Problems and Solutions in Structural Geology and Tectonics

By Elsevier Science

Problems and Solutions in Structural Geology and Tectonics, Volume 5, in the series Developments in Structural Geology and Tectonics, presents students, researchers and practitioners with an all-new set of problems and solutions that structural geologists and tectonics researchers commonly face. Topics covered include ductile deformation (such as strain analyses), brittle deformation (such as rock fracturing), brittle-ductile deformation, collisional and shortening tectonics, thrust-related exercises, rift and extensional tectonics, strike slip tectonics, and cross-section balancing exercises. The book provides a how-to guide for students of structural geology and geologists working in the oil, gas and mining industries.

• Provides practical solutions to industry-related issues, such as well bore stability
• Allows for self-study and includes background information and explanation of research and industry jargon
• Includes full color diagrams to explain 3D issues

• Language: English
• Publisher: Elsevier Science
• Release date: Feb 26, 2019
• ISBN: 9780128140499

Book preview

978-3-319-43318-0.

Part I

Integrating Observation and Interpretation To Understand Tectonics, Past and Present

Chapter 1

Cross-Section Construction and Balancing: Examples From the Spanish Pyrenees

Berta Lopez-Mir    CASP, Cambridge, United Kingdom

Abstract

Cross-sections are used by geologists to represent the structure of the Earth's subsurface. To construct a cross-section, observations from the Earth's surface and other available information are projected onto an underground vertical plane to interpret how data relate to each other. The main challenge is transforming the scattered data into a reasonable geological geometry. Interpretations are not unique, so geologists need validation rules to assess their results. A cross-section is considered valid if: (1) it fits the available data and geological knowledge (admissible cross-section); (2) it is retro-deformable (restored cross-section); and (3) the volume of material between the deformed and the restored state remains constant (balanced cross-section). This chapter explains the general methodology to construct a cross-section from a geological map and the standard techniques to balance it (bed length, area, and area-excess). For each method, a practical problem is presented. Problems are real examples from the Spanish Pyrenees.

Keywords

Cross-section construction; Balanced cross-sections; Balancing methods; Line-length restoration; Area-length restoration; Area-excess restoration

1 Introduction

Cross-sections are vertical slices through the Earth used to predict the structure of the ground subsurface. Besides the scientific interest of illustrating what you would see in a journey inside the Earth, geologists need cross-sections to find, evaluate, and extract hydrocarbons, minerals, groundwater, and other resources; as well as to store industrial wastes like CO2 and radioactive material. In general, cross-sections are constructed by projecting observations from a geological map onto an underground vertical plane, to interpret how these observations relate to each other (e.g., Davis et al., 2012; Fossen, 2016). Cross-sections can also be built or complemented by the interpretation of subsurface data (e.g., seismic reflection and refraction, magnetotellurics, gravity, or other geophysical data).

Accordingly, cross-sections rely on data interpretation. Interpretations are only approximations to reality, some of which are wrong (if they are inconsistent with data or violate a geological principle) and some of which are sound, but none is right. The acquisition of new data, the learning of new conceptual models, and/or scientific progress might change our interpretations. The more data and knowledge we have, the more constrained our interpretations will be, but outputs are not unique. This is known as uncertainty and is an inherent element on the interpretation of any geological data (Bond, 2015). Understanding uncertainty is important because the success of underground extractive activities or storage relies on those interpretations, and any mistake could lead to significant economic or environmental loss.

Structural geologists need validation rules to assess their interpretations (e.g., Groshong, 2006; Rowland et al., 2007). The fundamental requirements for a cross-section to be valid are:

(1)It honors the available data and is consistent with known geological concepts (admissible cross-section).

(2)It is retro-deformable. This means it can be recomposed into a plausible predeformational geometry (restored cross-section).

(3)There is no gain or loss of material between the deformed and the restored states (balanced cross-section). This means that if we compare the deformed and the predeformational restored strata geometries, there are no large gaps or overlaps. For this assumption to be correct, cross-sections must be parallel to the tectonic transport direction.

