# AAPG Bulletin

- Copyright ©2017. The American Association of Petroleum Geologists. All rights reserved.

## ABSTRACT

Confidently defining the trajectory of faults that control structural traps is a recurring challenge for seismic interpreters. In regions with fault-related folds, seismic and well data often constrain the upper fold geometry, but the location and displacement of the controlling fault are unknown. We present a generalized area–depth strain (ADS) analysis method that uses the observed depth variation in deformed horizon areas to directly estimate underlying fault depth, dip, displacement, and layer-parallel strain from a structural interpretation. Previously established ADS methods are only applicable to structures controlled by faults that sole into layer-parallel detachments. The new technique, referred to as the fault-trajectory method, generalizes ADS analysis to contractional and extensional structures controlled by fault ramps that cut across layers and displace the regional. For structures where area is conserved during deformation and shear is minimal, laterally shifting the analysis limits across the structure defines changes in fault orientation. We validate the method by applying it to numerical forward models, analog clay models, and seismically imaged structures from the San Joaquin basin in California, the Sierras Pampeanas in Argentina, and the North Sea. The fault-trajectory method is shown to be robust, because it exactly reproduces the prescribed fault trajectories and displacements used to construct the numerical and analog models. In the natural examples, the ADS-estimated fault trajectories are consistent with independent fault-location constraints such as earthquake focal mechanisms, seismic imaging, and forward modeling.

## INTRODUCTION

Fault-related folds are one of the most common structural traps for hydrocarbons. Frequently, folds develop in response to displacement along underlying faults that cut across stratigraphy at variable angles. Locating and defining such fault trajectories can be critical in assessing the trap and seal potential of a structure. Seismic reflection profiles and well data can often constrain the relatively shallow fold geometry in detail. These data sets can also provide information about the location and orientation of the shallow part of the controlling fault. However, the deep location and dip of the controlling fault are often unconstrained because of limited borehole depth and variable seismic image quality in deformed regions. To address this problem, tools such as kinematic forward modeling and area–depth strain (ADS) analysis have been developed to estimate likely fault geometries.

One method of estimating fault trajectories uses kinematic forward modeling of contractional and extensional fault-related folds (e.g., Suppe, 1983; Suppe and Medwedeff, 1990; Erslev, 1991; Xiao and Suppe, 1992; Eichelberger et al., 2015). In this method, the trajectory of the modeled controlling fault is varied (along with other parameters) until the modeled fold conforms well to the observed structure. This method works well in many circumstances but fails if the geometric assumptions in the forward model are not appropriate for the structure being analyzed, which is often the case for complex structures. Forward models can be applied in structurally complex areas with variable fault trajectories, but selecting the most appropriate model for a given setting requires understanding the inherent kinematic assumptions that differentiate the model types. At a basic level, these kinematic assumptions dictate whether or not bed thickness and/or length are maintained and how folding is accommodated between stratigraphic layers. Detailed observations of strata geometry and fault-displacement gradients are necessary to assess the permissibility of these assumptions and determine which model is most applicable to the given structure (Suppe et al., 1992; Shaw et al., 2005; Hughes and Shaw, 2014). The kinematic assumptions inherent in these models dictate the fault trajectory at depth and ultimately have a significant impact on estimates of displacement and strain.

A second method of analyzing faults beneath folded structures is ADS analysis, which gives estimates of detachment depth, fault displacement, and layer-parallel strain (LPS) within structures (Epard and Groshong, 1993; Groshong and Epard, 1994). This approach has the advantage of requiring fewer restrictive kinematic assumptions than the forward modeling approach, primarily relying on the assumptions of plane strain and conservation of area during deformation. In the modern graphical approach to ADS analysis, fold areas are measured for a series of horizons, and then the areas are plotted relative to the regional depths for the horizons on an area–depth graph (Epard and Groshong, 1993). In many circumstances, an area–depth graph is a simple but powerful means for displaying geometric information present within a structural interpretation that is not immediately apparent from the interpretation alone. This geometric information is the basis for computing parameters such as displacement, strain, and detachment depth directly from structural interpretations.

Variations in fold-area trends with depth can be used to identify fundamental structural features, including pregrowth and growth intervals and multiple detachment levels (Epard and Groshong, 1993; Groshong et al., 2012; Schlische et al., 2014; Groshong, 2015). Horizons that record constant displacement with depth, such as pregrowth intervals, are identified on an area–depth graph by a linear trend. The slope and depth intercept of the linear relationship give the horizontal boundary displacement and depth to the underlying detachment, respectively (Epard and Groshong, 1993). A central advantage of ADS analysis is that displacement is determined independently of constant bed-length or thickness assumptions. Bed-length variations relative to this linear relationship between area and shortening can lend insight into the amount of LPS present in a structure (Groshong and Epard, 1994) or, alternatively, detect imbalances in structural interpretations (Groshong, 2015). Gonzalez-Mieres and Suppe (2006) have described a similar method of analyzing detachment folds, involving flattening interpretations to an arbitrary stratigraphic horizon to work in the thickness domain rather than depth. This approach can make it easier to identify stratigraphic units that undergo regional thickness changes, but for the examples used in this paper, working in depth is accurate.

