All these models share a common foundation by using the radar equation e.g., which relates the energy received at the antenna to the energy emitted by the same antenna (or another in the case of interferometry), summing the reflections in all the points of the surface within the footprint, and possibly including the subsurface volume. They adopt a geometrical approach to energy transport (radiative transfer) and only account for first-order scattering (i.e. single reflection at the surface or in the volume), which is certainly sufficient when the surface scattering dominates. However, in the case of snow, particularly older, coarse-grained snow, multiple scattering may be significant at higher frequencies, such as in the Ka-band . Multiple scattering within the volume, between interfaces, and between layers and interfaces involves pathways longer than those of single scattering and primarily affects the trailing edge of the waveform.

Since these models are based on energy propagation, the phase information of the electromagnetic wave is discarded, making it impossible to calculate the coherence between signals acquired in different pulses. As a consequence, these models cannot fully simulate the phase information recorded by real altimeters. Since SAR processing relies on this information and the specific pulse sequence to synthetically compress the footprint in the azimuth direction, the output of these models cannot be used as input to SAR processing algorithms. Instead, they aim to simulate both electromagnetic-surface interactions and the effects of SAR processing, and produce delay-Doppler maps as output, which can eventually be converted into waveforms by summing the maps along the Doppler dimension. Despite this limitation, they remain valuable because they are based on physical principles that incorporate the characteristics of both the sensor and the surface properties, enabling investigations of waveform sensitivity to numerous geophysical and instrumental variables.

Another key difference between the reviewed models lies in how they represent variations in surface elevation within the footprint. The majority (8 out of 11) represent elevation variations using an isotropic random variable Z with a known distribution (“Surface elevation distribution” in Tables –). This random variable is assumed to be stationary within the footprint, meaning that the surface exhibits uniform statistical characteristics throughout. Consequently, these models cannot simulate heterogeneous terrain with spatial variations in roughness within the footprint; the roughness is assumed to be uniform, and Z is a true random variable rather than a random field Z(x,y). As a result of these approximations, the surface can be fully described by a 1D probability density function (PDF). Moreover, these models often also assume a Gaussian distribution, facilitating the analytical convolution and requiring only one parameter, the standard deviation. However, observed that a log-normal distribution might be more appropriate for sea-ice surfaces. This is an important consideration for any non-Gaussian target, as it affects the expected surface scattering response. For Gaussian surfaces, the local height and slope are independent Eq. 5.5.6 in, so the surface topography effects can be included by convolving the scattering response with the elevation distribution of scatterers. For non-Gaussian surfaces, the height and slope may not be independent , such that the surface topography-backscatter dependency cannot be accurately represented through a simple linear convolution. The bias introduced by nonuniform backscatter from a non-Gaussian surface is known as the “electromagnetic (EM) range bias” or “sea state bias” in ocean altimetry .

For the models using both this statistical representation of the topography and using the uni-dimensional convolution to represent UF-SAR processing, the signal can be formulated by extending Brown's equation. 1P(τ)=PDF(τ)∗FSR(τ)∗PTR(τ) to P(τ,f)=PDF(τ)∗FSR(τ,f)∗PTRtime(τ)2∗PTRDoppler(f) where P(τ,f) is the power received at the antenna at time (named gate) τ and Doppler frequency f, PDF is the statistical distribution of the topography, FSR is the flat surface response, and PTR is the point target response, corresponding to the shape of the compressed pulse emitted by the altimeter for conventional altimeters. PTRtime is the equivalent of PTR, and PTRDoppler describes the effect of UF-SAR processing. The operator ∗ denotes a convolution over time and/or Doppler frequency dimensions. The models by and also adopt the statistical representation of the topography, hence implementing a convolution with PDF(t) as in Eq. (). However, the effect of the UF-SAR processing is not represented by a convolution as in the latter term in Eq. ().

The three remaining models take a radically different approach by describing the surface with a deterministic function z(x,y), commonly referred to as a Digital Elevation Model (DEM). This approach offers greater control over the surface variation properties, allowing the use of any PDF, even without stationarity, and more importantly, enabling the application of real DEMs and stacking delay-Doppler maps along the satellite trajectory, as done in a real SAR processor. This method is particularly relevant for ice sheets, where the topography within the footprint is far from uniform and strongly affects waveform shape. Complex multiple-peaked waveforms are frequent and it is currently a major source of inaccuracy in estimated altimetric height . However, using a real DEM with a resolution of typically 5–10 m comes at a cost: the radar equation integration over the surface must be performed numerically, which is typically more computationally intensive than methods based on statistical representations.

The deterministic function z(x,y) can be provided on a Cartesian grid, with a recommended resolution of 5–10 m and an extent covering the effective footprint (≈20 km). This approach was adopted by and . Alternatively, employed a triangular mesh, which facilitates the calculation of the normal for each triangle, required to estimate the local incidence angle and account for variations in backscatter as a function of it. This is why model is referred to as a “facet-based model” and can simulate the EM bias effect.

Another significant difference among the models lies in their approaches to calculation, particularly in the balance between analytical derivation and numerical computation. In Tables –, we categorize these strategies into three types: purely analytical (A), semi-analytical (AN), and numerical (N). In practice, there is a continuum between these approaches, with some models favouring analytical derivation at the cost of substantial approximations e.g., while others prioritize calculation precision, resulting in higher computational costs e.g..

The analytical approach generally involves selecting specific shapes for the key functions in Eq. (), such as the Point Target Response or the surface elevation PDF. The Gaussian shape usually enables tractable analytical integration. Simplifying assumptions, like treating UF-SAR processing as a uni-dimensional convolution, can also make the problem more manageable. However, this assumption is not strictly necessary; for example, and have advanced the analytical development without relying on such simplification.

