Oct 10, 1989 - from time series of natural source electromagnetic field variations. These combine ... vertical and horizontal magnetic fields or quantities derived from them. A variety ..... to measure residual size by the magnitude of the complex residu- ..... be used with a row removed in turn to get the delete-one esti- mates ...
JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 94, NO. B10, PAGES 14,215-14,225, OCTOBER
10, 1989
SomeCommentson MagnetotelluricResponseFunctionEstimation ALAN D. CHAVE AND DAVID
J. THOMSON
AT& T Bell Laboratories,Murray Hill, New Jersey
A new setof computational procedures are proposedfor estimatingthe magnetotelluric response functions from time seriesof naturalsourceelectromagnetic field variations.Thesecombinetheremotereferencemethod, whichis effectiveat minimizingbiaserrorsin theresponse, with robustprocessing, whichis usefulfor removing contamination by outliersandotherdepartures from Gauss-Markov optimalityonregression estimates.In addition,a nonparametric jackknifeestimator for theconfidence limitson theresponse functionsis introduced.The jackknifeoffersmanyadvantages overconventional approaches, includingrobustness to heterogeneity of residualvariance,relativeinsensitivityto correlations inducedby the spectralanalysisof finite datasequences, and computational simplicity. Thesetechniques are illustratedusinglong-periodmagnetotelluric data from the EMSLAB Lincolnline. Thepaperconcludes witha cautionary noteaboutleverageeffectsby highpowerevents in thedependent variablesthatarenotnecessarily removableby anyrobustmethodbasedon regression residuals.
INTRODUCTION
bances can destroy conventional spectral estimates. This has motivatedthe developmentof methodswhich are robust,in the Magnetotelluricand geomagneticdepth soundingtheory is senseof beingrelativelyinsensitiveto a moderateamountof nonbasedon the assumptionthat the function spacerepresentingthe stationarityand outliersor to small inadequaciesof the model, externalsourceshaslow dimensionality.This is usuallyachieved and which reactgraduallyratherthan abruptlyto perturbations of by specifyinga form for the externalsourceelectriccurrents;the either. These methodsare sensitiveonly to outliers in the depenplane wave or zero wavenumbermodel is the most common dent variable (usually E in a magnetotelluric context), but example. A variety of studieshave shownthat this implies the responsefunctionscomputedwith real data will also be biased existenceof a set of linear relationsbetweenthe electromagnetic downwardby noisein the independentvariables. This led to the field componentsobservedat the Earth's surface[e.g., Berdichevremote reference method of Gamble et al. [ 1979] in which auxilisky and Zhdanov, 1984; Egbert and Booker, this issue]. In the ary observations from a secondlocation,usuallyof the horizontal absenceof noise, these linear relationshipsfor magnetotellurics magneticfield, are usedto minimizebiaseffects. The successof may be written remotereferencemethodologyis attestedto by its nearly univerE=ZB (1) sal adoption by magnetotelluricpractitioners. While a vast improvementover conventionalapproaches,outliersin the elecwhereE andB are frequencydomain,two vectorsof the horizon- tric field or noise coherence between the local and reference fields tal electricandmagneticfield components at a singlesiteandfre- can cause it to fail. quencyand Z is the secondrank responseor transfertensorconAn essentialfeatureof any statisticalmethodis the provisionof nectingthem. The solutionof (1) is both an estimateand a measureof its accuracy. The traditional Z = (EB") (BB")-• (2) accuracyestimates,or confidenceintervals,on spectralquantities are obtainedusingexplicit statisticalmodelswhich are ultimately where the superscriptH denotesthe Hermitian transposeand the based on a Gaussian distribution. Despite this simplifying terms in parenthesesare the exact cross-powerand autopower assumption,the probability distributionsassociatedwith spectra spectra.Similarresponsefunctionsmay be obtainedbetweenthe are complicated,especiallyfor multivariateproblems. Theseprovertical