Elsevier

Quaternary Geochronology

Volume 6, Issues 3–4, June–August 2011, Pages 369-382
Quaternary Geochronology

Research Paper
StalAge – An algorithm designed for construction of speleothem age models

https://doi.org/10.1016/j.quageo.2011.02.002Get rights and content

Abstract

Here we present a new algorithm (StalAge), which is designed to construct speleothem age models. The algorithm uses U-series ages and their corresponding age uncertainty for modelling and also includes stratigraphic information in order to further constrain and improve the age model. StalAge is applicable to problematic datasets that include outliers, age inversions, hiatuses and large changes in growth rate. Manual selection of potentially inaccurate ages prior to application is not required. StalAge can be applied by the general, non-expert user and has no adjustable free parameters. This offers the highest degree of reproducibility and comparability of speleothem records from different studies. StalAge consists of three major steps. Firstly, major outliers are identified. Secondly, age data are screened for minor outliers and age inversions, and the uncertainty of potential outliers is increased using an iterative procedure. Finally, the age model and corresponding 95%-confidence limits are calculated by a Monte-Carlo simulation fitting ensembles of straight lines to sub-sets of the age data.

We apply StalAge to a synthetic stalagmite ’sample’ including several problematic features in order to test its performance and robustness. The true age is mostly within the 95%-confidence age limits of StalAge showing that the calculated age models are accurate even for very difficult samples. We also apply StalAge to three published speleothem datasets. One of those is annually laminated, and the lamina counting chronology agrees with the age model calculated by StalAge. For the other two speleothems the resulting age models are similar to the published age models, which are both based on smoothing splines. Calculated uncertainties are in the range of those calculated by combined application of Bayesian chronological ordering and a spline, showing that StalAge is efficient in using stratigraphic information in order to reduce age model uncertainty.

The algorithm is written in the open source statistical software R and available from the authors or as an electronic supplement of this paper.

Highlights

StalAge, an algorithm for construction of speleothem age models is presented. ► StalAge is applicable to problematic datasets with outliers and age inversions. ► StalAge offers the highest degree of reproducibility and comparability. ► Robustness is demonstrated by application to synthetic and natural speleothem data. ► StalAge is written in the open source software R and applicable by non-expert users.

Introduction

Speleothems such as stalagmites and flowstones can be absolutely dated with unprecedented precision in the range of the last 650,000 a by U-series disequilibrium methods (Richards and Dorale, 2003, Scholz and Hoffmann, 2008). This is a great advantage in comparison to other climate archives, which are often difficult to directly date beyond the range of the 14C-method. The application of multi collector inductively coupled plasma mass spectrometry (MC–ICPMS) has further increased the precision of speleothem U-series ages in recent years (Hoffmann et al., 2007, Cheng et al., 2009). For this reason, and since a variety of climate proxies such as stable oxygen and carbon isotopes or trace elements can be measured at high spatial resolution, speleothems are increasingly used as archives of past climate variability (e.g., Asmerom et al., 2007, Wang et al., 2008, Drysdale et al., 2009, Fleitmann et al., 2009).

Although application of MC–ICPMS allows the use of smaller sample sizes and faster sample throughput, the spatial and, consequently, temporal resolution of the proxy analyses is usually much higher than that of U-series age determinations. Therefore, the age of the calcite between two adjacent U-series ages needs to be determined. In other words, a relationship between the distance of the proxy measurement along the growth axis of the speleothem and age must be established. This relationship is usually referred to as the age model.

Estimating the age of a sequence based on a limited number of dated depths/distances is a challenge that applies not only to speleothems but to all kinds of proxy archives (e.g., marine and lake sediments or ice cores). However, the details of the techniques and software used for age modelling are often poorly described. For instance, a recent literature review yielded that 65 out of 93 papers using age models did not indicate whether uncertainties for the age model were calculated (Blaauw, 2010).

For 14C-dating, a large number of age models and appropriate software are available (e.g., Bennett and Fuller, 2002, Blaauw and Christen, 2005, Heegaard et al., 2005, Bronk Ramsey, 2008, Christen and Pérez, 2009, Blaauw, 2010). For speleothems, in contrast, neither a standard approach to construct speleothem age models on the basis of U-series ages nor a general method to estimate the corresponding uncertainty has been developed yet. This is particularly surprising because the precise determination of the timing and duration of climatic events is considered as one of the major advantages of speleothems as climate archives. However, both the age of a climatic event and its uncertainty depend on the method used to calculate the age model. Thus, the comparability of speleothem time series from different studies is difficult, and a general approach is urgently needed. A variety of different methods for speleothems have been described in the literature. Some authors use linear interpolation between dated sub-sample distances (e.g., McDermott et al., 1999, Wang et al., 2005), others apply least squares polynomial fits (Spötl and Mangini, 2002). Other studies, in turn, use various kinds of splines (Beck et al., 2001, Spötl et al., 2006, Vollweiler et al., 2006, Hodge et al., 2008). In addition, more sophisticated methods based on the general growth mechanisms of speleothems (Drysdale et al., 2004, Genty et al., 2006) have been proposed. Finally, Bayesian statistics in order to include the stratigraphic order of the dated sub-samples have been used (Spötl et al., 2008). For 14C-age modelling Bayesian statistics have also been applied to identify outliers (Buck et al., 2003, Haslett and Parnell, 2008) or constrain the rate of accumulation (Blaauw and Christen, 2005, Bronk Ramsey, 2008). These latter approaches have not been carried over to speleothem age modelling yet.

