Wednesday, November 2, 2022

The Role Of Statistics In My Development Of A Geo-Effective (SID) Solar Flare Trigger

                  Multiple SID effects recorded over an interval from Slovakia

In my groundbreaking research on the SID flare the first thing I had to acknowledge was the need to separate this object of inquiry by definition, and statistics. It also meant admitting that arriving at publication  of research would by no means easy or straightforward.

First, I had to define exactly the secondary causal object of inquiry I was seeking: in this case a specific sunspot morphology, i.e. manifested in large and complex sunspot groups,  that produced solar flares yielding sudden ionospheric disturbances (SIDs) . In addition,  I had to identify the SID flares themselves by ensuring every spot group indexed could feasibly be tied (by proximity of heliographic location) to the site of a specific flare with a minimal soft x-ray flux (at SID threshold). This necessitated months of observations of sunspots using the telescope shown above, as well as x-ray, radio and other observations for flares assembled month by month in Solar Geophysical Data

One of the first realizations was that I would have to work with a limited sample set. It was absurd to attempt to look at every sunspot group and associated flare occurring over an entire  11 year cycle. So I limited the sample set to the Solar Maximum Year, which was reasonable and practical.. Other selection effects that entered included: use of Mt. Wilson magnetic classes for identifying magnetic morphology, which are coarse categorizations only; this use in turn introduces a further selection effect with respect to time, i.e. if there is some (SID) flare incidence that occurs in less than a day it would not have been exposed given the Mt. Wilson classes are identified daily only.  Also the use of vector magnetograms, such as:

                      Vector magnetogram made during one active interval in the study.

Other difficulties also emerged and were noted in the papers, including the uncertainties in making an SID flare identification based on its heliographic coordinates relative to the closest sunspot group. In other words, the greater the distance between heliographic coordinates the less secure the identity for the SID generating flare.

Given that optical images (H-alpha) were used to identify the nearest flares to suspect spot groups, the assumption was made that H-alpha flares could only be in the SID flare category if their emitted soft x-ray flux exceeded the magnitude: 2  x 10  -3  erg cm -2 s -1  (cf. Swider, W. 1979). This assumption necessitated close inspection of the soft x-ray records from the SMS-GOES satellite to ascertain the times for peak emission which later had to be tied to times of optical (H-alpha) flare occurrence. A typical soft x-ray flux record with the SID flares identified by SID and optical importance is shown below:

The Statistical Inputs and Aspects:

 SID flares vary appreciably in size (solar heliographic area) and power output, ranging from the barely resolvable subflare to major events capable of releasing up to 10  26 J of energy over period on the order of 10 3 s.    As shown in the soft x-ray flux capture,  these flares also vary in their impulsiveness, or the degree to which accelerated particles (electrons) are evident in the phase between onset and maximum power output (the rise phase). 

My first paper was mostly concerned with D-region ionizing flares, which generate SIDs in the form of: SWFs, or shortwave fadeouts (as the words imply actual fadeouts of high frequency radio waves occurring at the same time as the flare), SPAs, or sudden phase anomalies, generally caused by hard x-rays in the 0.5Å - 8Å band that engender a reduction in the reflection height for the incoming waves, and SEA or sudden enhancement of atmospherics- specifically, enhanced intensity of VLF or very low frequency waves.  

My second paper examined the specific statistics pertaining to frequency of occurrence and associated intensity.  This began with using the Poisson equation for probability:

P N     =   e -l    lN / N!

 Where  P N   is the probability corresponding to N flare days of the observed magnetic class (N = 0, 1, 2 etc.)  and  l  is the mean number of  flares per day per magnetic class.  Then the expected frequency of N flare days is found from:

E( d N )  = P N   å  d N   

Where E( d N )    is the expected number of N flare days, and the summation refers to the total number of recorded flare days for the particular magnetic class.   For any given magnetic class the extent of agreement between observed and expected flare days is calculated from:

 c2  =  å  [ O(d N ) - E( d N ) ] 2/   E( d N

A further index of goodness of fit was obtained by comparing the statistical moments  M n  with the predicted values for the Poisson theoretical distribution.   The moments about the mean ( l)are then given by:

Where  n = 2, 3. 4 etc.  and f j  (j= 1, 2, ...k)  =  f (No) denotes the observed distribution of N flare days  for the observed magnetic class.  For n = 2, for example, we obtain    =  s2   or the mean squared deviation from the mean (variance) which is a measure of the spread of   f (No) ;   for   n = 3  we obtain   =  d3   or the cubed deviation from the mean, i.e. the skewness of  f (No) 

.For a theoretical  Poisson distribution of form:    

P N     =   -l    lN / N! 

