Curve Number Approach to Estimate Monthly and Annual Direct Runoff

ion. The maximum potential retention, S, is related to the curve number, CN, an 63 empirical quantity that depends on land use and soil characteristics (NRCS 2004a,b): 64 Curve number approach to estimate monthly and annual direct runoff Guswa et al. page 4 of 21 . = 1000 () − 10 (S in inches) (2) . = 25 400 () − 254 (S in millimeters) (3) In application, curve numbers are either calibrated to rainfall-runoff data or estimated 65 from land-use and soil characteristics when streamflow data are unavailable (NRCS 2004a,b; 66 Hawkins et al. 2009). Both cases require a choice for l. The National Engineering Handbook 67 indicates a value of 0.2 for l, and tabulated values of curve numbers for different hydrologic soil 68 groups and land covers are based on this value (NRCS 2004a,b). Recent results, however, 69 indicate a smaller value of l, closer to 0.05 (Jiang 2001; Hawkins et al. 2009; Shaw and Walter 7


Introduction
Changes to the landscape affect many hydrologic processes and ecosystem services (Daily 1997;National Research Council 2004;Martin-Ortega et al. 2015).Estimates of those effects, even when uncertain, benefit land-management decisions.With respect to water resources, effects of interest include changes to total streamflow, to flooding potential, and to the availability of baseflow at monthly to annual and multi-annual timescales (Brauman et al. 2007;Guswa et al. 2014;Bremer et al. 2016;Ouyang et al. 2016).Some decisions may require a precise and detailed analysis.Other contexts may tolerate greater uncertainty in order to reduce the time and resources required; these include land-management decisions in ungaged and data-poor locales, or rapid assessments across many ecologic and hydrologic processes that may be followed up by more detailed studies.
The Natural Resources Conservation Service (NRCS) curve number method estimates the direct runoff that results from an individual rainfall event as a function of land cover and soil characteristics (Natural Resources Conservation Service 2004a,b).The concepts have been incorporated into many popular hydrologic models, such as HEC-HMS (U.S. Army Corps of Engineers 2000), SWMM (Rossman 2007), HydroCAD (HydroCAD Software Solutions LLC 2011), SWAT (Neitsch et al. 2011), WinTR-20 (Natural Resources Conservation Service 2015), and the InVEST seasonal water yield model (Sharp et al. 2016).The method also has a number of known challenges (Ponce and Hawkins 1996;Hawkins et al. 2009).Specifically, not all watersheds exhibit the asymptotic approach to a constant curve number (Hawkins 1993), curve numbers determined from rainfall-runoff data show significant variability (Hjelmfelt 1991;Shaw and Walter 2009;Hawkins et al. 2009), use of tabulated curve numbers in ungaged watersheds is highly uncertain (Titmarsh et al. 1995;Hawkins et al. 2009;Tedela et al. 2012), and the method page 3 of 21 is misused and misapplied (Walter and Shaw 2005;Ogden and Stallard 2013).Despite these challenges, Hawkins et al. (2009) recognize the potential for the curve number method to inform land-management decisions.
This study extends the application of the curve number method for land-management decisions when the uncertainty of the event-based method is tolerable but rainfall event data are unavailable.This investigation tests whether direct runoff accumulated over a month or year can be estimated, without an appreciable increase in the uncertainty, by approximating the distribution of actual rainfall depths with an exponential distribution.Rather than requiring a full description of event-by-event precipitation, this new approach requires only total rainfall and an estimate of the number of events over the defined period of interest.The sections that follow explain this new approach and present results from tests on 544 U.S. watersheds.

