GRAVITY GRADIENTS AND SPHERICAL HARMONICS - A NEED FOR DIFFERENT GOCE PRODUCTS?
J. Bouman and R. Koop
SRON National Institute for Space Research Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands
ABSTRACT
GOCE will deliver Level 2 products such as grids of geoid heights and gravity anomalies, along with a spherical harmonic model of the Earth’s gravity field. Furthermore, Level 1B calibrated gravity gradients will be available. The majority of these products may seem superfluous, as they are all linear functionals of one and the same quantity, the Earth’s gravitational potential. We will show why both observed gravity gradients and derived spherical harmonics might be useful. Their different information content will be discussed as well as the difficulties that occur when one product is to be transformed into another product.
1 INTRODUCTION
The main goal of the GOCE mission is to provide unique models of the Earth’s gravity field and of its equipotential surface, as represented by the geoid, on a global scale with high spatial resolution and to very high accuracy [5]. For this purpose, GOCE will be equipped with a GPS receiver for high-low satellite-to-satellite tracking (SST-hl) observations, and with a gradiometer for observation of the gravity gradients (SGG). The gradiometer consists of six 3-axes accelerometers mounted in pairs along three orthogonal arms. From the readings of each pair of accelerometers the so-called common mode (CM) and differential mode (DM) signals are derived. The DM observations are used to derive the gravity gradients.
The gradiometer is designed such as to give the highest achievable precision in the measurement bandwidth (MBW) ¢¡¤£¥£§¦¨¡ © © ¦¨¡
between 5 and 100 mHz. For the diagonal gravity gradients in the Gradiometer Reference Frame (GRF;
-axis on average in the velocity direction, the -axis approximately radially outward and the -axis complements the
¥§¤ ¤
right-handed frame) the errors range from 7 mE/ (1 E = s ) in the high frequency part of the MBW to 55 mE/ in the low frequency part [6].
The measurements will be contaminated with stochastic and systematic errors, see e.g. [4, 9]. For the GOCE gradiometer, systematic errors typically are due to instrument imperfections like misalignments of the accelerometers, scale factor mismatches etc. The CM and DM couplings, which are the result of such instrument imperfections, can be determined in
the on-ground calibration, using a test bench, to a relative accuracy level of ¥ ¤ ¤ ¥§ . A so-called internal calibration procedure has been proposed [5], by which the CM and DM couplings can be determined with an accuracy such that the gradients errors in the MBW stay below the required level. The values of the calibration parameters are measured by putting a known acceleration signal on the gradiometer in orbit using the thrusters (‘shaking’). After this procedure, the CM and DM read-outs of the gradiometer are corrected using the measured calibration parameters.
The gravity gradients are derived from the internally calibrated DM accelerations. The internal calibration, however, is not sensitive to all instrument imperfections, such as the read-out bias, and the accelerometer mispositioning. Therefore, in order to possibly correct for remaining errors after internal calibration (outside or inside the MBW), a third calibration step is proposed which is called external calibration (or ‘absolute’ calibration). It is performed during or after the mission and typically makes use of external gravity data. The calibrated GOCE gravity gradients in the GRF together with the SST measurements will be used in the Level 1 to Level 2 processing, that is, in the spherical harmonic analysis. The coefficients of a spherical harmonic series will be determined with a maximum degree and order of around 200 (or a resolution of approximately 100 km at the Earth’s surface). These coefficients may be used to compute geoid heights, gravity anomalies, etc., but also gravity gradients.
The motivation for this paper is to assess the usefulness of externally calibrated GOCE gravity gradients for Level 3 users such as geophysicists and oceanographers. To that end, we will first discuss the external calibration to get a feeling of the information content of the calibrated gravity gradients. Secondly, the spherical harmonic analysis is discussed briefly after which the calibrated gravity gradients and the spherical harmonics are compared. Finally, a few examples are given to illuminate the above comparison.
______Proc. Second International GOCE User Workshop “GOCE, The Geoid and Oceanography”, ESA-ESRIN, Frascati, Italy, 8-10 March 2004 (ESA SP-569, June 2004) 2 EXTERNAL CALIBRATION
Due to the mis-pointings, cross couplings, etc. the gravity gradient errors are coupled with the signal. For example, if the
¡!© © ¡
-axis of the gradiometer is not perpendicular to the § -plane, then part of the and signal is projected onto the ¡ £¥£ measurements, which is an error. Because the signals exhibit strong peaks at 0, 1 and 2 cpr (cycles per revolution),
the errors will have such characteristics as well. The GOCE satellite mainly rotates around the -axis. The rotational term
is removed from the gravity gradients as good as possible in the preprocessing, but additional errors may be introduced.
