JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 94, NO. B10, PAGES 14,201-14,213, OCTOBER 10, 1989
A Comparison of Techniquesfor Magnetotelluric
Response Function Estimation
ALANG. JONES,
1 ALAND. CHAVE,
2 GARYEGBERT,
3 DON AULD,4 AND KARSTENBAHR5
Spectral analysis of the time-varying horizontal magnetic and electric field components yields
the magnetotelluric (MT) impedance tensor. This frequency dependent 2x2 complex tensor
can be examined for details which axe diagnostic of the electrical conductivity distribution in the
Eaxth within the relevant (frequencydependent)inductivescalelength of the surfaceobservation
point. As such, precise and accurate determination of this tensor from the electromagnetictime
seriesis fundamental to successfulinterpretation of the derived responses. In this paper, several
analysis techniques are applied to the same data set from one of the EMSLAB
Lincoln Line sites.
Two subsetsof the complete data set were selected, on the basis of geomagnetic activity, to
test the methods in the presenceof differing signal-to-noiseratios for varying signalsand noises.
Illustrated by this comparison are the effects of both statistical and bias errors on the estimates
from the diversemethods. It is concludedthat robust processingmethodsshouldbecomeadopted
for the analysis of MT data, and that whenever possible remote reference fields should be used
to avoid bias due to runcorrelated
1.
noise contributions.
INTRODUCTION
EMSLAB has brought together in a cooperative ef-
fort manyelectromagnetic
(EM) inductionworkerswith
diverse backgrounds and experiences. That the EMSLAB project has many facets is well illustrated by the
breadth of the subject matter of the papers in this special section. One such topic has focussed interest on
the problemof determiningthe magnetotelluric(MT)
impedance tensor elements from measurements of the
time-varying components of the EM field as precisely
and as accurately as possible. The availability of synoptic observations of the time-varying EM field over
the
EMSLAB-Juan
de Fuca
area
motivated
examina-
tion of the many disparate spectral analysis methods
output linear system by analyses of the respective input and output time series is a problem that has received much attention over the past century. A tremendous
boon
occurred
with
the
advent
of fast
Fourier
transformation algorithms during the 1960s, and with
some exceptions, these transfer functions are now routinely estimated in the frequency domain. While this
is done mainly for computational reasons, direct estimation of the impulse response functions by crosscorrelation
methods
is unwise
because of bad statistical
propertiesfor the estimates[Jenkins and Walls, 1968,
pp. 422-429].
For the analysis of MT data, the linear system can
be thought of as having two inputs, the horizontal com-
ogous fashion to the objectives of the mini-EMSLAB
ponentsof the time-varyingmagneticfield (b•(t) and
by(t)), and two independentoutputs, the horizontal
components
of the time-varyingelectricfield (e•(t) and
ey(t)), with additivenoisecomponents
on eachchannel
project [Young e! al., 1988], we wishedto undertake a
(nbx(t), nb,(t), n,x(t), and n,, (t)) givingour observ-
comparison exercise to evaluate the relative efficaciesof
our analysis codesgiven the same data.
and •y(t)) (seeFigure 1). The true inputs and out-
usedto analyzesimilar (or identical)data and alsoled
to the developmentof new ways of computing MT re-
sponses(e.g., robustmethods,seebelow). In an anal-
The time-varying EM field components are, by
Maxwell's[1892]equations,related by linear differential
operators, and for certain classesof external sourcepotentials [Egbertand Booker,this issue],conceptsappropriate for multiple-input/multiple-outputlinear systems
can be appealedto. The estimation of the weightingre-
abletime-varying
fieldcomponents
(•(t), •y(t), •(t),
puts are related, by a convolutionoperation, to the four
lag-domainweightingfunctionsz•(r), Z•y(r), Zy•(r),
and z•(•'). In the frequencydomain,the complexfrequency dependent relation between these components
canbe written (dependence
on frequencyassumed)
sponsefunctions, or their frequency domain equivalent
E•
the transfer functions, for a multiple-input/multiple-
1Geological
Survey
of Canada,
Ottawa,Ontario.
where Z is the MT impedance tensor defined ini2AT&TBellLaboratories,
MurrayHill, NewJersey.
3College
of Oceanography,
Oregon
StateUniversity,
Gorvallis. tially by Berdichevsky[1960, 1964] and Tikhonov and
[1966] and where B•(w) is the Fourier
4pacificGeoscience
Genter,Geological
Surveyof Ganada,Sid- Berdichevsky
ney, British Golumbia.
transformof b•(t) and similarly for the other compo5InstitutfOxMeteorologie
undGeophysik,
FrankfurtUniver- nents.
sittlt, Frankfurt, Federal Republic of Germany.
Generally, we have no knowledgeof the true comCopyright 1989 by the American Geophysical Union.
Paper number 89JB00634.
0148-0227/89/89JB-00634505.00
ponents(or latent variables)E•, Eu, B•, and By or of
thenoisecontributions
onthesecomponents
(NE•, NE•,
Ns•, andNs,) but onlyof ourobservations
of these•,
14,201
14,202
nbx(t)
JONESET AL.: MAGNETOTELLURIC RESPONSEFUNCTION ESTIMATION
nex(t)
bxCt) ex(t)
Zxx(T) Zxy(T)
fieldcomponents
(Nr• and Nr•) but are unbiased
by
noiseonthemagnetic
fieldcomponents
(NB• andNB•).
Obviously,the downwardand upward biased estimatesgive an envelopewithin which the true transfer
functionshouldlie (to within the statisticallimits of the
bxCt)
ex(t)
estimates).Thesebiaseswerefirst discussed
for MT by
Simset al. [1971]but had beenknownof in otherfields
by(t)
ey(t)
muchearlier,particularlyeconometrics
(Gini [1921];reviewedby ReiersOl[1950]).
•
Zyx(T)Zyy(T) ,.-
To avoid these bias errors due to noise power, re-
mote referenceprocessingof MT data was introduced
nby(t)
'••y(t)
[Goubauet al., 1978a,b; Gambleet al., 1979]in which
)
hey(t)
the components
of the horizontalmagneticfieldrecorded
at a remote second site are correlated with the local
field components. The first use of such an estimator
Fig. 1. Magnetotelluric
linearsystemwith the twohorizontal for the transferfunction again comesfrom econometrics
magnetic
components
(bx(l•),by(l•))asinputs,
andthehorizontal(ReiersOl
[1941],andindependently
by Geary[1943];see
electric
components
asoutputs
(ex(•),ey(•))related
bythefour ReiersOl[1950]andAkaike[1967]),wherethe remoteref-
MT impedance
weighting
functions
in thelagdomain
(
erence fields were termed the "instrumental
Zxy(7'),
Zyx(7'),
Zyy(T)).
