© Copyright 2002, American Meteorological Society

**Is the January Thaw a Statistical Phantom?**

Daniel S. Wilks

Cornell University

Ithaca, New York

David
M. Schultz

*NOAA/National Severe Storms
Laboratory, Norman, Oklahoma*

Submitted to the *Bulletin of the American Meteorological Society*

12 June 2000

Revised 5 January 2001

Accepted September 2001

To be published in the January 2002 issue

DAVID SCHULTZ'S JANUARY THAW PAGE

Meteorological conditions that tend to occur on or near a specific calendar date more frequently than chance might suggest have been termed singularities (e.g. Huschke 1959) or calendaricities, a term introduced by Brier et al. (1963) to avoid confusion with the mathematical usage of 'singularities.' These features are commonly identified as consistently observed warm or cold departures from the annual march of temperature, or as wet or dry departures from some background annual precipitation trend.

Historically, the most studied singularity in European weather lore is an alleged cold period during 11-14 May, popularly known as the Ice Saints, named after the last killing frosts that seemingly tend to occur on days dedicated to Saints Mamertus, Pancras, Servatius, and Boniface on the ecclesiastical calendar (e.g., Buchan 1867; Talman 1919; Huschke 1959). Other hypothesized singularities have been discussed by Talman (1919) in a detailed bibliography of 144 worldwide publications from 1820 to 1917.

In the United States, a well-known example of a precipitation singularity is the onset of the monsoon in the southwest in early July (e.g., Bryson and Lowry 1955). The most widely recognized temperature singularity is the January thaw, a purported anomalous warming in the northeastern United States around 20-24 January. The January thaw may be particularly notable in weather folklore and the public imagination because it occurs around the time of expected minimum annual temperature and brings relief from below-freezing temperatures. It has been hypothesized to be associated with anomalies in other parameters such as sea-level pressure (e.g., Wahl 1952; Brier 1954; Lanzante and Harnack 1982; Kalnicky 1987), zonal and meridional indices (e.g., Brier 1954; Duquet 1963), upper-level geopotential height (e.g., Dickson 1959; Duquet 1963; Lanzante 1983), tornadoes (Dickson 1959), and ocean-wave directions on the East Coast of the United States (Hayden 1976). Table 1 summarizes previous studies of the January thaw, including the geographical coverage, the period of climatological record, meteorological variables used in the analyses, and the authors' conclusions.

Whether particular singularities such as the January thaw are dynamically important, statistically significant features of the atmosphere, or mere weather folklore, seems to excite perpetual controversy among forecasters and researchers. One argument against the existence of singularities is the absence of a viable physical mechanism for these phenomena. Indeed, singularities in general, and the January thaw in particular, could be given much more credence if a physical mechanism were determined. Explanations that have been offered for why departures of temperature or precipitation from a smooth annual cycle might occur include: meteor showers (e.g., Bowen 1956), sunspots (Newman 1965), melting snow and ice cover (e.g., Talman 1919, p. 556; Ruschy et al. 1991), atmospheric coupling to sea-surface temperatures in the Pacific Ocean (e.g., Lanzante 1983), and a relationship to the planetary-scale flow patterns such as the semipermanent centers of action (e.g., Talman 1919, p. 556). Unfortunately, few of these papers can confidently demonstrate the physical connection to the purported singularities.

Greely (1888, 117-119), Nunn (1927), Wahl (1952), and Lanzante and Harnack(1982) found that the warmth of the January thaw in the northeast United States was associated with southerly flow from a midlatitude cyclone tracking over the northern states. The January thaw was terminated by a shift to colder, northwesterly flow over New England associated with a continental anticyclone and an offshore trough (Wahl 1952). Aloft, Dickson's (1959) composite 500-hPa maps and Lanzante's (1983) composite 700-hPa maps are consistent with the results for the surface. Dickson's (1959) 500-hPa maps showed southwesterly flow over the eastern United States during the January thaw becoming more zonal after the January thaw, associated with rising heights centered over Louisiana and falling heights centered over Maine. Lanzante's (1983) 700-hPa maps showed the passage of a trough over the midwest United States (associated with the surface pressure trough) during the time of the January thaw and the arrival of a ridge from over Alaska into the western and central United States (associated with the surface anticyclone) that terminated the January thaw. Frederick (1966) provided further support for this evolution by showing the eastward progression of the warm spell across the United States (from 7-10 January in the Pacific Northwest to 19 January in Florida), suggesting that it may be related to eastward-moving offshoots of the Aleutian low, consistent with Duquet's (1963) analysis.

