Surveys in Geophysics: Author-created version of the article. The original publication is available at http://link.springer.com/article/10.1007/s10712-014-9309-8
Calibration/Data Assimilation Approach for Integrating GRACE Data into the WaterGAP Global Hydrology Model (WGHM) Using an Ensemble Kalman Filter - First Results Annette Eicker · Maike Schumacher · J¨ urgen Kusche · Petra D¨ oll · Hannes M¨ uller Schmied Received: date / Accepted: date
Abstract We introduce a new ensemble-based Kalman filter approach to assimilate GRACE satellite gravity data into the WaterGAP Global Hydrology Model. The approach (1) enables the use of the spatial resolution provided by GRACE by including the satellite observations as a gridded data product, (2) accounts for the complex spatial GRACE error correlation pattern by rigorous error propagation from the monthly GRACE solutions, and (3) allows to integrate model parameter calibration and data assimilation within a unified framework. We investigate the formal contribution of GRACE observations to the Kalman filter update by analysis of the Kalman gain matrix. We then present first model runs, calibrated via data assimilation, for two different experiments: The first one assimilates GRACE basin averages of total water storage and the second one introduces gridded GRACE data at 5◦ resolution into the assimilation. We finally validate the assimilated model by running it in free mode (i.e. without adding any further GRACE information) for a period of 3 years following the assimilation phase and comparing the results to the GRACE observations available for this period. Keywords data assimilation · GRACE · WaterGAP · ensemble Kalman filter · gain matrix
1 Introduction At the sub-continental to global scale, a number of land surface models (LSMs) and hydrological water balance models (HMs) find applications in diverse areas. LSMs basically represent the land-atmosphere interface in climate models and Annette Eicker, Maike Schumacher, J¨ urgen Kusche Institute of Geodesy and Geoinformation, Nussallee 17, 53115 Bonn, Germany Tel.: +49-228-733577 Fax: +49-228-732629 E-mail: eicker/schumacher/
[email protected] Petra D¨ oll, Hannes M¨ uller Schmied Institute of Physical Geography E-mail:
[email protected]
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numerical weather prediction, and aim at representing the energy and water fluxes by implementing surface energy and water balance equations, among others. HMs, in contrast, have mainly been developed for purposes such as simulating river streamflow, water resources assessment, and flooding prediction. The WaterGAP Global Hydrology Model WGHM ([3]) belongs to the second group. WGHM simulates continental water flows among all relevant water storage compartments, including anthropogenic groundwater and surface water abstractions. The model has been calibrated against mean annual measured runoff. However, despite being constantly improved, the model’s ability in reproducing observed large-scale water storage changes is limited due to errors in climate forcings and incomplete realism of process representations. This has been revealed in a number of studies involving independent water storage data derived from the GRACE (Gravity Field and Climate Experiment, [33]) satellite mission, and similar findings were reported for other HMs and LSMs. Since 2002, measurements of time variable gravity obtained with the twinsatellite mission GRACE allow for the determination of column-integrated terrestrial water storage (TWS) changes on the global scale with uniform data coverage. Beyond validation studies, which usually compare basin-averaged or gridded TWS from models and from GRACE, there are at least two possibilities of using GRACE data for the improvement of hydrological modeling: (1) through calibration, model parameters can be (sequentially or jointly) optimized to achieve a better agreement of model simulation and observations, (2) by data assimilation (DA) approaches, where the model states are updated toward the observations in order to provide more realistic model results. In fact, such studies have re-ignited the question about what the GRACE spatial resolution is, when compared to other remote sensing data that may be ingested in data assimilation frameworks. It is important to note that the concept of GRACE resolution must be related to mass, or volume, rather than surface area. Several studies have shown that while artefacts in GRACE monthly maps may be generally removed by smoothing over few 100 km radii, isolated large water height or level changes may be observed even over surface areas much smaller; e.g. [9] showed that GRACE is sensible to water level change in the Volta lake which has a surface area of less than 8500 km2 . [39], [37] and [38] developed a multi-objective calibration approach for tuning WGHM against basin averages of GRACE TWS and river discharge data. After carrying out a careful sensitivity analysis, for each large river basin a subset of six most sensitive parameters was identified and subsequently estimated within a Monte Carlo framework. An assimilation framework (see for example [26], [36], or [22]) aims at optimal estimation of water storages in different compartments (e.g. groundwater, surface water, soil moisture), integrating both model predictions and observations including their respective error estimate. As the model accuracy is usually difficult to assess, Monte-Carlo methods such as the ensemble Kalman filter (EnKF) and smoother (EnKS) are applied to estimate the model error information from an ensemble of runs. To our knowledge, [41] were the first to develop a method based on EnKS to assimilate GRACE TWS into the Catchment Land Surface Model (CLSM) for the Mississippi river basin. In their approach, GRACE observations were averaged over (sub-)catchments and a spatially and temporally uniform noise level was assumed for the GRACE TWS data. This study revealed an improved agreement of the data-assimilated model runs with (independent) in-situ ground-
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water observations. The same approach was then applied by [8] to the Mackenzie river basin to analyse its performance in a snow-dominated domain and by [13] for the determination of drought conditions in North America, while [20] used it for improved hydrological modeling in European river catchments. They proved the model states to be in better agreement with river discharge measurements after inclusion of GRACE observations. Other experiments were reported, e.g. in [32], [40], and [34]. In the present work, we suggest a different ensemble-based Kalman filter approach to convey the monthly GRACE gravity information through assimilation into hydrological models. This approach is designed to remedy three shortcomings of the published approaches (1) it enables to include GRACE TWS as a gridded product, with spatial information beyond the sub-catchment averages, (2) it is able to make full use of the representation of spatio-temporal GRACE TWS error and correlation patterns contained in the densely populated error covariance matrices of the gravity harmonics, which can be propagated rigorously from the GRACE gravity processing, (3) the approach is able to integrate both data assimilation and hydrological model calibration in the same framework. While a first sensitivity study was carried out in [30], here we will provide preliminary results of assimilating GRACE TWS grids into the WGHM model. Deriving maps of surface mass variation from spaceborne GRACE measurements, at any resolution useful for hydrological modeling, represents always a mathematically ill-posed problem; a fact that translates into ill-conditioned data error covariance matrices. On the other hand, mapping state, parameter and forcing uncertainty of HMs in an ensemble approach will inevitably lead to ill-conditioned, possibly rank-defect model covariances. Remedies to improve the conditioning of these matrices and/or the Kalman gains that follow from a relative weighting of them, include aggregation of GRACE data to (sub-) catchment averages, neglecting error correlations, or formal regularization techniques, but they may cause the resulting estimators to be biased with respect to the original assimilation problem. Obviously, this research has to involve assessing and understanding the intricate effects that anisotropically correlated