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Up and Down Through the Gravity Field

  • F. SansóEmail author
  • M. Capponi
  • D. Sampietro
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)

Zusammenfassung

Die Kenntnis des Schwerefeldes hat weitreichende Anwendungen in den Geo- wissenschaften, insbesondere in Geodsie und Geophysik. Unser Anliegen in diesem Beitrag ist die Beschreibung von Eigenschaften zur Fortpflanzung des Potentials oder seiner relevanten Funktionale nach oben und nach unten. Die Fortpflanzung des Potentials nach oben (“upward continuation”) ist stets ein wohlgestelltes Problem. “Downward Continuation” ist stets ein schlechtgestelltes Problem, nicht nur wegen der numerischen Instabilitten, sondern vor allem wegen der Nichteindeutigkeit der Bestimmung von Massenschichtung aus Potentialwerten.

Als Konsequenz fokussiert sich der Beitrag auf neuere Resultate aus dem Bereich geodätischer Randwertprobleme und zum anderen auf das inverse Gravimetrieproblem. Dabei machen wir den Versuch, die Bedeutung von mathematischer Theorie für numerische Anwendungen heraus zu streichen.

Das Paper ist vom mathematischem Anspruch her schlicht gehalten. Dazu bedienen wir uns oftmals der Rückführung auf sphärische Beispiele. Der größte Teil des Materials ist bereits in der Literatur vorhanden, bis auf Teile für die globalen Modelle und das inverse Gravimetrieproblem für Schichtungen.

Keywords

Upward continuation Downward continuation Geodetic boundary value problem Inverse gravimetric problem 

Abstract

The knowledge of the gravity field has widespread applications in geosciences, in particular in Geodesy and Geophysics. The point of view of the paper is to describe the properties of the propagation of the potential, or of its relevant functionals, while moving upward or downward. The upward propagation is always a properly posed problem, in fact a smoothing and somehow related to the Newton integral and to the solution of boundary value problems (BVP). The downward propagation is always improperly posed, not only due to its intrinsic numerical instability but also because of the non-uniqueness that is created as soon as we penetrate layers of unknown mass density.

So the paper focuses on recent results on the Geodetic Boundary Value Problems on the one side and on the inverse gravimetric problem on the other, trying to highlight the significance of mathematical theory to numerical applications. Hence, on the one hand we examine the application of the BVP theory to the construction of global gravity models, on the other hand the inverse gravimetric problem is studied for layers together with proper regularization techniques.

The level of the mathematics employed in the paper is willingly kept at medium level, often recursing to spherical examples in support to the theory. Most of the material is already present in literature but for a few parts concerning global models and the inverse gravimetric problem for layers.

1 Gravity Field: Why and How

The knowledge of the gravity field of the Earth is a scientific achievement of fundamental interest for almost all Earth sciences, but with distinct applications according to whether we need it on the Earth surface and in space above it, or below the Earth surface.

To the first category belong all geodetic applications, primarily the knowledge of the geoid and of various derivatives of the potential, which are essential in modern surveying, cartography as well as oceanography and in navigation, specially in space close to the Earth, where satellites and shuttles do their job. To the second category belong all solid Earth geophysical applications at different scales and depths, from the global ones, where different geological models of the Earth system can find a confirmation or a disproof, to the local ones where the presence of a short wavelength gravity anomaly can reflect the existence of a shallow density anomaly, corresponding to valuable natural resources.

For historical reasons, the manipulation of the gravity field in the geodetic reign has been considered as an (almost) well posed problem, dominated as it is by the work of transforming gravity anomalies into anomalous potential, or geoid undulations. Theoretically this is the domain of the Geodetic Boundary Value Problem (GBVP) (or better “Problems”) that has got a significant mathematical assessment in the very last decades (see [27] for a review) after the fundamental impulse imparted by Hörmander’s publication [8]. Essentially we know that if we want to determine the gravity potential, from the surface of the Earth, S, upward, from some kind of gravity anomalies, given all over S, this can be done under fairly general conditions, providing unique, stable solutions. Indeed things become different when we pose the problem of determining the gravity field on S from spatial observations, like those collected in recent years by the three gravity satellite missions CHAMP, GRACE and GOCE [5] and we will explain why, shortly.

On the contrary the use of the gravity field in the reign of solid Earth geophysics, namely inferring the mass distribution from the knowledge of the gravity field on S (or out of it) belongs to the domain of improperly posed problems. In fact its solution is generally not unique and its determination leads systematically to numerical instabilities even when the class of acceptable geological models is so restricted as to guarantee uniqueness, as it happens with the two constant density layers problem.

As a side remark on the above coarse and imprecise classification, let us notice that it is based on the concept: “surface of the Earth”. This is also a rather fishy object requiring a good deal of thought to be defined in an acceptable form (see [25]). Here we report the definition given in [28]: the surface of the Earth “is any surface S on which we can assume (after a limit process) to know both the value of the gravity potential W and the modulus of its gradient g to a predefined degree of accuracy”; in the present context one should add “solid (or liquid) masses above S should have a known density and a thickness below 100 m, such that their effects can be corrected for, at the above mentioned level of accuracy”. The limit of 100 m is not sharp, but not arbitrary either; it is related to the so called linearization band, of which one can read in [28].

Even with the above specifications one has to realize that it is not possible to avoid in a layer close to S a certain overlapping of the geodetic and geophysical use of gravity. This is the topographic layer (roughly between S and the geoid) where we assume to know the mass density too, though with unavoidable errors, that due to the thickness of such a layer, can be significantly reflected into our knowledge of the gravity field. This is not a big problem for geodetic applications, since we can remove and than restore the effects of sources below S, as far as linearization procedures are permitted. On the contrary errors in the density of the topographic layer can distort the subsequent geophysical gravity inversion. This is why terrain corrections, beyond numerical aspects, are still object of investigations.

Apart from this remark, let us now introduce the usual notation (B, S, Ω) to represent the body of the Earth B, which we assume to be simply connected and bounded, its surface S, that we shall assume star-shaped and Lipschitz, namely to be represented in spherical coordinates by the equation
$$\displaystyle \begin{aligned} r = R (\sigma) \equiv R(\lambda, \varphi) {} \end{aligned} $$
(1)
with (λ, φ) spherical angular coordinates, R(λ, φ) a function defined on the unit sphere with a gradient essentially bounded
$$\displaystyle \begin{aligned} \sigma \ \text{a. e.} \ , \qquad \vert \ \nabla_{\sigma} R(\sigma) \ \vert \le C {} \end{aligned} $$
(2)
and finally Ω is the space exterior to S (see Fig. 1). Indeed to determine the gravity field in Ω is the geodetic problem and to determine it, or (almost) equivalently the density field ρ, in B is the geophysical problem. Let us note en passant that due to known properties of the density ρ (it has to be measurable, positive and bounded) and to the assumption on S, it is also known that the gravity potential has to be continuous together with its first derivatives across S. However what we want primarily to reason on is that problems are well posed or not depending essentially from two other factors: whether we move upward or downward from the area where we have data to the area where we want to know the gravity field; whether along this path we meet or not sources of the gravity field, namely ρ ≠ 0.
Fig. 1

Notation for the geometry of the analyzed scenario

We will reason under a linearized, spherical approximation hypothesis, although our conclusions are valid under much more general conditions. So we assume that we can describe the anomalous gravity by means of the anomalous gravity potential u(P), that u is generated by an anomalous mass density distribution that we will call again ρ, obtained by subtracting to the actual density some density model compatible with the normal potential (see [18]). So the relation between the source anomaly ρ and the anomalous potential u, is just Newton’s integral:
$$\displaystyle \begin{aligned} u(P) = G \int_B \frac{\rho(Q)}{l_{PQ}} \ dB_{Q}. {} \end{aligned} $$
(3)
Likewise we shall assume that the gravity “anomalies” available are just the gravity disturbances δg, that in spherical approximation are related to u by (see [27])
$$\displaystyle \begin{aligned} \delta g = -\frac{\partial}{\partial r} \ u {} \end{aligned} $$
(4)

2 Principles of Upward Continuation

In this section we will recall the relations that bind a mass distribution in a thin layer to the gravity field, in terms of its potential u, generated on the layer itself and then the propagation of u, from the level of the layer, upward. In other words we shall study the operators that make us move up along the chain (Fig. 2). To keep things easy and evident we shall consider spherical layers or spherical surfaces to make our description self-evident, although our conclusions are valid under general conditions. So in this context up and down has to be understood in radial direction. The advantage of the use of a spherical geometry is the possibility of exploiting the spherical harmonic representation of functions, in L2(Sr),
$$\displaystyle \begin{aligned} f(r,\sigma) = \sum_{n = 0}^{+\infty} \sum_{m = -n}^{n} f_{nm}(r) \ Y_{nm}(\sigma); {} \end{aligned} $$
(5)
in particular when f is known to be harmonic in a spherical domain, BR ≡{r ≤ R}, we recall that Eq. 5 has the form (see [27] Cap. 3,4)
$$\displaystyle \begin{aligned} r \le R \quad , \qquad u(r, \sigma) = \sum_{n = 0}^{+ \infty} \sum_{m = -n}^{n} u_{nm} \bigg(\frac{r}{R}\bigg)^{n} Y_{nm}(\sigma) {} \end{aligned} $$
(6)
Fig. 2

Operators to move up along the chain

The functions
$$\displaystyle \begin{aligned} Si_{nm}(r, \sigma) = \bigg(\frac{r}{R}\bigg)^{n} Y_{nm}(\sigma) {} \end{aligned} $$
(7)
are called internal solid spherical harmonics.
Likewise a function harmonic in ΩR ≡{r ≥ R} can be expanded into the form
$$\displaystyle \begin{aligned} r \ge R \quad , \qquad u(r, \sigma) = \sum_{n = 0}^{+ \infty} \sum_{m = -n}^{n} u_{nm} \bigg(\frac{R}{r}\bigg)^{n+1} Y_{nm}(\sigma) {} \end{aligned} $$
(8)
and the functions
$$\displaystyle \begin{aligned} Se_{nm}(r, \sigma) = \bigg(\frac{R}{r}\bigg)^{n+1} Y_{nm}(\sigma) {} \end{aligned} $$
(9)
are called external solid spherical harmonics. The functions Ynm(σ) are as usual surface spherical harmonics of degree n and order m ( [27], Cap. 13.3).
Incidentally we note that
$$\displaystyle \begin{aligned} Si_{nm}(R,\sigma) \equiv Se_{nm}(R,\sigma) \equiv Y_{nm} (\sigma) {} \end{aligned} $$
(10)
which is, by a suitable choice of the normalization constants, an orthonormal complete basis of L2(σ) ( [27], Cap. 3.4), namely the space of f(σ) with bounded norm:
$$\displaystyle \begin{aligned} \Vert \ f \ \Vert^2_{L^{2}(\sigma)} = \frac{1}{4 \ \pi} \int f^2 (\sigma) \ d \sigma. {} \end{aligned} $$
(11)

To continue it is convenient to introduce a few definitions:

Definition 1

we say that an operator \(\mathscr {S}\) is a (isotropic) smoother in L2(σ) if
$$\displaystyle \begin{aligned} f \in L^2(\sigma) \qquad \mathscr{S} \ f = \sum_{n = 0}^{+\infty} \sum_{m = -n}^{n} \mathscr{S}_{n} \ f_{nm} \ Y_{nm} (\sigma) {} \end{aligned} $$
(12)
and
$$\displaystyle \begin{aligned} \mathscr{S}_{n} > 0 \quad , \qquad \lim_{n\rightarrow \infty} \mathscr{S}_{n} = 0 {} \end{aligned} $$
(13)
Here isotropic is related to the fact that the weights \(\mathscr {S}_{n}\) depend only on the degree n.

Definition 2

we say that \(\mathscr {S}\) has power strength N if
$$\displaystyle \begin{aligned} n^{N} \ \mathscr{S}_{n} \le C {} \end{aligned} $$
(14)

Definition 3

we say that \(\mathscr {S}\) has exponential strength if ∃ q > 1 such that
$$\displaystyle \begin{aligned} q^{n} \ \mathscr{S}_{n} \le C. {} \end{aligned} $$
(15)

We notice that, due to Eq. 13, the operator \(\mathscr {S}\) is always bounded in L2(σ) and even compact (see [32], X.1).

In addition if f ∈ L2(σ) and \(\mathscr {S}\) has power strength N, then all derivatives of f up to order N
$$\displaystyle \begin{aligned} \partial^{\alpha} f = \partial_1^{\alpha_{1}} \ \partial_2^{\alpha_{2}} \ \partial_3^{\alpha_{3}} \ f \in L^{2}(\sigma) \qquad \qquad \vert \ \alpha \ \vert = \alpha_1 + \alpha_2 + \alpha_3 \le N {} \end{aligned} $$
(16)
are also in L2(σ) and we say that f ∈ HN, 2(σ). Here and in the sequel Hn, 2 on any set A denotes the usual Sobolev Hilbert spaces of degree n and power 2 (see [14]).
If \(\mathscr {S}\) has exponential strength, then
$$\displaystyle \begin{aligned} \mathscr{S} \ f \in C^{\infty}(\sigma) {} \end{aligned} $$
(17)
i.e., equivalently the derivatives of f of any order are in L2(σ) and in fact also continuous. It is worth noticing that due to the first condition in Eq. 13 the operator \(\mathscr {S}\) is always invertible as:
$$\displaystyle \begin{aligned} \mathscr{S} \ f = 0 \Longrightarrow \mathscr{S}_{n} \ f_{nm} = 0 \Longrightarrow f_{nm} = 0 \Longrightarrow f = 0 {} \end{aligned} $$
(18)
indeed the inverse \(\mathscr {S}^{-1}\) is always an unbounded operator in L2(σ).
Now coming to the question posed at the beginning of the section, we only need to write the Newton integral for a thin layer of width Δ around a sphere of radius R0 and linearize the formula with respect to Δ:
$$\displaystyle \begin{aligned} u(r, \sigma) = G \int d \sigma' \int_{R_{0} - \frac{\varDelta}{2}}^{R_{0} + \frac{\varDelta}{2}} \frac{\rho(\sigma')}{l} \ r^{2} \ dr \simeq G \ R_{0}^2 \ \varDelta \int d \sigma ' \ \frac{\rho(\sigma ')}{l} {} \end{aligned} $$
(19)
with \( l = \left [r^2 + R_{0}^2 - 2 \ r \ R_{0} \ cos(\psi _{\sigma \sigma '})\right ]^{\frac {1}{2}} \) and \(\psi _{\sigma \sigma '}\) the spherical angle between the two directions, σ, σ′.

It is worth noting that u is essentially the potential of a single layer deposited on \(S_{R_{0}}\) with surface density ω(σ′) = Δ ρ(σ′).

