# Prediction at Ungauged Catchments through Parameter Optimization and Uncertainty Estimation to Quantify the Regional Water Balance of the Ethiopian Rift Valley Lake Basin

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## Abstract

**:**

## 1. Introduction

## 2. The Study Area

^{2}, providing water supply to a population of more than 15 million people who live mainly on subsistence agriculture (Figure 1). The RVLB is 84 km wide, and adjacent to it there are large, discontinuous Miocene-aged normal faults [35,36]. Within the Main Ethiopian Rift, there are a series of right-stepping, quaternary rift basins, which have faulted magmatic segments, extending about 20 km wide and 60 km long, that are embryonic oceanic spreading centers. The central part of the RVLB is formed by a Pliocene-aged, faulted caldera, caused by a fractured volcano [1]. Existing faults and repeated ground cracks at the floor of the caldera have increased the permeability of the rock. Within the basin, several small-to-medium-sized catchments drain into eight freshwater lakes. For most of these catchments, there is a lack of hydro-climatic data; what data are available contain gaps and are subject to human interventions. In recent decades, the RVLB has experienced major droughts and extreme flooding due to rainfall variability, making prediction difficult [37]. This has resulted in an uncertain analysis of high and low flows, factors that are important for quantifying the hydrological water balance components in this region.

#### Data and Catchment Properties

^{2}.

Variable | Spatial Resolution | Time Period | Temporal Resolution | Source | Reference |
---|---|---|---|---|---|

Precipitation | 0.1° | 1995–2007 | Daily | MSWEP V2 | Beck et al. [41] |

Potential evapotranspiration | 0.25° | 1995–2007 | Daily | GLEAM v3 | Martens et al. [43] |

HBV-parameters | 0.5° | - | - | www.gloh2o.org (access date 12 January 2020) | Beck et al. [30] |

Elevation | 30 m | - | - | SRMT V2.1 | https://earthexplorer.usgs.gov (access date 28 January 2020) |

Wetness index (P/PE) | Point scale | 1995–2007 | Daily | MSWEP V2 and GLEAM v3 | Beck et al. [41]; Martens et al. [43] |

Streamflow | Pont scale | 1995–2007 | Daily | MOWIE | - |

## 3. Methods

#### 3.1. Hydrological Model

_{C}, and L

_{P}, control the soil moisture dynamics. β controls the contribution (dQ) to the runoff response routing and the increase (dP-dQ) in soil moisture storage (Ssm) and F

_{C}is the maximum soil moisture storage in the model as shown by Equation (1). L

_{P}is the value of the soil moisture above which evapotranspiration (Ea) reaches its potential level (Ep). The actual evaporation from the soil moisture zone equals the potential evaporation if Ssm/F

_{C}is above L

_{P}* F

_{C}as shown by Equation (2).

^{−1}].

_{LZ}[mm], is filled by percolation from the upper reservoir (P

_{MAX}), and the outflow from this lower reservoir (Q

_{2}) is controlled by the recession coefficient K

_{2}[d

^{−1}]. However, the upper reservoir storage S

_{UZ}[mm] is drained by two recession coefficients, K

_{0}[d

^{−1}] and K

_{1}[d

^{−1}], draining the quick flow Q

_{0}[mm d

^{−1}] and slow flow component Q

_{1}[mm d

^{−1}] separated by a threshold V

_{UZL}[mm] (Equations (3)–(5)).

^{−1}] from the soil moisture routine exceeds the capacity, the upper reservoir will start to fill. This reservoir models the response during flood periods. The parameters calibrated from the runoff response function are P

_{MAX}, K

_{0}, K

_{1}, K

_{2}, and V

_{UZL}. Finally, the runoff is computed independently for each sub-basin by adding the contributions from the upper and the lower reservoir. To account for the damping of the runoff pulse in the river before reaching the basin outlet, a simple routing transformation is performed. This filter has a triangular distribution of weights with the base length and is expressed by the parameter M

_{MAXBAS}[d]. A detailed description of the model is shown by Bergstrom [56] and Seibert and Vis [49]. The ranges of the nine model parameters are derived from prior knowledge, provided through a global set of HBV model parameters [30].

