GEOPHYSICAL RESEARCH LETTERS, VOL. 20, NO. 19, PAGES 2023-2026, OCTOBER 8, 1993
SOME CONSEQUENCESOF THE SHIFT THEOREM FOR MULTISPACECRAFT MEASUREMENTS.
Sandra C. Chapman
SpaceScienceCentre(MAPS),Universityof Sussex,U.K.
Malcolm W. Dunlop
Spaceand AtmosphericPhysics,Imperial Collegeof Science,Technologyand Medicine,U.K.
Abstract. We show how shift theorem yields a routine
quantitative
testto determine
whethera structure
seenby
two or morespacecraft
is a quasistaticconvecting
object,
suchasa boundarylayer. The test indicatesthe frequency
rangeoverwhichthe data is consistent
with a structure
whichis coherentbetweenspacecraft,planar,and time in-
is known the techniquealso determinesthe dispersionrelation of wavesin the rest frame of the plasma(or vice
versa).
Generallyit is essentialto evaluatethe restrictionson a
giventechniquethat resultfrom the uncertaintiesin the
inter-spacecraftpositionsand timing, and from the char-
acteristicsof the individual instrument response.The formalismin w, k spaceusedherenaturally allowsthesecharacteristicsof the 'measurement'to be distinguishedfrom
the characteristicsof the event itself. In w, k spacewe are
foundto be consistentwith quasistaticconvectingobject,
this analysisof the data fromfournoncoplanarspacecraft ableto definethe frequencyrangeoverwhichthe technique
is robustagainstdetectorand inter-spacecraft
timing and
yieldsthe convection
velocityof the plasmagiventhe dispersionrelationor viceversa,as well as the wavevectors positional uncertainties.
The techniquediscussed
here has beendevelopedfor the
corresponding
to givenfrequency
components.
Thetestfor
particularcaseof four (or more) non coplanarspacecraft,
coherence,
planarityand time independence
is shownto
be robustagainstdetectorandsystematic
inter-spacecraft howeversomeaspectsare applicableto the analysisof data
from pairs of spacecraft.
timingand positionaluncertainties.
Randomerrorswill
dependent
in its restframe.A clusterof fournoncoplanar
spacecraft
isrequired
to determine
thevelocityofthestructure in the spacecraft
frame. Whetheror not the data is
affecta finite frequencyrange,in principlethis can be determinedto restrictthe frequencyrangeoverwhichthe test
The Ideal SpacecraftCluster
can be applied.
We begin by consideringa quasi-staticplanar structure
which convectsover the 'ideal' spacecraft cluster. This
can be treated as a coherent planar wavepacketwhich
One of the purposes
of missions
suchas Cluster['The has no dispersion,ie all frequencycomponentsmove at
the samevelocity acrossthe spacecraft,which samplethe
ClusterMission',1988],[Rolfe,1990]is to employmultispacecraft
measurements
to distinguish
betweenspatial wavepacket,or structure, at different times. This relative
motion between the structure and the spacecraft can be
and temporalchanges
in the field and plasma.There are
two approaches
to the orderingof setsof multispacecraft due either to wave motion in the plasma, or convection
data. The first is to usethe (four) spacecraft
to measure of the plasma acrossthe spacecraft,or a combinationof
the two; this analysis,along with knowledgeof the apspecificquantities,suchas y7A B, or low frequencymagneticwaves(the 'wavetelescope'),
thesetechniques
have propriatedispersionrelation, will allow theseto be distinbeendiscussed
elsewhere
(egin the caseof the magnetome- guished.We will then considerthe effectof departuresfrom
ter [Dunlop
et al., 1988],[Dunlop
et al., 1990a],[Neubauer these assumptionsof planarity, coherence,time independenceand non-dispersion,and the effectsof the properties
et al., 1990a],[Dunlopet al., 1990b],[Neubauer
et al.,
1990b]).The second
approach,
employed
here,is to de- of the detector, and positionaland timing uncertaintiesin
the spacecraftcluster.
veloptestsfor specificclasses
of events.As testsof this
For simplicitywe considerpairs of spacecraftat different
type are not directedat the measurement
of a particular
Introduction
quantity,they are not in principleinstrumentdependent.
