# Assessing Catchment Resilience Using Entropy Associated with Mean Annual Runoff for the Upper Vaal Catchment in South Africa

## Abstract

**:**

## 1. Introduction

## 2. Entropy, Resilience, Pseudo-Elasticity and Mean Annual Runoff

#### 2.1. Entropy and Mean Annual Runoff

_{j}, and shows the uncertainty associated with y

_{j}. This equation represents the entropy index or marginal entropy of y

_{j}, written as H(y

_{j}). In this specific case, y

_{j}can be MAR for a specific jth QC belonging to a given TC. The j values (j = 1, 2, 3, …, m) represent the j QCs in the TC and m varies from one TC to the other, in the Upper Vaal catchment. Each QC contributes to the total MAR of their TC:

_{j}). The unit is in bits if the base is 2, in Napiers if the base is e, and in decibels (dB) if the base is 10.

_{k}(Y):

_{j}) in Equation (2) have been defined previously and are associated with MAR in either the WR2005 or WR2012 data sets. The two extreme cases of the marginal entropy are the minimum entropy (written as H

_{min}) and maximum entropy (written as H

_{max}) [2]. The minimum entropy is reached when the value of the variable y

_{j}is 1 for j = i (i ≠ 0) and 0 for the remaining values, hence given by Equation (3a). Alternatively, the variable y

_{j}takes very few non-zero values while the remaining values are 0. The maximum entropy is reached when the variable y

_{j}takes the same value everywhere, i.e., ${y}_{j}=\frac{1}{m}$ [35], hence is given by Equation (3b):

_{min}is close to 0

_{max}shows a uniform/homogeneous distribution of the variable in the system, as opposed to H

_{min}.

_{j}), δH(y

_{j}) are the relative change/variation in MAR and relative change/variation in entropy for a specific jth QC respectively. The variable y

_{j}has been defined previously.

#### 2.2. Entropy and Resilience of Water Resources

_{min}) was a transition line between the lower and the middle zones, while maximum entropy (H

_{max}) was used to separate the middle zone from the upper zone. In the lower zone, the system is stable/resilient when it is subjected to disturbances or changes associated with relatively low entropy and becomes vulnerable when entropy increases. In the middle zone, the system is of acceptable resilience/sustainability as long as disturbances/changes are in the range between entropy thresholds. In the upper zone, the system becomes chaotic as disturbances have exceeded the maximum uncertainty. Zone 3 is the zone of high unpredictability or uncertainty. Zone 1 and 3 are states of vulnerability of the system, hence resilience becomes relatively low. The linear zoning of resilience via entropy could be similar to the linear resilience concept [18]; which implies that a system can only deal with one state/capacity at a time, whereas the holistic approach of resilience concept assumes the coexistence of multi-states/capacities (i.e., absorptive, adaptive and transformative capacities) within the system [18]. The linear resilience concept suggests that system resistance/absorption is required in a period of small perturbation/change; adaptation is required in a time of greater perturbation and finally transformability is required when conditions become unviable or cannot be sustained [7]. Linear resilience has been acknowledged to be an acceptable approach, but may not capture the complex interactions of different states within the system [18]. Since resilience is associated with sustainability or vulnerability of a system [7,17,25], which enables the link between entropy and resilience [17], entropy of MAR can be suggested as a surrogate measure of catchment resilience. This could also be supported from catchment resilience being associated with streamflow/runoff as hydrological response [13] or catchment hydrological flux [37,38] and this flux is always associated with a degree of uncertainty or entropy in space and/or in time [4,5,39]. Measuring systems’ resilience holistically using entropy could be appealing, however very challenging for hydrological systems, due to the complex interaction of variables, the occurrences of the different states and the selection of different thresholds. Thresholds are regarded as a range of values to accommodate fluctuation and statistical uncertainty or chaos associated with functioning states of the system [70]. Hence, the linear zoning approach of resilience was found preliminarily useful in this research, as a first step to determine, via entropy, catchment resilience in terms of hydrological response; i.e., MAR. While the resilience thresholds (i.e., H

_{min}and H

_{max}) were just parameterized [7], the current paper showed it is possible to quantify these thresholds for hydrological systems (i.e., catchments) as explained in the methods (Section 3). In principle, the development of thresholds is complex and normally requires insight in the availability of data and multiple disciplines to arrive at an acceptable range of minimum thresholds or indicators [19,28,29,71].