#### 3.2. Parameter Estimation in the Gauged Catchments

_{obs}and Q

_{sim}are monthly averages of observed and simulated discharges [m³ month

^{−1}], respectively, while ${\overline{Q}}_{0bs}$ is the mean observed discharge over the calibration or validation periods. Using monthly averages, it is focused on the seasonal, long-term behavior instead of daily, short-term fluctuations. In order to remove unrealistic parameter combinations, only parameter sets that produced NSE ≥ 0.5 in the calibration period were kept. Consequently, different catchments can result in a different number of behavioral parameter sets. A pre-analysis using NSE on a monthly time scale showed that the model performs well for all of the catchments. Comparing the mean and variability of model performance for the remaining catchments during calibration and validation allows to assess the predictive performance and uncertainty of the selected parameter sets. To prepare for regionalization, (1) the variability of each model parameter in the reduced parameter sample after NSE ≥ 0.5 (expressed by their coefficient of variation, CV), and (2) the best parameter set (largest NSE) of the calibration, NSE

_{CAL}, and the validation period, NSE

_{VAL}, for each of the catchments were extracted. We derived the behavioral parameter ranges (parRANGE) during calibration with NSE ≥ 0.5. Using the behavioral parameter sets, we ran the model for the validation period (2003–2007), and we selected the best validation parameter sets among the behavioral parameters. Among the behavioral parameter sets, there may be a best parameter set that performs differently from the best-calibrated parameter sets in some catchments. In addition, the most stable parameter set for each catchment was identified, i.e., the parameter set that showed the smallest difference in NSE values between calibration and validation periods, NSE

_{DIFF}. Again, it is considered that the parameter set with the highest NSE is selected. In this procedure, after sorting the NSE

_{DIFF}in ascending order, only 5% of the parameters with the lowest NSE

_{DIFF}were extracted, from which the parameter set with the highest NSE was selected. In the selection of the stable parameter sets, parameter sets of the calibration period with the highest NSE that show a minimum difference with the NSE of the validation period were preferred.

#### 3.3. Parameter Estimation in the Ungauged Catchments

#### 3.4. Evaluation and Uncertainty Estimation of the Regionalization Procedure

#### 3.5. Estimation of Regional Resilience of Streamflow to Precipitation Variability

## 4. Results

#### 4.1. Estimated Parameters in the Gauged Catchments

_{C}, K

_{2}, and L

_{P}were well identified in most of the catchments. Parameter β was highly identified towards the lower values as shown by the dark gray color (Figure 4) or by the narrow parameter range in catchment #6 (Figure S1). F

_{C}shows identifiability towards higher values in their parameter range for catchment #2. On the other hand, K

_{2}showed higher sensitivity towards the lower values for catchment #7. The confined ranges of model parameters (K

_{0}, and V

_{UZL}) result in a relatively uniform distribution in their median values for all catchments. These parameters (K

_{0}and V

_{UZL}) remained insensitive and were later excluded from the regression procedure due to less information contained in their parameter identifiability. Taking median values for unidentifiable parameters would be a better model for the regional model [10] (Figure 4, Figures S2 and S3).

#### 4.2. Performance of the Regionalization Procedure

_{0}and V

_{UZL}were unidentifiable during calibration, and their median values were taken for the regionalized model. The weighted regression shows acceptable performance in reproducing most of the parameters. For instance, the weighted regression reproduces well the parameters β, F

_{C}, and K

_{2}, whereas the remaining parameters are less reproducible by the regression model. Table S5 shows the performance of the weighted regression model during the leave-one-out cross-validation using a coefficient of determination (R

^{2}). It is seen that the weighted regression procedure does not always produce model parameters in their predefined range (Table 3). For example, the regressed F

_{C}of catchment #6 is above the maximum threshold of 700. In such cases, the outlier model parameter is assigned to its maximum value. Among the remaining parameters, most of them show acceptable correlations. However, some of them, such as K

