Estimation of Paleostress Orientation Within Deformation Zones Between Two Mobile Plates

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Estimation of paleostress orientation within deformation zones between two mobile plates

RUUD WEIJERMARS* Hans Ramberg Tectonic Laboratory, Geological Institute, University of Uppsala, Bcac 555, S-751 22 Uppsala, Sweden

ABSTRACT collision zone where part of the differential motion between the Eurasian and African plates is accommodated by strike-slip faults. The orientation of the principal stress axes within deformation zones between two mobile plates is modeled here analytically, using a INTRODUCTION thin-plate theory. The simple analytical approach helps to explain why plates cease to move after collision. Orogenic periods last only several Geologists have pieced together a picture which suggests that the tens of million years because the stress associated with a particular con- process of breakup, dispersal, and renewed aggregation of supercon- stant driving force (causing a constant strain rate) is no longer able to tinents is one of the key features of plate-tectonic evolution on Earth. maintain a significant horizontal displacement. In contrast, the uplift The mechanics of plate tectonics are complex, but the major forces rate increases rapidly as the horizontal velocity decreases, and this may acting on the lithosphere are due to slab-pull and ridge-push associ- explain why the termination of orogenic epochs are usually heralded by ated with the density variations due to surface cooling and various the rapid deposition of thick sequences of immature sedimentary rocks types of viscous forces and chemical differentiation in the mantle. The or flysch. fact that the surface plates of the Earth are in stress remains invisible The analytical model also elaborates the relationship between ho- unless leading to permanent distortion by brittle failure or ductile de- mogeneous bulk deformations driven by a constant stress orientation formation. Large deformations of the lithosphere are principally con- and those due to a fixed displacement direction of physical boundaries fined to plate boundaries, where large deviatoric stresses may accu- of the deformation zone. Two major kinematic possibilities are consid- mulate as manifested by a high earthquake frequency. Working out ered: (1) plates converging either orthogonally or obliquely, making the the physics and reconstructing the history of this concentrated de- deformation zone an analogue for orogenic collision; and (2) plates di- formation is a major research task. verging either orthogonally or obliquely, so that the deformation zone is Reconstructing the history of deformation structures is compli- dynamically similar to initiating rift basins. The theoretical investigation cated, because the deformation path followed depends principally led to the formulation of the following rules. Deformation zones between upon (1) the orientation of the stress field, (2) the rheology of the converging plates have the major axis of bulk deviatoric stress coinciding rocks, and (3) boundary conditions prevailing during the deformation. with the bisector of the acute angle between the relative plate velocity These three parameters may have been either constant or unsteady in vector and the normal to the deformation-zone boundary. In case of time. Despite all of these uncertainties, a significant amount of the- extension within a deformation or riftzon e separating diverging plates, oretical work and some practical methods have been developed for the bisector will outline the minor axis of the bulk deviatoric stress. inferring paleostress orientations from natural fault patterns (for ex- The deformation tensor of the analytical model yields a new ample, Anderson, 1951; Arthaud and Mattauer, 1969; Angelier, 1979). method for estimating the orientation of paleostress in natural examples, What has remained relatively unexplored is the relationship between here applied to the deformed wall rock of the Moroccan Border fault. stress and large finite deformations of more penetrative character. The marker used is a competent sequence of Devonian sandstone and Penetrative deformation is meant here to include creep either by crys- limestone asymmetrically folded adjacent to the dextral Border fault. talline flow or microfracturing and pressure solution, on a regional The steeply plunging Z-folds of the marker beds suggest that the prin- scale. cipal deviatoric paleostress, rv was oriented 37°-44° to the fault trace. The approach advocated here to understand better the mechan- Hie age of the Moroccan Border fault is poorly constrained and may be ics and history of large penetrative deformations involves two rigid Variscan or younger. The T1 orientation implies a major component of plates, separated by a deformation zone, with free-slip conditions at simple shear parallel to the strike-slip fault and a minor component of the top and the bottom surface (Fig. 1). Thin-plate approaches have extension perpendicular to the fault trace. The implied crustal move- been previously used both in analytical models (Stevenson and ment is compatible with the modern tectonics of the Eurasian-African Turner, 1975; McKenzie and Jackson, 1983; England and others, 1985; Wdowinski and others, 1989) and in numerical simulations of continental deformation (Vilotte and others, 1982; England and Mc- Kenzie, 1982, 1983; Sonder and England, 1986). Assumptions are 'Present address (leave of absence from Uppsala University): Earth Sci- ence Department, King Fahd University of Petroleum and Minerals, 31261 made to simplify the natural prototypes to the most essential features Dhahran, Saudi Arabia. so that the fundamental aspects of the mechanics are preserved. The

Data Repository item 9329 contains additional material related to this article.

Geological Society of America Bulletin, v. 105, p. 1491-1510, 18 figs., 2 tables, November 1993.

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Figure 1. Definition sketch of the thin- plate concept used in the analytical modeling. x3=-O1/2 The deformation is restricted to a zone separated by subvertical walls of differential slip from the adjacent rigid plates. The principal orientations of the total stress, or, are related to the relative movement direction of the plates, Vr, indicated by the arrows. The condition of plane strain implies that the intermediate de- viatone stress, T2 = 0, which in turn implies that the tectonic contribution to the confining a2=P=01/2

pressure P = TJ/2. Other cases are discussed T2=0 (plane strain) below.

analytical model developed here investigates the systematic relation- tion gradients. In nature, localized slip on narrow subvertical faults ship between plate motion, principal deviatoric stress direction, total bounding the foothills of an orogen may be accommodated either by stress, and confining pressure. This is valuable because experienced the development of mylonite zones (below ~7-km depth and typically field geologists commonly resort to mental models, invoking mech- strain-rate softening) or by weathering of the walls of brittle faults. anisms that could have led to the structures observed in the field. Bird (1984) has provided experimental support for the idea that major These models usually prompt allegations about the orientation of the brittle faults, weathered by hydrothermal circulation, develop ex- principal stresses. The accuracy of such mental models may benefit tremely low frictional resistance due to the hydration of clay minerals. from investigating and quantifying the relationship between stress, For mechanical modeling of deformation with negligible dilation progressive deformation, and tectonics. (no volume change), it is theoretically convenient and sufficient to Rather than forward modeling, finite deformation patterns in na- consider only deviatoric stresses. When this deviatoric stress is ture require inversion; that is, the final deformation stage is preserved, caused by lateral movements of plates adjacent to the deformation and the stress orientation is one of the unknown quantities. Estima- zone, however, these plates exert a total stress which may include tion of the orientation of paleostress is therefore only potentially both a deviatoric and confining component. Excluded from this anal- possible after determining how elements of the original deformation ysis is the obvious contribution to confining pressure arising from the tensor relate to normalized physical dimensions. The method is illus- lithostatic load. A tectonically induced confining component of stress, trated by application to fold structures adjacent to the Moroccan Bor- however, which would reduce or increase the actual confining pres- der fault, a strike-slip fault near the Eurasian-African plate boundary. sure at depth due to plate-tectonic forces, is carefully separated here Additionally, the critical parameters controlling the deformation pat- from the total tectonic stress. This approach not only helps to under- terns in the analytical model are suitable for determining paleostress stand variations in metamorphic gradients, a subsidiary interest of the orientation in a natural prototype only if the underlying assumptions present work, but is essential in coupling plate movements directly to hold for both. The analytical theory used here assumes (1) a constant deviatoric stresses. stress orientation, (2) an isotropic rheology, and (3) boundary conditions warranting homogeneous plane deformation without volume Deviatoric Stress change. Only if these conditions are fulfilled, is it possible to draw reliable conclusions concerning the paleostress orientation by the First consider deformations involving uniaxial compression or method introduced here. The validity of these assumptions is assessed tension in the X-direction with movement restricted to the vertical for the example of the Moroccan Border fault. XZ-plane. The Y-axis is taken parallel to the plate boundary and the XY-plane is horizontal. The stresses inside the deformation zone are MODELING STRATEGY, DEVIATORIC STRESS, TOTAL caused by surface forces transmitted by the tectonic plates, as por- STRESS, AND CONFINING PRESSURE trayed in Figures 2a-2d. Deformation may be either by shortening or extension, depending upon the relative movement direction of the The deformation zone modeled here is located between two rigid plates, here initially limited to movements perpendicular to the ver- plates and separated from them by two subparallel, vertical planes of tical boundaries of the deformation zone. In the ductile regime, all slip; it does not allow any gaps or voids to occur along these contact deformation will occur by pure shear, at strain rates controlled by the surfaces. The approach is three-dimensional, but the deformation is magnitude of the plate velocity (see below). The pure shear may lead limited to plane strain within either the XZ-plane or XY-plane (see to vertical thickening or thinning of the deformation zone (Figs. 2a and below). The rheology of the deformation zone is uniform, justified by 2b). For comparison, Figures 2c and 2d include Anderson's (1951) the model assumption of a thin plate. This work concentrates on the principal cases of plane deformation by reverse and normal faulting. ductile part of the deforming crust, but compares the results with These deformation mechanisms are related to the magnitude of the corresponding brittle deformations occurring at shallow depths. The deviatoric stress in an arbitrary fashion, but it has been customary to sample sections represented by the deformation zone thus consider link fault patterns to the orientation of the principal total stresses shallow and deeper crustal sections, with uniform frictional strength (Anderson, 1951). Reverse and normal faulting have the intermediate or uniform effective viscosity, respectively, and are sufiicientiy far principal stress axis parallel to the surface trace of the faults (Figs. 2c away from the brittle-ductile transition to justify neglecting deforma- and 2d). This stress orientation is the correct boundary condition for

