Scaling issues in snow hydrology

Abstract

The concept of scale can be used to quantify characteristic lengths of (a) a natural process (such as the correlation length of the spatial snow water equivalent (SWE) variability); (b) a measurement (such as the size of a snow density sample or the footprint of a satellite sensor), and (c) a model (such as the grid size of a distributed snow model). The different types of scales are denoted as process scale, measurement scale and model scale, respectively. Interpolations, extrapolations, aggregations, and disaggregations are viewed as a change in model scale and/or measurement scale. In a first step we examine, in a linear stochastic analysis, the effect of measurement scale and model scale on the data and the model predictions. It is shown that the ratio of the measurement scale and the process scale, and the ratio of the model scale and the process scale are the driving parameters for the scale effects. These scale effects generally cause biases in the variances and spatial correlation lengths of satellite images, field measurements, and simulation results of snow models. It is shown, by example, how these biases can be identified and corrected by regularization methods, At the core of these analyses is the variogram. For the case of snow cover patterns, it is shown that it may be difficult to infer the true snow cover variability from the variograms, particularly when they span many orders of magnitude. In a second step we examine distributed snow models which are a non-linear deterministic approach to changing the scale. Unlike in the linear case, in these models a change of scale may also bias the mean over a catchment of snow-related variables such as SWE There are a number of fundamental scaling issues with distributed models which include subgrid variability, the question of an optimum element size, and parameter identifiability. We give methods for estimating subgrid variability. We suggest that, in general, an optimum element size may not exist and that the model element scale may in practice be dictated by data availability and the required resolution of the predictions. The scale effects in distributed non-linear models can be related to the linear stochastic case which allows us to generalize the applicability of regularization methods. While most of the paper focuses on physical snow processes, similar conclusions apply and similar methods are applicable to chemical and biological processes.