Analysis of fault slip inversions: Do they constrain stress or strain rate?

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. B6, PAGES 12,205-12,222, JUNE 10, 1998 Analysis of fault slip inversions: Do they constrain stress or strain rate? Robert J. Twiss Geology Department, University of California, Davis jeffrey R. Unruh William Lettis & Associates, Walnut Creek, California Abstract. Fault slip data commonly are used to infer the orientations and relative magnitudes of either the principal stresses or the principal strain rates, which are not necessarily parallel or equal. At the local scale of an individual fault, the shear plane and slip direction define the orientations of the local principal strain rate axes but not, in general, the local principal stress axes. At a large scale, the orientations of P and T axes maxima for sets of fault slip data do not provide accurate inversion solutions for either strain rate or stress. The quantitative inversion of such fault slip data, however, provides direct constraints on the orientations and relative magnitudes of the global principal strain rates. To interprethe inversion solution as constraining the global principal stresses requires that (1) the fault slip pattern must have a characteristic symmetry no lower than orthorhombic; (2) the material must be mechanically isotropic; and (3) there must be a linear constitutive relationship between the global stress and the global strain rate. Isotropic linear elastic constitutivequations are appropriate to describe the local deformation surrounding an individual slip discontinuity. Fault slip inversions, however, constrain the characteristics of a large-scale cataclastic flow, which is described by constitutivequations that are probably, but to an unknown degree, anisotropic and nonlinear. Such material behavior would not strictly satisfy the requirements for the stress interpretation. Thus, at the present state of knowledge, fault slip inversion solutions are most reliably interpreted as constraining the principal strain rates. 1. Introduction Large-scale brittle deformation of the Earth's crust commonly is accommodated by brittle fracture and frictional sliding, and under some circumstances by slow progressive creep, on a multitude of differently oriented shear planes in the rock. Within a volume of rock that is much larger than the extent of any individual slip event, the net result of a large number of distributed slip events is a quasi-continuous deformation referred to by structural geologists as a cataclastic flow. The seismogenic component of this flow has been termed a "seismic flow" by Kostrov [1974] (referencing Riznichenko [1965]). Data on the orientations of the shear planes and their associated slip directions, which we refer to as "fault slip data," can be obtained either from field measurements of shear planes and slickenlines or from seismic focal mechanisms. Because the local slip events combine to produce the large-scale cataclastic flow, it is reasonable to assume that patterns of slip on faults are related in a systematic way to the characteristics of the global deformation. The interpretation of slip directions on shear planes has long been of interest to both structural geologists and seismologists. Among various ways of interpreting such data, two main hypotheses have been proposed that can account in detail for the slip directions on shear planes of different orientation. The stress hypothesis assumes that the slip direction Copyright 1998 by the American Geophysical Union. Paper number 98JB /98/98JB in any given plane is parallel to the direction of maximum resolved shear stress of a large-scale homogeneous stress tensor. Analysis of fault slickenlines in this manner was first proposed by Wallace [1951], and it has been applied by many workers, for example Bott [1959], Angelier and several of his coworkers (for references see review article by Angelier [1994]), Michael [1984, 1987], Michael et al. [1990], Gephart and Forsyth [1984], Lisle [1987], Gephart [1990], Zoback [1989, 1992], Zoback and Beroza [1993], Taboada [1993], and Seeber and Arrnbruster [1995]. Tarantola and Valette [1982], Angelier et al. [1982], and Yin and Ranalli [1993] have done work to put the inversion schemes on a sound statistical footing. The kinematic hypothesis, however, assumes that the slip direction on any given plane is parallel to the direction of maximum resolved shear rate of a largescale homogeneous strain rate tensor, modified in the micropolar theory [Twiss et al., 1991, 1993] by the relative rate of rigid block rotation. The kinematic approach has been used by Twiss and Gefell [ 1990], Marrett and Allrnendinger [ 1990], Cladouhos and Allrnendinger [1993], Taboada [1993], Twiss et al. [1991, 1993], Twiss and Unruh [1994], and Unruh et al. [1996, 1997], although in the first four of these references a formal fault slip inversion is not used. Both the stress and the kinematic hypotheses provide a theoretical basis for determining constraints on the orientation and relative magnitudes of either the principal stresses or the principal strain rates (and relative vorticity), respectively, from a sufficiently large set of fault slip data. The principal stresses and the principal strain rates, however, are not necessarily parallel or proportional in magnitude, and thus it is 12,205

2 12,206 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS important to understand which tensor is more directly constrained by the data. The literature does not contain a formal analysis of the relative merits of these interpretations, although Allrnendinger [1989] mentions some of the differences, and Schrader [1997] has recognized the difference between the stress and the kinematic hypotheses. Generally, either the difference is considered immaterial, or one interpretation is simply preferred over the other. In this paper we attempt to make a formal distinction between the stress and kinematic interpretations and to evaluate their relative merits. Section 2 introduces some defini- tions and conventions. Section 3 discusses various ways of interpreting fault slip data. Section 4 evaluates the interpretations of fault slip data and focuses on a comparison of the stress and kinematic hypotheses. We conclude that the stress interpretation requires some general assumptions about the constitutive characteristics of cataclastic flow which the kinematic hypothesis does not. Section 5 examines constitutive relations for distributed brittle deformation. We first dis- cuss the role of scale in choosing an appropriate continuum description and then focus on probable constitutive characteristics for cataclastic flow. We conclude that rocks may not satisfy the assumptions of the stress hypothesis. Section 6 reviews some observational constraints on fault slip interpretations. Definitive tests that can distinguish between the stress and the kinematic interpretations have not been performed. We present some evidence, however, that is at least consistent with our conclusion that fault slip data most reliably constrain the strain rate. Section 7 presents a summary and discussion of our conclusions. 2. Definitions and Conventions The data on which fault slip inversion schemes have been based consist of four pieces of information: (1) and (2) Two numbers, such as the strike and dip, define the orientation of the shear plane or fault surface. The orientation is observed directly in the field or inferred to be one of the nodal planes of a seismic focal mechanism. (3) The orientation of the slip line is defined, for example, by the rake of the line in the shear plane. In the field, the slip line is parallel to the slickenlines on fault surfaces; for focal mechanisms, it is the normal to the intersection line of the two nodal planes that lies in each nodal plane. (4) The shear sense on the shear plane is defined by a variety of shear sense criteria on a fault surface [e.g., Petit, 1987] or by the orientation of the seismic P and T axes relative to the pair of nodal planes (Figure 1). We use the terms "shear plane" and "slip line" or the collective term "fault slip" to refer to data obtained either directly from fault surfaces or from seismic focal mechanism solutions. We define two unit vectors, P and T, which are parallel to the seismic P and T axes, respectively, and are related to each fault slip datum by V- l V+ l P - T - (1) Iv- nl Iv + nl where xl is the unit normal vector to the shear plane and v is the unit vector parallel to the slip direction of the block into which xl points (Figure 1). Both P and T are unit vectors that lie in a plane containing the directions normal to the shear plane and parallel to the slip line, and both are oriented at 45 ø to the slip line. I I I I I v Figure 1. Definition of the P and T unit vectors parallel to the seismic P and T axes, respectively, in terms of the shear plane normal xl and the slip direction v of the block into which xl points. There is an inherent nonuniqueness in the choice of the shear plane from the two nodal planes of a seismic focal mechanism. Usually, the focal mechanism is determined, and then the nodal plane that gives the best fit to the deformation model is assumed to be the shear plane [Gephart and Forsyth, 1984; Gephart 1990; Unruh eta!., 1996, 1997]. Horiuchi et al. [1995] picked the focal mechanism solution and the shear plane simultaneously in the inversion of p wave first arrivals. Geologic criteria also may be used to pick the shear plane [Michael, 1987]. We do not concern ourselves further with this problem here, but Michael [1987] has explored the effects on inversions of the ambiguity in choosing the fault plane from the two nodal planes. Selecting the shear plane on the basis of minimum misfit tends to underestimate the true misfit, because this selection is not always correct. We also do not concern ourselves with the difference between focal mechanism data and shear plane/slickenline data. The latter probably accumulate over a much longer period of time than focal mechanism solutions from an aftershock sequence, for example, and because of temporal variations in the deformation characteristics, they can be expected to show larger misfits to the models. These two sources of difference in misfit between the focal mechanism data and the shear plane/slickenline data probably account for the observation that the misfits for the former data sets tend to be smaller than for the latter. In describing the kinematic variables, we often refer to the "deformation rate" rather than just the "strain rate." We include in the term "deformation rate" both the strain rate tensor and the relative vorticity W that arises in micropolar theory from an independent component of rigid fault-block rotation [Twiss et al., 1991, 1993], which we define in more detail in section 3.5. Although the kinematic theory is developed in terms of the deformation rates, in practice the measured data actually provide information only on small increments in the deformation and do not include any absolute timescale. The deformation rate, however, is nothing more than an infinitesimal increment of deformation divided by the infinitesimal increment of time over which it accumulates. Because time is a scalar quantity, dividing the components of the incremental strain tensor or incremental relative rotation by the same time increment does not change either the orientations of the principal axes or the relative magnitudes of the principal values. Thus, except for this scale factor, which is not determined by the

