Seismic attribute inversion for velocity and ... - Semantic Scholar

JOURNAL

OF GEOPHYSICAL

RESEARCH, VOL. 102, NO. B5, PAGES 9949-9960, MAY 10, 1997

Seismicattribute inversionfor velocityand attenuationstructure usingdata from the GLIMPCE Lake Superior experiment Michael P. Matheneyand RobertL. Nowack Departmentof EarthandAtmosphericSciences,PurdueUniversity,WestLafayette,Indiana

Anne M. Tr•hu Collegeof OceanicandAtmospheric Sciences, OregonStateUniversity,Corvallis

Abstract. A simultaneous inversionfor velocityandattenuationstructureusingmultiple seismicattributeshasbeenappliedto refractiondatafrom the 1986GLIMPCE Lake Superior experiment.The seismicattributesconsideredincludeenvelopeamplitude,instantaneous frequency,andtraveltime of first arrivaldata. Instantaneous frequencyis convertedto t* using a matchingprocedurewhichapproximatelyremovesthe effectsof the sourcespectra.The derivedseismicattributesare thenusedin an iterativeinversionprocedurereferredto as AFT inversionfor amplitude,(instantaneous) frequency,and time. Uncertaintiesand resolutionof the velocity and attenuationmodelsare estimatedusingcovariancecalculationsand checkerboard resolutionmaps. A simultaneous inversionof seismicattributesfrom the GLIMPCE dataresults in a velocity model similarto that of previousstudiesacrossLake Superior. A centralrift basin and a northernbasinare the mostprominentfeatureswith an increasein velocitynearthe Isle Royale fault. Althoughthereis an indicationof the centralandnorthernbasinsin the attenuationmodel for depthsgreaterthan 4 km, the separationis not evidentfor shallower depths.This may resultfrom microfractures maskingcompositional variationsin the attenuation modelfor shallowerdepths.AttenuationQ valuesrangefrom approximately60 nearthe surfaceto near500 at 10 km depth. A relationshipbetweeninverseQ and velocityof

Q'•=0.0210-0.0028*v wasfoundwitha correlation coefficient of-0.96. Thissuggests a nearly linear,inverse relationship between Q.landvelocity beneath LakeSuperior whichsupports previouslaboratoryresults.The invertedvelocityandattenuation modelsprovideimportant constraintson the lithologyandphysicalpropertiesof the Midcontinentrift beneathLake Superior.

sedimentaryunits. The refraction surveysprovide valuable informationon the velocity structureand lithologyof the rift The Midcontinentrift (MCR) is a 1.1 Gyr old featurewhich basin and underlying rocks. Early seismic refraction extendsfrom Kansasup throughIowa, Minnesota,Lake investigations by Berry and West[1966], Steinhartand Smith Superior,and into Michigan. The studyof the MCR is [1966], and Halls [1982] imaged lower velocities associated important for understanding riftingprocesses andreactivation with the uppersedimentaryrocks,highervelocitiesassociated of ancient rifts. The MCR is also of interest because of its with the underlyingvolcanic rocks, and a thickeningof the hydrocarbon potential[Dickas,1984] and mineralwealth crustbeneaththe MCR. More recenttomographicimagingand [LaBerge,1994]. Gravity studiesfirst revealedthe MCR forward modeling studies[Trdhu et al., 1991; Lutter et al., Introduction

[Woolard,1943;Hinze et al., 1975] and continueto provide 1993; Hamilton and Mereu, 1993] have delineated the insightintotherift's development [Allen,1994]. Themagnetic thickened crust, central and northern rift basins, an increase in properties of the volcanicrockshave also beenusefulin velocity around the Isle Royale fault, and higher velocities determining therift'sevolution [Hinzeet al., 1966;Chandler et associated with lava flows and intrusives. al., 1989; Mariano and Hinze, 1994].

In general, there have been fewer studies of in situ The most detailedimagesof the MCR have come from attenuationcomparedto studiesof velocity [Carpenter and seismicreflectionand refractionprofiles. Seismicreflection Sanford, 1985; Brzostowskiand McMechan, 1992]. This is surveysprovide the highestresolutionof the rift basin. mostly due to the difficulty in accuratelymeasuringseismic Substantial crustal reflections come from contrasts between

attributes

used

to

estimate

in

situ

seismic

attenuation.

volcanicflows and sedimentaryrocks [Behrendtet al., 1988; Laboratory measurementsof rocks, however, have provided Cannonet al., 1989; Chandleret al., 1989]. Additional seismic information on the attenuation associated with certain rock reflectors result from composition changes within the types[Toks6zet al., 1979; Wepferand Christensen, 1991;Best et al., 1994]. These laboratory attenuationestimates are typicallymadeat seismicfrequenciesin the megahertzrange, Copyright1997by theAmericanGeophysical Union. and it is not clear how these values relate to attenuation at

Papernumber97JB00332.

frequenciesusedin seismicrefractionstudies[Goldbergand

0148-0227/97/97JB-00332509.00

Yin, 1994]. 9949

Some attenuationstudies have used frequency-

9950

MATHENEY

ET AL.: SEISMIC ATTRIBUTE

92 ø

90 ø

..Ii,'•/i'I -:- .

''':-:

:

,-•.•,.,.,•,

•.,.,.,

-' '.'5 ....

' : ',',•mu,

2, •, ',',-,-;-7

, ", ',:; ",-,-; 48 ø '.-'",' ., . .•...•,•2•

86 ø

.R •-

':- /_-/.• .' '.,.,

':

•

- -

.c•-'/'

•

•Z

•, ' • G o • '

.....

:•1 ,'

'•c:

?-•;.""..

ß..

iI I•111111••'.•,

...

,,'_q •

-7'-' •-'/•,•• •½•,,.

•'"•••••••

•

88 ø

,. ,.-,-,-,.'•

,-,?,-,.•.'.-

4T

INVERSION

1•

Vz, :, ,,-,.-,- ,'. ','z ,-,

' ''•'"' _

•

70.

