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In this section, we introduce the Kirchhoff approximate integral
representation of the upward propagating response to a single
reflector, with separated source and receiver points. We then show
how the amplitude of this integrand is related to the zerooffset
amplitude at the source receiver point on the ray, making equal angles
at the scattering point with the rays from the separated source and
receiver. The Kirchhoff integral representation Bleistein (1984); Haddon and Buchen (1981)
describes the wavefield scattered from a single reflector. This
representation is applicable in situations where the highfrequency
assumption is valid (the wavelength is smaller than the characteristic
dimensions of the model) and corresponds in accuracy to the WKBJ
approximation for reflected waves, including phase shifts through
buried foci. The general form of the Kirchhoff modeling integral is
 
(1) 
where and stand for the source
and the receiver location vectors at the surface of observation; denotes a point on the reflector surface ; R is the
reflection coefficient at ; n is the upward normal to the
reflector at the point ; and U_{I} and G are the incident
wavefield and Green's function, respectively represented by their WKBJ
approximation,
 
(2) 
 
(3) 
In this equation,
and are the traveltime and the
amplitude of the wave propagating from to ; and are the corresponding
quantities for the wave propagating from to ; and
is the spectrum of the input signal, assumed to be
the transform of a bandlimited impulsive source.
In the time domain,
the Kirchhoff modeling integral transforms to
 
(4) 
with f denoting the inverse temporal transform of F. The
reflection traveltime corresponds physically to the
diffraction from a point diffractor located at the point on
the surface , and the amplitudes A_{s} and A_{r} are point
diffractor amplitudes, as well.
The main goal of this paper is to test the compliance of
representation (4) with the offset continuation differential
equation. The OC equation contains the derivatives of the wavefield
with respect to the parameters of observation (, and
t). According to the rules of classic calculus, these derivatives
can be taken under the sign of integration in formula
(4). Furthermore, since we do not assume that the trueamplitude
OC operator affects the reflection coefficient R, the
offsetdependence of this coefficient is outside the scope of
consideration. Therefore, the only term to be considered as a trial
solution to the OC equation is the kernel of the Kirchhoff integral,
which is contained in the square brackets in equations (1) and
(4) and has the form
 
(5) 
where
 
(6) 
 
(7) 
In a 3D medium with a constant velocity v, the traveltimes and
amplitudes have the simple explicit expressions
 
(8) 
 
(9) 
where and are the lengths of the incident and reflected
rays, respectively (Figure 1). If the reflector surface
is explicitly defined by some function , then
 
(10) 
cwpgen
Figure 1 Geometry of diffraction
in a constant velocity medium: view in the reflection plane.
We then introduce a particular zerooffset amplitude, namely the
amplitude along the zero offset ray that bisects the angle between the
incident and reflected ray in this plane, as shown in Figure
1. We denote the square of this amplitude as A_{0}. That
is,
 
(11) 
As follows from formulas (7) and (9), the amplitude
transformation in DMO (continuation to zero offset) is characterized
by the dimensionless ratio
 
(12) 
where is the length of the zerooffset ray (Figure 1).
As follows from the simple trigonometry of the triangles, formed by
the incident and reflected rays (the law of cosines),
 

 (13) 
where is the reflection angle, as shown in the figure.
After straightforward
algebraic transformations of equation (13), we arrive at the
explicit relationship between the ray lengths:
 
(14) 
Substituting (14) into (12) yields
 
(15) 
where is the zerooffset twoway
traveltime ().
What we have done is rewrite the finiteoffset amplitude in the
Kirchhoff integral in terms of a particular zerooffset amplitude.
That zerooffset amplitude would arise as the geometric spreading
effect if there were a reflector whose dip was such that the
finiteoffset pair would be specular at the scattering point. Of
course, the zerooffset ray would also be specular in this case.
Next: THE OFFSET CONTINUATION EQUATION
Up: Fomel & Bleistein: Offset
Previous: Introduction
Stanford Exploration Project
11/12/1997