Development of new event-based rainfall-runoff equations as an alternative to the curve number method in oceanic and semiarid climates

doi:10.21203/rs.3.rs-1879978/v1

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Development of new event-based rainfall-runoff equations as an alternative to the curve number method in oceanic and semiarid climates

https://doi.org/10.21203/rs.3.rs-1879978/v1

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Event-based rainfall-runoff models are practical tools commonly used to predict catchments' response to a rainfall event. However, one of the main concerns is that the characteristics of rain events are neglected in model development. This paper developed a novel event-based rainfall-runoff equation to incorporate rainfall characteristics into account. The performance of new equation is evaluated based on the root mean square error, Nash-Sutcliffe efficiency coefficient, and percent bias for 13,339 rainfall-runoff events between 2005 to 2020 over 23 catchments across New Zealand and Australia with oceanic and semiarid climates. Compared to the previous event-based models, the new equation shows an improvement in runoff estimation in almost all case studies. Considering the new equation is simple, efficient, and takes the rain event duration into account, the new equation has the potential to become a robust alternative method to the conventional curve number in hydrological engineering projects.

rainfall-runoff

rain duration

event-based

runoff coefficient

event average

rainfall rate

IDF

average rainfall rate

The hydrological response of a catchment to a rainfall event is commonly known as a rainfall-runoff process (Critchley et al. 1991). A rainfall-runoff process often consists of complex physical procedures, i.e. interception, depression storage, evapotranspiration, infiltration, subsurface flow, groundwater flow, overland flow, and channel flow (Kisi et al. 2013). Previous studies indicate that rainfall characteristics such as rainfall rate, duration, and distribution, besides the catchment characteristics such as size, soil type, land use, and slope, can impact the occurrence and volume of runoff (Hewlett et al. 1977; Hewlett and Bosch 1984; Howard et al. 2010). In addition, considering the physical condition of catchments is not homogenous, each catchment may respond differently to a rainfall event (Critchley et al. 1991).

Several physically-based mathematical expressions and computer simulation models are developed to explain the entire or part of the process. Sometimes, infiltration equations are implemented to predict runoff from a rainfall event. The most common Infiltration equations cited in the literature for rainfall-runoff modelling are Green-Ampt (Green and Ampt 1911) and Philip (2006), adopted from the Richards equation; Horton (1939), Holtan (Holtan et al. 1961), and Kostiakov (Ghosh 1985) methods developed from field data analysis as empirical equations. In addition, sometimes distributed surface water models are combined with distributed groundwater models to provide a better estimation (Maxwell and Miller 2005; Kollet and Maxwell 2006; Kollet, S. J. 2008; Kumar et al. 2009; Camporese et al. 2010; Shen and Phanikumar 2010; Brunner and Simmons 2012; Shokri and Bardsley 2016). However, these models are highly complex, data-hungry, and often require several spatial and temporal parameters, which in most cases are not readily available (Sivapalan 2005).

A simple alternative to physically based and continuous models is to estimate runoff from an individual rain event, often known as event-based models (Stephens et al., 2018). For example, the runoff coefficient has been used to estimate runoff from a rainfall event for many decades, which assumes that the depth of direct runoff is a percentage of the rainfall depth:

$$C=\frac{Q}{P}$$

where C is the runoff coefficient and Q is the total runoff depth in each rain event (Burch et al. 1987; Savenije 1996; McNamara et al. 1998; Iroumé et al. 2005), or sometimes Q is the total direct runoff depth for each event (Hewlett and Hibbert 1967; Woodruff and Hewlett 1970; Schellekens et al. 2004; van Dijk et al. 2005), where "depth" refers to the total volume of runoff or direct runoff for each event over the catchment area. The terminology of "runoff coefficient" can be confusing as it is not consistent throughout the literature (Blume et al. 2007). For example, the parameter is called response factor (Hewlett and Hibbert 1967), hydrologic response (Woodruff and Hewlett 1970), runoff ratio (McNamara et al. 1998), water yield (Sidle et al. 2000), conversion efficiency (Burch et al. 1987), direct runoff response ratio (Howard et al. 2010), and runoff coefficient (van Dijk et al. 2005; Savenije 1996). The other definition of runoff coefficients also can come from the rational method, which claims the peak flow is proportional to rainfall intensity for a given catchment (Chin 2000). For clarification, In the study, the runoff coefficient is defined as the ratio of total direct runoff depth over total precipitation for each rain event.

