A wavelet multiscale method for inversion of Maxwell equations

Transcription

1 Appl. Math. Mech. -Engl. Ed. 30(8), (2009) DOI:.07/s c Shanghai University and Springer-Verlag 2009 Applied Mathematics and Mechanics (English Edition) A wavelet multiscale method for inversion of Maxwell equations Liang DING ( ), Bo HAN ( ), Jia-qi LIU ( ) (Department of Mathematics, Harbin Institute of Technology, Harbin , P. R. China) (Communicated by Xing-rui MA) Abstract This paper is concerned with estimation of electrical conductivity in Maxwell equations. The primary difficulty lies in the presence of numerous local minima in the objective functional. A wavelet multiscale method is introduced and applied to the inversion of Maxwell equations. The inverse problem is decomposed into multiple scales with wavelet transform, and hence the original problem is reformulated to a set of sub-inverse problems corresponding to different scales, which can be solved successively according to thesizeofscalefromtheshortesttothelongest. ThestableandfastregularizedGauss- Newton method is applied to each scale. Numerical results show that the proposed method is effective, especially in terms of wide convergence, computational efficiency and precision. Key words Maxwell equations, wavelet multiscale method, inversion, regularized Gauss-Newton method, finite difference time domain method Chinese Library Classification O157.2, O Mathematics Subject Classification 35D20 Introduction The inversion of Maxwell equations has shown significant potential in oil and gas exploration. It also has important application in mineral, geothermal exploration and general geologic mapping. 3D-inversion has been used successfully to map subsurface transport pathways for contaminants [1] to delineate buried metallic waste, to define the extent of waste pits and to determine the safety of proposed long-term waste disposal sites [2]. There are currently many new methods applied to the inverse problems of Maxwell equations [3-6]. In spite of these successes, the inversion continues to be a cumbersome and complex process requiring significant time and computational resources [7-8], thus restricting its application. In most cases, such problems have been tackled by the Newton iterative method. However, the primary difficulty of this method is the presence of numerous local minima in the objective function. The presence of local minima at all scales prevent iterative methods from attaining a reasonable degree of convergence to the global minimum. Recently, with the development of wavelet theory, some wavelet frameworks have been applied to inverse problems [9-12]. This allows the idea of wavelet multiscale to be used for the inversion of Maxwell equations with little modification from the classical optimization methods. Received Dec. 27, 2008 / Revised Jun. 29, 2009 Project supported by the Program of Excellent Team of Harbin Institute of Technology Corresponding author Bo HAN, Professor, Ph. D., bohan@hit.edu.cn

2 36 Liang DING, Bo HAN, and Jia-qi LIU Wavelet is already recognized as a powerful new mathematical tool. The last few years have witnessed an intense activity and interest in the application of wavelet theory and its associated multiresolution analysis [13]. There are several papers that have involved the use of wavelets in Galerkin and other classical methods [14]. There have also been some developments in finite element and multigrid methods using wavelets. Recently, the multiscale method has been applied to distributed parameter estimation problems. The results showed that iterative inversion methods performed much better when employed with a decomposition by scale. All these works showed the effectiveness of the wavelet analysis method on inverse problems. In this paper, we investigate numerically the behavior of wavelet decomposition, in the case of the unknown conductivity and the wave fields are represented in the wavelet basis. This leads us to propose a multiscale optimization method to solve the parameter estimation problem. Once the inverse problem is decomposed to multiple scales by wavelet transform, the original inverse problem is reformulated to be a set of sub-inverse problems corresponding to different scales and is solved successively according to scale from the shortest to the longest. The longest scale component of the problem contains moderate varying features, and consequently, a regularized Gauss-Newton method can easily be used to find its global minimum. The global minimum of the longest scale component is in the neighborhood of the global minimum for the problem on the second longest scale. Therefore, the application of the regularized Gauss- Newton method initialized with the solution obtained from the long scale problem has a good chance of finding the global minimum of the original problem. This method turns out to be very robust (convergence is obtained for any initialization) and efficient (a good solution is obtained with a very small number of iterations). The multiscale method allows us to perform, for the problem under consideration, a global optimization. Moreover, the method of rescaling of the variables and the objective function for each optimization run can be easily performed to accelerate the convergence. The organization of this paper is as follows. In Section 1, we describe the forward model and the finite difference time domain (FDTD) method. In Section 2, we discuss the traditional inversion method. In Section 3, we discuss wavelet and multiresolution analysis. In Section 4, we discuss the wavelet multiscale method. We give numerous examples in Section 5 and the paper is summarized in Section 6. 1 Forward model and FDTD method The Maxwell TM equations in the 2D space are given as follows: with initial conditions ε E y t μ H x t μ H z t H x z + E y z =0, + E y x =0 + H z x + σe y = j y, (1) E t<0 = H t<0 =0, (2) and absorbing boundary conditions 2 E z t εμ 2 E t E εμ x 2 σ μ 2 ε 2 E x t εμ 2 E t E εμ z 2 σ 2 E t μ E ε t =0 (z = l), (3) =0 (x =0), (4)

