JOURNAL OF GEOPHYSICALRESEARCH,VOL. 99, NO. B10, PAGES20,123-20,140,
OCTOBER10, 1994
Flexural rigidity of the Basin and Range-Colorado
Plateau-Rocky Mountain transition from coherence
analysis of gravity and topography
Anthony R. Lowry and Robert B. Smith
Departmentof GeologyandGeophysics,Universityof Utah, Salt Lake City
Abstract. Stochasticinversionfor flexural loadsand flexural rigidity of the continentalelastic
layercanbe accomplished
mosteffectivelyby usingthe coherence
of gravityandtopography.
However,the spatialresolutionof coherenceanalysishasbeenlimitedby useof two-dimensional
periodogramspectrafrom very large(> 105km2)windowsthatgenerallyincludemultipletectonic
features.Using a two-dimensionalspectralestimatorbasedon the maximumentropymethod,the
spatialresolutionof flexuralpropertiescanbe enhancedby a factorof 4 or more,enablingmore
detailedanalysisat the scaleof individualtectonicfeatures.This new approachis usedto map the
spatialvariationof flexuralrigidity alongthe BasinandRangetransitionto the ColoradoPlateau
andMiddle Rocky Mountainsphysiographic
provinces.Largevariationsin flexuralisostaticresponseare found,with rigiditiesrangingfrom aslow as 8.7x102øN m (elasticthicknessT, = 4.6
km) in the Basin andRangeto ashigh as4.1x1024N m (T, = 77 km) in the Middle Rocky
Mountains.Theseresultscomparefavorablywith independent
determinations
of flexuralrigidity
in the region. Areasof low flexuralrigiditycorrelatestronglywith areasof high surfaceheat
flow, asis expectedfrom thecontingence
of flexuralrigidityon a temperature-dependent
flow law.
Also, late Cenozoicnormalfaultswith largedisplacements
are foundprimarilyin areasof low
flexuralrigidity, while deformationfrontsof Mesozoic/Tertiaryoverthrustsoccur0 to 100 km east
of the low-rigidityregion. The highestflexuralrigidity is foundwithin the ArcheanWyoming
craton,whereevidencesuggests
thatdeeplyrootedcratoniclithospheremay play a role in determining the distributionof tectonismat the surface.
Introduction
rains [Buck, 1988; Wernicke and Axen, 1988], as well as the
plateau-like expressionof orogens[Bird, 1991]. IsostaticreThe conceptof isostasyproposesthat the Earth is in hydro- sponseis essentialto our understandingof basin formation in
static equilibrium at depth, requiring topographyto be com- a variety of tectonic settings [e.g., McKenzie, 1978; Watts
pensated by lateral variations in crustal thickness [Airy,
and Steckler, 1981; Jordan, 1981]. Isostatic corrections to
1855] or crustal density [Pratt, 1855]. Flexural isostasyis an Bouguer gravity anomaly maps benefit from use of the true reextension of the concept in which topographyis supported lationship between gravity and topography [e.g., McNutt,
partly by deflection of density discontinuitiesin Airy iso1983] rather than the simplistic Airy approximationto isostatic fashion and partly by stressesin a strong elastic layer static behavior. Watts and Ribe [1984] suggestthat if the
overlying a weak ductile zone [Barrell, 1914; Gunn, 1943].
Earth's isostatic behavior is constrained, its effects can be reFlexural isostasyis modeledfrom theory of elastic plate flexmoved from geoid data to achieve a clearer picture of mantle
ure, governed by the fourth-order partial differential equation
convection processes.
[e.g., Turcotte and Schubert, 1982]:
Moreover, estimates of flexural rigidity contribute directly
to our understandingof the evolution and mechanicalbehavior
(1)
of the lithosphere.
Observations of flexural rigidity
(expressedin terms of an equivalent effective elastic thickwhere w is vertical deflection of the plate, AO is the density ness,Te) were pointed to by Watts [1978] as an affirmation of
differencefor material aboveand below the plate, g is gravita- the thermomechanicalplate-cooling model for oceanic lithotional acceleration,P is horizontalforce (per unit length), and sphere. Subsequentevaluation of perturbationalelastic thickq is vertical stressapplied to the plate. Thus flexural isostasy ness estimates have helped to illuminate anelastic behavior
[McNutt and Menard, 1982] and stress accumulation [Wessel,
is parameterizedby the resistanceto bendingor flexural rigid1992] in the oceanic elastic layer. Determinationsof flexural
ity D of the elastic layer.
rigidity
should provide similar insights into the formation,
An improved understandingof the Earth's isostaticbehavior
evolution, and seismic properties of the continental lithois desirable for a variety of applications. Investigators call
sphere.
upon isostasy to explain fault geometry in extensional ter-
V2*(D
V2w)
+PV2w+
Apgw=q
This
Copyright1994 by the AmericanGeophysical
Union.
Papernumber94JB00960.
0148-0227/94/94JB-00960505.00
discussion
will
first
describe
a new stochastic
inver-
sion for flexural rigidity of the elastic layer using a maximum
entropy-basedapproachto coherenceestimation. The method
is then demonstratedby application to the tectonically important transitionfrom the actively extendingBasin and Range to
20,123
20,124
LOWRY AND SMITH: RIGIDITY
OF THE BASIN-RANGE
the relatively stable Colorado Plateau and Rocky Mountains
physiographic provinces in the western United States.
Estimates of flexural rigidity are compared to relevant geologic and geophysicalinformation from the area and are found
to be strongly correlated with patternsof heat flow and tectonism.
The coherencefunction is the squareof the correlationcoefficient between two signals, a positive number ranging between zero and one. Coherencecan be thoughtof as a measure
of the fraction of the gravity field that can be predicted from
the topographyusing (2). Coherencehas a strongdependence
on flexural rigidity (Figure 1) and so is appropriatefor use in
its estimation.
Coherence
Method
The coherence
function
is estimated
from
the
autopowerspectraof topographyPhh(k)and gravity Pbb(k)and
the cross-powerspectrumof the two signals,Phl•(k),by
for Estimation
of Flexural Rigidity
Flexural rigidity is estimatedby assuminga model that describesloading of the elastic layer and then comparingtopography (or bathymetry) to gravitational potential (in the form
of a free air, Bouguer, or geoid anomaly) to solve for the loads
and the load response. One common methodologycompares
the observed flexural responseto an "obvious" loading processwith a modeledresponsefor variousassumptions
of flexural rigidity. Examples of obvious loading processesinclude
seamounts[e.g., Walcott, 1970a] and plate bending at subduction zones [e.g., Caldwell et al., 1976] in the case of the
oceanic elastic layer. Flexural rigidity of the continental elastic layer is determined from loading by sedimentarybasins
[e.g., Haxby et al., 1976], Pleistocene lakes [e.g., Walcott,
1970b], and mountain ranges formed by continental collisions [e.g., Karner and Watts, 1983; Lyon-Caen and Molnar,
1983]. Unfortunately, the paucity of sites in which suchspecific loading processesare found precludes the use of these
techniquesfor systematicmapping acrosslarge areas.
