Analytical procedures for weekly hydrological droughts: a case of Canadian rivers

Figures & data

Fig. 1 Spatial location across Canada of all hydrometric stations used in the analysis (source: Environment Canada).

Table 1  Summary of statistical parameters of flow sequences (weekly) of rivers across Canada

Station number	River name	Record (years)	Area (km^2)	μ (m^3/s)	cv	γ	ρ	ρ1	Model type
BC08KB001	Fraser River	1951–05	32400	816.9	0.90	1.30	0.92	0.73	AR-1
1**	at Shelley								(0.73)
NB01AQ001	Lepreau River	1919–05	239	7.24	1.09	2.91	0.60	0.51	AR-1
2	at Lepreau								(0.51)
NF02YC001	Torrent River	1960–05	624	24.50	1.12	2.95	0.73	0.59	AR-2
3	at Bristol Pool								(0.67, –0.12)
NF02YL001	Upper Humber River	1953–05	2110	79.38	1.09	2.25	0.71	0.49	AR-2
4	at Reidville								(0.53, –0.09)
NS01DG003	Beaverbank River	1922–00	96.9	2.99	1.09	2.20	0.55	0.46	AR-1
5	at Kinsac								(0.46)
NS01FB001	Northeast Margaree River	1929–05	368	17.11	0.97	2.44	0.65	0.46	AR-2
6	at Margaree Valley								(0.44, –0.05)
PE01CA003	Carruthers River	1962–05	46.8	0.94	1.32	4.19	0.66	0.53	AR-1
7	at St. Anthony								(0.53)
QC02PJ007	Beaurivage River	1951–96	709	13.95	1.46	3.24	0.68	0.48	AR-1
8	at Sainte-Etienne								(0.48)
QC02PL001	Becancour River	1923–68	1410	30.62	1.32	3.24	0.72	0.63	AR-2
9	at Lyster,								(0.69, −0.09)
ON02AB008	Neebing River	1954–05	187	1.63	1.88	3.91	0.67	0.67	AR-2*
10	at Thunder Bay								(0.63, –0.07)
ON02BB003	Pic River	1971–05	4270	52.70	1.19	2.68	0.77	0.76	AR-2*
11	at Marathon								(0.82, –0.07)
ON02BF002	Goulais River	1968–05	1160	18.46	1.29	3.37	0.74	0.73	AR-2*
12	at Searchmont								(0.83, –0.14)
ON02CF007	Whitson River	1961–05	272	3.04	1.53	4.36	0.67	0.69	AR-2*
13	at Chelmsford								(0.78, –0.11)
ON02DD013	La Vase River	1975–05	70.4	0.92	1.51	3.31	0.61	0.49	AR-1
14	at North Bay								(0.49)
ON02DD014	Chippewa Creek	1975–05	37.3	0.61	1.12	2.87	0.60	0.45	AR-1
15	at North Bay								(0.45)
ON02DD015	Commanda Creek	1975–05	106	1.71	2.07	2.88	0.71	0.60	AR-1
16	at Commanda								(0.60)
ON02EA010	North Magnetawan River	1969–05	149	2.75	1.29	3.27	0.63	055	AR-2*
17	at Pickerel Lake								(0.56, −0.02)
ON04JC003	Shekak River	1951–86	3290	36.17	1.20	2.58	0.87	0.85	ARMA(1,1)
18	at Highway #11								(0.76, −0.40)
The station identifier in the first column is prefixed by the provincial identifier (BC, NB, NF, NS, PE, QC and ON, respectively denoting the provinces of British Columbia, New Brunswick, Newfoundland, Prince Edward Island, Quebec and Ontario in which a particular river is metered). ** The number indicates the location of the station on Fig. 1. With the model type, parameters Φ1, Φ2 or θ1 are shown in parentheses; for AR-1 process only Φ1, for AR-2 Φ1 and Φ2, and for ARMA (1,1) Φ1 and θ1 are shown. * means AR-2 was still inadequate as the Portmanteau statistic exceeded marginally the critical value (at 5% level of significance and 23 degrees of freedom).

Table 2  Summary of observed and predicted weekly values of E(L _T ) with q = 0.5, gamma pdf of weekly flows

Station identifier*	Memory form	N (year)	e 0	LT -ob (week)	ρ	LT -pr (week)	ρ 1	LT -pr (week)
BC08KB001	AR(1)	55	−0.27	37	0.92	43	0.73	24
NB01AQ001	AR(1)	87	−0.32	29	0.60	21	0.51	18
NF02YC001	AR(2)	46	−0.32	27	0.73	23	0.59	19
NF02YL001	AR(2)	53	−0.32	31	0.71	23	0.49	17
NS01DG003	AR(1)	79	−0.32	29	0.55	19	0.46	17
NS01FB001	AR(2)	77	−0.29	23	0.65	22	0.46	17
PE01CA003	AR(1)	44	−0.36	38	0.66	20	0.53	17
QC02PJ007	AR(1)	46	−0.38	22	0.68	21	0.48	16
QC02PL001	AR(2)	46	−0.36	27	0.72	23	0.63	20
ON02AB008	AR(2)	52	−0.41	23	0.67	22	0.67	22
ON02BB003	AR(2)	35	−0.35	41	0.77	24	0.76	23
ON02BF002	AR(2)	38	−0.35	37	0.74	23	0.73	23
ON02CF007	AR(2)	45	−0.39	44	0.67	21	0.69	22
ON02DD013	AR(1)	31	−0.39	23	0.61	18	0.49	16
ON02DD014	AR(1)	31	−0.33	30	0.60	18	0.49	15
ON02DD015	AR(1)	31	−0.34	36	0.71	21	0.60	18
ON02EA010	AR(2)	37	−0.35	36	0.63	19	0.55	17
ON04JC003	ARMA(1,1)	36	−0.34	42	0.87	32	0.85	30
LT -ob and, LT -pr respectively mean observed and predicted T-week (T = 52 × N weeks) drought duration in weeks using a gamma pdf of weekly flows. *The station identifiers are the same as defined in Table 1.

