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.
frame of the ”RegGRAV I and II“ projects4, where the logo was introduced.
The following sections introduce the observable input functionals, but now, on the one hand from the physical perspective, and on the other hand from the modeling perspective. As mentioned, the gravitational potentialV is the fundamental quantity. Hence, the arrangement of the derived functionals follows the application of more and more field transformation operators, which corresponds to increasing spectral power in high frequencies, referring to Fig. 2.9. The selection of functionals can be supplemented by further (output) quantities, as e. g.
equivalent water heights (EWH), which are not part of this study. The corresponding measurement systems are described in Sec. 3; the observation equations of the modeling approach are presented in Tab. 4.7.
According to Fig. 2.9, gravitational potential differences∆V (Sec. 2.5.1) and gravity gradientsVab, a,b ∈ {x,y,z}, (Sec. 2.5.7) are functionalsY[ ˜V] =Y[V] of the gravitational potential; geoid heightsN(Sec. 2.5.2), quasigeoid heightsζ(Sec. 2.5.3), gravity disturbancesδg(Sec. 2.5.4), gravity anomalies∆g(Sec. 2.5.5), and deflections of the verticalη, ξ(Sec. 2.5.6) relate to the disturbing potential, i. e.Y[ ˜V] =Y[T], after reducing V by a normal potentialU.
For functionalsY[T] are further distinguished: The approximation of the gravitational by a normal potential allows to determine the differences between two equipotential surfaces, a spheropotential surface with poten-tialU and a geopotential surface of the Earth’s gravity field, e. g. the geoid with potentialW0. It is known as Stokes theory. As in most cases, the Earth’s surface does not coincide with the geoid, another theory, theMolodensky theory, has established. It considers the gravitational functionals as differences between the Earth’s surface and the telluroid. Both theories are needed in this work, as different observation data relate as well to the theories of Stokes and of Molodensky. They are briefly mentioned when describing the single functionals. For more details see e. g. Hofmann-Wellenhof and Moritz(2005); a comprehensive overview is given e. g. bySánchez(2015).
2.5.1 Gravitational potential difference
Modeling the gravitational potentialV of the Earth as accurately and highly resolving as possible, would in theory require the direct measurement ofVoutside all attracting masses. As mentioned before, no observation technique is able to realize this. However, the satellite mission GRACE, consisting of two spacecrafts, delivers gravitational potential differencesY[ ˜V] =∆V, i. e.
∆V =∆V (xi,xii) =V (xi)−V (xii), (2.61)
obtained from highly accurate distance measurements between the satellites (i),(ii) at positions P(xi) and P(xii). Several approaches have been established and realized in order to derive ∆V from the distance measurements, as e. g. the acceleration or the energy balance approach. Details are explained in Sec. 3.1.5.
Expressing Y[ ˜V] = ∆V in terms of spherical relations according to (2.56), only the scaling factor f0 is identical forV (xi) and V (xii). The summation parts fl(r )and fl,m(λ, ϑ) distinguish due to the different positionsP(xi)andP(xii)of the two satellites. Comparing the sensitivity to high frequencies with the one of other gravitational quantities, it is relatively low and can be arranged referring to ˜Vr within the Meissl scheme 2.7.
2.5.2 Geoid undulation
Equipotential surfaces of the Earth’s gravity field are suitable for height reference systems. The geoid for instance, defined as geopotential surface W = W0 according to Eq. (2.28), coincides with the undisturbed, continuously extended mean sea level of the oceans over the whole globe. The approximation of the geoid by an ellipsoid with normal potentialUand spheropotential surfaces defined in Eq. (2.43), is implicitly realized by the low-degree terms of global gravity field models. A selection is presented in Sec. 3.2.
4Projects funded by the Centre for Geoinformation of the German Armed Forces (ZGeoBw), Euskirchen, Germany: Generation of a software application for producing high precise regional gravity models as a height reference surface.
According to Fig. 2.10, the metric difference between a reference ellipsoid (with potentialU0) and the geoid (with potentialW0) can be described by thegeoid heightorgeoid undulationN. It is the distance between a certain point P0of the geoid and the – along the ellipsoidal normal n′ – projected point Q0 onto the normal potential surface with the same potentialU=U0 =W0 (Hofmann-Wellenhof and Moritz, 2005, p. 91). TheBruns formula
N = T
γ (2.62)
(e. g. Hofmann-Wellenhof and Moritz, 2005, p. 93) relates this geometric distance N to the physical quantityT via the normal gravity operator1/γ, cf. Tab. 2.2.
Figure 2.10: Geoid undulation N following Stokes theory.
