Geoff Laslett (1949-2010)

The fission track community will be very sad to learn of the untimely death of Geoff Laslett from cancer in January 2010. To many of us he was a highly respected colleague and good friend. His contribution to fission track analysis was immense.

Practically everyone working in this field will be routinely using ideas, methods and practices that derive directly from him.

Geoff was born in Mount Gambier, South Australia in 1949, and attended numerous schools around the state, as a result of his father’s work as a Methodist Minister in various rural parishes. He graduated from the University of Adelaide with first class honours and after completing his PhD in Statistics at the Australian National University, Geoff joined CSIRO (the Australian government scientific research organisation) in the division of Mathematics and Statistics, where he worked for most of his professional career. Below we give some personal appreciations of Geoff's work in fission track analysis.

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Paul Green writes:

Geoff’s involvement with fission track analysis began in the early 1980s, when he began a collaboration with geologists (Ian Duddy, Andy Gleadow, Paul Green and others) at the University of Melbourne. The way this collaboration came about is a prime example of serendipity in Science as well as typifying Geoff’s character. He was working at CSIRO on various applications of statistical modelling to problems in Earth Science, and came to the conclusion that there were a large number of problems that required detailed numerical analysis. So he rang the Earth Sciences Department to ask to be put on the distribution list for new publications. At this stage, fate intervened in a big way, because when he rang, the departmental secretary was out, and the phone was answered by none other than the Professor, John Lovering! John, being the character he was, asked Geoff who he was and why he was interested, and immediately saw an opportunity to take the still nascent field of fission track analysis to a new level.

At this early stage, the Fission Track Research Group had just revealed the classic curves of fission track age and length reduction with depth and temperature in the Otway Basin and realised that a quantitative understanding of the underlying processes would take the technique into a whole new world of thermal history analysis, which is where we find ourselves today. Geoff provided that quantitative understanding, and without his brilliant and incisive input, who knows where we would be now.

So John invited Geoff to visit the group and see if there was any common ground for collaboration. Indeed there was, and the results were brilliant. Early work included, among many other things, development of the concept of “length bias” and a method for dealing with it, a pioneering experimental study of fission track annealing and the associated statistical modelling, and a methodology for extending laboratory-derived models to geological field studies. The results of the annealing work were published in four joint papers in the late 1980s, along with the development of many practical procedures that are now used by laboratories all over the world.

One of the products of this work was the well known “Laslett et al (1987)” kinetic model for fission track annealing in apatite. This model tends to be regarded as flawed by many “fission trackers”, and the common view is that it produces a late stage cooling artefact. But since the model has remained in common use for over 20 years in the popular “MonteTrax” software package, it must have something going for it! We believe that this model is staggeringly good, particularly when you consider the nature of the data on which it is based. Geoff managed to fit a model to the laboratory annealing data which provides a more than passable description of the annealing behaviour in geological conditions, after extrapolation over up to 10 orders of magnitude in time! That is something that anyone could be proud of.

For further demonstration of the power of the model, take a look at Figure 13 of the recent paper by Spiegel et al. in Geochimica et Cosmochimica Acta (v. 71, 2007, 4512-4537). Mean track lengths predicted using Geoff’s model for apatites separated from ODP cores, based on modelled thermal histories, are within 1 micron and in most cases within 0.5 micron of the measured values. That is an astounding result, when you consider that the model is based solely on data from laboratory annealing experiments, mostly for durations of hours to days. In his subsequent work, Geoff moved on to improved formulations and new models. We can only speculate how these newer models might compare with the Laslett et al. (1987) model, as they have never been implemented. In this context, it is particularly sad that instead, a butchered version of Geoff’s model, renormalised to a lower L_o, has been implemented, for misguided reasons. (For the uninitiated, renormalising the model in this way changes the nature of the model at high temperatures as well — an unjustified and unwarranted result.) This type of senseless move represents everything that Geoff stood against.

The “Laslett et al (1987)” kinetic model illustrates many of Geoff’s qualities as a mathematician and a scientist. Arrhenius plots to describe annealing kinetics had been around for a long time, but usually based on a very small number of measurements of track density (or fission track age) over a limited range of temperature and time. No-one can remember now exactly who first thought of performing annealing experiments based on confined track lengths, in a monocompositional apatite, but if it wasn’t Geoff then the idea certainly came out of discussions in which he was involved. The ability to measure mean confined track length with greater precision, and elimination of possible compositional influences, were huge steps in taking the science of fission track analysis into a new quantitative realm. Then, to develop a quantitative description, Geoff began with the simplest form of model — the parallel Arrhenius plot. One of Geoff’s prime aims in this sort of work was not to make a model more complicated than it needs to be. He drilled the group with the philosophy that in estimating parameters, the fewer you need to estimate, the better the degree of precision that would be achieved. If the parallel Arrhenius plot model provided a good fit to the data, subsequent extrapolation to geological histories would be that much simpler. As we all now know, the parallel Arrhenius plot didn’t quite fit, and it quickly became clear that the slope of the “iso-annealing contours” increases with the degree of annealing. So Geoff then investigated the next simplest form, the “fanning Arrhenious plot” in which the contours extrapolate to a common point at 1/T_o=0. But he didn’t just assume this to be the case. Instead he tested this hypothesis and showed that the data were compatible with this model. Again, this makes the subsequent mathematics of extrapolation that much simpler. In subsequent work, this model was generalised to embody a finite value of 1/T_o.

