The effect of anomalous global lateral topographic density on the geoid-to-quasigeoid separation

Abstract

The geoid can be computed from the quasigeoid by applying the geoid-to-quasigeoid separation. The geoid-to-quasigeoid separation is also needed for a vertical datum unification. Information about the actual topographic density distribution is required to determine accurately the geoid and orthometric heights. In this study, we estimate the effect of (lateral) anomalous topographic density on the geoid-to-quasigeoid separation. This became possible after releasing the first global lateral topographic density model UNB_TopoDens. This model provides also information about topographic density uncertainties. According to our estimates by using the UNB_TopoDens model, the effect of anomalous topographic density on the geoid-to-quasigeoid separation globally varies mostly within ±0.02 m. In mountainous regions (particularly in the Himalayas and Tibet), this effect could reach (or even exceed) ±0.1 m. The analysis also reveals that the errors in computed values of the geoid-to-quasigeoid separation attributed to the UNB_TopoDens density uncertainties (provided in terms of standard deviations for representative lithologies) are globally mostly within a few centimeters. In parts of mountainous regions with large topographic density uncertainties, however, these errors might exceed ±0.1 m. There is another crucial aspect we address here. According to the UNB_TopoDens model, the average topographic density for the whole landmass (except for polar glaciers) is 2247 kg m⁻³. This average density is considerably smaller than the value of 2670 kg m⁻³ that is typically used in geodetic and geophysical applications. This new estimate, if confirmed independently, might have implications on Helmert orthometric heights (adopted in many countries for a vertical datum realization). Changes in Helmert orthometric heights due to adopting this new estimate of the average topographic density are systematic and reach several decimeters. We, therefore, propose to use a gravimetric geoid model for a practical realization of vertical datums that incorporates the topographic density information in countries where Helmert orthometric heights are adopted. This recommendation is fully compatible with modern concepts for a vertical datum realization based on using a gravimetric geoid model and geodetic (ellipsoidal) heights. We also address inconsistencies in computations of Helmert orthometric heights and gravimetric geoid models, and propose to use only accurately computed orthometric heights (including variable topographic density term) to combine (or fit) a gravimetric geoid model with geometric geoid heights at GPS-leveling benchmarks.

Appendices

Appendix A: disturbing potential

The disturbing potential $T$ (i.e., difference between values of the actual and normal gravity potential $W$ and $U$, respectively; $T = W - U$) at a location $\left( {r,\varOmega } \right)$ is computed from (e.g., Heiskanen and Moritz 1967)

$$ T\left( {r,\varOmega } \right) = \frac{\text{GM}}{{\text{R}}}\;\sum\limits_{n = 0}^{{\bar{n}}} {\sum\limits_{m = - n}^{n} {\,\left( {\frac{{\text{R}}}{r}} \right)^{n + 1} \,{{\text{T}}}_{{\text{n,m}}} \,Y_{{\text{n,m}}} \left( \varOmega \right)} } , $$

(A.1)

where ${\text{GM}} = 3 9 8 6 0 0 5\times 1 0^{ 8} \;{\text{m}}^{3} {\text{s}}^{ - 2}$ is the geocentric gravitational constant, ${{\text{R}}} = 6 3 7 1\times 1 0^{ 3}$ m is the Earth’s mean radius, ${{\text{T}}}_{{\text{n,m}}}$ are the disturbing potential coefficients, $Y_{{\text{n,m}}}$ are the surface spherical functions of degree n and order m, and $\bar{n}$ is the upper summation index of spherical harmonics. The 3D position in Eq. (A.1) and thereafter is defined in the spherical coordinate system $\left( {r,\varOmega } \right)$, where $r$ is a radius, and $\varOmega = \left( {\varphi ,\lambda } \right)$ is a spherical direction with latitude $\varphi$ and longitude $\lambda$. The coefficients ${{\text{T}}}_{{\text{n,m}}}$ are obtained from the Stokes coefficients of the global gravitational field after subtracting the coefficients describing the GRS80 (Moritz 2000) normal gravity field.

