Gravimetric Measurements, Gravity Anomalies, Geoid, Quasigeoid: Theoretical Background and Multiscale Modeling/Gravimetrische Messungen, Schwereanomalien, Geoid, Quasigeoid: Theoretischer Hintergrund und Multiskalenmodellierung

  • Gerhard Berg
  • Christian Blick
  • Matthias Cieslack
  • Willi FreedenEmail author
  • Zita Hauler
  • Helga Nutz
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)


Die methodischen Aspekte der Gravimetrie werden sowohl in messtechnischer als auch mathematisch/physikalischer Hinsicht untersucht. Lokale gravimetrische Datensätze werden genutzt, um Multiskalenmerkmale in geophysikalisch relevanten Signaturbändern von Gravitationsanomalien und Qua- sigeoidhöhen zu visualisieren. Wavelet-Dekorrelationen werden für ein bestimmtes Gebiet in Rheinland-Pfalz illustriert.


Gravimetrie Schwereanomalien Geoid versus Quasigeoid Multiskalendekorrelation 


The methodical aspects of gravimetry are investigated from observational as well as mathematical/physical point of view. Local gravimetric data sets are exploited to visualize multiscale features in geophysically relevant signature bands of gravity anomalies and quasigeoidal heights. Wavelet decorrelation is illustrated for a certain area of Rhineland-Palatinate.


Gravimetry Gravity anomalies Geoid versus quasigeoid Multiscale decorrelation 

1 Introduction

Concerning Earth’s gravity observation, it should be pointed out that the terrestrial distribution of Earth’s gravity data is far from being homogeneous with large gaps, in particular over oceans but also over land. In addition, the quality of the data is very distinct. As a matter of fact, a global terrestrial gravity data coverage now and in the foreseeable future is far from being satisfactory. This is one of the reasons why spaceborne measurements have to come into play for global gravity determination. Until now, however, the relatively poor precision of satellite-only spaceborne gravity measurements has hindered a wider use of this type of measurements, in particular for local purposes of geoidal modeling and exploration. Nonetheless, satellite models can be used as trend solution to avoid oscillation phenomena (Gibb’s phenomena) of terrestrial data modeling in data gaps and in the vicinity of the boundary of the local area under consideration. In fact, it must be emphasized that, in future, only a coordinated research of a horizontal as well as a vertical “zooming-in” approach will provide a breakthrough in local modeling to understand significant structures and processes of the Earth’s system.

As already pointed out in [17], the way forward in local modeling has to be based on two requirements:
  1. (i)

    Vertical multiscale modeling, i.e., “zooming-in downward continuation” of different data sources starting from globally available spaceborne SST and/or SGG data as means for an appropriate trend solution via more accurate (regional) airborne data down to (local) high-precision terrestrial gravimetric data sets (this aspect has been investigated in several publications, e.g., [18, 20, 24, 26, 27], and the references therein).

  2. (ii)

    Horizontal multiscale modeling, i.e., “zooming-in procedures” from rough to dense terrestrial data material from global to local areas, where certain geodetic features should be specified and investigated in more detail (see, e.g., [6, 29, 31, 32, 33, 34], and the references therein).

  3. (iii)

    Combining data from different sensors and sources, thereby observing that positioning systems are ideally located as far as possible from the Earth, while gravity field sensors are ideally located as close as possible to the Earth.

In this contribution, we briefly explain the status quo of gravimetric observation. We recapitulate the standard definitions of gravity anomaly, Bouguer anomaly, anomalous potential, geoid, and quasigeoid. On the basis of these results we present horizontal multiscale methods by means of geoscientifically relevant wavelets for the purpose of appropriate decorrelation of interpretable signatures inherent in the gravitational quantities under consideration (for an excursion to vertical multiscale modeling the reader is referred to, e.g., [35] (this handbook) and the references therein).

2 Gravity and Gravitation

The following terminology is standard in the geodetic context: The gravity acceleration (gravity)w is the resultant of the gravitation v and the centrifugal acceleration c such that
$$\displaystyle \begin{aligned} w=v+c. \end{aligned} $$
The centrifugal force c arises as a result of the rotation of the Earth about its axis. For purposes of local gravity exploration we are allowed to assume a rotation of constant angular velocity ω. The centrifugal acceleration acting on a unit mass is directed outward perpendicularly to the spin axis (see Fig. 1). Introducing the so-called centrifugal potential C, such that c = ∇C, the function C turns out to be non-harmonic. The direction of the gravity w is known as the direction of the plumb line, the quantity |w| is called the gravity intensity (often also just called gravity and denoted in the geodetic jargon usually by g). Altogether, the gravity potential of the Earth can be expressed in the form
$$\displaystyle \begin{aligned} W = V + C , \end{aligned} $$
and the gravity acceleration w is given by
$$\displaystyle \begin{aligned} w = \nabla W = {\underbrace{\nabla V}_{=v}} + {\underbrace{\nabla C}_{= c}}. \end{aligned} $$
Fig. 1

Gravitation v, centrifugal acceleration c, and gravity acceleration w

As already pointed out, the surfaces of constant gravity potentials, i.e., W = const., are designated as equipotential (level, or geopotential) surfaces of gravity (for more details, the reader is referred to the standard monographs in physical geodesy, e.g., [39, 40, 43, 64, 85]). The force of the gravity provides a directional structure to the space above the Earth’s surface (see Fig. 2). It is tangential to the vertical plumb lines and perpendicular to all (level) equipotential surfaces. Any water surface at rest is part of a level surface. Level, i.e., equipotential surfaces are ideal reference surfaces, for example, for changes in the Earth’s system. The geoid is defined as that level surface of the gravity field which best fits the mean sea level.
Fig. 2

Level surfaces and plumb lines for a homogeneous ball (left) and an Earth-like body (right) (taken from [20])

A gravity anomaly is understood as the difference between the observed acceleration on the Earth’s surface and the corresponding value originated by a model of the Earth’s gravity field. Historically, the model is constructed under simplifying assumptions, usually in such a way that the figure of an ellipsoid of resolution or a spheroidal surface are assumed (cf. Fig. 3). Gravity on the surface of this reference surface is given by a (simple) known formula. The subtraction from observed gravity at the same location provides the gravity anomaly. Of course, anomalies are much smaller than the values of gravity. A location with a positive anomaly typically shows more gravity than predicted by the model so that the presence of a subsurface positive mass anomaly is suggested. A negative anomaly exhibits a lower value than predicted so that a subsurface deficit is suggested. Thus, gravity anomalies are of substantial geophysical as well as geological interest.
Fig. 3

Geodetically relevant surfaces (sectional illustration from W. Freeden, M. Schreiner [28], Mathematical Geodesy – Its Role, Its Aim, and Its Potential, this handbook)

Once an artificial geopotential field U has been constructed matching the reference surface (in geodesy, usually, an ellipsoid/spheroid but, in principle, any surface close to the geoidal surface may be taken) with an equipotential surface, it is called a normal potential (see, e.g., [40]). The difference of the potential W and the normal potential U is known as the disturbing potential (or anomalous potential) T
$$\displaystyle \begin{aligned} T=W-U. \end{aligned} $$
In accordance with its construction, the disturbing potential T is smaller than U and W and captures the detailed variations of the true gravity field of the actual Earth from point-to-point, as distinguished from the global trend captured by the reference surface (e.g., ellipsoid/spheroid).

The direction of the gravity vector can be obtained, e.g., by astronomical positioning. Measurements are possible on the Earth’s surface. Observations of the gravity vector are converted into so-called deflections of the vertical by subtracting a corresponding reference direction derived from a simple gravity field model associated to, e.g., a reference surface. Deflections of the vertical constitute tangential fields of the anomalous potential. Due to the high measurement effort required to acquire these types of data compared to a gravity measurement, the data density of vertical deflections is much less than that of gravity anomalies. Gravitational field determination based on the observation of deflections of the vertical and combined with gravity is feasible in smaller areas with good data coverage.

The actual Earth’s surface (globally available from modern spaceborne techniques such as, GNSS, LASER, VLBI, etc.) does not coincide with an equipotential surface (i.e., a level surface). The force of gravity is generally not perpendicular to the actual Earth’s surface (see Fig. 3). We are confronted with the gravity intensity as an oblique derivative on the Earth’s surface. The gravity vector is an oblique vector at any point on the Earth’s surface and generally not the normal vector. The determination of equipotential surfaces of the potential W is strongly related to the knowledge of the potential V . The gravity vector w given by w = ∇W is normal to the equipotential surface passing through the same point. Once more, equipotential surfaces such as the geoid intuitively express the notion of tangential surfaces, as they are normal to the plumb lines given by the direction of the gravity vector.

3 Gravimeter and Gravimetry

Next, in some parts, we almost literally follow the explanations to be found in [7].

Gravimeters are typically designed to measure very tiny fractional changes of the Earth’s gravity, caused by nearby geologic structures or the shape of the Earth (Fig. 5). There are two types of gravimeters, viz. absolute gravimeters (cf. Fig. 4) and relative gravimeters (cf. Fig. 6 and Fig. 7). Absolute gravimeters measure the local gravity and are directly based on measuring the acceleration of free fall (for example, of a test mass in a vacuum tube). A new type of an absolute gravimeter is the atomic gravimeter, which measures the free fall of laser-cooled atoms. Atomic gravimeters promise an extremely increased precision of gravity measurements.
Fig. 4

The principle of an absolute gravimeter: The gravity intensity |w| can be obtained via the ordinary differential equation \(m \left (d/dt \right )^2 x = |w|\) by measuring the transition of the test mass through three time levels

Fig. 5

Relative gravimeter Scintrex-CG6 in action (State Office for Surveying and Geobase Information Rhineland-Palatinate (LVermGeo))

Fig. 6

Hooke’s law considers a simple helical spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is m |w(x)| at the location x. Hooke’s law is a first-order linear approximation to the real response of a spring. It tells us that the magnitude of the force is proportional to the extension of the spring, i.e., \( |w(x)| - |w(x')|= \frac {\kappa }{m} (L(x) - L(x'))\) (m represents the mass and κ is a constant characterizing the material of the spring)

Fig. 7

Mass implied gravitational effect obtained by a relative gravimeter (illustration with kind permission of Teubner-publishing taken from [46] in modified form)

Relative gravimeters compare the value of gravity at one point with another. They must be calibrated at a location, where the gravity is known accurately and measure the ratio of the gravity at the two points. Most common relative gravimeters are spring–based (cf. Fig. 6 and Fig. 7). By determining the amount by which the weight stretches the spring, gravity becomes available via Hooke’s law. The highest possible accuracy of relative gravity measurements are conducted at the Earth’s surface. Measurements on ships and in aircrafts deliver reasonably good data only after the removal of inertial noise.

Gravimetry relates the Earth’s mass density to the gravitational potential via Newton’s Law of Gravitation. By inverse gravimetry, we denote the determination of the Earth’s mass density distribution from data of the gravitational potential or related quantities. Clearly, the determination of gravity intensities as well as gravity anomalies of dimension very much larger than the gravity anomalies which are caused by regional structures is of less significance for purposes of local gravitational determination. More particularly, local gravimetric methods are based on the measurements of small variations.

It should be mentioned that gravity prospecting was first used in the case of strong density contrasts in a geological structure, and the usual approach is to measure differences in gravity from place to place. Today, the interpretation of gravimetric data is done by comparing the shape and size of gravity disturbances and anomalies to those caused by bodies of various geometrical shapes at different depths and differing densities.

The observed gravity intensity |w| on the Earth’s surface (see Fig. 7) depends on the following effects to be removed (for more detailed studies, see, e.g., [65, 72, 81]):
  • attraction of the reference surface (e.g., an ellipsoid/spheroid),

  • elevation above sea level,

  • topography,

  • time dependent (tidal) variations,

  • (Eötvös) effect of a moving platform,

  • isostatic balance on the lower lithosphere,

  • density variations inside the upper crust.

In more detail, certain corrections have to be applied to the data in order to account for effects not related to the subsurface: Drift corrections are necessary, since each gravimeter suffers mechanical changes over time, and so does its output measurement. This change is generally assumed to be linear. In case of acquisition on a moving platform, the motion relative to the surface of the Earth implies a change in centrifugal acceleration. The Eötvös correction depends on the latitude and the velocity vector of the moving platform. It should be observed that free air anomaly does not correct for the first two effects which could mask the gravity anomalies related to the Bouguer density contrasts in the crust. Complete Bouguer correction effectively remove the gravity anomalies due to bathymetry, but still contain the gravity effect of the Moho. Isostatics contain the gravity effect of the Moho. Special methods such as Poincare-Bey corrections are in use within boreholes or for special geoid computations. For more information the reader is referred to geodetic textbooks such as [40, 43, 84, 85] and to the literature concerned with prospecting and exploration (see, e.g., [65, 66, 81] and the references therein).

As a consequence, to isolate the effects of local density variations from all other contributions, it is necessary to apply a series of reductions (Fig. 8):
  • The attraction of, e.g., the reference ellipsoid/spheroid has to be subtracted from the measured values.
    Fig. 8

    Illustration of the components of the gravity acceleration (ESA medialab, ESA communication production SP–1314)

  • An elevation correction must be done, i.e., the vertical gradient of gravity is multiplied by the elevation of the station and the result is added. With increasing elevation of the Earth, there is usually an additional mass between the reference level and the actual level. This additional mass itself exerts a positive gravitational attraction.

  • Bouguer correction and terrain correction are applied to correct for the attraction of the slab of material between the observation point and the geoid.

  • A terrain correction accounts for the effect of nearby masses above or mass deficiencies below the station. Isostatic correction accounts for the isostatic roots (Moho).

Gravity observation (cf. Fig. 8) can be done over land or sea areas using different techniques and equipment. Terrestrial gravimetry in exploration was first applied to p͡rospect for salt domes, e.g., in the Gulf of Mexico (an example in the eastern part of Germany is shown in Fig. 9), and for looking for anticlines in continental areas (see, e.g., [65, 66], and the references therein).
Fig. 9

Top: Gravity effect in [μm s−1] of the salt dome Werle (Mecklenburg, Germany); bottom Geological vertical profile (with kind permission of Teubner-publishing taken from [46] in modified form)

Nowadays, gravimetry is in use all over the world in diverse applications, from which we list only a few examples:
  1. (1)

    Gravimetry is decisive for geodetic purposes of modeling gravity anomalies, geoidal undulations, and quasigeoidal heights.

  2. (2)

    Gravimetry is helpful in different phases of the oil exploration and production processes as well as in geothermal research.

  3. (3)

    Archaeological and geotechnical studies aim at the mapping of subsurface voids and overburden variations.

  4. (4)

    Gravimetric campaigns may be applied for groundwater and environmental studies. They help to map aquifers to provide formations and/or structural control.

  5. (5)

    Gravimetric studies give information about tectonically derived changes and volcanological phenomena.

All in all, nowadays the main applications of gravimetry can be listed as follows:
  1. (i)

    determination of geodetic key observables for modeling gravity anomalies and definition of geological structural settings, such as hotspots and plumes,

  2. (ii)

    faults delineation,

  3. (iii)

    recovery of salt bodies,

  4. (iv)

    metal deposits,

  5. (v)

    forward modeling, inversion (i.e., inverse gravimetry),

  6. (vi)

    postprocessing to assist seismic modeling, geomagnetic interpretation, etc. for explorational purposes (e.g., in geothermal research).


