Source synthesis for waveform inversion

Seismologists have long used extended sources, both physical and synthetic, to enhance the effectiveness of data acquisition and processing in various ways. Recently, extended synthetic sources have been suggested as a means to to substantially reduce the computational complexity of waveform inversion. Most proposed synthetic sources for waveform inversion are random in some way. In contrast, I suggest a deterministic choice of extended source, one that maximizes the difference between the data predicted by the current inversion iterate and the data synthesized from the target point source records. Optimal source synthesis in this sense has proven effective in several non-seismic inverse problems. I will describe the concept, and present some preliminary evidence concerning its performance in inversion of reflection seismograms. INTRODUCTION Construction of data from extended sources by superposition of data from impulsive localized sources, either physically in the field or synthetically in data processing, has a long history in reflection seismology. Simultaneous recording of signals from more than one shot can reduce acquisition costs and enhance source sampling and aperture (Beasley et al., 1998; Berkhout, 2008; Blaquiere et al., 2009). Migration with simultaneous sources can reduce the cost of shot-record imaging by the number of sources taken together (Romero et al., 2000; Verschuur and Berkhout, 2009). Plane wave migration is a special case of migration with extended sources, with an even longer history (Treitel et al., 1982). Recent intense interest in full waveform inversion has led several groups to propose the use of simultaneous or extended synthetic sources in that context. For example, Krebs et al. (2009) propose synthesis of all shots in a survey (or at least many of them, see the caveat below) by filtered stack, with a different filter for each shot. In common with many other recent source synthesis approaches, Krebs et al. (2009) choose the synthesis filters randomly. They find that random ±1 length-one filters, that is, weighted stack with no dephasing, work as well as more complicated random filters with phase delays. They also note that convergence of the iterative time-domain least-squares method used in their work is much improved when the weights are chosen anew for each iteration. Synthesis of all shots in this fashion dramatically decreases the inversion workload, essentially by the number of shots combined in the synthesis. This note proposes an alternative, deterministic method for source synthesis in waveform inversion. In common with Krebs et al. (2009), the method presented here updates the synthesis filters at each model update. Also in common with Krebs et al. (2009), the synthesis filters may have length 1, that is, parametrize a simple weighted stack, although more complex filters with phase delays could also be employed. In contrast to other synthesis