From NOAA news: NOAA Scientist Finds Clue to Predicting Solar Flares
Forecasters at NOAA’s Space Weather Prediction Center in Boulder, Colorado.
High resolution (Credit: NOAA)
For decades, experts have searched for signs in the sun that could lead to more accurate forecasts of solar flares — powerful blasts of energy that can supercharge Earth’s upper atmosphere and disrupt satellites and the land-based technologies on which modern societies depend. Now a scientist at NOAA’s Space Weather Prediction Center and her colleagues have found a technique for predicting solar flares two to three days in advance with unprecedented accuracy.
The long-sought clue to prediction lies in changes in twisting magnetic fields beneath the surface of the sun in the days leading up to a flare, according to the authors. The findings will be published in Astrophysical Journal Letters next month.
“For the first time, we can tell two to three days in advance when and where a solar flare will occur and how large it will be,” said lead author Alysha Reinard, a solar physicist at NOAA’s Space Weather Prediction Center and the Cooperative Institute for Research in the Environmental Sciences, a partnership between NOAA and the University of Colorado.
Twisting magnetic fields beneath the surface of the sun erupt into a large solar flare, as shown above.
High resolution (Credit: NSF)
The new technique is already twice as accurate as current methods, according to the authors, and that number is expected to improve as they refine their work over the next few years. With this technique, reliable watches and warnings should be possible before the next solar sunspot maximum, predicted to occur in 2013. Currently, forecasters see complex sunspot regions and issue alerts that a large flare may erupt, but the when-and-where eludes them.
Solar flares are sudden bursts of energy and light from sunspots’ magnetic fields. During a flare, photons travel at the speed of light in all directions through space, arriving at Earth’s upper atmosphere—93 million miles from the sun—in just eight minutes.
Almost instantly the photons can affect the high-orbiting satellites of the Global Positioning System, or GPS, creating timing delays and skewing positioning signals by as much as half a football field, risking high-precision agriculture, oil drilling, military and airline operations, financial transactions, navigation, disaster warnings, and other critical functions relying on GPS accuracy.
“Two or three days lead time can make the difference between safeguarding the advanced technologies we depend on every day for our livelihood and security, and the catastrophic loss of these capabilities and trillions of dollars in disrupted commerce,” said Thomas Bogdan, director of NOAA’s Space Weather Prediction Center.
Reinard and NOAA intern Justin Henthorn of Ohio University pored over detailed maps of more than 1,000 sunspot groups, called active regions. The maps were constructed from solar sound-wave data from the National Science Foundation’s Global Oscillation Network Group.
Reinard and Henthorn found the same pattern in region after region: magnetic twisting that tightened to the breaking point, burst into a large flare, and vanished. They established that the pattern could be used as a reliable tool for predicting a solar flare.
“These recurring motions of the magnetic field, playing out unseen beneath the solar surface, are the clue we’ve needed to know that a large flare is coming—and when,” said Reinard.
Rudi Komm and Frank Hill of the National Solar Observatory contributed to the research.
NOAA understands and predicts changes in the Earth’s environment, from the depths of the ocean to the surface of the sun, and conserves and manages our coastal and marine resources. Visit us on Facebook.
Note to Editors: The paper has been accepted for publication in Astrophysical Journal Letters in February: “Evidence that temporal changes in solar subsurface helicity precede active region flaring,” by Alysha Reinard, Justin Henthorn, Rudi Komm, and Frank Hill.
Leif Svalgaard (20:40:59) :
“No, but what does that” [movie] “show”
[This movie of a flare and mass ejection that the TRACE satellite recorded from solar Active Region AR 9143 on 28 August 2000):
http://tinyurl.com/y9sobnu
[The Trace camera used 171 Å filters to see emissions from iron ions, Fe (IX) and Fe (X).]
“other than what everyone knows: the CME and flare is triggered from above because the loops have become too twisted and now carry too much energy, wanting to [and doing it] relax to a lower energy state.”
What an imagination: Twisted magnetic fields fall from the sky and trigger the CME and flare from above!
Leif, the movie shows absolutely no twisted magnetic fields that fall on the solar surface and trigger the CME and flare from above.
