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

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x Preface

for surface uplift of mountain ranges during this timespan and the widespread

nature of the increase in sediment flux suggest the driving mechanism for this

increase to be climatic; thus, the uplift of mountain peaks may be an isostatic

response to, rather than a trigger of, increased denudation rates. Such a mechanism

requires denudation rates to be highly spatially variable: valley bottoms must

erode much more rapidly than mountain peaks (i.e., relief must increase) in order

for isostatic rebound to be effective in uplifting the peaks (e.g, Montgomery, 1994;

Small and Anderson, 1998). The resolution of this debate will thus depend in part

on our ability to constrain temporal and spatial variations in exhumation rates

within eroding mountain belts effectively.

A related question has arisen out of the realisation that tectonics and erosion

are not independently operating processes but must be strongly coupled (e.g.,

Beaumont et al., 1992; Zeitler et al., 2001). In effect, while tectonics controls

erosion rates through the creation of surface relief, erosion in turn also affects

tectonic patterns through its role in displacing mass at the surface, thereby influ-

encing the thermal (and hence rheological) stress and potential-energy fields of

actively eroding regions. The question is that of whether this coupled system is

fundamentally controlled by an internal (tectonics) or external (climatic) driving

force. Authors who have attempted to address this question through the com-

parison of spatial patterns of thermochronological ages with present-day climate

(precipitation) data have come to conflicting conclusions (e.g., Burbank et al.,

2003; Reiners et al., 2003a; Thiede et al., 2004; Wobus et al., 2003), and its

resolution will depend in part on a better comprehension of the significance of

spatial variations in thermochronological ages across orogenic systems.

A third and final example may be drawn from the debate about the relative

timescales of tectonic versus erosional processes and the significance of the con-

cept of topographic steady state. From a theoretical viewpoint, it is relatively

simple to demonstrate that active tectonic systems that are subject to continuous

uplift at a constant rate should tend towards flux steady state, where the tec-

tonic influx of material is balanced by the erosional outflux (e.g., Beaumont

et al., 1999; Willett et al., 2001). However, it is not clear whether such systems

also reach denudational and topographic steady state (where denudation rates

and surface topography, respectively, are constant in time). Thermochronologi-

cal data can be used to test thermal and denudational steady state in mountain

belts (Willett and Brandon, 2002), but attempts to demonstrate the existence of

denudational steady state in natural settings from such data (e.g., Bernet et al.,

2001) have been criticised under the argument that they were inconsistent with

other thermochronological datasets (e.g., Carrapa et al., 2003; Cederbom et al.,

2004). Again, resolution of this question will depend on our understanding of the

significance of spatial and temporal patterns in thermochronological data.

Preface xi

Technological advance has seen the accuracy and precision of thermochrono-

logical techniques rise steadily over the past 15 years. In parallel with this,

understanding of the fundamental behaviour of isotopic chronometers on geolog-

ical timescales has progressed to the point where the thermal sensitivity of key

systems can be constrained on a routine basis. These advances have brought pro-

gressively greater benefit to quantitative interpretation as a way of adding value

to what remain costly analytical techniques.

The purpose of this book is to provide tools to help the Earth scientist undertake

such interrogation of thermochronological data in a rigorous and quantitative

manner, and, in particular, to extract the information such data contain on the

tectonic (i.e., internal deformation) and geomorphic (i.e., surface evolution) history

of a given region.

Although working from generally applicable principles, the techniques we

develop and describe here are targeted most directly at the interpretation of

thermochronological data in environments where (a) tectonic uplift and erosion

combine to bring rocks to the Earth’s surface, where they are collected and anal-

ysed by the geologist and (b) the amplitude of surface relief is important. We

focus in particular on the effect of rock advection and finite-amplitude surface

relief on the temperature structure in the Earth’s crust and consider a range of

situations from the simplest, where tectonic advection is so slow that the heat

is mainly transported by conduction, to the most complex, where rapid vertical

advection of heat through a convoluted, cold upper surface produces a complex

three-dimensional thermal structure that must be considered. We also consider

a range of tectonic settings, ranging from situations where rocks are actively

uplifted by ongoing tectonic processes and erosion counteracts the resulting uplift

of the surface to situations where, in the absence of any tectonic forcing, rock

uplift is solely driven by the isostatic rebound caused by erosional unloading.

