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Keywords:

  • snow water equivalent;
  • data assimilation;
  • remote sensing

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Modeling and Assimilation Scheme
  6. 4. Experiments and Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] Hydrologists regularly utilize data assimilation (DA) techniques to merge remote sensing measurements with physical models in order to characterize hydrologic reservoirs. DA methods generally require an estimate of the uncertainty of the various inputs to the models; in practice, however, the uncertainty of these quantities is often unknown. This paper explores the effects of the unknown uncertainty on the efficiency of a multifrequency, multiscale hydrologic DA scheme for snowpack characterization. Synthetic passive microwave (PM) measurements at 25 km and near-infrared (NIR) measurements at 1 km were assimilated, and both snow water equivalent (SWE) and grain size were estimated at 1 km resolution. It is found that the uncertainty magnitude had a significant effect on the efficiency of both SWE and grain size estimation, but that the uncertainty magnitude had very different effects on these two variables because of the different PM and NIR measurement scales. Secondly, it was found that the uncertainty accuracy had a very important role in this DA scheme and that the filter may degrade the estimate of SWE and grain size if key model inputs are misspecified. Finally, four metrics were used to assess the difference between the PM and NIR measurement innovations and their expected values. It was shown that these metrics could potentially be used in an adaptive filtering scheme to correct misspecified uncertainty. More investigation will be required before the feasibility of such an adaptive filtering scheme is established. These findings have important ramifications for snowpack estimation since it implies that in the context of DA schemes, better use will be made of remote sensing products when better physical characterization of the uncertainty of modeled estimates of snow states is available.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Modeling and Assimilation Scheme
  6. 4. Experiments and Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[2] Accurate methods to characterize the states of hydrologic reservoirs have become more important as the reliability of historically dependable water supply has become increasingly suspect because of changing temperature and precipitation trends [e.g., Service, 2004; Mote et al., 2005]. Temporal variability in seasonal water fluxes between major reservoirs is often poorly understood and may be affected by future climate change [Oki and Kanae, 2006]. On the other hand, the amount of information for characterizing hydrologic reservoirs has increased as more and more satellites provide global remote sensing measurements of the land surface at a range of spatial scales, spectral wavelengths, and temporal frequencies [Beven and Fisher, 1996]. Furthermore, increasingly sophisticated land surface model (LSM) schemes provide physical process models for predicting the dynamics of hydrologic reservoirs. Remote sensing measurements and LSM estimates of hydrologic states have important drawbacks, however; it is not trivial to directly invert the remotely sensed signals, and LSM accuracy is necessarily limited to the accuracy of the inputs used to force and/or parameterize the models.

[3] Data assimilation (DA) methodologies may be used to merge any number of independent estimates of hydrologic states (or other quantities of interest) at different scales, allowing for better use of the voluminous data streams, and thus show great promise for characterization of land surface states. In order to weigh the uncertainty of the various sources of information, DA methods require specification of the joint probability distribution function of all uncertain inputs. In practice, hydrologists often assume that inputs follow normal or lognormal distributions, which are completely defined by the first two statistical moments (i.e., the mean and the uncertainty). Although the effects of the magnitude and the accuracy of the input uncertainty on DA schemes can be quite significant, these uncertainty estimates are generally unknown. By “uncertainty accuracy,” we refer to how accurately the estimate of the uncertainty of each DA input represents the true uncertainty. Indeed, Dee [1995] argues that the characterization of input uncertainty forms the primary crux of the DA scheme. However, the sensitivity of assimilation schemes to uncertainty estimates is not often investigated.

[4] This paper explores the effects of input uncertainty magnitude and accuracy in a multiscale ensemble Kalman filter (EnKF) [Evensen, 2003] DA scheme designed to merge radiometric remote sensing measurements with a physically based snow scheme (in an LSM) in order to characterize snow water equivalent (SWE) and grain size. We also investigate the misspecification of input uncertainty and its propagation to certain metrics of filter performance as evidence to suggest that adaptive filtering, where the uncertainty of the state (or measurements) is simultaneously estimated along with the states, may be feasible in the context of this multiscale DA environment. This paper does not investigate the application of an adaptive filtering scheme, however. Instead, we attempt to address several crucial questions in the context of this synthetic test. First, assuming that the uncertainty is known, how sensitive are the filter results to input uncertainty? Second, relaxing the assumption that the uncertainty is known, how sensitive are the filter results to misspecification of the input uncertainty? Third, does the potential exist for application of an adaptive filtering scheme to correct misspecified uncertainty?

[5] Section 2 reviews recent work done to characterize snowpack on the one hand, and to estimate uncertainty in adaptive filtering schemes on the other. In section 3, the models and measurements utilized in this study are described. In section 4, results are presented to demonstrate the central role of input uncertainty magnitude and accuracy in this DA study and to investigate the potential to diagnose input uncertainty misspecification using several metrics of filter performance.

2. Background

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Modeling and Assimilation Scheme
  6. 4. Experiments and Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[6] Accurate characterization of SWE at high spatial resolution is made difficult by significant uncertainty associated with the available data and models. DA methods allow for treating this uncertainty in a meaningful way, but estimates of input uncertainty are themselves often very uncertain. Adaptive DA methods have thus been developed, although they have not been widely implemented in hydrologic DA studies.

2.1. Snowpack Characterization

[7] According to Mote et al. [2003], the key in modeling SWE accumulation at a point is to force the model with accurate precipitation data. Many sources of uncertainty affect the modeled SWE [e.g., Zuzel and Cox, 1975]. At midlatitudes, however, sublimation processes generally remove less than 15% of the snowpack [Hood et al., 1999]. For continental snowpacks [Armstrong and Armstrong, 1987] at high elevations, there is a distinct accumulation and ablation season, and there is limited melt during the ablation season. Under these conditions, we hypothesize that modeled SWE uncertainty during the accumulation season may be approximated on the basis of precipitation uncertainty. Traditional snowfall estimation methods are highly uncertain [Goodison et al., 1981; Groisman and Legates, 1994] and satellite-based remote sensing approaches are still immature [e.g., Skofronick-Jackson et al., 2004; Bindschadler et al., 2005]. Thus all snowfall precipitation products are characterized by high levels of uncertainty. The difficulties associated with LSM application and remote sensing inversion for snowpack characterization (as discussed by Durand and Margulis [2006, 2007]) has motivated the application of DA schemes. Assimilation methodologies merge a priori estimates of the states (e.g., an LSM state variable) with a measurement of the state (e.g., a radiometric brightness temperature) by weighing the associated uncertainties.

[8] Several authors report inverting remote sensing measurements to obtain SWE estimates, then assimilating the SWE into an LSM. Sun et al. [2004] report such a synthetic test (using the extended Kalman filter), and Dong et al. [2007] extend this to assimilate satellite-derived SWE estimates. These “snow data assimilation” methods are impaired by the limitations of the linear inversion of the radiometric quantity to obtain the SWE measurement in the first place. Pulliainen [2006] reports assimilating PM radiometric measurements directly, and considers the relative uncertainty of each data stream in a meaningful way in a Bayesian scheme. In that study, in situ measurements are regional a priori SWE and grain size estimates, and PM data are considered as measurements on this prior. A one layer snowpack radiative transfer model (RTM) [Pulliainen et al., 1999], and RTMs of the vegetation and atmosphere are employed to relate the radiometric quantity to the states. Durand and Margulis [2006] demonstrate the 1-d assimilation of a suite of radiometric quantities using the EnKF and a three-layer snow and radiative transfer model of the snowpack. They showed that all twelve Advanced Microwave Scanning Radiometer-Earth Observing System (AMSR-E) frequencies contained useful information about the snowpack in that context. Durand and Margulis [2007] demonstrate the application of the EnKF to assimilate radiometric quantities in a synthetic test in a 625 km2 area in Colorado. First-order sources of snow modeling uncertainty were identified and modeled explicitly. The multiscale, multifrequency measurements had a synergistic effect in estimating the true SWE and grain size states. The present study is an extension of that work, demonstrating the crucial role that uncertainty plays in DA schemes and investigating the potential for using certain diagnostics of filter performance for identification of input uncertainty misspecification.

