Integral Equation Model Research Articles

Fast integral equation algorithms for the solution of electromagnetic wave propagation over general terrains

In this paper a fast numerical algorithm to solve an integral equation model for wave propagation along a perfectly conducting two-dimensional terrain is suggested. It is applied to different actual terrain profiles and the results indicate very good agreement with published work. In addition, the proposed algorithm has achieved considerable saving in processing time. The formulation is extended to solve the propagation over lossy dielectric surfaces. A combined field integral equation (CFIE) for wave propagation over dielectric terrain is solved efficiently by utilizing the method of moments with complex basis functions. The numerical results for different cases of dielectric surfaces are compared with the results of perfectly conducting surface evaluated by the IE conventional algorithm.

An Approximate Hybrid Method for Electromagnetic Scattering From an Underground Target

A hybrid method for modeling marine controlled-source electromagnetics, simplified integral equation (IE) (SIE) modeling, has shown very promising results in 2-D. The computational gain of SIE is very large for large problems in 3-D. We assess the accuracy and range of validity of SIE modeling in 3-D through order-of-magnitude analysis and through an extensive numerical comparison with rigorous IE modeling. A previously proposed order-of-magnitude analysis results in a dimensionless parameter which is easy to use as an a priori indicator for when SIE is valid. Unfortunately, this parameter is found to be sometimes inconsistent with our numerical results. Order-of-magnitude analysis for Maxwell's equations is then reassessed in an attempt to rectify the shortcomings of the parameter adapted from the work “Electrical Impedance Tomography” by Cheney The dimensionless parameter resulting from the novel order-of-magnitude analysis is found to have generally good predictive capability. Unfortunately, this parameter is not suitable for deciding a priori if the use of SIE is justified for a particular case, since it depends on numerical results from IE modeling. For future use of SIE, it is therefore recommended to compare the problem characteristics of the case at hand with those covered by the extensive numerical comparison in this paper. From the numerical investigation, it is found that the accuracy of SIE is very good for resistive targets and for frequencies lower than about 10 Hz. For conductive targets, the accuracy is mostly very good for frequencies lower than about 5 Hz but somewhat more dependent also on other problem characteristics.

Applying integral equation modeling technique in determination of charge distribution on conducting structures

We use, in this article, the integral equation technique for modeling the problem of determination of charge distribution on conducting structures. Such problems may be modeled by Fredholm integral equations and, in general, have no analytical solution. Hence, an appropriate computational approach should be used for obtaining the approximate solutions. For this purpose, an effective collocation method is used and two conducting structures are analyzed by it. Numerical results are given to illustrate the computational efficiency of the method.

Comparison of the multi-layer HUT snow emission model with observations of wet snowpacks

Information on snow properties plays an important role in hydrological, meteorological and climatological applications. Passive microwave remote sensing is an effective method to retrieve snowpack parameters; however, the observations can be obscured if there is wet snow in the satellite footprint. To study the emission properties of wet snow and check its response to snow wetness, this paper applies the multi-layer Helsinki University of Technology (HUT) snow emission model coupled with the Advanced Integral Equation Model to simulate the low-wetness snowpack observed at Luancheng in November 2009, and the high wetness snowpack observed at Weissfluhjoch in June 1995. Input parameters are acquired by the in-situ snow pit measurements, while the snow grain size is fitted by comparing model predictions with the observed passive microwave signals at a range of observing angles. Results show that the application of a multi-layer model is capable to consider the distribution pattern of the snow wetness along the snow profile and the refrozen ice crust of the snow surface. The multi-layer HUT model is able to reproduce the wet snow emission properties, with an rms error of 4.4 K (at Luancheng) and 5.7 K (at Weissfluhjoch) at vertical polarization, and an rms error of 7.9 K (at Luancheng) and 11.4 K (at Weissfluhjoch) at horizontal polarization. Copyright © 2012 John Wiley & Sons, Ltd.

A Potential Use for the C-Band Polarimetric SAR Parameters to Characterize the Soil Surface Over Bare Agriculture Fields

The objective of this study was to analyze the potential of the C-band polarimetric synthetic aperture radar (SAR) parameters for the soil surface characterization of bare agricultural soils. RADARSAT-2 data and simulations using the integral equation model were analyzed to evaluate the polarimetric SAR parameters' sensitivities to the soil moisture and surface roughness. The results showed that the polarimetric parameters in the C-band were not very relevant to the characterization of the soil surface over bare agricultural areas. Low dynamics were often observed between the polarimetric parameters and both the soil moisture content and the soil surface roughness. These low dynamics do not allow for the accurate estimation of the soil parameters, but they could augment the standard inversion approaches to improve the estimation of these soil parameters. The polarimetric parameter α1 could be used to detect very moist soils (>; 30%), while the anisotropy could be used to separate the smooth soils.

