Contribution of mass density heterogeneities to the quasigeoid-to-geoid separation

Abstract

The geoid-to-quasigeoid separation is often computed only approximately as a function of the simple planar Bouguer gravity anomaly and the height of the computation point while disregarding the contributions of terrain geometry and anomalous topographic density as well as the sub-geoid masses. In this study we demonstrate that these contributions are significant and, therefore, should be taken into consideration when investigating the relation between the normal and orthometric heights particularly in the mountainous, polar and geologically complex regions. These contributions are evaluated by applying the spectral expressions for gravimetric forward modelling and using the EIGEN-6C4 gravity model, the Earth2014 datasets of terrain, ice thickness and inland bathymetry and the CRUST1.0 sediment and (consolidated) crustal density data. Since the global crustal density models currently available (e.g. CRUST1.0) have a limited accuracy and resolution, the comparison of individual density contributions is—for consistency—realized with a limited spectral resolution up to a spherical harmonic degree 360 (or 180). The results reveal that the topographic contribution globally varies between $-$0.33 and 0.57 m, with maxima in Himalaya and Tibet. The contribution of ice considerably modifies the geoid-to-quasigeoid separation over large parts of Antarctica and Greenland, where it reaches $\sim$0.2 m. The contributions of sediments and bedrock are less pronounced, with the values typically varying only within a few centimetres. These results, however, have still possibly large uncertainties due to the lack of information on the actual sediment and bedrock density. The contribution of lakes is mostly negligible; its maxima over the Laurentian Great Lakes and the Baikal Lake reach only several millimetres. The contribution of the sub-geoid masses is significant. It is everywhere negative and reaches extreme values of $-$4.43 m. According to our estimates, the geoid-to-quasigeoid separation globally varies within $-$4.19 and 0.26 m while the corresponding values computed according to a classical definition are only negative and reach extreme values of $-$3.5 m. A comparison of these results reveals that inaccuracies caused by disregarding the terrain geometry and mass density heterogeneities distributed within the topography and below the geoid surface can reach $\pm$2 m or more in the mountainous regions.

Appendix: The potential coefficients \({ }_\mathrm{e}V_{{n,m}}^{{\delta \uprho }} \) and \({ }_\mathrm{i}V_{{n,m}}^{{\delta \uprho }} \)

The potential coefficients ${ }_\mathrm{e}V_{{n,m}}^{{\delta \uprho }}$ and ${ }_\mathrm{i}V_{{n,m}}^{{\delta \uprho }}$ of the volumetric mass density contrast layer (in Eq. 13) are given by (Tenzer et al. 2015a)

$$\begin{aligned}&{ }_\mathrm{e}{V}_{{n,m}}^{{\delta \uprho }} =\frac{3}{2n+1}\frac{1}{{\bar{\uprho }}^{\text {Earth}}}\sum \limits _{i=0}^I \, \left( {{ }_\mathrm{e}\mathrm{Fu}_{{n,m}}^{{(i)}} -{ }_\mathrm{e}\mathrm{Fl}_{{n,m}}^{{(i)}} }\right) ,\end{aligned}$$

(23)

$$\begin{aligned}&{ }_\mathrm{i}{V}_{{n,m}}^{{\delta \uprho }} =\frac{3}{2n+1}\frac{1}{{\bar{\uprho }}^{\text {Earth}}}\sum \limits _{i=0}^I \, \left( {{ }_\mathrm{i}\mathrm{Fu}_{{n,m}}^{{(i)}} -{ }_\mathrm{i}\mathrm{Fl}_{{n,m}}^{{(i)}} }\right) . \end{aligned}$$

(24)

The coefficients {${ }_\mathrm{e}\mathrm{Fl}_{{n,m}}^{{(i)}} ,\;{ }_\mathrm{e}\mathrm{Fu}_{{n,m}}^{{(i)}} :\;i=0,\,1,\ldots ,I$} in Eq. (23) read

$$\begin{aligned} { }_\mathrm{e}\mathrm{Fl}_{{n,m}}^{{(i)}} =\sum \limits _{k=0}^{n+2} {\left( {{\begin{array}{ll} {n+2} \\ k \\ \end{array} }}\right) } \frac{1}{k+1+i}\frac{{L}_{{n,m}}^{( {k+1+i})} }{R^{k+1}}, \end{aligned}$$

(25)

$$\begin{aligned} { }_\mathrm{e}\mathrm{Fu}_{{n,m}}^{{(i)}} =\sum \limits _{k=0}^{n+2} {\left( {{\begin{array}{ll} {n+2} \\ k \\ \end{array} }}\right) } \frac{1}{k+1+i}\frac{{U}_{{n,m}}^{( {k+1+i})} }{R^{k+1}}. \end{aligned}$$

(26)

