Second-order Small Perturbation Method Research Articles

Analysis of the Effect of Soil Roughness in the Forward-Scattering Interference Pattern Using Second-Order Small Perturbation Method Simulations

Soil moisture (SM) is a key geophysical variable that can be estimated at regional scales using remote sensing techniques, by making use of the known relationship between soil reflectivity and the dielectric constant in the microwave regime. In this context, the exploitation of available illuminators of opportunity that currently emit large amounts of power at microwave frequencies (compared to typical synthetic aperture radar systems) is promising. Some published techniques estimate SM by analyzing the interference pattern (IP) between direct and reflected signal as measured by a single antenna (i.e., IP technique). In this letter, a new approach to simulate the IP is proposed, in which the soil roughness is modeled straightforwardly using the second-order small perturbation model. Results illustrate that the “notch” in the VV-polarization IP (related to the Brewster angle) can only be directly observed for very low values of soil rms roughness ( $s cm). For typical values of soil roughness ( $s\sim 1.2$ cm), the notch disappears and only a minimum in the IP is observed near the Brewster angle.

3-D Electromagnetic Scattering From Multilayer Dielectric Media With 2-D Random Rough Interfaces Using <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math> </inline-formula>-Matrix Approach

The translation matrix ( $T$ -matrix) solution to the 3-D problem of scattering of the electromagnetic waves from a dielectric layered medium with random rough interfaces is presented. The solution is based on the coupled vector integral equation of the electric field using the dyadic periodic Green’s function as a kernel. It is shown that the $T$ -matrix solution conserves energy, the bistatic scattering pattern is regular even in cases, where guided modes within the layered medium are excited, and also that the method’s results coincide with those of the second-order Small Perturbation Method (SPM2) in the small height limit. One of the advantages of the $T$ -matrix method compared to the SPM2 is the wide validity range of the solution, which is also studied.

Microwave Thermal Emission Characteristics of a Two-Layer Medium With Rough Interfaces Using the Second-Order Small Perturbation Method

The second-order small perturbation method is applied to investigate brightness temperature corrections caused by the rough interfaces of a two-layer medium. The spectral weighting functions of the two rough interfaces are extracted from the solution, and their properties examined. It is found that the functions are identical for the two interfaces as the spectral variable approaches zero, indicating an identical weighting of the surface height variance on each interface and an additive effect on the brightness temperature at nadir. Sample results for some realistic scenarios show that surface roughness in a two-layer medium can increase or decrease the observed brightness temperature at shallower angles, and in the case of a wideband measurement, can shift the interference pattern in frequency.

Scattering of electromagnetic waves from 3D multilayer random rough surfaces based on the second-order small perturbation method: energy conservation, reflectivity, and emissivity.

A theoretical investigation of energy conservation, reflectivity, and emissivity in the scattering of electromagnetic waves from 3D multilayer media with random rough interfaces using the second-order small perturbation method (SPM2) is presented. The approach is based on the extinction theorem and develops integral equations for surface fields in the spectral domain. Using the SPM2, we calculate the scattered and transmitted coherent fields and incoherent fields. Reflected and transmitted powers are then found in the form of 2D integrations over wavenumber in the spectral domain. In the integrand, there is a summation over the spectral densities of each of the rough interfaces with each weighted by a corresponding kernel function. We show in this paper that there exists a "strong" condition of energy conservation in that the kernel functions multiplying the spectral density of each interface obey energy conservation exactly. This means that energy is conserved independent of the roughness spectral densities of the rough surfaces. Results of this strong condition are illustrated numerically for up to 50 rough interfaces without requiring specification of surface roughness properties. Two examples are illustrated. One is a multilayer configuration having weak contrasts between adjacent layers, random layer thicknesses, and randomly generated permittivity profiles. The second example is a photonic crystal of periodically alternating permittivities of larger dielectric contrast. The methodology is applied to study the effect of roughness on the brightness temperatures of the Antarctic ice sheet, which is characterized by layers of ice with permittivity fluctuations in addition to random rough interfaces. The results show that the influence of roughness can significantly increase horizontally polarized thermal emission while leaving vertically polarized emissions relatively unaffected.

