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 Climatic teleconnections are often used to interpret and sometimes to predict precipitation temporal variability at various time scales. However, the teleconnections are intertwined between the effects of multiple large-scale climate signals which are often interdependent. Each climate signal is composed of multitemporal components, which may result in different teleconnection patterns. The time lags of precipitation response may vary with climate signals and their multitemporal components. In order to effectively address these problems, a multiresolution analysis (MRA) with a discrete wavelet transform is utilized, and a stepwise linear regression model based on MRA and cross correlation analysis is developed in this study. The method is applied to examine monthly precipitation teleconnections in South Australia (SA) with five large-scale climate signals. The MRA first decomposes each of original monthly precipitation anomaly and climate signals into several component series at different temporal scales. Then the hierarchical lag relationships between them are determined for regression modeling using cross-correlation analysis. The results indicate that the MRA-based method is able to reveal at which time scale(s) and with what time lag(s) the teleconnections occur, and their spatial patterns. The method is also useful to examine the time-scale patterns of the interdependence between climate signals. These altogether make the MRA-based method a promising tool to address the difficulties in the climate teleconnection studies. The multiple linear regression based on MRA-decomposed climate signals is expected to better interpret monthly precipitation temporal variability than that based on the original climate signals.
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 Precipitation provides water on land for natural ecosystems, and agricultural, industrial, and domestic activities. A reliable prediction of regional rainfall on monthly and seasonal time scales is important for water resource management and agriculture planning, particularly in Australia where the hydroclimatic variability is high [Gallant et al., 2012; Kirono et al., 2010]. However, due to the high variability of precipitation in both space and time, it is difficult to predict precipitation with a lead time beyond a few days. This difficulty is exacerbated with the global climate change [Trenberth et al., 2003]. Fortunately, it is observed that when precipitation is summed over a certain period, such as a season, it tends to be correlated with some large-scale climatic indices, many of which represent the oscillation of ocean-atmospheric systems [Mantua and Hare, 2002; Meehl et al., 2010; Meneghini et al., 2007; Nicholls, 1988; Risbey et al., 2009; Saji and Yamagata, 2003; Timbal and Drosdowsky, 2012]. These correlation relationships can be used to predict rainfall on the monthly, seasonal, and annual scales if a lag structure can be found between precipitation and the climatic indices, or if these climatic indices can be forecasted with a certain lead time. They also provide information toward a better understanding of these large-scale precipitation response mechanisms.
 The variability of Australia rainfall has been linked to several large-scale climate signals. They include the El Niño Southern Oscillation (ENSO) [McBride and Nicholls, 1983], Indian Ocean Dipole (IOD) [Ashok et al., 2003; Nicholls, 1989], Southern Annular Mode (SAM) [Meneghini et al., 2007], Interdecadal Pacific Oscillation (IPO), or Pacific Decadal Oscillation (PDO, an index based on the northern Pacific, representing a similar oscillation as IPO) [Power et al., 1999], and Subtropical Ridge (STR) [Cai et al., 2011b; Timbal and Drosdowsky, 2012]. Correlation analysis is commonly used to reveal such “teleconnections” [Wallace and Gutzler, 1981]. ENSO is found to be correlated with precipitation in the eastern, northern, and central parts of Australia with more precipitation occurring in La Nina years [Chiew et al., 1998; McBride and Nicholls, 1983; Power et al., 1998; Risbey et al., 2009; Verdon and Franks, 2006]. IOD is correlated with precipitation in the southwest and southern parts of Australia, with more precipitation occurring in negative IOD years [Ashok et al., 2003; Risbey et al., 2009; Ummenhofer et al., 2009]. The effects of IPO (PDO) on precipitation are often reported in its modulations of ENSO effects [Cai and van Rensch, 2012; Kiem and Franks, 2004; Power et al., 1999; Verdon and Franks, 2006]. Modeling exercises suggest that the spatial pattern of the IPO effect on Australian precipitation is similar to ENSO [Meehl et al., 2010], with more precipitation occurring in negative IPO years [Verdon et al., 2004]. Knowledge of the spatial pattern of SAM effect is provided in previous studies [Meneghini et al., 2007; Risbey et al., 2009; Williams and Stone, 2009]. Precipitation in the southeast and southwest parts of Australia is commonly reported being influenced by SAM. The variation of the STR position is reported in part responsible for the interannual variability of cool season rainfall in southeastern Australia [Drosdowsky, 2005; Williams and Stone, 2009]. The STR intensity has recently been found to be more useful in interpreting rainfall variability in Australia [Timbal and Drosdowsky, 2012].
