Abstract:A method for using remotely sensed snow cover information in updating a hydrological model is developed, based on Bayes' theorem. A snow cover mass balance model structure adapted to such use of satellite data is specified, using a parametric snow depletion curve in each spatial unit to describe the subunit variability in snow storage. The snow depletion curve relates the accumulated melt depth to snow-covered area, accumulated snowmelt runoff volume, and remaining snow water equivalent.The parametric formulation enables updating of the complete snow depletion curve, including mass balance, by satellite data on snow coverage. Each spatial unit (i.e. grid cell) in the model maintains a specific depletion curve state that is updated independently. The uncertainty associated with the variables involved is formulated in terms of a joint distribution, from which the joint expectancy (mean value) represents the model state. The Bayesian updating modifies the prior (pre-update) joint distribution into a posterior, and the posterior joint expectancy replaces the prior as the current model state.Three updating experiments are run in a 2400 km 2 mountainous region in Jotunheimen, central Norway (61°N, 9°E) using two Landsat 7 ETMC images separately and together. At 1 km grid scale in this alpine terrain, three parameters are needed in the snow depletion curve. Despite the small amount of measured information compared with the dimensionality of the updated parameter vector, updating reduces uncertainty substantially for some state variables and parameters. Parameter adjustments resulting from using each image separately differ, but are positively correlated. For all variables, uncertainty reduction is larger with two images used in conjunction than with any single image. Where the observation is in strong conflict with the prior estimate, increased uncertainty may occur, indicating that prior uncertainty may have been underestimated.
[1] The effect of calibrating spatially distributed snow model parameters using satellite data is evaluated by a cross-validation approach in a 26,000 km 2 mountainous region in Norway. From 6 years of data and 13 to 15 Moderate Resolution Imaging Spectroradiometer (MODIS) images per melt season, parameters of local snow depletion curves are estimated annually for each grid cell. The estimated values are averaged over 5 years and evaluated by the sixth year, using each year in turn for validation. The parameters are the subgrid snow storage coefficient of variation cv, the premelt snow-covered fraction A 0 , the premelt snow storage m, and the time sequence of accumulated melt depth {l}. The likelihood is formulated in terms of the Normalized Difference Snow Index (NDSI), rather than fractional snow-covered area, in order to avoid highly skewed distributions for values close to 0 or 1. The 5 year averaged values for cv, A 0 , and a bias-correcting snow storage multiplier m corr were applied in rerunning the precipitation-runoff model. For 22 subcatchments within the region, validation-year standard error in snow melt runoff volume was reduced from 21% to 13%. Standard error in NDSI on the grid cell level was reduced from 0.34 to 0.27. The stationarity of individual parameters was also evaluated, comparing the 5 year calibrated values for each of cv, A 0 , and m corr to the validation-year estimates, after normalizing for the prior information. On average, the calibrated maps for cv, A 0 , and m corr predicted 32%, 46%, and 56%, respectively, of the spatial variance in the validation year's change from prior to posterior estimates.Citation: Kolberg, S., and L. Gottschalk (2010), Interannual stability of grid cell snow depletion curves as estimated from MODIS images, Water Resour. Res., 46, W11555,
Abstract.A method for assimilating remotely sensed snow covered area (SCA) into the snow subroutine of a grid distributed precipitation-runoff model (PRM) is presented. The PRM is assumed to simulate the snow state in each grid cell by a snow depletion curve (SDC), which relates that cell's SCA to its snow cover mass balance. The assimilation is based on Bayes' theorem, which requires a joint prior distribution of the SDC variables in all the grid cells. In this paper we propose a spatial model for this prior distribution, and include similarities and dependencies among the grid cells. Used to represent the PRM simulated snow cover state, our joint prior model regards two elevation gradients and a degree-day factor as global variables, rather than describing their effect separately for each cell. This transformation results in smooth normalised surfaces for the two related mass balance variables, supporting a strong inter-cell dependency in their joint prior model. The global features and spatial interdependency in the prior model cause each SCA observation to provide information for many grid cells. The spatial approach similarly facilitates the utilisation of observed discharge.Assimilation of SCA data using the proposed spatial model is evaluated in a 2400 km 2 mountainous region in central Norway (61 • N, 9 • E), based on two Landsat 7 ETM+ images generalized to 1 km 2 resolution. An image acquired on 11 May, a week before the peak flood, removes 78% of the variance in the remaining snow storage. Even an image from 4 May, less than a week after the melt onset, reduces this variance by 53%. These results are largely improved compared to a cell-by-cell independent assimilation routine previously reported. Including observed discharge in the updating information improves the 4 May results, but has weak Correspondence to: S. Kolberg (sjur.kolberg@sintef.no) effect on 11 May. Estimated elevation gradients are shown to be sensitive to informational deficits occurring at high altitude, where snowmelt has not started and the snow coverage is close to unity. Caution is therefore required when using early images.
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