Then taking for instance r > R0 and using the well known representation (see [18], Cap. 1.15)
$$\displaystyle \begin{aligned} \frac{1}{l} = \sum_{n = 0}^{+ \infty} \sum_{m = -n}^{n} \frac{R_{0}^{n}}{r^{n+1}} \frac{1}{2 \ n + 1} Y_{nm} (\sigma) \ Y_{nm} (\sigma') {} \end{aligned} $$
(20)
one gets
$$\displaystyle \begin{aligned} r \ge R_{0} \ , \qquad u(r, \sigma) = 4 \ \pi \ G \ R_{0} \ \varDelta \sum_{n = 0}^{+ \infty} \sum_{m = -n}^{n} \bigg( \frac{R_{0}}{r}\bigg)^{n+1} \rho_{nm} \ \frac{Y_{nm} (\sigma)}{2 \ n + 1} {} \end{aligned} $$
(21)
where
$$\displaystyle \begin{aligned} \rho_{nm} = \frac{1}{4 \ \pi} \int d \sigma ' \ \rho(\sigma') \ Y_{nm}(\sigma'). {} \end{aligned} $$
(22)
Letting r → R0, we get the spectral relation
$$\displaystyle \begin{aligned} u_{nm} (R_{0}) = 4 \ \pi \ G \ R_{0} \ \varDelta \ \frac{\rho_{nm}}{2 \ n + 1} {} \end{aligned} $$
(23)
where unm(R0) are the harmonic coefficients of u(R0, σ). As we see the Newton operator \(\mathscr {N}\) that transform ρ into u at the same level R0, is a smoother with power strength N = 1, i.e., it transforms a ρ ∈ L2(σ) into a potential u ∈ H1, 2(σ). This is confirmed by the well known formula:
$$\displaystyle \begin{aligned} \delta g \lvert_{S_{R_{0}}} = - u' \lvert_{S_{R_{0}}} = 4 \ \pi \ G \ R_{0} \ \varDelta \sum_{n = 0}^{+ \infty} \sum_{m = -n}^{n} \frac{n+1}{2 \ n + 1} \ \rho_{nm} \ Y_{nm} (\sigma) {} \end{aligned} $$
(24)
or, in spectral form,
$$\displaystyle \begin{aligned} \delta g _{nm} (R_{0}) = 4 \ \pi \ G \ R_{0} \ \varDelta \ \frac{n+1}{2 \ n +1} \ \rho_{nm} {} \end{aligned} $$
(25)
showing that if ρ ∈ L2(σ), then δg = −u′∈ L2(σ) too.

Let us note that Eq. 24 gives the limit from outside (r → R0+) of the gravity anomaly generated by the single layer (Eq. 19); as we know such a function has a sharp jump across \(S_{R_{0}}\), equal to − 4 π G ω(σ) = −4 π G Δ ρ(σ) (see [18], Cap. 1,3).

Among other things Eq. 25 means that giving the gravity anomaly δg on the upper surface of the thin layer, one can retrieve at once the body density ρ, apparently without numerical instabilities; namely the operator \(- \frac {\partial }{\partial r} \mathscr {N}( \cdot ) \lvert _{S_{R_{0}}}\) is not a smoother and its inverse is bounded.

Now to conclude our analysis we have to propagate the potential u from the level R0 to the level R (> R0), because we know that it is only on the surface SR that we can get information on u.

Since for r > R0, u has the form of Eq. 21, by using Eq. 23 we find
$$\displaystyle \begin{aligned} u(R, \sigma) = \sum_{n = 0}^{+ \infty} \sum_{m = -n}^{n} u_{nm}(R_{0})\bigg( \frac{R_{0}}{R}\bigg)^{n+1} Y_{nm} (\sigma) {} \end{aligned} $$
(26)
or
$$\displaystyle \begin{aligned} u_{nm} (R) = u_{nm}(R_{0}) \bigg( \frac{R_{0}}{R}\bigg)^{n+1}. {} \end{aligned} $$
(27)
As we see the upward continuation operator \(\mathscr {U}\) is a smoother with exponential strength. An analogous computation for δg leads to
$$\displaystyle \begin{aligned} \delta g (r, \sigma) = \sum_{n = 0}^{+ \infty} \sum_{m = -n}^{n} \frac{n+1}{R_{0}} \bigg( \frac{R_{0}}{R}\bigg)^{n+2} u_{nm} (R_{0}) \ Y_{nm}(\sigma) {} \end{aligned} $$
(28)
or
$$\displaystyle \begin{aligned} \delta g_{nm} (R) = \frac{n+1}{R_{0}} \bigg( \frac{R_{0}}{R}\bigg)^{n+2} u_{nm}(R_{0}) = 4 \ \pi \ G \ \varDelta \ \frac{n+1}{2 \ n +1} \bigg( \frac{R_{0}}{R}\bigg)^{n+2}\rho_{nm}. {} \end{aligned} $$
(29)

In this case the density coefficients ρnm(R0) (or the single layer coefficients ωnm = ρnm Δ) can still be uniquely retrieved from δgnm(R), but the numerical instability, due to the exponential factor \(\bigg ( \frac {R}{R_{0}}\bigg )^{n+2}\) present in \(\mathscr {D} = \mathscr {U}^{-1}\), is unbeatable, unless we use a regularization method, e.g., of the Tikhonov type (see for instance the recent paper [7]).

Remark 1

We can observe that, combining Eqs. 23 and 27 one can write, for any R′, R0 < R′ < R,
$$\displaystyle \begin{aligned} \begin{aligned} u_{nm} (R) &= 4 \ \pi \ G \ R_{0} \ \varDelta \frac{\rho_{nm}}{2 \ n+1} \bigg( \frac{R_{0}}{R}\bigg)^{n+1} =\\ &= u_{nm} (R_{0}) \bigg( \frac{R_{0}}{R'}\bigg)^{n+1} \bigg( \frac{R'}{R}\bigg)^{n+1} = u_{nm} (R') \bigg( \frac{R'}{R}\bigg)^{n+1} \end{aligned} {} \end{aligned} $$
(30)
This is the spectral representation of a property of the family of upward continuation operators. If we denote by \(\mathscr {U}(R:R_{0})\) the upward continuation of the harmonic function u from the level r = R0 to the level r = R, Eq. 30 can be written as
$$\displaystyle \begin{aligned} R_{0} < R' < R \ , \qquad \mathscr{U}(R, R_{0}) = \mathscr{U}(R, R') \ \mathscr{U}(R',R_{0}) {} \end{aligned} $$
(31)
highlighting the semi-group property of this family. But this means also that once we know the potential u(R′, σ), we don’t need anything else to propagate it at higher levels and in particular the relation between the density ρ(R0, σ) at level R0 and the field u(r, σ), for any r > R′, is broken. In fact the same u(r, σ) in {r > R′} could be generated by a single layer, at the same level R′, with density
$$\displaystyle \begin{aligned} \omega'(\sigma) = \omega(\sigma) \bigg( \frac{R_{0}}{R'}\bigg)^{n+1} = \varDelta \ \rho_{nm} \bigg( \frac{R_{0}}{R'}\bigg)^{n+1} {} \end{aligned}$$
which is nothing else but the famous Poincaré’s sweeping out (see [33]).

Notice that by suitably tuning the surface density ω″(σ) we could have substituted the layer on \(S_{R_{0}}\) with any other layer on \(S_{R''}\) with R0 < R″ < R′.

An analogous reasoning holds for δg(r, σ) too, with the spectral upward continuation factor \(\bigg (\frac {R_{0}}{R'}\bigg )^{n+2}\).

It is time now to add the effects of many thin layers, going to the continuous limit, which is obtained from Eqs. 21, 22, and 23 by assuming that ρ is a function varying with r, ρ(r, σ), so that also ρnm result to be functions of r, ρnm(r), and taking Δ = dr; we get then
$$\displaystyle \begin{aligned} \begin{aligned} r \ge R \qquad u(r, \sigma) &= 4 \ \pi \ G \sum_{n = 0}^{+\infty} \sum_{m = -n}^{n} \frac{Y_{nm}(\sigma)}{2 \ n+1}\int_{0}^{R} \bigg(\frac{r'}{r}\bigg)^{n+1} \rho_{nm}(r')r' dr' = \\ &= 4 \ \pi \ G \sum_{n = 0}^{+\infty} \sum_{m = -n}^{n} \bigg(\frac{R}{r}\bigg)^{n+1} \frac{Y_{nm}(\sigma)}{2 \ n+1}\int_{0}^{R} \bigg(\frac{r'}{R}\bigg)^{n+1} \rho_{nm}(r') \ r' \ dr' {} \end{aligned} \end{aligned} $$
(32)
where
$$\displaystyle \begin{aligned} \rho_{nm}(r) = \frac{1}{4 \pi} \int \rho(r, \sigma') \ Y_{nm}(\sigma') \ d \sigma' {} \end{aligned}$$
and its spectral counterpart, in terms of potential, is
$$\displaystyle \begin{aligned} u_{nm}(r) = \frac{4 \ \pi \ G}{2 \ n +1 } \int_{0}^{R} \bigg(\frac{r'}{R}\bigg)^{n+1}\rho_{nm}(r') \ r' \ dr'. {} \end{aligned} $$
(33)

Notice that Eq. 32 is valid for r ≥ R, since if we wanted u(r, σ) in {r < R} we should have used the development of \(\frac {1}{l}\) in terms of internal spherical harmonics in the region {r ≤ r′≤ R}.

In any event we underline that due to the previous Remark 1 for any R0, 0 ≤ R0 ≤ R, we can split the effect of the mass density into two parts, one inside \(B_{R_{0}} \equiv \{ r \le R_{0}\}\), which generates the internal potential uI(R0, σ) that is then upward continued up to SR ≡{r = R}, the other one expressed by the Newton integral in spherical form, namely:
$$\displaystyle \begin{aligned} \begin{aligned} r \ge R \qquad u(r,\sigma) &= \mathscr{U} (R, R_{0}) \ \left[ u_{I}(R_{0}, \sigma)\right] + \\ &+ 4 \ \pi \ G \sum_{n = 0}^{+\infty} \sum_{m = -n}^{n} \bigg(\frac{R}{r}\bigg)^{n+1} \frac{Y_{nm}(\sigma)}{2 \ n+1}\int_{R_{0}}^{R} \bigg(\frac{r'}{R}\bigg)^{n+1} \rho_{nm}(r') \ r' \ dr' \end{aligned} {} \end{aligned} $$
(34)

Obviously analogous formulas could be worked out for δg.

So, summarizing we could say that by the above, rather rudimentary example, we have understood the two main rules of moving “upward” through the gravity field:
  1. (a)
    if we start from a surface S0 (in the example the sphere \(S_{R_{0}}\)) and want to bring the gravity field to the level of the surface S encompassing S0 (in the example the sphere SR, R > R0), we need to know:
    1. (i)

      the potential uI of the masses inside B0 (∂B0 = S0) on S0, or any other boundary data on S0 that determine uniquely \(u_{I} \vert _{S_{0}}\), like for instance \(\delta g \vert _{S_{0}}\)

       
    2. (ii)

      the mass distribution between S0 and S

       
    3. (iii)

      that no other masses exist outside S

       
     
  2. (b)
    the gravity field from S outside (in Ω) can be obtained:
    1. (i)

      by solving a boundary value problem with the given data on S, computing a regular harmonic solution and than restricting it to S (in our example by using the upward operator \(\mathscr {U}(R, R_{0})\))

       
    2. (ii)

      by computing the Newton integral for the known density ρ between S0 and S (in our example the second term in Eq. 34)

       
     

From the comments done in the section we already know that, the harmonic upward continuation, namely the operator \(\mathscr {U}\), is one to one, while the Newton integral is not one to one, ρ(r, σ)⇔u|S, because we have already shown by the spherical example, that layers could be moved up and down by suitably changing their (surface) density without changing the potential on S. More on the upward continuation operator can be found in [28], Appendix A.

3 Geodetic Boundary Value Problems (GBVP’s)

This is the geodetic part of the use of gravity data for the determination of the gravity field on S (surface of the Earth) and Ω (exterior space), i.e., in \(\overline {\varOmega }\). The significance of GBVP’s stems from the fact that we can determine what is the form of a minimal information to be given on S in order to find the corresponding (anomalous) gravity potential u in a unique and stable mode. Speaking of the anomalous potential implies that we can safely go to the linearized versions of the GBVP’s (see [27], 15 and [28], 2); when we say that we can find u in a unique and stable mode, it means that we are able to identify spaces Y , for the data, and X for the unknowns such that to any fY  we can find one and only one uX satisfying the BVP and that u depends continuously on f with respect to the corresponding topologies. So we can say that a GBVP has to be a typical well posed problem, according to Hadamard, for the determination of u in \(\overline {\varOmega }\).

The standard form of a liniarized BVP is:
$$\displaystyle \begin{aligned} \left\{ \begin{array}{l l} \varDelta u = 0 \qquad \text{in }\varOmega \\ B \ u \vert_{S} = f \\ u \rightarrow 0 \quad , \quad r \rightarrow \infty\\ \end{array} \right. {} \end{aligned} $$
(35)
where B u represents the linear operator linking u to the observations, f, taken on the boundary S. The two terms f and S depend on the kind of data we consider as given; the same happens to the shape of the boundary operator B. In any event we shall assume S to be star-shaped, i.e., it can be described by the equation r = R(σ) = Rσ.
We notice that maybe the most “natural” GBVP would be the so called Altimetry Gravimetry (AG) problem, in which the data are splitted in two parts:
  1. (a)
    on the ocean it is supposed that the stationary height of the sea (SSH) is known from satellite altimetry, namely the height h(σ), with respect to the ellipsoid, of the foot of radar pulses, reflected by the sea. Furthermore it is assumed that oceanographers can provide the so called dynamic height, namely the height ξD of the stationary sea surface with respect to the reference equipotential surface, the geoid (G). Such dynamic height is supported by the steady circulation in the oceans. So if we call N, the geoid undulation, i.e., the height of the geoid on the ellipsoid, we have the notable relation
    $$\displaystyle \begin{aligned} h(\sigma) = \xi_{D}(\sigma) + N(\sigma) {} \end{aligned} $$
    (36)
    As obvious, h, ξD, N are function of horizontal coordinates σ = (λ, φ). Since N(σ) is related to u by Brun’s relation N(σ) = γ(σ)−1 u(σ), where γ is as usual the modulus of normal gravity, Eq. 36 implies that we can pretend to know on the ocean directly u(σ); in this case S is just the projection of the ocean on the corresponding part of the ellipsoid;
     
  2. (b)
    on land we assume to know the modulus of the gravity vector g(P) at any point, and some kind of altimetric information. If we know hP, the ellipsoidal height of P, then the linearization leads to the land boundary relation:
    $$\displaystyle \begin{aligned} \delta g (\sigma) = g(\sigma) -\gamma(\sigma) = - \frac{\partial u}{\partial h} {} \end{aligned} $$
    (37)
    δg is known as gravity disturbance and the surface S in this case is directly the Earth surface for the land part. If on the contrary we assume to know the total gravity potential at P, W(P) = U(P) + u(P) (with U(P) the normal potential), then the linearization process leads to the boundary relation:
    $$\displaystyle \begin{aligned} \varDelta g (P) = g(P) - \gamma(P^{*}) = - \frac{\partial T}{\partial h} - \frac{\frac{\partial \gamma}{\partial h}}{\gamma} \ T {} \end{aligned} $$
    (38)
    also known as fundamental equation of physical geodesy; Δg is known as free air gravity anomaly, P is a known point at a height related to W(P), and in this case the surface S is the telluroid, i.e., the surface swept by P. Such linearization processes are described in details in [18], 2–13 and in [27], 2.
     