Measured
are•x(t),•y(t),
However, Gamble et al. essentiallyrediscoveredthis
variables."
•y(t), i.e.,thesums
ofthese
truefieldcomponents
perturbed
by techniqueand gave it the now ubiquitousterm "remote
noise
components
(rtbx(i),rtby(i),flex(i),trey(i)).
reference."
It should
be noted
that
remote
reference
methodsare not as efficientas single-stationmethodsin
that for a givenlength data set, remotereferenceresults
•y, fix, and/•y. The problem
wehavethenis given will alwayshave larger associatedstatistical errorsthan
theseobservations,
how do we estimatethe elementsof standard LS ones. Also, correlated noise components
betweenthe local and the remotefields (in the caseof
Z in some optimum manner?
in the magneticfield)
Ignoringcertaintypesof instrumentation
error(non- MT thesecouldbe nonuniformities
are
not
removed
by
remote
reference
processingand can
linearclockdrift, erroneous
electrode
linelengths),there
aretwo typesof errorinherentin the estimationof trans- cause bias effects.
In this paper we comparevariousschemes(seeTafer functions;statisticalerrorand biaserror. Statistical
error,whichgivesa quantitativemeasureof the preci- ble 1) for estimatingthe elementsof the MT impedance
sionof an estimate,generallywill be reduced(for any tensor. Three of these schemes are based on standard
reasonableprocessing
scheme)by analyzingmore data LS spectral analysis without remote referenceprocessor by usingrobustmethods(seebelow)whicheliminate ing; anotherthree are robustschemesfor data processerrors due to non-Gaussian residuals. Bias error, by
ing(oneincorporating
remotereference
processing)
that
whichthe accuracyof an estimatemay be judged,is a arelessstronglyaffectedby outliersin the output(elecmoredifficultquantityto estimate,but techniques
exist tric field), while the other two are a nonrobustremote
(seebelow)foritselimination
undercertainconditions.referencecode and a cascadedecimation procedurewith
proStandardleastsquares
(LS) estimationof thetransfer weightedaveraging.We definea robustprocessing
functions Z leads to estimates that are biased downward
by uncorrelated
noiseon the inputs,i.e., noiseon the
magneticfield components
(NB• and NB•), but that
are unbiasedby uncorrelated
noiseon the outputs,i.e.,
noiseon the electricfield components
(Nr• andNr•).
Alternatively,the complex,frequencydependent,212
MT admittance
tensor A
cedureas one which is relatively insensitiveto the presence of a moderate amount of bad data or to inadequacies in the statistical model and that reacts gradually
ratherthan abruptlyto perturbations
of either. Much
literature already existson robustmethods,and an exhaustive review is outside the bounds of this paper.
In order to ascertain the abilities of the different
methods to extract reliable transfer function estimates
in the presenceof both low and high noiselevels,compared to the signallevels,we identifiedtwo time segEy
ments of data of 5 days duration each; sincethe sampling interval was 20 s, there are 21,600 samplesper
can be estimated which assumes that the electric
acfieldsare inputs to the linear systemwith the magnetic segment.One of thesewasfrom a geomagnetically
fieldsas outputs. This tensorrelationshipwasfirst rec- tive period(K indextypically>3), whilethe otherwas
ognizedandutilizedby Neves[1957],andalternativees- froma quietperiod(K indextypically1).
Although the schemesdescribedin this paper are
timates of the elements of the MT impedance tensor are
givenby invertingthe admittancetensor.StandardLS probablyrepresentativeof the majority of analysistechestimates of these elementshave the property that they niquesusedby the inductioncommunity,many other
are upwardbiasedby uncorrelatednoiseon the electric schemesexist for not only estimationof the impedance
[Bx
By][AxxAxy
JONES ET AL.: MAGNETOTELLURIC
RESPONSE FUNCTION ESTIMATION
14,203
E
1
JULY
lS
1
AUGUST
SEPTEMBER
Fig. 2. Time variations of five components of the electromagnetic field observed during the interval June 18 to
September
23, 1985. H, magnetichorizontalnorth-south;D, magnetichorizontaleast-west;
N, tellurichorizontal
north-south;E, tellurichorizontaleast-west;Z, magneticvertical. The full scaledeflectionis 750 nT and 750
mV/km for the magnetic and electric components,respectively.
elements
from
time
series but
also for bias reduction.
[Wannamakeret al., this issue). Instrumentationconsistedof an EDA flux gate magnetometer[Trigg et al.,
ment schemesof Kao and Rankin [1977]and Lienerr et 1970],Triggtelluricamplifiers[Trigg,1972],activetwoal. [1980],the iterative bias reductionweightingscheme pole Butterworth filters (magneticchannels:low-pass
of Gundel[1977],the cross-frequency
method of Dekker -3-dB point nominally 40 s, no high-passfilters; telluric
and Hastie [1981],bispectralanalysis[Haubrich,1965; channels: low-pass-3-dB points nominally 10 and 40 s
Hinich and Clay, 1968],frequency-time
analysis[Welch, of the two cascadedfilters, high pass-3-dB point nomi1967; Jonesand Hutton, 1979; Joneset al., 1983],com- nally 30,000s), and a 12-bit Datel cassettedata logger,
plex demodulation[Binghamet al., 1967;Banks,1975], all housed in an insulated aluminium case for thermal
singularvalue decompositionanalysis[Park and Chave, protection.The electrodelineswere55 m and 65 m long
1984],Cayley'sfactorizationtheorem[Spitz,1984],L• for the N-S and E-W (geomagnetic)
lines,respectively,
norm analysis [Turk et al., 1984], coherencesorting and power was provided by five 1.25-V, 2000-A h air
[Stodt,1986],maximumentropyanalysis[Tzanis and ceil batteries. The five componentsof the time-varying
Beamish,1986],and the mostone-dimensional
response electromagneticfield were sampled every 20 s with an
[Larsen,1989]amongstothers.We suggestto all authors identifying hour mark to facilitate error detection, and
For example, the iterative signal-to-noiseratio enhance-
of processingcodesthat they processour data with their
timing was generally accurate to better than a few sec-
schemes
to determinethe advantagesand disadvantages ondsas noted at the weeklycassette-changing
visits.
compared to the methods discussedherein.
For two of the analyses,magnetic field measurements
2.
DATA
from identically instrumented locations some 30 and 136
km farther to the east (long-periodsites4 and 13 on the
2.1.