Numerical methods become essential when using a DEM as input, as previously mentioned. However, this is not the only reason. For instance, opted for an analytical development of the functions in Eq. () in the Fourier domain, transforming the convolution into multiplication, and ultimately relied on numerical inverse Fast Fourier Transform (FFT) to convert results back to the real space. In this case, the computational cost is minimal due to the FFT algorithm's efficiency, making this model more efficient overall than some semi-analytical models that rely on traditional (i.e. non-FFT) numerical integration.

Both approaches offer complementary advantages. The analytical approach is better suited for running intensive sensitivity analysis, to understand how the altimetric signal varies with the various parameters and for developing physical retracker algorithms that fit observed waveforms across a large parameter space without lookup tables. In contrast, DEM-based, numerically intensive models are more effective as forward models for exploring the effects of complex terrain . However, this does not preclude the use of such models in retrackers, as demonstrated by in their retracker named AMPLI.

In the following sections, we examine other less prominent differences among the models.

The antenna gain pattern describes the angular response of the altimeter antenna and particularly influences waveform decay.

In analytical models, the antenna gain pattern is typically assumed to decrease with a Gaussian shape as a function of the angle or the sine of the angle (both are nearly equivalent for small angles as relevant in space-borne altimetry). This assumption effectively represents the main lobe of the antenna gain , but it completely neglects the secondary lobes. Early models e.g. also assumed a circular antenna, leading to an axisymmetric pattern. In this simple case, the Gaussian shape is characterized by a single parameter – typically the beamwidth angle (around 1°) – where the gain decreases by 3 dB (nearly a factor of 2). However, with the introduction of the elliptical antenna onboard CryoSat, this simple representation became inadequate. Under the assumption of the separability of the pattern in the across and along track directions, a more complex Gaussian pattern – decreasing at different rates in the along-track (1.06° for CryoSat-2) and cross-track (1.1992°) directions – was adopted. The Gaussian assumption remains convenient because it is well-suited for analytical calculations. For this reason, complex antenna patterns can be described as a sum of Gaussian assumption, as in , an advanced version of .

Incorporating the actual antenna diagram, including its irregularities and secondary lobes, can significantly enhance model accuracy . This approach requires numerical integration of the tabulated gain function. For example, do not require any particular assumption for the gain function, making their model adaptable to any antenna diagram. Additionally, their calculation framework is devised so that this integration involves only sensor parameters, which can be pre-calculated. This makes their model efficient and well-suited for use as physical retrackers. and also use the measured antenna pattern to achieve greater precision.

Most models, especially the most recent ones, explicitly account for the satellite's imperfect attitude, which leads to antenna mispointing. This factor significantly affects the analytical expressions, as seen in Eq. (26) in . The impact of the antenna mis-pointing on the waveforms is illustrated in in Figs. 9 and 10 with 0.2° of pitch and roll, respectively. Although do not consider mis-pointing, subsequent work not listed in Table  introduces this effect through an integral that incorporates a series of Bessel functions.

The PTRtime represents the shape of the compressed pulse, typically characterized by a sinc2 function or better, using the real PTR of the instrument obtained from the internal calibration in order to increase the retracking performance. However, the analytical integration involving these functions is generally intractable, leading many models to adopt a Gaussian approximation instead . It is indeed possible to fit a Gaussian to the sin⁡2(Nx)/sin⁡2(x) and sinc2 functions by neglecting their secondary oscillations. Yet, there is no single method for this fit, resulting in various amplitudes proposed in the literature, ranging from 0.425 to 0.513, as reported by and cited by . suggested a value of 0.443, while our own fit yielded a lower value of 0.36. These different options are available in SMRT.

It is worth noting that any model limited to a Gaussian pulse can in principle be extended by first calculating the delay-Doppler map without the PTRtime – that is, by setting the Gaussian width zero – and then reintroducing the PTRtime via numerical convolution. This approach may introduce numerical errors (due to the zero-width Gaussian) and increase computational cost, depending on the models.

The effect of UF-SAR processing in the azimuth direction is typically represented by a response function (PTRDoppler) in the majority of models, analogous to the time domain. Although the terminology and equations vary across different studies, this response ideally follows a sin⁡2(Nx)/sin⁡2(x) function (i.e. the Fourier Transform of a sampled rectangular signal), sometimes modified by the influence of a Hamming tapering window if it was applied during UF-SAR processing. The Hamming window aims to reduce the secondary lobes by apodisation at the expense of a slightly decreased resolution in the along-track direction, e.g. from 300 to 450 m for the CryoSat-2 mission. Some studies employ the sinc2 function instead (i.e. Fourier Transform of a rectangular signal), corresponding to the previous case when N, the number of Doppler frequencies is very large. At last, for analytical tractability, many models also employ a Gaussian approximation for the response function e.g.. This approximation is particularly justified and effective when the Hamming window is applied for the UF-SAR processing, as it significantly reduces secondary lobes compared to the untapered sin⁡2(Nx)/sin⁡2(x) function. For sensors with a low number of Doppler frequencies N such as Sentinel-3 at C-band (N=2), only models using sin⁡2(Nx)/sin⁡2(x) are suitable. For the other existing space-borne sensors with N=64, the sinc2 and Gaussian approximations are acceptable.

In models that use a statistical representation of surface elevation, the vertical distribution is typically described by a probability density function (PDF). This function is often assumed to follow a normal distribution, which is commonly used in various fields to represent rough surfaces due to its simplicity and the tractability it offers for analytical calculations. However, introduced a skewed Gaussian distribution to accommodate more general cases, while strongly suggested that a log-normal distribution is more appropriate for modelling sea-ice surfaces. The disadvantage of using a more complex distribution to represent the surface elevation is that each distribution parameter adds another dimension to the solution space for physical retracking.

The Fourier domain-based model proposed by is not limited to normal distributions. In fact, it can efficiently handle any distribution that has a known analytical form in Fourier space. Additionally, as mentioned for the point target responses (PTR), the models that assume a normal distribution can, in principle, be adapted to other distributions by applying numerical convolution with the PDF.

