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1 % *UDYLW\)LHOG7XWRULDO The formulae and derivations in the following Chapters 1 to 3 are based on Heiskanen and Moritz (1967) and Lambeck (1990). ([SDQVLRQRIWKHJUDYLWDWLRQDOSRWHQWLDOLQWRVSKHULFDOKDUPRQLFV The stationary part of the (DUWK VJUDYLWDWLRQDOSRWHQWLDO8DWDQ\SRLQW3UMO on and above the Earth s surface is expressed on a global scale conveniently by summing up over degree and order of a spherical harmonic expansion. The spherical harmonic (or Stokes ) coefficients represent in the spectral domain the global structure and irregularities of the geopotential field or, more generally spoken, of the gravity field of the Earth. The equation relating the spatial and spectral domain of the geopotential is as follows: Uϕ U U + ( FRVP + 6 VLQP) VLQϕ where Uϕ  spherical geocentric coordinates of computation point (radius, latitude, longitude)  reference length (mean semimajor axis of Earth)  gravitational constant times mass of Earth OP  degree, order of spherical harmonic 3  fully normalized Lengendre functions 6  Stokes'coefficients (fully normalized) (1.1) The 00 term is close to 1 and scales the value. The degree 1 spherical harmonic coefficients 6 are related to the geocentre coordinates and zero if the coordinate systems origin coincides with the geocentre. The coefficients connected to the mean rotational pole position that is a function of time. 6 are Subtracting from the lowdegree zonal coefficients (order 0) the corresponding Stokes coefficients of an HOOLSVRLGDOQRUPDOSRWHQWLDO 9UM leads to the mathematical representations of the GLVWXUELQJSRWHQWLDO7UMOin spherical harmonics, related to a conventional ellipsoid of revolution that approximates the Earth s parameters. At the Earth surface with U (in spherical approximation) the disturbing potential reads: 7 ϕ 8 ϕ 9 ϕ (1.2a) ( FRV P + 6 VLQ P)! " 7 ϕ 3 VLQ + ϕ (1.2b) with #$ $ and T defined on the geoid. Note, that '' is close to zero. 1
2 The maximum degree O(*) + of the expansion in Equation (1.1) correlates to the spatial resolution at the Earth surface by (*,  NPO(.) + (1.3) where (*,  is the minimum wavelength (or twice the pixel side length) of gravity field features that are resolved by the O(/) + O(.)0+ O(/) + 1 parameters Equation (1.1) contains the upwardcontinuation of the gravitational potential at the Earth s surface for U > and reflects the attenuation of the signal with altitude through the factor (U) 4 6. Figure 1.1 gives examples for the three different kinds of spherical harmonics 37 8 VLQϕ FRV P : (a) zonal with O P (b) tesseral with O P Oand (c) sectorial harmonic with O P Amplitudes and phase of the individual spherical harmonics then are determined by multiplication with the 9 : and 6 ; < coefficients. zonal: l6, m0 tesseral: l16, m9 sectorial: l9, m9 )LJXUH ([DPSOHV IRU VSKHULFDO KDUPRQLFV 3 OP VLQϕ FRV P >IURP ± EOXH WR )XQFWLRQDOVRIWKHGLVWXUELQJJUDYLWDWLRQDOSRWHQWLDO The JHRLG XQGXODWLRQ 1 (Figure 2.1) is the distance between the special equipotential surface U(ϕ) const that is close to the mean sea level and the surface of the conventional ellipsoid of revolution. As such the geoid is derived from the disturbing potential 7 applying %UXQVIRUPXOD 7 1 (2.1) γ where γ is 'normal'gravity on the surface of the ellipsoid. With γ 1 in spherical approximation, the geoid undulations (or geoid heights) can be computed from the spherical harmonic coefficients in Equation (1.2) by 1 ϕ 7 ϕ (2.2a) 2
3 _ ] K V H [ > > Q B F F Y Y? J > O ^ A V Z Y \ G F I P O R W W Y H ] Q [ ( >? FRV P + 6 VLQ P) C D E AA 1 ϕ 3>? VLQ >? + ϕ (2.2b) 7 The negative of the vertical derivative of the disturbing potential δ J is called U JUDYLW\ GLVWXUEDQFH GJ (Figure 2.2) that is equal to gravity at a point 3 (negative of vertical derivative of 8) minus normal gravity at point 3 (negative of vertical derivative of 9). On the geoid and in spherical approximation U the gravity disturbance then is expressed by ( F G FRV P + 6F G VLQ P) L M N II δ J ϕ O 3F G + + VLQϕ (2.3) The difference between gravity at a point 3 on the geoid and normal gravity at the corresponding point 4 on the ellipsoid is called JUDYLW\ DQRPDO\ 'J (Figure 2.3) and related to the disturbing potential by 7 J 7 (2.4) U U On the geoid this becomes (note: no degree 1 terms appear in Equation 2.) ( O P FRV P + 6O P VLQ P) O O P PTSU RR J ϕ + O 3 VLQϕ (2.a) thus J δj 7 (2.b) The second derivatives of the disturbing potential leads to the JUDYLW\JUDGLHQW WHQVRU. The most important vertical gradient represented as J 7 of the tensor component can be U J U \\ + ` a b O + O + U + 3Y Z VLQϕ ( Y Z FRV P + 6Y Z VLQ P) (2.6) Once the spherical harmonic coefficients c d 6c d of a global gravity field model are given, the quantities of the various functionals described above can be computed in its geographical distribution. If computed in terms of gravity disturbances or anomalies and gravity gradients, the higher frequency regional to local content is emphasised through the degreedependent factors OO and OO respectively, whereas the potential 3