3 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS 12,207 inversion, the geometries of the strain rate tensor and the incremental strain tensor are identical, as are the geometries of the relative vorticity and the relative incremental rotation. The increments of seismogenic deformation that are represented by individual earthquakes, when averaged over the scale of the deforming crust, are effectively infinitesimal. Thus, for the purposes of this analysis the deformation rate and the incremental deformation can be used interchangeably. In the subsequent discussion, it becomes necessary to refer to two relative length scales, which we term the "local scale" and the "global scale." The characteristic length for the local scale is the dimension of the volume that contains a single slip event and provides a single fault slip datum. For a focal mechanism, the local characteristic length is adequately defined by the rupture radius r; for a shear plane/slickenline field measurement, it is defined by a characteristic dimension of the local fault block. The global scale is large relative to the local scale. It is the dimension of the volume at which the accumulated deformation from many slip events of roughly the same rupture radius r on a variety of different local shear surfaces can be adequately described as a homogeneous continuum deformation. From a global volume we need a sufficiently large set of fault slip data in order to find an inverse solution for the homogeneous pri.- cipal deformation rates or the homogeneous principal stresses. Thus the local and global scales are well defined relative to each other, but they do not have an absolute definition because the rupture radius of individual slip events, or the scale of a fault block, has a large range. We discuss the implications of scale in more detail in section 5.1. We present fault slip data on stereograms as tangentlineation diagrams, on which the orientation of each shear plane and its slip line are plotted as a single arrow (Figure 2a). The fault-and-stfiae diagram is the alternative, and more common, method of plotting fault slip data on a stereogram (Figure 2b). The plot of the two orthogonal conjugate strikeslip faults in Figures 2a and 2b can be compared to the associated focal mechanism solution shown in Figure 2c. If numerous data are plotted in the same diagram, a comparison of these two plotting methods (Figures 3a and 3b) shows that the faultand-striae diagrams become unintelligible "string balls" that convey little if any information, whereas the tangentlineation diagrams are clear and uncluttered. Note for comparison that the data plotted in Figure 3 are only one third of those plotted in Figure 4a. The tangent-lineation diagram has the added advantage that it reveals clearly the symmetry of the slip line pattern, which is important to our arguments in this paper. In our view, the tangent-lineation diagram communicates essential information about the deformation more effectively and efficiently. 3. Review of Fault Slip Interpretation 3.1. Interpretation of Local Deformation: Slip on a Single Fault causing the slip event [e.g., Raleigh et al., 1972]. This is equivalent to assuming a yon Mises failure criterion for which the principal stresses are at 45 ø to the failure plane and the failure stress is independent of the confining pressure. Experimental data indicate that brittle or frictional failure is not independent of the pressure, and therefore this assumption does not seem well justified. In other cases, the principal stress orientations are inferred from the fault and slip orientations by assuming the applicability of the Coulomb fracture criterion, which includes the pressure sensitivity and is consistent with experiment. An alternative approach, however, is to interpret the direction of slip to be the direction of maximum resolved rate of shear on the fault surface. We examine these different interpretations in detail. The slickenlines on a shear plane quite clearly record the relative displacement of the opposite sides along the shear plane. The two perpendicular nodal planes of a seismic focal mechanism are constructed from data that define the radiation pattern from the seismic event, which, in turn, is directly related to the displacement on the fault. Thus these data most directly provide information about the displacement, not the In the interpretation of individual fault slip observations, continuum deformation. The assumption of global-scale the orientations of the principal stresses sometimes are smoothing is implicit in the inversion schemes of fault slip assumed to be uniquely determined by the orientations of a data. Regardless of whether the stress or the kinematic fault and its slip vector. In some cases, the P and T axes of a seismic focal mechanism are taken to be parallel to the maxihypothesis is assumed, the inversion schemes are based on three fundamental assumptions: (1) Slip accumulates on premum 6' 1 and minimum t 3 compressive stress, respectively, existing fractures having a wide diversity of orientations. stress. Considering a deformation accommodated by shearing on a particular shear plane, the direction of shearing is the direction of maximum rate of shear on the plane. Moreover, the local strain rate that is accommodated by that shearing necessarily has principal axes oriented at +45 ø to the slip direction and lying in the plane that contains both the normal to the shear plane and [he slip,, direction. Thus the local principal strain rate axes d 1 and d 3 (lengthening positive) inherently have the same orientation relative to a shear plane and its orthogonal conjugate plane as the seismic T and P axes, respectively, have relative to the nodal planes, so the two pairs of axes are equivalent. The association between the P and T axes and the local prin- cipal stress axes, however, is less obvious and less direct. Experiments show that in brittle fracture the maximum compressive stress 6'1 is oriented at less than 45 ø to the brittle fracture plane, but as McKenzie [1969] has pointed out for brittle faulting on preexisting fractures the maximum compressive stress can, in principle, lie anywhere in the quadrant that would be defined by dilational first arrivals. This same relationship would also apply to the brittle fracture of anisotropic materials. Thus the P and T axes generally do not provide an accurate measure of the orientation of the local principal stresses Interpretation of Global Deformation: Slip on a Multitude of Faults If the deformation we are considering is accommodated by a large number of slip events on a multitude of different shear planes and if the scale at which we choose to describe that deformation is large compared to the scale of the faults on which the deformation is accommodated, then we can smooth the local discontinuous slip events on individual faults over the large volume and describe the resultant deformation as a

4 12,208 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS Thus, in general, the shear planes are not in the optimum hypothesis adopted. (3) The local slip direction on each indiorientations with respect to the principal stresses that are vidual shear plane is for the stress hypotheses parallel to the predicted by the Coulomb failure criterion. (2) The global direction of the maximum resolved shear stress of the global volume of rock from which the data are taken for an inversion stress tensor; for the kinematic hypothesis, it is parallel to is large compared to the scale of the local slip discontinuities the direction of maximum resolved rate of shear for the combithat contribute to the deformation, and on that global scale the nation of the global strain rate tensor and the relative vorticstress or the deformation rate tensors are necessarily homoge- ity vector. neous. Thus the local slip direction on each individual shear The second assumption implies that although the local plane reflects, on average, the characteristics of a homoge- deformation can be described as a discontinuous slip on a disneous global tensor, either the global stress tensor or the crete shear plane, we can infer a global stress or a global global strain rate tensor (possibly with a relative vorticity, deformation rate only if the local slip discontinuities integrate which we define in section 3.5), according to the particular over a global volume to a homogeneous continuum deformation. Pollard et al. [1993] investigated theoretically the extent to which local slip discontinuities and their interaca Normal tions affect the deviation of principal stress orientations from I the ideal homogeneousolution. They found that under most conditions the predicted deviations were not excessive and that therefore the assumption of homogeneity was adequately strike slip tt strike slip Thrust -!- Thrust justified. Michael [1991] has shown that this assumption can be justified as long as the uniform component of the stress field is at least as large as the spatially varying component. Large-scale inhomogeneities in the global fields often can be analyzed by subdividing the domain into smaller more homogeneous subdomains [e.g., Michael et al., 1990; Twiss and Dextral,, Sinistral strike slip strike slip Normal Figure 2. Plotting conventions for fault slip data showing the idealized AndersonJan fault types. Diagrams are lower hemisphere equal-area projections. (a) Tangent-lineation plot. The fault plane orientation is plotted as the pole (normal) on the lower plotting hemisphere; it can be thought of as the point where the fault plane orientation is tangent to the external surface of the lower plotting hemisphere. The slip direction is plotted on the stereogram as an arrow through the shear plane pole, and it is parallel to the slip line in the tangent plane. The arrows indicate the direction of slip of the material outside the plotting hemisphere relative to the fixed hemisphere [Twiss and Gefell, 1990]. This is a generalization of the convention for characterizing strike-slip faults which is based on the slip direction of the block on the opposite side of the fault from the observer. For inclined faults plotted on a lower hemisphere projection therefore the arrows indicate the slip direction of the footwall block. This convention permits the diagram to be viewed as a plot of the direction in which material moves past the outside of a fixed lower hemisphere. The opposite convention was used in the original introduction of this diagram [Goldstein and Marshak, 1988; Hoeppener, 1955]. (b) Fault-and-striae diagram of the same fault slip data as is plotted in Figure 2a. The shear plane is plotted as a great circle on which lies a point indicating the lineetlon orientation. The arrows attached to the lineetlon orientation indicate the slip direction of the hanging wall block; for vertical faults, however, the arrow indicates the slip direction of the block to the right of the fault when viewed facing in the direction of the lineetlon trend. If the lineetlon is horizontal, as for the strike slip faults, there are two equivalent options for plotting its trend, leading to two consistent possibilities for showing the slip direction. (c) Focal mechanism "beach ball" diagram corresponding to the pair of strike slip faults plotted in Figures 2a and 2b. b Normal Sinistral slip Dextral strike sli Thrust C Sinistral strike slip t t Normal p P Figure 2. (continued) strike slip.t

