ABSTRACT
The measurement of a stream flow is a vital component of water quality monitoring, geomorphology, and flooding investigations. However, the direct measurements of stream flow can be costly and challenging. To obtain a continuous record of discharge, typically the recorded stage and discharge are computed from the correlation of stage and discharge in the form of a rating curve. This study focused on the estimation of flow discharges using a rating curve equation based on data recorded at the Erer-Jijiga Bridge hydrometric station at the outlet of the upper Erer Sub-basin in Ethiopia for 2020–2021. The fitted rating curve was then used to estimate discharges from continuously recorded stages. The installation and operation of a gauging station in the Upper Erer Sub-basin will provide local and regional water sector administrators with more reliable estimates of discharge than would be available if the sub-basin was ungauged. This will also assist the local water sector to undertake water resource planning in the Upper Erer River Basin to better use the limited water resources in the region.
HIGHLIGHTS
A rating curve was developed for the Upper Erer River from on-site recordings at the hydrometric station.
The installation and operation of a gauging station in the Upper Erer Sub-basin will provide local and regional water sector administrators with more reliable estimates of discharge.
This will also assist the local water sector in undertaking water resource planning in the Upper Erer River Basin.
INTRODUCTION
The measurement of stream flow is a major component of water quality monitoring, geomorphology, and flooding investigations (Barbetta et al. 2017; Gore & Banning 2017; Sikorska & Renard 2017). Steam flows are usually recorded once a day during normal periods and on an hourly basis during floods. Nevertheless, the direct measurements of stream flow can be costly and challenging (Patra 2008; WMO 2010; Othman et al. 2019). Hence, it has been a known practice to estimate discharges from the water level record using a rating curve that is less expensive and an easier approach. Both stage and discharge in a stream vary in time and space, in order to obtain a continuous record of discharge, typically the stage is recorded and the discharge is computed from the correlation of stage and discharge (Aytac et al. 2012). The stage–discharge relationship is described by a rating curve, which is a commonly used technique and has been developed for gauging stations across the world (Guerrero et al. 2012). The rating curve is a significant technique in flow discharge estimation (Asgeir 2006; Maghrebi & Ahmadi 2017; Kiang et al. 2018; Mansanarez et al. 2018). The availability and accuracy of rating curves are important for various hydrological applications, including irrigation, water resource management, sediment control, and hydrologic modeling (Sefe 1996).
When establishing a rating curve, an appropriate gauging site needs to be selected. The gauging site needs to be permanent and not subject to erosion or deposition. So, the elevation of the water surface corresponding to a given discharge is consistent (Sharma & Sharma 2008). The stream flow level can be measured directly or indirectly. Water level gauges can be categorized as direct examples, vertical staff gauges, inclined gauges, hook gauges, or indirect examples, self-recorder gauges or a crest-stage gauges (Subramanya 2007; Adegbola & Olaniyan 2012). On-site discharge measurement can be also direct or indirect. The velocity–area method using a current meter and floats is a direct measurement. While the indirect measurement includes the slope–area method using flow-measuring structures, such as flumes, weirs, and gated structures. Typically, the velocity recorded by a current meter at the site and the water level recorded on a staff gauge in combination with a known cross-section geometry is used to estimate the discharge (Alfa et al. 2018).
The accuracy of discharge estimated from the rating curve depends on the accuracy of stage and discharge measurements and the development of the rating curve (Muzzammil et al. 2018). Stage–discharge relationship has continued to be an area of attention for hydrologists and several efforts have been taken to develop acceptable rating curves using different techniques. Muzzammil et al. (2018) used an Excel solver technique when establishing a rating curve. Alfa et al. (2018) created a rating curve for the Ofu River in Nigeria using linear regression analysis and an analysis tool in Microsoft Excel. McGinn & Chubak (2002) and Braca (2008) suggested polynomial models for stage–discharge relationships. Li et al. (2017) used multiple linear regression analysis (MLRA) when creating the Yellow River diversion model. Ghimire Bohla & Janga Reddy (2010) used different MLR methods when establishing a stage–discharge relation in rivers. Appropriate rating curve approaches and extra measurements relative to different test sites were required (Di Baldassarre & Claps 2011) to minimize the uncertainty in the flood discharge data.
