This study investigated the correlation between western US streamflow and two of the most important oceanic–atmospheric indices having significant effects in this region, namely, El Niño Southern Oscillation (ENSO) and Pacific Decadal Oscillation (PDO). Data from 61 streamflow stations across six different hydrologic regions of the western USA were analyzed, using a study period of 60 years from 1951 to 2010. Continuous wavelet transformation along with cross wavelet transformation and wavelet coherence were used to analyze the interaction between streamflow and climate indices. The results showed that streamflows have changed coincidentally with both ENSO and PDO over the study period at different time-scale bands and at various time intervals. Both ENSO and PDO showed correlation with streamflow change behavior from 1980 to 2005. ENSO showed a strong correlation with streamflow across the entire study period in the 10–12 year band. PDO showed a strong correlation in bands of 8–10 years and bands beyond 16 years. The phase relationship showed that both ENSO and PDO preceded streamflow change behavior; in some instances, the variables were found to be moving in opposite directions even though they changed simultaneously. The results can be helpful in understanding the relationship between the climate indices and streamflow.

INTRODUCTION

Understanding the behavior of streamflow change can be considered one of the most important parameters used to trace changes that have occurred in the hydrologic cycle. Since streamflow measures the flow in natural streams, a change in the behavior consequently can threaten the entire water supply system. The hydrologic cycle, along with the mass balance mechanism associated with it, plays an important role in transporting mass and energy throughout the hydrosphere (Rice et al. 2015). Intensification of parameters in the hydrologic cycle can cause extreme events that bring about enormous loss and, subsequently, can endanger the entire water resource system (Lins & Slack 1999; Cayan et al. 2001; McCabe & Clark 2005). Studies have strongly suggested that proper documentation and understanding of the hydrologic variables can be used as effective tools to evaluate changes occurring in the hydrologic cycle (Clark 2010; Birsan et al. 2012). Hydrologic processes are directly related to climate conditions, and changes in hydrologic processes can be attributed as a major cause behind the spatiotemporal patterns of hydrologic events as well as their severity and recurrences (Burn et al. 2010; Dawadi & Ahmad 2012; Zhang et al. 2014). Change in the hydrologic cycle has been considered one of the crucial results of climate warming (Ampitiyawatta & Guo 2009; Durdu 2010).

Many previous studies have determined relationships among hydro-climatic parameters (i.e., temperature, precipitation, streamflow, etc.) and climate variability (McCabe & Wolock 2002; Birsan et al. 2005; Hamlet & Lettenmaier 2007; Durdu 2010). Temporal variability of climate change has been found to be related with the change in hydrologic variables as well (Burn & Elnur 2002). Recent works have studied the relationship between secondary hydrologic parameters, such as streamflow and climate variability (Kalra & Ahmad 2011; Carrier et al. 2013, 2016; Tamaddun et al. 2016). The need to understand the relationship between a change in climate and the consequent change in hydrologic variables (i.e., streamflow) is increasing since it is of utmost interest to efficiently manage sustainable water resources, especially with the increase in population and with the continuous and growing demand in the energy sector (Kalra & Ahmad 2012; Shrestha et al. 2012; Wu et al. 2013). Besides listing the potential dangers that can occur as a result of climate change (Bates et al. 2008; IPCC 2014), studies constantly have emphasized the importance of spatio-temporal scales on the change behaviors observed in the hydrologic variables (Weider & Boutt 2010).

Besides understating the relationship between climate change and hydrologic variables, studies have focused on finding correlations among climate indices, which represent various oceanic–atmospheric systems, and hydrologic variables; this is because climate indices can be a very effective tool for forecasting hydrologic cycle behavior. El Niño Southern Oscillation (ENSO) and Pacific Decadal Oscillation (PDO) are two of the most important oceanic–atmospheric indices found to have a great influence on the climate variability in the western United States (Barnett et al. 1999; Taylor & Hannan 1999; Beebee & Manga 2004). ENSO, an index associated with sea-surface temperature (SST) fluctuation, has been identified as one of the most dominant oceanic–atmospheric patterns found in the tropics of the Pacific Ocean; in addition, it is considered to be one of the prominent factors affecting the western US hydrology (Barnett et al. 1999; Cayan et al. 1999; Taylor & Hannan 1999; Beebee & Manga 2004). ENSO is a natural cycle that occurs on a scale of 2–7 years, which alters between a warm phase (El Niño, positive index) and a cold phase (La Niña, negative index). PDO, an index that represents SST fluctuations on a decadal scale, is another important oceanic–atmospheric pattern found in the North Pacific Ocean, and has a larger area of influence than ENSO (Hamlet & Lettenmaier 1999; Miles et al. 2000; McCabe & Dettinger 2002; Beebee & Manga 2004; Trenberth & Fasullo 2007). Similar to ENSO, PDO has two full phases, i.e., warm and cold, and these phases alter with a cycle of around 25 to 50 years (Hamlet & Lettenmaier 1999; Mantua & Hare 2002; Beebee & Manga 2004). The fluctuations of SST have been found to be a good predictor of hydrologic parameters – such as the formation of snowpack, precipitation, soil moisture, streamflow, etc. – since SST affects the air pressure and the wind dynamics above the influencing zone; this, in turn, affects the hydrology of the surrounding area.

