Topographic gravitational potential up to second-order derivatives: an examination of approximation errors caused by rock-equivalent topography (RET)

Abstract

In gravity forward modelling, the concept of Rock-Equivalent Topography (RET) is often used to simplify the computation of gravity implied by rock, water, ice and other topographic masses. In the RET concept, topographic masses are compressed (approximated) into equivalent rock, allowing the use of a single constant mass–density value. Many studies acknowledge the approximate character of the RET, but few have attempted yet to quantify and analyse the approximation errors in detail for various gravity field functionals and heights of computation points. Here, we provide an in-depth examination of approximation errors associated with the RET compression for the topographic gravitational potential and its first- and second-order derivatives. Using the Earth2014 layered topography suite we apply Newtonian integration in the spatial domain in the variants (a) rigorous forward modelling of all mass bodies, (b) approximative modelling using RET. The differences among both variants, which reflect the RET approximation error, are formed and studied for an ensemble of 10 different gravity field functionals at three levels of altitude (on and 3 km above the Earth’s surface and at 250 km satellite height). The approximation errors are found to be largest at the Earth’s surface over RET compression areas (oceans, ice shields) and to increase for the first- and second-order derivatives. Relative errors, computed here as ratio between the range of differences between both variants relative to the range in signal, are at the level of 0.06–0.08 % for the potential, $\sim$3–7 % for the first-order derivatives at the Earth’s surface ($\sim$0.1 % at satellite altitude). For the second-order derivatives, relative errors are below 1 % at satellite altitude, at the 10–20 % level at 3 km and reach maximum values as large as $\sim$20 to 110  % near the surface. As such, the RET approximation errors may be acceptable for functionals computed far away from the Earth’s surface or studies focussing on the topographic potential only. However, for derivatives of the functionals computed near the Earth’s surface, the use of RET introduces very spurious errors, in some cases as large as the signal, rendering it useless for smoothing or reducing of field observation, thus rigorous mass modelling should be used for both spatial and spectral domain methods.

Appendix

To analytically prove that the RET concept formulated in Sect. 2.2 holds for both masses above and below MSL we first show the derivation for ocean water masses located exclusively below MSL and then for ice masses that can be located partly above and partly below MSL (cf. Fig. 11). With respect to an upper crust with the mean density $\rho _{0}$, the ocean water masses are considered as mass deficiencies with the density difference $\Delta \rho _{\mathrm{OCN}}=\rho _{0} - \rho _{\mathrm{OCN}}$ where $\rho _{\mathrm{OCN}}$ is the density of ocean water. In this case, mass equivalence between anomalous ocean water masses and compressed RET masses is expressed by the condition (cf. Eq. 3)

$$\begin{aligned} \rho _0 \left[ {R_{\mathrm{up}}^3 -( {R_{\mathrm{low}} +\Delta H_c })^3} \right] =\Delta \rho _{\mathrm{OCN}} \left[ {R_{\mathrm{up}}^3 -R_{\mathrm{low}}^3 } \right] \end{aligned}$$

(9)

which can be re-arranged to

$$\begin{aligned} \rho _0 \left[ {( {R_{\mathrm{low}} +\Delta H_c })^3-R_{\mathrm{low}}^3 } \right] =\rho _{\mathrm{OCN}} \left[ {R_{\mathrm{up}}^3 -R_{\mathrm{low}}^3 } \right] \end{aligned}$$

(10)

by introducing $\Delta \rho _{\mathrm{OCN}}=\rho _{0} - \rho _{\mathrm{OCN}}$. Solving Eq. 10 for $\Delta H_c$, measured above the bedrock, results into

$$\begin{aligned} \Delta H_c =\root 3 \of {\frac{\rho _{\mathrm{OCN}} }{\rho _0 }\left( {R_{\mathrm{up}}^3 -R_{\mathrm{low}}^3 }\right) +R_{\mathrm{low}}^3 }-R_{\mathrm{low}}, \end{aligned}$$

(11)

which is identical to the general formula derived in Sect.  2.2 for the general RET concept (cf. Eq. 4).

Fig. 11

RET principle in spherical approximation for ocean water masses (left) and ice masses (right)

When considering ice masses, in general, we are dealing with anomalous masses above and below MSL with the respective mass densities $\rho _{\mathrm{ICE}}$ and $\Delta \rho _{\mathrm{ICE}}=\rho _{0} - \rho _{\mathrm{ICE}}$ (cf. Fig. 11). As mass anomalies above and below MSL represent mass excess (positive) and deficiency (negative), respectively, the height $H_{\mathrm{RET}}$ describing the RET masses (cf. Fig. 11) is obtained by the mass equivalence between the difference in anomalous ice masses above and below MSL and compressed RET masses expressed by the condition

$$\begin{aligned} \rho _0 \left[ {( {R+H_{\mathrm{RET}} })^3-R^3} \right]= & {} \rho _{\mathrm{ICE}} \left[ {R_{\mathrm{up}}^3 -R^3} \right] \nonumber \\&-\Delta \rho _{\mathrm{ICE}} \left[ {R^3-R_{\mathrm{low}}^3 }\right] \end{aligned}$$

(12)

which can be re-arranged to

$$\begin{aligned} \rho _0 \left[ {( {R+H_{\mathrm{RET}} })^3-R_{\mathrm{low}}^3 } \right] =\rho _{\mathrm{ICE}} \left[ {R_{\mathrm{up}}^3 -R_{\mathrm{low}}^3 } \right] \end{aligned}$$

(13)

by introducing $\Delta \rho _{\mathrm{ICE}}=\rho _{0} - \rho _{\mathrm{ICE}}$. Solving Eq. 13 for $H_{\mathrm{RET}}$, measured above/below MSL, results in

$$\begin{aligned} H_{\mathrm{RET}} =\root 3 \of {\frac{\rho _{\mathrm{ICE}} }{\rho _0 }\left( {R_{\mathrm{up}}^3 -R_{\mathrm{low}}^3 }\right) +R_{\mathrm{low}}^3 }-R. \end{aligned}$$

(14)

Based on $H_{\mathrm{RET}}$, the height of all compressed masses (above and below MSL) $\Delta H_c$ measured above the bedrock is obtained by

$$\begin{aligned} \Delta H_c= & {} H_{\mathrm{RET}} -H_{\mathrm{low}} \nonumber \\= & {} \root 3 \of {\frac{\rho _{\mathrm{ICE}} }{\rho _0 }\left( {R_{\mathrm{up}}^3 -R_{\mathrm{low}}^3 }\right) +R_{\mathrm{low}}^3 }-R_{\mathrm{low}}, \end{aligned}$$

(15)

which again is identical to the general formula given by Eq. (4). Similarly, the RET height can be obtained for lake water masses, which also can be located above and below MSL.