and horizontalmagneticfields or quantitiesderivedfrom cedures also require auxiliary information such as the correct them. numberof degreesof freedomthat is often difficult to obtaindue A variety of methodshave been proposedfor the numerical to estimator-induced correlationsor the presenceof nonstationary computationof electromagnetic responsefunctionsandtheir assosignals.These(and other)problemswith distribution-based error ciatederrorsfrom a finite realizationof the inductionprocessand estimateson variableswith complicatedpropertieshave led to the in the presenceof noise. Most of these are basedon classical developmentof nonparametricestimatorswhich require few resspectralanalysisproceduresand least squaresregression,and are trictive conditions. The most widely used of the nonparametric ultimatelyderivedfrom simple,wide-sensestationary,Gaussian estimatorsis thejackknife which is reviewedin this work. models. It is generallyrecognizedthat naturalsourceelectromagIn this paper, new proceduresfor the analysisof electromagnetic data exhibit grossdeparturesfrom this simple situation, netic inductiondata are proposedwhich combinesomeof the best includingnonstationary phenomenasuchas geomagneticstorms features of the robust and remote reference methods and use the and outlierscausedby both measurementerrorsand sourcefield jackknife to provideerror estimates.In the next section,a brief inhomogeneity,and it is equally well-known that these disrutreview of the principlesof robust statisticsand its applicationto responsefunction estimation is given. This is followed by a descriptionof a hybrid robustremotereferenceestimatorand its Copyright1989by theAmericanGeophysicalUnion. implementation. A review of the jackknife and its use for the computationof confidencelimits on the responsefunctionsis then Papernumber88JB03910. 0148-0227/89/88JB-03910505.00 presented.Examplesfrom the long-periodEMSLAB magnetotel14,215
14,216
CHAVEANDTHOMSON: MAGNETOTELLURIC RESPONSE FUNCTION ESTIMATION
luric data are used to illustratethe advantagesof this approach. The paper concludeswith a cautionarynote aboutthe effect of leverageby high powereventsin the magneticfield that may not be removableby robustestimation,alongwith a suggested diagnosticto detectits presence.
in (3) are uncorrelatedand share a common variance; this holds independentof any assumptions abouttheir statisticaldistribution exceptthat it must have a variance. In addition,if the residuals are drawn from a multivariate normal probability distribution, then the least squaresresult is a maximum likelihood, fully efficient, minimum variance estimate. It should be noted that the
ROBUST RESPONSE FUNCTION ESTIMATION
latter conditionis essentialto the computationof confidencelimapproachbut is not necessaryto When E and B are actual measurements,(1) and (2) do not its using a distribution-based hold exactly due to the presenceof noise from samplingerrors, obtainthe responsefunctionsthemselves. With natural sourceelectromagnetic data, the Gauss-Markov violations of the induction model assumptions,and variability aboutthe error structureare rarely tenable. First, it is producedby a finite realizationof an infiniteprocess.It becomes assumptions necessaryto estimatethe responsefunctionsZ from imperfect oftentrue that the errorvariancedependson the signalpowerover at leastpart of a data series. A large portionof the misfit of the data,andthe problembecomesstatistical. In the sequel, it is assumedthat simultaneous,finite time datato the model in (3) is due to sourcefield complications.It is
sequencesof the electric and magneticfield from one or more sitesare available. It is alsopresumedthat the time seriesare collected without aliasingand that any necessarypreprocessing to removetrendsand grossdataerrorshasbeenperformedcorrectly. This may includethe removalof periodicphenomena like ocean tidesin the caseof seafloordata, and robustleastsquaresmethods are essentialfor this stepin the datapreparation.The time series may optionallybe prewhitenedusing an autoregressive or differentiation
filter.