According to this, an admissible, retro-deformable, and balanced cross-section can be considered as a possible sound interpretation. This does not mean that it is correct, but our solution is reasonable and consistent with the data we have. Correspondingly, an admissible cross-section that does not balance is not plausible, and we need to readdress our interpretations or find an explanation for loss and/or gain of material (e.g., material transport in and out of the section plane). To summarize, with the exception of additional data collection, balancing may be the most practical criteria to validate and understand the uncertainty associated with our structural interpretations (Groshong et al., 2012). Beyond cross-section validation, balancing and restoration are also used by geologists to study the kinematic evolution of geological structures and to predict the structural geometry at depth (e.g., Hossack, 1979; Boyer and Elliot, 1982; Suppe, 1983; White et al., 1986).

The main goal of this chapter is to explain the general methodology for balanced cross-section construction by hand. The first part explains how to construct a cross-section from information contained on a geological map. The second part focuses on the basic concepts of section balancing and explains three standard techniques (bed length, area, and excess-area) to produce reasonable, valid structural interpretations. The third part presents four practical problems and solutions that enable you to apply the explained cross-section construction and balancing methods.

The student will need the following materials:

–Pencil, color pencil, and eraser

–Millimeter graph paper

–Triangular rulers (or straight edge)

–Protractor

Problems can also be addressed using specific software for cross-section construction and balancing. The most common ones are Move (Midland Valley) and Lithotect (Geo-Logic Systems). To date, both Move and Lithotect provide academic licenses for free, upon a license agreement, and tutorials to describe the basic features of the programs. Alternatively, Q-GIS is a free and open-source geographic information system (GIS) application that supports viewing, editing, and analysis of geospatial data and has specific plugins for cross-section construction (e.g., Open qProf). Q-GIS does not implement specific cross-section balancing features, but it allows measuring line lengths and areas to facilitate structural restorations.

2 Cross-Section Construction

The methodology to produce a cross-section from the information of a geological map is summarized in Fig. 1. It consists of three basic processes: cross-section line selection, data projection, and interpretation of the structure.

Fig. 1 General methodology to construct a cross-section from a geological map by hand. (A) Cross-section line selection. (B) Data projection from the map onto the vertical section. (Above) Projection of the topographic contour intersections with the cross-section line. (Below) Projection of the geological intersections with the topographic profile and of the dip-data. Projection lines are indicated in a dashed gray pattern for the topography, solid blue line for the dip-data, red for the faults, pink for the fold axial trace, and the corresponding map color for the lithological contacts. Note that the apparent dips change significantly if the beds are parallel to the cross-section line (e.g., 35 degrees dip on the right-hand side of the cross-section line). (C) Interpretation of the structure. The main fault follows the fault intersections with the topography. The geometry of the strata is interpolated following the Kink Method (see Fig. 3). Colors used in the cross-section are the same as in the map legend. This example is located in the Peña Montañesa area, in the Spanish Pyrenees. Modified from Lopez-Mir, B., Muñoz, J.A., García-Senz, J., 2016. 1:25.000 geological map of the Cotiella thrust sheet, southern Pyrenees (Spain). J. Maps.

2.1 Cross-Section Line Selection

The first process to construct a cross-section is to choose the cross-section line wisely. This should run across the areas where data are more abundant, of better quality, and where the structural uncertainty is smaller (unless we want to study this uncertainty). Additionally, to make sure that the cross-section balances (there is no gain or loss of material between the deformed and the restored states), the cross-section must be parallel to the tectonic transport direction. For this reason, cross-sections are usually oriented perpendicular to the major structural trends (e.g., fold axial traces, major faults, or bedding in unfolded strata). Choosing an appropriate orientation for the cross-section line is also important to give the most representative view of the geometry (Groshong, 2006).

