Half-Graben Structures: Balanced Models of Extensional Fault-Bend Folds

Half-graben structures: Balanced models of extensional fault-bend folds

RICHARD H. GROSHONG, JR. Department of Geology, University of Alabama, Tuscaloosa, Alabama 35487-1945

ABSTRACT The models proposed by Eyidogan and Jackson (1985) and Jackson (1987) for seismically active normal faults in Turkey have the same One common structural style in extensional regimes is a half general form. They found earthquakes on the horizontal to gently dipping graben bounded on one side by a master normal fault and on the other lower detachments, as well as on the ramps. A half graben is considered by side by a domain of beds dipping toward the master fault. This geome- Rosendahl (1987) to be the fundamental cross-sectional geometry in the try is modeled as being caused by a bend in the master fault. The East African rifts. The most basic geometries shown on seismic lines 206 hanging-wall beds dipping toward the master fault are bounded by and 212 of Rosendahl (1987) resemble those in Figure 1. axial surfaces formed as the hanging wall moves past the bend, result- There is more than one style of extensional faulting. A tilted—faulting in an extensional fault-bend fold. The footwall beds remain unde- block or domino style was proposed by Morton and Black (1975) for fault formed and unrotated. The major assumptions used in the derivation blocks in the Afar depression. In this style, both the faults and the fault are that the geometry is area balanced and that the axial surfaces in the blocks rotate during extension, a result that has been produced experimen- hanging wall have dips equal and opposite to the dip of the master tally by McClay and Ellis (1987) and Vendeville and others (1987). Listric fault above the bend. Important consequences of the model include a faults (Gibbs, 1983) and low-angle planar normal faults (Wernicke and relationship between master-fault dip above and below a bend and the Burchfiel, 1982) are other important styles that can be treated as generali- amount of hanging-wall dip. The horizontal width of the dipping beds zations of the model to be presented herein. in the hanging wall is twice the heave on the master fault above the In another type of geometric model, the geometry of a single hanging- bend. The asymmetry of the axial surfaces in the hanging wall requires wall bed is used to predict the shape of the underlying fault (Verrall, 1981; that the rotated beds be strained. Hanging-wall beds dipping toward Gibbs, 1983; Davison, 1986; White and others, 1986; Wheeler, 1987; the master fault are predicted for a fault bend wherein the dip de- Williams and Vann, 1987). Different fault shapes are predicted, depending creases downward. Hanging-wall beds dipping away from the master upon the assumptions of the model (Williams and Vann, 1987). The fault are predicted for a bend wherein the dip increases downward. geometry of a hanging-wall bed can be defined by the location and dip The model fits measurements taken from a published ramp-and-flat change at a sequence of hinges along the bed. Hinges in successive beds clay model by E. Cloos and is used to develop a balanced and restor- generate an axial surface. None of the models has explicitly used the able cross section of the Schell Creek fault, a Basin and Range axial-surface geometry in the hanging wall, which is, in fact, a critical structure in Nevada. aspect of the structure. The model developed herein relates the axial- surface geometry to the fault geometry and in this respect is analogous to INTRODUCTION the balanced kinematic models of reverse faults by Suppe (1983) and Jamison (1987). The model of Suppe (1983) also applies to normal faults, The basic balanced geometric and kinematic model for extensional but the axial-surface geometry is quite different from that in Figure 1 or the faulting and related bending of the half-graben style developed in this clay model of Cloos (1968) because the axial surfaces in the Suppe model paper is shown in Figure 1A. The essential elements are a flat-bottomed dip in the same direction as the master fault. The hanging-wall hinges are graben bounded by a normal fault on one side and by a continuous sequence of dipping beds on the other. The purpose of the model is to provide a set of mathematical relations that allow an interpreter to create a section that is balanced and restorable without going through a long proc- ess of trial and error. The relations developed herein are tested on a clay model for which the geometry is known and are used to develop a balanced cross section for the Schell Creek fault, a young Basin and Range structure in Nevada. In the extensional style treated herein, the footwall does not deform or rotate during deformation, and rotation in the hanging wall is related to a bend in the fault (Fig. 1). A clay model of this style was produced by Cloos (1968) by extending a clay cake above a horizontal detachment. Several authors have commented upon the resemblance between the clay c I model of Cloos and the younger structures in the Basin and Range province. Stewart (1971) may have been the first to note the similarity, and Anderson and others (1983) made a point of it. The review by Anderson and others (1983) shows several Basin and Range structures having a form Figure 1. Kinematic model of extension above a horizontal de- similar to that in Figure 1; Diamond Valley seems to be the most similar. tachment, showing the result of increasing displacement.