Previously published ADS techniques only apply to structures that form above faults that sole into a horizon parallel to the regional dip (the regional as defined by McClay, 1991; Figure 1A) and cannot be applied to structures that form above faults that crosscut the regional (Figure 1B). This paper describes a generalization of the ADS method that applies to such crosscutting faults, making it applicable to a much wider range of geological structures. We call this generalized method the fault-trajectory ADS method to distinguish it from the previously published ADS method, referred to here as the flat-detachment method. As with prior ADS methods, the fault-trajectory method assumes that pregrowth horizons experience constant displacement with depth and that area is maintained during deformation. These assumptions allow us to equate horizon areas (*A*, Figure 2) to the area displaced during deformation (*A* = *A*′, Figure 2). To apply ADS analysis to faults that dip relative to stratigraphy, horizon areas must be measured relative to their regional footwall depth (*z*_{f}, Figure 2) and then plotted on the area–depth graph at the average regional depth (average of the footwall and hanging-wall regional depths; *z*_{av}, Figure 3). The generalized area–depth theory presented here makes it possible to estimate both the depth and dip of the controlling fault in addition to displacement and LPS (Figure 3). In appropriate conditions, these estimates can be made at multiple points across a structure, mapping the fault trajectory and determining the variation of displacement along the fault. Similar to the flat-detachment method, the fault-trajectory method also applies to extensional and contractional faults.

In this paper, we explain the theory behind the fault-trajectory ADS method and validate it using forward models, analog models, and natural examples. In addition, we demonstrate how the fault-trajectory method can be used alongside kinematic forward models to build robust interpretations that are independently verified by both techniques. In contrast to prior ADS-related studies where area–depth graphs are plotted separately from the interpreted sections, the area–depth graphs in this paper are plotted directly over the interpreted sections to compare the computed results with the data and interpretation (Figure 3). In addition, the displacements for the underlying fault and the pregrowth and growth horizons are shown as vectors at the hanging-wall regional limit of each plot (Figure 3). Because the displacement vectors are plotted at the scale of the interpretation, they effectively represent a restored loose line for the hanging-wall regional limit of the structure (Figure 3). The ADS analyses were performed using interactive software. The interactive approach facilitated investigating the sensitivity of the computed fault trajectories, displacements, and strains to different analysis limits. This approach is particularly useful for structures controlled by curved faults or fault flat-to-ramp transitions.

## FAULT-TRAJECTORY–METHOD THEORY

### Defining Fold Area and Regional Depths

Figure 1 compares contractional folds formed above a fault with a flat detachment (Figure 1A) and above a continuous ramp (Figure 1B), illustrating the key differences between the fold-area measurement for the flat-detachment and fault-trajectory methods, respectively. In both cases, ADS analysis is carried out between two vertical pin lines: a hanging-wall regional limit and a footwall regional limit. In both cases, the footwall regional limit is situated beyond any footwall deformation associated with the fault. Regional depths for the footwall (*z*_{f}) and hanging wall (*z*_{h}) are measured relative to an overlying horizontal reference line (Figure 1).

For flat detachments, the hanging-wall regional limit is situated above the distal flat fault segment where the horizons return to the regional depth observed in the footwall. In this case, an individual horizon has the same regional depth, *z*, at the hanging-wall and footwall regional limits, because fault slip at the hanging-wall regional limit is purely horizontal (Figure 1A). The fold area created by the slip on the fault is measured relative to a horizontal datum at depth *z*.

For a fault with a ramp at the hanging-wall regional limit, any boundary-displacement results in a fault slip vector with both a horizontal component (heave) and a vertical component (throw, Figure 1B). The throw on the deep part of the fault lifts the entire hanging wall relative to the footwall regional depth and produces excess area in addition to that produced by fault-related folding on the upper curved part of the fault. Accordingly, in the fault-trajectory ADS method we use the footwall regional depth (*z*_{f}) of a folded horizon as the horizontal datum for fold-area measurements. In this case, in contrast to the flat-detachment case, the magnitude of the fold area increases or decreases as the hanging-wall regional limit is moved to the left or right, respectively. This effect is fully accounted for by the fault-trajectory method and will be demonstrated below.

All area–depth measurements are made with respect to the regional elevations of the horizons. Regional tilting or warping unassociated with fault slip can potentially result in erroneous fault-trajectory and displacement estimates if not correctly accounted for. In this paper, regional dip is restored prior to applying the fault-trajectory method by rotating the structural profile being analyzed.

### Determining Fault Depth, Dip, and Displacement

Figure 2 shows more detail for a contractional fold formed above a fault ramp. The structure has formed because of a horizontal boundary displacement toward the footwall of −*d* (by convention, contractional displacements are numerically negative, and extensional displacements are positive). Assuming conservation of area during deformation, the excess fold area (*A*, Figure 2) generated by displacement on the underlying fault is equal to the area that was displaced along the fault from outside the hanging-wall regional limit (trapezoidal area *A*′ in Figure 2). The following theory also applies to folds formed above extensional faults, in which case horizontal displacements are numerically positive and fold areas are negative (measured downward from the footwall regional depth).

The trapezoidal area *A*′ is defined by the horizontal displacement (*d*), the depth of the fault at the hanging-wall regional limit (*z*_{D}), and the depth of the fault at the predeformation hanging-wall regional limit (*z*_{D}′, Figure 2). Because the hanging-wall displacement vector is parallel to the deep part of the fault, we see that(1)We equate the measured fold area (*A*) to the displaced area (*A*′) for an individual horizon using the standard formula for the area of a trapezoid:(2)We simplify this equation significantly by introducing the average of the hanging-wall and footwall regional depths:(3)With this substitution, equation 2 can be rearranged as(4)For a series of horizons that all record the same displacement (e.g., pregrowth horizons with little to no layer-parallel simple shear, as in Figure 3), area will vary linearly as a function of average regional depth (equation 4). The slope of *A* with respect to *z*_{av} gives the horizontal displacement for the horizons(5)and the intercept of the area–depth graph at zero area gives the depth to the fault at the hanging-wall regional limit, *z*_{D}. In practice, these two values are found by calculating a least-squares linear regression through the area–depth points (with average regional depth as the independent variable).

