Studies On The Direct And Inverse Scattering Problems Of Unbounded Rough Surface With Tapered Electromagnetic Waves Incidence

Scattering and inverse scattering of unbounded rough surface is an importantproblem in scientific research and engineering practice, especially in geoscience remotesensing, target recognition, optical difraction and other areas. The theory of scatter-ing and inverse scattering of acoustic and electromagnetic waves plays an importantrole in modern scientific fields. The direct scattering problem, given the informationof the incident wave and the nature of the scatterer, is to find the scattered waveand in particular its behavior at large distances from the scatterer. The inverse scat-tering problem is to find the boundary or physical parameters of the scatterer withgiven incident field and scattered field (or far-field). The scattering problems can besolved numerically by integral equation method, finite element method and infiniteelement method, etc. Roughly speaking, numerical methods for solving inverse scat-tering problems can be classified into three groups: iterative methods, optimization methods and sampling methods. In this dissertation, the study focused on problem of the unbounded rough surface scattering and inverse scattering with tapered wave incident. First, based on the model problem of scattering by unbounded rough surface with tapered wave incident, we strictly deduced the corresponding integral equation and studied the properties of the integral operators. Then, according to the specific characteristics of the corresponding integral equation, we used FMM and RCGM for the numerical solution of the corresponding integral equation. A strategy for the se-lection of the regularization parameter is obtained by estimating the double integral as a perturbed right-hand side of the integral equation. Finally, we discussed the shape reconstruction of the sound soft (perfectly conducting) unbounded rough surface with tapered wave incident. The concrete work is as follows:I. Boundary integral equation methods for scattering by an un-bounded rough surface with tapered wave incidenceLet F:={(x, f(x)) E R2|x E R} denotes the unbounded rough surface, Ω:={r=(x,z)∈R2|z> f(x)} denotes the propagation domain above F, where K=2π/A is the fundamental surface wave number, b(b>1) is the frequency scaling parameter, h is the root mean square height, D(1<D<2) is the fractal dimension, φn is the specified phases, is a normalization factor. The schematic of the unbounded rough surface scattering with tapered wave incidence is shown in Fig.1.The closed region Dr with boundary (?)Dr=Hr U Sr U Tr, where Hr is a large semicircle of radius r and center O in z> f(x), Sr={(x, z)|z=f(x),-r<x<r} is a truncated rough surface, and Tr consists of line segments joining the ends of Hr and Sr. We also define the closed region Dr with boundary (?)Dr=Hr U Sr U Tr, where Hr is a large semicircle of radius r and center O in z<f(x), and Tr consists of the line segments joining the ends of Hr and Sr. Fig.1A schematic of the unbounded rough surface scattering problem with the tapered wave incidenceConsidering the Thorsos tapered wave incident, and the incident field Uinc satisfies nonhomogeneous Helmholtz equation where and additional phase term w(r) is and where kgcosθinc>>1, is the angle of incidence (it is the angle between the direction of propagation and the negative z axis), and g is the parameter that controls the tapering, k=2π/λ is the wave number, λ is the wavelength of incident wave, The tapered incident wave limit is the plane wave as g�'+∞. The scattered field us satisfies the Helmholtz equation thus full field u satisfies the equation and Dirichlet boundary conditionsDefinition0.1Let function us∈C2(Ω) is said to satisfy the angular spectrum representation radiation condition (ASRC), if where uDir(x)∈L1(R)∩L2(R), and (Fus)(ζ,l0) is the Fourier transform of uDir(x)=us(x,l0)(x∈R).In this dissertation, we consider the mathematical model of the scattering prob-lem with tapered wave incidence by an unbounded sound soft surface or perfectly conducting surfaces:Problem0.1Give f∈C (R), and an incident field uinc is defined by (2), find us∈C2(Ω)∩C(Ω), such that u:=us+uinc satisfies the equation (4) and boundary condition (5), where us satisfies the Helmholtz equation (3) and the ASRC (6). i. The derivation of boundary integral equationLet the two-dimensional Green’s function satisfies where H0(1)(·) is the first class of zero-order Hankel function, r=(x,z) and r’=(x’,z’) are points in R2,δ(·) is Dirac-δ function. We introduceFor scattering problem0.1, in order to obtain its corresponding boundary integral equation, we firstly prove the following lemmas:Lemma0.1If v(r,θ;α)=exp(ikr cos(θ-α)), where|α|≤π/2, thenLemma0.2If us satisfies the angular spectrum representation radiation con-dition(6), thenLemma0.3Let plane wave up(r,θ;θinc):=exp(-ikr cos(θ+θinc)), where|θinc|<π/2,|θ|<π/2, thenAccording to lemma0.1, lemma0.2and lemma0.3, the corresponding boundary integral equation is derived.Theorem0.1If the total field u satisfies the Helmholtz equation (4), and the scattered field us satisfies the angular spectrum representation radiation condition (6), then u satisfies the integral expression as follows ii. The properties of integral operators We will introduce a space equipped with the corresponding norm where We define the operator P form X to Y:=PX by thus, the integral equation can be written in operator form as whereIn the following, we proved the boundedness of operator P:Theorem0.2For all f∈C2(R), the operator P:X�'Y is a bounded linear operator.For sufficiently large L, we define the operator PT byIn the following, we studied the properties of operator P and PT:Theorem0.3The integral operators PT:X�'Y is a bounded linear operator, and we haveⅡ. Regularized conjugate gradient method based on the fast multi-pole acceleratedi. Numerical discretization of the boundary integral equationSpace XT is the restriction of X for x on [-L, L], and YT:=PTXT. The relevant norm areLet S:XT�'YT as Definition0.2We recall that a kernel K(x,y) is called weakly singular on dD x dD if K(x,y) is denned and continuous for {(x,y)\x,y E dD,x≠y}, and there exist constants M>0and α∈(0,m-1] such that where dD is of class C1.Lemma0.4C((?)D) is a Banach space equipped with the corresponding norm‖·‖∞. Let the mapping A: C(?)D)�'C((?)D) be defined by where K(x,y) is continuous on dD x(?)D or weakly