Even though the concept of estimating runoff for each event from runoff is attractive, the runoff coefficient is highly variable as the runoff coefficient is proportional to catchment and rainfall characteristics (Hewlett et al. 1977; Howard et al. 2010).

The most popular event-based model by far is the Soil Conservation Service (now the Natural Resources Conservation Service) Curve Number (SCS-CN) expression, which often is famous for its simplicity and applicability (Ponce and Hawkins 1996; Soulis et al. 2009; Beven 2012). This method is widely used in practice and applied in many hydrologic applications. Some examples are flood prediction, water quality modelling, soil moisture balance, and sediment yields (Steenhuis et al. 2002; Woodward et al. 2003; Garen and Moore 2005; Mishra et al. 2006; Singh et al. 2008; Van Dijk 2010; Abon et al. 2011; Soulis and Valiantzas 2013; Hawkins 2014; Soulis et al. 2017; Soulis 2018). Also, several lumped and semi-distributed models are designed based on an adapted form of the SCS-CN equation, among which are CREAMS (Kinsei W.G 1980), GLEAMS (Knisel et al. 2000), AGNPS (Young et al. 1989), EPIC (Williams and Sharply 1989; Sharpley and Williams 1990; Williams 1995), WinTR-55 (Karl Visser and Claudia Scheer 2013), HEC-HMS (Feldman 2000), EPA-SWMM (Rossman 2015), TOPNET (Clark et al. 2008) and SWAT (Neitsch et al. 2009).

The SCS-CN method originally comes from a lumped-based approach that calculates total direct runoff from a storm event (Hawkins, R., Ward, T., Woodward, D., and Van Mullem 2008). For deriving the SCS-CN expression, the proportionality between retention and runoff is assumed to be

$$\frac{F}{S}=\frac{Q}{P}$$

where F is the actual retention, S is the maximum potential retention, Q is the total direct runoff during a rainfall event, and P is s the cumulative rainfall

Assuming:

$$F=P-Q-{I}_{a}$$

and solving Eq. 2 for Q yields

$Q=\frac{{\left(P-{I}_{a}\right)}^{2}}{P+S-{I}_{a}}$ when $P\ge Ia,$ otherwise $Q=0$(4)

where I_a is the initial abstraction, which is the minimum amount of rainfall required before overland flow begins. However, the physics behind this assumption is not clear. Regardless of physical meaning, usually, a linear correlation is assumed between S and I_a

$${I}_{a}=\lambda S$$

where λ is the initial abstraction ratio. In the original formula, λ is considered 0.2 (Woodward et al. 2003; Shi et al. 2009; Yuan et al. 2014). However, the background of this assumption is known (Hawkins, R., Ward, T., Woodward, D., and Van Mullem 2008).

In the SI units, S is defined by a Curve Number (CN) parameter through:

$$S=\frac{25.4}{CN}-0.254$$

where CN value is defined from hydrological soil group, land use, hydrological surface condition, and soil moisture condition (Hawkins 2014). Theoretically, the CN may vary between 0 and 100.

The SCS-CN method is simple, transparent, appealing, and only requires one empirical parameter, "CN" from a hydrologic engineering perspective. The SCS is popular because SCS-CN databases can easily link to distributed soil and vegetation layers stored within a GIS (Beven 2012). Hawkins (2014) reported that there is no alternative with the benefits of the curve number method available, and within its group, the SCS-CN method is monotypic. However, besides many advantages, too many ambiguities raise questions about this method's accuracy (Ponce and Hawkins 1996; Garen and Moore 2005; Beven 2012; Hawkins 2014). For example, Eq. 2, as the central assumption of the SCS-CN method, has no physical justification (Beven 2012).

