Assimilating remotely sensed snow observations into a macroscale hydrology model

Transcription

1 Advances in Water Resources 29 (2006) Assimilating remotely sensed snow observations into a macroscale hydrology model Konstantinos M. Andreadis *, Dennis P. Lettenmaier Land Surface Hydrology Research Group, Department of Civil and Environmental Engineering, Box , University of Washington, Seattle, WA , United States Received 1 February 2005; received in revised form 29 July 2005; accepted 2 August 2005 Available online 25 October 2005 Abstract Accurate forecasting of snow properties is important for effective water resources management, especially in mountainous areas like the western United States. Current model-based forecasting approaches are limited by model biases and input data uncertainties. Remote sensing offers an opportunity for observation of snow properties, like areal extent and water equivalent, over larger areas. Data assimilation provides a framework for optimally merging information from remotely sensed observations and hydrologic model predictions. An ensemble Kalman filter (EnKF) was used to assimilate remotely sensed snow observations into the variable infiltration capacity (VIC) macroscale hydrologic model over the Snake River basin. The snow cover extent (SCE) product from the moderate resolution imaging spectroradiometer (MODIS) flown on the NASA Terra satellite was used to update VIC snow water equivalent (SWE), for a period of four consecutive winters ( ). A simple snow depletion curve model was used for the necessary SWE SCE inversion. The results showed that the EnKF is an effective and operationally feasible solution; the filter successfully updated model SCE predictions to better agree with the MODIS observations and ground surface measurements. Comparisons of the VIC SWE estimates following updating with surface SWE observations (from the NRCS SNOTEL network) indicated that the filter performance was a modest improvement over the open-loop (un-updated) simulations. This improvement was more evident for lower to middle elevations, and during snowmelt, while during accumulation the filter and open-loop estimates were very close on average. Subsequently, a preliminary assessment of the potential for assimilating the SWE product from the advanced microwave scanning radiometer (AMSR-E, flown on board the NASA Aqua satellite) was conducted. The results were not encouraging, and appeared to reflect large errors in the AMSR-E SWE product, which were also apparent in comparisons with SNOTEL data. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Data assimilation; Land surface modeling; Snow 1. Introduction Snow plays a key role in the hydrologic cycle over large areas of the mid latitudes, through its effects on water storage and surface albedo. In the western United * Corresponding author. Tel.: addresses: kostas@hydro.washington.edu (K.M. Andreadis), dennisl@u.washington.edu (D.P. Lettenmaier). States snowmelt accounts for about 75% of the annual runoff [3]. Consequently, accurate estimation and monitoring of snow properties, such as snow coverage and water equivalent, have important implications for water resources management. Surface observations, such as snow courses and automated in situ measurement devices like snow pillows, are unable to capture fully the considerable spatial and temporal variability in snow properties over large areas. For this reason, large scale strategies for observing snow properties rely heavily on remote sensing [47] /$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi: /j.advwatres

2 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) The contrast in reflectance between snow-covered and snow-free areas makes the estimation of snow extent straightforward using visible wavelength sensors. Indeed, operational snow cover maps have been produced since 1966 using the NOAA polar orbiting and the Geostationary Orbiting Environmental Satellite (GOES) satellites [20]. The major limitation to these methods are that they require cloud-free conditions, and, depending on their spatial resolution, can have difficulty identifying snow under vegetation, especially trees. With the advent of the Earth Observing System (EOS) era and launch of the NASA Terra satellite in 1999, research focus has shifted to snow extent products based on the moderate resolution imaging spectroradiometer (MODIS) which has higher spatial and spectral resolution, and hence improved cloud discrimination and snow detection under vegetation canopies as compared with AVHRR and GOES-based products [19,33]. Validation studies have shown that although MODIS snow covered area products are significantly improved relative to operational snow mapping products, misclassification errors still exist [33,23]. Furthermore, visible wavelength images lack any information about hydrologically more useful variables, like water equivalent, while cloud cover precludes snow mapping thus introducing temporal discontinuities in the data. Passive microwave remote sensing can partially overcome these limitations. These sensors are not restricted by weather conditions and microwave brightness temperature can be related to snow parameters [7]. Operational sensors that have been used to date for mapping snow properties include the SMMR on board the Nimbus-7 satellite and the DMSP SSM/I [47]. The AMSR-E instrument, launched on the EOS-AQUA satellite in late 2003, provides better spatial resolution and a wider range of wavelengths as compared with SMMR and SSM/I. However, several problems exist with passive microwave snow retrieval algorithms. Snow metamorphism, forest cover, liquid precipitation and complex topography all affect the microwave emission characteristics making it difficult to accurately extract values of snow properties. Moreover, the presence of wet snow prohibits retrieval of any parameter, since wet snow is not a scatterer [18]. Finally, several validation studies have found large errors in passive microwave estimates when comparing with surface snow water equivalent and depth measurements [41]. Additional information about snow state variables can be