Using wavelets, statistically significant interannual and interdecadal oscillations that occurred haphazardly have been detected in southwestern (SW) Canadian seasonal precipitation anomalies. At interannual scales, station precipitation anomalies show unstable relations with large-scale climate anomalies such as the El Niño–Southern Oscillation (ENSO), Pacific Decadal Oscillation (PDO), Pacific/North America (PNA), East Pacific (EP) and West Pacific (WP) patterns, and the Central North Pacific (CNP) index. Not all significant precipitation activities could be matched by similar activities in one or more climate anomalies considered. Inconsistent wavelet coherence and phase difference between the leading principal components (PC) of regional precipitation anomalies and climate indices as well as weak Pearson's correlations between band-passed precipitation PCs and climate indices for the 2–3 year and 3–8 year scales provide supporting evidence for unstable precipitation climate relationships at the interannual scale. On the other hand, interdecadal precipitation variability is mainly associated with low-frequency variability in CNP, PDO and ENSO. Composite analysis of winter precipitation shows that ENSO, PDO, PNA and WP offer better separation of positive and negative precipitation anomalies than EP and CNP. However, the effect of ENSO is found to be stronger than the others. Precipitation power spectrum plots mostly reveal two linear decay regions of different slopes separated by a breakpoint located approximately at 20 to 30 days, while empirical probability plots reveal power law behavior and hyperbolic intermittency in these data, whose correlation dimensions (D2) are between 8 and 9. Different multifractal behaviors are observed among stations because the amount of different rainfall generating mechanisms vary from station to station, as reflected by the haphazard nature of oscillations detected in most precipitation data. Although the leading PCs of winter regional precipitation show modest correlations at zero- to three-season lead times with ENSO and PDO indices, the high D2 values and absence of consistent interannual precipitation activities suggest that prediction of SW Canadian seasonal precipitation by teleconnection with climate indices is likely limited. Adding other predictor fields such as sea surface temperature and/or sea level pressure may be useful.
 Although few patterns of low-frequency variability have been identified from instrumented records, interdecadal precipitation variability has been found in western North America, e.g., regional dipole or seesawing of precipitation pivoting near 40°N [Dettinger et al., 1998; Cayan et al., 1998], Canadian Prairies (CP) [Bonsal et al., 1999], and others. Low-frequency fluctuations at decadal and higher timescales could give rise to large floods and extreme droughts that bring severe damages and grave economic losses. Examples of such extremes in western North America include droughts in the CP [Gan, 1998] and California [Rogers, 1994], and flooding in the west coast [Lott et al., 1997]. Western North America experiences low-frequency climate variability partly because of the Pacific Ocean that has sufficient storage for long-term memory.
 Precipitation processes are notoriously variable spatially and temporally and their properties, such as the intensity-duration-frequency relationship, are often based on the probability theory of stochastic processes that assume an infinite fractal dimension. However, recent discoveries of correlation dimensions (D2) of precipitation [e.g., Islam et al., 1993], and multifractal features of precipitation [Lovejoy and Schertzer, 1985, 1990; Svensson et al., 1996], suggest that precipitation more likely follows a chaotic rather than a stochastic system. Precipitation exhibits multifractal properties whenever its D2 is associated with certain density functions h(D2), which alone does not describe its spatial or temporal singularities. Similarly, the classical Fourier power spectrum can identify the dominant frequencies in a time series but cannot provide any information about the temporal locations of the dominant events. The wavelet transform (WT) decomposes a time series in the time and frequency domains simultaneously, which can reveal the temporal and/or frequency changes of its dominant oscillations [Torrence and Compo, 1998].
 This study has four objectives: (1) to identify the dominant oscillations of precipitation data from southwestern (SW) Canada and their temporal variations using WT; (2) to relate the detected precipitation signals to some large-scale climate anomalies or prominent teleconnection patterns found over the Pacific using both frequency and time domain analyses; (3) to identify the scaling and multifractal properties of precipitation data; and (4) from the above findings, attempt to address issues such as whether different precipitation timescales correspond to different teleconnection patterns, whether decadal or higher-level precipitation variations arise from climate dynamics that are separate from interannual or lower-level variations, and the feasibility of seasonal precipitation predictions by teleconnection with climate indices.
 The precipitation data used in the wavelet analysis were obtained from the Adjusted Historical Canadian Climate Data (AHCCD) database, which contains station observations statistically adjusted for known problems such as missing values, instrument changes, trace events, wind undercatch, evaporation and wetting loss [Mekis and Hogg, 1999]. Monthly precipitation data for 21 stations with long historical records were extracted from the AHCCD database (see Table 1 and Figure 1). To avoid getting results that suffer from low elevation bias, we selected stations with elevations ranging from 8 m at Quatsino, British Columbia (BC) to 1073 m at Calgary, Alberta (see Table 1). The wavelet analysis was carried out on standardized seasonal precipitation anomalies of the 21 stations for 1914–2001. For scaling and multifractal analyses, weekly totals from seven daily station precipitation observations taken from Environment Canada's National Climate Data and Information Archive CD-ROM were used to ensure sufficient data size. Since our data cover both the CP and BC, we can compare results from precipitation under the direct influence of the Pacific westerly wind and that under the rain shadow effect of the Western Cordillera (Canadian Rockies).
Table 1. Summary of the Precipitation Stations Used in the Study
Station Name and Province
Mean Annual Precipitation, mm
Prince Rupert A., BC
Bella Coola, BC
Fort St James, BC
Victoria Phyllis, BC
Quesnel A., BC
Agassiz CDA, BC
Princeton A., BC
Calgary Int'l A., AB
Edmonton M. A., AB
Medicine Hat A., AB
Prince Albert A., SK
Regina A., SK
Indian Head, SK
Brandon CDA, MB
Winnipeg Int'l A., MB
 The CP consists of undulated grassy or cultivated plains with countless small ponds and few low hills. BC generally experiences much wetter climate because of the wet westerly flow from the Pacific. Some parts of BC receive over 1500 mm of annual precipitation while the Prairies generally get less than 500 mm of precipitation because cyclonic precipitation rarely reaches these places from either west or east coasts, and partly because of the frequent visit of dry Arctic air. Some Pacific air streams succeed in penetrating past the Western Cordillera, bringing a considerable fraction of the water vapor and heat from the Pacific to the Prairies. Precipitation is usually light in winter and maximum in late spring or in summer when a thermal low is often present over the warm, sunny Prairies, e.g., the Alberta lows (“Alberta clippers”), where there could be up to 30 days of thunderstorms per year [Phillips, 1990].
 Among the 21 climate stations selected, Princeton, BC, located in the Okanagan River basin receives the lowest mean annual precipitation of about 370 mm. Annual precipitation increases north and westward to well over 2000 mm at the Pacific coast stations of Quatsino, BC and Prince Rupert, BC and eastward to 598 mm at Winnipeg, Manitoba. Agassiz and Bella Coola, BC, which are under the influence of moist Pacific westerly winds, also receive well over 1000 mm of annual precipitation (Table 1). In winter, the northern Arctic air sweeps down over the Prairies. At a relatively high altitude, southern Alberta, has relatively less of the Arctic air than the Chinook (a foehn-type wind that occurs several times a year). While winter and autumn are wet in parts of BC, February is usually the driest month in the Prairies, with precipitation amounting to less than 25 mm per month. The eastern Prairies receive higher annual precipitation than the western parts because of a near humid type of climate resulting from frequent influxes of moist air from the Gulf of Mexico [Hare and Thomas, 1974].