 These teleconnections between rainfall and the large-scale climatic indices, if a lag rainfall response exists and is revealed, will be useful for the prediction of monthly, seasonal, or even longer-period rainfall. Characterization of the temporal and spatial patterns of these teleconnections is crucial to the successful development of precipitation predicting methods for water resource management. This is also important to understanding the physical mechanisms of the large-scale climate systems. However, the nature of precipitation variation in response to the large-scale climate systems represented by the indices is far more complex than that shown in the simple linear correlation coefficients. The relationship between precipitation and the climate indices often vary seasonally [Hendon et al., 2007; Meneghini et al., 2007; Risbey et al., 2009; Timbal and Drosdowsky, 2012; Williams and Stone, 2009], interannually [Wang and Hendon, 2007], and decadally [Cai et al., 2001; Nicholls et al., 1996; Power et al., 1999, 1998; Risbey et al., 2009]. The effect of one climatic index on precipitation is often nonsymmetric between the two extreme phases [Cai et al., 2010; King et al., 2013; Power et al., 2006; Ummenhofer et al., 2009].
 These complexities, which should be addressed before these climate indices are used properly for teleconnection-based precipitation analyses, are partly associated with the following problems (or difficulty). The first one is that the large-scale climate signals are often interdependent (or intercorrelated). The correlation between precipitation and one climate signal is likely modulated by other signals, or includes the effects of other signals, and thus will change with time when other signals vary. The interdependence effects have been widely reported for ENSO and PDO [Cai and van Rensch, 2012; Gershunov and Barnett, 1998; Guan et al., 2005; Kiem et al., 2003; Risbey et al., 2009], ENSO and IOD [Cai et al., 2011a, 2012; Risbey et al., 2009], and ENSO and SAM [Pohl et al., 2010]. For easy discussion, this is referred to as the difficulty of interdependent explanatory variables (or interdependence) difficulty hereafter.
 The second difficulty is that each climate signal is composed of signals at different temporal scales [Jevrejeva et al., 2003; Jiang et al., 2003]. The relationship between precipitation anomaly and a climate signal is very likely different at different temporal scales. With direct comparison of original time series of precipitation anomaly and the climate signal, only a part of relationship is revealed. The relationship between precipitation anomaly and the climate signal may become insignificant if the effect of different temporal components go against each other. This problem is referred to as the multitemporal component difficulty hereafter.
 The third difficulty is that the maximum correlation between precipitation anomaly and climate signals often occurs with certain time lags, and these time lags may vary for different signals and for different time-scale components of a signal. This is referred to as the nonuniform time-lag difficulty hereafter.
 Simple correlation analysis between precipitation anomaly and an individual climate signal may, to some degree, address the time-lag problem, but does not address the interdependence and multicomponent difficulties. Partial correlation [Ashok et al., 2003; Risbey et al., 2009] and the composite method (comparing the mean of selected phase combinations) [Guan et al., 2005; Risbey et al., 2009] have been applied to disentangle the interdependence effects on precipitation between two climatic indices, but they are inconvenience and thus seldom used to examine the interdependence between multiple climatic indices. Multiple linear regression, including a good set of potential predictor variables, can somehow take care of the interdependence and time-lag problems, but it does not address the multicomponent problems in each climate signal. Because the interdependence between different climate signals may vary with temporal scales, and the time lags between precipitation anomaly and each climate signal may be different at different temporal scales, without temporal decomposition, the interdependence and nonuniform time-lag problems cannot be well addressed.
 The wavelet transform (WT) has been demonstrated useful to resolve patterns of the multiple temporal components of climatic indices [Jiang et al., 2003], and to define the relationship between precipitation and the climatic index at various temporal scales [Kulkarni, 2000; Penalba and Vargas, 2004]. However, the lag relationships between temporal components of rainfall anomaly and multiple climate signals at different time scales (i.e., multitemporal analysis of precipitation teleconnections) have not been examined in literature.