One result of all the exciting research that Geoff was involved in during the 1980’s was the establishment of Geotrack International, first as a commercial arm of the Fission Track Research Group and from 1987 as a private company. Provision of thermal history analyses to the oil exploration industry was only possible because of the quantitative treatment of the apatite fission track “system” to which Geoff contributed so much. Geoff continued to work with Geotrack on a commercial basis, and played a central role in development our multi-compositional annealing model. He also developed methods for extracting thermal history solutions from AFTA data, based on this model, and these remain in routine use today.

In another amazing example of Geoff’s qualities, he decided that in order to do justice to the problems involved, he needed a fuller appreciation of the nature of the data. So he joined Geotrack on a part time basis as a microscopist (1989 to 1993), counting and measuring tracks and generating the raw AFTA data to which his algorithms would be applied. Geoff’s data was of the highest quality, and he set exacting standards for quality control that “raised the bar” for everyone. During this time, Geoff was a member of the Geotrack cricket team that won the PESA cricket tournament in consecutive years. Geoff loved sport, particularly cricket and Aussie Rules. Rex and I were amazed one day at the MCG when Geoff demonstrably “came out” as an Adelaide Crows fan, when they won a place in the 1996 AFL Grand Final!

Sadly, changes in the CSIRO management structure made it more difficult for Geoff to collaborate with Geotrack and his energies were diverted into other areas, where he again made significant impact, for example in studying Bluefin Tuna stocks in the Southern Ocean. At Geotrack, we keenly anticipated Geoff’s retirement from CSIRO, so that we could again utilise his imaginative and innovative flair for problem solving that played so much of a role in our development. Sadly, Geoff’s health problems meant that this was not to be, and although at one stage we installed Geoff in an office with all his papers and books around him, soon afterwards a new illness meant that he devoted his energies to seeking remedies and spending time with his family.

Geoff will be sadly missed by us all, but his legacy will stay with us forever.

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Rex Galbraith writes:

I first met Geoff at the 1984 Fission Track Workshop in Troy, New York. We were two (mild) applied statisticians amongst many (seemingly aggressive) geologists and geochronologists, and we formed a permanent alliance.

Geoff's contributions to fission track analysis range from the fundamental theoretical to the highly practical. He formulated the “line segment model” as a mathematical and statistical description of latent fission tracks inside a crystal. This is a stochastic process of line segments in three dimensional space, having varying lengths and random orientations. This model provides a means of understanding many important aspects of fission track measurements as well as a basis for quantitative analysis. It shows, at a detailed quantitative level, how fission track dating works and what inherent variation is to be expected in track counts. It reveals the relevance and role of the mean track length to the density of tracks intersecting a plane surface, and it shows how this parameter is generalised to the equivalent isotropic length when the probability distribution of track lengths depends on orientation, as is the case when tracks have been heated.

From the line segment model Geoff was able to derive theoretical probability distributions for various quantities relating to tracks intersecting a plane, including projected lengths and angles. This revealed the presence of a number of geometrical and observational biases in fission track observations — particularly length bias, but also orientation bias due to anisotropic chemical etching, and other more subtle effects relating to fracture thickness when measuring tincles and proximity of confined tracks to the crystal surface.

Geoff did not stop at simply deriving formulae. He knew how to understand formulae through analysis and computation. It was typical of him to start with a simple case — for example, when all tracks have the same length, or when the length distribution is isotropic — before analysing the more complicated anisotropic case. This both aids understanding and helps to avoid mistakes. He also learnt how to etch, count and measure fission tracks, which gave him a unique grasp of this very confusing subject.

Theory derived from the line segment model can be used formally in modelling and simulation studies. But it also provides an appreciation of what features are relevant and important in practice. A characteristic (but far from unique) example was the idea of measuring confined track lengths just for horizontal tracks. In theory, when measuring the length of an etched confined track, the sampling bias factor is proportional to the volume of the region in which the track's associated uranium atom can lie. For horizontal tracks this volume is proportional to the track's length, whereas for non-horizontal tracks it is more complicated. Measuring just horizontal tracks means that a simple length-bias correction is sufficient. Again, it was not enough to think of this. It was necessary to verify the theory with empirical data and to develop a practical procedure, including criteria for judging whether a track is horizontal. Furthermore this measurement is simple, requiring just the identification of the (x,y) coordinates of each end of the etched track.

