Presented at IUGG meeting 2003 (2003) #:1–6
Calibration and Validation of GOCE Gravity Gradients 3 J. Bouman 1 , R. Koop1, R. Haagmans2 , J. Muller ¨ , N. Sneeuw4 , C.C. Tscherning5 , and P. Visser6 1
SRON National Institute for Space Research, Utrecht, The Netherlands European Space Agency, ESTEC, Noordwijk, The Netherlands 3 Institut f¨ur Erdmessung, University of Hannover, Hannover, Germany 4 Department of Geomatics Engineering, University of Calgary, Calgary, Canada 5 Department of Geophysics, University of Copenhagen, Copenhagen, Denmark 6 DEOS, Delft University of Technology, Delft, The Netherlands 2
[email protected] Received: ??? – Accepted: ???
Abstract. GOCE will be the first satellite ever to measure the second derivatives of the Earth’s gravitational potential in space. With these measurements it is possible to derive a high accuracy and resolution gravitational field if systematic errors have been removed to the extent possible from the data and the accuracy of the gravity gradients has been assessed. It is therefore necessary to understand the instrument characteristics and to setup a valid calibration model. The calibration parameters of this model could be determined by using GOCE data themselves or by using independent gravity field information. Also the accuracy or error assessment relies on either GOCE or independent data. We will demonstrate how state-of-the-art global gravity field models, terrestrial gravity data and observations at satellite track crossovers can be used for calibration/validation. In addition we will show how high quality terrestrial data could play a role in error assessment.
For the diagonal gravity gradients in a Local Orbital Reference Frame (LORF, -axis in the velocity direction, the axis approximately radially outward and the -axis complements the right-handed frame) this precision will not exceed 4 mE/ (1 E = s ) in the MBW (ESA, 1999). The observations will be contaminated with stochastic and systematic errors. 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 with a relative accuracy of prior to the mission by the so-called pre-flight calibration on ground using a test bench. In orbit, a so-called internal (in-flight) calibration procedure will be used (ESA, 1999), by which the CM and DM couplings can be determined to an accuracy level at which their effect onthe gradients in the MBW stays below the required 4 mE/ . The gravity gradients are derived from the internally calibrated DM accelerations. The internal calibration, however, is not sensitive to all instrument imperfections. The readout bias, for example, and the accelerometer mis-positioning cannot be accounted for. Therefore, in order to possibly correct for remaining errors after internal calibration, a third calibration step is required which is called external calibration (or absolute calibration). It is performed during or after the mission and typically makes use of external gravity data, e.g. terrestrial gravity data or global gravity field models (Arabelos and Tscherning, 1998; Koop et al., 2002). Along with the external calibration of the observations, the error of the observations needs to be assessed. For this purpose we could again use terrestrial gravity data. Alternatively, we could use the GOCE data themselves and perform an internal assessment, see e.g. (Albertella et al., 2000; Koop et al., 2002). Bouman and Koop (2003a) used along track interpolation of gravity gradients, which seems to be an adequate tool for error assessment in the MBW. Validation can be defined as the process of assessing, by independent means, the quality of the data products derived from the system outputs. In contrast to calibration, there
1 Introduction The main goal of the GOCE mission (expected to be launched early 2006) 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 an accuracy of 1 cm at 100 km resolution (ESA, 1999). To this end, GOCE will be equipped with a GPS receiver for high-low satellite-tosatellite tracking (SST-hl), 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. The sum of the accelerations of two accelerometers along one arm gives the so-called common mode (CM), which is the signal that is common to both accelerometers, atmospheric drag for example, while the difference between the accelerations of two accelerometers along one arm gives the differential mode (DM), which includes, for example, gravity gradients. The accelerometers are designed such as to give the highest achievable precision in the measurement bandwidth (MBW) between 5 and 100 mHz, which corresponds to a resolution along track of 40-8000 km. 1
Bouman et al.: Calibration and Validation of GOCE Gravity Gradients
The calibration model we use should be related to the gravity gradient errors. Both accelerometer scale factor errors and biases result in equivalent gravity gradient errors, whereas both accelerometer non-orthogonality and misalignment erroneously project part of the signal (gravity gradients plus satellite rotation) onto the measured accelerations. Because the signal has most of its power at 1 and 2 cycles per revolution (cpr), the DM acceleration errors and the gravity gradient errors will show relatively large 1, 2, ... cpr errors. The time series of GOCE gravity gradients may therefore be written as
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Fig. 1. PSD of gravity gradient errors before (external) calibration (left panel). Errors after calibration with OSU91A for a calibration window of 30 days and windows of 5 days (right panel).
