Starting Thoughts

Franco Zavatti
April 4, 2017

My general comments about pre-built sites and CMS

I have been forced by WordPress to create this site in order to post comments at the Judith Curry’s blog “Climate Etc.”.

I don’t like pre-build sites and also CMSs.

I don’t know why, but these structures do always what they want and never what I want!

So, I prefer to write my html pages by a text editor (I use joe under Linux), following the html 3.0 and some of 4.0.
I don’t like HTML v5.0 with its configuration style sheets (CSS) also if cannot discuss its usefulness in other situations.
I use (and like so much) html because it is an easy and clear way to take (very readable) notes if, and only if, I can use a plain version of it, without the impressive amount of tags introduced by a html editor or a CMS: after them I no more can see (and read) the text without a browser and that is not a good practice for me.

My own site was originally published on November 16, 1995 and from that date I have written thousands html pages, all of them by the above mentioned text editor. Among these pages there are a course of Astronomy (named IperAstro, ~2GB) and a 30-hour lecture on Didactics of Geography  (DDG, ~1.2GB).

As you can easily see, my pages don’t follow the actual “golden” rules for html pages. They never include author’s name, date, language, and so on, between the head and the /head tags; also, I widely use “deprecated” tags (such as u, /u or b, /b ), so my pages are quoted very low by the search engines,  but I don’t need to sell anything: the site serves as an easily accessible repository of stuff which meets my interests and then the quotation of my pages doesn’t matter.

I am a retired astronomer and teacher (1991-2010 Physics Lab for students of the Astronomy degree and 1996-2000 Probability and Statistics for Informatics, Cesena) at the Bologna University  and  also gave my lectures on Didactics of Geography at the Free University of Bolzano [Bozen],  (2001-2013). After my retirement (1st January 2011), I dedicated myself to camping (more than 4-months per year), food (I like anything, mainly pasta and pork but vegetables, fish and cakes are also welcome) and climate within the skeptical side (in the sense that I am skeptic about a predominant human influence on climate with respect to natural variations).

Actually, I am a regular contributor (94 articles) of the climate blog  Climate Monitor owned by Guido Guidi. Of course my posts are (and have been) directly written in html with joe.

I am really proud of the 2017 publication of a scientific paper on Climatology, published in a peer-review journal (IF 4.9), because this research field is not the one of my original training. A second paper will be submitted to the same journal in the next few days  and two (may be three) more are in preparation.

I discovered that can write the html directly here, also if with some limitation (e.g. cannot write the attribute size=”+1″ of the font tag) and with some arbitrary addition or changes of tags by the CMS (e.g. b, /b  changed to strong,  /strong, with a space and characters waste. Also, cannot use pre, /pre tags, that I always found useful for my scopes.

Perhaps, in spite of the above limitations, I will add more frequently some new stuff here.

My best regards to all
Franco Zavatti



Hurst Exponent, Persistence and Spectra

Franco Zavatti
February 17, 2018

The Italian, original version of this paper has been published at ClimateMonitor.it as Part I, Part II and Part III. The present version includes new applications and a “proof” of the goodness of my choices. fz.

Sostituisco il dataset iniziale con le sue differenze prime (o con le derivate numeriche in caso di passo variabile) e verifico sperimentalmente se l’esponente di Hurst H, cioè il livello di persistenza cambia nel senso che si avvicina maggiormente al valore 0.5 mentre il dataset trasformato mantiene l’informazione spettrale del dataset originale. Applico la modifica alle medie annuali NOAA di anomalia di temperatura, all’ultimo dataset mensile, sempre NOAA, al livello del lago Vittoria i cui dati non sono a passo costante e ad altri sette dataset di varia natura.

I change the original dataset with its differences or its numerical derivatives in case of a variable step and look at the Hurst exponent H. If its value is lowered by this procedure, I verify if the new dataset contains again the spectral information of the original one by comparing their spectra. Such a procedure has been applied to the yearly global temperature anomaly and to the last available monthly data from the NOAA GHCN cag-site. Also the lake Victoria levels (at variable time step) has been used to test the procedure.

After this paper had been written, I did find the site https://terpconnect.umd.edu/~toh/spectrum/Differentiation.html, part of A Pragmatic Introduction to Signal Processing by Prof. Tom O’Haver , Professor Emeritus, Department of Chemistry and Biochemistry, The University of Maryland at College Park.

From this site I quote the phrase: “The derivative of a periodic signal containing several sine components of different frequency will still contain those same frequencies, but with altered amplitudes and phases.

So, I assume that this statement is a proof of the conservation of spectral information, widely discussed in what follows, within the paper, during the transition from observed to difference/derivative series also if I did not check it in a Signal Theory textbook.

It also justify the ratio between observed and difference spectral peaks.

We have noted many times that the persistence affects several datasets we normally use in climatology (and not only in it). Persistence concerns measures that tend to reproduce previous results, shows autocorrelation among them and a probable dependence. The autocorrelation function at lag 1 (i.e. acf(1)) assumes values greater than 0.5 and denotes that “normal” statistics can no more be used, being based on independent data.

I always remember, mainly to myself, that independent data are uncorrelated, while the opposite is untrue: correlated data are not necessarily dependent. If data are correlated, then their “physical” dependence must be proven by another method or in another way.

As an example, the standard deviation of the mean of a sample

stddev(Xn)=σ/sqrt(n)=σ/n0.5 (1)

computed from a n-dimensional sample, σ being the (common) standard deviation of sample elements, becomes, in face of persistence (Koutsoyiannis, 2003):

stddev(Xn)=σ/n(1-H)      (2)

where H is the Hurst exponent (or coefficient). If H=0.5 equations (1) and (2) become the same, so we can say that, if H=0.5; the series we derived H from, does not show persistence and -at some extent and not being correct at all- that random variables (rv), whose values represent the actual series, are
independent from each other (we could prove the independence e.g. by verification that series, are the probability density of both the rv is given by the product of the single densities

where x,y are the rv and f(), g(), h() the probability densities).

