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## Cheop's pyramid for VTI media

To find the maximum of equation (8), we take its derivative with respect to ph and px, and set these derivatives to zero. The stationary point along the ph-px plane is obtained by solving the two new nonlinear equations in terms of px and ph. Since the source rayparameter, ps, and the receiver rayparameter, pg, are linearly related to px and ph, as follows

ps = px - ph,

and

pg = px + ph,

we can find the stationary point solution by solving for ps and pg, instead of solving for px and ph. Solving for ps and pg yields two independent nonlinear equations corresponding to the source and receiver rays, that can be solved separately.

The stationary point solutions (Appendix A) are then given by
 (9)
where y is either the lateral distance between the image point and source, given by 2(x-x0-h) for ps, or the lateral distance between the image point and receiver, given by 2(x-x0+h) for pg.

For isotropic media, and equation (9) reduces to
 (10)
where
 (11)
and

As a result,
 (12)

For traveltime calculation, equation (9) for ps and pg is inserted into
 (13)
where ys =2(h - x + x0) and yg=2(h + x - x0).

Equation (13) is the offset-midpoint (Cheop's pyramid) equation for VTI media. The derivation included the stationary phase (high frequency) approximation, as well as approximations corresponding to small . For =0, equation (13) reduces to the exact form (high-frequency limit) for isotropic media. However, for large the equation, as we will see later, is extremely accurate.

pyr3
Figure 2
Traveltime, in seconds, as a function offset, X, and midpoint, x, both in km, for, from left to right, an isotropic media (), a VTI media with , and a VTI media with , respectively. The velocity is 2 km/s and the vertical time is 1 s, for all three pyramids.

Figure 2 shows the traveltime calculated using equation (13) as a function of offset and midpoint for three values. The shape of the traveltime function resembles Cheop's pyramid, and as a result was given the name. Unlike the isotropic medium pyramid, the VTI ones include nonhyperbolic moveout along the offset and midpoint axis. Clearly, the higher horizontal velocity in the VTI media resulted in faster traveltime with increasing offset and midpoint than the isotropic case.

The stationary phase method also provides an amplitude factor given by the second derivative of the phase function [equation (8)] with respect to ps and pg. Specifically, the amplitude is proportional to the reciprocal of the square root of the second derivative of the phase evaluated at the stationary point (Appendix C).

Next: Vertical velocity variation Up: The midpoint-offset traveltime equation Previous: Prestack phase-shift migration
Stanford Exploration Project
7/5/1998