by the amount of precipitated salts (Dadey et al., 1992; Iurian et al., 2021): ρ * b= ρ bϕs;ϕs=m* s ms=(1−s mws/ms 1−s),(2) where * denotes salt-corrected values and s is the fraction (in weight) of salts in the pore fluid. Shear stress during sediment coring and extrusion can promote vertical displacement of sediments adjacent to the internal surface of the sampler. This perturbation can be minimised in the lab by using a subsampler cylinder of thin wall and wide cross section. Method II estimates the bulk volume from a ‘sediment model’ with several constituents. Its simplest version considers only two constituents: solids and pore water. The weight fraction of water is fw= (mws −ms)/mws. The bulk density can be estimated (assuming uniform values for the density of solids, ρ s, and water, ρ w, and additivity of partial volumes) as ρ b=1−fw 1−fw ρ s+fw ρ w .(3) The density of solids, ρ s, can be measured by (gas) picnometer for each sediment slice, although the most common approach assumes a reference value, typically ρ s =2.65 gˑcm −3 , the density of quartz and clay minerals. Equation (3) needs a correction for saline pore fluids (as in Eq. (2)). Note that corrections by precipitated salts must be consistently applied to bulk densities and to radionuclide mass activity concentrations. Method II can be refined by accounting for the organic matter content (usually determined by LOI). This is particularly important for vegetated coastal sediments, where the organic matter content can exceed 50 % of the sediment dry weight. However, here the assumption of additivity of partial volumes is questionable. Comparison between procedures I and II is recommended. Under field and/or laboratory conditions, some sediments can be partially unsaturated, and the volume of pores occupied by air produce the failure of the second method. The comparison allows estimating such a void volume, which provides some insight into the sediment structure close to the SWI. A high LOI value due to plant roots also challenges the twoconstituent conceptual model, since a significant fraction of the water lost by evaporation is not pore water but vegetal tissue water. An example of a polyphasic model of porous media including plant roots in a vegetated soil can be seen in Taieb Errahmani et al. (2022). The mass depth m(z) is the dry mass per unit area accumulated from the SWI until the sediment horizon that at the slicing operation is at depth z, and it can be estimated from the bulk density: m(z)= ∫z 0 ρ b(zʹ)dzʹ.(4) In sediments that met the model of a two-phase porous medium, steadystate bulk density profiles are produced by gravitationally driven early compaction (solids tend to replace the pore water below them) and they approach the mathematical expression: ρ b(z)= ρ b.∞−Δ ρ e−γz,(5) where γ is a scaling factor and ρ b.∞ is the asymptotic value for bulk Fig. 1. Examples of bulk densities versus metric depth profiles in: A) a riverine sediment (data from Lima et al., 2005 a); B) a lacustrine sediment (Lake 111, data from Crusius and Anderson, 1995); and C) a sediment core from a reservoir (Abril et al., 2018), in the secondary y axis. The continuous lines are fits by Eq. (5). Irregularities in profile C correlate with abrupt changes in environmental conditions. J.M. Abril-Hern´ andez Journal of Environmental Radioactivity 289 (2025) 107749 3 density, referred to as the early-compaction limit; ρ b.0 is its value at the SWI and Δ ρ = ρ b.∞− ρ b.0. Some examples are shown in Fig. 1 (cases A and B). In the theory for early diagenesis, Berner (1980) inconsistently used an advection-diffusion equation for the mass conservation of solids in terms of the bulk density. The revised theory for early compaction (Abril, 2003a) introduced a specific potential energy for solid particles, ψ , which decreases when the water pores are occupied by solids. It is defined as energy per unit weight and referred to as the compaction potential. From a reference frame anchored at the SWI of a sediment that accretes with a metric velocity v, the mass flow of solids through a sediment horizon at depth z is w= ρ b(v+q), where q is the relative velocity at which solids displace due to early compaction. q ρ b= − k(z) ∂ψ ∂ z.(6) k(z)is a conductivity function that decreases with depth and vanishes at the early compaction limit (Abril, 2011). The continuity equation that expresses the mass conservation of solids then writes: ∂ρ b ∂ t= − ∂ ∂ z( ρ bv+ ρ bq),or ∂ρ b ∂ t= − ∂ ∂ z(w).(7) As at the early-compaction limit q vanishes, w= ρ b.∞v.(8) The steady-state solution of Eq. (7) implies that the mass flow is uniform through the sediment column. At the SWI w corresponds to the mass sediment accumulation rate, referred to as SAR. It is worth noting that in most applications, we usually report mass flow or sediment accumulation rates, w, but in some cases, such as those when the major concern is their comparison with the rise in sea level, the relevant magnitude is the linear accretion rate of the SWI, v, given by Eq. (8) (many works incorrectly use w= ρ bv). Abril (2011) relates the gradient of the compaction potential with the buoyancy (β= ( ρ s− ρ w)/ ρ s) as ∂ψ ∂ z= − β. To obtain Eq. (5) as the steady-state solution of Eq. (7), it is necessary to explicitly note that conductivity decreases with depth, its rate of change being proportional to the default bulk density, defined as ρ D(z) = ρ b.