The dips and thicknesses observed in a cross-section perpendicular to the strike of the beds (alpha and Tt in Fig. 2A) represent true dips and thicknesses (i.e., Rowland et al., 2007). The true dip of a structure is defined as the amount of maximum inclination of the bedding plane with respect to the horizontal, measured perpendicular to the strike of the strata. The true thickness is defined as the distance between two beds, also measured perpendicular to the bedding. Contrastingly, a cross-section oblique to bedding (Fig. 2B) represents an apparent view of the structures, and both the resulting dips and thicknesses are apparent (they are not displayed perpendicular to the layers, as is the case for true measurements). A special type of apparent thickness, sometimes used to describe geological observations, is called the vertical thickness and represents the distance between two beds measured along a vertical line.

Fig. 2 Block diagrams showing different views of a cross-section. (A) True dips and thicknesses in a cross-section perpendicular to the strike of the beds. (B) Apparent dips and thicknesses in a cross-section oblique to bedding. The section parallel to the strike of the beds, (A), shows the end-member example where the apparent dips are horizontal. (C) Angular relationships between the true dip and the apparent dip. (D) Projected triangles to calculate apparent dips. (E) Projected dips using two different vectors: one following the strike of the beds (green) and the other one following the dip direction (blue) . The sketch to the right shows the variation between the original and the projected height of the blue dip . Note that the height of the dip projected using a vector parallel to bedding (in green ) remains the same as the original one. Capital letters are used for line segments, lower case for points, and Greek letters for angles. Dashed line patterns are used to indicate lines in the hidden planes of the block diagrams.

Apparent dips and thicknesses can be easily calculated using trigonometric algebra, if necessary (Fig. 2C and D). When choosing the cross-section line, it is important to understand the concepts of true and apparent views to avoid interpreting unrealistic geometries.

2.2 Data Projection

The main objective of this process is to transfer the data from the geological map into the cross-section (the plane extending vertically from the cross-section line drawn on the map). In particular, the following features from the map are projected onto the vertical section: topographic heights, geological contacts, and dip-data. For this purpose, the vertical section is placed parallel to the cross-section line on the map so that it is possible to construct projection lines perpendicular to both the cross-section line on the map and the vertical section (Fig. 1B). The vertical section should also be at the same scale as the map, otherwise both the dips and the thicknesses will be exaggerated (e.g., Groshong, 2006; Rowland et al., 2007).

The general methodology consists of the following operations (Fig. 1B):

(1)Transfer the topographic data (heights of the contour line intersections with the cross-section line, as well as the hilltops or valley bottoms) onto the vertical section and join the projected points to define the topographic profile (dashed gray lines in Fig. 1B).

(2)Transfer the intersections between stratigraphic contacts, fold axial traces, and faults, with the cross-section line, from the map to the topographic profile (colored lines, except blue ones in Fig. 1B).

(3)Transfer the dip-data from the map onto the vertical section;

a.First, transfer dip-data from different areas on the map onto the cross-section line. This is done using projection vectors parallel to fold axes or to the strike of the layers of unfolded strata. The definition of fold axes requires a detailed structural analysis beyond the scope of this chapter (Groshong, 2006). For the examples and problems provided, the used projection vectors will follow the strike of the layers or will be considered perpendicular to the section (for layers subparallel to the section). This simplification is acceptable because the data are located close to the cross-sections and the structures are not deformed intensely (blue lines in Fig. 1B). Otherwise, the structural analysis previously mentioned would be necessary. In any case, avoid projecting dips across faults or fold axial surfaces because geological structures usually terminate or change their dip across these elements.

b.Next, transfer the projected dips on the cross-section line from the map onto the vertical section. Remember that if the cross-section is not perfectly perpendicular to the strike of the beds, the dips in the cross-section will be apparent instead of true (Fig. 2A–D). Similarly, the height of the projected dip will only be the same as the original height if the projection vector is horizontal (e.g., when it follows the strike of the beds). Otherwise, it can be recalculated using trigonometric algebra (Fig. 2E).

(4)Plot any other constraints (e.g., seismic data, well data, or stratigraphic columns), if available.

