JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 102, NO. B12, PAGES 27,553-27,573, DECEMBER
10, 1997
Least squares restoration of Tertiary thrust sheets
in map view, Tajik depression, central Asia
O!ivier Bourgeois,Peter Robert Cobbold, Delphine Rouby,
and Jean-Charles Thomas 1
GdoscicnccsRennes, CNRS (UPR 4661), Rennes, France
Vassili
Shein
All RussiaResearchGeologicalOil Institute(VNIGNI), Moscow, Russia
Abstract. The Tajik depression,locatedwest of the Pamirsand southof the Tien Shan, is a
compressionalintermontanebasin,boundedby basementoverthrustsand filled with Mesozoic and
Cenozoicsediments.The internalstructureis typical of a thin-skinnedtbld and thrustbelt.
Kinematicdata available in the literaturesuggestthat indentationof the Pamirsinto Asia durino
the Cenozoiccollision of India and Asia hasbeenaccommodatedin variousways within the
depression,includingwestwardextrusion,thickeningcombinedwith wrenchingalongN-S tblds
and thrusts,and counterclockwise
rotations.Thesevariousdetbrmationprocesses
can be analyzed
and quantifiedby reconstruction
of thepredeformedstateof the depression.
However,the combination of thrusting,wrenchingandblockrotationsimpliesa nonplanedeformation,whichcannotbe
restoredproperlyusingbalancedcrosssectionsalone.We havethereforedevelopeda numerical
methodfor restorationof stratigraphicsurfaces,designedfor regionsof nonplanecompressional
tectonics.The deformedregionis represented
in map view as a mosaicof fault-boundedblocks,
overlappingeachotheralongthe faults.Blocksare separatelyunfoldedand thennumerically
packedtogetherby leastsquaresminimizationof overlaps,yieldingfieldsof finite horizontal
translationsand rotationsaboutverticalaxes.To analyzethe deformationpostdatingthe collision
of India andAsia, we have restoreda stratigraphicsurfaceat the baseof the Cenozoic:First, in
order to testthe numericalmethod,we restoreda map that had previouslybeenrestoredby a purely
manualmethod.Restorationof a secondmap, drawnfrom newly availablesubsurfacedata, leadsto
geometricalinconsistencies:
overlapsand gaps,whichcannotbe reduced,remainin the restored
state.After correctionof theseinconsistencies,
restorationyields a complexmodeof deformation
for the depression.
Individualthrustsliceshaveundergonecounterclockwise
rotationsaboutvertical axes,the magnitudesincreasingfrom west to east,up to a maximum of 40 ø near the Pamirs.
Horizontal shorteningis 150 km (35%) in the centerof the depressionand 240 km (85%) in the
northeasternpart, betweenthe Pamirsand the Tien Shan.Strike slip motions,commonlynot
revealedby balancingcrosssections,are associated
with thrustingon faultsstrikingN-S. Independentpaleomagnetic
dataandslip directionson small-scalefaultsprovidepositivecheckson the
results of our restoration.
1. Introduction
India collided with the southernmargin of Asia 50 Myr ago
[Patfiat a•d Achache, 1984] and, since then, some 2500 km of
shorteninghas occurred within both continents[Dewey eta!.,
1989; Molnar and Tapponnier, 1975; Minster and Jordan,
1978]. In Asia, shortening has been accommodated within a
triangular area about 3000 km large and 4000 km long,
bounded by the Himalayan belt in the south and by the Tien
Shan-Baikal wrench zone in the northwest [Cobbold and
Davv, 1988] (Figure 1).
Various mechanisms of accommodation have been recognized: crustal thickening within the Tibet and the Himalayas
[Dett'ev eta!., 1989; England and McKenz,ie, 1982], subduction of continentalcrust beneaththe Pamirs [Hamburger et al.,
1992; Burt, ta• a•d Molnar,
1993], eastward lateral extrusion
of southernAsia and southernChina [Molnar and Tapponnier,
1975: Tappo•ier eta!., 1982, 1986] and westward lateral
extrusion of Afghanistan [Tapponnier et al., 1981].
Associated with lateral extrusion, rotations about vertical axes
of blocks bounded by strike slip faults, domino style, have
been suggested [Cobbold and Davy, 1988; England and
Mo!•ar, 1990]. The relative partitioning between thickening,
subduction,
extrusion
and rotation
has been much
discussed
Nowat Laboratoire
de Gdophysique
Interneet Tectonophysique,[Cobbold aud Davy, 1988; Davy and Cobbold, 1988; Peltzer
CNRS (UMR 5559), Grenoble, France.
(t•d Tappo,•ier, 1988; Dewey eta!., 1989; England and
Mo!•ar, 1990; A•,ouac and Tapponnier, 1993; Buttman and
Copyright 1997 by the American GeophysicalUnion.
Papernutuber97JB02477.
() 148-0227/97/97J B-02477509.00
Moh•ar, 1993; Houseman and England, 1993].
At the southern tip of the Tien Shan-Baikal wrench zone, to
the west of the Pamirs, is the Tajik depression, a compres-
27,553
27,554
BOURGEOIS
ET AL.:RESTORATION
OFTHE TAJIKDEPRESSION
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Figure 1. Structuralcontext of the Tajik depression(modified after Cobbold and Davy [1988]).
sional sedimentary basin (Figure 1). On the basis of paleo-
magnetic and fault-slip data, together with a manual
reconstruction of the pre-Tertiary shape of the depression,
Th.omas et al. [i995,
Finally. we compare the results of these restorations with
paleomagneticrotations, shorteningdirections and kinematic
models proposed by various authors.
1996a, b] concluded that Tertiary
deformation of the depressionwas associatedwith indentation
of the Pamirs into the Asian continent during the India-Asia
collision. For Th,omas et al., the indentation has been accommodated by (1)westward displacement of the depression
between a right-lateral wrench zone within the Tien Shan to
the north and a left-lateral wrench zone to the south, (2)leftlateral wrenching and thrusting along N-S trending ranges
2. Geological Setting
The Tajik depression is a lozenge-shaped intermontane
basin, situated west of the Pamirs, on the eastern border of the
Turan platform (Figure 1). To the east, the depression is
separatedfrom the PaNits by the Darvaz-Karakul left-lateral
strike slip fault (Figure 2a). At the northernedge of the depreswithin the depression and (3) counterclockwise block sion is the southernmost belt of the Tien Shan range, the
rotations within the depression.
Gissar belt, which is a major right-lateral strike slip zone
Recently, more detailed subsurfacedata for the Tajik depres- [Babaev. 1975; Trifonov, 1978]. Between the Pamirs and the
sion have become
available
from the All Russia Research
Tien Shan, the northeasterncorner of the Tajik depressionhas
Geological Oil Institute (VNIGNI). We have used these new been strongly shortened,forming the Peter the First range, a
fold and thrust belt [Skobolev, 1977; Leith and Alvarez, 1985;
data to numerically restore the depression to its pre-Tertiary
shape, in order to test the model proposed by Thomas et al.
Hamburger et al., 1992]. At its western edge the depression
[ 1996b].
has been overthrustby the SW Gissar range, which separatesit
Within the depression, the combination of thrusting, fi'om the Turan platform. Finally, the Alburz fault forms the
wrenching and block rotations implies a nonplane deforma- southern edge of the depression.
tion, which cannot be restored properly using balanced cross
Within the depressionis a seriesof anticlinal ranges,striksections alone. However, a method of restoration of nonplane ing nearly N-S (Figure 2a). Toward the north, these ranges
deformation, based on least squaresnumerical fitting of fault- progressively swing into parallelism with the Gissar belt.
bounded blocks in map view, has been developed for regions Most anticlinal ranges are associatedwith thrust faults which
of strike slip faulting [Audibert, 1991] and normal faulting root into a major decollement level in Upper Jurassic
[Rottb3, et al., 1993a, b, 1996; Rouby, 1995; Rouby and
evaporites (Figure 2b) [Gubin, 1960; Babaev, 1975; Leith,
Cobbold, 1996]. We have adapted this method to conditions 1982]. Thus the Tajik depression has an internal structure
of reversefaultingand haveappliedit to the Tajik depression. typical of a thin-skinned fold and thrust belt, surroundedby
From the current (deformed) state and the restored(undeformed) basement overthrusts [Leith and Alvarez, 1985]. These boundstate, we have calculated
fields of finite
deformation
ary thrusts control the internal structure of the basin. There is
(translations, rotations, strain rates) that can be compared a westerndomain, where eastwardverging thrustsand folds are
with field data.
synthetic to the Gissar thrust, and an easterndomain, where
After discussingthe geological setting of the Tajik depres- westward verging thrustsand folds are syntheticto the Pamirs
sion and presenting the method of restoration, we describe a
thrust. Associated depocenters are due to footwall flexure
numerical restoration of the structural map that Thomas et al.
beneaththe SW Gissar and Pamirs thrusts.A third depocenter
[1996b] had previouslyrestoredmanually, and we comparethe lies in a central pop-down, between basement-involvedthrusts
manual and numerical restorations. Then, we describe the
of opposite vergences. In the depocenters,the Mesozoic and
restoration of two new maps compiled from VNIGNI data. Cenozoic sedimentary infill reaches 10 km in thickness.
ß
BOURGEOIS
ET AL.:RESTORATION
OFTHETAJIKDEPRESSION
27,555
27,556
BOURGEOIS
ET AL.:RESTORATION
OFTHETAJIKDEPRESSION
Elsewhere, it is 8 km thick on average [Burrman and Molnar,
1993; Thomas eta!., 1994, 1996a, b].
