Molodensky’s and Helmert’s Theories: Two Equivalent Geodetic Approaches to the Determination of the Gravity Potential and the Earth Surface

Abstract

A fundamental problem of physical geodesy is the determination of the “Surface of the Earth” and its gravitational potential from various types of observations performed on the Earth surface S itself or in the outer space. When data are derived from gravimetry on S we speak of Molodensky’s problem. Since the gravity field depends linearly on its source, i.e. the mass distribution, it follows that we can manipulate the (unknown) internal density in a known way and still return to the same external solution once the effects of the manipulation have been eliminated (restored). This is used, in the frame of Molodensky’s theory, with the Residual Terrain Correction that is removed (and then restored) before approximating the solution by some regularized (collocation or other) approach. Differently, Helmert’s approach shifts the masses of the topographic layer, compressing them to some internal surface and substituting their effects on gravity by that of a single layer. Data are thus lowered to some internal ellipsoid or sphere and a solution is then easily computed. The effects of the internal changes are then inverted and added back to the solution. Despite the apparent completely different approach one can prove that the final solutions, when data are given continuously on the boundary and the errors are made to tend to zero, converge to the true potential on the surface S and then in the outer space. So the two solutions are geodetically equivalent and do not create any scientific conflict. Different is what happens inside S, down to the geoid level. Here Helmert’s approach, that introduces the density of the topographic layer as data, is certainly less erroneous in approximating the true potential. Yet due to the imperfect knowledge of the density and even more to the ill-posedness of the downward continuation operator, the internal potential can have large errors, unless the solution is duly regularized and an appropriate tuning is introduced between the regularization parameter and the size of data errors.

1 The Gravimetric Surface of the Earth

The purpose of physical geodesy is to find the potential W of the gravity field of the Earth in the space outside the masses and possibly even a little below their surface S to be used in geological and geophysical interpretation. This has to be done by exploiting all the information we have on W, primarily observations of potential differences between couples of points on S, the gravity modulus at points on the surface and many others, including satellite altimetry on the oceans, satellite gravimetry and gradiometry, satellite tracking, etc.

One fundamental feature of geodetic problems is that the surface S itself has to be considered as unknown. This holds despite satellite observations of the Earth seem to supply us with a geometric determination of S, for instance by radar altimetry on the ocean and by SAR or photogrammetry on land. Yet, apart from the fact that the accuracy of the geometry is e.g. on land of about 1 m and of a few centimeters for the quasi stationary sea surface, the true issue is that such a surface is not the same we are looking for, from the gravity field view angle.

For instance a circular tower with a radius of 30 m and a height of 40 m produces an attraction of less than 0.03 mGal on top of its roof, on account of the fact its concrete occupies less than 1% of its volume. On the other hand, the same tower is well determined by space photogrammetry, i.e. it enters into the geometric surface of the Earth in photogrammetric sense, but not in the gravimetric sense.

For these reasons we have to accept that the gravimetric surface S is an unknown characterized by the fact that at its points we know the minimal information on the gravity field necessary to solve our problem. We will not repeat the long road leading from observations to the scalar Molodensky Problem (or scalar GBVP) for which we rather send to the literature (Heiskanen and Moritz 1967; Moritz 1980; Krarup 2006; Sacerdote and Sansò 1986; Sansò 1997; Heck 1989, 1997; Martinec 1998; Sansò and Sideris 2013). We only recall that the above definition of S within the scalar GBVP, the same given in Sansò and Sideris (2017), needs some more discussion. In fact we can observe that we have two sources of information on S; one is the coordinates of points at which we make gravimetric measurements, the other is the information coming from Digital Terrain Models (or Ocean Models) derived from space. So we can first of all smooth our gravimetric signal by applying a residual terrain modelling (Forsberg 1984; Forsberg and Tscherning 1997), namely by subtracting the gravimetric effect on W and g of a layer between some reference surface and the actual surface of the Earth, as it is known by a DTM, and then use the smoothed signal to produce a “local interpolation” to some surface that is only locally known with respect to the measurement stations, but globally a part of the unknowns of the GBVP. Of course in doing so we will significantly smooth the actual geometric S, particularly in mountain regions, avoiding sharp vertical discontinuities, ravines and especially overhangs, that would destroy the results of the mathematical analysis of the GBVP. In fact, it is well known that the oblique derivative BVP for the Laplace operator, in our case the direction of the vertical, the direction of the derivative should never be tangent to the boundary to guarantee existence and uniqueness theorems (Yanushauakas 1989; Miranda 1970). So we can imagine that S is at least a Lipschitz surface and that lines orthogonal to the Earth ellipsoid $\mathcal {E}$ cross S at one point only. In other words, we can assume the surface S to be in one-to-one correspondence with the ellipsoid $\mathcal {E}$ by orthogonal projection. This implies that S has an equation

$$\displaystyle \begin{aligned} h = H\left(\sigma\right) \ , {} \end{aligned} $$

(1)

with h indicating the ellipsoidal height, $\sigma$ the couple of angular ellipsoidal coordinates, $\sigma =\left (\theta ,\lambda \right )$, and H a Lipschitz function of the point P on $\mathcal {E}$ with coordinates $\sigma$.

So we can summarize our problem of knowing W outside the masses as the solution of a free BVP for the Laplace equation that can be formalized as follows (scalar Molodensky’s problem)

$$\displaystyle \begin{aligned} W\left(P\right) = V\left(P\right) + \frac{1}{2}\omega^2\left(x_P^2+y_P^2\right) \ , {} \end{aligned} $$

(2)

$$\displaystyle \begin{aligned} \left\{ \begin{array}{ll} \triangle V = 0 & \quad \text{outside} \,\, S \\ W\left(H\left(\sigma\right),\sigma\right) = W_0\left(\sigma\right) & \quad \text{on} \,\, S \\ \left\vert \nabla W\left(H\left(\sigma\right),\sigma\right) \right\vert = g_0\left(\sigma\right) & \quad \text{on} \,\, S \\ V \rightarrow 0 & \quad \text{for} \,\, h \rightarrow \infty \ . \end{array} \right. {} \end{aligned} $$

(3)

where the unknowns are $W\left (P\right )$, i.e. $V\left (P\right )$, in $\Omega \equiv \left \{h>H\left (\sigma \right )\right \}$, and $H\left (\sigma \right )$ itself.

Remark 1.1

Why do we insist so much to transform our “exterior” problem into a BVP? This is because, although non linear and difficult, the problem can be mathematically analyzed and proved to have a unique solution in functional Hölder spaces under “reasonable” conditions on the datum $\left \{W_0\left (\sigma \right ),g_0\left (\sigma \right )\right \}$ and, even more important, such a solution is continuously dependent on data (Sansò 1976, 1989; Hörmander 1976). Further on, the linearized version of (3) has a well understood behaviour and its solution is well posed in quite general Sobolev Spaces.

