water

Article Derivation of Soil Moisture Recovery Relation Using Soil Conservation Service (SCS) Curve Number Method

Jungho Kim 1,2, Lynn E. Johnson 1,2, Rob Cifelli 2, Jeongho Choi 3,* and V. Chandrasekar 1

1 Cooperative Institute for Research in the Atmosphere (CIRA), Colorado State University, Fort Collins, CO 80523, USA; [email protected] (J.K.); [email protected] (L.J.); [email protected] (V.C.) 2 NOAA Earth System Research Laboratory, Physical Sciences Division, Boulder, CO 80305, USA; [email protected] 3 Department of Mechatronics Engineering, Chosun College of Science and Technology, Gwangju 61453, Korea * Correspondence: [email protected]; Tel.: +82-31-838-7801



Received: 1 May 2018; Accepted: 20 June 2018; Published: 23 June 2018


Abstract: Soil moisture retention (SMR) capacity plays a key role in estimating the direct runoff when a multi-pulse storm event occurs. It is very important to know how much SMR will be recovered during the intervals of no rain of a multi-pulse storm. This study developed a new approach for derivation of the SMR recovery curve (R-curve) at sub-daily time-scales using the Curve Number (CN) method. The methodology was applied using complex storm events in the Napa River basin, California. The R-curve is classified into three sections depending on the recovery rate of SMR during the inter-storm interval of no rain (INR), and this study defines the characteristics. The first section of the R-curve (INR 0–21 h with 0.97 mm/h) is described as gradually recovering SMR, since water is being infiltrated and the upper soil layer is not fully saturated. The second section (INR 21–36 h with 2.11 mm/h) is defined as steeply recovering S due to downward drainage (sub-surface/inter flows) and evaporation without infiltration. The third section (INR 36–68 h with 0.34 mm/h) is described as gradually decreasing recovery dependent on evaporation since percolation and drainage have almost stopped.

Keywords: soil moisture retention; recovery curve; CN method; runoff curve number; antecedent moisture condition

1. Introduction The state of soil moisture retention (SMR) plays a key role for estimating the antecedent moisture content (AMC) at storm initiation, which has a large effect on the overall runoff response [1–6]. SMR is the amount of water a soil column can retain. Continuously estimating SMR, however, is a challenge at sub-daily time resolution due to the recovery of SMR during non-rain intervals of multi-pulse storms [1,7]. The recovery is attributed to soil water drainage by subsurface flow, interflow, and evapotranspiration. Various terms have been described in the literature focusing on the recovery of infiltration capacity to account for the water balance in soil [1,7,8]. Bauer [7] proposed a modification of the Horton’s equation by introducing a soil drainage term into the equation to overcome a lack of the recovery of infiltration capacity during the interval of no rain (INR). Aron [9] suggested a modification of Soil Conservation Service Curve Number (CN) method by including a drainage rate decay factor [6]. Pitt et al. [8] conducted laboratory experiments on infiltration rates, which supported the previous two studies. Also, there are a variety of soil moisture accounting models considering percolation,

Water 2018, 10, 833; doi:10.3390/w10070833 www.mdpi.com/journal/water Water 2018, 10, 833 2 of 21 drainage, evaporation, and evapotranspiration to reflect the recovery of SMR, examples include the SMA-NWSRFS [10], SAC-SMA [2,11,12], and a number of modified CN methods [3,4,13,14]. The CN method, which is based on Horton’s overland flow mechanism [15–17], is an appropriate option to track a change of the SMR and its recovery. This method employs a soil moisture accounting concept using a relationship between AMC and SMR. The CN method was developed by the United States Department of Agriculture (USDA) Soil Conservation Service in 1954, and is now referred to as the Natural Resource Conservation (NRCS) CN method [5,18–20]. The CN method parameter, the curve number (CN), represents the surface runoff potential using a numerical value ranging from 30 to 100; it is typically based on a combination of land-use and soil type [21,22]. The CN approach is useful and efficient to find SMR as CN can be simply transferred to a potential maximum moisture retention, S, using a CN-S relationship. S conceptually represents a potential maximum amount of water that a soil column can retain, and it depends on AMC by the water balance assumption. For the strengths and weaknesses of the CN method, Mishra and Singh [6] emphasized that it is easy to understand and apply for most of the runoff reflecting the watershed characteristics, such as antecedent moisture retention condition, soil, land-cover, and hydrologic condition. However, they mentioned that it is required to consider climatic indices (e.g., evaporation and temperature) and physical factors (e.g., deep storage, recovery rate of S), which could handle a change of the Ss for long-term hydrologic simulation. Verma et al. [23] addressed its advantages for the applications of remote sensing and geographical information system for ungauged watersheds. Verma et al. [23] and Moglen et al. [24] pointed out the uncertainty of the standard initial abstraction ratio generally used in the CN approach and suggested alternatives that are based on their analysis. For more details, the readers can see Mishra and Singh [6] and Verma et al. [23], as they summarized the advantages and limitations of the CN method according to research results from many previous studies. A number of CN-based hydrologic models reflect the recovery of SMR using evaporation, evapotranspiration, and a depletion coefficient at the daily time step resolution for a long-term simulation period [4,6,13,14]. However, the temporal resolution of those studies is too coarse to identify hourly SMR trends. While being useful for hydrologic events that span multiple days, the daily temporal resolution is too coarse for short-term applications involving flash flood events that occur in less than 6 h. Moreover, it is difficult to apply the average monthly lake evaporation and depletion coefficients that are estimated from daily and monthly rainfall-runoff data [6], for multi-pulse storms occurring within a day. To overcome this limitation, a new approach to track the state of SMR at sub-daily time scales is needed. In this paper, a new approach is developed to derive a recovery curve of SMR (R-curve) for estimating time-variable retention at sub-daily time-scales. This approach adopts a simple concept based on a relation of SMR change and the interval of no rain (INR) between the rainfall pulses in a complex storm event. The S in the CN method is used as SMR. We applied the approach for the upper Napa River basin in central California using the streamflow records for the gage near St. Helena (USGS station #11456000). Sixteen rainfall events from 2011 to 2012 are applied in order to derive the R-curve. This paper has six sections, including the Introduction and Conclusion sections. Sections2 and3 provide the background for the CN method, SMR, and the methodology that is developed in this study. Section4 presents the application data. Section5 explains the results related to preprocessing and computation of S. Section6 provides an analysis results from the final R-curve and an application instance in terms of flood forecasting.

2. Curve Number Method and Soil Moisture Retention The CN method is based on a water balance equation and two fundamental assumptions [6]. In Equation (1), the water balance statement equates total rainfall (P) to the sum of the amount of direct runoff (Q), the initial abstraction (Ia), and the amount of infiltration (F). The first assumption is the ratio of the amount of actual infiltration (F) to the amount of S (Equation (2)). The second assumption relates the initial abstraction (Ia) to S (Equation (3)). Water 2018, 10, x FOR PEER REVIEW 3 of 22

Water 2018Ratio, 10 assumption, 833 : 3 of 21 QF  (2) Water balance equation: PIS a P = Ia + F + Q (1) ISa  assumption: Ratio assumption: QIS  F a = (3(2)) P − Ia S Here,  is the initial abstraction coefficient, historically 0.2 [16]. Physically, this means that for a Ia − S assumption: given storm, 20% of S is the initial abstraction before runoff begins [24]. Ia = λS (3) The CN method calculates direct runoff from a rainfall depth, as shown in Equations (4)–(6). Here, λ is the initial abstraction coefficient, historically 0.2 [16]. Physically, this means that for a (PI) 2 given storm, 20% of S is the initial abstractiona before runoff begins [24]. Q  for PI , and Q0 for PI (4) The CN method calculates direct runoff from aa rainfall depth, as showna in Equations (4)–(6). PISa

Equation (4) becomes (P − I )2 Q = a for P > I , and Q = 0 for P ≤ I (4) P − I + S a a a (P  S)2 Q  (5) Equation (4) becomes P (1 )S (P − λS)2 Q = (5) S is expressed in terms of the dimensionless CNP + taking(1 − λ ) valuesS from 0, when S , to 100, when S0 . S is expressed in terms of the dimensionless CN taking values from 0, when S → ∞ , to 100, when S = 0. 1000 S1000 10 (6a) S = − 10 (6a) CNCN This definition definition was was originally originally applied applied to tothe the English English metric metric system system (with (with S in Sinches). in inches). In the In SI the units SI units(with (withS in mm) S in, the mm), following the following definition definition should shouldbe used: be used: 25,25, 400 400 SS=− 254254 (6b (6b)) CNCN

InIn the CN CN method, method, S Srepresents represents a storage a storage capability capability of water of water infiltrated infiltrated into soil into [9 soil]. Figure [9]. Figure1 shows1 showsa physical a physical (a) soil (a) profile, soil profile, (b) soil (b)- soil-water-airwater-air layer, layer, and and (c) (c) a a conceptual conceptual diagram diagram of of soil soil moisture,moisture, accounting to understand the meaning of S using volumetric terms since S depends on the volumesvolumes of thethe soil, water, water, and and air air layer layer and and soil soil textures textures in in the the soil soil profile. profile.