∞− ρ b(z): ∂ k ∂ z= − Cz ρ D. Cz is a parameter that encloses information on the granulometry of the solids, being a constant when the latter is uniform through the core. Numerical solutions of the above equations for early compaction under varying sedimentary conditions can be found in Abril (2011). The above Eqs. (6)–(8) refer to sediment under natural conditions, while the plot of Fig. 1 captures the situation after coring, storage, extrusion, and slicing the core. However, Fig. 1 provides information on the early-compaction limit, useful for reporting metric accretion rates. Deviations from the steady-state trend can reveal some abrupt changes in environmental conditions, some of them likely related to episodic sedimentation of coarser materials, upslope soils, or erosive events, among others, such as Example C in Fig. 1 (see details in Abril et al., 2018). Intuitively, early compaction is driven by the gravitational forcing and the perturbations (micro-earthquakes, pressure waves, bottom search stress, etc.) that promote the reallocation of solids and pore fluid. Early compaction can work with a progressive decrease of the pore space, but also with the relative downward transport of colloids and very small size fractions. However, this view does not apply in vegetated sediments, where rhizospheres represent a distinct phase with belowground production and decay, conform the sediment structure, and have a distinct advection relative to the mineral phase. However, some basic features relative to an early compaction limit below the rhizospheres still hold (see Iurian et al., 2021). 3.2. The continuity equation for a particle-bound radiotracer in porous and accreting sediments under early compaction For radionuclides of interest in radiometric dating ( 210 Pb, 137 Cs, 241 Am, 239+240 Pu), the partition coefficient, k d (defined as the ratio between mass concentrations in solids and in solution at equilibrium), is of the order of 10 3 to 10 7 (IAEA, 2010). Consequently, almost 100 % of these radionuclides are expected to be bound to the solid phase at equilibrium. This justifies considering sediments as continuous media. The continuity equation, in terms of the metric depth, z, is (Abril, 2003a): ∂ ( ρ bA) ∂ t= − λ ρ bA+ ∂ ∂ z(Db ρ b ∂ A ∂ z)− ∂ (wA) ∂ z; (9) where λ is the radioactive decay constant, A is the mass activity concentration of the studied radiotracer, and Db is a diffusion coefficient. By using Eq. (7) and that dm = ρ bdz, Eq. (9) can be written in terms of mass depth, m: ∂ A ∂ t= − λA+ ∂ ∂ m(Dm ∂ A ∂ m)−w ∂ A ∂ m,(10) where Dm=Db ρ 2 b is an effective coefficient for diffusion, and w denotes SAR. Note that A and Dm are functions of m and time. Solving Eq. (10) requires the provision of initial and boundary conditions. At the SWI (m =0), the continuity of fluxes is commonly imposed as a boundary condition: F(t)= − Dm ∂ A(m,t) ∂ m⎸m=0+w(t)A(0,t); (11) where F(t)is the flux of tracer at the SWI. This is, the time series of 210 Pb exc fluxes and SARs must be introduced as input information, as well as the particular parameterization of Dm. At the bottom boundary, for →∞ A(m,t)→0. Note that Eqs. (10) and (11) apply for tracers bound to particles of solids; thus, the diffusion requires a downward and upwards displacement of solids with a net null mass flow. The upward displacement is opposite to the gravitational forcing and requires the existence of a forcing agent. Bioturbation can provide such a forcing (e.g., Robbins and Edgington, 1975; Robbins, 1978). Spatial variability of mass flow through the sediment cross section results in a tiny diffusion term, similar to turbulent diffusion in the Reynold’s equations for fluids (Abril, 2011), but it can be neglected in most cases. When Dm takes very high values in a top zone of the sediment and vanishes below, the effect in A(m)is usually referred to as a mixing. The tidal cycles of resupension and deposition in some coastal sediments can result in a mixed surface layer that can be formally described by a depth-dependent Dm. Depth distributions of radiotracers promoted by kinetic reactive transport have been described by depth-dependent Dm, but this description fails for cores sampled at the same site after an elapsed time, as discussed further below. 3.3. Composite fluxes of tracers and kinetic reactive transport in the porous media of aquatic sediments Fig. 2 shows the physical environment of the SWI. Stacked solids of varying sizes and mineralogical nature conform to the structure of the porous medium, with the pore spaces occupied by water. Trace elements reaching the SWI can be associated to a wide diversity of carriers (large structural particles, fine grained and colloidal fractions, organic compounds, etc.), which may have reversible and irreversibly bound components, and they can also be present as free ions into solution. A fraction of the free surface of solids remains inaccessible to the pore water due to stacking. Solids have rugose surfaces where uncompensated electric charges and more specific reaction sites are distributed J.M. Abril-Hern´ andez Journal of Environmental Radioactivity 289 (2025) 107749 4 (Eisma, 1993). The uptake of trace elements