## Abstract

The total amount of deformation between two converging bodies is described by the three components of the displacement field: translation, rotation, and strain. Translations along faults and folding strain are the most common elements of the displacement field incorporated into estimates of tectonic shortening across orogenic systems. Determinations of vertical-axis rotations through paleomagnetic and structural analyses are keys for deciphering the rotational component of shortening within an orogenic system, and they can have a substantial effect on the amount of tectonic shortening in such systems. Accommodation structures observed in orogenic systems are typically noncoaxial and/or noncylindrical geometries (e.g., oblique and lateral ramps, superposed folding). These structures suggest that vertical-axis rotations have taken place, can aid in determining the relative timing of rotation with respect to translation, and may help constrain the location of the rotation axis. In this paper, we define the components of the total displacement field, describe the diagnostic and suggestive features associated with vertical-axis rotations, and apply trigonometric map-view calculations to estimate the amount of shortening contributed by such rotations. An error function relating shortening with vertical-axis rotation has been calculated and predicts values up to 50% for a 60° rotation if the rotation is not taken into account. Finally, we apply our approach to the Wyoming salient and show that previous estimates of shortening there may contain up to 14% error.

## INTRODUCTION

For an initial reference object, translation, rotation, and strain (distortion) all comprise the term “deformation” and together make up the total displacement field (Means, 1976; Cobbold, 1979), which represents the change in shape and position of a deformed object. Shape change (strain) can be achieved through simple shear, pure shear, and/or general shear, and a position change is achieved through translation and/or rotation (Fig. 1). While tectonic investigations have become increasingly concerned with three-dimensional transport and orogenic curvature, (e.g., see Weil and Sussman, 2003, and references therein; Soto et al., 2006; Weil et al., 2010), the most widely used tool to measure shortening in fold-and-thrust belts is the balanced cross section (e.g., Dahlstrom, 1969; Hossack, 1979), which assumes plane strain.

The most recent three-dimensional restoration methods (Gratier et al., 1991; Williams et al., 1997; Erickson et al., 2000; Rouby et al., 2000; Dunbar and Cook, 2003; Thibert et al., 2005; Muron et al., 2005; Moretti et al., 2005; Moretti, 2008; Guzofski et al., 2009) do not adequately consider vertical-axis rotations. Work by Pueyo et al. (2004), Oliva-Urcia and Pueyo (2007), and Arriagada et al. (2008) has attempted to quantify the magnitude and spatial distribution of vertical-axis rotations to achieve viable map-view restorations. Recently, Ramón et al. (2012) developed a restoration tool that incorporates paleomagnetic vectors to unfold surfaces in three dimensions. Here, we suggest an approach that incorporates all components of the total displacement field into shortening estimates. Although we emphasize paleomagnetic analysis as a primary method for determining the distribution, direction, and magnitude of rotations, since it provides absolute values for rotation (direction/sense and magnitude), we note that structural techniques, such as calcite twin analysis (e.g., Groshong, 1972), can also be used but are more limited in their ability to quantify magnitudes due to a lack of a stable reference frame. Finally, we present an example of the effects of vertical-axis rotations on shortening estimates from the Wyoming salient of the Sevier orogen in the western United States. This example is used to show the impact that vertical-axis rotations have on tectonic shortening estimates and, thus, the potential impact that vertical-axis rotations have on our understanding as to the structural evolution of contractional orogens.

## TOTAL DISPLACEMENT FIELD

_{i}=

*X*′

_{i}–

*X*, where

_{i}*X*=

_{i}*X*

_{1},

*X*

_{2},

*X*

_{3}is the initial location of a point in the orogen and

*X*′

_{i}=

*X*′

_{1},

*X*′

_{2},

*X*′

_{3}is the final location of the point (e.g., finite deformation of Ramsay, 1967). This notation becomes significantly more complicated when considering deformation of objects of dimension, such as lines, planes, and constant volumes. Whereas the deformation of a point can be described by a vector, the deformation of other objects, such as lines, planes, and constant volumes, requires elements of rotation, translation, and internal strain. For an individual point, ω

_{i}represents the total translation. The displacement gradient, or incremental deformation tensor, is defined by

*D*with

_{ij}For a distinct body or region, and for small/incremental displacements, ω_{i} is unique only if rotation and strain are confined to discrete surfaces. Rigid body rotation is satisfied when ε_{ij}*=* 0 and *r _{ij}* ≠ 0, and pure shear is achieved when ε

_{ij}≠ 0 and

*r*= 0. In cases where ε

_{ij}_{ij}≠ 0,

*r*≠ 0, and both are continuous, vertical-axis rotations will vary systematically (e.g., along the strike of a curved orogen), so long as the entire curvature is due to rotation and strain.

