Optimum parametrization of the soil conservation service (SCS) method for simulating the hydrological response in arid basins

Abstract The use of the soil conservation service (SCS) curve number (CN) model for estimation of the rainfall excess and the SCS-unit hydrograph (UH) model are common tools for flood studies in arid regions. In this research, we are investigating the capability of these models to simulate flood events in the arid region under the common parametrization provided by the SCS model (SCS-CN, the initial abstraction ratio, λ, and UH theory) and the optimum parameterization for best simulating the hydrologic response. A case study is performed in Al-Lith basin in the west of Saudi Arabia (SA). The study simulates measured rainfall-runoff events in the area using seven scenarios (various SCS-CN estimation methods: least-squares method (CNLSM), asymptotic fitting method (CN∞), SCS-CN tables (CNdesign), and antecedent moisture content CN (CNI, CNII, and CNIII), λ = 0.2 and 0.01, and SCS-UH and UH derived from streamflow data) and a comparison is made between the observations and model results under the common parameterization of the SCS model and parameterization estimated from the Saudi arid environment. The comparison between simulated and observed peak flow and runoff volume of the studied events shows high scatter which is a common feature in arid regions due to the inherent uncertainties in the hydrological processes which are not yet resolved due to the lack of detailed measurements of the rainfall-runoff processes. Statistical analysis showed that λ = 0.01 provides a minimum root mean square error (RMSE) in the peak flow (24.8 m3/s) and the runoff volume (0.31 million m3) with CNLSM obtained by LSM. CN∞ is bad to simulate the hydrologic response. The SCS-CN Tables cannot be used for hydrological simulation. They can rather be used for the design purposes of mitigation structures. HIGHLIGHTS The high scatter between observations and simulations in arid regions is due to the inherent uncertainties in the hydrological processes that are not yet resolved. The asymptotic CN (CN∞) is not a good estimation for curve numbers to simulate the hydrologic response. The SCS-CN Tables cannot be used for hydrological simulation. They can rather be used for the design purposes of mitigation structures. The dry condition (AMC I) is the best for the hydrological simulation in arid regions since it is the predominant condition in the region.


Introduction
Rainfall-runoff processes in arid regions are not yet fully understood. The reason for that is the lack of rainfall-runoff data in these regions due to the rare rainfall events and high cost of installation and maintenance of rainfall-runoff gauging stations, soil moisture measurements, etc. However, when rain falls in arid basins, it often leads to flash floods and cause loss of lives and severe damage to the infrastructures. Saudi Arabia is among many countries worldwide that frequently face such flash floods (; Elfeki and Bahrawi 2017;Kamis, et al. 2018;Abdulrazzak et al. 2018aAbdulrazzak et al. , 2018bMarko et al. 2019;Azeez, et al. 2020). Therefore, a deep understanding of measured rainfallrunoff events in the arid basins is essential to develop models that suit the arid and semi-arid regions in general and the Saudi arid environment in particular. The common model used to analyze rainfall-runoff processes and design of flood mitigation structures is the Hydrologic Engineering Center-Hydrological Modelling System (HEC-HMS) developed by the US Army Corps of Engineers (US Army Corps of Engineers HEC-HMS). Enormous studies are available in the literature based on HEC-HMS since it is being a public domain software and has strong publicity (e.g. Basahi et al., 2016;Masoud 2016;Al-Wagdany et al. 2020;Bamufleh et al., 2020).
Most of the hydrological studies in Saudi Arabia and even worldwide are using the HEC-HMS software, which has many options for the application of the unit hydrograph (UH) theory, the time lag equations, and the loss models (USDA-SCS 1985). One of the most used UH is the SCS-UH model which is widely applied in many catchments worldwide (e.g. Oleyiblo and Li 2010;Martin et al. 2012, Olayinka & Irivbogbe 2017, Din et al. 2019, and Hamdan et al. 2021. Also, the HEC-HMS software has many loss models to estimate the initial abstraction and infiltration losses and therefore predicts the direct runoff at the catchment outlet (the storm flood hydrograph of a given rainfall storm).