Appendix B: contribution of constant topographic density

From generalized expressions for the gravitational contribution of volumetric mass density contrast layer defined by Tenzer et al. (2015), the expression for computing the topographic potential difference for the constant topographic density in Eq. (7) is obtained in the following form (cf. Foroughi and Tenzer 2017)

$$ \begin{aligned} V_{g}^{{T,{{\rho }}^{{\text{T}}} }} - V_{t}^{{T,{{\rho }}^{{\text{T}}} }} & = \frac{\text{GM}}{{\text{R}}}\,\sum\limits_{n = 0}^{{\bar{n}}} {\left[ {1 - \left( {1 + \frac{H}{{\text{R}}}} \right)^{ - n - 1} } \right]} \\ &\quad \times \sum\limits_{m = - n}^{n} {{{\text{V}}}_{{\text{n,m}}}^{{{{\text{T}},\rho }^{{\text{T}}} }} \,{{\text{Y}}}_{{\text{n,m}}} \left( \varOmega \right)} \\ & \quad - \frac{\text{GM}}{{\text{R}}}\,\sum\limits_{n = 0}^{{\bar{n}}} {\sum\limits_{m = - n}^{n} {{{\text{V}}}_{{\text{n,m}}}^{\text{bias}} \,{{\text{Y}}}_{{\text{n,m}}} \left( \varOmega \right)} } . \\ \end{aligned} $$

(B.1)

The topographic ${{\text{V}}}_{{\text{n,m}}}^{{{{\text{T}},\rho }^{{\text{T}}} }}$ and topographic-bias ${{\text{V}}}_{{\text{n,m}}}^{{{{\text{T}},\rho }^{{\text{T}}} }}$ coefficients are defined by

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

(B.2)

$$ {{\text{V}}}_{{\text{n,m}}}^{\text{bias}} = 3\frac{{{{\rho }}^{{\text{T}}} }}{{{\bar{{\rho }}}^{\text{Earth}} }}\sum\limits_{k = 1}^{2} {\frac{1}{k + 1}} \frac{{{{\text{H}}}_{{\text{n,m}}}^{{\left( {{\text{k}} + 1} \right)}} }}{{{{\text{R}}}^{k + 1} }}, $$

(B.3)

where $\bar{\rho }^{\text{Earth}} = 5500$ kg m⁻³ is the Earth’s mean density, and ${{\rho }}^{{\text{T}}}$ is the average topographic density. The Laplace harmonics ${{\text{H}}}_{\text{n}}$ of orthometric heights $H$ ($H \ge 0$ on land and $H = 0$ offshore) are defined in terms of the following integral convolution (e.g., Sjöberg 1995)

$$ \begin{aligned} H\left( \varOmega \right) & = \sum\limits_{n = 0}^{\infty } {{{\text{H}}}_{\text{n}} \left( \varOmega \right)} ,\quad \\ {{\text{H}}}_{\text{n}} \left( \varOmega \right) & = \frac{2n + 1}{{4\uppi}}\iint\limits_{\varPhi } {H^{\prime}\,{\text{P}}_{\text{n}} \left( t \right)\;d}\varOmega^{\prime} = \sum\limits_{m = - n}^{n} {{{\text{H}}}_{{\text{n,m}}} \,{{\text{Y}}}_{{\text{n,m}}} \left( \varOmega \right)} , \\ \end{aligned} $$

(B.4)

where ${{\text{H}}}_{{\text{n,m}}}$ are the height coefficients, $H^{\prime}$ is the orthometric height, $d\varOmega^{\prime} = \cos \varphi^{\prime}\,d\varphi^{\prime}\,d\lambda^{\prime}$ is the infinitesimal spherical surface element, and $\varPhi = \left\{ {\;\varOmega^{\prime} = \left( {\varphi^{\prime},\lambda^{\prime}} \right):\varphi^{\prime} \in \left[ { - \pi /2,\,\pi /2} \right] \wedge \lambda^{\prime} \in \left[ {\left. {0,\,2\pi } \right)} \right.\;} \right\}$ is the full spatial angle. The Legendre polynomials ${\text{P}}_{\text{n}}$ are defined for the argument $t = \cos \psi$, where $\psi$ is a spherical distance between points $\left( {r,\varOmega } \right)\;$ and $\left( {r^{\prime},\varOmega^{\prime}} \right)$. The corresponding higher-order harmonics { ${{\text{H}}}_{\text{n}}^{{\left( {\text{k}} \right)}} :\;\,k = 2,\,3, \ldots$ } read (e.g., Tenzer et al. 2015)