The fault dip, *α*, at the hanging-wall regional limit can be estimated for a single horizon from the horizon heave (*d*) and the regional horizon throw (*z*_{f} *− z*_{h}):(6)If multiple pregrowth horizons have the same displacement, they should also have the same regional throw. So, in practice, the regional throws for the pregrowth horizons used in the least-squares linear regression are averaged and used in equation 3. A coefficient of variation (the ratio of the standard deviation to the mean) is computed for the pregrowth throws and used for quality control. Too much variation in throw between horizons indicates a problem in the structural interpretation or a breakdown in the assumptions of the ADS analysis.

Finally, the signed magnitude of the computed fault-displacement vector (Figure 3) is given by(7)The critical equation 4 is the same as the area–depth relationship for a flat detachment, with the substitution of the average of the hanging-wall and footwall regional depths, *z*_{av}, for the single regional depth, *z*, that applies in the flat-detachment case. If the fault-trajectory method is applied to a structure with a flat detachment, then all of the relationships above simplify appropriately.

An implicit assumption in the discussion above is that the overall vergence of the structure is known so that the hanging-wall side of the structure can be determined. This involves assessing whether the underlying fault is contractional or extensional. If the fault is contractional, then the hanging-wall side of the structure is the structurally upthrown side, and if the fault is extensional, then the hanging-wall side of the structure is the downthrown side. In the absence of observable faults with defined offsets, a structure can reasonably be assumed to be generally contractional if the measured bed lengths are significantly greater than the section length. Similarly, we can assume a structure is extensional if the measured bed lengths are significantly less than the section length. In some cases, such automatic geometric criteria fail and the nature of the underlying fault must be constrained by independent considerations.

Using a linear regression to determine displacement and fault depth from the depth variation in the area of structural relief requires that the boundary displacements for the horizons included in the regression are constant with respect to depth (Epard and Groshong, 1993). Selecting the stratigraphic interval to include in the regression takes some consideration, because horizons within growth sections have displacements that vary with depth, complicating the underlying area-balance theory behind ADS analysis. In general, growth horizons are characterized by fold areas that decrease toward the surface, because the displacement they record typically diminishes as a function of stratigraphic age and therefore depth (light-gray horizons, Figure 3). As such, they are not included in the linear fit applied to the pregrowth area–depth data. However, the displacement recorded by individual growth horizons can be determined using the best-fit fault depth based on the pregrowth area–depth relationship (Groshong et al., 2003). Displacement for an individual growth horizon is given by the slope of a line connecting the growth horizon’s area–depth point to the fault depth estimated from the pregrowth area–depth intercept line (*z*_{D}, Figure 3).

### Layer-Parallel Strain

The graphical approach to ADS analysis provides an independent measurement of displacement, making it possible to estimate changes in individual bed lengths caused by LPS (Groshong and Epard, 1994; Groshong et al., 2012). As discussed above, this circumvents the common assumption that bed lengths remained constant during deformation. For comparison, the ADS analyses in this paper display both the area–depth displacement and constant bed-length displacement (referred to here as the nominal displacement) as vectors and values (Figure 3). The nominal displacement vector is calculated using the fault dip estimated from the fault-trajectory method. In ADS analysis, the restored length of a horizon (*l*_{i}) is determined using the graphically determined horizontal displacement (*d*) and the curvimetric bed length between the regional limits (*l*_{f}):(8)The difference between the modern bed length (*l*_{f}) and the restored bed length (*l*_{i}) reflects the change in bed length during deformation. This is reported as the percent change in bed length or LPS:(9)Negative LPS values represent decreases in bed length or contractional strain, whereas positive LPS values indicate increases in bed length and extensional strain. Here, LPS for each horizon is reported numerically next to the nominal displacement vector associated with that horizon (Figure 3).

### Fault Curvature

The fault dip, depth, and displacement calculated using the fault-trajectory method are specific to the location where the hanging-wall regional limit intersects the fault (Figure 3). If the position of the hanging-wall regional limit is shifted laterally above a deep fault segment with constant dip, then a constant area is added to or subtracted from the area of each pregrowth horizon. This does not affect the slope of the area–depth graph, and hence the estimated displacement remains constant. The constant change in fold area will change the zero-area intercept depth such that the estimated fault depth correctly matches the fault depth at the new hanging-wall regional limit.