One of the commonly used models in loss estimation is the SCS Curve Number method (USDA-SCS 1985). The SCS-CN model is empirically derived from daily observations, and its application at a sub-daily scale is not accurate, however, it has been applied for various time scales (Garen, et al. 2005, Woodward et al. 2010. Abundant studies are devoted to the SCS-CN method due to its popularity and simplicity in rainfall-runoff modelling. Mishra and Singh (2013) reviewed many research work on the topic. Some of the studies are given below. Hawkins (1979, and1993), and Hjelmfelt et al. (2001) studied the relationship between SCS-CN value and the rainfall depth (P) using rainfall-runoff data from experimental watersheds in the United States. They distinguished three CN-P relationships namely: complacent, standard, and violent behaviors. Hawkins et al. (1985) studied the antecedent moisture content (Dry, Normal, and Wet) on SCS-CN estimation and relating that to that to probability (10%, 50%, and 90%) respectively. Mishra et al. (2004) evaluated the antecedent moisture content of the SCS-CN model, the modified SCS-CN model proposed by the authors performs far better than the existing SCS-CN model. Soulis et al. (2009) investigated the applicability of the SCS-CN method for direct runoff generation mechanism in a Mediterranean experimental watershed in Greece. Their analysis showed that the SCS-CN method fails to predict runoff and a linear rainfallrunoff formula provides better results than the SCS-CN method. Soulis and Valiantzas (2012) developed a two-CN system approach from the traditional SCS-CN method for parameter determination of a heterogeneous watershed in Greece. The results indicate that the determination of CN values from rainfall-runoff data using the proposed two-CN system approach overperforms the method that is based on a single asymptotic CN value. Soulis and Valiantzas (2013) continued their previous work (Soulis and Valiantzas, 2012) to identify the SCS-CN spatial distribution in heterogeneous watersheds based on rainfall-runoff data. Their methodology proposed the estimation of CN values for specific soil and cover complexes in heterogeneous watersheds, which may facilitate future studies aiming at the adaptation and the documentation of the SCS-CN method in various regions and for various land use. Psomiadis, et al. (2020) performed a comparative evaluation of the impacts of urbanization, forest fires, and their combined effect on runoff response using earth observation and the SCS-CN method. They concluded that the SCS-CN method proved to be a valuable tool that allows the calculation of the direct runoff for each soil, land cover and land management complex in a simple but efficient way.Most of the work in the literature, as stated above, on the SCS-CN method considers the SCS-CN parameter and the initial abstraction as the parameters for the validation of runoff depth (Hjelmfelt, 1980, Bonta, 1997, Hawkins, et al. 2008, Soulis et al. 2009, Soulis et al. 2012, Farran and Elfeki, 2020a, b, c, Farran, et al. 2021). However, the use of SCS-CN, the initial abstraction, and UH as validation parameters in a full simulation of the hydrologic response of a basin with its subbasins (semi-lumped) is not considered in arid regions. This paper addresses this point together with other important issues in rainfall-runoff modeling in the arid region of Saudi Arabia.
The main objective of this research is to investigate the applicability of the SCS-CN method for flood predictions in Saudi Arabia. More specifically, the following research questions need to be answered: 1. Is the SCS-UH applicable in simulating the hydrologic response in Saudi Arabia's arid basins?
2. What are the discrepancies in the predictive results between the SCS-UH theory and the UH derived from streamflow data of the basin? 3. What are the parameters to be optimized in the SCS-CN method as a loss model, for predicting the runoff hydrograph at the basin outlet? 4. Is the time lag equation of the SCS method can be considered as the appropriate equation for estimating the actual time lag in the observation? 5. What is the difference between using SCS-curve number tables (USDA-SCS, 1985) and the curve number obtained from rainfall-runoff data (by inverse method) for simulating the hydrological response of the basin? 6. How would the antecedent moisture content influence the flood predictions?
Which antecedent moisture content is appropriate in simulating the hydrologic response in arid basins?
To answer the above questions, the SCS method is applied to a representative basin in the southwestern part of Saudi Arabia where there are rainfall-runoff measurements to get the optimum parameterization of the method for the application in the Saudi arid environment. Seven scenarios are designed to investigate the aforementioned research questions. The scenarios are explained later in the methodology section.
There are two main new contributions besides some secondary ones in the current research. The first issue is the comparison between the use of the SCS-UH theory and UH derived from the field observations in arid basins and the consequences on the storm hydrograph of the events. This issue is hardly made in arid regions especially in the Saudi arid environment since the lack of runoff data. The second main issue is the optimum parametrization of the SCS-CN method in the arid region in general and in the Saudi environment in particular. The engineering practice in KSA normally uses the SCS-CN tables (USDA-SCS 1985). However, these tables are not suited to Saudi Arabia. Therefore, we applied several results obtained from the recent work Elfeki (2020a, 2020b) and Farran et al. (2021) to estimate the errors between using the SCS-CN tables and the new developments of Elfeki (2020a, 2020b) and Farran et al. (2021).
