Braun J., van der Beek P., Batt G. Quantitative Thermochronology: Numerical Methods for the Interpretation of Thermochronological Data

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198 Thermochronology in active tectonic settings

(a) Predicted P–T–t paths

Age

Distance

(b) Rock paths

Fig. 13.4. (a) Particle trajectories in P T  space as they travel through an

orogen as depicted in (b). (c) Predicted age distributions for two arbitrary ther-

mochronological systems.

In the vicinity of the retro-shear, rock particles from either side of the orogen

are brought together. This feature of most numerical models explains nicely the

formation of inverted metamorphic terrains by tectonic accretion (Jamieson et al.,

1996).

The times at which particles exposed at the surface of this model cooled through

two arbitrary temperatures are plotted in Figure 13.4(c). This could be viewed

as the predicted distribution of cooling ages for a pair of thermochronological

systems of different closure temperatures. There is a relatively broad distribution

of ages with a minimum in the region of maximum exhumation rate. Ages increase

on both sides of this minimum, reaching values that pre-date the onset of the

current tectonic event. These ages are those of rock particles that have entered

the orogen at a relatively shallow depth and have therefore never experienced

sufficiently high temperatures for the particular system to be reset. These particles

13.3 The Alpine Fault, South Island, New Zealand 199

consequently retain elements of their inherited age structure. Their ages correspond

to a previous tectonic event or, simply, to their crystallisation age, which, for the

sake of the argument, we will assume is much older than the age of onset of

tectonic activity. It follows from these simple geometrical arguments that the zone

of reset ages is broader for systems characterised by a lower closure temperature.

Furthermore, the higher the closure temperature, the older the ages. These various

factors combine to produce an age distribution made of a series of nested ‘smiles’,

one for each thermochronological system.

It is evident from these considerations that the age distribution will be a function

not only of the exhumation rate, and thus the horizontal tectonic convergence

velocity, but also of the dip of the retro-shear zone and the thickness of the

orogen. This demonstrates the potential that thermochronological datasets have

to constrain not only the timing of tectonic movements but also the geometry of

the deformation patterns they engender (Batt and Braun, 1997). In the following

section we will illustrate these points through an example of a rather complete

age dataset collected across an active orogen in the South Island of New Zealand.

13.3 The Alpine Fault, South Island, New Zealand

The South Island of New Zealand is the site of an active collision between two

continental fragments attached to the Australian and Pacific plates (Figure 13.5).

Most of the relative movement between the two plates is constituted by oblique

reverse movement along the steeply dipping Alpine Fault (Walcott, 1998).

Figures 13.6 and 13.7 summarise thermochronological data from the study

by Batt (1997), which were subsequently published in Batt et al. (2000). They

include a large range of thermochronometers, ranging from fission tracks in apatite

to Ar–Ar in biotite and muscovite. These studies are geographically extensive,

comprising a series of transects running perpendicular to the plate boundary and

spread laterally over 200 km of the central orogenic region.

Batt and Braun (1997, 1999) developed a fully coupled thermo-mechanical

model of the orogenic system to aid in interpreting the dataset and extracting

useful information on the development of the tectonic system from it. Using a

finite-element approach, the force-balance equation was solved assuming that the

crust consists of a complex, non-linear material where brittle failure is active at

low pressure and temperature and thermally activated dislocation creep is active

at high pressure and temperature. The mechanical model predicted the defor-

mation (and thus velocity) field in the collision zone from a set of imposed

velocity boundary conditions that represent subduction of the underlying mantle

lithosphere similar to that shown in Figure 13.1. A simple one-dimensional ero-

sion model represents the transport of mass by surface processes. The predicted

200 Thermochronology in active tectonic settings

Haast

Australian

plate

Franz

Josef

Pacific

plate

Hope Fault

Alpine Fault

Metamorphic Grade

Garnet–Oligoclase zone

Biotite zone

Chlorite zone

50 km 100 km0

M′

Fig. 13.5. Major geological features of the Southern Alps. The indicated regions

of exposed biotite zone and higher metamorphic grade approximately coincide

with areas experiencing rapid uplift and exhumation in the ongoing orogeny.

The stars labelled Haast and Franz Josef correspond to the locations of published

surface heat-flow measurements (Shi et al., 1996) indicating a strong perturbation

of the conductive geotherm by advection and, following (5.10), a minimum value

for Pe of 3.

velocities were used to compute the temperature field by solving the transient

heat-conduction/advection/production equation in two dimensions. Particle trajec-

tories were also computed, from which P–T –t paths were calculated. These paths

were finally used to compute synthetic ages for a range of thermochronometers.

These ages are shown in Figures 13.8 and 13.9.

The predicted ages were shown to display the typical ‘nested-smile’ pattern

described in the previous section: they increase from very young ages adjacent to

the surface trace of the retro-shear zone (the equivalent to the Alpine Fault in the

Southern Alps) to much older ages within the centre of the orogen. As expected,

the minimum age and width of the reset zone vary with the chronometer, as is

observed in the data (Figure 13.6).