• ' •

..... ..;/

,'.,

roc•

,

i

EXPLANATION

Upper Keweenawansedimentaryrocks (BayfieldGroupandJacobsvilleSandstone)

UpperKeweenawan sedimentary rocks(OrontoGroup) Middle

Keweenawan

(Basaltflows andsedimentaryrocks/Gabbroic intmsives) Lower

Keweenawan

(Basaltflowsandunderlyingsedimentary rocks) Lower Keweenawan? (SibleyGroup) ArcheanandLower Proterozoiccrystallinerocks Figure 1. Map of the Lake Superiorareashowingthe probablelithologiesand the locationof line A of the GLIMPCE experiment. SeismometerlocationsS4 and C1 are land-basedseismometers and A2, C4, C9, and C3

are lake bottomseismometers. The dashedline showsthe axis of the Midcontinent rift (MCR). (Geology adaptedfrom Cannonet al., [1989])

independent attenuation operators [Futterman, 1962; Kjartansson, 1979] to estimate attenuation[Johnston,1981],

while other studies have used frequency-dependent Q

Geologic Overview The crustalrocksimagedusingthe GLIMPCE reflectionand

mechanisms[Mason et al., 1978]. In all cases,in situ estimates refraction data are rift volcanics and sediments of middle

Proterozoic (~1.1 Ga) age. At the startof thisriftingevent, upwellingof mantlematerialcauseddomingacrossthe Lake attenuation estimates can also be used in seismic reflection data Superiorregion[Allenet al., 1992;CannonandHinze, 1992]. processingto provide improved images of the subsurface Resultingextensionalforces,due to the uplifting,initiated [Sollieand Mittet, 1994;Brzostowski and McMechan,1992]. rifting. The earliestfloodbasaltscoveredlargeareasin and This studyusesseismicrefractiondata from the 1986 Great adjacentto thecentralrift withflowsvaryingin thickness from Lakes International Multidisciplinary Program on Crustal severalmetersto over100m [Green,1989]. Duringthetimeof Evolution (GLIMPCE) experiment. Part of the GLIMPCE volcanism,1109Ma to 1084Ma, periodsof quiescence allowed and conglomerates betweenthe experimentincludedthe recordingof a 250 km long, wide- for depositionof sandstones anglerefractionprofile whichextendedacrossLake Superior lava flows. Furthersubsidence duringvolcanism, especially from northto south(Figure1). Data wererecordedby four alongthecentralrift, allowedlavaflowsto accumulate upto 19 lake bottom and two land-based seismometers. km in thickness[Behrendtet al., 1988; Cannonet al., 1989]. of volcanicandsedimentary rocksis calledthe The seismicattributesusedin this studyincludeenvelope This sequence of attenuationare important for the interpretationof the

physical state of the subsurface[ToksOzet al., 1979]. In situ

amplitude,instantaneous frequency,and travel time of the first

PortageLake Volcanicsequence.

As the igneousactivitydecreased alongtherift around1089 arrivals. The instantaneous frequencies are converted to t* by using a matchingproceduregiven by Matheneyand Nowack Ma, the depositionof sedimentsbecame the dominant rock[1995] which approximatelyremovesthe effectsof the source formingactivity. The OrontoGroupis a large sequence of spectra. The trace attributesare then utilized in an iterative,

sedimentaryrocks, with intermittent basaltic flows, which

inversion procedure which simultaneouslyimages the subsurfacevelocity and attenuationstructure. This is referred to as AFT inversion for the attributes amplitude, (instantaneous)frequency, and time. Uncertainty and resolution estimates are obtained through covariance calculationsandcheckerboard resolutionplots.

overlies the volcanicrocks aroundLake Superior. These sedimentsare similar in compositionto the volcanicflows, indicatingthat a significantpart of the originalbasalticlava flowswaserodedto formtheOrontoGroup.TheOrontogroup has a maximum thickness of about 8000 m near the center of

the rift basin. The BayfieldGroupis a younger,undeformed

MATHENEY

ET AL.: SEISMIC ATI'RIBUTE

INVERSION

9951

1.7-

0.8-

O0

-60.0

-•,9.2

-38.3

-27.5

-16.7

-5.8

5.0

15.8

26.7

37.5

•,8.3

59.2

Distance (km)

Figure2. GLIMPCELakeSuperior recordsection A2 withamplitudes for eachtracenormalized by the maximum amplitude overa 0.4second window afterthefirstarrival.Everyfifthtraceisplotted.

sequenceof sandstones. Thesesandstones vary up to 2000 m in thickness[Hallsand West,1971]anddid notundergothe tilting andwarpingassociated with theLake SuperiorSyncline. The GrenvilleOrogenysubjectedthe MCR to compressional forces which causedthrust faulting and horst development. Thesefaultsincludethe Keweenawfault, Isle Royale fault and the St. Croix Horst. With the uplift associatedwith faulting, some reworking of earlier sedimentsoccurred. This was the last significantrock-formingepisodein the Lake Superior region.

Becauseof the larger uncertainties that amplitudeshave relative to the travel times, an independentmeasureof attenuation is importantfor constraining the attenuation model. The approachused here is to utilize severalseismicattributes

includingamplitude,instantaneous frequencyandtraveltimes. The instantaneous frequencies are convertedto t* usinga matchingprocedure whichapproximately removesthe effects of the sourcespectra[MatheneyandNowack,1995]. The AFT inversion algorithm then uses the attributesto determinethe

velocityandattenuation models[Nowack andMatheney, 1997]. In order to extract seismicattributes,the travel times for the

first arrivalP wavesare first estimatedusingan interactive Data Analysis computer pickingroutine. To matchreciprocity andprovidea The initial dataanalysisof this studyrequiredthe extraction self-consistent data set betweencommon-receiver gathers, of seismic trace attributes, travel time, amplitude, and interpolationand smoothingof the data are performed. instantaneousfrequency, from the seismic record sections. Seismic attenuationcausedby intrinsic attenuation,as well as scattering,in the subsurfaceresultsin a loss of amplitudeas well as a lowering of the frequency content. The use of instantaneous frequencycan be used to estimatethe pulse frequencyfor specificphases. Figure2 is a typicalcommonreceivergatherfrom the GLIMPCE Lake Superiorexperiment. Every fifth traceis plottedwith amplitudesnormalizedfor each trace. The seismicprofilesare thereverseof typicalprofilesin that they have a singlereceiverandmultiplesources. In typicalstudiesof crustalstructure,only the traveltimesof reflectedand refractedarrivalsare used to obtain a velocity model [Zelt and Smith, 1992; Lutter et al., 1993]. To determine

in thedatarepresent variations notaccountable by themodeling andarethereforesmoothed.Smoothing alsoimprovesstability

the anelastic nature of the medium, additional information is

of the inversion and eliminates outliers which can affect the

Interpolation is performed using a splines-under-tension algorithm[Cline, 1974] to interpolatethe travel timesonto a

uniform0.2 km distance grid. An 11-pointboxcaraveraging filter witha lengthof 2.2 km is usedto smooththeinterpolated traveltimevalues.Thetraveltimesarethenreinterpolated to a 1.0km grid(Figure3a). Althoughthedataareresampled to a

uniform grid/the essential aspect isthesmoothing ofthedatato longer spatial wavelengthsconsistentwith the broadscale

featuresmodeled by theinversion algorithm.Forexample,the initial lateral nodespacingof the modelis 65.0 km, and it is 16.2 km in the final model. Lateral variations less than 4-5 km

required. Amplitudesare sometimesusedto determineseismic final model [Tarantola, 1987]. Sincethe lateralblock sizesare attenuationby initially using the travel times to invert for a muchlargerthanthe lengthof the smoothing filter, smoothing velocity model. The velocity model is usedto determinethe will haveonly a limited effect on the derivedmodel. amplitudedecaydue to geometricspreading.The differences After travel time picking,the amplitudesand instantaneous betweenthe observedamplitudesand the computed,geometric frequenciesare extracted from the P wave first arrival data. spreadingamplitudesare then used to determinethe seismic The amplitudes of the first arrivalP waveare calculated by

attenuation of themedium[Bregrnan et al., 1989].