Another concern is that the rainfall characteristics, such as rain duration, are neglected in the equation (Mishra S.K., Singh V.P. 2018). For example, the SCS-CN method does not count for the differences between 40 cm rainfall in 1 day and 10 days, while infiltration and runoff would be considerably different (Chin 2000; Hawkins, R., Ward, T., Woodward, D., and Van Mullem 2008). However, several studies statistically indicated that the runoff is proportional to the rainfall duration (Hewlett et al. 1977; Hewlett and Bosch 1984; Howard et al. 2010).

This study aims to introduce a novel, robust, and simple equation for estimating runoff from a rain event by taking the rain event duration into consideration. Similar to the SCS-CN method, the new equation only has one empirical parameter. However, it more precisely represents the physics of the rainfall-runoff process by incorporating rainfall characteristics.

2.1 Hydro-Meteorological Data

For this study, 23 catchments ranging from 0.3 Km² to 564 Km² are selected from Australia and New Zealand catchments. The sites in New Zealand have a subtropical climate, with warm-humid summers and mild winters, whereas the case studies in Australia have a range of climates from subtropical to semiarid areas. Commonly, No snow is recorded in any sites during the study periods.

Rain events are defined based on the minimum-interevent-time method (Dunkerley 2008), in which the beginning and end of each event are defined by a minimum rainless period of at least 24 hours. The available data set is divided into the calibration and validation periods. In total, 13,339 events are observed across 23 study areas. In all cases, rainfalls are recorded either at the outlet or in the vicinity of the outlet. A summary of catchment size, calibration and validation periods and the number of events are listed in Table 1.

Table 1

A summary of catchment size, study period and the number of events for each catchment in New Zealand (NZ) and Australia (AUS)
Site No.	Site name	Location	Area (km2)	Calibration		Validation
Site No.	Site name	Location	Area (km2)	Period (years)	No. of events	Period (years)	No. of events
1	West Hoe Heights	Auckland, West Hoe Heights, NZ	0.3	2008–2015	349	2015–2017	119
2	Mangemangeroa	Auckland, East Tamaki Heights, NZ	4.4	2008–2015	357	2015–2017	121
3	Pukekohe Ngakoroa	Auckland, Pukekohe, NZ	4.5	2009–2015	283	2015–2019	236
4	Arkins Creek	Victoria, Wyelangta, AUS	4.5	2005–2015	475	2015–2019	186
5	Wainui Te Whara	Bay of Plenty, Whakatane, NZ	5.4	2008–2015	331	2015–2020	267
6	Waingaehe	Bay of Plenty, Rotorua, NZ	9.5	2008–2015	329	2015–2019	243
7	Orewa	Auckland, Millwater, NZ	9.6	2007–2015	439	2015–2020	263
8	Maungaparerua	Northland, Kerikeri, AUS	11.7	2008–2015	357	2015–2018	184
9	Whangarei Ngunguru	Northland, Whangarei, AUS	12.5	2009–2015	313	2015–2018	193
10	Rutherford Creek	New South Wales, Brown Mountain, AUS	14.4	2005–2015	515	2015–2019	235
11	Opanuku	Auckland, Henderson, NZ	15.7	2007–2015	364	2015–2017	72
12	Johnson River	Queensland, Central Mill, AUS	16.2	2005–2015	455	2015–2019	187
13	Fisher Creek	Queensland, Innisfail, AUS	16.2	2005–2015	367	2015–2019	146
14	Raumanga	Northland, Whangarei, AUS	16.3	2008–2015	359	2015–2017	120
15	Scott Creek	South Australia, Southern Mount, AUS	29.0	2005–2015	385	2015–2018	170
16	Gellibrand	Victoria, Melborne, AUS	52.7	2005–2015	455	2015–2019	186
17	Elizabeth Valley	Northern Territory, Darwin, AUS	95.6	2005–2015	315	2015–2019	138
18	Tinana Creek	Queensland, north of the Sunshine coast, AUS	101.8	2005–2015	367	2015–2019	148
19	Mannus Creek	New South Wales, Tumbarumba, AUS	194.3	2005–2015	420	2015–2019	186
20	Bass River	Victoria, Melborne, AUS	240.2	2005–2015	501	2015–2019	205
21	Ashley Gorge	Christchurch, Canterbury, NZ	472.0	2005–2015	503	2015–2020	293
22	Wakefield	South Australia, Adelaide, AUS	500.9	2005–2015	360	2015–2019	146
23	Quenbeyan	New South Wales, Canberra, AUS	563.7	2005–2015	506	2015–2019	190
	Total				9,105		4,234

