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
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INVERSION
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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:matheney@geo.purdue.edu;
nowack@geo.purdue.edu)
A. Tr6hu, OregonState UniversityCollegeof Oceanicand
AtmosphericSciences, Corvallis, OR 97331.
(e-mail:
Tr6hu@oce.orst.edu)
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.)