However, if the hanging-wall regional limit is shifted between parts of a structure controlled by fault segments with differing dips, say from a flat to a ramp, then the change in the measured horizon area will not be constant for each horizon included in the linear regression. As long as the change in underlying fault dip is reflected by corresponding changes in horizon throw and shear is negligible at each hanging-wall regional limit location, the different computed fault trajectories will effectively constrain the ramp-to-flat transition. Figure 4 shows a series of fault-trajectory ADS analyses performed on an area-balanced forward model with multiple changes in the underlying fault dip. By progressively shifting the hanging-wall regional limit across the folded horizons, the estimated fault trajectory changes as a function of the fold area and horizon depths within each new set of analysis limits (Figure 4). At each hanging-wall regional limit, the method accurately estimates the fault depth, dip, and displacement for the extent of the model within the analysis limits (bold horizons, Figure 4). Collectively, each fault-trajectory estimate maps out the expected variation in fault shape. Because fault dip is calculated using the horizontal area–depth displacement (equation 3), the fault-trajectory method integrates changes in fault shape over a distance equal to the horizontal fault displacement. Incremental changes in fault dip that occur over distances less than the best-fit displacement will be aggregated into a single best-fit dip estimate. As a result, a series of best-fit fault planes determined by the fault-trajectory method will represent an average fault trajectory. This composite fault-trajectory estimate can be tested by forward modeling the structure based on the composite fault path and comparing the modeled and observed fold geometry. This process is explored in detail using a natural example in the Defining Fault Curvature: Inner Moray Firth, North Sea section.

## VALIDATION USING NUMERICAL FORWARD MODELS

We have extensively tested the fault-trajectory method using a wide variety of area-balanced numerical forward models, and in all cases the method accurately estimates fault depth, dip, displacement, and aggregate LPSs. Figures 3 and 4 are contractional forward models created using a combined trishear and inclined-shear algorithm after Xiao and Suppe (1992) and Zehnder and Allmendinger (2000). In both figures, the computed fault trajectories exactly match the models. In Figure 3, the model boundary-displacement vector for the pregrowth was set at 1095 m (3593 ft), and ADS analysis estimated 1090 m (3576 ft). For the bottom four pregrowth horizons, surface-parallel strains (comparable with LPS) in the forward model range from −0.6% to −1.5% for the bottom four pregrowth horizons, matching ADS-estimated LPS values exactly (dark-gray horizons, Figure 3). The model surface-parallel strain for the top pregrowth horizon (Figure 3) was slightly higher than LPS estimated from ADS analysis (8% compared with 5.3%). In Figure 4, the ADS-estimated displacement for each segment of the total model matches the model boundary displacement within 5 m (16 ft).

## VALIDATION USING ANALOG MODELS

Contractional (Figure 5A) and extensional (Figure 5B) analog models are used to demonstrate the effectiveness of the fault-trajectory method. The models consist of mechanically homogenous clay on top of a plywood base containing a built-in fault plane. The fault dip is constant, continuing to the base of the model. Prior to deformation, initially horizontal marker horizons and strain circles were inscribed on the cross-sectional free surface of the clay. Contraction and extension were simulated by moving the hanging-wall part of the plywood base. Both models were designed to be stylistically similar to basement-involved forced folds. The models were chosen because they demonstrate the applicability of the fault-trajectory method to structures where there is no direct constraint on the deep fault location or shape. Digital photos of the models were calibrated based on scale bars included in the photos. To demonstrate the predictive power of the fault-trajectory method, ADS analysis was limited to the horizons that were folded but not directly offset by the underlying fault. The folded marker horizons were manually digitized, and the resulting area–depth plot and the computed fault trajectory were overlain on the image of the model (Figure 5).

The structural relief areas for the horizons in both models decrease linearly with depth. For the footwall and hanging-wall regional limits used in each model, the intercept of the linear fit with the depth axis gives a fault depth below the imaged region (right-side insets in Figure 5). The computed fault is projected upward from the best-fit depth intercept and the dip determined from the best-fit horizontal displacement and the average horizon throws. The coefficient of variation between the horizon throws is less than 5% in both models, indicating that the horizons record a consistent amount of throw and that the computed fault dip is reliable. The fault trajectories estimated based on the best-fit depth intercepts and computed fault dips match the exact location of the plywood faults in the models (Figure 5). Although the computed fault depth does not fall within the model, it represents the projected depth the model fault would extend to if the model continued deeper (right-side insets in Figure 5). The computed fault displacement (white vectors parallel to the computed faults in Figure 5) also matches the offset of the plywood–clay interface (red horizon in Figure 5). The area–depth point for the plywood interface was not included in the ADS analysis. Therefore, the slip recorded at the plywood–clay interface independently verifies the ADS displacement estimate.

In both the contractional and extensional models, the nominal vector displacements significantly underestimate the computed vector displacement from area–depth analysis. In the contractional model, the horizons have nominal vector displacements ranging from −0.6 to −1.4 cm (−0.24 to −0.55 in.), at least 40% less than the −2.5 cm (−0.98 in.) computed vector displacement (Figure 5A). The nominal vector displacements in the extensional model are more consistent (0.2–0.3 cm [0.08–0.12 in.]) but underestimate the computed vector displacement by 75% (Figure 5B). Given that the fold geometry is known in both cases, there are no alternative interpretations that would produce line-length balanced sections with displacements that match the ADS analyses. Ruling out interpretation error, the LPS distribution indicates that bed lengths must have changed during deformation. The negative LPS values for the contractional model horizons indicate that horizon lengths decreased to varying degrees (Figure 5A). In the predeformation model state, the beds were 30.5 cm (12 in.) long between the regional limits. In the deformed state (Figure 5A), the bed lengths decreased from 30.5 to 30.0 cm (12 to 11.8 in.) at the orange horizon, 29.7 cm (11.7 in.) at the blue horizon, and 29.5 cm (11.6 in.) at the green horizon. The observed bed-length changes of −1.6%, −2.6%, and −3.2% are consistent with the LPS magnitudes from ADS analysis (Figure 5A). In fact, the general increase in LPS magnitude from −1.8% to −3.2% away from the final fault tip (Figure 5A) is consistent with the expected LPS distribution in trishear fault-propagation folds (Eichelberger et al., 2015). The horizons in the extensional model were 29.7 cm (11.7 in.) long prior to extension and extended to 30.7 cm (12.1 in.) during deformation. The observed 3.3% increase in bed lengths is consistent with the extensional LPS values from the ADS analysis (3.3%–3.5%, Figure 5B). The horizon lengths increased enough that the nominal vector displacements (0.3 cm [0.12 in.]) were only a small fraction of the true vector displacement (1.9 cm [0.75 in.]). So, although one could easily create a line-length balanced cross section of the structure in Figure 5B, the estimated extension would significantly underestimate the magnitude of displacement necessary to explain the observed folding.