obtained from land surface hydrologic models that are forced with surface meteorological and radiative variables (precipitation, air temperature, wind, downward solar and longwave radiation) and represent the effects of topography, soil, and vegetation on snow accumulation and ablation processes. Nonetheless, this information is imperfect because of uncertainties in forcing data and model parameters and the nonlinearity and scaling effects of the processes modeled [40,4]. Ideally, a system that optimally combines snow information from both, remote sensing and modeling predictions and at the same time accounts for the limitations of each, should provide estimates that are superior to those derived from either models or remote sensing alone. This method is commonly known as data assimilation [35]. Data assimilation methods have been applied in hydrology with increasing frequency in recent years. There are several different data assimilation techniques, but the most often used are variational assimilation and variants of the Kalman filter (KF). The former is essentially an optimization procedure that adjusts uncertain variables and/or parameters to obtain the best fit to observations [42,21]. Snow data assimilation studies have generally applied direct insertion of snow observations into land surface models [45,34]. Brasnett [5] used statistical interpolation to assimilate global synoptic snow depth observations into a simple snow model. Finally, Sun et al. [51] used an extended Kalman filter to assimilate synthetically generated snow water equivalent (SWE) observations into a catchment-based land surface model. They demonstrated the multivariate capabilities of the EKF in updating snow depth and temperature, as well as the positive effects on the water and energy balance. In this study, we use the ensemble Kalman filter (EnKF), which is a Monte Carlo implementation of the traditional Kalman filter [15]. It has been successfully applied to soil moisture estimation [43,10] and has been shown to have several advantages over other assimilation methods. The EnKF is easy to implement, very robust and computationally efficient, making it well suited for operational assimilation of real-time observations [22]. The objective of this study is to evaluate the performance of an EnKF method that assimilates satellite snow cover extent (SCE) and SWE data into the variable infiltration capacity (VIC) land surface model. We explore two options for assimilating remote sensing data. In the first, MODIS snow extent data are assimilated into the VIC model, resulting in adjustments to both the model-predicted SCE and snow water storage. The second option we explore is direct assimilation of AMSR-E passive microwave SWE estimates. Both these experiments are performed in a filtering context; that is satellite observations are assimilated sequentially as they become available, with the model propagating the system state between observation times. An example of such an application could be in streamflow forecasting. Assimilation of measurements is used up to the start of the forecast period to provide an optimal estimate of the initial conditions, and then running the model into the future. In the next section, we describe the EnKF method and the essential aspects of the VIC hydrologic model. We follow with details of the implementation of the data

3 874 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) assimilation system and the experimental design. The paper concludes with results and interpretation. 2. Methods 2.1. Ensemble Kalman filtering The Kalman filter solves the optimal estimation problem for linear processes, i.e. the state estimation of a process. The state estimation problem is described by a linear stochastic model, which incorporates observations of the process to minimize estimation errors [17]. Its nonlinear counterpart is the extended Kalman filter (EKF), which updates a model state estimate whenever an observation is available, and at the same time takes into account errors in model dynamics and observations [53]. The EnKF works essentially the same way as the EKF; the most important difference is that it uses Monte Carlo sampling from an ensemble of model states to propagate the error information, whereas the Kalman filter and EKF use a dynamic equation to explicitly propagate estimation errors. Consider a vector that is comprised of the state variables of interest (e.g. SWE) y. The model state equation can then be written as dy ¼ F ðy; a; u; vþ dt ð1þ where the model operator F is, in our case, the nonlinear hydrologic model, a represents time-invariant model parameters, u represents time-dependent forcing data and v is the model error term, which can be additive or multiplicative and temporally or spatially correlated. The EnKF is flexible in that it can represent complex error structures, as compared with other assimilation methods which use additive model errors. Its modularity is a further advantage in that it can be used with virtually any type of land surface model. The measurement model equation can be written as z ¼ Hðy; wþ ð2þ where z is the vector of observations, w is the measurement error with defined statistics, and H is the observation operator which relates the state variables y to the observations z. Like the hydrologic model, the observation operator need not be linear. The algorithm starts with the generation of an ensemble of model states y i, where i is the ensemble member index. This is done by treating model parameters, variables and/or forcing data as random variables and then propagating each ensemble member with Eq. (1) until a measurement becomes available. The filter works sequentially from one measurement time to the next. At the time of the measurement and prior to updating, the state vector is termed the state forecast y f i. Ideally, if the propagated state and the measurement vector are jointly Gaussian, the optimal updating of y f using the measurement at that time is given by y a i ¼ y f i þ K½ðz þ w iþ Hðy f i ÞŠ K ¼ P f H T ðhp f H T þ RÞ 1 ð3þ ð4þ where y a is the state vector after the update (analysis), P f is the error