Curve number method applied to event rainfall
The curve number method estimates the depth of direct runoff from a specified rainfall event (NRCS 2004b).Direct runoff refers to the water that reaches a stream quickly without specification of the pathway or origin of that water (Hawkins et al. 2009).For a given rainfall depth, Pi, the depth of direct runoff, Qi, is calculated as where the subscript i refers to an individual event, S is maximum potential retention with dimensions of depth, and lS is the rainfall depth needed to initiate runoff, also called the initial  (Jiang 2001;Hawkins et al. 2009;Shaw and Walter 2009;Dahlke et al. 2012).This study uses l = 0.05.Using that value requires a modification of the curve numbers given in the handbook tables, and Jiang (2001) provides the following relationship: () 9.9; = 0.0054 • (() 9./ ) / + 0.46 • () 9./ (4) where CN0.2 represents a tabulated curve number developed under the presumption that l = 0.2 and CN0.05 represents the curve number for use with l = 0.05.
While the curve number method and tabulated values were developed to estimate runoff from large events, the method has been applied to a wide range of event magnitudes (Hawkins et al. 2009).Hawkins (1993), however, showed that estimates of curve numbers derived from rainfall and runoff data vary with event depth; curve numbers are typically larger for smaller events and approach constant values for larger events, though there are exceptions (e.g., Tedela et al. 2012).Thus, an estimate of runoff for a single, small event could have a large relative error.Runoff accumulations over multiple events, however, are dominated by large events, and the non-linearity of Eq. ( 1) represents this phenomenon.For example, the MacLeish Field page 5 of 21 Station in West Whately, MA experienced seven rain events between 3 June and 27 June 2009 with magnitudes of 3.6, 4.0, 8.4, 11.2, 22.9, 38.6, and 58.7 mm (Guswa and Spence 2011).
Applying Eq. (1) to each event, with a curve number appropriate for pasture (CN0.05= 59), gives runoff estimates of 0.0, 0.0, 0.0, 0.0, 1.0, 4.3, and 11.0 mm, respectively.Over 93% of the total 16.4 mm of runoff is generated by the two largest events, and the contribution of the small events to the total error in accumulated runoff is small.Consequently, when event data are available, accumulated runoff over a longer time period (QN) can be estimated by direct application of the curve number method to n events over a period of N days, (5)

Curve number approach for monthly and annual runoff
This study presents an approach to estimate monthly and annual direct runoff when rainfall data are not available.This approach requires a tabulated curve number based on landscape characteristics (NRCS 2004a), total rainfall (PN) over the period of interest (N days), and an estimate or measurement of either the mean event depth (a) or the frequency (h) of rainfall events (events per day).This new approach approximates the actual distribution of rainfall depths with an exponential distribution, where p is rainfall depth and a is the mean event depth, which can be estimated as The exponential distribution is a recognized model of rainfall depths (e.g., Eagleson 1978;Richardson 1981;Rodriguez-Iturbe et al. 1999;Laio et al. 2001).Additionally, the exponential distribution is fully characterized by a single parameter, mean rain depth, making it useful in applications with limited data.
Combining event-based runoff (Eq. 1) with an exponential distribution of rainfall depths gives an expression for the mean runoff per event from the new approach, where angle brackets indicate expected value.Substituting Eqs. ( 1) and ( 6) into (8) gives Solving Eq. ( 9) results in the following expression for the mean runoff: where E1(x) is the exponential integral (Abramowitz and Stegun 1972), Cumulative runoff over the period of interest is The strength of this new approach lies in its approximation of the distribution of large events.
For the earlier example of seven rainfall events (3.6, 4.0, 8.4, 11.2, 22.9, 38.6, and 58.7 mm), event-by-event application of Eq. (1) results in an estimate of 16.4 mm of total runoff (CN0.05= 59).If Eq. ( 1) were applied directly to the mean rainfall depth of 21.1 mm, the estimate of page 7 of 21 cumulative runoff from seven such events would be just 5.6 mm; if Eq. ( 1) were applied directly to the total 147.4 mm of rainfall, estimated runoff would be 61.0 mm.Application of the new approach with an exponential distribution of depths results in an estimate of cumulative runoff of 17.4 mm, very close to the 16.4 mm estimated by application of the curve number method to each event individually.