It is also worth mentioning that the recent studies by Elfeki (2020a, 2020b) and Farran et al. (2021) considered the effect of SCS-CN on runoff depth only. However, in the current study, we focused on the hydrograph as a whole through the application of a flood routing scheme between the sub-basins and considering the hydrograph's significant parameters (peak flow and runoff volume). To the best of the authors' knowledge, there are no such studies made in the literature.

Study area
Wadi Al-Lith basin is situated in the middle portion of the Tihamah plain (Red Sea Coastal Plain). It runs from Hijaz mountains series in the East to the Rest Sea in the west at Al-Lith governorate. Wadi Al-Lith basin is restricted between 40 10 0 and 40 50 0 longitudes and 20 00 0 and 21 05 0 latitudes with a catchment area is about   3262 Km 2 , and basin length is about 111.0 Km as shown in Figure 1. Wadi Al-Lith basin is composed of many sub-basins. Bajabaa et al. (2014) classified it into 10 subbasins. This study focused on four subbasins of Al-Lith basin where the Saudi Arabia Dames and Moore (1988) company has performed a detailed study of the basin with dense rainfall and runoff gauges. Figure 2a shows the shape of the basin and its subbasins. Wadi Al-Lith basin is receiving a moderate amount of rainfall with an average annual of 100 mm over 55 years of record at the Red Sea coastal zone (downstream of the basin) to 165 mm at the upstream portion of the basin as shown in Figure 2b.

Geomorphology and geology of the study area
Geomorphologically, Wadi Al-Lith basin could be classified into three zones as shown in the Digital Elevation Model "DEM" (Figure 3a) as follows: High elevated zone, this zone represents the upper portion where the initial orders of the streams are initiated and slopes from the Al Hijaz Mountains series of elevation ranges from 2600 m to 1000 m above mean Sea level (AMSL). This zone is characterized by a very rugged region, steep slopes, and more rainfall than the rest of the basin. Medium elevated zone: this zone represents the middle part of the basin (named as, which includes the main tributaries. This zone is characterized by a mediumrange of elevation which ranges from 1000 m to 500 m AMSL with a medium gradient. In this zone, the main sub-basins and the main valley begin to form. Low land zone: this zone represents the flooding plain area, which is characterized by low elevation values ranging from 500 m to 0 m AMSL. This zone considers the main sedimentation process and is characterized by Quaternary deposits which are covered by the seasonal floods and consider the main shallow aquifer in this basin. Geologically, Wadi Al-Lith basin is apart from the Arabian Shield and composed of Precambrian rocks, Tertiary and Quaternary deposits as shown in (Figure 3b). The Precambrian rocks are covering most of the basin (about 80% of the total area of the basin, as shown in Figure 4b, which is composed of igneous and metamorphic rocks. The Precambrian rocks included Basih group (Pallister 1986), which is composed of basalt, volcanic, and some granite, diorite, and gabbro intrusions. The tertiary rocks included some exposures of dikes and magmatic volcanic rocks in the center of the  downstream basin. The Tertiary rocks consist of mafic-felsic minerals with transitional sub-alkaline to alkaline minerals (Pallister 1986). The Quaternary deposits which cover about 19% of the study basin composed of Wadi deposits which consist of gravel, coarse sand, fine sand, mud, and sabkha deposits.
The storm data was collected from the measurements by Saudi Arabia Dames and Moore (1988) company reports. Table 1 shows the storms considered in the analysis and the locations of the runoff stations ( Figure 2a) that recorded floods due to the corresponding rainfall. The rainfall data is recorded from a dense network distributed in the basin. The weighted average of the rainfall depth is considered in the hyetographs that are estimated over a time step of one hour. The details of some storms are presented in Figure 4. Figure 4 shows maps of samples of the rainfall-runoff events on the subbasins. This data is considered in the analysis of the simulation study.

Methodology
In this section, the methodology used in the analysis to achieve the goals of the article can be summarized as: 1. Estimation of the unit hydrograph of Al-Lith basin from streamflow data. 2. Brief introduction of the SCS theory (USDA-SCS 1985) used in HEC-HMS. 3. The curve number (CN) estimation method available in the literature for gauged and ungauged basins i.e., least squares method, asymptotic fitting method, and SCS-CN table method. 4. The effect of antecedent moisture content is also considered. 5. Performing hydrological simulation scenarios to achieve the goals.
The following scenarios are studied to achieve the goals of the article. Table 2 summarizes these scenarios and the values of CN and k, the method of UH and the Tlag estimation method used in each simulation.
