## Abstract

This study aims to assess various empirical synthetic unit hydrograph (SUH) methods and find the best method. Ideally, each river should have a definite rain gauge station (RGS) to get sufficient rainfall data that is available for carrying out meaningful analysis. The provisions of Indian Standard (IS) 4987:1994 determined the optimum number of RGS. In the absence of RGS, the SUH is recommended. SUHs have been developed using various methods such as Snyder's, Taylor and Schwarz, Soil Conservation Service, Mitchell's and Central Water Commission (CWC). In the present study, the Rel River Basin (RRB) is considered as the study area which has two existing RGS. IS 4987:1994 suggested that four RGS are required for more reliable rainfall data. Various efficiency criteria such as Correlation Coefficient, Pearson Coefficient, Nash Sutcliffe Efficiency, Index of Agreement, Normalized Root Mean Square Error, Mean Absolute Error, Root Mean Square Error and Kling-Gupta Efficiency have been used to compare SUH methods. The ranking of SUH methods was reported based on the compound factor (CF) through efficiency criteria. The 1.125 CF was observed as the minimum for the CWC method and recommended for determining peak discharge and timing for the study area.

## HIGHLIGHTS

An optimum number of RGS has been examined at the initial level of the study.

Various empirical methods of SUH have been evaluated.

Numerous efficiency criteria have been used to compare different SUH methods.

The compound factor (as an MCDM technique) has been used to find the best SUH method, which is a new concept in SUH.

## INTRODUCTION

Rainfall is one of the most essential and complex hydrological modelling parameters (Mishra & Singh 2010). After Bangladesh, India is an another important country frequently affected by floods (Gupta *et al.* 2003). 12.5% of India's geographical area is prone to flooding (Patel & Dholakia 2010). Due to this large area of submergence in India, flood analysis has become an essential feature (Kumara & Mehta 2020; Mehta *et al.* 2020). Rainfall measurement for any basin is the primary requirement for flood analysis (Marchi *et al.* 2010). RGS must be distributed equally in the basin to achieve reliable results (Berne *et al.* 2004). The installed RGS in the basin should be adequate to measure the rainfall. If the RGS is higher, the economy will be affected, and if the RGS is lower, the results will be unreliable (Buytaert *et al.* 2006).

In the ungauged and semi-gauged basins, flood estimates and predictions are among the significant issues resulting from heavy rainfall over the basin. There are many approaches in existence to the assessment of floods (Shaikh *et al.* 2018). Few approaches are based on statistics, while others based on the characteristics of the basin are designed to meet this objective. The Unit Hydrograph (UH) concept has been implemented in the runoff estimation for the last four decades. Sherman (1932) was the originator of the UH theory. UH is defined as a surface runoff hydrograph in a given basin due to effective rainfall for the unit duration.

Previous studies related to UH evaluation were devoted to Sherman's famous principle in 1932, which ended in a tremendous style of empiric and semi-empiric techniques used by hydrologists for many years. This technique is popular among hydrologists and is defined in several popular hydrology texts (Mujumdar *et al.* 2014). Many hydrologists have used various UH approaches such as deterministic, probabilistic, statistical and mathematical, and many others (Tunas *et al.* 2019). The UH is a conventional means of representing the basin's linear system response to rainfall over the basin (Maidment *et al.* 1996). The UH concept assumes that the basin acts linearly and time-invariantly with the introduction of effective rainfall to provide an instant output of storm runoff (Bruen & Dooge 1992). The UH form shows the runoff traits of the watershed (Wesley *et al.* 1987). In an ungauged or semi-gauged basin, the lack of observed rainfall and runoff data gave rise to the SUH's perception of the basin's characteristics rather than the rainfall-runoff data. Some of the SUH methods have been suggested for hydrologists and scientists by Snyder (1938), Mitchell (1948), Taylor & Schwarz (1952) and SCS (2002). However, these techniques are specific to the region and being simple, and require limited information for their use.

Snyder (1938) used an empirical approach to evaluate the salient hydrograph features such as lag time, peak time, base time, peak discharge, UH widths at 50, and 75% peak flow. Taylor & Schwarz (1952) derived SUH for 20 basins ranging from 50 to 4,000 km^{2} of basin area, all in North and Middle Atlantic States. The most significant features considered are the area, the length of the longest path, the main stem's length from the centre of the basin, and average slope. The United State Soil Conservation Service (SCS) (1957) approach determined SUH through an average dimensionless hydrograph and avoided physical fitting. SCS dimensionless UH is assumed to be an invariant of the shape, size and location of the catchment area, which may not be justified (Singh 2000). Mitchell (1948) had examined fifty-eight watersheds in Illinois. He had developed the summation curves and grouped them based on the watershed area. The error in peak discharge has been reported at 39.0 per cent and 37.5 per cent during the peak time. CWC has provided SUH for distinctive sub-zones to be used for flood assessment of 25, 50 and 100 years return period for small to medium-sized drainage basins (Central Water Commission 1993). This method provides seven points of one-hour SUH. Seven points are peak time, peak discharge, 50 per cent width of peak discharge (two points), 75 per cent width of peak discharge (two points) and the base time. The traditional methods like Snyder, Mitchell, Taylor & Schwarz and SCS method are the beneficial methods for SUH generation all over the globe. The CWC method is newly developed for India for SUH generation. So, in this study, these five methods have been compared and identified which method is the best method suitable for the study area. Some recent studies are listed here in which different SUH methods have been used. Prajapati & Subbaiah (2005) identified Snyder as the best suitable method for the Rawat Sagar Basin, India. Velez & Botero (2011) concluded that Taylor & Schwarz method also gave similar results to Snyder's method for San Luis Creek Basin, Manizales. Salami *et al.* (2009) had summarised that the SCS method was best for eight rivers in South-West, Nigeria. Patel & Thorvat (2016) found that Mitchell's method gives an acceptable percentage error for the Upper Kumbhi Basin and the Dhamani Basin, Maharashtra, India. Salami *et al.* (2017) again recommended the SCS method to estimate the ordinate required to develop peak runoff hydrograph for Ogun-Osun river catchment, south-west, Nigeria. Reddy *et al.* (2019) had identified that the CWC is also a suitable method for the ungauged catchments of the West coast of Karnataka, India. Pradipta & Nurhady (2019) concluded the SCS method was determined as the representative SUH in Juana Watershed, Indonesia Idfi *et al.* (2020) identified SCS has a high level of accuracy for Ngotok River watershed, Indonesia. Hence, every method is not suitable for every river basin. So, it is essential to find which method is best for the selected study area.