In previous studies, ENSO has been identified as a major factor affecting the atmospheric anomalies (extreme conditions) both globally and regionally (Ropelewski & Halpert 1986; Kahya & Dracup 1993). Studies have found PDO to have an influence on such parameters as snowpack formation, precipitation, and streamflow in the western USA, especially in such regions as the Colorado River Basin (CRB) and California (Dettinger & Cayan 1995; Hidalgo & Dracup 2003; Cañón et al. 2007; Sagarika et al. 2015a). Besides understanding the relationship between ENSO and PDO with the various hydrologic parameters, many studies have focused on understanding the coupling effect of ENSO and PDO. According to Praskievicz & Chang (2009) on the Willamette Valley of Oregon, La Niña was found to affect the intensity of November precipitation, while El Niño affected the intensity of April precipitation. This study revealed an inverse relationship between PDO and the intensity of precipitation. A study on the Upper Colorado River Basin (UCRB) by McCabe et al. (2007) found strong correlation between UCRB streamflow and temporal SST fluctuations. Hamlet & Lettenmaier (1999) observed the effect of lead time of ENSO and PDO on a forecasting model for the Columbia River. Sagarika et al. (2014) studied the shifts (step changes) for streamflow patterns in 240 streamflow stations across the continental USA, and observed the coupled effect of the PDO warm and cold phases with the change in ENSO indices. Kalra & Ahmad (2012) concluded that climate signals significantly influenced annual precipitation behavior in the CRB; PDO was found to be more influential on the upper CRB, whereas ENSO was more successful in predicting precipitation behavior in the lower CRB. Beebee & Manga (2004) studied the relationship between runoff generated from snowmelt with ENSO and PDO, and suggested some historical time intervals that were found to be more correlated compared to other intervals. Hoerling & Kumar (2000) provided an explanation on how change in pressure occurs in the Pacific, the subsequent change in tracks of cyclonic storms, and the effects of moisture on the western USA. These studies reveal some important insights regarding how ENSO and PDO are changing with respect to each other; however, they do not clarify whether one or both of these indices influences certain parameters in the same way.

Hydrologic and geophysical time series are very complex to analyze as they are non-stationary in nature and they do not follow normally distributed probability functions (Jevrejeva et al. 2003; Önöz & Bayazit 2003; Grinsted et al. 2004; Milly et al. 2008; Villarini et al. 2009; Sagarika et al. 2015b). As a result, predicting the trend patterns and periodicities of these time series has drawn much attention in recent times (Grinsted et al. 2004). The most traditional mathematical method used to examine periodicities in the frequency domain is Fourier analysis (Polikar 1996). The underlying drawback of Fourier analysis is it implicitly assumes a stationarity in time (Polikar 1996; David & Rajasekaran 2009); however, this cannot be a useful assumption for a time series of hydrologic variables, such as streamflow. Wavelet transformation has been suggested as a powerful tool for analyzing processes that occur over finite spatio-temporal domains and are non-stationary in nature, sometimes containing multiscale resolution (Lau & Weng 1995). Wavelets allow determination of the most significant periodicities (frequencies) of a time series and can explain how it has changed over time (Kumar & Foufoula-Georgiou 1997; Percival & Walden 2000). As a result, wavelet transformation emerged as a better alternative since it could provide information about time and frequency at the same time. By altering time and scale variations, wavelet analyses can produce graphs that can show how the frequency changes over amplitude with the change in time (Echer et al. 2007). Other studies have suggested using wavelet analyses as a successful statistical tool for analyzing trends and other properties of a time series (Nakken 1999; Kang & Lin 2007). A more detailed description of the history of wavelets, classification of wavelets, and how wavelets work can be found in Lau & Weng (1995), Torrence & Compo (1998), Torrence & Webster (1999), Grinsted et al. (2004), and David & Rajasekaran (2009).