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TABLE 1. LIST OF SYMBOLS

Symbol Quantity SI units Equation introduced

(T Total stress Pa (1) T Deviatoric stress Pa (1) P Confining pressure Pa (1) ¥ Stream function m2 s-1 (6) è Strain rate s"1 1 (6) 7 Engineering strain rate s" (7) V Velocity m s~] (7) t Time s (8) s Stretch (10) e Extension (10) 1 Viscosity Pas (13) 6 Inclination of T] (18) a Inclination of vr (19b) F Defoimation tensor (23) Ri Particular nondimensional time (24) ß Angle of rotated nonnal (25) e Strain ellipse orientation (27) L Fold arc length (28)

T1 = Txx = °xx ~ ? (la)

T2 = Tyy = CTyy - P (lb)

= T3 TZZ = CTZZ - P (lc)

with confining pressure P = (l/3)(cr1 +CT 2 + c3). The boundary condition of controlled plane strain implies that ayy = P, so that

P = Kx + ctzz)/2. (2a) Body forces contributing only to the build-up of confining pressure but no deviatoric stress are excluded from this analysis, so that crzz = 0. The tectonic contribution to the confining pressure is

P = °W2 = cti/2. (2b) The deviatoric stresses are then Txx = °xx - P = °xx/2 = ffi/2 (3a) lyy Jyy P = 0 (3b)

Figure 2. Block diagrams of ductile deformation zone (a) shorten- = —ct T77 P XX/2 = —cri/2. (3c) ing in compressional stress field between two orthogonally converging plates, and (b) extending in tension due to diametrically diverging plates. The expressions corresponding to uniaxial extension in the X-direc- Extension and shortening directions are recorded by the folding and tion are (Fig. 2b)

boudinaging of single competent marker layers as indicated. Diagrams = ct (c) and (d) show the corresponding brittle deformations. A more detailed P xx/2 = —cri/2. (4) account on the deviatoric stress trsuectories and the associated stress The deviatoric stresses are functions is available from the GSA Data Repositoiy (see footnote 1). Txx = o-xx - P = Cxx/2 = -CTJ/2 (5a)

iyy J - P = 0 (5b) plane deformation, although measurements in the rift zones of Iceland yy suggest that the minimum principal stress is sometimes oriented par- P = — graben structures (Price and Cosgrove, 1990). The analysis of the stress employed here takes a for total stresses The regional stress field may best be characterized by the stress tra- and T for deviatoric stresses, using subscripts to distinguish normal jectory pattern, which in turn may be expressed mathematically in and shear components (Table 1 gives a symbol list). Compression is terms of a stress function. If the stress is three dimensional, the stress commonly taken to be positive in geologic literature, but this is some- trajectory pattern of three orthogonal planes may be described by a times impractical (see section 4). The engineering convention, fol- set of three stress functions. These functions are given in the Data lowed here, assigns a negative sign to compressive stresses, and the Repository1, specifying the state of stress for the plate movements principal stress subscripts are ranked according to j o-j | > | cr21 > |

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tion of Navier-Stokes for laminar flows. This solution may be derived either from the equation of motion (Turcotte and Schubert, 1982, p. 243; compare with England and others, 1985) or the vorticity = - e x z transport equation (Malvern, 1969, p. 476), using the velocity con- X straints at the boundaries of this particular isochoric deformation. The velocity components in the X- and Z-directions at the boundary z = 0 (Fig. 3) are vx = è^ x and vz = 0, and boundary conditions at z = are ,vx = e x Zb vx = x and vz = ézz zb. Hyperbolae of v|i = constant show particle-movement paths or streamlines relative to the origin (Fig. 3). The velocities anywhere in the flow may now be obtained by differentiation of >(/ according to the stream function definition; that is, vx = 3i|i/dz and vz = — 9i]i/dx. Ap- -X plication of this differentiation to expression 6 yields

vx = 2-yx = éxx x (7a)

vz = -2yz = -èxx z = èz (7b)

with engineering strain rate 2y = exx, which should be spatially uniform. It is now apparent that the geologic practice to assign compressive deviatoric stresses a positive sign causes problems when dis- \|/ = constant cussing velocities of boundary displacement. The stress is directly related to the strain rate by a constitutive flow law (equation 13, below), and this would give positive strain rates and velocities in the direction leading to shortening. This results in an inconsistent direction of coordinate displacement. It was therefore decided to follow the Figure 3. Velocity components of initially rectangular volume in engineering convention for stress: compression is negative. the deformation zone deformed by pure shear with shortening along Expressions 7a and 7b describe the instantaneous velocity com- the X-axis and extension along the Z-axis. The actual velocity vectors ponents at any particular fixed position and do not track the move- are tangents to the streamlines indicated. All velocity components are ment of any particular particle (Fig. 3). The important conclusion is zero in the origin, and increase linearfy with distance according to ex- that the asymptotes (or traces of the eigenvector planes) to the stream- pressions 7a and 7b. This flowpatter n indicates that the m^jor principal line pattern show the movement direction of the plate boundaries, as deviatoric stress is parallel to the X-axis. is the case when the flow occurs in a pure shear box. Neither the plate velocity nor the strain rate needs to remain constant for the above arguments to be valid. Two extreme cases outlined below consider Tectonic Overpressure and Underpressure the variation in plate velocity necessary to maintain a constant strain rate, and vice versa. The tectonic stress a, in uniaxial compression regimes, associated with crustal shortening, may increase the confining pressure (ful- Constant Plate Velocity ly accounted for by the lithostatic load only in the absence of any tectonic stress field)b y P = aJ2. Similarly, the confining pressure will The actual variation of the strain rate (in terms of negative, com- be lowered by P = (rJ2 in uniaxial (extension. As tectonic stresses pressional exx) in a pure shear deformation driven by a constant plate are generally on the order of 100-500 MPa (Bott and Kuznir, 1984), velocity vx may be evaluated as follows. In fluidmechanics , the strain confining pressure may be altered by 50-250 MPa. Adopting a rep- rate is defined in Eulerian coordinates in terms of the spatial deriva- resentative lithostatic load of 25 MPa km-1, this implies that the ap- tives of the velocity components, that is, ¿y = (1/2) [(3V;/dXj) + (dVj/ parent depth of metamorphism may be misjudged by 2 to 10 km if the 3Xj)]. In the case of homogeneous deformation, Lagrangian and Eu- effect of the tectonic background stress were to be neglected. lerian descriptions are identical, and strain rate may be developed from the time derivative of strain: PLATE VELOCITY AND STRAIN RATE ¿xx = de^/dt = [(dx/x)/dt] = [(l/x)(dx/dt)] = vx/x, (8a) In pure shear deformations of Figures 2a and 2b, the principal which is similar to expression 7b. This may be rewritten as (recall: axes of all stresses remain aligned and orthogonal to the vector of : Vx = Xo ¿o) relative plate motion. The relationship between the magnitude of the deviatoric stress and the magnitude of the plate velocity is examined ¿xx = fvx/(xo + vx t)] = l/[(xo/vx) + t] = l/[(l/e0) + t] (8b) here. The pure shear flow of Figure 2a may most concisely be characterized by the following stream function (Weijermars and Poliakov, where XQ is the initial coordinate of a reference point in the deforma- 1993): tion zone, and e0 is the initial strain rate. Expression 8b specifies the strain rate e at any time t as a function of the initial strain rate e . «I» xz. (6) xx 0 Written in nondimensional form, using vx and x,, for normalization: The stream function of equation 6 is a valid solution of the biharmonic 4 equation V i|> = 0, and automatically satisfies the force-balance equa- ¿xx* = ¿xx xo/vx = Xo/[(vx/e0) + vxt] = 1/(1 + e0t). (9)