Remark 1.2

What about estimating $W\left (P\right )$ inside the masses, for instance in the topographic layer, which has a maximum width of roughly 9 km, between the ellipsoid $\mathcal {E}$ and the surface S? The problem, crossing S from outside towards the inner body, changes its mathematical nature. First of all in the body, $B=\left \{h<H\left (\sigma \right )\right \}$, the gravitational potential $V=W-\frac {1}{2}\omega ^2\left (x^2+y^2\right )$ is not any more harmonic, but it rather satisfies the Poisson equation

$$\displaystyle \begin{aligned} \triangle V\left(P\right) = -4 \pi G \rho\left(P\right) \ \quad \mbox{in} \ B \ , {} \end{aligned} $$

(4)

with G the Newton constant and $\rho \left (P\right )$ the density of topographic masses. Once S has been determined, and assuming we know $\rho \left (P\right )$ between S and $\mathcal {E}$, the solution of (4) complemented with the boundary conditions

$$\displaystyle \begin{aligned} \left\{ \begin{array}{ll} &\left. V + \displaystyle\frac{1}{2}\omega^2\left(x^2+y^2\right)\right\vert {}_S = W_0\left(\sigma\right) \\ &\left\vert \nabla\left(W + \displaystyle\frac{1}{2}\omega^2\left(x^2+y^2\right)\right)\right\vert {}_S = g_0\left(\sigma\right) \end{array} \right. \end{aligned} $$

(5)

has the form of an “initial” or Cauchy problem for the Laplace operator. Quoting Miranda (1970, page 60) “In this case we do not usually think of an extension of the Existence Theorem for Cauchy’s problem, inasmuch as this problem is not to be considered well posed for the elliptic equation in other than the analytical field. Hadamard in fact observed that the problem does not admit in general a solution, and that when it does, this solution does not depend continuously on the data”. We will return on this point later, showing that despite the above statements, the calculus of W in a thin layer like the topographic one is approximately possible, although with an error which might become unacceptable the deeper we go inside S.

2 The Ambiguous Gravimetric Surface S and the Linearization Band

As we have seen from the discussion of Section 1, the gravimetric surface S is not exactly fixed, but it rather depends on a number of preliminary operations we apply to the measurements to derive $W_0\left (\sigma \right )$ and $g_0\left (\sigma \right )$. Once this step is fixed, the surface $S=\left \{h=H\left (\sigma \right )\right \}$ comes from the solution of the scalar Molodensky problem. Then a natural question is: since $W_0\left (\sigma \right )$ and $g_0\left (\sigma \right )$ might be referred to the same surface in a layer of width $\pm \delta H_0$, can we claim that we have different solutions of the GBVP? Of course this should not be the case, because our physical gravity field is only one.

The answer comes from a sensitivity analysis to the shift of data between two surfaces, S and $S'$, with respect to the gap

$$\displaystyle \begin{aligned} \delta H\left(\sigma\right) = H^{\prime}\left(\sigma\right) - H\left(\sigma\right) \ . {} \end{aligned} $$

(6)

This resembles strictly the reasoning done to evaluate the linerization band for the nonlinear problem (3); in fact the idea is to make an evaluation of the second order terms arising in a Taylor formula and verify that they can be neglected. Such a reasoning has been developed in a very fine numerical analysis in Heck (1989, 1997) as well as by some rougher estimates in Sansò and Sideris (2017), in any way leading to the conclusion that the linearization band has a width somewhere between 100 m and 200 m. Note that the more optimistic estimate in Sansò and Sideris (2017) with respect to those in Heck (1989, 1997) is due to the fact that it is based on the mean square value of the curvature of equipotential surfaces (see Sansò and Sideris 2013) and since this is a quite oscillating function, especially in mountainous areas, it can easily reach as much as 10 times the value of the r.m.s.

We aim to show that if we stay in such a band we can safely switch from one surface S to another $S'$, shifting our data $W_0 \rightarrow W_0^{\prime }$ and $g_0 \rightarrow g_0^{\prime }$ by a linear Taylor formula, without changing substantially the solution of (3). Let us note that our shift has to be considered in free air since we assume that we have already cleared the masses in the space around S by the residual terrain correction.

We have, for some $t\left (\sigma \right )$, $\tau \left (\sigma \right ) < 1$,

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle W^{\prime}_0\left(\sigma\right) - W_0\left(\sigma\right) = W\left(H^{\prime}\left(\sigma\right),\sigma\right) - W\left(H\left(\sigma\right),\sigma\right) = \\ & &\displaystyle \displaystyle\frac{\partial W\left(H\left(\sigma\right),\sigma\right)}{\partial h} \delta H\left(\sigma\right) +\\ & &\displaystyle \frac{1}{2} \frac{\partial^2 W\left(H\left(\sigma\right)+t\left(\sigma\right) \delta H\left(\sigma\right),\sigma\right)}{\partial h^2} \delta H\left(\sigma\right)^2 {} \end{array} \end{aligned} $$

(7)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle g^{\prime}_0\left(\sigma\right) - g_0\left(\sigma\right) = g\left(H^{\prime}\left(\sigma\right),\sigma\right) - g\left(H\left(\sigma\right),\sigma\right) = \\ & &\displaystyle \displaystyle\frac{\partial g\left(H\left(\sigma\right),\sigma\right)}{\partial h} \delta H\left(\sigma\right) + \\ & &\displaystyle \displaystyle\frac{1}{2} \frac{\partial^2 g\left(H\left(\sigma\right)+\tau\left(\sigma\right) \delta H\left(\sigma\right),\sigma\right)}{\partial h^2} \delta H\left(\sigma\right)^2 \ . {} \end{array} \end{aligned} $$

(8)

For the sake of an estimate of orders of magnitude, we can substitute in (7) and (8)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & \left(\mu = G M \right) &\displaystyle \qquad \frac{\partial^2 W}{\partial h^2} \sim -2 \frac{\mu}{r^3} \sim - \frac{2}{r} g_0 {} \end{array} \end{aligned} $$

(9)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \qquad \frac{\partial^2 g}{\partial h^2} \sim 6 \frac{\mu}{r^4} \sim - \frac{6}{r^2} g_0 \ . {} \end{array} \end{aligned} $$

(10)

So that we get, with $O\left [ \cdot \right ]$ denoting the orders of magnitude,

$$\displaystyle \begin{aligned} O\left[\frac{1}{2}\frac{\partial^2 W}{\partial h^2} \delta H^2\right] = \frac{\delta H^2}{r} g_0 {} \end{aligned} $$

(11)

$$\displaystyle \begin{aligned} O\left[\frac{1}{2}\frac{\partial^2 g}{\partial h^2} \delta H^2\right] = 3 \left( \frac{\delta H}{r} \right)^2 g_0 \ . {} \end{aligned} $$

(12)

With $\delta H = 200$ m and $r \sim 6 \cdot 10^6$ m, dividing (11) by $g_0$ to transform the variation of potential into a shift of equipotential surfaces, we get

$$\displaystyle \begin{aligned} O\left[\frac{1}{g_0}\left(\frac{1}{2}\frac{\partial^2 W}{\partial h^2} \delta H^2 \right)\right] \sim 0.6 \ \mathrm{cm} {} \end{aligned} $$

(13)

$$\displaystyle \begin{aligned} O\left[\frac{1}{2}\frac{\partial^2 g}{\partial h^2} \delta H^2\right] \sim 3 \ \upmu \mathrm{Gal} \ . {} \end{aligned} $$

(14)

We consider these figures negligible and we can conclude that we can shift from any surface to another by a simple linear Taylor formula in the linearization band, getting substantially equivalent nonlinear scalar Molodensky problems.

