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Satellite Gravitational Gradiometry: Methodological Foundation and Geomathematical Advances/ Satellitengradiometrie: Methodologische Fundierung und geomathematische Fortschritte

  • Willi FreedenEmail author
  • Helga Nutz
  • Reiner Rummel
  • Michael Schreiner
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)

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 decorrelation 

Zusammenfassung

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∕s2. 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∕s2, 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.

Satellite Gravitational Gradiometry (SGG) is the measurement of the gradients of the three components of the gravitational vectors – all nine of them, or a few, or a linear combination of some components. Gradiometry is therefore equivalent to the measurement of second-order derivatives of the gravitational potential. The principle is illustrated in Fig. 1.
Fig. 1

The principle of a gradiometer, i.e., the measurement of variations in the acceleration due to gravity

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.

With the x1-axis pointing in flight direction, the x2-axis orthogonal to the orbit plane and the x3-axis (almost) radially towards the Earth’s center the complete set of nine measurable components becomes (with \(V_{x_ix_j}= \partial ^2 V/ \partial x_i \partial x_j, i,j =1,2,3\)):
$$\displaystyle \begin{aligned} \begin{array}{rcl} && {} \underbrace{\left( \begin{array}{ccc} V_{x_1x_1} & V_{x_1x_2} & V_{x_1x_3} \\ V_{x_2x_1} & V_{x_2 x_2} & V_{x_2x_3} \\ V_{x_3x_1} & V_{x_3 x_2} & V_{x_3 x_3} \end{array} \right)}_{\mbox{gravitational tensor}} \\ && \quad + \underbrace{\left( \begin{array}{ccc} -\omega_{x_2}^2-\omega_{x_3}^2 & \omega_{x_1} \omega_{x_2} & \omega_{x_1} \omega_{x_3} \\ \omega_{x_2} \omega_{x_1} & -\omega_{x_3}^2- \omega_{x_1}^2 & \omega_{x_2} \omega_{x_3} \\ \omega_{x_3} \omega_{x_1} & \omega_{x_3} \omega_{x_2} & -\omega_{x_1}^2- \omega_{x_2}^2 \end{array} \right)}_{\mbox{angular velocities}} \\ && \quad + \underbrace{\left( \begin{array}{ccc} 0 & \frac{\partial}{\partial t} \omega_{x_3} & - \frac{\partial}{\partial t} \omega_{x_2} \\ {} - \frac{\partial}{\partial t} \omega_{x_3} & 0 & \frac{\partial}{\partial t} \omega_{x_1} \\ {} \frac{\partial}{\partial t} \omega_{x_2} & -\frac{\partial}{\partial t} \omega_{x_1} & 0 \end{array} \right)}_{\mbox{angular accelerations}}. \end{array} \end{aligned} $$
(1)
While the gravitational tensor and the angular velocity part are symmetric, the angular acceleration part is skew-symmetric. This allows the separation of the rotational terms from the gravitational part (see [110] for more details).

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 the spherically reflected case (see Fig. 2) the relation between the known tensorial measurements v = (∇⊗∇)V, i.e., the gradiometer orbital data on Γ and the corresponding potential V outside the sphere \(\mathbb {S}^2_R\) is equivalently expressible in the following ways:
  • (in frequency domain) by a pseudodifferential equation of the form
    $$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hspace{-1pc}\sum^\infty_{n=0} \sum^{2n+1}_{m=1} {V}^{\wedge_{L^2(\mathbb{S}^2_R)}} (n,m) \sqrt{\tilde{\mu}_n^{(1,1)}} \ {\mathbf{h}}^{R;(1,1)}_{n,m} \ = \ (\nabla \otimes \nabla) {V}(x) = \mathbf{v}(x), \end{array} \end{aligned} $$
    (2)
    where the coefficients \(\tilde {\mu }_n^{(1,1)}\) are 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)
    \({\mathbf {h}}^{R;(1,1)}_{n,m}\) 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
    $$\displaystyle \begin{aligned} \begin{array}{rcl} {V}^{\wedge_{L^2(\mathbb{S}^2_R)}} (n,m) = \langle V, H^R_{n,m}\rangle_{L^2(\mathbb{S}^2_R)}= \int_{\mathbb{S}^2_R}V(y) \ H^R_{n,m}(y) \ dS(y) \end{array} \end{aligned} $$
    (4)
    with \(H^R_{n,m}\) as scalar outer harmonic of degree n and order m.
    Fig. 2

    The spherically reflected geometric situation of satellite gravitational gradiometry (SGG)

  • (in space domain) by a linear integral equation of the first kind
    $$\displaystyle \begin{aligned} \int_{\mathbb{S}_R^2} \,V(y) \ \underbrace{ (\nabla_x \otimes \nabla_x) \frac{1}{4 \pi R} \frac{\vert x\vert^2 - R^2}{\vert x-y\vert^3}}_{= :{\mathbf{k}}_R (x,y)} \, dS(y) = \ (\nabla \otimes \nabla) {V}(x) = \mathbf{v}(x), \end{aligned} $$
    (5)
    where, 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 KR(⋅, ⋅) given by
    $$\displaystyle \begin{aligned} K_R(x,y) := \frac{1}{4 \pi R} \frac{\vert x\vert^2 - R^2}{\vert x-y\vert^3}, \ \ x, y \in {\overline{\mathbb{S}_R^{2;\mathrm{ext}}}}, x \neq y, \end{aligned} $$
    (6)
    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 ).

SGG-Operator Formulation

The SGG-problem in its spherically reflected form is a typical member of a “downward continuation problem” in the mathematical discipline of “Inverse Problems”. It essentially requires the determination of the gravitational potential V on and outside the sphere \(\mathbb {S}^2_R\) from the Hesse tensor v=∇ ⊗ ∇V on the satellite orbit Γ. Because of the affecting effects of gravi- tational datasets at LEO’s altitude (particularly in the presence of noise), downward continuation by SGG inevitably leads to an inverse problem of ill-posed character in the sense of Hadamard’s classification [73]. As a consequence, our first interest are regularization methods, which are relevant as approximate SGG-solution strategies. In fact, it turns out that Eqs. (2) and (5) define the same operator equation
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \Lambda \ V = \mathbf{v}, \end{array} \end{aligned} $$
(7)
so that, in accordance with well-known results of the mathematical theory of inverse problems, the operator Λ : X → Y in (7) constitutes an invertible mapping between scalar and tensorial Hilbert spaces X, Y of Sobolev type (under the assumption that the data are not erroneous). Unfortunately, by virtue of functional analytic arguments (see, e.g., [21, 81, 101, 108, 129] for more details), the inverse of Λ turns out to be unbounded, hence, Λ−1 needs regularization even if the SGG-data are free of errors. As a matter of fact, a large variety of regularization techniques may be applied to the SGG-problem.
In this work, however, our purpose is to work out multiscale approaches for regularization of the inverse operator Λ−1 of Λ, namely
  • 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

Let X and Y be two Hilbert spaces with inner products 〈⋅, ⋅〉X and 〈⋅, ⋅〉Y, respectively. Let
$$\displaystyle \begin{aligned} \Lambda:\mathrm{X} \longrightarrow \mathrm{Y} \end{aligned} $$
(8)
be a linear bounded operator. Given y ∈ Y, we are looking for a solution of the operator equation
$$\displaystyle \begin{aligned} \Lambda x = y. \end{aligned} $$
(9)

Four (mutually exclusive) situations arise (cf. [45]) in the discussion of the range \(\mathscr {R}( \Lambda )\) and the null space \(\mathscr {N}( \Lambda )\) of the operator Λ (and its adjoint operator Λ).
  1. (i)

    \(\mathscr {R}( \Lambda )\) is dense in Y, hence, \(\mathscr {N}(\Lambda ^*) = \{0\}\), and \(y \in \mathscr {R}( \Lambda )\);

     
  2. (ii)

    \(\mathscr {R}( \Lambda )\) is dense in Y, and \(y \notin \mathscr {R}(\Lambda )\);

     
  3. (iii)

    \(\overline {\mathscr {R}(\Lambda )}\) is a proper subspace of Y, and \(y \in \mathscr {R}(\Lambda ) + \mathscr {R}( \Lambda )^\perp \);

     
  4. (iv)

    \(\overline {\mathscr {R}(\Lambda )} \neq \mathrm {Y}\), and \(y \notin \mathscr {R}(\Lambda ) +\mathscr {R}(\Lambda )^\perp \).

     
In case (i) one has, of course, a solution in the classical sense; in case (ii) and (iv) a classical solution does not exist, while in case (iii) a solution need not exist.
We say x is a “least-squares solution” of (9) if
$$\displaystyle \begin{aligned} \inf \{\Vert \Lambda u-y \Vert : u \in \mathrm{X}\} = \Vert \Lambda x - y \Vert . \end{aligned} $$
(10)
Since
$$\displaystyle \begin{aligned} \Vert \Lambda u-y \Vert^2 = \Vert \Lambda u-Qy \Vert^2 + \Vert y-Qy \Vert^2,\end{aligned} $$
(11)
where Q is the orthogonal projector of Y onto \(\overline {R(\Lambda )}\), it is clear that a least-squares solution exists if and only if
$$\displaystyle \begin{aligned} \begin{array}{rcl} y \in \mathscr{R}(\Lambda) + \mathscr{R}(\Lambda)^\perp, \end{array} \end{aligned} $$
(12)
where \(\mathscr {R}(\Lambda ) + \mathscr {R}(\Lambda )^\perp \) is a dense set in Y . For such y the set of all least-squares solutions of (9), denoted by \(\mathscr {L}(y),\) is a nonempty closed convex set (indeed \(\mathscr {L}(y)\) is the translate of \(\mathscr {N}(\Lambda )\) by a fixed element of \(\mathscr {N}(y)\)), hence, it has a unique element of minimal norm, denoted by Λy.
The generalized inverse (or pseudoinverse)Λ is the linear operator which assigns to each \(y~{\in }~\mathscr {D} (\Lambda ^\dagger )~{:=}~\mathscr {R}(\Lambda )~{+}~\mathscr {R}(\Lambda )^\perp \), the unique element in the set \(\mathscr {L}(y) \cap \mathscr {N}(\Lambda )^\perp \), so that \(\mathscr {L}(y) = \Lambda ^\dagger y+ \mathscr {N}(\Lambda )\). It is easy to show that Λy is the minimal norm solution (equivalently the unique solution in \(\mathscr {N}(\Lambda )^\perp \)) of the normal equation
$$\displaystyle \begin{aligned} \begin{array}{rcl} \Lambda^* \Lambda x = \Lambda^* y \end{array} \end{aligned} $$
(13)
(the equation obtained by setting the first variation of ∥Λx − y2 equal to zero). It also follows that
$$\displaystyle \begin{aligned}\begin{array}{rcl} \Lambda^\dagger =(\Lambda|\ \mathscr{N}(\Lambda)^\perp)^{-1} Q \end{array} \end{aligned} $$
(14)
so that Λ can be characterized as the linear operator with the function-theoretic properties:
$$\displaystyle \begin{aligned} \mathscr{D}(\Lambda^\dagger)= \mathscr{R}(\Lambda) + \mathscr{R}(\Lambda)^\perp, \quad \mathscr{N}(\Lambda^\dagger) = \mathscr{R}(\Lambda)^\perp = \mathscr{N}(\Lambda^*) \end{aligned} $$
(15)
and
$$\displaystyle \begin{aligned}\begin{array}{rcl} \mathscr{R}(\Lambda^\dagger) = \mathscr{N}(\Lambda)^\perp. \end{array} \end{aligned} $$
(16)
The equivalence of these characterizations of Λ is established, e.g., in [100] and a large amount of subsequent contributions (see also [45, 102] for a lucid exposition and [103] for the generalization to unbounded operators).

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.

In accordance with Hadamard [73] we call an operator problem (9) well–posed, if the following properties are valid:
  • For all “admissible” data, a solution exists.

  • For all “admissible” data, the solution is unique.

  • The solution depends continuously on the data.

In our setting, these requirements can be translated more accurately into
  • 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.

If one of the three conditions is not fulfilled, the problem (9) is called ill–posed in the sense of Hadamard (note that, in the case of the SGG–problem, which is ill–posed, the most crucial problem is the unboundedness of the inverse operator Λ−1).

As canonical evolution of Hadamard’s classification, M.Z. Nashed [100] called the operator equation (9) well-posed in the least-squares (relative to X and Y) if for each y ∈Y the equation has a unique least-squares solution (of minimal norm), which depends continuously on y; otherwise the problem is ill-posed. The advantage of adopting this notion of well-posedness is that it focuses on infinite-dimensional problems (e.g., an inconsistent finite system of linear algebraic equations will not be ill-posed in above sense, while it is ill-posed in the sense of Hadamard). It follows immediately from the open mapping theorem in functional analysis (see, e.g., [103]) that the following statements are equivalent:
  • the problem (9) is well-posed;

  • \(\mathscr {R}(\Lambda )\) is closed;

  • Λ is bounded.

Summarizing we are led to the following conclusion (see [103]): The problem \(\left (\Lambda ;\mathrm {X},Y\right )\) is called well-posed in the sense of Nashed, if \(\mathscr {R}\left (\Lambda \right )\) is closed in Y. If \(\mathscr {R}\left (\Lambda \right )\) is not closed in Y, the problem \(\left (\Lambda ;\mathrm {X}, \mathrm {Y}\right )\) is called ill-posed in the sense of Nashed.

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).

Finally it is helpful to discuss the following situation: We assume that \(y \in {\mathscr {R}}(\Lambda )\), but only a perturbed right hand side yε is known. Furthermore, we suppose that
$$\displaystyle \begin{aligned} \Vert y-y^\varepsilon \Vert_{{}_{\mathrm{Y}}} < \varepsilon . \end{aligned} $$
(17)
Our aim is to solve
$$\displaystyle \begin{aligned} \Lambda x^\varepsilon = y^\varepsilon. \end{aligned} $$
(18)
Since yε might not be in \({\mathscr {R}}(\Lambda )\), the solution of this equation might not exist, and we have to generalize what is meant by a solution. xε is the least–squares solution of (18), if
$$\displaystyle \begin{aligned} \Vert \Lambda x^\varepsilon - y^\varepsilon \Vert_{{}_{\mathrm{Y}}} = \inf \{ \Vert \Lambda z - y^\varepsilon \Vert_{{}_{\mathrm{Y}}} : z \in \mathrm{X}\}. \end{aligned} $$
(19)
xε is the best approximate solution of Λxε = yε, if xε is a least–squares solution and
$$\displaystyle \begin{aligned} \Vert x^\varepsilon \Vert_{\mathrm{X}} = \inf \{ \Vert z \Vert_{\mathrm{X}} : z\ \mbox{is a least--squares solution of }\Lambda z = y^\varepsilon\} \end{aligned} $$
(20)
holds.

2.2 Regularization Strategies and Error Behavior

A serious problem for ill–posed problems occurs when Λ−1 or Λ are not continuous (as in the SGG-context). That means that small errors in the data or even small numerical noise can cause large errors in the solution. In fact, in most cases the application of an unbounded Λ−1 or Λ does not make any sense. The usual strategy to overcome this difficulty is to substitute the unbounded inverse operator
$$\displaystyle \begin{aligned} \Lambda^{-1}:{\mathscr{R}}(\Lambda) \longrightarrow \mathrm{Y} \end{aligned} $$
(21)
by a suitable bounded approximation
$$\displaystyle \begin{aligned} R: \mathrm{Y} \longrightarrow \mathrm{X} . \end{aligned} $$
(22)
The operator R is not chosen to be fix, but dependent on a regularization parameterα. According to the conventional approach in the theory of ill–posed problems we are then led to introduce the following definition:
A regularization strategy is a family of linear bounded operators
$$\displaystyle \begin{aligned} R_\alpha : \mathrm{Y} \longrightarrow \mathrm{X}, \ \alpha > 0, \end{aligned} $$
(23)
so that
$$\displaystyle \begin{aligned} \lim_{\alpha \rightarrow 0} R_\alpha \Lambda x = x \end{aligned} $$
(24)
for all
$$\displaystyle \begin{aligned} x \in \mathrm{X}, \end{aligned} $$
(25)
i.e., the operators \({\mathscr {R}}_\alpha \Lambda \) converge pointwise to the identity.
From the theory of inverse problems (see, e.g., [102, 103]) it is also clear that if Λ : Y →Y is compact and X has infinite dimension (as it is the case for the SGG-application we have in mind), then the operators \({\mathscr {R}}_\alpha \) are not uniformly bounded, i.e., there exists a sequence {αj} with limjαj = 0 and
$$\displaystyle \begin{aligned} \Vert R_{\alpha_j} \Vert_{\mathrm{L}({\mathrm{Y}, \mathrm{X}})} \rightarrow \infty \ \ \mbox{for }j \rightarrow \infty. \end{aligned} $$
(26)

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.

Let us discuss the error of the solution. For that purpose, we let \(y \in {\mathscr {R}}(\Lambda )\) be the (unknown) exact right–hand side and yε ∈Y be the measured data with
$$\displaystyle \begin{aligned} \Vert y-y^\varepsilon \Vert_{{}_{\mathrm{Y}}} < \varepsilon. \end{aligned} $$
(27)
For a fixed α > 0, we let
$$\displaystyle \begin{aligned} x^{\alpha,\varepsilon} = R_\alpha y^\varepsilon, \end{aligned} $$
(28)
and look at xα, ε as an approximation of the solution x of Λx = y. Then the error can be split as follows:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \Vert x^{\alpha,\varepsilon}-x \Vert _{\mathrm{X}} &\displaystyle =&\displaystyle \Vert R_\alpha y^\varepsilon -x \Vert _{\mathrm{X}} \\ &\displaystyle \le&\displaystyle \Vert R_\alpha y^\varepsilon - R_\alpha y \Vert _{\mathrm{X}} + \Vert R_\alpha y - x\Vert_{\mathrm{X}} \\ &\displaystyle \le&\displaystyle \Vert R_\alpha \Vert_{\mathrm{L} (\mathrm{Y},\mathrm{X})}\ \Vert y^\varepsilon-y \Vert_{{}_{\mathrm{Y}}} +\Vert R_\alpha y - x \Vert_{\mathrm{X}}, \end{array} \end{aligned} $$
(29)
such that
$$\displaystyle \begin{aligned} \Vert x^{\alpha,\varepsilon}-x \Vert_{\mathrm{X}} \le \varepsilon \Vert R_\alpha \Vert_{\mathrm{L} (\mathrm{Y},\mathrm{X})} +\Vert R_\alpha\Lambda x - x \Vert_{\mathrm{X}}. \end{aligned} $$
(30)
We see that the error between the exact and the approximate solution consists of two parts: The first term is the product of the bound for the error in the data and the norm of the regularization parameter Rα. This term will usually tend to infinity for α → 0 if the inverse Λ−1 is unbounded and Λ is compact (cf. (26)). The second term denotes the approximation error ∥(Rα − Λ−1)yX for the exact right–hand side y = Λx. This error tends to zero as α → 0 by the definition of a regularization strategy. Thus, both parts of the error show a diametrically oriented behavior. A typical picture of the errors in dependence on the regularization parameter α is sketched in Fig. 3. Thus, a strategy is needed to choose α dependent an ε in order to keep the error as small as possible, i.e. we would like to minimize
$$\displaystyle \begin{aligned} \varepsilon \Vert R_\alpha \Vert _{\mathrm{L} (\mathrm{Y},\mathrm{X})} +\Vert R_\alpha\Lambda x - x\Vert_{\mathrm{X}}. \end{aligned} $$
(31)
Fig. 3

Typical behavior of the total error in a regularization process

In principle, we distinguish two classes of parameter choice rules: If α = α(ε) only depends on ε, we call α = α(ε) an a–priori parameter choice rule. Otherwise α depends also on yε and we call α = α(ε, yε) an a–posteriori parameter choice rule. It is usual to say that a parameter choice rule is convergent, if for ε → 0 the rule fulfills the limit relations
$$\displaystyle \begin{aligned} \lim_{\varepsilon\rightarrow 0} \sup \{ \Vert R_{\alpha(\varepsilon, y^\varepsilon)} y^\varepsilon - \Lambda^{\dagger} y \Vert _{\mathrm{X}} \ : y^\varepsilon \in \mathrm{Y}, \ \Vert y^\varepsilon - y \Vert_{{}_{\mathrm{Y}}}\le \varepsilon \} = 0 \end{aligned} $$
(32)
and
$$\displaystyle \begin{aligned} \lim_{\varepsilon \rightarrow 0} \sup \{\alpha(\varepsilon, y^\varepsilon) \ : y^\varepsilon \in \mathrm{Y}, \Vert y-y^\varepsilon \Vert_{{}_{\mathrm{Y}}} \le \varepsilon\} = 0. \end{aligned} $$
(33)
We are stopping the discussion of parameter choice rules here. For more material the interested reader is referred to any textbook on inverse problems, e.g., [20, 81, 84, 108].