5 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS 12,209 7/= 0.0 D = 0.5 b W=0.0 D = 0.5 define characteristics of the global stress tensor. Three of these parameters define the orientations of the principal stress axes, designated 6l > 2 > 3 (compression positive). Note that we use a circumflex on all components that are given in principal coordinates. The fourth parameter, either R or p is a scalar invariant of the stress tensor that constrains the shape of the stress ellipsoid, ' 2-6' 3 ' O'1 - '3 R- = l_ 3 Ip-- r, R=I- p. (2) The values of these parameters are not affected either by the multiplication of each of the principal stresses by the same scalar constant or by the addition of any isotropic stress (a pressure) to the state of stress. If we know the slip directions on a large number of shear planes (roughly 20 or more) having a wide distribution of orientations, then we can search for the values of these four parameters that minimize an average over all the data of a specified measure of misfit between each observed fault slip datum and the closest fault slip orientation predicted theoretically from the model parameters. The exact measure of misfit used and the search procedure for finding a solution vary among different researchers [e.g., Angelier, 1994; Michael, 1984; Gephart and Forsyth, 1984; Gephart, 1990; Yin and Ranalli, 1993, Horiuchi et al., 1995], but the principle remains the same Conventional Kinematic Hypothesis Figure 3. Comparison of a tangent-lineation plot with a fault-and-striae plot for the same set of 200 fault slip data and for fi deformation geometry defined by a plane strain (D = 0.5) with a zero relative vorticity of rigid fault blocks (W = 0.0). (See text for definitions of the deformation parameters D and W). These are a subset of the 600 fault slip data in Figure 4a. Diagrams are lower hemisphere, equal-area projections: (a) Tangent-lineation plot and (b) Fault-and-striae diagram of the same data as in Figure 3a. Unruh, 1994; Amelung et al., 1994; Seeber and Armbruster, 1995; Gephart, 1995, 1997; Unruh et al., 1996, 1997]. Implicitly, however, the scale of the local slip events must be small relative to the scale of the global domain used for analysis Stress Hypothesis To predict the slip direction on any given plane using the stress hypothesis, one must make the three assumptions listed above, and one must specify four model parameters, which To predict the slip direction on any given plane using the conventional kinematic hypothesis, one must again make the three fundamental assumptions discussed above, with the proviso that the relative vorticity be zero. In addition, one must specify four characteristics of the strain rate tensor, analogous to the case of the stress hypothesis. Three of these parameters define the orientation of the principal strain rate axes, desig- nated 1 >- 2 >- 3 (lengthening positive). The fourth parame- ter D is a scalar invariant strains the shape of the strain rate ellipsoid, of the strain rate tensor that con- D t l _t 3. (3) The value of D is unaffected either by the multiplication of each of the principal strain rates by the same scalar constant or by the addition of any component of volumetric strain rate to the deformation. The inversion for the four parameters characterizing the strain rate tensor proceeds in an identical manner to the inversion for the stress tensor: We seek the values of the parameters that minimize an average over all the data of the misfit between each fault slip datum and the closest fault slip orientation predicted theoretically using these four parameters Micropolar Kinematic Hypothesis The conventional kinematic fault slip inversion method requires that any local deviations from the homogeneous global kinematics should be random and should integrate to zero over the global volume. Local motions such as fault block rotations, however, can cause deviations of the slip directions from those predicted by the conventional kinematic model that are both systematic and independent of the global strain rate. Under these conditions, the local motion of a fault

6 12,210 ß TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS W=0.0 D = T axes 0.5 Figure 4. The relation between tangent-lineation diagrams of fault slip data, P and T axes diagrams, and the associated symmetry axes for a deformation model with zero relative vorticity of rigid fault blocks (W = 0.0) and four plane strain(d = 0.5) and with the principal strain rate axes dl and d 3 horizontal. (a) Tangent-lineation diagram for 600 nearly uniformly distributed shear planes in a lower hemisphere, equal-area projection. (b) P and T axes preferred orientations for the fault slip data in Figure 4a. Notice the maxima correspond with the principal strain rate axes. Kamb contours are at intervals of 3 standard deviations in a lower hemisphere, equal-area projection. (c) Symmetry elements that characterize the orthorhombic fault slip pattern in Figure 4a. The "m" indicates a mirror plane of symmetry; the ellipses mark opposit ends of twofold axes of rotational symmetry. block is described by the motion of its centroid, which defines the global strain rate, and by an independent component of rotation about its centroid. The fault block rotation rate does not contribute to the global strain rate, but it does affect the slip directions on the local shear planes because these directions are determined by the local motions. The effect of a systematic local rotation rate therefore must be accounted for to ensure a correct inversion for the global strain rate. The micropolar model [Twiss et al., 1991, 1993; Unruh et al., 1996] includes the effect of the local block rotation rate in a continuum description of the global deformation rate by means of the relative vorticity. The relative vorticity is in fact an axial vector, but for a restricted kinematic model in which the relative vorticity vector is assumed parallel to 2, it is characterized by the scalar relative vorticity parameter W defined by W = &13 -½ ( l 3), (4') where the principal coordinates implied by the circumflexes on the components in (4) are those of the strain rate tensor, where '13 is the spin component (the ant,,isymmetric part) of the global velocity gradient tensor about d 2, and where 13 is the independent spin component describing the local rigid block rotation rate about d 2. The denominator is the maximum possible shear strain rate obtainable from the global strain rate tensor. W represents an extra degree of kinematic freedom that describes a normalized measure of the difference between the rotation rate of the rigid fault blocks 13 and the average rotation rate of global material lines in the global deformation of the continuum ½13. Thus there are five model parameters in the restricted micropolar kinematic model, four being the same as for the conventional kinematic model (orientations of 1-->t 2-->t 3 and the value of D (equation(3)) and the fifth being the relative vorticity parameter W (equation(4)). Finding a solution requires finding those values of the five model parameters that minimize the average over all the data of a measure of misfit between each fault slip datum and the closest fault slip orientation predicted theoretically using the model parameters. Thus

7 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS 12, d2 d3 Figure 5. The relation between tangent-lineation diagrams of fault slip data, P and T axes diagrams, and the associated symmetry axes [or a deformation model with W = 0.5, D = 0.5, and the d I and d 3 axes horizontal. See caption to Figure 4 for plotting conventions. (a) Tangent-lineation diagram for the same 600 shear planes used in Figure 4. (b) P and T axes preferred orientations for the fault slip data in Figure 5a. Notice the maxima are not orthogonal and do not correspond with the principal strain rate axes, an effect of the nonzero value of W. Kamb contours are at intervals of 3 standard deviations in a lower hemisphere, equal area projection. (c) Symmetry elements that characterize the monoclinic fault slip pattern in Figure 5a. it is similar to solving the problem for the stress hypothesis or the conventional kinematic hypothesis, except that in the micropolar case there is one additional model parameter. The particular grid search procedure and the definition of the misfit that we minimize are described by Unruh et at. [1996]. In many cases, inversion based on the micropolar model gives values of W equal to or near zero. In these cases, the results of the inversion are no different from those using the conventional kinematic model. In other cases, however, the fit of the inversion is markedly improved by a nonzero value for W, and we have found decreases in misfit of up to 20% for focal mechanism data and up to 50% for shear plane/slickenline data Characteristic Symmetry of Fault slip Data The observed fault slip data are assumed to fit a specific theoretical pattern that can be calculated on the basis of either the stress or the kinematic hypotheses. These theoretical patterns can be conveniently displayed on tangent-lineation diagrams (Figures 4a and 5a). The specific pattern varies with the values of the parameters that define the relative magnitudes of the principal deformation rates (D and W) or the stress (R), and the orientations of the axes and planes of symmetry that characterize that pattern are related to the orientations of principal axes of either the homogeneous global deformation rate or the homogeneous global stress. It is the uniqueness in the association between such fault slip patterns and the parameters describing the stress or deformation rate that allows the parameters to be inferred from fault slip data. The characteristic symmetry of a fault slip pattern is the idealized symmetry shown by the slip directions on a set of shear planes whose poles are uniformly distributed on a stereogram and therefore have an isotropic distribution of orientations in space [Twiss et at., 1991]. The symmetry elements that define the symmetry include rotation axes, mirror planes, and centers of symmetry. For example, the characteristic symmetry of the fault slip pattern in Figure 4a is orthorhombic, which implies there are three orthogonal twofold axes of rotation and three mirror planes of symmetry, one perpendicular to each of the twofold rotation axes (Figure 4c). The characteristic symmetry of the fault slip pattern in Figure 5a is monoclinic, which implies there is one twofold axis of rotation perpendicular to one mirror plane of symmetry (Figure 5c).