The movie shows rigid, mountainous iron-rich structures that vent solar flare ejecta upward, triggered by deep-seated magnetic fields that arise from the Sun’s highly magnetic (~10^12 Gauss) core [JFE 21 (2002) 193-198: http://arxiv.org/pdf/astro-ph/0501441%5D
Can you simply address the observations recorded in the movie?
With kind regards,
Oliver K. Manuel
Clive E Burkland (22:28:22) :
Considering you left out the unknown vital RM factor, it should not be hard to work out my lack of understanding. Thank for your time Dr. Svalgaard.
So are you saying that you understand it now?
tallbloke (14:16:47) :
It’s been done by NASA scientist Ching Cheh Hung
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.121.9361&rep=rep1&type=pdf
The link fails.
(a) Large solar flares were forecasted to start between late June 3 and early June 5, 2007 […] large sunspot 960
Flares are always expected when a large sunspot rotates into view. The predictor is simple the presence of a large spot.
tallbloke (00:47:33) :
No, and I’ve never seen any equally successful predictions come out of the dynamo theory either. [Have you?]
Dynamo theory has nothing to do with flares and cannot be used a predictor. Dynamo theory has to do with the generation of the sunspot itself deep within the Sun, but not with the subsequent happenings.
why are you trying to dismiss Ching Cheh Hung’s successful predictions this way?
First, I don’t know it was a public prediction made ahead of time [link was broken]. Second, the presence of any large sunspot [especially early on] is in itself a good predictor, no need for Hung. Third, at solar maximum there are flares every day, so every prediction will be successful [like predicting sunshine in July in California]. Did Hung predict this one http://hirweb.nict.go.jp/sedoss/solact3 ?
Of the nine flares shown here http://hirweb.nict.go.jp/sedoss/solact3/do?d=2005,09,04
some might be in ‘range’, but for successful prediction everyone must be accounted for. Or here: http://hirweb.nict.go.jp/sedoss/solact3/do?d=2000,03,16
NOAA did not set up the prediction scheme advocated by Hung. If I had been Hung and had been snubbed by NASA/NOAA and believed that my result was solid, I would have put up my own website with real predictions based on my theory and shown them that there was something to this. Did Hung do that? Perhaps. I don’t know about any such, and your link didn’t work. The predictions would also have to deal with when flares didn’t happen. If I predict a flare every day, regardless, every flare that occurs would have been a successful prediction. Failures are actually more important here. Flares are not supposed to happen until predicted.
tallbloke (02:33:20) :
As you keep telling us Leif, we have to follow the observations and successful predictions.
There are well-established rules for calculating skill-scores and I have not seen any for Hung’s prediction website [see above].
I think there will eventually be an amalgamation of the dynamo theory and solar-planetary theory, since there are clearly worthwhile elements in both.
First, dynamo theory has to do with the overall level of sunspots, but the specific occurrence of a spot and later its flare activity is not the domain of dynamo theory. And [although it would be nice – from a prediction standpoint] there does not seem to be anything worthwhile or solid in solar-planetary speculations [they do not rise to the level of a ‘theory’ e.g. http://wilstar.com/theories.htm ]
So, let’s discuss the valid results in scientifically definable terms.
It would, indeed, be nice if you would adhere to that.
jinki (03:58:28) :
“Perhaps you are missing that TSI has a rotational modulation [and we don’t know why – actually]”
To be blunt Dr. Svalgaard you lost it right there. The level of solar knowledge is not impressive.
Perhaps you could explain it to me, then. TSI has a strong rotational modulation that seems unrelated to sunspots and solar activity. And solar physicists [not just me] do not have a good explanation [yet] for this.
Leif Svalgaard (07:12:07) :
tallbloke (14:16:47) :
It’s been done by NASA scientist Ching Cheh Hung
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.121.9361&rep=rep1&type=pdf
The link fails.
Hmm,still works for me. pops up a box asking if I want to save or open with acrobat. Check your file associations. Anyway, here’s another link direct to NASA
http://gltrs.grc.nasa.gov/Citations.aspx?id=330
The link to the full .pdf is near the bottom of the page. I’ll wait until you’ve had a look and had the chance to revise your other comments.
Oliver K. Manuel (05:47:29) :
What an imagination: Twisted magnetic fields fall from the sky and trigger the CME and flare from above!