Continuous wavelet transform (CWT), which is best suited for feature extraction, has been used in previous studies as a useful tool to extract a low signal-to-noise ratio (s/n) in a time series (Grinsted et al. 2004). In a time series, CWT can analyze intermittent oscillations that are localized; this method performs much better than traditional transformation tools (Foufoula-Georgiou & Kumar 1995; Holschneider 1995; Grinsted et al. 2004). As mentioned earlier, the coupling of two time series can provide information regarding their changing pattern with respect to each other. However, sometimes it becomes important to understand which of these time series affects a third time series more dominantly. The application of cross wavelet transform (XWT) and wavelet coherency (WTC) analysis are useful methods to examine multiple time series that might be linked in certain ways (Jevrejeva et al. 2003; Grinsted et al. 2004; Tang et al. 2014). XWT, which reveals a common power (covariance) and a relative phase relationship in a wavelet spectrum, is constructed from two separate CWTs that are supposedly linked in some way (Torrence & Compo 1998; Grinsted et al. 2004). By observing the XWT, the correlation as well as the phase relationship between the parameters can be assessed. To further quantify the correlation between the parameters, WTC can detect significant coherence even at a lower common power. This technique shows how confidence levels can be calculated against red noise backgrounds (Grinsted et al. 2004). Through the process of using CWT, XWT, and WTC, a one-dimensional time series is transformed into a two-dimensional time–frequency wavelet spectrum. This spectrum can show the amplitude of a signal (in this case time series) at different times and frequencies at the same time (Torrence & Webster 1999). Studies also suggest the use of wavelets as a better alternative compared to other traditional methods for analyzing oceanic–climatic fluctuations, since wavelets can follow the gradual changes occurring in a natural frequency with better accuracy (Meyers et al. 1993; Yiou et al. 2000).

Previous literary works motivated this current study to use CWT as an analysis tool to evaluate the correlation between parameters that other studies have found to be related somehow. Acknowledging some of the limitations of previous research, the current study endeavors to address some of the suggestions that were presented in those works. With this motivation in mind, this research focused on applying CWT, along with XWT and WTC, on data for 61 unimpaired streamflow stations (unimpaired stations are free from any sort of modifications in terms of flow path and condition) located in the western USA for a period of 60 years (i.e., 1951 to 2010). The primary objective of the study was to evaluate significant periodicities that have simultaneously triggered changing patterns of streamflow and climate signals (i.e., ENSO and PDO). Besides observing simultaneous change patterns, this study quantified the correlations present in the change patterns. Each station was transformed with CWT to their wavelet spectrum in order to observe their variability (higher power in the wavelet spectrum represented higher variance in data). A combined streamflow continuous wavelet spectrum was constructed using principal component analysis (PCA) of the data obtained from each station, and was used to construct the corresponding XWTs with ENSO and PDO CWTs. The XWTs revealed the common power of streamflow and ENSO/PDO over the study period. Finally, WTC was performed to quantify the correlation between streamflow and ENSO/PDO.

STUDY AREA AND DATA

Out of the 18 hydrologic regions delineated by the United States Geological Survey (USGS), this study focused on six regions representing the western USA: Rio Grande (13), Upper Colorado (14), Lower Colorado (15), Great Basin (16), Pacific Northwest (17), and California (18). A detailed description of the regions can be found in the hydrologic unit map provided by the USGS (http://water.usgs.gov/GIS/regions.html). Out of the 704 streamflow stations listed by USGS, published in 2012 as the Hydroclimatic Data Network (HCDN) 2009 (Lins 2012), 61 stations were selected based on the availability of continuous water-year data for 60 years from 1951 to 2010. A single station was chosen from each stream to remove spatial bias from the data. Additionally, the streamflow stations were free from any sort of modification or alteration in terms of controlling the flow behavior. Figure 1 shows the chosen regions, with spatial distribution of the stations in each region. Geospatial Attributes of Gages for Evaluating Streamflow, Version II (Falcone et al. 2010) provides details about the stations having the data as well. Upper Colorado was excluded from the analyses, since there were no stations in that region that met the time period of historic data needed in this study (Figure 1).
Figure 1

Map showing the selected regions of the continental USA and the stations within each region. The table at top right shows the number of stations in each region.

Figure 1

Map showing the selected regions of the continental USA and the stations within each region. The table at top right shows the number of stations in each region.

The climate indices datasets used in this study were ENSO and PDO. The data used in this study for ENSO and PDO had the same length as the streamflow data. For both ENSO (http://www.cpc.ncep.noaa.gov) and PDO (http://research.jisao.washington.edu), an increase in the index value refers to the warm phase and a decrease in index value refers to the cold phase.

METHODOLOGY

In the following sections, brief descriptions of CWT, XWT, and WTC are provided, based on Torrence & Webster (1999), Grinsted et al. (2004), and Tang et al. (2014). Interested readers may refer to Torrence & Compo (1998), Jevrejeva et al. (2003), Souza et al. (2007), and Beecham & Chowdhury (2009) for further details and clarification.

The steps followed in the current study are:

  • 1. decomposition of the original time series using CWT;

  • 2. construction of XWT from two CWTs;

  • 3. WTC analysis between two CWTs.

The following sections describe each step and explain how they were used to analyze the relationship between two different time series that supposedly are correlated.

