Deducing Subsurface Property Gradients from Surface Wave Dispersion Data

My talk deals with the same subject as that of the previous speaker, Dr. Szabo, except I am not going to assert that it involves residual stresses necessarily. I consider some kind of a perturbation of the elastic properties and/or density due to some kind of surface treatment. As in all problems of this kind I assume, at least for the time being, some kind of an ansatz about mutual proportionality between the various kinds of perturbed physical properties as they vary with depth. There are, perhaps, four ways of classifying approaches to this kind of problem. One is a parametric approach where one assumes that the candidate profile is defined by a finite set of parameters which are adjusted to give the best fit to the experimental data. Another is a nonparametric approach in which one does not have a finite set of parameters but an essentially infinite set of parameters. Another dimension of classification is whether one is using a probabilistic or a nonprobabilistic approach. In the nonparametric case one is forced to use a probabilistic approach. Here one treats every conceivable profile as being present in a statistical ensemble but with probability weightings reflecting one's a priori knowledge of what is more or less reasonable. Disciplines Materials Science and Engineering | Structures and Materials This 10. residual stresses is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/cnde_yellowjackets_1975/ 28 DEDUCING SUBSURFACE PROPERTY GRADIENTS FROM SURFACE WAVE DISPERSION DATA John Richardson Science Center, Rockwell International Thousand Oaks, California ~~y talk deals \tJith the same subject as that of the previous speaker, Dr. Szabo, except I am not going to assert that it involves residual stresses necessarily. I consider some kind of a perturbation of the elastic properties and/or density due to some kind of surface treatment. As in all problems of this kind I assume, at least for the time being, some kind of an ansatz about mutual proportionality between the various kinds of perturbed physical properties as they vary with depth. There are, perhaps, four ways of classifying approaches to this kind of problem. One is a parametric approach ~here one assumes that the candidate profile is defined by a finite set offarameters which are adjusted to give the best fit to the experimental data ,2. Another is a nonparametric approach in which one does not have a finite set of parameters but an essentially infinite set of parameters. Another dimension of classification is whether one is using a probabilistic or a nonprobabilistic approach. In the nonparametric case one is forced to use a probabilistic approach. Here one treats every conceivable profile as being present in a statistical ensemble but with probability weightings reflecting one•s ~priori knmvledge of what is more or less reasonable. Dense Data Case I•m going to start off by considering the dense data case in which the dispersion data is assumed to be given everywhere at all wave lengths or at least sufficiently dense on the wave length axis that interpolation between the points is not serious. I•m going to first proceed to solve the problem as though the continuous input data were exact and then we will see what kinds of problems are encountered. I will start off with the same integral equation that Dr. Szabo used, but written in slightly different notation, namely 00 g ( k) = k f dz K ( kz•) f ( z ) 0 (l) where g(k) is the relative change of the Rayleigh velocity at wave number k due to the subsurface structure described by the profile f(z) giving a scalar measure of the perturbed material properties as a function of the depth z. The kernel function K(kz) derived by Tittman and Thompson 3, is a relatively complex function of the unperturbed material properties. A detailed discussion of the kernel and its derivation may be found in Appendix A of a recent paper by myself~·.