Application of Generalized Likelihood Uncertainty Estimation (GLUE) at different temporal scales to reduce the uncertainty level in modelled river flows

Figures & data

Figure 1. Overview map of the UK showing the case study catchments.

Table 1. Number of iterations and ranges of NSE for monthly percentiles.

Catchment	Total no. of iterations	No. of iterations with monthly NSE > 0.50	Range of monthly NSE	Range of monthly NSE > 0.50
Eden	60	60	0.50–0.85	0.50–0.85
Frome	490	396	−0.25–0.93	0.50–0.93
Ebbw	108	103	0.27–0.95	0.51–0.95
Pang	648	342	−0.92–0.91	0.50–0.91
Don	336	225	−0.35–0.89	0.50–0.89

Table 2. Key model parameter ranges and number of iterations.

	Model parameters
	Base percentage of flow routed to stream		Exponent function of flow routed to stream		Catchment storage/time lag		Stream storage/time lag		Baseflow factor
Catchment	Range	Iterations	Range	Iterations	Range	Iterations	Range	Iterations	Range	Iterations
Frome	9–90	10	0.02–0.04	7	45	1	15	1	0.2–0.8	7
Pang	0.2–1.8	9	0 to 0.004	3	0.1–1	8	0.015	1	1.75 10^−8–3.3 × 10^−7	3
Ebbw	10–90	9	0.02–0.4	2	2	1	20	1	2.2 × 10^−9–2.2 × 10^−7	6
Eden	40–95	4	1.0 × 10^−5	1	315	1	15–55	3	0.14–0.75	5
Don	10–90	8	0–0.3	6	143	1	18	1	0.02–0.95	7

Table 3. Model performance for the calibration and validation stages of the five catchments studied.

Catchment	Period	NSE	lnNSE	R^2	Square root of R^2	Modelled flow (m^3 s^−1)	Observed flow (m^3 s^−1)	% Error
Eden	2012†	0.90	0.95	0.89	0.94	5.03	5.04	–0.19
	1971–1980	0.79	0.89	0.79	0.89	3.60	3.54	1.69
	1971–2012	0.79	0.90	0.80	0.89	4.11	4.13	–0.48
Ebbw	2000–2003†	0.91	0.88	0.92	0.96	7.19	7.23	–0.55
	1971–1980	0.87	0.82	0.88	0.94	6.70	6.53	2.56
	1961–2012	0.87	0.82	0.88	0.93	6.98	7.21	–3.17
Don	2011–2012†	0.92	0.86	0.91	0.95	5.41	5.32	1.81
	2001–2012	0.87	0.73	0.87	0.93	4.86	4.73	2.61
	1971–1980	0.82	0.66	0.83	0.91	4.68	4.90	– 4.63
	1966–2012	0.83	0.73	0.84	0.91	5.06	5.08	–0.60
Frome	2001–2012†	0.86	0.83	0.86	0.93	1.79	1.78	0.94
	1971–1980	0.80	0.78	0.81	0.90	1.56	1.48	4.93
	1962–2012	0.82	0.80	0.83	0.91	1.71	1.74	–2.25
Pang	2001–2003†	0.92	0.89	0.94	0.97	0.79	0.81	–2.14
	2000–2012	0.90	0.83	0.90	0.95	0.66	0.64	2.96
	1971–1980	0.78	0.79	0.78	0.88	0.62	0.61	1.76
	1971–2012	0.81	0.80	0.83	0.91	0.66	0.64	3.47

†calibration period.

Figure 2. Ebbw River catchment – model calibration for the period 2000–2003.

Table 4. Results of daily GLUE prediction bounds for all the studied catchments (See the Appendix for description of the parameters).