The terrain slope at the footprint-scale (>10 km) is particularly relevant for ice sheets, which is why it is explicitly accounted for in some models using the statistical representation of the surface elevation and dedicated to this type of surface e.g.. Despite this useful addition, using a real Digital Elevation Model (DEM) as input can implicitly incorporate this large scale terrain slope through the DEM data as well, and many more details . Note also that a constant terrain slope can be approximated by adjusting the pitch and roll parameters (with opposite sign).

The simplest approach for backscatter modeling is to only consider the surface echo and assume a constant backscatter value, independent of the incidence angle as in and . However, it is well established that the reflectivity of a smooth surface decreases sharply with increasing incidence angle, i.e. the angle between the radar beam and the surface normal, which is vertical for a horizontal surface. Although the incidence variations seem small in altimetry (<1–2°), they are large enough to have a first-order impact on the trailing edge of the waveform.

Assuming a Gaussian decrease in backscatter with incidence angle is advantageous for analytical tractability and aligns with the physics of the Geometrical Optics approximation . However, this theory is only applicable to very rough surfaces, where the Root Mean Square (RMS) height and the horizontal correlation length are much larger than the wavelength. Other empirical or physical rough surface scattering models can be used. For example, assumes a linear dependency between backscatter and incidence angle. In contrast, some models, such as , rely on numerical integration and thus can handle any backscatter function, including the antenna gain. and incorporate the Integral Equation Model IEM, or its variants e.g., which are valid for a wide range of moderate roughness conditions. accounts for the local incidence angle in each facet of the DEM, which is crucial to simulate the EM bias caused by skewed slope distributions, for instance, when flatter, more reflective facets are more likely at the base of the terrain rather than equally distributed .

Volume scattering is less commonly addressed in the models reviewed here, primarily because many were developed for oceanic applications, where microwave penetration beneath the surface is negligible. However, this is not the case for dry snow or ice. The simplest approach to account for volume scattering is to assume a homogeneous semi-infinite layer with user-prescribed empirical values for the backscatter and attenuation rates . For sea ice, assumed a finite single snow layer overlying the ice interface. Snow scattering and absorption were calculated using Mie theory . However, Mie theory is not suitable for snow due to its high density . In , the volume is discretized into cells, allowing for a fully 3D calculation of volume backscatter, potentially enabling the description of complex medium structures.

Radar measurements are inherently influenced by a physical fluctuation known as speckle, caused by wave interference from different scatterers within the radar footprint. The models reviewed here, which are based on the radar equation, describe only the propagation of the power and do not account for the wave nature, thereby implicitly averaging out the effects of speckle. However, it is possible to simulate speckle by ad hoc addition of a random noise in each cell of the delay-Doppler map. This noise is typically modelled using an exponential distribution, as described by . When averaging the delay-Doppler map to compute the waveform, this noise distribution approximates to a gamma distribution under stationarity approximation, which, with a large number of Doppler beams, tends to resemble a Gaussian distribution . However, this approximation is in general not valid for a moving target, as on the ocean or for complex terrains, as on the ice-sheet margins . The averaging process (stacking) is more complex and advanced strategies can be deployed. This is not considered in our new development.

The model presented by provides equations for two antennas separated by a baseline, making it suitable for SARIn altimetric simulation, as described by . Although this capability is an asset, it is not explored in this review.

The model developed by is the most computationally intensive for three reasons. First it performs the summation on the x,y grid of the surface as instead of relying on analytical or semi-analytical integrations. Second, it accounts for satellite motion during the acquisition of the 64 beams, a unique feature among the models. However, this requires 64 distinct summations over the x,y grid. At last, the convolution with Point Target Response (PTR) in time is performed using basic numerical integration in the original model. At first glance, choosing the sinc2 function imposes such an ineffective approach. However, we found that the “sinc2 transform” proposed by is appropriate to perform the specific convolution with this PTR. This transformation, by leveraging the Fast Fourier Transform (FFT), is significantly more efficient. Moreover, this transformation was implemented in a Python package (https://github.com/gauteh/fsinc, last access: 3 December 2025) that also accommodates non-uniformly spaced samples. This is essential given that the summation over the grid involves irregularly spaced ranges. The use of this transformation significantly accelerates computation by a factor of 50–100 based on our implementation (not tested in the original code). Nevertheless, our implementation of model remains the most computationally expensive despite this improvement.

In contrast, the models by and are the most computationally efficient. The former relies solely on analytical functions. Despite requiring the unconventional Exponentially Scaled Modified Bessel function, it benefits from an efficient implementation directly available in the mainstream SciPy Python package . The second model is mostly numerical but was designed for fast execution, suitable for physical retracking. It applies Fourier transforms to convert convolutions into products, but unlike other models, it directly derives analytical expressions for the flat impulse response and PTRs in Fourier space, thereby avoiding several FFT calculations. This approach results in a substantial reduction in computational cost. The other models fall between these two extremes, employing either slower integration methods or faster FFT-based convolutions.

The time to compute one delay-Doppler map is given in Table . It is important to note that detailed benchmarking of these models is beyond the scope of this review, as performance can vary significantly with implementation details and hardware. Additionally, our primary objective being to provide a readable Python implementation of the models that closely adheres to the published equations, aggressive code optimizations were excluded.

The delay-Doppler models reviewed in Sect. , despite sharing the common aim of predicting waveforms from radar and surface characteristics, exhibit a wide range of approaches. This diversity is multi-dimensional as illustrated by the numerous rows in Tables –. Some of these dimensions are fundamental, driving the model derivation and features (e.g. how to represent the topography), while others are specific to a study and could easily be modified (e.g. whether to take terrain slope and tilt into account). Nevertheless, while the most recent models are generally more capable than the earliest ones, there is no unique best model, mainly because they target different applications. For this reason, eight of these models have been implemented in SMRT version 1.7 , allowing users to easily compare them and select the most suitable for their application. The list is: Wingham04, Halimi14, Ray15, Boy17, Buchhaupt18, Dinardo18, Wingham18, and Landy19.

