Snow depletion curves (SDC) define the relationship between changes in snow cover area (SCA) and the snow pack, which can impact, for example, snow-albedo feedback in global climate models [1
] and the amount of water storage and melt for hydrological models [2
]. SDCs typically involve functions that are fit or tuned to a given set of snow-based observations or theoretical conditions. Currently, many hydrological, land surface models (LSMs), and climate models use simple to complex schemes to define this snow depth-cover relationship, with many models still using very simple schemes, which do not account for even regional or temporal changes [4
]. Some approaches to estimating snow depth-snow cover relationships involve statistical approaches, from using simple fitted functions to more shape and scale parameter-based gamma and beta distributions [4
]. Much of the snowpack depletion in mountainous regions relates to late winter and early spring peak snow water equivalent (SWE) melt energy and radiation spatial variations (e.g., [2
]), even though SWE can decrease without decreasing areal snow cover.
Several SDC schemes have been used to map the predicted SWE or snow depth states to snow cover fraction (SCF) (or vice versa) for different snow cover data assimilation (DA) approaches [10
]. The SDC scheme can be used as the observation operator, which relates the model variables to the observations. In this case, the SDC acts as the observation operator and converts snow-based model estimates (e.g., SWE) to be in the same units and similar value range as the snow cover observation estimates (e.g., snow cover fraction), for calculating the innovation and assimilation update. The innovation step is simply the difference taken between model-generated snow cover and the observed snow cover estimates. Some studies have utilized the Ensemble Kalman Filter (EnKF) to assimilate snow cover fraction or area observations [10
], as it has been shown to perform overall better than simpler methods, e.g., direct insertion [15
]. EnKF relies on an ensemble of model forecasts generated using error covariances. Snow cover assimilation using the EnKF method has been applied at different scales, including global [16
], continental [13
], and regional [10
]. Other ensemble-based snow cover DA studies have used the ensemble square-root filter [18
] and the particle filter [19
A variety of SDC functions have been used as the observation operator in these past snow-cover DA studies. Cumulative density functions (CDFs) of beta distributions for varying conditions have been applied regionally [10
], or tuning of a hyperbolic tangent formulation, using shape parameters and a snow density parameter [20
], has been applied also for different scales [13
]. Other studies have used, for example, a two-parameter based lognormal probability distribution function, where snow cover area was represented as a summation of areal SWE and the SDCs were tuned by varying different coefficient of variation (CV) parameters for melt and accumulation periods [18
]. Such SDC-based observation operator schemes have accounted for varying conditions, such as elevation, vegetation and accumulation and ablation phases [10
]. Accounting for the accumulation and ablation phases, also referred to as the hysteresis characteristic of snow, can be a very important aspect of the SDC scheme, since different curves can represent these phases and are reflected in the curve parameters (e.g., [10
Previous studies that have assimilated satellite-based snow cover to improve model snow states and other hydrological variables (e.g., streamflow) have utilized snow observations to tune the snow depletion curves. Despite these previous efforts, many of the SDCs used in snow cover assimilation may have been considered theoretically simplistic [12
] or tuned with very coarse spatial scale snow observations [20
], and these SDCs may not be suitable enough for assimilating snow cover data for finer scale, mountainous regions. Also, what if the snow observations themselves were used to derive the snow depletion curves to be used as the observation operators for snow cover assimilation? One recent study used satellite-based snow cover and five snow depth stations to derive SDC functions, based on the hyperbolic tangent formulation, to derive such observation operators for a mountainous catchment in China [17
]. Xu et al. [17
] generated curves separately for each station and the two snow phases, accumulation and ablation. Assimilating snow cover into an LSM using these highly tuned observational based observation operators improved the modeled snow depth states at those few sites [17
]. Another recent study, which focused on generating SDCs at different scales by using observed SWE and satellite-based snow cover area, showed how using such observational information can help estimate better SWE conditions (e.g., peak SWE) when 100% snow cover occurs (for an area or grid cell), given the location or area being represented [2
In this study, we present a new approach to estimating snow depletion curves and their application for assimilating snow cover fraction observations, using an EnKF data assimilation approach and a land surface model with a multi-layer snow physics scheme. Building upon these recent studies, we use observed SWE and snow cover fraction estimates to derive new SDCs, using a larger array of observations, spanning two different mountainous regions in the United States (U.S.). We refer to these new SDC-type observation operators as “observation-based” and benchmark their skill against the default, model-based SDC function. These new SDC observation-based observation operators are hypothesized to improve the model-based SCF forecasts and snow state analysis. A secondary goal in applying this SDC approach is to see how accounting for varying vegetation, elevation, and temporal conditions may better capture heterogeneous features related to the snowpack and snow cover patterns when assimilating snow cover observations. Finally, the new SDC-based observation operators are used to derive the observational errors, which are used in the EnKF method. Accounting for different errors related to the varying conditions provides different weighting of the snow cover observations against that of the model in the EnKF innovation and update steps.
The primary research question is: How do these new observation-based, SDC-type operational operators perform relative to a default model-based scheme when assimilating snow cover estimates? The secondary research question is: How does accounting for different temporal and physiographic conditions in relation to the new observation-based SDC impact the assimilation of the snow cover estimates? We address these questions throughout the paper within the following sections. Section 2
provides an overview of the study regions, and Section 3
provides several details of the snow observations, the multi-layer snow model, and data assimilation method used in this study. Section 4
describes the method on how the new observation-based SDCs are derived and applied in the EnKF assimilation approach, and Section 5
presents the results of the new SDCs applied as the observation operator when assimilating the snow cover observations. Finally, we discuss the results, benefits and challenges of this new SDC approach in the Discussion and Conclusions section.
