How to Calculate Bouguer Gravity Data in Planetary Studies

Abstract

In terrestrial studies, Bouguer gravity data is routinely computed by adopting various numerical schemes, starting from the most basic concept of approximating the actual topography by an infinite Bouguer plate, through adding a planar terrain correction to account for a local/regional terrain geometry, to more advanced schemes that involve the computation of the topographic gravity correction by taking into consideration a gravitational contribution of the whole topography while adopting a spherical (or ellipsoidal) approximation. Moreover, the topographic density information has significantly improved the gravity forward modeling and interpretations, especially in polar regions (by accounting for a density contrast of polar glaciers) and in regions characterized by a complex geological structure. Whereas in geodetic studies (such as a gravimetric geoid modeling) only the gravitational contribution of topographic masses above the geoid is computed and subsequently removed from observed (free-air) gravity data, geophysical studies focusing on interpreting the Earth’s inner structure usually require the application of additional stripping gravity corrections that account for known anomalous density structures in order to reveal an unknown (and sought) density structure or density interface. In planetary studies, numerical schemes applied to compile Bouguer gravity maps might differ from terrestrial studies due to two reasons. While in terrestrial studies the topography is defined by physical heights above the geoid surface (i.e., the geoid-referenced topography), in planetary studies the topography is commonly described by geometric heights above the geometric reference surface (i.e., the geometric-referenced topography). Moreover, large parts of a planetary surface have negative heights. This obviously has implications on the computation of the topographic gravity correction and consequently Bouguer gravity data because in this case the application of this correction not only removes the gravitational contribution of a topographic mass surplus, but also compensates for a topographic mass deficit. In this study, we examine numerically possible options of computing the topographic gravity correction and consequently the Bouguer gravity data in planetary applications. In agreement with a theoretical definition of the Bouguer gravity correction, the Bouguer gravity maps compiled based on adopting the geoid-referenced topography are the most relevant. In this case, the application of the topographic gravity correction removes only the gravitational contribution of the topography. Alternative options based on using geometric heights, on the other hand, subtract an additional gravitational signal, spatially closely correlated with the geoidal undulations, that is often attributed to deep mantle density heterogeneities, mantle plumes or other phenomena that are not directly related to a topographic density distribution.

Appendices

Appendix 1: Free-Air Gravity Disturbance

The free-air gravity disturbance δg^FA at a location (r, Ω) is computed as follows (e.g., Heiskanen and Moritz 1967)

$$\delta g^{\text{FA}} \left( {r,\Omega } \right) = \frac{\text{GM}}{{R^{ 2} }}\sum\limits_{n = 0}^{{\bar{n}}} {\sum\limits_{m = - n}^{n} {\left( {\frac{R}{r}} \right)^{n + 2} \left( {n + 1} \right)T_{n,m} Y_{n,m} \left(\Omega \right)} } ,$$

(7)

where GM is the central gravitational constant (i.e., Newton’s gravitational constant G multiplied by the total mass of a planetary body M), R is the mean radius of a planetary body, Y_n,m is the surface spherical function of degree n and order m, T_n,m are the (fully normalized) coefficients of the disturbing potential T (i.e., difference between the actual W and normal U gravity potentials, T = W−U), and $\bar{n}$ is the upper summation index of spherical harmonics. The 3-D position in Eq. (7) and thereafter is defined in the spherical coordinate system (r, Ω), where r is the radius of a computation surface point, and Ω = (ϕ, λ) is the spherical direction with a spherical latitude ϕ and longitude λ.

It is worth mentioning that the spectral expression for computing the free-air gravity anomaly Δg^FA comprises a term (n − 1), instead of (n + 1) for δg^FA in Eq. (7).

Appendix 2: Topographic Gravity Correction

The topographic gravity correction g^T (for a uniform topographic density distribution) is defined by Tenzer et al. (2015a)

$$g^{\text{T}} \left( {r,\Omega } \right) = \frac{\text{GM}}{{R^{ 2} }}\sum\limits_{n = 0}^{{\bar{n}}} {\left( {\frac{R}{r}} \right)^{n + 2} } \left( {n + 1} \right)\sum\limits_{m = - n}^{n} {V_{n,m}^{\text{T}} Y_{n,m} \left(\Omega \right)} .$$

(8)