The knowledge of horizontal/oblique/vertical derivatives of the gravity potential is a useful addendum to prospecting and exploration. This is the reason why we are interested in discussing derivatives later on in more detail (based on ideas and concepts developed in [20]).

4 Geoid: Physical Nature, Geomathematical Context, and Multiscale Decorrelation

Knowing the gravity potential, all equipotential surfaces (including the geoid at mean sea level) are given by an equation of the form W(x) = const (Fig. 2). By introducing U as the normal gravity potential corresponding to a reference domain, usually an ellipsoidal domain, the disturbing potential T is already known to be the difference of the gravity potential W and the normal gravity potential U, i.e., we are led to a decomposition of the gravity potential in the form W = U + T. According to the concept developed by [41, 54, 82], and [68, 69] we may assume that
  1. (a)

    the center of the reference surface (ellipsoid) coincides with the center of gravity of the Earth,

  2. (b)

    the difference of the mass of the Earth and the mass of the reference body (in today’s geodesy usually an ellipsoidal body, but in future more complicated bodies are definitely senseful) is zero.

A point x of the geoid can be projected onto its associated point y of the reference surface (e,g., ellipsoid) by means of the surface normal. The distance N(x) between x and y is called the geoidal height or geoidal undulation in x (cf. Fig. 10). The gravity anomaly vector a(x) at the point x of the geoid is defined as the difference between the gravity vector w(x) and the normal gravity vector u(y), i.e.,
$$\displaystyle \begin{aligned} a(x) = w(x) - u(y). \end{aligned} $$
Another possibility is to form the difference between the vectors w and u at the same point x such that we get the gravity disturbance vector d(x) defined by
$$\displaystyle \begin{aligned} d(x) = w(x) - u(x). \end{aligned} $$
In geodesy, several basic mathematical relations between the scalar fields |w| and |u| as well as between the vector fields a and d are known. In the following, we only describe the fundamental relations heuristically (see, for example, [39, 40] for more details):
Fig. 10

Illustration of the gravity vector w(x), the normal gravity vector u(x), and the geoidal height N(x). Here, ν and ν′ denote the normal to the geoid and the reference surface (ellipsoid), respectively (following [40])

The gravity disturbance vector d(x) at the point x on the geoid can be written as follows:
$$\displaystyle \begin{aligned} d(x) = w(x) - u(x) = \nabla\left(W(x) - U(x)\right) = \nabla T(x). \end{aligned} $$
According to Taylor’s formula of multivariate analysis, \(U(y) + \frac {\partial U}{\partial \nu '}(y)N(x)\) is the linearization of U(x), i.e., by expanding the potential U at the point x and truncating the Taylor series at the linear term, we obtain (cf. Figs. 10 and 11)
$$\displaystyle \begin{aligned} U(x) \simeq U(y) + \frac{\partial U}{\partial\nu'}(y)N(x), \end{aligned} $$
$$\displaystyle \begin{aligned} \nu'(y) = -\frac{u(y)}{\vert u(y)\vert}\end{aligned} $$
is the ellipsoidal normal at y and the geoidal undulation N(x) is the aforementioned distance between x and y (note that the symbol ‘≃’ means that the error between the left and the right hand side may be assumed to be insignificantly small). Using the fact that T(x) = W(x) − U(x) and observing the relations
$$\displaystyle \begin{aligned} \vert u(y)\vert = -\nu' (y)\cdot u(y) = -\nu'(y)\cdot\nabla U(y) = -\frac{\partial U}{\partial\nu'}(y), \end{aligned} $$
we find under the assumption of (8) that
$$\displaystyle \begin{aligned} N(x) = \frac{U(y)-U(x)}{\vert u(y)\vert} = \frac{T(x)-\left(W(x)-U(y)\right)}{\vert u(y)\vert}. \end{aligned} $$
Finally, considering U(y) = W(x) = const. = W0, we end up with the so-called Bruns formula (cf. [8])
$$\displaystyle \begin{aligned} N(x) = \frac{T(x)}{\vert u(y)\vert}.\end{aligned} $$
This formula relates the physical quantity T(x) to the geometric quantity N(x) for points x on the geoid.
Fig. 11

Geodetically relevant heights (from W. Freeden, M. Schreiner, [28] Mathematical Geodesy – Its Role, Its Aim, and Its Potential, this handbook)

It is helpful to study the vector field ν(x) in more detail:
$$\displaystyle \begin{aligned} \nu(x) = -\frac{w(x)}{\vert w(x)\vert}. \end{aligned} $$
Due to the definition of the normal vector field (13), we obtain the following identity
$$\displaystyle \begin{aligned} w(x) = \nabla W(x) = \ - \ \vert w(x)\vert \ \nu(x). \end{aligned} $$
In an analogous way we obtain
$$\displaystyle \begin{aligned} u(x) = \nabla U(x) = \ - \ \vert u(x)\vert \ \nu'(x) . \end{aligned} $$
The deflection of the vertical Θ(x) at the point x on the geoid is understood to be the angular (i.e., tangential) difference between the directions ν(x) and ν′(x) (see, e.g., [29] for more details). The deflection of the vertical is determined by the angle between the plumb line and the normal of the reference surface through the same point (see Fig. 10):
$$\displaystyle \begin{aligned} \varTheta(x)=\nu(x)-\nu'(x)-\left(\left(\nu(x)-\nu'(x)\right)\cdot\nu(x)\right)\nu(x). \end{aligned} $$
According to its construction, the deflection of the vertical Θ(x) at x is orthogonal to the normal vector field ν(x), i.e., Θ(x) ⋅ ν(x) = 0. Since the plumb lines are orthogonal to the equipotential surfaces of the geoid and the reference surface, respectively, the deflection of the vertical gives briefly spoken a measure of the gradient of the equipotential surfaces (cf. [40]).
From (14), in connection with (16), it follows that
$$\displaystyle \begin{aligned} w(x) = -\vert w(x)\vert \left( \varTheta(x) + \nu'(x) + \left(\left(\nu(x) - \nu'(x)\right)\cdot\nu(x)\right)\nu(x) \right). {} \end{aligned} $$
Using Eqs. (15) and (17) we finally obtain for the gravity disturbing vector d(x) at the point x
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} d(x) &\displaystyle =&\displaystyle \nabla T(x) =w(x) - u(x)\\ &\displaystyle =&\displaystyle -\vert w(x)\vert\left(\varTheta(x) + \nu'(x) + \left(\left(\nu(x) - \nu'(x)\right)\cdot\nu(x)\right)\nu(x)\right)- \left(-\vert u(x)\vert \nu'(x)\right)\\ &\displaystyle =&\displaystyle -\vert w(x)\vert\left(\varTheta(x) + \left(\left(\nu(x) - \nu'(x)\right) \cdot \nu(x) \right) \nu(x)\right)-\left(\vert w(x)\vert - \vert u(x)\vert\right)\nu'(x). {} \end{array} \end{aligned} $$
The quantity
$$\displaystyle \begin{aligned} D(x)=\vert w(x)\vert - \vert u(x)\vert \ \end{aligned} $$
is called the gravity disturbance, whereas
$$\displaystyle \begin{aligned} A(x)= \vert w(x)\vert - \vert u(y)\vert \ \end{aligned} $$
is called the gravity anomaly.
Splitting the gradient ∇T(x) of the disturbing potential T at x into a normal part (pointing into the direction of ν(x)) and an angular (tangential) part (using the representation of the surface gradient ∇), we have
$$\displaystyle \begin{aligned} \nabla T(x) =\nu(x) \frac{\partial T}{\partial\nu}(x)+\frac{1}{\vert x\vert} \nabla^*T(x). \end{aligned} $$
Since the gravity disturbances represent at most a factor of 10−4 of the Earth’s gravitational force (for more details see [40]), the error between \(\nu (x)\frac {\partial T}{\partial \nu }(x)\) and \(\nu '(x)\frac {\partial T}{\partial \nu '}(x)\) has no (computational) significance. Consequently, we may assume
$$\displaystyle \begin{aligned} d(x) \simeq\nu'(x) \frac{\partial T}{\partial\nu'}(x)+\frac{1}{\vert x\vert} \nabla^* T(x). \end{aligned} $$
Moreover, the scalar product \(\left (\nu (x)-\nu '(x)\right )\cdot \nu (x)\) can also be neglected. Thus, in connection with (18), we obtain
$$\displaystyle \begin{aligned} d(x) \simeq -\vert w(x)\vert \ \varTheta(x) - D(x) \nu'(x). \end{aligned} $$
By comparison of (22) and (23), we therefore obtain
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} D(x) &\displaystyle =&\displaystyle -\frac{\partial T}{\partial\nu'}(x) = - \nu'(x) \cdot d(x), \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \vert w(x)\vert\ \varTheta(x) &\displaystyle =&\displaystyle -\frac{1}{\vert x\vert} \nabla^* T(x). \end{array} \end{aligned} $$
In other words, the gravity disturbance D(x), beside being the difference in magnitude of the actual and the normal gravity vector, is also the normal component of the gravity disturbance vector d(x). In addition, we are led to the angular differential equation (25).
Applying Bruns formula (12) to Eqs. (24) and (25) we obtain
$$\displaystyle \begin{aligned} D(x) = \vert w(x)\vert - \vert u(x)\vert= -\vert u(y)\vert \ \frac{\partial N}{\partial\nu'}(x) {} \end{aligned} $$
for the gravity disturbance and
$$\displaystyle \begin{aligned} \vert w(x)\vert \ \varTheta(x) = -\frac{1}{\vert x\vert} \ \nabla^* T(x)= -\frac{1}{\vert x\vert} \vert u(y)\vert \ \nabla^*N(x) {} \end{aligned} $$
for the deflection of the vertical (note that Θ(x) may be multiplied (without loss of (computational) precision) either by |w(x)| or by |u(x)| since it is a small quantity).
Turning over to the gravity anomalies A(x), it follows from the identity (24) by linearization that
$$\displaystyle \begin{aligned} -\frac{\partial T}{\partial\nu'}(x) =D(x)\simeq A(x) - \frac{\partial |u(y)|}{\partial\nu'} N(x). \end{aligned} $$
Using Bruns formula (12), we obtain for the gravity anomalies that
$$\displaystyle \begin{aligned} A(x) =-\frac{\partial T}{\partial\nu'}(x)+\frac{1}{|u(y)|}\frac{\partial |u(y)|}{\partial\nu'} T(x). \end{aligned} $$
Summing up our results (24) for the gravity disturbance D(x) and (29) for the gravity anomaly A(x), we are led to the so-called fundamental equations of physical geodesy:
$$\displaystyle \begin{aligned} \begin{array}{rcl} D(x) &\displaystyle =&\displaystyle |w(x)| - |u(x)| = -\frac{\partial T}{\partial\nu'}(x), {} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} A(x) &\displaystyle =&\displaystyle |w(x)| - |u(y)| = -\frac{\partial T}{\partial\nu'}(x) + \frac{1}{|u(y)|}\ \frac{\partial |u(y)|}{\partial\nu'}\ T(x). {} \end{array} \end{aligned} $$
Equations (30) and (31) show the relation between the disturbing potential T and the gravity disturbance D and the gravity anomaly A, respectively, on the geoid (see, for example, [39, 40, 57]). They are used as boundary conditions in boundary-value problems.

Following [40], the geoidal heights N, i.e., the deviations of the equipotential surface on the mean ocean level from the reference ellipsoid, are extremely small (Fig. 11). Their order is of only a factor 10−5 of the Earth’s radius (see [40] for more details). Even more, the reference ellipsoid only differs from a sphere \(\mathbb {S}^2_R\) with (mean Earth’s) radius R in the order of the flattening of about 3 ⋅ 10−3. Therefore, since the time of [82], it is common use that, in theory, a reference, e.g., an ellipsoidal surface or a more appropriate surface should be taken into account. However, in numerical practice for reasons of numerical economy and practical efficiency, the reference surface is adequately treated as a sphere, and the Eqs. (26) and (27) are solved in spherical approximation. In doing so, a relative error of the order of the flattening of the Earth’s body at the poles, i.e., a relative global error of 10−3, is accepted in all equations containing the disturbing potential. Considering appropriately performed reductions in numerical calculations, this error seems to be quite permissible (cf. [40, 43], and the remarks in [37, 38] for comparison with ellipsoidal approaches), and this is certainly the case if local features are under consideration. In addition, Runge’s approach (see, e.g., [3, 27] for more details on the Runge context) allows the calculation of gravitational quantities on arbitrary surfaces, in particular ellipsoids, spheroids, telluroids, just by suitably operating with spherically based equipment such as multipoles (i.e., (outer) spherical harmonics).

In other words, for computational purposes in gravitational theory we are not required to use, for example, ellipsoidal/spheroidal framework. Instead we are allowed to perform calculations involving gravitation on an ellipsoid/spheroid just in an appropriate spherical framework.

Moreover, in geoscience, it is common numerical practice for local approximations to replace the reference surface by a sphere or even by a plane.

In what follows we first use the classical (global) spherical approach of physical geodesy (see, e.g., [40, 43]) for subsequent application in regional/local approximation: According to the classical Pizzetti assumptions (see [68, 69]), it follows that the first moment integrals of the disturbing potential vanish, i.e.,
$$\displaystyle \begin{aligned} \int_{\mathbb{S}^2_R} T(y) H_{-n-1,k}^R (y) \, dS(y) = 0,\end{aligned} $$
for n = 0, 1, k = 1, …, 2n + 1, where \(\{ H_{-n-1,k}^R \}\) denotes the system of outer spherical harmonics, dS is the surface element in \(\mathbb {R}^3,\) and \(\mathbb {S}^2_R\) is the sphere in \(\mathbb {R}^3\) around the origin with radius R. More concretely, the Pizzetti assumptions tell us that
  • if the Earth’s center of gravity is the origin, there are no first-degree terms in the spherical harmonic expansion of T,

  • if the mass of the spherical Earth and the mass of the reference surface (ellipsoid) is equal, there is no zero term.