2.2. Large-Scale Climate Indices
 The El Niño–Southern Oscillation (ENSO) is by far the most widely documented source of interannual climate variability. ENSO has its origin in the equatorial Pacific. The warm phase of ENSO (El Niño) is associated with an eastward spreading of warm ocean waters due to weakening and/or reversal of trade winds and ocean currents. El Niño occurrences affect fishing along the coast of Peru because the warm waters displace nutrient-rich cold water off the west coast of South America. There are also significant changes in precipitation patterns because of an eastward shift in the location of the equatorial convective loop. During the cold phase of ENSO (La Niña), anomalously cool ocean waters move from east to west and the convective loop shifts further westward.
 ENSO occurrences are now understood as global climate phenomena because their effects have been linked to various climatic anomalies over the globe, e.g., precipitation patterns in western and continental USA, and east of the Rockies [Ropelewski and Halpert, 1986], Caribbean and tropical America [Rogers, 1988], and northeast Brazil [Kane, 1997]. ENSO teleconnections to the extratropical Northern Hemisphere occur during winter, when its “mature phase” is often associated with anomalously deep central North Pacific lows [Rasmusson and Carpenter, 1982]. Albeit ENSO occurs every 3 to 7 years, it also undergoes mysterious decadal variations, with a relatively large intensity in the late 1800s to early 1900s, followed by a relatively low intensity between 1920 and 1950, and then an increased activity after 1960 [Torrence and Compo, 1998].
 In this study, we investigate the effects of ENSO through teleconnection of the Nino3 index, a time series of equatorial Pacific SST anomalies averaged over the equatorial Pacific (5°S–5°N, 150°W–90°W), and the Southern Oscillation Index (SOI), a time series of normalized monthly differences in sea level pressure (SLP) at Tahiti (≈150°W, 18°S) and Darwin (≈130°E, 13°S) [Allan et al., 1991], which are commonly used as a measure of the strength of ENSO. We also study the effects of extratropical Northern Hemisphere ocean/atmosphere circulation through teleconnections of the Pacific/North America pattern (PNA), West Pacific pattern (WP), East Pacific pattern (EP), Central North Pacific index (CNP) and Pacific Decadal Oscillation (PDO). The PNA pattern represents a quadripole of 700 mbar geopotential height anomalies, with opposite anomalies centered over the Aleutian Low and western Canada, and between the Hawaiian Islands and southeastern US [Wallace and Gutzler, 1981],
 PNA exhibits a wide range of variability in the Northern Hemisphere extratropics, and has been teleconnected to the Great Salt Lake levels [Moon and Lall, 1996] and streamflow along the US west coast [Cayan and Peterson, 1989].
 The WP pattern consists of a north-south dipole of anomalies centered over Kamchatka Peninsula, portions of southeastern Asia and the lower part of western North Pacific. The EP pattern reflects a north-south dipole of height anomalies over the eastern North Pacific, with its northern center located around Alaska and the west coast of Canada [Wallace and Gutzler, 1981]. The CNP index is SLP anomalies averaged over central north Pacific window (35°N–55°N, 170°E–150°W) [Cayan and Peterson, 1989]. The PDO, which represents interdecadal oscillations in the extratropical north Pacific climate system, is represented by a time series of the leading principal component of the North Pacific SST anomalies poleward of 20°N [Mantua and Hare, 2001].
 Some of the selected climate indices are significantly correlated to one another, e.g., winter season PDO, CNP, PNA and WP indices are all significantly correlated to SOI and Nino3 (Table 2). Warm ENSO episodes are associated with negative midtropospheric geopotential height anomalies over the North Pacific, which should lead to strong SLP anomalies in the central and North Pacific regions [Horel and Wallace, 1981]. In winter strong SLP anomalies at the central North Pacific consistently correspond to ENSO events with Nino3 ≥ 2 [Cayan and Peterson, 1989]. The PNA index is significantly correlated to CNP for all seasons and to PDO except during summer. Although PDO is an oceanic (SST) variability mode, its signature extends through the depth of the troposphere and is manifested in the midtroposphere as persistence in the PNA [Mantua and Hare, 2001]. WP is significantly correlated to Nino3 and SOI in winter and spring, to CNP in spring and autumn and to PNA in spring. EP is linked to Nino3, SOI and CNP in spring.
Table 2. Cross Correlations Between Winter Season Climate Indicesa
Statistically significant correlations at the 5% level are indicated in bold text.
Data period used for correlation analysis is 1914–1997.
Data period used for correlation analysis is 1914–1999.
Data period used for correlation analysis is 1914–2001.
Data period used for correlation analysis is 1914–1990.
Data period used for correlation analysis is 1948–2001.
Data period used for correlation analysis is 1950–2001.
3. Research Methodology
 To extract the dominant oscillations for SW Canadian precipitation, we used the continuous Morlet wavelet. The relations between the detected oscillations and some large-scale climate anomalies that are known to exert influences on the climate of western North America were investigated using principal components, wavelet coherence and multiscale correlation analyses. The relations between precipitation and climate indices were further explored on seasonal basis using composite and correlation analysis. The wavelet analysis is briefly described below.
3.1. Wavelet Analysis
 In a WT, a signal x(t) is expanded using a wavelet g(t) constructed from a single real or complex function by dilation/contraction and translation. The former means scaling a particular wavelet member up and down with a flexible window width while the latter means sliding the window center along the time axis with x(t) projected over the wavelet. g(t) is a “packet” of waves of certain amplitude and scale, satisfying a general admissibility condition g(t)dt = 0. The WT of x(t) based on g(t) is
where g* is the complex conjugate of g, ξ is the scale or “dilation” parameter for controlling the window width and often taken as multiples of the lowest possible frequency (1/δt), δt is the time interval, γ is the translation parameter to slide the wavelet along the time axis, and ξ−1/2 is a normalization factor to keep the total energy constant. The wavelet power spectrum is defined as ∣Wt(ξ, γ)∣2. By varying ξ and γ, one can construct a picture of how the wavelet power spectrum of a signal varies in the time-frequency domain, thus making it suitable for studying nonstationarity in time and in frequency.
Figure 2 shows the WT for the seasonal precipitation anomaly (Figure 2a) at Calgary, Alberta. The contour lines in Figure 2b enclose regions of statistically significant wavelet power in the time-frequency space at the 5% significance level of a white noise process. Significant interannual (≈2–8 year) oscillations at Calgary occurred in the early 1900s, 1930s and 1950s, while interdecadal (≈10–32 year) oscillations were active from 1900s to 1960s. Since the wavelet transform is performed in the Fourier space, the ends of the time series are padded with zeros to bring it to the next higher power of two to reduce wraparound effects [Torrence and Compo, 1998]. The wavelet power outside the cone of influence (dashed line in Figure 2b) is suppressed because of the zero-padding and should be interpreted with caution.
 Various quantities can be derived from the wavelet transform so as to condense the vast quantity of information contained in the wavelet spectrum and enhance further interpretation and statistical analysis. One such quantity is the global wavelet spectrum, which is the time average of all the local wavelet power spectra for each scale,
 The global wavelet spectrum only shows dominant scales with no temporal information. The global wavelet spectrum for Calgary (Figure 2c) shows statistically significant oscillations at 1, 12 and about 22 year scales.
 We also used the scale-averaged wavelet power (SAWP) to examine the spatial and temporal fluctuations of the wavelet power in a given scale band, defined as,
where δj is a factor for scale averaging, Cδ is a reconstruction factor, δt is the sampling interval and j1 and j2 are the upper and lower cutoff scales.