#### 2.3. Elasticity and Mean Annual Runoff

_{p}) was derived from hydrological modelling, i.e., sensitivity of the modelled streamflow to changes in rainfall [56], as shown in Equation (6):

_{k}(y

_{j}) is the relative change in MAR between WR90 and WR2005 and is dimensionless, δH

_{k}(y

_{j}) is the relative change in entropy between WR90 and WR2005 and is dimensionless, k

_{1}is the intercept of entropy axis at the origin of MAR and j is defined as previously.

_{k}defining the rate of the relative change in entropy with respect to the relative change in MAR, can be roughly used as the MAR elasticity of the entropy associated with MAR of QCs. Moreover, its computation departs slightly from previous approaches on the usual rainfall elasticity. This justifies the prefix pseudo-elasticity. That is from streamflow elasticity [49,50,52,53,54,56], the regression coefficient (ε

_{k}) is generally interpreted as MAR elasticity of entropy associated with MAR.

_{k}is the first derivative of δH

_{k}(y

_{j}) with respect to δ

_{k}(y

_{j}), as displayed in Equation (8):

## 3. Data Availability and Methods

#### 3.1. Study Area and Data Availability

#### 3.2. Methods

#### 3.2.1. Marginal Entropy Computation

_{min}, H

_{max}] for an acceptable resilience level [7] of the hydrological system (i.e., catchment). This constitutes the basis of the linear resilience zoning approach as outlined earlier and was adapted to hydrological systems.

#### 3.2.2. Linear Resilience Zoning Determination

_{min,k}and the upper limit H

_{max,k}for the k-th TC of the Upper Vaal catchment. The former is determined using Equation (9) and is not necessarily 0 according to [7]. The latter is determined according to Equation (10), which has been adapted from Equation (3b). Since both entropy and resilience are properties of the system (i.e., catchment), so are the thresholds H

_{min,k}and H

_{max,k}, which are given by Equations (9) and (10) respectively:

_{k}is the number of QC in the k-th TC and H(y

_{j})(j = 1,2,…,m

_{k}) are entropy values of m

_{k}QCs in the given TC, using data from WR90; H

_{lowest,k}(Equation 9) is viewed as a local minimum entropy related to the specific TC, whereas Equation (3a) could be considered as the global minimum entropy for catchments.

_{min,k}values. Firstly, WR90 is the first historic and comprehensive appraisal of water resources of South Africa and was considered arbitrarily as the baseline for hydrological data. It includes 70 years of records, which could be fairly reasonable for hydrological studies, planning and management of water resources in a developing country. The issue of lack of data, missing data, or short length of records has been common in developing countries [2]. Hence 70 years of records were considered relatively sufficient and acceptable for South Africa and included hydrological events of low, frequent and high occurrences. The baseline for data (i.e., length of records) is very important in the determination of resilience thresholds and may influence resilience analysis [19,28,71]. Secondly, WR90 played a significant role in making available hydrological data and information for planning, design, development, research and management of water resources of South Africa [75].

_{k}(y

_{j}) < H

_{min}_{,k}. This is the zone of low resilience and where water resources in the catchment could likely present a risk of vulnerability and instability, if subjected to disturbances associated with relative changes in MAR. Even relatively small changes in MAR may impact catchment resilience. In this situation, water resources sustainability and hence the resilience of the catchment could be compromised. The ability for the catchment to adapt and transform recognised to be pivotal for resilience [18], could be very low, due to adverse hydrological impacts on the catchment. The recovery process of water resources from impactful absorptive shocks could take longer. Therefore, the adaptive capacity of the catchment [18] could be very low. This zone could be highly dispersive in terms MAR since the entropy is very low [76].