3. Method
of Restoration
In areas of compressional tectonics, the traditional method
for restoring the deformation is to use balanced cross sections
l Dahlstrom, 1969; Hossack, 1979; Jones and Linsser, 1986;
De Pact, 1988; Geiser et al., 1988; Moretti and Larrkre, 1989].
Such restoration takes into account both folding and faulting.
The main assumption is conservation of lengths or surface
areas of the layers in cross section. This assumption is not
valid
if there are differential
motions
of material
in directions
perpendicular to the planes of section, such as occur during
strike slip faulting or block rotations about vertical axes. This
restriction
makes the methods
of doubtful
use in areas of non-
plane deformation, such as the Tajik depression.
Some three-dimensionalunfolding programs,avoiding this
restriction, have been developed for restoration of non-
cylindricalfolds [Gratier, 1988;Gratier eta!., 1991;Gui!!ier,
1991: Gratier aud Gui!!ier, 1993]. They allow unfolding of
individual
blocks.
Restoration
of
fault
heaves
between
previously unfolded blocks may be carried out manually
[Cobbold and Davv, 1988; Dokka and Travis, 1990; Gratier
a•d Gui/lier, 1993; Thomas eta!., 1996b]. Numerical proce-
dures for fitting blocks in plan view, applicableto nonplane
deformation, have been developed for regions of continuous
deformation [Cobbold, 1979; Perce•,au/t and Cobbold, 1982;
Cobhold a•d Percevault, 1983]. They have been adapted to
discontinuousdeformation of regions dominated by strike slip
motions [Audibert, 1991] or normal faulting [Rouby, 1995;
Rottbr el a/., 1993a, b, 1996; Roubv and Cobbold, 1996]. We
haveadaptedthem to regionsdominatedby reversefaulting.
3.1.
Principle
of the Method
The ainl of the inethod is to reconstruct the undeformed
initial stateera single stratigraphiclayer, currentlyfolded and
offset by a populationof dominantlyreversefaults. First, we
representin map view the present,deformedstateof the stratigraphic surfaceto be restored,as a mosaicof folded blocks
boundedby faults. On this mosaic (fault block map), blocks
may be contiguous,separatedby gaps, or overlappingeach
other, accordingto the natureof the faults(strikeslip, normal,
or reverse, respectively). The width of the gap or overlap
betweenblocks is proportionalto the heave of the associated
fault. Second,unfolding of the blocks yields changesin their
shapesand in the width of overlapsat their boundaries.Third,
the programpacksthe unfolded blocksusingrigid translations
and rotations so as to minimize the total area of gaps and over-
laps. Tinis yields a restoredmosaic. By comparisonwith the
n{)nrestored(deformed) mosaic, we can then computedclbrmation fields, including finite translations and rotations, or
azimuths of slip on faults.
3.2.
Main
Assumptions
The main assumptionsof the method and their validity have
already been discussedby Audibert [1991] and by Roubv et al.
11993al. We briefly review and discuss them in the specific
context of compressional tectonics.
1. Wc assume that the surface to be restored was planar,
h{•rixontal and continuous before deformation. This assumpti{•n is reasonable for shallow marine sediments, where sedi-
mentary irregularities are generally small in comparisonwith
the scale at which we work. Under these conditions, restora-
tion of a stratigraphicsurfacetakes into accountall the deforlnation it has encountered since deposition.
2. We assumethat it is reasonableto divide the region to be
restored into a mosaic of blocks completely bounded by faults.
Since natural fault arrays are seldom totally connected,the
mosaic has to be completed by introducing artificial block
boundaries. Criteria for locating artificial block boundaries
will
be discussed in section 3.3.
3. Blocks so defined are assumedto be rigid or to consistof
folded layers. The asumptionis reasonableif surfacestrains
are negligible. In general, the layers will require untilting or
unfolding,after whichthe blockscan be consideredrigid.
3.3.
Preliminary
Data
Processing
We briefly describe the preparation of a fault block map
prior to its restoration. A fuller description of such preliminary data processing, for regions of strike slip and normal
faulting, is given by Audibert [1991] and Rouby et al.
[1993a]. Here we insist on specific aspects, applicable to
regions of reverse faulting and folding.
1. We start with a structure-contour map (Figure 3a) and
cross sections(Figure 3b) of the surface to be restored.Along
the cross sections we project the hanging wall and footwall
cutoffs of every fault vertically onto a map (Figure 3b). We
then interpolate cutoff traces between cross sections,followin,, the contours of the structure-contourmap This procedure
yields a preliminary fault heave map (Figure 4a).
2. On the fault heave map (Figure 4a), we extrapolateeach
fault trace until it meets another fault trace (Figure 4b). This
yields a temporary fault block map where each block is
boundedby faults or artificial block boundaries.
3. Wc unfold each block separately. In general, for noncylinclrical folds, it is convenient to use automatic unfolding
mothotis l G•'atier e! a/., 1991]. For the Tajik depression,where
folds are nearly cylindrical, we have performed this task
manually. On a seriesof cross sections,drawn perpendicularto
•.
the
.
main
structul-eS
we
have
conserved
bed
len,,ths
This
process yields an extension of the blocks in map view,
parallel to the directions of the cross sections. We have
distinguishedtwo situations.For folds of long wavelengthand
low amplitude (Figure 3c, fold B) and folds associatedwith
faults (Figure 3c, folds C and E), unfolding leads to slnall
changes in line shapes of the predefined blocks. Usually, it
increasesthe width of the overlaps between blocks (Figure 3c;
see als{>G•'•tl[•,•'rt•t! Gui!!ier[1993, Figure 5bl). We can thus
modify the tempol'aryfault block map accordingly(Figure 4c).
In contrast, f{>ldsof short wavelength and high amplitude can
be c{•nsidered as regions where the deformation is concentrated.
as for faults.
Therefore
we can draw
an artificial
block
bot•ndary alone the axis of each of these folds We represent
the f{•ld-related hori/,ontal shortening as a horizontal Incaveat
the artificial block boundary (Figure 3c, fold F). We extrapolate each block boundary so defined until it meets another
block boundary (Figure 4c).
4. To allow some bending within blocks that are far from
cquant, we subdivide them into smaller ones, using new
al'tificial block boundaries (Figure 4c). These artificial block
m
boundal'ies
are
ß
drawn
where
internal
strain
is known
or
suspected,on the basis of topography, shapeof the blocks, en
echelon minor faults, and so on.
BOURGEOISET AL.:RESTORATIONOFTHE TAJIKDEPRESSION
27,557
a) Structure-contour map
of the surface
to be restored
b) Projection
offaultcutoffs
Cross
sections
•
ofthesurface
to
f be
restored.•
Fault
heave
map
l•
c)Unfolding
heave due
••.heave
due
to fault D
tofault
A
artificial
'::•
':•i•
'"•
::•i::
i,,,,',,,i}iii',•i??:•-":':••
heave
due
'•'
to fold E
heavedue -•'
to fold B
l•
•
fault
heave
due
•.heavedue totbld
F
to fold C
Figure 3. Construction of a fault heave map. (a) Structure-contourmap of the surface to be restored. This
representsthe initial data. (b) Projection of fault cutoffs. The surfaceto be restoredis the top of the grey layer,
visible on the sections. For faults (A and D) the footwall and hanging wall cutoffs are projected onto a
horizontal plane. The width of the lens so obtained is equal to the heave of the fault. (c) Unfolding. For a/'old
o1'long wavelength and small amplitude (B), the whole block is unfolded and the changesin size and shapeare
transferredto its external boundaries.For folds (C and E) associatedwith a fault (D), the apparentheave due to
the folds is addedto the heaveof the fault and projectedonto the map. For a told of shortwavelengthand large
amplitude (F), an artificial fault is drawn along the axis of the Ibid. The apparentheave is projected along this
artificial fault onto the cutoff map.
27,558
BOURGEOISET AL.' RESTORATIONOF THE TAJIKDEPRESSION
/•
a) faultheavemap
•
e) neighbor seeking
b) preliminary fault block map
>•
f) adjustment
c) definitive fault block map
/
fault-related cutoff
':•:...•fold-related cutoff
g) restored map
d) digitization, segmentation
Figure 4. Principle of the method of restoration.(a) From a structure-contour
map and crosssectionsof the
surface to be restored,heavesof the faults are projectedonto a preliminary fault heave map. (b) Fault traces are
extrapolatedto constructfault-boundedblocks. (c) Each block is unfolded and the fault block map is modified
accordingly. (d) Boundaries of blocks are digitized and their sides are automatically subdivided into line
elementsof unit length. (e) To each elementz the program associatesa neighboringelement zn so that z and
zn face each other across a cutoff lens; d is the distance between z and zu. (f) Blocks are then packed by
translations and rotations so as to minimize the sum D of the squaresof all distancesd. (g) The neighborseekingand packingproceduresare repeateduntil D becomesstabilizedat an acceptablysmall value.
not covered, displacements inferred from the restoration can
only be considered as relative to the stationary block.
6. We finally digitize the fault block map, representingthe
bl(•cks as polygons. An identity number is assigned to each
bl(•ck. At each iteration the numerical procedurewill adjust the
blocks in the order of their identity numbers. In order that the
restoration propagates away from it, the stationary block
types.