2.2. Adaptive Filtering Data Assimilation Methods

2.2.1. Ensemble Kalman Filter

[9] Before surveying some of the important work that has been done in the field of adaptive filtering, the EnKF is explained. The EnKF [Evensen, 2003] DA framework is often used in atmospheric and hydrologic studies since it is a flexible approach for dealing with the various nonlinearities and uncertainties in the systems [e.g., Reichle et al., 2002a; Margulis et al., 2002; Slater and Clark, 2006]. An ensemble of model inputs is generated on the basis of the assumed statistical distribution and the corresponding moments of key model inputs, e.g., forcing data and the model parameters. The model is then run for each of the input realizations. During this propagation step, the LSM is used to forecast the whole ensemble of state estimates forward until the measurement time. At the update time each state is updated on the basis of its correlation to all of the measurements predicted by the RTMs. Though linear, this update analysis takes into account the first two moments of the joint distribution between the states and measurements as estimated from the ensemble. The Kalman update equation for each ensemble member or replicate, r, is given by

  • equation image

where yprior is the state estimate prior to the incorporation of the new measurement, z is the measurement at the current time, w is random error that is added to the measurements to prevent the introduction of correlations among the replicates [Burgers et al., 1998], zpredicted is the RTM prediction of the measurement as a function of the prior, yposterior represents the state estimate conditioned on the new information available from the current measurement, and K is the Kalman gain, which effectively determines the weight given to the states and measurements in the posterior estimate. The Kalman gain is computed from the assumed error covariance of the measurements Λv and two sample statistics obtained from the ensemble: the covariance of the states with the predicted measurements, Cyz, and the covariance of the predicted measurements Czz:

  • equation image

As described by Reichle and Koster [2003], when this method is applied to a vector of states consisting of different states at different spatial locations, interpixel correlations for pixels separated by large distances are suppressed to avoid spurious correlations that often result from small ensemble sizes [Houtekamer and Mitchell, 2001].

2.2.2. Adaptive Ensemble Kalman Filtering

[10] In this study, we investigate the potential use of the filter innovations to diagnose misspecification of model input uncertainty. The filter innovation for each measurement channel m, at each time t, and for each replicate r is defined as the difference between the measurement and the predicted measurement

  • equation image

where nm indicates the number of measurements (i.e., different measurement frequencies or locations), nt is the number of measurement times, and nr is the number of replicates or ensemble members. If the various assumptions made in deriving the EnKF are satisfied, then the innovation sequence should have a mean of zero, be temporally uncorrelated, and have a covariance equal to Czz + Λv; thus the following two equalities should be satisfied at each time, t:

  • equation image

where Cvv,t is the innovation covariance, which is a square matrix with dimension nm and where an element at row i and column j of this matrix could be calculated directly from the ensemble, by:

  • equation image
2.2.3. Application of Adaptive Filtering in Atmospheric and Hydrologic Studies

[11] Mehra [1972] and Moghaddamjoo and Kirlin [1993] describe several methods for using measurement innovation sequences to estimate the model uncertainty parameters in the context of Kalman filtering. In the contexts described in those papers, state and measurement models can be adequately represented as linear operators. This is done by prognosing the state error covariance (i.e., the state uncertainty) explicitly; a “model error matrix” is added to the state error estimate at each time step. Thus an estimate of the state uncertainty is prognosed directly, and adaptive schemes may be used to calculate the optimal additive model error matrix. In the context of the atmospheric sciences, Dee [1995] demonstrates an approach to model uncertainty parameter estimation which is an example of a “covariance matching” approach: the optimal input uncertainty is calculated so that the actual innovations are consistent with the expected value of the innovation covariance (Czz + Λv). This is done by assuming that the innovations follow a Gaussian distribution with zero mean and the correct covariance, treating the expression for the Gaussian distribution as a likelihood function for the unknown inputs, inserting the actual innovations and the expected covariance into the likelihood function, and then maximizing to find the optimal input parameters. Dee [1995] applies this approach successfully to estimate uncertainty inputs in a simplified Kalman filter scheme. Mitchell and Houtekamer [2000] use the method developed by Dee [1995] and apply an adaptive EnKF to estimate both the model error matrix and correlation length, in the context of atmospheric data assimilation.

[12] In the context of hydrologic data assimilation (the topic of this paper) it is often neither attractive nor feasible to represent the model uncertainty (i.e., the model error matrix) as additive noise as described above. The uncertainty in the hydrologic model estimates represents many different sources of error; errors in the forcing data often determine the state variable uncertainty via complex, nonlinear relationships. When the state uncertainty is estimated from the ensemble instead of being prognosed directly using an additive model error matrix, the system is potentially better represented, but adaptive filtering is not straightforward, although Mitchell and Houtekamer [2000] applied such a scheme successfully, in the context of atmospheric DA. Reichle et al. [2002b], Reichle and Koster [2002], and Crow [2003] do necessary preliminary work to investigate the potential for applying an adaptive EnKF for soil moisture characterization by showing that the filter innovations contain valuable information for estimation of the model and forcing input error. Crow and Van Loon [2006] diagnose the potential for an adaptive filter to degrade the filter estimate in the presence of multiple sources of error if the error source is misdiagnosed. Although in the context of the EnKF the actual innovation covariance Cvv,t in equation (5) can be explicitly estimated from the ensemble, this approach is not taken by Crow and Van Loon [2006]; instead, the ensemble mean of the innovations is used with a metric proposed by Dee [1995] to diagnose input misspecification. Zhang et al. [2004] characterize soil moisture by prognosing soil temperature, exploiting the fact that the specific heat capacity of soil is a function of soil moisture. In their adaptive approach they tune the unknown input uncertainty based on both the mean and covariance (calculated over a sample obtained from a moving window subset of the time series) of the innovation.

[13] In this study, we investigate several different metrics for evaluating the difference between the innovation mean and covariance calculated from the ensemble, and the expected values of the mean and covariance. The metrics will be used to explore the information content of the innovations in the context of evaluating the misspecification of model inputs. The metrics, along with the rest of the modeling and assimilation scheme, are described in section 3.

3. Modeling and Assimilation Scheme

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Modeling and Assimilation Scheme
  6. 4. Experiments and Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[14] As explained in section 2, sequential DA schemes involve two steps: the development of a prior estimate of the state variables and the conditioning of the prior on the measurement of the states to obtain a posterior estimate. In this section we describe the prior estimates and synthetic measurements, both of which are identical to those used by Durand and Margulis [2007]. We then describe the metrics we propose to use to evaluate the potential of the innovations for use in an eventual adaptive DA application.

