Scale invariance of water stress and scarcity indicators: Facilitating cross-scale comparisons of water resources vulnerability

Several indicators are commonly used to measure the degree of water resources vulnerability (e.g., water stress and scarcity) in different populations and regions. Little is known, however, about how these indicators respond to changes in the scale of data used to derive them. Two of the most widely used water resources vulnerability metrics, conventionally computed for mean annual values at the country level are Falkenmark Index (FI) for per capita water availability and the Criticality Ratio (CR) for water use to availability. This study computes FI and CR values at a wide range of scales and tests for trends with scale in three river basins: Missouri (North America), Danube (Europe) and Ganges (South Asia) Basins. Gridded sub-continental hydro-climatic data sets at 0.5° resolution are used and aggregated at multiple scales from 0.5° to 5.0°.Analytical logic and empirical evidence show that mean grid-cell values of these vulnerability metrics are in fact scale-independent (scale-invariant) for a given basin. When unscaled variables like water availability and use are ratioed to variables that depend on area, such as population, their dependency on scale may be lost and they become spatially scaled variables. For example, grid-cell mean values of water availability are scale dependent, but grid-cell mean values of the ratio of water availability to population (i.e. FI) are not. This implies that, for a particular river basin, average water resources vulnerability computed by FI and CR at one scale should apply to all scales. This has tremendous implications to applied geographic studies of water resources, and is especially interesting since the unscaled variables used to derive the two indices are scale dependent and vary greatly with scale. The paper and findings highlight the multi-scale complexities of water resources and the geographic nature of water resources and vulnerability metrics.

Research highlights
► This paper is a component of a broader study on multi-scale assessments and spatial and statistical variability using water resources and population data. The variables chosen for this study is also used in the computation of WRV globally. Data, methods, and scaling functions developed for changes in water resources and population data with scale for the three basins, are described in detail in Perveen (2008) and Perveen and James (2009).
► Statistical trend analysis of changes in mean values of variables with increasing scale (coarser grid-cell sizes) has demonstrated two classes of variables with distinctly different spatial behaviors – “unscaled” variables that tend to increase with scale (e.g., water availability, use) and “scaled” variables as ratios of an area-dependent factor (e.g., water availability per capita, population density) that decrease with scale. It is hypothesized that scaled and unscaled variables will behave differently due to differences in two conflicting theories: either variability increases due to increasing magnitudes of cell values or variability decreases due to spatial averaging within cells (Perveen, 2008).
► The objective of this paper is to extend the research on effects of spatial heterogeneity and scale variability in water resources and population data on the metrics computed for WRV. In particular, the impacts of changing scale on two key WRV indices (FI and CR) will be demonstrated. Given that the FI and CR are ratios of two unscaled variables, it may seem intuitive they will change with scale. It is however shown that the influence of scale on the two variables cancels out when they are expressed as a ratio.
► The findings presented here demonstrate that the grid-cell means of two commonly used water resources vulnerability (WRV) indices – Falkenmark index and Criticality Ratio – are scale-independent. Furthermore, this paper presents a theoretical proof that grid-cell means of these indices are independent of scale. This has tremendous implications to applied geographic studies of water resources, and is especially interesting since the variables used to derive the two indices are scale-dependent. This study further provides an empirical and theoretical basis on which detailed study of the adapting capacity of the locale to WRV can be done.