Comparison of antecedent precipitation based rainfall-runoff models

SCS-CN

Soil Conservation Service Curve Number

AMC

antecedent moisture condition

ARC

antecedent runoff condition

CN

curve number

S

potential maximum retention

NEH-4

National Engineering Handbook-Section 4

I_a

initial abstraction

λ

initial abstraction coefficient

USDA-ARS

United States Department of Agriculture-Agricultural Research Service

RMSE

root mean square error

NSE

Nash-Sutcliffe Efficiency

PBIAS

percentage bias

RGS

rank and grading system

In surface water hydrology, the correct estimation of runoff is arguably the most important assignment for hydrologists and water scientists. This is useful in water supply, water allocation, hydrological analysis, watershed management, flood forecasting, flood protection work, integrated water management, control of non-point source pollution, etc. The runoff amount depends on rainfall intensity and its amount, wetness condition of the soil, watershed slope, various abstractions, land use, land cover, and many more factors. Hewlett et al. (1977) developed a rainfall-runoff relationship for a forest catchment and found negligible effects of rainfall intensity on peak discharge and runoff volume. Several studies have been conducted at event scale on various catchments of Europe to understand surface runoff generation process (Jordan 1994; Meyles et al. 2003; Latron et al. 2008). In case of events having both low-intensity rainfall and low-rainfall magnitude, the antecedent wetness state plays a major role in controlling runoff amount than rainfall intensity or its magnitude and a more precise and correct estimate can be achieved by using antecedent rainfall characteristics. If the joint dependence between rainfall event and antecedent precipitation is not taken into account properly, the runoff amount is consistently underestimated and when only one or two days' antecedent rainfall is considered, the magnitude of design flood can be as much as 30% less than actual (Pathiraja et al. 2012). The impact of prior rainfall of 48 hrs to 120 hrs of a rainfall event is more significant in determining the resulting runoff compared to other factors such as average and maximum rainfall intensity, event magnitude, and rainfall duration.

The relationship of intensity of storm event with antecedent moisture condition has become an increasing research subject in the context of anthropogenic climate change. To understand the role of antecedent moisture in rainfall runoff modelling, a tank experiment has been conducted based on high resolution observation in Hohai University, China, and a non-linear relationship was observed between soil moisture index and total runoff (Song & Wang 2019). If a model is calibrated without considering the impact of prior rainfall on the land surface, there will be a high likelihood that calibrated parameters will not produce the correct runoff yield. The water content in the upper soil surface may be an important factor in the rainfall-runoff relationship. A relatively lower sensitivity of antecedent rainfall on surface runoff has been found, maybe, due to the relatively dry upper surface in semiarid small watersheds located in Southeastern Arizona in the USA, contrary to other reported results (Zhang et al. 2011). The surface runoff is a threshold process and is governed by the runoff coefficient, which increases abruptly when a certain moisture threshold is exceeded. Similar results were found with changing values of moisture threshold by Radatz et al. (2013) and Zhao et al. (2015). A soil moisture accounting procedure was proposed by Michel et al. (2005) but criticized by Sahu et al. (2007) for not circumventing an undesirable sudden jump. Verma et al. (2020) conceptualized a theory of activation soil moisture (ASM) by coupling the infiltration component with the Soil Moisture Accounting (SMA) concept. The rainfall intensity, state of antecedent moisture and process of surface runoff are the three main reasons which evolve spatial and temporal non-linearity and complicate the watershed response. Most of the research shows that the rainfall-to-runoff transformation process is non-linear and the prime factor of nonlinearity is the antecedent moisture condition (Radatz et al. 2013; Zhao et al. 2015). This antecedent moisture, along with soil characteristics, regulates the runoff process and affects the capability to store new water due to the occurrence of rainfall as well as the infiltration capacity of the soil (Merz & Blöschl 2009). It is a challenging task to evaluate the impact of antecedent rainfall on runoff at a watershed scale and it is required in the planning and management of a watershed.