Specifically
wedevelopa technique
to determineto what
extent an event can be described as a static structure con-
vectingoverthe spacecraft
cluster. This techniquerelies
on the analysisof data in co,k space.If thisis possible,
the
techniqueyieldsthe frequencyintervalof the data consistent with a structure which is quasistaticand convecting
over the cluster and the degreeof spatial and temporal
coherenceof the structure. In principlethis definesthe
filter which allowsthis segmentof the data to be seperated for further analysis.If the plasmaconvection
velocity
locationsrj and rj+• whichare at restw.r.t. eachother.
We assumethat the spacecrafttake data over a period
ATM whichis longerthan the time takenfor the structure
to convectover all of the spacecraft. The jth spacecraft
seesa signalS(r•, t) whichhasFouriertransform
F(r•, w):
$(rj, t) •
F(rj,w)= •(rj,w)eiø(r3
,•)
(1)
whereß and 0 are real. If the wavefieldS(r, t) represents
a planar, static structure convectingover, and coherent
between,the jth and j + lth spacecraftthen the j + lth
spacecraftseesthe same structure as the jth spacecraft,
Copyright1993by theAmericanGeophysical
Union.
but at a time5tj later,sothat
Papernumber93GL02053
(2)
0094-8534/93/93 GL-02053503.00
2023
2024
Chapman and Dunlop: MultispacecraftMeasurementAnalysis
If the wavepacket
is dispersive,
however,R• will still
From shift theorem
(3)
at the differentspacecraft,but at differenttimes. The dis-
persivewavepacket
will not of coursehavea staticenvelope,
and will not representa static convecting
structure.Informationon thesetime delaysis givenin the phasespectrum.
Then at the j + lth spacecraft(1) and (2) give
=
be independentof frequency,as the componentsof the
wavepacket
will all still arrivewith unchanged
magnitudes
=
(4)
The amplitude spectrumat the j + lth spacecraftis just
that at the jth spacecraft;
Aj+• = F(rj+,,co)tF(rj+•,co)
=
Phase Spectrum
From(6) the delaybetweenthe arrivaltimesof the static
convecting
structure5t• is simply
%=
-
(S)
andis independent
of the time delay5t•. The phasespectrum at the j + lth spacecraft
= O(r,
-
(6)
contains
the time delay5tj.
The time period ATM overwhicha givenspacecrafttakes
data then correspondsto a lower limit to the frequency
rangeof informationcontainedin the A• and 0, which
2,, . The
is justWmi,•
-_ ATM
time interval between suces-
sire samplesATs (whichdefinesthe samplingfrequency
%) corresponds
to an upper limit to the frequencyrange
•
"
are measured
Wmaz= • ws = •s's' wherethesefrequencies
in the spacecraftrest frame. The amplitude and phase
spectra at the j q- lth spacecraftcontain different information describingthe behaviourof the wavepacketor convecting structure.
The significanceof this time delaycan againbe understood
if we treat the movingstructureas the envelopeof a propagating wavepacket.The wavepacketwill have a number
of frequencycomponents,and if all componentsmove at
the samevelocitythen 5t• will be independent
of w and
will simply yield the velocity with which the envelopeof
the wavepacket(the structure)movesw.r.t. the spacecraft. The componentsof the velocity along each spacecraft seperationvector are then givenby
-
(9)
so that four spacecraft
at non coplunarrj are neededto
determinev. This velocity is just the phasevelocity of
the wavesin the wavepacketin the spacecraftrest frame.