2.2 Baseflow separation

Baseflow and direct runoff (quick-flow) are separated using a "one-parameter recession digital filter" in BFI + software (Lyne and Hollick 1979; Chapman and Maxwell 1996). As an example, the base flow separation for the Opanuku river is shown in Fig. 1. Then, the direct runoff depth for each event is calculated by dividing the total direct runoff for each event over the catchment area.

3.1 Development of the new equation

The event-average rainfall rate is calculated for each event:

$${i}_{av}=\frac{P}{t}$$

where i_av is the average rainfall rate in each event, P is the total rainfall depth (mm), and t is the event duration (hr). Existing a correlation between runoff coefficient (Q/P) and average rainfall rate (Critchley et al. 1991) has been suggested in the previous studies. As two examples, the relationships between runoff coefficient and average rainfall rate in Opanuku and Pukekohe Ngakoroa rivers are investigated and illustrated in Fig. 2.

Considering the duration varies in each event, the proportionality between runoff and average rainfall rates (I_av) is not substantial.

Intensity-duration-frequency (IDF) curves are implemented to enhance the correlation by unifying rain events durations. The IDF curves concept was established in early 1930, and many mathematical expressions for IDF curves were suggested subsequently (Chin 2000).

IDF curve can be expressed in a general form

$${i}_{av}=\frac{f\left(T\right) }{g\left(t\right)}$$

where t is the rain storm duration, T is the storm event return period, f (T) is a function of the return period, and g(t) is a function of the storm duration t. f (T) and g(t) are unique functions to each site and can be estimated directly from previous rainfall records.

For this study, following a method suggested by (Chin 2000; Sun et al. 2019), the annual maximum rainfall for each site is rank-ordered, and the return period for each annual maximum rainfall value is estimated. Then, the rank-ordered rainfall value is divided by the event duration for each return period, resulting in the average intensity plotted versus the duration. As an example, the IDF curves of Pukekohe for return periods ranging from 0.01 to 100-year in Fig. 3.

For storms with a return period of T, Eq. 8 gives the following relation between the equivalent rainfall rate in 24 h and the rainfall rate in each rainstorm with a duration of t (Chin 2000):

$$f\left(T\right)={i}_{av}g\left(t\right)={i}_{24}g\left(24\right)$$

where i₂₄ is the equivalent rainfall rate in 24 (hr).

Rearranging for i₂₄ yields:

$${i}_{24}=\frac{g\left(t\right)}{g\left(24\right)}{i}_{av}$$

Assuming a power function form for g(t) after (Koutsoyiannis et al. 1998)

$$g\left(t\right)={at}^{n}$$

where a and n are local constant variables obtained from the past rainfall records for each site.

Therefore

$$g\left(24\right)={a\left(24\right)}^{n}$$

Substituting g(t) and g(24) in Eq. 10 yields:

$${i}_{24}={i}_{av}({\frac{t}{24})}^{n}$$

Then, Eq. 13 is used to calculate i₂₄ for all events. With the new adjustment on rainfall rate, the runoff coefficient shows a stronger correlation to i₂₄ than i_av in all case studies.

For all sites, the n value is calculated using the IDF curves. For example, from the rainfall record of the Opanuku and Pukekohe Ngakoroa sites, the IDF curves are generated, and then the n value is calculated as 0.37 and 0.56, respectively. In the next step, i₂₄ is calculated, and the relationships between the runoff coefficient and i₂₄ are plotted in Fig. 4.