_{min}< H

_{k}(y

_{j}) < H

_{max,k}. In this zone, catchment resilience is perceived to be acceptable with reference to water resources. It is the zone of tolerable uncertainty or changes in hydrological response. This is the zone of relatively acceptable resilience of the catchment. The catchment may still recover from disturbances or shock associated with changes in MAR. Water resources in the catchment present a relatively low risk of vulnerability. Hence the adaptive capacity of the catchment from disturbances could still be relatively good. For example, water resources can be sustainably managed to satisfy the different water requirements; extreme hydrological events such as droughts or floods could be manageable or mitigated and the catchment could still recover quite rapidly from the disturbances, perturbations or changing climatic conditions.

_{k}(y

_{j}) > H

_{max,k}. This zone is of relatively low resilience, which can be the result of rapid and significant changes in MAR, outsides acceptable margins. Relatively high values of entropy can be very demanding [7] since there is a huge pressure/stress on the catchment in terms of water resources and water infrastructures. For example, in the Upper Vaal, as the South African economic hub, rapid urban growth, new settlements (both formal and informal), could contribute to increasing water stress. The water resource/system might be unable to satisfy the different requirements. The water resources may not be sustainable and may present a high risk of vulnerability. The level of transformation and adaptation of water management in the catchment may require external water sources and additional infrastructures. In addition, extreme events (floods or droughts) or any climatic shifts could exacerbate the situation and have serious impacts on the catchment, if adaptive and transformative measures were not being put in place.

#### 3.2.3. Determination of MAR Pseudo-Elasticity of Entropy

- -
- The robust standard error of the sample of the mean of elasticity values (the standard deviation of the sampling distribution) was computed.
- -
- The 95% class interval was used for the mean elasticity (e.g., limits of critical values times the standard error to/from the mean were computed, using a t-test. This was to assess the significance of the difference in mean of sampling distribution). Hence, the significance level of variability of MAR elasticity estimates is 0.05.
- -
- The assessment of goodness-fit of regression models was carried out using the coefficient of determination R
^{2}(correlation coefficient R). Generally, R^{2}between 0.5 and 1 can be considered good for the linear regression. However, the statistical t-test was used to carry out the significance of R. The level of significance used was 0.05.

## 4. Results and Discussion

#### 4.1. Entropy for Linear Resilience Zoning

_{min}and upper limit H

_{max}, as depicted in Figure 2a–i.

_{min}but very close to this limit) whereas none of entropy values of QC were in the chaotic zone (i.e., above H

_{max}). This could imply that very few QCs could be highly vulnerable, in terms of their water resources since they are close to the uniformity resilience zone. In general, the majority of the entropy values for QCs were between H

_{min}and H

_{max}(i.e., the intermediate zone, which is the zone of tolerated disorder or zone of acceptable catchment resilience/vulnerability). In this zone, the uncertainty associated with changes in MAR could be tolerable and any recovery process from perturbations could be manageable. It could also mean that, although the majority of QCs in the different TCs displayed generally a certain level of sustainability/resilience for water resources, they could be vulnerable when exposed to disturbances causing significant changes in MAR. The changes in MAR for QCs of the Upper Vaal catchment, occurring between 1990 and 2004; and between 2005 and 2012 could not compromise the overall acceptable catchment resilience level and hence the water resources sustainability. But the management plan in the Upper Vaal could be reinforced, should catchments be subjected to perturbations such as extreme hydrological events, land use change, economic growth, increasing population growth, industrial, agricultural, climatic variability and climate change and ecological/ecosystem changes. This could also signal that, although catchment planning, development, and management in the Upper Vaal are within the South African National Water Strategy (NWS) [58,73], adjustments/revision to this strategy could be needed at QC level. In fact, the national water strategy has been revised recently [73], but the implementation of the new water strategy has not been effective. An integrated approach within the water management strategy of the Upper Vaal river system was already proposed since this river system plays an important role in the economy of South Africa [58]. At the quaternary catchment level, within the water strategy, adaptive and transformative measures could be put in place to anticipate any adverse impacts associated with significant changes in MAR. For instance, additional water supply could be needed in the Upper Vaal for economic and population growth, yet water requirements/demands have exceeded the current water supply [77]. This situation could be exacerbated from the geographical position of South Africa as a water scarce country associated with very high evaporation rates and high variable rainfall. (In the case of Upper Vaal region, MAE exceeds MAP, as shown in Table 1). Moreover, climate change/variability is likely to have adverse effects on the runoff in South Africa [19,78] and specifically a decrease in MAR and yield in the South Hemisphere [60]. This situation presents a risk of vulnerability to South Africa, in particular to the Upper Vaal catchment.