_{ij}For finite deformation, [D] = [T] [R] or [D] = [R] [U], where D is the total deformation, [T] and [U] are strain, [R] is rotation, and [T]≠[U] for the same [R]. Because multiplication is not commutative, the order in which rotation and strain are incorporated into the total deformation changes the results (Elliott, 1976b; Means, 1976; Ramsay, 1976).

The total amount of contractional deformation that takes place in a fold-and-thrust belt is used to: (1) compare contractions in different regions, (2) balance crustal masses in isostatic calculations, (3) generate palinspastic restorations of fold-and-thrust belts to their predeformed state, and (4) describe the geometry, kinematics, and dynamics of fold-and-thrust belt evolution. We define the total amount of shortening between two converging tectonic elements (i.e., plates, thrust sheets) as “tectonic shortening” to distinguish it from the amount of shortening contributed independently from translation, strain, and rotation. Since extension can take place in contractional regimes, which complicates shortening estimates, the term “contractional deformation” is defined as all those translations, rotations, and strains occurring in contractional regimes. The overall effect of tectonic shortening can be described by a finite rotation about an Euler pole on the surface of Earth.

Balanced cross sections have been used since the 1970s to measure amounts of contractional deformation in fold-and-thrust belts. For sections to fulfill the viability and admissibility criteria as defined by authors such as Dahlstrom (1969), Elliott (1972, 1983), Hossack (1979), Price (1981), and Woodward et al. (1989), the line of section must be drawn parallel to the transport direction and located at the apex of the curved map trace of a target thrust fault or orogen. In addition, cross sections through oblique zones or oblique ramps may be admissible, but they are not viable due to out-of-plane motions, if appropriate corrections are not considered (Cooper, 1983). The transport direction, internal strain, and rotations are all important factors that require consideration in the construction of cross sections and the way in which cross sections are used to quantify shortening.

For regional studies that rely on serial cross sections to understand an orogen in three dimensions, such sections either converge to a point in the hinterland (as in salients) or converge to a point in the foreland (as in recesses) (Woodward et al., 1986) (for examples, see Dixon, 1982; Kley and Monaldi, 1998). The resultant “fanning” of cross-section lines is also common when several local studies are compiled together, due to a generalized assumption that transport direction is perpendicular to strike. Price (1981), Cooper (1983), and Woodward et al. (1989) determined the errors in shortening estimates when cross sections are not drawn parallel to the transport direction. For example, an error of ≈15% occurs when a line of section is constructed 30° to tectonic transport, with principal errors resulting from overprojection of structures along-strike (Price, 1981; Woodward, 1989). Techniques by Cooper (1983) and DePaor (1988) help to resolve projection errors for map-scale structures in cross sections drawn oblique to the transport direction. Since transport directions can change temporally and spatially, such error corrections might involve several steps. Changes in the transport direction may be identifiable in the geologic record; for instance, Kwon and Mitra (2004) used structures associated with layer-parallel shortening and elastico-frictional deformation features to determine the early- and late-stage transport directions (respectively) in the Provo salient of the Sevier fold-and-thrust belt.

To produce as accurate a balanced section as possible, contributions from small-scale penetrative strains, as well as map-scale data (e.g., translations along faults and folding strains) need to be included to avoid errors that become cumulative across an orogen and can lead to inaccurate restored fold-and-thrust belt geometries (Mitra, 1994). The kinematic indicators associated with strains and the mechanisms that produce strains can also influence the kinematic admissibility of a section and ultimately the interpretation of an orogen’s evolution (Geiser, 1988; Ferrill and Dunne, 1989; Mitra, 1994; Beaumont et al., 2000; Ismat and Mitra, 2005a). This is especially true when considering the range of deformational scales, such that it is important to define as precisely as possible the scale of the features or phenomena being described. For example, the definition of “discrete” (for describing a fault) is a scale-dependent phenomenon, as exemplified by cataclastic flow, which is continuous at outcrop and larger scales but is discontinuous at smaller scales (e.g., Ismat and Mitra, 2005b).

