3.1.1. Study Locations and Instrumentation
 Temperature data sets from Loch Vale in the Colorado Rocky Mountains [Lundquist and Rochford, 2007] (available at http://faculty.washington.edu/jdlund/home/publications.shtml), from the Pyrenees, France [Pepin and Kidd, 2006], and from Yosemite National Park, Sierra Nevada, California [Lundquist and Cayan, 2007] were used to test the algorithm (Figure 1, Table 1). The Colorado study used Maxim 1922L iButtons [Hubbart et al., 2005], and the Pyrenees and Sierra Nevada studies used Onset Tidbits and HOBOs [Whiteman et al., 2000]. Instruments were deployed in evergreen trees approximately 2 m above the ground, a deployment method that compared well (root-mean-square error (RMSE) < 1°C) with nearby standard Gill-shielded temperature sensors on poles [Lundquist and Huggett, 2008]. These instruments have been successfully used in many studies [Whiteman et al., 2001; Taras et al., 2002; Lookingbill and Urban, 2003; Lundquist et al., 2003; Mahrt, 2006; Pepin and Kidd, 2006; Tang and Fang, 2006; Lundquist and Cayan, 2007; Marshall et al., 2007]. Table 1 details instrument specifications, sampling intervals, and accuracy, as well as topographic information, for each study area.
Figure 1. Maps of geographic locations of temperature sensors in (a) Loch Vale, Rocky Mountain National Park, Colorado, (b) Pyrenees, France, and (c) Yosemite, Sierra Nevada, California.
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Table 1. Study Site Characteristics
| ||Rocky Mountains||Pyrenees||Sierra Nevada|
|Area of study (km2)||15||400||7500|
|Elevation range (m)||3100–3400||1400–2200||1200–3200|
|Resolutions examined (m)||10, 20, 50, 100, 500||100, 500||100|
|Number of sites||17||26||51|
|Dates examined||Aug 2005 to Jul. 2006||May 2002 to May 2005||Jul 2002 to Jul 2005|
|Radius (half the average peak-to-peak distance) (m)||375||3500||1500|
|Instruments used||Dallas Semiconductor Maxim iButtons (DS-1922L, purchased 2005)||HOBO Pro Series RH and Temp (purchased 2002)||HOBO Pro Series RH and Temp (purchased 2003, 2004); onset StowAway TidbiT Temp Logger (purchased 2002, 2003, 2004)|
|Sampling interval (min)||60||15||30|
|Instrument response time (min)||<10||<10||<10|
|Instrument temperature range (°C)||−35–85||−30–50||−30–50 (HOBO), −20–50 (Tidbit)|
|Radiation shield design||upside-down white funnel, placed in tree||white PVC tubes, hung in evergreen trees at a 45° angle, with the top end facing north||rain shield from Onset Computer Corporation, painted brown, or Gill radiation shield; all shields placed in trees|
|Specified instrument accuracy (°C)||±0.5||±0.2||±0.3 (for HOBO Pro), ±0.5 (for Tidbit)|
|Deployed instrument/shield accuracy||better than ±1.0°C||better than ±1.0°C||better than ±1.0°C|
|Sensor microscale deployment/location||in evergreen trees, 2 m above the ground||in forested areas, away from paths and avoiding local topographic hollows||in evergreen trees, 2 m above the ground, generally along roads, trails, or streams|
 The Loch Vale watershed [Campbell et al., 2000; Clow et al., 2003] in Rocky Mountain National Park, Colorado (Figure 1a), is a glaciated U-shaped valley. Site locations included flat valley bottoms, steeper stream-cut valleys, and the steep sidewalls of the valleys, all within the drainage area of the Loch (Figure 1a). A terminal moraine at the outlet of the Loch causes a terrain constriction which blocks cold-air drainage and leads to pooling above.