All methods currently applied have specific advantages and disadvantages. The simplest and most commonly used method is linear interpolation between individual data points. However, this method has several disadvantages: (i) The age model between two subsequent ages is only based on these two data points and not on the whole distribution of ages. Hence, linear interpolation does not make the most efficient use of the data and may put too much weight on individual data points. (ii) Age inversions (i.e., ages violating the constraint that the stalagmites and flowstones must become progressively older with increasing distance from top), even within the range of the age uncertainties, result in negative growth rates and such ages must be discarded. This is especially critical when it is not possible to decide whether one point is too old or the adjacent point is too young. (iii) Although statistical methods to calculate the uncertainty of linear interpolation age models are available (Bennett, 1994, Blaauw, 2010), most speleothem age models based on linear interpolation are quoted without an estimate of the uncertainty.

Smoothing splines and polynomials do take into account the whole age distribution, provide an estimate of the uncertainty of the age model and are, in principle, able to handle age inversions. However, polynomials and most splines are not necessarily monotonic. Thus, the resulting age models may potentially violate the above-mentioned constraint of increasing age with depth. Furthermore, a variety of different smoothing splines are available, and most of them offer the user to adjust several parameters, with some of them having a strong influence on the resulting age model. In some cases this may actually be necessary since application of automatic splines is not always possible, depending on the variability and scatter of the underlying dating results. For instance, restriction of the degrees of freedom is sometimes required in order to derive a monotonic age model (Hoffmann et al., 2010). The possibility to adjust several parameters may also be considered as an advantage since it allows the user to ‘tune’ the age model to some extent. However, not using an automatic spline algorithm makes the result less reproducible and includes the risk of constructing a ‘personally biased’ or subjective age model. In terms of comparability of speleothem age models from different studies, the variety of available splines is problematic.

Using Bayesian statistics in order to include information based on the constraint of increasing age with depth (Drysdale et al., 2004, Spötl et al., 2008) is advantageous. (i) Outliers can be detected on a statistically sound basis, (ii) age inversions within the range of uncertainties are corrected, and (iii) inclusion of stratigraphic information may considerably reduce the uncertainty of individual ages. However, after pre-treatment of the data, one of the methods discussed above must be used to construct an age model. Thus, the individual disadvantages of these methods also apply to the combination with Bayesian chronological ordering.

In order to make the most efficient use of speleothem U-series age data and to ensure the greatest possible comparability between different studies, a general and reproducible technique for the construction of speleothem age models and the corresponding uncertainty is of particular interest. Here we present the algorithm StalAge, which is especially designed for this purpose. The algorithm is able to detect and account for outliers and age inversions, includes stratigraphic information in addition to a monotonicity constraint, and provides 95%-confidence limits for the uncertainty of the age model. The code is written for use with the open source statistical software R (R Development Core Team., 2010) and available from the authors or as an electronic supplement of this paper.

Section snippets

Simulation of synthetic stalagmite growth models and U-series dating

In order to test and demonstrate the performance of the new StalAge algorithm, a numerical model was developed simulating (i) speleothem growth, (ii) incorporation and temporal evolution of U-series isotopes and (iii) mass spectrometric analysis (compare Fig. 1). This approach has two major advantages: Firstly, the ‘true’ or, in this case, simulated age model is known and can be compared to the age model calculated by the algorithm. Secondly, extreme scenarios resulting in sections of the

Basic philosophy

As described in the introduction, various methods are currently used to construct speleothem age models. Thus, the age model and its uncertainty do not only depend on the underlying age data but also on the method of choice. Even if the same method is used in different studies, the resulting age models are not necessarily identical because many methods offer the user several adjustable parameters, with some of them having a strong influence on the age model. Adjusting parameters may be

Application to natural speleothem U-series data

We apply StalAge to published speleothem U-series data to compare the results with published age models. First, we fit the U-series data of speleothem SPA 121 from Spannagel Cave, Austrian Alps (Spötl et al., 2008). The application of StalAge to this dataset is particularly interesting because the published age model was obtained by combined application of Bayesian modelling for chronological ordering and a spline. Second, we apply StalAge to stalagmite GB89-25-3 from the Bahamas (Hoffmann

Performance of the algorithm

In the two previous sections we have applied StalAge to one synthetic example (Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6) and three published speleothem records (Fig. 7, Fig. 8, Fig. 9, Fig. 10). Whereas the synthetic example was designed to include several problematic features, such as outliers, hiatuses and periods of a very different growth rate (Fig. 2), SPA 121 and GB89-25-3 do not exhibit extremely problematic regions. For these two examples, the age models and uncertainties calculated by

Conclusions and outlook

We have presented here a robust and user-friendly algorithm (StalAge) that is designed to construct age models for speleothems on the basis of U-series ages. The algorithm starts with the U-series ages and their corresponding age uncertainties and then uses additional stratigraphic information to further constrain and improve the resulting age model. StalAge is able to handle problematic datasets including outliers, age inversions, hiatuses and large changes in growth rate, as shown by

Acknowledgements

Denis Scholz was funded by DFG grant SCHO 1274/1-1. We thank Ronny Boch for exhaustive application of the algorithm and stimulating discussions. We are also thankful to Claudia Fensterer, Marc Luetscher, and Christoph Spötl for testing and discussing our algorithm. Thorough reviews by Maarten Blaauw and an anonymous reviewer were extremely helpful.

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