Soil Profile Soil-Water-Air Soil Moisture n (a) (b) (c) o i Layer Accounting t a r t l i f

n Surface soil I V1

n Root Air Va o i

t S a ) l 1

o V ( Topsoil V2 v c r e P V V Water V Sabs n w o i t a ) l 2 o ( c

r Subsoil V3 A e

P Water M Soil Vs C

Bedrock

Figure 1. Schematic diagram of soil profile, soil-water-air layer, and a concept of soil moisture accounting.Figure 1. Schematic In this diagram figure, of V asoilis profile, the volume soil-water of air,-air and layer V, wandis a the concept volume of soil of water.moisture The accounting sum of. airIn this and figure, water representsVa is the volume as the of total air, volume and Vwof is voids,the volume Vv. S absof water.is the absoluteThe sum capacityof air and of waterS. V inrepresents Figure1

is the total volume: (a) Generalv soil profile; (b) Structure of soil-water-air layer; and, (c) Conceptual as the total volume of voids, V . Sabs is the absolute capacity of S . V in Figure 1 is the total volume: (a) diagram of soil moisture accounting. General soil profile; (b) Structure of soil-water-air layer; and, (c) Conceptual diagram of soil moisture accounting.

Water 2018, 10, 833 4 of 21

In general, a soil column consists of surface soil, topsoil, subsoil, and bedrock, as shown in Figure1a. Depending on the depth of each soil layer and their soil type, S of each soil layer may differ. However, for the CN method, S indicates the total capacity of SMR in a soil column because the method does not consider the soil profile with various soil layers. In the soil column, infiltration and percolation rates are different since their mechanism differ, and they depend on AMC and air entrapment [25,26]. Also, the percolation rates from surface soil to topsoil and from topsoil to subsoil are different depending on the soil texture and the depth in each layer [27,28]. Generally, the upper layer has a loose texture, while the lower layer has a relatively lower hydraulic conductivity [29–31]. Therefore, a retardation effect on the boundary between the upper and lower layers could occur. Each soil layer in the soil profile is classified among solids, water, and air, as shown in Figure1b. In this figure, soil indicates the total volume of the soil particles, including its minerals and organics. The pore space of soil contains the liquid and gas phases of soil, i.e., everything but the solid phase that contains mainly minerals of varying sizes as well as organic compounds. Water means that the total water amount currently stored in the soil pore space, and represents AMC from previous precipitation [32]. Air can be part of the void space when not saturated with water. The volume of void space is determined by a degree of soil saturation under the assumption the total pore space volume of the soil column is constant. So, the void space is equal to the current potential maximum moisture retention. The sum of air and water are the absolute potential maximum moisture retention (Sabs)[14]. The total volume of voids and the absolute potential maximum moisture retention are theoretically equal, as shown by Equation (7). Sabs = Vv = Va + Vw (7)

In soil moisture accounting (Figure1c), the sum of S and AMC is S abs, and the same as Vv in the soil-water-air layer. An approach to estimating the current S uses a relationship between the absolute potential maximum retention and AMC (Equation (8)). According to this equation, S depends on AMC as Sabs is constant. When the soil is very dry, AMC is reduced by drainage and evapotranspiration, and S approaches Sabs. S = Sabs − AMC (8)

S(t + α) = Sabs − (AMC(t) − ds/d(t + α)) (9) Using a time variable concept, Equation (8) becomes Equaiton (9), which represents the change of S after an INR. S(t + α) is the time variable S after the interval, α. However, Sabs is a constant. AMC(t) is the soil moisture content at the end of antecedent rainfall and it depends on ds/d(t + α), indicating a change of S for INR. Also, it tracks the recovery of SMR. From the equation, it is evident that S depends on the recovery amount for the INR.

3. Methodology to Derive R-Curve This study developed a methodology to derive the R-curve, which involved several preprocessing steps and a computation of S. Figure2 shows the flowchart representing details of the workflow. (a) First, we collected data for a number of complex storm events having multiple short-term independent rainfall pulses (Figure2(A-1)). This study employed a concept of Inter-Event Time Duration (IETD) in order to separate rainfall pulses from a complex rainfall event [33–35]. In this study, 6 h was used as the minimum IETD to classify an independent rainfall event. Other analysts have used 6 h as a proper IETD [1,36–39]. Sixteen rainfall events were identified for the study period. (b) Next, the complex hydrograph [40] from the multi-pulse rainfall event was separated into individual direct runoff hydrographs (A-2 in pre-processing in Figure2). In the hydrograph separation process, the total direct runoff was calculated from the separated hydrograph to estimate an event-based CN for each rainfall pulse. This study used the N-day method [41] to identify the start of the event direct runoff from the complex hydrograph. Cubic spline interpolation was used to estimate the pulse event discharges on the regression limb. Water 2018, 10, x FOR PEER REVIEW 5 of 22

Water 2018runoff, 10 from, 833 the complex hydrograph. Cubic spline interpolation was used to estimate the pulse event5 of 21 discharges on the regression limb.

A. Pre-processing

A-1 Select a complex storm event with multiple peak A-3 Make a map of the initial runoff curve number (CN) flows in a complex hydrograph Landuse Soil 6000.0 0.0

) Rainfall

) c 5.0

s e

/ Streamflow

3 s /

t

f 3

( 4000.0 10.0 m (

e

g

e 15.0

r

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a r a h 2000.0 h

c c

s

i s i

D D 0.0 02/14/11 0:00 02/16/11 0:00 02/18/11 0:00 02/20/11 0:00 02/22/11 0:00 Time (MM/DD/YY hh:mm) ` A-2 Separate a complex hydrograph into independent hydrographs CN map

6000.0 0.0 )

) Hydrograph01

c 5.0

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/ e Hydrograph02

3 s

t /

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( 4000.0 10.0 m

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D D 0.0 02/14/11 0:00 02/16/11 0:00 02/18/11 0:00 02/20/11 0:00 02/22/11 0:00 Time (MM/DD/YY hh:mm)

B. Computation of S

150.0 0.0 ) B-1 Estimate S at the beginning of an antecedent rainfall pulse m b1 m

5.0 (

IETD * l l · Convert initial CN map to initial S (Si) map using the eq. (8b) a f n ) 10.0 i c

B-1. At the beginning of a e R s 100.0 · Using eq. (7) and initial S, estimate Sb1. At the eq. (7), gradually / antecedent rainfall pulse B-3. At the beginning of 3 15.0 m post rainfall pulse change the initial S by 1.0% until the best objective function is (

e found, which is the smallest difference of Q and P in the eq. (8b). g r

a Also assume that the initial abstraction is 20% of initial S h c s i 50.0 D 2 B-2. At the end of (PSS 0.2ib1 ) Gradually change Sb1 by 1.0% to find antecedent rainfall pulse Q  objective function: PSS 0.2 ib1 0.0 12/21/12 0:00 12/22/12 0:00 12/23/12 0:00 Ti me (mm/dd/yy HH:MM) QP0

B-2 Calculate Sa1 at the end of antecedent rainfall pulse B-3 Estimate Sb2 at the beginning of post rainfall pulse

· To calculate Sa1, using Sb1 and direct runoff of the antecedent · Using the post hydrograph separated to calculate direct runoff hydrograph separated because Sb1 represents AMC at the begging of the antecedent rainfall pulse and the infiltrated rainfall · Using the same process B-1, estimate Sb2 can be calculated by direct runoff with equations as below

SSFab11

FPQS   0.2 b1

Figure 2. R-curve pre-processing steps and computation of S: (A) Description of each pre-processing; and, Figure 2. R-curve pre-processing steps and computation of S: (A) Description of each pre-processing; (B) Description of each progress to estimate S. and, (B) Description of each progress to estimate S. (c) To estimate an event-based CN, an initial 1 km by 1 km CN map based on the land-use and soil (c)maps To was estimate used (A an-3 event-based in pre-processing CN, anin initialFigure 12). km To bydetermine 1 km CN the map appropriate based on event the-based land-use CN andfor soil mapseach was rainfall used (A-3pulse, in CN pre-processing values in the CN in map Figure were2). changed To determine uniformly the and appropriate incrementally event-based by 1.0% until CN a for each rainfallvolume of pulse, simul CNated valuesrunoff flows in the by CN CN map method were matches changed a volume uniformly of the and observed incrementally runoff flows. by Areal 1.0% until a volumeaverage of simulatedCN from the runoff best appropriateflows by CN event method-based matches CN map a was volume then ofused the to observed calculate runoffS as the flows. representative value for the event. In addition, it should be emphasized that the initial abstraction is Areal average CN from the best appropriate event-based CN map was then used to calculate S as the important to estimate CN values as it is related to the capacity of soil moisture retention. In this study for representativea developing value purpose for, the it is event. assumed In to addition, be 20%. itHowever, should many be emphasized studies have that suggested the initial various abstraction initial is importantabstractions to estimate depending CN on values region as [6,23,42,43 it is related]. to the capacity of soil moisture retention. In this study for a developing(d) Three purpose,S values for it three is assumed time steps to are be required 20%. However, to derive manythe R-curve. studies The have process suggested computation various initialof abstractions S in Figure 2B depending involves the on definition region of [6 ,three23,42 S,43 values]. at the specified times (Sb1, Sa1, and Sb2). Sb1, Sa1, (d)and Three Sb2 are S associated values for with three the time antecedent steps are and required post rainfall to derive pulses. the The R-curve. first S The value, process Sb1, represents computation antecedent SMR for the antecedent rainfall pulse. Sb1 was calculated from the event-based CN in the of S in Figure2B involves the definition of three S values at the specified times (S b1,Sa1, and process (c) and Equation (8b) (Figure 2(B-1)). The second S value, Sa1, represents SMR after the event Sb2). Sb1,Sa1, and Sb2 are associated with the antecedent and post rainfall pulses. The first S value, rainfall pulse. Sa1 was calculated using the water balance Equation (3) and a relation of S, AMC, and Sb1, represents antecedent SMR for the antecedent rainfall pulse. Sb1 was calculated from the event-based infiltrated water (Figure 2(B-2)). In the computation process, infiltrated water was calculated from the CN in the process (c) and Equation (8b) (Figure2(B-1)). The second S value, S a1, represents SMR after the event rainfall pulse. Sa1 was calculated using the water balance Equation (3) and a relation of S, AMC, and infiltrated water (Figure2(B-2)). In the computation process, infiltrated water was calculated from the water balance equation using direct runoff from a separated hydrograph, total rainfall, and 20% initial abstraction coefficient. The third S value, Sb2, represents SMR after the INR; it is the antecedent Water 2018, 10, 833 6 of 21 Water 2018, 10, x FOR PEER REVIEW 6 of 22