CWT

Wavelets are functions with a zero mean; unlike Fourier transforms, which are localized only in frequency, wavelets have the ability to be stretched and translated in both time and frequency (Jevrejeva et al. 2003). Studies suggest that using CWT is more appropriate for analyzing a time series that has a non-normal distribution (Grinsted et al. 2004). Non-normal distributions are frequently found in non-stationary parameters, for example, such hydroclimatic variables as precipitation and streamflow. The advantage of using a wavelet transformation is that it allows the analysis of non-stationary time series at different frequencies (periodicities) (Foufoula-Georgiou & Kumar 1995), and can be used effectively to observe how the frequencies have changed over time. The Morlet wavelet has been suggested in previous studies (Torrence & Compo 1998; Percival & Walden 2000) as the most appropriate wavelet function to be used for analyzing geophysical signals; accordingly, it was used in this study. A combined streamflow CWT was obtained using PCA; the first principal component, which explained 71.21% variance of the data obtained from all the stations, was used to represent the overall variance in data.

XWTs and cross wavelet phase angle

An XWT was constructed from two CWTs to observe their high common power (covariance) and relative phase relationship in time–frequency space (Grinsted et al. 2004). The cross wavelet spectrum, which shows the covariance of two time series, occurs from a complex conjugation of the two time series. It produces a cross wavelet power spectrum that is used to observe the correlation between the two time series. The phase angle of the cross wavelet power shows how the two time series are related in terms of their phase relationship in time–frequency space (Jevrejeva et al. 2003). The presence of a statistically significant covariance was determined against red noise background (Torrence & Compo 1998).

WTC analysis

The presence of high common power across two different CWTs could be observed by means of the XWT constructed from them, as mentioned in the previous section. In order to observe the coherency of two CWTs in the time–frequency space, WTC is considered to be more useful (Grinsted et al. 2004). WTC analysis shows the common frequency bands and the time intervals of two CWTs that were found to be correlated (Tang et al. 2014). The advantage of using WTC is that it quantifies the correlation and shows the presence of significant coherence at lower common powers as well. It explains how to calculate confidence levels alongside red noise backgrounds (Grinsted et al. 2004). In this study, the Monte Carlo approach (Wallace et al. 1993) was used to calculate the significance of the wavelet coherence and a 5% significance level was chosen against red noise to calculate the statistical significance.

RESULTS

In this study, standardized streamflow data of 61 stations across six western US hydrologic regions were decomposed using CWT. A combined CWT for the standardized streamflow data was obtained using PCA, which represented the entire time series and the amount of variance in the data. CWTs of standardized ENSO and PDO data were obtained for the chosen study period. Figure 2 shows the standardized time series of the combined streamflow, ENSO, and PDO along with their CWTs and their respective global wavelet spectrums. Figures 3 and 4 show the XWT and WTC, respectively, of the combined streamflow with both ENSO and PDO.
Figure 2

Standardized time series, CWT, and global wavelet spectrum of (a) combined streamflow, (b) ENSO, and (c) PDO. Red and blue represent stronger and weaker powers, respectively. A thick black contour line delineates a 5% significance level against the red noise. The full color version of this figure is available in the online version of this paper, at http://dx.doi.org/10.2166/wcc.2016.162.

Figure 2

Standardized time series, CWT, and global wavelet spectrum of (a) combined streamflow, (b) ENSO, and (c) PDO. Red and blue represent stronger and weaker powers, respectively. A thick black contour line delineates a 5% significance level against the red noise. The full color version of this figure is available in the online version of this paper, at http://dx.doi.org/10.2166/wcc.2016.162.

Figure 3

Cross wavelet spectrum between a standardized combined streamflow with standardized (a) ENSO and (b) PDO. A thick black contour line delineates a 5% significance level against the red noise (red and blue represent stronger and weaker powers, respectively). The cone of influence (COI), which potentially can distort the picture around the edges, is shown by lighter shades. The arrows represent the relative phase relationship between the two time series. Right (left) pointing arrows show an in-phase (anti-phase) relationship, while vertically upward arrows show that ENSO and PDO leads streamflow by 90 °. The full color version of this figure is available in the online version of this paper, at http://dx.doi.org/10.2166/wcc.2016.162.

Figure 3

Cross wavelet spectrum between a standardized combined streamflow with standardized (a) ENSO and (b) PDO. A thick black contour line delineates a 5% significance level against the red noise (red and blue represent stronger and weaker powers, respectively). The cone of influence (COI), which potentially can distort the picture around the edges, is shown by lighter shades. The arrows represent the relative phase relationship between the two time series. Right (left) pointing arrows show an in-phase (anti-phase) relationship, while vertically upward arrows show that ENSO and PDO leads streamflow by 90 °. The full color version of this figure is available in the online version of this paper, at http://dx.doi.org/10.2166/wcc.2016.162.

Figure 4

Wavelet coherence spectrum between a standardized combined streamflow with standardized (a) ENSO and (b) PDO. A thick black contour line delineates a 5% significance level against the red noise (red and blue represent stronger and weaker powers, respectively). The COI, which potentially can distort the picture around the edges, is shown by lighter shades. The arrows represent the relative phase relationship between the two time series. Right (left) pointing arrows show an in-phase (anti-phase) relationship, while vertically upward arrows show a lag between ENSO and PDO with streamflow. The full color version of this figure is available in the online version of this paper, at http://dx.doi.org/10.2166/wcc.2016.162.

