Author's personal copy · Author's personal copy A comparative study of methods to reconstruct a periodic time series from an environmental proxy record Anouk de Brauwerea,, Fjo De

Author's personal copy

A comparative study of methods to reconstruct a periodic time series from anenvironmental proxy record

Anouk de Brauwere a,⁎, Fjo De Ridder a,b,c, Rik Pintelon b, Johan Schoukens b, Frank Dehairs a

a Department of Environmental and Analytical Chemistry, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgiumb Department of Fundamental Electricity and Instrumentation; Team B: System Identification and Parameter Estimation, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgiumc Current position: Vlaamse Instelling voor Technologisch Onderzoek (VITO), Boeretang 200, B-2400 Mol, Belgium

a b s t r a c ta r t i c l e i n f o

Article history:Received 5 December 2007Accepted 24 April 2009Available online 3 May 2009

Keywords:time seriesproxyaccretion rategrowth rateperiodicseasonality

The past environment is often reconstructed by measuring a certain proxy (e.g. δ18O) in an environmentalarchive, i.e. a biogenic or abiogenic accreting structure which gradually accumulates mass and records thecurrent environment during this mass formation (e.g. corals, shells, trees, ice cores, speleothems, etc.). Proxyanalysis usually yields a record along a distance axis. However, to relate the data to environmental variations,the date associated with each data point has to be known too. This transformation from distance to time isnot straightforward to solve, since accretion mostly proceeds at a varying and unknown rate. To solve thisproblem some hypotheses about the growth rate or the time series must be made. Depending on theapplication, different assumptions may be appropriate, resulting in or requiring a particular method toperform this transformation. The actual method used can hugely influence the final result and hence theinterpretation of the data in terms of frequency and timing of events. However, no comparative study hasbeen made so far, and most of the existing methods haven't been thoroughly assessed. Therefore, this paperaims to evaluate and compare the most popular methods. To keep the review manageable the scope waslimited to those records where it can be assumed that the time series is periodic. Examples of periods includetidal, seasonal and ENSO (El Niño Southern Oscillation) cycles, and even cycles of thousands of years could beconsidered, as long as they are resolved in the measured record. Six methods to reconstruct the time base forperiodic proxy records are compared in this review. Their performance in the presence of stochastic andsystematic errors is tested on simulations and linked to the methods' underlying assumptions. As a finalcomparison, all methods are applied to a real world example. The goal of this overview is to provide anobjective structure and comparison of the methods mostly used, so that the users are aware of the underlyingassumptions and their consequences.

© 2009 Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

2. Classification of the methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013. Test datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.1. Stochastic errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.2. Model errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.2.1. Periodic signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.2.2. Amplitude modulated signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.2.3. Trended signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.2.4. Hiatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4. Mapping methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.1. Anchor point method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.1.1. Method description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.1.2. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.1.3. Tuning parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Earth-Science Reviews 95 (2009) 97–118

⁎ Corresponding author. Tel.: +32 2 629 32 64; fax: +32 2 629 32 74.E-mail address: [email protected] (A. de Brauwere).

0012-8252/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.earscirev.2009.04.002

Contents lists available at ScienceDirect

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4.1.4. Performance in the presence of stochastic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.1.5. Performance in the presence of potential model errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.1.6. Conclusion on the anchor point method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.2. Correlation maximization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.2.1. Method description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.2.2. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.2.3. Tuning parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.2.4. Performance in the presence of stochastic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.2.5. Performance in the presence of potential model errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.2.6. Conclusion on the correlation maximization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.3. Martinson et al.'s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.3.1. Method description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3.2. Improvements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3.3. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3.4. Tuning parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3.5. Performance in the presence of stochastic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3.6. Performance in the presence of potential model errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3.7. Conclusion on Martinson et al.'s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5. Class 2: signal model methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.1. Time domain method developed by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1.1. Method description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.1.2. Improvement by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.1.3. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.1.4. Tuning parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.1.5. Performance in the presence of stochastic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.1.6. Performance in the presence of potential model errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.1.7. Conclusion on the time domain method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2. Frequency domain method: a phase demodulation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2.1. Method description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2.2. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2.3. Tuning parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2.4. Performance in the presence of stochastic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2.5. Performance in the presence of potential model errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2.6. Conclusions on the phase demodulation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3. Parametric time base distortion method, coupled to an automated model selection procedure . . . . . . . . . . . . . . . . . . . 1135.3.1. Method description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.3.2. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.3.3. Tuning parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.3.4. Performance in the presence of stochastic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.3.5. Performance in the presence of potential model errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.3.6. Conclusion on the parametric time base distortion method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6. Comparison of the methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.1. Robustness to stochastic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2. Use of tuning parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.3. Assumptions and model errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7. Real world example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.1. Dataset description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.2. Tuning parameters for all methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

1. Introduction

Past climate or environmental conditions that are no longerdirectly measurable can sometimes be reconstructed throughproxies. These are geochemical or physical signals captured inaccreting biological or geological structures during their formation.If accretion occurred more or less gradually through time and if theproxy is linked to the environment at the time of incorporation, anindirect time series of the past environment can thus be obtained.Examples of structures from which such records were extractedinclude tree rings, sclerosponges, speleothems, corals, molluscshells, fish otoliths, mammal teeth, sediment and ice cores (Gröckeand Gillikin, 2008).

Several problems arise when trying to interpret a proxy recordsin terms of an environmental time series. The one problem under

study here concerns the transformation of the distance axis, alongwhich the proxies are actually measured, to the underlying timeaxis. This translation is needed to obtain the actual time series of theproxy which is generally the record of interest. The time series ismuch more interpretable than a “distance series” and can becompared with other proxy or instrumental records. However, dueto variations in the accretion rate of the medium, this transforma-tion is generally not a simple linear mapping. The transformation ofthe distance series into a time series requires knowledge about thegrowth or accretion rate (Fig. 1). Unfortunately, the accretion rate isgenerally unknown too.

This problem can only be solved by including additional informa-tion or assumptions about the time series or the growth rate. Anexample of such extra information are growth bands, present e.g. insome corals, bivalves or trees, which can be used to date some points

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of the record. However, this approach still overlooks variations inaccretion rate on time scales smaller than one growth cycle, and thusneeds additional assumptions to fill this gap. It is clear thatappropriate assumptions depend on the species, proxy and processesunder study and therefore no universal set of “good assumptions” canbe given. However, the aim of this study is to provide some generalstatements about the consequences of certain assumptions made toreconstruct a proxy time series. If the assumptions are too stringent,the resulting time series is a mere translation of these assumptions.Instead, assumptions should be used that not fully determine theresult, but leave some freedom for the data to speak. This study doesnot aim at telling which concrete assumptions should be chosen, asthis is indeed a far from straightforward task for real applications.Neither is it the authors' intention to mark any method by “good” or“bad”. Rather, this study intends to provide a systematic classification,description (with emphasis on the underlying assumptions) andquantitative comparison of the methods mostly used. This informa-tion should make the user aware of the consequences of using a givenmethod, such that his/her results are certainly not interpreted todegree that is no longer relevant.