Equivalently, the coefficients {${ }_{i}\mathrm{Fl}_{{n,m}}^{{(i)}} ,\;{ }_{i}\mathrm{Fu}_{{n,m}}^{{(i)}} :\;i=0,\,1,\ldots ,I$} in Eq. (24) are defined by

$$\begin{aligned} { }_{i}\mathrm{Fl}_{{n,m}}^{{(i)}} =\sum \limits _{k=0}^\infty {( {-1})^k\left( {{\begin{array}{ll} {n+k-2} \\ k \\ \end{array} }}\right) } \frac{1}{k+1+i}\frac{L_{{n,m}}^{( {k+1+i})} }{{R}^{k+1}},\nonumber \\ \end{aligned}$$

(27)

$$\begin{aligned} { }_{i}\mathrm{Fu}_{{n,m}}^{{(i)}} =\sum \limits _{k=0}^\infty {( {-1})^k\left( {{\begin{array}{ll} {n+k-2} \\ k \\ \end{array} }}\right) } \frac{1}{k+1+i}\frac{U_{{n,m}}^{( {k+1+i})} }{{R}^{k+1}}.\nonumber \\ \end{aligned}$$

(28)

The coefficients in Eqs. (25–28) utilize the spherical lower-bound and upper-bound functions $L_{n}$ and $U_{n}$ of a volumetric mass density contrast layer and their higher-order terms (cf. Tenzer et al. 2012a, b)

$$\begin{aligned}&{L}_{n}^{( {k+1+i})} ( \Omega )\nonumber \\&\quad =\left\{ {{\begin{array}{ll} {\begin{array}{l} \frac{2n+1}{4\pi }\int \!\!\!\int \limits _\Phi {\delta \rho ( {{H}'_U ,{{\Omega }}'})\,H_L^{k+1} ( {{\Omega }'}){P}_{n} ( t)\;\mathrm{d}{\Omega }'} \\ =\sum \limits _{m=-n}^n {{L}_{{n,m}}^{( {k+1})} Y_{{n,m}} ( \Omega )\;} \quad \quad \quad \quad \quad \quad \quad \quad i=0\quad \;\quad \\ \end{array}} \\ {\begin{array}{l} \\ \\ \end{array}} \\ {\begin{array}{l} \frac{2n+1}{4\pi }\int \!\!\!\int \limits _\Phi {\beta ( {{\Omega }'})\;\alpha _i ( {{\Omega }'})\,H_L^{k+1+i} ( {{\Omega }'}){P}_{n} ( t)\;\mathrm{d}{\Omega }'} \\ =\sum \limits _{{m}=-n}^{n} {{L}_{{n,m}}^{( {k+1+i})} Y_{{n,m}} ( \Omega )\;} \quad \quad \quad \quad \quad \quad \quad i=1,\,2,\ldots ,I \\ \end{array}} \\ \end{array} }} \right. \nonumber \\ \end{aligned}$$

(29)

and

$$\begin{aligned}&{U}_{n}^{( {k+1+i})} ( \Omega )\nonumber \\&\quad =\left\{ {{\begin{array}{ll} {\begin{array}{l} \frac{2n+1}{4\pi }\int \!\!\!\int \limits _\Phi {\delta \rho ( {{H}'_U ,{\Omega }'})\,H_U^{k+1} ( {{\Omega }'}){P}_{n} ( t)\;\mathrm{d}{\Omega }'} \\ =\sum \limits _{m=-n}^n {{U}_{{n,m}}^{( {k+1})} Y_{{n,m}} ( \Omega )\;} \quad \quad \quad \quad \quad \quad \quad \quad i=0\quad \;\quad \\ \end{array}} \\ {\begin{array}{l} \\ \\ \end{array}} \\ {\begin{array}{l} \frac{2n+1}{4\pi }\int \!\!\!\int \limits _\Phi {\beta ( {{\Omega }'})\;\alpha _i ( {{\Omega }'})\,H_U^{k+1+i} ( {{\Omega }'}){P}_{n} (t)\;\mathrm{d}{\Omega }'} \\ =\sum \limits _{m=-n}^n {{U}_{{n,m}}^{( {k+1+i})} Y_{{n,m}} ( \Omega )\;} \quad \quad \quad \quad \quad \quad \quad i=1,\,2,\ldots ,I \\ \end{array}} \\ \end{array} }} \right. \nonumber \\ \end{aligned}$$

(30)

The density contrast distribution $\delta \rho$ in Eqs. (29) and (30) is taken with respect to the reference topographic density ${\uprho } ^\mathrm{T}$. We then write

$$\begin{aligned} \delta \rho ( {r,\Omega })= & {} {\uprho } ^\mathrm{T}-\rho ( {r,\Omega }) \nonumber \\= & {} \delta \rho ( {H_U ,\Omega })+\beta ( \Omega )\;\sum \limits _{i=1}^I {\alpha _i ( \Omega )\;( {r-R})^i} \,,\nonumber \\&for\quad R+H_U ( \Omega )\ge r>R+H_L ( \Omega ), \end{aligned}$$

(31)

where $\delta \rho ( {H_U ,\Omega })$ is the (nominal) value of the lateral density contrast at the upper bound $H_U$ and location $\Omega$. The radial density change with respect to $\delta \rho ( {H_U ,\Omega })$ is described by the parameters $\beta$ and {$\alpha _i :\;\,i=1,\,2,\ldots , I$}.