#### 4.2. MAR Pseudo-Elasticity of Entropy

^{2}) values were relatively good (>0.5) for a goodness-of-fit of the linear regression. The correlation coefficient (R) values were derived from R

^{2}values and the test conducted showed that all R values were not rejected at 0.05 significance level. The values of the MAR elasticity of uncertainty associated with MAR were in the interval [0.47–0.71], between WR2005 and WR90 data sets. Between WR2012 and WR2005, MAR elasticity ranged from 0.48 to 0.74. This could be interpreted that, for 1% increase in MAR, the uncertainty associated with MAR, varied nearly between 0.5% and 0.7%. Hence 10% increase in MAR, would correspond to the increase in entropy of 5% to 7%. The ranges of elasticity were very close; hence they could show the same level of sensitivity (stability) of entropy when MAR changes in the tertiary catchments. The level of sensitivity (elasticity) is relatively low compared with the changes in MAR. These variations are relatively smaller and below the threshold value of 2% change of uncertainty associated with MAR. These variations were confirmed from statistical analysis of MAR elasticity. For that, the standard error of the sampling distribution of the mean of MAR elasticity for both WR90/WR2005 and WR2005/WR2012 data sets was found to be the same 0.03. The means of samples corresponding to the two data sets were 0.56 and 0.60 respectively. The t-test at 95% confidence interval (at 0.05, significance level) of the mean of MAR elasticity showed there was no significant difference in the mean of MAR elasticity. Based on low sensitivity [13], this could mean that the Upper Vaal catchment displayed a certain level of resilience or elastic stability for its water resources since the linear regression models were established within the zone of tolerable uncertainty or resilience. Tertiary catchments in the Upper Vaal displayed a relatively good level of resilience such that they could be able to keep a good functioning [13], when subjected to light changing runoff. Similar to [52] and considering WR2005 and WR90 data sets, the tolerable limits of variations of entropy values due to change in MAR were within 10% and 25% increase/decrease in runoff, as shown in Table 2.

^{2}values were 12% and 14% for WR90/WR2005 and W2005/W2012 respectively. Hence R values were 0.34 and 0.37 respectively; which implied that the derived R values were not being significant statistically. This translated in a weak relationship between MAR elasticity of entropy and MAP, despite MAP impacting positively on elasticity. Figure 4b, MAR elasticity displayed a very small decreasing variation (between 0.5 and 0.7), when MAE varied between 1351 and 1673 mm.

