Acoustic 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 multisource gradient of the misfit function. Synthetic tests on 2D Marmousi model data show that multi-source waveform inversion using an encoded multi-source deblurring filter as a preconditioner can provide a good inverted model but with less computational cost. We showed successful inversion results using multi-source gathers compositing 10 shot gathers in our midyear report. In this report, we blended 100 shot gathers and successfully inverted the data. However, this required 300 iterations compared to just 50 iterations for the 10-shot multi-source 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 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 the crosstalk noise. 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 the 2D Marmousi model shows that multi-source waveform inversion with an encoded multi-source deblurring filter can provide a good inversion result and reduce the computational time by two orders of magnitude. 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 obtained using synthetic 2D 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):