One-dimensional Rough Surface Research Articles

Surface Profile Recovery from Electromagnetic Fields with Physics-Informed Neural Networks

Physics-informed neural networks (PINN) have shown their potential in solving both direct and inverse problems of partial differential equations. In this paper, we introduce a PINN-based deep learning approach to reconstruct one-dimensional rough surfaces from field data illuminated by an electromagnetic incident wave. In the proposed algorithm, the rough surface is approximated by a neural network, with which the spatial derivatives of surface function can be obtained via automatic differentiation, and then the scattered field can be calculated using the method of moments. The neural network is trained by minimizing the loss between the calculated and the observed field data. Furthermore, the proposed method is an unsupervised approach, independent of any surface data, where only the field data are used. Both transverse electric (TE) field (Dirichlet boundary condition) and transverse magnetic (TM) field (Neumann boundary condition) are considered. Two types of field data are used here: full-scattered field data and phaseless total field data. The performance of the method is verified by testing with Gaussian-correlated random rough surfaces. Numerical results demonstrate that the PINN-based method can recover rough surfaces with great accuracy and is robust with respect to a wide range of problem regimes.

Direct Electromagnetic Wave Scattering Calculation Using Methods of Moments through Layered Rough Surface

This thesis focuses on the direct calculation of electromagnetic wave scattering through layered rough surfaces using the Method of Moments. The study aims to contribute to existing knowledge by addressing the problems associated with accurately predicting scattered electromagnetic fields from rough surfaces with layered structures. By using the Method of Moments, this research seeks to provide insights into the behavior of electromagnetic waves when interacting with multi-layered rough surfaces, offering valuable implications for practical engineering applications and the development of advanced computational tools for electromagnetic wave analysis. In practical terms, the surface is typically divided into small patches, and the electromagnetic field at each patch is determined using MoM. This involves solving integral equations describing the interaction between the induced currents on the surface and the incident field. The layered structure may require additional equations to model the reflection of waves and the transmission at the boundaries between the layers. The direct problem is also examined in a two-dimensional picture, where the rough surface profile and the medium are known parameters, and the electromagnetic waves scattered from the surface to the entire space are the unknowns. The rough surface is considered to have a finite length and can be deterministic or arise from a stationary stochastic random process, with a focus on Gaussian-type rough surfaces. The roughness is considered as two basic types: as a perfect electric conductor (PEC) in free space and as a boundary between two dielectric media. The equations of integration obtained are based on the scalar solution of Green's theory, Helmholtz equations, and using Hankel type green functions as kernels due to the one-dimensional nature. The MoM is extensively used to solve these integral equations and is compared to asymptotic approaches and analytical. Various simulations are conducted to test the feasibility of the algorithms, and the results are thoroughly discussed. The study of electromagnetic wave scattering in one-dimensional rough surfaces involves the direct problem, where the field distribution in the whole space is determined using the equation of integration. These equations take into account the surface roughness and medium parameters they are solved using the Method of Moments (MoM) by expressing the field distribution on the surface using basis functions. The direct calculation of electromagnetic wave scattering through a layered rough surface using the Method of Moments (MoM) is a computationally intensive process and complex and. It contains simulating the interaction of EM waves with a complex surface, which presents challenges due to the multiple layers with different properties, and a roughly surface causing and diffraction of incident waves and scattering. When working with a layered rough surface increases the electromagnetic wave interaction complexity. Additional multiple layers introduce considerations, such as layer interfaces and accounting for different material abilities. This requires careful modeling and analysis to exactly capture the behavior of electromagnetic waves as they traverse through and interact with layers. MoM, a commonly used numerical method for solving electromagnetic scattering problems, discreteness the surface into small segments and solves integral equations to determine the fields that is scattered. This approach allows for a maximum analysis of electromagnetic interactions at a grained level.

Resolution dependence of rough surface scattering using a power law roughness spectrum.