	Daily prediction bounds
Catchment	CR	B	RB	S	T	D	RD	R factor	Period
Eden	61.70	2.29	0.74	0.50	0.98	0.88	0.31	0.72	1975–1976
	61.97	2.82	0.82	0.42	0.89	0.97	0.36	0.71	1976–1977
	62.61	2.51	0.87	0.46	0.94	0.96	0.38	0.88	1971–1980
	69.69	2.94	0.84	0.39	0.86	1.06	0.32	0.87	1981–1990
	67.78	2.91	0.87	0.40	0.87	1.05	0.35	0.90	1991–2000
	63.43	2.99	0.80	0.46	0.93	1.15	0.34	0.82	2001–2012
	65.76	2.85	0.84	0.43	0.90	1.06	0.35	0.86	1971–2012
Ebbw	75.83	4.54	0.83	0.40	0.87	1.50	0.29	0.67	1971–1980
	81.22	5.25	0.77	0.35	0.82	1.74	0.21	0.61	1981–1990
	85.60	5.34	0.82	0.30	0.76	1.64	0.21	0.57	1991–2000
	83.87	4.96	0.83	0.33	0.79	1.57	0.23	0.59	2001–2010
	79.87	4.97	0.81	0.36	0.82	1.63	0.24	0.62	1961–2012
Don	55.49	4.17	0.83	0.56	1.05	1.76	0.42	0.68	2000–2012
	57.35	3.88	0.74	0.61	1.10	1.67	0.36	0.68	1971–1980
	58.38	4.24	0.79	0.55	1.04	1.82	0.39	0.70	1981–1990
	59.51	4.16	0.78	0.53	1.02	1.72	0.38	0.64	1991–2000
	57.31	4.10	0.79	0.60	1.10	1.76	0.39	0.68	1967–2012
Frome	47.50	1.23	0.93	0.88	1.41	0.60	0.49	0.50	1971–1980
	48.60	1.37	0.96	0.86	1.40	0.63	0.49	0.52	1981–1990
	48.15	1.38	0.88	0.93	1.48	0.71	0.47	0.49	1991–2000
	48.48	1.34	0.93	0.85	1.38	0.64	0.48	0.49	1971–2012
Pang	35.40	0.22	0.37	1.02	1.57	0.16	0.25	0.51	2000–2012
	38.18	0.22	0.37	0.99	1.54	0.15	0.24	0.60	1993–1999
	36.37	0.22	0.37	1.01	1.56	0.15	0.25	0.53	1993–2012

Table 5. Results of monthly GLUE prediction bounds for all the studied catchments.

	Monthly prediction bounds
Catchment	CR	B	RB	S	T	D	RD	R factor	Period
Eden	70.83	4.72	0.58	0.38	0.85	1.37	0.20	0.75	1975–1976
	75.00	5.23	0.62	0.33	0.81	1.48	0.24	0.70	1976–1977
	70.00	5.09	0.68	0.38	0.85	1.62	0.28	0.93	1971–1980
	71.67	5.63	0.61	0.34	0.80	1.77	0.21	0.95	1981–1990
	68.33	5.40	0.63	0.38	0.85	1.89	0.25	0.87	1991–2000
	66.67	5.57	0.59	0.42	0.90	2.13	0.25	0.87	2001–2012
	69.05	5.43	0.63	0.38	0.85	1.87	0.25	0.89	1971–2012
Ebbw	65.83	8.39	0.80	0.53	1.02	3.34	0.34	0.65	1971–1980
	75.83	10.11	0.84	0.41	0.88	3.86	0.24	0.64	1981–1990
	73.33	10.17	0.82	0.37	0.84	3.75	0.26	0.64	1991–2000
	84.17	9.32	0.84	0.34	0.81	3.11	0.28	0.66	2001–2010
	72.44	9.27	0.78	0.42	0.90	3.51	0.27	0.65	1961–2012
Don	68.59	7.76	0.70	0.39	0.86	2.83	0.29	0.79	2000–2012
	72.50	7.28	0.62	0.38	0.85	2.54	0.23	0.84	1971–1980
	68.33	7.99	0.67	0.39	0.86	2.83	0.28	0.83	1981–1990
	75.00	7.91	0.68	0.35	0.82	2.62	0.26	0.73	1991–2000
	70.47	7.70	0.66	0.38	0.85	2.76	0.27	0.82	1967–2012
Frome	46.67	2.18	0.74	0.66	1.15	1.20	0.37	0.60	1971–1980
	50.83	2.36	0.76	0.57	1.07	1.21	0.34	0.60	1981–1990
	50.83	2.36	0.68	0.68	1.19	1.40	0.33	0.53	1991–2000
	51.19	2.33	0.73	0.59	1.08	1.21	0.33	0.58	1971–2012
Pang	45.51	0.67	0.43	0.69	1.19	0.38	0.25	0.61	2000–2012
	39.29	0.65	0.44	0.69	1.19	0.37	0.25	0.72	1993–1999
	43.33	0.66	0.44	0.69	1.19	0.38	0.25	0.64	1993–2012