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is no active correction applied to the data. With validation one could test the success of the calibration. One could validate the gravity gradients themselves (validation on Level 1), or one could validate derived products such as harmonic coefficients, gravity anomalies or geoid heights (validation on Level 2). In Section 2 we give some typical examples of external calibration, while validation examples are given in Section 3.
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with scale factor , bias ( , trend * and Fourier coefficients 0 , : at GFIH , ... cpr as9 calibration parameters. The trend is included to model, for example, slow bias variations in time. In addition, is time, frequency and D noise. The true gravity gradients 6# are of course unknown, but we will use calibration gradients instead, that is, approximate gradients # are computed using, for example, an existing global gravity field model. Because the errors at long wavelengths of gravity gradients computed with a global gravity field model are expected to be smaller than the errors in the GOCE gravity gradients, it is fair to expect that external calibration using global models could remove systematic long wavelength errors from the gravity gradient time series. Hence, the calibration parameters can be estimated using measured gradients and calibration gradients # . Of course, the GOCE gradients are expected to be more accurate at short wavelenghts.
300 and a ‘GRACE’ gravity field model, that is, EGM96 but complete up to degree and order 120. In Figure 1 (left) the PSD of the NOO errors is shown. Clearly, the errors at 1, 2, 3 and 4 cpr are large (peaks between 8 and P Hz), and in addition the errors on the low frequencies between 0 and 1 cpr are large. The former are caused by the coupling between the error and the signal, which is large at 1 and 2 cpr due to the Earth’s flattening and the orbital eccentricity, while the latter are due to the accelerometer instability for lower frequencies. The NQIQ errors show similar behaviour, while the NR4R errors at 3 and 4 cpr are relatively small (not shown). The estimated calibration parameters are a scale factor, a bias and a trend for the total data set (30 days) for each of the three diagonal gradients, see model (1). In addition, Fourier coefficients at 1, 2, 3 and 4 cpr for NO O FMN QIQ , and at 1 and 2 cpr for N R4R were estimated. The error reduction of the calibrated gradients with respect to the uncalibrated measurements is shown in Table 1. Calibration with the ‘GRACE’ model is only marginally better then calibration with OSU91A. Although the gravity gradient errors have been reduced, we are now confronted with at least two limitations of our calibration model. The first is that errors at long wavelengths between 0 and 1 cpr other than a scale factor error and a trend are uncalibrated because in (1) only a scale factor and a trend have effect at these low frequencies. Secondly, the gravity gradient errors are correlated with the orbital height variation, which changes slowly but significantly with time. Hence, the signal and the errors at 1, 2, ... cpr vary in time as well, and the estimated Fourier coefficients are not representative for the whole 30 day period. The error reduction at 1,2, ... cpr is therefore modest (see Figure 1, right). One way to deal with these problems is to divide the calibration period
2.1 Global gravity field models The results presented in this Section are based on the IAG SC7 test data set1 , see also (Bouman and Koop, 2003b). A 30 day GOCE gravity gradient time series was computed using EGM96 complete up to degree and order 300, and simulated errors were added. The interval between two consecutive measurements is 5 s. Calibration gradients were computed using OSU91A also complete up to degree and order 1 See
www.geod.uni-bonn.de/SC7/index.html
Table 1. Standard deviation of the errors after calibration with OSU91A. The numbers give the fraction with respect to the standard deviation of the errors before calibration. The numbers for calibration with ‘GRACE’ gradients are given between brackets. Besides a scale factor, a bias and trend, and are calibrated with 4 cpr coefficients, while is calibrated with 2 cpr coefficients.