Second-Order Perturbative Solutions for 3-D Electromagnetic Radiation and Propagation in a Layered Structure With Multilayer Rough Interfaces

We present a fast method for deriving a general explicit closed-form expression of second-order perturbative solutions of the problem of radiation and propagation electromagnetic (EM) fields in a hybrid layered structure with an arbitrary number of rough interfaces and planar stratified medium based on the classical small perturbation method (SPM). The second-order SPM expression of the electric and magnetic fields in each region of the rough-layered structure is provided by theoretical analysis and fully formula derivation instead of considering any other approximation and equivalent process. In addition, we give the complete understanding of physical radiation and propagation mechanism of double-bounce second-order SPM solutions. Finally, the second-order SPM solutions derived by employing the proposed method are verified with existing methods and numerical results. What is more, we investigate completely the influence of various parameters of layered structure and configuration of radar system on EM scattering fields.

Numerical Simulations and Analysis of Wide-Band Range-Resolved HF Backscatter From Evolving Ocean-Like Surfaces

Low-grazing wide-band HF radar echoes from evolving ocean-like surfaces in 2-D space are simulated employing a first-principles boundary integral equation technique. The synthesized signals achieve range resolution of about 15 m and correspond to observations with a hypothetical narrow-beam system having a 10-MHz effective bandwidth. Range-time plots of backscatter magnitude appear to contain a pattern consistent with the long surface waves. The double Fourier transform of the signal magnitude or power in range and time reveals strong harmonics located along the dispersion curve of the propagating gravity waves. The effect is discernable due to the high range resolution employed and is not predicted by the first-order Small Perturbation Method (SPM) approximation. It is explained by considering the next, second order of the SPM under certain simplifying assumptions such as perfect surface conductivity (with the Norton attenuation factor set to 1). The derived analytical expression is then used to investigate the feasibility of the surface profile retrieval from such time-varying range-resolved HF backscatter (gathered in the course of the “numerical experiment” through the exact scattering simulations). For the case of perfectly conducting sea-like surface developed under 10-m/s wind conditions, the results are very encouraging. The approach is sensitive to the surface harmonics with wavelengths far exceeding the Bragg resonance scales of the HF band. Extending this type of analysis to cases with realistic finite conductivity and/or rougher surface (assuming that the second-order SPM is still valid) will require proper accounting for ground-wave propagation effects in perturbation expressions. The study offers a glimpse into ocean observation possibilities that the wide-band HF systems can provide if and when they become a reality.

Random Rough Surface Effects in Waveguides Using Mode Matching Technique and the Method of Moments

To address the rough surface effects in high speed interconnects on printed circuit boards and microelectronic packages, we study the electromagnetic wave propagation in a parallel plate waveguide with rough surface walls, using a mode matching technique in combination with the method of moments (MoM). In the formulation, a line source in the smooth section of the waveguide excites a summation of modal fields. For each waveguide modal excitation, the modal response fields in the rough surface section of the waveguide are obtained by the solution of a set of coupled surface integral equations through MoM. Results are illustrated for a variety of roughness types including the very rough case based on exponential correlation functions. The results are compared with the previously developed analytical second-order small perturbation method (SPM2) result and the finite element method (FEM). It is shown that for rough surfaces with large slopes, the mode matching MoM model gives a smaller enhancement factor than SPM2. It is also shown that MoM can give accurate solutions even for the very rough case of exponential correlation functions. The MoM method of rough surface simulation in waveguide geometry requires much less computational resources than FEM.

