Handbuch der Geodäsie pp 1-54 | Cite as

# Up and Down Through the Gravity Field

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## 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.

*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

*λ*,

*φ*) spherical angular coordinates,

*R*(

*λ*,

*φ*) a function defined on the unit sphere with a gradient essentially bounded

*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.

*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:

*δg*, that in spherical approximation are related to

*u*by (see [27])

## 2 Principles of Upward Continuation

*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

*L*

^{2}(

*S*

_{r}),

*f*is known to be harmonic in a spherical domain,

*B*

_{R}≡{

*r*≤

*R*}, we recall that Eq. 5 has the form (see [27] Cap. 3,4)

*Ω*

_{R}≡{

*r*≥

*R*} can be expanded into the form

*Y*

_{nm}(

*σ*) are as usual surface spherical harmonics of degree

*n*and order

*m*( [27], Cap. 13.3).

*L*

^{2}(

*σ*) ( [27], Cap. 3.4), namely the space of

*f*(

*σ*) with bounded norm:

To continue it is convenient to introduce a few definitions:

### Definition 1

*L*

^{2}(

*σ*) if

*n*.

### Definition 2

*N*if

### Definition 3

*q*> 1 such that

We notice that, due to Eq. 13, the operator \(\mathscr {S}\) is always bounded in *L*^{2}(*σ*) and even compact (see [32], X.1).

*f*∈

*L*

^{2}(

*σ*) and \(\mathscr {S}\) has power strength

*N*, then all derivatives of

*f*up to order

*N*

*L*

^{2}(

*σ*) and we say that

*f*∈

*H*

^{N, 2}(

*σ*). Here and in the sequel

*H*

^{n, 2}on any set

*A*denotes the usual Sobolev Hilbert spaces of degree

*n*and power 2 (see [14]).

*f*of any order are in

*L*

^{2}(

*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:

*L*

^{2}(

*σ*).

*Δ*around a sphere of radius

*R*

_{0}and linearize the formula with respect to

*Δ*:

*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′*).

*r*>

*R*

_{0}and using the well known representation (see [18], Cap. 1.15)

*r*→

*R*

_{0}, we get the spectral relation

*u*

_{nm}(

*R*

_{0}) are the harmonic coefficients of

*u*(

*R*

_{0},

*σ*). As we see the Newton operator \(\mathscr {N}\) that transform

*ρ*into

*u*at the same level

*R*

_{0}, is a smoother with power strength

*N*= 1, i.e., it transforms a

*ρ*∈

*L*

^{2}(

*σ*) into a potential

*u*∈

*H*

^{1, 2}(

*σ*). This is confirmed by the well known formula:

*ρ*∈

*L*

^{2}(

*σ*), then

*δg*= −

*u′*∈

*L*

^{2}(

*σ*) too.

Let us note that Eq. 24 gives the limit from outside (*r* → *R*_{0+}) 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 *R*_{0} to the level *R* (> *R*_{0}), because we know that it is only on the surface *S*_{R} that we can get information on *u*.

*r*>

*R*

_{0},

*u*has the form of Eq. 21, by using Eq. 23 we find

*δg*leads to

In this case the density coefficients *ρ*_{nm}(*R*_{0}) (or the single layer coefficients *ω*_{nm} = *ρ*_{nm} *Δ*) can still be uniquely retrieved from *δg*_{nm}(*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

*R′*,

*R*

_{0}<

*R′*<

*R*,

*u*from the level

*r*=

*R*

_{0}to the level

*r*=

*R*, Eq. 30 can be written as

*u*(

*R′*,

*σ*), we don’t need anything else to propagate it at higher levels and in particular the relation between the density

*ρ*(

*R*

_{0},

*σ*) at level

*R*

_{0}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

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 *R*_{0} < *R*″ < *R′*.