Rainfall and runoff for U.S. watersheds
To test the new approach, this work used a dataset of daily meteorology and streamflow for 671 watersheds throughout the contiguous United States (Newman et al. 2014;Newman et al. 2015).
Watersheds range in size from 1 to 25 000 km 2 , with a median size of 335 km 2 and two-thirds of the watersheds between 100 and 1000 km 2 (Newman et al. 2015).Streamflow data are from the U.S. Geological Survey and the Daymet dataset is the source of meteorological data (Newman et al. 2015).The dataset includes precipitation and streamflow records from 1/1/1980 through 12/31/2010.Some of the records were eliminated or modified for this analysis after quality assurance checks; Appendix A includes details.
Runoff and baseflow were computed for two time scales of analysis: monthly and annual.
Because the curve number method is not appropriate for snowmelt, analyses were limited to snow-free months and years.For the monthly analysis of each watershed, this study eliminated all months for which the snow-water equivalent was non-zero for some time during the month.
Similarly, for the annual analysis, all years that were influenced by snow were removed.To ensure an adequate sample size of monthly runoff values for each watershed, monthly analyses were restricted to watersheds with more than ten months (total, not per year) of snow-free page 8 of 21 observations, and annual analyses were limited to watersheds with more than ten years of snowfree observations.Figure 1 presents a map of the watersheds used to test the approaches.Open circles represent watersheds included in the monthly analysis; filled circles represent watersheds included in both monthly and annual analyses.
Daily streamflow was separated into baseflow and direct runoff with a one-parameter recursive digital filter (Nathan and McMahon 1990) with a filter parameter of 0.925.This automated method of baseflow separation is objective, repeatable, and gives results similar to the smoothed minima method (Nathan and McMahon 1990).Summing direct runoff over each month and year produced records of observed monthly (Qm obs ) and annual (Qa obs ) direct runoff for each watershed.

Curve numbers determined from daily records
The objective of this investigation is to test whether the accumulated runoff estimated by using an exponential distribution of rainfall depths is equivalent to that determined by applying the curve number method directly to a record of daily rainfall depths.To separate the uncertainty introduced by the use of tabulated curve numbers from the uncertainty due to the approximation of the rainfall distribution, a curve number for each watershed was determined through calibration.Consistent with the intent of estimating accumulated runoff, the curve number for each watershed was determined by matching the cumulative direct runoff, estimated by applying the curve number to daily rainfall, to the cumulative observed runoff over the entire period of record.This calibration ensures that the average bias in the daily application of the curve number method is zero, i.e., the mean error between observed (monthly or annual) runoff and the runoff estimated by application of the curve number method to daily rainfall is zero.page 9 of 21 Accumulated runoff is dominated by large events, and the largest events of the period of record strongly influence the calibration of the curve number.
With a calibrated curve number for each watershed, this study applied Eq. ( 1) to daily rainfall to compute daily runoff, which was then summed to create records of monthly and annual direct runoff.Monthly and annual errors were quantified by taking the difference between the monthly and annual estimates and observations: where Qm daily and Qa daily represent the monthly and annual direct runoff, respectively, estimated by applying the curve number method to daily rainfall.By design, the mean values of em daily and ea daily are zero for each watershed, as noted previously.

Application of the new approach
In the new approach, the actual, empirical distribution of daily rainfall depths is replaced with an exponential distribution, defined by a mean event depth, a, for each month or year.This average depth was calculated in two ways.One variation computed the mean rainfall depth by dividing the cumulative rainfall by the actual number of days with rain in each month or year.A second variation evaluated the utility of the new approach when information on number of events is approximate.In the monthly application, mean rainfall depth was computed with the average number of events for that month over all years in the dataset for that watershed (for example, the average number of events for all Septembers).Similarly, the average number of events per year was used in the annual application.The resulting two variations of the exponential distributions were used with calibrated curve numbers in Eqs.(10-12) to estimate monthly and annual runoff page 10 of 21 for each watershed.Thus, each watershed is associated with four records of monthly (and annual) runoff: observed runoff, runoff estimated by application of the curve number method to daily rainfall, runoff estimated from an exponential distribution of rain depths with mean rainfall depth determined by the actual number of events in each month (and year), runoff estimated from an exponential distribution of rain depths with mean rainfall depth determined by the average number of events.