The admittance and coherence techniques [Lewis and
Dorman, 1970; Forsyth, 1985] compose another class of
methods that exploit the stochastic relationship between
gravity and topography signals to determine flexural response. Because sinusoidsare eigenfunctionsof the differential operator(1) that governsplate flexure, the Fourier amplitudes of topographyH(k) and of the completeBouguergravity
anomaly B(k) can be linearly related by
B(k) = Q(k)H(k)
TRANSITION
(2)
[Dorman and Lewis, 1970], where Q is the linear isostatic responsefunction (or admittance) and k is the two-dimensional
=I
'
(4)
In practice, power spectra are averagedto reduce bias introducedby uncorrelatednoise processes.Averaging of the twodimensionalspectra,denotedby angle brackets,is performed
within annular wavenumberbins basedon the assumptionthat
flexural responseis isotropic. In that case coherenceis onedimensionaland expressedas
where k is the modulus of the two-dimensional wavenumber,
a=
=
Coherence analysis of isostatic response [Forsyth, 1985]
first
uses the observed
admittance
and an assumed
flexural
rigidity to solve for load structurein the Earth. Fourier amplitudesof the topographyH and the Bouguergravity B are used
to solve algebraically for the amplitudesof topographyH• and
a subsurfaceload horizon W• prior to flexural compensation
(Figure 2b). Once the initial loads have been determined,the
topographyand gravity amplitudescan be deconvolvedinto
their respective components,Hrand Br due to surfaceloading
and H s and Bs due to interior loading of the elastic plate
(Figure 2c). The coherenceof the deconvolvedsignalsis then
estimatedby assumingthe surfaceand subsurfaceloading processesare statistically uncorrelated:i.e., they have some ranWavelength(km)
wavenumber:
100
I I I
k=
,
ß
I
I..• •
Ill,
10
, , ,
I
[ I i
(3)
When flexural rigidity is estimated from the admittance
function,the model chosento parameterizeQ(k) is critically
important. Early investigationsassumedlocal or Airy compensationof surfacetopography
andinvertedfor the compensating density anomaliesat depth [Lewis and Dorman, 1970;
Dorman and Lewis, 1972]. The Airy isostaticparameteriza-
tion was superseded
by thatof an isostaticresponse
to topographicloadingat the top of an elasticplate [e.g., McKenzie
and Bowin, 1976; Banks et al., 1977]. Lewis and Dorman
0.2 i
[1970] recognized,
however,that muchof the surfaceexpreso
sion of topographyrepresents
the Earth'sresponse
to density
0.1
1
O.Ol
variationsat depthratherthan loadsemplaceduponthe surface. The further recognitionthat flexural rigidity would be
Wavenumber
(km-1)
systematicallyunderestimatedif interior or bottom loading
contributedto a flexuralresponse
modeledas a plateloadedat Figure 1. Coherenceof gravity and topographyat specific
the top providedthe impetusfor developmentof the coherence wavelengthsis strongly dependenton flexural rigidity and so
method by Forsyth [1985].
is well suited for use in its estimation. After Forsyth [1985].
LOWRY AND SMITH: RIGIDITY
OF THE BASIN-RANGE
2.4
2.8
3.2
Surface
0
2o,125
[e.g., Kay, 1988]. Windowing effects can be reducedby using
a data sequencethat is very large relative to the wavelengthsof
interest, and so previous researchers[e.g., Bechtel, 1989]
have chosento mirror the data (i.e., repeat the data symmetrically across a window edge). However, the wavelengthsof
transition from high to low coherenceare very long, -100 to
Densityp
(Mg m-3)
a. Prior to
TRANSITION
loading
Subsurface
800 km in the North American continent [Bechtel
load interface
1990], and even when data are mirrored, the spectralproperties
et al.,
mustbe computed
for verylargegeographic
areas,of order105
Loadingat the surface
Subsurfaceloading
HI
loading
to 106 km2, to be accuratelyresolved. Coherenceanalysisassumesuniform flexural rigidity, so the flexural rigidity which
gives the best fit to the coherencefunction will be a weighted
average of the rigidity throughoutthe sampled region. As a
result,periodogramcoherenceanalysishas not been useful for
smaller-scale
WI
over
c. After
flexure
tectonic
features
or at boundaries
between
tec-
tonic features where elastic strength may vary significantly
HB
HT
WB
WT
short
distances.
To test the periodogramcoherenceestimator,we generateda
pair of data setswith known coherence.We first Fourier transformed a large area (1600 by 1200 km) of topographicdata
from the western United States to the frequency domain. We
then generatedsynthetic "gravity" amplitudesB• using an as-
sumedcoherence
functionq'aby addingrandom-phase
noise
with amplitudeN to the topographicamplitudes,where
H
d. Final load
distribution
IN(k)
I=1H(k)I
1-T•(k)
W
Figure 2. Fourier isostaticmodel [after Forsyth, 1985]. (a)
An elastic layer of thicknessh, having arbitrary density variation with depth. (b) Sinusoidalloading with Fourier amplitude H I is appliedat the top; loadingwith amplitudeWi occurs
at depthzL. (c) Flexure resultsin topographicamplitudesHr
due to surface loading and H a due to subsurface loading.
Deflections of the subsurfaceload interface have amplitude Wr
and WB,respectively. (d) ObservedtopographyH and internal
deflection W sum the surfaceand subsurfaceresponses.
dom difference in phase. In that case,the predictive coherence
•h•,(k)iscalculated
from
(6)
h
+
k
•,
+ Pt;t,
where superscriptst and b denote top and interior (or "basal")
loading, respectively. The predictive coherencefunction (6)
is estimatedfor severaldifferent assumedflexural rigidities and
then comparedto the observedcoherence(4) to determine the
flexural rigidity that best modelsthe load responseof the elastic layer.
Periodogram Spectral Estimation
Previous investigationsof flexural rigidity using the coherence method have employed a classical Fourier transform
technique,the periodogrammethod,to estimatepower spectra.
Unfortunately, the periodogramis plaguedby spectralbias and
leakage resulting from the implicit windowing of finite data
The syntheticgravity amplitudeswere then inverse Fourier
transformedto the spatial domain, and a smaller data window
was extracted from the center of each data set for estimation of
the coherenceusing (5). In spite of the use of mirrored data,
the periodogrammethodusedby previousresearchers
provides
a poor estimate of coherencewhen the data window is small
(Figure 3a). Figure 3a, like all subsequentfigures displaying
coherencefunctions, superimposescoherenceestimatescalculated for several realizations of the spectral averaging operation, corresponding to different numbers n of annular
wavenumber bins, n = 20 to 30, to indicate variance. The periodogram coherenceestimate exhibits a great deal of scatter.
Also, the true coherencefunction changesfrom zero to one at
wavelengthsclose to the dimensionof the data window, and
that transition is virtually unresolved. Increasingthe size of
the data set improves the periodogramestimate of coherence
(Figure 3b), but the method still yields a poor approximation
to the true coherencefunction at transitional wavelengths. A
still larger data set would further improve the coherencefunction, but given the nonstationarityof flexural properties,it is
preferableto minimize the size of the data sequence.
Maximum Entropy Spectral Estimation
Several alternative spectralestimatorshave been developed
that attemptto reduceor remove the effects of windowing of
data [e.g., Kay and Marple, 1981]. Maximum entropyspectral
estimation (MESE) is one such method that has enjoyed a measure of successfor a wide range of applicationsin scienceand
engineering[e.g., Smith and Grandy, 1985]. The motivation
for MESE is similar to that for mirroring: namely, to reduce
the effectsof data windowingvia a reasonableextrapolationof
the information. Specifically, MESE seeks to describe a
20,126
LOWRY AND SMITH: RIGIDITY OF THE BASIN-RANGE TRANSITION
Wavelength(km)
100
I
10
,
,
,
,
,
,
,
ß
I
,
,
,
1i,
a. 100 by 100 km window
0.8
1
0.7
0.6
c
0.•
0.4
o
0.3
0.2
0.1
x
x
0
x
x
0.01
Wavenumber
(km-1)
Wavelength(km)
100
10
1
•m• 12)ø
20012)-y200knlwi
0.9
0.8
0.7
0.6
0.$
0.4
0.3
8
0.2
Xx
0.1
x
0
0.1
0.01
1
Wavenumber
(km-1)
Figure 3. Coherence of synthetic data. Solid line is the coherencefunction used to generate the data.
Periodogramcoherenceis indicatedby crosses;maximum entropy-basedcoherenceis indicatedby open circles. (a) Estimatesfor a 100x100 km window. (b) Estimatesfor a 200x200 km window.
power spectrum exhibiting minimum bias, correspondingto
the Fourier transform of the extrapolatedcorrelationfunction
having maximum entropy [Burg, 1975].