Comparing Figs. 2 and 4 shows that using i₂₄ instead of i_av strongly enhances the coefficient of determination (R²), around 47% (from R² = 0.38 to 0.56) in Opanuku and 104% (from R² = 0.27 to 0.55) in Pukekohe Ngakoroa sites. The same pattern is observed in the other case studies. A scatterplot and boxplot in Fig. 5 compare the correlation (R²) of Q/P vs i_av and Q/P vs i₂₄ for all case studies.

Considering all points in the scatterplot located above the 1:1 line indicates that using the IDF curves concept to unify the event durations effectively improved the runoff coefficient and rainfall rate significantly. A comparison between medians in the boxplot shows that the median of R² is increased by around 100% from R² = 0.22 in Q₀/P - i_av to R² = 0.44 in Q₀/P - i₂₄.

Then, it is assumed that the runoff coefficient (Q/P) is directly proportional to i₂₄,

$$\frac{Q}{P}\propto {i}_{24}$$

Then by multiplying the right side of the proportionality by a constant, α, the proportionality is replaced by an equation:

$$Q=\alpha {i}_{24}P$$

where α (hr/mm) is a local constant and varies for each site.

Substituting i₂₄ from Eq. 13 in Eq. 15 yields

$$Q=\alpha {i}_{av}{\left(\frac{t}{24}\right)}^{n}P$$

and substituting i_av from Eq. 7 in Eq. 16 yields

$$Q=\frac{{\alpha P}^{2}}{t}{\left(\frac{t}{24}\right)}^{n}$$

From this point, Eq. 17 is known as the "new equation" in this paper.

For all case studies, α and CN values are calculated using the least squares fitting technique to minimise the sum of the squared differences between the observed data and computed runoff from the new equation and SCS-CN methods.

$$\sum _{Events}{\left\{{Q}_{0}-{Q}_{C}\right\}}^{2} =\sum _{Events}{\left\{{Q}_{0}-\frac{{\alpha P}^{2}}{t}{\left(\frac{t}{24}\right)}^{n}\right\}}^{2} \underset{\Rightarrow }{by varying \alpha }Minimum \left(15\right)$$

$$\sum _{Events}{\left\{{Q}_{0}-{Q}_{C}\right\}}^{2} =\sum _{Events}{\left\{{Q}_{0}-\frac{{\left(P-{I}_{a}\right)}^{2}}{P+S-{I}_{a}}\right\}}^{2} \underset{\Rightarrow }{by varying CN }Minimum \left(16\right)$$

where Q₀ is the observed runoff depth, and Q_C is the calculated runoff depth from the new equation and SCS-CN method. The optimised CN and α values from the calibration process are listed in Table 2.

Also, as some examples, the observed and estimated runoff by the new equation and SCS-CN method versus the total rainfall for each event for Opanuku and Pukekohe Ngakoroasites are illustrated in Fig. 6.

The SCS-CN method predicts only a single value for each rainfall event regardless of the rainfall characterises. However, the new equation estimates various runoff for each rainfall event depending on the rainfall durations. For example, looking at three extreme events around P = 250 mm in Pukekohe Ngakoroa (Fig. 5) the SCS-CN method predicts 71 mm for the three events. In contrast, the new equation predicts 42, 51 and 82 mm, respectively. Similarly, three different runoffs are also observed at the outlet, which is 29, 36 and 83 mm, respectively).

As a way of comparison, the calculated runoff by the new equation and SCS-CN method is plotted against the observed runoff for the Opanuku river in Fig. 7.

The R² in the SCS-CN plot is around 23% lower than the R² in the new equation (from R² = 0.79 to 0.64). Also, the slope of the linear trendline in the new equation is closer to the 1:1 line than the SCS-CN. So, overall, the new equation shows a better performance in the runoff depth estimation than the SCS-CN method in the Opanuku river for the study period.