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

*ρ*is a function varying with

*r*,

*ρ*(

*r*,

*σ*), so that also

*ρ*

_{nm}result to be functions of

*r*,

*ρ*

_{nm}(

*r*), and taking

*Δ*=

*dr*; we get then

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*}.

*R*

_{0}, 0 ≤

*R*

_{0}≤

*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

*u*

_{I}(

*R*

_{0},

*σ*) that is then upward continued up to

*S*

_{R}≡{

*r*=

*R*}, the other one expressed by the Newton integral in spherical form, namely:

Obviously analogous formulas could be worked out for *δg*.

- (a)if we start from a surface
*S*_{0}(in the example the sphere \(S_{R_{0}}\)) and want to bring the gravity field to the level of the surface*S*encompassing*S*_{0}(in the example the sphere*S*_{R},*R*>*R*_{0}), we need to know:- (i)
the potential

*u*_{I}of the masses inside*B*_{0}(*∂B*_{0}=*S*_{0}) on*S*_{0}, or any other boundary data on*S*_{0}that determine uniquely \(u_{I} \vert _{S_{0}}\), like for instance \(\delta g \vert _{S_{0}}\) - (ii)
the mass distribution between

*S*_{0}and*S* - (iii)
that no other masses exist outside

*S*

- (i)
- (b)the gravity field from
*S*outside (in*Ω*) can be obtained:- (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 \(\mathscr {U}(R, R_{0})\)) - (ii)
by computing the Newton integral for the known density

*ρ*between*S*_{0}and*S*(in our example the second term in Eq. 34)

- (i)

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 *f* ∈ *Y* we can find one and only one *u* ∈ *X* 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 \(\overline {\varOmega }\).

*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*) =

*R*

_{σ}.

- (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*ξ*_{D}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 relationAs obvious,$$\displaystyle \begin{aligned} h(\sigma) = \xi_{D}(\sigma) + N(\sigma) {} \end{aligned} $$(36)*h*,*ξ*_{D},*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; - (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*h*_{P}, 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:also known as fundamental equation of physical geodesy;$$\displaystyle \begin{aligned} \varDelta g (P) = g(P) - \gamma(P^{*}) = - \frac{\partial T}{\partial h} - \frac{\frac{\partial \gamma}{\partial h}}{\gamma} \ T {} \end{aligned} $$(38)*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.

*S*

_{0}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].

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

*L*

^{2}(

*S*) for the data

*f*(

*σ*) with norm

*H*

^{1, 2}(

*S*) for

*u*, with norm

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* |^{2}*dS* = 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*.

*Ω*and using the divergence theorem as well as an integration by parts on

*r*between

*R*

_{σ}and +

*∞*, we arrive at the identity (see [8], [26])

*cosI*=

**n**⋅

**e**

_{r}),

*I*is the inclination of the surface

*S*with respect to the main direction of the vertical, approximated here by

**e**_{r},

*R*

_{±}are the max and min of

*R*

_{σ},

*I*

_{+}is the maximum inclination of

*S*which is supposed to be less than 90

^{∘}, so that

As we can see the stability of the solution of this GBVP in *H*^{1, 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

*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

*u*is such, to determine

*v*from its boundary values

*v*(

*r*,

*σ*) by means of a double layer potential, namely

*l*

_{xy}= |

**x**−

**y**|). Then taking the limit for

**x**→

**x**

_{S}≡ (

*R*

_{σ},

*σ*), i.e., approaching the boundary

*S*from outside along the normal to

*S*, one gets the well known relation (see [18], 1-4)

*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:

*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

**e**

_{ψ}):

*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

*J*

_{+}, now becomes:

*J*

_{+}= (

*cosI*

_{+})

^{−1}, shows that Eq. 63 is satisfied if the maximum inclination of the topography

*I*

_{+}, satisfies

## 4 Global Models as Approximate Solutions of the GBVP

*u*

_{N}(

*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)

*R*is just a matter of convenience. In the present context we choose

*R*=

*R*

_{0}, a Bjerhammar radius, namely a radius such that the sphere

*S*

_{0}≡{

*r*=

*R*

_{0}} is totally inside the body

*B*of the Earth (Fig. 4).

