1 Introduction

The hydrological response of a catchment to a rainfall event is commonly known as a rainfall-runoff process [1]. 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 [2]. 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 [3,4,5,6]. In addition, considering the physical condition of catchments is not homogenous, each catchment may respond differently to a rainfall event [1].

Several physically based mathematical expressions and computer simulation models have been developed that fully or partially represent the hydrological processes. 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 [7], Philip [8], Horton [8], Holtan [9], and Kostiakov [10] methods. In addition, sometimes, distributed surface water models are combined with distributed groundwater models to provide a better estimation [11,12,13,14,15,16,17,18,19]. However, these models are highly complex, data-hungry, and often require several spatial and temporal parameters, which in most cases are not readily available [20].

A simple alternative to physically based and continuous models is to estimate runoff from an individual rain event, often known as event-based models [21]. 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:

$$C=\frac{Q}{P},$$
(1)

where C is the runoff coefficient, P is the total runoff depth, and Q is the total runoff depth in each rain event [22,23,24,25], or sometimes Q is the total direct runoff depth for each event [26,27,28,29], 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 [30]. For example, the parameter called response factor [26], hydrologic response [27], runoff ratio [23], water yield [31], conversion efficiency [24], direct runoff response ratio [5], and runoff coefficient [6, 22, 28]. 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 [32]. For clarification, in the current study, the runoff coefficient means the total direct runoff 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 [3, 5].

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 [33,34,35]. 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 [36,37,38,39,40,41,42,43,44,45,46]. Also, several lumped and semi-distributed models are designed based on an adapted form of the SCS-CN equation, among which are CREAMS [47], GLEAMS [48], AGNPS [49], EPIC [50,51,52], WinTR-55 [53], HEC-HMS [54], EPA-SWMM [55], TOPNET [56], and SWAT [57].

The SCS-CN method originally comes from a lumped-based approach that calculates the total direct runoff from a storm event [58]. For deriving the SCS-CN expression, the proportionality between retention and runoff is assumed to be:

$$\frac{F}{S}=\frac{Q}{P},$$
(2)

where F is the actual retention, S is the maximum potential retention, and Q is the total direct runoff during a rainfall event. Assuming:

$$F=P-Q,$$
(3)

and subtracting a certain amount of rainfall, referred to as initial abstraction from the total runoff depth P and solving Eq. 2 for Q yields:

$$Q=\frac{\left(P-I_a\right)^2}{P+S-I_a},\mathrm{when}\;P\geq Ia,\mathrm{otherwise}\;Q=0,$$
(4)

where Ia is the initial abstraction. Commonly, a linear correlation is assumed between S and Ia,

$${I}_{a}=\lambda S,$$
(5)

where λ is the initial abstraction ratio. λ is considered 0.2 in the original formula [40, 59, 60]. However, the background of this assumption is known [58].

In the SI units,

$$S=\frac{25.4}{CN}-0.254,$$
(6)

where CN value is defined by hydrological soil group, land use, hydrological surface condition, and soil moisture condition [41]. Theoretically, the CN may vary between 0 and 100.

From a hydrologic engineering perspective, the SCS-CN method is simple, transparent, appealing, and only requires one empirical parameter, the CN. The SCS is popular because SCS-CN databases can easily link to distributed soil and vegetation layers stored within a GIS [34]. Hawkins [41] 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 [33, 34, 41, 42]. For example, Eq. 2, as the central assumption of the SCS-CN method, has no physical justification [34].

Another concern is that the rainfall characteristics, such as rain duration, are neglected in the equation [61]. 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 [32, 58]. However, several studies statistically indicated that the runoff is proportional to the rainfall duration [3,4,5].

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. The new equation only has one empirical parameter. However, compared to the other event-based models, it more precisely represents the physics of the rainfall-runoff process by incorporating rainfall characteristics.

2 Materials

2.1 Hydro-Meteorological Data

For this study, 23 catchments ranging from 0.3 to 564 km2 are selected from catchments in New Zealand and Australia. The sites in New Zealand have an oceanic climate, with cool but not cold winters and warm summers. In contrast, the case studies in Australia have a range of climates, including Oceanic, Mediterranean, tropical, subtropical, and semi-arid climates. Commonly, no snow is recorded in any sites during the study periods.

Rain events are defined based on the minimum-interevent-time method [62], in which the beginning and end of each event are defined by a minimum rainless period of at least 24 h. 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, climate type, 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)
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2.1.1 Baseflow Separation

The quick flow (direct runoff), which is the direct response to a catchment to rainfall, is separated from the base flow, water seeping into the stream from groundwater. Therefore for the development of a rainfall-runoff model often, the quick flow is separated from the runoff. The BFI + , which is a powerful and user-friendly software with eleven methods for baseflow and quick flow separation methods, is selected [63, 64]. The one-parameter recession digital filter method in the BFI + software is used for baseflow quick-flow separation for the current study. As an example of baseflow separation, 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.

