This study examines the microwave emission properties of snow-covered sea ice, modeled as a layer with rough top and bottom boundaries. The emission model is based on the first-order solution to the radiative transfer equation (RTE). This equation describes the brightness temperatures that propagate both upward and downward in the layer, and it is solved numerically using the eigenvalue method. Additionally, the model takes into account contributions from irregular boundaries as well as surface and volume scattering interactions. The study looks at the variability of brightness temperature and the growth of different ice types. It uses modulation theory to consider the roughness of the boundaries, showing how this affects the emission from various types of snow-covered sea ice. We used polarization ratio-gradient ratio (PR-GR) space to analyze ice-grown transitions quantitatively. The thresholds of PR were found to be sensitive to changes in ice concentration. It was observed that the threshold of PR did not change with the roughness of the top boundary. We also found that the roughness of the ice surface has a more significant impact on emission than that of the snow-covered ice. It also observed that roughness effects are more potent in H-polarization than in V-polarization for each of the seven ice types. The roughness effects are weaker at large look angles, indicating stronger volume scattering. The study concludes that the presence of roughness in the ice layer’s boundaries leads to noticeable variations in brightness temperature.

This article employs the modulated correlation function (MCF) model to describe the ocean surface in radar backscattering. Surface correlation functions derived from wind wave spectra: Apel, Elfouhaily, Kudryavtsev, and Hwang models are examined. The spectral properties of the above spectra are also analyzed, including the height, slope, and saturation spectra. The results suggest that the MCF model is applicable in depicting spatial correlation of ocean surface. At the same time, the spectrum of MCF may not be proper in describing the energy cascade of ocean waves. The comparisons of backscatter are made among the MCF model and wind wave spectra, with regard to dependences of radar frequency at L-, C-, X-, and Ku-bands, wind vector, and incidence angle. The results indicate that the MCF model yields the overall minimum errors in radar backscatter against geophysical model functions (GMFs). We validate the model with the airborne and spaceborne radar measurements and show that the MCF model gives good agreement at the C-band but only moderate at the Ku-band. Meanwhile, the backscatters calculated based on different wind wave spectra can significantly deviate from each other, related to the magnitudes of the short-wave spectrum. Besides, the breaking wave effect on radar backscatter is also discussed.

Surface parameters (roughness and soil permittivity) are crucial for characterizing backscattering from bare soil. However, the estimation of roughness parameters (root-mean-square (rms) height and correlation length) depends on the sample surface size. The conversion between dielectric constant and soil moisture is disturbed by the dielectric model. These estimation biases significantly compromise the reliability of backscattering coefficients derived from analytic modeling, numerical simulations, and experimental measurements. In this study, we illustrate the statistical relationship between sample surface size and estimation bias of surface roughness at varied accuracy levels. To quantify the estimation bias of surface roughness with sample surface size and the estimation bias of soil permittivity, we analyze the propagation from the estimation bias of surface parameters to the backscattering coefficient error by the advanced integral equation model (AIEM) model. By comparing it with measurement data, we quantitatively confirm the impact of roughness parameter estimation bias. Ultimately, quantifying the backscattering coefficient error as a function of sample surface size and incident angle allows for selecting the optimal sample surface sizes suitable for L-band synthetic aperture radar (SAR) simulation and soil moisture retrieval, along with their applicability over various incident angles. This study suggests sample surface sizes for estimating roughness parameters and backscattering coefficients at various levels of accuracy.

We analyze the bistatic scattering of a gamma-distributed rough surface to explore the scattering properties in the context of polar angular patterns and specular angular beamwidth as a dependence on the degree of non-Gaussianity. We focus on the incoherent scattering because of its importance in remote sensing of rough surfaces, e.g., soil and sea ice. We use a 3 dB angular width, the width of the range of scattering angle, as a measure of angular broadening and dropping-off of the scattered power. Some insights into the physical significance are perceived. For the co-polarized scattering in the incident plane, as the skewness and kurtosis of Gamma height probability density (HPD) surface decreases, the scattering becomes narrower and concentrates more at the specular direction and has a faster decay than in the Gaussian surface at large angles. As the shape parameters increase, the incoherent scattering concentrates more at 0° of scattering angle for the cross-polarized scattering in the cross-plane. The narrow 3 dB angular width is located at the larger shape parameters (or low skewness and kurtosis) with a small to moderate root mean square (rms) height. The overall 3 dB angular width of the exponential power spectral density (PSD) surface is smaller than the Gaussian PSD surface. This study reveals that angular broadening is narrower, and the dropping-off of the scattered power is more significant for Gamma height distributed and exponential-correlated surfaces than the Gaussian surface.

