Regional models of flow-duration curves of perennial and intermittent streams and their use for calibrating the parameters of a rainfall–runoff model

Figures & data

Fig. 1 The Brazilian states, Minas Gerais and Ceará, in which the basins employed in the study are located.

Fig. 2 Flow chart of the runoff production module of the Rio Grande model, which is similar to the Xinanjiang model described by Zhao et al. (1980).

Table 1 Streamflow gauging stations and physical, climatological and morphological characteristics of the basins in Minas Gerais (A: drainage area of basin, L: length of main stream in basin, H_max: watershed maximum elevation, H_min: watershed minimum elevation, k_c: watershed compactness coefficient, k_f: watershed shape coefficient, P: mean annual rainfall, C_esc: runoff coefficient of the basin, given by the relationship between runoff volume and rainfall volume, DD: drainage density, S_eq: equivalent slope).

Stationcode	A(km^2)	L(km)	Hmin(m)	Hmax(m)	kf	kc	P(mm)	Cesc	DD(junc./km^2)	Seq(m/km)
40130001	1620	75.8	759	1160	0.30	1.47	1533	0.33	0.114	1.92
40300001	1558	119.1	656	1100	0.11	1.70	1432	0.31	0.099	1.82
40330000	7378	219.4	609	1160	0.15	1.66	1456	0.30	0.036	0.75
40450001	11302	293.1	589	1160	0.13	1.71	1431	0.29	0.054	0.46
40549998	461	52.0	877	1140	0.17	1.52	1400	0.38	0.098	2.69
40680000	486	47.3	863	1100	0.22	1.34	1369	0.44	0.136	1.25
40710000	2760	118.9	794	1140	0.20	1.28	1408	0.40	0.137	1.59
40740000	3939	187.4	736	1140	0.11	1.49	1422	0.35	0.134	1.21
40800001	5680	236.3	703	1140	0.10	1.64	1449	0.33	0.141	1.00
40865001	10192	419.8	633	1140	0.06	1.87	1414	0.33	0.133	0.60

Fig. 3 L-moment ratio diagram for FDCs in Minas Gerais.

Table 2 Regression equations for the parameters of the extended Burr XII distribution for FDCs (R²: coefficient of determination; R²_adj: adjusted coefficient of determination).

Regional equation	R^2	R^2adj.	Standard error	Ftotal
λ = 0.02003A^0.97077kc^-1.01951	0.99	0.99	0.055	1680
α = –3.23919 + 0.0008Hmax + 0.4406kc + 0.00075P + 2.40647Cesc	0.88	0.78	0.024	8.8
β = –78.24 + 11.409ln(Hmax) + 0.434ln(kf) + 5.558ln(kc) + 5.327ln(Cesc)	0.91	0.83	0.141	12.2

Table 3 Regression equations for the parameters of the extended Burr XII distribution for median AFDCs (R²: coefficient of determination; R²_adj: adjusted coefficient of determination).

Regional equation	R^2	R^2adj.	Standard error	Ftotal
λ = 0.018A^0.98kc^-1.15	0.99	0.99	0.129	312.5
α = 0.55 + 0.17(1/Seq) – 590.40(1/(Hmin) – 0.003(1/DD) + 0.126(1/Cesc)	0.90	0.82	0.020	10.9
β = –2.08 –0.15(1/DD) + 1.97(1/Seq) – 0.14(1/kf) + 437.08(1/A)	0.89	0.80	0.420	10.0

Fig. 4 Scatter plots of CRM and JKM parameters in comparison to MML estimates.

Fig. 5 Comparison between synthetic and observed FDCs at Jaguaruna station (40300001).

Table 4 Performance indices for the three parameter estimation methods for FDCs (MML: method of L-moments; CRM: complete regional model; JKM: jack-knife model).