Tests of the new approach
Both across watersheds and for each individual watershed, this study evaluated the performance of the new monthly and annual approaches by assessing 1) the mean error in monthly and annual runoff relative to observations, 2) the difference in squared errors of monthly and annual runoff between the new approach and the application of the curve number method to daily rainfall, and 3) the error in runoff relative to the uncertainty attributed to the use of tabulated curve numbers in ungaged watersheds.The descriptions that follow refer to monthly runoff, and the same tests apply to annual estimates as well.All tests were restricted to months (and years) with non-zero observed direct runoff.
The first tests assessed the mean error between observations and estimates from the new approach.A non-parametric bootstrap technique (Efron and Tibshirani 1993) was used to test the null hypothesis that the mean error in monthly runoff is indistinguishable from zero.
Sampling (with replacement) the m monthly errors m times for all months and all watersheds generated a bootstrap estimate of the mean error.This process was repeated to generate 10 000 estimates of the mean error.A 95%-confidence interval for the mean error in monthly runoff was created from the 2.5% and 97.5% quantiles of the bootstrap estimates.The null hypothesis that the mean error is indistinguishable from zero was accepted if the confidence interval contained zero.Estimates of the mean monthly runoff were also regressed against the observed means for all watersheds.To assess the mean error for each individual watershed, 10 000 bootstrap estimates of the mean error were generated by sampling (with replacement) the M months of errors M times for each watershed.A 95%-confidence interval for the mean error was created from the 2.5% and 97.5% quantiles of the bootstrap estimates.
Even when Eq. ( 1) is applied to daily data and curve numbers are calibrated to ensure no bias in the mean monthly runoff, model structural error leads to uncertainty in estimated runoff for any given month.Approximating the rainfall depths with an exponential distribution further increases this uncertainty.While it is desirable for monthly errors in the new approach to be small, more important for this study is to test whether the errors from the new approach are comparable to those from the application of the curve number method to daily rainfall, i.e., to test whether the additional error due to the exponential approximation is small relative to the structural error of the curve number method.For each watershed, the square of the error between estimated and observed monthly runoff was determined, and the difference in squared-error between the daily method and the new approach computed: This statistic is positive when the squared error in monthly runoff is larger for the new approach and negative when the error is larger for the daily application.To test whether the mean of squared errors from the new approach are significantly larger than those from the daily application of the curve number, 10 000 bootstrap samples of the mean difference in squarederror were generated.The null hypothesis that the error of the new approach is no larger than the error in the daily method (one-sided test) was rejected if the 5%-quantile of the mean difference page 12 of 21 in squared error was greater than zero.A linear regression of the square root of the meansquared error (RMSE) from the new approach to the RMSE from the daily application of the method quantified the difference in uncertainty between the approaches.
A third test compared the mean error in runoff estimates with the uncertainty due to the use of tabulated curve numbers for ungaged basins.Tabulated curve numbers are a function of land-cover and soil characteristics and are reported for average antecedent runoff conditions, ARC II (NRCS 2004a).Titmarsh (1995) and Hawkins andWard (1998, reproduced andcited in Hawkins et al. 2009) showed that the uncertainty in using tabulated curve numbers is large and comparable to the envelope of uncertainty created by using curve numbers that correspond to antecedent runoff conditions ARC I and ARC III (NRCS 2004b).Hjelmfelt (1991) showed that this envelope created by ARC I and ARC III represents the 10% and 90% exceedance probabilities for runoff.Runoff estimates from the new approach were tested against this envelope of uncertainty that resulted from the application of Eq. ( 1) to daily rainfall with curve numbers corresponding to ARC I and III for the calibrated curve numbers.