A Comparison of IEM and SPM Model for Oil Spill Detection Using Inversion Technique and Radar Data

The purpose of this work is to verifying L band wave behavior for oil spill detection over oceans from satellite remote sensing sensors. Moreover, evaluation of the optimum theoretical backscattering model for this application is considered. This objective was realized by performing an inversion technique which uses neural networks and backscattering model to estimate ocean surface parameters like roughness and dielectric constant. The normalized measure of the radar signal that backscattered to the antenna (sigma zero) includes specific characteristics of the target such as roughness and dielectric constant. Although, in radar imagery, surface roughness is the dominant factor in determining the amplitude of the return signal, this study, most often focus on the dielectric properties because there is a significant difference between dielectric constant of water and oil and therefore they can be distinguished easily by estimating dielectric constant parameter. The neural networks were first trained with a simulated data set generated from the integral equation model (IEM) and second with metadata from small perturbation model (SPM). It used a fast learning algorithm for training a multilayer feed forward neural network. A theoretical database was used for the learning stage. The proposed approach was applied with ALOS-PALSAR images from the Philippine Sea and output are presented in binary images.

Spatial and temporal soil moisture estimation from RADARSAT-2 imagery over Flevoland, The Netherlands

► Assess the ability of RADARSAT-2 for soil moisture retrieval over Flevoland. ► Evaluate both spatial and temporal soil moisture retrieval techniques. ► Circumvent surface roughness field measurements. ► Spatial and temporal retrievals have accuracy around 4 vol% for medium rough soils. ► Temporal retrieval performs better for smoother fields covered by stubbles. Both spatial and temporal soil moisture dynamics have a large impact on hydrologic processes in agricultural catchments. As acquiring ground measurements of soil moisture is labor intensive, it is often limited in space (to a few locations) and time (to a small number of campaigns). Remote sensing offers a useful alternative, as it allows for observing both across time and space. This study analyses the potential to retrieve spatial and temporal soil moisture information from a series of RADARSAT-2 HH- and VV-polarized C-band (5.405 GHz) backscatter observations over a large number of (nearly) bare soil agricultural fields in Flevoland, The Netherlands, acquired in the framework of AgriSAR 2009. The RADARSAT-2 backscatter observations are found to be strongly correlated with continuous soil moisture measurements recorded at two agricultural sites during the 2009 growing season, as well as with spatial soil moisture measurements conducted during three intensive field campaigns in August and September 2009. Furthermore, two different soil moisture retrieval approaches have been evaluated: a physically-based modeling approach and change detection technique. The physically-based approach makes use of the Integral Equation Model with input of effective surface roughness parameters that are updated every acquisition for each field. The soil moisture retrieval accuracy of this technique is calculated based on leave-one-out cross-validation, yielding an accuracy (RMSE) about 4 vol% for fields with medium surface roughness at both polarization schemes. For smoother fields covered by stubbles, the retrieval accuracy slightly deteriorates. The change detection technique relies on a normalization of the backscatter between dry and wet reference soil moisture measurements, yielding accuracies about 4 vol% both for fields characterized by medium roughness and stubble conditions. This study confirms the large potential of RADARSAT-2 data for the retrieval of spatial and temporal soil moisture dynamics.

Estimation of soil parameters over bare agriculture areas from C-band polarimetric SAR data using neural networks

Abstract. The purpose of this study was to develop an approach to estimate soil surface parameters from C-band polarimetric SAR data in the case of bare agricultural soils. An inversion technique based on multi-layer perceptron (MLP) neural networks was introduced. The neural networks were trained and validated on a noisy simulated dataset generated from the Integral Equation Model (IEM) on a wide range of surface roughness and soil moisture, as it is encountered in agricultural contexts for bare soils. The performances of neural networks in retrieving soil moisture and surface roughness were tested for several inversion cases using or not using a-priori knowledge on soil parameters. The inversion approach was then validated using RADARSAT-2 images in polarimetric mode. The introduction of expert knowledge on the soil moisture (dry to wet soils or very wet soils) improves the soil moisture estimates, whereas the precision on the surface roughness estimation remains unchanged. Moreover, the use of polarimetric parameters α1 and anisotropy were used to improve the soil parameters estimates. These parameters provide to neural networks the probable ranges of soil moisture (lower or higher than 0.30 cm3 cm−3) and surface roughness (root mean square surface height lower or higher than 1.0 cm). Soil moisture can be retrieved correctly from C-band SAR data by using the neural networks technique. Soil moisture errors were estimated at about 0.098 cm3 cm−3 without a-priori information on soil parameters and 0.065 cm3 cm−3 (RMSE) applying a-priori information on the soil moisture. The retrieval of surface roughness is possible only for low and medium values (lower than 2 cm). Results show that the precision on the soil roughness estimates was about 0.7 cm. For surface roughness lower than 2 cm, the precision on the soil roughness is better with an RMSE about 0.5 cm. The use of polarimetric parameters improves only slightly the soil parameters estimates.

