NRCS Curve Number Method: Comparison of Methods for Estimating the Curve Number from Rainfall-Runoff Data

Rainfall-runoff methods from the Natural Resources Conservation Service (NRCS) have served the hydrologic community for many decades. These methods estimate runoff from actual and synthetic rainfall for purposes ranging from flood forecasting to hydrologic design. The first step in using these methods is to determine the curve number (CN), which is a value between 1 and 100, for the modeled watershed. The CN modulates the conversion of rainfall volume into runoff volume, with larger CNs estimating greater runoff. Forested watersheds and other natural systems that allow considerable infiltration may have CNs as low as the 40s and 50s, whereas highly urbanized systems with impervious surfaces may have CNs in the 80s or 90s. Determination of the NRCS CN is often the first of several calculations required in rainfall-runoff modeling, with subsequent modeling seeking to quantify the timing of the runoff peak and/or the shape of the overall runoff hydrograph resulting from a rainfall event. If the CN poorly models the runoff volume, the subsequent modeling of the peak discharge will be poor.

Although tables quantifying CNs as a function of land use and underlying soil type have long existed (SCS 1973, 1986), there is new interest in CN quantification, resulting primarily from efforts over the last decade to revisit the long-standing initial abstraction ratio, $\lambda$ (Fu et al. 2011; Galbetti et al. 2022; Lal et al. 2017; Lim et al. 2006; Menberu et al. 2015; Yuan et al. 2014). The observation commonly made to support a smaller initial abstraction ratio is that the long-standing value of $\lambda =0.20$ underpredicts runoff for small precipitation events. The initial abstraction ratio and CN are not independent. A smaller (larger) initial abstraction ratio will result in a smaller (larger) calibrated CN value for a given rainfall-runoff data set.

This paper examined multiple elements that collectively contribute to the calibration of a CN value from a set of rainfall-runoff observations. Decisions about what elements to employ and how to apply them influence the CN that is calibrated. Goodness-of-fit, bias, and defensibility of the methods used were analyzed. This paper discusses the core principles that should guide a recommended approach for future CN calibration efforts.

The NRCS curve number method dates to 1954, and was developed and calibrated using observations of rainfall-runoff data gathered from small, primarily agricultural watersheds instrumented by the Agricultural Research Service (ARS).

The central equation of the NRCS rainfall-runoff method iswhere $Q$ = runoff (mm); $P$ = precipitation, i.e., rainfall depth, originally and commonly taken over a 24-h duration (mm); and $S$ = storage (mm). The term $\lambda S$ represents the initial abstraction, which is the depth of rainfall captured by depression storage, leaf litter, and watershed surfaces. The parameter $\lambda$ is the initial abstraction ratio. This fraction of the precipitation is not available for runoff. If $P$ is less than $\lambda S$, then Eq. (1) does not apply, and $Q$ is zero. The long-standing value of $\lambda$ is 0.20, although studies have proposed that $\lambda$ is smaller, with a value of 0.05 the most commonly suggested alternative (D’Asaro et al. 2014; Galbetti et al. 2022; Hawkins et al. 2009; Lal et al. 2017). Proposals to formally change $\lambda$ to 0.05 also include a redefinition of $S$ (and its directly associated CN value), which partially compensates for the reduced value of $\lambda$ (USDA 2017a, b).

The value of CN is related directly to storage through

Values of CN were determined and tabulated (SCS 1973, 1986) by calibration methods that often were poorly documented (Hawkins et al. 2009). These tables present curve numbers as a function of land use and underlying soil type. With the advent of GIS, such tables commonly are used to infer pixel-scale CN values from commonly available sources of land use–land cover and soil maps.

The NRCS method for estimating runoff has been criticized (e.g., Ogden et al. 2017). It is empirical, and generally is applied in a coarse or lumped approach to determine runoff. However, this method remains popular because of its ease of application, its ability to quantify effects caused by land-use change, and the fact that it satisfies the basic engineering need to estimate watershed response at ungauged locations for purposes of hydrologic and hydraulic design. Owing to its widespread application and ongoing use as a design tool, an examination of the elements selected when performing a CN calibration is valuable and needed.

This paper explored four elements that comprise the CN calibration for a specific set of observed rainfall and runoff data: calibration approach, precipitation event threshold, data ordering approach, and initial abstraction value. This paper demonstrated how these different elements influence the CN value that is calibrated and the goodness-of-fit between the observed and predicted runoff data set.