^{2}values of 0.1 and 0.51 for WR90/WR2005 and W2005/W2012, respectively. The corresponding R values were −0.31 and −0.71, respectively. The test conducted on R values showed that only −0.71 was significant. Therefore, it showed a weak relationship between MAR elasticity and MAE for WR90/WR2005 and a good or significant relationship for W2005/W2012. This could be translated in MAE negatively affecting over time entropy associated with MAR. Catchments in South Africa are dominated by higher evaporation rates rather than precipitation levels, as can be seen in Table 1. In particular, close to the lower zone of resilience, the increase in MAE could have adverse effects on catchment hydrology. Overall, the small variations in MAR elasticity of entropy showed that entropy was not sensitive/elastic to the change in MAR, for both MAP and MAE. Similarly, previous studies showed that small variations of elasticity relate to insensitivity of changes of inputs (climate factors) to catchment generating runoff [50,52,53]. Hence, catchments in the Upper Vaal region could be associated with a certain level of stability or resilience [13], when they were subjected to changes in MAP or MAE. The effect of these meteorological parameters [56,57] on MAR elasticity was analysed independently in this preliminary study. However, the combined effect of hydro-climatic parameters [53], together with anthropogenic parameters [50] could impact on MAR elasticity. For future climate change [53], the bivariate parameter MAR elasticity may not be used, as it does not depend on the temperature (a proxy for evapotranspiration) and precipitation, which are determinant factors for runoff generation. On a spatial basis, multiple parameter elasticity [72] of entropy could be influenced positively or negatively. Multivariate regression methods could enable an easy integration of the effects of various climate factors [55] on elasticity.

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Quaternary and tertiary catchments of the Upper Vaal catchment: upstream of Vaal Dam covers from C12 to C13; downstream of Vaal from C21 to C23 and Wildge from C81 to C83.

**Figure 2.**Entropy associated with mean annual runoff for each quaternary catchment in its respective tertiary catchment, H

_{min}and H

_{max}are entropy thresholds for: (

**a**) tertiary C11; (

**b**) tertiary C12; (

**c**) tertiary C12; (

**d**) tertiary C21; (

**e**) tertiary C22; (

**f**) tertiary C23; (

**g**) tertiary C81; (

**h**) tertiary C82 and (

**i**) tertiary C83.

**Figure 3.**Entropy associated with mean annual runoff (MAR) for each tertiary catchment in the Upper Vaal catchment.

Tertiary Catchment (TC) | Area (km^{2}) | MAP (mm) | MAE (mm) | MAR (WR90) MCM | MAR (WR2005) MCM | MAR (WR2012) MCM | Number of QC in TC |
---|---|---|---|---|---|---|---|

C11 | 8791 | 685 | 1449 | 548.2 | 546.28 | 527.34 | 12 |

C12 | 6498 | 647 | 1552 | 296.7 | 231.32 | 211.96 | 11 |

C13 | 5182 | 699 | 1416 | 291.8 | 322.49 | 343.05 | 8 |

C21 | 3541 | 686 | 1614 | 141.5 | 89.88 | 98.98 | 7 |

C22 | 5110 | 657 | 1633 | 131.5 | 150.02 | 157.51 | 10 |

C23 | 8273 | 615 | 1673 | 238.7 | 185.68 | 219.00 | 11 |

C81 | 6167 | 723 | 1351 | 450.6 | 515.02 | 529.08 | 12 |

C82 | 4471 | 645 | 1443 | 198 | 214.82 | 151.72 | 8 |

C83 | 7529 | 650 | 1478 | 283.8 | 236.76 | 252.56 | 12 |

**Table 2.**Relative changes in entropy associated with mean annual runoff (MAR) for TCs in the Upper Vaal for WR90, W2005 [46] and WR2012.

Tertiary Catchment | Relative Change in H (WR2005/WR90) | Relative Change in H (WR2012/WR2005) |
---|---|---|

C11 | −1.13 | 0.97 |

C12 | −3.78 | −1.04 |

C13 | 0.30 | 0 |

C21 | 4.45 | 0 |

C22 | −2.39 | 0 |

C23 | −14.35 | −2.35 |

C81 | −2.50 | 0 |

C82 | 4.40 | 3.5 |

C83 | 0.68 | 0 |

**Table 3.**Models for tertiary catchments (TCs) in Upper Vaal: relative change in entropy δH(y

_{j}) associated with MAR versus relative change in MAR (δ(y

_{j}) for quaternary catchments.