Geological Society of America Bulletin, v. 101, p. 96-105, 11 figs., 1 table, January 1989.

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directly related to bends in the fault, and so they are "fault-bend folds" in therefore, the terminology of Suppe (1983). A balanced cross section is defined as one in which volume remains Area AIJE = (d sin (180 - 0) = id sin 0. (2) constant during deformation (Dahlstrom, 1969). The model developed herein is area balanced, which means that the cross-sectional area is con- Equations 1 and 2 are identical, showing that the assumed geometry is area stant. It is based on the geometry of a planar normal fault that joins a balanced. The total displacement on the flat to the right of H2 is 2d = W, planar detachment at depth (Fig. 1A). As a result of extension, a half the standard width of the antithetic dip domain. graben develops, bounded on one side by the master fault and on the other The relationship between the antithetic dip, a, and the dip of the fault by a zone of dipping beds, causing the total structure to be asymmetric. is found by using the law of sines on triangle IJE. The side opposite a is I "Reverse drag" (Hamblin, 1965) is a commonly used term for hanging- = d/cos 0, angle IJE = 0 - a, and the side opposite, IE, = 2d; therefore, wall beds that dip toward a normal fault. The dip is not related to the d _ 2d mechanics of "drag" on the fault but rather is interpreted as the conse- quence of rotation above a bent or curved fault; consequently, this term sino cos0 sin (0 - a) will not be used. Hanging-wall dips opposite in direction to that of the Using the relationship sin (x - y) = sin x cos y - cos x sin y, the previous master fault are herein termed "antithetic dips," and dips in the same expression can be reduced to direction as the master fault are termed "synthetic dips." Regions of uniform dip between axial surfaces or faults are referred to as "dip domains" tana = (tan 0)/3. (3) (Groshong and Usdansky, 1988). As displacement increases on the master fault (Figs. IB and 1C), the flat bottom of the graben gradually disappears The antithetic dip is related to the dip of the master-fault ramp by equation as it drops down and shifts laterally into the domain of antithetic dip. The 3, plotted in Figure 3. The width of the domain is W = 2d, or if h, rather antithetic dip domain is bounded by two parallel axial surfaces that dip at than d, is known (Fig. 2), defining a length D equal to 2d, an angle equal and opposite to that of the master fault. First, it will be 2h demonstrated that the geometry of Figure 1 is area balanced. The model D = W = . (4) will then be extended by considering a finite initial width at the base of the tan 0 graben and by deriving the relationships for a dipping lower detachment. The model is quantitatively tested against the clay model of Cloos (1968) WIDE ZONE OF ANTITHETIC DIP and is then applied to the Schell Creek fault as an example of the method. The antithetic-dip domain may be wider than the standard-width domain from the first instant of displacement. This will be seen in the clay STANDARD-WIDTH DOMAIN OF ANTITHETIC DIP model and is presumably related to the difficulty of bending a thick single layer and to the extension that occurred prior to formation of the through- going master fault. This is treated as an initial offset, W , between axial What is herein termed the "standard-width domain of antithetic dip" 0 surfaces HI and H2 (Fig. 4A). After displacement on the master fault, the has a width W equal to the displacement on the lower detachment C-G as antithetic dip, a', will be less than the dip within the standard-width zone shown in Figure 2 for a horizontal lower detachment. The exact geometry (Fig. 4B). The displacement, D, along the lower flat is the same as for the is established by applying the area-balance criterion. In Figure 2, a given standard-width zone, but the width of the zone of antithetic dip, W', is hanging-wall horizon is displaced to the right an amount d on the ramp significantly wider, by an amount x + W ', where W ' is the final (dip 0) of the master fault. The resulting vertical displacement of the bed is 0 0 (strained) width of W . It will now be shown that the geometry is area h. At the base of the ramp, axial surface HI forms with a dip equal and 0 balanced for the proper value of x. opposite to that of the ramp. The entire original triangle ABC is imagined to be displaced down the ramp a vertical distance h to a new position To be area balanced, the area of the original triangle, A0, must equal DEF. The right side of triangle DEF defines the position of axial surface that of the deformed triangle Aj. Because the horizontal width of triangle x H2, giving the standard width, W, of the zone of antithetic dip. This A0 is D, the area A0 = h Dh. Note that h can be the thickness of a single establishes the relationship between displacement and axial-surface width. bed or the thickness of the entire sequence. The area A] is found from the As displacement increases, axial surface HI remains at the base of the relationship A = Vi ab sin C, where a = D + x, b = i', and C = 180 - 0. The ramp, and axial surface H2 migrates to the right, fixed to the hanging-wall length £' is obtained from the law of sines as beds. The triangle ABC is not actually displaced below the level of the flat; {' D + x instead, the material moves laterally into the zone of antithetic dip. This = . (5) sin«' sin(0 - a') geometry is area balanced if the area of ABC equals the area of region DIJGC. The areas of triangles ABC and DEF are equal by the original Equating A0 and A], we obtain construction. The quadrilateral DEGC is common to both DEF and re- 1 sina' sin 0 (D2 + 2Dx + x2) gion DIJGC. Area balance thus can be proved by demonstrating that CGF - hD = . 2 2 sin (0 - a') has same area as triangle IJE. In addition, the relationship between 0 and a, the angle of antithetic dip, will be determined. Solving for x by substituting for D from equation 4 and rearranging, we find The area of any triangle is h ab sin C, where a and b are adjacent , 4hx 2h2 / 3 tan a - tan 0 \ sides and C is the angle between them. In CGF, angle GCF = 0, length CF x2 + -— + —( ) = 0 , tan0 tan20 v tan a' 7 = I, and length CG = 2d; therefore, a quadratic equation with a general solution of Area ACGF = fd sin 0. (1) - b ± (b2 - 4c)* x = , (6) In IJE, angle IEJ =180-0, length IE = CG = 2d, and length JE = AD = £;