The (AG) problem is not an easy one, as for its quantitative analysis, so the actual state of the art is still an analysis for a spherical surface S0 divided in ocean and land; this can be found in [24].

Fortunately, after some years, it was understood that marine geoid data can be manipulated to produce also on the ocean a dataset of gravity anomalies (see [1]) to provide a high resolution gravity map. This thanks to old formulas of Vening Meinesz [18], 2–22 combined with the calculation of the horizontal gradient of u and the physical fact that the geoid on the ocean is a quite smooth function.

Therefore we could arrive at GBVP’s of two types according to whether we consider as known the gravity disturbance δg, in which case S is directly the surface of the Earth, or the free air gravity anomaly Δg, in which case S is the telluroid; this second problem is also known under the name of scalar Molodensky’s problem. The analysis of such problems has been carried out to a satisfactory point; the most recent results are collected in [26].

Here we shall concentrate on the first BVP, namely the so called Fixed Boundary gravimetric BVP:
$$\displaystyle \begin{aligned} \left\{ \begin{array}{l l} \varDelta u = 0 \qquad \qquad \text{in }\varOmega \\ \frac{\partial u}{\partial h} \vert_{S} = f(\sigma) \qquad \text{on }S\\ u = O(\frac{1}{r}) \quad , \quad \quad r \rightarrow \infty\\ \end{array} \right. {} \end{aligned} $$
(39)

This is to avoid the technicalities related to the Molodensky’s problem and its non-unique solution.

Even more, since our purpose is to discuss the relation of the GBVP to global gravity models, we shall further simplify Eq. 39, transforming the boundary relation into the corresponding spherical approximation formula:
$$\displaystyle \begin{aligned} \frac{\partial u}{\partial r} \bigg \vert_{S} = u'(\sigma) \vert_{S} = u' (R_{\sigma}, \sigma) \equiv f(\sigma) {} \end{aligned} $$
(40)
For such a simplified problem:
$$\displaystyle \begin{aligned} \left\{ \begin{array}{l l} \varDelta u = 0 \qquad \qquad \text{in }\varOmega \\ u' \vert_{S} = f(\sigma) \qquad \text{on }S\\ u \rightarrow 0 \ , \qquad \qquad r \rightarrow \infty\\ \end{array} \right. {} \end{aligned} $$
(41)
the classical analysis leading to existence, uniqueness and stability of the solution is of an overwhelming simplicity. For this reason we sketch it here, in a slightly modified form; for more details you can consult [27], 15,4 where the extension to the non-spherical approximation, i.e., to the boundary relation of Eq. 37, is performed.
The functional spaces involved are L2(S) for the data f(σ) with norm
$$\displaystyle \begin{aligned} \Vert \ f \ \Vert^2_{L^{2}(S)} = \int_{S} f^2(\sigma) \ dS {} \end{aligned} $$
(42)
and H1, 2(S) for u, with norm
$$\displaystyle \begin{aligned} \Vert \ u \ \Vert^2_{H^{1,2}(S)} = \int_{S} \vert \ \nabla u \ \vert^2 dS. {} \end{aligned} $$
(43)

Note that Eq. 43 gives a true norm for the harmonic function u (i.e., it is 0 if and only if u = 0). If ∫S| ∇u |2dS = 0 then | ∇u ||S = 0, almost everywhere, namely both the tangential gradient of u on S and its normal derivative should be zero. From the first statement one derives that u should be constant on S. But then S is an equipotential surface of u, and \(\frac {\partial u}{\partial n} = 0\) implies, by the unique solvability of the exterior Neumann problem (e.g., [31], Cap. 7 and 9), that u has to be zero everywhere on S and in Ω.

Now we can start from the differential identity
$$\displaystyle \begin{aligned} \nabla \cdot [r \ u' \ \nabla u] = \vert \ \nabla u \ \vert^2 + \frac{1}{2} \ r \ \frac{\partial}{\partial r } \vert \ \nabla u \ \vert^2 {} \end{aligned} $$
(44)
that the reader is invited to verify directly, recalling that \(r \frac {\partial }{\partial r} = \sum x_i \frac {\partial }{\partial x_i}\). Integrating Eq. 44 on Ω and using the divergence theorem as well as an integration by parts on r between Rσ and + , we arrive at the identity (see [8], [26])
$$\displaystyle \begin{aligned} \int_{\varOmega} \vert \ \nabla u \ \vert^2 d \varOmega + \int_S R_{\sigma}^3 \ \vert \ \nabla u \ \vert^2 d \sigma \equiv 2 \int_S R_{\sigma} \ u' \ \frac{\partial u}{\partial n} \ dS. {} \end{aligned} $$
(45)
Then we use the following inequalities:
$$\displaystyle \begin{aligned} \int_{\varOmega} \vert \ \nabla u \ \vert^2 d \varOmega \ge 0 {} \end{aligned} $$
(46)
$$\displaystyle \begin{aligned} \bigg \vert \int_S R_{\sigma} \ u' \ \frac{\partial u}{\partial n} \ dS \bigg \vert \le R_{+} \bigg [ \int_S f^2 (\sigma) \ dS \bigg ]^{\frac{1}{2}} \bigg [ \int_S u_{n}^2 \ dS\bigg ]^{\frac{1}{2}} \le R_{+} \Vert \ f \ \Vert_{L^2(S)} \Vert \ u \ \Vert_{H^{1,2}(S)} {} \end{aligned} $$
(47)
and, noticing that \(cos I \ dS = R_{\sigma }^2 \ d \sigma \) (cosI =n ⋅er),
$$\displaystyle \begin{aligned} \int_S R_{\sigma}^3 \ \vert \ \nabla u \ \vert^2 d \sigma \ge R_{-} \int_{S} \vert \ \nabla u \ \vert^2 cos I \ d S \ge R_{-} \ cos I_{+} \Vert \ u \ \Vert^2_{H^{1,2}(S)}, {} \end{aligned} $$
(48)
where I is the inclination of the surface S with respect to the main direction of the vertical, approximated here by er, R± are the max and min of Rσ, I+ is the maximum inclination of S which is supposed to be less than 90, so that
$$\displaystyle \begin{aligned} (cos I_{+})^{-1} = J_{+} < + \infty. \end{aligned}$$
Wrapping up Eqs. 46, 47, and 48 in Eq. 45 we find
$$\displaystyle \begin{aligned} R_{-} \ cos I_{+} \Vert \ u \ \Vert^2_{H^{1,2}(S)} \le 2 \ R_{+} \ \Vert \ f \ \Vert_{L^2(S)} \ \Vert \ u \ \Vert_{H^{1,2}(S)} \end{aligned}$$
namely
$$\displaystyle \begin{aligned} \Vert \ u \ \Vert_{H^{1,2}(S)} \le 2 \ \frac{R_{+}}{R_{-}} \ J_+ \ \Vert \ f \ \Vert_{L^2(S)}. {} \end{aligned} $$
(49)

As we can see the stability of the solution of this GBVP in H1, 2(S) depends essentially on the inclination of the terrain with respect to the vertical. Uniqueness obviously depends from Eq. 49; existence requires some further reasoning for which we send to literature (e.g., see [26]).

Remark 2

Naturally we would like not only to know that given δg there is the one and only one harmonic u such that u′|S = −δg, but also to know how to compute it, e.g., at the level of the boundary S itself. This can be done, as we will shortly see, by global modelling u with finite combinations of external spherical harmonics, but also by the representation of u by means of boundary layers, that has occupied geodesists for several decades. Here we want to return to the so called Prague method introduced in Geodesy by T. Krarup in the first of his famous letters on Molodensky’s problem [10]. We build on Krarup’s ideas, adapting them to the case of the simple, (i.e., spherical) Fixed Boundary GBVP. The concept is that, noticing that
$$\displaystyle \begin{aligned} v = - r \ u' = - \sum x_i \ \frac{\partial u}{\partial x_i} {} \end{aligned} $$
(50)
has to be harmonic, when u is such, to determine v from its boundary values
$$\displaystyle \begin{aligned} v \vert_{S} = f(\sigma) = R_{\sigma} \ \delta g ( \sigma) {} \end{aligned} $$
(51)
means solving a Dirichlet problem. A classical approach to Dirichlet is to represent v(r, σ) by means of a double layer potential, namely
$$\displaystyle \begin{aligned} \underline{x} \in \varOmega, \quad v(\underline{x}) = \int_S \mu ({\mathbf{y}}) \ \frac{\partial}{\partial n_{y}} \ \frac{1}{l_{xy}} \ d S_{y} {} \end{aligned} $$
(52)
with (lxy = |x −y|). Then taking the limit for x →xS ≡ (Rσ, σ), i.e., approaching the boundary S from outside along the normal to S, one gets the well known relation (see [18], 1-4)
$$\displaystyle \begin{aligned} 2 \ \pi \ \mu ({\mathbf{x}}_S) + \int_S \mu ({\mathbf{y}}) \ \frac{\partial}{\partial n_{y}} \ \frac{1}{l_{xy}} \ dS_{y} = f(\sigma) {} \end{aligned} $$
(53)
that can serve as an integral equation for the unknown μ(y) (y ∈ S). Once μ(y) is obtained, Eq. 52 allows to compute v in Ω. Once v is known, one can observe that, from Eq. 50 we have:
$$\displaystyle \begin{aligned} \textbf{x} \equiv (r, \sigma) \in \varOmega; \qquad u(r, \sigma) = \int_{r}^{+\infty} \frac{1}{s} \ v(s, \sigma) \ ds; {} \end{aligned} $$
(54)
notice that in fact, when v(s, σ) is a regular harmonic function, it is \(v(s, \sigma ) = O(\frac {1}{s})\), so that the integral in Eq. 54 is convergent and in fact it provides a regular potential \(u = O(\frac {1}{r})\). The Prague method consists essentially in making the two operations Eqs. 52 and 54 in one shot only, namely setting
$$\displaystyle \begin{aligned} u(r, \sigma) &= \int_S dS_{y} \ \mu({\mathbf{y}}) \ {\mathbf{n}}_{y} \cdot \nabla_{y} \ K({\mathbf{x}}, {\mathbf{y}}) {} \end{aligned} $$
(55)
$$\displaystyle \begin{aligned} K({\mathbf{x}}, {\mathbf{y}}) &= \int_{r}^{+\infty} \frac{1}{s_{\xi}} \ \frac{1}{l_{\xi y}} \ ds_{\xi}; {} \end{aligned} $$
(56)
in the above formula the following notation has been used
$$\displaystyle \begin{aligned} {\mathbf{x}} = r \ {\mathbf{e}}_r \qquad {\mathbf{y}} = \rho \ {\mathbf{e}}_{\sigma'} \qquad {\boldsymbol{\xi}} = s_{\xi} \ {\mathbf{e}}_{\sigma} \end{aligned}$$
$$\displaystyle \begin{aligned} l_{\xi y} = \sqrt{s_{\xi}^2 + \rho^2 - 2 \ \rho \ s_{\xi} \ cos \psi} \qquad cos \psi = {\mathbf{e}}_{\sigma} \cdot {\mathbf{e}}_{\sigma'}. \end{aligned}$$
A remarkable point is that the integral of Eq. 56 has an explicit form, namely,
$$\displaystyle \begin{aligned} K({\mathbf{x}}, {\mathbf{y}}) = K (r, \rho, \psi_{\sigma \sigma'}) = \frac{1}{\rho} \ \text{log} \frac{l_{xy} + \rho - r \ cos \psi}{r (1 - cos \psi)}. {} \end{aligned} $$
(57)
Therefore one has (see Fig. 3 for the meaning of eψ):
$$\displaystyle \begin{aligned} \nabla_{y} K({\mathbf{x}}, {\mathbf{y}}) = \frac{\partial K}{\partial \rho} \ {\mathbf{e}}_{\sigma'} + \frac{1}{\rho} \ \frac{\partial K}{\partial \psi} \ {\mathbf{e}}_{\psi} {} \end{aligned} $$
(58)
Fig. 3

Unit vectors for the computation of \(\nabla _{y} K({\mathbf {x}}, {\mathbf {y}}) \ ; \ {\mathbf {e}}_{\psi } = \frac {\partial }{\partial \psi } {\mathbf {e}}_{\sigma '}\)

It is easy to see that
$$\displaystyle \begin{aligned} {\mathbf{e}}_{\psi} = - \frac{1}{sin \psi} \ {\mathbf{e}}_{\sigma} + cotg \psi \ {\mathbf{e}}_{\sigma'} {} \end{aligned} $$
(59)
with \({\mathbf {e}}_{\sigma } = \frac {1}{r} \ {\mathbf {x}}\), \({\mathbf {e}}_{\sigma '} = \frac {1}{\rho } \ {\mathbf {y}}\). Moreover
$$\displaystyle \begin{aligned} \frac{\partial K}{\partial \rho} = - \frac{1}{\rho} \ K + \frac{1}{\rho \ l_{xy}} {} \end{aligned} $$
(60)
$$\displaystyle \begin{aligned} \frac{\partial K}{\partial \psi} = - \frac{sin \psi}{\rho} \ \frac{1}{l_{xy} + \rho - r \ cos \psi} \bigg[ \frac{l_{xy} + \rho - r}{1 - cos \psi} - \frac{r}{l_{xy}} \bigg]. {} \end{aligned} $$
(61)
It is interesting to observe that
$$\displaystyle \begin{aligned} \frac{\partial K}{\partial \rho} \ , \ \frac{\partial K}{\partial \psi} = O \bigg (\frac{1}{l_{xy}} \bigg ) \end{aligned}$$
so they are integrable functions on S.

Therefore the relation in Eq. 55 can be used even if we take x ∈ S; so once μ(y) has been reckoned by solving Eq. 53, u(x)|S is retrieved by Eq. 55 with Eqs. 60 and 61. As a final comment on this Remark, one can raise the question of the computability of a solution in two steps; μ from Eq. 53 and u from Eq. 55. In fact if the solution method goes through the representation of μ in spherical harmonics, then a direct approximation method can only be superior. On the other hand the solution of Eq. 53 can also be pursed by a multiscale method ( [6], Cap. 3) and the corresponding discretization that in principle is capable of reducing the calculations taking advantage of the large areas of the Earth surface (e.g., the oceans) on which the gravity field is smoother. In such a case the Prague method could be taken into consideration.

Remark 3

it is interesting to observe that the above results valid for the simple fixed boundary GBVP, can be easily carried on by a perturbative technique to the true linearized fixed boundary GBVP, namely to Eq. 39. The analysis can be found in [27], Cap. 15,4 and in synthesis one can say (with a slight different definition of the norms) that the stability constant, that in Eq. 49 was about 2J+, now becomes:
$$\displaystyle \begin{aligned} \left\{ \begin{array}{l l} C \cong 2 \ J_{+} (1 - 2 \ \epsilon_{+} \ J_{+}) \\ \epsilon_{+} \cong \frac{1}{2} \ e^2 \cong 6.72 \cdot 10^{-3}\\ \end{array} \right. {} \end{aligned} $$
(62)
on condition that
$$\displaystyle \begin{aligned} 2 \ \epsilon_{+} \ J_{+} < 6.72 \ J_{+} < 1. {} \end{aligned} $$
(63)
An easy calculation, recalling that J+ = (cosI+)−1, shows that Eq. 63 is satisfied if the maximum inclination of the topography I+, satisfies
$$\displaystyle \begin{aligned} I_{+} < 89.6^{\circ} \end{aligned}$$
certainly not a very restrictive condition. The above can be generalized to the Molodensky linearized problem, though with a much more involved analysis.