8 12,212 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS Both the stress hypothesis and the conventional kinematic hypothesis predict characteristic fault slip patterns with orthorhombic symmetry or higher, because orthorhombic symmetry is the minimum possible symmetry of the ellipsoids that characterize the stress and strain rate tensors. Monoclinic fault slip patterns are predicted only by the micropolar theory and only when the relative vorticity parameter W is nonzero [Twiss et al., 1991, 1993]. Triclinic patterns having only a center of symmetry can also be accounted for with micropolar theory if the relative vorticity vector is nonzero and is not parallel to d 2. Attempts to fit a fault slip pattern having monoclinic or lower symmetry with a theoretical pattern that can have a symmetry no lower than orthorhombic must result in deviations from the correct orien- tations of the principal axes and in higher misfit values than would occur if the theoretical pattern could be monoclinic. It is important to distinguish between the characteristic symmetry of a fault slip pattern and the actual symmetry displayed by a particular data set. The observed fault slip data are understood to be a subset of the associated ideal theoretical pattern, and they have the characteristic symmetry of that pattern if the orientations of the observed slip lines and shear planes match, within reasonable error, the orientations of the slip lines in similarly oriented shear planes of the theoretical pattern. It is not necessary, however, for the actual distribution of the observed shear plane orientations to be uniform or even symmetrical with respect to the principal axes found by the inversion solution. Thus the shear planes in the data set could have a preferred orientation such that the actual symmetry of the tangent-lineation plot for the data has no apparent relation to the characteristic symmetry of the theoretical fault slip pattern that best fits the data. 4. Evaluation of Fault Slip Interpretations 4.1. Interpretation of P and T Axis Patterns P and T axes are easily calculated for any set of fault slip data (equation (1) and Figure 1). When plotted on a lower hemisphere stereogram, these axes tend to duster in concentration maxima. A simple technique for interpreting the fault slip data is to assume that the P and T concentration maxima themselves provide a solution for the orientation either of the global principal stresses 6'1 an 6'3, respectively, or for the global principal strain rates d 3 and d 1, respectively [e.g., Marrett and Allmendinger, 1990; Ring and Brandon, 1994]. We evaluate this assumption as it relates only to the principal strain rate axes, as we have already argued that the P and T axes are more directly related to these axes than to the principal stress axes. A theoretical P and T axes pattern for the conventional kinematic hypothesis, or for the micropolar hypothesis with W - 0, calculated for an isotropic distribution of shear plane orientations is shown in Figure 4b. The P and T concentration maxima are m utually^ orthogonal, and they essentially coincide with the d 3 and d 1 axes, respectively. Comparing Figure 4b with Figure 4a shows that the symmetry of the P and T axes plot is the same as the characteristic symmetry of the tangentlineation diagram for the same data. For this case, the P and T axes concentration maxima would provide a good solution for the principal strain rate axis orientations. The P and T axes pattern for a case in which W is nonzero is shown in Figure 5b. In this case, the P and T axes concentra- tion maxima are not orthogonal, and they do not coincide with the ' 3 and, l axes. Comparing Figure 5b with Figure 5a, however, shows again that the symmetry of the P and T axes pattern is the same as the characteristic symmetry of the tangent-lineation diagram for the same data. Nevertheless, the requirement that the principal strain rate axes be mutually orthogonal is inconsistent with the actual P and T axes concentration maxima, and these maxima would at best p vide only an approximate guide to the location of the d 3 and dl axes. Equating the P and T axes maxima with the inversion solution also leads to problems if there exists a preferred orientation of shear planes. Figures 6a and 6c show an artificial example of such a preferred orientation derived by taking a subset of the fault slip data shown in Figures 4a and 5a, respectively. For these two diagrams, the shear planes and principal deformation rate axes have the same orientations, but the values of W, and hence the slip directions, are different. The P and T axes patterns derived from these two examples are shown in Figures 6b and 6d. Comparing these patterns with those in Figures 4b and 5b shows the following, for both zero and nonzero values of W: The P and T concentration maxima for the preferred orientation of shear planes are displaced from the maxima for the full isotropic set of shear pla ne orien tations; the P and T maxima do not coincide with the d 3 and d axes, respectively, for the deformation; and the symmetry of the P and T axes plot is not the same as the characteristic symmetry of the fault slip data. Because the distribution of shear plane orientations is rarely if ever isotropic, the P and T axes concentration maxima provide only an approximate guide to the orientations of the global principal strain rates but not, in general, an accurate solution Comparison of the Stress and Kinematic Hypotheses The mathematics is identical for the calculations of the direction of maximum resolved shear stress and the direction of maximum resolved shear strain rate, because both the stress and the strain rate tensors are symmetric second-rank tensors. Therefore the fault slip patterns calculated using each hypothesis would be identical if (1) the relative vorticity were zero, (2) the principal axes of stress and strain rate were parallel, and (3) the relative magnitudes R and D were identical, R=D. (5) Under these conditions, the directions of maximum resolved shear stress and maximum resolved rate of shear on any given plane are parallel. Thus the inversion of fault slip data would give the correct solution for both the stress and the strain rate. The stress and strain rate tensors, however, are not the same; they need have neither the same orientation of principal axes nor the same relative magnitudes of the principal values, and the relative vorticity need not be zero. Thus, in general, the predicted slip directions for any given plane would be different for the two hypotheses, and both interpretations of the inversion cannot be correct except under very specific conditions. Therefore we ask the following questions: (1) Does a fault slip inversion most directly constrain the stress or the deformation rate? (2) Under what circumstances will the above three conditions be satisfied? With regard to the first question, we argued in section 3.1 that fault slip data are a direct record of the displacement on each shear plane. The displacements that accumulate on a mul-

9 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS 12,213 A A W=0.0 D= 0.5 D dz A A b,, d D W=0.0 = :.'.; _. - -,:,1,:o,, <ø:ø:ø:':' Taxes D=0.5 A Taxes ß,..,:.-o /e',e,,.o:_ o'. o','o.'. yx...,.ili,. " p ",. * ';!; ø i.' axes,,',.\,0, N,,b,_... :..!i i i.!!! i" :::!':.:!!!'.".." i!i..'.:..',. :.::.:'!!::-'.::'.:':is.::"."':':' ::: A,,:,,':,,,,,.::,:...,,.,,,,,,,,o :...,.,,'-.,,,,,m '""'ø'"'"'"'" iilli}i{ii... " "-"'""":" '"' Figur 6. Tangent lineation and P and T axes diagrams for 30 nonuniformly distributed shear planes. The fault slip data in Figures 6a and 6c are considered to have the same characteristic symmetry as the theoretical patterns in Figures 3a and 4a, respectively, even though that symmetry is not evident from the plot because of the preferred orientation of shear planes. See caption to Figure 4 for plating conventions. Note that' the P and T maxima do not coincide with the principal strain rate axes, d and d respectively, because of the preferred orientation of the shear planes. Kamb contour intervals are at 3 standard deviations in a lower hemisphere, equal area projection. (a) Tangent-lineation diagram for a subset of the fault slip data in Figure 4a. (b) P and T axes diagram showing axes and Kamb contours for the subset of fault slip data in Figure 6a. Kamb contours are at intervals of 3 standardeviations in a lower hemisphere, equal-area projection. (c) Tangent-lineation diagram for a subset of the fault slip data in Figure 5a. (d) P and T axes diagram showing axes and Kamb contours for the subset of fault slip data in Figure 6c. Kamb contours are at intervals of 3 standard deviations in a lower hemisphere, equal-area projection. tiplicity of individual shear surfaces in a volume, over a particular increment of time, add tensorially to produce a global strain rate tensor (the symmetric part of the velocity gradient tensor [cf. Kostrov, 1974]) possibly plus a relative vorticity within that volume. Because the displacements on distributed shear planes are the direct mechanism by which a global deformation rate is accommodated, the association between a set of fault slip data and the global deformation rate is direct and unequivocal, and we conclude that the inversion of the fault slip data most directly constrains the global deformation rate tensor, not the global stress tensor. The second question is more complex. Stress is a quantity

10 12,214 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS that can never be observed directly. The measurement of stress always involves the direct measurement of some intermediate quantity and the implicit or explicit application of some constitutive relation that relates the intermediate quantity to the stress. For example, the determination of stress commonly involves the measurement of a displacement in an elastic material and the determination of the stress from the equations of elasticity, either by calibration or by direct calculation. Examples of this type of relation include the elastic displacement of a spring scale, the elastic strain of a load cell, or the elastic distortion of the sensing element in an oil pressure gage. The same principle applies to the inference of stress from fault slip data, which are observations or records of local deformation. The fault slip inversion provides direct information about the orientations of the principal strain rates and about D and W. The characteristics of the stress tensor must be causes; in other words, the symmetry of the effects must be no lower, but may be higher, than the symmetry of the combined causes. The causes of a mechanical process are determined by the material properties and by the boundary conditions that govern the process [Twiss et al., 1991, 1993]. If stress boundary conditions are appropriate, then stress is a cause of the process and the deformation rate is an effect; if velocity boundary conditions are appropriate, then the deformation rate is a cause of the process and the stress is an effect. Regardless of whether the deformation rate is a cause or an effect, however, the characteristic symmetry of the fault slip data necessarily determines the characteristic symmetry of the cataclastic deformation rate, because brittle shearing on faults directly accommodates the cataclastic flow. The relationship between the causes and effects of a mechanical process described by Neumann's symmetry principal places important restrictions on the conditions under which the principal axes of the global stress and strain rate tensors may be parallel. This relationship is analyzed in detail in the appendix, but the results are easily summarized. Regardless of the exact constitutive equation that describes the cataclastic flow, the principal axes of global stress can be parallel to those of the global strain rate only if two specific conditions are satisfied: (1) The characteristic symmetry of the fault slip pattern must be orthorhombic or higher [see also Schrader, 1991, 1997] and (2) the symmetry of the combined causes must be orthorhombic or higher. As shown in our discussion of characteristic symmetry in section 3.6, the first condition is satisfied if the relative vorticity W is zero (compare Figures 4a and 5a). The second condition requires that the deforming material must be mechanically isotropic, except for very special orientations of material anisotropies relative to the stress Implicit Requirement of the Stress Hypothesis for a Linear Constitutive Relation In order that the shape factor for the stress ellipsoid (R) be the same as that for the strain rate ellipsoid (D), as defined by (5), the material must be rheologically linear. To demonstrate this, we write the most general possible form for a linear isotropic constitutive relation between the principal strain rates and the principal stresses, k=l:3, (6) related to this inversion solution by a constitutive relationship between the deformation rate and the stress. In the fol- where the negative signs and the (4-k) subscript on the stress lowing two subsections, we determine the specific constraints are required because of our assumptions that compressive under which a fault slip inversion also provides a solution for stress and lengthening strain rate are both positive. C, a, and the orientations of the principal stresses and for R. /3 are scalar constants, l(a ) is the first scalar invariant of the stress tensor (3 times the mean normal stress), and t5 k are the principal diagonal components of the kronecker delta and thus 4.3. Symmetry Constraints: Implicit equal to 1. If we substitute (6) into (3) and then use the first Requirements of the Stress Hypothesis for Zero equation in (2), we find that (5) is satisfied. Relative Vorticity and for Mechanical Isotropy If the material is nonlinear, then (6) cannot be true because the constant of proportionality would depend on the stress. Physical processes are characterized by fundamental sym- Thus in (3), the differences in the magnitudes of the principal metry relations that are very general but that, nonetheless, strain rates would not be directly proportional to differences in provide powerful constraints on the system. Neumann's symthe magnitudes of the principal stresses, and the value of D metry principle [Paterson and Weiss, 1961] states that the inferred from the inversion of fault slip data, which charactersymmetry elements that characterize the effects of a process must at least include those elements that are common to all the izes the shape of the strain rate ellipsoid, would not correctly define the value of R that characterizes the shape of the stress ellipsoid Discussion Fault slip patterns have been observed in nature with monoclinic characteristic symmetry, as indicated either by the plotted lineation pattern [Schrader, 1988, 1991, 1997; Twiss and Gefell, 1990] or by highly nonzero values of the relative vorticity parameter W [Fontaine, 1992; Unruh et al., 1996]. We have found decreases in misfit due to nonzero values of W of up to 20% for seismic focal mechanism data and up to 50% for shear plane/slickenline field data. Lineation patterns showing even triclinic characteristic symmetry have been observed [Schrader, 1991]. Such patterns are not consistent with the requirement for zero relative vorticity, and thus we must conclude that for these deformations, the micropolar kinematic hypothesis would provide a more reliable inversion than the stress hypothesis. 5. Constitutive Relations for Brittle Crustal Deformation We consider, now, whether or not the conditions of isotropy and linearity also are likely to be satisfied in the Earth. First, however, we must understand the effect of scale on the description we use of crustal deformation, and then we can consider the likely characteristics of the constitutive relations.