Even the second figure on the thread shows where the explosion takes place. Here you can learn more about flares and CMEs: http://cse.ssl.berkeley.edu/SEGwayed/lessons/exploring_magnetism/in_Solar_Flares/s4.html#sf pay special attention to Figure 4.5.
The movie shows rigid, mountainous iron-rich structures that vent solar flare ejecta upward, triggered by deep-seated magnetic fields
The structures are not iron-rich [less than 0.1% is iron]. They are not rigid [only on the short time scale of the movie do they appear not to be changing]. Come back a few hours or a day later and the structures are gone and replaced with other structures. The triggering takes place high above the spot, not deep seated and no trillion Gauss fields.
A somewhat more technically demanding description of flares can be found here:
http://solarphysics.livingreviews.org/Articles/lrsp-2008-1/
From the paper:
“the flare energy is released in the corona by reconnecting magnetic fields. The process heats the plasma in the reconnection region to temperatures of tens of millions of degrees Kelvin (MK), but also efficiently accelerates electrons to super-thermal energies peaking below some 20 keV and extending sometimes to several tens of MeV. […] The energy then propagates from the corona into the dense chromosphere along a magnetic loop by thermal conduction or free-streaming non-thermal particles, depending on the flare and the flare phase. The chromospheric material is heated to tens of million degrees and expands into the corona. The upward motion fills up existing coronal loops, but the motion may continue in an expansion of these loops.”
Paul Vaughan (21:57:13) :
“This is just numerology.”
Not at all. The calculations objectively summarize the geometry of NASA Horizons output.
That is what numerology is, unless there is a physical basis for doing that [which you do not claim]
The period has been worked out theoretically by 3 different methods (none of which are “eyeball” or “statistical”).
Still just numerology.
Not at all. I draw no conclusions about the physics.
Then your numerology is void of meaning. Only when you can and do connect to something physical or a physical process does it acquire meaning.
Paul Vaughan (21:57:13) :
Not at all. The calculations objectively summarize the geometry of NASA Horizons output.
One can perform numerology on the same data and produce correlations with Jupiter+Saturn+(Uranus+Neptune) as we have seen numerous times on the blog. The people peddling those claim that Venus+Earth are not effective and you do not involve Saturn [not to speak about U and N – which would be silly]. Then there are others that invoke Mercury [e.g. tallbloke (14:16:47) : sunspot group 960 was rotated to overhead point of Mercury and Venus] and so on. It is all numerology until we connect it with physics.
After few days of absence I am trying to catch-up on up to date discussion on the two latest solar threads. What do I find; beleaguered Dr. Svalgaard fighting on all fronts. Despite the past, he has my sympathy not that he needs it, or even less that he is expecting it from me. One image comes to my mind; Don Quixote and the windmills.
Now to some serious science:
Leif Svalgaard (07:12:07) :
TSI has a strong rotational modulation that seems unrelated to sunspots and solar activity. And solar physicists [not just me] do not have a good explanation [yet] for this.
Although a bit odd and unexpected, I am not entirely surprised. Some 30+ years ago Dr. Svalgaard and his colleague discovered what appears to be kind of Bartels rotation modulation within the solar wind. In recent years this was confirmed by Dr. Marcia Neugebauer at NASA’s Jet Propulsion Laboratory as: “the repetition interval at 27 days and 43 minutes and shows that the Sun has kept this steady rhythm, much like a metronome, for at least 38 years.”
It just may show that there is subtle connection to theTSI.
Question: Dr. Svalgaard do you have any numerical data relating to this TSI effect.
Thanks, and keep it up !
tallbloke (07:44:37) :
The link to the full .pdf is near the bottom of the page. I’ll wait until you’ve had a look and had the chance to revise your other comments.
The report [not a peer-reviewed paper that you like so much] has a date of July 2007, so is not a prediction of a flare in June 2007.
Simple as that.
vukcevic (09:00:42) :
One image comes to my mind; Don Quixote and the windmills.
I doubt that they would like to be characterized as mere Windmills. Some of them have quite a high opinion about themselves.
It just may show that there is subtle connection to the TSI.
You sound as you think none of us have thought of that. Another one, perhaps, with a [too] high opinion of himself.
do you have any numerical data relating to this TSI effect.
Yes, one can compute its power spectrum and, unfortunately it does not show a strong peak at 27 days.
Leif Svalgaard (08:47:23) :
“It is all numerology until we connect it with physics.”