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t/Ma t /Ma 0 5 10 1 1 1 1 1 1 1 1 1 1 1 0 5 10 15 20 25 30 t*

Figure 4b. Plot of strain (in terms of stretches S, and S3) versus nondimensional time t* (= — t exx) in hypothetical collision at constant strain rate eM. The strain in such pure shear flowsi s not accumulating exponentially according to expression 17. Also plotted is the variation in S3 (=1+exx) the nondimensional plate velocity, vx*, with time for maintaining the constant shear rate in the deformation zone. Dimensional quantities Figure 4a. Variation of strain rate with time in a pure shear flow may be obtained from vx = vx* e^ x„ and t = — t*/^. The dimensional if deformation is maintained by a constant plate velocity vx. Axes are velocities at the vertical right-handscal e are for a hypothetical orogen graphed with nondimensional strain rate e^*, nondimensional time t* of initial width x„ = 1,000 km deforming at a strain rate of -0.1 Ma-1 -1 -1 and minor principal stretch S3 for universal application. Dimensional so that vx = -100 [km Ma or -10 cm a ] x vx*. The uplift rate of strain rates may be obtained from exx = exx* vx/x„, and dimensional the orogen is indicated by vz (isostasy is neglected). The dimensional times from t = -t* xjvx. The dimensional times, t, are scaled for a times at the upper horizontal axis are for the same orogen so that t — hypothetical orogen of initial width x<, = 1,000 km and vx = -100 km 10 [Ma] x t*. Plotted on the basis of equations 16a and 16b. Ma-1, so that t = 10 [Ma] x t*. The dimensional strain rates indicated at the left vertical axis are for the same orogen so that e^ = -0.1 [Ma-1] 14 x e^*. For comparison, the characteristic geological strain rate 10" 1 + e, = 1 - t*. (12) s~' = 0315 Ma-1. Plotted on the basis of equations 11 and 12. The dimensional times, t, at the top of Figure 4a are scaled for a hypothetical orogen of initial width Xq = 1,000 km and vx = -100 km _1 The shortening in terms of the stretch S3 = 1 + e^ is (recall that vx Ma ,sothatt = 10[Ma] x t*. The variation in the dimensional strain is negative): rates indicated at the left vertical axis are for the same orogen, so that e = —0.1 [Ma-1] x e *. The parameters used here are arbitrarily 1 + e,« = (xo + v t)/xo = 1 + (v t/x ) = 1 + e t (10) xx M x x 0 0 fixed at convenient integer values for convenience of computation. Combining expressions 9 and 10 yields an expression quantifying the For comparison, the characteristic geological strain rate 10~14 s_1 = -1 variation of the normalized strain rate exx* with amount of strain, 0.315 Ma is also indicated. Estimates of actual collisional velocities assuming constant plate velocity: reconstructed on the basis of the spreading record are slightly lower than the 100 km Ma-1 used here. More specifically, estimates of ¿XX* = 1/(1 + exx) = 1/S3 = Sj. (11) global mean subduction velocities time averaged for the 180 Ma record since Pangea breakup vary between 42 and 63 km Ma-1, al- The increase of the absolute strain rate with progressive strain (or though the Circum-Pacific trenches alone would yield velocities of 1 time) is graphed in Figure 4a. The nondimensional time scale t* in- 52-79 km Ma" (Weijermars, 1989). cluded in Figure 4a is related to the dimensional time t by t* = -vx Examination of Figure 4a reveals that the nondimensional strain t/Xo; the minus sign in the nondimensionalization accounts for the rate increases twentyfold and more than one order of magnitude if negative value vx so that t* will be positive. Substitution in expression strain accumulates from 0 to 95% shortening (S3 = 1 + exx = 0.05) at 10 gives the strain accumulation in terms of stretch and nondimen- constant plate velocity. The variation remains within a factor of 2, sional time: however, during the first 50% shortening (S3 = 0.5) in the deformation

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zone. The variation in the magnitude of the principal deviatoric stresses Th and Tzz may be estimated from the effective kinematic viscosity ti of the deformation zone and the variable strain rate eH = —¿j^ of the deformation:

T^ = 2r| ¿xx. (13) Expression 13 implies that for any viscosity value, the deviatoric stress increases proportionally to the strain rate. Hence, if the strain rate in a deformation zone were to increase twentyfold after 95% shortening, the stresses required to keep the deformation going at that increased rate need to be 20 times larger than at the onset of the deformation. This may explain why orogenic strains typically reach finite strains with stretches Sx of about 3 (Pfiffner and Ramsay, 1982), corresponding to S3 = 0.66 in Figure 4a, well below the threshold value requiring significant increase of driving forces. This result is critically dependent on the assumption that the rheology be Newto- nian. If the average rheology is non-Newtonian with a power-law dependence on the stress (compare Weijermars and Schmeling, 1986), then a twentyfold increase in strain rate requires a stress increase of only (20)1/3 or 2.7. This strain-rate softening, however, is likely to be partly offset by stress increases associated with thermally induced hardening due to the downwarp of isotherms in compressional regimes.

Constant Strain Rate

The relative plate velocity vx needs to continually decrease if the strain rate e^ in the deformation zone is to remain constant. Alter- natively, plate velocity needs to continually increase to maintain a constant strain rate in regions of extension. The manner in which this velocity needs to change is outlined in what follows, concentrating on compressional regimes (Fig. 4b):

vx(x,t) = XQ ¿xx S3 = xn ¿xx (14a)

where XQ and x„ are the initial and final coordinates of a reference point in the deformation zone, and S3 is the minor principal stretch (1 + e.^). If a particular e^ is to be maintained, it is useful to deter- Figure 5. Block diagrams of ductile deformation zone deforming by mine how X,, changes with time: (a) horizontal pure shear with shortening of deformation zone, and (b) horizontal simple shear. Diagrams (c) and (d) show the corresponding xn = xo exp^ t). (14b) brittle deformations. Deviatoric stress tnyectories and the associated stress functions are discussed in the GSA Data Repository. Combining expressions 14a and 14b yields

vx(x,t) = xo ¿xx exp(exx t). (14c) vz* = exp(t*). (16b) This algorithm shows how the plate velocity vx should vary with time t for maintenance of any particular strain rate e^. Expression 16a specifies how the plate velocity must decrease with For universal application it is useful to nondimensionalize ex- time in order to maintain a constant strain rate. Figure 4b graphs this pression 14c using the following scaling rules: relationship; dimensional numbers may be retrieved by applying scaling rules (equations 15a, 15b). The time required to achieve a short- vx* = vx/(xo e,«) ^ (15a) ening stretch of 1 + e^ is (recall S, = S3):

t* = -t ¿xx. (15b) S3 = 1 + exx = xn/xo = exp(—t*). (17) The minus sign in the nondimensionalization 15b accounts for the The nondimensional time scale is indicated along the bottom of negative value of e^, so that t* will be positive. Substitution of equa- Figure 4b. The dimensional velocities at the vertical right-hand scale tions 15a and 15b in equation 14c yields are for a hypothetical orogen of initial width Xq = 1,000 km deforming -1 _1 at a strain rate of -0.1 Ma so that vx = -100 [km Ma ]xvx*. The

vx* = exp(-t*). (16a) dimensional times at the upper horizontal axis are for the same orogen, so that t = 10 [Ma] X t*. The uplift rate of the orogen is indicated

Because vz = -vx and ezz = -e^, it follows that by vz. The model of Figure 3 may be envisaged to include the buildup

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a) Moving plate

Vr\ 1.1 Normal

Tzz, èzz tzx, èzx Stretching fault

Deformation eXz zone exx

Stretching fault T7 Stagnation point/^v^

Stationary plate Stationary plate

Figure 6. a. Deviatoric stress components arising in a rock volume subjected to a nonhydrostatic stress component T] at an arbitrary angle i; with the normal to the stable base of the deformation zone. The rock deforming between the plates, converging at relative velocity, Vr, is separated from the rigid wall rock by stretching faults, b. Streamlines or particle movement paths relative to a reference frame fixedi n an arbitrary stagnation point at one wall of the deformation zone between obliquely converging plates. The locking point is a point of no-slip at the wall opposite to the stagnation (or flow attachment) point.