Hurst exponent derives from an estimate in a simplified procedure, described e.g. in Koutsoyiannis (2002, 2003), not so simple for me. So, as an estimate of H, I use equation (5) of Koutsoyiannis (2003) or, being the same, equation (17) of Koutsoyiannis (2002)

ρj(k)j=H•(2H-1)•j2H-2 (3)

(it can be proved this equation is independent from k)

where ρj is the autocorrelation function at lag j or acf(j), j>0.

So, if I fix lag 1, equation(3) becomes

acf(1)=2H2-H or 2H2-H-acf(1)=0,

from which H can be derived as:

H=(1+sqrt(1+8•acf(1)))/4. (4)

In such a way I can get an estimate of H from the acf(1) (what of course implies the acf computation). I need to note acf is a positive function between 0 and 1; if a numerical procedure produces a negative values for acf(1), equation (4) gives an indefinite result (NaN, not a number). We can assume negative results as fluctuations around zero and assign to them an average zero-value, so equation (4) gives H=0.5 (i.e. uncorrelated data).

In what follows, I will define a procedure that, hopefully, reduces or cancels the persistence in a climate variable and apply spectral analysis to the “corrected” dataset (i.e. the one at reduced/null persistence). The last operation implies the “corrected” data will conserve the information content of the original data (or, at least, the spectral information): I am not able to prove such an hypothesis in a general case, so, after the first example, I’ll apply the same procedure to several datasets in order to verify for any single situation the so-called conservation of the (spectral) information content after a transformation.

Annual NOAA-NCDC Temperature Anomaly

The first example where I apply the above mentioned procedure is the annual average anomaly of NOAA global (earth+ocen) temperature. I own data from 2011 through 2017 and show here the 2017 ones.

Computing the acf and the Hurst exponent via equation (4) gives a Hobs=0.975, a large value implying a strong persistence (autocorrelation) among the data.
An article by Roman Mureka at WUWT shows that the differences between successive values of a dataset can reduce dependence (his words are:
“… it might not be unreasonable to assume that the annual changes are independent of each other and of the initial temperature”) and I add that the difference dataset looks like “more casual” or “less structured” (whatever such terms could mean) than the original data.

The method used by Mureka (by the way, it was applied to the same annual anomaly but in another context) appeared interesting to me and the procedure itself easy to be implemented, so decided to apply it to the 2017 annual NOAA anomaly (hereafter noaa-17). Result is in figure 1

Fig.1. upper panel: 2017 annual average anomaly, NOAA. central panel: Differences d(i)=t(i+1)-t(i). bottom panel: Detrended values, computed from the line in the upper panel compared to a fixed sine-wave.

The comparison in figure 2 between the autocorrelation functions (original vs. differences) shows an impressive improvement (reduction) of the persistence.

Fig.2. Comparison of observed and difference acfs. Hobs=0.975; Hdiff=0.5.

Now I can apply spectral analysis (say MEM [Childer 1978; Press et al. 2009] or LOMB [Lomb, 1976; Scargle, 1982] methods, which is my final scope) to the difference in order to obtain a more reliable spectral structure for annual anomaly if and only if the differences conserve the (spectral, at least) information content of the original data. As stated above, I’m not able to prove the hypothesis, so I’ll compare the spectra of both datasets shown in the next two plots (figure 3 and figure 4)

Fig.3. Original NOAA global anomaly and its MEM spectrum.


Fig.4. As in figure 3, for the differences of anomaly.

A direct comparison between spectra shows they are very similar in spectral peaks positions (periods), the only variety being the ratio among peaks height and the better definition of the long-period maxima in the difference spectrum. This one cannot be a proof, but surely may be a strong suggestion about the conservation of the information content of the differences dataset. Also, the above plots indicate the persistence does not affect the spectrum, at least in this case of annual global anomaly.

Actually, this supposed conservation of the information must be experimentally confirmed for any dataset, before whatever conclusion can be drawn.

A synthetic summary of the section is:

  1. I can extract from a autocorrelated dataset a new series without persistence, from which
  2. I can derive a more reliable spectral analysis, possibly not distorted by an high Hurst exponent.
  3. For a given dataset I must demonstrate the conservation of the information by comparing both (original and differences) spectra.

After this first example, I can apply the procedure to a variety of climate data in order to verify its reliability (and also if trust/untrust the statements listed in the first section of this paper).

Lake Victoria level

The lake Victoria series has a Hurst exponent H=0.962, so it is good choice for the actual procedure, the main difference with the NOAA data being the variable data step. This implies the differences must be computed per unit of the abscissa, i.e. it needs the ratio Δy/Δx, or the numerical derivative of the dataset (the same computed above, but with Δx=1).

The present one being the first time I use the procedure in a derivative/difference context, I do include both the transformations of the lake Victoria series, so that they can be compared with each other, as “obs”, “deriv” or “der” and “diff” outputs.
In the meanwhile I note that Hderiv=0.781 and Hdiff=0.638, large enough to push toward the little effectiveness of the method in reducing or nullify persistence. Nevertheless, the spectral analysis could be affected in a positive sense by the lower Hurst exponents.
Figure 5 shows the comparison among the acfs of the series.
Fig.5. acfs of lake Victoria. black Original data. blue: Absolute differences. red: Numerical derivatives. Both transformations show well visible improvement of the original autocorrelation. It should be noted that at lag 1 the acf of the derivatives is more than the double of the differences acf. Here Hobs=0.962; Hderiv=0.781; Hdiff=0.638.

Fig.6. Original serie of lake Victoria level and its LOMB spectrum. In what follows (figures 7 and 8) the main spectral feature at about 78 year appears as a macroscopic spurious shape due to the persistence, while the lowest periods remain also in the “transformed” spectra.

Fig.8. Absolute difference (i.e. not referred to a time-base) between lake Victoria levels. The peak at 34.4 year doesn’t appear in figure 7 and some minor variety is visible around 3-4 year.

Also in lake Victoria (strong) differences among power ratios of near-period peaks appear.