## APPLICATION TO NATURAL EXAMPLES

To show how the fault-trajectory method is applied to typical oil-field data, we selected fault-related folds where the shallow structural geometry is constrained by seismic and well data but the geometry and location of the controlling fault at depth are unknown. In each case, the deformed horizons feature differing structural depths on either side of the controlling fault (Figures 6–8), so the flat-detachment method of ADS analysis is not applicable. The fact that structural relief (unrelated to regional dip) is maintained across the imaged extent of these structures suggests that the underlying fault continues to dip rather than merging into a flat detachment. The ADS analyses were performed directly on the interpretations, providing rapid assessments of structural balance in each interpretation as well as instantaneous metrics on fault location, displacement, strain, and deformation history.

### Locating Deep Basement Faults: Bermejo Basin, Argentina

The first example is located in the Bermejo Basin adjacent to the Sierras Pampeanas (Figure 6). Here, ADS analysis is used to independently asses the structural consistency between the interpreted fold geometry and the reflections in basement that may image the fault trajectory at depth. The structure is a contractional fault-related fold that involves Paleozoic and Cenozoic sediments overlying crystalline basement comprised of the Precambrian metamorphic and Cenozoic plutonic rocks. Similar structures in the Sierras Pampeanas have been characterized as basement-involved fault-propagation folds (Zapata and Allmendinger, 1996). The interpretation shown here was originally published in Hughes and Shaw (2014). That study showed that the displacement variation along the fault length is consistent with a breakthrough fault-propagation fold (Hughes and Shaw, 2014). The seismic image clearly resolves horizons within the sedimentary section, but the resolution degrades with depth, presumably within the basement rocks (Figure 6A). Below the well-imaged sedimentary section, there is a dipping event that has been interpreted as the controlling fault plane (Shaw et al., 2005; Hughes and Shaw, 2014; Figure 6A). The ADS analysis of the structural interpretation gives a computed fault trajectory that is consistent with the depth and orientation of the dipping seismic reflection in the basement. The ADS analysis of this structure is only possible with the fault-trajectory method because of the sustained regional uplift of the hanging wall (Figure 6A). The imaged horizons show a consistent 4°–5° tilt in the distal footwall and hanging wall, and this was removed prior to ADS analysis by rotating the image and the interpretation 4° (Figure 6B).

Area–depth analysis of the interpretation gives a best-fit fault trajectory that matches the imaged fault-plane depth within 100 m (328 ft) and the dip within 2° (Figure 6B). The calculated fault trajectory at depth is determined using only the best-imaged part of the structure where interpretation uncertainties are the lowest. Even if the deep fault was not directly imaged, the fault-trajectory method indicates that fault dip should decrease by approximately 16° into the deep basement section to explain the interpreted fold geometry. A fault-bend–fold model of the backlimb syncline in the hanging wall (Shaw et al., 2005) gives similar results, estimating a 15° change in the underlying fault based on the interlimb angle.

The area–depth regional limits were chosen to include as much of the structure as possible within the analysis aperture but exclude any folding unrelated to the controlling fault. The hanging-wall regional limit was set at the distal end of the structure, whereas the footwall regional limit was set inward toward the fault. The location of the footwall regional limit was chosen to exclude moderate folding observed in the lower footwall horizons that is unassociated with the structure of interest (mainly in horizons A and B; Figure 6B).

The LPSs estimated for the interpreted section of the Bermejo fault-propagation fold are all negative, reflecting layer-parallel shortening within the structure (Figure 6B). This means that bed lengths generally decreased during deformation. In detail, the magnitude of contraction increases stratigraphically upward from horizon A (0%) to G (−7%) then remains between −7% and −8% for horizons H and I (Figure 6B). The distribution of LPS within a fault-propagation fold has been shown to be a function of fold kinematics (Eichelberger et al., 2015). This pattern is similar to the expected strain distributions for fixed-axis or breakthrough trishear fault-propagation folds (Eichelberger et al., 2015). Rather than assuming a particular kinematic style a priori, the results of area–depth analysis can be used to independently assess the structural style, providing hypotheses that can be independently tested using other models. In general, ADS analysis shows that the interpreted fold geometry is consistent with the orientation of the imaged fault at depth.