4 Global Models as Approximate Solutions of the GBVP

A global model uN(r, σ) is an approximate representation of the actual anomalous potential by means of a truncated series of external spherical harmonics, i.e., (cfr. Eq. 9)
$$\displaystyle \begin{aligned} u_N (r, \sigma) = \sum_{n = 0}^{N} \sum_{m = -n}^{n} u_{nm} \ Se_{nm} (r, \sigma) = \sum_{n = 0}^{N} \sum_{m = -n}^{n} u_{nm} \bigg( \frac{R}{r}\bigg)^{n-1} Y_{nm}(\sigma); {} \end{aligned} $$
(64)
since the summation in Eq. 64 is finite, the choice of R is just a matter of convenience. In the present context we choose R = R0, a Bjerhammar radius, namely a radius such that the sphere S0 ≡{r = R0} is totally inside the body B of the Earth (Fig. 4).
Fig. 4

The geometric setting of Runge-Krarup’s theorem

In order to say that uN is an approximation of u, we need a norm to measure the residual u − uN.

Given the frame of the theory of the GBVP presented in Sect. 3, in our application we shall use the norm in H1, 2(S) in a form essentially equivalent to Eq. 43 namely
$$\displaystyle \begin{aligned} \Vert \ u \ \Vert^{2}_{H^{1,2}(S)} = \int_S \vert \ \nabla u \vert_S \ \vert^2 \ d\sigma {} \end{aligned} $$
(65)
We underline that here u ∈ H1, 2(S) means also that u is harmonic in Ω (Fig. 1), namely if we call \(\mathscr {H} (\varOmega )\) the space of all functions harmonic in Ω we understand
$$\displaystyle \begin{aligned} u \in H^{1,2} (S) \Longleftrightarrow \Vert \ u \ \Vert_{H^{1,2}(S)} < + \infty \quad , \quad u \in \mathscr{H}(\varOmega) {} \end{aligned} $$
(66)

An analogous notation will be used for functions in H1, 2(S0) that are also harmonic in Ω0.

Remark 4

We have given to the norms of Eqs. 43 and 65 the same name because the two are equivalent from the functional point of view. In fact from
$$\displaystyle \begin{aligned} dS = \frac{R_{\sigma}^2}{cos I} \ d \sigma = R_{\sigma}^2 \ J \ d \sigma {} \end{aligned} $$
(67)
one sees that, with obvious symbolism,
$$\displaystyle \begin{aligned} R_{-}^2 \ \Vert \ u \ \Vert_{(65)} \le \Vert \ u \ \Vert_{(43)} \le R_{+}^2 \ J_{+} \ \Vert \ u \ \Vert_{(65)}. {} \end{aligned} $$
(68)

Nevertheless the two norms assign, so to say, different weights to different areas; in view of Eq. 67 the norm in Eq. 43 gives more weight to rugged areas where cosI can become smaller. This may not be wise from the approximation point of view.

To give a mathematical basis to our statement that uN is an approximative representation of u, in this case in H1, 2(S), we just need to know
$$\displaystyle \begin{aligned} H_0 = \text{Span} \{ Se_{nm} (r, \sigma)\}, {} \end{aligned} $$
(69)
of finite linear combinations of the external spherical harmonics is dense in H1, 2(S), denoted
$$\displaystyle \begin{aligned} H_0 \subset_d H^{1,2} (S) {} \end{aligned} $$
(70)
and that the coefficients unm of uN are judiciously chosen according to some convergent approximation principle.
This can be seen in two steps. At first consider the two spaces H1, 2(S0) and H1, 2(S) and the restriction operator RΩ, that restricts to Ω a function harmonic in Ω0; it is then obvious that
$$\displaystyle \begin{aligned} R_{\varOmega}(H^{1,2}(S_0)) \subset H^{1,2}(S). {} \end{aligned} $$
(71)
On the other hand, in view of the very general Runge-Krarup theorem ( [11], Cap. 9) we know that, as a particular case, Eq. 71 holds densely, i.e.,
$$\displaystyle \begin{aligned} R_{\varOmega} (H^{1,2}(S_0)) \subset_d H^{1,2}(S). {} \end{aligned} $$
(72)
As a second step, we can see that
$$\displaystyle \begin{aligned} H_0 \subset_d H^{1,2}(S_0); {} \end{aligned} $$
(73)
in fact since S0 is a sphere we have
$$\displaystyle \begin{aligned} \forall u \in H^{1,2}(S_0) \ , \qquad u = \sum_{n = 0}^{+\infty} \sum_{m = -n}^{n} u_{nm} \ Se_{nm} (r, \sigma), {} \end{aligned} $$
(74)
$$\displaystyle \begin{aligned} \begin{aligned} \Vert \ u \ \Vert ^2 _{H^{1,2}(S_0)} &= \int_{S0} (u' \ ^2 + \frac{1}{R_0^2} \ \vert \ \nabla_{\sigma} u \ \vert^2) \ d \sigma = \\ &= \frac{4 \ \pi}{R_0^2} \sum_{n=0}^{+\infty} \sum_{m = -n}^{n} u_{nm}^2 (n+1) (2 \ n+1). \end{aligned} {} \end{aligned} $$
(75)
This formula is easily explained by taking into account that
$$\displaystyle \begin{aligned} \int_{\sigma} \vert \ \nabla_{\sigma} u (R_0, \sigma) \ \vert ^ 2 \ d \sigma = - \int_{\sigma} u(R_0, \sigma) \ \varDelta_{\sigma} \ u(R_0, \sigma) \ d \sigma = \sum_{n=0}^{+ \infty} \sum_{m = -n}^{n} n \ (n+1) \ u_{nm}^2. \end{aligned}$$
From Eq. 75 it is obvious that Eq. 73 holds because \(\Vert \ u - u_N \ \Vert _{H^{1,2}}^2\) is just the residual of a convergent series of positive terms. But then, combining Eqs. 72 and 73,
$$\displaystyle \begin{aligned} R_{\varOmega} \ H_0 = Span \{ R_{\varOmega} \ Se_{nm} (r, \sigma) \} \subset_d H^{1,2} (S), {} \end{aligned} $$
(76)
which is basically what we wanted to prove.
Now notice that the theory of Sect. 3 tells us that the operation
$$\displaystyle \begin{aligned} B : H^{1,2} (S) \longrightarrow L^2 (S) \end{aligned}$$
is in fact one to one, bounded and also with a bounded inverse B−1 : L2(S)→H1, 2(S). In this case we know ( [32] Cap. VII,1) that we can define a transpose operator BT,
$$\displaystyle \begin{aligned} B^T : L^2(S) \longrightarrow H^{1,2} (S), \end{aligned}$$
which is also bounded, invertible and with bounded inverse.
As a consequence, identifying H0 and RΩ(H0), we can claim that
$$\displaystyle \begin{aligned} B(H_0) \subset_d L^2(S). {} \end{aligned} $$
(77)
In fact if φ ∈ L2(S) is such that
$$\displaystyle \begin{aligned} \forall u \in H_0 \qquad < \varphi \ , \ B \ u >_{L^2(S)} = 0 \end{aligned}$$
we have as well, thanks to Eq. 76,
$$\displaystyle \begin{aligned} \forall u \in H_0 \qquad < B^T \varphi \ , \ u >_{H^{1,2}(S)} = 0 \Rightarrow B^T \varphi = 0 \Rightarrow \varphi = 0, \end{aligned}$$
and Eq. 77 is proved.
At this point it is expedient to set up an approximation procedure by using the data f(σ) = B u|S and setting up, for instance, a least squares principle
$$\displaystyle \begin{aligned} \min_{\{u_{nm}\}} \ \Vert \ f(\sigma) - B \ u_N ( \sigma) \ \Vert^2_{L^2 (S)} = \epsilon^2 (N) {} \end{aligned} $$
(78)
allowing to determine the set of coefficients {unm} that give the “best” approximation of f among the linear combinations of {B Senm} complete up to degree N.
Indeed Eq. 77 implies that
$$\displaystyle \begin{aligned} \lim_{N \rightarrow \infty} \varepsilon^2 (N) = 0, {} \end{aligned} $$
(79)
i.e., a l.s. estimate is such B uNB u. But then, as a consequence of Eq. 49, we have as well
$$\displaystyle \begin{aligned} \lim_{N \rightarrow \infty} \Vert \ u - u_N \ \Vert_{H^{1,2}(S)} = 0 \ , {} \end{aligned} $$
(80)
i.e., uN is converging to the exact solution u. Note be taken that in this context the least squares principle has nothing to do with stochastic errors that always affect observations, like δg; in our case the principle has a pure deterministic meaning as functional approximation criterion. Another important remark is that the functions B Senm(Rσ, σ) are not orthogonal in L2(S), nor they are if we take a modified norm substituting dS with .

Therefore the least squares principle (Eq. 78) will lead to estimates of the coefficients that change if we change the maximum degree N, unm(N). Two issues related to this non-orthogonality are worth to be discussed.

The first is of numerical character; in fact a least squares solution requires the computation of a very large normal matrix with entries
$$\displaystyle \begin{aligned} (A^T A)_{nmjk} = \int_S B \ Se_{nm} (R_{\sigma}, \sigma) \ B \ Se_{jk} (R_{\sigma}, \sigma) \ dS {} \end{aligned} $$
(81)
which for a maximum degree 2000 requires some 1013 integrals.

Indeed there are methods for a direct iterative solution of the normal system and among them, accepting some approximations, we can count the actual geodetic solution passing through an approximate downward continuation of the datum δg (or Δg in case of Molodensky’s problem) down to the ellipsoid, followed by numerical quadrature with functions that are orthogonal on such a surface. The approach is discussed in both [27], Cap. 15,5 and [28], Cap. 5.

Beyond the method in fact employed to compute the actual global model EGM 2008 (see [19]), this “change of boundary” approach has been systematically studied, from the numerical point of view, in a recent paper ( [4]), where clear improvements in terms of decrease of biases are shown with respect to more traditional approaches.

A second important question is what is the significance of unm(N) when N; are such limits existing? Are the limits related somehow to the internal moments of the mass distribution? To answer this question we can make the following reasoning. Let us take a sphere Se (with radius Re) external to S and consider for our u ∈ H1, 2(S), the function of σ only u(Re, σ); such a function has a convergent harmonic series:
$$\displaystyle \begin{aligned} u(R_e, \sigma) = \sum_{n = 0}^{+\infty} \sum_{m = -n}^{n} u_{nm} (R_e) \ Y_{nm}(\sigma) {} \end{aligned} $$
(82)
and since it is a quite smooth function (on the bounded set Se), we certainly have
$$\displaystyle \begin{aligned} u_{nm} (R_e) = \frac{1}{4 \ \pi} \int u (R_e, \sigma) \ Y_{nm} (\sigma) \ d \sigma {} \end{aligned} $$
(83)
and
$$\displaystyle \begin{aligned} \Vert \ u(R_e, \sigma) \ \Vert^2_{L^2 (\sigma)} = \sum_{n = 0}^{+\infty} \sum_{m=-n}^{n} u_{nm} (R_e)^2 < +\infty. {} \end{aligned} $$
(84)
Since the norm squared of Eq. 84 is controlled by \(\Vert \ u \ \Vert ^2_{H^{1,2}(S)}\), i.e.,
$$\displaystyle \begin{aligned} \Vert \ u(R_e, \sigma) \ \Vert_{L^2(\sigma)} \le C \ \Vert \ u \ \Vert_{H^{1,2}(S)}, \end{aligned}$$
the linear functionals of Eq. 83 are continuous on H1, 2(S) and therefore there must be functions fnm ∈ H1, 2(S) such that
$$\displaystyle \begin{aligned} u_{nm} (R_e) = < f_{nm} \ , \ u >_{H^{1,2}(S)}; {} \end{aligned} $$
(85)
the reasoning is identical to that presented in [27], Cap. 15,2 Prepositions 1,2.
Now consider applying the same functionals to the least squares solution (see Eq. 64 with R = R0); one obviously hase
$$\displaystyle \begin{aligned} < f_{nm} \ , \ u_N >_{H^{1,2}(S)} = u_{nm} (N) \bigg( \frac{R_0}{R_e} \bigg)^{n+1}. {} \end{aligned} $$
(86)
On the other hand since uN is known to converge to u in H1, 2(S), one must have too
$$\displaystyle \begin{aligned} \lim_{N \rightarrow \infty} u_{nm} (N) = \bigg( \frac{R_e}{R_0} \bigg)^{n+1} u_{nm}(R_e). {} \end{aligned} $$
(87)

So we see that individual coefficients of the approximation do converge to harmonic coefficients that represent u(r, σ) from the level r = Re up.

Notice however that although uN → u in H1, 2(S) and \(u_{nn}(N) \rightarrow \bigg ( \frac {R_e}{R_0} \bigg )^{n+1} u_{nm} (R_e)\), this does not mean that the truncated series has a limit in the form of such a series convergent down to the surface S (or even worse down to the surface S0).

This is due to the fact that one cannot presume that
$$\displaystyle \begin{aligned} \sum_{n = 0}^{+ \infty} \sum_{m = -n}^{n} \bigg( \frac{R_{e}}{R_0} \bigg)^{2 \ n+2} u_{nm} (R_e)^2 \end{aligned}$$
be bounded, unless the original function u(r, σ) can be continued down to the level r = R0. As matter of fact every time we see an expression like
$$\displaystyle \begin{aligned} u(R_{\sigma}, \sigma) = \sum_{n = 0}^{+ \infty} \sum_{m = -n}^{n} u_{nm} \ Se_{nm} (R_{\sigma}, \sigma) \end{aligned}$$
we are using just a symbol, devoid of a precise mathematical meaning.

This argument might be important when we want to combine gravity measurements from satellite with boundary information given on S. In fact assume that the satellite surveying at the altitude r = Re is transformed into estimates of the coefficients unm(Re), as it has been done in the so called spacewise approach for the GOCE mission [15]; then we would like to know how to write “observation equations” for unm(Re), to be combined with the “terrestrial” global model.

This in principle should be done in the form of Eq. 85; namely we should know explicitely the functions fnm. This is difficult to do if we want to preserve one of the forms of the scalar product in H1, 2(S) that we have already introduced, namely Eqs. 43 and 65.

Nevertheless, as proved in Appendix A, it is not difficult to find functions {Fnm(σ)}, {Gnm(σ)} such that the identity holds
$$\displaystyle \begin{aligned} u_{nm} (R_e) = \frac{1}{4 \ \pi} \int F_{nm} (\sigma) \ (u' \vert_{S} ) \ d \sigma + \frac{1}{4 \ \pi} \int G_{nm} ( \sigma) \ (u \vert_{S}) \ d \sigma; {} \end{aligned} $$
(88)
the explicit analytical form of Fnm(σ) and Gnm(σ) is also given in Appendix A.