 Scale bands and time periods within which precipitation exhibits covariance with climate indices can be identified from wavelet coherence, which is defined as [Torrence and Webster, 1999],
whereWtXY(ξ, γ) is the cross-wavelet spectrum of X and Y, 〈.〉 is a smoothing operator and 0 ≤ Rt2(ξ) ≤ 1. Equation (5) is an accurate representation of the normalized covariance between two time series since the wavelet transform conserves variance. It has been used to study the relationships between Indian rainfall and ENSO [Torrence and Webster, 1999] and Baltic Sea ice conditions and the Arctic and North Atlantic oscillation indices [Jevrejeva et al., 2003].
 We also assess correlations between the principal components (PC) of band-pass filtered signals of precipitation and climate indices at multiple scale bands. The band-passed signal x′(t) over a subset of scales can be recovered from the wavelet coefficients by the equation,
 To study the scaling properties of precipitation, we used the traditional Fourier power spectrum [S(f) where f is the frequency] and an empirical probability distribution function (PDF). A power spectrum that takes the form S(f) ∼ f−β is an indicator of scaling behavior. Likewise, a PDF that obeys a power law, i.e., Pr(R > r) = r−α, α > 0, where r is the precipitation intensity, is termed hyperbolic, with statistical moments valid for orders less than α. A hyperbolic behavior is a characteristic of multifractal processes, which have been used to study scaling in geophysical phenomena [Mandelbrot, 1974; Tessier et al., 1993; Svensson et al., 1996]. In our analysis, weekly precipitation data is divided into nonoverlapping windows of width n (where n is equal to the number of data points in the window), where the average precipitation intensity, r(n, j) at window j was estimated and raised to a power q, and subsequently summed to obtain the statistical moment, M(n, q) from
where N is the number of nonoverlapping windows each of width n, and q is any real number. Since negative and larger positive values of q involve much bias, we chose q from 0.5 to 4.0. From scaling theory, M(n, q) is related to n by a momentum function τ(q) so that,
where τ(q) can be regarded as a characteristic function of either multi or monofractal behavior. If τ(q) is a convex (linear) function of q, precipitation is multifractal (monofractal) [Frisch and Parisi, 1985]. Lovejoy and Schertzer  proposed several equations relating τ(q) with multifractal parameters estimated by the double trace moment technique. However, Gupta and Waymire  questioned the generality of the forms of their equations.
4. Discussions of Results
4.1. Wavelet Analysis of Precipitation
 The wavelet power spectra for all 21 stations are presented using SAWP and time-longitude plots (also called Power Hovmöller) in Figure 3. The SAWP was computed for three scale bands (2–3 year, 3–8 year and 8–30 year). The precipitation variability in the first two bands could be related to the high- and low-frequency components of ENSO and extratropical modes of low-frequency climate variability whereas the last band corresponds to decadal to interdecadal oscillations that could be associated with decadal variations in the North Pacific climate. The solid contours in Figure 3 enclose statistically significant SAWP at the 5% level of a white noise process.
 At individual stations, the 2–3 year scale (Figure 3a) accounts for up to 38% of the variance (e.g., in the 1910s at Regina and Indian Head, Saskatchewan; in the 1980s at Vavenby, BC). At the regional scale, there appears to be some spatial coherency in the early part of the record, with the activities around the 1920s and 1930s occurring at several stations across the Prairies and BC (see Figure 3a). This is followed by a period of little activity during 1940s to late 1950s with the exception of Victoria, BC in 1950. Significant activities since the late 1950s are observed for stations west of 117°W and east of 105°W, while those in eastern BC, Alberta, and most of Saskatchewan show little activity. Considering the space-averaged power, the proportion of variance explained by the 2–3 year band ranges from as low as 5% in the 1950s to about 15% in the 1910s (Figure 3d). This is far below the maximum variance explained at individual stations, showing that significant precipitation activities tend to be not well organized in both space and time. Hence significant precipitation activities in the 2–3 year scale band generally have short lifetime and seem to be haphazard in nature.
 The 3–8 year band (Figure 3b) accounts for up to 30% of the precipitation variance at some stations (e.g., Prince Albert, Saskatchewan around 1920; Quatsino, BC around 1950). As opposed to the 2–3 year band, the 1950s is the time during which the precipitation across the study area shows the largest significant power for the 3–8 year band. There are also scattered activities before and after 1950s, such as those in Saskatchewan in the 1920s, and in BC in the 1960s and 1990s. Similar to the 2–3 year scale, these fluctuations have short lifetimes and show less spatial coherence (see Figure 3e). The variability at the 8–30 year scale (Figure 3c) accounts for up to 20% of the variance at a few stations (e.g., from 1910 to 1930 at Calgary, Alberta; from 1920 to 1950 at Victoria, BC; from 1910 to 1940 at Agassiz, BC; and from 1985 to end of record at Creston, BC). Although there appears to be no spatial coherence except for that at Agassiz and Victoria, BC during 1920–1940, this scale shows persistent variability at those few stations exhibiting significant precipitation activity.
4.2. Regional Precipitation Activities
 Visual inspection of the spatial climate pattern of the Hovmoller diagrams for the 2–3 year and 3–8 year scales in Figure 3 suggests that the 21 precipitation stations can be grouped into 3 regions: (1) west of 117°W (9 stations), (2) 117°W to 107°W (6 stations), and (3) east of 105°W (6 stations). For simplicity, these three regions will be referred to as western, central and eastern regions, respectively in subsequent discussions (see Table 1 for description of stations in each region). Stations in the western region tend to show more variability in the second half of the record period (i.e., after 1950) than in the first half of the record period whereas those in the central region essentially exhibit an opposite pattern, with very little activity during the second half of the record, especially for the 3–8 year scale. Stations in the eastern region appear to exhibit moderate activity in both halves of the record period.
4.3. Influence of Large-Scale Climate Anomalies
4.3.1. Wavelet Analysis of Climate Indices
 The wavelet power spectra of Nino3, SOI, CNP, PDO, PNA and WP indices are shown in Figures 4a–4f. In agreement with previous studies [Torrence and Compo, 1998], the Nino3 index exhibits interannual (2–8 year scale) oscillations of large amplitude during pre-1920 and post-1960 periods, and a reduced level of activity in between (Figure 4a). Nino3 also shows interdecadal oscillations in the early 1900s that persisted with decreasing power until 1940s. Similarly, SOI shows interannual variability of 2–8 year scale during pre-1920s, and post-1970s, and few significant oscillations in between (Figure 4b). There is also some interdecadal variability in SOI that persisted up to the 1920s, and resumed in the 1980s.
 CNP shows scattered interannual oscillations in the 1910s, 1940s, 1950s and 1980s, and a strong 15 year oscillation mode during 1916–1947 (Figure 4c). Even though PDO shows relatively strong power near the 20 year scale, most of its power is concentrated at scales of 42 and 62 years, which are outside the cone of influence (Figure 4d). PNA shows strong oscillations of 8 to 12 year modes between 1960 and 1970s, and a 4 year oscillation from early 1970s to mid 1980s (Figure 4e). The PNA pattern changed from a negative phase in 1964–1967 to a positive phase in 1976–1988, which probably agrees with the southward shift and intensification of the Aleutian low noted in the mid-1970s [Graham, 1994]. Its negative phase again dominated during 1989–1990, followed by a prolonged positive phase from fall 1991 to spring 1993. WP shows significant interannual oscillation during 1950–1965, and 1985–1990s (Figure 4f).
