Integral Equation Model Research Articles

A Simulation Study of GNSS-R Polarimetric Scattering from the Bare Soil Surface Based on the AIEM

In the past two decades, global navigation satellite system-reflectometry (GNSS-R) has emerged as a new remote sensing technique for soil moisture monitoring. Some experiments showed that the antenna of V polarization is more favorable to receive the reflected signals, and the interference pattern technique (IPT) was used for soil moisture and retrieval of other geophysical parameters. Meanwhile, the lower satellite elevation angles are most impacted by the multipath. However, electromagnetic theoretical properties are not clear for GNSS-R soil moisture retrieval. In this paper, the advanced integral equation model (AIEM) is employed using the wave-synthesis technique to simulate different polarimetric scatterings in the specular directions. Results show when the incident angles are larger than 70°, scattering at RR polarization (the transmitted signal is right-hand circular polarization (RHCP), while the received one is also RHCP) is larger than that at LR polarization (the transmitted signal is RHCP, while the received one is left-hand circular polarization (LHCP)), while scattering at LR polarization is larger than that at RR polarization for the other incident angles (1°∼70°). There is an apparent dip for VV and VR scatterings due to the Brewster angle, which will result in the notch in the final receiving power, and this phenomenon can be used for soil moisture retrieval or vegetation corrections. The volumetric soil moisture (vms) effects on their scattering are also presented. The larger soil moisture will result in lower scattering at RR polarization, and this is very different from the scattering of the other polarizations. It is interesting to note that the surface correlation function only affects the amplitudes of the scattering coefficients at much less level, but it has no effects on the angular trends of RR and LR polarizations.

Soil roughness retrieval from TerraSar-X data using neural network and fractal method

The purpose of this study is to estimate the surface roughness (rms) using TerraSar-X data in HH polarization. Simulation of data is carried out at a wide range of moisture and roughness using the Integral Equation Model (IEM). The inversion method is based on Multi-Layer Perceptron neural network. Inversion technique is performed in two steps. In the first step, the neural network is trained using synthetic data. The inputs of the first neural network are the backscattering coefficient and incidence angle, and the moisture is the output. In the next step, three neural networks are built based on a prior and without prior information on roughness. The inputs of three neural network are backscattering coefficient, estimated moisture in the first step and incidence angle and the roughness is output. The validation of the proposed methods is carried out based on synthetic and real data. Ground roughness measurements are extracted from Digital Terrain Model (DTM) using the fractal method. The accuracy of moisture from synthetic data is 6.1 vol% without prior information on moisture and roughness. The roughness (rms) accuracy of synthetic datasets is 0. 61 cm without prior information and is 0.31 cm and 0.38 cm for rms lower than 2 cm and rms between 2 and 4 cm, with prior information on roughness. The result's analysis of the simulated data showed that the prior information on roughness strongly improves the accuracy of roughness and moisture estimates. The accuracy of rms estimates for the TerraSar-X image in the HH polarization is about 0.9 cm in the case of no prior information on roughness. The accuracy improves to 0.57 cm for rms lower than 2 cm and 0.54 cm for rms between 2 and 4 cm with prior information on roughness. An overestimation of rms for rms lower than 2 cm and an underestimation of rms for rms higher than 2 cm are observed. The results of the accuracy of the synthetic and real data showed that the X band in HH polarization has a very good potential to estimate the soil roughness.

Effect of vegetation index choice on soil moisture retrievals via the synergistic use of synthetic aperture radar and optical remote sensing

