Improved runoff curve numbers for a large number of watersheds of the USA

Figures & data

Figure 1. Locations of United States Department of Agriculture, Agricultural Research Service (USDA-ARS) experimental watersheds used to develop the relationships

Figure 2. Location of ARS experimental watersheds used to validate the developed relationships

Figure 3. Scatterplots of CN derived from NEH-4 (CN_T) versus CN derived from Verma et al. (CN_V) for (a) linear, (b) exponential, (c) logarithmic, (d) power and (e) polynomial relationships

Figure 4. Coefficient of determination (R²)-based comparative performance of (a) tabular, (b) linear, (c) exponential, (d) power, (e) logarithmic and (f) polynomial models for watershed ID 44018. P: rainfall; Qc: computed runoff; Qo: observed runoff

(Continued)

Table 1. Mathematical description of models under study. CN: curve number

Model	Method of CN estimation	Method of runoff computation
Tabular	NEH-4 table	Equation (4)(4) $Q=\{\begin{array}{l}\frac{{\left(P-I_a\right)}^2}{\left(P-I_a+S\right)}\text{for}P>I_a\\ 0\text{for}P\le I_a\end{array}$(4)
Linear	Equation (16)(16) $\mathrm{C}{\mathrm{N}}_{\mathrm{L}\mathrm{i}\mathrm{n}}=2.2632\mathrm{C}{\mathrm{N}}_{\mathrm{T}}-130.99$(16)	Equations (12(12) $\mathrm{I}\mathrm{f}P\le \left(0.12S-0.19S^{\ast }\right)\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}Q=0$(12) –15(15) $S^{\ast }=\sqrt{P_{5}S}$(15) )
Exponential	Equation (17)(17) $\mathrm{C}{\mathrm{N}}_{\mathrm{E}\mathrm{x}\mathrm{p}}=4.1915\mathrm{e}\mathrm{x}\mathrm{p}\left(0.0317\right)$(17)	Equations (12(12) $\mathrm{I}\mathrm{f}P\le \left(0.12S-0.19S^{\ast }\right)\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}Q=0$(12) –15(15) $S^{\ast }=\sqrt{P_{5}S}$(15) )
Power	Equation (18)(18) $\mathrm{C}{\mathrm{N}}_{\mathrm{P}\mathrm{o}\mathrm{w}}=0.000200{\mathrm{C}\mathrm{N}}_{\mathrm{T}}^{2.8354}$(18)	Equations (12(12) $\mathrm{I}\mathrm{f}P\le \left(0.12S-0.19S^{\ast }\right)\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}Q=0$(12) –15(15) $S^{\ast }=\sqrt{P_{5}S}$(15) )
Logarithmic	Equation (19)(19) $\mathrm{C}{\mathrm{N}}_{\mathrm{L}\mathrm{o}\mathrm{g}}=202.45ln\mathrm{C}{\mathrm{N}}_{\mathrm{T}}-838.2$(19)	Equations (12(12) $\mathrm{I}\mathrm{f}P\le \left(0.12S-0.19S^{\ast }\right)\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}Q=0$(12) –15(15) $S^{\ast }=\sqrt{P_{5}S}$(15) )
Polynomial	Equation (20)(20) $\mathrm{C}{\mathrm{N}}_{\mathrm{P}\mathrm{o}\mathrm{l}}=0.0154{\mathrm{C}\mathrm{N}}_{\mathrm{T}}^2-0.5039\mathrm{C}{\mathrm{N}}_{\mathrm{T}}-7.1365$(20)	Equations (12(12) $\mathrm{I}\mathrm{f}P\le \left(0.12S-0.19S^{\ast }\right)\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}Q=0$(12) –15(15) $S^{\ast }=\sqrt{P_{5}S}$(15) )

Table 2. Model performance based on coefficient of determination (R²). Exp.: exponential; Log.: logarithmic; Poly.: polynomial

Reference	Category	Range	Model
			Linear	Exp.	Power	Log.	Poly.	Tabular
Santhi et al. 2001	Good	>0.60	48	49	46	48	48	37
	Poor	<0.60	9	8	11	9	9	20
Donigian 2002 and Moriasi et al. 2012	Very good	>0.80	13	12	14	13	13	0
	Good	0.70–0.80	19	20	19	18	20	23
	Acceptable	0.60–0.70	16	17	13	17	15	14
	Unsatisfactory	<0.60	9	8	11	9	9	20
Moriasi et al. 2015	Very good	>0.85	0	0	0	0	0	0
	Good	0.75–0.85	25	25	24	25	25	12
	Acceptable	0.60–0.75	23	24	22	23	23	25
	Unsatisfactory	<0.60	9	8	11	9	9	20

Figure 5. Box-and-whisker plot of Nash-Sutcliffe efficiency (NSE) of the models under study