Following the Stokes theory, assuming that there are no masses outside the geoid, the disturbing potential T =T (P0)from Eq. (2.54) is computed at the geoid pointP0, while the normal gravityγ =γ0(Q0)according to Eq. (2.48), is computed at the ellipsoidal pointQ0. The mathematical relations between the geometrical and physical quantities are given in Appendix A. ModelingT (P0)in terms of SHs, the spherical relationsY[ ˜V]
=V from Tab. 2.3, and the spherical approximations∂/∂r ≈ ∂/∂n′from cf. Eq. (2.59) apply in Eq. (2.54).
Arranging geoid undulationsY[ ˜V] = N in the order of the Meissl scheme (Fig. 2.7) at the Earth’s surface, this functional would be located along with ˜V.
2.5.3 Quasigeoid undulation
As mentioned in Sec. 2.3.5, the disturbing potential satisfies the Laplace equation∇2T =0 outside attracting masses. However the geoid, as described in Sec. 2.5.2, generally does not coincide with the Earth’s surface:
especially on land, the topography causes masses above or below the geoid. Consequently, the determination of geoid heights from Eq. (2.62) following Stokes theory, requires the reduction of measured gravity quantities at the Earth’s surface down to the geoid. One method applied in this work is presented in Sec. 2.7.
Following the Molodensky theory, another option is the introduction ofquasigeoid heights orquasigeoid undulations
ζ = T
γ , (2.63)
applying Bruns formula according to Eq. (2.62), but now considering a pointPat the Earth’s surface. Pro-jecting Pfrom the Earth’s surface along the ellipsoid normaln′onto a surface with potentialU=UQ =WP, results in the geometric distanceζ = PQ, see Fig. 2.11.
Physically, this quasigeoid undulation between the geopotential surface withW = WP through a point P at the Earth’s surface and the spheroidal surface with U = UQ = WP is defined analogously to N. The quantities in Eq. (2.63) are computed at geopotential surface forT =T (P), and at the spheroidal surface for γQ=γQ(Q), see Appendix A.
Figure 2.11: Quasigeoid undulation following Molodensky w.r.t. the ellipsoid and analog w.r.t. the telluroid. Dashed lines denote non-equipotential surfaces.
Geometrically, the projections of surface pointsP1,P2, ...with different potential valuesW1,W2, ...onto points Q1,Q2, ... with the same normal potential valuesW1 = U1,W2 =U2, ... describe a new surface, denoted as telluroidΣ, see Fig. 2.11. It is an artificial point-wise defined surface. Counting the metric distancesζ vice versa, w.r.t. the ellipsoid (potentialU = U0), results in the surface called quasigeoid. Over the ocean, it is practically identical with the geoid (e. g.Hofmann-Wellenhof and Moritz, 2005, p. 299), due to the absence of topographic masses.
ForY[ ˜V] =ζ, the same spherical approximations and field transformations account as forN, discussed in the previous section. Hence, the spectral power of quasigeoid undulations in high frequencies and the arrangement in the Meissl scheme (Fig. 2.7) corresponds to the ones of geoid undulations.
2.5.4 Gravity disturbance
Physically, the gravity disturbance δg describes the difference
δg=gP−γP (2.64)
between the magnitude gP of the gravity vector gP along the plumb linenand the magnitudeγPof the nor-mal gravity vectorγPalong the ellipsoidal normaln′, in one and the same pointPat the Earth’s surface follow-ing Molodensky, see Fig. 2.12. The gravity disturbance vector δg is oriented alongn′ and expresses the gra-dient ofT: δg = [∂T/∂x,∂T/∂y,∂T/∂z]T =gradT. For modeling purposes, its magnitude (2.64) is expressed by
δg=−∂T
∂h′ . (2.65)
Figure 2.12: Gravity disturbance following Molodensky w.r.t. the Earth’s surface.
Note, within the physical relation according to Eq. (2.64) the direction of γ = γP is oriented inwards the ellipsoid, while in Eq. (2.65) the ellipsoidal height h′ is counted positively outwards. According to the Molodensky theory, the disturbing potential and the normal gravity are computed at the Earth’s surface with T =T (P),γ = γP(P). In the spherical approximation according to Eq. (2.59) the gravity disturbance from Eq. (2.65) becomes
δg=−∂T
∂r . (2.66)
and the spherical relationsY[ ˜V]= ∂V˜/∂r = ∂T/∂r from Tab. 2.3 can be applied, neglecting the difference of the directions ofr andn′.
