The impulse response of a Maxwell Earth

Abstract

An extended form of the correspondence principle is employed to determine directly the quasi‐static deformation of viscoelastic earth models by mass loads applied to the surface. The stress‐strain relation employed is that appropriate to a Maxwell medium. Most emphasis is placed on the discussion of spherically stratified self‐gravitating earth models, although some consideration is given to the uniform elastic half space and to the uniform viscous sphere, since they determine certain limiting behaviors that are useful for interpretation and proper normalization of the general problem. Laplace transform domain solutions are obtained in the form of ‘s spectra’ of a set of viscoelastic Love numbers. These Love numbers are defined in analogy with the equivalent elastic problem. An efficient technique is described for the inversion of these s spectra, and this technique is employed to produce sets of time dependent Love numbers for a series of illustrative earth models. These sets of time dependent Love numbers are combined to produce Green functions for the surface mass load boundary value problem. Through these impulse response functions, which are obtained for radial displacement, gravity anomaly, and tilt, a brief discussion is given of the approach to isostatic equilibrium. The response of the earth to an arbitrary quasi‐static surface loading may be determined by evaluating a space‐time convolution integral over the loaded region using these response functions.