5. Boundary conditions on displacementsare conveniently bears number 1: the ()thers are numbered sequentially away
introduced by making one block stationary. Against this, the froin it (see, for example, Figure 5b). Blocks are groupedinto
i•ulnerical procedurewill pack all other blocks. The choice of strips: those that are adjacent to the stationary one form the
lh½ stationary block does not affect the result of the restora- first strip; the next ones form the second strip; and so on. To
tion; but, when choosing it, one must keep in mind that the CI'ISurcnumerical convergenceof the solution, it is important
displacementswill be calculated with respectto the stationary that the blocks of the second strip bear identity numbers
block. In order that the calculated displacements can be greater than those of the first strip. Within each strip,
directly compared with geological data, a suitable stationary however, the order of block numbers is not critical.
Numerical
Procedure
block is one representinga platform region, known to have 3.4.
undergone little or no motion with respect to a surrounding
The entry data for the numericalprocedureform a file, where
continent, during the studied deformation. If such a region is each block is represented by its identity number and the
At this stage(Figure 4c), blocks have four different types of
boundaries:real faults, artificial block boundariessimulating
folds, arlificial block boundaries due to the extrapolation of
both preceding types, and artificial block boundaries introduced to subdivide elongate blocks. Notice that there are no
initial gaps or overlaps along boundaries of the last two
BOURGEOISET AL.: RESTORATION OF THE TAJIK DEPRESSION
digitized coordinates of its vertices. The computer program
automatically subdivides the sides of the blocks into line
segments of equal length, hereinafter referred to as line
elements (Figure 4d). The length of these line elements is that
o1' fine smallest
block,
that is, the size of the smallest
irregularity taken into account during digitization. Typically,
this nncans about 100 line elements per block, or about
1(),()()() line elements for the whole mosaic. The use of smaller
clements slightly improves the fit of the restoration, but the
calculation time increases considerably. Each line element is
del'incd by the coordinates of its center, its azimuth and the
identity number of the block to which it belongs.
Tlnen begins a series of iterations, designed to minimize
overlaps (or gaps, in the case of normal faults). Iterations
include two stages.
1. To each line element z of a given block B, the program
automatically assigns a neighboring element zn chosen
amongst tlneelements of blocks which are surroundingB. Line
elements z.and •.n face each other acrossthe overlap (or gap)
between the blocks to which they belong. The line elements
are separatedby a distance, d (Figure 4e). This procedureis
repeatedfor every line element of every block.
2. Blocks are then successivelypacked, hsing cycles of
rigid translations and rotations, so as to minimize the sum D
of the squaresof all distancesd (Figure 4t).
Iterations are repeated until D reachesa minimum; we then
consider that the fault block map is restored (Figure 4g).
Further details of the numerical procedureare given in the
appendix.
4. Restoration of the Tajik Depression
To take into account
the deformation
that has accumulated
since the India-Asia collision, and to exclude earlier deforma-
tions, we have restored a stratigraphic surface at the base of
tlne Buklnara Formation.
This is a Paleocene
marine shelf
sequenceof limestonesand evaporites,a few hundredsmeters
tlnick [Gubin, 1960; Skobolev, 1977]. In the area of the Tajik
depression,it lies with slight angular unconformity on eroded
Cretaceous strata. It is readily identified in the field and on
satellite images, and it has a characteristic seismic signature
on reflection profiles.
4.1.
Map 1
4.1.1. Data. Thomas et al. [1996b] compiled published
data, producing a structure-contour map on the base of the
Tertiary. They then drew up a series of six cross sections,
perpendicular to major structures. They unfolded the cross
sections manually, conserving bed lengths. From borehole
27,559
Paleocenebeds to the surface in the Peter the First range [Leith
and AIvare•., 1985]. In some places, they are completely
eroded and the fault heaves cannot be estimated.
Therefore
the
area to be restored goes no farther than the southerntip of the
Peter the First range. The northern edge is the Iliac fault. To
the west, palcomagnetic work in the SW Gissar has demonstrated no significant rotation with respect to the Turan
platform since tlneCretaceous[Bachenovet al., 1994; Thomas
eta/., 1994]. We therefore assumedthe block representingSW
Gissar to be stationary. All other blocks and boundarieswere
rnobile.
4.1.2.
Quality of restoration.
The overall fit of
tlne restored fault block map (Figure 5c) is visually satisfactory. The area of gaps and overlaps in the restored state is
().8% of tlne total area of the blocks, which is much reduced
from fine4% in the deformed state. Displacementson artificial
block
boundaries
are small.
These
artificial
block
boundaries
mainly allowed the blocks on each side to undergo differential
rotations, witIn no significant translation. We interpret tlnese
displacements as representing internal bending of elongate
blocks about vertical
axes. However,
in the southeastern
corner of tlne map, restoration involves significant strike slip
motions along tlne tlnree southernmost artificial block
boundariesstriking E-W. This problem is discussedbelow.
4.1.3.
Boundary displacements.
Comparison of
the fault block maps before restoration (Figure 5b) and after
restoration (Figure 5c) yields the following regional displacements. (1) Between the Pamii-sand the SW Gissar, shortening
of the basin is 60 km (18%) on average. (2) At the northern
edge of the restored area, between the depression and the
Gissar range, a N-S shorteningof 20 km is associatedwith an
E-W riglnt-lateral strike slip displacement of 25 km. (3)
Between the Pareits and Tien Shan, the displacement is 120
kin, implying 70% of shortening. (4) Along the southernedge
of finerestoredarea, left-lateral displacementsrange from 0 km
in the west to 10 km in the east.
4.1.4.
Displacement field.
We have computed the
Field of horizontal finite displacementsat the nodesof a Cartesian grid drawn on the restoredmap (Figure 6a). The displacements increase from west to east, because we assumed the
westernmost block to be stationary. The azimuth of the
displacement vectors changesprogressively,from nearly E-W
in the westernpart of the map to NNW-SSE in the east. Vector
leagrinsalso increase away from the SW corner of the map,
indicating a global counterclockwise rotation of the depression about a pole near its SW corner.
4.1.5. Block rotations. All blocks have undergone
counterclockwise rotations about their centroids (Figure 6b).
The western half of the map shows uniformly small rotations
(()ø-10ø).Tlne easternhalf is more complex but, on the whole,
shows a gradient of rotations, which ranges fi'om 15ø in the
and seismic data, they estimatedthat the dips of major faults
ranged fi-om 45 ø to 55ø and so calculatedthe heaves.Finally,
they interpolated between the heaves calculated on each cross west to 30 ø in the east.
section, obtaining a fault heave map (Figure 5a).
4.1.6.
Azimuths of slip on faults.
Azimuths
of
From this same fault heave map, we have drawn up a fault
slip on t2qults,computed from the restoration, are subperpenblock map with 55 blocks (Figure 5b). Elongate blocks have dicular to fault traces (Figure 6c). Thus most faults show an
been subdivided using artificial straight block boundaries almost pure reverse slip with little strike slip component.
trendingE-W to NW-SE.
Because of a lack of data from Afghanistan, the area to be
restored goes no further south than the 37th parallel. To the
east, because we are interested only in the deformation within
tiaoTajik depression,we assumedthe Pareits to be rigid and
representedthem by a single block. In the NE, intense crustal
tlnickeningbetweenthe Pamirs and the Tien Shan has brought
4.1.7.
Comparison
with
manual
restoi'ation.
Independently of our numerical restoration, Thomas et al.
[1996b] manually restored the same map by cutting a paper
copy along each fault trace and pulling out the paper from west
to east, in order to invert horizontal displacements along
faults (Figure 7a). On the whole, the results of the manual
restoration (MR) and of the numerical restoration (NR). are
27,560
BOURGEOISET AL.: RESTORATIONOF THE TAJIK DEPRESSION
68 ø E
70 ø E
Tien
a)
•han
I
38øN i
!
!
I
I
I
m
Border
ofthe
i
restored
area
37øN m-
-.:-
i 50km
i(
N
•/ i
_
68 ø E
b)
70 ø E
[_.__•
.......
':.......
2.......
_,•:
.......
•......
:•z
.....
i:',_-___
39øN
I...........
:
n.........
!--•,•o•.............
-E--|
'• i en Sha ''• N• / I
36(masked
SW
Gissar
- •"••-• • + by
29
and
42)
.
/
(s'mtiøna•:•:•':•-•?':'-"/.
] block)::?:•-:.-.:?..?'-
...:.
:•.•-::::::::•:•.
•..
....
1•-.:..•.::•:•..•::.::
•........
....
38øN........•..........
7<'
' •• +
.;::•.•:•:•,•:•,•:•,•:•:•,•,•:•:•
............ -.........
. •..
37øN
overlapped
/ block
c)
I
•-
I
SWG•sar;:
.:.:.:...:.
,-
•
•
-'-
t•
......
......
.
T i
t
e
n
,
•
Sh
....
•
-
•
•
a
n
•
,
/
%
,•
.
/
/
•
I
undeformed
(restored)
border
10km
•x
•
/'
strike
slip%%
displacements
o• • '//
•
•,•
aaificial
block
bound•ies
' /
/
BOURGEOIS
ET AL.'RESTORATION
OFTHETAJIKDEPRESSION
27,561
wlnicinsinewssubsidiaryE-W and NW-SE faults. Elsewhere, we
compatible. However, four significant differences are to be
noted. (1) In the MR, displacement vectors trend on average added E-W artificial block boundaries. We obtained a mosaic of
90 blocks (Figure 9a). The area covered by this map is slightly
NNW-SSE, wlnereasin the NR, they are closer to NW-SE. As a
consequence,the block representing the Pamirs restores to a
larger tinan map 1. Nevertheless, the boundary conditions are
1neronol'tinel'nposition in the NR. (2) The NR shows smaller fine same: fine block representing the SW Gissar is stationary,
block rotations. (3) In the NR, some left-lateral displacements wineteas line otiners are mobile.