Table 3  Summary of observed and predicted weekly values of E(L _T ) with q = 0.5, gamma pdf of weekly flows and second-order Markov chain model

Station identifier	N (year)	e 0	LT -ob (week)	LT -pr (week)
				ρ1 and qqq , q qp (ei series)	ρ (original) and qqq , qqp (ei series)	ρ and qqq , q qp (original flow series)
1	2	3	4	5	6	7
BC08KB001	55	−0.27	37	31	39	67
NB01AQ001	87	−0.32	29	26	27	35
NF02YC001	46	−0.32	27	24	24	33
NF02YL001	53	−0.32	31	20	21	29
NS01DG003	79	−0.32	29	24	24	37
NS01FB001	77	−0.29	23	24	24	36
PE01CA003	44	−0.36	38	25	26	40
QC02PJ007	46	−0.38	22	20	20	27
QC02PL001	46	−0.36	27	22	23	32
ON02AB008	52	−0.41	23	21	21	54
ON02BB003	35	−0.35	41	29	32	63
ON02BF002	38	−0.35	37	29	29	49
ON02CF007	45	−0.39	44	26	29	45
ON02DD013	31	−0.39	23	19	20	34
ON02DD014	31	−0.33	30	19	19	28
ON02DD015	31	−0.34	36	25	26	36
ON02EA010	37	−0.35	36	23	24	38
ON04JC003	36	−0.34	42	34	36	62
L T -ob and, LT -pr respectively means observed and predicted T-week (T = 52 × N year) drought duration in weeks using a gamma pdf of weekly flow sequences. The station identifier in column 1 is the same as defined in Table 1.

Table 4  An illustration for computing the observed and predicted weekly values of E(L _T ) at q = 0.5, and using the second-order Markov chain model

Station identifier	N (Year)	Period of sub-sample	qq , qp and e 0*	qqq, , qqp and e 0**	LT -ob (week)	LT -pr (week)
1	2	3	4	5	6	7
Fraser River at Shelley, BC08KB001	55	1955–05	0.87, 0.13 and –0.27	0.703, 0.888, and –0.20	37	43
	30	1951–80			37	38
	27	1951–77			34	37
	28	1978–05			31	37
Upper Humber at Reidville, NF02YL001	53	1953–05	0.75, 0.25, and −0.32	0.704, 0.776 and −0.31	31	24
	27	1953–79			17	21
	26	1980–05			20	21
	30	1953–82			16	21
	40	1953–92			20	23
* the value of e 0 was computed from equation (4)(4) with q = 0.5
** the value of e 0 was obtained from the SHI sequences (i.e. ei series) which were obtained from the observed flow series such that q = 0.5.

Fig. 2 Comparison of predicted and observed drought durations, E(L_T ).

Table 5  An illustration of the table for computing observed and predicted E(M _T ) at the median level of truncation based on the truncated normal distribution of I and gamma pdf of E(L _T )

Station identifier	N	Sub-sample	e 0, q, I (eq.16)	LT -pr	MT -pr	MT -ob	LT -pr-1*	MT -pr-1**	Remark
1	2	3	4	5	6	7	8	9	10
BC08KB001	55	1951–05	−0.27,0.39, 0.71	42.86	30.43	24.66	39.13	27.78	AR-1 to be used
	30	1951–80		37.76	26.81	29.09	34.69	24.63
	27	1951–77		36.87	26.18	26.31	34.18	24.27
	28	1978–05		37.18	26.40	24.77	34.30	24.35
NF02YL001	53	1953–05	−0.32,0.37, 0.69	23.69	16.35	13.76	21.34	14.72	AR-1 to be used
	27	1953–79		21.02	14.50	9.24	18.99	13.10
	26	1980–05		20.87	14.40	11.75	18.11	12.49
	30	1953–82		21.44	14.79	9.46	11.00	7.59
ON02CF007	38	1968–05	−0.39,0.36, 0.67	35.05	23.48	21.29	35.05	23.48	AR-2 to be used
	30	1968–97		32.62	21.85	18.20	32.62	21.85
	30	1976–05		32.62	21.85	21.15	32.62	21.85
ON04JC003	36	1951–86	−0.34,0.37, 0.69	45.5	31.40	38.02	31.40	31.40	AR-2 to be used
	30	1951–80		43.90	30.29	31.12	30.295	30.29
*Predicted values of LT with AR-1 or AR-2 type model as shown in column 10.
**Predicted values of MT based on values of LT as given in column 8.

Fig. 3 Comparison of predicted and observed magnitudes (L_T , second-order Markov chain).

Fig. 4 Comparison of predicted and observed magnitudes (L_T , first- and second-order Markov chain).