$$ {{\text{H}}}_{\text{n}}^{{\left( {\text{k}} \right)}} \left( \varOmega \right) = \frac{2n + 1}{{4\uppi}}\iint\limits_{\varPhi } {H^{{\prime }{k}} \,{\text{P}}_{\text{n}} \left( t \right)\;d}\varOmega^{\prime} = \sum\limits_{m = - n}^{n} {{{\text{H}}}_{{\text{n,m}}} \,{{\text{Y}}}_{{\text{n,m}}} \left( \varOmega \right)} , $$

(B.5)

where the superscript k denotes the k-th integer power of orthometric heights.

Appendix C: contribution of anomalous lateral topographic density

The topographic potential difference $V_{g}^{{T,\delta \rho^{T} }} - V_{t}^{{T,\delta \rho^{T} }}$ for the anomalous lateral topographic density (in Eq. 7) is computed from

$$ \begin{aligned} &\mathop { \lim }\limits_{{r \to {{{\rm R}}}^{ - } }} V_{\text{i}}^{{T,\delta \rho^{T} }} \left( {r,\varOmega } \right) - V_{{\rm e}}^{{T,\delta \rho^{T} }} \left( {r_{t} ,\varOmega } \right) \\ & \quad = \frac{\text{GM}}{{\text{R}}}\;\sum\limits_{n = 0}^{{\bar{n}}} {\,\sum\limits_{m = - n}^{n} {\,\left[ {{}_{{\rm i}}{{\text{V}}}_{{{\rm n,m}}}^{{T,\delta \rho^{T} }} - \left( {1 + \frac{H}{{\text{R}}}} \right)^{ - n - 1} {}_{{\rm e}}{{\text{V}}}_{{\rm n,m}}}^{{{T,\delta \rho^{T} }} } \right]\,Y_{{{\rm n,m}}} \left( \varOmega \right)} } . \\ \end{aligned} $$

(C.1)

The potential coefficients ${}_{{\text{e}}}{{\text{V}}}_{{\text{n,m}}}^{{T,\delta \rho^{T} }}$ and ${}_{\text{i}}{{\text{V}}}_{{\text{n,m}}}^{{T,\delta \rho^{T} }}$ of the anomalous topographic density contrast in Eq. (C.1) are given by

$$ {}_{{\text{e}}}V_{{\text{n,m}}}^{{T,\delta \rho^{T} }} = \frac{3}{2n + 1}\frac{{{}_{{\text{e}}}Fu_{{\text{n,m}}}^{{T,\delta \rho^{T} }} }}{{{\bar{{\rho }}}^{\text{Earth}} }}, $$

(C.2)

$$ {}_{\text{i}}V_{{\text{n,m}}}^{{T,\delta \rho^{T} }} = \frac{3}{2n + 1}\frac{{{}_{\text{i}}Fu_{{\text{n,m}}}^{{T,\delta \rho^{T} }} }}{{{\bar{{\rho }}}^{\text{Earth}} }}, $$

(C.3)

where the numerical coefficients ${}_{{\text{e}}}Fu_{{\text{n,m}}}^{{T,\delta \rho^{T} }}$ and ${}_{\text{i}}Fu_{{\text{n,m}}}^{{T,\delta \rho^{T} }}$ read

$$ {}_{{\text{e}}}Fu_{{\text{n,m}}}^{{T,\delta \rho^{T} }} = \sum\limits_{k = 0}^{n + 2} {\left( {\begin{array}{*{20}c} {n + 2} \\ k \\ \end{array} } \right)} \frac{1}{k + 1}\frac{{{{\delta \rho }}^{{\text{T}}} {\text{K}}_{{\text{n,m}}}^{{\left( {{\text{k}} + 1} \right)}} }}{{{{\text{R}}}^{{{\text{k}} + 1}} }}, $$