from the dissolved phase can be described by box models with parallel or consecutive reversible reactions (Barros and Abril, 2008). Different carriers show different depositional patterns; those bound to structural particles are ideally deposited at the top region of the SWI, while the mobile fractions (dissolved and colloidal) are transported through the pore fluid. Even for weakly particle-reactive radionuclides, such as 137 Cs, the typical percolation velocities do not result in significant penetration depths; and the same is true for molecular diffusion (Abril and Barros, 2022). However, in those ‘energetic’ systems where the overlying water column has a noticeable eddy diffusivity (e.g., because of tidal or wind-driven currents), this does not drop to zero at the SWI, but perturbs the pore space, progressively vanishing with depth. The interplay of the kinetic uptake with this transitional eddy diffusivity can result in noticeable penetration depths for mobile radionuclides, while those others highly particle-reactive, such as Pb and Pu isotopes, are retained in the first few millimetres of the sediment column (Abril and Barros, 2022; Abril, 2023b). Shortly after the peak in atmospheric bomb-fallout, or after the Chernobyl accident, artificial Cs isotopes reached the SWI mostly in mobile form, reaching penetration depths of up to more than 10 cm in some energetic sedimentary systems while in others, free of eddy diffusivity, they were trapped in a very thin surface layer. In the former systems, Cs isotopes continued reaching the SWI but now bound to solids of different provenances and undergoing ideal deposition (Abril and Barros, 2022). In general, the mobility downcore of a fraction of the input tracer may coexist with the ideal deposition of another fraction bound to structural particles. 241 Am, Pu isotopes and 210 Pb exc , even in cases where a fraction of their inputs may be in dissolved form and transitional eddy diffusivity is present, do not show any significant depth penetration. Exceptions can be made in those cases where the colloidal fraction is particularly relevant. The set of governing differential equations and the application of the kinetic reactive transport models in real cases for 137 Cs, 7 Be, 210 Pb, 239+240 Pu and 236 U can be found in Abril and Barros (2022) and Abril (2023b). Fig. 2 also highlights the composite nature of the 10 Pb exc fluxes. Different contributions may have varying weights as responses to seasonal changes, episodic events, and human impacts. When the particlebound fraction of the 210 Pb exc inputs is dominant, intensification in the mass flow can result in an increase in global 210 Pb exc flux. This must be understood in terms of statistical correlation more than in terms of a physical law, since the intensification of the mass flows may involve varying weights of components with different provenances, resulting in different aggregated A0. Abril and Brunskill (2014) reported the reconstructed palaeorecords of SARs, A0 and 210 Pb exc fluxes in varved sediments from a variety of lacustrine, riverine, and marine environments. They found large but independent variability in SARs and initial activity concentrations (roughly approached by normal or log-normal distributions), resulting in a positive correlation of 210 Pb exc fluxes and SARs (Fig. 3). 4. The empirical dataset For a sediment core that has been sliced into n sections, the 210 Pb method needs n pairs of values of 210 Pb exc mass activity concentrations with their analytical uncertainties, and the mass thickness of each slice (Ak± σ k,Δmk);k=1,2….n. This will be referred to as the primary data. The mass thickness allows for the construction of the mass depth scale, it is needed for estimating the inventories required in some models, and relates with the bulk density of each slice. Independent chronostratigraphic markers must be used when possible; examples are artificial 137 Cs, 241 Am , and 239+240 Pu, tephras, pollen markers, anthropogenic pollutants with known deposition histories, etc. They can serve for validating the 210 Pb-based chronology, or as integral part of the method, as discussed further below. As common analytical methods involve gamma measurements, other γ-emitter radionuclides can be recorded simultaneously. Thus, the cosmogenic 7 Be (half-life 53.3 days), when found in the upper layer, serves as a quality control of a coring procedure without loss of the upper material. When found distributed in depth, it informs on non-ideal deposition, either for transitional eddy diffusivity or for the lack of definition of the SWI, what alerts of similar effects with 137 Cs in both cases, and with 210 Pb exc in the latter. 