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0.64 (0.64) 0.29 (0.29)
0.84 (0.80) 0.54 (0.42)
0.54 (0.53) 0.32 (0.27)
Bouman et al.: Calibration and Validation of GOCE Gravity Gradients
2.2 Terrestrial data Gravity gradients in the LORF may be predicted from terrestrial gravity data using, for example, least-squares collocation (LSC), (Tscherning, 1993). We use the GRAVSOFT package in our computations (Tscherning, 1974). The idea is to use gravity anomalies (ed to predict gradient anomalies f-OO,FUf R4R and f QIQ in the LORF in a region with extension jlg$mh$i , h$H6inj)obj)p,qGi ). ( g H$h6ikj The SC7 data set, however, is not particularly suited for regional applications. We would like to have more high frequency signal and we therefore generated gravity gradients using EGM96 up to degree and order 360. In addition, the sampling was increased from 5 s to 1 s. The selected area contained 55 GOCE tracks, with approximately 1500 points. Simulated errors were added to the GOCE gradients, and the
Table 2. Statistics of the errors in the LSC predicted gravity gradients (1528 data points): predicted minus true. Gravity anomalies with 1 mGal error were used in the prediction (8601 data points). Between brackets are the LSC predicted standard deviations. Units are [mE]. gradient mean min max +0.1 0.6 (0.6) -1.6 1.7 +0.2 0.5 (0.7) -1.3 1.2 -0.3 0.9 (1.0) -2.0 2.4
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error RMS is 3-4 mE for the three different gradients. Secondly, ‘true’ gravity anomalies were computed using EGM96, Dt h uv`6p$ . Because regional data cannot represent long wavelengths, we subtracted D jwhG , which is on the safe side given the size of the region (corresponds to Dt x ). The gradient prediction becomes more reliable by using data in a larger region, so we selected gravity anomaly data points on a grid with g`h6ij jyg h$i , zH$i{j|o)j}q$q$i , and (~v m$H i F ( o Lmz i (8601 points in total). Errors were added to these data which were uncorrelated with zero mean and a standard deviation of 1 mGal. An empirical covariance function of the noisy gravity data was determined and an analytical model was fitted to this empirical function (Knudsen, 1987). The variance of the gravity data is 207 mGal , so the signal in this region is rather smooth. Finally, we predicted fO O,FUf RR and f QUQ in the LORF in the inner region. These LSC predicted gradients may be compared with true gradients (without errors) computed with EGM96, D h `$p$ . Differences between these gradients are due to the 1 mGal error in the terrestrial data in combination with model errors such as spherical approximation in LSC and the use of a limited region in the computations (Table 2). The errors in the predicted gradients are small with almost zero mean and an error standard deviation of 1 mE. The predicted standard deviations from LSC are given between brackets and they closely resemble the true standard deviations (column 3 of Table 2). It is therefore feasible to calibrate the GOCE gradients using LSC and our simulated ground data. In addition, these data may be used for error assessment, see Figure 2. Given the GOCE gradients, we would like to determine global calibration parameters, that is, calibration parameters valid for the whole time series of gravity gradients. However, it is not possible to discriminate between, e.g. a bias and 1 and 2 cpr effects in limited regions. In this study, we therefore estimated one bias per track, and could add, for example, one scale factor for the whole region. Also from
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into shorter calibration windows and to estimate new calibration parameters for each observation window. If the window is small enough, then the estimated Fourier coefficients are representative for this window. In addition, this windowing will remove part of the long wavelength errors below 1 cpr: if calibration gradients are very accurate at the long wavelengths, then the window size determines the longest possible wavelength of the errors in the calibrated gradients. The total observation period of 30 days has been divided into six windows of five days. For each of these windows we determined the full set of calibration parameters, that is, scale factor, bias, trend and up to 2 cpr Fourier coefficients for N8R4R and up to 4 cpr Fourier coefficients for the other two gradients. Results for N O O are shown in Figure 1, while Table 1 summarizes the reduction of the standard deviation of the errors with respect to the original uncalibrated gradients. As expected the long wavelength errors are reduced using shorter calibration periods. A period of 5 days, for example, corresponds to H _ `ba) c Hz, and it is clear from Figure 1 (right panel) that the errors below this frequency have indeed become smaller. Furthermore, the errors at 1-4 cpr have been largely reduced. Calibration with the ‘GRACE’ model is better then calibration with OSU91A, except for NO O . The latter is probably due to our accurate knowledge of the zonal harmonics to which especially N O O is sensitive. The above examples indicate that the long wavelength error of existing high resolution global gravity field models such as OSU91A or EGM96 is small enough to be used in the external calibration of GOCE gravity gradients. Furthermore, the calibration results may improve when more accurate CHAMP or GRACE based high resolution global gravity field models become available. The CHAMP-only and GRACE-only models that are available today are not suited for external calibration of the GOCE gradients. These models have full power approximately up to degree and order 65 and 90 respectively, whereas a proper calibration model needs to be complete, with full power, up to degree and order 120 at least. Otherwise the omission error is too large and there will be no improvement in the external calibration compared to, for example, EGM96.
