Autocorrelation in Power Spectra, continued

This is a short comment pointing to an addition to a previous post

The context is looking for autocorrelations in the frequency domain of a time-series. Although not as common as performing autocorrelations in the time domain, it is equally as powerful.

The earlier idea was to look for harmonics in the periodicity of the ENSO signal, and the chart described in the post showed clear annual and higher harmonics in the time series. This was via a straightforward sliding autocorrelation in the power spectra.

As an additional technique, we can look for symmetric sidebands of the annual fundamental and harmonics frequencies by folding the spectra over about the annual frequency and performing a direct correlation calculation.

Lower x-axis is the lower sideband interval (blue) and
upper x-axis is the symmetric upper sideband interval (red) shown in reverse

This correlation is painfully obvious and is well beyond statistically significant in demonstrating that an annual impulse signal is modulating another much more complex forcing signal (likely of tidal origin). This is actually a well-known process known as a double-sideband suppressed carrier modulation, used most commonly in facilitating broadcast transmissions. As shown in the equations below, the modulation acts to completely suppress the carrier (i.e. annual) frequency.

Read the previous post for more detail on the approach.

Ordinarily, the demodulation is straightforward via a standard mixing approach, as the carrier signal is a much higher frequency than the informational signal, but since annual and long-period tides are of roughly similar periods, the demodulation will only complicate the spectrum. This is not a big deal as we need to fit the peaks via the LTE formulation in any case.

This is a new and novel finding and not to be found anywhere in the ENSO research literature. Why it hasn’t been uncovered yet is a bit of a mystery, but the fact that the annual signal is completely suppressed may be a hint. It may be that we need to understand why the dog didn’t bark.

Gregory (Scotland Yard detective): “Is there any other point to which you would wish to draw my attention?”
Holmes: “To the curious incident of the dog in the night-time.”
Gregory: “The dog did nothing in the night-time.”
Holmes: “That was the curious incident.”

— “The Adventure of Silver Blaze” by Sir Arthur Conan Doyle

If that is not the case, and this has been published elsewhere, will update this post.

Wind Energy Dispersion

In Chapter 13 of the book, we derive the distribution of wind speeds and show what role the concept of maximum entropy plays into the formulation. It’s a simple derivation and one that can be extended by layering more dispersion on the variability, in effect superposing more uncertainty on the Rayleigh or Weibull distribution that is typically used to quantify wind speed distribution. This is often referred to as superstatistics, first described by Beck, C., & Cohen, E. (2003) in Physica A: Statistical Mechanics and Its Applications, 322, 267–275.

A recent article uploaded to arXiv [1] gives an alternate treatment to the one we described. This follows Beck’s original approach more than our simplified formulation but each is an important contribution to understanding and applying the math of wind variability. The introduction to their article is valuable in providing a rationale for doing the analysis.

“Mitigating climate change demands a transition towards renewable electricity generation, with wind power being a particularly promising technology. Long periods either of high or of low wind therefore essentially define the necessary amount of storage to balance the power system. While the general statistics of wind velocities have been studied extensively, persistence (waiting) time statistics of wind is far from well understood. Here, we investigate the statistics of both high- and low-wind persistence. We find heavy tails and explain them as a superposition of different wind conditions, requiring q-exponential distributions instead of exponential distributions. Persistent wind conditions are not necessarily caused by stationary atmospheric circulation patterns nor by recurring individual weather types but may emerge as a combination of multiple weather types and circulation patterns. Understanding wind persistence statistically and synoptically, may help to ensure a reliable and economically feasible future energy system, which uses a high share of wind generation. “

[1]Weber, J. et al. “Wind Power Persistence is Governed by Superstatistics”. arXiv preprint arXiv:1810.06391 (2019).

Chandler Wobble according to Na

In Chapter 13 of the book, we have a description of the mechanism forcing the Chandler Wobble in the Earth’s rotation. As a counter to a recent GeoenergyMath post suggesting there is little consensus behind this mechanism, a recent paper by Na et al provides a foundation to understand how the lunar forcing works. 

Chandler wobble and free core nutation are two major modes of perturbation in the Earth rotation. Earth rotation status needs to be known for the coordinate conversion between celestial reference frame and terrestrial reference frame. Due mainly to the tidal torque exerted by the moon and the sun on the Earth’s equatorial bulge, the Earth undergoes precession and nutation.

Na, S.-H. et al. Chandler Wobble and Free Core Nutation: Theory and Features. Journal of Astronomy and Space Sciences 36, 11–20 (2019).
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AO, PNA, & SAM Models

In Chapter 11, we developed a general formulation based on Laplace’s Tidal Equations (LTE) to aid in the analysis of standing wave climate models, focusing on the ENSO and QBO behaviors in the book.  As a means of cross-validating this formulation, it makes sense to test the LTE model against other climate indices. So far we have extended this to PDO, AMO, NAO, and IOD, and to complete the set, in this post we will evaluate the northern latitude indices comprised of the Arctic Oscillation/Northern Annular Mode (AO/NAM) and the Pacific North America (PNA) pattern, and the southern latitude index referred to as the Southern Annular Mode (SAM). We will first evaluate AO and PNA in comparison to its close relative NAO and then SAM …

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The Indian Ocean Dipole

In the book, we modeled the ENSO and QBO climate behaviors. Based on the approach described therein we have since extended this to the PDO, AMO, and NAO indices, with the IOD the focus of this post.

