Developing and adapting a regional modeling approach for the use of real data is one of the main objectives in this work. As the gravitational potential cannot be observed directly, different techniques established for
determining a variety of gravitational functionals. In order to combine these observations, field transforma-tions are indispensable.
In the sequel, ”horizontal“ and ”vertical“ transformations are distinguished, according to the so-called Meissl scheme in Fig. 2.7. For geodetic applications, both the horizontal transformations from one functional to another, as well as vertical transformations of a functional, observed or to be modeled in a certain altitude to another altitude, denoted ”up-/ downward continuation“, have to be considered. The field transformations apply two times in the modeling approach, see Fig. 2.9: (1) Different measurement systems observe various functionals of the Earth’s gravity field in different heights. The input quantities, i. e. observables as well as modeled quantities, have to be homogenized for their combination. The method, realized in a software called
”RegGRAV“ is presented in Sec. 5. (2) The aim is to compute any gravitational quantity in any height on or above the Earth’s surface. Consequently, field transformations are essential for computing the output model.
The different transformations can be applied to gravitational quantities, which are, for instance, expanded in a series of SHs according to Eq. (2.34), by using several operators. Referring toKeller (2003), most of the functional relations are described by four operators, listed in Tab. 2.2.
Table 2.2: Field transformation operators.
operator description mathematical solution
Xl+i =(R/r)l+i up-/ downward continuation operator (for the i-1thderivative afterr),
Poisson’s integral (e. g.Hofmann-Wellenhof and Moritz, 2005, pp. 28)
∂/∂r radial derivative operator (any parameter w.r.t.r),
Green’s representation theorem, (Keller, 2003)
gl(r )=r/l+i gravity anomaly operator (to apply to the(i−1)thderivative),
Poisson’s integral (e. g.Hofmann-Wellenhof and Moritz, 2005, pp. 99)
1/γ normal gravity operator Stokes (and Bruns) formula, (e. g.Hofmann-Wellenhof and Moritz, 2005, pp. 102).
The first three operators depend on the radial distancer. They are applied in the Meissl scheme, cf. Fig. 2.7.
The scheme is adapted in the following and expanded by further operators in order to derive a variety of gravitational quantities needed in this work. As mentioned above, the observed quantities relate either to the gravitational, or to the disturbing potential. In order to apply the Meissl scheme and the field transformations as flexibly as possible, an initial parameter is defined as (differential) gravitational potential ˜V. It summarizes
V˜ =V−U+Z =T disturbing potential,
V˜ =V−Vback gravitational potential reduced by the gravitational potential of a background model, V˜ =V−0 full gravitational potential.
Functionals derived from the (differential) gravitational potential are in the following denoted asY[ ˜V]. The most relevant operator is hereby the (partial) derivative∂/∂rw.r.t. a certain radial distancer.
2.4.1 Meissl scheme
Rummel and van Gelderen(1995) adapted the Meissl scheme (Meissl, 1971a,b) for coupling gravitational functionals depending on the radial distancer. It is appropriate for functionals related toT, stemming from near-Earth observations, as e. g. from terrestrial, air- and shipborne gravimetry. However, in this work, satellite data are used as well and they deliver information related to the gravitational potentialV. Thus, a more general scheme is developed for the functionalsY[ ˜V], see Fig. 2.7, based on the principles ofRummel and van Gelderen(1995): The transitions (arrows) can be interpreted as eigenvalues ¯λl, i. e. integral relations in the spatial domain. The SH coefficientsFl,mare the appropriate eigenfunctions.
... w.r.t. radial derivatives
Using the operators from Tab. 2.2, depending on radial distancer and degreel, the construction in Fig. 2.7 connects all vertical transformations of the initial (differential) gravitational potential ˜V by the up-/downward
continuation operator Xl+i=(R/r)l+i, and all horizontal transformations by the normal derivative (∂/∂r), and the gravity anomaly (gl(r ) = r/l+i) operators. Starting exemplarily from ˜Vr, the (differential) gravitational potential in any height (radial distancer > R), or from ˜V at the surface of a sphereΩR with constant radius R, e. g. approximately the Earth’s surface (r = R), all quantitiesY[ ˜V] related to the first derivatives∂V˜/∂r,
∂V˜r/∂r, can be obtained. Examples for ˜V = T are gravity anomalies or gravity disturbances. Pursuing in horizontal direction, all quantities related to the second derivatives ∂2V˜/∂r2, ∂2V˜r/∂r2, can be received (e. g.
vertical gravity gradients). The transformations based on the spherical approximation of ˜V =V, Eq. (2.40), or ˜V =T, Eq. (2.54), thus only depend on degreeland radial distancer.
The normal gravity operator1/γfurther connects the disturbing potentialT with (quasi)geoid undulations N (ζ), according to Bruns formula, see Eq. (2.62). Naturally this transformation is only applied at the Earth’s surface, computing the normal gravityγ =γ0from Eq. (2.48) referring to a rotational ellipsoid with h= 0.
(It is neglected in Fig. 2.7.)
