Accelerating the T‐matrix approach to seismic full‐waveform inversion by domain decomposition

Abstract

A seismic variant of the distorted Born iterative inversion method, which is commonly used in electromagnetic and acoustic (medical) imaging, has been recently developed on the basis of the T‐matrix approach of multiple scattering theory. The distorted Born iterative method is consistent with the Gauss–Newton method, but its implementation is different, and there are potentially significant computational advantages of using the T‐matrix approach in this context. It has been shown that the computational cost associated with the updating of the background medium Green functions after each iteration can be reduced via the use of various linearisation or quasi‐linearisation techniques. However, these techniques for reducing the computational cost may not work well in the presence of strong contrasts. To deal with this, we have now developed a domain decomposition method, which allows one to decompose the seismic velocity model into an arbitrary number of heterogeneous domains that can be treated separately and in parallel. The new domain decomposition method is based on the concept of a scattering‐path matrix, which is well known in solid‐state physics. If the seismic model consists of different domains that are well separated (e.g., different reservoirs within a sedimentary basin), then the scattering‐path matrix formulation can be used to derive approximations that are sufficiently accurate but far more speedy and much less memory demanding because they ignore the interaction between different domains. However, we show here that one can also use the scattering‐path matrix formulation to calculate the overall T‐matrix for a large model exactly without any approximations at a computational cost that is significantly smaller than the cost associated with an exact formal matrix inversion solution. This is because we have derived exact analytical results for the special case of two interacting domains and combined them with Strassen's formulas for fast recursive matrix inversion. To illustrate the fact that we have accelerated the T‐matrix approach to full‐waveform inversion by domain decomposition, we perform a series of numerical experiments based on synthetic data associated with a complex salt model and a simpler two‐dimensional model that can be naturally decomposed into separate upper and lower domains. If the domain decomposition method is combined with an additional layer of multi‐scale regularisation (based on spatial smoothing of the sensitivity matrix and the data residual vector along the receiver line) beyond standard sequential frequency inversion, then one apparently can also obtain stable inversion results in the absence of ultra‐low frequencies and reduced computation times.

Taming the divergent terms in the scattering series of Born by renormalization

“…A comparison of equations (26)(27) and (30)(31) clearly suggests that we have generalized the convergent Born series of . We can construct a family of convergent Born series similar to the convergent Born series of if we set H = γ but use different values of the convergence control parameter h. The CBS is based on the use of a dissipative reference medium, which makes the Green's function finite and localized Huang et al, 2019c;Jakobsen et al, 2019a).…”

“…This method should be possible, due to the computational efficiency, to perform a 4-D seismic full waveform inversion. Future works will be also directed toward to a 4-D inversion of the time-lapse waveform data and extended by the domain decomposition method [31].…”

“…where V (r 1 ,r 2 ) = ω 2 χ(r 1 )δ(r 1 −r 2 ), V is a local scattering potential operator, which can be represented by a diagonal matrix in coordinate representation [29,31].…”

“…The computation of the T-matrix by matrix inversion can be very costly. The domain decomposition was proposed by Jakobsen and Wu [29,31] to accelerate the T-matrix approach to seismic full-waveform inversion. Also, if the perturbation is small, we can use an approximated form of the T-matrix [24] to update the background Green's function for every iteration.…”

“…These features make the distorted Born iterative T-matrix method more applicable to seismic full waveform inversion. Additional works on this method can be found in Jakobsen and Wu [29,31] and Wang et al [32]. Recently, the integral equation formulations were applied to the time-lapse seismic data and to estimate the uncertainty [33].…”