TECTONICS, VOL. 17,NO. 5, PAGES780-801, OCTOBER 1998
Kinematic history of the Laramide orogeny in latitudes 35ø-49øN, western United States
Peter Bird Departmentof Earthand Space Sciences, University of California,Los Angeles
Abstract. The kinematic history of the Rocky Mountain First, it occurredin a plate interior,so it hasno simpleexpla- forelandand adjacent areas is computedback to 85 Ma, using nationin termsof platetectonics. Second, it occurredin a re- virtually all the structural,paleomagnetic, and stressdata in gion with a continuouscover of Phanerozoicsection whose the literature.A continuousvelocity field is fit to the data in surfacewas initially near sea level, so that all strainsand up- eachtime stepby weightedleast squares, and this velocity is lifts are especiallywell recorded.Third, the literatureon this integratedback throughtime. As proposedby Hamilton subject is mature, and the chance for future discoveriesof [1981], the net movementof the ColoradoPlateau was a first-orderstructures seems small. The Laramideorogeny is clockwiserotation about a pole in northernTexas; but the ro- the primaryfocus of this paper,although the calculationnec- tationwas less(3 ø) than somehave inferredfrom paleomag- essarily includesthose parts of the Sevier orogeny and the netism.The Laramide orogenyoccurred during 75-35 Ma, Tertiary extensionthat overlappedit. withpeak Colorado Plateau velocities of 1.5 mm yr -1 during During this time, easternNorth America was stable.Oce- 60-55 Ma. The mean azimuth of forelandvelocity and mean anic lithospherewas subductingat the westernmargin of directionof forelandshortening was stableat 40ø for mostof North America, as it had been since the Jurassic.The Kula the orogeny,increasing to 55ø in 50-40 Ma; the counter- plate was subductingalong the north part of the margin,and clockwiserotation of shorteningdirections proposed by some the Farallon plate was subductingalong the southpart; the previousauthors is incorrect.Comparing the computed histo- past history of the triple junction betweenthem is controver- riesof forelandflow speedand direction with the knownmo- sial [Engebretsonet al., 1985]. This subductionmay have in- tions of the Kula and Farallon plates confirmsthat the fluenced the Rocky Mountain foreland by generating Laramideorogeny had a differentmechanism from the early horizontalstresses in the lithosphereat the margin.Alterna- Sevierorogeny: it was drivenby basaltraction during an in- tively, there may have been an episodeof horizontalsubduc- terval of horizontal subduction,not by edge forces due to tion of one or both oceanic plates, allowing direct stress coastalsubduction or the spreadingof the westerncordillera transferto the baseof the lithospherein the foreland[Dickin- or by accretionof terranesto the coast.Tentatively, a minor son and Snyder, 1978]. A major goal of this paperis to refine clockwiserotation of shorteningdirections at 50 Ma may rec- the kinematicsso asto permittesting of thesehypotheses. ordthe passageof an activeKula-Farallon transform within In this paper, I apply a new algorithm to computethe the subducted slab. kinematicsof the orogenyin time stepsof 5 m.y. back to 85 Ma. The detailsof the methodare describedin AppendixA. Broadly,however, the algorithmhas the five followingsteps: 1. Introduction (1) Geologic and paleomagneticdata concerningfinite dis- The west centralpart of the United States(Figure 1) has placements,rotations, or strainsover long time periodsare been deformedby three overlappingevents since Cretaceous convertedto rate estimates.An uncertaintyis assignedto each time. The Sevier orogenylasted from about 119 Ma [Heller rate estimate.(2) The velocitiesof all the nodes in a finite and Paola, 1989] to 50 Ma [DeCellesand Mitra, 1995] and elementgrid are determinedby solvinga linear system,which resultedin displacementof thick platesof sedimentaryrock is based on weighted least squaresfitting of the velocity eastwardfor tensof kilometerson bedding-planethrusts with modelto the tentativerates. (3) Thesevelocities are integrated westdipping ramps. The Laramideorogeny lasted from about backwardover time, using 5 m.y. steps.The programmoves 75 to 35 Ma [Dickinsonet al., 1988] and involvedthrusting the nodesof the finite elementgrid, the presentstate lines, or of the Precambrianmetamorphic basement in a varietyof di- otherfiducial markersand the positionsof all dataconcerning rectionson faults of 25o-30ø dip with throwsup to 13 km. earlier times. In particular,paleostress indicators are restored Extensionbegan about 49 Ma [Constenius,1996] in the for- to their original azimuths.(4) When the historyis complete, mer Sevierorogenic belt, and after29 Ma it alsoaffected the eachgeologic and paleomagneticdatum on finite strain,dis- Rocky Mountain forelandin New Mexico and Colorado placement,or rotationis comparedto the historypredicted by where the Rio Grande rift was formed. the model. In general,the model rate of strain,displacement, The Laramideorogeny did not involvelarge strains or dis- or rotation will not be uniform over the time window of the placements,but it is of particularinterest for severalreasons. datum, as initially assumed.New target rates are now as- signed,based on the time-historyfrom the previousmodel but adjustedby a factor to achieve the correct total strain, dis- Copyright1998 by the AmericanGeophysical Union. placement,or rotation.(5) The entirecomputation is now re- Papernumber 98TC02698. peated, beginning with step 2. In all, 50 iterationsof the 0278-7407/98/98TC-02698 $12.00 historywere performed.
780 BIRD' KINEMATIC HISTORY OF THE LARAMIDE OROGENY 7g I
150' 140' 130' 120' 110' 100' 90' 80' I •
Canadian Rocky Mountains
NORTH
Laramide orogen
RGR AMERICA
Colorado Plateau
PLATE
130' 120' 110' 100' Figure 1. Locationof the Laramideorogen in relationto otherCretaceous-Tertiary tectonic provinces in pre- sentcoordinates. The GreatPlains is the upwarpedmargin of the stablepart of the North Americaplate. The Laramideorogen (or Rocky Mountainforeland) is a regionof basementthrusts overlain by forcedfolds. The ColoradoPlateau has similarstructure but underwentless strain. The Sevierorogen (or Sevierbelt or Over- thrustbelt) containseast vergent ramp/flat thrusts in thick sedimentarysequences. The CanadianRocky Mountainsare similar in structureto the Sevierorogen. The Hidalgo orogenis intermediatein characterbe- tweenthe Laramideand the Sevierand may be structurallyconnected to both (dashedlines). The zone of Tertiaryextension includes the Basinand Range province and the Rio Granderift (RGR). The zoneof dis- placedterranes includes Baja Californiaand allochthanousterranes in BritishColumbia and Washington. The Explorer,Juan de Fuca,Gorda, and Cocos plates are remnants of the largeFarallon plate. Bold rectangle showsthe studyarea and locationof Figures2-6.