 The objectives of this study are (1) to develop a wavelet-based method to examine the multitemporal lag correlations between precipitation and climatic indices and (2) to interpret precipitation temporal variability with these multitemporal components of climatic indices, using multiple linear regressions. Our hypotheses are that (1) the interdependence between climatic indices varies with temporal scales, (2) more details of the relationship between precipitation and a climatic index can be revealed when the climatic index is decomposed into various temporal scales, and (3) precipitation temporal variability can be better interpreted by the decomposed climatic indices.
 We will examine these hypotheses for precipitation in South Australia (SA). Overall, precipitation teleconnections in South Australia are not as strong as those in the eastern and northern Australia [Risbey et al., 2009]. Nevertheless, precipitation in this region is reported being correlated to multiple climatic indices, including ENSO, IOD, SAM [Risbey et al., 2009], STR, IOD, PDO, and ENSO [van DijK et al., 2013]. Thus, a study based on South Australia allows us to examine precipitation teleconnections with multiple climatic indices, and to test the performance of the proposed wavelet-based methods. The indices we will examine include STR, ENSO, IOD, SAM, and PDO.
2. Data and Methodology
 Monthly precipitation data are downloaded from the high-quality Australian monthly rainfall data set. It is available from ftp://ftp.bom.gov.au/anon/home/ncc/www/change/HQmonthlyR. There are 54 stations in South Australia. Eight stations with data missing over 12 months (∼2%) were excluded. This results in 46 stations with data available in 1959–2008 (Figure 1), among which stations 16005, 16022, 16032, 16055, 16083, 22020, 18023, 18063, 18069, 19061, 20005, 20020, 22009, and 22801 have missing data for 1–12 months. Most of the missing values are filled with data downloaded from the Bureau of Meteorology (BOM) Australia website http://www.bom.gov.au/climate/data/index.shtml, and the remaining few missing values are substituted by mean of the two neighboring months. Monthly time-series data are first transformed to standardized monthly anomaly data for teleconnection analysis.
 Selected large-scale climate indices are described in the following. To represent ENSO conditions, a variety of indices are available based on sea surface temperature (SST), sea level pressure, and combinations of different oceanic and atmospheric variables [Kurtzman and Scanlon, 2007]. Southern Oscillation Index (SOI) was chosen for this study because it has been successfully used in previous hydroclimatic analyses in Australia [e.g., Kirono et al., 2010]. (Niño3.4, a sea surface temperature based ENSO index, is also examined, but no significant difference has found.) Monthly SOI index data were obtained from BOM at http://www.bom. gov.au/climate/glossary/soi.shtml. The PDO index is based on a projection of sea surface temperature anomaly onto a pattern defined by the leading principal component of monthly SST anomaly in the North Pacific to the north of 20°N. PDO index is available from http://jisao.washington.edu/pdo/PDO.latest. The SAM is the dominant mode of atmospheric variability in the southern hemisphere [Marshall, 2003]. The SAM index data from 1957 to present are available from http://www.antarctica.ac.uk/met/gjma/sam.html. The IOD index represents the anomalous SST gradient between the western equatorial Indian Ocean (50°E–70°E and 10°S–10°N) and the south eastern equatorial Indian Ocean (90°E–110°E and 10°S–0°N). The IOD index (monthly from 1958 to 2010) derived from HadISST data set [Rayner et al., 2003] was selected for this study due to its relatively long period of record. The data of this index were obtained from http://www.jamstec.go.jp/frcgc/research/d1/iod/e/iod/dipole/mode/index.html. Following Timbal and Drosdowsky , we used the Drosdowsky  index for STR which measures position and intensity of the local maxima in monthly surface pressure along the zonal profiles for a 5° longitude band around 150°E between 10°S and 44°S. In this study, STR intensity, provided by Drosdowsky, was used to examine the STR relationship with precipitation in South Australia according to Timbal and Drosdowsky . Standardized monthly anomaly of STR intensity was calculated for the analysis described later.
2.2.1. Wavelet Transform
 The wavelet transform is a mathematical technique introduced for signal analysis in the early 1980s [Goupillaud et al., 1984]. It is a method based on expressing signals as a sum of “little waves.” The wavelet analysis is a multiresolution analysis in time and frequency domains. It has applications in many areas such as climatology, geophysics, and hydrology [e.g., Jiang et al., 2003; Kulkarni, 2000; Penalba and Vargas, 2004; Ozger et al., 2009]. The basic idea of WT is the decomposition of a signal into different time scales with a set of basis functions. The set of basis functions can be generated by translating and scaling the wavelet function ψ(t) called the mother wavelet, according to Daubechies 
where a is the scale parameter which adjusts the dilation of the wavelet and b determines the location of the wavelet. The mother wavelet ψ(t) satisfies two basic properties: (i) and (ii) which mean that this function oscillates around zero and is localized in a finite-width interval.
