Amplitude preservation for offset continuation: Confirmation for Kirchhoff data |

where and stand for the source and the receiver location vectors at the surface of observation; denotes a point on the reflector surface ; is the reflection coefficient at ; is the upward normal to the reflector at the point ; and and are the incident wavefield and Green's function, respectively represented by their WKBJ approximation,

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

with denoting the inverse temporal transform of . The reflection traveltime corresponds physically to the diffraction from a point diffractor located at the point on the surface , and the amplitudes and 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
). 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 true-amplitude
OC operator affects the reflection coefficient , the
offset-dependence 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

In a 3-D medium with a constant velocity , the traveltimes and
amplitudes have the simple explicit expressions

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

cwpgen
Geometry of diffraction in a
constant velocity medium: view in the reflection plane.
Figure 1. |
---|

We then introduce a particular zero-offset 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 .
That is,

where is the length of the zero-offset ray (Figure 1).

As follows from the simple trigonometry of the triangles, formed by
the incident and reflected rays (the law of cosines),

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:

Substituting (14) into (12) yields

where is the zero-offset two-way traveltime ( ).

What we have done is rewrite the finite-offset amplitude in the Kirchhoff integral in terms of a particular zero-offset amplitude. That zero-offset amplitude would arise as the geometric spreading effect if there were a reflector whose dip was such that the finite-offset pair would be specular at the scattering point. Of course, the zero-offset ray would also be specular in this case.

Amplitude preservation for offset continuation: Confirmation for Kirchhoff data |

2013-03-03