4.3.2. Wavelet Coherence and Phase Difference
 A visual comparison of Figure 3 and Figures 4a–4f indicates that the relationships between the wavelet powers of precipitation anomalies of individual stations and climate indices appear to be highly unstable. Using the wavelet coherence analysis, we could measure the links between regional precipitation signals and climate indices. Since the leading PCs (hereinafter referred to as PC1) of precipitation anomalies for the western, central and eastern regions account for a considerable percentage of the variance in regional precipitation (41.8%, 42.7% and 53.7%, respectively), we used the PC1s as surrogates for regional precipitation signals. The wavelet power spectra of the PC1s for the three regions are shown in Figures 4g–4i.
Figures 5a–5f show the wavelet coherence between the western region PC1 and six climate indices. The thick contours in Figure 5 enclose periods of statistically significant coherence of a red noise process as determined by a Monte Carlo experiment [Jevrejeva et al., 2003]. The phase differences between the two signals for coherences greater than 0.5 are plotted as vectors in Figure 5, where a right pointing arrow indicates that the two signals are in phase while a left pointing arrow indicates an antiphase relationship. At the interannual scale, the western region PC1 and Nino3 show scattered coherences of over 0.8 in the 1950s, 1970s, 1980s and 1990s (Figure 5a). Note that the existence of significant coherence between the two signals does not necessarily depend on the existence of significant wavelet power in both signals. For instance, both Nino3 (Figure 4a) and western region PC1 (Figure 4g) do not have significant power near the 5-year scale in the 1950s but still show significant coherence. Similarly, the western region PC1 power near the 2 year scale in the 1980s, albeit relatively strong, is not statistically significant but the coherence with Nino3 for that time period and scale is statistically significant. The phase difference shows that there is generally an antiphase relationship between Nino3 and western region PC1 for time periods of significant coherence. The phase distribution outside periods of significant coherence is less consistent. For instance, while the phase difference near the 15 year scale is antiphase before 1950s, it is close to in phase after 1980s.
 For the post-1950 period, the coherence between SOI and western region PC1 (Figure 5b) mostly mirrors that of Nino3. Periods of scattered but high coherence with SOI are also observed near the 2-year scale during 1910s to 1920s, 1930s and late 1940s. For time periods with significant coherence, the western region PC1 and SOI are generally in phase. For CNP, the strongest coherence with the western region PC1 occurs at the interdecadal scale centered near 15 year (Figure 5c). This is not surprising since both CNP and western region PC1 have significant wavelet power at that scale (Figures 4c and 4g). For time periods of significant coherence, the CNP and western region PC1 are generally in phase, the only exception being the antiphase relationship in the 2–3 year scale in the 1980s. The strength of the 15 year scale relationship between western region PC1 and the North Pacific ocean-atmosphere system is also supported by the relatively strong coherence with the PDO (Figure 5d).
 Most of the high coherence between PNA and western region PC1 is found in the 1–2 year scale (Figure 5e). However, interannual-scale coherence values of over 0.8 are also observed during the early 1970s, 1980s and late 1990s. A notable feature of the relation between western region PC1 and PNA is the phase change from antiphase in the 1970s to in phase in the 1980s and then back to nearly antiphase in the 1990s. Statistically significant coherence between WP and western region PC1 is observed during the early 1970s, 1980s and late 1990s (Figure 5f), with the relation being out of phase for all three periods.
 The central region PC1 shows very weak interannual-scale coherence with Nino3 and SOI, with the only noticeable periods of significant coherence being the 1920s and 1980s (Figures 6a and 6b). Nino3 shows strong coherence with precipitation in this region in the 1–2 year scale. The CNP index also has weak interannual-scale coherence with the central region PC1 except in the 1920s and 1960s (Figure 6c). An interesting feature of Figure 6c is the antiphase (in-phase) relationship between CNP (PDO) and central region PC1 for the interdecadal scale centered near 15 year for the earlier part of the record. This observation is in direct contrast to the phase distribution for the western region PC1 (Figures 5c and 5d). The PNA and WP indices appear to show better overall interannual-scale coherence with the central region PC1 than the remaining climate indices (Figures 6e and 6f). A phase shift in the coherence between WP and central region PC1 around 1980 is evident from Figure 6f.
 Significant interannual-scale coherence between the ENSO indices and eastern region PC1 exists only in the 1920s, 1950s and 1970s (Figures 7a and 7b). CNP and PDO lead the eastern region PC1 for the interdecadal scale centered near 15 year (Figures 7c and 7d). Again PNA appears to exhibit better interannual-scale coherence with the eastern region PC1 compared to the other indices (Figure 7e). For WP, the only periods of significant coherence occurred in the 1970s and 1980s, with the phase difference showing antiphase relations between the two signals (Figure 7f).
 The above results show that the strength and consistency of interannual-scale relations between SW Canadian seasonal precipitation anomalies and large-scale climate indices changes in both time and frequency domains. This is partly due to the haphazard nature of interannual oscillations in the seasonal precipitation anomalies (Figures 3 and 4g–4i). In other words, as compared to climate indices like Nino3 and SOI, seasonal precipitation anomalies have low overall signal-to-noise ratio, leading to scattered periods of high coherence with climate indices.
 Notwithstanding variations in phase differences seen in the coherence maps, decadal to interdecadal precipitation variability in SW Canada appears to be linked mainly to CNP and PDO. Low-frequency precipitation variability should be linked to SST variations since the latter has distinct decadal fluctuations [Trenberth and Hurrell, 1994; White et al., 1997]. Cayan et al.  found that the Prairies' precipitation fluctuations are associated with extensive shifts of SLP and SST anomalies, which they suggested are components of low-frequency precipitation variability from global-scale climate processes. Weaver et al.  showed that freshwater flux could excite decadal and interdecadal oceanic variability that may be important in the observed decadal/interdecadal variability in our climate system. Ghil and Vautard  suggested that interdecadal oscillation of global surface air temperature could be related to changes in the extratropical ocean circulations. From a coupled ocean-atmosphere model and observations, Latif and Barnett  found that about one third of the low-frequency variability (mode of 20 year) over the North Pacific Ocean and North America can be attributed to unstable air-sea interactions between the subtropical recirculation in the North Pacific and the Aleutian low system. They showed that a correlation of −0.5 to −0.6 exists between anomalies of atmospheric pressure south of the Aleutians and Prairies' air temperature.
4.3.3. Correlations at Multiple Scales
 We also explored the relations between the PC of band-passed seasonal precipitation and band-passed climate indices at multiple scales. If a climate index consistently exerts significant influence on the regional precipitation at a given scale, we expect the band-passed climate index and the band-passed precipitation PC scores to show strong correlations. Table 3 shows Pearson's correlations between the first three band-passed precipitation PC scores of each region and climate indices for the 2–3 year, 3–8 year and 8–30 year scale bands, with statistically significant correlations at the 5% level indicated in bold text. Confidence intervals for the correlations were estimated by the bootstrap resampling approach [Efron and Tibshirani, 1993]. Given N data points, the bootstrap procedure for a single realization involves randomly resampling N rows of precipitation PC and climate index pairs with replacement, and computing the correlation coefficient. We estimated the 5th and 95th percentiles of 10,000 bootstrap correlations. Positive (negative) correlations for which the 5th (95th) percentile is greater (less) than the 5% significance level from the standard significance test for correlation were deemed significant. Correlations significant at the 5% level are indicated in bold text in Table 3.
Table 3. Pearson's Correlations Between the PC Scores of Band-Passed Precipitation and Band-Passed Climate Indices for Selected Scale Bandsa
Correlations significant at the 5% level based on 10,000 bootstrap samples are indicated in bold text.
