1. Introduction
Addressing continental-to-global-scale hydrologic problems that are at the core of the World Climate Research Programme (WCRP) Global Energy and Water Cycle Experiment (GEWEX; Schaake 1994) and the National Aeronautics and Space Administration (NASA) Energy and Water Cycle Study (NEWS; Houser and Entin 2006) requires the use of remote sensing data (McCabe et al. 2008; Rodell et al. 2004). However, remote sensing observations are insufficient to understand the spatial and temporal mean and variability in the water cycle variables because of varying sensor spatial scales, different observing times for different variables, and remote sensing retrieval errors. This results in inconsistencies in the water and energy cycle budgets when estimated solely using remote sensing retrievals (McCabe et al. 2008; Pan et al. 2008). Although land surface models (LSMs) have consistency by construct, they require inputs that are often unavailable in many areas of the globe (e.g., Africa and the Arctic); therefore, there is a need to merge at continental scales remote sensing retrievals into land surface models.
There has been considerable work in assimilating remote sensing data into land surface models (e.g., Crow and Wood 2003; Reichle and Koster 2005; Margulis et al. 2006; Slater and Clark 2006). To date, the assimilation of actual remote sensing retrievals have been done grid by grid, ignoring the spatial structure in the retrieval errors. Pan et al. (2009) developed a multiscale data assimilation system to dynamically merge continental-scale remotely sensed retrieved variables into an LSM. The focus of Pan et al. (2009) is the implementation of the multiscale filtering algorithm (Zhou et al. 2008; Zhou 2006; Frakt and Willsky 2001; Willsky 2002) for applications in land surface hydrology by developing the necessary techniques to automatically build a proper multiscale tree that is 1) suitable for an arbitrary computing grid, 2) topologically balanced, and 3) structurally efficient for assimilating measurements at different resolutions. The goal of Pan et al. (2009) is primarily to prove the concept of multiscale assimilation for land surface hydrology—that is, its feasibility, applicability, and strength. Pan et al. (2009) used precipitation ensembles from a climate model to force the land surface model, which worked well to test the multiscale ensemble assimilation; however,the feasibility of implementing such a system that relies on remotely sensed precipitation has not been investigated. Here in this paper, we use remotely sensed precipitation retrievals from the TRMM satellite in the multiscale assimilation system presented in Pan et al. (2009), resolving issues in generating the ensemble from the satellite data and evaluating the proposed assimilation system for regional-scale hydrologic studies.
The major challenge in constructing such an ensemble assimilation system is the generation of input forcing ensembles (e.g., for hydrology, the rainfall) given a single map of remotely sensed values. These input ensembles are fed into an LSM to produce the prior (forecast) state estimate and the information on state errors (magnitude and correlation structure). The quality of the input ensemble determines the quality of prior state ensembles—that is, its error correlation structure—and thus the quality of the assimilation. The multiscale approach works best with large-scale problems—that is, when the state vector has a very high dimension (equal to the number of pixels in the computing grid times the number of states at each pixel) that may reach 104–106. For ensemble data assimilation, the same high-dimensional spatial rainfall fields need to be generated as ensembles. There are two major requirements for the ensembles: 1) ensembles must contain a proper multiscale cross-member spatial correlation structure and 2) each ensemble member must present a realistic rainfall process in both space and time. The latter means that there should be a reasonable time evolution of storms. This property is important because if storms that exist in one time step are unrelated to the next then the LSM would recharge the soil at random places in different time steps and eventually the moisture would be smeared across the domain as the model integrates forward, making ensemble members indistinguishable from each other. In the sections to follow, the ensemble generation will be first discussed with a novel generation method proposed, and the ensembles so generated will be used to force the assimilation experiments. The multiscale spatial correlation structure in the rainfall ensemble will be tested in the experiments, as well as how this correlation structure is translated into the correlation structure of the soil moisture state errors by the LSM and how it affects the assimilation of soil moisture observations.