3 A wavelet multiscale method for inversion of Maxwell equations 37 2 E x t + εμ 2 E t E εμ z 2 + σ μ E =0 (x = h). (5) 2 ε t An additional condition is E(x, 0,t)=f(x, t), (6) where f(x, t) is the observed data, σ(x, y) is the electrical conductivity, ε(x, y) is the electric permittivity, and μ(x, y) is the magnetic permeability. Furthermore, we take ε(x, y) andμ(x, y) to be constant. E(x, y, t) andh(x, y, t) are the electric and magnetic fields, respectively. The symbol E x denotes the electric field in the x-direction. If the parameters σ(x, y),ε(x, y), and μ(x, y) are known, the direct problem is to solve E and H. When only the observed data, ε(x, y), and μ(x, y) are known, the inverse problem is to reconstruct the parameter σ(x, y). We can discretize (1) by using Yee grid [15] as follows: Ey k+1 2ε(i, j) σ(i, j)τ (i, j) = 2ε(i, j)+σ(i, j)τ Ek y (i, j) k+ 2τ ε(i, j)+σ(i, j)τ [H x (i, j 1 2 ) 1 Hk 2 x (i, j 1 2 ) z (i + 1 2,j) 1 Hk 2 z (i 1 2,j) ], h x (7) H k+ 1 2 x (i, j )=Hk 2 x (i, j )+ τ [Ey k μ(i, j)l (i, j +1) Ek y (i, j)], z (8) Hk+ 1 2 H k+ 1 2 z (i ,j)=Hk 2 z (i + 1 2,j)+ τ [Ey k μ(i, j)h (i +1,j) Ek y (i, j)] (9) x with initial conditions E t<0 = H t<0 = 0. () An additional condition is discretized as l z E k i,0 = f k i, (11) where h x and l z are the step sizes in the x- andz-direction respectively, τ is the step size of time, m is the grid number in the x-direction, n is the grid number in the y-direction, and p is the grid number of time. We denote a grid point of the space as and for any function of the space and time we denote 2 Traditional inversion methods (i, j) =(ih x,jl z ), (12) E(ih x,jl z,kτ)=e k (i, j). (13) A main impediment to inverting time domain electromagnetic data is the size of the problem and the number of computations to be carried out. To be more specific, it is assumed that the forward problem (in continuous space) is of the form A(m)u b =0, (14) where A(m) is a version of Maxwell equations (including boundary conditions), m = lg(σ) is the log conductivity, u stands for the fields, and b represents sources and boundary values. We