Figure 4

Wavelet coherence spectrum between a standardized combined streamflow with standardized (a) ENSO and (b) PDO. A thick black contour line delineates a 5% significance level against the red noise (red and blue represent stronger and weaker powers, respectively). The COI, which potentially can distort the picture around the edges, is shown by lighter shades. The arrows represent the relative phase relationship between the two time series. Right (left) pointing arrows show an in-phase (anti-phase) relationship, while vertically upward arrows show a lag between ENSO and PDO with streamflow. The full color version of this figure is available in the online version of this paper, at http://dx.doi.org/10.2166/wcc.2016.162.

CWT

The time series for the standardized combined streamflow of all the stations, along with the continuous wavelet power spectrum, is shown in Figure 2(a). Significant variabilities in the wavelet power spectrum were found in 2–4 years’ band from 1970 to 1977, in 6–16 years’ band from 1970 to 2010, and in 3–4 years’ band from 1998 to 2002. From observing the wavelet power spectrum, the highest power (which represents the variance of data) was observed near the bands of 2–3 years and 12–14 years. The global wavelet spectrum showed that the highest peak was located near 12–14 years’ band.

ENSO has been identified as one of the dominant oceanic–atmospheric patterns in the tropics of the Pacific Ocean, with a period of 2–7 years. From the wavelet spectrum of ENSO (Figure 2(b)), from 1976 to 2003, the presence of significant high power in 3–7 years’ band was observed. The presence of significant high powers was also observed in 5–7 years’ band from 1953 to 1962 and in 3–5 years’ band from 1966 to 1975. From the wavelet power spectrum, the highest power was observed from 1982 to 1990 near 3–5 years’ band. In addition, the global wavelet spectrum showed the highest peak near 3–5 years’ band. The presence of higher power was observed near 12–14 years’ band in the global wavelet spectrum as well; however, they were not statistically significant.

PDO is another oceanic–atmospheric pattern found in the Pacific Ocean with a time period of 25–50 years. From the wavelet power spectrum of PDO (Figure 2(c)), a substantially high power was found at a 5% significance level in 3–7 years’ band from 1951 to 1962, in 4–6 years’ band from 1986 to 2001, in 3–4 years from 1982 to 1988, and in 8–12 years’ band from 1993 to 2005. From the global wavelet spectrum, 8–12 years’ band was found to have the highest power among the statistically significant regions. Higher powers even were observed in 16 years’ band and above; however, they were not found to be statistically significant.

The exact correlation between ENSO/PDO with streamflow variations was found to be quite difficult to observe from their respective CWTs. However, the comparison of the wavelet power spectra suggested that higher powers (higher variance) near bands of 3–7 years and 8–12 years were found to be statistically significant. Higher powers near 3–7 years’ band were found to be present in both the combined streamflow power spectrum and the ENSO power spectrum. Both the combined streamflow power spectrum and the PDO power spectrum showed higher powers in 8–12 years’ band.

XWT

To understand the correlations between ENSO/PDO with streamflow variations, XWT analysis was performed. From the XWT of combined streamflow and ENSO (Figure 3(a)), it was found that they shared common power in 2–4 years’ band from 1968 to 1976, in 3–4 years’ band from 1981 to 1986, in 3–4 years’ band from 1995 to 2001, in 6–7 years’ band from 1992 to 2002, in 8–12 years’ band from 1997 to 2006, and in 12–16 years’ band from 1972 to 2005. The arrows in the figure indicate the phase angle relationship between the two time series. In the lower time-scale bands, arrows mostly pointed left, which indicated an anti-phase relationship between streamflow and ENSO; this meant they were moving at the same time but in the opposite direction. Anti-phase can be interpreted as an increase (decrease) in streamflow and decrease (increase) in ENSO index, which means a colder (warmer) phase. As the time-scale band increased, arrows were observed to have a greater tendency to point straight up, indicating a time lag between ENSO and streamflow variation. Arrows indicating straight up indicated that ENSO led streamflow by 90 °. The phase relation can be used to calculate the exact time lag; however, since it depends on the specific wavelength of the signal, this step was not performed in this study.

XWT analysis of the combined streamflow and PDO (Figure 3(b)) showed common power in 2–3 years’ band from 1974 to 1981, in 3–4 years’ band from 1972 to 1978, in 5–7 years’ band from 1991 to 1998, around 3 year band during 2000, and in 7–14 years’ band from 1983 to 2008. Common powers observed at lower time-scale bands were lower compared to higher time-scale bands. At lower time scales (in 2–4 years’ band), arrows indicating phase relationship were found to point towards both the right and left during various time intervals across the study period; this indicated an in-phase and anti-phase relationship, respectively. In 5–7 years’ band, arrows pointed downward and slightly towards both the left and right. In higher time scales (in 6–14 years’ band), arrows mostly were found to point straight up, indicating a phase difference of 90 °; this referred to a lag between PDO and streamflow variations. Common powers at time scales higher than 16 years’ band were observed; however, they were not found to be statistically significant.

