Keywords

1 Introduction

The topographic corrections in gravimetric geoid determination can be decomposed into the corrections for the Bouguer shell/plate and the terrain (e.g., Heiskanen and Moritz 1967, Sect. 3-3). If one assumes that the mass distribution of the topography is rather random, it is suitable to define the Bouguer shell as spherically symmetric with density distribution as that given along the radius vector at the computation point. This implies that there are terrain corrections to be considered all-over the Earth except along the radius at the computation point. Here we assume that the computation point is located at the Earth’s surface, and we will consider the locations of masses causing possible terrain bias in analytical downward continuation (DWC) of the (disturbing) potential to sea-level. (The Bouguer shell bias can be found, e.g., in Sjöberg 2007 and in Sjöberg and Bagherbandi (2017, Sect. 5.2.5).

In the DWC process of the surface disturbing potential to sea level, it is only the topographic potential that may cause a bias. Below we will divide the study of the possible bias caused by the terrain masses located in the exterior zone (i.e. the zone exterior to the Bouguer plate) as well as in the remote and near zones inside the Bouguer plate. In each zone we search for the answer to the question whether there could be a source of mass causing a bias.

Fig. 1
figure 1

The shaded area describes the dome Ω generated by Eq. (14)

Full size image

The method we use to answer the question is to compare the effect of each surface potential (dV P) generated by a point mass in the zone when downward continued to sea level \( \left({dV}_P^{\ast}\right) \) and its true potential at sea level (dV g). The DWC is formulated by the Taylor series:

$$ {dV}_{\mathrm{P}}^{\ast }=\sum\limits_{\mathrm{k}=0}^{\mbox{\textyen}}\frac{{\left(-{H}_P\right)}^{\mathrm{k}}}{\mathrm{k}!}\frac{\partial^{\mathrm{k}}\mathrm{d}{V}_{\mathrm{P}}}{\partial {H}_P^k}, $$
(1)

where H P is the orthometric height of the topography at point P. If \( {dV}_P^{\ast}\ne {dV}_{\mathrm{g}} \) (which may be due to that the series diverges or converges to the wrong value), there is a bias, otherwise not.

Assuming that sea-level is located on a flat Earth, the terrain potentials at the surface point P and at sea-level generated by a point mass of density dμ (more precisely, denoting the gravitational constant times the density) located at lateral distance s from P and at height h become (see Fig. 1)

$$ {dV}_P= d\mu /D\kern1em \mathrm{and}\kern1em {dV}_g= d\mu /{D}_0, $$
(2a)

where

$$ D=\sqrt{s^2+{\Delta}^2},\kern1em {D}_0=\sqrt{s^2+{h}^2}\kern1em \mathrm{and}\kern1em \Delta ={H}_P-h. $$
(2b)

2 The Terrain Correction for Masses Located in the Remote Zone of the Bouguer Shell

Let us define the remote zone as the location of all points at lateral distance s exceeding the height of the computation point, i.e. s > s 0 = H P.

Then the potential at sea-level of the point mass can be developed in the series

$$ {dV}_g^t=\frac{\mu }{s}\sum \limits_{k=0}^{\infty}\left(\begin{array}{c}-1/2\\ {}k\end{array}\right){d}^k=\frac{\mu }{s}\left[1-\frac{d}{2}+\frac{3{d}^2}{8}-\right]. $$
(3)

where d = (h/s)2.

Also, for H P ≥ h the surface potential at P becomes

$$ {dV}_P^t=\frac{\mu }{s}\sum \limits_{k=0}^{\infty }{\left(\begin{array}{c}-1/2\\ {}k\end{array}\right)}^k{t}^k=\frac{\mu }{s}\left[1-\frac{t}{2}+\frac{3{t}^2}{8}-\right], $$
(4)

where t = (Δ/s)2.

We note that each term t k is a polynomial in H P, implying that it is downward continued by simply putting H P = 0.

Hence, inserting (4) into (1) with (t k)∗ → d k for all k > 0, it follows that

$$ {\left({dV}_P\right)}^{\ast }={dV}_g, $$
(5)

i.e. there is no terrain correction needed in this zone.

Note that the exterior part of the far-zone (where H P < h) is not yet included. See the next section.