Results
Removing months and years with snow from the analyses left 544 watersheds with more than ten months of monthly runoff observations and 97 watersheds with more than ten years of annual data (Fig. 1).The total number of observations of monthly runoff across all watersheds and all months is 127 927; the number of total observations of annual runoff is 2270.Estimates of mean monthly runoff from the new approach show good agreement with the observed runoff (Fig. 2).
Though the mean errors are statistically different from zero (95% confidence), they are small: 1.2 mm/month and 2.9 mm/month for use of the actual and average number of events, respectively (Table 1).  1 and Fig. 2).Considering each watershed separately, the error in mean monthly runoff is indistinguishable from zero (95%confidence interval) for 65% of the 544 watersheds when the actual number of rain events is used in the new approach (Table 2).When the average number of events per month is used, the error in mean monthly runoff is indistinguishable from zero (95%-confidence interval) for 26% of the 544 watersheds.For both monthly approaches, estimates of mean monthly runoff for all (100%) of the 544 watersheds fall within the envelope of uncertainty associated with using tabulated curve numbers (x's in Figure 2).
The RMSE of monthly runoff for the application of the calibrated curve number method to daily rainfall quantifies the structural error of the method.Fig. 3 indicates that this structural error is increased only slightly by the introduction of the exponential approximation.Regression slopes of 1.02-1.10indicate that the RMSE of monthly runoff determined via the new approach is approximately 5-10% larger than the RMSE for monthly runoff determined via application of the curve number method to daily data (Table 1 and Figure 3).Mean monthly errors from the approach using the average number of events per month are larger than those from the approach that uses the actual number of events.The paired test of differences in monthly squared errors (Eq.15) found that monthly squared errors from the new approach are not significantly larger than the errors from the daily application of the curve number method for 80% and 65% of the watersheds (actual and average number of events, respectively, 95%-confidence, 1-sided test, Table 2).
Tables 1 and 2 and Figs. 4 and 5 present results for the annual approaches.Fig. 4 indicates a good match in annual runoff between the new approach and observations.The mean errors in annual runoff are statistically different from zero (95% confidence, Table 1), and they page 14 of 21 are small: 10 mm/year and 8 mm/year for use of the actual and average number of events, respectively.Across the watersheds, mean annual runoff estimated via the new method is approximately 7% less than observed, as evidenced by a regression slope of 0.93 (Table 1).
Mean error in annual runoff is indistinguishable from zero for 64% (actual number of events) and 65% (average number of events) of watersheds.Errors in annual estimates of direct runoff with the new approach are comparable to the errors associated with employing the curve number method to daily data (Fig. 5).The RMSE of annual runoff determined via the new approach is approximately 4 mm larger than the RMSE of annual runoff determined via application of the curve number method to daily data, indicated by regression slopes of 1.0 and intercepts of 4 mm (Table 1 and Fig. 5).The paired tests indicate that squared errors from the new approach are not significantly larger than the errors from the daily application of the curve number method for 74% and 88% of the watersheds (actual and average number of events, respectively, 95%confidence, 1-sided test).Estimates of mean annual runoff for all (100%) of the 97 watersheds fall within the uncertainty envelope associated with use of tabulated curve numbers (x's in Fig. 4).

Discussion
Figs. 2-5 indicate that the new approach presented in this work estimates monthly and annual direct runoff with a similar degree of certainty as the application of the curve number method to daily data for ungaged watersheds.The overestimation of runoff in the monthly results (Fig. 2 and Table 1) may indicate a deviation from the simplification of an exponential distribution of rainfall events.If actual rain events within a month are more similar to each other, i.e., if the empirical distribution has a smaller variance than the exponential, then the approach based on the exponential distribution would overestimate runoff, consistent with what is seen in Fig. 2. page 15 of 21 Month-to-month and year-to-year errors in estimates from the new approach are similar to errors from the application of the curve-number method to daily rainfall (Tables 1 and 2 and Figs. 3 and    5).Most importantly, mean monthly and annual estimates of direct runoff lie well within the confidence interval attributed to uncertainty in the curve number (Figs. 2 and 4).This is consistent with earlier findings that estimated runoff is more sensitive to the selection of the curve number than to the precipitation depth (Hawkins 1975) and indicates that the approximation of an exponential distribution of rainfall depths does not appreciably increase the uncertainty associated with the application of the curve number method in ungaged watersheds.
The large uncertainty in estimates of monthly and annual runoff for ungaged watersheds suggests that runoff estimates should be used with care.
While the new approach does not require daily rainfall data, it does require an estimate of the number of rain events within a given period of interest.Tables 1 and 2 and Figs Many monthly (and annual) water-balance models have as a first step the partitioning of precipitation into direct runoff and retention (e.g., Ponce and Shetty 1995;Zhang et al. 2008;Sivapalan et al. 2011;Kirby et al. 2013;Chen and Wang 2015).These incorporate a relationship between monthly rainfall and direct runoff as a function of landscape characteristics (such as slope, soil type, land use) and state variables of the system (such as soil moisture and streamflow).The approach presented here provides a means for estimating or eliminating model parameters in these models.For example, the Dynamic Water Balance Model (DWBM; Zhang page 16 of 21 et al. 2008), relies on a parameter, a1, to partition monthly precipitation into direct runoff and retention.This parameter must generally be determined via calibration, as attempts to relate the parameter to measurable watershed characteristics have proved challenging (Zhang et al. 2017).
The approach presented here, with knowledge of the curve number and typical number of precipitation events, is another way to determine the amount of direct runoff from monthly precipitation.
Estimates of annual runoff from this new approach enable the partitioning of annual streamflow into direct runoff and baseflow.For example, a Budyko-type approach can estimate average annual streamflow based on average annual precipitation and potential evapotranspiration (e.g., Budyko 1974;Porporato et al. 2004;Szilagyi and Jozsa 2009;Hamel and Guswa 2015).Based on rainfall data from Monteverde, Costa Rica (Guswa et al. 2007), the Budyko curve predicts an increase in annual streamflow of 160 mm/yr following the conversion of forest to pasture (Table 3).The new approach presented in this study complements this result by estimating changes to direct runoff and, by subtraction, baseflow.For two soil groups (B and D), the new approach indicates a decrease in baseflow (40 mm/yr or 210 mm/yr for soil groups B and D, respectively), despite the increase in total streamflow.The large uncertainty associated with using a tabulated curve number (characterized by ARC I and III), however, prevents a definitive statement, as the confidence intervals for the change in baseflow include zero (Table 3).Nonetheless, the interpretation that baseflow is more likely than not to decrease when forest is converted to pasture may be sufficient to inform land-management decisions.

