Open Mind

Gregory XIII

November 23, 2008 · 30 Comments

On February 24th, 1582, Pope Gregory XIII instituted a new calendar system which is now known as the Gregorian calendar. It replaced the former Julian calendar in which each year was 365 days long, except for every 4th year (”leap years”) which were 366 days long. As a result, in the Julian calendar the average length of the calendar year was precisely 365.25 days. The problem was that the tropical year (which determines the passing of the seasons) isn’t precisely 365.25 days long; it’s actually about 365.2424 days (it varies). (Actually, it’s 365.24219 days when “day” means exactly 86,400 seconds as measured by atomic clocks — but it’s about 365.2424 days when “day” means the average duration of the day-night cycle.) In consequence, the average Julian calendar year is too long to match the seasons, by 0.0076 days.

That may not seem like a lot, but in the 1500+ years since the institution of the Julian calendar the differences added up. More relevant to Christian Europe, in the 1200+ years since the Council of Nicea in 325 A.D., which established conventions for determining the timing of religious holidays like Easter which had no fixed date, the accumulated difference between the calendar and the seasons amounted to about 10 days. Since the timing of easter was determined by astronomical aspects and therefore followed the seasons, Easter was slowly getting earlier and earlier in the “year,” a fact which vexed clerics no end.

With the Gregorian calendar, we still have 365 days in ordinary years and 366 days in leap years, but we no longer have a leap year every four years. Years that are a multiple of 100 are not leap years, unless they’re also a multiple of 400. Because of this, every 400 years we omit 3 leap days that would have occured in the Julian calendar. This makes the average length of the year in the Gregorian calendar 365.2425 days — much closer to the 365.2424 average of the tropical year. The calendar excess is only 0.0001 days/yr, so for the calendar to get out of synchronization with the seasons by 1 day will take 10,000 years. Not to worry.

What does this have to do with climate science? As many of you are aware, a large number of data sets used in climate studies are monthly averages. Nothing wrong with that. But because the months are defined as whole numbers of days with respect to the calendar, while the tropical year is not a whole number of days and is not perfectly synchronized with the calendar, the “timing” of any given month with respect to the cycle of seasons is actually variable. This can affect the numbers when the quantity under study shows a seasonal cycle, because the seasonal cycle is governed by the tropical year, not the calendar year. And that’s the subject of a recent paper by Cerveny et al. (2008, Geophysical Research Letters, 35, L19706, doi:10.1029/2008GL035209). Their conclusions are expressed in the abstract:

In this study we address a systematic bias in climate records that manifests due to the establishment of the Gregorian calendar system and exerts a statistically significant effect on monthly and seasonal temperature records. The addition of one extra day in February normally every fourth year produces a significant seasonal drift in the monthly values of that year in four major temperature datasets used in climate change analysis. The addition of a ‘leap year day’ for the Northern Hemisphere creates statistically significantly colder months of July to December and, to a lesser degree warmer months of February to June than correspondingly common (non-leap year) months. The discovery of such a fundamental bias in four major temperature datasets used in climate analysis (and likely present in any dataset displaying strong annual cycles, e.g., U.S. streamflow data) indicates the continued need for detailed scrutiny of climate records for such biases

I’ll state my overall opinion at the outset: although they’ve identified an interesting phenomenon, I think they’ve done a very poor job analyzing it. While the calendar year has an effect on monthly averages of data, they’ve represented the effect in a rather naive and essentially incorrect fashion. I also suspect they may not have done the statistics right.

First let’s look at how they characterize the effect. We don’t even have to get beyond the abstract to see one of the problems:

The addition of one extra day in February normally every fourth year produces a significant seasonal drift in the monthly values of that year…

This thesis, that the real issue is the difference between leap years and non-leap years (which is the very foundation of their analysis), is deeply flawed. The fact is that with or without leap years, there’s a “seasonal drift in the monthly values.” With no leap days (if all years were 365 days long), the calendar year is 0.2424 days short of a tropical year so calendar dates drift backward with respect to the seasons by about 1 day every four years. The whole reason we have leap years is to correct for that drift. The journal should have included an astronomer among the referees!

