1
On Planning and Exploiting Schumann
Resonance Measurements for Monitoring
the Electrical Productivity of Global
Lightning Activity
Vadim MUSHTAK, Earle WILLIAMS
Massachusetts Institute of Technology,
Parsons Laboratory, Cambridge, MA 02139 USA
2010 AGU Fall Meeting, San-Francisco, USA
2
OBJECTIVES
1. Presenting an algorithm for monitoring worldwide
lightning activity from electromagnetic observations in
the Schumann resonance (SR) frequency range.
2. Choosing a method of processing SR observations and
analyzing the applicability of its results to the monitoring
procedure.
3. Choosing an optimal inversion algorithm for the
monitoring procedure.
4. Testing the monitoring procedure on the basis of actual
observations from a world-wide net of ELF stations.
3
GENERAL CONCEPT OF MONITORING
PROCEDURE
The power spectrum of an electric/magnetic component F ( f ;UT ) of
the background natural ELF electromagnetic field is formulated via the
distribution of lightning sources  (; UT ) as
M
| F ( 0 ; f ; UT ) |  
2
2

(

;
UT
)
|
P
(
f
;

,

)
|
d 
0

m 1  m (UT )
M
 Am (UT )
m
2
|
P
(
f
;

,

)
|
d
0

(1)
 m (UT )
where  m are the symbolically limited territories of the major global
thunderstorm “chimneys”, P( f ; 0 , ) is the propagation factor for the
given component observed at a station with a location  0 (all locations
are formulated in the spherical coordinates   {a,  , } of a temporally
4
dynamic system whose pole coincides with under-the-zenith point on the
Earth’s surface r  a ), and the sought-for factor
Am (UT ) | QdS |2  N m (UT ) [ C 2  km2 / s ]
is the change in the integrated vertical electric charge moment over the
territory of the given “chimney” (for short, the “chimney’s” electric
activities) expressed via the statistically averaged charge moment of an
2
individual lightning source | QdS |  and the average number of
lightning discharges N m (UT ) occurring per 1 second over the
“chimney’s” territory.
In this formulation, the general concept of the procedure is to
monitor the spatial (the geographical coordinates) and temporal (the UT
variations of activities) dynamics of the major thunderstorm “chimneys”
from the background electromagnetic fields within the Schumann
resonance frequency range observed at a global network of ELF stations.
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OBSERVATIONS
The initial observational material presents continuous
electromagnetic time series recorded at a world-wide net of ELF stations
with working frequency bands limited by or including the SR range. In
this study, use has been made of magnetic observations from five stations
shown by blue-yellow pentagrams in Figure 1: BLK = Belsk, Poland;
MOS = Moshiri, Japan; RID = Rhode Island, USA; SHL = Shillong,
India, and SYO = a Japanese-team-operated station in Antarctica. Due to
diversity of recording conventions at different stations, the initial time
series have been reformatted to a unified format: 120 of 12-minute
periods per day (Figure 2A). In order to clean the data from nonbackground elements (of impulse or man-made nature), each period has
been divided into 144 of 5-sec segments, each segment’s power spectrum
and a 5 to 29 Hz energy content (EC) have been computed. The energy
6
contents for the given 12-min period have been presented as sampling
distributions for the east-west (EW) and north-south (NS) magnetic
components (Figure 2B), from which sampling mean values (shown by
red pentagrams) and standard deviations (SD; red triangles showing 1 to
5 SDs) of the ECs have been calculated. From the distributions it can be
seen, for instance, that in period #3 observed on January 1, 2009 at the
MOS station, the EW segments with ECs exceeding 65 pT2/Hz do
certainly contain non-background elements (seen clearly in Figure 2A)
and their contributions are not be added to the period’s spectrum. In this
way, after an additional median filtering against narrow-band
interferences, rectified EW and NS spectra for each period have been
constructed (Figure 2B).
While the standard procedure at the RID station is to approximate
a background power spectrum by a N-mode Lorentzian functional
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N
L ( f )   Ln ;
2
n 1
L (f)
2
n
Pn