In order to say that *u*_{N} is an approximation of *u*, we need a norm to measure the residual *u* − *u*_{N}.

*H*

^{1, 2}(

*S*) in a form essentially equivalent to Eq. 43 namely

*u*∈

*H*

^{1, 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

An analogous notation will be used for functions in *H*^{1, 2}(*S*_{0}) that are also harmonic in *Ω*_{0}.

### Remark 4

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.

*u*

_{N}is an approximative representation of

*u*, in this case in

*H*

^{1, 2}(

*S*), we just need to know

*H*

^{1, 2}(

*S*), denoted

*u*

_{nm}of

*u*

_{N}are judiciously chosen according to some convergent approximation principle.

*H*

^{1, 2}(

*S*

_{0}) and

*H*

^{1, 2}(

*S*) and the restriction operator

*R*

_{Ω}, that restricts to

*Ω*a function harmonic in

*Ω*

_{0}; it is then obvious that

*S*

_{0}is a sphere we have

*B*

^{−1}:

*L*

^{2}(

*S*)→

*H*

^{1, 2}(

*S*). In this case we know ( [32] Cap. VII,1) that we can define a transpose operator

*B*

^{T},

*H*

_{0}and

*R*

_{Ω}(

*H*

_{0}), we can claim that

*φ*∈

*L*

^{2}(

*S*) is such that

*f*(

*σ*) =

*B*

*u*|

_{S}and setting up, for instance, a least squares principle

*u*

_{nm}} that give the “best” approximation of

*f*among the linear combinations of {

*B*

*Se*

_{nm}} complete up to degree

*N*.

*B*

*u*

_{N}→

*B*

*u*. But then, as a consequence of Eq. 49, we have as well

*u*

_{N}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*

*Se*

_{nm}(

*R*

_{σ},

*σ*) are not orthogonal in

*L*

^{2}(

*S*), nor they are if we take a modified norm substituting

*dS*with

*dσ*.

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

^{13}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.

*u*

_{nm}(

*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

*S*

_{e}(with radius

*R*

_{e}) external to

*S*and consider for our

*u*∈

*H*

^{1, 2}(

*S*), the function of

*σ*only

*u*(

*R*

_{e},

*σ*); such a function has a convergent harmonic series:

*S*

_{e}), we certainly have

*H*

^{1, 2}(

*S*) and therefore there must be functions

*f*

_{nm}∈

*H*

^{1, 2}(

*S*) such that

*R*=

*R*

_{0}); one obviously hase

*u*

_{N}is known to converge to

*u*in

*H*

^{1, 2}(

*S*), one must have too

So we see that individual coefficients of the approximation do converge to harmonic coefficients that represent *u*(*r*, *σ*) from the level *r* = *R*_{e} up.

Notice however that although *u*_{N} → *u* in *H*^{1, 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 *S*_{0}).

*u*(

*r*,

*σ*) can be continued down to the level

*r*=

*R*

_{0}. As matter of fact every time we see an expression like

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* = *R*_{e} is transformed into estimates of the coefficients *u*_{nm}(*R*_{e}), 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 *u*_{nm}(*R*_{e}), 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 *f*_{nm}. This is difficult to do if we want to preserve one of the forms of the scalar product in *H*^{1, 2}(*S*) that we have already introduced, namely Eqs. 43 and 65.

*F*

_{nm}(

*σ*)}, {

*G*

_{nm}(

*σ*)} such that the identity holds

*F*

_{nm}(

*σ*) and

*G*

_{nm}(

*σ*) is also given in Appendix A.