### Determining Fault Trajectory in Underconstrained Settings: San Joaquin Basin, California

In complexly faulted petroleum basins, well and seismic data often constrain only part of the structural system, resulting in an “empty space problem” where the geology is unconstrained. The ADS analysis by the fault-trajectory method makes it possible to use the best-constrained part of an interpretation to extrapolate structural geometry at depth. In this example from the White Wolf Fault at the southern edge of the San Joaquin basin (Figure 7A), well tops, dip meter data, geophysical logs, and seismic data constrain the upper part of the upturned foothill structure and the deep footwall down to the Miocene Monterey Formation (Figure 7B). The well bores locate the shallow extent of the White Wolf and Wheeler Ridge Faults, but the deeper structural relationships are unknown. The ADS analysis using the fault-trajectory method on the Monterey, Etchegoin, and San Joaquin Formation tops (as interpreted by Gordon and Gerke, 2009) gives a computed fault trajectory that favors the White Wolf Fault as the basal fault for this part of the upturn fold (Figure 7B). Depending on the regional limits used, the computed fault dip ranges from 50° to 60°. This dip range is consistent with the focal mechanism for the magnitude 7.3 1952 Kern County main shock on the White Wolf Fault, which had a rupture plane dipping at 65° (Figure 7A; Dreger and Savage, 1999). Time-dependent moment tensor inversion revealed two subevents that indicate that the rupture initiated at a depth of 20 km (66,000 ft; first subevent) and propagated upward to 5 km (16,000 ft; Junkyoung, 1989). The second subevent at 5 km is shown in Figure 7.

In addition, the area–depth graph accurately identifies the Etchegoin and San Joaquin Formations as growth horizons, deposited while the White Wolf Fault was active. Regional geology suggests that the White Wolf Fault initiated in the late Miocene, during deposition of the Reef Ridge Shale Member of the upper Monterey Formation (Reef Ridge is not indicated in Figure 7B but is located just above the top Monterey horizon, as drawn in Figure 7B; Graham and Williams, 1985). The area–depth graph shows that the digitized horizons within the Monterey Formation have a strong linear trend (square of the correlation coefficient for the best-fit line [*R*^{2}] = 0.989) with fold areas that increase stratigraphically upward. In contrast, the Pliocene Etchegoin and San Joaquin Formations have fold areas that decrease stratigraphically upward, suggesting that they are well within the growth section (Figure 7B).

### Defining Fault Curvature: Inner Moray Firth, North Sea

To demonstrate the ability of the fault-trajectory method to map out curved fault planes, we apply it to a basin-bounding normal fault from the Inner Moray Firth rift basin in the North Sea (Figure 8A). The data used here are available on the Virtual Seismic Atlas along with multiple structural interpretations by professional geologists that vary in terms of how the fault framework is defined (Butler, 2008). Although the shallow structural framework is fairly well imaged, all of the interpretations are limited by a lack of data constraining the major basin-bounding normal fault (Figure 8A). The upper segment of the normal fault is well resolved, but the deeper extent is not included in the data. The objective here is to estimate the most likely trajectory for the basin-bounding normal fault using area–depth analysis of the pregrowth interval defined in the existing interpretations of the structure. The pregrowth interval is well imaged in the hanging wall of the fault, showing progressive folding and variable structural relief. We can apply the fault-trajectory method to the folded pregrowth interval to estimate the likely trajectory of the normal fault at depth. The ADS analysis aperture was limited to the large normal fault by placing the footwall regional limit immediately adjacent to the shallow extent of the fault (Figure 8B). This was done to exclude the structural influence of shallow faults. Consequently, the estimated displacements and fault trajectory only pertain to the main basin-bounding normal fault rather than the entire extensional system.

Using ADS analysis, individual fault segments were computed by the fault-trajectory method for a series of hanging-wall regional limits (see Figure 4), with each analysis point spaced at horizontal intervals based on the computed vector displacement (gray circles with white centers, Figure 8). Horizontal displacements ranged from 2400 m (7900 ft) in the far field (segment 1; Figure 8B) to 2000 m (6600 ft) (segment 5; Figure 8B), where the imaged fault steepens near the base of the data. The fault trajectory was estimated every approximately 1200 m (3900 ft) for the far-field segments and approximately 1000 m (3300 ft) for the near-field segments. Collectively, the computed fault segments outline a listric-style fault trajectory (Figure 8B) that is conceptually consistent with rollover-style growth faulting (Xiao and Suppe, 1992). The computed fault segments show dips that decrease from 25°–30° near the imaged fault (segments 4 and 5, Figure 8B) to 10°–20° at the far edge of the image (segments 1 and 2, Figure 8B). The change in estimated fault dip manifests as an approximately 50% decrease in regional throw from 1200 m (3900 ft) at segment 5 to 600 m (2000 ft) at segment 1. Horizontal displacement between the two areas differs by only approximately 15%.

To independently test the fault trajectory estimated from ADS analysis, we created an inclined-shear forward model (Xiao and Suppe, 1992) of the folded pregrowth horizons using the ADS-computed fault segments as a guide (Figure 8C). Given the stylistic similarity between the interpreted structure and a rollover growth fold, the kink axes that define folding in inclined-shear models were oriented at 67° to approximate the Coulomb failure angle (Xiao and Suppe, 1992; dashed lines, Figure 8C). The modeled pregrowth surfaces were fixed at the interpreted footwall locations. Hanging-wall horizons were fixed at a depth where the hanging-wall cutoff locations matched the interpretation. The fault model in the shallow, well-imaged region follows the interpreted fault trajectory from Figure 8A. Below the lower limit of the data, the structure was initially modeled by setting the deep fault at the midline of the ADS-estimated fault-trajectory envelope (Figure 8B, C). The model fault geometry was then interactively adjusted to achieve the best visual fit between the model fold surfaces and observed hanging-wall fold shape (Figure 8C). Note that the interpreted pregrowth horizons (cyan, blue, and purple; Figure 8A) are still shown in Figure 8C, but the kinematically modeled fold surfaces (dark purple) reproduce the observed geometry so closely that the underlying interpretation is nearly covered.