A comprehensive exploration of the transformer and dual-receiver system-based direction-of-arrival (DOA) estimation is presented in the context of sea surface scattering, particularly under varying sea conditions. A bidirectional encoder representation from transformer (BERT) with a physics-based loss function is utilized to process two individual channel radars. The datasets are the radar scattering coefficients of sea surface simulated at C-band for copolarizations and cross-polarizations. Through detailed analysis of simulated datasets and root mean square error (RMSE) evaluations, the model’s performance is investigated across different observation modes, namely, the co-polar (CP), co-azimuth (CA), full-bistatic (FB), and Beaufort wind scale from 3 to 5. Our study demonstrates that the bidirectional encoder representation from transformer model, employing a physics-based loss function, outperforms the baseline long short-term memory (LSTM) model, especially under high noise levels and with larger datasets. Significant correlations between wind conditions and DOA accuracy are observed, highlighting the bidirectional encoder representation from transformer model’s adaptability to dynamic environmental factors, particularly under increased wind scales. The choice of observation mode, with CP and FB consistently outperforming CA, proves pivotal. Precise simulation of speckle variations and optimized observation mode selection are identified as crucial avenues for enhancing the model’s practical utility.

This paper examines the radar penetration into a rough soil surface with a vertical moisture profile. Numerical analysis shows that the penetration depth occurs at a larger dynamic range of incident angle in V polarization but in H polarization. The maximum polarization difference occurs at low frequencies (e.g., L band) and at a large incident angle (e.g.,70°~85°), which are rarely used due to low backscattering returns. Of the two roughness parameters, the RMS height significantly influences on the penetration depth more than the correlation length. The dependence of penetration depth on the wave polarization moderates when the surface becomes rougher. Results also suggest that the penetration depth is sensitive to the inhomogeneity of moisture profiles due to the temporal evaporation process, indicating that the penetration depth is difficult to quantitate and an equivalent model to estimate it may be inappropriate, or at least it is difficult to establish.

In this paper, we proposed a modulated correlation function to characterize the multiscale property of the sea surface and adopted it in the AIEM (Advanced Integral Equation Model) to calculate the radar backscattering. Comparisons of NRBCS (Normalized Radar Backscattering Cross Section) with GMFs (Geophysical Model Functions) predictions and radar measurements are conducted in various wind conditions. Good consistency and accuracy confirm the proposed model’s accuracy and applicability in predicting radar backscattering. In addition, the relations between two modulation parameters and wind vectors are analyzed at C-band. The effective correlation lengths determined from the modulated correlation function show strong wind dependence, besides the incident angle and frequency in the context of radar backscattering.

This study examines the speckle properties of SAR (synthetic aperture radar) images of non-Gaussian rough surfaces. Three height probability density functions were considered: Gaussian, exponential, and Weibull forms, while two forms of the power spectral density were considered: self-affine fractal and exponential. Computer realizations of the rough surface ensemble samples were done, followed by a physics-based simulation of SAR complex images of the rough surface. Speckle distributions were analyzed against the fully-developed speckle model. Illustrative examples with various surface roughness scales in the context of SAR imaging are given. When the surface HPD is non-Gaussian, image speckle is no longer following the Rayleigh distribution, regardless of the PSD. The speckle of exponential HPD surface profoundly away from Ravleigh model, especially under self-affine fractal PSD.

This paper analyze polarimetric signatures SAR image of an electrically large cargo ship sitting over a sea surface. For the systematic and illustrative purpose, we applied a fully coherent SAR image simulation in streamlining the data flow from backscattered field to complex single look images recently developed to generate high-resolution full polarization SAR images. Three scenes of the ship heading parallel, orthogonal, and 45° to the azimuth direction were simulated. An attempt is made to exploit the polarimetric analysis via the Y4R decompositions. Results reveal that the polarimetric features of sea surface and its interactions with the cargo ship at different headings are quite distinct. For headings parallel to the SAR azimuth direction, richer and diverse polarimetric signatures are presented, but they lose or diminish when the heading is 45° to the azimuth, where surface scattering becomes predominant. A large look angle may attribute this change.