The XWT analyses of the combined streamflow with ENSO and PDO revealed that common powers of ENSO (coincidence with streamflow variation) were found to be higher compared to PDO. These results were consistent with the CWTs of ENSO and PDO, where ENSO had more regions of significance compared to PDO (Figure 2(b) and 2(c)). The time-scale bands with significant common powers were in agreement with what was observed in individual CWTs. Even though 12–14 years’ band was not found to be significant in ENSO in the CWT (Figure 2(b)), the global wavelet spectrum showed the presence of higher power in 12–14 years’ band; this justified the relationship found from the XWT of combined streamflow and ENSO. To be certain that these relationships were not by mere chance, and to quantify the correlation, WTC analyses were performed on combined streamflow CWT and ENSO/PDO CWT.

WTC analysis

CWT and XWT analyses provided important information regarding the correlation between the two time series. However, to quantify the correlation between the two variables, WTC analysis was performed in this study, in which the Monte Carlo approach was used to compute the significance of correlation.

From the WTC of combined streamflow and ENSO, areas of significance were observed in the band of 10–16 years across the entire study period of 60 years, from 1951 to 2010 (Figure 4(a)). The time-scale bandwidths were observed to decrease at both ends of this time period. From 1968 to 1995 in the 10–12 years’ band, the correlation coefficient in this area of significance varied from 0.8 to approximately 1.0. Arrows indicating phase relationships mostly pointed upward; this suggested a lag between ENSO and streamflow variations, with ENSO leading streamflow by 90 °. High correlation values, ranging from 0.6 to 0.8, were observed in the band of 2–6 years from 1952 to 1978 and from 1987 to 2004. Higher correlation values were observed as well in the 16 years’ band and above across the study period; however, they were not found to be statistically significant.

The WTC of combined streamflow and PDO (Figure 4(b)) showed less areas of significance compared to the areas of significance observed in the WTC of combined streamflow and ENSO. Statistically significant areas were found in 10–12 years’ band at the beginning of 1950s, in 8–10 years’ band from 2003 to 2010, and above 16 years’ band from 1986 to 2010. Correlation values in these regions were found to be in the range of 0.7 to approximately 1.0; higher correlation values were found in 10–12 years’ band during the 1950s and in the 16 years’ band and above from 1986 to 2010. High correlations, in the range of 0.6 to 0.8, were observed from 1975 to 1995 in 12–14 years’ band and in 8–14 years’ band from 1995 to 2010, although they were found to be statistically insignificant. Presence of regions having higher correlation values – in the range of 0.6 to 0.9 – but not statistically significant were observed in some of the other intervals in the study period at lower time scales, near the band of 2–5 years from 1968 to 2005 with intervals in between.

The WTC analyses between combined streamflow and ENSO/PDO showed that ENSO had a much more pronounced correlation with streamflow compared to PDO, as ENSO showed the presence of more significantly correlated areas (high common power). For both ENSO and PDO, the band of 8–16 years was found to be most significantly correlated. For PDO, regions with high correlation were observed in the 16 years’ band and above; however, due to the limitation of data, the study could not detect the entire band length.

DISCUSSION

To understand how streamflow in the western USA has changed with the change in ENSO/PDO, CWT along with XWT and WTC were used in this study. The most significant periodicities that triggered simultaneous variations in the change patterns were observed to understand the correlation between climate indices and streamflow. By observing high common power in the wavelet spectrum at various time scales through the study period of 60 years (i.e., 1951–2010), the study investigated the correlation between ENSO/PDO and streamflow variations across the western USA.

In order to analyze two time series at the same time, XWT and WTC between two CWTs were performed in this study. An XWT constructed from two different CWTs showed a common power of the wavelet spectrum, and suggested a phase relationship between the time series under inspection. By using WTC, which was helpful in quantifying the correlation, significant coherence was found at lower common power. The results showed ENSO to have a higher correlation than PDO during the study period. The most influential periodicities varied from 8–12 years for both ENSO and PDO. The interval of 1980 to 2005 showed the presence of higher correlation with streamflow for both ENSO and PDO. Presence of significant regions in the 16 years’ band and above indicated that more areas of significance (at higher periodicities) could have been explored if a longer study period were chosen.

CWT analysis of the combined streamflow along with the CWTs of ENSO and PDO indices were formed to observe their individual significant variance (high power in the wavelet spectrum) across the study period. Significant high power in streamflow wavelet spectrum was found in bands of 2–4 years, 3–4 years, and 6–16 years at different historical time intervals (Figure 2(a)). The global wavelet spectrum revealed that the highest power for streamflow variation occurred in the band of 12–14 years from 1980 to 2000. For ENSO, significantly high power was observed in bands of 3–5 years, 3–7 years, and 5–7 years (Figure 2(b)), with the highest power in the 3–5 years’ band from 1982 to 1990. For PDO, significantly high power was observed in bands of 3–4 years, 3–7 years, 4–6 years, and 8–12 years (Figure 2(c)). The highest power was observed in the 8–12 years’ band from 1993 to 2005.

