Cross Mean Annual Runoff Pseudo-Elasticity of Entropy for Quaternary Catchments of the Upper Vaal Catchment in South Africa

Abstract

This study focuses preliminarily on the intra-tertiary catchment (TC) assessment of cross MAR pseudo-elasticity of entropy, which determines the impact of changes in MAR for a quaternary catchment (QC) on the entropy of another (other) QC(s). The TCs of the Upper Vaal catchment were used preliminarily for this assessment and surface water resources (WR) of South Africa of 1990 (WR90), of 2005 (WR2005) and of 2012 (WR2012) data sets were used. The TCs are grouped into three secondary catchments, i.e., downstream of Vaal Dam, upstrream of Vaal dam and Wilge. It is revealed that, there are linkages in terms of mean annual runoff (MAR) between QCs; which could be complements (negative cross elasticity) or substitutes (positive cross elasticity). It is shown that cross MAR pseudo-elasticity can be translated into correlation strength between QC pairs; i.e., high cross elasticity (low catchment resilience) and low cross elasticity (high catchment resilience). Implicitly, catchment resilience is shown to be associated with the risk of vulnerability (or sustainability level) of water resources, in terms of MAR, which is generally low (or high). Besides, for each TC, the dominance (of complements or substitutes) and the global highest cross MAR elasticity are determined. The overall average cross MAR elasticity of QCs for each TC was shown to be in the zone of tolerable entropy, hence the zone of functioning resilience. This could assure that water resources remained fairly sustainable in TCs that form the secondary catchments of the Upper Vaal. Cross MAR pseudo-elasticity concept could be further extended to an intra-secondary catchment assessment.

Shannon Entropy for Measuring Spatial Complexity Associated with Mean Annual Runoff of Tertiary Catchments of the Middle Vaal Basin in South Africa

“…RCE differs essentially from relative entropy and redundancy in the sense that it is defined between 2 hydrological phases occurring respectively at different times. The concept was used by [ 13 , 15 ] and a similar concept was used by [ 3 ]. RCE shows the uncertainty change involved when a system moves between phases 1 and 2.…”

“…Unpredictable, unstructured and complex systems can be described by information entropy [ 12 ]. The distribution of water resources over a basin is associated with entropy and usually present some dispersed configuration [ 5 , 13 ].…”

“… The relative change in entropy (RCE) of a hydrological system could be defined from one phase to the other and these phases occur in different periods of time. Information gain/loss between sets of data [ 13 ] was implicitly approached as the relative change of marginal entropy between two phases. In essence, the difference in entropies between two consecutive phases [ 3 ] is compared with the entropy of the initial phase.…”

“…The Lower Vaal together with the Upper Vaal and Middle Vaal basins forms the current Vaal management area. Like [ 13 , 15 ], surface water resources (WR90) of South Africa 1990 (WR90), 2005 (WR2005) and 2012 (WR2012) data sets were used for MAR, specifically for the Middle Vaal. WR90, WR2005 and WR2012 data sets were considered in this study as the different phases of the water resources of South Africa.…”

Entropy Applications in Environmental and Water Engineering

“…Foroozand et al [ 26 ] combined entropy with ensemble filter method to evaluate model performance, which can mitigate the computational cost of the bootstrap aggregating method. The entropy concept was linked to the notion of elasticity to assess catchment resilience, which determined the changes in mean annual runoff [ 27 ].…”