At the end of the computation,we have estimatedhistories This methodis a type of "inverse"tectonic modeling, of the slip on eachfault and the drift and rotationof eachpa- which computesvelocities from geologicdata. It is concep- leomagneticsite. More important,we have an estimatedve- tually and procedurallydistinct from previous efforts to locity field throughtime for the whole region,which can be modelthe Laramideorogeny [Bird, 1988, 1989, 1992],which examinedfor insightsinto the ultimatecauses of the orogeny were "forward"or "dynamic"models based on physicsand and its relationto nearby eventsin North America and/orthe assumedrheologies. Inverse tectonic modeling is relatively Pacific basin. new. Saucier and Humphreys[1993] modeledvelocities in 782 BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY
Californiafrom fault slip ratesand geodesy. Holt and Haines eastern North America on the eastern side and on the eastern [1993] usedseismicity to solvefor a continuousvelocity field part of the northernside. The history back to 85 Ma was in Asia, while Avouac and Tapponnier[1993] divided Asia computedusing 5 m.y. time steps.This historywas iterated into four rigid blocks,and Peltzer and Saucier [1996] mod- 50times, using initial rate uncertainties ( cr* ) of 4x 10 -12 m eled its fault network in more detail. In comparison,this s-1 (0.12 mm yr -1) for fault offsets and balanced cross-section methodis uniquein its ability to handlelong histories,finite extensions,2x10 -lø m s-• (0.06ø m.y.-1)for paleolatitude strain,many faults, paleomagneticdata, and stress-direction shifts,and 1.4x10 -•6 radians/s(0.25 ø m.y.-1)for vertical- data. axis rotations.In a typical iteration, about 30% of the rate histories were automatically adjusted; cumulatively, 64% were adjusted.(As discussedin AppendixA, somerate histo- 2. Data and Computation riescannot be adjustedbecause of the dangerof numericalin- The first applicationof this methodis a solutionfor the stability.) The rate uncertaintiesin the final iterationswere Rocky Mountain foreland province of the westernUnited basedonly on the individualuncertainties in displacementor States(latitudes 35ø-49øN, longitudes 103ø-113øW) since 85 rotationtabulated in the files namedin AppendixB. The un- Ma (Santonian).I chosethis regionfor an initial trial because certaintyin the nominally zero strainrate of regionswithout the tectonicsare relatively simple and noncontroversialand activefaults was 5x 10 -17 s-1 (0.16% m.y.-•) in theRocky becausethey are describedin a matureliterature of manage- Mountain foreland and Great Plains but slightly larger able size. (lx10-•6 s-1) in theSevier belt, and was largest (2x10 --•6 I attemptedto collect all the significantinformation from s-1) in areasnow obscured byvolcanic cover of theYellow- the geologicliterature through the end of 1997. While read- stone-SnakeRiver Plain hotspot(blank area in Figure2). ing, I excludedestimates of age or strainthat were basedon At the conclusionof the 50 iterations,all modelpredictions regionalconsistency arguments, so that the requirementof in- were comparedto correspondingoffset, strain, or rotation dependenterrors for each datum might be fulfilled. Useful data to check the size of residuals.These were generallyof data were found in 262 papers;their bibliographiccitations acceptablesize, measuredin terms of the assigneduncertain- are listed in one the files describedin AppendixB. This data ties of the data. Fault offset data were fit with an RMS resid- set is rich in fault offsets(307 faults)and has a goodnumber ual of 0.91 standarddeviation and an averageresidual of 0.39 of paleomagneticsites (220), but has fewer stressindicators standard deviation; 5% of residuals exceededtwo standard (71 sites)and only a few balancedcross sections (11). Most deviations. Paleolatitude anomalies were fit with an RMS re- of the fault offsets(Figure 2) are dip-slip;although I include sidual of 0.81 standarddeviation and an averageresidual of all the dextral strike-slip faults proposedby Chapin and 0.67 standarddeviation; none exceededtwo standarddevia- Cather [1981] or by Chapin [1983], their joint offsetis lim- tions. Vertical-axisrotations were not fit quite as well, with ited to about20 km by the stratigraphicconstraints of Wood- an RMS residualof 1.34 standarddeviations and an average ward et al. [1997]. Where the literaturespecifies amounts of residual of 1.05 standard deviations; 12% of residuals ex- crustal shorteningor extensionacross dip-slip faults, these ceededtwo standarddeviations. Overall, the built-in "reason- figures are used directly. When only the stratigraphicthrow ableness" constraints of consistent stress direction and was available,it is convertedto horizontalmotion by assum- minimum deformationin unfaultedareas did not preventthe ing fault dipsof 25ø for thrustsand 65 ø for normalfaults. modelfrom fitting the datawithin its assigneduncertainties. Paleomagneticpaleolatitude anomalies and vertical-axis rotations are derived from the databaseof McElhinny and 3. Results Lock [1995], usingonly rocksmagnetized since 85 Ma. I ex- cludesamples with secondarymagnetizations, those with un- Velocities and fault slip rates from selectedtimes in the known structural corrections, and all sedimentaryrocks solutionare shownin Figures3 through6. As expected,the (becauseof the possibilityof compaction-inducedinclination continuingSevier orogeny in the OverthrustBelt of Montana- anomalies).The North America polar wander path used to Idaho-Wyoming-Utahis the only activity during the 85-75 computeanomalies is from Van Alstineand de Boer [1978]. Ma time steps.The Laramideorogeny began during 75-70 None of the paleolatitudeanomalies exceeds twice the size of Ma with thrustingconcentrated in westernWyoming. It ex- its standarddeviation, and probablynone of them is signifi- panded eastward to the Black Hills of South Dakota- cant. Most of the computedvertical-axis rotations are less Wyomingby 55 Ma. Ratesof thrustingin the forelandpeaked than 34ø (except in some thrust plates of the Sevier belt). during60-55 Ma andthen declined smoothly to an endduring Thereforeit is not surprisingthat there are few featuresof the 40-35 Ma. Thesegeneral patterns are well known and can be solutiondriven entirelyby paleomagneticresults. inferredwithout modeling;therefore the rest of this discus- Becauseof the shortageof conventionalstress indicators sion will focus on details of the velocity field, which is the for someof the time stepsin this application,I treat faultsas most novel result. additionalstress indicators (in their first phaseof movement), One of the most notable features is a clockwise rotation of with the greatesthorizontal principal compressionnormal to the ColoradoPlateau region during 75-50 Ma, the time of the thrustsand parallelto normalfaults. Each of theserather un- highest velocity magnitudes.Since the Plateau simultane- reliable stressindicators is assigneda 90% confidencerange ouslymoved NE, its net motionwith respectto stableNorth of+45 ø. America since 85 Ma can be described as a clockwise rotation The finite element grid has 787 elementsof mean area of 2.6ø-3.1ø aboutan Euler pole near (34øN, 103øW) in the 1.6x 109m 2. It is fixedto thevelocity reference frame of northernpart of Texas. This is almostexactly the result of BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY 783
116' 115' 114' 113' 112' 111" 110" 109' 108' 107' 106' 105' 104' 103' 102' 'l' ' I I I I I I I I i I I I
MT
• o
.::
o
NM
AZ ateau \
114" 113" 112" 111" 110" 109" 108" 107" 106" 105" 104"
Figure 2. Locationmap with tracesof 307 Late Cretaceous-Tertiaryfaults used in this computation.Medium shadedline is the outer boundaryof the finite elementgrid and model region. Wide shadedlines approxi- mately separatethe Sevierbelt, ColoradoPlateau, and Rocky Mountainforeland regions for purposesof dis- cussion.The area with no faults around (44øN, 112øW) is obscuredby volcanic cover of the Snake River Plain-Yellowstonehotspot track. Abbreviationsfor statesare as follows: AZ, Arizona; CO, Colorado; ID, Idaho; MT, Montana;ND, North Dakota; NE, Nebraska;NM, New Mexico; NV, Nevada; SD, SouthDakota; UT, Utah; WY, Wyoming.This is a transverseMercator projection with prime meridian109øW. 784 BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY
116' 114' 112' 110' 108' 106' 104' 102'
Change in horizontal velocity across fault (mm/a): 0.78
Velocity (x 50 Ma):
.3 mm/a
ID
.co
.AZ
I 114' 112' 110' 108' 106' 104' Figure3. Paleotectonicsduring 80-75 Ma (Late Cretaceous: Campanian). Velocity vectors are relative to easternNorth America. (Note that velocities aremultiplied by50 Ma, not 5 Ma,for legibility.) Width of fault tracesisproportional tothe magnitude ofthe horizontal component ofthe velocity change across the fault, andtraces are labeled with this velocity change inmm y r-I . (Thefast mowng . P•oneer-Kelly-Grasshopper. thrustin Montanaisshown shaded toimprove legibility.) State lines and grid outline are restored; latitude andlongitude ticks in themargin show the undeformed reference frame of easternNorth America. At this time,the only activity is theSevier orogeny in theOverthrust Belt. BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY 785
108' 106' 104 ø 102 ø I I I I
Change in horizontal MT velocity across fault (mm/a)' O.78
Velocity (x 50 Ma)'
.3 mm/a lz80'0 / '
8t'0 •
,,,- w
ID 0.27 / ,,'"•""'
// / O.Oc?s ø
/ ß . /UT / /'/ ' I ' I / •.•o o/ / / / / 'coi
114" 112" 110 ø 108" 106" 104"
Figure 4. Paleotectonicsduring 75-70 Ma (Late Cretaceous:Campanian-Maastrichtian). Conventions are as in Figure 3. The Sevierorogeny continues. In this earliestpart of the Laramideorogeny, activity was cen- teredin westernWyoming but alsospread south to centralColorado-New Mexico and northto centralMon- tana.(The ageof monoclinesin the ColoradoPlateau is poorlyknown; therefore these structures have similar activityin all time stepsuntil 35 Ma.) 786 BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY
116 ø 114ø 112ø 110ø 108ø 106ø 104ø 102 ø
Change in horizontal velocity across fault (mm/a): 0.78
Velocity (x 50 Ma)' ,3 mm/a
ID
, wY
O '
o (.,3
' co
! /'
114ø 112ø 110ø 108ø 106ø 104ø
Figure5. Paleotectonicsat60-55 Ma (late Paleocene-early Eocene). Conventions areas in Figure 3. Thisis thetime of highestmean velocity in theRocky Mountain foreland and Colorado Plateau. Laramide shorten- ingin all regions (eastward tothe Black Hills of South Dakota) was simultaneous withlate Sevier orogeny in thewestern parts of the region. Note the rotational component Ofthe motion of the Colorado Plateau. BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY 787
116 ø 114 ø 112 ø 110 ø 108 ø 106 ø 104 ø 102 ø
Change in MT horizontal velocity across fault (mm/a): 0.78
Velocity (x 50 Ma)'
.3 mm/a
wY / / / / /o / / / /oo / // / o / / /
/ / / / //
114 ø 112 ø 110 ø 108 ø 106 ø 104 ø
Figure 6. Paleotectonicsduring 45-40 Ma (middle Eocene).Conventions are as in Figure 3. The Sevier orogenyis over. Forelandvelocities are only half as large as in Figure 5, and velocity vectorshave rotated clockwiseabout 15ø. The late Laramidestructures are generallythose farthest to the eastand south.Note the beginningof extensionin northwestMontana. 788 BIRD' KINEMATIC HISTORY OF THE LARAMIDE OROGENY
Hamilton [1981], who estimatedthe rotationas 2o-4ø during siteswhere the rockshave beenmagnetized during the time the Laramideorogeny alone, with a very similar pole posi- spanof the computation(0-85 Ma), and there are only four tion. such sites on the ColoradoPlateau (none with statistically This net rotationresult may help to resolvethe controversy significantrotations). It is also probably more precisethan that has grown up aboutthe interpretationof paleomagnetic any of the paleomagneticstudies since it is basedprimarily on data from the Colorado Plateau. Originally, Steiner [1986] net fault offsets,which have mostlybeen measured to preci- found post-Triassicclockwise rotation of 11ø+4ø and attrib- sionsof 1 km or better.(It is true that convertingsome fault utedthe excessover Hamilton'sfigure to post-Laramiderota- throws to fault heavesrequires the assumptionof fault dip; tion. Bryan and Gordon [1990] interpretedall available however,in orderto increasethis rotationresult by a factorof Jurassicand earlier data as showingless post-Jurassic clock- 2, it would be necessaryto changethe assumeddip of thrusts wise rotation: 5.0o+2.4ø. Bazard and Butler [1991] reviewed from 25 ø to 13%which is implausiblefor an averagedip of all the literature and preferredvalues of 3.5o-6.3ø . However, Laramidethrusts throughout the brittleupper crust.) Kent and Witte[1992] reopenedthe controversyby amending In the rest of the computedhistory, rotation is minimal. the North America polar wander path to one that implies The velocity field of the Rocky Mountain forelandis simple Colorado Plateau rotation of 13.5ø+3.5 ø. Molina Garza et al. enoughin most time stepsto be reasonablydescribed by a [1998] respondedwith a calculationshowing only 5.1ø+3.8ø meanazimuth and a meanvelocity, and this is done in Fig- of rotation when data from the Triassic rift basins of eastern ures 7 (azimuth history) and 8 (velocity history).The mean North America are excluded. azimuthis computedby summingthe velocitycomponents v This new result (3 ø) is largely independentof the data (southward)and w (eastward)separately over all nodes of quotedin thesepapers, since my methodis only able to use the Colorado Plateau, Rocky Mountain foreland, and Great
160 Gries [1983] X Livaccari [1991] 140 ( ) --e-- Farallon 120 ( --• - Kula/Pacific lOO --l-- Flow azimuth z= 8o
E 60 N • 4o
2o
-20 ,A -4O 90 85 80 75 70 65 60 55 50 45 40 35Ma
Figure 7. Computedhistory of the meanazimuth of crustalflow in the Rocky MountainForeland and Colo- rado Plateau(squares), compared to the azimuthhistories expected for possiblecauses. (Crustal flow azi- muthsare in parenthesesuntil 75 Ma becausevelocities are very low and theseazimuths are probablynot reliable.) Curveslabeled "Farallon" and "Kula/Pacific" are the azimuthsof the velocitiesof thoseplates with respectto stableNorth Americaat (38øN, 109øW) accordingto stagepoles from Engebretsonet aL [1985]. Curvelabeled "Gries [1983]" showsthe inferredhistory of shorteningdirection that sheattributed to changes in the directionof the absolutevelocity of North America.Curve labeled"Livaccari [1991]" showsthe in- ferredhistory of shorteningdirection that he attributedto the rise and fall of segmentsof the westerncordil- lera. No model is satisfactoryfor all times. Probablythe shorteningdirection was controlledby slip partitioningand slumpingof the cordillerabefore 75 Ma (early Sevierorogeny) but was then controlledby couplingto one or bothsubducted oceanic plates during 75-35 Ma (Laramideorogeny). There is a suggestion that azimuthwas controlledby the Kula platebefore 50 Ma andby the Farallonplate after 50 Ma. BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY 789
Farallon lOO% Kula/Pacific Contact area RMS velocity
O% 80 70 60 50 40 30 20 10 0Ma
Figure 8. Computedhistory of the RMS velocity of crustalflow in the Rocky Mountain Forelandand Colo- rado Plateau(squares), compared to the historiesof the magnitudesof possiblecauses. All curvesare nor- malizedto a maximumof 100%.The maximum of RMS velocitywas 1.06 mm yr -1. Curveslabeled "Farallon"and "KuladPacific"are the velocityhistories of thoseplates with respectto easternNorth America; theirmaxima were 155 and 190 mm yr -1, respectively. Curve labeled "contact area" is the inferred area of contactbetween North Americaand subductedoceanic slab(s), in the latituderange of the United States,with maximum1.85x106 km 2, according toFigure 31 of Bird [1992] which was derived from volcanic-arc posi- tionsmapped by Dickinsonand Snyder[1978] and slabwindow areasfrom Dickinsonand Snyder[1979]. The areaof contactis the bestpredictor of the rateof Laramidedeformation, especially if theywere linked by a power law relationship.