covariance matrix of the forecast, R is the error covariance matrix of the measurements and K is the Kalman gain that weighs the effect of the observations to the state update. These covariance matrices are sampled from the ensemble of model state and perturbed measurement vectors and should, theoretically, be Gaussian. Eq. (3) implies that the measurements are treated as an ensemble of random variables too, which verifies that for an infinite ensemble size the analysis of the EnKF is the same as the EKF [6]. The Gaussian assumption about the states and measurements is usually invalid for nonlinear hydrologic processes. This leads to sub-optimal updating since the probability density functions are not going to be Gaussian due to nonlinearities. However, this can be partly ameliorated by the fact that the error covariance, and hence the analysis, includes the full effect of the nonlinear terms, offering an advantage over the linearization scheme of the EKF [30] Land surface model In hydrologic data assimilation, a land surface model (LSM) is used as the forward operator F defined above. The LSM used in this study is the variable infiltration capacity (VIC) model [24,25,9]. VIC is a macroscale hydrology model that has been successfully applied to many continental scale river basins in various climates [1,37,31,27]. Essentially, VIC solves an energy and water balance over each model grid cell at each time step. Its distinguishing feature from other LSMs is the parameterization of subgrid variability in soil moisture, precipitation, topography and vegetation. Each grid cell can have multiple soil layers and be partially covered by different vegetation types in a mosaic-type representation, while it is subdivided into a maximum of five elevation bands. Soil moisture storage capacity is represented by a spatial probability distribution, and also precipitation can be spatially nonuniform. Baseflow is computed as a nonlinear function of the lower soil layer moisture. Moisture and energy fluxes are computed separately for each vegetation class and elevation band within each grid cell and then area-weighted and summed over the grid cell. Streamflow is then simulated by routing subsurface and surface runoff using the method of Lohmann et al. [26]. Snowpack dynamics are simulated in VIC using a two-layer energy and mass balance model [9]. The snow-

4 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) pack is modeled as two layers of variable thickness. The surface layer is used to solve the energy balance between the atmosphere and snowpack, while the lower layer is used to simulate deeper snowpacks, and acts as storage for the excess snow mass from the thin surface layer [55]. Snowfall can be intercepted by an overstory canopy and then released to the ground snowpack through meltwater drip, mass release or throughfall. Snow densification is accounted for through snowpack aging and compressing the pack when there is new snowfall. The latter allows for direct snow depth estimation [9]. The model also includes options for simulating frozen soil and heat exchange between the pack and ground surface [8], however these options are not implemented in this study. Currently, VIC only represents snow areal extent variations indirectly. When solving for a subgrid tile of certain vegetation type and elevation band, it assumes that any snow fully covers that tile, so the snow cover fraction is just the summed percent area of the snow covered tiles [49]. 3. Experimental design Our data assimilation experiment begins with a VIC simulation that does not use any remotely sensed snow observations, referred to as the prior estimate (or open-loop simulation). Two similar simulations are then performed which use the EnKF algorithm to assimilate remotely sensed snow observations. These are referred to as the filter estimate, which is the mean of the ensemble values. In both of these, SWE is updated by assimilating areal extent data from MODIS and water equivalent data from AMSR-E respectively. The first experiment was conducted for four consecutive winters (October 1999 June 2003) while the second was carried out for a shorter period (October 2003 April 2004). These two periods reflect the availability of the MODIS SCE and AMSR-E SWE data Study area description The study area selected for this application is the Snake River basin (Fig. 1), a major tributary of the Columbia River basin. Within the Snake River basin, as in most of the Columbia, snow accumulation and ablation exert strong controls over streamflow. Taken along with the complex topography of the region, the Snake provides a challenging backdrop for snow hydrology modeling. The Snake River basin extends from eastern Oregon to Wyoming and drains an area of over 280,000 km 2 and includes much of Idaho as well as parts of four additional states (Washington, Montana, Nevada and Utah). Mean annual precipitation ranges from less than 350 mm, in the lower basin, to more than 1500 mm in some headwater tributaries. Most of the Fig. 1. Map of the Snake River basin. NCDC COOP and SNOTEL stations are indicated by % and s respectively. precipitation falls during the winter and much of that is stored as snow. Hence, high streamflows occur mostly during spring and early summer snowmelt. Water resources have been intensively developed within the basin. Over 25 reservoirs have an active capacity of about 25 million m 3 and serve a variety of purposes that include hydropower, irrigation, recreation and flood control Model implementation Since the primary focus of this study is the estimation of snow properties, VIC simulations were performed in its water balance mode, which computes surface energy fluxes indirectly with the exception of the snow model that still solves the energy balance explicitly. The only required meteorological inputs were daily minimum and maximum air temperature, daily precipitation and wind speed. Also, several other parameters having to do with soil and vegetation characteristics, and topography have to be specified. Precipitation and air temperature data were obtained by spatial interpolation of NOAA Cooperative Observer data using the synergraphic mapping system algorithm (SYMAP) [54]. Details about the forcing dataset and model parameters can be found in Maurer et al. [32] and Nijssen et al. [37]. The model was applied at a spatial resolution of one-eighth degree latitude/longitude, and daily timestep. The state variables are SWE at each VIC model subgrid tile. The ensemble of model states propagated by the EnKF at each timestep is intended to represent the uncertainties of the prior estimates. This is accomplished by treating model variables and/or parameters as stochastic variables [16]. For this application, precipitation

