1. Introduction
The atmospheric dynamics underlying precipitation variability over the Midwest have received increasing attention, in part because of this region’s role in the upward trend in annual precipitation over the contiguous United States (Karl and Knight 1998; Groisman et al. 2004, 2005). The Midwest lacks direct influence from tropical systems and complex terrain, and is collocated with a maximum in cyclone frequency centered over the Great Lakes (Zishka and Smith 1980; Reitan 1974; Wernli and Schwierz 2006). In winter, the maximum in cyclone frequency over the Midwest is part of two cyclone trajectories that originate in Colorado and Alberta in the lee of the Rocky Mountains and merge over the Great Lakes (Zishka and Smith 1980; Isard et al. 2000), and associated trends in cyclone count have been investigated in several studies (e.g., Angel and Isard 1998; Key and Chan 1999; Konrad 2001).
Precipitation variability in the Midwest can also be understood in the context of large-scale standing-wave teleconnections typically resolved via empirical orthogonal function (EOF) analysis (e.g., Small et al. 2010). The positive phase of the Pacific–North American pattern (PNA) depicts a wave train consisting of an anomalously strong Aleutian low, positive height anomalies over the western United States, and negative height anomalies over the southeastern United States (Wallace and Gutzler 1981). Rodionov (1994) found that composites of 700-hPa geopotential heights for low and high Great Lakes precipitation closely resembled the positive and negative phases of the PNA pattern, respectively. An inverse relationship between precipitation in winter months and the PNA index was documented for the Ohio River valley (ORV) and the Midwest in several studies (Leathers et al. 1991; Serreze et al. 1998; Coleman and Rogers 2003). Great Lakes cyclones occurred more frequently over Canada during the positive phase of the PNA and more frequently over the southwest United States in the lee of the Rockies during the negative phase. Angel and Isard (1998) found that the PNA pattern was anticorrelated with strong cyclone occurrence over the Great Lakes during November, December, and January.
Midwest precipitation is also influenced by the El Niño–Southern Oscillation (ENSO), where El Niño events, characterized by warm SST anomalies in the eastern equatorial Pacific, are associated with reduced precipitation over the Midwest, primarily in the vicinity of the ORV during winter (e.g., Gershunov and Barnett 1998; Mo and Schemm 2008; Becker and Berbery 2009; Zhang et al. 2010). Eichler and Higgins (2006) showed changes in winter storm tracks corresponding to ENSO, with the Midwest experiencing an increase in surface cyclone frequency during La Niña events. The results of Becker and Berbery (2009) revealed a relative increase in the intensity of winter daily precipitation over the upper Midwest during El Niño events and a decrease in intensity over the ORV.
Variations in moisture transport are one way that teleconnections translate into precipitation variability. EOF analysis of moisture transport over North America identified a leading standing-wave pattern that was linked to the PNA and ENSO, and its Pacific center of action markedly impacted precipitation over the west coast of the United States (Dominguez and Kumar 2005). This same leading EOF also had a circulation center over the southeastern United States that regulated moisture flux from the Gulf of Mexico into the central United States, and significantly impacted precipitation variability over the Midwest.
The objective of this study was to determine how Midwest daily winter precipitation variability depended on leading patterns of propagating atmospheric variability over eastern North America. To accomplish this objective, we performed a complex Hilbert empirical orthogonal function (HEOF) analysis on four atmospheric fields, we used correlation analysis to evaluate the importance of the HEOFs to Midwest precipitation, and we used a novel HEOF phase shift method to enable comparison of the HEOFs and high-precipitation atmospheric composites. This set of analyses revealed the role of each propagating pattern in daily precipitation variability, as well as the role of each propagating pattern in daily variability of the field from which the pattern was derived. Section 2 describes data and methods, section 3 presents the propagating patterns and their relationship with precipitation and teleconnections, and summary and discussion are given in section 4.
2. Data and methods
a. Data
The Midwest was defined as the area bounded by 36°–46°N and 83°–95°W (bold box, Fig. 1)—essentially the region encompassing the upper Mississippi and Midwest subregions in Groisman et al. (2004) and Groisman et al. (2005). This region had a relatively dense precipitation measurement network with data extending back to the 1950s and earlier for most stations. The Midwest daily (0000–2359 UTC) total precipitation in mm from 1 December 1957 to 28 February 2009 was obtained from the Global Historical Climate Network (GHCN) daily dataset available from the National Climatic Data Center (NCDC; http://www.ncdc.noaa.gov/). Station data was inspected to ensure that no more than 10% of the measurements were missing from any record, and a total of 150 stations within the Midwest were retained after the inspection (indicated by dots within the bold box in Fig. 1).
