Multi-source Waveform Inversion with Deblurring

The theory of preconditioned multi-source waveform inversion is presented where many shot gathers are simultaneously back-propagated to form the gradient of the misfit function. The implication is that, relative to standard waveform inversion, an order of magnitude increase in computational efficiency can be achieved by multi-source waveform inversion. INTRODUCTION Time-domain waveform inversion has the potential to provide estimates of velocity models with significantly higher resolution compared to traveltime tomography. However, waveform inversion is computer intensive due to the multiple iterations of forward modeling and residual wavefield back-propagation. As a partial remedy to the expense of reverse time migration (RTM), Morton (1998) proposed phase-encoding of shot records to simultaneously migrate a number of shot gathers within a single migration. This results in an increase in computational efficiency but the penalties are additional noise in the misfit gradient and inaccuracy in the inverted velocity model (Romero et al., 2000). In this procedure, each shot gather is encoded with a unique random time series and the result is summed together to form an encoded multi-source gather. Here, the unique time series assigned to a shot gather is approximately orthogonal to any of the other random time series. In theory, only a single phase-encoded back-propagation operation should be needed to generate the misfit gradient for velocity updating. The problem is that a phaseencoded finite-difference (FD) simulation with insufficient temporal duration yields noticeable artifacts in the misfit gradient and so it is not widely adopted in the industry. To overcome this limitation, we develop an encoded multi-source deblurring filter to limit these cross-terms. Recent work by Aoki (2008), Aoki and Schuster (2008) and Dai and Schuster (2009) have shown that the use of deblurring filters as preconditioners in migration deconvolution (MD) and least squares migration (LSM) reduces migration artifacts and accelerates convergence. Here we successfully apply it to multi-source waveform inversion to provide a more accurate misfit gradient with fewer artifacts and thus accelerate the inversion process. Synthetic tests on 2D Marmousi model show that multi-source waveform inversion with an encoded multi-source deblurring filter can provide nearly the same result as single-source waveform inversion, but with an order of magnitude increase in computational efficiency. This paper is organized into three sections. First, the theory of multi-source waveform inversion is introduced followed by an application of the encoded multi-source deblurring filter. Then the multi-source waveform results are compared with single-source inversion results using synthetic Marmousi data. Finally, a summary is presented. THEORY Waveform inversion updates the 2D velocity model V (x, z) by matching the calculated seismograms Pcal(s, r, ω) to the observed seismograms Pobs(s, r, ω), where s and r denote the source and receiver vectors, respectively. This can be accomplished by minimizing the waveform misfit function (Lailly, 1983; Tarantola, 1984): f = 1 2 ∑