In this study, we have explored how different observation-based snow depletion curve (SDC) characteristics can be derived and used as observation-based observation operators in EnKF assimilation updates when assimilating MODIS, or other satellite-based, snow cover fraction estimates. Different SDC logarithmic functions, representing different physiographic and temporal conditions, were also explored as observation-based observation operators in a full suite of EnKF experiments. Using logarithmic functions with a y-axis intercept value, not set to 0, means that MODIS SCF values may not be assimilated below that value (e.g., 40% SCF). In a way, this acts as a cutoff for lower MODIS SCF values (e.g., less than 30%), which could contain contaminated SCF values, due to unresolved patchiness in a pixel and NDSI algorithm pixel discrimination issues [53
]. Also, the fact that many of the logarithmic curves never fully reach 100% SCF could allow MODIS SCF pixels that are near 100% to have an impact on the snow analysis, when the model SCF forecasts are much lower than 100%. This situation can work well when the LSM consistently underestimates SWE, relative to the in situ SWE observations. Therefore, for LSMs with a low SWE bias or precipitation input bias (e.g., an undercatch issue), this SDC approach can partly compensate for the low bias, especially when assimilating snow cover in higher mountain catchments locations. However, if an LSM or snow model has a high SWE bias compared to the observations, then this observed type SDC could contribute to overestimating and adding too much snow to the model.
In other snow cover data assimilation studies, most curves are designed to reach 100% SCF, even for coarser grid scale experiments [13
]. If model SWE forecast conditions remain high enough that the predicted SCF ensemble forecasts remain at 100%, with little ensemble spread in SCF, then there can be very little to no impact on the SWE analysis if the observed SCF is much less than 100%, e.g., partial coverage for the gridcell [14
]. Another minor downside to having an SDC-type observation operator reach 100% snow cover, like the case for many LSM curves, the EnKF-based model SCF forecast ensemble can become underestimated, if perturbations force the members to fall below 100%, even if both the MODIS and model predicted SCF were originally at 100%. This can also occur when perturbing the MODIS SCF observations. When the observed SCF is at 100%, this can also reduce that value when perturbed, leading to underestimated SWE analysis [14
]. Rules can be applied in the EnKF method as to how the ensemble members, e.g., within the observation-perturbed ensemble, get distributed or updated near the 100% SCF point. This could include a set of rules for reducing the SWE analysis by a certain fraction when there is a partial amount of observed SCF or by modifying the observation error covariance, σtotal
, that controls the ensemble or uncertainty spread [based on 14]. One slight advantage to the SNOTEL-MODIS observation-based observation operators, obs
, is that they reflect an average SCF percentage of what the satellite detects for a given range of conditions. Thus, if the predicted SCF is 80% and is the upper threshold in the SDC function, then 20% of the area could be considered exposed vegetation cover or average SCF conditions, for say, lower elevation regions.
Based on the filter statistics shown in Table 2
and Table 3
, there still remains some large differences between the magnitudes of the model predicted SCF and the MODIS SCF observations. One approach to address these innovation biases would be to scale the MODIS SCF values to reflect higher ones, bringing them into closer agreement with the predicted SCF values [54
]. Another option would be to place additional constraints on the observation operators or SCF observations themselves, or to further tune the observations or the curves towards the other, so that the filter statistics reflect less bias between observations and the model. In this study, by deriving the observation-based observation operators with the MODIS SCF observations, it helps to bring the predicted SCF values closer in agreement to the observations. With almost all data assimilation approaches, tuning may be the unavoidable course to be taken, so the final analysis may reflect the best that both the model and observations have to offer.
In this study, we presented a new approach to estimating snow depletion curves and their application for assimilating snow cover fraction observations, using an EnKF approach and a land surface model with a multi-layer snow physics scheme. We use observed SWE and snow cover fraction estimates to derive new SDCs, using a larger array of observations than previous studies, spanning two different mountainous regions in the U.S., in Washington and Colorado. We refer to these new SDC observation operators as “observation-based” and benchmarked their skill against the default, model-based SDC function. The SDC observation-based observation operators showed improvement over the default model-based SCF forecasts and snow state analysis. A secondary goal of this study was to apply this SDC-type approach to see how accounting for varying vegetation, elevation, and temporal conditions may better capture heterogeneous features related to the snowpack and snow cover patterns when assimilating snow cover observations. Vegetation-based curves showed improvement over the lumped annual observation-based SDC, especially for Colorado. Finally, the new SDC-based observation operators are used to derive the observational errors, which were used in the EnKF method’s snow cover observation perturbations. Accounting for different errors related to the varying conditions provides different weighting of the snow cover observations against that of the model in the EnKF innovation and update steps.
These results along with other previous SCA DA studies do show promise in that there are positive impacts without much tuning, but additional testing with the observation operators, model, and observations is still needed [10
]. In future work, more a priori information about the MODIS SCF predictions could be included to deriving and tuning functions, like the ones developed here for this DA application. For example, determining error information with regard to MODIS SCF 100% estimates in connection with the meteorological forcing fields, e.g., ability to estimate snowfall for same observed 100% SCF events or the role that temperature can play in MODIS SCA errors [55
]. Also, more adaptive type SDC schemes, which could be more representative at a point, could be developed that adjust relative to changing conditions. Different approaches could be applied to optimize these observational based SDCs and improve the overall performance of the EnKF system.