Scenario 1: hydrological simulation with UH of Al-Lith basin derived from streamflow data based on the work by Albishi, et al. (2017). Scenario 2: hydrological simulation with SCS-UH using observed lag time. Scenario 3: hydrological simulation with SCS-UH using estimated lag time from SCSlag time equation. Scenario 4: hydrological simulation with UH of Al-Lith basin derived from streamflow data using the asymptotic CN (CN 1 ) based on natural sorting of data pairs (P: R) based on the work by Farran and Elfeki (2020b). Scenario 5: hydrological simulation with UH of Al-Lith basin derived from streamflow data using asymptotic CN (CN 1 ) based on ordered sorting of data pairs (P: R) based on the work by Farran and Elfeki (2020b). Scenario 6: hydrological simulation with UH of Al-Lith basin derived from streamflow data using CN design values based on the work by Farran et al. (2021) Scenario 7: hydrological simulation with UH of Al-Lith basin derived from streamflow data using AMC conditions of the CN values CN I , CN II , and CN III based on the work by Farran and Elfeki (2020c). The measure of the model performance in each scenario is made through: the comparison between the observed and predicted hydrographs of three events that were not used in the derivation of the UH of the basin, A visual measure by a scatter plot between the observed and predicted peak flow and runoff volume of the events under the various scenarios. A statistical measure using the correlation coefficient, r, between the observed and predicted peak flow and runoff volume, and the root mean square error, RMSE, for each scenario.
The aforementioned steps are explained in detail in the following subsections and Figure 5 summarizes the methodology in a flowchart.

Unit hydrograph of Al-Lith basin from streamflow data
The unit hydrograph (UH) of the Al-Lith basin is derived from the streamflow data (Albishi, et al. 2016). Runoff events with a single peak are selected to calculate the UH. The methodology to calculate UH from streamflow data are presented in Albishi, et al. (2016), and Albishi, et al. (2017). Figure 6 shows the one-hour unit hydrograph derived from seven rainfall-runoff storms that occurred in the four subbasins (J-415, J-416, J-417, and J-418) (Figure 6a), and the mean UH is averaged over the storms. The mean UH is transferred to a dimensionless UH (DUH) by dividing the time by time to peak (t/t p ) and the discharge by the peak discharge (Q/Q p ). Figure 6b shows a comparison between the SCS dimensionless UH (SCS DUH) and the DUH from stream flow data. Both hydrographs are used to study the suitability of SCS-UH to model the hydrologic response of the Saudi arid environment and regions alike. Figure 6c shows the one-hour UH of each subbasin used in HEC-HMS software as user-defined UH. In the case of using the SCS-UH in HEC-HMS, the time lag must be estimated. Therefore, two simulations are implemented: (1) using the observed lag time from the events, and (2) using the lag time from the equation of the SCS method as given below, where, Tlag is the lag time in hours, L is the hydraulic length of the catchment in km, CN is the SCS curve number, and Y is the average catchment slope in percent.

The SCS methodology
The SCS-CN equations are introduced by the USDA-SCS (1985) as, where is P is the precipitation depth, R is the direct runoff depth, k is the initial abstraction ratio, which is commonly used as 0.2 (USDA-SCS 1985), and S is the soil maximum potential retention within the basin, which is given by, where, CN is the curve number.
In the case of gauged basins where P and R are known, one may estimate CN for a given value of k by combining Equations (2) and (3) to get, CN as, In the case of ungauged basins, CN is commonly estimated using the CN Tables from USDA-SCS (1985), and then R is estimated giving k ¼ 0.2 as a common value.

Methods of SCS-curve number estimation
The least square method (LSM) Hawkins et al. (2008) used the LSM to estimate CN in gauged basins. The method determines the parameters k and CN that minimizes the difference between the observed direct runoff and the estimated runoff. The CN based on the LSM is called CN LSM . The objective function that needs to be minimized is given by, where P i and R i are the observed rainfall and runoff depths respectively, of an event i (mm), and n is the no. of events. Farran and Elfeki (2020a) have used the LSM to obtain CN LSM and k for many basins and their sub-basins and found out that the best value for k is 0.01. In the current study, both the value of k ¼ 0.01 and the common value of k ¼ 0.2 (USDA-SCS 1985; Mishra and Singh 2013) are used for comparison. The values used in the simulation are presented in Table 2 Scenarios 1, 2, and 3.
The values of CN based on k ¼ 0.2 are estimated by substituting the value of k ¼ 0.2 in Equation (4) for the given values of P and R and therefore the CN is evaluated simply as: Asymptotic fitting method (AFM) The asymptotic fitting method (AFM) is developed by Hawkins (1993) to study the storm precipitation behavior (complacent, standard, and violent) via plotting the precipitation and CN data (the so-called CN-P relationship) and fitting an exponential model to the data given by, where, CN 1 and b are fitting parameters. The CN 1 and b are estimated by solving a minimization problem to find out the best values that fit the data (CN-P relationship).