As for the linearization of (3), we do not need to repeat the procedure well established in literature (see Sacerdote and Sansò 1986, Heck 1997, Sansò and Sideris 2013, Sansò and Sideris 2017); we rather underline some aspects of the almost equivalence between linearized and nonlinear BVP by the following remarks.

Remark 2.1

Let us recall that the linearization of (3) is done, following the lesson of T. Krarup (2006), by choosing an approximate surface $\tilde S = \left \{ h=\tilde H\left (\sigma \right ) \right \}$, such that $\delta H\left (\sigma \right ) = H\left (\sigma \right ) - \tilde H\left (\sigma \right )$ is known to belong to the linearization band, and on the same time splitting the actual potential W into normal plus anomalous potential

$$\displaystyle \begin{aligned} W\left(h,\sigma\right) = U\left(h,\sigma\right) + T\left(h,\sigma\right) \ ; {} \end{aligned} $$

(15)

$\delta H$ and T are considered as first order infinitesimals. Then the equations linearized starting from $\tilde S$ result to be

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle DW_0\left(\sigma\right) = W_0\left(\sigma\right) - U\left(\tilde H\left(\sigma\right),\sigma\right) = \\ & &\displaystyle T\left(\tilde H\left(\sigma\right),\sigma\right) - \gamma\left(\tilde H\left(\sigma\right),\sigma\right) \delta H\left(\sigma\right) \ , {} \end{array} \end{aligned} $$

(16)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle Dg_0\left(\sigma\right) = g_0\left(\sigma\right) - \gamma\left(\tilde H\left(\sigma\right),\sigma\right) = \\ & &\displaystyle \gamma'\left(\tilde H\left(\sigma\right),\sigma\right) \delta H\left(\sigma\right) - T'\left(\tilde H\left(\sigma\right),\sigma\right) \ ; {} \end{array} \end{aligned} $$

(17)

here we have denoted by a prime the derivative of a function $f\left (h,\sigma \right )$ with respect to h, and, as usual,

$$\displaystyle \begin{aligned} \gamma\left(h,\sigma\right) = \left\vert\nabla U\left(h,\sigma\right)\right\vert \ . {} \end{aligned} $$

(18)

Typically (16) is used to derive $\delta H\left (\sigma \right )$

$$\displaystyle \begin{aligned} \delta H\left(\sigma\right) = \frac{1}{\gamma\left(\tilde H\left(\sigma\right),\sigma\right)} \left[T\left(\tilde H\left(\sigma\right),\sigma\right) - DW_0\left(\sigma\right) \right] \ , {} \end{aligned} $$

(19)

known as generalized Bruns equation; then we substitute it into (18) to get a boundary condition on $\tilde S$ for the sole unknown T, namely

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle - T'\left(\tilde H\left(\sigma\right),\sigma\right) + \frac{\gamma'\left(\tilde H\left(\sigma\right),\sigma\right)}{\gamma\left(\tilde H\left(\sigma\right),\sigma\right)} T\left(\tilde H\left(\sigma\right),\sigma\right) = \\ & &\displaystyle Dg_0\left(\sigma\right) + \frac{\gamma'\left(\tilde H\left(\sigma\right),\sigma\right)}{\gamma\left(\tilde H\left(\sigma\right),\sigma\right)} DW_0\left(\sigma\right) \ . {} \end{array} \end{aligned} $$

(20)

The new boundary conditions (16) and (17) defined on $\tilde S$ neglect second order terms; among them, that of the Eq. (17) is the most worrying. It reads (Sansò and Sideris 2017)

$$\displaystyle \begin{aligned} &Q_2(g) = -T''\left(\tilde H\left(\sigma\right),\sigma\right) \delta H\left(\sigma\right) + \\ &\frac{1}{2\gamma\left(\tilde H\left(\sigma\right),\sigma\right)}\left[\left\vert\nabla T\left(\tilde H\left(\sigma\right),\sigma\right)\right\vert^2 - T{''}^2\left(\tilde H\left(\sigma\right),\sigma\right) \right] \ ; {} \end{aligned} $$

(21)

it has been shown (see Heck 1997 and references therein) that locally in mountainous areas $Q_2\left (g\right )$ can reach the level of 0.3 mGal, which is too large to be neglected. Yet, by using (19) and (21) with T substituted by some global model $T_M$, even with a moderate maximum degree, we can reduce the neglected part of $Q_2$ by at least an order of magnitude. In other words, computing $Q_2\left (g\right )$ with a model $T_M$ and subtracting it to $Dg_0\left (\sigma \right )$ brings the linear boundary value problem (20), complemented by Bruns equation (19), to be practically equivalent to the nonlinear (3).

3 Equivalent Linearized Molodensky Problems

Summarizing the discussion of the previous section, we can say that we can accomplish the aim of physical geodesy by solving the linearized Molodensky problem. This boils down to: choose an approximate $\tilde S$ in the linearization band around the gravimetric surface S and then solve the oblique derivative BVP

$$\displaystyle \begin{aligned} \left\{ \begin{array}{ll} \triangle T = 0 \qquad \text{in} \ \tilde{\Omega} \equiv \left\{ h\geq \tilde H\left(\sigma\right) \right\} \\ -T' + \left. \displaystyle\frac{\gamma'}{\gamma} T \right\vert {}_{\tilde S} = Dg_0 + \displaystyle\frac{\gamma'}{\gamma} DW_0 \end{array} \right. \ ; {} \end{aligned} $$

(22)

the generalized Bruns relation (19) will then provide us with the height anomaly $\delta H\left (\sigma \right )$ that allows to reconstruct S, i.e. $H\left (\sigma \right ) = \tilde H\left (\sigma \right ) + \delta H\left (\sigma \right )$.

As we know, typically in the asymptotic development of T in spherical harmonics, the terms of degree 0 and 1, i.e. the coefficients $T_{0,0}$ and $T_{1,k}$ ($k=-1, 0, 1)$, are skipped as they are related to the total mass of the Earth, accounted for the normal potential, and to the coordinates of the barycentre which are chosen as origin of the coordinates (see Sansò and Sideris 2013). This implies that usually (22) is complemented by the asymptotic relation

$$\displaystyle \begin{aligned} T = O\left(\frac{1}{r^3}\right) \ , \quad r \rightarrow \infty \ . {} \end{aligned} $$

(23)

Since this item is not central in the following discussion, we do not dwell on it.

