A wavelet multiscale–homotopy method for the inverse problem of two-dimensional acoustic wave equation

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Abstract

By introducing the wavelet multiscale method and the homotopy method to the inversion process for the velocity estimation problem of two-dimensional acoustic wave equation in seismic prospecting, a joint inversion method—wavelet multiscale–homotopy method is designed, which is stable, globally convergent, and has the ability of noise suppression. The results of numerical simulations and noise suppression tests indicate our method’s effectiveness.

Introduction

The inverse problems of wave equation is the key point in seismic prospecting. Traditional linearized inversion methods like Newton method and Newton-like methods which were effected by problem’s nonlinear and ill-posed properties may diverge if a good initial estimate cannot be provided. In order to improve the performance and efficiency of inversion, a wavelet multiscale–homotopy method is designed in this paper by introducing the wavelet multiscale method and the regularization–homotopy to the inversion process.

Wavelet multiscale inversion method is a newly developed inversion strategy to accelerate convergence, enhance stability of inversion, overcome disturbance of local minima, and by which we can search out the global minimum. For distributed parameter estimation problem of the one-dimensional elliptic equation, wavelet multiscale inversion was shown to be very effective [1]. Several authors have also used wavelet to analyze linear ill-posed problems [2], [3]. Recently, wavelet has been applied to solve nonlinear ill-posed problems. For instance, Chiao and Liang [4] presented a wavelet multiresolution procedure for identifying the parameter of nonlinear geophysics. Fu and Han [5] considers the problem of estimating the velocity in a two-dimensional acoustic wave equation. All these works showed the effectiveness of wavelet on inverse problems.

Homotopy method is a powerful tool for solving nonlinear problems due to its widely convergent properties [6], [7], [8]. It is a very interesting research to apply homotopy method to solve inverse problems of differential equations. Keller and Perozzi [9] may be the first work to use homotopy method to solve inverse problems. Later, several related researches appeared to combine a homotopy path-following formalism with singularity to track multiple solutions of a geophysical inverse problem [10], to develop a widely convergent homotopy method to determine the distributed parameters of an elliptical equation [11], to construct a widely convergent generalized pulse-spectrum technique for the inverse problem of one-dimensional wave equation [12], and to construct a homotopy–regularization method to solve inverse scattering problems with multi-experimental limited aperture data [13], and to construct a homotopy regularization method to solve the inverse problem of a two-dimensional acoustic wave equation [14], and to construct a homotopy method to solve general nonlinear inverse problems [15].

In fact, either multiscale method or homotopy method may fail to obtain the global minimum when they are used singly to solve the inverse problem of acoustic wave equation. The main reason lies in that we can not use the homotopy method to obtain the true solution when there are multi-solutions or obtain the global minimum when there are many local minima in the corresponding objective function; and we cannot use the multiscale method to obtain the global minimum at the longest scale, since the objective function may also not a convex one. Therefore, a natural idea is to make the combination of the two methods. The whole inversion process is conducted by the wavelet multiscale method, and the inversion at the longest scale is carried out by the homotopy method. This strategy may give a stable and globally convergent method.

Section snippets

Mathematical model

The inverse problem of acoustic wave equation attempts to find the velocity model that minimizes the difference between observed and modeled data. The modeled data are generated by the acoustic wave equation2ux2+2uz2-1v2(x,z)2ut2=s,(x,z)Ω=[0,L]×[0,H],t>0with boundary conditionsu(x,z,t)xx=0=u(x,z,t)xx=L=0,u(x,z,t)zz=H=0,and initial conditionsu(x,z,0)=u(x,z,0)t=0.Here the wave field u and the source signal s are functions of the space variables x,z and the time variable t, and the

A regularization–homotopy method

Eq. (2.8) is a nonlinear and ill-posed problem, therefore has to be regularized. Tikhonov regularization [16] is the most well-known regularization. Here we use it to regularize (2.8). Instead of directly solving (2.8), we solve the minimization problemJα(v)=T(v)2+αv-v02,where · is the 2-norm; α are regularization parameters; v0 is a prior estimate of the solution v. If v0 is close enough to v, the iteratively regularized Gauss–Newton methodvk+1=vk-T(ck)T(vk)+αI-1T(vk)T(vk)+α(vk-v0)

A wavelet multiscale–homotopy method

The theory and practical applications of the inverse problems of wave equation in seismic prospecting indicate that local extremum and ill-posedness are the bottlenecks restricting the development of the inversion theory. Therefore, we try to use wavelet to make decompositions. At every scale, use the iteratively regularized Gauss–Newton method as the optimization algorithm except the longest one, and at the longest scale, we use the regularization–homotopy method (3.6), (3.2) to find the

Numerical simulations

In this section, a synthetic example is tested to show the performance of our inversion algorithm. The exact velocity model are shown in Fig. 1, with the source of dominant frequency of 80 Hz, equispaced receivers on the surface along the x-axis (one receiver for each grid point in the x-direction).

In this numerical test, parameter N in formula (3.6) is chosen to be 10. Let hx = 50 m, hz = 50 m, τ = 0.01 s, by using the classic second-order, finite-difference scheme, (2.1), (2.2), (2.3), (2.4) and (2.6)

Conclusions

For the inverse problem of wave equation in seismic prospecting, we have designed a wavelet multiscale–homotopy method which takes advantage of the strengths of wavelet multiscale method and homotopy method, and realized the numerical inversion of velocity parameter of two-dimensional wave equation in seismic prospecting. The numerical results indicate our method’s effectiveness. And we have conducted numerical computation on some more complicated models, which also get good results. The method

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This work is supported by the National Science Foundation of China under Grant No. 40544016 and supported by Science and Technology Item of Education Department of Heilongjiang Province under Grant No. 11513041.

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