We expect:    = l , and  a  =  3 /(2 )1/2

But if these are appreciably different from the observed values a modified form of the theoretical Poisson distribution must be used, i.e.

Where  x  /h  =    and   ( l  +  l/h )   =  .  As with the theoretical Poisson form the goodness of fit may be assessed by using the  c2  distribution.

Suffice it to say, the preceding statistical aspects were critical in disclosing the need to incorporate a flare trigger to account for the different SID effects. One of the major findings on analysis was that: i) Subflares - with typical energy  1029  erg, were the major producers of SID flares, and (ii) 35% of the major SID flares (greatest geo-effective impacts) were optical subflares.

These results in turn disclosed the basis for a Poisson-based "delay time" and magnetic free energy (MFE) buildup preceding geo-effective solar flares, paving the way for a flare trigger.  Thereby it was shown how the flare distribution actually corresponds to a time-dependent Poisson process of the form:

P(t) =   -l   lt  / t!, 

where theoretically the Poisson mean rate of occurrence is: lm =   l Dt, with Dt  = t,  assuming the time interval Dt = 1d.  Since magnetogram measurements referred to solar active regions -sunspot groups will not generally be made at the exact same time each day this ensures  Dt ¹ 1d, so D¹  t thereby introducing a selection effect variability.  It is this inherent variability which opens the door as it were to the need for the modified Poisson distribution.

If MFE buildup was large, but the energy release (triggering)  'premature'  (t <<t', time of prediction) a subflare could then occur but with terrestrial effects (e.g. short wave fadeouts or SWFs). If the MFE buildup was large and triggering delayed enough to discharge most of all of it, then major impacts occurred, such as powerful magnetic (auroral) substorms.

These consequences were first postulated by me (Proceedings of the Second Caribbean Physics Conference, Ed. L.L. Moseley, pp. 1-11.) to account for the intermittent release of magnetic free energy in large area sunspots,  using:

/ t [ ò V    B 2/ 2m  dV] =   

1/m  ò V   div ( v X B) X B )dV -    ò V   han  Jms 2]

Where han  is the anomalous resistivity given by Chen (1974)[i]:

h an  =  4pneff/ we

where neff  is the effective collision frequency and we is the electron plasma frequency.  And  Jms  the current density at marginal stability of the magnetically unstable region.   Bear in mind that v X B) X B  reference relative footpoint motion within the large active region.

The plasma response to the rotary motion is accounted for by a (-J·E) term (or the  E·J  term, since -J·E = E·J). The change in total energy over a defined volume V may then be written (using appropriate identities of curl, div):

òv  [ e /t] dV = òv  [E curl HH curl E] dV -  òv  [J·E] dV

This work led directly to one of the first semi-successful uses of the Brier P-score to predict flare occurrence [ii] followed by publication of the key statistical results in the Meudon Solar-Terrestrial Predictions Proceedings [iii].

There also followed confirmation of an earlier hypothesis:

[i] Chen, Francis, T.: 1974, Introduction to Plasma Physics, Plenum Press, p. 158. 

 [ii] Stahl, P.A.: 1983, J. Royal Astron. Soc. Can., Vol. 77, p. 203.

[iii]Stahl, P.A.: 1986, ‘Limitations of Empirical-Statistical Methods of Solar Flare Prognostication’, in Solar –Terrestrial Predictions Proceedings,  Eds. P.A. Simon, G. Heckman, and M.A. Shea,  Meudon (France), p. 276.

See Also:

Example of Solar Loop Motions In A Magnetically Complex Active Region & The Existence Of A Solar Flare Trigger 

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