Handbuch der Geodäsie pp 1-71 | Cite as
Satellite Gravitational Gradiometry: Methodological Foundation and Geomathematical Advances/ Satellitengradiometrie: Methodologische Fundierung und geomathematische Fortschritte
Abstract
Satellite Gravitational Gradiometry (SGG) is an observational technique of globally establishing the fine structure and the characteristics of the external Earth’s gravitational field. The “Gravity field and steady-state Ocean Circulation Explorer” GOCE (2009–2013) was the first satellite of ESA’s satellite programme intended to realize the principle of SGG and to deliver useful SGG-data sets. In fact, GOCE was capable to provide suitable data material of homogeneous quality and high data density.
Mathematically, SGG demands the determination of the gravitational potential in the exterior of the Earth including its surface from given data of the gravitational Hesse tensor along the satellite orbit. For purposes of modeling we are led to invert the “upward continuation”-operator resulting from the Abel–Poisson integral formula of potential theory. This approach requires the solution of a tensorial Fredholm integral equation of the first kind relating the desired Earth’s gravitational potential to the measured orbital gravitational gradient acceleration. The integral equation constitutes an exponentially ill-posed problem of the theory of inverse problems, which inevitably needs two regularization processes, namely “downward continuation” and (weak or strong) “error regularization” in the case of noisy data.
This contribution deals with two different SGG-multiscale regularization methods, one in space domain and the other in frequency domain. Both procedures provide the gravitational potential as derived from tensorial SGG-data along the satellite orbit on the real Earth’s surface as required from the view point of geodesy.
Keywords
Satellite gravitational gradiometry (SGG) Tensorial pseudodifferential equation ‘‘Up- and downward continuation” Invertibility Exponential ill-posedness Multiscale regularization Space/frequency decorrelationZusammenfassung
Gravitationsgradiometrie mittels Satelliten (SGG) ist eine Meßtechnik zur globalen Bestimmung der Feinstruktur und der Eigenschaften des Gravitationsfeldes im Außenraum der Erde samt Erdoberfläche. Der ESA-Satellit GOCE (2009–2013) war der erste, dessen Aufgabe es war, das Prinzip SGG umzusetzen und brauchbare SGG-Daten zu liefern. GOCE gelang es in der Tat, Datenmaterial in einheitlicher Qualität und hoher Datendichte bereitzustellen.
Mathematisch erfordert SGG die Bestimmung des Gravitationspotentials im Außenraum der Erde unter Einschluss der Erdoberfläche aus vorgegebenen Daten des Hesse-Tensors des Gravitationspotentials auf dem Satellitenorbit. Die Modellierung führt auf die Inversion des ,,upward continuation“-Operators, der aus der Abel-Poissonschen Integralformel der Potentialtheorie resultiert. Dieser Zugang erfordert die Lösung einer tensoriellen Fredholmschen Integralgleichung erster Art, die das Gravitationspotential im Außenraum der Erde zu entlang des Orbits gemessenen Gravitationsgradientbeschleunigungen in Beziehung setzt. Zur Lösung der Integralgleichung ist ein exponentiell schlecht-gestelltes Problem der Therorie inverser Probleme zu bewältigen, das unvermeidlich zweier Regularisierungprozesse bedarf, nämlich ,,downward continuation“ und (schwache oder starke) ,,Fehler-Regularisierung“ im Falle verrauschter Daten.
Dieser Beitrag beschäftigt sich mit zwei verschiedenen SGG-Multiskalen-Regularisierungsmethoden, eine im Ortsbereich und eine andere im Frequenzbereich. Beide Zugänge liefern das Gravitationspotential – in der Tat wie aus geodätischer Sicht gefordert – auf der tatsächlichen Erdoberfläche aus tensoriellen, entlang von Satellitenorbits gemessenen SGG-Daten.
1 Introductory Aspects
The Earth’s gravitational field provides the natural orientation in our living environment, level (or equipotential) surfaces defining the horizontal and plumb lines the vertical. In first approximation, seen from the outer space, all level surfaces seem to be spherical and plumb lines close to the Earth’s surface look like straight lines directed radially towards the Earth’s center of mass with the gravitational acceleration being close to the well-known 9.8 m∕s^{2}. A closer look reveals, however, the slight oblateness of the level surfaces as well as a small increase of gravitational attraction towards the poles by 0.05 m∕s^{2}, corresponding to 1∕200 of the gravity intensity, both caused by the flattening of the Earth’s figure and by its rotation. The oblateness of the level surfaces causes the plumb lines to be somewhat curved, as they are orthogonal trajectories of them. Topography and density variations in the Earth’s crust and mantle produce gravity anomalies primarily in the range of one thousandth to one millionth the gravity intensity, temporal variations are much smaller, typically less than one millionth of the gravity intensity. The equipotential surface at mean sea level has a special relevance, as will be discussed below, and is denoted “geoid” since Listing [83]. When compared to a best fitting ellipsoid with a flattening of f = 1∕298.3 deviations become visible, so-called geoid undulations with an elevation range between − 100 and + 80 m. Undulations relative to a hydrostatic equilibrium figure (flattening f = 1∕299.63) are larger with maximum values of about 200 m (cf. [74]).
1.1 Geodetic Aspects
The geoid plays a prominent role in physical geodesy and oceanography. Ocean topography is the deviation of the actual mean ocean surface from the geoid. The geoid is the hypothetical surface of the oceans at rest [65]. Nowadays, the shape of the actual ocean surface is continuously monitored by satellite radar or laser altimetry. Ocean topography, the difference between geoid and mean ocean surface is typically about ± 30 cm only with maximum values of 1 − 2 m in the centers of the major circulation systems. Ocean topography can be translated to geostrophic ocean surface velocities. They serve as input to numerical ocean modeling with focus on ocean mass and heat transport. In geodesy the geoid is the classical global reference surface of heights. The most direct measure of height differences are gravity potential differences. In the past potential differences were measured by geodetic leveling in conjunction with gravimetry. Now, in space age, the geometric position differences of terrain points are given by Global Navigation Satellite Systems (GNSS). A global gravity field model is needed, in addition, to determine their potential difference. Reference of all potential differences to the geoid results in heights above mean sea level, so-called geopotential numbers [75]. In geophysics, gravity is of fundamental importance for the study of the composition and dynamics of the Earth (or of Moon and planets). Short scales gravity anomalies indicate density jumps in the subsurface and are employed for exploring ores, salt domes, fractures or sediment layers. Crustal anomalies are used for studying isostatic compensation of topographic masses or the elasticity of lithospheric plates. Subduction processes, faulting or mantle plumes are reflected in medium wavelength gravity or geoid anomalies (see [29, 135]). The dominating temporal variation of the Earth’s field is the tidal signal due to the attraction of Sun, Moon and planets and its effect on the solid Earth and oceans. Generally tides are well known and understood, but there remain some uncertainties in the models of the ocean tides. More interesting are gravity changes due to mass variations and mass transport in the global water cycle: the melting of the ice shields of Greenland and Antarctica and of the many glaciers and ice caps, the mass component of sea level change and changes of continental water storage. Other important contributions come from glacial isostatic adjustment processes and from ocean bottom pressure variations (cf. [17, 133]). Terrestrial absolute and relative gravimetry is the classical method of gravity field determination [130]. However, even after more than 100 years of terrestrial gravimetry the global coverage is incomplete, inhomogeneous and in large parts inaccurate. The modern answer to these deficiencies is satellite gravimetry. Only with satellites it is possible to cover the entire Earth with measurements within reasonable time and all measurements are done with one and the same set of instruments. Furthermore, as global sampling takes only between ten days to two months, depending on the requested sampling density, important time series of temporal changes can be established. Nevertheless, it remains almost paradoxical to try to measure the Earth’s gravity field with satellites. Why? Because of the great height of the satellites above the Earth’s surface. From Newton’s Law of Gravitation it is well-known that the force of gravitation caused by a mass and sensed by a test mass decreases with the square of their distance. Any satellite orbiting the Earth is such a test mass, circling the Earth in free fall in its gravitational field. The dominating part of the motion is the Keplerian (elliptic) orbit about the almost spherical Earth. Superimposed is a slight precession of this ellipse and of the orbit plane caused by the Earth’s oblateness. In the present context more important is the sequence of accelerations and decelerations due to mountains, valleys and any other mass anomaly on and inside the Earth. One could refer to it as a gravitational code that has to be deciphered. The higher the orbit altitude the “quadratically” smaller these perturbations and the more difficult it becomes to read this signal code. One obvious counter measure is to choose an orbit altitude as low as possible. Thus, dedicated satellite gravimetry missions fly at extremely near-Earth orbits (NEO’s). A second strategy is, instead of trying to decipher the orbit’s gravitational code, to apply the principle of satellite gravitational gradiometry for the measurements.