The scarcity of the observed stage (depth) and flow hydrograph and the lack of decent quality surveyed data are the main reasons for the restriction of river hydraulics studies in India (Mehta & Yadav 2020; Mehta *et al.* 2021). The Rel river flows west and has a total basin area of about 1,455 km^{2}, was flooded in July 2017 due to an unprecedented inflow of water. The Dhanera town received rainfall of 800 mm within 48 hours, i.e., 24/7/2017 and 25/7/2017, while the average annual rainfall in the basin was 600 mm. Historical data shows that similar rainfall occurred on the 24th and 25th of July 2017, which amounted to 112 years before. This high rainfall was triggered by the Bay of Bengal's simultaneous activation and the Arabian Sea's low-pressure system (a rare phenomenon).

There are only two RGS and one Stage Discharge Station (SDS) in the Rel River Basin (RRB). So, this region falls under scarce region as far as data is concerned. The depth of the river is almost negligible. As a result, the carrying capacity decreases from the confluence of the river downstream. The nearby villages and the riverbed are almost on the same elevation. Water in this region tends to spread in the lateral direction, resulting in severe flooding (Shaikh *et al.* 2021a). In the July 2017 flood, the total payable relief for agriculture, horticulture, cattle and land damage was INR 16.53 billion from the post-flood survey results. Nearly 14,300 cattle died, while more than 200 human lives were lost during the monsoon season of 2017 (Shaikh *et al.* 2020). Five National Highways, 156 State Highways and 550 Panchayats roads were heavily affected by the 2017 flood. Therefore, it is necessary to study flood behaviour and thereby develop flood inundation maps in order to select appropriate mitigation measures. This study focuses on flood analysis in the case of data scarcity. The selection of the best SUH technique has been made using multiple criteria approaches is employed in this study in order to make appropriate recommendations.

This study evaluates SUH's various empirical methods for the RRB and finds the best method. An optimum number of RGS has been examined in order to obtain sufficient and meaningful data at the initial level of this study. Various SUH methods have been developed. Numerous efficiency criteria have been used to compare different SUH methods. Finally, the MCDM technique has been used to find the best SUH method for the RRB. The Compound factor (as an MCDM technique) has been used to find the best SUH method, which is a new concept in SUH. Hence, an evaluation among the SUH methods has been made under the current effort to identify the study area's practicability. The discharge obtained may be used for the design of hydraulic structures within the river basin.

## STUDY AREA AND DATA COLLECTION

The RRB is selected for this study (Figure 1). It is the Luni River branch, which is one of the major rivers in the Rajasthan state of India. RRB has the latitude 24°23′–24°45′N, longitude 71°59′–72°30′E and watershed area was approximately 1,455 km^{2}. In The Spatial Analyst tool has been used in the present work. Spatial Analyst is an extension for ArcGIS that includes tools for raster-based (cell-based) spatial modelling and analysis. ArcGIS Spatial Analyst software may be found at: www.esri.com. The watershed ridge line and all the drainage lines were mapped using digitisation at the scale of 1:50,000. The , , , , and Survey of India (SOI) toposheets were used for mapping. Strahler (1957) method was used for stream ordering. According to the stream order, it was found that the RRB had the sixth order (Shaikh *et al.* 2021b). In the GIS environment, each stream was digitised, and the total length of each stream was evaluated.

In the current study, dataset types such as SOI toposheet, DEM, precipitation and discharge were used. SOI toposheets were downloaded from the Soinakshe (Survey of India) portal based on the basin's location. A Shuttle Radar Topography Mission (SRTM) DEM with a resolution of 30×30 m^{2} was downloaded from the website earthexplorer.usgs.gov. The records of hourly rainfall at the Bapla RGS were collected from the State Water Data Centre (SWDC), Gandhinagar, Gujarat, India. The hourly discharge from the SDS at Dhanera Highway Bridge was collected from the River Gauging (RG) section, Palanpur, Banaskantha, Gujarat, India.

## METHODOLOGY

The current study aims to determine the optimum number of RGS, and generation of UH for an isolated storm event (22/07/2017–31/07/2017). They have been selected to generate UH for the isolated flood event in the RRB. The rainfall is recorded at Bapla RGS, and the depth-discharge is recorded at the Dhanera Highway Bridge SDS. Due to the availability of rainfall and discharge data on an hourly basis, it could be possible to select the UH duration as one hour, which is generally appropriate for all the basins. Hyetograph and the Hydrograph are then plotted.