of an orogenic root by assuming mirror-image symmetry about the cipal axes of stress (both deviatoric and total). If the plate boundaries horizontal XY-plane. Syntectonic uplift rates are likely to be even higher move obliquely with respect to the deformation zone, the principal than estimated because the contribution of isostasy has been neglected. stress axes tend to be no longer parallel to the movement direction. Their angular relationship is derived in this section, concentrating on Implications for Orogeny obliquely converging plate boundaries leading to narrowing of the deformation zone. Oblique extension is discussed in a subsequent Evaluation of the above results reveals that geologic extensions section. One simplifying assumption made is that all deformation re- of about 0.66 observed in orogens (Pfiffner and Ramsay, 1982) would mains in the horizontal plane, and any combinations of dip-slip and take about 4 Ma to develop in collisions with a constant plate velocity strike-slip motion—geometrically modeled as transtension and of -100 km Ma~1 (Fig. 4a). There would be no mechanism to stop the transpression structures (Sanderson and Marchini, 1984)—are ex- orogeny, however. A slower collision rate of —10 km Ma-1 would cluded. This probably is an oversimplification, but it forms the basis increase the "orogenic" time by a factor of 10, to 40 Ma. Conversely, for a more complex approach, including triaxial strains (work in the reference strain of 0.66 would require about 10 Ma if built in a preparation). collision with a constant strain rate of -0.1 Ma-1 (Fig. 4b). A typical For convenience, first consider a pure shear deformation occur- geological strain rate of —10"14 s-1 would shorten the orogeny to ring in the horizontal plane. The reference frame is now rotated so that about 4 Ma if constant. the XZ-plane still coincides with the (horizontal) plane of deformation In any case of constant strain-rate collision, the plate velocity (Fig. 5a). The intermediate axis of the strain ellipsoid is always per- drops exponentially with time according to equation 17a, and asymp- pendicular to the plane of deformation and remains unchanged. This totically goes to zero (Fig. 4b). Actually, the plate motion never really implies that the magnitude of the intermediate principal deviatoric ceases, but becomes negligible, even on geological time scales, after stress, T2, also perpendicular to the plane of deformation, is always about 30 Ma. This may explain why orogenic periods in classic studies zero. Figure 5b illustrates an extreme situation where the plates are using the sedimentary record (fiysch and molasse deposits) are usu- moving parallel but in opposite directions, so that the deformation ally estimated to last 30 to 60 Ma. The exponential growth of the zone is subjected to true simple shear. This ductile deformation zone orogenic uplift with time (vz, Fig. 4b) may explain why the amount of corresponds to Anderson's (1951) wrench faulting which also has a sedimentary detritus derived from an orogen increases so dramati- vertically oriented intermediate axis of the total stress (Fig. 5d). cally toward the end of the orogenic period. Orogenic periods last only Anderson's (1951) condition of wrench faulting simply assumed that several tens of million years because the stress associated with a par- the major and minor principal stress axes are both in the horizontal ticular constant driving force (causing a constant strain rate) is no plane, but at 45° with respect to the fault trace(s) (Fig. 5d). This is longer able to maintain a significant horizontal displacement. similar to the stress orientation required to maintain the simple shear deformation illustrated in Figure 5b. For comparison, Figure 5c illus- OBLIQUELY CONVERGING PLATE MOTION trates a case of strike-slip faulting corresponding to the ductile deformation portrayed in Figure 5a. It was demonstrated above that in pure shear deformation the The next question is how the stress axes are oriented if the plates movement direction of the plate boundaries coincides with the prin- converge obliquely with respect to the plate boundaries. Figure 6a

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illustrates the balance of forces on a unit volume within the deforma- The direction of boundary displacement is indicated by vector vr in tion zone. The XY-axes of the coordinate system are fixed to the Figures 6a and 6b. The stream function of equation 20 is innovative nonrotating boundary of the deformation zone. The noncoaxial planar because the particle displacement patterns are described in terms of flow arising in the XZ-space of the ductile deformation zone is shown a boundary velocity orientation and bulk strain rate, without any ge- in Figure 6b. The flow is separating about two points of no slip: a netic assumption about the processes involved. The bulk strain rate stagnation point on the stationary plate and a locking point on the e1 is therefore not necessarily limited to ductile deformations. It is mobile plate. These points are aligned with the direction of plate sufficient that the bulk strain is homogeneous, implying isotropic rhe- movement, otherwise plate movement would be deflected toward the ology on the macroscopic scale. Particle movements of linear elastic streamline in the deformation zone. The stagnation and locking points materials and even for regions accommodating deformation by pen- separate regions of unconstrained lateral slip along the boundaries of etrative faulting may obey the same stream function. Expressions 18 the deformation zone. Only under these boundary conditions will the and 22 are valid only for the particular orientation of the coordinate major principal stress axis and plate movement remain constantly oriented and maintain homogeneous deformation.

Here the relationship between the angles a and £, is determined, Vr characterizing the orientations of the relative plate vector, vr, and that .¡; = 5° of the major deviatoric stress axis, T15 with respect to the plate a = 80î boundary as defined in Figures 6a and 6b. The flow in the deformation a) Moving plate zone can be represented by a general stream function, i|/ (Weijermars and Poliakov, 1993):

4» = è] (xz cos + z sin 2£). (18) 1 1m. The streamline pattern implied by this particular stream function Stationary plate is fully described by the orientation, of the principal deviatoric stress T, (Fig. 6a). The principal strain rate e, is only scaling the rate Vr ?l Z of material flux across sections of the flowline patterns described 10° byi|>. b) a=7oAr The streamline patterns of Figure 7 have been mapped using the stream function of expression 18 with fixed values of £ and arbitrary ¿j. If 0° < £ < 45°, the horizontal boundary of the moving plate wall obliquely approach the stationary plate. The streamline patterns pos- sess a unique set of two straight streamlines. The extensional flow n asymptote coincides with the X-axis, the other asymptote to the hyperbolae is parallel to the direction of relative plate motion. These two straight streamlines have tj; = 0 because they cross the origin. Sub- Vr z stitution of »|» = 0 in expression 18 yields •w S = 20° tan 2£ = -x/z. (19a) C) a = 50°/\_ \ It is obvious from Figure 6b that -x/z is equal to tan (90° - a), so that a = 90° - 2i, (19b) X which is similar to an expression obtained independently by using the M kinematic vorticity number (Weijermars, 1991a, equation 18b). The implication of expression 19b is that estimates of the direction of relative plate motion, a, are sufficient to estimate the orien- Vr z tation, of the principal deviatoric stress axis in deformation zones Ç = 30° between two obliquely converging plates. More specifically, the ma- d) a = Stretching fault jor axis of deviatoric stress will coincide with the bisector of the acute \ angle between the velocity vector and the normal to the boundary of Deformation zone the deformation zone (Figs. 7a-7d). This conclusion is critically de- 1 N/S. pendent on the assumption of unconstrained lateral slip along the Locking point \ \ \ boundaries of the deformation zone together with the development of Té X separation points of no slip. ^^ Stretching fault Stagnation point- The scalar role of the strain rate in this analytical model may be mathematically demonstrated by substituting equation 19b into the Figure 7. a-d. Solutions of the stream function given in equation stream function of expression 18. This stream function can then be 18 for stress orientations £ = 5°, 10°, 20°, and 30°, with £ measured as rewritten using the direction of the boundary displacement, a defined in Figure 6a. The streamlines contoured are nondimensional, (Fig. 6b): using arbitrary length scales and strain rates. The inclination of the compressional flow asymptote traces the motion of the moving plate. i|> = ¿1 (xz sin a + z2 cos a). (20) Relationship between angles Ç and at is given by expression 19b.