4 38 Liang DING, Bo HAN, and Jia-qi LIU assume for simplicity of exposition that A is invertible for any relevant m. That is,there is a unique solution to the forward problem. In the inverse problem we measure some function of the fields and desire to recover the model m. Let us write the measured data as d obs = Qu + ε, (15) where Q is a measurement operator which projects the fields (or their derivatives or integrals) onto the measurement locations in 3D space and time, and ε is the measurement noise. The data are finite in number and contaminated with noise, and therefore there is no unique solution. To obtain a single model which depends stably on the data, we incorporate a-priori information and formulate the inverse problem (in continuous space) as a constrained optimization problem of the form subject to min m,u 1 2 Qu dobs 2 +βr(m) (16) A(m)u b =0. (17) Here, β>0istheregularization parameter, and R( ) is a regularization operator reflecting apriori information. Typically, we know that m is a piecewise smooth function over the spatial domain Ω in 3D, so we assume that R( ) involves some norm of m over Ω, e.g., weighted L 2 or L 1 or a Huber combination [16-17]. This type of regularization can be written as R(m) = ρ( m )dx + γ 1 (m m ref ) 2 dx, (18) Ω 2 Ω where ρ is given by the Huber function, m ref is some background reference model, and γ is a user-specified constant that adjusts the relative influence of the two terms in the regularization functional. The choice of this functional ensures that the weak form has Dirichlet boundary conditions and that in regions far away from where we have data, the model converges to a known background. This choice also guaranties that the Hessian of the regularizer is invertible. Next, the problems (16) and (17) are discretized by using some finite difference method over a finite grid representing the domain in space and time. This yields the finite dimensional optimization problem subject to min m,u 1 2 Qu dobs 2 +βr(m) (19) A(m)u b =0, (20) where u, m,and b are grid functions ordered as vectors corresponding to their continuous counterparts as above; Q and A are large, sparse matrices; the matrix A depends on m and is nonsingular. In this work we aim to deal with very large scale problems. For such problems one cannot store all the fields on computer hardware that is currently available to us, and we therefore turn to other avenues for the solution to the problem. A simple way to reduce storage is to use a reduced space method, which eliminates Maxwell equations and solves the unconstrained optimization problem min m This leads to the Euler-Lagrange system 1 2 QA(m) 1 b d obs 2 +βr(m). (21) g(m) =J(m) T (QA(m) 1 b d obs )+βr m (m) =0, (22)

5 A wavelet multiscale method for inversion of Maxwell equations 39 where J(m) = QA(m) 1 G(m), (23) G(m) = [A(m)u] m, (24) and J(m) is the sensitivity matrix [18] and R m (m) = R m. Newton-type methods are generally far superior to descent methods when solving (22). In this paper the regularized Gauss-Newton method is used, then at each iteration we solve the linear (but dense) system (J(m) T J(m)+βR mm )s = g(m), (25) where s is a model perturbation. This system is never formed explicitly but is solved by using conjugate gradient (CG) methods. At each CG iteration J(m) T and J(m) are applied to a vector and hence a forward and adjoint problem must be solved. Even with reasonably good preconditioners, tens of CG iterations are likely to be required to get a good estimate of s. 3 Wavelet multiresolution analysis by Definition 1 Its inverse transform is For any function f(x) L 2 (R), its continuous wavelet transform is defined f(x) =C 1 ψ W f (a, b) = a f(x) ψ a,b (x)dx. (26) 1 W a f (a, b)ψ( )dadb. (27) x b Definition 2 A multiresolution analysis (MRA) for L 2 (R) is a nest of closed subspaces {V j,j Z} of L 2 (R), satisfying (1) V j V j+1, j Z; (2) V j = {0}; j Z (3) ( V j )=L 2 (R); j Z (4) f(x) V j f(2x) V j+1 ; (5) there is a function φ V 0 such that {φ(x n),n Z} is an orthonormal basis for V 0. The basis function φ jk (x) in subspace V j consists of the dilates and translates of the scaling function φ(x), and the basis function ψ jk (x) in subspace W j consists of the dilates and translates of the wavelet function ψ(x). From the definition of MRA, we can obtain V J = V J 1 W J 1 = V J 2 W J 2 W J 1 = V 0 W 0 W J 1. (28) A given function can be expressed in terms of scaling and wavelet functions f = k c Jk φ Jk (x)+ k J 1 d jk ψ jk (x). (29) j=0 When the problem is decomposed to scale J 1fromscaleJ, since the space V J 1 only consists of the lower frequency part, the solution in the space V J 1 should be an approximation to the original one, which does not contain higher frequency part, it is only a local trend near