In this way, together with the indicated processes in gravitational modeling, formulas and structures are obtained that are valid for the sphere.
In the well-known spherical nomenclature, involving a sphere \(\mathbb {S}^2_R\) as reference surface for purposes of computation (R being the mean Earth’s radius) with a mass M distributed homogeneously in its interior, we are simply led to (cf. [40])
$$\displaystyle \begin{aligned} U(y) =\frac{\gamma M}{\vert y\vert},\qquad u(y) =\nabla U(y)=-\frac{\gamma M}{\vert y\vert^2} \frac{y}{\vert y\vert},\end{aligned} $$
where γ is the gravitational constant (γ = 6.6742 ⋅ 10−11 m3 kg−1 s−2). Hence, we obtain
$$\displaystyle \begin{aligned} \vert u(y)\vert &=\frac{\gamma M}{\vert y\vert^2}, \end{aligned} $$
$$\displaystyle \begin{aligned} \frac{\partial |u(y)|}{\partial\nu'} &=-\frac{u(y)}{\vert u(y)\vert}\cdot\nabla|u(y)|=-2\frac{\gamma M}{\vert y\vert^3}, \end{aligned} $$
$$\displaystyle \begin{aligned} \frac{1}{|u(y)|}\frac{\partial|u(y)|}{\partial\nu'} &=-\frac{2}{\vert y\vert}. \end{aligned} $$
Furthermore, in spherical nomenclature, i.e., \(x \in \mathbb {S}^2_R\), we obviously have
$$\displaystyle \begin{aligned} -\frac{\partial T}{\partial\nu'}(x) =-\frac{x}{\vert x\vert}\cdot\nabla T(x). \end{aligned} $$
Therefore, we end up with the formulation of the fundamental equations of physical geodesy in terms of a spherical context:
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} D(x) &\displaystyle =&\displaystyle -\frac{x}{\vert x\vert}\cdot\nabla T(x), \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} A(x) &\displaystyle =&\displaystyle -\frac{x}{\vert x\vert}\cdot\nabla T(x) - \frac{2}{\vert x\vert} T(x). \end{array} \end{aligned} $$
In addition, in a vector spherical context (see also [30]), we obtain for the differential equation (25)
$$\displaystyle \begin{aligned} -\nabla^*T(x) =\frac{\gamma M}{R} \varTheta(x), \end{aligned} $$
and, by virtue of Bruns formula (12), we finally find that
$$\displaystyle \begin{aligned} -\nabla^* N(x)=R \ \varTheta(x). \end{aligned} $$
In physical geodesy, a componentwise scalar determination of the vertical deflection is usually used (see, e.g., [40], as well as the paper by [43]). Our work prefers the vectorial framework, i.e., the vector equation (40). In doing so, we are concerned with an isotropic vector approach by means of the fundamental solution with respect to the Beltrami operator (see also [29, 30]) instead of the conventional anisotropic scalar decomposition into vector components due to [89].
The disturbing potential enables us to make the following geophysical interpretations (for more details, see, e.g., the work by [38, 50, 73, 80], and the references therein):
  • Gravity disturbances D and gravity anomalies A (see Fig. 12) represent a relation between the real Earth and a reference (e.g., ellipsoidal) Earth model. In accordance with Newton’s Law of Gravitation they therefore show the imbalance of forces in the interior of the Earth due to the irregular density distribution inside the Earth. Clearly, gravity anomalies and/or gravity disturbances do not determine uniquely the interior density distribution of the Earth. They may be interpreted as certain filtered signatures, which give major weight to the density contrasts close to the surface and simultaneously suppress the influence of deeper structures inside the Earth.
    Fig. 12

    Gravity anomalies (above) and gravity disturbances (below) (from [30])

  • Geoid undulations provide a measure for the perturbations of the Earth from a hydrostatic equilibrium (see Fig. 13). They form the deviations of the equipotential surfaces at mean sea level from the reference surface (e.g., an ellipsoid). Geoid undulations show no essential correlation to the distributions of the continents. They seem to be generated by density contrasts much deeper inside the Earth.
    Fig. 13

    Geoidal undulations (from [30])

As already explained, the task of determining the disturbing potential T from gravity disturbances or gravity anomalies, respectively, leads to boundary-value problems usually corresponding to a spherical boundary. Numerical realizations of such boundary-value problems have a long tradition, starting from [82] and [67]. Nonetheless, our work presents some recent aspects in their potential theoretic treatment by proposing appropriate space-regularization techniques applied to the resulting integral representations of their solutions. For both boundary-value problems, viz. the Neumann and the Stokes problem, we are able to present two solution methods: The disturbing potential may be either solved by a Fourier (orthogonal) expansion method in terms of spherical harmonics or it can be described by a singular integral representation over the boundary \(\mathbb {S}^2_R\).

So far, much more data on gravity anomalies A(x) = |w(x)|−|u(y)| are available than on gravity disturbances D(x) = |w(x)|−|u(x)|. However, by modern GNSS-technology (see, e.g., [87]), the point x on the geoid is rather determined than y on the reference ellipsoid. Therefore, in future, it can be expected that D will become more important than A (as [43] pointed out in their monograph on physical geodesy). This is the reason why we continue to work with D. Nevertheless, the results of our (multiscale) approach applied to A are of significance. Therefore, the key ideas and concepts concerning A can be treated in parallel (see [13, 20, 92] for explicit details).

In order to formulate some results in the language of potential theory, we first introduce the potential space \(Pot^{(1)}\left ({\mathbb {R}^3 \backslash \mathbb {B}_R^3}\right )\), where \(\mathbb {B}_R^3\) is the (open) ball of radius R around the origin 0. More concretely, we let \(Pot(\mathbb {R}^3 \backslash {\overline {\mathbb {B}_{R}^3}})\) be the space of all functions \(F: \mathbb {R}^3 \backslash {{\mathbb {B}_{R}^3}} \to \mathbb {R}\) satisfying
  1. (i)

    \( F|{ }_{\mathbb {R}^3 \backslash \overline {\mathbb {B}_{R}^3}} \) is a member of \(C^{(2)}(\mathbb {R}^3 \backslash {\overline {\mathbb {B}_{R}^3}})\),

  2. (ii)

    F satisfies Laplace’s equation ΔF(x) = 0, \(x \in \mathbb {R}^3 \backslash {\overline {\mathbb {B}_{R}^3}}.\)

  3. (iii)

    F is regular at infinity, i.e., F(x) = O(|x|−1), |x|→.

\(Pot^{(1)}(\mathbb {R}^3\backslash \mathbb {B}_{R}^{3})\) is formally understood to be the space
$$\displaystyle \begin{aligned} Pot^{(1)}({\mathbb{R}^3 \backslash \mathbb{B}_R^3})= C^{(1)}({\mathbb{R}^3 \backslash \mathbb{B}_R^3}) \cap Pot(\mathbb{R}^3 \backslash {\overline{\mathbb{B}_{R}^3}}). \end{aligned} $$
In the language of potential theory, the exterior Neumann boundary-value problem corresponding to known gravity disturbances D (compare (38)) reads as follows: (ENPPG) Let D be a continuous function on \(\mathbb {S}^2_R=\partial \mathbb {B}_{R}^3\), i.e., \(D \in {C^{(0)}}(\mathbb {S}^2_R)\) with
$$\displaystyle \begin{aligned} \int_{\mathbb{S}^2_R} D(y) H_{-n-1,k}^R (y) \, dS(y) = 0, \end{aligned} $$
for n = 0, 1, k = 1, …, 2n + 1. Find \( T \in Pot^{(1)}\left ({\mathbb {R}^3 \backslash \mathbb {B}_R^3}\right )\), such that the boundary condition \(D = \frac {\partial T} { \partial \nu } \big |{ }_{\mathbb {S}^2_R}\) holds true and the potential T fulfills the conditions
$$\displaystyle \begin{aligned} \int_{\mathbb{S}^2_R} T(y) H_{-n-1,k}^R (y) \, dS(y) = 0 \end{aligned} $$
for n = 0, 1, k = 1, …, 2n + 1.
It is known (see, e.g., [20]) that the solution of the boundary-value problem (ENPPG) can be represented in the form
$$\displaystyle \begin{aligned} T(x) = \frac{1}{4 \pi R} \int_{\mathbb{S}^2_R} D(y) \; N (x,y) \ dS (y), \; x\in \mathbb{R}^3\backslash{\mathbb B}_R^{3}, \end{aligned} $$
where the Neumann kernel N(⋅, ⋅) in (45) possesses the spherical harmonic expansion
$$\displaystyle \begin{aligned} N(x,y)= \sum_{n=2}^\infty \left(\frac{R^2}{\vert x \vert \vert y \vert}\right)^{n+1}\frac{2n+1}{n+1} P_n\left(\frac{x}{\vert x \vert}\cdot \frac{y}{\vert y \vert}\right) . \end{aligned} $$
By well-known manipulations, the series in terms of Legendre polynomials can be expressed as an elementary function leading to the integral representation
$$\displaystyle \begin{aligned} T(x) =\frac{1}{4\pi R}\int_{\mathbb{S}^2_R} D(y)\left(\frac{2R}{\vert x-y\vert}+\ln\left(\frac{\vert y\vert + \left\vert y-\frac{R^2}{\vert x\vert^2}x \right\vert - \frac{R^2}{\vert x\vert}}{\vert y\vert + \left\vert y-\frac{R^2}{\vert x\vert^2}x \right\vert + \frac{R^2}{\vert x\vert}}\right)\right) \, dS(y). \end{aligned} $$
It is not difficult to see that for \(x \in \mathbb {S}^2_R\), the integral (47) is equivalent to
$$\displaystyle \begin{aligned} T(x) =\frac{1}{4\pi R}\int_{\mathbb{S}^2_R} D(y)\left(\frac{2R}{\vert x-y\vert}+\ln\left(\frac{\vert y\vert + \vert x - y\vert - R}{\vert y\vert + \vert x - y\vert + R}\right)\right) \, dS(y). \ \end{aligned} $$
Written out in spherical nomenclature \(x=R \frac {x}{\vert x \vert }\), \(y=R \frac {y}{\vert y\vert }\), x ≠ y on \(\mathbb {S}^2_R,\) we find
$$\displaystyle \begin{aligned} N\left(R\frac{x}{\vert x\vert},R\frac{y}{\vert y\vert}\right) =\frac{2}{\left\vert \frac{x}{\vert x\vert}- \frac{y}{\vert y\vert}\right\vert}+\ln\left(\frac{R\left\vert \frac{x}{\vert x\vert}- \frac{y}{\vert y\vert}\right\vert}{2R+R\left\vert \frac{x}{\vert x\vert}-\frac{y}{\vert y\vert}\right\vert}\right). \end{aligned} $$
If we use
$$\displaystyle \begin{aligned} \left\vert \frac{x}{\vert x\vert}-\frac{y}{\vert y\vert}\right\vert =\left({2-2\frac{x\cdot y}{\vert x\vert \ \vert y\vert}}\right)^{\frac{1}{2}}, \end{aligned} $$
then, for x ≠ y, we are led to the identity
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} N\left(R\frac{x}{\vert x\vert},R\frac{y}{\vert y\vert}\right)&\displaystyle =&\displaystyle N\left(\frac{x}{\vert x\vert},\frac{y}{\vert y\vert}\right) \\ &\displaystyle =&\displaystyle \frac{\sqrt{2}}{\sqrt{1-\frac{x}{\vert x\vert}\cdot\frac{y}{\vert y\vert}}}-\ln\left(1+\frac{\sqrt{2}}{\sqrt{1-\frac{x}{\vert x\vert} \cdot\frac{y}{\vert y\vert}}} \right). \end{array} \end{aligned} $$
Consequently, for points \(x\in \mathbb {S}^2_R\), we (formally) get the so-called Neumann formula, which constitutes an improper integral over \(\mathbb {S}^2_R\):
$$\displaystyle \begin{aligned} T\left(R\frac{x}{\vert x\vert}\right) = \frac{1}{4\pi R}\int_{\mathbb{S}^2_R} D\left(R\frac{y}{\vert y\vert}\right)N\left(\frac{x}{\vert x\vert}, \frac{y}{\vert y\vert}\right)\, dS(y), \end{aligned} $$
where the Neumann kernel constitutes radial basis function due to (50).
Once more, in accordance with the conventional approach of physical geodesy, the Neumann formula (52) is valid under the following constraints (see also [31, 40, 63]):
  1. (i)

    the mass within the reference ellipsoid is equal to the mass of the Earth,

  2. (ii)

    the center of the reference ellipsoid coincides with the center of the Earth,

  3. (iii)

    the formulation is given in the spherical context to guarantee economical and efficient numerics.

Note that we are able to set N(, ) = N(ξ, η) = N(ξ ⋅ η) in terms of the unit vectors \(\xi =\frac {x}{\vert x\vert }\) and \(\eta =\frac {y}{\vert y\vert }\) which simplifies our notation: If we define the single-layer kernel \(S:[-1,1) \to \mathbb {R}\) by
$$\displaystyle \begin{aligned} S(t) = \frac{\sqrt{2}}{\sqrt{1-t}}, \; t \in [-1,1), \end{aligned} $$
the Neumann kernel is the zonal function of the form
$$\displaystyle \begin{aligned} N(\xi\cdot\eta) =S(\xi \cdot \eta)-\ln\left(1+S(\xi \cdot \eta)\right),\; 1- \xi\cdot \eta > 0. \end{aligned} $$
An equivalent formulation of the improper integral (52) over the unit sphere \({\mathbb {S}^2}(=\mathbb {S}^2_1 = \partial \mathbb {B}_1^3)\) is then given by
$$\displaystyle \begin{aligned} T(R\xi)=\frac{R}{4\pi}\int_{\mathbb{S}^2} D(R\eta)N(\xi\cdot\eta)\, dS(\eta). \end{aligned} $$
It should be remarked that the exterior Stokes boundary value problem of determining the disturbing potential from known gravity anomalies can be handled in a quite analogous way (see [13, 20, 92]), providing the so-called Stokes integral associated to the radially symmetric Stokes kernel as an improper integral on \(\mathbb {S}^2_R\).
Next we deal with the vertical deflections Θ (cf. [20, 29]). Suppose that T fulfills the conditions (44). We consider the differential equation (compare Eq. (40))
$$\displaystyle \begin{aligned} \nabla^*_\xi T(R\xi) =-\frac{\gamma M}{R}\ \varTheta(R\xi), \end{aligned} $$
where T(R⋅) represents the disturbing potential and Θ(R⋅) denotes the vertical deflection (cf. (40)). The differential equation (56) can be solved in a unique way by means of the fundamental solution with respect to the Beltrami operator
$$\displaystyle \begin{aligned} T(R\xi) =\frac{\gamma M}{R}\int_{\mathbb{S}^2}\varTheta(R\eta)\cdot\nabla^*_\eta G\left(\varDelta^*;\xi\cdot\eta\right)\, dS(\eta),\end{aligned} $$
where \((\xi , \eta ) \mapsto G\left (\varDelta ^*;\xi \cdot \eta \right ), 1-\xi \cdot \eta \neq 0\), is the fundamental solution of the Beltrami equation on the unit sphere \({\mathbb {S}^2}\) (see, e.g., [16]) given by
$$\displaystyle \begin{aligned} G\left(\varDelta^*;\xi\cdot\eta\right) = \frac{1}{4 \pi} \ln(1-\xi \cdot \eta) + \frac{1}{4 \pi}(1- \ln(2)).\end{aligned} $$
The identity (57) immediately follows from the Third Green Theorem (cf. [20, 30]) for ∇ on \({\mathbb {S}^2}\) in connection with (44). By virtue of the identity
$$\displaystyle \begin{aligned} \nabla_\eta^*G\left(\varDelta^*;\xi\cdot\eta\right) =-\frac{\xi-(\xi\cdot\eta)\eta}{4\pi(1-\xi\cdot\eta)}, \; \xi \ne \eta,\end{aligned} $$
the integral (57) can be written in the form
$$\displaystyle \begin{aligned} T(R\xi)=\frac{R}{4\pi}\int_{\mathbb{S}^2} \varTheta(R\eta)\cdot g\left(\varDelta^*;\xi,\eta\right)\, dS(\eta), \end{aligned} $$
where the vector kernel g(Δ;ξ, η), ξη, is given by
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} g\left(\varDelta^*;\xi,\eta\right)&\displaystyle =-\dfrac{\gamma M}{R^2}\dfrac{\xi-(\xi\cdot\eta)\eta}{1-\xi\cdot\eta} \end{array} \end{aligned} $$
(note that we write \(\nabla ^*_\eta \) to indicate that the operator ∇ is applied to the η-variable). Again we are confronted with a representation of the disturbing potential T as an improper integral over the sphere \(\mathbb {S}^2\).
All our settings leading to the disturbing potential on the sphere \(\mathbb {S}^2_R\) turn out to be improper integrals. As we have shown they have either the singularity behavior of the single-layer kernel S (cf. Eq. (53)) or the characteristic logarithmic singularity of the fundamental solution with respect to the Beltrami operator G(Δ;⋅, ⋅) (cf. (58)). Indeed, the fundamental solution and the single-layer kernel are interrelated (see [20]) by the identities
$$\displaystyle \begin{aligned} S(\xi\cdot\eta)=\sqrt{2} \ e^{-2\pi G(\varDelta^*;\xi\cdot\eta)+\frac{1}{2}} \end{aligned} $$
$$\displaystyle \begin{aligned} G(\varDelta^*;\xi\cdot\eta)= - \frac{1}{2 \pi} \ln(S(\xi \cdot \eta)) - \frac{1}{4 \pi} (1 - 2 \ln(2)).\end{aligned} $$
Therefore, we are confronted with the remarkable situation that a (Taylor) regularization of the single-layer kernel implies a regularization of the fundamental solution, and vice versa.