The development of the UH and storm hydrograph has been performed for this study. Hydrograph that results from one-unit rainfall over the entire basin with uniform intensity is called UH. Geographical characteristics of the basin have been used in the generation of UH. First of all, it identified the gauged neighbouring basin, which has approximately the same geographical characteristics. Then, generate the UH for the gauged basin and use it for the ungauged basin. In the generation of storm hydrographs, rainfall data has been used to obtain direct runoff. This study addressed the empirical approaches like Snyder, TS, SCS, Mitchell and CWC methods for UH generation for the RRB. The authors have been used these empirical approaches to determine peak discharge, peak time, base period, and widths at 50 and 75% of SUH's peak discharge. Figure 2 shows the methodology used in this analysis.

### Optimum number of RGS

### SUH methods

This section is divided into two parts, namely, the UH development and the storm hydrograph development. The geographical features of the basin were used in UH generation. First of all, identify the neighbouring gauge basin, which has about the same geographical features. Then, derive the UH from the gauged basin and use it for the ungauged basin. The rainfall data is used to get direct runoff while developing storm hydrographs. The Sub-Watershed 1 (SW1) corresponds only to RGS, SW2 has neither RGS nor SDS and SW3 has only SDS. The UH of SW1 was developed, and the SUH of Sub-Watershed 2 (SW2) and Sub-Watershed 3 (SW3) were developed too (Figure 3).

### Snyder's method

^{3}/s) () are Equations (6)–(9).where, = Non-dimensional constant which varies from 1.8 to 2.2, = Main stream length (miles)(mi), = Distance from watershed outlet to the centre of watershed along the mainstream (mi).where, = Unit duration (h)where, = Area of the watershed (mi

^{2}) and = Non-dimensional constant which varies from 0.56 to 0.69.

As a result, a smooth curve can be drawn through seven points (two points of and two points of ) instead of three points (and ). The widths are generally distributed in such a way that one-third is placed before the and two-thirds placed after. Snyder's method is very monotonous, involving a higher level of subjectivity and error due to human fitting points and synchronised modifications for the area under SUH.

### Taylor and Schwarz (TS) method

^{2}and were situated in the North and Middle Atlantic States. Here, we used the same basin characteristics that were used in Snyder's method, with the average slope factor () of the main flow path being an additional factor introduced and derived by Equation (12). is also referred as vertical distance/horizontal distance. Then the simplification parameters ( and ) of this model are calculated by Equations (13) and (14). Finally, Peak time (h) (), Peak discharge rate (ft

^{3}/s) () and Base time (h) () are calculated by Equations (15)–(17).where, = Total numbers of reaches along the main flow path, = slope of

^{th}reach of the main flow path.where, = Mainstream length (mi), = Distance from watershed outlet to the centre of watershed along the mainstream (mi).where, = Unit duration (h).where, = Area of the watershed (mi

^{2}).

In the TS method, and The equation suggested by USACE (1940) was estimated for the smooth drawing of SUH, such as the Snyder method. The base time appeared to be approximately five times the actual peak time in this method (Ponce 1989). The fundamental incompatibility associated with the TS model is similar to Snyder's method (Singh *et al.* 2014).

### Soil conservation service (SCS) method

^{3}/s) () were calculated sequentially by Equations (18)–(21).where, = Length of the longest flow path (feet), Average watershed land slope (%), = Curve number (30<<100).where, Excess-rainfall unit duration (h).where, = Watershed area (mi

^{2}).

Curve number values range from 100 to 30. For water bodies, CN is nearer to 100 and approximately 30 for permeable soils with high infiltration rates (SCS 1986). Thus, a SUH can be easily derived with the help of known . A triangular shape is generally used to calculate the UH using SCS method.

### Mitchell's method

^{2}. Several approaches were offered for developing SUH under each of the three situations, wherein: (1) lag was known (2) lag was computed from time to peak (3) lag is computed from the drainage area. The 3rd condition applies when a UH for a watershed area must be derived using only the information to be created on a topographic map. Mitchell also developed the three-summation curve based on the basin's size: 175 mi

^{2}, 175 mi

^{2}< < 750 mi

^{2}and > 750 mi

^{2}. Mitchell suggested Equations (23)–(27) for the determination of Lag time (h) (), Peak discharge (ft

^{3}/s) (), Time to peak (h) () and Base time (h) ().where, = Watershed area (mi

^{2}).where, = number of intervals per daywhere, Excess-rainfall unit duration (h)

### Central water commission (CWC) method

^{3}/s) (), Width at 50% of maximum (peak) discharge ordinate (h) (), Width at 75% of peak discharge ordinate (h) (), Width of the rising side at 50% of peak discharge ordinate (h) (), Width of the rising side at 75% of peak discharge ordinate (h) (), Base time (h) (), Time to peak (h) (), and Peak discharge (m

^{3}/s) () are determined by Equations (28)–(36).where, = Watershed area (km

^{2}), = Equivalent stream slope (m/km).where, Excess-rainfall unit duration (h).

### Efficiency criteria

The Presentation and evaluation of efficiency criteria are carried out in this part of the study. These are a few efficiency criteria: Pearson Coefficient (), Correlation Coefficient (), Nash Sutcliffe Efficiency (), Index of Agreement (), Kling-Gupta Efficiency (), Root Mean Square Error (), Normalized Root Mean Square Error () and Mean Absolute Error (), which are used in the present study (Table 1).