A total of $12(3\times 2\times 2)$ experiments were performed in this study, defined by all permutations of three calibration approaches, two precipitation thresholds, and two initial abstraction fractions (Table 2). Furthermore, these 12 experiments were repeated for both naturally ordered and frequency-matched data. Within an experiment, individual calibrations were performed for each of the 31 study watersheds. For all naturally ordered data experiments, Figs. 6 and 7 present box-and-whisker plots of the distribution of calibrated CNs and $R_{\mathrm{SE}}$ values, respectively. Figs. 8 and 9 provide box-and-whisker plots of calibrated CNs and $R_{\mathrm{SE}}$ values for the frequency-matched data sets for the same watersheds and experiments. Finally, Figs. 10 and 11 show side-by-side box-and-whisker plots for direct comparison of the LS calibration findings for CNs and $R_{\mathrm{SE}}$, respectively, for naturally ordered and frequency-matched data. The box part of these plots shows the first and third quartiles, with the median represented by the horizontal line within the box. Outliers, defined as being more than 1.5 times the interquartile range, are graphed individually. The whisker part of these plots shows the extremes of the observations that were not determined to be outliers. In Figs. 7 and 9, the box plots for $R_{\mathrm{SE}}>2$ are truncated to show more-important detail for $R_{\mathrm{SE}}<2$.

The major findings deriving from the naturally ordered data experiments are summarized in Table 3. The largest CNs were calibrated from the NEH median method, and the smallest CNs were calibrated from the ASYMP method. Similarly, the largest $R_{\mathrm{SE}}$ values resulted from the NEH median method and the smallest $R_{\mathrm{SE}}$ values resulted from the ASYMP method. Although the LS method ranked in the middle for both measures, it produced both CN and $R_{\mathrm{SE}}$ values that are much more comparable in magnitude to those of the ASYMP method than to those of the NEH median method. If the quality of a calibration method is determined from the magnitude of the $R_{\mathrm{SE}}$ values it produces, the ASYMP method performed best and the NEH median method performed the worst. If robustness of a calibration method is determined from the outlier results it produces, the LS method was the most robust, and the NEH median method was the least robust. Again, whether quality or robustness is the measure, the ASYMP and LS methods were very comparable, and the NEH median method performed much more poorly.

A universal result was that the calibrated CNs were larger for $\lambda =0.20$ than for $\lambda =0.05$. This was true regardless of calibration method or precipitation threshold. The effect of precipitation threshold varied in strength as a function of calibration method: for LS, $P_T$ essentially had no effect; for the NEH median, $P_T$ had a strong effect, with larger CNs resulting from $P_T=0\mathrm{mm}$; for ASYMP, $P_T$ had a modest impact, with larger CNs resulting from $P_T=0\mathrm{mm}$. Concerning $R_{\mathrm{SE}}$, $\lambda$ caused modestly larger $R_{\mathrm{SE}}$ values for the LS and NEH median methods, but essentially had no effect for the ASYMP method. Precipitation threshold produced mixed results leading to modestly larger $R_{\mathrm{SE}}$ for $P_T=25.4\mathrm{mm}$ than for $P_T=0\mathrm{mm}$ for both the LS and ASYMP methods. However, the opposite was observed for the NEH median method, for which $P_T$ produced much larger $R_{\mathrm{SE}}$ values for $P_T=0\mathrm{mm}$ than for $P_T=25.4\mathrm{mm}$ (Fig. 5).

An additional point regarding Fig. 7 is merited. Hawkins et al. (2009) identified three different potential behaviors of the ASYMP fitting method: standard, violent, and complacent. The analyses presented in the present paper did not censor watersheds according to these behaviors, although Hawkins et al. (2009) expressly described complacent watersheds as not being “suitable for CN definition.” A rigorous definition of complacent watershed behavior could not be located, but seven watersheds in the data set were identified that did not reach their respective ${\mathrm{CN}}_{\infty }$ values over the range of observed rainfall values. A reduced 24-watershed data set was created that removed these watersheds and repeated all analyses. A uniform reduction of the $R_{SE}$ values in all distributions in Fig. 7 was observed—indicating that the relative standard error for the 24-watershed data set improved by about 0.1 compared with that of the overall 31-watershed data set. All other changes to the reported findings were small and were not specific to any of the four elements of the calibration processes that were investigated. That the changes in $R_{\mathrm{SE}}$ were observed across all experiment permutations indicates that the reported findings with regards to calibration methods, $\lambda$ values, and precipitation thresholds were insensitive to the absence or presence of the seven censored watersheds. The remainder of this paper presents findings for the full 31-watershed data set.