The book is built around a progressive increase in complexity both of the tec-

tonic scenarios considered and of the interpretative tools consequently required

to interpret thermochronological data. Along the way, a set of Fortran programs

developed by the authors is progressively introduced, together with an accom-

panying series of tutorials designed to assist the reader in understanding how

to use these tools to interpret the increasingly complex scenarios. The Fortran

programs can be downloaded from the web page; their application will allow the

reader to solve the tutorial problems. More importantly, however, the programs

are designed to be applied to the interpretation of real thermochronological data

in different settings. The Fortran program Pecube, which is presented towards

the end of the book, is a very general solver of the heat-transport equation that

includes conduction, production and advection of heat, as well as the option of

varying the geometry of the surface to represent geomorphic processes. It has

xii Preface

been designed for ease of use by any geologist interested in the interpretation of

thermochronological datasets. In this context, the book can be regarded as a long,

and, we hope, not too painful, introduction to this powerful modelling engine,

providing a numerical ‘toolkit’ for a geoscientist looking to interpret such data.

In designing and writing this book, we are aiming at an audience of upper-level

undergraduate students and graduate students who are commencing their studies.

We have included a set of slides (in PDF format) (see the website) for use during

lecturing and tutorials, if one wishes to use this book as a teaching tool. We assume

that our audience is acquainted with the basic principles of geochronology, as well

as the basics of geodynamics and heat transport in the Earth’s crust. Although we

review these basics in early chapters, readers who are unfamiliar with this material

may wish to consult textbooks developing these issues in more detail, such as

Faure (1986), Turcotte and Schubert (1982) and Fowler (2005). Developments

in the field of thermochronology and, more widely, the study of the interaction

between tectonics and surface processes are rapid; while we have aimed this book

to be up to date as it appears, it is inevitable that we fail to include the most

recent studies, however important their implications may be.

The idea for this book was triggered by a short course given by Jean Braun

during his appointment as visiting professor at the Université Joseph Fourier in

2002. Most authors who have written a textbook will admit that it has taken them

about two to three times as long as they had initially planned. This book is no

exception to the rule and we thank the staff at Cambridge University Press for

their patience and understanding. Writing of the book was facilitated by mutual

visits of the authors; the Research School of Earth Sciences at the Australian

National University and the Observatoire des Sciences de l’Univers de Grenoble

at the Université Joseph Fourier provided logistical support during these stays.

Our thinking on the interpretation of thermochronological datasets was shaped

in part through thought-provoking discussions and exchanges with colleagues,

collaborators and (former) graduate students. We would like to acknowledge in

particular Matthias Bernet, Mark Brandon, Roderick Brown, Jim Dunlap, Kerry

Gallagher, Frédéric Herman, Barry Kohn, Erika Labrin, Ian McDougall, Peter

Reiners, Xavier Robert, Malcolm Sambridge and Michael Summerfield. Many of

the ideas presented in this book were developed during research projects funded

by the Australian Research Council (ARC), the Institute of Advanced Studies of

the Australian National University, the Institut National des Sciences de l’Univers

(INSU) of the Centre National de la Recherche Scientifique (CNRS) in France

and the National Environmental Research Council (NERC) in Britain. Finally but

definitely not least, we would like to thank our partners Myriam, Gianna and

Victoria as well as our families for their support and understanding as we became

more and more submerged in this undertaking.

Introduction

Thermochronology is a technique that permits the extraction of information about

the thermal history of rocks. It is based on the interplay between the accumu-

lation of a daughter product produced through a nuclear decay reaction in the

rock (whether this daughter product be an isotope or some sort of structural

damage to the mineral lattice) and the removal of that daughter product by ther-

mally activated diffusion. Because temperature increases with depth in the Earth’s

lithosphere, this temperature information can be translated into structural infor-

mation – thermochronological data thus contain a record of the depth below the

surface at which rocks resided at a given time. For eroding basement terrains,

where rocks have been brought to the surface from depths of several to several

tens of kilometres, thermochronology is the only technique that will provide such

information and permit one to constrain the timing of rock exhumation towards

the surface quantitatively.