Monthly NOAA-NCDC Temperature Anomaly
We can suppose a resemblance between annual and monthly NOAA data but it is better to directly verify such possible common behaviour. So, I use here the last available monthly dataset at NOAA cag (climate at a glance) site: the series referred to December 2017 (here named 1712t.dat), from which I computed the differences and, from both, the acfs of figure 9.
Fig.9. acf of observed (black) and differences (blue) of December 2017 NOAA monthly anomaly. The enhancement of the persistence is evident. Here Hobs=0.983 and Hdiff=0.5.

The persistence has been totally reseted by the transformation and the spectra in figures 10 and 11 cleanly show that the spectral information has been mantained through the transformation procedure. In short, we can read here the same novel as above, for the annual data: we observe the same spectral structure and a sharpening of the ~60-year peak.
Fig.10. Global monthly anomaly through December 2017. A comparison with the black line of figure 9 shows how much the persistence is strong here, much more than in annual data. In the central frame we can note the weakness of the ~60-year peak identification.
Fig.11. Differences of 1712t.dat monthly anomaly and its MEM spectrum. The peak at ~60 year is well visible here.

From the spectral analysis of monthly data we can derive the same
spectral structure as the annual data and the confirmation that, with the
generic NOAA dataset, the persistence has little (if not none at all) effect
on the spectrum; correcting autocorrelations acts only on a better
definition of the ~60-year spectral maximum. Again, the differences works
well in nullifying the persistence.
In some a way the above three tests define a fixed point within the present work, so allow some

Intermediate thoughts.

While the above statements for NOAA data hold also in the general case, I must outline that the enhancement of the persistence is not the same in any dataset and for any climate variable. Lake Victoria levels shows the differences and derivatives did not cancel the persistence at all, but in any case give rise to a noticeable restoration of the spectral structure with very good resemblance between the spectra and the not-significant variety of the periods.
In the same time, the applied transformation allows to cancel spectral peaks (like the ~40 and ~78 year ones) whose significance has been discussed without understand their origin.
It should be also noted that, in spite of a diversity for longer periods of the “observed” spectrum of lake Victoria with respect of the other two ones, the shortest periods are common to all spectra, perhaps suggesting the persistence acts differently along the spectrum.

I think the present procedure, i.e. the use of differences/derivatives, generates uncorrelated data which contains the information (at least the spectral one) of the original data, allowing a more reliable spectral analysis. At the same time, I think we need to verify the improvement in any single situation.

Nile: annual minimum level, 622-1469 CE
The Nile river series is linked to teh lake Victoria leveò because tha lake is the source of the White Nile, while the Blue Nile takes its origin from the Ethiopian Highlands. They converge near Khartoum, Sudan and the two arms become “the Nile”.

The series of the Nile annual minimum level (site visited 20 November, 2017) has a Hurst exponent Hobs=0.833, high enough to justify the use of the differences. Figure 12 shows how the transformation can nullify the persistence.
Fig.12. Observed and difference acfs of the
Nile minimum level, 622-1469 CE. Hobs=0.833; Hdiff=0.5.

In the next figures 13 and 14 “observed” and “difference” spectra will be compared.

Fig.13. Observed annual minimum level of the Nile river and its MEM spectrum.


Fig.14. Differences between successive values of the Nile series and their spectrum.

A comparison between the spectra show that:

  1. An almost invisible “wave” in the “observed” spectrum at 320-340 year is enhanced in the differences as a peak at 308 year.
  2. An “observed” maximum at 242 year is present in the differences as decomposed in a main maximum at 199 year and two minor ones at about 225 and 260 year.
  3. “observed” maxima at 82.6 and 99.6 year become 89.1 and 96.7 (the last one not marked in figure 14). The main 89.1 maximum is better defined than the “observed” one.
  4. The shortest-period maxima are the same in both spectra, but I note the lack of a 37.2 year “observed” period, apparently substituted by a 40.3 year peak in the differences.
  5. The ratios between the power of “observed” ad “differences” peaks appear larger than previously.

TPW: Total Precipitable Water

The climate variable TPW is strongly linked to temperature and gains its largest values along the Pacific equatorial belt, mainly in the areas of the Indonesian sea, named the “warm pool”, where the Pacific water, pushed by the Trade Winds, accumulate during El Niño events. The above-mentioned strong link is outlined in figure 15

Fig.15. The HadCrut4 global temperature anomaly compared with a scaled TPW. Within any evidence, the plots refer to the same

TPW data is available for two latitude belts: ±20° and ±60°. Here, only the wider belt has been used.

The shapes of figure 15 and the NOAA anomaly of figure 1 push to think about persistence and the use of the difference, so figure 16 appears a natural process

Fig.16. TPW Observed and difference acfs. No need to highlight the enhancement. Here Hobs=0.968 and

Spectral analysis of observed and difference data confirms again the little or null influence of pthe persistence on the spectral periods of such a kind of data. Also a ratio among the powers (heights) of observed and differences spectral peaks appears, as usual. Here, some doubt can rise, due to the short time extension of the dataset (30 years) but I think they can be resolved in the best way by a comparison with the above NOAA data analysis.

Fig.17. TPW observed data for the ±60° belt and its MEM spectrum.

Fig.18. TPW differences and their MEM spectrum.

OHC: Ocean Heath Content (0-700m)

I use here only the data relative to the global ocean (0-700m). The constant data step is 1 year and the range 1955-2015 is covered.

Hurst exponent for observed data is Hobs=0.970 and becomes Hdiff=0.468 after the transformation, as in figure 19.

Fig.19. acfs of OHC and its differences. Hobs=0.970 and Hdiff=0.468.

This is another situation where the persistence is high in the observed data and null after the differences. Both spectra, figures 20 and 21, appear similar in their structure, with some minor variety in evidence: 4.1 year not present in the observed spectrum and the shallow 30.5 year in the observed is a “desaparecido” in the differences.

Fig.20. Observed OHC (0-700m) and its MEM spectrum


Fig.21. Differences of OHC and their MEM spectrum
Dendrology: tree rings, russ243, 1540-2004 CE

Here the so-called “observed data” is the average over the 45 available cronologies measured at the Skahalin Island (Russia).
Its acf, along with that of the differences is in figure 22.

Fig.22. acf of the dendrological series russ243mm and the one of its differences. Observed acf(1) tells us about a weak persistence (Hobs=0.809 that, nevertheless, is totally cancelled by the differences (Hdiff=0.5).