We can correctly characterize the impact of the calendar year by noting that the phase of any given month, with respect to the cycle of the seasons, changes every year because calendar years are necessarily a whole number of days while the tropical year is not. Ordinarily, the phase of a given month decreases by 0.2424 days (out of 365.2424) every year, but during leap years months after February advance in phase by 0.7576 dyas. In the year after a leap year, the phase of the month of January advances by 0.7576 days. Here’s the phase (in days) with respect to the cycle of the seasons, of the month of January, relative to the year 1850 (for which I’ve arbitrarily set the phase to zero):


Most other months follow a pattern just like that for January, but with “high” phase occuring in leap years rather than the years after leap years. February is an oddball because it’s not always the same length. If we compute the average phase, we find that it advances for two years, then recedes for two years. In any case, we can plainly see that the phase of the months follows a pattern which is periodic, with periods 4 yr, 100 yr, and 400 yr (the plot doesn’t cover a long enough time span to reveal the 400-yr pattern).

While it’s certainly true that leap years have later phase than non-leap years (for most months), it’s equally true that the years before leap years have earlier phase by just as much. To characterize the impact of the calendar as being a difference between leap years and non-leap years is only partly correct; in fact it misses the essential behavior of the system, losing the opportunity to characterize the effect most correctly.

Nonetheless, most months in leap years do, on average, have a later phase relative to the seasons than those in non-leap years. So we might expect that in the northern hemisphere, for about the first half of the year a shift to later phase will represent warmer temperatures while the effect will be reversed during the second half of the year. One of the data sets studied by Cevreny et al. is the CRUTEM3v series (analyzing the hemispheres separately), so I took that data for the northern hemisphere and averaged the temperature anomaly for each month, separating leap years from non-leap years. Here’s the result:


The difference is statistically significant only for the months of August through November. While these late-year months show the expected pattern, that leap years are colder than non-leap years, early-year months show a pattern opposite to expectation, although that result is not statistically significant.

Cevreny et al. also compute averages for February through June, and for July through December, referring to the earlier period as “rising-sun” months for the northern hemisphere while the later period is called “sinking-sun” months; the labels are reversed for the southern hemisphere. I did the same thing, and found a statistically significant difference between leap years and non-leap years for the “sinking-sun” period but not for the “rising-sun” period.

With this analysis I suspect they may have gone astray statistically. First, the average temperature anomalies they report are different than those I computed. This may be due to the fact that I used the hemispherically-averaged CRUTEM3v data sets, while they use the CRUTEM3v gridded data set and later composite the results. The big difference is that they report a tremendously significant t-test value of -19.31, while I got a significant but hardly tremendous value of -2.52. I can imagine one explanation: I worked with the averages for each year, while they may have worked with the individual monthly values when computing the statistical significance. However, the autocorrelation in monthly temperature data has a strong impact on the significance of such a comparison; if not corrected for, it will greatly inflate the apparent significance.

Given that the true impact of calendric phase drift on monthly averages is not encompassed in the difference between leap and non-leap years, and that non-leap years also show phase changes due to the calendar, how should one look for its effect in monthly temperature data? It’s straightforward to compute the phase of each month for each year, as I graphed above for January months from 1850 to 2007. I therefore computed the phase, in days, relative to the phase for that same month in 1850; this is the “phase difference,” positive values indicating the month came later in the tropical year than in 1850, negative values indicating the month came ealier in the tropical year than during 1850. We can then see whethere there’s any correlation between the phase difference and the average temperature anomaly reported for that month.

So I took the northern-hemisphere CRUTEM3v data, and computed, for each month, the correlation between phase difference and temperature anomaly, expressed as “degrees per day”:


The result is statistically significant for May, June, July, October, and November, and for each of those months the direction of the change is what we would expect: warming as phase advances for May through July, cooling as phase advances for October and November.

I did the same thing using GISS data for the northern hemisphere, based on met stations only, giving this:


This time, the result is statistically significant only for April and May, although the numbers generally follow expectation rather well throughout the year.