However, the relationships among vertical motions, temperature history and

the accumulation of daughter product (the present-day abundance of which will

provide a thermochronological age) are highly non-linear and depend on many

parameters. To the Earth scientist seeking to constrain geological history, this

presents a dichotomy. On the one hand, such thermochronological datasets have

the potential to offer insight into how parameters that influence the thermal history

of a geological sample vary over time. On the other hand, however, to extract

information on any one aspect of the system requires an explicit understanding of

the interplay of all of the variables. In particular, one will need to worry about how

the accumulation of daughter products is affected by varying temperatures (and

thus understand solid-state diffusion in minerals), but also about how tectonics

and surface processes affect the thermal structure of the lithosphere, in order

to comprehend the links between the thermal and structural frames of reference

(Figure 1.1). As an example, consider the thermochronological dataset that is

plotted in Figure 1.2 as the age of a sample versus the elevation at which it was

2 Introduction

Tectonics

Surface Processes

Exhumation

Cooling

Spontaneous Nuclear Reaction

Solid-State Diffusion

Time

Temperature

Temperature History

Fig. 1.1. Interpretative steps in relating thermochronological ages to structural

history. The measured age is controlled by the accumulation history of daughter

products in the host mineral, which depends on the thermal history. The thermal

history is controlled on the one hand by the exhumation history of the rock, and

on the other hand by the thermal structure of the crust. These two factors are

interdependent: owing to the advective component of heat transport, the thermal

structure depends on the exhumation history, which itself may be guided by

differences in thermal (and therefore rheological) structure.

(U–Th)/He apatite ages

Age (Myr)

40 50 60 70 80 90 100

Elevation (km)

1.6

1.8

2.0

2.2

2.4

Fig. 1.2. Apatite (U–Th)/He ages from the Sierra Nevada, California, plotted

versus the elevation at which the samples were collected. Data are from House

et al. (2001).

1.1 Thermal history 3

collected. The ages are for the apatite (U–Th)/He system, which is sensitive to

temperatures of ∼40–70



C, which are typically encountered at depths of 1–3 km

within the crust. Plotted in this manner, the data are slightly bewildering – they

appear very scattered and no obvious relationships appear. There can be many

reasons for this: the individual samples may be sensitive to varying temperatures

and therefore not record passage of a single isotherm in the crust; exhumation

rates and/or thermal structure may vary laterally; the surface relief may influence

thermal structure and this influence may vary with time. The challenge is to find

out what reason or combination of reasons fundamentally controls the observed

spatial distribution of ages and thereby constrain the geological history of the

region studied.

The purpose of this book is to provide tools to help the Earth scientist undertake

such interrogation of thermochronological data in a rigorous and quantitative

manner, and, in particular, to extract the information it contains on the tectonic

(i.e., internal deformation) and geomorphic (i.e., surface evolution) history of a

given region.

Alternatively, this textbook may be regarded as an exercise in quantitative or

numerical modelling. The problems addressed are relatively well posed and we

can therefore make use of a range of mathematical methods, ranging between two

end-members:

• deriving the exact analytical solution to a simplified form of a general equation, which

must then be regarded as an approximation of the real Earth problem; and

• deriving a numerical solution to the general form of an equation, which might be a

better representation of the Earth problem but generally provides less insight into the

behaviour of the physical system.

Both approaches are developed here, where possible and practical.

1.1 Thermal history: the accumulation of thermochronological age

Rather than reflecting a specific geological event, the isotopic ages of rocks and

minerals are fundamentally determined by their integrated thermal history. Every

isotopic system will behave as an open system at high temperatures, at which

the daughter product is removed by diffusion more rapidly than it is produced by

nuclear decay, and as a closed system at low temperatures, at which removal is so

slow that all daughter product is retained within the host mineral over geological

timescales (Figure 1.3).