Since
thephase
velocity
v = • where
wisalso
measured
in thespacecraft
restframe,•[etermining
A(w) andv also
AmplitudeSpectrum
determinesthe k modes presentin the structure. If the
wavescan then be identified, so that the true propaga-
For a coherent,quasi-staticplanar structurethen
tion velocityof the wavesin the plasmarestframevp is
known,thenthe convection
velocityvc (ie the bulkvelocity
of the plasma)can be distinguished
from the propagation
velocityas v = vp + vt. This convection
velocitycan be
independentlydeterminedby the plasmainstrument,givsothat a plot of Rj versus
w provides
a routinetestfor ing a consistencycheck. Alternatively, if the convection
coherence,
planarity and time independence
of the strucvelocityv• (plasmabulk velocity)is knownindependently,
ture. This is most easilyenvisagedif we treat the signal
the frequencyin the plasmarest frame w' can be obtained
as a movingwavepacket,
the envelopeof whichis the conby dopplershiftingthe frequencymeasuredin the spacevectingstructure.ForR• to be independent
of frequency craft rest frame, ie w' = w - vc.k yieldingthe propagation
Rj= Aj+•(rj+•,w)
Aj(rj,co)
=1
(7)
then:
1. both the jth and the j + lth spacecraftmust sample
the wavepacket
(ie it is coherentbetweenthem).
2. all frequencycomponents
must be time independent
sothat they havethe samemagnitudesat timest and
t + 5tj. Thisis consistent
with the envelope
of the
wavepacket,the structure,beingtime independent.
3. the wavemustbe planarat all frequencies;
otherwhise
the amplitudewill changewith distance(eg a spherical waverepresenting
an expandingbubble)
If Rj(w) oversomerangeof frequencies
only,thenthe
data can be filtered to removethosefrequencycomponents;
the remainingsignalcan then be analysedas a static convectingstructure. However,if there is an overlapin the
frequencyrangescontainingthe static convectingcomponent of the signal,and other components,
the information
carriedby the wavemodesin the regionof overlapin frequencycannotbe retrievedby this process.
velocityvp = •-k.
If 5tj = 5t•(w)thenv = v(w) is thephasevelocityof individualcomponents
of a wave-packetwhichhasdispersive
behaviour. Again this velocityis measuredin the spacecraft, not the plasma,rest frame so that knowledgeof the
dispersion
relation(ie vp(w'))hould, with the measured
v(w), in principledeterminethe plasmarestframe,or vice
versa. A dispersive,but coherent,wavepacketwill havethe
sameamplitudespectraat the differentspacecraft.
Again,a plotof5t• versus
w wouldrevealthosefrequency
ranges(if any) overwhichthe wavepacket
is non-dispersive;
if the wavepacket
doesrepresenta convecting
staticstructure a correspondingfrequencyrange shouldappear in a
Rj versusw plot as discussed
above.
ComparisonWith Modelsand SimulatedData
The valuesof Wma
x and Wmi
n neededto includethe information in cospacethat characterizesa givenstructurewill
dependupon the spacecraftvelocity w.r.t. that structure,
ChapmanandDunlop'Multispacecraft
Measurement
Analysis
2025
sincew•.x andw,•i,•are measured
in the spacecraft
rest then at the j q- lth spacecraft
frame.In addition,themultispacecraft
analysis
techniques
FM(r•+•,w)- F(r•+•,w)Ga,•+•(w)eiG*o+
'(co)
discussed
hererequireall spacecraft
to makemeasurements
(13)
of an eventover an intervalwhichis longerthan the time
taken for the structure to convectover all spacecraft,ie,
andsinceF(rj+l,W)= q•(rj,w)ei(ø(rJ,co)-co%)
then
for:
ATM
> (rJ+l
- rj)'v
v2
(10)
Henceto determineATM for a givenspacecraft
seperation,
andthe desirablesamplingintervalAts, beforea measurementis maderequiresa prioriknowledge
of the 'bandwidth'that characterizes
an event(ie w•,x andw•in) in
someknownframe,andthevelocityof thespacecraft
w.r.t.
that frame.This canbe providedto someextentby models,or selfconsistent
simulations
of givenevents.