The Burg [1975] formulation of MESE, commonly used for
one-dimensional signals, is equivalent to autoregressivespectral estimation [e.g., Kay and Marple, 1981], and the two follow a similar procedure. In each case, "extrapolation"of the
correlation function to larger lags is implicit rather than explicit. However, the two-dimensional extension of Burg's
one-dimensionalmaximum entropy formulation requiressolution of a large systemof nonlinearequations. Fortunately,the
implied computationalnightmarecan be avoidedby reformu-
lating the problem in terms of an iterative algorithm constrained such that the power spectrumwill converge to the
maximumentropysolution[Lim and Malik, 1981].
The problemis posedas follows: given a processh[m,n]
with correlationfunction rhh[m,n]that is known or estimated
over some windowed areaA, determinethe extrapolationof the
known correlation function having maximum entropy. The
extrapolatedcorrelationshouldof courseequalthe knowncorrelation
within
A:
•hh[m,rt]
=F-l{I3hh(kl,k2)
}= rhh[m,t•] for [m,n]• A
(8)
LOWRY AND SMITH: RIGIDITY
OF THE BASIN-RANGE
TRANSITION
20,127
whereF•[ o} represents
the inverseFouriertransformoperator
well as deconvolution
andrhh[m,
rt] is the extrapolatedcorrelationfunction. It can be
shown [e.g., Staylie et al., 1973] that if the processis
Gaussian,the ShannonentropyE is given by
tions) deservessome discussion. This practice was first used
by Lewis and Dorman [1970] and is still commonlyemployed
by researchersusing periodogramspectra[e.g., Blackman and
Forsyth, 1991] to reduce the spectralbias and leakagethat result from finite data. Figure 4a depictsthe periodogramcoherence estimated from mirrored and unmirrored synthetic data,
indicating that while the coherencefunctions are differently
biased at long wavelengths,the estimatesare comparablyac-
E=
log
•h
h(k
l,k2)
]dk
ldk2.
of surface and subsurface load distribu-
(9)
curate.
Maximization of (9) has the trivial solution
A more intriguing dissimilarityis apparentwhen mirroring
is applied to real gravity and topography data (Figure 4b).
Here the greatestdifference between the mirrored and unmir= 0for[m,n]
• A.
(1O) rored coherence functions is at short wavelengths, where the
unmirroredcoherenceactually increaseswith decreasingwavelength insteadof leveling off near zero. This feature is an artefact
of the homogeneous
reductiondensityusedto calculatethe
Lira and Malik [1981] developan iterative MESE algorithm
Bouguer
anomaly.
While
an averagedensity is adequatefor
basedon the constraints(8) and (10), which we have adapted
Bouguer
reduction
at
long
wavelengths, locally contiguous
to the determinationof autopowerand crosspowerspectraof
gravityandtopographyfor coherence
analysis.However,it is bodies which deviate significantlyfrom the reductiondensity
importantto note that whereasthe autospectra
are maximum and are exposedin surface topographywill generateBouguer
entropyspectra,in the caseof a crosspowerspectrumthe ap- "anomalies"which will correlate with topographybecausethe
plicationof the constraint(10) doesnot correspond
to maxi- reduction density was unrepresentativeat that location [e.g.,
mizationof entropy(a proof may be foundin AppendixA of Black, 1992]. The coherenceincreaseswith decreasingwaveLowry [1994]), and the information quantity that is being length becausethe likelihood that such bodies will be continminimaxedhas not yet been identified. Hence this studydoes uously represented at the surface decreaseswith increasing
••hh(kl'k2))
scale. When the data are mirrored, however, a skein of inco-
not actuallyemploya maximumentropycoherenceestimator;
althoughbasedon the maximumentropymethod,this is an ad herent noise is introducedinto the spectrawhich is apparently
hoc coherenceestimation technique.
The coherence function was reduced to one dimension via
averagingwithin annularwavenumberbins using
-'2
PhhPbbl
(11)
in placeof the averagingschemegiven by (6), simplybecause
(11) was found to perform better in tests of the method. The
resultingcoherencefunctionsexhibit a positivebias at wavelengthsfor which the true coherenceapproacheszero (Figure
3), perhapsbecausemaximumentropyspectratend to represent noise processespoorly. Nevertheless,the maximum entropy-basedestimatesof coherenceusedin this studyimprove
greatly upon those determinedusing the periodogrammethod
for the samesyntheticdata, particularlyat the importantwavelengths of transition.
of large enough amplitude to dominate the coherencyof signals at short wavelengths.
Despite the difference in coherencefunctions, mirrored data
yield a virtually identical value for the best fit flexural rigidity
as unmirrored data. For example, the coherencefunctions in
Figure 4b correspondto rigidities of 2.9x102• N m and
2.6x102•N m for the mirroredandunmirroredcases,respectively. To arrive at these estimates,the same type of Fourier
amplitudes (i.e., mirrored or unmirrored) were used to determine the predictive coherencefunctions as were used to estimate the observed
coherence.
Hence
the estimates
of flexural
rigidity are virtually identical becausethe bias and noise processesintroduced in the predictive coherencefunctions were
similar
to those in the observed
coherence
in each case.
The
artificial noise processesintroducedby mirroring causesevere
degradation of maximum entropy-basedcoherence functions,
however(Figure 5), so data were not mirroredfor this study.
Windowing
for
Deconvolution
Other Adjustments
Deconvolution of surface and subsurfaceload responsesis
performedin a relatively large window for this study. Spectral
In addition to the alternative choice of spectral estimator,
bias and leakagehave been greatly reducedin the observedcothe following minor modificationsof previous practice are
herencefunction by using a maximum entropy-basedmethod,
made for this study: (1) the data are not mirrored prior to
so it would be counterproductiveto introducethese effects into
stochasticanalysis; (2) surface and subsurfaceload responses
the predictive coherencefunction by performing the deconvoare deconvolvedwithin a much larger data segmentthan that lution within a small data window. Instead the deconvolution
for which power spectralestimationis performed;and (3) relauses the amplitudesfrom a much larger data segment(at least
tively complex one-dimensionaldensityprofiles of the elastic
12 times the area for which power spectraare estimated). The
layer, as inferred from Nafe-Drake regressionof P wave velociamplitudes are then inverse transformedto the spatial domain,
ties from seismic refraction profiles, are included in the isoand the correspondingsubset of the deconvolved data is sestatic model.
lected for use in estimatingthe predictive coherencefunction.
Mirroring
of Data
Density Model From P Wave Velocities
Mirroring of data prior to calculationof Fourier amplitudes
The assumed density variation with depth in the elastic
(which are utilized in calculation of periodogramspectra as layer was also modified. Previous investigationshave incor-
20,128
LOWRY AND SMITH: RIGIDITY OF THE BASIN-RANGE TRANSITION
Wavelength(km)
100
1
I
I
I_•
•M- 'J-'
ß
10
I•• • • I
•x-••.
0.9
x
0.8
I
I
I
III I
a.$•ynthetie
data
• x•x"
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.01
1
Wavenumber
(km-•)
Wavelength(km)
'
x
3,•4x
100
Im
I
•
m I
0.9
I
I
I
10
I I I
I
I
b. Real data
0.8
0.7
Xo
0.6
ß
0.5
ß ß
0.4
X Xx•X
I øø
ø
0.3
0.2
I.
el
ß
oo
•.
,•.
0.1
0
i
i
i
i
i
I ............
0.1
0.01
......
i
i
i
--.--
________
1 .... r•-
•
1
Wavenumber(km-1)
Figure 4. Mirrored and unmirroredperiodogramcoherencefor a 600x600 km window. (a) For synthetic
data, mirroring (crosses)producesslight negativebias; unmirroreddata (solid circles) are positivelybiasedat
transitionalwavelengths. (b) Unmirroredcoherenceis high at shortwavelengthsfor real data.
porated geologically reasonabledensity values for one or at
most two layers over a half-space,with depthsinferred either
from the gravity power spectrum or from seismic data.
However, the deconvolutionof gravity and topographycan be
accomplishedusing any arbitrary one-dimensionaldensity
model via the relations describedin the appendix. Density
distributionsfor this study are approximatedusing a NafeDrake regression of P wave velocities [Ludwig et al., 1970]
from crustal seismicrefraction surveysin the area of interest.