Leif,
Would you encourage a qualified physicist to attempt to make such a connection?
Leif Svalgaard (09:21:31) :
tallbloke (07:44:37) :
The link to the full .pdf is near the bottom of the page. I’ll wait until you’ve had a look and had the chance to revise your other comments.
The report [not a peer-reviewed paper that you like so much] has a date of July 2007, so is not a prediction of a flare in June 2007.
Simple as that.
NASA scientist Ching Cheh Hung would have written the report some time before it was published by NASA, and states in the report that the predictions were made before the events. So unless you are calling him a liar [are you?], I think we should take him at his word.
He also notes that his look-backs on the data also showed the solar flares conforming to his three rules, so it seems pretty good. If these flares are consistently occurring when big sunspot groups pass directly beneath or exactly opposite one of the nearest four planets, or when they are around 30 degrees away, then there is good reason to consider what the mechanism of causation could be.
Please could we discuss that, rather than red herrings about dates of publication?
Leif Svalgaard (09:21:31) :
do you have any numerical data relating to this TSI effect?
Yes, one can compute its power spectrum and, unfortunately it does not show a strong peak at 27 days.
Don’t be coy! Tell us what the periodicity of the peak is!!
Leif Svalgaard (09:21:31) :
“Yes, one can compute its power spectrum and, unfortunately it does not show a strong peak at 27 days.”
Thanks, perhaps you might direct me to a data-file, I am numerology enthusiast.
“You sound as you think none of us have thought of that. Another one, perhaps, with a [too] high opinion of himself.”
To paraphrase lovely Judy: I always try being a first-rate version of myself, instead of a second-rate version of someone else.”
pochas (09:35:18) :
Would you encourage a qualified physicist to attempt to make such a connection?
Absolutely, yes. There would be eternal glory to him’her if success. So, why don’t we all fall over each other to do this? Because qualified physicists usually only attempt to explain something if there is a reasonable chance of success, and the first thing he/she would do would be to look at the statistical significance, then at the energies and couples available. The current state of these are such that it will encourage most people from wasting their time on this.
tallbloke (09:43:07) :
I think we should take him at his word.
On page 29 he explains that it is a hindcast. His “Large solar flares were forecasted to start …”. Where? in what public report? What he means is that if he had made the forecast, then he would have …
As I said whenever a ‘very large sunspot rotates into view’ it is a good bet that there will be flares in any case.
Please could we discuss that, rather than red herrings about dates of publication?
When you are in the prediction business, dates are king and is the only thing that matters. The ‘red herring’ is a needless accusation.
tallbloke (09:45:11) :
“Yes, one can compute its power spectrum and, unfortunately it does not show a strong peak at 27 days.”
Don’t be coy! Tell us what the periodicity of the peak is!!
Who said there is a peak? 🙂
Well, there is weak one at 30 days: http://www.leif.org/research/FFT-TSI.png showing the PMOD long-term composite.
There is also a small peak at one year [365 days], which is artificial [having to do with PMOD not calculating the distance to the Sun correctly]. Finally there is the strong solar cycle peak at 4000 days.
But no marked peak near 27 days. The rotational modulation is not coherent in phase and therefore we see no single peaks near 27 days, but just a blur of power between 20 and 30 days.
vukcevic (09:47:53) :
Thanks, perhaps you might direct me to a data-file, I am numerology enthusiast.
certainly: on my website click on ‘B Download of data links (Links to downloadable data)’, then on “TSI from PMOD Composite [Total Solar irradiance since 1978]”
To paraphrase lovely Judy: I always try being a first-rate version of myself, instead of a second-rate version of someone else.”
Not the same as that version actually being first-rate.
Leif Svalgaard (10:20:21) :
pochas (09:35:18) :
The current state of these are such that it will not encourage most people from wasting their time on this.
—
I seem to forget the ‘nots’ a lot 🙂
Leif Svalgaard (10:20:21) :
tallbloke (09:43:07) :
If these flares are consistently occurring when big sunspot groups pass directly beneath or exactly opposite one of the nearest four planets, or when they are around 30 degrees away, then there is good reason to consider what the mechanism of causation could be.
Please could we discuss that, rather than red herrings about dates of publication?
When you are in the prediction business, dates are king and is the only thing that matters. The ‘red herring’ is a needless accusation.