All referenced studies are done in developed countries with sufficient data. In developing countries, the preservation of a watercourse gauge network is poor due to the scarcity of funds (Hamilton et al. 2019), stage–discharge measurements are rare (Abate et al. 2017; Zimale et al. 2018) and rating curves are not updated after major storms (Negatu et al. 2022). Similarly, a lack of flow discharge data (Abebe et al. 2020) highly affected a number of water resource studies in various regions of Ethiopia (Goshime et al. 2021). In the Upper Wabisheble Basin, including the Upper Erer Sub-basin, locating recent river discharge measurement data was difficult (Wudineh et al. 2021). Consequently, this study focused on the discharge estimation using a rating curve equation based on data recorded for 2020–2021 at the outlet of the Upper Erer Sub-basin. In this study, the number of recordings per day was greater than usually recorded when recording stream flow conditions in various parts of the country.
MATERIALS AND METHODS
Study area description
Upper Erer River sub-basin showing the location of the outlet and the staff gauge.
Upper Erer River sub-basin showing the location of the outlet and the staff gauge.
Data sources and collection
Upper Erer River at the Erer-Jijiga Road Bridge: (a) staff gauge installed in 2020, (b) flow-measuring process on-site using crane movement for vertical current meter in 2020, and (c) the Erer River at full stage during the peak flow in August 2020.
Upper Erer River at the Erer-Jijiga Road Bridge: (a) staff gauge installed in 2020, (b) flow-measuring process on-site using crane movement for vertical current meter in 2020, and (c) the Erer River at full stage during the peak flow in August 2020.
2.3. Discharge estimation



Determining the stage–discharge relation
The development of the rating curve involves two steps: first, measuring stages and velocities and then calculating the corresponding discharges in the river and establishing a relationship (rating curve), second, measuring water level only continuously and estimating the discharge using the established relationship (rating curve). The stage–discharge relationship that does not change with time is a permanent control and a stage–discharge relationship that changes with time is shifting control. Shifting control occurs when there is erosion or deposition of sediment at the measuring station.
A practical relationship between stage and discharge can be developed by the field measurement of stage and velocities across a section and, thereafter, can be expressed as a rating curve. While the literature presents a number of mathematical expressions for relating water levels to discharges in a given cross-section (Petersen-Overleir 2004; Franchini & Ravagnani 2007), in this study, the power-law was preferred (Equation (5)), in light of its simplicity and various application (e.g. Petersen-Overleir 2004; Schmidt & Yen 2009).
Fundamentally, rating curve equations are nonlinear. Hence, a nonlinear Microsoft Excel Solver was used to obtain the optimal values of the rating curve coefficients. The programing of the Microsoft Excel Solver was demonstrated for specific applications by scholars such as Muzzammil et al. (2018).
Performance evaluation of rating curves



RESULTS AND DISCUSSION
The rating curve at the gauging station
The values of a, c, and n were calculated by regression analysis using the analysis tool pack of Excel Solver available in Microsoft Excel, as shown in Table 1. Table 1 denotes that H represents the height of the water level, Qo is the observed discharge, and Qs is the simulated flow discharge.
Regression analysis for stage and observed discharge readings.
