Wave-field representations with Green's functions, propagator matrices, and Marchenko-type focusing functions

Abstract

Classical acoustic wave-field representations consist of volume and boundary integrals, of which the integrands contain specific combinations of Green's functions, source distributions, and wave fields. Using a unified matrix-vector wave equation for different wave phenomena, these representations can be reformulated in terms of Green's matrices, source vectors, and wave-field vectors. The matrix-vector formalism also allows the formulation of representations in which propagator matrices replace the Green's matrices. These propagator matrices, in turn, can be expressed in terms of Marchenko-type focusing functions. An advantage of the representations with propagator matrices and focusing functions is that the boundary integrals in these representations are limited to a single open boundary. This makes these representations a suitable basis for developing advanced inverse scattering, imaging and monitoring methods for wave fields acquired on a single boundary.

Propagator and transfer matrices, Marchenko focusing functions and their mutual relations

“…with F v (x, x F ) denoting the particle velocity counterpart of the focusing function Wapenaar (2022), to facilitate the derivations below). The focusing functions F p (x, x F ) and F v (x, x F ), which together form the right column of matrix Y(x, x F ), are illustrated in the lower frame of Figure 1.…”

“…Whereas there is ambiguity in the normalization of the matrices L± 1 and L± 2 , the matrix D± 1 is uniquely defined. Some examples of matrix D± 1 (for acoustic, electromagnetic and elastodynamic waves) are given by Wapenaar (2022). In Appendix A we derive for any wave phenomenon…”

“…Using these definitions and Parseval's theorem, we rewrite equation (10) as q(x) = ω 2 4π 2 R 2 W(x, s, x 3,F )q(s, x 3,F )d 2 s, for wavefield component q 2 , which focuses at x = x F and continues as an upgoing field in the homogeneous upper half-space (note that the definition of F2 is different from that in Wapenaar (2022), to facilitate the derivations below). The focusing functions F1 (x, s, x 3,F ) and F2 (x, s, x 3,F ) together form the right column of matrix Ỹ(x, s, x 3,F ).…”

“…The N 2 × N 2 Green's matrix G 12 (x, x S ) stands for the q 1 -type field observed at x, in response to this source. The spatial Fourier transform of the downgoing component at ∂D F (i.e., just below the source) is proportional to the upper-right submatrix of the decomposition operator of equation ( 104), according to G+ 12 (x F , s, x 3,S ) = { ∆1 (s, x 3,F )} −1 exp{iωs • x H,F } (109) (Wapenaar 2022). To compensate for the effects of the inverse matrix { ∆1 (s, x 3,F )} −1 , we define a modified Green's matrix as Γ12 (x, s, x 3,S ) = G12 (x, s, x 3,S ) ∆1 (s, x 3,F ), (110) such that its downgoing component at ∂D F is given by…”

Data-driven suppression of short-period multiples from laterally varying thin-layered overburden structures

“…Here, I BL is the identity operator I with the frequency-wavenumber (f − k x ) support matching that wavelet-free reflection response R. As a result, V þ e ( Vþ e ) contains the interference between wI BL and V þ m ( Vþ m ), which leaked into Θ þ due to band limitation. Finally, * denotes the time/path reversal, which in this paper we will carry out with complex conjugation in the frequency domain (see Dukalski et al [2022aDukalski et al [ , 2022b and Wapenaar [2022] for the most recent discussion on this topic). Note that equations 4a and 4b are not properly constrained, as the retrieval of the correct V þ and Vþ is possible only if their early-time waveforms are correct.…”