There are distinct mathematical models available in hydrology with various merits, demerits and complications. The SCS-CN method is the simplest and universally accepted model for event-based rainfall-runoff modeling and integrated with various hydrological, water quality, or erosion estimation models. After a long experimental work, this method was developed for conditions prevailing in the USA. The model requires only maximum potential storage (S, mm) of the watershed in the form of a curve number (CN) to calculate runoff depth. Generally, soil type, land use and adopted practices are not much changed, but antecedent moisture condition status is changed frequently with the antecedent rainfall amount, which affects the runoff flow pattern. Antecedent moisture condition (AMC) classes were re-labelled as antecedent runoff condition (ARC) classes (NRCS 2004). The observation and monitoring of soil moisture are difficult; that is why antecedent precipitation, which can address initial soil moisture status by using five days of rainfall (Brocca et al. 2008), is in common use. This term is used as a predictor to decide the antecedent runoff condition, which is categorized in three levels i.e. ARC I (dry), ARC II (normal), and ARC III (wet). These levels form a discrete relation between antecedent rainfall and curve number and are responsible for an undesirable sudden jump in runoff estimation (Mishra et al. 2008). Generally, the CN value obtained from NEH-4 table or calculated from the P-Q dataset is CN_II (normal) and can be converted into CN_I (dry) or CN_III (wet) as per the antecedent rainfall amount. The CN model uses the antecedent five-day rainfall to categorize it into either dry, normal, or wet conditions to account for watershed initial losses (Sahu et al. 2007).

To obviate the error in predicting runoff calculations due to the sudden jump in curve number value, consideration of pre-storm rainfall is required in event-based runoff modelling, which minimizes the error and tries to correct the runoff value. This study has been done to evaluate the relative significance of antecedent precipitation (P₅) on the calculated runoff amount. Very few studies have been done to investigate the effect of antecedent rainfall on runoff behaviour. This is assessed using six variants of models, which are introduced here.

The conversion of S_II to S_I or S_III depends on the previous 5 days' rainfall. The value of P₅ < 35.6 mm, 35.6 mm ≤ P₅ ≤ 53.3 mm, and P₅ > 53.3 mm assumes dry, normal, or wet conditions, respectively, for any storm event.

It is to be noted that in both the above equations, Equations (8) and (9), S is the only unknown parameter. These equations are unique in nature and also exclude λ. Their performance were found to be significantly better than the SCS-CN model for the studied catchments. Equations (8) and (9) directly used P₅ value instead to decide antecedent conditions as dry, normal, or wet before rainfall and to some extent were able to overcome the sudden jump in runoff computation. To check its applicability in other watersheds, similar equations are to be used in this study by utilizing the P-P₅ dataset of 114 US watersheds.

The previous research suggested antecedent rainfall as a key component. The original model considers antecedent rainfall only to classify the land condition as dry, normal, or wet and ignored the exact amount of 5 days' prior rainfall (P₅) which creates an undesirable sudden jump in runoff computation. This can be explained by an example. If we assume a rainfall amount of 100 mm (which occurs in the growing season) over different land conditions with CN values as 60, 70, 80, and 90; and the antecedent five days rainfall variation from 1 to 100 mm. These CN values have been assumed, since for most of the watersheds its value falls in between 55 to 95 (Hawkins et al. 1985).

From Figure 1 (assuming λ = 0.2 and using Equation (3), it can be seen that the calculated values of runoff are 1.29 mm for ARC I (P₅ < 35.6 mm, CN = 60 or S = 385.30 mm), 18.57 mm for ARC II (P₅ = 35.6 mm to 53.3 mm, CN = 60 or S = 169.33 mm), and 46.13 mm for ARC III (P₅ > 53.3 mm, CN = 60 or S = 72.81 mm). It means if five days' prior rainfall value changes the antecedent rainfall condition from ARC I to ARC II, the runoff value suddenly jumps from 1.29 mm to 18.57 mm for 100 mm rainfall. Similarly, shifting the antecedent state from ARC II to ARC III, runoff jumps from 18.57 mm to 46.13 mm. Similar jumps are illustrated in Figure 1 at CN = 70, 80, and 90 for ARC I, ARC II and ARC III. This undesirable jump is due to the indirect use of P₅ and it reduces the model efficiency.

This equation is a version of the SCS-CN equation with incorporating P₅ and represents a simplified form of model M₆. The different models using P₅ value either directly or indirectly and related equations are also summarized in Table 1.

This study has been done over 114 US watersheds having areas varying from 0.17 ha to 30,351.45 ha. Data is taken from the United States Department of Agriculture-Agricultural Research Service (USDA-ARS) database which is a collection of storm depth (P mm), runoff depth (Q mm), and antecedent five days rainfall (P₅ mm) and generated from the breakpoint rainfall-runoff dataset. The information of watershed ID with its serial number, situated location, and states have been provided in Table 2. The average storm depth (P), average runoff depth (Q) and average antecedent rainfall value (P₅) in mm unit for different watersheds are illustrated in Figure 2(a). In addition, the drainage area of different watersheds (ha) in log scale and number of available and selected events for this study are presented in Figure 2(b).