3.1. Prior Snowpack Estimates

[15] Forcing an LSM requires estimates of meteorologic, topographic, and vegetation data. Estimates of the state variables are then generated on the basis of numerical solutions to mass and energy balance equations. Neither the inputs nor the parameterized equations are known perfectly; to avoid modeling the uncertainty associated with the states in an arbitrary manner, we hypothesize the primary sources of uncertainty in our estimation problem. As pointed out by Dee [1995], it is necessary to parameterize the input uncertainty, i.e., describe the spatial uncertainty relationships as some function of distance. These topics are briefly reviewed here; see Durand and Margulis [2007] for more details.

3.1.1. Study Area and Data

[16] In this scheme, the prior estimate of the snowpack states is obtained on the basis of geophysical data of our study area in northwestern Colorado; the dimensions of our 625 km2 study area matches the nominal resolution of a PM measurement (25 km). In order to drive the LSM, we obtain meteorological forcing data based on North American Land Data Assimilation System estimates [Mitchell et al., 2004], elevation data from the Shuttle Radar Topography Mission, fractional vegetation data from Powell et al. [1993], and vegetation type data from the U.S. Geological Survey Colorado Land Cover data set (http://seamless.usgs.gov/). The study period is 1 November 2002 through 1 March 2003.

3.1.2. Land Surface Model

[17] These data are used to force the simple snow-atmosphere-soil transfer (SAST) model, which is an intermediate complexity, three-layer energy-balance snow scheme [Sun et al., 1999]. The state variables are produced at 1 km spatial resolution and include the snow depth and ground temperature, as well as the three-layer snow density, snow temperature, liquid water content, and snow grain size, for a total of 14 state variables per pixel. The grain growth is parameterized on the basis of the scheme of Jordan [1991], by

  • equation image

where d is the grain diameter with units of meters, Uv is the water vapor diffusing through the snowpack with units of kg m−2 s−1, and α1 is an uncertain parameter. The default grain size parameters from Jordan [1991] are used; this model is a highly simplistic description of actual snow grain physics. Snowfall grain size is based on Jordan's [1991] parameterization, where

  • equation image

where dnew is the snowfall grain size with units of meters, ρnew is the snowfall density (treated as a constant in this study) with units of kg m−3, and α2 is an uncertain parameter. After snowfall, the grain size in each layer is recomputed on the basis of a mass weighted average that incorporates the new grain size into the snowpack in a manner consistent with the SAST snowpack layering scheme.

3.1.3. First-Order Sources of Uncertainty

[18] In general, uncertainty in the snowpack states derives from many sources, including uncertain meteorological forcing data, incorrect model parameters, and incorrect model structure. State variable uncertainty in this scheme is estimated by making assumptions about forcing and model uncertainty, then propagating ensembles of forcing and model parameters through the LSM. In our previous study, four first-order sources of uncertainty were identified with this modeling setup. These include precipitation quantity, grain size model dynamics, disaggregation of the coarse precipitation product to the model scale, and vegetation leaf area index. In this study, we focus only on the first two of these.

[19] We parameterize the precipitation uncertainty at each pixel using a spatial covariance function with a specified coefficient of variation and the correlation length. These parameters are estimated from empirical data as described by Durand and Margulis [2007]. We hypothesize that precipitation is the most important source of uncertainty in this study (see section 2.1). Arguably the second most important source of uncertainty in this DA scheme is the grain size estimates. The grain size model parameter coefficient of variation and correlation length for both α1 and α2 are estimated on the basis of empirical data as described by Durand and Margulis [2007]. The nominal uncertainty parameters are given in Table 1. In the generation of an ensemble of these inputs to drive the land surface model, all inputs for a given replicate are perturbed in a temporally constant way. For other simplifying assumptions see Durand and Margulis [2007]. The methods used to derive these estimates are admittedly ad hoc and thus the uncertainty estimates themselves are quite uncertain. The effect of these uncertainty estimates on filter performance is the subject of this study.

Table 1. Nominal Input Uncertainty Values Used in the Experiments
 Coefficient of VariationCorrelation Length, km
Precipitation0.5150
Parameter governing grain growth, α11.010
Parameter governing new grain size, α21.010
3.1.4. Uncertainty Models

[20] As pointed out by Dee [1995] sufficient data are not available to estimate the spatial distribution of the uncertainty explicitly; thus the uncertainty must be parameterized. In our approach, a spatial covariance matrix relating the strength of the correlation between each pixel is specified (precipitation and grain size inputs are assumed to be independent). We make the assumption that the covariance function is stationary and model it on the basis of the exponential variogram [Isaaks and Srivastava, 1989]:

  • equation image

where C(h) is the covariance between points separated by distance h, σ2 is the variance of the variable at a point in space, and L is the correlation length, or distance beyond which the uncertainty parameterization predicts that there is negligible correlation between states. Thus σ and L must be specified for each uncertain input. While these types of uncertainty models may reasonably describe uncertainty, the a priori input parameters are inevitably misspecified to some degree in real applications.

3.2. Synthetic Measurements

[21] The above models are applied in a synthetic DA framework where synthetic measurements generated using the measurement models with specified error characteristics are assimilated instead of satellite observations. The purpose of a synthetic test is to verify that the assimilation scheme is feasible before applying it with real data. Nighttime PM measurements corresponding to the 12 AMSR-E channels are generated at 25 km spatial resolution and daytime near-infrared (NIR) measurements corresponding to Moderate Resolution Imaging Spectroradiometer (MODIS) reflectance band 5 (1230–1250 μm) are generated at 1 km. In this study, daytime PM measurements are not used, to avoid the more complex measurement issues when liquid water is present in the snowpack. The measurement error standard deviation associated with AMSR-E and MODIS measurements is assumed to be 2 K and 5%, respectively. All 14 states at all of the 625 pixels are updated on the basis of all of the measurements simultaneously at a given measurement time; thus the measurement vector z in equation (1) has a length which alternates from 625 for daytime NIR measurements to 12 for nighttime PM measurements. See Durand and Margulis [2007] for further details.

3.3. Metrics for Evaluating Innovation Usefulness

[22] For a DA scheme such as the one described in sections 3.1 and 3.2, the accuracy of the posterior estimates depend on how well the model and the model uncertainty characteristics represent actual snow processes and actual model uncertainties. The innovations may be able to be exploited to tune the model uncertainty parameters, and ultimately improve the accuracy of the posterior estimates. In this paper, we evaluate four metrics for the information content of the innovations. For each metric, we give the mathematical definition at a snapshot in time; in general, the mean or time correlation (e.g., whiteness), or some other assessment of the time series could be used in an adaptive scheme to tune the uncertainty inputs. In this paper, we use the mean of the time series of each metric, which will be indicated throughout by an overbar. The first metric we evaluate is the ensemble mean of the innovations,

  • equation image

This time series should theoretically be temporally uncorrelated, and the mean of the time series equation image should be zero. We also examine three measures of the difference between the expected and observed innovation covariance. Following Crow and Van Loon [2006] we compute the metric proposed by Dee [1995] based on the nm-dimensional vector of the innovation mean averaged across the ensemble at each time t, vt where each element m of this vector is defined as

  • equation image

where the bold-faced type indicates that vt is a vector quantity. The metric proposed by Dee [1995] is then given by:

  • equation image

If all of the assumptions made in the derivation of the filter are satisfied in the formulation, then the time series χt will be temporally uncorrelated and the mean of the time series equation image will have a mean of unity.