6 40 Liang DING, Bo HAN, and Jia-qi LIU the parameter of the model. But since the subspace W J 1 of V J contains the higher frequency part, we cannot get the information of higher frequency at a smaller scale in space V J 1, but we can obtain it through the inversion in space V J. Similarly, we cannot get the information of further higher frequency in space V J, but can obtain it through the inversion in space V J+1. Therefore, the resolution can increase gradually. 4 Wavelet multiscale method The Newton-type method is fast and stable for the inversion of Maxwell equations when the initial electrical conductivity model is in the neighborhood of the global minimum of the objective function. However, the direct application of this inversion technique to real and synthetic data has been disappointing when the initial data is far away from the true data or a good knowledge of the background electrical conductivity field is unavailable. This is because numerous local minima in the objective function prevent convergence. The multiscale method is based on decomposing a problem by scale, followed by the resolution of each scale component by a suitable relaxation operator(the relaxation operator is the regularized Gauss-Newton algorithm in this paper). This approach has two advantages. First, it accelerates convergence and reduces computational cost of the relaxation operator on longer scale. This is because on a longer scale, the objective function shows stronger convexity and has less minima. Second, for the nonlinear aspects of the problem the solution of the longer scale component is followed by a fining-up procedure on shorter scales. This approach is more effective on finding the global minimum of the problem. When the objective functional is decomposed into different scales from the shortest to the longest, the longest scale component of the problem contains little varying features. Consequently, the regularized Gauss-Newton method can easily be used to find the global minimum of this component of the problem. The global minimum of the longest scale component is in the neighborhood of the global minimum for the problem at the second longest scale. Thus, an application of the regularized Gauss-Newton method initialized with the solution obtained from the longest scale problem has a good chance of finding the global minimum of the second longest scale problem. Successively fining upward in this manner succeeds in finding the global minimum of the original fine scale problem. In this paper, we try to use wavelet (Daubechies 4) to decompose the object function on each scale. The source and electrical field are transformed by wavelets on different scales. Accordingly the initial objective function is decomposed into a sequence of nonlinear minimization problems on different scales. The regularized Gauss-Newton method is carried out on each scale. Then we can successively find the global minimum of the inverse problem with the scale changing from the longest to the original scale. The wavelet multiscale method requires three elements illustrated in Fig. 1. The first element is an operator that restricts the original problem to longer scales. The second element is an operator that performs relaxation on each scale. The third element is an operator that injects the solution available on a longer scale into the problem setting on a shorter scale. In Fig.1, S(0) represents the scale 0; it is on this scale that the original problem is defined. The operator H(0, 1) which maps the original problem from S(0) to S(1) is a restriction operator, it is wavelet decomposition algorithm H(0, 1) : S(0) S(1). (30) Once the problem has been restricted to longer scale S(1), the solution on this scale is obtained by relaxation(the regularized Gauss-Newton method (R(1))). Once the relaxation operator has been used to solve the inverse problem on scale S(1), the solution is then injected into the scale S(0) with the injector H(1, 0), it is the wavelet reconstruction algorithm H(1, 0) : S(1) S(0). (31)