Figure 1 shows a satellite orbiting the Earth and four mass probes in its interior. All five masses, the satellite and the four probes, are in free fall around the Earth in its gravitational field. We assume the satellite’s mass to be concentrated in its center of mass and the probes to be located around the center of mass of the spacecraft. Thus, the lower of the four mass probes will be slightly more attracted by e.g. a mountain on Earth than the satellite itself and even more than the mass probe on top. Also, the front probe is attracted by the mountain in a direction slightly different from that of the mass probe in the middle and from that in the rear. Measuring these tiny differences in gravitational attraction is (1) greatly attenuating the contribution of the main signal which is that of the spherical Earth and thereby (2) emphasizing the shorter scale signal due to a mountain or any other small scale signal. One could compare it to a gravitational magnifying glass. With a set-up like in Fig. 1 the mass probes are not only subject to the gravitational field of the Earth. If each of them remains Earth-pointing they also experience the effect of centrifugal and angular accelerations caused by the once-per-revolution rotation of the satellite around the Earth.
Let us say a few words about the history of satellite gravitational gradiometry. What were the essential steps? Already the radio signals of the first two artificial satellites, Sputnik 1 and 2, both launched in 1957, were used to determine the oblateness of the Earth’s gravitational field [13, 89]. This great success was the beginning of a continuous effort of gravitational model improvement and refinement based on the analysis of various kinds of orbit tracking data from a large number of satellites [76].
First ideas about gravitational gradiometry were published shortly after the Sputnik satellites (cf. [15, 18, 27, 28, 121]). From these ideas several alternative lines of hardware development emerged, as discussed in [136]. Some of these concepts entered into the NASA programme discussions of the workshop at Williamstown [79] and shortly afterwards into the NASA Earth and Ocean Physics Applications Program [99]. They were further elaborated in a workshop on satellite gravimetry at the National Research Council [104]. In Europe it started with a discussion of the theoretical and practical concepts of satellite gravimetry at a summer school in Lannion, sponsored by the French space agency CNES [1]. The European Space Agency began with a programmatic discussion at the Space, Oceanography, Navigation and Geodynamics workshop at Ellmau in 1978 [22]. As a result of these activities on both sides of the Atlantic, NASA finally concentrated on the realization of satellite-to-satellite tracking in the low-low mode (SST l-l), while in Europe the focus was on gravitational gradiometry.
SST l-l is a differential technique alternative to gradiometry or one should say: It is not a differential but a difference technique, because the test masses are typically apart at a distance as large as 200 km (the test masses of a gradiometer instrument have a distance of only 50 cm). This makes it less suitable than gradiometry for a very detailed determination of the static gravity field of the Earth, but much more suitable for measuring the temporal variations, albeit less detailed. The mass probes in the case of SST are two satellites themselves, which follow each other in the same orbit at a distance of 200 km. The gravimetric signal are the distance changes which are measured with a precision of a few micrometers. The first mission of this type was GRACE, a NASA mission with German participation, in orbit from 2002 to 2017 and highly successful [128]. It has recently been succeeded by the follow-on mission GRACE FO, again with contributions from the German DLR and GFZ. In Europe, encouragement for gradiometry came from the positive experience with the high precision micro-accelerometer CACTUS on-board of the French mission CASTOR (D5B) [8]. A proposal was addressed to ESA in the context of the science program Horizon-2000 for a gradiometer experiment denoted GRADIO [7]. It lead to a further proposal of a joint gravity and magnetic field mission denoted ARISTOTELES [111]. Neither GRADIO nor ARISTOTELES were approved. Technology was not yet mature for a complex mission such as this one, and, probably more importantly, it did not fit in any of the existing ESA programs. In a new attempt in the nineties the dedicated gravitational gradiometry mission proposal GOCE (Gravity and steady-state Ocean Circulation Explorer) was approved in 1998. It was the first mission of the newly established ESA program “Living Planet” [23]. GOCE was launched on March 17, 2009; it was in orbit till November 11, 2013. Its orbit was as low as 255 km and was further lowered to 224 km in the mission’s final phase. This was only possible with an active drag compensation system on board. GOCE carried the first satellite gradiometer instrument, a three axis device with arm lengths of 50 cm and centered at the satellite’s center of mass. Each gradiometer arm held two high precision three-axis accelerometers at its ends. Thus, it was a full tensor instrument, in theory, as described by (1). In practice, it was impossible to build accelerometers with three ultra sensitive axes in a laboratory on Earth (under the influence of the gravity intensity), and one axis had to be constructed much more robust. As a consequence, high precision measurements were only possible for the gradiometer components \(V_{x_1x_1}\), \(V_{x_2x_2}\), \(V_{x_3 x_3},\) and \(V_{x_1x_3}\) as well as for the angular rate \(\omega _{x_2}\). The determination of the angular rates was supported by star sensor measurements. Based on the results of GOCE the most detailed and accurate global static gravity models were derived. Fundamental theoretical work on satellite gradiometry goes back to [92, 95], and to [88], see also [109, 110]. In several fascinating publications Marussi discussed the geometric structure of the local gravitational field and its interpretation as local tidal field. The tidal effect is in this case not that of Sun, Moon and planets but of the Earth’s gravitation in a local triad on a test mass not exactly placed at the triad’s center [85, 86] and [87]. Some years later the mathematical foundations of gravitational gradiometry were elaborated and refined in the dissertations [105, 122, 123] and in the research notes [46, 47, 51, 53] of the Geomathematics Group, University of Kaiserslautern.
The great drawback of acquiring gravitational data at LEO’s altitude is that the upward continuation of the gravitation amounts to an exponential smoothing of the potential coefficients in terms of outer harmonics with increasing height. In other words, satellite measurements do not contain the same signal information at LEO’s height (i.e., 200–250 km) as on the Earth’s surface. This is the reason why the gravitational potential is obtainable from satellite data only in an attenuated form when continued to the Earth’s surface. Even more, it may happen in downward continuation that the noise in the measurements is amplified. Nevertheless, for satellite gravitational gradiometry (SGG), as provided by GOCE, advantage can be taken from the fact that second derivatives instead of the potential itself are used as observations on LEO’s orbit. Mathematically, this means that the exponential decay of the outer harmonic coefficients is reduced polynomially by two degrees. In other words, SGG takes advantage of the fact that second derivatives produce a rougher data set than the potential itself such that the resolution of the gravitational structure is much finer. In addition, in the frequency context of outer harmonics, the Meissl scheme (see, e.g., [88, 105, 115], and [47]) enables us in spectral nomenclature to relate the orthogonal coefficients at LEO’s height to the orthogonal coefficients at the surface of the Earth, at least in the context of a spherical model and under the restrictive assumption of bandlimited outer harmonic modeling without observational errors.