The global wavelet spectrum of PDO also showed the presence of higher power in bands higher than 16 years; however, they were not found to be statistically significant. Observation of individual CWTs revealed information regarding their changing patterns; however, it was difficult to formulate any strong correlation between them from sight only. From observing the individual CWTs, nevertheless, it could be concluded that both ENSO and PDO had some effect on the variation of streamflow, since high power bands overlapped in certain regions. Similar to previous works (Grinsted et al. 2004; Jevrejeva et al. 2003), results of the current study reinforced the choice of CWT as a better feature extraction tool, as CWT produced visible high power to represent variance in data.

To understand the correlation between the time series with greater precision, XWTs were constructed from two individual CWTs. These XWTs provided information regarding high common power (covariance) with consistent phase relationships as well as information regarding temporal lags between the two time series. The XWT between the combined streamflow and ENSO (Figure 3(a)) revealed that common high power was present in bands of 2–4 years, 3–4 years, 6–7 years, 8–12 years, and 12–16 years at different historical time periods. Highest power was observed in 2–4 years’ band from 1968 to 1973 and in 12–16 years’ band from 1972 to 2002. At lower time scales, in the 2–5 years’ band, arrows indicating the phase relationship mostly pointed to the left, which suggested an anti-phase relationship between streamflow and ENSO, suggesting the streamflow mirrors the behavior of ENSO. In other words, since they share common power, they both moved at the same time but in opposite directions. At higher time scales, in the 6–16 years’ band, arrows mostly pointed upwards, indicating a lag between ENSO and the variability of streamflow. Arrows pointing exactly upward suggested that ENSO leads streamflow by 90 ° at those points in time.

It was possible to calculate exact lag times from the phase relationships obtained from XWTs, but they were specific to a certain wavelength. As a result, calculation of exact lag times was not considered to be within the scope of this study. Previous studies have investigated the lag response of ENSO and streamflow, and also observed variable lags between oceanic oscillations and streamflow variations. The overall response time, which can be up to several months, is the result of all the lags that occur from oceanic fluctuations, precipitation events, the time required for snowmelt, and delays in streamflow response (Cayan et al. 1999; Hanson et al. 2004). Use of lags and their effects can be found in Trenberth & Hurrell (1994), Pozo-Vázquez et al. (2001), and Jevrejeva et al. (2003). Similar to the results of the current study, McCabe & Dettinger (1999) and Beebee & Manga (2004) found that ENSO had less correlation with mean annual flow from 1920 to 1950, and observed an increased correlation after 1950. In the current study, all the significant correlations observed at 5% significance level occurred after 1968 across all time-scale bands.

High common power between combined streamflow and PDO was found in bands of 2–3 years, 3–4 years, 5–7 years, and 7–14 years (Figure 3(b)) across different historical periods. The highest common power was observed in 7–14 years’ band from 1983 to 2008. The arrows indicating a phase relationship in the highest power region mostly pointed upward, which indicated a lag between PDO and streamflow (PDO leaded streamflow by 90 ° at the points where the arrows were pointing exactly upward). Phase relationship at lower time scales were observed to be not showing any uniform pattern.

Similar to ENSO, calculation of exact lag time between PDO and streamflow variation was not a focus for this current study. However, previous studies have investigated the lag response of PDO and streamflow, and found a delay of several months between oceanic oscillations and streamflow fluctuations. Hanson et al. (2004) studied the relationship between different climate variabilities and southwestern US discharge flows, and suggested that the lag time between the PDO index and flow change could vary between 1.5 and 5 years, depending on the type of flow. Although they were not found to be statistically significant, the XWT of combined streamflow and PDO from the current study revealed the presence of high common power at time scales greater than 16 years. The limited data restricted the confidence for bands beyond 16 years. Since PDO has a multi-decadal time period (25–50 years), it is probable that the presence of more common powers for bands at time scales greater than 16 years were missed.

WTC assisted in quantifying the correlation between the wavelet spectra and helped to detect significant coherence at low common powers found during the analyses with XWTs. From the WTC between combined streamflow and ENSO (Figure 4(a)), the continuous presence of common power was observed in the 10–16 years’ band across the entire study period. The correlation values in the 10–16 years’ band were in the range of 0.8 to as high as approximately 1.0 around the 10–12 years’ band from 1968 to 1995. The reason behind such strong common power at this range of the time scale could be because ENSO itself has a periodicity of 2–7 years, and the results might have occurred when two ENSO cycles joined together. In addition, the presence of high common power was observed with a correlation ranging from 0.6 to 0.8 at lower time scales in the 2–6 years’ band, though they were not found to be statistically significant. The phase relationships found in the significant regions were consistent with what was observed in the XWT between ENSO and the streamflow of the stations. ENSO was found to lead streamflow variation by 90 ° in most of the significant regions.