What is important is that since the solution of (22) on a practical ground is still a quite challenging task, according to different approaches, many efforts are done to reduce (22) to a problem on a Bjerhammar sphere (i.e. a sphere totally internal to the masses) to make it numerically tractable.

In fact:

• 1. the old classical Stokes solution (Heiskanen and Moritz 1967, section 3.9) was based, as a matter of fact, on reducing the data from $\tilde S$ (in fact in this case from S) to the geoid, then approximated by a sphere,

• 2. the classical solution given by Molodensky et al. (1962), further adapted to the scalar Molodensky problem as in Moritz (1980, section 8.6), refers to the use of a single layer, which seemingly is spread on the approximate surface $\tilde S$ but since it is derived by the Molodensky series, which is based on a Bjerhammar sphere, in reality provides a solution computed term by term by a spherical solution,

• 3. modern global models are based by a downward continuation of the observations, down to the ellipsoid and then solved for by spherical formulas, with suitable ellipsoidal corrections (Pavlis 2013),

• 4. collocation solutions are based on the remark that functional spaces of functions harmonic out of a Bjerhammar sphere are “dense” in the space of functions harmonic out of $\tilde S$ (Krarup 2006; Sansò and Sideris 2013); then, considering the discrete (pointwise) character of actual measurements, a formal solution can be written down with the help of spherical reproducing kernels. When these are tailored to the empirical covariance function of data, always exploiting the hypothesis of spherical isotropy of the field, the method becomes least squares collocation, and it is always framed into the family of spherical methods.

Since all the mentioned methods extend the harmonicity domain of T at least down to the ellipsoid or a sphere and, as we shall discuss in the next section, this operation is improperly posed basically because high frequency errors tend to be amplified, it is convenient to perform computations on data as smooth as possible. To this purpose we can use to our best the freedom that we have in designing the BVP (22).

The choices that we can exploit are two: one is the choice of $\tilde S$, always remaining in the linearization band, the other is to change in a known way the mass distribution below S and modify accordingly the data. This known change can then be retrieved at the end of the computation procedure.

Remark 3.1

Let us notice that already an early use of the Residual Terrain Correction to modify our data goes exactly in the wanted direction.

3.1 The Classical Molodensky Choice

This is to introduce as $\tilde S$ the so called telluroid defined by the relation

$$\displaystyle \begin{aligned} \tilde S \equiv S^* \equiv \left\{ h=h^*\left(\sigma\right); \quad U\left(h^*\left(\sigma\right),\sigma\right) = W_0\left(\sigma\right)\right\} \ . {} \end{aligned} $$

(24)

The height $h^*$ of a point, $P = \left (h,\sigma \right )$, identified by the relation

$$\displaystyle \begin{aligned} U\left(h^*,\sigma\right) = W\left(h,\sigma\right) {} \end{aligned} $$

(25)

is called normal height of P; the set of points $\left \{Q\right \}$ that have an ellipsoidal height equal to the normal height of P, when it runs on S, is the telluroid.

Remark 3.2

Let us notice that the original choice of Molodensky was to use isozenithal lines instead of the ellipsoidal normal. This leads to the formulation of a free BVP which is now called the vector Molodensky problem (Heck 1997; Sansò and Sideris 2013), the linearization of which has been first rigorously done in Krarup (2006). On the contrary, the above exposition is along the lines of the scalar Molodensky BVP, which has been developed by the Russian School (Brovar 1964; Pellinen 1982) and presented in Heiskanen and Moritz (1967), Moritz (1980) as well as theoretically systematized in Sacerdote and Sansò (1986).

Returning to our object, we notice that the choice (25) implies

$$\displaystyle \begin{aligned} DW = W_0\left(\sigma\right) - U\left(h^*\left(\sigma\right),\sigma\right) \equiv 0 \ . {} \end{aligned} $$

(26)

Therefore, in this case the known term of the oblique derivative problem (22) becomes

$$\displaystyle \begin{aligned} \begin{array}{ll} Dg_0 + \dfrac{\gamma'}{\gamma}DW_0 &= Dg_0 = g\left(H\left(\sigma\right),\sigma\right) - \gamma\left(h^*\left(\sigma\right),\sigma\right)\\ &= \Delta g\left(\sigma\right) \ , \end{array} {} \end{aligned} $$

(27)

namely the ordinary free air geodetic gravity anomaly. On the same time $\delta H\left (\sigma \right )$ becomes

$$\displaystyle \begin{aligned} \delta H\left(\sigma\right) = H\left(\sigma\right) - h^*\left(\sigma\right) = \zeta\left(\sigma\right) \ , {} \end{aligned} $$

(28)

also called height anomaly, and (19) reduces to the ordinary Bruns relation

$$\displaystyle \begin{aligned} \zeta\left(\sigma\right) = \frac{T\left(h^*\left(\sigma\right),\sigma\right)}{\gamma\left(h^*\left(\sigma\right),\sigma\right)} \ . {} \end{aligned} $$

(29)

It turns out that physically the height anomaly is everywhere less than $\sim$ 120 m in absolute value, so confirming that (22), maybe applying the correction $Q_2\left (g\right )$ (see 21) for mountainous areas, is fully in the linearization band, i.e. equivalent to the original nonlinear problem.

In conclusion the linearized Molodensky problem writes

$$\displaystyle \begin{aligned} \left\{ \begin{array}{l@{\quad }l} \triangle T = 0 & \text{in} \, \tilde{\Omega} \\ -T' + \displaystyle\frac{\gamma'}{\gamma} T = \triangle g_0 & \text{on} \, \tilde{\Omega} \end{array} \right. \ . {} \end{aligned} $$

(30)

3.2 The Helmert Approach

The Helmert idea was to combine both possibilities, first dislocating the topographic masses from their 3D distribution to a single layer on a condensation surface $S_0$, the condensation process, and then substituting in W the attraction of the column with that of the single layer element at its basis (Vaníček and Martinec 1994). In formula, accepting a spherical approximation where the ellipsoid is substituted by a sphere of radius $R_0$, this reads

$$\displaystyle \begin{aligned} dM = d\sigma \int_{R_0-D\left(\sigma\right)}^{R\left(\sigma\right) = R_0 + H\left(\sigma\right)} r^2 \rho\left(r,\sigma\right) dr = \kappa\left(\sigma\right) d S_0 \ , {} \end{aligned} $$

(31)

where $D\left (\sigma \right )$ is the depth of the condensation surface, $dS_0$ is its area element and dM is the mass contained in the column.

