Determination of gravity gradients from terrestrial gravity data for calibration and validation of gradiometric GOCE data M.Kem Institute of Navigation and Satellite Geodesy, TU Graz, Steyrergasse 30, 8010 Graz, Austria R. Haagmans Science and Applications Department, ESA/ESTEC, Keplerlaan 1, 2200 AG Noordwijk ZH, The Netherlands

Abstract. The satellite mission GOCE (Grav­ity field and steady-state ocean explorer) is the first gravity field mission of ESA's Living Planet Pro­gramme. Measurement principles are satellite-to-satellite tracking (SST) and, for the first time, satel­lite gravity gradiometry (SGG). To meet the mission goal of a 1-2 cm geoid at a spatial resolution of about 100 km, the satellite instruments will be calibrated in pre-flight mode and prior to the measurement phases (in-flight mode). Moreover, external calibration and validation of the measurements is performed using gravity information over well-surveyed areas.

In this paper, all components of the gravity tensor are determined from terrestrial gravity data. Integral formulas based on the extended Stokes and Hotine formulas are used. It is shown that the entire tensor can be computed with an accuracy of 1.5-2.5 mE in the local North-East-Up coordinate system. In addi­tion, the efi'ect of white noise and a bias in the terres­trial data is studied.

Keywords. Calibration/Validation • GOCE mission • Gradiometry • Upward continuation

1 Introduction

The GOCE mission is the first satellite mission with a gradiometer on board. The gradiometer consists of six three-axis accelerometers mounted in pairs along three orthogonal arms. The accelerometer readings allow for the determination of the common-mode and differential-mode signals, which are used to de­rive the gravity gradients. The highest precision is achieved in the measurement bandwidth (MBW) be­tween 5 and lOOmHz (Cesare 2002).

The observations will be burdened with (system­atic and stochastic) errors and to meet the mission goal, the measurements have to be calibrated and val­idated. Besides internal calibrations (pre-flight and in-orbit), a number of external calibration and val­idation concepts for SGG measurements have been

suggested: Existing gravity field models are used in Visser et al (2000), Bouman et al (2004). Terres­trial gravity data over well-surveyed areas are pro­posed in Arabelos and Tscheming (1998), Haag­mans et al. (2002), Pail (2002), Miiller et al (2004). Finally, cross-over techniques for (internal) calibra­tion/validation are presented in Miiller et al. (2004).

The upward continuation of terrestrial gravity data using least-squares collocation has been investigated in Arabelos and Tscheming (1998) and Bouman et al. (2004). Denker (2002) uses integral formulas for the upward continuation and transformation and finds re­sults for the radial component at the level of a few mE (1E== 10~^s~^). First results for all components of the gravity gradients based on Stokes's formula are shown in Reed (1973). In this paper. Reed's results are extended by deriving the kernel functions in the spectral domain. Also, the derivations are done for Hotine's formula. Note that covariance expressions for least-squares collocation are given in Tscheming (1993).

2 Extended Stokes and Hotine formula

The following derivations are based on the extended Stokes and Hotine formulas. The extended Stokes formula is given as

n

where T is the disturbing potential, /\g are grav­ity anomalies and R is the radius of the reference sphere. The extended (Pizzetti-) Stokes function S =r S(%l), r) is (Heiskanen and Moritz 1967)

2 / l-tx+D

= E 2 + l j + i

1=2 - 1 t'^'Pdx

with

t = R/r and D=%/\-2tx +1^

(2)

(3)

95

/ is the degree and x = cos '0 is the cosine of the spherical distance between the running point and the computation point (Heiskanen and Moritz 1967)

X = sin 99 sin 99'+cos 9? cos (/?'cos (A' - A) (4)

r, 99, A are the geocentric radius, spherical latitude and longitude and H stands for the pair of angular spherical coordinates (v?, A). Pi is the /^^-degree Leg-endre polynomial. The extended Hotine formula for gravity disturbances 6g can be written as

nr, ^) = -^l^ H{ij, r)5g{r, W) dQ.' (5)

where the extended Hotine function H (Picketal. 1973, eq. 1572)

i f (^ , r IS

H 2t f V

^=0

(6)

and V = D + t — X. Note the difference in the de­nominator / + 1 and / — 1 in Eq. (2) and (6) and the summation start I = 2 and / = 0, respectively. Band-limited kernel functions can easily be derived by truncating the spectral kernel functions. Also, appropriate weighting functions may then be intro­duced (see e.g. Haagmans et al. 2002).