Remark 5

As a matter of fact Eq. 88 suggests that we should inquire a third equivalent version of the norm in H1, 2(S), namely
$$\displaystyle \begin{aligned} \Vert \ u \ \Vert_{\tilde{H}^{1,2}(S)} = \frac{1}{4 \ \pi} \int [(u' \ ^2 \vert_S ) + (u^2 \vert_S) ] \ d \sigma. {} \end{aligned} $$
(89)
However, given the result of Remark 2 and considering the way in which it has been proved, the equivalence of Eq. 89 with Eqs. 43 and 65 is immediate. In any event Eq. 88 is already an answer to our question, and it can be used in a l.s. approximation process, e.g., in combination with ground observations δg = −u′|S. Alternatively one could use the above relation to transform Eq. 88 into
$$\displaystyle \begin{aligned} u_{nm} (R_e) + \frac{1}{4 \ \pi} \int F_{nm} (\sigma) \ \delta g (\sigma) \ d \sigma \equiv \frac{1}{4 \ \pi} \int G_{nm} (\sigma) \ (u \vert_S) \ d \sigma, {} \end{aligned} $$
(90)
which is again an observation equation for u, maybe in a simpler form. Is has to be underlined however that if we want to use Eq. 90 in a l.s. process in Gauss-Markov sense, one should properly propagate the covariance of the observations, taking into account the cross-covariances between δg and the left hand side of Eq. 90. Whatever Eq. 88 or Eq. 90 is more convenient, depends on the way in which we approximate u. For instance when u is approximated by a global model uN, certainly the use of Eq. 88 is easier; if on the contrary we use boundary elements to approximate u, maybe Eq. 90 could be convenient.

5 Principles of Downward Continuation

In this section we invert the considerations of Sect. 2, since now we want to move downward through the gravity field from an upper surface SU, to a lower surface SL. Given that we assume that there are no masses above SU, so that the exterior set ΩU is by definition a domain of harmonicity of u, the downward continuation is dominated by two principles:
  1. (a)
    as far as we move in the harmonicity domain of u, namely if we assume that SL coincides with or encloses the Earth surface S, the downward continuation (D.C.) operator
    $$\displaystyle \begin{aligned} \mathscr{D} (S_L, S_U) \equiv \mathscr{U}(S_U, S_L)^{-1} {} \end{aligned} $$
    (91)
    is unbounded, but as inverse of a one to one operator, it is one to one too;
     
  2. (b)
    when we dive into the masses there are two alternatives:
    1. (i)

      either we know the mass density ρ, as a function of the position, in which case the D.C. can be reconducted to the rules of (a),

       
    2. (ii)

      as soon as we penetrate an unknown spatial mass distribution, a strong non uniqueness of the D.C. starts, so that we cannot say anymore that there is a D.C. operator, like Eq. 91, but rather there is a whole family of mass distributions that can generate the same field given on SU. In fact in this case the same concept of D.C. is lost.

       
     
We start with the first principle, (a). The existence of the operator \(\mathscr {D} = \mathscr {U}^{-1}\) for the spherical case, as well as its unboundedness has already been commented in Sect. 2. In fact if SU and SL are two spheres with radius respectively RU and RL (RU > RL), inverting Eq. 27 we see that
$$\displaystyle \begin{aligned} u_{nm} (R_L) = u_{nm} (R_U) \bigg( \frac{R_U}{R_L} \bigg)^{n+1}, {} \end{aligned} $$
(92)
which means that the coefficients unm(RL) are uniquely identified by the coefficients unm(RU), although the latter are increasingly magnified with the degree n because \(\frac {R_U}{R_L} > 1\).

If we want to get out of the spherical example, we get the same result (at least for the uniqueness) by recalling the “identity principle” of harmonic functions (see [27], 13).

In fact two harmonic functions that coincide in a neighborhood of a point of their harmonicity domain, coincide everywhere in it. Then if u is given on SU and therefore on ΩU, any harmonic function coinciding with u on SU will also coincide with it in ΩU and therefore everywhere in the harmonicity domain of u; since SL is by hypothesis in such a domain, the D.C. of u from SU to SL is unique, i.e., there exists the operator \(\mathscr {D} = \mathscr {U}^{-1}\). That such operator is unbounded comes from the fact that \(\mathscr {U}\) is compact, although we will not dwell on the proof that can be found in [27], Appendix A.

Remark 6 (uniqueness of the inverse single layer problem with fixed geometry)

Given the discussion of Sect. 2, it is interesting too to know whether given that u can be downward continued to SL we can assume too that the potential u is generated by a single layer supported by SL. The answer is in the positive under general conditions; here however, to simplify matters, we stipulate that SL is fairly smooth and fully contained into the harmonicity domain of u, so that \(u_L = u \vert _{S_L}\) is a smooth function too. In other words we would like to know whether, once u is continued down to SL, the equation
$$\displaystyle \begin{aligned} P \in S_L \ , \qquad u_L(P) = \int_S \frac{\omega (Q)}{l_{PQ}} \ dS_Q. {} \end{aligned} $$
(93)
has a solution for the density ω(Q). We show how to construct ω(Q) from u and the procedure automatically proves existence and uniqueness of the solution. Let us consider that from u we can compute not only uL but also \(\frac {\partial u }{\partial n} \vert _{S_L}\) and this is a smooth function too, given the hypothesis done on SL. So from uL(P), (P ∈ SL) we can find as well the solution of the Dirichlet problem in BL (SL ≡ ∂BL), namely a function uI harmonic inside SL and coinciding with uL on SL. Since uL is smooth, so is uI too; so we can readily compute the function \(\frac {\partial u_I}{\partial n} \vert _{S_L}\). But then, by applying the well known jump relations of potentials of single layers (e.g., see [27], 1.5). We have
$$\displaystyle \begin{aligned} Q \in S_L \qquad \omega(Q) = - \frac{1}{4 \ \pi} \ \bigg[ \frac{\partial u}{\partial n} \bigg \vert_{S_L} (Q) - \frac{\partial u_I}{\partial n} \bigg \vert_{\overline{S}_L} (Q) \bigg]. {} \end{aligned} $$
(94)
Now such a surface density generates an outer potential, in ΩL, that has a normal derivative on the upper face of SL equal to \(\frac {\partial u}{\partial n}\vert _{S_L}\). But then by the uniqueness of the solution of the exterior Neumann problem (see [22] n. 81) such a potential coincides with u, i.e., it satisfied Eq. 93. Note be taken that in this reasoning SL is fixed. As claimed before, the above reasoning can be generalized to the case that SL is at the boundary of the harmonicity domain of u, so much so that we could roughly summarize the Remark by stating that the inverse problem for a single layer with fixed geometry SL has a unique solution if we are able to downward continue u from SU to SL.

We can come now to the principle (b); the case (i) is particularly simple. In fact if we know the mass distribution between SU and SL, we can apply the Newton integral to it, subtract the so derived potential uT from u, and then apply the rules of principle (a) to the remaining potential v = u − uT, that has now a D.C. down to SL.

This is the solution typically applied in Geodesy to subtract the influence of topographic masses; whence the index T used above.

So we are left with principle (b), the case (ii). Remember that S is the boundary of B, where ρ≠0; if SU is exterior to S, we have already described the operator \(\mathscr {D}\) that downward continues the potential u from SU to S. Since we want to know what happens when we penetrate inside S, to simplify the setting of the problem we can assume that SU ≡ S and u is given directly on such a surface.

Notice that if instead of u|S we assume that δg = −u′|S is given, we can always invoke the GBVP solution to transform such a datum into u.

On the other hand we have to notice that as for the inner surface SL, if we don’t know the mass distribution internal to it, or at least the potential uL, generated by such internal masses on SL (see comments in Remark 1) we are compelled to shrink SL to the origin, i.e., to consider the problem of inverting u|S into the whole B.

So our general setting is as in Fig. 5 in which we mean that SL can be reduced to the origin, namely to disappear as a surface. When SL does not coincide with O, we assume to know the internal density or equivalently that the body BL (internal to SL) is hollow, since we can always subtract from the overall picture the potential generated by the known masses in BL.
Fig. 5

The setting of the problem of inverting u|S into a mass density in the layer \(\mathscr {L}\) and several other ancillary surfaces

So basically we would like to study the solution ρ of the equation
$$\displaystyle \begin{aligned} P \in S \qquad u(P) = G \int_{\mathscr{L}} \frac{\rho(Q)}{l_{PQ}} \ dB_{PQ} \ , {} \end{aligned} $$
(95)
when we know that there exist a \(\overline {\rho } \in L^2 (\mathscr {L})\) that really generates u|S and we have fixed the geometry of \(\mathscr {L}\).

The first comment that we have to do on Eq. 95, is that indeed such an equation has not a unique solution.

The spherical example in particularly clear.

Example 1

Assume S is a sphere of radius R and SL a sphere of radius RL < R; decide further to look for ρ constant in \(\mathscr {L}\). Then (see [27], 1.3) if u(R) = u0, \(\forall \ \rho > \frac {u_0}{G \ \frac {4}{3} \ \pi \ R^2}\) we get a RL
$$\displaystyle \begin{aligned} R_L = \bigg[ R^3 - \frac{u_0 \ R}{G \ \frac{4}{3} \ \pi \ \rho} \bigg]^{\frac{1}{3}} \end{aligned}$$
such that \(u_0 = G \ \frac {4}{3} \ \pi \ \frac {(R^3 - R_L^3)}{R} \ \rho \).

But even more generally we can use Remark 6 to show that masses inside \(\mathscr {L}\) can be shifted without changing the field outside S. In fact if we take a thin layer around S′ (see Fig. 5) and we consider part of the masses in this layer and the associated potential, we can use Remark 7 to show that such a potential can be as well generated by a suitable thin layer around S″ (⊃ S′), at least outside S″ itself. So using repeatedly such “sweeping out” for many layers we can produce quite different density functions that generate the same u outside S.

Given the above reasoning, the task we have is double: to characterize the full set H0 of densities {ρ0} that produce a zero outer potential, namely the null space of the operator of Eq. 95; to show how to construct at least one solution of Eq. 95.

The problem, dating back one century with the works of [12, 20], has been solved long ago under the hypothesis that u is in fact generated by an \(L^2(\mathscr {L})\) density: we quote only [23] and [2]. A more recent and quite general analysis of the problem can be found in [29]. Another slightly different approach can be found in [30]. Here, nevertheless, we follow a more elementary approach, which has the merit to highlight that when we deal with a true layer (\(\mathscr {L} \ne B\)) the minimum L2-norm solution, that we are going to study, belongs to a space more restricted than that of functions harmonic in \(\mathscr {L}\) and, in particular, there are functions harmonic in \(\mathscr {L}\) that can generate a zero outer potential. This is not possible when \(\mathscr {L} \equiv B\), as is the case studied in the quoted literature.

So let us start from the fact that Eq. 95, as a linear equation between two Hilbert spaces, has generally as solution a linear manifold
$$\displaystyle \begin{aligned} \left\{ \begin{array}{ll} \mathscr{S} = \{ \overline{\rho} + \rho_0 \}\\ \rho_0 \in H_0; \quad \int_{\mathscr{L}} \frac{\rho_0 (Q)}{l_{PQ}} \ dB_0 = 0. \end{array} \right. {}\end{aligned} $$
(96)
The two Hilbert spaces in question are precisely
$$\displaystyle \begin{aligned} \rho \in L^2 (\mathscr{L}) \ ; \qquad u(P) \vert_S \in H^{\frac{3}{2}} (S) {}\end{aligned} $$
(97)
The situation is described in more detail in [23] and [29], nevertheless to grasp the idea of \(H^{\frac {3}{2}} (S)\) we suggest the reader to develop the following spherical example.

Example 2

let S be a sphere with radius R, SL a sphere with radius RL < R and \(\mathscr {L}\) the layer between the two. By using the same notation as in Eq. 33, we can write
$$\displaystyle \begin{aligned} u_{nm} (R) = \frac{4 \ \pi \ G}{(2 \ n +1)} \int_{R_L}^{R} \bigg( \frac{r'}{R} \bigg)^n \rho_{nm} (r') \ r' \ ^2 \ dr' {}\end{aligned} $$
(98)
or, by Schwartz inequality, putting \(q = \frac {R_L}{R}< 1\),
$$\displaystyle \begin{aligned} \vert \ u_{nm}(R) \ \vert^2 \le \frac{(4 \ \pi \ G)^2 }{(2 \ n +1)^2} \ \frac{R^4 (1 - q^{2n +3})}{(2 \ n + 3)} \int_{R_L}^{R} \rho_{nm}^2 (r') \ r' \ ^2 \ dr'. {}\end{aligned} $$
(99)
The above inequality is not strict, namely it can become an equality when \(\rho _{nm}(r) = C_{nm} \bigg ( \frac {r}{R} \bigg )^n\), we see that, for some constant C, with the possibility of an equality. So for functions u(R, σ) defined on S if we define the norm in \(H^{\frac {3}{2}}\) by
$$\displaystyle \begin{aligned} \Vert \ u \ \Vert^2_{H^{\frac{3}{2}} (S)} = \sum \vert \ u_{nm} \ \vert^2 (2 \ n +1)^2 (2 \ n +3) {} \end{aligned} $$
(101)
we find that \(\rho \in L^2(\mathscr {L}) \Rightarrow u \in H^{\frac {3}{2}} (S)\).

Remark 7

From the Eq. 101 it is particularly clear that, with an obvious notation,
$$\displaystyle \begin{aligned} L^2(S) \subset H^{\frac{1}{2}} (S) \subset H^1(S) \subset H^{\frac{3}{2}} (S); {} \end{aligned} $$
(102)
each embedding is in fact not only continuous but even compact. Moreover we notice that if \(u \in H^{\frac {3}{2}} (S)\) and \(f \in H^{-\frac {3}{2}} (S)\), namely
$$\displaystyle \begin{aligned} \Vert \ f \ \Vert^2_{H^{- \frac{3}{2}}(S)} = \sum \vert \ f_{nm} \ \vert^2 (2 \ n +1)^{-2} (2 \ n +3)^{-1} < + \infty, {} \end{aligned} $$
(103)
then indeed the following bilinear expression and the subsequent inequality makes sense
$$\displaystyle \begin{aligned} \bigg \vert < f \ , \ u >_{L^2(S)} \bigg \vert^2 = \bigg \vert \sum f_{nm} \ u_{nm} \bigg \vert^2 \le \Vert \ f \ \Vert^2_{H^{- \frac{3}{2}}(S)} \cdot \Vert \ u \ \Vert^2_{H^{\frac{3}{2}}(S)} . {} \end{aligned} $$
(104)
Actually \(f \in H^{- \frac {3}{2}}(S)\) represents a general continuous functional on \(H^{\frac {3}{2}}(S)\), by means of the L2(S) product. The above statements generalize to any surface with a reasonable degree of smoothness, e.g., so called Lipschitz surfaces (see [14] pag 96).
At this point, let us notice that Eq. 95 can be written in the form
$$\displaystyle \begin{aligned} G < \frac{1}{l_{PQ}} \ , \ \rho >_{L^2(\mathscr{L})} = u(P) \ ; \qquad \forall \ P \in S {} \end{aligned} $$
(105)
in fact, ∀ P ∈ S, \(\frac {1}{l_{PQ}} \in L^2(\mathscr {L})\), since, calling Δ the diameter of S,
$$\displaystyle \begin{aligned} \int_{\mathscr{L}} \ \frac{1}{l_{PQ}^2} dB_Q \le \int d \sigma \int_{0}^{\varDelta} \frac{r^2}{r^2} \ dr \le 4 \ \pi \ \varDelta. {} \end{aligned} $$
(106)
See Fig. 6 for a graphic explanation
Fig. 6

For every P the volume element \(dB_Q = d \sigma \ L^2_{PQ} \ dl\) along the radius in direction eσ(P) runs at most from l = 0 to L = Δ

From Eq. 105 we easily read out that the subspace H0, defined by Eq. 96, is closed in \(L^2(\mathscr {L})\). In fact if ρ0n ∈ H0 and \(\rho _{0 n} \rightarrow \overline {\rho _{0}}\) in \(L^2(\mathscr {L})\) it is obvious that
$$\displaystyle \begin{aligned} \overline{u}_0 (P) = \ < \overline{\rho}_0 \ , \ \frac{1}{l_{PQ}} >_{L^2(\mathscr{L})} = \lim_{n \rightarrow \infty} < \rho_{0 n} \ , \ \frac{1}{l_{PQ}} >_{L^2(\mathscr{L})} \equiv 0, {} \end{aligned} $$
(107)
namely \(\overline {\rho }_0 \in H_0\). So the picture of the set \(\mathscr {S}\) (see Eq. 96) is like that in Fig. 7, with \(\mathscr {S}\) a closed linear manifold in \(L^2(\mathscr {L})\)
Fig. 7

The closed manifold of all solutions of Eq. 95, \(\mathscr {S}\), and the minimum norm solution ρh

In such a situation, when we know that \(\overline {\rho } \in L^2(\mathscr {L})\) exists but we don’t know it, it becomes only natural to look for a geometrically significant element as a representative of \(\mathscr {S}\); this is the minimum norm element \(\rho _{h} \in \mathscr {S}\), which does coincide with the pseudosolution of our problem (look for instance into [7] for the definition of pseudoinverse of a continuous operator with non trivial null space).

