Corresponding author: T. Y. Gan, Department of Civil and Environmental Engineering, University of Alberta, Edmonton, Alberta, Canada. (email@example.com)
 Changes to the North American snowpacks for 1979–2007 were detected from snow water equivalent (SWE) retrieved empirically from horizontally polarized brightness temperature (TB) of a scanning multichannel microwave radiometer (18 and 37 GHz) and special sensor microwave imager (19 and 37 GHz) passive microwave data using the nonparametric Kendall's test. The predominant SWE trends detected agree with negative anomalies in snow cover observed in Northern Hemisphere since the 1980s, and both the SWE and snow cover results should be related to the significant increase in the surface temperature of North America (NA) observed since the 1970s. About 30% of detected decreasing trends of SWE for 1979–2007 are statistically significant, which is three times more than the significant increasing trends of SWE detected in NA. Significant decreasing SWE trends are more extensive in Canada than in the United States. The mean trend magnitudes detected for December–April are −0.4 to −0.5 mm/yr, which means an overall reduction of snow depth of about 5–8 cm in 29 years (assuming a snowpack density between 200 and 250 kg/m3), which can impact regions relying on spring snowmelt for water supply. From detected increasing (decreasing) trends of gridded temperature (precipitation) based on the North American Regional Reanalysis data set and the University of Delaware data set for NA, their respective correlations with SWE data and other findings, such as global scale decline of snow cover, longer rainfall seasons, etc., it seems the extensive decreasing trends in SWE detected mainly in Canada are more caused by increasing temperatures than by decreasing precipitation. However, climate anomalies could also contribute to the detected trends, such as PC1 of NA's SWE, which is found to be correlated to the Pacific Decadal Oscillation index and marginally correlated to the Pacific North American pattern.
 Recent studies on detecting changes in snowpacks have been based on snow cover [e.g., Brown and Robinson, 2011; Dyer and Mote, 2007; Hantel and Hirtl-Wielke, 2007; Frei and Robinson, 1999; Brown and Goodison, 1996], or snow depths, than on snow water equivalent (SWE), which is the amount of water a snowpack would give after complete melting, and it is estimated by multiplying the snow depth with the average snow density. SWE is more useful than snow cover for the management and planning of snow-dominated water resources. As a snowpack ages, it undergoes compaction as ice crystals settle and metamorphose, which is partly due to the temperature gradient of snowpack [Colbeck, 1983], partly due to increasing overburden load as snowfall occurs. Due to compaction, snow depth will decrease but snow density will increase, and so unless there is new snowfall or the snowpack is reduced by sublimation (direct transition from ice to water vapor) or melting, the SWE of a snowpack should remain unchanged. Conversely, new snowfall (sublimation) will cause a thicker (thinner) snow cover, or increased (decreased) SWE, even though the snow cover area (SCA) could remain unchanged or could change marginally. Even though SCA is useful for water resources management, from SCA, we cannot quite know the amount of freshwater stored in a snowpack.
 Seasonal snow mass variations at midlatitude to high latitude are the largest signals in the changes of terrestrial freshwater storage each year [Barry and Gan, 2011; Niu et al., 2007]. However, we cannot rely on snow gauges or ground-based snow course measurements alone for accurate estimation of the amount of SWE at the regional scale since snowpack generally exhibits substantial spatial variability. Other than being point measurements, it is well known that snow gauges, even when mounted with shields such as the Nipher shield, suffer from under-catch problems especially under windy conditions. For example, the catch ratios of the Wyoming Fence to World Meteorological Organization (WMO)-Double Fence Inter-Comparison Reference gauges were 89% and 87% at Regina (Canada) and Valdai (Russia), respectively. Yang et al.  found that the mean catch of snowfall for the U.S. 8 in. gauge at Valdai was 44%. For the Tretyakov and Hellmann gauges, the mean catch of snowfall was 63%–65% and 43%–50%, respectively, at the northern test sites of the WMO experiment.
 Another major problem concerning snow gauge data is the lack of representative snowpack density. We expect properties of snowpack found in North America to vary widely in terms of snow depth, density, and grain size. Newly fallen snow normally has a density less than 100 kg m−3, an albedo of 90% or higher, and grain size of 50 µm to about 1 mm, but the grain size and density will increase as snow ages, e.g., wild snow less than 30 kg/m3 to firn snow greater than 500 kg/m3. From analyzing 848 stations across Canada that reported daily snowfall and daily precipitation from October 2004 to February 2005, Cox  found that the histogram of the frequency of snowfall events by snow depth/SWE ratio is dominated by a spike at the 10:1 ratio which means that the snow density is 1/10th that of water, a bias caused by the 10:1 approximation being used not just in forecasting but in place of actual snowpack measurements. Recognizing the inadequacy of this 10:1 ratio, for climate stations only equipped with a snow ruler, Mekis and Hopkinson  proposed an alternative for more accurately estimating the SWE of snowfall recorded at a station based on a factor called the Snow Water Equivalent Adjustment Factor (SWEAF), which can range from 0.6 to 1.8. SWEAF generally increases with latitude. The province of British Columbia of Canada tends to have SWEAF less than 1.