Plains (but not the Sevier belt). I then define the mean azi- ure 7) do not show any counterclockwiserotation after the muthas (•y) -- ATAN2(Y'.w,-• v), whereATAN2 is the two- beginningof the Laramide at 75 Ma; in fact, there is a 15ø argumentinverse tangent of Fortran.Since the easternedge of clockwiserotation at about 50 Ma. The profounddifference the model is fixed, the mean velocity azimuth is also the betweenour resultsstems from the differentage rangesthat mean shorteningdirection. The mean forelandvelocity is the we assignedto individual structures;using more complete root-mean-square(RMS) value acrossthe same area (the data, Dickinsonet al. [1988] also found that someof Gries' whole modelregion, except the Sevierbelt). age assignmentswere in needof revision. These historiesare importantnew informationwhich can The subductionof an oceanicplateau [Livaccari et al., be usedto chooseamong the modelsthat have been proposed 1981; Hendersonet al., 1984] may have occurred,but it can- for the driving mechanismof the Laramideorogeny. At least not be the solecause of the Laramideorogeny because the du- five differentconcepts have been proposedin the literature: ration of the orogenywas too great. As theseresults show, (1) Laramidecompression was parallel to the absolutemotion continuousorogeny persisted at leastfrom 75 until 40-35 Ma. of North America [Gries, 1983]; (2) the Laramideorogeny During this time, any featureattached to the Farallonplate or was causedby the subductionof an oceanicplateau under the Kula plate moved about4900 km with respectto North North America [Livaccari et al., 1981; Henderson et al., America.In contrast,the distancefrom the continentalmargin 1984]; (3) the orogenywas causedby the accretionof the (formertrench) to the Black Hills was probablyno morethan "Baja British Columbia" superterraneand its subsequent the presentdistance of 1550 km. Thereforethe hypothetical northwarddrift [Maxsonand Tikoff, 1996]; (4) the driving plateaucould not have remainedlong enoughin a position force was transmitted from subduction zones on the western where it couldapply compressionto the foreland.If subduc- margin, mediated by a hot mobile cordillera [Livaccari, tion of an oceanicplateau occurred, it might have beenan in- 1991]; or (5) the orogenywas causedby the basal drag of directcause of the orogenyby initiatinga periodof horizontal horizontallysubducting oceanic plates [Dickinsonand Sny- subduction(discussed below). der, 1978; Bird, 1988]. A relatedidea was presented by Maxsonand Tikoff[1996]: Gries' [1983] conceptwas basedon a patterninferred from the Laramideorogeny was due to lateral forcesfrom the ac- a regionalsynthesis: that shorteninguntil 65 Ma was along cretion and northwarddrift of the "Baja British Columbia" azimuthsof 72ø-90ø, while shorteningduring 55-40 Ma was superterraneon the westernmargin of North America.They along azimuthsof 0ø-12ø. During the transition,a rapid did not explainwhy theseevents should affect the stressstate counterclockwiserotation of stresswas inferred.This history of the continentalinterior. However, one might reasonably was explainedas a resultof the changein absolutemotion of expectthe greateststress pulse at the time of initial accretion, North Americafrom westwardto southward.My results(Fig- at 94 Ma, and one might expectit to radiatefrom the region 790 BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY of accretionin presentnorthern Mexico. However,my results basisfor estimatingthe changingarea of contact.The curve show that NE directedshortening occurred strictly after 75 shown in Figure 8 is computedfrom Figure 31 of Bird Ma. Furthermore,there was no steadyand systematicrotation [1992], which, in turn, was basedon Figures3-5 of Dickinson of shorteningdirections, as onewould expect if the sourceof and Snyder [1978]. It showsthat contactarea probably in- the stresswere moving northwardfrom Mexico to Canada creasedby a factor of 4 very rapidly during 80-75 Ma and during 94-40 Ma. Thereforethe accretionand motion of that the area remainedlarge until 55 Ma and declinedcon- "Baja British Columbia"may have occurred,but they were tinuouslythereafter. Of the curvesconsidered, this is the best not the causesof the Laramide orogeny. matchto the velocity historycomputed in this paper.It is true Like Gries [1983], Livaccari [1991] inferred a rapid that the velocity history is more strongly peaked than the changein shorteningdirection during the Laramideorogeny, historyof contactarea. However, this is an expectedconse- from azimuths of 680-76 ø before 65 Ma to azimuths of 20 ø- quenceof the nonlinearrheology of continentallithosphere. If 28ø afterward.Livaccari interpretedthe patternhe saw as the total horizontalforce transmission was proportionalto contact resultof successivecollapse of first the northernand then the area, then all velocities and strain rates within North America southernpans of a westerncordillera with an elevated,weak, would be expectedto be proportionalto somepower of the extendingcore. Thus he was extendingthe mechanicalideas contact area. of Burchfieland Davis [1975]. Actually,any modelin which The directionsof forelandvelocity and shortening(Figure the lateral force for the Laramide orogeny is transmitted 7) are also reasonablyconsistent with the directionsof oce- througha high, weak cordillerais also implicitly a model in anic plate motion after 75 Ma. (Of course,there are possible which this force is generatedby North America/Farallonor circumstances,such as anisotropyand heterogeneity,which North America/Kulacoupling in a coastalsubduction zone. could causethe surfacevelocity to divergefrom the direction (If it werenot, the cordillerawould be unconfinedon thewest of basaldrag, so this coincidenceis not proof.) The foreland andwould collapserapidly in an enormouslandslide.) Fortu- velocity and shorteningazimuth of 40ø that was maintained nately,while the elevationhistory of the cordillerais elusive during 75-50 Ma is very close to the predictedazimuth of and debatable,the history of relativeplate motionswith re- Kula plate motion. During 50-35 Ma, however,the azimuth spectto North America is known [e.g., durdy, 1984; Enge- of forelandvelocity bettermatches the azimuthof the Faral- bretson et al., 1985]. If we temporarily postpone lon plate. There is a model alreadypublished that (implicitly) considerationof horizontalsubduction, the othermost plausi- makes a similar prediction: Engebretsonet al. [1985] at- ble reasonsfor increasedtransmission of horizontalcompres- temptedto explain northwardtransport of coastalterranes by sion across the subduction zone would be a decrease in trench their "southernoption" model, in which the Kula plate was depthor an increasein viscousshear coupling. The firstfactor subductingbeneath the United Statesfrom 85-59 Ma. During was undoubtedlypresent, since the agesof the pansof the re- 59-45 Ma (in their model), a long Kula-Farallontransform constructedoceanic plates that were enteringthe trench be- was subductedat the coast and moved graduallyunder the came graduallyless throughoutthe Tertiary. However, this Rocky Mountains,changing the identity and azimuthof the youngingwas continuous,and it is difficult to see how it flat slab from Kula to Farallon (Figure 9). The transitionin could have given rise to an orogenywith a well-definedbe- velocity azimuth that I see around 50 Ma is an excellent ginningand end. Therefore, in thisclass of modelsit is neces- matchto