The fundamental difficulty lies in the fact that the accretion rate israrely constant and moreover unknown. For instance, bivalves arereported to grow slower as they become older, but “superimposed onthis ontogenetic pattern are growth increment patterns controlled bydiurnal, monthly and/or annual variations in the environmentalconditions experienced by the individual mollusc” (Bice et al., 1996).Even during one day, growth variations and even growth stops havebeen observed (Schöne et al., 2006). These growth variations can berelated to internal physiological effects (Rodland et al., 2006) and age(Schöne et al., 2002) or to many environmental factors such astemperature (Goodwin et al., 2001 and references therein; Schöneet al., 2002; Chauvaud et al., 2005), salinity (Chauvaud et al., 1998;Schöne et al., 2003a), hydrodynamical conditions like tidal regime(Schöne et al., 2002; Clark II, 2005), precipitation (Schöne et al.,2007), food availability (Schöne et al., 2003b; Schöne et al., 2005) orthe occurrence of toxic algal blooms (Chauvaud et al., 1998).Sedimentation and snow accumulation do not occur at constantrates either. In addition, the depth profile can be further distorted bybioturbation, diffusive-like processes, natural compaction, focusing,winnowing and other events removing parts completely and/or

disturbances related to the core recovery (Raymo, 1997; Huybers andWunsch, 2004). As a result, the accretion rate can sometimes be verycomplex. However, the inferences made from the derived time serieshighly depend on a correct distance–time mapping or accretion ratereconstruction, or as Huybers and Wunsch (2004) highlight thisdifficulty and state: “It is not an exaggeration to say that under-standing and removing these age–depth (or age model) errors is oneof the most important of all problems facing the paleoclimatecommunity”.

Failing to accurately take into account variations in accretion ratecan induce a considerable bias e.g. when estimating annual averagesor seasonal ranges of a measured proxy. Wilkinson and Ivany (2002)already noticed that “the practice employed by many studies is tocalculate mean annual δ18O (or temperature) from microsample dataas the arithmetic mean of all analyses in the data set. This approach,however, does not account for the fact that most organisms have apreferred range of temperature over which they grow more rapidlyand produce more skeletal material. Arithmetic means of all analysesare therefore biased toward values attained during seasons ofmaximum growth. Even worse, many organisms cease to growaltogether over certain temperature ranges, further skewing meancomposition.”

Another practice that should be appliedwith care is spectral analysison the reconstructed time series. The assumptions made to transformtheoriginal distance series into the time series are prone to influence theresults of the spectral analysis. If themappingdoesnot correspond to thetrue (unknown)variations in accretion rate, the spectrumwill be alteredas some (real) peaks will disappear and other (artificial) peaks willappear. Interpretation of this spectrum thus very much relies on theconfidence put in the original data axis transformation. For instance, ifthe distance series contains a clear cyclicity, the time series can bereconstructed by assuming that this cycle corresponds to a given, e.g.annual period. By imposing this period to be present in the data, thisperiod will obviously be found in a subsequent spectral analysis, but itcannot be further interpreted as a result found in the data. Huybers andWunsch (2004) already pointed to this pitfall in their investigation forclimatic periodicities in sediment cores: “To avoid circular reasoning, anage model devoid of orbital assumptions is needed”.

The above examples were an attempt to convince the reader thatassumptions have implications concerning the interpretation range ofthe results. But even if this fact is recognised, it may not be thatobvious to identify which are exactly the assumptions underlying agiven method. This is especially true because the time basereconstruction method often does not receive much attention withina study. Accordingly, during the last 25 years only a handful ofmethods are described in literature and no real assessment, let alonecomparison of these methods has been made to this date. The currentwork aims at filling this gap by presenting an in-depth comparison ofthe most popular/relevant methods.

In order to keep this comparative study manageable, the scope waslimited to those applications where it can be assumed that the timeseries is periodic. This classwas chosen because periodicity is an evidentassumption for many applications. The periods can range from tidal(Clark II, 2005) to thousands of years (deMenocal et al.,1991), includingthe more frequently observed seasonal (Khim et al., 2000; Swart et al.,2002; Ivany et al., 2003; Lorrain et al., 2004; Gillikin et al., 2008) andmulti-annual cycles (related to the El Niño Southern Oscillation (ENSO)or North Atlantic Oscillation (NAO): Felis et al., 2000; Carré et al., 2005;Schöne et al., 2007). Also, it is important to realise that “periodic” is notrestricted to sinusoidal signals, i.e. signals with only one periodicity.Indeed, much more complex signals can be considered, as long as theycontain at least one periodic component. Nevertheless, some recordsindeed do not exhibit such a cyclicity at all, and our conclusions shouldnot be extrapolated to these kinds of records.

Summarising, the core of this study is the classification, descrip-tion, assessment and intercomparison of a variety of existing methods

Fig. 1. Conceptual graph showing the transformation from a record along a distance grid(dots on horizontal axis) to a time series (dots on vertical axis). The full black line couldbe a target function. The dashed diagonal line represents a constant growth, while thecurved gray line along the diagonal represents an optimized accretion. Mirroring thedistance series on the diagonal curve results in the time series.

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to transform the distance axis of a proxy record into a time axis. Thiswork aims to

(1) bring some structure in the variety of existing methods(although we acknowledge that this study can be infinitelyextended by additional method subtleties, test cases and/orevaluation criteria);

(2) enable users of a specificmethod to “look up” its characteristics,with emphasis on the underlying assumptions such that thefinal results are not over-interpreted;

(3) illustrate the effect of the different underlying assumptions on anumber of artificial and one real data set.

In brief, the text is structured as follows. In Section 2 the methodsare classified. In order to assess their performance, all methods areapplied to the same five test datasets and one real dataset. In Section 3the artificial test datasets are described in more detail. In Sections 4and 5 the two classes of methods are discussed. For each class, first thecommon rationale behind the method class is introduced, followed bya detailed discussion of each method. Some of the methods have beenupdated compared to their original description, such that eachmethod can be evaluated based on the assumptions made, ratherthan on the way the underlying algorithm was implemented. Thisresults in the following topics considered for each method:

(i) a formal description of the method (possibly followed byalgorithmic improvements);

(ii) a listing of the assumptions underlying the method;(iii) a list of the tuning parameters present in the method, since

these are subjective inputs;

(iv) results when the method is applied to the first test dataset withonly stochastic errors;

(v) results when the method is applied to the test datasets withpotential model errors;

(vi) conclusion, summarizing the main facts and observations.

Note that the method description is kept as concise as possible. It ismeant as an introduction of the main ideas behind the method andshould enable the interpretation of its performance on the test datasets.For those readers interested in the implementation of the method, werefer to the original publications. To facilitate cross-reference the samenotation is used as in the original publications; Table 1 summarises thesymbols used and theirmeaning. The strict organisation of themethods'sections is intended tomake intercomparisons simple. Also, each sectionhas been written such that it can be read independently, and thestructure should make it easier to look up one method or one topic.

Before going into more detail about the methods, it is useful todefine the most important concepts within the scope of this study.Most concepts are illustrated in Fig. 1.

1.1. Definitions

(1) Distance series: observations as function of their position on thephysical specimen (horizontal record in Fig. 1).

(2) Time series: observations as function of time (vertical record inFig. 1).

(3) Target function: mathematical function for the time series,usually defined a priori (full black line along the time axis inFig. 1).

Table 1Symbols list.