The recent launch of the Sentinel-1 A and Sentinel-1B synthetic aperture radar (SAR) satellite constellation has provided high-quality SAR data with fine spatial and temporal sampling characterizations (6˜12 revisit days at 10 m spatial resolution). When combined with high-resolution optical remote sensing, this data can potentially be used for high-resolution soil moisture retrieval over vegetated areas. However, the suitability of different vegetation index (VI) types for the parameterization of vegetation water content in SAR vegetation scattering models requires further investigation. In this study, the widely-used physical-based Advanced Integral Equation Model (AIEM) is coupled with the Water Cloud Model (WCM) for the retrieval of field-scale soil moisture. Three different VIs (NDVI, EVI, and LAI) produced by two different satellite sensors (Moderate Resolution Imaging Spectroradiometer (MODIS) and Landsat) are selected to examine their impact on the parameterization of vegetation water content, and subsequently, on soil moisture retrieval accuracy. Results indicate that, despite the different sensitivity of estimated surface roughness parameters to various VIs (i.e., this sensitivity is highest when utilizing MODIS EVI and lowest in the LAI-based model), the optimum roughness parameters derived from each VI exhibit no discernible difference. Consequently, the soil moisture retrieval accuracies show no noticeable sensitivity to the choice of a particular VI. Generally, meadow and grassland sites with small differences in VI-derived roughness parameters exhibit good performance in soil moisture estimation. With respect to the relative components in the coupled model, the vegetative contribution to the scattering signal exceeds that of soil at VI about 0.6∼0.8 [-] in NDVI-based models and 0.4∼0.6 [-] in EVI-based models. This study provides insight into the proper selection of vegetation indices during the use of SAR and optical imagery for the retrieval of high-resolution surface soil moisture.

Momentum Conservative Scheme for Simulating Wave Runup and Underwater Landslide

This paper focuses on the numerical modelling and simulation of tsunami waves triggered by an underwater landslide. The equation of motion for water waves is represented by the Nonlinear Shallow Water Equations (NSWE). Meanwhile, the motion of underwater landslide is modeled by incorporating a term for bottom motion into the NSWE. The model is solved numerically by using a finite volume method with a momentum conservative staggered grid scheme that is proposed by Stelling & Duinmeijer 2003 [12]. Here, we modify the scheme for the implementation of bottom motion. The accuracy of the implementation for representing wave runup and rundown is shown by performing the runup of harmonic wave as proposed by Carrier & Greenspan 1958 [2], and also solitary wave runup of Synolakis, 1986 [14], for both breaking and non-breaking cases. For the underwater landslide, result of the simulation is compared with simulation using the Boundary Integral Equation Model (BIEM) that is performed by Lynett and Liu, 2002 [9].

Nonlocal nonlinear model of Bernoulli–Euler nanobeam with small initial curvature and its application to single-walled carbon nanotubes

Although the small-scale effect and initial curvature have significant influence on the nonlinear mechanical property of the nanobeam, there are few researches to consider both factors. On the basis of the nonlocal differential constitutive relation, this paper, for the first time, presents a nonlinear partial differential–integral equation model of Bernoulli–Euler nanobeam with a small initial curvature. Then the model is applied to research the static bending and the frequency of free vibration for the single-walled carbon nanotubes (SWCNTs). This study indicates that the static deformations are relate to the load direction due to the initial curvature. In addition, the initial curvature complicates the relation between the free vibration frequencies and vibration amplitudes.

Fractal algorithm for surface roughness parameters retrieval using multiband/polarization AIRSAR data

Surface roughness is an important characteristic in analyzing synthetic aperture radar images and three-dimensional (3-D) surfaces that influence the backscattering of electromagnetic waves. We propose an algorithm based on a fractal method for retrieving parameters of a surface. To estimate surface roughness parameters, 10,000 different 3-D surfaces with different rms-height and correlation length are simulated. 3-D simulation of surfaces is carried out based on fractal Brownian motion. The results show that the fractal method represents a good relationship between roughness parameters and fractal dimension. It can be seen that, in some cases, surfaces with different roughness have the same fractal dimension. Roughness index (RI) can be used as a complement to fractal dimension. We present empirical relationships among fractal dimension, roughness parameters, and RI. The method is implemented on L and C bands with HH and HV polarizations of AIRSAR data at different dates. The comparison between field measurement and estimated roughness showed that the accuracy of soil surface roughness estimation for band L with HH polarizations is better than bands L and C with polarizations HV and HH, respectively. The results of this method also are compared with the roughness estimates using the integral equation model (IEM). The analysis of outputs shows that the roughness estimation using the fractal and IEM is very similar in the low moisture at L band in HH polarization. The root mean square error of roughness for data at L band in HH polarization on July 1, 2002, is 0.54 and 0.53 cm for fractal and IEM, respectively.

