The geometry of AVO spaces

The most commonly used amplitude variation with offset (AVO) space is (A,B) space. When a collection of data points is displayed in this space, it is referred to as an intercept-gradient crossplot. At times, however, alternative AVO spaces have been proposed, using for example the reflectivities of Kρ and µρ or of λ and µ as coordinate axes instead of A and B. It is shown here that these and other AVO spaces are mathematically equivalent, and it is shown how to convert from one to another. Properties that are preserved in the conversion are identified, as well as some that are not. One property that is not preserved is the angle, in (A,B) space known as the χ angle, associated with a particular rock property or fluid effect. Projections in intercept-gradient crossplots are often referred to as rotations. A rotation keeps the length of a vector the same, whereas a projection changes it. The χ angle, commonly referred to as a rotation angle, is in fact a projection angle. It is the angle of a line onto which points are projected. To clarify the process, a fairly comprehensive description is included in this paper. There are an infinite number of possible AVO spaces. All are mathematically equivalent, and it is easy to convert between them. It is not a given that (A,B) space is always the best for a particular goal. Several other AVO spaces are discussed.