Instrumentation
The data analyzed and discussedin this paper were
recordedat one of the EMSLAB Lincoln Line long-
LincolnLine [Wannamaker
et al., thisissue])weretaken
to facilitate remote referenceprocessingof these data.
2.2.
Time
Series
period land MT sites. The site was 10.6 km from the
Figure 2 illustrates the 10 weeks of data from site
oceanandlocatedon the CoastRangesediments(site i
i during the EMSLAB observationperiod, July 18 to
14,204
JONESET AL.: MAGNETOTELLURIC RESPONSEFUNCTION ESTIMATION
H
D
N
E
Z
4
2
1
0
0
6
12
13
18
0
6
12
14
18
0
6
12
18
6
12
16
18
0
6
12
18
17
Fig. 3. Timevariations
of fivecomponents
of the electromagnetic
fieldobserved
duringthe quietintervalof
September
2-6, 1985.Thefull scaledeflection
is350nT and350mV/km forthemagnetic
andelectriccomponents,
respectively.
The 3-hourVictoriaMagneticObservatory
K indices
areplottedin histogram
format thebaseof
the figure.
September23, 1985. These data have already been are 21,600 samplesin this window, and telluric noise
treatedfor grosserrorsusingan objectivescheme.Short spikesbetween approximately 1100 and 1500 UT of
(< I min, i.e., three data points)missingor erroneous •200 mV/km amplitudecompletelycontaminateand
data segmentswere interpolatedbasedon a five-point dominate the data. The causeof thesespikesis unknown
median filter approach;that is, data gapswere infilled but may possiblybe relatedto suddendischargingof the
with the median of the two points on either side of the capacitorsin the high-passfilter stagesfollowedby the
gap. Longersectionsof missingor erroneousdata were rechargingwhichrequiresa time of order the high-pass
marked and left untreated. Single-pointsteps (boxcar -3-dB cutoff of 30,000 s, or 8.33 hours. Also apparent
shifts)wereremovedby despikingfirst differences
and from Figure 3 is a daily event, most obviousin the By
then reconstitutingthe time seriesusingfive-point me- component(D), of two baylikefeatureseachof 1 hour
duration separatedby approximatelyI hour. This podian interpolation.
earlier and with
The quiet time diurnal variation (Sq) is apparenton lar substormevent comesprogressively
diminishingamplitudeeachday and posall components,and storm modulations of this pattern progressively
are particularly evident at the end of July, mid-August, sibly resultedin contaminationof the MT impedance
and mid-September. Even on the coarsescaleof Figure elementsin someof the analysesat these periods (see
2 it is possibleto detect several errors and problems section4.2). Shownon the baseof Figure3 in histogram
still presentin the data as releasedto all participantsin form are the Victoria Observatory 3-hour K indices.
this comparison.In particular, the spikeson the telluric K indices are a local quasi-logarithmicmeasureof geocomponents
on July 30, August8, and September5 (see magneticactivity [Maynaud,1980]and have 10 classes
andK=9 (magnetic
alsoFigure 3) obviouslyhave no magneticcounterpart betweenK=0 (magneticquietness)
storm)with the upperthresholdfor K=0 correspondand are potentially a substantial noisesource.
The quiet period chosen(Q) was the 5 daysbegin- ing to a range of 6.5 nT and the lower thresholdfor
ning 0000 UT on September2, 1985, and the five time- K=9 to 650 nT for Victoria. These K indices confirm
varying EM componentsobservedat the site are illus- the visual observationthat apart from the latter part of
trated in Figure 3 (note that the data are plotted at the day on September6, there is little activity during
more than twice the sensitivityof Figure 2). There the interval. The 24 hours beginning 0600 UT Septem-
JONES ET AL.: MAGNETOTELLURIC
RESPONSE FUNCTION
ESTIMATION
14,205
H
D
N
E
Z
2
2
0
I
6
I
12
I
18
2
0
......
'"'1
.......
'.....
I..........
I.........
6
12
18
0
I
6
I
12
3
I
18
0
4
I
6
I
12
I
18
0
I
6
5
I
12
I
18
0
6
Fig. 4. Time variations of five components of the electromagnetic field observed during the active interval of
September 13-17, 1985. The full scale deflection is 500 nT and 500 mV/km for the magnetic and electric com-
ponents,respectively.The 3-hour Victoria MagneticObservatoryK indicesare plotted in histogramform at the
base of the figure.
ber 4 are defined as an Extremely Quiet Period accord-
deriveestimatesout to 10,000 s (in 5 days there are 43
ing to the International Associationof Geomagnetism cyclesat 10,000s).
and Aeronomy(IAGA) definitionby havingplanetary
3.1. Method 1: Single Station Conventional Spectral
3-hourKp indicesthat do not exceed1+. (The Kp index is scaled in 28 classesfrom 0o, 0+, 1-, to 90, and
Analysis
The time series were visually inspected and nonoveris a weighted average of a selectionof local K indices
with the weightsreflectinggeomagneticlatitude and lo- lapping intervals selected with sufficiently high signalto-noise ratio. Intentionally, outliers and spikes were
cal time.)
In contrast,the active period chosen(A) was the 5 not removed. For each section,(1) the first and last
days commencing0000 UT on September 13, 1985, and 10% were cosinetapered, (2) the windowedserieswere
the data and Victoria Observatory K indices are illus- transformedinto the frequencydomain (using a stancomputed,(3) a
trated in Figure 4. Midday on September 16 is partic- dard FT algorithm),and cross-spectra
ularly active,with K indicesof 5 and Iip indicesof 7- Parzen window in the frequency domain was applied to
and 60. Obvious in the data, particularly in the telluric smooth the crossspectra such that there were typically
east-westcomponent(E), is the ssc(suddenstormcom- seven estimates per decade.
mencement)at 0601 UT on September14, 1985. Some
telluric spikesare evident in the data, e.g., •0730 UT on
September 13, •0200 UT on September 17, but these
are of much smaller amplitude than those during the
quiet interval.
3. ANALYSIS TECHNIQUES
Brief descriptions are given here of the processing
stepstaken for eachof the eight analyses(seeTable 1).
As the-3-dB high-passcutoff period for the telhlric fields
was nominally 30,000 s, the aim of the schemeswere to
TABLE
1. Short Descriptions of the Methods Used
Method
D escription
Source
1
2
3
4
conventional spectral analysis
conventional spectral analysis
conventional spectral analysis
weighted cascade decimation
Bahr
Auld
Jones
Jones
5
6
remote
robust
Chave
Jones
7
robust
8
robust
reference
cascade decimation
Egbert
remote
reference
Chave
14,206
JONES ET AL.: MAGNETOTELLUP•IC
From these ensemble estimates of smoothed spectra,
RESPONSE
FUNCTION
ESTIMATION
function from the averaged spectra, whereas method 2
averagedestimateswere derivedby stacking(without involvesaveragingthe estimatesof the transferfunctions
weightingbasedon somequality measure)and the MT from each sectionwith 15% trimming.