 If satisfies equation (1), for a time series or energy finite signal f(t), the continuous wavelet transform (CWT) of f(t) is defined as
where is the mother wavelet complex conjugate. For each scale, the WT result is a set of coefficients associated with different time locations [Lindsay et al., 1996].
 The continuous wavelet is often discretized to deal with discrete signals in real application. The discrete wavelet transform (DWT) can be thought as dyadic sampling of Wψ(a, b), in which the mother wavelet is scaled by powers of two, a = 2j and, within a given scale, translated by integers, b = k2j, where k is the location index and j is referred to as the decomposition level. Thus from equation (1), a discretely scaled and translated wavelet is expressed as
and the discrete wavelet transform of f(t) can be written as
 The characteristics of the original time series f(t) at the decomposition level j and time location index k are reflected by Wψ(j, k). At a lower level j, the smaller and finer component of the signal is accessed.
 The signal can be reconstructed from the wavelet coefficients Wψ(j, k) by means of the inverse discrete wavelet transform, as shown in
2.2.2. Multiresolution Analysis
 Multiresolution analysis based on DWT is to decompose a signal with different frequencies into a certain number of component time series at different temporal scales. In order to perform MRA, DWT is implemented in a hierarchical algorithm, well known as the pyramid algorithm [Mallat, 1989]. MRA decomposes the signal into different scales by successively translating and convolving the elements of a high-pass filter and low-pass scaling filter associated with the mother wavelet [Burrus et al., 1998; Percival and Walden, 2000]. These filters retain small-scale and large-scale components of the signals also known as detail (D) and approximation (A) subseries. In the first level of the decomposition, , the signal has a low-frequency part A1 and a high-frequency part D1. The same procedure is performed on A1 in order to obtain a decomposition on a coarser scale, . The process is continued in this way to compute for . A recursive decomposition for the low-frequency parts is illustrated in the following diagram:
 The relationship gives us information about the portion of the signal that can be attributed to variations between the scales [Martínez and Gilabert, 2009]. This diagram illustrates a wavelet decomposition of an original time series into J levels, as shown in
where J is the highest decomposition level that is considered.
 The detail component Dj of the discretely sampled signal for a particular decomposition level j is given by
 And the smoothed representation of the signal on the scale 2j, also known as the approximation component Aj, is given by [Percival and Walden, 2000]
where is a scaled and translated basis function called the scaling function, which is given together with the wavelet function when a wavelet is chosen, and is the scaling coefficient calculated from in a similar way for the wavelet coefficient Wψ(j, k) from .
 In the present study, the Meyer orthogonal discrete wavelet was chosen for the multiscale wavelet decomposition of all monthly rainfall anomaly and climate signal time series following the suggestion by Martínez and Gilabert . Five resolution levels were employed in the study. Thus, for a time series with a sampling period of 1 month, the time scales a of the wavelet decomposition are 2, 4, 8, 16, and 32 months, respectively, for the resolution levels j = 1, 2, 3, 4, and 5. Figure 2 shows an example of the MRA decomposition with five resolution levels for the standardized monthly rainfall anomaly time series of station 23721 and for the SOI index. This figure clearly shows how DWT decomposes the original series into its detail (D's) and approximation (A5) subseries. According to the interpretation of equation (6), the detail components at levels j = 1, 2, 3, 4, 5 can be attributed to variations on the time scales of 2, 4, 8, 16, and 32 months. The detail component of the first decomposition level contains the high-frequency part of the signal, thus can be used to identify the local features of a short duration in the signal. The detail components of higher levels contain the lower frequency parts. The approximation component A5 is associated with the average over the scale of 32 months and longer and, therefore captures the slowly varying portion of the original time series. From Figure 2, we can observe that as the decomposition level decreases, the time-domain resolution increases and the smaller and finer components of the signal can be retained in detail subseries, and a smoother aspect of the signal is given by the approximation subseries.