In practice, an approximation is applied to (31), namely $\rho$ is considered as constant along the column, implying that (31) can be written as

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \kappa\left(\sigma\right)=\rho \left[H\left(\sigma\right)+D\left(\sigma\right)\right] \\ & &\displaystyle \cdot \left[\!R_0^2+R_0\left(H\left(\sigma\right)-D\left(\sigma\right)\right){+}\frac{1}{3}H\left(\sigma\right)^2{+}\frac{1}{3}D\left(\sigma\right)^2\!\right] \frac{d\sigma}{dS_0}\ ; {}\\ \end{array} \end{aligned} $$

(32)

no doubt that the approximation $\rho \left (r,\sigma \right )$ = constant is one of the weakest points of Helmert’s approach. Investigations on the impact of using density models with lateral or even 3D variations has been done in Kingdon et al. (2009). Yet, this is not to the extent of invalidating the approach, as far as what we subtract at the beginning we add back at the end. Only the conclusion that all masses above the geoid are so removed has to be regarded as an approximate statement.

The term $\frac {d\sigma }{dS_0}$ depends on what condensation surface $S_0$ is chosen; in literature two choices are considered (see Heck 2003). The first condensation method corresponds to $D\left (\sigma \right ) = 0$; the second method, the one mostly applied in recent literature (see for instance Vaníček et al. 1999), consists in putting $D\left (\sigma \right ) = -N\left (\sigma \right )$ (the geoid undulation), namely in choosing the geoid itself as $S_0$. This has the effect of transforming the term $H\left (\sigma \right ) - D\left (\sigma \right )$ into

$$\displaystyle \begin{aligned} H_0\left(\sigma\right) = H\left(\sigma\right) - N\left(\sigma\right) \ , {} \end{aligned} $$

(33)

i.e. the orthometric height of the point on S with horizontal coordinates $\sigma$. $H_0\left (\sigma \right )$ is considered as known on S, although this statement is not so firm because once more to know $H_0\left (\sigma \right )$ one would need to know as well $\rho \left (h,\sigma \right )$ below the point $P \equiv \left (h,\sigma \right )$ to compute the necessary orthometric corrections (see Heiskanen and Moritz 1967, Sansò et al. 2019).

Yet, by taking the geoid as $S_0$, one has

$$\displaystyle \begin{aligned} dS_0 = \frac{R_0^2}{\cos \delta_0} d\sigma {} \end{aligned} $$

(34)

with $\delta _0$ the deflection of the vertical. Since we have at most $\delta _0 \sim 3 \cdot 10^{-4}$, (34) can be safely written as $dS_0 = R_0^2 \, d\sigma$ and (32), after one further simplification, becomes just

$$\displaystyle \begin{aligned} \kappa\left(\sigma\right) = \rho \, H_0\left(\sigma\right) \ . {} \end{aligned} $$

(35)

The Helmert potential correction is then defined as

$$\displaystyle \begin{aligned} \delta V^H\left(h,\sigma\right) = V_T\left(h,\sigma\right) - V_C\left(h,\sigma\right) \ , {} \end{aligned} $$

(36)

where $V_T$ is the potential of the topographic masses, i.e. the masses between the geoid and S, and $V_C$ is the potential of the single layer with density (35).

Remark 3.3

The topographic potential $V_T$ is strictly speaking not computable as it is given on S by the formula (in spherical approximation)

$$\displaystyle \begin{aligned}{} V_T\left(H\left(\sigma\right),\sigma\right) = G \rho \int d\sigma' \int_{R_0+N\left(\sigma'\right)}^{R_0+H_0\left(\sigma'\right)+N\left(\sigma'\right)} \frac{r^2 dr}{\ell_{\sigma\sigma'}} \ , \end{aligned} $$

(37)

where $\ell _{\sigma \sigma '}$ is the distance between $P \equiv \left (R_0+H\left (\sigma \right ),\sigma \right )$ and the running point $\left (r,\sigma '\right )$. As we see, $V_T$ on S depends on $N\left (\sigma \right )$ which is in fact unknown. Yet, as proved in Sansò and Sideris (2017), (37) can be substituted, with a small error, by

$$\displaystyle \begin{aligned} V_T\left(H\left(\sigma\right),\sigma\right) \cong G \rho \int d\sigma' \int_{R_0}^{R_0+H_0\left(\sigma'\right)} \frac{r^2 dr}{\ell_{\sigma\sigma'}} {} \end{aligned} $$

(38)

which is then computable with the available information and may be then refined by iterating on (37). The same holds therefore for $\delta V^H\left (h,\sigma \right )$ in (36).

One fundamental statement of empirical nature we will need in the next development is that

$$\displaystyle \begin{aligned} O\left(\max\frac{\delta V^H\left(\sigma\right)}{\gamma\left(\sigma\right)}\right) = 2 \ \text{m} {} \end{aligned} $$

(39)

(see Vaníček et al. 1999). Since this is two orders of magnitude smaller than the width of the band, where two nonlinear formulations of Molodensky’s problem can be transformed linearly one into the other, we can safely assume that the use of the corrective term $\delta V^H\left (h,\sigma \right )$ will not change the problem yielding the sought potential.

By definition the Helmert potential $W^H$ is given by

$$\displaystyle \begin{aligned} \begin{array}{ll} W^H\left(h,\sigma\right)&=W\left(h,\sigma\right) - \delta V^H\left(h,\sigma\right)\\ &=W\left(h,\sigma\right) - V_T\left(h,\sigma\right) + V_C\left(h,\sigma\right) \ , \end{array} {} \end{aligned} $$

(40)

namely it is the actual potential where the topographic part $V_T$ is substituted by the potential of the same masses squeezed on the geoid $S_0$. As such, $W^H$, apart from its centrifugal component, is harmonic down to the geoid, at least if the hypothesis of constant density $\rho$ has to be considered correct.

Since $\delta V^H\left (h,\sigma \right )$ is computable at the level of S, we come to know $W_0^H\left (\sigma \right )$ on it. With the same accuracy we have now to transform $g_0\left (\sigma \right )$ into $g_0^H\left (\sigma \right )$. Denoting with $\underline {n}^H$ the vertical direction of the Helmert field, according to (39) one has for $P \in S$

$$\displaystyle \begin{aligned} \begin{array}{rcl} g\left(P\right) & =&\displaystyle \left\vert \nabla W^H\left(P\right) + \nabla \delta V^H\left(P\right) \right\vert \\ & \cong&\displaystyle g^H\left(P\right) - \underline{n}^H \cdot \nabla \delta V^H\left(P\right) \\ & \cong&\displaystyle g^H\left(P\right) - \frac{\partial}{\partial h} \delta V^H\left(P\right) \ . {} \end{array} \end{aligned} $$

(41)

All approximations here are easily justified on the basis of Remark 3.3.

Now we have transformed our original nonlinear problem for W into an equivalent nonlinear problem for $W^H = W - \delta V^H$ by claiming that $W^H$ minus the centrifugal term has to be harmonic outside S and on (the unknown) S one must have

$$\displaystyle \begin{aligned} W_0^H\left(\sigma\right) = W_0\left(\sigma\right) - \delta V^H\left(\sigma\right) {} \end{aligned} $$

(42)

$$\displaystyle \begin{aligned} g_0^H\left(\sigma\right) = g_0\left(\sigma\right) + \frac{\partial \, \delta V^H}{\partial h}\left(\sigma\right) \ . {} \end{aligned} $$

(43)

The two scalar nonlinear Molodensky problems for W and $W^H$ are equivalent by construction. So if we go to the respective linearized version for the anomalous potential T and $T^H$, i.e.

$$\displaystyle \begin{aligned} T = W - U \ , {} \end{aligned} $$

(44)

$$\displaystyle \begin{aligned} T^H = W^H - U = W - U - \delta V^H = T - \delta V^H\ , {} \end{aligned} $$

(45)

we would not need to make any computation to claim that also the two linearized problems are equivalent since each of them is equivalent to its nonlinear version. Yet, for the sake of clarity we proceed to verify this statement.