4.2.2.
Quality of restoration.
In the restored state
appear aion,, the E-W faults of the SE corner and along the
soutincrnedge of the map. They do not appear on the MR.
(Figure 9b), tlne area of gaps and overlaps is 2.2% of the total
area of tlne blocks, whereas it is 10.7% in the deformed state.
(4) In fine MR, paper overlaps in the southeasterncorner of
fine map are necessary to accommodate the inversion of
However, tlne fit of the blocks is visually unsatisfactory. In
displacements(Figure 7a). The NR does not show such over- particular, significant overlaps remain for the central parts of
laps.
faults 1-15 (Figure 9b), whereas the fit is good in their northWe explain these differences as follows (Figure 7b). To
ern parts and gaps appear in their southernparts. Overlaps in
draw up tine fault block map for the NR, we extrapolated the tineircentral parts could not be further reducedwithout opening
m•[ior N-S faults until they intersectedthe southernedge of the extra gaps in tlnenorth and in the south.There is a geometrical
map. We also introduced artificial E-W block boundaries to
incompatibility between the initial overlaps and block shapes
allow some "bending" of elongate blocks trending N-S. As a in tinis area.
result, blocks located in the southeasternpart of the map were
These problems may result partly from the use of a onefree to undergo differential E-W translation with respect to dimensional unfolding method, where the shortening
11neirneiglnboring blocks to north and south, leading to (1)
directions are imposed by the directionsof the unfolded cross
strike slip motions on artificial E-W block boundaries, (2)
sections. Tlne use of a three-dimensional unfolding method
smaller block rotations and (3) displacementvectors trending would prevent this bias. Nevertheless, because we unfolded
NW-SE.
cross sections that are perpendicular to nearly cylindrical
In fine MR, such translations in the SE did not occur,
structures, errors generated by one-dimensional unfolding
becauseN-S faults were not extrapolatedtoward the south and sinouldbe small. The bad fit of the restoration suggeststhat
elonoate
•
blocks
were
not
subdivided
into
smaller
ones
ß
fine incaves
of faults
1-15
are overestimated
in their
central
Instead, paper overlaps occurred in this area. As emphasized partsaccordingto the VNIGNI data. Indeed, fault heaveson the
by Thomas et ai. [1996b], these paper overlaps would imply
VNIGNI cross sectionsare poorly constrainedby wells in this
Cenozoic extension in the southern part of the depression, area. According to satelite images and field data [Thomas et
winicinis not compatible with field data. They thus considered a/., 1996b], fault lneavesdecreasesignificantly from north to
tinat a left-lateral strike slip displacement along the southern soutin.
border of tine depression would prevent the overlaps. In the
Map 3
NR, finestrike slip motions on the southernborder and on the 4.3.
E-W artificial block boundariesof the map are indeed left4.3.1.
Data.
In order to test our conclusion
that the
lateral.
4.2.
Map
2
4.2.1. Data. The All Russia Research Geological Oil
Institute (VNIGNI) supplied a new structuralmap of the same
stratigraphic horizon and a series of four E-W cross sections
tlnrougln the Tajik depression. According to these data,
constrainedby regional seismic surveys and numerous wells,
sino1-tening
across tinrustand folds is approximately 2 times
larger tinan on the cross sections compiled by Thomas eta/.
[1996b] (Figure 8). In order to test the quality of the new data,
we attelnpted a furtlnerrestoration.
Because Paleozoic formations outcrop to the east of tlne
Darvaz-Karakul
fault,
the heave of this fault
cannot be
estimatedwith accuracy.We have estimateda minimum heave,
equal to the sum of the heaves of a series of imbricate thrust
slices, tinrough the sedimentary cover in this area. The real
ineave,including the thrust of the Pamirs, is probably much
greater.
To subdivide elongate blocks located in the west, we used
line structural map of Yachminnikov and Nikelayer [1990],
incavesof faults 1-15 are overestimated in their central parts,
we lnave restored a modified version of map 2. Foliowine
classical balancing techniques [Dahistrom, 1969], we have
produced this modified version (map 3) by subtracting from
tlne initial overlaps along faults 1-15 the values of the overlaps remaining after restoration of map 2. Other parts of
map 3 are identical to map 2, and the boundaryconditionsare
also the same (Figure 10a).
4.3.2. Quality of restoration.
The area of gaps and
overlaps in the restored state (Figure 10b) is 1.8% of the total
area of tlneblocks, in comparisonwith 10.4% in the deformed
state. Tinis 1.8% value, averaged over the whole map, is only
sliglntly smaller than the 2.2% value obtained from the
restoration of map 2. Yet, the area of remaining gaps and
overlaps is 5.2% on faults 1-13, in comparisonwith 8.8% in
tln½same areas on map 2. The fit of the blocks is visually far
better tinantinatof map 2, especially in the southernpart of the
map. Displacements on artificial block boundariesare mostly
small. They show no significant strike slip components,gaps
or overlaps. Bending about vertical axes is significant only
for elongate blocks.
Figure 5. Map 1 and its restoration.(a) Cutoff lens map. The widtlnof each black lens is equal to the heave
of tiaofault or fold that it represents.(b) Fault block map in currentdeformed(nonrestored)state.Along real or
fold-simulating faults the darker block is overlappedby the lighter one (see insert). Artificial block boundaries
appear as straiglntlines without overlap. For restoration,only shadedblocks are taken into account.They are
numbered sequentiallyaway l¾omthe stationaryblock, which representsthe SW Gissar (block 1). The Pamirs,
consideredrigid, are representedby a single block (block 55). (c) Fault block map in restoredstate.
27,562
BOURGEOIS
ETAL.:RESTORATION
OFTHETAJIKDEPRESSION
ements
:
68 ø E
,
•
0o
32ø
o
70øE
::
,, oo•••••i:iii
i •_.o
-•• b)
Block
rotation
5
o
.............
•'"•
............
i"-.'5:
:"•:'::'"'•v'
'
__
38øN
37ON
Counterclockwise
0ø
rotations
predictedby the restoration
isindicated
by
thewidth
of
the
black
angular
sector.
•.__•90ø magnitude
is
indicated
by
the
white
line.
Error
bar
90ø
,
0o
.
10o
i
20 ø
Measured
paleomagnetic
rotations.
Rotation
.
30 ø
counter-
clockwise
clockwise
. 'in Tertiarysediments
o ' inCretaceous
sediments
68 ø E
38ON
37ON
--I 50kmI
Figure6. Deformation
field for map1. (a) Fieldof finitedisplacements.
Eachvectorspansthedistance
between
points
of a material
Cartesian
gridin therestored
andcurrent
states.
Vectors
arenullwithinstationary
block(topleft).Compare
withmanual
restoration
(Figure7). (b) Finiteblockrotations.
Blocksareshaded
according
to theamount
of rigidrotation
calculated
fromtherestoration.
Paleomagnetic
data[Thomas
et al.,
1994]aresuperimposed,
for comparison.
(c) Azimuths
of sliponfaults.Blackarrowsindicate
thesense
and
dircction
of finitedisplacements,
calculated
fromtherestoration.
Principal
shortening
directions
interred
from
ananalysis
of faultpopulations
[Thomas
et al., 1996a]aresuperimposed,
forcomparison
(whitearrows).
BOURGEOISET AL.: RESTORATIONOF THE TAJI'KDEPRESSION
27,563
Manual
restoration
be restored
re•naining
paperoverlap
Numerical
restoration
tN
50 km
a)
Figure 7. (a) Manual restorationof map I (modiffied after Thomaset al. [1996b]). This restorationwas
performedby cuttinga structuralmapalongeach/Sqult
traceandpullingout the papel'fi'omwestto east,in order
to invert horizontaldisplacements
alongfaults.Stippledareasin the southindicateoverlapsin the paperafter
restoration.Vectors representdisplacements
of lines of latitudeand longitude,consideredas material lines.
Comparewith Figures5c and 6a. (b) Schematicdiagramexplainingthe differencebetweenthe resultsof the
lnanual and the numerical restoration.Deformed (nonrestored)state is representedon the left, restoredstate on
the right. The darkerblockB is overlappedby the lighteroneA. For the manualrestoration,fault tracesare not
prolongedto the south,and elongateblocksare not subdivided.The restorationof the overlapis performedby
a singlerotationof blockB, yieldingpaperoverlapin the south.For the numericalrestoration,fault tracesare
prolongeduntil they meetotherfault traces,andelongateblocksare dividedinto subblocks.
The restorationof
the overlapis performedby E-W translations
and smallrotationsof the subblocks.Strike slip displacements
between subblocksallow internal deformation within block B due, for example, to noncylindrical folding.
4.3.3. Boundary displacements. A comparison of
deformedstate (Figure 10a) and restoredstate(Figure 10b) for
map 3 yields the following boundary displacements. (1)
Between the Pamirs and the SW Gissar, the shorteningis 150
becauseof lack of precisionon the heavesof the faults in this
strongly deformed area. The eastern half of the map shows a
nearly uniform counterclockwiserotation of about 40ø in
km (35%) in a WNW-ESE direction. (2) At the northern border
of the restored area, between the depressionand the Gissar
of slip on faults.