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Bouman et al.: Calibration and Validation of GOCE Gravity Gradients
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the simulated GOCE gradients the contribution of EGM96, D jh$ was subtracted. Gravity gradient biases were estimated for each track, for all three diagonal components, using LSC with parameters (Moritz, 1980). Different combinations of observations were used, with an assigned uncertainty of 5 mE for the gravity gradients and 1 mGal for the ground data. The bias errors per track (that is, the differences between the LSC predicted bias and the true bias) are small in all cases, that is, the bias error RMS is 1 mE or less. The ‘true’ bias is simply the mean simulated gravity gradient error per track, which is a lumped bias (sum of several instrumental errors). The statistics of the errors in the calibrated gradients for the different cases are given in Table 3. The mean error of the calibrated gradients is almost zero for all cases, while the error standard deviation is at the mE-level. The error ’s have become smaller compared to the error in the observed gradients, also when only the gradients were used in the calibration. This is due to the fact that the LSC takes the signal correlation into account and smooths the data. 3 Validation 3.1 Band-limited kernels In the operational phase of the GOCE mission there is a need to check the validity of the calibration. Here a method for monitoring and validation is proposed to be used during the measurement phases. The idea is to compare regional gravity and satellite gradiometer data from GOCE in a spectrally band-limited sense, where the band-limits need to be tuned to keep most of the gradiometer signal in the MBW. In case we isolate a bandpass filtered satellite gradiometer contribution and apply the same bandpass filter to modify the Stokes kernel, we are able to compare and possibly validate the results directly in a limited area in terms of geoid heights or disturbing potential, or even gravity or gravity gradients (cf. Arabelos and Tscherning, 1998; Koop et al., 2001). The initial focus will be on the potential at the Earth’s surface or geoid heights, because the main interest is to analyze the effect of a possible build up of systematic errors or change in calibration characteristics of the accelerometers in the final end products. There is no need to consider the low frequency part of the signal because this is for the most part determined by SST, whereas the high frequency part is determined mainly by the ground data. For gradiometer monitoring purposes it is therefore sufficient to study a bandpass filtered signal. See also (Haagmans et al., 2003).