The above explanation illustrates another aspect of Geoff's insight. He always thought about the underlying uranium atoms (in three dimensional space) that had fissioned and that might have produced a particular measurement. We tend to think in terms of the tracks themselves, but more clarity can often be gained by thinking about the uranium atoms that produced them.

Stochastic geometry (or geometrical probability) is a subject notorious for the ease with which fallacious arguments and incorrect formulae can be produced. Not only is it easy to make a mistake but also it is often hard to see even that a mistake has been made. This can happen especially when looking at one and two dimensional measurements that were generated by a three dimensional process. Geoff exposed several such fallacies, usually (but not always) before they were published.

One of our projects, largely led by Geoff, was a study of what information can be obtained from projected semi-track lengths (i.e., the projections onto the plane of observation of tracks that intersect that plane). Theory suggests that both the lengths and the angles to the c-axis of these projections are in principle informative, so it was of interest to quantify this. It transpired that such information is limited in practice — for example, much more limited than measurements of horizontal confined tracks — but this study did identify situations where projected semi-track measurements could be useful. The associated estimation problems were difficult because of the complex relation between the underlying length distribution of latent tracks in the crystal and that of semi-tracks projected onto a plane. The study used parametric modelling and maximum likelihood fitting of both simulated data and real data from designed laboratory experiments. A further practical complication with the laboratory data was the difficulty in measuring very short projected lengths reliably. We dealt with this by using just lengths greater than a prescribed minimum value (one or two microns) and adjusting the theoretical distribution accordingly. This made the likelihood function yet more complicated but it needed to be done. A further message that came out of this study was that ad hoc estimation methods (for example, based on the sample means and variances) were not good enough here, even for samples of 500 or 1000 tracks, and more efficient methods based on the likelihood function were necessary.

The projected lengths study illustrates other features of Geoff's work. He was able to perform calculations that most others would find formidable, and which needed considerable ingenuity in working out how to program efficiently. He was very concerned about the reliability of the statistical methods he used and his computer programs, and he went to great lengths to verify both. In the projected lengths study he obtained precisions of the parameter estimates in two different ways (by computing profile log-likelihood functions and by fitting a quadratic surface to the log-likelihood function itself) in order to ensure they were reliable. He usually avoided using easier methods based on the Hessian or information matrix, particularly for models with non-linear parameters or more parameters than could be supported by the data.

Geoff will be well remembered for his annealing models — mathematical descriptions of how fission track lengths depend on temperature and time. He formulated a class of models in which contours of equal mean length reduction, and hence equal mean length, were straight lines on an Arrhenius plot (log time against reciprocal absolute temperature); and he fitted such models to experimental data obtained from laboratory experiments in which apatite samples were heated at varying temperatures for varying times. These models, and modifications of them, have been highly successful both directly in providing predictions and indirectly in providing a quantitative understanding of the annealing process. They are now in routine use as a research tool in fission track analysis.

Paul has alluded to the “Laslett et al 1987” model, which has been very successful and has also come under criticism. Geoff regarded his main achievement here as determining the form of the model, as well as providing a quantitative understanding of the annealing process

— indeed, he regarded the function and purpose of laboratory annealing experiments as doing just this — rather than as making direct (un-calibrated) predictions. As an experienced statistician and scientist, Geoff was well aware of the differences between laboratory and field data and the types of inferences that can be made from each. So he was somewhat bemused at the way that particular model was sometimes applied in practice.

The nature of the relation between track length, temperature and time is highly non-linear. For tracks held at a fixed temperature, the rate of track shortening decreases with time and becomes very slow; while for tracks heated for a fixed time, the rate of shortening increases with temperature and becomes very rapid. Contour lines on the Arrhenius plot (for equal changes in mean length) thus become very close together as temperature increases. This has implications both for fitting models and for designing annealing experiments. Geoff wrote some very instructive notes on designing annealing experiments, which are too detailed to include here.

Geoff found fairly strong empirical evidence that the slopes of the contour lines increased with the amount of annealing, and modelled the lines as all "fanning" out from a distant single point (located at an extremely low log time and reciprocal temperature). But he also gained much insight by first considering the case where the contour lines are parallel — effectively fanning from minus infinity. This lead to a more tractable model form, both mathematically and conceptually, which allowed much simpler analysis — especially when using the models in thermal history calculations, where temperature changes over time.