SMRwater for the balance post rainfall equation pulse. using S directb2 was runoff estimated from a using separated the same hydrograph, process total for estimatingrainfall, and S 20%b1. Depending initial on theabstraction rainfallpulse, coefficient. Sb2 couldThe third be SSb1 value,of the Sb2 post, represents rainfall SMR pulse. after the INR; it is the antecedent SMR for (e)the Thepost INRrainfall between pulse. S twob2 was rainfall estimated pulses using was the usedsame asprocess the time for estimating interval to Sb1 recover. Depending S. In on this the study, the intervalrainfall pulse, was assumed Sb2 could be as S theb1 of timethe post period rainfall enabling pulse. S to recover through the downward movement of soil moisture(e) The INR and between evapotranspiration. two rainfall pulses The was INR used was as calculated the time interval while to considering recover S. In thethis timestudy, duration the interval was assumed as the time period enabling S to recover through the downward movement of soil from the end of antecedent rainfall pulse to the beginning of the post rainfall pulse (Figure2). moisture and evapotranspiration. The INR was calculated while considering the time duration from the (f)end The of antecede R-curvent wasrainfall derived pulse to from the beginning the relationship of the post between rainfall pulse the change(Figure 2). of S values and the INR between rainfall(f) The R pulses.-curve was The derived change from of S the values relationship was calculated between asthe thechange difference of S values between and the Sa1 INRand Sb2. It indicatesbetween the rainfall change pulses. of S The during change the of INR S values enabling was calculated S to recover. as the difference between Sa1 and Sb2. It Basedindicates on the the change methodology of S during explainedthe INR enabling above, S to the recover. relationship between the change of S and the duration asBased R-curve on the is methodology derived for the explained sixteen above, rainfall the events. relationship between the change of S and the duration as R-curve is derived for the sixteen rainfall events. 4. Data 4. Data 4.1. Watershed 4.1. Watershed The Napa River basin in Napa County, CA, was used as an application watershed. The gage at St. HelenaThe (USGS Napa 11456000),River basin in Napa Napa County, County, CA, was was used the as primary an application site to watershed. collect hydrographs The gage at St. as it is Helena (USGS 11456000), Napa County, CA, was the primary site to collect hydrographs as it is located located in the upper portion of the basin and has minimal water management influences. Figure3 in the upper portion of the basin and has minimal water management influences. Figure 3 shows the showslocation the location of the Napa of the watershed Napa watershed and the land and-use theand land-usethe soil map ands. The the drainage soil maps. area The is 204.8 drainage km2 (79.3 area is 2 2 204.8mi km2). Annual(79.3 mi average). Annual precipitation average is 685 precipitation mm, of which is 68580% mm, occurs of mainly which during 80% occurs the rainy mainly season during from the rainyNovember season from to March. November to March.

FigureFigure 3. Land-use 3. Land-use map map (left (left),), digitaldigital elevation elevation with with the watershed the watershed location location (middle (),middle and soil), map and (right soil) map (rightfor) for the the application application watershed, watershed, St. Helena St. area, Helena Napa area, County, Napa California County,. California.

Table 1 lists the fractions (%) of land-use type and the hydrologic soil group in the basin (Figure 3). TableThe sum1 lists of theforest fractions and grassland/herbaceous (%) of land-use type areas and exceed thes hydrologic 65%, followed soil by group cultivated in the crops basin (9.9%), (Figure 3). The sumpasture/hay of forest (8.4%), and grassland/herbaceousand shrub/scrub (7.1%). Based areas on exceeds the hydrologic 65%, followedsoil group, by the cultivated drainage capability crops (9.9%), pasture/hayis relatively (8.4%), low as and the shrub/scrubwell-drained group (7.1%). A area Based is only on 1.5%, the hydrologic and there are soil large group, areas of the poorly drainage capabilitydrained is group relatively C (28.6 low%) asand the D (29.5 well-drained%) soils. group A area is only 1.5%, and there are large areas of poorly drained group C (28.6%) and D (29.5%) soils.

Water 2018, 10, 833 7 of 21

Table 1. Watershed land-use and hydrologic soil group area fractions, and min/max curve number (CN) values.

NRCS Cover Type Description (Cover Type CN Values in Hydrologic Soil Group (fraction) Primary Landuse Fraction(%) Max./Min. and Hydrologic Condition) A (1.5%) B (40.4%) C (28.6%) D (29.5%) Wood-Good Min. 30 55 70 77 Forest 39.6 Wood-grass combination-Poor Max. 57 73 82 86 Meadow-continuous grass, protected from Min. 30 58 71 78 Grassland/herbaceous 25.7 grazing and generally mowed for hay-Good Pasture, grassland, or range continuous forage Max 68 79 86 89 for grazing-Poor Row crops: Contoured & terraced-Good Min. 62 71 78 81 Cultivated crops 9.9 Row crops: Straight row-Poor Max 72 81 88 91 Meadow-continuous grass, protected from Min. 30 58 71 78 Pasture/hay 8.4 grazing and generally mowed for hay-Good Pasture, grassland, or range continuous forage Max 68 79 86 89 for grazing-Poor Brush-brush-forbs-grass mixture with brush the Min. 30 48 65 73 Shrub/scrub 7.1 major element-Good Desert shrub-major plants (salt brush, grease Max 63 77 85 88 wood, creosote bush, black brush, etc.)

The NRCS cover type influences the CN mapping since Min and Max values depend on the cover type. Table1 shows Min and Max values that are associated with the NRCS cover types [ 16]. For example, the CN can range from 70 to 82 in group C of the forest category. Thus, CN maps resultant from the land-use and soil maps could differ depending on the analyst’s choices. Also, it is important for estimating the event-based CN when the map is used as an initial condition.

4.2. Complex Storm Event Complex storm events having many rainfall pulses in succession were used to derive the R-curve, with the S value of each pulse being linked to the previous and the post soil moisture contents. The rainfall pulses are established as independent of each other through hydrograph separation and a 6 h period as the minimum IETD. Figure4 shows the time series of rainfall and streamflow for the four complex storm events that were disaggregated into 16 independent rainfall pulses. Table2 shows the characteristics of the independent events (complex storm #01-three independent rainfalls; #02-seven independent rainfalls; #03-three independent rainfalls, #04-three independent rainfalls). In Table2, a complex storm event consists of independent rainfalls with their total rainfall, the duration with no rain, and runoff coefficient, which is a dimensionless coefficient related to representation of a proportion of the amount ofWater direct 2018 runoff, 10, x FOR to PEER the amount REVIEW of rainfall. 8 of 22

Figure 4.4. RainfallRainfall h hyetographsyetographs and and hydrographs hydrographs of of the the four four complex complex storm storm events events addressed addressed in this in thisstudy study..

Table 2. Characteristics of the applied storm events.

Complex Independent Period Total Rainfall Runoff Interval with No Rain (h) Storm Event No. Rainfall No. (Pacific Standard Time, PST) (mm) Coefficient 1 14 February 2011 03:00–15 February 2011 00:00 19.2 over 72 0.05 1 2 15 February 2011 05:00–16 February 2011 01:00 46.3 21 0.43 3 17 February 2011 02:00–18 February 2011 15:00 34.4 29 0.60 4 13 March 2011 10:00–14 March 2011 00:00 21.4 over 72 0.17 5 15 March 2011 02:00–16 March 2011 15:00 36.3 28 0.51 6 18 March 2011 01:00–19 March 2011 04:00 23.8 59 0.37 2 7 19 March 2011 05:00–20 March 2011 19:00 61.0 14 0.82 8 22 March 2011 22:00–23 March 2011 10:00 21.9 66 0.27 9 24 March 2011 05:00–24 March 2011 14:00 26.1 23 0.99 10 25 March 2011 18:00–26 March 2011 15:00 22.0 36 0.28 11 29 November 2012 08:00–01 December 2012 10:00 164.8 17 0.23 3 12 01 December 2012 16:00–02 December 2012 12:00 80.8 10 0.50 13 04 December 2012 15:00–06 December 2012 05:00 21.7 56 0.17 14 20 December 2012 18:00–21 December 2012 19:00 61.7 over 72 0.38 4 15 22 December 2012 00:00–22 December 2012 13:00 25.3 11 0.10 16 23 December 2012 03:00–23 December 2012 14:00 77.8 21 0.61

4.3. Radar-Based Rainfall Data Multi-Radar Multi-Sensor (MRMS) was used as rainfall data in this study. MRMS is a National Weather Service (NWS) operational system having automated algorithms that integrate data streams from multiple radars, surface and upper air observations, lightning detection systems, and satellite and forecast models [44]. MRMS uses the Next-Generation Radar (NEXRAD) and rain gauge network data to produce a suite of Quantitative Precipitation Estimation (QPE) products, including radar-only, bias adjusted radar using rain gauges, and gauge-only. It generates gridded rainfall fields at 1 km resolution every hour. The MRMS archive can be downloaded for a user-selectable area at the public website (https://www.nssl.noaa.gov/projects/mrms/). It is possible to estimate the CN value of each grid of the rainfall field since MRMS is a radar-based gridded rainfall product. Moreover, based on Willie et al. [45], the MRMS QPE was generated using the Vertical-Profile-of-Reflectivity (VPR) and the Mean Field Bias (MFB) correction method [46,47]. Figure 5 shows (a) a sample MRMS hourly rainfall map in Northern California and (b) the same sample MRMS hourly rainfall map for the application basin.

