Abstract:The importance of the mean annual runoff (MAR)-hydrological variable is paramount for catchment planning, development and management. MAR depicts the amount of uncertainty or chaos (implicitly information content) of the catchment. The uncertainty associated with MAR of quaternary catchments (QCs) in the Upper Vaal catchment of South Africa has been quantified through Shannon entropy. As a result of chaos over a period of time, the hydrological catchment behavior/response in terms of MAR could be characterized by its resilience. Uncertainty (chaos) in QCs was used as a surrogate measure of catchment resilience. MAR data on surface water resources (WR) of South Africa of 1990 (i.e., WR90), 2005 (WR2005) and 2012 (W2012) were used in this study. A linear zoning for catchment resilience in terms of water resources sustainability was defined. Regression models (with high correlation) between the relative changes/variations in MAR data sets and relative changes in entropy were established, for WR2005 and WR2012. These models were compared with similar relationships for WR90 and WR2005, previously reported. The MAR pseudo-elasticity of the uncertainty associated with MAR was derived from regression models to characterize the resilience state of QCs. The MAR pseudo-elasticity values were relatively small to have an acceptable level of catchment resilience in the Upper Vaal catchment. Within the resilience zone, it was also shown that the effect of mean annual evaporation (MAE) was negatively significant on MAR pseudo-elasticity, compared to the effect of mean annual precipitation (MAP), which was positively insignificant.

Hydrological data (e.g. rainfall, river flow data) are used in water resource planning and management. Sometimes hydrological time series have gaps or are incomplete, or are not of good quality or are not of sufficient length. This problem seems to be more prevalent in developing countries than in developed countries. In this paper, feed-forward artificial neural networks (ANNs) techniques are used for streamflow data infilling. The standard back-propagation (BP) technique with a sigmoid activation function is used. Besides this technique, the BP technique with an approximation of the sigmoid function by pseudo Mac Laurin power series Order 1 and Order 2 derivatives, as introduced in this paper, is also used. Empirical comparisons of the predictive accuracy, in terms of root mean square error of predictions (RMSEp), are then made. A preliminary case study in South Africa (i.e. using the Diepkloof (control) gauge on the Wonderboomspruit River and the Molteno (target) gauge on Stormbergspruit River in the River summer rainfall catchment) was then done. Generally, this demonstrated that the standard BP technique performed just slightly better than the pseudo BP Mac Laurin Orders 1 and 2 techniques when using mean values of seasonal data. However, the pseudo Mac Laurin approximation power series of the sigmoid function did not show any substantial impact on the accuracy of the estimated missing values at the Molteno gauge. Thus, all three the standard BP and pseudo BP Mac Laurin orders 1 and 2 techniques could be used to fill in the missing values at the Molteno gauge. It was also observed that a linear regression could describe a strong relationship between the gap size (0 to 30 %) and the expected RMSEp (thus accuracy) for the three techniques used here. Recommendations for further work on these techniques include their application to other flow regimes (e.g. 4-month seasons, mean annual extreme, etc) and to streamflow series of a winter rainfall region.

This study evaluates essentially mean annual runoff (MAR) information gain/loss for tertiary catchments (TCs) in the Middle Vaal basin. Data sets from surface water resources (WR) of South Africa 1990 (WR90), 2005 (WR2005) and 2012 (WR2012) referred in this study as hydrological phases, are used in this evaluation. The spatial complexity level or information redundancy associated with MAR of TCs is derived as well as the relative change in entropy of TCs between hydrological phases. Redundancy and relative change in entropy are shown to coincide under specific conditions. Finally, the spatial distributions of MAR iso-information transmission (i.e., gain or loss) and MAR iso-information redundancy are established for the Middle Vaal basin.

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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.