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4h 3 tan a - tan0 Substituting I = h/sin 0, and f from equation 5 with D = 2h/tan 0, we where b = and c = tan (v mn—26 )' tan a obtain 1 (2h cos 0 + x sin 0) sin a \ The positive root of this equation gives a geometry that is area balanced. ( y (7) -2 + 2h sin (0 - a'1) )' , As a result of the deformation in the zone of antithetic dip, the

original width W0 is increased to W0'. The value of W0 is found from the To solve this equation, the value of W0' must be determined from a area-balance criterion. The original area of the antithetic-dip domain prism measurement of W', as

(Fig. 4A) is W0h; the final area of the prism is W0'h minus the area of the small dashed triangle in Figure 4B. The area of the small dashed triangle is W ' = W' - D - x. (8)

Vi (« - V) W0' sin (180 - 0), so that If W' and h can be measured, then a can be found by using the law

W0h = W0'h - Vi (i - {') (W0') sin 0. of sines in the triangle with sides W' and t (Fig. 4), giving h tan 0 tan a' = (9a) h + W' tan 0 or, if a is measured, h (tan 0 - tan a) W (9b) tan a tan 0 For small values of a, equation 9 is quite sensitive to measurement errors. To show this, Figure 5 is a plot of the normalized variable W'/h versus a. At values of a less than 20°, very small changes in a lead to large changes in W'. The same is true for the relationship between a and W as shown by the large points in Figure 5. In a natural example, it is best to measure h, a, and W' and check them for consistency, using equation 9. After con-

sistent values are obtained, x and W0 can be computed.

INCLINED LOWER DETACHMENT

Figure 2. Geometric elements of the extensional model with a A bend in the fault is treated as a lower detachment that dips at an horizontal lower detachment and a standard-width zone of antithetic angle y in the same direction as the master fault (Fig. 6). To area-balance dip. The reference point D on the diagram is not the same as length D the geometry, it is assumed that the displacement is parallel to the lower used in the text. detachment and that the axial surfaces HI and H2 dip equally and opposite to the master fault. As in the previous derivation, it is convenient to consider the geometry of a bed of thickness h that is displaced entirely onto 90 • the lower detachment.

CO 80 • LLI LU en 70- U LU o 60- z 50- o Q. a 40- o i- 30- w T 1- 20- 1- 7 < 10- Figure 4. Geometric elements of the extensional model with a 0- horizontal lower detachment and a wide zone of antithetic dip. The 0 10 20 30 40 50 60 70 80 90 initial width of the zone of antithetic dip, W0, results in a lower dip of DIP OF MASTER FAULT Q IN DEGREES bedding within the zone of antithetic dip when compared to the geometry in Figure 2. A. Hypothetical geometry at the initiation of Figure 3. Relationship between dip of the master fault and anti- extension. B. Geometry after sufficient extension to drop the cutoff of thetic dip for a standard-width zone of antithetic dip above a horizon- the top of bed h onto the flat. Because of the low antithetic dip, W' is tal detachment, from equation 3. significantly longer than D + W0.

