Reflection, transmission, and the generalized primary wave

Summary A primary reflection event on a seismic record is often considered to be the result of a process in which a downgoing signal travels without dispersion to a geologic interface, reflects, and returns to the surface. For a one‐dimensional continuous medium, we generalize this point of view in a fashion which properly accounts for multiple scattering. This leads to a definition of a “generalized primary reflection”, which consists of the classical primary from a reflector and all multiples whose propagation paths are confined to the region above the reflector. In the frequency domain, the generalized primary can be written as a product of three factors: T down (x, ω), the transfer function from the surface to a point at traveltime x ; ρ( x ), the reflection coefficient at x; and T up (x, ω), the transfer function from x to the surface. Both T down ( x , ω) and T up ( x , ω) are computed as though the medium contained an absorbing boundary at x. The reflectance, R (ω), for a medium of total travel time L , is then the sum of all the generalized primary reflections: Hubral (1980) gave an analogous description for the discrete case. We refer to T down ( x , ω) as the “generalized incident wave”. Because of multiple scattering, T down ( x , ω) disperses as x increases. A natural way of approximating T down ( x , ω) leads to a deterministic derivation of a result originally obtained by O'Doherty and Anstey in a stochastic framework. The approximate form of T down ( x , ω) is expressed in terms of the autocorrelation of the reflection coefficients. Numerical studies suggest that this accurately represents those interbed multiples with an extra time delay small compared to the propagation time. Corresponding approximations are also obtained for the reflectance of the medium and for the group delay of the generalized incident at low frequencies.