Tracking the Teletherms: The spatiotemporal dynamics of the hottest and coldest days of the year
Peter Sheridan Dodds, Lewis Mitchell, Andrew J. Reagan, Christopher M. Danforth
TTracking climate change through the spatiotemporal dynamics of the Teletherms, thestatistically hottest and coldest days of the year
Peter Sheridan Dodds, ∗ Lewis Mitchell, † Andrew J. Reagan, ‡ and Christopher M. Danforth § Computational Story Lab, Department of Mathematics and Statistics, Vermont Complex Systems Center,& the Vermont Advanced Computing Center, University of Vermont, Burlington, VT, 05401 School of Mathematical Sciences, North Terrace Campus, The University of Adelaide, SA 5005, Australia (Dated: November 5, 2018)Instabilities and long term shifts in seasons, whether induced by natural drivers or human activi-ties, pose great disruptive threats to ecological, agricultural, and social systems. Here, we propose,measure, and explore two fundamental markers of location-sensitive seasonal variations: the Summerand Winter Teletherms—the on-average annual dates of the hottest and coldest days of the year. Weanalyse daily temperature extremes recorded at 1218 stations across the contiguous United Statesfrom 1853–2012, and observe large regional variation with the Summer Teletherm falling up to 90days after the Summer Solstice, and 50 days for the Winter Teletherm after the Winter Solstice.We show that Teletherm temporal dynamics are substantive with clear and in some cases dramaticshifts reflective of system bifurcations. We also compare recorded daily temperature extremes withoutput from two regional climate models finding considerable though relatively unbiased error. Ourwork demonstrates that Teletherms are an intuitive, powerful, and statistically sound measure oflocal climate change, and that they pose detailed, stringent challenges for future theoretical andcomputational models.
Logline:
This paper introduces, formalizes, andexplores two fundamental climatological and season-al markers: the Summer and Winter Teletherms—the on-average hottest and coldest days of the year.Across the contiguous United States, the variation of theTeletherms—in date, extent, and temperature—is foundto be highly variable spatiotemporally with local coher-ence. The Teletherms reveal complex climate changehistories over many scales, including bifurcations andinstabilities, and provide stringent, detailed challengesto models and theory.
INTRODUCTION
Day length and temperature are two of the most impor-tant driving factors for life on Earth and for human cul-ture. While evidently strongly coupled, their relationshipis not a simple one in detail.Due to the regularity of celestial and planetary motionand the relative ease with which sun position can berecorded, the Solstices and Equinoxes have been deter-mined and commemorated by cultures around the worldfor thousands of years (e.g., Stonehenge), long beforebeing scientifically understood. We thus know with greatprecision when the longest and shortest day of the yearwill be, but what about the on-average hottest and cold-est days?Temperature behaves stochastically with highs andlows on a specific date potentially differing greatly rel- ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ative to surrounding dates and across years. Compound-ing temperature’s unevenness is that reliable measure-ment has only been realized in the last few hundred years.Indeed, widespread, systematic recording in the UnitedStates, which we study here, only began in the late 1800s.We are only now in a position to capitalize on sufficientlylarge data sets to give a reasonably solid answer to ourquestion.We propose to call the dates of on-average extremetemperature the Teletherms , using the Greek roots tele for distant and therm for heat. This construction is pat-terned after the Latin origin of Solstice with sol for sunand stit for stationary.As we will find, the Teletherms and their temperaturesare not fixed but vary in both space and time. In par-ticular, we will show that across the United States, thedynamics of the Teletherms are locally coherent but over-all highly variable, revealing intricate patterns includ-ing bifurcations in dates and both warming and cool-ing. For many regions, we will also demonstrate that theTeletherm is more appropriately acknowledged as occur-ring over a range of dates rather than a single one. Wewill therefore also speak both of each location’s singleday Teletherm and its
Teletherm Period which we definebelow in a pragmatic fashion.Our conception of the Teletherm is related to but dif-fers from existing meteorological quantities drawn fromstations around the United States. The National Oceanicand Atmospheric Administration (NOAA) captures ‘cli-mate normals’: 30 year averages at day, month, season,and year resolutions for a range of quantities includingmean, maximum, and minimum temperatures; precipita-tion; and snowfall [1]. Climate normals are made avail-able to and used broadly by the public. For example,monthly averages are of great use to people travelingto new areas. NOAA and the National Weather Ser-Typeset by REVTEX a r X i v : . [ phy s i c s . a o - ph ] M a r vice provide the Local Climate Analysis Tool (LCAT)at http://nws.weather.gov/lcat/home for people toexplore historical and recent climate dynamics. As weexplain below, we estimate the Teletherms’ aspects—date, temperature, extent, and period—from a dailymaximum and minimum temperature data set, and assuch our contribution can be seen as building a new lensfor the United States’ rich meteorological data set. Weaccompany our paper with an interactive site at http://panometer.org/instruments/teletherms to enablethose interested to examine climate dynamics throughthe Teletherms. If the notion of the Teletherm becomesstandard, we would hope a version of this site might even-tually be incorporated into the LCAT.Despite the evident imperative of quantifying climatechange, the task has proven to be both scientifically com-plex [2–5] and politically fraught and controversial [6, 7].Tied as they are to the changing of the seasons [8–12], Teletherm dynamics matter for ecological stabili-ty, agriculture, the Earth’s water cycle, the livability ofcities [13], and cultural and religious observances.By formalizing these annual turning points in temper-ature we hope to help advance our collective understand-ing of and ability to discern climate change. While wewill make a number of general observations regardingTeletherm dynamics, the central objective of our presentwork is the introduction of a statistically sound quantifi-cation of these two fundamental aspects of the annualclimate cycle, with the hope of both expanding and chal-lenging future work on climate dynamics.We structure our paper as follows. We first makesome basic observations about the historical weather dataset which we build our analysis around, along with afew details about our approach. We then present ourmain findings, describing and testing our approach todetermining Teletherms and Teletherm Periods at specif-ic locations, highlighting a few of the extreme locationssuch as the hottest Summer Teletherm and coldest Win-ter Teletherm. Moving out from individual stations, wethen examine a range of results for the contiguous Unit-ed States. We first show that the Winter and SummerTeletherms vary strongly according to geographic loca-tion. We then explore the temporal dynamics of region-al Teletherms, and discuss their relationship to climatechange. Finally, we compare empirical Teletherm dateswith those produced by two Regional Climate Models(RCMs). To close, we put forward a few concludingremarks, contemplating future directions.We provide a complete set of figures and code as part ofthe paper’s online appendices at http://compstorylab.org/share/papers/dodds2015c . DATA
We consider daily records of maximum and mini-mum temperatures for 1218 stations distributed acrossthe contiguous United States for the time period 1853–
FIG. 1. Locations of 1218 weather stations represented in theUnited States Historical Climatology Network (USCHN) dataset (version 2.5 through 2012) [14]. The distribution indicatesrelatively uniform coverage of the 48 contiguous states.