singular on dD x dD, Then A defined by (0.4) is compact as an operator from C((?)D) into C((?)D).Theorem0.4Integral operator S: XT�'YT is a compact linear operator.Note that the truncated integral equation is that and its operator formApplying the point matching method, and taking into account the singularity of Hankel function and theorem0.1, we get the discretization form of the corresponding integral equation: Applying the small argument approximation of the Hankel function, the equations (12) becomes In the actual numerical solution, for ρm is often ignored in discrete form of operator equation (11), that is solving the following equations: For obtain the estimates of the right-hand member ρm, we proved the following theo-rem.Theorem0.5For the right side of (13), the following estimates hold: where C is a constant which is dependent on k, L,θinc, H1, M1, M.ii. Numerical discretization based on the fast multi-pole acceleratedWhen solving equations (13), the main problem of the above method is that the computational complexity grows dramatically as the size of the scattering surface increases. So we consider to use the FMM, we get the discretization form of the corresponding integral equation: The linear systems (15) can be expressed as whereiii. Regularization conjugate gradient methodCurrently, most of the researches on rough surface scattering problem with tapered wave incidence had omitted the double integral∫Ωk2R(r’)G(r, r’)dσ in the correspond-ing integral equation. Although it is a small perturbation, it is known that a small perturbation for ill-posed problem may bring some troubles. More importantly, it has an impact on the selecting of regularization parameter for RCGM. So we will introduce the noise level S. The RCGM is terminated with m=m(δ, bδ) when where AUmδ—bδ is the residual of the mth iteration, bδ is the perturbed right-hand side of linear equations (16), τ(τ>1) is a priori chosen constant. For scattering of acoustic waves from sound soft surfaces or scattering of horizontally polarized electromagnetic waves from perfectly conducting surfaces, the received power by the rough surface equals to the scattered power. So we also use the energy relation as a posteriori iteration stopping criterion to ensure the effectiveness of computing, i.e. energy index where σ(θs) is the BSC. RCGM is given below.Algorithm0.1.(RCGM)1. Input the noise level S and the largest admissible number of iteration steps mmax, chose constant τ (τ>1), set m=1;2.Input the initial guess U0δ, compute r0=bδ-AU0,δ,p1=s0=A*r0;3.Compute the received power by the rough surface4.Input and the energy stopping tolerance e;5.do while6.if (m≥mmax or‖sm‖∞≤τδ<‖|sm-1‖∞), break; else compute 7. compute8. Set m=m+1;It should be noted that the steps3,4,5,7in algorithm0.1can ensure the energy relation for scattering calculation. In order to obtain the stopping rule, we given the estimate of p(x): Hence, we obtained the noise level.iv. Numerical results and their analysisIn this dissertation, we numerically calculated the BSC for one-dimensional fractal rough surface with tapered wave incidence by using the RCGM for several numerical examples. The influence of different parameters of fractal rough surface on scattering coefficient is considered, the accuracy and efficiency of the RCGM is also studied. Numerical results show that:(1) The stopping criteria of RCGM is effective for controlling the number of it-erations. The RCGM can be performed accurately and efficiently to deal with the scattering problem of large size rough surface with tapered wave incident, which can not be easily accomplished by MOM or classical CGM for their disadvantages of large memory dependence or lack of appropriate stopping criteria.(2) The parameters of rough surface often have nonlinear effects on the distribution of scattering field, we can see that h has significant effect on the distribution of BSC and the other parameters also have some effects on the distribution of BSC for a certain incident wave. In fact, some information of the rough surface itself, such as parameters h, D or K ect., can be retrieved from the data of BSC(or the scattering field). Ⅲ. The iterative method for multi-parameter of unbounded fractal rough surfacesIn this section, we study the parameters reconstruction and shape inversion of the rough surface based on the scattered field. It is a simplification of ground surface (or sea level) imaging via ground-penetrating radar(GPR). In GPR systems, arrays of above-ground transmitters and receivers illuminate surfaces of interest and receive scattered data from the rough surface(see Fig.2), and then using the received data for surface imaging or target recognition. Here, we investigate the parameters inversion and shape reconstruction of unbounded rough surface with tapered wave incidence.Fig.2The schematic of the rough surface inverse scattering with the ta-pered wave incidencei. Fast iterative algorithm of the inverse scattering problemWe presented a new inversion algorithm for the reconstruction of unbounded frac-tal rough surfaces from a set of scattered field measurements for an illumination by tapered wave. The unknown parameters of the surfaces are estimated by minimizing the cost function which is defined as some norm between simulation scattered field data and measured scattered field data. The approach used the fast forward solver-RCGM based on tapered wave incidence combined with the BFGS optimal technique. The flowchart of the inverse scattering algorithm is shown in Fig.3.ii. Numerical implementation and examplesIn general, the cost function is a nonlinear function of the parameters. It is difficult to predict the behavior of cost functions with respect to variations in these parameters. Fig.3Flowchart of inverse scattering algorithmIt is one of the important reasons for the convergence of the iterative method which is difficult to prove. Therefore, in the next study, we examined the nature of the inverse algorithm convergence by numerical examples. The influence of measurement error, initial data, survey position and different rough surfaces on the inversion results were studied. The incidence strategy of multi-angle and multi-frequency can achieve the better results of reconstruction than single incidence within much less iteration steps. The numerical results show that the proposed algorithm is accurate, and the inversion results can be obtained with reasonable computing times.