GRACE total water storage errors and correlated WGHM compartmental water storage errors create in an optimal state estimator. This is closely related to the question whether upscaling the (correlated) GRACE data to coarser grids and catchment averages prior to assimilation may cause the assimilated solution to be biased with respect to a statistically optimal estimate. In [7], the effect of spatial aggregation of GRACE TWS anomalies in a DA framework was investigated through a twin-experiment with truth model and simulated forcing/parameter and data errors. In line with our findings, the authors suggest that aggregation to sub-basin scale performs superior compared to when basin-scale TWS anomalies is assimilated. They conclude that DA appears most enhanced when GRACE is assimilated at the smallest scale “at which they can be reasonably resolved”, this scale is thought as “the smallest scale at which the observations are uncorrelated”. However, [7] consider GRACE errors to be primarily caused by leakage and truncation effects, or proximity of incorrectly corrected ocean mass, and they suggest that aggregation of GRACE gridded TWS to larger areas may mitigate this error correlation. In fact, error correlation information was retrieved from the GRACE Tellus website; we believe this is a crucial assumption and much more realistic correlation information is obtained by evaluating full error
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covariance matrices from the GRACE level 1-2 analysis as we do here, since these do account for anisotropic correlation effects like orbital geometry and measurement system sensitivity. Also, the [7] DA framework did not allow accounting for spatial data error correlation in the update step, instead they carried out experiments with modeling isotropic model error correlation at different spatial scales. In contrast, our study does not rely on such workarounds since our EnKF can ingest any model of error correlation. We will first provide an analysis of the Kalman filter gain matrix. After assessing the numerical conditioning of the GRACE observation error covariance matrix, depending on the spatial aggregation of GRACE data after upscaling and with rigorous error propagation, we study the filter gain in the hypothetical reference case that GRACE observations suggest a basin-wide uniform water surplus compared to modeling. Systematic analysis of the gain matrix tells how much of this surplus total water storage is accepted by the assimilating model as a function of season and the saturation of the filter, and whether the vertical and spatial disaggregation through the EnKF appears reasonable. Secondly, we design an experiment with real data, where, after a 2-year spin-up phase, one year of GRACE data is used for assimilating into WGHM and calibrating several model parameters. Then, the model runs forward for three more years and its output in terms of total water storage is compared to real GRACE data, both in terms of basin average and of spatial grid values. In lieu of a truly independent validation method, in this way we are able to test the ability of the standard model and the (via data assimilation) calibrated versions to predict total water storage without using any additional GRACE data. Next to the standard model run, we compare assimilation of GRACE data after basin averaging and, for the first time, after averaging to five degree grids. We find distinct spatial differences between the two assimilated runs in the validation phase, that remain hidden when one considers the performance in terms of basin averages only. For our test area we choose the Mississippi river basin as in [41], including the heavily managed and extensively irrigated region of the High Plains aquifer (HPA). WGHM does incorporate human water abstraction both from surface water bodies and groundwater, however, in addition to errors introduced by the incomplete representation of “natural” model physics and errors in the input climate data, these anthropogenic effects constitute a large source of uncertainties. Therefore, the Mississippi basin with the HPA presents a challenging test region, where GRACE is indeed expected to add real new information to hydrological modeling.
2 Data and Model 2.1 WaterGAP Global Hydrology Model WaterGAP, a global water resources and use model, consists of the WaterGAP Global Hydrology Model WGHM ([3], [1], [4]) and a number of water use models for irrigation, livestock, manufacturing, cooling of thermal power plants and households. With a spatial resolution of 0.5◦ × 0.5◦ and a daily time step, WGHM simulates water flows among all continental water storage compartments except glaciers. Model compartments comprise canopy, snow, soil, groundwater, lakes, man-made reservoirs, wetlands and rivers. Net groundwater and surface water ab-
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stractions are taken from groundwater storage and river, lake and reservoir storage, respectively (see Fig. 1 in [2]). Computed water flows include evapotranspiration, total runoff, groundwater recharge, base flow and river discharge. Due to uncertainties mainly in the climate input, we consider daily WGHM output values as less reliable and monthly output is used instead. The model version applied here is WaterGAP 2.2 ([23]); this model is controlled by 23 steering parameters, all of which may be improved through calibration. In the standard version, WaterGAP is calibrated against mean annual average river discharge, currently at 1319 gauging stations, by adjusting one of these parameters (the runoff coefficient) in each upstream basin ([14]). The other 22 parameters will then be calibrated from GRACE data in the approach presented here. Snow storage in WGHM is represented by a simple degree-day algorithm. The soil is modeled as one single storage compartment; soil water capacity being a function of soil and land cover type. In most regions, surface water bodies constitute important water storage compartments. Surface water reduces markedly the spatial correlation lengths of water storage fields ([11]). The location of lakes, including man-made reservoirs, and wetlands as well as their areas are provided by the Global Lakes and Wetlands Database GLWD ([19]). Finally, groundwater storage is represented by a linear storage approach, where the change in storage is modeled as the difference between groundwater recharge, net groundwater abstractions and baseflow to the surface water bodies. The contribution of the different storage compartments to the seasonal differences of total water storage varies strongly with the climate zone (Table 3 in [11]). In the WaterGAP 2.2 application used in this study, the CRU climate dataset is applied as input; temperature, cloud cover and number of wet days are taken from the global dataset of monthly climate CRU TS 3.2 ([12]), while for precipitation the monthly monitoring product GPCC v6 ([29]) is chosen. Monthly climate data are downscaled to daily values, in the case of precipitation taking into account information on the number of wet days per month. The distribution of wet days within a month is modeled as a two-state, first-order Markov chain, with parameters chosen according to [10]. Monthly precipitation is distributed evenly to all wet days in the month, while other climate variables like temperature or cloudiness are interpolated to daily values. WGHM has been frequently used for comparison to GRACE data, see for example [28], [18], [2] and [1]. Such comparisons show, in general, a good agreement between model and observations for many regions; WGHM appears therefore well suited for a combination with GRACE within a data assimilation framework. Furthermore, we have extended experience with the model on the coding level. However, many investigations also reveal the potential of GRACE for further model improvements, especially regarding the annual amplitude, which is in many regions underestimated by the model, and the representation of secular trends of TWS. In the study area of the Mississippi basin, modeled water storage changes show a strong annual signal with amplitudes of around 60mm TWS and a maximum storage around February but no signficant trend, when averaged over the whole basin. The annual signal as modeled by WGHM has in more detail been studied in [1], the results being in general agreement with other models, see for example [27].