Symbol Method (section) Meaning Dimension

A0 M (4.3), PTBD (5.3) Offset in target function Same as measurementAi M (4.3), PTBD (5.3) Coefficients of sines and cosines in target function (i {1,…,2h}) Same as measurementAmp0 WI (5.1) True amplitude of the signal Same as measurementAmp(m) WI (5.1) Amplitude within window m Same as measurementb M (4.3), PTBD (5.3) Number of basis functions = number of unknown TBD parameters Scalarbmax PTBD (5.3) Maximal number of TBD parameters to be considered during model selection ScalarBp M (4.3), PTBD (5.3) Basis function coefficient = unknown parameters (p {1,…,b}) Scalarc WI (5.1) Width of the window expressed in number of observations ScalarC(ω) PD (5.2) Spectral window Scalarh M (4.3), PTBD (5.3) Number of harmonics in target function or signal model Scalarhmax PTBD (5.3) Maximum number of harmonics in signal model to be considered during model selection ScalarK(B) M (4.3) Cost function Depending on definitionm WI (5.1) Window number m {0,…,N−1−c} Scalarn All Sample identifier, n {1,…,N} Scalarnθ PTBD (5.3) Number of unknown parameters ScalarN All Total number of observations ScalarPer WI (5.1) Period DistancePer0 WI (5.1) True period of the sine signal TimePhs WI (5.1) Phase as defined by Wilkinson and Ivany (2002) DistancePhs0 WI (5.1) True phase of the sine signal ScalarPos WI (5.1) Mean position or offset Same as measurementPos0 WI (5.1) True mean position or offset Same as measurementS(t) or s(n) M (4.3), PD (5.2) Measured signal as a function of time or sample number (distance) Same as measurementsmodel(t) M (4.3) Target function Same as measurementS(ω) PD (5.2) Fourier transform of s(n)SWI(m) WI (5.1) Signal model for window m Same as measurementt(n) M (4.3), PD (5.2) Time instance of sample n TimeTime WI (5.1) True time instance TimeTS M (4.3), PD (5.2) Sample period (time between two subsequent samples) TimeWmid WI (5.1) Middle of the window DistanceYsin WI (5.1) Assumed true signal Same as measurementδ(n) M (4.3), PD (5.2) Mapping function or time base distortion Scalarδ(ω) PD (5.2) Part of the spectrum S(ω) Scalarϕp(n) M (4.3) Value of basis function p(p {1,…b}) for sample n Scalarφ(.) PD (5.2) Phase of signal between brackets Timeθ PTBD (5.3) Vector containing all unknown parameters (length nθ) Scalar vectorω M (4.3) Angular frequency in target function Time−1

ω0 PD (5.2) Fundamental angular frequency (2π/age) Time−1



(4) Mapping function: relation between the position and the date ofthe observations (gray line in Fig. 1).

(5) Growth/accretion rate: derivative of the positionwith respect totime or derivative of the mapping function.

(6) Time scale: difference between the date of the first and the lastobservation, i.e. absolute dimension of the time axis.

(7) Time base: dates of the observations divided by time scale, i.e.relative dates of the observations with respect to each other.

2. Classification of the methods

The methods used to reconstruct time bases are divided into twoclasses (Fig. 2):

1. Mapping methods generate an accretion profile by assuming asimilarity between the distance record and a postulated targetfunction. In this review, the focus lies on annually resolved proxyrecords with seasonal (i.e. periodic) variations. Therefore, mostoften a sinusoidal target function is postulated. In a next step,this similarity is used to construct a time base, which can beperformed in many different ways. We discuss three strategies,i.e. the anchor point method (Paillard et al., 1996), thecorrelation maximization (Yu and Ding, 1998; Lisiecki andLisiecki, 2002) and Martinson et al.'s method (Martinson et al.,1982a,b, 1987).

2. Periodic Signal Model Methods exploit the (assumed) annualtranslation symmetry of the time series to estimate the time baseand the time series. In contrast to the mapping methods theexact shape of the target function is not known or assumed to beknown in advance, but a model for the target function ispostulated by the user, in which some model parameters are still

free to be optimized. As mentioned in the introduction, we willfocus on periodic functions. Three methods are discussed: a timedomain method (Wilkinson and Ivany, 2002; Ivany et al., 2003),a frequency domainmethod (phase demodulationmethod by DeRidder (2004)) and a parametric time base distortion approach(de Brauwere et al., 2008), which is actually a generalization ofMartinson et al.'s method in which not only the modelparameters but also the model itself is not fixed but must beoptimized.

The methods can also be classified in parametric (number ofparameters is independent of the amount of data) and non-parametric methods (Fig. 2). Parametric models generally perform

Fig. 3. Simulated dataset to test the methods' performance in the presence of stochasticnoise.

Fig. 2. Schematic classification of the different approaches, including their interrelations, limitations and solutions.



better in the presence of stochastic noise, but are also more sensitiveto particular model errors, as will be illustrated in this study.

3. Test datasets

The precision and accuracy of a reconstructed time series willlargely depend upon the assumptions made to enable the transforma-tion from the distance record. For this reason, special attention isgiven to the robustness of the methods with respect to theseassumptions. This is done by testing each method on five artificialdatasets containing different frequently encountered errors.

3.1. Stochastic errors

In the first test dataset only stochastic errors are present, in order toevaluate the sensitivity of each method to measurement noise. Anyreal measured record is disturbed by this kind of errors. A precisemethod is able to discard these randomvariations in the data such thatthe stochastic errors are not propagated into the estimated time base.

The ‘true’ time series used for this test is shown in Fig. 3. It issinusoidal, without variations in accretion rate (constant accretionrate). Consequently, the distance series is sinusoidal as well and allvariations in the reconstructed accretion rate are errors generated bythe particular method. The signal spans seven years and is sampled50 times, has a frequency of one period per year, a sample period of7/50 year and a distance grid interval of 7 mm, corresponding to aconstant accretion rate of one millimeter per year. The samples takenfrom this true signal are disturbed by normally distributed whitenoise with a signal-to-noise-ratio (SNR) of 11.6. This value waschosen based on the conditions found in the real world exampletreated at the end of this study, adapted from Gillikin et al. (2005b).Admittedly, the SNR may be lower in some applications. But in thepresence of high noise levels some methods appeared to performvery poorly, making a reasonable comparison less evident. Never-theless, the influence of increasing the SNR is briefly discussed foreach method.

3.2. Model errors

Four additional test datasets were designed to check the sensitivityof the methods to a violation of their assumptions concerning theshape of the true time series. If this time series is different from theassumed target function, a model error is introduced. A more flexible

method can handle more possible shapes of the time series and willthus be less disturbed by model errors. These simulations were set upto verify the effect of not having a sinusoidal time series, since this isthe basic assumption often made.

In order to clearly differentiate the influence of the noise propagationandmodel errors, no randomerrors are added in these test datasets. Fig. 4shows the different simulated signals (full lines) and the sinusoidalreference signals assumed by most mapping methods (dotted lines). Allthe considered model errors can be found in real world datingapplications. However, one should keep in mind that much more modelerrors can be imagined. Consequently, the four cases considered here arenot meant to represent a complete overview of all possible model errors,but rather to form a sample of potential and realistic problems that can beused as an objective basis to assess and intercompare the performance ofthe sixmethods. On the other hand, if it iswell known that a given “modelerror” (i.e. feature present in the data but not in the model/targetfunction) is present, it is good scientific practice to incorporate thisknowledge into themodel or target function. To put it bluntly: you cannotexpect fromaclearlywrongmodel toperformwell. In this perspective, the

Fig. 5. Visualization of the anchor point method. Horizontally the proxy is shown asfunction of distance; vertically the target function is shown as function of time. The dateof some observations is known (in this example the first and last observation and themaxima, shown by ‘o’) and between these anchor points a linear accretion is assumed(dotted line).

Fig. 4. Test datasets with potential model errors: (a) a periodic signal (full line) instead of a sinusoidal signal (dotted line), (b) an amplitude modulated signal, (c) a signal with atrend and (d) a signal with a hiatus.



model errors considered here are supposed to be rather “mild” errors thatcould occur in the data without the investigator noting them.

The signal parametersused togenerate these test datasets are identicalto those used in the stochastic simulation (Section 3.1). The accretion ratewas againconstant, except for thehiatusdataset (Fig. 4d). Soexcept for thelatter, any variation in the reconstructed accretion rate is a bias.