[23] We argue that metrics based solely on the ϕt and χt metrics do not utilize all of the available information in the innovations, since they are based only on the first statistical moment across the ensemble as shown in equations (9) and (10). We hypothesize that the second moment of the innovations should also contain valuable information about misspecified uncertainty parameters, and that utilizing this information may be useful in adaptive filtering schemes. It is possible that an adaptive scheme using the innovation covariance-based metrics (below) instead of the mean-based metrics could identify misspecified uncertainty parameters more efficiently, and converge more quickly to the true parameters. Moreover, it is possible that an adaptive scheme utilizing both types of metrics could identify more uncertainty parameters than if only one metric were used. We propose two new metrics based on a direct comparison of the filter innovation covariance matrix Cvv defined in equation (5) and the expected innovation covariance. We chose the equation image1-norm (maximum absolute column sum norm), since it has been used in a previous study [Zhang et al., 2004], and we chose the trace of the difference between the matrices, since it addresses a potential weakness of the equation image1-norm, as will be explained below. The equation image1-norm was used in evaluating the change in the innovation covariance matrix in a covariance-matching DA study by Zhang et al. [2004]. The relative equation image1-norm of the matrix difference is given by

  • equation image

While this metric has been used in a previous study, it ignores by construction any aggregate measure of the measurement variances (diagonal elements of the innovation covariance matrix), focusing instead on a single column. We postulate that the sum of the measurement variances of the matrices may contain more information in some cases; the relative trace of the innovation covariance difference is given by

  • equation image

Both the time series of ψt and ωt should theoretically be temporally uncorrelated, and the mean values equation image and equation image should both be zero, in order to satisfy equation (4). It should be noted that in previous work on adaptive filtering in the EnKF context, the innovation covariance is not calculated directly from the ensemble of innovations, and only the ensemble mean of the innovations is used to diagnose input misspecification [e.g., Houtekamer and Mitchell, 2001; Crow and Van Loon, 2006].

4. Experiments and Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Modeling and Assimilation Scheme
  6. 4. Experiments and Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[24] We present results of three experiments designed to answer several scientific questions: (1) How sensitive are the filter results to input uncertainty, assuming that the uncertainty is known? (2) How sensitive are the filter results to misspecification of the input uncertainty? (3) Is there potential for application of an adaptive filtering scheme to correct misspecified uncertainty? In general, we are interested both in SWE and grain size; thus we examine these questions in the context of three different estimation problems. The first is a SWE estimation problem in which precipitation is uncertain and PM measurements are assimilated. The second is a grain size estimation problem in which only the parameters α1 and α2 governing the grain size estimate are uncertain and only NIR measurements are assimilated. We explore the effects on estimation efficiency of both coefficient of variation and correlation length magnitude and accuracy in both of these experiments. In the third experiment, we investigate the effects of coefficient of variation magnitude and accuracy on the update efficiency of both SWE and grain size. The potential synergy between the PM and NIR measurement innovations for simultaneously diagnosing misspecified SWE and grain size uncertainty is also explored.

[25] All three experiments are carried out in the context of a synthetic test. In a synthetic test the truth (a particular model realization) is known, thus enabling exact evaluation of filter performance. In order to precisely evaluate the effect of assimilating the multiscale measurements under different uncertainty scenarios, the filter results are compared to a model run without the benefit of the observations. This “open loop” model run provides the basis for calculation of the fraction of error corrected, which is used throughout to evaluate the filter performance. All experiments are performed with an identical setup to the synthetic tests described by Durand and Margulis [2007]. An ensemble size of 80 was used throughout; this size provided stable results for this setup. The precipitation and grain size inputs were chosen to be statistical outliers, introducing biases into the DA system, similar to what one might expect in a real application.

[26] We use three metrics to evaluate the results: (1) the root-mean-square error (RMSE) of the areal mean state variable time series (domain-averaged RMSE); (2) the RMSE computed across the time series at each pixel, then averaged across the domain (pixelwise mean RMSE); and (3) the RMSE calculated across the pixels at one point in time (spatial RMSE). The PM measurements were found to be able to correct the first SWE metric far more efficiently than the second [Durand and Margulis, 2007].

4.1. SWE Estimation

[27] In the first experiment, precipitation is uncertain and no grain size uncertainty is added. Only PM measurements are assimilated. Here we investigate the effects of uncertainty magnitude, spatial distribution, and accuracy. The spatial distribution component is especially interesting since the PM is a coarse measurement (25 km sensor resolution) on state variables represented at a much finer scale (1 km resolution).

4.1.1. Effect of Uncertainty Magnitude

[28] To investigate the effect of uncertainty magnitude, we use identical uncertainty parameters for both the filter and the truth in each case. We perform two separate experiments in which first the coefficient of variation and second the correlation length of the precipitation are varied over a range of values. In Figures 1a and 1b the domain-averaged and pixelwise mean fraction of corrected open loop RMSE is shown. From Figure 1a, the PM measurements are able to correct around 70% of the domain-averaged SWE RMSE regardless of the coefficient of variation within this range. On the other hand, as the coefficient of variation increases, the fractional improvement in the pixelwise mean RMSE decreases dramatically: this was not expected, since increasing model uncertainty generally tends to increase fractional improvement in DA schemes.

image

Figure 1. Sensitivity of the filter results to correct the domain-averaged SWE RMSE (dashed line) and pixelwise mean RMSE (solid line) shown as a function of (a) magnitude of precipitation coefficient of variation and (b) magnitude of precipitation correlation length.

Download figure to PowerPoint

[29] We hypothesize that the unexpected behavior seen in Figure 1a is due to the fact that (1) the 25 km PM measurements can resolve only the mean, and not the spatial pattern of the 1 km state variables and (2) less and less of the error is resolved as the coefficient of variation increases. We demonstrate this using the update on 25 November 2002: let ɛt,p be the difference between the true and filter estimate of snow depth at time t for all pixels p. One measure of error is the “spatial RMSE” at a point in time, RMSEt, which is defined by:

  • equation image

The spatial RMSE can be expressed in terms of the spatial standard deviation σe,t and the spatial mean μe,t (i.e., bias) of ɛt,p evaluated over the np pixels [Walpole et al., 1998]:

  • equation image

These values have been tabulated in Table 2 for the measurement on 25 November 2002, as a function of the precipitation coefficient of variation. For this update, we found that (1) as expected, as the coefficient of variation increases, a larger fraction of μe,t (spatially averaged bias) is corrected; (2) σe,t is not corrected by the filter (σe,t is identical for the prior and for the posterior filter estimate); and (3) as the precipitation coefficient of variation increases, the posterior RMSEt is dominated more and more by σe,t (rather than μe,t). Thus the coarse PM measurement is able to resolve less and less of the total error as the precipitation coefficient of variation increases.

Table 2. Prior and Posterior Error Statistics Shown for the Update on 25 November 2002 as a Function of the Precipitation Coefficient of Variationa
Precipitation Coefficient of VariationRMSEtAbsolute Value of μɛ,t for Posterior Estimateσɛ,t for the Prior Estimateσɛ,t for the Posterior EstimateFraction of μɛ,t CorrectedRatio of Posterior σɛ,t and Posterior RMSEt
  • a

    In explanation of the unexpected results of Figure 1a, the fractional improvement in the spatial mean of the error μɛ,t, and the ratio of the posterior standard deviation and the posterior spatial RMSE of the error are shown in the sixth and seventh columns (see text for further details).