2.3 Interpretation of the Structure

The last process on cross-section construction is to interpret how the data relate to each other. The best approach is to start with the youngest elements such as faults and unconformities against which the rest of the structures terminate (e.g., red fault in Fig. 1). The geometry of strata on either side of faults or angular unconformities should be interpreted separately.

The geometry of the geological structures can be determined from the intersections between geological contacts and the topography, and from the dip-data. The main challenge is transforming these scattered data into a reasonable geological geometry. Except in simple cases such as the red fault in Fig. 1C (the geometry of which is interpreted from joining the two intersections of the fault with the topography, as projected from the geological map), the use of geometric models is necessary. These models will guide the interpolation between data and the extrapolation beyond the data to simplify the structure into a simple geometry. Geometric models will additionally facilitate the construction of balanced cross-sections and, more importantly, will reduce the degree of uncertainty.

There are two popular geometric models used to simplify data for the construction of balanced cross-sections (i.e., Groshong, 2006; Rowland et al., 2007). The Busk Method (Busk, 1929) assumes that folds are parallel and concentric, that bed segments are portions of circular arcs, and that the arcs are tangent at their ends. The Kink Method (Faill, 1973) assumes that folded geologic surfaces can be simplified into planar dip-domains (portions of the structure where the dip of the strata can be considered the same and simplified into a planar surface), which change their dip across axial surfaces, unconformities, or faults. This method assumes that the thickness of beds remains constant across axial surfaces and that the axial surfaces bisect the angle between the planar dip-domains. If the cross-section is not perpendicular to the strike of the beds, thicknesses will be apparent, axial surfaces will not be bisectors, and therefore the Kink Method is not valid. This is another reason why the cross-section line should be as perpendicular as possible to the structural trends.

The Busk Method is more appropriate to reconstruct concentric folds. The Kink Method is more appropriate in general, because any curve can be subdivided into straight segments. Structures reconstructed using the Kink Method are more manageable for constructing balanced cross-sections, and this model is therefore the most commonly used by geologists. This chapter will only address cross-sections constructed using the Kink Method (Fig. 3).

Fig. 3 Standard methodology to construct the geometry of folded strata using the Kink Method. (A) Planar dip-domains (blue circles) are areas where bedding presents the same dip. Depending on the structure and the amount of data, a tolerance between 5–10 degrees can usually be accepted. Dip-data at either side of unconformities and faults constitute distinct dip-domains. Roman numbers are used to differentiate each dip-domain. Dip-domain III is an interpretation to provide a more realistic view of the fold geometry. (B) Bisector lines between adjacent dip-domains are calculated using a drawing compass. First, draw bedding guidelines that follow the orientation of the dip-data in each dip-domain. Next, place the needle of the compass where two bedding guidelines meet. Draw a tick mark at the other end of the compass across each bedding guideline; place the sharp end on one of the two ticks and draw a semicircle. Move the sharp end to the other point and draw another semicircle. Where the two circles intersect is the angle bisector. The bisector line is the line that joins the intersection between the bedding guidelines and the angle bisector. Note that where two bisectors intersect, a dip-domain disappears and a new bisector is formed. (C) Dip-domain boundaries consist of faults, unconformities, and bisector lines across which beds change their dip. The bisector line will be located in the midpoint between dip-data, unless we have other constraints such as a fold axial trace (pink cross) . (D) The geological interpretation follows the geological contact intersections with the topography and the dip-domains defined in the previous operations. Use triangular rulers to depict beds parallel to the bedding guidelines, which intersect the topographic profile in the geological contact intersections. Sometimes you will need to move the bisector lines up and down to make sure that the resulting geometry honors the geological contacts and the known stratigraphic thicknesses. The result is achieved by trial and error.

The last process on cross-section construction is to color and/or pattern each stratigraphic level (Fig. 1C). Do not forget to make a legend with a color, pattern, and symbol explanation, and to indicate the orientation and scale of the cross-section, as well as its location on the map.