 In view of the aforementioned problems associated with snow gauge data, which are point measurements, spatially distributed SWE data have been estimated from the brightness temperature (TB) in K (Kelvin) of passive microwave platforms, such as the SMMR (Scanning Multichannel Microwave Radiometer) from 25 October 1978 to 20 August 1987 and the SSM/I (Special Sensor Microwave Imager) since 7 September 1987 to April 2007 for the F13 satellite that carried the SSM/I sensor that failed in 2007. The SMMR sensor flew on NASA's Nimbus 7, while SSM/I are mounted on the Defense Meteorological Satellite Program satellites of the United States. Since May 2002, TB retrieved from the Advanced Microwave Scanning Radiometer–EOS (AMSR-E) sensor aboard the Aqua satellite has also been used to estimate SWE. The frequencies of SSM/I are 6.6, 19 GHz (69 × 43 km), 22, 37 GHz (37 × 29 km) to 85 GHz (15 × 13 km), while that of AMSR-E are 18.7 GHz (27 × 16 km), 36.5 GHz (14 × 8 km), and 89 GHz (7 × 4 km). These TB values are either horizontally (H) or vertically (V) polarized.
 There is a conflation between the frequency specifications of the SMMR, SSM/I, and AMSR-E gridded products, and the actual spatial measurements controlled by the instantaneous field of view of the respective sensors. However, NSIDC (National Snow and Ice Data Center) resamples all TB to the 25 km EASE-Grid. A good summary of these instruments can be found in Kelly et al. . Field measurements at large scales are logistically difficult and expensive, but there have been some intensive field studies conducted to validate remotely sensed SWE with observation such as the Cold Land Processes Experiment [Cline et al., 2003, 2009], the SnowSTAR2002 Transect [Shi et al., 2009], and others [e.g., Langlois et al., 2010; Yueh et al., 2009; Derksen, 2008; Derksen and Walker, 2003; Mote et al., 2003; Chang et al., 1991]. Using measurements from snow course transects in the former Soviet Union, Armstrong and Brodzik  found that nearly all retrieval algorithms (e.g., horizontal-based polarization algorithm of Chang et al.  and vertical-based counterpart of Goodison et al. ) of passive-microwave data tend to underestimate SWE, especially as the forest cover density begins to exceed 30%–40%. Takata et al.  assessed NSIDC's global monthly SWE data (a sample size of 11,730) over Russia and former Soviet Union with the independent International Association for the promotion of co-operation with scientists from New Independent States of former Soviet Union - Snow cover changes over Northern Eurasia (INTAS-SCCONE) snow course observations from the period of 1979–2000 for March only. On the basis of the snow course observations, they found a 12 mm bias in the NSIDC data, which means a bias of 10% or less if the mean SWE is 120 mm or higher.
 To analyze continental scale SWE trends of North America (NA), it will not be feasible to rely on limited observed SWE measurements such as snow course data, which are inadequate to provide the spatial coverage needed at such a scale of analysis. On the other hand, such a scale of analysis is possible with passive microwave SWE data albeit such data are subjected to uncertainties caused by spatiotemporal variations in snowpack properties [Brown and Mote, 2009; Brown and Robinson, 2011]. Over time, due to compaction and settling of snow particles partly due to increasing overburden load as new snowfall occurs, a snowpack will metamorphose from low-density fine grains to high-density coarse grains, isothermal snowpack with higher liquid permeability and thermal conductivity. This evolution of snow properties directly modifies TB, which poses a problem for SWE retrieved from microwave radiometers. Volumetric scattering is the dominant loss mechanism for microwave radiation greater than 15 GHz incident on a snowpack. In theory, the greater the depth of snowpack, the greater should be the volumetric scattering of the microwave radiation incident on the snowpack, and so, the measured TB should be lower. On the other hand, snowpack of coarser grains and higher density is also expected to incur more scattering of microwaves and therefore a lower TB, and different diurnal and seasonal metamorphism will also add variations to TB [Mätzler, 1987; Tait and Armstrong, 1996; Pulliainen, 2006; Pardẻ et al., 2007; Durand et al., 2008; Brown et al., 2010]. The presence of lakes (frozen or unfrozen) and heterogeneity caused by mountain terrains could further complicate TB. However, unless the snowpack is wet or the regions are dominated by thick vegetation which could complicate the microwave emissions of the snowpack, reasonably accurate SWE at regional scale can generally be estimated from passive microwave data [Mätzler, 1987; Singh and Gan, 2000; Gan et al., 2009; Pampaloni, 2004; Derksen and Walker, 2003; Kruopis et al., 1999; Ferrazzoli and Guerriero, 1996; Chang et al., 1996; Paloscia, 1995; Karam et al., 1992]. Sun et al.  had characterized the errors of SWE data, and Foster et al.  had attempted to improve their algorithm with these errors in mind. Even though SWE retrieved from TB data are subjected to uncertainties, e.g., some parts of the Canadian Arctic stretching from the western Hudson's Bay to the North Slope of Alaska show a persistent pattern of high SWE [Armstrong and Brodzik, 2002; Andreadis and Lettenmaier, 2006], it should generally be reasonable to use such data to analyze SWE trends at the continental scale of NA using a nonparametric approach unless such uncertainties vary significantly from year to year, which is unlikely.
