Reconstruction algorithm for limited‐view angle in crosshole seismic diffraction tomography

Abstract

The limited-view angle problem is one of the difficulties in seismic diffraction tomography. In the configuration of seismic data acquisition systems, the exploring wave cannot probe the object from all directions. This limitation will lead to incomplete coverage in the wavenumber domain. If we ignore this incomplete data by assigning zero to the blind area and use the ordinary reconstruction algorithms, poor resolution will be introduced. Thus, it is important to estimate the missing data in the spectrum in order to improve the resolution. In this paper, POCS (Projections Onto Convex Sets) is applied to the cross-hole seismic diffraction tomography in order to solve the limited-view angle problem in this configuration. The algorithm that combines POCS with direct Fourier transform and bilinear interpolation in the wavenumber domain is proposed. POCS can incorporate many kinds of convex-type a priori knowledge about the object function as constraints and its estimate is superior to that of G-P extrapolation. INTRODUCTION In cross-hole seismic tomography, the sources and geophones are deployed in the two bore-holes respectively and it is very difficult to probe the object in the subsurface from all directions. This problem is known as the limited-view angle problem and it will lead to large patches of “blind area” in the wavenumber domain. Up to now, most seismic diffraction tomography methods have bypassed this problem by assigning zero to the “blind area” where data are not available. The reconstruction directly from the limited-view data is the so-called zero-order or “naive” solution and the image is often blured and distorted because the data in the blind area are not actually zero. In this paper, we combine the convex projections and direct Fourier transform to estimate the blind area data for cross-hole seismic diffraction tomography. The limited-view angle problem is an underdetermined problem in which only a small number of projections is available. Artifacts such as streaking and geometric distortion are inevitable in the reconstructed image. This lack of information, however, may be made up in part by adequate a priori knowledge about the image. Such knowledge may be used to impose constraints on the solutions or the reconstruction procedure in an attempt to regularize the ill-posed problem. POCS offers two note-worthy advantages: 1) it enables any number of a priori convex-type constraints to be incorporated in the algorithm; and 2) it guarantees convergence: weak in general, strong in practice. It is a recursive technique that finds a solution consistent with the measured data and a priori known constraints in both the space and Fourier domain. Because of these advantages, POCS has found applications in image restoration and X-ray CT image reconstruction, etc. In this paper, it is used in seismic diffraction tomography. BASIC EQUATIONS OF SEISMIC DIFFRACTION TOMOGRAPHY Consider the case of the acoustic wave equation with constant density. The object is described by the velocity distribution where is the position vector. The background medium has a velocity The wave equation in the source-free region is