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.

  • 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.

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.

Study area description

The Upper Erer Sub-basin, which drains the Harar highlands within the Wabisheble River basin, is located about 15 km southeast of Harar Town. The Harar town is located on geographic grid reference longitude 42°6′59.71″E, latitude 9°18′55.434″N. The Upper Erer Sub-basin is located between 9°13′26.4″ N and 9°31′26.4″ N latitude and 42°4′40.8″ E 42°20′38.4″ E longitude. The catchment area of the Upper Erer River is 466 km2 with an elevation varying from 1,306 to 3,019 m above sea level. The outlet of the Upper Erer River is found at 9°14′10.339″N latitude and 42°15′2.301″E longitude. The Upper Erer river is a source of water for both domestic and agricultural purposes. A staff gauge was installed at the road crossing at the Erer-Jijiga Road junction which was adopted as the outlet point for the study catchment. Figure 1 shows the catchment area, streamlines, and the location of the staff gauge.
Figure 1

Upper Erer River sub-basin showing the location of the outlet and the staff gauge.

Figure 1

Upper Erer River sub-basin showing the location of the outlet and the staff gauge.

Close modal

Data sources and collection

Stage–discharge measurements at the Upper Erer gauging station (on the Erer-Jijiga Road bridge crossing) were done by on-site observations. The selected site was stable and satisfied all the standards for installing a gauging station, as suggested by Meals & Dressing (2008) and Sauer & Turnipseed (2010). The location of the gauging station is shown in Figure 1. For this study, the stage readings were recorded using a calibrated staff gauge for the 18 months from 13 May 2020 until 16 October 2021 while velocity measurements were recorded using an ultrasonic current meter during base-flow, low-flow, and high-flow periods, respectively, in May 2020, August 2020, and October 2021 that were used to calculate the observed discharge corresponding to the stage (see Figure 2). A stage time series was assembled by recording the stage once a day at noon under base-flow conditions and four times a day during low-flow periods at 06:00 AM, 10:00 AM, 02:00 PM, and 06:00 PM. High flows during the summer season were recorded seven times a day at 06:00 AM, 08:00 AM, 10:00 AM, 12:00 PM, 02:00, PM 04:00 PM, and 06:00 PM.
Figure 2

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.

Figure 2

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.

Close modal

2.3. Discharge estimation

Discharge estimation, using an indirect method, can be implemented more accurately by employing the velocity–area method using the measured depth and velocity. There are two approaches to the indirect measurement of discharge: the mid-section and mean-section (Subramanya 2007). In this study, the mid-section approach was selected (see Equation (1)).
(1)
The method used to calculate the total discharge flowing through a section was the summation of the average discharge values calculated for each strip, as shown in the following equations.
(2)
(3)
where Q is the discharge (m3/s), is the velocity (m/s), is the span length (m), is the depth (m), and n is the number of intervals across the section. The velocity of the flow increases from the bottom to the water surface across the channel section.

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).

If H represents the water level for a given flow discharge Q, then the relationship between H and Q (Equation (4)) can be approximated using the equation given by Kennedy (1984) (Equation (5)). The determination of the datum correction a for Q = 0, (H + a) = 0, where a = −H.
(4)
where f(H) is an algebraic function of water level.
(5)
where Q is the flow in m3/s, H is the measured water level (m), c and n are rating curve parameters, and a is a constant representing the gauge reading which corresponds to zero discharge.

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

In this study, the performance of the rating curve model was verified using the evaluation matrices given in Equations (6)–(8). The performance measures include the Nash–Sutcliffe measure of efficiency (NSE), the root-mean-square error (RMSE), and the coefficient of determination (R2). The NSE shows how well the simulation coincides with the observation with values ranging between –∞ and 1.0, NSE = 1 represents a perfect fit to the observations. The higher the NSE value, the more reliable the model. The coefficient of determination (R2) value is between 0 and 1, if the R2 equals 1, then a perfect fit to the observations is achieved. And, the smallest values of RMSE mean a good agreement between observed and simulated data (Nash & Sutcliffe 1970; Hayden 2009).
(6)
(7)
(8)
where is the estimated discharge, and the average of the observed data.

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.

Table 1

Regression analysis for stage and observed discharge readings.