Higher correlation at one season lead time than at no lead time.
Higher correlation at one year lead time than at no lead time.
 From the correlations in Table 3, it is clear that no single climate index can explain more than 30% of interannual precipitation variability in SW Canada. The rather weak correlations for interannual-scale bands corroborate the inconsistencies observed in the wavelet coherence and phase difference presented in section 4.3.2. In general, it appears that regional precipitation shows better correlation with climate indices in the 2–3 year and 8–30 year scale bands than the 3–8 year band.
4.3.4. Composite Analysis
 Here, we use compositing to explore the impacts of the extreme phases of ENSO, PDO, CNP, PNA, WP and EP on the winter (December-January-February; DJF) precipitation of SW Canada. The El Niño and La Niña composites were based on years during which the 5-month moving average of SOI remained in the lower (higher) 25% of the distribution for a period of 5 months or longer [Shabbar et al., 1997]. High (low) PNA, WP and EP phases were defined using a threshold of ±0.5 standard deviations based on the DJF standardized index. High (low) CNP phases were defined as those years where the standardized DJF CNP index is above (below) ±0.75 standard deviations. The years used for ENSO, CNP, PNA, WP and EP composites are listed in Table 4. Observational studies indicate that 20th century PDO regime shifts have occurred around 1924/1925, 1946/1947, 1976/1977 and 1998/1999 [Mantua and Hare, 2001]. Therefore our warm PDO composites are for 1925–1946 and 1977–1998 while the cool PDO composites are for 1914–1924 and 1947–1976.
Table 4. Years Included in Composite Analysis of Winter Precipitation for the Extreme Phases of ENSO, CNP, PNA, WP, and EP Patterns
 The composite precipitation (e.g., for El Niño) for a given station was computed as the ratio of the mean of winter precipitation for anomalous years (e.g., the 21 El Niño years in Table 4) relative to the long-term mean winter precipitation of that station. The long-term means were computed from the period 1914–2001 for ENSO and PDO composites, 1914–1990 for CNP composites, 1948–2001 for PNA composites, and 1950–2001 for WP and EP composites. Figure 8 shows the composites for all 21 stations, where a composite value of greater than unity means that the climate anomaly is associated with positive winter precipitation anomaly, and vice versa. Albeit the composite magnitudes show considerable variation from station to station, La Niña (El Niño), cool PDO (warm PDO), low PNA (high PNA) and low WP (high WP) are typically associated with positive (negative) winter precipitation anomalies across SW Canada (Figures 8a–8c and 8e). In addition, high (low) EP years are associated with positive (mainly negative) precipitation anomalies in the western region, though lately changes to low EP years is unclear (Figure 8d).
 In general, ENSO, PDO, PNA and WP appear to distinguish negative and positive precipitation anomalies better than CNP and EP. Low CNP winters (anomalously low central North Pacific SLP) are mostly associated with negative precipitation anomalies across the study area (Figure 8f). On the other hand, high CNP winters are associated with weak positive precipitation anomalies in the eastern half of the study area but mostly (seven of the nine stations) with negative precipitation anomalies in the western half.
 The composites for stations in the western, central and eastern regions were also averaged separately and for the entire study area in order to assess the characteristic response of regional precipitation to the extreme phases of the climate anomalies (see Table 5). Prince Rupert and Quatsino, BC were excluded from the regional aggregate composites since their ENSO and PNA composites had opposite signs to those of the remaining seven stations in the western region (see stations 1 and 2 of Figures 8a and 8b). Considering the average of the remaining 19 stations, El Niño (La Niña) is associated with a 14% decrease (20% increase) of winter precipitation, high (low) PNA is associated with a 12% decrease (9% increase), warm (cool) PDO is associated with a 8% decrease (9% increase), high (low) CNP is associated with a 11% decrease (4% increase), high (low) WP is associated with a 8% decrease (9% increase), and high (low) EP is associated with a 12% increase (1% decrease) relative to the long-term mean winter precipitation. In general, the responses to the opposite phases of a climate index are opposite but not symmetrical.
Table 5. Aggregate Composites of Winter Precipitation for the Western, Central, and Eastern Regions
 The precipitation ENSO/PNA relations detected in this study generally concur with previous studies; for example, Shabbar et al.  showed that El Niño (La Niña) events are associated with below (above) normal precipitation anomalies across western Canada. Cayan et al.  indicated that ENSO could explain about 10–20% of the annual precipitation variance for the CP, with the variance explained increasing to 20–30% toward the BC coast. Moore and McKendry  indicated that winters dominated by an enhanced PNA pattern are associated with below normal spring snowpack in BC. Hsieh and Tang  showed that El Niño (La Niña) and high (low) PNA years are associated with lighter (heavier) than normal 1 April snow water equivalent in the Columbia River basin, BC.
 Provided that winter precipitation in SW Canada is controlled by large-scale climate anomalies, atmospheric circulation patterns that prevailed during anomalous winter seasons should provide supporting physical evidence for the relations revealed by the composite analysis. For example, during warm PDO phases, the SST field tends to be anomalously cool over the central North Pacific basin, but anomalously warm SST prevails along the west coast of North America [Mantua and Hare, 2001]. These SST anomalies are accompanied by variation in the placement and strength of the Aleutian low during winter [Bond and Harrison, 2000]. A deeper than normal Aleutian low favors enhanced cyclonic winds, which could reinforce the cool SST anomalies over the central North Pacific and inhibit latent and sensible heat release. Bond and Harrison  showed that surface latent and sensible heat fluxes in the central North Pacific basin are suppressed (enhanced) by about 15 W/m2 during warm (cool) PDO phases for both ridge and trough events. They also showed that the meridional temperature gradient in the planetary boundary layer across the Pacific roughly north of 40°N is suppressed (enhanced) during warm (cool) PDO phases. These conditions would likely be associated with disturbances in the location and intensity of moisture bearing westerly flows across western North America.
 Mature El Niño winters are typically associated with deepening of the Aleutian low and an amplification of the western Canadian ridge. The ridge causes enhanced anticyclones and a northward shift in the midlatitude jet stream, leading to relatively dry conditions in western Canada [Shabbar et al., 1997]. On the other hand, mature La Niña winters are typically associated with an erosion of the western Canadian ridge and enhanced westerly flow, resulting in relatively wet conditions in western Canada. Like El Niño winters, high PNA winters are associated with deeper than normal Aleutian low and an enhanced ridge over western Canada. However, the height anomalies over North America during El Niño winters exhibit pronounced meridional gradients whereas the gradients during high PNA winters are mainly zonally oriented [Straus and Shukla, 2002].
Figure 9 shows the DJF 300-mbar geopotential and wind anomaly patterns associated with anomalous CNP winters. During low CNP winters (Figure 9a), the orientation of height anomalies around the Aleutian low has some superficial resemblance to that during mature El Niño winters (not shown). A closer scrutiny of the composite fields reveals that the Aleutian anomaly center during El Niño winters is shifted eastward by 5 to 10° relative to that during low CNP winters. In addition, the highs over western North America extend further south during low CNP winters than during El Niño winters. These differences are evident from Figure 9b, which is a difference field between El Niño and low CNP composites. The positive (negative) height fields over the central North Pacific (western North America and eastern Pacific) in Figure 9b shows that the anomalies near the centers of the Aleutian low and the western Canadian high are much stronger during low CNP winters than during El Niño winters. The difference in the wind patterns in Figure 9b suggests a more enhanced meridional flow during El Niño winters than during low CNP winters though both types of winters appear to be devoid of westerlies; hence both are associated with negative precipitation anomalies over SW Canada.