 f

1  2Qn   1
 fn


2
via the modal frequencies f n , intensities Pn , and quality factors Qn , it
was found that some modes at some stations are irrevocably spoiled by
interferences that are neither identified by the rectifying procedure nor
effectively removed by the filtering one. For this reason, it was decided
2
L
to present the SR observations, instead of the full Lorentzian’s ( f ) ,
by the modal characteristics of isolated Lorentzian terms (I-Lors, for
2
L
short) n ( f ) approximating the spectra in the vicinities of corresponding
modes (Figure 3).
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Some examples of the diurnal variations of the SR characteristics are
shown in Figure 4 (modal frequencies) and Figure 5 (modal intensities).
In this material, it can be seen signs of both stability (reflecting the
general dynamics of global lightning activity, which allows for
constructing its general models) and variability (due to the activity’s dayto-day individual patterns, which makes it sensible to monitor it for the
patterns’ details). Since along with rather regular behavior on the 24hour scale, both frequencies and intensities show certain, sometimes
significant, variations within 5 periods of each hour, the periodical
diagrams of the kind presented in Figures 4 and 5 has been recalculated
into hourly mean values and standard deviations of the SR characteristics
(Figure 7), which material has been actually exploited in the monitoring
procedure.
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PROPAGATIONAL MODEL
Since some preliminary simulations had shown that the major –
day/night – electrodynamic non-uniformity of the Earth-ionosphere
waveguide plays a significant role in monitoring efficiency, in this study
use is being made of the two-dimensional telegraph equation (TDTE) [1]
1 

1
2
1
2 2
1
H L  sin 
U
U

k
a
H
C  U  U ст   0
2
2
sin  

H L  sin  
 (   S ) (   S )
uст  P0 ( f )
 0 a 2 sin 
where the pole of the spherical system of coordinates O  {a, , }
coincides with the solar zenith, the source’s location is S  {a,  S ,  S } ,
a is the Earth’s radius, P0 ( f ) is the source’s spectral dipole moment, the
proper propagation parameters are two complex frequency-dependent
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characteristic ionospheric altitudes H C and H L [1], and the
electromagnetic field’s components are expressed via the potential
U ( f ; ,  ) as
Er ( f ; S  O) ~ ifP0 ( f )
H  ( f ; S  O) ~ ifP0 ( f )
H L (S )
1
U (S  O)
H C ( S ) H C (O)
H L ( f ;S)
1
U ( f ; S  O)
[
]
,
H C ( f ; S ) H L ( f ; O)

H L ( f ;S)
1
U ( f ; S  O)
H  ( f ; S  O) ~ ifP0 ( f )
[
]
H C ( f ; S ) H L ( f ; O)

(3)
The model’s propagation parameters – tested in other studies - have been
constructed basing on the results of both recent research [2] and still
actual full-wave computations presented in the classical monograph by
Galejs [3].
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INVERSE PROCEDURE
Generally, the inverse procedure is based on an iterative
minimization of some metrics of discrepancies between the set of
experimentally determined modal SR characteristics E  {E k }
[k=1,…,K] and analogous characteristics T(ρ)  {Tk (ρ)} theoretically
computed via a propagational model in dependence on the set
ρ  { n } [n=1,…,N] of the source model’s parameters - the positions
and electric activities of the modeled global “chimneys”.
This general concept has been realized in two versions:
NELSON’S (linear, deterministic) ALGORITHM [4] is based on a
linearized presentation
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 k (ρ)
 n
 n ρi
n 1
N
 k (ρ )   k (ρ )  
i 1
i
of the deviations Λ(ρ)  E  T(ρ) as dependent on the changes in the
i 1
i






model’s parameters
n
n between iterations i  1 and i .
n
i 1
Assuming Λ(ρ )  0 , the next iteration is obtained as
ˆ 1 (ρ i ) Λ(ρ i ) (5)
ρ i 1  ρ i  D
via the theoretically computed sensitivity matrix with elements
D̂ k ,n (ρ i ) 
Tk (ρ)
 n ρi .
(6)
GOLTZMAN’S (quadratic, statistic) ALGORITHM [5] is based on a
quadratic presentation
N
k (ρ)
1 N  2k (ρ)
i 1
i
k (ρ )  k (ρ )  
 n  
 n  n1
2 n , n1  n  n1
n 1  n
ρi
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T ˆ

(
ρ
)