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a) Figure 8. a-c. Sketches through the deforming unit volume explaining how deformation tensor elements Fn, F13, and F33 relate to physical dimensions. Indicated are the defini- tions of the constant angle, of the arbitrary • Unit length —I Stretching fault principal deviatoric stress, T„ with respect to the normal to the reference plane. Also indicated are the deformation parameters used in Deformation tensor: the calculations. These are the angle p between the reference plane and its rotated normal, the Fu 0 Fi3 principal stretches S, and S3 of the strain el- 0 I 0 lipsoid, and its inclination 0. 0 0 F33

axes shown in Figures 6a and 6b. A stream function for the same flows Fu = exp (Rt cos 2£) (24a) but with a reference frame always oriented at 45° to the principal deviatoric stress axis has been discussed elsewhere (Weijermars, F22 = 1 (24b)

1991a). 1 F33 = Fn (24c) The velocity components, vx and vz in XZ-space are partial dif- ferentials of the stream function: v = di|j/dz and v = -di|//dx. Dif- 1 x z F = (F„ - Fn" ) tan 2g (24d) ferentiation of the stream function (18) therefore yields the velocity 13 components: where the only two independent variables are the angle between the v = di|f/dz = ¿j x cos + 26] z sin 2£ (21a) smallest principal deviatoric stress T, and the normal to the stable x reference plane (Fig. 8b), and the nondimensional time R, = t e,.

vz = -dty/dx = —èi z cos 2£ (21b) Arbitrary quantities are time t and principal strain rate ex. Positive angles of £ are measured counterclockwise from the normal to the Expressions (21a and 21b) reduce to (7a, 7b) for £ = 0°. The alternative reference plane. Note that y0 does not change in planar flows, so that expressions in terms of the imposed direction of the boundary veloc- y = y0 in all cases. Expressions 24a-24c were obtained by time-in- ity, a (compare with equation 20) are tegrating the Eulerian rate-of-displacement equations. Expressions 24a-24d are practical for forward modeling of pro- v = d'bldz = èi (x sin a + 2z cos a) (22a) x gressive deformation, using arbitrary stress orientations and normalized times Rt. Figures 8a and 8b explain how the matrix elements vz = -di|

(x,y,z) = Fjj (xo,yo,z0) (23) Sj = [0.5(K + [K2 - 4]1/2)]1/2 (26a)

2 1/2 1/2 with Fjj = (dUj/dXj) + Sy, the deformation tensor, and (x^y^Zg) and S3 = [0.5(K - [K - 4] )] (26b) (x,y,z) the position vectors of an arbitrary material particle before and 2 : after deformation. In the case of homogeneous plane strain in the with K = Fu + F13 , provided F31 = 0 as assumed here by XZ-plane, only four of the nine elements of the deformation tensor are choice of reference frame. The angle 8 between the finite strain el- non-zero if the X-axis is chosen parallel to a nonrotating, lipsoid's major axis and the X-axis parallel to the stable reference slip-boundary to the deforming volume (Fig. 8a). The four non-zero plane is tensor elements may then be expressed in the following dynamic 2 2 terms (Weijermars, 1991a): 0 = 0.5 atan [(2 F13 F33)/(Fn 13 F33 )]. (27)

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Strain rate 10"14 s"1

z] JY

'z] Tv

\ I ^ ^—

Figure 10. Block diagrams of ductile deformation zone deforming by (a) horizontal pure shear with extension of deformation zone, and (b) horizontal simple shear, (c) and (d): the corresponding brittle deformations. Deviatoric stress trajectories and the associated stress functions Figure 9. a-d. Progressive deformation of a unit volume of ductile are discussed in GSA Data Repository material. rock within the deformation zone between two plates converging obliquely as outlined by the flowline s in Figures 7a-7d. The snapshots the difference in efficiency of pure and simple shear deformations). are essentially nondimensional, and the time steps of 2 Ma indicated are The finite strains in Figures 9a-9d, however, are close to 3, corre- for a characteristic geological strain rate of 10"14 s-1 or 0.315 Ma-1. sponding to those considered characteristic for orogenic strains Other cases have been investigated in an analytical computer simulation (Pfiffiier and Ramsay, 1982). fWeiiermars, 1991a). There is no difference between the homogeneous bulk deformations driven by a constant stress orientation and that due Obliquely Diverging Plate Motion to a constant direction of relative boundary velocity vr. More specifically, the major axis of the deviatoric stress, T„ coincides with the bi- Deformation zones in areas of intracontinental rifting may also be sector of the acute angle between the relative velocity vector and the understood better if modeled analytically by the simple mechanical normal to the boundary of the deforming zone. model adopted here. Consider a new plate boundary formed by two vertical faults at either side of a narrow rift or deformation zone with subparallel boundaries. The assumptions made above still apply, lim- Equations 24-27 have been used to reconstruct the finite defor- iting all deformation to the horizontal plane. Figures 10a and 10b show mation patterns in Figures 9a-9d. These are the deformations that the two extreme cases of extension by pure shear and parallel plate would occur in the flow fields of Figures 7a-7d after 2 and 4 Ma motion accommodated by simple shear, respectively. The corre- assuming a characteristic geological strain rate of 10"14 s_1. The max- sponding brittle cases are compiled in Figures 10c and lOd. Figure 11 imum stretches at 4 Ma vary with the direction of plate motion and are outlines the extensional flow field, governed by the stream function largest for smaller £ (see Weijermars, 1991a, for detailed discussion of (18) using Rvalues, 45° < £, < 90°.

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The study area selected here is located in a zone of distributed Moving plate continental deformation between the Eurasian and African plates. The deformed wall rock of the Moroccan Border fault is suitable for illustrating the new method of paleostress estimation, because it is extremely well exposed and is composed of relatively simple structural elements. The geology of the area is outlined below, before dis- 'J" cussing the details of the method for estimation of paleostress //// orientations. ¿-I Locking point \ *tft t \ Geological Setting Stretching ) / / //// / // I \ Deformation Detailed geologic mapping has established that strike-slip motion faults^ I l V zone is a characteristic feature of the Neogene deformation between the I ^ x Stagnation point

Stationar y Plate ^

Figure II. Streamlines or Bowlines relative to a reference frame fixed in an arbitrary stagnation point at one wall of the deformation zone between two obliquely diverging plates. The locking point is a point of no-slip at the wall opposite to the stagnation (or flow separation) point.

Figures 12a-12d illustrate the streamline pattern for various relative spreading directions of the rift-zone walls. The asymptote angle a' in Figures 12a-12d is measured counterclockwise from the X-axis. The direction of the major principal stress axis, T15 is still coinciding with the bisector of the angle between the normal to the plate boundary and the inverse of the relative velocity vector shown in Figures 12a-12d. This implies that the minor deviatoric stress axis, T3, always perpendicular to is bisecting the acute angle between the velocity vector and normal to the plate boundary. The finite deformation patterns in Figures 13a-13d correspond to those occurring in the flow fields of Figures 12a-12d after 2 and 4 m.y., again assuming a Characteristic geological strain rate of 10~14 s-1. The flows of Figures 12 and 13 also may be interpreted, looking at the diagrams with the Z-axis pointing downward, as corner flow models of subduction zones (compare with McKenzie, 1979; Ribe, 1989) dipping at various angles as outlined by the extensional flow asymptotes. Figures 13a-13d would then illustrate vertical sections orthogonal to the strike of the collision zone. These images help to visualize how a unit volume of an oceanic plate, or the root of a continental plate, deforms when sinking into the mantle.

APPLICATION OF PALEOSTRESS ESTIMATION TO MOROCCAN BORDER FAULT

The description of deformation patterns in rocks and associated reconstruction of the tectonic evolution of the terrain has been one of the principal aims of structural analysis. An additional step toward understanding a terrain's structural development can be made by at- tempting to determine the stress field responsible for the deformation. One complicating factor is that orogens usually expose rock units Figure 12. a-d. Solutions of the stream function given in equation deformed at different crustal levels and at different times, displaying 18 for stress orientations £ = 85°, 80°, 70% and 60°, with £ measured as an intricate pattern of polyphase folds, allochthonous nappes, and defined in Figure 6a. The streamlines contoured are nondimensional, core complexes, all transected and dislocated by various types of late using arbitrary length scales and strain rates. The inclination of the brittle faults. compressional flow asymptote traces the motion of the moving plate.