Figure 9c shows the 300-mbar level circulation patterns associated with high CNP winters. Again, the orientation of the height anomalies around the Aleutian low during high CNP winters has some similarity to that during La Niña winters (not shown). On the contrary, the orientation of the height anomalies over western Canada during high CNP winters is considerably different from that during La Niña winters. First, the upper level flow during high CNP winters is predominantly meridionally oriented, whereas that during La Niña winters is mostly zonally oriented. Second, there is a southwestward displacement of the western Canadian anomaly center during high CNP winters compared to La Niña winters. The difference field between the wind patterns during La Niña and high CNP winters (Figure 9d) suggests an enhanced westerly flow over SW Canada during La Niña winters relative to high CNP winters, possibly due to a southward shift in the position of geotropic westerlies during the latter. To substantiate this argument further, we examined the 300-mbar zonal (U-) wind anomaly patterns during ENSO and CNP winters in Figure 10.
 The zonal wind patterns during El Niño (Figure 10a) and La Niña (Figure 10b) winters show the familiar ENSO related upper level circulation (i.e., a southward displacement of the subtropical jet stream during El Niño winters; enhanced westerlies over western North America during La Niña winters). The zonal wind patterns associated with low CNP winters (Figure 10c) are broadly similar to those during El Niño winters, with the only major difference being an apparent weakening of the jet stream over the subtropical eastern Pacific during low CNP winters compared to El Niño winters. On the other hand, there is a clear southward displacement of the jet stream over western North America during high CNP winters (Figure 10d) compared to La Niña winters. In addition, the jet stream axis during high CNP winters is aligned in the southwest-northeast direction, as opposed to a northwest-southeast orientation during La Niña winters. Hence, with much of the storm track expected to be to the south, the western part of our study area would experience negative precipitation anomalies during high CNP winters. A significant implication of the nonlinearity of western Canadian precipitation responses to high and low CNP winters is that anomalous atmospheric circulation over the central North Pacific alone is not a sufficient condition for the prevalence of moisture deficiency (or surplus) over western Canada.
 As noted earlier, the impacts of the EP pattern are more likely confined to the western parts of the study area. This result is also supported by composites of geopotential height and wind anomalies for high EP (Figures 11a) and low EP (Figure 11b) winters. Figure 11a exhibits the signature of a strong positive phase of the EP pattern, with a deeper than normal trough in the vicinity of the Bering Sea/Alaska, and positive height anomalies to the northeast of Hawaii [Horel and Wallace, 1981]. This phase of EP is associated with enhanced westerly flow over southwestern BC and the US Pacific Northwest.
 During low EP winters, negative height anomalies cover a large portion of the subtropical Pacific extending to the west coast of the US (Figure 11b). A similar anomaly center to that of the Pacific is also observed over a large portion of the polar region and the high latitudes (not shown). Positive height anomalies are centered over the Gulf of Alaska. The flow over the eastern central North Pacific is dominated by easterlies, which could lead to a split and/or displacement of the jet stream over North America (Figure 11b). This would lead to drier than normal conditions over southern BC. A similar circulation pattern dominated the region during the winter of 1992–1993, leading to drier than normal conditions across Canada, and wetter than normal conditions across southwestern and central US [Bell and Basist, 1994].
4.3.5. Correlations With Raw Precipitation
 To assess the usefulness of large-scale climate anomalies for predicting seasonal precipitation, we computed Pearson's correlations between climate indices at zero to 3 season lead times and the dominant modes of regional precipitation. The regional precipitation signals for the western, central and eastern regions were separately represented by the leading PCs (PC1) of seasonal precipitation anomalies from stations in each region. Because of strong intersite correlations among stations in each region, the leading PCs explain a large portion of the variance of regional precipitation. For the western region, the PC1 of each season (excluding data at Prince Rupert and Quatsino, BC) explains 50% of the variance for winter, 37% for spring, 53% for summer and 54% for autumn. The amount of variance explained by PC1 of the central region stands at 49% for winter, 34% for spring, 42% for summer and 49% for autumn. For the eastern region, PC1 explains 55% of the variance for winter, 52% for spring, 47% for summer and 58% for autumn.
 The winter season correlations, when teleconnections in the extratropical Northern Hemisphere are known to be relatively strong [Wallace and Gutzler, 1981; Shabbar et al., 1997], are given in Table 6 (where correlations significant at the 5% level are indicated in bold text). The correlations between Nino3 and PC1 of the eastern and central regions increase for one season lead time, and those between SOI and the PC1 of all three regions increase up to one or two-season lead times. Comparatively, the PDO and PNA indices show the strongest contemporaneous (lag-0) correlations with PC1 of all three regions. The strongest lag-0 correlation of PDO and PNA is with the central region PC1 (−0.54 for PDO and −0.52 for PNA). These results are in agreement with the composite analysis, i.e., El Niño (La Niña), warm PDO (cool PDO), high PNA (low PNA), low CNP (high CNP), high WP (low WP) and low EP (high EP) are generally associated with negative (positive) winter precipitation anomalies. However, for prediction purposes, only ENSO and PDO seem to provide useful information at one- to three-seasons lead time.
Table 6. Pearson's Correlations at Zero to Three-Season Lags Between Selected Climate Indices and Winter Precipitation PC1 Time Series of Western, Central, and Eastern Regionsa
Statistically significant correlations at the 5% level are indicated in bold text.
 During spring, only CNP and PNA show significant lag-0 correlations with the central region PC1 (ρ = 0.23 with CNP and ρ = −0.38 with PNA) and the eastern region PC1 (ρ = 0.25 with CNP and ρ = −0.41 with PNA). During summer, CNP shows significant correlation at lag-0 with the central region PC1 (ρ = 0.25), and at lag-1 with the western region PC1 also (ρ = 0.36). In autumn, significant lag-0 correlations exist between the western region PC1 and PNA (ρ = −0.37), the western region PC1 and CNP (ρ = 0.22), the central region PC1 and PNA (ρ = 0.44) and the eastern region PC1 and WP (ρ = 0.30). It appears that ENSO events affect SW Canadian precipitation mainly during winter but the influence of CNP, PNA and WP could extend to spring and autumn.
 It seems that the proportion of precipitation variance explained by individual climate indices is generally not strong enough to achieve consistently accurate prediction of precipitation anomalies in SW Canada. The unstable relations between climate indices and precipitation may partly be because teleconnection patterns mainly capture the large-scale features of variability while local changes in their anomaly centers can result in large differences in western North American surface climate [Yarnal and Diaz, 1986]. In addition, some climate indices (e.g., CNP), being of limited area averages, may be too simplistic to capture the (nonlinear) dynamical links between large-scale circulation and SW Canadian precipitation.
 Inconsistencies in the strength of the links between a climate index and precipitation may also reflect the complicated relations between the tropical Pacific SST forcing and the internal dynamics of the North Pacific during the mature phase of ENSO [Bell and Basist, 1994]. In particular, precipitation activities may be affected by constructive (destructive) interactions between two or more climate patterns operating at different characteristic timescales. For instance, using coupled general circulation model (GCM) experiments, Yeh and Kirtman  argued that decadal-scale central North Pacific SST variability can influence the midlatitude ENSO response on interannual timescales. Using historical data, several studies have shown that western North American surface climate and ENSO relationships exhibit interdecadal variations that are synchronized with PDO [McCabe and Dettinger, 1999; Gobena and Gan, 2006]. Similar PDO-ENSO interactions are evident in SW Canadian precipitation response, with El Niño (La Niña) winters during warm (cool) PDO regimes being associated with enhanced negative (positive) precipitation anomalies relative to the mean El Niño (La Niña) response (Figure 12). On the other hand, precipitation anomalies during El Niño (La Niña) winters conditioned on cool (warm) PDO regimes remain close to normal, suggesting that the opposite phases of these two climate anomalies could negate each other. The differences between the conditioned and unconditioned ENSO responses in Figure 12 are all statistically significant at the 5% level of the two sample t test.