A second argument against singularities was raised by Marvin (1919) who claimed that since the annual march of surface temperature should be dominated primarily by incoming solar radiation, which is a smooth annual curve, the annual march of surface temperature should also be a smooth annual curve (i.e., harmonics beyond second-order, annual and semiannual, should be small). If higher-order harmonics were large, a preferred periodicity to singularities within the year may be indicated. At 24 stations across the United States, Marvin found that the first two harmonics of the weekly mean temperature were generally adequate to represent the total time series. Oftentimes, the residual temperature trace indicated 13-14 features that Marvin claimed may have been singularities, but these residual anomalies were not explained easily by higher-order harmonics. Guttman and Plantico (1989) also tested this hypothesis with 16 eastern U.S. stations and found similar results. Therefore, singularities, if they exist, are not likely to be explained by a particular harmonic of the annual cycle; instead, they are likely to be isolated anomalies apart from the annual cycle.

A third argument is that the apparent temperature anomalies are a result of too short a data record (i.e., given enough data, these anomalies would be smoothed out). Using tests on the daily temperatures at Boston, Wahl (1952) and Newman (1965) were able to show at the 93% and 99% confidence levels, respectively, the existence of above-normal temperatures around the time of the January thaw, although Newman (1965) dismissed the likelihood of a singularity because the probability of obtaining this result by chance was high. Guttman (1991) found a significant warming trend on 25 January (and a significant cooling trend on 8 January) at Central Park, New York at the 95% level. Finally, Guttman and Plantico (1987) examined 74 stations across the eastern United States and found, at the 95% confidence level, a significant warm period in New England during 22-27 January and in a band from Michigan to Virginia during 20-26 January and a significant cold period in New England during 30 January-2 February, results consistent with earlier studies that failed to perform tests of statistical significance.

Consequently, without an identifiable physical basis for presumed climatic singularities such as the January thaw, a competing hypothesis must be that they have arisen simply by chance in the averaging of finite climate records. In this paper, we investigate this possibility for the January thaw in the northeastern United States.

Denote by (t = 1, . . . , 365) the raw daily means, and let mt be
the climatological mean for each day defined by the smooth Fourier
function. The standard deviation of for each day of the year, st, is
given by

where n is the number of years in the period of record with a valid
temperature observation xi,t on day t in year i. Note that since Equation
1 pertains to the standard deviation of the daily sample means , it is
smaller by a factor of ˆn than the standard deviation of the daily
observations xi,t (e.g., Wilks 1995). These standard deviations are larger
in winter and smaller in summer, reflecting the fact that the underlying
daily temperatures, xi,t, comprising the means are also more variable in
winter (e.g., Figure 1). Accordingly it is not surprising that deviations
from the smooth annual cycle of daily mean temperature, such as the January
thaw, stand out most strongly in winter (a point also made by Bingham,
1961), and this effect has probably contributed to the popularity of the
January thaw as an object of speculation and study. However, a fair
analysis of the unusualness of excursions from the annual cycle must not be
confounded by differences in the intrinsic variability of the atmosphere in
different seasons. Thus, in the following we analyze nondimensional mean
temperature data.

Define a dimensionless standardization, zt, for each day of the
year, t, by

Here st are the standard deviations st (Equation 1), smoothed with a 6-wave
Fourier function (e.g., the smooth curve in Figure 1). Time series of the
standardized values zt exhibit excursions around the mean ( = 0), at least
some of which derive from sampling variations within the finite climate
record. That is, suppose the true annual cycle of the daily temperature
means is smooth, in a manner that is well approximated by a six-wave
Fourier function, and contains no high-frequency (periods shorter than a
week or so) components such as a January thaw. Any finite (e.g., 80-year)
series of observations from such a climate will exhibit variations of the
raw daily means or their standardized counterparts (Equation 2) around the
smoothly varying climatological march of the seasons. How large are the
reported instances of the "January thaw" in relation to these sampling
variations?