•ransfer functions were estimated from these averages
of the spectra. The confidenceintervals of the trans-
fer functionswereestimated(assuming
Gaussiannoise)
3.4. Method •: Single-Station-Weighted Cascade Decimation
The cascadedecimationschemeof Wightet al. [1977]
for 5% error probability. Both upward and downward
(seealso Wight and Bostick[1980];codepublishedby
biased estimates were computed.
Bostickand Smith [1979]),wasmodifiedto incorporate
3.2.
Method 2: Single-Station Conventional Spectral ministacks of eight discrete Fourier transform harmon-
ics. A 32-pointbasewasused,with two estimates(sixth
The processingsequencesusedfor method 2 are de- and eighth harmonics)from eachseries,with a decimascribed by Law et al. [1980]. The data were plotted tion factor of 2. This yielded the first estimate at 4 times
and nonoverlapping8-hour time sectionsof 1440 points the sampling interval, or 80 s for these data, with 6-7
were selectedon the basisof suitable(moderateactiv- points/decadeand sevendecimatesto coverthe periods
;_ty)geeraag_netic
activity. The selectedtime sections up to 10,000 s. The ministacks were averagedinto the
Analysis
were edited to correct for spuriouserrors. Then, for each total stack using the inversesof the geometric means
time section,(1) the meanandtrend wereremoved,and of the variances of the off-diagonal elements of Z as
end effectsminimized by useof a cosinetaper on the first weights. This stacking cascade decimation scheme is
andlast 10%, (2) the 1440point data serieswerepadded thus identical to the in-field processingschemeof the
with zeroesto expandeachsampleto 2048 points,(3) MT systemfrom Phoenix GeophysicsLtd. (Toronto).
a standard FFT algorithm was used to transform the Both upward and downward biased estimates were com-
data to the frequency domain, cross-spectracomputed, puted.
Note that the use of inverse variances as weights inand a Parzen frequency window applied to neighboring
corporates
not only signal criteria but also downweights
Fourierharmonics,(4) the MT transferfunctionswere
then derived from these smoothed crossspectra.
These estimates
of the transfer
functions
from each
time sectionwere then averagedand the mean and standard deviationscalculatedto give the final transferfunction estimates. For the 5-day intervals, typically 510 time sections were analyzed, whereas 25 time sections were taken for the total data set. Additionally,
for the total data set, the 15% highestand lowestestimates of the transfer functions, at each center frequency,
were eliminated prior to averaging. Note that as only
1440-pointlength serieswere analyzed,estimatescould
only be obtained out to a maximum possibleperiod of
m3000s (assuming10 cyclesare neededin the time interval).
eventsfor high coherence
betweenH and D (i.e., no in-
dependent
information),'/•rD, anddownweights
events
for low multiple coherence between the output electric
component,
7}HD and7•HD, andthe inputmagnetic
components
(low correlationsignal).The latter two are
equivalent to requiring a high partial coherencebetween
the output electric component of interest and the input
magnetic component of interest, removing the effect of
the other magnetic component on the electric compo-
nentin an LSsense,
e.g.,•[}D.HfortheZxycomponent.
It hasbeenthe experienceof oneof us (A.G.J.), and of
Phoenix, that inversevariancesasweightsgivessuperior
estimates of Z than using multiple coherencesalone as
weights.
3.3. Method 3: Single-Station Conventional Spectral 3.5. Method5: Remote
Reference
Conventional
Spectral
Analysis
Analysis
The processingschemeis the nonrobustmethod deThe processingsequencesused for method 3 are describedin detail by Joneset al. [1983]. For the 5-day scribedby Chaveand Thomson
[thisissue].The data
intervals, the data were split into five 24-hour long time werefirst plottedand inspectedfor grosserrorswhich
sequencesof 4320 points which were then padded with wereeithercorrected
(if of shortduration)or notedfor
zeroes to 8192 points. The smoothed crossspectra from exclusion
in subsequent
processing.A subsetlength
each of the 5 days were normalized by the power in the of 12 hours (2160 points) was selectedand a timehorizontal magnetic field components,then averagedin bandwidth
4 prolatedatawindow[Thomson,
1977]was
a weightedmanner usingthe coherencefunctionsdefined appliedto eachsubsetwith 70%overlapbetween
adjaby Jones[1981]as weights.Note that spuriousdata are cent sections. The discrete Fourier transform was then
not rejected subjectively but are downweightedin the takenfor all data series,includingthe remotehorizontal
averaging stage.
magneticfield. A set of centerfrequencies
wereselected
Although methods 1, 2, and 3 represent "conven- to give eight estimatesper decade,and arithmeticsectional" techniques, it should be appreciated that they tion andbandaveraging
withoutoverlapwasappliedin
differ in one important aspect; methods 1 and 3 involve the usualway to givethe remotereference
impedances.
averagingspectra(unweightedaveragingfor method 1, The jacknife[Chaveand Thomson,
this issue],a nonweightedaveragingfor method3) to obtainthe transfer parametricerror estimator which is relatively insensi-
JONES ET AL.: MAGNETOTELLURIC
RESPONSE
FUNCTION
ESTIMATION
14,207
rive to departuresfrom the usualGaussianassumption yieldingsmootherimpedances.The jacknife is used to
implicit in parametricapproaches,
was usedto obtain get error estimates.
error estimates.
4.
ANALYSES
3.6. Method 6: Single-StationRobustCascadeDecima-
For the purposesof comparison,we present the MT
apparent resistivitiesand phasesfor the analysesof all
Method 4 was modified by incorporating the trans- 67 days, and the apparent resistivitiesonly for the analfer functionimprovementschemeof Jonesand Jb'dicke ysesof the two 5-day time segments.Obviously,badly
[1984].The eightharmonics
that composed
eachmini- scatteredmagnitudeswith large errorswill have associstack were removedand replaced,in turn, in a jacknile ated badly estimated phases. It shouldbe remembered
approachto determinewhichharmonic,whenomitted, that under the limitation that the noise components on
led to a minimum in the variances of the off-diagonal the channelsare uncorrelated, the phasesare unaffected
MT impedances.This was repeatediteratively for pro- by bias errors.
tion
gressively
fewerharmonicsuntil the variancescouldnot
4.1.