Here the climate variable is the ring width (in microns) and NOT the temperature, due to large and well known problems of the calibration process, because ring growth doesn’t depend only on temperature but on many meto-climatic and geological factors.

Comparing the observed, figure 23, and the difference, figure 24, spectra tell us that the persistence can be corrected also through a 450-year time range and that, again, the information content doesn’t change after the transformation.

Fig.23. Average dendrology of russ243 and its MEM spectrum.


Fig.24. Differences d(i)=w(i+1)-w(i) of the average dendrology russ243 and its MEM spectrum.

The actual spectrum shows the most severe difference of all the up-to-now transformation processes, also if some spectral maxima appear in both spectra and some period variety doesn’t seem to be significative.

Kinderlinskaya Cave, Russia, Souther Urals

This is a δ18O series, spanning all over the Holocene for 11 ka (ka=kiloyears=1000 years). Data are available from NOAA paleo site and its reference paper is Baker et al., 2017. The time line is referenced as BP2k (i.e. the present is the year 2000) and the series is the longest one I did apply the difference method to. Also, the Hurst exponent has the larger value among those actually available Hobs=0.995. Comparison between observed and differences acfs is
in figure 25, where an amazing enhancement of the persistence clearly appears.

Fig.25. acf of δ18O, Kinderlinskaya Cave, and its differences. We have a very high persistence in the observed data, totally nullified by the differences. Hobs=0.995 and Hdiff=0.484.

From the spectra in figures 26 and 27 the following information can be derived:

  1. Differences show structures which correspond to the “jumps” in δ18O values, so that they don’t appear uncorrelated, also if the acf says the opposite.
  2. Diversity between spectral maxima are real but the time extension of the data (almost 12000 year) make the 500-year maximum difference poorly significative.
  3. Several spectral peaks are present in both spectra, in that confirming as the information propagates from observed to difference data, also through a so-long time range.


Fig.26. Observed δ18O data and its LOMB spectrum.


Fig.27. Difference δ18O data and its LOMB spectrum. Several maxima are common to both obs and diff spectra.

Stockholm tide gauge 1774-2000 CE

The Stockholm tide gauge is the longest series in the world; data, available at the PSMSL site (Permanent Service for Mean Sea Level), include monthly values and the annual means I use here. They show some breaks at the beginning of the dataset, so derivatives have been used as transformation function. The respective acfs are in figure 28.

Fig.28. Observed and derivatives acfs. ACF(1) of “deriv” is zero. Persistence has been cancelled by derivatives also if with large oscillations. Hobs=0.950 and Hdiff=0.523.

Spectral comparison (figures 29 and 30) shows again the above-mentioned power ratio among peaks, also if the spectra are very similar. An exception is the maximum at 94.2 year, present oncly in the derivatives, without any signal in the observed data.

Fig.29. Sea level at Stockholm, annual means, and its LOMB spectrum.


Fig.30. Numerical derivatives of the Stockholm tide gauge and their LOMB spectrum. Longest periods (94, 140-160 year) show some differences while the shortest ones, mainly those “El Niño” like, are mantained in both spectra.

Concluding remarks

Actually, mainy due to the lack of opposite proofs, the use of the differences/derivatives with the scope to eliminate the persistence and bring the Hurst exponent to H=0.5 (i.e. uncorrelated data) appears a really effective method.

I’m not able to prove the general statement that differences conserve the information content of the original data, so tried at least to verify that this is true in a variety of concrete situations which covered various: steps, time extension, climate variables and persistence content.

The enhancement of the persistence pushes me to think the actual procedure allows to overcome the problem of the autocorrelation, at least as far as spectral analysis is concerned.

After the due tests, I can suggest the differences/derivatives spectrum gives the best available results.

All plots and data relative to this article can be found at the support site here.


  1. Alexander W.J.R., Bailey F., Bredenkamp D.B., van der Merwe A. and Willemse N., 2007. Linkages between solar activity, climate predictability and water resource development Journal of the South African Institution of Civil Engineering, 49(2), 32-44, 2007. Full text available at https://saice.org.za/downloads/journal/vol49-2-2007/vol49_n2_e.pdf
  2. Jonathan L. Baker, Matthew S. Lachniet, Olga Chervyatsova, Yemane Asmerom and Victor J. Polyak: Holocene warming in western continental Eurasia
    driven by glacial retreat and greenhouse forcing
    , Nature GeosciencePUBLISHED ONLINE: 22 MAY 2017. doi:10.1038/NGEO2953
  • Childers, D.G. (Ed.), 1978. Modern Spectrum Analysis. IEEE Press, New York (chapter II).
  • Koutsoyiannis D.: The Hurst phenomenon and fractional Gaussian noise made easy, Hydrological Sciences-Journal-des Sciences Hydrologiques47:4, 573-595, 2002. doi:10.1080/02626660209492961
  • Koutsoyiannis D.: Climate change, the Hurst phenomenon, and hydrological statistics, Hydrological Sciences-Journal-des Sciences Hydrologiques,48:1, 3-24, 2003. S.I. doi:10.1623/hysj.481.3.43481
  • Koutsoyiannis D.: Nonstationarity versus scaling in hydrology, Journal of Hydrology, 324, 239-254, 2006. doi:10.1016/j.jhydrol.2005.09.022
  • Lomb, N.R., 1976. Least-squares frequency analysis of unequally spaced data. Astrophys. Space Sci., 39, 447-462, 1976.
  • Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 2009. Numerical Recipes in Fortran. Cambridge University Press India Pvt. Ltd., New Delhi, 2009.
  • Scargle, J.D., 1982 Statistical aspects of spectral analysis of unevenly spaced data. Astrophys. J., 263, 835-853.


In Italy also the unborn children know that: CO2 is not so good.