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A band-limited solution for the potential at the Earth’s surface fZ , either from gravity anomalies (ed or from vertical gravity gradients of GOCE fk> at satellite altitude is:
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with the average Earth radius, the geocentric radius of the GOCE orbit, the spherical distance between points and , and in which the kernels for surface gravity and vertical gravity gradients for GOCE are
1365 . H D& Z (4) F D ) G e G ¤"¥ . H¦ D & F 1 4 3 5 % D§& 4¨D{& H a G u¡ £¢ (5) Z with D the spherical harmonic degree, Legendre function, Uª ¥ ª ¥ª Uª and band pass filter g© g© , where «© and g©
are two lowpass filters (see Figure 3). Eq. (2) and (3) are the global solutions for gravity and satellite gradient data to obtain the medium wavelength part of the disturbing potential. Next a regional solution will be derived starting from the global ones. For this purpose one needs to consider the following steps:
1. Divide the solutions for gravity and gravity gradients in inner and outer zone components; 2. Design adequate bandpass filter; 3. Derive the corresponding omission errors; 4. Estimate the truncation capsizes from omission error analysis; 5. Apply these parameters in regional solutions. We can evaluate the integral components in Eq. (2) and (3) within a certain integration cap and omit the part outside this cap. Therefore we need to define spherical caps k¬ and ¥ for the integration of gravity and gravity gradient data by fixing one value for the omission error variance for geoid heights or the potential. An adequate choice of the truncation caps should imply that the regional contribution for any selected area solely based upon gravity or on
Bouman et al.: Calibration and Validation of GOCE Gravity Gradients satellite gravity gradients has exactly the same truncation error, which means that the commission parts are the same for a ‘deterministic case’ with continuous signals without measurement errors, and the omission error parts as well. Based upon a degree-variance model ®¯ for gravity anomalies, i.e. Tscherning/Rapp’s model (Tscherning and Rapp, 1974), both geoid height omission error variances can be estimated (see Heck, 1979). at respectively The spatial truncation caps Z¬ , ¥ of the ‘modified’ kernel functions and in Eq. (2) and (3) depend on the choice of the weights , and on the admis sible size of the truncation error variance. So, by analyzing the omission error variance for various capsizes given chosen weight functions a selection is done for a fixed variance. In case the omission error is evaluated at the Earth’s surface, the downward continuation term is dominant, especially for higher degrees resulting in an unstable operator. For a meaningful practical application one has to create a bandpass filter that counteracts the spectral¥ characteristics A° G¤"¥ of the downward continuation operator G¤(º¹ ¥ _
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The derivative of the gradients in the height direction N ²@²@µ and the derivative with respect to the satellite orientations N³²@²
¸ is computed from the EGM96 model up to degree and order 300 by simple numeric differentiation together with the height ( (~± ¥ ) and orientation ( (~¹ ¥ ) differences. Using EGM96 in each calculation step as well as for the computation of the observations, a closed-loop study has been carried out. The resulting differences in the in-line components ( N ²
² are less than 0.6 mE (see Figiure 5), in agreement with the accuracy of the interpolation of the input parameters (XO position, satellite height and orientation differences and measured gradients); similar results are obtained with other
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Bouman et al.: Calibration and Validation of GOCE Gravity Gradients
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against a regionally derived disturbing potential from gravity anomalies, also bandpass filtered. The logical next step is to insert LS prediction into the integral equations. Finally, we have studied one example of ‘internal’ validation, that is, we validated GOCE gravity gradients at satellite track crossovers. The method may benefit from the use of LSC, but this requires further study. Acknowledgement. We acknowledge the review by Thomas Gruber which improved the paper.
References Fig. 5. Consistency check: gradient differences in XOs of IAG SC7 data.