This article presents the computation of the backscattered field in simulation of synthetic aperture radar (SAR) raw data. We develop an improved Kirchhoff approximation to estimate the surface fields induced by the incident waves. The improved Kirchhoff approximation is a second-order iterative solution of the integral equations governing the targets’ electric and magnetic current densities. In computing the second-order re-radiated fields, we applied the shoot-bouncing-ray (SBR) technique to enhance propagation path tracking efficiency. The geometrical theory of diffraction (GTD) was employed to account for the diffraction fields from the edges and wedges, which constitute the total scattered fields collected by SAR in synthetic aperture. As the SAR moves along the azimuth direction, the backscattered signal computation is repeated as the antenna beam crosses the targets. This procedure demands heavy computational resources but requires no priori assumptions of radar cross-sections (RCSs) nor speckle statistics. The raw data is generated as an output signal of the SAR system response into which the backscattered signal is input. We chose three types of the target to demonstrate our approach to confirm the effectiveness. The first set of targets are three dihedral reflectors – two of them were rotated to constitute three unique scattering matrices. The results show that the polarimetric information, both relative amplitudes and phases, are well–preserved. The second target is a rough surface with exponential power spectral density and Gaussian height probability density. We examined image speckle statistics of multi-looking image. The third target type is an electrically large cargo ship (container) sitting over a sea surface. Polarimetric analysis via the Pauli and Y4R decompositions reveals that the polarimetric features are well preserved. Simulation results demonstrate that the present approach is fully coherent in streamlining the data flow from backscattered field to complex single look images within the SAR imaging scene.

We present a generalized power-law roughness spectrum to account for the spatial anisotropy effects on the radar scattering of a rough surface, where both the correlation anisotropy and the scaling anisotropy are accounted for. The spatial anisotropy is essential to correctly interpret the radar scattering from an agriculture field where both plow and sow are practiced. We investigate the dependence of the backscattering coefficient on the correlation anisotropy and the scaling anisotropy through a model simulation. A drastic change in backscattering strength is observed due to the anisotropy. The correlation anisotropy and the scaling anisotropy generate similar backscattering angular behavior, implying that in the context of spatial anisotropy, merely using correlation length in scattering modeling is insufficient. Equivalently, the correlation length retrieved from the backscattering coefficients perhaps is not unique. Fair use of the generalized anisotropic power-law roughness spectrum in conjunction with the scattering model is illustrated by comparing the backscattering coefficients with experimental measurements. However, the anisotropy complicates the roughness description in terms of surface parameters retrieval because we can generate similar backscattering angular patterns by combining different correlation anisotropy and scaling anisotropy. When the soil moisture is of primary interest, a more suitable radar observation geometry to minimize the spatial anisotropy influence is desirable.

This paper presents a physics-based machine learning (ML) approach to estimate the direction-of-arrival (DOA) over sea surfaces. We designed a recurrent neural network (RNN) to accept two receiving channel radar data to ensure angular coherence between the receiving signals from different directions, given the scattered signal by rough surfaces having angular memory. Specifically, the radar scattering coefficients of sea surfaces were simulated at C-band and for both co- and cross polarizations; investigations show that the sea speckles significantly and negatively impact the dual receiver’s DOA estimation performance; to mitigate the speckle interferences, we further designed an optimal configuration of dual-channel observation for DOA estimations over the sea surface, i.e., the co-polar (CP) observation mode performed best compared to that of the co-azimuth (CA) and the full-bistatic (FB) observation mode, and at moderate sea speckle impacts, the root-mean-square error of CP observation mode was about 1° for incident angle and 5° for incident azimuth angle estimations, results demonstrate the superior performances of the proposed physics-based neural DOA estimator.

We illustrate the wave scattering from a multiscale rough surface modeled by a modulated correlation function. The modulation ratio, defined as the ratio of baseband correlation length and the modulated length, determines the degree of multiscale roughness. The dependence of bistatic scattering on the modulation ratio are investigated. Numerical results show that without considering the multiscale roughness, the scattering coefficients are overestimated at a small incident angle region but underestimated at a large scattering region. Radar wave scattering from multiscale rough surface is highly frequency selective. As an application example, we compare the model predictions with two independent sets of measurement data. The results demonstrate that the model predictions with modulation effects are in good agreement with the measurement data.

There are few studies on predicting fully bistatic scattering from the rough surface of Mars, though some bistatic radar observations have been made, such as in the MARS EXPRESS mission. To better understand the interaction of radar signals with a planetary surface in bistatic radar observations, the topographic-scale roughness of Mars, characterized by a two-dimensional power spectrum density (2D-PSD), is examined in view of its global roughness variations and scale dependence on geological units. The analysis shows that the Martian 2D-PSD is strongly dependent on the geological units and that it lies between Gaussian and exponential functions, with a power index equal to 1.9. The bistatic scattering coefficients are calculated by an advanced integral equation model (AIEM) with the 2D-PSD as the input. It shows that the specific surface roughness spectrum and the dielectric inhomogeneity should be taken into account in interpreting the bistatic radar scattering response.