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2.2 GRACE We use monthly sets of GRACE spherical harmonic coefficients of the ITG-Grace2010 time series ([21]), which are provided with full variance-covariance information for the time span 08/2002 to 08/2009. Temporal gravity field variations include primarily hydrological mass changes, as background models for ocean, Earth and pole tides, atmospheric and oceanic mass variations, glacial isostatic adjustment and geocenter motion were corrected either directly during the data processing or in a post-processing step. The GRACE solutions were filtered using a 500 km Gaussian filter, evaluated up to spherical harmonic degree n = 60 and converted to gridded values of TWS following [35]. A full variance-covariance propagation is carried out to obtain the error covariance matrices for the gridded values. Due to the GRACE measurement concept and orbit configuration, GRACEderived TWS grids are highly correlated in space, and their error variances depend on latitude (due to the orbits converging toward the poles) and, to less extent, on time due to changing orbital repeat patterns. Propagating errors from the GRACE monthly spherical harmonic solutions to gridded TWS values, therefore, leads to a dense error covariance matrix of the gridded values. We use the full error covariance matrices of the potential coefficients provided individually for each month, as it can be shown that using only the formal errors of the spherical harmonic solutions underestimates the GRACE errors of the gridded values. The resulting gridded standard deviations for the Mississippi test area and a map of correlations in the basin are displayed in Fig.1.
Fig. 1 Standard deviations (a) of gridded GRACE observations derived by rigorous error propagation of the full variance-covariance matrix of the spherical harmonic coefficients and corresponding correlations (b) of one point (denoted by the black dot) and every other point in the Mississippi basin.
To be flexible regarding the aggregation scale of the TWS observations, we initially map the GRACE data to the 0.5◦ WGHM grid. However, since, especially after 500 km Gaussian smoothing, GRACE does not provide such a high spatial resolution, we then average the GRACE observations (and the corresponding covariance information) to larger grid cells. The choice of the aggregation scale
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always represents a trade-off between exploiting as much spatial GRACE resolution as possible and keeping the GRACE error covariance matrices reasonably well conditioned, see also Section 4.2. In the results presented below, we compare assimilation runs using 5◦ averages of GRACE TWS with runs that assimilate only the basin average of GRACE. To account for the signal attenuation effect of the filtering process, an average re-scaling factor of 1.1 was determined for the Mississippi basin from analysis of filtered vs. unfiltered monthly WGHM model output from one year. This factor was applied to the gridded TWS values and was also used in establishing the error covariance matrices.
3 Ensemble Kalman Filter Approach 3.1 Motivation The central hypothesis underlying data assimilation is that both, numerical simulation of a process and the measurement of at least some of its observable states, provide useful information. Data assimilation combines these two pieces of information in an optimal way, usually defined through an optimization functional and weighted by error statistics. From an observational point of view, data assimilation is useful in assessing variables that are not directly observed, and in interpolating fields between observations. Important for GRACE, data assimilation may be viewed as a tool for downscaling its coarse-resolution information guided by processes in the model structure. From a hydrological perspective, the assimilation of observation data consists in constraining the realism of model runs. In this sense, data assimilation pursues the same aim as model calibration in traditional hydrological modeling. In geosciences, most data assimilation systems have been developed to integrate data in atmospheric or oceanic general circulation models. Assimilation techniques may be briefly categorized into simple approaches like direct insertion, variational approaches such as 3DVAR and 4DVAR, and Kalman filter and smoother methods. Since the latter two classes both may be formulated through minimizing an optimization functional, there is a certain overlap between approaches. Variational algorithms employ advanced numerical methods to minimize an optimization functional directly, while they usually require the generation of adjoint code to do so ([16]). Kalman filter methods ([15]) are essentially sequential schemes which combine, at any time when observations of the system state are available, the model output and the observations using a weighting scheme based upon second moments of the respective probability density functions. To this end, they explicitly compute error covariances through an additional matrix equation, not required in variational approaches, that propagates error information from one update time to the next, subject to possibly uncertain model dynamics. Like variational methods, the Kalman filter can be derived from an objective function, given a number of additional assumptions about the error structure, including model and observation errors. The Ensemble Kalman filter (EnKF), that we choose as our data assimilation method, replaces the costly computation of model and error covariances by
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a Monte Carlo approach ([6]). It is thus easy to implement and suitable for parallelization. The EnKF can accommodate for optimal estimation of model calibration parameters or data biases by state vector augmentation. In this study we integrate the calibration of several model parameters into our approach. This includes parameters such as a net radiation multiplier, a river roughness multiplier and a precipitation multiplier, that were found in [31] to be most sensitive toward GRACE in the Mississippi basin. However, since water storage simulation in hydrological models involves nonlinearities and non-Gaussian probability density functions (pdfs, see also [17]), the EnKF is not strictly optimal in a formal statistical sense. In addition, the performance of the EnKF is known to depend strongly on tuning such as ensemble generation and variance inflation strategy. Some of these aspects are investigated in the following.