their suggestion. saryto appealto increasedviscous shear stress transmission In conclusion,the bestexplanation for Cretaceous-Tertiary acrossthe subductionthrust, probablycaused by increased orogeniesin the westernUnited Statesseems to be a compos- relative plate velocity. Comparingthe normalizedvelocity ite one. In the early Sevier orogeny(before 75 Ma), driving historiesfrom Engebretsonet al. [1985] in Figure 8, we see forcewas transmittedthrough a weak, elevatedcordillera, as that increased Farallon/North America velocity at 70 Ma arguedby Burchfiel and Davis [1975] and Livaccari [1991]. comes a little too late to explain the beginning of the Perhapscoast-parallel shear componentswere absorbedin Laramideorogeny, and increasedKula/North America veloc- coastalstrike-slip systems and cordillerandeformation [e.g., ity comes20 m.y. too late. There is alsoa problemabout the Tikoffand de Saint Blanquat, 1997], and only normal stresses end of the Laramide, since both Farallon and Kula velocities perpendicularto the cordillerawere transmittedinland (Figure with respectto North America peakedat 55-45 Ma, a time 9). However,the Laramideorogeny was a distinctand differ- whenthe orogenywas waning.Using a differentplate recon- ent event,during which the bulk of the driving forcewas ap- structionscheme (imposing Antarctic deformationand ig- plied to the baseof North Americaby horizontallysubducting noringhotspots), Jurdy [1984] proposeda slightlydifferent slabsof oceaniclithosphere. Consequently, the azimuth of history of Farallon/NorthAmerica relative velocities.This shorteningwas controlled by the azimuth of relative plate alternatevelocity history has velocitiesexceeding 100 mm motion, while the rate of shorteningwas controlledby the yr-1 only at 60 Ma andpeaking at 50Ma. Thus it alsopeaks changingarea of contactbetween plates. More tentatively,it too late to correlatewell with my historyof velocity. appearsthat the change in stressdirections that occurred The remainingproposal is that the Laramideorogeny was around 50 Ma may have marked the passageof an active causedby increasedshear coupling between North America Kula-Farallon transform within the horizontal subducting and the Farallon and/or Kula plate(s) causedby increased slabs. contact area during an episode of horizontal subduction. 4. Conclusions Horizontal subductionwas originally invoked to explain the inland migration of the volcanic arc [Snyder et al., 1976; A new paleotectonicand palinspasticmethod was applied Dickinsonand Snyder, 1978], and this gives an independent to the Rocky Mountain foreland region, using reasonably BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY - 7 91
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120 • 110 . 120 • 110 • Fisure 9. Proposedhisto• of plateconfigurations based on thisstudy, on platereconstructions of En•ebret- ßon et •1. []985], and on volcanicarc frontsfrom Dickinson•nd Snyder[]978] and Urr•ti•-F•c•chi []986]. Shadin[shows the forearcarea of contactbc•ccn the baseof No•h Americaand the topsof sub- ductin[ slabs. Dia[onai pa•crn showsre[ions of oro[cny or extension.At 80 Ma, obliquesubduction of the Kula platecaused paditioncd deformation: coastal terranos moved noah, while the volcanichi[bland trans- miEcdonly cas•ard pressureto the CanadianRocky Mountainsand Sovietoro[cn. By 65 Ma, horizontal subductJonunder a hu[c areadrove the LaramJdcoro[cny (and the contJnuJn[Canadian Rock• Mountains- Sovietoro[cny). About 50 Ma, a subductedKul•Farailon transformbound• may have passedunder the forearc,chan[Jn[ the directionof basaltractions. The laststale of the LaramJdcoro[cny, around 40 Ma, was drivenby horizontalsubduction oft he F•allon slabin the southwesternUnited Statesand noahwest•exico. 792 BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY completedata from the literaturethrough publicationyear eration,the elapsedtime is held constant,giving a Gaussian 1997. Velocities, strain rates, and fault-slip historieswere distributionfor the rate.) computedback to 85 Ma in 5 m.y. time steps.As expected, Finally,I assumethat there is a probability0 < q• < 1 that the solution displaysthe last stagesof the Sevier orogeny eachdatum is relevantto thetime step. Normally, q• is unity during 85-50 Ma, the Laramide orogenyduring 75-35 Ma, becauseirrelevant data are simply omitted from the list andNeogene extension spreading southward after 50 Ma. The k = 1,...,K. However,in the casesof certainpaleostress data, solution also provides estimatedslip historiesof each fault fractionalq• are required. The naturallogarithm of the den- duringeach event; 64% of theseare differentfrom the simple sity of the joint probabilitythat the velocitymodel matches constant-ratehistories that were assumedat the beginningof all the relevantrates is thenformed from the individualprob- the computation.However, the most interestingresult is the abilitydensities (O) as estimateof the horizontalvelocity field overtime. Forelandvelocities peaked at 1.5mm yr -1 during60-55 S--In [cI)(p/•= r/• = •qk ln[cI)(p/,= r/• )]= Ma, when the Colorado Plateau was rotating clockwise k=l aroundan Euler pole in the northernpart of Texas, as pre- dictedby Hamilton [1981]. The computedtotal rotationof the plateauabout a vertical axis since85 Ma is only 3 ø, which is - k=lY" q• 2cr•+ ln(cr•)+ In2xf• . (1) less than claimed by some authorsworking with paleomag- netic poles of older rocksbut is the sameas the lower limit I referto this quantityS as the "score"of the velocitysolu- quotedby Bryan and Gordon [ 1990]. tion,which is to be maximized.That is, thesum of squaresof During the Laramideorogeny in the foreland,the mean the relevantprediction errors (each divided by the varianceof shorteningazimuth was steadyat 40ø until 50 Ma and then the correspondingrate) is to be minimized. rotated clockwise to 55ø; the counterclockwisestress rotation On the surfaceof a sphericalplanet with radiusR, I define proposedby Gries [1983] and by Livaccari [1991] did not a coordinatesystem of colatitude(0) measuredsouthward occur.The mean velocity directions(with respectto eastern from the north pole and longitude(q•) measuredeastward North America)are similarto the knownrelative velocities of fromthe primemeridian. The unknownsin eachvelocity so- the Kula and Farallonplates, which were subductingbeneath lutionare the horizontal0 componentsand • componentsof North America. However, the history of velocity and strain thevelocity of thesurface. The predicted rates pk canbe ex- rate magnitudesdoes not matchwell with the historiesof the pressedas a linearcombination of the velocitycomponents v relativevelocities of theseoceanic plates. Instead, the ratesof (southward)and w (eastward)at eachof thed nodesof a finite Laramide deformation were more closely related to the elementgrid: changingarea of contactbetween North Americaand the oce- d anic plates,confirming that the most likely driving mecha- Pl•= cl• + Z (fuvj + gini wj) (2) j=l nism for the orogeny was the transmissionof basal shear stressduring an episodeof horizontalsubduction. With this linearrelation, S is a quadraticform in the nodal- velocity-componentvalues vj andwj, soit ismaximized by Appendix A' Algorithm findingthe single stationarypoint in multi dimensionalve- locity spacewhere A1. Least Squares Formalism &S &S •=0=•; i=l,...,d. (3) I assumethat all geologicand paleomagnetic data that con- straindisplacement, strain, and/or rotation in a particulartime Algebraically,this leadsto a 2J x 2J linearsystem, which stephave been transformed to scalarrate estimates r[. (The can be thoughtof as being partitionedinto four submatrices subscriptk = 1,...,K identifiesthe datum,and the superscript timestwo subvectorsequaling two subvectors' n identifiesthe time step.) Let the correspondingscalar rate predictionsderived from the velocity field of the finite ele- mentmodel in a particulartimestep be called p•. (In there- mainder of this section, I will discuss only a single I••]D0. jL•y-yj= [-•1 (4) computationaltime step, so the superscriptn will be sup- usingthe abbreviations pressed.)I furtherassume that each scalar rate rk hasan un- certaintythat can be approximatedby a Gaussianprobability distributionwith standarddeviation crk andthat the errorsin theserates are independent.