Following the Meissl scheme (Fig. 2.7), gravity disturbancesδgcontain more power in high frequencies than TorN, as they relate to the first derivative ofT. For instance, computing geoid undulationsNfrom observed gravity disturbances δgat the Earth’s surface consequently smooths the high-frequent signal contents. With the advent of the Global Navigation Satellite System (GNSS) measurements, the determination ofδgbecomes more and more prominent: GNSS delivers directly ellipsoidal coordinatesλ, β,hof a pointP, which enables the computation of the normal gravityγP =γh(P)according to Eq. (2.50) of this pointPat ellipsoidal height h′.
2.5.5 Gravity anomaly
In contrast to gravity disturbancesδg, the difference of the gravity vectorgPat a pointPat the Earth’s surface with potentialWPand the vectorγQat a pointQof a spheropotential surfaceU =UQ, see Fig. 2.13, is defined as gravity anomaly vector∆g= gP−γQ. The corresponding magnitude∆gof the vector is denotedgravity anomaly
∆g=gP−γQ (2.67) in the sense of Molodensky. It can be determined eas-ily by measuring the magnitudeg = g(P) of the total gravity point-wise at the Earth’s surface, and reducing it by appropriate normal gravity values γ = γQ(Q), computed at the telluroid pointsQby Eq. (2.48). Fol-lowing Stokes, the gravity anomaly in a point P0 on the geoidW0is defined as differenceg(P0)−γ0(Q0).
Vice versa to the projection of a point P0 onto a pointQ0, the normal gravityγ0(Q0) can be continued from the reference ellipsoid to the geoid with increas-ing height h′ in order to express the according value γ(P0) =γ0(Q0)+∂γ/∂h′N.
Figure 2.13: Gravity anomaly following Molodensky w.r.t.
the Earth’s surface, and following Stokes w.r.t. the geoid.
Extending the relation from Eq. (2.65) by an up-/ downward continuation operator (depending on the surface of the normal potential, if it is below or above the geoid), the gravity anomaly typically is expressed by
∆g=−∂T
∂h′ + ∂γ
∂h′N . (2.68)
With Bruns formula according to Eq. (2.62) and the spherical approximation∂T/∂h ≈∂T/∂r (cf. Sec. 2.4.2), the gravity anomaly yields
∆g=−∂T
∂r − 2
RT (2.69)
and the relationY[ ˜V] = ∂V˜/∂r from Tab. 2.3 applies. The signal content in the high frequencies of ∆g is equivalent to that ofδgand can be related to other gravitational quantities by the Meissl scheme (Fig. 2.7).
The connection of gravity anomalies ∆g and geoid undulations N is given by the Stokes operator; the connection to gravity disturbanceδgby the fundamental equation of physical geodesy, which is for instance used as boundary condition for the first boundary-value problem according to Eq. (2.41). The relations are described in Appendix A. However, they are not applied in this work, as all quantities are directly derived from the (differential) gravitational potential ˜V.
2.5.6 Deflection of the vertical
As mentioned before, the gravity anomaly vector∆gis defined by the difference between the gravity vector gP pointing along the plumb line n at a point P at the Earth’s surface, and the normal gravity vector γQ oriented along the ellipsoidal normaln′in the projected pointQon the telluroid, following Molodensky. The normal gravity vectorγP in pointP, as it is used for describing gravity disturbance vectorsδg, points in the same direction asγQ.
The difference in direction of both vectors∆gandδg, i. e. of the according unit vectorsnandn′, is displayed in Fig. 2.14 and denoted asdeflection of the vertical.
The magnitude is expressed by a north-south (ξ) and an east-west (η) component. Hence, physically they express the direction of g, i. e. of the gradient gradW of the potential fieldW, in longitude (north-south) and latitude (east-west) direction with
ξ=− 1 γR
∂T
∂ϕ
η=− 1
γRcosϕ
∂T
∂λ , (2.70)
(Hofmann-Wellenhof and Moritz, 2005, p. 117).
Figure 2.14: Deflection of the vertical in north-south (ξ) and east-west (η) direction, with n′throughQfollowing Molodensky, and withn′throughPfollowing Helmert.
Geometrically, the quantitiesξ,ηdescribe the angle between the direction of the plumb line and the ellipsoidal normaln′in a pointP, expressed by astronomical (Λ,Φ) and ellipsoidal (λ,β) coordinates, i. e.