Monitoring Soil Moisture to Support Risk Reduction for the Agriculture Sector Using RADARSAT-2

Monitoring the amount of moisture held in the soil is critical in the management of risk for the agriculture sector. Extremes in soil moisture can lead to devastating consequences. Early assessment of soil moisture reserves, and monitoring of changes in available soil moisture, could assist in risk reduction strategies for the agriculture sector and effective delivery of government programs. Agriculture and Agri-Food Canada has been acquiring RADARSAT-2 data since 2008 to evaluate the accuracy with which this sensor can provide soil moisture to assist with implementing risk reduction strategies for the Canadian agriculture sector. The calibrated Integral Equation Model (IEM) was used to estimate soil moisture for 15 RADARSAT-2 data sets acquired over an eastern and western Canadian test site. Using this approach, field level soil moisture was estimated to a mean average error of 7.95%, although considerable scatter in the results was observed. Removing fields which had significant residue cover improved site specific soil moisture errors, but only for the fall campaign prior to spring tillage and seed bed preparation. Higher errors were also observed for data sets where angles between the RADARSAT-2 look direction and field tillage structures were largest. When soil moisture estimates were evaluated at a regional scale, mean errors fell to 3.14%. The IEM was also able to detect increases and decreases in soil moisture which followed periods of rainfall and drying.

The IEM2M rough-surface scattering model for complex-permittivity scattering media

The integral equation model (IEM) was developed in the late 1980s and arguably became the most cited and implemented rough-surface scattering model in the field of radar remote sensing for Earth observation. It was derived by applying a second-order iteration in the incident electromagnetic field to the integral equations of the surface fields as given by Poggio and Miller. It is thus an extension of the first-order, Born approximation of these equations that produce the classical Kirchhoff approximation. The IEM has been subject to numerous amendments and variations over the last 20 years due to the imperfect introduction and handling of the Weyl representation of the spherical wave in its first version. The work presented here is a further development of the contribution made by the same author in 2001 (IEM2M), which was the first version of IEM able to blend analytically both the Kirchhoff and the small-perturbation approximations for the bistatic case. The improvement reported in this article is concerned with the inclusion of evanescent waves in the formulation of the model and the extension of the range of applicability of the second-order scattering terms to interfaces with complex-permittivity scattering media.

Structure and thermodynamics of nonaqueous solvation by integral equation theory

Electronic structure theory under the influence of apolar solvents suffers from substantial methodical difficulties since in this case the solvent-induced solute polarization originates mainly from specific directional interactions and higher electric multipoles. Continuum solvation models based on the dielectric solvent response such as the PCM approach ignore such interactions and can therefore not adequately model solvation effects in nonaqueous environments. The “embedded cluster reference interaction site model” (EC-RISM) [1] retains the granularity of the solvent and represents a microscopically more detailed and therefore improved approach towards solvation modeling. EC-RISM is based on a self-consistent solution of solvent distribution functions described by a 3D integral equation theory and solute electronic structure by mapping the solvent charge distribution onto discrete, solute-embedding point charges. In aqueous solution EC-RISM theory is capable of calculating pKa shifts [1] and tautomer ratios relatively fast and with high accuracy [2]. Here we outline the strength of the integral equation model by studying benzene and hexafluorobenzene solutions. In particular, the thermodynamics of differential solvation is quantified for organic compounds dissolved in these media. Moreover, it is shown that the respective solvent structures around particular solutes differ strongly, possibly leading to changes in the thermodynamic stability scale of various isomers which are not reproduced by the PCM model.