That such ρh exists (and is unique) as the orthogonal projection of the origin O, on \(\mathscr {S}\), descends from the fundamental projection theorem in a Hilbert space (see for instance [27], 12.3). Nevertheless we use the knowledge of the existence of ρh, to derive its peculiar shape by the principle of Lagrange multipliers.

Namely we know that ρh, the solution of the principle
$$\displaystyle \begin{aligned} \min_{\rho \in \mathscr{S}} \Vert \ \rho \ \Vert^2 _{L^2(\mathscr{L})}, {} \end{aligned} $$
(108)
is also the unconstrained solution ρλ of the other principle
$$\displaystyle \begin{aligned} \min \frac{1}{2} \ \Vert \ \rho \ \Vert^2 _{L^2(\mathscr{L})} + < \lambda \ , \ \bigg [ u - G \int_{\mathscr{L}} \frac{\rho}{l_{PQ}} \ dB_Q \bigg ] >_{L^2(S)} {} \end{aligned} $$
(109)
on condition that the multiplier λ satisfies the equation
$$\displaystyle \begin{aligned} G \int_{\mathscr{L}} \frac{\rho_{\lambda}}{l_{PQ}} \ dB_Q = u(P) \qquad \qquad P \in S {} \end{aligned} $$
(110)

In Eq. 109 the scalar product \(< \ , \ >_{L^2(S)}\) is meant as a continuous linear functional of the member to the right. Since \(u-G \int _{\mathscr {L}} \frac {\rho _{\lambda }}{l_{PQ}} \ dB_Q\) has to belong to \(H^{\frac {3}{2}}(S)\), thanks to Remark 7 we know that λ, also as a solution of Eq. 110, has to be sought in \(H^{-\frac {3}{2}}(S)\).

The free variation of Eq. 109 with respect to ρ gives
$$\displaystyle \begin{aligned} Q \in \mathscr{L} \qquad \rho_{\lambda}(Q) - G \int_S \frac{\lambda(P)}{l_{PQ}} \ dS_P = 0, {} \end{aligned} $$
(111)
namely ρλ(Q) is a function harmonic in \(\mathscr {L}\), generated by a single layer with density λ(P) deposited only on the upper surface S. Substituting Eq. 111 into Eq. 110 we get the equation for λ
$$\displaystyle \begin{aligned} \left \{ \begin{array}{l l} G \int_S dS_P \ \lambda(P) \ G(P,Q) = u(Q) \qquad Q \in S \\ G(P,Q) = \int_{\mathscr{L}} \frac{1}{l_{PP'}} \ \frac{1}{l_{P'Q}} \ dB_{P'}\\ \end{array} \right. {} \end{aligned} $$
(112)
Recurring to the spherical case, as in Example 5.1, we can easily compute, when S is a sphere of radius R and SL a sphere of radius RL,
$$\displaystyle \begin{aligned} \bigg ( q = \frac{R_L}{R} \bigg) \quad G(P,Q) = 4 \ \pi \ R \sum \frac{(1 - q^{2 \ n +3})}{(2 \ n +1)^2 (2 \ n +3)} \ Y_{nm}(\sigma_P) \ Y_{nm} (\sigma_Q) {} \end{aligned} $$
(113)
showing clearly that G is a strictly positive definite kernel, with power strength N = 3. This is compliant with the fact that if \(u \in H^{\frac {3}{2}} (S)\), then \(\lambda \in H^{- \frac {3}{2}} (S)\).
Again by using our spherical harmonic representation we can further compute, from Eq. 111, the
$$\displaystyle \begin{aligned} \Vert \ \rho_{\lambda} \ \Vert^2_{L^2(\mathscr{L})} = \frac{4 \ \pi}{R^2} \sum \lambda_{nm}^2 \ \frac{1}{(2 \ n +1)^2 (2 \ n +3)} <+ \infty {} \end{aligned} $$
(114)
showing that the solution ρh = ρλ so found does belong to \(L^2(\mathscr {L})\) as it should.
At this point we have a coherent scheme for the solution of the inverse gravimetric problem for the layer \(\mathscr {L}\):
  • we are given the potential u|S and we assume it to belongs to \(H^{\frac {3}{2}}(S) \),

  • then we solve for λ Eq. 112 and we know that there is one and only one solution in \(H^{- \frac {3}{2}}(S)\),

  • finally we compute the minimum norm solution ρh = ρλ from Eq. 111, obtaining in fact an \(L^2(\mathscr {L})\) bounded function; ρh results to be equal to the harmonic potential of a single layer on S; we call
    $$\displaystyle \begin{aligned} H_{h} = \bigg \{ \rho_{h} = \int \frac{\lambda}{l_{PQ}} \ dB_Q \qquad \lambda \in H^{- \frac{3}{2}} (S) \bigg \}; {} \end{aligned} $$
    (115)
    it is obvious that Hh is a closed subspace of \(L^2(\mathscr {L})\),
  • the space H0, such that \(\mathscr {S} = \{ H_{h} \bigoplus H_{0} \}\), is just the orthogonal complement of Hh in \(L^2(\mathscr {L})\).

Two remarks are in order to conclude the above analysis.

Remark 8

The subspace Hh is not the full subspace of functions in \(L^2(\mathscr {L})\), also harmonic in \(\mathscr {L}\), that we call \(HL^2(\mathscr {L})\). In fact this full subspace can be proved to be generated by the sum of all single layer potentials with density on S plus all single layer potentials with density on SL.

If \(H_{h} \subset H L^2 (\mathscr {L})\) strictly then we should be able to find a harmonic function h0 which is also in H0, namely that generates a zero outer potential; in other words \(H L^2(\mathscr {L}) \cap H_0 \ne 0\). For the usual spherical example, this is the case if one takes
$$\displaystyle \begin{aligned} \rho = h_0 = \frac{3}{2} \ \frac{1}{R} \ \frac{1-q^2}{1 - q^3} - \frac{1}{r} \quad , \qquad R_L \le r \le R, \end{aligned}$$
as the reader is invited to verify.

Remark 9

When we have
$$\displaystyle \begin{aligned} \mathscr{L} \equiv B, \end{aligned}$$
i.e., the inversion domain is the whole body B, we have one further characterization of H0, which has proved to be useful in many instances.
First of all we observe that if we eliminate the internal surface SL, then the densities of the form of Eq. 111 with \(\lambda \in H^{- \frac {3}{2}}(S)\) are in fact spanning HL2(B), namely Hh ≡ HL2(B). In this case therefore H0 is characterized by functions ρ0 such that
$$\displaystyle \begin{aligned} < \rho_0 , h >_{L^2(B)} = 0 \ , \ \forall \ h \in HL^2(B). {} \end{aligned} $$
(116)
To proceed further, notice that ∀ ρ0 ∈ H0 ⊂ L2(B) we can define a φ such that
$$\displaystyle \begin{aligned} \left \{ \begin{array}{l} \varDelta \varphi = \rho_0 \\ \varphi \vert_S = 0\\ \end{array} \right. {} \end{aligned} $$
(117)
It turns out that the solution of Eq. 117 is a φ ∈ H2, 2(B). But in this case one must have form of Eq. 116 and from Δh = 0,
$$\displaystyle \begin{aligned} \int_B \rho_0 \ h \ dB_P = \int_B \varDelta \varphi \ h \ dB_P = \int_S \frac{\partial \varphi}{\partial n} \ h \ dS_P = 0; {} \end{aligned} $$
(118)
since Eq. 118 has to hold ∀ h ∈ HL2(B) one may conclude that it has to be too
$$\displaystyle \begin{aligned} \frac{\partial \varphi}{\partial n} \bigg \vert_S \equiv 0 {} \end{aligned} $$
(119)
our statement then becomes:
$$\displaystyle \begin{aligned} \rho_0 \in H_0 \Longleftrightarrow \rho_0 = \varDelta \varphi \ ; \ \varphi \in H_0^{2,2} \equiv \{ \varphi \in H^{2,2} \ , \ \varphi \vert_S = 0 \ \varphi_n \vert_S = 0 \} {} \end{aligned} $$
(120)
i.e., ρ0 ∈ H0 if it is the laplacian of a function square integrable together with its second derivatives, and such that it goes to zero, together with its normal derivative at the boundary S. Not such a similar characterization is possible when \(\mathscr {L} \subset B\).

6 The Constant Density Layer with Unknown Geometry

In the previous section we have analyzed the inversion of the Newton operator, with a family of densities supported by a layer, \(\mathscr {L} \equiv \{ R_{L \sigma } \le r \le R_{\sigma } \}\), contiguous to the known outer surface S ≡{r = Rσ}, that generate the same potential u in Ω. By hypothesis the shape of \(\mathscr {L}\), namely S and SL, was supposed to be given while the density ρ was considered as unknown. In this section we will discuss the inversion of Newton’s operator, by exchanging the above hypotheses; namely we shall assume that ρ is a given function of the point Q ∈ B and viceversa the layer surface SL, cutting the domain where ρ has to be considered different from zero, is unknown.

For the sake of simplicity we will assume that ρ is constant, although for instance some ρ depending on r in the form of a decreasing function could be treated very much in the same way (see for instance [9]). In any event the problem of one layer only, is less cumbersome than one might believe, from the geophysical point of view. As a matter of fact this is an idealization of the so called Moho problem. On such items one can consult [13, 17].

Although this is not completely exact, we will consider the Moho as the boundary between crust and mantle. If we idealize the interior of the Earth as a distribution of layers of constant density like in Fig. 8, and if we further assume that the layer interfaces are known, we could write:
$$\displaystyle \begin{aligned} \begin{aligned} u(P) \vert_S &= G \ \rho_c \int d \sigma' \int_{0}^{R_{C \sigma'}} \frac{r^2 \ dr}{l_{PQ}} + G \ \rho_{M} \int d \sigma' \int_{R_{C \sigma'}}^{R_{M \sigma'}} \frac{r^2 \ dr}{l_{PQ}} +\\ &\quad + G \ \rho_{Cr} \int d \sigma' \int_{R_{M \sigma'}}^{R_{\sigma'}} \frac{r^2 \ dr }{l_{PQ}}. {} \end{aligned} \end{aligned} $$
(121)
Fig. 8

A planetary model with 3 layers of constant density, core, mantel and crust: interfaces are star-shaped

Equation 121 can be rearranged as
$$\displaystyle \begin{aligned} \begin{aligned} u(P) \vert_S &= G \ \rho_{C} \int d \sigma' \int_{0}^{R_{C \sigma '}} \frac{r^2 \ dr}{l_{PQ}} + G \ \rho_{M} \int d \sigma ' \int_{R_{C \sigma'}}^{R_{\sigma'}} \frac{r^2 \ dr}{l_{PQ}} + \\ &\quad - G \ (\rho_M - \rho_{Cr}) \int d \sigma' \int_{R_{M \sigma'}}^{R_{\sigma'}} \frac{r^2 \ dr}{l_{PQ}} \end{aligned} {} \end{aligned} $$
(122)
and finally
$$\displaystyle \begin{aligned} \begin{aligned} G \ (\rho_{M} - \rho_{Cr}) \int d \sigma' \int_{R_{M \sigma'}}^{R_{\sigma'}} \frac{r^2 \ dr}{l_{PQ}} &= - u(P) + G \ \rho_{C} \int d \sigma' \int_{0}^{R_{C \sigma'}} \frac{r^2 \ dr}{l_{PQ}} + \\ &\quad + G \ \rho_M \int d \sigma' \int_{R_{C \sigma'}}^{R_{\sigma'}} \frac{r^2 \ r}{l_{PQ}} \equiv f(\sigma). \end{aligned} {} \end{aligned} $$
(123)

Equation 123 shows the relation existing between a known function f(σ), a known constant density ρM − ρCr, that in the sequel we will denote simply as ρ, and the unknown function r = R ≡ R, representing the Moho surface. The above model can be indeed complicated with many layers as far as we keep the hypothesis that all the other interfaces, beyond R, and all the constant densities of the layers are known.

One might argue that indeed the interfaces of the various layers in the Earth interior are not precisely known, nor the density of the layers are really constant. Nevertheless it is also known that such perturbations of our simplistic model, do have a visible effect on a global gravity model, only in low degrees, so that one could think that considering the model only for degrees higher than a certain Lmax one could consider it mainly due to the effects of the first layer, the crust. Naturally it is unrealistic to think of cutting a global model precisely in two at degree Lmax, nevertheless there are arguments saying that if we choose Lmax somewhere between 36 and 72, we get a model for which Eq. 123 becomes sufficiently representative to make of its solution an interesting geophysical problem.

So we are left with the task of solving the equation
$$\displaystyle \begin{aligned} G \ \rho \int_{\mathscr{L}} \frac{dB_{Q}}{l_{PQ}} = f(\sigma) \qquad P \in S \ , \ \mathscr{L} \equiv \{ S, S_L \} {} \end{aligned} $$
(124)
where the unknown is SL.

Notice that Eq. 124 is slightly more general than Eq. 123 in that in Eq. 124 we don’t need to make the hypothesis that SL is star-shaped, although in the sequel we will always accept such a restriction, in particular to simplify the result concerning the uniqueness of the solution.