For this purpose we notice that we can define a Helmert telluroid $S^{*H} = \left \{h=h^{*H}\left (\sigma \right )\right \}$ by stating that

$$\displaystyle \begin{aligned}{} W^H\left(H\left(\sigma\right),\sigma\right) = U\left(h^{*H}\left(\sigma\right),\sigma\right) \ . \end{aligned} $$

(46)

Then (46) suitably linearized gives us the Helmert-Bruns relation

$$\displaystyle \begin{aligned} \zeta^H\left(\sigma\right) = H\left(\sigma\right) - h^{*H}\left(\sigma\right) = \frac{T^H\left(h^{*H}\left(\sigma\right),\sigma\right)}{\gamma\left(h^{*H}\left(\sigma\right),\sigma\right)} \ . {} \end{aligned} $$

(47)

We observe that from (44) one has too

$$\displaystyle \begin{aligned} \zeta^H = \frac{T^H}{\gamma} = \frac{T}{\gamma} - \frac{\delta V^H}{\gamma} = \zeta - \frac{\delta V^H}{\gamma} {} \end{aligned} $$

(48)

and from (28) and (46)

$$\displaystyle \begin{aligned} h^{*H}\left(\sigma\right) - h^*\left(\sigma\right) = \zeta\left(\sigma\right) - \zeta^H\left(\sigma\right) = \frac{\delta V^H}{\gamma} \ . {} \end{aligned} $$

(49)

Finally the linearized boundary condition for $T^H$ reads

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \frac{\partial}{\partial h} T^H\left(h^{*H}\left(\sigma\right),\sigma\right) + \frac{\gamma'}{\gamma} T^H\left(h^{*H}\left(\sigma\right),\sigma\right) = \\ & &\displaystyle g_0^H\left(\sigma\right) - \gamma\left(h^{*H}\left(\sigma\right),\sigma\right) = \Delta g _0^H\left(\sigma\right) \ . {} \end{array} \end{aligned} $$

(50)

Now let T be the solution of (28) with boundary condition on $S^*$

$$\displaystyle \begin{aligned} &- \frac{\partial T}{\partial h}\left(h^*\left(\sigma\right),\sigma\right) + \frac{\gamma'}{\gamma} T\left(h^*\left(\sigma\right),\sigma\right)\\ &\quad = g_0\left(\sigma\right) - \gamma\left(h^*\left(\sigma\right),\sigma\right) {} \end{aligned} $$

(51)

and $T^H$ the solution of the same problem on $S^{*H}$ with boundary condition (50). We prove that $T-T^H=\delta V^H\left (\sigma \right )$ so that (49) holds too and then

$$\displaystyle \begin{aligned} H\left(\sigma\right) = h^*\left(\sigma\right) + \zeta\left(\sigma\right) = h^{*H}\left(\sigma\right) + \zeta^H\left(\sigma\right) \ , {} \end{aligned} $$

(52)

i.e. the gravimetric surface reconstructed from the two problems is the same and on the same time

$$\displaystyle \begin{aligned} W_0\left(\sigma\right) = W_0^H\left(\sigma\right) + \delta V^H\left(\sigma\right) \ , {} \end{aligned} $$

(53)

i.e. the potential of the gravity field is the same outside S. All that in the usual linear approximation since we are moving well inside the linearization band.

If we take (51) minus (50) and consider that $h^{*H}\left (\sigma \right ) - h^*\left (\sigma \right )$ is of the order of few meters, we can write

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \left. - \frac{\partial}{\partial h}\left( T-T^H \right) + \frac{\gamma'}{\gamma} \left( T-T^H \right) \right\vert {}_{S^*} = \\ & &\displaystyle g_0\left(\sigma\right) - g_0^H\left(\sigma\right) - \gamma\left(h^*\left(\sigma\right),\sigma\right) + \gamma\left(h^{*H}\left(\sigma\right),\sigma\right) = \\ & &\displaystyle - \frac{\partial \delta V^H}{\partial h} + \gamma' \cdot \left(h^{*H}\left(\sigma\right) - h^*\left(\sigma\right)\right) \ . {} \end{array} \end{aligned} $$

(54)

On the other hand, by the very definition of $S^*$ and $S^{*H}$, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta V\left(\sigma\right) & =&\displaystyle W_0\left(\sigma\right) - W_0^H\left(\sigma\right) \\ & =&\displaystyle U\left(h^*\left(\sigma\right),\sigma\right) - U\left(h^{*H}\left(\sigma\right),\sigma\right) \\ & =&\displaystyle -\gamma \cdot \ \left(h^*\left(\sigma\right) - h^{*H}\left(\sigma\right)\right), {} \end{array} \end{aligned} $$

(55)

namely

$$\displaystyle \begin{aligned} h^{*H}\left(\sigma\right) - h^*\left(\sigma\right) = \frac{\delta V\left(\sigma\right)}{\gamma} \ . {} \end{aligned} $$

(56)

Substituting (56) into (54) we see that

$$\displaystyle \begin{aligned} \left. - \frac{\partial}{\partial h}\left(T-T^H\right)+\frac{\gamma'}{\gamma}\left(T-T^H\right) \right\vert {}_{S^*} = -\frac{\partial \, \delta V^H}{\partial h} + \frac{\gamma'}{\gamma} \delta V^H {} \end{aligned} $$

(57)

and therefore, by the uniqueness of the solution of Molodensky’s problem, also recalling that we do not consider here the first degree harmonics problem, we find

$$\displaystyle \begin{aligned} T - T^H = \delta V^H {} \end{aligned} $$

(58)

as we wanted to prove.

The conclusions of this section are that:

• the gravimetric surface S, once assigned by giving both $W_0\left (\sigma \right )$ and $g_0\left (\sigma \right )$ on it, is achievable either by Molodensky’s approach, starting from the telluroid $S^*$, or by Helmert’s approach and the Helmertized data $W_0^H\left (\sigma \right )$ and $g_0^H\left (\sigma \right )$, starting from the Helmertized telluroid $S^{*H}$; the two solutions are equivalent within second order approximation errors, i.e. at centimetric level, in terms of shift of the equipotential surfaces;

• the two potentials determined either by Molodensky’s method or by Helmert’s method, after restoring $\delta V^H$, are also equivalent from S outward.