Franco Zavatti

Written on the wave of two articles on Climate Monitor here (in Italian) and on WUWT here, where the fall of sexual wish caused by anthropogenic global warming is commented. Well aware of some kind of link between sexual desire and birthrate, I asked myself if the last one could show some link with meteo or climatic parameters and look for such a supposed relationship from Italian census office and international climate data.
In detail, I tried to pair births in Italy, month by month and averaged over the 6 years 2009-2014, with monthly precipitation of 5 northern Italian cities, averaged over time ranges between 200 and 212 years (Padua, Pavia, Udine, Turin, Milan). Yes, I’m aware that whole-Italy births and only the northern precipitations have been used here, but would not give origin to a … last-biberon war for an article whose ultimate vein is a sarcastic or also a funny one.
As a further step I used the monthly Mauna Loa CO2 data between 1959 and 2015, averaged over the observed 57 years and did compute the mean value of all Januaries, all Februaries, and so on (it must be remenbered that CO2 is a well-mixed gas in the atmosphere, so I can refer Hawaii data to Italy).
All information is summarized in figure 1 where in a) right scales refer to precipitation (red) and CO2 (blue) and the left scale to the number of births (black). The upper scale shows the month of conception.
In the b) frame the Cross-Correlation Functions (CCFs) between born and precipitation are plotted (red) and
born and CO2 (blue).born-ave Fig.1. a) Thick line (black)– births for month of the year, averaged over the 6 years 2009-2014; Dashed line and full circles (red)– average of monthly precipitations of 5 Northern Italy cities, over periods between 200 and 212 years; Full line and full diamonds (blue) -monthly CO2 concentration (Mauna Loa, from NOAA, in ppm) averaged over 57 years (1959-2015). b) Cross-Correlation Function (CCF) between average monthly births and average precipitation (red line); CCF between averaged monthly births and CO2 monthly average concentration (blu line).

Frame b) tells us that births are uncorrelated with precipitation, i.e. babies born (decide to born?) independently from rain or snow fall (CCF at lag zero, the Pearson correlation coefficient, is 0.26).
On the other hand, from the strong inverse correlation (at lag=0 CCF is yet -0.72) between birth and CO2 it can be derived that less babies do born when the yearly wave brings the CO2 to higher-than-average values and, vice-versa, more babies do born when CO2 values are below the average. May be we can derive that the Italian unborn babies -surely aware of the validity of the AGW theory- decide to born in the best available conditions.
Our babies are very, very smart … Really!
At the moment we are not able to know if such a consciousness includes the fatidic 97% of the “clients” (babies): at first sight it would seem this is not the case but we need further studies, also with the support of global models, of course.

In conclusion two important highlights:

  1. Italian babies appear to be more “sharp” in their choices than their
    parents. If we observe the month of conception (December-January for the maximun of the birth rate and July as an intermediate month during the minimum of birth rate) we can see a preference for mean values of CO2 and then a minor consciousness about the risks from AGW. So, a question arises: would the Italian parents be a little skeptical about the AWG? And, following the new political tendencies, could they be notified to the authority for such a deviance?
  2. A long list exists which shows the risks due to the climate change but here we are speaking about a benefit, added to a greener world and improved crops production: smarter babies (then, what a pity, they grow, deviate and became parents).

Originally written: April 16, 2016.

Spectral Analysis of Proxy Data

Franco Zavatti
January 27, 2018

Riassunto: Viene mostrata l’analisi spettrale di tre serie proxy tra quelle citate da Abbot e Marohasy(2017) in un lavoro di applicazione a reti neurali artificiali (ANN) per confrontare il contributo antropico rispetto a quello naturale nel cambiamento climatico.
Abstract: Spectral analysis of three proxy series quoted by Abbot and Marohasy (2017) is presented. These authors use artificial neural network (ANN) in order to compare antropogenic and natural contribution in climate change.

Abbot e Marohasy (2017, hereafter AM17) use an artificial neural network (ANN) to simulate 20th century temperatures starting from the analysis (also the spectral one) of the available temperatures through 1830. These temperatures are also derived from proxy data, mainly from tree rings in various places of the Globe. Also data from a stalagmite in China and information from archaeological reserch and oral tales from north-east Greenland have been used.

When the (ANN) model has accumulated the information, it serves to extrapolate the 20th century temperatures. The difference between the observed and computed temperatures is then used as  a signature of the human contribution to the global warming. Authors find the natural anthropogenic contribution to be much more important than the anthropogenic one.

In summary: the neural network receives as input the spectral parameters -period (or frequency), amplitude and phase derived from the spectra of the proxy series- and its output is a series of air temperatures “forecast” for the 20th century, in essence a model of the temperatures to be compared to what is really observed.

As it is well known, IPCC AR5 report defines that the average global surface temperatures rise of 0.85 °C from 1880 through 2012 and that the warming from 1900 depends on the concentration grow of greenhouse gases caused by human activities.

Such an attribution has been based on GCMs (Global Climate Models) from which  an Equilibrium Climate Sensitivity (ECS) of 3.2°C derives.  In practice, a doubling of the concentration of (mainly) the CO2 would bring to an average global temperature rise of 3.2°C (the estimate has an time horizon, i.e. that of the models, and refers to the year 2100. Here we can assume that this horizon always exists).

The concept of ECS was born from a 1896 work by Arrhenius forecasting a 5-6°C rise. ECS estimate went to more and more diminish but the one derived from global models is clearly higher than what derives from experimental spectroscopy methods whose values are set to about 0.5-0.6°C (and, in some cases, also lesser than it). From ECS definition appears that, if the estimates would be those from spectroscopy, AGW and more its catastrophic version or CAGW could not exist at the political desk. The social consequences would be deep for any of us.

In AM17 air temperature -both regional and global- models from neural networks give estimated values of ECS=0.6-0.8°C. This result is similar to the spectroscopic one and to that from energetic imbalance models (0.4-2°C) and far from GCMs’ and paleoclimate approach(2-3°C).


Use of spectral analysis in order to obtain sine waves to be the input for a neural network include both use of proxies and selection of the available periodicities in literature: M17 lists  huge references of spectral results and characteristic periods referred to natural behaviours (Sun, Oceanic oscillations as AMO, NAO, El Niño and their internal relationships).

I tried to verify the real existence of the spectral periods quoted in literature but have faced myself with the difficulty to acquire the datasets used by the respective authors.