All
Data
be reducedfurther by additional rejection. These minThe MT transfer functions from analyses of all 10
istacks, which containeddiffering numbersof harmonweeks
of data availablefor three of the methods(2, 6,
ics, werethen combinedusingthe samejacknifescheme
so as to minimize the variances of the final estimates of
and 8), and for the first 5 weeksof data for one of the
the off-diagonalelementsof Z. Error estimateswereob- methods(7) are illustratedin Figure 5. The horizontal
tained using standard statistical methodsthat assume magnetic fields observedat site 4 were used as the rethe noiseis Gaussian[Goodman,1965;BendatandPier- mote reference fields for method 8. It is apparent that
sol, 1971, pp. 204-207]Both upward and downwardbi- even with all 10 weeks of data, conventional spectral
ased estimates were computed.
analysisschemes(top left), representedby method 2,
3.7. Method 7: Single-Station Robust Processing
may not necessarilygive smooth estimates with low associated statistical error. In contrast, the three robust
The processingscheme,an extensionof the regression schemes,methods6 (top right), 7 (bottom left), and
M-estimate, is describedin detail by Egbert and Booker 8 (bottom right) all give extremely smooth estimates
[1986]. Single-pointoutlierswere cleanedup by a me-
with very low standard errors (generally(1%) in the
dian and median absolute deviation seven-point filter
scheme. Then, the data were Fourier transformed by an
approachsimilar to cascadedecimationwith a 128-point
100-1000 s range.
Other points to note are as follows'
monicswith power in the horizontal magnetic fields less
than a certain minimum, chosen on instrument noise
considerations,were rejected. In the estimation of the
transfer functions, a combination of band and section
the field componentitself, or of a correlating noisesource
1. The Pxy upward and downwardbiasedestimates
lengthbaseand a decimationfactorof 4. The 25% over- for method 6 appear to be separated by a virtually frelapping 128-point data segmentswere conditionedby quency independent multiplicative factor. This could
prewhiteningwith a first differencefilter and window- be explained in terms of noise sourceson either the D
ingby a time-bandwidthI prolatedata taper [Thomson, magnetic component or the E telluric component that
1977]prior to fast Fouriertransformation.Fourierhar- varies with period proportionally with the strength of
betweenthe components.The downwardbiasedPxyestimates of method 6 are, to within its statistical estimators, virtually identical to those of method 7 which
would imply that the noise source is on the E telluric
component. However, when compared to method 8 the
upward biased estimates are "correct" at short periods
averagingwas used with a bandwidth of 25% of the
center frequencyusing a regressionM estimate implementedas describedby Egbertand Booker[1986]. The
errors were derived using the standard asymptotic ap- (<400 s), whereasthe downwardbiasedestimatesare
proachas describedby Egbertand Booker[1986].
"correct"at the longerperiods(>1000 s). This would
imply that the uncorrelatednoiseis most significantin
the magneticfields at short periodsand in the electric
This method is the robust counterpart of method 5. fields at longer periods, which is consistent•vith the
It differsonly in the additionalstepof iterative reweight- results obtained from a multiple station analysis of a
ing of the LS response,
as describedin detail by Chave separateset of five long-period EMSLAB MT stations
3.8. Method 8: Remote ReferenceRobust Processing
et al. [1987]and ChaveandThomson
[thisissue],andis [Egbertand Booker,this issue].
2. All of the single-stationB-field referenceestimates
very similar to method 7. An initial LS solutionis applied to get a set of regressionresidualswhich are com- (methods2 and 7 and the downwardbiasedestimatesof
pared to a Gaussianmodel. Residualswhich are larger method6) appearto givesignificantlybiasedestimates
than expectedyield weightson the correspondingdata of apparent resistivities at the short periods.
sections which reduce their influence.
This continues
3. Method 7 appearsto give somewhaterratic Pxy
iteratively until the residual sum of squares does not and qb•yestimatesat periodscloseto I hour. This may
changesignificantly.The final residualsare Gaussian, be due to the nonuniform and energetic source fields
14,208
JONES ET AL.- MAGNETOTELLURIC
RESPONSE
FUNCTION
ESTIMATION
I
• Os
10
lO
9O
9o
75
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60
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45
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15
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Fig. 5. MT analysesof all data usingmethods2 (upper left), 6 (upper right), 7 (lowerleft), and S (lowerright).
Illustrated
arethePxyandPyxapparent
resistivities
andtheirphases
(notethatthe•Syxphases
havebeenrotated
into the first quadrant for clarity), with associatedstandard errors. Note that for method 6 there are both upward
and
downward
biased
estimates.
from magnetic bay disturbances,whosecharacteristic
periodsare in this range, dominatingthe responseover
the uniformbackgroundfield contributions,or it may be
due to only analyzing the first 5 weeksof data instead
of the whole 10 weeks as did the other three methods.
inherent assumptions for remote reference processing
have become invalid, i.e., that there existed a correlat-
ing "noise"sourceon the H componentover a distance
of some 30 km, although one would expect that singlestation processingwould also exhibit the same effects.
4. Method 8 givesthe smoothestresultsat longperiods,but a direct comparisonbetweenthe estimatesfrom
the different techniquesis difficult due to variations in
6. The results for method 8 at the shorter periods
appear to be more scattered than those for methods 6
bandwidth between the methods at these periods.
5. The estimates for method 8 at the longest periods
relative inefficiency of remote reference processing,or
and 7, particularly in ½•v' Does this just reflect the
does it reflect the effectsof outliers in the input (or
(>6000 s), particularlyof Pvo•,alsoappearto "droop" reference?) channels. The presenceof noise, and the
and be downward biased when compared to the cor-
possibility of outliers, in all channelsmakes this a very
respondingestimatesof methods6 and 7. The actual nonstandard robustnessproblem. Further work will be
causeof this droop is unclear. It could imply that the required to elucidate the causeof this problem.
JONES ET AL.: MAGNETOTELLURIC
7. The larger error bars at the shortest periods for
method 8 compared to methods 6 and 7 may be due
to the inherent inefficiencyof remote referencemethod
RESPONSE
FUNCTION
ESTIMATION
14,209
riods shorter than 100 s). Note that the long-period
"droop" in pyx is not evident and that theseestimates
of both pxyand pyxare moreprecise(smallererrors)at
oversingle-station
ones(seeintroduction)or maybe due the shorter periods, than those for the all data analysis
to erroneousassumptionsbeing made about the nature (Figure 5).
of the noise contributions by methods 6 and 7, whereas 4.3.
jacknileerrors(method8) shouldbe relativelymorero-
bust to violation of several assumptions. This difference
is particularly noticeablein comparisonwith method 7
and suggeststhat the standard asymptotic error estimates used for method 7 are overly optimistic.