Table 3. Model performance based on Nash-Sutcliffe efficiency criterion (NSE). Exp.: exponential; Log.: logarithmic; Poly.: polynomial

Reference	Grade	Range	Model
			Linear	Exp.	Power	Log.	Poly.	Tabular
Santhi et al. 2001	Satisfactory	>0.50	55	55	54	55	56	43
	Unsatisfactory	≤0.50	2	2	3	2	1	14
Moriasi et al. 2007	Very good	>0.75	23	23	24	23	23	5
	Good	0.65–0.75	15	15	15	15	16	21
	Acceptable	0.50–0.65	17	17	15	17	17	17
	Unsatisfactory	<0.50	2	2	3	2	1	14
Ritter and Mu~noz-Carpena 2013	Very good	>0.90	0	0	0	0	0	0
	Good	0.80–0.90	10	11	12	10	11	0
	Acceptable	0.65–0.80	28	27	27	28	28	26
	Unsatisfactory	< 0.65	19	19	18	19	18	31
Moriasi et al. 2015	Very good	> 0.80	10	11	12	10	11	0
	Good	0.70–0.80	21	20	19	21	20	19
	Acceptable	0.50–0.70	24	24	23	24	25	24
	Unsatisfactory	< 0.50	2	2	3	2	1	14

Figure 6. Box-and-whisker plot of root mean square error (RMSE) of the models under study

Table

WS ID	Area (ha)	Number of storms	CNT	CNV	CNLin	CNExp	CNPow	CNLog	CNPol
42016	8.4	293	86.35	67.31	64.44	64.74	61.82	64.40	64.18
42017	7.5	237	91.09	75.78	75.16	75.23	71.93	75.22	74.74
42023	1.31	80	92.79	80.63	79.01	79.40	75.80	78.97	78.70
42024	1.2	252	93.54	82.10	80.71	81.31	77.55	80.60	80.47
42028	1.2	263	92.8	79.47	79.03	79.43	75.83	78.99	78.72
42035	1.3	163	91.21	76.18	75.44	75.52	72.20	75.49	75.02
42036	1.3	197	91.63	77.35	76.39	76.53	73.15	76.42	75.99
42037	4.6	181	88.77	73.16	69.91	69.90	66.86	70.00	69.49
42038	2.3	158	89.3	73.92	71.11	71.08	67.99	71.21	70.67
42039	4	237	87.96	69.47	68.08	68.13	65.14	68.14	67.69
42040	4.6	226	87.98	69.17	68.13	68.17	65.18	68.19	67.73
44001	194.7	407	89.86	71.73	72.38	72.36	69.21	72.47	71.94
44002	166.3	482	86.65	65.69	65.12	65.36	62.43	65.11	64.83
44003	844.2	235	87	66.75	65.91	66.09	63.15	65.92	65.59
44004	1412.4	349	84.85	62.61	61.04	61.73	58.82	60.86	60.98
44007	1.5	524	94.13	82.11	82.05	82.85	78.95	81.87	81.88
44008	1.5	515	93.73	80.86	81.14	81.80	78.00	81.01	80.93
44009	1.6	537	91.21	75.04	75.44	75.52	72.20	75.49	75.02
44010	1.6	537	91.96	76.68	77.13	77.34	73.90	77.15	76.76
44011	1.7	486	90.63	74.03	74.12	74.15	70.91	74.20	73.69
44012	1.59	355	88.79	69.63	69.96	69.94	66.90	70.05	69.53
44014	1.6	262	89.22	70.55	70.93	70.90	67.82	71.02	70.49
44015	1.6	295	92.6	77.92	78.58	78.92	75.36	78.55	78.25
44016	1.5	309	93.23	79.23	80.01	80.52	76.83	79.92	79.74
44017	1.4	276	92.02	76.99	77.27	77.49	74.03	77.28	76.90
44018	1.4	274	92.71	78.50	78.83	79.20	75.62	78.79	78.51
44019	1.5	303	93.6	79.96	80.85	81.47	77.69	80.73	80.62
44020	1.4	281	93.23	79.73	80.01	80.52	76.83	79.92	79.74
44021	1.6	321	93.18	79.64	79.89	80.39	76.71	79.82	79.62
44022	1.5	320	88.1	68.95	68.40	68.43	65.44	68.47	68.00
44023	1.7	258	92.63	78.93	78.65	79.00	75.43	78.62	78.32
44024	1.6	264	90	73.33	72.70	72.68	69.52	72.79	72.25
44025	1.6	238	91.21	75.91	75.44	75.52	72.20	75.49	75.02
44026	1.5	241	92.34	77.08	77.99	78.28	74.76	77.98	77.64
44028	1.8	269	92.68	78.05	78.76	79.12	75.55	78.73	78.44
61003	157.8	463	85.08	62.06	61.56	62.18	59.27	61.41	61.47
62001	809.37	236	78.4	54.62	46.44	50.32	47.01	44.85	48.01
62002	404.7	136	90.74	76.00	74.37	74.40	71.15	74.44	73.94
62003	2237.9	28	89.03	69.86	70.50	70.48	67.41	70.59	70.07
62004	9226.9	22	89.82	69.10	72.29	72.27	69.12	72.38	71.84
62005	12 990.5	92	78.83	59.33	47.42	51.01	47.74	45.96	48.84
62010	8093.7	104	88.65	71.12	69.64	69.63	66.60	69.73	69.22
62012	3055.4	35	87.58	67.30	67.22	67.31	64.35	67.27	66.85
62014	0.6	134	94.93	84.82	83.86	84.97	80.86	83.58	83.81
62017	1295	26	89.04	67.56	70.53	70.50	67.43	70.62	70.09
62018	441.1	59	89.35	70.97	71.23	71.20	68.10	71.32	70.78
67005	11 116.4	247	79.01	55.01	47.83	51.30	48.05	46.42	49.19
69030	7.2	161	83.57	62.30	58.15	59.28	56.34	57.78	58.31
69032	17.9	198	89.7	72.00	72.02	71.99	68.86	72.11	71.57
69033	12.1	156	88.64	70.07	69.62	69.61	66.58	69.70	69.20
69035	5.26	116	85	64.06	61.38	62.03	59.12	61.21	61.30
69036	10.7	113	84.84	62.87	61.02	61.71	58.80	60.83	60.96
69042	9.6	85	76.94	53.30	43.14	48.04	44.57	41.05	45.26
69044	7.8	225	90.81	74.80	74.53	74.57	71.31	74.60	74.10
69045	11.1	250	86.91	67.26	65.70	65.90	62.96	65.71	65.39
70008	3.5	23	87.71	68.38	67.52	67.59	64.62	67.57	67.14
71002	33.51	1120	79.85	56.00	49.73	52.68	49.52	48.56	50.82

