Module 7 - Statistical models used to calculate luminescence ages

To calculate a luminescence age, an estimate of the absorbed radiation dose of the sample (aka the equivalent dose, or D_e), must be calculated from a distribution of D_e values measured from individual grains or multi-grain aliquots from the sample. The most straightforward samples to date are those where i) all grains have been exposed to sufficient sunlight to completely deplete their luminescence signal prior to burial, and ii) the sample site contains a homogeneous dose rate field with minimal grain-to-grain variations in dose rate.

Such samples often include sediments that have been transported and deposited by wind (i.e., dune sands). In these cases, the D_e distribution is typically tightly clustered around a single weighted mean value and the measured overdispersion is fairly low (Fig. 1).

Many samples, however, include grains that contain a residual, unbleached signal, and their D_e distributions exhibit a much more broad, or skewed age distribution (Fig. 2). They may even have a distribution that exhibits multiple modes. In these latter cases, a calculated weighted mean or average D_e value may severely overestimate the true D_e (and hence, age) of the deposit. It’s these scenarios that require a more sophisticated approach to D_e calculation.

Statistical models and approaches have been developed to address challenges with calculating the sample D_e from samples with complex depositional or bleaching histories. Below, we discuss four models commonly used today.

The Central Age Model

The central age model (CAM), also known as the central dose model, assumes that the sample has been completely reset by sunlight prior to burial. This model is often applied to wind-blown sediments that are well bleached prior to deposition.

If this model is applied to samples of several hundred years in age or older, it is typically applied to log-transformed D_e values to take into account the fact that the D_e standard errors typically increase with the size of the D_e estimates. Take for example the Colorado River sample plotted in Figure 2A above. The empirical distribution function superimposed on the KDE curve allows visualization of individual D_e values (points) and their errors (error bars on the points). You’ll notice that the width of the error bars increases with D_e value. Such a trend is commonly negligible or absent in very young or modern samples, and in these cases, CAM is applied to unlogged data.

The CAM yields two parameters:

1. the central dose (δ), which is essentially the geometric mean of the D_e distribution (in Gy), and

2. and the overdispersion (σ), or the relative standard deviation of this distribution (in percent or Gy).

The overdispersion parameter (OD or σ) represents the “spread” in the D_e distribution that remains after all measurement errors have been taken into account. When OD is high, we may infer that: i) some grains have been insufficiently bleached prior to deposition, ii) there has been mixing between sedimentary layers of different ages, and/or iii) there are grain-to-grain variations in environmental dose rate not taken into account in our dose rate measurements.

Example - Dating the Pine Barrens Dunes

The CAM was used to calculate ages for sands collected from the Pine Barrens Dunes in New Jersey, USA (Wolfe et al., 2023) (Fig. 3). These sands were deposited by wind near the end of the Last Glacial Maximum in North America, just as the Laurentide Ice Sheet began retreating north.

Wind-transport of grains increased their sun-exposure so that any prior luminescence signal stored in them was largely depleted prior to their deposition in the dunes. D_e distributions from these samples are unimodal and symmetrical with low OD values ranging between 7 and 18% (Table 1, Fig. 4). For this reason, the final D_e value of these samples was calculated using the CAM.

Table 1. Luminescence OD values and ages for Pine Barrens Dune samples (Wolfe et al., 2023).

The Minimum Age Model

The minimum age model (MAM), also known as the minimum dose model, is applied to samples thought to have been partially bleached prior to deposition (such as in a river). These samples commonly show a truncated De distribution with a positive skew, where a tail of higher De values record residual, unbleached signal (Fig. 5).

The MAM assumes that the measured (log-transformed) D_e values are drawn from a truncated normal distribution, where the lower truncation point (γ) corresponds to the average true log D_e of the population of fully bleached grains. Just as is the case with the CAM, the MAM may be applied to unlogged data if the sample is young, or shows no correlation between D_e values and their errors.

The MAM yields a series of parameters:

1. the minimum dose (γ) in Gy, which corresponds to the minimum age of the deposit,

2. the proportion (p) of grains/aliquots in the D_e distribution that comprise the minimum age,

3. the mean (µ) and standard deviation (σ) of the truncated normal distribution from which the log-transformed D_e values are assumed to be derived.

A limitation of the MAM is that it can be sensitive to low outliers in the D_e distribution leading to age underestimates (e.g., Firla et al., 2024). Consequently, D_e distributions should be screened for such low outliers.

In samples that show minimal evidence of partial bleaching (i.e., low OD and symmetrical D_e distributions), the calculated CAM D_e will be close to the MAM D_e. Where partial bleaching is evident, however, the CAM D_e will be higher than the MAM D_e, and will likely overestimate the true age of the deposit (Fig. 5).