Next, we present multiscale representations for the Neumann kernel N (cf. Eq. (54)). Note that all modern multiscale approaches have a conception of wavelets in common as constituting multiscale building blocks, which provide a fast and efficient way to decorrelate a given signal data set.

The properties (basis property, decorrelation, and efficient algorithms) are common features of all wavelets, so that these attributes form the key for a variety of applications (see, e.g., [17, 24], as well as [20]), particularly for signal reconstruction and decomposition, thresholding, data compression, denoising by, e.g., multiscale signal-to-noise ratio, etc. and, in particular, decorrelation.

We follow the space (taylorized) regularization methods presented in [31] for linear regularization of the single-layer kernel S and [29] for linear regularization of the fundamental solution G(Δ;⋅) of the Beltrami equation. For higher order approximations, the reader is referred to the Ph.D.-theses [13] and [92].

The essential idea is to regularize the single-layer kernel function
$$\displaystyle \begin{aligned} S(t) = \frac{\sqrt{2}}{\sqrt{1-t}} \end{aligned} $$
by replacing it by a Taylor linearization. To this end, we notice that the first derivative of the kernel S is given as follows
$$\displaystyle \begin{aligned} S'(t) = \frac{1}{\sqrt{2}(1-t)^{\frac{3}{2}}}, \; t \in [-1,1). \end{aligned} $$
Consequently, we obtain as (Taylor) linearized approximation corresponding to the expansion point \(1-\frac {\tau ^2}{2R^2}, \ \tau \in (0, 2R],\)
$$\displaystyle \begin{aligned} S(t)=S\left(1-\frac{\tau^2}{2R^2}\right) +{S'\left(1-\frac{\tau^2}{2R^2}\right)} \left(t-(1-\frac{\tau^2}{2R^2})\right)+\ldots. \end{aligned} $$
In more detail, the kernel S is replaced by its (Taylor) linearized approximation Sτ at the point \(1-\frac {\tau ^2}{2R^2}, \ \tau \in (0, 2R],\) given by
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} S^\tau(t)=\begin{cases}\frac{R}{\tau}\left(3- \frac{2R^2}{\tau^2} (1-t)\right),&\displaystyle \qquad 0\leq 1-t\leq\frac{\tau^2}{2R^2}, \\ \frac{\sqrt{2}}{\sqrt{1-t}},&\displaystyle \qquad \frac{\tau^2}{2R^2}<1-t\leq 2.\end{cases} \end{array} \end{aligned} $$
Note that the expansion point 1 − τ2∕(2R2), τ ∈ (0, 2R], is chosen in consistency with the notation in the initial paper [29] and the subsequent papers [33] and [31]. A graphical illustration of the original kernel S(t) and a τ-scale dependent version of its linear space-regularized kernel Sτ(t) is shown in Fig. 14.
Fig. 14

Single-layer kernel S(t) (continuous black line) and its Taylor linearized regularization Sτ(t), for R = 1 and \(\tau = \frac {1}{2},1,2\) (dotted lines)

Clearly, the function Sτ is continuously differentiable on the interval [−1, 1], and we have
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \left(S^\tau\right)'(t) =\begin{cases}\frac{2R^3}{\tau^3},&\displaystyle \qquad 0\leq 1-t \leq \frac{\tau^2}{2R^2}, \\ \frac{1}{\sqrt{2}\left(1-t\right)^{\frac{3}{2}}},&\displaystyle \qquad \frac{\tau^2}{2R^2} < 1-t \leq 2.\end{cases}\vspace{-4pt} \end{array} \end{aligned} $$
Furthermore, the functions S and Sτ are monotonically increasing on the interval [−1, 1), such that S(t) ≥ Sτ(t) ≥ S(−1) = Sτ(−1) = 1 holds true on the interval [−1, 1). Considering the difference between the kernel S and its linearly regularized version Sτ, we find
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} S(t)-S^\tau(t)=\begin{cases}\frac{\sqrt{2}}{\sqrt{1-t}}-\frac{R}{\tau} \left(3 - \frac{2R^2}{\tau^2}(1-t)\right),&\displaystyle 0 < 1-t \leq \frac{\tau^2}{2R^2}, \\ 0,&\displaystyle \frac{\tau^2}{2R^2} < 1-t \leq 2.\end{cases}\vspace{-4pt} \end{array} \end{aligned} $$
By elementary manipulations of one-dimensional analysis we readily obtain
$$\displaystyle \begin{aligned} \int_{-1}^1\left|S(t) - S^\tau(t)\right|\, d t \ = \ O(\tau). \end{aligned} $$
As a consequence, we have

Satz 1

For \(F \in {C}^{(0)} ({\mathbb {S}^2})\) and Sτ defined by (67),
$$\displaystyle \begin{aligned} \lim_{\tau\to 0+} \sup_{\xi \in{\mathbb{S}^2}} \left| \int_{\mathbb{S}^2} S(\xi \cdot \eta) F(\eta) \ dS (\eta) -\int_{\mathbb{S}^2} S^\tau(\xi \cdot \eta) (\xi \cdot\eta) F(\eta) \ dS (\eta)\right| = 0. \end{aligned} $$
To study the surface gradient \(\nabla _\xi ^*\), we choose F to be of class \({C}^{(1)}({\mathbb {S}^2})\). Letting \({\mathbf {t}}_\xi \in \mathbb {R}^{3\times 3}\) be the orthogonal matrix (with det(tξ) = 1) leaving ε3 fixed such that tξξ = ε3, we obtain
$$\displaystyle \begin{aligned} \nabla_\xi^*\int_{\mathbb{S}^2} S(\xi\cdot\eta) F(\eta) \ dS(\eta)= \int_{\mathbb{S}^2} S(\eta_3)\nabla_\xi^*F({\mathbf{t}}_\xi^T\eta) \ dS(\eta) \end{aligned} $$
for \(\xi \in {\mathbb {S}^2}\) and η = (η1, η2, η3)T. By regularizing the single-layer kernel, we obtain
$$\displaystyle \begin{aligned} \int_{\mathbb{S}^2} \nabla_\xi^* S^\tau(\xi\cdot\eta) F(\eta)\ dS(\eta)= \int_{\mathbb{S}^2} S^\tau(\eta_3)\nabla_\xi^*F({\mathbf{t}}_\xi^T\eta) \ dS(\eta) \end{aligned} $$
for \(\xi \in {\mathbb {S}^2}.\) Finally, Lemma 1 leads us to the following limit relation (see [30]).

Satz 2

Let F be of class \({C}^{(1)}({\mathbb {S}^2}).\) Let Sτ be given by (67). Then
$$\displaystyle \begin{aligned} \lim_{\tau\to0+} \sup_{\xi\in{\mathbb{S}^2}} \left|\int_{\mathbb{S}^2} \nabla^*_\xi S^\tau(\xi\cdot\eta) F(\eta)dS(\eta)- \nabla_\xi^*\int_{\mathbb{S}^2} S(\xi\cdot\eta) F(\eta)dS(\eta)\right|=0. \end{aligned} $$
Following Eq. (63) we introduce modified kernels Gτ(Δ;⋅) by
$$\displaystyle \begin{aligned} G^\tau(\varDelta^*;t)= - \frac{1}{2 \pi} \ln(S^\tau(t)) - \frac{1}{4 \pi} (1 - 2 \ln(2)), \ -1 \le t \le 1. \end{aligned} $$
These kernels Gτ(Δ;⋅) are “single-layer kernel regularization” of the fundamental solution G(Δ;⋅), which fulfill the following integral relations (cf. [30]).

Satz 3

For \(F \in {C}^{(0)} ({\mathbb {S}^2})\) and Gτ(Δ;⋅) defined by (75), we have
$$\displaystyle \begin{aligned} \lim_{\tau\to 0+} \sup_{\xi \in{\mathbb{S}^2}} \left| \int_{\mathbb{S}^{2}} G(\varDelta^*;\xi \cdot \eta) F(\eta) \ dS (\eta) -{\int_{\mathbb{S}^2}} { {G}^\tau} (\varDelta^*;\xi \cdot\eta) F(\eta) \ dS (\eta)\right| = 0, \end{aligned} $$
$$\displaystyle \begin{aligned} \lim\limits_{\tau\to 0+}\sup_{\xi \in{\mathbb{S}^2}}\left| {\int_{\mathbb{S}^{2}}} \nabla^*_\xi {G}^\tau(\varDelta^*;\xi\cdot\eta) F(\eta) \ dS (\eta) - \nabla^*_\xi\int_{\mathbb{S}^2} G (\varDelta^* ;\xi \cdot\eta) F(\eta) \ dS (\eta)\right| = 0. \end{aligned} $$

Numerical implementations and computational aspects of the Taylor regularization techniques as presented here have been applied (even for subsets of \(\mathbb {S}^2_R\)) to different fields of physical geodesy (see, e.g., [19, 20, 29, 30, 31, 33] and the references therein).

The space-regularization techniques enable us to formulate multiscale solutions for the disturbing potential from gravity disturbances or vertical deflections. Note that we need higher-order regularizations whenever gravitational observables containing second or higher order derivatives come into play. An example is gravity gradiometry, which will not be discussed here. The interested reader is referred to the contribution about Satellite Gravity Gradiometry within this handbook [35].

As point of departure for our considerations serves the special case study of the linear regularization of the single-layer kernel in the integral representation of the solution of the Neumann boundary-value problem (ENPPG).

As we already know, the solution of the (Earth’s) disturbing potential \( T \in Pot^{(1)}\left ({\mathbb {R}^3\backslash \mathbb {B}_R^3}\right )\) from known vertical derivatives, i.e., gravity disturbances \(D = \frac {\partial T} { \partial \nu }{\big |}_{\mathbb {S}^2_R},\) satisfying the conditions (44) on the sphere \(\mathbb {S}^2_R\), can be formulated as an improper integral (see Eq. (55))
$$\displaystyle \begin{aligned} T(R\xi) = \frac{R}{4\pi} \int_{\mathbb{S}^2} D(R\eta) \ N(\xi\cdot\eta)\, dS(\eta), \ \xi\in\mathbb{S}^2,\end{aligned} $$
with the Neumann kernel (54). Our interest is to formulate regularizations of the disturbing potential T by use of the (Taylor) linearized approximation of the single-layer kernel \({S}^\tau :[-1,1]\to \mathbb {R} ,\ \tau \in (0,2R] ,\) introduced in (67). As a result, we obtain the regularized Neumann kernels
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle {}{}{}&\displaystyle {N^\tau(\xi\cdot\eta)} {}\\ &\displaystyle =&\displaystyle \begin{cases}{S}^\tau(\xi\cdot\eta) - \ln\left(1+{S}^\tau(\xi\cdot\eta)\right),&\displaystyle \qquad 0\leq 1-\xi\cdot\eta\leq\frac{\tau^2}{2R^2}, \\ S(\xi\cdot\eta) - \ln(1+S(\xi\cdot\eta)),&\displaystyle \qquad \frac{\tau^2}{2R^2}< 1-\xi\cdot\eta\leq 2,\end{cases} \\ &\displaystyle =&\displaystyle \begin{cases}\frac{R}{\tau}\left(3- \frac{2R^2}{\tau^2} (1-\xi\cdot\eta)\right)\\ -\ln\left(1+\frac{R}{\tau}\left(3- \frac{2R^2}{\tau^2} (1-\xi\cdot\eta)\right)\right),&\displaystyle \quad 0\leq 1-\xi\cdot\eta\leq\frac{\tau^2}{2R^2}, \\[1.5ex] \frac{\sqrt{2}}{\sqrt{1-\xi\cdot\eta}}-\ln\left(1+\frac{\sqrt{2}}{\sqrt{1-\xi\cdot\eta}}\right),&\displaystyle \quad \frac{\tau^2}{2R^2}< 1- \xi\cdot\eta\leq 2.\end{cases} \vspace{-4pt} \end{array} \end{aligned} $$
In doing so, we are immediately led to the regularized representation of the disturbing potential T corresponding to the known gravity disturbances: The representation (87) is remarkable, since the integrands of T and Tτ only differ on the spherical cap \(\varGamma _{{\tau ^2}/(2R^2)}(\xi )=\{\eta \in {\mathbb {S}^2}: \ 1-\xi \cdot \eta \leq \frac {\tau ^2}{2R^2}\}\).

