A non-uniqueness problem in the identification of power-law spectral scaling for hydroclimatic time series

Figures & data

Table 1 Summary of records used in this study. “Start” and “end” are the first and last years of the time series. Acronyms are as follows. NOAA: US National Oceanic and Atmospheric Administration; JISAO: Joint Institute for the Study of the Atmosphere and Ocean; AHCCD: Adjusted Historical Canadian Climate Data; WSC: Water Survey of Canada; RHBN: Reference Hydrometric Basin Network; INSTAAR: Institute of Arctic and Alpine Research, University of Colorado at Boulder.

Data set	Start	End	Source
Circulation indices—instrumental
Atlantic Multidecadal Oscillation (AMO) Index, DJF mean	1857	2011	NOAA Earth System Research Laboratory
North Atlantic Oscillation (NAO) Index, DJF mean	1951	2011	NOAA Climate Prediction Center
Pacific Decadal Oscillation (PDO) Index, DJF mean	1901	2011	JISAO, University of Washington
Oceanic Niño Index (ONI), DJF mean	1950	2011	NOAA Climate Prediction Center
Circulation indices—palaeoclimatological
Dendrochronologically reconstructed PDO index	1661	1991	NOAA National Climatic Data Center; Biondi et al. (2001)
Dendrochronologically reconstructed PDO index	1700	1979	NOAA National Climatic Data Center; D’Arrigo et al. (2001)
Dendrochronologically reconstructed PDO index	993	1996	NOAA National Climatic Data Center; MacDonald and Case (2005)
PDO index reconstructed from historical documents	1470	1998	NOAA National Climatic Data Center; Shen et al. (2006)
Meteorological station data
Annual mean central England air temperature, UK	1659	2011	UK Met Office (HadCET); Parker et al. (1992), Manley (1974)
Annual total precipitation, Penola Post Office, Australia	1863	2010	Bureau of Meteorology “high-quality climate change datasets”
Annual mean air temperature, Toronto, Canada	1841	2010	Environment Canada 2nd generation AHCCD
Sep-Oct mean sea-level pressure, Madras (Chennai), India	1796	2002	University of East Anglia Climate Research Unit; Allan et al. (2002)
Water resources time series
Nile River annual minimum level, Cairo, Egypt	622	1284	Roda Nilometer: Beran (1994), Toussoun (1925)
Naturalized annual mean flow, Thames River, Kingston, UK	1883	2011	UK National River Flow Archive
Annual maximum daily flow, Bow River, Banff, Canada	1911	2010	Environment Canada/WSC/RHBN; Harvey et al. (1999), Brimley et al. (1999)
Annual mean groundwater elevation, Dalton Holme, UK	1889	2011	British Geological Survey National Groundwater Level Archive
Annual net glacier mass balance measurements
Storglaciären (Kebnekaise, Lapland, Sweden)	1946	2004	INSTAAR: Dyurgerov and Meier (2005)
Tsentralnyj Tuyuksu (Tien Shan, Kazakhstan)	1957	2003	INSTAAR: Dyurgerov and Meier (2005)
South Cascade (Cascade Range, Washington, USA)	1953	2003	INSTAAR: Dyurgerov and Meier (2005)
Echuarren Norte (Central Andes, Chile)	1976	2003	INSTAAR: Dyurgerov and Meier (2005)

Fig. 1 Illustrative examples of numerical power spectra. Thin, solid, dark grey line gives best-fit regression between ln(f) vs ln(PSD); thick, dashed, light grey vertical line is ln(f *). (a) PDO index (circulation pattern, outcome 2). (b) Central England mean temperature (climate station, outcome 3). (c) Bow River maximum flow (water resources, outcome 1a) (f * undefined and not shown; see Table 2). (d) Storglaciären net balance (glacier mass balance, outcome 2).

Table 2 Analysis results. “Log regression” gives regression slope, β₁, and significance of correlation, p, corresponding to potential linear associations between ln(f) vs ln(PSD) for each time series. “Heuristic rule” gives data set duration, T; observed serial correlation coefficient of the observational time series, ρ; critical timescale from heuristic rule, τ*; natural logarithm of Rayleigh frequency corresponding to T, ln(f_Ray); and logarithm of frequency corresponding to τ*, ln(f *). “Outcome” summarizes the assessment of background spectrum form for each time series, using both quantitative information and visual spectrum assessments; the numbers correspond to the numbered outcomes discussed in the text.

Data set	Log regression		Heuristic rule					Outcome
	β1	p	T	ρ	τ*	ln(fRay)	ln(f *)
Instrumental climate indices
AMO index	–0.71	<0.01	155	0.61	141	–5.0	–5.0	Possible but insufficient data (2)
NAO index	–0.65	<0.01	61	0.35	67	–4.1	–4.2	Possible but insufficient data (2)
PDO index	–0.70	<0.01	111	0.39	73	–4.7	–4.3	Possible but insufficient data (2)
ONI	0.23	>0.10	62	–0.13	n/a	–4.1	n/a	Unlikely (1a)
Pre-instrumental PDO indices
Biondi et al. (2001)	–1.71	<0.01	331	0.85	414	–5.8	–6.0	Unlikely (1b)
D’Arrigo et al. (2001)	–0.46	<0.01	280	0.37	69	–5.6	–4.2	Possible but insufficient data (2)
MacDonald and Case (2005)	–1.19	<0.01	1004	0.75	236	–6.9	–5.5	Strong evidence (3)
Shen et al. (2006)	–0.16	0.05	529	0.14	35	–6.3	–3.6	Moderate evidence (3)
Station data
Central England temperature	–0.40	<0.01	353	0.24	49	–5.9	–3.9	Strong evidence (3)
SW Australia precipitation	–0.17	>0.10	148	–0.01	n/a	–5.0	n/a	Unlikely (1a)
Eastern Canada temperature	–0.31	0.03	170	0.20	43	–5.1	–3.8	Possible but insufficient data (2)
SE India sea-level pressure	–0.22	>0.10	207	0.16	38	–5.3	–3.6	Unlikely (1a)
Water resources
Nile annual minimum level	–0.67	<0.01	663	0.54	114	–6.5	–4.7	Strong evidence (3)
Thames annual mean flow	–0.35	0.07	129	0.25	50	–4.9	–3.9	Possible but insufficient data (2)
Bow annual maximum flow	–0.03	>0.10	101	–0.12	n/a	–4.6	n/a	Unlikely (1a)
Dalton Holme groundwater level	–0.49	<0.01	123	0.25	50	–4.8	–3.9	Possible but insufficient data (2)
Glacier mass balance
Storglaciären	–0.52	0.03	59	0.19	41	–4.1	–3.7	Possible but insufficient data (2)
Tsentralnyj Tuyuksu	–0.46	>0.10	47	0.26	51	–3.9	–3.9	Possible but insufficient data (2)
South Cascade	0.43	0.04	51	–0.19	n/a	–3.9	n/a	Unlikely (1a)
Echuarren Norte	0.16	>0.10	28	–0.30	n/a	–3.3	n/a	Unlikely (1a)

Note: n/a denotes sample lag-1serial correlation coefficient is (at least nominally) negative, so that τ* and f * are undefined.

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