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dextral strike-slip faults. A splay pattern of smaller strike-slip faults, 4 Ma all consistently sinistral, segments the Betic Zone of southeast Spain and the coast of northwest Africa (for details, see Wildi, 1983). Ductile deformation patterns adjacent to the Palomares shear zone indicate that part of this motion is aseismic (Weijermars, 1987b, 1987c). Most of the aforementioned faults, however, are seismically active and ac- commodate part of the differential motion between the Eurasian and African plates. Figure 14 does not follow McKenzie's (1970, 1972) earlier con- 0 Ma tention locating the approximate plate boundary as an eastward con- Ç = 85° tinuation of the Azores transform through the Straits of Gibraltar (see a) Xi_ Fig. 28 in Weijermars, 1991b). D. P. McKenzie (1990, personal com- mun.) presently believes that his early sketch map of the plate 5° boundary has been interpreted too literally by subsequent workers: "I Vr 4 Ma r~j believe that the motion from about -15° W to Sicily is distributed on numerous faults and east of Gibraltar corresponds to shortening in a N-S direction" (quoted with permission). Seismicity in the western Mediterranean for the past 70 years can account for only 10%-25% of the 5 mm a-1 shortening rate inferred Jd from plate rotations (Jackson and McKenzie, 1988). Recent estimates suggest a smaller convergence rate of 2 mm a-1, implying that only MP 0 Ma J ¡J! 50% aseismic shortening may occur (Westaway, 1990). The seismic ç = 80 V- record is so short that quantitative conclusions may be biased, but \J J L 70° M T1 s 7 aseismic creep seems to contribute significantly to the bulk deforma- b) 10 — tion. Detailed reconstructions of the relative positions of Africa and Eurasia using updated aeromagnetic patterns of Atlantic spreading show that the separation between Iberia and Africa during Pangea's 200 Ma of dispersal has never exceeded 200 km (Klitgord and Schouten, 1986; Srivastava and Tapscott, 1986; compare with Vogt and Tucholke, 1989). This configuration implies even long-term convergence rates in the order of several mm a-1, extremely slow com- pared to convergence rates of about 5 cm a-1 estimated for the Indian- Eurasian collision zone (Tapponnier and others, 1982). Updated sea-floor-spreading data further suggest that Iberia moved as an independent plate for the major part of the Cretaceous magnetic quiet period until 84 Ma (chron 34), when Iberia became attached to the African plate (Srivastava and others, 1990; Roest and Srivastava, 1991). The plate boundary at 84 Ma was located in the Bay of Biscay which had previously opened by the sinistral rotation of Iberia away from Europe (Van der Voo, 1969). This agrees also with the closure of a subduction suture in the metamorphic core of the Betic Cordilleras, southeast Spain, tentatively dated to about 85 Ma Stagnation point (De Jong, 1987, 1990). Iberia moved mainly with Africa between 84 Figure 13. a-d. Progressive deformation of a unit volume of ductile and 42 Ma, mainly with Eurasia between 35 Ma and the present, and rock within the deformation zone between two plates diverging was an individual plate between 42 and 35 Ma (Roest and Srivastava, obliquely as outlined by the flow lines in Figures 12a-12d. The photo- 1991). The Gloria-Azores Fracture Zone became active about 42 Ma graphs are essentially nondimensional, and the time steps of 2 Ma in- (chron 18) and has stayed active until the present, whereas displace- dicated are for a characteristic geological strain rate of 10~14 s_1 or ment ceased along the boundary in the Bay of Biscay about 35 Ma 0.315 Ma-1. The direction of maximum stretch is parallel to the stretch- (Roest and Srivastava, 1991). Slow, but active, southwest-northeast ing fault or the X-direction for principal deviatoric stress orientations 0° spreading of 1 mm a"1 is occurring at the Terceira Rift near the Azores < i- < 45°. The direction of maximum stretch, however, is not parallel triple junction (Searle, 1980; Grimison and Chen, 1986). Modern seis- to the X-axis but aligned with the orientation of the extensional flow mic fault-plane solutions show right-lateral strike-slip faulting near the asymptote for deformations where 45° < £ < 90° (compare with Wei- Gloria fault, and thrust faulting with northwest-southeast contraction jermars, 1993). near Gibraltar (Grimison and Chen, 1986; Argus and others, 1989).

Moroccan Border Fault stable forelands of Europe and Africa in the western Mediterranean (see references in Weijermars, 1987a, 1991b). Figure 14 shows the This study concentrates on a major dextral strike-slip fault found surface traces of the major strike-slip faults. The east-northeast-strik- south of the Anti-Atlas Mountains (Fig. 14). This fault was first men- ing Guadalquivir, Crevillente, and South Atlas faults are primary, tioned explicitly by John Everett (pi. T-30 in Short and Blair, 1986; see

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Figure 14. Geotectonic map of the western Mediterranean showing the location of the m^jor strike-slip faults with respect to the tectonic provinces. Mapped are the Guadalquivir fault (1), Crevillente fault (2), Palomares fault (3), Almería fault zone (4), Jedha fault (5), Nekor fault (6), South-Atlas fault (7), and the Moroccan Border fault (8). (Compiled using data from Manspeizer and others, 1978; Her- mes, 1978; Bousquet, 1979; Van de Fliert and others, 1980; Vegas and Banda, 1982; LeBlanc and Olivier, 1984; Harmand and Cantagrel, 1984; De Smet, 1984; We^ermars, 1987a, 1987b, 1987c, 1988; De Larouziére and others, 1988.)

also pi. 323 in Short and others, 1976), but was left nameless and is 1958). The middle Rich is most pronounced and forms the transition referred to here as the "Moroccan Border fault." The fault trace between the Siegenian and Emsian stages. The competent sequences transects the Bani Mountains (or Jebel Bani, in transcribed Arabic), of the Rich are embedded in incompetent Devonian shales which are which are composed of Paleozoic sedimentary cover surrounding the stratigraphically underlain by similarly incompetent Gothlandian Proterozoic core of the Anti-Atlas. I am unaware of any detailed graptolite shales (Choubert, 1963). structural data from ground mapping in the vicinity of the Moroccan Choubert (1963), who did not recognize the Moroccan Border Border fault, the scene of Polisario guerrilla operations after Morocco fault, suggested that the Paleozoic cover of the Bas du Dra was folded assumed the protectorate of western Sahara from Spain in 1976. The during the Carboniferous. A 1:200,000 geologic map of the Dra region excellent exposure in this area facilitates geologic mapping by remote (Choubert and others, 1956), however, reveals that the Paleozoic sensing. Figure 15a is a Landsat image of the detailed area outlined in cover of the Bas du Dra basement dome is only gently and symmet- Figure 14; the corresponding geologic map is given in Figure 15b. This rically folded, whereas the tight Z-folds in the Rich are localized to the map is based on the 1:1,000,000 regional mapping of Choubert (1963, zone directly north of the Moroccan Border fault. The Paleozoic Fig. 8) completed in 1948 with the aid of stereoscopic aerial photo- cover along the south margin of the Tindouf Basin and in the south- graphs. All inferences made here are subject to modification if future east part of the Anti-Atlas are essentially nonfolded, subhorizontal ground studies allow refinement of the structural data. strata. The Z-folds in the Rich therefore are not attributable to a re- The Moroccan Border fault forms the contact between Devonian gional orogeny. Their age therefore may be different from the Vari- and Carboniferous sedimentary rocks younging toward the south. scan deformation responsible for the regional folds in the Paleozoic The dextral sense of shear is apparent from the asymmetry of folds of cover. a competent unit in the Devonian sequence. The folded marker beds Direct fieldmeasurement s of hinge-line plunges of the Z-folds are are exposed in three subparallel topographical ridges (the "Rich" of not available, but the outcrop pattern of the marker horizon is unaf- Choubert, 1963), each composed of a regressive marine cycle of fected by steep valleys outlined by height contours on the geologic oolitic limestone covered by shale and competent sandstone. The map of Choubert and others (1956). Stratum contour construction Devonian Rich ridges make up several hundred meters near the base using this map suggests that both flanks of the Z-folds are very steep, of a Devonian sequence, with a total thickness of 4 km (Hollard, and this implies that their intersection, and therefore the fold hinges,

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Figure 15a. Part of a color composite image of southern Morocco, 150 km wide, taken from 817-km height by Landsat space- craft on March 27,1973. This arid region, with only 10-15 cm of annual precipitation, exposes a geological history from the Precambrian to the present. The semicircular Precambrian basement inlier of Sidi Ifni near the Atlantic in the northwest (black) is covered by gently folded sedimentary rocks of Paleozoic age. In the south, the meandering Draa valley (white) covers the dextral Moroccan Border fault, which separates the subhorizontal Carbonifer- ous sequence of the Tindouf basin in the south wall from the tightly folded Devonian limestone and sandstone ("Rich") in the north wall. The image is orthographic, north is up and the scale of resolution is 30 to 80 m. For scale, 1 cm of the print matches 10 km on the ground. A full original of this cutout is pub- lished in NASA's Special Publication 360 (Short and others, 1976).