(C.4)

$$ {}_{\text{i}}Fu_{{\text{n,m}}}^{{T,\delta \rho^{T} }} = \sum\limits_{k = 0}^{\infty } {\left( { - 1} \right)^{k} \left( {\begin{array}{*{20}c} {n + k - 2} \\ k \\ \end{array} } \right)} \frac{1}{k + 1}\frac{{{{\delta \rho }}^{{\text{T}}} {\text{K}}_{{\text{n,m}}}^{{\left( {{\text{k}} + 1} \right)}} }}{{{{\text{R}}}^{{{\text{k}} + 1}} }}. $$

(C.5)

The combined density-heights coefficients $\left\{ {{{\delta \rho }}^{{\text{T}}} {\text{K}}_{\text{n}}^{{\left( {\text{k}} \right)}} :\;\,k = 1,2,\,3, \ldots } \right\}$ in Eqs. (C4) and (C.5) are given by

$$ \begin{aligned} {{\delta \rho }}^{{\text{T}}} {\text{K}}_{\text{n}}^{{\left( {\text{k}} \right)}} \left( \varOmega \right) & = \frac{2n + 1}{{4\uppi}}\iint\limits_{\varPhi } {\delta \rho^{T} \left( {\varOmega^{\prime}} \right)K^{\prime k} \,{\text{P}}_{\text{n}} \left( t \right)\;{\text{d}}}\varOmega^{\prime} \\ & = \sum\limits_{m = - n}^{n} {{{\delta \rho }}^{{\text{T}}} {\text{K}}_{{\text{n,m}}} \,{{\text{Y}}}_{{\text{n,m}}} \left( \varOmega \right)} , \\ \end{aligned} $$

(C.6)

where the height $K$ is equal to the orthometric height $H$ except for areas covered by polar glaciers and lakes. In these two cases, the height $K$ is defined as the orthometric height minus either the ice thickness or the inland bathymetric depth. The anomalous lateral topographic density $\delta \rho^{T}$ in Eq. (C.6) is taken with respect to the constant topographic density ${{\rho }}^{{\text{T}}}$. We then write

$$ \delta \rho^{T} \left( \varOmega \right) = {{\rho }}^{{\text{T}}} - \rho^{T} \left( \varOmega \right). $$

(C.7)

Appendix D: contribution of lakes and glaciers

The potential differences $V_{g}^{L} - V_{t}^{L}$ and $V_{g}^{I} - V_{t}^{I}$ in Eq. (7) of the gravitational contributions of lakes and polar glaciers are computed using the following generalized expression (cf. Foroughi and Tenzer 2017)

$$ \begin{aligned} & \mathop {\lim }\limits_{{r \to {{\text{R}}}^{ - } }} V_{\text{i}}^{\delta \rho } \left( {r,\varOmega } \right) - V_{{\text{e}}}^{\delta \rho } \left( {r_{t} ,\varOmega } \right) \\ & \quad = \frac{\text{GM}}{{\text{R}}}\;\sum\limits_{n = 0}^{{\bar{n}}} {\,\sum\limits_{m = - n}^{n} {\,\left[ {{}_{\text{i}}{{\text{V}}}_{{\text{n,m}}}^{{{{\delta \rho }}}} - \left( {1 + \frac{H}{{\text{R}}}} \right)^{ - n - 1} {}_{{\text{e}}}{{\text{V}}}_{{\text{n,m}}}^{{{{\delta \rho }}}} } \right]\,Y_{{\text{n,m}}} \left( \varOmega \right)} } . \\ \end{aligned} $$

(D.1)