In the westcrnmost part of the map, faults show only pure reverse slip,
without strike slip components(Figure 11c). In the central and
eastern parts of the map, faults show some left-lateral slip
associatedwith thrusting. Curiously, two of the N-S faults
situated near the Pareits show right-lateral slip, apparentlyin
contradiction with the regional shortening direction. As
explained in section 5.3, this may be due to insufficient
range,60 km of N-S shorteningis associated
with 20 km of EW right-lateral strike slip displacement. (3) Between the
Pareits and the Tien Shah, the shorteningis 240 km (85%) in a
NW-SE direction. (4) Along the southernborderof the restored
area, !eft-lateral displacementsrange from 0 km in the west to
about 40 km in the east.
4.3.4.
Displacement field.
Finite
displacement
vectors(Figure 11a) trend NNW-SSE on averageand indicatea
olobal counterclockwise rotation of the basin about a pole
near the SW corner of the map.
4.3.5
Block rotations.
Restoration
yields
two
distinct domains of block rotations (Figure lib). In the western half of the map, blocks have undergone clockwise or
counterclockwiserotations of up to 10ø in magnitude.There is
a gradient, from counterclockwise values in the south to
clockwise
values
in
the
north.
However,
the
clockwise
rotations of the northernmost blocks are not significant,
magnitude.
4.3.6.
Azimuths
boundary constraints in this area.
5. Discussion
of the Results
Because of the bad fit after restoration, map 2 is not an
admissible geometrical model for the Tajik depression.
Maps I and 3 show better fits, but the resultsare significantly
dilTerent.
For map 3 the displacementof the mobile easternboundary
is about 130 kin, nearly twice that of map 1. This is because
the fault heavesare systematicallyabout twice as big as those
27,564
BOURGEOISET AL.' RESTORATIONOF THE TAJIK DEPRESSION
of map 1. In general, systematicerrors or increasesin fault
heaves cause correspondingincreasesin boundary displacemerits(appendix). Also for map 3, rotationof the mobile eastern boundaryis about 32¸, whereasit is 28¸ for map 1. This is
because the heave gradients along faults are systematically
greater for map 3. In general, systematicerrors or increasesin
heave gradients cause corresponding increases in boundary
rotations (appendix). We now discuss the restorations of
maps l and 3, comparing them with geological and geophysical data.
5;.1.
Boundary
Displacements
The Gissar range, located at the northern edge of the Tajik
depression, is a major zone of right-lateral thrusting
[Tr(fonov, 1978; Leith and Airarea, 1985]. Both maps l
and 3 yield such motion on their northern boundaries. Leith
[1985] noted a marked discontinuity in thicknessesof postJurassic
sediments
across
the
Iliac
fault.
He
attributed
this
discontinuity to deposition on faulted blocks along a passive
margin. The large thrust component (60 km) obtained by
restoration of map 3 confirms the alternative explanation
proposedby Hamburger et al. [1992] for this discontinuity:it
results from the tectonic juxtaposition of sediments initially
depositedin areas 60 km apart (Figure 10b).
At the southernedge of the Tajik depression,Thomas et al.
[1996b] inferred a !eft-lateral strike slip motion along the
eastern part of the Alburz fault. The restorationsof maps 1
and 3 both yield such displacements.
In the NE, from a balanced cross section through the Peter
the First range, Hamburger et al. [1992] calculated a minimal
shortening of 60 km in the sedimentary cover between the
Panairs and the Tien Shah. Our restorationsof maps I and 3
yield larger shortening in this area: 120 and 240 km, respectively. We suspect that this discrepancy reflects internal
deformation
of the northern Pamir, a feature not taken into
account in the cross section of Hamburger et al. [1992].
Indeed, displacementsand shorteningrates between the Pamirs
and the Tien Shah have beenestimatedby Buttman and Molnat
[1993], using geological evidence of three types. First, they
noted that a Paleozoic suture crossingAfghanistan and Tibet is
deflected
northward
300 km
around
the
northern
Pamirs.
Second, they inferred from paleomagnetic data a model of
bending of the outer arc of the Pamirs, associatedwith at least
300 km of northward displacement, mainly since the
Paleogene [Bar.h.enov and Burrman, 1986]. Third, Cretaceous
and Paleogenefacies belts, trendingeast-westwithin the Tajik
depression are abruptly truncated at lhe western edge of the
Pareits, their eastward continuationslying a minimum of 200
km farther north at the northern edge of the Pamirs [Burtman
and Molnar, 1993]. The shortening of these sediments has
been estimated to be more than 100 km [Burtman and Molnar,
1993]. Together, displacementand deformation of these sediments imply a total of more than 300 km of convergence
between
the northern
Pamirs
and the Tien
Shan.
From
these
data, Burrman and Molnar [1993] also inferred that the Tajik
basin
must
have extended
eastward
100 km or more
into the
territory now occupied by the Pamirs. These estimates are
remarkably close to the results from restorationof map 3. But,
because we assumed
a conservative
value for the heave of the
Darvaz-Karakul fault, the shorteningobtainedl¾omrestoration
is a minimum
value.
o
BOURGEOIS
ETAL.:RESTORATION
OFTHETAJIKDEPRESSION
27,565
,
':• I
.
•
! e n
/
n a n•
••4
(smtlona• (
block)
}t7.•.
'
•
-
/•
-
_
',
' •:•
ß :
q
..:•::,:•=:
:: ..................................................
........
//
•:•:•:•:•:
•.........
'?::::•:?•:•:3•:•:•:•:3•:3•:•:•:•:•:•:•:•:•:•:•:•:3•:•:•:•:3•:•:3•....
4--
38ON--. ...............:,,
:•::•
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
.
:.:.:.
::
.:.:::.::.:.:.-.:.:::.:::::::::::::::::::::::.
::.:::::::::::::
:::::::S::
v[:[:[:
:::•:•:•:•:•:•:[:[•:•:•:•:•:333•:•:•:[:•:[:•:[:•:•:[:•:•:•:[:
½3:::•:•:3•:•:S•:3
.........
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::•::::::?•
37ON ..........
67øE
r
i
-i-
-F
68øE
-'-•-
4-
-i-
69øE
4-
+
+
-i.-
70øE
i
i
71øE
i.
i-
....
T i e n
4-
SW Gissar-
-t-
S h a n
F
4-
-I-t.-
4
F
i
i
,/,
•
deformed(non-
•
f
(stationary
block)
restored)
border
!
I
uncleformed
I
(restored)border II
I
I
I
I
I
,
/
/
ß
-.
/
/
/
/
150km
• IN
/
/
/
/
/
/
/
Figure9. Restoration
ofmap2. a)Faultblock
mapincurrent
deformed
state.
Same
legend
asFigure
5b.The
Pamirs
arerepresented
bya singleblockP. b) Faultblockmapin restored
state.
Notepoorfit in thesouthern
halfof themap.Significant
overlaps
remain
in thecentral
parts
of faults1-15,whereas
thefit isgoodin their
northern
parts,andgapshaveappeared
in theirsouthern
parts.
5.2.
Block
Rotations
measured
paleomagnetic
rotationsandrotationspredicted
by
the restoration
of map 3 (Figure lib). Althoughrotations
predicted by the restoration generally lie within the
Froma palcomagnetic
study,Thomas
et al. [1994]showed confidence
limitsof the paieomagnetic
measurements,
they
ihat
thewestern
and
eastern
parts
oftheTajik
depression
haveare systematically smaller than the measured rotations. This
differingpatterns
of blockrotations
(Figures6b and l lb). difference
maybe dueto errorsgenm'ated
by preliminary
Durino the Cenozoic,the westernpart underwentsmall unfolding along cross sections,or to deformationwithin the
rotations(10ø-20ø),whereasrotationswere largerin the northeastern
partof theSW Gissar,whichweneglected.
It may
eastern
part(40ø-50ø).Thisis in verygoodagreement
withthe alsobe becauserotationsobtainedby restoration
are valid for
restorationof map 3, where there are two distinctdomainsof blocksof kilometricsize, whereaspaleomagnetic
rotations
block rotations. There is a remarkablecorrelationbetween are measuredon samplesof centimetric size.
27,566
BOURGEOIS
ET AL.:RESTORATION
OFTHETAJIKDEPRESSION
•9N--I/
::
'""<•. +::T
i e n Si, h a, n, •/ /::z
-%,..•
( ::,.
•I
':•"-•
i....
!
'"([%/
:: , I
/• ......
•,
•••4.
"'/-:::•-•:••:...:.• +
+
•:
I
!....+
',
' '
'
I (Smtlonory
I
.....
+ /
__•
/
',
•
-,-
"i:, ' -,- I
',
' •
I
---J--
-,,
•?:•::•i•::?:?:?:E•?:??:•:::•?'":•?:?•?:•?:•:?:?:•3
' I
....::::::::-:-::: ::::::::::::::..
"':--'•:-: :::::::::::::-g:
/
::
'•/
:
'
38oN...............
•'
/
/
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
====================================================
===============================
I
..•::•:•E:E:•:•:3•:E:3•
::::2•
:•:•:•:[:•:-.[:[:•E:33•:•:•:•:•:•:½E:•E:E:•:E:•:•E:•:E:•:3•:•:•
-t- I
.-::3333•:ErE:E:Ez•:EEE:I:3EzEzE½3E:•
EE:E:
'
::E:33•:•:•:•½E:33EE:E:3•:3E:E:E:E:E:3•:E:E:½33E:•E
I
I
_
37•N
.......
50kin
i••
67 ø E
68 ø E
69 ø E
70 ø E
71 ø E
deformed (nonrestored)border
SW Gissar
(stationary
block)
undeformed
) border
•
50
km t
•
-i-
-i-
-i.
i
t
/
i/
!
N
.............
.....
-i
!
!
!
Figure 10. Restorationof map 3. (a) Fault block map in currentdel:ormed
state.Samelegendas Figure 5b.