### Remark 5

*H*

^{1, 2}(

*S*), namely

*δg*= −

*u′*|

_{S}. Alternatively one could use the above relation to transform Eq. 88 into

*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

*u*

_{N}, 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

*S*

_{U}, to a lower surface

*S*

_{L}. Given that we assume that there are no masses above

*S*

_{U}, so that the exterior set

*Ω*

_{U}is by definition a domain of harmonicity of

*u*, the downward continuation is dominated by two principles:

- (a)as far as we move in the harmonicity domain of
*u*, namely if we assume that*S*_{L}coincides with or encloses the Earth surface*S*, the downward continuation (D.C.) operatoris unbounded, but as inverse of a one to one operator, it is one to one too;$$\displaystyle \begin{aligned} \mathscr{D} (S_L, S_U) \equiv \mathscr{U}(S_U, S_L)^{-1} {} \end{aligned} $$(91) - (b)when we dive into the masses there are two alternatives:
- (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), - (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

*S*_{U}. In fact in this case the same concept of D.C. is lost.

- (i)

*S*

_{U}and

*S*

_{L}are two spheres with radius respectively

*R*

_{U}and

*R*

_{L}(

*R*

_{U}>

*R*

_{L}), inverting Eq. 27 we see that

*u*

_{nm}(

*R*

_{L}) are uniquely identified by the coefficients

*u*

_{nm}(

*R*

_{U}), 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 *S*_{U} and therefore on *Ω*_{U}, any harmonic function coinciding with *u* on *S*_{U} will also coincide with it in *Ω*_{U} and therefore everywhere in the harmonicity domain of *u*; since *S*_{L} is by hypothesis in such a domain, the D.C. of *u* from *S*_{U} to *S*_{L} 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)

*u*can be downward continued to

*S*

_{L}we can assume too that the potential

*u*is generated by a single layer supported by

*S*

_{L}. The answer is in the positive under general conditions; here however, to simplify matters, we stipulate that

*S*

_{L}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

*S*

_{L}, the equation

*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

*u*

_{L}but also \(\frac {\partial u }{\partial n} \vert _{S_L}\) and this is a smooth function too, given the hypothesis done on

*S*

_{L}. So from

*u*

_{L}(

*P*), (

*P*∈

*S*

_{L}) we can find as well the solution of the Dirichlet problem in

*B*

_{L}(

*S*

_{L}≡

*∂B*

_{L}), namely a function

*u*

_{I}harmonic inside

*S*

_{L}and coinciding with

*u*

_{L}on

*S*

_{L}. Since

*u*

_{L}is smooth, so is

*u*

_{I}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

*Ω*

_{L}, that has a normal derivative on the upper face of

*S*

_{L}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

*S*

_{L}is fixed. As claimed before, the above reasoning can be generalized to the case that

*S*

_{L}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

*S*

_{L}has a unique solution if we are able to downward continue

*u*from

*S*

_{U}to

*S*

_{L}.

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

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 *S*_{U} is exterior to *S*, we have already described the operator \(\mathscr {D}\) that downward continues the potential *u* from *S*_{U} 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 *S*_{U} ≡ *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 *S*_{L}, if we don’t know the mass distribution internal to it, or at least the potential *u*_{L}, generated by such internal masses on *S*_{L} (see comments in Remark 1) we are compelled to shrink *S*_{L} to the origin, i.e., to consider the problem of inverting *u*|_{S} into the whole *B*.

*S*

_{L}can be reduced to the origin, namely to disappear as a surface. When

*S*

_{L}does not coincide with

*O*, we assume to know the internal density or equivalently that the body

*B*

_{L}(internal to

*S*

_{L}) is hollow, since we can always subtract from the overall picture the potential generated by the known masses in

*B*

_{L}.

*ρ*of the equation

*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

*S*is a sphere of radius

*R*and

*S*

_{L}a sphere of radius

*R*

_{L}<

*R*; decide further to look for

*ρ*constant in \(\mathscr {L}\). Then (see [27], 1.3) if

*u*(

*R*) =

*u*

_{0}, \(\forall \ \rho > \frac {u_0}{G \ \frac {4}{3} \ \pi \ R^2}\) we get a

*R*

_{L}

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 *H*_{0} of densities {*ρ*_{0}} 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 *L*^{2}-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.