The black arrows in Fig. 2.7 symbolize decreasing spectral power in the high frequencies, i. e. a smoothing of the quantities as well as of their noise in case of observables – here, in total, green displayed: from gravity gradients ∂2V˜/∂r2at the Earth’s surface to (differential) gravitational potential ˜Vr in heighth. AfterRummel and van Gelderen(1995), the Meissl scheme can be interpreted as follows: On the one hand, it provides a kind of ”tool box“ in modeling approaches for deriving gravitational functionals from ˜V. On the other hand, it allows to propagate the signal and noise behavior of different observed gravity quantitiesY[ ˜V] and their relationships. At the Earth’s surface for instance, gravity anomalies (related to ∂V˜/∂r) have more power in the high frequencies than the disturbing potential (or geoid undulations derived from ˜V = T). The higher the derivatives w.r.t. r of a functional, the finer are the gravitational structures which can be determined by appropriate observation techniques, taking into account the measurement altitude.
By applying the scheme the other way round, inverse transformations amplify both signal and noise in the upper part of the frequency spectrum (red arrow). This is important for the coupling of functionals in different heights: All satellite measurements have to deal with increasing errors in downward continuation applications. However, e. g. for the gravity gradient∂2V˜r/∂r2, the effect is partly compensated by the sequence of horizontal operators, making satellite gradiometry a very attractive measurement technique for globally gaining high-frequent gravitational information.
Figure 2.7: Modified Meissl scheme after Rummel and van Gelderen(1995), representing field transformations based on the operators fl(r )of radial-depending quantities related to the (differential) gravitational potential ˜V, observed or to be determined in any heighth>0 above or at the Earth’s surface (h=0). The black arrows symbolize the smoothing behavior of both signal and noise (in case of observables) in high frequencies.
... w.r.t. Cartesian xyz-derivatives
The Meissl scheme originally relates gravitational functionals depending on the radial derivatives ofT. It is further extended in this work by derivatives w.r.t. Cartesian xyz-coordinates, see Fig. 2.8, as the functional tensor of gravity gradients typically refers to those.3 Hereby, the direction of the z-axis is assumed to point in radial direction. Consequently, for the first and second derivatives ∂V˜/∂z, ∂2V˜/∂z2 at the Earth’s surface, respectively∂V˜r/∂z,∂2V˜r/∂z2above the surface, the same horizontal and vertical field transformation operators yield as for radial derivatives. For derivatives w.r.t. x, y, as well as for mixed derivatives and their linear combinations, the horizontal operators consist of several terms, cf. Fig. 2.8. An analogy to Fig. 2.7, the black arrows indicate smoothing behavior in the high frequencies.
Figure 2.8: Modified Meissl scheme afterRummel and van Gelderen(1995), representing field transformations between (combined) xyz-elements of the gravity tensor related to the (differential) gravitational potential ˜V, and compared with the first- and second-order derivatives w.r.t.r(bottom line) according to Fig. 2.7. Horizontal operators represent the singular values. The green color indicates smoothing behavior in high frequencies from lower right to upper left, the red color amplification vice versa.
2.4.2 Spherical derivatives of the (differential) gravitational potential in terms of SHs
The gravitational potentialV is the main quantity of the Earth’s gravity field; any gravitational functional Y[ ˜V] can be derived from it or from the disturbing potentialT. In this section, a variety of derived quantities is expressed in terms of SHs to give a general overview. The basis functions then are replaced by SBFs in the sequel of this thesis.
The common series expansion in terms of SHs for describing ˜V=Vwas given in Eq. (2.40), ˜V=Tin Eq. (2.54) respectively. As ˜V depends on the spherical coordinatesλ, ϑ,r, the series expansion can be split into several parts (e. g.Hofmann-Wellenhof and Moritz, 2005, pp. 32): a scaling factor f0=G M/R, a first summation part fl(r ) depending on the radial distancer, and a second summation part fl,m(λ, ϑ) depending on spherical longitudeλand co-latitudeϑ. For an arbitrary functionalY[ ˜V] it yields
Y[ ˜V]= f0
X∞ l=0
fl(r ) Xl
m=0
F˜l,mfl,m(λ, ϑ), (2.56)
by introducing (difference) SH coefficients ˜Fl,m = Fl,m,∆Fl,m according to Eqs. (2.41), (2.55). As men-tioned above, the summation parts can be represented by diverse basis functions, i. e. for example by a linear
3The adaption to (mixed) derivatives w.r.t.λ, ϑis neglected, because according quantities play a minor role.
combination of solid SHsHl,m(λ, ϑ,r ), cf. Eq. (2.35), or later by a linear combination of SBFs.