must also be steep. It was thus inferred that hinge lines are subver- quired to activate crystalline creep if stress can be relieved by fric- tical, and so the map pattern gives an orthogonal section through the tional slip (Byerlee, 1968, 1978; Goetze and Evans, 1979; Brace and fold profile. Kohlstedt, 1980; and refs. in Weijermars and others, 1993). The only The geometry of the Z-fold array can be used to estimate the processes that may modify rocks at shallow crustal levels are com- direction of the principal deviatoric stress responsible for this struc- paction, penetrative fracturing, and pressure solution (Gratier, 1987). tural pattern. The assumptions of plane strain, isotropic rheology, and These deformation mechanisms are related to the magnitude of the homogeneous deformation were considered before applying the tech- deviatoric stress in an arbitrary fashion, but equations 24a~24d de- nique of paleostress estimation to this particular area. The steeply scribe deformation patterns in terms of a stress orientation and bulk dipping Devonian beds of the Rich are interpreted to have been turned strain rate, without any genetic assumption about the processes in- subvertical by Hercynian regional homoclinal folding around the Dra volved. It is sufficient that the bulk rheology is isotropic on the mac- dome before deposition of the Carboniferous cover to the south of the roscopic scale, which is here assumed to have been the case during Moroccan Border fault. The subsequent development of the steeply the formation of the asymmetric Z-folds. The relationship between plunging fold hinges is confined to the north wall and fully attributed stress orientation and progressive deformation in anisotropic rocks to motion on the fault plane with a large component of simple shear has been elaborated elsewhere (Weijermars, 1992). (Fig. 15b). Such a model seems to preclude any significant material Laboratory experiments on folding of competent single layers transport along the fold-axes directions, thus excluding crustal thick- with imposed homogeneous bulk deformation suggest that the bulk ening and warranting plane strain. strength is largely controlled by the rheology of the matrix (Abbassi The folded competent marker beds are embedded in an incom- and Mancktelow, 1990). Although strain will be distributed inhomo- petent shale sequence. The rheology of these shales at the time of geneously around fold hinges, the bulk strain may be considered ho- folding is difficult to determine, but it is unlikely that the folds adjacent mogeneous at the scale larger than that of the characteristic fold- to the Moroccan Border fault are due to crystalline creep. Rock me- amplitude (see, for example, shear box experiments by Ghosh, 1966; chanics strongly suggest that the deformation mechanism which re- Abbassi and Mancktelow, 1990). Strain homogeneity at various scales quires the least deviatoric stress to work at shallow depths is brittle has been discussed elsewhere (Schwerdtner, 1973; Bell, 1981; Talbot, faulting. Deviatoric stress therefore cannot attain the magnitude re- 1987; Weijermars, 1993).

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Figure 15b. Detailed geologic map of area A outlined in Figure 14 and of the image of Figure 15a. The Moroccan Border fault separates Devonian and Carboniferous sedimentary rocks. The dextral sense of displacement is obvious from asymmetric folds in the Devo- nian sequence in the northern wall of the fault. The outlined folds are traced and used for the stress analysis in Figures 17a and 17b. Map- ping data are after Choubert and others (1956), Choubert (1963), and Everett (in Short and Blair, 1986).

Paleostress Estimation surfaces during rotation, because there is no material transport across these planes. The method outlined below is suitable for analyzing both stress An additional characteristic feature of folds of single competent orientation and finite strain magnitude from structural fold patterns at layers, which develop only at high viscosity contrasts, is that layer any scale, provided the bulk strain is homogeneous. The method of parallel shortening may amount to only 10%-20% (Hudleston, 1973; strain analysis outlined here is novel and complementary to other Abbassi and Mancktelow, 1990). Consequently, Frl may be deter- techniques developed mainly for application on outcrop scale (Ram- mined by measuring the arc length L^, of the folded competent layer, say and Huber, 1983, 1987; Lisle, 1988). Figures 8a and 8b illustrate and subsequently normalizing the spacing L of axial surfaces as meas- how the normalized lengths of a deformed unit volume correspond to ured along the reference plane: Fn = L/L,, (Fig. 16b). In general, the deformation tensor components Fn, F13, and F33. Figures 16a and folding of the competent layer occurs only if Fn < 1, and boudinage 16b outline how the normalized lengths Fn, F13, and F33 can be de- (not elaborated here) results ifFu > 1. F13 may be calculated from Fu termined for the practical situation involving steeply plunging folds and the angle p between the detachment surface and the axial surface, near a vertical detachment surface or stretching fault. Laboratory or any other rotated marker initially perpendicular to the detachment _1 experiments investigating the kinematic development of asymmetric surface using F13 = Fu /tan (3 (equation 25). folds have established that the axial planes of such folds are initially Combining expressions 24d and 25 yields equations allowing the perpendicular to the enveloping surface of the folded layer (Fig. 16a; determination of the stress orientation responsible for the bulk finite compare with Ghosh, 1966). The axial surfaces progressively rotate strain of the rock volume comprising the folded single competent toward the direction of shear as the amplitudes of the folds increase layer, using only Fu and (3: and the wavelengths decrease (Fig. 16b). The same experiments have established that fold hinges do not migrate through material planes 2 during fold amplification, and axial planes occupy the same material € = 0.5 atan {l/[(Fn - 1) tan (3]}, for Fu > 1 (28a)

Geological Society of America Bulletin, November 1993 1505

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(Fig. 17b). The calculated principal deviatoric paleostress orientation 2 ? = 0.5 I atan {l/[(Fn - 1) tan ß]} + 180° £ for locations A-F varies between 46° and 53° with respect to the normal to the subvertical Moroccan Border fault. These values cor- for Fn < 1. (28b) respond to angles of 37° to 44° with respect to the trace of the fault itself (Table 2). Incorporating a maximum value of 20% layer parallel The alternative equations (28a and 28b) arise from the condition in shortening of the marker horizon would modify the stress angles by equation 24d that F is always positive, because F > 1 for 0° < £ 13 n only 2°. An oblique section would affect the ratio L/Lg in Fn, as both < 45° and 0 < F < 1 for 45° < g < 90°. u L and L0 are likely to change with the profile. Additionally, the true P might be smaller than the apparent angle measured in any section Results oblique to the hinge line, and oblique sections would overestimate The predicted orientation for strike-slip faults based on laboratory The orthographic satellite image of Figure 17a illustrates the sur- tests of brittle failure is between 30° and 45° with respect to (Ander- face trace of the asymmetric folds in the Bani Mountains. Figure 17b son, 1951; Byerlee, 1968, 1978). The laboratory results suggest that the state of stress inferred from the structures in the northern wall of the Moroccan Border fault is consistent with stress orientations required to maintain strike-slip motion.