¡ © © ¡ £¥£ ¡
As a result, the errors will be smaller than the and errors, and will have a somewhat different characteristic.
¡ © © ¡¤£¥£ ¡
The errors exhibit peaks at 0-2 cpr, whereas the and errors show peaks at 0-4 cpr [3].
%$& %$&
# "
The GOCE observed gravity gradients " are related to the true gravity gradients as
:3> ? @ A B :3@EDGFA B
%$&+*-,/.10 %$&3254 2 $!298;:=< %$&125C %$&IH
')(
# " " "16§7
" (1)
: :
. 4 < ¦¨C
B
" "
where the scale factor , the bias , the trend 6 as well as the Fourier coefficients are the unknown calibration
,KJLM$ENPO $ O
parameters to be determined, and , is the time, is the mean orbital period. The gravity gradient errors
%$&
#
" R are described by the error matrix Q . Gravity gradients derived from a global model are related to the true gravity
gradients as
%$&+*-, %$&
')(
" "
R (2) R
with Q the corresponding error matrix. Note that, e.g., the global model omission error is neglected in (2). Since the #
systematic errors in " are at low frequencies and the global gravity field is well known at these frequencies, we used, in
,
R " previous studies, " in (1) and solved for scale factor, bias, trend and Fourier coefficients with satisfactory results [3, 7]. Here, however, the error of the global model is taken into account, see also [2].
Subtracting (2) from (1) gives a combined non-linear model. The least squares solution of the linearised model is
,VU WYX3Z W\[ WYX^Z`_aX1b
S T
M] (3)
_ b W
where X represents the difference observations and represents the linear relation between these observations and the
Zd,e %_ _)
X
T
# R
Q-c M] Q Q
calibration parameters . The observation error matrix Qac is composed of the matrices and , and .
b 25J;h
f g
The vector contains f GOCE observations and calibration gradients. The number of unknowns is with e.g. hi,kj
cpr. In general Qlc is full and the least squares solution can not be directly computed (there are for example half a
million observations in 30 days with 5 s sampling). We will assume, however, that Q)c is diagonal (no correlation between Z
observations). Then M] is diagonal with elements
%$onI12/ ¢. p ¦rq^, ¦tststs ¦ %$onIE
m m
R
# f (4) and we can compute the least squares solution, that is, calibration parameters can be estimated and the gravity gradients
can be calibrated. ¡!£¥£ As an example, consider Fig. 1 which shows the PSD of simulated errors before and after external calibration. In total 30 days of measurements were simulated with a sampling interval of 5 s (or 40 km along track). Calibration parameters as in Eq. (1) were estimated for calibration windows of 5 days. Obviously, the gravity gradient long wavelength error is reduced, but the errors at these wavelengths remain relatively large compared to the MBW where gradients are relatively good.
3 SPHERICAL HARMONIC ANALYSIS
A model of the Earth’s gravity field can be derived from the calibrated Level 1B gradients in combination with the SST-hl observations. This is called Level 1 - Level 2 analysis: coefficients of a spherical harmonic series are derived from GOCE
gravity gradients in the GRF and SST-hl observations. As an example, consider the expansion in a series of spherical
| M
harmonics of the 2nd radial derivative of the potential | (radial gravity gradient):
| |
¡ u+uv ¢wM¦¨.!¦+x;y,{z8 | }!~r % 2 % 25Jv 8 } |t ¢wM¦¨.
x
(5)
Error before calibration Calibration windows of 5 days 1 1 10 10
] 0 ] 0
1/2 10 1/2 10
−1 −1 10 10
−2 −2 10 10 Power [E/Hz Power [E/Hz
−3 −3 10 10 −7 −5 −3 −1 −7 −5 −3 −1 10 10 10 10 10 10 10 10
Frequency [Hz] Frequency [Hz] ¡ £¥£
Figure 1: errors before (left panel) and after (right panel) external calibration.
|
,/J
|
v
with model coefficients, is the maximum degree which determines the resolution (typically for GOCE), m
wM¦¨.!¦+x
is the order, are the coordinates of the evaluation point, is the Earth’s semi-major axis, and are the spherical ¡
harmonic base functions. Smoothing is caused by the upward continuation, with respect to the potential , whereas the
% 2 % 25Jv
2nd derivatives, gravity gradients, augment the higher degrees and orders (here ).