Table

WS ID	RMSE						NSE
	Tabular	Linear	Exponential	Power	Logarithmic	Polynomial	Tabular	Linear	Exponential	Power	Logarithmic	Polynomial
42016	7.37	6.21	6.19	6.52	6.22	6.24	61.35	71.80	72.04	68.93	71.77	71.57
42017	9.14	8.02	8.02	8.34	8.02	8.04	69.05	75.26	75.28	73.28	75.28	75.13
42023	12.76	11.71	11.67	12.35	11.71	11.75	71.92	74.23	74.41	71.35	74.20	74.05
42024	11.23	10.78	10.76	11.10	10.79	10.80	58.60	60.97	61.17	58.65	60.92	60.86
42028	10.32	9.39	9.39	9.64	9.40	9.41	59.64	65.36	65.42	63.55	65.35	65.28
42035	13.16	11.79	11.79	12.07	11.79	11.81	49.12	58.21	58.24	56.24	58.23	58.07
42036	11.54	10.36	10.35	10.64	10.36	10.38	49.62	58.10	58.15	55.83	58.11	57.94
42037	7.81	7.32	7.32	7.75	7.31	7.37	70.37	72.38	72.37	69.05	72.45	72.01
42038	10.77	10.10	10.11	10.59	10.09	10.16	63.91	66.62	66.59	63.31	66.69	66.26
42039	9.72	8.03	8.03	8.29	8.02	8.05	52.79	66.04	66.06	63.76	66.07	65.83
42040	11.01	9.06	9.06	9.29	9.06	9.08	44.11	60.22	60.24	58.23	60.24	60.05
44001	5.07	4.25	4.25	4.38	4.25	4.26	66.80	76.46	76.46	75.02	76.47	76.42
44002	4.23	3.41	3.40	3.60	3.41	3.42	74.26	83.14	83.24	81.19	83.14	83.01
44003	5.26	4.54	4.53	4.85	4.54	4.56	72.92	79.52	79.63	76.65	79.53	79.29
44004	4.66	3.90	3.85	4.10	3.91	3.90	71.19	79.64	80.09	77.40	79.50	79.59
44007	4.92	4.56	4.56	4.74	4.56	4.56	74.19	77.64	77.65	75.82	77.61	77.61
44008	5.32	4.77	4.78	4.92	4.77	4.77	75.32	79.90	79.86	78.62	79.89	79.88
44009	4.33	3.80	3.80	3.93	3.80	3.80	75.82	81.17	81.18	79.81	81.17	81.12
44010	4.92	4.22	4.22	4.34	4.22	4.23	59.72	70.02	70.02	68.40	70.02	69.98
44011	5.01	4.34	4.34	4.46	4.34	4.34	57.14	67.49	67.50	65.66	67.51	67.39
44012	4.61	4.13	4.13	4.24	4.13	4.13	64.60	71.20	71.20	69.58	71.21	71.11
44014	5.44	4.48	4.48	4.57	4.48	4.49	40.24	58.20	58.19	56.47	58.21	58.11
44015	4.93	4.31	4.31	4.43	4.31	4.31	68.78	75.57	75.54	74.21	75.57	75.57
44016	5.89	5.02	5.03	5.11	5.02	5.02	54.91	66.56	66.48	65.41	66.57	66.57
44017	4.84	3.66	3.66	3.83	3.66	3.67	69.44	81.86	81.86	80.17	81.86	81.81
44018	4.92	3.93	3.94	4.10	3.93	3.94	71.77	81.31	81.30	79.69	81.31	81.28
44019	5.11	4.15	4.16	4.23	4.15	4.14	61.97	74.17	73.99	73.07	74.18	74.19
44020	4.48	3.99	3.99	4.16	3.99	3.99	75.95	80.08	80.06	78.35	80.08	80.06