Example - Landscape dynamics along Rangitikei River

Luminescence ages from feldspar minerals in river terraces were used to reconstruct incision rates of the Rangitikei River, New Zealand (Bonnet et al., 2019) (Fig. 6). Twenty samples collected from a range of terrace elevations along a ~20 km reach of the river revealed a non-steady pattern of incision characterized by an initial fast incision phase commencing ~11,000 years ago followed by a much slower incision period.

Samples were collected from sandy sediments encased in gravels (e.g., sample RO_04) deposited during higher energy flow regimes, as well as silts and sands (e.g., RO_10) deposited in lower energy conditions (Fig. 7). Sample D_e values were modeled using a modified version of the MAM, called the bootstrapped Minimum Age Model (BS MAM), that incorporates uncertainty from age-model parameters and plausible variation in the input data (Fig. 8).

Bonnet et al. observed an interesting correlation between the number of unbleached grains in a sample and river incision rate. High incision rates were associated with samples with the highest number of unbleached grains, suggesting that rapid incision was accompanied by a large input of unbleached material and limited bleaching opportunities for sediment during river transport. These results show how luminescence data not only provides information on sample age, but also earth surface processes that inform our understanding of landscape dynamics.

The Average Dose Model

The average dose model, or ADM, is applied to samples that: i) are thought to be fully bleached prior to burial, and ii) are affected by a heterogeneous dose rate environment at the grain-scale (Guérin et al., 2017). These heterogeneities are typically associated with "hot" spots generated by K-rich feldspar grains or heavy minerals and are likely to affect all sediment samples to some degree.

The ADM yields 2 parameters:

1. the average dose (𝚫) in Gy of the sample, and

2. the relative standard deviation of single-grain dose rates (σ_d).

D_e distributions exhibiting low OD and symmetrical KDE curves will yield CAM D_e values that are similar to, or statistically in line with ADM D_e values. On a positively skewed D_e distribution with high OD however, the calculated ADM D_e will always be higher than the CAM D_e (Fig. 9).

Dose rate modeling suggests that hot spots from K-feldspars or heavy minerals should lead to log-normal dose rate distributions to single grains and hence, some positive skew in D_e distributions (Mayya et al., 2006). However, current technology only allows estimation of an average dose rate (D_r) for all grains of a sample, rather than the calculation of dose rates specific to each individual grain. For this reason, the developers of the ADM advocate its use over the geometric mean calculated by the CAM for well-bleached deposits, as the CAM is likely to underestimate the true age of the sample when the age is calculated using a sample-averaged dose rate.

Example - Dating partially bleached glacial sediments

Glacial sediments deposited by the last British-Irish Ice Sheet during the last Ice Age were dated using single grains of both quartz and feldspar (Smedley et al. 2019). These sediments were deposited by glacial meltwater streams emanating from the ice sheet during its retreat. Such sediments commonly exhibit high overdispersion and evidence of incomplete sun-exposure of grains during transport (Thrasher et al., 2009). Also, luminescence signals from feldspar grains are typically expected to bleach at slower rates than those from quartz grains (Godfrey-Smith et al., 1988). Smedley et al. examined both quartz and feldspar D_e distributions and compared their ages with independent age control. Samples that showed evidence of partial bleaching were analyzed using the MAM, while those that appeared to be well-bleached were analyzed using the ADM.

Smedley et al.’s results demonstrated that the accuracy and precision of their ages from feldspar were similar to those from quartz when the appropriate statistical model was applied. However, feldspars were 5–18 times more efficient than quartz at determining the population of interest for age calculation as a larger proportion of feldspar grains emitted a detectable luminescence signal in comparison to quartz (Fig. 10). These findings contradicted common perceptions, and showed that feldspar luminescence ages can provide efficient and accurate chronologies in environments where partial bleaching is a problem.

Example - Comparing the CAM and ADM

There is still debate concerning the applicability of various statistical models, even for deposits thought to have been fully bleached prior to deposition. In cases where there is uncertainty as to which model is most appropriate, ages are often calculated using multiple models and a comparison is made. Two such studies from archaeological sites in South Africa compare ages calculated using the CAM and ADM models (Table 2). The results show that all but two samples (UBB10 and LOV5) overlap within error at 1σ.

Table 2. Comparisons between CAM and ADM ages from two archaeological sites in South Africa.

The Finite Mixture Model

The Finite Mixture Model (or FMM) is applied to samples with mixtures of grains derived from two or more sedimentary units of different ages (Roberts et al., 2000). Sediment mixing can occur during soil forming processes, bioturbation, translocation of finer grained material into coarser grained deposits, or human activities such as trampling, ploughing or digging. In these situations, the FMM is used to distinguish between individual D_e components in the distribution. The absolute number of components present in a D_e distribution is not always obvious, but can be inferred using statistical tests in conjunction with field observations at the sample site (e.g., Neudorf et al., 2014; Gliganic et al., 2015).