1.2 Mathematical Aspects
The literature dealing with the solution procedures of satellite gravitational gradiometry can be divided essentially into two classes: the timewise approach and the spacewise approach. The former one considers the measured data as a time series, while the second one supposes that the data are given in advance as dataset on the satellite orbit. This contribution is part of the spacewise approach (see, e.g., [57, 109, 110, 112, 113, 114, 116, 119, 122, 123] for some earlier work).
The goal of this work is a potential theoretically based and numerically reflected approach to satellite gravitational gradiometry corresponding to tensorial SGG–data by use of multiscale regularization methods, both in frequency as well as space domain.
Spherically Reflected Formulation
For simplicity, we start with the mathematical SGG-description for the outer space of a sphere \(\mathbb {S}^2_R\) in Euclidean space \(\mathbb {R}^3\) of radius R around the origin (thereby using [46, 47, 53] as basic material for the discussion in the frequency domain; in space domain a novel methodology will be presented).
- (in frequency domain) by a pseudodifferential equation of the formwhere the coefficients \(\tilde {\mu }_n^{(1,1)}\) are given by$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hspace{-1pc}\sum^\infty_{n=0} \sum^{2n+1}_{m=1} {V}^{\wedge_{L^2(\mathbb{S}^2_R)}} (n,m) \sqrt{\tilde{\mu}_n^{(1,1)}} \ {\mathbf{h}}^{R;(1,1)}_{n,m} \ = \ (\nabla \otimes \nabla) {V}(x) = \mathbf{v}(x), \end{array} \end{aligned} $$(2)\({\mathbf {h}}^{R;(1,1)}_{n,m}\) denotes a tensor outer harmonic of type (1, 1) of degree n and order m , and \({V}^{\wedge _{L^2(\mathbb {S}^2_R)}} (n,m)\) are the orthogonal (Fourier) coefficients given by$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu}_n^{(1,1)}:= (n+2) (n+2) (2n-3) (2n-1), \end{array} \end{aligned} $$(3)with \(H^R_{n,m}\) as scalar outer harmonic of degree n and order m.$$\displaystyle \begin{aligned} \begin{array}{rcl} {V}^{\wedge_{L^2(\mathbb{S}^2_R)}} (n,m) = \langle V, H^R_{n,m}\rangle_{L^2(\mathbb{S}^2_R)}= \int_{\mathbb{S}^2_R}V(y) \ H^R_{n,m}(y) \ dS(y) \end{array} \end{aligned} $$(4)
- (in space domain) by a linear integral equation of the first kindwhere, as already stated, v(x) = (∇_{x} ⊗∇_{x})V (x) with locations x on the satellite orbit Γ designates the Hesse tensor of V at x, and K_{R}(⋅, ⋅) given by$$\displaystyle \begin{aligned} \int_{\mathbb{S}_R^2} \,V(y) \ \underbrace{ (\nabla_x \otimes \nabla_x) \frac{1}{4 \pi R} \frac{\vert x\vert^2 - R^2}{\vert x-y\vert^3}}_{= :{\mathbf{k}}_R (x,y)} \, dS(y) = \ (\nabla \otimes \nabla) {V}(x) = \mathbf{v}(x), \end{aligned} $$(5)is the Abel–Poisson kernel in the outer space \( {\overline {\mathbb {S}_R^{2;\mathrm {ext}}}}\) of the sphere \({\mathbb {S}_R^{2}}~{\subset }~\mathbb {R}^3\) (providing “upward continuation” from the sphere \(\mathbb {S}^2_R\) to its outer space by forming the convolution against V ).$$\displaystyle \begin{aligned} K_R(x,y) := \frac{1}{4 \pi R} \frac{\vert x\vert^2 - R^2}{\vert x-y\vert^3}, \ \ x, y \in {\overline{\mathbb{S}_R^{2;\mathrm{ext}}}}, x \neq y, \end{aligned} $$(6)
SGG-Operator Formulation
SGG using frequency regularizing Tikhonov and truncated singular integral techniques,
SSG using space regularizing “downward continuation” techniques involving Abel–Poisson kernels.
Alternative wavelet methods and multiscale realizations can be found in several publications. It started with [35, 49, 57, 122]. Further related notes are [24, 25, 26, 42, 46, 47, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 66, 69, 77, 90, 91, 105, 106, 125, 137]. Tree algorithms are available if SGG-data within the Hesse tensor (for example, second radial derivatives) are used for multiscale approximation. The trace of the Hesse tensor (which is equal to zero because of the harmonicity of the gravitational potential) offers the possibility to validate tensorial multiscale methods. Moreover, observational errors can be handled by filtering techniques within the tree algorithm (see [35, 40, 41]). Since all these techniques and procedures are well–documented in the literature, they will be not discussed in this approach.
1.3 Layout of the Paper
The layout of the paper is as follows: As a preparation we first present a brief overview on relevant aspects of the theory of ill-posed inverse problems (Sect. 2) and tensorial potential theory (Sect. 3). Then, based on (2), we propose multiscale frequency regularization methods in Sect. 4. Finally, based on (5), we come to multiscale space regularization methods in Sect. 5.
Both techniques are based on a “Runge argument” to allow for a multiscale regularization of the Earth’s gravitational potential V on and outside the actual Earth’s surface.
2 SGG–Aspects of Inverse Ill-Posed Problems
For the convenience of the reader, we present a brief course of basic facts on regularization in Hilbert space settings, which are useful to understand solution strategies in the framework of pseudodifferential equations. The explanations are based on functional analytic tools (see, e.g., [45], where much more additional material and references can be found).
2.1 Ill-Posed Problems in Hilbert Spaces
- (i)
\(\mathscr {R}( \Lambda )\) is dense in Y, hence, \(\mathscr {N}(\Lambda ^*) = \{0\}\), and \(y \in \mathscr {R}( \Lambda )\);
- (ii)
\(\mathscr {R}( \Lambda )\) is dense in Y, and \(y \notin \mathscr {R}(\Lambda )\);
- (iii)
\(\overline {\mathscr {R}(\Lambda )}\) is a proper subspace of Y, and \(y \in \mathscr {R}(\Lambda ) + \mathscr {R}( \Lambda )^\perp \);
- (iv)
\(\overline {\mathscr {R}(\Lambda )} \neq \mathrm {Y}\), and \(y \notin \mathscr {R}(\Lambda ) +\mathscr {R}(\Lambda )^\perp \).
In case (i) above, Λ^{†} gives the minimal-norm solution of (3). In case (iii), (9) has a least-squares solution (which is unique if and only if \(\mathscr {N}(\Lambda )= \{0\}\)). In both cases the infimum in (10) is attained and is equal to zero and ∥y − Qy∥, respectively. Case (ii) and (iv) are pathological and usually not under discussion in generalized inverse theory, since in both cases \(y \notin \mathscr {D}(\Lambda ^\dagger )\), and the infimum in (10) is not attained. For gravimetric synthesis and identification [44] as well as in satellite problems in geodesy [35], cases (i), (ii), (iii) are of practical interest. Thus, for the identification problem, case (i) corresponds to identification with complete and exact information. Case (ii) may correspond to identification in the presence of contamination in the measurement. In either of these cases, it is theoretically possible to approximate the infimum (whether actually attainable or not) to within any desired degree of accuracy. For practical reasons, it may be necessary to limit the accuracy of the approximation in order to insure that certain a priori conditions are met. Such constraints on the input (or control) functions are characteristic of control systems and often lead to sets of state which are unreachable; the corresponding control problem is not even approximately controllable. The case (iii) and (iv) arise for more general synthesis problems if \(\mathscr {N}(\Lambda ^*)\) is nontrivial.
For all “admissible” data, a solution exists.
For all “admissible” data, the solution is unique.
The solution depends continuously on the data.
Λ is injective, i.e., \({\mathscr {R}}(\Lambda ) = \mathrm {Y}\).