3 Cross-Section Balancing

The fundamental assumption of balancing is that there is no gain or loss of material during deformation. This means that deformation simply redistributes rock volumes in space, and volume does not change during deformation (e.g., by metamorphism, compaction, or pressure solution). In a two-dimensional cross-section profile, balancing assumes that deformation occurs by plain strain: there is no movement of material in or out the section plane. However, in some cases, surface processes may also affect the reconstructions by erosion or resedimentation of material. This should be implemented or at least taken into account in structural restorations.

The standard approach for balancing a cross-section is to restore the deformation (return the current structure to a plausible predeformational geometry) and check that the amount of rock remains constant between the predeformational and restored states. The technique used for restoring depends fundamentally on the way deformation has occurred and evolved. In general, deformation occurs following four kinematic models or a combination between them (i.e., Fossen, 2016). Rigid-body displacement involves translation and/or rotation of an object. Bed lengths, areas, and shape remain constant (Fig. 4A). Flexural slip involves deformation on closely spaced, parallel planes. Bed length and thickness remain constant in a direction perpendicular to the slip planes (Fig. 4B). Pure shear involves deformation along a vertical axis. Area is maintained constant, but bed length and thickness change (Fig. 4C). Simple shear involves slip parallel to the boundary of the shear zone. Areas and thickness remain constant parallel to the shear direction (Fig. 4). Considering this kinematic typology, balancing techniques are based on the assumption that either the bed length and/or the bed-areas remain constant through time.

Fig. 4 Kinematic models to describe deformation. (A) Rigid-body translation or rotation, (B) flexural slip, (C) pure shear, and (D) simple shear. Arrow depicts the main deformation direction. Pure shear kinematics are described in detail by Mukherjee (this volume), and simple shear in Mukherjee (2012). Modified from Fossen, H., 2016. Structural Geology, second ed. Cambridge University Press. 480 pp.

3.1 Bed-Length Balancing

Bed-length balancing assumes that the length, thickness, and angles between beds remain constant between the deformed and the restored states. Bed-length restorations were developed and have been largely used in fold and thrust belts (e.g., Dahlstrom, 1969; Hossack, 1979; Elliott, 1983). However, the technique is adequate to restore any type of structure where the length, thickness, and angles between beds remain constant through time (e.g., parallel folds).

Restorations by bed-length balancing are based on the initial and final positions of reference lines that ideally will restore to a simple geometry, usually a rectangle. Therefore restored cross-sections are bounded by four main lines (i.e., Groshong, 2006; Rowland et al., 2007):

–The uppermost boundary is the reference horizon (green lines in Fig. 5). This line will be retro-deformed to the shape it had before deformation and will define the positions of the underlying horizons. The original position of this horizon, including both its shape and elevation, is known as the datum or regional (McClay, 1992). The datum is usually a horizontal line in the restored cross-section (dashed gray lines in Fig. 5).

Fig. 5 Line-length balancing techniques for parallel folds. (A) Fold affecting parallel strata, (B) folded unconformity, and (C) folded parallel strata affected by two faults. The bed length, bed thickness, and angular relationships between beds remain constant between the deformed and the restored states. Red lines, with filled and unfilled heads, represent pin and loose lines, respectively. Gray dashed lines are used for datums and bisector lines. Thick black lines represent faults. The reference horizon is represented in green and bedding with plain black lines . Lf , length of the deformed section; Lo , length of the undeformed section. Roman numbers indicate different steps of restoration. Modified from Groshong, R.H., 2006. 3-D Structural Geology A Practical Guide to Quantitative Surface and Subsurface Map Interpretation, second ed. Springer-Verlag, Berlin, Heidelberg/New York, 400 pp. and Rowland, S.M., Duebendorfer, E.M., Schiefelbein, I.M., 2007. Structural Analysis and Synthesis: A Laboratory Course in Structural Geology, third ed. Blackwell publishing, Oxford, 300 pp.

–The pin line is a line that does not change between the deformed and the restored state (red vertical lines with a filled head in Fig. 5). It should fulfill the following conditions: be located in the less deformed (or preferably undeformed) portion of the cross-section, cross the most complete stratigraphic section, and be perpendicular to bedding and not cross a fault.

