FetKin: Couplingkinematic restorations andtemperature to predictthrusting, exhumation histories,and thermochronometric agesAriel Almendral, Wilmer Robles,Mauricio Parra, Andrés Mora,Richard A. Ketcham, and Michael Raghib

ABSTRACT

FetKin is a C++ program for forward modeling thermochrono-logical ages on a two-dimensional geological cross section.Modeled ages for various thermochronometers are computed fromtime–temperature histories that result from coupling the modeledkinematics of deformation obtained from commercial softwarefor balanced reconstructions (2DMove) and a finite elementcomputation of temperatures. Additional capabilities include theability to accommodate (1) a smooth change of topological relief;(2) the influence of variation in rock physical properties; and(3) multikinetic modeling of fission-track ages and length distri-butions, as well as apatite and zircon (U-Th)/He and muscovite40Ar∕39Ar systems. A joint first-order analysis of the impact oferosion parameters and material properties improves age predic-tions and allows for a more complete analysis of observed coolingages based on their modeled thermal histories. Thus, this paperpresents a new software tool that has been developed as a basicsupport for the methodological approach used to build thekinematic restorations shown in this volume, which are the basicinput for petroleum systems modeling and prediction in theColombian Eastern Cordillera and Llanos foothills basin.

INTRODUCTION

Knowledge about the timing of deformation of geological struc-tures is critical to understanding virtually all geological processes

Copyright ©2015. The American Association of Petroleum Geologists. All rights reserved.

Manuscript received August 1, 2011; provisional acceptance December 10, 2012; revised manuscriptreceived August 2, 2013; revised manuscript provisional acceptance October 2, 2013; 2nd revisedmanuscript received March 7, 2014; final acceptance July 7, 2014.DOI: 10.1306/07071411112

AUTHORS

Ariel Almendral ∼ Instituto Colombianodel Petróleo, Piedecuesta, Santander,Colombia; present address: NorskRegnesentral/Norwegian Computing Center,P.O. Box 114, Blindern, NO-0314 Oslo,Norway; Ariel.Almendral.Vazquez@nr.no;aalmendral@gmail.com

Ariel Almendral is a senior researcher atNorwegian Computing Center in Oslo,Norway, where he develops geostatisticalsoftware for the O&G industry. He has alsocontributed to the development of the Petrelframework in the area of three-dimensionalgeology and modeling. His Ph.D. work fromthe University of Oslo (2005) dealt withnumerical computation of financialderivatives.

Wilmer Robles ∼ Instituto Colombiano delPetróleo, Piedecuesta, Santander, Colombia;wilmerangel@gmail.com

Wilmer Robles is an external consultinggeophysicist at Instituto Colombiano delPetróleo-Ecopetrol, with an M.Sc. degree ingeophysics from the Universidad Nacionalde Colombia. His interests include fault-bendfolding, thermochronology, and theacquisition, processing, and interpretation ofgeophysical data.

Mauricio Parra ∼ Instituto Colombiano delPetróleo, Piedecuesta, Santander, Colombia;Jackson School of Geosciences, TheUniversity of Texas at Austin, Austin,Texas 78759; m.parra@iee.usp.br

Mauricio Parra is an external researchconsultant for Instituto Colombiano delPetróleo-Ecopetrol. He received a Ph.D. ingeosciences from the University of Potsdamin 2009, followed by a postdoctoral stay atthe Department of Geosciences, Universityof Texas at Austin. He has worked in thetectonic evolution of Colombian orogensusing thermochronometry and basinanalysis.

Andrés Mora ∼ Instituto Colombiano delPetróleo, Piedecuesta, Santander, Colombia;andres.mora@ecopetrol.com.co

Andres Mora is a senior geologist at InstitutoColombiano del Petróleo-Ecopetrol. He

AAPG Bulletin, v. 99, no. 8 (August 2015), pp. 1557–1573 1557

including those related to petroleum systems. Usually, determina-tion of the timing of deformation relies on features directlyobserved on folds and faults. This is particularly the case withgrowth strata (Anadón et al., 1986; Medwedeff, 1989; Suppe et al.,1991; Vergés et al., 1996; Suppe et al., 1997; Novoa et al., 2000;Shaw et al., 2004; Strayer et al., 2004) where the geometry, rates,and kinematics of folding can be directly assessed when preserva-tion and exposure permit. However, in many situations, thegrowth strata record has been erased by erosion, and it becomesnecessary to use alternative, indirect methods to assess the timingand rates of deformation for individual structures. This is the caseof the settings like the Colombian Eastern Cordillera where resto-rations have to rely on limited to absent growth strata, whichposes a challenge for accurate petroleum system modeling inmature fold-and-thrust-belt settings. Therefore, an alternativeapproach incorporating thermochronometric methods has to bedeveloped. If those methods are correctly used, a more preciseprediction of the timings of oil migration, generation, and trappingcan be achieved.

One approach is to evaluate cooling histories resulting fromhot material at depth being exhumed toward the cool surface ofthe earth. By keeping track of this cooling history with the helpof thermochronometers, it is possible to obtain approximate esti-mates of the times over which rocks remained at temperaturescooler than a certain threshold temperature. The basic dataobtained from thermochronology are, therefore, cooling ages.Ideally, from cooling ages, it is possible to obtain exhumation his-tories or the timing when thrusting began in a given fault block orstructural domain (Gans et al., 1991; Stüwe et al., 1994; Ehlersand Chapman, 1999; Ring et al., 1999; Ehlers et al., 2003;Ehlers, 2005; Stockli, 2005, Eichelber et al., 2013). Simple calcu-lations are possible by assuming a paleogeothermal gradient(Sobel and Strecker, 2003), but this is not straightforward in situa-tions where there are rapid shortening rates or strong contrasts inlocal topographic relief. In such cases, isotherms markedly deviatefrom the simplest planar shape, resulting in perturbed patterns thatare not easy to predict (Mancktelow and Grasemann, 1997; Huertaand Rodgers, 2006; Lock and Willett, 2008).