2.3 Singular Systems and Resolution Methods

If Λ : X →Y is compact (see, e.g., [45] for more details), a singular system (σn;vn, un) is defined as follows: \(\{\sigma _n^2\}_{n\in \mathbb {N}}\) are the nonzero eigenvalues of the self–adjoint operator ΛΛ (Λ is the adjoint operator of Λ), written down in decreasing order with corresponding multiplicity. The family \(\{v_n\}_{n \in \mathbb {N}}\) constitutes a corresponding complete orthonormal system of eigenvectors of ΛΛ. We let σn > 0 and define the family \(\{u_n\}_{n \in \mathbb {N}}\) via \( u_n = { \Lambda v_n}/{\Vert \Lambda v_n \Vert _{{ }_{\mathrm {Y}}}}. \) The sequence \(\{u_n\}_{n \in \mathbb {N}}\) forms a complete orthonormal system of eigenvectors of ΛΛ, and the following formulas are valid
$$\displaystyle \begin{aligned} \begin{array}{rcl} \Lambda v_n &\displaystyle =&\displaystyle \sigma_n u_n, \end{array} \end{aligned} $$
(34)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \Lambda^* u_n &\displaystyle =&\displaystyle \sigma_n v_n, \end{array} \end{aligned} $$
(35)
so that
$$\displaystyle \begin{aligned} \begin{array}{rcl} \Lambda x &\displaystyle =&\displaystyle \sum_{n=1}^\infty \sigma_n \langle x, v_n\rangle_{\mathrm{X}} \ u_n, \ \ x \in \mathrm{X}, {} \end{array} \end{aligned} $$
(36)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \Lambda^* y &\displaystyle =&\displaystyle \sum_{n=1}^\infty \sigma_n \langle y,u_n\rangle_{{}_{\mathrm{Y}}} \ v_n, \ \ y \in \mathrm{Y} {}. \end{array} \end{aligned} $$
(37)
The convergence of the infinite series is understood with respect to the Hilbert space norms under consideration. The identities (36) and (37) are called the singular value expansions of the corresponding operators. If there are infinitely many singular values, they tend to 0, i.e., limnσn = 0.

An important result in the theory of singular value expansions should be presented, which can be found in all standard textbooks on Inverse Problems:

Let (σn;vn, un) be a singular system for the compact linear operatorΛ, y ∈Y. Then we have
$$\displaystyle \begin{aligned} y \in {\mathscr{D}}(\Lambda^\dagger) \end{aligned} $$
(38)
if and only if
$$\displaystyle \begin{aligned} \sum_{n=1}^\infty \frac{|\langle y, u_n\rangle_{{}_{\mathrm{Y}}} |{}^2}{\sigma_n^2} < \infty, \end{aligned} $$
(39)
and for\(y \in {\mathscr {D}}(\Lambda ^\dagger )\)we have
$$\displaystyle \begin{aligned} \Lambda^\dagger y = \sum_{n=1}^\infty \frac{\langle y, u_n\rangle_{{}_{\mathrm{Y}}}}{\sigma_n}\ v_n. \end{aligned} $$
(40)

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.

The representation (40) of the best–approximate solution motivates a method for the construction of regularization operators, namely by damping the factors 1∕σn in such a way that the series converges for all y ∈Y. We are looking for filters
$$\displaystyle \begin{aligned} q :(0, \infty) \times (0, \Vert \Lambda \Vert _{\mathrm{L} (\mathrm{X},\mathrm{Y})}) \longrightarrow \mathbb{R} \end{aligned} $$
(41)
such that
$$\displaystyle \begin{aligned} R_\alpha y := \sum_{n=1}^\infty \frac{q(\alpha, \sigma_n)}{\sigma_n} \langle y, u_n\rangle_{{}_{\mathrm{Y}}}\;v_n, \ \ y \in \mathrm{Y}, \end{aligned} $$
(42)
is a regularization strategy. The following statement is, e.g., known from [81].
LetΛ : X → Y be compact with singular system (σn;vn, un). Assume thatqfrom (41) has the following properties:
  1. (i)

    |q(α, σ)|≤ 1 for allα > 0 and 0 < σ ≤∥ΛL(X,Y).

     
  2. (ii)

    For everyα > 0 there exists ac(α) so that |q(α, σ)|≤ c(α)σfor all 0 < σ ≤∥ΛL(X,Y).

     
  3. (iii)

    limα→0q(α, σ) = 1 for every 0 ≤ σ ≤∥ΛL(X,Y).

     
Then the operatorRα : Y →X, α > 0, defined by
$$\displaystyle \begin{aligned} R_\alpha y := \sum_{n=1}^\infty \frac{q(\alpha, \sigma_n)}{\sigma_n} \langle y, u_n\rangle_{{}_{\mathrm{Y}}}\;v_n, \ \ y \in \mathrm{Y}, \end{aligned} $$
(43)
is a regularization strategy withRαL(Y,X) ≤ c(α).
The function q is called a regularizing filter for Λ. Two important examples should be mentioned:
$$\displaystyle \begin{aligned} q(\alpha, \sigma) = \frac{\sigma^2}{\alpha+\sigma^2} \end{aligned} $$
(44)
defines the Tikhonov regularization, whereas
$$\displaystyle \begin{aligned} q(\alpha,\sigma) = \left\{\begin{array}{ll} 1 &,\ \sigma^2 \ge \alpha \\ 0 &, \ \sigma^2 < \alpha\end{array}\right. \end{aligned} $$
(45)
leads to the regularization by truncated singular value decomposition.

2.4 Summary Excursion to Regularization Techniques

The strategy of resolution and reconstruction of ill-posed problems involves one or more of the following intuitive ideas (cf. [45, 101, 102], and the references therein):
  • 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]).

From the standpoint of mathematical and numerical analysis one can roughly group “regularization methods” into six categories (cf. [45, 48]):
  1. 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. 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. 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. 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. 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. 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.

     
The philosophy of resolution leads to the use of algebraic methods versus function space methods, statistical versus deterministic approaches, strong versus weak noise, etc.

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

According to the classical Newton’s Law of Gravitation (1687), knowing the density distribution inside the Earth, the gravitational potential V can be computed everywhere in \({\mathbb {R}}^3\). In the Earth’s exterior Σext, the Newtonian Earth’s gravitational volume potentialV convolving the fundamental solution x↦(4π)−1|x − y|−1 of the Laplacian Δ against the mass density distribution ρ inside the Earth Σint, i.e.,
$$\displaystyle \begin{aligned} V(x) = \int_{\varSigma^{\mathrm{{int}}}}\frac{1}{4 \pi} \frac{1}{|x-y|} \ \rho(y) \ dV(y) \end{aligned} $$
(46)
is harmonic in Σext:
$$\displaystyle \begin{aligned} \varDelta V(x) = 0,\ \quad x \in \varSigma^{\mathrm{{ext}}} \end{aligned} $$
(47)
(we omit the gravitational constant for our theoretical purposes, which must be observed, however, in numerical obligations). In the Earth’s interior Σint, V is related to the Earth’s density ρ (provided that ρ is assumed to be Hölder continuous) via the Poisson differential equations
$$\displaystyle \begin{aligned} \varDelta V(x) = - \rho (x), \ \quad x \in \varSigma^{\mathrm{{int}}}. \end{aligned} $$
(48)
Moreover, the gravitational potential V is regular at infinity. In fact, it can be readily seen that
$$\displaystyle \begin{aligned}|y| \le \frac{|x|}{2}, \ \ y \in \overline{\varSigma^{\mathrm{{int}}}},\end{aligned} $$
(49)
implies
$$\displaystyle \begin{aligned} |x-y| \geq ||x| -|y||\geq \frac{1}{2} |x|, \end{aligned} $$
(50)
i.e.,
$$\displaystyle \begin{aligned} \vert V(x) \vert = O \left( \vert x \vert^{-1}\right), \quad \vert x \vert \to \infty . {} \end{aligned} $$
(51)
Collecting the results known from Newtonian potential theory on the Earth’s gravitational field v for the outer space Σext we are confronted with the following characterization: The vector field v is real-analytic in Σext such that
(v1)

div v = ∇⋅ v = 0, curl v = L ⋅ v = 0 in Σext,

(v2)

v is regular at infinity: \(|v(x)| = O \left ( \vert x \vert ^{-2} \right ) \), |x|→.

The properties (v1) and (v2) imply that the Earth’s gravitational field v in the exterior of the Earth Σext is a gradient field
$$\displaystyle \begin{aligned} v(x)=\nabla V(x), \ \ x \in \varSigma^{\mathrm{{ext}}}. \end{aligned} $$
(52)
Moreover, the gradient field of the Earth’s gravitational field (i.e., the Jacobi matrix field) v = ∇v, obeys the following properties: v is a real-analytic tensor field in the Earth’s exterior Σext such that
(v1)

div v = ∇⋅v = 0, curl v = L ⋅v = 0 in the Earth’s exterior Σext,

(v2)

v is regular at infinity: \({\vert \mathbf {v} (x) \vert } = O \left ( \vert x \vert ^{-3} \right )\), |x|→.

Combining our identities we finally see that v can be represented as the Hesse tensor of the scalar field V , i.e.,
$$\displaystyle \begin{aligned} \mathbf{v} (x)= \left( \nabla \otimes \nabla\right) V(x), \ \ x \in \varSigma^{\mathrm{{ext}}}. \end{aligned} $$
(53)

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 determiningVon 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.

In conclusion, seen from the potential-theoretic context, SGG is a non-standard problem. The reasons are obvious:
  • 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.
    Fig. 4

    The (real) SGG-geometric situation as interplay between Earth Σ and 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.

Let us begin with the introduction of some auxiliary material: We understand the Earth’s surface \(\varSigma \subset \mathbb {R}^3\) to be a regular surface, i.e., Σ is a surface with the following properties:
  1. 1.

    Σ divides the Euclidean space \(\mathbb {R}^3\) into the (open) bounded region Σint (inner space) and the (open) unbounded region Σext (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. ∅ = Σint ∩ Σext,

     
  2. 2.

    Σint contains the origin 0,

     
  3. 3.

    Σ is a closed and compact surface free of double points,

     
  4. 4.

    Σ is locally of class C(2), i.e. Σ is locally C(2)–smooth

     
(see [36] for more details concerning regular surfaces).

The function spaces C(2)(Σext) and c(2)(Σext) etc. are defined in canonical way.

We define Pot(Σext) as the scalar space of potentials harmonic in Σext and regular at infinity:
$$\displaystyle \begin{aligned} {\mathrm{Pot}(\varSigma^{\mathrm{{ext}}})} = \{ F \in \mathrm{C}^{(2)}(\varSigma^{\mathrm{{ext}}}): \, \varDelta F = 0 \mbox{ in } \varSigma^{\mathrm{{ext}}},\, F(x)=O\left(|x|{}^{-1}\right), \ \ |x| \to \infty \}. \end{aligned} $$
(54)
\(\mathrm {Pot}^{(0)}(\overline {\varSigma ^{\mathrm {{ext}}}})\) is the space of continuous functions \(F: \overline {\varSigma ^{\mathrm {{ext}}}} \to \mathbb {R}\) whose restrictions \(F |{ }_{\varSigma ^{\mathrm {{ext}}}}\) are members of Pot(Σext).
In brief (but not quite mathematically accurate),
$$\displaystyle \begin{aligned} \mathrm{Pot}^{(0)}(\overline{\varSigma^{\mathrm{{ext}}}}) = \mathrm{Pot}(\varSigma^{\mathrm{{ext}}})\cap \mathrm{C}^{(0)}(\overline{\varSigma^{\mathrm{{ext}}}}), \end{aligned} $$
(55)
and, in similar way,
$$\displaystyle \begin{aligned} \mathrm{Pot}^{(k)}(\overline{\varSigma^{\mathrm{{ext}}}}) = \mathrm{Pot}(\varSigma^{\mathrm{{ext}}})\cap \mathrm{C}^{(k)}(\overline{\varSigma^{\mathrm{{ext}}}}), \ \ \; 0 \le k \le \infty . \end{aligned} $$
(56)
We introduce the tensorial counterpart pot(Σext) of Pot(Σext) as follows:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{pot} (\varSigma^{\mathrm{{ext}}}) = \{ \mathbf{f} \in {\mathbf{c}}^{(1)}(\varSigma^{\mathrm{{ext}}}): &\displaystyle &\displaystyle \nabla \cdot \mathbf{f}=0, \quad \nabla \wedge \, \mathbf{f}=0 \ \mathrm{in} \ \varSigma^{\mathrm{{ext}}}, \notag \\ &\displaystyle &\displaystyle |\mathbf{f}(x)|= O\left(|x|{}^{-3}\right), \, |x| \to \infty \} . \end{array} \end{aligned} $$
(57)
Similarly, we let
$$\displaystyle \begin{aligned} \mathbf{pot}^{(0)}(\overline{\varSigma^{\mathrm{{ext}}}}) = \mathbf{pot}(\varSigma^{\mathrm{{ext}}}) \cap {\mathbf{c}}^{(0)}(\overline{\varSigma^{\mathrm{{ext}}}}), \end{aligned} $$
(58)
and
$$\displaystyle \begin{aligned} \mathbf{pot}^{(k)}(\overline{\varSigma^{\mathrm{{ext}}}}) = \mathbf{pot}(\varSigma^{\mbox{ext}}) \cap {\mathbf{c}}^{(k)}(\overline{\varSigma^{\mathrm{{ext}}}}), \ \ \; 0 \le k \le \infty. \end{aligned} $$
(59)

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

For all \(x \in {\mathbb {R}}^3\), x = (x1, x2, x3)T, different from the origin 0, we have x = , \(r=\vert x \vert = \sqrt {x_1^2+x_2^2+x_3^2},\) where ξ = (ξ1, ξ2, ξ3)T is the uniquely determined directional unit vector of \(x \in {\mathbb {R}}^3\). As usual, the unit sphere in \({\mathbb {R}}^3\) will be denoted by \(\mathbb {S}^2=\mathbb {S}_1^2\), while \(\mathbb {S}_\alpha ^2\) designates the 2-sphere around the origin with radius α. The unit 3-ball around the origin is denoted by \(\mathbb {B}^3=\mathbb {B}_1^3,\) and \(\mathbb {B}_\alpha ^3\) is the ball around the origin with radius α. If the vectors ε1, ε2, ε3 form the canonical orthonormal basis in \({\mathbb {R}}^3\), we may represent \(\xi \in \mathbb {S}_1^2\) in polar coordinates by
$$\displaystyle \begin{aligned} \xi &= t \varepsilon^3 + \sqrt{1-t^2} \left( \cos \varphi \varepsilon^1 + \sin \varphi \varepsilon^2 \right), \\ &\quad - 1 \le t \le 1, \, 0 \le \varphi < 2 \pi,\; t = \cos \theta. \end{aligned} $$
(60)
Inner, vector, and dyadic (tensor) product of two vectors \(x,y \in {\mathbb {R}}^3\), respectively, are denoted by x ⋅ y, x ∧ y, and x ⊗ y.

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 Fij of f are defined by Fij = εi ⋅ (fεj) = (εi)T(fεj), so that y = fx is equivalent to \(y \cdot \varepsilon ^i = \sum ^3_{j=1} F_{ij} (x \cdot \varepsilon ^j)\). We write fT 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| = (ff)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.

Let Hn be a homogeneous harmonic polynomial of degree n in \(\mathbb {R}^3\), \(n \in \mathbb {N}_0\), i.e. \(H_n \in \mathrm {Harm}_n(\mathbb {R}^3)\). The restriction \(Y_n = H_n|{ }_{\mathbb {S}^2}\) is called (scalar) spherical harmonic of degreen. The space
$$\displaystyle \begin{aligned} \begin{array}{rcl}\{Y_n = H_n|{}_{\mathbb{S}^2} : H_n \in \mathrm{Harm}_n(\mathbb{R}^3) \}\end{array} \end{aligned} $$
(61)
of all (scalar) spherical harmonics of degree n is denoted by \(\mathrm {Harm}_n(\mathbb {S}^2)\).
\(\mathrm {Harm}_n(\mathbb {S}^2)\) is known to be of dimension 2n + 1. Spherical harmonics of different degrees are orthogonal in \(\mathrm {L}^2(\mathbb {S}^2)\)–sense, that is
$$\displaystyle \begin{aligned} \left\langle Y_n, Y_{\tilde{n}} \right\rangle _{\mathrm{L}^2(\mathbb{S}^2)} = \int _{\mathbb{S}^2} Y_n(\xi) Y_{\tilde{n}} (\xi) \hspace{0.5mm} d S (\xi) = 0, \quad n \neq \tilde{n}. \end{aligned} $$
(62)
Throughout this text a capital letter Y followed by one or two indices always denotes a spherical harmonic of the degree given by the first index and order given by the second index. Two indices mean that the function, for example Yn,m, is a member of an \(\mathrm {L}^2(\mathbb {S}^2)\)–orthonormal system of functions \(\{Y_{n,1},\ldots , Y_{n,2n+1} \}_{n \in \mathbb {N}_0}\).
By use of the scalar spherical harmonics every function \(F~{\in }~\mathrm {L}^2(\mathbb {S}^2)\) can be written as an orthogonal (Fourier) series
$$\displaystyle \begin{aligned} F = \sum \limits _{n=0} ^{\infty} \sum \limits _{m=1} ^{2n+1} F^{\wedge_{\mathrm{L}^2(\mathbb{S}^2)}} (n,m) Y_{n,m} \end{aligned} $$
(63)
(in \(\mathrm {L}^2(\mathbb {S}^2)\)–sense) with Fourier coefficients
$$\displaystyle \begin{aligned} F^{\wedge_{\mathrm{L}^2(\mathbb{S}^2)}} (n,m) = \langle F,Y_{n,m}\rangle_{\mathrm{L} ^2(\mathbb{S}^2)} = \int _{\mathbb{S}^2} F(\xi) Y_{n,m} (\xi) \ dS(\xi). \end{aligned} $$
(64)
The system {Yn,m}n=0,1,…;m=1,…,2n+1 is closed in \(\mathrm {C}^{(0)} (\mathbb {S}^2)\) with respect to the norm \(\|\cdot \|{ }_{\mathrm {C}^{(0)} (\mathbb {S}^2)}\), i.e., for any number ε > 0 and any function \(F \in \mathrm {C}^{(0)}(\mathbb {S}^2)\), there exists a linear combination
$$\displaystyle \begin{aligned} F_N = \sum _{n=0} ^{N} \sum _{m=1} ^{2n+1} d_{n,m} Y_{n,m}, \end{aligned} $$
(65)
such that \(\|F - F_N\|{ }_{\mathrm {C}^{(0)}(\mathbb {S}^2)} \le \varepsilon \). The system {Yn,m}n=0,1,…;m=1,…,2n+1 is, furthermore, complete in \(\mathrm {L}^2(\mathbb {S}^2)\) with respect to \(\langle \cdot , \cdot \rangle _{\mathrm {L}^2(\mathbb {S}^2)}\), i.e., \(F~{\in }~\mathrm {L}^2(\mathbb {S}^2)\) with \(F^{\wedge _{\mathrm {L}^2(\mathbb {S}^2)}} (n,m) = 0\) for all n = 0, 1, …;m = 1, …, 2n + 1 implies F = 0 (see, e.g., [58, 98]).