Table 6. Results of seasonal GLUE prediction bounds for all the studied catchments.

	Seasonal prediction bounds
Catchment	CR	B	RB	S	T	D	RD	R factor	Period
Eden	87.50	3.64	0.41	0.29	0.73	0.67	0.10	0.61	1975–1976
	75.00	4.19	0.40	0.32	0.79	1.23	0.14	0.71	1976–1977
	70.00	3.77	0.49	0.37	0.84	1.25	0.19	0.84	1971–1980
	80.00	4.12	0.43	0.35	0.82	1.37	0.15	0.95	1981–1990
	60.00	3.95	0.45	0.44	0.92	1.68	0.19	0.82	1991–2000
	65.96	4.00	0.41	0.43	0.92	1.67	0.17	0.95	2001–2012
	68.86	3.96	0.44	0.40	0.88	1.50	0.18	0.88	1971–2012
Ebbw	57.50	6.73	0.62	0.52	1.01	2.85	0.29	0.67	1971–1980
	52.50	7.50	0.59	0.53	1.00	3.32	0.19	0.63	1981–1990
	80.00	7.52	0.57	0.38	0.86	2.87	0.19	0.61	1991–2000
	71.79	7.27	0.58	0.32	0.77	2.29	0.19	0.70	2001–2010
	63.77	7.04	0.56	0.45	0.92	2.85	0.21	0.64	1961–2012
Don	72.55	6.44	0.59	0.36	0.82	2.30	0.23	0.92	2000–2012
	82.50	6.24	0.52	0.36	0.82	2.16	0.18	0.89	1971–1980
	72.50	6.58	0.56	0.37	0.85	2.36	0.23	0.87	1981–1990
	77.50	6.56	0.56	0.35	0.83	2.32	0.21	0.80	1991–2000
	75.96	6.43	0.56	0.36	0.83	2.29	0.21	0.90	1967–2012
Frome	52.50	1.90	0.66	0.55	1.04	0.97	0.27	0.65	1971–1980
	57.50	2.10	0.63	0.47	0.93	0.94	0.24	0.71	1981–1990
	50.00	2.08	0.56	0.54	1.03	1.15	0.24	0.61	1991–2000
	56.29	2.05	0.61	0.48	0.96	0.95	0.23	0.67	1971–2012
Pang	52.94	0.67	0.44	0.55	1.04	0.33	0.22	0.64	2000–2012
	48.15	0.66	0.46	0.60	1.10	0.35	0.23	0.78	1993–1999
	50.63	0.66	0.44	0.57	1.06	0.34	0.22	0.68	1993–2012

Table 7. Results of annual GLUE prediction bounds for all the studied catchments.