takingthepeakof thetraceenvelope for thedesiredpulse.The

9952

MATHENEY ET AL.: SEISMICATTRIBUTE INVERSION SHOT C4 TRAVEL

TIMES

the effects of the sourcespectra. A near-sourcereference pulse Pr(t) is first selectedfor an observedseismicgather. The referencepulseis thenattenuated resultingin att

Pr (t) = 1FFT[Pr(•)A(•)]

(1)

whereIFFT refersto theinverseFouriertransform,Pr (to) is the Fourier transformedreferencepulse, and A(to) is the attenuation operator.The causalattenuation operatorusedhere

I Observed data pt.

is given by ---t*ln

I

50

100

150

200

Distance (krn)

'•'

2

GATHER C4 NORMALIZED

o•

e

2

,

(2)

where t = • (Q-l(s)! c(s))ds, c(s)is thevelocity, tor is

AMPLITUDES

the referenceradialfrequency, Q is the seismicqualityfactor,

b)

Z

A(to) -- e

•r

an s is the lengthalongthe ray path [Aki and Richards,1980]. The t* values are obtainedby matchingthe instantaneous frequencyof the observedpulse with that of the attenuated reference pulseasdescribed by Matheney andNowack[ 1995]. The qualityof theinstantaneous frequencies is determined in the sameway as for the amplitude. The amplitudeafter the first arrival peak must decreaseby at least 20% for the instantaneous frequencyto be accepted. For recordsections A2, C4, C9, and C3, the samereferencepulse taken from

-10 50

100

150

200

Distance (km)

GATHER C4 DIFFERENTIAL

0.08

'

ATTENUATION .

0.06

0.04

[3

0.02

0.00 i

is usedto obtaint* valuesfrom the instantaneous frequencies. For thetwoendrecordsections, $4 andC 1, reference pulsesare takenfrom their respectivegathersat source-receiver distances

of 23.7 km and 13.5 km, respectively.The larger sourcereceiveroffsetsare the result of the experiment arrangement

c)

50

receivergatherA2, with a source-receiver distanceof 1.76 km,

,

.

.

i

i

!00

i

150 Distance (km)

i

i

!

,

200

with no shortoffsettracesbeingavailable. Oncethe relativet* valuesare extracted,they are interpolatedand smoothedto a uniformspacingof 1 km (Figure3c). Becauseof the multipleshotlayoutof the Lake Superior experiment andtheinterpolation andsmoothing procedure used to processthe data,reciprocityof traveltimes,amplitudes, and t* canbe usedto checkfor consistency andaccuracy of thedata picks. Figure 4 shows plots of seismic attributesversus midpointfor the differentrecordsections. In theseplots, reciprocitypointsfall halfwaybetweenreceiverlocations.This

Figure 3. (a) Observedand interpolated traveltimesusedin provides a simplegraphical checkfor reciprocity.Reciprocity the inversionroutinefor recordsectionC4. (b) Normalized, locationsare shownon Figure 4 by vertical dashedlines

naturallogarithm of the observed andinterpolated amplitudes halfway between the different receiver locations. For the for gatherC4. (c) Observedand interpolated differential resulting travel times, reciprocityis satisfied given an attenuationvaluesfor gatherC4.

uncertaintyof 0.06 s. This uncertaintyis determined interactivelyduringthepickingof thefirst arrivaltraveltimes. Reciprocityplotscan alsobe usedto checkthe amplitude quality of the amplitudecalculationis determinedby the and t* valuesfor consistency.However,while the travel times amounttheamplitudedecreases afterthepeak. If theamplitude are absolute values, the amplitudes and t* values are doesnot dropat least20% after the peak,thenthe amplitudeis normalizedto the referencepulsedistance. Thereforeany consideredto be corrupteddue to interferencefrom later differencesin amplitudeor t* betweenthe shotlocationandthe arrivals, and the amplitudevalue is not included. The natural referencepulse(in this casea source-reference distanceof 1.76 logarithmof the amplitudesfor gatherC4, normalizedby a km) would not be accounted for in the reciprocityplot. near-source referencepulse,is shownin Figure3b. Although Consideringthat the four central shot gathersare all lake the air gun sourcestrengthis known, slight variationsin the bottom instrumentsand that the sourceis close to the reference sourceand energy penetrationcan affect the amplitudeand pulse,the reciprocity plotswill still be usefulfor checking frequencyestimates.As a result,smoothing of the amplitude consistency andaccuracy in thedatapicksin thisexperiment. and t* estimateshasbeen appliedin a similar fashionas to the Figure4b showstheIn-amplitude with midpoint.The vertical travel times.

dashed linesshowthereciprocity locations.Reciprocity is still

The instantaneous frequencyvaluesareconverted to t* using approximatelysatisfiedwithin the _+0.5data uncertaintiesin an instantaneousfrequency matching proceduregiven by the In-amplitudes.Figure4c showst* versusmidpoint.

Matheney andNowack[1995],andthisapproximately removes Reciprocityis also approximately satisfiedgiven a t*

MATHENEYET AL.: SEISMICATTRIBUTEINVERSION LAKE

SUPERIOR

MID-POINT

-VS-

TIME

9953

GATHER

a) 4

50

100

150

200

250

•/•d-point(kin) MID-POINT

-lO

VS AMPLITUDE

i

50

i

100

i

150

200

.

,

.

250

Mk-point MID-POINT 0.06

i

....

i

....

i

VS

T*

i

!

c) 0.04

0.02

0.00 ß

,

50

i

i

100

150

i

200

i

.

.

