GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L17102, doi:10.1029/2009GL039564, 2009
Deriving daily snapshots of the Earth’s gravity field from GRACE L1B data using Kalman filtering Enrico Kurtenbach,1 Torsten Mayer-Gu¨rr,1 and Annette Eicker1 Received 12 June 2009; revised 14 July 2009; accepted 21 July 2009; published 11 September 2009.
[1] Different GRACE data analysis centers provide temporal variations of the Earth’s gravity field as monthly, 10-daily or weekly mean fields. These solutions are derived independently for each time span, i.e., no correlation between the analyzed batches is considered. Following this procedure, an increase in temporal resolution is accompanied by a loss in accuracy. To avoid this problem, a new approach is followed, which takes into account the temporal correlations of the gravity field variations thus enabling the enhancement of the temporal resolution up to daily snapshots. The GRACE Level-1B (L1B) instrument data processing is performed within the framework of a Kalman filter estimation procedure, where the information about the temporal correlation patterns can be derived from geophysical models. The WaterGAP hydrological model was analyzed to derive the required information in terms of an empirical auto-covariance function. First results are presented and compared to GFZ-RL04 monthly and weekly gravity field solutions. Citation: Kurtenbach, E., T. Mayer-Gu¨rr, and A. Eicker (2009), Deriving daily snapshots of the Earth’s gravity field from GRACE L1B data using Kalman filtering, Geophys. Res. Lett., 36, L17102, doi:10.1029/2009GL039564.
1. Introduction [2] For more than 7 years the twin satellite mission GRACE [Tapley et al., 2004] has measured the Earth’s gravity field and its temporal variations. The official product of the GRACE project to describe these temporal variations is a series of monthly mean gravity field solutions. This concept has the disadvantage that these fields cannot represent the gravity field variations within the monthly time span. This leads to aliasing errors due to unmodeled high frequency gravity field changes resulting in the wellknown error stripe pattern. Furthermore, these aliasing errors are accepted to be one of the reasons that the predicted accuracy of the GRACE mission (baseline) has not been achieved yet. Several analysis centers try to overcome this drawback by reducing the time span to 10-daily [Lemoine et al., 2007] or even weekly [Flechtner et al., 2009] batches. [3] An increase in temporal resolution, however, results in less observations per time span and therefore a reduced redundancy in the parameter estimation process. This leads to a decreasing accuracy of the estimated parameters with decreasing time span. When processing individual batches, no further information about the temporal behavior of the 1
Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany. Copyright 2009 by the American Geophysical Union. 0094-8276/09/2009GL039564
gravity field is exploited. It can be assumed, however, that the gravity field does not change arbitrarily from one time step to the next. Utilizing this knowledge, the temporal resolution can be enhanced without losing spatial information. This leads to the concept of regarding the gravity field variations of the Earth as a dynamic process. [4] In the approach presented here, modeled temporal correlations are introduced into GRACE L1B data processing, which enables the calculation of high temporal resolution (in the following daily) gravity field solutions. An efficient tool to model dynamic processes and to fit observations at the same time is the Kalman filter approach. To our knowledge, it is here for the first time adapted to GRACE gravity field analysis.