 According to Lemke et al. , the observed Northern Hemisphere snow cover since the 1920s had remained quite steady until around the 1980s, since when a significant decrease in snow cover was observed. Various researchers detected substantial decreases in snow cover extent and snow depth over NA [Dyer and Mote, 2007] and the Northern Hemisphere [Brown, 2000; Frei and Robinson, 1999]. Dyer and Mote further demonstrated positive trends in the frequency of snow ablation events during March, but a significant negative trend in May, which again indicates an earlier onset of ablation. They attributed the March trend to higher fluxes of sensible heat into the snowpack caused by warmer air, an increase in the frequency of dry moderate air masses over central Canada, and a decrease in dry polar air masses. Hantel and Hirtl-Wielke  showed that in the Alpine region, snow cover duration could decrease by a maximum of 4 weeks per °C of warming in winter at an elevation of 700 m amsl. However, above the 700 m level, the reduction rate of snow cover duration will fall rapidly. On the basis of the observed 1 April SWE, Pierce et al.  found that about half of the SWE/P (P = water-year-to-date precipitation) reductions observed in the western United States from 1950 to 1999 are the result of climate changes forced by anthropogenic greenhouse gases, ozone, and aerosols. With 29 years of continuous SWE data from SMMR (late 1978–1987) and SSM/I (summer of 1987 until 2007), it is practical to perform a trend analysis of SWE over NA for the winter and spring seasons to detect possible changes.
 From the perspective of climatic change, relative to the 1906–1970 surface temperature of NA, both observed and 58 cases of GCMs' modeled data show drastic increase in the surface temperature of NA in recent decades [Forster et al., 2007]. At higher elevations/latitudes, where it is still sufficiently cold for snowfall to occur, higher temperatures may increase the amount of snowfall because of the nonlinear increase in saturation vapor pressure with temperature. However, in lower elevations/latitudes, a significant increase in the surface temperature probably means more rainfall at the expense of snowfall and hence less snow on the ground. With the above statement of problems, the research objectives are given in section 2, description of data in section 3, Kendall's trend analysis in section 4, results in section 5, influence of climate anomalies in section 6, SWE-climate data relationships in section 7, and conclusions in section 8.
2 Research Objectives
 With the background information given in section 1, this study has the following objectives:
 to analyze for monthly monotonic trends for the October–April SWE over NA using the 1979–2007 SWE data retrieved from the passive microwave, SMMR/SSMI brightness temperature data;
 to compute trend magnitudes, trend homogeneity, and the spatial distributions of trends and variability in SWE across NA from 1979 to 2007;
 to perform principal component analysis (PCA) on SWE, and to correlate PCAs of SWE with climate anomalies and to relate SWE to major climate variables;
 from all the results, identify factors responsible for the detected changes of the snowpacks of NA.
2.0.1 SWE Retrieved From SMMR and SSM/I Passive Microwave Data
 On the basis of volumetric scattering, the dominant loss mechanism for microwave radiation greater than 15 GHz incident on a snowpack, it is possible to empirically relate the brightness temperature (TB) of a certain frequency and polarization (H or V) to the SWE of the snowpacks [e.g., Armstrong et al., 2001; Gan et al., 2009]. The National Snow and Ice Data Center (NSIDC) at the University of Colorado Boulder provides such passive microwave TB data at Equal-Area Scalable Earth, or EASE Grid format. The 1979–1987 SMMR SWE data of NSIDC were retrieved from equation ((1)),
 Where TB18H and TB37H are the TB at 18 and 37 GHz, horizontally polarized, respectively. The 1987–2007 SSM/I SWE data of NSIDC were retrieved from equation ((2)) [Armstrong and Brodzik, 2002],
which is slightly different from that of equation ((1)). The daily SWE is adjusted for the surface forest cover using the Boston University-MODIS (Moderate Resolution Imaging Spectroradiometer) land cover data [NSIDC, 2005]
 Any pixels with forest percentages higher than 50% are set to the 50% threshold, so that the forest correction by equation ((3)) is limited to a maximum factor of 2. Further, to ensure snowpacks are detectable by passive microwave data, SWE less than 7.5 mm is considered unreliable and set to zero. Figure 1 shows the latitudinal distributions of SWE of NA for January–April, 1979–2007. Given that the SMMR and SSM/I data are at 25 × 25 km resolution, we expect the microscale spatial variability of snowpack to be mostly averaged out, and so we can mainly expect to analyze mesoscale to regional scale variability of snowpack from these data. For NA, the 5 month (December–April) running mean (solid curve) and the monthly (dotted curve) SWE anomalies based on SMMR and SSM/I data of 1979–2007 generally show negative anomalies since the late 1980s, even though for individual years SWE anomalies could either be negative or positive (Figure 2, top). From weekly snow cover area (SCA) maps produced primarily from daily visible satellite imagery of National Oceanic and Atmospheric Administration-advanced very high resolution radiometer (NOAA-AVHRR) at 1 km resolution by the Rutgers Global Snow Lab between November 1966 and December 2006, and using a 12 month running mean, these monthly SCA over NA, with the exception of 1997, mainly show negative snow cover anomalies since the late 1980s (Figure 2, bottom) [Robinson, 2008]. Despite the differences between the periods of NOAA-AVHRR and SSMR and SSM/I data, overall, both SCA and SWE of NA predominantly show negative anomalies since the late 1980s. More detailed spatial analysis of SWE of NA is given below.