The CN approaches an asymptotic value called CN 1 as the precipitation approaches infinity ðP ! 1Þ: Farran and Elfeki (2020b) applied this method to some basins in KSA and Al-Lith basin was included in their study and derived the CN 1 of the Al-Lith basin. Table 3 shows the equations of the standard behavior of Al-Lith basin for estimation of CN 1 under natural sorting of P:R (physical or cause and effect, i.e., the runoff is produced from the corresponding rainfall event) and ordered sorting of P:R (statistical or frequency-based, that is, runoff doesn't need to be produced from the corresponding rainfall event) . Both approaches are available in the literature (see e.g. Hjelmfelt 1980, Hawkins 1993Bonta 1997;Sh. Chandramohan 2000) for more details.
In the current study, a comparison is made between the CN values based on ordering and natural sorting of data on the hydrological response of the basin at both k is 0.2 and 0.01. The values used in the simulation are presented in Table 2 Scenarios 4 and 5.

SCS-CN tables
For ungagged basins, the traditional approach to estimate CN is based on the SCS-CN table (NRCS, 2004). This approach is mainly used for the design of flood protection schemes; therefore, we call the CN based on this approach CN design . The SCS-CN table values are given based on the abstraction ratio, k ¼ 0.2 (NRCS, 2004). Farran et al. (2021) applied remote sensing and geographical information system (RS-GIS) techniques for the classification of the land cover of the basins into alluvium, rock, and vegetation and estimated of the proportion of each of these classes in the basins. They were able to derive CN design from the traditional SCS-CN table and also were able to derive SCS-CN table based on k ¼ 0.01. Therefore, in the current study, we compared the results of CN design based on k ¼ 0.2 and 0.01. The values used in the simulation are presented in Table 2 Scenario 6.

Antecedent moisture content conditions for curve number (AMC-CN)
The equations of AMC-CN for dry (AMC I) and wet (AMC III) conditions and at different values of k for the Saudi arid environment are developed by Farran and Elfeki (2020c) from field observations of five basins in KSA. These equations are provided in Table 4. The CN I is for dry condition, CN II is for the normal condition, and CN III is for wet condition. The values for the three conditions are available in Table  2, Scenario 7 and used in the hydrological simulations for comparison reasons to find out the best condition that fits the runoff data.

Results and discussions
In this section, the results of the methodology are presented for the seven considered scenarios. Three rainfall-runoff events are simulated (see Table 1). The events used in the simulations are different from the events used in the derivation of the UH to avoid any bias on the results. The evaluation of the simulation results is presented by comparing the predicted hydrographs and the observed ones. Also, visual measures are used by making a scatter plot between observed and predicted peak flow and runoff volume together with the statistical measure using the correlation coefficient (r) and root mean square error (RMSE) as shown in Table 5.
Scenario 1: hydrological simulation with UH of Al-Lith basin derived from streamflow data and CN LSM : In this scenario, a comparison is made between the hydrological simulation with UH derived from streamflow data of Al-Lith basin and using CN LSM at k ¼ 0.2 and 0.01 respectively. The results in terms of predicted and simulated hydrographs of the rainfall event are presented in Figure 7 leftmost column for a sample event (Event 2, 14th of May 1985) at stations J-415 top image, J-417 middle image, and J-418 bottom image. The visual inspection shows that the simulation at k ¼ 0.01 is better than at k ¼ 0.2. The comparison of the three events in terms of peak flow and runoff volume is displayed in the scatter plot in Figure 8 (leftmost column) with the line of a perfect fit at 45 . The figure shows a general scatter of the points which are common in arid regions (e.g. Farran et al., 2021). The reason for such scatter is the inherent uncertainties in the hydrological processes in arid regions which are not yet resolved. However, the scatter is less for at k ¼ 0.01 (triangular symbols) when compared with at k ¼ 0.2 (circular symbols). Also, the peak flow and the runoff volume are overestimated at k ¼ 0.2. The statistical measures presented in Table 5 shows that RMSE ¼ 38.91 m 3 /s and r ¼ 0.33 for peak flow at k ¼ 0.01 while it is RMSE ¼149.85 m 3 /s and r ¼ 0.12 at k ¼ 0.2. For runoff volume, RMSE ¼ 0.34 MCM and r ¼ 0.57 at k ¼ 0.01 while the RMSE ¼1.42 MCM and r ¼ 0.36 at k ¼ 0.2. These discrepancies are because k ¼ 0.01 is obtained based on the LSM technique as explained in the methodology (Farran and Elfeki 2020a) however, k ¼ 0.2 is the common value in the SCS method that is derived for basins that are not arid or semi-arid. Therefore, it is recommended to use k ¼ 0.01 in the Saudi arid environment and regions alike.