3.2.1. Periodic signalIn the first case (Fig. 4a) themethods are tested in the presence of a

second harmonic. Such signals could appear in tropical climate, wheretwo rainy seasons occur in addition to the annual variation (Lazarethet al., 2003). If the time base reconstruction method does not includethe possibility of several harmonics, a model error is introduced.

3.2.2. Amplitude modulated signalFig. 4b shows an amplitude modulated sinusoidal signal. Such

amplitude modulation acts as an envelope changing the annual

amplitude of the proxy. Examples include phenomena like El Niño orage effects in bivalves where the amplitude decreases with ontogenicage. For an accurate (and thus interpretable) reconstruction of thetime series, both the time axis (time base) and the amplitude axismust be truthful. This test will assess the ability of the methods tocope with variations in the amplitude axis.

3.2.3. Trended signalIn this test dataset a linear trend was added to a sinusoidal signal

(Fig. 4c). Red noise in ocean proxies is an example of such disturbingtrends.

3.2.4. HiatusFig. 4d shows a signalwith a hiatus. Technically, the hiatus is caused by

removing the 19th, 20th and 21st observations and adding threeobservations at the end of the record. It simulates a sudden discreteaccretion stop during the time that these observations should have been

Fig. 6. Accretion rate estimated by the 6 methods (full line) for the test dataset with stochastic errors, compared to the true constant accretion rate (dotted line).



formed. This kind of growth stop can occur if the recording species isexposed to extreme conditions (high or low temperatures, lack of food,etc.) for a period of time.

4. Mapping methods

For these methods the investigator starts from

(i) a proxy record, as a function of the distance, and(ii) aknownprofile for the timeseries, the so-called target or template

function (see Fig. 1). Note that this assumption is not necessarilyneeded for theanchor pointmethod. Instead, thismethoddirectlyassumes the dates of some observations to be known.

The accretion rate relates both and is not known in advance. In general,this last function is called the mapping function. Three sub-classes ofmethodswill be described, differing in how exactly themapping functionis constructed to relate distance record with time series, i.e. the anchorpoint method (Section 4.1), the correlation maximization methods(Section 4.2) and Martinson et al.'s method (Section 4.3).

4.1. Anchor point method

4.1.1. Method descriptionThis approach is themost intuitive of all, but has actually never been

described before and has never been given a name. Paillard et al. (1996)provided a free toolbox to perform the computations, which probablyhas helped to make it by far the method most frequently used (e.g.Dickens et al., 1995; Charles et al., 1997; Felis et al., 2000; Wurster andPatterson, 2001; Swart et al., 2002; Schöne et al., 2003b; Dettman et al.,2004; Gillikin et al., 2005a; Moses et al., 2006). In this non-parametricmethod the date of some observations is (assumed to be) known. Theseobservations are called the anchor, tie or control points. The dates of theobservations between these anchor points can be estimated employingan interpolation technique. Usually a linear interpolation is used, whichmeans a linear accretion is assumed between the anchor points. Thedating of the anchor points can be based upon the target function or onother events registered in the proxy, like growth bands, well datedvolcanic eruptions, etc. To our knowledge thismethod is very frequently

used, but, surprisingly enough, we were unable to find a description ofthe properties of this method in literature.

Fig. 5 schematizes the method: horizontally a proxy record asfunction of a distance grid is shown; vertically the target function isshown. An example of proxy record could be the δ18O signal recordedin a bivalve and the target function could be the sea surfacetemperature, directly measured as function of time. The annualmaxima can be used as anchor points (shown by the open circles inFig. 5) and in-between a constant accretion rate is assumed (shown bythe straight dotted lines in Fig. 5). Compared with the other mappingmethods, this method tunes the time series by the judgment of theinvestigator who chooses the position and the number of anchorpoints—which makes the procedure subjective. On the other hand, anadvantage of thismethod is that it can date any record, as long as someanchor points are known and a stepwise constant accretion rate is areasonable assumption.

4.1.2. Assumptions

Unlike the other mapping methods, the anchor point method doesnot need to assume the actual shape of the time series in advance.Instead, the following assumptions are made:

1. The date of each anchor point is known, with absolute certainty.This means that no procedures are implemented in order to reducethe influence of stochastic noise. Note that the anchor points maybe chosen and dated based on an implied target function. Forinstance, it may be implicitly assumed that the time series issinusoidal, such that the observed maxima in the distance seriesare given summer/winter dates. Also note that for thewhole recordto be dated (i.e. transformed in a time series) the first and the lastobservation should be anchor points, because only observationsbetween two anchor points can be assigned a date.

2. Between two subsequent anchor points the accretion rate is constant.

4.1.3. Tuning parameters

The number and position of anchor points is chosen by theinvestigator. For the example of Fig. 5, one investigator can decide to

Fig. 7. Accretion rate estimated by the anchor point method (full line), compared to the true accretion rate (dotted line), for the test datasets with potential model errors: (a) aperiodic signal instead of a sinusoidal signal, (b) an amplitude modulated signal, (c) sinusoidal signal+a linear trend and (d) a hiatus.



use the annual maxima, while another one would use the annualminima, and yet another would use both the maxima and minima.

4.1.4. Performance in the presence of stochastic errorsFig. 6a shows the accretion rate estimated by the anchor point

method when applied to the artificial record disturbed by stochasticnoise (Fig. 3). Because the first and last observations are dated, thenumber of annualmaxima is known. The dotted line at 1mmyear−1 isthe true (constant) accretion rate. Clearly, the reconstructed accretionrate is not constant. The root-mean-square (RMS) of the error is 9%.This error has several reasons:

(i) Suppose a record without disturbing noise. Because this record issampled at discrete positions, the exact maxima will probably fallbetween two subsequent samples. This causes a discretizationerror, i.e. an error related to the finite sampling resolution. So evenin the theoretical case of no stochastic noise, the anchor pointmethod will not work perfectly.

(ii) Besides this discretization error, the disturbing noise changes thenumerical value of the observations and consequently a wrongobservation can be selected as the maximum, which can cause anadditional error.

(iii) The linear interpolation between anchor points can also inducean error, but not in this artificial example because the realgrowth rate is constant.

If the stochastic noise level were higher, this would probably notaffect the outcome much, since it would largely be absorbed by thediscretization error.

To conclude, because the dates of the anchor points are defined inan absolute way by the user, noise on the measurement is ignored.This makes the method vulnerable to stochastic fluctuations presentin themeasured data. In addition, a discretization error is unavoidable.The magnitude of this error depends on the sampling density.

4.1.5. Performance in the presence of potential model errorsThe reconstructed accretion rates for the four potential model

errors (see Section 3.2) are shown in Fig. 7.Periodic signal. The yearly maxima were chosen as anchor points

and the reconstructed accretion rate is shown in Fig. 7a. Note that

(i) The record does not start or end with a maximum. As only theobservations between two anchor points can be dated(assumption 1), in this example the first and the last fourobservations could not be dated and hence were dropped.

(ii) Even without stochastic noise the anchor point method cannotestimate the accretion rate exactly, because of the discretiza-tion error. To clarify this, the number of observations betweentwo subsequent maxima is counted. For the first five years,seven observations were made each year; while the sixth yearcounts eight observations. Consequently, a higher accretionrate is estimated for that year.

(iii) The RMS error is 5%. So, despite the inevitable discretizationerror the anchor point method is relatively insensitive to thepresence of a second harmonic.

Amplitude modulated signal. Each year the maximum value waschosen as anchor point and the reconstructed accretion rate is shownin Fig. 7b. The RMS error is 7%, so again the anchor point methodperforms relatively well. Most of the mismatch is due to thediscretization error, but on the whole the anchor point methodseems to perform well on amplitude modulated signals.

