More fun with random boundaries

Introduction

Time domain finite difference is the most computationally efficient algorithm for both RTM (Baysal et al., 1983) and waveform inversion (Woodward, 1990). The kernel's simplicity, Single Instruction Multiple Data (SIMD), and high level of data reuse has led to high-performance implementations on Field Programmable Gate Arrays (FPGA) (Nemeth et al., 2008) and General Purpose Graphics Processing Units (GPGPU) (Micikevicius, 2008) along with conventional CPUs.

Both waveform inversion and RTM require source and receiver wave-fields at equivalent time to be correlated. Unfortunately, the source wave-field is propagated forward in time while the receiver wave-field is propagated backwards in time. As a result, the wave-fields are computed sequentially and one of the two wave-fields must be saved to disk. To avoid having to save the entire wave-field to disk, checkpointing Symes (2007) or boundary reinjection (Clapp, 2008; Dussaud et al., 2008) have been suggested. These schemes result in lower IO requirements at the cost of 50% more computation. Given that disk IO bandwidth is increasing at a much slower rate than computational power, all of these approaches seem likely to face an IO bottleneck. Clapp (2009) proposes to change the boundary condition on the computation kernel from one that damps to one that randomizes the wave-field hitting the boundary. Unlike the damping boundary, a randomizing boundary, which only modifies the velocity within the boundary region, is time reversible. As a result the source wave-field can be propagated from $t=0$ to $t_{\rm max}$ , and then the source and receiver wave-field can be propagated simultaneously backwards. This eliminates the need to write the entire wave-field to disk in RTM and requires only two time steps of the source wave-field to be saved to disk in the case of waveform inversion. Clapp (2009) showed that the RTM image produced using a random boundary was nearly identical to one using a damping boundary except at shallow depths.

In this paper, I propose several ways to improve the imaging result using a random boundary. I begin by proposing a modified random velocity boundary that not only becomes more random as energy propagates into it, but whose average velocity decreases, as a result causing larger time delays and less coherent reflections. In addition, I show for VTI propagation that if the horizontal and NMO velocities increase within the boundary region, energy can be turned parallel to the boundary, producing more randomness.

More fun with random boundaries