These annealing models, and their use, did not come easily but were the result of much effort and empirical study. The actual fitting of them to data was also far from trivial and required a high level of expertise in statistical modelling. Non-statisticians are probably not aware of some of the computational difficulties and pitfalls that can arise. His original models were applied to “reductions” of horizontal confined track lengths (i.e., mean annealed track length divided by mean unannealed track length) which were transformed by two Box-Cox power transformations. This was done partly for statistical reasons (to achieve approximately constant error standard deviations) but mainly because it provided a formal means of assessing the adequacy or otherwise of some earlier models — and indeed it showed that those were deficient. The parallel and fanning Arrhenius models were a great advance, but Geoff was still keen to improve the methodology and to rectify some technical deficiencies. Firstly, the practice of dividing all mean lengths by the measured mean length for an unannealed control sample, while expedient and certainly a reasonable initial approach, has both logical and statistical difficulties. There were good reasons, too, for avoiding Box-Cox transformations and for using a more realistic error model. Another aspect was how to deal with experiments in which no confined tracks are observed (because of the high level of annealing) so no confined length measurements are available. It is clearly incorrect to assume that the underlying mean length is zero, but it is also wrong to ignore such experiments. I worked on these aspects with him and we published an improved method in 1996.

In relation to Paul’s comment above concerning L_o, it is worth noting that one feature of the new approach was the introduction of a parameter m_max that was estimated from data along with the other parameters. This parameter did not represent a real physical quantity but was necessary to define the model, in much the same way as often arises with the intercept parameter in a linear regression model. Geoff was worried (rightly as it tuned out) that

m_max might be mis-interpreted as the “real” mean length of unannealed tracks. In fact it allowed the same form of model to produce excellent descriptions of all annealing data (based on confined track lengths) available at that time. I suspect that this approach would be a good starting point for future development of annealing models.

As the above examples illustrate, Geoff was concerned about using appropriate statistical methods. He was particularly concerned that his work should stand up to scrutiny and criticism from the statistical community as well as being scientifically correct. He had a high reputation as applied statistician as well as a substantive scientist.

With colleagues at Melbourne University and Geotrack International, Geoff developed a method for identifying features of thermal histories undergone by a geological field sample, and estimating key parameters. The annealing models needed to be extended in several ways because tracks form continuously over time, temperature varies over time and field data are affected by differing chemical compositions and other natural processes not present in laboratory experiments. Much of this work is computationally complex and some of it is unpublished, but it has been used successfully, and is still used, in commercial applications. My own role in this was largely to act as a “sounding board” for Geoff to discuss his ideas and methods. Needless to say, they were sound! But it was characteristic of him to try to avoid fallacy and pitfall by all means.

In general, he subjected his own work to close scrutiny before showing it to others, and he applied rigorous and time consuming tests to his computer programs. I'm sure this is one reason why his work has lasted and will continue to last. In our joint work he insisted that we both derived the mathematical results independently, and checked each other’s numerical results also, where feasible.

I also worked with Geoff in developing statistical models for mixed fission track ages —particularly the “minimum age” models for estimating the timing of uplift and erosion in certain types of thermal history, or the age of the youngest population in a mixture. The basic idea was simple enough, but there were computational problems (and some distributional ones too) that needed to be solved, particularly when spontaneous track counts were relatively small. Largely at Geoff's insistence we undertook an extensive simulation study of the estimation procedure before advocating it to practitioners. Data were simulated for both 3- and 4-parameter models, with a good practical range of parameter values and sample sizes; and the quality of both the estimates and their precisions was assessed when fitting right and the wrong models — particularly, in the latter case, when fitting the 3-parameter model to data simulated from the 4-parameter model. The minimum age models have also been adapted and applied successfully in the field of optical dating, which has allowed correct dates to be obtained from partially bleached samples — previously a stumbling block in luminescence dating applications. So our rigorous study of them has proved to be more widely beneficial.

A few years ago I was invited by Chapman and Hall to write a book on the mathematical and statistical theory underlying fission track analysis for their Interdisciplinary Statistics series. I asked Geoff to join me as a co-author and we did a considerable amount of work planning the book and drafting the early chapters. To my regret, Geoff's work commitments prevented him from continuing and he encouraged me to complete it without him. The resulting book is not as extensive as we had planned, particularly in the areas of annealing models and thermal history estimation, but I hope it well reflects part at least of his contribution to this subject.

Geoff made major contributions too in other areas of science, including the theory of measurement, environmental monitoring, soil science and marine science.

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Geoff passed away on the 9^th of January 2010. We all miss him not only as a colleague, but also as a mate, whose company everyone enjoyed and whose wit and amazing knowledge of so many subjects would always enliven the conversation. It is unlikely that we will enjoy the opportunity to work with anyone of Geoff’s abilities in the future, and not only the fission track community but many others are the worse for his passing. Our deepest sympathies go to Kay, Katia and Mikki.

Paul Green and Rex Galbraith, May 2010.