not included in the statistical analyses).
-30
-20
Teo = -a/b = –5.7 ºC
slope = b/4 = –0.06 ºC-1
-10
degrees C
0
10
0
10
20
30
Relation between SNOW vs NO-SNOW and Tmin
Teo = -a/b = +6.5 ºC
slope = b/4 = –0.07 ºC-1
without overplotting (strictly as a graphic device,
more of the 3377 daily observations to be visible
been perturbed randomly in the vertical to allow
Observations of snow (=1) vs no-snow (=0) have
-10
0
10
20
Relation between SNOW vs NO-SNOW and Tmax
windows.
no-snow (=0) in 100-observation
Percentages of snow (=1) vs
Logistic regression
Jittered observations
Percentage with snow
40
30
The slope of the logistic relation at Teo which indicates how abrupt is
the transition from snowfall to no-snow (with temperature) [ = b/4]
The even-odds temperature (Teo) at which temperature half of the wet
days yield snowfall and half yield only rain [ = -a/b]
These figures show that not all wet days with temperatures below freezing
yield snowfall and not all wet days with temperatures (even Tmins) above
freezing yield no-snow. Very warm wet days almost always yield no snow
(probability of snowfall is near zero); very cold wet days almost always yield
snowfall (probability of snowfall is near one). At intermediate temperatures,
probabilities range from zero to one in ways that are reasonably well captured
by the fitted logistic relations.
Two important attributes of the temperature dependence of snowfall
occurrence emerge immediately from the fitted logistic relations:
0
0.2
0.4
0.6
0.8
1
-10
0
0.2
0.4
0.6
0.8
1
-20
0
0.2
0.4
0.6
0.8
1
COOP Station: Fort Valley, AZ (111.73W 35.27N 2240m)
Relation between SNOW(=1) vs NO-SNOW (=0) and Tavg
This distribution is the logistic relation and looks like the blue,
reversed-S-shaped curves in the figures below.
-1
Probability of snowfall = [ 1 + exp( -Lo) ] , where Lo = a + b * Temperature
Logistic regression is a maximum-likelihood method for estimating the
dependence of the probability of a binomial (”heads or trails”) random
variable upon one or more continuous independent variables (Pampel,
2000). If the occurrence of snowfall (vs no snow) on a given wet day is
considered to be random, with odds of snowfall that are dependent on the
temperature that day, then an application of logistic regression allows us to
estimate the probability of snow according to the following distribution:
LOGISTIC REGRESSION
-8
-6
-4
120ºW 0
2
DEGREES C
-2
4
6
8
105ºW -8
-6
-4
120ºW 0
2
DEGREES C
-2
4
6
8
105ºW -8
-6
-4
120ºW -2
0
2
DEGREES C
4
6
8
105ºW 30ºN 3000
4000
0
5
10
0
1000
2000
Altitude, in meters
3000
4000
West of 114W
East of 114W
0
1000
2000
Altitude, in meters
3000
4000
West of 114W
East of 114W
Warmer Teo to the east and at higher altitudes
in the far West.
Daily-mean temperatures are typically near or
above freezing at even odds of snowfall.
-2
0
2
4
6
AVERAGE TEMPERATURES AT WHICH
THERE ARE EVEN ODDS OF SNOW
Daily TAVG’s
Maximum temperatures are typically well above
freezing at even odds of snowfall, except
in PNW.
A strong (and poorly understood) increase in
Teo with altitude and to the east.
-0.15
-0.10
-0.05
Less Abrupt
0.00
105ºW del_PROBABILITIES / DEGREES C
-0.20
More Abrupt
-0.25
120ºW 0
1000
2000
Altitude, in meters
3000
4000
West of 114W
East of 114W
Similar patterns for TMAXs and TAVGs.
In Far West, transition is more abrupt
at lower altitudes. (Little altitude
dependence in Rockies.)
A more abrupt transition from always
snowfall to always rainfall in the Northwest than in most of Rockies.
-1
-0.8
-0.6
-0.4
-0.2
SENSITIVITY OF PROBABILITY OF SNOW
TO MINIMUM TEMPERATURE
Corresponding slopes (b/4) of logistic relations are shown in the following map.
The more negative the slope, the more abrupt (reliable) is the probability
transition from favoring snowfall to favoring rainfall near Teo.
SENSITIVITY OF ODDS (near Teo) TO
MINIMUM-TEMPERATURE DIFFERENCES
45ºN 2000
Altitude, in meters
HOW ABRUPT IS TRANSITION FROM
SNOWFALL TO RAINFALL WITH WARMING?