226 Ra serves to estimate the supported fraction of the total of 210 Pb, but its depth profile is useful in identifying changes in sedimentary conditions. This last can be reinforced when such changes simultaneously occur in the depth profiles of 40 K and 228 Ra (e.g., see Abril et al., 2018). For vegetated sediments, the analysis of LOI and in situ concentration ratios must be considered (see Iurian et al., 2021). Granulometry, mineralogy, magnetic properties, concentrations of trace metals and other anthropogenic pollutants, etc., conform a wide set of data for the holistic analysis of sediment cores, along with the geochronological data, and can support the best model choice. This work focusses on the dating models built from the primary data for 210 Pb exc and the artificial radionuclides that can support the chronology. In this work, the use of the models will be illustrated with real cases from the literature. The raw data sets needed are available in the referenced papers, so they are not replicated here. 5. Overview of 210 Pb-based dating models Analytical solutions of Eq. (10) can be found only in some simple cases, but numerical solutions are always possible. The solution is unique for each particular setting of all the required input data. However, the reverse is not true. 210 Pb-based models can be seen as particular solutions of Eq. (10), but they can also be directly derived from their basic assumptions as originally formulated. A summary of the models developed during the last five decades is presented in Fig. 4. This section provides a general view, and the models will be treated in more detail in the subsequent sections. The first and most populated group of models share the following assumptions: i) Tracers are particle-bound, which allows for the simplification of considering a continuous medium characterised by its bulk density. ii) For 210 Pb exc , penetration depths are negligible, so the assumption of ideal boundary conditions (continuity of fluxes at the SWI) can be adopted. iii) There is no postdepositional redistribution (Dm=0 in Eq. Fig. 2. The physical environment of the SWI region and the composite nature of the mass flows of solids and trace elements. Geometric figures entering the SWI represent different carriers that conform the composite flux of the tracer; among them, dots represent ions, in a dynamical exchange among the dissolved and solid-bound phases. Adapted from Abril and Barros (2022). J.M. Abril-Hern´ andez Journal of Environmental Radioactivity 289 (2025) 107749 5 (10)). Implicitly, the continuity of the sedimentary sequence is also assumed, which is not disrupted by erosion nor by episodic slump events. These assumptions may hold in a large proportion of lacustrine systems and marine basins. Exceptions are sediments with bioturbation and sites with energetic hydrodynamics (with wind or tidal forcing) that may result in a mixed sediment layer. In deep-sea sediments uniform diffusion has been considered in several studies, the reason being their common very low sedimentation rates, so that tiny physical redistribution processes may gain relevance comparatively. Within the above group of models, a first distinction is whether the sedimentary conditions prevail for the entire range of the chronology of the core, or whether there are two or more transects with distinct conditions. This can be elucidated from a cluster analysis (Abril, 2019), as illustrated in Fig. 5. Briefly, the logarithmic plot Ln[A(m)] can reveal slope and/or jump discontinuities, in which case a piecewise analysis must be conducted. In the absence of such discontinuities, the simplest version of the models can be used. With these assumptions, the boundary condition writes: F(t) = w(t)A(0,t). A first group of models assumes the flux F to be constant in time, a second group assume a constant sedimentation rate, w, a third group considers a constant initial activity concentration, A0, while a four group considers the three magnitudes varying with time. After getting familiar with the above set of dating models, it is possible to discuss model errors and studying the methods for extending their application to multitransect problems. Validation of the chronology with independent time markers is necessary in all the cases. However, some models can use the reference dates provided by the time markers as an integral part of the model formulation, rather than as an external validation. The use of artificial radionuclides such as 137 Cs and Pu isotopes as time markers is not so straightforward since their fluxes reaching the SWI may noticeably differ from the histories of their atmospheric deposition. Thus, Fig. 4 considers some specific models for these predepositional processes (labelled as ‘Models for time-marker RNs’). Another group of 210 Pb-dating models accounts for diffusion or mixing but still keeping the assumption of a continuous medium and the continuity of fluxes at the SWI. The simple case of steady-state solutions under constant fluxes and SARs allows for analytical solutions. The manmade 137 Cs is particularly prone to undergo post-depositional redistribution, which complicates its use as a time mark. Although these processes will be better understood from the kinetic reactive transport, there are a series of models accounting for diffusion and mixing that can be adapted for man-made radionuclides with their time-dependent fluxes. Numerical solutions of Eq. (10) can face any kind of problems with depth-dependent diffusion and temporal variability in fluxes and SARs, when keeping the assumption of a continuous medium and the boundary condition of Eq. (11). The assumption of ideal deposition at the SWI can be broken when a fraction of the fluxes undergoes a fast depth penetration. However, this problem can be treated keeping the formalism of a continuous medium by introducing a depth-dependent source