Comparison of antecedent precipitation based rainfall-runoff models

The Soil Conservation Service Curve Number (SCS-CN) method is one of the popular methods for calculating storm depth from a rainfall event. The previous research identified antecedent rainfall as a key element that controls the non-linear behaviour of the model. The original version indirectly uses five days antecedent rainfall to identify the land condition as dry, normal or wet. This leads to a sudden jump once the land condition changes. To obviate this, the present work intends to improve the performance of antecedent rainfall-based SCS-CN models. Two forms of SCS-CN model (M1 and M2), two recently developed P-P5 based models (M3 and M4), and an alternate approach of considering P5 in the SCS-CN model (M5 and M6), as proposed here, were investigated. Based on the evaluation of several error metrics, the new proposed model M6 has performed better than other models. The performance of this model is evaluated using rainfall-runoff events of 114 watersheds located in the USA. The median value of Nash Sutcliffe Efficiency was found as 0.78 for the M6 model followed by M5 (0.75), M3 (0.73), M4 (0.72), M2 (0.63) and M1 (0.61) model.


INTRODUCTION
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. () 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 ; Meyles et al. ; Latron et al. ). 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. ). 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 ). 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. ). 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. () and Zhao et al. Most of the research shows that the rainfall-torunoff transformation process is non-linear and the prime factor of nonlinearity is the antecedent moisture condition (Radatz et al. ; Zhao et al. ). 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 ). 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 ). 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. ), 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. ). 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. ).
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 (P 5 ) 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.

SCS-CN MODEL (M 1 AND M 2 )
The original SCS-CN equation (USDA ) calculates runoff depth directly using rainfall data. This universally accepted equation is well documented, easy to understand, and useful due to its simplicity. This water balance equation, based on two assumptions, is expressed, respectively, as follows: where P and Q are rainfall and runoff depth in mm, I a is an initial abstraction, F is cumulative filtration which is equal to P-I a -Q and S is maximum potential retention in mm. After simplification of the above-stated assumptions, the expression is as follows: , if P > I a or λS, otherwise Q ¼ 0 λ value can vary from 0 to 1 but assumes as 0.2, while S value in mm varies from 0 to ∞. A dimensionless curve number (CN) for any watershed can be used to estimate S by using the following equation: In addition, S can be directly obtained (λ ¼ 0.2) using a gauged dataset of P and Q by solving Equation (3) with the quadratic equation as: This Equation (5) produced a data-derived value of S (normal condition or S 2 ). Hawkins () suggested a median CN value gives a better representation of the curve number (λ ¼ 0.2) of any watershed using the P-Q dataset.
Two mathematical equations proposed by Mishra et al.
() can be used to convert CN II or S II to CN I (S I ) and CN III (S III ) for dry (ARC I) or wet conditions (ARC III), respectively. These are: The conversion of S II to S I or S III depends on the previous 5 days' rainfall. The value of P 5 < 35.6 mm, 35.6 mm P 5 53.3 mm, and P 5 > 53.3 mm assumes dry, normal, or wet conditions, respectively, for any storm event.

AJMAL & KIM MODEL (M 3 AND M 4 )
Ajmal & Kim () proposed two new equations and investigated runoff prediction over 15 South Korean catchments.
These two equations are as follows: 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)   In the proposed model, instead of only selecting antecedent rainfall conditions (ARC), P 5 is incorporated directly as a model component. This is done by replacing S with S P (P þ P 5 ) and placing it in the original SCS-CN equation with the sole objective of improving runoff prediction. In this fashion, the replacement of S will ensure the change in watershed characteristics (S) with different rainfall depth and P 5 value; and will avoid the chances of an unexpected jump in runoff calculation. The proposed model gives the following equation as: After simplification, Equation (10) yields λ ¼ 0.2 makes Equation (11) as: This expression provides variation in the S value and such replacement changes the runoff value only for those events for which P 5 is greater than zero. With increasing or decreasing P 5 , the S value changes significantly. If there is no capacity to store water in the soil, whole rain will convert into runoff i.e. S → 0. Similarly, if the S value tends to infinity (CN → 0), all water goes for storage and there will no runoff. Thus, these two boundary conditions are satisfied using this empirical model. For improving the model performance, instead of a fixed λ value, it can also be obtained after optimization. The best λ value obtained was zero in the tested watersheds, which leads to further simplification of Equation (11) as: This equation is a version of the SCS-CN equation with incorporating P 5 and represents a simplified form of model M 6 . The different models using P 5 value either directly or indirectly and related equations are also summarized in Table 1.

Study area and data selection criteria
This study has been done over 114 US watersheds having areas varying from 0.17 ha to 30,351.45 ha. Data is taken and antecedent five days rainfall (P 5 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 reformed into the following form as: Here, P/S ¼ 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.

Parameter estimation
To obtain the best possible results with different models, optimization has been carried out by minimizing the sum of the square difference between observed and computed runoff employing Microsoft Excel (Solver) and using Equation (15). The main intention of optimization is to obtain a realistic and conceptually unique value of model parameter (Lal et al. ).
Model M 1 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 M 2 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 M 6 optimized both the S and λ value.
In the M 6 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.

Performance evaluation
The statistical indices-based assessment has been done to analyze the performance of different models. The root mean square error (RMSE), Nash-Sutcliffe efficiency (NSE), percent bias (PBIAS), and n(t) criteria have been used on 114 US watersheds to evaluate the performance of different models. The mathematical equation of these criteria are respectively given below: where Q o , Q c and SD OR are observed runoff, computed runoff and variation in observed runoff given by standard deviation, respectively and i is an integer varying from 1 to N. The n(t) value depicts the number of times the cumulative variation in mean observation is more than the mean error.   The higher n(t) value reflects the appropriateness of the model for efficient runoff computation.
To select the best model for a particular watershed, NSE based rank and grading system (RGS) was applied on different models and ranks (I to VI) were assigned according to their NSE value (Verma et al. ). Their awarded score was added to select the best model among all. The percentage improvement achieved by the best model over rest models can be addressed by r 2 criteria. The mathematical expression of r 2 is as follows:

RESULTS AND DISCUSSIONS
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 M 6 model and lowest for the M 1 model. The higher S value also increases the standard deviation (SD) and standard error (SE) value of the M 6 model.
The RMSE values for all models are presented in   Like the previous three indices, the n(t) value was also found to be best in most of the watersheds using the M 6 The superiority of the M 6 model over the rest of the models was judged by making a cumulative frequency distribution curve for percentage improvement (r 2 ) criteria ( Figure 5). The M 6 model performed better than the M 1 , M 2 , and M 5 models for all watersheds while for the M 3 and M 4 models, only 17 and 14% watersheds respectively performed better than M 6 . To visually assess the performance  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 M 6 model (Rank I,  (Figure 7). Therefore, on the basis of RGS, the M 6 model can be also rated as the best model. After the M 6 model, the next better performing models were found to be M 3 , M 5 , M 4 , M 2 , and M 1 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 P 5 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 P 5 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 P 5 value and provides a more rational basis to M 5 and M 6 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.

CONCLUSION
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 M 5 and M 6 models, M 5 performance was poor due to the fixed λ value. The M 5 model indicates that a fixed λ value of 0.2 does not fit well and some other value should be used. Model M 6 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 M 6 can be used with λ value as zero (Equation (11)) simplifying the M 5 model.

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