3.2 Concepts Following the notation of [5], model prediction of the n-dimensional state vector x from time tk−1 to tk can be written as xk = f (xk−1 ) + qk−1 ,
(1)
in which the nonlinear dynamics f is implemented through a model simulation code. In Eq. (1), qk−1 represents the model error which may originate from missing physics, model discretization, and/or uncertainties among the model forcing fields. Constant calibration or bias parameters related to forcing fields or to the observations can be included straightforwardly in the above formulation by augmenting the state vector. We assume that at time tk , observations of the model state combined in the m-dimensional vector dk are available, and that they can be related to xk through the linear or linearized observation equation dk = Hxk + k ,
(2)
with the m × n design matrix H and the observation error vector k . The relative weighting of data and model simulation is determined from the respective error covariance matrices. However, if n is large and the model operator f involves nonlinearities, a straightforward computation of the model error covariance matrix is usually prohibitive. Instead, ensemble Kalman filters (EnKF) estimate the model covariance emprically from an Ne -sized ensemble of model predictions. With the column vector of each model prediction x(i) , i = 1 . . . Ne , ¯ containing the ensemble mean in each of its arranged in the matrix X, and X columns, the empirical model error covariance matrix is generally estimated from Ce =
1 1 ¯ ¯ T = (X − X)(X − X) X0 X0T , Ne − 1 Ne − 1
(3)
¯ The update where we have introduced the ensemble perturbations X0 = X − X. or analysis step, correcting the model ensemble prediction from the observations, is then facilitated through X+ = X + K(D − HX),
(4)
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with the Kalman gain matrix K = Ce HT (HCe HT + R)−1 .
(5)
In the above, R is either the known error covariance matrix of the observations or replaced by an ensemble approximation Re = Ne1−1 EET to it, with R being the matrix of simulated observation errors. In the original formulation of the EnKF, the column vectors of D contain the observations dk plus random perturbations drawn from R, in order to maintain a correct ensemble representation of the error covariance matrix of the updated state. In any practical situation, the performance of the EnKF depends on the ability of the model ensemble to represent the most relevant directions of the true model covariance C through Ce ; this must be guaranteed through a suitable choice of the ensemble members x(i) . In addition, the filter should accommodate for the uncertainties introduced by model errors q, e.g. by explicitly ingesting simulated forcing field errors or mimicking these using variance inflation techniques. In summary, due to the limited ensemble size, the EnKF update state can never convey the same information from the data to the model state vector as in the hypothetical case where we would know C; but this may not be a severe loss since the model states follow from a mathematically dictated discretization of continuous and correlated physical fields.
4 Implementation of the EnKF for WaterGAP and GRACE 4.1 General In the following, we will describe the tailoring of the EnKF approach to the objective of assimilating GRACE data into WGHM. A flow chart of the procedure is shown in Fig. 2. The model always runs globally, but before entering the Kalman filter, the storages are masked out for one specific river basin. This results in a model state vector x according to Eq. (1), which contains the WGHM water storages in 10 individual model compartments (e.g. canopy, soil, surface waters, groundwater) for each grid cell of the river basin of interest. In order to jointly calibrate the model parameters within the assimilation procedure, the state vector is augmented by the WGHM calibration parameters. This means we run the model parameter calibration in a joint step with the assimilation of model states. The state vector then has the dimension n × 1 with n = #cells × #compartments + #parameters with one value being assumed for each parameter per river basin. The measurement operator H establishes the relation between the model state and the total water storages observed by GRACE by accumulating the 10 individual storage compartments for each cell. The m × 1 observation vector d according to Eq. (2) contains the TWS variations derived from the spherical harmonic expansion of monthly GRACE solutions, evaluated at m grid cells (m = #cells). Since the GRACE observations are evaluated on the same 0.5◦ grid cells used in the model, the number of GRACE and WGHM grid cells is equal. Within the Kalman filter, the GRACE TWS anomalies are then upscaled to larger grid cells (see Section 4.2), in order to better match the spatial resolution of the observations and therefore to enable more stable computations.
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Ensemble Generator: - Parameter Model Ensemble X
- Forcing Data
Error Models: - Parameter - Forcing Data
- Water States WGHM Integration
No
Observations Available? Yes Model Prediction Ensemble X-
Measurement Operator H GRACE observations D: Ensemble Kalman Filter
Mapping Operator B
- Gaussian smoothing - full covariance matrix
EnKF Update Ensemble X+
Reinitialization
Fig. 2 Flow chart of the EnKF procedure
4.2 Choice of Spatial Resolution of GRACE Data As mentioned above, the error covariance matrix R of the 0.5◦ GRACE TWS data does not appear well-conditioned for the test region because of the limited resolution of the observations and the applied smoothing. Therefore, it is reasonable to upscale the GRACE observations to J larger grid cells, prior to introducing them into the Kalman filter. As a limit case, for J = 1 the GRACE TWS basin average is used and all sub-basin spatial information is discarded. For this upcaling purpose, we introduce a mapping operator B of dimension J × m into Eq. (2): each row of B has elements a1i with i = 1 . . . nj containing the area fraction for each of the nj gridded values on theP 0.5◦ grid belonging to the larger cell j. The total area of cell j is given by Aj = ai . This results in the following modification of the Kalman gain matrix: ¯ = Ce HT BT (BHCe HT BT + BRBT )−1 B. K
(6)
¯ has the same dimension as K from Since the aggregation is absorbed in Eq.(6), K Eq. (5); we therefore use the same GRACE observation vector (provided on the 0.5◦ cells) for each model run and modify the upscaling by choosing the matrix B accordingly. This is mathematically equivalent to using a shorter J × 1 GRACE
