PARAMETER ANALYSIS

The 2.5-D Kirchhoff inversion can be viewed as a weighted Kirchhoff depth migration. In other words, if there is no middle row in equation kirchhoff-integral, the final result will be a 2-D Kirchhoff depth migration in V(x,z) media. In this section, we investigate the relationship between the two key components in equation kirchhoff-integral in the homogeneous medium.

Weighting function

The weighting function determines the contribution of each data sample to the image. The weighting function depends on the locations of source, receiver, and diffractor.

$$w({\bf x}_s,{\bf x}_r;{\bf x}) = [A_sA_r]^{-1}\sqrt{\frac{1... ...\cdot {\bf p}_{r_0}A^2_r({\bf x},{\bf x}_r)\sigma_{r_0}\right ]$$ (25)

Double-square-root (DSR) equation

The DSR equation is the kinematic relation between source, receiver, and diffractor in the homogeneous media.

$$\begin{eqnarray} \tau ({\bf x}_s,{\bf x}_r;{\bf x}) & = & \tau_s + \tau_r \nonum... ...( \sqrt{(x_s-x)^2+(z_s-z)^2} + \sqrt{(x_r-x)^2+(z_r-z)^2} \right )\end{eqnarray}$$ (26)

It is worth investigating the relationship between these two components and other parameters, such as image depth, integral aperture, velocity, and offset, etc. In order to simplify the problem, we assume a homogeneous media and discuss the dependence of weighting function and DSR equation on other parameters, such as offset, depth, and velocity.

The DSR equation is a function of imaging depth, velocity, and offset. As shown in Figure 1, with increasing imaging depth, the hyperbolic curve becomes flatter. Therefore, anti-aliasing requirements in the deep zone are not as severe as it is in the shallow zone.

Similarly, high velocity corresponds to a flattened hyperbola. Large offset has a similar effect. Actually, if we view the offset response in 3-D, it is the famous "Cheops pyramid" Claerbout (1985). We then take the first and second derivatives of the hyperbolic curves. As show in Figure 2, with the increase of offset, the first derivative has two inflection points. Correspondingly, two peak values show up in the second derivative for non-zero offset.

 

cheops
Figure 1 Cheops pyramid changes with different parameters. (L) From top to bottom, the hyperbolic moveout curves become flatter when velocity increases. (M) From bottom to top, the curves become flatter with the increase of depth. (R) From bottom to top, the hyperbolic curves change from zero to nonzero offset.

 

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Figure 2 Cheops pyramid, first, and second derivatives. (L) Cheops pyramid changes from zero to nonzero offset. (M) The first derivative has two inflections points near the middle of the panel in the case of non-zero offset. (R) The second derivative has a corresponding double-peaked shape in non-zero offset.

In a constant velocity medium, the weighting function depends only on imaging depth and offset. As shown in Figure 3, the weighting function has a double-peaked shape in non-zero offset. This feature is very interesting. Intuitively, it is very natural to think that the data value located right in the middle of the panel should have the largest contribution to the image. The double-peaked weighing function in the case of common-offset configuration suggests that the largest contribution to the image is not from the middle of the integral curve, but from the two flanks. Therefore, it is very important to include the two peaks to get a true-amplitude image when choosing the integral aperture.

 

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Figure 3 The weighting function changes with depth and offset. (L) With increasing depth, the double peaks smear out. (R) From zero to nonzero offset, the weighting function goes from single-peaked to double-peaked shape.