(A Gaussianprobability distri- (5) bution is reasonablyappropriate for the numeratorin each rate, which is an amountof strainor displacement.It is not appropriatefor the denominator,the elapsed time, which typicallycomes with a hardupper limit basedon cross-cutting relationsbut without any lower limit. Becauseof this, I will k=1 iteratethe solutionof the entire history in a way that allows (6) additional degrees of freedom in each rate history; this Fi:•gla(rl•-cl•) method is described below. However, in each individual it- k=1 BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY '/93
A2. Boundary Conditions In this notation,the superscriptjon the vectornodal function The equationsstated above are singularin the absenceof •xj ornodal function component GxJ,y identifies thenode boundaryconditions. Some edge(s) of the model must be thathas unit velocity (all othernodes having zero velocity in fixed (or movedin a predeterminedway) to prov;.dea veloc- this particularnodal function). Subscri,•t x- 1 indicatesthe ity referenceframe. I replacethe row equationsthat statethat nodalfunction associated with unit southward velocity v; sub- S is stationarywith respectto variationsin thesenodal veloc- script x - 2 indicatesthe nodal functionassociated with unit ity componentswith simplerequations stating the desired eastwardvelocity w. Subscripty- 1 indicatesthe southward valuesof thesecomponents. orO component of the vector nodal function (•xj , andsub- This methodpermits only velocity boundaryconditions scripty - 2 indicatesthe eastward or qbcomponent. not stressboundary conditions. Along eachaxis (O or •b),one Thecoefficients of thelinear system are augmented by boundaryshould be constrainedand one left free (because integratingstrain information to find velocitiesis like solving first-orderdifferential equations). When one margin of a con- 1,1+ CSC0 1,2+• 1,1+ tinent is facing a subductionzone, it is best to leave that boundaryfree.
W L cot067G['1• GJ•,•+ •1,1 •-- + A3. A Priori or Pseudo Data R t=l An essentialcontext for all the geologicdata showing lo- tttae2csc0 ø•i;2+tan,0/csc0 •2+t•0) cally intensestraining is that they shouldbe overlaidon a set -'-//cscO •,•+ 1,2 1,2 of a prioridata (or "pseudodata") stating that in otherplaces [2( • • tan0)( • • tan0 the strainrate is closeto zero.An appropriateformalism is to I•CSC0•[,14 •1,2 G[2 •( •' •' G') assigna zerotarget strain rate, with a statisticaluncertainty. A 2,1+ csc0 2,2 largerstandard deviation should be attachedto this null hy- pothesisin complexregions where unknownfaults and oro- cot0 G ,1+ • + genicphases might havebeen buried or overlooked. W L To implementthese constraints, the scoreS of anyvelocity lit at solution,which is to be optimized,is first augmentedby a 2 csc06T•1'2 + csc02,2 + 2,1+ term 8• tanOil 8• tanO AS=-WJJ ø6020+&OO&44 +&•q• +&a2q• dU (7) 8• 80tan 0•CSC 08• 2,1 4 80 2,2 tanO2,2 U 3/2 2,1+C$C0 6•2'1 2,2 + 2,1+ where W is the weightcoefficient for all pseudodata (in unitsof m'2),la is the standard deviation ofthe (nominally W L zero)strain rate (per second),and U is the areawithout active R faults.In practice,any finite elementthat has no active fault •tt at G2,1)•cscO67G2,2+G2J,11+ crossingit is a part of U. Weightingby areais usedto make 2Icsc0-•+ i tan0•,t • j tan0) thisnew term roughly independent of localvariations in finite CSC00•2'1461•'2 G•'2- csc0 2,1 + 2,2 2,2 elementsize. However, an appropriatevalue for the weight d4 dO tan0 d4 dO tan0 coefficientW is approximatelythe inverseof the meanof the areasof all finite elements,as explained below. (9) This term has the effectof causingunfaulted areas to be- haveas Newtonianviscous sheets of lithosphere.The algo- rithmwill adjustthe velocityin eachtime stepto minimize where• = 1,...,L identifiesthe "nonfaulting"elements com- the area integralof squaredstrain rates for theseelements; prisingU, withindividual areas a t . In practice,area integrals this is exactlythe resultone obtains by beginningfrom the withineach element are performed numerically, using seven momentumequation (in the absenceof horizontalforces), Gausspoints with associated weights [Zienkiewicz, 1971 ]. adoptinga linearrheology, and solvingfor velocitywith in-
homogeneousboundary conditions. ß The 2 x 2 strainrate tensor • on the sphericalsurface is A4. Use of Balanced Cross Sections calculatedby summingspatial derivatives of the nodalfunc- Many structuralgeologists publish restored cross sections tions.The nodal functions that I usewere introduced by Kong fromwhich they estimate the amount of shorteningor exten- and Bird [1995] and shownto satisfythe requirementsof sionalong the line of section.Dividing the amountof exten- horizontality,continuity, and completeness: sion(compression is negative extension) by thetime available givesthe relative velocity component (along the line of sec- s ' tion)for theendpoints, which is therate estimate rk . If the (8) endpointsof the section are marked by positionvectors bk =r i, G2(O,4)lLwj]' and d•, then c• = 0 and 794 BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY
f/q.=q1,1j (b/,)cos - ?(b_ k ) - qj1,1 (d/,)cos-' ?(d/, ( boo= bN$,bo4 = bSE,b44 = hEW) as a one-subscriptvector (bin; m= 1,2,3 ), permiRingme to write the covarianceof G1,2j (b/,)sin- r(b • ) + Gj (•)sin?(•/,) strainrates as a 3x3 matrix. If all the activefault segments (10) that cut (even pa•ay) throughone finite elementare num- gkj= G2,1j ([•)cos?(•) - G2,1j (•)cos bered z = 1,...,Z, then I expressthe strainrate vectorin the elementas a linearcombination of theirscalar slip rates Sz: G2,2j ([•k)sin?(/•/,) + G2,2j (•)sin?