Second-order small perturbation method for transmission from dielectric rough surfaces

Although the small perturbation method (SPM) for rough surface scattering has been studied extensively in problems in optics, remote sensing and propagation, there are fewer studies on rough surface transmission by the SPM. In this paper, from Huygens’ principle and the extinction theorem, the SPM is used to derive the transmitted field to the second order, and expressions for the bidirectional transmission coefficient and the total surface transmittance to the second order are developed for the random rough surface. The refined expressions can be applied to the situations where the transmission characteristics of a random rough surface need to be more accurately calculated. For example, to calculate the brightness temperature of stratified rough media by the incoherent method, we have to know the bidirectional transmission coefficient or transmittance of random rough surface as accurately as possible. The accuracy of the presented expressions is verified through the conservation of energy. It is shown that the transmission characteristics calculated by SPM to the first order violate conservation of energy, whereas solutions to the second order conform to energy conservation much better. This is particularly important for the calculation of transmittance or emissivity.

Electromagnetic Scattering and Inverse Scattering of Layered Media with a Slightly Rough Surface

An electromagnetic (EM) scattering model for layered media covered by a 3D infinite rough surface and the corresponding inversion technique are investigated. The work aims at remote sensing the surface roughness and dielectric constant for different depths of bear soil through radar measurement data. The forward problem is carried out by the wave decomposition method. The small perturbation method (SPM) and EM boundary conditions are employed to solve the integral equations introduced by the wave decomposition method. The second-order SPM solution of the scattering field is involved in the computation of the forward problem for the first time. The backscattering coefficients of multiple frequencies, multiple angles and multiple polarizations are employed to create a nonlinear optimization problem. A genetic algorithm is introduced to help the inversion procedure approach to the global minimum of the cost function. Examples are carried out to validate the inversion technique. The inversion results show good agreement with the forward problem with given parameters and pose good tolerance to the input data with the additive white Gaussian noise.

Random Rough Surface Effects on Wave Propagation in Interconnects

To address the rough surface effects in high-speed interconnects on printed circuit boards (PCBs) and microelectronic packages, we study the electromagnetic wave propagation in a rough surface environment. In our model, the rough surface is characterized by a stochastic random process with correlation function or spectral density. This paper reviews the analytical theory, numerical simulations and experimental results based on such a model. We describe the rough surface characterization and the extraction of roughness parameters from 3D profile measurements. Initially we study the 2D case with the rough surface height function varying in only one horizontal direction and consider the case of plane wave incidence. Analytic second-order small perturbation method (SPM2) was used to obtain simple closed-form expressions for the absorption enhancement factor. The numerical transfer matrix (T-matrix) method and the method of moments (MoM) were also used. We next consider the case of the 3D problem with the rough surface height varying in both horizontal directions. We also used SPM2 to obtain a simple closed form expression for the enhancement factor. In interconnect problems, electromagnetic (EM) waves propagate in a guided wave environment. Thus, we next considered a waveguide model to study the effects of random roughness on wave propagation and compare with results from the plane wave formulation. Analytic SPM2 and numerical finite element method (FEM) with mode matching were used to obtain the enhancement factor. We also describe experimental results and correlation with the theoretical models. Finally, we explain how the enhancement factor concept used throughout lends itself to direct inclusion of rough surface effects in a wide variety of modeling problems.

Wave Propagation in a Randomly Rough Parallel-Plate Waveguide

We study the effects of random rough surface on electromagnetic wave propagation in a parallel-plate waveguide excited by a line source. The second-order small perturbation method is applied to derive a closed-form expression for the coherent wave propagation and the power loss. The derived result is expressed in terms of a double Sommerfeld integral. The double Sommerfeld integral is evaluated by direct numerical integration and by approximate methods. The results of the two methods are shown to be in good agreement with each other. Numerical results of coherent wave propagation and power loss are illustrated as a function of roughness characteristics and waveguide thickness. The enhancement factors are compared with previous results of plane-wave scattering. The waveguide model and the plane-wave model are in good agreement for moderate to large waveguide thicknesses. For small waveguide thickness, the waveguide model shows significantly different power loss as compared to the plane-wave model. Full-wave simulations by the finite-element method are used to validate the second-order small perturbation in waveguide model.