In order to investigate this question quantitatively, an event
definition is necessary. Define the index

where z is a threshold anomaly (i.e. cutoff parameter) defining extreme
values of zt, and the summation is taken over any sequence of consecutive
days with zt more extreme than z, provided that there are at least t such
consecutive days in the particular run. The first date of an excursion
that exceeds z in absolute value is denoted by d, and the last such
consecutive date is denoted by d'. If the difference between d' and d is
at least t, the excursion is assigned a nonzero singularity index value y.

Figure 2 illustrates the application of Equation 3, assuming z = 1.2 and t = 3 days. The singularity index for excursion (a) in Figure 2 is zero, because this excursion does not persist for at least t = 3 consecutive days. Similarly, y = 0 for excursion (b) because it does not exceed the threshold anomaly, z. Excursions (c) and (d) each exceed the threshold anomaly in absolute value for at least three days and thus are each assigned nonzero y-values. Geometrically, Equation 3 approximates the area enclosed by the excursion between zt and ±z over a period of at least t consecutive days (shaded). Equation 3 differs from the thaw index defined by Lanzante and Harnack (1982), which considers exactly two days in late January and is quantified by dimensional residuals.

Note that Equation 3 contains two adjustable parameters, z and t. Different values for these parameters will produce different y-values for a given sequence of consecutive mean temperature anomalies. For each of the observed time series, the values for these parameters were found which resulted in the y-value for the late January warm anomaly being ranked as highly as possible among all such excursions from the annual cycle. For the 1920 - 1999 Boston average daily temperature data, letting z = 0.6 and t = 5 days leads to a y-index of 3.01 and a rank of 8 for the purported January thaw centered on January 24 (Figure 3): the departure from the smooth climatic mean (equivalent to zt = 0) centered on this date is the eighth largest such excursion in terms of y, and no other choices for z and t result in a better-ranked January thaw for this series. Similarly, optimized assignments for the z and t parameters were made for the other temperature series considered here, as shown in Table 2. The dates of the observed thaw at these locations range from 22-25 January, a result consistent with results from previous papers that the thaw falls on or around these dates (e.g., Nunn 1927, Wahl 1952, Lautzenheiser 1957, Lanzante and Harnack 1982).

Here, mt is the mean temperature on day t, fk is the autoregressive parameter (lag-1 autocorrelation coefficient) for month k, and the et are independent Gaussian random numbers. While the autoregressive parameters f were allowed to vary by month and location, they are generally close to 0.6 as is typical for daily temperature data (e.g., Richardson 1981, 1982). The random numbers are produced by a Gaussian random number generator using the Box-Muller method (Bratley et al. 1983), and have mean zero and standard deviation

where s2x,t is the temperature variance on day t. Both mt and s2x,t are specified by smooth, six-wave Fourier functions. This is a conventional and reasonable model for daily temperature variations (Wilks 1995, Wilks and Wilby 1999), and it produces synthetic series with statistical characteristics similar to those of the real data. Each n-year realization of daily temperatures from this model was averaged for each of the 365 days in the year, and six Fourier harmonics were fit to each synthetic annual cycle of mean temperature and its standard deviation; exactly as for the real temperature data, as described previously. The synthetic mean series show excursions from their climatological mean that are similar in both character and magnitude to excursions in the observed mean series (Figure 4).

Realizations of synthetic daily average and maximum temperature series were analyzed in the same manner as were the real data, except that apparent climate "singularities" were defined using Equation 3 and the z and t parameters optimized for the observed data. Frequency distributions of y-indices having the same rank as the January thaw in the observations were then tabulated in each case. This procedure was repeated 10000 times for each city and data type, producing, for example, 10000 second-ranked y-indices for Portland daily average temperature, 10000 first-ranked y-indices for Central Park daily average temperature, 10000 eighth-ranked y-indices for 1920 - 1999 Boston daily average temperature, and so on (Table 2). In comparison to the observed anomalies, the synthetic series produce many apparent events of similar and larger magnitudes, and these occur randomly throughout the year and are equally divided between warm and cool deviations.