If v is differentbetweendifferentspacecraftpairs then
in generalthe structureis not coherent
overthe spacecraftseperation,
although
special
cases
of nonplanar,but
coherent
structures
(expanding
spherical
wavefronts
forexample)willbeconsistent
withdifferent
values
ofv. These
special
cases,
forwhichRj(w)and5tj(w)canstillbeused
to identifyproperties
of the structurecanonlybe investigatedby usingmodels
or 'data'generated
by numerical
simulations
to calculate
thebehaviour
of Rj and5tj which
canthenbe compared
with spacecraft
data. Forexample,
for the simplemodelof an expanding
spherical
wavefront,
weexpecttheamplitude
of waves
to fall with ! dependeuce,wherer0 is distance
fromthe centreof the sphere.
If thej + lth spacecraft
is at a distance
r• fromthecentre,
r0
and
thejthspacecraft
isatadistance
r0,then
Rj= • is
Fm(rj+•,w) = •(rj, O3)GR,j+
1(03)½i(O(rj'co)-coStJ+Ge),J
+1(co))
(14)
then
G•+•
R•M:R•(G•,•)2
(15)
and
5tjM- 5tj +
Gqs,j(o3
) - Gqs,j+l
(16)
Thefunctions
R•M(W)and5tiM(CO
) will havea systematic
dependenceon w which is dependenton the differences
in instrument responsebetweenthe spacecraft. This has
significancein particular for the Cluster mission,where
all spacecraftcarry the sameinstrumentation,with similar
G(w). If the instrumentresponse
as a functionof w is not
well known, calibration couldin principletake placein-situ
by event comparison. A systematic error in interspacecraft timing for a givenspacecraftshouldonly appearas a
systematic,
frequency
independent
factorin G•,•/w which
then contributes
to 5t•. Systematic
timinguncertainties
will then not affect conclusions as to whether
the event is
quasistatic,but will give a corresponding
uncertaintyin v,
as will positionaluncertainties
in the r•.
Randomerrorsproduceeffectswhichwill be moresubtle.
not unity, but is independentof w.
A randomerrorin 5tjM will directlyaffectthe determination of v, as will random errors in spacecraftseperation.
Instrument Response
Thiswill not yieldan uncertainty
in R•M directly,but will,
We will now consider some effects of instrument
re-
sponseand spacecraft
uncertainties,
non coherence
and
non-quasistatic
structureson theseconclusions.
The response
of a givendetector,and systematic
timinguncertaintiesbetweenspacecraft,
canbe represented
by a 'filter'
in frequencyspace:
G(w)= GR(w)e
iG*(co)
(11)
wherethe(real)Ga(w)givestheamplitude
response
ofthe
detector(ie the rangeof frequencies
overwhichmeasurementscan be made)and the (real) G•(w) givessystematic phaselagsintroduced
both by the detector,and by
inter-spacecraft
timingerrors.Timingand positionaluncertainties
(in thespacecraft
rj) willbe manifested
in the
determinationof the convectionvelocityv from (9). Offsetsmeasuredby the instrument(for example,the spacecraft magneticfieldwhichis seenat the magnetometer
in
additionto the localfield)canalsobe shownnot to invalidate this analysisprovidedthat the offsetis either small
compared
to the signal,or approximately
constantoverthe
timescaleof the measurement
(ie the frequencyrangecorresponding
to the offsetis muchlowerthan that required
to resolvethe signalstructure).
The measuredtime seriesat the jth spacecraftwill be
a convolution
betweenthe fouriertransformof G(w) •
g(t) andthe timeseries
S(rj, t), i.e.
(12)
unlike systematicerrors,lead to an uncertaintyin measurementsof the signalw and hencefor examplethe test for
whether the eventis quasistatic.This uncertaintyin w will
becomeincreasinglysignificantat higherfrequencies,
as we
wouldexpect,effectivemeasurements
can only be made of
frequencycomponentsfor which the corresponding
period
is longer than the random error in t. If the size of the
random error in t is known, then the frequencyrange over
which the test is valid can be determined.