4 )LJXUH *HRLG XQGXODWLRQV 1 UHVROXWLRQ NP UPV 1 FRV ϕ P )LJXUH *UDYLW\ GLVWXUEDQFHV δj UHVROXWLRQ NP UPV δj )LJXUH *UDYLW\ DQRPDOLHV J UHVROXWLRQ NP UPV J FRV ϕ PJDO FRV ϕ PJDO 4
5 and geoid representations of the gravity field show the broad and generalized features of the gravity field. Vice versa, a gradiometer measuring gravity gradients is capable to better resolve detailed structures of the gravity field rather than the long wavelength part. The fully normalized spherical harmonic coefficients in Equation (1.1) are related to the mass distribution within the Earth by O + e f e 0 g* h ikj l U 3e f VLQ ϕ FRV PG0 (2.7a) O + 6 m n m 0 o* p qkr s U 3m n VLQ ϕ VLQ PG0 (2.7b) e m with the mass element G0 G0 U ϕ Figure 2.4 depicts the geopotential distribution of gravity anomalies over Europe derived from spherical harmonic coefficients complete to Ot.u v equal to 10, 0, 100, 300, respectively, in order to demonstrate the relation between spectral and spatial resolution according to Equation (1.1). )LJXUH *HRJUDSKLFDO GLVWULEXWLRQ RI JUDYLW\ DQRPDOLHV RYHU (XURSH ZLWK GLIIHUHQW VSHFWUDO Owyx z DQG VSDWLDO UHVROXWLRQ SL[HO VL]H w { }
6 ƒ Š Ž Œ ˆ 7KHSRZHUVSHFWUXPRIWKH(DUWK VJUDYLW\ILHOG Given the fully normalized Stokes coefficients ~ 6~ of a specific degree O over orders m P O the VLJQDO GHJUHH DPSOLWXGHV V (or square root of power per degree O) of functions of the disturbing potential 7ϕ at the Earth s surface are readily computed by ƒ ƒ σ ƒ + 6 in terms of unitless coefficients (3.1a) σ 7 σ in terms of disturbing potential values (m 2 /s 2 ) (3.1b) σ ˆ 1 σ in terms of geoid heights (m) (3.1c) σ δj O + σ in terms of gravity disturbances (m/s 2 ) (3.1d) σ J O σ in terms of gravity anomalies (m/s 2 ) (3.1e) σ J O + O + σ in terms of vertical gravity gradients (1/s 2 ) (3.1f) where the 6 are related to the normal potential. The SI units of the physical gravitational quantities are given in parenthesis. Following Kaula s 'rule of thumb' (Kaula, 1966) the power law follows approximately σ 2O + 1. (3.2) 4 O Examples for signal degree amplitudes are given in Figure 3.1. If the estimation errors of the Stokes coefficients in a global gravity field model are known, the HUURUGHJUHHDPSOLWXGHV (error spectrum) are computed accordingly replacing the coefficients in Equation (3.1) by their standard deviations. 'LIIHUHQFH GHJUHH DPSOLWXGHV, representing the agreement of two different gravity field models per degree, are readily computed replacing the coefficients in Equation (3.1) by the coefficients differences between the two models. Examples for difference degree amplitudes are given in Figure
7 )LJXUH 6LJQDO GHJUHH DPSOLWXGHV IRU JHRLG XQGXODWLRQV UHG JUDYLW\ GLVWXUEDQFHV EOXH DQG JUDYLW\ DQRPDOLHV JUHHQ LQ PHWHU DQG PJDO UHVSHFWLYHO\ The GHJUHH DPSOLWXGHV DV D IXQFWLRQ RI PLQLPXP DQG PD[LPXP GHJUHH O displays the power (signal, error, difference) spectrum accumulated over a spectral band from O to O : σ DFFXPXODWHG σ (3.3) Usually O 0 or 2 is taken to display the increase in overall power with increasing degree O. Recall that the spectral degree O is related to the spatial extension or wavelength of features in the gravity field according to Equation (1.3). Examples for difference amplitudes as a function of maximum degree l (successive accumulation of the curves in Figure 3.1) are given in Figure 3.2. Equations (3.1) again demonstrate that the higher degree terms, i.e. the shorter wavelengths in the signal spectra, are enhanced by factors proportional to degree O for gravity anomalies and disturbances and proportional to O for gravity gradients compared to the signals in the geoid and gravitational potential. )LJXUH 'LIIHUHQFH GHJUHH DPSOLWXGHV *$( 6 YV ( LQ WHUPV RI JHRLG XQGXODWLRQV UHG JUDYLW\ GLVWXUEDQFHV EOXH DQG JUDYLW\ DQRPDOLHV JUHHQ LQ PHWHU DQG PJDO UHVSHFWLYHO\ 7
8 )LJXUH 'LIIHUHQFH GHJUHH DPSOLWXGHV *$(6 YV ( DV D IXQFWLRQ RI PD[LPXP GHJUHH LQ WHUPV RI JHRLG XQGXODWLRQV UHG JUDYLW\ GLVWXUEDQFHV EOXH DQG JUDYLW\ DQRPDOLHV JUHHQLQPHWHUDQGPJDOUHVSHFWLYHO\ HIHUHQFHV Heiskanen, W.A. and H. Moritz, Physical Geodesy, W.H. Freeman and Co., San Francisco. Kaula, W.M., Theory of Satellite Geodesy, Blaisdell Publ. Company, Waltham, Mass. Lambeck, K., Aristoteles An ESA Mission to Study the Earth s Gravity Field, ESA Journal 14:
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