3 Derivatives of the extended Stokes and Hotine formula

Gravity gradients at satellite altitude can be derived using the derivatives of the extended Stokes and Ho­tine formulas. The first derivatives of the disturb­ing potential with respect to the spherical coordinates can be transformed to the local cartesian system NEU (x, y, z; X pointing north, y pointing east and z point­ing radially outwards) as follows (e.g. Tscheming 1976)

T, =

Ty =

T. =

IT

r cos 95 •Tx (7)

Note that T^ stands for dT/dx and so on. Similarly, the second derivatives are given as (Koop and Stelp-

stra 1989)

•T .2 fP

xy

r^ • • r

SUKf (pX r^ cos^ (f

Tx

n. =

yy

r ' cos 9?

IT - I T ^-^^^ ^2-^'f

1 _ tan(^ 1 r"^ COS^ if rp/. -^ rp

(8)

T, y2 r cos If Tx,

r^ cos (f Tx

T.

Eq. (8) assembles the tensor T^^^. The computation ofTy^x requires the determination of T ^ ^ and T^. For the other components, other derivatives are needed. Hence, partial derivatives of the extended Stokes and Hotine integral with respect to r, (p and A have to be determined. The first partial derivatives of the ex­tended Stokes function are given as (Ecker 1969)

R

n

(9)

R rr „ di)

Analogously, the second derivatives are given as (Reed 1973)

R_

47r ^^ Srr^g'dQ.'

Txx

= TzJJP^^l — ) +V

R_

47r

s n

s

' * * > dip J

+S-

^"^^d^ipdX^^^difdXl

Ag'dn'

Ag'dn'

Ag'dft'

TrX —

47r JJ n

If R_

ATT

^rip o

^rtp

dip

dx

Ag'dQ'

Ag'dW (10)

where Ag{r,Vt') is abbreviated with Ag'. For Ho-tine's function, Eqs. (9) and (10) contain gravity disturbances 5g. Partial derivatives of the spherical distance with respect to the geocentric latitude and longitude can be derived using Eq. (4) and formulas

96

from spherical trigonometry, see e.g Heiskanen and Moritz(1967,pg. 113):

[mGal]

dcpdX

cos a 'dX = — cos cp sm a

= sin^ a' cot tp

COS(/?

sin-^ ^cos(/?'cos(A' — A)

— cos (f sin^ a' cos ip]

sin a' (sin ^ — cos (^ cos a ' cot t/;)

where a is the azimuth as defined in Heiskanen and Moritz (1967). After some operations, the deriva­tives of the closed Stokes kernel with respect to r and ip can be derived, see Table 1 (Pi^i and Pi^2 are unnormalized Legendre ftmctions for order 1 and 2, respectively). Some of them can already be found in Reed (1973) and Witte (1970). The formulas have been checked successfully against the spectral ker­nel ftmctions by calculating the kernel ftmction for various r and x. Analogously, the kernel functions for the extended Hotine formula are listed in Table 2. They may be used for the upward continuation and transformation of airborne gravity disturbances.

Gravity gradients can be obtained using the fol­lowing steps:

i) Remove long-wavelength components from the local data using a global geopotential model.

ii) Compute first (Eq. (9)) and second derivatives (Eq. (10)) of the disturbing potential for the area where data are available (near zone). Deter­mine estimates of the far-zone contribution as de­scribed in Thalhammer (1994).

in) Assemble the gravity tensor T^^^ in the NEU system using Eq. (8).

iv) Restore the long-wavelength components.

v) If necessary, rotate the tensor into a different co­ordinate system T = RT^^^R^, where J^ is a rotation matrix.

4 Numerical results

The following numerical tests provide insights into the numerical accuracy of the integral formulas. All tests are performed in a closed-loop based on grav­ity data from the global geopotential model GPM98A (Wenzel 1998). Gravity anomalies and gravity dis­turbances, complete up to degree and order 1800, have been computed at the Bjerhammer sphere and

120

100

80

60

40

20

0

-20

-40

-60

-80

•100

•120 5° 10° 15° 20° 25°

Fig. 1 Gravity anomalies from GPM98A at the reference sphere [mGal]

the full tensor at 250 km height. A regular grid of 0.25° X 0.25° has been used. Note that this grid is coarse and some aliasing might occur. The test area is given as:

(70 : (^,A) e [38°, 54°] x [0°,28°: (11)