(•/,) z bm= ZHzmsz;m= 1,2,3. (11) where7(b•) and 7(dk) arethe forward azimuths at eachend of the directedgreat circle arc b•-• d• (each measured clockwisefrom north). The covariancematrix of the strainrate componentsis com- posedof •o pa•s: the continuumcompliance common to all A5. Use of Fault-Slip Data pa•s of the lithosphere(see section A3) andthe terms arising fromthe standard deviations 6s z of thescalar slip rates Sz: A large fraction of the available data concernoffsets on faults. While offset is actually a vector, I use only the larger of the dip-slip or strike-slipcomponents and treat this as a scalar datum. This is becausethe strike-slipcomponent of 0, dominantlydip slip faults is rarely known, while any dip-slip k-2/3 0 4/3 on strike-slipfaults is irrelevantto relativehorizontal veloci- To find H z (the pa•ial derivativeof elementstrain rate ties. After divisionby the time available,this scalaroffset be- with respectto slip rate of one activefault), I makethe sim- comesa scalarrelative velocity component across the fault. pli•ing restrictionthat no nodelies exactlyon a fault. Also,I When a fault is long enough to cross several finite ele- slightlystraighten the tracesof any hult segmentthat crosses ments,I imposethe sameslip and slip rate in eachelement. In the sameelement bounda• more than once.Then, eachhult the caseof rigid microplatetectonics, where each fault con- segment(with its prQectedextensions, if necessa•) must nects to other faults at triple junctions, this method is rea- separateone node of the elementfrom the other•o. Let uz sonablyaccurate. The other end-memberis the casewhere no bethe index number of theisolated node. If nodeu z is onthe faults connect, but all terminate within the domain. In that right sideof the hult segment(when lookingalong its azi- case,each fault might be expected(on the basisof crackthe- muth D, measuredclockwise from noah), then I definethe ory for linear materials)to have an ellipsoidalprofile of slip variable•z as +1; othe•ise, it is -1. Let r z be the fraction versuslength. Such "elliptical" faultswould have a mean slip of the widthof the elementthat is cut by the fault segment: which is only 79% (x/4) of their maximum slip. Thus my 0 'C sin7z + cos7z , •', g'rnem+ •Hzmpz-•m (16) m=l3 / z=l 6•2,1COS2' z oT•,•sin7_•_z + • • + sin7z+ where •s z is the standarddeviation of eachslip rate accord- rZ -- qzrz l • • sinO• sin0•0 , ing to the input data.Then, to find a local solutionthat hasall •2,2• G• sin?z + G2, • 2cOS?z hult ratesas closeas possibleto their goals,while the contin- COS•z - • t•O uum strain rate is close to zero and the total strain rate is cor- sinrz •2,2u• cOSrzG• sin•z +G2, u•• cOSrz rect, find the stationa• point of S' with respectto variations sinO • sinO t•O in the Pz, the .c andthe Cmin turn,leading to a linear system.The sliprate p• thatis finallyrecorded (for hult k in time step n) is the averageof the rates Pz in all the ele- (13b) ments the fault passesthrough, where the averagingweights The nextstep i•s to findthe threepositive eigenvalues (Ah; arethe segmentlengths Pez ß h = 1,2,3) of V and their correspondingunit eigenvectors The method describedhere for incorporatinghults effec- (Ahm). Theseeigenvectors indicate strain rate patternsthat tively multiplies each known hult offset into roughly 1•2 are statisticallyuncorrelated; they have targetamplitudes of scalardata per finite elementcrossed by the trace.In addition, •m Ahm and standarddeviations of X/Ah, respectively. 1•2 pseudodata per elementcrossed are usedto expressthe a Eachof the threetargets is now imposedas a scalardatum in priori assumptionof continuumstiffness. Therefore, for pari• the global systemof equations.The correspondingcoeffi- betweenhulting elementsand nonhulting elements,the sug- cients of the nodal velocities are gestedvalue of W is the inverse of the mean area of finite elements. 1,1 1,1 + 1,2 1,2,Ah2 A6. Use of Paleomagnetic Data c?0 tan0 61GjGjI I assumethat the paleomagneticdata set is restrictedto fo.=I Ahl+ csc06¾ sitesthat includesome geologic or geochemicalindication of TMAh3 the orientationof the paleohorizontalplane at the time of +csc0 1,2+tan 0 magnetizationand that have beenproperly corrected for local structure.It is also best to excludecertain sedimentaryrocks 2,1 2,1 + 2,2 2,•2Ah 2 that are known to be especiallyprone to postmagnetization c?0Ahl + csc0c¾ c?0 tan0 compaction,which can produce a nontectonicinclination 6T]jGjI anomaly. gkj=• 6T]j The interpretationof paleomagneticinclination and decli- 2,1Ah 3 nation data in terms of north-southdisplacement and rotation tan0 +[csc0 2,2+ requiresthe definition of a referencepolar wander path. It is necessaryto use the samevelocity referenceframe for polar (14) wanderand for velocity boundaryconditions. The inclinationof a sampleyields its magneticpaleolati- (Note that k equalsh plusa constantthat indicateshow many tude accordingto a simple dipole model for the paleofield. datahave previously been incorporated into the system.) This is convertedto a paleolatitudeanomaly by comparison Once the global velocity solutionhas been found for any with the polar wander path. Multiplying this paleolatitude time step,it is necessaryto do a localoptimization calculation anomalyby the radiusof the planetand dividing by the length within each faulting elementto find the predicted(model) of time available results in the mean velocity componentin ratesPz at whicheach fault ( z = 1,...,Z ) is slipping,as well asthe residual strain rate •cm whichis dueto deformationof the paleodirectionof magneticsouth (at the time of magneti- the continuum around the faults. The total strain rate of the zation). This becomesthe rate estimate r/•. For comparison, element must be the sum of the continuum and the fault con- the model predictionis formedusing tributions: f lq'= Gj 1,1 ½os•,- G1,2j sin 7 z (17) ern+ Y',Hzmpz = •rn' (15) g•,j- G2,1½øs]/-j G 2,2j sin • andc• = 0, wherethe nodal functions GxJ,y are evaluated at This problemis differentfrom the globalproblem because the datum location and y is the azimuth (measuredclock- the •rn vectoris known.Because of this constraint,it is an wise) of north pointing paleomagneticdeclinations (at the algebraicconvenience to usethe Lagrangemultiplier method time of magnetization)with respectto present geographic with threetemporary weight variables ( g'l, g'2,and g'3). De- north (in the referenceframe of the undeformedpart of the fine the local score(in oneelement) that is to be optimizedas continent)at the datumlocation. 796 BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY While the magneticdeclination of a sampleis clearly re- is geologicallyrecorded as the strikeof igneousdikes or other lated to its historyof rotationabout a local verticalaxis, the verticalveins that breakthrough any laterallyhomogeneous, relationshipis nonuniqueand model-dependent.In general, isotropicrock. In some cases,a populationof faults with the declinationanomaly can only be convertedto a vertical- slickensidescan be statisticallyanalyzed to determinethe axis rotationif the displacementof the sampleis negligibleor stressdirection [e.g., Gephart,1990]. approximatelyknown. Here I assumethat these questions To relate this informationabout stressto my kinematic have beenresolved by the assumptionor approximationmost model,I approximatethe lithosphere as horizontally isotropic, appropriateto the particularproblem and that estimatednet so that the principaldirections of stressare the sameas the rotationsabout the local vertical axis (A 7) are available as principaldirections of strainrate. There may be an errorof up inputdata. I alsoassume that the senseof largerotations has to 35ø associatedwith this assumption;even so, I believethat beendecided in advancebased on regionaltectonics. the solutionswill typically be more accurateand reasonable These values determine the average rotation rates thanthose that ignorestress data. rk = CO(kdatum) = Ay/t, where A7 is the vertical-axis rotation Oncewe knowthe azimuth of •31h,we canuse this as the in goingfrom pastto present(counterclockwise positive) and direction of a new local horizontal axis d and also define a t is the age of the magnetization.The interpretationof these perpendicularhorizontal axis /3' (right-handed;d x/3'= ;). rotationsrequires some assumptions about the shapeand the In thesecoordinates, the requirement that d is themost com- stiffnessof the bodiescarrying the remnantmagnetization; I pressivehorizontal principal strain rate direction can