Amplitude vs offset

Usually referred to as AVO, amplitude vs offset is a general euphemism for the angle dependence of seismic reflectivity. Strictly speaking, this behaviour is AVA, or amplitude vs angle. But AVO has become a very generally used 'word' in geophysics.

The non-literality of seismic
Seismic reflections come from geology, so it seems reasonable to interpret them as geology. But the reflections are waves, and waves are slippery things: they have to travel through kilometres of imperfectly known geology; they can interfere and diffract; they emanate spherically from the source and get much weaker quickly.

Despite this complexity, the strength of seismic reflections can have real, quantitative, geological meaning. For example, amplitude is related to changes in acoustic impedance, sometimes called Z, which is equal to the bulk density ρ and P-wave velocity V, itself related to lithology, fluid, and porosity:


 * $$Z = \rho V_\mathrm{P} $$

Offset and incidence angle
The idea that we collect data from different angles is not complicated. Remember the classic cartoon of a seismic survey. It's clear that some of the ray paths bounce off the geological strata at relatively small incidence angles, closer to straight down-and-up. Others, arriving at receivers further away from the source, have greater angles of incidence. The distance between the source and an individual receiver is called offset, and is deducible from the seismic field data because the exact location of the source and receivers is always known.

The basic physics behind AVO analysis is that the strength of a reflection does not only depend on the acoustic impedance—it also depends on the angle of incidence. Only when this angle is 0 (a vertical, or zero-offset, ray) does the simple relationship above hold.

Angle dependence and the Zoeppritz equations
Though it may be unintuitive at first, angle-dependent reflectivity is an idea we all know well. Imagine an ordinary glass window: you can see through it perfectly well when you look straight through it, but when you move to a wide angle it suddenly becomes very reflective (at the so-called critical angle). The interface between water and air is similarly reflective at wide angles.

Karl Zoeppritz (German, 1881–1908) was the first seismologist to describe the relationship between reflectivity and angle of incidence. In this context, describe means write down the equations for. Not two or three equations, lots of equations.

The Zoeppritz equations are very good model for how seismic waves propagate in the earth. There are some unnatural assumptions about isotropy, total isolation of the interface, and other things, but they work well in many real situations. The problem is that the equations are unwieldy, especially if you are starting from seismic data and trying to extract rock properties—trying to solve the so-called inverse problem. Since we want to be able to do useful things quickly, and since seismic data are inherently approximate anyway, several geophysicists have devised much friendlier models of reflectivity with offset.

Two pre-eminent examples of such approximations are the Aki–Richards equation and the Shuey equation.

Links to Agile

 * What is AVO? — Blog post