By aid of Lemma 1, we obtain

Theorem 1

Suppose that T is the solution of the Neumann boundary-value problem (ENPPG) of the form (78). Let \(\mathcal {T}\,\,^\tau ,\) τ ∈ (0, 2R], represent its regularization (87). Then
$$\displaystyle \begin{aligned} \lim_{\tau\to0 +}\sup_{\xi\in{\mathbb{S}^2}}\left\vert T(R\xi) - \mathcal{T}\,\,^\tau(R\xi) \right\vert=0. \end{aligned}$$
For numerical applications, we have to go over to scale-discretized approximations of the solution to the boundary-value problem (ENPPG). For that purpose, we choose a monotonically decreasing sequence \(\{\tau _j\}_{j\in \mathbb {N}_0},\) such that
$$\displaystyle \begin{aligned} \lim_{j \to \infty} \tau_j = 0, \; \tau_0 = 2R. \end{aligned} $$
A particularly important example, that we use in our numerical implementations below, is the dyadic sequence with
$$\displaystyle \begin{aligned} \tau_j=2^{1-j}R, \ \; j\in \mathbb{N}_0. \end{aligned} $$
It is easy to see that \(2\tau _{j+1}\,{=}\,\tau _j,\ j\,{\in }\,\mathbb {N}_0,\) is the relation between two consecutive elements of the sequence. In correspondence to the sequence \(\{\tau _j\}_{j\in \mathbb {N}_0}\), a sequence \(\{N^{\tau _j}\}_{j\in \mathbb {N}_0}\) of discrete versions of the regularized Neumann kernels (79), so-called Neumann scaling functions, is available. Fig. 15 (left) shows a graphical illustration of the regularized Neumann kernels for different scales j.
Fig. 15

Illustration of the Neumann kernel N(t) (left, continuous black line) and its Taylor linearized regularization \( N^{\tau _J}(t)\), J = 0, 1, 2, τJ = 21−JR and R = 1 (left, dotted lines). The corresponding Taylor linearized Neumann wavelets \( WN^{\tau _J}(t)\) for scales J = 0, 1, 2, are shown on the right

The regularized Neumann wavelets, forming the sequence \(\{{WN}^{\tau _j}\}_{j\in \mathbb {N}_0}\), are understood to be the difference of two consecutive regularized Neumann scaling functions, respectively,
$$\displaystyle \begin{aligned} {WN}^{\tau_j} ={N}^{\tau_{j+1}}-{N}^{\tau_{j}}, \; j \in \mathbb{N}_0. \end{aligned} $$
The Neumann wavelets are illustrated in Fig. 15 (right). These wavelets possess the numerically important property of a local support. More concretely, \(\eta \,{\mapsto }\, {WN}^{\tau _j}(\xi \cdot \eta ), \ \eta \,{\in }\, {\mathbb {S}^2},\) vanishes everywhere outside the spherical cap \(\varGamma _{{\tau _j^2}/(2R^2)}(\xi )\).
Let \(J\in \mathbb {N}_0\) be an arbitrary scale. Suppose that \({N}^{\tau _{J}}\) is the regularized Neumann scaling function at scale J. Furthermore, let \({WN}^{\tau _j},\) j = 0, …, J, be the regularized Neumann wavelets as given by (83). Then we obviously have
$$\displaystyle \begin{aligned} {N}^{\tau_{J}} ={N}^{\tau_{0}}+\sum_{j=0}^{J-1}{WN}^{\tau_j}. \end{aligned} $$
The local support of the Neumann wavelets within the framework of (84) should be studied in more detail: We start with the globally supported scaling kernel \({N}^{\tau _{0}}= {N}^{2R}\). Then we add more and more wavelet kernels \({WN}^{\tau _j},\) j = 0, …, J − 1, to achieve the scaling kernel \({N}^{\tau _{J}}\). It is of particular importance that the kernel functions \(\eta \mapsto {WN}^{\tau _j}(\xi \cdot \eta ),\) \(\xi \in {\mathbb {S}^2}\) fixed, are ξ-zonal functions with local support (spherical caps).

In conclusion, a calculation of an integral representation for the disturbing potential T starts with a global trend approximation using the scaling kernel at scale j = 0 (of course, this requires data on the whole sphere, but the data can be rather sparsely distributed since they only serve as a trend approximation). Step by step, we are able to refine this approximation by use of wavelets of increasing scale. The spatial localization of the wavelets successively allows a better spatial resolution of the disturbing potential T. Additionally, the local supports of the wavelets provide a computational advantage since the integration has to be performed on smaller and smaller spherical caps. In consequence, the presented numerical technique becomes capable of handling heterogeneously distributed data.

All in all, keeping the space-localizing property of the regularized Neumann scaling and wavelet functions in mind, we are able to establish an approximation of the solution of the disturbing potential T from gravity disturbances D in form of a “zooming-in” multiscale method. A lowpass filtered version of the disturbing potential T at the scale j in an integral representation over the unit sphere \({\mathbb {S}^2}\) is given by (compare Eq. (87))
$$\displaystyle \begin{aligned} {T}^{\tau_j}(R\xi) =\frac{R}{4\pi}\int_{\mathbb{S}^2} D(R\eta) \ {N}^{\tau_j}(\xi\cdot\eta)\, dS(\eta),\; \xi\in{\mathbb{S}^2}, \end{aligned} $$
while the j-scale bandpass filtered version of T leads to the integral representation by use of wavelets
$$\displaystyle \begin{aligned} {WT}^{\tau_j}(R\xi) = \frac{R}{4\pi}\int_{ \varGamma_{{\tau_j^2}/(2R^2)} (\xi)}D(R\eta) \ {WN}^{\tau_j}(\xi\cdot\eta)\, dS(\eta),\; \xi\in{\mathbb{S}^2}. \end{aligned} $$

Theorem 2

Let \({T}^{\tau _{J_0}}\) be the regularized version of the disturbing potential at some arbitrary initial scale J0 as given in (85), and let \({WT}^{\tau _{j}},\) j = 0, 1, …, be given by (86). Then, the following reconstruction formula holds true:
$$\displaystyle \begin{aligned} \lim_{N \to \infty}\sup_{\xi\in{{\mathbb{S}^2}}} \left\vert T(R\xi) - \left( {T}^{\tau_{J_0}}(R\xi) + \sum_{j=0}^N \ {WT}^{\tau_{J_0+j}}(R\xi)\right)\right\vert=0. \end{aligned}$$
The multiscale procedure (wavelet reconstruction) as developed here can be illustrated by the following scheme As a consequence, a tree algorithm based on the regularization in the space domain has been realized for determining the disturbing potential T from locally available data sets of gravity disturbances D.
In order to get a fully discretized solution of the Neumann boundary-value problem (ENPPG), approximate integration by use of appropriate cubature formulas is necessary (see, e.g., [32, 42] for more details about approximate integration on the (unit) sphere). The fully discretized multiscale approximations have the following representations
$$\displaystyle \begin{aligned} {T}^{\tau_{j}}(R\xi) \ \simeq \ \frac{R}{4\pi} \sum_{k=1}^{N_j} w_k^{N_j} D\left(R\eta_k^{N_j}\right) \ {N}^{\tau_{j}}\left(\xi\cdot\eta_k^{N_j}\right), \; \xi \in {\mathbb{S}^2},\end{aligned} $$
$$\displaystyle \begin{aligned} {WT}^{\tau_{j}}(R\xi) \ \simeq \ \frac{R}{4\pi} \sum_{k=1}^{N_j} w_k^{N_j} D\left(R\eta_k^{N_j}\right) \ {WN}^{\tau_{j}}\left(\xi\cdot\eta_k^{N_j}\right), \; \xi \in {\mathbb{S}^2},\end{aligned} $$
where \(\eta _k^{N_j}\) are the Nj integration knots and \(w_k^{N_j}\) the integration weights.

Whereas the sum in (87) has to be calculated on the whole sphere \(\mathbb {S}^2\), the summation in (88) has to be computed only for the local supports of the wavelets (note that the symbol ≃ means that the error between the right and the left hand side can be neglected).

Figs. 16, 17, and 18 present a decomposition of the Earth’s disturbing potential T in lowpass and bandpass filtered parts via Neumann wavelets for data sets of increasing data width leading to the hotspot of the Galapagos Islands. Seen from the geodetic reality, the figures are remarkable in the following sense: For getting a better accuracy in numerical integration procedures providing the (global) solution of the boundary-value problem (ENPPG) as illustrated in Fig. 16a, we need denser, globally over the whole sphere \(\mathbb {S}^2_R\) equidistributed data sets (most notably, in the sense of Weyl’s Law of Equidistribution). However, in today’s reality of gravitational field observation, we are confronted with the problem that terrestrial gravitational data (such as gravity disturbances, gravity anomalies) of sufficient width and quality are only available for certain parts of the Earth’s surface (for more details concerning the observational aspects see, e.g., [10, 11, 12, 74, 75]). As a matter of fact, there are large gaps, particularly at sea, where no data sets of sufficient quality are available at all. This is the reason why the observational situation implies the need for specific geodetically oriented modeling techniques taking the heterogeneous data situation and the local availability of the data (usually related to latitude-longitude data grids) into consideration. In this respect, the “zooming-in” realization based on single-layer space-regularization is a suitable efficient and economic mathematical answer.
Fig. 16

Lowpass filtered version \(T^{\tau _4}\) of the disturbing potential T in \([ \frac {m^2}{s^2}]\) and the corresponding bandpass filtered versions \({WT}^{\tau _{j}}\) for scales j = 4, 5 of the magenta bordered region in 16(a) calculated from different numbers of data points (from the Ph.D.-thesis [92], Geomathematics Group, University of Kaiserslautern). (a) Low pass part \(T^{\tau _4}\) calculated from 490, 000 data points distributed over the whole sphere \( \mathbb {S}^2_R\). (b) Details \(WT^{\tau _4}\) at scale 4 from 281, 428 data points distributed within the black bordered region in (a). (c) Details \(WT^{\tau _5}\) at scale 5 from 226, 800 data points distributed within the gray bordered region in (a)

Fig. 17

Lowpass filtered version \({T}^{\tau _{6}}\) of the disturbing potential T in \([ \frac {m^2}{s^2}]\) of the magenta bordered region in 16(a) and the corresponding bandpass filtered versions \({WT}^{\tau _{j}}\) for scales j = 6, 7 (from the Ph.D.-thesis [92], Geomathematics Group, University of Kaiserslautern). (a) Low pass part \({T}^{\tau _{6}}\) of the magenta bordered region in (a) computed by the sum of \({T}^{\tau _{4}}\) (a), \({WT}^{\tau _{4}}\) (b), and \({WT}^{\tau _{5}}\) (c) in this region. (b) Details \(WT^{\tau _6}\) at scale 6 from 71 253 data points distributed within the black bordered region in (a). (c) Details \(WT^{\tau _7}\) at scale 7 from 63 190 data points distributed within the gray bordered region in (a)

Fig. 18

Lowpass filtered version \({T}^{\tau _{8}}\) of the disturbing potential T in \([ \frac {m^2}{s^2}]\) of the magenta bordered region in 17(a) and the corresponding bandpass filtered versions \({WT}^{\tau _{j}}\) for scales j = 8, 9 (from the Ph.D.-thesis [92], Geomathematics Group, University of Kaiserslautern). (a) Low pass part \({T}^{\tau _{8}}\) of the magenta bordered region in (a) computed by the sum of \({T}^{\tau _{6}}\) (a), \({WT}^{\tau _{6}}\) (b), and \({WT}^{\tau _{7}}\) (c) in this region. (b) Details \(WT^{\tau _8}\) at scale 8 from 71 253 data points distributed within the black bordered region in (a). (c) Details \(WT^{\tau _9}\) at scale 9 from 63 190 data points distributed within the gray bordered region in (a)

As already known from (60), the solution of the surface differential equation (see Eq. (40))
$$\displaystyle \begin{aligned} \nabla^* T(R\xi) = -\frac{\gamma M}{R} \ \varTheta(R\xi),\; \xi\in{\mathbb{S}^2}, \end{aligned} $$
determining the disturbing potential T from prescribed vertical deflections Θ under the conditions (44) is given by
$$\displaystyle \begin{aligned} T(R\xi) = \frac{R}{4\pi} \int_{\mathbb{S}^2} \varTheta(R\eta)\cdot g\left(\varDelta^*;\xi,\eta\right) \, dS(\eta), \end{aligned} $$
where the vector kernel \(g\left (\varDelta ^*;\xi ,\eta \right ), \ 1-\xi \cdot \eta > 0,\) reads as follows (see Eq. (61))
$$\displaystyle \begin{aligned} g\left(\varDelta^*;\xi,\eta\right) &=-\frac{1}{2} \frac{\gamma M}{R^2}\frac{2}{1-\xi\cdot\eta}(\xi-(\xi\cdot\eta)\eta) \\ &= -\frac{1}{2} \frac{\gamma M}{R^2} (S(\xi\cdot\eta))^2(\xi-(\xi\cdot\eta)\eta). \end{aligned} $$
Analogously to the calculation of the disturbing potential T from known gravity disturbances D (i.e., the Neumann problem (ENPPG)), the numerical calamities of the improper integral in (90) can be circumvented by replacing the zonal kernel S(ξ ⋅ η) by the regularized kernel Sτ(ξ ⋅ η). This process leads to space-regularized representations Tτ of the disturbing potential T calculated from vertical deflections Θ within a multiscale “zooming-in” procedure analogous to the approach for gravity disturbances as input data. To be more concrete, the kernel function g(Δ;⋅, ⋅) is replaced by the space-regularized function using Eq. (67) for τ ∈ (0, 2R]. This leads to the following approximative representation of the disturbing potential T:
$$\displaystyle \begin{aligned} \mathcal{T}\,\,^\tau(R\xi) =\frac{R}{4\pi}\int_{\mathbb{S}^2}\varTheta(R\eta) \cdot {g}^{\tau}\left(\varDelta^*;\xi,\eta\right)\, dS(\eta), \end{aligned} $$
with gτ(Δ;⋅, ⋅) given by (92).

Theorem 3

Suppose that T is the solution (90) of the differential equation (89), with Θ being a member of the class of continuous vector valued functions \({c}^{(0)}(\mathbb {S}^2_R)\). Let Tτ, τ ∈ (0, 2R], represent its regularized solution of the form (93). Then
$$\displaystyle \begin{aligned} \lim_{\tau\to0+} \sup_{\xi\in{{\mathbb{S}^2}}}\left\vert T(R\xi) - \mathcal{T}\,\,^\tau(R\xi)\right\vert=0. \end{aligned} $$
By restricting \(\{{g}^{\tau }\left (\varDelta ^*;\cdot ,\cdot \right )\}_{ \tau \in (0,2R]}\) to the sequence \(\{{g}^{\tau _j}\left (\varDelta ^*;\cdot ,\cdot \right )\}_{j\in \mathbb {N}_0}\), corresponding to a set of scaling parameters \(\{\tau _j\}_{j\in \mathbb {N}_0}\) satisfying τj ∈ (0, 2R] and limjτj = 0, we are canonically led to regularized vector scaling functions such that a scale-discrete solution method for the differential equation (89) can be formulated. The vector scaling function \({g}^{\tau _{j+1}}(\varDelta ^*;\cdot ,\cdot )\) at scale j + 1 is constituted by the sum of the vector scaling function \({g}^{\tau _{j}}(\varDelta ^*;\cdot ,\cdot )\) and the corresponding discretized vector wavelet \({wg}^{\tau _{j}}(\varDelta ^*;\cdot ,\cdot )\), given by
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} {wg}^{\tau_{j}}\left(\varDelta^*;\xi,\eta\right) &\displaystyle = {g}^{\tau_{j+1}}\left(\varDelta^*;\xi,\eta\right) - {g}^{\tau_{j}}\left(\varDelta^*;\xi,\eta\right). \end{array} \end{aligned} $$

Hawaii: Ritter and Christensen [71] believe that a stationary mantle plume located beneath the Hawaiian Islands created the Hawaii-Emperor seamount chain while the oceanic lithosphere continuously passed over it. The Hawaii-Emperor chain consists of about 100 volcanic islands, atolls, and seamounts that spread nearly 6000 km from the active volcanic island of Hawaii to the 75–80 million year old Emperor seamounts nearby the Aleutian trench. With moving further south east along the island chain, the geological age decreases. The interesting area is the relatively young southeastern part of the chain, situated on the Hawaiian swell, a 1200 km broad anomalously shallow region of the ocean floor, extending from the island of Hawaii to the Midway atoll. Here, a distinct gravity disturbance and geoid anomaly occurs that has its maximum around the youngest island that coincides with the maximum topography and both decrease in northwestern direction. The progressive decrease in terms of the geological age is believed to result from the continuous motion of the underlying plate (cf. [60, 91]).