Here, P/S = 0.46 put into Equation (14), and finds Q/P as 0.12. It mean we can replace P/S ≥ 0.46 by Q/P ≥ 0.12. Thus, this study excluded lower runoff producing rainfall events (runoff coefficient value is less than 0.12) and sorted only those events for which C > 0.12. After adopting this criterion out of 28,849 events of 114 US watersheds, 11,784 events were selected for this study.

Model M₁ used a fixed value of S (using Equation (3)) while the rest of the models obtained their S value through the optimization process. The initial estimate of S in the M₂ model was taken as 125 mm and allowed to vary in between 1 and 500, while in other models S was allowed to vary in the range of 1 to 1,000 with an initial estimate of 250 mm. Model M₆ optimized both the S and λ value. In the M₆ model, the initial estimate of λ was assumed as 0.05 and permitted to vary between 0 and 1. The statistical summary of S and λ value for different models are presented in Table 3.

The performance of all 114 US watersheds resulting from applications of each of the models has been compared using RMSE, NSE, PBIAS, n(t), rank and grading system, and by visual assessment. The performance of all models for individual watersheds was compared by using a scatter plot (Figure 3,) and the collective performance of models can be seen in a Box and Whisker plot for all statistical indices (Figure 4,). The best model suggestion was based on total score obtained using rank and grading system (RGS) for all 114 US watersheds. The value of S was optimized for all six models which are presented in Table 3. The mean and median value of S was found to be higher for the M₆ model and lowest for the M₁ model. The higher S value also increases the standard deviation (SD) and standard error (SE) value of the M₆ model.

The RMSE values for all models are presented in Figure 3(a). The lower the RMSE, the better the performance of the model, and vice versa. It can be seen that the existing SCS-CN model shows a worse performance than the other models. The median value of RMSE for M₁, M₂, M₃, M₄, M₅, and M₆ models were found to be 7.54, 7.23, 5.70, 5.88, 5.85 and 5.32 mm, respectively, which is lowest for the M₆ model. Its mean value was also lowest (5.53 mm) for the M₆ model. When we compared all models based on RMSE value, out of 114 watersheds the M₆ model performs best with minimum RMSE values in 95 watersheds. Next is the model M₃, which had the lowest RMSE values in the remaining 19 watersheds. From the RMSE value, it can be inferred that the M₆ model performs well followed by M₃, M₅, M₄, M₂, and M₁. The higher RMSE values exhibit the worst performance and are found mostly in the M₁ and M₂ models. Figure 4(a) exhibits the overall RMSE performance of all models by using a Box and Whisker plot. Noteworthy, the M₃ and M₄ models follow the same trend and the RMSE value of the M₃ model is better than the M₄ model.

The NSE value indicates the model efficiency and a high value means a better model. The NSE value was found to be better in the M₆ model for most of the watersheds (Figure 3(b)). NSE values fall in a good range (0.80 ≤ NSE < 0.90) in 41 watersheds followed by 30, 25, 19, 15, and 11 for M₅, M₃, M₄, M₂, and M₁, respectively. The cumulative median value for all watersheds was found to be highest for the M₆ model as 0.78 followed by M₅ (0.75) > M₃ (0.73) > M₄ (0.72) > M₂ (0.63) > M₁ (0.61). The mean value was very low, as 0.62 for the M₅ model, since some of the watersheds' NSE values were found to be very low. In some cases it was negative, which indicates model prediction was worse than the mean observed value. From Figure 4(b), it is evident that model M₆ performed superiorly, with higher NSE than other models and model M₃ and M₄ depict nearly the same performance.

PBIAS quantifies a model tendency either to overestimate or underestimate. A PBIAS value of zero indicates perfect fit, its negative value indicates overestimation and vice versa. Models M₃ and M₄ underestimated (positive PBIAS) the runoff in all 114 watersheds and the mean PBIAS value was also as high as 14.13% and 17.12% respectively compared to the other models. Models M₁, M₂, and M₅ underestimated the runoff in 80, 107, and 101 watersheds and overestimated (negative PBIAS) in 34, 7, and 13 watersheds respectively. The mean and median PBIAS were both lowest and nearer to zero for the M₆ model with 4.77% and 2.62% respectively. The PBIAS results indicated that the M₆ model performance was very good in 91 watersheds and unsatisfactory in 4 watersheds only. By using the M₆ model, most of the watersheds fall either in the very good or good range (Figure 3(c),). The PBIAS statistical index prefers the M₆ model because it gives a more consistent and better PBIAS value (Figure 4(c),). The overall statistics of RMSE, NSE and PBIAS values for all models are presented in Table 4. The overall SD and SE values of RMSE, NSE, and PBIAS for 114 watersheds were found to be less for the M₆ model. For the M₅ model, SD and SE values are quite high because in some watersheds the performance of the model was very poor, which increases dispersion from its mean value, lowers the mean and maximizes their range.