2. Rainfall ensemble generation
a. Background and previous approaches
Because of the nonstationary and intermittent nature of rainfall processes, ensemble generation of rainfall over large domains for assimilation purposes has been a difficult problem, and no standard or de facto approach exists. Usually, two classes of approaches are taken: generating rainfall fields using stochastic models or using dynamic models. The first approach assumes a probabilistic behavior of rainfall and uses Monte Carlo simulation to generate random realizations (e.g., Sivapalan and Wood 1987). The probabilistic models can be parametric or nonparametric. The parametric models are often complicated, with space–time correlations that capture the multiscale structure of synoptic scales/mesoscales with embedded convective cells that are often represented as occurring as a Poisson process in space (Sivapalan and Wood 1987). For example, storm dynamics can be modeled with an exponential decay of rainfall in time and Gaussian clustering of storms in space (Chatdarong 2006), the conditional rain rate probability can be based on a multiple-point process (Wójcik et al. 2008), and the spatial correlation of rain rates can be distance dependent (Villarini et al. 2009). Nonparametric rainfall models often resort to random sampling of historical fields and rescaling of these rain fields to adjust the distribution of rainfall totals. Examples of this approach are the Schaake shuffle (Clark et al. 2004), cumulative distribution function (CDF) matching, and similar procedures being used in downscaling research (Luo and Wood 2008).
The second approach for generating rainfall ensembles is to run multiple realizations of a dynamic atmospheric model—for example, a weather model—with perturbed initial conditions or forcing inputs to produce ensembles of rainfall forecasts. Such an approach provides rainfall ensembles that follow the physics of the underlying model and thus should be physically realistic both in the spatial distribution of rainfall depths and in the temporal evolution of the storms. This approach should provide very good spatial correlation among ensemble members because the ensemble spread and its intermember correlation come directly from the uncertainties that are built into the initial conditions, forcing fields, and model physics. However, it is computationally very expensive to generate ensembles using a regional atmospheric model. Ensemble simulations of atmospheric models are being carried out at all weather and seasonal climate centers, such as the National Oceanic and Atmospheric Administration (NOAA) National Centers for Environmental Prediction. The spatial resolution of seasonal climate models like the Climate Forecast System (Saha et al. 2006) or the NOAA global weather model (the Global Forecast System) are too coarse relative to the needs of LSM remote sensing applications, and their higher temporal fields are often not archived for application users. Regional weather models do provide a comparable resolution—for example, the Weather Research and Forecasting Model (Skamarock et al. 2008)—though the ensemble simulations are computationally expensive at large scales. Even when ensemble dynamic modeling is possible, it is so far impossible to condition the predictions upon the satellite observations—that is, to force the dynamic model to produce forecasts that are scattered around the observations.
After a review of these previous studies on ensemble generation, we determine that the approach of random sampling from historical rainfall fields offers for our current study the best physical consistency of rainfall storms in individual ensemble members. The main challenge with this approach is related to the high dimensionality of the rainfall field that must match the modeling grid, which makes the requirement on sample size very high and consequently the sampling process slow.
b. Rainfall ensemble generation with pattern-based sampling and CDF matching
In this study, the remote sensing assimilation experiment is carried out over the same domain as was used in Pan et al. (2009)—the Red–Arkansas River basin in the central United States. Figure 1 shows the latitude and longitude range of the basin. This is a relatively large basin that covers ∼645 000 km2—a size that is ideal for this study because it is large enough to observe multiscale phenomena in hydrology and to test the efficiency of the assimilation algorithm; however, it is not too large so that no more than one mesoscale precipitation system is expected to exist at any one time. Satellite-derived rainfall fields from the Tropical Rainfall Measuring Mission (TRMM) project are used—specifically the real-time TRMM product version 3B42RT. This product combines multiple satellite data sources (both microwave and infrared) in the estimation of the rainfall fields (Huffman et al. 2007). TRMM-3B42RT is available at a 0.25° spatial resolution and 3-h temporal resolution.
The LSM used in this study is the Variable Infiltration Capacity Model (VIC; Liang et al. 1994, 1996), and it is configured to run at a 0.25° computing grid that results in 1062 pixels over the Red–Arkansas River basin study domain (see Fig. 1) The model time step is set to hourly because the model parameters available are calibrated at hourly step (Troy et al. 2008), with the assimilation of the 24-hourly soil moisture fields occurring at 1900 UTC, which matches the ascending overpasses of the Advanced Microwave Scanning Radiometer on board the Earth Observing System (AMSR-E) for soil moisture retrievals.
