The application of diffraction tomography to cross‐hole seismic data

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Abstract
Previously published equations for diffraction tomography do not solve the “two and one‐half dimensional problem” (point source illumination of two‐dimensional geology) if sources and receivers are confined to linear arrays. In spite of this lack of a formal solution, useful images can be formed by the application of two‐dimensional formulas to such problems. The estimation of difference fields, of crucial importance in diffraction tomography, reduces to the problem of estimating the source function. Using assumptions about the consistency of the source behavior, we extract the source function in a statistical fashion from cross‐hole data. Using this technique, the difference fields are computed directly from the recorded wave fields for two experiments and diffraction tomographic images are obtained. In the first experiment, the data are generated using a two‐dimensional finite‐difference modeling algorithm. In the second, a physical scale model of a crosshole experiment is performed in an ultrasonic modeling tank. Images are obtained within both the first Born and Rytov’s approximation. Our results indicate that Rytov’s approximation achieves good resolution of the lower wavenumber components of the object, whereas the first Born approximation is more successful where the object is discontinuous.

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