[
E

T
(
ρ
)]
R[E  T(ρ)] (the function of
of the function
maximum likelihood) with a subsequent application of Le Cam’s
suggestion to average the second derivatives over the random
realizations of experimental data via a covariance matrix R̂ . In this case:
ˆ 1 (ρ i )g(ρ i ) ,
ρ i 1  ρ i  G
(7)
where
TT (ρ) ˆ 1
TT (ρ) ˆ 1 T(ρ)
i
Ĝ n ,n1 (ρ )  [
R
] i g n (ρ )  [
R {E  T(ρ)}] i
ρ
ρ (8)
,
 n
 n
 n
i
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INVERSION: TESTS
At the previous stage of this project, it was shown [6] that a
monitoring procedure exploiting the modal frequencies f n ( n  1,...,4)
alone localizes the global African and American “chimneys” in a
reasonable agreement with general knowledge about global activity’s
properties, which cannot be told about the Maritime “chimney”. At the
present stage, the modal intensities are incorporated into the sets of the
measured E and T characteristics, though not directly, but – due to
some calibration problems yet to be solved – as their ratios rm  Pm / P1
( m  2,3,4 ) to the first modal intensity. As at the previous stage, the set
of estimated parameters ρ includes the geographical coordinates of two
most active at the given time “chimneys”, the pair being selected on the
basis of a general geophysical information (Figure 8). (The activities
themselves, the ultimate objective of this project, are to be included into
the set at the next stage).
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The relative stabilities of the two above inversion algorithms have
been tested as their abilities to provide the same positions of the modeled
“chimneys” by various initial conditions. The results of the test are
shown in Figure 9 (Nelson’s algorithm) and 10 (Goltzman’s algorithm).
Not surprisingly, the more sophisticated – and statistically founded –
quadratic algorithm turns out to be much less vulnerable to the danger of
local minima than the linear one. For this reason, Goltzman’s algorithm
(with a diagonal covariance matrix) has been exploited in the actual 24hour inversions.
INVERSION: RESULTS AND DISCUSSION
The hour-to-hour results of the application of the above-selected
inversion algorithm to the data recorded by 5 ELF stations on January 1,
2009 are shown in Figures 1 (global map) and 11 (continental areas).
Not surprisingly, the results for the African “chimney” look most
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consistent with general geophysical knowledge: this region is directly
(let it be remotely) circled by 4 stations and being “serviced” much better
than the other two.
The Maritime “chimney” is in a less benign position, but still shows
rather a compact dynamics (the 12UT “confusion” yet to be carefully
inspected). A possible shortcoming here is the obvious westward shift of
the active area in comparison with the general knowledge, which could
be the result of complicated along-the-terminator propagation to one of
two most informative – for this area - stations, MOS, without enough
support from other stations.
The American area is being “served” mostly by the RID station
alone, and the along-the-terminator propagation definitely plays its not
too friendly role here, too. The 16Ut and 17UT accidents could be
interpreted as results of American activity being “masked” by African
one (see Figure 8) without enough support from other – too remote –
stations. Interesting scenarios are those at 06UT and 07UT: any one of
them could be considered as the inversion’s “confusion”, but their being
close to each other looks like a reflection of some objective reality.
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INVERSION: PROSPECTS
The above considerations can be confirmed – or undermined – by
exploiting a more sophisticated, better informatively founded inversion
procedure. The obvious ways to improve it are:
1) increasing the number of participating stations (good prospects);
2) incorporation of electric measurements where available (in
progress);
3) including the “chimneys’” activities into the set of estimated
parameters (next step);
4) constructing and exploiting the full covariance matrix instead of its
diagonal simplification (some research required).
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ACKNOWLEDGEMENTS
We are extremely thankful to all – too numerous to be listed –
experimentalists who have provided their observations, knowledge, and
expertise for this project in numerous discussions and continuous e-mail
correspondence.
REFERENCES
[1] Kirillov, V.V. (2002) Solving a two-dimensional telegraph equation
with anisotropic parameters. Radiophysics and Quantum
Electronics 45, 929-941.
[2] Greifinger, P. S., V. C. Mushtak, and E. R. Williams (2007). On
Modeling the Lower Characteristic ELF Altitude from Aeronomical
Data, Radio Science, 42, RS2S12, doi:10.1029/2006hRS003500.
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[3] Galejs, J. (1972). Terrestrial Propagation of Long Electromagnetic
Waves. Pergamon Press, Oxford-New York-Toronto-SydneyBraunschweig.
[4] Nelson, P.H. (1967). Ionospheric Perturbations and Schumann
Resonance Data, Project NR-37-401, Geophysics Laboratory,
Massachusetts Institute of Technology (PhD dissertation).
[5] Goltzman, F.M. (1982). Physical Experiment and Statistical
Conclusions. Leningrad State University, 192p. [in Russian].
[6] Mushtak, V., E. Williams, R. Boldi, and T. Nagy (2010). On
Estimation Strategies in an Inverse ELF Problem. European General
Assembly 2010, Vienna, Austria.