The results from forward modeling and ADS analysis using the fault-trajectory method are remarkably consistent (Figure 8C). Following the fault-trajectory envelope, the final model reproduces the interpreted fold geometry within 100 m (328 ft). Both the model and ADS analysis report total horizontal displacements of approximately 2400 m (7900 ft) at the far-field limit of the data. The interpreted pregrowth lengths shown in Figure 8 differ by no more than 180 m (590 ft), suggesting that the interpretation is nearly balanced with respect to line length. However, both the nominal horizontal and vector displacements are 40%–45% less than the horizontal and vector displacements reported by ADS analysis and modeling (Figure 8B). Although small adjustments could be made to the interpretation in the vicinity of the minor antithetic fault, unwarranted revisions would be required to account for the increase in bed length implied by the 14%–17% extensional LPSs computed by ADS analysis (Figure 8B). Furthermore, the LPSs from ADS agree well with the average surface-parallel strains from the hanging wall of the forward model (12%–14%). The small difference in strain is likely resulting from the four minor synthetic faults included in the interpretation and ADS analysis that are omitted by the forward model. In any case, these strains imply that bed lengths increased by at least 10% in the hanging wall of the normal fault during deformation.

The ADS analysis of the normal fault also gives insights into variations in fault displacement with time (Figure 8B). The growth section in the interpretation suggests that the main normal fault has been continuously active, whereas the shallow faults were only active during the onset of extension. The interpreted growth horizon at the top of the section in Figure 8A is offset by the main normal fault but not the shallow faults. All of the interpreted faults feature growth sequences below this growth horizon, but only the main normal fault offsets that horizon (Figure 8A). This suggests that extension was initially accommodated by both the shallow faults and the major normal fault, but subsequently only the major normal fault was active. The computed displacement vector for the pregrowth interval on the main normal fault (2431 m [8000 ft]) integrates both these events (Figure 8B). Assuming the interpreted growth horizon does indeed represent the stratigraphic boundary between the two extensional episodes, the displacement vector for the growth horizon records the second phase of extension that was accommodated on the normal fault (749 m [2500 ft], Figure 8B). If true, then displacement accommodated during the first phase of extension is the difference between the total extension and the second phase of extension, or 1682 m (5500 ft).

## DISCUSSION

The flexibility of the fault-trajectory method allows interpreters to apply ADS analysis to more complex structural settings, but care must be taken to assure reliable results. For an individual structure, the accuracy of the fault-trajectory method is dependent on meeting the kinematic assumptions implicit in estimating underlying fault geometry from observed structural geometry. In this section, we highlight specific structural circumstances in which these assumptions are not strictly valid and result in decreased fault-trajectory estimate accuracy.

A fundamental assumption of the fault-trajectory method is that the measured variation in structural relief with depth is related to slip on a single, controlling fault underlying the analysis region. The fault-trajectory method can allow for structures with multiple subsidiary faults that contribute to the total structural relief as long as they fall completely within the analysis limits and share the same basal detachment (e.g., Figures 5B, 8). However, in structurally complex settings where faults deform areas that were previously or contemporaneously folded because of slip on other structures, the estimated fault trajectory may not directly correspond to a single fault.

A subtler assumption made by ADS-analysis methods is that the horizons used in the linear regression record a constant horizontal boundary displacement (heave, Figure 2) with depth. To calculate fault dip (equation 6), the fault-trajectory method makes two additional assumptions. First, the horizon throws at the hanging-wall regional limit (Figure 2B) are assumed to be constant with depth. Second, horizon throw at the hanging-wall regional limit is assumed to closely approximate throw on the underlying master fault at the point where the hanging-wall regional limit and fault intersect (Figure 2B). Master fault throw is calculated as the mean difference in horizon depths at the hanging-wall and footwall regional limits (*z*_{f} − *z*_{h} in equation 6; Figure 2B). Because the regional limits are vertical lines, placing the hanging-wall regional limit at locations where horizon displacement varies along a vertical profile will result in less accurate estimates of fault dip, depth, and displacement (Figure 9).

A common structural setting in which heave and throw vary with depth is within the limbs of fault-bend folds (Figure 9). For a vertical regional limit located within a fold limb, the measured heave and throw at each dipping horizon will vary as a function of height because of the angular discordance between an orientation of constant displacement and the vertical limit line. Figure 9 illustrates the difference in horizon throws along a hanging-wall regional limit compared with a true constant displacement profile. In fault-bend folds, the orientation of the active axial surface (heavy, dashed gray line, Figure 9A) defines the orientation of lines of constant displacement. Any line within the backlimb that is parallel to the axial surface will record constant horizon displacements where it intersects the folded horizons (circles in Figure 9B). Both the hanging-wall regional limit and the constant displacement profile line start at the same point on the fault (Figure 9A), so heave and throw are the same for both lines at that point (Figure 9B). The fault-trajectory method significantly underestimates fault dip at the hanging-wall regional limit location in Figure 9A. This is because throw and heave vary with respect to a depth along the vertical profile at the hanging-wall regional limit (Figure 9B). The variation in horizon throw is reflected by a high coefficient of variation (20%), and the average horizon throw (362 m [1200 ft]) overestimates the actual fault throw at the hanging-wall regional limit (230 m [755 ft]). The variation in heave along the hanging-wall regional limit results in slight underestimation of the fault depth, although the effect is minor relative to the dip-estimate error. The magnitude of the fault-trajectory estimate error is primarily related to the angle between the vertical regional limit and the axial surface. The larger the angle, the larger the difference between the horizon displacements measured at the hanging-wall regional limit compared with the underlying fault.