Water 2018, 10, 833 8 of 21

Table 2. Characteristics of the applied storm events.

Complex Storm Independent Total Rainfall Interval with Runoff Period (Pacific Standard Time, PST) Event No. Rainfall No. (mm) No Rain (h) Coefficient 1 14 February 2011 03:00–15 February 2011 00:00 19.2 over 72 0.05 1 2 15 February 2011 05:00–16 February 2011 01:00 46.3 21 0.43 3 17 February 2011 02:00–18 February 2011 15:00 34.4 29 0.60 4 13 March 2011 10:00–14 March 2011 00:00 21.4 over 72 0.17 5 15 March 2011 02:00–16 March 2011 15:00 36.3 28 0.51 6 18 March 2011 01:00–19 March 2011 04:00 23.8 59 0.37 2 7 19 March 2011 05:00–20 March 2011 19:00 61.0 14 0.82 8 22 March 2011 22:00–23 March 2011 10:00 21.9 66 0.27 9 24 March 2011 05:00–24 March 2011 14:00 26.1 23 0.99 10 25 March 2011 18:00–26 March 2011 15:00 22.0 36 0.28 11 29 November 2012 08:00–01 December 2012 10:00 164.8 17 0.23 3 12 01 December 2012 16:00–02 December 2012 12:00 80.8 10 0.50 13 04 December 2012 15:00–06 December 2012 05:00 21.7 56 0.17 14 20 December 2012 18:00–21 December 2012 19:00 61.7 over 72 0.38 4 15 22 December 2012 00:00–22 December 2012 13:00 25.3 11 0.10 16 23 December 2012 03:00–23 December 2012 14:00 77.8 21 0.61

4.3. Radar-Based Rainfall Data Multi-Radar Multi-Sensor (MRMS) was used as rainfall data in this study. MRMS is a National Weather Service (NWS) operational system having automated algorithms that integrate data streams from multiple radars, surface and upper air observations, lightning detection systems, and satellite and forecast models [44]. MRMS uses the Next-Generation Radar (NEXRAD) and rain gauge network data to produce a suite of Quantitative Precipitation Estimation (QPE) products, including radar-only, bias adjusted radar using rain gauges, and gauge-only. It generates gridded rainfall fields at 1 km resolution every hour. The MRMS archive can be downloaded for a user-selectable area at the public website (https://www.nssl.noaa.gov/projects/mrms/). It is possible to estimate the CN value of each grid of the rainfall field since MRMS is a radar-based gridded rainfall product. Moreover, based on Willie et al. [45], the MRMS QPE was generated using the Vertical-Profile-of-Reflectivity (VPR) and the Mean Field Bias (MFB) correction method [46,47]. Figure5 shows (a) a sample MRMS hourly rainfall map in Northern California and (b) the same sample MRMS hourly rainfall map for the application basin. Water 2018, 10, x FOR PEER REVIEW 9 of 22

(a) (b)

Figure 5. Radar-based gridded rainfall fields from Multi/Radar Multi/Sensor data: (a) Northern California Figure 5. Radar-basedMulti-Radar Multi gridded-Sensor (MRMS rainfall) precipitation fields from; (b) MRMS Multi/Radar precipitation field Multi/Sensor for St. Helena Napa data: basin, (a ) Northern California Multi-RadarCA. Multi-Sensor (MRMS) precipitation; (b) MRMS precipitation field for St. Helena Napa basin, CA. 5. Pre-Processing

5. Pre-Processing5.1. Complex Hydrograph Separation Hydrograph separation of multiple peak flows arising from a multi-pulse rainfall event requires the 5.1. Complex Hydrographcalculation of direct Separation runoff for each pulse. It should be emphasized that the complex hydrograph separation is important as the complex hydrograph separation methods are used to determine the total Hydrographrunoff separationvolume. It, also, of might multiple have an peak effect flowson the R arising-curve derivation from a depending multi-pulse on the rainfallmethods. eventTo requires the calculationseparate of direct the multiple runoff hydrograph for each into pulse. its components It should (i.e., bebase emphasized and direct flows) that we used the two complex methods, hydrograph N-day and spline interpolation. The main reasons for selecting the two methods are as follows: (1) separation is importantN-day method as is theconvenient complex to estimate hydrograph the end of separationa hydrograph methodsusing the drainage are used area to and determine (2) the the total spline interpolation could reflect the trend of recession limb of hydrographs to a separated hydrograph. However, it is able to apply various hydrograph separation methods for deriving the R-curve [48]. When considering various methods, it is required to analyze the sensitivity of the hydrograph separation methods to examine their effect on the R-curve. The N-day method estimates the duration between a time-to-peak and the runoff flow recession at the end of a hydrograph [41]; this is an adaptation of an equation like Equation (10). For the Napa River near Helena site N-days is approximately 2.4 days (58 h), when substituting the drainage area 205.4 km2.

0.2 Ndays  0.386 A (10)

Here, N is the number of days runoff ceases after a storm and A is the drainage area of the application basin (km2). Figure 6 shows the process to separate a multiple peak hydrograph into its pulse components, described as follows: (1) Estimate the duration from a time to peak to the end time of runoff using the N-day method, as shown in Figure 6A. (2) Use the spline interpolation method to estimate the pulse discharges on the regression limb, as shown in Figure 6B. When applying the interpolation method, this study used the discharges for the recession limb and the end time of runoff flow from the N-day method. Finally, the separated hydrograph is shown in Figure 6C. A similar process is done for the remaining peaks.

Water 2018, 10, 833 9 of 21 runoff volume. It, also, might have an effect on the R-curve derivation depending on the methods. To separate the multiple hydrograph into its components (i.e., base and direct flows) we used two methods, N-day and spline interpolation. The main reasons for selecting the two methods are as follows: (1) N-day method is convenient to estimate the end of a hydrograph using the drainage area and (2) the spline interpolation could reflect the trend of recession limb of hydrographs to a separated hydrograph. However, it is able to apply various hydrograph separation methods for deriving the R-curve [48]. When considering various methods, it is required to analyze the sensitivity of the hydrograph separation methods to examine their effect on the R-curve. The N-day method estimates the duration between a time-to-peak and the runoff flow recession at the end of a hydrograph [41]; this is an adaptation of an equation like Equation (10). For the Napa River near Helena site N-days is approximately 2.4 days (58 h), when substituting the drainage area 205.4 km2. 0.2 Ndays = (0.386 × A) (10) Here, N is the number of days runoff ceases after a storm and A is the drainage area of the application basin (km2). Figure6 shows the process to separate a multiple peak hydrograph into its pulse components, described as follows: (1) Estimate the duration from a time to peak to the end time of runoff using the N-day method, as shown in Figure6A. (2) Use the spline interpolation method to estimate the pulse discharges on the regression limb, as shown in Figure6B. When applying the interpolation method, this study used the discharges for the recession limb and the end time of runoff flow from the N-day method. Finally, the separated hydrograph is shown in Figure6C. A similar process is done for the remainingWater 2018 peaks., 10, x FOR PEER REVIEW 10 of 22

Figure 6. Procedure for hydrograph separation: (A) Schematic diagram of N-day method; (B) Schematic Figure 6. Procedure for hydrograph separation: (A) Schematic diagram of N-day method; (B) Schematic diagram of spline interpolation method; and, (C) Sample result of separated hydrograph. diagram of spline interpolation method; and, (C) Sample result of separated hydrograph. 5.2. Estimation of Soil Moisture Retention 5.2. Estimation of Soil Moisture Retention A gridded CN for each storm event was estimated using an initial gridded CN map, as described at ASection gridded 3. The CN initial for eachgridded storm CN eventmap was was estimated estimated from using Table an 2 with initial both gridded minimum CN and map, maximum as described at SectionCN values3. The to determine initial gridded the difference CN map of CNs was depending estimated on from NRCS Table cover2 types.with Figure both minimum7 shows the and maximumresult of CN the values initial gridded to determine CN map the and difference its corresponding of CNs dependingS values. As onshown NRCS in Figure cover types.7, the areal Figure 7 showsmean the CN result using of the the minimum initial gridded values is CN 67; mapthe maximum and its correspondingmean CN value is S 81. values. As shown in Figure7, the areal mean CN using the minimum values is 67; the maximum mean CN value is 81.

(a) (b)

Figure 7. Maps of runoff curve number (CN) and potential maximum moisture retention (S) in Napa River basin: (a) The estimated runoff curve number; and, (b) The converted potential maximum retention from the runoff curve number.

In Figure 7b, S and CN are inversely proportional, so the maximum S is from the minimum CN value. The maximum S is twice the minimum S, because the CN-S relationship is non-linear (Equation (6)). Areal mean S using minimum CN is 134.3 mm and the areal mean S using maximum CN is 61.4 mm. These S values indicate the amount of SMC the soil column could hold at any time. The minimum value in Table 1 was used to map the initial CN since it is better than the maximum value to estimate event-based CN. The non-proper values resulted in a large difference of the observed direct runoff and the calculated direct runoff from the estimated CN by the maximum value. Results from the minimum value have relatively small difference. The resulting S values (Sb1, Sa1, Sb2, and a change of S) are tabulated in Table 3. This table also lists the event-based CNs and INR for deriving the R-curve for each event. In Table 3, INR is calculated from the time interval between antecedent rainfall pulse and post rainfall pulse with no rain, the recovery of S is a recovered capacity of SMR for the duration. For the independent rainfall events No. 2 and 3 in Table 3, the recovery of SMR, 8.0 mm, occurred over a duration of 29 h. The first S value, Sb1, is estimated at the time just before the onset of the first rainfall. The second S value, Sa1, is representative of the time just after the first rainfall period ends. The third S value, Sb2, represents the first response of the second hydrograph arising from the second rainfall period.