ANALYSIS AND RESULTSTeletherms of Individual stations
Our goal is to identify the Teletherms and TelethermPeriods and their respective dynamics in as straightfor-ward a fashion as possible. Because of the stochasticityof temperature, our analysis necessarily involves severalsteps.We first compute the mean maximum and minimumtemperature for each day of the year at each station. Weaverage over all error-free data points, acknowledging thevariability of both length and completeness of each sta-tion’s temperature time series. In the following sectionon Teletherm maps, we will only include averages for sta-tions for which we have data for at least 80% of the dateswithin a given window.To enable us to illustrate and explain our treatmentin full, we will use a selection of six extreme Telethermlocations in the contiguous U.S. In Figs. 2A–B and 3A–D, we present diagnostic plots for the following specificTeletherms: • Fig. 2A. Hottest Summer Teletherm: Death Valley,California, (Station ID: 042319). • Fig. 2B. Coldest Winter Teletherm: Willow City,North Dakota, (Station ID: 329445). • Fig. 3A. Earliest Summer Teletherm: Alpine, Texas(Station ID: 410174). • Fig. 3B. Earliest Winter Teletherm: Anaconda,Montana (Station ID: 240199). • Fig. 3C. Latest Summer Teletherm: Santa Cruz,California. (Station ID: 047916). • Fig. 3D. Latest Winter Teletherm: Chatham ExpFarm 2, Michigan (Station ID: 201486).Each figure has the same format: a main plot showingaverage and smoothed maximum or minimum temper-ature (explained below), and three subplots across thetop. We will address the main plots first.Taking the example of the Death Valley station, in themain plot in Fig. 2A, the black dots represent the aver-age maximum temperature for each day of the year. Wesmooth these points by convolving the average maximumtemperature time series with a Gaussian kernel of width15 days, resulting in the red curve, and we elaborate onthis choice below.After smoothing the data, we assign the day ofthe most extreme value of the resultant curve as theTeletherm for that station. In all plots, we indicateTeletherms with a gray vertical line and for reference,we locate the Summer or Winter Solstice with a dashedgray vertical line.The left inset in each main plot shows the fraction ofdays with error-free data as a function of year. In the case of Death Valley, we see the data set contains records from1961 on, and that these are fairly complete. For WillowCity in Fig. 2B, the period of record begins before 1900but shows an imperfect collection rate; we generally seethat winter temperatures, especially minima, are (unsur-prisingly) more error prone.Turning to the Teletherms themselves, for Death Val-ley, we estimate that the Summer Teletherm occurs onJuly 29 (day 210), a considerable 38 days after the Sum-mer Solstice (Fig. 2A). The coldest Winter Telethermoccurs on January 27 in Willow City, North Dakota,a similarly lengthy 37 days after the Winter Solstice(Fig. 2B). While we define Teletherms as the date, eachone has of course an associated effective temperature aris-ing from our analysis. For Death Valley, this temperatureis 117 ◦ F (47 ◦ C ) and for Willow City, we find − ◦ F ( − ◦ C ). Death Valley also has the maximum tempera-ture recorded in the data set: 129 ◦ F (54 ◦ C ).The earliest Teletherms occur in Alpine, Texas for thesummer (Fig. 3A) and Anaconda, Montana for the winter(Fig. 3B). These Teletherms precede the adjacent Solsticeby two days and one day respectively following a long lin-ear change in temperature, and both display an initiallyslow return afterwards.The Teletherms occurring latest in the year have differ-ent stories. For the summer, Santa Cruz’s Teletherm isexperienced extremely late on September 23—essentiallythe Autumnal Equinox—around three months (94 days)after the Summer Solstice. As Fig. 3C shows, the aver-age maximum temperature for Santa Cruz rises to afalse peak (a localized Teletherm) at the Summer Sol-stice, drops slightly and then climbs again to the trueTeletherm. We find similar behavior for stations alongthe west coast but not to any extent inland, a feature weexamine further in the following section.We estimate that the latest winter Teletherm takesplace on February 11—a remarkable 52 days after theWinter Solstice and 9 days after Groundhog Day—at theChatham Exp Farm 2 station in Michigan’s Upper Penin-sula (Fig. 3D).We note that many of the smoothed average maximumand minimum temperature curves we observe exhibit asmall periodic behavior as they climb and fall. Not beinga focus of our present work, we suggest a more detailedanalysis may uncover the source, if any, of these apparentpulsings in the time series.Continuing with our explanation of our analysis, wemove to the three diagnostic subplots marked i , ii , and iii in each of Figs. 2A–B and 3A–D. The first subplot i summarizes the distribution of maximum or minimumtemperature for each station. The black curve gives themedian for each day of the year, the yellow region rep-resents the inter-quartile range, and the blue and redregions show the rest of the range. For example, the topof the red region for a Summer Teletherm figure indicatesthe hottest maximum temperatures, the bottom of theblue the lowest maximum temperatures. The stochastic-ity of the extreme temperatures measured at the levels A. Hottest Summer Teletherm: B. Coldest Winter Teletherm: S u mm e r S o l s t i ce J un e D a y S u mm e r T e l e t h e r m J u l y D a y + d a y s Day of Year ◦ F DEATH VALLEY, CA
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec iii
Kernel width (days) D a y o f Y e a r i Day of Year ◦ F
190 200 210 220111112113114115116117118 ii Day of Year ◦ F Year C o v e r a g e
182 213 244 274 305 335 1 32 60 91 121 152−1001020304050 W i n t e r S o l s t i ce D ece m b e r D a y W i n t e r T e l e t h e r m J a nu a r y D a y + d a y s Day of Year ◦ F WILLOW CITY, ND
Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun iii
Kernel width (days) D a y o f Y e a r
182 274 1 91 181−70−4003272105 i Day of Year ◦ F
20 30−14−13−12−11−10−9−8−7−6−5−4 ii Day of Year ◦ F Year C o v e r a g e FIG. 2. Plots establishing Teletherm date and Teletherm Periods for the examples of A: the hottest summer Teletherm(Death Valley, California) and B: the coldest winter Teletherm (Willow City, North Dakota). The main plots in A and B showthe average daily maximum and minimum temperature (black dots) along with a smoothed curve formed using a GaussianKernel (solid red). For all minimum temperature analyses, we wrap the year from July 1 to June 30. The main plots’ insetsshow the fraction of error-free recording for each year. Subplot i: Representation of the spectrum of maximum/minimumtemperatures per day of the year. The black curve indicates the median, the blue area indicates lowest to first quartile, yellowthe inter-quartile range, and red the fourth quartile. Subplot ii:
Expansion of the inset around the Teletherm in the mainplot. The dark gray vertical line indicates the Teletherm and the lighter gray region the Teletherm Period which we defineas the days for which the smoothed maximum/minimum temperature curve is within 2% of the Teletherm’s temperature,relative to the dynamic range of the smoothed curve over the entire 365 days. Subplot iii:
Robustness diagnostic showinghow the Teletherm date varies as a function of Kernel width. We use 15 days, marked in red. See the main text for furtherdetails. See Fig. 3 for four more extreme Teletherm examples. We provide Teletherm plots for the maximum and minimumtemperatures for all 1218 stations in the Supporting Information (Files S22 and S23) and in the paper’s online appendices at http://compstorylab.org/share/papers/dodds2015c . of day is readily apparent in these subplots.The second subplot ii is an expanded and rescaledmatch of the inset in the main plot around the Teletherm.As for the main plot, the black dots show the averagemaximum or minimum temperature for each day of theyear, and the red curve the smoothed version. The grayshaded region shows the full Teletherm Period for a sta-tion which we describe below.The third subplot iii shows how the Teletherm variesas a function of the width of Gaussian kernel, providinga measure of robustness. To smooth the data, we usedthe Matlab command gausswin with Kernel width W andstandard deviation σ = ( W − /
4. For the examples inFigs. 2A–B and Figs. 3A–D, we see that the estimateddate of the Teletherm varies relatively little—typically 2to 4 days—for Kernel widths ranging from 7 to 31.Our choice of a Gaussian kernel with a width of 15 isa defensible, reasonable, and practical one, well withinwhat is a range of widths producing similar outputs andinterpretable as spanning a week to the side of each date.We observe that very narrow kernels may however give quite different results as for the station Chatham ExpFarm 2 in Fig. 3D. Such jumps may occur when two ormore localized Teletherms are present which we addressin the next section.
Teletherm Periods for Individual stations
In looking more closely at the behavior of average max-imum and minimum temperatures, we are obliged to aug-ment our definition of Teletherms beyond single days ofthe year. Being able to assign one date to a locationmakes for a simple story but we must acknowledge threeaspects: (1) We are working with a statistically speakingsmall number of samples for each station; (2) The choiceswe have made in our statistical analysis mean that thespecific Teletherm date is subject to minor error; and (3)Fundamentally, some locations undergo on-average max-imum or minimum temperatures that hold over a rangeof dates.We define the Teletherm Period for a location to be
A. Earliest Summer Teletherm: B. Earliest Winter Teletherm: S u mm e r S o l s t i ce J un e D a y S u mm e r T e l e t h e r m J un e D a y d a y s Day of Year ◦ F ALPINE, TX
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec iii
Kernel width (days) D a y o f Y e a r i Day of Year ◦ F
160 170 1808990919293 ii Day of Year ◦ F Year C o v e r a g e
182 213 244 274 305 335 1 32 60 91 121 152101520253035404550 W i n t e r S o l s t i ce D ece m b e r D a y W i n t e r T e l e t h e r m D ece m b e r D a y d a y s Day of Year ◦ F ANACONDA, MT
Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun iii
Kernel width (days) D a y o f Y e a r
182 274 1 91 181−70−4003272105 i Day of Year ◦ F
350 3609101112131415 ii Day of Year ◦ F Year C o v e r a g e C. Latest Summer Teletherm: D. Latest Winter Teletherm: S u mm e r S o l s t i ce J un e D a y S u mm e r T e l e t h e r m S e p t e m b e r D a y + d a y s Day of Year ◦ F SANTA CRUZ, CA
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec iii
Kernel width (days) D a y o f Y e a r i Day of Year ◦ F
260 2707475767778 ii Day of Year ◦ F Year C o v e r a g e
182 213 244 274 305 335 1 32 60 91 121 152510152025303540455055 W i n t e r S o l s t i ce D ece m b e r D a y W i n t e r T e l e t h e r m F e b r u a r y D a y + d a y s Day of Year ◦ F CHATHAM EXP FARM 2, MI
Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun iii
Kernel width (days) D a y o f Y e a r
182 274 1 91 181−70−4003272105 i Day of Year ◦ F