The region within the black box defines the Midwest. Dots show the locations of precipitation stations, and thin lines indicate Voronoi polygons. Light gray shading indicates Voronoi polygons included in the Midwest domain, and dark gray shading denotes the Midwest polygons that were used to define a lake effect snow index.
Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-0111.1
Atmospheric fields described in section 2b were based on daily values derived from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (NNR; Kalnay et al. 1996) provided by the Physical Sciences Division (PSD) of the National Oceanic and Atmospheric Administration (NOAA) Earth Systems Research Laboratory in Boulder, Colorado (http://www.esrl.noaa.gov). Six-hourly values were developed for each field, and daily means were then calculated to align with the daily mean precipitation data.
Monthly mean values of two teleconnection indices were obtained from the Climate Prediction Center (http://www.cpc.ncep.noaa.gov): the North Atlantic Oscillation index (NAOI; Barnston and Livezey 1987) and the PNA index (PNAI; Wallace and Gutzler 1981). The NAOI and PNAI were computed from the rotated principal components (e.g., Horel 1981; Barnston and Livezey 1987) of standardized 500-hPa height anomalies based on the 3-month period centered on each month. The multivariate ENSO index (MEI; Wolter and Timlin 1993, 1998) was defined using the first unrotated principal component of combined sea level pressure, surface wind velocity, sea surface temperature (SST), surface air temperature, and cloud cover over the central Pacific. Bimonthly MEI values were obtained from the NOAA/OAR/ESRL PSD (http://www.esrl.noaa.gov).
b. Derived fields
Jet stream–level wind speeds (
To analyze horizontal moisture transport, the vector field qv ≡ (qu, qυ) was calculated at 850 hPa. The vector field qv proved to be more informative than scalar moisture advection (−v · ∇q) because qv indicated the magnitude and direction of moisture transport. Use of vertically integrated horizontal moisture flux would have had a minimal impact on the findings because most atmospheric water vapor was concentrated in the lower troposphere.
To gain insight into the dynamics of vertical motion, 850-hPa temperature advection (TA; K s−1) and the difference in vorticity advection between 850 and 250 hPa (VA; s−2) were analyzed. The geostrophic counterparts of VA and TA are associated with large-scale vertical motion according to the traditional omega equation (e.g., Holton 2004).
c. Statistical methods
Bootstrapping (e.g., Wilks 1995) was used to test the statistical significance of all Pearson correlation coefficients (r) reported here. Bootstrapped samples were constructed by selecting n pairs of values with replacement from a sample of n pairs, and r was calculated for each bootstrapped sample. One-thousand bootstrapped samples were developed for each correlation, and results were tested for significance at the α = 0.05 level.
For the estimate of power spectra, a Hanning window was applied to each winter, and the spectral estimates for each of the m = 52 winters were averaged. The resulting spectral estimate resolved periods from 2 to 90 days, and had 2m = 104 degrees of freedom (Welch 1967). The null red noise spectrum followed Gilman et al. (1963), and the 95% confidence limit was based on the right-tail quantile of the chi-squared distribution with 2m degrees of freedom as given in Welch (1967).
Traditional EOF analysis is useful for identifying leading variability patterns within a dataset, but its inability to resolve propagating wavelike structures is a limitation (Barnett 1983). Complex HEOF analysis resolves propagating patterns by complexifying the input data so that its imaginary part is the original dataset phase shifted in time by π/2 (Barnett 1983; Horel 1984). As discussed in Hannachi et al. (2007), HEOF analysis is distinct from “complex EOF analysis” in which the two components of a vector field are cast as real and imaginary parts to enable analysis of standing-wave patterns in vector fields (e.g., Kaihatu et al. 1998).
Together, the real and imaginary parts of an HEOF provide a parsimonious representation of a propagating pattern that would otherwise appear as two degenerate patterns in quadrature in traditional EOF analysis (Hannachi et al. 2007). Here, HEOF analysis was applied separately to the detrended 
For each HEOF (notation defined in text), the fraction of field variance accounted for by the HEOF.
(a) The real and (b) the imaginary part of the first HEOF of qv prior to any phase shift.
Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-0111.1
d. HEOF phase shift formulation
Research context may render one phase of a complex HEOF more informative or convenient than another (e.g., Klink and Willmott 1989; Merrifield and Winant 1989; von Storch et al. 1988). Here, to determine the location where a propagating pattern (HEOF) was maximally related to precipitation variability over a fixed region (the Midwest), we phase shifted the HEOF so that its real part was maximally correlated with P0.25 and its imaginary part had zero correlation with P0.25. Following the phase shift, the total (modulus) correlation with P0.25 was consolidated in the real part of the HEOF, and the real part of the HEOF depicted the propagating pattern in its phase most relevant to Midwest precipitation.