Water 2018, 10, x FOR PEER REVIEW 10 of 22

Figure 6. Procedure for hydrograph separation: (A) Schematic diagram of N-day method; (B) Schematic diagram of spline interpolation method; and, (C) Sample result of separated hydrograph.

5.2. Estimation of Soil Moisture Retention A gridded CN for each storm event was estimated using an initial gridded CN map, as described at Section 3. The initial gridded CN map was estimated from Table 2 with both minimum and maximum CN values to determine the difference of CNs depending on NRCS cover types. Figure 7 shows the Water 2018result, 10, 833of the initial gridded CN map and its corresponding S values. As shown in Figure 7, the areal 10 of 21 mean CN using the minimum values is 67; the maximum mean CN value is 81.

(a) (b)

Figure 7. Maps of runoff curve number (CN) and potential maximum moisture retention (S) in Napa Figure 7. Maps of runoff curve number (CN) and potential maximum moisture retention (S) in Napa River basin: (a) The estimated runoff curve number; and, (b) The converted potential maximum retention River basin:from the (a runoff) The curve estimated number. runoff curve number; and, (b) The converted potential maximum retention from the runoff curve number. In Figure 7b, S and CN are inversely proportional, so the maximum S is from the minimum CN value. The maximum S is twice the minimum S, because the CN-S relationship is non-linear (Equation In Figure7b, S and CN are inversely proportional, so the maximum S is from the minimum (6)). Areal mean S using minimum CN is 134.3 mm and the areal mean S using maximum CN is 61.4 mm. CN value.These The S values maximum indicate the S isamount twice of theSMC minimum the soil column S, becausecould hold the at any CN-S time. relationship is non-linear (Equation (6)).The Arealminimum mean value S in using Table minimum 1 was used CNto map is 134.3the initial mm CN and since the it is areal better mean than the S usingmaximum maximum CN is 61.4value mm. to estima Thesete event S values-based indicate CN. The thenon- amountproper values of SMC resulted the in soil a large column difference could of holdthe observed at any time. Thedirect minimum runoff and value the incalculated Table1 directwas usedrunoff to from map the the estimated initial CNCN sinceby the it maximum is better value. than theResults maximum value tofrom estimate the minimum event-based value have CN. relatively The non-proper small difference. values resulted in a large difference of the observed The resulting S values (Sb1, Sa1, Sb2, and a change of S) are tabulated in Table 3. This table also lists the direct runoff and the calculated direct runoff from the estimated CN by the maximum value. Results event-based CNs and INR for deriving the R-curve for each event. In Table 3, INR is calculated from the from thetime minimum interval between value antecedent have relatively rainfall smallpulse and difference. post rainfall pulse with no rain, the recovery of S is a Therecovered resulting capacity S values of SMR (S forb1 ,Sthea1 duration.,Sb2, and For the a changeindependent of S)rainfall are tabulatedevents No. 2 inand Table 3 in Table3. This 3, table also liststhe the recovery event-based of SMR, 8.0 CNs mm, and occurred INR over for derivinga duration theof 29 R-curve h. The first for S value, each S event.b1, is estimated In Table at the3, INR is calculatedtime fromjust before the timethe onset interval of the betweenfirst rainfall. antecedent The second rainfallS value, S pulsea1, is representative and post rainfall of the time pulse just with no rain, theafter recovery the first of rainfall S isa period recovered ends. Thecapacity third S of value, SMR S forb2, represents the duration. the first For response the independent of the second rainfall hydrograph arising from the second rainfall period. events No. 2 and 3 in Table3, the recovery of SMR, 8.0 mm, occurred over a duration of 29 h. The first

S value, Sb1, is estimated at the time just before the onset of the first rainfall. The second S value, Sa1, is representative of the time just after the first rainfall period ends. The third S value, Sb2, represents the first response of the second hydrograph arising from the second rainfall period.

Table 3. Results of three S estimated and the recovery amount of S for the duration with no rain using the proposed methodology.

Soil Moisture Retentions (mm) Independent (3)–(2) Recovery CN INR (h) (1) (2) (3) Rainfall Event No. b1 (mm) Sb1 Sa1 Sb2 - - 1 78.2 70.9 49.9 40.2 21 −9.7 2 86.3 40.2 14.0 22.0 29 8.0 3 92.0 22.0 9.1 - Over 72 - 4 84.7 45.7 28.7 27.3 28 −1.3 5 90.3 27.3 11.3 32.2 59 20.8 6 88.7 32.2 17.1 3.9 14 −13.2 7 94.8 3.9 0.1 27.9 66 27.8 8 90.1 27.8 17.2 8.6 23 −8.5 Water 2018, 10, 833 11 of 21

Table 3. Cont.

Soil Moisture Retentions (mm) Independent (3)–(2) Recovery CN INR (h) (1) (2) (3) Rainfall Event No. b1 (mm) Sb1 Sa1 Sb2

9 96.7 8.6 3.7 37.3 36 16.5 10 87.2 37.3 22.1 - 17 - 11 65.9 131.0 51.8 25.4 10 −26.3 12 90.9 25.4 6.6 31.3 56 24.6 13 89.0 31.3 20.6 - Over 72 - 14 79.6 64.8 25.4 45.0 11 −25.3 15 84.9 45.0 30.5 11.3 21 −19.1 16 95.7 11.3 2.5 -

6. Results

6.1. Derivation of the Recovery Curve The R-curve is derived from the relationship between the change of S values (the recovery in Table3) and INR, as shown in Figure8a. The R-curve is divided into three sections, according to the recovery rate (mm/h) of SMR. Root Mean Square Error (RMSE) is used as a criterion to determine the best linear regression line to estimate the recovery rate in each section. For the three sections, (1) the first section is in the range 10 to 21 h and exhibits a gradually recovering S, (2) the second section is in the range 21 to 36 h and exhibits a steeply recovering S, and (3) the third section is in the range 36 to 66 h and it exhibits a gradually decreasing recovery rate. Water 2018, 10, x FOR PEER REVIEW 12 of 22

Figure 8. Derivation of R-curve for the Napa River basin, based on the sixteen rainfall events. In the figure Figure 8. Derivation of R-curve for the Napa River basin, based on the sixteen rainfall events. In the 8a, R-curve Sections 1–3 represent the gradually recovering soil moisture retention (SMR) section, the figure 8a,rapid R-curve recovery Sections of SMR1 section,–3 represent and the the gradually gradually decreasing recovering recovery soil section, moisture respectively. retention In the (SMR) figure section, the rapid8b, recovery five R-curves of SMRderived section, depending and on the the gradually conditions decreasingin the N-day recoveryand the initial section, abstraction respectively. (Ia) are In the figure 8b,compared. five R-curves derived depending on the conditions in the N-day and the initial abstraction (Ia) are compared. However, ΔS is negative because S at the end of first storm event is larger than S after an INR. a1 b1 From the Napa River events, the negative S values continue to 27 h, which means that soil moisture contents are being increased without additional rainfall for 27 h. In other words, actual SMR is decreased instead of increasing. In general, SMR is decreased in the short-term by drainage arising from the movement of water flowing to lower soil layers and this process is commonly finished within 2–3 days. This physical process is difficult to describe using the CN method because it regards a soil column as a single layer in accounting for the total value of SMR. The results that are described above can be explained conceptually when dividing a soil column into two or more layers reflecting the soil profile and taking consideration of physical phenomenon occurring within each layer. Figure 9 shows a conceptual diagram of a soil column consisting of two soil layers with S and SMC, and showing infiltration, percolation, and drainage by arrows. In this study, the upper layer was defined as a soil layer that is saturated by infiltration, and the lower layer is defined as a soil layer saturated by downward percolation.

Water 2018, 10, 833 12 of 21

However, ∆S is negative because Sa1 at the end of first storm event is larger than Sb1 after an INR. From the Napa River events, the negative S values continue to 27 h, which means that soil moisture contents are being increased without additional rainfall for 27 h. In other words, actual SMR is decreased instead of increasing. In general, SMR is decreased in the short-term by drainage arising from the movement of water flowing to lower soil layers and this process is commonly finished within 2–3 days. This physical process is difficult to describe using the CN method because it regards a soil column as a single layer in accounting for the total value of SMR. The results that are described above can be explained conceptually when dividing a soil column into two or more layers reflecting the soil profile and taking consideration of physical phenomenon occurring within each layer. Figure9 shows a conceptual diagram of a soil column consisting of two soil layers with S and SMC, and showing infiltration, percolation, and drainage by arrows. In this study, the upper layer was defined as a soil layer that is saturated by infiltration, and the lower layer is definedWater 2018 as, 10 a, soilx FOR layer PEER saturated REVIEW by downward percolation. 13 of 22

Infiltration (f)

S 1 SMC = AMC + Infiltrated Water Upper Soil water lagged Layer SMC1 because of p < f

Wetting front

Lower S2 Percolation (p) Layer

SMC2

Bedrock Interflow and subsurface outflow

FigureFigure 9.9. Conceptual diagram of a soilsoil columncolumn consideringconsidering thethe soilsoil layerslayers inin thethe firstfirst section:section: ThisThis studystudy assumedassumed that that the the soil soil column column consists consists of two of two layers, layers, an upper an upper layer layer(saturated (saturated by infiltration) by infiltration) and a lower and alayer lower (saturated layer (saturated by percolation). by percolation). S indicates the S indicatessoil moisture the retention soil moisture (SMR) retention and SMC (SMR)means andSoil Water SMC meansContent Soil consisting Water Content of antecedent consisting moisture of antecedent content ( moistureAMC) and content the infiltrated (AMC) andwater. the The infiltrated wetting water. front Thebased wetting on Green front and based Ampt on [4 Green9] represents and Ampt the boundary [49] represents between the two boundary layers causing between the two retardation layers causing effect theby the retardation difference effect of infiltration by the difference and percolation of infiltration rates. and percolation rates.

