Investigation of Schumann resonance polarization parameters

\PERGAMON Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 0668Ð0678

0253Ð5715:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reservedPII] S 0 2 5 3 Ð 5 7 1 5 " 8 7 # 9 9 0 4 1 Ð 6

Investigation of Schumann resonance polarizationparameters

Daniel Labendz�Institut fu�r Geophysik\ Universita�t Go�ttingen\ Herzberger Landstra)e 079\ 26964 Go�ttin`en\ Germany

Received 00 August 0887^ received in revised form 0 December 0887^ accepted 1 December 0887

Abstract

The EarthÐionosphere cavity is characterized by a number of disturbances which cause departures from the behaviourexpected for the spherical symmetric case usually applied in model calculations[ The main e}ects are the inhomogeneityof the upper atmosphere which is characterised by di}erent conductivity pro_les during day and night\ and theconductivity anisotropy introduced by the Earth|s magnetic _eld[ Standing waves\ excited by the global thunderstormactivity\ with an eigenfrequency of ¼7 Hz develop in this cavity[ This phenomenon\ which is called Schumann resonance\su}ers a frequency splitting due to the removal of the spherical symmetry[

At the Institut fu�r Geophysik we measured the electromagnetic _eld components for a whole year at frequencies upto 19 Hz[ We present the results of the analysis of the polarization of individual events of the horizontal magnetic _eldcomponents[ With the help of the polarization parameters as indicators\ we show the existence of the aforementionedsplitting[ A frequency di}erence of 9[0Ð9[1 Hz between the two elliptical modes has been estimated[ Þ 0888 ElsevierScience Ltd[ All rights reserved[

0[ Introduction

The spherical EarthÐionosphere cavity forms a reson!ator in which standing electromagnetic waves can occur[A condition for this occurrence of resonances is that thewavelength l is on the order of the dimension of theresonator "l¼39\999 km#[ A subset of such resonancesare the Schumann resonances in the ELF "Extremely!Low!Frequency#!range "2 HzÐ2 kHz#[

The _rst theoretical discussion of this electromagneticresonance phenomenon was presented by Schumann"0841#\ whilst Ko�nig "0848# and Balser and Wagner"0859# presented the _rst experimental evidence[Schumann calculated resonance frequencies fn for then!modes\ yielding values of fn � 09[5\ 07[3 [ [ [ Hz for ahomogeneous spherical cavity with perfectly conductingwalls[ However\ the _nite conductivity of the lower iono!sphere causes the resonance frequencies to decrease tothe observed values of fn � 7\ 03\ 10 [ [ [ Hz[ Many authors"e[g[ Wait\ 0869^ Galejs\ 0861^ Tran and Polk\ 0868# sub!

� Tel[] 9938 440 28 6358^ fax] 9938 440 28 6348[E!mail address] danÝwilli[uni!geophys[gwdg[de "D[ Labendz#

sequently developed more realistic models\ which takeaccount of the external in~uences of the system\ such asvariations in the conductivity of the upper atmosphere[

The excitation of these resonances is connected withglobal lightning activity\ the discharges "called sferics#producing a power spectrum which extends into the ELF!range[

1[ Re~ections of the eigenfrequencies

The _rst investigation of eigenfrequencies bySchumann suggested that the periods of the standingwaves in the EarthÐionosphere cavity depend only on themode!number and are constant in time[ However\ severalauthors "e[g[ Balser and Wagner\ 0851b^ Fu�llekrug\ 0881#have documented frequency changes[ These relate toinhomogeneities\ and temporal and local variations inthe conductivity of the ionosphere as well as the dis!tribution of sources\ which are ignored in most modelcalculations[ The inhomogeneities discussed here arisefrom the asymmetry of the day! and night!time iono!sphere\ which leads to two hemispheres with di}erentimpedances[ Moreover\ the magnetic _eld of the Earth

D[ Labendz:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 0668Ð06780679

causes an anisotropy in the ionospheric conductivity\which further in~uences the wave propagation in thecavity[ Some disturbances exist regularly\ whereas extra!terrestrial in~uences\ like PCA "Polar!Cap!Absorption#events\ occur irregularly "Cannon and Rycroft 0871^Holtham and McAskill\ 0877#[