2. Kalman Filter Approach [5] The processing of GRACE measurements (K-band range data, GPS and accelerometer observations) for one day can be formulated in a Gauss Markov Model according to lt ¼ At xt þ vt ;
ð1Þ
where lt is the vector containing the GRACE measurements, xt are the unknown gravity field coefficients, At is the design matrix and vt is the measurement noise vector for each day t. When calculating monthly or weekly solutions, these daily observation equations are accumulated to the corresponding system of normal equations. This accumulation results in temporal averaging leading to an abandonment of information regarding the temporal variations within the time span. Solving the normal equation for each day individually provides unsatisfactory results due to insufficient data coverage. Instead, to reach such a high temporal resolution, another piece of information is needed to stabilize the solution. Assuming that the gravity field parameters cannot change within an arbitrary range, the solution xt on the current day t can be predicted from the previous at t 1 according to xt ¼ Bt xt1 þ wt :
ð2Þ
Here, Bt describes the temporal behavior of the solutions and wt defines the prediction noise vector. Equations (1) and (2) together constitute a first order Markov process with the state vector xt, which can be estimated, for example, within a Kalman filter approach [Kalman, 1960; Kalman and Bucy, 1961]. It will be shown in section 3 that the assumption of a first order Markov process is valid for the approach at hand. [6] It is expected that, as a first approximation, the gravity field coefficients remain nearly constant from one day to the next. Therefore, Bt can be chosen as the identity
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applied, which has been kindly provided by Prof. Do¨ll from the University of Frankfurt as daily equivalent water heights (EQWH) over the continents on a 0.5° 0.5° grid. Concerning details on the calculation of these daily values see Do¨ll et al. [2003] and Hunger and Do¨ll [2008]. These EQWH (reduced by a temporal mean) are in a first step expanded into a spherical harmonic series and then converted to gravitational potential by taking into account the indirect effects caused by loading. The result is a time series of potential coefficients, which can be interpreted as a stochastic process in time as well as on the surface of the sphere. From this time series the covariance matrix described in section 2 has to be derived from the corresponding empirical covariance function. In the approach presented here, the gravitational potential has to be a signal of a stationary stochastic process on the sphere. Stationarity in time implies that the covariance function only depends on the temporal distance Dt = ti+1 ti. Furthermore, homogeneity and isotropy are presumed resulting in a dependency only on the spherical distance between two points. The expectation value of a homogeneous, isotropic, and stationary stochastic process on the sphere is given by the average [see, e.g., Moritz, 1980], Figure 1. (top) Empirical auto-covariance function of WGHM daily gridded data sets as function of time for different spatial distances. A dominant annual signal is obvious. (bottom) Empirical covariance function of the residual stochastic part of the signal (annual and semi-annual signal reduced). matrix. Whereas a strict application of this assumption would lead to a static field, the introduction of a prediction noise vector allows variations within a certain range. Within a Kalman filter approach it is not necessary to know the noise vector itself, but only its stochastical characteristics are required. Therefore, some prior information about the covariance matrix S(wt) of the prediction noise vector has to be introduced. In practice, S(wt) will be replaced by an empirical version derived from the analysis of geophysical models. It has to be pointed out that in the GRACE estimation process only this empirical covariance matrix is taken into account, but not the model results themselves. Thus the estimated gravity field solution will not be biased towards the applied geophysical model. The model and inaccuracies in the model are therefore less critical.
3. Prior Information From Hydrological Model [7] As described before, the use of the Kalman filter approach requires prior information about the expected daily gravity field variability which can be derived from geophysical models. Within the GRACE gravity field recovery, short-term mass redistributions such as atmospheric variations or ocean tides are reduced before the estimation process [Flechtner et al., 2009] so that the residual signal is mainly describing the variations in global continental hydrology and, in specific regions, long-term processes such as postglacial rebound or seismic activities [see, e.g., Gu¨ntner, 2008]. Therefore, we extract the necessary prior information from hydrological models. In this study, the WaterGAP Global Hydrology Model (WGHM) is
2p Z Z N Z 1 X C ðy; Dt Þ ¼ 2 f ðr; ti Þf ðr0 ; ti þ Dt ÞdWda: ð3Þ 8p N i¼1 0
W
Here r and r0 are the position vectors of two points on the sphere W; r depends on r0 due to the fact that the two points on the sphere are separated by the spherical distance y and that the second point is located at azimuth a from the first point. N denotes the number of time steps available in the time series. The integral can be solved as a closed expression in terms of spherical harmonics according to Moritz [1980]. The resulting auto-covariance function is a two-dimensional function depending on time and on the spatial distance. In Figure 1 (top) the temporal correlations are displayed for different spatial distances (indicated by different colors) on the surface of the sphere. As a dominant annual signal is obvious, a separation of the signal into a deterministic part representing the annual and semi-annual period and a residual stochastic part appears to be reasonable. The covariance function of the stochastic part is shown in Figure 1 (bottom) for the same spatial distances. The residual empirical covariance function can sufficiently well be approximated by an exponential function, which in the following is used as analytical covariance function. This dynamic process described by an exponential covariance function can be approximated as first order Markov process [see, e.g., Gelb, 1974]. The covariance matrix of the prediction noise can now be derived from the empirical covariance function. Since for daily snapshots the variations of the Earth’s gravity field from one day to the next are essential, the covariance function is evaluated at the time step Dt = 1 day.
