### Scale independent hydrologic metrics

A short discussion of common metrics used to characterize the hydrology of watersheds.

Scale independent hydrologic metrics, or sometimes called hydrologic signatures, are quantities that capture watershed hydrologic responses or characteristics that do not depend on the size of the watershed. Put another, way they are useful ways to measure hydrologic response that do not scale with watershed area. However, even these metrics generally breakdown when used at the very fine or large scales (and definiately at pore or continental scales!). Such metrics are useful in relating hydrologically similar watersheds of different extent to one another to identify fundamental hydrologic properties of a watershed. For example the relation between storage and discharge of a watershed can be described using scale invariant recession metrics. Below is a table I put together of four basic and well known scale independent hydrologic metrics that can be calculated for a watershed using commonly available data like streamflow, precipitation, and temperature.

Metric name Quantity Description
Runoff ratio $\huge{\frac{\overline{Q}}{\overline{P}}}$ $\overline{Q}$ and $\overline{P}$ are long-term average streamflow and precipitation for the watershed. The runoff ratio represents the fraction of precipitation that has contributed to streamflow in a watershed assuming the only hydrologic input is direct precipitation. Depending on the time period used to calculate the runoff ratio it can be an indicator of surface properties in a watershed (impervious to highly permeable) or over longer periods it indicates losses to evapotranspiration.
Snowfall/Precipitaion-day ratio $\huge{\frac{N_{S}}{N_{P}}}$ $N_{S}$ is the number of days in a given year that had snowfall and $N_P$ is number of days that exhibited precipitation. Sometimes these are measured using a threshold amount of of snow or precipitation e.g. number of days with snowfall above 1 cm or rainfall above 1 mm, $N_S$ may also be calculated as the number of days when the temperature was below freezing. Also called the snow day ratio this metric quantifies the dominance of snow as the major form of precipitation for a watershed.
Streamflow elasticity median$\huge{\left(\frac {\frac{dQ}{\overline{Q}}} {\frac{dP}{\overline{P}}} \right)}$ $\overline{Q}$ and $\overline{P}$ are long-term average streamflow and precipitation for the catchment. The derivative of $Q$ or $P$ is likely not available due to the non-continuous measurements of these variables and these quantities are estimated by differencing each day from the previous day's value, alternatively the difference can calculated by subtracting each day by the annual mean value. Streamflow elasticity indicates how sensitive the watershed is to precipitation, that is for a given change in $P$ how much will $Q$ change. A high value indicates an elastic or sensitive watershed whereas a low vaule indicates an inelastic or insensitive watershed.
Baseflow index $\huge{ \frac{\overline{Q_{bf}}} {\overline{Q_{sf}}} }$ $\overline{Q_{bf}}$ is long-term average of the baseflow component of streamflow, usually estimated using a digital filter and $\overline{Q_{sf}}$ is the long-term average streamflow for the catchment. Baseflow index or BFI represents the long-term average proportion of the baseflow component (i.e. groundwater discharge) of total streamflow in a stream. A BFI close to one means that nearly all flow is from slow subsurface discharge to a stream whereas a BFI near zero means that the stream gets most of its input from fast overland flow or preferential flow from precipitation events. Thus the BFI can be related to land cover, soil/aquifer permeability, and other watershed characteristics.

The concept of scale independent metrics for hydrology is important; they can give insight into fundamental hydrologic processes in a watershed. This is an abbreviated list of straightforward metrics that can easily be calculated for a gaged watershed. Using scale independent metrics to investigate hydrologic change and variability is important because the metrics relate to key hydrologic processes or important watershed characteristics. They are also commonly used to classify hydrologically similar watersheds. Can you think of other scale independent metrics? Feel free to post your thoughts in the comments.