The switch from open- to closed-system behaviour is not instantaneous, but

rather takes place over a discrete temperature interval known as the partial-

retention zone. Somewhere within this temperature range lies the closure

temperature, formally defined by Dodson (1973) as the ‘temperature of a

4 Introduction

Closed System Open System

decay

Parent

Daughter

Fig. 1.3. Hourglass analogies to open- and closed-system behaviours. In a closed

system (left), the daughter product accumulated by nuclear decay of a parent

isotope is completely retained within the host mineral; an isotopic age builds up

linearly with time. In an open system, the daughter product is removed effectively

immediately upon its production; no isotopic age is built up within the system.

thermochronological system at the time corresponding to its apparent age’.

The closure temperature provides the simplest conceptual entity to which a

thermochronological age can be related; from its definition, however, it is a

mathematical rather than a physical concept. Closure temperature will vary not

only among different thermochronological systems, but also within any single

system as a function of grain size, chemical composition and cooling rate.

Moreover, it is strictly applicable only in the simplifying case of monotonic

cooling. For a more general solution, or to deal with more complicated histories,

the equations describing the accumulation and loss of daughter product need to

be solved numerically. In Chapter 2, we review the basics of solid-state diffusion,

derive an analytical mathematical description of the closure temperature that

illustrates the dependence on cooling rate, and present a numerical method to

resolve the isotopic age equation for systems that suffer diffusive loss of the

daughter product.

The effective closure temperature of any isotopic system will determine what

type of process can be addressed by its study: the time and length scales of

constraint will scale directly with the closure temperature so that, in any region,

higher-temperature systems will constrain deeper, larger-scale processes than will

lower-temperature systems. Within the context of this book, we focus on low-to-

intermediate-temperature systems, with closure temperatures ≤350–400



C, that

record the exhumation history of rocks from upper to mid-crustal levels. Three

systems will be considered in detail:

• Ar-based dating methods, utilizing the decay of

Kto

Ar;

• the (U–Th)/He system that exploits the -decay of

235238

U and

232

Th; and

• fission-track thermochronology, which is based on the analysis of damage zones in the

mineral lattice that result from fission of

238

1.1 Thermal history 5

Table 1.1. Estimated closure temperatures for some commonly used

thermochronometers

Method Mineral

Closure

Temperature



C Reference

K–Ar Hornblende 500 ±50 Harrison (1981)

K–Ar Muscovite 350 ±50 Hames and Bowring (1994)

K–Ar Biotite 300 ±50 Harrison et al. (1985)

K–Ar K-feldspar 150 −350 Lovera et al. (1989)

(U–Th)/He Zircon 200−230 Reiners et al. (2002)

(U–Th)/He Titanite 150 −200 Reiners and Farley (1999)

(U–Th)/He Apatite 75±5 Wolf et al. (1998)

Fission track Zircon 240±20 Brandon et al. (1998)

Fission track Titanite 265 −310 Coyle and Wagner (1998)

Fission track Apatite 110±10 Gleadow and Duddy (1981)

The theoretical and practical bases of these methods are treated in Chapter 3,

together with the currently available constraints on their thermal sensitivity. First-

order estimates of the closure temperatures for these systems in various minerals

are presented in Table 1.1.

There are three possible ways to retrieve information on cooling paths from

thermochronological data. The kinetics of

Ar–

Ar diffusion in feldspar and

fission-track annealing in apatite are sufficiently well understood that inverse

numerical modelling techniques can give direct constraints on the thermal

history from a single sample for these systems. These numerical models will

be examined in Chapter 3. Other isotopic dating systems lack sufficiently

detailed characterisation to allow such modelling, although progress is being

made for the zircon fission-track system as well as for the (U–Th)/He system in

apatite.