The observed January thaw y-indices were compared to the distributions of synthetic y-indices (e.g., Figure 5), and p-values pertaining to the null hypothesis that the January thaw occurs by chance were evaluated according to the magnitudes of the observed y-indices in relation to these synthetic distributions. The large frequency of zero in Figure 5 for Boston, 1920 - 1999, results from the numerous (3012) synthetic series in which the standardized zt did not exceed z = 0.6 in absolute value for at least t = 5 consecutive days, at least eight times in the annual cycle, so that the eighth-ranked y-value had magnitude zero. Note that, for cases where Table 2 indicates that the observed January warm period could not be made into the largest singularity in the annual cycle, comparison to distributions of synthetic y-indices of the same rank essentially allows for the possibility of other, more prominent, singularities elsewhere in the year. Since these y-indices are necessarily no larger than the first-ranked y-indices in each n-year realization, the procedure used here is as favorable as possible to detection of a January thaw.

The results of these analyses are summarized in Table 2 for all
cities and temperature variables considered. Of the locations in this
study, only the results for the Washington, D.C. maximum temperatures show
nominal statistical significance, with a computed p-value of 0.02 (i.e.,
the observed y-value in this case was larger than 98% of those generated
randomly). However, when evaluating the results of multiple hypothesis
tests, it is likely that a small fraction of the results will appear to be
statistically significant, even if all null hypotheses are true. If each
test result is independent, the probability that at least r = 1 result of N
= 12 tests would have a p-value of p = 0.02 or smaller can be evaluated
quantitatively using the binomial distribution,

where the R indicates the random variable (in this case, the number of
nominally significant hypothesis tests) whose precise value is unknown, and
r denotes a specific, particular value that the random variable can take on
(e.g., Livezey and Chen 1983, Wilks 1995). The result in this case is that
Pr{R„1} = 1 - Pr{R=0} = 0.215. While it seems superficially plausible that
a p-value of 0.02 provides evidence for the January thaw, the binomial
distribution indicates that this result could have occurred by chance with
a sufficiently high probability (0.215) that the test results in aggregate
cannot be considered to have achieved statistical significance. To the
extent that these twelve cases have been chosen on the basis of the same
data that are analyzed in the tests, and that the results of the twelve
tests may be correlated, the evidence for the January thaw is even weaker.
It is also important here to note that, since in each case the definition
of the y-index has been tuned to emphasize the January warm excursions in
the observed data, the p-values reported in Table 2 are smaller than the
true probabilities for the individual tests, which further weakens any case
for the January thaw.

It is interesting to consider how extreme a departure from the smooth annual cycle would be necessary in order to reject the null hypothesis in the tests presented above. Addressing this question is complicated by the fact that, assuming the January thaw exists as a physically real phenomenon, we do not know its true structure (e.g., its threshold magnitude z, duration t, or overall amplitude y). However, assuming the reasonableness of Equation 3 as an index for the January thaw and the optimized z and t values in Table 2, one can calculate how much the observed January warm excursions would need to be inflated in order to reject the null hypothesis (and thus "detect" the January thaw) at specified test levels. For example, in order to reject the null hypothesis of a purely sampling source for average temperature excursions at the 5% level, the observed y-index would need to be larger than all but 500 of the 10000 maximum synthetic y-indices. If z and t are fixed, the y-index is increased by magnifying the excursions of the observed temperature means from their smooth annual cycle.

Figure 6 shows the resulting hypothetical excursions corresponding to the 5% and 1% test levels (gray), relative to the observed temperature means (black) and smooth annual cycle (heavy smooth curve) for the first 60 days of the annual cycle of average daily temperature at Boston, 1873 - 1952. That is, the black curves are reproduced from the first 60 days of Figure 4 (upper curves), and the gray excursions indicate magnitudes of January thaws inflated sufficiently to reject the null hypothesis of a random source for them, at the indicated test levels. This figure is typical of the other eleven cases listed in Table 2 (not shown). The observed excursion is roughly half the magnitude that would be necessary to provide convincing evidence. Note that, when plotted on this expanded scale, it can be seen that the late January warm excursion is only slightly larger than another apparent warm excursion in mid-February.