Tertiary Catchment | Linear Model (WR2005/WR90) | Linear Model (WR2012/WR2005) |
---|---|---|

C11 | $\delta H({y}_{j})=0.65\delta ({y}_{j})-0.36$ (R ^{2} = 98.36%, $\epsilon $ = 0.65 ) | $\delta H({y}_{j})=0.65\delta ({y}_{j})+2.50$ (R ^{2} = 99.12%, $\epsilon $ = 0.65) |

C12 | $\delta H({y}_{j})=0.55\delta ({y}_{j})+9.41$ (R ^{2} = 91.8%, $\epsilon $ = 0.55) | $\delta H({y}_{j})=0.62\delta ({y}_{j})+4.35$ (R ^{2} = 98.2%, $\epsilon $ = 0.62) |

C13 | $\delta H({y}_{j})=0.47\delta ({y}_{j})-4.82$ (R ^{2} = 99.00%, $\epsilon $ = 0.47 ) | $\delta H({y}_{j})=0.74\delta ({y}_{j})+36.47$ (R ^{2} = 98.40%, $\epsilon $ = 0.74 ) |

C21 | $\delta H({y}_{j})=0.63\delta ({y}_{j})+24.7$ (R ^{2} = 99.10%, $\epsilon $ = 0.63) | $\delta H({y}_{j})=0.48\delta ({y}_{j})-4.81$ (R ^{2} = 99.1%, $\epsilon $ = 0.48) |

C22 | $\delta H({y}_{j})=0.48\delta ({y}_{j})-8.89$ (R ^{2} = 98.36%, $\epsilon $ = 0.48) | $\delta H({y}_{j})=0.58\delta ({y}_{j})-2.91$ (R ^{2} = 99.8%, $\epsilon $ = 0.58) |

C23 | $\delta H({y}_{j})=0.46\delta ({y}_{j})-1.002$ (R ^{2} = 95.40%, $\epsilon $ = 0.46) | $\delta H({y}_{j})=0.5\delta ({y}_{j})+12.94$ (R ^{2} = 96.9%, $\epsilon $ = 0.5) |

C81 | $\delta H({y}_{j})=0.63\delta ({y}_{j})-10.13$ (R ^{2} = 99.0%, $\epsilon $ = 0.63) | $\delta H({y}_{j})=0.65\delta ({y}_{j})-1,77$ (R ^{2} = 99.88%, $\epsilon $ = 0.65) |

C82 | $\delta H({y}_{j})=0.49\delta ({y}_{j})-4.25$ (R ^{2} = 98.85%, $\epsilon $ = 0.49) | $\delta H({y}_{j})=0.74\delta ({y}_{j})+21.07$ (R ^{2} = 99.57%, $\epsilon $ = 0.74) |

C83 | $\delta H({y}_{j})=0.71\delta ({y}_{j})+11.52$ (R ^{2} = 98.00%, $\epsilon $ = 0.71 ) | $\delta H({y}_{j})=0.52\delta ({y}_{j})-3.5$ (R ^{2} = 99.41%, $\epsilon $ = 0.52) |

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**MDPI and ACS Style**

Ilunga, M.
Assessing Catchment Resilience Using Entropy Associated with Mean Annual Runoff for the Upper Vaal Catchment in South Africa. *Entropy* **2017**, *19*, 147.
https://doi.org/10.3390/e19050147

**AMA Style**

Ilunga M.
Assessing Catchment Resilience Using Entropy Associated with Mean Annual Runoff for the Upper Vaal Catchment in South Africa. *Entropy*. 2017; 19(5):147.
https://doi.org/10.3390/e19050147

**Chicago/Turabian Style**

Ilunga, Masengo.
2017. "Assessing Catchment Resilience Using Entropy Associated with Mean Annual Runoff for the Upper Vaal Catchment in South Africa" *Entropy* 19, no. 5: 147.
https://doi.org/10.3390/e19050147