The phase shift for the first HEOF of qv is presented as an example. Prior to phase shift, the real part of the first HEOF of qv depicted a cyclonic circulation centered over the Midwest (Fig. 2a), and the imaginary part depicted northeasterly qv over the Midwest (Fig. 2b). These two patterns lacked clear relevance to Midwest precipitation dynamics, and the modulus correlation between P0.25 and this unshifted HEOF was partitioned into real and imaginary parts.
The phase shift φ1 = −0.33π [from Eq. (7)] placed the real part of the HEOF pattern optimally to account for Midwest precipitation, with a southwesterly moisture fetch associated with an upstream surface cyclonic circulation and downstream surface anticyclonic circulation (Fig. 3a). Because the imaginary part had zero correlation with P0.25 after the phase shift, it was rendered statistically irrelevant, serving only to indicate the pattern’s direction of propagation (eastward in this case; not shown).
The real part of the following phase-shifted first HEOFs: (a) horizontal moisture transport (qv1), (b) temperature advection (TA1), (c) vorticity advection (VA1), and (d) jet stream–level wind speed (
Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-0111.1
To illustrate that the phase shift φ1 = −0.33π maximized the correlation between P0.25 and the real part of the first HEOF of qv, Fig. 4 shows how the phase and real part of the first HEOF of qv varied as a midlatitude cyclone passed through the region during 1–5 December 1989. As the system entered the west edge of the HEOF domain on 1 December 1989 (Fig. 4b), the phase of the first HEOF of qv was close to −π/2 (Fig. 4a), indicating that the system was one-quarter wavelength upstream, rendering the real part of the first HEOF of qv close to zero (Fig. 4a). When the system arrived to the study region during 3 December 1989 (Fig. 4c), the phase of the first HEOF of qv approached zero (Fig. 4a), the real part of the first HEOF of qv maximized (Fig. 4a), and qv maximized at 2.25 mm0.25 (not shown). As the system slowed, the phase of the first HEOF of qv fluctuated around zero (Fig. 4a), and the magnitude of its real part weakened (Fig. 4a) as the moisture delivery weakened (Fig. 4d). The correlation between P0.25 and the real part of the first HEOF of qv for 1–5 December 1989 was r = 0.84.
(a) For 1–5 Dec 1989, the phase and real part of the leading HEOF of horizontal moisture transport (qv1). (b)–(d) The daily mean horizontal moisture transport vector field (qv) for 3 days during 1–5 Dec 1989.
Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-0111.1
3. Results
Over the period of analysis (1958–2009), the linear trend of P was small (order 10−2 mm decade−1), so the results section focuses on oscillatory variability rather than linear trends. In sections 3a–b, the real part of propagating patterns of moisture transport (qv), temperature advection (TA), vorticity advection (VA), and jet stream–level wind speed (
a. Propagating atmospheric system
The nth HEOF of a field is denoted by the field name with a subscript n (e.g., the first HEOF of 
The first HEOF 
Phase-shifted 
Of all the HEOFs analyzed, qv1 had the strongest correlation with P0.25 (r = 0.67; Fig. 5a; Table 2). Correlations between qv1 and precipitation at individual stations tended to be weaker than the correlation of qv1 and P0.25, and generally increased in strength from northwest to southeast across the study region (Fig. 5b). Various correlation and composite analyses indicated that the qv1 pattern was part of a single archetypal precipitation system. Specifically, the 
(a) Scatterplot of P0.25 (mm0.25) vs the real part of the phase-shifted first HEOF of 850-hPa moisture transport (qv1). (b) Correlation between qv1 and precipitation at each station in the Midwest domain.
Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-0111.1
Correlation matrix for daily P0.25 and five HEOFs. All nonzero values are significant at α = 0.05.
Finally, composites of 
For days with Midwest precipitation (P0.25) greater than or equal to the 90th percentile, anomalies of (a) jet stream–level wind speed (
Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-0111.1
To further illustrate the role of the jet stream, Fig. 6b shows the jet streak configuration associated with the upper-level trough, indicating enhanced likelihood of jet stream cores (
Spectral analysis of the qv1 time series revealed statistically significant energy for variations with periods from 3 to 7 days (above the dashed curve, Fig. 7), which is consistent with a midlatitude cyclone. Although almost indistinguishable from red noise at the 95% confidence level, qv1’s energy for periods longer than 10 days was nonnegligible. In the next section, some of this lower-frequency energy is shown to be tied to planetary-scale teleconnections.
Power spectrum of the real part of the phase-shifted first HEOF of horizontal moisture transport (qv1, bold curve). A null red noise spectrum is shown (solid curve) along with its associated 95% confidence limit (dashed curve). Vertical dotted lines bound periods from 3 to 7 days.
Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-0111.1
b. Relationship with teleconnections
Because teleconnections are most often based on monthly or seasonal mean data, daily qv1 and P0.25 values were averaged over each winter (
For December–February (DJF) mean values, correlations between 
The NAO’s centers of action were downstream from the Midwest (not shown), and the NAO had only weak influences on eastern North America’s propagating atmospheric patterns [e.g., 
Of the teleconnections analyzed here, the PNA had the strongest correlation with 
The moisture transport pattern in Fig. 3a has been linked to the PNA as well. In studying the PNA and monthly mean precipitation in the ORV, Coleman and Rogers (2003) found that the meridional component of 850-hPa moisture flux was significantly greater over the ORV for winters in the uppermost quintile of their ORV index of precipitation compared to winters in the lowest quintile of their ORV index; their moisture flux pattern shares similarities with the spatial pattern of qv1, featuring an axis oriented southwest–northeast extending from the Gulf of Mexico to the eastern United States.
c. Propagating pattern relevance to lake effect snow
Lake effect snow from the Great Lakes region is an important component of Midwest precipitation, accounting for at least one-third of lakeshore snow totals (e.g., Eichenlaub 1970). Lake effect snow often occurs following the passage of an upper-level trough, which typically results in low-level north or northwesterly flow over the surfaces of one or more of the Great Lakes. When the lake is unfrozen, and the surface to 850-hPa lapse rate is approximately 10°–13°C (e.g., Holroyd 1971; Niziol et al. 1995), cold advection over the lake results in upward heat and moisture fluxes, favoring the development of clouds and precipitation (e.g., Rothrock 1969; Braham 1983).
The HEOF phase shift φ1 = −0.33π (section 2d) maximized the correlation between the real part of the first HEOF of qv and Midwest precipitation, and the resulting phase-shifted qv1 pattern depicted a cyclone upstream of the Midwest (Fig. 3a). In contrast, the phase shift that maximized the real correlation between the first HEOF of qv and the lake effect snow index 
For days with the lake effect snow index (
Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-0111.1
To verify the lake effect snow atmospheric pattern in Fig. 8c, we calculated composite qv anomalies for days when 
The modulus correlation between 
4. Summary and discussion
Propagating patterns of atmospheric variability relevant to Midwest winter precipitation were detected over eastern North America via HEOF analysis. A novel method was used to phase shift each HEOF so its real part was maximally correlated with Midwest precipitation, meaning the real part of the HEOF showed geographically where the propagating features maximally impacted Midwest precipitation variability. The phase shift also rendered the imaginary part of the HEOF statistically irrelevant (zero correlation with Midwest precipitation), meaning it served only to indicate the pattern’s direction of propagation.
The first HEOF of 
The strongest predictor of Midwest precipitation was the first HEOF of qv (r = 0.67), and phase shifting positioned the qv cyclonic circulation upstream from the Midwest. The propagating qv pattern was predominantly synoptic, but had nonnegligible energy at weekly-to-monthly time scales related to hemispheric teleconnections. The propagating qv pattern’s relevance to lake effect snow was uncovered by phase shifting the pattern an additional one-half wavelength downstream.
The propagating atmospheric patterns were detected independent of precipitation, and the fact that HEOFs 
Empirical orthogonal functions have important roles in predicting climate variability over multimonth time scales (e.g., outlooks provided by the NOAA Climate Prediction Center). The results reported here indicate that, for regions where propagating synoptic systems strongly impact precipitation, projection of forecast ensembles onto leading propagating modes (phase-shifted HEOFs) may provide forecast skill beyond analysis of standing waves detected by traditional EOF analysis. For shorter-term forecasting, the patterns identified here show statistically important propagating synoptic conditions in their phase most important for moisture and precipitation delivery to the Midwest.
The phase-shifting methods used here are applicable to other seasons and locations, but compared to Midwest winters, the association between Midwest summer precipitation variance and leading propagating patterns of atmospheric variability are relatively weak (e.g., the first HEOF of June–August qv accounted for 16% of P0.25; not presented). During Midwest summers, other processes become important including mesoscale convective complexes (e.g., Fritsch et al. 1986) and Caribbean moisture sourcing into the Central Plains low-level jet stream (Dirmeyer and Kinter 2010). In other regions of North America, the diversity of precipitation mechanisms expands further beyond the scope of classic propagating midlatitude cyclones to include monsoonal regimes and systems of tropical origin.
Acknowledgments
Comments from anonymous reviewers helped to improve the manuscript. J. Liptak was partially supported by National Science Foundation Grant ARC-1022485.
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