Fast Microwave Through Wall Imaging Method With Inhomogeneous Background Based on Levenberg–Marquardt Algorithm

In this paper, a fast solution for microwave through wall imaging (TWI) with nonlinear inversion is proposed to reconstruct the unknown targets embedded in an inhomogeneous background medium. We treat inhomogeneous background, i.e., the wall around bounded in a finite domain as a known scatterer, which has the advantage of avoiding the time-consuming calculation of inhomogeneous background Green's function. Under this scheme, a new approach under the framework of difference integral equation model, i.e., difference Lippmann-Schwinger integral equation, with modified enhanced Levenberg-Marquardt algorithm is proposed. In particular, we used a hybrid regularized technique, i.e., generalized cross-validation and truncated singular value decomposition, to stabilize the inversion. It is shown that the proposed method runs fast and is stable in presence of noise. Also, it is able to alleviate the nonlinearity and reconstruct unknown scatterers of high contrast with respect to the background. Both the numerical and experimental TWI tests validate the efficiency of the proposed inversion method.

An Improved Approach for Soil Moisture Estimation in Gully Fields of the Loess Plateau Using Sentinel-1A Radar Images

As an essential ecological parameter, soil moisture is important for understanding the water exchange between the land surface and the atmosphere, especially in the Loess Plateau (China). Although Synthetic Aperture Radar (SAR) images can be used for soil moisture retrieval, it is still a challenge to mitigate the impacts of complex terrain over hilly areas. Therefore, the objective of this paper is to propose an improved approach for soil moisture estimation in gully fields based on the joint use of the Advanced Integral Equation Model (AIEM) and the Incidence Angle Correction Model (IACM) from Sentinel-1A observations. AIEM is utilized to build a simulation database of microwave backscattering coefficients from various radar parameters and surface parameters, which is the data basis for the retrieval modeling. IACM is proposed to correct the deviation between the local incidence angle at the scatterer and the radar viewing angle. The study area is located in the Loess Plateau of China, where the main land cover is mostly bare land and the terrain is complex. The Sentinel-1A SAR data in C-band with dual polarization acquired on October 19th, 2017 was adopted to extract the VV&VH polarimetric backscattering coefficients. The in situ measurements of soil moisture were collected on the same day of the SAR acquisition, for evaluating the accuracy of the SAR-derived soil moisture. The results showed that, firstly, the estimated soil moisture with volumetric content between 0% and 20% was in the majority. Subsequently, both the RMSE of estimation values (0.963%) and the standard deviation of absolute errors (0.957%) demonstrated a good accuracy of the improved approach. Moreover, the evaluation of IACM confirmed that the improved approach coupling IACM and AIEM was more efficient than employing AIEM solely. In conclusion, the proposed approach has a strong ability to estimate the soil moisture in the gully fields of the Loess Plateau from Sentinel-1A data.

Soil Moisture Retrival Based on Sentinel-1 Imagery under Sparse Vegetation Coverage.

Soil moisture is an important aspect of heat transfer process and energy exchange between land-atmosphere systems, and it is a key link to the surface and groundwater circulation and land carbon cycles. In this study, according to the characteristics of the study area, an advanced integral equation model was used for numerical simulation analysis to establish a database of surface microwave scattering characteristics under sparse vegetation cover. Thus, a soil moisture retrieval model suitable for arid area was constructed. The results were as follows: (1) The response of the backscattering coefficient to soil moisture and associated surface roughness is significantly and logarithmically correlated under different incidence angles and polarization modes, and, a database of microwave scattering characteristics of arid soil surface under sparse vegetation cover was established. (2) According to the Sentinel-1 radar system parameters, a model for retrieving spatial distribution information of soil moisture was constructed; the soil moisture content information was extracted, and the results were consistent with the spatial distribution characteristics of soil moisture in the same period in the research area. (3) For the 0–10 cm surface soil moisture, the correlation coefficient between the simulated value and the measured value reached 0.8488, which means that the developed retrieval model has applicability to derive surface soil moisture in the oasis region of arid regions. This study can provide method for real-time and large-scale detection of soil moisture content in arid areas.