Contemporary high-resolution sonar systems use broadband pulses and long arrays to achieve high resolution. It is important to understand effects that high-resolution sonar systems might have on quantitative measures of the scattered field due to the seafloor. A quantity called the broadband scattering cross section is defined, appropriate for high-resolution measurements. The dependence of the broadband scattering cross section, σbb, and the scintillation index, SI, on resolution was investigated for one-dimensional rough surfaces with power-law spectra and backscattering geometries. Using integral equations and Fourier synthesis, no resolution dependence of σbb was found. The incoherently averaged frequency-domain scattering cross section has negligible bandwidth dependence. SI increases as resolution increases, grazing angle decreases, and spectral strength increases. This trend is confirmed for center frequencies of 100 and 10 kHz, as well as for power-law spectral exponents of 1.5, 2, and 2.5. The hypothesis that local tilting at the scale of the acoustic resolution is responsible for intensity fluctuations was examined using a representative model for the effect of slopes (inspired by the composite roughness approximation). It was found that slopes are responsible in part for the fluctuations, but other effects, such as multiple scattering and shadowing may also play a role.

A spectral domain integral equation technique for rough surface scattering problems

ABSTRACT In this paper, a novel formulation for scattering from one-dimensional (1D) rough surface problem has been presented. The formulation is based on the representation of the scattering field in spectral domain with an unknown spectral coefficient which is solved by applying Taylor series and Fourier Transforms techniques. It is also shown that forward–backward (FB) acceleration technique can be adapted to the spectral formulation to improve the effectiveness in a limited region of convergence. The obtained results are compared with the conventional method of moments (MoM) and the limitations of the formulation are analyzed in detail.

A Fast Algorithm for Electromagnetic Scattering from One-Dimensional Rough Surface

In this paper, the Adaptive Modified Characteristic Basis Function Method (AMCBFM) is proposed for quickly simulating electromagnetic scattering from a one-dimensional perfectly electric conductor (PEC) rough surface. Similar to the traditional characteristic basis function method (CBFM), Foldy-Lax multiple scattering equations are applied in order to construct the characteristic basis functions (CBFs). However, the CBFs of the AMCBFM are different from those of the CBFM. In the AMCBFM, the coefficients of the CBFs are first defined. Then, the coefficients and the CBFs are used to structure the total current, which is used to represent the induced current along the rough surface. Moreover, a current criterion is defined to adaptively halt the order of the CBFs. The validity and efficiency of the AMCBFM are assessed by comparing the numerical results of the AMCBFM with the method of moments (MoM). The AMCBFM can effectively reduce the size of the matrix, and it costs less than half the CPU time used by the MoM. Moreover, by comparing it with the traditional CBFM, the AMCBFM can guarantee the accuracy, reduce the number of iterations, and achieve better convergence performance than the CBFM does. The second order of the CBFs is set in the CBFM. Additionally, the first order of the CBFs of the AMCBFM alone is sufficient for this result.

Fast Simulations of Electromagnetic Scattering From One-Dimensional Rough Surface Over a Frequency Band Using Hybrid AMCBFM-Maehly Method

In this paper, the hybrid AMCBFM-Maehly method is presented to fast simulate the electromagnetic scattering from one-dimensional rough surface over a frequency band. In this approach, the traditional Maehly technique is applied to get the rational function of the induced electric current along the rough surface over a frequency band. The adaptive modified characteristic basis function method (AMCBFM) is utilized to fast compute the induced electric current along the rough surface at the Chebyshev nodes of the Maehly technique. Moreover, in the AMCBFM, the size of the matrix is reduced. And a current criterion is defined to adaptively halt the order of the characteristic basis functions (CBFs). The validity and efficiency of the hybrid AMCBFM-Maehly are assessed by comparing the numerical results of the hybrid AMCBFM-Maehly with the method of moments (MoM). The hybrid AMCBFM-Maehly can effectively reduce the size of the matrix, and it costs nearly half the CPU time used by the MoM. Moreover, by comparing with the traditional Maehly technique, the hybrid AMCBFM-Maehly costs less time. Additionally, in most cases, the first order of the CBFs of the hybrid AMCBFM-Maehly is sufficient for this result.

Modified PO–PO hybrid method for scattering of 2D ship model on the rough sea surface

The hybrid physical optics-physical optics (PO-PO) method was introduced in the previous works to compute the electromagnetic scattering from an object above a random rough surface. Here, first, the conventional PO-PO method is applied to find the scattering from a simple smooth two-dimensional (2D) perfect electric conductor (PEC) object on a one-dimensional (1D) rough PEC surface. Then, this method is modified to be used to calculate the scattering from a complex sharp-cornered 2D PEC target on the 1D random rough PEC surface. In addition, here, the physical theory of diffraction is utilised to take into account the diffraction from the edges of the target. The accuracy and efficiency of the proposed method are validated by the method of moments.