So we write Eq. 124 in the form
$$\displaystyle \begin{aligned} Q \equiv (r, \sigma') \ , \qquad G \ \rho \int d \sigma' \int_{R_{L \sigma}}^{R_{\sigma}} \frac{r^2 \ dr }{l_{PQ}} = f(\sigma) {} \end{aligned} $$
(125)
and we would like to know whether solving such an equation constitutes a properly posed problem or not and whether its solution is unique or not. We will respond to the first question on the ill-posedness, so that once we have stated that the solution is unstable it is more important to know whether in any way it is unique or not. In fact for an improperly posed problem it makes little sense to try to prove conditions for the existence of the solution, because known data will not in general satisfy these conditions, but we would rather like to know whether some approximation method, like Tikhonov regularization (see [7]), will provide a quasi solution that ultimately, when the perturbation of data is going to zero, converges to the exact solution.

We proceed to the analysis of Eq. 125 in two steps: first we will consider its linearized version, then we will derive some results for the non linear form. At least we will discuss the most appropriate form of a Tikhonov principle for the Eq. 124.

So assume that we know an approximate solution \(\tilde {R}_{\sigma }\) of Eq. 125, in the sense that if we compute
$$\displaystyle \begin{aligned} \tilde{f} (\sigma) = G \ \rho \int d \sigma' \int_{\tilde{R}_{\sigma'}}^{R_{\sigma'}} \frac{r^2 \ dr}{l_{PQ}} {} \end{aligned} $$
(126)
we find an \(\tilde {f}(\sigma )\) that we consider close to f(σ). Then, assuming that
$$\displaystyle \begin{aligned} R_{L \sigma} = \tilde{R}_{\sigma} + \delta R_{\sigma} {} \end{aligned} $$
(127)
we develop to the first order the integral of Eq. 125, namely we put
$$\displaystyle \begin{aligned} \int d \sigma' \int_{R_{L \sigma '}}^{R_{\sigma '}} \frac{r^2 \ dr}{l_{PQ}}\cong \int d \sigma' \int_{\tilde{R}_{\sigma'}}^{R_{\sigma'}} \frac{r^2 \ dr}{l_{PQ}} + \int d \sigma' \frac{\tilde{R}_{\sigma '}^2 \delta R_{\sigma'}}{l_{P \tilde{Q}}} \ , \ \quad \tilde{Q} = (\tilde{R}_{\sigma'}, \sigma') {} \end{aligned} $$
(128)
From the geodetic point of view this is nothing more than mass coating, namely squeezing along the radius the masses in the layer \((\tilde {S}\ , \ S_L )\) onto the surface \(\tilde {S}\); from the mathematical point of view this is computing the Gâteaux differential which is in fact coinciding with the Frechet differential if the hypothesis that
$$\displaystyle \begin{aligned} Dist (\tilde{S} , S) = \min_{Q \in \tilde{S}, P \in S} \overline{PQ} = \delta > 0 {} \end{aligned} $$
(129)
is satisfied.
If we recall that
$$\displaystyle \begin{aligned} cos \tilde{I}_{\sigma'} \ d \tilde{S}_{Q} = \tilde{R}_{\sigma'} \ d \sigma' \qquad \quad (cos \tilde{I}_{\sigma '} = {\mathbf{e}}_{\sigma '} \cdot {\tilde{\textbf n}}_{\sigma '}) \end{aligned}$$
and we put
$$\displaystyle \begin{aligned} \omega (\sigma ') = cos \tilde{I}_{\sigma'} \ \delta \tilde{R}_{\sigma'} {} \end{aligned} $$
(130)
we see that Eq. 125 combined with Eq. 128, can be written as
$$\displaystyle \begin{aligned} \delta f (\sigma) = f (\sigma) - \tilde{f}(\sigma) = G \ \rho \int_{\tilde{S}} d \tilde{S}_{Q} \frac{\omega(\tilde{Q})}{\tilde{l}_{P \tilde{Q}}} \ , \qquad (P \in S \ , \ \tilde{Q} \in \tilde{S}). {} \end{aligned} $$
(131)
But this is exactly the problem of inverting a single layer with fixed geometry, upward continued from \(\tilde {S}\) to S. This problem has already been discussed in Sect. 5 and we have stated that it is improperly posed, but that the solution ω(σ′), when it exists, is unique (see Remark 6).
If we further assume that
$$\displaystyle \begin{aligned} cos \tilde{I} \ge C \ge 0 \ , {} \end{aligned} $$
(132)
thanks to Eq. 130 we can conclude that \(\delta R_{\sigma '}\) is also the unique solution of the linearized problem (Eq. 128).

Naturally we have to assume that \(\tilde {R} + \delta R_{L}\) in any case is completely inside S to guarantee a reasonable solution. This concludes the analysis of the linearized problem.

Now we come to the non linear problem (Eq. 125). First of all we would like to know whether also this problem could be transformed into that of inverting a single layer but with unknown geometry. To this aim we set up the following reasoning. Consider that f(σ) is as a matter of fact the trace on the boundary S of a potential v harmonic in Ω,
$$\displaystyle \begin{aligned} v \vert_S = f(\sigma). \end{aligned}$$
Now by solving the exterior Dirichlet problem, when f(σ) and S are sufficiently smooth, we are in condition to compute v′ and its trace on S; analogously we can compute
$$\displaystyle \begin{aligned} g(\sigma') = (r \ \frac{\partial}{\partial r} - 2) \ u \vert_S. {} \end{aligned} $$
(133)
On the other hand the following identity holds
$$\displaystyle \begin{aligned} r \ \frac{\partial}{\partial r} \ \frac{1}{l_{PQ}} - \frac{2}{l_{PQ}} = \frac{1}{s^2} \ \frac{\partial}{\partial s} \ \frac{s^3}{L_{PQ}} \ , {} \end{aligned} $$
(134)
where
$$\displaystyle \begin{aligned} \left\{ \begin{array}{l} P \equiv (r, \sigma) \ , \ Q \equiv (s, \sigma') \ , \ cos \psi = {\mathbf{e}}_{\sigma} \cdot {\mathbf{e}}_{\sigma'} \\ l_{PQ} = [ r' + s^2 - 2 \ r \ s \ cos \psi ]^{\frac{1}{2}} \\ \end{array} \right. {} \end{aligned} $$
(135)
Therefore it is
$$\displaystyle \begin{aligned} \begin{aligned} g(\sigma) &= \bigg ( r \ \frac{\partial}{\partial r} - 2 \bigg ) \ G \ \rho \int d \sigma' \int_{R_{L \sigma'}}^{R_{\sigma'}} \frac{s^2 \ ds}{l_{PQ}} \bigg \vert_{r = R_{\sigma}} = \\ &= G \ \rho \int d \sigma' \int_{R_{L \sigma'}}^{R_{\sigma'}} \frac{\partial}{\partial s} \bigg ( \frac{s^3}{l_{PQ}} \bigg ) ds \bigg \vert_{r = R_{\sigma}} = \\ &= G \ \rho \int d \sigma' \bigg ( \frac{R_{\sigma}^3}{l_{PQ_e}} - \frac{R^3_{L \sigma'}}{l_{PQ_L}}\bigg) \end{aligned} {} \end{aligned} $$
(136)
with
$$\displaystyle \begin{aligned} Q_e = (R_{\sigma'}, \sigma') \ , \ Q_L = (R_{L \sigma'}, \sigma'). {} \end{aligned} $$
(137)
Equation 136 can be arranged as
$$\displaystyle \begin{aligned} G \ \rho \int d \sigma' \ \frac{R_{L \sigma'}^3}{l_{PQ_L}} = G \ \rho \int d \sigma' \ \frac{R_{\sigma'}^3}{l_{PQ_e}} - g(\sigma) = \overline{g}(\sigma). {} \end{aligned} $$
(138)
This is a non linear Fredholm equation of the first kind, with R as unknown and \(\overline {g}(\sigma )\) as known term. To see its relation to a single layer potential it is enough to recall that \(dS_L \ cos I_L = R_{L \sigma '}^2 d \sigma '\), to realize that Eq. 138 can be written in the form
$$\displaystyle \begin{aligned} G \ \rho \int_{S_L} d S_L \ \frac{cos I _{L \sigma'} \ R_{L \sigma'}}{l_{PQ_L}} = \overline{g}(\sigma), {} \end{aligned} $$
(139)
showing that \(\overline {g}(\sigma )\) is the boundary value (on S) of the potential of a single layer with unknown geometry (SL), with density \(\omega (\sigma ') = \rho \ cos I_{L \sigma '} R_{L \sigma '}\), upward continued from SL to S. As always the upward continuation implies a smoothing with exponential strength, so solving Eq. 139 is a (strongly) improperly posed non-linear problem.
As for the uniqueness of the solution we return to the problem in the form of Eq. 123 that we rewrite
$$\displaystyle \begin{aligned} G \ \rho \int_{\mathscr{L}} \frac{dB_Q}{l_{PQ}} = \delta f(\sigma) \ , \quad P \equiv (R_\sigma , \sigma) \ , \quad \mathscr{L} \equiv \{ R_{L \sigma} \le r \le R_{\sigma} \}. {} \end{aligned} $$
(140)

The first proof of uniqueness of the solution of Eq. 140 is an old one, due to P.S. Novikov, dating back to 1938 (see [9], 3.1).

Many years later one of the authors of this paper, unaware of Novikov’s work, has found exactly the same proof, as one can read in [3]. The authors are glad to restore the correct attribution of the proof.

Here we will follow a different path, taking advantage of our Eq. 136 and of Remark 6.

Assume to have two solutions R = R1σ , R = R2σ producing the same potential, for the respective layers \(\mathscr {L}_1\), \(\mathscr {L}_2\), on the surface S
$$\displaystyle \begin{aligned} u_1(P) = G \ \rho \int_{\mathscr{L}_1} \frac{dB_Q}{l_{PQ}} \equiv u_2(P) = G \ \rho \int_{\mathscr{L}_2} \frac{dB_Q}{l_{PQ}} \ , \ P \in S. {} \end{aligned} $$
(141)
But then, by the unique continuation principle
$$\displaystyle \begin{aligned} u(P) = u_2 (P) - u_1 (P) \equiv 0 {} \end{aligned} $$
(142)
for every P in the common domain of harmonicity.
So if we define
$$\displaystyle \begin{aligned} R_{+ \sigma} = \max \ R_{1 \sigma} \ , \ R_{2 \sigma} \ , {} \end{aligned} $$
(143)
and correspondingly Ω+, B+ the exterior and the interior of S+ ≡{r = R+σ} (see Fig. 9), Eq. 142 has to hold in P ∈ Ω+.
Fig. 9

In this figure are represented the lower surfaces S1, S2 (continuous lines) of the two layers \(\mathscr {L}_1\), \(\mathscr {L}_2\) that generate the same potential (on and outside) S; the differential body δB, segmented into subset δBk where the density is + ρ or − ρ; the external and internal boundaries S± (dashed lines) of δB. Notice that the number of δBk could even be + 

If we call the differential body δB,
$$\displaystyle \begin{aligned} \delta B = \mathscr{L}_1 \div \mathscr{L}_2 = (\mathscr{L}_{1} \cup \mathscr{L}_2) \ominus (\mathscr{L}_{1} \cap \mathscr{L}_2) {} \end{aligned} $$
(144)
and S± its boundaries
$$\displaystyle \begin{aligned} \left\{ \begin{array}{l} S_{+} \equiv \{ r = R_{+ \sigma} \} \ , \ R_{+ \sigma} = \max R_{1 \sigma} R_{2 \sigma} \\ S_{-} \equiv \{ r = R_{- \sigma} \} \ , \ R_{- \sigma} = \min R_{1 \sigma} R_{2 \sigma} \\ \end{array} \right. {} \end{aligned} $$
(145)
and, referring to Fig. 9,
$$\displaystyle \begin{aligned} \begin{array}{l} \delta B_{k+} = \text{subset of }\delta B \text{ where }R_{1 \sigma} > R_{2 \sigma} \\ \delta B_{k-} = \text{subset of }\delta B \text{ where }R_{1 \sigma} < R_{2 \sigma} \\ \end{array}, \end{aligned}$$
we find that the difference potential (Eq. 141) u(P) can be written as
$$\displaystyle \begin{aligned} \begin{aligned} u(P) &= G \ \rho \int_{\mathscr{L}_1 \div \mathscr{L}_2} \frac{( \pm \rho )}{l_{PQ}} dB = \\ &= G \ \rho \sum_k \int_{\delta B_k} \frac{\pm \rho}{l_{PQ}} dB = \\ &= G \ \rho \bigg ( \sum_k \rho \int_{\delta B_{k+}} \frac{1}{l_{PQ}} dB - \sum_j \rho \int_{\delta B_{j-}} \frac{1}{l_{PQ}} dB \bigg). \end{aligned} {} \end{aligned} $$
(146)
So the density
$$\displaystyle \begin{aligned} \rho_0 (Q) = ( \pm \rho) = \left \{ \begin{array}{l l} \rho \qquad \quad P \in \delta B_{k+} \ , \ \forall k \\ - \rho \qquad P \in \delta B_{j-} \ , \ \forall j \\ \end{array} \right. {} \end{aligned} $$
(147)
must be such as to guarantee a zero Newtonian potential outside S+, i.e., in Ω+.
We observe that from the definition of Eq. 147 ρ0(Q) = ±ρ depends basically on σ but not on r, so that inside δB the identity holds:
$$\displaystyle \begin{aligned} \frac{\partial \rho (Q)}{\partial r} \equiv 0 \qquad \quad Q \in \partial B. {} \end{aligned} $$
(148)
Now we want to return for a minute to our formula of Eq. 136. It is obvious from the way in which it has been derived, that if ρ is a function of σ; it can be modified taking into account that here the layer is between S and S+, as
$$\displaystyle \begin{aligned} \bigg ( r \ \frac{\partial }{\partial r} - 2 \bigg) \ u(P) \equiv G \int d \sigma' \ \rho (\sigma' ) \bigg [ \frac{R_{+ \sigma'}^3}{l_{PQ_+}} - \frac{R_{- \sigma'}^3}{l_{PQ_-}} \bigg] {} \end{aligned} $$
(149)
$$\displaystyle \begin{aligned} P \equiv (R_{+ \sigma}, \sigma) \in S_+ \ , \quad Q_- \equiv (R_{- \sigma'}) \in S_- \ , \quad Q_+ \equiv (R_{+ \sigma'}, \sigma') \in S+. \end{aligned}$$

On the other hand u(P) is continuous with its first derivatives in Ω+S+, because it is generated by a bounded mass density in δB, i.e., | ρ | = ρ.