Given the complications engendered by even these limiteddepartures from an ideal situation, it is easy to imagine that it iseven more difficult to derive deformation rates from exhumationhistories in yet more complex, realistic geological contexts.Prime examples are thrust belts with complex structural geom-etries, such as the Canadian Rockies (Price, 1981), Taiwan(Suppe, 1980a, b), and the sub-Andean basins of South America(Baby et al., 1997; Martínez, 2006; Mora et al., 2006; Espurt et al.,2008).

earned a B.Sc. degree in geology from theUniversidad Nacional de Colombia in 1999and a Ph.D. in geological sciences from theUniversity of Potsdam (summa cum laude)in 2007. His research interests includestructural geology, thermochronology, andbasin analysis.

Richard A. Ketcham ∼ Jackson School ofGeosciences, The University of Texas atAustin, Austin, Texas 78759; ketcham@jsg.utexas.edu

Richard Ketcham is an associate professor inthe Jackson School of Geosciences at theUniversity of Texas at Austin. He received hisB.A. degree in geology and computerscience from Williams College in 1987 andhis Ph.D. in geological sciences from theUniversity of Texas, Austin in 1995. His activeresearch interests include thermochronologyand geological applications of high-resolution x-ray computed tomography.

Michael Raghib ∼ Instituto Colombiano delPetróleo, Piedecuesta, Santander, Colombia;michael.raghib@gmail.com

Michael Raghib is an external seniorresearch consultant at Instituto Colombianodel Petróleo-Ecopetrol. He received his Ph.D.in applied mathematics from the Universityof Glasgow in 2006, followed bypostdoctoral stays at the PrincetonEnvironmental Institute at PrincetonUniversity and the Applied Mathematicsgroup (T-5) at Los Alamos NationalLaboratory. His interests includemathematical biology and geosciences.

ACKNOWLEDGEMENTS

This study was a part of the project“Cronología de la deformación en lasCuencas Subandinas,” a project of theInstituto Colombiano del Petróleo, theresearch division of Ecopetrol S.A. We thankW. F. Jaimes for his help with validation ofexamples of varying rock properties usingthe Finite Element software ANSYS.

1558 Coupling Kinematic Restorations and Temperature

The many challenges in deriving well-constrained shortening rates from exhumation ratesmay explain why such studies are quite rare andwhy it is, in fact, very unusual to have thermochrono-metric data fully incorporated into balanced crosssections and the subsequent construction of kinematicrestorations. However, in such geological settings,cooling ages could represent the best data availableto derive past deformation rates. In previous studieswhere thermochronological data were used to pro-duce kinematic restorations (Mora et al., 2008,2010), derivation of shortening rates relied on planarisotherms and simplified assumptions concerningpast topographic relief. This approach, though usefulin interpreting the deformation history in certain sim-ple structural settings, might become misleading insituations where isotherms bend due to mass move-ment and where the sample path experiences reburialor lateral movement due to complex and intercon-nected structures (Batt and Braun, 1999; Batt andBrandon, 2002; Bollinger et al., 2004; Hermanet al., 2010).

A more detailed computational model that calcu-lates heat conduction and advection due to develop-ment of structures and/or erosion and landscapeevolution can enable more precise interpretation ofmeasured ages. Early and simple examples tying ther-mal computations to thermochronometric data includemodels of erosion and landform periodicity (Stüweet al., 1994) and detachment faulting (Ketcham,1996). As computational power has increased, suchefforts have become more ambitious and powerful.The PeCube software package (Braun, 2003) simu-lates development of complex landscapes using athree-dimensional thermal solution for erosion, whichcan also incorporate simple structures in the subsur-face. To allow simulation of more complex anddetailed structures, Lock and Willett (2008) linked athermal solver to a commercial structural recon-struction package (2DMove by Midland Valley,Ltd.) to examine the thermal development along crosssections of the Taiwan thrust. For a survey of addi-tional related resources, see Ehlers et al. (2005).

FetKin expands upon the approach of Lock andWillett (2008) by being designed so that it can beincorporated into a section balancing workflow usingany commercial balancing software (i.e., 2DMove,

LithoTect®, Dynel, etc.) and thermochronologicalfield data for calibration. It enables a more detailedinterpretation of data and geology that provides aquantitative and accessible means to support or refutethe expert but subjective interpretation of the geolo-gist. FetKin enables calculation of the changes in thethermal configuration of an area under rapid, com-plex deformation and evolving topographic relief(understood as in Huerta and Rodgers, 2006) and,therefore, makes the calculation of past exhumationrates easier and more precise. The workflowassociated with FetKin is iterative, based on two-dimensional balanced cross sections, using as a start-ing point a kinematic restoration obtained classically,that is, without the constraints of a detailed thermalmodel (i.e., assuming planar isotherms). This pro-vides an initial velocity field that is subsequently fedinto the finite element method (FEM) thermal modelthat advects the isotherms according to the geologicaldisplacements. In addition, FetKin enables predictionof cooling ages at any user-specified position fordifferent thermochronometers in a given structuralcross section. Thus, with the possibility of forwardmodeling ages, FetKin may provide hints on whereto sample when looking for specific timing informa-tion like activation of faults, duration of the deforma-tion, and rate of motion, provided that previousthermochronological data are available.