30ºN 45ºN 1000
MAXIMUM TEMPERATURES AT WHICH
THERE ARE EVEN ODDS OF SNOW
Daily TMAX’s:
DAILY-AVERAGE TEMPERATURES
WITH EVEN ODDS OF SNOWFALL
30ºN 45ºN 0
West of 114W
East of 114W
No clear dependence on geography or
altitude.
DAILY-MAXIMUM TEMPERATURES
WITH EVEN ODDS OF SNOWFALL
30ºN -8
-6
-4
-2
0
MINIMUM TEMPERATURES AT WHICH
THERE ARE EVEN ODDS OF SNOW
Minimum temperatures must be below
freezing to obtain even odds of snowfall.
degrees C
degrees C
degrees C
45ºN DAILY-MINIMUM TEMPERATURES
WITH EVEN ODDS OF SNOWFALL
30ºN 45ºN -0.6
-0.4
120ºW 0.0
0.2
Correlation with Year
-0.2
0.4
0.6
105ºW TRENDS IN FRACTION OF WET DAYS
WITH TMIN > LOCAL Teo, 1949-2001
-0.4
-0.2
0
0.2
Correlation with Year
86% of sites
have increasing
trends
0.4
0.6
Statistically
significant
trends
At a large majority of sites,
wet days warmer than Teo
have become more common.
1960
1970
1980
Water Year
1990
2000
Overall, the annual fraction of
wet days warmer than local
Teo’s has increased by +6%
during the past 53 years (r =
0.10; p << 0.01).
-0.6
-0.4
120ºW 0.0
0.2
Correlation with Year
-0.2
0.4
0.6
105ºW 25
30
0
5
10
15
20
-0.4
-0.2
0
0.2
Correlation with Year
0.4
83% of sites have warmed
0.6
Distribution of WET-DAY TMIN Trends
Cayan, D.R., Kammerdiener, S., Dettinger, M.D., Caprio, J.M., and Peterson, D.H., 2001, Changes in the
onset of spring in the western United States: Bulletin, American Meteorological Society, 82,
399-415.
Knowles, N., Dettinger, M., and Cayan, D.R., 2004, Trends in snowfall versus rainfall for the Western
United States: American Geophysical Union Fall 2004 Meeting, San Francisco.
Pampel, F.C., 2000, Logistic regression--A primer: SAGE Publications, 96 p.
For more information:
30ºN 45ºN TRENDS IN WET-DAY TMINS, 1949-2001
These trends are in agreement with trends in water-year average wet-day
minimum temperatures, both geographically and in terms of their (relative)
magnitudes (see map and histogram below).
0
1950
0.2
0.4
0.6
0.8
1
Fraction of Wet Days with Tmin > Teo
Compiling the numbers of wet days with temperatures warmer than Teo each
water year (October-September) from all stations, we obtain the following time
series.
0
10
20
30
Distribution of Trends in Fraction of Wet Days > Teo
Thus wet-day minimum temperatures have warmed across most of the region
and this trend has translated into widespread increases in the fraction of wet
days with better than even odds of yielding NO SNOWFALL.
Fraction of Wet Days
-30
-20
-10
0
TMIN, in degrees C
1949
Intervening yrs
2001
% snow in 20-obs
windows, 1949
% snow in 20-obs
windows, 1996
10
20
35 40 125ºW 115 110 105 DEGREES C
-4.0 -2.0 -1.0 -0.5 0.0 0.5 1.0 2.0 4.0
120 100 Same general pattern
obtained from OND, DJF,
or FMA subsets (not shown)
Same general pattern
obtained using TMAX and
TAVG (not shown)
In Rockies, warmer temps
now more likely to yield
snowfall than in the past!
In the Northwest, cooler
temps now are required
for snowfall!
–1.2ºC
125W
0
+1.2ºC
100W
Potential Temperature Changes
–8 %
125W
+8 %
850mb
500mb
300mb
30ºN 45ºN -0.6
-0.4
120ºW -0.2
0.0
0.2
Correlation with Year
0.4
0.6
105ºW TRENDS IN FRACTION OF WET DAYS
WITH TMIN > TIME-VARYING LOCAL Teo, 1949-2001
0
100W
Relative Humidity Changes
Now that we know how Teo has trended during
the study period, we can improve the figure at top
of 3rd panel (slightly) by showing mapping trends
in the fraction of wet days with Tmins > the
time-varying Teo’s. The pattern of trends is
generally the same as detected earlier. Thus, wet
days warm enough to strongly favor rainfall
rather than snowfall have become more
common across the West, even while the
temperature dependence of precipitation form
has changed.