models like OSU91A, too. In a further investigation, the coefficient variances of EGM96 have been taken into account. Their error propagation shows a (¦± -dependent accuracy for the crossover reduction of (max) 2.5 mE for (¦± ’s of 12 km. In the majority (over 65%) of the crossovers the reduction errors are below 1 mE (Jarecki and M¨uller, 2003). Considering a GOCE accuracy of about 2 mE, the used algorithms turn out to be sufficient for the internal validation of the gradients. 4 Conclusions and discussion We discussed a number of methods for the calibration and validation of GOCE gravity gradients. The difference between calibration, including error assessment, and validation is that no corrections are applied to the data in the validation procedure. We have shown that (existing) global gravity field models may be used for external calibration of the GOCE gradients. Although these models are not as accurate as the gravity field information we may derive from GOCE, they are expected to be more accurate at longer wavelengths below the MBW. However, if the GOCE gravity gradient scale factors, for example, are frequency independent, then we are also able to (indirectly) calibrate in the MBW with global models. A large part of the systematic errors at long wavelengths can be removed. Terrestrial gravity data may be used for external calibration error assessment and validation (see also Denker, 2003). The number of regions, however, with high quality terrestrial data is limited and it may therefore not be possible to calibrate the full, global GOCE gradient data set. With LSC it is possible to accurately calibrate or validate the gradients at satellite level. In addition, the upward continued and converted terrestrial data may be used for error assessment. The validation of Level 2 data, such as the disturbing potential, is possible in a band-limited integral equation approach. Gravity gradients are downward continued and bandpass filtered. In a regional solution, with limited integration area, the disturbing potential can be computed which may be compared
Albertella, A., Migliaccio, F., Sans`o, F., and Tscherning, C., Scientific data production quality assessment using local space-wise pre-processing, in From E¨otv¨os to mGal, Final Report, edited by H. S¨unkel, ESA/ESTEC contract no. 13392/98/NL/GD, 2000. Arabelos, D. and Tscherning, C., Calibration of satellite gradiometer data aided by ground gravity data, Journal of Geodesy, 72, 617–625, 1998. Bouman, J. and Koop, R., Error assessment of GOCE SGG data using along track interpolation, Advances in Geosciences, 1, 27–32, 2003a. Bouman, J. and Koop, R., Geodetic methods for calibration of GRACE and GOCE, Space Science Reviews, 108, 293–303, 2003b. Denker, H., Computation of gravity gradients for Europe for calibration/validation of GOCE data, in Gravity and Geoid 2002; 3rd Meeting of the IGGC, edited by I. Tziavos, pp. 287–292, Ziti Editions, 2003. ESA, Gravity Field and Steady-State Ocean Circulation Mission, Reports for mission selection; the four candidate earth explorer core missions, ESA SP-1233(1), 1999. ESA, From E¨otv¨os to mGal, Final report, ESA/ESTEC Contract No. 13392/98/NL/GD, 2000. Haagmans, R., Prijatna, K., and Omang, O., An alternative concept for validation of GOCE gradiometry results based on regional gravity, in Gravity and Geoid 2002; 3rd Meeting of the IGGC, edited by I. Tziavos, pp. 281– 286, Ziti Editions, 2003. Heck, B., Zur lokalen Geoidbestimmung aus terrestrischen Messungen vertikaler Schweregradienten, Reihe C No. 259, Deutsche Geod¨atische Kommission, 1979. Jarecki, F. and M¨uller, J., Validation of GOCE gradients using crossovers, in GEOTECHNOLOGIEN: Observation of the System Earth from Space, GEOTECHNOLOGIEN Science Report No. 3, 2003. Knudsen, P., Estimation and modelling of the local empirical covariance function using gravity and satellite altimeter data, Bulletin G´eod´esique, 61, 145–160, 1987. Koop, R., Visser, P., and Tscherning, C., Aspects of GOCE calibration, in International GOCE user workshop, vol. WPP-188, ESA/ESTEC, 2001. Koop, R., Bouman, J., Schrama, E., and Visser, P., Calibration and error assessment of GOCE data, in Vistas for Geodesy in the New Millenium, ´ am and K.-P. Schwarz, vol. 125 of International Associaedited by J. Ad´ tion of Geodesy Symposia, pp. 167–174, Springer, 2002. Moritz, H., Advanced physical geodesy, Wichmann, 1980. M¨uller, J., Jarecki, F., and Wolf, K., External calibration and validation of GOCE gradients, in Gravity and Geoid 2002; 3rd Meeting of the IGGC, edited by I. Tziavos, pp. 268–274, Ziti Editions, 2003. Tscherning, C., A FORTRAN IV program for the determination of the anomalous potential using stepwise least squares collocation, Report No. 212, Department of Geodetic Science and Surveying, Ohio State University, 1974. Tscherning, C., Computation of covariances of derivatives of the anomalous gravity potential in a rotated reference frame, Manuscripta Geodetica, 8, 115–123, 1993. Tscherning, C. and Rapp, R., Closed covariance expressions for gravity anomalies, geoid undulations, and deflections of the vertical implied by anomaly degree variance models, Report No. 208, Department of Geodetic Science and Surveying, Ohio State University, 1974.