NoH (m)Qo (m3/s)H + a (m)Qs = c (h + a)n (m3/s)(QoQs)2
0.23 2.8000 1.38 2.5496 0.0627 
0.24 2.9400 1.39 2.6463 0.0863 
0.28 2.5900 1.43 3.0629 0.2236 
0.30 2.4070 1.45 3.2902 0.7800 
0.35 3.2560 1.50 3.9183 0.4387 
0.35 3.2700 1.50 3.9183 0.4204 
0.36 4.0000 1.51 4.0549 0.0030 
0.40 5.3300 1.55 4.6398 0.4763 
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 
NoH (m)Qo (m3/s)H + a (m)Qs = c (h + a)n (m3/s)(QoQs)2
0.23 2.8000 1.38 2.5496 0.0627 
0.24 2.9400 1.39 2.6463 0.0863 
0.28 2.5900 1.43 3.0629 0.2236 
0.30 2.4070 1.45 3.2902 0.7800 
0.35 3.2560 1.50 3.9183 0.4387 
0.35 3.2700 1.50 3.9183 0.4204 
0.36 4.0000 1.51 4.0549 0.0030 
0.40 5.3300 1.55 4.6398 0.4763 
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).

In this study, a limited number of stage–discharge measurements (34) were done within the period for the development of the rating curve (see Table 1). The maximum water level observed was 1.8 m while the minimum water level was 0.23 m (see Figure 3). Figure 3 indicates that the maximum stage levels were observed during August and May in the rainy seasons of Summer and Spring in the region, respectively, while the minimum stage level was also observed in May. This was attributed to the starting period of the rainy season in the region and the local communities harvesting the river water and the first drop of rainfall was not approaching the river.
Figure 3

Average daily stage hydrograph for the Upper Erer River at the Erer Bridge Gauge Station.

Figure 3

Average daily stage hydrograph for the Upper Erer River at the Erer Bridge Gauge Station.

Close modal

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.

Figure 4 shows converting recorded stages into discharges using the fitted rating curve. It discloses that major flows occurred on 12th August 2020 and 15th August 2020.
Figure 4

Discharge hydrograph for Upper Erer River based on the fitted rating curve.

Figure 4

Discharge hydrograph for Upper Erer River based on the fitted rating curve.

Close modal
The Excel solver method is used to compute the minimum square of errors for different iterations and the constant values are calculated. The final iteration output for the values of a is 1.1533 m, for c is 0.4769 and for n is 5.1656, and the rating curve equation is determined as follows
(9)
The developed rating curve represents the actual discharge crossing the station with the corresponding water level. Figure 5 shows the observed, estimated discharge with a developed rating curve (refer to Table 1).
Figure 5

Observed and predicted with the fitted stage–discharge rating curve at Upper Erer River.

Figure 5

Observed and predicted with the fitted stage–discharge rating curve at Upper Erer River.

Close modal

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.

Table 2

Parameters for the validation of the curve fitting process.

ParameterValue
RMSE 4.49 
NSE 0.97 
R2 0.98 
Slope 0.94 
ParameterValue
RMSE 4.49 
NSE 0.97 
R2 0.98 
Slope 0.94 

The result indicates that the statistical performance evaluation gave a strong NSE coefficient of 0 0.97, an RMSE value of 4.49, and a coefficient of determination value of 0.98. In addition, the result for the observed and simulated discharge in Figure 6 shows the slope of the equation (0.94). From this result, the Excel solver is a promising tool for estimating rating curve parameters. It has been found that the Excel solver minimizes the time consumption during trial and error for formulating stage–discharge relations. This result supports that the equation represents the best fit.
Figure 6

Relationship between simulated and observed discharge for the rating curve.

Figure 6

Relationship between simulated and observed discharge for the rating curve.

Close modal

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).

Figure 7 shows the observed stage data from 13 May 2020 to 16 October 2021. About 2020 stage readings were collected. During the rainy season, the stage level is maximum and after the rainy season, the stage becomes zero. This occurs in December, January, February, and March. During these months, the stage reading was recorded once a day since this was the zero-discharge level. As shown from the graph, some readings have stage levels greater than 1.5 m and most of the readings fall between 1 and 1.5 m, while some observations exceed 2 m.
Figure 7

Observed stages (m) during the data collection period.

Figure 7

Observed stages (m) during the data collection period.

Close modal
Figure 8 plots the flows at the gauging station which were estimated using Equation (9) based on the observed stages. The estimated peak discharge approaches 250 m3/s.
Figure 8

Discharges (m3/s) estimated from observed stage readings using the fitted rating curve.

Figure 8

Discharges (m3/s) estimated from observed stage readings using the fitted rating curve.

Close modal

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.

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.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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