_{1}and P

_{MAX}, are poorly reflected through the regional regression in a few catchments. For instance, K

_{1}in catchment #11 is poorly modeled. Therefore, parameter K

_{1}is not identified in catchment #11. The regression model using parameter P

_{MAX}is poorly represented in four catchments (#09, #10, #13, and #14). This shows that for catchments #09, #10, #13, and #14, the parameter P

_{MAX}is not identifiable for the carefully selected catchment properties. Furthermore, for catchment #09, model parameters F

_{C}, and P

_{MAX}were poorly identified. However, the weighted regression procedure sufficiently represents the parameters in the remaining catchments. The best performing (#08) and most poorly performing (#09) catchments were also shown by the black and red colors, respectively. Catchment #08 primarily shows a stable prediction for all parameter-sampling procedures during calibration, validation, and stable relationships. Identifiable parameters in this catchment are also reproduced well from the regression model, whereas in catchment #09 (red scatter), most parameters were poorly identified and poorly reproduced by the regression model.

#### 4.3. Estimation of Regional Resilience of Streamflow to Precipitation Variability

## 5. Discussion

#### 5.1. Reliability of the Regionalization Approach

#### 5.2. Parameter Sensitivity and Spatial Variability

_{C}, and L

_{P}) and the recession coefficient in the lower reservoir (K

_{2}) are readily identifiable in most catchments. However, the insensitivity of the parameters in some catchments may be due to interactions with other parameters. Abebe et al. [70] show the interaction between parameter K

_{2}and the percolation rate (P

_{MAX}), where the increment of K

_{2}beyond the optimum rate of percolation may not show any sensitivity. In addition, the insensitivity of model parameters is related to the poor identifiability of model parameters in the catchments. For instance, parameters such as K

_{0}and V

_{UZL}were poorly identified and remained insensitive for any parameter value throughout the catchment (Figure 4).

_{2}, which is a recession coefficient in the lower reservoir, and it is well identified in catchments #07 towards the lower parameter value. Catchment #07 is characterized by sloppy catchment with a high drainage channel slope (catchment index) (Table 2) which facilitates the drainage of available water by surface runoff. Thus, for the little remaining water in the lower reservoir, lower values of K

_{2}will become more sensitive. Parameter K

_{1}, which is a recession coefficient from the upper reservoir, is well identified in catchment #05. A lower value of the recession coefficient results in a relatively low drainage density in catchment #05 (Table 2), which facilitates a relatively higher percolation (P

_{MAX}) in the catchment.

_{1}) controlling the water flow might not affect the outflow conditions. This is also shown by the negative correlation of the slope with K

_{1}(Table S1). Furthermore, the insensitivity of the parameters in the upper reservoir can be affected by low precipitation amounts. In low precipitation conditions (such as in catchments #08, #10, and #13) the resulting soil moisture from the soil profile and the upper reservoir will be much less, resulting in less runoff. The adjustment of the runoff-controlling parameters (K

_{0}and V

_{UZL}) might not have any influence (remain insensitive) on the resulting runoff. Parameter K

_{0}only functions when the cumulative precipitation exceeds the threshold of the V

_{UZL}value. Other than climatic properties, the insensitivity of the parameters can result from the interaction of the parameters, and this influence will be more pronounced for complex models. Other studies also showed that the insensitivities of the model parameters could result from the mismatch between the model complexity and the available data used to parameterize the model [71,72,73,74].

#### 5.3. Estimation of Regional Resilience of Streamflow to Precipitation Variability

#### 5.4. Transferability of the Approach to Other Catchments and Models

_{0}, and V

_{UZL}) could add bias to the regionalized models, thus taking their median values from the estimated parameters can be a better model [10]. The stability and resilience of the model parameters from the regionalization procedure would minimize the error and bias in parameter transfer within the model component [62]. However, success in the parameter transfers will be influenced by the dominant catchment properties that are identified regionally [32]. Despite this, a sufficient correlation between the catchment property and model parameters can be a good indicator of the predictive power of the selected catchment properties.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The study basin showing 14 gauged catchments for the regional model development and 35 ungauged catchments that are draining to the respective lakes through the river networks.