So we have
$$\displaystyle \begin{aligned} u(P) \equiv 0 \qquad \quad \frac{\partial u}{\partial r} (P) \equiv 0 \ , \qquad \quad P \in S_+ {} \end{aligned} $$
(150)
and Eq. 149 becomes
$$\displaystyle \begin{aligned} P \in S_+ \qquad \quad 0 \equiv \int d \sigma' \ \rho (\sigma') \bigg [ \frac{R_{+ \sigma'}^3}{l_{PQ_+}} - \frac{R_{- \sigma'}^3}{l_{PQ}} \bigg]. {} \end{aligned} $$
(151)
Now we multiply Eq. 151 by some function ω(P) and integrate on S+; moreover we denote
$$\displaystyle \begin{aligned} v(Q) = \int dS_P \ \frac{\omega (P)}{l_{PQ}} \qquad \quad Q \in S_+ \cup B {} \end{aligned} $$
(152)
and we note that as a single layer on S+, v(P) is a harmonic function in B. So when Q ∈ S+, v(Q+) is just the boundary value on S+ of a function harmonic in B.
We have already discussed in Remark 6 that the correspondence between \(v \vert _{S_{+}}\) and ω is one to one and we have shown how to derive one from the other. So we can rewrite Eq. 151 as
$$\displaystyle \begin{aligned} 0 \equiv \int d \sigma' \ \rho (\sigma') \bigg [ R_{+ \sigma'}^3 \ v(Q_+) - R_{- \sigma'} \ v(Q_-) \bigg] \ , {} \end{aligned} $$
(153)
having at our disposal the trace of v on S+. So we can choose, with the same notation as in Eq. 147
$$\displaystyle \begin{aligned} v(Q_+) = ( \pm 1) = \left \{ \begin{array}{l l} 1 \qquad \quad Q_+ \in \delta B_{k+} \ , \ \forall k \\ -1 \qquad Q_+ \in \delta B_{j-} \ , \ \forall j \\ \end{array} \right. {} \end{aligned} $$
(154)
and we find, since v(Q+) ρ(σ′) = ρ,
$$\displaystyle \begin{aligned} 0 \equiv \int d \sigma' \bigg [ R_{+ \sigma'}^3 \ \rho - R_{- \sigma'}^3 \ \rho (\sigma') v(Q_-) \bigg ]. {} \end{aligned} $$
(155)
On the other hand, by the maximum principle that all harmonic functions satisfy,
$$\displaystyle \begin{aligned} \vert \ v (Q_-) \ \vert \le \sup \vert \ v (Q_+) \ \vert = 1 {} \end{aligned} $$
(156)
and so
$$\displaystyle \begin{aligned} - \rho (\sigma') \ v (Q_-) \geqq - \rho. {} \end{aligned} $$
(157)
Therefore Eq. 155 now becomes
$$\displaystyle \begin{aligned} 0 \geqq \int d \sigma' \ \rho \bigg [ R_{+ \sigma'}^3 - R_{- \sigma'}^3 \bigg ] \ , \ \rho > 0. {} \end{aligned} $$
(158)
Since
$$\displaystyle \begin{aligned} R_{+ \sigma'} \geqq R_{- \sigma'} \ , \end{aligned} $$
(159)
Eq. 158 can be true only if
$$\displaystyle \begin{aligned} R_{+ \sigma'} \equiv R_{- \sigma'} \qquad \quad \sigma \ a. \ e. {} \end{aligned} $$
(160)
On the other hand
$$\displaystyle \begin{aligned} \vert \ R_{1 \sigma'} - R_{2 \sigma'} \ \vert = \vert \ R_{+ \sigma'} - R_{- \sigma'} \ \vert {} \end{aligned} $$
(161)
so Eq. 160 implies that \(R_{1 \sigma '}\) and \(R_{2 \sigma '}\) are equal almost everywhere, i.e., SL1 ≡ SL2, namely uniqueness is proved.
Now that we have achieved a uniqueness result, we can pass to discuss the application of some regularization method to get an approximate solution of Eq. 124. Instead of the original Eq. 124, we can use the equivalent formulation of Eq. 138, that we write as (recall, \(P \in (R_{\sigma }, \sigma ) \ , \ Q_L = (R_{L \sigma '}, \sigma ')\))
$$\displaystyle \begin{aligned} \left \{ \begin{array}{l l} F [ R_L \ ; \ \sigma ] = \overline{g} (\sigma ) \\ F [ R_L \ ; \ \sigma ] = G \ \rho \int d \sigma' \ \frac{R_{L \sigma'}^3}{l_{PQ_L}}.\\ \end{array} \right. {} \end{aligned} $$
(162)
A classical regularization of Eq. 162 has the form of the Tikhonov variational principle, (see [7]), namely we search RL in such a way that the minimum of the functional
$$\displaystyle \begin{aligned} T (R_L) = \Vert \ \overline{g} - F [ R_L ] \ \Vert^2_{L^2 (\sigma)} + \lambda J (R_L) {} \end{aligned} $$
(163)
is achieved. Usually the regularizing term in T(RL) is taken to be the square of a norm or a semi-norm like
$$\displaystyle \begin{aligned} J_0 (R_L) = \int d \sigma \ R_{L \sigma}^2 {} \end{aligned} $$
(164)
$$\displaystyle \begin{aligned} J_1 (R_L) = \int d \sigma \vert \ \nabla_{\sigma} R_{L \sigma} \ \vert^2 {} \end{aligned} $$
(165)
$$\displaystyle \begin{aligned} J_2 (R_L) = \int d \sigma ( \varDelta_{\sigma} \ R_{L \sigma} )^2. {} \end{aligned} $$
(166)

While J0(RL) is nothing but the L2(σ) norm, J1(RL), J2(RL) are semi-norms, i.e., they enjoy all the properties of norms, but for strict positivity, namely \(J_2 (R_L) = 0 \nRightarrow R_L = 0\) and the same is true from J1(RL).

Moreover the set C of admissible solutions R, in our case is quite naturally
$$\displaystyle \begin{aligned} 0 \le R_{L \sigma} \le R_{\sigma} \ , {} \end{aligned} $$
(167)
because the surface SL should be wrapped by S. Indeed T(RL) defined as in Eq. 163 is never negative, so we can always define
$$\displaystyle \begin{aligned} m = \inf_{R_L \in C} T (R_L) {} \end{aligned} $$
(168)
and it will be m ≥ 0. Nevertheless to see whether m is a real minimum of T on C, i.e., whether there is a \(\overline {R}_L \in C\) such that
$$\displaystyle \begin{aligned} m = T (\overline{R}_L) \ , {} \end{aligned} $$
(169)
is a different story requiring a deeper mathematical analysis.
We list here sufficient conditions for m to be a true minimum:
  1. (1)

    C has to be bounded and closed in H,

     
  2. (2)

    F[RL] has to be continuous in H,

     
  3. (3)
    the sets
    $$\displaystyle \begin{aligned} C \cap \{ T(R_L) \le a \} \ , {} \end{aligned} $$
    (170)
    a > m, have to be compact in C, i.e., ∀ {RLn ∈ C} we can find a subsequence \(\{ R_{L n_k} \}\) that is convergent in H, indeed the limit of \(R_{L n_{k}}\) belongs to C because of (1),
     
  4. (4)
    J(RL) has to be lower semi-continuous, i.e.,
    $$\displaystyle \begin{aligned} R_{L n }{\overrightarrow{H}} \overline{R}_L \Rightarrow J (\overline{R}_L) \le \lim J(R_{Ln}). \end{aligned}$$
     

That such conditions are satisfied in our case, is a problem examined in detail in the Appendix B, concentrating on the choice of J1(RL) as a regularizer. In fact we can argue that J0(RL) is too weak, for our purposes; in other words solutions with this choice can oscillate too much. In particular condition (3) is not satisfied because spheres in L2(σ) are certainly not compact, nor are their intersections with C.

As for the choice of J2(RL), which is however successfully present in literature (see [21], 4.2), we consider it too strong; in other words solutions, with such a choice, become too smooth and loose the kind of look that is expected for geological interfaces. So we shall concentrate on the classical choice of J1(RL).

To have a pictorial perception of the above discussion we provide in Fig. 10 an example of bounded surfaces which satisfy J0 < a, J1 < a, J2 < a. Once the existence of the minimum m is granted, we would also like to know how to compute the solution \(\overline {R}_L\).This can be done by direct methods, namely using the variational principle (Eq. 168) or by solving, with some discretization technique, the corresponding Euler equation.
Fig. 10

Examples of functions f(σ) which satisfy respectively f ∈ L2(σ), ∇f ∈ L2(σ), Δf ∈ L2(σ); in case (a) f can make jumps, (b) n exists a.e. but can make jumps, (c) curvature exists a.e. but can make jumps

Observing that the following two variational relations holds:
$$\displaystyle \begin{aligned} \left \{ \begin{array}{l l} \delta F [ R_L \ ; \ \sigma ] = \int K[ (R_L ; \sigma, \sigma' ] \ \delta R_{L \sigma'} \ d \sigma' \\ K [R_L ; \sigma, \sigma'] = R_{L \sigma'}^2 \ \frac{2 \ R_{L \sigma'}^2 + 3 R_{\sigma}^2 - 5 R_{L \sigma'} R_{\sigma} cos \psi}{l_{PQ}^3},\\ \end{array} \right. {} \end{aligned} $$
(171)
with
$$\displaystyle \begin{aligned} P \equiv (R_{\sigma}, \sigma) \ , \ Q \equiv (R_{\sigma'}, \sigma') \ , \ cos \psi = \underline{e}_{\sigma} \cdot \underline{e}_{\sigma'}; \end{aligned}$$
$$\displaystyle \begin{aligned} \begin{aligned} \delta \int \vert \ \nabla_{\sigma'} R_{L \sigma'} \ \vert^2 d \sigma' &= 2 \int \nabla_{\sigma'} R_{L \sigma'} \cdot \nabla_{\sigma'} \delta R_{L \sigma'} d \sigma' = \\ &= - 2 \int (\varDelta_{\sigma'} R_{L \sigma'}) \delta R_{L \sigma'} d \sigma', \end{aligned} {} \end{aligned} $$
(172)
it is not difficult to derive the Euler equation from Eq. 168, with J(RL) ≡ J1(RL), namely
$$\displaystyle \begin{aligned} \int d \sigma F [ R_L; \sigma ] \ K [ R_L; \sigma, \sigma' ] - \lambda \ \varDelta_{\sigma'} R_{L \sigma'} = \int d \sigma \ \overline{g}(\sigma) \ K [R_L; \sigma, \sigma']. {} \end{aligned} $$
(173)

Remark 10

It is interesting to try to linearize Eq. 173, to get perturbation equations for a spherical layer, namely setting
$$\displaystyle \begin{aligned} R_{\sigma} \equiv R (\text{const}) \qquad \quad R_{L \sigma} = R_0 + \delta R_{L \sigma} \ \quad (R_0 = \text{const}). {} \end{aligned} $$
(174)
Without entering into the detailed computation we observe that with this spherical symmetry, calling
$$\displaystyle \begin{aligned} l_{0} = [R^2 + R_0^2 - 2 \ R \ R_0 \ cos \psi]^{\frac{1}{2}}, {} \end{aligned} $$
(175)
one has
$$\displaystyle \begin{aligned} F [ R_0; \sigma ] &= G \ \rho R_0^3 \int \frac{d \sigma'}{l_0} = 4 \ \pi \ G \ \rho \frac{R_0^3}{R} \ , {} \end{aligned} $$
(176)
$$\displaystyle \begin{aligned} K [ R_0; \sigma, \sigma' ]& = \frac{R_0^2 (2 \ R_0^2 + 3 \ R^2 - 5 \ R_0 \ R \ cos \psi)}{l_0^3} \ , {} \end{aligned} $$
(177)
$$\displaystyle \begin{aligned} \left \{ \begin{array}{l l} K [R_L; \sigma, \sigma'] \cong K [ R_0; \sigma, \sigma' ] + G(\sigma, \sigma') \ \delta R_{L \sigma'}\\ G(\sigma, \sigma') = \frac{R_0 [2 \ R_0^4 + 5 \ R_0^2 \ R^2 + 6 \ R^4 - 10 \ R_0^3 \ R \ cos \psi - 18 \ R_0 \ R^3 \ cos \psi + 15 \ R_0^2 \ R^2 \ cos^2 \psi]}{l_0^5} \\ \end{array} \right. {} \end{aligned} $$
(178)
so that the linearized version of Eq. 173 reads
$$\displaystyle \begin{aligned} \begin{aligned} &\int d \sigma'' \ \delta R_{L \sigma''} \bigg [ \int d \sigma \ K[ R_0; \sigma, \sigma''] \ K[R_0; \sigma, \sigma'] \bigg ] +\\ &\quad + \bigg [\int d \sigma F [R_0; \sigma] \ G(\sigma, \sigma') \bigg] \ \delta R_{L \sigma'} - \bigg [ \int d \sigma \ \overline{g}(\sigma) \ G (\sigma, \sigma') \bigg] \ \delta R_{L \sigma'} + \\ &\quad + \lambda \ \varDelta_{\sigma'} \ \delta R_{L \sigma'} = \int d \sigma' \ \overline{g}(\sigma) \ K[R_0; \sigma, \sigma'] - \int d \sigma \ F[R_0; \sigma ] \ K [ R_0; \sigma, \sigma']. \end{aligned} {} \end{aligned} $$
(179)
Despite its awful appearance, in reality taking into account Eq. 176 and the definition of \(\overline {g} (\sigma )\), Eq. 138, it turns out that
$$\displaystyle \begin{aligned} \int d \sigma \ \bigg ( F[R_0, \sigma] - \overline{g}(\sigma) \bigg ) \ G (\sigma, \sigma') = \int d \sigma g (\sigma) \ G (\sigma, \sigma') {} \end{aligned} $$
(180)
with
$$\displaystyle \begin{aligned} g(\sigma) = r \ \partial_r \ u - 2 \ u \vert_S. {} \end{aligned} $$
(181)
Since in the left hand side of Eq. 179, such a term multiplies \(\delta R_{L \sigma '}\) , neglecting second order terms, we can substitute g(σ) with the constant
$$\displaystyle \begin{aligned} \overline{g}_0 = (r \ \partial_r - 2) \ G \ \frac{4}{3} \ \pi \ \rho \frac{(R^3 - R_0^3)}{r} \bigg \vert_{r = R} = - 4 \ \pi \ G \ \rho \ \frac{R^3 - R_0^3}{R}. {} \end{aligned} $$
(182)
Moreover we have too
$$\displaystyle \begin{aligned} \int d \sigma \ G (\sigma, \sigma') = G_0 \quad (\text{constant}) {} \end{aligned} $$
(183)
so that the second and third terms put together are just a constant multiplying \(\delta R_{L \sigma '}\). Therefore Eq. 179 becomes an equation, where all integral kernels are isotropic, namely depending on (σ, σ′) only through \(cos \psi _{\sigma \sigma '}\), so that its solution is not difficult if we look for it in terms of a series of spherical harmonics. We leave the exercise to the patient reader.

7 Some Conclusions

The paper has examined, after recurring to spherical examples, how to move up and down through the gravity field.

To move up the theory of Newton’s integral accompanied by the solution of the GBVP has to be applied.

To move down, one finds a unique continuation of the field as far as we stay outside its source, namely the mass density. When we start penetrating the masses, a strong non uniqueness of the downward continuation starts and a choice has to be made in order to guarantee the uniqueness of the solution. The classical cases of the minimum L2-norm solution and that of a layer with constant density, known upper surface and unknown lower surface, are discussed.

In any way when we come to numbers and try to downward continue the gravity field, we always have to apply a regularization method. Some discussion on applying the Tikhonov principle, has been presented, although this is not yet finished since the need to find a regularization stronger that L2 but milder that H1, 2 suggests to perform an analysis in the space of functions of bounded variation. This however will be object of a forthcoming work.

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Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringPolitecnico di MilanoMilanItaly
  2. 2.Department of Civil, Constructional and Environmental EngineeringUniversità di Roma La SapienzaRomeItaly
  3. 3.Department of Civil and Environmental EngineeringPolitecnico di MilanoMilanItaly
  4. 4.Geomatics Research & Development s.r.l.ComoItaly

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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