EXAMINATION COMMITTEE PAGE
7.1 Seismic Redatuming by Inversion
Chapter 7
Seismic Redatuming by Inversion Using Batched
TLR-MVM
78 equations:
−g− g+∗
=
I −R
−R∗ I
f− f+
. (7.2)
Note that due to the extremely large size of these matrices, whilst the problem is written in a compact matrix-vector formulation, its numerical solution is per- formed using matrix-free operators and iterative solvers such as conjugate gradient least-squares (CGLS) or LSQR [1]. Focusing now our attention on the multi- dimensional convolution (MDC) integral operator, which represents the most ex- pensive computations in the overall chain of operations, its inner working can be written more explicitly as follows:
y=Rx: y(t,xB,xA) =Fω−1max Z
δD
R(ω,xB,xR)Fωmax(x(t,xR,xA))dxR
. (7.3) Similarly, the adjoint of such an operator can be written as:
x=RHy: x(t,xR,xA) = Fω−1max Z
δD
R∗(ω,xB,xR)Fωmax(y(t,xB,xA))dxB
, (7.4) where F and F−1 represent the forward and inverse Fourier transforms, ω is the angular frequency, xA, xB and xR represent spatial locations with the latter two spanning the integration domain δD. ωmax is used to indicate that the output of the forward Fourier transform is truncated to contain only frequencies where the signal spectrum resides. Finally,R(ω,xB,xR) represents the kernel of the integral operator in the frequency-space domain and can be created upfront by applying the Fourier transform along the time axis of the physically recorded seismic data R(t,xB,xR). Moreover, once the spatial integral is discretized, the kernel sim- ply becomes a stack of matrices (one for each frequency ω within the specified frequency spectrum of the seismic data) and the integral can be interpreted as a batched matrix-vector multiplication (MVM) operations.
Alongside with advances in processing algorithms, the size and scale of seismic
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surveys have increased since the late 20th century. Nowadays, large-scale high- resolution 3D surveys are routinely acquired where the recorded data can easily be on the order of several Terabytes. Recently, the implementation of the 3D Marchenko equations has been discussed in [1] and [99]. In both cases, special attention has been placed on the implementation of the integral operator and the handling of kernels that cannot directly fit in the main memory of single compute node. In the former approach, the embarrassingly parallel nature of the batched MVM is leveraged by reading different frequency batches in the main memory of multiple compute nodes only once prior, to solving the inverse problem in Eq. 7.1.
The latter approach, on the other hand, utilizes the ZFP-based compression algo- rithm of [100] to reduce the size of the reflection response to be stored on disk and the read-in time to memory. The authors report a compression factor of four for this lossless compression when applied to their frequency-space reflection seismic data. In fact, even when the data is compressed, on-the-fly decompression is still required to be able to perform the computations in Eqs. 7.3 and 7.4. In both im- plementations, however, no attempt is made to expedite the dense MVM required in both the forward and adjoint processes.
Whilst the redatuming Eq. 7.1 is relatively new to the field of geophysics, integral operators of the kind of R (Eq. 7.3) and RH (Eq. 7.4) are common to a number of other wave-equation-based seismic processing methods, such as surface- related multiple elimination (SRME – [101]), estimation of primaries by sparse inversion (EPSI – [102]), and up/down deconvolution [103] just to name a few.
This work may therefore have a broader impact beyond the Marchenko redatuming technique.
7.1.1 Impact of Matrix Reordering
A seismic reflection response, used here as the kernel of the MDC operator, can be conceptually visualized as a three-dimensional tensor of sizeNf×Ns×NrwhereNf frequency matrices are arranged along the slowest axis, each of them composed
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Figure 7.1: Single frequency matrix (f = 33Hz) with default (left) and Hilbert space- filling (right) matrix reordering techniques.
of Ns sources (i.e., rows) and Nr receivers (i.e., columns). When performing MDC with such dense frequency matrices, the arrangement of sources (along the rows of each matrix) and receivers (along with the columns of each matrix) is in principle arbitrary as long as it is consistent with that of the input ˜x. The arrangement is, however, important within the context of TLR compression as it may lead to better or worse block compressibility. A variety of reordering strategies that aim at reducing the overall distance between sources (and receivers) within the same tile have been investigated. We display a single frequency matrix (f = 33 Hz) in Fig. 7.1 (a) and (b), obtained with the original and distance- aware matrix reordering methods shown in Fig. 7.1 (c) and (d), respectively. We consider this matrix as representative of general trend in the dataset since its total rank summation fall in the interval between the median and third quartile of the whole stack of frequency matrices. Based on Hilbert space-filling curves [104], the distance-aware reordering permits to the cluster of the highest interactions
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Phase 1
LD LD
LD
Nb
Phase 3
Phase 1
Phase 3 Basic TLR-MVM
Transpose TLR-MVM
U basis V basis
Transposed V basis Transposed U basis
Nb Nb
Nb
Nb
Nb
Nb
Nb
Nb LD
Figure 7.2: Memory layout for Phase 1 and Phase 3 of the standard and transpose TLR-MVM. “nb” is the block size of the tiles and “LD” is the leading dimension of the matrix.
between nearby sources-receivers mostly around the matrix diagonal and avoids a diffusive pattern, as seen for the naive ordering. Hilbert reordering provides the best compression rate by better leveraging the data sparsity of the frequency matrices. Space-filling Morton (or “Z”) curves [105] have also been considered and are not as effective.
7.1.2 Impact on SRI Computational Stages
Whilst in [67], the authors focused only on the implementation of the forward batch MVM operation, here we cover the entire SRI workflow where an iterative solver is used to solve the Marchenko equations. This requires the application of four batch MVM operations. We can proceed with the standard TLR-MVM for Rx. For the conjugate TLR-MVM R∗x, we can simply apply R to x∗ and then conjugate the output. Indeed, the conjugate of the product of two numbers is the product of the conjugates and the conjugate of the sum of two numbers is the sum of their conjugates; therefore, element-by-element, the conjugate of (R(x∗))∗
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is R∗x. Using similar reasoning, the conjugate transpose of RHy is ((RT)y∗)∗. In other words, we do not need to explicitly form the conjugate transposeRH, rather we can simply apply the transpose RT, which also covers the last operation. In summary, we need to store onlyRand access it in two different ways, i.e., standard and transpose, as illustrated in Fig. 7.2 with a 3x3 tile frequency matrix. The U and V bases can be safely swapped but accessed with a different data layout (from row-major to column-major). By conjugating only the vector and not the compressed matrix generated from the denseR, we do not require any data storage beyond what is required for the original TLR-MVM.