250

Mk-point(km) Figure 4. (a) Travel timesplottedwith the midpointbetweensources andreceiversto bettershowreciprocity. Dashed,verticallinesare the reciprocitylocationsbetweengathers.(b) Amplitudeswith midpoint.(c) t* with midpoint. Note thatreciprocityis matchedat all locationsfor eachattributewithindatauncertainties.

uncertaintyof _+0.004 s. Both the amplitudeand t* refraction data, a simultaneous inversion of the seismic uncertainties are obtainedby estimatingthe magnitude of the attributesis used. Separateinversionsof amplitudeand t* scatteraboutthe interpolatedand smoothed datapointsshown would need to accountfor the velocity componentin these in Figures 3b and 3c.

Record sectionsS4 and C1 are not

parameters.The modelparameters, slowness and Q-l, are

includedin thereciprocityplotsfor amplitudeandt* because of thelargersource-receiver offsetof thereference pulse.

specifiedat node locationswith a splineinterpolationof both parameters.The model parametersare relatedto the seismic attributesthroughthe linearizedrelation

Inversion

Method

We presenta brief descriptionof the inversionmethod. A more detailed accountis given by Nowack and Matheney [1997]. Owing to curved rays associatedwith seismic

• * --I i•tlieui•t* /i•Q -l

lnA L•l lnAli}u•}lnAli}Q -•

(3)

9954

MATHENEY ET AL.' SEISMIC ATFRIBUTE INVERSION

where T is the travel time, t* is the attenuation factor, and

REDUCED TIMES (STARTING MODEL)

ln A is the In-amplitude. The calculated travel times, amplitudes,andt* areobtainedby kinematicanddynamicraytracing. The amplitudesinclude geometricspreadingand attenuation. The geometric spreadingcomponentof the amplitudeis computedusingdynamicray methods[Cerven• and Hron, 1980] where the validity of ray methodsrequiresa smoothlyvaryingmedium[Ben-Menahern and Beydoun,1985]. The travel time, amplitudeand t* partialsare obtainedfrom perturbation analysis[Nowackand Lutter, 1988a;Nowackand Lyslo 1989;NowackandMatheney,1997]. The solutionof (3) is obtainedby iterative, dampedleast

5.00 4.00

-

+ observed

-

- calculated

-

3.00 _

2.001.00

-

0.00

,

I

,

Data error= 0.060

NORMALIZED

squares whichatthen'hiteration solves 0.00

d - •'(•n) = Gn(• - •n)

(4)

I

!0o,Distance (km) 200,RMS error= 0.775

North

AMPLITUDES

-

•ø-2O0 '•-4 .oo-

whered is the datavector,•('•n) is the solutionof the

forward problem at thenthiteration, Gn is thesensitivity matrix, and .• is the modelparametervector. Normalization

of the data and model residual vectors is

-6.00

-

-8.oo

-

accomplishedby weightingthedataresiduals by theestimated

b)

data covariance matrixCd andthe modelresiduals by a North

weighting matrixCx,. Thedatacovariance matrix Ca is assumedto be diagonalwith the diagonalelementsgivenby the squareddata uncertainties.The diagonalcomponentsof the

,

I

100.

,

I

,

200, RMS error=2.334

Distance (kin)

Data error= 0.500

t* 0.04

modelweighting matrixCx, are proportional to prior

0.03

estimates of the squared model errors and inversely proportionalto the block size of the parameterization of the model. The variable block weighting removesthe effects of unequalblock sizesin the discretizationof the model [Nolet, 1987]. The prior errorsof the modelparametersusedare 0.15

0.02 0.01 0.00

km/s for the velocityand 0.0020for Q". These weights

result in the vectors•' =C•112 (t•- •('•n))

and

North

100,

200, RMS error= 0.006

Distance (km)

Data error= 0.004

.•, _-1/2

Figure 5, (a) Observed and calculated travel times for a = Cx n (•n+l- Xn)ßThe solution ofthelinearized problem laterally homogeneousstarting model with 35 nodes. (b)

is then

,T

,

.•': (Gn Gn +I)

-1

, -'

Gnd'

(5)

_ _1/2.•, , =c_-1/2_ _1/2 and Xn+ 1--'•n 4-C'xn . where Gn d CinCxn Results

Observedand calculatedamplitudesfor the startingmodel. (c) Observedandcalculatedt* valuesfor thestartingmodel.

modelis 5.5 km/s. Thisincreases to 6.0 km/sat a depthof 5 km and6.5km/sat 13km depth.Thestarting modelprovides a preliminaryfit to the travel timesof recordsectionsS4 and C1

nearthe edgesof the model(Figure5a). The velocityof the startingmodelis clearlyoverestimated nearthecentralportion We apply the AFT inversion,using travel time, envelope of themodel. The startingattenuation modelis specified as a amplitude, and instantaneous frequencyconvertedto t* to one-dimensional Q modelwhich approximately matchesthe obtain a velocity and attenuationmodel that best fits the relativet* valuesof the end gathers,S4 andC1 (Figure5c). Velocity and Attenuation Models

observed data. The starting velocity and Q" models are laterally homogenouswith five horizontal node positions spaceduniformlyin distance,andsevennodesin depthat 0 km, 1 km, 2.5 km, 5 km, 8.5 km, 13 km and 20 km. This gives a startingmodelwith 35 nodes. A variablenodespacingin depth is usedso that the large velocity gradientsnear the surfacecan be matched. Also, variablenodespacingallowsfor largernode spacingat depthwhere there are fewer rays. The numberof horizontalnodepositionsis increasedin subsequent inversions until the data are fit to within the observational uncertainty. The startingvelocity model is basedon estimatesof in situ seismic velocities of the rocks along the edge of the Lake Superiorrift structure[Trdhuet al., 1991; Shay and Trdhu, 1993; Allen, 1994]. The initial velocity at the surfaceof the

The selectedmodel has a Q of 150 near the surface,330 at a

depthof 5 km, and 1000at a depthof 13 km. Two iterations areperformed on the35 nodestartingmodel in anattemptto matchthelongwavelength featuresof thedata. The travel time RMS error is reduced from 0.775 s to 0.184 s after the two iterations, but without additional nodesthe travel

times cannotbe matchedwithin data uncertainties. Also, successive iterations on the 35 node model cause unrealistic

velocitiesbecauseof the sparselateralnodespacingin the model.Aftertwoiterations, amplitude andt* RMSmismatches are reducedfrom 2.334 to 0.672 and 0.006 s to 0.003 s, respectively.

To allowformorelateralheterogeneity, ninehorizontal node locations arelinearlyinterpolated in themodel.Thisresultsin

MATHENEY

ET AL.: SEISMIC ATTRIBUTE

REDUCEDTIMES (119 NODE MODEL)

9955

modelerrorsareestimated fromtheresulting covariance matrix

=Cxn Cx CXn , where Cxn istheprior model

5.00 •- + observed ' - calcul 4.00

INVERSION

weightingmatrix, andCx, =(.G,T n G• +I) -1 is obtained

.

from the final iteration[Tarantola,1987;Nowackand Lutter, 1988b]. Thismeasure of modelerroris dependent on boththe errorspropagated from thedata,as well as theprior error. The outputerrormapsarethencomputed by takingthesquareroot

A

3. O0 .

2.00

-

1 .oo .

ofthediagonal elements ofthecovariance matrixCx .