Given the above experiment setting, an ensemble of rainfall fields needs to be generated for every 24-h period for input into the LSM. A particular random sample (i.e. ensemble member) consists of 24-hourly fields covering a 16° × 9° rectangular area that encompasses the basin, because rainfall features are not limited by the basin boundary. This results in each rainfall field consisting of 64 × 36 = 2304 pixels (0.25°), with one 24-h random sample having the dimension 2304 × 24 = 55 296. Such a high-dimensional sample not only makes it difficult to compare among ensemble samples or to search for good samples but it requires a very large rainfall database from which to sample. However, rainfall events do not take arbitrary shapes, and the effective dimension is not that high, because most events follow specific patterns. Therefore, it is feasible to reduce the dimensionality of the sample by aggregation. Also, by categorizing rainfall events into different patterns, a long rainfall history can be divided into different groups according to their patterns, which can be used to limit the number of days from which samples are drawn.
1) Spatial/temporal aggregation to reduce dimensionality
2) Pattern classification to limit sample size
Figure 2 shows an example of pattern classification for NP = 16, together with the fraction of the total records that belongs to each of the patterns. These 16 patterns in Fig. 2 basically summarize the possible location/distribution of storms in this area. Note that the eighth pattern, the no-rain scenario, makes up ∼60% percent of the days, which is reasonable because most days are nonrain days. Figure 3 shows the daily rainfall patterns for some typical days that are classified as the fourth pattern, labeled with dates.
3) CDF matching to adjust rainfall intensity distribution
4) Ensemble spread control, correlation structure, and other issues
A fundamental concern is what kind of uncertainty/error this pattern-based sampling method generates and how to control it. Some generation schemes (Wójcik et al. 2008; Villarini et al. 2009) try to parameterize and simulate the sensor errors, whereas others target model forecast uncertainties (Luo and Wood 2008). The sampling approach presented here makes no comparison between retrievals or forecasts to any “truth” reference, and therefore it is unable to simulate any type of errors. Instead, it focuses on the uncertainty of the rainfall process itself, with the long-term record providing the rainfall climatological values. If the sampling is performed over the entire record, then the ensemble spread will reflect the climatological uncertainty of the rainfall process over the study domain. When sampling is constrained within a specific pattern, the ensemble spread reflects the conditional uncertainty of the rainfall process given one event type. The number of patterns NP determines the sampling spread across events from one classification. We can estimate the spread of one pattern (one rainfall event type) as 1/NP times the climatological spread. Note that the spread differs from one type to another; for example, the no-rain type would have a much smaller spread than the spread for a heavy-storm type. Here the ensemble spread is measured relative to the climatological uncertainty only; therefore, if a certain amount of uncertainty is needed in the ensemble, then we need to translate it to the corresponding NP or 1/NP first. Once NP is determined, then the ensemble spread is fixed no matter how large or small an ensemble will be drawn from it (i.e., it is not related to ensemble size NE), because samples are always drawn from within the same pattern. The correlation structure should be fixed for a specific pattern. Different from some parametric methods, in which stationarity and isotropy are assumed for the errors, our sampling approach generates a correlation structure that is event-type specific and nonstationary. Figure 5 gives the correlation matrices—that is, normalized covariance ρij = σij/(σiiσjj)1/2—computed for each pattern (using all the samples in that pattern; NP = 16). All of the matrices are 144 × 144 with the 144 pixels in the 2D field stretched along the side (i.e., at 1° resolution). Although it is difficult to match the matrices to what happens in the field, we can still see that high/low correlation happens at different places among different patterns, indicating the nonstationarity in time (event) and space.
The many advantages mentioned above rely on a long record length for a complete sampling of the rainfall climatological values. However, for such a high-dimensional problem, 10 591 days is still relatively short. Therefore, in practice, NP cannot be too large; otherwise some low-probability event types get very few samples and the sampling becomes unstable. This poses a lower limit of the ensemble spread that can be achieved. To obtain tighter ensembles, one can set a maximum distance measure dmax such that all the days sampled are not farther than dmax away from the TRMM-3B42RT observations. The tightest ensemble so generated would be the NE nearest neighbors in the historical record—the hard limit for pure sampling. For even tighter ensembles, additional manipulations must be performed—for example, linear or nonlinear rescaling to shrink the ensemble toward TRMM-3B42RT.