The altimetry module, newly implemented in SMRT, aims to simulate UF-SAR processing and to output Doppler maps given the scattering and extinction properties (Ks(z) and Ke(z) respectively) and the effective permittivity ϵeff(z) of each layer, and the snowpack interface reflectivity and transmittivity, as calculated in other pre-existing SMRT modules. Instead of developing a complete module for each DDM model, we first identified the parts of the altimetric computation that are common across models and those that are specific to each model.

The key assumption is to limit the computation to first–order scattering, which implies independence between the different scattering mechanisms. It follows that the total power P(τ,f) received by the sensor (i.e. the DDM) is the sum of the contributions from each elementary layer and each interface independently. In general, the role of the DDM models is to compute the power Π(τ,f,σ0) received as a function of delay τ and Doppler frequency f, given the backscattering coefficient σ0(θ) of a surface. The other existing modules in SMRT compute the backscatter σinterfacei0(θ) of each interface i (i.e. surface, internal inter-layer interfaces, and substrate) and the profile of backscatter σvolume0(z,θ) coming from the layers. In addition we compute the delay and attenuation at depth z relative to the surface, represented by the generalized function Δ(z,τ) using the extinction Ke(z), transmittivity of the interfaces and the wave speed deduced from the refractive index ϵeff(z). It follows that: P(τ,f)=∫zΠ(τ,f,σvolumeo(z,θ))∗Δ(τ,z)dz3+∑iΠ(τ,f,σinterfacei0(θ))∗Δ(τ,zi) where the integral applies to the whole snowpack and i runs over the interfaces (surface and inter-layer interfaces). A second key assumption is considering volume backscatter is virtually independent of the zenith angle within the footprint of near-nadir altimeters (typical range ≈1–3°). This simplification is not applicable to the interfaces because their backscatter varies rapidly for off-nadir angles, especially the coherent backscatter component. With this simplification, the integral over the snowpack becomes a linear convolution Eq. 17 in yielding: P(τ,f)=σvolume0(z)∗Δ(τ,z)∗Π(τ,f,1)4+∑iΠ(τ,f,σinterfacei0(θ))∗Δ(τ,zi). Hence, the computation can be performed in three independent steps: (1) calculate the vertical profile of volume backscatter and the backscatter values of each interface, (2) compute the delay-Doppler maps for each interface Πinterfacei(τ,f,σinterfacei0(θ(τ))) and for the volume Πinterfacei(τ,f,1), (3) combine the vertical profile of backscatter and the DDMs to provide the final DDM P(τ,f) using Eq. () and eventually the waveforms (Fig. ) as follows.

Step 1 computes the backscattering coefficients of each layer and each interface according to the type of medium (snow, ice, …) and its properties, and the electromagnetic theories (Improved Born Approximation, Geometrical Optics Approximation, …) selected by the user (Fig. ). This step uses existing SMRT functions that are not specific to altimetry and were developed in early stages of SMRT development . Only the coherent backscatter (Sect. ) was added specifically for this study. The volume backscatter of each layer is then interpolated onto a sub-grid, regular in propagation time τ (“gate”) following , leading to a backscatter as a function of delay σvolume0(τ). In the LRM altimetry module, the interface backscatter was treated similarly to the volume backscatter because the authors assumed that all interfaces had the same roughness and the same angular variations in backscatter. However, this assumption is not satisfactory especially for flat surfaces with peaked backscatter, so Π depends on the diagram of backscatter σ0(θ). In the SAR altimetry module, each interface can have a different roughness, which requires computing and storing the backscatter for each interface at all incidence angles within the illuminated footprint. The output of step 1 is thus the volume backscatter time series σvolume0(τ) and a list of backscatter σinterfacei0(θ) for all interfaces.

Step 2 calls the user-selected DDM model (Fig. ). Our approach for implementing the DDM models was to follow the equations of the original studies as closely as possible. Most equations in the code are traceable to their respective studies. Nevertheless, some models required minor adaptations. does not provide sufficient equations; the gaps were inferred from . In the case of the full model was not implemented, as SMRT already included many of the necessary components (e.g. interface and volume backscatter). As with many other modules in SMRT, these DDM modules are fairly independent of the rest of the code and can be tested and used outside of SMRT. New DDM models can also be added easily thanks to the plug-in framework.

In this step 2, the selected DDM model is called for each value of interface roughness, providing a DDM for each interface Π(τ,f,σinterfacei0(θ)) along with the delay and attenuation generalized function Δ(τ,zi) at the depth of the interface zi. For the models using an explicit DEM (Boy17 and Landy19), the backscatter dependence on incidence angle for each interface is accounted for at this stage, since the local incidence angle can differ across facet slopes and orientations. In addition, a DDM is calculated for the volume Πvolume(τ,f,1) assuming volume scattering angular dependency is negligible over the range of incidence angle typical of altimeters (0–2°). The output of this step is a list of DDMs. These DDMs are calculated with a finer sampling interval in the time and Doppler dimensions to reduce numerical errors in the integrations and Fourier transforms. This results in oversampled DDMs compared to those DDM produced for altimeters (e.g. 64 bins in Doppler and 128 in time for Sentinel-3). An oversampling factor of 4 in both dimensions is typically sufficient for most models. This value is adjustable by the user. The final output DDM is optionally downsampled to match the real sensor's resolution. Figure  illustrates where the calculation of the impulse function is effectively performed in the case of Wingham04 (dotted circles), with an oversampling of 4 in the time dimension.