11 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS 12, Effect of Scale on the Description of Brittle Crustal Deformation The empirical proportionality of seismic moment (in dyne centimeters) to the rupture radius r (in meters) for earthquakes having r roughly between 100 m and 15 km can be determined from the plot of these data given by Scholz [1990, Figure 4.10] logm 0 = 31ogr (8) Combining these equations gives to a first approximation logr = 0.5M s (9) This relation indicates that events of magnitude M s = 1.5 to 3, for example, have a rupture radius of the order of r = 45 m to 250 m. Combining such events over volumes with dimensions of several kilometers, roughly an order of magnitude larger than r, is in our experience sufficient to yield a good approximation to the global continuum deformation of the volume. Using a volume of dimension of the order of r would not include sufficient slip events to give a representative integration. Using a volume of much larger dimension, for example, 100r or 1000r, could introduce heterogeneity that we wish to avoid. In fact, using the range of earthquake magnitudes and the crustal volumes suggested by the approximate 1 Or scale factor, we have been able to resolve heterogeneity in the global deformation within the aftershock volumes of major earthquake such as the Landers funrub et al., 1996], the Loma Prieta [Twiss and Unruh, 1994], and the Northridge earthquakes funrub et al., 1997]. Similar approaches have been used by others [e.g., Michael et al., 1990; Gephart, 1995, 1997; Antelung et al., 1994; Seebet and Arntbruster, 1995]. Thus for the sake of argument, we use orders of magnitude as a convenient distinction among the different scales at which we describe brittle deformation in the crust, with the caveat that this scale factor is decidedly not precise, unique, or definitive. fault. We relate these scales to the rupture radius r of a slip event on the particular fault of interest. First, the largest scale is a global-scale volume having a dimension of the order of at least 10r. A slip event on any given fault is just one of many such events on many different faults in that volume, and these events, when integrated over the volume, accommodate a cataclastic or quasi-ductile flow of the Earth's crust [e.g., Kostrov, 1974]. An elastic deformation presumably exists at this scale as well, in which case the constitutive behavior is actually elastic-ductile. The set of earth- Modeling of stresses in the Earth's crust and the effect of slip discontinuities on the distribution of those stresses typically is done by assuming an isotropic, linear elastic constitutive equation for the crust [e.g., Pollard and Segall, 1987; Stein et al., 1992; King et al., 1994]. Elastic deformation, however, is intrinsically a recoverable deformation in the sense that if the stress were removed, the deformation would disappear. Permanent deformation, however, accumulates in quakes within the volume, however, accommodates the quasithe Earth's crust, and mechanisms operate to convert recoverductile component of the deformation, and therefore it is this able elastic deformation into a permanent deformation. These aspect of the deformation at this scale that is relevant to our facts imply that constitutive equations other than those for discussion. elastic deformation also must be relevant. The understanding Second, the local scale is the neighborhood of the particular of the different constitutive equations relevant to brittle fault having a dimension of the order of the rupture radius r. At crustal deformation and the circumstances under which they this scale, the local deformation around a slip discontinuity on apply comes from considering the effects of scale on the the fault can be described strictly as an elastic deformation description of deformation. (e.g., the dislocation modeling of elastic strain associated For seismogenic deformation, the scale of the slip event is with the 1994 Northridge earthquake [Hudnut et al., 1996]). indicated by the magnitude of the associated earthquake. A Third, at a sublocal scale, which is a scale of the order of the first-order relation between the magnitude M s and the seismic width of the fault zone, perhaps 0.1r or less, slip on the fault moment M 0 (in dyne centimeters) is [Scholz, 1990, equation can itself be considered a cataclastic flow accommodated by 4.26] slip on pervasive subsidiary shear planes whose rupture radii logm 0 = 1.5M s (7) are much smaller than the rupture radius of the fault in question and less than the width of the fault zone, for example <0.01r. Thus the deformation that occurs by cataclastic flow at a sublocal scale relative to r also can be described as occurring at a global scale of 0.1r relative to the subsidiary fractures that have a rupture radius of the order of 0.0 l r. The distinctions between the elastic deformation at the second (local) scale and the cataclastic deformation at the first (global) and third (sublocal) scales are as follows: (1) The elastic deformation is a recoverable deformation that would disappear if the stress were removed, whereas the cataclastic flow is a nonrecoverable deformation that is permanent even if the stress were removed. (2) The elastic deformation presumably builds up gradually in the crust and is then released locally when the stress is relaxed by a discontinuou slip event on a fault, whereas the cataclastic flow reflects a continual global accumulation of deformation due to the integral of individual local slip events on a multitude of faults within a volume that is large relative to the rupture radii of the contributing events. The different absolute scales at which both the cataclastic and the elastic descriptions of fault-related deformation can apply is a consequence of the fractal nature of faults. A fault is actually a zone of finite width that includes a network of smaller-scale shear planes, and this description applies in a self-similar manner over a wide range of scales from at least the outcrop scale to a scale of hundreds of kilometers. Thus the terms "global" and "local" must be understood as being defined relative to the rupture radius of the pertinent slip events. Of course, there are obvious dimensional limits above and below which this scale invariance fails. The description of fault-related deformation using the constitutive equations for elastic behavior at the local scale and the constitutive equations for cataclastic flow at the global scale is analogous to the description of deformation associated with dislocations and with dislocation creep in a crystal lattice. The dislocations themselves can be described as local ß With these ideas in mind, we can consider three scales at which to describe the deformation associated with a particular elastic distortions of the crystal lattice, and the movement of