(b) Fault blockmap in restoredstate.Comparethe fits of the blocksin the southernpartof the map with those
of Figure 9 b.
5.3.
Azilnuths
of Slip on Faults
An analysis of fault-slip data for small-scalefaults [Thomas
et ai.. 1996a, b] indicatesleft-lateral slip on most N-S faults,
in very good agreementwith the restorationof map 3 (Figure
11c). In contrast• restoration of map 1, where horizontal
displacemen•azimuths are everywhere perpendicularto fault
iraces, does not show such strike slip components(Figure 6c).
The small right-lateral strike slip displacementsthat appear
along two of the N-S faults situatednear the Pamirson map 3
is an artifact of the method of restoration: for a perfectly
straightor circularfault, the restorationmethodcannotpredict
strike slip displacements(see the apendix). In the easternpart
of the map, fault traces are indeed nearly straight lines.
Boundary constraints in this area are therefore insufficient to
accuratelypredict strike slip displacements.
BOURGEOISET AL.: RESTORATION OF THE TAJIK DEPRESSION
Finally, earthquake focal mechanisms [Roecker etal.,
1980; Leith and Simpson, 1986; Lukk etal., 1995] and in situ
strain measurements [Soboleva etal., 1992] in the Garm area,
in the NE of the Tajik depression (located on Figure 2a)
indicate a predominance of subhorizontal compression,
manifested in a mixture of thrust and strike slip deformation.
The maximum horizontal compressionaxis trends N-S to NWSE. These resultsare comparablewith the azimuthsof horizontal displacementsdeducedfor this area from restorationof map
3.
to 4()ø). Field data of various kinds are available in this area,
and they are compatiblewith the resultsof the restoration.
Thereforethis area providesan excellenttest /'or the validity
andthe accuracyof our methodof restoration
in mapview. In
particular,the restorationhas predictedrotationsand strike
slip offsetsthat correlatewell with field dataand that would
not have been revealed by separatelybalancing individual
crosssections.In the partsof the map where fault tracesare
straight lines, supplementaryboundaryconditionsare necessaryto predictmoreaccuratestrikeslip displacements.
Such
constraintscould be provided by imposingblock rotations
accordingto paleomagnetic
data [Audibert,1991].
As with rotations, azimuths of slip along faults could be
6. Conclusions
6.1.
27,567
Deformation
of the Tajik
Depression
After comparing the results of the various restorationswith
publisheddata, we favor map 3 as the most reasonablegeometrical model for the Tajik depression. Fault heaves and fold
sizes are greatly underestimated on the map compiled by
Tho,•as etal. [1996b]. According to the restorationof map 3,
in responseto a 240 km displacementof the Pamirs toward the
northwest, 150 km (35%) of WNW-ESE shortening occurred
between the Pa.mirs and the SW Gissar during the Tertiary.
Shortening within the depression is associated with a rightlateral strike slip motion along its northern border (the Tien
Shah) and a left-lateral strike slip motion along its southern
border (the Alburz fault). Within the depression,shorteningis
accommodatedby individual thrust slices trending NNE-SSW.
These slices have undergonecounterclockwiserotationsabout
vertical axes, increasingin magnitudefrom west to east, up to
a maximum of 40 ø near the Pamirs. Left-lateral strike slip
motions are associated with thrusting along the NNE-SSW
tlsedas carry data for the restoration,insteadof being used
only for comparison
at the end.This wouldhavethe advantage
of using all availabledata to calculatethe best undeformed
state. Tiao aim would then be to solve an inverse problem
[Taran•o/a, 1987] by minimizationof a weightedaverageof
the fault heaves and the differences
between measured and
computedrotationsand azimuthsof slips.Undersuchconditions, the solution should be heavily overconstrained.
Appendix:Details of the Numerical Procedure
A1.
Method
of
Solution
Our numerical procedure solves an inverse problem by
minimizinga singlequantity:the sumD of the squaresof a set
of distancesd. These distancesare the moduli of the neighbor
vectors,which span the centersof neighboringline elements
on the footwalls and hangingwalls of the faults. Becausethe
interveningfault blocksare assumedto be rigid, the unknown
quantities,to be solvedfor, are the translationsand rotations
faultstinatbQt{nd
theslices.
of the blocks. More specifically, for each fault block, the
Though showing greater displacements and shortening unknownsare the two Cartesiancomponentsof the translation
rates, these results are compatible with the kinematic model vectorand the magnitudeof the finite rotationabouta vertical
proposed by Thomas etal. [1995, 1996a, b]. They also axis throughthe block centreid. Minimization of D yields a
confirm their assumption of left-lateral strike slip motion systemof simultaneous
equationsin termsof the unknown
along the Alburz fault. This modelaccountsremarkablywell translationsand rotations. These equations are strongly nonfor the values of displacementsand shorteningrates proposed linear in terms of rotations. Also, the calculated translations
by Burt,.'•anand Molnar[1993] betweenthe Pamirs and the are strongly sensitiveto these rotations.We thereforesolve
Tien Shan.
the equationsby an iterativeprocedure,designedfor restoring
Becausewe representedthe Pamirsby a singlerigid block, ductile deformation [Cobbold, 1979; Cobbold and Percevault,
we did not take into account its internal deformation; thereIbm
1983}and later adaptedto strikeslip faulting[Audibert, 1991]
the values of displacementand rotation computed for this and normal faulting [Roubyetal., 1993a]. As in the iterative
block are valid only for its western boundary. Within the Gauss-Seidel procedure for solving simultaneous linear
Pareits, deformation is indeed very large: shortening is equations,each calculatedparameteris immediatelyre-used,
estimated to be as much as 300 km in a N-S direction [Buttman
for calculating the next parameter.
and Moluar, 1993].
Because,in the Tajik depression,Upper Jurassicevapetites A2.
Neighbor Seeking
provide a major decollement level, we emphasizethat our
Where two line elements face each other, across an overlap
model represents the Cenozoic deformation of the postbetweentwo blocks, it may seemobviousto associatethem as
Jurassicsedimentarycover. Deformation in the basementmay
neighbors.However,it is lesssimpleto find objectivecriteria
occur elsewhere and the style may be different [Lukk etal.,
so tinatthis task can be performedrapidly by a computer.
19951.
For restoration of strike slip domains, where blocks are
contiguous, Audibert [1991] chose as neighboring line
6.2.
Method
The restoration
of
Restoration
of the VNIGNI
clement
data has shown the value of
the method as a balancing technique. A restorable map has
been obtainedby removing geometricalinconsistencies
from
the initial data set. Restorationof this map yields a model of
complex deformationfor the Tajik depressionwith very large
magnitudesof shortening(up to 85%) and block rotation (up
•.n the closest one to line element z.. However,
in
reøions of normal faulting, where blocks are separatedby
gaps,Roubvetal. [1993a] and Rouby [1995] showedthat the
closest line element is not necessarilythe one for which translations and rotations minimize the gaps.
In regionsof reversefaulting,an objectiveidentificationof
neighboringline element is even less obvious.We consider
27,568
BOURGEOIS ET AL.: RESTORATION OF THE TAJIK DEPRESSION
a) Finite displacements
50
km
ß •
N
67 ø E
39øN
[' i•
68øE
i
.....
'"";•'••
ßi 14
b) Block rotations
32•43
ø
"15'-.
½•:'::;;=;
......... ,,
y
'i•
= 50 ø-.•
40o'• • 30ø• • 20ø
g-el0o
o
0o
-10ø
•50
km• •N'
• 6½
øE
Rotationspredictedby the restoration:
[
!] counterclockwise
rotations
[!iii:iiii!11
ii•i
illclockwise
rotations
. '......................
i........................
71:
......... .
Oø
10 ø
20 ø
30 ø
0ø
0ø 10ø 20ø 30ø .40ø 50ø
paleomagneticrotation
regression
line' y = 0.84x - 10ø
37ON ....
correlation coefficient'
R = 0.9
Measured
paleomagnetic
rotations.
Confidence
limits
are
indicated
by
the
width
of
p•_•90ø Rotation
magnitude
is
indicated
by
the
white
line.
90ø
counter- clockwise
theblackangular
sector.
clockwise
ß 'in Tertiarysediments
o 'in Cretaceous sediments
40 ø
39ON ......................
':......
i•iiii!iiii!i•'•'"'"ii
•ii:i:iiiii;ii!;•i:;•i""::"
..... i i .:::ii:i
c)
Azimuths
of
slip
on
faults
• I I I I I I I J I I I I I I I..::½:"
..•
$ .?":"
.-::2'
..:..s..
..:?!.•'
1...........-.:.:..•....•?•i5
::i:! .:
37øN
---
0
50
km
•
N90øE
N 180øE
directionof shortening
inferredfrom
an analysisof faultpopulations
N
67 ø E
68 ø E
69 ø E
BOURGEOIS
ET AL.:RESTORATION
OFTHETAJIKDEPRESSION
a) Before restoration
27,569
b) First iterations
[---I Given block B
1---I Given block B
':"•::•
Surrounding
blocks
......
•:::•:•,•,'•::::!::i•iii•ii!iiiiiiii?j.:.-'iii
I--]Non
surrounding
blocks
9:::: $.'-I:i:
::•:!:i:!:
:•:•-•i:i:•:!:i:i:?-:..'::-..':i:
•
.............
[---I Given block B
• Admissibleneighboringblocks
.....
