Global earth isostatic model using smoothed Airy-Heiskanen and Vening Meinesz hypotheses

Abstract

Isostatic hypotheses are used for different purposes in geophysics and geodesy. The Erath crustal thickness modelling is more complicated than the classical isostatic models. In this study we try to modify Airy-Hesiskanen model, utilizing a smoothing factor, to a model with regional or global isostatic model through a modern solution of the gravimetric-isostatic Vening Meinesz model and CRUST.0. In Airy-Hesiskanen’s theory there is no correlation between neighbouring crustal columns, while this must be the case in reality due to the elasticity of the Earth. The idea is to keep the simplicity of the Airy-Heiskanen model, because it needs only the topographic data, and change the model which becomes to a model with regional/global isostatic model. The isostatic assumption for compensating the topographic potential is incomplete, as there are other geophysical phenomena which should be considered. Using the isostatic hypothesis for determining the depth of crust causes some disturbing signals in the gravity anomaly (approximately 285 mGal), which influence the crustal thickness determination. In this paper a simple method use for removing these effects. Spherical harmonic potential coefficients of the topographic compensation masses are used for modifying Airy-Heiskanen’s model in a least-square adjustment procedure by estimating smoothing factor. The numerical analysis shows that below degree 10, the modified Airy-Hesiskanen and Vening Meinesz models are close together. Smoothing factors for modifying the Airy-Hesiskanen model vary from 0.75 to 0.64 between degrees 200 and 2159.

Appendix

Spectral and closed form of the kernel $K(\psi, s)$

Here we summarize Sjöberg’s (2009) derivation of the spectral and closed forms of $K\left( {\psi, s} \right)$. In order to obtain the spectral form of $K\left( {\psi, s} \right)$the polynomial expression of the inverse distance should be inserted in Eq. (8a). It is defined by the following series in Legendre’s polynomials, P _n (t):

$$ l_P^{{ - 1}} = \sum\limits_{{n = 0}}^{\infty } {\frac{{{{r}^n}}}{{r_P^{{n + 1}}}}} {{P}_n}(t);\,\,t = \cos \psi, \,\,{\text{for}}\,\,{{r}_P} \geqslant R \geqslant r $$

(A.1)

Residual compensating potential based on Eq. (A.1) can be written (Sjöberg 2009)

$$ \begin{gathered} d{{V}_C}(P) = k\sum\limits_{{n = 0}}^{\infty } {\iint\limits_{\sigma } {\int\limits_{{R - T}}^R {\frac{{{{r}^{{n + 2}}}}}{{r_P^{{n + 1}}}}} }} dr{{P}_n}(t)d\sigma \hfill \\ { } = k\sum\limits_{{n = 0}}^{\infty } {\frac{{{{R}^2}}}{{n + 3}}} {{\left( {\frac{R}{{{{r}_P}}}} \right)}^{{n + 1}}}\iint\limits_{\sigma } {\left[ {1 - {{s}^{{n + 3}}}} \right]}{{P}_n}(t)d\sigma . \hfill \\ \end{gathered} $$

(A.2)

After radial differentiation of Eq. (A.2) and performing some simplifications the spectral form of the kernel function of Eq. (9) becomes

$$ K\left( {\psi, s} \right) = \sum\limits_{{n = 0}}^{\infty } {\frac{{n + 1}}{{n + 3}}} \left( {1 - {{s}^{{n + 3}}}} \right){{P}_n}(t). $$

(A.3)

$K(\psi, s)$ can be rewritten in a closed form (Sjöberg 2009):

$$ \begin{gathered} K\left( {\psi, s} \right) = \frac{1}{{{{L}_0}}} - \frac{{{{s}^3}}}{L} + 3t(1 - s) + \left( {{{s}^2} + 3st} \right)L - \left( {1 + 3t} \right){{L}_0} \hfill \\ { } + \left( {3{{t}^2} - 1} \right)\left( {s\ln \frac{{s - t + L}}{{1 - t}} - \ln \frac{{1 - t + {{L}_0}}}{{1 - t}}} \right), \hfill \\ \end{gathered} $$

(A.4)

where we have introduced

$$ L_0^{{ - 1}} = \sum\limits_{{n = 0}}^{\infty } {{{P}_n}(t) = } \frac{1}{{\sqrt {{2(1 - t)}} }} = \frac{1}{{2\sin \frac{\psi }{2}}}, $$

(A.5)

and

$$ {{L}^{{ - 1}}} = \sum\limits_{{n = 0}}^{\infty } {{{s}^n}{{P}_n}(t)} = \frac{1}{{\sqrt {{1 - 2st + {{s}^2}}} }}. $$

(A.6)

As s < 1, Eq. (A.4) shows that $K\left( {\psi, s} \right)$ is a regular function all over the sphere. In addition, s is close to 1, the function is close to zero, but for t = 1 strong singularity occurs.