Amplitude-variation-with-angle behavior of self-similar interfaces

Amplitude-variation-with-angle (AVA) analysis is generally based on the ass umption that the medium parameters behave as step functions of the depth co ordinate z, at least in a finite region around the interface. However, outl iers observed in well logs often behave quite differently from step functio ns. In this paper, outliers in the acoustic propagation velocity are pa ram eterized by functions of the form c(z)= c(1) \ z/z(1)\(alpha). The wavelet transform of this function reveals properties similar to those of several o utliers in real well logs. Moreover, this function is self-similar, accordi ng to c(beta z)= beta(alpha)c(z), for beta > 0. Analytical expressions are derived for the acoustic normal incidence reflection and transmission coeff icients for this type of velocity function. For oblique incidence, no expli cit solutions are available. However, by exploiting the self-similarity pro perty of the velocity function, it turns out that the acoustic angle-depend ent and frequency-dependent reflection and transmission coefficients are se lf-similar as well. To be more specific. these coefficients appear to be co nstant along curves described by p(1-alpha)omega(-alpha) = constant, where p is the raypath parameter and omega the angular frequency. The singularity exponent alpha that is reflected in these curves may prove to be a useful indicator in seismic characterization.