3. Data assimilation experiments and results
Many assimilation experiment settings have been described in section 2, and some of them are similar to those used in Pan et al. (2009). The same domain (Red–Arkansas River basin), computing grid (1062 pixels at 0.25°), and hydrologic model (VIC) are used. Major differences include using satellite-retrieved rainfall for the estimation experiment and that no “synthetic” truth is assumed. Instead, the LSM soil moisture fields, driven by ground-based rainfall, are used as the reference for a truth check. The ground-based rainfall comes from the retrospective forcing fields prepared for phase 1 of the NLDAS project (Mitchell et al. 2004), named “NLDAS” as opposed to “NLDAS-2”, which is a combination of gauges and radar (Cosgrove et al. 2003). These rainfall data draw from more than 13 000 daily gauge reports and gauge-corrected hourly Weather Surveillance Radar-1988 Doppler (WSR-88D) radar estimates (for time disaggregation) over the contiguous United States. It is considered to be the best ground truth we can obtain in our study and should be sufficient to provide a truth check for this remote sensing assimilation experiment. VIC simulation forced with NLDAS-based rainfall is called a “reference” run here instead of a truth. Other significant differences in this experiment relative to Pan et al. (2009) are that the model predictions are hourly instead of daily to work with the remote sensing measurements and that the measurements are assimilated every 24 time steps (once per day) instead of every step as in the earlier study. The experiment period is the three summer months, 1 June–31 August 2004. Ensembles of 20 members (NE = 20 and NP = 25) are generated and used in the assimilation experiment. Two kinds of initializations are used prior to the assimilation: LSM spinup and an ensemble spinup. VIC is first run using a single observation-based forcing from 1 January to 31 May 2004 to minimize the effect of inaccurate initial conditions, and then it is run in ensemble form from 1 June to 30 June 2004 without any assimilation to achieve a stable ensemble spread. Measurements for the top-layer soil moisture (10 cm deep) are generated by adding Gaussian perturbations (zero mean and 3% standard deviation) to the reference simulation forced with NLDAS-based rainfall. These measurements are then assimilated into the model at 1900 UTC every day (at the AMSR-E ascending overpass) from 1 July to 31 August. Although no synthetic truth is assumed, the experiments are not purely “real” either, because no real AMSR-E soil moisture estimates are used.
Because one important goal of the study is to test the multiscale spatial correlation structure in the rainfall and consequently in soil moisture state errors, and how it affects the assimilation, we first look at the magnitude of the error correlations in the top-layer soil moisture ensemble. Figure 6 shows an example of the error correlation in top-layer soil moisture versus interpixel distances. Because the error correlation matrix is a large (1062 × 1062) symmetric matrix, only the statistics (median, 25th, and 75th percentiles) are computed from the lower triangle (diagonal included) of the correlation matrix using 200-km bins. The distance-lagged correlation structures before and after assimilation—that is, both the prior correlation (solid line/boxes) and posterior correlation (dashed line/boxes)—at 1900 UTC 30 July 2004 are computed. An immediate observation is that a significant amount of spatial correlation exists in the prior soil moisture errors. The median correlation at interpixel distance of ∼600 km is about 0.2, large enough to have an effect on the assimilation. This relatively strong horizontal coupling in soil moisture errors suggests that a measurement taken as far as 600 km away may still carry some information about the pixel being updated. Thus, such horizontal coupling should be considered in soil moisture assimilation practice, which low-dimensional uncoupled filters may fail to do, as was also found in Pan et al. (2009). Second, the overall strength of the spatial correlation drops sharply after the update—the mean error correlation at all distances drops from 0.35 to 0.05. This suggests that the spatial correlation in soil moisture errors is more associated with the rainfall errors than with the intrinsic characteristics of the soil moisture field, and once the assimilation update reduces the ensemble spread around the measurement, the error correlation also diminishes.
