A GENERALIZED ODOHERTY-ANSTEY FORMULA FOR WAVES IN FINELY LAYERED MEDIA

We investigate the angle-dependent plane wave transmissivity of a pres sure wave in a random, multilayered, acoustic, variable velocity and v ariable density medium. The main result of our consideration is a simp le, explicit analytic description of the influence of such a medium on the transmissivity kinematics and dynamics for the whole frequency ra nge. We assume that the velocity and density dependencies on depth are typical realizations of random stationary processes. Moreover, the fl uctuations in both values must be relatively small compared to their c onstant mean values (of the order of 30 percent or smaller). In our de rivation, we combine the small perturbation technique with the localiz ation and self-averaging theory. We obtain the attenuation and the pha se of the time-harmonic transmissivity, as well as the pulse form of t he transient transmissivity from an angle-dependent combination of the auto- and crosscorrelation functions of both the sonic and density lo gs. Our results for the kinematics of the transmissivity yield the wel l-known ''Backus averaging'' in the low-frequency limit. Likewise, the y provide the ray theory result as the high-frequency asymptotic value . The analytic expression for the transmissivity can be viewed as a ge neralization of the O'Doherty-Anstey formula. Numerical computations o f the actual transmissivity show fluctuations around the theoretical p rediction given by our formula, which is strictly valid only in the ca se of infinitely thick media. The larger the layered medium, the small er are these fluctuations. They can be well estimated with a formula w hich we derive to describe the deviations between the analytic and the exact transmissivity obtained for a layered medium of finite thicknes s.