An airborne acoustic method to reconstruct a dynamically rough flow surface.

Currently, there is no airborne in situ method to reconstruct with high fidelity the instantaneous elevation of a dynamically rough surface of a turbulent flow. This work proposes a holographic method that reconstructs the elevation of a one-dimensional rough water surface from airborne acoustic pressure data. This method can be implemented practically using an array of microphones deployed over a dynamically rough surface or using a single microphone which is traversed above the surface at a speed that is much higher than the phase velocity of the roughness pattern. In this work, the theory is validated using synthetic data calculated with the Kirchhoff approximation and a finite difference time domain method over a number of measured surface roughness patterns. The proposed method is able to reconstruct the surface elevation with a sub-millimeter accuracy and over a representatively large area of the surface. Since it has been previously shown that the surface roughness pattern reflects accurately the underlying hydraulic processes in open channel flow [e.g., Horoshenkov, Nichols, Tait, and Maximov, J. Geophys. Res. 118(3), 1864-1876 (2013)], the proposed method paves the way for the development of non-invasive instrumentation for flow mapping and characterization that are based on the acoustic holography principle.

Accuracy improvement of the radar backscatter simulation from sea surface covered by oil slick using fetch-dependent waveheight spectrum: Comparison with the 2007 Heibei Spirit Case in the Yellow Sea

In this paper, results are presented on the comparison of X-band radar backscattering coefficient (RBC) from an oilcovered sea surface that features the Elfouhaily and Durden-Vesecky waveheight spectra. The Durden-Vesecky spectrum applies to a fully-developed sea, while the Elfouhaily spectrum accounts for the fetch of arbitrary length. Using these two waveheight spectra, a one-dimensional random rough surface is simulated by the Monte Carlo method, and the method of moments (MoM) is applied to yield the RBC. Comparison of the results with TerraSAR-X synthetic aperture radar (SAR) data acquired over the coastal waters polluted by the Hebei Spirit oil tanker shows that the Elfouhaily spectrum yields better agreement than the Durden-Vesecky spectrum for the fully-developed sea, and that the fetch-dependent Elfouhaily spectrum improves the agreement with SAR data in comparison with the fetch-independent spectrum for the fully developed sea. A possible application to estimate the amount of spilled oil is also suggested.

Reflection of diffuse light from dielectric one-dimensional rough surfaces.

We study the reflection of diffuse light from 1D randomly rough dielectric interfaces. Results for the reflectance under diffuse illumination are obtained by rigorous numerical simulations and then contrasted with those obtained for flat surfaces. We also explore the possibility of using perturbation theories and conclude that they are limited for this type of study. Numerical techniques based on Kirchhoff approximation and reduced Rayleigh equations yield better results. We find that, depending on the refractive index contrast and nature of the irregularities, the roughness can increase or decrease the diffuse reflectance of the surface.

Wide-band composite electromagnetic scattering from the earth soil surface and multiple targets shallowly buried

Wide-band electromagnetic scattering from multiple objects shallowly buried beneath rough earth soil surfaces has been an important research topic in recent years because of its extensive applications in detecting the buried objects such as mines, pipes, and tunnels. Due to the advantages of finite-difference time-domain (FDTD) method in simulating wide-band electromagnetic scattering from rough surface in the presence of multiple objects, the FDTD method under Gaussian differential pulse wave incidence is utilized in the present study to analyze the frequency response of rough soil surfaces with shallowly buried objects, which serves as a basis for the detection and discrimination of objects buried below rough soil surfaces. The Topp equation model that can predict the dielectric constant of soil-water mixture is adopted in the present study to properly describe the dielectric property of earth soil with water. The actual rough land surface is modeled as the realization of a Gaussian random process with exponential spectrum by using Monte Carlo method. Simulation results show that the variation of composite scattering coefficient with frequency is oscillatory. It is also shown that the composite scattering coefficient versus frequency increases with the increase of root-mean-square of soil surface, water ratio of soil, the target section height, and the separation distance of target. However, simulation results indicate that the composite scattering coefficient versus frequency decreases with the increase of target section width. In summary, the variation of wide-band scattering coefficient is very complicated and is very sensitive to the incidence angle of electromagnetic wave. However, the wide-band scattering coefficient under Gaussian differential pulse wave incidence is less sensitive to the correlation length of rough soil surface, the depth of buried objects, and the dielectric constant of target. These qualitative results relating to the frequency response of rough soil surfaces in the presence of multiple objects are potentially valuable for detecting and discriminating the objects buried below rough soil surfaces by utilizing a wide-band ground penetrating radar system, although the present study is limited to one-dimensional rough soil surface due to the severe computational burden encountered in the large-scale Monte Carlo simulations. In addition, compared with frequency-domain numerical methods, the FDTD method has significant advantages in calculating wide-band composite scattering from rough surfaces in the presence of multiple objects, and thus has extensive applications in radar imaging simulation of multiple objects below or above rough surfaces, which goes beyond the scope of this paper.