Figure 7 summarizes the assimilation in time series. Because of the relatively long length of the experiment (2208 time steps), only a period of approximately three weeks (26 July–13 August) is illustrated. The period covers a couple of significant rainfall events and a long dry-down. The top panel in Fig. 7 shows the basin-averaged ensemble-mean top-layer soil moisture and the basin-averaged ensemble standard deviation (STD). The growth of the ensemble spread during the 24-h periods between every two filter updates is very clearly expressed by the error bars (STD), as is the reduction of the soil moisture ensemble spread at every update time (1900 UTC). The ensemble spread grows between assimilation periods and during large rainfall events (e.g., 28–30 July and 10–11 August). The ensemble spread becomes much smaller during the dry-down period in the beginning of August. This behavior reflects the nonstationary nature of the rainfall ensemble generation, which determines the extent of spread based on event type and climatological values (i.e., a large spread for large storm events and small spread for dry days), and the response of the LSM to rainfall or to dry conditions.
The middle panel compares the open-loop simulation (no measurements assimilated; thick dashed line), the ensemble mean (thick solid line), and the NLDAS rainfall–driven reference simulation (thin solid line). The improvement in the soil moisture estimation made by the assimilation against the open-loop run can be seen in this panel. Such an improvement is significant, especially during the period from 25 July to 4 August, when the biased low soil moisture is well corrected through assimilation.
The bottom panel draws the time series of mean spatial correlation in soil moisture errors (over all interpixel distances). The correlation magnitude grows in the 24-h periods between updates and decreases significantly after assimilating new measurements—confirming the observation that the spatial correlation in soil moisture errors comes primarily from the rainfall-forcing ensemble and will diminish as the soil moisture ensemble spread decreases upon the filter update.
The basin-averaged root-mean-square errors (RMSE) in top-layer soil moisture computed against the NLDAS-driven reference run is 2.93% for the open-loop run and 2.29% for the assimilation run (22% error reduction). The results are based on the one-overpass-per-day configuration, whereas AMSR-E can provide two overpasses per day. The assimilation experiment with two overpasses per day (0700 and 1900 UTC) brings the RMSE down to 2.09% (29% error reduction). All of the absolute RMSE values—2.93%, 2.29% and 2.09%—are small, as are the 3% error added to create the synthetic soil moisture measurements when compared with the error levels reported for soil moisture retrievals (Bindlish et al. 2003; Gao et al. 2006). Note that all these small numbers are not unreasonable because they are all relative to the soil moisture dynamics of VIC LSM simulations. The VIC model drains its 10-cm top layer only by gravity and evapotranspiration, and its soil moisture value is usually between 20% and saturation (∼45%). The dynamic range of VIC surface moisture is considerably smaller than remotely sensed, X-band (10.67 GHz), thin (∼1 cm) top-layer moisture, which sometimes reaches <5% (Gao et al. 2006). The RMSEs observed in this study should be rescaled by a factor of 1.5–2 to obtain the equivalent values in remote sensing terms.
4. Conclusions
We design and test a nonparametric rainfall ensemble generation scheme for a multiscale land surface hydrologic assimilation system such that remotely sensed rainfall data can be used. The random sampling/CDF matching approach proposed here provides a computationally efficient way for generating realistic rainfall ensembles. The rainfall uncertainties (i.e., errors) so generated are nonstationary in both time and space, and the uncertainty in the rainfall errors are no longer assumed to be normal or lognormal but are measured relative to the climatological uncertainties of the rainfall in the region. Assimilation experiments conducted show that this generation method works reasonably well with the remote sensing rainfall products being used (TRMM-3B42RT). In addition to using remotely sensed rainfall, all of the assimilation experiments are configured to work with the soil moisture retrievals from an operational sensor like the AMSR-E onboard NASA’s EOS Aqua satellite. Although no real AMSR-E retrievals are used in this paper, the experiments with the LSM-created measurements illustrate good potential of having a remote sensing–based multiscale assimilation system for improving soil moisture estimations.