	Annual prediction bounds
Catchment	CR	B	RB	S	T	D	RD	R factor	Period
Eden	50.00	20.32	0.19	0.52	0.94	9.02	0.09	3.20	1975–1976
	100.00	23.46	0.19	0.09	0.49	2.12	0.02	1.28	1976–1977
	90.00	22.42	0.23	0.25	0.71	5.01	0.05	0.89	1971–1980
	100.00	24.11	0.20	0.23	0.68	5.65	0.04	1.14	1981–1990
	90.00	23.64	0.20	0.27	0.74	6.11	0.05	1.12	1991–2000
	91.67	24.13	0.19	0.24	0.69	5.48	0.04	0.89	2001–2012
	92.86	23.60	0.20	0.25	0.71	5.56	0.05	0.93	1971–2012
Ebbw	60.00	34.62	0.19	0.45	0.92	17.05	0.11	0.71	1971–1980
	40.00	30.73	0.13	0.72	1.24	21.50	0.08	0.80	1981–1990
	70.00	32.29	0.13	0.33	0.78	10.38	0.04	0.71	1991–2000
	90.00	39.24	0.19	0.28	0.75	10.37	0.05	0.74	2001–2010
	65.38	36.10	0.17	0.44	0.91	15.15	0.07	0.73	1961–2012
Don	76.92	44.27	0.30	0.25	0.70	10.26	0.08	0.75	2000–2012
	80.00	41.60	0.28	0.35	0.84	14.43	0.11	1.00	1971–1980
	100.00	44.18	0.28	0.19	0.65	8.04	0.05	1.35	1981–1990
	80.00	43.91	0.28	0.25	0.69	10.34	0.08	0.96	1991–2000
	84.78	43.20	0.28	0.26	0.72	10.76	0.08	1.02	1967–2012
Frome	100.00	13.99	0.29	0.21	0.67	2.97	0.06	1.09	1971–1980
	100.00	14.34	0.26	0.17	0.62	2.44	0.05	1.57	1981–1990
	90.00	14.42	0.25	0.25	0.71	3.74	0.06	0.92	1991–2000
	97.62	14.55	0.27	0.19	0.65	2.88	0.05	0.99	1971–2012
Pang	76.92	7.22	0.37	0.32	0.80	2.28	0.12	0.81	2000–2012
	100.00	8.74	0.43	0.27	0.76	2.37	0.12	1.38	1993–1999
	85.00	7.75	0.39	0.30	0.78	2.31	0.12	0.97	1993–2012

Figure 3. Model output uncertainty boundaries (5th and 95th percentiles) when performing the GLUE analysis on model calibration (2000–2004) and validation (1961–2012) periods for the Ebbw catchment river flow (monthly values).

Figure 4. Model output uncertainty boundaries (5th and 95th percentiles) when performing the GLUE analysis on model calibration (2001–2012) and validation (1967–2012) periods for the Don catchment river flow (seasonal values).

Figure 5. Model output uncertainty boundaries (5th and 95th percentiles) when performing the GLUE analysis on model calibration (2001–2012) and validation (1971–2000) periods for the Frome catchment using annual observed and simulated data.

Figure 6. Uncertainty band of the DiCaSM parameters for the period 1962–2012 for the Don and Ebbw catchments. The solid (red) vertical line represents the average value of the measured river flow for the simulated period.

Figure 7. Cumulative probability plot of flows for the Don and Ebbw catchments (1962–2012).

Figure 8. Containment ratio (CR) at different timescales for the Frome, Eden, Don and Ebbw catchments.

Figure 9. Asymmetric degree S for the Frome, Eden, Don and Ebbw catchments.

Figure 10. Asymmetric degree T for the Frome, Eden, Don and Ebbw catchments.

Figure 11. Average relative band width, RB, for the Frome, Eden, Don and Ebbw catchments.

Figure 12. Average relative deviation, RD, for the Frome, Eden, Don and Ebbw catchments.

Figure 13. R factor values for the Frome, Eden, Don and Ebbw catchments.

Figure A1. Flowchart for the uncertainty analysis procedure.