In the spatial and frequency domain, the 3-D acoustic wave equation can be formulated as
where can be approximated by the WKBJ Green's function
where is the traveltime from source to an arbitrary point . Using the WKBJ Green's function, Beylkin 1985 gave an inversion formula in 3-D media
In the above formula, is the perturbation to the background velocity . The updated velocity model is given by
S0 is the 2-D integral surface. is introduced by Beylkin 1985, which is associated with the ray curvature. and are the WKBJ Green's function. is a high-pass filter determined the source. represents the observed data at due to the source .
Bleistein et al.1987 specialize the 3-D formula to the 2.5-D geometry using the method of stationary phase. The corresponding 2.5-D inversion formula is
Here, and are the slowness vectors at the imaging location pointing to the source and receiver respectively. and are the parameters defined by the following equations
and are unit downward normals at the source and receiver points respectively. and are the slowness vectors at the source and receiver points respectively.
This inversion formula is only valid in the high-frequency limit. Under such circumstances, it is better to process data for the upward normal derivative at each discontinuity surface of . is a sum of weighted singular functions with peaks on the reflectors. Therefore, actually provides an image of the subsurface. Using the Fourier transform, we can obtain the following 2.5-D formula for .
Bleistein et al.1987 also shows that can be related to the reflection coefficient on the interface by
in the singular function of the model space. is determined by the changes of velocity and density above and below the interface and the incident angle on the interface
In order to determine from , we have to determine . In their paper, Bleistein et al.1987 proposed another inversion operator with a kernel slightly modified from that in .
There is a simple relation between , , and , that is
With known, we can use and to calculate the reflection coefficient . From and , we can further estimate the AVO coefficients: intercept and slope.
Instead of using and , we propose another pair of inversion operators that can determine in a similar, but more straightforward and physically meaningful manner. The first operator gives the reflection coefficient at the specular incident angle
The second gives the reflection coefficient multiplied by
From and , we can easily calculate
In order to reduce the sensitivity of to noise in the data, we use a least-squares procedures to estimate . First, we define a small window (nx nz). Within the window, we can get a series of equations
the least-squares sense estimate of is then