The WTC between the combined streamflow and PDO revealed the presence of high correlation in bands of 8–10 years, 10–12 years, and beyond 16 years at different intervals across the study period. The correlation values were found to be as high as approximately 1.0 in 8–10 years’ band during the 1950s and beyond 16 years’ band from 1986 to 2010. The region found beyond the 16 years’ band suggested that this was likely to continue at even greater time scales. Since the relatively short study period of 60 years could not generate a wavelet spectrum beyond this time scale, it was not possible to investigate beyond this point. PDO has a time period of multiple decades, which explains the presence of common power at higher time scales. A correlation in the range of 0.6 to 0.8 was observed at lower time scales, but was not found to be statistically significant. PDO was found to lead streamflow by 90 ° at some points in time in higher bands. In other regions having a higher common power, PDO and streamflow were mostly found in an anti-phase relationship. Similar anti-phase or inverse relationship was found by Lins (1997) and Dettinger et al. (2001), which supports the results of the current study.

ENSO was found to have a higher correlation with the change in streamflow compared to PDO. Similarly, Beebee & Manga (2004) found a higher correlation between ENSO and the mean annual discharge compared to PDO while studying snowmelt and consequential runoff in Oregon. They found mean annual discharge to be more correlated than temperature and precipitation, and concluded that the underlying reason might be because the discharge represents the spatial average of a much smaller area compared to broader climatic zones of temperature and precipitation (Beebee & Manga 2004). This phenomenon, that flow behavior can represent a change occurring in a localized area with better accuracy, influenced the current study to work with streamflows of a particular region – in this case, the western region – rather than working with the entire United States.

A longer study period would have allowed the current study to investigate the wavelet spectrum at time scales beyond 16 years. For oceanic–atmospheric patterns, such as PDO, which has a periodicity (recurrence interval) of multiple decades, a longer study period would have resulted in a better understanding of the correlation between the parameters in hand. Analyses of a longer period of data are important as well for regions that are currently going through extreme scenarios; for example, the western USA has been experiencing drought for several years. Inclusion of a larger number of stations would have provided results having more reliability, but that would have minimized the minimum temporal length of the data since many of the stations do not have longer data records.

CONCLUSION

In this study, data from 61 unimpaired streamflow stations with 60 years of continuous data (i.e., 1951–2010) were obtained across six hydrologic regions in the western USA to evaluate the correlation between streamflow and two major oceanic–atmospheric patterns, also known as climate signals, of the Pacific Ocean, namely, ENSO and PDO. To understand these relationships, CWT along with XWT and WTC were applied. The study investigated the correlation between the parameters and also provided some insights regarding the significant frequencies (periodicities) of the multiple time series that were analyzed.

The results of this study indicated the presence of multiple significant time scales (bandwidths), which are important in understanding the relationships between streamflow and the oceanic–atmospheric patterns (i.e., ENSO and PDO). The results indicated that both ENSO and PDO had significant correlation with the streamflow variation in the 8–16 years’ band during the study period. In addition, ENSO showed the presence of significant correlation at lower time scales, i.e., 2–5 years’ band. The presence of high correlation was found with PDO in bands of 16 years and above. Limitations due to the length of data prevented the current study analyzing results beyond the bands of 16 years.

The major contributions of this study are as follows:

  • A continuous wavelet-based analysis for unimpaired streamflow stations across the entire western USA to evaluate the coupled effect of streamflow change with oceanic–atmospheric patterns (ENSO and PDO).

  • Application of cross wavelet and WTC analyses to understand the relationship between the parameters chosen (streamflow, ENSO, and PDO variations).

  • Evaluation of the most significant periodicities (frequencies) that affect the streamflow change patterns.

  • Quantification of the correlations observed between the parameters.

  • Conforming to the results of previous works using a comparatively recent approach.

The scope of the current study can be extended by analyzing data of greater lengths. A greater length of data could be obtained by using various reconstruction methods that have been found effective in extrapolating data in previous studies. Reconstruction could be helpful in interpolating missing data (data dropout) or in cases of data irregularities. As for the record, similar methods could be applied to climate signals’ data as well to obtain data of greater length. Incorporation of reconstructed (interpolated) data in wavelet analysis has not been well explored in the field of hydrologic time series analyses. Potentially, this can be an opportunity for further research since there has been some work in signal processing dealing with similar techniques. Analyzing other oceanic–atmospheric indices could be possible as well by applying the methods used in this study. Another plausible extension of this work could be the calculation of precise lag times at specific wavelengths.

The results of this study provided insights regarding the coupled behavior of streamflow in the western USA with the changes in ENSO and PDO indices. The study focused on formulating a correlation between the parameters in hand. The results provided information about the periodicities of the fluctuation patterns and presented insight regarding their effects over the historical time series of streamflow. These findings can be helpful to water managers to get a better understanding of the relationships between oceanic–atmospheric patterns and streamflow.

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