The phenomenon of phase shift of the months due to the calendar, and its impact on monthly average temperature data, is certainly real. But it should definitely be treated by analyzing temperature anomalies as a function of phase difference rather than simply viewing the issue as one of “leap years” vs. “non-leap years.” Furthermore, I should analyze a larger number of data sets and do so for both hemispheres; this would enable us to quantify the impact on important data sets with much greater confidence.

It’s also clear that the results of simply averaging temperature anomalies may be contaminated by the long-term trends present due to genuine climate change. Therefore there’s probably merit in de-trending the data (on a suitably long time scale) in order to reduce the variance of the anomalies; however, since phases are trending upward throughout the 20th century, this should be done with care. Furthermore, phases were generally higher before the 20th century due to the fact that the year 1900 is not a leap year in the Gregorian calendar. I’m working on all these ideas, and just might submit the results to GRL as a comment on the Cerveny et al. paper.

Categories: Global Warming

30 responses so far ↓

  • David B. Benson // November 23, 2008 at 8:15 pm

    Tamino — Encourage more that “might just”.

  • Gavin's Pussycat // November 23, 2008 at 11:50 pm

    Tamino — yes, go for publication. And while what you have written is interesting and true, don’t forget to address the elephant in the room, i.e., the authors’ claim of a ’significant issue’ with climatic data sets, more precisely their implication that — well, you know what they want to imply, don’t you? And you know how false it is. Rub it in, by showing how the effect propagates into long term trends when combining monthly into annual values — the correct way, using the numbers of days as weights — and both hemispheres into global values.

    Good luck!

    [Response: Indeed their claims of "significant issue" are tremendously overblown.]

  • DrC // November 24, 2008 at 12:43 am

    Huh. It certainly is interesting phenomenon. But the shift is easily modeled as you point out. This is cool to think about!

  • Ray Ladbury // November 24, 2008 at 1:28 am

    I have a colleague who occasionally makes exaggerated claims about his research. I’ve taken to saying that there is less there then meets the eye.

  • crf // November 24, 2008 at 5:31 am

    There are, occasionally, leap seconds added to a year. So, with leap years and leap seconds, the calendar will not get of sychronisation in even ten thousand years. According to wikipedia, they are added, when needed, on december 31 or june 30. So those months will sometimes be one second longer, and get so much more solar energy.

    [Response: Sorry, but leap seconds don't help keep the calendar synchronized with the seasons, they're used to keep the clock synchronized with the day-night cycle.]

  • Sekerob // November 24, 2008 at 10:00 am

    Thought it was funny when mention was made of this paper in another thread and shared it elsewhere. Elephant deflated and back in fridge.

    That said, there are too many data sets that give annuals/quarterlies as simplest division 12 or 3. E.g. on eagle eye watched sea ice extent/area it makes 10k km square and more difference for a quarter or a year just by using the exact days. It’s the difference for the GW refuters to claim “it was not a record” when 2008 barely missed 2007 lows [deep laughter]

  • Bob North // November 24, 2008 at 2:01 pm

    Tamino - would basing the annual average anomalies on the daily values instead of the monthly averages get rid of this apparent phase issue? Would going solstice to solstice help?

    [Response: As far as I know, annual averages *are* based on daily values (or averaging monthly values weighted by the number of days, which is the same thing).

    Going solstice-to-solstice is logical, but wouldn't really help with the phase variation problem. Data that occur on a daily basis (like high/low temperature) require a whole number of days, but the tropical year isn't a whole number of days. The best you can do is never get more than 1 day out of sync -- which is one of the purposes of the Gregorian calendar.]

  • void{} // November 24, 2008 at 5:37 pm

    And how are these situations handled in the GCMs, most of which, I think, use a constant 365 day/year model. Seems like the GCM model year needs to be worked into the calculated numbers whenever they are compared with the measured data sets.

    And, how does numerical-solution-method induced phase shift fit in?

    Didn’t Real Climate have some info on this?

  • Barton Paul Levenson // November 24, 2008 at 7:33 pm

    I want to echo Gavin’s Pussycat on this, Tamino — you should definitely submit a paper to this journal rebutting the other paper. You might have to take down this blog post, though, if the journal considers it “publication” (I know some magazines are that way about stories).