**Figure 3.**The ranges of monthly NSE values derived from the confined parameter sets during calibration and the corresponding NSE ranges during the validation period derived from the confined parameter sets; and the monthly NSE derived from parameter sets of best calibration, best validation, and most stable parameter sets for each catchment.

**Figure 4.**Cumulative distributions of parameters in the 14 catchments that show the probability of identifiability for each parameter on their ranges. The ranges between the most identifiable and poorly identified parameters are indicated through dark gray to light gray colors. The respective identifiability is also shown by the weight (1/CV) of parameter distribution. Where the dark gray color shows the catchment number with the most identifiable parameter. Parameters such as K

_{0}and V

_{UZL}are not identified through their parameter ranges, therefore the median of the parameter was used for the regression model.

**Figure 5.**Evaluation of the regression procedure on the calibration (1995–2002) and validation period (2003–2007), showing (

**a**,

**b**) the relationship of the NSE of the best-estimated parameters (from calibration, validation, and stable parameters set) and the parameters from the regression. The black and red scatter indicate the best and most poorly performing catchments, respectively. (

**c**) Shows the performances of best-estimated parameters (from calibration, validation, and stable sets) and the performance of the three regression models developed using the best parameters of calibration (REGcal), validation (REGval), and stable sets (REGstable) while evaluated during the calibration and validation period.

**Figure 6.**Relationships between best parameters estimated from stable sets and parameters derived from the regression model. The black scatters (catchment #8) represent the catchment with most parameters that are highly identified during parameter estimation and its corresponding parameters during regression. The red scatters (catchment #9) represent a catchment with most parameters that are poorly identified during parameter estimation and its corresponding parameters during regression. For unidentifiable parameters (K

_{0}and V

_{UZL}), their median value was taken for the regionalized model.

**Figure 7.**Prediction interval derived from the 14 regression models using the best parameter sets from stable parameter sets and uncertainty interval; (

**a**) for best−performing catchment (#08) and mean of the 14 simulations, and (

**b**) uncertainty interval for most poorly performing catchment (#09) and mean of the 14 simulations.

**Figure 8.**(

**a**–

**c**) Elasticity calculated for the median, wettest, and driest years for both gauged and ungauged catchments and (

**d**–

**f**) their corresponding simulation uncertainty expressed by the coefficient of variation CV.

**Table 2.**Descriptions and values of properties for gauged catchments in the RVLB that were used for the development of the regional model.

Cat No and Names at the Gauge Location | Catchment Properties | Drainage Area [km^{2}] | Drainage Density [km km^{−2}] | Mean Slope [%] | Mean Elevation [m] | Catchment Index [m km^{−1}] | Permeability [log_{10} m^{2}] | Porosity [-] | Wi [-] | P [mm] |
---|---|---|---|---|---|---|---|---|---|---|

Description of properties | Index of catchment area | The ratio of catchment stream length to the drainage area | Mean of the percentage slope for each terrain unit | Index describing the mean of catchment elevation | Mean of all inter-nodal slopes in a catchment | Index describing the nature of water flow in the shallow aquifer | The fraction of the volume of voids in the shallow aquifer | Wetness index (Wi) as the ratio of precipitation (P) to potential evapotranspiration (PE) | Annual average precipitation (1995–2007) | |

#01-Bilate@Tena | 3821.2 | 0.075 | 16.22 | 2037.1 | 10.07 | −12.194 | 0.07 | 0.85 | 923.6 | |

#02-Gelana@Tore bridge | 506.4 | 0.124 | 24.17 | 2084.5 | 10.39 | −12.5 | 0.09 | 1.17 | 1309.1 | |

#03-Gidabo@Measso | 2590 | 0.113 | 20 | 1805.4 | 14.54 | −12.248 | 0.097 | 0.85 | 942.07 | |

#04-Gedemso@Langano | 241.5 | 0.67 | 18.1 | 2759.3 | 28.11 | −12.5 | 0.09 | 0.88 | 919.19 | |