A (non-)linear observation model such as (5) can be established for all GOCE observations. The linearised model is
*,/W1¦ *-,
')(P (P
Q- (6)
W
with the SGG and SST observations, the spherical harmonic coeffiecients to be determined, the linearised model,
,/v§s
p and Q- the error matrix of the observations. Because of the satellite altitude and its orbit with an inclination of , a standard least-squares solution of (6) is unstable. The details of the Earth’s gravity field are damped at satellite altitude,
whereas there are no GOCE observations at the poles. It is therefore necessary to regularise the solution
, %WYX W923h WYX
S
u
M] M] M]
Q Q
(7)
h S with u the regularised solution, a non-negative number and a positive definite matrix.
The regularised solution is biased, that is,
M*, %WYX W523h 3h,
S
')(
u
M] M]
Q
(8)
,
unless . The larger is, the larger the bias. In practice, is determined by some regularisation parameter choice rule and the more severe the instability of the inverse problem is, the larger must be to obtain a stable solution. For GOCE, the unsurveyed polar caps have a severe negative effect [1], but this effect may be counteracted by filling the gaps with airborne and/or terrestrial gravity data. The effect of the downward continuation, however, remains and regularisation is necessary. As a consequence of the bias in the spherical harmonic coefficients, all gravity field quantities will be biased when they are predicted using these coefficients. For example, if gravity gradients, in the GOCE observations points, are
computed with the model coefficients, then these predicted observations are
,kWa %W W923h W ,/W
X X
S S
u
M] M] M]
Q Q
(9)
S
and therefore , that is, the predicted observations have less power than the original observations. The power difference may be small, but this has to be estimated once real observations become available. One of the advantages of a spherical harmonic model is that it is relatively easy to compute gravity gradients or other functionals of the gravitational potential. In addition, the errors at long wavelengths will be reduced compared to the GOCE measured gradients because SST data is included in a spherical harmonic model as well.
4 COMPARISON OF GOCE GRAVITY GRADIENTS AND SPHERICAL HARMONICS
Given the expected gravity gradient signal along the GOCE orbit and the expected measurement error, it is estimated that
,=J v the maximum degree and order of a GOCE-only spherical harmonic model is . Such a maximum degree corre- sponds to a resolution of approximately 100 km. GOCE gravity gradients, however, have a sampling of 1 s but they will
be low-pass filtered such that there is no signal below 5 s sampling. This corresponds to an along track resolution of 40
, v km (or ) given the satellite velocity of 8 km/s. Of course, if the gravity gradient signal would be above the mea- surement noise globally up to a resolution of 40 km, then it would make sense to choose a much higher maximum degree and order for the spherical harmonic model. This will not be the case, however, as the average numbers above indicate. Nevertheless, it may be that regionally the GOCE gravity gradients contain information down to 40 km resolution.
Even if one would solve for a spherical harmonic degree up to degree and order 500, then it could make sense to use the original GOCE gravity gradients. The higher the spherical harmonic degree of a GOCE model, the higher the resolution, and the more the instability of Eq. (6) becomes apparent. The downward continuation and the polar gaps become more and more visible for higher degrees. It is therefore necessary to regularise more and more for higher degrees, which means that the bias in the solution will become larger. Of course, it may be necessary to apply regularisation regionally using GOCE gravity gradients, but this regularisation can be adapted to the local situation which is not true for the global solution.
5 EXAMPLES
High Resolution Gravity Gradient Signal. The GOCE gravity gradients along the orbit directly provide three dimen-
sional information about the gravity field. Consider the along-track and cross-track gravity gradient signal near Indonesia,
O1£¥£ O!© © Fig. 2. A GOCE-like orbit was simulated and the and anomalous gravity gradients are shown (sampling interval is 5 s). The gradients were generated using the EGM96 model from degree and order 200 to 360 [8], which is part of the additional signal that is present in the GOCE gravity gradients compared to a GOCE derived gravity field model with a maximum degree of 200. Obviously, the along-track and cross-track gradients ‘see’ a different gravity field.
T L=200−360 1.5 T L=200−360 1.5 xx yy 20 20
15 1 15 1
10 10 0.5 0.5 5 5
0 0 0 0 latitude latitude −5 −5 −0.5 −0.5 −10 −10
−15 −1 −15 −1
−20 −20 100 110 120 130 140 150 100 110 120 130 140 150
longitude −1.5 longitude −1.5
O1£¥£ OM© © J v
Figure 2: Gravity gradient signal, in mE, along-track ( ) and cross-track ( ) for ¢¡ at satellite altitude.