5 876 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) and air temperature are probably the most sensitive time-dependent forcing variables regarding snow processes. Therefore, precipitation and minimum and maximum air temperature forcing fields were perturbed from their nominal value, creating a different forcing dataset for each of the ensemble members. Log-normally distributed precipitation values were generated and implemented as in [38] 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi bp ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi exp½ E 2 þ 1ŠP ð5þ E 2 þ 1 where bp is the perturbed precipitation value, P is the nominal precipitation value, E is the relative error which was chosen as 25%, and is a spatially correlated Gaussian random field with zero mean and unit variance. The assumption of log-normally distributed precipitation errors was made because it precludes negative precipitation values, and has been used frequently in past studies. Minimum and maximum air temperature values were generated by perturbing the daily air temperature mean and range, using the equation below bt min;max ¼ðT þ 1 Þ T max T min 2 ð6þ 2 where bt min;max are the generated maximum and minimum air temperatures, T is the nominal daily mean air temperature and 1, 2 are spatially correlated Gaussian random fields with zero mean and standard deviations of 2 C and 0.5 C respectively. The spatially correlated random fields were generated using a 2-D Turning Bands algorithm and an, arbitrarily chosen, isotropic exponential correlation model, with spatial correlation decreasing by 1/e at a distance of one degree [46]. It should be noted that the error models used are approximate and do not fully characterize all aspects of the error structure that will be encountered in practice. However, in the context of Ensemble Kalman filtering, the forcing errors are primarily used for the generation of the ensemble, and our implementation is similar in many respects to previous studies [42,30], where they have performed satisfactorily. Temporal correlation could be readily incorporated in the scheme but we chose not to for simplicity. After generating the initial ensemble, the model was integrated without any updates for a spin-up time period to ensure that the system developed the proper multivariate correlations and reached dynamic balance. Because no observations were available until early 2000, we started our model simulations in October 1999 to allow for a spin-up period of approximately four months. The VIC model is formulated in such a way that is solves the energy and mass balance for each grid cell independently and does not take into account horizontal interactions among grid cells (other than through the channel routing model, which was not implemented here). Consequently, even though model states and inputs are spatially correlated, the EnKF analysis step uses a block diagonal covariance matrix and updates each grid cell state vector independently. The implication is that a small ensemble size may prove adequate for solving the analysis equation. The dimension of the single grid cell state vector is much smaller than the total dimension of model states, and thus can be solved for in a relatively smaller ensemble space [36,16]. For this reason, we use an ensemble of 25 model replicates Observation datasets MODIS is an imaging spectroradiometer that provides information about snow cover, amongst many other land surface features, using 36 discrete spectral bands. The snow mapping algorithm is based on two indices (normalized difference vegetation index and normalized difference snow index) and various threshold values that classify each image pixel. Additionally, a thermal and cloud mask facilitates the classification process [19]. The dataset we used is designated MOD10A1, which is a daily gridded snow cover data product at a spatial resolution of 500 m. This product has been available since 24 February After retrieving the data, a pre-processing step that included (1) geo-referencing a 30-m DEM map of the model domain to the snow cover data map, and (2) re-projecting the geo-referenced map to a VIC elevation band coverage map, was performed. The end product was a fractional snow cover map of the VIC model elevation bands. One of the limitations of MODIS data is cloud cover. For our application, a fractional cloud cover threshold of 20% was used to decide whether to use the observation or not. If less than 20% of the grid cell was covered by cloud, SCE data were assimilated. Alternative cloud cover thresholds were used (up to 50%), and while the days on which assimilation was feasible increased by about 6% on average, the SCE mean and variance for each model grid cell were almost identical. The AMSR-E operational snow mapping algorithm uses an empirical relationship to estimate SWE from surface brightness temperature [7] and provides SWE estimates at a spatial resolution of 25 km. For the purposes of our study, we reconstructed the official mapping algorithm and extended the SWE dataset back to October 2003, by using the brightness temperature swath datasets that have been produced since July The reconstructed SWE images were practically identical to the official product, with slight differences occurring because of our use of a slightly different forest cover dataset than the one used in the official estimates. The algorithm uses a set of simple temperature threshold criteria to identify wet snow and precipitation signals, which hinder SWE retrievals.