Yet here we are, still agonizing over dates instead of taking Ching Cheh Hung at his word when he says:
“Large solar flares were forecasted to start…”
and getting on with the more interesting questions about causation his paper raises. 🙁
Quick question about TSI. Is it a measure of the activity of the whole sun or just the side facing Earth, or more accurately facing the satellite? Is the satellite always between Earth and the Sun?
Interesting little peak around 400 days… (E+J synodic period)
Leif Svalgaard (10:22:22) :
Leif Svalgaard (10:20:21) :
pochas (09:35:18) :
The current state of these are such that it will not encourage most people from wasting their time on this.
Anyone who is willing to invest some of their time on this interesting voyage of discovery will find useful info on my site here:
http://tallbloke.wordpress.com
tallbloke (09:43:07) :
I think we should take him at his word.
Look at the first Figure at:
http://www.leif.org/research/Most%20Recent%20IMF%2C%20SW%2C%20and%20Solar%20Data.pdf
Note the blue line marked B. It shows the magnitude of the interplanetary magnetic field measured at the Earth since 2003. You will note several spikes: in late 2003, early 2005, late 2006. Each of these correspond to times with powerful flares and CMEs. These pump up the solar wind speed [the red curve marked V]. In fact, if you look really close, the B spikes precede the flares and I now predict that when the interplanetary field goes up, that a powerful flare will occur shortly thereafter. This is actually also borne out by data before 2003. There is even a peak in B just before the July 2007 flare! So, my scheme works! This I have known for many years.
So why doesn’t NOAA pay me a lot for my prediction service [and why don’t I make a website with that]? Because, B goes up when a lot of extra magnetic flux appears on the Sun because a big active region is developing and that development is the predictor in any case.
Leif Svalgaard (11:00:30) :
So why doesn’t NOAA pay me a lot for my prediction service [and why don’t I make a website with that]? Because, B goes up when a lot of extra magnetic flux appears on the Sun because a big active region is developing and that development is the predictor in any case.
The point of interest which I’d like to discuss is that the trigger for the flares seems to be the angular position of the planets in relation to the sunspot group. The increase/decrease in overall flare numbers you are able to predict is of great interest, but a separate issue.
tallbloke (10:52:19) :
and getting on with the more interesting questions about causation his paper raises. 🙁
I don’t think his analysis is valid is it peer-reviewed – your standard of validity, it seems], so why bother.
Quick question about TSI. Is it a measure of the activity of the whole sun or just the side facing Earth, or more accurately facing the satellite? Is the satellite always between Earth and the Sun?
TSI is measured at the satellite which is always between Earth and Sun, so only measures what comes our way. and is not just of activity [which is only a fraction of a percent of the total], but of the total energy we [the instrument – which by the way is of the size of your thumb] receive from the Sun.
Interesting little peak around 400 days… (E+J synodic period)
Typical pseudo-science reaction. The peak is completely artificial. Only shows in PMOD’s data. Not in SORCE that does compute the distance correctly [now], partly due to my analysis of the calculation leading to discovery of their neglect of a Special Relativity correction ( ! ) and to not calculating the distance at the time the photons left the Sun rather than at the time the photons were observed [the difference of 8+ minutes means that the distance has changed while the photons were in flight and more importantly the relative speed between Sun and Earth has changed, and hence the Doppler shift correction comes out slightly wrong- TSI measurements are now so sensitive that this matters].
tallbloke (11:11:08) :
The point of interest which I’d like to discuss is that the trigger for the flares seems to be the angular position of the planets in relation to the sunspot group.
‘seems’ is much too strong. ‘is claimed to’ is more appropriate. There are thousands of sunspots and thousands of flares in each cycle. Hung’s analysis is even hampered with his events not being independent. There were only five events not nineteen [or whatever the number claimed was]. Flares often have a tendency to repeat in the same group over several days, basically because the photospheric magnetic configuration does not change much during a flare, so the twisting will often resume.
The increase/decrease in overall flare numbers you are able to predict is of great interest, but a separate issue.