Active
Period
The MT apparent resistivities from analyses of the
data for the activeinterval (Figure 4) are illustratedin
Figure 7, and once more, the pxy estimateshave been
shifted downward by one decadefor clarity. Again, also
displayed on Figure 7 are the estimates from method 8
(excludingthe first and last estimates)for all data as
4.2. Quiet Period
The MT apparent resistivities from analyses of the
a reference,and the horizontal magnetic fields observed
data for the quiet interval (Figure 3) are illustratedin at Site 13 (136 km distant) were used as the remote
Figure6. Note that the pxyestimateshavebeenshifted referencefields for method 5 and 8. The following points
are worthy of note:
1. Again, where both upward and downward biased
the figure are the estimatesfrom method 8 (excluding
estimates
have been computed the two estimates appear
the first and last estimates)for all data as a reference.
downward by one decadefor clarity. Also displayedon
The horizontalmagneticfieldsobservedat site 13 (136
km distant)wereusedas the remotereferencefieldsfor
method 5 and 8. The following points are worthy of
note:
to bracket
the
"truth."
2. The conventionalschemes(methods1, 2, and 3)
do a fairly reasonablejob for the pyx estimatesbut exhibit large bias and randomerrorsfor the pa:yones.The
1. Generally, where both upward and downwardesti- error estimates are substantially larger than for the robust methods.
mates have been computed for single-stationprocessing
3. All methods appear to have difficulty estimating
(methods1, 4, and 6), the two estimatesbracketthe
the pxy apparentresistivitiesbetween2000 and 4000 s.
"truth" as given by the referencelines.
2. The standardLS schemes
(methods1, 2, and 3)
performedparticularly badly with biasedestimatesthat
would lead to erroneous interpretations. This is particularly true for method 3 where, becauseof the apparent
consistency,one might attempt an interpretation. The
That this is evident on the remote reference processed
results(methods5 and 8) as well as on the upwardbiased results of method 6 is indicative perhaps of nonuniform source field problems rather than noise contributions.
4. Either remotereferenceprocessing
(method5), or
Pyxestimatesare smoothlyvaryingwith periodbut ex(methods6 and 7), or both (method
hibit a steepergradient than is real, and the pxy es- robustprocessing
8) can extract excellentestimateswith small random
timates are all some one third of a decade downward
errors and low bias errors from just 21,600 samples of
3. Remotereferenceprocessing
(method5) obviously data. However, the long-period error estimatesare much
aided correct estimation of the MT apparent resistivi- smallerfor the robust schemes,especiallyfor the pxy
ties, but bias effectsare apparent at the shortest periods component.
biased.
5. Again, the long-period"droop"in pyxfor method
8 is not evident,and the estimatesof both pxy and pyx
4.
Non robust cascade decimation processing are moreprecise(smallererrors)at the shorterperiods,
(method4) givesreasonablepyx estimates,but the ro- than those for the all data analysis(Figure 5). This
bust equivalent(method6) performedfar better for pxy would lead one to believe that the method 8 analysesof
(< 100s) and alsoin the pxyestimatesat around2000s.
The scatter
of the estimates
is still substantial
however.
all the data became contaminated at the longest pe-
with the lower inherent signal-to-noiseratio.
5. Many methods appear to give obviouslybiasedresults in the period range 1000-3000 s, particularly in the
km, but not over 135 km (all data analysisused site
6. Methods 5, 6, 7, and 8 all give reasonably inter-
contaminated to a larger degree by source-fieldeffects
riods by noise sources which were correlated over 30
pxy estimates.This is possiblydue to the nonuniform 4's magnetic components for remote reference, whereas
daily polar substormsdiscussed
above(Figure 3) which the 5-dayintervalsusedsite 13'smagneticcomponents).
Alternatively, the estimates from all of the data may be
appeargenerallyto upwardbias the pxy estimates.
pretableresponses,
particularlyof the pyxestimates,but
the robust methods 7 and 8 are superior even though
there is evident bias error in the pxy estimates for
[Egbertand Booker,this issue].
5.
CONCLUSIONS
AND OTHER
REMARKS
method 7 for periods in the 100-1000 s range.
7. Remote reference robust processing gave the
"best" estimates lying within a few percent of the
1. Travassosand Beamish[1988, p. 390] s•.ate "If
the (coherence-based)
selectionprocedurecanbe termed
"truth" (with the exceptionsof thoseestimatesat pe-
adequate then simple spectral stacking of individual so-
Four conclusionsare obvious from this comparison.
14,210
JONES ET AL.: MAGNETOTELLLTRIC
RESPONSE
FUNCTION
ESTIMATION
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JONES ET AL.: MAGNETOTELLURIC
RESPONSE FUNCTION
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ESTIMATION
14,211
14,212
lutions
JONES ET AL.: MAGNETOTELLURIC
works
well and there
is no need to resort
to a
statistically more robust treatment such as described
RESPONSE FUNCTION
ESTIMATION
Acknowledgments. The authors thank John Booker for his enthusiasm and leadership of the EMSLAB-Juan de Fuca project.
Many others are thanked for their contributions to EMSLAB -the list is endless and we refer the reader to the authorship list of
by Egbert and Booker[1986]." In this paper we have
demonstratedthat spectralstacking(methods1 and 3) the EMSLAB article [EMSLAB, 1988]. AGJ prepared and procan not "be termed adequate," and we have shown that
robust schemes,such as methods 6, 7, or 8, give superior
results. This is particularly true of short piecesof data
of low signal-to-noiseratio, such as the quiet 5 days.
2. To minimize bias errors, whenever possible, remote reference processingshould be undertaken. If the
remote components are not available, then both the upward
and
downward
biased
estimates
should
thesebiasedestimatesare the truth, but that (hopefully) they bracketthe truth (see,e.g., "all data", point
1). Robust singlestation estimatescan still be significantly biased by noise in the input channels,particularly
during periods of low activity.
3. Remote reference processing can still lead to
bias errors and is not the panacea once perhaps beThe
noise correlation
distances
in remote
refer-
enceMT havebeeninvestigatedby Goubauet el. [1984]
and Nichols et el. [1988], but obviouslyfurther work
is necessary on the sources of noise contributions and
the effects of possible nonuniform fields that are coherent between
the local and remote
sites. Coherent
noise
sourcesare possiblythe explanation for the long-period
"droop"in the p•x estimatesin the analysesof all the
data for method8 (Figure5) whichusedmagneticcomponents30 km distant as the remote references,whereas
in the analysesof the 5-dayintervals(Figures6 and 7)
method 8 used magnetic components 135 km distant.