It is a recognized characteristic of human psychology that people will find patterns in the world around them, whether or not those patterns result from coherent underlying causes. "The tendency to impute order to ambiguous stimuli is simply built into the cognitive machinery we use to apprehend the world. It may have been bred into us through evolution because of its general adaptiveness . . ." (Gilovich 1993, Ch. 2). While this powerful human capacity to find order in nature has served and continues to serve us extremely well, it also sometimes leads us to falsely impute meaning to chance events. Gilovich nicely illustrates this problem using the statistics of consecutive hit or missed shots in basketball (the "hot hand"), where statistical independence can reasonably be assumed. When dealing with the non-independent statistics of the atmosphere, the problem of "detecting" spurious patterns is amplified by the statistical relatedness of data that are nearby in time or space or both (see Livezey and Chen, 1983, for a good example), and here our instinctive tendency to read too much into apparent patterns must be guarded against especially strongly. In the case of the January thaw, what superficially appear to be coherent singularities in the observed data can be adequately explained as products of time dependence, spatial dependence, and chance weather occurrences.

It is important to recognize that we have not proved, and cannot prove, that the January thaw does not exist. We have failed to reject the null hypothesis that mere sampling variations have produced the warm excursion in northeastern U.S. temperature records during late January, despite having biased the test toward rejecting a random source for this feature of mean temperature records in this region. Note, however, that no dynamical basis, or even a plausible physical mechanism, has been advanced in the literature to explain why a warming in northeast U.S. temperatures should occur during this particular narrow time window. To date, studies noting relationships between the January thaw and atmospheric circulation features have been wholly empirical; and the coincidence of warm surface temperatures with, for example, southerly flow, ridging, or poleward movement of the jet (at any time of year) are hardly surprising. Indeed, the warm surface temperatures are mainly consequences of such flow features, which are themselves subject to chaotic variations. If a dynamical basis for their phase-locking to late January were to be found, then the statistical analysis presented here would be of no interest. However, in the absence of such a physical rationale, our results leave one with little reason to look beyond simple statistical sampling variations as the cause of the January thaw. This is the same conclusion reached long ago by Marvin (1919), who wrote that "each striking feature on a long record is, therefore, no evidence of the persistent recurrence of peculiar irregularities, but is simply the residual scar or imprint of some unusual event, or a few which have been fortuitously combined at about the time in question."

*Acknowledgements.* We thank Dave Robinson for supplying
historical data for New Brunswick; Dave Henry of the National Weather
Service in Taunton, Massachusetts for providing historical data and a
detailed record of observing locations for Boston; the Northeast
Regional Climate Center for furnishing historical data and computing
resources; and Chuck Doswell for useful discussions.

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Table 1. Brief summary of studies of the January thaw.

Study | Geographical Coverage | Period of
Record | Analyzed Variables | Conclusions |

Esten & Mason (1910) | Storrs, CT | 1888-1909 | record max and min; max, min and avg temps | singularity |

Marvin (1919) | Continental U.S. | 1778-1865 | avg. weekly temps | no strong singularity |

Nunn (1927) | Northeast U. S. | 1873-1925 | avg. temps | singularity |

Slocum (1941) | Northeast U.S. | 1871-1939 | avg. temps | no conclusion |

Wahl (1952) | Northeast U.S. | 1873-1952 | avg. temps, sea-level pressures | singularity |

Wahl (1953) | Boston | 1873-1952 | avg. temps | singularity |

Brier (1954) | N. Hemisphere | 1899-1939 | sea level pressure | no conclusion |

Lautzenheiser (1957) | Boston, MA | 1911-1950 | avg. temps | no strong singularity |

Dickson (1959) | Nashville, TN | 1871-1950 | avg. temps, tornadoes | singularity |

Bingham (1961) | Northeast U.S. | 1896-1956 | avg. weekly temps | no strong singularity |

Duquet (1963) | Northeast U.S. | 1872-1961 | weekly avg. temps | singularity |

Newman (1965) | Boston, MA | 1872-1964 | max and min temps | no strong singularity |

Frederick (1966) | U.S. and SW Canada | 1897-1956 | avg., max, and min temps | singularity |