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.

Let {Yn,m}m=1,…,2n+1 be an \(\mathrm {L}^2(\mathbb {S}^2)\)–orthonormal system in \(\mathrm {Harm}_n(\mathbb {S}^2)\). Then, for any pair \((\xi ,\eta ) \in \mathbb {S}^2 \times \mathbb {S}^2\), the addition theorem reads
$$\displaystyle \begin{aligned} \sum \limits _{m=1} ^{2n+1} Y_{n,m}(\xi) Y_{n,m}(\eta) = \frac {2n+1} {4 \pi} P_n(\xi \cdot \eta), \end{aligned} $$
(66)
where Pn : [−1, 1] → [−1, 1] is the Legendre polynomial of degree n.

Tensor Spherical Harmonics

We recapitulate the introduction of tensor spherical harmonics as proposed by [52]: By \({\mathbf {l}}^2(\mathbb {S}^2)\) we denote the Hilbert space of square–integrable tensor fields \(\mathbf {f}: \mathbb {S}^2 \to {\mathbb {R}}^3 \otimes {\mathbb {R}}^3\) with the inner product
$$\displaystyle \begin{aligned} \langle \mathbf{f},\mathbf{g}\rangle_{{\mathbf{l}}^2(\mathbb{S}^2)} = \int _{\mathbb{S}^2} \mathbf{f}(\xi) \cdot \mathbf{g} (\xi) \hspace{0.5mm} dS (\xi), \quad \mathbf{f}, \mathbf{g} \in {\mathbf{l}}^2(\mathbb{S}^2), \end{aligned} $$
(67)
and associated norm \(\| \cdot \|{ }_{{\mathbf {l}}^2(\mathbb {S}^2)}\) (note that the space \({\mathbf {l}}^{2} (\mathbb {S}^2)\) is the completion of \({\mathbf {c}}^{(0)}(\mathbb {S}^2)\) with respect to the norm \(\| \cdot \|{ }_{\mathbf {l} ^2(\mathbb {S}^2)}\)).
We now introduce the operators \({\mathbf {o}}^{(i,k)}:\mathrm {C}^{(\infty )}(\mathbb {S}^2) \rightarrow {\mathbf {c}}^{(\infty )}({\mathbb {S}^2})\), i, k = 1, 2, 3, which transform scalar functions into tensor fields:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mbox{\hspace{-0.5cm}} {\mathbf{o}}^{(1,1)} F(\xi) &\displaystyle :=&\displaystyle \xi \otimes \xi F(\xi), \end{array} \end{aligned} $$
(68)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{o}}^{(1,2)} F(\xi) &\displaystyle :=&\displaystyle \xi \otimes \nabla^*_\xi F(\xi), \end{array} \end{aligned} $$
(69)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{o}}^{(1,3)} F(\xi) &\displaystyle :=&\displaystyle \xi \otimes L^*_{\xi} F(\xi), \end{array} \end{aligned} $$
(70)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{o}}^{(2,1)} F(\xi) &\displaystyle :=&\displaystyle \nabla^*_{\xi} F(\xi) \otimes \xi, \end{array} \end{aligned} $$
(71)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{o}}^{(2,2)} F(\xi) &\displaystyle :=&\displaystyle {\mathbf{i}}_{\mathrm{tan}}(\xi) F(\xi), \end{array} \end{aligned} $$
(72)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{o}}^{(2,3)} F(\xi) &\displaystyle :=&\displaystyle \left(\nabla^*_{\xi}\otimes\nabla^*_\xi - L^*_{\xi} \otimes L^*_\xi\right) F(\xi) + 2 \nabla^*_{\xi} F(\xi) \otimes \xi, \end{array} \end{aligned} $$
(73)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{o}}^{(3,1)} F(\xi) &\displaystyle :=&\displaystyle L^*_{\xi} F(\xi) \otimes \xi, \end{array} \end{aligned} $$
(74)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{o}}^{(3,2)} F(\xi) &\displaystyle :=&\displaystyle \left(\nabla^*_\xi\otimes L^*_\xi + L^*_\xi\otimes\nabla^*_\xi\right) F(\xi) + 2 L^*_{\xi} F(\xi) \otimes \xi, \end{array} \end{aligned} $$
(75)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{o}}^{(3,3)} F(\xi) &\displaystyle :=&\displaystyle {\mathbf{j}}_{\mathrm{tan}}(\xi) F(\xi), \end{array} \end{aligned} $$
(76)
\(F \in \mathrm {C}^{(2)}(\mathbb {S}^2)\), \(\xi \in \mathbb {S}^2\). The tensors itan = i − ξ ⊗ ξ and jtan = ξ ∧i are the surface identity tensor and the surface rotation tensor, respectively. The adjoint operators O(i, k) to o(i, k) satisfying
$$\displaystyle \begin{aligned} \langle {\mathbf{o}}^{(i,k)} F, \mathbf{f}\rangle_{{\mathbf{l}}^2(\mathbb{S}^2)} = \langle F, O^{(i,k)} \mathbf{f}\rangle_{\mathrm{L}^2(\mathbb{S}^2)} \end{aligned} $$
(77)
for \(F \in \mathrm {C}^{(2)}(\mathbb {S}^2)\) and \(\mathbf {f} \in {\mathbf {c}}^{(2)}(\mathbb {S}^2)\) are given by
$$\displaystyle \begin{aligned} O^{(1,1)} \mathbf{f}(\xi) &:= \xi^T \mathbf{f}(\xi)\xi, \end{aligned} $$
(78)
$$\displaystyle \begin{aligned} O^{(1,2)} \mathbf{f}(\xi) &:= -\nabla^*_\xi\cdot p_{\mathrm{tan}} \left(\xi^T \mathbf{f}(\xi)\right), \end{aligned} $$
(79)
$$\displaystyle \begin{aligned} O^{(1,3)} \mathbf{f}(\xi) &:= -L^*_\xi\cdot p_{\mathrm{tan}} \left(\xi^T \mathbf{f}(\xi)\right), \end{aligned} $$
(80)
$$\displaystyle \begin{aligned} O^{(2,1)} \mathbf{f}(\xi) &:= -\nabla^*_\xi\cdot p_{\mathrm{tan}} \left(\mathbf{f}(\xi)\xi\right), \end{aligned} $$
(81)
$$\displaystyle \begin{aligned} O^{(2,2)} \mathbf{f}(\xi) &:={\mathbf{i}}_{\mathrm{tan}}(\xi) \cdot \mathbf{f}(\xi), \end{aligned} $$
(82)
$$\displaystyle \begin{aligned} O^{(2,3)} \mathbf{f}(\xi) &:=\nabla^*_\xi \cdot p_{\mathrm{tan}}\left( \nabla^*_\xi \cdot {\mathbf{p}}_{\mathrm{tan,*}} \mathbf{f}(\xi)\right) -L^*_\xi \cdot p_{\mathrm{tan}} \left( L^*_\xi \cdot {\mathbf{p}}_{\mathrm{tan,*}}\mathbf{f}(\xi)\right) \\ & \quad -2 \nabla^*_\xi \cdot p_{\mathrm{tan}} \left(\mathbf{f}(\xi)\xi\right), \end{aligned} $$
(83)
$$\displaystyle \begin{aligned} O^{(3,1)} \mathbf{f}(\xi) &:= -L^*_\xi\cdot p_{\mathrm{tan}}\left(\mathbf{f}(\xi) \xi\right), \end{aligned} $$
(84)
$$\displaystyle \begin{aligned} O^{(3,2)} \mathbf{f}(\xi) &:=L^*_\xi \cdot p_{\mathrm{tan}} \left( \nabla^*_\xi \cdot {\mathbf{p}}_{\mathrm{tan}} \mathbf{f}(\xi)\right) +\nabla^*_\xi \cdot p_{\mathrm{tan}} \left( L^*_\xi \cdot {\mathbf{p}}_{\mathrm{tan}} \mathbf{f}(\xi)\right) \\ & \quad -2 L^*_\xi \cdot p_{\mathrm{tan}}\left(\mathbf{f}(\xi)\xi\right), \end{aligned} $$
(85)
$$\displaystyle \begin{aligned} O^{(3,3)} \mathbf{f}(\xi) &:={\mathbf{j}}_{\mathrm{tan}}(\xi) \cdot \mathbf{f}(\xi), \end{aligned} $$
(86)
where
$$\displaystyle \begin{aligned} \begin{array}{rcl} p_{\mathrm{tan}} f(\xi) &\displaystyle :=&\displaystyle f(\xi) - (\xi \cdot f(\xi))\xi, \end{array} \end{aligned} $$
(87)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{p}}_{\mathrm{tan,*}} \mathbf{f}(\xi) &\displaystyle :=&\displaystyle \mathbf{f}(\xi) - \xi \otimes ((\mathbf{f}(\xi)) ^T \xi), \end{array} \end{aligned} $$
(88)
\(\xi \in \mathbb {S}^2\).
With the help of the operators o(i, k) we are able to define a set of tensor spherical harmonics\(\{ \mathbf {y} _{n,m} ^{(i,k)} \}_{i,k = 1,2,3; \ n= 0_{ik}, \ldots ; \ m=1, \ldots , 2n+1}\) by
$$\displaystyle \begin{aligned} \mathbf{y} _{n,m} ^{(i,k)} := \left( \mu_n^{(i,k)} \right) ^{-1/2} {\mathbf{o}}^{(i,k)} Y_{n,m}, \end{aligned} $$
(89)
where the normalization constants \(\mu _n^{(i,k)}\) are given by
$$\displaystyle \begin{aligned} \hspace{-.5pc} \mu_n^{(i,k)} := \left\{ \begin{array}{lll} 1, & \; (i,k) = (1,1), \\ {} 2, & \; (i,k) \in \{ (2,2), (3,3) \}, \\ {} n(n+1), & \; (i,k) \in \{ (1,2), (1,3), (2,1), (3,1) \}, \\ {} 2n(n+1)(n(n+1)-2), & \; (i,k) \in \{ (2,3), (3,2) \}. \end{array} \right.\hspace{-.5pc} \end{aligned} $$
(90)
For simplicity, we use the abbreviation
$$\displaystyle \begin{aligned} 0_{ik} := \left\{ \begin{array}{lll} 0, & \; (i,k) \in \{ (1,1), (2,2), (3,3) \}, \\ {} 1, & \; (i,k) \in \{ (1,2), (1,3), (2,1), (3,1) \}, \\ {} 2, & \; (i,k) \in \{ (2,3), (3,2) \}. \end{array} \right. \end{aligned} $$
(91)
By \(\mathbf {harm}_n^{(i,k)}(\mathbb {S}^2)\) we denote the space of all tensor spherical harmonics of degree n and kind (i, k).

Addition Theorem for Tensor Spherical Harmonics.

Let \(\{ \mathbf {y} _{n,m} ^{(i,k)} \}_{m=1, \ldots , 2n+1}\) be an \(\mathbf {l} ^2 (\mathbb {S}^2)\)–orthonormal basis of \(\mathbf {harm}_n^{(i,k)}(\mathbb {S}^2)\). Then, for any pair \((\xi ,\eta ) \in \mathbb {S}^2 \times \mathbb {S}^2\), the tensorial addition theorem reads
$$\displaystyle \begin{aligned} \sum \limits _{m=1} ^{2n+1} \mathbf{y} _{n,m} ^{(i,k)} (\xi) \otimes \mathbf{y} _{n,m} ^{(p,q)} (\eta) = \frac{2n+1} {4 \pi} \mathbf{P} _n ^{(i,k,p,q)} (\xi, \eta), \end{aligned} $$
(92)
i, k, p, q ∈{1, 2, 3}, where \(\mathbf {P} _n ^{(i,k,p,q)}: \mathbb {S}^2 \times \mathbb {S}^2 \to \mathbb {R} ^3 \otimes \mathbb {R} ^3 \otimes \mathbb {R}^3 \otimes \mathbb {R} ^3\) denote the Legendre tensors of degreen defined by
$$\displaystyle \begin{aligned} \mathbf{P} _n ^{(i,k,p,q)} := \left( \mu _n ^{(i,k)} \right) ^{-1/2} \left( \mu _n ^{(p,q)} \right) ^{-1/2} \mathbf{o} _{\xi} ^{(i,k)} \mathbf{o} _{\eta} ^{(p,q)} P_n(\xi \cdot \eta), \quad \xi, \eta \in \mathbb{S}^2. \end{aligned} $$
(93)
Note that, for sufficiently smooth tensor fields \(\mathbf {f}: \mathbb {S}^2 \to \mathbb {R}^3 \otimes \mathbb {R}^3\) of the form
$$\displaystyle \begin{aligned} \mathbf{f} (\xi) = \sum \limits _{i,k=1} ^{3} F_{i,k} (\xi) \varepsilon^i \otimes \varepsilon ^k, \quad x \in \mathbb{S}^2, \end{aligned} $$
(94)
we set
$$\displaystyle \begin{aligned} \mathbf{o} _{\xi} ^{(p,q)} \mathbf{f} (\xi) = \sum \limits _{i,k=1} ^{3} \left( \mathbf{o} _{\xi} ^{(p,q)} F_{i,k} (\xi) \right) \otimes \varepsilon ^i \otimes \varepsilon ^k. \end{aligned} $$
(95)
Explicit representations of the Legendre tensors can be found in [52].

By \(\mathbf {harm}_n(\mathbb {S}^2)\) we denote the space of all tensor spherical harmonics of degree n.