4.1 The equations performance evaluation strategies

Three statistical criteria are used to evaluate the new equation and SCS-CN performances

Root Mean Square Error (RMSE):

$$RMSE=\sqrt{\frac{1}{m}\sum _{Events}{\left({Q}_{O}-{Q}_{C}\right)}^{2}}$$

where m is the total number of events. RMSE = 0 indicates the perfect model fit, and the closer the RMSE values are to zero, the better model performance.

Nash-Sutcliffe model Efficiency (NSE)

The Nash–Sutcliffe model efficiency coefficient (Nash et al. 1970) that is often used for evaluation of the performance of hydrologic models (Deshmukh et al. 2013; Ajmal et al. 2015) is calculated for both methods

$$NSE=1-\frac{\sum _{Events}{\left({Q}_{O}-{Q}_{C}\right)}^{2}}{\sum _{Events}{\left({Q}_{O}-\overline{{Q}_{O}}\right)}^{2}}$$

where $\overline{{Q}_{O}}$ is the mean of observed flow depth, and NSE is the Nash–Sutcliffe model efficiency coefficient. The NSE values range from -∞ to 1, where 1 shows a perfect prediction, whereas NSE ≤ 0 suggests that a simple average of observed values is better than the model estimations. Therefore, the model performance is unacceptable.

Percent Bias (PBIAS)

PBIAS measures the average tendency of the predicted flow depth to be larger or smaller than the observed data (Gupta et al. 1999).

$$PBIAS=\left[\frac{\sum _{Events}\left({Q}_{O}-{Q}_{C}\right)}{\sum _{Events}\left({Q}_{O}\right)}\right]\times 100$$

A low-magnitude value indicates an accurate model simulation where 0.0 is a perfect fit. Positive values suggest model underestimates, and negative values indicate model overestimates.

4.2 Calibration

RMSE, NSE and PBIAS are calculated for the calibration periods for all case studies. A summary of the calibrated values (CN and α) and performance evaluation of the new equation and SCS-CN models in the calibration process are listed in Table 2.

Table 2

a summary of the calibrated values, CN and α; and the new equation (NE) and SCS-CN models' performance in the calibration process based on NSE, RMSE and PBIAS for all case studies
Site No.	Site name	Calibrated values		NSE		RMSE (mm)		PBIAS
Site No.	Site name	CN	Alpha (hr/mm)	SCS	NE	SCS	NE	SCS	NE
1	West_Hoe_Heights	77	0.2522	0.62	0.73	10.1	8.5	38%	30%
2	Mangemangeroa	61	0.0634	0.67	0.75	4.1	3.5	46%	30%
3	Pukekohe_Ngakoroa	52	0.0574	0.46	0.74	3.8	2.6	59%	31%
4	Arkins Creek	36	0.0148	0.35	0.67	0.6	0.4	86%	34%
5	Wainui_Te_Whara	66	0.0769	0.78	0.81	5.1	4.7	36%	27%
6	Waingaehe	23	0.0087	0.08	0.61	0.9	0.6	93%	39%
7	Orewa	67	0.1074	0.59	0.66	6.9	6.3	45%	38%
8	Maungaparerua	76	0.1092	0.81	0.79	10.3	10.6	26%	30%
9	Whangarei Ngunguru	68	0.1307	0.82	0.92	10.6	7.1	26%	19%
10	Rutherford Creek	55	0.0452	0.90	0.88	2.5	2.7	25%	16%
11	Opanuku	62	0.2016	0.61	0.80	8.0	5.7	41%	22%
12	Johnson River	30	0.0134	0.56	0.63	17.3	15.8	50%	39%
13	Fisher Creek	27	0.0831	0.62	0.73	25.7	21.9	47%	31%
14	Raumanga	59	0.0606	0.88	0.94	4.2	2.9	37%	20%
15	Scott Creek	49	0.0106	0.23	0.48	2.2	1.8	70%	44%
16	Gellibrand	36	0.0253	0.81	0.89	3.5	2.6	50%	18%
17	Elizabeth Valley	33	0.0601	0.72	0.81	15.9	13.0	47%	25%
18	Tinana Creek	48	0.0388	0.85	0.84	8.8	8.9	26%	23%
19	Mannus Creek	59	0.0929	0.34	0.50	3.1	2.7	69%	45%
20	Bass River	61	0.0536	0.49	0.53	3.8	3.6	53%	43%
21	Ashley Gorge	68	0.0743	0.73	0.75	5.0	4.8	43%	34%
22	Wakefield	50	0.0001	0.23	0.60	0.7	0.5	80%	41%
23	Quenbeyan	41	0.0005	0.91	0.92	1.0	0.9	38%	30%
	* SCS-CN method, **NE: new equation
	MAX	77	0.2522	0.91	0.94	25.70	21.88	93%	45%
	Average	52	0.0687	0.61	0.74	6.69	5.74	49%	31%
	Median	55	0.0601	0.62	0.75	4.22	3.62	46%	30%
	Min	23	0.0001	0.08	0.48	0.60	0.44	25%	16%
	SD	16	0.0715	0.24	0.14	7.18	6.21	21%	9%
* SCS-CN method, **NE: new equation