Efficiency criteria . | Formula . | |
---|---|---|

Pearson coefficient () | Where, = observed value, = predicted value = average of observed values and = average of predicted values. | (37) |

Correlation Coefficient () | (38) | |

Nash Sutcliffe Efficiency () | (39) | |

Index of Agreement () | (40) | |

Kling-Gupta efficiency () | Where; = Pearson coefficient, = standard deviation of predicted values, = standard deviation of observed values, = mean of predicted values and = mean of observed values. | (41) |

Mean Absolute Error () | (42) | |

Root Mean Square Error () | (43) | |

Normalised Root Mean Square Error () | (44) |

Efficiency criteria . | Formula . | |
---|---|---|

Pearson coefficient () | Where, = observed value, = predicted value = average of observed values and = average of predicted values. | (37) |

Correlation Coefficient () | (38) | |

Nash Sutcliffe Efficiency () | (39) | |

Index of Agreement () | (40) | |

Kling-Gupta efficiency () | Where; = Pearson coefficient, = standard deviation of predicted values, = standard deviation of observed values, = mean of predicted values and = mean of observed values. | (41) |

Mean Absolute Error () | (42) | |

Root Mean Square Error () | (43) | |

Normalised Root Mean Square Error () | (44) |

### Compound factor () model

*et al.*(2012) for water harvesting structure positioning, and Kadam

*et al.*(2017) for plant growth potential. However, the model has been used for the first time in the field of SUH. This is the main highlight of the current study.where; R = rank of the efficiency criteria.

## RESULTS AND DISCUSSION

In the present study, the optimum number of RGS and UH derived for an isolated storm event (22/07/2017–31/07/2017) was carried out for RRB. As rainfall and discharge data were available hourly, selecting the UH duration as one hour was possible. Hourly rainfall and hourly discharge data were analysed by the single-event method to derive the UH. Results showed considerable variation in peak discharge and the peak time for the storm event.

### Optimum number of RGS

According to Indian Standard 4987:1994, a single RGS is required within 750 km^{2}, 100–250 km^{2}, and 1,500–10,000 km^{2} for plain, hilly and arid regions. The basin area of the RRB is 1,455 km^{2}, and the total available RGS are two only (Table 2 and Figure 3). So, the average area per RGS is 727.5 km^{2}. RRB lies between the flat as well as the hilly region. The average elevation is 400 m from the mean sea level. Thus, the existing two RGS are not enough for estimating the average rainfall of the basin within an allowable error of 10%. According to Equations (1)–(5), , , , and , the optimum number of RGS should be at least four. Hence, two extra RGS are required within the basin.

No. . | Station name . | Taluka . | Average annual rainfall (mm) . |
---|---|---|---|

1 | Bapla | Dhanera | 504 |

2 | Naroli | Tharad | 387 |

No. . | Station name . | Taluka . | Average annual rainfall (mm) . |
---|---|---|---|

1 | Bapla | Dhanera | 504 |

2 | Naroli | Tharad | 387 |

The isohyetal map has been plotted at an equal interval. The developed isohyetal map was also compared with the annual rainfall map of the India Meteorological Department Pune (2012). The isohyetal map was very similar to IMD. So, this isohyetal map can be used for further study. Isohyetal lines for 387 mm, 411 mm, 445, 479 and 504 mm are shown in Figure 4. The proportionate area is determined by the area of that zone divided by the basin's total area. The proportionate area then determines the required RGS for the zone into the optimum number of RGS for the entire basin. Bapla and Tharad RGS are already situated in Zones X4 and X1, respectively. Tables 3 and 4 show that new RGS are provided in X2 and X3. In both zones, one RGS will be provided for each.

Zone . | Area (km^{2})
. | Proportionate area . | RGS required . |
---|---|---|---|

X1 | 454 | 0.31 | 1.25 |

X2 | 372 | 0.26 | 1.02 |

X3 | 240 | 0.17 | 0.66 |

X4 | 388 | 0.27 | 1.07 |

Total | 1,455 | 1.00 | 4.00 |

Zone . | Area (km^{2})
. | Proportionate area . | RGS required . |
---|---|---|---|

X1 | 454 | 0.31 | 1.25 |

X2 | 372 | 0.26 | 1.02 |

X3 | 240 | 0.17 | 0.66 |

X4 | 388 | 0.27 | 1.07 |

Total | 1,455 | 1.00 | 4.00 |

Zone . | Existing RGS . | Additional RGS . | Total RGS . |
---|---|---|---|

X1 | 1 | 0 | 1 |

X2 | 0 | 1 | 1 |

X3 | 0 | 1 | 1 |

X4 | 1 | 0 | 1 |

Total | 2 | 2 | 4 |

Zone . | Existing RGS . | Additional RGS . | Total RGS . |
---|---|---|---|

X1 | 1 | 0 | 1 |

X2 | 0 | 1 | 1 |

X3 | 0 | 1 | 1 |

X4 | 1 | 0 | 1 |

Total | 2 | 2 | 4 |

Thi Hong *et al.* (2016) had established an optimum RGS network on the Vu Gia-Thu Bon river basin to develop an up-to-date flood warning system in real-time in Vietnam. According to the investigation, the study area had nine RGS, which showed that the existing network's error was approximately 7.47%. If the error decreases from 7.47% to 5%, the additional RGS should be twenty. Similarly, Patel (2018) had also analysed the optimum number of RGS in the Sabarmati River Basin, India. According to the analysis, the study's optimal number of RGS was thirty-seven, while the existing number of RGS was thirty-one. So, six additional RGS are required.