### Example 2

*S*be a sphere with radius

*R*,

*S*

_{L}a sphere with radius

*R*

_{L}<

*R*and \(\mathscr {L}\) the layer between the two. By using the same notation as in Eq. 33, we can write

*C*,

*u*(

*R*,

*σ*) defined on

*S*if we define the norm in \(H^{\frac {3}{2}}\) by

### Remark 7

*L*

^{2}(

*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).

*P*∈

*S*, \(\frac {1}{l_{PQ}} \in L^2(\mathscr {L})\), since, calling

*Δ*the diameter of

*S*,

*H*

_{0}, defined by Eq. 96, is closed in \(L^2(\mathscr {L})\). In fact if

*ρ*

_{0n}∈

*H*

_{0}and \(\rho _{0 n} \rightarrow \overline {\rho _{0}}\) in \(L^2(\mathscr {L})\) it is obvious that

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 *ρ*_{h} 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 *ρ*_{h}, to derive its peculiar shape by the principle of Lagrange multipliers.

*ρ*

_{h}, the solution of the principle

*ρ*

_{λ}of the other principle

*λ*satisfies the equation

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 \(H^{-\frac {3}{2}}(S)\).

*ρ*gives

*ρ*

_{λ}(

*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*

*S*is a sphere of radius

*R*and

*S*

_{L}a sphere of radius

*R*

_{L},

*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 \(\lambda \in H^{- \frac {3}{2}} (S)\).

*ρ*

_{h}=

*ρ*

_{λ}so found does belong to \(L^2(\mathscr {L})\) as it should.

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
*ρ*_{h}=*ρ*_{λ}from Eq. 111, obtaining in fact an \(L^2(\mathscr {L})\) bounded function;*ρ*_{h}results to be equal to the harmonic potential of a single layer on*S*; we callit is obvious that$$\displaystyle \begin{aligned} H_{h} = \bigg \{ \rho_{h} = \int \frac{\lambda}{l_{PQ}} \ dB_Q \qquad \lambda \in H^{- \frac{3}{2}} (S) \bigg \}; {} \end{aligned} $$(115)*H*_{h}is a closed subspace of \(L^2(\mathscr {L})\), the space

*H*_{0}, such that \(\mathscr {S} = \{ H_{h} \bigoplus H_{0} \}\), is just the orthogonal complement of*H*_{h}in \(L^2(\mathscr {L})\).

### Remark 8

The subspace *H*_{h} 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 *S*_{L}.

*h*

_{0}which is also in

*H*

_{0}, 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

### Remark 9

*H*

_{0}, which has proved to be useful in many instances.

*S*

_{L}, then the densities of the form of Eq. 111 with \(\lambda \in H^{- \frac {3}{2}}(S)\) are in fact spanning

*HL*

^{2}(

*B*), namely

*H*

_{h}≡

*HL*

^{2}(

*B*). In this case therefore

*H*

_{0}is characterized by functions

*ρ*

_{0}such that

*ρ*

_{0}∈

*H*

_{0}⊂

*L*

^{2}(

*B*) we can define a

*φ*such that

*φ*∈

*H*

^{2, 2}(

*B*). But in this case one must have form of Eq. 116 and from

*Δh*= 0,

*h*∈

*HL*

^{2}(

*B*) one may conclude that it has to be too

*ρ*

_{0}∈

*H*

_{0}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* = *R*_{σ}}, that generate the same potential *u* in *Ω*. By hypothesis the shape of \(\mathscr {L}\), namely *S* and *S*_{L}, 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 *S*_{L}, 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].