Overall the new equation shows a better performance compared to the SCS-CN model. For example, the average NSE of the new equation for 23 case studies is 0.74, which is 25% higher than the NSE of the SCS-CN. Likewise, the average RMSE and PBIAS improved to about 14% and 38%, respectively. Scatterplots in Fig. 8 compare the NSE and PBIAS of the new equation with the SCS-CN for each case study.

In the NSE scatterplot, points above 1:1 lines indicate the new equation performs better, whereas, in the PBIAS scatterplot, points below the 1:1 line mean the new equation performs better on that specific site. In the NSE scatterplot, apart from 3 sites where the SCS-CN method performs slightly better, the new equation has supremacy over the SCS-CN model in predicting flow. Similarly, the PBIAS indicator also shows that the new equation predominantly performs better in all case studies apart from one site.

4.3 Validation:

For validation purposes, the calculated values for CN and α from the calibration process are directly used in the SCS-CN and the new equation on the validation data sets, defined in Table 1. A summary of NSE, RMSE and PBIAS for the validation process is listed in Table 3.

Table 3

a summary of validation of the new equation (NE) and the SCS-CN model based on performance evaluation techniques, including NSE, RMSE and PBIAS for all sites
Site No.	Site name	NSE		RMSE (mm)		PBIAS
Site No.	Site name	SCS	NE	SCS	NE	SCS	NE
1	West_Hoe_Heights	0.06	0.09	27.9	27.6	48%	42%
2	Mangemangeroa	0.67	0.72	6.5	6.0	49%	38%
3	Pukekohe_Ngakoroa	0.41	0.85	7.0	3.6	37%	23%
4	Arkins Creek	0.11	0.38	0.5	0.4	82%	16%
5	Wainui_Te_Whara	0.58	0.72	4.9	4.0	31%	18%
6	Waingaehe	0.02	0.44	1.2	0.9	94%	37%
7	Orewa	0.59	0.66	6.7	6.1	48%	39%
8	Maungaparerua	0.76	0.77	10.0	9.7	29%	33%
9	Whangarei Ngunguru	0.70	0.63	11.5	12.8	25%	26%
10	Rutherford Creek	0.92	0.90	3.4	3.8	34%	28%
11	Opanuku	-0.26	0.68	15.5	7.7	30%	10%
12	Johnson River	0.66	0.72	19.1	17.1	45%	36%
13	Fisher Creek	0.65	0.68	31.0	29.8	31%	28%
14	Raumanga	0.29	0.57	4.6	3.5	48%	20%
15	Scott Creek	0.16	0.25	5.1	4.8	86%	73%
16	Gellibrand	0.17	0.65	4.1	2.7	87%	35%
17	Elizabeth Valley	0.98	0.98	5.3	6.3	25%	10%
18	Tinana Creek	0.81	0.83	3.5	3.3	20%	10%
19	Mannus Creek	0.00	0.16	2.7	2.5	77%	56%
20	Bass River	0.73	0.65	1.8	2.0	46%	28%
21	Ashley Gorge	0.44	0.49	6.0	5.8	45%	35%
22	Wakefield	-0.01	-0.01	4.0	4.0	100%	98%
23	Quenbeyan	0.88	0.70	1.0	1.6	49%	51%
	MAX	0.98	0.98	31.0	29.8	100%	98%
	Average	0.45	0.59	8.0	7.2	51%	34%
	Median	0.58	0.66	5.1	4.0	46%	33%
	Min	-0.26	-0.01	0.5	0.4	20%	10%
	SD	0.36	0.27	9.3	8.9	26%	24%