- a)

0.00

I

For thevariablegridsizeparameterization used,theoutput

100. Distance (km) 200'RMS erro:: 0.052 errorsare also scaledby the variableblock sizesto obtainthe

North

Data error= 0.060

NORMALIZED

•

AMPLITUDES

velocityand0.002for Q-i. Figures 8a and8b showthefinal modelerrorsweighted by blocksizefor velocityandQ-l. The

0.00

•.•-2.00

resulting model errors are small near the surfaceat distance

rangesfrom 50 km to 260 km andthroughmostof the central portionof the modeldownto 8.5 km in depth. The smallest

_• -4.00

• -6.O0 •

finalmodelerrorsperunitvolume.Thepriormodelparameter errorsusedin the prior weightingmatrix are 0.15 km/s for the

model errors are located near the receiver locations at the

surface. This is due to the denseray coveragenear the

-8. O0

receivers. The edgesof the modelare not as well constrained

200.RMS error= 0.347 andthisis shownby thelargererrorsin boththevelocityand

100.

North

Distance(kin)

Data error= 0.500

Q-l. Themodel errors closely correlate withtheraycoverage shownin Figure9a. The mostdenseray coverage is nearthe surfacein the centralpart of the model,and this is wherethe

.

model errors are the smallest.

0.04

-

0.03

-

resolution diagrams.Forthiscalculation, thefinalvelocityand

0.02

-

Q-imodels areslightly perturbed byalternately increasing and

Model resolutionis estimatedby using checkerboard A

decreasingthe values of each node by a small amount. The 0.01 perturbedmodelis thenusedto computea syntheticdata set. A one step inversionfrom the initial, unperturbedmodel is then 0.00 performed,and the amounteach node moved is plotted. If a m I I I • has perfect resolution,the perturbedmodel would be 100 'Distance (km) 200.Data RMS error= 0.002 model error= 0.004 North recoveredand when the amounteachnode movedis plotted, a Figure 6. (a) Observedand calculatedtravel times for the checkerboardappearancewould be viewed. However, due to laterallyvarying119 nodefinal model. (b) Observedand damping and variable ray coverage,some node points in the i

calculated amplitudes for the final model. (c) Observed and

model

are

not

as well

resolved.

For

the

checkerboard

calculated t* values for the final model.

resolutionplots, the perturbedvaluesare chosento be small so that nonlineareffects are minimized and a one iteration step would recoverthe perturbedmodel in the well resolvedareas. a 63 node model with nine horizontalnode locationsspaced Figures 9b and 9c show the amount each node moved after a equallyacrossthe model. One iterationis performedon the 63 one stepinversion. The centralportionand near the surfaceof themodelthevelocityandQ-inodesmovedby approximately node model which reducesthe travel time, amplitude,and t* mismatchesto 0.078 s, 0.427, and 0.002 s, respectively.Once the amountof the perturbation.Along the edgesof the model again, the horizontalnode locationswere linearly interpolated and at the deeper nodes, the ray coverageis too sparseto to give 17 horizontalnodesand 119 nodestotal. One iteration constrainthe modelparameters,and thesenodesdo not recover of the 119 node model reduces the travel time RMS error below the starting,unperturbedmodel. The well-resolvedareasin the checkerboardresolutionplots correspondwell with the lower the dataerrors(Figure6a). In general,the RMS mismatchwill not be less than the data uncertainties for all the attributes at the error regionsin the covariancecomputations. same iteration.

For this reason, the iterations have been

continued until the RMS

mismatches were less than the data

uncertainties for all the attributes.

The final RMS errors are

0.052 s for the travel times,0.347 for the naturallogarithmof the amplitudes,and 0.002 s for t* (Figure 6). The iterative

procedure is stopped at thispointandfinal velocityand Q-1

Finally,the crosscorrelation betweenthe velocityand parameterscanbe obtainedfrom the modelcovariancematrix

through therelation PvQ_ 1=CvQ_ 1/ •CvvCQ_iQ_i, where at each node PvQ_ 1isthecross correlation between velocity

modelsare obtained(Figure7).

and Q-l, CvQ_ 1is thecovariance between velocity and

Error Analysis

Cvv is the velocityvariance,and CQ-IQ-1 is the

To analyze the model uncertaintyand resolution,model covarianceestimatesand checkerboardresolutiondiagramsare computedfor the last iterationof the inversionsequence.The

variance. For this study, the crosscorrelationswere in the rangeof-0.12 to 0.08 with an averageof-0.020 indicatingthat velocity and inverse Q are uncorrelatedfrom one anotherat each node.

9956

MATHENEYET AL.: SEISMICATrRIBUTEINVERSION

VELOCITYMODEL (110 NODE) a)

4.4 •:•

4.8'• 5.6 ;> 6 6.4

lO

20

lOO

200

280

Distance (krn) INVERSE-Q MODEL (119 NODE) b)

$4 v

A2 v

CA v

C9 v

C3 v

Cl v

0.0165

0,015

•'"'"'•'"'•• 0.0135 i!•!'-.' :•I:?: 0-012

4

!'?• 0,0075 0,006 •-

0,0045 0,003

100

200

0.0015

Disanc e (km) Figure 7. (a) Velocitymodelfor the 119 nodefinal modelfor GLIMPCE line A. (b) Attenuationmodelfor the 119 node final model.

Discussion

The final velocity model (Figure 7a) obtainedthroughthe AFT inversion is comparable to previous velocity models obtainedby forward modelingand inversionof travel times alone [Lutteret al., 1993; Shayand Trdhu,1993;Hamiltonand Mereu, 1993]. A large central rift basin, a smaller northern basin,and an increasein velocity betweenreceiverlocationsA2 andC4 are the mostprominentfeaturesof thesemodelsand are also the most prominentfeaturesof the velocity model in this study(Figure 7a). Near the surface,the sedimentaryrocksof the Bayfield and OrontoGroupform a low-velocitycap across the seismicprofile. Thesesedimentaryrocksvary in thickness up to about 2 km [Cannon et al., 1989], with the thickest sectionof Oronto and Bayfield Group rocks located in the central basin. Previous seismicrefraction studies[Lutter et al.,

indicative of the volcanic and interflow sedimentsof the lower

Oronto and PortageLake Volcanics[Daniels, 1982]. The increase in velocityjust southof theIsle Royalefaultbetween shotpointsA2 andC4 at distancerangesof 100km to 120 km is evidenton previousrefractionstudies.This featurehasbeen

explainedby a thinning of the Bayfield and Oronto's sedimentaryrocksnear the Isle Royale fault [Lutter et al., 1993]andby highlyinduratedsedimentary rocksnearthe fault [ShayandTrdhu,1993]. To thenorthof theIsle Royalefault, at distances between20 km and 100 km on Figure7a, the sequence of middle Keweenawan volcanics and interflow

sedimentary rocksis absentand,instead, thesedimentary rocks of the Bayfield and Oronto Group overlie older, lower Keweenawan volcanicrocksof theOsierGroup[Cannonet al., 1989].