Wilson et al., 2007

I really appreciated  the work of Wilson et al., 2007, an accurate spectral analysis of temperature series derived from tree  rings of 22 sites (and 9 ones analised but not used) in the Gulf of Alaska (GOA). The paper does not give the initial data (air surface temperature along January to September) also if they are a complex combination , followed by calibration and detrending, of what is observed at the selected sites. Wilson limits himself to show plots I cannot manage and spectral periods obtained by MTM (Multi Taper Method): 10-11, 13-14 and 18-19 years and by SSA (Singular Spectrum Analysis): 14.1, 15.3, 24.4, 38, 50.4, 91.8 years. So, it is impossible to verify if the series, which ranges from 700 to 1999 CE, shows periods shorter or longer than the quoted ones.

Rob Wilson appears as an accurate scientist, well aware of the “fake news” which can derive by the use of the proxy data. He is very critical, in particular, of the Mann’s Hockey Stick. In the Bishop Hill blog  I found this witness by Andrew Montford -the site’s owner- who listened to a 2-hours Wilson’s lecture in 2013.

While I was writing this article, instead of cry for the lack of data, I thought I could directly write to Dr. Wilson and ask for his dataset.  After my mail, within three hours I received  the link to the data and to a next analysis not used here. In his reply, Rob Wilson suggests that, due to the detrending, periods longer than 100 year would be difficult to find out (his words:  The detrending of the data will make it difficult to find a strong centennial signal I think.).

In spite of that advice, in figure 1 three spectral maxima above 100 years can be observed (not identified in Wilson et al., 2007) and one of them has  the maximum power of the spectrum.


Fig.1: Plot of the detrended temperature (with a 20-yr low-pass filter in gray) and MEM spectrum of the GOA07 data made available by Rob Wilson . In the spectrum are visible all the peaks listed in Wilson et al, 2007, the maxima over 100 years and, on the other side, a  6-yr El Niño signature. The 2.87-yr peak is an "old friend"  I do (and did) find it in many datasets, mainly not related to the Sun but, it may be, to the Ocean (El Niño? But not sure of that).

Given the large uncertainty in tree rings measures and in the causes of the tree grow (not only temperature, of course), I think it can be assumed that the main spectral peak (222 years) is associated to the de Vries (or Suess) 208 years solar cycle. Within the spectrum, the  14.1, 15.4, 18.1, 23.8, 38 and 92.8 years maxima are clearly visible, coincident or widely compatible with the one listed in AM17 (and above). The three period ranges obtained by Wilson with MTM are also visible.

The analysis confirms Wilson’s (2007) results and add a strong signal, solar with some probability, and a El Niño signature, not really powerful but well visible. As a bottom line, I note the presence of the 2.87 years peak, a character of some spectra of series not strongly linked to the Sun but, it seems to me, to the oceanic oscillation.

Tan and Liu, 2003

Another available dataset, among those listed by AM17, is Tan and Liu, 2003. Here the temperature derives from a stalagmite in the cave of Shihua, Bejing (China). The series spans over 2650 years, 650 BCE to 2000 CE. The plot is shown in figure 2.


Fig.2: Series from Tan and Liu, derived from a stalagmite. Blue line is a 20-yr low-pass filter.

For AM17 this series allows the definition of two significative periods, 206 and 325 years while the authors add  also 750 and 900 years.

The Lomb spectrum of figure 3, with little differences, confirm  these four periodicities but shows a different, much more complex, story: for example, all the GOA7 (Wilson) periods are present along with the El Niño (3-6 years) ones;  a 9.8 year maximum, linked to Sun-Planets interaction and a 342 year peak, all of them present in the Wilson’s data spectrum.


Fig.3: Lomb spectrum of figure 2 data, where the complex relationships concurring in the formation of the stalagmite are highlighted.

Moffa-Sanchez et al., 2014

uses  sea sediments cores in order to analyse the hydrographic variability of the North Atlantic Current (NAC) during the last 1000 years. The NAC is important for salt transport toward high latitudes and the salt is  a necessary element in the formation of deep waters, in their turn essential for AMOC (Atlantic Meridional Overturning Circulation) and then for the global climatic system. Moreover, the warm produced by NAC – carried out by the westerlies- contributes to a milder climate in Europe. Authors of this research are interested in comparison among temperature, salinity , Mg/Ca ratio and to their relationship with solar irradiance (TSI, Steinhilber et al, 2009).
I received the data concerning this work in March, 2014 from Paola Moffa-Sanchez and did analise at  that time but never published. In figure 4 the comparison among the variable  and their relation with TSI is shown.


Fig.4:  Comparison between temperature, salinity, Mg/Ca ratio and their relation with TSI. In c) the direct comparison between temperature and TSI to which a lag of 12.42 years, according to cross-correlation computed in Moffa-Sanchez et al., 2014 have been applied. This figure is similar to the fig.3 of the original paper.

From the wavelets spectrum the authors derive significative periods in the range 135-225 years but also find 14, 16 and 30 years.

My spectrum, in figure 5, does not confirm or only faintly confirms the above spectral findings. In detail, out of the three shortest periods I can find only the 14 years one, and the range 135-225 years is not precise and does not include the second more powerful maximum at 261.5 years also if it is reminescent of two out of three longest period in the Wilson data spectrum (fig.1). My most powerful maximum at 50.3 years, present also in Wilson and Tan-Liu spectra, is not quoted by Moffa-Sanchez. Please also note that, in spite of the appearance of fig.4d, the spectral analysis shows a almost-non existent relationship  with the total solar irradiance (TSI).


Fig.5:  Spectrum of the temperature (black) and comparison with the spectrum of TSI (red).


I can draw the conclusion that the A.I. (artificial intelligence) techniques allow good hope for a really interesting future, but actually they use only a minor part of the complex interactions among the climate variables outlined by the spectral analysis.


