Handbuch der Geodäsie pp 164  Cite as
Gravimetric Measurements, Gravity Anomalies, Geoid, Quasigeoid: Theoretical Background and Multiscale Modeling/Gravimetrische Messungen, Schwereanomalien, Geoid, Quasigeoid: Theoretischer Hintergrund und Multiskalenmodellierung
Zusammenfassung
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. WaveletDekorrelationen werden für ein bestimmtes Gebiet in RheinlandPfalz illustriert.
Schlüsselwörter
Gravimetrie Schwereanomalien Geoid versus Quasigeoid MultiskalendekorrelationAbstract
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 RhinelandPalatinate.
Keywords
Gravimetry Gravity anomalies Geoid versus quasigeoid Multiscale decorrelation1 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 satelliteonly 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 “zoomingin” approach will provide a breakthrough in local modeling to understand significant structures and processes of the Earth’s system.
 (i)
Vertical multiscale modeling, i.e., “zoomingin 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) highprecision terrestrial gravimetric data sets (this aspect has been investigated in several publications, e.g., [18, 20, 24, 26, 27], and the references therein).
 (ii)
Horizontal multiscale modeling, i.e., “zoomingin 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).
 (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.
2 Gravity and Gravitation
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 socalled 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].
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.
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 PoincareBey 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).
 The attraction of, e.g., the reference ellipsoid/spheroid has to be subtracted from the measured values.
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).
 (1)
Gravimetry is decisive for geodetic purposes of modeling gravity anomalies, geoidal undulations, and quasigeoidal heights.
 (2)
Gravimetry is helpful in different phases of the oil exploration and production processes as well as in geothermal research.
 (3)
Archaeological and geotechnical studies aim at the mapping of subsurface voids and overburden variations.
 (4)
Gravimetric campaigns may be applied for groundwater and environmental studies. They help to map aquifers to provide formations and/or structural control.
 (5)
Gravimetric studies give information about tectonically derived changes and volcanological phenomena.
 (i)
determination of geodetic key observables for modeling gravity anomalies and definition of geological structural settings, such as hotspots and plumes,
 (ii)
faults delineation,
 (iii)
recovery of salt bodies,
 (iv)
metal deposits,
 (v)
forward modeling, inversion (i.e., inverse gravimetry),
 (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
 (a)
the center of the reference surface (ellipsoid) coincides with the center of gravity of the Earth,
 (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.
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.
if the Earth’s center of gravity is the origin, there are no firstdegree 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.
 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.
 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.
As already explained, the task of determining the disturbing potential T from gravity disturbances or gravity anomalies, respectively, leads to boundaryvalue problems usually corresponding to a spherical boundary. Numerical realizations of such boundaryvalue 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 spaceregularization techniques applied to the resulting integral representations of their solutions. For both boundaryvalue 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 GNSStechnology (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).
 (i)
\( F{ }_{\mathbb {R}^3 \backslash \overline {\mathbb {B}_{R}^3}} \) is a member of \(C^{(2)}(\mathbb {R}^3 \backslash {\overline {\mathbb {B}_{R}^3}})\),
 (ii)
F satisfies Laplace’s equation ΔF(x) = 0, \(x \in \mathbb {R}^3 \backslash {\overline {\mathbb {B}_{R}^3}}.\)
 (iii)
F is regular at infinity, i.e., F(x) = O(x^{−1}), x→∞.
 (i)
the mass within the reference ellipsoid is equal to the mass of the Earth,
 (ii)
the center of the reference ellipsoid coincides with the center of the Earth,
 (iii)
the formulation is given in the spherical context to guarantee economical and efficient numerics.
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 signaltonoise ratio, etc. and, in particular, decorrelation.
We follow the space (taylorized) regularization methods presented in [31] for linear regularization of the singlelayer 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].
Satz 1
Satz 2
Satz 3
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 spaceregularization techniques enable us to formulate multiscale solutions for the disturbing potential from gravity disturbances or vertical deflections. Note that we need higherorder 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 singlelayer kernel in the integral representation of the solution of the Neumann boundaryvalue problem (ENPPG).
By aid of Lemma 1, we obtain
Theorem 1
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.
Theorem 2
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).
Theorem 3
Hawaii: Ritter and Christensen [71] believe that a stationary mantle plume located beneath the Hawaiian Islands created the HawaiiEmperor seamount chain while the oceanic lithosphere continuously passed over it. The HawaiiEmperor 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]).
All in all, by the spacebased 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 W_{0}. As pointed out, the determination of such a horizontal surface, that best approximates the (mean) sea level leads to the geoid.
 (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.
 (ii)Geodetic height (or ellipsoidal height): h is the height above the biaxial “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 satelliteobserved heights (geodetic) and practical heights (orthometric). These three heights are consequently related by the equation:$$\displaystyle \begin{aligned} H=hN.\end{aligned} $$(96)
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.
Beyond the theoretical problem posed by the unknown topographic density, downward continuation is an illposed problem, which is known as inverse gravimetric problem (for more details and mollifier regularizations, see, e.g., [25]).
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.
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.
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
 (i)orthometric heights,Table 1
Geodetic height type classification
Orthometric
H
Measured from the geoid
Geodetic
h
Measured from the reference surface (ellipsoid)
Geoidal
N
Calculated as the difference N = h − H
 (ii)
normal heights,
 (iii)
geodetic heights,
 (iv)
geopotential heights.
Each has its advantages and disadvantages. Geopotential heights are physical measures of potential energy (in [m^{2}s^{−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 GNSSinstruments (such as GPS) give heights above the reference surface (ellipsoid).
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.
Summarizing all the steps of the linearization procedure we are led to discuss the following type of a boundaryvalue problem in potential theory (note that a constructive Runge solution is given, e.g., in [3, 23]).
It should be remarked that, in the case that \(\partial \mathcal {T}\) is a sphere, the problem becomes the wellknown 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 socalled 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 socalled 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:

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

German Main Height Network 2016 (DHHN2016) and

German Main Gravity Network 2016 (DHSN2016).

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.
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 Federal Agency for Cartography and Geodesy (BKG),
the Institute of Geodesy of the Leibniz University Hannover (IfE).
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.
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 RhinelandPalatinate (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 satellitesupported altitude determination using the official satellite positioning service SAPOS [76, 77]. For the computation of such an accurate quasigeoid amongst others precise, uptodate, 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].
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 remeasuring 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 fieldsuitable 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 RhinelandPalatinate, the secondorder 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 RhinelandPalatinate the gravity values of a special kind of existing fixed points, the socalled superordinate fixed points (ÜFP), have been determined at a later time. As a result, surface gravity values measured for the entire territory area of RhinelandPalatinate with a density of approx. 1 point per 8 km^{2} 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∕s^{2}(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.
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 areacovering 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 RhinelandPalatinate (LVermGeo), between August 2014 and November 2015, around 1200 further gravity points were determined, affiliated to the secondorder 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 areacovering gravity measurements on unmarked points, approximately 9000 SW were available for the computation of the GCG2016 for the subarea of RhinelandPalatinate.
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 highresolution 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 areacovering gravity measurements are already taking place in the State Office for Surveying and Geobase Information RhinelandPalatinate (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
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 SWNE and overthrusts (see [93]). During the Variscan orogeny, the Rhenish Slate Mountains represented a part of the Rhenohercynian zone as a foldandthrust 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 SWNE with NWfacing folds and overthrusts (NWvergence), see [51, 58].
The main tectonostratigraphic units are from north to south separated by the Siegen Main Thrust, the BoppardGörgeshausen 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 folddominated Eifel and thrustdominated Hunsrück situated in the south of the Rhenish Slate Mountains are separated by a Permian graben, the socalled 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 redcolored, fluviatile sandstone. The Muschelkalk transgression results in marine conditions, which comprise various sandymarlydolomitic 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 sandyclayeymarly 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 SaarNaheBasin with a sequence of PermianCarboniferous 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 SaarNahe Basin, a pronounced fault zone, the Southern Hunsück Boundary Fault represents a large southwarddipping detachment fault, and is responsible for the SaarNahe Basin being a halfgraben (see, [55]).
The southern part of RhinelandPalatinate – 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 SaarNahe 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 siltymarly layer of the Oligocene and the overlying calcareous Miocene sediments. The preTertiary rocks of the Mainz Basin consist mainly of Permian (Rotliegend) rhyolithes and siliciclastics of the SaarNahe Basin.
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 SWNE striking Variscan structures below the Rhenish Slate Mountains, the SaarNahe 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 tectonostratigraphic 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 SaarNahe 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 SWNE 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 SaarNahe 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 SSWNNE oriented faults between the Trier Basin and the Rhenish Slate Mountains, is indicated by a line of minima.
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).
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 “zoomingin/zooming out” multiresolution process.
10 Conclusion
 (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.
 (ii)
Based on the excellent geodetic prework thus far (such as GCG2016 for Germany), a vertical as well as horizontal “zoomingin” 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 highprecision local data sets.
 (iii)
For data supplementation and numerical stabilization, spaceborne data are indispensable even for local purposes because of their a priory trend predetermination. 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.
 (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.
Notes
Acknowledgements
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” (PtJcorporate 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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