6 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) Observation operator When updating model predicted SWE by assimilating snow areal extent data, a nonlinear functional, the observation operator (H) that interpolates and/or converts state to observation variables, is necessary. This operator is used to compute the model predicted observation, when the state variable is indirectly observable. When this functional is nonlinear the analysis equation (3) must be modified, usually by linearization of H. However this can be avoided in the EnKF by augmenting the state vector with the model predictions of the measurements and modifying the analysis equation accordingly [16]. In this study, a snow depletion curve (SDC) parameterization scheme developed by Anderson [2] is used, which relates areal average SWE to the snow covered area of the model element. This scheme is currently used by the National Weather Service River Forecast System to estimate snow spatial distributions during snowmelt. The fractional snow covered area is modeled as W A f ¼ A dc ð7þ min ðw max ; SIÞ where A dc is the functional that represents the spatial distribution of snow as it increases in depth, W is the mean areal water equivalent, W max is the maximum water equivalent since accumulation began, and SI is a preset value of SWE above which there is always 100% snow cover. When new snow occurs over a partially snow covered area during snowmelt, A f is regressed to 100% and follows a rescaled depletion curve until it reaches the point before the snowfall [2,28]. It is generally difficult to obtain direct observations of both areal extent and water equivalent of snow from which the depletion curve can be estimated [2]. The approach we used makes use of the available MODIS data and SWE data from ground measurements. The latter are obtained from the US Natural Resources Conservation Service SNOpack TELemetry (SNOTEL) station network dataset, which provides daily observations of SWE and other climatic variables. There are about 600 SNOTEL stations in the western US, most of which have been operational since the early to mid 1980s. These stations are located in relatively high elevation mountainous areas, where snow water storage is greatest. The SNOTEL data used include years from 1980 to 1999, thus keeping the data just for validation purposes. It is known that snow spatial variability is affected by topography, land cover type, and wind exposure [40]. Therefore, each model elevation band of every grid cell in the basin was grouped according to elevation and land cover (Table 1). A separate depletion curve was developed for each of these physiographic classes. The parameter SI in Eq. (7) for each class was determined by examining the snow extent time series and estimating an average day of year that snowmelt began. Then the SNOTEL SWE values (expressed as fractions of the maximum seasonal SWE) that corresponded to full snow coverage immediately before the onset of snowmelt, were averaged for That average fraction was then applied to the VIC SWE climatology ( ) of each model grid cell in order to estimate an average SI value for each physiographic class (Table 1). Parametric distributions have been extensively used to represent the shape of the depletion curve [14,13]. Luce and Tarboton [29] suggested that the curve should be concave downward near the origin and showed that the curve shape is primarily sensitive to the coefficient of variation rather than the choice of the fitted parametric distribution. We chose to fit standard beta distributions (Table 1), i.e. the A dc functional is the beta distribution cumulative density function. Snow depletion functionals were developed for each physiographic class and separate parameters were estimated for snow accumulation and ablation periods, since the snow spatial patterns are different for these two periods. It should be noted that the MODIS data were not used in the SDC parameter estimation procedure. An example plot of a SDC for a particular class is shown in Fig. 2 for both accumulation and ablation. Table 1 Physiographic classes and parameters for the snow depletion curve models Class Elevation z (m) Vegetation type SI (mm) Shape parameter p Shape parameter q Accumulation Ablation Accumulation Ablation A z Forest B 1500 < z Forest C z > 2000 Forest D z Shrubland E 1500 < z Shrubland F z > 2000 Shrubland G z Grassland H 1500 < z Grassland I z > 2000 Grassland

7 878 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) Snow Cover Extent Accumulation Ablation SWE Fraction Fig. 2. Example of a snow depletion curve (Forest low elevation class), showing the relationship between SCE, and SWE divided by the minimum of the maximum seasonal SWE and SI parameter. 4. Results and discussion Both our data assimilation experiments utilize real observations. In contrast with synthetic experiments that have been widely used in past studies [44,51,56], the true state of the system is unknown, and hence the performance of the EnKF must be evaluated either by comparison with independent data or qualitatively. Large scale validation data for SWE are generally unavailable. Consequently, surface observations are the only practical option for independent evaluation. Several studies have successfully used surface point observations for estimation of snow properties [48,33, 39]. We used two surface datasets of SWE and snow depth, the first being the SNOTEL SWE dataset. The second dataset used is derived from the NOAA Cooperative Observation station network, which measures various surface climatological variables daily, including snow depth. These stations tend to be located at lower elevations. By combining both networks, we have a sufficiently large and representative coverage of the river basin (Fig. 1) for a reasonably comprehensive evaluation of the performance of the EnKF estimates. The data were quality controlled using a procedure described by Serreze et al. [48], to eliminate outliers and implausible values MODIS data assimilation The EnKF accommodates both model and observation errors. Observation errors can come both from errors in the MODIS retrievals, and from the algorithm and assumptions used in developing the snow depletion curves. Maurer et al. [33] compared MODIS images to ground observations and found that the fraction of misclassified snow-covered and snow-free pixels for the Columbia River basin was about 20%. In a similar study Klein and Barnett [23] found misclassification errors ranging from 10% to 15% for the upper Rio Grande river basin. From these results an expected error in fractional snow coverage would range from 30% to 30% (for a 20% misclassification error). Here, we choose a normally distributed, spatially uncorrelated random variable with zero mean and 10% standard deviation to represent the observation errors. To prevent implausible perturbed observation values, the ensemble observations were constrained to be between zero and unity. In order to account for the observation operator errors, arbitrary normally distributed errors were selected for the SI, and SDC shape and scale parameters with zero mean and 10% and 1% (of the nominal value) standard deviations respectively Snow cover extent We first examine the impact of assimilating