No . | H (m) . | Qo (m3/s) . | H + a (m) . | Qs = c (h + a)n (m3/s) . | (Qo–Qs)2 . |
---|---|---|---|---|---|
1 | 0.23 | 2.8000 | 1.38 | 2.5496 | 0.0627 |
2 | 0.24 | 2.9400 | 1.39 | 2.6463 | 0.0863 |
3 | 0.28 | 2.5900 | 1.43 | 3.0629 | 0.2236 |
4 | 0.30 | 2.4070 | 1.45 | 3.2902 | 0.7800 |
5 | 0.35 | 3.2560 | 1.50 | 3.9183 | 0.4387 |
6 | 0.35 | 3.2700 | 1.50 | 3.9183 | 0.4204 |
7 | 0.36 | 4.0000 | 1.51 | 4.0549 | 0.0030 |
8 | 0.40 | 5.3300 | 1.55 | 4.6398 | 0.4763 |
9 | 0.47 | 5.3326 | 1.62 | 5.8262 | 0.2437 |
10 | 0.50 | 5.3338 | 1.65 | 6.4042 | 1.1458 |
11 | 0.51 | 5.3342 | 1.66 | 6.6069 | 1.6197 |
12 | 0.52 | 5.3346 | 1.67 | 6.8146 | 2.1905 |
13 | 0.54 | 5.3353 | 1.69 | 7.2460 | 3.6506 |
14 | 0.55 | 5.3357 | 1.70 | 7.4697 | 4.5541 |
15 | 0.58 | 5.3369 | 1.73 | 8.1747 | 8.0532 |
16 | 0.60 | 5.3376 | 1.75 | 8.6738 | 11.1303 |
17 | 0.63 | 5.3388 | 1.78 | 9.4683 | 17.0524 |
18 | 0.66 | 5.3436 | 1.81 | 10.3204 | 24.7683 |
19 | 0.68 | 5.9145 | 1.83 | 10.9220 | 25.0751 |
20 | 0.70 | 6.2880 | 1.85 | 11.5516 | 27.7059 |
21 | 0.75 | 7.9340 | 1.90 | 13.2545 | 28.3081 |
22 | 0.78 | 17.1400 | 1.93 | 14.3697 | 7.6744 |
23 | 0.84 | 21.1790 | 1.99 | 16.8272 | 18.9381 |
24 | 0.85 | 25.2770 | 2.00 | 17.2679 | 64.1464 |
25 | 0.86 | 25.8400 | 2.01 | 17.7178 | 65.9707 |
26 | 0.88 | 25.5800 | 2.03 | 18.6459 | 48.0813 |
27 | 0.88 | 25.6140 | 2.03 | 18.6459 | 48.5539 |
28 | 0.94 | 29.9840 | 2.09 | 21.6683 | 69.1512 |
29 | 0.96 | 28.5600 | 2.11 | 22.7592 | 33.6497 |
30 | 1.18 | 34.9140 | 2.33 | 37.9596 | 9.2759 |
31 | 1.20 | 34.9426 | 2.35 | 39.6706 | 22.3542 |
32 | 1.24 | 35.0000 | 2.39 | 43.2793 | 68.5462 |
33 | 1.72 | 105.2350 | 2.87 | 111.2601 | 36.3022 |
34 | 1.80 | 134.1200 | 2.95 | 128.2174 | 34.8409 |
No . | H (m) . | Qo (m3/s) . | H + a (m) . | Qs = c (h + a)n (m3/s) . | (Qo–Qs)2 . |
---|---|---|---|---|---|
1 | 0.23 | 2.8000 | 1.38 | 2.5496 | 0.0627 |
2 | 0.24 | 2.9400 | 1.39 | 2.6463 | 0.0863 |
3 | 0.28 | 2.5900 | 1.43 | 3.0629 | 0.2236 |
4 | 0.30 | 2.4070 | 1.45 | 3.2902 | 0.7800 |
5 | 0.35 | 3.2560 | 1.50 | 3.9183 | 0.4387 |
6 | 0.35 | 3.2700 | 1.50 | 3.9183 | 0.4204 |
7 | 0.36 | 4.0000 | 1.51 | 4.0549 | 0.0030 |
8 | 0.40 | 5.3300 | 1.55 | 4.6398 | 0.4763 |
9 | 0.47 | 5.3326 | 1.62 | 5.8262 | 0.2437 |
10 | 0.50 | 5.3338 | 1.65 | 6.4042 | 1.1458 |
11 | 0.51 | 5.3342 | 1.66 | 6.6069 | 1.6197 |
12 | 0.52 | 5.3346 | 1.67 | 6.8146 | 2.1905 |
13 | 0.54 | 5.3353 | 1.69 | 7.2460 | 3.6506 |
14 | 0.55 | 5.3357 | 1.70 | 7.4697 | 4.5541 |
15 | 0.58 | 5.3369 | 1.73 | 8.1747 | 8.0532 |
16 | 0.60 | 5.3376 | 1.75 | 8.6738 | 11.1303 |
17 | 0.63 | 5.3388 | 1.78 | 9.4683 | 17.0524 |
18 | 0.66 | 5.3436 | 1.81 | 10.3204 | 24.7683 |
19 | 0.68 | 5.9145 | 1.83 | 10.9220 | 25.0751 |
20 | 0.70 | 6.2880 | 1.85 | 11.5516 | 27.7059 |
21 | 0.75 | 7.9340 | 1.90 | 13.2545 | 28.3081 |
22 | 0.78 | 17.1400 | 1.93 | 14.3697 | 7.6744 |
23 | 0.84 | 21.1790 | 1.99 | 16.8272 | 18.9381 |
24 | 0.85 | 25.2770 | 2.00 | 17.2679 | 64.1464 |
25 | 0.86 | 25.8400 | 2.01 | 17.7178 | 65.9707 |
26 | 0.88 | 25.5800 | 2.03 | 18.6459 | 48.0813 |
27 | 0.88 | 25.6140 | 2.03 | 18.6459 | 48.5539 |
28 | 0.94 | 29.9840 | 2.09 | 21.6683 | 69.1512 |
29 | 0.96 | 28.5600 | 2.11 | 22.7592 | 33.6497 |
30 | 1.18 | 34.9140 | 2.33 | 37.9596 | 9.2759 |
31 | 1.20 | 34.9426 | 2.35 | 39.6706 | 22.3542 |
32 | 1.24 | 35.0000 | 2.39 | 43.2793 | 68.5462 |
33 | 1.72 | 105.2350 | 2.87 | 111.2601 | 36.3022 |
34 | 1.80 | 134.1200 | 2.95 | 128.2174 | 34.8409 |