The invertedattenuationmodel acrossLake Superioris

1993; Shayand Trdhu,1993] haveestimatedthe sedimentary shownin Figure 7b. The overall structurepresentin the rock velocitiesat between2.0 km/s to 4.6 km/s. This is in good agreementwith the velocitiesobtainedin this paper which rangefrom 2.1 km/s and4.7 km/s in the centralbasin'supper2

attenuation model is a basin which extends across the central

portionof the model. The Q valuesrangefrom 60 (Q" = 0.0167)at thesurface to approximately 250 (Q-I-'0.004)at a km (Figure10a). Beneaththesesedimentary rocks,the velocity depthof 5 kmandupto Q near500(Q-I=0.002) at a depth of increasesto 5.0 km/s up to a maximum of 6.5 km/s at 9 km 10 km in thecentralpartof themodelat distance rangesfrom depth. These velocitiesare higher than the velocitiesfor the 80 km to 200 km. Figure10bis a plotof average Q-Iversus sedimentaryrocks of the Bayfield and Oronto Group but are depthacross themodel.NodeswithQ" errorslessthan0.0016

MATHENEY ET AL.: SEISMIC ATYRIBUTE INVERSION

9957

LINEARIZED VELOCITY ERROR (119 NODE) . ::.---;•.:._ •'•,-.:-•','•:f.•'."• .............. i:!•:i.i! -'. ........... • :.:.:.;.•.•:;•:•.?!.....::;:::?•. 0.149 0.136 0.123 0.11 0.097

ß..-• 0.084 :;:

•:• 0.071

'"•0.058 ':

0.045

0.032

0.019

8

0.006

20

100

200

280

Distance (km)

LINEARIZED INVERSE-Q ERROR (119 NODE) ; '. 0.00.185

2

o,oo,,

4

" o.oo,,

0.00155

•

0.00'11

(1.)6

O.(XX)95 0.00065

20

100

200

280

Distance (km) Figure8. (a) Scaled velocity errorsforthe119nodefinalmodel.(b) Scaled inverse-Q errorsforthe119node final model.

are shown.This plot showstheincreasein Q with depth,or a decreasein attenuationwith depth. Similar increasesin

attenuation decrease in theQ'•model.Thevelocityincrease has beenexplainedby induratedsedimentary rocks[Shayand

apparent Q withdepth,or confining pressure, havebeenshown Trdhu,1993] or thinnedor absentsedimentaryrocks[Lutteret the centralrift in numerouslaboratory studies [Winkler and Nur, 1979; al., 1993]nearthe Isle Royalefault separating Johnstonet al., 1979; Wepferand Christensen, 1991] where basin from the northern rift basin. Although there is an theattenuation decreases with increasing pressure andlevelsoff indicationof the two basinsin the attenuationmodel for depths at highpressures. For oceanic basalts[Wepfer,1989]andfor saturatedsandstonesamples[Johnstonet al., 1979], the

greater than4 km(Figure7b),theseparation is notapparent for

attenuation values level off between 150 and 200 MPa.

resolution due to the use of relative t* and amplitude

The

shallower depths.Thismayresultfroma lossof nearsurface

rateof changeof apparent Q with confiningpressure depends measurementswith a near-offset reference distance of 1.74 km. resolutionmap(Figure9c) suggests ontherocktype,amountof saturation, andcrackporosity.The However,thecheckerboard is adequate.A second alternative is that dominant mechanismcontrolling the increasein Q with theshallowresolution modelis apparentattenuation resultingfrom pressure is theassociated closing of microfractures [Peacock et the attenuation a!., 1994;Wepfer,1989]. For the caseof the Lake Superior both the effects of microfractures,as discussedabove, as well

ascompositional differences [Besteta!., 1994;Peacock et al., at greater porosity changes withpressure occurwithdepthresulting in the 1994;Wepfer,1989]. The closingof microfractures depthwouldcause thecompositional variations tobecome more observedvelocityandQ variations. in theattenuation model.As a result,thenorthand Whencomparing the velocityandattenuation models,the apparent modelfor velocityincrease neartheIsleRoyalefault(at a rangefrom100 centralrift basinsarebetterimagedin theattenuation attenuationstructure,both compositionalchangesand crack

km to 120 km in Figure7a) doesnot havea correspondingdepthsgreaterthan4 km (Figure7b).

9958

MATHENEY

ET AL.: SEISMIC ATFRIBUTE INVERSION

RAY-DIAGRAM(119 NODE MODEL) 20`O

70,0

120,0

170,0

220,0

.'"' '':.. ' o.o •!•i. •'.:i:': ..!• .... .'•-.-', .

'...!11 i'"•" .'.• ß

.•%.

1o•

VELOCITY CHECKERBOARD

RESOLUTION

AREA WEIGHTED DISTANCE IN KM

b)

270.0

....

................ •%•-•.•!•

'"' •'"'•?:*•"•'"•',/**•?•'•11 ............ • 'i::i• •:•'•:/ '•' •"

.

o, oooo

.0.0006

:

.

.

lO,O

Q 4CHECKERBOARD RESOLUTION AREA WEIGHTED

c)

DISTAHCE IN KM

20.0

70.0 :'.......

120•0

:::zzY.'". ........ I

170`O

..

220.0 . "'z":.;;• z

270.0 :.

ß .'":;•;•":;•i I :'::'" :.......' '. " ::. !::.i•?:': . . .. i;* *; .......................

... ..::.;--..*.'.'.....,...•;': ....... •,:;;.

..

:

.

..... .......

.

..

:,:.!;

:,.. .:

. .

... ,. :

;':':' i :'" ',;•. :.

; ..:

•

--.a;......,.%. ::.... .......... ......

;

:-..

.

:.

lo.o

Figure9. (a) Raydiagram forthefinaliteration of theinversion. (b) Checkerboard resolution forvelocity from the last iterationof the inversion. (c) Checkerboard resolutionfor inverse-Qfrom the last iterationof the inversion.

