小波多分辨分析方法在波动方程正演模拟中的应用

THE APPLICATION OF WANELET MULTIRESULUTION ANALYSIS IN SEISMIC MODELING OF WAVE EQUATION

:初步探讨小波多分辨分析理论在地震波动方程正演模拟中的应用 ,给出一种基于地震波场局部变化性质而自适应调整空间网格点的波动方程数值算法 .目前求解波动方程所用的有限差分法、有限元法以及伪谱法都不能根据波场的局部变化性质而动态选择空间网格点的大小 .小波基函数在空间域和频率域中都具有局部性特征 ,它的性质优于有限差分法、有限元法和伪谱法中所用的基函数 .通过阀值运算 ,地震波场的多分辨表示变得非常稀疏 ,同时地下介质中的主要信息又不会受到损害 .本文将声波方程的矩阵表示形式在小波多分辨分析的框架下进行了展开 .通过对算子矩阵和地震波场矩阵进行多分辨分解和压缩 ,得到了小波域中地震波场正演模拟算法 .

This paper studies the application of wavelet theory in seismic modeling. The motivation of this work is to develop a numerical method, which an dynamically adapt computationally grid to local structures of the wave solution. Until now, however, the conventional numerical methods such as finite difference method, finite element method and pseudo spectral method can not dynamically adapt computationally grid to local feature of wave solution. Good wavelet locallzation properties in physical and wavenumber spaces are to be contrasted with the spectral approach, which employs infinitely differentiable functions but with global support and small discrete changes in the resolution. On the other hand, finite difference and finite element methods use bases with small compact support but poor continuity properties. Through thresholding, the multiresolution representation of seismic data is very sparse with little damage to the main information of the earth. Based on the above mentioned facts, this paper represents the matrix form of sonic wave equation in the frame of wavelet multiresolution analysis. Through the multiresolution decomposition and compression of operator matrix and seismic wave field matrix, we proposed a new method of seismic modeling in wavelet domain.