Scenario 2: hydrological simulation with SCS-UH using observed lag time and CN LSM : In this scenario, a comparison is made between the hydrological simulation with SCS-UH, observed lag time from rainfall-runoff data, and using CN LSM at k ¼ 0.2 and 0.01, respectively. The results in terms of predicted and simulated hydrographs of the rainfall event are presented in Figure 7 middle column for a sample event (Event 2, 14th of May 1985) at stations J-415 top image, J-417 middle image, and J-418 bottom image. The visual inspection shows that the simulation at k ¼ 0.01 is better when compared with at k ¼ 0.2. The comparison of the three events in terms of peak flow and runoff volume is displayed in the scatter plot in Figure 8 (middle column) with the line of a perfect fit at 45 . One may still notice that the points at k ¼ 0.01 (triangular symbols) are less scatted when compared with k ¼ 0.2 (circular symbols). Also, the peak flow and the runoff volume are overestimated at k ¼ 0.2. Table 5 shows that RMSE ¼ 24.81 m 3 /s and r ¼ 0.68 for peak flow at k ¼ 0.01 while it is RMSE ¼53.71 m 3 /s and r ¼ 0.68 at k ¼ 0.2. For runoff volume, RMSE ¼ 0.31 MCM and r ¼ 0.6 at k ¼ 0.01 while the RMSE ¼1.41 MCM and r ¼ 0.39 at k ¼ 0.2. The results of this scenario indicate that the use of SCS-UH with measured Tlag from the observation and CN LSM can provide the minimum RMSE and the highest r among other scenarios. The results still suggest the use of k ¼ 0.01 for the simulation of the hydrologic response in the Saudi Arabian arid basins Scenario 3: hydrological simulation with SCS-UH using estimated lag time from SCSlag time equation and CN LSM : In this scenario, a comparison is made between the hydrological simulation with SCS-UH, lag time estimated from the SCS-lag time equation (Equation (1)) and using CN LSM at k ¼ 0.2 and 0.01, respectively. The results in terms of predicted and simulated hydrographs of the rainfall event are presented in Figure 7 rightmost column for a sample event (Event 2, 14th of May 1985) at stations J-415 top image, J-417 middle image, and J-418 bottom image. The visual inspection shows that the simulation is poor, however at k ¼ 0.01 still looks better than k ¼ 0.2. The comparison of the three events in terms of peak flow and runoff volume is displayed in the scatter plot in Figure 8 (rightmost column) with the line of a perfect fit at 45 . For peak Figure 9. Comparison between the generated storm hydrographs using CN 1 (based on natural and ordered sorting of data pairs P: R) at k ¼ 0.2 and 0.01 and observed hydrograph based on UH of Al-Lith basin derived from streamflow data: AMC I (leftmost column), AMC II (middle column), AMC III (rightmost column) for Event 2 (11th April 1985) at stations J-415 top row, J-417 middle row, and J-418 bottom row. flow results, both k ¼ 0.01 and 0.2 are highly scattered when compared with the previous two scenarios. It can also be seen that at k ¼ 0.01 the peak flow is underestimated, while for k ¼ 0.2 the peak flow is overestimated. For runoff volume, it is noticeable that the points at k ¼ 0.01 (triangular symbols) are less scatted when compared with k ¼ 0.2 (circular symbols). Table 5 shows that RMSE ¼ 39.48 m 3 /s and r ¼ 0.55 for peak flow at k ¼ 0.01 while it is RMSE ¼38.37 m 3 /s and r ¼ 0.61 at k ¼ 0.2. For runoff volume, RMSE ¼ 0.36 MCM and r ¼ 0.4 at k ¼ 0.01 while the RMSE ¼1.39 MCM and r ¼ 0.37 at k ¼ 0.2. The results of this scenario indicate that applying SCS-UH with k ¼ 0.01 could simulate the hydrologic response better than at k ¼ 0.2 when using the SCS T lag equation.