The frequency-matched data analyses summarized in Figs. 8 and 9 mirror the preceding statements for the naturally ordered data and the findings summarized in Table 3. Across calibration methods, precipitation thresholds, and $\lambda$ values, the calibrated CN values resulting from frequency-matched observations consistently were 5–10 points larger, depending on the experiment.

Notable findings followed from comparing like experiments for naturally ordered versus frequency-matched data. Figs. 10 and 11 provide a visual, side-by-side comparison of the LS experiment outcomes. Focusing on the data marked (a) ($P_T=0.\mathrm{mm}$, $\lambda =0.05$) in Fig. 10, the median CN increased from 56 for the naturally ordered results to 61 for the frequency-matched results. The interquartile ranges similarly shifted from $45<\mathrm{CN}<63$ (naturally ordered) to $55<\mathrm{CN}<72$ (frequency-matched). The $R_{\mathrm{SE}}$ for the LS experiments marked (a) had medians of 0.8 (naturally ordered) and 0.29 (frequency-matched). A similar shift in the quantile boundaries is also observed (Fig. 11). This result is not surprising, given the data manipulations associated with the frequency-matching process, and the consequences for the goodness-of-fit [Figs. 1(a and b)].

The findings presented here suggest best practices for CN calibration and operational use of the CN method.

Regarding calibration approach selection, the LS and ASYMP approaches performed comparably and both were demonstrably superior to the NEH median approach in terms of their ability to produce smaller $R_{\mathrm{SE}}$ values and robust results across many different data sets. Based on the results in this analysis, a clear choice between the LS and ASYMP approaches cannot be made; however, the spurious correlation associated with the ASYMP approach is problematic. Therefore, the authors recommend the LS approach for calibrating $S$, and thus CN, directly from $P\text{-}Q$ data.

Regarding precipitation threshold, two of the three approaches (NEH median and ASYMP) had a bias of smaller calibrated CNs for $P_T=25.4\mathrm{mm}$ than for $P_T=0\mathrm{mm}$. The LS method was unaffected by $P_T$. The two recommended calibration approaches (LS and ASYMP) exhibited slightly larger $R_{\mathrm{SE}}$ values for $P_T=25.4\mathrm{mm}$ than for $P_T=0\mathrm{mm}$. The NEH median approach produced the opposite result. No clear recommendation is possible based on these findings, but the effects of choosing to impose a $P_T$ value, both on the calibrated CN and on the goodness-of-fit of that CN, should be considered.

Given the long-standing history of $\lambda =0.20$ and arguments made more recently in favor of $\lambda =0.05$, examining these results is critical and also must be more nuanced. As noted previously, the chosen value of $\lambda$ universally leads to trends in the calibrated curve number, with larger CNs resulting from $\lambda =0.20$ versus $\lambda =0.05$. This result follows naturally from the runoff equation [Eq. (1)] in which, for a given $P\text{-}Q$ pair, if $\lambda$ decreases, then $S$ must increase for the equation to hold. Because $S$ and CN are inversely related, a smaller $\lambda$ means that a smaller CN is calibrated, and a larger $\lambda$ means that a larger CN is calibrated. Although this result is important to understand, it does not indicate which value of $\lambda$ is superior. In terms of the goodness-of-fit indicated by $R_{\mathrm{SE}}$, the LS approach produced slightly better (smaller) $R_{\mathrm{SE}}$ values for $\lambda =0.05$ than for $\lambda =0.20$. For the NEH median approach, the opposite was true, and for the ASYMP approach, $\lambda$ had little effect on $R_{\mathrm{SE}}$. These mixed results indicate a draw as to which value of $\lambda$ should be selected.

On this last point regarding $\lambda$ selection, one critical additional fact must be considered. If the proper $\lambda$ value were being selected de novo, then the draw between the goodness-of-fit performance of the two $\lambda$ values described previously would be difficult to resolve. However, that is not the case. The CN method has been in use for more than 60 years, and considerable infrastructure has been erected in support of $\lambda =0.20$ in the form of widely published tables of CN values and existing computer programs (NRCS 2017; SCS 1986). Additionally, $\lambda =0.20$ is understood widely by practicing engineers not as a parameter, $\lambda$, but simply as the value, 0.20 in Eq. (1). If a recommendation is to be made based on the findings presented here, the recommendation would be to remain with $\lambda =0.20$, with the rationale that the performance of $\lambda =0.05$ relative to $\lambda =0.20$ is not sufficient to merit the effort necessary to change this value. Considering the potential for confusion and possible erroneous use of CN tables derived for one value for $\lambda$ while using the other value of $\lambda$ in Eq. (1), the argument for remaining with $\lambda =0.20$ becomes even stronger.