0.10.030.0240.020.020.290.46
0.30.060.0230.060.060.760.88
0.50.100.0100.100.100.930.99
0.70.140.0010.140.150.991.00
0.90.180.0030.170.190.991.00

[30] In Figure 1b, when the correlation length is less than 100 km, the filter scheme is able to correct less than 40% of the pixelwise mean RMSE. It should be noted that the resolution of the sensor is 25 km. As the correlation length of the precipitation error increases, the 25 km observation can resolve a greater fraction of the error, since the state variables become more and more homogenous. It should be noted, however, that even for a precipitation correlation length of 150 km (a factor of six larger than the domain size) with a precipitation coefficient of variation of 0.5, there is significant spatial variability in the precipitation perturbations. The difference between the open loop and true snow depth on 1 March (not shown) varies from −0.2 m to 1.2 m across the domain; the open loop snow depth varies from 0.25 to 1.25 m at that time.

4.1.2. Effect of Uncertainty Spatial Distribution

[31] The radically different results from the two RMSE metrics in Figure 1a point to the fact that the spatial distribution of SWE uncertainty is an important factor in these results. Since we have assumed that precipitation uncertainty is constant in time in setting up these experiments, it is adequate to consider a single PM update (on 25 November 2002) in investigating the effects of the spatial distribution of uncertainty on the filter. Figures 2a and 2b show the true and filter ensemble mean prior snow depth at this update time, respectively. Figure 2c shows the prior snow depth error, calculated as the difference between the true and prior snow depth; in almost all parts of the domain, the filter underestimates the truth, especially in the south-central area. Figure 2d shows the posterior error, calculated as the difference between the true and posterior snow depth; although the RMSE has been reduced by 36% by the update, the filter now overestimates snow depth in the northeastern area but still underestimates in the south-central area. In Figure 2e, the change in the snow depth (state increment) due to the update is shown; as expected, the spatial pattern is very similar to that of the state uncertainty estimate, which is shown in Figure 2f, and is calculated as the standard deviation across the prior ensemble. The scatterplot between these two quantities is shown in Figure 2g; the correlation coefficient is 0.72. Note that the spatial pattern of prior estimated uncertainty (Figure 2f) is different than that of the actual prior error in Figure 2c; hence the filter will update the state suboptimally as long as the actual spatial distribution of the error is unknown, which is why the posterior estimate is too high in some areas (see Figure 2d). Similar to other studies [e.g., Crow and Van Loon, 2006], we model the precipitation error as a lognormal variable; this means that the ensemble standard deviation of the state will be, to some degree, a linear function of the ensemble mean of the state, as shown in Figure 2h. The implication of this relationship between the ensemble mean and uncertainty is that when modeling state variables at a finer resolution than the measurements, the spatial pattern of the update will be strongly influenced by the spatial pattern of the mean.

image

Figure 2. Fields of (a) true snow depth, (b) filter prior (ensemble mean) snow depth, (c) prior snow depth error (i.e., difference between true snow depth and prior filter ensemble mean), (d) posterior snow depth error, (e) snow depth increments (i.e., difference between prior and posterior snow depth), and (f) ensemble standard deviation of the prior filter estimate, along with scatterplots of (g) snow depth increments (Figure 2e) versus prior snow depth standard deviation (Figure 2f) and (h) the prior snow depth standard deviation (Figure 2f) versus the prior snow depth mean (Figure 2b) at a single update, on 25 November 2002. All units are meters. The x and y axis labels (omitted) are degrees Longitude and Latitude, respectively, for Figures 2a–2f.

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[32] The mechanism behind this update can be visualized by noting that the spatial pattern of the change in state variables is due to the spatial pattern of the Kalman gain in equation (1), since each 25 km measurement is constant in space. Figure 3 shows maps of the Kalman gain for all twelve measurements with units of meters per kelvin (m K−1) brightness temperature. Several of the channels show a maximum or minimum value for the Kalman gain in the east-central area (where the prior estimate of state uncertainty is high), including the 6.925, 10.65, 36.5 GHz channels for both polarizations (Figures 3a, 3b, 3e, 3g, 3h, and 3k). The relative weight given to the measurements versus the prior states can be assessed by calculating a Kalman gain for the model-predicted brightness temperatures:

  • equation image

where Kz represents the Kalman gain for the model-predicted brightness temperatures, and is a square matrix with dimensions of 12 for this update; that is, it is not spatially variable. If the values of Kz approach unity, then the implication is that maximum weight is given to the measurement, and the Kalman update reduces to a linear interpolation as described by Durand and Margulis [2006]. If the values of Kz approach zero, on the other hand, then no weight is given to the measurement, and there is no update. For this update, the values of the diagonal of the Kz matrix varied between 0 and 0.4, with larger values for the higher frequencies.

image

Figure 3. Kalman gain for the snow depth state variable with units of m K−1, for the 6.925, 10.65, 18.7, 23.8, 36.5, and 89 GHz frequencies for the (a–f) horizontal polarization and (g–l) vertical polarization. The x and y axis labels (omitted) are degrees longitude and latitude, respectively, for each panel.

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[33] The product of the Kalman gain and the difference between the predicted and actual (spatially uniform) measurement produces the update due to each channel [Durand and Margulis, 2006], shown in Figure 4. These maps reveal which measurements were the most influential on a given update. In Figure 4, it is clear that the same spatial pattern holds for the update due to each measurement, and that two channels (36.5 H and 23.8 V in Figures 4e and 4j) are most effective in correcting the estimate. The commonly used retrieval algorithm by Chang et al. [1987] is based on a similar combination of the 36.5H and 19.0H channels. The fact that the 36.5H and the 23.8V channels were most useful in correcting the snow depth confirms the physical basis of using retrieval algorithms such as Chang et al. [1987] to characterize shallow snow. The 37.0 V and 89.0V channels (Figures 4k and 4l) were also very helpful in this update; indeed, most of the channels proved to be useful throughout the course of a winter in a 1-D test of this methodology [Durand and Margulis, 2006].

image

Figure 4. Change in snow depth due to the 6.925, 10.65, 18.7, 23.8, 36.5, and 89 GHz frequencies for the (a–f) horizontal polarization and (g–l) vertical polarization. The x and y axis labels (omitted) are degrees longitude and latitude, respectively, for each panel.

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[34] If the spatial distribution of the uncertainty were accurate (i.e., matched the spatial distribution of the actual error), the filter would be able to almost fully recover the truth with time. Thus it is clear that the estimate of the spatial distribution of the uncertainty is as important as the spatial estimate of the mean for the case of a spatially coarse (i.e., PM) measurement. In this DA context, the spatial distribution of precipitation uncertainty is essentially controlled by the spatial distribution of the mean precipitation and by the random number schemes. In reality, the uncertainty varies in space because of other factors, e.g., distance to the nearest precipitation gage. Development of more sophisticated methods of estimating precipitation uncertainty will be an important step in application of land surface data assimilation methods. However, the spatial uncertainty estimate will always itself be somewhat uncertain; the effects of uncertainty accuracy are examined in section 4.1.3.