Fig. 1
figure 1

An example of the baseflow separation for Opanuku using one parameter recession digital filter

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3 Methods

3.1 Development of the New Equation

The event average rainfall rate is calculated for each event:

$${i}_{av}=\frac{P}{t},$$
(7)

where iav is the average rainfall rate (mm/h) in each event, and t is the event duration (h). The existing correlation between runoff coefficient (Q/P) and average rainfall rate [1] has been suggested in previous studies. The correlation between the runoff coefficient and average rainfall rate in Opanuku and Pukekohe Ngakoroa rivers is illustrated in Fig. 2.

Fig. 2
figure 2

Correlation between runoff coefficient (Q0/P) and average rainfall rate in Opanuku and Pukekohe Ngakoroa rivers

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

Intensity–duration–frequency (IDF) curves were implemented to enhance the correlation by unifying rain event durations. The IDF curves concept was established in early 1930, and many mathematical expressions for IDF curves were suggested subsequently [32].

IDF curve can be expressed in a general form:

$${i}_{av}=\frac{f(T) }{g(t)},$$
(8)

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 for each site and can be estimated directly from previous rainfall records.

For this study, following a method suggested by [32, 65], the annual maximum rainfall for each site was rank-ordered, and the return period for each annual maximum rainfall value was estimated. Then, the rank-ordered rainfall value was 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 years are given in Fig. 3.

Fig. 3
figure 3

IDF curves of Pukekohe for return periods ranging from 0.01 to 100 years

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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 [32]:

$$f\left(T\right)={i}_{av}g\left(t\right)={i}_{24}g\left(24\right),$$
(9)

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

Rearranging for i24 yields:

$${i}_{24}=\frac{g\left(t\right)}{g\left(24\right)}{i}_{av}.$$
(10)

Assuming a power function form for g(t) after Koutsoyiannis et al. [66]:

$$g\left(t\right)={at}^{n},$$
(11)

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

Therefore,

$$g\left(24\right)={a(24)}^{n}$$
(12)

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

$${i}_{24}={i}_{av}({\frac{t}{24})}^{n}$$
(13)

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 iav 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.

Fig. 4
figure 4

Correlation between runoff coefficient (Q/P) and i24 in Opanuku and Pukekohe Ngakoroa sites

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A comparison between Figs. 2 and 4 shows using i24 instead of iav increases the R2 by 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 iav and Q/P vs i24 for all case studies.

Fig. 5
figure 5

Scatterplot and boxplot compare the correlation (R2) of Q/F vs i24 and Q/P vs iav for all case studies

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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 − iav to R2 = 0.44 in Q0/P − i24.

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

$$\frac{Q}{P}\propto {i}_{24}$$
(14)

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

$$Q=\alpha {i}_{24}P,$$
(15)

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

Substituting i24 from Eq. 13 in Eq. 15 yields:

$$Q=\alpha {i}_{av}{\left(\frac{t}{24}\right)}^{n}P,$$
(16)

and substituting iav from Eq. 7 in Eq. 16 yields:

$$Q=\frac{{\alpha P}^{2}}{t}{\left(\frac{t}{24}\right)}^{n}$$
(17)

From this point, Eq. 17 is known as the “new equation” in this paper. Considering quick flow data is used to develop the new equation, the concept of initial abstraction is already included in the P/Q correlation.

4 Results and Discussion

For all case studies, α and CN values were 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.

$${\sum\nolimits_{\mathrm{Events}}}\left\{Q_0-Q_C\right\}^2={\sum\nolimits_{\mathrm{Events}}}\left\{Q_0-\frac{{\alpha P}^2}t\left(\frac t{24}\right)^n\right\}^2\overset{\text{by varying}\;\alpha}\Rightarrow\mathrm{Minimum}$$
(18)
$${\sum\nolimits_{\mathrm{Events}}}\left\{Q_0-Q_C\right\}^2={\sum\nolimits_{\mathrm{Events}}}\left\{Q_0-\frac{\left(P-I_a\right)^2}{P+S-I_a}\right\}^2\overset{\mathrm{by\;varying\;CN}}\Rightarrow\mathrm{Minimum}$$
(19)

where Q0 is the observed runoff depth, and QC 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.

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
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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.