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observation vector and including the mapping operator into the observation operator H. In order to devise a reasonable choice for the upscaled grid size, we ¯ = BRBT of the have investigated the condition of the error covariance matrix R gridded GRACE values for the Mississippi test region. Table 1 shows the matrix ¯ that uses singudimension, rank (as determined by the Matlab function rank(R) lar value decomposition), and condition number for different choices of J. For grid sizes smaller than 2◦ , the error covariance matrix clearly exhibits a rank deficit. Starting from 2◦ , the matrix has full rank, however, the condition numbers are very poor for small grid sizes, rendering stable computations very difficult. Therefore, observing the trade-off between spatial resolution and stability of the matrices, we choose an averaging grid size of 5◦ . This also relates well to the 500 km Gaussian smoothing which we applied to the GRACE solutions. cell size 0.5◦ 1◦ 2◦ 3◦ 4◦ 5◦ 10◦
matrix dim. 1382 357 91 38 25 17 5
rank 128 128 91 38 25 17 5
condition 4.6 · 1020 3.6 · 1019 1.3 · 1014 2.2 · 1007 2.5 · 1005 6.4 · 1003 4.6 · 1001
Table 1 GRACE error covariance matrix: rank and condition depending on the size of the averaging cell
4.3 Generation of the Model Ensemble The initial ensemble to be used within the EnKF is created by Latin-Hypercube sampling of the 22 model parameters (cf. Section 2.1) according to a realistic pdf assumed for each parameter and by perturbation of the monthly forcing data (precipitation and temperature); at the time being we work with N = 30 ensemble members. To perturb the precipitation input fields, a multiplicative error of ±30% was assumed following a triangular distribution centered at one, whereas for the uncertainties of the temperature values we used an additive error of ±2◦ C with a triangular distribution about zero. We have, in [31], studied the sensitivities of the model to variations in individual parameters and regarding the uncertainty ranges introduced for each of the parameters. Variations of the parameters are here considered to account for internal uncertainties of the model as well; in follow-on work we will seek to improve the representation of model uncertainties. The model is run for a spin-up phase of two years (2003-2004) and the resulting model outputs at the end of the second year are stored as initial samples for the assimilation run. The spread of the ensemble during the spin-up phase is shown in Fig. 3 for basin averages of TWS. Afterwards, data assimilation is performed for the year 2005 by running the model for each of the ensemble members and introducing GRACE observations in the framework of the EnKF. The ensemble members differ regarding their initial start values and the different sets of parameters as well as by different sets of
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Original WGHM Samples Ensemble Mean
total water storage [mm]
300
250 200
150
100 50
0 2003
2004 time [years]
2005
Fig. 3 Ensemble spread for two year spin-up phase. The ensemble is generated by sampling of 22 calibration parameters and perturbed forcing data (precipitation and temperature).
perturbed climate forcing data. Monthly averages of water states in the individual storage compartments simulated on the 0.5◦ grid cells within the river basin, plus the model parameters, are stored in the state vectors x for each of the samples. These vectors are then combined in the prediction matrix X. Subsequently, the empirical ensemble model error covariance matrix Ce is derived according to Eq. (3). Together with the observation vector and the corresponding GRACE error covariance matrix of the gridded values, the model results enter the EnKF to calculate the filter update for each of the samples according to Eq.(4). The ensemble mean of the update then represents the assimilated model result for each month, representing the monthly mean assimilated storage values. To restart the model for the next month, the updated state is needed for the last day of the month. Therefore, the bias between prediction and update, for the monthly averages, is added as offset to the model output on the last day of the month for each sample, in order to generate the start ensemble for the next month of model prediction. Furthermore, the model parameters for each sample are replaced by the respective updates of the parameters.
5 Results We have carried out two different versions of assimilation runs. For the first run, we aggregate GRACE to a basin average, in the second run we use the GRACE values averaged to 5◦ grid cells. The latter approach results in 17 larger grid cells covering the Mississippi basin and thus allows for a significantly higher spatial resolution compared to the first approach. The ensemble is created as described in Section 4.3 and assimilation runs are carried out for a one-year assimilation phase during the year 2005. Afterwards, we validate the results by running the model in free mode (i.e. without adding any GRACE information) for a validation phase of three years (2006-2008). In fact, validating data assimilating experiments as we report here represents a non-trivial issue. Validation of the (DA-) calibrated model by predicting the same
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quantity that was assimilated in the DA phase in free runs does not provide an overall verification of the realism of individual storages and fluxes; it solely adds confidence that the model does better in simulating total water storage (in our case). Due to the “equifinality”, or in other words the non-uniqueness of partitioning integral quantities, we cannot exclude that we “get the right result due to the wrong reason”, i.e. that the assimilation fails to mitigate or even causes unphysical internal redistribution of water. The only way out would be to add independent observations of water storage change in compartments such as groundwater, soil moisture, snow, or others. This type of validation has been applied in some of the earlier GRACE-DA studies. However, there are also unsolved problems associated with this, see the discussion in [7]: (1) in-situ observations such as from groundwater wells or river discharge stations are sparse, even for a well-observed region as we consider here, their locations have been chosen based on various criteria, and they may be representative for an area of few km2 only, (2) from direct validation of a single model field (e.g. groundwater storage) it is equally difficult to draw more general conclusions, i.e. regarding the reliability of other unobserved fields or fluxes. In other words, it cannot be ruled out that DA improves only one particular compartment of the model unless we could observe all fields. Very preliminary results with our DA system indicate that post-DA river discharge simulation improves for a number of stations but not for all; a more complete account will be presented in a follow-on publication. Finally, one may resort to observation system simulation experiments (OSSE), where true data are generated from a reference truth model run, perturbed by realistic noise, and the performance of the ensemble DA in reconstructing storages and fluxes can be evaluated. However, whether the outcome of such an experiment can be transferred to the real-data DA depends on both our understanding of the real data noise characteristic and of the model realism; both are still limited in the case of GRACE assimilation into hydrological models.