12 12,216 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS an individual dislocation is affected by and, in turn, affects the local elastic strain field in a well-defined manner. This is analogous to the description of the elastic deformation around a slip discontinuity on a fault in the crust. The net result of a multitude of dislocations propagating through a crystal lattice, however, is a time-dependent nonlinear power law creep of the solid crystal for which the strain rate depends on the stress raised to the power 3 or more. This is analogous to the cataclastic flow that results from an integration of slip events on a multitude of brittle faults in the crust. In both cases an elastic deformation can also occur at the global scale, of course, but that aspect of the deformation is not directly reflected by the dislocation motion or the fault slip events. It is tempting to argue that the inversion of fault slip data from within the deformation zone of a slip discontinuity should reflect the tectonic stress associated with elastic distor- tion of the material outside that zone. If this argument were correct, then because the elastic equations that are usually applied are both isotropic and linear, one could conclude that the solutions from the inversion of fault slip data having an orthorhombic characteristic symmetry justifiably could be interpreted in terms of the stress. However, the inversion of fault slip data from within a fault zone, such as the focal mechanism data from an aftershock sequence, provides information about the characteristics of the permanent quasi-ductile deformation at a global scale that is defined relative to the small aftershocks; it does not characterize the recoverable localscale elastic deformation associated with the much larger main shock event [Unruh et al., 1997]. Note that here the terms "global" and "local" are defined relative to different absolute scales of slip event. Referred to the same scale of the fault zone, the "global" deformation is the same as the "sublocal" scale discussed above. Thus, depending on the scale at which we wish to consider fault slip events, there are two legitimate classes of constitutive behavior that are appropriate for describing brittle deformation of the Earth's crust. Because fault slip inversion is based on a global-scale continuum description of a multitude of locally discontinuouslip events, it must be evaluated from the perspective of a global cataclastic (quasi-ductile) flow rather than a local elastic deformation Constraints on the Constitutive Relation for Cataclastic Flow Cataclastic flow of the Earth's crust is accommodated largely by brittle fracture and frictional sliding. The global constitutive equations that describe such flow can be expected to reflect the characteristics of the local processes by which the deformation is accommodated. The exact form of these equations is not known, but our current state of knowledge permits some general inferences based on observations both of seismogenic deformation and of the large-scale material properties of the crust. Brittle fractures in the crust associated with distributed brittle deformation typically have a strong preferred orientation. This observation is consistent with the understanding that a preferred orientation of the shear planes introduces a bias in the orientation of the P and T maxima (Figures 6b and 6d) and with the observation that even when W is zero, the best fit principal strain rate axes commonly do not lie at the center of the P and T concentration maxima. Because cataclas- tic flow is accommodated by shear on brittle fractures, we can expect, in general, that the brittle crust in the vicinity of fault zones is not mechanically isotropic with regard to cataclastic flow and that therefore the constitutive equations describing cataclastic flow should be anisotropic. Moreover, the frequency of aftershocks following a large earthquake typically exhibits a time-dependent rate of decay. If this behavior is an expression of the material properties of the brittle crust itself, rather than a reflection of the ductile properties of a deeper layer of crust, then that behavior also must be a characteristic of the large-scale cataclastic flow. Phenomenological constitutive equations for one-dimensional frictional sliding on an individual fault have been inferred from laboratory experiments [e.g., Dieterich, 1979a, b, 1981, 1994; Ruina, 1983; Tullis and Weeks, 1986]. These equations are characterized by a dependence of the frictional shear stress on the normal stress across the shear plane and on a nonlinear function of the sliding velocity. Tse and Rice [1986] have argued that this type of constitutive equation can lead to the time-dependent behavior observed with faulting, and Dieterich [ 1994] has developed a model based on this type of rheology that accounts for aftershock frequency. Scholz [1998] reviews the characteristics of these friction laws and their application to earthquake mechanics. A constitutive equation describing cataclastic flow presumably should be based on the characteristics of frictional sliding on shear surfaces. The normal stress on the fault should generalize in a continuum description to a term involving the first invariant of the stress tensor, because that is a measure of the mechanical pressure. The sliding velocity on a fault is a nonobjective quantity, and therefore, according to the axioms of constitutive theory [Eringen, 1967, section 2.10], it is not admissible in a continuum constitutive equation. If the sliding velocity on discrete surfaces is appropriately smoothed over a volume that contains many shear planes, however, we expect the relevant constitutive variable to be the velocity gradient tensor across a global volume, which is objective. We also expect the nonlinear relationship between the surface shear stress and the sliding velocity to appear as a nonlinear dependence between the deviatoric stress tensor and the strain rate tensor, which is the symmetric part of the velocity gradient tensor. Thus we can expect that a constitutive equation for cataclastic flow would express the strain rate as an anisotropic function that depends on the first stress invariant and a nonlinear function of the deviatoric flow stress. A more accurate consti- tutive representation of crustal rocks probably would be an elastic-ductile material for which the ductile component of the response is an anisotropic nonlinear cataclastic flow. If this were the appropriate relation for interpreting fault slip data, then the implicit constitutive assumptions of isotropy and linearity that justify the use of the stress interpretation of fault slip inversions would not be supported. If the constitutive description of the deformation inside and outside a fault zone could be so different, then the stress also could be different in these two zones. The only necessary relation between the stress in the two zones is defined by the compatibility conditions at the fault zone boundary: The normal stress component acting perpendicular to the boundary and the shear stress components acting parallel to the boundary must be equal on both sides of the boundary. The other normal and shear stress components of the stress tensor, however, can be different on opposite sides of the boundary [cf. Rice, 1992]. On a Mohr diagram for two-dimensional stress, the compatibility conditions amount to the requirement that

13 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS 12,217 IJs ls Possible shear zone tresses that satisfy the boundary condition Surface stress on shear zone boundary Elastic stress Shear zone boundary (P) Elastically O n P) 0e x Trajectories deforming of elj 1 country rock ' l /._ "I '" I b Surface stress on shear zone boundary Possible cataclastic flow stress I I i c 3 I c l Elastic stress I e 3 elj 1 Figure 7. Possible stress relations across the boundary of a shear zone. (a) Mohr diagrams with axes of normal stress O'n and shear stress e s illustrating a state of stress in an elastic material outside a shear zone and several possible states of stress in a constitutively different material within the shear zone (shaded circles). All circles shown represent states of stress that satisfy the boundary condition that the surface stress components on the shear zone boundary be continuous across the boundary. The normal and shear stress components of that surface stress ((yn, (y?)) plot as a unique point on the Mohr diagram. (b) Angular relations on the Mohr diagram for one of the shaded circles in Figure 7a between the normal com )onent of the surface stress on the shear zone boundary o'(n PI and the maximum compressive stresses in the elastic material (eo'l) and in the cataclastic material (co'l) defined by angles 20 e and 20 c, respectively. Note that in general, the magnitudes of the principal stresses are not the same on opposite sides of the shear zone boundary. (c) Trajectories in physical space of the maximum compressive stress across the shear zone boundary defined by the angles 0 e and 0 c from Figure 7b. Note that, in general, the orientations of the principal stresses are not the same on opposite sides of the shear zone boundary. the Mohr circles representing the states of stress in the elastic material outside the shear zone and in the cataclastically flowing material inside the shear zone need have only one point in common (Figure 7a). Clearly, neither the magnitude (Figure 7b) nor the orientation (Figure 7c) of the stress in the cataclas- tically deforming shear zone need be the same as in the intact elastic crust bounding the shear zone. Because the fault slip data reflect a cataclastic flow within the fault zone, it is not, in general, correct to associate an elastic stress outside the fault zone with the cataclastic flow within that zone. Thus, within the fault zone, the deformation results from a multitude of brittle slip events smoothed over the volume of the fault zone and is a cataclastic flow. In the neighborhood surrounding the fault zone, however, the deformation results from a slip discontinuity on the fault and can be modeled as an elastic deformation. Because the stresses inside and outside the fault zone could have different orientations and magnitudes, the linearity and isotropy of the elastic behavior outside the fault zone cannot be used as a justification for inferring the stress from the inversion of fault slip data within the fault zone. The stress within the fault zone is probably related to the strain rate by a nonlinear anisotropic constitutive equation, and if this is true, then the stress could not accurately be inferred directly from the inversion of fault slip data. The degree of inaccuracy would depend on the degrees of anisotropy and nonlinearity, which at present are unknown.. Given this uncertainty, it seems more secure to interpret the fault slip inversions kinematically. 6. Observational Constraints on the Stress and Kinematic Hypotheses There are few observational constraints available at present that permit a direct test of whether the stress or the kinematic hypothesis provides a better interpretation of fault slip data. Generally when a stress interpretation is used for fault slip data, the orientations of the principal stresses are assumed a priori to be given by the solution to the inverse problem. Although the results are often consistent with general expectations inferred from the orientations of major structures, independent measurements of the actual orientations of the principal stress are not available, so any possible difference between principal axes of stress and strain rate (or incremental strain) cannot be evaluated. Some examples exist, however, in which a kinematic interpretation of faulting works better than a stress interpretation, and although these studies are not