With seismic tomography, several features of the Hawaiian mantle plume are gained (cf. [71] and the references therein). They result in a Low Velocity Zone (LVZ) beneath the lithosphere, starting at a depth of about 130–140 km beneath the central part of the island of Hawaii. So far, plumes have just been identified as low seismic velocity anomalies in the upper mantle and the transition zone, which is a fairly new achievement. As plumes are relatively thin with respect to their diameter, they are hard to detect in global tomography models. Hence, despite novel advances, there is still no general agreement on the fundamental questions concerning mantle plumes, like their depth of origin, their morphology, their longevity, and even their existence is still discussed controversially. This is due to the fact that many geophysical as well as geochemical observations can be explained by different plume models and even by models that do not include plumes at all (e.g., [15]). With our space-localized multiscale method of deriving gravitational signatures (more concretely, the disturbing potential) from the vertical deflections, we add a new component in specifying essential features of plumes. The vertical deflections of the plume in the region of Hawaii are visualized in Fig. 19.
Fig. 19

Illustration of the vertical deflections in the region of Hawaii (from the Ph.D.-thesis [13], Geomathematics Group, University of Kaiserslautern)

From the bandpass filtered detail approximation of the vertical deflections (Fig. 20) and the corresponding disturbing potential (Fig. 21), we are able to conclude that the Hawaii plume has an oblique layer structure. As can be seen in the lower scale (for which numerical evidence suggests that they reflect the higher depths), the strongest signal is located in the ocean in a westward direction of Hawaii. With increasing scale, i.e., lower depths, it moves more and more to the Big Island of Hawaii, i.e., in eastward direction.
Fig. 20

Approximation of the vector valued vertical deflections Θ in [ms−2] of the region of Hawaii (compare Fig. 19). A rough lowpass filtering at scale 6 is improved by several bandpass filters of scale j = 6, …, 11, the last picture shows the multiscale approximation at scale j = 12 (from the Ph.D.-thesis [13], Geomathematics Group, University of Kaiserslautern)

Fig. 21

Multiscale reconstruction of the disturbing potential T in [m2s−2] from vertical deflections Θ for the Hawaiian (plume) area using the scaling function gτ. A rough lowpass filtering at scale j = 6 is improved by several bandpass filters at scales j = 6, …, 11, the last illustration shows the approximation of the disturbing potential T at scale j = 12 (from the Ph.D.-thesis [13], Geomathematics Group, University of Kaiserslautern)

All in all, by the space-based multiscale techniques initiated by [29, 30] in gravitation we are able to come to interpretable results involving geological obligations in relation to hotspots/mantle plumes based on “surface interpretations” and just by looking at the anomalous behavior in terms of surface integrals without using the framework of Newton’s volume integrals.

5 Geoid Versus Quasigeoid

In what follows, we are strongly influenced by the work of [88]. As a matter of fact, we use parts of his highly instructive contribution almost literally to compare the conceptional background of geoid and quasigeoid. Nevertheless, seen from a modern mathematical point of view, we are not allowed to share all the conclusions of this contribution.

Following [88] we wish to argue that the classical, physically meaningful surface, the geoid, as introduced by [54] and mathematically discussed in the preceding chapter, is still not only the most natural surface to refer to, but also an artificial surface, called the quasigeoid should be used and computed from data on the surface of the Earth. In fact, it is well known in surveying practice that heights have to be referred to mean sea level. However, to obtain some heights of practical value, the mean sea level underneath the continents has to be known. The mean sea level anywhere more or less follows the gravity equipotential surface to a certain constant W0. As pointed out, the determination of such a horizontal surface, that best approximates the (mean) sea level leads to the geoid.

Two conceptually different kinds of height systems are commonly used in geodesy (cf. Fig. 22):
Fig. 22

Relation between geoidal, geodetic, and orthometric heights

  1. (i)

    Orthometric height: H is the “practical height” above the sea level used in mapping and engineering practice. The orthometric height of a point of interest is measured along the plumbline, a line always tangent to the gravity vector, form the geoid to the point of interest.

  2. (ii)
    Geodetic height (or ellipsoidal height): h is the height above the bi-axial “geocentric reference surface (ellipsoid)”, measured along the normal to the reference surface (following the concept of [88]). It can be readily determined from observations from satellites, but it is of very little practical use on their own. However, if the departure of the geoid from the geocentric reference surface (ellipsoid) N is subtracted from a geodetic height h, as we can see in Fig. 22, we get the orthometric height H, which subsequently can be used in practice. Geoidal heights are useful as an intermediary between satellite-observed heights (geodetic) and practical heights (orthometric). These three heights are consequently related by the equation:
    $$\displaystyle \begin{aligned} H=h-N.\end{aligned} $$
Fig. 23 shows that the geodetic height h can be computed from the satellite determined position (given in the Cartesian coordinates x1, x2, x3) exactly, if specific values for the size and shape (flattening) of the geocentric reference ellipsoid are adopted. The calculation is simply a matter of applying general geometrical principles. In Fig. 23, the center of the ellipsoid is coincident with the center-of-mass of the Earth by definition.
Fig. 23

Relation between Cartesian and curvilinear coordinates

Orthometric heights can be determined by a simple differential procedure which is quite accurate, but it is also slow, expensive, and prone to systematic errors. This classical process, the terrestrial levelling, has been used all around the world for more than a century. For economical reasons, the tendency today is to replace this process by satellite methods, which provide geodetic height differences. Satellite methods are almost as accurate as terrestrial levelling, particularly for larger distances, and much cheaper to use. If this approach is used, accurate knowledge of the geoidal heights on land becomes a prerequisite for converting geodetic heights to orthometric heights. Orthometric heights and geoidal heights are widely used around the world, particularly in America and in portions of Africa and Asia. More recently, there has been the decision in Canada and in the US to adopt orthometric heights and a geoidal model as their national systems of heights.

The determination of the geoid is a purely physical problem: if we knew the mass density distribution within the Earth we could compute the gravity field, including the gravity potential and thus the geoid, to any accuracy anywhere by calculating Newton’s volume integral. We would then get the geoid by simply connecting all the points of the same required (constant) value W0 of the potential. Unfortunately, we do not know the density distribution within the Earth to a sufficient accuracy to do this, so this approach cannot be used in practice. The only viable alternative is to use gravity values, which are cheap, plentiful and sufficiently accurate. If we have these, we can take advantage of the relation between gravity and gravity potential, as shown in Fig. 24.
Fig. 24

The relation between gravity w, its potential W, and local surfaces

Beyond the theoretical problem posed by the unknown topographic density, downward continuation is an ill-posed problem, which is known as inverse gravimetric problem (for more details and mollifier regularizations, see, e.g., [25]).

The fact that the topographic density was not known with an adequate accuracy back in the 1960s (and this problem lingers on still today) led Molodensky to declare the geoid impossible to determine to a sufficient accuracy and to introduce an alternative quantity known as the quasigeoid (see [59]). Methods of determining the quasigeoid have since been somewhat refined, especially by the formulation in terms of analytical continuation as described by [4], but also by numerous other mathematical and theoretical developments (e.g., [45, 49, 62] and [44]). The interplay of the quasigeoid with the geoid and the reference ellipsoid, is shown in Fig. 25.
Fig. 25

The relation between quasigeoid, geoid and reference ellipsoid (following [88])

The vertical distance between the quasigeoid and the reference surface (ellipsoid) is called the quasigeoidal height (also known as height anomaly) ζ. For the determination of the quasigeoid, it would not be necessary to know the topographic density as all the computations are done not on the geoidal surface but on the surface of the Earth (or at an almost identical surface to it, called the telluroid – see the definition below). Molodensky’s theory deals with the gravity potential outside the Earth’s surface. Molodensky’s approach does not require any knowledge of topographic density, as it deals only with the external field and needs only to know the geometry of the external field. On the other hand, as the approach is based on geometry, it requires integration over the surface of the Earth, or more precisely over the telluroid. The telluroid is a surface that looks like the Earth surface except that it is displaced from the Earth surface by the quasigeoidal height.

For the quasigeoid to have use in practice, a meaningful height system must be associated with it. This system is called normal heights and it is used in the countries of the former Soviet Union and nine other European countries (Germany, France, Sweden, Poland, Czech Republic, Slovak Republic, Hungary, Romania and Bulgaria). The normal height HN of a point on the topographical surface is defined as the height of the corresponding point on the telluroid above the reference surface (ellipsoid), measured along the normal plumbline. However, normal heights may equivalently be seen as heights of the topographical surface above the quasigeoid, also measured along the normal plumbline. The relation among the normal height HN, height anomaly ζ and geodetic height h is exactly the same as that among orthometric height, geoidal height and geodetic height (cf. Figs. 22 and 25):
$$\displaystyle \begin{aligned} H^N\approx h-\zeta. \end{aligned} $$
Following this concept (cf. [88]), normal heights and orthometric heights at open sea are exactly the same, while they may differ by up to one and a half meters on land.
The difference between the two surfaces – the geoidal surface and the telluroid – over which the integration for the geoid or quasigeoid determination respectively is carried out is as follows:
  • The geoid is a fairly smooth surface without any kinks, edges or other irregularities (as seen in Fig. 22).

  • The telluroid, or the Earth surface for that matter, is much rougher.

It is the common belief in geodesy that the Earth’s surface is not sufficiently smooth to allow approximate integration (as [88] pointed out). As a matter of fact, although the calamities should not be underestimated, the progress in numerical integration is high (see, e.g., [22]), so that the mathematical apparatus required by Molodensky’s approach should be realizable in its original meaning, in future.

Altogether, we are allowed to come to the following conclusion: The knowledge of topographic density is still a problem, but it can be resolved to an accuracy of a few centimeters if the geological formation of the crust is reasonably well known. To obtain the total uncertainty of the geoid, the uncertainty in the effect of irregular topographic density must be added to the uncertainty in geoid determination that comes from the approximations in the theory and the numerical computations. Molodensky was right 50 years ago and today, where the substantial increase in the knowledge of topographical density distribution have changed the situation substantially.

Once again, Molodensky’s approach does not require any knowledge of topographic density, but it needs the surface integration to be carried out over the surface of the Earth, or over the telluroid to be accurate. Up till now, geodesists’ opinion is that this cannot be done with sufficient accuracy. However, it may be questioned that this opinion is true in view of the tremendous progress in numerical integration.

6 Quasigeoid: Geometric Nature, Geomathematical Context, and Multiscale Decorrelation

As pointed out earlier, heights in geodesy come in the following variants (see also Table 1):
  1. (i)
    orthometric heights,
    Table 1

    Geodetic height type classification



    Measured from the geoid



    Measured from the reference surface (ellipsoid)



    Calculated as the difference N = h − H

  2. (ii)

    normal heights,

  3. (iii)

    geodetic heights,

  4. (iv)

    geopotential heights.


Each has its advantages and disadvantages. Geopotential heights are physical measures of potential energy (in [m2s−2]). Both orthometric and normal heights are geometrically defined. They are heights in meters above the sea level. Orthometric and normal heights differ in the way in which mean sea level is conceptually continued under the continental masses. The reference surface for orthometric heights is the geoid. Since measurements cannot be taken from the geoid, levelling is in use. It produces the practically most useful heights above sea level directly, the more economical use of GNSS-instruments (such as GPS) give heights above the reference surface (ellipsoid).

Once more, it should be mentioned that the relation among the “normal height” HN and the geodetic height h is exactly the same as that among orthometric height, geoidal height and geodetic height:
$$\displaystyle \begin{aligned} H^N=h-\zeta.\end{aligned} $$
The original problem leading to the Molodensky approach (in its rigorous formulation discussed, e.g., by [45, 48]) can be briefly described as follows:

Given, at all points on the Earth’s surface \(\partial \mathcal {G}\), the gravity potential W and the gravity vector w = ∇W, then the aim is to determine the quasigeoidal surface.

The quasigeoidal height determination is based on the fact that the Earth’s surface \(\partial \mathcal {G}\) is approximated by the boundary \(\partial \mathcal {T}\) of a (regular) region \(\mathcal {T}\), i.e., the telluroid (close to the Earth’s surface) with known gravitational potential U in \(\overline {\mathcal {T}\,\,^c}\), i.e. on the surface \(\partial \mathcal {T}\) and in the exterior of \(\partial \mathcal {T}\) (cf. Fig. 26). We assume that there exists a one-to-one correspondence between \(\partial \mathcal {G}\) and \(\partial \mathcal {T}.\) W is the actual potential and U is an approximation of W called the normal potential. As usual, we let u = ∇U which is called the normal gravity and w = ∇W called the actual gravity which is given on \(\partial \mathcal {G}\). Assume that, for given \(x\in \partial \mathcal {T}\), the point \(y\in \partial \mathcal {G}\) is the one associated to x by the one-to-one correspondence between \(\partial \mathcal {G}\) and \(\partial \mathcal {T}\) (cf. Fig. 26). The two points are connected by the normal height vector n = y − x.
Fig. 26

Earth’s surface \(\partial \mathcal {G}\), telluroid \(\partial \mathcal {T}\), and their one-to-one correspondence

Hence, a substitute formulation of the classical Molodensky problem is to determine the length of |n| = HN, i.e., the distance of \(\partial \mathcal {G}\) and the approximating telluroid along the one-to-one correspondence between \(\partial \mathcal {G}\) and \(\partial \mathcal {T}\). To this end we introduce
$$\displaystyle \begin{aligned} \delta W&=W|{}_{\partial \mathcal{G}}-U|{}_{\partial \mathcal{T}}, \end{aligned} $$
$$\displaystyle \begin{aligned} \delta w&=w|{}_{\partial \mathcal{G}}-u|{}_{\partial \mathcal{T}}, \end{aligned} $$
where δW is called the potential anomaly and δw is called gravity anomaly (see [45, 48, 49, 61]). Furthermore, we use the disturbing potential T by
$$\displaystyle \begin{aligned} T=W-U \end{aligned} $$
in \(\overline {\mathcal {G}},\) so that we have
$$\displaystyle \begin{aligned} \delta W&=T|{}_{\partial \mathcal{G}}+U|{}_{\partial \mathcal{G}}-U|{}_{\partial \mathcal{T}},{} \end{aligned} $$
$$\displaystyle \begin{aligned} \delta w&=w|{}_{\partial \mathcal{G}}-u|{}_{\partial \mathcal{T}}\,.{} \end{aligned} $$
Using the Taylor expansion of u and U in terms of n and neglecting terms of higher order in n (which represents no substantial loss of accuracy if a sufficiently close telluroid is chosen) we finally arrive at the approximations
$$\displaystyle \begin{aligned} \delta W (x)&=T(x)+u(x)\cdot n, \end{aligned} $$
$$\displaystyle \begin{aligned} \delta w(x)&=w(y)-u(y)+ \mathbf{m}(x) \ n, \end{aligned} $$
\(x \in \partial \mathcal {T}\), \(y \in \partial \mathcal {G}\), where we set
$$\displaystyle \begin{aligned} \mathbf{m} = \nabla u = \left( \frac{\partial ^2 U}{\partial x_i \partial x_j}\right)_{i,j=1,\ldots,3}. \end{aligned} $$
Observing the relations
$$\displaystyle \begin{aligned} w(y)-u(y) =(\nabla W)(y)-(\nabla U)(y) =(\nabla T)(y) =(\nabla T)(x) \end{aligned} $$
we arrive at
$$\displaystyle \begin{aligned} \delta W (x)&=T(x)+u(x)\cdot n,{} \end{aligned} $$
$$\displaystyle \begin{aligned} \delta w(x)&=(\nabla T)(x)+\mathbf{m}(x) \, n.{} \end{aligned} $$
Equation (108) is a counterpart to the (already known) Bruns formula. Actually it connects the disturbing potential T on the telluroid \(\partial \mathcal {T}\) with the anomalies between \(\partial \mathcal {G}\) and the telluroid \(\partial \mathcal {T}\). If we assume that m(x) is invertible for all \(x\in \partial \mathcal {T}\), we obtain by virtue of (107)
$$\displaystyle \begin{aligned} n=\mathbf{m}(x)^{-1}(\delta w(x)-(\nabla T)(x)), \end{aligned} $$
so that
$$\displaystyle \begin{aligned} |n|=|\mathbf{m}(x)^{-1}(\delta w(x)-(\nabla T)(x))|. \end{aligned} $$
Inserting the identity (110) into Eq. (108), we end up with
$$\displaystyle \begin{aligned} T(x)-u(x)\cdot(\mathbf{m}(x))^{-1}(\nabla T)(x)=\delta W(x)-u(x)\cdot\mathbf{m}(x)^{-1}\delta w(x).\end{aligned} $$
This is the so-called fundamental boundary condition for the Molodesky problem. It is formulated exclusively for points on the telluroid, and it does not need information about the topographic density.
Following [49] (see also [61]) the vector u(x)(m(x))−1 can be seen in first order to be oriented in the direction of the exterior unit normal field ν on the telluroid \(\partial \mathcal {T}\). More specifically,
$$\displaystyle \begin{aligned} u(x)(\mathbf{m}(x))^{-1}=-\frac{|x|}{2}\,\nu(x). \end{aligned} $$
Inserting expression (113) into Eq. (112) therefore results in the identity
$$\displaystyle \begin{aligned} \nu(x)\cdot(\nabla T)(x)+\frac{2}{|x|}\,T(x)=F(x),\end{aligned} $$
where we have used the abbreviation
$$\displaystyle \begin{aligned} F(x)=\nu(x) \cdot \delta w(x)+\frac{2}{|x|} \,\delta W(x) \end{aligned} $$
(note that the boundary condition (114) can be seen to be equivalent to (112) transformed in an appropriate coordinate system).