FetKin is written in C++ with an object-oriented approach to facilitate its maintenance andfuture development. At the time of writing of thispaper, the program is invoked from the command linewith the input being an extensible markup language(XML) file. This text file contains a set of keywordsthat specify initial and boundary conditions, sampledata, topography data, and kinematic data in the formof a sequence of files containing the interpreted gridevolution in time. The FetKin first performs a qualitycontrol test of the input grids to detect inconsistenciessuch as large velocity jumps or nonmatching gridpoints. Once the moving grid is accepted as a validkinematic model, the program proceeds to computetemperatures in a time-stepping loop. Finally, agesfor specified thermochronometers are determinedfrom the time–temperature paths for the input sam-ples. An important feature is that velocity is an inputparameter that the user can vary at each geological

ALMENDRAL ET AL. 1559

time step. In particular, we include the concept ofgrid-timing or geological time for each particular gridin the time-stepping process so we may simulate sce-narios of tectonic activity or quiescence. Moreover, aset of user-defined topographies can be incorporatedinto the computations to account for the topographiceffect on the temperature. The time–temperaturepaths produced by FetKin can be used as an initialguess in HeFTy (Ketcham, 2005) for inverse model-ing and data fitting.

The calculated thermochronometric ages inFetKin use state-of-the-art algorithms and kineticsfor various thermochronometers. These include mod-eling of apatite fission-track (AFT) annealing thataccounts for compositional variation in annealingkinetics (e.g., Ketcham et al., 2007), a detailneglected in most modeling efforts that can have apivotal effect on data interpretation (e.g., Mora et al.,2010).

For visualization purposes, several MATLABroutines have been implemented which automaticallyload the finite element grids, the 2DMove sections,and the sample locations to produce a report sequencefor each time step. This permits a more complete andefficient analysis of the effect of material properties,erosional relief changes, and sedimentation into thecalculated age profile.

In this paper, we show how critical it is to incor-porate in a standard kinematic restoration workflow,completely or partially based on thermochronology,the behavior of the isotherms during deformation. Ina companion paper (Mora et al., 2015, this issue),we show the precise workflow used to apply FetKinwith actual geological data during a kinematic resto-ration. In an additional paper (Sánchez et al., 2015,this issue), cross sections where the FetKin approachwas used are the base for petroleum system modelingin Petromod.

FETKIN’S TEMPERATURE MODEL

One-dimensional thermal models have proven tohave shortcomings in translating low-temperaturecooling ages into exhumation rates. (1) Topographyis known to vertically bend the low-temperature iso-therms for moderate exhumation rates (>0.5 mm/yr),

affecting the interpretation of observed ages (Stüweet al., 1994; Mancktelow and Grasemann, 1997).(2) Lateral advection of rocks causes lateral tempera-ture variations. (3) Local contrasts in material proper-ties are usually encountered associated with faultplanes. These factors call for a more advanced model-ing of the thermal regime. We have, thus, implementedthe well-studied linear diffusion–advection equation intwo dimensions (2-D) that has consistently provenuseful in modeling the temperature distribution withinthe earth’s crust. This equation needs to be solvednumerically in time and space to provide a completedescription of the temperature field. In this section,we introduce our notation and then proceed to explainthe connection with a kinematic model.

The Two-Dimensional Diffusion–AdvectionEquation

The FetKin solves numerically the transient advec-tion–diffusion equation in two dimensions (Carslawand Jaeger, 1986):

ρc�∂T∂t

− vx∂T∂x

− vy∂T∂y

�=

∂∂x

k∂T∂x

+∂∂y

k∂T∂y

+ ρH (1)

where Tðx; y; tÞ is the temperature at location ðx; yÞ attime t, ρðx; yÞ is the (space-dependent) rock density,kðx; yÞ defines the rock thermal conductivity tensor,cðx; yÞ is the specific heat, v = ðvx; vyÞ is the velocityof the moving grid, and H is the radioactive heatproduction. For this equation to be well defined,one needs to specify initial and boundary conditions.The boundary conditions on the top and bottomof the grid are defined as Dirichlet-type fixed-temperature conditions; see Braun et al. (2006) for asimilar notation:

Tðt; x; y = SðxÞÞ = TSðyÞ = TMSL + βðy − ymaxÞ (2)

Tðx; y = yminÞ = TBasal (3)

where SðxÞ is the topography line, TMSL is themean sea level (MSL) temperature, β is the surfacetemperature–elevation gradient (temperature changewith elevation), ymax is the MSL level, ymin is the baselevel, TBasal is the basal temperature and TSðyÞ is thetop surface temperature. We assume no heat flux onthe left-hand and right-hand sides of the grid, that is,

1560 Coupling Kinematic Restorations and Temperature

∂T∕∂x = 0 (Neumann-type boundary conditions). Forthe initial condition,

Tðt = 0; x; yÞ = T0ðx; yÞ (4)

we use the steady-state solution of equations 1–3,assuming no velocity, no heat production, andnegligible temperature gradient in the x-direction.This can be summarized by means of the linearrelationship:

T0ðx; yÞ = TSðyÞy − ymin

ymax − ymin+ TBasal

ymax − yymax − ymin

(5)

This initial condition assumes no heat production, aswell as all subsequent modeling in this paper.Although heat production can be an important varia-ble for crustal-scale thermal modeling, for petroleumsystems where the depths of interest are relativelyshallow, the difference between heat generated withinthe crust and heat that flows up from below is asecond-order consideration. Although it is notreported in this paper, a Neumann-type (given by atemperature flux instead of a constant temperature)boundary condition for the bottom boundary is alsoavailable in FetKin.