It has relatively large gravity variations (RMS = 29.7 mGal), see Fig. 1. Since the long-wavelengths are not well represented by the local data, the long-wavelength components (using GPM98A) up to de­gree and order I = 90 have been removed. After the estimation of the gradients at satellite height, the long-wavelengths are restored again. A spherical cap of 6° has been used. Results of this test are summa­rized in Table 3. Diftbrences of the estimated ten­sor components to the ones directly obtained from GPM98A are listed. The values in brackets are for the Hotine integration using gravity disturbances. Note that the differences are only computed for the output area ai:

0-1 : (< ,A) G [44°, 48°] x [6°, 22 (12)

All components can be computed with a numerical accuracy of about 1.3-2.5 mE (Std). Best results are obtained for the radial component. The larger errors on the other components could be due to some alias­ing. However, this has to be studied in more detail. The largest errors occur at the edges of the compu­tational area where missing data deteriorate the solu­tions.

The differences are shown graphically in Fig. 2. Due to the symmetry of the tensor, the components yx, zx and zy are equivalent to the components xy, yz and yz, respectively. Therefore, they are shown in grey.

As mentioned before, the long-wavelength grav­ity field is not well represented by the local data

97

Table 1 Extended Stokes kernel function and its derivatives. The summation of the series is taken from two up to infinity

Kernel Spectral form Closed kernel expression

5

^rip

^ipip

i: 2i + l , i + i (' + ! ) ( ' + 2) l - l

Pl(.x)

2 / 1 -tx + D \- 1 - SD - tx { 5 + 3ln

D V 2

R

1-t 4 / ^ 1-tx + D _ _ + _ + l _ 6 D - t a . f l 3 + 61n

•t^ sin ip 2 6 1-tx- D l-tx + D

i?2 <'-'"i^-^)-^-S— l-tx + D\ 24 4 ( 1 - t ^ ) !

. 3 t e ( 1 5 + 61n j + - + A _ _ ^

t^ sin ip

R 6tx{D + 1)

Mi^,A,l_,3-„.l^i|±£.

r>(i - ta: + r>) 2 6 6 ^ „ l - t c c - L > ^, 1-tx + D

-— + 8 - 3 o 31n D3 D Dsin^T/; 2

+ 4 + 3 i ? ^ + 3 . ( + ) 1-tx- D r 2x 1 "

3 „ • o . ( • o . + D sin^ -0 \t sin^ t/? D^

Table 2 Extended Hotine kernel function and its derivatives. The summation of the series is taken from zero up to infinity

Kernel Spectral form Closed kernel expression

H

Hr

Hip

Hrr

rLfip ^ ^ , . ( i ± i ) , , , . ,

?^-ln D \ l - x j

2R 2t^{x-t) t{x-t)

r'^D rD^ Drv

R

2t^ sin T/J t sin "ip sin -0 sin i/;

D^ Dv V 1 — X

2^2 t \ t ft 2

t(a; -t) r 8t Qt'^{x-t) 2 t{x - t) t{x - t) 2t + Dr2 \^~^ 5 4 " ' " ^ ^ D2^ + /)^2 ~ ^2

4^2 sin?/; 6t^ sin'0(x — t) tsinip t^ smip{x — t) j.£)3 jjbrp rDv rD'^v

t^ simlj{x — t) t'^sinip tsmip{x — t) tsinip

^j^2y2 rDv"^ rDv"^ rip"

—2t^x ^t^ SVC? Ip tx t"^ sir? ip t'^ sii? ip tsii? ip

D3 " D^ 'DV'^ D^V " D2z;2 " Dv^ v ^ 2 Z + l , , 1 , , r. / X r. / xN -2t2a: et^ sin' E T ^ ^ ( -co t^P , , i ( a . ) + P,,2(x)) - ^ ^ + — ^

07 t sin^ ?/; sin^ ip x sin^ -j/; 1 1. ^ z_ _| 1__ V Dv^ f 2 1 — X (1 — x)"^

and the contribution of a global gravity field is sub­tracted. After the estimation of the gravity tensor, the removed effects are restored. The previous tests involved long-wavelength information up to degree and order 90. For a comparison, a reference field of

degree and order 60 produces a RMS error of about 2.6 mE and a 360 field an error of about 0.7 mE for the radial component. Thus in this comparison the use of a higher reference field model improves the solution. In any case, the results are optimistic since

98

Table 3 Differences between Tij directly from GPM98A and estimates from terrestrial data [mE]. Values in brackets are from the extended Hotine formula

5 mE. Obviously, this also limits the selection of data areas as they have to be sufficiently large.