Like the previous three indices, the n(t) value was also found to be best in most of the watersheds using the M₆ model (Figure 3(d),). Similar to the NSE, the cumulative mean of the n(t) value was found to be high for the M₆ model at 1.19, followed by 1.04 for model M₃. It was lowest for the M₁ model at just 0.58 followed by the M₂ model at 0.70. Table 5 summarises the guidelines for classifying the model performances according to their range as unsatisfactory, satisfactory, good and very good. This table suggests the usefulness and better performance of the M₆ model compared to other models. Model M₆ found exemplary performance in maximum cases, while poor performance was found comparatively in fewer watersheds.

The superiority of the M₆ model over the rest of the models was judged by making a cumulative frequency distribution curve for percentage improvement (r²) criteria (Figure 5,). The M₆ model performed better than the M₁, M₂, and M₅ models for all watersheds while for the M₃ and M₄ models, only 17 and 14% watersheds respectively performed better than M₆. To visually assess the performance of different models a scatter plot (Figure 6 ) has been drawn between observed and computed runoff for WS ID- 9004, and 33005. The visual assessment indicates almost equal performance of M₆ and M₅ models for WS ID 9004 and much better performance of the M₆ model than the rest of the models for WS ID 33005.

A NSE-based rank and grading based system (RGS) was adopted to select the preference of the best model. A rank was assigned to each model for the individual watershed. A high NSE value has been given to the best rank as 1 and credit 6 point score in its account. From Table 2, for WS ID 9004, the NSE value was highest for the M₆ model (Rank I, Score 6) followed by M₅ (Rank II, Score 5), M₄ (Rank III, Score 4), M₃ (Rank IV, Score 3), M₂ (Rank V, Score 2), and M₁ (Rank VI, Score 1) as 0.88, 0.76, 0.67, 0.69, 0.26 and a negative NSE respectively. Model M₆ gained 6, 5, 4, 3, 2, and 1 score in 95, 3, 16, 0, 0, and 0 watersheds respectively. The cumulative sum of the score was highest for the M₆ model at 649 (6 × 95 + 5 × 3 + 4 × 16 + 3 × 0 + 2 × 0 + 1 × 0 = 649) followed by 504 for M₃, 471 for M₅, 368 for M₄, 264 for M₂ and 138 for the M₁ model (Figure 7 ). Therefore, on the basis of RGS, the M₆ model can be also rated as the best model. After the M₆ model, the next better performing models were found to be M₃, M₅, M₄, M₂, and M₁ respectively.

The results indicate that the indirect approach to categorize runoff conditions on the basis of antecedent five days rainfall as dry, normal, or wet does not reflect the best performing potential. Better is to incorporate P₅ directly in the runoff model and this certainly helps in improved model performance. Using trial and error methods, Ajmal and Kim proposed two models for South Korean catchments that directly consider the P₅ value in model formulation. Both model performances were found to be better than the SCS-CN model and increase the model efficiency of US watersheds. In this study, the SCS-CN model was modified by incorporating P₅ value and provides a more rational basis to M₅ and M₆ models. Despite being empirical in nature, it is needless to reiterate that the new model proposed in this work shows the promise of better performance using several error metrics.

Based on this study, the following can be inferred:

• 1.

  The proposed study obviates the problem of the sudden jump and improves the performance of the model. The proposed model changes the maximum potential retention value (S, mm) due to the antecedent 5 days' rainfall values.

• 2.

  All statistical criteria were found to be better for the proposed model with less RMSE, high NSE, better PBIAS, high n(t) value, and better overall rank and grading based score.

• 3.

  Between M₅ and M₆ models, M₅ performance was poor due to the fixed λ value. The M₅ model indicates that a fixed λ value of 0.2 does not fit well and some other value should be used. Model M₆ includes variation in λ value and it increases the efficiency of the model. For 97 watersheds, the λ value was found to be zero. The median value of parameter λ was found to also be zero. Therefore, model M₆ can be used with λ value as zero (Equation (11)) simplifying the M₅ model.

A special thanks to Prof. R.H. Hawkins, Emeritus Professor, University of Arizona for providing rainfall-runoff dataset of US watersheds and Prof. C.S.P. Ojha for supervising and giving useful suggestions to complete this research work.

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