Prediction of hydrological drought durations based on Markov chains: case of the Canadian prairies

Figures & data

Fig. 1 Location of hydrometric stations used in the analysis.

Table 1  Particulars of the rivers chosen for the study with salient statistical features based on annual flows (see Fig. 1 for locations)

No.	Name of river and location	Period of record	Area (km^2)	Mean (m^3/s)	cv	γ	ρ1 (Model)
1	Athabasca River at Athabasca (54°43′ 20″N, 113°17′10″W)	1941–2008 (N = 57)	74 600	426.21	0.23	0.97	0.16 (Random)
2	Beaver River at cold Lake Reserve (52°21′18″N, 10°13′2″W)	1956–2008 (N = 53)	14 505	18.76	0.76	1.33	0.37 (AR-1)
3	Bow River at Banff (51°10′ 30″N, 115°34′10″W)	1911–2008 (N = 98)	2 210	39.23	0.13	0.09 (normal)	0.02 (Random)
4	Churchill River above (56°38′ 47″N, 104°44′ 05″W)	1964–2008 (N = 45)	119 000	275.61	0.39	0.88	0.62 (AR-1)
5	Sturgeon–Weir River at outlet of Amisk Lake (54°26′20″N, 103°10′30″ W)	1961–1995 (N = 35)	14 600	46.97	0.43	0.97	0.62 (AR-1)
6	Gods River below Allen Rapids (55°01′35″N, 93°50′10″W )	1948–1993 (N = 46)	25 900	161.41	0.27	0.54 (normal)	0.31 (AR-1)
7	Islands Lake River near Island Lake (54°03′ 34″N, 94°39′34″ )	1934–1994 (N = 61)	14 000	86.37	0.28	0.23 (normal)	0.27 (AR-1)
8	English River at Umfreville, Ontario (49°52′ 30″N, 91°27′30″)	1922–1908 (N = 87)	6 230	58.24	0.34	0.33 (normal)	0.19 (Random)
cv: coefficient of variation, γ: skewness, and ρ1: lag-1 serial correlation of annual stream flow sequences.

Table 2  Summary of autocorrelations of SHI and salient statistical parameters of monthly and weekly flow (non-stationary) sequences

River	μ* (m^3/s)	Monthly flows					Weekly flows
		cv ^†	γ	ρ1	ρ2	ρ ^‡	cv ^†	γ	ρ1	ρ2	ρ^‡
Athabasca	424.56	0.35	1.25	0.61	0.45	0.73	0.42	1.26	0.819	0.644	0.901
Beaver	18.63	1.02	1.83	0.81	0.68	0.64	1.11	1.92	0.921	0.814	0.871
Bow	39.09	0.24	0.71	0.49	0.27	0.66	0.31	0.92	0.726	0.521	0.916
Churchill	275.27	0.44	0.87	0.96	0.90	0.93	0.45	0.87	0.996	0.986	0.994
Sturgeon	46.91	0.49	1.13	0.97	0.91	0.94	0.49	1.14	0.997	0.990	0.995
Gods	161.29	0.38	0.55	0.96	0.87	0.93	0.34	0.52	0.995	0.986	0.996
Islands	86.19	0.38	0.89	0.88	0.72	0.81	–	–	–	–	–
English	58.10	0.52	0.55	0.78	0.58	0.65	0.54	1.24	0.968	0.899	0.961
*μ is the average of 365 daily values.
^†cv is computed as the ratio of the average of 12 monthly or 52 weekly standard deviations to their means of the respective periods; likewise γ is the average value of 12 months or 52 weeks.
^‡ρ is lag-1 autocorrelation of the original non-stationary series of monthly or weekly flows, whereas ρ1 and ρ2 are lag-1 and lag-2 autocorrelations from the SHI (standardized) series.

Table 3  Summarized values of threshold levels, conditional probabilities, observed (L _T -ob) and predicted, E(L _T ) at the threshold level corresponding to q = 0.5 for annual drought lengths

River	T (year)	Flow ratio *	e 0 -ob	Counting method		LT -ob (year) ^†	E(LT ), (year)		Model used
				q	q q		Markov-1 or random ^‡	Extreme ^§
Athabasca	57	0.97	−0.15	0.51	0.59	6(1998−2003)	6(5)	6	Random
Beaver	53	0.77	−0.29	0.51	0.78	16(1980–1995)	11(8)	10	Markov-1
Bow	98	1.01	0.04	0.50	0.49	7(2000–2006)	7(6)	6	Random
Churchill	43	0.94	−0.15	0.51	0.78	8(1988–1995)	10(8)	9	Markov-1
Sturgeon	35	0.88	−0.27	0.49	0.59	7(1987–1993)	6(5)	6	Markov-1
Gods	41	0.94	−0.04	0.50	0.61	7(1987–1993)	7(5)	6	Markov-1
Islands	61	0.96	−0.20	0.51	0.48	6(1939–1944)	6(5)	5	Markov-1
English	87	0.98	−0.05	0.51	0.57	6(1979–1984)	6(6)	6	Random
*ratio of median flow to mean flow (annual).
^†values in parentheses represent drought duration.
^‡values in parentheses are based on the Weibull plotting position formula; those without parentheses are based on the Hazen plotting position formula.
^§values were computed using the extreme number theorem.

Fig. 2 Comparison of L_T -ob and E(L_T ) on an annual time scale by Markov chain-1 or random model: (a) Hazen plotting position, and (b) Weibull plotting position.

