Scaling issues in snow hydrology GuÈnter BloÈschl* Institut fu Èr Hydraulik, Gewa Èsserkunde und Wasserwirtschaft, Technische Universita Èt Wien, Austria 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 dierent 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 ®rst step we examine, in a linear stochastic analysis, the eect 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 eects. These scale eects generally cause biases in the variances and spatial correlation lengths of satellite images, ®eld measurements, and simulation results of snow models. It is shown, by example, how these biases can be identi®ed 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 dicult 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 identi®ability. 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 eects 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. Copyright # 1999 John Wiley & Sons, Ltd. KEY WORDS scale; scaling; aggregation; snow cover patterns; regularization; distributed snow models; fractals; eective parameters; representative elementary area; subgrid variability; snow water equivalent INTRODUCTION This paper addresses two questions: (1) How can we measure and represent snow processes at dierent scales? and (2) How can we aggregate and disaggregate spatial snow data?; These are very broad questions indeed and have rami®cations in three main areas of snow hydrology: (i) In the quest for revealing the true nature of snow processes, more speci®c questions include: What is the nature of spatial snow variability? CCC 0885±6087/99/142149±27$1750 Received 23 September 1998 Copyright # 1999 John Wiley & Sons, Ltd. Revised 8 January 1999 Accepted 11 March 1999 HYDROLOGICAL PROCESSES Hydrol. Process. 13, 2149±2175 (1999) *Correspondence to: Assoc. Prof. GuÈnter BloÈschl, Institut fuÈr Hydraulik, GewaÈsserkunde und Wasserwirtschaft, Technische UniversitaÈ t Wien, Karlsplatz 13/223, A-1040 Wien, Austria. E-mail: g.bloeschl@email.tuwien.ac.at Contract grant sponsor: United States Army, European Research Oce. Contract grant number: R&D 8637-EN-06.