The Nafe-Drake relationshipmay be poorly suitedto the pre-
undertaken here the functional dependence is on vertical
changesin densityAp and the depthz at whichthey occur:
exp(-kz)dz.
Jones et al. [1992] suggest that variations in density for a
given P wave velocity are primarily the result of variationsin
dictionof densitystructures(i.e., lateral variationsin density, the mean atomic number of crustal rocks; in which case a Nafesee, for example,Barton [1986]); however,in the modeling Drake density profile should be representative(and probably
LOWRY AND SMITH: RIGIDITY
OF THE BASIN-RANGE TRANSITION
20,129
Wavelength(km)
100
10
0.9
0.8
0.7
0.6
0.5
0.4
0.3
o
o
0.2
o
0.1
o
O.Ol
o.1
1
Wavenumber
(km-1)
Figure 5. Maximum entropycoherencefunctionsfrom unmirroreddata (open circles) and mirrored data
(crosses)for a 200x200 km window. Solid line is true coherence.
will give the mostaccurateprofile from amongthe geophysiThe coherenceestimatoremployedhere is basedon a maxical optionsavailable). Velocitiesfrom seismicrefractionpro- mum entropy spectral estimator, which also has some minor
files are preferred over thosefrom reflection data becauserefraction averagesout lateral variationsin velocity structure,
yielding a more representativeone-dimensionaldistribution.
The densitymodificationis ultimatelya minor improvement,
however: testsindicatethat complicatingthe densitystructure
as well as changingthe subsurfaceload depth changethe resulting estimate of flexural rigidity by no more than about a
factor of 2 (correspondingto a factor of 1.3 changein elastic
thickness).
Limitations
of the Method
As with all Earth models,the coherencemethodis subjectto
the validity of its implicit assumptions.Includedamongthese
are the fundamentalapproximationof the behaviorof the elas-
tic layer as that of a perfectlyelasticplate,incorporation
of a
thin plate approximationto flexural behavior,assumption
of
a crustaldensityvariationwith depth,and restrictionof loading processes
to either the surfaceor a singlesubsurface
density interface. However,it is expected(basedon empirical
limitations. The appropriateprobability density distributions
for gravity and topographyprocessesare not explicitly solved
for; rather the entropy description (9) implicitly assumesa
Gaussian (correspondingto the most entropic [e.g., Jaynes,
1985]) probability density distribution. The latter methodology is commonly referred to as a Gibbsian approach. Some
minor problemshave been recognizedin other Gibbsian maximum entropy methods such as the Burg algorithm, including
the exhibition of spurious peaks, spectral line splitting, and
phase dependenceof sinusoidalpeak location [e.g., Kay and
Marple, 1981]. Malik and Lim [1982] observe similar difficulties with the two-dimensionalalgorithm used in this study.
It is unclear how (or if) these problems may translate into the
coherenceestimate,particularly given the ad hoc nature of the
cross-powerspectral estimator. Nevertheless,the maximum
entropy-basedestimatesof coherenceafford a clear improvement over periodogram estimates for the topographic and
gravity data used to estimateflexural rigidity (Figure 3).
Flexural Properties of the Basin and RangeColorado Plateau-Rocky Mountain Transition
[1987], andBechtel [1989]) that theseassumptions
introduce
testsand other arguments,e.g., Forsyth [1985], Bechtel et al.
acceptably
smallerrorsinto the estimateof rigidity.
Coherence analysis was applied to the eastern Basin and
On the other hand,the supposition
that surfaceand subsur- Range province at its transition to the Middle Rocky
face loadingprocesses
are uncorrelated
is criticallyimportant. Mountains and northwesternColorado Plateau physiographic
Macario et al. [1992] report that the coherencemethodunderestimatesflexural rigidity if surfaceand subsurfaceloads are
stronglycorrelated. One can envisioncircumstances
(suchas
provinces(Plate 1). Flexural rigidity dependsstrongly on the
composition and tectonic evolution of the lithosphere, and
these provinces reflect very different formative and tectonic
the combinationof volcanicand thermalloadingin volcanic histories that are likely to manifest as abrupt variations in
provinces)in whichsuchcorrelationmay indeedbe expected. lithosphericcharacter. The Middle Rocky Mountains physioHence the possible bias of coherenceanalysis should be graphic province lies within the Wyoming isotopic age
heeded,particularlygiven that the possibilityof load correla- province, a > 2500-m.y.-old cratonic province that is approxtion may increaseif one samplesfewer independent
geologic imately defined by surface expressionof basementrocks with
features in smaller data windows.
K-Ar biotite ages greater than 1.4 b.y. [Condie, 1981]. The
20,130
LOWRY AND SMITH: RIGIDITY
OF THE BASIN-RANGE
b. BouguerGravity
a. Topography
113 ø
112 ø
111 ø
TRANSITION
110 ø
113 ø
43.•
112 ø
110 ø
Elevation Bouguer
gravity
f 43 ø
(m)
r---'l
(mGal)
3600
\3 O'
-150
3400
-160
D
3200
-t- 42 ø
-170
3000
-180
1.4 G•
'
I
!
[MRM
2800
-190
2600
-200
2400
-210
2200
-220
40
2000
40'
-230
B
1800
-240
['
1600
CP
39 •
1400
q-
•
112'
+
111ø
CP
-250
-260
+
110ø
39 ø
112ø
Longitude
111ø
110 ø
Longitude
0
100
Scale (km)
Plate 1. The data usedfor this study: (a) Topography. (b) CompleteBouguergravity. Grey lines are K-Ar
isotopic age contours of the Archean Wyoming craton; physiographic province abbreviations are CP,
ColoradoPlateau;BR, Basin and Range;MRM, Middle Rocky Mountains;ESRP, easternSnakeRiver Plain.
ColoradoPlateau is thoughtto have originatedwhen island arc
material accreted to the Wyoming province circa 1800-1600
Ma [Karlstrom and Houston, 1984], but a lack of surfaceexposures of Precambrian
basement
hinders
confident
assessment
of the age and genesisof the province. The modern Basin and
Range was initially the site of miogeoclinal depositionsubsequent to a late Precambrianrifting event.
The tectonic history of the Cordillera is complex and likewise varies according to province. During the Paleozoic and
up to Jurassictime, a hinge line separatedthe cratonic shelf of
the Middle Rocky Mountains-ColoradoPlateau from the passive margin basin deposition in the Basin and Range
[Burchfiel and Davis, 1975; Stewart, 1978]. From late
Jurassic to Eocene, western North America experienced
Andean-type tectonism associatedwith low-angle subduction
of the Farallon plate [e.g., Lipman et al., 1972]. Exact timing
varies from place to place, but within the study area, contraction occurred primarily in Cretaceous-Tertiary time (-90-50
Ma [e.g., Wernicke et al., 1987]). East-west shorteningin the
Basin and Range was accommodated by low-angle reverse
faults and associated folding that inverted the miogeoclinal
structure during the Sevier and Laramide orogenies [e.g.,
Wernicke et al., 1987]. In the Middle Rocky Mountains, contraction was expressed in the form of higher-angle reverse
faults and monoclines, generating broad basins and basement
uplifts during the Laramide. Contraction of the Colorado
Plateau in Laramide time occurred principally in the form of
monoclinal
structures.
Tectonism
reversed to extension
in the
Basin and Range beginningaroundOligocene(-30 Ma [e.g.,
Anderson, 1989]) and continuingto the presentday. Normal
faults exhibiting significant displacement are generally restricted to the Basin and Range, but various indicators suggest
the stress regime is currently extensional in the Colorado
Plateau and Middle Rocky Mountains provinces as well
[Zoback and Zoback, 1989].