Techniques for incorporating small-scale strains into balanced cross sections have been explained by various workers including: Woodward et al. (1986), Geiser (1988), Protzman and Mitra (1990), McNaught and Mitra, (1996), and Von Winterfeld and Oncken (1995). Though horizontal-axis rotations (due to folding) have been addressed by Mitra (1994) and Mukul and Mitra (1998), in this paper, we consider the impact of vertical-axis rotations on shortening estimates.

## VERTICAL-AXIS ROTATIONS IN OROGENIC SYSTEMS

### Diagnostic Measures

We refer to rotations of a volume about a vertical axis as a “coherent-body” rotation (a scenario in which is continuous and reasonably uniform) because we recognize that deformation can also occur internal to the body. For this reason, we avoid the term “rigid-body” unless the only accommodation of a vertical-axis rotation is along the margins of the body such that internal strain ε_{ij}*=* 0. For rotating thrust sheets or tectonic plates, the distribution, magnitude, and direction of the rotations can be determined primarily through paleomagnetic analyses and subordinately through structural methods.

Paleomagnetic techniques can yield a quantifiable angle of rotation, and they allow for the identification of coherent-body vertical-axis rotations by comparing observed paleomagnetic declinations for a given time interval with expected paleomagnetic declinations from undeformed rocks magnetized during the same time interval. This assumes that the age of magnetization can be accurately determined, and that there is sufficient well-dated paleomagnetic data from the area of reference. In this sense, it is worth mentioning that paleomagnetic data must honor several reliability criteria to be considered robust (Van der Voo, 1990; Pueyo, 2010). A complete laboratory isolation of magnetic remanence components must be performed to avoid potentially large errors (Rodriguez et al., 2011). Evidences of internal deformation in the rock are enough to reject or to reevaluate paleomagnetic measurements (Lowrie et al., 1986; van der Pluijm, 1987; Kodama, 1988). Accurate restoration of paleomagnetic data to the appropriate reference frame also must be achieved, which is critical in complex portions of fold-and-thrust belts where noncoaxial axes of deformation have acted (e.g., conical and plunging folds, superimposed folding, forced folds, fold closures, oblique thrust ramps, etc.) (Stewart, 1995; Pueyo et al., 2003a, 2003b; Weil et al., 2000; Weil and Van der Voo, 2002).

Paleomagnetic data have been used to measure vertical-axis rotations in fold-and-thrust belts for nearly five decades (Norris and Black, 1961); some recent examples of investigations that have used paleomagnetic techniques for measuring vertical-axis rotations in orogenic systems include: (1) the Bolivian arc of the central Andes (e.g., Butler et al., 1995; Roperch et al., 2000; Arriagada et al., 2003, 2008); (2) the Pennsylvania salient in the central Appalachians (e.g., Schwartz and Van der Voo, 1984; Kent, 1988; Stamatakos and Hirt, 1994; Cioppa and Kodama, 2003); (3) the Southern Pyrenees (e.g., Dinarès et al., 1992; Pueyo, 2000; Sussman et al., 2003; Oliva-Urcia et al., 2008; Oliva et al., 2012; Mochales et al., 2012); (4) the Cantabrian-Asturian arc (e.g., Van der Voo et al., 1997; Weil et al., 2001; Weil, 2006); or (5) the Alps (Thöny et al., 2006; Pueyo et al., 2007).

Structural techniques typically document rotations by measuring components of the incremental strain field in different thrust sheets. For instance, Ferrill and Groshong (1993) used the temporal and spatial change of principal strain orientations from calcite twins to explain fold-and-thrust belt curvature in the Subalpine chain of France. In the Wyoming salient, Craddock et al. (1998), Apotria (1995), and Holl and Anastasio (1995) used similar data to document local thrust sheet rotations and variations in strain-axis orientations at oblique ramps. Using fault lineations and fold and axial trace geometries, respectively, Allerton (1998) and Oldow et al. (1990) determined relative thrust sheet rotations in the eastern Subbetic zone of southern Spain. Gray and Mitra (1993) and Kwon and Mitra (2004) conducted similar investigations in the Pennsylvania and Provo (Utah) salients, respectively. Hnat et al. (2008) and Pueyo-Anchuela et al. (2012) used, apart from paleomagnetism, the anisotropy of magnetic susceptibility (AMS) as an auxiliary tool to estimate vertical-axis rotation magnitudes. As such, structural techniques can provide constraints on the rotation history of a region by allowing for determination of the transport direction at different time intervals.