AccordingAccording to Thomas et al. al. [ [5050]],, there there is is a agap gap be betweentween the the infiltration infiltration rate rate (92 (92–267–267 mm/day) mm/day) and and the thepercolation percolation rate rate (40– (40–109109 mm/day). mm/day). If percolation If percolation is is slower slower than than the the infiltration infiltration rate, rate, a a retardationretardation boundaryboundary between two layers occur occurss because because water water is is not not readily readily flowing flowing to to the the lower lower layer layer,, as asshown shown in inFigure Figure 9. The9. The boundary boundary could could be described be described as the aswetting the wetting front. Thus front., w Thus,hen the when upper the layer upper is saturated layer isand saturated there is the and retardation there is the effect retardation, it appears effect, as if itthe appears total SMR as ifis thedecreased total SMR without is decreased additional without rainfall additionalduring the rainfallINR, even during though the the INR, actual even total though SMR theis increased actual total by the SMR drainage is increased. by the drainage. However,However, it isis hardhard toto concludeconclude thatthat aa soilsoil columncolumn hashas onlyonly oneone wettingwetting frontfront sincesince thethe columncolumn consistsconsists of numerous numerous soil soil layers.layers. Percolation at the layers layers belowbelow thethe upper upper layer layer depends depends onon the the thicknessthickness and and saturated saturated hydraulic hydraulic conductivity conductivity of ofeach each layer layer [29,30 [29]., 30Generally,]. Generally, the upper the upper layer layerhas a hasloose a texture loose texture,, while the while other the layer other below layer the below upper the layer upper has a layer relatively hasa lower relatively hydraulic lower conductivity hydraulic conductivity[31]. Thus, it is [31 highly]. Thus, possible it is highly that a possiblesoil column that has a soil several column wetting has fronts. several wetting fronts. As mentioned above, this result is a limitation of the CN method since the method assumes that entire soil column is a single layer. To overcome this limitation and to derive the appropriate R-curve, a correction is needed. This study examined the effects of the complex hydrograph separation and the initial abstraction (Ia) on the derivation of the R-curve. Figure 8b shows the derived R-curve results from various N-days and Ia. For applying the various N-days, this study changed ±10% of the estimated N-day, 58 h for the application watershed. Two Ia, 0.05 and 0.10, were used for the effect analysis. As a result, the changes of the N-days and the Ia have no obvious effect on the derivation of the R-curve. In the three sections of the R-curve, the recovery rates of R-curves with various conditions have only a little change under 5.0% against to the R-curve derived from the N-day with no change and Ia of 0.20. The result is natural because the R-curve is based on the relation of INR and a change of the Ss. Since INRs are constant for all cases, a change of the Ss is important. A change of the Ss is defined as the difference between Sb2 and Sa1 in this study. When changing the N-days or the Ia, Sb2 and Sa1 are changed together as the total runoff flows are increased or decreased. For example, the independent rainfall No. 1 in Table 3 has 1.0 mm of the recovery difference between the results from 0.20 and 0.05 Ia. It is same when changing the N-days. In this study, the R-curve was corrected assuming two conditions: (1) a trend of SMR between 0 to 10 INR follows the trend between 10 to 21 INR. Based on this assumption, the regression line in the first section is extended to the time zero, and (2) the difference between rates of infiltration and percolation,

Water 2018, 10, 833 13 of 21

As mentioned above, this result is a limitation of the CN method since the method assumes that entire soil column is a single layer. To overcome this limitation and to derive the appropriate R-curve, a correction is needed. This study examined the effects of the complex hydrograph separation and the initial abstraction (Ia) on the derivation of the R-curve. Figure8b shows the derived R-curve results from various N-days and Ia. For applying the various N-days, this study changed ±10% of the estimated N-day, 58 h for the application watershed. Two Ia, 0.05 and 0.10, were used for the effect analysis. As a result, the changes of the N-days and the Ia have no obvious effect on the derivation of the R-curve. In the three sections of the R-curve, the recovery rates of R-curves with various conditions have only a little change under 5.0% against to the R-curve derived from the N-day with no change and Ia of 0.20. The result is natural because the R-curve is based on the relation of INR and a change of the Ss. Since INRs are constant for all cases, a change of the Ss is important. A change of the Ss is defined as the difference between Sb2 and Sa1 in this study. When changing the N-days or the Ia,Sb2 and Sa1 are changed together as the total runoff flows are increased or decreased. For example, the independent rainfall No. 1 in Table3 has 1.0 mm of the recovery difference between the results from 0.20 and 0.05 Ia. It is same when changing the N-days. In this study, the R-curve was corrected assuming two conditions: (1) a trend of SMR between 0 toWater 10 2018 INR, 10 follows, x FOR PEER the trend REVIEW between 10 to 21 INR. Based on this assumption, the regression line14 of in22 the first section is extended to the time zero, and (2) the difference between rates of infiltration and percolation,which cause whichthe retardation cause the effect retardation, is constant effect, from is time constant zero. from Given time these zero. assumptions, Given these then assumptions, the R-curve thencan be the adjusted R-curve until can ΔS be is adjusted zero at the until time∆ Szero. is zero Figure at the10 shows time zero. the result Figure of the10 shows adjusted the R- resultcurve. ofIn thethe adjustedNapa River R-curve. case, the In correction the Napa Riverwas 33.6 case, mm. the correction was 33.6 mm.

FigureFigure 1 10.0. DiagramDiagram of ofthe theR-curve R-curve adjustment adjustment process process:: In this figure, In this the figure, arrows the represent arrows the represent adjustment the adjustmentprocess of the process uncorrected of the R uncorrected-curve. R-curve.

Figure 11 shows the derived R-curve consisting of three sections with a hydrograph for event No. Figure 11 shows the derived R-curve consisting of three sections with a hydrograph for event No. 16. The R-curve, recovered SMR, and CN curves are plotted on the hydrograph to show a potential 16. The R-curve, recovered SMR, and CN curves are plotted on the hydrograph to show a potential application instance for flood forecasting. The first section for low INR periods is characterized by application instance for flood forecasting. The first section for low INR periods is characterized by gradually recovering SMR values. For the Napa River watershed, the time period of this section is 0–21 h gradually recovering SMR values. For the Napa River watershed, the time period of this section is with 0.97 mm/h (23.28 mm/day). In this section, surface water is being infiltrated into soil, so the rate is 0–21 h with 0.97 mm/h (23.28 mm/day). In this section, surface water is being infiltrated into soil, slower than for the second section. Also, downward percolation is a primary contributor to the recovery of SMR in a short-term (Figure 12a).

Water 2018, 10, 833 14 of 21 so the rate is slower than for the second section. Also, downward percolation is a primary contributor to the recovery of SMR in a short-term (Figure 12a). Water 2018, 10, x FOR PEER REVIEW 15 of 22

Water 2018, 10, x FOR PEER REVIEW 15 of 22

Figure 11. Final R-curve and its application with the recovery rate in Napa River basin: In this figure, Figure 11. Final R-curve and its application with the recovery rate in Napa River basin: In this figure, circles indicate the hydrograph, the red line represents the R-curve with recovery rates, and the blue circlesFigure indicate 11. theFinal hydrograph, R-curve and its the application red line representswith the recovery the R-curve rate in N withapa River recovery basin: rates, In this and figure, the blue circlesgreed lines indicate are the the value hydrograph S and CN, the of redCN linemethod represents at a time the step. R-curve From withthis figure, recovery it is ratespossible, and to theidentify blue greed lines are the value S and CN of CN method at a time step. From this figure, it is possible to greedthe amount lines are of therecovered value S SMRand CN at aof time CN methodstep (hourly) at a time and step. a variation From this of figure,S and itCN is possibledepending to identify on the identify the amount of recovered SMR at a time step (hourly) and a variation of S and CN depending theinterval amounts of no of rainrecovered (INRs) .SMR at a time step (hourly) and a variation of S and CN depending on the on theinterval intervalss of no of rain no rain(INRs (INRs).). (a) First section (b) Second section (c) Third section Infiltration (a) FirEsvta speocratitoionn (b) Second section (c) Third section Infiltration Evaporation

Upper Recovered SMR Layer AMC+infiltrated water Recovered SMR Upper Recovered SMR AMC + infiltrated water Layer AMC+infiltrated water Recovered SMR AMC + infiltrated water

Percolation w

Lower o l f Percolation Water by percolation Layer e w c

Lower o a l f f r Water by percolation e Layer u s AMC c AMC a b AMC+Water by percolation f u r s u / r AMC s AMC e

