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

We present new and exact mathematical formulas for the P-SV particle displa cement reflection and transmission coefficients for elastic plane waves inc ident upon a nonwelded contact (or linear slip) interface separating two so lid half-spaces. We represent the nonwelded contact at the interface with d isplacement discontinuity boundary conditions, that is, the traction is con tinuous across the interface, but the particle displacement is not: the dis continuity in the displacement is proportional to the traction. The formula s are derived by algebraically solving these boundary conditions. The formu las 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 prov ide insight into the nature of the coefficients and can be used to derive a pproximate formulas which have applications in the inversion and interpreta tion 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 includ es 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, re flected elastic waves are produced, unlike the case of welded contact, in w hich no reflections are produced because of the lack of an impedance contra st. This effect may partially explain occasional reports in the literature of anomalously large seismic amplitudes observed in areas where impedance c ontrasts are small. We present some numerical examples illustrating the eff ects of nonwelded contact on reflection and transmission coefficients. We s how that energy is conserved at all incidence angles at a nonwelded interfa ce between two different elastic solids. We also present some examples of a pplications of the exact formulas: (1) we derive approximate formulas for a weakly nonwelded interface between two identical solids, (2) we use them t o study the sensitivity of the coefficients to the normal and parallel spec ific compliances (measures of the amount of nonweldedness), finding that th ey are generally more sensitive to the normal than the parallel compliance (except for the SS reflection), that the sensitivities vary considerably wi th incidence angle, and that the PP and SS transmission coefficients are ba sically insensitive to the compliances for a weakly nonwelded interface, an d (3) we further approximate the formulas in example 1 to the case of small propagation angles, resulting in formulas that could be useful in amplitud e versus offset studies.