Λ is surjective, i.e., \({\mathscr {N}}(\Lambda ) = \{0\}\).
Λ^{−1} is bounded and continuous.
the problem (9) is well-posed;
\(\mathscr {R}(\Lambda )\) is closed;
Λ^{†} is bounded.
Let us discuss the consequences of the violations of the above requirements for the well–posedness of (9): The lack of injectivity of Λ is perhaps the easiest problem. From theoretical point of view, the space X can be replaced by the orthogonal complement \({\mathscr {N}}(\Lambda )^\perp \), and the restriction of the operator Λ to \({\mathscr {N}}(\Lambda )^\perp \) yields an injective problem. However, in practice, one is normally confronted with the problem that \({\mathscr {R}}(\Lambda ) \neq \mathrm {Y}\), since the right hand side is given by measurements and is, therefore, disturbed by errors (note that this calamity is also inherent in the SGG-problem).
2.2 Regularization Strategies and Error Behavior
Note that the convergence of \({\mathscr {R}}_\alpha \Lambda x\) in (23) is based on y = Λx, i.e., on unper- turbed data. In practice, the right hand side is affected by errors and then no convergence is achieved. Instead, one is (or has to be) satisfied with an approximate solution based on a certain choice of the regularization parameter.
2.3 Singular Systems and Resolution Methods
An important result in the theory of singular value expansions should be presented, which can be found in all standard textbooks on Inverse Problems:
The condition (38) is the so-called Picard criterion. It says that a best–approximate solution of Λx = y exists only if the Fourier coefficients of y decrease fast enough relative to the singular values.
- (i)
|q(α, σ)|≤ 1 for allα > 0 and 0 < σ ≤∥Λ∥_{L(X,Y)}.
- (ii)
For everyα > 0 there exists ac(α) so that |q(α, σ)|≤ c(α)σfor all 0 < σ ≤∥Λ∥_{L(X,Y)}.
- (iii)
lim_{α→0}q(α, σ) = 1 for every 0 ≤ σ ≤∥Λ∥_{L(X,Y)}.
2.4 Summary Excursion to Regularization Techniques
change the notion of what is meant by a solution, e.g., an ε-approximate solution: \(\Vert \Lambda \tilde x-y\Vert \le \varepsilon \), where ε > 0 is the prescribed error value; quasi-solution: \(\Vert \Lambda \tilde x-y\Vert \le \Vert \Lambda x-y \Vert \) for all \(x\in \mathscr {M}\), a prescribed subset of the domain of A; least-squares solution of minimal norm, etc,
modify the operator equation or the problem itself,
change the spaces and/or topologies,
specify the type of involved noise (“strong” or “weak” noise as discussed, e.g., in [19]).
- 1.
Regularization methods in function spaces is one category. This includes Tikhonov-type regularization, the method of quasi-reversibility, the use for certain function spaces such as scale spaces in multiresolutions, the method of generalized inverses (pseudoinverses) in reproducing kernel Hilbert spaces, and multiscale wavelet regularization.
- 2.
Resolution of ill-posed problems by “control of dimensionality” is another category. This includes projection methods, moment-discretization schemes. The success of these methods hinges on the possibility of obtaining approximate solution while keeping the dimensionality of the finite dimensional problem within the “range of numerical stability”. It also hinges on deriving error estimates for the approximate solutions that is crucial to the control of the dimensionality.
- 3.
A third category is iterative methods which can be applied either to the problem in function spaces or to a discrete version of it. The crucial ingredient in iterative methods is to stop the iteration before instability creeps into the process. Thus iterative methods have to be modified or accelerated so as to provide a desirable accuracy by the time a stopping rule is applied.
- 4.
A fourth category is filter methods. Filter methods refer to procedures where, for example, values producing highly oscillatory solutions are eliminated. Various “low pass” filters can, of course, be used. They are also crucial for the determination of a stopping rule. Mollifiers are known in filtering as smooth functions with special properties to create sequences of smooth functions approximating a non-smooth function or a singular function.
- 5.
The original idea of a mollifier method (see, e.g., [45] and the references therein) is of interest for the solution of an operator equation, but we realize that the problem is “too ill-posed” for being able to determine the (pseudo)inverse accurately. Mollifiers are known as smooth functions with special properties to create sequences of smooth functions approximating a non-smooth function. Thus, we compromises by changing the problem into a more well-posed one, namely that of trying to determine a mollified version of the solution. The heuristic motivation is that the trouble usually comes from high frequency components of the data and of the solution, which are damped out by mollification.
- 6.
The root of the Backus–Gilbert method (BG-method) was geophysical (cf. [4, 5, 6]). The characterization involved in the model is known as moment problem in the mathematical literature. The BG-method can be thought of as resulting from discretizing an integral equation of the first kind. Where other regularization methods, such as the frequently used Tikhonov regularization method (see, e.g., [45] and the references therein), seek to impose smoothness constraints on the solution, the BG-method instead realizes stability constraints. As a consequence, the solution is varying as little as possible if the input data were resampled multiple times. The common feature between mollification and the BG-method is that an approximate inverse is determined independently from the right hand side of the equation.
A regularization-approximation scheme refers to a variety of methods such as Tikhonov’s regularization, projection methods, multiscale methods, iterative approximation, etc., that can be applied to ill-posed problems. These schemes turn into algorithms once a resolution strategy can be effectively implemented. Unfortunately, this requires the determination of a suitable value of a certain parameter associated with the scheme (e.g., regularization parameter, mesh size, dimension of subspace in the projection scheme, specification of the level of a scale space, classification of noise, etc.). This is not a trivial problem since it involves a trade-off between accuracy and numerical stability, a situation that does not usually arise in well-posed problems.
3 SGG-Aspects of Potential Theory
In what follows we list some (known) potential theoretic tools to be needed for the study of SGG. For more details and proofs the reader is referred, e.g., to [30, 36], and the references therein.
3.1 Gravitation
- (v1)
div v = ∇⋅ v = 0, curl v = L ⋅ v = 0 in Σ^{ext},
- (v2)
v is regular at infinity: \(|v(x)| = O \left ( \vert x \vert ^{-2} \right ) \), |x|→∞.
- (v1)
div v = ∇⋅v = 0, curl v = L ⋅v = 0 in the Earth’s exterior Σ^{ext},
- (v2)
v is regular at infinity: \({\vert \mathbf {v} (x) \vert } = O \left ( \vert x \vert ^{-3} \right )\), |x|→∞.
Hence, the potential theoretic situation for the SGG–problem can be formulated briefly as follows:
Suppose that satellite data\( \mathbf {v}~{=}~\left ( \nabla \otimes \nabla \right ) V \)are known on the orbitΓ, the satellite gravitational gradiometry problem (SGG-problem) amounts to the problem of determiningV on and outside the Earth’s surfaceΣ, i.e., in the set\({\overline {\varSigma ^{\mathrm {{ext}}}}} = \varSigma ^{\mathrm {{ext}}} \cup \varSigma ,\)from (discrete data of)\(\mathbf {v} = \left ( \nabla \otimes \nabla \right ) V \)on the satellite orbitΓ.
Tensorial SGG-data (or a scalar manifestation of them such as the second radial derivative on the orbit) do not form the standard equipment of potential theory (such as, e.g., Dirichlet’s or Neumann’s boundary data). Thus, uniqueness cannot be deduced in the way as known (cf. [80]) for boundary data in classical boundary value problems (such as Dirichlet’s or Neumann’s boundary value problem). Nevertheless, “upward continuation” via Dirichlet’s problem as the inverse to “downward continuation” plays an important role in the mathematical treatment of SGG.
SGG–data have its natural limit because of the exponential damping of the frequency parts (i.e., the orthogonal coefficients) of the gravitational potential with increasing satellite heights.