Station	MML			CRM			JKM
	εs	σe,s	Es	εs	σe,s	Es	εs	σe,s	Es
40130001	0.0006	0.059	0.973	–0.0264	0.059	0.950	–0.0039	0.085	0.979
40300001	–0.0141	0.080	0.985	0.0072	0.077	0.962	0.0822	0.129	0.823
40330000	–0.0246	0.090	0.955	–0.0613	0.084	0.946	–0.0228	0.083	0.963
40450001	–0.0014	0.080	0.957	0.0148	0.127	0.944	0.0712	0.193	0.884
40549998	0.0015	0.053	0.985	–0.0812	0.051	0.964	–0.0878	0.059	0.947
40680000	–0.0028	0.050	0.990	–0.0072	0.065	0.989	0.0390	0.108	0.981
40710000	–0.0092	0.076	0.990	–0.0108	0.069	0.978	0.0214	0.077	0.958
40740000	–0.0117	0.068	0.986	–0.0782	0.050	0.976	–0.0514	0.057	0.974
40800001	0.0004	0.061	0.978	–0.0345	0.062	0.938	0.0053	0.069	0.891
40865001	–0.0098	0.080	0.979	–0.1036	0.074	0.945	–0.0870	0.081	0.931
p descriptors	p1	100.00		p1	100.00		p1	100.00
	p2	0.00		p2	0.00		p2	0.00
	p3	0.00		p3	0.00		p3	0.00

Table 5 Performance indices for the three parameter estimation methods for median AFDCs.

Station	MML			CRM			JKM
	εs	σe,s	Es	εs	σe,s	Es	εs	σe,s	Es
40130001	–0.030	0.034	0.987	–0.026	0.048	0.982	–0.125	0.088	0.965
40300001	0.107	0.091	0.989	–0.032	0.192	0.705	–0.002	0.357	0.257
40330000	–0.093	0.093	0.967	0.115	0.106	0.945	0.063	0.105	0.951
40450001	0.106	0.118	0.977	–0.065	0.098	0.927	–0.040	0.087	0.940
40549998	–0.012	0.064	0.954	–0.108	0.137	0.941	–0.169	0.138	0.875
40680000	0.028	0.092	0.984	–0.009	0.096	0.977	–0.003	0.050	0.989
40710000	0.001	0.055	0.992	0.073	0.058	0.980	0.202	0.064	0.889
40740000	–0.003	0.062	0.990	0.073	0.132	0.874	0.090	0.154	0.811
40800001	–0.004	0.075	0.964	–0.006	0.094	0.831	–0.003	0.105	0.770
40865001	0.026	0.088	0.975	0.005	0.077	0.962	–0.039	0.076	0.947
p descriptors	p1	100.00		p1	100.00		p1	90.00
	p2	0.00		p2	0.00		p2	0.00
	p3	0.00		p3	0.00		p3	0.00

Table 6 Performance indices for hydrographs simulated by the Rio Grande model, as calibrated with long-term flow-duration curves (FDCs).

Station	RMSE	AAPE	E	K	R^2
40130001-2	28.642	0.301	0.197	1.193	0.861
40300001	14.258	0.181	0.641	1.061	0.766
40330000	76.695	0.200	0.541	1.113	0.884
40450001	115.092	0.184	0.365	1.226	0.893
40549998	7.447	0.304	0.232	1.034	0.692
40680000	14.219	0.338	0.642	0.860	0.659
40710000	43.080	0.322	0.380	1.049	0.698
40740000	72.844	0.243	-0.246	1.150	0.832
40800001	61.514	0.221	0.563	1.053	0.801
40865001	84.103	0.207	0.781	1.069	0.890
Mean	51.789	0.250	0.482*	1.081	0.798

*Excluding station 40740000.

Table 7 Performance indices for hydrographs simulated by the Rio Grande model, as calibrated with median annual flow-duration curves (AFDCs).

Station	RMSE	AAPE	E	K	R^2
40130001-2	26.479	0.311	0.313	1.184	0.872
40300001	19.293	0.234	0.598	1.159	0.784
40330000	61.564	0.198	0.762	1.099	0.875
40450001	79.639	0.251	0.739	1.126	0.868
40549998	6.900	0.312	0.395	1.032	0.681
40680000	10.482	0.302	0.729	0.871	0.839
40710000	43.986	0.312	0.569	1.048	0.713
40740000	51.653	0.235	0.569	1.089	0.799
40800001	59.632	0.184	0.592	1.041	0.823
40865001	86.913	0.298	0.682	1.121	0.840
Mean	44.654	0.264	0.595	1.077	0.809

Fig. 6 Comparison between the hydrographs simulated by the model calibrated with the FDC and the median AFDC, and the observed hydrograph at Porto Pará gauging station for the hydrological year 1994/95.

Table 8 Streamflow gauging stations and physical, climatological and morphological characteristics of the basins in Ceará (A: drainage area of basin, L: length of the main stream in basin, CTD: total length of drainage in basin, P: mean annual rainfall, C_esc: runoff coefficient of the basin, C_ris: watershed percent area located on fractured rock aquifer).