  • dko // November 25, 2008 at 2:12 am

    All the figuring I’ve done has always given each month equal weight. Guess I really should give January credit for being 31/365 or 31/366 of a year, February 28/365 or 29/366, etc.

  • Bob North // November 25, 2008 at 5:01 pm

    Tamino and DKO - From reviewing the downloadable GISS station data, it appears that they do not weight each month according to the number of days when computing the annual averages. They do use Dec-Nov as the default for the annual average however, which confused me until I figured out what they were doing.

  • Stuart Harmon // November 25, 2008 at 5:50 pm

    Talk about clutching at straws. Statistics is a tool. The emperical evidence is overwhelming:-
    1 Temperatures last century are not unusual.
    2 CO2 Levels are not unusual
    3 The rate of change in so called global temperatures is not unusual

  • dhogaza // November 25, 2008 at 6:17 pm

    And the empirical evidence is … a post by some random denialist on a blog?

    Is overturning ALL of science this easy in your mind?

    “The empirical evidence is overwhelming - the earth is flat!”

    Wow, that’s easy! I like this game.

  • Gavin's Pussycat // November 25, 2008 at 7:58 pm

    Bob North, yes indeed… inspecting toANNanom.f (routine annav) in STEP2 tells the same story. First, months are simply averaged (without weighting) into seasons, then, seasons into years.
    Would be interesting to fix, if only to see how it does not noticably change any trends ;-)

  • David B. Benson // November 25, 2008 at 8:05 pm

    Stuart Harmon // November 25, 2008 at 5:50 pm — All three of your points are wrong. Begin by reading “The Discovery of Global Warming” by Spencer Weart:

    Review of above:

  • tamino // November 25, 2008 at 8:33 pm

    I took monthly data from GISS and averaged (over the calendar year Jan-Dec) in two ways: a “simple” average by just averaging the monthly values, and a “weighted” average weighting each month by its number of days (including using 28d for non-leap year Feb. and 29 days for leap-year Feb.). I then computed the trend rate indicated by these annual averages for the modern global warming era, from 1975 to 2007 (the last complete year available). The results: for the simple average the rate is 0.01817318 deg.C/yr, for the weighted average the rate is 0.01815870 deg.C/yr. The difference in the estimated trend rates is 0.00001448 deg.C/yr, not nearly enough to worry about.

    The average difference between the simple and weighted annual averages is 0.000006 deg.C, the standard deviation is 0.00095 deg.C. Again, not nearly enough to worry about.

    Nonetheless, it’s still a good idea to compute a weighted average rather than a simple average.

  • Dave A // November 25, 2008 at 10:54 pm


    I’m sure I have asked you this before but are you Spencer Wearts agent?

  • aliunde // November 25, 2008 at 11:25 pm

    Gregory XIII: “The journal should have included an astronomer among the referees!”

    And by the current analysis, the peer reviewers also needed a statistician.

    Doesn’t that say something more important about the peer review process than the paper?

  • David B. Benson // November 26, 2008 at 12:27 am

    Dave A // November 25, 2008 at 10:54 pm — Not in the sense that he pays me, or even asked me. I’ve never even had so much as an e-mail from him.

    It is simply when folks demonstratrate their ignorance of matters climatological, I like to try to send them to a good read.

    Its a starter.

  • Ray Ladbury // November 26, 2008 at 2:31 am

    Aliunde, what it says about peer review is that it is a threshold, not a ceiling. It is not a guarantee of correctness–merely an indication that some of our peers thought it would be of interest to some portion of the community. In other words, it tells those of us who actually understand science what we already knew.

    Like it or not, science works.

  • aliunde // November 26, 2008 at 8:59 am

    Given my post grad work in NWP, I’d certainly agree that science works. Politicized science, however, is another matter (and another debate).