term. This is the NDI model considered in Fig. 4, as a transition to models that surpass the assumption of a continuous medium. Within them, Fig. 4 considers models for the reactive transport of tracers into solution, but this can be faced with the approach of a local equilibrium, described by a partitioning kd coefficient, or by solving the kinetics of the uptake. The assumption of a continuous medium also breaks down in vegetated sediments, where the rhizospheres represent a distinct phase, with achronical below-ground production and decay, a distinct uptake of tracers and a quite different physical transport. Faceting this problem requires updating both the analytical methods and the models. The above classification does not encompass the full diversity of sedimentary conditions that have been studied in the scientific literature. Thus, a group of some special cases will be considered in this work. Fig. 3. Frequency distribution for initial 210 Pb exc concentrations and SAR (w) (left panels) and statistical correlations (from bivariate correlated errors and intrinsic scatter) of 210 Pb exc fluxes and initial concentrations with SARs (** in R 2 means significant at 99 % confidence level; N.S. is not statistically significant). The analysis uses normalised (to the arithmetic mean of each core) values from 11 varved sediment cores. Data from Abril and Brunskill (2014). J.M. Abril-Hern´ andez Journal of Environmental Radioactivity 289 (2025) 107749 6 6. Models for continuous media, ideal deposition, and non-postdepositional redistribution 6.1. A single transect in the Ln[A(m)] plot 6.1.1. Models assuming a constant flux 6.1.1.1. CFCS model. Within the models assuming a constant flux, additional assumptions are still necessary. The CFCS model assumes a constant SAR, which implies an also constant A0=F/w, allowing for the analytical formulation: A(m)=A0e−λm/w(12) A least squares fit to the empirical data (mk, Ak) allows solving w and A0 and then the flux F; the chronology being T(m) = m/w. However, there are different possibilities to perform the fit others that the least squares (e.g., weighted fit by σ k, minimum distance, etc.). 6.1.1.2. χ -CFCS model. The χ -mapping version of the CFCS model uses a numerical technique that tests a very large number of possible pairs of values for w and A0 to find the pair that minimised a χ 2 function (see details in Abril, 2023c). 6.1.1.3. CF (CRS) model. The constant flux model (CF in the formulation by Robbins, 1978; and CRS in the equivalent version by Appleby and Oldfield, 1978) assumes a steady-state total inventory below the SWI. The inventory below the sediment horizon at mass depth m is Σm= ∫∞ mA(mʹ)dmʹ, and the total inventory corresponds to m=0. This assumption allows for the analytical solution for the chronology: T(m)=1 λLn (Σo Σm)(13) The SAR history can be obtained from Δmk and the above chronology, although alternative analytical formulations can be handled (see Abril, 2023c). Note that steady-state inventories are not met, even under conditions of a constant flux, in sedimentary systems younger than a century, as may be the case of some reservoirs and harbours. In some cases, the core length is not long enough to recover the total inventory, which prevents the application of the CF model. However, when the trend line of A(m)in the deepest section of the core is well defined, it is possible to use the local value of SAR (inferred from a CFCS model) to extrapolate the missing tail and estimate in this way the missing part of the inventory. This is the reference-SAR method, which can be considered as a version of the CF (or CRS) model that uses only the primary data set. It is worth noting the extreme sensitivity of the model to the accurate estimation of Σo. 6.1.1.4. PLUM model. Aquino-L´ opez et al. (2018) presented a Bayesian formulation of the CF model using as an additional assumption the Fig. 4. 210 Pb-based dating models grouped by their involved assumptions on the functioning of the sedimentary system. Commonly used short names are indicated. Fig. 5. Example of a cluster analysis. Ln[A(m)] plot for a varved sediment from Lake Zabinskie (data from Tylmann et al., 2016). Two jump discontinuities define three transects (r 1 , r 2 , and r 3 ). The continuous lines are the linear fits from a piecewise CFCS model (see text). The records of palaeofluxes (plotted in the secondary y-axis) have been reconstructed from the empirical (Ak,Δmk) and the chronology from varves. Note that models that assume a constant flux within each transect are affected by model errors. Note in Transect 1 how a continuous trend of decrease in fluxes is not distinguished from an exponential decay in the fit r 1 . A study of this core with the piecewise version of several dating models can be seen in Abril (2019, 2020). J.M. Abril-Hern´ andez Journal of Environmental Radioactivity 289 (2025) 107749 7 adoption of a particular Bacon age-depth model. The reader is addressed to the above reference for details. 