In Chapter 11, we concentrated on the Pacific ocean dipole referred to as ENSO (El Nino/Southern Oscillation).  A dipole that shares some of the characteristics of ENSO is the neighboring Indian Ocean Dipole and its gradient measure the Dipole Mode Index.

The IOD is important because it is correlated with India subcontinent monsoons. It also shows a correlation to ENSO, which is quite apparent by comparing specific peak positions, with a correlation coefficient of 0.2.  This post will describe the differences found via perturbing the ENSO model …

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North Atlantic Oscillation

In Chapter 11 of the book, we derived an ENSO standing wave model based on an analytical Laplace’s Tidal Equation formulation. The results of this were so promising that they were also applied successfully to two other similar oceanic dipoles, the Atlantic Multidecadal Oscillation (AMO) and the Pacific Decadal Oscillation (PDO), which were reported at last year’s American Geophysical Union (AGU) conference. For that presentation, an initial attempt was made to model the North Atlantic Oscillation (NAO), which is a more rapid cycle, consisting of up to two periods per year, in contrast to the El Nino peaks of the ENSO time-series which occur every 2 to 7 years. Those results were somewhat inconclusive, so are revisited in the following post:

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Commenting at PubPeer

For our Mathematical GeoEnergy book, there is an entry at for comments (one can also comment at, but you need to be a verified purchaser of the book to be able to comment there)

PubPeer provides a good way to debunk poorly researched work as shown in the recent comments pertaining to the Zharkova paper published in Nature’s Scientific Reports journal.

An issue with the comment policy at Amazon is that one can easily evaluate the contents of a book via the “Look Inside” feature or through the Table of Contents. Often there is enough evidence to provide a critical book review just through this feature — in a sense, a statistical sampling of the contents — yet Amazon requires a full purchase before a review is possible. Even if one can check the book out at a university library this is not allowable. Therefore it favors profiting by the potential fraudster because they will get royalties in spite of damaging reviews by critics that are willing to sink money into a purchase.

In the good old days at Amazon, one could actually warn people about pseudo-scientific research. This is exemplified by Curry’s Bose-Einstein statistics debacle, where unfortunately political cronies and acolytes of Curry’s have since purchased her book and have used the comments to do damage control. No further negative comments are possible since smart people have not bought her book and therefore can no longer comment.

PubPeer does away with this Catch-22 situation.


In Chapter 12 of the book, we concentrated on the mechanism behind the QBO of stratospheric equatorial winds. In a related topic (but only briefly touched on in the book), there is interesting data from a presentation on the equatorial-only Semi-Annual Oscillation (SAO) of the upper stratosphere and lower mesosphere wind pattern [1]. The distinction between QBO and the SAO is that the QBO has a longer periodic cycle and exists at altitudes lower in the stratosphere than the SAO.

[1] T. Hirooka, T. Ohata, and N. Eguchi, “Modulation of the Semiannual Oscillation Induced by Sudden Stratospheric Warming Events,” in ISWA2016, Tokyo, Japan, 2016, p. 16.

— presentation slides from International Symposium on the Whole Atmosphere

What’s interesting at the core fundamental level is that the SAO is understood by consensus to be forced by a semi-annual cycle (a resonant condition happening to match 1/2 year is just too coincidental) whereas there is no consensus behind the mechanism behind the QBO period (the tidal connection is only available from Chapter 12). To make the mathematical connection, the following shows how the SAO draws from the QBO tidal model.

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Peak Oil Barrel

The first part of our book Mathematical GeoEnergy deals with the mathematics behind the depletion of fossil fuels, and specifically crude oil. One of the co-authors, Dennis, helps maintain and moderate the Peak Oil Barrel blog. Recently, Dennis posted a blog entry on Oil Shock model scenarios, which is based partly on the mathematics described in Chapter 5 (and elsewhere in the book, as the shock model is a fundamental aspect of modeling oil depletion).

There’s lots of commentary on the POB blog, including climate science topics on the Non-Petroleum comment threads, so worthwhile to have it bookmarked.

Teleconnections vs Common-mode mechanisms

The term teleconnection has long been defined as interactions between behaviors separated by geographical distances. Using Google Scholar, the first consistent use in a climate context was by De Geer in the 1920’s [1]. He astutely contrasted the term teleconnection with telecorrelation, with the implication being that the latter describes a situation where two behaviors are simply correlated through some common-mode mechanism — in the case that De Geer describes, the self-registration of the annual solar signal with respect to two geographically displaced sedimentation features.

As an alternate analogy, the hibernation of groundhogs and black bears isn’t due to some teleconnection between the two species but simply a correlation due to the onset of winter. The timing of cold weather is the common-mode mechanism that connects the two behaviors. This may seem obvious enough that the annual cycle should and often does serve as the null hypothesis for ascertaining correlations of climate data against behavioral models.

Yet, this distinction seems to have been lost over the years, as one will often find papers hypothesizing that one climate behavior is influencing another geographically distant behavior via a physical teleconnection (see e.g. [2]). This has become an increasingly trendy viewpoint since the GWPF advisor A.A. Tsonis added the term network to indicate that behaviors may contain linkages between multiple nodes, and that the seeming complexity of individual behavior is only discovered by decoding the individual teleconnections [3].

That’s acceptable as a theory, but in practice, it’s still important to consider the possible common-mode mechanisms that may be involved. In this post we will look at a possible common-mode mechanisms between the atmospheric behavior of QBO (see Chapter 11 in the book) and the oceanic behavior of ENSO (see Chapter 12). As reference [3] suggests, this may be a physical teleconnection, but the following analysis shows how a common-mode forcing may be much more likely.

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