 As explained in section 1, SWE retrieved from passive microwave data of 25 km resolution are subjected to errors, such as a possible mixture of deep snow on north facing slopes but almost snow free on south facing slopes particularly in mountainous areas with large topographic variability or mixed microwave emissions from trees, snow canopy, and ground surface in forested areas. Pixels of melting snow or wet snowpack could return low or zero SWE values [Matzler, 1994]. Singh and Gan  employed screening procedures to eliminate potentially erroneous SWE data retrieved from such pixels.
3 Non-Parametric, Man-Kendall's Test and Trend Magnitude (β)
 There are various parametric methods available to detect trends, e.g., the linear least squares [e.g., Dyer and Mote, 2007]. However, the nonparametric Mann-Kendall test [Kendall, 1975; Mann, 1945] for testing trends in seasonal time series should be more robust because it can handle nonnormality, censoring (data reported as values “less than”), missing values, and seasonality [Hirsch et al., 1982]. To safeguard against nonrepresentative results, only data sets with few missing values were tested. According to Berryman et al. , who gave a comprehensive review on the practical limits of nonparametric tests for monotonic trends, among 12 types of such tests, Kendall and Spearman's tests, have the highest asymptotic efficiency of 0.98. Kendall's statistic Sk for season k (where k = 1, 2, .… q) is given as
where n is the sample size, Sk tends to be Gaussian normal asymptotically, and
Xjk − Xjk
sgn(Xjk − Xjk
 Using a two-sided test at a significant level α, the trend is statistically significant at α if the absolute value of the standard normal variate, lZkl > Zα/2, where Zk is given as
and σk is the standard deviation of Sk. Details of Kendall's test are given in Gan  and Hirsh et al. . Since Sk is not based on the magnitudes of individual SWE, it is expected to be more robust than say, the simple linear regression. Results of this nonparametric approach should not be much affected by SWE retrieved from passive microwave data of two different sensors. There have been past studies conducted on the TB inhomogeneity between SMMR and SSM/I [Jezek et al., 1993; Derksen and Walker, 2003; Derksen et al., 2003]. To circumvent this problem of inhomogeneity between SMMR and SSM/I data, we did the trend analysis of SWE data for NA involving both SMMR (data before 1987) and SSM/I data (data between 1987 and 2007) using the nonparametric approach of Mann-Kendall, which does not depend on the underlying probability distribution of the SWE data, because it simply counts the recurrence frequency. Furthermore, equation ((4)) shows that Sk is not based on the magnitudes of individual SWE but on the number of times SWE has been increasing minus the number of times SWE has been decreasing as we move, for a particular month, of 1 year to the next until the study period (29 years) is completed. On the other hand, because the metamorphism of the snowpack could change from year to year, which means that grain size and other snowpack properties could also change from year to year, and since changes to TB also depend on snowpack properties (not just the SWE), the trend estimated by the nonparametric algorithm may still (or may not) be biased.
4 Discussions of Results
 The snow cover area (SCA) over NA has been shown to increase during autumn and early winter (November–January), but decrease over early spring [Frei and Robinson, 1999; Dyer and Mote, 2006], which likely indicates an earlier onset of the spring snowmelt, but which could also be partly caused by the declining SWE trends detected in December, January, and February (Figure 5). In terms of SWE, both statistically significant decreasing and increasing trends detected at α/2 = 0.05, α/2 = 0.025, and α/2 = 0.01 are shown in Figure 3, which also shows the approximate monthly snow cover areas for NA (light colors). These SCAs are likely more extensive than a typical year since they represent the areal extent of the snowpacks of NA averaged between 1979 and 2007. About 30% of detected decreasing trends in SWE for 1979–2007 are statistically significant at α/2 = 0.05, which is about three times the detected increasing trends in SWE (see Table 1 and Figure 4). Significant decreasing trends in SWE are more extensive in Canada than in the United States, where such decreasing trends are mainly found along the American Rockies (see Figure 3d that shows the elevation map of NA prepared from a 1 min arc digital elevation map and resampled to 25 km resolution). Scattered increasing trends are found mainly near Hudson Bay in Canada (see Figure 3e), but not near the Great Lakes where large increases in lake-effect snowfall since 1951 had been reported [Ellis and Johnson, 2004]. Figures 3a, 3b, 3c, 3f, and 3g show that in Canada, statistically significant decreasing trends are mainly detected east of the Canadian Rocky and within the Canadian boreal forest. Because a forest cover amplifies the snowfall/temperature variations on the snow cover (open versus forested areas), the contrast in SWE evolution between open and forested areas is a function of the magnitude of the differences between open and forest snow accumulation (i.e., input terms) versus differences between open and forest ablation (snowmelt and sublimation, i.e., loss terms) [Nakai et al., 1999]. Therefore, the pattern of SWE trends and variations detected within the Canadian boreal forest is likely linked to the effect of boreal forest cover.
Table 1. Results of Kendall's Nonparametric Test on Statistically Significant Increasing (+ to + ****) and Decreasing (− to −****) Trends of Monthly SWE for 1979–2007 for January–April and 1979–2006 for December at Different Significant Levels
α/2 < 0.95
α/2 > 0.95
α/2 < 0.975
α/2 < 0.99
α/2 < 0.995
Significant decreasing trends are about three or more times that of statistically significant increasing trends.
