Gravity effect of spreading ridges: comparison of 2D and spherical models

Abstract

Because of its importance to many Earth science analyses, it is worth assessing whether gravity modelling can be simplified depending on the intended purpose and required precision. While it is obvious that large-scale gravity studies should account for the sphericity of the Earth, each case should be examined on its own merits. Demonstrations are useful for providing estimates of the errors in much simpler 2D modelling. The example of the Mid-Atlantic Ridge serves to compare “large” 2D and spherical 3D models. My model extends horizontally ±2,000 km (±18°) from the model profile across and along the straight ridge axis (along a great circle) and to a depth of 82 km across the axis. 3D modelling would generally be considered obligatory, but this is not clearly necessary from this study. The density structure is highly idealised, the asthenospheric uplift or lithosphere thinning is simplified. The Bouguer anomaly is fitted by least-squares for the density contrast, and the 2D–3D difference of the results is taken as the error. A lithosphere–asthenosphere density contrast of 86.56 kg/m³ was computed for the 2D model, and 84.14 kg/m³ for the spherical model. The difference is small, in the order of 3%, well within all the other uncertainties. My study shows that despite the significant sphericity of the structure, 2D models are well suited for such ridge studies, or generally for models with a laterally extended layered structure, and that spherical modelling can be applied discriminately.

Appendix

Calculation of the potential and the gravity effect

The 2D calculations were carried out with the widely known Talwani method (Talwani et al. 1959). For the 3D calculations the program INVGRA. for was written by the first author as part of his PhD thesis (Çavşak 1992). The parametrization of arbitrarily shaped uniform mass bodies is based on triangulated polyhedra, a classical method in gravity calculations (e.g. Chapman 1979; Holstein et al. 1999; Holstein 2002a, b; Pohánka 1988). A brief outline of the method follows below as it had been derived in a non-traditional way.

Given is the terrestrial x, y, z coordinate system with z pointing downward in g direction. The basic unit of the massive polyhedron is defined by the tetrahedron expanded from the observation point O to any of the planar triangles. Their orientation is arbitrary in (x, y, z). A coordinate transformation is carried out (by vector operations) to the triangle-oriented coordinate system (ξ, η, ζ), such that the triangle is in the ξ–η plane and one of its edges (A–B) is parallel to ξ, see Fig. A-1.

Fig. A-1

(a) Coordinate transformation with vector operation. (I) $\vec{\zeta } = \vec{c} \times \vec{a}$ and (II) $\vec{\eta } = \vec{\zeta } \times \vec{c}$ and (b) illustration of the parameters. The triangle in its transformed coordinate system

The potential effect is given by

$$ \Updelta U = \frac{G \cdot \rho }{h}\int\limits_{{\eta _{\text{A}} }}^{{\eta _{\text{C}} }} {\int\limits_{{\xi ^{\left( 1 \right)} }}^{{\xi ^{\left( 2 \right)} }} {\int\limits_{\zeta = 0}^{h} {\frac{\zeta \cdot d\zeta \cdot d\xi \cdot d\eta }{{ (\xi ^{ 2} + \eta ^{2} + h^{2} )^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} }}} } } $$

(A\hbox{-}1)

where h is the height of tetrahedron (Fig. A-1)

Analytical integration renders $\Updelta U = \frac{1}{2}G \cdot \rho \cdot h \cdot F\left( {\eta ,\xi } \right)$ with

$$ F\left( {\eta ,\xi } \right) = \left| \begin{gathered} \eta \cdot \ln \left[ {\xi + \sqrt {\xi ^{2} + \eta ^{2} + h^{2} } } \right] \hfill \\ + \xi _{2} \cdot \cos \beta \cdot \ln \left[ {\sqrt {\xi ^{2} + \eta ^{2} + h^{2} } + \frac{\eta }{\cos \beta } + \xi_{2} \cdot \sin \beta } \right] \hfill \\ + \zeta \cdot \arctan \left[ {\frac{{h^{2}\cdot \tan \beta - \xi _{2} \cdot \eta }}{{h \cdot \sqrt {\xi ^{2} + \eta ^{2} + h^{2} }}}} \right] \hfill \\ \end{gathered} \right|_{{\xi ^{{(1)^{2} }} ,\eta _{\text{A}}}}^{{\xi ^{{(2)^{2} }} ,\eta _{\text{C}} }} $$

(A\hbox{-}2)

(Chapman 1979), from which follows:

$$ Y = F_{1} \left( {\eta _{\text{C}} ,\xi ^{\left( 2 \right)} } \right) - F_{2} \left( {\eta _{\text{A}} ,\xi ^{\left( 2 \right)} } \right) - F_{3} \left( {\eta _{\text{C}} ,\xi ^{\left( 1 \right)} } \right) + F_{4} \left( {\eta _{\text{A}} ,\xi ^{\left( 1 \right)} } \right) $$

(A\hbox{-}3)

The gravity effect is

$$ \Updelta g = \frac{\partial }{\partial z}\left( {\Updelta U} \right) $$

(A\hbox{-}4)

The formal differentiation leads to a lengthy expression, not reproduced here. It reduces to

$$ \Updelta g = \frac{1}{2}G \cdot \rho \left\{ {\frac{\partial }{\partial z}\left( h \right) \cdot Y + \frac{\partial }{\partial z}\left( Y \right) \cdot h} \right\} $$

(A\hbox{-}5)

With the z component of the normal unit vector of the respective triangle $\hat{\zeta }_{z} = \frac{\partial }{\partial z}\left( h \right)$ and $Y^{\prime} = \frac{\partial }{\partial z}\left( Y \right)$ we can write

$$ \Updelta g = \frac{1}{2}G \cdot \rho \cdot \sum\limits_{i = 1}^{n} {\left( {\hat{\zeta }_{{z_{i} }} \cdot Y_{i} + Y^{\prime}_{i} \cdot h_{i} } \right)} $$

(A\hbox{-}6)