Migration Of The Full Wave Equation In The Wavelet Domain

The traditionary Kirchoff and wave migration all decompose wavefield with a sort of base functions, which are simple and global solutions to the wave equation. For example, after operated with Fourier transformation, Helmholtz equation becomes a simple equation with a solution to plane wave, which describes propagation of sine wave in the uniform media. Anther example is Green function of point source, which also has a simple form in the uniform media. The wavefield initiated by point source is full of all of space and directions. Hence, its propagation in the heterogeneous media is very complicated. In a view of mathematics, this difficulty is due to the global character of the base function, like Fourier transformation, so it is very hard to obtain the effective equation that describes the wave propagation in the heterogeneous media. Therefore, the global character leads to low resolution of image, for instance, Kirchoff migration based on Green function of point source may not work for the complex structure. Moreover, the global character results in low efficiency of computation, like the Fourier migration adopting sine or cosine functions as base functions, which is time-consuming and has intensive memory for large-size problems. The newly developed fast wavelet transform is considered to be revolutionary breakthrough in signal processing, which is applied in many fields for its local analysis and multiscale analysis. On the basis of predecessors' work, a new migration method in the wavelet domain is presented in this paper. According to the standard form of multiscale representation of operator, we deduced multiscale representation of wave equation and initial and boundary conditions in the wavelet domain, and then obtain formulations for migration. The wave equation in the wavelet domain has two merits that matrices are very sparse and represented on multiscale. Considering matrices' sparseness, we use the linear list to contain them. What's more, we can cut coefficients on the small scale, corresponding to signal with the very high frequency, for its absolute value is tiny. Numerical tests show that cutting half of original numbers almost has no influencet on image resolution. The migration results of the zeros offset profiles for common geological structures proved its correction and effectiveness. Finally, the application of this method to real examples produced excellent results.