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Q. Q O

LL O LU Z o N 0U- 1 I- Q Figure 6. Geometric elements of the extensional model with a Q dipping lower detachment. LU N <_) cr which is solved for f to give O 2d sin 0 f = (10) 0 10 20 30 40 50 60 70 sin (0 + 7)

ANTITHETIC DIP d' IN DEGREES The horizontal width, W, of the zone of antithetic dip (Fig. 6) is obtained by using the law of sines on the triangle with sides W and f and included Figure 5. Width of the zone of antithetic dip above a horizontal angle y as detachment as a function of the dip of beds within the zone. Curves f W W are for the wide zone of antithetic dip; large points are for the sine sin [180 - (0 + 7)] sin (0 + 7)' standard-width zone of antithetic dip. and by substituting f from equation 10, it is seen that

The area of triangle A0 will be equal to the area of triangle A! for the W = 2d. (11) proper choice of f, thereby establishing that the geometry is area balanced.

A0 = dh, and A| is found from the relationship h\ = Vi ab sin C, where a = This result demonstrates that W is the same as for the standard-width zone

f, b = f = h/sin 0, and C = 180 (0 + y). Equating A! and A0, of antithetic dip with a horizontal lower detachment. The magnitude of the antithetic dip a (Fig. 6) is found by using the law of sines on the triangle fh sin (0 + 7) • = dh, with sides W and f 1 with the angle opposite W being 6 - a; 2 sin

Figure 7. Hanging-wall dip a related to fault bend geometry from equation 13. The dip of the upper fault segment is d and of the lower segment is 7.

0 10 20 30 40 50 60 70 80 DIP OF LOWER FAULT SEGMENT 7 IN DEGREES Hangingwall dip a related to fault shape.

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il W sin a sin (8 - a)

If we substitute 11 = (h - li2)/sin 6, h = d tan 8 and I12 = f sin 7, and use trigonometric substitutions as in the derivation of equation 3, we obtain

tan 8 sin (8 - y) tan a • (12) 2 sin (6 + y) + sin (0 - 7)

which reduces to equation 3 for 7 = 0. If 0 = 7, then a equals zero in equation 12, indicating that the fault is straight and that there is no zone of antithetic dip. The dip of the lower fault segment can be found as a Figure 9. Thinning of a bed between asymmetric axial surfaces. function of 0 and a by solving equation 12 for 7;

tan 8 (tan 8 - 3 tan a) tan 7 = • (13) tan 8 + tan a Strain is computed using the definition The hanging-wall dip is related to the geometry of the fault bend in Figure 7, plotted from equation 13. The left-hand vertical axis represents a e = (Li - L0)/L0, (15) horizontal lower detachment. Fault dips decreasing downward result in antithetic dip (+a) in the hanging wall, whereas fault dips increasing for which extension is positive and contraction negative. Substituting equa- downward result in synthetic dip (-a) in the hanging wall. There are two tion 14 into this relationship gives possible values of 8 that produce the same synthetic dip for each value of 7. Some representative geometries are illustrated in Figure 8. e± = (ti/O - 1 (16)

STRAIN WITHIN HANGING-WALL DIP DOMAIN where e is the strain of a line normal to bedding. The strains will usually be large; consequently, the bedding-parallel strain must be determined Beds located within a hanging-wall dip domain will strain as a result from the constant-area relationship. Given a segment of undeformed bed

of the deformation (Fig. 9). Length £ along the axial surface can be of length £0 and thickness t„ that is deformed to £j and tj, then for

expressed as either t0/sin 0 or tj/sin (6 - a'). Equating the two gives constant area £0t0 = £ iti. If we substitute this relation into equation 15, we sin (8 - a) obtain ti/to = (14) sin 8 e„ =(£,/£„)- l=(to/t,)-l, (17) Only for a symmetric axial surface, 0 = -a, will the beds maintain constant thickness. For +a', the beds extend, and for -a, the beds shorten, where ej| is the strain parallel to bedding. as can be seen in Figure 8. The angle a in equation 14 can be replaced by a if it is computed from equation 3 or 12. TOTAL EXTENSION The length £ along the axial surface (Fig. 9) is constant in length and direction throughout the zone of dipping beds, and so the deformation is The total horizontal extension across the width of the original graben, equivalent to simple shear parallel to the axial surfaces HI and H2. This en, can be computed from equation 15. For a graben having the standard

shows that there is a close relationship between this model and the original width at a reference horizon, G, L0 = G, and in the case of the

oblique-simple-shear model of White and others (1986). wide domain of antithetic dip, L0 = G + W0. For the graben of width G, L|

x: Figure 8. Fault geometries that result in hanging- 39 \7 wall dips of 15°, all having the same heave. In A and B, the bend in the fault causes a domain of antithetic (positive) bedding dip, and in C and D, the bedding dip is synthetic (negative) with respect to the fault dip. The dashed arrow is the trajectory of a point originally at the upper corner of the deformed region; restoration of displacement will move the point in the opposite direction.