O1£¥££¦+O!© © O! PSD’s of the anomalous gravity gradients and are shown in Fig. 3. A PSD was computed for each track
in the Indonesian area and these are plotted in one figure. The signal PSD’s show a sharp power decrease just before
,
¤)¥
¥§¤ g
Hz which in this case is the frequency that corresponds to the maximum degree, , of the global model
J ¥
that was used in the simulation. The minimum frequency of approximately ¥ Hz corresponds to 500 s or, given O^£¥£
the satellite velocity, 4000 km which is the maximum track length in the given region. The signal is very small, while
O!© © O! and are larger. Also shown is the error PSD (black solid line). This error PSD, however, is expected to be valid
for observed gradients with a sampling of 1 s. The observations will be low-pass filtered with a cut-off frequency of 0.1
Hz and we are allowed to average the observations down to 5 s. This will decrease the noise with a factor roughly. The corresponding error PSD is also shown. Apparently, the anomalous GOCE gravity gradient signal is above the noise for a few instances only. This is also to be expected because otherwise the maximum degree of a GOCE gravity field model could be much higher. Nevertheless, the example shows that there may be regions where it is of interest to look at
the gravity gradients themselves. ¡!£¥© ¦¨¡ ©
Frame Transformations. The accuracy of the off-diagonal components is expected to be a factor of thousand ¡!£¥ worse than that of the diagonal components, whereas the accuracy of is a factor of ten worse [6]. In general it is therefore true that a point-wise rotation from the GRF, in which the gravity gradients have been measured, to any other reference frame results in a degradation of the accuracy of all gravity gradients. The errors in the off-diagonal compo- nents are ‘projected’ onto the diagonal components by the frame transformation. Nevertheless, it may be of interest to have gravity gradients available in a north-west-up frame, for example. The latter is an Earth-fixed frame, whereas the PSD of T along tracks, L=200−360 PSD of T along tracks, L=200−360 PSD of T along tracks, L=200−360 xx yy zz −1 −1 −1 10 10 10
Error Error Error
Error/51/2 Error/51/2 Error/51/2 ] ] ] 1/2 1/2 1/2
−2 −2 −2 10 10 10 Power [E/Hz Power [E/Hz Power [E/Hz
−3 −3 −3 10 10 10 −3 −2 −1 −3 −2 −1 −3 −2 −1 10 10 10 10 10 10 10 10 10
Frequency [Hz] Frequency [Hz] Frequency [Hz]
J v Figure 3: PSD of gravity gradient signal and errors for each track in the region of Fig. 2 for ¢¡ at satellite altitude.
GRF is not. A point-wise rotation should therefore be avoided and luckily alternatives are available. One alternative is to use the GOCE spherical harmonic model to compute gravity gradients in any desired frame, with the advantages and disadvantages as described above. Another alternative is to look at gravity gradients in cross-overs, that is, where ascend- ing and descending tracks cross eachother. The actual measured gravity gradients along the ascending and descending track can be interpolated to the cross-over point. For the sake of simplicity, let’s assume that there is no height difference
between ascending and descending track in the cross-over. Furthermore, consider a rotation around the -axis only. Then
Wn ¦ n ¦
the gradients along the ascending track, , are related to the gradients along the descending track, , as
%<§&X %<§&WYn ¦ §n ¦,
(10) < with the angle between ascending and descending track. This angle may be derived once the orbit is known. After precise orbit determination, the orbit should be determined with high accuracy, and therefore errors in the angle are not
considered here. (If the angle is derived from the velocity vectors, then a rough computation shows that a velocity error ¤¨
of 1 cm/s leads to an error of ¥ rad.) From (10) we have §£¥£l,ª©^ %< ¦WY£¥££¦WY£¥© ¦WY© ©;s
(11)
§£¥£§¦< ¦WY£¥£ WY© © W-£¥© Because and are known with high accuracy, we can use (11) to determine with higher accuracy as
compared to the measurement accuracy. Specifically, we have
> ? @ @EDGF
@EDGF > ? @
WY£¥©-, %§£¥£ WY£¥£ < WY© © <§
J < <
(12)
£¥© WY£¥©
which gives for the error standard deviation m of
> ? @ @EDGF @EDGF > ? @
£¥©-, 2 <2 <§NJ <
m m¬«
(13)
, £¥£, © © £¥©
m m m
assuming that m . Thus is a function of the cross-over angle, which is shown in Fig. 4. For cross- ¡¤£¥©
over angles close to 0 or 90 degrees, the error in is much larger than that of the diagonal components. An angle of
£¥© £¥£ m
0.5 degrees, for example, leads to an error standard deviation m that is eighty times larger than . However, this is ¡M£¥© considerably smaller than the original measurement error which is a factor of thousand larger for .
