Exact and approximate formulas for P‐SV reflection and transmission coefficients for a nonwelded contact interface

Abstract

We present new and exact mathematical formulas for the P–SV particle displacement reflection and transmission coefficients for elastic plane waves incident upon a nonwelded contact (or linear slip) interface separating two solid half‐spaces. We represent the nonwelded contact at the interface with displacement discontinuity boundary conditions, that is, the traction is continuous across the interface, but the particle displacement is not: the discontinuity in the displacement is proportional to the traction. The formulas are derived by algebraically solving these boundary conditions. The formulas can be expressed in the form of those for the welded contact case, with some terms being modified due to nonwelded contact, plus additional terms due purely to nonwelded contact. Such formulas are useful because they provide insight into the nature of the coefficients and can be used to derive approximate formulas which have applications in the inversion and interpretation of seismic data. The exact formulas can also be applied to the case of a viscous nonwelded interface by simply replacing one of their parameters (the tangential specific compliance) with a modified parameter which includes the specific viscosity. We also apply the exact formulas to the specific case in which the incidence and transmission media are identical (e.g., a joint, fault, or fracture in a single homogeneous medium). In this case, reflected elastic waves are produced, unlike the case of welded contact, in which no reflections are produced because of the lack of an impedance contrast. This effect may partially explain occasional reports in the literature of anomalously large seismic amplitudes observed in areas where impedance contrasts are small. We present some numerical examples illustrating the effects of nonwelded contact on reflection and transmission coefficients. We show that energy is conserved at all incidence angles at a nonwelded interface between two different elastic solids. We also present some examples of applications of the exact formulas: (1) we derive approximate formulas for a weakly nonwelded interface between two identical solids, (2) we use them to study the sensitivity of the coefficients to the normal and parallel specific compliances (measures of the amount of nonweldedness), finding that they are generally more sensitive to the normal than the parallel compliance (except for the SS reflection), that the sensitivities vary considerably with incidence angle, and that the PP and SS transmission coefficients are basically insensitive to the compliances for a weakly nonwelded interface, and (3) we further approximate the formulas in example 1 to the case of small propagation angles, resulting in formulas that could be useful in amplitude versus offset studies.