### SUH methods

These methods were used to calculate the lag time, peak discharge, base time and the widths of peak flow with the help of basin characteristics, which were acquired from the topographic map of the river basin (Ramirez 2000). The parameters were obtained in a GIS environment. The UH of SW1 and the SUH of SW2 and SW3 was developed (Figure 3). SW1 has RGS only. SW2 has neither RGS nor SDS, and SW3 has SDS only (Table 5). Table 6 and Figure 5 show the UH property for each SW and method.

Parameters . | SW1 . | SW2 . | SW3 . |
---|---|---|---|

RGS availability | Yes | No | No |

SDS availability | No | No | Yes |

(km) | 37.70 | 19.47 | 18.11 |

(km) | 27.66 | 13.98 | 13.55 |

Average slope of the main channel (S) | 0.0035 | 0.00344 | 0.00226 |

(km^{2}) | 174.45 | 138.61 | 94.09 |

Parameters . | SW1 . | SW2 . | SW3 . |
---|---|---|---|

RGS availability | Yes | No | No |

SDS availability | No | No | Yes |

(km) | 37.70 | 19.47 | 18.11 |

(km) | 27.66 | 13.98 | 13.55 |

Average slope of the main channel (S) | 0.0035 | 0.00344 | 0.00226 |

(km^{2}) | 174.45 | 138.61 | 94.09 |

Method . | Component . | SW1 . | SW2 . | SW3 . |
---|---|---|---|---|

Snyder | (hr) | 13.88 | 10.27 | 10.05 |

(hr) | 104.65 | 93.82 | 93.15 | |

(m^{3}/s) | 77.39 | 92.01 | 64.43 | |

(hr) | 14.11 | 9.13 | 8.83 | |

(hr) | 8.06 | 5.22 | 5.05 | |

TS | (hr) | 11.75 | 12.98 | 16.14 |

(hr) | 64.07 | 70.80 | 88.05 | |

(m^{3}/s) | 74.46 | 87.52 | 59.98 | |

(hr) | 14.71 | 9.64 | 9.54 | |

(hr) | 8.41 | 5.51 | 5.45 | |

SCS | (hr) | 17.18 | 11.48 | 12.80 |

(hr) | 45.86 | 30.64 | 34.18 | |

(m^{3}/s) | 53.74 | 63.91 | 38.89 | |

(hr) | 20.92 | 13.54 | 15.23 | |

(hr) | 11.96 | 7.73 | 8.70 | |

Mitchell | (hr) | 16.13 | 14.44 | 12.07 |

(hr) | 43.06 | 38.55 | 32.21 | |

(m^{3}/s) | 102.57 | 81.50 | 55.32 | |

(hr) | 10.41 | 10.41 | 10.41 | |

(hr) | 5.95 | 5.95 | 5.95 | |

CWC | (hr) | 4.15 | 3.79 | 2.85 |

(hr) | 13.90 | 13.07 | 10.62 | |

(m^{3}/s) | 118.95 | 103.48 | 95.10 | |

(hr) | 3.78 | 3.47 | 2.63 | |

(hr) | 2.29 | 2.12 | 1.66 |

Method . | Component . | SW1 . | SW2 . | SW3 . |
---|---|---|---|---|

Snyder | (hr) | 13.88 | 10.27 | 10.05 |

(hr) | 104.65 | 93.82 | 93.15 | |

(m^{3}/s) | 77.39 | 92.01 | 64.43 | |

(hr) | 14.11 | 9.13 | 8.83 | |

(hr) | 8.06 | 5.22 | 5.05 | |

TS | (hr) | 11.75 | 12.98 | 16.14 |

(hr) | 64.07 | 70.80 | 88.05 | |

(m^{3}/s) | 74.46 | 87.52 | 59.98 | |

(hr) | 14.71 | 9.64 | 9.54 | |

(hr) | 8.41 | 5.51 | 5.45 | |

SCS | (hr) | 17.18 | 11.48 | 12.80 |

(hr) | 45.86 | 30.64 | 34.18 | |

(m^{3}/s) | 53.74 | 63.91 | 38.89 | |

(hr) | 20.92 | 13.54 | 15.23 | |

(hr) | 11.96 | 7.73 | 8.70 | |

Mitchell | (hr) | 16.13 | 14.44 | 12.07 |

(hr) | 43.06 | 38.55 | 32.21 | |

(m^{3}/s) | 102.57 | 81.50 | 55.32 | |

(hr) | 10.41 | 10.41 | 10.41 | |

(hr) | 5.95 | 5.95 | 5.95 | |

CWC | (hr) | 4.15 | 3.79 | 2.85 |

(hr) | 13.90 | 13.07 | 10.62 | |

(m^{3}/s) | 118.95 | 103.48 | 95.10 | |

(hr) | 3.78 | 3.47 | 2.63 | |

(hr) | 2.29 | 2.12 | 1.66 |

The ordinates of UH were used to generate the direct runoff hydrographs (DRH) due to total precipitation over the basin. DRH is derived by multiplying the UH ordinates by incremental precipitation excess, adding and lagging in an arrangement to generate a storm hydrograph. This is also called hydrograph convolution (Salami *et al.* 2009). The incremental precipitation was determined by consecutively deducting the precipitation excess from the earlier time events. The combined storm hydrograph of SW1, SW2 and SW3 are shown in Figure 6. The rainfall of Bapla RGS is used for SW1. Bapla village is located under SW1, which has RGS. However, for SW2 and SW3, the same rainfall data was used because SW2 and SW3 did not have RGS. Nearly all the SW have similar characteristics. These three hydrographs were generated for the upstream part of the entire basin. At some distance from downstream, the flows of two tributaries of SW1 and SW2 meet at a junction, namely Mevada village. Mevada was the most affected area of the 2017 flood because of this confluence. Now, the total flow from Mevada to Dhanera is the summation of these three hydrographs.