20 30 40 50456789101112 ii Day of Year ◦ F Year C o v e r a g e FIG. 3. Teletherm plots for four extremes for the contiguous U.S.: the earliest Summer and Winter Teletherms and the latestSummer and Winter Teletherms. See the caption of Fig. 2 for full details. the range of dates, possibly non-contiguous, for whichthe smoothed maximum/minimum temperature curvelies within 2% of the Teletherm’s temperature as mea-sured with respect to the dynamic range of the smoothedcurve. We chose 2% as a cutoff, asserting that the human-experienced temperature would be roughly similar tothat of the Teletherm. An alternate approach would be touse an absolute difference (e.g., within 1 ◦ F ); the results will not differ substantially.Returning to Figs. 2A–B and Figs. 3A–D, we nowidentify the gray shaded region in the inset around theTeletherm in the main plot (reproduced in the subplot ii )as the Teletherm Period. Across all stations, we see sub-stantial variation in duration and continuity of TelethermPeriods. For Death Valley (Fig. 2A) the Teletherm Peri-od lasts an unpleasant 34 days with a smoothed maxi-mum temperature of at least 115 . ◦ F (46 . ◦ C ) (July 9thto August 11th, day numbers 190 to 223). The WinterTeletherm Period for Anaconda, Montana is compara-tively brief running 8 days with smoothed minimum tem-peratures below 13 . ◦ F ( − . ◦ C ) (December 18th to25th, numbers 352 to 359).The station Chatham Exp Farm 2 in Michigan(Fig. 2D) shows how our definition may lead to twoor more Teletherm Periods surrounding minor coolingor warming periods. In looking across all stations, wesee that Winter Teletherms for stations in the Northeastmay present a statistically sound early spring thaw, andBurlington WSO AP, Vermont (Station ID: 431081) isanother clear example (see Supporting Information filesS22 and S23 and http://compstorylab.org/share/papers/dodds2015c ). Evidently, if we used a thresh-old of, say, 5%, some separated Teletherm Periods wouldcoalesce, but we believe the threshold should be suitablystrict.For the whole data set, we observe considerable thoughlocally coherent variation in dynamics with temperaturesrising and falling, and Teletherm periods expanding,dividing, and coalescing, and Teletherm dates switch-ing. In Fig. 4, we show example behavior for 25 yearTeletherms for Aberdeen, MS (Summer), Uniontown, PA(Winter), and Kennewick, WA (Winter). For Kennewick,we see the 25 year Winter Teletherm moves sharply toan early date around the middle of the 20th century.See Supporting Information Files S28, S29, and S30 forthe complete set of stations for 50, 25, and 10 yearTeletherms. Teletherm maps
We move now to exploring how the Teletherms varyacross the contiguous U.S. through maps. Once againdrawing on the full data set, we plot the SummerTeletherms in Fig. 5A and the Winter Teletherms inFig. 5B. We present accompanying maps of the Telethermtemperatures in Figs. S1 and S2, and a map showing thenumber of days separating the two Teletherms at eachstation in Fig. S3.On all maps here and in the Supplementary Infor-mation, we indicate the Teletherm’s day of year by anarrow on a clock. We orient the angle 0 radians upwardsand assign days of the standard year to multiples of1 / × π (December 31st then corresponds to angle 0).To reinforce the visibility of variation, we color pointsand arrows per the color wheel in the bottom right cor-ner of all maps. The black arrows in these color wheelsmark the Summer and Winter Solstices as appropriate.We visually supply information about the TelethermPeriod by linearly scaling the size of the marker for eachlocation. For stations with multiple Teletherm Periods,we use what we call the Teletherm Extent—the numberof days from the start of the first Teletherm period to theend of the last one (inclusive). We provide a histogram of the Teletherm days of theyear in the bottom left corner of each map, again usingthe same color scheme. The inverted black triangle iden-tifies the relevant Solstice.A number of observations stand out. For the SummerTeletherm, we see considerable but largely smooth vari-ation. From Figs. 2A and 2C, we had identified that therange of dates for the Summer Teletherm spans 96 days(June 19 in Alpine, Texas to September 23 in Santa Cruz,California), but we now see that the bulk of Telethermsfall between July 15 and August 1 (dark blue). Thesesecond-half-of-July Teletherms hold in the north of thecontiguous U.S. and extend down into California on thewest and Georgia on the east.The variant Summer Teletherms span several regions.The earliest summer Teletherms occur in Arizona, NewMexico, and the west of Texas (purple/red). In mov-ing from west to east, we see a longitudinal disconti-nuity in Texas with a switch to relatively late Sum-mer Teletherms, which remain apparent in Oklahoma,Arkansas, Louisiana, Mississippi, and over to Florida.These August Teletherms form a noticeable minor peakin the histogram (light blue). The gulf coast shows someirregularity in the Teletherm but more clearly exhibitsthe longest Teletherm Extents.Stations along the west coast show how exposure to thePacific and incoming weather patterns make them breakstrongly with the nearby inland Teletherm “directions”,moving to generally later in the year as per example ofSanta Cruz we examined earlier (Fig. 3C). By contrast,stations along the east coast are consistently aligned withtheir inland counterparts.For the Winter Teletherm, we see a different over-all pattern with the contiguous U.S. dividing into tworegions: the west, midwest, and south with largely ear-ly January Winter Teletherms (blue), and the mid-northand northeast showing Teletherms in late January andearly February (green). In the northeast’s winter, thetemperature continues to fall well beyond the shortestday of the year in the northeast, typically 5 to 6 weeks.We venture that a possible source of this clear regionalseparation might lie in the jet streams dynamics acrossNorth America, with snowfall leading to increased albedoin the northern section coupled with a continental-scaleshadow of the Rockies. Beyond the scope of the presentanalysis, future modeling would be needed to properlytest such an hypothesis. Teletherm dynamics
In order to discern Teletherm dynamics and theirpotential value in quantifying and studying climatechange, we carry out the same smoothing we have per-formed for the full time range for sliding windows of 50years in duration. We also now make our data require-ments more stringent and estimate the Teletherm andthe Teletherm Period(s) for only those time ranges for
