Geophysical Prospecting,1995' 43, 859-872 Point-to-curve ray tracing in complicated geological modelsl Andrzej Hanyga2 and Jan Pajchet3 Abstract Boundary-value problems (BVPs) for seismic rays generally have multiple solu- tions. In practical applications the number of solutions can be large. The algorithm presented below solves a one-parameter family of BVPs and makes it easy to obtain all the solutions of a BVP. lntroduction The necessity of coping with caustics is a serious disadvantage of ray-tracing methods. A concavity of the reflector or varying gradient of the propagation speed generates a pair of caustics. Each caustic implies an additional solution of the two- point ray-tracing problem. As a result, numerical determination of all the solutions of the two-point problem becomes a formidable task. In particular, an iterative solver of a two-point problem for seismic rays converges to one of the solutions; the outcome depends unpredictably on the stafting approximation and the very number of solutions is a priori unknown. The same observation applies to other boundary-value problems (BVPs) for seismic rays, including normal rays. The important problem is how to compute all the solutions or how to pick out the desired ones. The method described below addresses the problem of findirig all the solutions of the two-point ray-tracing problem for a linear array of receivers such as a bore- hole axis. One end of the ray is constrained to lie on a curve, called the receiver curve, while the other one can be fixed or allowed to be a normal-incidence point on a fixed reflector. In addition the ray can undergo an arbitrary number of reflec- tions, transmissions or edge diffractions. An appropriate algorithm determines all the rays with a fixed ray code satisfying the above boundary conditions. The output is a dense sequence of rays whose receiver ends sweep the receiver curve or selected derived data such as traveltimes. In addition, more complete data, includ- ing amplitudes, phase shifts, etc., can be generated at selected receivers on the curye, e.g. for individual geophones in the borehole. Paper presented at the 54th EAEG meeting, June 1992, Paris. Received February 1993, revision acceptedJanuary 1995. 2 University of Bergen, Institute of Solid Earth Physics, All6gt. 41, N-5007 Bergen, Norway. 3 Norsk Hydro Research Center, Sandsli, Norway. @) 1995 European Association of Geoscientists & Engineers