be stated assumethat the outcropsselected for paleomagneticsampling in twoparts: ha? = 0 and•aa < •flfl' In termsof theglobal were rigid inclusionsof equidimensionalshape embedded in coordinatesystem, the first becomes a deformingcontinuum. Therefore the rotationrate that the magnetizationwould be predictedto recordis •o•cos(2r) +o•oo -2 b•sin(2r) =o. (20) In termsof derivativesof velocity,this is Pk--w? ødel) z 2R •+•-csctan0 c*0 . (18) Consequently,the modelpredictions are formed using c•: = 0 •+ and 2R c¾W0 t0' cos(2r)+ (21) f/q= •-tanO 1,2+ dO1,2_CSC 0 1,1 (c•dO csc0 c•d• tan0 v )sin(2r)} =0, (19) sothe coefficientsof the linearsystem can be computedfrom 2,2+ 2,2-csc0 2,1 the factors gkJ=• tan0 RO as/' The methodsdescribed in sectionsA3 and A5 effectively multiply the continuum-stiffnessassumption into one scalar datum per finite elementand multiply each fault offset into 1-2 scalardata per finite elementtraversed. If the finite ele- mentgrid has many elements,the paleomagneticdata will re- quire a similar weighting in order to avoid being overwhelmedin the global solution.Accordingly, both f and g functionsof (17) and (19) shouldbe multipliedby a dimen- csc0 2,1+ 2,2 2,2 cos(27)+ sionlessweight factor P which is commonto all paleomag- 1 d4 dO t-•nnb netic data.A suggestedvalue for P is the squareroot of twice (22) the number of finite elements, which is the mean number of 2,1_ csc0 2,•2 2,1sin(27) elementstraversed by a fault crossingthe whole grid. Then, a dO d4 tan0 paleomagneticdatum indicating an exoticterrane will receive the sameprocedural weight as a fault-offsetdatum on the ter- if we use c•:= 0 anda rateestimate r•: = 0. The difficultyis rane-boundingfault, and the outcome will depend on the in decidingwhat standarddeviation cr k to associatewith this relative values and their uncertainties. Different values of P constraintha? = 0, sincewe have transformed the constraint can be usedto investigatesolutions in which paleomagnetic from one concerningan angleto one concerninga strainrate dataare eitherless or moreprominent. component.When the calculationis first startingand there are no strain ratesknown as yet, a purely arbitrarysmall strain A7. Use of Stress Directions rateuncertainty ( • ) mustbe assignedas cry:.However, when One principalstress direction must alwaysbe perpendicu- strainrate estimatesare availablefrom a previoustime step lar to the free surfaceof the Earth or approximatelyvertical. (or from a previousiteration of the currenttime step),it is Thus the orientationof the stresstensor is describedby the bettertouse cry: =2(87)(b• +(boo - b• )2/4) •/2 , wherethe azimuth (7; measuredclockwise from north) of the most symbol g7 indicatesthe standarddeviation (in radians)of compressivehorizontal principal stress (•lh)' Thisdirection theazimuth ¾ of thedirection 31h' Thissuggests that the ve- BIRD: KINEMATIC HISTORY OF THE LARAMIDE OROGENY 797 locity solutionshould be iteratedwithin eachtime step;in this as a result of solving the momentumequation. In practice, projectI have usedfour iterationsper time step. there are rarely more than a dozen relevant paleostressindi- The secondrequirement was the inequalitybaa < catorsin any given time step.A relatedproblem is that if each During each iterationof the solution,I evaluatethe strain paleostressindicator were only comparedto the strain rate ratesbaa and b?? to seeif thisis true. If not,then in future tensorin a single finite element,then its influenceon the so- iterationsI impose an additionalconstraint b?? = baa+ lution would decreaseas the finite elementgrid was refined. I where • is a small (positive) strain rate differencewhich attempt to solve both problemsby interpolatingpaleostress mustbe arbitrarilychosen. In termsof the globalcoordinates, directions(with associateduncertainties) for every finite ele- ment in the grid, basedon the relevantpaleostress data. The (• - •oo)COS(2r)+2bo• sin(2?) =•. (23) interpolationis by nonparametricstatistics based on the spa- tial autocorrelationof the presentstress field as represented This can be expressedin termsof velocitycomponents as by the World StressMap [Zoback, 1992]. The interpolation methodis given by Bird and Li [1996]; I use the simplerof their two methodsin which the dataare consideredindepend- (24) ent, becausethis permitsthe weightingof the original data by theirrelevance ( q• ) valuesduring the interpolation. In caseswhere stress-directiondata are sparse,it may be desirableor necessaryto use active fault segmentsas addi- so the coefficientsof the linear systemcan be computedfrom tional indicators,assuming •lh to be perpendicularto the factors thrusts, etc. Only the first phase of movement on a fault should be used to indicate stress,because in later times the fault is an inheritedplane of weakness. CSC01,2 -I 1,1 1,1.COS(2y)+ •Y• tan0 < cT•lGl 6T•j / A8. Integration Over Time I use the "predictor/corrector"method of time integration, cscO 1,!+ 1,2 1,2,sin(2y) as in earlier forwarddynamic models of the historyof North c¾ c?O tanO America [Bird, 1988; 1992]. Each time step beginswith an explicit "prediction"of new node locations.Using these,all nodal function derivatives,coefficients, and velocities are re- csc0 2,2+ 2,1 2,1.cos(2y)+ computedfor the sametime step.Then, the "predicted"ve- 1 (25) locity for that time step is "corrected"by addingone half of gkj= •' the (vector)change between the solutions.The nodelocations cscO 2,1+ 2,.•_2 2,2.sin(2y) are correctedaccordingly. Bird [1989] presentedstudies of •{b c*O tanO the accuracyof this method;for practicalpurposes, it is suffi- . cient to usetime stepsof-l-5 m.y. and ck = 0 if we createa new rateestimate r• = •. The same value of • is also usedto set the standarddeviation for this A9. Iterative Revision of Rate Histories constraintas cr• = (0.83)•, so that the Gaussiandistribution Becausethe events in geologic history that can be dated (which my methodforces me to use) will best approximate are not alwaysthose we would choose,many dataabout strain the desiredHeaviside distribution near the origin. or displacementcome with loosetime windows,bracketing Paleostressdata are different from structuraland paleo- but not specifyingthe true durationof deformation.However, magneticdata becausethey are not integral constraintsover adjacentdata with better constraintsshould cause strain rates time but are momentarysamples. A thin igneousdike may to rise in the correctperiod. If the model-predictedrate for form in a day. Thereforeit is necessaryto distinguishbetween any datumis largerthan the tentativegoal rate, this is proba- two types of publishedreferences on paleostress.The more bly an indicationthat the goal ratesfor that datumshould be desirabletype summarizesmany paleostressindicators of dif- revisedto permit more rapid deformationin a portionof the ferentages to showthat •'lh has remainedconstant over time window. sometime intervalfrom age t2 to age q. Thesedata should Mymethod isto assign newtarget rates(r/• r• (for datum k be applied in each time step of that interval in eachtime step n), basedon the predicted rates p] fromthe ( t1/At < n _ References Avouac,J. P., and P. Tapponnier,Kinematic model magneticpoles, Geophys. Res. Lett., 17, 1501- ics of the Laramide orogeny, in Laramide of activedeformation in CentralAsia, Geophys. 1504, 1990. Folding Associated with Basement Block Res. Lett., 20, 895-898, 1993. Burchfiel, B. C., and G. A. 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