2.2.3. Lag-Time Estimates
 For climatic teleconnections, it is assumed that precipitation responds to large-scale climate signals with or without a lag time. A common method for the lag relationship is to examine the cross correlation between the two sets of signals. The time lag is determined by finding the offset in time of the two time series which results in the maximum correlation coefficient. In this study, all monthly time series are decomposed using MRA into a certain number of component time series (or subseries) at different temporal scales. Cross correlation analysis is implemented to identify the lag relationships between rainfall subseries versus each climate signal subseries at the corresponding time scale. The lag correlation coefficient (LCC) between the two sets of subseries is used for this purpose. Let and denote, respectively, the rainfall anomaly subseries and large-scale climate signal subseries, where t0 is the offset, and Δt is the sampling interval. The lag correlation coefficient between and is defined as
where and stand for the mean of and , respectively, and . The lag time between two subseries and is found from the peak of LCC (τ). The lag time between the original rainfall anomaly series and each climate signal series can be found similarly. A maximum lag time of 12 months is applied in this study to examine optimal time lags for the correlation between monthly precipitation anomaly and individual climate signals (and their MRA decomposition subseries).
2.2.4. Linear Regression
 In order to quantify how much rainfall temporal variability can be explained by the large-scale climate signals, multiple linear regression is used. After the lag relationships between rainfall anomaly and large-scale climate signals are identified, regression analyses are performed based on the original time series (equation (10)) and the decomposed time series (equations (11)-(13)).
where is the estimated rainfall anomaly, is the ith selected original climate signal, τi is the lag time between the rainfall anomaly and the ith signal, and ai is the regression coefficient. To avoid the regression including nonsignificant predictor variables, a stepwise regression algorithm [Draper and Smith, 1998] is performed with a cutoff statistical significance value (e.g., α = 0.01) for entering predictor variables and removing them. For easy discussion, we refer to equation (10) as the traditional multiple linear regression (TMLR) model.
where is the estimated detail component of the rainfall anomaly series, is the detail component of the selected climate signal xi at the decomposition level j, and dj,i and τj,i are the regression coefficient and the time lag, respectively.
where is the estimated approximation component of the rainfall anomaly series, is the approximation component of the selected climate signal xi at the highest decomposition level J, aJ,i is the regression coefficient, and ςJ,i is the lag time between the rainfall subseries and climate variable subseries .
 From equations (6), (11), and (12), the final wavelet-based multiple linear regression (WMLR) model for the rainfall anomaly is written as
where Similar to TMLR, the stepwise regression algorithm is performed to only include significant predictor variables.
2.2.5. Statistical Significance Testing
 The statistical significance of the correlation coefficient r between the two time series is judged with the t test statistic, which is determined by
where Neff is the effective sample size taking into account the autocorrelation of the time series. The effective sample size Neff is estimated using the relationship outlined by Bretherton et al. :
where N is the sample size and r1 and r2 are the lag-one autocorrelations of the two time series.
 The proposed algorithm is programmed and performed based on the Wavelet and Statistics Toolboxes provided in MATLAB2012. The stepwise regression algorithm is implemented by calling a Function stepwisefit in the Statistics Toolbox. These stepwise linear regression analyses (TMLR and WMLR) are performed for all monthly data. Given that the teleconnections vary with seasons, regression analyses are also performed for the months in each season, i.e., seasonal stratification data, for each station in the study area. Hindcast testing is used to examine the performance of both TMLR and WMLR models. To do this, the regression models are trained in the 1959–1995 period and tested in the 1996–2008 period for all months for each station. Owing to a short data length, hindcast testing is not performed for seasonal stratification data.
3. Results and Discussion
3.1. Time-Scale Patterns of the Intercorrelations Between Climate Signals
 Large-scale climate signals are often intercorrelated. This phenomenon makes precipitation teleconnection analysis difficult. Among the five climate signals, with the original time series, STR is correlated with IOD, SOI, and SAM at and above 99% confidence level. IOD and SOI, and PDO and SOI are significantly correlated as well (Table 1). It is not known, however, at which time scales these climate signals are correlated. MRA decomposition will help to address this question. Before we come to this point, one interesting finding from the correlation matrix of decomposed subseries (Table 2) is that for each of the five climate signals, no significant correlation is found between its decomposed detail subseries across different time scales. The approximation subseries of a climate signal is also independent of its detail subseries. It suggests that the MRA of one climate signal series results in a set of decomposition subseries which are mostly independent of each other. This finding is very significant for the regression analysis discussed in the next section.