Trended signal. Again the annual maxima are taken as anchorpoints and the same accretion rate was found (Fig. 7c). So, the sameRMS error of 7% is achieved, mostly caused by the discretization error.

Hiatus. Because the hiatus did not remove a maximum, the samemaximawere selected by the anchor point method and the hiatus wasreconstructed within the precision allowed by the number of anchorpoints. The position of the hiatus was identified, but the magnitudewas averaged over the period between the two neighboring anchorpoints. This results in a RMS error of 13%. Obviously, the results wouldbe much poorer if the choice of the anchor points was obstructed bythe hiatus.

4.1.6. Conclusion on the anchor point methodThe anchor point method is not very robust against stochastic

noise, but it performs reasonably well in the presence of model errors,at least those which were tested here. However, it has three majorweaknesses. First, a discretization error limits its precision in all cases.This discretization error can be decreased by lowering the number ofanchor points. However, the accuracy will be lower as well. Secondly,the assumption of a constant growth rate between anchor points is inmost cases unrealistic and will thus induce an error. Thirdly, themethod is highly dependent on the user's decisions regarding whichand how many observations are used as anchor points and to whichdates these anchor points correspond. Any wrongly assigned date willgenerate an error in the reconstructed time series. Yet, this userdependence alsomakes themethod very flexible and applicable to anykind of underlying signal (not only periodic).

4.2. Correlation maximization methods

The two disadvantages of the anchor point method can becircumvented by optimizing the time instances of the anchor points,instead of fixing them according to the subjective choice of the user.

Fig. 8. Schematic representation of the dynamic programming procedure, used byLisiecki and Lisiecki (2002). (a) Steps 1–4; (b) step 5; (c) step 6 (see Section 4.2.1 formore explanation). (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article.)



The correlation maximization methods (this section) and Martinsonet al.'s method (Section 4.3) perform such optimizations.

4.2.1. Method descriptionThe correlation maximization methods also work with anchor

points. The method proposed by Yu and Ding (1998) (hereafter:YD98) uses all observations as anchor points, whereas Lisiecki andLisiecki (2002) (hereafter: LL02) suggest to use a lower, predefinednumber of anchor points. The date corresponding to each anchor pointis adapted, so that the correlation between themeasured proxy recordand a given target function is maximized. Consequently, discretizationerrors can be reduced. Both methods were originally proposed to dateor align sediment cores, and it is only in this area that they have beenapplied so far (YD98: e.g. Tian et al., 2002; Ding et al., 2002; Sun andAn, 2004; Sun et al., 2006; Bloemendal et al., 2008; LL02: e.g. Lisieckiand Raymo, 2005; Milkovich and Head, 2005; Lauwrence et al., 2006;Lisiecki and Raymo, 2007; Channell et al., 2008; Xuan and Channell,2008).

The method proposed by LL02 uses dynamic programming, apowerful technique to find optimal values for problems by breaking itdown into smaller subproblems which can be solved sequentially

(Bertsekas, 2005). The procedure has important properties: theoptimum found is guaranteed to be the global optimum, meaningthat no better solution can be found. LL02 recognised that datinglongitudinal proxy records is a problem that can indeed be dividedin subproblems, i.e. the dating (or alignment) of the individualsamples, and developed a dynamic programming algorithm to solvethis problem.

LL02 illustrate the procedure on some very simple examples,which can be of pedagogical use to the interested reader. To explainthe general principle, first assume that the samples are ordered fromthe youngest to the oldest. Then the following steps are performed(see Fig. 8):

(1) First a time scale is defined, i.e. a time interval [tstart, tend] thatshould contain the dates of all the samples. It is in this timeinterval that the sample dates will be optimised, so it shouldnot be chosen too narrow (if possible).

(2) This time scale is discretized, i.e. not all (i.e. infinite number)dates (t∈ [tstart, tend]) in this time interval, but a discretenumber (say M) of them (t∈ {tstart,tstart+1, … tend}) areconsidered.

(3) The date of the first sample is varied over all these discretevalues (t1= tstart, t1= tstart+1, …, t1= tend), and for all thesedatings the correlation with the target function is calculated: C[t1= tstart], C[t1= tstart+1], …, C[t1= tend].

(4) Next, the date of the second sample is also varied over alldiscrete time values (t2= tstart, t2= tstart+1, …, t2= tend). Thecorrelation with the target function is calculated for eachpossible (t1, t2) combination (see dashed lines in Fig. 8a), i.e. fort1b t2. For each t2 only the (t1, t2) combination associated withthe highest correlation is stored. For instance, for the last dateof sample 2 this results in: C[t2= tend]=max {C[t1= tstart;t2= tend], C[t1= tstart+1; t2= tend], …, C[t1= tend−1; t2= tend] }(see red line in Fig. 8a)). The advantage of doing this, is areduction of the number of possible combinations (only M) tobe considered.

(5) For the third sample the same procedure is repeated. Only the(t1, t2, t3) combination associated with the highest correlationis stored (see blue line in Fig. 8b), remembering that not all (t1,t2) combinations are considered anymore. Instead, one t2 is

Fig. 9. Estimated accretion rate by the correlationmaximizationmethod proposed by Yuand Ding (1998) in the presence of a hiatus: dotted line is the true accretion rate; thefull line is the estimated accretion rate.

Fig. 10. Accretion rate estimated by Lisiecki and Lisiecki's method (full line), compared to the true constant accretion rate (dotted line), for the test datasets with potential modelerrors: (a) a periodic signal instead of a sinusoidal signal, (b) an amplitude modulated signal, (c) a sinusoidal signal+a linear trend and (d) an hiatus.



now linked to only one t1 (which we could denote by t1t 2). So,

e.g. C[t3= tstart]=max {C [t3= tend; t2= tstart; t1=t1t2= tstart], C[t3=tend; t2=tstart+1; t1=t1t2= tstart+ 1],…, C[t3=tend; t2=tend−1;t1=t1t2= tend− 1]}. Again, onlyM(t1, t2, t3) combinations remain.

(6) This procedure is repeated for all samples. At the end,M(t1, t2,…,tN) combinations remain and the best one is the associated withthe highest overall correlation (see thick grey line in Fig. 8c).

Both LL02 and YD98 basically use this approach, but they differ inexactly which samples are dated. YD98 maximize the overallcorrelation by varying all observations' dates. If the number ofobservations increases, the number of parameters (time instances tobe estimated) changes, so this method is non-parametric.

LL02 sub-divide the proxy record until the sub-records havereached a predefined length (tuning parameter). In other words, thenumber of points to be dated is fixed in advance. Consequently, thisadaptation makes the approach parametric. Further, their methodincludes a number of refinements. For instance, a mismatch betweenthe sub-record and target function is allowed, depending on user-defined penalty functions, and may change from sub-record to sub-record in order to tie both more closely in certain regions.

4.2.2. Assumptions

1. The target function is known. (In LL02 the target function is really asecond proxy record, since they are in fact matching or aligningpaleoclimate records i.e. with respect to one another, and not to anabsolute chronology.)

2. Two observations must be dated (e.g. the first and last one). This isnecessary to establish a scale for the time grid: is one period100 kyear or 1 year?

3. The proxy and the target function are related by a static linearrelation (to allow the use of the correlation as criterion).

4. The order of the observations is not altered during the procedure.

4.2.3. Tuning parametersNo tuning parameters are used in the method described by YD98.

For LL02's algorithm the number of observations within one sub-record needs to be specified by the investigator, i.e. he/she fixes thenumber of anchor points to be optimized in advance.