Domain decomposition based integral equation modeling of 3-dimensional topography in frequency domain for well electromagnetic field

As an efficient geophysical exploration technology, well electromagnetic method is particularly applicable to oil and gas exploration in China's complex terrain areas (deserts, mountains, etc.). A serious influence of topographic relief area on the electromagnetic response of well is inevitable but challenging. To the best of our knowledge, there is no literature on modeling the electromagnetic response of three-dimensional (3D) topography with well electromagnetic method. Based on the domain decomposition, an integral equation method is presented to simulate the electromagnetic response of 3D topography in frequency domain via the well electromagnetic method. Compared with the finite difference and finite element method based on partial differential equation, this method is very efficient in simulating topographic response without huge computation or truncation boundary error accumulation or special boundary condition requirements. Firstly, an induction coefficient is defined according to the topographic relief situation. Then the computational domain consisting of the target body, background medium and 3D topography is divided into reference model, background medium and the distribution of target body medium area. According to the characteristics of each sub-region, Anderson algorithm is an analytic solution based on Gaussian filtering, which is used to provide the primary field from the excited sources in surface. And then, the stable double conjugate gradient-fast Fourier transform is incorporated into integral equation algorithm to obtain the fast 3D terrain shaft frequency domain electromagnetic responses. By comparing the calculation results using the new algorithm presented in this paper with the analytical solutions of Anderson algorithm for half-space model with surface electromagnetic method, the precision and the efficiency of this new algorithm are demonstrated. And the ability to model the electromagnetic responses of 3D topography is shown by comparing with the published results of 3D boundary integral equation. Thus, the high accuracy and high efficiency of the new algorithm presented in this paper are validated. Finally, the influence of 3D valley topography on electromagnetic field response of surface to borehole electromagnetic (SBEM) observation system is presented and analyzed. It is observed that the response of SBEM is seriously disturbed by the field of 3D valley topography which is necessarily removed. The research results presented in this paper are of significance for guiding the identification and correction of electromagnetic topographic effect from 3D SBEM.

A Coupling Model for Soil Moisture Retrieval in Sparse Vegetation Covered Areas Based on Microwave and Optical Remote Sensing Data

Soil moisture is an important component of natural water cycles and plays a crucial role for the healthy development of ecological environment in arid and semiarid areas. Microwave remote sensing techniques are promising to rapidly monitor regional soil moisture. However, major difficulties associated with retrieving soil moisture via microwave remote sensing are attributed to effects of surface roughness and vegetation cover. The objective of this paper is to investigate the potentials of combined roughness parameters and develop a model that mostly relies on the satellite data and requires minimum a priori information for soil moisture inversion over sparse vegetation covered areas through combing Radarsat-2 synthetic aperture radar (SAR) and GF-1 data. For the purpose, the impacts of vegetation on the radar backscattering coefficient were removed by the water-cloud model (WCM) and a new roughness parameter $Zs$ ( $Zs = S^{3}/L$ is defined as a combined roughness parameter) was proposed by simulating relationships between surface roughness parameters, soil moisture, and backscattering coefficients using advanced integral equation model (AIEM). On the basis, a coupling model of soil moisture inversion for blown-wind areas of the Uxin Banner in Inner Mongolia, China, was in turn developed with Radarsat-2 HH and VV polarization data, and the soil moisture content (SMC) values for sparse vegetation covered surfaces were retrieved with limited use of in situ roughness parameters. The in situ measurements and satellite data set were used to validate the reliability of the developed model. The results showed that the retrieved soil moisture levels fell below in situ soil moisture levels before the vegetation effect was removed, and the precision of retrieved soil moisture was effectively improved when the vegetation effect was corrected by the WCM, with the root-mean-square error and mean absolute error decreasing from 7.45% and 6.24% to 5.12% and 3.44%, respectively. According to the map of retrieved soil moisture levels generated for the study area, most of SMCs were below 35% in accordance with field observations. These study results can serve as a foundation for monitoring soil moisture and water environments in arid and semiarid regions.