In this step 2, the slant range correction (a.k.a delay migration) is part of the analytical formulation from some models, yielding directly the corrected DDM. However, in other cases (Halimi14, Wingham04, Boy17, Landy19), the model first computes the uncorrected DDM, and we then migrate each cell in delay, with a left translation as depicted Fig. . The delay depends on the square of the Doppler frequency f as τ0(f)=f2αHλ2/(8V2) Eq. 7 in with α the Earth curvature correction, H and V the satellite altitude and velocity, and λ the radar wavelength. This ensures that, in the slant range corrected DDM, the echoes for all the Doppler frequencies are aligned at the range of the zero-Doppler echo (dark green line) by compensating for the additional range sensed at different along-track distances from the target on ground (dark red parabola).

The third step combines the backscatter σ0 and DDMs Π using Eq. (). This step returns the final DDM P(τ,f). Optionally, the terms in Eq. () can be returned separately (called volume DDM and interface DDM hereinafter) to enable comparison of the different contributions.

Steps 1 and 3 are common to all DDM models and implemented in the module “nadir_sar_altimetry”, while step 2 is implemented in independent modules (the files in the directory “delay_doppler_model”) that provide the Π function, one for each DDM model.

Backscatter from a rough surface is commonly composed of a diffuse component and a coherent one . The latter is only significant in the specular direction (also known as quasi-specular reflection). For this reason, incoherent backscatter is the only significant component in monostatic side-looking radars such as scatterometers and Synthetic Aperture Radars, and all existing SMRT functions neglect coherent backscatter. However, in nadir-looking altimetry, the specular direction is vertical, and coherent backscatter can become significant at near-nadir angles in the first time gates, known as the Fresnel region.

A specificity of coherent backscatter is to depend not only on surface roughness but also on the altitude of the sensor when the angle subtended by the coherent scatterers at the surface is not small compared to the antenna beamwidth . For SAR altimeters, “antenna beamwidth” is understood to be the pulse-limited, focused beamwidth, which is very small (300 m in the Doppler direction). Similarly, for rough surfaces, “coherent scatterers” are understood as objects of size comparable to the correlation length, typically less than a meter. It results in the coherent backscatter having a strong impact on the interpretation of differences between in-situ radar and airborne altimeters. Space-borne sensors are only concerned when the surface is flat and smooth, such as on sea ice .

A coherent backscatter formulation was added directly in the “nadir_sar_altimetry” module, following Eq. 6,. This approximation is considered valid for a wide range of surface roughness .

In-situ measurements relevant to perform microwave simulations have been collected through several intensive campaigns in Antarctica, namely Vulnerability of ANtarctic Ice SHeet and its atmosphere (VANISH) in 2012, BI-POle in 2012 (BIPOL), Accuracy of the Surface Mass balance of Antarctica (ASUMA) in 2016, and East Antarctic International Ice Sheet Traverse (EAIIST) in 2019. The visited sites are reported in Table  with their main characteristics. Most of these data were already used in previous work where detailed information is provided. Only a brief description is provided here.

In a first set of simulations, SMRT was driven with the measured vertical profiles of density, layer thickness, temperature, Specific Surface Area (SSA), surface roughness, and height distribution. At every site except VANISH, a ≈8 m core was extracted with a 10 cm diameter drill and processed on-site in a cold laboratory. Density was measured on ≈10 cm long slices. Each slice is represented as a layer in SMRT, with its measured thickness. Specific surface area (SSA) was continuously measured along the core side using the Alpine Snowpack Specific Surface Area Profiler (ASSSAP) , and averaged over each layer. The SSA and density profiles were then extended to 100 m depth by repeating the last 1 m of the ≈8 m measured profiles. A linear trend estimated over the measured profile was applied to the density to simulate the densification. This rough approximation has little impact because the altimetric signal usually comes from the upper part, where measurements were actually taken for most sites. The snow temperature was measured with a Pt100 sensor at the bottom of the drilled hole for 24 h. At this depth, the temperature is close to the annual mean temperature. For the simulations, the temperature profile was assumed uniform. assessed the impact of seasonal temperature variations.

The protocol applied during the earliest campaign, VANISH, was less elaborate. The cores extracted in January 2012 were shipped back to France and cut into 10 and 5 cm pieces at S2 and S4, respectively, in July 2012, in a cold chamber, to measure density and SSA.

Key measurements for altimetry are the characteristics of the surface roughness at the radar wavelength scale and the large scale height distribution within the footprint. Both are related to height variations of the surface, but differ in the spatial scales. The former (hereinafter called radar-scale roughness) controls the backscatter intensity of the surface (and the interlayer interfaces), while the latter (hereinafter large scale roughness) controls the spread of the delay of the echo and influences the local incidence angle. The radar-scale roughness is quantified by the Mean Square Slope and was estimated at some of the sites (Table ) using a centimetre-resolution DEM of 5m×5m areas. The technique uses manual photogrammetry with ground-level photographs. The measurement protocol and processing are detailed in . The large scale roughness, quantified by the standard deviation of height (σsurf), was obtained at each site using the Reference Elevation Model of Antarctica (REMA) tiles at 2 m resolution . The selected area was a 700 m diameter circle around the in-situ measurement point. These data are reported in Table .

For Ku-band (and independently for C band), the closest 20 Hz waveform of each site was extracted from the BC-005 ESA Land Ice Thematic Products after discarding the low quality waveforms (“waveform_qual_ice_20_Ku” parameter different from 0)

The waveforms were scaled with the parameter “scale_factor_20_Ku” (and C, respectively, for the Ku and C band), also provided in the L2 product. This operation is necessary to enable cross-site comparison of waveform amplitudes. However, it is not sufficient to obtain absolute power due to the lack of absolute calibration data. Therefore, the comparison with SMRT can only be relative. addressed this issue by estimating a single factor α(f,S) for each band f and sensor S to scale all SMRT simulations at all sites to match the observations. Using this normalization factor, the site-to-site amplitude variations between the observations and the model are preserved and are safely comparable, while the absolute values are meaningless.