Fig. 6
figure 6

Observed and calculated direct runoff estimation with the new equation and SCS-CN method in Opanuku and Pukekohe Ngakoroa rivers

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The SCS-CN method predicts only a single value for each rainfall event regardless of the rainfall characterises. However, the new equation estimates various runoffs 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, 29, 36 and 83 mm, are also observed at the outlet.

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

Fig. 7
figure 7

Calculated runoff depth from the new equation and SCS-CN versus observed runoff depth in the Opanuku river

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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 Performance Evaluation Strategies

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

4.1.1 Root Mean Square Error (RMSE)

$$\mathrm{RMSE}=\sqrt{\frac1m{\textstyle\sum_{\mathrm{Events}}}\left(Q_O-Q_C\right)^2}$$
(20)

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 the model performance.

4.1.2 Nash–Sutcliffe Model Efficiency (NSE)

The Nash–Sutcliffe model efficiency coefficient [67] that is often used for evaluation of the performance of hydrologic models [68, 69] is calculated for both methods:

$$\mathrm{NSE}=1-\frac{\sum_{\mathrm{Events}}{\left({Q}_{O}-{Q}_{C}\right)}^{2}}{\sum_{\mathrm{Events}}{\left({Q}_{O}-\overline{{Q}_{O}}\right)}^{2}}$$
(21)

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.

4.1.3 Percent Bias (PBIAS)

PBIAS measures the average tendency of the predicted flow depth to be larger or smaller than the observed data [70].

$$\mathrm{PBIAS}=\left[\frac{\sum_{\mathrm{Events}}\left({Q}_{O}-{Q}_{C}\right)}{\sum_{\mathrm{Events}}\left({Q}_{O}\right)}\right]\times 100$$
(22)

A low-magnitude value indicates an accurate model simulation where 0 is a perfect fit. Positive values suggest model underestimates, and negative values indicate model overestimates.

4.2 Calibration

RMSE, NSE, and PBIAS were 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.

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 improved to 14%. Scatterplots in Fig. 8 compare the NSE and PBIAS of the new equation with the SCS-CN for each case study.

Fig. 8
figure 8

A comparison between NSE and RMSE of the new equation (NE) and SCS-CN for the calibration process

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In the NSE scatterplot, points above 1:1 lines indicate the new equation performs better, whereas, in the RMSE 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 RMSE indicator also shows that the new equation predominantly performs better in all case studies apart from one site. Figure 9 shows a scatterplot that compares the α and CN for 23 case studies.

Fig. 9
figure 9

A comparison between the α and CN for all case studies

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The scatterplot shows a relatively weak to moderate correlation between α value and CN with an R2 of 0.42. It means the higher the α value in a catchment, the less rainfall infiltrates the groundwater system and generates more runoff.

4.3 Validation

For validation, the calculated values for CN and α from the calibration process were 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 the validation of the new equation (NE) and the SCS-CN model based on the performance evaluation techniques, including NSE, RMSE, and PBIAS for all sites
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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 in the new equation is higher by 9. 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 perform 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.

Figure 10 shows no correlation between the new equation performance (NSE) and the α value. Considering that the α value represents the capacity of catchments to generate runoff, the lack of a correlation means the performance of the new equation is independent of the α value and can be used in a wide range of catchments.

Fig. 10
figure 10

A comparison between the new equation performance (NSE) and the α value

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In general, it can be concluded that the new equation is a robust alternative to the conventional models in oceanic climate conditions. This research also studied a few catchments in Mediterranean, tropical, subtropical, and semi-arid classmates. However, no advantage or disadvantage is observed in the performance of the new equation in different climates. However, the numbers of samples in each climate are relatively small. Therefore, further investigation with a broader range of climates is required to conclude whether the new equation performs reasonably in all climate conditions.

4.4 Discussion on the Core Assumption of the New Equation and the SCS-CN Method

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 R2 of Q0/P vs F/S, and Q0/P vs i24 for all case studies are calculated. Equations 3, 6, 8, and 13 are used to calculate F, S, iav, and i24, respectively. A scatterplot and a boxplot in Fig. 11 compare the R2 of Q0/P vs F/S and i24 for all case studies.

Fig. 11
figure 11

Scatterplot and boxplot compare R2 of Q/F -i24 and Q/P -F/S as the central assumption of the new equation and the SCS-CN methods, respectively

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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 more reliable than the central assumption of the SCS-CN method (Eq. 2).

5 Conclusion

A new empirical event-based rainfall-runoff equation was developed. The IDF curve concept was implemented to introduce a 24-h 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 semi-arid 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 event-based model in oceanic and semi-arid climate conditions. Also, there is a possibility that the new equation will be valid in other climate conditions. However, it needs further investigation.