Where the simplifying assumptions of monotonic and relatively rapid cooling

are justified, thermochronological data can be interpreted through the elementary

closure-temperature model, as discussed in Section 2.4. In these cases, a thermal

history can be retrieved by combining the ages obtained by applying a range of

thermochronological systems (with correspondingly varied closure temperatures)

to a single sample. Further tectonic and structural insight can also be derived

by comparing the ages of multiple samples in close, well-constrained geographic

proximity to each other. This approach is commonly applied along local sam-

ple transects spanning a large range in altitude (often referred to as ‘vertical

transects’), and serves to assess at what times samples that are now at different

elevations cooled through the closure temperature of the system being studied. As

6 Introduction

Apatite fission-track age (Ma)

1000

2000

3000

Elevation (m)

Sim

plon

Monte Rosa

otthard

Ticino

ergell

Age (Ma)

100

300

400

500

200

Temperature (

°C)

Rb–Sr (muscovite)

K–Ar (muscovite)

K–Ar (biotite)

fission track (zircon)

fission track (apatite)

Sample KAW 1884

Ticino, Switzerland

12.4 ± 1.6 µm

0 5 10 15 20 25 30

10 16 µm

0246810121416

Fig. 1.4. An example of the use of different strategies to obtain time–temperature

history data. (a) Multiple-method thermochronology performed on a single sam-

ple from the Ticino region, European Alps. Ages (with errors) are plotted against

the best estimates of closure temperature for various thermochronometers. Also

shown is the apatite fission-track length distribution of the sample. (b) Apatite

fission-track age plotted versus elevation for a series of profiles from different

locations in the European Alps. Modified from Hurford (1991). Reproduced with

permission from Springer-Verlag.

an example, Figure 1.4 (after Hurford (1991)) shows a time–temperature (cooling)

path obtained using multiple thermochronometers on a single sample from the

European Alps, as well as results from various apatite fission-track age–elevation

profiles in the European Alps. Note that these approaches are strictly valid only

1.2 Cooling, denudation and uplift paths 7

when thermochronological ages can be interpreted as reflecting the time at which

the sample passed through its closure temperature (i.e. monotonic cooling). Where

samples have experienced more complex thermal histories, numerical diffusion

or annealing modelling approaches are required when one wants to assess the

measured ages quantitatively.

1.2 Cooling, denudation and uplift paths

Thermochronology provides information only on the thermal history of rocks,

which, among other processes, depends on the rate at which rocks are brought

towards the surface by exhumation. Fully extracting the tectono-geomorphic his-

tory, i.e., the rate at which rocks are exhumed to the surface by the combined

processes of tectonic rock uplift and surface erosion, from the time–temperature

results obtained using multiple methods requires an understanding of the inter-

play among the various physical processes that determine the thermal structure

of the lithosphere. These processes are, in turn, related to the various ways

by which heat is transported and produced in the crust, namely by conduction,

tectonic advection and the decay of radioactive elements. This aspect of inter-

preting thermochronological data forms the core of this book and is treated in

Chapters 4 and 5.

The altitudinal profiling technique has the advantage that no a-priori knowledge

of the thermal structure of the crust is required, because the ages of different

samples relate to the time of their passage through the same isotherm or thermal

interval. However, because such ‘vertical’ profiles in nature are never truly ver-

tical (except in the case of samples retrieved from drillholes, e.g., Brown et al.

(2002)), the steepness of the observed age–elevation gradient will be influenced

by topographic controls on near-surface thermal structure. Geothermal gradients

are also influenced by the rate at which rocks are advected towards the surface,

which again may vary regionally. Together with introducing further complexity

into the basic interpretation of thermochronological ages, this phenomenon also

provides an opportunity to constrain the relief development of a study area, as

discussed in Chapters 6 and 7.

As illustrated by Figure 1.4, either the observed thermochronological age can

increase more or less linearly with elevation or there can be a distinct break in

slope in the age–elevation relationship. Such breaks in slope result from the non-

linear increase of annealing or diffusion rates with temperature and reflect the

exhumation of partially reset samples. In the limiting case of a stable crustal block

(no denudation), a characteristic age profile will develop, whereby rocks situated

above the partial-retention zone (PRZ) retain information about an earlier tectonic

episode during which they traversed their closure temperature. Their age provides