- “Downward continuation” by SGG as the inverse of “upward continuation” (cf. Fig. 4) leads to an ill–posed problem, since the data are not given on the boundary of the domain of interest, i.e., on the Earth’s surface Σ, but on locations along the orbit Γ.
3.2 Potential Spaces
Next we are interested in characterizing the essential players involved in the SGG-match as members of infinite-dimensional potential spaces.
- 1.
Σ divides the Euclidean space \(\mathbb {R}^3\) into the (open) bounded region Σ^{int} (inner space) and the (open) unbounded region Σ^{ext} (outer space) so that \( \varSigma ^{\mathrm {{ext}}} = \mathbb {R}^3 \backslash \overline {\varSigma ^{\mathrm {{int}}}}\), \(\varSigma = \overline {\varSigma ^{\mathrm {{int}}}} \cap \overline {\varSigma ^{\mathrm {{ext}}}} \), i.e. ∅ = Σ^{int} ∩ Σ^{ext},
- 2.
Σ^{int} contains the origin 0,
- 3.
Σ is a closed and compact surface free of double points,
- 4.
Σ is locally of class C^{(2)}, i.e. Σ is locally C^{(2)}–smooth
The function spaces C^{(2)}(Σ^{ext}) and c^{(2)}(Σ^{ext}) etc. are defined in canonical way.
3.3 Spherical Harmonics
Since tensor spherical harmonics do not belong to the standard equipment of geodesists as well as mathematicians a short introduction should be given. A more detailed study of our approach can be found in [52]; similar SGG-relevant aproaches are presented in [112, 114, 115].
Spherical Nomenclature
As usual, a second order tensor \(\mathbf {f} \in {\mathbb {R}}^{3} \otimes {\mathbb {R}}^{3}\) is understood to be a linear mapping that assigns to each \(x \in {\mathbb {R}}^3\) a vector \(y \in {\mathbb {R}}^{3}\). The (cartesian) components F_{ij} of f are defined by F_{ij} = ε^{i} ⋅ (fε^{j}) = (ε^{i})^{T}(fε^{j}), so that y = fx is equivalent to \(y \cdot \varepsilon ^i = \sum ^3_{j=1} F_{ij} (x \cdot \varepsilon ^j)\). We write f^{T} for the transpose of f. The dyadic (tensor) product x ⊗ y of two elements \(x,y \in {\mathbb {R}}^3\) is the tensor that assigns to each \(u \in {\mathbb {R}}^3\) the vector (y ⋅ u)x. More explicitly, (x ⊗ y)u = (y ⋅ u)x for every \(u \in {\mathbb {R}}^3\). The inner product f ⋅g of two second order tensors \(\mathbf {f}, \mathbf {g} \in {\mathbb {R}}^{3} \otimes {\mathbb {R}}^{3}\) is defined by \(\mathbf {f} \cdot \mathbf {g} = \sum ^3_{i,j=1} F_{ij} G_{ij},\) and |f| = (f ⋅f)^{1∕2} is called the norm of f.
Note that, throughout this paper, scalar valued (resp. vector valued, tensor valued) functions are denoted by capital (resp. small, small bold) letters. A function \(F:\mathbb {S}^2\to {\mathbb {R}}\) (resp. \(f: \mathbb {S}^2 \to \mathbb {R}^3\), \(\mathbf {f}:\mathbb {S}^2\to {\mathbb {R}}^{3} \otimes {\mathbb {R}}^{3}\)) possessing k continuous derivatives on the unit sphere \(\mathbb {S}^2\) is said to be of class \(\mathrm {C}^{(k)}(\mathbb {S}^2)\) (resp. \(\mathrm {c} ^{(k)}(\mathbb {S}^2)\), \({\mathbf {c}}^{(k)} (\mathbb {S}^2)\)). \(\mathrm {C}^{(0)}(\mathbb {S}^2)\) (resp. \(\mathrm {c}^{(0)} (\mathbb {S}^2)\), \({\mathbf {c}}^{(0)} (\mathbb {S}^2)\)) is the class of real continuous scalar–valued (resp. vector–valued, tensor–valued) functions on the unit sphere \(\mathbb {S}^2\).
Spherical Differential Operators
In terms of polar coordinates (60) the gradient ∇ in \({\mathbb {R}}^3\) allows the representation \(\nabla _x = \xi {\partial }/{\partial r} + ({1}/{r}) \nabla ^*_\xi ,\) where ∇^{∗} is the surface gradient of the unit sphere \(\mathbb {S}^2~{\subset }~{\mathbb {R}}^3\). The operator Δ^{∗}=∇^{∗}⋅∇^{∗} is called the Beltrami operator of the unit sphere \(\mathbb {S}^2\). Obviously, it can be understood as the angular part of the Laplace operator. For \(F \in \mathrm {C}^{(1)} (\mathbb {S}^2)\) we introduce the surface curl gradient\(L^*_{\xi }\) by \(L^*_{\xi } F(\xi ) = \xi \wedge \nabla ^*_\xi F(\xi )\), \(\xi \in \mathbb {S}^2,\) while \(\nabla ^*_\xi \cdot f(\xi )\), \(\xi \in \mathbb {S}^2\), and \(L^*_\xi \cdot f(\xi )\), \(\xi \in \mathbb {S}^2\), respectively, denote the surface divergence and surface curl of the vector field f at \(\xi \in \mathbb {S}^2\).
Scalar Spherical Harmonics
Scalar spherical harmonics are defined as restrictions of homogeneous harmonic polynomials to the unit sphere \(\mathbb {S}^2\). In all geosciences interested in global modeling, spherical harmonics are the functions which are usually taken to represent scalar fields in spectral way on a spherical surface such as the Earth’s (mean) sphere.
We conclude this short introduction to the theory of scalar spherical harmonics with the so–called addition theorem which formulates the correlation between the spherical harmonics and the corresponding Legendre polynomials.
Addition Theorem for Scalar Spherical Harmonics.
Tensor Spherical Harmonics
Addition Theorem for Tensor Spherical Harmonics.
By \(\mathbf {harm}_n(\mathbb {S}^2)\) we denote the space of all tensor spherical harmonics of degree n.
3.4 Outer Harmonics
We begin with the well-known scalar theory (see, e.g., [35]).
Scalar Outer Harmonics
\(H_{n,m}^{R}\) is of class \(\mathrm {C} ^{(\infty )} (\mathbb {S}_R^{2;\mathrm {ext}})\),
\(H_{n,m} ^R\) is harmonic in \(\mathbb {S}_R^{2;\mathrm {ext}}\): \(\varDelta _x H_{n,m}^R (x) = 0\), \(x \in \mathbb {S}_R^{2;\mathrm {ext}}\),
\(H_{n,m}^R|{ }_{\mathbb {S}_R^2} = \frac {1}{R} Y_{n,m}\),
\(\langle H_{n,m}^R, H_{l,s}^R\rangle _{\mathrm {L}^2(\mathbb {S}_R^2)} = \int _{\mathbb {S}_R^2} H_{n,m}^R (x) H_{l,s}^R (x) \ dS(x) = \delta _{n,l} \delta _{m,s}\),
\(|H _{n,m} ^{R}(x)| = O \left (|x|{ }^{-1} \right )\), |x|→∞.