  • John Abbot and Jennifer Marohasy: The application of machine learning for evaluating anthropogenic versus natural climate change, GeoResJ, 14, 36-46, December 2017. doi:10.1016/j.grj.2017.08.001.
  • Arrhenius S : On the influence of carbonic acid in the air upon the tempera- ture of the ground. Philos Mag, 41(5), 237-76, 1896 .
  • Ming Tan and Tungsheng Liu: Cyclic rapid warming on centennial-scale revealed by a 2650-year stalagmite record of warm season temperature , GRL, 30, 191-194, 2003. doi:10.1029/2003GL017352
  • Paola Moffa-Sánchez, Andreas Born, Ian R. Hall, David J. R. Thornalley and Stephen Barker: Solar forcing of North Atlantic surface temperature and salinity over the past millennium, Nature Geoscience , 7, 275-278, 2014. doi:10.1038/NGEO2094
  • Steinhilber, F., J. Beer, and C. Fröhlich. Total solar irradiance during the Holocene. Geophys. Res. Lett., 36, L19704, 2009. doi:10.1029/2009GL040142
  • Rob Wilson, Greg Wiles, Rosanne D’Arrigo and Chris Zweck: Cycles and shifts: 1,300 years of multi-decadal temperature variability in the Gulf of Alaska , Clim Dyn, 28, 425-440, 2007. doi:10.1007/s00382-006-0194-9.

Interglacials between 1 and 2.7 Ma

Franco Zavatti
April 16, 2017

Data in the range 0-2.7 million years (Ma) BP, as reported in the Tzedakis et al. (2017) paper, have been used in order to verify if, going back in time, the separation among interglacial periods from 41 to 100 ka (also known as the Mid-Pleistocene Transition) can be put in evidence. Such separation could not be observed with the 0-800 ka data used in the two earlier post of the series. It is observed that the spectrum of the δ18O within the range 1.5-2.7 Ma, shows the main spectral peak at 41 ka, while the ones at ~100, 72, 51 ka have a very low spectral power, also if with a high 99% significance level (white noise). The spectrum computed over the entire range of 0-2.7 Ma shows the main periods are both 41 and 100 ka. The spectrum of the intermediate period 0.6-1.5 Ma outlines a half-way structure.
The fact that the 100 ka spectral maximum can be some kind of mix of other peaks and not the direct influence of the orbital eccentricity should be also considered.

Introduction and analysis
After the publication at CM of two earlier posts (here and here) I noted that at the end of February, 2017 a work by Tzedakis et al., 2017 (hereafter T2017) was published, concerning the same argument, with the analysis extended through 2.7 Ma BP (Ma=millions years; ka=thousands years).

T2017 is paywalled, but I found a possibility, described in 02readme.html, I would like to share with the readers of the post.

The work by Tzedakis and colleagues is really important and noticeable: in a simple way they can separate Interglacials (IG) of the whole Pleistocene from Interstadials (IS) and continuous Interglacials (CIG, IGs apparently ending but then revitalized by some new strength in a sort of continuity with the preceding IG. By definition, they are located near the IGs and ISs border).

By the way, I show in fig.1 the series δ18O between 0.6 and 2.7 Ma with IGs labels (T2017; from Lisiecky and Raymo, 2015).

Fig.1: Plot of 0.6-2.7 Ma IGs from benthic δ18O. The data has a variable step of 2, 2.4 and 2.5 ka along the series. Red labels in the upper part of the plots are the MIS IGs coding. The figure is similar to the figure 2 of T2017, where the IG labeled (105) does not appear. Note as the δ18O range is wider after (on the left side of) 1.5 Ma BP, also the date before which T2017 computed the detrended section of the series.

The 0-800 ka series is partially shown because it has been already used in the two earlier posts. I outline the presence of an IG, labeled (105), at about 6.63 Ma, which does not appear in T2017, fig.2.

From the figure, we note that the overall range of glacial-interglacial evolutions between 1.5 and 2.7 Ma appears to be shallower than the one between 0-1.5 Ma. Relatively to the same period, T2017 decided to compute a detrendig of data.

Authors’ model is based on the “effective” energy, derived from the summer peak insolation (Gj/m2); it is defined as

E(Ipeak,Δt)=Ipeak+bΔt           (1)

where Ipeak is the summer insolation peak at 65°N; Δt the time (in ka) lasted from the preceding glacial and b the slope of the line (GJ/m2•ka) reported in their figure 4.

Data of T2017 are available at the site of one of the authors, Michel Crucifix, whose interest here was also data analysis, software and plotting. I downloaded the data and use them to reproduce their results, so the reference to “Crucifix data” used in the support site must be intended as “T2017 data”. From that data I can plot again T2017’s figure 5 into fig.2 which

Fig.2: Rebuilt of T2017 fig.5. Model (1) capability to separate different “warm situations”. Dashed line has been computed from the model maximum a posteriori probability. The diagonal section, the “ramp”, is the Medium Pleistocene Transition. Two IGs (red, 59 and 63) and one CIG (black, 7a) have been marked with the respective MIS code. No IS (aquamarine)
can be found above the dashed line. MIS 1 and MIS 5e (left top) are the Holocene and the Eemian, labelled only as reference.

shows in a very precise way the model’s (1) capability to distinguish among the glacial-interglacial sequences of the last 2.7 million years, the whole Pleistocene. Three cases “out of the choir” are indicated by their respective MIS (Marine Isotope Stage) code, while MIS 1 and MIS 5e, Holocene and Eemian, are identified only as reference.

The δ18O complete series from T2017 is plotted in fig.3, along with the detrended series from 1.5 to 2.7 Ma.

Fig.3: Run of δ18O between 0 and 2.7 Ma. (black) Original data is here indicated as “smoothed” because it is sampled in three steps, from 2 to 2.5 ka, compared to the 1 ka used in the earlier posts. (red) Detrended series which begins at 1.5 Ma BP. With respect to fig.1, here the different amplitude of glacial-interglacial exchanges after 1.5 Ma BP is better evidenced.

A slow but constant decline of the isotopic ratio from the beginning of Pleistocene to about 0.6-0.7 Ma, followed by a weak raise -or maybe a constant phase which follows a “break-point” at about 0.6-0.7 Ma- can be noted as an overall behaviour of the plot.

LOMB spectrum of both fig.3 data and data used in the previous posts are shown in fig.4 where two main maxima at ~100 ka and at ~41 ka clearly appear.