MODIS data on the SCE of the basin. As expected, the EnKF SCE estimates are closer to the observations (1958 out of 1968 model grid cells), with the mean absolute difference (averaged over all the model grid cells) being and for the open-loop and the filter estimate respectively. The effect of the assimilation is similar for each physiographic class on average (2 3%). Fig. 3 shows spatial snapshots of the simulated SWE for two days in The filter estimates do not match the MODIS observations exactly because the EnKF accounts for errors in both model predictions and observations. The upper and middle rows show SWE estimates with and without assimilation respectively, while the bottom row shows the MODIS fractional snow coverage. The EnKF updates SWE in a consistent manner as indicated by the spatial coherency of the updates and accordant addition/removal of snow. Next we compare SCE from MODIS and the two model simulations with ground observations. The comparison is made in terms of the presence or absence of snow, expressed in percentage agreement. The comparisons were limited to the model pixels where a ground station was present; in order to account for elevation differences, we screened all stations where the absolute elevation difference from the corresponding model elevation band was greater than 200 m. This reduced the available number of stations to 123 from 257, with elevations ranging from 114 to 3139 m. The days included in the comparison were selected based on whether MODIS imagery was missing or excessively cloud covered. Because the comparison only reflects the presence of snow, the cloud cover threshold for the comparison was set to 50%. Table 2 shows the percentage agreement between the simulations and MODIS, and SNOTEL/ COOP measurements, i.e. the percentage of days that each data source agreed (snow versus no snow) with

8 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) Fig. 3. SWE simulation comparisons between VIC (upper row) and EnKF (middle row), on 7 January 2001 (left column), and 27 February 2001 (right column). The bottom row shows MODIS SCE observations for the same dates. Table 2 Summary of SCE comparisons with ground observations Data source Overall Lower elevation Mid elevation Higher elevation Number of stations with minimum misclassification MODIS Open-loop EnKF The values shown are percentages of pixels classified correctly over all pixels that contained a ground observation, for the entire simulation period (October 1999 June 2003). the station measurement. The results are averaged over all the available stations, and for each elevation zone separately. The number of stations with minimum misclassification is the number of stations for which the particular data source (MODIS, VIC and EnKF simulations) has the highest percentage agreement (or minimum misclassification). Both simulations and MODIS show good agreement with ground observations; in fact, the open-loop simulation is already superior to the satellite observations. However none of them is perfect and some misclassification occurs. Although the comparison of a point observation to an areal estimate can be problematic, on average the EnKF simulation shows a small improvement (statistically significant

9 880 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) at a 95% level) over both the open-loop simulation and MODIS Snow water equivalent Validation of model grid estimates against point measurements is hindered by scale differences and representativeness problems. Each model SWE estimate corresponds to a 1/8 1/8 grid cell areal average. Therefore, the difference of scales between point measurements and model grid estimates must be addressed. If a relatively large number of stations is available and they are well spread geographically, the large-scale variability of snow processes can be captured by aggregating the station data over larger time and spatial scales (e.g. seasonal or regional averages) [39]. Another way to compensate for this problem is by examining both grid and point estimates relative to their respective climatologies. That is, the SWE data can be expressed as percentiles computed from a relatively long record. We employed the latter method for our SWE comparisons, and use independent 20-year SWE climatologies ( ) for SNOTEL and VIC estimates. Both the open-loop and filter SWE percentiles are calculated from the same model SWE distribution, to account for the nonavailability of an EnKF climatology. We only included stations that had a sufficiently long data record and used the same procedure as previously for screening, resulting in a set of 65 stations. In terms of hydrologic forecasting an interesting variable is the peak seasonal SWE. Fig. 4 shows scatterplots of the seasonal maximum SWE (expressed as percentiles) for SNOTEL and the two simulations. Even though the results are similar the EnKF reduces the scatter slightly (R values of 0.61 and 0.63 for the open-loop and filter simulations respectively), and the filter-estimated peak SWE is closer to the corresponding SNOTEL value for 49 of the available 65 stations. However, the average improvement is rather small (2%). Fig. 5 shows the relative mean squared error (RMSE) between simulated and SNOTEL SWE percentiles, for each winter season of the simulation period. The left and right columns show the prior (VIC) and filter (EnKF) estimates respectively. On average the RMSE for both simulations tends to be the same with 46 of the available stations having a lower RMSE, however that reduction is very small (ranging from 1% to 10%). It is interesting to examine the RMSE variations in terms of time and elevation. Table 3 shows the RMSE averaged over all stations in different elevation zones for accumulation and ablation periods separately. The elevation zones are the same as those used for developing the snow depletion curves, while March 1st was taken as the separating date for the snow accumulation/ablation periods. The table shows that during snow accumulation the assimilation had the smallest impact. This can be explained by the fact that during wintertime, when snow coverage is usually full, MODIS observations and model predicted snow covered fraction often are both one, thus the assimilation has minimal effect on the ensemble of model states. As a result, some ensemble members, may diverge (because of the EnKF representation of forcing errors, from which the ensembles are derived) and produce erroneous SWE estimates. On the other hand, during snowmelt when snow coverage is partial, the filter innovations (difference between model predicted and actual observations, see Eq. (3)) are large enough to reduce the ensemble spread and constrain its replicates closer to the observations. Snow coverage tends to be 100% most of the time at the mid and higher elevations during wintertime, and therefore the updates have a smaller effect on SWE as elevation increases. However, during snowmelt at the lower to mid elevations the impact of the assimilation is larger reducing the RMSE. On the contrary, at higher elevations snow accumulation patterns are different, as Fig. 4. Comparison of mean seasonal maximum SWE between model simulations and SNOTEL measurements. SWE values are expressed as percentile values.