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= G + W + D*, where D* is the displacement on the hanging-wall cutoff of TABLE 1. CLAY MODEL BED DIP AND THICKNESS DATA the reference horizon along the lower flat, if this occurs. For the graben of Unit Dip at top (°) Thickness (mm) width G + W0, L[ = G + W + D*. This leads to the general equation •o 'I <1*0 W' + D* - W0 e = ———(18) H ; 6 14.5 9.0 7.0 0.78 G + W0 5 14.5 13.0 11.0 0.85 4 13.0 11.0 9.0 0.82 where for a standard-width dip domain, W0 = 0 and W' = W. For 3 13.0 7.0 5.5 0.79 2 13.5 3.5 3.0 0.86 example, en for Figure 8A is the horizontal width of the domain of 1 14.5 14.0 9.5 0.68 antithetic dip divided by the original (restored) width of the graben and equals 0.29 or 29%, and for 8B it equals 67%. The /? value of McKenzie

(1978) is equal to 1 + eH. (Table 1) that average 13.8°. The axial surfaces marking the boundaries of The linear increase of total extensional strain with displacement the zone of antithetic dip are nearly parallel to each other and dip 56° and makes an interesting contrast to the strain in the dip domains which is a 58°. The geometry is reasonably close to the straight-line approximations function of the dip change of the master fault, not the displacement (equa- of the analytical model. tion 17). In interpreting a natural example, it might be tempting to assume For reference purposes, six units between bedding markers are de- that the total horizontal extension is the sum of ejj and some function of ey, fined in Figure 10. Thickness and length measurements are in millimeters but the model shows that they are independent; the strain in a zone of syn- at the scale of the original published photograph. The maximum vertical thetic or antithetic dip does not add to or subtract from the total horizontal displacement is h = 13 mm and was measured as the offset of the top of extension across the graben. unit 6 in the center of the graben. Because of normal-drag bending adjacent to the master fault, the displacement there is less. The width of the

APPLICATION TO EXPERIMENTAL MODEL zone of antithetic dip is 40 mm. The original unit thicknesses (t0) given in Table 1 were measured at the left side of the model. The experimental clay model of Cloos (1968) provided a partial Given the master fault dip of 53°, the standard-width zone of anti- inspiration for this analysis, and the computations are quantitatively tested thetic dip should be 19.6 mm wide (equation 4) and contain beds dipping using the model (Fig. 10). For measurement purposes, the master fault, 23.9° (equation 3). The zone is significantly wider, and the average dip of "bedding" markers, and axial surfaces are approximated as straight lines, the units is significantly less. The initial stage of the model shown by Cloos shown superimposed in Figure 10. The clay was deformed by sliding the has an antithetic dip domain of finite width present well before a through- basal metal sheet to the left, out from under the sheet beneath the right side going master fault is developed. In the experiment, this represents an initial of the model, as indicated by the arrows. The master fault developed above ductile sag stage. Both the experimental information as well as the com- the fixed side of the model and is relatively planar with a dip of 0 = 53°. A puted geometry indicate that the zone of antithetic dip initiated with a zone of antithetic dip developed, within which the marker horizons (or the finite width, resulting in what is herein computed as a wide zone of medial lines of the faulted marker horizons) have nearly uniform dips antithetic dip.

Figure 10. Clay model of ramp-and-flat extension with straight-line geometry superimposed. The clay model is from Cloos (1968, his Fig. IS). The offset of bedding markers at the left-most axial surface is a result of the straight-line approximation; the actual bedding markers are continuous.