 Precipitation climate relations could also be influenced by interactions between interannual modes such as ENSO and PNA. While there is a higher tendency for anomalous PNA years to be associated with mature ENSO phases [Trenberth and Hurrell, 1994], Table 4 shows that extreme phases of PNA can also occur without the ENSO forcing. Using ensemble GCM experiments, Straus and Shukla  showed that tropical SST anomalies during El Niño events primarily force midlatitude circulation patterns that are distinctly different from the PNA pattern. From the analysis of SW Canadian streamflow anomalies, Gobena and Gan  found that streamflow responses to high PNA years conditioned on non–El Niño years were not symmetrical with streamflow responses to low PNA years conditioned on non–La Niña years. While the former produced responses that are similar to those during El Niño years, the latter were associated with streamflow responses that were significantly weaker than those during La Niña years.
 Precipitation processes are also affected by local factors such as topography, surface heating and friction, which will contribute significantly to precipitation variability. In particular, orographic controls could exert significant influence on stations located in the valleys of the rugged terrains of inland BC, where the Pacific air stream has to descend to reach the valley stations. Western Canada's precipitation has been found as a high-dimensional chaotic process with D2 of 8 to 9 [Gan et al., 2002], implying that many independent variables are needed to describe the precipitation process. Here we analyzed the scaling and multifractal properties of some precipitation data at weekly time steps to ensure sufficient amount of data for estimating D2.
4.4. Scaling Properties
 The skewness, coefficient of variation (CV) and Hurst exponent of 6 of the 21 stations plus one station from Northwest Territories (NWT) at weekly time step (Table 7) show that precipitation in SW Canada are highly skewed with short-term memory, for they come from a wide spectrum of processes, ranging from large-scale climatic dynamics to microscale droplet formation. The power spectra for daily precipitation at Calgary, Victoria, Winnipeg and Bella Coola stations are plotted on a log-log scale in Figure 13. The power spectra reveal a significant fluctuation of power over a wide range of frequencies, especially on the high-frequency parts, which implies that precipitation at those stations is obviously not a white noise process. Further, the power spectra show two linear decay regions of different slopes separated by a breakpoint approximately at a timescale of 20 to 30 (≈101.4 to 101.5) days (Figures 13a–13d), which has also been confirmed by the multifractal detrended analysis method of Kantelhardt et al. .
Table 7. Statistical Measures, D2, and Dominant Cycles of 7 Selected Stations
Data Length, weeks
Coefficient of Variation
Hurst Exponent, H
Correlation Dimension, D2
52.74, 26.10, 17.78
 The breaking point at around 20 to 30 days is typical for synoptic processes, such as the cold and warm frontal systems [Fraedrich and Larnder, 1993]. This breakpoint distinguishes the timescales of synoptic and climatic dynamics that are quite different from each other (as shown by the change of slopes in the plot), possibly with the latter representing the teleconnection of Calgary data with climate anomalies like SOI, PNA, and CNP. From the power spectrum plots, it seems that these scale invariant behaviors within each linear decay region, and the abrupt change from one region to the next are typical of the precipitation in SW Canada.
 Besides power spectrum, empirical PDFs of the form Pr (R > r) where r is daily precipitation intensity, are also useful for studying the scaling properties of precipitation processes. The PDFs for Bella Coola, Victoria, Calgary, Edmonton, Regina, Winnipeg and Norman Wells are shown in Figure 14a. We find that asymptotically the probability distributions for all seven precipitation data show a power law or hyperbolic behavior with some intermittency, e.g., Pr (R > r) = r−α, which has been shown to be a general feature of the atmosphere [Mandelbrot, 1974]. According to Lovejoy and Schertzer , empirical values for a similar hyperbolic exponent β based on Pr (ΔX > Δx) = Δx−β, where ΔX is the fluctuation of variable X, are 5/3 for rainfield, 5 for temperature, 1 for radar reflectivity, and 2 for rain and cloud drop volumes. They argued that fractal measures and hyperbolic or flat-tailed probability distributions are characterized by extremely erratic fluctuations. The smaller the values of β, the more extreme will be the fluctuations. When β < 2, the fluctuations could be drastic like the biblical flood of Noah [Mandelbrot and Wallis, 1968] and the largest event could be of the same order in magnitude as the sum of all others in the sample.
 Virtually all the PDFs of daily precipitation from four stations in the Prairie Provinces, two in BC and one in NWT in Figure 14a show two distinct regions. For the 4 Prairies stations, the separation point for those two regions lies around r = 20 mm/day; for Bella Coola, the separation point is around r = 40 mm/day, while that of Victoria lies somewhere between r = 20 and 40 mm/day. For both BC and the Prairies, the regions to the left of the separation points generally correspond to stratiform systems since such storms are of smaller precipitation intensity, while regions to the right of the separation points generally correspond to convective storms of larger precipitation intensities.
 The slopes (exponent α) of the regions to the left of the separation points are slightly larger for the Prairies than for Bella Coola, BC. This implies that small rainfall events (intensities generally less than 20 mm/day) occur less frequently and in smaller intensities in the Prairies than in Bella Coola, BC. In contrast, the slopes of the regions right of the separation points for the Prairies are slightly smaller than that of Bella Coola, BC. This means that even though Prairies are generally drier than BC, they can be slightly more susceptible to storms of large magnitudes (e.g., floods in Southern Alberta in 1995 and Red River in 1997) or precipitation extremes than BC. Since correlations with climate indices such as ENSO, PNA, etc., predominantly happen in the winter, and occasionally in autumn and spring (with PNA), it seems that on the whole more stratiform than convective storms are affected by these large-scale teleconnections. Further analysis is needed to confirm this speculation.
 From the perspective of atmospheric physics, where the atmosphere becomes progressively “flatter” as the scale increases, the fractal dimension of the atmosphere is only about 2.55 [Lovejoy and Schertzer, 1986]. Therefore in the beginning it seems that low fractal dimensions to precipitation data found by earlier researchers seem reasonable; for example, Rodriguez-Iturbe et al.  obtained a D2 of 3.8. In the 1990s, such dimensionality began to appear low [Jeong and Rao, 1996; Zeng and Pielke, 1993] because climate variables have been found to be loosely coupled set of fairly high-dimensional subsystems [Gan et al., 2002] and D2 should reflect the complexity of the climate dynamics. We expect precipitation to exhibit high dimensionality because it should be linked to the water vapor, general circulation, terrain and land use features, air temperature, pressure, solar energy, and possibly other variables.
 However, as most precipitation data are highly skewed, this means clustering of data in some localized areas but leaving other areas virtually empty, which resulted in significant negative bias on the D2 estimated [Wang and Gan, 1998]. For example, for the weekly precipitation data of Canada and Uganda, Gan et al.  obtained a D2 that varies from 1.4 to 3.2, but after bias correction, the D2 becomes 7.8 to 9.0. They also found that temperate climate seems to have precipitation D2 higher than D2 of river basins located at hot or cold climates because the former has four distinct seasons while the latter mainly has two distinct seasons, wet versus dry or warm versus cold seasons. Also, Gan et al.  found that streamflow and precipitation data show comparable D2 values, which implies that terrain and land use characteristics of river basins likely directly influence the dynamics of precipitation.
