Physical interpretation of hydrologic model complexity revisited
Authors: Pande Saket , Moayeri Mehdi 
Published: March 2020

&lt;p&gt;It is intuitive that instability of hydrological system representation, in the sense of how perturbations in input forcings translate into perturbation in a hydrologic response, may depend on its hydrological characteristics. Responses of unstable systems are thus complex to model. We interpret complexity in this context and define complexity as a measure of instability in hydrological system representation. We use algorithms to quantify model complexity in this context from Pande et al. (2014). We use Sacramento soil moisture accounting model (SAC-SMA) parameterized for CAMEL data set (Addor et al., 2017) and quantify complexities of corresponding models. Relationships between hydrologic characteristics of CAMEL basins such as location, precipitation seasonality index, slope, hydrologic ratios, saturated hydraulic conductivity and NDVI and respective model complexities are then investigated.&lt;/p&gt;&lt;p&gt;Recently Pande and Moayeri (2018) introduced an index of basin complexity based on another, non-parameteric, model of least statistical complexity that is needed to reliably model daily streamflow of a basin. This method essentially interprets complexity in terms of difficulty in predicting historically similar stream flow events. Daily streamflow is modeled using k-nearest neighbor model of lagged streamflow. Such models are parameterised by the number of lags and radius of neighborhood that it uses to identify similar streamflow events from the past. These parameters need to be selected for each time step of prediction &amp;#8217;query&amp;#8217;. We use 1) Tukey half-space data depth function to identify time steps corresponding to &amp;#8217;difficult&amp;#8217; queries and 2) then use Vapnik-Chervonenkis (VC) generalization theory, which trades off model performance with VC dimension (i.e. a measure of model complexity), to select parameters corresponding to k nearest neighbor model that is of appropriate complexity for modelling difficult queries. Average of selected model complexities corresponding to difficult queries are then related with the same hydrologic characteristics as above for CAMEL basins.&lt;/p&gt;&lt;p&gt;We find that complexities estimated on SAC-SMA model using the algorithm of Pande et al. (2014) are correlated with those estimated on knn model using VC generalization theory. Further, the relationships between the two complexities and hydrologic characteristics are also similar. This indicates that interpretation of complexity as a measure of instability in hydrological system representation is similar to the interpretation provided by VC generalization theory of difficulty in predicting historically similar stream flow events. &amp;#160;&lt;/p&gt;&lt;p&gt;Reference:&lt;/p&gt;&lt;p&gt;Addor, N., Newman, A. J., Mizukami, N., and Clark, M. P. (2017) The CAMELS data set: catchment attributes and meteorology for large-sample studies, Hydrol. Earth Syst. Sci., 21, 5293&amp;#8211;5313, https://doi.org/10.5194/hess-21-5293-2017.&lt;/p&gt;&lt;p&gt;Pande, S., Arkesteijn, L., Savenije, H. H. G., and Bastidas, L. A. (2014) Hydrological model parameter dimensionality is a weak measure of prediction uncertainty, Hydrol. Earth Syst. Sci. Discuss., 11, 2555&amp;#8211;2582, https://doi.org/10.5194/hessd-11-2555-2014.&lt;/p&gt;&lt;p&gt;Pande, S., and Moayeri, M. (2018). Hydrological interpretation of a statistical measure of basin complexity. Water Resources Research, 54. https://doi.org/10.1029/2018WR022675&lt;/p&gt;