There are several regions in which structural and paleomagnetic techniques have both been used to understand the magnitude, direction, and distribution of rotations in fold-and-thrust belts. In some cases, the data from these different techniques are comparable, and in other cases, the techniques yield different results. For example, the predicted thrust rotations determined by Kollmeier et al. (2000) matched within error rotations determined from paleomagnetic analysis for the Cantabrian-Asturian arc (Weil et al., 2001). In the Subbetic zone, Allerton (1994) ascertained that thrust lineation rotations were constant relative to paleomagnetic rotations. Similarly, Oldow et al. (1990) and Channell et al. (1990), using fold geometries and paleomagnetism respectively, determined comparable rotation history for the Sicilian fold-and-thrust belt. On the other hand, in the Wyoming salient, the calcite twin data from Craddock et al. (1988) and paleomagnetic data from Grubbs and Van der Voo (1976) did not directly correspond. A similar discrepancy can be seen in the Swiss Jura, where rotational strain directions were not consistent with paleomagnetically measured vertical-axis rotations (Kempf et al., 1998; Hindle and Burkhard, 1999). Finally, paleomagnetic investigations in the Provo/Charleston-Nebo salient (Conder et al., 2003) are at odds with structural work by Kwon and Mitra (2004) in the same area. Such varied results reveal the complexity involved in incorporating all components of the total displacement field in developing reliable kinematic models (i.e., relationships among translation, strain, and rotation). The causes of these discrepancies are not fully understood, and they need to be defined for individual cases. Like Hindle and Burkhard (1999), who used varying data sets to successfully propose an integrated model for the Jura arc, Gray and Stamatakos (1997) also combined seemingly opposing data to develop a compelling integrated model for the Pennsylvania salient. Finally, consistent results from paleomagnetic, AMS, and calcite twinning allow rejection of any rotation in the interpretation of the kinematics of the Jones Valley thrust in the Appalachians (Hnat et al., 2008).

Since the rotational velocity and rotational ages (Pueyo et al., 2002; Mattei et al., 2004; Mochales et al., 2011) of thrust sheets are key variables, the effect of vertical-axis rotations must be removed at the proper time intervals during restoration to accurately determine the regional transport direction as well as the amount of rotational shortening. Integration of high-resolution paleomagnetic and structural data sets from strata of different ages and at the scale of individual thrust sheets is required to more accurately decipher the complex development of orogens that have undergone vertical-axis rotations.

### Structures Accommodating Out-of-Plane Motion

Investigations from orogens worldwide show that a number of structures form to accommodate out-of-plane motion. For example, in the western Transverse Ranges of California, Onderdonk et al. (2005) demonstrated that a zone of deformation, populated by folds, refolded folds, and reverse faults trending oblique and parallel to the two main bounding faults, accommodated the rotation between the bounding faults. Bayona et al. (2002) established the relationship between several transfer zones and rotations in the Southern Appalachians. Additionally, Berberian et al. (2000) suggested that a strike-slip fault connecting thrusts with opposite vergence took up vertical-axis rotations in the Sefidabeh area of the Sistan ranges in eastern Iran. Out-of-plane motion can also be accommodated by “thrust segmentation,” such that an otherwise continuous thrust sheet is broken up by secondary structures, and/or variation in the along-strike nature of a fault or fault system, such as the Altyn Tagh in China (Rumelhart et al., 1999) or the southwestern Pyrenees in Spain (Pueyo et al., 1997). These investigations, as well as recent analogue modeling data (Soto et al., 2006), suggest that if the nature of along-strike shortening changes due to vertical-axis rotations, geologic structures that accommodate those changes will develop. Observations of accommodation structures raise several important questions:

(1) How do coherent bodies accommodate vertical-axis rotations?

(2) What features or group of features develop along the boundaries of such rotating bodies?

(3) How do these features change depending on the location of the rotation axis and timing relationships between translation and rotation?

To address these questions, we present several models for coherent-body rotations.

