A Born-WKBJ Inversion Method For Acoustic Reflection Data
, by Robert W. Clayton and Robert H. Stolt
A method is presented for determining density and bulk-modulus variations
in the earth from standard reflection surveys. Explicit formulas are given
which utilize the amplitude-versus offset infrormation present in the
observed wave fields. The method automatically accounts for dipping
reflectors, but since it is based on a Born approximation of the scattering
equation, it is restricted to subcritical reflections.
For the inversion, the medium is considered to be composed of a known
low-spatial frequency variation (the background) plus an unknown high-spatial
frequency variation in bulk modulus and density (the reflectivity). The division
between the background and the reflectivity depends on the frequency content
of the source.
For constant background parameters, the computations are done in the Fourier
domain, where the first part of the algorithm includes a frequency shift
identical to than in an F-K migration. The modulus and density variations
are then determined by observing in a least-squares sense amplitude verses
offset wavenumber.
For a spatially variable background WKBJ Green's operators that model the
direct wave medium with a smoothly varying background are used. A downward
continuation with these operators removes the effects of the variable velocity
from the problem, and consequencly the remainder of the inversion essentially
proceeds as if the background were constant. If the background is strictly
depth dependent, then the WKBJ's Green's operators are analytic, and consequently
the inversion can be expressed in closed form.
STANFORD