25 ye ar Su mme r Te le th e rm d yn amic s for ABERDEEN, MS:
Ju nJu lAu gSep D a y o f y e a r A ◦ F B
25 ye ar W inte r Te le the rm dynamic s for UNIONTOW N 1 NE, PA:
N ovD ecJanFeb D a y o f y e a r C ◦ F D
25 ye ar W inte r Te le the rm dynamic s for K ENNEW ICK , WA:
N ovD ecJanFeb D a y o f y e a r E ◦ F F FIG. 4. Dynamics of 25 year Teletherm dates, periods, extents, and temperatures for three example locations displayingabrupt switchings in time and gradual increases and decreases of temperature. We provide sets of these plots for 50, 25, and 10year Teletherms for all 1218 stations in the Supporting Information (Files S28, S29, and S30). The same plots are also availableat http://compstorylab.org/share/papers/dodds2015c/places.html . A. Summer Teletherms for 1853–2012: S ep − S e p − S e p − A ug − A ug − J u l − J u l − J un − J un J u l J u l A u g1 A u g15 S e p S e p S e p C o un t B. Winter Teletherms for 1853–2011: − F eb − F eb − J an − J an − D e c D ec J a n J a n F e b F e b C o un t FIG. 5. A: Summer Teletherms and B: Winter Teletherms across the contiguous United States based on all data recordedfrom 1853 to 2012. Arrows point in the direction of the Teletherm’s day of year mapped into angles traveling clockwise withDecember 31st aligned upwards. The sizes of the markers (discs) represent the duration of a location’s Teletherm Period. Inthe case of multiple Teletherm Periods, sizes correspond to the full extent. Colors map to Teletherm dates as indicated bythe partial color wheel in the bottom right corner of each map. The black arrows in the color wheels show the location of theSolstices. The histograms shows the distributions of the Summer and Winter Teletherm, using the same colors.
A. 50 year Summer Teletherm shifts for 1963–2012 relative to 1913–1962: S u mm e r T e l e t he r m s h i ft ( da ys ) −35−20−100102035 −35 −20 −10 0 10 20 35050100150 Summer Teletherm shift ( C oun t B. 50 year Winter Teletherm shifts for 1962–2011 relative to 1912–1961: W i n t e r T e l e t he r m s h i ft ( da ys ) −35−20−100102035 −35 −20 −10 0 10 20 35050100 Winter Teletherm shift ( C oun t FIG. 6. A: Summer Teletherm shifts comparing the 50 year period 1963–2012 relative to 1913–1962. We show a total of 837out of 1218 (68.7%) which have ≥
80% error-free data in both 50 year spans. See Figs. S4 and S5 for maps of the SummerTeletherms for each period. B: Winter Teletherm shifts comparing 1962/1963–2011/2012 relative to 1912/1913–1961/1962. Atotal of 835 out of 1218, 68.6%, stations have ≥
80% error-free data. See Figs. S6 and S7 for maps of the Winter Telethermsfor each period. http://compstorylab.org/share/papers/dodds2015c and http://panometer.org/instruments/teletherms .In Fig. 6A, we show how the Summer Teletherm hasmoved between these two half century time periods. Wecompute the “Teletherm shift” in days (see Figs. S4and S5 for plots of the respective Summer Teletherms)and use a color map to present the results. The lowerleft histogram in Fig. 6A represents the distribution ofshifts. Now, if change was random, we would expect tosee a normal distribution centered around a shift of 0. Weinstead find two peaks separated from zero shift, mean-ing very few locations experienced no change. The largerpeak (blue) means the Summer Teletherm has moved toearlier in the year, connecting more strongly with the Sol-stice, and corresponds generally to the northeast extend-ing across and down into the midwest and south. Sta-tions in the west are reflected in the histogram’s smallerpeak (green/yellow) indicating the Summer Telethermhas moved to later in the year for that area’s stations.We show shifts in the Winter Teletherm in Fig. 6B (seeFigs. S6 and S7 for the Teletherms themselves). We againfind a texture different to that of the Summer Teletherm.The dominant change is that the Winter Teletherm hasadvanced to earlier dates in the year across the north-ern half of the contiguous U.S., and down along the westcoast (blue). As the histogram shows, the spread of for-ward shifts peaks in the range 10 to 20 days. Going inthe other direction, we see that the Winter Teletherms forthe states of Georgia and the two Carolinas have experi-enced a delay of around 25 days (red). For the rest of thecontiguous U.S., from New Mexico across to Florida, theWinter Teletherm has remained fairly constant. Com-paring the histograms for the Winter Teletherm dates inFigs. S6 and S7, we see that three distinct peaks havemerged into one grouping over time. In sum, the WinterTeletherm has become more homogeneous in the east-ern half of the contiguous U.S., with both early and lateWinter Teletherms moving into the first half of January,while largely moving to earlier dates in the west.In Figs. S8A–B and S9A–B, we present the shifts inTeletherm Temperatures and Extents for the same pair of50 year periods. The changes in temperature are milderfor the Summer Teletherm ( ± ◦ F , Fig. S8A) than forthe Winter ( ± ◦ F , Fig. S9A). The Extents howev-er have maximally increased or decreased by 30 to 40days, with the Summer Teletherm seeing the most flux.