Bedrock b Bedrock Bedrock t AMC+Water by percolation u n s I / r

Bedrock e Bedrock Bedrock t n Figure 12. Conceptual diagram of soil columnI water movement depending on the specified section by the FigureFigurerecovery 12. Conceptual 12 rate. Conceptual: In this diagram figure, diagram AMC ofof soil indicates column column the water antecedent water movement movement moisture depending content depending on and the thespecified on thickness the section specified of arrows by the section denotes the magnitude of the process. It is assumed that evaporation is constant for the short-term by therecovery recovery rate rate:: In this In thisfigure, figure, AMC AMCindicates indicates the antecedent the antecedent moisture content moisture and the content thickness and of the arrows thickness interval with no rain: (a) Hydrological processes in a soil column at the first section; (b) at the second of arrowsdenotes denotes the magnitude the magnitude of the process. of the It process. is assumed It is that assumed evaporation that is evaporation constant for theisconstant short-term for the intervalsection; and,with ( cno) a raint the: t(hirda) Hydrological section. processes in a soil column at the first section; (b) at the second short-term interval with no rain: (a) Hydrological processes in a soil column at the first section; section; and, (c) at the third section. (b) at theThe secondsecond section;section of and, the (Rc-)curve at the involves third section. more rapid recovery of SMR. For the Napa River events, the tiTheme period second of section this section of the isR -21curve–36 hinvolves (Figure more11). The rapid recovery recovery rate of is SMR. 2.11 mm/hFor the (50. Napa64 mm/day), River events, and Thetheis the secondtime fastest period section rate of amongthis of section the three R-curve is sections21–36 involves h of(Figure the R more1-curve1). The rapid (the recovery rate recovery is rate about is of 2.11 two SMR. mm/h times For (50. faster the64 Napamm/day), than the River firstand events, section and over seven times faster than the third section). From Figure 12, the main reasons for the faster the timeis the period fastest of rate this among section three is 21–36sections h of (Figure the R- curve11). The (the recoveryrate is about rate two is 2.11times mm/h faster than (50.64 the mm/day),first recovery rate in Section 2 are: (1) infiltration is stopped, (2) the downward percolation rate in the second and issection the fastest and over rate seven among times three faster sections than the third of the section). R-curve From (the Figure rate 1 is2, the about main two reasons times forfaster the faster than the recoverysection is rate faster in thanSection the 2 first are: section(1) infiltration since the is stopped,lower layer (2) isthe more downward saturated percolation than the firstrate layer,in the andsecond (3) first section and over seven times faster than the third section). From Figure 12, the main reasons for sectionthe downward is faster percolationthan the first continues section sincewith time.the lower layer is more saturated than the first layer, and (3) the fasterthe downward recoveryThe third rateRpercolation-curve in Sectionsection continues 2involves are: with (1) a infiltrationgraduallytime. decreasing is stopped, recovery (2) the rate downward, as shown in percolation Figure 11. In rate in the secondthis section,The section third the R is -cu downward fasterrve section than movement theinvolves first a sectionof gradually water since is decreasing provisionally the lower recovery completed,layer rate is, more as and shown saturated evapo in transpirationFigure than 11. In the first layer,andthis section, (3) the thedownward downward percolation movement continues of water is with provisionally time. completed, and evapotranspiration

Water 2018, 10, 833 15 of 21

The third R-curve section involves a gradually decreasing recovery rate, as shown in Figure 11. In this section, the downward movement of water is provisionally completed, and evapotranspiration might be a dominant factor to recover SMR (Figure 12c). Thus, the recovery rate in the third section is 0.34 mm/h (8.16 mm/day), which is the slowest among the three sections. However, total capacity of SMR in this section is the largest of the three sections as the entire soil column is involved. The third section is initiated at 36 h and ends around 68 h. Ultimately, the recovery rate of SMR after the completion of third section is going to converge on a constant value that is close to zero because the evaporation approaches zero. Also, the remaining water in soil could be decreased by transpiration in the long-term [51–53].

6.2. Application Instance A rainfall-runoff simulation was implemented as an example to demonstrate the usefulness of the new R-curve methodology for continuous runoff simulation. The HEC-HMS model developed by the US Army Corps of Engineers is used as the rainfall-runoff model. Among many of the rainfall-runoff methods in HEC-HMS, the ModClark approach that is based on Clark’s unit hydrograph [54] was selected as a transform method for runoff routing. There are two reasons for selecting the ModClark method: (1) The model is a semi-distributed hydrologic model which can use gridded rainfall inputs such as a radar-based rainfall data; and, (2) It is convenient to apply the gridded CN method as the ModClark is based on a grid representation. For the detailed methodology of the ModClark, readers can see references [55–57]. Also, many previous studies have used the ModClark for various hydrologic applications [58–64]. In most of the studies cited above, the CN method was used with the ModClark approach. The ModClark method requires two parameter inputs, time of concentration (Tc) defined as a time needed for runoff flow from the most remote location to the outlet in a watershed and storage coefficient (K) defined as a lag time due to storage effects. Using independent storm events in Table2, the parameters are estimated. This study employs the HEC-HMS optimization tools [65] to estimate the proper parameters for each storm event. Table4 shows the estimated parameters with error indices (peak flow, total runoff flow, correlation coefficient (CC), bias (BS), and Nash-Sutcliffe efficiency coefficient (NSE)). Except for two events (1 and 4), all of peak flow error (%) and total runoff flow error (%) were under 25%. The mean error values of peak flow, total runoff flow, and time to peak are 6.4%, −0.2%, and −0.1 h, respectively. CC, BS, and NSE, also, confirm that the simulation results using the estimated parameters are acceptable. For the application instance, the mean values from the estimated parameters are used as the representative parameters (Tc = 8.5 h and K = 6.5 h). Figure 13 shows a scatterplot of the observed runoff flows and the calibrated runoff flows using the representative parameters. The data are for the same sixteen storm events in Table4. The error indices, NSE, BS, and CC, showed similar results, as above. Water 2018, 10, x FOR PEER REVIEW 17 of 22 Water 2018, 10, 833 16 of 21 Table 4. Estimated results of parameters with error indices between the observed and simulated streamflows: In this table, Errors in peak flow and total runoff flow are calculated using the equation, Tableerror 4. Estimated(%) = (cal. −results obs.)/obs. of × parameters 100. Correlation with coefficient error indices (CC), bias between (BS), and the Nash observed-Sutcliffe and efficiency simulated streamflows:coefficient In (NSE this) table,indicate Errors correlation in peak coefficient, flow and bias total (ratio), runoff and flow Nash are-Sutcliffe calculated efficiency using coefficient the equation,, errorrespectively (%) = (cal.. − obs.)/obs. × 100. Correlation coefficient (CC), bias (BS), and Nash-Sutcliffe efficiency coefficient (NSE) indicate correlation coefficient, bias (ratio), and Nash-Sutcliffe efficiency Parameter Error Indices coefficient, respectively.Event Tc K Peak Total Runoff Flow Time to CC BS NSE (hr) (hr) Flow (%) (%) Peak (h) 1 8.3Parameter 9.2 105.6 39.1 Error Indices0.0 0.95 1.39 −0.56 Event Tc K Peak Total Runoff Time to 2 6.5 8.4 −10.8 −0.9 0.0 0.98CC 0.99 BS 0.95 NSE (hr) (hr) Flow (%) Flow(%) Peak (h) 3 8.2 4.9 −8.7 1.7 −1.0 0.98 1.02 0.96 41 9.5 8.3 8.0 9.2 86.1 105.6 − 39.12.3 − 0.01.0 0.92 0.95 0.98 1.39 −0.14−0.56 2 6.5 8.4 −10.8 −0.9 0.0 0.98 0.99 0.95 53 7.9 8.2 8.5 4.9 −−14.48.7 − 1.77.4 −0.01.0 0.96 0.98 0.93 1.02 0.89 0.96 64 7.6 9.5 9.3 8.0 8.9 86.1 −9.82.3 −7.01.0 0.96 0.92 1.10 0.98 0.79−0.14 75 9.9 7.9 6.1 8.5 −−22.514.4 −8.47.4 0.00.0 0.95 0.96 1.08 0.93 0.90 0.89 86 7.4 7.6 9.7 9.3 19.4 8.9 − 9.83.3 7.00.0 0.96 0.96 0.97 1.10 0.72 0.79 7 9.9 6.1 −22.5 8.4 0.0 0.95 1.08 0.90 9 7.8 5.7 −15.0 −23.1 0.0 0.95 0.77 0.86 8 7.4 9.7 19.4 −3.3 0.0 0.96 0.97 0.72 109 9.3 7.8 6.0 5.7 −5.815.0 −9.823.1 0.00.0 0.99 0.95 1.10 0.77 0.98 0.86 1110 7.7 9.3 2.5 6.0 −−3.45.8 2.7 9.8 0.00.0 0.95 0.99 1.05 1.10 0.81 0.98 1211 9.5 7.7 2.3 2.5 3.4−3.4 − 2.75.6 − 0.01.0 0.99 0.95 0.94 1.05 0.97 0.81 12 9.5 2.3 3.4 −5.6 −1.0 0.99 0.94 0.97 13 9.0 9.7 −9.5 −23.1 −4.0 0.96 0.77 0.86 13 9.0 9.7 −9.5 −23.1 −4.0 0.96 0.77 0.86 1414 8.9 8.9 5.7 5.7 −−18.118.1 −15.115.1 1.01.0 0.97 0.97 0.85 0.85 0.92 0.92 1515 8.1 8.1 3.0 3.0 −−5.85.8 4.3 4.3 −−1.0 0.95 0.95 1.04 1.04 0.90 0.90 1616 9.7 9.7 4.5 4.5 −−7.17.1 2.0 2.0 −−2.0 0.98 0.98 1.02 1.02 0.96 0.96 MinMin 6.5 6.5 2.3 2.3 −−22.522.5 −23.123.1 −−4.0 0.92 0.92 0.77 0.77 −0.56−0.56 MaxMax 9.9 9.9 9.7 9.7 105.6 105.6 39.1 39.1 7.07.0 0.99 0.99 1.39 1.39 0.98 0.98 Mean 8.4 6.5 6.4 −0.2 −0.1 0.96 1.00 0.74 Mean 8.4 6.5 6.4 −0.2 −0.1 0.96 1.00 0.74