5.1 Ensemble Error Covariance of WGHM-Derived Storages The EnKF update depends both on the model errors, represented by the ensemble error covariance matrix Ce , and the observation error covariance matrix R. An analysis of Ce helps to understand the uncertainties of the model output for each of the individual storage compartments for each month of the assimilation run. Fig. 4 shows the standard deviations of the model output for five compartments, for different months of the assimilation year 2005 and for the model run assimilating the GRACE values averaged to 5◦ grid cells. Additionally, standard deviations are displayed for TWS, derived by rigorous error propagation. Obviously, the represented uncertainties vary among the different storage compartments, among the different grid cells and also in time. Relatively large uncertainties are revealed for simulated soil moisture throughout the year. For the snow compartment, it is obvious that large uncertainties persist in the winter months (here March and December) in the northern mountainous regions of the Mississippi basin, while no further model spread occurs in the summer months due to lack of snow fall. For simulated surface water bodies, large uncertainties are represented, but for a small number of grid cells only. For example, the course of the Mississippi river is well visible in the ensemble spread for river storage. Considering groundwater, large uncertainties
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appear in the region of the High Plains aquifer, where extensive irrigation and groundwater abstraction is common but poorly represented in the model. In general, those storages where uncertainties are large will most likely dominate in the vertical and horizontal disaggregation of the GRACE TWS. This will be investigated in more detail in Section 5.2. The aggregation of the uncertainties of the individual compartments can be observed in the standard deviations of simulated TWS, as visible in the last row of Fig. 4. Large error bars in the north of the basin and towards the river outlet in the south can be attributed to uncertainties in wetland storage and simulated snow in mountainous regions. Large standard deviations in the HPA region arise from the ensemble spread in the groundwater storage and a certain amount of variability throughout almost the whole basin originates from the soil water compartment. In general, it can be observed that the WGHM model uncertainties tend to decrease from the beginning to the end of the one-year assimilation period. This can be associated with the fact that the ensemble spread reduces with time, as each month each of the ensemble members is nudged towards the GRACE observations. At the same time, the calibration of the model parameters leads to a smaller spread of the parameters with increasing number of calibration months. Therefore, the ensemble spread reduces with time; whether this should be countered by involving ensemble inflation techniques needs to be investigated.
5.2 GRACE Contribution Analysis Within the EnKF, simulations of water distribution in various storages based on the WaterGAP conceptual realization of physical and anthropogenic processes, are merged with actual observations of GRACE TWS variations. Like with all EnKF methods, the relative weighting of the GRACE observations and the model prediction is controlled by their error covariance representations. It is vital for the understanding of the merged storage estimates to be able to quantify the contribution of GRACE within the EnKF. For example, since the EnKF does not conserve mass by definition, the contribution of GRACE also hints to the degree to which the merged storage estimates may violate the mass balance that is prescribed within the original WaterGAP simulation (that, however, may miss physics that is observed by GRACE). In Eq. (4), the gain matrix K controls the actual contribution of GRACE to a merged storage estimate. A formal sensitivity study can thus be based on the time-variable properties of K. For the following investigations we consider the gain ¯ which follows from the aggregation of the GRACE observations to 5◦ matrix K ¯ ij weights the contribution averages as described in Section 4.2. Each element K of the TWS observation to one particular storage compartment of one WGHM ¯ to the gridded GRACE signal reduced by the prediction grid cell. Applying K e = d − Hx, tells how this TWS surplus or deficit is actually distributed, ¯ k = Ke.
(7)
For example, with e containing 1 cm for just one 0.5◦ ×0.5◦ grid cell and zero otherwise, k describes how this localized water volume of (50 km)2 by 1 cm thickness will be distributed vertically and horizontally. However, for ease of interpretation
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we will consider the fate of a basin-wide uniform layer of thickness 1 cm surplus seen by GRACE within the EnKF merging (update) process em×1 = [1 . . . 1]T .
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With this choice of e, k as in Eq. (7) contains the row sums of the Kalman gain matrix, i.e. the sum of the contributions from all GRACE cells to the individual storage compartments of each WGHM cell. Note that adding an observed uniform surplus layer of water would by no means result in a uniform change to the simulated storage within the different storage compartments, nor to the same maps for all compartments, due to both spatial model dynamics and spatially varying error covariances. The effect that a uniform layer creates for each individual element of the WGHM state vector can be mapped for each compartment separately. Fig. 5 shows the sum of the respective contributions (Eq. (7)) of all GRACE grid cells in the Mississippi basin to individual elements of the WGHM state vector for five storage compartments (and summed up for TWS) and for different months of the assimilation run. As expected, contributions show pronounced variability in the space domain, as well as across the different storage compartments. A comparison of Fig. 5 with the standard deviations of the model ensemble (Fig. 4) shows very similar patterns which confirms the assumption that GRACE contributes especially in those cells and storage compartments, where the model uncertainties are large. The ratio between the contributions to the individual compartments also changes in time: For example, in winter large model uncertainties in the estimation of snow mass persist, resulting in a significant part of the GRACE water surplus being transferred to the snow compartment. These results suggest that it appears possible and sensible to disaggregate the integral GRACE TWS observations both spatially and vertically into the individual hydrological storage compartments, as the GRACE information is for each month of the assimilation procedure directed towards the model cells and compartments in which the largest uncertainties exist. As mentioned in Section 5.1, model uncertainties decrease during the course of the year due to a decrease in the spread of the model ensemble. This results in GRACE having less influence on the EnKF update, as can be concluded from the decreasing GRACE contributions visible in Fig. 5 towards the end of the year. Ensemble inflation techniques might be helpful to maintain the influence of the GRACE data throughout the assimilation runs. Indeed it appears that some of the contributions (or increments) resulting from a uniform 1cm surplus (observation-minus-forecast residual) are negative, for example, groundwater in June in the southeastern portion of the Mississippi River basin. Whether this is an artefact of the filter, possible due to the finite number of ensemble members, will require further investigation.
5.3 First Assimilation Results In the following we will compare the results of the two different assimilation runs, one using basin averages from GRACE and one which introduces TWS averages in 5◦ cells as observations. Fig. 6 shows time series of basin-averaged TWS for the one-year assimilation phase (2005): GRACE observations (blue), ensemble mean of the EnKF update (red) for the model run which uses a GRACE basin mean and in purple for the one which uses 5◦ GRACE grid cells. For comparison, the standard
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Fig. 5 GRACE contributions to different WGHM storage compartments for different months of the year 2005.