(Figs. S8B and S9B). Some of the other trends we seeare that (1) the Summer Teletherm Temperature hasdropped in the middle of the contiguous United States while remaining neutral or increasing elsewhere; (2) Sum-mer Teletherm Extents have increased most stronglythroughout the south; (3) The Winter Teletherm Tem-perature has lowered in the South East and increased inthe central and western areas of the north; and (4) Win-ter Teletherm Extents have decreased in the south andincreased in areas around the Great Lakes.Finally, we observe that the transitions in Telethermfeatures between these two adjacent 50 year periods isnot linear, and that window length matters [15]. Toshow this, we break the same century (1913–2012) intofour 25 year periods. First, we see a strengthened ver-sion of the same general overall changes to the Telethermdates as for the 50 year analysis in comparing the last 25years to the first 25 years (1988–2012 relative to 1913–1937) (Fig. S10). The three transitions between the four25 year periods show accelerations, stasis, and reversals(see Figs. S11, S12, and S13). For example, in Louisianaand Mississippi, the Summer Teletherm has shifted tolater dates but through an advance, retreat, advancemovement (Figs. S11A, S12A, and S13A). Much of theshift toward an earlier Winter Teletherm across the northoccurred in the 50 year period 1937–1986 (Fig. S12B),and the southeast first saw the Winter Teletherm advanceand then start to fall back to later dates (Figs. S11B,S12B, and S13B).We show the corresponding maps for shifts inTeletherm temperatures and extents in Figs. S14, S15,S16, S17, S18, S19, S20, and S21. The transition fromthe first 25 years to the last 25 years of 1913–2012 sees anaverage drop in the 25 year Summer Teletherm temper-ature (mainly due the interior states) but an increase inthe Winter Teletherm temperature (concentrated morealong the north and down into Utah, Colorado, and Ari-zona). Of many notable details, we see a dropping of theWinter Teletherm’s temperature, in the eastern half ofthe contiguous U.S. between 1937–1961 and 1962–1986,followed by a reverse swing upwards over the next 25years (Figs. S16B and S17B).Interpreting the dynamics of the Teletherms is not aneasy task and we limit our assertions in this initial work.We might suspect the jet stream may have played a partin the transition of the Winter Teletherm in the south-east. Even without a clear understanding, we can seethat impact of these changes is potentially dramatic. Themovement of the Winter Teletherm for example alters thelocal advent of spring, a strong driver of ecological sys-tems. COMPARISON TO MODELS
We end our analysis with a comparison of esti-mated Teletherm data to output from two RegionalClimate Models (RCMs) from the North Ameri-can Regional Climate Change Assessment Program(NARCCAP) [7]. Specifically, we analyze out-put of the WRF model nested within both the1 −60 −40 −20 0 20 40 6000.020.040.060.080.10.120.14 ∆ Summer Teletherm date N o r m a li z e d f r e qu e n c y WRFG-CCSM1968 to 1999 A −60 −40 −20 0 20 40 6000.020.040.060.080.10.120.14 ∆ Winter Teletherm date N o r m a li z e d f r e qu e n c y WRFG-CCSM1968 to 1999 B −60 −40 −20 0 20 40 6000.020.040.060.08 ∆ Summer Teletherm date N o r m a li z e d f r e qu e n c y WRFG-NCEP1979 to 2004 C −60 −40 −20 0 20 40 6000.020.040.060.080.10.12 ∆ Winter Teletherm date N o r m a li z e d f r e qu e n c y WRFG-NCEP1979 to 2004 D FIG. 7. Distributions of errors in days for the Summerand Winter Teletherms at all stations when comparing mea-sured temperatures to the Community Climate Systems Mod-el (CCSM) (version 3) [16] and the National Centers forEnvironmental Prediction Climate Forecast System Model(NCEP) [17, 18]. The differences are to be interpreted ashow many days the models are “off” from the real data. Apositive ∆ means the model’s Teletherm occurs later in theyear than the measured one. −60 −40 −20 0 20 40 6000.010.020.030.040.05 T max , model − T max , observed ( ◦ F) N o r m a li z e d f r e qu e n c y WRFG-CCSM1968 to 1999 A −60 −40 −20 0 20 40 6000.010.020.030.040.05 T min , model − T min , observed ( ◦ F) N o r m a li z e d f r e qu e n c y WRFG-CCSM1968 to 1999 B −60 −40 −20 0 20 40 6000.010.020.030.040.05 T max , model − T max , observed ( ◦ F) N o r m a li z e d f r e qu e n c y WRFG-NCEP1979 to 2004 C −60 −40 −20 0 20 40 6000.010.020.030.040.05 T min , model − T min , observed ( ◦ F) N o r m a li z e d f r e qu e n c y WRFG-NCEP1979 to 2004 D FIG. 8. Comparison between predicted daily maximumand minimum temperatures generated by two climate mod-els (CCSM and NCEP) relative to real measurements for allstations. stations.We find the average absolute error in estimating theSummer and Winter Teletherm are 12.88 and 10.05 daysfor the CCSM, and 12.24 and 7.57 days for the NCEPmodel. Spearman correlations are mixed with a best val-ue of 0.85 for the CCSM’s Winter Teletherm ( p -valueeffectively 0) and a worst case of 0.059 for CCSM’s Sum-mer Teletherm ( p -value 0.039). At the level of stations,the worst errors for both models are for the SummerTeletherm with spans 78 and 59 days too early and 44and 48 too late for the CCSM and the NCEP modelrespectively.In Fig. 8, we step back from Teletherms, and plot thedistribution of errors at the day level between the outputof both models and measured maximum and minimumtemperatures. This is an exacting test: how does a mod-el fair with predicting the maximum temperature, say,in Death Valley on March 3, 1982, along with all otherstations and all other dates over several decades? Withapproximately 10,000 points per panel, we see a muchsmoother distribution and the form is now Gaussian-like.We find that the Spearman correlations betweenthe models’ outputs and measured daily temperatureextremes are good, ranging from 0.796 (CCSM, dailyminimum temperature) to 0.875 (NCEP, daily maximumtemperature) ( p -values effectively 0). The NCEP modelis on average more accurate with an average differencefor the daily minimum of 0 . ◦ F . The average abso-lute error varies from 7.32 (NCEP, daily minimum tem-perature) to 12.1 (CCSM, daily maximum temperature).The potential for wild inaccuracies remain with CCSM’sworst prediction being 95 ◦ F below the real measurementof a minimum temperature.2In testing these climate models for Teletherm timingand daily temperature extremes, we are certainly askingfor more than they have been intended to deliver. Indeed,if these models were integrating in Numerical WeatherPrediction mode, with initial values updated throughdata assimilation, the errors would be much smaller.Nevertheless, understanding the successes and limita-tions of any model, whether aimed for or not, shouldbe of benefit to future refinements [21]. CONCLUDING REMARKS