Finite strain parameters such as the stretches S , and S3 of a strain ellipsoid (Figs. 8a, 8b) are intrinsic to the deformation tensor and are Initial orientation a) of axial planes straightforward if the matrix elements Fu, F13, and F33 are known (equations 26a, 26b). Table 2 includes estimates of Sls S3, and 6 for the .Competent locations labeled A-F in Figure 17b. The finite strains are quite large layer with a principal stretch varying between 4.2 and 6.3. These strain estimates could be used to predict that a cleavage fabric might ac- Initial separation company the fold structures in the Rich. between axial planes Stretching fault The age of the Moroccan Border fault is poorly known, and it may have been in existence since the Carboniferous, according to the Lo" youngest strata transected by the faults. Expression 24a can be rear- ranged to estimate the time required to develop the asymmetric folds Final orientation of in the Bani Mountains: b) axial planes • / / t = (In Fn)/(0! cos 2£). (29) / Competent layer An average deviatoric stress orientation of £ = 50°, Fn = 0.67, and a /„ L0S / characteristic range of geological strain rates from 10"14 to 10~15 s_1 .z,^ P / / Stretching fault implies that the folds could have developed within 7.3 to 73 Ma. is The similarity in both the strike and sense of displacement between the seismic South Atlas fault and the Moroccan Border fault suggests that the two are genetically related. Geologic mapping has established that the collision zone between the Eurasian and African Figure 16. a and b. Sketches explaining how the normalized lengths plates in the western Mediterranean comprises a complicated pattern Lfl and L, the initial and final separation between axial planes, define the of active strike-slip faults (Fig. 14). The pattern of right-lateral strike- deformation tensor element Fu (= IVL«) portrayed as a physical length l slip faults in southern Morocco maybe inherited from the Alleghanian in Fjgure 8b. Deformation tensor element F33 is equal toFjj~ (equation (Variscan) orogeny because the Anti-Atlas was not detached from 24c). The deformation tensor component F13 follows fromF 33 and angle North America until dispersal of Pangea about 200 Ma (Mattauer and P through equation 25. See text. others, 1972). A major system of Alleghanian dextral strike-slip faults has also been mapped subparallel to the structural fabric of the Ap- palachians (Gates and others, 1986, 1988; Goldstein, 1989). The penetrative deformation patterns adjacent to the Moroccan Border fault indicate that part of the motion has been aseismic. Both the stress orientation and the structural features of the Moroccan Border fault maps the trace of both the Z-folds and the Moroccan Border fault. The are compatible with the modern picture of deformation in a broad traces of the axial planes were obtained by connecting the hinge points zone of coseismic and aseismic strike-slip motion. in the upper and lower contact of the Z-folds, simultaneously aiming for a best fit with the bisectors of their interlimb angles. The slight DISCUSSION deviation from parallelism between the fault trace and the enveloping surface of the folds was neglected. Deformation patterns are formed in the Earth's crust when stress Paleostress orientations were calculated for six locations (A-F) concentrations are sufficiently large to cause permanent distortion by by substituting estimates of Fu and 0 in equation 28b. The stress brittle faulting, ductile creep, or both. Studying such deformation pat- orientations inferred from asymmetric folds in the northern wall of the terns is important to develop a better understanding of the way in Moroccan Border fault are near 45° with respect to the fault trace which continents break apart, grow, and remold. Earthquake hazard

1506 Geological Society of America Bulletin, November 1993

Downloaded from http://pubs.geoscienceworld.org/gsa/gsabulletin/article-pdf/105/11/1491/3381722/i0016-7606-105-11-1491.pdf by guest on 27 September 2021 Figure 17a. Detail of Landsat image showing the trace of the folded competent single layer outlined in the map of Figure 15b and image of Figure 15a.

Figure 17b. Working map traced from Figure 17a outlining the Z-folds north of the Moroccan Border fault. Parameters used to determine the principal deviatoric paleostress orientations are listed in Table 2.

Geological Society of America Bulletin, November 1993 1507

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TABLE 2. PARAMETERS USED FOR ESTIMATING PALEOSTRESSES AT LOCATIONS deformation zones is homogeneous and isotropic. The subsequent A-F IN FIGURE 17b analysis explains, assuming differential slip at the subparallel bound- A B C D E F aries of the deformation zone, why uplift rates tend to accelerate during an orogenic period. Conversely, thinning and subsidence as- lfl(km) 11.6 11.2 7.6 7.4 30.4 6.0 sociated with crustal extension tend to decelerate during a stretching L (km) 7.2 7.6 7.2 4.8 15.2 3.6 ß 15° 15° 15° 15° 20° 20= episode. Careful separation of total stress, deviatoric stress, and con- Fu (IVLo) 0.62 0.68 0.95 0.65 0.5 0.6 fining pressure offers explanations for unexpectedly closely spaced FU (Fn-;/tan ß) 6.01 5.49 3.93 5.74 5.50 4.58 F33(F,r') 1.61 1.47 1.06 1.54 2.00 1.67 metamorphic isograds in collisional areas. The confining pressure be- ? 50° 49° 46° 49° 53° 52° S, 6.25 5.72 4.17 5.98 5.87 4.91 tween converging plates will exceed the normal lithostatic value of s3 0.16 0.18 0.24 0.17 0.17 0.20 e 15° 15° 14° 15° 20° 20° nondeforming areas because there is a tectonic contribution to the confining pressure. The structural patterns within the idealized deformation zones control benefits from the movement that history recorded by defor- can be related to the orientation of the principal stress axes, using a mation structures exposed in the wall rock of seismically active faults. fundamental deformation tensor. The associated regional flow pattern The reconstruction of the tectonic history of deformation structures within the deformation zones, described by a concise stream function, exposed in rocks therefore is an important activity of structural ge- is then used to relate the orientation of the major principal stress axis ologists, field geologists, and tectonicians. One possible approach is to the orientation of the relative velocity vector of the juxtaposed backward modeling, meant here to reconstruct from a finite defor- plates. In deformation zones between converging plates, the major mation pattern mapped in nature, the entire deformation history in axis of deviatoric stress, T1? will coincide with the bisector of the acute logical steps back to the undistorted situation. It is not advisable, angle between the velocity vector and the normal to the boundary of however, to focus research interest on backward modeling only, be- the deformation zone (Figs. 7a-7d, 9a-9d). In deformation zones because forward modeling helps to develop a better understanding of tween diverging plates, it is the minor deviatoric stress axis, T3, which the physical parameters and boundary conditions which control tec- is bisecting the acute angle between the velocity vector and normal to tonic deformations. the plate boundary (Figs. 12a-12d, 13a-13d). In view of the geometric variety of deformation patterns exposed The forward modeling advocated here considers the progressive in the Earth's tectonic provinces, it seems worthwhile to start with a development of deformation patterns in an analytical model using a simple model of deformation zones between two mobile rigid plates. set of algorithms to describe the deformation history. The deforma- One major simplification made here is that the bulk rheology of the tion tensor used relates to physical dimensions in a simple fashion and

Figure 18. a and b. Progressive deformation by pure shear and simple shear can be distinguished if a physical boundary of the deforming rock volume can be established. The orientation of the principal stress axis remains (a) at 45° to the stretching fault at the base of the rock volume deforming in simple shear and (b) perpendicular to the fault plane in pure shear deformation. The marked unit volume deformed in Figure 18b is congruous to that in Figure 18a, but only if its initial orientation is chosen such that one side is at an angle of 56° with the stretching axis of the incremental strain ellipse. This holds true only for a pure shear of 66% flatteningan d a simple shear of 0.83, and they can still be told apart. Further insight may be generated by comparison with figure 1.12 in Hobbs and others (1976, p. 25).

1508 Geological Society of America Bulletin, November 1993

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therefore can be used to backward model the paleostress orientation ACKNOWLEDGMENTS which led to deformation features exposed in the wall of the Moroc- can Border fault. The Moroccan Border fault seems to be one location The calculations on the relationship between plate velocity and where the paleostress directions determined from megascopic, asym- strain rate and stress functions were elaborated while staying as a metric fold patterns visible on Landsat images are consistent with the Visiting Research Fellow at the Bureau of Economic Geology dextral strike-slip motions on the faults separating the deformation (BEG) of the University of Texas at Austin (1990). The theory of finite zone from the adjoining crustal segments (that is, Moroccan Border deformation was completed during an earlier stay at the Federal fault arid South Atlas fault). This agreement suggests that two of the Institute of Technology (ETH) in Zürich, Switzerland (1989). The principal assumptions made (homogeneous plane deformation with- illustrations were designed working as a Research Scientist at the out significant volume change, and an isotropic bulk rheology) are Hans Ramberg Tectonic Laboratory (HRTL) of the University of valid for the wall rock of the Moroccan Border fault at the time of the Uppsala, Sweden (1991). The final text was written, and later revised, Z-folding in the Rich. in between teaching semesters at King Fahd University of Petro- The method of stress and strain analysis outlined here also sug- leum and Minerals (KFUPM) in Dhahran, Saudi Arabia (early 1992). gests that progressive deformation by pure shear and simple shear can I am indebted to all hosts and friends at the various institutions men- be distinguished if the field relationships indicate that rock has de- tioned. Particularly thanked for unreserved support of my work in formed with one face adjacent to a stable rigid boundary. This would progress are John Ramsay (ETH), Martin Jackson (BEG), Hans be an exception to the common understanding among field geologists Ramberg and Hans Annersten (HRTL), and Abdulwahab Abokho- that it is impossible to determine whether a particular deformation dair (KFUPM). Win Means is thanked for constructive criticism in an pattern has been formed by pure shear or simple shear, for which the early stage of this work. Research grants came principally from the principal deviatoric stress is perpendicular to a fixed reference plane Natural Science Research Council of Sweden (NFR). in the material, and at 45°, respectively. This caution is because "the only differences between such a pure shear and a simple shear are a rigid body rotation and a rigid body translation" (Hobbs and others, REFERENCES CITED 1976; following Jaeger, 1956). Abbassi, M. R., and Mancktelow, N. 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