4 The Problem of the Internal Potential

The question we want to focus on is not so much the equivalence between Molodensky’s and Helmert’s solutions in the topographic layer (i.e. between S and the geoid), but rather to clarify what are the error sources that limit the knowledge of W in this layer. Such errors are then reflected on the accuracy of the determination of internal equipotential surfaces, in particular of the geoid, defined as the equipotential surface corresponding to some conventional value $\overline {W}_0$ of W.

In fact the problem has to be set in the following way, as already recalled in Sect. 1:

(59)

$$\displaystyle \begin{aligned} \text{and} \quad \left\{ \begin{array}{ll} W\left(R\left(\sigma\right),\sigma\right) = W_0\left(\sigma\right) \\ \left\vert\nabla W\left(R\left(\sigma\right),\sigma\right)\right\vert = g_0\left(\sigma\right) \end{array} \right. {} \end{aligned} $$

(60)

and given the mass density $\rho =\rho \left (r,\sigma \right )$ in a layer $L=\left (S_0,S\right )$ below S, we have to determine W satisfying the conditions (60) on S and the Poisson equation

$$\displaystyle \begin{aligned} \triangle V = \triangle\left(W-\displaystyle\frac{1}{2}\omega^2\left(x^2+y^2\right)\right) = -4 \pi G \rho {} \end{aligned} $$

(61)

in the layer L.

There are two possible versions of the problem:

• (a) one is to fix $S_0 \equiv \left \{ r = R_0\left (\sigma \right )\right \}$ and then try to determine W in the fixed L,

• (b) the other one is to try to solve the problem in a layer with $S_0$ as deep as possible and then to determine $\overline {S}_0$ in such a way that

  $$\displaystyle \begin{aligned} W\left(R_0\left(\sigma\right),\sigma\right) = \overline{W}_0 {} \end{aligned} $$

  (62)

  with $\overline {W}_0$ a constant such that

  $$\displaystyle \begin{aligned} \overline{W}_0 > \max_\sigma W\left(R\left(\sigma\right),\sigma\right) \ . {} \end{aligned} $$

  (63)

It is clear that (a) includes (b) as soon as we are able to fix a surface $S_0$ such that $\overline {S}_0$ is placed in an intermediate position between $S_0$ itself and S; this can be verified a posteriori if the solution W found in L is such that on $S_0$

$$\displaystyle \begin{aligned} \min_\sigma W\left(R\left(\sigma\right),\sigma\right) > \overline{W}_0 \ . {} \end{aligned} $$

(64)

For the Earth, for instance, an $S_0$ inside the ellipsoid, down some 150 m, will do if $\overline {S}_0$ has to be the geoid. The reason for stating the condition (63) or the check (64) is that W is increasing going downward, so $\overline {W}_0$ on the geoid is larger than W at any point with positive height and W is larger than $\overline {W}_0$ at any point on $\overline {S}_0$.

Remark 4.1

In any event an important remark is necessary here to interpret the initial conditions (60). In fact we have to recall that as far as $V = W - \frac {1}{2}\omega ^2\left (x^2+y^2\right )$ is a Newtonian potential generated by a bounded density, we expect this function to be globally continuous together with its first derivatives, which are even Hölder continuous for any exponent $\lambda < 1$. This is an easy combination of a majorization of W and $\nabla W$, derived from Newton’s integral, when $\rho$ is in $L^p\left (B\right )$, $\forall p$ (what is true because the Earth body B is bounded and $\rho$ is bounded too) (Miranda 1970) and well known embedding theorems of Sobolev spaces in Hölder spaces (see e.g. Adams 1975). Physically it means that any sensible solution for the potential W cannot have a discontinuity in the first derivatives across S because otherwise we would have a single layer (i.e. an unbounded $\rho$) on this surface. Therefore, we can expect that $W_0$ and $g_0$ are pointwise well defined functions, as they are traces of spacewise Hölder functions on a surface S which, as a minimum requirement, is Lipschitz continuous.

At this point it is convenient to observe that it is useless to carry on (60), i.e. a nonlinear condition on $\nabla W$ on S. In fact from $W_0\left (\sigma \right )$ and $g_0\left (\sigma \right )$, we can reconstruct any first derivative of W on S. Assume for instance we want to have $\left . W^{\prime }\right \vert { }_S = \left . \frac {\partial W}{\partial r}\right \vert { }_S$; after defining

$$\displaystyle \begin{aligned} \underline{\partial}_\sigma W\left(R\left(\sigma\right),\sigma\right) = \left.\nabla_\sigma W\left(r,\sigma\right)\right\vert {}_{r=R\left(\sigma\right)} \ , {} \end{aligned} $$

(65)

we can write the system

$$\displaystyle \begin{aligned} \nabla_\sigma W_0\left(\sigma\right) = W^{\prime}\left(R\left(\sigma\right),\sigma\right) \nabla_\sigma R\left(\sigma\right) + \underline{\partial}_\sigma W\left(R\left(\sigma\right),\sigma\right) {} \end{aligned} $$

(66)

$$\displaystyle \begin{aligned} g_0^2\left(\sigma\right) = W^{\prime}\left(R\left(\sigma\right),\sigma\right)^2 + \frac{1}{R\left(\sigma\right)^2} \left\vert \underline{\partial}_\sigma W\left(R\left(\sigma\right),\sigma\right) \right\vert^2 \ . {} \end{aligned} $$

(67)

Deriving $\underline {\partial }_\sigma W$ from (66) and substituting it into (67), we get an easily solvable quadratic equation in $W^{\prime }\left (R\left (\sigma \right ),\sigma \right )$. In a similar fashion, by using $T\left (r,\sigma \right ) = W\left (r,\sigma \right ) - U\left (r,\sigma \right )$ and $T'\left (r,\sigma \right ) = W^{\prime }\left (r,\sigma \right ) - U'\left (r,\sigma \right )$, we can get hold on S of the initial values

$$\displaystyle \begin{aligned} T_0\left(\sigma\right) = T\left(R\left(\sigma\right),\sigma\right) {} \end{aligned} $$

(68)

and

$$\displaystyle \begin{aligned} \delta g_0\left(\sigma\right) = - T'\left(R\left(\sigma\right),\sigma\right) {} \end{aligned} $$

(69)

or even, equivalently,

$$\displaystyle \begin{aligned} \Delta g_0\left(\sigma\right) = -T' - \frac{2}{R\left(\sigma\right)} T \ . {} \end{aligned} $$

(70)

Clearly the symbols $\delta g$ and $\triangle g$ refer to gravity disturbance $\delta g$ and gravity anomaly $\triangle g$, here in spherical approximation.

In any way the problem we are facing now is a Cauchy problem for the Poisson equation for T, which, as recalled in Sect. 1, in general has a solution which, when it exists, is not continuously dependent on data.

Referring to the formulation a) of a fixed layer L and to the initial data (68) and (69) on S, it seems natural as a first step to subtract the known “topographic” potential $V_T$ (see (31)) to data and then solve the downward continuation of $T-V_T$, which has now to be harmonic in L.