Summarizing all the steps of the linearization procedure we are led to discuss the following type of a boundary-value problem in potential theory (note that a constructive Runge solution is given, e.g., in [3, 23]).

Exterior Molodensky Problem (EMP): Find \(T \,{\in }\, {Pot}^{(1)} (\overline {\mathcal {T}\,\,^c}),\) i.e., \(T\,{\in }\, {C}^{(2)}(\mathcal {T}\,\,^c)\cap {C}^{(1)}(\overline { \mathcal {T}\,\,^c})\) with ΔT = 0 in \(\mathcal {T}\,\,^c\) and |T(x)| = O(|x|−1), |x|→, such that
$$\displaystyle \begin{aligned} \frac{\partial T}{\partial \nu}(x)+\mu(x) T(x)=F(x), \; x\in\partial \mathcal{T}, \end{aligned} $$
where μ, \(F\,{\in }\, {C}^{(0)}(\partial \mathcal {T})\) are known functions on the boundary surface \(\partial \mathcal {T}\) of the regular region \(\mathcal {T}\). Obviously, in mathematical nomenclature, the exterior Molodensky problem (EDP) forms a special Robin problem (cf. [47]).

It should be remarked that, in the case that \(\partial \mathcal {T}\) is a sphere, the problem becomes the well-known Stokes problem (see [40] or [61]) and in the case of an ellipsoid it is called ellipsoidal Stokes problem (see, e.g., [36, 56, 61]). Locally reflected multiscale solutions of the Stokes’ problem are due to [31, 34] (see also the references in [20]).

7 The German Quasigeoid GCG2016

A geodetic realization of a quasigeoid is the German Combined Quasigeoid 2016 (GCG2016). The GCG2016 is the official height reference surface of the Surveying Authorities of the Federal Republic of Germany. It corresponds to the reference ellipsoid of the Geodetic Reference System 1980 (GRS80). Among others the GCG2016 is well suited to transform ellipsoidal heights determined by GNSS observations into normal heights, and vice versa.

The GCG2016 is a core component of the so-called Integrated Geodetic Spatial Reference 2016 (IGSR2016), which has been introduced in 2016 as the official geodetic spatial reference in Germany. The IGSR2016 pursues an holistic approach of the geometrically and the physically defined components of geodetic spatial reference. This approach has been implemented by the national surveying authorities in Germany as the result of the extensive project “Renewal of the DHHN”.

Over the course of the project nearly the complete German first order leveling network has been remeasured between 2006 and 2012 to an accuracy (standard deviation) of only 3–8 mm. In addition, in 2008 an elaborate GNSS campaign over six weeks with GNSS observation times of at least two times 24 h at each station has been performed. In this process 250 so-called geodetic basic network points (“Geodätische Grundnetzpunkte” – GGP) have been determined to an accuracy of a few millimeters throughout Germany. Each of the GGP has been integrated into the remeasured leveling network. Furthermore, from 2009 to 2015 for the total number of GGP high precision gravimetric measurements have been performed, partially by absolute and partially by relative gravimetry.

Finally in 2016 the project led to a complete revision and upgrade of the authorized geodetic spatial reference systems and their realizations in Germany with its significant components as follows:

Geometrical Component:
  • European Terrestrial Reference System 1989 (ETRS89) in its official German realization: ETRS89/DREF91, realization 2016.

Physical Components:
  • German Main Height Network 2016 (DHHN2016) and

  • German Main Gravity Network 2016 (DHSN2016).

Components Linking the Geometrical and the Physical Copmponents:
  • GGP, each with high precision coordinates in ETRS89/DREF91, realization 2016 as well as precise normal heights in DHHN2016 und gravity values in DHSN2016,

  • Quasigeoid GCG2016 as physically defined height reference surface in relation to the mathematically defined ellipsoid GRS80.

These components in common constitute the IGSR2016 in Germany. They are of a piece and therefore completely consistent to each other. Thus, the result of the project is unique:
  • A significantly improved authorized geodetic spatial reference for Germany, where the approach of integrated geodesy (and thus strictly speaking the Bruns formula, cf. Sect. 4) is accomplished in practice for the first time for a complete country.

The modeling of the GCG2016 has been executed by order of the Working Committee of the Surveying Authorities of the states (Laender) of the Federal Republic of Germany (AdV) by two independent institutions:
  • The Federal Agency for Cartography and Geodesy (BKG),

  • the Institute of Geodesy of the Leibniz University Hannover (IfE).

Both institutions processed the extensive data provided by the federal states and by other sources with basically different approaches. The results of both institutions were alike and the GCG2016 (Fig. 27) finally has been computed by averaging the two independent solutions of BKG and IfE.
Fig. 27

German Combined Quasigeoid GCG 2016

The GCG2016 is provided as a grid over the whole area of Germany. The grid resolution is 30″ by 45″ (latitude by longitude), which means in metric system about 0.9 km by 0.9 km. The standard deviations of the height anomalies are specified as follows: In lowlands and uplands 1 cm, in the alpine mountains 2 cm and in marine areas 2–6 cm.

For more details concerning the respective approaches, the computation, the properties and the providing of the GCG2016 see the publications of BKG:

8 Acquisition and Provision of Gravity Data for the Quasigeoid Modeling

An essential goal of the survey of gravity data at the State Office for Surveying and Geobase Information Rhineland-Palatinate (LVermGeo) is the derivation of a consistent height reference surface (quasigeoid) with an accuracy in the centimeter range and better. This is a mandatory prerequisite for satellite-supported altitude determination using the official satellite positioning service SAPOS [76, 77]. For the computation of such an accurate quasigeoid amongst others precise, up-to-date, and sufficiently dense gravity values are necessary.

This task can not be achieved with the data of current satellite gravity field missions alone, since the spatial resolution is not sufficient. Therefore, precise terrestrial measurements of the gravitational field at the Earth’s surface with measuring point distances of at most 1–2 km can not be dispensed with in the near future.

Furthermore, a uniform geodetic spatial reference of all measurement data sets (coordinates of the gravimetric measurement points, ellipsoidal and physical heights, digital terrain model (DTM)) is a basic requirement for quasigeoid modeling. This requirement is fulfilled in Germany since 2016 with the unitary Integrated Geodetic Spatial Reference 2016 (IGSR2016) with its components ETRS89/DREF91 (Realization 2016), German Main Height Network 2016 (DHHN2016), German Main Gravity Network 2016 (DHSN2016) and German Combined Quasigeoid 2016 (GCG2016) [1, 14].

In order to be able to estimate the quality of the terrestrial gravity data used for the modeling of the quasigeoid, it is important to know the origin of this data [90].

To ensure the German national gravity standard, the Federal Agency for Cartography and Geodesy (BKG) maintains the German Basic Gravity Network (DSGN). The DSGN forms the overall reference framework for the German Main Gravity Network (DHSN), for which the German federal states are responsible. In the course of time, various implementations of the DSGN and the DHSN emerged:
  • German Basic Gravity Network 1976 (DSGN76) and German Main Gravity Network 1982 (DHSN82) On the basis of a few DSGN76 points (datum points) determined with absolute gravimeters, the DHSN82 has been realized by the official gravity fixed points of 1st order (SFP 1.O.). These have been determined by the states of the Federal Republic of Germany in the years 1978 to 1984 by gravity measurements with relative gravimeters [83].

  • German Basic Gravity Network 1994 (DSGN94) and German Main Gravity Network 1996 (DHSN96) After the reunification of Germany, the DSGN94 has been created by re-measuring the benchmarks of the DSGN76 with absolute gravimeters and extending the network to the eastern part of Germany (see [70, 86]). Subsequently, the countries introduced the DHSN96. In the eastern part of Germany, the DHSN96 has been established by direct connection to the new absolute gravity stations of the DSGN94. In the western federal states, the DHSN82 was adapted to the DSGN94 by a constant level shift around the mean level difference between DSGN94 and DSGN76 of − 19 μGal.

  • German Basic Gravity Network 2016 (DSGN2016) and German Main Gravity Network 2016 (DHSN2016) The DSGN2016 has been established in the context of the generation of the Integrated Geodetic Spatial Reference 2016. The DSGN2016 consists of the fixed points of the previous DSGN94 as well as selected benchmarks of the Integrated Geodetic Reference Network (GREF) of the BKG. New gravimatric values have been determined by high precision gravimetry for all benchmarks of the DSGN. The accordance of the applied field-suitable absolute gravimeters with the International Gravity Standardization Network 1971 (IGSN71) is ensured by periodic measurements on gravimetric reference stations. The current DHSN2016 has the same level and scale as the DHSN96, but it differs in terms of accuracy, reliability and currency.

The densification of the German Main Gravity Network (DHSN) below the SFP of first order is in the responsibility of the federal states. Thus, in Rhineland-Palatinate, the second-order SFP network was created, which in turn served as a reference network for the subsequent measurements (SFP of 3rd order). The primary goal of 3rd order gravity measurements was to determine the gravity values of height fixed points in order to calculate the physically defined normal heights of the official height fixed points.

In order to close the remaining gaps and to realize a considerable better coverage of the territory of Rhineland-Palatinate the gravity values of a special kind of existing fixed points, the so-called superordinate fixed points (ÜFP), have been determined at a later time. As a result, surface gravity values measured for the entire territory area of Rhineland-Palatinate with a density of approx. 1 point per 8 km2 were available. The density of the data was much higher along the leveling lines due to the SFP of 3rd order. However, there remain already some extended larger areas of significantly lower density, such as, e.g., military training areas, an airport, large forest areas and peripheral areas along the state border.

A certain disadvantage for the quasigeoid modeling is undoubtedly that the actually available gravity data meanwhile have an average age of 20–25 years, and that intermediate gravity changes are thus not sufficiently reflected in the data.

The accuracy of the SFP depends on the order and amounts in the range between 10 and 30 ⋅ 10−8 m∕s2(μGal) with respect to the sensitive point of the gravimeter and to the time of the measurement. The standard use of the theoretical vertical gravity gradient to reduce the measured gravity values from the sensitive point of the gravimeter to the Earth’s surface additionally reduces the accuracy of the gravity values. It should also be kept in mind that, for reasons of economy, parts of the SFP network of the 3rd order could not be determined by double measurements but only by uncontrolled simple measurements.

The requirements for the gravity data for the computation of the GCG2016 [53] are described as follows:
  • Current gravity values (SW),

  • Density of the SW as a function of the roughness of the gravitational field, i.e., in regions with larger horizontal gradients of Bouguer anomalies, the density of the measurements should be higher, since in these areas the quasigeoid is structured more irregularly,

  • uniform density of at least one SW per 4 × 4 km to max. two SW per 2 × 2 km,

  • Accuracy of the gravity measurement 50–100 μGal,

  • Accuracy of georeferencing the height better than 0.3 m,

  • Accuracy of georeferencing the location better than 3 m,

  • Points must be representative of the surrounding topography,

  • suitable conditions for GNSS and gravity measurement.

To meet these requirements, the federal states were asked by resolution AK RB 09/20 of the Working Group “Spatial Reference” of the Working Committee of the Surveying Authorities of the states of the Federal Republic of Germany (AdV) to carry out supplementary area-covering gravity measurements and to make these additional gravity data available to the BKG for quasigeoid modeling. As a result, in the State Office for Surveying and Geobase Information Rhineland-Palatinate (LVermGeo), between August 2014 and November 2015, around 1200 further gravity points were determined, affiliated to the second-order SFP. These points are locally unmarked. They have been georeferenced by GNSS measurements by assistance of SAPOS. The gravimetric measurements have been carried out with modern relative gravimeters [79] with an accuracy better than 50 μGal.

Depending on the necessary travelling time, the location of the affiliating points, the existing road infrastructure, the applied measuring method and the respective local conditions, it is possible to measure approximately 6 to 8 points per working day in this way. The measurement in extensive forest areas is particularly complex, since the GNSS measurement under these conditions as is known is limited in space (clearances, clearings) and time (season with little foliage). In large cities, due to the strong anthropogenic microseismics, extended measuring times must be expected.

With the official gravity fixed points (SFP) and additional area-covering gravity measurements on unmarked points, approximately 9000 SW were available for the computation of the GCG2016 for the subarea of Rhineland-Palatinate.

Subsequently, in cooperation with the BKG, a comprehensive evaluation of the existing gravity data with regard to suitability for quasigeoid modeling was carried out according to the following criteria and limits:
  • Plausibility check of the gravity values by means of gravity prediction on the basis of Bouguer anomalies (maximum permissible difference: 3 mGal).

  • Comparison of the heights of the measured points with a current high-resolution terrain model. For this purpose the official Digital Terrain Model (DGM25) was used. Larger deviations (over 5 m) indicate unrepresentative or not in the DGM25 resolved point locations such as bridges, towers or steep slopes. Also, erroneous or inaccurate georeferencing of the measured gravity points (e.g., caused by poor digitization) may be the cause of larger deviations.

  • Sighting the survey sketches, point descriptions and photos of SFP located on or in the immediate vicinity of rock or massive buildings, thus influencing the measured gravity value through the gravitational effect of their masses. By calculations with different block models it could be estimated that the gravity value, e.g., at a height bench mark, which typically is mounted in the exterior wall of a church building, depending on the wall thickness, material and construction, is measured too low by a few 100 μGal.

Detected points with differences greater than the predetermined limits were systematically examined to identify the cause of the differences. Erroneous gravity data or SFP unsuitable for quasigeoid modeling were identified and thus excluded from the GCG2016’s further calculations.