## MODELS OF VERTICAL-AXIS ROTATIONS ASSOCIATED WITH THRUST SHEETS

Given a finite portion of a thrust surface, there are two scale-independent models that can be used to describe the effects of a rotational component on the total deformation field: (A) an internal pivot-point model (Figs. 2A and 2B) and (B) an external and/or termination pivot-point model (Figs. 2C and 2D), these models correspond to the near-field and far-field concepts defined by Allerton (1998). Our analysis assumes that the pivot remains stationary throughout the motion of the body; if a rotation is caused by motion due to the presence of a preexisting structure (i.e., buttresses or lateral ramps), this simple approach is likely sufficient. On the other hand, if a rotation is the result of differential movement within a growing structure (e.g., thrust sheet or fold), then the pivot may move, and/or its position relative to the structure can change. As such, geologically meaningful pivots, such as a thrust termination/tip, represent only the pivot’s final location such that a finite deformation is implied. If sufficient evidence is available to determine the incremental/continuous deformation and the stepwise location of the pivot, an approach similar to that proposed by Tikoff and Fossen (1995) can be applied.

In areas where thrust terminations are unavailable (e.g., eroded or covered), or to determine more precise pivot locations, high-resolution arrays of paleomagnetic vertical-axis rotation measurements need to be collected and merged with structural studies that yield transport directions. We propose that, rather than having a unique geological marker, suites of geologic structures can help identify whether or not a rotation occurred (e.g., Allerton, 1998; Rumelhart et al., 1999; Onderdonk et al., 2005), when the rotation took place with respect to translation, and the approximate location of the rotational pivot.

When a stationary pivot point is located inside a thrust sheet, the final geometry of the thrust sheet and its footwall is the same, but the geologic features that form during the deformation stages will be different. It is important to note that this scenario implies that all deformation is concentrated along the thrust sheet boundaries, rather than distributed within the sheet, probably a rare scenario to find in nature. In any case, timing effects are important. When translation is followed by a rotation such that [D] = [R] [U], part of frontal thrust boundary will be replaced by normal motion (Fig. 2A). The lateral lengthening of the contractional and extensional fronts will depend on the geometry of the hanging wall.

On the other hand, if a rotation occurs first such that [D] = [T] [R], the final feature to form will be a frontal thrust (Fig. 2B). Thus, if the pivot point is inside the block or thrust sheet (e.g., domino styles; McKenzie and Jackson, 1983; Ron et al., 1984; Garfunkel, 1989) and is in a fixed position throughout the active deformation, it may be possible to constrain timing relationships between translation and rotation by looking at geological features developed along the boundaries of the rotating object. A larger-scale demonstration of the concept is available in the North American plate (a plate-scale rotational block), which has a pivot within its body with respect to the Pacific plate/rotational block, such that the plate margins have similar boundaries to the example above. Likewise, with known timing relationships, the geological features might allow for the pivot location to be approximated.

When a thrust sheet undergoes translation and rotation with a pivot point located outside the sheet (Fig. 2C), the entire thrust sheet moves in the same direction, creating fewer space problems associated with bounding structures (Allerton, 1998). The farther a pivot point is from a thrust sheet, the more the effect from rotation is minimized such that displacement (deformation) trajectories can be simplified as a translation. In this case, the geologic features that form may include normal faults that are later reactivated as thrust faults (rotation followed by translation; Fig. 2C, left), or thrust faults reactivated as normal faults (translation followed by rotation; Fig. 2C, right). Likewise, a scenario with simultaneous rotation and translation will produce similar features depending on the deformation rates of the various processes (Fig. 2C).

A pivot point that exists at the end of a fault can be called a “termination pivot” (Fig. 2D); we propose a termination pivot model to be the most applicable for developing thrust structures (and listric normal faults). A useful concept that explains our rationale is Elliott’s “bow and arrow rule” (1976a), in which the normal bisector of a straight line, drawn between the terminations of a curved thrust, yields the maximum displacement of that thrust with a decrease of displacement toward the tips, so long as the transport direction is parallel to the bisector. Self-similar “bow and arrow” thrust growth, in which thrust length and displacement along the normal bisector increase proportionately, requires nonzero strain and rotation tensors, where ε_{ij} ≠ 0 and *r _{ij}* ≠ 0. It is at the tips of these thrusts, however, where strain and rotation are both zero, that pivots are located at the terminations on either end of the thrust. Allerton (1994) documented an along-strike increase in shortening from a pivot located on the end of a “bow and arrow” and identified the pivot point of a rotating structure at its termination by measuring folding strain and paleomagnetically determining vertical-axis rotations in the Subbetics of southern Spain. Similar results are found in the south Pyrenean sole thrust in the Central Pyrenees (Millán et al., 1995, 2000; Pueyo et al., 1999; Oliva-Urcia et al., 2010), where the westernmost termination of the External Sierras thrust front ends in a detachment anticline followed by a more steeply plunging fold (the San Marzal fold-closure).