Application of the FMM to a D_e distribution is an iterative process that determines:

1. the number (k) of discrete De components in a distribution,

2. the relative proportions of grains (π) in each component,

3. and the mean (µ) and standard deviation (σ) of each component.

The FMM assumes that the log D_e estimates of individual grains are randomly sampled from a mixture of k dose populations that are normally distributed with a common standard deviation σ, but different proportions and mean log doses (Galbraith and Roberts, 2012; Peng et al., 2023). Sometimes the FMM will identify populations of grains that are not age related, but rather a factor of their distinct luminescence properties. In these cases, the FMM is useful for understanding the structure of D_e distribution data.

Limitations of the FMM include i) the fact that the overdispersion of all detected components is assumed to be the same, ii) the dose rate of all sedimentary units, prior to mixing, are assumed to be the same as the measured dose rate of the sample used for age calculation, iii) the model does not account for the possibility that mixing may have occurred long after deposition of the original sedimentary units, and iv) the true age of the deposit may not be accurately determined if serious post-depositional mixing has occurred (Guérin et al., 2017; Peng et al., 2023). As with all statistical models, the FMM is most effective when its results are interpreted alongside the geomorphic and stratigraphic context of the sample site.

Example - Reconstructing the history of the formation of Mt Chambers Alluvial Fan

An example of the application of the FMM is shown in a study that reconstructs the history of alluvial fan growth along the eastern footslopes of the Flinders Ranges, Australia (Gliganic et al., 2015). Alluvial fans are sensitive to changes in precipitation and temperature and their sediments record thousands of years of changing climate.

Thirteen luminescence samples were collected from representative sedimentary units across the fan surface (Fig. 11). FMM modeling of the sample D_e distributions showed multiple populations of grains in most samples, suggesting that samples suffer from beta microdosimetry, partial bleaching, and/or post-depositional mixing (Fig. 12).

The majority of the scatter in D_e values was attributed to mixing processes, where grains from some sedimentary units translocate down or laterally into older or younger sedimentary units via roots, insects, water and wetting (swelling of the substrate) and drying (cracking and shrinking of the substrate) processes. This was based on a number of observations including the generally structureless nature of the deposits (indicative of homogenization), and the disagreement between luminescence ages and radiocarbon ages from the same sites.

FMM modeling of samples CF373-6A and CF373-6B identified 3 components, one of which comprised the vast majority of the grain population (93% and 77% for -6A and -6B, respectively) (Fig. 12). The authors inferred that the component with the most grains is representative of deposition of the sedimentary unit and that significant younger minor components represent intrusive grains from overlying units by mixing and soil formation (pedogenic) processes. Components that are older than the depositional component were inferred to represent either intrusive grains from underlying units or grains that were incompletely bleached during transport.

Analysis of all 13 samples from the fan found that most depositional ages (the largest sample FMM components) cluster around 11.0 ± 0.4 ka, 6.2 ± 0.2 ka, and 3.4 ± 0.1 ka (Fig. 13). Three samples dated to >30 ka (Marine Oxygen Isotope Stage 3), and these were sampled from older weathered units found below Unit 6. These results point to periods of alluviation (or sediment accumulation) along the southern palaeochannel and mid-fan floodouts at ~11 ka, on the modern (northern) channel and in mid-fan floodouts ~ 6 ka, and along the modern channel and mid-fan ~3.4 ka.

Despite what one might expect, younger and older identified FMM components derived from intrusive grains were not viewed as spurious data without meaning. Across the fan surface, these formed clusters at ~18 ka, ~6 ka and ~1.5 ka (Fig. 13) and were suggestive of mixing processes induced by regional environmental factors. These intervals of mixing were inferred to represent discrete phases of enhanced pedogenic processes affecting the Mt Chambers Creek fan region at ~18 ka, ~6 ka and ~1.5 ka.

Other statistical models

How best to analyze luminescence data is still an area of research and new statistical models are continually being developed. Some additional approaches not discussed above are listed below and include alternative approaches for calculating ages from well bleached or partially bleached deposits, novel statistical approaches for new depositional scenarios as well as Bayesian models.

Table 3. Other statistical models applied in luminescence dating.

Summary

The main characteristics and assumptions behind the models discussed above are illustrated in Figure 14. It is important to realise that each model is designed for samples with a specific depositional history, and that we infer this depositional history from several lines of evidence (e.g., the type of sediment or artefact we are measuring, its D_e distribution shape, OD, the geomorphic and archaeological context of the sample site and any independent age control). If our inferences with regard to the sample’s depositional history are incorrect or flawed, the statistical model will yield erroneous results. For this reason, we are explicit about the assumptions we make when we apply any statistical model.