The instrumentresponse
G(w), and the randomtime errorsdiscussed
abovedefinea 'window'(rangeof w) within
which measurementscan be made, wherew is in the spacecraft rest frame. Given v this correspondsto an w, k win-
dow,sinceforanyfrequency
component
v(w) = •k in the
spacecraftrest frame also.
Conclusions
We have examined how shift theorem yields a simple techniquewhich can be used to order multispacecraft
datasets. By first characterizingthe propertiesof the frequencyspectraof data from pairs of 'ideal' spacecraftin
termsof two quantitiesR• and5t•, whicheffectively
compare their amplitude and phasespectrarespectively,it has
been shown that:
1. The data, or specificallya subsetover somew range,
is consistentwith a singlecoherentquasistaticplanar
structureconvecting
overa spacecraft
pair if both R•
and5tj areindependent
of w overthat w range.
2026
ChapmanandDunlop:Multispacecraft
Measurement
Analysis
2. If Rj isindependent
ofw oversomew rangethenover
that range the structureor wavepacketenvelopeis
coherentand planar, and may be time independent.
3. If 5tj is independent
of w oversomew rangethen
overthat rangethe wavepacket
is non dispersiveand
may havea time independentenvelopethat definesa
conveeringstructure.
4. As each spacecraftpair yieldsthe componentof velocity of the envelope(or structure)relativeto the
spacecraft,four noncoplunarspacecraftare neededto
determine the velocity of the structure in the spacecraft frame.
5. SinceSty(w)computed
fromthe phasespectrafrom
four non coplunarspacecraftdeterminesthe velocity of a givenfrequencycomponentin the spacecraft
frame, at a frequencywhich is alsomeasuredin the
spacecraftframe, knowledgeof the convectionvelocity of the plasmadeterminesthe dispersionrelation,
or alternatively,knowledgeof the dispersionrelation
determinesthe convectionvelocity. In any frame, determination of the velocityof a givenfrequencycomponentalsodeterminesthe corresponding
wavevector.
The quantities
R• and5t•, andhenceconclusions
(1)-(3)
are found to be robust against the effectsof instrument
responseand systematic uncertaintiesin the spacecraft
seperationand timing. The effectof randomerrorsis more
subtle, howeverthese shouldonly affect a clearly identifiable rangeof w so that it shouldbe possibleto determine
the range over which the test for a quasistaticconvecting
structure can be applied.
The shift theoremtechniquesrequirethe time seriesmeasuredat the spacecraftto be transformedinto w space;the
comparisonbetweensimulationgenerated,and spacecraft
data require the data to be examinedin w, k space. It is
thereforesuggestedthat routine analysistools shouldinclude the ability to manipulate data in w, k, as well as r,
t space.
References
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the Instruments,ESA SP-1103, 1988.
Proceedings
of an International Workshopon SpacePlasma
Physics Investigationsby Cluster and Regatta: Graz,
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Dunlop, M. W., D. J. Southwood,K. -H. Glassmeierand
F. M. Neubauer,Analysisof multipoint magnetometer
data, Adv. SpaceRes., 8, 273, 1988.
Dunlop, M. W., Reviewof the Clusterorbit and separation
strategy: consequence
for measurements,Proceedings
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an International Workshopon SpacePlasmaPhysicsInvestigations
by Clusterand Regatta,Graz, Austria, ESA
SP-306, 1990.
Dunlop, M. W., A. Balogh, D. J. Southwood,R. C. E1-
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sattelitesas an array telescope,J. Geophys.Res., 95,
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S.C. Chapman,SpaceScienceCentre(MAPS),Univ. of
Sussex,BrightonBN1 9QU, U. K.
M. W. Dunlop, Spaceand AtmosphericPhysicsGroup,
BlackerrLaboratory,ICSTM, Prince ConsortRd., London
SW7 2BZ, U. K.
(RecievedApril 19, 1993;
revisedJune 16, 1993;
acceptedJuly 22, 1993.)