The behaviour of the cavity directly in~uences thepropagation of electromagnetic waves in the form ofdi}erent phase velocities "Bliokh et al[\ 0879#\ varying thefrequencies of the signals[ The result is that the Schumannresonances exhibit a di}erent frequency compared to thatexpected for the homogeneous and spherical case[ Inaddition to the temporary variations\ frequency!splittingof the di}erent modes is expected[ This can be shown byconsidering the excitation for the Schumann resonancesdue to an electrical dipole[ The generated electromagneticwaves then orientate omni!directionally along the hori!zontal plane[ The waves with the eigenfrequencies of theresonator will be _ltered out leaving standing\ eastwardand westward travelling waves[ If we take into accountinhomogeneities\ the phase velocity will be in~uenced andwe expect that the resonance frequencies will change[ Areceiver placed on the surface of the Earth should registeran additional splitting] the _ne structure of the Schumannresonances[ This _ne structure\ comparable with theZeemann e}ect in nuclear physics\ represents the inhom!ogeneities and anisotropies of the resonator system[

Bliokh et al[ "0879# made analytical calculations toinvestigate how strongly the di}erent disturbances coulda}ect the splitting of the _rst eigenfrequency[ Theyshowed that the _rst mode splits into a triplet\ if thedi}erent in~uences of the cavity resonator are considered\where the Earth|s magnetic _eld has the strongest con!tribution[ For the submodes m � −0\ 9\ 0\ which belongto the _rst spherical harmonic n � 0\ they calculatedvalues of f "0# � 7[05 Hz\ f "9# � 7[9 Hz and f "−0# � 6[4 Hz[

A statistically signi_cant proof of this _ne structurehas not yet been presented[ One reason is the relativelylow Q!factor of Schumann resonances\ which is not con!stant over a long time duration[

Sentman "0876\ 0878# developed an idea how the modesplitting of the Schumann resonances could be madevisible by inclusion of the polarization characteristics[ Heassumed a spherical model "r\ f\ u#\ which is excitedat u � 9>[ The potential\ derived from the Helmholtzequation\ was in accordance with the product of a radialfunction and a spherical harmonic[ In a spherical polarcoordinate system\ ignoring the time!dependence\ we get

c"r\ u\ f# � s�

n�9

sn

m�−n

AnmPmn "u\ f#\

in which Anm are expansion coe.cients determined by theboundary conditions[ Rotation of this potential leads tothe magnetic!_eld

B"u\ f#mn � 9×"r¼c#

� u¼imsinu

Pmn −f¼

1

1uPm

n \

where "r¼\ u¼\ f¼ # is the right!handed triad of unit vectorsin spherical coordinates[ If we substitute the sphericalharmonic of order n � 0 we get the di}erent modes withthe magnetic _eld in the form

B00"u#�−i ei"vt¦f# B0

0"f#� cos u ei"vt¦f#

B90"u# � 9 B9

0"f#� sin u eivt

B−00"u#�−i ei"vt−f# B−0

0"f#�−cos u ei"vt−f#[

The west! and eastward travelling waves correspond tothe side multiplets m � 20\ while the m � 9!mode rep!resents the central submode of the standing waves[ Theside!multiplets are elliptically polarized in di}erent direc!tions\ while the central mode is linearly polarized[ Sent!man proposed that a proof of the _ne structure inSchumann resonances is possible if the parameters descri!bing the polarization act as indicators[

2[ The polarization parameters

The polarization of Schumann resonances can bedescribed in terms of their ellipticity and the sense of theellipse[ The determination of the polarization parametershas been described by several authors "e[g[ Paulson andEgeland\ 0853^ Fowler et al[\ 0856^ Rankin and Kurtz\0869#[ The polarization describes the temporary har!monic variation between two orthogonal signals at con!stant phase[ Considering the two horizontal magneticcomponents described by the planar wave equation

Bx"t# � A0"t# ei"vt¦80"t##

By"t# � A1"t# ei"vt¦81"t##

it is possible to form a coherency!matrix J from whichall polarization parameters can be determined "Fowler etal[\ 0856#]

J � 0ðBx\ B�xŁ ðBx\ B�yŁ

ðBy\ B�xŁ ðBy\ B�yŁ1� 0

Jxx Jxy

Jyx Jyy1[The asterisk denotes the complex conjugate and the angu!lar brackets indicate a time average[ The diagonalelements of J are the autospectra\ while the o}!diagonalelements are the cross spectra between the twocomponents[