The observed stage and respective discharge at a particular point of reading were used to develop rating curves based on the least square method as per the general equation given by Equation (5).
Average daily stage hydrograph for the Upper Erer River at the Erer Bridge Gauge Station.
Average daily stage hydrograph for the Upper Erer River at the Erer Bridge Gauge Station.
Figure 3 indicates that the water level drops in October and reaches zero; this is the beginning of the winter (dry) season in the region when local communities divert water from the river for local consumption and the river flow reaches zero. In this river gauging site, the zero-stage level is recorded for most of the year except for rainy seasons.
Discharge hydrograph for Upper Erer River based on the fitted rating curve.
Observed and predicted with the fitted stage–discharge rating curve at Upper Erer River.
Observed and predicted with the fitted stage–discharge rating curve at Upper Erer River.
The goodness of fit
The developed rating curve was evaluated using the performance measures. Table 2 indicates the statistical measures of RMSE, Nash–Sutcliffe efficiency (NSE), and coefficient of determination (R2) to evaluate the quality of the adjusted dataset on different time scales. The trendlines for power, exponential, and polynomial function indicate that the R2 values nearly approach unity. The result indicates that the observed and simulated discharges were well fitted for the rating curve under the study area.
Parameters for the validation of the curve fitting process.
Parameter . | Value . |
---|---|
RMSE | 4.49 |
NSE | 0.97 |
R2 | 0.98 |
Slope | 0.94 |
Parameter . | Value . |
---|---|
RMSE | 4.49 |
NSE | 0.97 |
R2 | 0.98 |
Slope | 0.94 |
Relationship between simulated and observed discharge for the rating curve.
Estimating discharge using the developed rating curve
The rating curve is a very important tool in surface hydrology because the reliability of discharge data values is highly dependent on a satisfactory stage–discharge relationship at the gauging site. It can be used to obtain an estimate of the discharge of a large flood where only the stage data are available, by an extension of the rating curve (Patra 2008).
Discharges (m3/s) estimated from observed stage readings using the fitted rating curve.
Discharges (m3/s) estimated from observed stage readings using the fitted rating curve.
CONCLUSIONS
A rating curve was developed for the Upper Erer River using on-site recordings of the stage and discharge at the Erer-Jijiga Bridge hydrometric station. The fitted rating curve was then used to estimate discharges from continuously recorded stages. The installation and operation of a gauging station in the Upper Erer Sub-basin will provide local and regional water sector administrators with more reliable estimates of discharge than would be available if the sub-basin was ungauged. This will also assist the local water sector in undertaking water resource planning in the Upper Erer River Basin to better use the limited water resources in the region.
Future studies, to improve the rating curve and discharge estimation, could include continuous stage readings and an assessment of the possible impact of sediment transport and geomorphologic characteristics of the Erer River on the fitted rating curve.
ACKNOWLEDGEMENT
The authors would like to acknowledge the Ministry of Water and Energy of Ethiopia for providing the hydrometric gauging instrument and experts for field data collection.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.