A fast PO–PO hybrid method for analysing the Doppler spectrum of a plasma-coated object above a rough sea surface: horizontal polarization

A fast hybrid method based on physical optics (PO) and the reciprocity theorem is proposed to analyse the Doppler spectrum of a two-dimensional, moving plasma-coated object above a one-dimensional time-evolving lossy dielectric rough sea surface. This hybrid method eliminates the need for numerical solution of the polarization currents on the object and the time-evolving sea surface, thereby greatly improving the computing speed. The method is verified by conventional numerical methods. The characteristics of the Doppler spectrum of a plasma-coated object above a time-evolving sea surface are analysed in detail, including the effects of several parameters such as the incident angle and the velocity of the moving object. The effects of some key parameters on the stealth performance are also examined.

Sub-Domain Decomposition Iterative Method Combined With ACA: An Efficient Technique for the Scattering From a Large Highly Conducting Rough Sea Surface

The sub-domain decomposition iterative method (SDIM) is presented to solve efficiently a large linear system obtained by sampling the boundary integral equations from the method of moments. Then, this new technique is tested on the electromagnetic scattering problem from a large highly conducting one-dimensional rough sea surface. The diagonal blocks are the local impedance matrices corresponding to the geometry sub-domains while the off-diagonal blocks are the coupling matrices describing the interaction between two different sub-domains. The principle of SDIM is to invert the impedance matrix by blocks reducing significantly the complexity in comparison to a direct LU inversion of the whole impedance matrix. In addition, to accelerate the matrix-vector products and to reduce the memory requirement, the adaptive cross approximation (ACA) is applied to compress the sub-domain coupling matrices. For microwave frequencies, the results show that SDIM converges rapidly and faster for the TM polarization. Moreover, the mean compression ratio of ACA is of the order of 98%, which makes this method very efficient.

Bidirectional reflectance distribution function modeling of one-dimensional rough surface in the microwave band

In this study, the bidirectional reflectance distribution function (BRDF) of a one-dimensional conducting rough surface and a dielectric rough surface are calculated with different frequencies and roughness values in the microwave band by using the method of moments, and the relationship between the bistatic scattering coefficient and the BRDF of a rough surface is expressed. From the theory of the parameters of the rough surface BRDF, the parameters of the BRDF are obtained using a genetic algorithm. The BRDF of a rough surface is calculated using the obtained parameter values. Further, the fitting values and theoretical calculations of the BRDF are compared, and the optimization results are in agreement with the theoretical calculation results. Finally, a reference for BRDF modeling of a Gaussian rough surface in the microwave band is provided by the proposed method.

Fast Hybrid Method for the Study on Monostatic Scattering from Plasma-Coated Target above a Rough Surface

A fast hybrid method combining the reciprocity theorem with high frequency approximation algorithm is presented to deal with the problem of the monostatic scattering from a two-dimensional (2D) plasma-coated target above a one-dimensional (1D) Gaussian rough surface illuminated by the tapered incident wave. Without numerical solution of the polarization currents on the target and the surface, this hybrid method does not only save computer resources but also improve the computing speed significantly in contrast to the numerical methods. The hybrid method based on equivalent principle and reciprocity theorem, which is an improved and generalized version of the traditional multipath technique, can deal with the interactions between plasma-coated target and underlying surface much more accurately. Numerical results are given to verify the validity of the hybrid method, and then the hybrid method is employed to investigate the monostatic scattering from a plasma-coated airfoil above a Gaussian rough surface, including the effects of several key parameters on stealth performance, such as plasma angular frequency and electron collision frequency.