Hayden (1976) | East Coast U.S. | 1954-1970 | means of surf heights | singularity |

Logan (1982) | Portland, Maine | 1965-1979 | avg. temps | no conclusion |

Lanzante & Harnack (1982) | New Brunswick, NJ | 1858-1981 | max temps | singularity |

Lanzante (1983) | N. Am., Atl. & Pac. Oceans | 1947-1976 | 700-mb heights | singularity |

Kalnicky (1987) | N. Hemisphere | 1899-1969 | sea-level pressure | no conclusion |

Guttman & Plantico (1987, 1989) | Eastern U.S. | 1951-1980 | max and min temps | singularity |

Guttman (1991) | Central Park, NY | 1876-1987 | max and min temps | no strong singularity |

Table 2: Observed January thaw statistics for each of the five northeastern cities; parameters z, t and y from Equation 3; and p-values pertaining to the null hypothesis that each has arisen through chance sampling variations as evaluated through 10000 realizations of n-year synthetic temperature series.

Data Set | Variable | Rank | "Thaw" Date | z | t
(days) | y-index | p-value |

Boston, MA (1920-1999) | Avg. temps | 8 | 24 Jan | 0.6 | 5 | 3.01 | 0.34 |

Max. temps | 7 | 24 Jan | 1.1 | 3 | 1.63 | 0.22 | |

Boston, MA (1873-1952) | Avg. temps | 1 | 22 Jan | 1.2 | 3 | 2.35 | 0.84 |

Max. temps | 1 | 22 Jan | 1.4 | 3 | 1.62 | 0.79 | |

Central Park, NY | Avg. temps | 1 | 23 Jan | 1.3 | 5 | 2.46 | 0.40 |

Max. temps | 2 | 22 Jan | 1.1 | 3 | 3.42 | 0.33 | |

New Brunswick, NJ | Avg. temps | 1 | 23 Jan | 1.4 | 4 | 2.21 | 0.47 |

Max. temps | 1 | 24 Jan | 1.1 | 7 | 4.53 | 0.15 | |

Portland, ME | Avg. temps | 2 | 24 Jan | 1.0 | 5 | 2.12 | 0.56 |

Max. temps | 1 | 24 Jan | 0.8 | 6 | 3.22 | 0.87 | |

Washington, D.C. | Avg. temps | 4 | 25 Jan | 0.6 | 6 | 4.88 | 0.23 |

Max. temps | 4 | 25 Jan | 1.2 | 5 | 2.23 | 0.02 |

Figure 1. Portland, Maine (1920-1999) average daily temperature standard deviations st, smoothed with a six-wave harmonic function st. These standard deviations are larger in winter and smaller in summer, reflecting the fact that the underlying daily temperatures are intrinsically more variable in winter.

Figure 2. Hypothetical standardized anomalies zt (Equation 2), and their transformation to y-values according to Equation 3, using z = 1.2 and t = 3 days. Excursions (a) and (b) do not meet the duration or magnitude requirements, respectively, and are assigned zero y-values. Excursions (c) and (d) meet the requirements to be classified as putative singularities, and are assigned y-values approximately equal to the shaded areas.

Figure 3. Standardized anomalies zt (Equation 2) for the annual cycle of average daily temperatures at Boston, 1920-1999. Dashed lines show ±z = ±0.6. The January thaw is the eighth largest singularity according to Equation 3, with z = 0.6 and t = 5 days.

Figure 4. a) Observed annual cycle of mean (n = 80 years of record) average daily temperatures (heavy) and corresponding smooth mean series mt (light) for Boston, 1873-1952. b) Corresponding results for four example 80-year synthetic series (temperature scales suppressed) with statistical properties derived from the 1873-1952 Boston record.

Figure 5. Histogram of y-indices produced by ten thousand 80-year series generated using daily average temperature statistics for Boston, 1920 - 1999. Arrow indicates the actual y-index for the observed series.

Figure 6. First 60 days of the raw and smoothed annual cycles of the mean of average daily temperatures at Boston, 1873-1952 (as in Figure 4), with hypothetical January thaws (gray) large enough to allow rejection of the null hypothesis of a purely sampling source for this excursion, at the indicated test levels.

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