A New Model for Rough Surface Scattering

A new model for rough surface scattering is presented; the model has a form similar to the small slope approximation (SSA) of Voronovich, but with modified kernel functions. As with the SSA, when including two field series terms in the solution the model matches the first- and second-order small perturbation method in the low-frequency limit. Unlike the SSA, the model also achieves agreement with the Kirchhoff approximation in the high-frequency limit even for penetrable surfaces. It is also shown that the new model achieves first-order tilt invariance for first-order SPM predictions. The new model is derived based on a previous extension of the local curvature approximation (LCA) to third order; the new model is termed the "reduced local curvature approximation of third order" (RLCA3) for this reason. Sample results for scattering from dielectric surfaces are presented to illustrate the new model and its relationship with other theories of rough surface scattering.

Bistatic Scattering and Emissivities of Lossy Dielectric Surfaces With Exponential Correlation Functions

Bistatic scattering and emissivities of surfaces with exponential correlation functions are studied numerically for 2-D geometries in a numerical Maxwell model with 2-D simulations. Surfaces with exponential correlation functions are important for the active and passive microwave remote sensing of land surfaces. Because of the fine-scale features with large slopes of such surfaces, numerical accuracy, which is particularly important for the calculation of emissivity in passive remote sensing, is ensured by a variety of procedures in this paper. The rooftop function and Galerkin's method with numerical integration of near-field impedance matrix elements are used. Cubic spline interpolation is employed to connect knots on random rough surfaces. Numerical accuracy convergence tests are performed for numerical solutions of Maxwell equations by varying the number of points from 13 to 103 points per wavelength in the dielectric medium corresponding to 50-400 points per free wavelength. Surface lengths of up to 100 and 200 free wavelengths and root mean square heights of up to 0.4 and 0.8 free wavelengths, respectively, are used at 5 and 10 GHz to capture all the essential features. Because of the large number of surface unknowns (up to 80 000), the multilevel UV method is further used to accelerate the matrix equation solver. Numerical results are illustrated for both bistatic scattering and emissivities as functions of frequencies and incidence and scattering angles for cases of interests in microwave remote sensing. Comparisons are made with the second-order small perturbation method and Kirchhoff's approximation to reestablish the regimes of validity of these methods

Extension of the local curvature approximation to third order and improved tilt invariance

The second-order local curvature approximation (LCA2) is a theory of rough surface scattering that reproduces fundamental low and high frequency limits in a tilted frame of reference. Although the existing LCA2 model provides agreement with the first order small perturbation method up to the first order in surface tilt, results reported in this paper produce a new formulation of the model that achieves consistency with perturbation theory to first order in surface height and arbitrary order in surface tilt. In addition, extension of the modified LCA to third order is presented, and allows the theory to match the second-order small perturbation method to arbitrary order in surface tilt. Crucial to the development of the theory are a set of identities involving relationships among the small perturbation method (i.e. low frequency) and Kirchhoff approximation (i.e. high frequency) kernels; a set of new identities obtained in our derivations is also presented. Sample results involving 3D electromagnetic scattering from penetrable rough surfaces, as well as 2D scattering from Dirichlet sinusoidal gratings, are provided to compare the new results with the existing LCA2 model and with other rough surface scattering theories.

Polarimetric passive microwave remote sensing of wind vectors with foam‐covered rough ocean surfaces

In this paper, polarimetric microwave emissions from wind‐generated foam‐covered ocean surfaces are investigated. The foam is treated as densely packed air bubbles coated with thin seawater coating. The absorption, scattering and extinction coefficients are calculated by Monte Carlo simulations of solutions of Maxwell equations of a collection of coated particles. The effects of boundary roughness of ocean surface are included by using the second‐order small perturbation method (SPM) describing the reflection coefficients between foam and ocean. An empirical wavenumber spectrum is used to represent the small‐scale wind‐generated sea surfaces. The iterative method is employed to solve dense media radiative transfer (DMRT) equations, and is applied to calculate results of all four Stokes parameters of rough ocean surfaces. The theoretical results of four Stokes brightness temperatures with typical parameters of foam in passive remote sensing at 10.8 GHz, 19 GHz and 36.5 GHz are illustrated. The azimuth variations of polarimetric brightness temperature are calculated. Emission with various wind speed and foam layer thickness is studied. The results are also compared with those based on Quasi‐Crystalline Approximation (QCA).