4. GRACE L1B Data Processing and First Results [8] For the results to be presented in this section, two years of GRACE L1B data were processed, i.e., the years 2006 and 2007. The analysis procedure equals the one
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applied in the calculation of the ITG-Grace03s time series, which is based on the formulation of Newton’s equation of motion as a boundary value problem for short arcs of the satellite’s orbit (approximately 30 min.). For further information see Mayer-Gu¨rr [2006] and Mayer-Gu¨rr et al. [2007]. In the Kalman filter approach the same background models (static gravity field, ocean tides etc.) were introduced as for ITG-Grace03s. Additionally the annual and semi-annual signal were estimated from the ITG-Grace03s time series and were removed from the data. This corresponds to the assumption that the residual signal can be modelled by a first order Markov process. From these reduced observations, the Kalman filter solution was calculated for daily time steps and the annual and semi-annual signal was restored afterwards. The output are daily gravity field models up to a spatial resolution of a spherical harmonic degree n = 40, assuming that this resolution is
Figure 3. Time series averaged over (top) the Amazon basin and (bottom) the Congo basin. Displayed are the daily Kalman solution (red), the monthly GFZ-RL04 series (green) and the weekly GFZ solution (blue).
Figure 2. Temporal variability in terms of RMS values of different GRACE solutions. (top) Daily Kalman solutions. (middle) Weekly GFZ-RL04 solutions. (bottom) Monthly GFZ-RL04 solutions. Both GFZ solutions filtered with nonisotropic decorrelation filter (DDK).
higher than the temporal variations that can be detected from GRACE at the moment. During the Kalman filter process an implicit filtering has taken place, therefore subsequent users do not have to apply any filtering in a post-processing step. [9] Results of the Kalman solution are presented in Figures 2 and 3. An animation showing the temporal evolution of the daily snapshots can be found in the auxiliary material.1 In Figure 2, the spatial distribution of the temporal variances of the daily solutions are displayed for the time span 2006 –2007 (Figure 2, top). They are compared to the corresponding variances of the GFZ-RL04 weekly (Figure 2, middle) and monthly (Figure 2, bottom) gravity field models. The weekly models are provided up to degree and order 30 and the monthly models up to degree and order 120. For further details on theses two solutions, see Flechtner et al. [2009]. Because of the implicit filtering taking place during the Kalman filter estimation, the unfiltered GFZ solutions had to be filtered as well. Therefore, to both GFZ-RL04 solutions a non-isotropic decorrelation filter (DDK) [see Kusche, 2007] was applied. Even though a direct comparison of the results is difficult due to the different filtering techniques, all three illustrations depict very similar spatial patterns. In the large river basins the daily solutions show a variability comparable to the weekly or monthly solutions. This indicates that by using the
1 Auxiliary materials are available in the HTML. doi:10.1029/ 2009GL039564.
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Table 1. Temporal Variability in Terms of wRMS Values Averaged Over the Amazon River Basin, Greenland, and the Sahara Area in cm Equivalent Water Heighta Kalman daily GFZ weekly (DDK) GFZ weekly (500 km) GFZ weekly (700 km) GFZ weekly (1000 km) GFZ monthly (DDK) GFZ monthly (500 km) GFZ monthly (700 km) GFZ monthly (1000 km)
Amazon
Greenland
Sahara
SNR
17.6 18.9 15.2 12.4 9.3 18.2 14.5 12.1 9.1
7.1 7.9 5.8 4.7 3.7 7.1 5.3 4.3 3.3
1.5 4.2 3.8 2.7 2.3 2.4 2.5 2.0 1.9
11.7 4.5 4.0 4.6 4.0 7.6 5.8 6.1 4.8
a Signal-to-noise ratio (SNR) calculated as ratio Amazon/Sahara. Nonisotropic decorrelation filter (DDK) and different radii of Gaussian filter applied to GFZ solutions.