A Higher-Order Polynomial Method for SPECT Reconstruction.

Existing single-photon emission computed tomography (SPECT) reconstruction methods are mostly based on discrete models that may be viewed as piecewise constant approximations of a continuous data acquisition process. Due to low accuracy order of piecewise constant approximations, a traditional discrete model introduces irreducible model errors which are a bottleneck of the quality improvement of reconstructed images in clinical applications. To overcome this drawback, we develop a higher-order polynomial method for SPECT reconstruction. Specifically, we represent the data acquisition of SPECT imaging by using an integral equation model, approximate the solution of the underlying integral equation by higher-order piecewise polynomials leading to a new discrete system and introduce two novel regularizers for the system, by exploring the a priori knowledge of the radiotracer distribution, suitable for the approximation. The proposed higher-order polynomial method outperforms significantly the cutting edge reconstruction method based on a traditional discrete model in terms of model error reduction, noise suppression, and artifact reduction. In particular, the coefficient of variation of images reconstructed by the piecewise linear polynomial method is reduced by a factor of 10 in comparison to that of a traditional discrete model-based method.

GEOPHYSICAL PARAMETER RETRIEVAL FOR MICROWAVE C BAND SYNTHETIC APERTURE RADAR (SAR) DATASET USING INTEGRAL EQUATION MODEL

Abstract. The microwave remote sensing is highly useful, as it provides synoptic observation of the Earth’s surface or planetary bodies, regardless of day or night and the atmospheric conditions, propagation through ionosphere with minimum loss. One of the best microwave technology for imaging system is the Synthetic Aperture Radar (SAR) remote sensing. The microwave SAR currently represents the best approach for obtaining spatially distributed geophysical parameter present on the Earth’s surface or planetary bodies. In the present work, geophysical parameters viz., Soil Moisture, Surface Roughness, Dielectric Constant (ε) and Backscattering Coefficients (σ0) will be retrieved. The modelling makes the process of estimating information beyond the real observation range for data interpretation. In the present paper most widely used modelling techniques for the microwave SAR dataset is an Integral Equation Model (IEM) which is implemented for above said geophysical parameters retrieval. The aim of the present work is to estimate accurate, reliable and skillful measurements of geophysical parameters from the microwave SAR dataset. In the present study microwave C band SAR dataset is used. The overall processing was done by using PolSARPro Ver. 5.0 software. In the present work, geophysical parameters are measured with the help IEM modelling, the statistical parameter and occurrence plane, estimated from the microwave SAR image, which was very helpful for retrieving geophysical parameters. From the overall paper work, it was concluded that the IEM modelling is a one of the realistic modelling methods for retrieving geophysical parameters for microwave C band SAR dataset.

Theoretical Study of Global Sensitivity Analysis of L-Band Radar Bistatic Scattering for Soil Moisture Retrieval

This letter explores the optimal bistatic radar configurations for bare soil moisture retrieval at L-band using a global sensitivity analysis method, the extended Fourier amplitude sensitivity test (EFAST) algorithm. Complete sets of bistatic scattering, covering a wide range of geometric parameters and ground surface conditions, are simulated by the well-established advanced integral equation model. The sensitivity of radar bistatic signals to soil moisture and surface roughness, and the interactions among the parameters are quantified using the EFAST algorithm. The results show that in bistatic scattering, VV polarization has notably higher sensitivity to soil moisture than HH polarization, particularly at large incident angles. For VV polarization, as incident angle increases, the sensitivity zone of soil moisture expands and shifts toward the forward direction, specifically at small azimuth scattering angles and large scattering angles, thereby becoming promising configurations for soil moisture retrieval. For HH polarization, in contrast, the sensitive zone gradually moves to the backward direction as incident angle increases, and an intermediate incident angle (e.g., 40°) is recommended for retrieving soil moisture by considering both sensitivity strength and parameter interaction effects.