Tensor Outer Harmonics
\({\mathbf {h}}_{n,m}^{R;(i,k)}\) is of class \(\mathbf {c} ^{(\infty )} (\mathbb {S}_R^{2;\mathrm {ext}})\),
\(\varDelta _x \mathbf {h} _{n,m} ^{R;(i,k)} (x) = 0\), \(x \in \mathbb {S}_R^{2;\mathrm {ext}}\), i.e., the component functions of \(\mathbf {h} _{n,m} ^{R;(i,k)}\) fulfill the Laplace equation,
\(\mathbf {h} _{n,m} ^{R;(i,k)} |{ }_{\mathbb {S}_R^2} = \frac {1}{R} \tilde {\mathbf {y}}_{n,m} ^{(i,k)}\),
\(\langle \mathbf {h} _{n,m} ^{R;(i,k)}, \mathbf {h} _{l,s} ^{R;(p,q)}\rangle _{{\mathbf {l}}^2(\mathbb {S}_R^2)} = \int _{\mathbb {S}_R^2} \mathbf {h} _{n,m} ^{R;(i,k)}(x) \cdot \mathbf {h} _{l,s} ^{R;(p,q)} (x) d S(x) = \delta _{i,p} \delta _{k,q} \delta _{n,l} \delta _{m,s}\), where \({\mathbf {l}}^2(\mathbb {S}_R^2)\) is the space of square–integrable tensor fields on \(\mathbb {S}_R^2\),
\(|\mathbf {h} _{n,m} ^{R;(i,k)}(x)| = O \left (|x|{ }^{-1} \right )\), |x|→∞.
3.5 Runge-Walsh Concept
In the theory of harmonic functions related to regular surfaces, a result first motivated by [117] in one-dimensional complex analysis and later generalized by [134] and [132] to potential theory in three-dimensional Euclidean analysis is of basic interest. For geodetically relevant obligations, the reader is referred to, e.g., [3, 12, 30, 67, 78, 82, 93, 94, 96, 97, 118].
The Runge–Walsh approximation property (135) justifies the approximation of the Earth’s gravitational potential on and outside the regular surface by a linear combination of scalar outer harmonics, i.e., by harmonic “trial functions of polynomial nature” showing a “harmonicity domain” \(\mathbb {S}_R^{2;\mathrm {ext}} \supset \overline {\varSigma ^{\mathrm {{ext}}}}\). It should be remarked that the same property holds true, for example, for outer ellipsoidal harmonics outside a regular surface Ξ located totally in Σ^{int} with dist(Σ, Ξ) > 0. However, once again, for reasons of numerical economy and efficiency we restrict ourselves to outer spherical harmonics and Runge (Bjerhammar) spheres. Moreover, the Runge-property can be verified for so-called fundamental systems of monopoles inside Σ^{int} (cf. [30, 42]), certain kernel representations as well as harmonic splines and wavelets (cf. [35, 36, 42]).
The calamity of evaluating the gravitational potential \(\tilde V\) (for example, the Earth’s gravitational potential) by a potential V in terms of outer harmonics is that these basis functions are globally supported. This is the reason why they do not show any space localization but ideal frequency (momentum) localization (for a more detailed description see [34, 52, 63]). This property makes outer harmonics laborous to use for high resolution modeling at local scale. As a matter of fact, the uncertainty principle (see [63] and the references therein) leads us to the conclusion that outer harmonics are well suited to resolve low–frequency problems at global scale, i.e. to represent trend approximations.
As a well promising compromise, “sum conglomerates” of outer harmonics, i.e., so-called (outer) harmonic kernel functions, can be handled (see [63] offering a limited but appropriately balanced range of frequency as well as space localization). Even better, we can construct families of kernels which control the increase of space localization at the cost of the decrease of frequency localization by specifying a scale parameter. This leads to the multiscale philosophy by wavelets in space as well as frequency context as discussed later on.
- (in space domain) by a linear Fredholm integral equation of the first kind$$\displaystyle \begin{aligned} \int_{\mathbb{S}_R^2} \,V(y) \ \underbrace{ \nabla_x \otimes \nabla_x \frac{1}{4 \pi R} \frac{\vert x\vert^2 - R^2}{\vert x-y\vert^3}}_{= {\mathbf{k}}_R (x,y)} \ dS(y) = \ (\nabla \otimes \nabla) {V}(x) = \mathbf{v}(x) \approx \tilde{\mathbf{v}}(x) , \end{aligned} $$(140)
- (in frequency domain) by a pseudodifferential equation of the form$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \sum^\infty_{n=0} \sum^{2n+1}_{m=1} {V}^{\wedge_{\mathrm{L}^2(\mathbb{S}^2_R)}} (n,m) \sqrt{\tilde{\mu}_n^{(1,1)}} \ {\mathbf{h}}^{R;(1,1)}_{n,m} \ = \ (\nabla \otimes \nabla) {V}(x) = \mathbf{v}(x) \approx \tilde{\mathbf{v}}(x) \\ \end{array} \end{aligned} $$(141)
SGG Uniqueness
Some words about the uniqueness of spherically reflected SGG should be made: Keeping in mind that any solution of the SGG-problem in tensor spherical framework can be expressed as a series of outer harmonics due to the Runge–Walsh approximation property and observing the closure and completeness of the spherical harmonics in the space of square-integrable functions on spheres, we are able to conclude (cf. [123, 124], and [125]) that the SGG–problem is uniquely solvable (up to some low order spherical harmonics) involving the O^{(1, 1)}, O^{(1, 2)}, O^{(2, 1)}, O^{(2, 2)}, and O^{(2, 3)} components.
More concretely, we are able to formulate the following results:
- 1.\(O^{(i,k)} \nabla \otimes \nabla V|{ }_{\mathbb {S}_\gamma ^2}= 0\) if (i, k) ∈{(1, 3), (3, 1), (3, 2), (3, 3)},
- 2.
\(O^{(i,k)} \nabla \otimes \nabla V|{ }_{\mathbb {S}_\gamma ^2} = 0\) for (i, k) ∈{(1, 1), (2, 2)} if and only if \(V|{ }_{\mathbb {S}_\gamma ^2} = 0\),
- 3.
\(O^{(i,k)} \nabla \otimes \nabla V|{ }_{\mathbb {S}_\gamma ^2}= 0\) for (i, k) ∈{(1, 2), (2, 1)} if and only if \(V|{ }_{\mathbb {S}_\gamma ^2}\) is constant,
- 4.
\(O^{(2,3)} \nabla \otimes \nabla V|{ }_{\mathbb {S}_\gamma ^2} = 0\) if and only if \(V|{ }_{\mathbb {S}_\gamma ^2}\) is linear combination of spherical harmonics of degree 0 and 1.
In this respect, it should be noted that the ESA-satellite GOCE had been injected into its orbit with some additional height so as to perform the complex commissioning activities. In fact, the orbit was left to decay naturally due to atmospheric drag from the initial height of 278 km to the desired height of 254 km.
4 SGG in Frequency-Based Framework
The considerations of this chapter use some pre-work to be found in [31, 35, 46, 53, 58]. Our goal is to formulate the SGG-problem in terms of a pseudodifferential operator equation involving suitable Sobolev reference spaces.
4.1 Sobolev Spaces
The scalar case merely serves as a preparation of the tensorial context, which should be studied subsequently.
Scalar Case
The space \(\mathscr {H}_0(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) may be understood as the space of all solutions of the Dirichlet boundary value problem in \(\overline {\mathbb {S}_R^{2;\mathrm {ext}}}\) corresponding to \(\mathrm {L}^2(\mathbb {S}_R^2)\)-boundary values on \(\mathbb {S}_R^2\) (note that the potential in \(\mathscr {H} _0 (\overline {\mathbb {S}_R^{2;\mathrm {ext}}}) \) corresponding to the \(\mathrm {L}^2(\mathbb {S}_R^2)\)-(Dirichlet) boundary conditions on \(\mathbb {S}_R^2\) is uniquely determined).
Sobolev Lemma
Assume that the sequences \(\{A_n\}_{n \in \mathbb {N}_0}, \{B_n\}_{n \in \mathbb {N}_0} \in \mathscr {A}\) are given in such a way that \(\{ B_n^{-1} A_n\} _{n \in \mathbb {N}_0}\) is summable. Then each \(F \in \mathscr {H} \left ( \{B_n^{-1} A_n \}; \overline {\mathbb {S}_R^{2;\mathrm {ext}}} \right )\) corresponds to a potential of class \(\mathrm {Pot} ^{(0)} (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\).