Kalman filter process the signal is not damped more strongly than in case of applying the DDK filter. [10] On the oceans, on the other hand, the results vary considerably when comparing the three different solutions. As the oceanic mass variations were considered by models during the analysis process, the remaining variability indicates, besides residual errors in the background models, the noise level of the gravity field solutions. Another region where no significant variability can be expected is the Sahara area. It can be observed that the daily solutions show the least variance in these regions. Therefore, it can be concluded that the noise level can be kept small during the Kalman filter process without losing too much information by smoothing in areas where large temporal variations are expected. This statement is also confirmed by the results listed in Table 1. It shows the variabilities averaged over the Amazon and the Sahara area as examples for regions featuring strong signals and regions dominated by noise, respectively. The ratio of both values can be interpreted as signal-to-noise ratio (SNR), and a considerably larger SNR value can be observed for the Kalman result compared to the GFZ solutions. In order to investigate the influence of filtering, Gaussian filters with different radii were applied to the GFZ solutions besides the DDK filter. As can be concluded from Table 1, the SNR changes only slightly while the signal is strongly dampened with increasing radius. [11] It has to be pointed out that no prior information about the spatial distribution of the signal has been included into the Kalman filter due to the use of an homogeneous and isotropic covariance function. To indicate that besides the hydrological information other signals such as glaciological mass variations can be detected as well, a column referring to the signal in the Greenland area has been included in Table 1. [12] Figure 3 shows time series of the temporal variations averaged over two different river basins, the Amazon (Figure 3, top) and the Congo basin (Figure 3, bottom). The red line represents the daily Kalman filter steps, whereas the GFZ-RL04 weekly (blue line) and monthly solutions (green line) are printed for reasons of comparison. Here the same DDK filter has been applied to the GFZ solutions as described above. It has to be mentioned that the scales for the signal amplitudes differ when comparing the two river basins. In the Congo basin the noise in the weekly
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solutions emerges more dominantly than is the case in the Amazon basin. Besides the weaker signal in the Congo basin, the smaller basin area is responsible for a decreased signal-to-noise ratio, too. A comparison of the different curves shows that in general the solutions agree very well in amplitude as well as in phase. The Kalman filter exhibits a very smooth temporal evolution with a much smaller noise level than the weekly solutions. On the other hand it can be observed that in the Kalman solution as well as in the weekly solutions significant gravity field changes within one month can be detected. This is especially true for the Amazon basin. Figure 3 (bottom), displaying the Congo results, reveals that during the first approximately six months the lines do not match as well as for the rest of the time series. A possible reason might be the fact that the start values for the Kalman filter are not known and therefore a warm-up phase is needed in the Kalman filter estimation process.
5. Summary and Outlook [13] For the first time, daily snapshots of the Earth’s gravity field have been derived from GRACE L1B data analysis using a Kalman filter approach. This enhancement in temporal resolution without loss of spatial resolution was possible as prior information about the temporal behavior of the gravity field was introduced into the analysis process. As within the GRACE gravity field recovery short-term mass redistributions such as atmospheric variations or ocean tides are reduced before the estimation process the residual signal is mainly dominated by variations in continental hydrology. Therefore, the desired prior information was derived from the hydrological model WGHM. Here it is important to mention that the model has not been applied itself, but only its stochastical correlation patterns have been used. Even though no spatial distribution of the signal was specified beforehand, the Kalman filter solution appears small and smooth at locations where no signal is expected (e.g., over the oceans), and features high variability in large river basins. The daily Kalman filter time series exhibits a very good agreement to the GFZ-RL04 weekly and monthly solutions (DDK filter applied) in amplitude as well as in phase. [14] In the future an extended validation of the daily gravity field snapshots is intended and the warm-up behavior of the Kalman filter has to be further investigated. Furthermore, the Kalman filter procedure will be applied to the regional gravity field analysis techniques described by Eicker [2008]. Here a special focus will be put on the adaption of the approach to the characteristics of specific river basins. [15] Acknowledgments. The authors would like to thank Petra Do¨ll, University of Frankfurt, for providing the WaterGAP hydrological model. The support of the German Research Foundation within the framework of the special priority program SPP1257 ‘‘Mass Transport and Mass Distribution in the Earth System’’ is gratefully acknowledged. Helpful comments from two anonymous reviewers led to improvements in the manuscript.