An Integral Equation Model for a Pile in a Layered Transversely Isotropic Saturated Soil

Objective:In this paper, an integral equation model is established to predict the time-dependent response of a vertically loaded pile embedded in a layered Transversely Isotropic Saturated Soil (TISS).Methods:Based on the fictitious pile method, the pile-soil system is decomposed into an extended saturated half-space and a fictitious pile. The extended half-space is treated as a layered TISS, while the fictitious pile is considered as a 1D bar. The pile-soil compatibility is accomplished by requiring that the axial strain of the fictitious pile be equal to the vertical strain of the extended layered TISS along the axis of the pile. The second kind Fredholm integral equation of the pile is then derived by using the aforementioned compatibility condition and the fundamental solution of the layered TISS, which is equivalent to the solution of the layered TISS subjected to a uniformly-distributed load acting vertically over a circular area with the radius equal to that of the pile. The fundamental solution of the layered TISS is obtainedviathe Reflection-Transmission Matrix (RTM) method for the layered TISS. Applying the Laplace transform to the Fredholm integral equation, and solving the resulting integral equation, the transformed solution is obtained. The time domain solution of the pile-soil system is retrievedviathe inverse Laplace transform.Results and Conclusion:Numerical results of this paper agree with existing solutions very well, validating the proposed pile-soil interaction model. A parametric study is carried out to examine the influence of some parameters on the response of the pile-soil system.

Combined Use of GF-3 and Landsat-8 Satellite Data for Soil Moisture Retrieval over Agricultural Areas Using Artificial Neural Network

Soil moisture is the basic condition required for crop growth and development. Gaofen-3 (GF-3) is the first C-band synthetic-aperture radar (SAR) satellite of China, offering broad land and ocean imaging applications, including soil moisture monitoring. This study developed an approach to estimate soil moisture in agricultural areas from GF-3 data. An inversion technique based on an artificial neural network (ANN) is introduced. The neural network was trained and tested on a training sample dataset generated from the Advanced Integral Equation Model. Incidence angle and HH or VV polarization data were used as input variables of the ANN, with soil moisture content (SMC) and surface roughness as the output variables. The backscattering contribution from the vegetation was eliminated using the water cloud model (WCM). The acquired soil backscattering coefficients of GF-3 and in situ measurement data were used to validate the SMC estimation algorithm, which achieved satisfactory results (R2 = 0.736; RMSE = 0.042). These results highlight the contribution of the combined use of the GF-3 synthetic-aperture radar and Landsat-8 images based on an ANN method for improving SMC estimates and supporting hydrological studies.

Soil Moisture Retrieval from the Chinese GF-3 Satellite and Optical Data over Agricultural Fields.

Timely and accurate soil moisture information is of great importance in agricultural monitoring. The Gaofen-3 (GF-3) satellite, the first C-band multi-polarization synthetic-aperture radar (SAR) satellite in China, provides valuable data sources for soil moisture monitoring. In this study, a soil moisture retrieval algorithm was developed for the GF-3 satellite based on a backscattering coefficient simulation database. We adopted eight optical vegetation indices to determine the relationships between these indices and vegetation water content (VWC) by combining Landsat-8 data and field measurements. A backscattering coefficient database was built using an advanced integral equation model (AIEM). The effects of vegetation on backscattering coefficients were corrected using the water cloud model (WCM) to obtain the bare soil backscattering coefficient (). Then, soil moisture retrievals were obtained at HH, VV and HH+VV combination respectively by minimizing the observed bare soil backscattering coefficient () and the AIEM-simulated backscattering coefficient (). Finally, the proposed algorithm was validated in agriculture region of wheat and corn in China using ground soil moisture measurements. The results showed that the normalized difference infrared index (NDII) had the best fit with measured VWC values (R = 0.885) among the eight vegetation water indices; thus, it was adopted to correct the effects of vegetation. The proposed algorithm using GF-3 satellite data performed well in soil moisture retrieval, and the scheme combining HH and VV polarization exhibited the highest accuracy, with a root mean square error (RMSE) of 0.044 m3m−3, followed by HH polarization (RMSE = 0.049 m3m−3) and VV polarization (RMSE = 0.053 m3m−3). Therefore, the proposed algorithm has good potential to operationally estimate soil moisture from the new GF-3 satellite data.