850mb
500mb
300mb
(NCAR/NCEP REANALYSIS 1, averaged from 35N to 45N)
1976–2001 minus 1949-1974
SOME HYPOTHESES RE: TIME-VARYING DEPENDENCES: In the Far West, do
these changes reflect warming & drying that has been greater aloft than
near surface? The changes shown below could stabilize the atmosphere &
yield shallower warmer precipitation and would mean that a given surface
temperature is (on average) associated with warmer temperatures aloft
recently. Over the Rockies, have these trends destabilized the atmosphere,
facilitating deep clouds and snowfall at warmer station temperatures?
30 NO SIGNIFICANT CHANGES
COOLER CONDITIONS
REQUIRED FOR SNOWFALL
RECENTLY
SNOW FALLS AT WARMER
TEMPERATURES RECENTLY
50 CHANGES IN Teo DURING THE 1949 - 2001 PERIOD
Temperature dependence of snowfall occurrence has changed significantly
(at 95% level) at 62% of western coop stations in recent decades. A simple
measure of these changes is the difference in Teo from beginning to end of the
study period, which is given by ∆Teo = –53 yrs * c / b.
0
-40
0.2
0.4
0.6
0.8
COOP Station: Fort Valley, AZ (111.73W 35.27N 2240m)
Relation between Snow-vs-NoSnow Probabilities and Tmin,Year
1
long-term changes in the effects of temperature on the odds of snowfall can be
detected and mapped. Fitting the relation above to snowfall reports and
daily-minimum temperatures, long-term changes in probabilities of snowfall like
those shown below have been detected at many coop stations. At the site
shown, a given minimum temperature now yields snow less often than in the
past.
Probability of snow = [ 1 + exp( -Lo) ] -1 ,
Lo = a + b * Temperature + c * WaterYear,
To interpret the impact of these temperature changes on precipitation form
(snowfall vs rainfall), we also need to know whether the dependence of
precipitation form on temperature has changed during the decades of warming.
By extending the logistic regression analysis to fit the following distribution
A number of studies have shown that winter-spring temperatures have warmed
by about +2ºC across the western US since the 1950s (e.g. Cayan et al., 2001). In
order for such trends to significantly affect the occurrences (and amounts) of
snowfall (vs rainfall), warming would have to have raised temperatures to or
above the snow-rain transition (Teo) during more wet days recently. If warming
was not sufficient to raise temperature above the transitions, or if iwarming was
on dry days, then snowfall-vs-rainfall mixes might not have changed.
The map below shows linear trends in the fraction of wet days with minimum
temperatures warmer than each station’s even-odds temperature (Teo).
Performing similar logistic regressions on the long-term daily weather records
from 172 western coop stations yields the following maps of the long-term
even-odds temperatures (Teo) across the western US.
Precipitation can be either a boon to western water, flood, and land
management or a flood-generating disaster, depending on whether it falls as
rain or snow. Long-term warming trends across the western US have already
resulted in changes in snow-to-rain ratios (Knowles et al, 2004), snowpack
amounts, snowmelt timing, and streamflow timing (e.g., Cayan et al., 2001). In
order to determine the relative contributions to streamflow-timing trends of
snow-to-rain changes vs. snowmelt-timing changes, we will need to better
understand conditions under which snowfall occurs and will need to determine
whether the relations between temperature and snowfall have changed. Coop
weather stations routinely report the occurrence of snowfall and, although
these reports can be quantitatively suspect, the occurrence of snow (vs no
snow) is readily discerned. Logistic regression of daily snowfall reports from 172
coop weather stations over a 53-yr period with the corresponding temperatures
provides an opportunity to explore the geography and history of the
dependence of snowfall on local temperatures.
del_probability/degree C
Percent of Sites
Daily TMIN’s:
TRENDS IN TEMPERATURE DEPENDENCE
TRENDS IN TEMPERATURES RELATIVE TO Teo
HOW COLD MUST IT BE FOR SNOWFALL?
MOTIVATION
NOAH KNOWLES, U.S. Geological Survey, Menlo Park, CA; & DAN CAYAN, USGS, Scripps Institution of Oceanography
TRENDS IN THE TEMPERATURE DEPENDENCE OF SNOW- VS RAINFALL
IN THE WESTERN US, 1949-2001 MICHAEL DETTINGER, U.S. Geological Survey, Scripps Institution of Oceanography, La Jolla, CA;
Percent of Sites
Probability of Snowfall