Zoeppritz and Shuey
Learning objectives
- Write the Shuey form: intercept R0 plus gradient G times sine-squared
- Identify the intercept and gradient and what each carries
- See the two-term approximation track Zoeppritz to about 30 degrees
- Know when the exact equations are needed
From Exact to Usable
The exact Zoeppritz equations are correct at every angle, but as a four-by-four system they bury the physics in algebra. Shuey rearranged the small-to-moderate-angle behaviour into a form built from quantities an interpreter can name and use:
is the intercept, the normal-incidence reflection you already know. is the gradient, how fast the amplitude changes as the angle opens, and it is the term that carries the fluid signature. is a curvature term that only matters at far angles. Dropping gives the famous two-term Shuey, .
Where the Approximation Lives
Overlay the exact Zoeppritz curve with the two-term Shuey and the three-term Aki-Richards and the result is reassuring: they track each other almost perfectly out to about thirty degrees, then part company. Inside the usable offset range of most surveys, the two-term approximation is excellent, which is why practical AVO is done on just two numbers, the intercept and the gradient. Beyond thirty degrees the error grows and the exact equations are needed, which is why far-offset and wide-azimuth work carries the full Zoeppritz.
This is the same fit-for-purpose logic as the rest of the course, now applied to a formula rather than an engine: use the simplest description that is accurate enough for your angles. Switch the lower rock among a gas sand, a brine sand, and a hard sand and watch the gradient, and the whole curve, change character. That change is the fluid and lithology signal, and the next section names the shapes it makes.