Improved continuous wavelet analysis of variation in the dominant period of hydrological time series

Figures & data

Table 1 Criteria for evaluation of the de-noising results of hydrological time series

Series' type	Characteristic numbers
		σ	Cs	r 1
Original series		σ o	Cs o	r 1o
Main series		σ m < σ o	Cs m ≈ Cs o	r 1m > r 1o
Noise		−	−	r 1n ≈ 0
Notes: For the original series, main series and separated noise: , and are their mean; σ o , σ m and σ n are their standard deviation; Cs o , Cs m and Cs n are their coefficient of skewness; and r 1o , r 1m and r 1n are their lag-1 auto-correlation coefficient, respectively.
− means the characteristic number of noise is unknown.

Fig. 1 Three synthetic series used (upper) and their de-noising results using different wavelets (lower).

Table 2 Synthetic series used in this paper and their de-noising results using different wavelets

Wavelet used	Series	S1					S2					S3
		Characteristic numbers				SNR	Characteristic numbers				SNR	Characteristic numbers				SNR
			σ	Cs	r 1			σ	Cs	r 1			σ	Cs	r 1
	S	0.000	1.001	0.000	0.999	−1.958	0.000	0.708	0.000	0.996	−7.603	0.013	0.409	−0.031	0.971	−2.819
	O	−0.001	1.495	−0.248	0.467		−0.004	1.284	0.018	0.367		0.001	0.615	−0.105	0.408
Haar	S^∼	0.005	0.958	−0.484	0.978	−3.447	−0.003	0.604	−0.176	0.972	−12.342	0.001	0.384	−0.235	0.851	−4.469
	N	−0.006	1.139	−0.119	0.091		−0.001	1.120	0.095	0.160		0.000	0.481	−0.123	−0.022
Db8	S^∼	−0.002	1.016	−0.305	0.998	−1.554	−0.015	0.628	0.086	0.997	−10.805	0.004	0.423	−0.165	0.968	−1.084
	N	0.001	1.098	−0.139	0.049		0.011	1.078	0.041	0.104		−0.003	0.447	−0.109	−0.098
Sym6	S^∼	−0.003	1.001	−0.197	0.998	−1.967	−0.007	0.648	0.075	0.996	−10.942	0.003	0.432	−0.169	0.967	−0.333
	N	0.002	1.105	−0.128	0.025		0.003	1.076	0.098	0.110		−0.002	0.439	−0.076	−0.134
Coif4	S^∼	0.006	1.003	−0.347	0.998	−1.949	−0.003	0.626	0.037	0.997	−10.153	0.003	0.424	−0.152	0.968	−0.983
	N	−0.007	1.106	−0.152	0.030		−0.001	1.082	0.053	0.100		−0.002	0.445	−0.100	−0.105
Bior3.1	S^∼	−0.027	1.147	−0.162	0.980	−0.016	−0.019	1.063	0.085	0.978	−0.413	0.002	0.482	−0.270	0.958	0.221
	N	0.026	1.148	0.092	0.119		0.015	1.085	0.164	0.157		−0.001	0.477	−0.163	0.161
Note: S, O, S^∼ and N: true series, original series, de-noised series and separated noise, respectively.

Fig. 2 Analytical process of hydrological time series by the CWT procedure proposed. CWT: continuous wavelet transform; TSR: temporal scale range; CTS: central temporal scale.

Table 3 The eight observed hydrological series used in this study. Precipitation series data (P1–P5) are from the China Meteorological Data Sharing Service System. Runoff series data sources: R1—Hydrology Bureau, Yellow River Conservancy Committee, Ministry of Water Resources; R2—Zuo and Gao (2004); R3—National Natural Science Foundation of China

Series	Basin	Hydrological station	Climatic condition	Length
P1	Liaohe	Shenyang	Temperate continental monsoon climate	41 years (1961–2001)
P2	Haihe	Heilongdong	Temperate East-Asian monsoon climate	47 years (1956–2002)
P3	Zhujiang	Shenzhen	Subtropical climate	41 years (1961–2001)
P4	Changjiang	Chengdu	Subtropical climate	41 years (1961–2001)
P5	Huaihe	Zhumadian	Temperate continental monsoon climate	366 days (2000)
R1	The Yellow River	Lijin	Temperate East-Asian monsoon climate	54 years (1950–2003)
R2	Kaiduhe	Dashankou	Inland climate	20 years (1978–1997)
R3	Baixi	Baixi	Subtropical East-Asian monsoon climate	144 months (1987–1998)