14 12,218 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS specifically based on a formal inversion of fault slip data, they nevertheless lend observational support to the theoretical preference for the kinematic approach. In the study of fluid pressure control of faulting at Rangely, Colorado [Raleigh et al., 1972], both the seismic P and T axes from local focal mechanism solutions and from the hydraulic fracturing of a borehole were used to constrain the orientations of the principal stresses. The maximum horizontal compressive stress deduced from the hydraulic fracturing is at a distinctly smaller angle to the main fault plane and the slip direction than the maximum compressive stress inferred from the orientation of the concentration maximum in the plot of seismic P axes [Raleigh et al., 1972, Figures 3 and 8). If we take these results at face value but reinterpret the orientations derived from the P and T axes as the principal strain rate orientations, then the result is consistent with the principal shortening strain axes being at a higher angle to the major fault plane than the principal compressive stress axis. However, a formal inversion of the focal mechanism data was not done, and as we argued above, the concentration maxima of P and T axes are not a reliable solution for the principal strain rate axes, so the observed difference is not conclusive. Anderson's [1951] theory of faulting applies the Coulomb fracture criterion to situations in the Earth's crust in which one of the principal stresses is vertical. It predicts that 'l should be oriented at +30 ø to the fault planes, that conjugate strikeslip faults should be vertical and should intersect at 60 ø, and that normal faults should have dips of about 60 ø. Thatcher and Hill [ 1991 ], however, documented examples of strike-slip faults in California and Japan that are mutually orthogonal, and they noted that the worldwide catalogue of moderate- to large-magnitude (M 5.5 or greater) earthquakes on normal faults shows that the mean hypocentral dip of seismogenic fault planes is approximately 45 ø. They attempted to reconcile their observations with the predictions of the Anderson theory by suggesting three possibilities for such faults: (1) the faults initially form as nonorthogonal sets in a geometry consistent with Coulomb theory and subsequently rotate to their present mutually orthogonal orientations through progressive deformation; (2) the faults nucleate as weak, relatively friction-free orthogonal surfaces in the shallow crust; or (3) the faults nucleate as orthogonal ductile shear zones in the lower crust and maintain their geometry as they propagate upward into the brittle crust. Rather than presume a constitutive behavior for crustal deformation such as Coulomb failure in the shallow crust or plastic failure in the lower crust, a more direct approach is simply to focus on the strain in the crust that is accommodated by the slip on the various faults. For individual faults, the local principal strain rate axes always are oriented 45 ø from the shear plane and the slip direction regardless of the constitutive behavior of the rocks, and the shear strain rate on planes in this orientation is a maximum. For groups of faults in which fault orientation is highly variable, the global prin- cipal deformation rate axes are defined by the pattern of maximum shear directions. Conjugate normal faults that dip about 45 ø are orthogonal and are oriented so that the local principal strain rate axes would be parallel to the global principal strain rate axes for a horizontal extensional strain. Similarly, for conjugate strike-slip faults that are vertical and orthogonal, the local principal strain rate axes are parallel to the global principal strain rate axes for a horizontal plane strain. Thus, from a kinematic point of view, such fault orientations are optimal. An example of this approach is the work of Unruh et al. [1996], who determined the orientations of the principal strain rate axes in the Mojave block, southern California, by inverting seismic focal mechanisms from both the background seismicity and the aftershocks of the 1992 Landers earthquake. They found that fault slip inversions from seismically active faults define consistent regional trajectories of principal strain rate axes that are approximately 45 ø to the traces of the major faults in the area, consistent with the findings of Thatcher and Hill [ 1991 ], and that these axes, in general, are parallel to the principal axes of incremental strain determined from geodetic analysis of secular deformation. Seen from this perspective, the observations actually constrain the pattern of strain in the Earth's crust, and the stress can be deduced only if we know the constitutive characteristics of cataclastic flow in the crust at that scale of deformation. 7. Discussion and Conclusions Our purpose in this paper is to evaluate carefully the rationales behind the various hypotheses on which the interpretations of fault slip data have been based, and in particular to examine the differences between the kinematic and the stress interpretation of such data. The basis of our analysis is the fact that fault slip data are fundamentally displacement data and that the net result of many small displacements on faults of varying orientations integrated over a global-scale volume is just an increment in the global continuum deformation of the volume, which is best described as resulting from a cataclastic flow. An inversion of a set of fault slip data thus gives direct information about the characteristics of the global deformation rate not the global stress. The characteristics of the global stress are related to the global deformation rate and thereby to the fault slip data through the rheologic behavior of the material, which is described by a constitutive equation for cataclastic flow. At a local scale, the seismic P and T axes are the minimum and maximum strain rate axes, respectively, not the maximum and minimum compressive stress axes. At a global scale, the concentration maxima for P and T axes do not provide an accurate solution to the inversion problem for either the stress or the strain rate (1) because the location of the maxima are biased by preferred orientations of shear planes that accommodate the cataclastic deformation in the rock and (2) because the orientations of the concentration maxima of P and T axes need not be orthogonal, whereas the principal axes of both the strain rate and the stress must be orthogonal. The stress interpretation and the kinematic interpretation of inversions of fault slip data are equivalent only if the directions of maximum resolved shear stress and maximum resolved rate of shear on any given plane are parallel. This condition is satisfied only if the relative vorticity W is zero, if the principal axes of stress and strain rate are parallel, and if the ellip- soid shape parameters R and D are equal. These conditions, in turn, are satisfied only if (1) the fault slip pattern has.an orthorhombic characteristic symmetry, because only then is the relative vorticity W equal to zero; (2) the material is inechanically isotropic, because only then can the principal axes of global stress be parallel to those of global strain rate; and (3) the constitutive equation relating deviatoric stress to deviatoric strain rate is linear because only then can the shape factors for the global stress ellipsoid (R) and the global strain rate ellipsoid (D) be equal (equation(5)). Justification of the interpretation of fault slip inversions in

15 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS 12,219 terms of the stress requires the assumption only of the very general characteristics of the constitutive equation, namely, that it be isotropic and linear. Other constitutive properties are completely unconstrained by the fault slip data. If any one of these conditions is not satisfied, then the direction of maximum resolved shear stress of the global stress on any given plane would not, in general, be parallel to the direction of maximum resolved rate of shear of the global deformation rate. In this event, the stress hypothesis and the kinematic hypothesis could not both be accurate. Because the fault slip data most directly constrain the deformation rate, we would conclude that the stress interpretation would not be accurate. Linear elastic equations used for modeling the deformation associated with a slip discontinuity in the crust apply if the scale of the volume within which the deformation is to be modeled is comparable to the rupture radius of the slip discontinuity. The constitutive equations for modeling cataclastic flow apply if the scale of the volume within which the flow occurs is roughly an order of magnitude or more larger than the rupture radius of the slip discontinuities that contribute to the cataclastic flow of the volume. These scales are not the same. Thus the equations of elasticity that model the deformation in the neighborhood of a major slip discontinuity on a fault cannot be applied to the cataclastic deformation represented by the much smaller aftershocks in that same fault zone, and the principal stresses at the two scales will, in general, have different magnitudes and orientations. Application of the stress interpretation therefore cannot be justified on the basis of the isotropic and linear character of the equations of elasticity because these equations do not describe the deformation that is reflected by the fault slip data. It seems probable that the conditions of isotropy and linearity are not, in general, satisfied for cataclastic flow in the Earth's crust. The shear fracturing of brittle rock generally produces a material with a strong preferred orientation of fractures which would result in a mechanically anisotropic material for cataclastic flow. Experimental work has shown that frictional sliding on shear planes is represented by a nonlinear state-and-rate-dependent constitutive relation. To the extent that cataclastic flow is governed by this process, its constitutive relation is most likely nonlinear as well. These arguments provide a strong rationale for inferring that the principal axes of stress are unlikely to be parallel to those of the strain rate and that the shape parameters for the ellipsoids of stress (R) and strain rate (D) are unlikely to be the same. There are two general effects that affect the accuracy of the stress hypothesis relative to that of the micropolar kinematic hypothesis. First is the effect of a fault slip pattern that has a monoclinic characteristic symmetry. Second is the effect of anisotropy and nonlinearity of the constitutive equation for cataclastic flow of the Earth's crust. The effect of data whose fault slip pattern has a monoclinic characteristic symmetry is that application of the stress hypothesis to the inversion produces a larger misfit than application of the micropolar kinematic hypothesis. This result occurs because the stress hypothesis attempts to fit the monoclinic fault slip data with a pattern having an orthorhombic characteristic symmetry, whereas the micropolar kinematic model can account for the monoclinic symmetry with a nonzero value of W. The inability of the stress hypothesis to model accurately the monoclinic characteristic symmetry can also result in inaccurate orientations for the principal axes. Of course, in general, it is to be expected that increasing the number of model parameters will lead to an improvement in the fit of the model to the data, and at present we cannot definitively state that the improvements we find for some data sets are statistically significant. This is part of our ongoing research. Nevertheless, we often find both zero and nonzero values for W, consistent with both the absence and presence, respectively, of relative fault block rotations in fault zones. Along the Kickapoo fault, a fault segmenthat ruptured during the 1992 Landers earthquake, inversion provides a nonzero value of W that is consistent with the mapped geometry of rotating fault blocks [Unruh et al., 1996]. Thus the significance of the improvement in misfit due to the relative vortic- ity seems worthy of continued research. The constitutive characteristics of anisotropy and nonlinearity only determine how accurately the stress can be inferred from the inversion solution, they do not affect the accuracy of the solution itself and thus do not show up in the magnitude of the misfit for the inversion. The inversion directly constrains the global strain rate tensor. By inferring characteristics of the stress from this solution, one implicitly assumes that the crust is an isotropic linear material. If, however, the crust is significantly anisotropic and/or nonlinear, then the inversion solution will still correctly constrain the orientations and relative magnitudes of the principal strain rates, but the difference between true characteristics of the stress and those inferred from the inversion solution will depend on the magnitudes of the anisotropy and nonlinearity, which are not known. Existing data do not provide independent constraints on the orientation and relative magnitudes of the global principal stresses that can be directly compared with the inversions of fault slip data. Thus, at present, we cannot use fault slip inversions to test whether the expected anisotropy and nonlinearity of the Earth's crust exist and, if so, whether they are significant. The arguments presented here, however, indicate the importance of developing and testing an appropriate constitutive equation describing large-scale cataclastic flow in the upper crust, for only then will we have a sound theoretical basis from which to evaluate the accuracy of inferring constraints on the state of stress from inversion of fault slip data. In summary, we submit that fault slip data most directly constrain the global deformation rate; that we have insufficient knowledge of the constitutive equations and material properties through which the global stress is determinable from the global deformation rate; and that we should therefore be circumspect in accepting as accurate the stress solutions inferred from the inversion of fault slip data. The many inversions for stress that have been published in the literature are, we believe, more confidently interpreted as solutions for the principal strain rate axes. Whether there is a resolvable difference between the orien- tations and relative magnitudes of principal global strain rates and the principal global stresses depends on how anisotropic and nonlinear the cataclastic flow behavior of the crust is. At present, we do not have answers to these questions because it is difficult to find independent determinations of stress characteristics against which the inversion solutions can be checked. The questions do not even arise, however, if we do not understand the assumptions implicit in using the stress interpretation for fault slip inversions. In answering the questions, we ultimately will improve our understanding of the mechanical behavior of the crust.