Ultimately, the most accurate and internally consistent fault-trajectory estimates will be obtained if the hanging-wall regional limit is positioned where horizons are nearly flat (Figure 4) or have relatively low dips (Figure 8). In these areas, variations in displacement with depth at the hanging-wall regional limit will be relatively minor. In practice, both the *R*^{2} value and the coefficient of variation for the average horizon throws can be used to generally assess how reliable any given fault-trajectory analysis point may be (Figure 9A).

Structures that detach on basal shear zones, such as shear fault-bend folds (Suppe et al., 2004), represent another situation in which horizon displacements vary with depth. Layer-parallel shear confined within a fixed stratigraphic interval above a basal detachment is often invoked to explain contractional structures characterized by long backlimbs and hanging-wall dips lower than the underlying fault ramp (Suppe et al., 2004; Shaw et al., 2005). Epard and Groshong (1993) demonstrated that shear intervals can be recognized as nonlinear trends on area–depth plots but that if shear is small relative to the total boundary displacement, the error in estimated fault depth and displacement will also be minor. With regard to the fault-trajectory method, the observation that horizon shape and fault geometry are decoupled above shear zones indicates that horizon throw may not accurately reflect fault throw. In published examples of naturally occurring shear fault-bend folds, the basal shear intervals are often interpreted to terminate above fault ramps that follow a flat-to-ramp transition (Suppe et al., 2004; Shaw et al., 2005). The practice of positioning the hanging-wall regional limit away from fault bends as well as excluding suspected shear intervals from the ADS analysis will minimize the effects of depth-dependent displacement on the estimated fault trajectory.

If there are depth-dependent variations in horizon displacement, an interpreter can adjust the hanging-wall regional limit and observe where the coefficient of variation for horizon throws is at a minimum to determine the most robust fault-trajectory estimate. Points on the area–depth graph that depart from a linear relationship may indicate stratigraphic intervals where displacement varies with depth. Finally, the uncertainties in a fault trajectory at a given set of regional limits can be estimated using a bootstrap approach to examine the variance in fault trajectory as horizons are iteratively removed from the linear fit (Eichelberger et al., 2015).

## CONCLUSION

The fault-trajectory method expands the applicability of ADS analysis to structures controlled by dipping faults. In general, the underlying theory and graphical approach to ADS analysis using the fault-trajectory method is the same as the flat-detachment method established by Epard and Groshong (1993). The central difference is that the fault-trajectory method measures displaced fold area for a given horizon relative to the horizon’s footwall depth. By accounting for structural relief between the hanging wall and footwall of a structure, the fault-trajectory method can estimate fault dip in addition to fault depth, displacement, and LPS. For situations where the depth difference between the footwall and hanging wall at the regional limits becomes negligible, the fault-trajectory method is equivalent to the flat-detachment method. As a result, the fault-trajectory method is capable of estimating progressive fault curvature in structures overlying curved and ramp-flat master faults. The graphical approach can be applied directly to interpretations to assess internal consistency and provide independent estimates of displacement, strain, and quantitative guidance on fault depth and geometry.

## ACKNOWLEDGMENTS

Discussions with Chris Connors helped refine the nomenclature for the fault-trajectory method. Stuart Gordon and Stefano Mazzoni provided insight on regional San Joaquin basin geology. Ian McGregor provided assistance with figure drafting and proofreading. Comments from J. Steven Davis and Sandro Serra improved the manuscript during the review process. Amanda N. Hughes thanks the management of the Chevron Energy Technology Company for the opportunity to participate in this study. The kinematic forward models and area–depth strain analyses shown in the paper were created using StructureSolver 2.1.

## Footnotes

- Manuscript received April 15, 2016.
- Revised manuscript received July 25, 2016.
- Final acceptance July 7, 2016.
- Final acceptance August 23, 2016.

Nathan W. Eichelberger is a structural geologist at StructureSolver, a geological software and consulting company. He holds a Ph.D. in geosciences from Princeton University and a B.S. in geology from Bates College. Nate’s research focuses on defining the evolution of structural geologic systems at scales ranging from single faults to entire mountain belts.

Alan Nunns is the president of StructureSolver, a geological software and consulting company. Alan has a B.Sc. (Hons) degree in geology from the University of Auckland and a Ph.D. in geological sciences from the University of Durham. Alan worked for 24 years for Chevron in research, operations, and technology management. His interests include the development of interactive structural geological techniques.

Rick Groshong, professor emeritus from the University of Alabama, is an independent researcher. His 2014 paper with Dave Hale, *Conical Faults Apparent in a 3-D Seismic Image*, won the best paper award in *Interpretation*, and his 2006 book, *3-D Structural Geology*, second edition, won the 2016 Best Seminal Publication award from the Petroleum Structure and Geomechanics Division of AAPG.

Amanda N. Hughes is a research scientist in the Department of Geosciences at the University of Arizona. Hughes received her Ph.D. from Harvard University (advisor: John Shaw) and subsequently worked for Chevron’s Exploration Technology Company in Houston, Texas. Her research combines geological observations, seismic interpretation, and kinematic and mechanical modeling in understanding structural growth in the context of petroleum systems.