APPLICATION OF THE POCS INVERSION METHOD TO CROSS-BOREHOLE IMAGING

Cross-borehole tomography suffers from a well-known problem of data in completeness: the limited ray coverage dictated by the poor experiment al geometry implies that certain features of the velocity field are no t determined by the data. Construction of a tomographic image of the v elocity field therefore requires the addition of prior constraints to the inversion. In the Fourier wavenumber domain (assuming straight-lin e rays), the process of adding prior constraints is equivalent to spec ifying unmeasured wave-number coefficients. The projection onto convex sets (POCS) algorithm can impose physically plausible constraints tha t allow high quality tomographic images to be produced. Each constrain t is viewed as defining a set (in function space) of images that satis fy that particular constraint. The POCS method finds one or more image s in the intersection of the constraining sets, which is equivalent to finding an image that simultaneously satisfies a number of constraint s including the observed data. The sets of images that we employ inclu de: those that satisfy the data in the sense of having certain known w avenumber components, those that have bounded energy in certain unmeas ured wavenumber components, those that have seismic velocity bounded e verywhere (e.g., non-negative), and those in which the velocity struct ure is confined to the region between the boreholes. An advantage of t he POCS algorithm is that it allows both space-domain and wavenumber-d omain constraints to be imposed simultaneously. In our implementation of the POCS algorithm, we make use of the fast Fourier transform to ra pidly iterate between the space and Fourier-wavenumber domains. We tes t the method on synthetic data, and show that it significantly reduces the artifacts in the image, when compared to other methods. We then a pply it to data from a cross-borehole experiment in Manitoba, Canada, that were previously studied by others. We achieve a tomographic image of the velocity field that is similar in many respects to the results of others, but which possesses fewer artifacts.