10 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) Fig. 5. Mean seasonal RMSE (%) between simulated SWE percentiles (VIC and EnKF) and SNOTEL at point locations. Table 3 RMSE comparison of the simulated SWE percentiles with SNOTEL averaged over stations in different elevation zones for snow accumulation and ablation periods Lower elevation Mid elevation Higher elevation Accumulation Ablation Accumulation Ablation Accumulation Ablation Open-loop EnKF shown by the very small improvement in performance when assimilating MODIS data. This problem is somewhat similar to the very dry/wet conditions in soil moisture assimilation [44].

11 882 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) Fig. 6. Comparison of SWE percentiles between prior estimate (red line), EnKF (green line) and SNOTEL (blue line) for the Dollarhide Summit SNOTEL station (upper figure). The lower figure displays the model-predicted SCE for VIC (red) and EnKF (green), and actual MODIS observations (blue circles), for the same station. (For interpretation of colours in this figure legend the reader is referred to the web version of this article.) Fig. 7. Comparison of SWE percentiles between prior estimate (red line), EnKF (green line) and SNOTEL (blue line) for the Galena Summit SNOTEL station (upper figure). The lower figure displays the model-predicted SCE for VIC (red) and EnKF (green), and actual MODIS observations (blue circles), for the same station. (For interpretation of colours in this figure legend the reader is referred to the web version of this article.)

12 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) Further insight can be obtained by examining the SWE time series for selected stations. Figs. 6 and 7 show time series of SWE percentiles and the model predicted observation compared with the MODIS data for two SNOTEL stations. The upper subplot shows the SWE percentiles for the two simulations and SNOTEL, while the lower subplot shows the SCE predicted from the snow depletion curve. The circles represent MODIS data and assimilation times. The SWE estimates do not differ very much near the beginning of the first winter season for both stations because of the limited observation frequency. The first station (Dollarhide Summit, Fig. 6) is at an elevation of 2426 m; the RMSE for this station is 0.187, whereas the open-loop RMSE was slightly smaller, The EnKF successfully updated the snow areal extent, which verifies our assumption about the small ensemble size used. However, the filter underestimates SWE when compared to the SNOTEL value, during the winters of 2001 and The filter removed snow much earlier in the 2001 winter, while MODIS observations of a melt event during snow accumulation led to the underestimation of SWE during the 2003 winter. During the third winter (2002), the model-predicted SCE and MODIS observations were identical for almost the entire winter. Subsequently, no updates occurred, and SWE was adjusted entirely based on the VIC model physics. This result suggests that better performance might result from use of a more sophisticated error modeling than the one used here. Furthermore, the observation operator can play an important role, and any simplifications might have a detrimental impact on system performance. The second station (Galena Summit, Fig. 7) is at an elevation of 2634 m; the RMSE for this station was (versus the open-loop estimate of 0.194). During the melt season, the EnKF predicted a deeper snowpack (similar to SNOTEL), tracking the spatial distribution of snow better than the open-loop estimate. In the snow accumulation period the filter SWE is almost identical to the open-loop estimate. The assimilation has almost no effect because of the open-loop and filter predicted SCE being equal to 100%. In the final simulation year, the EnKF was again closer to the SNOTEL value including the peak seasonal SWE. Another diagnostic of the performance of the Kalman filter is provided by the innovation sequence (actual minus predicted measurement). If the statistical and linearity assumptions are fully met, the innovation sequence should be temporally uncorrelated for the filter to be optimal. We calculated the sample autocorrelation of the innovation time series and used a 95% significance level for the statistical test. The null hypothesis (white innovation sequence) is valid for almost 70% of the grid cells (1362 out of 1968). This suggests that the filter does not operate entirely optimally, which is to be expected because of the high nonlinearity of the problem, and the imperfect representation of model errors. An important issue arises when assimilating snow observations for streamflow prediction. The data assimilation updates SWE and SCE by compensating for errors in temperature and precipitation forcings. Assuming that the model temperature is biased positively, the model will tend to melt the snowpack earlier. The assimilation of a snow observation will restore the snowpack to its true state but at the same time it will overestimate streamflow. Similarly, in a cold biased simulation, the model snowpack will persist as long as no observation is available to correct the SWE estimate. The magnitude of the water balance errors will depend on the assimilation frequency. Therefore, it is essential to remove such biases for an operational snow data assimilation application. This could be achieved by constraining the model error (namely precipitation and temperature forcings) as well as the model states [11]. Such an approach is beyond the scope of this work, but could be an important future contribution AMSR-E data assimilation All the