4.2.4. Performance in the presence of stochastic errorsThe YD98method reconstructs an accretion rate (not shown) with

an RMS error of 35%. The reason for this high error is that by using allobservations as anchor points the presence and effect of stochasticnoise is neglected. By doing this, the noise in the measurements isinterpreted as significant variations and is propagated into theaccretion rate, which maximizes the correlation between the (noisy)signal and target function as demanded by the algorithm. As anunwanted result the error on the accretion rate will inflate.

A remedy is proposed by LL02: lower the number of anchor points.Fig. 6b shows the results when seven anchor points are used (one foreach year, equal to the number of anchor points used in the anchorpoint method). The error is now quite acceptable (RMS error=1%)and the difference with the true accretion rate falls within theuncertainty bounds. This result will not drastically change if the noiselevel is increased, since the dynamic programming algorithm averagesout those stochastic components.

4.2.5. Performance in the presence of potential model errorsTested on the simulation exampleswith potential model errors, the

correlation maximization method proposed by YD98 performs poorlyon most model errors, i.e. on the periodic signal (RMS error=107%),the amplitude modulation (206%) and on the trended signal (65%).Only for the hiatus, its performance is very good (RMS error=0.35%).The target functionwas sinusoidal, while the true signalwas not for thefirst three cases. Therefore, maximizing the correlation in these cases

will never provide accurate results. In Fig. 9 the reconstructedaccretion rate is shown only for the hiatus simulation. The 0.35%error that remains is a numerical error and can be further reduced byincreasing the number of iterations.

The correlation maximization method proposed by LL02 performsbetter in the presence of model errors, as can be seen in Fig. 10. In thepresence of overtones (a) the error is significant (19%), while it isnegligible in the presence of amplitude modulations (b) and zero inthe trends example (c). In the presence of a hiatus (d), the error (16%)is slightly larger than for the anchor point method, which may besurprising, because here no discretization error is made. Thisdifference is due to the position of the anchor points, which arenow further away from the hiatus.

4.2.6. Conclusion on the correlation maximization methodsWhereas the anchor point method fixes each anchor point in

advance, the correlation maximization methods optimize their dates.The price for this optimization is that a target function must be givenexplicitly. The method proposed by YD98 optimizes the date of eachindividual observation. Consequently, the unavoidable stochasticmeasurement noise of one single observation can have a largeinfluence on the estimated accretion rate. Therefore, this method isnot recommended. This problem is remedied by LL02, who limited thenumber of anchor points. In addition, the method proposed by YD98does not perform well in the presence of model errors, while thecorrelation maximization of Lisiecki and Lisiecki (2002) achieves areasonable performance in reconstructing accretion rates.

4.3. Martinson et al.'s method

In the previous methods, the accretion rate between anchor points isassumed to be constant, which is unlikely to be a realistic representationof a natural growth process. The procedure proposed by Martinson et al.(1982a) avoids this simplification by employing a parametric time basemodel. Thismethod starts fromameasuredproxy record along adistancegrid and a known target function along a time grid. The mappingfunction, related to the accretion rate, is unknown and is expanded in aset of basis functions. This means that the mapping function isrepresented as a linear combination of some other, usually simple,

Fig. 11. Visualization of Martinson et al.'s method. Horizontally the proxy is shown asfunction of a distance grid; vertically the target function is shown as function of time.The accretion rate is expanded in a set of basis functions with unknown coefficients.These coefficients are estimated employing a least squares estimator. Starting from aconstant accretion rate (dotted line), the parametrized accretion rate can be estimated(full diagonal line). Additionally two observations need to be dated (e.g. ‘o’s in the firstand last observation).



functions. Examples of frequently used basis functions are sines andsplines. Each of these functions is multiplied by a coefficient and thesecoefficients can be optimized to obtain the best fit with the observations.Because the number of parameters (coefficients) is smaller than thenumber of observations, the influence of stochastic noise is reduced. If, inaddition, the model is complex enough, model errors will be suppressedtoo. Originally, Martinson et al.'s method was not developed for archiveswith annual periodicities, but to date sediment cores, by comparison tothe insolation over typically several hundreds of thousands of years. It hasalso been used to align different sediment cores (deMenocal et al., 1991;Sowers et al.,1993;Hagelberg et al.,1995; Peck et al.,1996;Williams et al.,1997; Pisias et al., 2001). To construct a mapping function in theframework of this study we assume that the annual signal is sinusoidal.

4.3.1. Method descriptionFor reasons of future convenience, first consider a periodic target

function, described by

smodel tð Þ = A0 +Xhk=1

Ak sin kωtð Þ + Ak + h cos kωtð Þ; ð1Þ

where t is a time variable, Ai (i=0,…, 2 h) are known coefficients,ω is aknown radial frequency and h is the number of harmonics, which isdefined by the user. By analogywith the previousmethods, it is assumedhere that the target function is sinusoidal, hence h=1 (see also remark1 in Section 4.3.2).

The measured signal, s(t), is sampled at time instances

t nð Þ = n + δ nð Þ½ �Ts; ð2Þ

with n the sample number (n=1,…,N, withN the number of samples),Ts the sample period and δ(n) the unknownmapping function. The latteris parameterized by a sum of b basis functions, ϕp(n) (p=1,…, b) withtheir unknown coefficients Bp (Martinson et al., 1982a),

δ nð Þ =Xbp=1

Bp/p nð Þ: ð3Þ

Depending on prior knowledge, different classes of basis functionscan be chosen. In the next section this choice is further discussed(remark 2). Regardless of which basis functions are chosen, it can bestated that the higher b is, the more flexible the mapping function is,but also the higher the sensitivity for noise will be. So, the actual valueof b will also influence the quality of the final results and should bechosenwith care (cf. remark 3 in Section 4.3.2). Note that b can also bezero (no variation is significant).

Once the class and the number of basis functions is determined,the unknown parameters Bm, stored in vector B, can be estimated byminimizing a non-linear least squares cost function

B = arg minB

K Bð Þ ð4Þ

with

K Bð Þ = 12

XNn=1

s nð Þ−smodel t nð Þð Þ½ �2: ð5Þ

Fig. 11 illustrates Martinson et al.'s method:

1. A target function must be postulated. Within the scope of thisstudy, sinusoidal target functions are considered.

2. Two observations need to be dated (e.g. the first and last one,shown by the ‘o’s), in order to establish a time scale.

3. An initial estimate of the time base (mapping function) is neededto initialize the parameters: e.g. the dotted line in Fig. 11.

4. A least squares cost function is minimized to optimize theparameters B. The corresponding accretion rate is shown by thefull line in Fig. 11.

4.3.2. ImprovementsIn the previous section, the general idea proposed byMartinson et al.

(1982a) has been presented. However, it appears that for an efficientapplication of this method to periodic, annually resolved records anumber of adaptations are useful.

4.3.2.1. Choice of target function. Martinson et al. originallydeveloped this method to match proxy records on orbital forcingparameters on longer time scales. Here, the method is used on a sub-annual scale, which necessitates a different kind of target function tobe assumed. Within the scope of this study, it seems logical to assumethe target function to be periodic (see Eq. (1)). However, to have afully defined target function, all amplitudes and the radial frequencyshould be known or (in practice) assumed. In order to reduce thenumber of assumptions and analogous to the correlation maximiza-tion methods, we have further assumed a sinusoidal signal (targetfunction), i.e. h=1. In other words, only four parameters remain to bechosen: the mean, the amplitude, the frequency and the phase of thesine. These can either be decided by the user, based on priorexperience or on independent information about the system. Forthose cases when this knowledge is not available, we suggest thefollowing data-based procedure to determine suitable values for thesine parameters: find a first guess target function by mapping thedistance series and assuming a constant accretion rate. Thenapproximate this first guess target function by a sine, by tuning theunknown parameters (amplitude, frequency and phase) until anoptimal match has been found. These sine parameters now fullydetermine the target function.