6.1.1.5. χ -CF model. Abril (2023c) presented a χ -mapping version of the CF model. The solution is a set of n pairs of values (A0.i.wi), i =1, 2, …n, accomplishing the model and that, when conveniently sorted downcore produces the best fit to the empirical profile (mk, Ak) – this is, minimize the χ 2 function. A first version of the model assumes that the n A0.i values follow a normal distribution (see Fig. 3) with mean value A0 and relative standard deviation sA= σ A/A0. The model uses the so-called canonical representative sample of size n with zi values from a normal typified distribution N (0,1). It needs a first estimate of A0 and sA to generate the n values A0.i=A0(1+sAzi). As the flux is assumed to be constant, the n wi values must satisfy wi=F A0.i=w η (1+sAzi)−1, with w η =F/A0. The model defines a three-parametric domain with a wide range of values (A0, sA, w η ) that is discretised with a regular grid. The number of grid points is typically of the order of 10 6 , each one defining a solver, a set of three values that allows generating n pairs (A0.i.wi) accomplishing the model assumptions. The numerical code solves the best sorting downcore of these pairs and generates the theoretical profile, which is quantitatively compared against the empirical one with the χ 2 function. The model repeats the process for all the solvers and identifies the one that leads to the absolute minimum of χ . The treatment of propagated uncertainties can be seen in Abril (2023c). In the same reference, a second version of the model is presented, now using a log-normal distribution for A0.i. A comprehensive review of these five families of models assuming constant flux, along with performance tests using real data from varved sediments, can be found in Abril (2023c). Fig. 6 shows an example of the application of the CFCS, χ -CF and CRS models to a varved lacustrine sediment core, which allows comparing model and varve ages in the whole range of the chronology. 6.1.2. Models assuming a constant SAR 6.1.2.1. CSAR model. This model lacks an analytical counterpart, but the χ -mapping technique allows for a numerical solution. The constantSAR (CSAR) model was first presented by Abril (2023d). The strategy is similar to that of the χ -CF model. As SAR is constant, it is denoted as w , and the solution consists of n pairs of values (A0.i.w). The three-parametric domain is now (A0, sA, w). For normal-distributed initial activity concentrations A0.i=A0(1+sAzi). The CSAR model can be alternatively formulated using a log-normal distribution for A0.i. Details on the model and applications to real cores with a varve chronology can be found in Abril (2023a, 2023d). An example of application of this model is shown in Fig. 7. 6.1.3. Models assuming a constant initial activity concentration, A0 6.1.3.1. CIC model. The analytical solution for the chronology is trivial (Robbins,1978): T(m)=1 λLn (Ao A(m))(14) There is no unique method for estimating A0, and the model is prone to age reversals. Some authors computes Eq. (14) for the n empirical pairs of data and use the trend line as the best estimate of the chronology. 6.1.3.2. χ -CIC model. The χ -mapping version of this model needs to assume a certain distribution for the sedimentation rates. On the basis of empirical data (Fig. 3), normal or log-normal distributions are acceptable proxies. The solution strategy can be easily adapted from CSAR. The three-parametric domain is (A0, w, sw). For normal-distributed SARs wi=w(1+swzi). The χ -CIC model can be alternatively formulated using a log-normal distribution for wi. Details on the model and applications to real cores can be found in Abril (2024). An example of application of this model is shown in Fig. 7. 6.1.4. Model errors Although this topic affects all models, it can be insightful to introduce it at this point. In view of Fig. 3, which shows that the available empirical evidence on sedimentary conditions involve varying fluxes, SARs and initial activity concentrations, the reader can think that the above models, which assume constant some of these magnitudes, must be useless. However, these models have shown their use in many cases, providing agreement with independent time marks. Moreover, the above-provided references include full validations using varve chronologies, as the cases shown in Fig. 6. In addition to the propagated uncertainties, which at least ideally could be reduced as much as desirable, the model chronology can show deviations from the true solution due to the poor or null accomplishment Fig. 6. Example of application of a set of models assuming constant flux to a varved sediment core from Kevoj¨ arvi Lake with a complex 210 Pb exc profile (raw data from Haltia et al., 2021). The exponential fit (least squares) to Eq. (12) corresponds to the CFCS model. The χ -CF model closely fits the empirical data. The chronologies from varves, the above two models, and the CRS model are depicted on the secondary y-axis. Propagated errors are not depicted for the sake of clarity. See Abril (2023c) for a more detailed discussion. Fig. 7. Example of application of three χ -mapping models to the varved sediment core from Kevoj¨ arvi Lake: χ -CIC, CSAR and TERESA (see Section 6.1.5). Normal distributions have been used in the three cases. The corresponding chronologies are depicted on the secondary y-axis, along with the ages from the varves. J.M. Abril-Hern´ andez Journal of Environmental Radioactivity 289 (2025) 107749 8 of some of the model assumptions. Model errors are non-quantifiable since the true solution is unknown; exceptions are