3 The Terrain Correction for Masses Located Outside the Bouguer Plate

In the exterior zone (outside the Bouguer plate at height H P) it holds that H P < h (the height of the point mass) and D 0 > D (see notations in Fig. 1). Then the inverse distances in Eq. (2b) are related by the sequence

$$ {\displaystyle \begin{array}{l}A=\frac{1}{D}=\frac{1}{\sqrt{s^2+{\Delta}^2}}=\\ {}\frac{1}{\sqrt{B^2+\left({\Delta}^2-{h}^2\right)}}=\frac{B}{\sqrt{1+{B}^2\left({\Delta}^2-{h}^2\right)}}=\frac{B}{\sqrt{1+q}}\end{array}}, $$
(6a)

where

$$ \begin{array}{l}B=1/{D}_0=1/\sqrt{s^2+{h}^2}\kern1em \mathrm{and}\kern1em q={B}^2\left({\Delta}^2-{h}^2\right)\\\qquad\qquad\qquad ={B}^2\left({H}_P^2-2{H}_Ph\right). \end{array}$$
(6b)

Hence, the potential at P can be written

$$ {dV}_P=\mu A=\mu B/\sqrt{1+q}, $$
(7)

and, if |q| < 1, the inverse square-root can be expanded as a power series in q as for t in Eq. (4). However, outside the Bouguer shell h > H P, so that

$$ \left|q\right|=\frac{H_P\left(2h-{H}_P\right)}{s^2+{h}^2}<1, $$
(8)

and, accordingly, one can expand A as a convergent power series of q. Then, after applying Eq. (1) follows

$$ {\left({dV}_P\right)}^{\ast }=\mu {A}^{\ast }=\mu B={dV}_g, $$
(9)

which means that there is no geoid bias generated in the DWC process by the masses in the exterior to the Bouguer plate.

4 The Terrain Correction Due to Masses in the Near-Zone Inside the Bouguer Plate

Next, let us replace s 0 = H P by s 1 < s 0 in Eq. (2b). Then the DWC effect on the inverse distance

$$ {\displaystyle \begin{array}{l}C=\frac{1}{D}=\frac{1}{\sqrt{s_1^2+{\left({H}_P-h\right)}^2}}=\\ {}\frac{1}{\sqrt{s_1^2+{h}^2}}\frac{1}{\sqrt{1+p}}=\frac{1}{\sqrt{s_1^2+{h}^2}}\sum \limits_{k=0}^{\infty}\left(\begin{array}{c}-1/2\\ {}k\end{array}\right){p}^k,\end{array}} $$
(10)

will approach \( 1/\sqrt{s_1^2+{h}^2} \), if

$$ \left|\left({H}_P^2-2{hH}_P\right)/\left({s}_1^2+{h}^2\right)\right|<1, $$
(11)

which yields

$$ {dV}_P^{\ast }=\mu {C}^{\ast }=\mu /\sqrt{s_1^2+{h}^2}={dV}_g. $$
(12)

Recalling that inside the Bouguer plate H P > h, so that inequality (11) becomes

$$ \left({H}_P^2-2{hH}_P\right)/\left({s}_1^2+{h}^2\right)<1, $$
(13a)

or

$$ \frac{s_1^2+{\left({H}_P+h\right)}^2}{2{H}_P^2}>1. $$
(13b)

This inequality is met for all masses located at points (s, h) outside the dome Ω (with a hole of radius 0 < s 1 < s 0 = H P in its center) generated by the circle

$$ \frac{s^2+{\left({H}_P+h\right)}^2}{2{H}_P^2}=1 $$
(14)

within the sector s 1 ≤ s ≤ H P and height \( 0\le h\le \left(\sqrt{2}-1\right){H}_P \) , rotating round its vertical axis. See Fig. 1. Inside Ω (s,h) violates (13b). (See the shaded area in Fig. 1 with vertex at point Q 0 .)

5 Conclusions

If the Bouguer shell is modelled with a spherically symmetric density distribution that changes radially according to the topographic density distribution along the radius vector r through the computation point on the Earth’s surface, there will remain terrain masses all over the Earth, except along r. Assuming that the disturbing potential is known at the Earth’s surface at height H P, the study shows that only terrain mass inside a dome Ω of height \( \left(\sqrt{2}-1\right){H}_P \) along r with its base of radius H P at sea level, is likely to cause a bias. Masses located outside Ω cannot produce a bias in the geoid determination, and if there are no terrain masses inside Ω, the only topographic bias will be that caused by the Bouguer plate.