Assessment of SAR-retrieved soil moisture uncertainty induced by uncertainty on modeled soil surface roughness

The Integral Equation Model (IEM) is frequently used to retrieve moisture content of bare soils from synthetic aperture radar (SAR) images. This physically-based backscatter model requires surface roughness parameters, generally obtained by in situ measurements, which unfortunately often result in inaccurately retrieved soil moisture contents. Furthermore, when the retrieved soil moisture contents need to be used in data assimilation applications, it is important to also assess the retrieval uncertainty. Therefore, in this paper a regression-based method is developed that allows for the parameterization of roughness and that provides an estimation of its uncertainty by means of a probability distribution. By further propagating this distribution through the inversion of the IEM, a probability distribution of soil moisture content is obtained. It was found that 70% of the thus obtained distributions are skewed and non-normal. Furthermore, it is shown that their interquartile range varies depending on soil moisture conditions. Comparison of soil moisture measurements with the retrieved median values of the soil moisture histograms results in a root mean square error (RMSE) of approximately 3.5vol%.

Soil moisture monitoring in agricultural lands via active-passive microwave remote sensing

Soil moisture is a critical element for monitoring crop growth and drought in agricultural lands.Microwave remote sens-ing is a reliable mode of retrieving soil moisture in real time and under all weather and physiographic conditions.Retrieving soil moisture via combined active and passive microwave remote sensing has the advantage of high spatial resolution and real-time mea-surement at regional scale.Although it has practical significance for agricultural lands,this technique has largely been previously applied in retrieving soil moisture under natural vegetation.Here,the technique was used to retrieve soil moisture in agricultural lands.The TRMM Microwave Imager(TMI) emissivities at different soil moisture and surface roughness conditions were simulated using the Advanced Integral Equation Model(AIEM).After emissivity sensitivity analysis of soil moisture and roughness,a soil moisture inversion model for bare surfaces via passive microwave remote sensing was established based on the parameterized mi-crowave radiation model(Qp model).The inversion algorithm for vegetated surface was established using the microwave radiation transfer model(ω-τ model).Then a soil moisture variation inversion algorithm that combines active and passive microwave remote sensing(using TRMM Microwave Imager and Precipitation Radar data) was developed based on the Simple Scattering Model(SSM) and Geometry Optical Model(GOM).The daily soil moisture variations in agricultural lands between 30o-38oN and 110o-120oE was estimated for January through April of 2008.The retrieved daily soil moisture variations were compared with interpolated daily pre-cipitation variations at the grid which is located at 34.45oN and 119.25oE in the croplands of Lianyungang,Jiangsu Province.The study showed that microwave remote sensing was practicable in monitoring and managing soil moisture in agricultural lands.

Using RADARSAT-2 polarimetric and ENVISAT-ASAR dual-polarization data for estimating soil moisture over agricultural fields

There are several challenges in estimating soil moisture from radar remote sensing over agricultural fields in eastern Canada. To begin, snow cover or frozen ground is observed from November to April. From April to May, agricultural activities (e.g., ploughing and sowing) change the surface roughness from week to week thereby limiting the applicability of multitemporal and multi-incidence angle approaches. Techniques using a priori information on surface roughness are difficult to apply since the type of crop often changes from year to year. Here, we present an approach using effective roughness parameters (i.e., effective root mean square height and effective correlation length) that are obtained using an empirical relationship (independent of the crop type) between the root mean square height and the correlation length. The effective parameters allow us to resolve the Integral Equation Model for observed incidence angle and backscattering coefficient in HH and VV polarizations (σ°HH and σ°VV) using a look-up table. An additional challenge is posed by the growth of vegetation that begins in May. Three-component decompositions and radar vegetation indices were used to characterize the vegetation in agricultural fields. Surface backscattering coefficients in HH and VV polarizations (σ°SURF_HH and σ°SURF_VV) were calculated using the decompositions. An improvement in estimates of soil moisture was observed with the use of surface backscattering coefficients for bare soil and sparsely vegetated fields instead of the total backscattering.

DETERMINATION OF CLOSED-FORM EXPRESSIONS FOR RAYLEIGH SCATTERING OF POLARIZED LIGHT FROM ADSORBED PARTICLES ON OR BELOW A SUBSTRATE

An integral equation formulation describing the scattered fleld from a distribution of optically small Rayleigh objects of arbitrary shape adsorbed onto a planar dielectric substrate is presented. When certain approximations are introduced concerning the scatterers' permittivity contrast and small size compared to the wavelength, simple closed-form expressions are obtained for the ellipticity ratio and re∞ectivity which can be readily related to the surface coverage and average height of the surface layer. The formulation is an alternative to thin-island fllm theory often used to describe electromagnetic scattering from such conflgurations. Results derived from the integral equation model are compared with previously published measurements of ellipticity ratio and re∞ectivity and are found to be in good agreement with observation.