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The geometry of the wide zone of antithetic dip can be computed Gans and others (1985) and McCarthy (1986) show the associated reflec- from different sets of the data available. If one uses 6, a' (13.8°), and h in tors to be approximately planar. The bend below shot point 2200 is based equation 9b, the width of the zone W' = 43.0 mm, or if one uses 6, h, and on poorly imaged seismic reflectors at the level of the NSRD and well- the measured W' (40 mm) in equation 9a, the dip of the units should be a' imaged reflectors in the lower valley-fill sequence above the NSRD. = 14.6°. The differences between the measured and predicted values are The axial-surface geometry in Figure 11 relates only to beds depos- small but greater than the uncertainty in the measurements. This confirms ited prior to faulting. These axial surfaces represent the loci of the instan- the inference from Figure 5, namely that when a is small, small dip taneous rotation axes. They form the boundaries of dip domains oriented changes lead to large changes in W'. If one uses the measured value of a' = as in Figures 1 and 8 only in the pre-growth sequence. Growth-sequence 13.8° in equation 14, the predicted value of t[/t0 = 0.79, very close to the beds are deposited across the already-formed hinges in older beds. The average value of 0.80 obtained from Table 1. Using the computed value of growth-sequence beds are rotated as they cross an instantaneous rotation a = 14.6° gives a predicted value of t|/t0 = 0.78, not quite as close. If one axis, forming a growth axial surface that is not parallel to the loci of uses the value of tj/t0 of 0.79, the bed-thinning strain is -21% (equation instantaneous rotation axes. A growth axial surface is parallel to the trajec- 16) and the layer-parallel extension is +27% (equation 17), in good agree- tory of the growth-sequence beds as they cross the loci of instantaneous ment with the measured values of -20% and +25%, respectively. The rotation axes. The geometry of growth axial surfaces is beyond the scope original width of the zone of antithetic dip can be predicted from equations of this paper. 6,7, and 8, giving x = 9.5, W0 = 10.9, and W0' = 9.2, using the measured A simple ramp-and-flat geometry for the Schell Creek fault in the values of 40 mm for W' and 13.8° for a. style of Figure 1 is possible but requires the Schell Creek fault ramp to dip The over-all agreement between the geometric model and the clay 46° (Fig. 1 IB). This is very close to the dip seen on the time-migrated model confirms the general validity of the analytical model and checks the seismic profile of Gans and others (1985) and approximates their interpre- equations. The small discrepancies are probably caused by the approxima- tation as well as the oblique simple-shear solution by White and others tion of the bedding markers as planes, an unavoidable lack of precision in (1986). Two objections to this solution are (1) the evidence that the dip locating the axial surfaces of the zone of antithetic dip because the hinges on the Schell Creek fault is lower and (2) the fact that restoration of the are curved, and the approximation of the axial surfaces as having the same geometry does not return the NSRD to the level at which the associated dip as the master fault. Even with these approximations, the greatest stratigraphic units (Pioche Shale/Prospect Mountain Quartzite) are found discrepancy is in the predicted width of the zone of antithetic dip and is in the footwall in the Schell Creek Range. only 8% of the measured width. The computed strains are even closer to The multiple fault bends shown in Figure 11C provide a possible the measured values. solution. Figure 7 shows that a variety of fault-bend angles will give the observed antithetic dip of 19° for an upper segment dip greater than 47°. APPLICATION TO THE SCHELL CREEK FAULT In order to fit with the near-surface fault dip of 37.5°, a steeper segment must occur beneath Spring Valley. The location of the bend from 37.5° to The Schell Creek fault forms the eastern border of the Schell a steeper dip is placed below the zone of rotated valley fill indicated on Creek Range, a young Basin and Range structure in east-central Nevada Figure 11 A. The location of the western axial surface is in the zone of (Fig. 11 A). This example is selected because of the high-quality seismic rotation of the basal valley-fill sediments. The specific fault dips are chosen and geologic data available (Gans and others, 1985; McCarthy, 1986) and to give a suitable geometry and a 19° antithetic dip. The 54° segment of because it has been interpreted by White and others (1986) using the the Schell Creek fault occurs within a region of crossing reflectors on the oblique simple-shear model. The purpose herein is not to prove that an seismic lines where the fault dip is not closely constrained and the 15° interpretation in Figure 11 is the best possible for the Schell Creek fault, lower segment is below the level of interpretable data. A 7° synthetic but is rather to show that the model can be applied and that it results in a rotation is computed for the upper bend. Synthetic dips in the valley fill in cross section that is balanced, kinematically feasible, and restorable. the range of 12° are seen on the seimic lines, but some of this dip is The most critical factor in the application of the model is the bending presumed to be primary. of the Northern Snake Range detachment (NSRD) between the Snake The 12° antithetic dip is the result of beds having a prior synthetic dip Range and Spring Valley. This establishes the continuity between the being rotated as they cross the axial surface into the region of 19° antithetic rotated and unrotated blocks and rules out a tilted-fault-block or domino- dip. The geometry in Figure 11C slightly decreases the depth to the NSRD style model at this boundary. The NSRD is a possibly Oligocene exten- in the Yelland well, which could be compensated by moving the projected sional detachment that in this area, was once approximately horizontal at location of the well slightly downdip. the stratigraphic level of the upper Prospect Mountain Quartzite and over- The major success of the model in Figure 11C is that restoration of lying Pioche Shale (Gans and others, 1985). The NSRD is the horizon that the fault slip returns the NSRD to the same elevation as the Pioche-Pros- is used to define the geometry of the zone of antithetic dip. pect Mountain contact in the footwall, as shown by the line labeled The shape of the NSRD provides key facts needed to define the "restoration." After this restoration, the NSRD would be continuous and geometry of the model. The eastern axial surface must reach ground level horizontal from the Schell Creek Range to the Snake Range. The magni- in the vicinity of shot point 100. The dip of the NSRD in its nearest tude of the displacement is a function primarily of the 15° dip of the lower exposure to the valley-fill sediment cover is about 20° (Gans and others, fault segment and the width of the domain of antithetic dip above it. For 1985). The Yelland well penetrated valley-fill sediments and volcanics, this style of structure, only a restoration along a dipping lower fault can entered a phyllitic fault zone at 1,733 m, below which is 67 m of Pioche raise the elevation of the NSRD in the hanging wall to that of the footwall, Shale, and bottomed in the Prospect Mountain Quartzite at 2,000 m and the width of the dip domain must be correct to achieve the proper (McCarthy, 1986). Correlating the fault in the well to the NSRD (Gans amount of uplift. and others, 1985; McCarthy, 1986) and connecting it to the nearest surface The steep dip of the beds in the footwall of the Schell Creek fault is exposure of the NSRD gives an average dip of 19°. The seismic lines in not explained by the models in Figure 11. To be consistent with the