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WGHM model run without any use of GRACE is displayed in green. Comparing GRACE with the original model run reveals that for the year 2005 the amplitude of the simulated TWS changes is significantly larger than the one observed by GRACE, see also [1]. Both assimilation experiments match the GRACE observations considerably better than the original WGHM model. The spatio-temporal RMS of the differences can be found in the first column of Table 2: assimilation of the GRACE observations reduces the RMS between model and GRACE by more than 70%. For validation, free runs of the model are carried out for three years (20062008) following the assimilation phase. These predictions are then compared to the GRACE observations for the same period. Fig. 7 shows basin averaged TWS derived from the model predictions by the standard model version (green) and by the two versions in which GRACE data has been assimilated during 2005 (black: basin mean assimilated, gray: 5◦ gridded averages assimilated) compared to the GRACE observations (blue). Again it can be observed that the original model overestimates the annual amplitude of water storage changes when compared to GRACE. The amplitude of the data assimilated models fits better to GRACE. When the time series of model predictions and observations are compared, it can be concluded that the RMS of the data assimilated models is smaller by almost one third than the RMS for the original model, even though no additional GRACE information has been included during the validation period (see second column of Table 2). We attribute these differences mainly to an improvement of the model parameters achieved through the calibration, but also to improved start values for the storages in the WGHM compartments due to the assimilation of GRACE the year before. Since GRACE can only observe TWS anomalies, a temporal mean has been reduced both from the observations and the model runs. Therefore, no offset can be detected between the curves for model and observation results. When estimating a trend over the complete (assimilation and validation) period, the original model results have a slight positive trend (4mm/year), whereas for GRACE the
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trend in the Mississippi is almost zero. In the two data assimilated model runs, the trend of the model was reduced to 1.6mm/year (assimilation of basin mean) and 2.6mm/year (assimilation of 5◦ ×5◦ grid values) fitting closer to the observed trend than the original model. However, a river basin with a more pronounced trend signal would be required to make valid assumptions on how the assimilation of GRACE data stabilizes the long-term trend in the model results. Some intraannual variations, as for example the water mass anomalies in September/October 2006 and 2007, that are visible in the GRACE time series, are apparently not well predicted by the model in the free run. The reasons might be model deficiencies that have not been accounted for by the calibration of the parameters after one year of data assimilation. It can be expected that a continuous assimilation of GRACE observations (beyond the year 2005) will make the model more realistic and allow to better capture such short-term variations. model run original WGHM assimilation of basin mean assimilation of 5◦ grid values
assimilation phase 28 (-1.72) 8 (0.60) 9 (0.49)
validation phase 27 (0.37) 20 (0.63) 19 (0.67)
complete period 27 (0.27) 15 (0.66) 17 (0.69)
Table 2 Temporal RMS values in millimeter equivalent water height (EWH) of the differences between GRACE and the original model runs and the two assimilated model runs for the assimilation phase (2005), the validation phase (2006-2008) and the complete period (20052008). In brackets the Nash-Sutcliffe model efficiency coefficient (NSC) applied to total water flux dTWS/dt is shown.
When comparing basin averaged TWS values for the whole Mississippi basin, almost no difference is observable between using basin averaged GRACE observations and exploring the higher GRACE resolution provided by the 5◦ gridded averages. The respective curves (red/purple in Fig. 6 and black/gray in Fig. 7) show a very similar behavior and the corresponding RMS values in Table 2 differ only slightly. In addition, we provide the Nash-Sutcliffe model efficiency coefficient
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WS NSC ([24]) applied to total water flux, dTdt , derived from central differences of TWS anomalies. The NSC measures the fit of modeled water flux to observed data. The NSC ranges between −∞ and 1: in case the NSC is equal to 1, the modeled flux matches perfectly to the observed flux; whereas when it is equal to 0, the modeled quantity is as accurately reproduced by the mean of the observed data as by the data themselves. A coefficient less than zero indicates that the observed mean represents the flux better than the model prediction. Here, we observe that the data assimilated model indeed represents the observed fluxes very well, in line with the RMS; this is indicated by an NSC > 0 in Tab. 2, whereas in case of the standard model the observed mean value is more accurate than the model prediction (NSC < 0) within the assimilation phase.
To better investigate the impact of the higher spatial resolution of the GRACE observations, we then calculated spatial RMS values for the differences between the 0.5◦ ×0.5◦ values of the different TWS model outputs and the (i) basin mean GRACE observations or (ii) the 5◦ ×5◦ grid GRACE observations, respectively, for each month. Fig. 8 illustrates the evolution of these spatial RMS values for the assimilation and the validation phase. Green and blue curves show the spatial
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Fig. 8 Temporal evolution of spatial RMS of model differences compared to GRACE observations for original WGHM (green: compared to basin average of GRACE, blue: compared to 5◦ gridded GRACE averages) and data assimilated model (red/black: assimilation and validation phase when the basin mean was used for assimilation, purple/gray: assimilation and validation phase when 5◦ GRACE observations were used for assimilation.).
RMS of the 0.5◦ TWS output of the original model run compared to (i) one uniform GRACE basin average value (green) and (ii) compared to the 5◦ gridded GRACE averages (blue). Here, no obvious difference can be observed, which is also confirmed by the spatio-temporal RMS of these curves (first row compared to third row of Table 3).
Assimilation of GRACE data into WGHM model run original WGHM - GRACE BM assimilation of basin mean original WGHM - GRACE 5◦ assimilation of 5◦ grid values
assimilation phase 130 98 126 66
21 validation phase 106 64 104 61
complete period 112 74 110 62
Table 3 Spatio-temporal RMS values in millimeter equivalent water height (EWH) of the differences between GRACE and the original model runs and the two assimilated model runs for the assimilation phase (2005), the validation phase (2006-2008) and the complete period (2005-2008).
The red/black curves show the spatial RMS of the model run using basin averaged GRACE observations for the assimilation and for the comparison and the purple/gray curve shows the same for 5◦ GRACE observations. We find that the spatial RMS of both assimilated model runs with respect to GRACE is significantly lower than for the original model version. This is true during the assimilation phase as well as during the validation phase and it is confirmed by the spatiotemporal RMS of the respective curves (second/fourth compared to first/third row of Table 3). The second result is the strong influence of the higher spatial resolution provided by the 5◦ gridded averages during the assimilation phase. The 5◦ assimilated solution is much closer to GRACE during the assimilation phase compared to a basin-average assimilated solution. A temporal RMS of 98 mm, i.e. an improvement of 25% to the original model in contrast to 66 mm (improvement of 48% to original model) demonstrate that the use of higher resolution GRACE observations helps to improve the assimilation results.