This article presents a numerical study of the wave scattering from a modulated rough surface characterized by a modulated correlation function that shows zero crossings along the lag distance. The zero crossings imply the presence of multiscale roughness. In such a way, we treat the rough surface to include a baseband correlation length and a modulation length. The modulation ratio, the ratio of baseband correlation length, and the modulated length determine the degree of multiscale roughness. We then examine the dependence of bistatic scattering, both in level and angular trends, on the modulation length. The results indicate that as the modulation ratio increases, the scattering coefficients reduce in the incident plane. Comparing the model predictions with two independent sets of measurement data demonstrates a good agreement of the scattering coefficients between the experimental data and the model predictions with modulation effects. The proposed modulated rough surface concept offers higher flexibility to model the radar scattering of multiscale rough surface.

This article presents a new index, polarization-conversion ratio (PCR) to characterize depolarized bistatic scattering from rough surfaces with dielectric inhomogeneity and spatial anisotropy. We then investigate the dependence of PCR on both surface and radar parameters. Numerical results show that the distribution of PCR on the scattering plane varies with the polarization state of the incident wave and incident angle. The PCR clusters more in the cross-plane for horizontally polarized incidence. However, for vertically polarized incidence, the PCR disperses as “triangular shape” on the whole scattering plane with a sharp valley occurring in the incident plane. The following points can be drawn: 1) the inhomogeneity effectively enhances the PCR in the cross-plane; 2) the effect of anisotropy on the PCR is relatively weak, because the scattering is less affected by correlation length; 3) the impacts of surface rms height on the PCR are negative on the whole scattering plane; and 4) as the background permittivity increases, at the horizontally polarized incidence, the PCR is enhanced in the backward and forward regions, while at vertically polarized incidence, it is enhanced in the incident plane and the forward region. As is demonstrated, the PCR is an effective measure of the sensitivity of depolarization, making it potentially useful as a new reliable index for surface parameter inversion.

This article attempts to examine the Brewster effect of incoherent scattering from a lossy inhomogeneous rough surface with a vertical dielectric profile. Five typical dielectric profiles are selected for the purpose of illustration. In calculating the reflection coefficients, a transition model is used to convert the angle of incidence to local incidence, which accounts for the surface roughness. Numerical results show that the Brewster angle gradually moves to the large incident angle with an increase in the background dielectric constant and in the surface root-mean-squared (rms) height. The angular dependence of reflection coefficients, both the level and the trend, is slightly affected by the correlation length. The scattering strength is much more sensitive to the rms height than to the correlation length. The results could be useful in the retrieval of vertical soil moisture content when proper radar observation is available.

Radar wavelength and observing geometry together determine the radar probing spectral width within which the Bragg resonance occurs. In this paper, we present numerical study of radar bistatic scattering from sea surface, and analyze the performance of bistatic radar on observing ocean surface. Bistatic radar in sensing the wave number range, wind speed, and wind direction are comprehensively discussed for general bistatic geometry. Moreover, the frequency dependence is also demonstrated. The results show that, compared to the mono-static radar, bistatic radar exhibits wider covering of the spectrum width, and can better catch the multi-scale marine dynamic process. In addition, bistatic radar, with receiver a slight shift off the specular angle, can capture the scattering of wind-generated high-frequency components of wave spectrum. Therefore, it is expected that bistatic observation provides greater sensing capability of wind field.

This paper presents full polarization wave scattering from a spatially anisotropic rough surface with inhomogeneous dielectric profile modeled by a transitional layer as a function of depth. The spatial anisotropy is described by directional correlation function. In this study, both linearly and circularly polarized scattering are investigated in light of azimuthal dependence, degree of anisotropy, and inhomogeneity. Numerical results show that the backscattering at small scattering angle exhibits strong dependence on degree of anisotropy, but turns to a strong azimuthal dependence at larger scattering angle. The bistatic scattering in the incidence plane reveals strong dependence on degree of anisotropy near the specular direction. In virtue of dielectric inhomogeneity, the scattering pattern reveals several distinct features including: HH polarization increases significantly in the backward region but decreases slightly in the forward region; the forward and backward scattering of VV polarization are enhanced; HV polarization can be greater than VH polarization; both LR and RR polarizations on the whole scattering plane are enhanced. Furthermore, backward scattering is stronger at 0°or 180° of azimuthal angle, but at 90° of azimuthal angle the forward scattering is stronger.