Table 1. Correlation Matrix of the Monthly Time Series of the Five Large-Scale Climatic Indices
The correlation coefficients italicized are statistically significant at the 99% confidence level, others are not significant at the 95% confidence level.
Table 2. The Significant Correlation Matrix for the Decomposed Subseries of the Five Large-Scale Climatic Indices
Only the correlation coefficients that are statistically significant are shown (the ones italicized are statistically significant at the 99% confidence level, others are significant at the 95% confidence level). The abbreviation “STR-D1” stands for the detail component D1 of STR signal and other abbreviations are used similarly.
 After the MRA decomposition, the correlation matrix of statistical significance (Table 2) reveals more details of the correlation between the selected climatic variables. STR is now shown to be correlated with all other four climatic indices, while with the original signals, STR and PDO are not significantly correlated. This is consistent with the notion that STR is an atmospheric phenomenon influenced by various coupled ocean and atmospheric systems.
 Time scales at which these climate signals intercorrelate are clearly shown after the MRA decomposition. STR correlates with SAM at the first, second, and third decomposition levels with the corresponding time scales of 2, 4, and 8 months. The correlation between STR and SOI occurs at the second, third, and fourth decomposition levels (corresponding to the 4, 8, and 16 month scales). STR correlates with PDO at the second and third decomposition levels, and with IOD at the fourth decomposition level (Table 2).
 SOI and PDO correlate at the third and fourth decomposition levels (corresponding to 8 and 16 month scales). This is consistent with the PDO modulations of ENSO impacts on precipitation which has been commonly reported in literature [Cai and van Rensch, 2012; Guan et al., 2005; Verdon and Franks, 2006]. IOD and SOI are negatively correlated at the fourth decomposition levels. The physical mechanism of this negative relationship between IOD and SOI has been recently discussed in Cai et al. .
 In addition to the time-scale structure of the intercorrelations, the MRA decomposition provides more information on the signs of these intercorrelations. Most of the signs between two climatic indices are consistent across time scales, such as those between STR and SOI, STR and SAM; while those between PDO and STR, and PDO and SOI are not. This indicates the complexity of PDO modulation effects on other climatic indices, which would otherwise be hidden without the MRA decomposition.
 These results test the first hypothesis that the interdependence between climatic indices varies with temporal scales. We find that the intercorrelations between the indices occur only at some time scales, and the signs between indices may vary across time scales.
3.2. Details of Teleconnection Patterns From WMLR and TMLR
 Because large-scale climate signals are intercorrelated (section 3.1), multiple linear regression of precipitation with all significant influencing variables was performed to cancel out the overlapping effects from the intercorrelated variables. The results of these regressions with all five climate signals with (equation (13)) or without (equation (10)) MRA decomposition are summarized for all monthly data (1959–2008) in Table 3. For regressions with all monthly data, both WMLR and TMLR results suggest similar significant correlated large-scale climate signals. The number of stations with suggested significant teleconnection is ordered decreasingly from STR, IOD, SOI, PDO, to SAM (Figure 3a). After MRA decomposition, WMLR reveals that STR and IOD are significant at 16% and 47% more stations than TMLR in interpreting the variability of monthly precipitation anomaly (Figure 3a). WMLR also tells at which time scales these climate signals are correlated with precipitation in the study area (Table 3 and Figure 3b). STR tends to be more significantly influencing precipitation at the first, second, and fourth decomposition levels (corresponding to the 2, 4, and 16 month scales), IOD at the fourth, SOI at the fourth and fifth (32 month), and PDO at the first and second, and SAM at the second and fifth decomposition levels. The approximation subseries of all five climate signals, however, are not statistically significant to be associated with precipitation.