The system \(\{ \mathbf {y} _{n,m} ^{(i,k)} \}\) of tensor spherical harmonics was introduced by concentrating on the fact that the decomposition into normal and tangential tensor fields is fulfilled (cf. [105]). However, one disadvantage of this set of tensor spherical harmonics is that these functions are not eigenfunctions of the (scalar) Beltrami operator. Nonetheless, this property enables us to define so–called outer harmonics in such a way that they fulfill the Laplace equation in the outer space. This is the reason why we introduce another set of operators \(\tilde {\mathbf {o}}^{(i,k)}: \mathrm {C}^{(\infty )}(\mathbb {S}^2) \rightarrow {\mathbf {c}}^{(\infty )}({\mathbb {S}^2})\), i, k = 1, 2, 3, based on the operators o(i, k) by
$$\displaystyle \begin{aligned} \left( \begin{array}{c} \tilde{\mathbf{o}} ^{(1,1)} Y_n \\ \tilde{\mathbf{o}} ^{(1,2)} Y_n \\ \tilde{\mathbf{o}} ^{(2,1)} Y_n \\ \tilde{\mathbf{o}} ^{(2,2)} Y_n \\ \tilde{\mathbf{o}} ^{(3,3)} Y_n \end{array} \right) = {\mathbf{a}}_D \left( \begin{array}{c} Y_n \\ Y_n \\ Y_n \\ Y_n \\ Y_n \end{array} \right), \quad \mathrm{and} \quad \left( \begin{array}{c} \tilde{\mathbf{o}}^{(1,3)} Y_n \\ \tilde{\mathbf{o}} ^{(2,3)} Y_n \\ \tilde{\mathbf{o}} ^{(3,1)} Y_n \\ \tilde{\mathbf{o}} ^{(3,2)} Y_n \end{array} \right) = {\mathbf{b}}_D \left( \begin{array}{c} Y_n \\ Y_n \\ Y_n \\ Y_n \end{array} \right), \end{aligned} $$
(96)
where the matrix operators aD and bD are defined by (see [52])
$$\displaystyle \begin{aligned} { {\mathbf{a}}_D := \left( \begin{array}{c@{\ }c@{\ }c@{\ }c@{\ }c} {\mathbf{o}}^{(1,1)} (D+1)(D+2) & - {\mathbf{o}}^{(1,2)}(D+2) & - {\mathbf{o}}^{(2,1)}(D+2) & -\frac{1}{2} o^{(2,2)} (D+2)(D+1) & \frac{1}{2} {\mathbf{o}}^{(2,3)}\\ {} {\mathbf{o}}^{(1,1)} D^2 & {\mathbf{o}}^{(1,2)} D & - {\mathbf{o}}^{(2,1)}(D-1) & -\frac{1}{2} {\mathbf{o}}^{(2,2)} D(D-1) & -\frac{1}{2}{\mathbf{o}}^{(2,3)}\\ {} {\mathbf{o}}^{(1,1)} (D+1)^2 & - {\mathbf{o}}^{(1,2)}(D+1) & {\mathbf{o}}^{(2,1)} (D+2) & \frac{1}{2}{\mathbf{o}}^{(2,2)} (D+2)(D+1)& -\frac{1}{2}{\mathbf{o}}^{(2,3)}\\ {} {\mathbf{o}}^{(1,1)} D(D-1) & {\mathbf{o}}^{(1,2)} (D-1) & {\mathbf{o}}^{(2,1)} (D-1) & - \frac{1}{2} {\mathbf{o}}^{(2,2)} D(D-1) & \frac{1}{2} {\mathbf{o}}^{(2,3) }\\ {} 0 & 0 & {\mathbf{o}}^{(2,1)}& -\frac{1}{2}{\mathbf{o}}^{(2,2)}D(D+1) & -\frac{1}{2}{\mathbf{o}}^{(2,3)} \end{array} \right)} \end{aligned} $$
(97)
and
$$\displaystyle \begin{aligned} {\mathbf{b}}_D := \left( \begin{array}{c@{\ }c@{\ }c@{\ }c} {\mathbf{o}}^{(1,3)} (D+1) & {\mathbf{o}}^{(3,1)} & - \frac{1}{2} {\mathbf{o}}^{(3,2)} & - \frac{1}{2}{\mathbf{o}}^{(3,3)} D(D+1) \\ {} {\mathbf{o}}^{(1,3)} D & - {\mathbf{o}}^{(3,1)} & \frac{1}{2} {\mathbf{o}}^{(3,2)} & \frac{1}{2} {\mathbf{o}}^{(3,3)} D(D+1) \\ {} 0 & {\mathbf{o}}^{(3,1)} (D+2) & - \frac{1}{2} {\mathbf{o}}^{(3,2)} & \frac{1}{2} {\mathbf{o}}^{(3,3)}(D+2)(D+1) \\ {} 0 & {\mathbf{o}}^{(3,1)} (D-1) & \frac{1}{2} {\mathbf{o}}^{(3,2)} & -\frac{1}{2} {\mathbf{o}}^{(3,3)} D(D-1) \end{array} \right),\end{aligned} $$
(98)
and D is the (pseudodifferential) operator \(D = (-\varDelta + \frac {1}{4}) ^{1/2} - \frac {1}{2}\) of order 1 characterized by
$$\displaystyle \begin{aligned} \begin{array}{rcl} D Y_n := D^{\wedge} (n) Y_n = n Y_n\end{array} \end{aligned} $$
(99)
for all \(Y_n \in \mathrm {Harm} _n(\mathbb {S}^2)\).
The adjoint operators \(\tilde {O}^{(i,k)} : \mathbf {c} ^{(\infty )} (\mathbb {S}^2) \to \mathrm {C}^{(\infty )} (\mathbb {S}^2)\), i, k = 1, 2, 3, to the operators \(\tilde {\mathbf {o}}^{(i,k)}\) satisfying the equation
$$\displaystyle \begin{aligned} \begin{array}{rcl} \langle \tilde{\mathbf{o}}^{(i,k)} G, \mathbf{f} \rangle_{{\mathbf{l}}^2(\mathbb{S}^2)} = \langle G,\tilde{O}^{(i,k)}\mathbf{f}\rangle_{\mathrm{L}^2(\mathbb{S}^2)},\end{array} \end{aligned} $$
(100)
\(\mathbf {f}\in {\mathbf {c}}^{(\infty )}(\mathbb {S}^2)\), \(G\in \mathrm {C}^{(\infty )} (\mathbb {S}^2)\), are obviously given by
$$\displaystyle \begin{aligned} \left( \begin{array}{c} \tilde{O} ^{(1,1)} Y_n \\ \tilde{O} ^{(1,2)} Y_n \\ \tilde{O} ^{(2,1)} Y_n \\ \tilde{O} ^{(2,2)} Y_n \\ \tilde{O} ^{(3,3)} Y_n \end{array} \right) : = {\mathbf{a}}_D \left( \begin{array}{c} Y_n \\ Y_n \\ Y_n \\ Y_n \\ Y_n \end{array} \right), \quad \mathrm{and} \quad \left( \begin{array}{c} \tilde{O}^{(1,3)} Y_n \\ \tilde{O} ^{(2,3)} Y_n \\ \tilde{O} ^{(3,1)} Y_n \\ \tilde{O} ^{(3,2)} Y_n \end{array} \right) : = {\mathbf{b}}_D \left( \begin{array}{c} Y_n \\ Y_n \\ Y_n \\ Y_n \end{array} \right). \end{aligned} $$
(101)
After these preliminaries we are now in a position to introduce the tensor spherical harmonics
$$\displaystyle \begin{aligned} \tilde{\mathbf{y}}_{n,m}^{(i,k)} := \left( \tilde{\mu}_n^{(i,k)} \right)^{-1/2} { \tilde{\mathbf{o}}}^{(i, k)} Y_{n,m}, \end{aligned} $$
(102)
\( n = \tilde {0}_{ik}, \ldots ; m=1, \ldots , 2n+1\), with
$$\displaystyle \begin{aligned} \tilde{0}_{ik} := \left\{ \begin{array}{lll} 0, & \; (i,k) \in \{ (1,1), (2,1), (3,1) \}, \\ {} 1, & \; (i,k) \in \{ (1,2), (1,3), (2,3), (3,3) \}, \\ {} 2, & \; (i,k) \in \{ (2,2), (3,2) \}, \end{array} \right. \end{aligned} $$
(103)
and normalization constants
$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu} _n ^{(1,1)} &\displaystyle :=&\displaystyle (n+2)(n+1)(2n+3)(2n+1), \end{array} \end{aligned} $$
(104)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu} _n ^{(1,2)} &\displaystyle :=&\displaystyle 3n^4, \end{array} \end{aligned} $$
(105)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu} _n ^{(1,3)} &\displaystyle :=&\displaystyle n(n+1)^2 (2n+1), \end{array} \end{aligned} $$
(106)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu} _n ^{(2,1)} &\displaystyle :=&\displaystyle (n+1)^2(2n+3)(2n+1), \end{array} \end{aligned} $$
(107)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu} _n ^{(2,2)} &\displaystyle :=&\displaystyle n(n-1)(2n+1)(2n-1), \end{array} \end{aligned} $$
(108)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu} _n ^{(2,3)} &\displaystyle :=&\displaystyle n^2 (n+1) ^2, \end{array} \end{aligned} $$
(109)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu} _n ^{(3,1)} &\displaystyle :=&\displaystyle n^2(n+1)(2n+1), \end{array} \end{aligned} $$
(110)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu} _n ^{(3,2)} &\displaystyle :=&\displaystyle n(n+1) ^2 (2n+1), \end{array} \end{aligned} $$
(111)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu} _n ^{(3,3)} &\displaystyle :=&\displaystyle n^2 (n-1)(2n+1). \end{array} \end{aligned} $$
(112)
In contrary to the system (89), each member of the system \(\{ \tilde {\mathbf {y}}_{n,m} ^{(i,k)} \}\) is an eigenfunction of the Beltrami operator. More explicitly, we have the following result.
Assume that {Yn,m}n=0,1,…; m=1,…,2n+1 is an \(\mathrm {L}^2(\mathbb {S}^2)\)–orthonormal set of scalar spherical harmonics. Then, the set
$$\displaystyle \begin{aligned} \left\{ {\tilde{\mathbf{y}}}_{n,m}^{(i,k)} \right\}_{i,k = 1,2,3; \ n= \tilde{0}_{ik}, \ldots; \ m=1, \ldots, 2n+1}, \end{aligned} $$
(113)
as defined by (102), forms an \({\mathbf {l}}^2(\mathbb {S}^2)\)–orthonormal set of tensor spherical harmonics which is closed in \({\mathbf {c}}^{(0)}(\mathbb {S}^2)\) and \({\mathbf {l}} ^2(\mathbb {S}^2)\) with respect to \(\| \cdot \|{ }_{\mathbf {c}(\mathbb {S}^2)}\) and \(\| \cdot \|{ }_{{\mathbf {l}}^2(\mathbb {S}^2)}\), respectively, and complete in \(\mathbf {l} ^2(\mathbb {S}^2)\) with respect to \(( \cdot , \cdot )_{{\mathbf {l}}^2 (\mathbb {S}^2)}\). Furthermore, we are able to verify that
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta ^{\ast} _{\xi} {\tilde{\mathbf{y}}}_{n,m}^{(1,1)} &\displaystyle =&\displaystyle -(n+2)(n+3) {\tilde{\mathbf{y}}}_{n,m} ^{(1,1)}, \end{array} \end{aligned} $$
(114)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta ^{\ast} _{\xi} {\tilde{\mathbf{y}}}_{n,m}^{(1,2)} &\displaystyle =&\displaystyle -n(n+1) {\tilde{\mathbf{y}}}_{n,m} ^{(1,2)}, \end{array} \end{aligned} $$
(115)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta ^{\ast} _{\xi} {\tilde{\mathbf{y}}}_{n,m}^{(1,3)} &\displaystyle =&\displaystyle -(n+1)(n+2) {\tilde{\mathbf{y}}}_{n,m} ^{(1,3)}, \end{array} \end{aligned} $$
(116)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta ^{\ast} _{\xi} {\tilde{\mathbf{y}}}_{n,m}^{(2,1)} &\displaystyle =&\displaystyle -n(n+1) {\tilde{\mathbf{y}}}_{n,m} ^{(2,1)}, \end{array} \end{aligned} $$
(117)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta ^{\ast} _{\xi} {\tilde{\mathbf{y}}}_{n,m}^{(2,2)} &\displaystyle =&\displaystyle -(n-1)(n-2) {\tilde{\mathbf{y}}}_{n,m} ^{(2,2)}, \end{array} \end{aligned} $$
(118)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta ^{\ast} _{\xi} {\tilde{\mathbf{y}}}_{n,m}^{(2,3)} &\displaystyle =&\displaystyle -n(n-1) {\tilde{\mathbf{y}}}_{n,m} ^{(2,3)}, \end{array} \end{aligned} $$
(119)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta ^{\ast} _{\xi} {\tilde{\mathbf{y}}}_{n,m}^{(3,1)} &\displaystyle =&\displaystyle -(n+1)(n+2) {\tilde{\mathbf{y}}}_{n,m} ^{(3,1)}, \end{array} \end{aligned} $$
(120)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta ^{\ast} _{\xi} {\tilde{\mathbf{y}}}_{n,m}^{(3,2)} &\displaystyle =&\displaystyle -n(n-1) {\tilde{\mathbf{y}}}_{n,m} ^{(3,2)}, \end{array} \end{aligned} $$
(121)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta ^{\ast} _{\xi} {\tilde{\mathbf{y}}}_{n,m}^{(3,3)} &\displaystyle =&\displaystyle -n(n+1) {\tilde{\mathbf{y}}}_{n,m} ^{(3,3)}, \end{array} \end{aligned} $$
(122)
where the application of the Beltrami operator is understood component-by-component.
Because of the completeness of the tensor spherical harmonics (see [37, 38]) every tensor field \(\mathbf {f} \in {\mathbf {l}}^2(\mathbb {S}^2)\) can be written as an orthogonal (Fourier) series
$$\displaystyle \begin{aligned} \mathbf{f}= \sum \limits _{i,k=1} ^{3} \sum \limits _{n=\tilde{0}_{ik}} ^{\infty} \sum \limits _{m=1} ^{2n+1} {\mathbf{f}}^{(i,k)\wedge_{{\mathbf{l}}^2(\mathbb{S}^2)}} (n,m) \ \tilde{\mathbf{y}} _{n,m}^{(i,k)} \end{aligned} $$
(123)
(in \(\| \cdot \|{ }_{{\mathbf {l}}^2(\mathbb {S}^2)}\)–sense), where the Fourier coefficients are given by
$$\displaystyle \begin{aligned} {\mathbf{f}}^{(i,k)\wedge_{{\mathbf{l}}^2(\mathbb{S}^2)}}(n,m) = \langle \mathbf{f}, \tilde{\mathbf{y}} _{n,m}^{(i,k)}\rangle_{{\mathbf{l}}^2(\mathbb{S}^2)} = \int _{\mathbb{S}^2} \mathbf{f}(\xi) \cdot \tilde{\mathbf{y}} _{n,m}^{(i,k)} (\xi) \ dS(\xi). \end{aligned} $$
(124)

3.4 Outer Harmonics

We begin with the well-known scalar theory (see, e.g., [35]).

Scalar Outer Harmonics

The scalar outer harmonics are defined by
$$\displaystyle \begin{aligned} H_{n,m}^{R} (x) := \frac{1}{R} \left( \frac{R}{|x|} \right) ^{n+1} Y_{n,m} \left( \frac{x}{|x|}\right), \quad x \in \overline{\mathbb{S}_R^{2;\mathrm{ext}}}, \end{aligned} $$
(125)
n = 0, 1, …, m = 1, …, 2n + 1, where, as usual, \(\mathbb {S}_R^{2;\mathrm {ext}}\) denotes the outer space of the sphere \(\mathbb {S}_R^{2}\) in \(\mathbb {R}^3\) with radius R around the origin.
The following properties are valid:
  • \(H_{n,m}^{R}\) 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

We introduce an associated class of tensor outer harmonics (cf. [52, 58, 105]) based on the definition of tensor spherical harmonics as defined in (102):
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \mathbf{h} _{n,m} ^{R;(1,1)}(x) &\displaystyle :=&\displaystyle \frac{1}{R} \left( \frac{R}{|x|}\right)^{n+3} \tilde{\mathbf{y}}_{n,m} ^{(1,1)} \left( \frac{x}{|x|}\right), \end{array} \end{aligned} $$
(126)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{h} _{n,m} ^{R;(1,2)}(x) &\displaystyle :=&\displaystyle \frac{1}{R} \left( \frac{R}{|x|}\right)^{n+1} \tilde{\mathbf{y}}_{n,m} ^{(1,2)} \left( \frac{x}{|x|}\right), \end{array} \end{aligned} $$
(127)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{h} _{n,m} ^{R;(1,3)}(x) &\displaystyle :=&\displaystyle \frac{1}{R} \left( \frac{R}{|x|}\right)^{n+2} \tilde{\mathbf{y}}_{n,m} ^{(1,3)} \left( \frac{x}{|x|}\right), \end{array} \end{aligned} $$
(128)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{h} _{n,m} ^{R;(2,1)}(x) &\displaystyle :=&\displaystyle \frac{1}{R} \left( \frac{R}{|x|}\right)^{n+1} \tilde{\mathbf{y}}_{n,m} ^{(2,1)} \left( \frac{x}{|x|}\right), \end{array} \end{aligned} $$
(129)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{h} _{n,m} ^{R;(2,2)}(x) &\displaystyle :=&\displaystyle \frac{1}{R} \left( \frac{R}{|x|}\right)^{n-1} \tilde{\mathbf{y}}_{n,m} ^{(2,2)} \left( \frac{x}{|x|}\right), \end{array} \end{aligned} $$
(130)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{h} _{n,m} ^{R;(2,3)}(x) &\displaystyle :=&\displaystyle \frac{1}{R} \left( \frac{R}{|x|}\right)^{n} \tilde{\mathbf{y}}_{n,m} ^{(2,3)} \left( \frac{x}{|x|}\right), \end{array} \end{aligned} $$
(131)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{h} _{n,m} ^{R;(3,1)}(x) &\displaystyle :=&\displaystyle \frac{1}{R} \left( \frac{R}{|x|}\right)^{n+2} \tilde{\mathbf{y}}_{n,m} ^{(3,1)} \left( \frac{x}{|x|}\right), \end{array} \end{aligned} $$
(132)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{h} _{n,m} ^{R;(3,2)}(x) &\displaystyle :=&\displaystyle \frac{1}{R} \left( \frac{R}{|x|}\right)^{n} \tilde{\mathbf{y}}_{n,m} ^{(3,2)} \left( \frac{x}{|x|}\right), \end{array} \end{aligned} $$
(133)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{h} _{n,m} ^{R;(3,3)}(x) &\displaystyle :=&\displaystyle \frac{1}{R} \left( \frac{R}{|x|}\right)^{n+1} \tilde{\mathbf{y}}_{n,m} ^{(3,3)} \left( \frac{x}{|x|}\right), \end{array} \end{aligned} $$
(134)
\(x \in \overline {\mathbb {S}_R^{2;\mathrm {ext}}}, \ n=\tilde {0}_{ik},\ldots ; \ m=1,\ldots ,2n+1\).
It is not difficult to show that the following properties are satisfied:
  • \({\mathbf {h}}_{n,m}^{R;(i,k)}\) 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|→.

It must be emphasized that the spherically reflected formulation of the tensorial SGG-problem exclusively uses the tensor outer harmonics of kind (1, 1) specified by (126).

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].

In our approach (cf. Fig. 5) we use the Runge concept in the formulation [30]. It tells us that
$$\displaystyle \begin{aligned} \mathrm{Pot}^{(0)}(\overline{\varSigma^{\mathrm{{ext}}}}) = \overline{\mathrm{span} _{\stackrel{\scriptstyle{n=0,1,\ldots;}}{m=1,\ldots,2n+1}} (H_{n,m}^R ) |{}_{\overline{\varSigma^{\mathrm{{ext}}}}}} ^{\| \cdot \|{}_{\mathrm{C}^{(0)}(\overline{\varSigma^{\mathrm{{ext}}}})}}, \end{aligned} $$
(135)
where Σ is a regular surface (e.g., sphere, ellipsoid, telluroid, geoid, or real Earth’s surface) and \(\mathbb {S}_R^2\) is a sphere inside Σint, such that the “Runge condition” R < σ = infxΣ|x| is valid (note that \(\mathbb {S}_R^2\) in (135) may be replaced by any regular surface Ξ (cf. [3]) located totally in Σint with dist(Σ, Ξ) > 0, but we restrict ourselves to the spherical (Bjerhammar) case).
Fig. 5

The spherically reflected Runge (Bjerhammar) situation. (For more details see [30] and [3])

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 Σint 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 Σint (cf. [30, 42]), certain kernel representations as well as harmonic splines and wavelets (cf. [35, 36, 42]).

From [105] we know that
$$\displaystyle \begin{aligned} \mathbf{pot}^{(0)}(\overline{\varSigma^{\mathrm{{ext}}}}) = \overline{\mathrm{span} _{\stackrel{\scriptstyle{n=0,1,\ldots;}} {m=1,\ldots,2n+1}} ({\mathbf{h}}_{n,m}^{R;(1,1)} ) |{}_{\overline{\varSigma^{\mathrm{{ext}}}}}} ^{\| \cdot \|{}_{{\mathbf{c}}^{(0)}(\overline{\varSigma^{\mathrm{{ext}}}})}}. \end{aligned} $$
(136)
Again, it should be remarked that the same property holds true for other trial systems, such as tensor outer ellipsoidal harmonics outside an internal ellipsoid, fundamental systems of monopole Hesse tensors inside Σint, certain tensor kernel representations as well as tensor harmonic splines and wavelets.

From the scalar Runge concept (135) it can be deduced additionally (see, e.g., [30, 36, 42]) that, for an arbitrarily small number ε > 0 and any given potential \(\tilde V \in \mathrm {Pot}^{(0)}(\overline {\varSigma ^{\mathrm {{ext}}}}),\) there exists a potential \(V \in {\mathrm {Pot}}({\mathbb {S}_R^{2;\mathrm {ext}}})\) with R < σ = infxΣ|x| such that
$$\displaystyle \begin{aligned} \mathrm{sup}_{x \in \overline{\varSigma^{\mathrm{{ext}}}}} |\tilde{V}(x) - V(x)| \leq \varepsilon \end{aligned} $$
(137)
and
$$\displaystyle \begin{aligned} \mathrm{sup}_{x \in \varGamma} |{\underbrace{(\nabla \otimes \nabla)\tilde{V}(x)}_{=\tilde{\mathbf{v}}(x)}} - {\underbrace{(\nabla \otimes \nabla) V(x)}_{={ \mathbf{v}(x)}}}| \leq C \ \varepsilon \end{aligned} $$
(138)
for some positive constant C (depending on the geometries of Σ and Γ).

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.

Summarizing our considerations we are led to the following conclusion: A viable way to model SGG may be based on the Runge property that there exists a potential V outside a Runge (Bjerhammer) sphere \(\mathbb {S}_R^2\) inside the Earth in ε-accuracy (ε > 0, arbitrarily small) to the Earth’s gravitational potential \(\tilde V\) such that
$$\displaystyle \begin{aligned} \mathrm{sup}_{x \in \varGamma} |{\underbrace{(\nabla \otimes \nabla) \tilde{V}(x)}_{=\tilde{\mathbf{v}}(x)}} - {\underbrace{(\nabla \otimes \nabla) V(x)}_{={ \mathbf{v}(x)}}}|=\mathrm{sup}_{x \in \varGamma} |\tilde{\mathbf{v}}(x)- \mathbf{ v}(x)| \leq C \ \varepsilon. \end{aligned} $$
(139)
As a consequence of our excursion to the Runge theory, the relation between the tensorial measurements \(\tilde {\mathbf {v}},\) i.e., the gradiometer orbital data of \(\tilde V\) on Γ and the auxiliary potential V outside the sphere \(\mathbb {S}^2_R\) may be explained for points x on the satellite orbit Γ as follows (see (5), (2)):
  • (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)
(note that “ ≈ ” means “approximately equal”, so that the input data \(\tilde {\mathbf {v}}(x)\) may be replaced by v(x)). In other words, SGG in frequency as well as space domain is appropriately modeled in the spherically reflected nomenclature of V, thereby taking as input data the tensor measurements \(\tilde {\mathbf {v}}(x)\) of the (actual) Earth’s potential \(\tilde V\) on the orbit Γ.

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:

Let V be of class \(\mathrm {Pot}^{(0)} (\mathbb {S}_\gamma ^{2;\mathrm {{ext}}}),\) with γ < infxΓ|x| (cf. Fig. 6). Then the foll- owing statements may be listed:
  1. 1.
    \(O^{(i,k)} \nabla \otimes \nabla V|{ }_{\mathbb {S}_\gamma ^2}= 0\) if (i, k) ∈{(1, 3), (3, 1), (3, 2), (3, 3)},
    Fig. 6

    The geometric situation of satellite gravitational gradiometry as discussed in our frequency framework

     
  2. 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. 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. 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.