For a comprehensive analysis of the effect that a termination pivot point has on shortening, imagine a surface element deformed in a discontinuous fashion (Fig. 3A), first by translation (ε), with the transport direction perpendicular to the hanging wall thrust trace, followed by a coherent body rotation about a vertical axis (β). For equal amounts of rotation and translation, the sequential order of the operative mechanism will play a definitive role in the final location of the initial element and, thus, is noncommutative (Fig. 3A). Also, note the centripetal migration of the rotation pivot point. Regardless of the first motion, the location of the final pivot point will be controlled by the sequence and magnitude of both deformational processes. In any case, a circular path will be able to relate the initial (nondeformed) and the final (deformed) stages.

Given a continuously developing thrust system, we illustrate several merged increments of translation and coherent body rotations (Fig. 3B). In the general case, an ellipse best relates the initial to the final deformation processes, including noncommutative translations and rotations. However, except for extreme and complex translation/rotation ratios that need to be individually calculated, the eccentricity of the ellipse will be very low, and so a circular path is employed in our model for mathematical simplicity. The elliptical trajectory defining the movement of a hanging-wall point depends on the translation and rotation magnitudes and their temporal relationship. In extreme translation/rotation ratios, this mathematical function may be very complex and would have to be calculated for each individual case. In most other cases, however, the eccentricity of the ellipse will be very low and can be easily approached by the circular paths represented in our model.

A map-view model displaying the trigonometric relations between the footwall and hanging-wall cutoffs in a rotational setting was proposed by Pueyo (2000) and Pueyo et al. (2004). This model (Fig. 4A) relates the true shortening (S and S′, assuming a stationary pivot) with the measured shortening from a balanced cross section (Sm), if the rotation (β) has not been considered.

A graph of this trigonometric function (Fig. 4C) calculates the percent error in shortening given the above conditions. For example, vertical-axis rotations of 80° will produce an apparent shortening almost double that of the true shortening, if the cross section is drawn in a straight line perpendicular to the thrust trace. For β = 50°, the measured shortening will contain a 30% error. Rotations below 30° generate errors <10%. These calculations will need an additional correction, if the cross section is not drawn parallel to the transport direction (Cooper, 1983; DePaor, 1988; Pueyo et al., 2004).

## APPLICATION TO THE WYOMING SALIENT

The Sevier fold-and-thrust belt of Idaho, Utah, and Wyoming (Armstrong, 1968) defines the eastern margin of the Cordilleran orogen in western North America. Proterozoic, Paleozoic, and Mesozoic sedimentary rocks in this area underwent shortening and translation during the late Jurassic to early Eocene Sevier orogeny (Burchfiel and Davis, 1975), with successive thrusts generally forming eastward in time with transport in that direction (e.g., Armstrong and Oriel, 1965; Royse et al., 1975; Jordan, 1981) and continued reactivation in the back of the orogen (DeCelles and Mitra, 1995). The Sevier fold-and-thrust belt is divided into a number of salients (Smith and Bruhn, 1984; Lawton et al., 1994), each convex to the foreland. The Wyoming salient, located in western Wyoming, southeastern Idaho, and northeastern Utah, is bounded to the north by the Gros Ventre and Wind River Ranges, to the south by the Uinta Arch, and to the east by the Hoback and Green River basins. From the hinterland to the foreland, the major thrusts in the northern Wyoming salient are: the Paris, Meade, Absaroka, Darby, and Prospect thrusts (Fig. 5).