    I would disagree with your characterization of the purpose of the peer review process. It certainly is meant as a “ceiling” for bad science. That’s its primary purpose. If the referees do not have the requisite qualifications to identify scientific error in the paper being reviewed, it says a great deal about the present state of the peer review process and the quality of what is printed in the pages of the various journals, whose reputation the peer process is intended to protect.

  • Gavin's Pussycat // November 26, 2008 at 10:20 am

    Tamino, thanks… I didn’t mean it as a literal request. But still… intuition good, computation better. This too belongs in your GRL comment.

  • Ray Ladbury // November 26, 2008 at 3:40 pm

    Aliunde , Horse Puckey. Bad papers have always gotten past peer review. What matters is whether they get past the community and wind up being cited frequently in subsequent research. This one didn’t.

    Now as to the politicization of science. Care to cite an example? Maybe scientists being subpoenaed to appear before Congressional committees? Oh wait! That was those whose research supported anthropogenic causation! The science has gone on despite these distractions and continues to improve despite them. It is leaving fewer and fewer dark crevices into which denialists can scurry.

    Science! It still works, Bitches!

  • Former Skeptic // November 26, 2008 at 7:11 pm


    Given that this paper was published in October 2008, don’t you think it would be rather disingenuous to suggest that this paper hasn’t been cited much because it is bad, and not because it is only a month old? :P

    As for the quality of this paper…I will await to see Tamino’s comment in GRL on this before commenting further.

  • aliunde // November 27, 2008 at 5:21 pm

    Ray Ladbury // November 26, 2008 at 3:40 pm

    Aliunde , Horse Puckey. Bad papers have always gotten past peer review.

    I don’t think you’re helping your case.

  • dhogaza // November 27, 2008 at 5:45 pm

    I don’t think you’re helping your case.

    He doesn’t have a “case”. He’s explaining reality.

  • Ray Ladbury // November 27, 2008 at 5:48 pm

    Aliunde, it is interesting that for someone who claims to have studied science, you utterly fail to grasp how it works. Peer review is one of many tiers in the defense against bad science. If a paper gets past peer-review, it still faces many other tests :
    1)Does the community AS A WHOLE find it correct and cogent?
    2)Is it sufficiently useful that others incorporate its methods, cite it, etc.?
    And so on. In the end, you wind up with the best approximation of truth humans can devise–and best of all, it is self correcting. Maybe you ought to go back and review what you claim you learned.

    Former: I was speaking in general terms about how science works. However, as I said, there’s less to the paper than meets the eye.

  • Gavin's Pussycat // November 27, 2008 at 6:10 pm

    > Ray Ladbury // November 26, 2008 at 3:40 pm
    >> Aliunde, Horse Puckey. Bad papers have always gotten past peer
    >> review.
    > I don’t think you’re helping your case.

    In related news, spam filters the world over are letting through thousands of unsolicited bulk emails — like they have always done, and will do.

    To rub it in: it’s a compromise between errors of the first and of the second kind. Having to read junk now and then in reputable journals is the price we pay for “unconventional” or non-mainstream but valid results making it to publication. Well, most of the time anyway.

    And there is the “comment” mechanism for dealing with the junk.

  • RomanM // November 30, 2008 at 3:02 pm

    I would suggest that you have not looked at January properly. The effect of an extra day in leap year affects January of the following year and not the year in which the extra day occurs. The difference is larger than shown in your second graph.

    Similarly there are three “types” of February. Normal, leap year (an extra “warmer” day) and the year after leap year (shifted, like January).

    Using the “variance adjusted” CRU data artificially reduces the estimates of the actual variability of the data while possibly introducing bias due to the manipulation of the data.

    [Response: I'm aware of the effect, as stated in the post that "Most other months follow a pattern just like that for January, but with “high” phase occuring in leap years rather than the years after leap years." But I was reproducing the analysis of Cerveny et al., and apparently they were not aware of it.

    The right way is to analyze the relationship between monthly mean temperature and phase relative to the seasons, for which I've correctly computed the phases of each month (including the fact that January phase isn't shifted forward due to leap years until the year after a leap year).]

  • Eli Rabett // November 30, 2008 at 3:32 pm

    Probably the best thing to do is to decompose the record using a sawtooth function.

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