The divergent histories of these provinces are reflected in
their geophysical signatures. As observed from seismic refraction data, the Middle Rocky Mountains crust is about 40
km thick, and upper mantle compressionalwave (P, velocity
is in the range7.9 to 8.0 km s-• [e.g.,Smithet al., 1989]. The
LOWRY AND SMITH: RIGIDITY
OF THE BASIN-RANGE
TRANSITION
20,131
ColoradoPlateaualsohas-40 km crustalthickness,but Pn velocity is 8.1 km s-] [Beghouland Barazangi,1989]. Most of
60 km east of the Basin and Range boundaryin Plate l a) with
lower, -1.5 km, elevation in the province interior.
the Basin and Range has a 28-30 km thick crust with a 7.8
Topographic features analogous to the Colorado Plateau rim
km s-• Pn velocity,but interpretation
of refractiondatain the are commonly found adjacent to continental rifts. These rift
easternmost Basin and Range is ambiguous [Smith et al.,
flank uplifts appear to be independentof the local uplift asso1989]. Crustal thickness there has been modeled from 25 to
ciated with normal faulting [e.g., King and Ellis, 1990] and
45 km, with corresponding
Pn velocitiesof 7.4 to 7.9 km s-], have been modeled respectively as a flexural isostatic rerespectively. Tomographicimaging of upper mantle P veloc- sponse to lithospheric necking [Chery et al., 1992] and an
ity from teleseismictravel time residuals [Humphreys and erosional process which is concentrated by the flexural reDueker, 1994] generallyindicateshigh velocity in the Middle sponse to denudational unloading [Tucker and Slingerland,
Rocky Mountains and northernpanhandleof Utah, intermedi- this issue].
ate velocityin the ColoradoPlateauinterior,and low velocity
Coherenceanalysis was applied to the companiontopograin the eastern Snake River Plain, easternBasin and Range phy and complete Bouguer gravity data sets(Plate 1) employed
south of the Great Salt Lake, and around the Colorado Plateau
by Simpson et al. [1986] for development of an isostatic
periphery.
residual gravity map. The gravity is gridded at a 4 by 4 km
The topographicexpression(Plate l a) is dominatedby rela- spacing from similarly distributed measurementsthat were
tively low, <1.5 km, average elevation in the easternmost processedto remove outliers [e.g., O'Hara and Lyons, 1983].
Basin and Range and high, slightly >2.0 km, averageeleva- The topographyis an identically parameterizedgrid generated
tion in the Middle Rocky Mountains. The regional Bouguer from 5 arc min data specifically for the purposeof tandem siggravity field (Plate lb) is negativelycorrelatedwith topogra- nal processing[Simpson et al., 1986]. Flexural rigidity was
phy at province-widescales,as is expectedfrom the isostatic estimated for 200x200 and 400x400 km windows with centers
model. Interestingly, the Colorado Plateau exhibits higher, spaced50 km apart, and rigidity estimateswere then interpo-2.0 km, averageelevationaroundits rim (for example,-40lated to a 10-km spacingfor the purposeof contouringusing a
b. Heat Flow
a. Flexural Rigidity
113 ø
112 ø
111 ø
110 ø
113 ø
112 ø
111 o
110 ø
..•
$q• •c. I+WI•Cl•"• Flexural
Elastic
Rigidity Thickness
Heat Flow
43'-•
•
•
(Nm)
m
r--n (km)
2.9x1023
42
, Bon•evi/_•
1.0x1023
.MRM
. 1.4G•
(mWm-2
\3O',
I
32.0
42
ID
-
----.
I
lOO
22.6
1.4 G•
41'n
i•'
3.6x1022
90
16.0
80
&&
1.3x 1022
11.4
i
40
70
40
4.6x1021
&
8.0
I-I
R
P
+
60
R
1.6x1021
.+
5.7
39'
&
I
112 ø
111 ø
Longitude
110 ø
112 ø
0
100
111 ø
&
110 ø
Longitude
Scale (km)
Plate2. Flexuralrigidityandregional
surface
heatflow. (a) Interpolated
colorcontour
of flexuralrigidity.
Contourintervalvarieslogarithmically.Boxesare centerlocationsof windows(largerboxes,400x400km
windows;smallerboxes,200x200km windows). MT denotespalinspastic
locationof the Meadethrustmodeled by Jordan [1981]; white contourapproximatesthe highstandshorelineof PleistoceneLake Bonneville.
(b) Regionalsurfaceheatflow [Blackwelland Steele,1992]. Solidtrianglesindicateconstraining
measurements. Age contoursand province abbreviationsare as in Plate 1.
20,132
LOWRY AND SMITH: RIGIDITY OF THE BASIN-RANGETRANSITION
minimum curvaturealgorithm. We note that the close spacing
requires significantoverlap of data windows for neighboring
estimates of flexural rigidity, implying some smoothing of
the rigidity distribution. In order to rigorouslydeterminethe
resolution, the method should be applied to synthetic data
with a known variation of rigidity, and we have not yet attempted suchtesting. However, we hope to demonstratefrom
comparisonto geologic information that coherenceanalysis
does indeed resolve changesin flexural strengthat the spatial
scalesused in this study.
Flexural rigidity of the study area is presentedin Plate 2a.
Conversion to the equivalent effective elastic thicknessTe is
function is less than a single standarddeviation of the coherence estimate. Error limits determinedin this manner are particularly useful for assessingwhether a data window is large
enoughto completelyresolvethe transitionalwavelengthsof
coherence: When the data window is too small, the error func-
tion flattensand the error rangeis unreasonablybroad. Hence
a 200x200 km window centered at the northern boundary of
the Colorado Plateau province may exhibit an error range of
3x1022to lx1026N m (Te= 15 to 250 km), whereas
the corresponding
400x400km windowhaserrorlimitsat 2x1022to
6x1023N m (T• = 13 to 41 km), indicatingthat the latteris a
preferable size.
also given, assuminga Young's modulusof 10TMPa and
Poisson's ratio 0.25.
The P wave velocities
that were used to
constrainthe densitydistributionare describedin Table 1. We
chose the shallowest first-order density/velocity discontinuity
to serve as the subsurfaceload depth, becausethe downward
continuationoperation implicit in the load deconvolutionis
most stable for shallow depths. Examples of observed and
best fit predicted coherence functions, as well as plots of
residualerror (the L 2 norm of observedminuspredictedcoherence), are given in Figure 6 for two locations.
The residual error of the coherence functions (Figure 7) is
used to assessreliability of the coherenceestimate. Where the
best fit residualerror significantlyexceedsthe standarddeviation of the coherence estimates (~ 0.03), some failure of the
modeling assumptionsis indicated. The bulk of the misfit is
likely to occurin areaswherethereis correlationof surfaceand
subsurfaceload processes[Macario et al., 1992], implying
possibleunderestimationof rigidity at locationswhere residual error is high. Residualerror functionsare alsousedto assessstandarderror limits of the rigidity estimate,as defined by
the range of flexural rigidities for which the differencebetween
residual error and the global minimum of the residual error
Very large variationsin flexural isostaticresponseare ob-
served within the 125,000 km2 study area (Plate 2a).
Estimates
of rigidityrangefromas low as 8.7x102ø
N m (T•=
4.6 km) to as high as 4.1x1024N m (Te= 77 km). To place
these values in perspective, consider that flexural rigidities
reportedfor the oceanicelasticlayerrangefromabout2x102ø
N m (T• = 3 km) for seamounts
formedat mid-oceanridgesto
around3x1024N m (T• = 70 km) in thecaseof a few studiesof
very old (75 to 140 Ma) lithosphereat oceanictrenches(see,
e.g., the compilationof Wessel [1992]). The Bechtel et al.
[1990] coherence study of the North American continent,
which assessedrigidity at scalesof 400 km or more, docu-
mentedrigiditiesrangingfromlx102•N m (Te= 4.9 km)in the
BasinandRangeto 2x1025N m (Te = 123km) in theCanadian
shield.
Large-scalevariationsof flexural rigidity broadly correspondto physiographicprovince (Plate 2a). The Basin and
Rangeexhibitsrelativelylow flexuralrigidityof 8.7x102ø
to
3.6x1022N m (Te = 4.6 to 16 km), with a meanrigidityof
9x102•N m (Te - 10 km) andmedianerrorlimits lx102•to
4x1022N m (T•= 5 to 16 km). The northwestern
part of the
study area, within the eastern Snake River Plain volcanic
Table 1. Density Models Usedto Constrainthe Inversion
Western Middle Rocky Mountainsa
Eastern Basin and Rangea
Layer
Depth to Top, Velocity, m/s
Density,
Depth to Top,
kg/m3
m
m
1
2
3
4
5
0
1700
8400
14700
24700
3400
6000
2280
2700
5500
6500
2610
2800
7400
3050
0
3000
16000
40000
......