### Snyder's method

The (h), (h) and (m^{3}/s) were calculated for all the sub-watersheds using Snyder's formula (Equations (6)–(9)). In this method, was calculated from . The was introduced by Horner (1936). The was assumed constant for a particular watershed. The main affecting factors for are the area and the shape of the basin. The takes care of the slopes and storage variations and has not changed significantly in the basin. After a lot of trial-and-error processes, the values of and were fixed for all the sub-watersheds as 1.8 and 0.69, respectively. Generally, and have an inverse relation; that is, if was maximum then would become minimum or vice versa. A similar relationship was observed in this study too. The primary purpose of fixing the values of and was to get simulated discharge closer to the observed discharge. For SW1, the values of , and were 13.88 h, 104.65 h and 77.39 m^{3}/s. For SW2, the values of , and were 10.27 h, 93.82 h and 92.01 m^{3}/s. For SW3, the values of , and were 10.05 h, 93.15 h and 64.43 m^{3}/s. Established UH and SUH ordinates were used to develop the storm hydrograph with actual precipitation over the basin. Prajapati & Subbaiah (2005) had also compared the different SUH models and identified Snyder as the best suitable method due to having the highest efficiency (88.03%) for the Rawat Sagar Basin.

### Taylor and Schwarz (TS) method

No big streams or other unusual features in the basin are shown in the existing toposheet. Therefore, the specified method can be applied without modification. The values of , and were taken from the SOI toposheet for all sub-watersheds (Table 5). The average slope () of the main flow path was determined from Equation (12). The values of were 0.0035, 0.00344 and 0.00226 for SW1, SW2 and SW3 respectively. The (h) and (m^{3}/s) were calculated for all the sub-watersheds using the TS formula (Equations (12)–(16)). First of all, was obtained in ft^{3}/s. Then, it was converted into m^{3}/s for comparison with other methods. For SW1, the values of and were 11.75 h and 74.46 m^{3}/s. For SW2, the values of and were 12.98 h and 87.52 m^{3}/s. For SW3, the values of and were 16.14 hand 59.98 m^{3}/s. The UH was completed by , , , and . The values of were 64.07, 70.80 and 88.05 for SW1, SW2 and SW3, respectively. Velez & Botero (2011) had estimated the time of concentration and the lag time using different empirical methods at San Luis Creek Basin, Manizales, and concluded that the TS method also gave similar results to Snyder's method.

### Soil conservation service (SCS) method

In this method, first, (in percentage) and were determined. The values of were 0.35, 0.34 and 0.23 for SW1, SW2 and SW3, respectively. After much trial-and-error process, the values of were fixed for all the sub-watersheds. The (h), (h) and (m^{3}/s) were calculated for all the sub-watersheds using the TS formula (Equations (18)–(22)). For SW1, the values of , and were 17.18 h, 45.86 h and 53.74 m^{3}/s. For SW2, the values of , and were 11.48 h, 30.64 h and 63.91 m^{3}/s. For SW3, the values of , and were 12.80 h, 34.18 h and 38.89 m^{3}/s. Salami *et al.* (2009) had also evaluated different SUH methods for the development of storm hydrographs for eight rivers in south-west Nigeria. They summarised that the SCS method was best due to the lowest percentage difference for the study area.

### Mitchell's method

Equations (23) and (24) are used generally for the calculation of . But in this study, was not known so Equation (24) was used for the calculation of . The values of were 13.13, 11.44 and 9.07 for SW1, SW2 and SW3, respectively. The (h), (h) and (m^{3}/s) were calculated for all the sub-watersheds using the TS formula (Equations (23)–(27)). For SW1, the values of , and were 16.13 h, 43.06 h and 102.57 m^{3}/s. For SW2, the values of , and were 14.44 h, 38.55 h and 81.50 m^{3}/s. For SW3, the values of , and were 12.07 h, 32.21 h and 55.32 m^{3}/s. Patel & Thorvat (2016) had worked on the different SUH methods at the Upper Kumbhi Basin and the Dhamani Basin, Maharashtra, India. They found that Mitchell's method was provided with a percentage error of 11.361% (still within the range of error) for the study area.

### Central water commission (CWC) method

The (h), (h) and (m^{3}/s) were calculated for all the sub-watersheds using the CWC formula (Equations (28)–(36)). The results indicate that the UH parameter values are determined with the CWC method. The relation of these parameters with basin characteristics is presented in Tables 5 and 6. For SW1, the values of , and were 4.15 h, 13.90 h and 118.95 m^{3}/s. For SW2, the values of , and were 3.79 h, 13.07 and 103.48 m^{3}/s. For SW3, the values of , and were 2.85 h, 10.62 h and 95.10 m^{3}/s. Reddy *et al.* (2019) had worked on the different SUH methods for flood estimation at the ungauged catchments of the west coast of Karnataka, India. They identified that the CWC is also a suitable method for the study area.