     
The SGG-uniqueness list gives detailed information, which tensor components of the Hesse tensor are suitable to guarantee the uniqueness of the SGG-problem. In fact, for a potential of class \(\mathrm {Pot}^{(0)} (\mathbb {S}_\gamma ^{2;\mathrm {{ext}}})\) with vanishing spherical harmonic moments of degree 0 and 1 such as the Earth’s disturbing potential (see, e.g., [75, 78, 97]), uniqueness is assured for all cases listed above.
Figure 7 gives graphical illustrations of the disturbing potential and its second order radial derivative at the height of the Earth’s surface (0 km) and at the (orbital) height of 250 km.
Fig. 7

Disturbing potential data generated from the EGM 2008 model [107], evaluated up to degree and order 720. From top to bottom: heights 250 and 0 km. From left to right: disturbing potential in m2/s2, negative first radial derivative in 10−6 m/s2 and the second order radial derivative in 10−12/s2 (taken from [63])

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

We let \(\mathscr {A}\) be the linear space consisting of all sequences \(\left \{ A_{n} \right \} _{n \in \mathbb {N}}\) of real numbers An ≠ 0, \(n \in \mathbb {N}_0\):
$$\displaystyle \begin{aligned} \mathscr{A}:= \left\{ \{A_{n} \} \hspace{0.5mm} : \hspace{0.5mm} A_{n} \in \mathbb{R}, \, A_{n} \neq 0, \, n \in \mathbb{N} _0 \right\}. \end{aligned} $$
(142)
We consider the set \(\mathscr {E} = \mathscr {E} ( \{A_n\} ; \overline {\mathbb {S}_R^{2;\mathrm {ext}}} )\) given by
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathscr{E}: = \Big\{ F \in \mathrm{Pot}^{(\infty)} (\overline{\mathbb{S}_R^{2;\mathrm{ext}}}) \hspace{2mm} &\displaystyle : \hspace{2mm} &\displaystyle \sum \limits _{n = 0} ^{\infty} \sum \limits _{m=1} ^{2n+1} A_n^2 \left( F ^{\wedge_{\mathrm{L}^2(\mathbb{S}_R^2)}} (n,m) \right)^2 < \infty \Big\}, \end{array} \end{aligned} $$
(143)
where
$$\displaystyle \begin{aligned} F^{\wedge_{\mathrm{L}^2(\mathbb{S}_R^2)}} (n,m) = \int _{\mathbb{S}_R^2} F(y) H^R_{n,m} (y) \ dS(y). \end{aligned} $$
(144)
Note that we assume that R is chosen in accordance to the Runge concept (cf. Fig. 6). On \(\mathscr {E}\) we introduce an inner product \(\langle \cdot ,\cdot \rangle _{\mathscr {H} (\{ A_n \}; \overline {\mathbb {S}_R^{2;\mathrm {ext}}})}\) by
$$\displaystyle \begin{aligned} \langle F,G \rangle _{\mathscr{H} (\{A_n\}; \overline{\mathbb{S}_R^{2;\mathrm{ext}}})} = \sum \limits _{n = 0} ^{\infty} \sum \limits _{m=1} ^{2n+1} A_n^2 F^{\wedge_{\mathrm{L}^2(\mathbb{S}_R^2)}} (n,m) G^{\wedge_{\mathrm{L}^2(\mathbb{S}_R^2)}} (n,m), \quad F,G \in \mathscr{E}. \end{aligned} $$
(145)
The associated norm is given by
$$\displaystyle \begin{aligned} \| F \|{}_{\mathscr{H} (\{A_n\}; \overline{\mathbb{S}_R^{2;\mathrm{ext}}})} = \left( \langle F,F \rangle _{\mathscr{H} (\{A_n\}; \overline{\mathbb{S}_R^{2;\mathrm{ext}}})} \right) ^{1/2}. \end{aligned} $$
(146)
The (scalar) Sobolev space\(\mathscr {H}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) (\(= \mathscr {H} (\{A_n\}; \overline {\mathbb {S}_R^{2;\mathrm {ext}}})\)) is the completion of \(\mathscr {E}\) under the norm \(\| \cdot \| _{\mathscr {H} (\{ A_n \}; \overline { \mathbb {S}_R^{2;\mathrm {ext}}})}\):
$$\displaystyle \begin{aligned} \mathscr{H} (\{ A_n \}; \overline{\mathbb{S}_R^{2;\mathrm{ext}} } ) := \overline{\mathscr{E}} ^{\| \cdot \| _{\mathscr{H} (\{ A_n \}; \overline{\mathbb{S}_R^{2;\mathrm{ext}}})} }. \end{aligned} $$
(147)
\(\mathscr {H} (\{ A_n \}; \overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) equipped with the inner product \(\langle \cdot , \cdot \rangle _{\mathscr {H} (\{ A_n \}; \overline {\mathbb {S}_R^{2;\mathrm {ext}}}) }\) is a separable Hilbert space. The system \(\{ H_{n,m}^{\ast \{A_n\}} (R; \cdot ) \}\) given by
$$\displaystyle \begin{aligned} H_{n,m} ^{\ast\{A_n\}} (R;x) := A_n ^{-1} H_{n,m}^R(x),\quad x \in \overline{\mathbb{S}_R^{2;\mathrm{ext}}}, \end{aligned} $$
(148)
is a Hilbert basis (we simply write \(H_{n,m}^{\ast } (R;\cdot )\) instead of \(H_{n,m}^{\ast \{A_n\}} (R;\cdot )\) if no confusion is likely to arise). Any function \(F \in \mathscr {H}(\{A_n\};\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) can be expanded as a Fourier series in terms of the basis \(H_{n,m}^{\ast } (R;\cdot )\):
$$\displaystyle \begin{aligned} F = \sum \limits _{n = 0} ^{\infty} \sum \limits _{m=1} ^{2n+1} F^{\wedge_{\mathscr{H}(\{A_n\};\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}(n,m) H_{n,m}^{\ast\{A_n\}} (R;\cdot), \end{aligned} $$
(149)
where
$$\displaystyle \begin{aligned} F^{\wedge_{\mathscr{H}(\{A_n\};\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}(n,m) = \langle F, H_{n,m}^{\ast}(R;\cdot)\rangle_{\mathscr{H}(\{A_n\};\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}. \end{aligned} $$
(150)
If no confusion is likely to arise we also use the notation
$$\displaystyle \begin{aligned} F^{\wedge_{\mathscr{H}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}(n,m) = F^{\wedge_{\mathscr{H}(\{A_n\};\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}(n,m). \end{aligned} $$
(151)
Furthermore, according to its construction, the space \(\mathscr {H}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) has the reproducing kernel function\(K_{\mathscr {H}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})}(\cdot , \cdot ): \overline {\mathbb {S}_R^{2;\mathrm {ext}}} \times \overline {\mathbb {S}_R^{2;\mathrm {ext}}} \to \mathbb {R}\) given by
$$\displaystyle \begin{aligned} K_{\mathscr{H}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}(x,y) = \sum \limits _{n=0} ^{\infty} \sum \limits _{m=1} ^{2n+1} H_{n,m}^{\ast\{A_n\}} (R;x) H_{n,m}^{\ast\{A_n\}} (R;y), \quad x,y \in \overline{\mathbb{S}_R^{2;\mathrm{ext}}}. \end{aligned} $$
(152)
Our next goal is the introduction of a class of scalar Sobolev spaces based on the spherical symbol \(\{ (\varDelta ^{\ast ;R})^{\wedge } (n) \} _{n \in \mathbb {N}_0}\) of the Beltrami operator Δ∗;R related to the sphere \(\mathbb {S}_R^2\). We observe that
$$\displaystyle \begin{aligned} \varDelta ^{\ast;R} Y_{n,m} = \frac{1}{R^2} \varDelta ^{\ast} Y_{n,m} = - \frac{1}{R^2}n(n+1) Y_{n,m}, \quad n \in \mathbb{N}_0; \, m=1,\ldots,2n+1.\end{aligned} $$
(153)
In particular, we have Δ∗;RY0,1 = 0, which requires a shift by a non-zero constant, for example \(\frac {1}{4R^2}\), to obtain invertibility. As a consequence we formally obtain
$$\displaystyle \begin{aligned} \left( -\varDelta ^{\ast;R} + \frac{1} {4R^2} \right) ^{s/2} Y_{n,m} = \left( \frac{n+1/2} {R} \right) ^s Y_{n,m} \end{aligned} $$
(154)
and
$$\displaystyle \begin{aligned} \left( \left( -\varDelta^{\ast;R} + \frac{1} {4R^2} \right) ^{s/2} F \right) ^{\wedge} (n,m) = \left( \frac{n+1/2}{R} \right) ^s F^{\wedge} (n,m), \end{aligned} $$
(155)
\(n \in \mathbb {N}_0\), m = 1, …, 2n + 1.
For \(s \in \mathbb {R}\) we let
$$\displaystyle \begin{aligned} \mathscr{H} _s (\overline{\mathbb{S}_R^{2;\mathrm{ext}}}): = \mathscr{H} \left( \left\{ \left( \frac{n+1/2}{R} \right)^s \right\}; \overline{\mathbb{S}_R^{2;\mathrm{ext}}} \right), \end{aligned} $$
(156)
such that the norm in \(\mathscr {H}_s(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) fulfills
$$\displaystyle \begin{aligned} \| F \|{}_{\mathscr{H} _s (\overline{\mathbb{S}_R^{2;\mathrm{ext}}})} = \| (- \varDelta_x ^{\ast;R} + \frac{1} {4 R^2} ) ^{s/2} F \|{}_{\mathrm{L} ^2(\mathbb{S}_R^2)}. \end{aligned} $$
(157)
For \(\mathscr {H} _0 (\overline {\mathbb {S}_R^{2;\mathrm {ext}}}) \) we identify the norm \(\| \cdot \|{ }_{\mathscr {H} _0 (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})}\) with the \(\| \cdot \|{ }_{\mathrm {L} ^2(\mathbb {S}_R^2)}\)–norm.

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).

Furthermore, if t < s, then we have
$$\displaystyle \begin{aligned}\| F \|{}_{\mathscr{H}_t (\overline{\mathbb{S}_R^{2;\mathrm{ext}}})} \le \| F \|{}_{\mathscr{H} _s (\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}\end{aligned} $$
(158)
and
$$\displaystyle \begin{aligned}{\mathscr{H} _s (\overline{\mathbb{S}_R^{2;\mathrm{ext}}})} \subset \mathscr{H} _t (\overline{\mathbb{S}_R^{2;\mathrm{ext}}}).\end{aligned} $$
(159)
In order to formulate some results (cf. [31, 35, 58]) about the convergence of the expansion in terms of outer harmonics to a function in ordinary pointwise sense (Sobolev Lemma) we need the concept of summable sequences \(\{ A_n \}_{n \in \mathbb {N}_0}~{\in }~\mathscr {A}\) satisfying
$$\displaystyle \begin{aligned} \sum \limits _{n = 0} ^{\infty} \frac{2n+1} {A_n^2} < \infty. \end{aligned} $$
(160)

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 \(\mathrm {Pot} ^{(0)} (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\).

For any F in \(\mathrm {L}^2(\mathbb {S}_R^2)\), there exists one and only one “harmonic continuation” \(U \in \mathscr {H}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) which is given by
$$\displaystyle \begin{aligned} U(x)= \sum \limits _{n=0}^{\infty} \sum \limits _{m=1}^{2n+1} A_n^2 F^{\wedge_{\mathrm{L}^2(\mathbb{S}_R^2)}}(n,m) H^{\ast}_{n,m}(R;x), \quad x \in \overline{\mathbb{S}_R^{2;\mathrm{ext}}}, \end{aligned} $$
(161)
where
$$\displaystyle \begin{aligned} F^{\wedge_{\mathrm{L}^2(\mathbb{S}_R^2)}}(n,m) = \int _{\mathbb{S}_R^2} F(y)\ H_{n,m}^R(y)\ dS(y) = \frac{1}{A_n^2} U^{\wedge_{\mathscr{H}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}(n,m), \end{aligned} $$
(162)
n = 1, 2, …;m = 1, …, 2n + 1.

Tensorial Case

In order to introduce Sobolev spaces for tensor fields we start from
$$\displaystyle \begin{aligned} \mathbf{e} := \Big\{ \mathbf{f} \in \mathbf{pot} ^{(\infty)}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}}) \, : \, \sum \limits _{n=0} ^{\infty} \sum \limits _{m=1} ^{2n+1} A _n^2 ({\mathbf{f}}^{\wedge_{{\mathbf{l}}^2(\mathbb{S}_R^2)}}(n,m))^2 < \infty \Big\}, \end{aligned} $$
(163)
where
$$\displaystyle \begin{aligned} {\mathbf{f}}^{\wedge_{{\mathbf{l}}^2(\mathbb{S}_R^2)}}(n,m)= \int_{\mathbb{S}_R^2} \mathbf{f} (y)\ \mathbf{h} ^{R;(1,1)} _{n,m} (y) \ d S(y). \end{aligned} $$
(164)
Equipped with the inner product
$$\displaystyle \begin{aligned} \langle \mathbf{f}, \mathbf{g}\rangle _{\mathbf{h}(\{A_n\}; \overline{\mathbb{S}_R^{2;\mathrm{ext}}})} = \sum \limits _{n = 0} ^{\infty} \sum \limits _{m=1} ^{2n+1} A_n^2\ {\mathbf{f}}^{\wedge_{{\mathbf{l}}^2(\mathbb{S}_R^2)}}(n,m)\ {\mathbf{g}}^{\wedge_{{\mathbf{l}}^2(\mathbb{S}_R^2)}}(n,m), \end{aligned} $$
(165)
f, g ∈e, the space e becomes a pre–Hilbert space. We define the Sobolev space\(\mathbf {h} (\overline {\mathbb {S}_R^{2;\mathrm {ext}}}) = \mathbf {h} ( \{ A_n \}; \overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) as the completion of e under the norm \(\| \cdot \|{ }_{\mathbf {h}(\{A_n\}; \overline {\mathbb {S}_R^{2;\mathrm {ext}}})}\), which denotes the norm associated to \(\langle \cdot , \cdot \rangle _{\mathbf {h}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})}\):
$$\displaystyle \begin{aligned} \mathbf{h} ( \{ A_n\}; \overline{\mathbb{S}_R^{2;\mathrm{ext}}}) := \overline{\mathbf{e}} ^{\| \cdot \|{}_{\mathbf{h} ( \{ A_n \}; \overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}. \end{aligned} $$
(166)
If no confusion is likely to arise we also use \(\mathbf {h} (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) instead of \(\mathbf {h} ( \{ A_n\}; \overline {\mathbb {S}_R^{2;\mathrm {ext}}})\). \(\mathbf {h} (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) equipped with the inner product \(\langle \cdot , \cdot \rangle _{\mathbf {h} ( \{ A _n\}; \overline {\mathbb {S}_R^{2;\mathrm {ext}}})}\) is a separable Hilbert space. The system \(\{{\mathbf {h}}_{n,m}^{\ast \{A_n\}}(R;\cdot )\}_{n \in \mathbb {N}_{0}; m=1,\ldots ,2n+1}\), given by
$$\displaystyle \begin{aligned} {\mathbf{h}}_{n,m} ^{\ast\{A_n\}} (R;x) := A_n^{-1} {\mathbf{h}}_{n,m}^{R;(1,1)} (x), \quad x \in \overline{ \mathbb{S}_R^{2;\mathrm{ext}}}, \end{aligned} $$
(167)
represents an \(\mathbf {h}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\)-orthonormal Hilbert basis in \(\mathbf {h}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) (note that we sometimes write \({\mathbf {h}}_{n,m}^{ \ast }(R;\cdot )\) instead of \({\mathbf {h}}_{n,m}^{\ast \{A_n\}}(R;\cdot )\)). As a consequence we can expand a function \(\mathbf {f} \in \mathbf {h}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) as a Fourier series in terms of the basis \({\mathbf {h}}_{n,m} ^{\ast \{ A_n \}}(R;\cdot )\)
$$\displaystyle \begin{aligned} \mathbf{f} = \sum \limits _{n =0} ^{\infty} \sum \limits _{m=1} ^{2n+1} \mathbf{f} ^{\wedge_{\mathbf{h}(\{A_n\};\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}} (n,m) \mathbf{h} _{n,m} ^{ \ast\{ A_n\}}(R;\cdot), \end{aligned} $$
(168)
where
$$\displaystyle \begin{aligned} \mathbf{f} ^{\wedge_{\mathbf{h}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}} (n,m) = \mathbf{f} ^{\wedge_{\mathbf{h}(\{A_n\};\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}} (n,m) = \langle\mathbf{f}, \mathbf{h} _{n,m} ^{\ast\{ A_n \}}(R;\cdot)\rangle _{\mathbf{h} (\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}. \end{aligned} $$
(169)
The space \(\mathbf {h}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) possesses the reproducing kernel \({\mathbf {K}}_{\mathbf {h}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})} (\cdot ,\cdot ): \overline {\mathbb {S}_R^{2;\mathrm {ext}}} \otimes \overline {\mathbb {S}_R^{2;\mathrm {ext}}} \to \mathbb {R}^3 \otimes \mathbb {R}^3 \otimes \mathbb {R}^3 \otimes \mathbb {R}^3\) given by
$$\displaystyle \begin{aligned} {\mathbf{K}}_{\mathbf{h}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})} (x,y) = \sum \limits _{n=0} ^{\infty} \sum \limits _{m=1} ^{2n+1} {\mathbf{h}}_{n,m}^{\ast\{A_n\}} (x) \otimes {\mathbf{h}}_{n,m}^{\ast\{A_n\}} (y), \end{aligned} $$
(170)
\(x,y \in \overline {\mathbb {S}_R^{2;\mathrm {ext}}}\), i.e.,
  • 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 \(x \in \overline {\mathbb {S}_R^{2;\mathrm {ext}}}\).

Finally, we set
$$\displaystyle \begin{aligned} {\mathbf{h}}_s (\overline{\mathbb{S}_R^{2;\mathrm{ext}}}) := \mathbf{h} \left( \left\{ \left(\frac{n+1/2}{R} \right)^s \right\}; \overline{\mathbb{S}_R^{2;\mathrm{ext}}}\right). \end{aligned} $$
(171)

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 \(\mathbf {pot}^{(0)}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\).

For any \(\mathbf {f} \in {\mathbf {l}}^2(\mathbb {S}_R^2)\), there exists one and only one tensorial “harmonic upward continuation” \(\mathbf {u} \in \mathbf {h}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) of the form
$$\displaystyle \begin{aligned} \mathbf{u}(x)= \sum \limits _{n=0}^{\infty} \sum \limits _{m=1}^{2n+1} A_n^2 \ {\mathbf{f}}^{\wedge_{{\mathbf{l}}^2(\mathbb{S}_R^2)}}(n,m)\ {\mathbf{h}}^{\ast}_{n,m}(R;x), \quad x \in \overline{\mathbb{S}_R^{2;\mathrm{ext}}}, \end{aligned} $$
(172)
where
$$\displaystyle \begin{aligned} {\mathbf{f}}^{\wedge_{{\mathbf{l}}^2(\mathbb{S}_R^2)}}(n,m) = \int _{\mathbb{S}_R^2} \mathbf{f}(y) \cdot {\mathbf{h}}_{n,m}^{R;(1,1)}(y)\ S(y) = \frac{1}{A_n^2} \mathbf{u} ^{\wedge_{\mathbf{h}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}} (n,m). \end{aligned} $$
(173)
Obviously, \(\mathbf {f} ^{\wedge _{\mathbf {l} ^2 (\mathbb {S}_R^2)}} (n,m) = \mathbf {u} ^{\wedge _{\mathbf {l} ^2 (\mathbb {S}_R^2)}} (n,m)\) holds true for all n, m. More detailed information about tensorial Sobolev spaces can be found in [105].

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

Let \(\{{\Lambda }^{\wedge }(n) \}_{n \in \mathbb {N}_0}\) be a sequence of real numbers. The operator \({\Lambda }: \mathscr {H}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}}) \to \mathscr {H}(\overline {\mathbb {S}_\tau ^{2;\mathrm {ext}}})\), τ ≥ R > 0, defined by
$$\displaystyle \begin{aligned} {\Lambda} F = \sum \limits _{n=0} ^{\infty} \sum \limits _{m=1} ^{2n+1} {\Lambda} ^{\wedge} (n) \ F ^{\wedge_{\mathscr{H}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}} (n,m)\ H_{n,m} ^{\ast}(\tau;\cdot), \end{aligned} $$
(174)
is called a scalar pseudodifferential operator of ordert if
$$\displaystyle \begin{aligned} \lim \limits _{n \to \infty} \frac {|{\Lambda} ^{\wedge} (n)|} {\left( n + \frac{1}{2}\right) ^{t}} = \mathrm{const} \neq 0 \end{aligned} $$
(175)
holds true for some \(t \in \mathbb {R}\). If the limit
$$\displaystyle \begin{aligned} \lim \limits _{n \to \infty} \frac {|{\Lambda} ^{\wedge} (n)|} {\left( n + \frac{1}{2}\right) ^{t}} = 0 \end{aligned} $$
(176)
holds true for all \(t \in \mathbb {R}\), the operator Λ is called a scalar pseudodifferential operator of exponential order. The sequence \(\{ {\Lambda } ^{\wedge }(n) \} _{n \in \mathbb {N}_0}\) is called the symbol of the scalar PDOΛ.