7 A wavelet multiscale method for inversion of Maxwell equations 41 S(0) H(0,1) S(1) R(0) H(1,0) S(J 1) S(J) Fig. 1 R(J 1) H(J 1, J) H(J, J 1) R(J) Description of the algorithm The algorithm is summarized as follows: Step 1 For Maxwell equations, given the true electrical conductivity model, the source function, an initial electrical conductivity and the observed data. Step 2 Decompositon of the source and electrical field into different scales (from 0 to J). Step 3 Carrying out the regularized Gauss-Newton iteration with the initial conductivity, we can obtain the solution of optimization on the longest scale. Step 4 Shortening one scale, reiterating the solution from Step 3 as the new initial conductivity and repeating Step 3. Step 5 Proceeding inductively until the global minimum of the original inverse problem (on scale 0) is obtained. 5 Numerical simulations In this section, we experiment with three model problems to demonstrate the effectiveness of our method. These problems are commonly used to test time domain electromagnetic inversion codes for geophysical applications. Example 1 We select a homogenous layer model for simulation, where permittivity ε = F/m, permeability μ =2.73 H/m, log electrical conductivity m =lg(σ) =2.35, the scale is 5. The results of inversion by the regularized Gauss-Newton method are shown in Table 1. The results of inversion by wavelet multiscale method are shown in Table 2. By comparing Table 1 and Table 2, we see that when the initial value is poor (as experiments 1, 2, 6, and 7), there is no convergence by the regularized Gauss-Newton method, while for the multiscale method, convergence and a good approximation can be achieved. Furthermore, from experiments 3, 4, and 5, we observed that the multiscale method is more efficient against the regularized Gauss-Newton method, and the approximate error is large. Therefore, we conclude that the wavelet multiscale method is widely convergent, accurate, and efficient. Example 2 In the second example we consider the model of three anomalies in a homogeneous media with conductivity of. The anomalies have the conductivity of 5, 2, and 1, respectively, and the conductive block is 30 m 30 m. The source is a square current loop on top of the earth. The measurements are taken in boreholes located at the corners of the loop. We place 6 receivers in each borehole. Each receiver measures 3 components of the magnetic field. The total data is 6 (recievers) 32 (times) 2 (boreholes) 3 (fields) = We discretize the problem on a grid size The size of the unknown discrete conductivities vector is A sketch of the model is plotted in Fig. 2(a). Other parameters are the same as in Example 1. In our experiment we solve the forward problem of this mesh on a dual core processor (dual 3.00 Ghz) and takes roughly 5 minutes. A typical inverse problem can involve as many as 30 forward/adjoint problems which takes about 0 minutes. For each scale the

8 42 Liang DING, Bo HAN, and Jia-qi LIU Gauss-Newton iteration takes about 5 minutes and the longest scale is usually related to the source and velocity of electromagnetic wave. The results of this experiment are presented in Fig. 2(b) and Fig. 2(c). Table 1 The results of inversion by the Gauss-Newton method Number Initial value Real value Solution CPU time t/s Error e/% No convergence No convergence No convergence No convergence Table 2 The results of inversion by the wavelet multiscale method Number Initial value Real value Solution CPU time t/s Error e/% y/(30 m) x/(30 m) x/(30 m) x/(30 m) (a) The true model (b) The inversion results (c) Inversion results with 25 db y/(30 m) y/(30 m) Fig. 2 Thetruemodelandinversionresults Example 3 In the third example we want to identify that wavelet multiscale method can be extended in the 3D case. We consider the 3D model of a single anomaly of 5 in a homogeneous media with conductivity of 90. Unlike the previous example, the conductivity contrast is high which makes the inverse problem highly nonlinear. The size of the conductive block is 20 m 20 m 20 m. The source is a square current loop on top of the earth. The model is plotted in Fig. 3 (left). To solve the problem we discretize on a grid of size The size of the unknown discrete conductivity vector is The recovered data is plotted in Fig. 3 (right). The numerical results demonstrate that although the conductivity contrast is much higher, the effectiveness of the method is unchanged.