Eastern Snake River Plain b
Layer
Depth to Top, Velocity, m/s
m
1
2
3
0
1500
5200
Density,
2470
2680
6750
7900
2870
3230
Plateau c
m
Density,
kg/m3
2270
2560
2730
0
1700
26000
3000
6200
6800
2210
2740
2880
7800
3190
10700
6510
2810
41000
5
6
18800
41500
6800
7920
2880
3240
......
......
bFromBraile et al. [1982].
CFromRoller [ 1965].
4600
5900
Depth to Top, Velocity, m/s
4
aFrom Braile et al. [1974].
Density,
kg/m3
Colorado
kg/m3
3330
5210
6130
Velocity, m/s
LOWRY
AND SMITH:
RIGIDITY
OF THE BASIN-RANGE
b©
Wavelength(km)
a©
100
10
1
1
0.9
0.9
0.8
TRANSITION
Wavelength(km)
100
0.8
BestFit T e = 5.6 km
0.7
20,133
BestFit T e = 17 km
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0'1
1
0.1
0
0
0.01
0.1
0.01
Wavenumber(km-1)
Wavenumber(I•Q'1)
0.25
0.5
O
0.45 -
0.2-
0.4-
o o
ooo
000
0.15-
O
0.3 0
oooø
0.25-
O
0.2
o
0.1-
o
0.15-
O
0.05 -
O
0.35-
0.1-
O
o
O
O
0000
...,-,•
0.05 0
0
2b
Elastic thickness(km)
0
30
0
Elastic thickness(km)
Figure 6. Examplesof observed
(opencircles)andbestfit predicted(solidline) coherence
functions,along
with plotsof residualerror,from (a) the northwestcornerof the studyareaand (b) the southeast
corner.
province,exhibitslow rigidity as well (1.6x1021to 7.4x1021 smallestestimatefrom the BasinandRange(D = 9x102{)
N m;
N m;Te= 5.6to9.4km).TheQolorado
Plateau
istypified
by
higherrigidity thanthe adjacentBasinandRange,3.0x1022to
4.1x1023
N m (Te=15 to 36 km), withmean9x1022N m (T, =
22 km) andmedianerrorlimits2x1022
to 4x1023
N m (T, = 13
T, TM4.6 km). Bechtel [1989] used periodogram coherence
analysis in windows designed to approximately isolate the
Basin and Range and Colorado Plateau provinces, finding
rigiditiesof 1x1021(4.9 km) and5x1022N m (18 km), respec-
to 36 km). The greatestflexural strengthis observedin the
Middle Rocky Mountains province, where estimatesare from
tively. Those results are virtually identical to the lowest
rigidities reportedhere for correspondingprovinces. Finally,
2.0x1022
to 4.1x1024
N m (T, = 13 to 77 km)andmeanrigidity modeling of local uplift on Basin and Range normal faults
is 3x1023N m (T•= 33 km) with medianerrorlimits4x10TMto [King and Ellis, 1990] indicatesthat associatedflexural rigid-
3x1024N m (T, = 16 to 68 km).
ity shouldbe of order7x1019to 6x102ø
N m (T, = 2 to 4 km),
These resultscomparefavorablywith most independentdeterminationsof flexural rigidity in the region. For example,
two-dimensional modeling of loading by low-angle thrust
faulting and associatedforeland basins [Jordan, 1981] indi-
for which the upperboundis consistentwith our lowestBasin
and Rangevalues. Modeling of the Wasatchfault in particular
[Zandt and Owens, 1980] suggestsflexural rigidity in the
with the Meade thrust event, denoted MT in Plate 2a. Because
area.
footwallof 6x1021to 4x1022N m (T• = 9 to 17 km); again,the
catesa flexuralrigidityof -1023N m (T• =- 22 km) associated upper bound is consistentwith our lowest values from that
Jordan [1981] modeledbasinsformedduringSevierthrusting,
her resultshoulddocumentthe rigidity of the elasticlayer dur- Discussion
ing the mid-Cretaceousrather than at present,but nevertheless,her estimateis consistentwith the rangeof flexural rigidThere is one independent assessmentof flexural rigidity
ity thatwe observein the area(6x1022to 4x1023N m; Te= 19 that is decidedly inconsistentwith our results. Several investo 36 km). Londe [1986] employedan admittancetechniqueto tigators [Walcott, 1970b; Passey, 1981; Nakiboglu and
assessflexural rigidity along the Basin and Range-Colorado Lambeck, 1983; Bills and May, 1987] have reported flexural
Plateautransitionat -102ø-1021N m (T, ~ 2.2-4.8 km).
rigiditiesrangingfrom 5x1022tO 2x1023N m (T• = 18 to 30
Bearing in mind that inversionof the admittancefunction emphasizesthe flexurally weakestportionsof a domain [Forsyth,
1985], Londe's [1986] upper bound is consistentwith our
km) for the rebound of the Pleistocene Lake Bonneville, as
modeled from measurements
of relative
elevation
of lake shore
terraces.Thesenumbersare muchgreaterthanthe rigiditieswe
20,134
LOWRY AND SMITH: RIGIDITY
113 ø
OF THE BASIN-RANGE
111 ø
112 ø
TRANSITION
110 o
Residual
Error
........
43 ø
43'
0.104
0.096
0.088
..-.:..-.:.:.:.:.x..-.:.:
0.080
0.072
0.064
::::::::::::::::::::::::::
0.056
0.048
:•:•:•:•:)3':•:?
0.040
,:::::::::..:.:.:.:.:.:...:.
0.032
0.024
112 ø
111 ø
110 ø
Longitude
Figure 7. Interpolatedcontourof residualerror (predictedminusobservedcoherence).
have determined for the same location of 9x102ø to 4x1022
N m; Te= 4.6 to 16 km (Plate 2a). There are two possibleexplanations for the discrepancy. The first possibility is that
crustaltemperatureswere much greaterin the past (say, for example, the mid-Miocene,-15 Ma), and the Basin and Range
load structurewas emplaced prior to that time; that is, there
has been negligible erosional-depositional
or tectonic redistribution in the time since. In that case, the flexural rigidity
indicated by the coherencemethod would reflect an ancient
rather than modern temperature field [e.g., Courtney and
Beaumont, 1983]. In effect, it would record a fossil flexural
rigidity. However, evidence for many tens of kilometers of
approximately east-west extension since 15 Ma in the easternmost Basin and Range [e.g., Wernicke et al., 1987] suggests that the load structure of the easternmostBasin and
Rangehas not remainedunchangedfor enoughtime to reflect a
radically different crustal geotherm.
Alternatively, one may note that all of the Lake Bonneville
rebound studies model viscoelastic isostatic responseof the
lithosphere, and all infer that the elastic layer achieved isostatic equilibrium both during loading (a < 15 kyr time span)
and after the load was removed(-11 ka). Althoughsomestudies usedmodelscapableof describinga rheologicallystratified
Earth suchas that postulatedby Smith and Bruhn [ 1984], none
of the investigationsexplored the possibility of a viscoelastic lower crust. The probablereasonfor omissionof the lower
crust from
the calculations
is that containment
of viscoelastic
fluid in a thin channel dramatically increasesthe isostatic responsetime [Nakiboglu and Lambeck, 1983]. Thus while the
lithosphereindeed achieved isostaticequilibrium with respect
to viscosities of the upper mantle ductile zone, it probably
was not loaded sufficiently long to achieve isostatic equilib-
rium with respectto a lower crustalductilezone sandwiched
between crustal and upper mantle elastic layers. Willett et al.