Tensorial Pseudodifferential Operator

Let \(\{ \boldsymbol {\Lambda }^{\wedge }(n) \}_{n \in \mathbb {N}_0}\) be a sequence of real numbers. The operator \(\boldsymbol {\Lambda }: \mathscr {H}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}}) \to \mathbf {h}(\overline {\mathbb {S}_\tau ^{2;\mathrm {ext}}})\), τ ≥ R > 0, defined by
$$\displaystyle \begin{aligned} \boldsymbol{\Lambda} F = \sum \limits _{n=0} ^{\infty} \sum \limits _{m=1} ^{2n+1} \boldsymbol{\Lambda} ^{\wedge} (n) F ^{\wedge_{\mathscr{H}\ (\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}} (n,m)\ {\mathbf{h}}_{n,m} ^{\ast}(\tau;\cdot), \end{aligned} $$
(177)
is called a tensorial pseudodifferential operator of ordert if
$$\displaystyle \begin{aligned} \lim \limits _{n \to \infty} \frac {|\boldsymbol{\Lambda} ^{\wedge} (n)|} {\left( n + \frac{1}{2}\right) ^{t}} = \mathrm{const} \neq 0 \end{aligned} $$
(178)
holds true for some \(t \in \mathbb {R}\). If the limit
$$\displaystyle \begin{aligned} \lim \limits _{n \to \infty} \frac {|\boldsymbol{\Lambda} ^{\wedge} (n)|} {\left( n + \frac{1}{2}\right) ^{t}} = 0 \end{aligned} $$
(179)
holds true for all \(t \in \mathbb {R}\), the operator Λ is called a tensorial pseudodifferential operator of exponential order. The sequence \(\{ \boldsymbol {\Lambda } ^{\wedge }(n) \} _{n \in \mathbb {N}_0}\) is called the symbol of the tensorial PDOΛ.

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

Let τ, R satisfy τR ≥ 0. Then any kernel \(Q_{R, R} (\cdot , \cdot ):\overline {\mathbb {S}_R^{2;\mathrm {ext}}} \times \overline {\mathbb {S}_R^{2;\mathrm {ext}}} \to \mathbb {R}\) of the form
$$\displaystyle \begin{aligned} Q_{R, R} (x,y) = \sum \limits _{n=0} ^{\infty} Q^{\wedge}(n) \sum \limits _{m=1} ^{2n+1} H_{n,m} ^{\ast}(R; x) H_{n,m} ^{\ast} (R; y) \end{aligned} $$
(180)
\(x,y \in \overline {\mathbb {S}_R^{2;\mathrm {ext}}}\) is called an \(\mathscr {H}_{R, R}\)–kernel. Any kernel \(\mathbf {q} _{R, \tau } (\cdot , \cdot ):\overline {\mathbb {S}_R^{2;\mathrm {ext}}} \times \overline {\mathbb {S}_\tau ^{2;\mathrm {ext}}} \to \mathbb {R}^3 \otimes \mathbb {R}^3\) of the form
$$\displaystyle \begin{aligned} \mathbf{q} _{R, \tau} (x,y) = \sum \limits _{n=0} ^{\infty} \mathbf{q} ^{\wedge}(n) \sum \limits _{m=1} ^{2n+1} H_{n,m} ^{\ast}(R; x) {\mathbf{h}}_{n,m} ^{\ast} (\tau; y) \end{aligned} $$
(181)
\((x,y) \in \overline {\mathbb {S}_R^{2;\mathrm {ext}}} \times \overline {\mathbb {S}_\tau ^{2;\mathrm {ext}}}\) is called an hR,τ-kernel. The sequence \(\{ Q^{\wedge }(n) \} _{n \in \mathbb {N}_{0}}\) is called the symbol of the\(\mathscr {H}_{R, \tau }\)-kernel, and \(\{\mathbf {q} ^{\wedge } (n)\}_{n \in \mathbb {N}_{0}}\) is called the symbol of thehR,τkernel.

Convolutions

An \(\mathscr {H}_{R, R}\)–kernel QR,R(⋅, ⋅) with the symbol {Q(n)}n=0,… is called admissible, if the following conditions are satisfied:
  1. 1.

    \(\sum _{n=0} ^{\infty } (Q ^{\wedge } (n))^2 < \infty \),

     
  2. 2.

    \(\sum _{n=0} ^{\infty } (2n+1)^2 \left ( \frac {Q^{\wedge } (n)} {A_n} \right )^2 < \infty \).

     
In analogy, an hR,τ–kernel qR,τ(⋅, ⋅) with the symbol {q(n)}n=0,… is called admissible, if the conditions (i) and (ii) are satisfied.
We define the convolution of an admissible \(\mathscr {H}_{R, R}\)-kernel against a function \(F \in \mathscr {H} (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) via the Parseval identity as follows:
$$\displaystyle \begin{aligned} ( Q_{R, R} \star_{\mathscr{H} (\overline{\mathbb{S}_R^{2;\mathrm{ext}}})} F) (x) = \sum \limits _{n=0} ^{\infty} \sum \limits _{m=1} ^{2n+1} Q^{\wedge} (n) F ^{\wedge_{\mathscr{H}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}(n,m) H_{n,m} ^{\ast} (R;x), \end{aligned} $$
(182)
\(x \in \overline {\mathbb {S}_R^{2;\mathrm {ext}}}\). In analogy, we introduce the convolution of an admissible hR,τ-kernel against \( F \in \mathscr {H} (\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) and \(\mathbf {f} \in \mathbf {h} (\overline {\mathbb {S}_\tau ^{2;\mathrm {ext}}})\), respectively, by
$$\displaystyle \begin{aligned} (\mathbf{q} _{R, \tau} \star_{{\mathscr{H}} (\overline{\mathbb{S}_R^{2;\mathrm{ext}}})} {F} ) (x) = \sum \limits _{n=0} ^{\infty} \sum \limits _{m=1} ^{2n+1} \mathbf{q} ^{\wedge} (n) F ^{\wedge_{\mathscr{H}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}(n,m) {\mathbf{h}}^{\ast}_{n,m} (\tau;x), \end{aligned} $$
(183)
\(x \in \overline {\mathbb {S}_\tau ^{2;\mathrm {ext}}}\), and
$$\displaystyle \begin{aligned} (\mathbf{q} _{R, \tau} \ast_{\mathbf{h} (\overline{\mathbb{S}_\tau^{2;\mathrm{ext}}})} \mathbf{f} ) (x) = \sum \limits _{n=0} ^{\infty} \sum \limits _{m=1} ^{2n+1} \mathbf{q} ^{\wedge} (n) \mathbf{f} ^{\wedge_{\mathbf{h}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}(n,m) H_{n,m} ^{\ast} (R;x), \end{aligned} $$
(184)
\(x \in \overline {\mathbb {S}_R^{2;\mathrm {ext}}}\).

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 Σint, i.e., R < infxΣ|x|. Furthermore, let γ be smaller than the lowest possible altitude of the satellite, i.e., γ < infxΓ|x| (cf. Fig. 9).

Consider a potential of class \(\mathscr {H}(\{A_n\};\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) of the form
$$\displaystyle \begin{aligned} F=\sum_{n=0}^{\infty} \sum \limits _{m=1}^{2n+1} F^{\wedge_{\mathscr{H}(\{A_n\};\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}(n,m) \ H_{n,m}^{\ast \{A_n\}}(R;\cdot). \end{aligned} $$
(185)
The upward continuation operator\(\Lambda _{\mbox{{up}}}^{R,\gamma }\) associates to \(F~{\in }~\mathscr {H}(\{A_n\};\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) the solution \(\Lambda _{\mbox{{up}}}^{R,\gamma }\) of the Dirichlet problem \(\Lambda _{\mbox{{up}}}^{R,\gamma } F \in \mathrm {Pot} ^{(0)}(\overline {\mathbb {S}_\gamma ^{2;\mathrm {ext}}})\) corresponding to the boundary values \((\Lambda _{\mbox{{up}}}^{R,\gamma } F)|{ }_{\mathbb {S}^2_{\gamma }}= F|{ }_{\mathbb {S}^2_{\gamma }}\). The upward continuation operator \(\Lambda _{\mbox{{up}}}^{R,\gamma }\) has the associated symbol
$$\displaystyle \begin{aligned} ({\Lambda_{\mbox{{up}}}^{R,\gamma}})^\wedge (n) = \left(\frac{R}{\gamma}\right)^n. \end{aligned} $$
(186)
The inverse of \({\Lambda _{\mbox{{up}}}^{R,\gamma }}\) is called the downward continuation operator, so that
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\Lambda_{\mbox{{down}}}^{R,\gamma}} = ({\Lambda_{\mbox{{up}}}^{R,\gamma}})^{-1}. \end{array} \end{aligned} $$
(187)
It brings down the gravitational potential at height γ to the height R and has the associated symbol
$$\displaystyle \begin{aligned} ({\Lambda_{\mbox{{down}}}^{R,\gamma}})^\wedge (n) = \left(\frac{\gamma}{R}\right)^n. \end{aligned} $$
(188)
It is obvious, that the upward continuation is well–posed, whereas the downward continuation leads to an ill–posed problem.

Pseudodifferential Operator of the First Order Radial Derivative

This operator associates to \(F~{\in }~\mathscr {H}(\{A_n\};\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) the solution ΛFRDF of the Dirichlet problem \(\Lambda _{\mbox{{FRD}}} F \in \mathrm {Pot} ^{(0)}(\overline {\mathbb {S}_\gamma ^{2;\mathrm {ext}}})\) corresponding to the boundary values \((\Lambda _{\mbox{{FRD}}} F)|{ }_{\mathbb {S}^2_{\gamma }}= -\frac {\partial }{\partial r} F|{ }_{\mathbb {S}^2_{R}}\). This is an operator of order 1 with the associated symbol
$$\displaystyle \begin{aligned} (\Lambda_{\mbox{{FRD}}})^\wedge (n) = \frac{n+1}{R}. \end{aligned} $$
(189)

Pseudodifferential Operator of the Second Order Radial Derivative

Analogous considerations applied to the operator \(\frac {\partial ^2}{\partial r^2}\) on F yields the operator of the second order radial derivative which associates to \(F~{\in }~\mathscr {H}(\{A_n\};\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) the solution ΛSRDF of the Dirichlet problem \(\Lambda _{\mbox{{SRD}}} F \in \mathrm {Pot} ^{(0)}(\overline {\mathbb {S}_\gamma ^{2;\mathrm {ext}}})\) corresponding to the boundary values \((\Lambda _{\mbox{{SRD}}} F)|{ }_{\mathbb {S}^2_{\gamma }}= \frac {\partial ^2}{\partial r^2} F|{ }_{\mathbb {S}^2_{R}}\). This is an operator of order 2 with the associated symbol
$$\displaystyle \begin{aligned} (\Lambda_{\mbox{{SRD}}})^\wedge (n) = \frac{(n+1)(n+2)}{R^2}. \end{aligned} $$
(190)

The Pseudodifferential Operator of the Hesse Tensor

Let us consider the operator of the second derivative (i.e., the Hesse tensor)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla \otimes \nabla: \mathscr{H}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}}) \to \mathbf{h}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}}),\end{array} \end{aligned} $$
(191)
which associates to \(F~{\in }~\mathscr {H}(\{A_n\};\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) the solution ΛHesseF of the Dirichlet problem of finding \(\boldsymbol {\Lambda }_{\mbox{{Hesse}}} F \in \mathbf {{h} }(\overline {\mathbb {S}_\gamma ^{2;\mathrm {ext}}})\) corresponding to the boundary values
$$\displaystyle \begin{aligned} \begin{array}{rcl} (\boldsymbol{\Lambda}_{\mbox{{Hesse}}} F)|{}_{\mathbb{S}^2_{\gamma}}= (\nabla \otimes \nabla) F|{}_{\mathbb{S}^2_{R}}.\end{array} \end{aligned} $$
(192)
This is an operator of order 2 given by the symbol
$$\displaystyle \begin{aligned} (\boldsymbol{\Lambda}_{\mbox{{Hesse}}}) ^{\wedge} (n,m) = \frac{\sqrt{(n+2)(n+1)(2n+3)(2n+1)}}{R^2}, \end{aligned} $$
(193)
n = 0, 1, …;m = 1, …, 2n + 1.

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 Σint 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

Let a function G of class \(\mathscr {H}(\{A_n\};\overline {\mathbb {S}_\gamma ^{2;\mathrm {ext}}})\) be known. Suppose that the symbol of the pseudodifferential operator is given by
$$\displaystyle \begin{aligned} (\Lambda_{\mbox{{SGG}}})^\wedge (n) = \left(\frac{R}{\gamma}\right)^n \ \frac{(n+1)(n+2)}{\gamma^2}.\end{aligned} $$
(194)
Find a potential \(F \in \mathscr {H}(\{A_n\};\overline {\mathbb {S}_R^{2;\mathrm {ext}}})|{ }_{\overline {\varSigma ^{\mathrm {ext}}}}\) such that
$$\displaystyle \begin{aligned} (\Lambda_{\mbox{{SGG}}} F) =G.\end{aligned} $$
(195)
The solution process of the SGG-problem of determining the gravitational potential of the Earth on the Earth’s surface from orbital values v of the Hesse tensor on Γ may be based on the Runge assumption that there exists a potential outside a Runge (Bjerhammar) sphere \(\mathbb {S}_R^2\) inside the Earth with ε-accuracy (ε > 0, but arbitrarily small) to the real gravitational potential outside and on the Earth’s surface Σ. Thus we are allowed to model the SGG-problem (see also [48]) in the following way:

The Tensorial SGG Pseudodifferential Operator

The SGG-operator
$$\displaystyle \begin{aligned} \begin{array}{rcl} \boldsymbol{\Lambda} _{\mbox{SGG}} ^{R,\gamma}: \mathscr{H}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}}) \to \mathbf{h}(\overline{\mathbb{S}_\gamma^{2;\mathrm{ext}}}) \end{array} \end{aligned} $$
(196)
expressed in terms of outer harmonics is given by
$$\displaystyle \begin{aligned} \begin{array}{rcl} \boldsymbol{\Lambda} _{\mbox{SGG}} ^{R,\gamma} H_{n,m}^{\ast} (R;x) &\displaystyle = \frac{\sqrt{(n+2)(n+1)(2n+3)(2n+1)}}{\gamma^2}\left( \frac{R}{\gamma} \right) ^{n} {\mathbf{h}}_{n,m}^{\ast}(\gamma;x), \quad x \in \overline{\mathbb{S}_\gamma^{2;\mathrm{ext}}}.\\ \end{array} \end{aligned} $$
(197)
The symbol of this operator is given by
$$\displaystyle \begin{aligned} \begin{array}{rcl} ( \boldsymbol{\Lambda} _{\mbox{SGG}} ^{R,\gamma}) ^{\wedge} (n,m) = \ \left( \frac{R}{\gamma} \right) ^{n} \ \frac{\sqrt{(n+2)(n+1)(2n+3)(2n+1)}} {\gamma^2}, \end{array} \end{aligned} $$
(198)
n = 0, 1, …;m = 1, …, 2n + 1.
The SGG–operator applied to \(F \in \mathscr {H}(\overline {\mathbb {S}_R^{2;\mathrm {ext}}})\) is representable as a outer harmonic series expansion as follows:
$$\displaystyle \begin{aligned} \boldsymbol{\Lambda} _{\mbox{SGG}} ^{R,\gamma} F(x) = \sum \limits _{n=0} ^{\infty} \sum \limits _{m=1} ^{2n+1} \left( \boldsymbol{\Lambda} _{\mbox{SGG}} ^{R,\gamma} \right) ^{\wedge} (n,m) F^{\wedge_{\mathscr{H}(\overline{\mathbb{S}_R^{2;\mathrm{ext}}})}}(n,m) \mathbf{h} _{n,m} ^{\ast}(\gamma;x). \end{aligned} $$
(199)
The interrelations between the potential F and the full Hesse tensor of F on the Earth’s surface and the satellite orbit can be represented in a so-called Meissl scheme (see Fig. 8). Meissl schemes both in the framework of outer harmonics and in multiscale nomenclature can be found in [47]. Table 1 presents a list of all the afore discussed pseudodifferential operators.
Fig. 8

The Meissl scheme for the Hesse tensor on the Runge reference sphere of the Earth’s surface and on the reference sphere of the satellite orbit (see also [47], and [112], [115])

Fig. 9

The geometric situation of satellite gravitational gradiometry as discussed in our “downward continuation” multiscale space regularization procedure

Table 1

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

\(\displaystyle \frac {R^n}{\gamma ^n}\frac {(n+1)(n+2)}{\gamma ^2}\)

ΛHesse

Hesse tensor

\(\displaystyle \frac {\sqrt {(n+2)(n+1)(2n+3)(2n+1)}}{R^2}\)

2

ΛSGG

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

Tensor spherical harmonics of type (1,1) allow to express the Hesse tensor applied to solid (outer) harmonics in the form
$$\displaystyle \begin{aligned} \begin{array}{rcl} (\nabla \otimes \nabla) \ H^R_{n,m} = \sqrt{\tilde{\mu}_n^{(1,1)}} \ \ {\mathbf{h}}^{R;(1,1)}_{n,m}, \end{array} \end{aligned} $$
where we remember
$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{\mu}_n^{(1,1)}= (n+2) (n+2) (2n-3) (2n-1). \end{array} \end{aligned} $$
(200)
This leads to the tensor-isotropic SGG-pseudodifferential equation
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \Lambda ({V}) = \sum^\infty_{n=0} \sum^{2n+1}_{m=1} {V}^{\wedge_{\mathrm{L}^2(\mathbb{S}^2_R)}} (n,m) \sqrt{\tilde{\mu}_n^{(1,1)}} \ {\mathbf{h}}^{R;(1,1)}_{n,m} \ = \ (\nabla \otimes \nabla) {V}= \mathbf{v}, \\ \end{array} \end{aligned} $$
(201)
as spectral (frequency) representation for the inversion of the SGG-integral equation ΛV = v. Equivalently, we have a representation as convolution
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \Lambda (V) = \int_{\mathbb{S}^2_R} {\mathbf{K}}^{\Lambda}_{R}(\cdot,y)\ V (y) \ dS(y)\ = \ (\nabla \otimes \nabla) V =\mathbf{v}, \end{array} \end{aligned} $$
(202)
where the tensorial kernel \({\mathbf {K}}^{\Lambda }_{R}(\cdot ,\cdot )\) is given by
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{K}}^{\Lambda}_{R}(x,y) \ = \ \sum^\infty_{n=0} \sum^{2n+1}_{m=1} \sqrt{\tilde{\mu}_n^{(1,1)}} \ {\mathbf{h}}^{R;(1,1)}_{n,m}(x) \ H^R_{n,m}(y) \end{array} \end{aligned} $$
(203)
in spectral way. By the completeness of the system \( \{ H^R_{n,m}\}\) this enables us to conclude in the framework \({\mathbf {l}}^2(\mathbb {S}^2_R)\) of square-integrable tensor fields on \(\mathbb {S}^2_R\) that
$$\displaystyle \begin{aligned} \begin{array}{rcl} \langle(\nabla \otimes \nabla )V , {\mathbf{h}}^{R;(1,1)}_{p,q}\rangle_{{\mathbf{l}}^2(\mathbb{S}^2_R)} &\displaystyle =&\displaystyle \int_{\mathbb{S}^2_R} (\nabla \otimes \nabla) V (y) \cdot {\mathbf{h}}^{R;(1,1)}_{p,q} (y) \ dS (y) \\ &\displaystyle = &\displaystyle V^{\wedge_{\mathrm{L}^2(\mathbb{S}^2_R)}} (p,q) \sqrt{\tilde{\mu}_{n}^{(1,1)}}. \end{array} \end{aligned} $$
(204)
Consequently, we obtain the following expansion for the potential V in \({\mathbb {S}_R^{2;\mathrm {ext}}}\)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} {V}(x)&\displaystyle =&\displaystyle \Lambda^{-1} (\nabla \otimes \nabla {V})(x)\\ &\displaystyle = &\displaystyle \ \sum^\infty_{n=0} \sum^{2n+1}_{m=1} \langle \underbrace{\nabla \otimes \nabla V}_{= \mathbf{v} }, {\mathbf{h}}^{R;(1,1)}_{n,m}\rangle_{\mathbf{ l}^2(\mathbb{S}^2_R)} (\tilde{\mu}_n^{(1,1)})^{-1/2} H^R_{n,m}(x), \end{array} \end{aligned} $$
(205)
i.e.,
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} V(x) ={\Lambda^{-1}} \mathbf{v}(x) = \int_{\mathbb{S}^2_R} {\mathbf{K}}^{\Lambda^{-1}}_{R}(x,y) \cdot \mathbf{v} (y) \ dS(y) \end{array} \end{aligned} $$
(206)
with
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{K}}^{\Lambda^{-1}}_{R}(x,y) \ = \ \sum^\infty_{n=0} \sum^{2n+1}_{m=1} (\tilde{\mu}_n^{(1,1)})^{-1/2}\ H^R_{n,m}(x) \ {\mathbf{h}}^{R;(1,1)}_{n,m}(y). \end{array} \end{aligned} $$
(207)
This formula expresses the gravitational potential V in terms of the tensor v on locations x of the satellite orbit Γ.