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-Gpi • Elevation PROJECTED NSRD + 3 -, SCHELL CREEK FAULT -3 km NSRD, { 35-38° DIP + 2 - - +2

+ 1 - - +1 UNROTATED 0 - Sea Level

-1 - -1

-2 - NO VERTICAL EXAGGERATION -2 -3 -3

Figure 11. Geologic data and geometric interpretations of the Schell Creek fault, Nevada. A. Geologic data and selected interpretations from seismic lines. The topography, surface geology, and shot-point locations are from Gans and others (1985). The Yelland well (Y) has been projected 13 km along strike onto the section by Gans and others (1985). The NSRD is the Northern Snake Range detachment, €pi is the Lower Cambrian Pioche Shale, €pm is the Prospect Mountain Quartzite, and p€mc is the upper Precambrian McCoy Creek group of bedded metasediments. The dip of the upper part of the Schell Creek fault is from the migrated seismic section of McCarthy (1986). The geologic interpretation of the Yelland well is from McCarthy (1986), and the hinge locations (clockwise arrows) are approximated from the seismic lines in Gans and others (1985) and McCarthy (1986). See text for further discussion. B. Hypothetical section, assuming a horizontal detachment and controlled primarily by the 19° antithetic dip. C. Hypothetical section, controlled primarily by the 37.5° fault dip, 19° antithetic dip, and the restoration amount.