Equation 123 shows the relation existing between a known function *f*(*σ*), a known constant density *ρ*_{M} − *ρ*_{Cr}, that in the sequel we will denote simply as *ρ*, and the unknown function *r* = *R*_{Mσ} ≡ *R*_{Lσ}, 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*_{Lσ}, 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 *L*_{max} 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 *L*_{max}, nevertheless there are arguments saying that if we choose *L*_{max} 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.

*S*

_{L}.

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 *S*_{L} 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.

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.

*f*(

*σ*). Then, assuming that

*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).

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.

*f*(

*σ*) is as a matter of fact the trace on the boundary

*S*of a potential

*v*harmonic in

*Ω*,

*f*(

*σ*) and

*S*are sufficiently smooth, we are in condition to compute

*v′*and its trace on

*S*; analogously we can compute

*R*

_{Lσ}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

*S*) of the potential of a single layer with unknown geometry (

*S*

_{L}), with density \(\omega (\sigma ') = \rho \ cos I_{L \sigma '} R_{L \sigma '}\), upward continued from

*S*

_{L}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.

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.

*R*

_{Lσ}=

*R*

_{1σ},

*R*

_{Lσ}=

*R*

_{2σ}producing the same potential, for the respective layers \(\mathscr {L}_1\), \(\mathscr {L}_2\), on the surface

*S*

*P*in the common domain of harmonicity.

*Ω*

_{+},

*B*

_{+}the exterior and the interior of

*S*

_{+}≡{

*r*=

*R*

_{+σ}} (see Fig. 9), Eq. 142 has to hold in

*P*∈

*Ω*

_{+}.

*δB*,

*S*

_{±}its boundaries

*u*(

*P*) can be written as

*S*

_{+}, i.e., in

*Ω*

_{+}.

*ρ*

_{0}(

*Q*) = ±

*ρ*depends basically on

*σ*but not on

*r*, so that inside

*δB*the identity holds:

*ρ*is a function of

*σ*; it can be modified taking into account that here the layer is between

*S*

_{−}and

*S*

_{+}, as

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*.

*ω*(

*P*) and integrate on

*S*

_{+}; moreover we denote

*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*.

*ω*is one to one and we have shown how to derive one from the other. So we can rewrite Eq. 151 as

*v*on

*S*

_{+}. So we can choose, with the same notation as in Eq. 147

*v*(

*Q*

_{+})

*ρ*(

*σ′*) =

*ρ*,

*S*

_{L1}≡

*S*

_{L2}, namely uniqueness is proved.

*R*

_{L}in such a way that the minimum of the functional

*T*(

*R*

_{L}) is taken to be the square of a norm or a semi-norm like

While *J*_{0}(*R*_{L}) is nothing but the *L*^{2}(*σ*) norm, *J*_{1}(*R*_{L}), *J*_{2}(*R*_{L}) 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 *J*_{1}(*R*_{L}).

*C*of admissible solutions

*R*

_{Lσ}, in our case is quite naturally

*S*

_{L}should be wrapped by

*S*. Indeed

*T*(

*R*

_{L}) defined as in Eq. 163 is never negative, so we can always define

*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

*m*to be a true minimum:

- (1)
*C*has to be bounded and closed in*H*, - (2)
*F*[*R*_{L}] has to be continuous in*H*, - (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., ∀ {*R*_{Ln}∈*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)
*J*(*R*_{L}) 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 *J*_{1}(*R*_{L}) as a regularizer. In fact we can argue that *J*_{0}(*R*_{L}) 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 *L*^{2}(*σ*) are certainly not compact, nor are their intersections with *C*.

As for the choice of *J*_{2}(*R*_{L}), 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 *J*_{1}(*R*_{L}).

*J*

_{0}<

*a*,

*J*

_{1}<

*a*,

*J*

_{2}<

*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.

*J*(

*R*

_{L}) ≡

*J*

_{1}(

*R*

_{L}), namely

### Remark 10

*g*(

*σ*) with the constant

*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 *L*^{2}-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 *L*^{2} but milder that *H*^{1, 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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