Abstract. Motivated by an important and challenging task encountered in material discovery, we consider the problem of finding K basis patterns of numbers that jointly compose N observed patterns while enforcing additional spatial and scaling constraints. We propose a Constraint Programming (CP) model which captures the exact problem structure yet fails to scale in the presence of noisy data about the patterns. We alleviate this issue by employing Machine Learning (ML) techniques, namely kernel methods and clustering, to decompose the problem into smaller ones based on a global data-driven view, and then stitch the partial solutions together using a global CP model. Combining the complementary strengths of CP and ML techniques yields a more accurate and scalable method than the few found in the literature for this complex problem.

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Shot gathers from the Parkfield, California, deep crustal seismic reflection line, recorded in 1977 by COCORP, reveal coherent events having horizontal to reverse moveouts. These events were migrated using a multioffset three-dimensional Kirchhoff summation method. This method is a ray-equation back projection inversion of the acoustic wave field. which is valid under the Born, WKBJ, and far-field assumptions. Migration of full-wave acoustic synthetics, having the same limitations in geometric coverage as the COCORP survey, demonstrates the utility of the imaging process. The images obtained from back projection of the survey data suggest that the Gold Hill fault carries ultramafic rocks from the surface to 3 km depth at a dip greater than 45 degrees. where it joins the San Andreas fault, which may cut through more homogeneous materials at shallow depths. To the southwest, a 2 km Tertiary sedimentary section appears to terminate against a nearvertical fault. The zone between this fault and the San Andreas may be floored at 3 km by flat-lying ultramatics. Lateral velocity inhomogeneities are not accounted for in the migration but, in this case, do not seriously hinder the reconstruction of reflectors.

The application of the Born approximation to the scattered wave field, followed by a WKBJ and far-field approximation on the propagation Green's function for a slowly varying background medium, leads to a simple integral relation between the density and bulk-modulus anomalies superimposed on the background medium and the scattered wave field. An iterative inversion scheme based on successive back-projections of the wave field is used to reconstruct the two acoustic parameters. The scheme, when applied to data generated using the direct integral relation, shows that the variations of the parameters can be reconstructed. The procedure is readily applicable to actual data, since every iterative step is essentially a prestack Kirchhoff migration followed by the application of the direct Born approximation and far-field operator.

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