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structural style of the model, the footwall dip could be the result of an removes the strain in the regions of synthetic and antithetic dip and re- earlier antithetic rotation on the Duck Creek fault which is about 15 km to stores the continuity of originally horizontal lines from hanging wall to the west (Gans and others, 1985). Alternatively, the dip could be the result footwall. of deformation that occurred prior to the formation of the NSRD. The standard-width models are equivalent to the White and others Layer-parallel extension in the region of 19° antithetic dip of the (1986) oblique simple-shear model for simple shear parallel to the axial model in Figure 11C is 41%, computed from equations 14 and 17. The surface of a single bend. Multiple bends using the model presented herein small normal faults seen at the surface and others suggested by the seismic require multiple directions of simple shear, in contrast to the single direc- data indicate layer-parallel extension in this zone but do not appear to tion assumed in the model of White and others (1986). produce sufficient extension. Presumably, most of the strain is below the Any physical model produced without volume changes is balanced resolution of the seismic lines. Extension strains greater than 125% have by definition. Any analytical model that is supposed to represent nature been measured for the NSRD by Miller and others (1983) and Gaudemer should fit physical models that employ the same boundary conditions. The and Tapponnier (1987) but are related to the formation of the NSRD, not analytical model presented in this paper is based on the same boundary the Schell Creek fault. A critical test of the model (or, in fact, of any model conditions as in the Cloos (1968) clay model. In the clay model, a certain with asymmetric axial surfaces) would be to demonstrate strain of the amount of strain precedes the formation of the master fault, and the thick proper magnitude and of the proper age. On the basis of the original width single layer does not form a sharp axial surface, two effects which are of the Spring Valley half graben at the restored level of the NSRD, the treated as an initial width of the graben. The analytical model fits this clay total extension on the Schell Creek fault is 66%. model (Fig. 10) in spite of such factors as slight curvature of the master The dip of the Schell Creek fault below 6 km could be greater or less fault, lack of perfect parallelism of hinges, and hinge dip not exactly than 15°. The seismic lines of Gans and others (1985) and McCarthy opposite to that of the master fault. This demonstrates that the analytical (1986) show strong horizontal reflectors from 4 s to greater than 6 s model is not critically dependent on the exact geometry and mechanics of two-way traveltime, representing depths starting at 10 km for an average the natural example. seismic velocity of 5 km/s. Gans and others (1985), McCarthy (1986), Two solutions for the geometry of the Schell Creek fault are pre- and White and others (1986) interpreted the Schell Creek fault as flatten- sented (Fig. 11). The ramp-and-flat solution fits the general geometry but ing into these reflectors. The more recent COCORP seismic line across the requires a steeper dip on the fault than indicated by the migrated seismic region (Hauser and others, 1987), however, shows a discontinuity in the line of McCarthy (1986) and implies a shallower depth to detachment reflectors below the Snake Range which is interpreted to be a fault that has than seems desirable. The two-bend solution provides a better fit to the an average dip of about 30° from the base of Spring Valley to the Moho. geologic and seismic data and implies a lower detachment that dips 15° to The fault in Figure 11C could bend at a depth of 6 km or deeper to the east. This detachment could steepen or shallow below 6 km without accommodate either possibility. The dip domain associated with such a affecting the geometry on the cross section. The analytical interpretation of deep bend would probably reach the surface in the eastern Snake Range or the cross section serves to highlight the critical data and illustrate the even farther east, well outside the area shown in Figure 11. consequences of certain assumptions. For example, the ramp-and-flat model fits reasonably well the data on the time-migrated seismic line of Gans and others (1985), according to the interpretation of this paper and CONCLUSIONS that of White and others (1986). The geometry, however, requires a relatively shallow and horizontal detachment. If this is not considered The extensional fault-bend fold model developed herein is based on reasonable, then the data must be reinterpreted. Depth migration the following assumptions: (1) constant area, (2) dip changes in the hang- (McCarthy, 1986) is found to change the angles in such a way as to change ing wall are the result of dip changes on the master fault, (3) axial- the character of the solution. Because not just the angles but also the surface dip is equal and opposite to that of the master fault, and (4) there is angular relationships and lengths are part of the model, the interpretations no rotation in the hanging wall except where the dip changes. The equa- of specific features are interconnected. The best-fitting solution must satisfy tions based on these assumptions relate the dip change of the master fault a significant number of constraints, and if it does so, confidence in the to the dip in the hanging wall, and the displacement on the master fault to interpretation should be high. the width of the dip domains in the hanging wall. Direct consequences of the model include (1) a vertical line in ACKNOWLEDGMENTS the hanging wall that is not between an axial-surface pair remains vertical; (2) the axial surfaces are not symmetric, and so beds change thickness in Previous work with Steven I. Usdansky on the related problems of domains between axial surfaces; (3) the horizontal width of a fault- compressive fault-bend folding made this project go much more smoothly. bend-related dip domain is two times the horizontal separation on the I thank Steve for his review. Thanks go to John Dennis for the suggestion master fault segment directly above the bend; (4) the horizontal compo- that the term "reverse drag" be abandoned and to R. H. Groshong III for nent of slip (heave) on the master fault increases downward on a listric the numbers used to construct . Figure 7. Cliff Ando, John Bartley, and bend and decreases downward on an anti-listric (steepening) bend (Fig. 8); Charles Kluth made many useful comments on the first version of this and (5) the vertical component of slip (throw) on the master fault de- manuscript. Discussions with Phil Gans and Jill McCarthy helped clarify creases downward across a listric bend and increases downward across an issues in the interpretation of the Schell Creek fault and in the interpreta- anti-listric bend (Fig. 8). tion of the seismic lines; McCarthy kindly provided me with prints of the The pre-fault geometry is restored by reversing the displacement on seismic lines from her paper, and Ernie Hauser provided a preprint of his the lower detachment. This displacement is equal to the spacing between paper. None of the aforementioned necessarily endorses the interpretations axial surfaces in the pre-fault stratigraphic sequence, measured parallel to presented herein. The research was partially supported by National the lower detachment (Fig. 8). This restoration technique automatically Science Foundation Grant EAR-8402915.

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Printed in U.S.A.

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