9 A wavelet multiscale method for inversion of Maxwell equations 43 Synthetic model Inverted model Z/m 12 Z/m Y/m 0 X/m 20 0 Y/m 0 X/m Fig. 3 True model (left) and the inversion results (right) recovered 6 Conclusions As seen in the examples, the wavelet multiresolutionmethodproposedinthispaperhas been successful in the inversion of Maxwell equations. All of these demonstrate that this method is effective in diminishing the influence of local minima and reduce the computational cost. Especially, Fig. 2(c) shows that this method is also stable. However, we also have a pendent problem. It is important for the wavelet multiresolution method to select a suitable decomposition scale. If we choose a scale that is too long, the information about the real solution in this scale will be too little, the convergent rate will be slower, and the approximate solution will not be obtained in some cases. Therefore, how long should the scale be for the inverse problem, and is there some necessary relation between scale and parameter? The issue will be considered in the future. References [1] Alumbaugh, D. L. and Newman, G. A. 3D massively parallel electromagnetic inversion II. analysis of a cross well experiment. Geophys. J. Int. 128, (1997) [2] Newman, G. A., Recher, S., Tezkan, B., and Neubauer, F. M. 3D inversion of a scalar radio magnetotelluric field data set. Geophysics 68(3), (2003) [3] Ascher, U. M. and Haber, E. A mutigrid method for distributed parameter estimation problem. Electron.Trans.Numer.Anal.15, 1 17 (2003) [4] Baboolal, S. and Bharuthram, R. Two-scale numerical solution of the electromagnetic two-fluid Plasma-Maxwell equations: shock and soliton simulation. Mathematics and Computers in Simulation 76(1-3), 3 7 (2007) [5] Haber, E. Quasi-Newton methods for large-scale electromagnetic inverse problems. Inverse Problems 21(1), (2005) [6] He, Sailing and Weston, V. H. Wave-splitting and absorbing boundary condition for Maxwell s equations on a curved surface. Mathematics and Computers in Simulation 50(5-6), (1999) [7] Dorn, O., Aguirre, H. B., Berryman, J. G., and Papanicolaou, G. C. A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields. Inverse Problems 15(6), (1999) [8] Farquharson, C. G., Oldenburg, D. W., and Li, Y. G. An approximate inversion algorithm for time-domain electromagnetic surveys. Journal of Applied Geophysics 42(2), (1999)

10 44 Liang DING, Bo HAN, and Jia-qi LIU [9] Cohen, A., Hoffmann, M., and Reiss, M. Adaptive wavelet Galerkin methods for linear inverse problems. SIAM J. Numer. Anal. 42(4), (2004) [] Dicken, V. and Maass, P. Wavelet Galerkin methods for ill-posed problems. J. Inverse and Ill- Posed Problems 4(3), (1996) [11] Fu, Chunli, Zhu, Youbin, and Qiu, Chunyu. Wavelet regularization for an inverse heat conduction problem. J. Math. Anal. Appl. 288(1), (2003) [12] Liu, Jun. A multiresolution method for distributed parameter estimation. SIAM J. Sci. Comput 14(2), (1993) [13] Fu, Hongsun and Han, Bo. A wavelet multiscale method for the inverse problems of a twodimensional wave equation. Inverse Problems in Science and Engineering 12(6), (2004) [14] Bunks, C., Saleck, F. M., Zaleski, S., and Chavent, G. Multiscale seismic waveform inversion. Geophysics 60(5), (1995) [15] Yee, K. S. Numerical solution of initial boundary value problems involving Maxwell s equations in isotropic media. IEEE Transactions on Antennas and Propagation 14(3), (1966) [16] Farquharson, C. and Oldenburg, D. Non-linear inversion using general measures of data misfit and model structure. Geophysics 134(1), (1998) [17] Huber, P. J. Robust estimation of a location parameter. Ann. Math. Stat. 35(1), 73 1 (1964) [18] Haber, E., Ascher, U., and Oldenburg, D. On optimization techniques for solving non-linear inverse problems. Inverse Problems 16(5), (2000)