The ensemble generation method proposed has its limitations. It relies on the existence of a long-term subdaily rainfall database, which poses a lower bound on the ensemble spread that can be achieved and also limits its application in some data-limited areas. The potential use of nonobservational rainfall databases—for example, the European Centre for Medium-Range Weather Forecasts “ERA-Interim” global reanalysis (Simmons et al. 2006) or the statistically interpolated 50-yr global surface meteorological records developed in Sheffield et al. (2006)—may help to alleviate this limitation. The length of the historical record also puts a limit on domain size (number of pixels), and our experiments are all at regional scale so far. Therefore, this method may only be applied to larger-scale (continental or global) applications on a region-by-region basis. The uncertainties built into the ensembles are relative to the regional rainfall climatological values and do not directly reflect the satellite sensor errors, which is sometimes expected in data assimilation. Overall, this study serves as a good proof of concept for the remote sensing–based multiscale assimilation system, even though additional issues—for example, the sensitivity of assimilation performance on retrieval accuracy, spatial availability, and revisit frequency—need to be resolved.
Acknowledgments
This research is supported by National Aeronautics and Space Administration (NASA) Grants NNG06GD79G (“Estimating Continental-Scale Water Balances through Modeling and Assimilation of EOS Terra and Aqua Data,” in collaboration with the University of Washington) and NNX07AK41G (“Assessment of the Predictive Skill of GPM-ERA Precipitation Estimates for Hydrologic Applications”).
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The 1062-pixel computing grid at 0.25° resolution for the study area (fine pixels within the basin) and the “multiscale tree” defined for the grid. The 16 × 9 rectangular coarse grid at 1° resolution used in rainfall ensemble generation is delineated by the thick gray lines.
Citation: Journal of Hydrometeorology 10, 6; 10.1175/2009JHM1155.1
Rainfall patterns identified in the experiment, with NP = 16. The percentage value next to the pattern number in each panel is the fraction of samples that are classified as that pattern. Note that the eighth pattern, the no-rain pattern, contains the most samples (59.38%).
Citation: Journal of Hydrometeorology 10, 6; 10.1175/2009JHM1155.1
Days in the historical record that are classified as the fourth pattern in the experiment, with NP = 16 (only 25 of them are shown in the figure because of limited space). The shading scale is the same as in Fig. 2.
Citation: Journal of Hydrometeorology 10, 6; 10.1175/2009JHM1155.1
(a) The daily total TRMM-3B42RT observations (center-top panel) and rainfall ensemble members (20 panels below) generated for the 24-h period from 1900 UTC 11 Jul to 1900 UTC 12 Jul 2004, using NP = 25 and NE = 20. (b) The time series of basin average rainfall corresponding to the same TRMM-3B42RT observation (center-top panel) and rainfall ensemble members shown in (a).
Citation: Journal of Hydrometeorology 10, 6; 10.1175/2009JHM1155.1
Correlation (normalized covariance) matrices computed for each different rainfall pattern (using all of the samples in that pattern) in the NP = 16 experiment. Each matrix is 144 × 144 with the 144 columns/rows corresponding to the 16 × 9 1° pixels.
Citation: Journal of Hydrometeorology 10, 6; 10.1175/2009JHM1155.1
Distribution of error correlation in top-layer soil moisture along different interpixel distances. The 1062 × 1062 error correlation matrix is computed for the model time step at 1900 UTC 30 Jul both before and after the update (prior/posterior), and the statistics in this figure are computed using the values in the lower triangle (with the diagonal included) of the 1062 × 1062 correlation matrix (symmetric) and 200-km bins. The lines and the upper/lower edges of the boxes indicate the median, 75th, and 25th percentiles of the distribution in that bin, respectively. Solid-line (dashed line) boxes are for the prior (posterior) error correlation. The mean error correlation at all distances is 0.35 and 0.05 for the prior and posterior, respectively. The histogram of interpixel distances is plotted in light gray in the background.
Citation: Journal of Hydrometeorology 10, 6; 10.1175/2009JHM1155.1
(top) Basin-averaged ensemble-mean top-layer soil moisture (thick solid line), basin-averaged ensemble STD (error bars), and basin-averaged ensemble-mean precipitation (gray bars). (middle) Basin-averaged ensemble mean (thick solid line) vs the open-loop run (thick dashed line) and NLDAS rainfall–driven reference run (thin solid line). (bottom) Mean spatial correlation in soil moisture errors, computed from the same lower triangle of error correlation matrices as was used in Fig. 6.
Citation: Journal of Hydrometeorology 10, 6; 10.1175/2009JHM1155.1