Table 3. Significant Predictor Variables in TMLR and WMLR, Resulting From the Stepwise Linear Regressions, With a Cutoff Value α = 0.01
STR_D2,3,4; SOI_D5; PDO_D1
STR; IOD; PDO
STR_D1,4; IOD_D5; PDO_D2
STR; IOD; PDO
STR; SOI; PDO
STR_D1,4; SAM_D2; PDO_D2
STR_D1,4; IOD_D4; SAM_D2; PDO_D2
STR_D2,4; IOD_D1; SAM_D2
STR; IOD; PDO
STR_D4; IOD_D3; SOI_D2
STR; IOD; SAM
STR_D1; IOD_D4; PDO_D1
STR; IOD; SAM; PDO
STR_D3, 4; IOD_D1; PDO_D1
STR; IOD; PDO
SOI_D4; PDO_D2; SAM_D5
IOD_D3; SOI_(D4, A5);
 Both TMLR and WMLR results tell similar spatial patterns of the significant teleconnections, with WMLR results being a little more spatially coherent (Figure 4). Resulting from both models, STR influences precipitation anomaly in the south and west parts of SA, consistent with previous studies [Timbal and Drosdowsky, 2012]. SOI influence is confined to the northeast and east parts of SA, in agreement with previous studies [Risbey et al., 2009]. WMLR reveals that IOD has significant influence at more stations than TMLR, and most of these stations are located near the coast. This spatial pattern is consistent with previous studies [Ashok et al., 2003; Risbey et al., 2009; Ummenhofer et al., 2009]. But TMLR results indicate that a quarter of stations with suggested IOD significant influence are located inland, which does not seem to be consistent with other studies and the WMLR results of this study.
 Spatial patterns of significant climate signal decomposition subseries are worth of some attention (Figure 5). The statistical influence of the first and second STR decomposition subseries is generally distributed at and near the coast, while that of the fourth decomposition subseries extends from the coast into central SA. The influence of the IOD fourth decomposition subseries is clearly confined to the coast. SOI tends to influence precipitation in the northeast SA at the 16 month time scale, and in the east at the 32 month time scale.
 Lag structures of significant influencing climate decomposition series are summarized in Table 4, together with the number of stations for such significant relationship. No lag is found for STR subseries. This finding is consistent with that STR serves as a bridge for remote climate drivers' influence on Australia's precipitation, such as the case in southeast Australia [Cai et al., 2011b]. IOD subseries lead precipitation from 4 to 9 months at the first three decomposition levels (2, 4, and 8 month scale), and 0 months at the fourth and fifth levels (16 and 32 month scale). SOI leads precipitation 12 months at the fifth levels, and 0 month at the fourth levels. Lag structures are also found for PDO and SAM influences at a few stations. Overall, except for STR, lag time between precipitation and climate signal decomposition subseries varies among time scales.
Table 4. Lag (in Months) of Precipitation Correlation With the Decomposition Subseries of Large-Scale Climate Signals at the Stations (the Number of Such Stations Is Shown in the Brackets) Where the Signals Are Significant in the WMLR With a Cutoff Value α = 0.01
 The above regression exercises were performed with all monthly data. The performance is likely dampened and often biased to wet season teleconnections when the teleconnection patterns vary with seasons. Using seasonal stratification data are a common way to avoid this problem. Both WMLR and TMLR were performed for monthly data for four seasons during the study period (1959–2008). Overall, the two models reveal similar teleconnections for each of four seasons (Figure 6). In spring, STR and IOD are the most influencing climate signals in the study area; in summer, no dominant influencing signals are found; in autumn, STR is the dominant signal influencing precipitation; and in winter, both STR and SAM are important signals interpreting the temporal variability of precipitation anomaly. Spatial patterns of these teleconnections from WMLR are shown in Figure 7. Overall, the teleconnection spatial patterns are similar for winter and spring, with stronger teleconnection at the coast, extending northward inland. In autumn, teleconnection tends to be restricted to the southern part of the study area, while in summer the contrast between inland and coastal areas becomes weaker.
 These results test our second hypothesis that more details of the relationship between precipitation and a climatic index can be revealed when the climatic index is decomposed into various temporal scales. The WMLR analysis results indicate the first three significant climate signals for interpreting monthly precipitation variability in the study area are STR at the 2, 4, and 16 month scales, IOD at the 16 month scale, and SOI at 16 and 32 month scales. SAM primarily influences winter precipitation. The spatial extents of these teleconnections are different between climate signals and between their subseries. It is also found that teleconnection spatial patterns and significant climate signals vary between seasons.