The potential coefficients $V_{n,m}^{\text{T}}$ in Eq. (8) read

$$V_{n,m}^{\text{T}} = \frac{3}{2n + 1}\frac{{\rho^{\text{T}} }}{{\bar{\rho }}}\sum\limits_{k = 0}^{n + 2} {\left( {\begin{array}{*{20}c} {n + 2} \\ k \\ \end{array} } \right)} \frac{1}{k + 1}\frac{{H_{n,m}^{{\left( {k + 1} \right)}} }}{{R^{k + 1} }} ,$$

(9)

where $\bar{\rho }$ denotes the mean mass density of a planetary body, ρ^T is the average topographic density, and H_n,m denote topographic coefficients for physical heights. The coefficients H_n,m as well as the corresponding topographic coefficients for geometric heights and nonnegative heights (in the case of the Bjerhammar-referenced topography) are defined in Appendix 3. We note that the potential coefficients $V_{n,m}^{\text{T}}$ in our numerical studies are computed using the topographic coefficients up to the fifth-order of binomial series in Eq. (9).

Appendix 3: Topographic Coefficients

The Laplace harmonics H_n of physical heights H are defined by the following integral convolution (e.g., Sjöberg 1997)

$$H\left(\Omega \right) = \sum\limits_{n = 0}^{\infty } {H_{n} \left(\Omega \right)} ,\quad H_{n} \left(\Omega \right) = \frac{2n + 1}{4\pi }\iint\limits_{\Phi } {H^{\prime } P_{n} \left( t \right){\text{d}}\Omega ^{\prime } } = \sum\limits_{m = - n}^{n} {H_{n,m} Y_{n,m} \left(\Omega \right)} ,$$

(10)

where H_n,m are the topographic coefficients generated from physical heights H, ${\text{d}}\Omega ^{{\prime }} = \cos \phi^{{\prime }} {\text{d}}\phi^{{\prime }} {\text{d}}\lambda^{{\prime }}$ is the infinitesimal spherical surface element, and $\Phi = \left\{ {\Omega ^{\prime } = \left( {\phi^{\prime } ,\lambda^{\prime } } \right):\phi^{\prime } \in \left[ { - \pi /2,\pi /2} \right] \wedge \lambda^{\prime } \in \left[ {\left. {0,2\pi } \right)} \right.} \right\}$ is the full spatial angle. The Legendre polynomials P_n are defined for the argument $t = \cos \psi$, where $\psi$ is the spherical angle between points (r, Ω) and (r′, Ω′). The corresponding higher-order terms $\left\{ {H_{n,m}^{\left( k \right)} :k = 2,3, \ldots } \right\}$ are given by

$$H_{n}^{\left( k \right)} \left(\Omega \right) = \frac{2n + 1}{4\pi }\iint\limits_{\Phi } {H^{\prime k} P_{n} \left( t \right){\text{d}}\Omega ^{\prime } } = \sum\limits_{m = - n}^{n} {H_{n,m}^{\left( k \right)} Y_{n,m} \left(\Omega \right)} .$$

(11)

By analogy with Eq. (10), we define the Laplace harmonics h_n (and their higher-order terms) of geometric heights h as follows

$$h_{n}^{\left( k \right)} \left(\Omega \right) = \frac{2n + 1}{4\pi }\iint\limits_{\Phi } {h^{\prime k} P_{n} \left( t \right){\text{d}}\Omega ^{\prime } } = \sum\limits_{m = - n}^{n} {h_{n,m}^{\left( k \right)} Y_{n,m} \left(\Omega \right)} \quad \left( {k = 1,2, \ldots } \right) ,$$

(12)

where the topographic coefficients $h_{n,m}^{\left( k \right)}$ are generated from geometric heights h.

Finally, we introduce the Laplace harmonics $\tilde{h}_{n}$ for (nonnegative) geometric heights that are taken with respect to the Bjerhammar sphere/ellipsoid. For this purpose, we define the depth D of the Bjerhammar sphere (below the height reference sphere/ellipsoid) that is equal to the largest negative value of geometric height of a particular planetary body, i.e., $D \equiv \hbox{max} \left| { - h} \right|$.

The Laplace harmonics $\tilde{h}_{n}$ (and their higher-order terms) of h + D then read

$$\tilde{h}_{n}^{\left( k \right)} \left(\Omega \right) = \frac{2n + 1}{4\pi }\iint\limits_{\Phi } {\left( {h^{\prime } + D} \right)^{k} P_{n} \left( t \right){\text{d}}\Omega ^{\prime } } = \sum\limits_{m = - n}^{n} {\tilde{h}_{n,m}^{\left( k \right)} Y_{n,m} \left(\Omega \right)} \quad \left( {k = 1,2, \ldots } \right) .$$

(13)