### Efficiency criteria

In this part, all the methods were evaluated through eight efficiency criteria, and bold text shows the best result for the efficiency criteria (Table 7).

Efficiency criteria . | Methods . | ||||
---|---|---|---|---|---|

Snyder . | TS . | SCS . | Mitchell . | CWC . | |

r | 0.69 | 0.74 | 0.76 | 0.77 | 0.89 |

r ^{2} | 0.47 | 0.54 | 0.58 | 0.59 | 0.79 |

E | −2.39 | −1.32 | 0.38 | −0.15 | 0.59 |

d | 0.60 | 0.69 | 0.85 | 0.80 | 0.92 |

KGE | −0.06 | 0.03 | 0.55 | 0.24 | 0.52 |

MAE | 870.75 | 656.43 | 323.16 | 438.15 | 243.33 |

RMSE | 1,268.79 | 1,050.20 | 543.40 | 737.80 | 427.11 |

NRMSE | 3.62 | 3.00 | 1.55 | 2.11 | 1.14 |

Efficiency criteria . | Methods . | ||||
---|---|---|---|---|---|

Snyder . | TS . | SCS . | Mitchell . | CWC . | |

r | 0.69 | 0.74 | 0.76 | 0.77 | 0.89 |

r ^{2} | 0.47 | 0.54 | 0.58 | 0.59 | 0.79 |

E | −2.39 | −1.32 | 0.38 | −0.15 | 0.59 |

d | 0.60 | 0.69 | 0.85 | 0.80 | 0.92 |

KGE | −0.06 | 0.03 | 0.55 | 0.24 | 0.52 |

MAE | 870.75 | 656.43 | 323.16 | 438.15 | 243.33 |

RMSE | 1,268.79 | 1,050.20 | 543.40 | 737.80 | 427.11 |

NRMSE | 3.62 | 3.00 | 1.55 | 2.11 | 1.14 |

Bold values indicate best result for the efficiency criteria.

### Pearson coefficient ()

This is the statistical test that quantifies the statistical relationship between two continuous variables. It is based on covariance. It gives the magnitude as well as the direction of the relationship (Equation (37)). It ranges between +1 and −1, where +1 is a total positive correlation, 0 is no correlation, and −1 is a total negative correlation. Pearson coefficients for Snyder, TS, SCS, Mitchell and CWC methods were 0.69, 0.74, 0.76, 0.77 and 0.89, respectively. The best Pearson coefficient was achieved for the CWC method because it is closer to 1, and the worst Pearson coefficient was achieved for Snyder's method because it is nearer to 0.

### Correlation coefficient ()

According to Bravais Pearson, it is the squared value of the Pearson coefficient. It can also be expressed as the squared ratio of covariance to multiplying with the standard deviation (Equation (38)). The range lies between 0 and 1. A zero value indicates no correlation at all, and a value of one means perfect correlation. One of the major disadvantages is that it quantifies only dispersion. A model which has systematic dispersion all the time will give a good r^{2} value (≈1). Correlation coefficients for Snyder, TS, SCS, Mitchell and CWC methods were 0.47, 0.54, 0.58, 0.59 and 0.79 respectively. The best correlation coefficient was achieved for the CWC method because it is closer to 1, and the worst correlation coefficient was achieved for Snyder's method because it is nearer to 0.

### Nash-Sutcliffe efficiency ()

This was proposed by Nash & Sutcliffe (1970). It is defined as one minus the ratio of the sum of the squared difference between observed and predicted values to the variance observed values (Equation (39)). The range lies between 1 and −∞. A value of one indicates a perfect fit for observed and predicted values. The biggest drawback is that the differences between observed and predicted values are determined as squared. So similar to r^{2}, E is not very sensitive to systematic dispersion. Nash-Sutcliffe efficiency for Snyder, TS, SCS, Mitchell and CWC methods was −2.39, −1.32, 0.38, −0.15 and 0.59. The best Nash-Sutcliffe efficiency was achieved for the CWC method because it is nearer to 1, and the worst Nash-Sutcliffe efficiency was achieved for Snyder's method because of the maximum negative value.

### Index of agreement ()

It was proposed by Willmott (1981) to overcome the insensitivity of r^{2} and E. It is the ratio of mean square error to the potential error (Equation (40)). The denominator represents the largest value that the squared difference of each pair can attain. The numerator is very sensitive to high values and insensitive for low values. The range of d is 0 (no correlation) to 1 (perfect fit). The Index of agreement for Snyder, TS, SCS, Mitchell and CWC methods was 0.60, 0.69, 0.85, 0.80 and 0.92. The best Index of agreement was achieved for the CWC method because it is nearer to 1, and the worst Index of agreement was achieved for Snyder's method because it is nearer to 0.

### Kling-Gupta efficiency ()

This was proposed by Gupta *et al.* (2009). It provides a diagnostically interesting decomposition of the E (and hence MSE), which helps to study the relative importance of its different components like correlation, bias, and variability (Equation (41)). The range lies between 1 and −∞. A value of one indicates a perfect fit for the observed and predicted values. KGE for Snyder, TS, SCS, Mitchell and CWC methods was −0.06, 0.03, 0.55, 0.24 and 0.52. The best KGE was achieved for the SCS method because it is nearer to 1, and the worst KGE was achieved for Snyder's method because of the negative value.

