Fast 3D frequency-domain full-waveform inversion with a parallel block low-rank multifrontal direct solver: Application to OBC data from the North Sea

Abstract

Wide-azimuth long-offset ocean bottom cable (OBC)/ocean bottom node surveys provide a suitable framework to perform computationally efficient frequency-domain full-waveform inversion (FWI) with a few discrete frequencies. Frequency-domain seismic modeling is performed efficiently with moderate computational resources for a large number of sources with a sparse multifrontal direct solver (Gauss-elimination techniques for sparse matrices). Approximate solutions of the time-harmonic wave equation are computed using a block low-rank (BLR) approximation, leading to a significant reduction in the operation count and in the volume of communication during the lower upper (LU) factorization as well as offering great potential for reduction in the memory demand. Moreover, the sparsity of the seismic source vectors is exploited to speed up the forward elimination step during the computation of the monochromatic wavefields. The relevance and the computational efficiency of the frequency-domain FWI performed in the viscoacoustic vertical transverse isotropic (VTI) approximation was tested with a real 3D OBC case study from the North Sea. The FWI subsurface models indicate a dramatic resolution improvement relative to the initial model built by reflection traveltime tomography. The amplitude errors introduced in the modeled wavefields by the BLR approximation for different low-rank thresholds have a negligible footprint in the FWI results. With respect to a standard multifrontal sparse direct factorization, and without compromise of the accuracy of the imaging, the BLR approximation can bring a reduction of the LU factor size by a factor of up to three. This reduction is not yet exploited to reduce the effective memory usage (ongoing work). The flop reduction can be larger than a factor of 10 and can bring a factor of time reduction of around three. Moreover, this reduction factor tends to increase with frequency, namely with the matrix size. Frequency-domain viscoacoustic VTI FWI can be viewed as an efficient tool to build an initial model for elastic FWI of 4C OBC data.

On the Complexity of the Block Low-Rank Multifrontal Factorization

“…because we know these are the values for which the result is meaningful for the application [2]. For both Poisson and Helmholtz, in all the following experiments, the backward error is in good agreement with the low-rank threshold used.…”

“…Assuming that p such clusters have been defined, and that a permutation P has been defined so that permuted variables of a given cluster are contiguous, a BLR representation S of a dense matrix S is shown in Equation (2). Subblocks B ij = (P SP T ) ij , of size m ij × n ij and numerical rank k ε ij , are approximated by a low-rank product B ij = X ij Y T ij at accuracy ε, where X ij is a m ij × k ε ij matrix and Y ij is a n ij × k …”

“…Note that pivoting may slightly perturb the initial clustering of the variables shown in Equation (2). These details are omitted in Algorithm 1 for the sake of clarity.…”

“…While its efficiency has been shown in practice on real applications [1,2], its theoretical complexity was unknown. Unlike hierarchical formats, it remained to be proved that the BLR format does not only reduce the computations by a constant, i.e., possesses a complexity in O(n 2 ) just like the full-rank solver.…”

Higher-order Source-wavefield Reconstruction for Reverse-time Migration from Stored Values in a Boundary Strip Just One Point Wide

“…Three-dimensional RTM typically uses higher frequencies than FWI, and storage of source wavefields on disk or solid-state drives or even in main memory may lead to performance bottlenecks, in particular on manycore or GPU hardware. Migration in the frequency domain avoids the storage problem and outperforms migration in the time domain in two dimensions (Marfurt and Shin, 1989;Pratt, 1990;Østmo et al, 2002;Mulder and Plessix, 2004) but not yet in three dimensions (Riyanti et al, 2006;Plessix, 2009;Wang et al, 2010Wang et al, , 2011Knibbe et al, 2014;Amestoy et al, 2016). For 3D applications, there are several ways to reduce or circumvent the storage of the source wavefields (Dussaud et al, 2008;Nguyen and McMechan, 2015).…”