It is now possible to calculate all the requiredparameters[ In our case we are only interested in the senseof polarization and the ellipticity\ which will later be used

D[ Labendz:Journal of Atmospheric and Solar!Terrestrial Physics 59 "0887# 0668Ð0678 0670

as indicators[ The sense of polarization can be expressedby the angle b "Jones\ 0868# given by]

sin 1b �1ImJxy

z"Jxx−Jyy#1¦3JxyJyx

[

For clockwise rotation\ b × 9\ which is called right handpolarization "RHP#\ whilst the other case with b ³ 9 istermed left hand polarization "LHP#[ If b � 9 we havelinear polarization[

The ellipticity is de_ned simply as

o � tan b\

with circular polarization at =o= � 0\ linear polarizationat o � 9 and elliptical polarization for −0³ o ³ 0\ o�9[

The analysis presented here is restricted to discreteevents rather than dealing with the complete time series[As a rule these individual events extend to approximatelyone period of the relevant resonance frequency[ For thiscase damping can be ignored and therefore only mono!chromatic signals will be considered\ in which amplitudeand phase are viewed as constants[ For these specialconditions the coherency matrix takes the form

J � 0A1

0 A0A1 ei"80−81#

A0A1 e−i"80−81# A11

1

Fig[ 0[ Detection of an elliptically polarized event "_rst row# and a linearly polarized event "second row#[

where A0\ A1\ 80\ 81 are time independent[ In Born andWolf "0854# the angle b is expressed by the simple term

sin 1b �1A0A1

A10¦A1

1

sin "81−80#\

in which "81−80# represents the phase di}erence[ There!fore the ellipticity can be determined from the two ampli!tudes and the phase di}erence\ which may be directlyderived from hodograms in the time domain[

Figure 0 shows two examples for the detection of indi!vidual events[ First of all we derived a _ltered time series"5Ð09 Hz\ Fig[ 0"a##\ from which trigger extracted seg!ments "Fig[ 0"b## complying to the following conditions]

, The average amplitude of an e}ect should lie within thelower and upper limitations 9[0Ð39 pT[

, The determined frequency of the two signals is limitedto lie between 6 and 8 Hz[

, The two horizontal magnetic _eld!components musthave nearly coinciding frequencies which form closedhodograms "Fig[ 0"c##\ rather than Lissajous _gures[

Figure 0 shows an individual event with elliptical pol!arization "o¼−9[14# and also a linearly polarized "o � 9#event[


3[ Data acquisition

The Institute of Geophysics at the University ofGo�ttingen recorded the electromagnetic _eld componentsbetween April 0884 and May 0885[ To minimize man!made noise\ all measurements were done at the test siteof the University of Silberborn:Solling "40>N\ 8[4>E#[The instrument\ developed by Wehmeier "0881#\ is ableto record unprocessed digital time series of 7Ð09 dayslength\ and covers a frequency range from 9[0Ð19 Hz[

Induction coils are used as sensors for the magnetic_eld components^ the horizontal electric _elds are derivedfrom potential measurements with AgÐAgCl electrodes

Fig[ 1[ The calculated power spectra for all electromagnetic _eld!components[

of the Filloux "0862# type[ The vertical electrical _eld isrecorded by a rod antenna of 1 m height\ with anextremely high impedance preampli_er[ The data are rec!orded simultaneously with a sampling rate of 099 Hzwith 05 bit resolution and precision[ The timer of theinstrument is controlled by a time code receiver\ whichused the German time signal transmitter DCF!66[ Thisalso controls the exact starting time of the recording[

For a _rst check of the readings\ power spectra werecalculated to make sure that the Schumann resonancesreally could be observed[ Figure 1 shows\ as an examplefor the whole measuring interval\ the one!hour spectrafor the registered horizontal components for June 0884[


For the vertical components we decided to analyse onlya selected one!hour spectrum\ because the data arecharacterised by strong transient and temporal dis!turbances[ Showing the power logarithmically\ the _rsttwo Schumann resonances can be recognized very well[It can be observed that the EastÐWest component "By#for this case has much less energy than the NorthÐSouthcomponent "Bx#[ This observation also holds for ampli!tudes recorded during the summer months[

The amplitude maximum at 05[56 Hz is produced bythe German railway network\ while the spectral peaks at5[14\ 09[9 and 01[4 Hz are identi_ed as subharmonics ofthe power lines[