We were initially motivated by the simple question ofwhen should we expect the on-average warmest and cold-est day of the year to occur at a given location. In thenortheast of the U.S. for example, the Winter Solsticepasses and as the days lengthen, the cold deepens andpeople begin to wonder when will the winter end. Tra-ditionally, prognosticators have used diverse methods todivine the length of winter such as, famously, how certainspecies of rodents react to their umbra. And in gener-al, people look for signs of all the seasons arriving suchas the emergence of daffodils in spring or the first leavesturning to their autumnal colors. We realized howeverthat a data-driven, less poetic path could be assayed.While the analysis promised to be initially straight-forward (as is often believed to be the case), we soonfound that we had to move beyond a single day ver-sion of the Teletherm to a Teletherm Period. Overall,we believe we have shown the spatiotemporal variabilityof the Teletherms and the surrounding Teletherm Peri-ods to be considerable, informative, and of general inter-est. Importantly, we have seen that the variations inTeletherm characteristics are not a reflection of randomnoise but rather linear movements, periods of stasis, andswitching reminiscent of bifurcations in dynamical sys-tems. Teletherms seem therefore to present a real facetof climate change, whatever the origin. A number of future directions are possible. Where datais available, our analysis could readily be carried out forother regions around the world. Beyond local interest,such efforts could lead to an effort to patch togethera global picture of the Teletherms. Online displays ofTeletherms could also eventually include the ability toadjust time frames for the analysis and to show the like-lihood that the warmest or coldest day has occurred asa function of day of the year. A global map would alsoafford more opportunities to test models and hypothe-ses regarding climate dynamics. For example, does thetemporal behavior of Teletherms correlate in an way tochanges or stationarity of average annual temperature?The stochastic nature of temperature could also be ofvalue in our collective general education regarding pre-diction for noisy systems.We close by venturing that a region’s Teletherm mayalso be acknowledged annually (using, say, the mostrecent 50 years), potentially with a set of associated food-based rituals or celebrations.
ACKNOWLEDGMENTS
The authors appreciate helpful discussions with IstvanSzunyogh, Linda Mearns, John Kaehny, Bill Gottesman,Bruce Shaw, and Andrew Gelman. The authors thankthe North American Regional Climate Change Assess-ment Program (NARCCAP) for providing the modeldata used in this paper. NARCCAP is funded by theNational Science Foundation (NSF), the U.S. Depart-ment of Energy (DoE), the National Oceanic and Atmo-spheric Administration (NOAA), and the U.S. Environ-mental Protection Agency Office of Research and Devel-opment (EPA). LM, AJR, and CMD were in part sup-ported by the Mathematics and Climate Research Net-work (MCRN), NSF Award [1] A. Arguez, I. Durre, S. Applequist, R. S. Vose, M. F.Squires, X. Yin, R. R. Heim, Jr., and T. W. Owen, Bull.Amer. Meteor. Soc. , 1687 (2012).[2] D. A. Stainforth, T. Aina, C. Christensen, M. Collins,N. Faull, D. J. Frame, J. A. Kettleborough, S. Knight,A. Martin, J. Murphy, et al. , Nature , 403 (2005).[3] T. R. Karl, A. Arguez, B. Huang, J. H. Lawrimore, J. R.McMahon, M. J. Menne, T. C. Peterson, R. S. Vose, andH.-M. Zhang, Science Magazine , 1469 (2015).[4] R. Barkemeyer, S. Dessai, B. Monge-Sanz, B. G. Ren-zi, and G. Napolitano, Nature Climate Change ,10.1038/nclimate2824 (2015).[5] C. M. O. et al., Geophysical Research Letters , 10,773(2015).[6] L. Antilla, Global Environmental Change , 338 (2005).[7] R. J. Lempert, Nature Climate Change , 914 (2015). [8] M. Mann and J. Park, Geophys. Res. Lett. , 1111(1996).[9] T. H. Sparks and A. Menzel, International Journal ofClimatology , 1715 (2002).[10] M. D. Schwartz, R. Ahas, and A. Aasa, Global ChangeBiology , 343 (2006).[11] A. R. Stine, P. Huybers, and F. I. Y., Nature , 435(2009).[12] A. K. Betts, Weather , 245 (2011).[13] K. K. R. Lai, “How much warmer was yourcity in 2015?” (2016), The New YorkTimes’s interactive weather chart: ; accessed February 20, 2016.[14] M. J. Menne, I. Durre, R. S. Vose, B. E. Gleason, andT. G. Houston, Journal of Atmospheric and Oceanic Technology , 897 (2012).[15] K. E. Trenberth, Science Magazine , 691 (2015).[16] W. D. Collins et al. , J. Climate , 2122 (2006).[17] L. O. Mearns, W. J. Gutowski, R. Jones, L.-Y. Leung,S. McGinnis, A. M. B. Nunes, and Y. Qian, EOS ,311 (2009).[18] L. O. Mearns et al. , “The North American RegionalClimate Change Assessment Program dataset, Nation-al Center for Atmospheric Research Earth System Griddata portal, Boulder, CO,” (2014), data downloaded2102-03-28. doi:10.5065/D6RN35ST.[19] L. O. Mearns et al. , Bull. Amer. Meteor. Soc. , 1337(2012).[20] T. A. Greasby and S. R. Sain, “Assessing uncertaintyin climate model ensembles via annual temperature pro-files,” (2012), Unpublished manuscript.[21] T. Greasby and S. Sain, Journal of Agricultural, Biolog-ical, and Environmental Statistics , 571 (2011). Supporting Information for
Tracking climate change through the spatiotem-poral dynamics of the Teletherms, the statistical-ly hottest and coldest days of the year
Peter Sheridan Dodds,Lewis Mitchell,Andrew J. Reagan,and Christopher M. Danforth.See also the paper’s online appendices at: http://compstorylab.org/share/papers/dodds2015c .2 S ummer Teletherm Temp eratures f or f ull data set: T e m p e r a t u r e ◦ F
63 70 80 90 100 110 117050100
S ummer Teletherm Temperature ◦ F C o un t FIG. S1. Summer Teletherm temperatures for the full data set (1853–2012). Teletherm temperatures are determined bysmoothing the average daily maximum and minimum temperatures; see main text for details.
