True amplitude wave equation migration arising from true amplitude one-waywave equations

Abstract

One-way wave operators are powerful tools for use in forward modelling andinversion. Their implementation, however, involves introduction of the square rootof an operator as a pseudo-differential operator. Furthermore, a simple factoringof 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 forhomogeneous media and for almost all points in heterogeneous media. Here, wepresent augmented one-way wave equations. We show that these equations yieldsolutions for which the leading order asymptotic amplitude as well as thetravel time satisfy the same differential equations as the correspondingfunctions for the full wave equation. Exact representations of the square-rootoperator appearing in these differential equations are elusive, except incases in which the heterogeneity of the medium is independent of thetransverse spatial variables. Here, we address the fully heterogeneous case.Singling out depth as the preferred direction of propagation, we introduce arepresentation of the square-root operator as an integral in which a rationalfunction of the transverse Laplacian appears in the integrand. This allows usto carry out explicit asymptotic analysis of the resulting one-way waveequations. To do this, we introduce an auxiliary function that satisfies a lowerdimensional wave equation in transverse spatial variables only. We prove thatray theory for these one-way wave equations leads to one-way eikonalequations and the correct leading order transport equation for the full waveequation. We then introduce appropriate boundary conditions at z = 0 togenerate waves at depth whose quotient leads to a reflector map and an estimateof the ray theoretical reflection coefficient on the reflector. Thus, these trueamplitude one-way wave equations lead to a ‘true amplitude wave equationmigration’ (WEM) method. In fact, we prove that applying the WEMimaging condition to these newly defined wavefields in heterogeneous medialeads to the Kirchhoff inversion formula for common-shot data when theone-way wavefields are replaced by their ray theoretic approximations.This extension enhances the original WEM method. The objective ofthat technique was a reflector map, only. The underlying theory did notaddress amplitude issues. Computer output obtained using numericallygenerated data confirms the accuracy of this inversion method. However, thereare practical limitations. The observed data must be a solution of thewave equation. Therefore, the data over the entire survey area must becollected from a single common-shot experiment. Multi-experiment data, suchas common-offset data, cannot be used with this method as currentlyformulated. Research on extending the method is ongoing at this time.