The description of the snowpack for the Antarctic simulations follows the theoretical approach recently introduced by . This new approach calculates the Porod length lp using in-situ measurements of snow microstructure, namely the specific surface area (SSA, m2kg-1) and density with: 5lp=4(1-ρ/ρice)/SSAρice and then the microwave grain size with: 6lMW=Klp where K is the microwave polydispersity. It was empirically found that a constant polydispersity K=0.625 is suitable for Antarctic snowpacks . The profiles of lMW, density and temperature are the only three snow properties required as inputs of the Symmetrized Strong Contrast Expansion SymSCE, to compute scattering and absorption in each layer.

In addition to these snow properties, we assumed the snowpack surface and internal interfaces to be very rough and thus applied the Geometrical Optics approximation (valid for radar-scale RMS height >5 mm at Ku-band). The same mean square slope (MSS) value was initially assumed for the surface and all interfaces and was taken from in-situ measurements.

The mean square slope (MSS) at the radar scale is not available at all sites and is certainly the least constrained by measurements, while it is the most important driving parameter . For this reason, we optimized this parameter for each site following . To do so, we take as a MSS initial guess, the value obtained from the global normalization factor α, itself determined from simulations using measured MSS. We also set the radar-scale horizontal correlation length to a constant l=10 cm and assume a surface characterized by a Gaussian autocorrelation function, to deduce the radar-scale RMS height s such that s2=MSSl2/2. The brentq function from the scipy.optimize library is then used to explore the range MSS=0.005–0.2 and find the best agreement between the simulated and observed Sentinel-3 amplitudes, defined in ICE-1 as: 7A=P(τ)4‾P(τ)2‾ where the overline denotes the average over τ. This function quickly finds the maximum, usually in fewer than 9 iterations, using a tolerance for the mean-squared slope of 0.001.

Figure  shows the waveforms calculated with all the delay-Doppler models for the Sentinel-3 Ku-band altimeter parameters for a Gaussian surface with a RMS height of 40 cm, without any volume underneath, the satellite perfectly at nadir with a circular antenna, no apodisation and the delay window widening of 2 (i.e. 256 gates are used before slant range correction). These simplified conditions are chosen to use only the common capabilities across all DDM models. For the models working with a deterministic DEM as input (Boy17, Landy19), independent samples of the Gaussian distribution were drawn for every 10m×10m cell in the footprint, thus allowing comparable behavior to the other statistics-based models. The waveforms are presented normalized to 1 at their maximum to facilitate intercomparison. The simulated waveforms have a typical shape of UF-SAR altimeter waveform on a flat surface with a sharp rise (leading edge) and quasi-exponential decrease (trailing edge). This decrease is quicker than in LRM waveforms because the surface area effectively contributing to each time gate (Fig. ) is decreasing with time instead of being constant for the LRM as explained in . All the waveforms peak within the same range (bin 44.8–45.2), with the surface theoretically at gate 44 for Sentinel-3, as prescribed in the sensor's parameters in SMRT.

The waveforms are very close, but there are small observable differences when overlapping all the waveforms in a single graph (Fig. ). The initial rise at the level of 5 % of the maximum is slightly earlier for Wingham18, Buchhaupt18, Landy19, Halimi14 than follows Wingham04 and Boy17 and finally Ray15 and Dinardo18 (Fig. a). The trailing edge at 20 % is earlier with Dinardo18 and Ray15, then Wingham04 followed by a group with Wingham18, Buchhaupt18 and Landy19, and finally Halimi14. These differences are likely due to a combination of theoretical and numerical approximations and the specificities of our implementation. As an example, the difference between Ray15 and Dinardo18 (e.g. the trailing edge is slightly later in Ray15) are likely numerical since the two models are in principle equivalent up to the final integration, which is performed numerically for the former and analytically for the latter. For Halimi14, we noticed that this model specifically requires a significant oversampling factor along the Doppler dimension (64 by default, vs. 4 for other models) to produce the same waveforms as the other models. This is due to the simplified integration over delay-Doppler cells, as exposed in their Eqs. (13)–(14) and the text around. This simplification is compensated here by oversampling.

This comparison of the DDMs shows general agreement among the eight models, and the small differences observed are likely to be nonsignificant in most applications. However, the conditions of this comparison were chosen to allow a fair comparison. Each application should first review the approximations made by each model (e.g. Gaussian antenna pattern, statistical representation of the surface, Gaussian approximation of the PTRs, etc.) to decide which model or group of models is most suited. For instance, if the terrain slope is critical or the antenna is elliptical, not all the models are suitable.

To further verify our implementation, Fig.  compares Landy19 implementation in SMRT with the original Matlab model available as open source . The medium is a slightly rough surface at the radar scale (2 cm in RMS height and 7 cm in correlation length) and has a large scale roughness of 40 cm, with no slope and no underlying volume. For a fair comparison, SMRT was run with an oversampling time of 1, while the default is 4, and the original model used the geometrical optics approximation, whereas its default is I2EM . We normalized the backscatter to its maximum for comparison. The results show high consistency between the two models, with the difference not exceeding ±1 %. We note a smooth rise before the leading edge, possibly due to difference in the convolution of the PTR (as no echo is coming earlier than the surface echo, only the PTR influences the first gates), and a noise component all along the trailing edge, likely to the resolution of the grid (5 m in both simulations) and rounding errors. It is worth noting that when using a time oversampling of 4 in SMRT (results not shown), we observed a much larger difference, up to +14 %, due to a slightly earlier peak in SMRT, which could be incorrectly interpreted as a difference of surface elevation. Such results suggest to always compare simulations with the same oversampling and more generally call for careful consideration of the selection options when conducting precise comparisons or sensitivity analyses.

A second comparison is performed to verify that pseudo-LRM waveforms calculated with DDM models is consistent with the output of the LRM module previously developed in SMRT . To simulate such pseudo-LRM waveforms it is necessary to disable the slant range correction. This is possible with most models, either because this correction is an independent geometric transformation applied directly to the DDM in the last stage of the simulation (Wingham04, Halimi14, Boy17), or because it appears as an explicit term in the equations, easy to disable (Wingham18, Buchhaupt18, Landy19). To our understanding, this is not possible with Ray15 and Dinardo18 because the correction is deeply embedded in their analytical derivation.