[1984, 1985] modeled the temporal evolution of viscoelastic
flexural responsein a rheologicallystratifiedlithosphereand
found that the apparentelasticthicknessstronglydependson
duration of the load. In that study, continental lithosphere
havingsurfaceheat flow 90 mW m-2 exhibitedrigidity of
2x1023
N m (Te= 30 km)after104yearsascompared
to 5x1022
N m (Te = 18 km)after 106 years, followingrelaxationof
stresses in the lower crust.
Since most of the load structure de-
scribedby coherenceanalysishas residedfor-106 to 108
years,as comparedto 104 yearsfor Lake Bonneville,the observeddiscrepancywould be expected.
Correlation
With
Heat
Flow
and
Tectonism
Flexural rigidity in the Basin and Range-ColoradoPlateauMiddle Rocky Mountains transition exhibits remarkable spatial correlation with a number of geologic and geophysical
features. There are strongrelationshipswith surfaceheat flow,
with age and tectonic history of the continental lithosphere,
and with locations of Cenozoic normal faults that have experienced large displacements.More subtlerelationshipsseem to
LOWRY AND SMITH: RIGIDITY
OF THE BASIN-RANGE
occur between rigidity and secondaryfault structuressuch as
segmentation of the late Cenozoic Wasatch normal fault zone
and displacement transfer structures associated with
Cretaceous/early-Tertiary
low-anglereversefaults.
Where regional heat flow is well determined, low flexural
rigidity correspondswith high heat flow (comparePlates 2a
TRANSITION
20,135
be explained in either of two possibleways: (1) major faults
are directly responsiblefor weakening of the elastic layer, or
(2) lithosphericcharacteristicsthat determinethe variation of
flexural rigidity independently control the geometry and distribution of faulting as well. As an example of the first case,if
fractures extend all the way through the elastic layer and can
and 2b). This resultis expectedbecauserigidity dependsmost serve as discontinuitiesin stressbehavior, they could reduce
heavily on power law creepof rocks,which dependsin turn on flexural rigidity by promoting "broken plate" behavior, for
temperatureand composition.The correlationis mostreadily which the isostatic response would resemble that of a much
apparentin the western half of the study area. Highest heat weaker continuous media [e.g., Walcott, 1970a]. However,
flow, > 100 mW m-2, characterizesthe eastern Snake River
fault-boundedblocks in the Basin and Range are supportedenPlain, where the lithospherewas thermallyperturbedby pas- tirely by the strength of the plate rather than rotational
isostasy or other isostatic mechanisms [Eaton et al., 1978;
sage of the Yellowstone hotspot around 10 Ma, and the
provincecorrespondinglyexhibits very low flexural rigidity. Bechtel, 1989], so "broken plate" behavior is deemed unHeatflow is alsohigh(90-100 mW m-2)in theflexurallyweak likely.
Great Salt Lake and Sevier Desert regions of the Basin and
Plate-bending stressesresulting from footwall flexure are
Range. Inside the 1.4 Ga isotopic age contour, Basin and likely contributorsto reducedflexural strength,however [e.g.,
Zandt and Owen, 1980]. In-plane (or tectonic) stressescan
Range heat flow is interpreted to be slightly less, 80-90
mW m-2, and flexural rigidity is correspondingly
greater. also reduce flexural strength as a result of failure within a
Heat flow and flexural rigidity also correlatewith post-mid- depth-dependentyield strengthenvelope[McNutt and Menard,
Miocene extensionof the easternBasin and Range;many tens 1982], and hence the correlation with faulting might also be
of kilometers of extension have occurred in the Great Salt Lake
traced to the stress regime which activated the faults in the
first place. On the other hand, the long-term persistenceof
and Sevier Desert regions,decreasingto less than 10 km farthe tectonic boundary implies that the relationship between
ther north [e.g., Wernicke et al., 1987].
Heat flow is much lower in the northwestern Colorado
rigidity and faulting is at least partially a function of independent control by other factors, such as variationsin bulk comPlateau and Middle Rocky Mountains than in the Basin and
Range, correspondingto much higher flexural rigidity, but position of the lithosphere(translatingto variations in the resmaller-scale correlation is not apparent within the eastern tardationof heat flow). Consequently,neither of the proposed
provinces.The cold thermalinteriorand transitionallyhigher explanationsfor the relationship of flexural strengthto faulting can be ruled particularlyunlikely.
heat flow thermal periphery of the Colorado Plateau, as deFlexural rigidity appearsto correlate with other characterisscribedby Bodell and Chapman[1982], is not apparentin the
flexural rigidity estimates. Moreover, few measurementsde- tics of faulting as well. For example,there is a correlationof
scribe heat flow in the Middle Rocky Mountains province rigid blocks in the Middle Rocky Mountains (Plate 3a) with
where flexural rigidity is greatest. However, the most rigid proposed segmentation of the 370-km-long, late Cenozoic
portion of the Middle Rocky Mountains province ends Wasatch fault [e.g., Schwartz and Coppersmith, 1984]. Also,
abruptlyat the 1.4 Ga K-Ar biotite age contourthat definesthe the easternmostexposureof low-angle thrust sheetsassociated
with the Laramide and Sevier orogenies(Plate 3b) occurs,in
Archean Wyoming craton (Plate 2a). Many investigatorsobserve low surface heat flow over cratons,which they suggest general, 0 to 100 km east of the flexural strengthtransition.
Other studies have suggestedthat the final location of thin
results from refraction of deep mantle heat around a deskin deformationfronts may be defined by mechanicalcouvolatilized Archean lithospheric root [e.g., Ballard and
pling to cratonic lithosphereof the Indian shield south of the
Pollack, 1988; Nyblade and Pollack, 1992]. Hence the
ArcheanWyoming cratonmight be expectedto have relatively Himalayas [Lyon-Caen and Molnar, 1985], and the Brazilian
low heat flow despite thermal overprinting by recent tecton- shield east of the Bolivian Andes [Lyon-Caen et al., 1985], at
ism.
distancesof 100 to 200 km from the leading edge of the craThe persistenceof the tectonicboundaryat the easternlimit ton. Displacement transfer structuresin thin skin thrusts also
of the Basinand Range,over hundredsof millionsof yearsand correspondto local maxima of flexural rigidity along a norththrough multiple tectonic events, arguesstrongly that inher- south transect. Note, however, that the Uinta arch transfer
ited characteristicsof the lithosphere influence the modern structures have been exaggerated by subsequent folding,
distributionof tectonism. Lower thermalconductivityis ex- thrusting and erosion during the Laramide compressional
pectedto causethermal refractionof deep mantle heat around event that formed the Uinta mountains [Bruhn et al., 1986].
the Archean lithosphereof the Wyoming province,and hence
It is not surprisingthat secondarystructuresof large faults
reducedheat flow. Devolatilizationof mantlerock probably should indicate a relationship with flexural rigidity. In the
contributesto lowered ductility of the upper mantle as well case of Wasatch fault segmentation,a differential isostatic re[Pollack, 1986], and in addition to increasingthe flexural sponseto unloading of the footwall block along the length of
rigidity, the resulting inhibition of advective heat transfer the fault should evolve a recognizable variation in geomormay furtherreduceheat flow. Hencethe relationshipof conti- phic and geometric expression over its-•18 m.y. history.
Similarly, in the case of low-angle thrust sheets advancing
nental lithospheregenesisto flexural rigidity and heat flow
propertiesmay be in large part a function of minor variations from an elasticallyweak accretionaryterraneto a strongerarea
in bulk composition.
of cratonic or epicratonic origin, differential isostatic reThe transitionfrom low to high flexural rigidity also ex- sponse of the craton to loading by the thrust sheets might
hibits an apparentcorrelationwith the easternmostoccurrence play a role in determining where thrustsadvance the furthest
of Cenozoicnormal faults with large displacements
(Plate 3a). and, consequently,where displacementtransfer structuresare
The coincidenceof major normalfaultsand low rigidity may likely to be found.