L.&KE SUPERIOR

VELOCITY

LAKE SUPERIORQ]-VS- DEPTH

-VS- DEPTH

0

0

2

2

8 []

223-271 km

10

0.020

Velocity (km/s)

Q•

Figure 10. (a) Velocityversus depthacross GLIMPCElineA. Nodeswithvelocityerrorslessthan0.10 km/s are shown.(b) Inverse-Qversusdepthfor thecentralbasin. Nodeswith inverse-Qerrorslessthan0.00!6 are shown.

MATHENEY

ET AL.' SEISMIC ATI'RIBUTE

INVERSION

9959

0.018 0.016

0.014 0.012

o.oo o.oo 0.004

0.002

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Velocity (km/s) Figure 11. Velocityversusinverse-Qfor the 119 nodefinal model.Nodeswith velocityerrorslessthan 0.10 km/s are shown.

A plotof velocity versus Q-• is shown in Figure11. Only

References

nodes with velocity errors of 0.11 km/s or less are shown

(Figure8a). Corresponding Q-•errorsfor thesenodes areless than 0.0016. A line fit throughthe data gives a relationship

Aki, K. andP. Richards,Quantitative$eismology, Theoryand Methods, W.H. Freeman, New York, 1980.

investigation of the Midcontinent between v and Q'• of Q'•=0.0210-0.0028*v witha correlation Allen, D.J., An integratedgeophysical

coefficient of-0.96.

Similar linear relationshipsbetween

rift system:WesternLake Superior,Minnesotaand Wisconsin,Ph.D. thesis,PurdueUniv., WestLafayette,Indiana,1994.

velocityandQ-'havebeenfoundin laboratory studies [Wepfer, Allen, D.J., W.J. Hinze,and W.F. Cannon,Drainage,topography, and 1989]. Sincethe relationshipbetweenvelocityand attenuation will dependon the compositionandcrackporositywith depth, in situ measurements of both velocity and attenuationprovide importantconstraints on thelithologyandphysicalpropertiesof the subsurface.The estimationof smoothlyvarying velocity and attenuationmodels has important applicationsto the processingand imaging of seismic reflection data. This includesinverse-Qfiltering [Hargreavesand Calvert, 1991], Qphase compensation [Bano, 1996], amplitude statics [Brzostowskiand McMechan, 1992] and the incorporationof attenuationin migrationalgorithms[Sollieand Mittet, 1994].

gravityanomaliesin the Lake Superiorregion:Evidencefor a 1100 Ma mantleplume,Geophys.Res.Lett., 19, 2119-2122, 1992.

Bano,M., Q-phasecompensation of seismicrecordsin the frequency domain, Bull. $eisrnol.$oc. Am., 86, 1179-1186, 1996. Behrendt,J.C., A.G. Green, W.F. Cannon,D.R. Hutchinson,M.W. Lee, B. Milkereit, W.F. Agena,and C. Spencer,Crustalstructureof the

Midcontinentrift system:Resultsfrom GLIMPCE deep seismic reflectionprofiles,Geology,16, 81-85, 1988. Ben-Menahem, A., andW.B. Beydoun,Rangeof validityof seismicray andbeammethodsin generalinhomogeneous media,Geophys.J. R. Astron. Soc., 82, 207-234, 1985.

Berry,M.J., and G.F. West,A time-terminterpolation of the first-arrival dataof the 1963 Lake Superiorexperiment,in The Earth Beneaththe Continents, editedby J.S.Steinhart andT.J. Smith, Geophys. Monogr. $er. ,vol. 10, pp. 166-180,AGU, Washington,D.C., 1966. Conclusions Best, A.I., C. McCann, J. Sothcott,The relationshipbetween the velocities,attenuationsand petrophysicalpropertiesof reservoir An AFT inversionof seismicattributeshas been applied to sedimentar-y rocks,Geophys.Prospect.,42, 151-178, 1994. refraction data from the 1986 GLIMPCE Lake Superior Bregman, N.D., C.H. Chapman,and R.C. Bailey, Travel time and amplitudeanalysisin seismictomography, J. Geophys. Res.,94, 7577Experimentto obtainvelocity and attenuationmodelsbeneath 7587, 1989. Lake Superior.The invertedvelocitymodelis similarto thatof Brzostowski,M.A., and G.A. McMechan,3-D tomographic imagingof previousstudies. Northernandcentralrift basinsare the most near-surface seismicvelocityand attenuation,Geophysics,57, 396403, 1992. prominentfeatureswith an increasein velocitynear the Isle Royale fault. The invertedattenuationmodel has attenuation Cannon, W.F., et al., The Noah American Midcontinent rift beneath Lake Superiorfrom GLIMPCE seismicreflectionprofiling,Tectonics,8, valueswhichrangefrom Q valuesof 60 at the surfaceto 250 at 305-332, 1989. 5 km andover 500 at a depthof 10 km. Althoughan indication Cannon,W.F., and W.J. Hinze, Speculations on the originof the Noah of the north and central basins is seen in the attenuation model

AmericanMidcontinentrift, Tectonophysics, 213, 49-55, 1992.

for depthsgreater than about 4 km, this separationis not Carpenter, P.J.,andA.R. Sanford,ApparentQ for uppercrustalrocksof the centralRio Granderift, J. Geophys.Res.,90, 8661-8674, 1985. evidentfor shallowdepths. This couldresultfrom the effects of microfracturesmasking the effects of compositional Cerven9,V., and F. Hron, The ray seriesmethodand the dynamicray tracing systemfor three dimensionalinhomogeneousmedia, Bull. differencesfor shallowerdepths. A linear, inverserelationship $eismol. $oc. Am., 70, 47-77, 1980.

hasbeenfoundbetween velocityandQ-ibeneath LakeSuperior supportingpreviouslaboratoryresults.