The Sobolev Lemma states that in the case of summability of the sequence \(\{ B_n ^{-1} A_n \} _{n \in \mathbb {N}_0}\), the Fourier series in terms of the basis functions \(H_{n,m}^{\ast }~{\in }~\mathscr {H} \left ( \{ B_n^{-1} A_n \} ; \overline {\mathbb {S}_R^{2;\mathrm {ext}}} \right )\) is continuous on the boundary \(\mathbb {S}_R^2\). In particular, we have the following statement: If \(F~{\in }~\mathscr {H} _s (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) with s > 1, then F corresponds to a function of class \(\mathrm {Pot} ^{(0)} (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\).
Tensorial Case
for all \(x \in \overline {\mathbb {S}_R^{2;\mathrm {ext}}}\), \(\tilde {O}_R ^{(1,1)} \mathbf {K} _{\mathbf {h} (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})} (\cdot , x) \in \mathbf {h} (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\), where the operator \(\tilde {O}_R ^{(1,1)}\) is the extension of the adjoint operator of \(\tilde {\mathbf {o}} _R ^{(1,1)}\) to tensor fields of rank four,
\(\tilde {O}_R^{(1,1)} {\mathbf {f}} (x) = \left \langle \tilde {O}_R^{(1,1)} \mathbf {K} _{\mathbf {h}(\overline {\mathbb {S}_R^{2,\mathrm {ext}}})} (\cdot , x), \mathbf {f}\right \rangle _{\mathbf {h}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})} \) for every \(\mathbf {f} \in {\mathbf {h}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})}\) and all \(x \in \overline {\mathbb {S}_R^{2;\mathrm {ext}}}\).
Tensorial Sobolev Lemma
Let the sequences \(\{A_n\}_{n \in \mathbb {N}_0}\), \(\{B_n\}_{n \in \mathbb {N}_0} \in \mathscr {A}\) be given such that \(\{ B_n^{-1} A_n \}_{n \in \mathbb {N}_0}~{\in }~\mathscr {A}\) is summable. Then each \(\mathbf {f} \in \mathbf {h}\left ( \{B_n^{-1} A_n\}; \overline {\mathbb {S}_R^{2;\mathrm {ext}}} \right )\) corresponds to a function of class \(\mathbf {pot}^{(0)}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\).
4.2 Pseudodifferential Equation
Since we know, at least in the spherically reflected context (as introduced by [127]), which conditions guarantee the uniqueness of a SGG-solution we can turn to the question how to find a solution from tensorial data and what we mean with a spectral solution, when we have to take into account the ill–posedness. This leads us to analyze the SGG-problem step by step by use of the concept of pseudodifferential operators.
We shortly introduce the framework of scalar and tensorial pseudodifferential operators (PDO’s).
Scalar Pseudodifferential Operator
Tensorial Pseudodifferential Operator
In the following we define scalar and tensorial kernel functions which are of basic importance for the consideration of the SGG–problem in terms of pseudodifferential operators.
Kernel Functions
Convolutions
- 1.
\(\sum _{n=0} ^{\infty } (Q ^{\wedge } (n))^2 < \infty \),
- 2.
\(\sum _{n=0} ^{\infty } (2n+1)^2 \left ( \frac {Q^{\wedge } (n)} {A_n} \right )^2 < \infty \).
Within the context of pseudodifferential operators we are able to formulate the (scalar or tensorial) SGG-problem (for more details on pseudodifferential operators the reader should consult [127], [35, 58] and the references therein).
Upward/Downward Continuation
Let \(\mathbb {S}^2_{R}\) be a Runge sphere inside the real Earth’s Σ^{int}, i.e., R < inf_{x ∈ Σ}|x|. Furthermore, let γ be smaller than the lowest possible altitude of the satellite, i.e., γ < inf_{x ∈ Γ}|x| (cf. Fig. 9).
Pseudodifferential Operator of the First Order Radial Derivative
Pseudodifferential Operator of the Second Order Radial Derivative
The Pseudodifferential Operator of the Hesse Tensor
Up to now, we assumed spherical geometry in connection with spherical harmonics, i.e., we presented spherical harmonics which are adequate for the common spherical approach, where the reference surface is supposed to be a sphere and the data are given on a spherical satellite orbit. Next, we make the first steps to a Runge concept which enables the application of arbitrary geometries. The basic idea underlying the Runge concept is to specify two spheres as illustrated in Fig. 6, thereby leading back to spherical basis functions and using the specific properties of outer harmonics.
Starting from the data given on the real satellite orbit Γ (which is not necessarily required here to be a closed surface) we pull down the tensorial information to a sphere \(\mathbb {S}_\gamma ^2\) of radius γ with \(\mathrm {dist}(\mathbb {S}_\gamma ^2, \varGamma )>0\). By virtue of “downward continuation” from \(\mathbb {S}_\gamma ^2\) to the sphere \(\mathbb {S}_R^2\) inside the Earth Σ_{int} such that \(\mathrm {dist}(\mathbb {S}_R^2, \varSigma )>0\) we are able to express the solution in terms of scalar outer harmonics, i.e., we obtain a representation of the gravitational potential on the real Earth’s surface Σ using data on the real orbit Γ.
Scalar Pseudodifferential Operator for Satellite Gravitational Gradiometry
The Tensorial SGG Pseudodifferential Operator
Pseudodifferential operators which play a certain role within the SGG-context
Operator | Description | Symbol | Order |
---|---|---|---|
\(\displaystyle {\Lambda _{\mbox{{up}}}^{R,\gamma }}\) | Upward continuation operator | \(\displaystyle \left (\frac {R}{\gamma }\right )^n\) | −∞ |
\(\displaystyle {\Lambda _{\mbox{{down}}}^{R,\gamma }}\) | Downward continuation operator | \(\displaystyle \ \left (\frac {\gamma }{R}\right )^n\) | ∞ |
\(\displaystyle \Lambda _{\mbox{{FRD}}}^{R}\) | First order radial derivative at the Earth’s surface | \(\displaystyle -\frac {(n+1)}{R}\) | 1 |
\(\displaystyle \Lambda _{\mbox{{SRD}}}^{R}\) | Second order radial derivative at the Earth’s surface | \(\displaystyle \frac {(n+1)(n+2)}{R^2}\) | 2 |
\(\displaystyle {\Lambda _{\mbox{{SGG}}}^{R,\gamma }}\) | Scalar pseudodifferential operator for satellite gravitational gradiometry | \(\displaystyle \frac {R^n}{\gamma ^n}\frac {(n+1)(n+2)}{\gamma ^2}\) | −∞ |
Λ_{Hesse} | Hesse tensor | \(\displaystyle \frac {\sqrt {(n+2)(n+1)(2n+3)(2n+1)}}{R^2}\) | 2 |
Λ_{SGG} | Tensorial pseudodifferential operator for satellite gravitational gradiometry | \(\displaystyle \frac {\sqrt {(n+2)(n+1)(2n+3)(2n+1)}} {\gamma ^2} \left ( \frac {R}{\gamma } \right ) ^{n}\) | −∞ |
4.3 Multiscale Frequency Regularization
Λ is a linear bounded injective compact operator (see, e.g., [35] for more details) so it follows (see, e.g., [45]) that the SGG-problem is ill-posed because of the unboundedness of Λ^{−1}. Hence, the SGG-problem needs regularization.
In principle, all regularization methods (mentioned in Sect. 2) are applicable to SGG. In what follows, however, we are only interested in two SGG-multiscale regularization strategies.