The potential coefficients ${}_{{\text{e}}}{{\text{V}}}_{{\text{n,m}}}^{{{{\delta \rho }}}}$ and ${}_{\text{i}}{{\text{V}}}_{{\text{n,m}}}^{{{{\delta \rho }}}}$ in Eq. (D.1) read

$$ {}_{{\text{e}}}V_{{\text{n,m}}}^{{{{\delta \rho }}}} = \frac{3}{2n + 1}\frac{{{{\delta \rho }}}}{{{\bar{{\rho }}}^{\text{Earth}} }}\left( {{}_{{\text{e}}}Fu_{{\text{n,m}}}^{{}} - {}_{{\text{e}}}Fl_{{\text{n,m}}}^{{}} } \right), $$

(D.2)

$$ {}_{\text{i}}V_{{\text{n,m}}}^{{{{\delta \rho }}}} = \frac{3}{2n + 1}\frac{{{{\delta \rho }}}}{{{\bar{{\rho }}}^{\text{Earth}} }}\left( {{}_{\text{i}}Fu_{{\text{n,m}}}^{{}} - {}_{\text{i}}Fl_{{\text{n,m}}}^{{}} } \right), $$

(D.3)

where

$$ \begin{aligned} {}_{{\text{e}}}Fl_{{\text{n,m}}}^{{}} & = \sum\limits_{k = 0}^{n + 2} {\left( {\begin{array}{*{20}c} {n + 2} \\ k \\ \end{array} } \right)} \frac{1}{k + 1}\frac{{L_{{\text{n,m}}}^{{\left( {k + 1} \right)}} }}{{{{\text{R}}}^{{{\text{k}} + 1}} }},\quad \\ {}_{{\text{e}}}Fu_{{\text{n,m}}}^{{}} & = \sum\limits_{k = 0}^{n + 2} {\left( {\begin{array}{*{20}c} {n + 2} \\ k \\ \end{array} } \right)} \frac{1}{k + 1}\frac{{{{\text{H}}}_{{\text{n,m}}}^{{\left( {k + 1} \right)}} }}{{{{\text{R}}}^{{{\text{k}} + 1}} }}, \\ \end{aligned} $$

(D.4)

$$ \begin{aligned} {}_{\text{i}}Fl_{{\text{n,m}}}^{{}} & = \sum\limits_{k = 0}^{\infty } {\left( { - 1} \right)^{k} \left( {\begin{array}{*{20}c} {n + k - 2} \\ k \\ \end{array} } \right)} \frac{1}{k + 1}\frac{{L_{{\text{n,m}}}^{{\left( {k + 1} \right)}} }}{{{{\text{R}}}^{{{\text{k}} + 1}} }},\quad \\ {}_{\text{i}}Fu_{{\text{n,m}}}^{{}} & = \sum\limits_{k = 0}^{\infty } {\left( { - 1} \right)^{k} \left( {\begin{array}{*{20}c} {n + k - 2} \\ k \\ \end{array} } \right)} \frac{1}{k + 1}\frac{{{{\text{H}}}_{{\text{n,m}}}^{{\left( {k + 1} \right)}} }}{{{{\text{R}}}^{{{\text{k}} + 1}} }}. \\ \end{aligned} $$

(D.5)

The coefficients defined in Eqs. (D.4) and (D.5) utilize the spherical lower-bound functions ${{\text{L}}}_{\text{n}}$ of a volumetric mass density contrast layer and their higher-order terms (cf. Tenzer et al. 2012a, 2012b)

$$ \begin{aligned} L_{\text{n}}^{{\left( {k + 1} \right)}} \left( \varOmega \right) & = \frac{2n + 1}{{4\uppi}}\iint\limits_{\varPhi } {H_{L}^{k + 1} \left( {\varOmega^{\prime}} \right)P_{n} \left( t \right)\;{\text{d}}\varOmega^{\prime}} \\ & = \sum\limits_{m = - n}^{n} {L_{{\text{n,m}}}^{{\left( {k + 1} \right)}} {{\text{Y}}}_{{\text{n,m}}} \left( \varOmega \right)} . \\ \end{aligned} $$

(D.6)

Since the upper bound of lakes and glaciers is identical with the topographic relief, the numerical coefficients ${}_{{\text{e}}}Fu_{{\text{n,m}}}^{{}}$ and ${}_{\text{i}}Fu_{{\text{n,m}}}^{{}}$ were generated directly from the height coefficients {${{\text{H}}}_{\text{n}}^{{\left( {\text{k}} \right)}} :\;\,k = 1,2, \ldots$ }; see Eqs. (B.4) and (B.5).