Figures a and b illustrate the DDM and waveform before and after slant range correction for Buchhaupt18 and Wingham18. The DDM without slant range correction show a typical parabolic shape explained by the fact that the distance from the satellite to reach the surface at a given look angle (i.e. Doppler bin) is increasingly further as the look angle increases, as illustrated in Fig. a. By summing this non-corrected DDM along the Doppler-frequency dimension yields the pseudo-LRM waveform (Fig. c) featuring a slowly-increasing leading edge and a decreasing trailing edge. In contrast, Fig. b shows the benefit of the slant range correction on the focusing of these high look angle echoes closer to the first echo (located at the nominal 44) and as a consequence, the resulting more peaked and narrower waveform.

Figure c compares these pseudo-LRM waveforms with the true-LRM calculation from the LRM module in SMRT based on . The simulated snowpack is homogeneous and infinitely deep with a rough surface. This comparison with the original, independent LRM module shows near-perfect agreement with Wingham18. This provides a validation of our implementation, as both the LRM and SAR modules are completely independent. The agreement with Buchhaupt18 is also excellent in the leading edge, but degrades progressively in the trailing edge. Despite starting from the same principles as Wingham18, the elaborate Fourier transform approach may introduce small numerical errors.

The comparison in previous Sect.  and were conducted assuming that the time window recorded by the sensor extends well beyond the 128 gates. This choice was motivated by the need to achieve a fair comparison between all models since some are unable to account for the limited window of real sensors. However, for a comparison with real observations, when the surface is sufficiently rough, it is crucial to account for this detail as highlighted in Fig. . The figure shows the delay-Doppler map before and after slant range correction when considering either 128 gates (real case of Sentinel-3) or 256 gates (ideal case) and the resulting waveforms in both cases.

The simulation with 128 gates (Fig. b) shows that no signal is recorded at the highest positive and negative Doppler frequencies (bins <10 and bins >54) before the slant range correction because the echoes are coming beyond the 128th gate. Once the slant range correction (Fig. d) is applied, the signal is non-null only in a parabolic-shaped area. Comparing with the ideal focused DDM (Fig. c) gives an idea of the area of the DDM with missing power in the real case (Fig. d). Once summed along the Doppler dimension to obtain the waveform, and after normalization to the maximum, the impact on the waveforms is mainly visible on their trailing edge. The 128-gate simulations miss some power from the longest delays. This difference obviously increases at larger gate numbers, where more power is missing compared to the ideal simulations.

This change in waveform shape is significant and should be considered when comparing with observations, especially if the first echo occurs after the expected nominal gate (44 by default for Sentinel-3). In fact, the missing power strongly increases as the gate of the first echo arrival shifts to the end of the recording window.

In SMRT, it is possible to control the delay window widening factor for all but Ray15 and Dinardo18 models. The default factor is 1 (no widening), suitable for a comparison with observations, while 2 is usually sufficient to record all echoes, as shown in Fig. . All simulations in the previous sections use this value of 2. The user must be aware that, to achieve a realistic, limited window (no widening), a trick has been implemented in the models using an analytical slant-range correction (Wingham18, Buchhaupt18, and Landy19). In this case, the analytical correction present in the original studies is disabled and replaced by the numerical method used in other models (Wingham04, Halimi14, Boy17). This trick is automatically enabled for a widening of 1 (or less) while the original analytical correction is applied for larger widening.

In addition to returning the total DDM, the SAR module can optionally return the contributions from each layer and each interface, and separate coherent reflections from incoherent ones. Figure  illustrates this feature for a snowpack with a rough surface (3 cm RMS height) and a smooth crust (0.5 cm RMS height, 450 kgm-3 and microwave grain size of 0.3 mm) buried at 2 m depth. The layer above the crust is composed of fresh snow (300 kgm-3 and microwave grain size of 0.1 mm) and the semi-infinite layer below of older snow (350 kgm-3 and microwave grain size of 0.2 mm).

The simulation considers a specific snowpack to illustrate the signal components. Although such crust layers exist, the particular shape of the simulated waveform may not be observed in reality due to speckle noise and the non-uniformity of the snowpack over the kilometre-wide footprint. The volume (orange) has the largest contribution but is delayed relative to the surface echo (interface 0, dark green). The surface has a strong incoherent contribution and virtually no coherent contribution, due to the high RMS height of 3 cm. The top and bottom sides of the crust layer (interfaces 1 and 2, respectively) have a moderate incoherent contribution (light green solid curves) and a coherent peak around gate 49 (light green dashed curve). This coherent peak is due to the layer's smoothness. It is worth noting that this idealized simulation assumes a sensor pointing precisely perpendicular to the crust layer. With a satellite pitch angle as small as 0.3°, the coherent contribution immediately vanishes (not shown).

To check our implementations, we performed several comparisons: first between the eight implemented DDM models; then between Landy19 and the original code in ; and, last, between Buchhaupt18 and Wingham18 in pseudo-LRM mode with respect to the original LRM implementation in . The two former considered a surface only, and the latter also included a volume. In general, these comparisons show good agreement, and the residual differences were attributed to numerical errors (particularly evident in Landy19) or to formulation and implementation details (between Ray15 and Dinardo18). However, to achieve such an agreement, it has been necessary to devise a simple medium configuration (statistical representation of the terrain with a Gaussian approximation, no slope, circular antenna) and to carefully adjust the model settings (widening factor =2). While this approach fits with our goal of cross-validation, these settings are not the default in SMRT (widening factor =1 except in Ray15 and Dinardo18), and are usually not the same as in the original studies (e.g. the topography representation, the Gaussian approximation of the terrain distribution and of the PTR). For future use of this SAR Altimetry module, it is recommended to (i) refer to the documentation and source code for the default settings and the specific capabilities of each DDM model (e.g. Landy19 using the accelerated sinc2 transform is incompatible with the Hamming window which the Matlab code can account for), (ii) prefer the most recent models (post 2017) that account for the terrain slope and antenna asymmetry, (iii) perform an initial cross-comparison with as many DDM models as possible in the conditions relevant for the application (for instance for using a real DEM, both Boy17 and Landy19 are suitable, and comparing Wingham18 and Buchhaupt18 may reveal numerical issues due to the Fourier transform), and (iv) carefully check and adapt the altimeter parameters (e.g. PRF and altitude) as well as understand the adequacy of the selected DDM model and the specific L1 ground processing settings (e.g. oversampling in range, selection of the beams in the stack, etc).