Fig.4: The spectrum of δ18O between 0 and 2.7 Ma, compared to the ones (LOMB and MEM) of the same series in the range 0-800 ka. Lomb power (cyan line) has been divided by 2. We know that 0-2.7 Ma data has a minimum 2 ka step, while the step of 0-800 ka data is 1 ka, so it is unclear the origin of the apparent major resolution of the T2017 data.

A couple of comments about the data used here:

  1. Detrended data, required by the Lomb method, used in place of the original ones does not produce noticeable differences in the spectrum.
  2. 0-2.7 Ma has 2, 2.4, 2.5 ka step, while the 0-800 ka ones, used in the comparison, have a 1 ka step. It is unclear why the spectra of T2017 data could show so much more details than more resolved data, mainly a double peak across 100 ka which does not appear in other spectra.

Given that also in this case (as in the previous posts) the spectrum over the whole range does not allow for checking the existence of a climatic regime change during the Pleistocene, the spectrum between 1.5 and 2.7 Ma has been also computed (fig.5).

Fig.5: δ18O spectrum between 1.5 and 2.7 Ma. (black) Original data. (red) Detrended data in the range 1.5-2.7 Ma. Only the peak at 41 ka appears, which confirms the hypothesis of a change around 1.5 Ma BP. Spectral maxima at about 50, 70, 90 ka are weak but their significance is as high as 99% (white noise).
To be noted that the spectrum of the detrended data (required by the Lomb’smethod) is in practice the same as the spectrum of the original data. The maximun at the extreme left side (period ~2.5 ka) is strong and also visible in fig.4. Its nature is not discussed here.

I don’t think the choice of 1.5 Ma must be considered as a cherry picking: it is the time when the δ18O amplitudes change with respect to nearby ages and also the beginning of the “ramp” of fig.2, namely the Transition of Medium Pleistocene which terminates at 0.6-0.7 Ma BP.

Fig.5 shows as in the first (ancient) section of the Pleistocene the main astronomical influence is the orbital obliquity (period 41 ka), with some possible, weak, contribution by other (one or more) orbital parameters.

A confirmation of that hypothesis comes from fig.6, showing the δ18O spectrum between 0.6 and 1.5 Ma (the “ramp” of fig.2).

Fig.6: δ18O spectrum between 0.6 and 1.5 Ma. The 41 ka peak is again the main spectral behaviour, but spectral maxima appear (at ~80 and ~122 ka) which in nearby ages could merge each other to give a ~100 ka peak. As a whole, the spectrum is less defined when compared to those of adiacent periods.

Here the spectrum is less defined than the ones in figs.4 and 5: the 41 ka maximum is again the dominant feature, but less powerful than in the previous time range; the 100 ka maximun is not yet present but noticeable peaks at 80 and 122 ka appear, which perhaps “promise” to combine themselves and became a 100 ka maximum after 0.6 Ma BP.

Final comments
Tzedakis and colleagues not only shows how it can be possible to separate the different climatic phases of the Pleistocene but also how to restore the position, with a 41 ka step, of all the more than 100 pleistocenic interglacials.
Figs. 5 and 6 show that the main maximum directly depends on orbital obliquity and allow the hypothesis that the 100 ka maximum be only the result of a combination among other different peaks and not a direct influence of changes in orbital eccentricity.

All plots and original/derived data concerning this post are available at the support site here, mainly in the last botton section, referred to as Crucifix’s data


  • L. E. Lisiecki, M. E. Raymo: A Pliocene-Pleistocene stack of 57 globally distributed benthic δ18O records. , Paleoceanography, 20, PA1003, 2005. doi: 10.1029/2004PA001071
  • P. C. Tzedakis, M. Crucifix, T. Mitsui & E. W. Wolff: A simple rule to determine which insolation cycles lead to interglacials, Nature, 542, 527-544, 2017. doi:10.1038/nature21364

Temperature of 45 Australian Stations

Franco Zavatti
March 8, 2017

At the Judith Currys’ blog Climate etc here, a comment here by Geoff Sherrington presents the link to a temperature dataset of 44 (45 for me) stations selected as the available most pristine ones (i.e. as less affected by human activity as possible). Data are available at the support site (DM40) and at Geoff’s site here.

The author of the dataset confirms he filled lost data by eyeballing a suitable mean and that data are not much useful apart the case where a comparison is needed with non-australian data.
I downloaded the dataset and used both my calculations and Geoff’s results (e.g. he gives slopes without errors or doesn’t compute spectra). For sake of clarity I divided all stations (alphabetically ordered) in 7 groups: 6 of them contain 7 stations and the last one the remaining 3.

In fig.1 (pdf) an example of the first group is shown, the other ones beeing available at the support site.

Fig.1. TMAX (top plot) and TMIN temperature for the first 7 stations listed on the right side.

The stations have been linearly fitted and the first 7 fits are in fig.2 (pdf)

Fig.2. Linear fit of the first 7 stations for both TMAX and TMIN. Numerical values are at the support site.
The MEM spectra have been also computed and, again, the first 7 stations
are shown in fig.3 (pdf). Numerical values are at the support site.

Fig.3. MEM spectra of the first 7 stations, both TMAX and TMIN. Spectral shape of top and bottom plots, also for the other groups, appears different.

The slopes of the stations have been set in relation (by Geoff) with Longitude, Latitude, Altitude asl, Distance from Sea and, as (almost) a joke, with WMO code and Ordinal (i.e. alphabetical order) of the stations, for both TMAX and TMIN. Results for TMAX are in fig.4 (pdf) and fig.5 (pdf)

Fig.4. Longitude, Latitude, Altitude asl vs. Slope of all stations.
Fig.5. Distance from Sea, WMO # and Station # vs. Slope of all stations. Numerical values available as max.app and min.app at the support site.

Some connection appears between slope – longitude and slope – altitude.
With surprise, slopes seems to be not related with the distance from sea shore.
The funny plot of slopes vs station ordinal doesn’t give any relation at all, of course. The slope shown a (weak) relation with the WMO code, depending, may be, on the choice of WMO (related to long and lat).

All plots and numerical data for this article are at the support site here

January 10, 2016