Modeling Bidirectional Reflectance Distribution Function of One-dimensional Random Rough Surfaces with the Finite Difference Time Domain Method

This article describes the development of a suite of programs that is capable of simulating the radiation properties of a random rough surface (RRS). The fundamental approach involves the generation, by fast Fourier transform (FFT) built with rigorous finite difference time domain (FDTD), as the theoretical basis for the simulation of a bidirectional reflectance distribution function (BRDF) of the RRS. The results are compared with the measurements and modeling of existing work to verify the feasibility of customized programming. It was found that the results of this study were a better match to the measurement data than those achieved in other modeling work.

Comparison of the finite element method with perturbation theory and the small-slope approximation for acoustic scattering from one-dimensional rough poroelastic surfaces

The finite element method is used to address the problem of acoustic scattering from one-dimensional rough poroelastic surfaces. The poroelastic sediment is modeled following the Biot-Stoll formulation. The rough surfaces are generated using a modified power law spectrum. Both backscattering strengths and bistatic scattering strengths are calculated. These results are compared with lowest order perturbation theory and the lowest order small-slope approximation, as extended to the case of scattering from poroelastic surfaces. It is known that these approximate methods are sufficient for the study of rough surface scattering in the case of sediments modeled as fluids. This work seeks to assess whether or not these methods are accurate when applied to the case of poroelastic sediments. [Work supported by the Office of Naval Research, Ocean Acoustics Program.]

A study of the reflection coefficients and backscattering effects of one-dimensional rough poroelastic surfaces using the finite element method

Acoustic reflection and scattering effects of one-dimensional rough poroelastic surfaces are studied using the finite element method. The poroelastic sediment layer is modeled following the classical work of Biot as extended by Stoll, which assumes that two attenuating compressional waves and one attenuating shear wave propagate in the sediment. The rough surfaces are generated using power-law type spectra and the incident wave used is a Gaussian tapered plane wave. This work seeks to assess how the reflection coefficients and backscattering effects of a poroelastic bottom vary as a function of frequency, roughness, and sediment type. Special consideration is given to the mesh required to accurately resolve the effects of the slow compressional and shear waves, which often have wave speeds slower than the fast compressional wave by an order of magnitude or more. [Work sponsored by the Office of Naval Research, Ocean Acoustics.]

A Transformation Media Based Approach for Efficient Monte Carlo Analysis of Scattering From Rough Surfaces With Objects

This paper presents a computational model that utilizes transformation-based metamaterials to enhance the performance of numerical modeling methods for achieving the statistical characterization of two-dimensional electromagnetic scattering from objects on or above one-dimensional rough sea surfaces. Monte Carlo simulation of the rough surface scattering problem by means of differential equation-based finite methods (such as finite element or finite difference methods) usually places a heavy burden on computational resources because at each realization of the Monte Carlo technique, a mesh must be generated anew for each surface realization. The main purpose of the proposed approach in this paper is to create a single mesh, without repeating mesh generation at each step, by introducing a transformation medium above the rough surface in the computational domain of the finite methods. Material parameters of the medium are obtained by the coordinate transformation technique, which is based on the form-invariance property of Maxwell's equations. At each realization, only the material parameters are modified with respect to the geometry of surface without changing the mesh. In this manner, a great reduction in CPU time is achieved. The proposed technique is analyzed and validated via various finite element simulations.

An analysis of the radar backscatter fromoil-covered sea surfaces usingmomentmethod andMonte-Carlo simulation: preliminary results

An analysis of the radar backscattering from the ocean surface covered by oil spill is presented using a microwave scattering model and Monte-Carlo simulation. In the analysis, a one-dimensional rough sea surface is numerically generated with an ocean waveheight spectrum for a given wind velocity. A two-layered medium is then generated by adding a thin oil layer on the simulated rough sea surface. The electric fields backscattered from the sea surface with two-layered medium are computed with the method of moments (MoM), and the backscattering coefficients are statistically obtained with N independent samples for each oil-spilled surface using the Monte-Carlo technique for various conditions of surface roughness, oil-layer thickness, frequency, polarization and incidence angle. The numerical simulation results are compared with theoretical models for clean sea surfaces and SAR images of an oil-spilled sea surface caused by the Hebei (Hebei province, China) Spirit oil tanker in 2007. Further, conditions for better oil spill extraction are sought by the numerical simulation on the effects of wind speed and oil-layer thickness at different incidence angles on the backscattering coefficients.