varved sediments, where a complete and independent chronology is available, and synthetic cores. With this methodology, model errors have been studied for the CFCS, CIC and CRS models (Abril, 2019, 2020). The results show that when the temporal variability in fluxes is randomly distributed along the time line, positive and negative deviations from the true chronology tend to mutually cancel out, so that the model chronology does not depart too much from the true solution. However, when fluxes show a continuous trend of increase or decrease with time, these models catastrophically fail. A similar analysis can be conducted in terms of SARs and initial activity concentrations. Model errors affect the results of SAR and A0 histories differently. There is an intrinsic asymmetry in the equations that makes the SAR very sensitive to model errors, resulting in false peaks and valleys that do not reflect any true changes in sedimentary conditions. This discourages the use of SARs to monitor historical changes in sedimentary conditions. In contrast, A0 records are very robust to model errors and can be used for the aforementioned purpose, although interpreting their temporal variability is not straightforward. A detailed analysis of this topic can be found in Abril (2022). The agreement in the chronologies often observed with models that separately assume constant F, A0, or w is not surprising, reason being that randomly distributed variability is consistent with mean values of SARs, fluxes and A0. However, as in most of the studies the history of sedimentary conditions is unknown, there is no way to know whether the chronology provided by these models is acceptable or not (that is, whether the studied site underwent continuous trend of changes). For this reason, validation with independent chronostratigraphic markers is necessary. 6.1.5. Models assuming varying fluxes, SARs and A0 Note that we are still considering the assumptions of continuous medium, ideal deposition, continuity of the sedimentary sequence and non-post-depositional redistribution, and a single transect. 6.1.5.1. TERESA model. Fig. 3 summarises the empirical evidence of a large and independent temporal variability of SARs and A0. Their values found in the n measured layers of a core are normal or log-normal distributed around their arithmetic mean. This independent variability results in fluxes being statistically correlated with SARs. This view is adopted by the TERESA model (time estimates from random entries of sediments and activities), which uses a χ -mapping technique (Abril, 2016). The model is defined in the four-dimensional parametric space (A0, sw,w, sw), large enough as to contain any plausible solution. Each point in the grid in the 4-D mesh defines a solver that generates the n pairs of values (A0.i, wi) using normal distributions A0.i=A0(1+sAzi), wi= w(1+swzʹ i)and two different sets of canonical representative samples (zi and zʹ i). The numerical code solves their best arrangement downcore and compares with the empirical profile through the χ 2 function. The absolute minimum in the whole domain is identified as the solution. The TERESA model can be alternatively formulated using log-normal distributions for A0, for w, or for both. The model is not free of model errors, since the used distributions are only proxies for the true but unknown ones. Validation against synthetic cores and varved sediments has been presented in several papers (see, e.g., Abril, 2016, 2020, 2022). An example of application is shown in Fig. 7, and other applications can be found in Botwe et al. (2017); Klubi et al. (2017); Abril (2024), 2025. 6.1.5.2. SIT model. The SIT model (Carroll and Lerche, 2003) is mentioned here for the sake of completeness. It claims the ability to establish chronologies with fluxes and SARs independently varying over time, without any restriction. Nevertheless, it makes a misuse of the Fourier series expansions (Abril, 2015). 6.1.5.3. RUS2023 model. Rusakov et al. (2024) presented their RUS2023 model, which is a modification of the CIC model. Instead of assuming a constant A0 for the whole core, they estimate a different value for each sediment slice as a linear function of its content of sand, silt, and clay, being the three proportionality constants the so-called sorption capacity of each grain size fraction. A key point is determining the ‘in situ’ values of the three sorption capacities. To this end the authors need handling the empirical value of A0 in the upper sediment slice in at least three different sediment cores from the same region and with similar sedimentation rates. The authors illustrated the use of this model with a set of sediment cores from the Laptev Sea. 6.2. Multiple transects Here, we consider the cases where the logarithmic plot Ln[A(m)] reveals slope and/or jump discontinuities. In the case of a single transect, empirical data are distributed around a linear trend in the Ln[A(m)] plot, with fluctuations arising from the random temporal variability in the sedimentary conditions. This is expected to reflect short-term climatic variability in relatively unperturbed and low-energetic aquatic environments. However, this can be interrupted by episodic events (storms, floods, etc.), the dynamic of sand bars, the silting and diverting