–The loose line is a straight line on the deformed-state section that may assume any configuration on the restored section state (red vertical lines with an unfilled head in Fig. 5). If in the restored state, the loose line is straight and parallel to the pin line; this is the best indicator of a correct restoration.

–The base of the restored section is usually either a stratigraphic marker or detachment horizon.

The general methodology to obtain the restored geometry consists of the following operations (Fig. 5A):

(1)Measure the bed length, thickness, and angular relationships between beds in the deformed section (Fig. 5A-I). Bed lengths are measured from the pin line to the loose line. Use a curvimeter to measure the length of the lines in the deformed state. Alternatively, use the edge of a sheet of paper to mark the length of each dip-domain, one after the other (possible dip-domains segments are indicated in the reference horizon in Fig. 5A). Note that the bisector lines between dip-domains are perpendicular to bedding and can therefore be used as thickness control points in different portions of the cross-section (dash-point gray lines in Fig. 5A).

(2)Unfold the reference horizon to the datum (dashed gray line in Fig. 5A) by drawing a straight line (assuming a straight datum). This line must have the same length as the reference horizon in the undeformed section (Fig. 5A-II).

(3)Determine the positions of the underlying horizons. Start transferring the bed thicknesses to the pin line, as this should not change during deformation. Next, transfer the bed lengths of each horizon. Angular relationships should also be transferred, if appropriate (alpha in Fig. 5B).

(4)Join the end of the restored stratigraphic horizons to define the restored geometry of the loose line (Fig. 5). In a valid restoration, the loose line should remain straight and parallel to the pin line. Otherwise, our interpretation needs to be re-evaluated (e.g., consider a different style of folding).

Assuming a horizontal datum and parallel beds, both the reference horizon and the underlying horizons result in straight horizontal lines with the same thickness and length as the measured lines in the deformed state (Fig. 5A). Contrastingly, if beds are not parallel (e.g., because of an angular unconformity), only the uppermost layers become horizontal (Fig. 5B).

The methodology to restore structures involving multiple faults is similar (Fig. 5C):

(1)Measure bed lengths of all the horizons from the pin line to the first fault interruption, and unfold them following the methodology previously described (Fig. 5C-I).

(2)Join the end of the restored stratigraphic horizons to define the restored geometry of the following fault (Fig. 5C-II).

(3)Transfer bed lengths on the following fault hanging wall (Fig. 5C-III). The geometry of the restored faults should be realistic and the loose line remains straight and parallel to the pin line in the restored cross-section. Otherwise, the present day cross-section needs to be re-evaluated (e.g., change the angles of the fault).

The difference in length between the deformed (Ld) and restored (Lu) cross-sections represents the amount of shortening or extension (S; Fig. 5C).

3.2 Area Balancing

Area balancing assumes that areas remain the same in the deformed and restored states. Equal-area techniques are subdivided into two methods: (1) equal-area restoration and (2) excess-area restoration (Mitra and Namson, 1989).

3.2.1 Equal-Area Restoration

Equal-area restoration is adequate to restore structures formed by simple or oblique shear, which might have changed their thickness, length, and/or angular relationships during deformation. It is therefore appropriate to restore cross-sections involving significant thickness changes and complex angular-relationships between beds that do not necessarily remain constant through time. The method has been largely developed to restore the geometry of extensional basins and, in particular, listric faults (e.g., White et al., 1986; Groshong, 1994). However, when combined with bed-length restoration of a key bed, equal-area restoration can also be used to construct balanced cross-sections in thrust and fold belts (e.g., Mitra and Namson, 1989).

Listric faults are curved normal faults in which the fault surface is concave upward. These faults are often associated with syntectonic strata (deposited during their deposition) that, in contrast to the fault, are usually convex upward. The equal-area restoration methods and problems explained in the following text are presented to restore these type of structures (but the methods can be used for any other type of structure deformed by vertical or oblique shear).