Improved discrimination of geological units via geomorphological classification of synthetic aperture radar images

Geological mapping usually requires field visits, but the processing and interpretation of hyperspectral satellite images can also be very beneficial for this task. Microwave or radar remote sensing can obtain surface morphologies using synthetic aperture radar (SAR) images and reduce the duration of field visits dramatically by discriminating geological units based on lithology and texture. This requires the surface roughness to be modeled against microwave signal backscattering. The integral equation model (IEM) is the most well-known rough scattering model, in which surface roughness is calculated using the roughness height statistical parameter (rms-height); however, this study uses an improved IEM based on power-law geometry. The roughness map of the Anaran anticline (located between Dehloran and Ilam in Iran) using TerraSAR images is computed and classified to generate a morphological map. Calculating the roughness map requires training the IEM model and the formation of the look-up-table for pure lithological sites. In situ microtopography measurement was performed on seven different sites containing the main lithologies in the study area, using total station surveying, to train the mathematical model and compare and evaluate the results. Comparing this roughness map with ground-truth data at test sites indicates that computations using the new IEM model results in a misclassification of <10 % of the samples in the map. This error is acceptable, indicating that the new model could, in many cases, reduce the duration of field visits.

A MODIFIED TWO-SCALE MICROWAVE SCATTERING MODEL FOR A GAUSSIAN-DISTRIBUTED CONDUCTING ROUGH SURFACE

Abstract. A modified two-scale microwave scattering model (MTSM) was presented to describe the scattering coefficient of natural rough surface in this paper. In the model, the surface roughness was assumed to be Gaussian so that the surface height z(x, y) can be split into large-scale and small-scale components relative to the electromagnetic wavelength by the wavelet packet transform. Then, the Kirchhoff Model (KM) and Small Perturbation Method (SPM) were used to estimate the backscattering coefficient of the large-scale and small-scale roughness respectively. Moreover, the ‘tilting effect’ caused by the slope of large-scale roughness should be corrected when we calculated the backscattering contribution of the small-scale roughness. Backscattering coefficient of the MTSM was the sum of backscattering contribution of both scale roughness surface. The MTSM was tested and validated by the advanced integral equation model (AIEM) for dielectric randomly rough surface, the results indicated that, the MTSM accuracy were in good agreement with AIEM when incident angle was less than 30° (θi &lt;30°) and the surface roughness was small (ks = 0.354).

A method to obtain soil-moisture estimates over bare agricultural fields in arid areas by using multi-angle RADARSAT-2 data

Soil moisture is an important parameter for agriculture, meteorological, and hydrological studies. This paper focuses on soil-moisture estimation methodology based on the multi-angle high- and low-incidence-angle mode RADARSAT-2 data obtained over bare agricultural fields in an arid area. Backscattering of the high- and low-incidence angles is simulated by using AIEM (advanced integral equation model), with the surface-roughness estimation model built based on the simulated data. Combining the surface-roughness estimation model with the backscattering model of the low-incidence-angle mode, a soil-moisture estimation method is put forward. First, the natural logarithm (ln) of soil moisture was obtained and then the soil moisture calculated. Soil moisture of the study area in Dunhuang, Gansu Province, was obtained based on this method; a good agreement was observed between the estimated and measured soil moisture. The coefficient of determination was 0.85, and the estimation precision reached 4.02% in root mean square error (RMSE). The results illustrate the high potential of the approach developed and RADARSAT-2 data to monitor soil moisture.