 Using the snow course data from the U.S. NRCS National Water and Climate Center, Pierce et al.  found that snowpack had generally declined across much of the western United States over 1950–1999. In our study, we did not detect much change to SWE along the coastal areas of NA partly because SWE data retrieved from passive microwave data suffer from mixing signals emitted from relatively dense vegetation along the coastal areas (Armstrong, personal communication). Even though few significant trends have been detected, we suspect that SWE along the western coastal areas of NA had also undergone decreasing trends, but we are not sure whether the decreasing trends are statistically significant or not. The SWE data along the American and Canadian Rockies (Figure 3e) could be affected by certain terrain characteristics of mountainous areas, such as the slope and aspects, but the net effects are likely not significant or more or less averaged out by the relatively coarse 25 × 25 km resolution of passive microwave data, and this is partly why significant trends in these areas are detected, but not along the western coastal areas of NA covered with relatively dense forest covers. However, regional studies suggested both changing river flows [Barnett et al., 2008] and losses of snowpack associated with warming trends in the western Unites States have been ongoing since the mid-20th century [Adam et al., 2009], which very likely reflect human-induced impact on the climate.
 Simple linear regressions fitted to the mean monthly SWE of NA show overall monthly decreasing trends in SWE ranging from about −0.4 (January) to −0.7 (March) mm/yr (Figure 5). In addition, a more rigorous approach to compute the trend magnitude βk in mm/yr was used. It was based on an estimator extended from that proposed by Sen  and defined as
where 1 < i < j < n.
 The overall mean trend magnitudes estimated using equation ((6)) are about −0.4 to −0.5 mm/yr, which are of smaller range than that of simple regressions (Figure 6), while the mean (median) of negative trends alone is about −1.3 (−1.0) mm/yr (Table 2). This means that the overall SWE of NA has decreased by about 12–15 mm or possibly more in 29 years, which if expressed in terms of snow depth will be about 5–8 cm (assuming an average snow density of about 200–250 kg/m3); this will have significant impacts on the spring snowmelt of some parts of NA such as the Canadian Prairies and the Washington Cascades. On the basis of negative trends alone, apparently the decrease in SWE mainly occurred in the area extending from the western part of the Canadian high Arctic and boreal forest to north of the five Great Lakes (January–April). There are also anomalous increasing trends in SWE as shown by βk (Table 2), which could be partly attributed to the effects of climatic anomalies discussed in section 6.
Table 2. Means and Medians of Trend Magnitudes (β in mm/yr) for North American SWE Between 1979 and 2007 for January–April and Between 1979 and 2006 for December
Overall Mean of Negative and Positive Trends
Mean of Negative Trends Only (mm/yr)
Median of Negative Trends Only
5 Influence of Climatic Anomalies on SWE
 Besides the possible impact of warming (section 7), climatic anomalies could also contribute to part of the detected changes, especially to increasing SWE trends, even though they occurred less frequently and were more isolated than decreasing trends. The possible effects of the Pacific Decadal Oscillation (PDO), Pacific/North American (PNA) teleconnection pattern, and the El Nino–Southern Oscillation (ENSO) are examined using the PDO, PNA, Southern Oscillation Index (SOI), and Nino3 indices (Tables 3-5) since these three climate anomalies have been shown to affect the precipitation and streamflow of western Canada [e.g., Gan et al., 2007; Gobena and Gan, 2006]. The PDO is represented by the leading principal component (PC1) of monthly Sea Surface Temperature (SST) anomalies in the North Pacific [Mantua and Hare, 2001]; Nino3 is a time series of monthly SST anomalies averaged over the window (5°N–5°S, 150°W–90°W) in the equatorial Pacific Ocean, Southern Oscillation Index (SOI) is a time series of the normalized monthly differences in sea level pressure at Tahiti (≈ 150°W, 18°S) and Darwin (≈ 130°E, 13°S); and the Pacific/North American (PNA) pattern represents a quadripole of 700 mb geopotential height anomalies, with opposite anomalies centered over the Aleutian Low and western Canada, and between the Hawaiian Islands and southeastern United States [Wallace and Gutzler, 1981].
Table 3. Pearson's Correlations (ρp) Between PC1 to PC4 With the Corresponding Monthly PDO (Columns 1 to 7), One-Season Lag, Seasonal PDO of OND (October–November–December), NDJ (November–December–January), DJF, and JFM
Numbers in bold mean statistically significant at α/2 = 0.05.
The ρp between PCs of October, November, and December and PDO-JAS (averaged of July–August–September JAS) are not shown because no statistically significant case is detected.
Table 4. Pearson's Correlations (ρp) and Spearman's Rank Correlation (ρs) Between PC1, PC2, PC3, and PC4 With the Corresponding Monthly PNA, Nino3, and SOI, Showing Less Consistent Influence of PNA Than PDO Over the SWE of North America in All Four PCs
Numbers in bold mean statistically significant at α/2 = 0.05.
The ρp between PCs of October, November, and December and PDO-JAS (averaged of July–August–September JAS) are not shown because no statistically significant case is detected.