This argument in favor of holding firm with the value of $\lambda =0.20$ disagrees with much of the more-recent curve number literature that has examined this question. Numerous studies and reports (D’Asaro et al. 2014; Galbetti et al. 2022; Hawkins et al. 2009; Lal et al. 2017) noted that $\lambda =0.05$ gives a better fit to observations from smaller rainfall-runoff events and claimed that $\lambda =0.20$ leads to overestimation of runoff and thus to unnecessarily costly engineering designs. This study argues that observed differences in goodness-of-fit are minimal at best, and therefore are not sufficient to justify a shift to $\lambda =0.05$.

A final consideration is one of the trade-offs between cost and safety in engineering design, as presented by Moglen et al. (2018). If the long-standing value of $\lambda =0.20$ produces less accurate overdesign, as proponents for shifting to $\lambda =0.05$ argue, the result is unnecessarily costly designs. However, if the opposite is true, that using $\lambda =0.05$ leads to underdesign (especially for storm depths corresponding to longer return periods), then potentially unsafe designs will follow. This is the classic engineering trade-off, and safety considerations favor remaining with $\lambda =0.20$.

This study used rainfall-runoff data from 31 Agricultural Research Service watersheds to examine how several different elements of the calibration process affect both the calibrated CN value and the goodness-of-fit of that value to observations.

Results showed that the calibration of CN values using a least-squares approach either to the NRCS runoff equation (LS) itself or to a decaying exponential (ASYMP) model of curve number as a function of precipitation both were superior to the NEH median approach. Based on concerns for spurious correlation associated with the ASYMP approach, this study recommends the use of the LS approach applied directly to rainfall-runoff ($P\text{-}Q$) data to calibrate the watershed CN.

For precipitation threshold, results were inconclusive, with a downward bias in the calibrated CN noted for the NEH median and ASYMP approaches when using rainfall-runoff events for which $P>25.4\mathrm{mm}$ rather than all observations. No bias in $P_T$ results was found for the LS method. In terms of goodness-of-fit, the relative standard error, $R_{\mathrm{SE}}$, was slightly larger when only the $P>25.4\mathrm{mm}$ events were used in the calibration rather than all events for both the LS and ASYMP approaches. For the NEH median approach, larger $R_{\mathrm{SE}}$ values were observed when the precipitation threshold was 0 mm rather than 25.4 mm.

Although not investigated as a valid element of CN calibration, this study did examine the consequences of using frequency-matched data in the calibration process. The primary findings were that frequency-matching results in calibrated CN values about 5–10 points larger and in much improved goodness-of-fit, quantified by much smaller relative standard error values. The reader should be aware of how frequency-matching influences the CN calibration process, but should not interpret the greatly improved goodness-of-fit as evidence that this method should be used. Although goodness-of-fit is improved, this improvement is the result of removing the causality link between observed rainfall and runoff.

Like the precipitation threshold, the use of $\lambda =0.05$ or 0.20 produced mixed results, and it was not possible to determine the clear superiority of one value of $\lambda$. The decades of use of $\lambda =0.20$ and associated publication of CN tables and software favors remaining with $\lambda =0.20$. Staying with $\lambda =0.20$ also avoids confusion or erroneous mismatching of CN values and their corresponding $\lambda$ values in the NRCS rainfall-runoff equation. Finally, erring on the side of safety favors remaining with $\lambda =0.20$ because $\lambda =0.05$ estimates smaller runoff volumes for storm depths corresponding to larger return periods.

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Some of the authors of this paper are members of the Curve Number Hydrology Committee, a task committee organized under the Watershed Management technical committee within the Environmental & Water Resources Institute (EWRI) of ASCE. The views presented here are those of the authors and do not necessarily represent the views of the entire Curve Number Hydrology Committee, EWRI, or ASCE. Additionally, this paper benefitted from the careful review and comments provided by three anonymous reviewers. The authors thank them for their efforts.