Exploration of parameter space in the model showed that the pattern of age

displayed across the Southern Alps is most consistent with a steeply dipping

13.3 The Alpine Fault, South Island, New Zealand 201

250

(a)

(b)

Zircon

fission track

Apatite

fission track

Muscovite

K–Ar

Biotite K–Ar

Enlarged

in (b)

200

150

Apparent age (Myr)

100

024681012

Distance southeast of the Alpine Fault (km)

40 60 80

Fig. 13.6. Isotopic age data from Batt (1997). (a) Apatite and zircon fission-track

ages from the Whataroa, Perth and Havelock Rivers (all located just north

of Franz Josef glacier), plotted against distance southeast of the Alpine Fault.

(b) Data plotted on an expanded scale to show variation in muscovite and biotite

ages from the same area. Similar patterns of variation across the Southern Alps

were reported by Tippett and Kamp (1993) (fission-track ages) and Adams and

Gabites (1985) (K–Ar ages). After Batt and Braun (1999). Reproduced with

permission from Blackwell Publishing.

(60



) Alpine Fault, the involvement of a crustal layer 25 km thick in the orogenic

deformation and convergence rates of 10−12 mm yr

−1

across the plate boundary.

Futhermore, the increase in age observed along the Alpine Fault from younger

ages in the Mount Cook area to older ages towards the southwest (Figure 13.7),

is consistent with varied degrees of convergence and thus exhumation along the

plate boundary. This variation was, in turn, easily understood by considering

the present-day variation in convergence rate predicted from the well-constrained

position of the relative pole of rotation between the Pacific and Australian plates

in the vicinity of the plate boundary (Batt and Braun, 1999). This led to the

202 Thermochronology in active tectonic settings

100 150 200

Apparent age (Myr)

250

Whataroa

Valley

Fox

Glacier

Haast

Pass

A′

Copland

Valley

Distance along Alpine Fault from Fiordland coast (km)

300 350 400

Fig. 13.7. Variation along the Southern Alps in the thermochronological age

of material from the zone of consistent young ages close to the Alpine Fault.

Muscovite K–Ar ages are indicated by dark grey symbols, biotite K–Ar ages by

light grey symbols and zircon fission-track ages by white symbols. Data from

Batt (1997) are indicated by circles, data from Adams and Gabites (1985) by

squares, data from Adams (1981) by triangles, data from Chamberlain et al.

(1995) by a diamond, data from Hawkes (1981) by stars and data from Tippett

and Kamp (1993) by inverted triangles. Note that the indicated data points for

the Tippett and Kamp (1993) study are summaries of a much larger data-set.

After Batt and Braun (1999). Reproduced with permission from Blackwell

Publishing.

conclusion that the onset of deformation, uplift and exhumation of material at

present exposed in the vicinity of the Alpine Fault around Fox Glacier took place

approximately 6 Myr ago, a result in very good agreement with independent

estimates of the onset of plate convergence (Sutherland, 1996; Walcott, 1998).

13.4 Application of the Neighbourhood Algorithm

to Southern Alps data

The results obtained by Batt and Braun (1999) required an extensive search

through parameter space to find a set of parameters that would produce a distri-

bution of ages similar to the observations. As seen in Section 8.5, this search can

be performed in a more automated and systematic manner by using a so-called

inverse method, such as the Neighbourhood Algorithm (NA). Here we will use

the dataset of Batt and Braun (1997) described in the previous section to illustrate

the use of the NA to constrain the geometry of the Alpine Fault as well as the

velocity of convergence between the two plates. We use a simpler model than the

13.4 Application of the NA to Southern Alps data 203

8.0

0102030

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0

0 200 400

Temperature (°C)

Pressure (kbar)

Model age (Myr)

600

010

(c)(b)

(a)

biotite K–Ar

muscovite K–Ar

amphibole K–Ar

apatite fission

track

Distance (km)

Depth (km)

30 40

Fig. 13.8. The P–T history of material exposed at the surface of the model

after 10 Myr. (a) Exhumation trajectories of selected points. The heavy dotted

line corresponds to the approximate location of the retro-shear zone in this

model. This is inferred to correlate with the Alpine Fault zone for the Southern

Alps. (b) P–T histories of the points indicated. The P–T history of material

currently exposed on Craig Peak (Craw et al., 1994) is marked by the bold

boxes and black arrows for comparison. (c) Modelled thermochronological ages

of the selected points across the surface of the model for a range of different

thermochronometers. After Batt and Braun (1999). Reproduced with permission

from Blackwell Publishing.

one described above, with a three-dimensional velocity field imposed to represent

rock movement along a curved thrust fault and used to compute the temperature

field within the crust and the temperature history of particles that travel through

the orogen. The computer code used for these calculations is a modified version

of the Pecube code (see Chapter 7) in which the three components of the advec-

tion term have been included, allowing full representation of a three-dimensional

velocity field.