Adding Kinematics to the Temperature Model

We have adopted the methodology set forth by Lockand Willet (2008), coupling the thermal solution witha kinematic model generated from 2DMove, to findthe temperature distribution that honors a givendeformation mechanism. The temperature is com-puted on an Eulerian or computational grid with thehelp of the FEM. This is a classical choice given itsversatility in coping with deformed geometries. Thenodes of the FEM grid are evenly positioned alongthe lateral direction. In the vertical direction, thenodes are evenly distributed from the topography lineto the bottom of the grid. The nodes are used to buildtriangular elements where linear basis functions aredefined. The temperature field is obtained as a linearcombination of linear basis functions whose coeffi-cients are found by solving a linear system wherethe matrix and right-hand side vector are assembledfrom element matrices. The time derivative has beenapproximated by the implicit Euler scheme. This

time-integration is of first-order accuracy in timeand is unconditionally stable, that is, no time-steprestriction is required for stability. We believe thatthese choices of a first-order method in space and intime suffice for our purposes of a first-order descrip-tion of thermochronological ages.

In a typical geological setting, the computationalgrid is bounded in the lateral direction by xmin andxmax and in the vertical direction by ymin and the topog-raphy surface. The advection component incorporatesthe direction of the velocity field at a given point ofthe computational grid, and the magnitude of thisvelocity affects the thermal regime. A sufficientlydense cloud of particles that move in time accordingto some kinematic law (fault parallel flow, trishear,flexural slip, etc.) is used to define a velocity field;see Figure 1 for a schematic view, where the particlesfrom the moving grid are represented as filled squares.

The purpose of the cloud of particles is manifold.By construction, this set of points is implicitly used inthe algorithms of 2DMove to describe a deformationrule. A sufficient number of frames or cross sectionsthroughout the tectonic activity will describe in detail

Figure 1. Interpolation from moving grid (blue squares) ontoclosest quadrature points in finite element method grid (reddiscs). The closest moving grid point with respect to a quadraturepoint could be located in a different triangle of the computationalgrid; therefore, we omit in this figure some connecting arcs.

ALMENDRAL ET AL. 1561

the velocity field associated with the deformation andwill, therefore, be a good approximation for theadvection component. In addition to the velocity,information on the thermal conductivity, the density,and the specific heat can be also extracted from theparticles in the moving grid (see equation 1). Thevelocity vector and physical properties from the clos-est particle are assigned to the closest quadraturepoint in the computational grid, represented as discsin Figure 1. Note that due to the nature of the FEMmethod, properties are sampled at quadrature pointsbecause the integration of inner-products of hat func-tions can be done exactly by quadrature rules (seeZienkiewicz et al., 2005). We believe that thisapproach is sufficient for this first-order method. Animportant by-product of this strategy is that the setof samples can be restored, yielding a set of trajecto-ries in time for subsequent time–temperature historycomputation. Numerical tests have been conductedto verify that our results coincide with publishednumerical examples from Lock and Willet (2008),as shown in the section “Numerical Experiments”.

SETTING UP A FETKIN PROJECT ANDIMPLEMENTATION DETAILS

In this section, we outline the main steps to prepare aFetKin project, with focus on the following aspects:(1) construct a valid velocity field, (2) build a set oftopography lines with corresponding times, (3) assigntimes to the moving grids, (4) define the sample loca-tions, and (5) specify the thermochronometers for ageand length calculations. A brief summary of the spa-tial discretization method and time-integration is alsoincluded.

Constructing 2DMove Moving Grids

The two primary tools used in 2DMove for tracingpositions throughout a geological history are lines,usually corresponding to stratigraphic boundaries,and points, which may be on lines or between them.Our experience from the use of FetKin on real sec-tions from the Colombian foothills has suggested thata number of rules must be followed for proper con-struction of a moving grid that can be successfully

processed. Preparing a set of grids using 2DMove istime-consuming, so one normally bypasses a numberof important details that are left to internal 2DMovealgorithms to sort out. However, for our purpose withFetKin, we need to go further in the number of condi-tions we demand from a set of grids to be eligible torun coupled with a thermal model. These rules helpin creating robust moving grids in 2DMove: (1) assigna unique name to each line; (2) keep the same numberof lines for all grids; and (3) keep the same number ofpoints in each line.

Once the grids have been created, they can beexported to an ASCII format and tested for consistencyusing the batch subprogram QCSECTIONS. Theimportance of using this tool lies mainly in detectingpossible human errors as well as locating and repairingthem. Note that consistency in the number of lines andpoints is an essential step for achieving a consistentvelocity description. In the case of sedimentation,where new lines are added to a cross section, the line/point consistency must still be respected. In this case,a dummy line can be placed outside the computationalgrid that will have no effect on the temperature compu-tation until it enters the grid. During the balancingprocess, it becomes practically impossible to strictlymaintain a constant number of points. This is part ofthe nature of the geological interpretation and work-flow. However, if the geologist is aware that her/hiscross sections are going to be part of the FetKinworkflow, she/he can resample the edited lines in eachbalancing step, preserving the geometry suggested, torestore the initial number of points.