Table 5. Monthly Pearson's Correlation Coefficients Between PDO, PNA, Nino3, and SOI
Numbers in bold mean statistically significant at α/2 = 0.05.
The ρp between PCs of October, November, and December and PDO-JAS (averaged of July–August–September JAS) are not shown because no statistically significant case is detected.
 To reduce the dimensionality and capture any dominant signals in the SWE data, a principal component analysis (PCA) was conducted. Scree plots (figure not shown) of the January–April 1979–2007 SWE show that in most months, about 65%–70% of the total variance is explained by the first four PCs. The PCA for March is based on over 9000 pixels for March, when the snowpack detectable by the passive microwave sensors extended to the upper states of U.S.
 Pearson correlation coefficients (ρp) of Table 3 essentially show statistically significant ρp between monthly PC1 of NA's SWE and monthly (columns 1–7) or seasonal [October–November–December (OND) to January–February–March (JFM)] PDO (columns 8–15), while PC2, PC3, and PC4 of NA's SWE do not have significant ρp with PDO (see Figure 3 for the window of data analyzed). No significant ρp was found between PDO-JAS and SWE of October–December, which means that winter to early spring SWE are more associated to PDO than the autumn SWE, and their relationships can vary widely from almost nil, e.g., ρp of PC1-PDO for December is −0.23 to fairly significant, e.g., ρp of PC1-December–January–February (DJF) for April is 0.52. The characteristics between PC1 and PDO for April are quite similar to each other (Figure 7), such that the PC1 score and PDO are positive until about 1988, after which the values are mostly negative. Because there are even less statistically significant Spearman's statistics ρs than statistically significant ρp between the first four PCs of NA's SWE and PDO, the corresponding statistics ρs are not shown. Apparently, PDO exerts some influence on the overall snowpack SWE of NA during winter months of February–April, but the effect is not consistent nor extensive and likely quite unpredictable. Although we have analyzed 29 years of snow data, which should provide some basis to relate changes to SWE to PDO, given the interdecadal feature of PDO, it will be better if longer snow data set are available to analyze their relationship.
 Compared to PDO, the influence of PNA is relatively modest with significant ρs and ρp in February (PC2) and March (PC3 and PC4) only, while ENSO represented by Nino3 and SOI exhibits only marginal influence over the snowpacks of NA (only March and April correlations are shown), e.g., no statistically significant ρp or ρs (Table 5), even though Hsieh and Tang  found ENSO to have affected the snowpack of British Columbia, Canada, the Canadian cryosphere had responded to the anomalous warm summer of 1998 [Atkinson et al., 2006], and the impact of ENSO on the precipitation of western NA has been established [Gan et al., 2007; Cayan et al., 1999; Shabbar et al., 1997; Ropelewski and Halpert, 1986], and Nino3 and SOI are found to be significantly related to PDO and PNA in some months (Table 5). Monthly PDO is most significantly related to the corresponding monthly PNA, except for October and December. PDO is more consistently related to Nino3 and SOI than PNA.
Sobolowski and Frei  found significant Spearman's ranked correlation between preceding seasonal (JAS or OND) ENSO, NAO, and PNA (PDO) indices with the PC1 (PC3) of North American winter SWE (JFM) of window (35°N–55°N)(130°W–50°W) over the 1980–1997 period. The PC1 relationship is located mainly in the north (south) central to western regions of USA (Canada), while the PC3 relationship stretches from the midwest to eastern USA and Atlantic Canada, e.g., these relationships are strongest at large spatial scales. They suggested that these relationships could vary over time, but there is not enough SWE data to explore how such relationships could change with time.
 To gain better insight into the influence of ENSO (in terms of Nino3 and SOI), PDO, and PNA on the snowpacks of North America, the first four principal components (PC1 to PC4) of monthly SWE are compared with these climate anomalies (Figure 7). The influence of PDO (see also Figure 8) is approximately represented by PC1 for November and February to April. Except for April, PC1 is mainly negative until about 1988 and is mainly positive thereafter, and it likely describes SWE variations of low frequency. From Figure 7g, it seems that PDO is more correlated to PC1 until 1988, and PDO's influence became fuzzy, but on a whole, its influence centered on February–April. PC2 to PC4 exhibit interannual fluctuations between positive and negative scores, and the characteristics vary from month to month. Furthermore, PC2 of November and April are of opposite phase to that of December–March. The relationships between the three PCs and PNA are relatively scattered and mostly not statistically significant.
 To gain some understanding on the spatial relationships between climate anomalies and SWE, some selected cases of the spatial correlation fields between PC1, PDO, PNA, and monthly SWE of individual pixels for February, March, and April are examined (Figure 9). They show a wide range of negative and positive relationships that are mainly centered along the northern parts of Canada above the Great Lakes on the east to Alaska of the United States on the west and in the midwest near the American Rockies. Apparently, PDO and PNA exerted some influence on the snowpacks of southern Canada and the United States for most of these selected cases exhibit either statistically significant Pearson (ρp) or Spearman rank (ρs) correlations, given in Tables 3 and 4. Given PDO is of interdecadal time scale, and with only 29 years of data analyzed, our understanding of PDO's influence on the snowpack of NA is somewhat limited.