16 12,220 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS Appendix Neumann's symmetry principle lays down necessary relations between the symmetry of causes and effects in a physical process, and we apply this principle to determine the conditions under which the stress hypothesis can be justified. The causes and effects of a physical process can be identified as the independent and dependent variables, respectively, in the equations that describe the behavior of the system. The equations themselves, however, are completely neutral as to which variables should be considered independent and which should be considered dependent. In the flow of a continuum, for example, the stress can be considered a cause, and the deformation rate can be considered an effect; conversely, the deformation rate can be considered a cause, and the stress can be considered an effect. The causes of a physical process must be identified by examining the boundary conditions on the physical system, because it is the boundary conditions that represent the externally imposed constraints on the system [Twiss et al., 1991, 1993]. Because there are several strands to this argument, we summarize them in a flow chart (Figure A1). In all cases, we assume the material properties are one of the basic causes, and by choosing either the stress or the deformation rate as the other cause, we explore the implications of a fault slip pattern whose characteristic symmetry is either orthorhombic or monoclinic. The symmetry of both the stress and the strain rate can be no lower than orthorhombic, since both can be represented by triaxial ellipsoids. The mirror planes and the twofold rotation axes of symmetry are thus identical in orientation to the principal planes and principal axes of stress or strain rate, respectively (Figure 4c). The symmetry of the deformation rate can be lower than that of the strain rate, however, because of the possibility in micropolar theory of a nonzero relative vorticity parameter W, which reflects an independent component of local rigid body rotation. In our argumen ts, we assume that the axis of relative vorticity is parallel to d 2, so that the minimum symmetry of the deformation rate can be no lower than monoclinic (Figure 5c). The characteristic symmetry of the fault slip pattern is a direct reflection of the symmetry of the deformation rate. A1. Stress as a Cause We consider first the case for which the material properties and the stress are the causes, and the deformation rate (and thus the fault slip pattern) is an effect. If the fault slip pattern has an orthorhombic characteristic symmetry, the relative vorticity parameter W must be zero, and the orthorhombic symmetry is that of the strain rate. By Neumann's symmetry principle, the symmetry of the combined causes can be no higher than the orthorhombic symmetry of the effects (Figure A1 conclusion A and path 1). Since, in general, the symmetry of the stress (cause) is orthorhombic, if the material properties are orthorhombic/ Effect:. monoclinic (W- - O) Strain rate and ( W O) =1- '" relative vorticity ' _a. u I,-sl i p pattern/.....:: i : i!?... monoclinic I [11ii::iiiii::i::i::iiii::i:::'.:..' : : i:i i: :'i!i: i!: deformation rate orthorhombic ::i::::i::iiii..'i.ie. 'i ::iiiiiiiiii?:?...: i...'!::: :.. mnnne. linie. ( W = 0 )...:: i!!! i! i!!!!5:::::::::::::::::::::::: :i.: :!!!!!' t"; i'fii i i "i : : : : : : :! i:: : :::::::::::! i! i i i!?ii i ::::..... fiji:'... i i :: i i i i iii ::... W O), /'"'"'"-'"'"'-'- --- '"'""'"' :':': :': :.:...: :::..",..,, l I l i ::::::::::::::::::::::::::::::::::::::::::: i " i : :.' :::: :::: i!: :' i ii:.:': ; "'i :' : I I I ::.,.' : '. ::::::-': '. : ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: :.a.'::; ' :o..: : i... [!i i i i i!i!::;:i:::::;:i:;:;:i:: ::. :::::::::::::::::::::::::::::::::::::::::::::::::: : :i: i: :i Conclusion I Principal stress axes parallel I the principal strain rate axes I justified I Stress interpretation I "... :i ' i Effect: symmetry of stress must be at least orthorhombic rincipal I, Conclusion stress axes parallel he princi?al strain rate axes I I Stress interpretation I I justified I Effects: symmetry of stress must include at least the monoclinic (or lower) symmetry elements of the combined causes Conclusion Not all principal stress axes must parallel strain rate axes Stress interpretation not justified A. b. C. D. Figure AI. Flow diagram summarizing the symmetry arguments that show the stress hypothesis justified only if the mechanical properties of the material are isotropic. Decision branch points are in hexagonal boxes. The circled numbers identify different paths to the same conclusion. Boxes relating to the causes of the process are shaded in gray. The symmetry of the combined causes enters the argument at places indicated by boxes with the double outline. Conclusions are at the bottom of each path in heavily outlined boxes labeled A through D. Extra-heavy outlines mark the conclusions that the stress hypothesis not justified. See appendix for discussion.

17 TWISS AND UNRUH: ANALYSIS OF FAULT SLIP INVERSIONS 12,221 isotropic, then the combined causes have the orthorhombic symmetry of the stress. Because those symmetry elements must also appear in the effects, which are also orthorhombic, we conclude that the principal axes of the stress and the strain rate must be parallel. Thus, for this case, the requirements justifying the stress interpretation of fault slip data are satisfied. One can imagine special cases which involve a symmetry of the material properties that is lower than isotropic and which can still result in an orthorhombic symmetry of the combined causes (e.g., Figure A1 conclusion A and path 2), but the result is unchanged because the orthorhombic symmetry elements of the combined causes must be symmetry elements of the stress. Of course, Neumann's symmetry principle allows the combined causes to have lower symmetry than the effects. Suppose therefore that for an orthorhombic fault slip pattern, the combined causes had monoclinic (or lower) symmetry (Figure A1 conclusion B and path 1). This could only happen if the material properties were anisotropic. The symmetry constraint requires only that the monoclinic (or lower) symmetry elements of the combined causes appear in the effects, so in this case, the principal stress axes are not, in general, all parallel to the principal strain rate axes, and in this case the requirements justifying the stress interpretation are not satisfied. The result is the same if the effect and thus the fault slip pattern have monoclinic symmetry (Figure A1 conclusion B and path 2). In this case, the symmetry of the combined causes can be no higher than monoclinic. Thus the principal axes of stress and strain rate are not, in general, all parallel, and the requirements justifying the stress interpretation are not satisfied. A2. Deformation Rate as a Cause Next, consider the case for which the material properties and the deformation rate are taken to be the causes and the stress is taken to be an effect. If the characteristic symmetry of the fault slip pattern is orthorhombic, then the relative vorticity parameter W is zero, and the characteristic symmetry is that of the strain rate. If the material properties are isotropic (Figure A1 conclusion C and path 1), then the symmetry of the combined causes is the orthorhombic symmetry of the strain rate. The symmetry of the effect must include the symmetry of the combined causes, and thus for a general orthorhombic stress, the principal axes of stress and strain rate must be parallel, and the requirements justifying the stress interpretation are satisfied. Special orientations could allow the material properties to have a symmetry as low as orthorhombic without destroying the orthorhombic symmetry of the combined causes (Figure A1 conclusion C and path 2), but the result is still that the requirements justifying the stress interpretation are satisfied. If the material properties are anisotropic and not in a special orientation, however, the combined causes will have monoclinic or lower symmetry (Figure A1 conclusion D and path 1). Since the symmetry of the effect can be higher than that of the combined causes and need only include the symmetry elements of the combined causes, the general orthorhombic symmetry of the stress need only include the monoclinic symmetry elements of the combined causes. Thus the principal axes of the stress and the strain rate are not, in general, all parallel, and the requirements justifying the stress interpretation are not satisfied. The results are the same if the characteristic symmetry of the fault slip pattern is monoclinic, in which case the relative vorticity is nonzero, and the relative vorticity axis is the unique twofold axis of rotational symmetry. Regardless of whether the material properties are isotropic or anisotropic (Figure A1 conclusion D and path 2), the symmetry of the combined causes is monoclinic or lower. The only constraint is that these monoclinic or lower symmetry elements must be included among the symmetry elements of the effect, which is the orthorhombic stress. Thus again the principal axes of stress are not, in general, all parallel to the principal strain rate axes, and the requirements justifying the stress interpreta- tion are not satisfied. A3. Summary These arguments show that even if the fault slip pattern has orthorhombic or higher symmetry, the stress interpretation can be justified only if the material is mechanically isotropic (Figure A1 conclusion A and path 1 and Figure A1 conclusion C and path 1) or if the special case obtains that the mechanical anisotropy has exactly the right orientation so that is does not decrease the symmetry of the combined causes (Figure A1 conclusion A and path 2 and Figure A1 conclusion C and path 2). In all other cases, the stress interpretation is not justified. Acknowledgments. We are indebted to Mark Brandon, Roland Btirgmann, and Andy Michael for their careful reading and helpful comments which sharpened our presentation and our thinking on the questions raised in this paper. David Castillo also reviewed an earlier version. 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Unruh, William Lettis & Associates, Inc., 1777 Botelho Drive, Suite 262, Walnut Creek, CA ( unruh@lettis.com) (Received September 30, 1997; revised January 19, 1998; accepted February 11, 1998.)