204 Thermochronology in active tectonic settings

Amphibole K–Ar

Muscovite

K–Ar

Biotite

K–Ar

Apatite fission track

Minimum apparent age exposed after

10-Myr model run (Myr)

6810

con

(mm yr

–1

)

12 14 16

Fig. 13.9. Minimum modelled ages of various isotopic chronometers exposed at

the surface of the model after 10 Myr plotted against the velocity of convergence

in the latter half of the experiment. After Batt and Braun (1999). Reproduced

with permission from Blackwell Publishing.

We first show the results of a single forward model in which the Alpine Fault is

assumed to be a listric fault with a surface dip of approximately 60



, soling out at

approximately 25 km depth; the rate of convergence is assumed to have abruptly

changed from 0.5 to 10km Myr

−1

5 Myr ago and to have remained constant

ever since (Batt and Braun, 1999); the mean, conductive surface heat flow is

67mW m

. We have used a 1-km-resolution DEM to constrain the geometry of the

surface landform. Assuming that rock trajectories are everywhere parallel to the

Alpine Fault leads to the surface velocity distribution shown in Figure 13.10(a).

The final temperature distribution is shown in Figure 13.10(c). The predicted ages

for a set of thermochronometric systems are compared with the data collected

along the Whataroa (Batt et al., 2000) and Godley (Tippett and Kamp, 1993)

rivers in Figure 13.10(b). The ability of the model to produce the core features

of spatial variation in the dataset demonstrates that the tectonic evolution of the

area is relatively simple and consistent with the behaviour expected for a simple

orogenic system.

The surface dip of the Alpine Fault (here, a simple function of the listricity

of the curved fault) and the rate of convergence are also investigated by system-

atically searching the parameter space by the NA to provide the best fit to the

thermochronological data. The listricity, , is a length scale that determines the

curvature and thus surface dip of the fault according to the following expression

for the geometry of the fault:

z = fx =z

1−e

−x/

 (13.1)

13.4 Application of the NA to Southern Alps data 205

DIstance from the Alpine Fault (km)

0 1020304050

Topography (km) and Velocity (km Myr

–1

)

Topography

Horizontal velocity

Vertical velocity

X (km)

0 10203040

Age (Myr)

Predicted K–Ar Biotite ages

Observed K–Ar Biotite ages

Predicted K–Ar Muscovite ages

Observed K–Ar Muscovite ages

Predicted Apatite FT ages

Observed Apatite FT ages

(a)

(c)

(b)

Fig. 13.10. (a) Velocity distributions assumed in order to compute the ther-

mochronological ages shown in (b), where they are compared with real ther-

mochronological data (FT, fission track). (c) The computed three-dimensional

temperature structure and surface topography used in the calculations.

where x is the horizontal distance away from the surface trace of the Alpine Fault

and z

is the soling depth of the fault.

The listricity was allowed to vary between 25 and 50 km, whereas the conver-

gence velocity was bounded between 0 and 15 km Myr

−1

. The solution of the NA

206 Thermochronology in active tectonic settings

5.0 10.0 15.0

Velocity

30.0

35.0

40.0

45.0

50.0

Listricity

070Percentile

Fig. 13.11. Results of a NA search expressed as locations in the parameter space

of the various models tested and their fits. Each dot represents a forward model

run; its position corresponds to the values of the parameters used and its colour

is proportional to the misfit between predicted thermochronological ages and

observations. The cross represents the location of the model run characterised

by the smallest misfit.

search is shown in Figure 13.11 and demonstrates that the thermochronological

dataset contains information that can be used to constrain the values of both

parameters. The best-fit values are approximately 8 km Myr

−1

for the convergence

rate and 35 km for the listricity (which corresponds to a surface dip of 54



for the

Alpine Fault).

Appendix 1

Forward models of fission-track annealing

A1.1 Variable temperature history

Equations (3.15)–(3.18) define isothermal annealing models; they can be applied

to a variable temperature history by approximating this as a series of isothermal

steps. The amount of annealing at the end of each step is calculated using the

‘equivalent-time’ algorithm (Duddy et al., 1988); for the fanning Arrhenius model

(Equation (3.16)):

gr

ab= C

lnt

+t

 +C

1/T

−C

(A1.1.1)

where r

is the reduced track length at the end of the jth isothermal step, t

the duration of the jth step and T

is the temperature during the jth step. t

the equivalent time, which is defined by

lnt

 =−

gr

j−1

ab−C



−1



(A1.1.2)

Equations (3.16) and (A1.1.1) are undefined over certain t−T intervals near r =0

and r = 1 because of the exponents in (A1.1.2). The regions where this occurs

are where

lnt ≤

−aC

−1



−C



−

(A1.1.3)

near r =1 and

lnt ≥

1/b

−aC

−1



−C



−

(A1.1.4)

near r = 0. This problem is circumvented by choosing a time step sufficiently

large to fall outside the range (A1.1.3) and stopping the calculations when t

+t

satisfies (A1.1.4). From there on, r = 0 is adopted.

207