6 SWE-Temperature and SWE-Precipitation Relationships
 Higher air temperatures in the last three decades in NA could be a major contributor to the fairly wide spread reduction in its snowpack, since warmer temperature contribute to melting or sublimation, and the ratio of rainfall over snowfall, especially when air temperature > 0°C. Similarly, possible changes to precipitation (e.g., snowfall) could be partly responsible for the detected changes in SWE. For example, annual precipitation for latitudes north of 50°N had been observed to increase by about 4% in the last five decades, though this increase varies spatially and temporally [Groisman et al., 2005]. In lower latitude regions, not much change in the annual precipitation had been observed, but less snowfall and more rainfall had been observed in some places, e.g., the lower Missouri River Basin [Berger et al., 2002] and in New England [Huntington et al., 2004].
 To find possible climatic factors leading to a general reduction in the snowpacks of NA especially in Canada and along the American Rockies in the west, a feasible approach is to correlate changes of SWE to gridded precipitation and air temperature data to explore the possible relationships between climate (temperature and precipitation) and SWE. Trend analysis of both the gridded, 2 m air temperature data of the University of Delaware (figure not shown) and that of the North American Regional Reanalysis (NARR) showed little agreement between areas of detected increasing temperature trends (Figure 10a) and that of decreasing SWE trends that are statistically significant based on passive microwave data (Figure 3). However, extensive areas of significant negative correlations between SWE and temperature exist both across the United States and Canada except in January, and the distribution of these areas of negative correlation closely follow the areas of the decreasing trends detected from the SWE data, which for Canada is mainly east of the Canadian Rocky while for the United States is mainly on the American Rocky (compare Figures 3d and 10c).
 There is limited agreement between areas of detected decreasing precipitation trends based on gridded data of the University of Delaware (figure not shown) and NARR (Figure 10b) and decreasing SWE trends that are statistically significant. Unlike SWE trends, only scattered significant precipitation trends are detected, which implies that extensive decreasing SWE trends detected are more likely caused by warming than by decreasing precipitation, which likely played a relatively minor role compared to temperature. There is more significant negative correlation (at 1% significant level) between NARR precipitation and SWE (Figure 10d, red color) than the decreasing trend of precipitation itself (Figure 10b). Unlike temperature and SWE data, there is less agreement between the areas of significant negative correlation of precipitation with SWE and areas of decreasing SWE trends that are significant.
 Even though climatic anomalies, such as PDO, could contribute to part of the detected changes, it seems that extensive decreasing trends in SWE detected in Canada and parts of United States are caused more by increasing temperatures than by decreasing precipitation, for rising temperatures have generally resulted in rain rather than snow especially in areas and seasons where the average air temperature is close to 0°C. Through three spring indicators—lilacs, honeysuckles, and streamflow, Cayan et al.  found earlier onset of the spring season by up to 3 weeks in the western North America since the 1970s. By simulating the snow energy balance to climatic changes projected by nine regional climate models to the end of the 21st Century in the Pyrenees, Moreno et al.  concluded that the most significant changes to future snowpack processes are related to temperature. The aforementioned observations, and the global scale decline of snow and ice cover since 1980 and increasing rate of decline during the past decade, as noted by Lemke et al.  in their International Panel on Climate Change 2007 report, likely support our conclusion that warmer temperature caused the decline in SWE.
7 Summary and Conclusions
 Since the late 1980s, both the snow water equivalent (SWE) retrieved from 1979 to 2007 SMMR and SSM/I Passive Microwave data and snow cover area (SCA) retrieved from 1966 to 2006 NOAA-AVHRR data of North America (NA) show negative anomalies. On the basis of the nonparametric Kendall's test applied to the SWE data of SMMR and SSM/I for NA, decreasing trends have been detected. About 30% of the detected decreasing trends in SWE for 1979–2007 are statistically significant at α/2 = 0.05, which is about three times that of detected increasing trends in SWE. Significant decreasing trends in SWE are more extensive in northern parts of Canada than in the United States, where such decreasing trends are mainly found along the American Rockies.
 The overall mean trend magnitudes are about −0.4 to −0.5 mm/yr and −0.4 to −0.5 mm/yr, which means an overall reduction of snow depth of about 5–8 cm in 29 years (assuming the average snowpack density was between 200 and 250 kg/m3), which can have significant impacts on regions relying on spring snowmelt for water supply. For the mean March SWE values of NSIDC for Northern Hemisphere, Takata et al.  found the snow mass of NH to have decreased by about 7% between 1982 and 2009, which is similar (or slightly lower) to a mean decreasing trend of about 0.5 mm of SWE/yr found in this study (about 15 mm over 30 years), given the mean SWE of the snow dominated part of NA (Canada and upper states of the United States) is about 150–200 mm (Figure 1). Similarly, Brown and Robinson , who analyzed changes to the snow cover extent (SCE) of Northern Hemisphere (NH) spring (March, April) over the 1922–2010 period, show that NH spring snow cover extent has undergone significant reductions over the past ~90 years and that the rate of decrease has accelerated over the past 40 years. The rate of decrease in March and April NH SCE over the 1970–2010 period had been 7% and 11% decrease in NH March and April SCE, respectively, from pre-1970 values. They also said that the observed trends in SCE are being mainly driven by warmer air temperatures.