4.1.3. Effect of Uncertainty Accuracy

[35] To investigate the effect of uncertainty accuracy, we use different uncertainty parameters for the filter when compared with the truth. We perform two separate experiments in which first the filter coefficient of variation and second the filter correlation length of the precipitation are varied over a range of values, while the true value stays constant. Figures 1c and 1d show the effect of incorrect precipitation error assumptions on the fraction of SWE RMSE corrected (evaluated pixelwise and over the entire basin) for different misspecification of the coefficient of variation and correlation length (Figures 1c and 1d, respectively). In Figure 1c, the domain RMSE is sensitive only to underestimation while the pixelwise RMSE metrics are sensitive to both overestimation and underestimation of the coefficient of variation. In Figure 1d, all error metrics are sensitive to underestimation, but not to the overestimation of the correlation length. Furthermore, very poor results are only observed for severe (greater than one order of magnitude) underestimation of the correlation length; in these cases the filter degrades the estimate with respect to the open loop. For the correlation length, error values are slightly lower for slight underestimation of the parameters than for the actual parameter. Indeed, it should be noted that the maximum efficiency roughly corresponds to the sensor resolution of 25 km. Whether this is due to some intrinsic property of the multiscale DA system is an interesting question that merits further investigation in future studies. Filter results are evidently more sensitive to misspecification of the coefficient of variation than to correlation length.

[36] The fact that the maximum estimation efficiency does not correspond to the true error statistics is somewhat unexpected, although not unprecedented in hydrologic DA studies [Crow and Van Loon, 2006]. In order to investigate whether or not this was due simply to the choice of random numbers in our ensemble, we reproduced this analysis for three additional ensemble seeds. In each of these cases (not shown), the same general pattern persisted: estimation efficiency decreases for both underestimation and overestimation of the coefficient of variation, and the maximum estimation accuracy varies slightly from run to run. Thus these additional runs bolster confidence in the reliability of the pattern in Figure 1c.

[37] In Figure 5, the potential for utilizing the innovation mean to diagnose incorrect precipitation coefficient of variation specification is explored using the same experiments analyzed in Figures 1c and 1d. Four different metrics (defined above) for the difference between the observed and expected innovations are assessed. For the innovation mean metric averaged across the time series, equation image (Figure 5a) there is clearly a minimum near the correct coefficient of variation. The metric proposed by Dee [1995], equation image (Figure 5b) is sensitive to underestimation of coefficient of variation, but less sensitive to overestimation of the coefficient of variation. Ostensibly because of the nonlinearities in the LSM and RTMs, equation image approaches unity asymptotically for overestimation of the coefficient of variation: according to Dee [1995], equation image should be less than unity for overestimation of the error covariance. A similar asymptotic behavior was seen in the soil moisture estimation study of Crow and Van Loon [2006], although in that study, equation image generally crossed unity and converged asymptotically to a value less than unity. The normalized equation image1-norm of the difference between expected and sample innovation covariance, equation image (Figure 5c) is a monotonic function of error covariance. The normalized trace of the covariance difference, equation image (Figure 5d), is clearly sensitive to both underestimation and overestimation of the error covariance. In this experiment, in which precipitation correlation length and grain size statistics are all known, both the equation image and equation image metrics could be used in an adaptive filtering scheme to tune the precipitation coefficient of variation. The equation image and equation image metrics, on the other hand, would not be of use in diagnosing input misspecification.

image

Figure 5. Four different metrics for the evaluation of the filter innovation, including the time average of: the innovation mean equation image, the metric proposed by Dee [1995]equation image, the normalized λ1-norm equation image, and the trace of the innovation covariance difference from expectation.

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[38] Figures 5e–5h show the potential for diagnosing misspecification of the precipitation correlation length using filter innovations from PM measurements. In this case, all of the innovation covariance statistics (equation image, equation image, and equation image) show a minimum value. The Dee metric equation image is not expected to show a minimum value; nonetheless, constraining equation image to be as close to unity as possible would (in this case) result in a correlation length close to the true value. None of the minima for equation image, equation image, and equation image correspond exactly to the true correlation length. This may be due to the particular ensemble used in this experiment.

[39] In general, the equation image metric appears most promising for correcting precipitation uncertainty parameters using PM measurement innovations. More importantly, however, all four statistics are far more sensitive to underestimation of the correlation length than to overestimation; each of the statistics has much higher values (not shown) corresponding to very low (less than 25 km) estimates of the correlation length. Thus, though our experiments indicate that minima are found for these statistics, it is doubtful that they could be used directly to correct a parameter if that parameter were overestimated because of the low sensitivity of these metrics to overestimation of correlation length. An underestimation of the correlation length, however, might be able to be diagnosed. It should be noted that in the presence of the bias introduced by using a statistical outlier to represent the true simulation, the assumptions governing the derivation of the expected values of the four metrics defined in section 3.3 no longer hold. For an adaptive application, it may be necessary to use a bias-aware filter [e.g., Dee, 2005] to estimate the input bias, before the innovations prove to be useful in correcting misspecified uncertainty parameters.

[40] All of these metrics should theoretically be temporally uncorrelated if the assumptions made in the EnKF derivation are satisfied. Crow and Van Loon [2006] found that an adaptive filter based either on the temporal correlation of the innovations or on the equation image metric would predict identical uncertainty parameters. The temporal autocorrelation function for each of these innovation metrics was also examined (not shown). The autocorrelation function of the innovation signal was not helpful in diagnosing the error, but tended to oscillate in time; this oscillatory behavior was also reported by Reichle et al. [2002b]. Indeed, the temporal sequence of the innovation metrics themselves exhibited oscillatory behavior. For this reason, the Fourier transform of the sequences was taken. The magnitude of the frequency coefficients identified using the Fourier transform showed very similar dependence on the input uncertainty misspecification as the time average and thus are not shown. It is possible that if the measurement uncertainty Λv in equation (2) were also unknown, we would need to utilize the temporal correlation of the innovation sequence in addition to the metrics introduced above in order to estimate both uncertainty parameters. In that case, use of the innovation temporal correlation to estimate the uncertainty parameters would need to be further investigated.

4.2. Grain Size Estimation

[41] In the second experiment, we treat the grain size as uncertain and assimilate NIR measurements. This experiment is quite different than the previous estimation problem because there is an NIR measurement corresponding to each state pixel (i.e., the measurement and state pixel scales are the same). In the tests involving the grain size coefficient of variation, both the coefficient of variation for α1 and α2 are perturbed by the same multiplicative factor. For simplicity, “grain size coefficient of variation” refers to the coefficient of variation of α1 hereafter.