As we see in the following elementary Example 4.1, however this leads to correct T with a term $V_T$ which can easily be 10 times larger than the former, especially in mountainous areas. Since this is never a good idea, as it multiplies by 10 as well the various model errors, Helmert has invented his “trick” of substituting $V_T$ by its condensed version $V_C$ on the geoid. In this way in fact the residual (Helmertized) potential is still close to T, but harmonic in L and the downward continuation problem is put in its pure form. To be precise, since we fixed $S_0$, the above refers to the first Helmert condensation method (see Heck 2003); this however has little relevance to the present discussion.

Example 4.1

Let us consider the rather simplistic case that S is a sphere of radius R, $S_0$ a sphere of radius $R_0$, with $R = R_0+1$ (in km), $R_0 \cong 6 \cdot 10^3$ km and the layer L between $S_0$ and S is filled with a mass of a constant density $\rho \sim 2.67$ $\mathrm {gcm}^{-3}$ so that $4 \pi G \rho \cong 0.2$ $\mathrm {Galkm}^{-1}$. Moreover let us assume that $\gamma \sim 10^3$ Gal, as it is approximately true in reality. Then

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{V_T\left(R_0,\sigma\right)}{\gamma}& =&\displaystyle \frac{\displaystyle\frac{4}{3} \pi G \rho \left[ \left( R_0+1\right)^3 - R_0^3 \right]}{R_0 \gamma} \\ & =&\displaystyle \frac{4 \pi G \rho}{\gamma} \left[ R_0+1-\frac{1}{3 R_0}\right]\\ & \cong&\displaystyle 1.2 \, \mathrm{km} + 0.2 \cdot 10^{-3} \, \mathrm{km} + 10^{-8} \, \mathrm{km} \ . \end{array} \end{aligned} $$

As we see, in terms of shifts of the equipotential surfaces, the first term is very large (10 times the order of magnitude of the geoid), the second term amounts to only 20 cm and the third term is negligible. But the first term is exactly the effect of the condensed Helmert layer, so the second term is essentially the Helmert correction to the potential. Indeed, in high mountain and with a more complicated geometry, this term can rise by an order of magnitude, as stated in (39).

At this point would we know already $\delta g\left (r,\sigma \right )$ or $\Delta g\left (r,\sigma \right )$ inside L, then we could simply integrate a first order differential equation, namely

$$\displaystyle \begin{aligned} \frac{\partial T}{\partial r} = \delta g \quad \rightarrow \quad T\left(r,\sigma\right) = T_0\left(\sigma\right) + \int_r^{R\left(\sigma\right)} \delta g\left(s,\sigma\right) ds {} \end{aligned} $$

(71)

or

$$\displaystyle \begin{aligned} \begin{array}{ll} &\dfrac{\partial T}{\partial r} + \dfrac{2}{r} T = - \Delta g \quad \rightarrow \quad T\left(r,\sigma\right)= T_0\left(\sigma\right)+\\[3pt] &\qquad \qquad \qquad \qquad \qquad \quad \displaystyle\int_r^{R\left(\sigma\right)} \dfrac{s^2}{r^2} \Delta g\left(s,\sigma\right) ds \ . \end{array} {} \end{aligned} $$

(72)

So the problem is to continue say $\delta g$, for the sake of definiteness, in L either in free air, if we refer to the Helmert version, or following the classical Bouguer theory (Heiskanen and Moritz 1967), if we leave the masses in L. In both cases we solve essentially the approximate Bruns equation

$$\displaystyle \begin{aligned} \frac{\partial \delta g}{\partial r} = - \frac{2}{r} \delta g - 4 \pi G \rho \ , {} \end{aligned} $$

(73)

the last term being absent in the Helmert approach.

In Sansò and Sideris (2013, appendix A2), (73) has been justified by using in the exact Bruns equation the estimate on the difference between the true equipotentials mean curvature $\mathcal {C}$ and the corresponding normal counterpart $\mathcal {K}$

$$\displaystyle \begin{aligned} \left\vert \mathcal{C} - \mathcal{K} \right\vert \sim \frac{10^{-3}}{2R} {} \end{aligned} $$

(74)

which has been derived as a mean square value for the high resolution global model EGM96 (Lemoine et al 1998).

To better understand the error committed in this approximation, one can simply compare (73) with Poisson’s equation, written in spherical coordinates, and recall that $\delta g = T'$; then one sees that in (73) the term

$$\displaystyle \begin{aligned} \mathcal{E}\left(\delta g \right) = \frac{1}{r^2} \triangle_\sigma T {} \end{aligned} $$

(75)

is lost. Although this can have a relative small r.m.s., satisfying (74), it is clear that it can become quite large in rugged areas. This suggests that a much better approximation could be obtained by substituting (73) with

$$\displaystyle \begin{aligned} \frac{\partial \delta g}{\partial r} = - \frac{2}{r} \delta g - 4 \pi G \rho -\frac{1}{r^2} \triangle_\sigma T_M \ , {} \end{aligned} $$

(76)

where $T_M$ is some high resolution global model of T. Calling

$$\displaystyle \begin{aligned} f = 4 \pi G \rho + \frac{1}{r^2} \triangle_\sigma T_M \ , {} \end{aligned} $$

(77)

(76) has the solution

$$\displaystyle \begin{aligned} \delta g\left(r,\sigma\right) = \frac{R^2\left(\sigma\right)}{r^2}\delta g_0\left(\sigma\right) + \int_r^{R\left(\sigma\right)} \frac{s^2}{r^2} f\left(s,\sigma\right) ds \ . {} \end{aligned} $$

(78)

This substituted in (71) gives the sought solution $T\left (r,\sigma \right )$ in L. After some computations the result is

$$\displaystyle \begin{aligned} \begin{array}{rcl} T\left(r,\sigma\right) & =&\displaystyle T_0\left(\sigma\right) + \delta g_0\left(\sigma\right) R^2\left(\sigma\right) \left( \frac{1}{r}-\frac{1}{R\left(\sigma\right)}\right) \\ & +&\displaystyle \int_r^{R\left(\sigma\right)} \left( \frac{1}{r}-\frac{1}{s} \right) f\left(s,\sigma\right) ds \ . {} \end{array} \end{aligned} $$

(79)

One has to underline that this line of thought is already contained in Heiskanen and Moritz (1967).

5 Conclusions

Two main conclusions can be drawn from the presented analysis:

• As for the determination of the surface of the Earth S and the outer potential W, i.e. the solution of the nonlinear, scalar Molodensky problem, the two approaches of Molodensky and Helmert are basically equivalent at centimetric level.

• As for the propagation of W (or T) inside the masses and then the computation of the geoid (not discussed here), both methods are affected by two errors:

  • the imperfect knowledge of $\rho$; this gives the same error in both approaches,

  • the downward continuation error; since this can be reduced by computing correction terms with the help of a global model $T_M$, the use of Helmert’s approach that removes discontinuities and of a Helmertized model $T^H_M$, is likely to produce better results.