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.

As described earlier, a regularization strategy for the SGG-problem is a sequence \(\{R_j \}_{ j \in \mathbb {N}_0}\) of linear bounded pseudodifferential operators Rj so that
$$\displaystyle \begin{aligned} \begin{array}{rcl} \lim_{j \to \infty} R_j\Lambda ({V}) = {V} \end{array} \end{aligned} $$
(208)
in the outer space of \(\mathbb {S}^2_R\), i.e., the operators RjΛ converge in pointwise sense to the identity operator.

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

This method makes use of the (non-bandlimited) isotropic Tikhonov-kernels (scaling functions) Φj, j = 0, 1, …, given by
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varPhi_j (x,y)\ = \ \sum^\infty_{n=0} \sum^{2n+1}_{m=1} (\varPhi_j)^\wedge (n) H^R_{n,m}(x) H^R_{n,m}(y) \end{array} \end{aligned} $$
(209)
with
$$\displaystyle \begin{aligned} \begin{array}{rcl} (\varPhi_j)^\wedge (n) = \frac{(\Lambda^\wedge (n))^2}{(\Lambda^\wedge (n))^2 + \mu_j^2}, \quad n=0,1, \ldots, \ j=0,1, \ldots, \end{array} \end{aligned} $$
(210)
where {μj}, j = 0, 1, …, is a sequence of real numbers satisfying
$$\displaystyle \begin{aligned} \begin{array}{rcl} \lim_{j\to \infty} \mu_j = 0.\end{array} \end{aligned} $$
(211)
Then, the operators Rj = Φj ∗ Λ−1 constitute a multiscale regularization strategy. More concretely,
$$\displaystyle \begin{aligned} \begin{array}{rcl} R_j\Lambda ({V})(x) = \sum^\infty_{n=0} \sum^{2n+1}_{m=1} (\varPhi_j)^\wedge (n) \langle \underbrace{\nabla \otimes \nabla V}_{= \mathbf{v} }, \mathbf{ h}^{R;(1,1)}_{n,m}\rangle_{{\mathbf{l}}^2(\mathbb{S}^2_R)} (\tilde{\mu}_n^{(1,1)})^{-1/2} H^R_{n,m}(x),\\ \end{array} \end{aligned} $$
(212)
i.e.,
$$\displaystyle \begin{aligned} \begin{array}{rcl} R_j\Lambda ({V})(x) = \int_{\mathbb{S}^2_R} \int_{\mathbb{S}^2_R} \varPhi_j (x,z)\ {\mathbf{K}}^{\Lambda^{-1}}_{R}(z,y)\ dS(z) \cdot (\nabla \otimes \nabla) V (y) \ dS(y).\\ \end{array} \end{aligned} $$
(213)

Truncated Singular Value Regularization Strategy

The point of departure is a one-dimensional function \(\varphi _0 : [0,\infty ) \to \mathbb {R} \) with the following properties (cf. [63]):
  1. (i)

    φ0(0) = 1,

     
  2. (ii)

    φ0 is monotonically decreasing,

     
  3. (iii)

    φ0 is continuous at 0,

     
  4. (iv)

    \( \varphi _0 : [0,\infty ) \to \mathbb {R} \) has a local support, i.e., supp φ0 ⊂ [0, 1].

     
Accordingly we are led to the isotropic scaling functions Φj, j = 0, 1, …, given by
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varPhi_j (x,y)\ = \ \sum^\infty_{n=0} \sum^{2n+1}_{m=1} \ \varphi_0 (2^{-j} n) \ H^R_{n,m}(x) H^R_{n,m}(y), \end{array} \end{aligned} $$
(214)
where
$$\displaystyle \begin{aligned} \begin{array}{rcl} 0 \leq (\varPhi_j)^\wedge(n) = \varphi_0 (2^{-j} n) \leq 1, \qquad n =0,1, \ldots \end{array} \end{aligned} $$
(215)
and
$$\displaystyle \begin{aligned} \begin{array}{rcl} \lim_{j \to \infty} (\varPhi_j)^\wedge(n)= \lim_{j \to \infty} \varphi_0 (2^{-j} n) = 1, \qquad n =0,1, \ldots \ . \end{array} \end{aligned} $$
(216)
In accordance with our construction, the compact support of φ0 implies that only finitely many (Φj)(n) are different from 0. Obviously, it follows that
$$\displaystyle \begin{aligned} \mathrm{{supp}} \; \varphi_j \subset [0, 2^j ] . \end{aligned}$$
So, the bandlimited kernels Φj, j = 0, 1, …, defined via a generator φ0 satisfying the properties (i) – (iv) as stated above define operators
$$\displaystyle \begin{aligned} \begin{array}{rcl} R_j = \varPhi_j *\Lambda^{-1}, \end{array} \end{aligned} $$
(217)
which constitute a regularization strategy in the following sense:
  1. 1.

    Rj is bounded,

     
  2. 2.
    the limit relation
    $$\displaystyle \begin{aligned} \begin{array}{rcl} \lim\limits_{j \to \infty} R_j \Lambda ({V})={V} \end{array} \end{aligned} $$
    (218)
    holds true in the outer space of \(\mathbb {S}^2_R\) .
     

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

In the spherically reflected Runge (Bjerhammar) framework of a potential V approximating arbitrarily close the Earth’s external gravitational potential \(\tilde V\) (cf. (140)), the SGG-integral equation
$$\displaystyle \begin{aligned} \int_{\mathbb{S}_R^2} \,V(y) \, \underbrace{ \nabla_x \otimes \nabla_x \frac{1}{4 \pi R} \frac{\vert x\vert^2 - R^2}{\vert x-y\vert^3}}_{= {\mathbf{k}}_R (x,y)} \ dS(y) = \ (\nabla \otimes \nabla) {V}(x) = \mathbf{v}(x) \approx \tilde{\mathbf{v}}(x) , \end{aligned} $$
(219)
holds true for all locations x along the satellite orbit Γ, where
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{v}(x) = (\nabla \otimes \nabla) V(x)\approx (\nabla \otimes \nabla) \tilde{V}(x), \ \ \ x \in \varGamma,\end{array} \end{aligned} $$
(220)
are the known tensorial input data, kR(⋅, ⋅) is as usual the Hesse tensor of the Abel–Poisson kernel given by (6), and, in accordance with our construction, the scalar function V |Σ, i.e., the potential on the (actual) Earth’s surface Σ obtained by “upward continuation” via the integral formula
$$\displaystyle \begin{aligned} {V}|{}_\varSigma = \int_{\mathbb{S}_R^2} {V} (y) \ K_R(\cdot ,y) \ dS(y)|{}_\varSigma \end{aligned} $$
(221)
represents the desired SGG-solution to the known Hesse tensor field v(x), x ∈ Γ, given by
$$\displaystyle \begin{aligned} \mathbf{v}(x) = \int_{\mathbb{S}_R^2} \ V(y) \ {\mathbf{k}}_R (x,y) \ dS(y) , \quad x\in \varGamma \end{aligned} $$
(222)
(note that we do not require that Γ is a closed surface in Euclidean space \(\mathbb {R}^3\)).

5.2 Space Solution in Preparatory Framework

By approximate integration over the sphere \(\mathbb {S}_R^2 \) (see, e.g., [39] for appropriate rules) we are led to a cubature formula of the form
$$\displaystyle \begin{aligned} \mathbf{v} (x) = \int_{\mathbb{S}_R^2} \ V(y) \ {\mathbf{k}}_R (x,y) \ dS(y)\approx \sum^{N_R}_{k=1} \underbrace{w_k^R V (y_k^R)}_{=: a_k^R} {\mathbf{k}}_R (x,y_k^R), \ \ x\in \varGamma , \end{aligned} $$
(223)
with appropriately given weights \(w_k^R{\in } \mathbb {R}, k{=}1,\ldots , N_R\), and nodes \(y_k^R {\in } \mathbb {S}_R^2\), k=1, …, NR ( ≈″ means that the continuous sum, i.e., the integral, on the left side is approximated by an associated discrete cubature formula on the right side).
In order to determine the coefficients \(a_k^R{\in } \mathbb {R}, k{=}1,\ldots , N_R\), we assume that tensor values v(xl), l = 1, …, LR, are known along the orbit Γ. Thus, SGG is reduced to a discrete problem, and a (discrete) solution can be obtained, e.g., by interpolatory requirements (in case of error-free data) or by smoothing and/or adjustment procedures (such as proposed, e.g., by [32, 55, 96] in case of error-affected data)
$$\displaystyle \begin{aligned} \mathbf{v}(x_l) \approx \sum^{N_R}_{k=1} a_k^R \ {\mathbf{k}}_R (x_l,y_k^R), \ \ \; l=1,\ldots, L_R . \end{aligned} $$
(224)
Once the coefficients of the linear system (224)
$$\displaystyle \begin{aligned} \begin{array}{rcl} a_k^R = w_k^R V(y_k^R), \ \ k=1,\ldots, N_R, \end{array} \end{aligned} $$
(225)
are available (note that the weights \(w_k^R, k=1, \ldots , N_R,\) are known from the integration rule), the Runge (Bjerhammar) potential V can be obtained from its discretization (223) as follows:
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} V(x) &\displaystyle = &\displaystyle \int_{\mathbb{S}_R^2}\ {V} (y) K_R(x ,y) \ dS(y)\\ &\displaystyle \approx &\displaystyle V^R(x) := \sum^{N_R}_{k=1} \underbrace{w_k^R V (y_k^R)}_{= a_k^R} K_R(x ,y_k^R), \quad x \in \mathbb{S}_R^{2, \mathrm{ext}},y_k^R \in \mathbb{S}_R^2. \end{array} \end{aligned} $$
(226)
As a consequence, the restriction VR|Σ of VR represents the desired approximate SGG-solution on the Earth’s surface Σ in space domain.

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 VR. 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 NR and LR. 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

Let us consider a monotonously decreasing sequence {γn}n=0,1,… of real numbers γn, i.e., γn > γn+1,n = 0, 1, …, satisfying the conditions (cf. Fig. 9)
  • the initial value γ0 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 {γn}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)
In what follows, we identify the members of the sequence {γn}n=0,1,… with the radii of spheres around the origin to act as the “scales” in an Abel–Poisson kernel filter-reflected low- and bandpass approach thereby observing that
$$\displaystyle \begin{aligned} K_{\gamma_n}(x,y) = \frac{1}{4 \pi {\gamma_n}} \frac{\vert x\vert^2 - {\gamma_n}^2}{\vert x-y\vert^3}, \ \ x, y \in {\mathbb{S}_{\gamma_n}^{2}}, \ x \neq y, \end{aligned} $$
(229)
shows increasing space localization with increasing “scale” n, i,e., with decreasing radii γn (for more details about the space localization of Abel–Poisson kernels see, e.g., [63] and the references therein).

Initial Step: Discrete Abel–Poisson Lowpass Filtering

Classical potential theory (see, e.g., [80]) tells us that the restriction \(V|{ }_{\overline {\mathbb {S}_{\gamma _0}^{2, \mathrm {ext}}}}\) of the potential V can be represented in the form
$$\displaystyle \begin{aligned} V(x) =V|{}_{\overline{\mathbb{S}_{\gamma_0}^{2, \mathrm{ext}}}}(x) = \int_{\mathbb{S}_{\gamma_0}^2} {V} (y) \ K_{\gamma_0}(x ,y) \ dS(y), \quad x \in \overline{\mathbb{S}_{\gamma_0}^{2, \mathrm{ext}}}. \end{aligned} $$
(230)
By approximate integration over the sphere \(\mathbb {S}_{\gamma _0}^2 \) we are led to a cubature formula\( V_{\gamma _0}\) approximating the integral representation (230) of V on \(\overline {\mathbb {S}_{\gamma _0}^{2, \mathrm {ext}}}\) in the form
$$\displaystyle \begin{aligned} V|{}_{\overline{\mathbb{S}_{\gamma_0}^{2, \mathrm{ext}}}}(x) \approx V_{\gamma_0} (x) := \sum^{N_{\gamma_0}}_{k=1} \underbrace{w_k^{\gamma_0} V (y_k^{\gamma_0})}_{=: a_k^{\gamma_0}} K_{\gamma_0} (x,y_k^{\gamma_0}), \ \ \ x \in \overline{\mathbb{S}_{\gamma_0}^{2, \mathrm{ext}}}, \end{aligned} $$
(231)
with appropriately given weights \(w_k^{\gamma _0}{\in } \mathbb {R}, k{=}1,\ldots , N_{\gamma _0}\), and prescribed nodes \(y_k ^{\gamma _0}{\in } \mathbb {S}_{\gamma _0}^2\), Open image in new window. Hence, we obtain by forming the Hesse operator
$$\displaystyle \begin{aligned} \mathbf{v} (x) \approx \mathbf{v_{\gamma_0}} (x) = ( \nabla_x \otimes \nabla_x ) V_{\gamma_0} (x) = \sum^{N_{\gamma_0}}_{k=1} \underbrace{w_k^{\gamma_0} V (y_k^{\gamma_0})}_{= a_k^{\gamma_0}} {\mathbf{k}}_{\gamma_0} (x,y_k^{\gamma_0}), \ \ x\in \varGamma.\end{aligned} $$
(232)
In order to determine the coefficients \(a_k ^{\gamma _0}{\in } \mathbb {R}\), \( k\,{=}\,1,\ldots , N_{\gamma _0},\) we assume in accordance with the observational SGG-situation that the tensor values \(\mathbf {v} (x_l^{\gamma _0}), l=1,\ldots , L_{\gamma _0}\), are known along the orbit Γ.
In other words, the problem of determining \(V|{ }_{\overline {\mathbb {S}_{\gamma _0}^{2, \mathrm {ext}}}}\) by a cubature based approximation \(V_{\gamma _0}\) is reduced to the discrete problem
$$\displaystyle \begin{aligned} \mathbf{v}(x_l^{\gamma_0}) \approx \mathbf{v_{\gamma_0}}(x_l^{\gamma_0}) = \sum^{N_{\gamma_0}}_{k=1} a_k ^{\gamma_0} \ \mathbf{ k}_{\gamma_0}(x_l^{\gamma_0},y_k^{\gamma_0}), \ \ \; l=1,\ldots, L_{\gamma_0} . \end{aligned} $$
(233)
Once the linear system (233) is solved for the input data \(\mathbf {v}(x_l^{\gamma _0}), l=1,\ldots , L_{\gamma _0}\), so that the coefficients \(a_k ^{\gamma _0}, k=1,\ldots , N_{\gamma _0}\), are available (once more, note that \(w_k^{\gamma _0}\) is known from the integration rule), the potential \( V_{\gamma _0}\) as a discrete version to \(V|{ }_{\overline {\mathbb {S}_{\gamma _0}^{2, \mathrm {ext}}}}\) can be obtained from
$$\displaystyle \begin{aligned} V_{\gamma_0}(x) = \sum^{N_{\gamma_0}}_{k=1} \underbrace{w_k^{\gamma_0} V (y_k ^{\gamma_0})}_{= a_k ^{\gamma_0}} K_{\gamma_0}(x,y_k ^{\gamma_0}), \quad x \in \overline{\mathbb{S}_{\gamma_0}^{2, \mathrm{ext}}}.\end{aligned} $$
(234)
Subsequently, (regularization by) “downward continuation” yields the “0-scale” potential \( V_{\gamma _0}^R\) in \(\mathbb {S}_R^{2, \mathrm {ext}}\) of the form
$$\displaystyle \begin{aligned} V_{\gamma_0}^R(x) = \sum^{N_R}_{k=1} w_k^R \ V_{\gamma_0} \left(\gamma_0 \frac{ y_k^R}{R} \right) K_R(x,y_k^R),\quad x \in \mathbb{S}_R^{2, \mathrm{ext}}, \; y_k^R \in \mathbb{S}_R^2,\end{aligned} $$
(235)
hence, the restriction \( V_{\gamma _0}^R\vert _\varSigma \) of the potential \( V_{\gamma _0}^R\) represents the “0-scale” SGG-lowpass solution on the (real) Earth’s surface Σ, where the weights \(w_k^R \in \mathbb {R}, k=1,\ldots , N_R\), and nodes \(y_k^R \in \mathbb {S}_R^2\), k = 1, …, NR, in (235) can be taken from (226) or another appropriate cubature formula.

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

An elementary calculation yields
$$\displaystyle \begin{aligned} &{\mathbf{k}}_{\gamma_0} (x,y) = (\nabla_x \otimes \nabla_x) K_{\gamma_0}(x,y) \\ &\quad = \left( \frac{\partial^2}{\partial x_i \partial x_j} K_{\gamma_0}(x,y) \right)_{i,j=1,2,3}, \quad y = (y_1, y_2, y_3)^T \in \mathbb{S}^2_{\gamma_0} , \end{aligned} $$
(236)
where
$$\displaystyle \begin{aligned}& \frac{\partial^2}{\partial x_i^2} K_{\gamma_0}(x,y) \\ &\quad = \frac{1}{4 \pi {\gamma_0}} \left( \frac{2}{\vert x-y\vert^3} - \frac{12 x_i (x_i - y_i)}{\vert x-y\vert^5} \right. \\ &\qquad - \left. \frac{3 (|x|{}^2 - {\gamma_0}^2)}{\vert x-y \vert^5} + \frac{15 (x_i - y_i)^2 (|x|{}^2 - {\gamma_0}^2)}{\vert x-y\vert^7} \right) {} \end{aligned} $$
(237)
and
$$\displaystyle \begin{aligned}&\frac{\partial^2}{\partial x_i \partial x_j} K_{\gamma_0}(x,y) \\ &\quad = - \frac{1}{4 \pi {\gamma_0}} \left( \frac{6 x_i (x_j - y_j)}{\vert x-y\vert^5} + \frac{6 x_j (x_i - y_i)}{\vert x-y\vert^5} \right. \\ & \qquad - \left. \frac{15 (|x|{}^2 - {\gamma_0}^2) (x_j - y_j) (x_i - y_i)}{\vert x-y\vert^7} \right) ,\quad i\neq j,{} \end{aligned} $$
(238)
for x ∈ Γ.
In order to solve the linear system (233) numerically, we can take advantage of the decorrelation capability caused by the substitution of the kernel of a “monopole” (cf. (237), (238)) by a linear combination of Gaussians, i.e.,
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{1}{\vert x-y\vert} \approx \sum^M_{m=1} \omega_m e^{-\alpha_m \vert x-y\vert^2}, \ \ \ \delta \leq \vert x-y\vert, \end{array} \end{aligned} $$
(239)
being valid for some sufficiently small δ > 0. The critical point concerning (239), however, is to find an a priori specification in our numerics for the coefficients αm, ωm, m = 1, …, M, depending on the integer M (note that the use of Gaussians also allows to choose γ0 equal to infxΓ|x| in the initial step, since |x − y|−1 is bounded away from the singularity at x = y by an arbitrarily small constant δ > 0).
Indeed, different approaches to attain suitable αm, ωm, m = 1, …, M, for appropriate choices δ > 0 can be found in the literature (not discussed here in more detail):
  • 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]).