Conclusions
This study developed a new approach to estimate monthly and annual direct runoff by combining the NRCS curve number method with an exponential distribution of rainfall depths.Evaluation of the approach with daily rainfall and runoff data from 544 U.S. watersheds indicates that the error introduced by the exponential approximation is small and lies well within the uncertainty associated with application of the curve number method in ungaged watersheds.The simplicity and robust performance of the approach indicate that it can inform planning and landmanagement decisions in data-poor contexts.
Tables Table 1: Assessment of mean error and root-mean-squared error of monthly and annual runoff across watersheds.
Regression of estimated mean runoff from new approach against observed (Figs. 2 and 4)     Mean monthly direct runoff estimated by new approach versus observed mean monthly direct runoff for 544 U.S. watersheds.Circles represent estimates for which the actual number of rain events per month were used; pluses represent estimates that use the average number of events per month.The uncertainty envelope associated with tabulated curve numbers is given by the x's.
Figure 3 Comparison of root-mean-squared error (RMSE) for estimates of monthly direct runoff from the application of the curve number method to daily rainfall data to RMSE from the new approach for 544 U.S. watersheds.Circles represent estimates using the actual number of rain events per month; pluses represent estimates that use the average number of events per month.

Figure 4
Mean annual direct runoff estimated by new approach versus observed mean annual direct runoff for 97 U.S. watersheds.Circles represent estimates using the actual number of rain events per year; pluses represent estimates that use the average number of events per year.The uncertainty envelope associated with tabulated curve numbers is given by the x's.
Figure 5 Comparison of root-mean-squared error (RMSE) in estimates of annual direct runoff from the application of the curve number method to daily rainfall data to RMSE from the new approach for 97 U.S. watersheds.Circles represent estimates using the actual number of rain events per year; pluses represent estimates that use the average number of events per year.
. 2-5 indicate that estimates based on an average number of events are almost as good as those that use the actual number of events.Local estimates of the number of rain events could be obtained from traditional knowledge, global precipitation datasets (e.g.,Gehne et al. 2016; The World Bank    Group 2016), or historical records.
Figure captions

Figure 2
Figure 2Mean monthly direct runoff estimated by new approach versus observed mean monthly direct runoff for 544 U.S. watersheds.Circles represent estimates for which the actual number of rain events per month were used; pluses represent estimates that use the average number of events per month.The uncertainty envelope associated with tabulated curve numbers is given by the x's.
annual runoff [mm], daily rain RMSE annual runoff [mm], annual rain Hawkins et al. 2009).Both cases require a choice for l.The National Engineering Handbook indicates a value of 0.2 for l, and tabulated values of curve numbers for different hydrologic soil abstraction.The maximum potential retention, S, is related to the curve number, CN, an empirical quantity that depends on land use and soil characteristics (NRCS 2004a,b): page 4 of 21 groups and land covers are based on this value (NRCS 2004a,b).Recent results, however, indicate a smaller value of l, closer to 0.05

Table 2 :
Mean error, magnitude of squared error, and mean error versus the uncertainty in curve number for each watershed.