4.2.1. Effect of Uncertainty Magnitude

[42] To investigate the effect of uncertainty magnitude, we use identical uncertainty parameters for both the filter and the truth. We perform two separate experiments in which first the coefficient of variation and second the correlation length of the grain size parameters are varied over a range of values. In these runs, only NIR measurements are assimilated and only grain size parameters are treated as uncertain (no precipitation error is added). From Figure 6a, it seems that the NIR measurements are increasingly more useful as the coefficient of variation increases. This is contrary to the situation in using PM measurements to correct precipitation error. This is mostly because the measurements are not averaged spatially over the domain (but available at each model pixel). Thus the increased variability of the surface grain size field does not come into play; instead, the uncertainty in the state estimate increases, and the filter scheme gives more weight to the measurements. It is also clear that only about 10% of the pixelwise RMSE is corrected even for the maximum coefficient of variation. This is likely because, as found by Durand and Margulis [2007], the thick vegetation in many parts of the region limits the filter efficiency. From Figure 6b, it is clear that as the correlation length increases, the results improve dramatically. This is partially due to the fact that as the correlation increases, pixels underneath vegetation (which are essentially unobserved by the NIR measurement) are able to be updated. When the correlation length is less than 20 km, the filter scheme is able to correct less than 40% of the domain-averaged error and less than 10% of the pixelwise error.

image

Figure 6. Sensitivity of the filter results to correcting the domain average (solid line) and pixelwise mean RMSE (dashed line) for the top layer grain size as a function of (a) magnitude of grain size coefficient of variation and (b) magnitude of grain size correlation length. Fraction of the pixelwise mean surface grain size error RMSE is shown for pixels with less than 20% vegetation cover (solid lines) and more than 95% vegetation cover (dashed lines), as a function of (c) grain size coefficient of variation and (d) correlation length. Sensitivity of the filter results to correct the pixelwise mean RMSE for the top layer grain size for pixels with less than 20% vegetation cover (solid lines) and more than 95% vegetation cover (dashed lines) is shown as a function of (e) accuracy of grain size coefficient of variation and (f) accuracy of grain size correlation length.

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[43] The effects of vegetation are investigated in Figures 6c and 6d where the RMSE is averaged over the six pixels with less than 20% vegetation cover (solid lines) and over the 23 pixels with greater than 95% vegetation cover (dashed lines). The fraction of corrected error is qualitatively much lower for the pixels with minimal vegetation. The filter estimates for pixels with the most vegetation are actually degraded for large grain size uncertainty.

4.2.2. Effect of Uncertainty Accuracy

[44] In this test, we use different uncertainty parameters for the filter when compared with the truth. We perform two separate experiments in which first the filter coefficient of variation and second the filter correlation length of the grain size parameters are varied over a range of values, while the true value is constant. The effects of the accuracy of the coefficient of variation and correlation length on grain size estimation are shown in Figures 6e and 6f, respectively. In Figure 6e, the sensitivity of the filter to misspecification of the coefficient of variation for pixels with less than 20% vegetation cover (solid line) and greater than 95% vegetation cover (dashed line) is shown. It is probable that grain size estimation efficiency is sensitive both to underestimation and overestimation of the coefficient of variation. Although the results for pixels with very little vegetation cover are noisy, perhaps because of having only been averaged over six pixels, it is clear that no improvement is shown for overestimation greater than a factor of two and underestimation less than an order of magnitude. Inside this range, however, results seem fairly stable with regard to misspecification. Pixels with full vegetation perform poorly regardless. In Figure 6f, the pixels with the least vegetation are sensitive to underestimation of the correlation length. Pixels with the most vegetation are more sensitive to overestimation of correlation length, presumably because the filter updates these unobserved pixels on the basis of the mistaken information that those unobserved pixels are correlated with observed pixels, when in fact no such correlation exists.

[45] The potential for diagnosing incorrect specification of grain size error coefficient of variation is shown in Figures 7a–7d. While equation image is apparently a monotonic function of the coefficient of variation and does not cross the expected value of zero and equation image decreases monotonically toward a minimum value as the grain size error increases, equation image crosses unity near the correct grain size, and equation image has a minimum value near the true value of 10 km. Thus equation image and equation image may be useful for diagnosing misspecified grain size error coefficient of variation, although the small range of values for equation image shown in Figure 7d could preclude its use as a diagnostic. Because of the bias in surface grain size (apparent in, e.g., Figure 7a) introduced by choosing a statistical outlier for the true simulation, a bias-aware filter [Dee, 2005] may have to be used in an adaptive scheme to correct the grain size bias, before tuning the input uncertainty.

image

Figure 7. As Figure 5 but for NIR measurement innovations.

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[46] The potential for diagnosing incorrect specification of grain size correlation length using the four error metrics proposed above is shown in Figures 7e–7h. The equation image metric approaches the expected value of zero asymptotically as the coefficient of variation increases, and equation image decreases monotonically toward a minimum value as the grain size error increases. The equation image metric, however, does not show much (if any) relationship to the correlation length. Thus these metrics will not be useful in diagnosing misspecified inputs. The innovation metric equation image is expected to have a value of unity; filter results with a correlation length less than the true value of 10 km have equation image less than 1.0, and results with a correlation length greater than the true value have equation image greater than 1.0. The total range of equation image across the wide range of correlation lengths is fairly small, suggesting that it would be difficult to use this metric to tune the grain size correlation length. Note that although underestimation of coefficient of variation leads to equation image greater than 1.0, underestimation of correlation length leads to equation image less than 1.0. This may be because lower correlation lengths will lead to greater domain-wide variability in the grain size field, which is also the effect of increasing the coefficient of variation. In general, then, the equation image metric is most promising for correcting grain size uncertainty parameters using NIR measurement innovations.

4.3. SWE and Grain Size Estimation

[47] In the third experiment, both precipitation and grain size parameters are treated as uncertain, and both PM and NIR measurements are assimilated. For brevity, only the effects of coefficient of variation magnitude and accuracy on estimation efficiency are discussed here.

4.3.1. Effect of Uncertainty Magnitude

[48] Figure 8 shows the effect of precipitation uncertainty and grain size uncertainty on the filter estimates. In Figure 8a, the grain size coefficient of variation is held constant and the precipitation coefficient of variation is varied. In Figure 8b, the precipitation coefficient of variation is held constant and the grain size coefficient of variation is varied. In both tests, it is clear that these results are quite similar to those shown in Figures 1a and 6a, except that the filter efficiency no longer increases monotonically with the precipitation for very low precipitation coefficient of variation Figure 8a.

image

Figure 8. Sensitivity of the correction of (a) pixelwise SWE error and (b) pixelwise grain size error as a function of the precipitation and grain size coefficient of variation, respectively. Both grain size and grain size parameters were uncertain, and both PM and NIR measurements were assimilated in both experiments.

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4.3.2. Effect of Uncertainty Accuracy

[49] Figure 9a shows contours of the fraction of mean pixelwise SWE error corrected for the various cases. It is clear that SWE error is sensitive to both grain size and precipitation coefficient of variation misspecification. The minimum error corresponds to inputs that are close to the true values. Underestimation of the grain size coefficient of variation produces worse results than overestimation, while both overestimation and underestimation of precipitation coefficient of variation of precipitation coefficient of variation lead to degradation of the filter estimation efficiency. It is clear that the minimum error is not exactly associated with the correct uncertainty input parameters. We could likewise consider contours of the fraction of the grain size error, which are shown in Figure 9b. As is the case for SWE error, grain size error is sensitive to both grain size and precipitation coefficient of variation misspecification. Better error results are generally associated with filter inputs close to the truth. Underestimation of the uncertainty inputs is generally less detrimental to the results than overestimation. Overestimation of precipitation error tends to degrade the filter estimate of the grain size, but underestimation is less detrimental. Additionally, the maximum correction efficiency is clearly associated with uncertainty inputs different than the true inputs. This may be due to the fact that only a single ensemble was used to generate the results; it was found in the SWE estimation experiment (section 4.1.3) that a single ensemble may show an RMSE minimum at values other than the actual input uncertainty, though the results averaged over several ensembles correspond to the actual input uncertainty in that experiment.

image

Figure 9. (a) Contours of the fraction of pixelwise SWE error corrected and (b) contours of the fraction of grain size error corrected shown as a function of the assumed precipitation and grain size parameter coefficients of variation. The solid white circles show the actual error statistics, and the white crosses in Figure 9a show the combinations of input parameters used to generate the contours.