Chandler,V.W., P.L. McSwiggen,G.B. Morey, W.J. Hinze, and R.R. Anderson,Interpretation of seismicreflection,gravityand magnetic

9960

MATHENEY

ET AL.: SEISMIC ATYRIBU'I•

INVERSION

data acrossmiddle ProterozoicMid-continentsystem,northwestern Matheney, M.P.,andR.L. Nowack,Seismicattenuation valuesobtained Wisconsin, easternMinnesota and central Iowa, AAPG Bull., 73, 261275, 1989.

frominstantaneous frequencymatchingandspectralratios,Geophys. J. Int., 123, 1-15, 1995.

andseismictomography, in Seismic Cline, A., Scalar-and planar-valued curve fitting usingsplinesunder Nolet,G., Seismicwavepropagation tension, Camm. ACM, 17, 218-223, 1974. Tomography, editedby G. Nolet,pp.1-23, Norwell,Mass.,1987. Daniels, P.A., Upper Precambriansedimentaryrocks: Oronto Group, Nowack,R.L., andW.J. Lutter,Linearizedrays,amplitudeandinversion, Michigan-Wisconsin, in Geologyand Tectonicsof the Lake Superior PureAppl. Geophys., 128, 401-421, 1988a. basin,editedby R.J. Wold and W.J. Hinze, Mere. Geol. Soc.of Am., Nowack, R.L., and W.J. Lutter,A note on the calculationof covariance 156, 107-134• 1982. andresolution,Geophys. J. Int., 95, 205-207, 1988b. Dickas, A.B., Midcontinentrift system:Precambrianhydrocarbontarget, Nowack,R.L., andJ.A.Lyslo,Frechet derivatives for curvedinterfaces in Oil Gas J., 82, 151-159, 1984. therayapproximation, Geophys. J., 97, 497-509,1989. Nowack,R.L., and M.P. Matheney,Inversionof seismicattributesfor Futterman,W.I., Dispersivebody waves, J. Geophys.Res., 67, 52575291, 1962. velocityandattenuation structure, Geophys. J. Int.,in press,1997. Goldberg,D., and C.S. Yin, Attenuationof P-waves in oceaniccrust: Peacock,S., C. McCann, J. Sothcott,and T. Astin, Experimental Multiple scatteringfrom observedheterogeneities,Geophys.Res. Lett.,21, 2311-2314, 1994.

Green, J.C., Physicalvolcanologyof mid-Proterozoicplateaulavas;the KeweenawanNorth Shorevolcanicgroup,Minnesota,Geol. Soc.Am. Bull., 101,486-500,

1989.

Halls, H.C., Crustal thicknessin the Lake Superiorregion, in Geology and Tectonicsof the Lake SuperiorBasin, editedby R.J. Wold and W.J. Hinze, Mere. Geol. Soc. Am., 156, 239-243, 1982.

Halls, H.C., and G.F. West,A seismicrefractionsurveyin Lake Superior, Can. J. Earth Sci., 8, 610-630, 1971.

Hamilton, D.A., and R.F. Mereu, 2-D tomographicimagingacrossthe North AmericanMidcontinentrift system,Geophys.J. Int., 112, 344-

measurements of seismicattenuationin microfracturedsedimentary rocks,Geophysics, 59, 1342-1351,1994.

Shay,J., andA.M. Tr6hu,Crustalstructure of thecentralgrabenof the Midcontinentrift beneathLake Superior,Tectonophysics, 225, 301335, 1993.

Sollie, R., and R. Mittet, Prestackdepthmigration;sensitivityto macro

absorption model,$EGAnnu.Meet.Expanded Tech.ProgramAbstr., 64, 1422-1425, 1994.

Steinhart,J.S., and T.J. Smith (eds.), The Earth Beneaththe Continents,

Geophys. Monogr.$er.,vol.10,Washington, D.C., 1966. Tarantola,A., InverseProblemTheory,Elsevier,New York, 1987. ToksOz,M.N., D.H. Johnston,and A. Timur, Attenuationof seismic

358, 1993. wavesin dry and saturatedrocks:I. Laboratorymeasurements, Geophysics, 44, 681-690, 1979. Hargreaves,N.D., and A.J. Calvert, InverseQ filteringby Fourier Tr6hu,A., et al., Imagingthe Midcontinent rift beneathLake Superior transform,Geophysics, 56, 519-527, 1991. Hinze, W.J., N.W. O'Hara, J.W. Trow, and G.B. Secor, Aeromagnetic usinglargeaperture seismic data,Geophys. Res.Lett.,18, 625-628, 1991. studiesof easternLake Superior,in the Earth beneaththe continents, of laboratory velocityandattenuation data editedby J.S.SteinhartandT.J. Smith,Geophys. Monogr.Ser.,vol. Wepfer,W.W., Applications to studies of the Earth's crust, Ph.D. thesis, Purdue Univ., West I0, pp. 95-110, AGU, Washington,1966. Lafayette,Indiana, 1989. Hinze, W.J., R.L. Kellogg,and N.W. O'Hara, Geophysicalstudiesof Q structure of the oceaniccrust, basement geologyof thesouthern peninsula of Michigan,AAPGBull., Wepfer,W.W., andN.I. Christensen, Mar. Geophys.Res.,13, 227-237,1991. 59, 1562-1584, 1975. Johnston,D.H., Attenuation:A stateof the art summary,in SeismicWave Winkler, K., and A. Nur, Pore fluids and seismicattenuationin rocks, Attenuation,D.H. Johnston,and M.N. Toks/3z,SEG Geophys.Reprint Geophys. Res.Lett.,6, 1-4, 1979. ser.no. 2, pp. 123-135,Tulsa,Oklahoma,Soc.of Explor.Geophys., Woolard,G.P., Transcontinental gravitational and magneticprofileof 1981. NorthAmericaandits relationto geologicstructure, Geol.Soc.Am. Johnston,D.H., M.N. Toks/3z, and A. Timur, Attenuation of seismic Bull., 54, 747-790, 1943. wavesin dry and saturatedrocks:II. Mechanisms, Geophysics, 44, Zelt, C.A., andR.B. Smith,Traveltime inversionfor 2-D crustalvelocity

691-711, 1979.

structure, Geophys. J. Int., 108, 16-34,1992.

Kjartansson, E., ConstantQ-wave propagationand attenuation, J. Geophys.Res.,84, 4737-4748, 1979.

LaBerge,G.L., Geologyof theLakeSuperiorRegion,Geoscience Press, Phoenix, Arizona, 1994.

Lutter, W.J., A.M. Tr6hu, and R.L. Nowack, Applicationof 2-D traveltime inversion of seismic refraction data to the mid-continent rift

beneath LakeSuperior, Geophys. Res.Lett.,20, 615-618,1993. Mariano, J., and W.J. Hinze, Gravity and magneticmodelsof the Midcontinent rift in easternLakeSuperior,Can.J. EarthSci.,31, 661-

M. Matheney andR. Nowack, Purdue University Department ofEarth & Atmospheric Sciences, 1397CivilBuilding, WestLafayette, IN 49707. (e-mail:[email protected]; [email protected]) A. Tr6hu, OregonState UniversityCollegeof Oceanicand AtmosphericSciences, Corvallis, OR 97331. (e-mail: [email protected])

674, 1994.

Mason,W.P., K.J. Marfurt, D.N. Beshers,and J.T. Kuo, Internalfriction in rocks,J. Acoust.Soc.Am., 63, 1596-1603, 1978.

(Received May 17,1996;revised November 5, 1996; accepted January31, 1997.)