4.5. Multifractal Properties
 The hyperbolic PDFs (Figure 14a) reveal that the exceedance probability of precipitation intensity follows a power law behavior. The D2 of precipitation data obtained using the Hill algorithm, and adjusted for the bias caused by the skewness of data distribution [Wang and Gan, 1998] (Table 7) reveals that trajectories of the precipitation dynamics stay on an attractor whose submanifold is fractal (noninteger D2). The estimated D2 indicates that the number of independent variables present in the evolution of precipitation is around 8 to 9. In other words, D2 of precipitation should be much higher than that of the atmosphere (about 2.55).
Figure 14b shows the plot of M(n, q) of the weekly precipitation data of Calgary. When q = 1, we get a horizontal straight line which means M(n, q) is independent of n. As q increases, we notice a slow but progressive increase in the curvature of the plot between M(n, q) and n. The slopes τ(q) for q = 0.5, 1.0 … 4.0 are shown in Figure 14c. The weekly precipitation data at all seven stations exhibit multifractal behavior since the plot of τ(q) versus q is a convex function. Further, the degree of multifractal behavior changes from one station to the other (Figure 14c), with a clear difference between the Prairies and BC. The higher degree of curvature for the Prairies stations shows that precipitation variations in the Prairies are more extreme than in BC.
 Our results concur with that of Svensson et al.  who found both the monsoon precipitation of China and temperate precipitation of Sweden to be multifractals. The multifractal approach shows that high rainfall intensities of short duration are embedded in lower intensities of longer duration, which in turn are part of even lower intensities of even longer duration. In contrast to a wide range of D2 estimated by various researchers, most researchers concluded that the precipitation time series they analyzed display multifractal properties [e.g., Tessier et al., 1993].
 These temporal, multifractal properties that vary from station to station complement the “haphazard” oscillations of various timescales (less than annual to interdecadal levels) detected in the wavelet transform of precipitation data of SW Canada. In some sense, the intermittency of precipitation is reflected through the wavelet spectrum (Figure 3) where significant oscillations appeared and disappeared over time and over a wide range of periodicity. The wavelet coherence plots between climate anomalies and precipitation data (Figures 5–7) further demonstrate the significant spatial and temporal variability of active interactions between climate anomalies and precipitation. Intermittent precipitation consists of weak and active periods, and the latter that accounts for most of the energy and moisture fluxes are often found to be irregularly distributed.
5. Summary and Conclusions
 From the wavelet, scaling, and multifractal analysis of 21 stations of long-term precipitation data of SW Canada, and their teleconnection to large-scale climate anomalies for atmospheric circulation and SST of the Pacific, the conclusions are as follows:
 1. Using the Morlet wavelet, statistically significant (α = 0.05) interannual oscillations that occurred haphazardly (even for stations located in the same region) have been detected in the precipitation data of SW Canada. Interdecadal fluctuations tend to be more persistent but significant oscillations were detected only at a few stations.
 2. On the basis of similarities in precipitation wavelet power at interannual scales, three regions were identified and the precipitation activities in each region were related to large-scale atmospheric circulation patterns (via climate indices). The wavelet coherence and phase difference between climate indices and the leading PC of precipitation for each region was found to be highly inconsistent in both time and frequency. In addition, Pearson's correlations between the PCs of band-pass filtered precipitation and climate indices were generally found to be weak, indicating that the strength of teleconnections changes in both time and frequency.
 3. In general, ENSO (Nino3 and SOI) exerted relatively strong influence on the winter precipitation data of SW Canada as compared to the other climate anomalies. At regional level, El Niño (La Niña) leads to a 14% decrease (20% increase) in mean winter precipitation. PNA, PDO, CNP, WP and EP also have their shares in forcing the precipitation over the region. Strong positive (negative) PNA leads to a 12% decrease (9% increase) in mean winter precipitation. The influence of PNA also extends to the spring and autumn seasons. PDO is associated with an 8% decrease (9% increase) whereas high (low) CNP is associated with a 11% decrease (4% increase) in mean winter precipitation. Strong positive (negative) WP leads to an 8% decrease (9% increase) in mean winter precipitation. The impact of EP tends to be confined to the western region of the study area only, with strong positive (negative) EP leading to a 14% increase (4% decrease) in mean winter precipitation in that region. The detected teleconnections could occur at interannual or interdecadal levels (depending on which anomaly index is used), and their strength changes in time and space, making their applications for seasonal precipitation prediction unreliable. Further, climate indices alone are too simplistic to capture the precipitation variability consistently.
 4. Power spectrum plots of precipitation show two linear decay regions of different slopes separated by a breakpoint approximately at 20 to 30 days, which distinguishes the timescales of synoptic and climatic dynamics that influence the precipitation processes, possibly with the latter representing teleconnection with climate anomalies like ENSO, PNA, CNP, etc. Conversely, it seems more stratiform than convective storms are affected by these anomalies. Empirical probability distribution plots reveal power law behavior and hyperbolic intermittency, a general feature of atmosphere. By the Hill algorithm, and adjusting for the bias caused by the skewness of data distribution, the estimated D2 ranges from 8 to 9, showing precipitation as a high-dimensional chaotic process.
 5. Weekly precipitation data exhibit multifractal behavior, which essentially means that high rainfall intensities of short duration are embedded in lower intensities of longer duration, which in turn are part of even lower intensities of even longer duration. Furthermore, different multifractal behavior is observed between stations because the proportions of rainfall generating mechanisms (such as stratiform and convective) vary from station to station, and different rainfall generating mechanisms give rise to different multifractal properties. These changes in multifractal properties should also be partly attributed to the haphazard oscillations detected in most precipitation data.
 In conclusion, given the high D2 of 8 to 9 and multifractal properties identified, we expect the climate systems for precipitation prediction to be very sensitive to the initial conditions, a basic characteristic of a chaotic system. In other words, any slight perturbation or errors will amplify exponentially in successive iterations. Furthermore, compounded by their haphazard low-frequency oscillations, it will be difficult to get consistent seasonal predictions of the highly nonlinear precipitation processes in SW Canada by only teleconnecting with climate indices. Some selected SST and/or SLP predictor fields from the Pacific Ocean could be useful. The prediction result will likely be season- and region-dependent, with better results expected during winter (or autumn) seasons. This is also supported by the fact that lagged correlations between climate indices and precipitation that are significant are mainly found during the winter.
 This project was partly funded by the NSERC of Canada. The AHCCD monthly precipitation data were obtained from the Climate Research Branch of the Meteorological Service of Canada (MSC). The daily precipitation data were taken from the Canadian daily temperature and precipitation data archive of MSC. The SOI data were made available by P. Jones of University of East Anglia, UK, while the Nino-3 SST data are from UK Meteorological Office, GISST2.3. The PNA, WP, and EP indices are obtained from the Web site of Climate Prediction Center of NOAA, and the CNP data were supplied by M. Dettinger of Scripps Institution of Oceanography. The upper level geopotential and wind data were obtained via anonymous ftp from the NOAA-CIRES Climate Diagnostics Center (CDC). The software codes for wavelet analysis were made available by Torrence and Compo (http://paos.colorado.edu/research/wavelets/) and A. Grinsted (http://www.pol.ac.uk/home/research/waveletcoherence/). Comments and suggestions provided by two anonymous reviewers led to improvement in the contents of the paper and are highly appreciated.