An application of the propagator matrix of dynamic ray tracing: The focusing and defocusing of body waves by three-dimensional velocity structure in the source region

Abstract

Summary. Since the dynamic ray tracing system can be written in the linear form dW/ds=SW, it permits the definition of fundamental matrix and propagator matrix solutions. The propagator matrix can be exploited to connect complex 3-D and 2-D regions to 1-D regions of a model. For example, if the three-dimensional variations of a model are confined only to the portion of the ray path between O’ and Oo, numerical integration of the linear system need only be performed between O’ and Oo to construct the propagator π(Oo, O'). Along the segment of the ray path between O’ and Os, traversing a 1-D or homogeneous portion of the model, analytic solutions exist for π(O', Os). Thus π(Oo, Os) can be simply obtained by multiplying the analytic and numerically obtained π matrices. With this procedure, quantities are calculated that are needed for the synthesis of teleseismic P waves by superposition of Gaussian beams. These synthetics are used to investigate the focusing and defocusing of teleseismic P-waves by 3-D structure in the vicinity of the source. A source region, having a fluctuation in velocity of 4 per cent over a characteristic scale length of 50 to 100 km, produces factors of two fluctuations in amplitude and several tenths of a second in travel time at teleseismic range in experiments in which either source location is varied at constant azimuth or in which azimuth is varied at constant source location. A model having a maximum fluctuation as small as 0.8 per cent is capable of producing caustics and multipaths at teleseismic range, depending on its distribution of scale lengths and its ratio of vertical with respect to horizontal scale length. The multipaths and caustics of such a model, however, occur over too small an area and are too closely spaced in arrival time to be resolved with seismograph systems in the 0.01 to 4 Hz band.