The Aki-Richards approximation comes in two forms, one involving the incidence angle and the other involving the average of the incidence and transmission angles. The first of these may be straightforwardly derived by expanding a matrix form of the Knott-Zoeppritz equation in series and truncating. The second is formally a linearization but is more reasonably interpreted as being nonlinear, and this can be quantified by expanding the average angle in series about the P-wave velocity perturbation. The Aki-Richards approximation is often discussed in terms of P-wave, S-wave, and density reflectivities. The average angle too may be expressed in terms of the incidence angle and the P-wave reflectivity, with the latter perturbing the former.
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