Separation of wavefields into directional components can be accomplished by an eigenvalue decomposition of the accompanying system matrix. In conventional pressurenormalized wavefield decomposition, the resulting one-way wave equations contain an interaction term which depends on the reflectivity function. Applying directional wavefield decomposition using flux-normalized eigenvalue decomposition, and disregarding interaction between up-and downgoing wavefields, these interaction terms were absent. By also applying a correction term for transmission loss, the result was an improved estimate of the up-and downgoing wavefields. In the wave equation angle transform, a crosscorrelation function in local offset coordinates was Fourier-transformed to produce an estimate of reflectivity as a function of slowness or angle. We normalized this wave equation angle transform with an estimate of the planewave reflection coefficient. The flux-normalized oneway wave-propagation scheme was applied to imaging and to the normalized wave equation angle-transform on synthetic and field data; this proved the effectiveness of the new methods.
SUMMARY
Gradient computations in full-waveform inversion (FWI) require calculating zero-lag cross-correlations of two wavefields propagating in opposite temporal directions. Lossless media permit accurate and efficient reconstruction of the incident field from recordings along a closed boundary, such that both wavefields propagate backwards in time. Reconstruction avoids storing wavefield states of the incident field to secondary storage, which is not feasible for many realistic inversion problems. We give particular attention to velocity–stress modelling schemes and propose a novel modification of a conventional reconstruction method derived from the elastodynamic Kirchhoff–Helmholtz integral. In contrast to the original formulation (in a previous related work), the proposed approach is well-suited for velocity–stress schemes. Numerical examples demonstrate accurate wavefield reconstruction in heterogeneous, elastic media. A practical example using 3-D elastic FWI demonstrates agreement with the reference solution.
We have constructed novel temporal discretizations for wave equations. We first select an explicit time integrator that is of second order, leading to classic time marching schemes in which the next value of the wavefield at the discrete time [Formula: see text] is computed from current values known at time [Formula: see text] and the previous time [Formula: see text]. Then, we determine how the time step can be doubled, tripled, or generally, [Formula: see text]-tupled, producing a new time-stepping method in which the next value of the wavefield at the discrete time [Formula: see text] is computed from current values known at time [Formula: see text] and the previous time [Formula: see text]. In-between time values of the wavefield are eliminated. Using the Fourier method to calculate space derivatives, the new time integrators allow larger stable time steps than traditional time integrators; however, like the Lax-Wendroff procedure, they require more computational effort per time step. Because the new schemes are developed from the classic second-order time-stepping scheme, they will have the same properties, except the Courant-Friedrichs-Lewy stability condition, which becomes relaxed by the factor [Formula: see text] compared with the classic scheme. As an example, we determine the method for solving scalar wave propagation in which doubling the time step is 15% faster than a Lax-Wendroff correction scheme of the same spatial order because it can increase the time step by [Formula: see text] only.
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