Scenarios 4 and 5: hydrological simulation with UH of Al-Lith basin derived from streamflow data using the asymptotic CN (CN 1 ) based on natural sorting (Scenario 4) and ordered sorting (Scenario 5) of data pairs (P: R): Figure 10. Scatter plot between observed and simulated peak flow (top row) and volume (bottom row) using CN 1 for natural (left) and ordered (right) sorting of data pairs (P: R) for the three rainfall-runoff events for k ¼ 0.2 and 0.01 at stations J-415, J-416, J-417, and J-418. In this scenario, a comparison is made between the hydrological simulation with UH of Al-Lith basin derived from streamflow data and the asymptotic CN for both natural and ordered sorting of data Pairs (P: R) respectively at k ¼ 0.2 and 0.01. The results in terms of predicted and simulated hydrographs of the rainfall event are presented in Figure 9 left column (natural sorting) and right column (ordered sorting) for a sample event (Event 2, 14th of May 1985) at stations J-415 top image, J-417 middle image, and J-418 bottom image respectively. The visual inspection shows that the simulation is relatively poor, however at k ¼ 0.01 still looks better than at k ¼ 0.2. The comparison of the three events in terms of peak flow and runoff volume is displayed in the scatter plot in Figure 10 (left column for natural sorting and right column for ordered sorting) with the line of a perfect fit at 45 . For peak flow results, both k ¼ 0.01 and 0.2 are highly scattered when compared with the previous scenarios. Table 5 shows for Scenario 4 (natural sorting of data pairs; P:R), the RMSE ¼ Figure 12. Scatter plot between observed and simulated peak flow (top row) and volume (bottom row) using CN desgin for the three rainfall-runoff events with k ¼ 0. 2 and 0.01, at stations J-415, J-416, J-417, and J-418. 36.03 m 3 /s and r ¼ 0.26 for peak flow at k ¼ 0.01 while the RMSE ¼39.10 m 3 /s and r ¼ 0.13 at k ¼ 0.2. For runoff volume, RMSE ¼ 0.35 MCM and r ¼ 0.37 at k ¼ 0.01 while the RMSE ¼0.38 MCM and r ¼ 0.26 at k ¼ 0.2. For Scenario 5 (ordering sorting of data pairs; P: R), the RMSE ¼ 29.34 m 3 /s and r ¼ 0.51 for peak flow at k ¼ 0.01 while the RMSE ¼41.5 m 3 /s and r ¼ 0.14 at k ¼ 0.2. For runoff volume, RMSE ¼ 0.34 MCM and r ¼ 0.33 at k ¼ 0.01 while the RMSE ¼0.38 MCM and r ¼ 0.27 at k ¼ 0.2. The results of these two scenarios indicate that the use of ordered sorting of data pairs provides the best simulation at k ¼ 0.01 since it gives a relatively low RMSE ¼ 29.34 m 3 /s and a relatively high r ¼ 0.51 for peak flow and the RMSE ¼ 0.34 MCM and r ¼ 0.33 for the runoff volume. The results indicate that the asymptotic CN is not a good estimation for curve number to simulate the hydrologic response in general. However, the use of k ¼ 0.01 could in some sense improve the simulation results. In this scenario, a comparison is made between the hydrological simulation with Al-Lith UH, and using CN estimated from SCS-Tables at k ¼ 0.2 (USDA-SCS, 1985), and CN estimated from the modified SCS-Tables at k ¼ 0.01 (Farran, et al. 2021) respectively. The results in terms of predicted and simulated hydrographs of the rainfall event are presented in Figure 11 for the three events (Event 1; 25th of November 1984 left-most column, Event 2; 14th of May 1985 middle column, and Event 3; 11th of April 1985 right column) at stations J-415, J-416, J-417, and J-418 from top to bottom respectively. The visual inspection shows that the simulation overestimates the observation at both k ¼ 0.01 and k ¼ 0.2. The comparison of the three events in terms of peak flow and runoff volume is displayed in the scatter plot in Figure 12 (top: peak flow and bottom: runoff volume) with the line of a perfect fit at 45 . The points at k ¼ 0.01 (triangular symbols) and k ¼ 0.2 (circular symbols) show overestimation in the predictions. Table 5 shows that RMSE ¼ 122.38 m 3 /s and r ¼ 0.30 for peak flow at k ¼ 0.01 while the RMSE ¼162.35 m 3 /s and r ¼ 0.17 at k ¼ 0.2. For runoff volume, RMSE ¼ 1.21 MCM and r ¼ 0.56 at k ¼ 0.01 while the RMSE ¼1.57 MCM and r ¼ 0.46 at k ¼ 0.2. Both cases show relatively high RMSE and poor correlation, however, k ¼ 0.2 is even worse than k ¼ 0.01. This means that SCS-CN tables at k ¼ 0.2 or its modification at k ¼ 0.01 cannot be used for hydrological simulation. They can rather be used for design purposes of mitigation structures since they can give a reasonable overestimation for safer design.  