The first summation part fl(r ) contains the eigenvalues referring to the Meissl scheme 2.7. It expresses field transformations in terms of the gravity anomaly operatorgl(r )=gl, forr = R, and the up-/ downward continuation operatorXl+ifrom Tab. 2.2, each depending on degreeland the latter additionally on the(i−1)th derivative w.r.t.r, i. e.
fl(r )=gl·X (r )l+i . (2.57)
Choosing SHs as basis functions, the second summation part fl,m(λ, ϑ)represents the λ, ϑ-depending terms of surface SHsHl,mR , according to Eq. (2.36). This summation part is further split into the longitude-depending sine fs,m and cosine fc,m functions, and the co-latitude depending Legendre function, cf. Eq. (2.32), or its derivatives, cf. Eqs. (A.1), (A.2) in Appendix A, collected in fP,l,m, by
fl,m(λ, ϑ) =
fc,m(λ) fP,l,m(ϑ)
fs,m(λ) fP,l,m(ϑ) . (2.58)
Table 2.3 lists the (differential) gravitational potential ˜V, developed in terms of SHs, and all first and second derivatives w.r.t. r, λ, ϑaccording to the splitting of Eqs. (2.56) – (2.58). Such a decomposition of the series expansion of a harmonic function, cf. Eq. (2.34), in its different dependencies allows an efficient and flexible implementation in software routines and a flexible replacement of various basis functions.
Multiplying the expressions from Tab. 2.3 with appropriate gravity field operators presented in context with the Meissl scheme enables to describe a large variety of gravitational functionals. This is shown in Sec. 2.5 in terms of SHs, and in Sec. 4.2.5 in terms of SBFs.
Table 2.3: Zero, first and second order derivatives of the (differential) gravitational potential ˜V in terms of SHs w.r.t. spherical coordinatesr, λ, ϑ, according to Eqs. (2.56) – (2.58).
Y[ ˜V] f0
fl(r ) fl,m(λ, ϑ)
gl Xl+i i fs,m(λ) fc,m(λ) fP,l,m(ϑ)
V˜ G MR 1 R
r
l+i
1 cos(mλ) sin(mλ) Pl,m(cosϑ)
∂V˜
∂r
G M
R −l+1R R
r
l+i
2 cos(mλ) sin(mλ) Pl,m(cosϑ)
∂V˜
∂λ G M
R 1 R
r
l+i
1 mcos(mλ) −msin(mλ) Pl,m(cosϑ)
∂V˜
∂ϑ G M
R 1 R
r
l+i
1 cos(mλ) sin(mλ) ∂Pl,m∂ϑ(cosϑ)
∂2V˜
∂r2
G M R
(l+1)(l+2) R2
R
r
l+i
3 cos(mλ) sin(mλ) Pl,m(cosϑ)
∂2V˜
∂r∂λ G M
R −l+1R R
r
l+i
2 mcos(mλ) −msin(mλ) Pl,m(cosϑ)
∂2V˜
∂r∂ϑ G M
R −l+1R R
r
l+i
2 cos(mλ) sin(mλ) ∂Pl,m∂ϑ(cosϑ)
∂2V˜
∂λ2
G M
R 1 R
r
l+i
1 −m2cos(mλ) −m2sin(mλ) Pl,m(cosϑ)
∂2V˜
∂ϑ ∂λ G M
R 1 R
r
l+i
1 mcos(mλ) −msin(mλ) ∂Pl,m∂ϑ(cosϑ)
∂2V˜
∂ϑ2
G M
R 1 R
r
l+i
1 cos(mλ) sin(mλ) ∂2Pl,m∂ϑ(cosϑ)2
Spherical approximations
Expressing gravitational functionals in terms of basis functions, as e. g. SHs or radial SBFs, related to spherical coordinates is very convenient from the modeling point of view. Due to the flattening of the Earth, the quantities would be better approximated by ellipsoidal functions. However, the computational effort is enormous compared withspherical approximations. The deviation between quantities, computed on a sphere
or on a reference ellipsoid which is treated as a sphere, is maximal in the order of magnitude of only 3×10−3 (Hofmann-Wellenhof and Moritz, 2005, p. 96). This corresponds to the order of magnitude of the flattening.
The deviations between ellipsoidal or spherical derivatives of the gravitational potential, which are needed for the computation of several functionals, are even smaller. Consequently, expanding series in terms of spherical functions suffices for many applications.
For the regional gravity field modeling approach presented in this thesis, the computations are related to a sphere with constant radiusr = R, and the functionals are projected in the end to ellipsoidal (or any other) coordinates. The following spherical approximations are applied, neglecting second (or higher) order terms (e. g.Hofmann-Wellenhof and Moritz, 2005, pp. 96,97):
∂
∂h = ∂
∂n′ ≈ ∂
∂r ≈ ∂
∂n, (2.59)
∂γ
∂h ≈ −2γ
R . (2.60)
Equation (2.59) approximates the normal derivative operator w.r.t. ellipsoidal heighth′by the radial derivative operator from Tab. 2.2. The normal vectorn on a sphere with magnitudenhas the same directionr as the radius vector r; and the normaln′ on an ellipsoid with the according vectorn′ defined in Eq. (2.18) is the direction of the ellipsoidal heighth′. The latter Eq. (2.60) describes the vertical gradient of the normal gravity γw.r.t. ellipsoidal heighth′, approximated by a term which only varies with the normal gravityγ. Depending on the application, either the spherical approximation suffices, or it is applied for upward continuation of gravity, i. e. forgravity reduction, discussed in Sec. 2.7.
Figure 2.9: Functionals and field transformations in the regional gravity modeling approach. The sensitivity to high frequencies is color-shaded according to Fig. 2.7.