This paper examines the properties of bistatic scattering from an inhomogeneous rough surface, which, in this paper, is modeled by the transitional layer as a function of depth. The lower medium of the rough surface is horizontally uniform but vertically inhomogeneous. Both linear and circular polarizations are investigated in light of the dependences of transition rate, background dielectric constant, and surface roughness. The presence of dielectric inhomogeneity generally leads to several features that do not appear in the homogeneous surface, such as the scattering coefficient on the whole scattering plane is enhanced; the dynamic range of HH and VV over the azimuth plane is reduced; HV can be greater than VH; and the difference of LR and RR is decreased. With the increasing transition rate, the scattering coefficients for both the linear and circular polarizations are enhanced. As the background dielectric constant increases, the scattering responses of the linear and circular polarizations are quite different. For the linear polarization, HH exhibits a stronger angular dependence; VV reduces in the forward region and enhances notably in the backward region; and HV decreases but VH increases. For circular polarizations, the cross-polarized LR increases in the backward region but decreases in the forward region, and the copolarized RR enhances on the whole scattering plane. With the increasing surface roughness, the scattering coefficient becomes more evenly distributed over the entire scattering plane.

This paper examines the polarized backscattering of spatially anisotropic rough surfaces. To better explore the physical mechanisms that control the azimuthal dependence of the backscattering from anisotropic surfaces, the effects of surface roughness [correlation length and root-mean-square (rms) height], dielectric constant, and radar parameters from anisotropic surfaces are studied. The advanced integral equation model (AIEM) is used to simulate both co- and cross-polarized backscattering coefficients, including the single and multiple scattering. Numerical results suggest that the multiple scattering exhibits a stronger azimuthal dependence for HH than VV polarization, especially more so at a larger incident angle. For weakly anisotropic surface, the azimuthal variation of backscattering tends to be a sinusoidal-like pattern. However, with the enhancement of anisotropy, such a scattering pattern is distorted, and the sharp dip appears at up/down direction. As the rms height and dielectric constant increase, the scattering is enhanced on the whole. The HH/VV ratio at lower dielectric constant is greater than that at higher one. In comparison, scattering shows stronger dependence on anisotropy at lower dielectric constant, especially at a larger incident angle. As an application example, we compare the model predictions with reported measurements from two different sites. Preliminary results are quite encouraging, and thus, the analysis presented in this paper is potentially useful to predict and interpret backscattering from crop field surface, where strong anisotropic surfaces commonly present due to plowing or raking practice.

This paper presents bistatic scattering from rough surface with an inhomogeneous layer where the permittivity is a transitionary function of depth. For reference, both homogeneous and inhomogeneous media are discussed in co- and cross polarization with various background dielectric constant and transition rate. Results show that the angular curves become more slowly in azimuthal direction in bistatic scattering, where the minimum dips are smearing out compared to those in homogeneous layer. The azimuthal dip appearing in the case of homogeneous surface starts to vanish for inhomogeneous surface, and a valley-like azimuthal curve is observed in both HH and VV polarizations, though the valley angles are different. To a great extent, the dynamic range of the scattering coefficient is reduced, particularly for VV polarization. Moreover, as for inhomogeneous medium, the HV polarized scattering coefficients are greater than VH polarization, which is opposite to the homogeneous medium. But for larger background dielectric constant, HV- and VH- polarized scattering coefficients look more symmetrical and VH polarization is relatively larger.

This paper presents a multiple scattering in the AIEM model. The derivation is mathematically intricate, though, the final expression is compact and easy for numerical implementation, which only involves series of two-dimensional integration. Some of special cases in backscattering are derived and compared with known analytical model to partly validate the update AIEM model in cross polarization. Finally, comparisons with numerical simulation and field measurements are made to show the model performance and accuracy, particularly in cross-polarized backscattering.

This paper presents a comprehensive reappraisal of the scattering, both backscattering and bistatic scattering, and emission of rough soil surface predicted by a well-established theoretical model, the advanced integral equation model (AIEM). Extensive numerical data simulated by approximate and numerically exact models as well as experimental datasets of well-characterized bare soil surfaces were used to evaluate the performance of AIEM in predicting the scattering coefficient and microwave emissivity over a wide range of geometric parameters and ground surface conditions. The results show that AIEM predictions are generally in good consistency with both of numerical simulations and experiment measurements in terms of angular, frequency and polarization dependences, except for some deviations in a few cases (e.g. at large incident angles and dry soil conditions). Possible explanations for the discrepancy between model prediction and data are given, together with suggestions for model usage and refinements.