### Mean absolute error ()

This is the most straightforward measure of prediction correctness. MAE is straightforward, as the name recommends, the mean of the absolute errors. The absolute error is the absolute difference between the predicted and the observed value (Equation (42)). MAE expresses how big an average error can be expected from the prediction without considering their direction. The MAE can range from 0 to ∞. A value of zero indicates a perfect fit for the observed and predicted values. MAE for Snyder, TS, SCS, Mitchell and CWC methods were 870.75, 656.43, 323.16, 438.15 and 243.33. Compared with each other, the best MAE was achieved for the CWC method because it is nearer to 0, and the worst MAE was achieved for Snyder's method because of the maximum value.

### Root mean square error ()

The RMSE is a frequently used specific metric between the predicted values and the observed values. These individual differences are also called residuals, and they are aggregated in a single predictive power measure by the RMSE (Equation (43)). RMSE measures the error level between two sets of data. The RMSE can range from 0 to ∞. A value of zero indicates a perfect fit for the observed and predicted values. RMSE for Snyder, TS, SCS, Mitchell and CWC methods were 1,268.79, 1,050.20, 543.40, 737.80 and 427.11, respectively. Compared with each other, the best RMSE was achieved for the CWC method because it is nearer to 0, and the worst RMSE was achieved for Snyder's method because of the maximum value.

### Normalized root mean square error ()

The NRMSE enables the comparison of models with distinct scales. Therefore, the NRMSE can be interpreted as a fraction of the overall range, which the model typically resolves (Equation (44)). The NRMSE can range from 0 to ∞. A value of zero indicates a perfect fit for the observed and predicted values. The NRMSE for Snyder, TS, SCS, Mitchell and CWC methods was 3.62, 3.00, 1.55, 2.11 and 1.14. The best NRMSE was achieved for the CWC method because it is nearer to 0, and the worst NRMSE was achieved for Snyder's method due to the maximum value.

### Compound factor () model

The was used to assess all methods’ priority ranks, determined by Equation (45). From this group of methods, the first rank was assigned to the method having the lowest and so on. Further, the ranked these methods into 1–5. The for Snyder, TS, SCS, Mitchell and CWC methods were 5, 4, 2.13, 2.75 and 1.13. So, the rankings for Snyder, TS, SCS, Mitchell and CWC methods were 5, 4, 2, 3 and 1, respectively. According to the ranking, the method of CWC is the best suitable method for predicting runoff for the RRB because of the minimum value (Table 8). The hydrograph shape of CWC is similar to the observed hydrograph. The peak time difference between observed and CWC is also minimum (Figure 5). So, due to these all factors, the CWC method is best for RRB for SUH generation.

Efficiency criteria . | Methods . | ||||
---|---|---|---|---|---|

Snyder . | TS . | SCS . | Mitchell . | CWC . | |

r | 5 | 4 | 3 | 2 | 1 |

r ^{2} | 5 | 4 | 3 | 2 | 1 |

E | 5 | 4 | 2 | 3 | 1 |

d | 5 | 4 | 2 | 3 | 1 |

KGE | 5 | 4 | 1 | 3 | 2 |

MAE | 5 | 4 | 2 | 3 | 1 |

RMSE | 5 | 4 | 2 | 3 | 1 |

NRMSE | 5 | 4 | 2 | 3 | 1 |

CF | 5 | 4 | 2.13 | 2.75 | 1.13 |

Ranking | 5 | 4 | 2 | 3 | 1 |

Efficiency criteria . | Methods . | ||||
---|---|---|---|---|---|

Snyder . | TS . | SCS . | Mitchell . | CWC . | |

r | 5 | 4 | 3 | 2 | 1 |

r ^{2} | 5 | 4 | 3 | 2 | 1 |

E | 5 | 4 | 2 | 3 | 1 |

d | 5 | 4 | 2 | 3 | 1 |

KGE | 5 | 4 | 1 | 3 | 2 |

MAE | 5 | 4 | 2 | 3 | 1 |

RMSE | 5 | 4 | 2 | 3 | 1 |

NRMSE | 5 | 4 | 2 | 3 | 1 |

CF | 5 | 4 | 2.13 | 2.75 | 1.13 |

Ranking | 5 | 4 | 2 | 3 | 1 |

## CONCLUSIONS

In the present study, the optimum number of RGS was determined. The empirical methods were used to establish the relationship between several variables that synthesise unit hydrograph for the RRB. Two RGS already existed, which was not sufficient. Indian Standard 4987:1994 suggests that two more RGS are required for carrying out meaningful rainfall analysis. This study also illustrates the -based ranking of methods. The is minimum (1.13) and maximum (5.0) for the CWC and Snyder methods. Minimum indicates the highest ranking. The simulated hydrograph shape is similar to the observed hydrograph shape, the peak time difference between observed and simulated is minimum and the CF value is also minimum for CWC method. This study recommended using the CWC method, which is found to be the most appropriate SUH method for the RRB. The computed flood discharges could be used as the primary input in the hydraulic model, which is seldom available along the semi-arid river basin. This study will help in the planning of flood mitigation and water conservation measures for data-scarce regions. The lack of flood data is a basic limitation for hydrological studies and hydrologic modelling in this study area. In fact, many past floods have not been recorded by hydrometric stations. So, this analysis is only subjected to the 2017 flood event. The present study can be further refined by considering more catchments of varying sizes and geomorphologic conditions.

## DATA AVAILABILITY STATEMENT

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

## REFERENCES

*2021*

*June*