In this scenario, a comparison is made between the hydrological simulation with UH of Al-Lith basin and using CN estimated at different AMC (Table 4) with k ¼ 0.2 and 0.01 respectively. The results in terms of predicted and simulated hydrographs of the rainfall event are presented in Figure 13 for the event (14th of May 1985) at stations J-415 top row, J-417 middle row, and J-418 bottom row respectively. The visual inspection shows discrepancies between the observations and the simulations at both k ¼ 0.01 and k ¼ 0.2. There is an obvious increase in the overestimation when the AMC changes from AMC I to AMC III. The reason for this increase is that CN increases from CN I to CN III and consequently the flood hydrograph increases due to less infiltration. The comparison of the three events in terms of peak flow and runoff volume is displayed in the scatter plot in Figure 14 (top row: peak flow and bottom row: runoff volume at AMC I, AMC II, and AMC III respectively from left to right) with the line of a perfect fit at 45 . The points at k ¼ 0.01 (triangular symbols) and k ¼ 0.2 (circular symbols) show an increase in the overestimation in the predictions from AMC I to AMC III (the data points tend to move to the right of the line of perfect fit). Table 5 shows that the minimum RMSE ¼ 34.6 m 3 /s and r ¼ 0.28 for peak flow at k ¼ 0.01 for AMC I, while the maximum RMSE ¼256.28 m 3 /s and r ¼ 0.17 at k ¼ 0.2 for AMC III. For runoff volume, the minimum RMSE ¼ 0.33 MCM and r ¼ 0.39 at k ¼ 0.01 while the maximum RMSE ¼2.62 MCM and r ¼ 0.5 at k ¼ 0.2 for AMC III. The results for AMCII are in between the two results. The results indicate that the dry condition (AMC I) is the best for the hydrological simulation in arid regions since it is the predominant condition in the region Elfeki 2020a, 2020c). Therefore, for the hydrological simulation, it is recommended to use AMC I. However, for the design of flood mitigation structures, AMC II would perhaps be appropriate and safer.

Conclusions
The following conclusions can be obtained from the current study: The comparison between simulated and observed peak flow and runoff volume of the studied events shows high scatter which is a common feature in arid regions and is supported by earlier studies (e.g. Bamufleh et al., 2020;Farran et al. 2021). The reason for such scatter is the inherent uncertainties in the hydrological processes in arid regions which are not yet resolved due to the lack of detailed measurements of the rainfall-runoff processes. Statistical analysis showed that k ¼ 0.01 provides a minimum root mean square error (RMSE) in the peak flow (24.8 m 3 /s) and in the runoff volume (0.31 million m 3 ) with CN LSM obtained by the method of least squares, which minimizes the difference between the observed and estimated runoff depths (Farran and Elfeki 2020a). Therefore, the value of k ¼ 0.01 is the best to simulate the hydrological response in the Saudi arid environment instead of the common value of k ¼ 0.2. The use of SCS-UH with k ¼ 0.01 could simulate the hydrologic response better than at k ¼ 0.2 when using the SCS Tlag equation. The asymptotic CN is not a good estimation for curve numbers to simulate the hydrologic response in general. However, the use of k ¼ 0.01 could, in some sense, improve the simulation results under the ordered sorting of data pairs. Since it is based on minimizing the difference between the observed and the estimated CN in the CN-P relationship (Farran and Elfeki 2020b). The SCS-Tables at k ¼ 0.2 (USDA-SCS 1985) or its modification at k ¼ 0.01 (Farran, et al. 2021) cannot be used for hydrological simulation of rainfall-runoff events. However, they can rather be used for design purposes of mitigation structures since they can perhaps provide a reasonable overestimation for safer design. The dry condition (AMC I) is the best for the hydrological simulation in arid regions since it is the predominant condition in the region. This is supported by earlier studies Elfeki 2020a, 2020c). Therefore, it is recommended for hydrological simulation in arid regions. However, for the design of flood mitigation structures, AMC II would perhaps be appropriate and safer.
It is recommended to apply these scenarios and other ones on other basins with more data to get a deep understanding of the rainfall-runoff modeling in the arid region. Also, a physically-based method such as Green-Ampt equation for calculating excess rainfall could be tested. For future directions, one may consider the use of stochastic rainfall generators (De Luca and Petroselli, 2021) for rainfall-runoff modeling when data is scarce. Also, the problems dealing with climate change projections, both in terms of rainfall and in terms of land cover changes (Wheater, et al. 2005) are foreseen.