Table 4 De-noising results of the eight hydrological series using the chosen wavelet basis functions (WBF)

Series	Chosen WBF	Series type	Characteristic numbers				SNR
				σ	Cs	r 1
P1	Db2	O	683.9	128.8	0.24	0.15
		S^∼	685.0	87.5	0.18	0.54	4.43
		N	−1.1	70.1	−0.01	−0.19
P2	Coif3	O	557.4	173.5	1.06	−0.14
		S^∼	555.4	152.42	1.16	0.56	−0.35
		N	2.0	155.13	0.10	−0.18
P3	Dmey	O	1944.4	399.7	−0.10	0.13
		S^∼	1948.2	326.5	0.23	0.56	5.25
		N	−3.8	251.1	0.17	−0.18
P4	Db4	O	889.3	157.9	0.68	0.004
		S^∼	889.1	123.2	1.36	0.46	5.60
		N	0.2	93.1	−0.31	−0.21
P5	Db3	O	4.3	16.2	−0.65	0.39
		S^∼	4.3	15.6	−0.65	0.40	49.70
		N	0.0	1.3	−0.67	0.21
R1	Db2	O	324.5	195.0	0.64	0.69
		S^∼	323.1	164.3	0.87	0.68	18.43
		N	1.4	65.4	−0.10	0.14
R2	Bior2.4	O	1218.7	145.17	0.35	−0.04
		S^∼	1218.5	104.9	−0.26	0.74	0.84
		N	0.2	100.6	0.20	−0.24
R3	Sym3	O	292.6	334.2	2.27	0.17
		S^∼	291.3	289.7	1.69	0.26	28.64
		N	1.3	69.2	−0.77	−0.21
Note: O, S^∼ and N are the original series, de-noised series (i.e. the main series) and separated noise, respectively.

Fig. 3 Locations, in China, of the eight observed hydrological series studied. P: precipitation; R: runoff.

Fig. 4 Eight observed hydrological series studied and their de-noising results. Thin lines: observed series, bold lines: de-noised series.

Fig. 5 Global wavelet spectra of the eight observed hydrological series used in this study.

Fig. 6 Variations in wavelet coefficients of the eight series at different central scales.

Table 5 The first main periods of six annual hydrological series with different lengths identified by the main series spectral analysis method (MSSA) and wavelet variance estimator (WVE)

Annual series	Method
	MSSA	WVE
P1	2.1 (20), 2.1 (25), 5.7 (30), 6.0 (35), 2.3 (40), 7.5 (41)	2.0 (20), 2.0 (25), 6.0 (30), 6.0 (35), 3.0 (40), 3.0 and 7.0 (41)
P2	3.2 (20), 8.8 (25), 8.8 (30), 3.2 (35), 5.1 (40), 4.8 (45), 4.8 (47)	3.0 (20), 9.0 (25), 9.0 (30), 7.0 (35), 5.0 (40), 5.0 (45), 5.0 (47)
P3	2.1 (20), 2.1 (25), 2.3 (30), 17.7 (35), 6.8 (40), 18.3 (41)	2.0 (20), 2.0 (25), 3.0 (30), 18.0 (35), 6.0 (40), 18.0 (41)
P4	9.8 (20), 3.0 (25), 2.1 (30), 2.1 (35), 2.3 (40), 10.3 (41)	10.0 (20), 2.0 (25), 2.0 (30), 2.0 (35), 2.0 (40), 10.0 (41)
R1	8.3 (25), 7.2 (30), 8.3 (35), 6.7 (40), 10.0 (45), 10.0 (50), 17.2 (54)	8.0 (25), 7.0 (30), 8.0 (35), 11.0 (40), 13.0 (45), 17.0 (50), 7.0 (54), 18.0 (54)
R2	2.8 (10), 7.1 (15), 7.1 (20)	3.0 (10), 8.0 (15), 8.0 (20)
A (B) in Table 5 means that the identified period of the extended series with the length of B is A.

Zuo , Q.T. and Gao , F. 2004 . “ Periodic overlapping prediction model and its three improved models of hydrological time series (in Chinese with English abstract) ” . In Journal of Zhengzhou University (Engineering Science) Vol. 25 , 67 – 73 .

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