#05-Woito@Bridge | 4528.2 | 0.07 | 29.09 | 1439.5 | 10.64 | −11.433 | 0.028 | 1.34 | 1319.6 | |

#06-Hamassa@Wajifo | 534.4 | 0.34 | 15.86 | 1655.5 | 15.22 | −12.306 | 0.076 | 0.94 | 1208 | |

#07-Hare | 196.5 | 0.36 | 33.08 | 2343.1 | 77.67 | −12.2 | 0.076 | 0.897 | 1107.5 | |

#08-Katar@Abura | 3241.1 | 0.115 | 8.7 | 2601.9 | 19.99 | −12.034 | 0.064 | 0.69 | 779.8 | |

#9-Kulfo@Arbaminch | 397.2 | 0.226 | 36.39 | 2249.9 | 76.55 | −12.283 | 0.08 | 1.52 | 1617.8 | |

#10-Meki@Meki village | 2033.1 | 0.111 | 19.27 | 2124.4 | 11.12 | −12.155 | 0.068 | 0.58 | 667.2 | |

#11-Gidabo@Bedesa | 144.2 | 0.341 | 30.18 | 2149.7 | 56.74 | −12.5 | 0.09 | 1.29 | 1397.7 | |

#12-Katar@Fete | 1940.9 | 0.117 | 14.49 | 2668.9 | 17.78 | −12.171 | 0.075 | 0.87 | 991.2 | |

#13-Katar@Timela | 205.2 | 0.65 | 18.29 | 2953.5 | 40.87 | −12.371 | 0.084 | 0.7 | 785.6 | |

#14-Gidabo@Aposto | 491.8 | 0.327 | 21.49 | 2012.6 | 26.76 | −12.483 | 0.089 | 1.18 | 1300.1 |

**Table 3.**HBV parameter ranges for the RVLB and their descriptions, derived from Beck et al. [30].

Parameter | Description | Global Range (Min to Max) |
---|---|---|

β [-] | Shape coefficient of recharge function | 1–6 |

F_{C} [mm] | Maximum water storage in unsaturated-zone store | 50–700 |

K_{0} [d^{−1}] | Additional recession coefficient of upper groundwater store | 0.05–0.99 |

K_{1} [d^{−1}] | Recession coefficient of upper groundwater store | 0.01–0.8 |

K_{2} [d^{−1}] | Recession coefficient of lower groundwater store | 0.001–0.15 |

L_{P} [-] | Soil moisture value above which actual evaporation reaches potential evaporation | 0.3–1 |

M_{MAXBAS} [d] | Length of equilateral triangular weighting function | 1–3 |

P [mm d_{MAX}^{−1}] | Maximum percolation to lower zone | 0–6 |

V_{UZL} [mm] | Threshold parameter for extra outflow from upper zone | 0–100 |

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## Share and Cite

**MDPI and ACS Style**

Abraham, T.; Liu, Y.; Tekleab, S.; Hartmann, A.
Prediction at Ungauged Catchments through Parameter Optimization and Uncertainty Estimation to Quantify the Regional Water Balance of the Ethiopian Rift Valley Lake Basin. *Hydrology* **2022**, *9*, 150.
https://doi.org/10.3390/hydrology9080150

**AMA Style**

Abraham T, Liu Y, Tekleab S, Hartmann A.
Prediction at Ungauged Catchments through Parameter Optimization and Uncertainty Estimation to Quantify the Regional Water Balance of the Ethiopian Rift Valley Lake Basin. *Hydrology*. 2022; 9(8):150.
https://doi.org/10.3390/hydrology9080150

**Chicago/Turabian Style**

Abraham, Tesfalem, Yan Liu, Sirak Tekleab, and Andreas Hartmann.
2022. "Prediction at Ungauged Catchments through Parameter Optimization and Uncertainty Estimation to Quantify the Regional Water Balance of the Ethiopian Rift Valley Lake Basin" *Hydrology* 9, no. 8: 150.
https://doi.org/10.3390/hydrology9080150