4.3.2.2. Choice of basis functions. Martinson et al. (1982a) proposedto use trigonometric functions as basis functions, which are mostsuitable if the variations in accretion are periodic. Alternative basisfunctions can be used depending on the available prior knowledgeabout the shape of the mapping function, without changing theessence of the method. For this study we decided to use splines(Dierckx, 1995) as basis functions, because

(i) Splines are appropriate to describe smooth variations (i.e. if forone observation the value is high, it is also quite high for theneighboring observations), which seems a more realisticassumption than periodic variations.

(ii) Splines are actually a generalization of the previous methods.Indeed, if zero-order splines are used, the accretion rate againbecomes stepwise constant.Wewill use third order (cubic) splines,in order to estimate a continuous and smooth accretion rate.

4.3.2.3. Choice of number of basis functions. How many times basecoefficients should be chosen (b in Eq. (3))? This problem is handledempirically in Martinson et al. (1982a): “The number of coefficientscould for example be equal to the number of periods in the proxy”. Notheoretical considerations were used to estimate the number of para-meters. In order to stay as close as possible to the original method, wefollowed this rule of thumb to select the number of coefficients.However, the method could be further improved if the number ofcoefficients (i.e. the complexity of the mapping function) was chosenusing an objective model selection criterion (see Section 5.3.1).

4.3.2.4. Initial values for optimization. Which initial values shouldbe used for the unknown coefficients, B? This appears to be a key-choice to eventually find realistic results. Indeed, the optimization iscomplicated by local minima in the cost function surface (Martinsonet al., 1982a) and the global minimum can only be found if good



initial values are chosen. Martinson et al. (1987) proposed analternative strategy to generate initial values, using an independentradiometric chronology, but this procedure introduces new problemswhich can only be solved by making additional hypotheses about thechronology. We propose to generate initial values for the time baseparameters by the phase demodulation technique (De Ridder et al.,2004), which is discussed in more detail in Section 5.2. In principle,the anchor point method or correlation maximization could be usedtoo.

4.3.2.5. Estimation of the average accretion rate. Martinson et al.estimated the time base in an absolute sense,whileweprefer to start fromanaverage constant accretion rate, disturbedbya timebasedistortion (seeEq. (2)). Theadvantageof thisprocedure is that themethod is less sensitiveto the choice of initial values—and thus numerically more reliable.

4.3.3. Assumptions1. The target function is known.2. The time base can be modeled by a limited number of ‘simple’

functions.3. The relation between the proxy and the target function is static and

linear. A generalization to linear dynamical models is made byBrüggeman (1992).

4. The noise has zero mean (assumption for the least squaresestimator to be consistent). If in addition the noise is Gaussianand identically and independently distributed (i.i.d.), the leastsquares estimator exhibits the additional property of giving thelowest parameter variance (maximum likelihood properties). If thenoise variances are not identical (i.e. each measurement isassociated with a different uncertainty) and known, a weightedleast squares cost function can be used.Two additional assumptions were added in order to apply thismethod to periodic archives:

5. The target function is sinusoidal (cf. remark 1 in the previoussection). If prior knowledge about the system indicates that moreharmonics must be included, and their associated amplitudes areknown, a more complex target function may be used.

6. Two observations are dated, in order to establish the time scale.

4.3.4. Tuning parametersThe number of coefficients used to parametrize the accretion rate

(b) has to be specified by the user (see also remark 3 in Section 4.3.2).

4.3.5. Performance in the presence of stochastic errorsThe number of parameters used to reconstruct the mapping

function and accretion rate was seven, resulting in a RMS error of 0.9%(see Fig. 6c). Compared with the method of Yu and Ding (1998) (RMSerror=35%) this value is low, while it is similar to the error obtainedwith Lisiecki and Lisiecki's method (1%). The small variability (i.e.non-constancy) present in the accretion rate is due to the relativelylarge number of basis functions. If Martinson et al.'s method had beenimplemented with a model selection criterion to estimate the desirednumber of parameters (cf. remark 4 in Section 4.3.2), this RMS errorwould have been even lower.

4.3.6. Performance in the presence of potential model errorsTested on the simulation examples shown in Fig. 4, Martinson et

al.'s method performs well: the RMS error of the accretion rate is 3%when the proxy record consists of two harmonics, even if the targetfunction is only sinusoidal. The estimated accretion rate is shown inFig. 12a. The method is also robust in the presence of an amplitudemodulated signal (RMS error of 4%—Fig. 12b). On the trended signal,Martinson et al.'s method performs slightly worse (RMS error of 6%—Fig. 12c). On a discontinuous record, Martinson et al.'s method is ableto reconstruct the hiatus with an RMS error of 18% (Fig. 12d). Thereason for this error is that discontinuity in the growth rate caused bythe hiatus cannot be accurately described by a smooth δ(n), except ifits complexity were specially adjusted to describe the hiatus. If thenoise level increases, this will not influence the results drastically.However, if the number of parameters is increased with an identicalnoise level, the stochastic error will increase. If, however, the numberof parameters were decreased, the stochastic error would decrease aswell, but so will the accuracy of the results.

4.3.7. Conclusion on Martinson et al.'s methodThis method is insensitive to the presence of stochastic noise and

performs well in the presence of most tested model errors. However,

Fig. 12. Accretion rate estimated by Martinson et al.'s method (full line), compared to the true constant accretion rate (dotted line), for the test datasets with potential model errors:(a) a periodic signal instead of a sinusoidal signal, (b) an amplitude modulated signal, (c) sinusoidal signal+a linear trend and (d) hiatus.



the performance of the method depends on the choice of the targetfunction and the complexity of the function used to model themapping function (i.e. the number of basis functions).

5. Class 2: signal model methods

Formany annually resolved archives it seems reasonable to assumethat they possess a large periodic component. However, it is muchharder to predict exactly how this component will look like. So far, wehave mostly assumed that it was sinusoidal (except for the anchorpoint method), with fixed amplitude, frequency and phase—and haveconcentrated on the mapping function (mapping methods). Thisprocedure is perfectly suitable as long as the time series is reallysinusoidal with known properties. Otherwise, model errors areintroduced.

Therefore, in the three remainingmethods, the assumptions on thetime series (target function) are loosened. Remember that recoveringtime series and accretion rate from a measured distance series (Fig. 1)is only possible if additional assumptions are made. In the signalmodel methods, a model class is assumed for the target function,implying the general shape of the function, but leaving a number ofmodel parameters free to be optimized in order to best describe thedata. In practice, these signal model parameters will be optimizedsimultaneously with the time base (mapping function). This makesthe procedure more flexible than the mapping methods, where theexact target function (or the exact dates of some observations, for theanchor point method) is to be known or assumed.

Two distinct non-parametric signal model methods exist inliterature. The first acts in the time domain (Wilkinson and Ivany,2002; Section 5.1), the second in the frequency domain (DeRidder et al.,2004; Section 5.2). Finally, a parametric method is discussed, which is ageneralization of Martinson et al.'s method (de Brauwere et al., 2008;Section 5.3).