The cause of the short-period "droop" in the estimates
from method 8 is unknown, but we do not ascribe this
to source effects.
4.
Look
at the data!
If the data
contain
obvious
noise,then interpolate short segmentsand removelarge
segments.
For in-field processingof data, adhoc robust schemes
that require relatively few computations, such as
method6 [Jonesand Jd'dicke,1984],couldbe appliedto
the data given today's computingtechnology.If more
advancedand more powerful computers are available in
the field, then rigorousrobust schemessuchas methods
7 and 8 [Egbert and Booker,1986; Chave e! al., 1987,
Chaveand Thomson,this issue],shouldbe adopted.
Although this work has concentratedon long-period
data, comparisons
by oneof us (A.G.J.) overthe last 4
years using a Phoenix MT data acquisition system has
shownthat the robustschememethod 6 alwaysgavesuperior results to the nonrobust scheme method 4. While
this doesnot in itself constitutea rigorouscomparative
study,it doessuggestthat notwithstandingthe very different noise sourcesat higher frequenciescompared to
thosein the data studied herein, robust methodsshould
always be used.
bution
26088.
be com-
puted to give a qualitative estimate of the magnitude
of the bias problem. Considering the possiblesources
of noise and their relative contributions in varying frequency bands, it should not be expected that either of
lieved.
cessed these data whilst at the Institute of Geophysics and Planetary Physics, University of California, San Diego, as a Green
Scholar. He wishes to express his thanks to the Green Foundation
and to all those at Scripps that made his stay there welcome, especially Steve and Cathy Constable. These data analyzed herein,
and the impedance tensor results, are available on application
to AGJ should others wish to test the performance of their own
algorithms and programmes. Geological Survey of Canada contriREFERENCES
Akaike, H., Some problems in the application of the cross-spectral
method, In Advanced Seminar on Spectral Analysis o] Time
Series, edited by B. Harris, pp. 81-107, John Wiley, New York,
1967.
Banks, R.J., Complex demodulation of geomagnetic data an the
estimation of transfer functions, Geophys. J.R. Astron. Soc.,
d3, 87-101, 1975.
Bendat, J.S. and A.G. Piersol, Random Data: Analysis and Measurement Procedures, Wiley-Interscience, New York, 1971.
Berdichevski, M.N., Principles of magnetoteluric profiling theory,
Appl. Geophys, 28, 70-91, 1960.
Berdichevski, M.N., Linear relationships in the magnetotelluric
cield, Appl. Geophys., 38, 99-108, 1964.
Bingham, C., M.D. Godfrey and J.W. Tukey, Modern techniques
for power spectrum estimation, IEEE Trans. Speech, Signal
Process., A U-15, 56-66, 1967.
Bostick, F.X. and H.W. Smith, Development of real-time, on-site
methods for analysis and inversion of tensor magnetotelluric
data, report, Electr. Geophys. Res. Lab., Univ. o] Tax. at
Austin, 1979.
Chave, A.D. and D.J. Thomson, Some comments on magnetotelluric response functions estimation, J. Geophys. Res., this
issue.
Chave, A.D., D.J. Thomson and M.E. Ander, On the robust estimation of power spectra, coherencies, and transfer functions,
J. Geophys. Res., 92, 633-648, 1987.
Dekker, D.L. and L.M. Hastie, Sourcesof error and bias in a magnetotelluric depth sounding of the Bown Basin, Phys. Earth
Planet. Inter., 25, 219-225, 1981.
Egbert, G.D. and J.R. Booker, Robust estimation of geomagnetic
transfer functions, Geophys. J.R. Astron. Soc., 87, 173-194,
1986.
Egbert, G.D. and J.R. Booker, Multivariate analysis of geomagnetic array data, 1, The response space, J. Geophys. Res., this
issue.
EMSLAB Group, The EMSLAB electromagnetic sounding experiment, Eos Trans. A GU, 89, 98-99, 1988.
Gamble, T.D., W.M. Goubau and J. Clarke, Magnetotellurics
with a remote reference, Geophysics, •, 53-68, 1979.
Geary, R.C., Relations between statistics: The general and the
sampling problem when the samples are large, Proc. R. Irish
Aced., •9, 177-196, 1943.
Gini, C., Sull'interpolazionedi una retta quando i valori della variabile indipendente SOhOaffetti da errori accidenteli, Metton,
1, 63-82, 1921.
Goodman, N.H., Measurement o] matrix ]requency response]unctions and multiple coherence ]unctions. Tech. rep. AFFDL
TR 65-56, Air Force Flight Dyn. Lab., Wright-Patterson AFB,
Ohio, 1965.
Goubau, W.M., T.D. Gamble and J. Clarke, Magnetotellurics using lock-in signal detection, Geophys. Res. Left., 5, 543-546,
1987a.
Goubau, W.M., T.D. Gamble, J. Clarke, Magnetetelluric data
analysis: Removal of bias, Geophysics, •3, 1157-1166, 1978b.
Goubau, W.M., P.M. Maxton, R.H. Koch and J. Clarke, Noise
correlation lengths in remote reference magnetotellurics, Geophysics, •9, 433-438, 1984.
Gundel, A., Estimation of transfer functions with reduced bias in
geomagneticinductionstudies, Acta Geod. Geophys. Montan.
Aced. Sci. Hung., 12, 345-352, 1977.
Haubrich, R.A., Earth noise, 5 to 500 millicycles per second, 1,
JONES ET AL.: MAGNETOTELLURIC
RESPONSE
FUNCTION
ESTIMATION
14,213
Spectral stationarity, normality and nonlinearity, J. Geophys. Thomson, D.J., Spectrum estimation techniques for characterizaRes., 70-, 1415-1427, 1965.
tion and developmentof WT4 waveguide,I, Bell Syst. Tech.
Hinich, M.J. and C.S. Clay, The applicationof the discreteFourier
J., 56, 1769-1815, 1977.
transform
in the estimation
of powerspectra,coherence
and Tikhonov,A.N., andM.N. Berdichevski,
Experience
in theuseof
bispectra
ofgeophysicaldata,
Rev.Geophys.,
6,347-363,1968. m&gnetotelluric
methods
to studythegeological
structures
of
Jenkins, G.M. and D.G. Watts, Spectral Analysis and Its Applications, Holden-Day, San Francisco, Calif., 1968.
Jones, A.G.,
Transformed
coherence functions for multivariate
studies. IEEE Trans. Acoust. SpeechSignal Process.,ASSP29, 317-319, 1981.