Equal-area restoration methods are based on the initial and final positions of strata. The latter are represented by polygons, bounded by strata bases and tops, or by faults. Pin and loose lines are not necessary, but the boundaries of the section should restore to a rectangle. The methodology to obtain the restored geometry consists of the following operations (Fig. 6A):

(1)Unfold the top of the polygon to the datum by using equidistant translation vectors parallel to the direction of shear (Fig. 6A-II). The shear angle can be determined from observations in the field (such as the orientation of secondary faults), by conceptual models, or by iterations.

(2)Displace the base and sides of the polygon by the same amount of distance using the same translation vectors (Fig. 6A-III). The more closely spaced the translation vectors are, the more accurate the result will be.

(3)Check that the restored geometry of the polygon has the same area as the deformed one. Calculate the areas counting squares on millimeter graph paper. If the areas are not the same, either the geometry of the strata or the shear angle needs to be changed.

(4)Translate the restored polygons to their original position, if appropriate (Fig. 6A-IV). This position should be next to the fault. The presence of gaps or overlaps between the fault and/or the restored polygons (e.g., Fig. 6B-IV) indicates that the shape of the fault or the strata is not consistent with the chosen shear angle (or that the latter is invalid to restore the given geometry).

Fig. 6 Equal-area restoration techniques for an extensional fault and associated strata. (A) General methodology to restore a single polygon using an antithetic shear angle of 70 degrees. (B) General methodology to restore a single polygon using vertical shear. The gap between the strata and the fault after translation indicates a problem in the restoration (possibly the chosen shear angle). (C) General methodology to restore two polygons, using the base of the uppermost restored polygon as the datum for the underlying polygon. Dashed gray lines represent the shear direction. Red lines are the translation vectors used to translate the polygon bases and tops. Opaque polygons indicate the deformed input and restored output. Transparent polygons indicate the different operations in each polygon. Roman numbers indicate different steps of restoration. (B) Modified from White, N.J., Jackson, J.A., McKenzie, D.P., 1986. The relationship between the geometry of normal faults and that of sedimentary layers in their hanging walls. J. Struct. Geol. 8, 897–910.

For structures involving with multiple polygons (Fig. 6C), use the uppermost polygon as a reference to restore the underlying one. This means that, once a polygon is unfolded, the base of the restored polygon will become the datum to unfold the underlying one. If the shape of the following polygons is not reasonable, either the shape of the beds or the shear angle needs to be modified. Sometimes, it is not possible to find a perfect fit for the strata polygons with the fault, and some error has to be accepted. For example, if the area of the overlaps with the fault compensates for the area of the gaps, the shape of restored polygons can be modified by hand, and the area will still be constant.

3.2.2 Excess-Area Restoration

Excess-area restorations were firstly used by Chamberlin (1910) to calculate the depth of the detachment below a detachment fold. Detachment folds form because of continued displacement along an underlying thrust in sedimentary successions composed of a basal ductile layer (the detachment) overlain by competent rocks (Fig. 7A-I). Ductile materials are mechanically weak and, under some stress conditions, they behave as a viscous fluid (e.g., salt; Hudec and Jackson, 2007, 2011). When this is the case, deformation involves flow or migration of material in different directions (e.g., toward the anticlinal core; Fig. 7). Assuming that there is no movement of material out of the cross-section plane, the area of the ductile material between the deformed (Fig. 7A-II) and restored states (Fig. 7A-III) must remain constant. Therefore any excess area above a regional line in the deformed state needs to have its counterpart in the restored state. This has a well-constrained solution, which relates the excess area, the horizontal displacement of the thrust, and the depth of detachment (Fig. 7B).

Fig. 7 (A) Excess-area restoration for a detachment fold. The polygon depicted in pink is made up of a ductile material. Any excess area above a regional line in the deformed state has its counterpart in the restored state. The folded competent layers above the pink polygon represent the same structure as in Fig. 5A. Roman numbers indicate different steps of restoration. (B) Formula to calculate the depth of detachment (D), of a structure with a known excess area (A) and a known thrust displacement