So far we assumed that in a cross-over the GRF of the ascending track could be transformed to the GRF of the descending track by a rotation around the -axis only. This would imply a perfectly circular orbit and it also implies that the -axis is
aligned with the velocity direction of the satellite. The orbit, however, has a small eccentricity and the -axis of the GRF
¡M£¥ ¡ © varies with respect to the velocity direction up to ®g degrees. This means that also the accuracy of and/or may be improved, at least in cross-overs. As an alternative, least-squares collocation may be used to improve the accuracy of the off-diagonal components, see [10].
6 SUMMARY
The comparison between GOCE gravity gradients and spherical harmonics is summarised in Table 1. The calibrated gradients could be useful for regional applications when high resolution is required, whereas spherical harmonics are more accurate, especially at long wavelengths, with a somewhat reduced resolution. Hence, we think that the answer to the question “gravity gradients and spherical harmonics - a need for different GOCE products?” is “yes”. 1/2 abs((1+cos(a)4+sin(a)4)1/2/sin(a)/cos(a)) 50
40
30
20
10
0 0 0.5 1 1.5 2 2.5 3
a
£¥© £¥£a, © © <¯0 ¦+L¤H
m m m
Figure 4: Error standard deviation in cross-overs relative to as function of the cross-over angle .
Table 1: Spherical harmonics (SH) and gravity gradients (GG): advantages & disadvantages and applications. advantages disadvantages global applications regional applications SH easy to compute other limited resolution and orbit determination; height datums; geophysics; gravity functionals; error solution is biased; orig- height datums; geoid oceanography (grids and reduction by combining inal GG observations computations; etc. along track geoid heights, GG and SST cannot be reproduced etc.) GG high resolution; direct Level 1 - Level 2 analy- none (?) resolution down to 40 km: detailed information sis may be complicated; geophysics, oceanography, about the Earth’s gravity errors remain at long local geoid computations field in three dimensions wavelengths
References
[1] J. Bouman, Quality assessment of satellite-based global gravity field models, Publications on geodesy. New series no. 48, Netherlands Geodetic Commission, 2000. [2] J. Bouman and R. Koop, Calibration of GOCE SGG data combining terrestrial gravity data and global gravity field models, Gravity and Geoid 2002; 3rd Meeting of the IGGC (I.N. Tziavos, ed.), Ziti Editions, 2003, pp. 275–280. [3] J. Bouman, R. Koop, C.C. Tscherning, and P. Visser, Calibration of GOCE SGG data using high-low SST, terrestrial gravity data, and global gravity field models, Accepted for publication in Journal of Geodesy, 2004. [4] S. Cesare, Performance requirements and budgets for the gradiometric mission, Issue 2 GO-TN-AI-0027, Prelimi- nary Design Review, Alenia, 2002. [5] ESA, Gravity Field and Steady-State Ocean Circulation Mission, Reports for mission selection; the four candidate earth explorer core missions, 1999, ESA SP-1233(1). [6] ESA, GOCE high-level processing facility: statement of work, GO-SW-ESA-GS-0079, 2003. [7] R. Koop, J. Bouman, E. Schrama, and P. Visser, Calibration and error assessment of GOCE data, Vistas for Geodesy in the New Millenium (J. Ad´ am´ and K.-P. Schwarz, eds.), International Association of Geodesy Symposia, vol. 125, Springer, 2002, pp. 167–174. [8] F.G. Lemoine, S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, E.C. Pavlis, R.H. Rapp, and T.R. Olson, The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96, TP 1998-206861, NASA Goddard Space Flight Center, 1998. [9] SID, GOCE End to End Performance Analysis, Final Report ESTEC Contract no 12735/98/NL/GD, SID, 2000. [10] C.C. Tscherning, Testing frame transformation, gridding and filtering of GOCE gradiometer data by Least-Squares Collocation using simulated data, Submitted proceedings IAG General Assembly, Sapporo July 2003, 2003.