The density contrast ${{\delta \rho }}$ is taken with respect to a constant topographic density ${{\rho }}^{{\text{T}}}$. We then write

$$ {{\delta \rho }} = {{\rho }}^{{\text{T}}} - {{\rho }}\quad for\quad {{\text{R}}} + H_{U} \left( \varOmega \right) \ge r > {{\text{R}}} + H_{L} \left( \varOmega \right), $$

(D.7)

Appendix E: sub-geoid mass density contribution

To evaluate the no-topography disturbing potential difference $T_{g}^{NT} - T_{t}^{NT}$ in Eq. (7), the complete topographic effect (including lakes and glaciers) is first subtracted from the disturbing potential $T$ on the topographic surface. This procedure yields the no-topography disturbing potential $T^{NT}$. The computation of $T^{NT}$ is realized according to (Tenzer et al. 2015)

$$ T^{\text{n,m}} \left( {r_{t} ,\varOmega } \right) = \frac{\text{GM}}{{\text{R}}}\;\sum\limits_{n = 0}^{{\bar{n}}} {\sum\limits_{m = - n}^{n} {\,\left( {\frac{{\text{R}}}{{{{\text{R}}} + H}}} \right)^{n + 1} {{\text{T}}}_{{\text{n,m}}}^{\text{n,m}} Y_{{\text{n,m}}} \left( \varOmega \right)} } . $$

(E.1)

The (fully normalized) coefficients ${{\text{T}}}_{{\text{n,m}}}^{\text{n,m}}$ are defined by

$$ {{\text{T}}}_{{\text{n,m}}}^{\text{n,m}} = {{\text{T}}}_{{\text{n,m}}} \, - {{\text{V}}}_{{\text{n,m}}}^{{{{\text{T}},\rho }^{{\text{T}}} }} + {}_{{\text{e}}}V_{{\text{n,m}}}^{{\text{L}}} + {}_{{\text{e}}}V_{{\text{n,m}}}^{\text{I}} + {}_{{\text{e}}}{{\text{V}}}_{{\text{n,m}}}^{{{\text{T,}}\delta \rho^{{\text{T}}} }} , $$

(E.2)

where ${}_{{\text{e}}}V_{{\text{n,m}}}^{{\text{L}}}$, ${}_{{\text{e}}}V_{{\text{n,m}}}^{\text{I}}$ and ${}_{{\text{e}}}{{\text{V}}}_{{\text{n,m}}}^{{{\text{T,}}\delta \rho^{{\text{T}}} }}$ are, respectively, the coefficients of lakes, glaciers and anomalous topographic density.

The coefficients ${{\text{T}}}_{{\text{n,m}}}^{\text{n,m}}$ are then used to compute the no-topography disturbing potential difference in Eq. (7) as follows (Tenzer et al. 2015)

$$ \begin{aligned} T_{g}^{NT} - T_{t}^{NT} & = T^{NT} \left( {r_{g} ,\varOmega } \right) - T^{NT} \left( {r_{t} ,\varOmega } \right) \\ & = \frac{\text{GM}}{{\text{R}}}\;\sum\limits_{n = 0}^{{\bar{n}}} \sum\limits_{m = - n}^{n} \left[ {\,1 - \left( {1 + \frac{H}{{\text{R}}}} \right)^{ - n - 1} } \right]\\ & \quad {{\text{T}}}_{{\text{n,m}}}^{\text{n,m}} \,Y_{{\text{n,m}}} \left( \varOmega \right) . \\ \end{aligned} $$

(E.3)

A downward continuation of $T^{NT}$ in Eq. (E.3) is permissible, because topographic masses above the geoid are mathematically removed.