I do not predict the overall number, but the single superflares that occurred at those times. Those are the ones dominating the data. But, let me be clear: my prediction is spurious in the sense [and that is true of Hung’s as well] that the real predictor is just the appearance of a ‘very large sunspot’ group.
tallbloke (10:52:19) :
Interesting little peak around 400 days… (E+J synodic period)
One can compare SORCE [left] and PMOD [right]
http://www.leif.org/research/FFT-TSI.png
To get a feeling for what is real and what is not repeatable. The SORCE data set is shorter so the noise is a bit higher.
The Figure shows [from the bottom up] TSI, FFT logarithmic scale, and FFT linear scale.
From a paper linked by Dr. Svalgaard: “…the flare energy is released in the corona by reconnecting magnetic fields.”
No, Dr. Svalgaard, the flare energy is released by exploding Electric Double Layers because too much electrical energy is flowing through the “circuit” and when double Layers breakdown, explode, the entire energy of the circuit is released, as Hannes Alfven, 1970 Nobel Prize winner, described — as he saw in electrical transmission circuits in the Swedish power transmission system.
So-called “magnetic reconnection” is a misnomer, the energy doesn’t come from impinging magnetic fields alone, but from electrical energy generated by the Electric Double Layer process, a process that encompasses magnetic fields, electric fields, plasma flows, and charged particle acceleration of free electrons and ions in opposite directions.
Even NASA doesn’t agree with your take:
http://science.nasa.gov/headlines/y2009/31aug_mms.htm
“It’s ubiquitous [“magnetic reconnection].
The problem is, researchers can’t explain it.”
“But how? How does the simple act of crisscrossing magnetic field lines trigger such a ferocious explosion?”
NASA can’t explain it because the “magnetic reconnection” concept ignores the electric fields, plasma flows, and charged particle acceleration, free electrons and ions accelerated in opposite directions.
NASA can’t explain it because the causation process is fundamentally an electromagnetic process and astronomy is in denial about the fundamental importance of electric fields and plasma flow in space plasma phenomenon.
To explain the physical processes of “magnetic reconnection” is to explain electro-magnetic processes and NASA hasn’t quite come to grips with that, or they have privately, but know any public clarification will upset the astronomy community — their most important constituency.
tallbloke (11:11:08) :
The point of interest which I’d like to discuss is that the trigger for the flares seems to be the angular position of the planets in relation to the sunspot group.
And Hung’s analysis seems less that kosher. For example, for the Nov 4th 2003 event he omits the X-flare on Oct. 22, omits one of the X-flares on Oct. 26, and omits two X-flares on Nov. 03. For his Apr. 15, 2001 event he omits the X-flares on Mar. 29, and April 2. Includes the M-flare on Apr. 9, but omits the large M-flares on Apr. 5. This wanton selection of what fits and omission of what does not, completely destroys his argument. No wonder NOAA is not suing his ‘method’.
Peddlers of pseudo-science are all to eager to uncritically accepting something like this, without minimally even checking the data.
Leif Svalgaard (11:27:07) :
Interesting little peak around 400 days… (E+J synodic period)
The peak is completely artificial. Only shows in PMOD’s data. Not in SORCE that does compute the distance correctly
Make your mind up. A couple of posts ago you said it was the peak at one year that was spurious. I’m talking about the separate peak at 400 days.
James F. Evans (12:07:46) :
To explain the physical processes of “magnetic reconnection” is to explain electro-magnetic processes
This is not an electric-magnetic process. And I thought we had already closed that discussion: The energy is a release of magnetic energy stored in the field. Every well-versed physicist knows this. Even NASA’s people. That we are still researching the exact mechanism is just a tribute to how much we already know about reconnection. The issue is to pin down the reason for the increase of resistivity that allows reconnection to occur fast. The Reconnection Experiment http://mrx.pppl.gov/ provides you with the details necessary to grasp this. You can forget about the double layers. There are sometimes observed in the debris from reconnection, but play no role otherwise. Even the double layers that are at times postulated to occur in the Earth’s ionosphere are far from where the reconnection takes place, way out in the magnetotail. You really should come to grips with modern space physics.
and NASA hasn’t quite come to grips with that, or they have privately, but know any public clarification will upset the astronomy community — their most important constituency.
I think there are other websites where you can peddle your conspiracy theories.
Leif Svalgaard (08:32:36) “[…] numerology […] numerology […] numerology […]”
Leif Svalgaard (11:49:02) ” http://www.leif.org/research/FFT-TSI.png “
Surely you don’t assume stationarity.