Finally, Fig. 9 shows, for different months of the assimilation period, grids of GRACE TWS (top), simulated TWS provided by the standard WGHM model (middle) as well as from the assimilation of 5◦ GRACE observations (bottom). The RMS of the differences between GRACE (averaged to the 5◦ cells) and the two model versions have been shown in Fig. 8. It can be observed that assimilation of GRACE effectively reduces the seasonal amplitude, as expected. As an example, in March the original model simulates large TWS in the northeast and in the south of the Mississippi basin. These are at the same time regions in which GRACE is expected to have a large contribution from the analysis of the gain matrix (Fig. 5) due to large uncertainties in the model (Fig. 4). In the assimilated model version TWS anomalies are reduced and the model prediction fits better to the GRACE observations. We find that GRACE informs the model at much smaller spatial scale than the GRACE resolution of few 100 km implies, as also described by. e.g., [41] and [20]. When interpreting Fig. 9, one should keep in mind that results of both the standard run and the assimilation run also depend on the evolution of the model states in the preceeding months, which are not shown here. As a consequence, the updated model state is computed from a combination of GRACE with the prediction of the assimilation run and not with the respective month of the standard model.
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Fig. 9 TWS observations by GRACE (top) compared to TWS simulated by the WGHM standard model (middle) and results from the Kalman filter update (bottom) for different months of the assimilation phase 2005.
6 Conclusions and Outlook We introduce a new EnKF-based approach to GRACE data assimilation into hydrological models. Unlike with previous studies, our approach 1) calibrates model parameters simultaneously with state assimilation, and 2) capitalizes on the spatial resolution of GRACE by utilizing gridded total water storage data, while 3) allowing for rigorous error propagation from the monthly GRACE spherical harmonic coefficients. We represent hydrological model uncertainties due to climate forcing, unknown model parameters and uncertain states. The approach is then
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used to integrate GRACE data into the global hydrological model WGHM, for the Mississippi river basin, and a validation experiment is carried out. Since the underlying problem is inherently ill-posed, fully populated, ill-conditioned model and data error covariance matrices have to be represented and combined in the EnKF. This requires some measure of numerical stabilization; here we investigate the effect that aggregation of GRACE data to catchment averages or coarser grid resolution (upscaling) prior to assimilation create. To this end, we first provide an analysis of the GRACE variance-covariance matrix for the Mississippi region. We find that for spatial aggregation of GRACE TWS anomalies to grids of 200-500 km resolution, the condition number of this matrix appears reasonable to facilitate informing a model; this may change of course when improved GRACE level 1 data reprocessings become available or when improved level 2 decorrelation procedures are employed. Contribution measures also reveal how the integrated GRACE signal is disaggregated spatially and vertically, i.e. to the different storage compartments simulated in the model. In our setup, we study the filter gain for the reference case that GRACE observations suggest a basin-wide uniform water surplus compared to modeling. We find that GRACE informs the model at much smaller spatial scale than the GRACE resolution of few 100 km implies, and that the GRACE impact strongly varies between the individual WGHM grid cells and across the storage compartments. In fact, since model calibration parameters are not observed directly, we hypothesize that correctly representing correlations between these parameters and the model states through the ensemble is critical for the success of the approach. Furthermore, the gains and the resulting disaggregation changes in time depending on the season of the year and the number of GRACE months ingested. This is not surprising, since in an optimal state estimator the GRACE information for each time step is bound to modify those model cells and compartments where, at this time, the largest model uncertainties persist. We are confident that our method provides a step towards an effective spatial and vertical disaggregation of GRACE TWS information. Furthermore, we find that spatial upscaling of GRACE TWS to coarser grids or basin averages effectively stabilizes the EnKF disaggregation scheme: generally, coarser grids lead to improved condition numbers. We find that assimilating gridded GRACE data (at 5◦ resolution) causes the assimilated runs to follow GRACE closer in spatial domain compared to assimilating GRACE-derived basin averages; this does not become visibile when one would consider basin averages post-assimilation only. With more extensive validations pending, we conclude that assimilation of GRACE at spatial resolution of a few hundred km appears numerically feasible and provides results that appear superior to the assimilation of basin averages in our test set-up; i.e. in predicting total water storage anomalies. To this end, we created a scenario with two years spin-up phase, one year of assimilating GRACE into the WGHM model while calibrating several hydrological parameters, and then simulating three more years of water storage variations with the calibrated model and comparing them to GRACE. Follow-on work will address the simulation and representation of model uncertainties, e.g. with respect to anthropogenic forcing, and we will work on optimizing the generation of ensembles. Moreover, the spatial resolution at which GRACE TWS grids are introduced, and the degree of spatial filtering or decorrelation that is required prior to assimilation, requires further investigation. A trade-off between
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numerically stable computation and loss of spatial resolution has to be balanced. However, we would like to point out that for this to be addressed, it appears indeed mandatory to implement the full GRACE spatial correlation structure as it was done in this study. Optimal state estimation, as implemented through the EnKF, does not conserve mass. The GRACE contribution measures analysed in this study indicate the magnitude of violation of the terrestrial water balance, i.e. to what extent the updated storages are not being balanced by precipitation, evapotranspiration and runoff fluxes. A possible remedy for this problem was suggested by [25], and more investigations are required to understand whether such a constraint would be helpful in the joint analysis of GRACE data and models. Finally we mention that, possibly most important, assimilated model results will be validated against independent data sets, such as river discharge, lake level or groundwater in-situ observations. This will be subject of an upcoming publication. Acknowledgements The support of the German Research Foundation (DFG) within the framework of the Special Priority Programme “Mass transport and mass distribution in the Earth’s system” (SPP1257) is gratefully acknowledged. Furthermore, we acknowledge two anonymous reviewers and the editor, Prof. Sneeuw, whose suggestions helped to improve the manuscript.
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