 The PC1 of NA's SWE are found to be significantly correlated to the Pacific Decadal Oscillation (PDO) index, marginally correlated to the Pacific North American (PNA) pattern and to El Niño–Southern Oscillation (ENSO). The PDO index for February to April was significantly correlated to the SWE of Canada, and the SWE PC1 scores and PDO indices show quite similar variation, such that the PC1 score and PDO are positive until about 1988, after which the values are mostly negative.
 Lastly, from the detected significant increasing (decreasing) trends of gridded temperature (precipitation) data of NARR and the University of Delaware for NA, and their respective correlations with SWE data, and related findings such as global scale decline of snow cover and warming temperature trends, longer rainfall seasons, etc., it seems that the extensive decreasing trends in SWE detected in Canada mainly east of the Canadian Rocky are more attributed to increasing temperatures than to decreasing precipitation, even though climate anomalies could also play a minor role on part of the detected trends.
8 Recommendations for Future Research
1.Several possible future studies that will contribute to passive microwave snow research are as follows: Derksen and Walker  found good agreements between in situ snow measurements and SWE derived from the Meteorological Service Canada (MSC) land cover sensitive algorithms of the form, SWE = K1 + K2(TB37V − TB19V), applied to vertically polarized SSM/I data of 37 and 19 GHz for northern boreal forests of northern Manitoba. The values of K1 and K2 depend on the land use, such as open, coniferous, deciduous, and sparse forests. They found that the retrieved SWE pattern agree with SWE reanalysis data set of Brown et al.  and SWE simulated by the Canadian Regional Climate Model [MacKay et al., 2003]. However, they found consistently lower SWE retrieved from SSM/I data compared to in situ measurements across denser boreal forest to the south and open tundra to the north, and the absolute SWE differences between them increase with larger mean snow depths. They also found a wide range of SWE values measured on the ground in each SSM/I pixel, but noted that a high percentage of SSM/I SWE agree within ±20mm of the median in situ measurements, even though such observations are somewhat limited by the large-scale mismatch between ground measurements and large SSM/I pixels (625 km2 in area). It will be interesting to do a similar trend analysis using the SWE data of Meteorological Service Canada (MSC), given that the SWE data of NSIDC are based on horizontally polarized, while SWE of MSC are based on vertically polarized passive microwave data and different retrieving algorithms, and possibly the latest database of Foster et al. .
2.Using AMSR-E and radiosonde data sets over two widely separated regions in the continental United States, Wang and Tedesco  found that even under a clear sky, the atmospheric absorption on passive microwave measurements near 19 and 37 GHz could account for as much as 25%–50% to the estimation of SWE. Conversely, an AMSR-E estimated SWE of 10 cm could turn out to be about 14 cm when measured at the ground level under the same atmospheric condition. It will be interesting to examine atmospheric contributions at regional scale to passive microwave TB. Atmospheric correction has been a standard correction procedure applied to optical sensor received multispectral data to selectively filter out scattered radiation to a target area. The atmospherically scattered radiance due to air particles can be estimated with atmospheric modeling programs like MODTRAN 3.5 or MODTRAN 4 [Kneizys et al., 1988, 1996] even though MODTRAN is seldom used in microwave data analysis (http://research.metoffice.gov.uk/research/interproj/nwpsaf/rtm/rttov_description.html). Moreover, as far as we know, there have only been limited examples of atmospheric corrections applied to passive microwave data partly because a general lack of appropriate observed climate data such as surface emissivity required to atmospherically correct microwave data of coarse resolution (25 km). For example, Tedesco and Wang  accounted for atmospheric effects to TB of AMSR-E at 18.7 and 36.5 GHz by means of a simplified radiative transfer model whose parameters were estimated using radiosonde data collected from several climate stations. They combined the surface emissivity estimated by the model and the surface temperature to compute TB. By comparing with SCA derived from MODIS data, they only found a marginal improvement to the SCA retrieved from corrected TB over the uncorrected TB by about 7%. Moreover, they only demonstrated how atmospheric correction could improve the retrieval of SCA, not SWE from TB data. Lately, with the availability of reanalysis data, such as NARR-ERA, and models, such as, Helsinki University of Technology (HUT), it may be possible to apply atmospheric corrections to SSM/I microwave data even though the return may only be marginal partly because of the coarse resolution of such data.
3.The simple equations ((2)) and ((3)), that likely do not provide proper assessment of vegetation contribution to the TB signal, can be improved by estimating the impact of stem volumes of coniferous and deciduous forests on TB signal [Karam et al., 1992; Mätzler, 1994; Paloscia, 1995; Ferrazzoli and Guerriero, 1996; Kruopis et al., 1999; Pampaloni, 2004]. A possible equation to account for the contribution of vegetation to TB signal is as follows:
where F is the fraction of forest cover, and τveg,p is the transmissivity of vegetation at frequency p.
4.Lastly, effects of snowpack geophysical properties such as grain size and correlation length on snow microwave emissions in terms of TB at 19 and 37 GHz for different classes of snow (Prairie, tundra, taiga, alpine, and maritime) can be simulated using electromagnetic models such as the Helsinki University of Technology (HUT) snow emission model, the microwave emission model of layered snowpacks, a dense-medium radiative-transfer theory model, and a strong fluctuation theory model. Such experiments can help us to understand interactions between electromagnetic waves and snow media and can benefit remote sensing of snow.