A subset size is then chosen based on the
lowest frequency of interest and a target value for the final degreesof freedom. Each subsetis taperedwith a data window, Fourier-transformed,and stored;the subsetsmay be overlapped by an amountthat dependson the correlationpropertiesof the data window. The superiorityof the discreteprolate spheroidal sequences as data windowsis now well-documented.See Thomson [1977] for a thoroughdiscussionof the propertiesof prolate data windows and their use. In the remainderof this paper, the Fourier transformsof windowedsubsetsof the electricand magnetic field will be referredto as the data. The implementationof robust processingrequiresthe use of the overlappedsectionaveragingmethodoutlinedhere and is not amenableto a more conventional,straight band-averagingapproach. The subsets neednot be contiguous,so that sectionsof poor quality data may be excludedif necessary.However, it is sometimesdesirableto combinebandand sectionaveragingto raisethe effectivedegrees of freedom or produceestimatesthat are evenly spacedin frequency.
Thereis a largebodyof literatureon response functioncomputation; in additionto standardreferenceson spectralanalysisand regression, it includesthe papersby Simset al. [ 1971], Gambleet al. [1979], Larsen [ 1980], and Egbert and Booker [1986], which explicitly addressthe magnetotelluricproblem. Theseprocedures are basedon least squaresmethodsin which the tensorequation (1) is replacedby an equivalentmatrix form e=bz+r
(3)
where there are N observations(i.e., the number of data sections
timesthe numberof frequenciesin each sectionis N) so that e and r are N-vectors, b is an Nx2 matrix, and z is a two vector. The
residualpowerin (3) is thenminimized,yielding
z = (bHb)-1(bHe)
(4)
Theelements of (bHb) and(bHe) aretheaveraged autopower and cross-powerspectralestimatesbasedon the availabledata.
The advantagesof least squaresincludesimplicityand the optimality propertiesestablishedby the Gauss-Markovtheorem [e.g., Kendall and Stuart, 1977, chapter19]. For example,linear regressionyieldsthe bestlinear unbiasedestimatewhenthe errors
well-known that the energeticearly phasesof magneticstorms correspondto times of source field complexity; other active eventsmay be producedby morphologicallycomplex,small spatial scale current systems. Second, finite duration, transient features in the geomagneticfield cause outliers to occur in patches, violating the Gauss-Markov independencecondition. Finally, the requirementthat the errorsbe normally distributedis often untenable.Due to markednonstationarity, departuresfrom the model that producevery large residualsare likely, and such outliersare poorly describedby a Gaussianmodel. This situation virtually guarantees problemswith conventionalleastsquares.As a consequence, data analystshave adopteda variety of screening techniquesrangingfrom inspectionof the data for outliersto ad hoc coherenceweighting or high power event rejection in an attemptto alleviate the limitations of conventionalapproaches. True robustmethodscan accomplishthis undermore generalcircumstances and in a more automatic fashion.