References Do¨ll, P. F., F. Kaspar, and B. Kaspar (2003), A global hydrological model for deriving water availability indicators: Model tuning and validation, J. Hydrol., 270, 105 – 134.
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Eicker, A. (2008), Gravity field refinements by radial basis functions from in-situ satellite data, Ph.D. dissertation, Univ. of Bonn, Bonn, Germany. (Available at http://hss.ulb.uni-bonn.de/diss_online/landw_fak/2008/ eicker_annette) Flechtner, F., C. Dahle, K. H. Neumayer, R. Koenig, and C. Foerste (2009), The release 04 CHAMP and GRACE EIGEN gravity field models, in Satellite Geodesy and Earth System Science—Observation of the Earth from Space, edited by F. Flechtner et al., Springer, Berlin, in press. Gelb, A. (1974), Applied Optimal Estimation, MIT Press, Cambridge, Mass. Lemoine, J.-M., S. Bruinsma, S. Loyer, R. Biancale, J.-C. Marty, F. Perosanz, and G. Balmino (2007), Temporal gravity field models inferred from GRACE data, Adv. Space Res., 39, 1620 – 1629, doi:10.1016/ j.asr.2007.03.062. Gu¨ntner, A. (2008), Improvement of global hydrological models using GRACE data, Surv. Geophys., 29, 375 – 397, doi:10.1007/s10712-0089038-y. Hunger, M., and P. Do¨ll (2008), Value of river discharge data for globalscale hydrological modeling, Hydrol. Earth Syst. Sci., 12, 841 – 861. Kalman, R. E. (1960), A new approach to linear filtering and prediction problems, Trans. ASME J. Basic Eng., 82, 35 – 45. Kalman, R. E., and R. S. Bucy (1961), New results in linear filtering and prediction theory, Trans. ASME J. Basic Eng., 83, 95 – 107.
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Kusche, J. (2007), Approximate decorrelation and non-isotropic smoothing of time-variable GRACE-type gravity field models, J. Geod., 81, 733 – 749. Mayer-Gu¨rr, T. (2006), Gravitationsfeldbestimmung aus der Analyse kurzer Bahnboegen am Beispiel der Satellitenmissionen CHAMP und GRACE, Ph.D. dissertation, Univ. of Bonn, Bonn, Germany. (Available at http:// hss.ulb.uni-bonn.de/diss_online/landw_fak/2006/mayer-guerr_torsten) Mayer-Gu¨rr, T., A. Eicker, and K. H. Ilk (2007), ITG-Grace02s: A GRACE gravity field derived from short arcs of the satellites orbit, paper presented at the 1st International Symposium of the International Gravity Field Service ‘‘Gravity Field of the Earth,’’ pp. 193 – 198, Int. Assoc. of Geod., Istanbul, Turkey. Moritz, H. (1980), Advanced Physical Geodesy, Wichmann, Karlsruhe, Germany. Tapley, B. D., S. Bettadpur, M. Watkins, and C. Reigber (2004), The gravity recovery and climate experiment: Mission overview and early results, Geophys. Res. Lett., 31, L09607, doi:10.1029/2004GL019920.
A. Eicker, E. Kurtenbach, and T. Mayer-Gu¨rr, Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, D-53115 Bonn, Germany. (
[email protected];
[email protected]; tmg@ geod.uni-bonn.de)
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