### Wyoming Salient: Geologic Features

The curved nature of the Wyoming salient has been well-studied by a number of workers and is generally considered to be the result of buttressing effects from the Teton–Gros Ventre and other basement uplifts to the east (Crosby, 1969; Grubbs and Van der Voo, 1976; Beutner, 1977; Blackstone, 1977; Dorr et al., 1977; Wiltschko and Dorr, 1983; Hunter, 1988; Weil and Yonkee, 2009; Yonkee and Weil, 2010; Weil et al., 2010). However, Woodward et al. (1986) and Royse (1993) suggested that stratigraphic controls play a larger role than buttressing in determining regional along-strike changes. Dixon (1982) used industry seismic data to construct ∼50 cross sections, providing insight into the along-strike variations of structures within the salient (Fig. 5A). In particular, the Prospect thrust shows significant variation in the amount of displacement along strike (Dixon, 1982). The differential transport implied by the bulk shear angles (up to 48°) in parts of the Prospect thrust predicts the presence of oblique ramps and tear faults along the thrust trace (Wilkerson, 1992). Similarly, Apotria (1995) modeled particle paths for oblique ramps in this region and found that out-of-plane strains are taken up by footwall geometries and varying amounts of thrust displacement, which manifest themselves as accommodation structures. In fact, the Prospect thrust is highly segmented (Fig. 5A) by tear, strike-slip, and normal faults oriented perpendicular to the thrust front (Dorr et al., 1977; Blackstone, 1977; Hunter, 1987). The geologic characteristics reported in these investigations suggest that vertical-axis rotations occurred in the Wyoming salient.

### Diagnostic Measures of Vertical-Axis Rotations in the Wyoming Salient

Variations in the transport direction and vertical-axis rotations within the Wyoming salient have been documented by paleomagnetic and structural studies. Several workers, including Craddock et al. (1988), Mitra (1994), Apotria, (1995), and Yonkee and Weil (2010), documented out-of-plane strains from small-scale structures. For example, Craddock et al. (1988) used calcite twin strains to suggest curved deformation paths for material within the Wyoming salient. The strain trajectories due to layer-parallel shortening determined by Craddock et al. (1988) vary from counterclockwise in the northern part of the salient to clockwise in the southern part of the salient. Data such as this allow us to reconstruct the early transport direction of material within the Wyoming salient.

In addition, vertical-axis rotations from paleomagnetic data were measured in the Wyoming salient by Grubbs and Van der Voo (1976), Eldredge and Van der Voo (1988), and Weil et al. (2010). McWhinnie et al. (1990) suggested that the vertical-axis rotations of the Prospect thrust sheet likely occurred contemporaneously with thrust motion. Based on Sussman et al. (2003), Weil and Sussman (2004), and Weil et al. (2010), this portion of the Wyoming salient can be considered a progressive arc, such that the pivot location may not have been fixed through time during deformation. The structures that developed along the thrust front reflect the expected accommodation styles based on an external pivot point model (Fig. 2C).

Several workers (e.g., Royse et al., 1975; Blackstone, 1977; Dorr et al., 1977; Dixon, 1982; Woodward et al., 1986; Craddock, 1992; Mitra, 1994; Yonkee and Weil, 2010) have calculated shortening across the northern Wyoming salient (or portions thereof). While most of these authors based their estimates on translational shortening and folding strain, Craddock (1992), Mitra (1994), and Yonkee and Weil (2010) also included internal strains in their estimates.

### Impact of Vertical-Axis Rotation on Shortening Estimates in the Wyoming Salient

Paleomagnetic analyses (Grubbs and Van der Voo, 1976; Weil et al., 2010) have shown that the Prospect thrust sheet has undergone extensive and systematic rotation (Fig. 5A); thus, we focus our example on the Prospect sheet to estimate the amount of shortening error due to neglecting the contribution of coherent body rotation. To incorporate the rotational component into total shortening estimates, we first examine the transport direction.

Curved strain trajectories from calcite twinning, cleavage, and other structures developed during layer-parallel shortening (Craddock et al., 1988; Yonkee and Weil, 2009), and paleomagnetic data (Grubbs and Van der Voo, 1976; Weil et al., 2010) have been used to constrain the early transport direction of the Prospect thrust sheet. Since layer-parallel shortening develops early in the deformational history of the thrust sheet, any later coherent body rotations would have passively rotated the preexisting strain trajectories, and their implied transport direction. For our analysis in the Wyoming salient, we have chosen a reference point (Fig. 5A, star) that represents the footwall cutoff of the Prospect thrust in Mitra’s (1994) cross section (Fig. 5B). We used the divergent calcite twin paths (Craddock et al., 1988) to interpolate the transport direction at the point indicated (Fig. 6A) in the footwall cutoff in the Prospect thrust. Since the transport direction recorded by the calcite twins at that point has been passively rotated by vertical-axis rotations (Grubbs and Van der Voo, 1976; Weil et al., 2010), the transport direction must be rotated to obtain the original direction of thrust motion. We have rotated this layer-parallel shortening transport direction by 33° in a counterclockwise direction to reflect the 33° clockwise rotation about a vertical axis as determined by Grubbs and Van der Voo (1976); likewise, vertical-axis rotation results from Weil et al. (2010) range from 20° to 40°.

