In numerical holography one constructs in a computer a map of the locations of reflecting interfaces. Any place in space where the material impedance undergoes a jump is called a reflector. Since it is the waves which we observe, not the material, we need an operational definition of a reflector in terms of waves. A simple such definition is that reflectors are the loci of points in space where an impulse of incident radiation is time coincident with the reflected radiation. This definition may be recast into the frequency domain, may be extended to multiple reflections, and may be elaborated on in a number of other ways. In many geometries this definition may not include all of the impedance contrasts simply because they are not illuminated. Reflector mapping formulas are taken up in more detail by Claerbout (1970). Here we intend to take up some practical aspects of computing the incident and scattered wave fields. Associated with a given spatial coordinate the scalar wave equation has two solutions. For use in holography it is essential to separate the two solutions, that is, it is of no use to solve the wave equation if the solution cannot be split into its two parts. In diffraction theory one manipulates exp(imr)/r and exp(-imr)/r, the two solutions of the wave equation. In seismology one speaks of down-going waves and up-going waves. We seek two equations, one to control the waves from the source to the reflectors and the other to integrate backwards from the area of observation back toward the reflectors.
KeywordsWave Equation Phase Velocity Group Velocity Coarse Grid Schroedinger Equation
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- 1.J. F. Claerbout, Toward a Unified Theory of Reflector Mapping, Geophysics, in press, perhaps 36(1)(1971).Google Scholar
- 2.J. F. Claerbout, Coarse Grid Calculations of Waves in Inhomogeneous Media With Application to Delineation of Complicated Seismic Structure, Geophysics, 35(3)(1970).Google Scholar