On the first of December 2016, the GCG2016 was published and officially introduced as part of the official German Integrated Geodetic Spatial Reference 2016 (IGSR2016). Thus, for the first time, there was a height reference surface valid throughout Germany which was completely consistent with the official reference frames with an accuracy (standard deviation) of 1 cm in the lowlands and in the low mountain ranges, 2 cm in the Alps and 2–6 cm in the sea area. Thus, the geometric and the physical components of the integrated geodetic spatial reference can now be linked with high accuracy.

It is planned to calculate and publish an updated and improved version of the GCG at the beginning of the 2020s. Further area-covering gravity measurements are already taking place in the State Office for Surveying and Geobase Information Rhineland-Palatinate (LVermGeo) with the aim of filling the gaps left by the evaluation mentioned above, and to adapt the density of the measured gravity data to the ever increasing demands. This is intended to further increase the accuracy, but in particular the reliability, of the future GCG. It is also being considered to systematically review the existing gravity data and, if necessary, to replace it with current gravity data. This is particularly useful where major gravity changes have occurred due to mass changes (e.g., mining of mineral resources, reservoirs, pumped storage facilities, etc.).

9 Multiscale Decorrelation of Gravimetric Data

Next we present a multiscale decomposition technique based on the Bouguer anomalies and the quasigeoidal undulations for subareas of Rhineland-Palatinate (see Fig. 28 for the landscapes and Fig. 29 for the geological subareas). As multiscale tools we use Haar wavelets and scaling functions for the decomposition (see [5, 19, 20] for more details).
Fig. 28

Landscapes in Rhineland-Palatinate (taken from [52] in modified form)

Fig. 29

Geological subareas in Rhineland-Palatinate (taken from [9] in modified form)

The geologic units of Rhineland Palatinate can be divided as follows (see [51]): In the northern part we have the Rhenish Slate Mountains (or Rhenish Massif) with anticlines and synclines striking SW-NE and overthrusts (see [93]). During the Variscan orogeny, the Rhenish Slate Mountains represented a part of the Rhenohercynian zone as a fold-and-thrust belt (see [58]). It is mainly composed of slates, siltstones, sandstones and quartzite from the Lower Devonian, where igneous rocks as keratophyres and diabases have penetrated. Subordinately, lime and dolomite rocks occur. Middle Devonian limestones can be found in the Eifel depression. The regional structures are generally trending SW-NE with NW-facing folds and overthrusts (NW-vergence), see [51, 58].

The main tectono-stratigraphic units are from north to south separated by the Siegen Main Thrust, the Boppard-Görgeshausen Thrust and the Southern Hunsrück Boundary Fault. The southern partly thrust sheet is the metamorphic zone of the southern Hunsrück (Northern Phyllite Zone), see [58]. It forms a multiply folded imbricate belt of partly mylonitic metasediments and metavolcanics. This thrust sheet is cut by the Southern Hunsrück Boundary Fault. The rather fold-dominated Eifel and thrust-dominated Hunsrück situated in the south of the Rhenish Slate Mountains are separated by a Permian graben, the so-called Wittlicher Senke bounded by normal faults (see [78]). From western direction Mesozoic layers (Buntsandstein, Muschelkalk, Keuper, and Lower Jura) overlap the Rhenish Slate Mountains in the Trier Basin. The Buntsandstein consists typically of red-colored, fluviatile sandstone. The Muschelkalk transgression results in marine conditions, which comprise various sandy-marly-dolomitic rock types. The carbonate sedimentation of the Muschelkalk is terminated by a regional regression (see [78]). The deposits in the follow age of the Keuper are sandy-clayey-marly and colored. The transgression of the Lias (Lower Jura) sea finished the continental and marginal marine shaped sedimentation of the Triassic. Marine marl abundant in fossils and limestone with interbedded sandstones occure.

Eifel and Westerwald are marked with Tertiary and Quaternary Volcanics from basaltic origin. In between is the subsidence area of the Neuwied Basin with sediments as gravel, sand and clay as well as Pleistocene volcanic deposits, e.g., pumice.

The southern edge of the Rhenish Slate Mountains is adjacent to the Saar-Nahe-Basin with a sequence of Permian-Carboniferous sedimentary rock (clay-, silt- and sandstone) and Volcanics (rhyolithe, dacite and andesite) of the Rotliegend of a thickness of some 1000 m shaped as trough in the north and as saddle in the south (see, [93]). At the northern margin of the Saar-Nahe Basin, a pronounced fault zone, the Southern Hunsück Boundary Fault represents a large southward-dipping detachment fault, and is responsible for the Saar-Nahe Basin being a half-graben (see, [55]).

The southern part of Rhineland-Palatinate – the Palatine Trough – is covered with rock of Triassic, mostly sandy conglomeratic deposits of the Buntsandstein. They can be found in a shallow syncline where Muschelkalk is widespread in its center in the southwest.

The Cenozoic Upper Rhine Graben crosses the eastern part of the Permocarboniferous Saar-Nahe Basin. Due to the subsidence in this graben structure, the top of the up to 2 km thick Permocarboniferous is located at a depth of 600–2.900 m and is overlain by Tertiary and Quaternary sediments (see [2]). The partly 4.000 m thick graben is filled up with mighty sediment sequences of marl- and limestone as well as gravels and sands (see, [93]). The Mainz Basin was formed as an extension at its northern end. It is essentially built from the sedimentary deposits of the Tertiary, which can be subdivided into the lying silty-marly layer of the Oligocene and the overlying calcareous Miocene sediments. The pre-Tertiary rocks of the Mainz Basin consist mainly of Permian (Rotliegend) rhyolithes and siliciclastics of the Saar-Nahe Basin.

Fig. 30 shows the decompositon of the Bouguer data with the lowpass filtered approximations on the left and associated bandpass filtered detail information on the right. The lowpass filtered version at scale j = 2 only provides the coarse structure of the Bouguer anomaly. The multiscale method justifies that, with increasing scales, the lowpass filtered Bouguer anomaly converges more and more to the input Bouguer anomaly data.
Fig. 30

Decorrelation of the Bouguer anomaly for a subarea of Rhineland Palatinate, Germany, via a multiscale approach for different scales. Bandpass filtered data on the right, Lowpass filtered data on the left. (a) Scale j = 2. (b) Scale j = 2. (c) Scale j = 3. (d) Scale j = 3. (e) Scale j = 4. (f) Scale j = 4. (g) Scale j = 5. (h) Scale j = 5. (i) Scale j = 6. (j) Scale j = 6. (k) Scale j = 7

The most important structures are in general well outlined by anomalies. Positive anomalies usually indicate thrusts or anticlines and negative anomalies synclines or depressions, respectively. The anomalies or the boundary between anomalies are often correlated with major faults observable at the surface. Starting with the decorrelation at scale j = 3 of the bandpass filtering of the area under investigation the geological structures can be roughly assigned. Especially the Mainz Basin sticks out with an almost concentric positive anomaly, whereas the Upper Rhine Graben has a minimum.

Lineaments and rough shape of the tectonical units and a structural direction can be observed at scale j = 4. The map reveals the main SW-NE striking Variscan structures below the Rhenish Slate Mountains, the Saar-Nahe Basin and the Wittlicher Senke. In contrast, the Mainz Basin and the Upper Rhine Graben are characterized by the relatively large anomaly. The shape of the basin is more clearly defined, as well as in the northwest the Trier Basin with a sligtly negative anomaly. The positive and negative anomalies occur due to greater fault zones. This is most obvious in the middle of the map in the Hunsrück Slate zone between quartzite/phillite and claystone.

At scale j = 5 a refinement of the tectono-stratigraphic units becomes visible. Thus, more structures can be seen as, e.g. the Mosel Basin. Furthermore, in addition to the maximum in the north of the Southern Hunsrück Boundary Fault a minimum in the northern part is constituted, which corresponds to the quartzite.The positive anomaly in the east correlates with the Northern Phyllite Zone. Between the Saar-Nahe Basin and the Mainz Basin the fault zone at the border fault is indicated by an axis of minima. Altogether, the illustration of the tectonical structures is refined.

The finest formation of the tectonical structure becomes visible at scale j = 6 and different geological relations have an effect onto the map. The structures which have been elaborated at scale j = 5 become even more obvious. The SW-NE oriented line of maxima in the middle of the map are sligtly shifted in southern direction compared to j = 5 and mark the Southern Hunsrück Boundary Fault. In the Mainz Basin, the contour of the positive anomaly is smaller and it shows a more local importance. This can be correlated to the mighty Permian and Tertiary magmatite below a thin layer of sediments. The anomaly at the edge of the Graben between the Saar-Nahe Basin and the Mainz Basin shows more exactly the course of the border fault. The Wittlich Basin is clearly defined by negative anomalies. The weakness area caused of many SSW-NNE oriented faults between the Trier Basin and the Rhenish Slate Mountains, is indicated by a line of minima.

In accordance with the quasigeoidal definition, the multiscale decomposition of the quasigeoid (Fig. 31) does not indicate in the same way a remarkable match to the geologically based structures as the Bouguer anomaly. Nonetheless, at scales j = 4, 5, we can still detect the Rhenish Slate Mountains, the Saar-Nahe Basin, and the Mainz Basin, but the border lines are less obvious. It seems that density contrasts much deeper inside the Earth play a particular role.
Fig. 31

Decorrelation of the quasigeoid (GCG2016) for a subarea of Rhineland Palatinate, Germany, via a multiscale approach. Bandpass filtered data on the right, Lowpass filtered data on the left. (a) Scale j = 3. (b) Scale j = 3. (c) Scale j = 4. (d) Scale j = 4. (e) Scale j = 5. (f) Scale j = 5. (g) Scale j = 6. (h) Scale j = 6. (i) Scale j = 7

Summarizing our multiscale results, we are led to the following conclusions:

The multiscale approach as proposed in this contribution breaks up a complicated signal such as the Bouguer anomaly field into “wave band signatures” at different scales, i.e., a certain resolution. To each scale parameter, a scaling function is defined leading to an approximation of the data at this particular resolution. The difference between two successive scaling functions, i.e., in the jargon of constructive approximation, the wavelets represent the corresponding wave bands. They yield desired geologically based detail information. With increasing scale, the approximation is getting finer and finer starting form a lowpass approximation and adding more and more wave bands. The multiscale approach guarantees that the lowpass information contained on a certain (coarse) level is also contained in the approximations of higher scales. Thus, it is advantageous that we are able to analyze the wave bands separately (decorrelation). In doing so, the multiscale concept helps to find adaptive methods to the particular structure of the input data. Additionally, the resolution of the model can be adapted to the spatial structures, i.e., for areas with coarse spatial structures, the resolution of the model can be chosen to be rather low, and for areas with complicated structures the resolution can be increased accordingly. Consequently, since most data show correlation both in space as well as in frequency, the multiscale technique is an appropriate method for a simultaneous space and frequency localization. As far as the numerical realization is concerned, fast wavelet methods (FWT) are applicable.

Considering especially quantities involving the disturbing potential field in the outer space via boundary values, we observe – from computational point of view – two major requirements: First, the field characterisics of geodetic features are usually of local character such that the use of local wavelets is evident. Second, in view of the physical relevance of the multiscale approach, we need wavelets which have a certain relation to the corresponding partial differential equation (here, the Laplace equation). Moreover, we have to be concerned with wavelet types which are manageable from mathematical point of view and, additionally, show a close relation to the physical model (for examples, the reader is referred to the following contributions of the Geomathematics Group, Kaiserslautern: [5, 6, 7, 17, 20, 21, 24, 27, 29, 30], and the list of references therein).

All in all, the main results and characteristics of our multiscale method involving the (quasi)geoidal model can be summarized as follows:
  • Physically based behavior and appropriate interpretability of detail information via the developed wavelet (band) structures.

  • Numerical efficiency and economy by virtue of the wavelets enabling an adaptive choice of the local support and resulting in fast algorithms.

  • Scale dependent decorrelation into wavebands and scale dependent detection of specific geodetic/geologic structures within a systematic “zooming-in/zooming out” multiresolution process.

10 Conclusion

Local knowledge of the gravity potential and its equipotential (level) surfaces have become an important issue not only in geodetic modeling but also for geological interpretation, e.g., for purposes of exploration and prospecting (see, e.g., [7] and the references therein). Indeed, the gravity field is a key component of future investigation. Seen from a numerical point of view, the way forward has to focus on two challenges:
  1. (i)

    It is commonly known that highly accurate sensors, when operating in an isolated manner, have their shortcomings. Combining globally available satellite data with regional airborne and/or local terrestrial observations within a physically founded and mathematically consistent multiscale process is therefore an essential step forward.

  2. (ii)

    Based on the excellent geodetic pre-work thus far (such as GCG2016 for Germany), a vertical as well as horizontal “zooming-in” detection of specific geological/geophysical attributes is an outstanding field of interest for validating the multiresolution method based on heterogeneous datasets and geophysically oriented multiscale “downward continuation” modeling of the different data sources starting from spaceborne data as a trend solution via more accurate airborne data down to high-precision local data sets.

  3. (iii)

    For data supplementation and numerical stabilization, spaceborne data are indispensable even for local purposes because of their a priory trend pre-determination. The unfortunate terrestrial situation with larger gaps in most countries causes particular mathematical attention for homogenization and unification to suppress undesired oscillation phenomena within the numerical modeling process of the data.

  4. (iii)

    The whole spectrum of spaceborne/airborne/ground data systems covers all verifiable “signature wave packages”. Actually, the advantage of satellite lower frequency band data at the ground is their availability everywhere, while (airborne) medium and (terrestrial) high(er) frequency bands are merely at the disposal for regional and local occurrence, respectively. Geologically relevant signatures (as discussed, e.g., in [25] for purposes of inverse gravimetry), however, presuppose an extremely dense and highly accurate gravitational input data material (better than the usual geodetic situations today), in particular if more detailed internal structures are of interest.


Altogether, the connecting link for all requirements is a vertical/horizontal multiscale philosophy including all data information, where the localization in space enables us to handle the data dependent of their space availability and density for a particular area and the localization in frequency bands provides appropriate decorrelation of specifically demanded features contained in the signatures.



The authors C. Blick, W. Freeden, Z. Hauler, and H. Nutz thank the “Federal Ministry for Economic Affairs and Energy, Berlin” and the “Project Management Jülich” (PtJ-corporate managers Dr. V. Monser, Dr. S. Schreiber) for funding the projects “GEOFÜND” (funding reference number: 0325512A, PI Prof. Dr. W. Freeden, University of Kaiserslautern, Germany) and “SPE” (funding reference number: 0324061, PI Prof. Dr. W. Freeden, CBM – Gesellschaft für Consulting, Business und Management mbH, Bexbach, Germany, corporate manager Prof. Dr. M. Bauer).


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Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  • Gerhard Berg
    • 1
  • Christian Blick
    • 2
  • Matthias Cieslack
    • 1
  • Willi Freeden
    • 3
    Email author
  • Zita Hauler
    • 2
  • Helga Nutz
    • 2
  1. 1.Landesamt für Vermessung und Geobasisinformation Rheinland-PfalzKoblenzDeutschland
  2. 2.CBM – Gesellschaft für Consulting, Business und Management mbHBexbachDeutschland
  3. 3.Mathematics DepartmentUniversity of KaiserslauternKaiserslauternDeutschland

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Arbeitsgruppe Geomathematik, Fachbereich MathematikTU KaiserslauternKaiserslauternDeutschland

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