The Prospect thrust rotation pivot (Fig. 6B, “pivot”) had to satisfy the following requisites: (1) A circle centered on this pivot must be tangent to the transport direction at the Prospect footwall cutoff point. This path represents the actual trajectory followed by material in the hanging wall of the Prospect thrust. Importantly, in any orogenic system where vertical-axis rotations have been documented, all cross sections should be drawn along curved lines reflecting the real movement of the particles. (2) The intersection of the circle in the hanging-wall cutoff must be tangential to the rotated transport direction (Fig. 6). From the thrust “pivot,” the measured angle between the footwall and hanging-wall cutoffs (Fig. 6B) must be equal to the paleomagnetic rotation. If other portions of the Prospect sheet, or areas of the Wyoming salient, are considered, different pivot locations will be found.

A trigonometric map-view model has been built for the Prospect thrust sheet of the Wyoming salient (Fig. 6B). The first observation is that the real tectonic transport trajectory (see previous explanation) is located far from the line of section drawn by Dixon (1982, labeled D). Through application of Equation 2, we find an overestimation of the shortening along the Prospect thrust of ∼14%. It is important to mention that such errors are cumulative and depend on the misorientation of the transport direction, as well as from having ignored vertical-axis rotation (not incorporating the effects of penetrative strain [Mitra, 1994] may add more error). For the portion of the Prospect thrust sheet investigated, the error introduced into these sections from neglecting the 33° clockwise vertical-axis rotation (Grubbs and Van der Voo, 1976) is ∼11%. The remaining error (∼3%) is due to the oblique orientation of the cross section with respect to the real transport direction.

Layer-parallel shortening data from transects farther to the south (Mitra, 1994) indicate uniform transport direction for those southerly transects (Figs. 5 and 6A). Thus, in this more southern area, Dixon’s other cross sections would yield a better approximation of shortening. Given this distinction, a change in the type of deformation could be indicated such that in the more curved portion of a thrust sheet, particle paths are radial, whereas in the straighter portions, particle paths are straight and parallel. Therefore, in the curved portions, restorations would result in excess volume at the point at which the radial sections converge.

## CONCLUSIONS

Tectonic shortening is the total amount of deformation between two converging bodies and can be completely described by the displacement field, which consists of three components of deformation: translation, rotation, and strain. Neglecting the contribution of vertical-axis rotations may lead to inaccurate estimations of tectonic shortening, causing misunderstandings as to the kinematic and mechanical development of orogenic systems. This investigation indicates that shortening estimates that take rotations into account will differ from those estimates that do not incorporate rotations. We have calculated errors for previous estimations of the tectonic shortening for the Prospect thrust sheet of the northern portion of the Wyoming salient. This was accomplished by combining structural analyses, which have independently calculated translation and strain shortening, with a trigonometric function that incorporates the contribution from rotational shortening. The rotational contribution toward tectonic shortening was determined by analyzing the paleomagnetic evidence for regional (coherent and/or rigid) vertical-axis rotations in conjunction with field and microstructural evidence of such rotations. The relative timing of these deformation processes plays a role in the development of various geological features. We propose that suites of geologic features (thrust segmentation, etc.) suggest that vertical-axis rotations have taken place, can aid in determining the relative timing of rotation with translation, and can provide constraints as to the location of the rotation axis, or pivot point. Collection of high-resolution data sets of vertical-axis rotations and further integration of paleomagnetic analyses and structural studies in other orogens are needed in order to more accurately measure tectonic shortening.

We thank Ruth Soto Marin and an anonymous reviewer for their suggestions and improvements on this paper. We also thank David Ferrill and Robert Butler for their comments on an earlier version of this manuscript. Support for this work was provided by the American Chemical Society to Chase and Sussman and from the Pmag3Drest (CGL2009-14214 MICINN) and 3DR3 (PI165/09 Aragonian government) to E.L. Pueyo. Finally, we are forever grateful for and remember Martin Burkhard.