The approach of the Geomathematics Group, Kaiserslautern, described in [14, 44], closely parallels the concepts presented in [10, 11]. This concept starts with an approximation obtained by the appropriate discretization of an integral expression of a monopole |xy|−1. Afterwards, in order to reduce the number M of terms of the Gaussian sum on the right side of (239), an algorithm is applied based on Prony’s method. An advantage is that one is able to work with the one-dimensional function rr−1, r ≥ δ > 0, sufficiently small:
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{1}{r} \approx \sum^M_{m=1} \omega_m e^{-\alpha_m r^2}. \end{array} \end{aligned} $$
(240)

Subsequent Steps: Discrete Abel–Poisson Lowpass and Bandpass Filtering

The potential \(V|{ }_{\overline {\mathbb {S}_{\gamma _1}^{2, \mathrm {ext}}}}\) admits the representation
$$\displaystyle \begin{aligned} V(x) =V|{}_{\overline{\mathbb{S}_{\gamma_1}^{2, \mathrm{ext}}}}(x) = \int_{\mathbb{S}_{\gamma_1}^2} {V} (y) K_{\gamma_1}(x ,y) \ dS(y), \quad x \in \overline{\mathbb{S}_{\gamma_1}^{2, \mathrm{ext}}}. \end{aligned} $$
(241)
By approximate integration over the sphere \(\mathbb {S}_{\gamma _1}^2 \) we are able to deduce a cubature formula\( V_{\gamma _1}\) of the integral representation (241) of \(V|{ }_{\overline {\mathbb {S}_{\gamma _1}^{2, \mathrm {ext}}}}\) in the form
$$\displaystyle \begin{aligned} V(x) = V|{}_{\overline{\mathbb{S}_{\gamma_1}^{2, \mathrm{ext}}}}(x) \approx V_{\gamma_1} (x) = \sum^{N_{\gamma_1}}_{k=1} b_k^{\gamma_1}\ K_{\gamma_1} (x,y_k^{\gamma_1}), \ \ \ x \in \overline{\mathbb{S}_{\gamma_1}^{2, \mathrm{ext}}}, \end{aligned} $$
(242)
with \( N_{\gamma _1}>N_{\gamma _0}\) and prescribed nodes \(y_k ^{\gamma _1} \,{\in }\, \mathbb {S}_{\gamma _1}^2\), \(k=1,\ldots , N_{\gamma _1},\) such that the weights satisfy the properties
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} b_k^{\gamma_1} := a_k^{\gamma_0}, \quad k= 1, \ldots, N_{\gamma_0}. \end{array} \end{aligned} $$
(243)
Hence, we obtain by forming the Hesse operator
$$\displaystyle \begin{aligned} \mathbf{v} (x) \approx \mathbf{v_{\gamma_1}} (x) = ( \nabla_x \otimes \nabla_x ) V_{\gamma_1} (x) = \sum^{N_{\gamma_1}}_{k=1} b_k^{\gamma_1} \ {\mathbf{k}}_{\gamma_1} (x,y_k^{\gamma_1}), \ \ x\in \varGamma. \end{aligned} $$
(244)
We consider the “wavelet potential” \(W_{\gamma _0}\) given by
$$\displaystyle \begin{aligned} W_{\gamma_0} := V_{\gamma_1}|{}_{\overline{\mathbb{S}_{\gamma_0}^{2, \mathrm{ext}}}} - V_{\gamma_0}. \end{aligned} $$
(245)
Because of (243), \( W_{\gamma _0}\) allows the discretization in the form
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} W_{\gamma_0} (x) &\displaystyle = &\displaystyle \sum^{N_{\gamma_1}}_{k=1} b_k^{\gamma_1} K_{\gamma_1} (x,y_k^{\gamma_1}) - \sum^{N_{\gamma_0}}_{k=1} a_k^{\gamma_0} K_{\gamma_0} (x,y_k^{\gamma_0})\\ &\displaystyle =&\displaystyle \sum^{N_{\gamma_1}}_{k={N_{\gamma_0}+1}} b_k^{\gamma_1} K_{\gamma_1} (x,y_k^{\gamma_1})\\ &\displaystyle \quad &\displaystyle + \sum^{N_{\gamma_0}}_{k=1} a_k^{\gamma_0}\ \left( K_{\gamma_1} (x,y_k^{\gamma_1}) - K_{\gamma_0} (x,y_k^{\gamma_0}) \right), \ \ \ x \in \overline{\mathbb{S}_{\gamma_0}^{2, \mathrm{ext}}}. \end{array} \end{aligned} $$
(246)
We use the canonical notation \( {\mathbf {w}}_{\gamma _0} := (\nabla \otimes \nabla )W_{\gamma _0}.\) Then, in connection with (246), the SGG-equations
$$\displaystyle \begin{aligned} {\mathbf{w}}_{\gamma_0} (x_l^{\gamma_1}) = {\underbrace{{\mathbf{v}}_{\gamma_1} (x_l^{\gamma_1})}_{\approx \mathbf{v}(x_l^{\gamma_1})}} -{\mathbf{v}}_{\gamma_0} (x_l^{\gamma_1}), \ \ x_l^{\gamma_1} \in \varGamma, \ l = L_{\gamma_0}+1, \ldots, L_{\gamma_1}, \end{aligned} $$
(247)
lead to the linear system
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} {\mathbf{w}}_{\gamma_0} (x_l^{\gamma_1}) &\displaystyle = &\displaystyle \sum^{N_{\gamma_1}}_{k={N_{\gamma_0}+1}} b_k^{\gamma_1}\ {\mathbf{k}}_{\gamma_1} (x_l^{\gamma_1},y_k^{\gamma_1})\\ &\displaystyle \quad &\displaystyle + \sum^{N_{\gamma_0}}_{k=1} a_k^{\gamma_0}\ \left( {\mathbf{k}}_{\gamma_1} (x_l^{\gamma_1},y_k^{\gamma_1}) - {\mathbf{k}}_{\gamma_0} (x_l^{\gamma_1},y_k^{\gamma_0}) \right) \end{array} \end{aligned} $$
(248)
in the unknowns \(b_k^{\gamma _1}\), \( k={N_{\gamma _0}+1}, \ldots , N_{\gamma _1}.\) Equivalently, we obtain
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \sum^{N_{\gamma_1}}_{k={N_{\gamma_0}+1}} b_k^{\gamma_1}\ {\mathbf{k}}_{\gamma_1} (x_l^{\gamma_1},y_k^{\gamma_1}) \approx \mathbf{v} (x_l^{\gamma_1}) - \sum^{N_{\gamma_0}}_{k=1} a_k^{\gamma_0}\ {\mathbf{k}}_{\gamma_1} (x_l^{\gamma_1},y_k^{\gamma_1}), \end{array} \end{aligned} $$
(249)
\( l = L_{\gamma _0}+1, \ldots , L_{\gamma _1},\) where \(a_k^{\gamma _0},\, k = 1, \ldots , N_{\gamma _0}\), are known (note that (i) the points \(x_l^{\gamma _1} \in \varGamma , \ l = L_{\gamma _0}+1, \ldots , L_{\gamma _1},\) should be chosen in an appropriate way to guarantee an improvement provided by the bandpass potential \(W_{\gamma _0}\) of scale 0 to the whole multiscale approach and (ii) the choice \(y_k^{\gamma _1} = y_k^{\gamma _0},\,\, k= 1, \ldots , N_{\gamma _0}\), is not excluded).

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 \({\overline {\mathbb {S}_{\gamma _1}^{2, \mathrm {ext}}}}\).

“Downward continuation” yields the “1-scale” potential \( V_{\gamma _1}^R\) in \(\mathbb {S}_R^{2, \mathrm {ext}}\) of the form
$$\displaystyle \begin{aligned} V_{\gamma_1}^R(x) := \sum^{N_R}_{k=1} w_k^R \ V_{\gamma_1} \left(\gamma_1 \frac{ y_k^R}{R} \right) K_R(x,y_k^R),\quad x \in \mathbb{S}_R^{2, \mathrm{ext}}, \; y_k^R \in \mathbb{S}_R^2, \end{aligned} $$
(250)
so that the restriction \( V_{\gamma _1}^R\vert _\varSigma \) of the potential \( V_{\gamma _1}^R\) represents the “1-scale” SGG-lowpass solution on the (real) Earth’s surface Σ (as pointed out earlier, for reasons of comparability, we use the same cubature formula for the “downward continued” 1-scale” SGG-lowpass solution \(V_{\gamma _1}^R \) as for the “downward continued” 0-scale” SGG-lowpass solution \(V_{\gamma _0}^R\) on \(\mathbb {S}_R^{2, \mathrm {ext}}\)).

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 \(V_{\gamma _{n+1}}^R\):

The potential \(V|{ }_{\overline {\mathbb {S}_{\gamma _{n+1}}^{2, \mathrm {ext}}}}\) admits the representation
$$\displaystyle \begin{aligned} V(x) = V|{}_{\overline{\mathbb{S}_{\gamma_{n+1}}^{2, \mathrm{ext}}}}(x) = \int_{\mathbb{S}_{\gamma_{n+1}}^2} {V} (y) \ K_{\gamma_{n+1}}(x ,y) \ dS(y), \quad x \in \overline{\mathbb{S}_{\gamma_{n+1}}^{2, \mathrm{ext}}}, \end{aligned} $$
(251)
Therefore, we obtain by applying the Hesse operator
$$\displaystyle \begin{aligned} \mathbf{v} (x) \approx \mathbf{v_{\gamma_{n+1}}} (x) = ( \nabla_x \otimes \nabla_x ) V_{\gamma_{n+1}} (x) = \sum^{N_{\gamma_1}}_{k=1} b_k^{\gamma_{n+1}} \ {\mathbf{k}}_{\gamma_{n+1}} (x,y_k^{\gamma_{n+1}}), \ \ x\in \varGamma. \end{aligned} $$
(252)
By approximate integration over the sphere \(\mathbb {S}_{\gamma _{n+1}}^2 \) we are able to deduce a cubature formula \( V_{\gamma _{n+1}}\) of the integral representation (251) of \(V|{ }_{\overline {\mathbb {S}_{\gamma _{n+1}}^{2, \mathrm {ext}}}}\) in the form
$$\displaystyle \begin{aligned} V_{\gamma_{n+1}} (x) = \sum^{N_{\gamma_{n+1}}}_{k=1} b_k^{\gamma_{n+1}}\ K_{\gamma_{n+1}} (x,y_k^{\gamma_{n+1}}), \ \ \ x \in \overline{\mathbb{S}_{\gamma_{n+1}}^{2, \mathrm{ext}}}, \end{aligned} $$
(253)
with \( N_{\gamma _{n+1}}>N_{\gamma _n}\) and given nodes \(y_k ^{\gamma _{n+1}} \in \mathbb {S}_{\gamma _{n+1}}^2\), \(k=1,\ldots , N_{\gamma _{n+1}},\) such that
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} b_k^{\gamma_{n+1}} = b_k^{\gamma_n}, \quad k= 1, \ldots, N_{\gamma_{n}}. \end{array} \end{aligned} $$
(254)
We consider the “wavelet potential” \(W_{\gamma _n}\) given by
$$\displaystyle \begin{aligned} W_{\gamma_{n}} := V_{\gamma_{n+1}}|{}_{\overline{\mathbb{S}_{\gamma_n}^{2, \mathrm{ext}}}}- V_{\gamma_n}. \end{aligned} $$
(255)
Because of (254), \( W_{\gamma _{n}}\) allows the discretization in the form
$$\displaystyle \begin{aligned} \begin{array}{rcl} W_{\gamma_{n}} (x) &\displaystyle = &\displaystyle \sum^{N_{\gamma_{n+1}}}_{k=1} b_k^{\gamma_{n+1}} K_{\gamma_{n+1}} (x,y_k^{\gamma_{n+1}}) - \sum^{N_{\gamma_{n}}}_{k=1} b_k^{\gamma_{n}} K_{\gamma_{n}} (x,y_k^{\gamma_{n}})\\ &\displaystyle =&\displaystyle \sum^{N_{\gamma_{n+1}}}_{k={N_{\gamma_{n}}+1}} b_k^{\gamma_{n+1}} K_{\gamma_{n+1}} (x,y_k^{\gamma_{n+1}})\\ &\displaystyle &\displaystyle + \sum^{N_{\gamma_{n}}}_{k=1} b_k^{\gamma_{n}}\ \left( K_{\gamma_{n+1}} (x,y_k^{\gamma_{n+1}}) - K_{\gamma_{n}} (x,y_k^{\gamma_{n}}) \right), \ \ \ x \in \overline{\mathbb{S}_{\gamma_{n}}^{2, \mathrm{ext}}},{} \end{array} \end{aligned} $$
(256)
where the coefficients \(b_k^{\gamma _{n}}, k = 1, \ldots , N_{\gamma _{n}},\) are known. By definition we set
$$\displaystyle \begin{aligned} \begin{array}{rcl}{\mathbf{w}}_{\gamma_{n}} := (\nabla \otimes \nabla)W_{\gamma_{n}}.\end{array} \end{aligned} $$
(257)
Then, in connection with (256), the SGG-equations
$$\displaystyle \begin{aligned} {\mathbf{w}}_{\gamma_{n}} (x_l^{\gamma_{n+1}}) \approx \mathbf{v} (x_l^{\gamma_{n+1}}) -{\mathbf{v}}_{\gamma_{n}} (x_l^{\gamma_{n+1}}), \ \ x_l^{\gamma_{n+1}} \in \varGamma, \ l = L_{\gamma_{n}}+1, \ldots, L_{\gamma_{n+1}}, \end{aligned} $$
(258)
lead to the linear system
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} {\mathbf{w}}_{\gamma_{n}} (x_l^{\gamma_{n+1}}) &\displaystyle =&\displaystyle {\mathbf{v}}_{\gamma_{n+1}} (x_l^{\gamma_{n+1}}) -{\mathbf{v}}_{\gamma_n} (x_l^{\gamma_{n+1}}) \\ &\displaystyle = &\displaystyle \sum^{N_{\gamma_{n+1}}}_{k={N_{\gamma_{n}}+1}} b_k^{\gamma_{n+1}}\ {\mathbf{k}}_{\gamma_{n+1}} (x_l^{\gamma_{n+1}},y_k^{\gamma_{n+1}})\\ &\displaystyle ~ &\displaystyle + \sum^{N_{\gamma_{n}}}_{k=1} b_k^{\gamma_{n}}\ \left( {\mathbf{k}}_{\gamma_{n+1}} (x_l^{\gamma_{n+1}},y_k^{\gamma_{n+1}}) - {\mathbf{k}}_{\gamma_{n}} (x_l^{\gamma_{n+1}},y_k^{\gamma_n}) \right) \end{array} \end{aligned} $$
(259)
in the unknowns \(b_k^{\gamma _{n+1}}, k={N_{\gamma _n}+1}, \ldots , N_{\gamma _{n+1}},\) i.e.,
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle {\sum^{N_{\gamma_{n+1}}}_{k={N_{\gamma_0}+1}} b_k^{\gamma_{n+1}}\ {\mathbf{k}}_{\gamma_{n+1}} ((x_l^{\gamma_1},y_k^{\gamma_{n+1}})}\\ &\displaystyle ~&\displaystyle \qquad \quad \approx \mathbf{v} (x_l^{\gamma_{n+1}}) - \sum^{N_{\gamma_n}}_{k=1} b_k^{\gamma_n}\ {\mathbf{k}}_{\gamma_{n+1}} (x_l^{\gamma_{n+1}},y_k^{\gamma_{n+1}}), \end{array} \end{aligned} $$
(260)
\( l = L_{\gamma _n}+1, \ldots , L_{\gamma _{n+1}}\) (again, \(x_l^{\gamma _{n+1}} \in \varGamma , \ l = L_{\gamma _n}+1, \ldots , L_{\gamma _{n+1}}\) should be chosen in an appropriate way to guarantee an improvement by the bandpass potential \(W_{\gamma _n}\)). Once the coefficients \(b_k^{\gamma _{n+1}}, k={N_{\gamma _n}+1}, \ldots , N_{\gamma _{n+1}}\) are calculated from (260), the potential \(V_{\gamma _{n+1}}\) is known from (242). The potential \(V_{\gamma _{n+1}}\) may be regarded as the sum of the lowpass potential \(V_{\gamma _n}\) and the bandpass potential \(W_{\gamma _n}\) of scale n in \(\overline {\mathbb {S}_{\gamma _n}^{2, \mathrm {ext}}}\), and \(V_{\gamma _{n+1}}\) represents the “(n + 1)-scale” SGG-lowpass solution in \({\overline {\mathbb {S}_{\gamma _{n+1}}^{2, \mathrm {ext}}}}\).
“Downward continuation” leads to the “(n + 1)-scale” potential \( V_{\gamma _{n+1}}^R\) in \(\mathbb {S}_R^{2, \mathrm {ext}}\) of the form
$$\displaystyle \begin{aligned} V_{\gamma_{n+1}}^R(x) := \sum^{N_R}_{k=1} w_k^R \ V_{\gamma_{n+1}} \left(\gamma_{n+1} \frac{ y_k^R}{R} \right) K_R(x,y_k^R),\quad x \in \mathbb{S}_R^{2, \mathrm{ext}},y_k^R \in \mathbb{S}_R^2, \end{aligned} $$
(261)
so that the restriction \( V_{\gamma _{n+1}}^R\vert _\varSigma \) of \( V_{\gamma _{n+1}}^R\) represents the “(n + 1)-scale” SGG-lowpass solution on the (real) Earth’s surface Σ.

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.

However, there is also no doubt that each method in approximation theory and numerics has its own aim and character. In fact, it is the essence of any numerical realization that it becomes optimal only with respect to especially specified criteria. For example, Fourier expansion methods with polynomial trial functions (such as spherical harmonics) offer the canonical “trend–approximation” of low-frequency phenomena (for global modeling), they provide an excellent control and comparison of spectral properties of the signal, since any spherical harmonic relates to one “frequency” (i.e., the degree of the polynomial). This is of tremendous advantage for relating data types under spectral aspects. Nonetheless, it is at the expense that the polynomials are globally supported such that local modeling usually results into serious problems of economy, efficiency, and stability. Bandlimited kernels can be used for the transition from long-wavelength to short-wavelength phenomena (global to local modeling) in the signal. Because of their excellent localization properties in the space domain, the non-bandlimited kernels can be used for the modeling of short-wavelength phenomena. Local modeling is effective and economic because of a decorrelation ability, but it must be emphasized that the information obtainable by kernel approximations such as wavelets is clustered in frequency bands so that spectral investigations are much more laborious and time consuming. All in all, for the numerical work to be done in constructive approximation, we have to make an a priori choice which feature should be pointed out and preferably handled. In particular, we have to reflect the different stages of space/frequency localization so that the modeling process can be performed under the localization requirements necessary and sufficient for relevant geodetic interpretation. A positive decision in one direction often amounts to a negative implication in an other direction. Ultimately, because of the uncertainty principle, it is impossible to provide a “cure-all methodology”. This “sine qua non” ingredient in any mathematical approximation method is also the reason, why we are indispensibly led to two essential calamities in the “step by step” SGG-approximation by scaling and wavelet potentials as proposed here:
  • 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.

In conclusion, a loss of information in SGG-modeling is unavoidable. An algorithm establishing an approximate solution for the inverse SGG-problem has to reflect as good as possible the intention of the geoscientific applicant. Geodetic a priori information today for the characterization of the right scaling and wavelet potentials for the SGG-solution process as well as a mathematically validated stopping strategy in multiscale regularization is an important challenge for future work.

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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Copyright information

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

Authors and Affiliations

  • Willi Freeden
    • 1
    Email author
  • Helga Nutz
    • 2
  • Reiner Rummel
    • 3
  • Michael Schreiner
    • 4
  1. 1.Department of MathematicsTU KaiserslauternKaiserslauternGermany
  2. 2.CBM – Gesellschaft für ConsultingBusiness und Management mbHBexbachGermany
  3. 3.Astronomical and Physical GeodesyTU MunichMunichGermany
  4. 4.Institute for Computational EngineeringUniversity of Applied Sciences of Technology NTBBuchsSwitzerland

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

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