Theory of True Amplitude One-way Wave Equations and True Amplitude Common-shot Migration
نویسندگان
چکیده
One-way wave operators are powerful tools for forward modeling and inversion. Their implementation, however, involves introducing the square-root of an operator as a pseudo-differential operator. A simple factoring of the wave operator produces one-way wave equations that yield the same traveltimes as the full wave equation, but do not yield accurate amplitudes except for homogeneous media and for some rare points in heterogeneous media. Here, we present augmented one-way wave equations. We show that these equations yield solutions for which the leading order asymptotic amplitude as well as the traveltime satisfy the same differential equations as do the corresponding functions for the full wave equation. Exact representations of the square-root operator appearing in these differential equations are elusive, except in cases in which the heterogeneity of the medium is independent of the transverse-spatial variables. Here, singling out depth as the preferred direction of propagation, we introduce a representation of the squareroot operator as an integral in which a rational function of the transverse Laplacian appears in the integrand. This allows for an explicit asymptotic analysis of the resulting one-way wave equations. We have proven that ray theory for these one-way wave equations leads to oneway eikonal equations and the correct leading order transport equation for the full wave equation. By introducing appropriate boundary conditions at † z = 0, we generate waves at depth whose quotient leads to a reflector map and estimate of the ray-theoretical reflection coefficient on the reflector. Thus, these true amplitude one-way wave equations lead to a “true amplitude wave equation migration (WEM)” method when we use the same imaging condition as is standardly used in WEM. In fact, we have proven that applying the WEM imaging condition to these newly defined wavefields in heterogeneous media leads to the Kirchhoff inversion formula for common-shot data. This extension enhances the original WEM. The objective of that technique was a reflector map, only. The underlying theory did not address amplitude issues. Computer output using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data over the entire survey area must be collected from a single common-shot experiment. Multi-experiment data, such as commonoffset data, cannot be used with this method as presently formulated. Research on extending the method is ongoing at this time.
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