Abstract
As in many fields, in seismic imaging, the data in the field is collected over a relatively large medium even though only a part of that medium is truly of interest. This results in significant waste in computation as a typical inversion algorithm requires many solutions of the wave equation throughout the entire domain, even if only a small part of the domain is being updated. One way to mitigate this is to use a numerically exact local wave equation solver to perform waveform inversions in an area of interest, where the idea is to compute accurate solutions of the wave equation within a subdomain of interest. Although such solvers exist, many require the computation of Green’s function matrices in the background domain. For large-scale seismic data acquisition, the computation of the Green’s function matrices is prohibitively expensive since it involves solving thousands of partial differential equations in the background model. To mitigate this, in this work, we propose to exploit the low-rank structure of the full subsurface Green’s function. Using carefully selected 2D stylized models, we first show that the full subsurface Green’s function tensor organized as a matrix exhibits the low-rank structure in a transform domain. We then propose a randomized singular value decomposition–based framework to compute the low-rank approximation of the Green’s functions, where the cost of wave equation solves depends on the rank of the underlying Green’s function matrix instead of the number of grid points at the surface of the background model and on the boundary of the local domain. Next, we validate the proposed framework by performing time-lapse waveform inversion using the 2D Marmousi model. Finally, we demonstrate a rank-minimization–based framework to compute the low-rank factorized form of the Green’s function matrices in large-scale 3D seismic data acquisition.
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Funding
The first author would like to thank the member organizations of the SINBAD II project and the SINBAD Consortium for supporting this work. This study was supported by Chevron, NSERC, InnovateNL, and the Hibernia Management and Development Corporation.
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Appendices
Appendix 1
To perform FWI in the truncated domain, we need to compute the forward and adjoint wavefields. In order to do so, we need to first simulate the Green’s function at the surface and on the boundary nodes in the background model. In this section, we explain the process of extracting the Green’s function matrices in the background model to efficiently solve the PDEs, i.e., Eq. 3 in the local domain. For a 2D background velocity model, the dimensions of the full subsurface Green’s function responses are NzNx × Ns and \(N_{z}N_{x} \times N_{b}^{\delta {\Omega }}\) from which we extract the four desired sets of Green’s function matrices, i.e., \(\mathbf {u}_{0}^{(s,r)}\), \(\mathbf {u}_{0}^{\delta {\Omega }}\), \({\mathbf {G}}^{\delta {\Omega }_{+1}}_{0}\), and \({\mathbf {G}}^{\delta {\Omega }}_{0}\). The matrix \(\mathbf {u}_{0}^{(s,r)}\) represents the Green’s function matrix computed in the background model between the sources and receivers at the surface, which we convolve with the source wavelet while computing the data residual during the inversion process. Moreover, we convolve \(\mathbf {u}_{0}^{\delta {\Omega }}\) with the source wavelet (data residual) while computing the forward (adjoint) wavefields using Eq. 3. The matrices \(\mathbf {u}_{0}^{(s,r)}\) and \(\mathbf {u}_{0}^{\delta {\Omega }}\) are easy to extract from the full subsurface Green’s function matrix via taking out the rows corresponding to the locations of the receivers at the surface and the location of the nodes on the boundary of the local domain, respectively. To form the other Green’s function matrices, we follow a node numbering scheme in the truncated domain where the nodes are numbered in a counter-clockwise inward spiraling fashion as shown in Fig. 14 a.
Under this numbering scheme, we obtain the matrix \(\mathbf {u}_{0}^{\delta {\Omega }}\) from the full subsurface Green’s function simulated using the sources at the surface of the model. Again, we select the rows corresponding to the locations of the boundary nodes δΩ for each source experiment. We illustrate the matrix structure of the Green’s function matrices \(\mathbf {G}_{0}^{\delta {\Omega }}\) and \(\mathbf {G}_{0}^{\delta {\Omega }_{+1}}\) used in the second block row of Eq. 3 by using the 5 × 5 node example from Fig. 14 a.
The foundation of the local solver is the equation for the scattered field. It is an essential component of the local solver in Eq. 3 (i.e., it forms the second block row) and also gives the ability to propagate the local wavefield solution to the receiver locations. This scattered field equation is derived in Appendix 1 of the work of [58] and the final result is restated here for convenience
where all the quantities are scalars, h is the grid spacing, and the summation goes all around the boundary but does not include the corner nodes as we will illustrate below. Equation 5 closely resembles a discretization of Green’s third identity. We simplify Eq. 5 by removing the common term \(u^{\delta {\Omega }}G_{0}^{\delta {\Omega }}\) to get
The second block row of Eq. 3 evaluates (6) for each boundary node i on δΩ, with i ∈{1,..., 16} in this 5 × 5 example. The matrices \(\mathbf {G}_{0}^{\delta {\Omega }_{+1}}\) and \(\mathbf {G}_{0}^{\delta {\Omega }}\) therefore have 16 rows in this example. Figure 14 b illustrates which Green’s function combinations are required when the boundary summation (6) is evaluated around δΩ for the case when i = 1. Evaluating (6) gives
where each term in square brackets represents a contribution from a single box in Fig. 14 b. Equation 7 is the multiplication of vectors uδΩ and \(\mathbf {u}^{\delta {\Omega }_{+1}}\) with the first row of matrices \(\mathbf {G}_{0}^{\delta {\Omega }_{+1}}\) and \(\mathbf {G}_{0}^{\delta {\Omega }}\) in Eq. 3. We now show the first five rows of these matrices to show how (7) continues for i = 1, 2, 3, 4, 5
In Fig. 14 b, we see that the corner nodes on Ω+ 1 are involved twice while the corner nodes on Ω are never evaluated. This explains the zero columns in Eq. 8 and the columns with sums in Eq. 9.