In the comparison in Antarctica, the SMRT model was able to predict the waveform shape and the spatial gradient in waveform amplitude using only in-situ measurements, and performed better when optimizing the radar-scale surface roughness. The shape is primarily controlled by the delay-Doppler map model, and secondly by the correct balance between the surface and volume (and interfaces) contributions. Despite the significant improvement brought by the surface roughness optimization, the results show irreconcilable discrepancies, as in stops 4a, 4b, 1, and 5. Part of this may be attributed to the following limitations of our approach.

The in-situ measurements are taken at a single point on a single date and used to represent a wide footprint across distant observations in time. The representativeness is a common issue in remote sensing and modelling, but one should mention that the skills of SMRT observed in the passive microwave with resolution >25 km are indicative that the large scale variations in Antarctic snow and surface properties are more important than the intra-footprint variability, though it could not be excluded. The Antarctic snowpack is also very stable far from the coasts. The most volatile variables are certainly the radar-scale surface roughness, grain size and density, with a dynamics related to storms and snowfall events . The exploration of the temporal dimension is left to future work.

The second main limitation of our approach is to consider a Gaussian topography and neglect the slope. It is now well demonstrated that the exact topography within the footprint has a considerable control on the waveform shape . It is interesting to note that obtained excellent results in terms of waveform shape while only considering the surface. However, neglecting the volume and the penetration of the wave is likely to bias the retrieved elevation. To explore this aspect further, the Boy17 and Landy19 delay-Doppler models are suitable, as a real DEM can be prescribed. This has not been tested and is also left to further work.

The newly implemented module and its delay-Doppler models have several limitations that may affect results in the present study and future investigations.

The medium is assumed homogeneous over the entire footprint, which typically extends to a dozen of kilometres. This limitation owes to the current implementation, and although the studies reviewed in Sect.  do not account for heterogeneity, the models using an explicit spatial grid are easily amendable (namely Landy19 and Boy17) as long as 3D effects remain negligible (i.e. no lateral wave propagation across different media). The main cost is to repeat the computation of the scattering properties of each medium within the footprint, which is moderate compared to the computation of the delay-Doppler map itself using these most numerically intensive models. On the ice-sheets, the benefit would be to account for the spatial variability, but this would require a realistic set of in-situ measurements. However, the main gain would be for simulating strongly heterogeneous targets like sea ice and frozen lakes in the presence of leads, as newly-forming ice in leads acts as a strong coherent radar reflector and thus shows a contrasted response to the sea or lake ice floes .

Another limitation arises from the single-scattering calculation of the echo: the wave is reflected by the surface, an interface, or the volume only once, while interactions involving double bounce and higher-order scattering are neglected. These interactions are susceptible to increasing late echoes, because of the longer travel, with an impact on the trailing edge and possibly double peaks. Overcoming this limitation is not straightforward and would practically require using intensive ray tracing techniques to compute interactions while recording time of flight . In general, multiple scattering in the volume is likely to affect the highest frequencies, i.e. the Ka-band altimeters, although the extinction is also high when multiple scattering exists, limiting the distance between two interactions, and so the delay. Reflection in layered media at low frequency and low temperature, where absorption is reduced, may also be subject to multiple scattering and relatively long distances between reflections. A first practical way to assess the amount of volume multiple scattering consists in computing the approximate fraction due to higher order scattering ω̃/(1-ω̃)), where ω̃ is the single scattering albedo. The typical distance between two interactions is estimated by the inverse of the extinction coefficient (i.e. the e-folding depth). Both are outputs of SMRT simulation for each layer. For instance, the average single scattering albedo over the top meter of the snowpack reaches 0.39 at ago5 and s2b for Sentinel-3 in Ku-band, with an e-folding depth of 22 m (about 50 range gates). This value rises to 0.76 at 35.75 GHz (Ka-band), the frequency of the future Copernicus CRISTAL , indicating strong higher-order scattering in the volume. However, the e-folding depth is significantly reduced to 1 m (about 3 range gates). Another approach consists of comparing the backscattering coefficient simulated using the first-order iterative solver (single scattering only) and the DORT solver, which accounts for all scattering orders . This approach also accounts for the between-interface multiple scattering. At ago5, we found a difference of 2 dB in Ku-band (63 % of the amplitude is due to single scattering), and 4 dB in Ka-band (43 % is due to single scattering). The comparison between observed and simulated waveforms may also bear the imprint of missing multiple scattering, with the simulated trailing edge underestimated and a secondary peak appearing in the observations.

Another point of attention is the approximate nature of the delay-Doppler models and the numerical inaccuracies arising from implementation details, as illustrated in the results (Sect. ). These models simulate the effect of the UF-SAR processing but are not simulators of the full complexity of modern SAR processing. In particular, except Landy19 they do not account for stacking multiple acquisitions as the AMPLI model does . The current implementation also misses FF-SAR and SARIn processing, for applications to Copernicus Sentinel 6 and CRISTAL over sea ice . However, it is suitable to extension, simply by implementing a new delay-Doppler map model for these processing e.g.. More complex would be the adaptation to airborne altimeters, because the assumption of small near nadir angles is used in the main nadir_sar_altimetry module as well as in the DDMs.