• Inadmissibleneighboringblocks
:U'rlDomainof seeking
ß
ß
ß
ß
ß
ß
e
ß
ß
ß
ß
Block centroids
,, Block centroidsoverlapped
by block B
c) Standard iterations
ß
::.'-•Admissibleneighboringblocks
'-::"-'---:'•
Inadmissibleneighboringblocks
:lWilDomain of seeking
Block centroids
d) After restoration
Representat
of
overlap
• ...............
overlapping
block
overlapped
block
ß
Figure A1. Proceduresfor seeking admissible neighboring blocks. The shape of the blocks is visible in the
restoredstate (Figure Ald). Insert showshow solid or dashedoutlinesare usedto representblock overlaps.(a)
Before restorationthe user defines surroundingblocks: these are the blocks amongstwhich the elements of a
given block B will be authorized to seek admissible neighboringblocks. Then, at each iteration, each element
•. born by B automatically seeks its admissible neighboring blocks (Figures Alb and Alc). (b) For the first
ite,'ations. the domain of seeking for line element • is bounded by th,'ee straight lines: the fi,-st one bears
element •.. the other two a,'e perpendicularto 7•and passthroughits ends. A surroundingblock of B becomesan
admissible neighboring block for • if at least one of its elements lies in this domain. (c) For standard
iterations the surroundingblocks of B, whose centroidslie in a half-spaceboundedby the line bearing element
- become admissible neighboring blocks for •..
seve,'al criteria to be satisfied. First, line elements z and :n
should lie on adjacent blocks; second,line elementsz and zn
shouldface each other acrossthe overlapthat separatesthese
blocks; third. they should be more or less parallel. The last
condition is especially critical near block corners. To ensure
the fi,-stcondition,we first seek the admissibleneighboring
blocks of line element z., that is, the blocks amongstwhich
line elementz. will be allowedto seekits neighboringelement
z.,. The other two conditions
A2.1.
Procedure
for
are considered later.
seeking admissible neigh-
boring
blocks.
1. In orderto reducecomputingtime, we define,onceand for
all before restoration, the blocks amongst which line
elements of a given block B will be authorized to seek
admissible neighboring blocks (Figure A la). These blocks
(hereinalterreferredto as surroundingblocksof B) are all the
blocksthat may comeinto contactwith B during restoration.
Becauseappreciablestrike slip motionsmay occur during
restoration,
we take into account more blocks than those
directly in contactwith B in the currentstate. Surrounding
blocksare definedfor every blockof the map.
2. At each iteration, the program automatically assignsto
each line element z. born by block B, a number of admissible
neighboring blocks from amongst the surroundingblocks of
B (Figure A lc). During the current iteration, these blocks
must be adjacentto the edge of B that bears line element •. To
ensu,-ethis. we define a domain of seeking for line element •.
It is the half-plane bounded by the line bearing z and not
containing the centroid of B. The blocks surrounding B,
whose cent,-oids lie within this half-plane, then become
admissible neighboring blocks for z.
At the beginning of the restoration,if overlaps are wide,
the edge of block B that bears element z may overlap the
centfolds of its surroundingblocks, making it impossiblefor
•. to find any admissible neighboringblock (Figure A lb). We
therefore use another procedure for the first iterations. We
define the domain of seekingof z with three lines: the first one
bears line elementz, the other two are perpendicularto z and
pass through its ends. A surrounding block of B becomes an
admissible neighboring block for •. if at least one of its
elements lies in this domain of seeking (Figure Alb). This
procedure being rather slow and not restrictive enough, it is
Figure 11. Deformationfield for map 3. (a) Field of finite displacements.
Lengthof eacharrow is one half
of calculated
displacement.
(b) Finiteblockrotations.
SamelegendasFigure6b. Correlation
betweenpredicted
and measuredrotations(right) is very good (R=0.9). The restorationpredicts85% of the measuredrotation
magnitude.
Paleomagnetic
errorbarsare indicated.(c) Azimuthsof slip on faults.Samelegendas Figure6c.
Azimuthsof slip predictedby the restorationare very closeto the shortening
directionsinferredfrom the
analysis of fault populations.
27,570
BOURGEOISET AL.: RESTORATIONOF THE TAJIK DEPRESSION
stoppedas soon as there are no more blocks which overlap the
2. We draw the normal vector v, perpendicular to z and
centfoldsof their surroundingblocks. We then use the standard pointino outward on block B and vector vn perpendicular to
seeking procedure. Both procedureswork for either overlaps :.n and pointing inward on the block bearing zn. The inside of
(reverse faults) or gaps (normal faults) between blocks.
a block is defined as the spacewhere its centroid lies. Because
A2.2. Procedurefor seeking neighboring elements. Line long curved blocks are subdivided into smaller ones to allow
clement: seeks its neighboring element zn amongst all the some bending, this definition is always valid. We then define
line elements of the admissible neighboringblocks assigned an angular sectorof magnitudeocabout the azimuth of vector v
by the previousprocedure(Figure A2). Only one neighboring (Figure A2b). Vector vn must lie within this angulhr sector.
At the beginning of the restoration, the magnitude ocof the
element is selected:it is the closestelement from amongstthe
admissible neighboring blocks which also satisfies the
angular sector is set to 90ø so that only those elementswhose
following two conditions, designed to ensure that elementsz
azimuths differ from the azimuth of z by less than 45 ø can be
and z.• face each other acrossthe overlap and are subparallel.
selected as neighboringelements. This relies on the assump1. Element :n must lie in a new domain of seeking,between tion that a line element whose azimuth differs from the
two straight lines perpendicularto z and a distance2l apart, l
azimuth of :. by more than 45ø is most probablya neighborfor
being the length of line element z (Figure A2a). The width 2I
a line element that is orthogonal to z. It implies that
of the domain of seeking allows a line element, that does not
differential rotations between two adjacent blocks, greater
strictly face line elementz, to be selectedas its neighboring than 45 ø in magnitude, cannot be restored. The magnitude (x
element, so that lateral displacementsbetween blocks (strike decreases, from one iteration to another, down to 40 ø in the
slip) are possibleat each iteration. Becausethe procedurefor
last iterations, in order to make the neighbor-seeking
seeking neighboringelements is repeated,defining new pairs procedure more and more restrictive as the overlaps close up,
of neighboring elements at each iteration, the width of the
that is, as the edges of adjacent blocks tend to becon•e
parallel.
domain of seeking does not affect the final values of strike
slip motions. However, wider domains of seeking (4/and 8/),
Our neighbor-seekingprocedure, although more complex
thouoh ,fivin• similar results along block edges, result in the
than the ones used by Audibert [1991] and Rouby et al.
selection of aberrant neighboring elements near block
[1993a1, has the great advantageof being applicableto block
corncrs.
configurationswith very large overlaps and even with gaps. It
a)
Given
'• "'•'"'•':•'":'"•::'.".",:.:.:•-.............
I•?dom
ainofseekingfor G•ven
. ½"...'i•i•::•:":•'"'•"':•i•::
•i::i::i::i::i::i".-i•i•iii?:
•iii::i-':iii•i•i•"
--'""-'•'"'"'"'""•
•-/-•element
zofblock
B
elements
ofblock
A lying
element
a :i•iii
i :":'":'""•?]
•' ,' /
9 :ii!i:•..'.:..'...'..:•!?
•"•
in
the
domain
of
seeking
9.........
•:'"::•7":"'""••
ed
angular sector
for vector vni
C)
•I elements
ofblock
Alying
inthe
domain
of
seeking,andtheirassociated
vectors,n
i
vn,•,•
elements
ofblock
Aibrwhich
vector
s'n
i lies
i
in the authorizedangularsector
Given
e le m ent -
boring
element
Figure A2. Procedurefor seekingneighboringelements.BlockB (white) overlapsblockA (grey). BlockA
hasalreadybeenrecognizedas an admissible
neighboring
blockfor elementz.of blockB (bottomleft, Figure
A2a). Elementz.then seeksa neighboringelementzn amongstthe elementsof blockA. (a) Elementsof block
A lying in a domainbetweentwo straightlines,perpendicular
to z anda distance2I apart(l beingthe lengthof
lineelement
:), arepreselected.
(b) Foreachpreselected
element
wedrawthevectorvni perpendicular
to the
clementandpointinginwardon blockA. We alsodefinean angularsectorocaroundvectorv, perpendicular
to z.
and pointing outward on block B. The magnitudecz is set to 90ø at the beginningof the restorationand
decreases
fromoneiterationto another.
Preselected
elements
for whichthevectorvni liesin thisangular
sectorare then selected.(c) The neighboringelementzn for elementz is the closestin distanceamongthe
elements
so selected.
BOURGEOISET AL.: RESTORATION OF THE TAJIK DEPRESSION
is possiblethat some of the elementsdo not find a neighboring element' indeed, this is so for all elements at the outer
edgeof the block mosaic.Theseelement•are not taken into
account during adjustmentof the blocks.
A3.
Block
Packing
Let the neighboring blocks of a given block B be those
blocks amongstwhich the elementsof B have found neighboring elements. Once pairs of neighboring elements have been
dellned, our program packs each block successivelyinto the
space defined by its neighboring blocks. The packing
procedureis fully describedby Audibert [1991] and Rouby et
•t/. [1993a]. Briefly, it computes,for each block B, a cycle of
rigid translations and rotations about the centroids of the
blocks• designed to minimize the sum of the squares of
distancesbetween the line elements on B and their neighboring line clements.
Translations are calculated according to the expression
originally derived by Etchdcopar [1974] for crystalline aggre-
gates.For eachblockB, let •i be the neighborvector,joining
the center of each element zi to the center of its neighboring
27,571
not taking into account the positions of those blocks that lie
on their right side. This type of iteration drastically accelerates the convergence,but it is less accuratethan the standard
type of iteration, in which a given block is adjusted with
respectto all its neighboringblocks.