results to this point have been based on assimilation of MODIS SCE. For hydrologic purposes, the state variable of greatest interest is SWE, the water content of the snowpack. The use of SCE data is essentially an indirect approach to updating SWE. Therefore we also conducted a preliminary assessment of the value of assimilating AMSR-E SWE estimates into the VIC model. As noted above, AMSR-E SWE estimates have only been available since mid-winter 2004, although we were able to extend them back to late fall, The procedure for generating the ensemble, model errors and general model setup were the same as for the MODIS assimilation experiment. Although there have not been many validation studies for AMSR-E products, mainly because they had only become generally available shortly before this writing, validation results from other passive microwave sensors, such as the SSM/I [52,50], can provide some insight about observation errors. Generally, passive microwave SWE retrievals perform reasonably well when comparing seasonal averages to in situ data, especially for shallow snowpacks. However, these retrievals become problematic for deeper snowpacks, for which SWE is consistently underestimated [12]. For our application the observation error is additive, normally distributed, and spatially uncorrelated with zero mean and standard deviation of 20% (of the nominal SWE value). The filter and prior estimates were compared with SNOTEL data using the percentile methodology described above. The RMS errors were computed for each of the simulations with the EnKF performing slightly better for only 18 stations (out of 53 stations that

13 884 K.M. Andreadis, D.P. Lettenmaier / Advances in Water Resources 29 (2006) Fig. 8. SWE percentile RMSE of VIC (red circles), EnKF (green stars), and EnKF with cutoff SWE value (blue diamonds) simulations against SNOTEL stations. The peak winter SWE for the SNOTEL stations is also shown (gray line). (For interpretation of colours in this figure legend the reader is referred to the web version of this article.) provided data for the 2004 water year). However, the average RMSE of the EnKF was much higher than the VIC (open-loop) errors for the rest of the stations (0.328 and respectively). Fig. 8 shows the RMS errors in terms of the peak SWE for each SNOTEL station. The magnitude of the improvement when assimilating the AMSR-E data was very small and appeared mostly when the peak seasonal SWE was relatively low. On the other hand, for increasingly deeper snowpacks the assimilation performance degraded. This result is rather intuitive considering the general validation results of passive microwave SWE estimation mentioned above. The underestimation of SWE for deep snowpacks, especially in a mountainous and topographically complex basin such as the Snake River, increases the magnitude of the EnKF update leading to erroneous filter estimates. A similar experiment was performed, where a cut-off SWE value was incorporated (relative to the microwave saturation effects). Whenever, the model-predicted SWE was over a snowpack saturation value of 240 mm, the AMSR-E observation was not assimilated. The results (Fig. 8) were somewhat better than previously with a station average RMSE of 0.227, however the majority of the stations still had a larger error. It should be noted, though, that this is a preliminary analysis of AMSR-E SWE assimilation, and a better quantification of the observation uncertainty will be necessary. 5. Conclusions We used an EnKF methodology to assimilate remotely sensed snow observations into the VIC hydrologic model, in two experimental settings. First, we used MODIS SCE data to update SWE model estimates for four consecutive winter seasons. A simple snow depletion curve scheme was developed from SNOTEL station SWE data and MODIS imagery, to form the observation operator of the data assimilation system. The EnKF successfully updated SCE over the study area and generally provided slightly improved estimates of SWE, when compared with independent SNOTEL ground measurements. The impact of the assimilation was more evident during snowmelt, while during snow accumulation the EnKF tended to be very similar to the open-loop estimate for some stations. This is due to the noncontinuity of snow areal extent (zero to one) and assumptions made by the snow depletion curve model (SWE value for full snow coverage). Furthermore, since the open-loop estimate is already using nominal forcing data, the additional information from the remotely sensed measurements was small on average. Examination of SWE time series at specific stations revealed that although the mean relative error was higher, the updates of the EnKF were consistent with MODIS observations. An exploratory evaluation was also undertaken in which AMSR-E SWE data were assimilated using the same dynamical core as in the MODIS SCE assimilation. The results were consistent with other validation efforts for passive microwave SWE retrieval, specifically the effect of the assimilation was only positive for very shallow snowpacks, while deeper snowpacks were significantly underestimated by AMSR-E and consequently by the EnKF. In an operational application setting, some of the assumptions made here should be reevaluated. The snow depletion curve has an important impact on the assimilation, thus better parameter estimation is required, e.g. using independent SWE datasets such as snow surveys. Another important aspect of data assimilation is the modeling of errors, both observation and forecast. Nonetheless, the work presented here is essentially a proof of concept which, while showing the potential