Changing Topography and ComputationalGrid Adaption

The need for a more realistic approach to numericallymodel the effect of a nonstatic topographic relief wasaddressed in Braun (2003). The method implementedin PeCube is based on an interpolation of the solutionobtained at each time step over a deformed, currentgrid caused by a displacement of the top nodes.PeCube makes use of an amplification factor in orderto augment or reduce the relief height. In FetKin,we specify a set of topography lines with its corre-sponding time. The topography within a given timeinterval is not available and has, therefore, to be

1562 Coupling Kinematic Restorations and Temperature

interpolated to deduce missing data. More precisely,for a given time interval where the topography at timetinit has height yinit and at time tfinal the height is yfinal,the height at any t ∈ ½tinit; tfinal�; is interpolated withthe help of the formula:

yðtÞ = yinit + αðtÞðyfinal − yinitÞ (6)

Here, αðtÞ is a function given by

αðtÞ =1 − exp

�− t−tinit

λtopo

�

1 − exp�− tfinal−tinit

λtopo

� (7)

where λtopo is a positive parameter used to modeltransitions between topographic lines. High valuesof λtopo will give an almost linear interpolationbetween two topography lines; whereas, low valueswill speed up erosion of peaks for the most ancienttopography.

Moving Grid Timing and Time Interpolation

As mentioned earlier, FetKin can model periods oftectonic activity and quiescence. When the practi-tioner constructs the sequence of sections, he/she isonly taking into consideration the displacement. Inorder to assign times to each particular event, onecan enter a list of section versus time via a text file.For the episodic example in the numerical experi-ments, such a list takes the form of Table 1.

The timing in Table 1 represents geologicaltimes. For the episodic tectonic example, sections 21and 22 are identical to simulate no motion over theperiod 22 to 12 Ma. Next, the sections labeled from1 until 21 simulate motion in the period from 12 to10 Ma. Finally, the sections labeled 0 and 1 are iden-tical to simulate negligible velocity in the intervalfrom 10 to 0 Ma. The number of time steps does notneed to be 22. Sections not defined for a time stepare interpolated linearly according to the closest mov-ing grids in time. The sections interpolation ensures asmoother transition of velocities across sections.

Sample Data

Data can be sampled at arbitrary locations in the com-putational grid, provided sufficient information on

time–temperature development can be derived fromthe moving grid. Given a data point, FetKin findsthe closest point from the final grid (most recent)and traces this point back. To obtain a smooth agecurve, it is most advantageous to simply include thesamples in the moving grid as markers and do a fullrestoration of these markers. Occasionally, sampletrajectories fall slightly outside of the computationalgrid where temperatures have not been computed.This happens, for example, when generating topogra-phies using a numerical method that tends to smoothout corners by adding artificial numerical diffusion.A cure for this is to vertically offset the topographyby a minor increment so that those shallow samplesgive meaningful temperatures. Another situation inwhich problematic age holes may arise is where theterrain has a hill slope angle greater than the angleof shallow velocity vectors. In this situation, pointswill necessarily drop out from the fixed grid duringa recent period and will reenter the grid at an olderstage, producing jumps in the age profile. Hence, asa rule of thumb, hill slope angles should not exceedthe dip of the steepest fault.

Thermochronometer Calculations

The design of FetKin allows incorporation of anynumber of thermochronometers. For the models pre-sented in this paper, apatite fission-track ages, as wellas length distributions, are calculated using theKetcham et al. (2007) calibration. Apatite (U-Th)/Heages are calculated based on the Wolf et al. (1998)and Farley (2000) calibration for Durango apatite.Other thermochronometers computations are alsoavailable in FetKin. Zircon fission-track (ZFT) and40Ar∕39Ar in muscovite (MAr) ages are calculatedaccording to the implementation by Braun (2003) thatuses the parameters from Brandon et al. (1998) forZFT and Hames and Bowring (1994) for MAr, basedon the framework by Dodson (1973).

Time Integration and the Finite ElementMethod

Equations 1–4 have been discretized in time bythe implicit Euler scheme in time and the FEM inspace. We now briefly outline the method; for a

ALMENDRAL ET AL. 1563

practical introduction, see Lynch (2004). Let u1 andu0 be the temperature for the next time step and thecurrent time step, respectively. For simplicity, denoteby F the operator that only involves spatial deriva-tives in equation 1. The implicit Euler scheme canbe compactly written as

ρcu1 − u0

Δt= Fðu1Þ (8)

The objective is to compute u1 implicitly, afterrearranging terms to arrive at an equation of the formu0 = Gðu1Þ. The next step, and most difficult, is todiscretize this equation in space. The method ofchoice for spatial discretization is the FEM. First,one needs a weak interpretation of the second-orderdifferential equation. Next, the region of interest, thatis, our computational 2-D grid, is triangulated. Oneach triangle, three linear 2-D basis functions definethe so-called test functions (a linear function thatequals one on one vertex and vanishes on the othertwo vertices). The numerical solution is set up as alinear combination of these test functions, where theunknown coefficients represent the temperature val-ues at each node. This leads to a linear system onthe form u0 = Gu1, where the matrix G is assembledfrom element matrices whose entries are inner prod-ucts of test functions for each element. The boundaryconditions are embedded in the matrix and the right-hand side. The solution to this system of equationsyields the sought temperature for the next time step.One then proceeds iteratively until all time steps havebeen computed. We mention that no automatic time-step detection is done by FetKin. This entails thatrapidly changing topography surfaces demand asmaller time step, which must be imposed by the user.