Winter Teletherm Temp eratures f or f ull data set: −11010203040506065 W i n t e r T e l e t h e r m T e m p e r a t u r e ◦ F −11 0 10 20 30 40 50 60650204060 Winter Teletherm Temperature ◦ F C o un t FIG. S2. Winter Teletherm temperatures for the full data set (1853–2012). Teletherm temperatures are determined bysmoothing the average daily maximum and minimum temperatures; see main text for details. N umb er of days f rom the Winter to S ummer Teletherm, f ull data set: W i n t e r t o S u mm e r T e l e t he r m ( da ys )
155 182.5 200 225 250 265050
Winter to Summer Teletherm ( C oun t FIG. S3. Number of days from the Winter to the Summer Teletherm. The vertical gray line in the histogram indicates halfof a standard 365 day year. The variation is substantial with the northeast showing as short a span as just over 5 months andthe west coast as much as 9 months. . S ummer Teletherm—50 year estimates: 1913 to 1962 . O c t − S e p − S e p − A ug − A ug − J u l − J u l − J un − J un − J un J un J u l J u l A u g1 A u g15 S e p S e p O c t C o un t S o l s t i ce FIG. S4. Map of the Summer Teletherms and Teletherm Extents estimated for the 50 year range 1913–1962, to be comparedwith the equivalent map for 1963–2012 in Fig. S5. Fig. 6A in the main text maps the changes in Summer Teletherms betweenthese two periods. Relatively few Summer Teletherms have remained stable with the majority shifting to an earlier date. Inthe bottom left histograms, the gray horizontal line shows the interquartile range and the inverted triangle the median. . S ummer Teletherm—50 year estimates: 1963 to 2012 . O c t − S e p − S e p − A ug − A ug − J u l − J u l − J un − J un − J un J un J u l J u l A u g1 A u g15 S e p S e p O c t C o un t S o l s t i ce FIG. S5. Map of the Summer Teletherms and Teletherm Extents estimated for the year ranges 1963–2012, to be comparedwith the preceding map in Fig. S4. Fig. 6A in the main text maps the changes in the Summer Teletherm between these twoperiods. . Winter Teletherm—50 year estimates: 1912 to 1961 . − M a r − M a r − F eb − F eb − J an − J an − D e c D ec J a n J a n F e b F e b M a r M a r C o un t S o l s t i ce FIG. S6. Map of the Winter Teletherms and Teletherm Extents estimated for the 50 year range 1912–1961, to be comparedwith the equivalent map for 1962–2011 in Fig. S7. Fig. 6B in the main text maps the changes in Winter Teletherms betweenthese two periods. In the bottom left histogram, the gray horizontal line shows the interquartile range and the inverted trianglethe median. . Winter Teletherm—50 year estimates: 1962 to 2011 . − M a r − M a r − F eb − F eb − J an − J an − D e c D ec J a n J a n F e b F e b M a r M a r C o un t S o l s t i ce FIG. S7. Map of the Winter Teletherms and Teletherm Extents estimated for the year ranges 1962–2011, to be comparedwith the preceding map in Fig. S6. See Fig. 6B for a map of the changes. A. 50 year Summer Teletherm Temperature shifts for 1963–2012 relative to 1913–1962: S u mm e r T e l e t h e r m T e m p e r a t u r e s h i f t ◦ F −505 −5 0 50100200300 Summer Teleth er m Temp er atur e shift ◦ F C o un t B. 50 year Summer Teletherm Extent shifts for 1962–2011 relative to 1912–1961: S u mm e r T e l e t h e r m E x t e n t s h i f t ( d a y s ) −35−20−100102035 −35 −20 −10 0 10 20 350204060 Su mmer Telether m Extent shift ( C o un t FIG. S8. Shifts for the Summer Teletherm for A: Temperature and B: Extent derived from Figs. S4 and S5. A. 50 year Winter Teletherm Temperature shifts for 1963–2012 relative to 1913–1962: W i n t e r T e l e t h e r m T e m p e r a t u r e s h i f t ◦ F −10−50510 −5 0 5050100150 Winter Telether m Temp er atur e shift ◦ F C oun t B. 50 year Winter Teletherm Extent shifts for 1962–2011 relative to 1912–1961: W i n t e r T e l e t h e r m E x t e n t s h i f t ( d a y s ) −35−20−100102035 −35 −20 −10 0 10 20 35050100 Win te r Telether m Extent shift ( C oun t FIG. S9. Shifts for the Winter Teletherm for A: Temperature and B: Extent derived from Figs. S6 and S7. A. 25 year Summer Teletherm shifts for 1988–2012 relative to 1913–1937: S u mm e r T e l e t he r m s h i ft ( da ys ) −35−20−100102035 −35 −20 −10 0 10 20 35050100 Summer Teletherm shift ( C oun t B. 25 year Winter Teletherm shifts for 1987–2011 relative to 1912–1936: W i n t e r T e l e t he r m s h i ft ( da ys ) −35−20−100102035 −35 −20 −10 0 10 20 350204060 Winter Teletherm shift ( C oun t FIG. S10. Teletherm shifts comparing the quarter centuries at the ends of the 1912 to 2012. A: Summer Teletherm shiftscomparing the 25 year periods 1988–2012 relative to 1912–1937. Out of all 1218 stations, 716 (58.8%) have ≥
80% error-freedata in both 25 year spans. B: Winter Teletherm shifts comparing 1987/1988–2011/2012 relative to 1912/1913–1936/1937. Atotal of 725 out of 1218, 59.5%, stations have ≥