The leastsquaresor Gaussianmaximumlikelihoodsolution(4)
to(3) wasobtained byminimizing theL 2 normrH'rof theresiduals. Some of the problemswith the L 2 norm treatmentof electromagneticdata have alreadybeen noted. The first of these,the occurrenceof unequalerror variance,is easily detectedby plotting the residualpower againstthe power in the magneticfield andnotingany correlations.It canbe treatedusingweightedleast squareswherethe rows of (3) are scaledby the inverseof the total power in the samerow of b. This is discussedin the contextof geomagneticdepth soundingby Egbert and Booker [1986], but hasrarelybeenfoundby the authorsto sufficefor magnetotelluric data,probablybecausethe errorvariancestructureis itselfnonstationary. In addition,the existenceof a fractionof residualswhose distributionis anomalouscomparedto the remainderseemsto be ubiquitousand requiresadditionaltreatmentto achieverobustness. However, in someinstancesit may be necessaryto prescale the data to equalize the error variance;it will be assumedthat suchpreprocessing is appliedwhenrequiredprior to utilizingthe robustestimatorsdescribedin this paper. The least absolutedeviationsor L • norm, where the sum of the absolutevaluesof the elementsof r is minimized, is a commonly usedrobustmeasure.This hasled to the suggestion to substituteit for leastsquaresin manygeophysicalapplications.Sucha course of actionis not recommendedfor reasonsdiscussedby Chave et al. [1987]. There are severalclassesof more generalrobustestimatesin currentuse;the most widely appliedare the M estimates which are motivatedby analogyto the statisticalconceptof maximum likelihood. For the presentapplication,M estimationis similarto leastsquaresin that it minimizesa normof the residuals, but the misfit measureis chosenso that a few extreme values cannotdominatethe answer. The M estimateis obtainedby solv-
CHAVE ANDTHOMSON: MAGNETOTELLURIC RESPONSE FUNCTION ESTIMATION
14,217
ing min{RH'R},whereR is an N vectorwhoseith entryis duals. If N real residuals{r,} are convenedto order statisticsby (p(ri/d))•, d is a scalefactor,andp(x) is calleda lossfunction. placingthemin theascending orderr(•)_ 10; this is generally not truefor the morecomplicated disb•econsidered. Denoteanestimate of 0 basedonall of thedataby 0. The data are then divided into N groupsof size N-1 each by tributionsused in parametricestimateson spectra. However, deletingan entry in turn from the whole set. Let the estimateof 0 asymptoticbehaviordoesnot guaranteecorrectresultsfor finite based on the ith subset,where the ith datum has been removed, be 0_i. The pseudovaluesare
•i - N• -(N-1)•_i
(20)
samples,and Hinkley [1977a] has shown that substantialerrors
can accrueif (23) is usedblindly on markedlynon-normalstatis-
ticswith,small samples. Thiscanbe corrected usingtransformationson 0 to geta moreGaussian form. In a magnetotelluric con-
functionsare roughlyGaussianwithoutmodifiand serveas substitutejackknife data in standardstatisticalpro- text, the response cation, and nonnormality is not a criticalconsideration.However, cedures.The jackknife mean is just the arithmeticaverageof the jackknifing of coherence estimates requiresan inversehyperbolic pseudovalues tangent transformation; see Thomson and Chave [1989] for details.
N ^
1•(i)i N^
- N•- N-1 ,•_,0_, N i=1
Some further complicationsoccur when the jackknife is appliedto theregression problems(3) or (6). First,the response z are vector-valued,so the scalarvariance(22) becomes (21)functions a covariance matrix. Second,regression is an unbalanced opera-
tion in the sensethat the overallsamplesizesand possiblythe >From (21), it is clear that the pseudovalues (20) are not really sample variancesare different for the vector e and the matrix b. necessary,and the delete-oneestimatescan be used directly. To apply the balancedjackknife (22) to (3) or (6), (4) or (7) can However,the pseudovalue form pervadesthe jackknife literature. be usedwith a row removedin turn to get the delete-oneestiThe quantity(21) was originally• introduced as a lowerbias mates,and (22) canbe applieddirectlyaftermodificationto allow replacementfor the regularmean 0; seeEfron [1982] for details. for the vectorform of 0 and replacementof