Behaviour of earth’s crust due to topographic loads derived by inverse and direct isostasy

The behaviour of the earth’s crust due to topographic loads can be derived by either inverse or direct approach. As for the inverse approach, it is postulated that the density anomaly is proportional to the earth’s radius vector so that it is linearly related to topography by a convolution of the topography and an isotropic kernel function. Accordingly, one can prove that the attraction of the compensating masses is also a convolution of the topography and an isotropic isostatic response function. Such an isostatic response function can be determined by deconvolution. This paper gives the derivation of such a deconvolution by means of spherical harmonics. A practical determination of the isotropic isostatic response function needs the harmonic analysis of both the topography and the attraction of the compensating masses. Applying the principle of inverse isostasy, by which we aim to achieve zero isostatic anomalies, then the attraction of the compensating masses equals the Bouguer anomalies with an opposite sign. The harmonic analysis of the Bouguer anomalies is thus a combination of the harmonic analysis of the topographic potential and the already existing global reference models. As for the direct approach, consider that the earth’s crust is an infinite thin plate subject to topographic loads. The solution of such a bent plate represents the displacement of the earth’s crust due to topographic loads. The paper illustrates that the exact solution of the bent plate is given by the Kelvin function keix. A practical application has been carried out for both approaches using EGM96 and GPM98CR geopotential earth models as well as TUG87 and TBASE digital height models. The results show that the estimated isotropic isostatic response functions derived by the inverse approach behave similarly as that given by direct approach represented by the Kelvin function keix.