Factorization and Path Integration of the Helmholtz Equation: Numerical Algorithms

SUMMARY In seismic tomography, the finite frequency content of broad‐band data leads to interference effects in the process of medium reconstruction, which are ignored in traditional ray theoretical implementations. Various ways of looking at these effects in the framework of transmission tomography can be found in the literature. Here, we consider inverse scattering of body waves to develop a method of wave‐equation reflection tomography with broad‐band waveform data—which in exploration seismics is identified as a method of wave‐equation migration velocity analysis. In the transition from transmission to reflection tomography the usual cross correlation between modelled and observed waveforms of a particular phase arrival is replaced by the action of operators (annihilators) to the observed broad‐band wavefields. Using the generalized screen expansion for one‐way wave propagation, we develop the Fréchet (or sensitivity) kernel, and show how it can be evaluated with an adjoint state method. We cast the reflection tomography into an optimization procedure; the kernel appears in the gradient of this procedure. We include a numerical example of evaluating the kernel in a modified Marmousi model, which illustrates the complex dependency of the kernel on frequency band and, hence, scale. In heterogeneous media the kernels reflect proper wave dynamics and do not reveal a self‐similar dependence on frequency: low‐frequency wave components sample preferentially the smoother parts of the model, whereas the high‐frequency data are—as expected—more sensitive to the stronger heterogeneity. We develop the concept for acoustic waves but there are no inherent limitations for the extension to the fully elastic case.

Section: The Wave Equation Evolution In Depthmentioning

confidence: 99%

Wave-equation reflection tomography: annihilators and sensitivity kernels

Hoop

Hilst

Shen

2006

“…in such a way that A is diagonal. For an extensive list of references on the theoretical and numerical aspects of these operators, see Fishman, McCoy & Wales (1987). Some recent references in the seismic context were given in the previous section.…”

Section: Decomposition Of the Two-way Operatormentioning

confidence: 99%

Reciprocity theorems for one-way wavefields

Wapenaar

Grimbergen

1996

Acoustic reciprocity theorems have proved their usefulness in the study of forward and inverse scattering problems. The reciprocity theorems in the literature apply to the two-way (i.e. total) wavefield, and are thus not compatible with one-way wave theory, which is often applied in seismic exploration. By transforming the two-way wave equation into a coupled system of one-way wave equations for downgoing and upgoing waves it appears to be possible to derive 'one-way reciprocity theorems' along the same lines as the usual derivation of the 'two-way reciprocity theorems'. However, for the one-way reciprocity theorems it is not directly obvious that the 'contrast term' vanishes when the medium parameters in the two different states are identical. By introducing a modal expansion of the Helmholtz operator, its square root can be derived, which appears to have a symmetric kernel. This symmetry property appears to be sufficient to let the contrast term vanish in the above-mentioned situation.The one-way reciprocity theorem of the convolution type is exact, whereas the oneway reciprocity theorem of the correlation type ignores evanescent wave modes. The extension to the elastodynamic situation is not trivial, but it can be shown relatively easily that similar reciprocity theorems apply if the (non-unique) decomposition of the elastodynamic two-way operator is done in such a way that the elastodynamic one-way operators satisfy similar symmetry properties to the acoustic one-way operators.

“…This equation quantifies the changes in the Green's matrix due to variations of the depth level [. In this sense, it resembles the method of invariant imbedding for initial-value problems (Bellman & Wing 1975;Fishman, McCoy & Wales 1987). To see this, we consider again the special situation for which the half-space x3 5 x3,0 is homogeneous and isotropic.…”

Section: Generalized Linear Representationmentioning

confidence: 99%

One-way representations of seismic data

Wapenaar

1996

A general one-way representation of seismic data can be obtained by substituting a Green's one-way wavefield matrix into a reciprocity theorem of the convolution type for one-way wavefields. From this general one-way representation, several special cases can be derived.By introducing a Green's one-way wavefield matrix for primaries, a generalized Bremmer series representation is obtained. Terminating this series after the first-order term yields a primary representation of seismic reflection data. According to this representation, primary seismic reflection data are proportional to a reflection operator, 'modified' by primary propagators for downgoing and upgoing waves. For seismic imaging, these propagators need to be inverted. Stable inverse primary propagators can easily be obtained from a one-way reciprocity theorem of the correlation type.By introducing a Green's one-way wavefield matrix for generalized primaries, an alternative representation is obtained in which multiple scattering is organized quite differently (in comparison with the generalized Bremmer series representation). According to the generalized primary representation, full seismic reflection data are proportional to a reflection operator, 'modified' by generalized primary propagators for downgoing and upgoing waves. Internal multiple scattering is fully included in the generalized primary propagators (either via a series expansion or in a parametrized way). Stable inverse generalized primary propagators can be obtained from the oneway reciprocity theorem of the correlation type. These inverse propagators are the nucleus for seismic imaging techniques that take the angle-dependent dispersion effects due to fine-layering into account.