44021	4.68	3.98	3.98	4.13	3.98	3.98	72.13	79.29	79.27	77.71	79.28	79.26
44022	5.19	4.16	4.16	4.30	4.16	4.17	50.78	68.01	68.03	65.90	68.04	67.85
44023	4.52	4.09	4.08	4.32	4.09	4.10	76.77	80.34	80.40	78.04	80.33	80.24
44024	5.52	4.69	4.69	4.83	4.69	4.70	41.28	56.22	56.21	53.59	56.25	56.01
44025	5.36	4.82	4.81	5.01	4.81	4.83	63.94	69.77	69.80	67.34	69.79	69.61
44026	5.44	4.33	4.34	4.43	4.33	4.33	65.78	77.37	77.32	76.29	77.37	77.40
44028	5.38	4.38	4.38	4.50	4.38	4.38	69.51	79.18	79.14	77.99	79.18	79.19
61003	3.38	2.93	2.92	3.00	2.93	2.93	43.06	56.76	57.04	54.79	56.67	56.70
62001	4.40	4.17	3.64	4.09	4.43	3.94	70.59	73.17	79.54	74.27	69.80	76.09
62002	8.34	7.90	7.90	8.38	7.90	7.94	77.42	79.23	79.24	76.64	79.26	79.00
62003	9.08	7.65	7.65	7.86	7.65	7.66	44.84	57.05	57.05	54.66	57.05	56.97
62004	7.34	5.91	5.90	5.79	5.91	5.86	39.74	54.30	54.34	56.00	54.15	54.97
62005	9.43	10.13	9.19	10.04	10.55	9.74	53.52	44.72	54.55	45.70	40.12	48.88
62010	9.93	8.82	8.82	9.26	8.81	11.05	70.26	75.81	75.81	73.30	75.85	61.98
62012	10.13	8.53	8.53	8.71	8.53	8.54	36.75	52.15	52.17	50.16	52.16	52.02
62014	7.33	6.76	6.71	7.18	6.78	6.76	73.08	76.67	76.95	73.61	76.52	76.64
62017	8.66	8.47	8.47	8.42	8.48	8.44	51.99	51.66	51.68	52.25	51.59	51.96
62018	8.08	6.43	6.43	6.59	6.43	6.44	43.55	62.95	62.95	61.15	62.96	62.85
67005	3.57	3.44	3.30	3.43	3.51	3.38	13.54	13.86	20.85	14.38	10.47	16.87
69030	4.62	3.84	3.76	4.00	3.87	3.83	55.15	68.62	69.96	65.98	68.13	68.82
69032	4.15	4.23	4.23	4.42	4.23	4.24	73.34	71.78	71.77	69.15	71.81	71.60
69033	4.84	4.94	4.94	5.14	4.94	4.96	67.95	65.73	65.73	62.87	65.77	65.50
69035	5.51	5.37	5.33	5.53	5.38	5.37	46.74	47.77	48.46	44.57	47.58	47.67
69036	5.15	4.78	4.74	4.95	4.79	4.78	49.58	55.59	56.31	52.34	55.37	55.52
69042	4.85	6.16	5.60	5.98	6.43	5.90	57.64	29.20	41.53	33.21	22.81	35.04
69044	4.45	3.66	3.66	3.92	3.66	3.68	73.38	81.69	81.70	79.04	81.71	81.52
69045	3.46	3.15	3.14	3.37	3.15	3.17	74.27	78.38	78.53	75.26	78.39	78.12
70008	7.85	8.48	8.48	8.69	8.48	8.50	58.48	47.61	47.65	44.94	47.63	47.38
71002	3.23	2.99	2.85	3.00	3.06	2.93	49.43	59.31	63.12	58.99	57.47	60.86

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