N-1 with N-p, For a statistic0 which is linear in the data,the jackknifevalue 0 wherep is thenumberof columnsin b, to correctthesamplebias. and the conventionalone 0 will be equivalent. Several studies However,Hinkley [1977b]examinedthe smallsampleproperties haveshown thatthevaria_bility of thejackknife meancanbelarg•e of thebalancedjackknifevarianceappliedto regression andsugfor some statistics, and 0 should be used as a substitute for 0 gestedthatit producesa consistently biasedresult. He proposed when it is distinctonly with caution. the useof a weightedpseudovalueto eliminatethe error. RewritA moreimportantapplicationfor the jackknifeis in the non- ing (20) in vectorform andincludingtheweightgives parametricestimationof the varianceof an arbitrarystatistic.The jackknifevarianceis just the standardsamplevarianceof the Pi= (N(1-hi)+l) •-N (l-h/)•)-i (24) pseudovalues, andmay conveniently be writtenin termseitherof the pseudovalues or of the delete-oneestimatesas •2
1
N
S=/(/--1),•_,(•i_•)2 i=1
_N-1 N ^
----•--/•(0-i --•)2 .=
where • isthevector statistical parameter replacing • and the{hi}
are the diagonalelementsof the hat matrix. Since the hat matrix is importantto severalpointscontainedin the remainderof this paper,a digression to examineits propertiesis in order. The hatmatrixis definedby considering theresidualr in (3) as the difference between the observed electric field e and that
(22)predictedbytheregression •, sothat
where
r = (I- H) e --
I
N
^
0= •0_ i i=i
(25)
whereI is theidentitymatrixandH is thehatmatrixgivenexplicitly by
14,220
CHAVEANDTHOMSON: MAGNETOTELLURIC RESPONSE FUNCTION ESTIMATION
•00
cult to showthat the magnitudesof the diagonalelementsof (27) are bounded by 0 and 1. Furthermore,numerical simulations showthat the diagonalelementsare nearlyreal when b and bR are highly correlatedand behavelike thoseof (26). In addition,the equivalenceof (26) and (27) can be verified usingthe errors-invariables model (17)-(18). While these are not rigorous arguments,they can be justified heuristicallyfor the applicationsconsideredhere. It will be assumedthat the magnitudeof the diagonal of (27) can be usedinterchangeably with the diagonalof (26) in applicationsusedin the remainderof thispaper. When hi=p/N in (24), the balancedpseudovalueis obtained; since this is actually the expectedvalue of hi , the unbalanced pseudovalueis in generaldifferentfor finite samples.The jackknife estimateof the regressioncovariancematrix is
-
•0-
•0 10 4
10 3
102
Per•',od (s)
102 • _
S=N(N_p-••.i•i[O-Pi][•)-pi]H (28)
_
_
_
_
•
where • isthearithmetic average ofthe{P,}asin (20).The
10 • _
•
diagonal terms of (28) give the variancesof the corresponding termsin z, while the off-diagonaltermsare the covariances;note that the errors on the real and imaginaryparts are identical. An importantpropertyof the jackknife regressionvarianceis robustnessin the presenceof inhomogeneityof error variance,in contrast to parametric estimators[Hinkley, 1977b]. Note also that (22) or (28) do not require an explicit accountingfor the effective degreesof freedom. In the presenceof the inevitablecorrelations
_ _
q.
-
I
I
10 4
I
I
IliJll
I
I
I
10 3
IIIltl
I
t
I
10 2
Per?;od (s) Fig. 1. The phase(top panel)andapparentresistivity(bottompanel)for the TE moderesponsefunctionat Lincoln line site4 remotereferencedto site 1. The TE mode has the electric field polarized geographicnorth-south. The bandsare the double-sided95% confidencelimits computedusingthe jackknifeestimatesof standarderrorand (29)-(30). Note the variabilityof the responsefunctionand the largeerror bars;the uncertaintyin the phase is typically 15ø andfor the apparentresistivityit oftenexceeds50%.
H = b(bHb)-1bH
80
•
60
•
40
(26)
For robust regressionusing (6)-(7), (26) must be modified by
a• 2o
replacing b" withbHwto account fortheweights.Thehatmatrix is a projection matrixwhichmapse onto•. Thelackof balance 10 4
in a regressionproblem is reflectedin its diagonal. Hoaglin and Welsch[ 1978] summarizesomeof the propertiesof the hat matrix diagonalthat are essentialto its interpretation.First, H is a pro-
10 3
10 2
Period (s)
jectionmatrix,henceis symmetricand idempotent (H2=H). These characteristicscan be used to show that 0_