Integral Equation Modeling of Doubly Periodic Structures With an Efficient PMCHWT Formulation

A surface integral equation modeling is described for complex doubly periodic structures. To avoid long computations of the slowly convergent pseudoperiodic Green's function, fictitious surfaces between translated unit cells are set in order to bound regions of the structure within the symmetry cell and use the free-space Green's function. The integral operators on top and bottom surfaces are computed with an algorithm originally used for planar frequency-selective surfaces. This approach uses a unique periodic PMCHWT formulation in all the regions, with two different kinds of Green's function. The method and its advantages are illustrated by two cases in the near-IR domain and in the radar domain. A frequency selective structure is studied, that shows a large flat-top bandwidth under oblique incidence and TM polarization.

Orientation Angle Calibration for Bare Soil Moisture Estimation Using Fully Polarimetric SAR Data

This paper focuses on assessing the effectiveness of applying orientation angle calibration to polarimetric synthetic aperture radar (PolSAR) data for soil moisture estimation. We employ Cloude-decomposition-based method to estimate the orientation angle because it can relate a scatter-distributed pixel to its major component of an equivalent "pure target," use the Jet Propulsion Laboratory/Airborne Synthetic Aperture Radar L-band fully polarimetric data to validate the proposed method, and observe results in good agreement after orientation angle compensation is employed. Specifically, root mean square errors of measured radar backscattering coefficients σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hh</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> and σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vv</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> and copolarization ratio versus advanced integral equation model predictions are reduced significantly from 1.95, 1.33, and 2.03 dB to 1.30, 1.15, and 1.43 dB, respectively. The compensated copolarized backscattering coefficients are also used as inputs to a novel inversion model to estimate the dielectric factor R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hh</sub> and volumetric soil moisture m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> . The results show that the estimation errors are reduced significantly from 0.075 to 0.054 and 0.056 to 0.041 for R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hh</sub> and m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> , respectively. This paper demonstrates the advantage of orientation angle calibration as a preprocessing for estimating bare soil moisture, particularly in agricultural areas, and the preponderance of fully PolSAR data on soil moisture estimation over dual and single polarizations.

An integral equation model for radiowave propagation over inhomogeneous smoothly irregular terrain

AbstractThis work introduces an integral equation formulation to model the radiowave propagation over inhomogeneous smoothly irregular terrain. The ground energy loss is estimated via the Leontovich boundary condition and the fast far‐field algorithm accelerating technique is applied in order to improve the computational performance when solving the integral equation. The proposed formulation was implemented in a prediction software named SPRad, which was numerically tested with two measurement campaigns performed in Denmark and Brazil. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:26–31, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26473

Evaluation of emission from snow-covered ground for passive microwave remote sensing

This article investigates emission behaviour at frequencies of 18.7, 36.5 and 89 GHz and an incidence angle of 55° over a snow-covered surface at the local scale observation site in Fraser, CO, USA, using both one-layer and two-layer emission models. The models employ the matrix doubling approach to implement the radiative-transfer equation based on dense media theory and the advanced integral equation model. When compared to Ground-Based Passive Microwave Radiometer (GBMR-7) observation on 21 February 2003, both the models could simulate the observed brightness temperature well, but the polarization difference between the observation and the models was smaller for the two-layer emission model than the one-layer model. In addition, we successfully interpreted the emission magnitude and polarization separation of a snow-removed surface by incorporating a Mie scattering transition layer above the soil medium. In this work, we also demonstrated the effect of snow fraction on the brightness temperature difference at 18.7 and 36.5 GHz over a snow-covered surface with the field observation. In conclusion, we demonstrate the snow impact on soil surface with snow depth (SD) and snow fraction variation through modelling and in situ data.

Evaluation of Radar Backscattering Models IEM, Oh, and Dubois for SAR Data in X-Band Over Bare Soils

The objective of this letter is to evaluate the surface radar backscattering models, namely, integral equation model (IEM), Oh, and Dubois, for synthetic aperture radar data in X-band over bare soils. This analysis uses a large database of TerraSAR-X images and in situ measurements (soil moisture “mv” and surface roughness “ h_rms”). Oh's model correctly simulates the radar signal for HH and VV polarizations, whereas the simulations performed with the Dubois model show a poor correlation between TerraSAR-X data and model. The backscattering IEM simulates correctly the backscattering coefficient only for h_rms ; 1.5 cm in using Gaussian function. However, the results are not satisfactory for the use of IEM in the inversion of TerraSAR-X data. A semiempirical calibration of IEM was done in X-band. Good agreement was found between the TerraSAR-X data and the simulations using the calibrated version of the IEM.