 The Eastern Pyrenees measurements [Pepin and Kidd, 2006] focused on transects across three river valleys draining a central plateau area (Figure 1b): the Conflent, which drains to the east/northeast and reaches the Mediterranean east of Perpignan, the Cerdagne, which drains southwest to Spain, and the Capcir, with flows north toward Carcassonne. The Cerdagne and Capcir are wide valleys with flat bottoms, the latter being restricted in its lower reaches and particularly prone to cold air pooling. The Conflent is a V-shaped canyon with steep sides (20°–30°) and a steep longitudinal profile gradient.
 The Yosemite National Park, Sierra Nevada, California data set included not only HOBO loggers deployed in trees, but also RAWS sites, CA DWR snow pillow sites, and cooperative observing sites, as described by Lundquist and Cayan . Site locations spanned the eastern and western slopes of the central Sierra Nevada and ranged from glacier-carved U-shaped valleys to steep gorges to flatter meadows at the limits of tree line. Sensors were distributed along road corridors in addition to along streams, sampling undulating topography and escarpments in addition to fluvial- and glacial-carved valleys.
3.1.2. Empirical Orthogonal Function (EOF) Technique for Identifying Cold-Air Pooling (CAP) Locations
 Sites with frequent nocturnal temperature depressions due to CAP were identified for each study area using the empirical orthogonal function (EOF) techniques developed by Lundquist and Cayan . For each data set, we analyzed the daily minimum temperature, calculated as the lowest temperature recorded within a 24-h period starting at midnight. We defined daily minimum temperature at a point to be a function of (1) the mean annual minimum temperature from each station, (x), primarily an elevation effect, (2) temporal deviations in the mean temperature across measurements within the domain, ′(t), primarily a synoptic-weather effect, (3) local spatial deviations that change through time, (x, t), and (4) local instrument error, ɛ. Thus
The first two terms are well represented by existing techniques, where the slope of (x) versus station elevation typically corresponds to the regional average lapse rate, and ′(t) corresponds to fluctuations in temperatures across all stations due to variations in large-scale weather patterns. Essentially, these are region-wide positive or negative temperature anomalies. We analyzed local temperature patterns by first removing each station's long-term mean, (x), and then removing the daily minimum temperature anomalies averaged across all stations, ′(t). The third and fourth terms were then decomposed into their principal spatial patterns of variation and their evolution through time using empirical orthogonal functions (EOFs) [Beckers and Rixen, 2003; Preisendorfer, 1988]. The EOFs are linear and orthogonal, such that a sum of each spatial component multiplied by its corresponding temporal score recreates the original temperature data set, and are normalized such that the variances of the spatial components sum to one, and the variances of the temporal components sum to the total variance of the original temperature record, (x, t) + ɛ. Within each of the three data sets examined, the dominant spatial mode of daily minimum temperature variations had a temporal component highly correlated with clear weather and weak winds, and a spatial component identifying locations with very low minimum temperatures during these events. Thus, the first EOF corresponded to CAP and accounted for 75%/59%/30% of the variance of (x, t) + ɛ in the Rockies, Pyrenees, and Sierra Nevada, respectively. The percentage of the variance explained decreased as the area examined increased, because other factors such as variable exposure to air mass advection and slope orientation became increasingly important at larger domain sizes. The present analysis focuses only on CAP, and other modes of variation are not discussed.
 To summarize, EOFs decompose temperature variability into spatial and temporal weights, which become space-time series of temperature variation when multiplied together. For the first EOF, Figure 2 illustrates the spatial weights (Figure 2a), the temporal variations, also called principal components (Figure 2b), and two representative minimum temperature time series for the Pyrenees (Figure 2c). The two flat-bottomed valleys, Cerdagne and Capcir, exhibited strong CAP while the steeper-sided Conflent valley did not (Figure 2a). High principal component (PC) values (Figure 2b) indicate time periods when cold-air pools (CAP) were prevalent, i.e., sites with negative spatial weights had minimum temperatures 2°–6°C colder than the regional average. Physically, sites with strong negative weights had greater temperature depressions than the regional average during a CAP event, while sites with strong positive weights had warmer temperature anomalies than the regional average on those same days (an example of each is shown in Figure 2c). Sites with near-zero weighting had temperature anomalies close to the regional average during CAP events or had unsystematic anomalies. This could occur because these sites experienced slight cold-air pooling or infrequent cold-air pooling, whereas sites with strong positive weights very seldom experienced CAP temperature depressions.
