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Rock properties and amplitude versus angle

Geo Design Services, Houston, Texas, U.S.

Corresponding author: Paul M. Krail, joy1@flash.net

The first 20% of the full text of this article appears below.

As every geophysicist knows, the reflection amplitude versus angle for reflection from an interface between two elastic rocks is given by the Zoeppritz equation. The Zoeppritz equation in the forward mode predicts the amplitude of the reflection, given the angle, the compressional and shear velocities, and densities of the rocks that form an interface. It is possible to turn the Zoeppritz equation around and ask, given the reflection amplitude and angle, compressional velocities and densities, what is the shear velocity of the lower rock? This turning around of the equation is called the inverse mode of the Zoeppritz equation.

A popular procedure for inverting Zoeppritz's equation is to first make the Bortfeld type approximation of the Zoeppritz equation as

Where R is the reflection amplitude, b and m are constants, relative to , containing the rock parameters, and is the angle of reflection. Then using this formula, if the amplitude is plotted versus the sin² , a straight line is obtained and b and m are determined from the graph since b is the zero angle-intercept and m is the slope. One limitation of this procedure is that the approximation should not be used beyond an angle of 30°. Another limitation of the approximation is that a 70% error in the derived shear- wave velocity can occur (Demirbag and Coruh, 1988).

One look at the Zoeppritz equation shows why approximations of its functional dependence on the rock parameters are made. The amplitude is related to the rock parameters via a quite complicated algebraic expression. Until now we have not been able to invert the algebraic expression exactly, but have to first make approximations. It's actually amazing that the Zoeppritz equation with its proliferation of trig functions of the reflection angle and the square root functions of the rock properties combined . . . [Full Text of this Article]