3.3. WMLR and TMLR Performance in Interpreting Precipitation Anomaly
 In terms of how much variability of monthly precipitation anomaly can be interpreted by the sum of these climate signals, WMLR, however, does not result in larger adjusted coefficient of determination (R2) than TMLR (not shown). This is not supportive to the third hypothesis. Hindcast testing was performed to further examine this issue, by training the models with data of 1959–1995, and testing the models with data of 1996–2008. Averaged over all stations, the hindcast statistics (correlation coefficient and relative absolute error) do not show much difference between the two models either (not shown). Nevertheless, when the hindcast correlation coefficients (between model estimated values and observations at each individual station) are plotted versus adjusted R2 of the regression models in the training period, the WMLR performance in the testing period clearly increases with its performance in the training period, while the relationship between testing and training performance is less clear for the TMLR model (Figure 8). This result indicates that WMLR is likely to capture more fundamental patterns of the precipitation teleconnections than TMLR. After it is trained with a good performance, indicating by the adjusted R2, it can be expected to perform well during the testing period. The previously shown nondifferentiated WMLR and TMLR performance when averaged over all stations either in the training or the testing period appears to be related to an overall weak teleconnection in the study area.
 For seasonal stratification data, the sample size at each station is not large enough to be split into two parts for hindcast experiments. The stepwise linear regressions with a cutoff value α = 0.02 were trained for both TMLR and WMLR for each season with all available data. The performance of both models is plotted for four seasons (Figure 9). WMLR performs strongly better in interpreting precipitation anomaly for summer, better for spring than TMLR, but similarly for autumn and winter as TMLR. If the relationship between training and testing performance in Figure 8 can be assumed, WMLR would interpret variability of monthly precipitation anomaly 45% in spring, 77% in summer, −20% in autumn, and 16% in winter, more than that interpreted by TMLR.
 These results support the third hypothesis that temporal variability of precipitation anomaly tends to be better interpreted by the decomposed climatic indices than the original ones. In the study area, the WMLR method is more efficient in revealing teleconnections for spring and summer.
 The above results and discussion demonstrate that the WMLR method is useful to address three difficulties in teleconnection analyses: interdependent climate signals, multitemporal components, and nonuniform time lags, and thus can be used to better reveal the details of teleconnection patterns. Having said this, it should be noted that the proposed MRA-based regression method does not address the nonsymmetric effects of some climatic indices. In this context, the composite method may be a good option [Hendon et al., 2007].
 In this study, the multiresolution analysis incorporating with stepwise linear regression was applied to examine precipitation teleconnections. The method decomposes original monthly precipitation anomaly and climate signals into independent component time series at different time scales, with stepwise regression being used to include significant influencing factors. The results indicate that the MRA-based WMLR method is able to reveal at which time scale(s) and with what time lag(s) the teleconnections occur with the large-scale climatic indices, and their spatial patterns. The MRA method is also useful to examine time-scale patterns of interdependence between climate signals. These altogether make the MRA-based method a promising tool to address the three difficulties in climatic teleconnection studies: interdependence between climate signals, multitemporal components of each signal, and nonuniform time lags between precipitation and the climate signals.
 The WMLR method was applied in South Australia with five large-scale climate signals, revealing teleconnection patterns between precipitation anomaly and each climate signal at various time scales, corresponding spatial patterns, and hierarchical lag structures. Based on all monthly data, of the five examined climate signals STR, IOD, and SOI are the first three most significant ones in the study area. STR (at 2, 4, and 16 month scales) and IOD (at 16 month scale) influence precipitation temporal variability more in the coastal area of South Australia. SOI (at 16 and 32 month scales) influences precipitation temporal variability more significantly in the northeast and east of South Australia. STR appears to be the most influencing signal, but lacks of a lead-time structure for precipitation. Analysis based on seasonal stratification data suggests that significant climate signals for precipitation are STR and IOD in spring, IOD in autumn, and STR and SAM in winter. In comparison to TMLR, WMLR reveals more details of teleconnection patterns and at more stations. It is expected to better interpret the temporal variability of precipitation anomaly than TMLR.
 Wasyl Drosdowsky kindly provided the STR index. Discussion with Xinping Zhang is appreciated. We thank three anonymous reviewers for many constructive comments which helped to improve the quality of the paper. The authors acknowledge funding supports from Hunan Bairen Program, the Construct Program of the Key Discipline in Hunan Province of China, China Scholarship Council, Scientific Research Fund of Hunan Provincial Education Department of China grant 10B065, Australia National Centre for Groundwater Research and Training, and Flinders University.