ξ =Φ− β ,
η =(Λ−λ)cosβ . (2.71)
Following Molodensky, the direction ofn′is defined throughQat the telluroid. Consequently,T =T (Q)and γ=γQ(Q)have to be computed in Eq. (2.70) w.r.t. the according spheropotential surface (potentialU =UQ, cf. Fig. 2.14). Another definition established for practical applications (Torge, 2003, p. 199): Helmert defined n′throughPat the Earth’s surface. Hence, the curvature of the plumb line between telluroid and the Earth’s surface (order of magnitude of a few 0.11′′) is neglected in favor of an easy determination of the direction of n′.T =T (P)andγ=γP(P)relate to the according spheropotential surface throughP(potentialU=UP).
The quantities Y[ ˜V] = {η, ξ} are referred to the first derivatives of the disturbing potentialT after ϕ and λ. In modeling approaches, the Eqs. (2.70) are described by the spherical approximations ∂T/∂ϕ and ∂T/∂λ
from Tab. 2.3. η, ξ cannot be directly arranged in the Meissl scheme in Fig. 2.7: according to Fig. 2.3, the functionals do not depend on the radial derivative, but on the operators fl,m(λ, ϑ).
2.5.7 Gravity gradients
While the three components of the gravity vectorgdescribe the change of gravity potentialW along the three axes of a Cartesian coordinate system,gravity gradientsY[ ˜V] =(Vab),a,b ∈ {x,y,z}quantify the change of gravity (in the three-dimensional spaceR3), i. e.
∆V =(Vab) = ∂2V
∂a∂b
!
, (2.72)
w.r.t. a Cartesian coordinate system. Physically they describe a Marussi tensor ∆V, referring to Antonio Marussi (1908-1984), of nine elements
∆V =(Vab) =
Vx x Vx y Vx z Vy x Vy y Vy z
Vz x Vz y Vz z
, (2.73)
containing all second-order derivatives ofV. The gravity gradients are typically given in the unit Eötvös (1 E
= 1×10−9s−2), named after the Hungarian physicist Loránd Eötvös (1848-1919).
FollowingRummel and van Gelderen(1992), the two most important characteristics of the Gravity Gradient (GG) tensor are: The gravitational potentialV is harmonic (∇2V = 0) and irrotational (∇ × ∇V =0) outside the attracting masses. Consequently, the Laplace equation (2.25) can be written in Cartesian coordinates as Vx x+Vy y+Vz z =0, i. e. the trace of the gravity tensor (2.73) is zero, and each two of the diagonal elements are linear dependent. Further, the tensor is symmetric with(Vab)= (Vba)so thatVx y =Vy x,Vx z =Vz x,Vy z =Vz y. In total, five independent components remain.
FollowingKoop(e. g. 1993, p. 32), the elements can be expressed w.r.t. spherical coordinatesλ,ϑ,r, i. e.
Vx x = 1 r
∂V
∂r + 1 r2
∂2V
∂ϑ2, (2.74)
Vx y = 1 r2sinϑ
∂2V
∂ϑ∂λ − cosϑ r2sin2ϑ
∂V
∂λ, (2.75)
Vx z = 1 r2
∂V
∂ϑ −1 r
∂2V
∂r∂ϑ, (2.76)
Vy y = 1 r
∂V
∂r + 1
r2tanϑ
∂V
∂ϑ + 1
r2sin2ϑ
∂2V
∂λ2, (2.77)
Vy z = 1 r2sinϑ
∂V
∂λ − 1
rsinϑ
∂V2
∂r∂λ, (2.78)
Vz z = ∂2V
∂r2 . (2.79)
Modeling the components (Vab) in terms of SHs, the first- and second-order derivatives of ˜V=Vfrom Tab. 2.3 are inserted in the Eqs. (2.74) - (2.79). In spherical approximation it yieldsVz z ≈ Vr r. According to the Meissl scheme (Fig. 2.7), the amplification of high frequencies is exemplarily demonstrated for the component Vr r. Although spaceborne observations, observed in a certain height r > R, contain less information at small wavelengths, the effect of downward continuation (vertically arranged operator to the power ofl +3 in Fig. 2.7) is partly compensated for the second derivatives by the multiplication of several horizontally arranged operators, as e. g. the gravity anomaly operatorgl(r ). Consequently, those differentiation operators counteract the attenuation operator such that gravity gradients are especially profitable for the detection of high frequencies (Rummel et al., 2002).
In analogy to the Meissl scheme in Fig. 2.7 referring to field transformations w.r.t. spherical coordinates, cf.
Tab. 2.3, a second scheme has been developed in Fig. 2.8 w.r.t. Cartesian coordinates, cf. Rummel and van Gelderen(1995). Herein, exemplarily the relation of thezz-derivative, Eq. (2.79), is derived according to the spectral power in high frequencies.