Figure 2. (a) First empirical orthogonal function (EOF), with negative weights corresponding to cold-air pooling (CAP) for the Cerdagne (“ce”), Capcir (“ca”), and Conflent (“co”) valleys. Sites not in any valley are marked “nv.” Vertical dashed lines at weights of −0.5 and 0.5 identify cutoffs classifying sites as CAP (<−0.5), no signal (−0.5 to 0.5), and no CAP (>0.5). (b) Two-month segment of the principal component (PC) time series corresponding to the spatial weights. (c) Original daily minimum temperature records from the two Cerdagne Valley sites marked with stars in Figure 2a for the same time period as in Figure 2b. The solid line has a negative weight and is prone to CAP, while the dashed line has a positive weight and does not experience CAP. Vertical dashed lines in Figures 2b and 2c identify example time periods with no CAP (17–21 January) and with strong CAP (2–7 February).
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 On average in the Pyrenees, some cold air pooling occurred, such that the site at 1500 m was colder than the site at 2100 m at times when the PC weight was near 0, i.e., when (x, t) was near 0 and T(x, t) ≈ (x) + ′(t) (Figures 2b and 2c). Times with negative PC weights, such as 17–21 January 2004, identify periods with strong gradient winds and no CAP, when higher elevations were cooler than the valley bottoms. Positive PC weights indicate times with stronger than normal CAP, when the temperature inversion between CAP-prone locations and higher elevations was much stronger than usual, with a greater temperature depression in the valleys.
 Within each of the three data sets, high values of the principal components (PCs) of CAP patterns (Figure 3) were correlated with large-scale patterns of high pressure, clear skies, and weak gradient winds, as found by many other studies [Barr and Orgill, 1989; Clements et al., 1989; Gudiksen et al., 1992; Lundquist and Cayan, 2007]. The Rocky Mountain site (Figure 3a) is on the eastern side of the Continental divide and because of its high mean elevation, is dominated by strong westerly winds, which generally prevent the formation of thermally forced air circulations. Thus, mean conditions describe a steep lapse rate and no CAP. The PC for this region hovers near zero except for distinct time periods of weak westerlies, when local circulations set up and result in significant cold-air pooling in flat valley bottoms. In both the Pyrenees (Figure 3b) and the Sierra Nevada (Figure 3c), such strong winds are not normal, as evidenced by PCs that oscillate between positive and negative values. Thus, moderate CAP occurs most nights, with time periods of no CAP or particularly strong CAP modulated by larger-scale circulation patterns.
Figure 3. Time series of temporal weights (PCs) for (a) the Rocky Mountain data set, (b) the Pyrenees data set, and (c) the Yosemite, Sierra Nevada, data set during one winter.
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 The standard deviation of the temporal variation in CAP was 0.75, 1.06, and 1.16°C for the Rocky Mountains, Pyrenees, and Sierra Nevada, respectively. Thus, for a site to experience ±0.5°C temperature oscillation when the temporal PC varies by one standard deviation, it would need a spatial weight magnitude of 0.67, 0.47, or 0.43, respectively, for the three study areas. For simplicity and generality, we used the average of these, 0.5, as a cutoff value for classifying CAP (Figure 2a) and then tested the sensitivity of the cutoff value for each study area (see section 4.4). We classified sites as “CAP” (EOF weight < −0.5), “no CAP” (EOF weight > 0.5), and “no signal” (EOF weights between −0.5 and 0.5). “No CAP” means sites show warmer temperatures than the regional average during CAP events, whereas “no signal” means that the EOF representing CAP has little or no influence on the temperature variance at these sites.
 To test the robustness of the EOF technique for identifying CAP sites, we ran the analysis for subsets of the sites with different time periods and groups of included sites. While the precise EOF weights changed slightly, the general classifications defined above (CAP, no CAP, and no signal) were consistent for most sites, with the exception of a few sites with weights very close to the cutoff value (see section 4.4).