Analyzing and classifying dynamical hydrological systems by uncertainty and complexity with the c-u-curve method

Recently, Ehret and Dey (2022) suggested the c-u-curve method to analyze, classify and compare dynamical systems of arbitrary dimension, deterministic or probabilistic, by the two key features uncertainty and complexity. It consists of subdividing the system’s time-trajectory into a number of time slices. For all values in a time slice, the Shannon information entropy is calculated, measuring within-slice variability. System uncertainty is expressed by the mean entropy of all time slices. System complexity is then defined as “uncertainty about uncertainty”, expressed by the entropy of the entropies of all time slices. Calculating and plotting uncertainty u and complexity c for many different numbers of time slices yields the c-u-curve. Systems can be analyzed, compared and classified by the c-u-curve in terms of i) its overall shape, ii) mean and maximum uncertainty, iii) mean and maximum complexity, and iv) its characteristic time scale expressed by the width of the time slice for which maximum complexity occurs.

In our contribution, we will briefly revisit the basic concepts of the c-u-curve method, and then present results from applying it to hydro-meteorological time series of 512 catchments from the CAMELS-US data set (Newman et al., 2015). We will show how c-u-curve properties i) relate to hydro-climatological features, ii) how they can be used for catchment classification, and iii) how the classes compare to existing classifications by Knoben et al. (2018) and Jehn et al. (2020).

References

Ehret, U., and Dey, P.: Technical note: c-u-curve: A method to analyse, classify and compare dynamical systems by uncertainty and complexity, Hydrol. Earth Syst. Sci. Discuss., 2022, 1-12, 10.5194/hess-2022-16, 2022.

Jehn, F. U., Bestian, K., Breuer, L., Kraft, P., and Houska, T.: Using hydrological and climatic catchment clusters to explore drivers of catchment behavior, Hydrol. Earth Syst. Sci., 24, 1081-1100, 10.5194/hess-24-1081-2020, 2020.

Knoben, W. J. M., Woods, R. A., and Freer, J. E.: A Quantitative Hydrological Climate Classification Evaluated With Independent Streamflow Data, Water Resources Research, 54, 5088-5109, https://doi.org/10.1029/2018WR022913, 2018.

Newman, A. J., Clark, M. P., Sampson, K., Wood, A., Hay, L. E., Bock, A., Viger, R. J., Blodgett, D., Brekke, L., Arnold, J. R., Hopson, T., and Duan, Q.: Development of a large-sample watershed-scale hydrometeorological data set for the contiguous USA: data set characteristics and assessment of regional variability in hydrologic model performance, Hydrol. Earth Syst. Sci., 19, 209-223, 10.5194/hess-19-209-2015, 2015.