The first iteration is systematically a Fast-Pack iteration.
Throughout the restoration process we introduce other FastPack iterations, separatedby more accuratestandarditerations.
Alternation
between
Fast-Pack
and standard
iterations
is auto-
maritally monitored by the parameterdescribedbelow.
A3.2.
Monitoring
parameter.
To monitor the
restoration process, we use a nondimensional parameter G
dcl'ined
as
G = So/S
b
where So is the total area of all gapsand overlapsand Sb is the
total area of all blocks. SO is given by
So= LEdi
whereL is the lengthof a line elementandZdi is the sumof
distances between neighboring elements. If the digitization
has been carefully performed, that is, if the smallest known
irregularity on the block sides has been taken into account,
clementzni.The translation
of blockB thatminimizesthe sum the valuesel' S,, and St, do not dependon the size of the
o1'the distancesbetween neighboringelementsis the mean of
clements.
all neighborvectorswi. The solutionis thereforeunique.
The convergence of the parameter G is tested at the end of
each iteration n in order to determine the type (standard or
Fast-Pack) o[ the next iteration n+l. After trying several test
procedures,we have adoptedone which is reliable and achieves
a good compromise between calculation time and accuracyof
adjustment. A Fast-Pack iteration is launched at iteration n+l
if the three t'ollowingconditionsare satisfied.(1) The gradient
Along a perfectly straight or circular fault, strike slip
displacements have no influence on the gaps and overlaps
betweencontiguousblocks [Audibert, 1991]. For two adjacent
blocks, isolated from any others, strike slip displacements
between them are controlled by irregularitiesof their common
boundary. Indeed, if the boundary is perfectly straight or circular, the solution for the strike slip displacementis not unique.
Should the two blocks be surrounded by others, strike slip
displacements between them are very sensitive to the
boundaryconditions at the tips of the fault.
Rotations are calculated accordingto the expressionderived
by Attdibert [1991]. In the presence of strike slip displacements, there is no simple analytical expressionfor rotations.
The solution is therefore approachedby a cycle of elementary
rotations, designed to minimize the distancebetween the tips
o[ each element z and their orthogonal projections on the
straight line bearing its neighboring element zn. The calculation is repeated, redefining the projection of the tips each
time, until further adjustmentsare negligible, in comparison
with the length of the elements.
The packing procedure takes into account the adjustments
made to blocks already processedduring the current iteration.
Within each cycle the procedureis repeated,maintainingthe
same pairs of neighboring elements, until translations and
rotations become smaller than a predefined critical value,
close to zero. Then new neighboringelementsare defined and
a new cycle is launched.
A3.1.
Accelerating
convergence.
In order to
accelerateconvergence,we use specific iterations, the "FastPack" iterations of Cobbold and Percevauh [1983], in which a
o[G
between
iteration
n-1
and iteration
n is less than one
[ourth o[ the gradient between iteration n-2 and iteration n-1.
(2) The value of parameter G at iteration n or at iteration n-1
is the smallest value reached since the beginning of the
restoration process. (3) The last Fast-Pack iteration was
perlereed at least three iterations ago. The restorationprocess
is stopped when the gradient of G becomes smaller than a
preset critical value, close to zero.
A4.
Geometrical
Significance
of the Calculations
The geometrical significance of our iterative procedure is
that each block in turn is packed againstits immediate neighbors. This involves a rigid translation and rotation of the
block.
To understand
how the translations
and rotations reduce the
fault heaves,consideran array of quasi-parallelfaults, as are
common in areas of compressionaltectonics(Figure A3). For
convenience,
let the first
block
on the left-hand
side be
stationary and the others mobile. Minimization of D yields
the rigid translations or rotations of the mobile blocks, so
that all fit together as well as possible.
Consider
first
the situation
where
the fault
heaves
have
averagevaluesH/and H 2 that do not vary alongthe faults
given block B is packed, taking into account only those (Figure A3a). Fitting involves translations only. For each
neighboring blocks that bear identity numbers smaller than block the calculated translationis simply the vector mean of
that of B. As mentioned previously, the blocks are numbered all neighborvectorsaround it [Etchdcopar, 1974; Cobbold and
sequentially away from the stationaryone. Say, for example, Pe•'ce•,attlt, 1983; Audibert, 1991]. Local variations in d,
that the stationary block lies on the left of the block mosaic representingirregular heaves, have no effect on the calculated
and that the blocks are numberedby strips, from left to right translation,becausethey contributeto the variance,not the
acrossthe block mosaic. Blocks will then be adjusted with mean, of the neighbor vectors. The calculated translationsare
respect to those neighboring blocks that lie on their left side, H/, for thesecond
blockrelativeto thefirstblock,andH2, for
27,572
BOURGEOISET AL.: RESTORATIONOF THE TAJIK DEPRESSION
calculated displacementfield. In particular, where faults have
variable curvatures or where some faults terminate against
others, strike slip componentsalong the faults will be well
constrainedby the numericalfitting [Audibert, 1991].
A5.
on
Errors
and
Boundary
Their
Cumulative
Effects
Displacements
in general, tracesof faults at the Earth's surfacetend to be
well defined, whereas their geometries at depth are less
constrained.An inadequatedensityof subsurfacedata may lead
to severe errors in interpolation. In determining footwall cut-
L0+HI+H2
offs beneath reverse faults, lack of seismic definition or inade-
quate borehole data may result in random or systematicerrors
in fault heaves.When the original data are available only as a
line drawing, the lines may be as wide as the fault heaves
themselves, leading to systematic errors.
Becausefault blocks are assumedto be rigid, random errors
in fault heave have little
Figure A3. Geometrical significance of the method and
sensitivity to errors. (a) Situation where the fault heaves (H l
and H2) do not vary, on average, along faults. Fitting
involves translations only. (b) Situation where the heaves
vary linearly, on average, along faults. Fitting involves
rotations (ct I and ct2). Discussionin the text.
the third
block
relative
to the second
block.
Notice
how the
displacements accumulate, from the fixed block toward the
right-hand end of the array, so that the deformed length L 0
restoresto an originallengthLo+ H t + H 2. Noticealsothat
strike slip displacementsbetween the blocks will be determinate by the minimization process, only if the faults have
variable
curvatures.
Now consider the situation where the heaves vary linearly,
on average, along each fault (Figure A3b). For each mobile
block, minimization of D yields, not only a translation, but
also a rigid rotation [Cobbold, 1979; Cobbold and Percevault,
1983]. The calculated rotations are proportional to the linear
gradients of the distances d along the faults, these gradients
representing
scissoringmotions.Thus the rotationsare
for the secondblockrelativeto the first block,and(•2, for the
third block relative to the secondblock. Local departuresfrom
the linear
trends have no incidence
on the calculated
rotations.
Thus finding the translation and the rotation of the blocks by
minimizing D is equivalent to finding the best fit linear variations of d along the faults. Notice again that the rotations
accumulate, from the stationary block toward the right-hand
end of the array. As before, strike slip displacementsbetween
the blocks will be determinate, only if the faults have variable
curvatures. Under these conditions, an expression for the
calculatedblock rotation was derived by Audibert [1991].
In general, there may be more than one array of faults and
some arrays may crosscutothers. A fault block may therefore
be bounded by more than two faults. Packing of this block
with respectto its immediate neighborswill involve a translation and rotation, which togetherminimize all the fault heaves
around the block. The final block rotation will be a compromise between clockwise
and counterclockwise
rotations,
dictated by different heave gradients. From one block to the
next, the way the displacements accumulate will be more
complex than in the more simple arrays of quasi-parallel
faults, previously discussed.In general, however, the more
complex is the fault network, the more determinate will be the
effect
on calculated
translations
and
rotations.The random errors may detractfrom the goodnessof
fit between the blocks, but they contibute little or nothing to
the average neighbor vectors, which determine the block
translations,
or to the best fit linear
distributions
of heaves
along faults, which determine the relative rotations between
blocks.
In contrast, systematic errors may build up toward the
boundaries of the array, especially if most faults are quasistraight and parallel, as is the case for many of the structures
in the Tajik depression. Under these conditions, if all
distances d are subject to an increase Ad, the boundary
displacementwill increase by had, where n is the numberof
parallel faults. Similarly, if all linear gradientsof fault heaves
alono faults increase by a quantity Ao•, then the rotation that
accumulatestoward the boundaryof the array will increaseby
.A or. Finally, where straight faults reach the external
boundaries of the array of blocks, there will be little or no
constraint on strike slip displacementsalong the faults, so
that significant strike slip errors may accumulate.
References
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Nttmdriqttes de Restauration, Mdm. Doc. Cent. Armoricain Etttd.
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1991.
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Bazhenov, M. L., H. Perroud, A. Chauvin, V. S. Buttman, and J. C.
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Bumnan, V. S., and P. Molnar, Geological and geophysicalevidencefor
deep subductionof continentalcrust beneaththe Pamir, Spec. Pap.
Geol. Soc. Am., 281, 1-76, 1993.
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France. (e-mail: obourgeo•univ-rennesl.fr;
cobbold•univ-
rennes
1.fr;drouby•univ-rennes
1.fr)
V. Shein,VNIGNI, 36 ShosseEntuziastov,Moscow,Russia.
J. C. Thomas, Laboratoirede G6ophysiqueInterne et
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(P,eceivedJanuary30, 1997; revisedJune30, 1997'
acceptedSeptember2, 1997.)