A comparison between the calibration and validation performances shows that in both models, all performance indicators are slightly dropped in the calibration process. For example, the average NSE in the SCS-CN equation is lower by 26% (from 0.59 in calibration to 0.45), and NSE in the new equation declined by 21% (from 0.74 in calibration to 0.59).

However, similar to the calibration, in most case studies, the new equation performs better than the SCS-CN method in the validation process. For example, on average, NSE in the new equation for all case studies is 34% higher than SCS-CN performance. Likewise, the average RMSE and PBIAS in the new equation are higher by 9% and 33%, respectively. However, there are a few cases where SCS-CN performs the same or marginally better than the new equation in the validation process.

There are some cases where both models are performed exceptionally well, for example, in Rutherford Creek and Elizabeth Valley case studies. But, there are some cases in which both models failed to predict runoff depth, for instance, Wakefield and West Hoe Heights.

The proportionality of Q/S vs F/S (Eq. 2) and Q/S vs i₂₄ (Eq. 14) are the central assumptions for SCS-CN and the new equation, respectively. In order to investigate the validity of these assumptions, the coefficient of determination of Q₀/P vs F/S, and Q₀/P vs i₂₄ for all case studies are calculated. Eq. 3, 6, 8 and 13 are used to calculate F, S, i_av and i₂₄, respectively. A scatterplot and a boxplot in Fig. 9 compare the R² of Q₀/P vs F/S and i₂₄ for all case studies.

From the boxplot, the median R² of Q₀/P vs i₂₄ is 0.44, which is 101% higher than the R² of Q₀/P vs F/S. Also, a one-to-one compression in the scatterplot shows that all points are above the 1:1 line in the advantage of the new equation. Therefore it can be concluded that the central assumption of the new equation (Eq. 14) is strongly more reliable than the central assumption of the SCS-CN method (Eq. 2).

A new empirical event-based rainfall-runoff equation was developed. The IDF curves concept was implemented to introduce a 24-hour event average rainfall (i₂₄) for each rain event. An acceptable correlation between the runoff coefficient (Q/P) and i₂₄ was observed. Based on this correlation, a new equation was developed. The new equation and SCS-CN equation were used to estimate runoff for 23 catchments across New Zealand and Australia with oceanic and semidry climates ranging from 0.3 to 584 Km². Statistical indicators such as NSE, RMSE and PBIAS were used to compare the predictive skill of the new equation and SCS-CN. It was shown that the new equations perform better in both calibration and validation processes. Therefore, it can be concluded that the new equation is a robust alternative to the conventional SCS-CN model in oceanic and semiarid climate conditions. Also, there is a possibility that the new equation will be valid in other climate conditions. However, it needs further investigation.

Statements and Declarations

The author declare that no funds, grants, or other support were received during the preparation of this manuscript. The author have no relevant financial or non-financial interests to disclose. All data that support the findings of this study are available from public wesites.

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Development of new event-based rainfall-runoff equations as an alternative to the curve number method in oceanic and semiarid climates

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Abstract

Figures

1 Introduction

2 Materials

2.1 Hydro-Meteorological Data

2.2 Baseflow separation

3 Methods

3.1 Development of the new equation

4 Results

4.1 The equations performance evaluation strategies

4.2 Calibration

4.3 Validation:

5 Discussion

Conclusion

Declarations

References

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