4[ Results

4[0[ Amplitude

Besides the polarization parameters and frequency\ wenow consider the amplitudes of the individual events ofSchumann resonances[ The amplitudes have already beeninvestigated at di}erent sites by Balser and Wagner"0851a# and Fu�llekrug "0883# but\ unlike our analyseswhich include selection criteria\ their analyses were basedon considerations of the entire time series[ Fu� llekrug"0883#\ for example\ found a good correlation betweenthe average amplitudes of the horizontal magnetic _eldcomponents and the global thunderstorm activity[ In ourcase we want to consider only the seasonal variationswithin the one!year!measuring interval[ The questionwhich must be asked is] is it possible that the long periodvariations of the thunderstorm centres will be recognized<To solve this question we investigate the daily variationsof the amplitude in the Bx! and By!components[ With thisobjective\ we calculate the quarter!hourly means for fourrepresentative days of the year "see Table 0#[ We onlyinvestigate days without local disturbances\ because suchevents could cause great changes within the data[ Figure2 shows that the daily ~uctuations in amplitude arecharacterised by three maxima located between 03]99Ð04]99 "UT# in the By!component\ and between 08]99Ð19]99 "UT# and 09]99Ð00]99 "UT# in the Bx!component[These three maxima could be connected with the three

Table 0Days of analysed data

Date Starting time "UT# Season

08Ð19 June 0884 09]04 Summer19Ð10 October 0884 09]04 Autumn1Ð2 February 0885 09]04 Winter3Ð4 April 0885 09]04 Spring

dominant thunderstorm centres in the world] Middle andSouth America ð08]99 "UT#Ł\ the equatorial area of Africað03]99 "UT#Ł and Australia:Polynesia ð8]99 "UT#Ł[ TheAfrican contribution is much less in the summer than inthe other seasons^ the American contribution is weak inautumn\ whereas in the summer it could not be recog!nized at all[ These observations show very clearly atwhich local time stimuli from the thunderstorm centresof the Earth are expected and their amplitudes for thedi}erent seasons[

4[1[ Ellipticity

For the statistical investigation of the amplitude wehave used mean values[ This approach is not possible forthe ellipticity\ because the range spans over −0 ³ o ³ 0[We therefore decided to investigate the frequency dis!tribution and calculated their maxima at quarter!hour!intervals[ The class width of the distribution was chosenas 9[94[ To ensure that we received a su.cient numberof events we investigated six days "08Ð12 June 0884# ofmeasurements and calculated a {standardized| day^ theresult is shown in Fig[ 3[ Events with linear polarizationare not considered here[ Strong daily variations are notapparent[ In fact\ the values are stable over many hours[Whereas during the day only negative senses are appar!ent\ the values change to positive in the late evening[ Thistransition occurs over a time interval of less than onehour[

The justi_cation for averaging over six days is partiallyprovided by the 37!h segment shown in Fig[ 4[ This 2!D_gure shows the number of events of a given ellipticityover this time interval[ The dominance of the positivesense during the night and the change to negative senseduring the day are prominent features[ We thereforebelieve our ellipticity estimates derived from our averageddata\ and we consider the changing sense of polarizationto be reliable[

4[2[ Mode splittin`

Our main aim was to prove mode splitting of the _rstSchumann resonance mode with the help of the ellipticityand the sense of polarization[ To investigate the associ!ated frequency splitting the frequencies of the individualevents will be determined[ Because of the low Q!factorand the existence of arti_cial and natural disturbances\we considered only short time intervals suitable for fre!quency determinations[ We therefore required numericalalgorithms with high resolution\ such as the MarquardtÐLevenberg!Routine "Marquardt\ 0851#\ which we used tocalculate the frequency from the very short timesegments\ of the order of a hundred ms[

As mentioned before\ it is essential that our two signalshave identical frequencies\ to ensure that we get closedhodograms[ However\ this cannot be expected to occur


Fig[ 2[ The daily ~uctuations in amplitude for four representative days[

very often from numerical estimates and we thereforeallowed a maximum di}erence between the frequenciesof the Bx! and By!components of Df � 9[1 Hz[ It will beshown later that this tolerance does not have a strongin~uence on the calculated peak frequencies of each sub!mode[ For our analysis of the frequency we now have allthe required parameters and we classify our observationswith respect to the relative frequency[ For the classes ofthe distributions we chose 9[94 for the ellipticity and 9[94Hz for the frequency[