Tikhonov Regularization Strategy
Truncated Singular Value Regularization Strategy
- (i)
φ_{0}(0) = 1,
- (ii)
φ_{0} is monotonically decreasing,
- (iii)
φ_{0} is continuous at 0,
- (iv)
\( \varphi _0 : [0,\infty ) \to \mathbb {R} \) has a local support, i.e., supp φ_{0} ⊂ [0, 1].
- 1.
R_{j} is bounded,
- 2.the limit relationholds true in the outer space of \(\mathbb {S}^2_R\) .$$\displaystyle \begin{aligned} \begin{array}{rcl} \lim\limits_{j \to \infty} R_j \Lambda ({V})={V} \end{array} \end{aligned} $$(218)
For more bandlimited as well as non-bandlimited regularization strategies and methodologies the reader is referred to, e.g., [45, 63].
5 SGG in Space-Based Framework
As already pointed out, the SGG-problem can be modeled by a tensorial Fredholm integral equation of the first kind in space domain. In what follows we are interested in a novel SGG-modeling method using exclusively arguments in space domain thereby involving Abel–Poisson kernels, where the geometric situation is illustrated in Fig. 9.
5.1 Fredholm Integral Equation
5.2 Space Solution in Preparatory Framework
Unfortunately, it should be noted that the linear system (224) with all tensorial input data situated in the “far field” Γ of the Runge (Bjerhammar) sphere \(\mathbb {S}_R^2\) represents a serious obstacle to establish a discrete SGG-solution V^{R}. In fact, it may be expected that the coefficient matrix of the linear system (224) tends to be ill–conditioned at least for larger integers N_{R} and L_{R}. This is the reason, why we propose a twofold regularization procedure, namely a “downward continuation” multiscale regularization-strategy to take advantage from the decorrelation property of multiscale structures and to suppress numerical instabilities as far as possible.
5.3 Space Solution in Multiscale Framework
- the initial value γ_{0} is equal to γ, i.e., we let$$\displaystyle \begin{aligned} \begin{array}{rcl} \gamma_0 = \gamma, \end{array} \end{aligned} $$(227)
- the limit of the sequence {γ_{n}}_{n=0,1,…} is equal to the radius R of the Runge (Bjerhammar) sphere, i.e.,$$\displaystyle \begin{aligned} \begin{array}{rcl} \lim_{n \to \infty} \gamma_n \ = \ R. \end{array} \end{aligned} $$(228)
Initial Step: Discrete Abel–Poisson Lowpass Filtering
It should be noted that, for reasons of comparability, we use the same cubature formula for all forthcoming “downward continued” n-scale SGG-lowpass solutions on the Runge (Bjerhammar) sphere \(\mathbb {S}_R^{2, \mathrm {ext}}\).
Gaussian Sum Mollification
In [71], the approximation is attacked by a Newton-type optimization procedure.
In [72], a Remez algorithm exploits specific properties of a certain error functional.
Fast multipole methods (see, e.g., [16, 68, 70]) also provide tools of great numerical significance. The application of the fast multipole method also allows the treatment of noisy data by specifying parameter choices with and without prior knowledge of the noise level (cf. [70]).
Subsequent Steps: Discrete Abel–Poisson Lowpass and Bandpass Filtering
Once the coefficients \(b_k^{\gamma _1}\), \( k={N_{\gamma _0}+1}, \ldots , N_{\gamma _1}\) are calculated from (249), the potential \(V_{\gamma _1}\) is known from (242). The potential \( {V}_{\gamma _1}\) may be approximately regarded as the sum of the lowpass potential \(V_{\gamma _0}\) and the bandpass potential \(W_{\gamma _0}\) of scale 0 in \(\overline {\mathbb {S}_{\gamma _0}^{2, \mathrm {ext}}}\), leading to the “1-scale” SGG-lowpass solution in \({\overline {\mathbb {S}_{\gamma _1}^{2, \mathrm {ext}}}}\).
Now, generally, assume that “downward continued” k-scale” SGG-lowpass solutions \(V_{\gamma _k}^R,\,\, k\,{=}\,1, \ldots n,\) are calculated on \(\mathbb {S}_R^{2, \mathrm {ext}}\) in the indicated way. Our purpose is to determine \(V_{\gamma _{n+1}}^R\):
Continuing our wavelet process we obtain “downward continued” k-scale” SGG-lowpass solutions \(V_{\gamma _k}^R\), k = 0, 1, …, calculated on \(\mathbb {S}_R^{2, \mathrm {ext}},\) hence, available on the real Earth’s surface Σ and its exterior space just by restriction.
All in all, we successively obtain lowpass and bandpass SGG-discretizations of the Earth’s gravitational potential from a tensorial (error-free) SGG-dataset, thereby basing our multiscale procedure exclusively on operations in space domain (note that the error-affected case can be handled scale-by-scale by obvious smoothing and/or adjustment manipulations as already mentioned above).
6 Conclusion
The great advantage of SGG is that gradiometer data are globally available in homogeneous quality and high density. These facts justify an intense study of SGG from geodetic as well as mathematical point of view.
Although an impressive rate of the Earth’s gravitational potential can be detected globally at the orbit of a satellite (like GOCE), the intrinsic drawback of satellite techniques in geoscientific research is that the measurements are performed at a certain altitude. Consequently, in satellite gravitational gradiometry, only the Hesse tensor of an Abel–Poisson “filtered version of the gravitational potential on the Earth’s surface” is available by measurements and a loss compared to gravitational field information detectable on the Earth surface is unavoidable. So, it naturally follows that a “downward continuation” process involving adapted regularization procedures must be applied in order to handle the filtered “portion of the signal” that is available from the Earth’s potential by taking gradiometer measurements on the orbit.
In this respect, multiscale techniques using regularizing wavelets as constructed in this contribution, indeed, represent an outstanding methodology by its particular ability to exhaust all specific features of the “portion of the signal” by a “zooming-in” process. In fact, different powerful techniques for regularization are at the disposal of the geodetic analyst in frequency as well as space domain from globally via regionally up to locally reflected scales. All these facts lead to the conclusion that multiresolution methods are superior to any other SGG-solution strategy. As an immediate consequence, this work on multiscale regularization may be rated without any doubt as a well-promising and far-reaching methodology in future SGG-research.
Similarly to the collocational spline and smoothing theory (see, e.g., [32, 33, 120, 126, 131]), the choice of the regularization scaling function is an important problem in SGG. Mathematically, all wavelet regularization strategies are equivalent, however, the right computational compromise between mathematical rigor and geodetic relevance in respect to the geometry of the orbit, data width, accuracy of the data, and occurring noise level and characteristics is a task for future research, which should not be underestimated.
From mathematical point of view, we are not confronted with a multiscale solution of a well-posed boundary value problem of elliptic partial differential equations, for which subsequently each detail information guarantees an improvement to come closer and closer to the solution. Boundary value problems (see, e.g., [2, 50]) do not require any stopping strategy of the multiscale process, since the boundary data are (generally) not given only as “portion of the original signal” in filtered form. An algorithm establishing an approximate solution for the inverse SGG-problem, however, has to take into account the requirement to stop at the right level of approximation in order to model appropriately the Abel–Poisson filter-nature of the measured data. Unfortunately, today, the amount of amplification for the gravitational potential is not yet suitably known on the orbit as an a priori state. As a consequence, a missing stopping strategy in the multiresolution regularization caused by manual input usually produces huge errors in the potential at the Earth’s surface even from extremely small errors in the measurements. Thus there is a strong need for a geodetically relevant and mathematically motivated SGG-stopping strategy in the near future.
Notes
Acknowledgements
W. Freeden and H. Nutz thank the “Federal Ministry for Economic Affairs and Energy, Berlin” and the “Project Management Jülich” for funding the project “SPE” (funding reference number 0324016, CBM – Gesellschaft für Consulting, Business und Management mbH, Bexbach, Germany) on gravimetric potential methods in geothermal exploration.
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