NUMERICAL EXPERIMENTS

We now conduct several numerical tests forwell-known geologically sound examples. Thepurpose is to demonstrate the impact of the followingfactors on the structure of the forward-modeledcooling age: (1) time and velocity of deformation,(2) rock properties, and (3) topography evolution.For each of these factors, we have considered twoextreme scenarios, running from 22 Ma to the

present. The first scenario assumes a period oftectonic quiescence from 22 to 12 Ma, followed by atectonically active period from 12 to 10 Ma, and con-cludes with a quiescent period from 10 until 0 Ma.For future reference, we denote this situation asepisodic tectonic. The second scenario assumescontinuous motion throughout the complete runningtime from 22 to 0 Ma. This situation will be referredto as continuous tectonic. Summaries of these twocases are shown in the columns of Figures 2 and 3,respectively.

Example 1: Fault-Bend Fold on a Ramp withFlat Topography and Homogenous PhysicalProperties

We start by comparing results from FetKin with pub-lished results. This example tests the thermo-kinematics of a fault-bend fold as described inSuppe (1980a, b, 1983). This model was used inLock and Willet (2008) for the fold-and-thrust-beltexample of the Taiwan Western Foothills, focusingon the determination of the longterm deformationrate. Note that unlike the example in Lock andWillet (2008) where a heat flux is specified as a bot-tom boundary condition, here a Dirichlet-type condi-tion as a basal temperature TBasal is assigned on thebottom of the grid. The results are very similar forthese two boundary conditions because of the largedistance from the grid bottom to the detachment.

The computational grid has 80 km (49 mi) later-ally and 30 km (18 mi) in depth. A single 30° dippingfault ramp extends from 0 to 15 km (0 to 9 mi). Thetop of the ramp is positioned at the central (zero) xposition and at a depth of 5 km (3 mi), the bottom isplaced at a depth of 15 km (9 mi). The rock layersride over the ramp from right to left, at a slip rate of8 mm/yr (0.31 in./yr). A fold is created abovethe upper kink of the ramp with lateral horizontaltransport of material. The rock-uplift rate equalsthe denudation rate in this case because no relief isbeing created. The material properties are assumedconstant throughout the grid and are as follows:specific heat 1000 J kg−1 K−1, thermal conductivity2.5 Wm−1 K−1, density 2500 kgm−3, basal tempera-ture 600°C (1112°F), MSL temperature 0°C (32°F)and no heat production. Finally, the grid spacing in

1564 Coupling Kinematic Restorations and Temperature

Figu

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12Ma,(B)1

1Ma,(C)1

0Ma,(D)0

Ma,(E)A

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lineindicatesthatthis

portion

hasbeen

eroded

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shales

(green)with

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ntaneous

removalofthetopography

viaerosion;thelastcolumnshow

sthecase

where

topography

evolveson

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escales

asthe

tectonicdeform

ationevents.The

circlesmarkedwith

A,B,

andCaresamples

locatedon

thesurfaceat

−3,5,and10

km(−1.8,3,and6mi).

ThepointsEat

(3,−

4)and

F(10,−4),m

arkedwith

squares,arereferencepoints.

Forthiscase,there

isno

relevantinfluence

ofthepaleotopographyon

theages.

ALMENDRAL ET AL. 1565

Figu

re3.

Exam

ples

1to3explorethesamesituationas

inFigure2,butassum

econtinuous

tectonics:(A)

22Ma,(B)1

5Ma,(C)8

Ma,(D)0

Ma,(E)A

FTandAH

eages.The

shaded

partabovethetopography

lineindicatesthatthisportion

hasbeen

eroded

away.

1566 Coupling Kinematic Restorations and Temperature

both directions is taken as 1 km (0.62 mi), and themodel has been run for 22 m.y. An inherited age of8 m.y. was chosen arbitrarily so that the differencesbetween fully reset and inherited ages could beclearly visualized.

The first column of Figures 2 and 3 shows theresults for the episodic and continuous tectonics,respectively, together with the corresponding(U-Th)/He and AFT ages. The cooling ages have adistinctive U-shape, where younger ages are locatedright above the ramp. The shape of the U variesdepending on the thermochronometer and on theduration of the tectonic scenario. For the followingdiscussion, we use samples A, B, and C, located atlateral coordinates −3, 5, and 10 km (−1.8, 3, and6 mi) and marked with circles, and samples E and F,at coordinates (3, −4) and (10, −4) marked withsquares, as reference points. Note the significant dif-ference in age for sample A in Figures 2-E1 and3-E1, caused by its earlier cooling in the continuousmodel. Summaries of calculated AFT ages (usingF-apatite kinetics) for all selected samples can befound in Tables 2 and 3.

Example 2: Fault-Bend Fold on a Ramp withFlat Topography and NonhomogenousPhysical Properties

This example shows how physical properties can beassigned to layers in the FetKin moving grid andhow these properties influence the relative positionsof the isotherms. The moving medium is consideredto be a sequence of alternating layers of sands andshales, with a uniform thickness of 5 km (3 mi) each.Note that these exaggerated layer dimensions arechosen to demonstrate that AFT ages may feature sig-nificant lateral variations if material properties featuresufficient contrast (Figures 2 and 3). The thermalparameters within each layer are assumed constant,

but their values vary from layer to layer accordingto mean depth and lithology. Material properties forthis example were computed with formulas inDemongodin et al. (1991), Allen and Allen (2005),and Nemcok et al. (2005) and are summarized inTable 4.