Figure 5 presents the results for a complete day "08June 0884#[ It can be recognized that the _gure shows

three dominant areas\ which represent the linearly pol!arized events "central mode# and the elliptically polarizedevents of opposite sense[ These zones\ where o¼29[14\can now be connected with the theoretical side multiplets[We can recognize that the greatest number of events isdetected near the resonance frequency at 7 Hz[ At higherand lower frequencies\ where resonance would not beexpected very often\ the mode splitting is weaker[ Thisindicates that only events with polarization frequenciesnear the peak of the resonance contribute signi_cantly tothe mode splitting[ The next step is to give an estimate ofthe frequency splitting between the di}erent modes[ The


Fig[ 3[ The daily variation of the ellipticity averaged over six days[

2!D!_gure presented is not su.cient for that deter!minations\ so that we decided to sum over all ellipticitiesbelonging to a single mode[ This leads to the result shownin Fig[ 6"a#[

To facilitate a better comparison of the di}erentmodes\ we calculated the relative frequency with classlength of 9[94 Hz[ The data are from a 13!h interval of12 June 0884[ From these graphs we infer that it wouldbe possible to make a splitting in the _rst Schumannresonance visible\ if we discriminate between the sensesof polarization[ The three distributions do not show thesame central value[ This fact is illustrated very well bythe two elliptical modes[

In Fig[ 6"b# the distributions are _tted by a polynomialto determine the frequencies of their respective maxima[This leads to the following values for the di}erent modes]

f "−0# � 6[71 Hz

f "9# � 7[94 Hz

f "¦0# � 6[87 Hz[

Comparable values can be determined for the other days[Note that we have allowed a tolerance of 9[1 Hz betweenour two components[ In spite of this we believe that thesplitting is signi_cant[ Figure 7 shows that the di}erencebetween the distributions of the two signals for all pol!arization senses is very small compared with the errorbars that result from the averaging[

In spite of these results it is necessary to make a com!parison with the theoretically predicted values[ In thisconnection it becomes noticeable that the values fromBliokh et al[ "0879# are in accord with those calculated[However\ the ascertained order is di}erent[ While thetheoretical consideration assumed thatf "−0# ³ f "9# ³ f "¦0# our values show f "−0# ³ f "¦0# ³ f "9#[An explanation for this cannot yet be given[

5[ Conclusions

The basis of the analysis presented were time intervalsfrom a one!year registration of the temporal changes in


Fig[ 4[ The absolute frequency of the ellipticity during a 37!h interval[

the magnetic _eld components at frequencies up to 19Hz[ The _rst Schumann resonance was analyzed by amethod which determined the polarization parameters ofindividual events[ With the parameters ellipticity and thesense of polarization we discriminate the three modes andhave estimated the _ne structure[ The _rst examination ofthe amplitudes shows a daily ~uctuation which is cor!related with the global thunderstorm activity[ Addition!ally\ we found seasonal di}erences in the amplitudes[

With the determination of the ellipticity and the fre!quency we could determine a 1!D!reference space inwhich the three modes of the _rst Schumann resonance\which are characterized by di}erent senses of pola!rization\ could be distinguished very signi_cantly[ Thehighest densities of splitting can be recognized close tothe resonance frequency[ With integration along the o!

axis we can estimate a relative splitting of 9[0Ð9[1 Hz[ Itwas thus possible to verify the _ne structure of the _rstSchumann resonance\ which is ascribed to the inhom!ogeneities and anisotropies of the resonator[

Acknowledgements

The author wishes to thank Fiona Simpson\ UlrichSchmucker\ Matthias Wehmeier\ Martin Fu�llekrug andKarsten Bahr for their useful suggestions and discussions\and also two anonymous reviewers[ Data collection wassponsored by the Deutsche Forschungsgemeinschaftunder Grant No[ Schmu 090:12!0[


Fig[ 5[ The mode splitting for the _rst Schumann resonance in the frequency range of 6Ð8 Hz[


Fig[ 6[ The frequency splitting for the _rst Schumann resonance "a# and the accompanying polynomial _t "b#[

Fig[ 7[ The di}erence between the distributions of the submodes for the two signals[

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Investigation of Schumann resonance polarization parameters