Table 4  Illustration of calculations for monthly and weekly drought lengths for Athabasca River, parameters estimated by counting method

q	LT -ob	Markov parameters									E(LT )
		e 0	qq	qp	qqq	qqp	α1	α2			1*	2^†
Monthly drought parameters (T = 684):
0.50	22	−0.17	0.77	0.23	0.81	0.62	1.05	0.77			18	21
0.45	22	−0.30	0.72	0.29	0.78	0.55	1.08	0.71			15	18
0.40	19	−0.44	0.71	0.19	0.74	0.63	1.04	0.85			14	15
0.35	14	−0.57	0.68	0.17	0.72	0.59	1.06	0.82			12	13
0.30	10	−0.66	0.64	0.15	0.66	0.60	1.03	0.91			11	11
0.25	9	−0.76	0.59	0.14	0.62	0.54	1.05	0.87			9	10
0.20	8	−0.86	0.55	0.11	0.55	0.57	1.00	1.04			8	8
0.15	8	−0.97	0.48	0.09	0.54	0.52	1.13	0.96			6	7
0.10	5	−1.06	0.41	0.07	0.43	0.39	1.05	0.91			5	5
0.05	5	−1.27	0.35	−0.04	0.34	0.36	0.97	1.06			4	4
		e 0	qq	qp	qqq	qqp	qqqq	qqqp	α1	α2	1*	2^†	3^‡
Weekly drought parameters (T = 2964):
0.50	44	−0.17	0.85	0.15	0.88	0.70	0.89	0.78	1.04	0.80	34	40	44
0.45	42	−0.32	0.85	0.13	0.86	0.78	0.87	0.78	1.01	0.91	33	35	37
0.40	42	−0.43	0.82	0.12	0.84	0.75	0.86	0.75	1.03	0.89	29	31	34
0.35	41	−0.53	0.81	0.11	0.83	0.73	0.84	0.76	1.03	0.88	26	28	30
0.30	35	−0.61	0.78	0.10	0.80	0.74	0.81	0.74	1.03	0.93	22	24	25
0.25	22	−0.71	0.76	0.08	0.77	0.71	0.79	0.73	1.01	0.92	20	21	22
0.20	20	−0.81	0.72	0.07	0.74	0.68	0.77	0.64	1.03	0.92	17	17	20
0.15	18	−0.91	0.68	0.06	0.70	0.64	0.72	0.63	1.03	0.91	14	15	16
0.10	16	−1.04	0.65	0.04	0.68	0.58	0.70	0.64	1.05	0.85	12	13	14
0.05	2**	−1.21	0.60	0.02	0.63	0.55	0.71	0.45	1.05	0.87	9	9	11
* Markov chain-1 model, ^† Markov chain-2 model, and ^‡ Markov chain-3 model.

Fig. 3 Comparison of monthly L_T -ob and E(L_T ) using (a) Markov chain-1 and (b) Markov chain-2 models.

Table 5  Illustration of calculations for monthly and weekly drought lengths for Athabasca River using revised estimates (DAR-1, variance based) of parameters

q	LT -ob	Markov parameters *					Monthly E(LT )
		e 0	qq′	qp′	qq	qp	1^†	2 ^‡
Monthly drought parameters (T = 684):
0.50	22	−0.17	0.81	0.19	0.77	0.23	21	25
0.45	22	−0.30	0.78	0.18	0.72	0.23	18	21
0.40	19	−0.44	0.75	0.17	0.71	0.19	16	18
0.35	14	−0.57	0.71	0.16	0.69	0.17	13	14
0.30	10	−0.66	0.65	0.15	0.64	0.15	11	12
0.25	9	−0.76	0.59	0.14	0.59	0.14	9	9
0.20	8	−0.86	0.53	0.12	0.54	0.11	7	8
0.15	8	−0.97	0.52	0.09	0.48	0.10	7	7
0.10	5	−1.06	0.39	0.07	0.41	0.07	5	5
0.05	5	−1.27	0.39	0.03	0.42	0.03	4	4
		e 0	qq′	qp′	qq	qp	Weekly E(LT )
Weekly drought parameters (T = 2964):							1^†	2^‡
	44	−0.17	0.89	0.11	0.85	0.15	46	57
	42	−0.32	0.88	0.10	0.85	0.13	41	50
	42	−0.43	0.87	0.09	0.82	0.12	36	42
	41	−0.53	0.85	0.08	0.81	0.11	32	36
	35	−0.61	0.81	0.08	0.78	0.10	26	28
	22	−0.71	0.78	0.07	0.76	0.08	21	23
	20	−0.81	0.75	0.06	0.72	0.07	18	19
	18	−0.91	0.70	0.05	0.68	0.06	15	16
	16	−1.04	0.68	0.04	0.64	0.04	13	13
	14	−1.21	0.64	0.02	0.60	0.02	10	10
*value of qq and qp as determined by the DAR (1) mean length method.
^†Markov chain-1 model, ^‡ Markov chain-2 model,
Remarks: for monthly droughts, α1 =1.04 and α2 =0.88; for weekly droughts, α1 =1.025 and α2 =0.88;
qqq ′ =α1 qq ′; qqp′ =α2 qqq ′

Fig. 4 Comparison of weekly L_T -ob and E(L_T ) using: (a) Markov chain-2 and (b) Markov chain-3 models.

Fig. 5 Comparison of (a) monthly and (b) weekly L_T -ob and E(L_T ) using Markov chain-1 model with revised estimates of parameters.

Fig. 6 Comparison of (a) monthly and (b) weekly L_T -ob and E(L_T ) using Markov chain-2 with revised estimates of parameters.