Figure 13 Scatterplot of the observed and calibrated runoff flows for the sixteen storm events in Table 4: Figure 13. Scatterplot of the observed and calibrated runoff flows for the sixteen storm events in Table4: In the figure, the acronyms are same as Table 4 and the values are in average. In the figure, the acronyms are same as Table4 and the values are in average. In the application example, a different complex storm event was simulated, which was not used in Inthe the derivation application of the example, R-curve and a different the estimation complex of the storm model event parameters. was simulated, The complex which storm was event not used in theoccurred derivation from of 01 the-Dec R-curveember-20 and14 to the 11 estimation-December-20 of14 the. During model the parameters. period, three The independent complex stormrainfall event occurredevents from generated 01-December-2014 three corresponding to 11-December-2014. streamflow responses During. The first the event period, is used three as independenta spin-up period rainfall eventsand generated as a starting three point corresponding of estimating streamflowdirect runoff responses.using the R- Thecurve. first The event MRMS is useddata was as a used spin-up as the period and as a starting point of estimating direct runoff using the R-curve. The MRMS data was used as the gridded rainfall data. An hourly time step was chosen as the temporal resolution of the model simulation. To apply the R-curve, a period for the model spin-up was needed. This process is common in continuous hydrologic models to set up initial conditions. During this spin-up period, a CN value Water Water2018, 201810, 833, 10, x FOR PEER REVIEW 18 of 2217 of 21

gridded rainfall data. An hourly time step was chosen as the temporal resolution of the model was thesimulation. initial conditionTo apply the for R the-curve, soil a moistureperiod for state,the model and spin the R-curve-up was needed. can start This to process estimate is common soil moisture retentionin continuous with S values hydrologic at the models end of to the set spin-up up initial period. conditions. In this During application this spin instance,-up perio thed, a spin-upCN value period encompasseswas the initial the condition rainfall eventfor the from soil moisture 01-December-2014 state, and the toR-curve 03-December-2014, can start to estimate and soil the moisture two rainfall eventsretention on 06-December-2014 with S values at the and end 11-December-2014 of the spin-up period were. In this used application to verify instance, the R-curve the spin performance-up period with encompasses the rainfall event from 01-December-2014 to 03-December-2014, and the two rainfall events the hydrologic model. on 06-December-2014 and 11-December-2014 were used to verify the R-curve performance with the Figure 14 shows the rainfall-runoff simulation results with streamflow that was observed by the hydrologic model. USGS streamFigure station. 14 shows In the Figure rainfall 14-,runoff there simulation are two simulations. results with streamflow One is the that result was of observed using theby the R-curve method,USGS and stream the otherstation. one In Figure (antecedent 14, there condition) are two simulations. represents One the is conventional the result of usingCN method the R-curve by using AMCmethod conditions, and [ the16]. other The AMC one (antecedent conditions condition) (AMC I, represents II, and III) the depend conventional on the accumulated CN method by precipitation using amountAMC over conditions a five-day [16]. The antecedent AMC condition periods (AMC (P5). I, ForII, and the III) 06-December-2014 depend on the accumulated event, theprecipitation results show thatamount the difference over a five of-day total antecedent runoff flows period between (P5). For the the 06 observed-December and-2014 R-curveevent, the is results only show 11%. that For the 11-December-2014the difference of event, total the runoff results flows show between that the the R-curve observed method and R- alsocurve performed is only 11%. well For with the errors in peak11-Dec flowember and-20 total14 event, runoff the flowresults of show 23% andthat the−6%, R-curve respectively. method also On performed the other well hand, with the errors simulation in peak flow and total runoff flow of 23% and −6%, respectively. On the other hand, the simulation from from the conventional CN method showed that the errors of peak flow and total runoff flow are the conventional CN method showed that the errors of peak flow and total runoff flow are −31% and − − 31%−42% and, respectively42%, respectively.. According According to the conventional to the conventional CN method criteria CN method used to criteria decide used the AMC to decide the AMCcondition conditionss (I, II, and (I, II II,I),and the 11III),-Dec theember 11-December-2014-2014 was classified was as classifiedAMC I (P5 as < AMC35.5 mm I (P5 in the < 35.5 rainy mm in the rainyseason season [16]), [which16]), which is a dry is a drycondition condition.. This classificationThis classification resulted resulted in a CN in a value CN value for the for the 11-December-201411-December-2014 event event that that was was underestimated.

Figure 14. Rainfall-runoff simulation results: In this figure, ‘obs’ indicates observed streamflow, ‘Sim Figure 14. Rainfall-runoff simulation results: In this figure, ‘obs’ indicates observed streamflow, [R-curve]’ indicates a simulation hydrograph resulting from the proposed R-curve method, and ‘Sim ‘Sim [R-curve]’ indicates a simulation hydrograph resulting from the proposed R-curve method, [Antecedent condition]’ indicates a simulation hydrograph resulting from the conventional antecedent and ‘Sim [Antecedent condition]’ indicates a simulation hydrograph resulting from the conventional condition method in CN method to determine the antecedent CN value. antecedent condition method in CN method to determine the antecedent CN value. 7. Discussion and Conclusions 7. Discussion and Conclusions This study developed a new approach for deriving a soil moisture retention recovery curve (RThis-curve). study The developed R-curve provides a new a approachmeans for tracking for deriving the time a-variable soil moisture state of retentionsoil moisture recovery retention curve (R-curve).(SMR) Theat hourly R-curve time provides-scales. Its aapplication means for can tracking improve the continuous time-variable flood forecasting state of soil when moisture a complex retention (SMR)storm at hourly event occurs time-scales. in a short Its time application frame. The can R- improvecurve method continuous was derived flood from forecasting a change whenof SMR a and complex the interval of no rain (INR). The data were obtained for the upper Napa River basin in California and storm event occurs in a short time frame. The R-curve method was derived from a change of SMR the sixteen rainfall events from 2011 to 2012 were used in estimating the event-based SMR. The and themethodology interval ofthat no israin proposed (INR). in Thethis study data werewas based obtained on the for CN the method, upper and Napa is considered River basin appropriate in California andto the apply sixteen for the rainfall other watersheds events from using 2011 observed to 2012 direct were runoff. used The salien in estimatingt results of the the study event-based include: SMR. The methodology that is proposed in this study was based on the CN method, and is considered appropriate to apply for the other watersheds using observed direct runoff. The salient results of the study include: We addressed the importance of considering the soil profile with multi-layers since interaction between soil layers by infiltration and percolation influences the change of SMR. In this study, the recovery of SMR had a negative value at the beginning of INR since the soil column is assumed Water 2018, 10, 833 18 of 21 to be a single layer in the CN method. Thus, SMR resulting from the CN method cannot account for the negative retardation effect arising from the difference of downward flows by infiltration and percolation. However, the retardation effect concept is theoretically the same as the Green and Ampt infiltration model due to the wetting front. When the infiltration rate is faster than the percolation rate, the boundary (wetting front) between the upper layer and the lower layer could develop a temporary stagnant water layer when the retardation effect occurs. Thus, when the upper layer is saturated, but not the lower layer, it appears as if the total SMR is decreased without additional rainfall for the INR, even though the actual total SMR is recovered. For the three sections of the R-curve, the first section is regarded as the infiltration period, so the recovery rate is relatively slower than the second section. Also, the downward percolation is the main contributor to recover SMR in this time period. The second R-curve section involves the rapid recovery of SMR. It is the fastest rate among three sections since most of soil layer is almost saturated. In the third section, the movement of water is provisionally complete, and evaporation is a potential dominant factor to the recovery of SMR. Thus, the recovery rate is the slowest among three sections. The recovery rate of this section converges on a constant value that is close to zero in the long-term. The HEC-HMS model was used to comparison the performance of the CN and R-curve methods for a storm event with multiple rain and streamflow response pulses. It was shown that the R-curve method outperformed the traditional CN method in terms of characterizing the peak flow and the total runoff volume. Based on the results above, this new R-curve approach could be used to improve the limitation of the traditional CN methodology, which regards the soil column as a single layer in accounting for the total value of SMR. Ultimately, the R-curve method could be used to estimate the state of SMR, which would improve flood forecasting during a complex storm event with multiple rainfall pulses without using complicated hydrologic models. Since the R-curve method can indicate the state of soil moisture using the recovery rates, it could be used to represent a critical time period that is vulnerable to flooding due to saturated soils, even by light rainfall.

Author Contributions: J.K., L.J., R.C., V.C. and J.C. conceived and designed the analysis; Conceptualization, J.K. and L.J.; Formal analysis, J.K. and J.C.; Project administration, R.C.; Software, J.K.; Supervision, L.J. and V.C.; Writing—original draft, J.K.; Writing—review & editing, L.J. and R.C., J.K. wrote the paper. Funding: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1A6A3A03011525). Acknowledgments: Funding for this study was provided by the California Department of Water Resources and the NOAA Physical Sciences Division. We appreciate the internal reviewer contribution from Ben Livneh. Conflicts of Interest: The authors declare no conflict of interest.

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