Snapshots are displayed in the second column ofFigures 2 and 3. Compared to the case of homog-enous materials, the temperature field in this case isinfluenced by different layer properties, which resultsin a different temperature field from 22 to 12 Ma.This difference is most clearly seen in A2–D2 as adownward shift in the 120°C (248°F) isotherm,which explains the older ages observed for samplesA and C, as compared to Example 1, and the narrow-ing of the U shape; see also Table 2. We note thatapatite samples are rarely found in shales, so thisexample is extreme. However, another scenario (notreported in this paper) that assumed shales with 10%sand gave similar results.

Material properties have a posttectonic thermalinfluence as well. To highlight this effect, we usesamples D and E, located at the same present-daydepth, which remain in the AFT PAZ after thrusting.Their ages differ by about 5 m.y. (see Table 3) as aresult of the different steady-state temperatures in

Table 1. Timing Table for Episodic Tectonic in Examples1 to 3

Grid Label 0 1 2 3 4 – 20 21 22

Timing(Ma)

22 12 11.9 11.8 11.7 – 10.1 10 0

Table 2. Apatite Fission Track Ages for Samples A, B, and Cat Lateral Coordinates −3, 5, and 10, Respectively*

Sample/Example 1epi 1cont 2epi 2cont 3epi 3cont

A 12.2 18.03 16.80 21.01 12.44 17.2B 11.45 12.28 11.87 12.11 11.93 17.89C 11.61 12.37 17.29 16.27 11.73 19.34

*The notation epi is for the episodic tectonic activity in the interval 12 to 10 Maand cont stands for continuous tectonics during the period 22 to 0 Ma. Theabbreviation 2cont is read as example 2 with continuous tectonics.

Table 3. AFT Ages for Samples D and E at Coordinates(2, −4) and (8, −4), Respectively*

Sample/Example 1epi 2epi

D 6.99 6.61E 6.87 1.87

*The notation epi is for the episodic tectonic activity in the interval 12 to 10. Theabbreviation 2epi is read as example 2 with episodic tectonics.

ALMENDRAL ET AL. 1567

the sands and shales caused by their contrastingmaterial properties and geometric arrangement.

These differences cannot be accounted for in thecase of Example 1 with homogenous materials, wherea uniform cooling rate for these samples givesapproximately the same age. This post-tectonic effectof materials is large in this case of episodic tectonics,in contrast to our second scenario of continuous tec-tonics where this effect is not apparent.

Example 3: Fault-Bend Fold on a Ramp withChanging Topography and HomogenousPhysical Properties

We now show how FetKin incorporates changingrelief. Here we have used the same moving grid fromExample 1. In the episodic case, topography creationtriggers erosion followed by 10 m.y. of only erosion.For our second continuous case, a combination oftopography creation and erosion occurs throughoutthe model execution. We mention that FetKin alsoallows the definition of a series of intermediate topog-raphies such that the missing ones are interpolated foreach time step. For details, see section “ChangingTopography and Computational Grid Adaption”.

The erosion causing topography change is simu-lated by a fluvial incision model (Whipple and Tucker,1999). The elevation changes (h) in time and spaceaccording to a model computed from the equation:

∂h∂t

= vy + vx∂h∂x

− kejxrjm���� ∂h∂x

����n

(9)

where xr is the distance from the drainage divide, ke isan erodibility parameter, vx and vy are the velocity com-ponents of the particles close to the topographic boun-dary, and m and n are parameters that need to be

calibrated depending on the situation. The last compo-nent in this equation represents the erosion rate resultingfrom a stream-power law. This law, although heuristic,is useful to capture the interconnection between erosionand orogeny formation, see Willett (1999) and Huertaand Rodgers (2006). Empirical studies suggest that theratio m∕n is likely to be about 0.5 (Whipple andTucker, 1999; Whipple, 2004). Our tests assumem = 0.5 and n = 1.

This model needs to be constructed in two steps.First, we build the sequence of eroded topographiesusing FetKin as a preprocessor only for topographysurface generation and not for temperatures. Oncethe elevation model has been constructed, we use2DMove’s restoration algorithm to find the trajecto-ries of the samples by restoring the samples at theirpresent topography. These samples are then added tothe moving grid as markers. We then prepare FetKinfor a second run that includes the previously com-puted topographies and the sample trajectories. Thedeformed grid, the eroded topography, and the com-puted temperatures are displayed in the right-handcolumns of Figures 2 and 3. The resulting agesequence now has a W-shape in the continuous case,where older ages are located around the relief top, asexpected. For the episodic tectonic case, there is norelevant influence of the paleotopography on theages. As proved by Stüwe et al. (1994), the topogra-phy has a large impact on the temperature distributiondepending on the topography amplitude, the wave-length, and the denudation rate. For denudation rateslarger than 1 mm/yr (0.03 in./yr) (our episodic casehas 4 mm/yr [0.15 in./yr]), a topographic amplitudeof 3 km (1.8 km), and a wavelength of about 20 km(12 mi), the apatite retention isotherm can be per-turbed by up to 1 km (0.6 mi). This phenomenon is

Table 4. Rock Properties for Example 2

Rock TypeInitial MeanDepth (km)

Densityðkg m−3Þ

Specific HeatðJ kg−1 K−1Þ

Thermal ConductivityðWm−1 K−1Þ

Diffusivityðkm2 myr−1Þ

Sandstone3 25 2649 973 2.19 26.82Shale2 17.5 2720 837 1.28 17.77Sandstone2 12.5 2622 993 3.24 39.21Shale1 7.5 2696 854 1.46 19.97Sandstone1 2.5 2238 1330 3.14 33.26

1568 Coupling Kinematic Restorations and Temperature

observed in our episodic case, just after motion hasceased. Fast erosion decreases the wavelength, whichadded to overall cooling results in almost flat station-ary isotherms, as exemplified in case D3 in Figure 2.