'i'j

XX

xy xz

yy yz zz

Min

-6.1 (-5.9) -5.0 (-4.7) -5.5 (-4.9) -8.2 (-7.9) -5.6 (-4.8) -3.4 (-3.6)

Max

8.7 (8.5) 4.3(4.1) 5.4 (4.5) 5.5(5.7) 4.4 (3.9) 3.8 (3.9)

Mean

0.0 (0.0) -0.2 (-0.2)

0.0 (0.0) 0.0 (0.0)

-0.1 (0.1) 0.0 (-0.0)

Std

2.5 (2.4) 1.9(1.8) 2.1 (1.8) 2.3(2.1) 1.8(1.6) 1.3(1.4)

0° 5° 10° 15° 20° 25° 0° 5° 10° 15° 20° 25° 0° 5° 10° 15° 20° 25°

[E]

Fig. 2 Gravity gradient differences (T^x - T||'^^^'', T^v -rpgpm98a rp rpgpm98a rp rpgpm98a rp _ rpgpm98a J-xy '>J-xz~-'-xz y-^yx -^yx •> ^yy •'-yy » rp rpgpm98a rp rpgpm98a rp rpgpm98a rp _ J-yz " J-yz y ^zx ~ ^zx •> ^ zy J-zy y J-zz

rjngpm98a ^^^ j ^ ^ ^ ^ ^^^^ ^^^ ^^ bottom) in [E], grey areas are equivalent components

the long-wavelengths are assumed to be errorless. Improved results may be possible by introduction of kernel modifications and/or the use of weighting functions. This is, however, outside the scope of this investigation.

In the previous tests a spherical cap of 6° was used, which requires a large data area such as Eu­rope, North America or Australia. The question arises if a smaller integration cap, in combination with a smaller data area, can produce similar results. Six test runs have been done using spherical caps of -jo _ go The differences to the values directly ob­tained from the GPM98A model are computed and summarized in Fig. 3. In all cases, the output area is cTi. Clearly, smaller spherical caps, such as 1° or 2°, produce large errors (up to 38 mE) and cannot be used for the problem at hand. A spherical cap of 3" is the minimum in order to keep the error well below

36

LU

£,24

Q^ 12

0 1 2 3 4 5 6 7 Spherical cap [deg]

Fig. 3 RMS differences using different spherical cap sizes for the radial component (extended Stokes function is used)

The last test involves errors in the input data. In particular, the effect of white noise and a regional bias is studied. Noise of 2 mGal standard deviation and a bias of 8 mGal on Austria is shown in Fig. 4. It is added on the input data and an integration with a spherical cap of 6° is performed. The results are shown in Fig. 5. Clearly, the bias directly af­fects the radial component Trr- It is also visible, yet with different sign and half of the magnitude, in the other diagonal components. Compared with the other components, the horizontal component Txy is less af­fected by the bias. It is worth noting that the effect of the noise (up to 2 mGal Std) is small due to the fact that the kernel functions have low-pass filter charac­teristics (large upward continuation height).

5 Summary and conclusions

Formulas have been presented for the estimation of the full gravity tensor at satellite height using gravity anomalies and gravity disturbances. They allow for an estimation of the gravity tensor in the North-East-

[mGal] ^

Fig. 4 Noise and bias [mGal]

99

0° 5° 10° 15° 20° 25° 0° 5° 10° 15° 20° 25° 0° 5° 10° 15° 20° 25°

Fig. 5 Gravity gradient differences [E], order as in Figure 2

Up coordinate system if a large data area is available. Closed-loop simulations have shown that best results are obtained with a spherical cap of 3* or larger. Fur­thermore, the use of a high-degree reference field (D/0 60 or higher) is advantageous. Then, the inte­gration error is between 1.3 mE and 2.5 mE, but ad­ditional tests using a finer grid spacing are necessary. A bias in the input data affects all components. In a relative sense, however, the off-diagonal components are less affected. The effect of (white) noise in the input gravity data is small since the kernel functions have low-pass filter characteristics.

The closed-loop simulation has demonstrated that the presented integral formulas can be used for the computation of gravity gradients at satellite height. Future refinements of the method are the introduc­tion of kernel modifications or weighting functions. Also, error propagation is necessary. The obtained reference gradients may then be used for calibra­tion and validation purposes of the satellite mission GOCE. The actual calibration and validation proce­dure is outside the scope of this paper and will be addressed in an upcoming contribution.

6 Acknowledgement

Financial support for the first author came from an external ESA fellowship. This is gratefully acknowl­edged.

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