As mentioned before, once we form these four sets of Green’s function matrices, we solve (3), where we compute the forward wavefield in the following three steps: (i) Evaluate the vector of unknowns by solving equation 3, (ii) extract the scattered field on the boundary of the local domain and one layer to the interior, (iii) project the scattered wavefield δu on the receiver locations in the seismic acquisition using Eq. 2, (iv) add the precomputed background Green’s function matrix \(\mathbf {u}_{0}^{(s,r)}\) and the projected scattered wavefield to compute the forward wavefield at the receiver locations. We thus compute the data residual and the objective function for the locally perturbed model using the local solver exclusively. To compute the adjoint wavefield, we now propagate the residual back to the truncated domain. We do this by using the same sets of Green’s functions that we used to project the scattered wavefield to the surface of the full domain. This is simply achieved by multiplying the Green’s function matrix \(\mathbf {u}_{0}^{\delta {\Omega }}\) with the data residual. We again solve (3) using the modified source function on the boundary of the truncated domain to compute the adjoint wavefield. Just like the forward wavefield, this adjoint wavefield is numerically exactly the same as would have been generated by a full domain solver on the perturbed model. Finally, we use the numerically exact forward and adjoint wavefields to compute the numerically exact gradients in the truncated domain, which is exactly the same FWI gradient as a full domain Helmholtz solver would have returned.
Appendix 2
Here, we illustrate a computationally efficient rank-minimization–based framework to solve the source-separation problem in Section 5.3.1, which is an important step to make the numerically exact local solver feasible for the large-scale 3D seismic data acquisition. Rank-minimization–based formulations are based upon the following three fundamental principles: (i) the underlying target matrix should exhibit low-rank structure in some transform domain, (ii) the subsampling-blending operator should increase the rank or slow down the decay of the singular values in some transform domain, (iii) a scalable and computationally efficient rank-minimization framework to handle large-scale data matrices. Under assumptions (i) and (ii), the goal of the rank-minimization problem is to find the matrix of lowest possible rank that agrees with the experimental observations. This is written as the following optimization problem:
where rank is defined as the maximum number of linearly independent rows or column of a matrix, b is a set of blended measurements, and \(\mathcal {A}\) represents the sampling-blending operator. Since rank-minimization problems are NP hard and therefore computationally intractable, [46] showed that solutions to rank-minimization problems can be found by solving the following nuclear norm minimization problem:
where ∥.∥∗ = ∥σ∥1 and σ is the vector of singular values for each monochromatic data matricization. To efficiently solve equation 10 for large-scale seismic data, we used an extension of SPGℓ1 solver [4] developed for basis-pursuit denoising (BPDNσ) in [2]. The resulting algorithm which is dubbed SPG-LR by [2] finds the solution to the BPDNσ by solving a sequence of robust LASSO (least absolute shrinkage and selection operator) subproblems:
where τ is updated by traversing the Pareto curve. The LASSO is a regularized regression formulation that seeks to do variable selection using a sparsity penalty, whereas the Pareto curve describes the tradeoff between the data fit and the nuclear norm of the solution vector. Interested readers can find a detailed explanation of the Pareto curve in [4]. Solving each robust LASSO subproblem requires a projection onto the nuclear norm ball ∥X∥∗≤ τ in every iteration by performing a singular value decomposition and then thresholding the singular values. In the case of large-scale seismic problems, it becomes prohibitive to carry out such a large number of SVDs. Therefore, we avoid the direct approach to the nuclear norm minimization problem and follow a factorization-based approach [30, 47, 48]. The factorization-based approach parametrizes each monochromatic data matrix X as a product of two low-rank factors \(\mathbf {L} \in \mathbb {C}^{n \times k}\) and \(\mathbf {R} \in \mathbb {C}^{m \times k}\) such that X = LRH, where k represents the rank of the underlying matrix and H represents the Hermitian transpose. The optimization scheme can then be carried out using the matrices L,R instead of X, thereby significantly reducing the size of the decision variable from n × m to k × (n + m) when k ≤ (n,m). Rennie and Srebro [48] show that the nuclear norm obeys the relationship
where \(\|\cdot \|_{F}^{2}\) is the Frobenius norm of the matrix (sum of the squared entries). Consequently, the LASSO subproblem can be replaced by
where the projection onto \(\frac {1}{2} \|\mathbf {L};{\mathbf {R}}\|_{F}^{2} \leq \tau \) is easily achieved by multiplying each factor L and R by the scalar \(2\tau /(\frac {1}{2} \|\mathbf {L};{\mathbf {R}}\|_{F}^{2})\).
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Kumar, R., Willemsen, B., Herrmann, F.J. et al. Enabling numerically exact local solver for waveform inversion—a low-rank approach. Comput Geosci 23, 829–847 (2019). https://doi.org/10.1007/s10596-019-09832-9
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DOI: https://doi.org/10.1007/s10596-019-09832-9