A comparison between the topography generatedin the episodic and continuous cases reveals a sym-metric topography in the first case and an asymmetrictopography in the second case. As implemented inour model, episodic tectonics create an almost sym-metrical shape after 12 m.y. of running time, andbecause no lateral change in erosion rate is assumed,the final result is symmetrical. This is in clear contrastwith the continuous case, where a horizontal velocitycomponent is always present throughout the modelrun, making erosion rates vary laterally. Even thoughthis is a simplified model that does not account formaterial properties versus erosion rates, it illuminatesthe influence of the topography creation process dur-ing tectonically active periods on the age profile.

The FetKin also allows straightforward produc-tion of age–elevation plots from model results(Figure 4). For the episodic case, almost the wholerelief was eroded away, and so it is not shown. Forthe continuous case, the slope in the age elevationplot varies as a consequence of the asymmetry of thetopographic surface. At low-angle hillslope, thetopography cuts the rock units in such a way thatyounger stratigraphic units or crustal levels may cropout at topographically higher elevations. Therefore, incertain cases, higher elevation does not correspond to

an older cooling age in the age–elevation plot. As weadvance uphill, this trend reverses to result in thefamiliar positive slope.

Example 4: Fault-Bend Fold on a Listric Fault

This example considers a listric fault detaching at adepth of 35 km (21 mi). This is a synthetic modelbased on the geometries and depths to detachmentof an actual cross section reported by Mora et al.(2008). The same model is tested in FetKin withactual data in the companion paper by Mora et al.(2015, this issue) because it appears to be the styleof intracrustal deformation that more closely resem-bles the deformation in the Colombian EasternCordillera. The rates of thrusting, exhumation, depthof erosion, and thickness of sedimentary rocks usedhere are also derived from the study by Mora et al.(2008). The fault moves at a rate of 0.5 km/m.y.(0.31 mi/m.y.) throughout a running time of 30 m.y.Again, fault-parallel flow is used to simulate themotion of the hanging wall to the left of the fault.The computational grid extends from 0 to 80 km(0 to 49 mi), and the cell size is 1 km (0.6 mi) in eachdirection. The modern topographic surface isextracted from a digital elevation model of theFarallones Anticline from the Cordillera Orientalin Colombia, along the trace of the Mora et al.(2008) cross section. The topography at the begin-ning of the run starts at sea level and amplifies lin-early in time to reach the modern topography.Samples have been placed along the topographic lineand restored using 2DMove for a full description ofthe sample trajectory. Figure 5 displays the evolutionof the thermal field at 30 and 0 Ma. The AFT andAHe ages are reset near the valley located at about50 km (31 mi) and closer to the emerging faultplane. One interesting feature of this profile is thegreater roughness of the AHe profile, as comparedto the AFT one, caused by the lower AHe closuretemperature, which is in some cases a desired prop-erty when capturing topographical effects usinglow-temperature thermochronometers.

These numerical experiments can be used torefine deformation rates in actual cross sections. Forexample, Mora et al. (2008) constrained their kin-ematic restoration based on cooling ages and planar

Figure 4. Age–elevation plot for Example 3 with changingtopography and continuous tectonics.

ALMENDRAL ET AL. 1569

isotherms. Using FetKin, the shortening rates in theinitial kinematic restoration can be adjusted until themodel matches as many observations/variables aspossible. One of the powers of a kinematic approachlies in the restoring properties of the acting deforma-tion operators. Complicated sample paths can beaccurately traced, which is a key tool when determin-ing the time in the past when samples reset ascompared to their computed forward ages. Theapproach created by FetKin thus combines the most

sophisticated kinematic modeling tools with thetime- and rate-resolving power of thermochronomet-ric data.

CONCLUSIONS

We have introduced a software package that is able tocalculate isotherm advection in complex geologicalsettings. The new software FetKin is especially suit-able for areas with rapid shortening rates (past andrecent), jagged topography, and can be used consider-ing numerous and complex kinematic paths for agrowing structure. To do that, FetKin couples kin-ematic restorations and a 2-D temperature model tomodel temperatures, from which it calculates coolingages for various thermochronometers. It includes thepossibility of testing realistic thermal parameters,changing relief, and episodic versus continuous tec-tonic events. These factors have observable effectson the age response. A careful modeling of each ofthese features provides a better basis for interpretingand understanding thermochronological data. Thedeviations observed in age response for changingmaterials warn us of the nonuniqueness of this com-plex problem and the uncertainty related to notalways having access to physical properties. One ofthe main implications of a careful 2-D modeling isthat we may estimate exhumation rates, denudationrates, and velocity of deformation by testing differentscenarios in actual structural cross sections. The pro-cedure is shown in more detail in a companion paperby Mora et al. (2015, this issue). In those cases, theprocedure includes finding a modeled solution thatfits more observations by trial and error, departingfrom an initial simple kinematic restoration whichassumes planar isotherms.

Forward modeling as done by FetKin allowsfor a more efficient and significant sampling of data.We may identify regions that do not contain a topo-graphical effect or that do not have significant localinfluence of materials. FetKin is designed to workclosely with the kinematic algorithms of 2DMove,providing a general framework for working with realgeological cross sections. However, the generalizeddesign and approach in FetKin should allow it to beadapted to other 2-D cross section restoration pro-grams, if desired.

Figure 5. Example 4, fault-bend fold on a listric fault: (A)30 Ma, (B) 0 Ma, (C) AFT and AHe ages.

1570 Coupling Kinematic Restorations and Temperature

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