Analytical comparison between the surface current integral equation and the second-order small-slope approximation

This paper is the third in a series discussing a new approximate bistatic model for electromagnetic scattering from perfectly conducting rough surfaces. Our previous approach supplemented the Kirchhoff model through the addition of new terms involving linear orders in slope and surface elevation differences that arise naturally from a second iteration of the surface current integral equation. This completion of the Kirchhoff was shown to provide the correct first-order small perturbation method (SPM-1) in the general bistatic context. The agreement with SPM-1 was achieved because differences of surface heights are no longer expanded in powers of surface slope. While consistent with SPM, our previous formulation fails to reconverge toward the Kirchhoff model, at some incidence and scattered angles, when the illuminated surface satisfies the high frequency roughness condition. This weakness is also shared with the first-order small slope approximation (SSA-1) which is structurally equivalent to our previous formulation where the polarization is independent of surface roughness. The second-order small slope approximation (SSA-2), which satisfies the SPM-1 and second-order small perturbation method (SPM-2) limits by construction, was shown by Voronovich to converge toward the tangent plane approximation of the Kirchhoff model under high frequency conditions. In the present paper, we show that, in addition to the linear orders in our previous model, one must now include cross-terms between slope and surface elevation to ensure convergence toward both high frequency and small perturbation limits. With the inclusion of these terms, our new formulation becomes comparable to the SSA-2 (second-order kernel) without the need to evaluate all the quadratic order slope and elevations terms. SSA-2 is more complete, however, in the sense that it guarantees convergence toward the second-order Bragg limit (SPM-2) in the fully dielectric case in addition to both SPM-1 and Kirchhoff. Our new generalization is shown to explain correctly extra depolarization in specular conditions to be caused by surface curvature and surface autocorrelation for incoherent and coherent scattering, respectively. This result will have large repercussions on the interpretation of bistatically reflected signals such as those from GPS.

Bistatic scattering and emissivities of random rough dielectric lossy surfaces with the physics-based two-grid method in conjunction with the sparse-matrix canonical grid method

Bistatic EM wave scattering from 2-D lossy dielectric random rough surfaces (3-D scattering problem) with large permittivity is studied. For media with large permittivities, the fields can vary rapidly on the surface. Thus, a dense discretization of the surface is required to implement the method of moments (MoM) for the surface integral equations. Such a dense discretization is also required to ensure that the emissivity can be calculated to the required accuracy of 0.01 for passive remote sensing applications. We have developed a physics-based two-grid method (PBTG) that can give the accurate results of the surface fields on the dense grid and also the emissivities. The PBTG consists of using two grids on the surface, the coarse grid and the required dense grid. The PBTG only requires moderate increase in central processing unit (CPU) and memory. In this paper, the numerical results are calculated by using the PBTG in conjunction with the sparse-matrix canonical grid (SMCG) method. The computational complexity and memory requirement for the present algorithm are O(N/sub scg/log(N/sub scg/)) and O(N/sub scg/), respectively, where N/sub scg/ is the number of grid points on the coarse grid. Numerical simulations are illustrated for root mean square (rms) height of 0.3 wavelengths and correlation length of 1.0 wavelength. The relative permittivity used is as high as (17+2i). The numerical results are compared with that of the second-order small perturbation method (SPM). The comparisons show that a large difference in brightness temperature exists between the SPM and numerical simulation results for cases with moderate rms slope.