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[50] The usefulness of the filter innovations for correcting the uncertainty estimates under multiple sources of error was investigated. Since the PM and NIR measurements respond very differently to the SWE and grain size, the innovations from these measurement sets are examined separately. Contours of the four metrics for the PM innovations are shown in Figure 10 as a function of precipitation and grain size parameter coefficients of variation. It is clear that the equation image metric is more sensitive to the grain size input uncertainty than to the precipitation input uncertainty, and that the metric is not as sensitive to overestimation of grain size parameter error (Figure 10a). The values of the equation image metric approach unity asymptotically, and thus an adaptive filtering scheme based on this metric would mistakenly increase both precipitation and grain size coefficient past the true values (Figure 10b). In Figures 10c and 10d, the metrics corresponding to the innovation covariance matrix are sensitive to both grain size and precipitation uncertainty, but they show a far greater sensitivity to underestimation of the uncertainty parameters than to overestimation, and thus would be of limited use in isolating the true input uncertainty values. One interpretation of these results is that the grain size uncertainty obscures the information contained in the PM innovations about the precipitation input uncertainty. Thus results from Figure 10 suggest that an adaptive filtering scheme in this DA scheme would be difficult when both grain size and precipitation are treated as uncertain, despite the promising results from the SWE estimation experiment when grain size was not treated as uncertain.

image

Figure 10. Contours of (a) equation image, (b) equation image, (c) equation image, and (d) equation image averaged over the PM measurements for different input values of the input error covariance for two sources of error. The solid white circles show the actual error statistics.

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[51] In Figure 11, the same four metrics are shown for the NIR measurement innovations. Figure 11a shows a contour of zero innovation mean values, but this contour does not correspond to the true input uncertainties, nor does it represent a single combination of input uncertainties. Furthermore, this metric has exceptionally small values (similar to Figure 7a), so it could prove difficult to implement in an adaptive filtering scheme. Although Figures 11c and 11d do not show minima in the innovation metrics, Figure 11b shows some promise for use in an adaptive scheme. For precipitation coefficient of variation above 0.25, overestimation of grain size coefficient of variation produces values of equation image less than one, and underestimation of grain size coefficient of variation produces values of equation image greater than one. Thus, corresponding with results in Figure 7b, the metric proposed by Dee [1995] could potentially be used to estimate grain size coefficient of variation in an adaptive filtering scheme even in the presence of precipitation error.

image

Figure 11. Contours of (a) equation image, (b) equation image, (c) equation image, and (d) equation image averaged over the NIR measurements for different input values of the input error covariance for two sources of error. The solid white circles show the actual error statistics.

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5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Modeling and Assimilation Scheme
  6. 4. Experiments and Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[52] Data assimilation methods are being regularly applied to hydrologic state estimation problems. These methods depend upon the uncertainty inputs which are used to weigh the various sources of information in order to obtain an optimal estimate. In general, however, these uncertainty estimates are not well known. Furthermore, very few studies delve into the effects of input uncertainty on hydrologic DA schemes. We have thus examined the sensitivity of a multifrequency, multiscale snowpack characterization scheme to the sensitivity of both the magnitude and the accuracy of key uncertainty inputs, including coefficient of variation and correlation length for both precipitation and grain size parameters.

[53] Specifically, we have performed three separate experiments. In the SWE estimation experiment it was found that SWE estimation efficiency varies monotonically with the magnitude and the accuracy of both the coefficient of variation and the correlation length. It was also found that the SWE estimation efficiency is dependent upon the accuracy of the input uncertainty, and SWE estimates from this filtering scheme may be inaccurate unless an adaptive filter is used. Several metrics of the PM measurement innovations were examined to assess the potential of the innovations to provide information about the error statistics; some of these could potentially be used to identify the correct precipitation uncertainty parameters in this filtering scheme.

[54] In the grain size estimation experiment, the estimation efficiency is sensitive to both the magnitude and the accuracy of the grain size uncertainty parameters. The same metrics were examined for the NIR measurement innovations. Although it is not clear whether these metrics will be useful for diagnosing input uncertainty misspecification due to the low sensitivity of the metric to the input uncertainty, this experiment suggests that one metric in particular could be used for diagnosing either correlation length or coefficient of variation misspecification. In the simultaneous grain size and SWE estimation experiment, results show that both SWE and grain size estimation efficiency is a function of both grain size and precipitation accuracy.

[55] It was found that some potential exists to correct the grain size input uncertainty even in the presence of precipitation error using the NIR measurements. In our experiments, however, the information about precipitation uncertainty misspecification contained in the PM innovations is obscured by grain size uncertainty. Thus, without accurate estimates of the grain size uncertainty inputs, it may be difficult to tune the precipitation inputs. Because the experiments were performed in the presence of grain size and precipitation biases and such biases are expected to occur in real applications, estimation of such biases (e.g., a bias-aware filter) may be a requisite component of an adaptive filtering scheme to characterize snowpack properties. We evaluated four metrics for assessing the difference between the innovations estimated from the ensemble, and expected value of the innovations. Two of the metrics were based on the innovation mean, and two were based on the innovation covariance calculated across the ensemble. We found that for the PM measurements, the most useful metric was based on the innovation covariance across the ensemble. For the NIR measurements, the must useful metric was based on the innovation mean.

[56] The entire set of uncertain inputs involved in applying this DA scheme in a real application include both the coefficient of variation and correlation length inputs for both the precipitation and grain size, combined with those not investigated in this study: vegetation, precipitation lapse rate uncertainty, and others. Furthermore, model dynamics such as the densification and ripening of the snowpack, and the measurement error covariance were assumed to be known perfectly in this study. A vast number of permutations of these potentially interdependent inputs exist, likely more than can be exhaustively investigated. More work is needed to investigate the complex relationships involved in the different sources of error for hydrologic models in general and in snowpack characterization schemes in particular. Despite the many difficulties, we conclude from this preliminary investigation that this filtering scheme is quite sensitive to both magnitude and accuracy of all four sources of input uncertainty investigated, and that in some circumstances, measurement innovations contain information about the true magnitude of the uncertainty. Furthermore, this study highlights the need for better characterization of both the mean and the covariance of precipitation in mountainous regions. Likewise, better models of snowpack grain size must be developed [e.g., Flanner and Zender, 2006]. Ultimately, the feasibility of snowpack characterization by radiance assimilation may be tied to the magnitude and accuracy of the grain size and precipitation uncertainty.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Modeling and Assimilation Scheme
  6. 4. Experiments and Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[57] This work has been sponsored by NASA Earth Science Systems fellowship NNG05GQ86H and by NASA New Investigator Program grant NNG04GO74G. We thank Yongkang Xue, Christian Mätzler, Jouni Pulliainen, Noah Molotch, and Mark Flanner for use of their codes or data sets, as well as their advice and support throughout this project. The authors wish to thank Rolf Reichle and two anonymous reviewers, whose input significantly improved the quality of the manuscript.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Modeling and Assimilation Scheme
  6. 4. Experiments and Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Modeling and Assimilation Scheme
  6. 4. Experiments and Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
jgrd13931-sup-0001-t01.txtplain text document0KTab-delimited Table 1.
jgrd13931-sup-0002-t02.txtplain text document1KTab-delimited Table 2.

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