Jones,A.G. and R. Hutton, A multi-station magnetotel]uricstudy
in southern Scotland, I, Fieldwork, data analysis and results,
Geophys, J. R. Soc., 56, 329-349, 1979.
Jones,A.G. and H. JSdicke,Magnetotelluric transfer function estimation improvement by a coherence-basedrejectiontechnique,
paper presentedat 54th Annual International Meeting, Soc. of
Expl. Geophys., Atlanta, Ga., Dec. 2-6, 1984.
sedimentary basins, Izv. Acad. Sci. USSR Phys. Solid Earth
Eng. Transœ, 2 34-41, 1966.
Travassos, J.M. and D. Beamish, Magnetotelluric data processing
- A case study, Geophys. J., 93, 377-391, 1988.
Trigg, D.G., An amplifier and filter system for tel]uric signals,
Publ. Earth. Phys. Branch Can., JJ, 1-5, 1972.
Trigg, D.G., P.H. Sersonand P.A. Camfield, A solid state electrical
recordingmagnetometer, Publ. Earth Phys. Branch Can., •il,
67-80, 1970.
Turk, F.J., J.C. Rogers and C.T. Young, Estimation of the mag-
netotel]uricimpedancetensorby the l! and 12 norms,paper
presented at 54th Annual International Meeting, Soc. of ExJones,A.G., B. Olafsdottir and J. Tiikkainen, Geomagneticinducplor. Geophys., Atlanta, Ga., Dec. 2-6, 1984.
tion studiesin Scandinavia, III, Magnetotelluric observations, Tzanis, A., and D. Beamish, E.M. transfer function estimation
J. Geophys., 5•i, 35-50, 1983.
using maximum entropy spectral analysis, paper presented
Kao, D.W. and D. Rankin, Enhancementof signal-to-noiseration
at Eighth Workshop on Electromagnetic Induction in the
in magnetotelluric data, Geophysics,•i•, 103-110, 1977.
Earth and Moon, sponsor, Internat. Assoc. Geomag. Aeron.,
Larsen, J.C., 1989. Transfer functions: Smooth robust estimates
Newchat•l, Switzerland, Aug. 24-31, 1986.
by least squaresand remote referencemethods, Geophys.J.,
in press.
Law, L.K., D.R. Auld and J.R. Booker, A geomagneticvariation
anomaly coincident with the Cascade Volcanic Belt, or. Geophys. Res., 85, 5297-5302, 1980.
Lieneft, B.R., J.H. Whitcomb, R.J. Phillips, I.K. Reddy and R.A.
Taylor, Long term variationsin magnetotel]uricapparentresistivities
observed near the San Andreas
fault
in southern
California, J. Geomagn. Geomagn. Geoelectr.,$•, 757-775,
1980.
Maxwell, J.R., A Treatiseon Electricity and Magnetism,3rd ed.,
Vozoff, K. (ed.), Magnetotelluric Methods,Reprint Set. 5, Society
of Exploration Geophysicists, Tulsa, Okla., 1986.
Warmamaker, P.E., et al., Magnetotel]uric observations across the
Juan de Fuca subduction system in the EMSLAB project, J.
Geophys. Res., this issue, 1988.
Welch, P.D., The use of fast Fourier transform for the estimation of
power spectra: A method based on time averaging over short,
modified periodograms, IEEE Trans. Acoust. Speech Signal
Process., A U-15, 70-73, 1967.
Wight,D.E.,andF.X. Bostick,
Cascade
decimationA technique
for real time estimation of power spectra, Proc. IEEE Intern.
Conf. Acoustic, Speech, Signal Processing, Denver, Co!orado,
Maynaud, P.N., Derivation, Meaning and Use off Geomagnetic
April 9-11,626, 629, 1980.
Indices, Geophys.Monogr. Set., vol. 22, AGU, Washington, Wight, D.E., F.X. Bostick and H.W. Smith, Real time
D.C., 1980.
Fourier transformation of magnetotelluric data, report, Electr.
Neves, A.S., The magnetotelluric method in two-dimensional
Geophs. Res. Lab., Univ. of Tex. at Austin, 1977.
structure,Ph.D. thesis,Mass. Inst. of Teclmol.,Cambridge, Young, C.T., J.R. Booker, R. Fernandez, G.R. Jiracek, M. Mar1957.
tinez, J.C. Rogres, J.A. Stodt, H.S. Waft and P.E. WannaNichols,E.A., H.F. Morrison and J. Clarke, Signalsand noisein
maker, Verification of five magnetotel]uric systems in the minimagnetotel]urics,J. Geophys.Res. 93, 13,743-13,754, 1988.
EMSLAB experiment, Geophysics, 53, 553-557, 1988.
2 vols., Clarendon Press, Oxford, 1892.
Park, J., and A.D. Chave,On the estimationof magnetotel]uricresponsefunctionsusing the singularvalue decomposition,Geophys. J. R. Astron. Sot., 77, 683-709, 1984.
Reiers½l,O., Confluenceanalysisby means of lag momentsand
other methodsof confluenceanalysis,Econometrica,9, 1-22,
1941.
Reiers½l, O. Identifiability of a linear relation between variables
which are subject to error, Econometrica, 18, 375-389, 1950.
Sims,W.E., F.X. BostickandH.W. Smith,The estimationof magnetotel]uricimpedancetensor element, Geophysics,36, 938942, 1971.
Spitz, S., Field data linearizationin magnetotel]urics,
paper presentedat 54th Annual International Meeting, Soc. of Explor.
D. Auld, Geological Survey of Canada, Pacific Geoscience Center, P.O. Box 6000, Sidney, B.C., Canada VSL 4B2.
K. Bahr, Institut fiir Meteorologie und Geophysik, Frankfurt
Universit//t, Feldbergstrasse 47, D-6000 Frankfurt am Main 1,
Federal Republic of Germany.
A.D. Chave, AT&T Bell Laboratories 1E444, 600 Mountain
Ave., Murray Hill, NJ 07974.
G. Egbert, College of Oceanography,Oregon State University,
Corvallis, OR 97331.
A.G. Jones, Geological Survey of Canada, 1 Observatory Crescent, Ottawa, Ontario, Canada KIA 0Y3
Geophys., Atlanta, Ga., Dec. 2-6, 1984.
Stodt, J.A., Weightedaveragingand coherencesortingfor leastsquaresmagnetotel]uric estimates,paper presentedat Eighth
Workshop on Electromagnetic Induction in the Earth and
(receivedJuly 18, 1988;
Moon,sponsor,Internat. Assoc.Geomag.Acton.,Neuchat•l,
revised March 28, 1989;
Switzerland, Aug. 24-31, 1986.
.acceptedMarch 28, 1989.)