of water channels, etc. Anthropogenic impacts can alter the flows of matter and 210 Pb exc activities with gradual or drastic and permanent changes (e.g., waterworks). Situations where the temporal variability in fluxes does not have a pure random character are: i) Changes in environmental conditions lead to a stepped shift on the mean value of F, over which a random variability is superposed. ii) Changes in environmental conditions lead to a continuous trend of increase/decrease in F. iii) Episodic events with abnormal high or low fluxes. iv) Any combination of the previous cases. Case i) can be observed as a jump and/or slope discontinuity in the plot of LN[A(m)]. Piecewise versions of the models can be used in these cases. The adaptation of the CFCS model is trivial (see, e.g., Robbins, 1978). Briefly, the mass depth of each discontinuity is identified, and a linear least squares fit is applied to each transect, which allows estimating the corresponding values of SAR, A0 and F (e.g., see Alonso-- Hern´ andez et al., 2006; Abril, 2019). The piecewise version of the CIC model is equally straightforward. This methodology is also possible for the CRS model, but it needs support of the dates of the discontinuities previously estimated by the CFCS model. The empirical inventory within each discontinuity, along with the preceding ages, serves to estimate the equivalent constant flux in each transect. This way, the piecewise CRS model transforms into the known problem of a CRS model with multiple reference dates, as seen in more detail in the following. A piecewise version of the TERESA model is also available. It has been formulated with two different approaches: i) Multimodal (M-) and ii) Fast-Multimodal (FM-). The former is conceptually simpler, since it applies separately TERESA to each transect and connects the solution to ensure the continuity of the chronology (by summing the last age of the previous transect to the first horizon of the following one) and correcting the initial activity concentrations by radioactive decay. The FM version generates separate normal distributions of A0 and SARs for each transect, but merges then into a single pack (solver) and applies the general method. The fundamentals of these models and their validation using real cases of varved sediments can be found in Abril (2020). As the other χ -mapping models are particular cases of TERESA (e.g., CSAR is TERESA with sw =0, and χ -CIC is TERESA with sA =0), their piecewise versions can be applied following the methodology described above. The case of continuous trends of change may have a similar fingerprint in the LN[A(m)] plot, lacking clear methods to distinguish this case (see the first transect in Fig. 5). It has been shown that with these sedimentary conditions all the models fail, but the TERESA, which has shown promising results (Abril, 2020). However, the key problem is how to recognise this type of continuous trend of change from the commonly J.M. Abril-Hern´ andez Journal of Environmental Radioactivity 289 (2025) 107749 9 the above mentioned cases. However, a large number of dating models apply to these conditions, attending to the temporal variability in fluxes, SARs and A0. Some of these models have analytical formulations (e.g., CFCS, CIC, CF) and others use numerical techniques (e.g., the χ -mapping family). Cluster analysis, using the Ln[A(m)] plot, can serve to decide whether to use the piecewise versions of the models. When the temporal variability in F.w and A0 is randomly distributed along the time line, models that partially ignore such variability still can produce chronologies close to the true solution, but this results in model errors that differently affect the model output for the histories of w and A0. The former is undermined by large model errors, while the latter is more robust and appropriate for tracking past environmental changes. These models fail for continuous trends of change in the sedimentary conditions, and TERESA is the only one that has shown some use in these cases. The agreement between chronologies from different models (e.g., CF, CSAR, TERESA, etc.) is indicative of random variability in the magnitudes at the SWI, and is a necessary but not sufficient condition for the reliability of the chronology, because the failure of the attractors in some complex profiles. In all the cases, an independent validation is necessary. Alternatively, the known reference dates can be used internally to adapt the model formulation. 137 Cs, 241 Am and 239+240 Pu, with a known history of atmospheric deposition, are widely used as time markers. However, predepositional and post-depositional processes may complicate the identification of the peaks. Understanding the whole shape of their profiles in the sediments makes their use more secure. A series of models has been reviewed for this goal. The above items summarise the lessons learnt after five decades of applying the 210 Pb dating method to a wide diversity of sedimentary systems throughout the wold. They are intended to provide guidance to researchers in this field to avoid misapplications of the method and improve the confidence and reliability of their studies. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 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