Station code	A (km^2)	L (km)	CTD (m)	P (mm)	Cesc	Cris (%)
34750000	18340	255	12526	945	0.066	85.1
35210000	1643	120	1624	780	0.189	100.0
36125000	3746	139	2353	738	0.053	67.2
36210000	1756	121	1156	1041	0.070	18.5
36250000	4372	157	3105	887	0.032	24.4
36290000	12680	176	11203	928	0.080	59.0
36470000	4838	142	4102	764	0.109	100.0
35260000	2818	155	2653	720	0.16	99.6

Table 9 Regression equations for the parameters of the extended Burr XII distribution function for FDCs.

Regional equation	R^2	R^2adj.	Standard error	Ftotal
λ = 0.1817CTD^0.5453Cesc^0.6463	0.93	0.91	0.576	34.7
α = 1.4916 – 0.0054CT + 0.0130Cris	0.92	0.88	0.209	27.5
β = –0.6849 – 0.000041CTD + 0.0043Cris	0.89	0.85	0.096	20.6
τ = 1.3601 – 0.0104P^0.5 + 0.0050CTD^0.5 – 0.0863Cris^0.5	0.87	0.78	1.163	9.2

Fig. 7 Scatter plots of CRM and JKM parameters vs MML estimates.

Fig. 8 Comparison between synthetic and observed FDCs at Sítio da Lapinha station (36210000) for estimated τ (a) and observed τ (b).

Table 10 Performance indices of FDC models for intermittent streams in Ceará state.

Station	MML models			CRM τobs			CRM τest			JKM τobs			JKM τest
	εs	σe,s	Es	εs	σe,s	Es	εs	σe,s	Es	εs	σe,s	Es	εs	σe,s	Es
34750000	0.081	0.27	0.96	0.064	0.30	0.82	0.514	1.88	0.07	1.426	6.23	0.85	3.992	20.11	0.09
35210000	–0.092	0.30	0.98	–0.113	0.27	0.95	1.038	3.50	0.96	–0.185	0.24	0.89	2.153	8.27	0.88
35260000	–0.075	0.22	0.95	–0.126	0.23	0.95	–0.612	0.29	0.89	–0.129	0.24	0.98	0.101	0.20	0.97
36110000	–0.089	0.29	0.97	0.178	0.35	0.92	–0.248	0.38	0.99	0.303	0.39	0.94	–0.481	0.36	0.97
36125000	0.179	0.51	0.98	0.189	0.72	0.93	1.299	3.54	0.95	0.051	0.41	0.95	2.414	6.77	0.89
36210000	0.389	1.27	0.98	0.257	1.28	0.99	1.053	3.55	0.98	–0.156	0.46	0.91	4.252	11.99	0.98
36290000	0.045	0.36	0.92	0.149	0.49	0.82	0.508	0.40	0.59	0.194	0.53	0.87	–0.549	0.42	0.46
36470000	–0.094	0.25	0.99	–0.135	0.24	0.94	2.648	4.34	0.91	–0.138	0.24	0.68	3.842	6.55	0.45
p descriptors	p1	100.0		p1	100.0		p1	75.0		p1	87.5		p1	62.5
	p2	0.0		p2	0.0		p2	0.0		p2	12.5		p2	0.0
	p3	0.0		p3	0.0		p3	25.0		p3	0.0		p3	37.5

Table 11 Performance indices of hydrographs simulated through parameter calibration with FDCs.

Station	RMSE	AAPE	E	K	R^2
34750000	120.335	7.094	0.684	0.657	0.722
35210000	35.612	15.937	0.599	0.581	0.639
35260000	33.274	6.001	0.612	0.882	0.597
36110000	16.290	6.413	0.453	0.773	0.649
36125000	10.122	5.013	0.801	0.897	0.788
36210000	6.201	2.142	0.522	0.834	0.570
36290000	148.399	18.021	0.357	0.523	0.401
36470000	65.061	18.767	0.602	0.721	0.556
Mean	54.412	9.924	0.579	0.734	0.615

Fig. 9 Comparison between the hydrograph obtained by calibration with the FDC and the observed hydrograph at Fazenda Boa Esperança station for the period 1 October 1981–30 September 1987.

Yu, P.S. and Yang, T.C., 2000. Using synthetic flow duration curves for rainfall–runoff model calibration at ungauged sites. Hydrological Process, 14 (1), 117–133.

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