20,136
LOWRY
ao
Cenozoic
113 o
AND SMITH'
Normal
112 o
OF THE BASIN-RANGE
Faults
111 ø
TRANSITION
b. Mesozoic/TertiaryThrustFaults
110 ø
s l
43
RIGIDITY
113ø
112 ø
111 ø
+ Flexural Elastic
110ø
4• \
Rigidity
Thickness
+
(Nm)
(kin)
"
k•
+ 43ø
EBL
2.9x10 23
32.0
M
42' ' C' )/ i
+ 42ø
,/
1.0x 1023
22.6
3.6x 1022
16.0
1.3x 1022
11.4
+
112 ø
111 ø
4.6x 1021
8.0
1.6x1021
5.7
110 ø
+
39 ø
112 ø
Longitude
40 ø
111 ø
110 ø
Longitude
0
1O0
Scale (km)
Plate 3. Relationship of flexural rigidity to Cenozoic normal faults and Cretaceous-Tertiary overthrusts. (a)
Flexural rigidity and surfacetracesof normal faults exhibiting late Quaternary (<500 ka) surfacerupture [after
Hecker 1993; Smith and Arabasz, 1991]. The easternmostfaults with significant (> 1 km) offset are indicated
in bold white pen. Faultsare CM, Crawford Mountains;EBL, East Bear Lake; EC, East Cache;SV, Star Valley;
WFZ, Wasatch fault zone. Segmentationof the Wasatch fault is after Schwartz and Coppersmith[1984]: C,
Collinston; L, Levan; N, Nephi; O, Ogden; P, Provo; SL, Salt Lake. (b) Flexural rigidity and
Cretaceous/Tertiaryoverthrusts[after Blackstone,1977;Hintze, 1980]. Locationsof displacementtransfer
structuresare emphasizedby dashedwhite lines.
Conclusions
ably with independentestimatesof flexural rigidity from modeling of individual features and periodogram stochasticinverIn this paper, a methodology is developed for stochastic sion using much larger windows, with the exception of deterinversionof flexural rigidity using a maximum entropy-based minations from modeling of the Lake Bonneville rebound.
mayindichte
a viscoelastic
lowercrustal
approach to coherence analysis. This methodology is em- The latterdiscrepancy
ployed to map large variations in the Earth's isostatic re- zone of flexural decoupling which did not respond isostatisponsewith greater resolution than was previously possible. cally to the short-lived Lake Bonneville load but has reFlexural rigidity in the easternmostBasin and Range is found spondedto more enduring loads.
The mapped variation of flexural rigidity is supportedby
to average9x1021N m (Te= 10 km) andvariesfrom 8.7x102ø
to 3.6x10TMN m (Te = 4.6 to 16 km). Rigidityof the north- correlations with independent geologic and geophysical inwesternColoradoPlateaurangesfrom 3.0x1022to 4.1x1023 formation. Low flexural rigidity is strongly correlated with
N m (T• = 15 to 36 km) andaverages
9x1022N m (T• = 22 km). high surface heat flow, as predicted by the relationship beFlexural rigidity of the Middle Rocky Mountains averages tween strength of the elastic layer and the temperature-depen3x1023N m (T• = 33 km) and can rangefrom 2.0x1022to dent ductile flow law. The elastic layer is very rigid in the
4.l x1024N m (T• = 13 to 77 km). Theseresultscompare
favor- Archean Wyoming craton, where heat flow is poorly con-
LOWRY
AND
SMITH:
RIGIDITY
OF THE BASIN-RANGE
strainedbut is probably relatively low as is observedin other
cratonic regions. Relationships are also apparent between
flexural rigidity and the locations of displacementtransfer
structuresin Mesozoic overthrusts,as well as segmentboundaries on the Wasatch fault.
In addition, the distribution of
Cenozoic normal faults with large displacementsis strongly
correlatedto regions of low flexural rigidity, implying a genetic relationshipbetweenfaulting and rigidity.
TRANSITION
H•(k)
=-
APr W•(k).
pl-•,?-D-V
•4
g
Following Forsyth's [1985] derivation, consider a thin
elastic plate having density p = P0 at the top (the Earth'ssurface, z = 0), p = p• at the base(depthz = h), and a one-dimensional (but otherwise arbitrary) density distribution p(z) between (Figure 2a). A topographicload atop the plate with initial FourieramplitudeH•(k) (Figure2b) will inducea deflection
Wr(k) to achieveflexural isostasy(Figure 2c). The resulting
topographywill have amplitudeHr(k) given by
Hr(k) = H•(k) + Wr(k).
(A6)
If loads at the surface and at depth are in phase, i.e., spatially correlated,the observedtopographicamplitude is thus
given by
H(k) = H7(k)+ Ha(k)
Appendix: Load Deconvolution for an Arbitrary
Density Profile
20,137
(A7)
and the Bougueranomalywill have Fourier amplitude:
B(k) = Br(k) + Ba(k).
(A8)
However, it is expectedthat surfaceand subsurfaceloading are
not in phase. Hence theseoperationsare performedseparately
for the real and imaginary partsof the complexFourier amplitudes. Substitutiongives the topographicamplitudeto be
(A1)
H(k)=
(A9)
The amplitudeof the BougueranomalyBr(k) corresponding
to
the flexure Wr(k) is given by
and the harmonicof the Bouguer anomalyis
•dp
exp
(-kz)
dz W•(k)
Gfh
2n
(A2)
where G is the gravitational constant. The relationship between amplitude of the resultingtopographyand amplitudeof
the compensatingdeflectionis given by
Wr(k)
=-
Po Hr(k),
hdp
exp(-kz)dz
-2nøPø
p•+_v
•4
g
(A3)
f{i
dzexp
• exp(-kz)
(-kzD}.
p•-p0+•
in which D is flexural rigidity of the plate and g is
gravitational acceleration.
If a subsurfaceload with initial amplitudeW•(k) and density
contrast Apt.is introduced at a depth zt`, flexural isostatic
responsewill producetopographyof amplitudeHa(k) (Figure
2c). The final expressionof the density contrast at zt`will
have amplitudeWa(k):
Wa(k)= Wz(k) + Ha(k).
(A4)
- 2riGAPr
' P•+•
Dk4
Apt,
(A10)
The above system of equationsis solved for the initial applied loadsH• and W•. Then the topographyand gravity amplitudes are deconvolvedinto their respectivecomponentsvia
D•
=
p•-p0+-•
'
Dk4
p•+•-
The resultingBouguergravity anomalywill be
Ha(k) = H(k) - Hr(k)
(All)
(A5)
with
•(,h
-'•-exp(-kz)dz
D
B•(k)
= 2nGApL
'
P•
+--k4-exp(-kzt`)
g
Br(k) = B(k) - Ba(k).
20,138
LOWRY AND SMITH: RIGIDITY OF THE BASIN-RANGE TRANSITION
Acknowledgments. Thoughtfuland critical reviewsby D. W. Forsyth,
C. D. Ruppel, M. Ellis, V. J. Mathews, J. M. Fletcher,and R. N. Harris
greatly improvedthis manuscript. Animated discussions
with J. O. D.
Byrd, R. L. Bruhn, and J. M. Bartley alsoaidedin crystallizationof some
of the ideaspresented.We would particularlylike to thankDon Forsyth
and Tim Bechtel for their developmentof the coherencemethod and
Maria Zuber for providing accessto Bechtel'soriginal computercode.
Thanksalsoto Dave Blackwell for providingthe regionalheatflow data
and to Rick Saltus for the USGS topography and Bouguer gravity
gridded data sets. Support was provided by the National Science
Foundation through grant EAR 89-04473. Support for reproduction
of color plates has been provided by NASA Grant 3338 to Michael
Braile, L. W., R. B. Smith, J. Ansorge,M. R. Baker, M. A. Sparlin,C.
Prodehl, M. M. Schilly, J. H. Healy, St. Mueller, and K. H. Olsen,
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Buck, W. R., Flexural rotation of normal faults, Tectonics, 7, 959-973,
1988.
Burchfiel, B.C., and G. A. Davis, Nature and controls of Cordilleran
A. Ellis.
orogenesis,
westernUnitedStates:Extensions
of an earliersynthesis,
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(ReceivedApril 6, 1993; revisedlanuary31, 1994;
acceptedApril 7, 1994.)