5.1. Time domain method developed by Wilkinson and Ivany (2002)

5.1.1. Method descriptionThe non-parametric approach proposed by Wilkinson and Ivany

(2002) was originally developed to enable the computation of annualaverages or seasonal ranges of measured proxies (cf. introduction;Ivany et al., 2003) but can be used to estimate the annual average ofany irregularly sampled time series (Dutton et al., 2005). The methodagain assumes that the time series is sinusoidal. A window movesstepwise over the proxy record, each step corresponding to an

observation. For each window the amplitude Amp, period Per, phasePhs, and position (offset) Pos are estimated by matching

SWI mð Þ = Amp mð Þ2

sin Wmid − Phs mð Þ½ � 2πPer mð Þ

� �+ Pos mð Þ ð6Þ

in a least squares sense on the samples in the window. In Eq. (6) thesame notation is used as in the original publication (Wilkinson andIvany, 2002) to facilitate reference. However, as the symbols are ratherunusual, their definition and dimension is presented in Table 1 andsome of them are illustrated in Fig. 13.

In those parts of the record where the accretion rate was large, thiswill be reflected in the frequency parameter, which will be low andvice versa. By moving the window over the observations, thefrequency is estimated over the total length of the record and thevarying accretion rate can be calculated. This is the basic idea of themethod proposed by Wilkinson and Ivany (2002).

5.1.2. Improvement by De Ridder et al. (2007)However, when inspecting the problem more closely, it becomes

clear that variations in accretion rate do not only modulate thefrequency, but also the phase. So, if wewant to reconstruct the accretionrate from a stepwise fitting of Eq. (6) to the observations, the phase hasto be taken in to account aswell. An adaptation to do this was presentedin De Ridder et al. (2007) and it is this refined method which is used inthe remainder of this paper. The new equations are briefly presentedbelow, but for more details we refer to De Ridder et al. (2007).

No expression for the time is present in Eq. (6). Therefore, we haveto assume that the time series is described by

Ysin =Amp02

sin2πPer0

Time + Phs0

� �+ Pos0: ð7Þ

For a definition of the symbols, again refer to Table 1. Note that nowthe method turns out to be a mapping method, since Eq. (7) has nodegrees of freedom left. The variable Time can be separated byequating Eqs. (6) and (7) and by assuming that the amplitude andoffset do not change with time

Amp = Amp0 and Pos = Pos0; ð8Þ

which leads to

Time = Per0Wmid − Phs

Per− Phs0

2π

� �: ð9Þ

In this equation, the assumed period Per0, acts as a scaling factor: ifthe basic period increases, the total length of the record will increase.The assumed phase Phs0, acts as an offset in the time base: increasingthis phase will translate the signal in time.

5.1.3. Assumptions1. The time series of the proxy record within a window of width c is

modeled by a sine.2. The variations in accretion rate are small within the width of the

window, such that it is acceptable to approximate the accretion rateto be constant within this window.

3. At least two observations are dated, in order to establish a time scale.4. For the improved method, it must be assumed that no amplitude

modulation is present in the signal.5. The noise has zero mean (cf. assumption 4 in Section 4.3.3).

5.1.4. Tuning parametersThewidthof thewindowis theonly tuningparameterused. Enlarging

the window will decrease the noise influence on the parameters, but atthe same time it will lower the time resolution of the variations inaccretion rate. Decreasing the window width will improve the time

Fig. 13. Illustration of the parameters used to estimate the amplitude, frequency andphase in Wilkinson and Ivany's method. The observations are shown by the ‘+’, thedotted line is the model and the rectangle represents the time window.



resolution of the reconstructed accretion rate, but the estimatedaccretion rate will be more sensitive to stochastic errors.

5.1.5. Performance in the presence of stochastic errorsThe accretion ratewas estimated with awindow of ten observations

and the resulting RMS error was 0.9% (see Fig. 6d). If the noise levelincreases, this will have a large effect on the results, since this method isnonparametric, stochastic noise will be propagated into the computedresults.

5.1.6. Performance in the presence of potential model errorsTested on the simulation examples with potential model errors,

the adapted time domain method performs quite poorly, as can beseen in Fig. 14. The RMS error for the periodic signal is 8% (Fig.14a), forthe amplitude modulated signal 9% (Fig. 14b), for the trended signal

7% (Fig. 14c) and for the hiatus 14% (Fig. 14d). The window width wasagain ten samples. Decreasing this would lower the errors for theamplitude modulated signal, because the amplitude variation wouldbe smaller in a smaller window. It would also reduce the error for thehiatus, because fewer windows would cover the hiatus.

5.1.7. Conclusion on the time domain methodThe adapted time domain method (Wilkinson and Ivany, 2002; De

Ridder et al., 2007) performs well in the presence of noise, but can besensitive to model errors. In addition, the method seems veryvulnerable to the choice of the window width.

5.2. Frequency domain method: a phase demodulation approach

Before the phase demodulation approach is discussed, it is useful toarguewhy an approach in the frequency domain has advantages. Fig. 15shows the Fourier spectra of the different simulated records withmodelerrors (Fig. 4 showed them in the time domain). In the time domain, themodel errors are ‘distributed’ over thewhole record, which complicatestheir separation from the relevant information. On the contrary, in thefrequency domain the model errors in cases (a) (overtone) and (c)(trend) appear at different frequencies than the first harmonic, andhence it is straightforward to correct for them. Unfortunately, for theremaining model errors (amplitude modulation and hiatus), the modelerror appears around the first harmonic and can thus still influence thetime base. Notice that many noise properties are defined in thefrequency domain, like white noise (power density is constant over afinite frequency range). To summarize, in order to identify the differentcomponents of a signalmodel (especiallywhenconcernedwithperiodicsignals), the spectral interpretation can be beneficial.

5.2.1. Method descriptionThe phase demodulation method works in the frequency domain to

identify the time base. As will be shown, the robustness againststochastic noise and model errors is larger than the similar method inthe time domain, however it is certainly less intuitive. An extendeddiscussion of this method can be found in De Ridder et al. (2004) and ithas been applied in a few studies on bivalves (Gillikin et al., 2005b,c,2008). It is assumed that the time series is periodic and bandwidth

Fig. 15. The spectra of the different proxy records (full line) and the assumed targetfunctions (dotted line) are shown: (a) a periodic signal instead of a sinusoidal signal,(b) an amplitude modulated signal, (c) sinusoidal signal+a linear trend and (d) hiatus.

Fig. 14. Accretion rate estimated by Wilkinson and Ivany's method (full line), compared to the true constant accretion rate (dotted line), for the test datasets with potential modelerrors: (a) a periodic signal instead of a sinusoidal signal, (b) an amplitude modulated signal, (c) sinusoidal signal+a linear trend and (d) hiatus.



limited. This means that it can be modeled by a limited number ofharmonics in the frequency domain. The method is based on theobservation that when a periodic signal is distorted along the time axis,this is translated in the frequency domain by a broadening of theharmonic peak (see De Ridder et al. (2004) for an example). Thismeansthat somehow the information on the time base distortion is hidden inthese parts of the spectrum. In fact, this feature is similar to what isobserved when a periodic signal is subject to a phase modulation. Inother words, variations in accretion rate act as a phase modulation onthese harmonics. Hence a phase demodulation technique can recon-struct the accretion rate, according to the following procedure.

1. In order to transform the measurements from the time domain tothe frequency domain, we need an initial time base. For that reason,we assume a constant accretion rate

t nð Þ = nTs ð10Þ

with Ts the mean period between to subsequent samples.2. The complete proxy record s(n) is transformed into the frequency

domain by a Discrete Fourier Transform (DFT). These transformeddata S(ω) now depend on frequency instead of time

S ωð Þ = DFT s nTsð Þ½ �: ð11Þ

3. So far, we made a mistake, since the true accretion rate is notconstant. The clue of this method is that it is possible to estimatethis mistake. It will appear in the spectrum of the data as ghostpeaks, often found symmetrically around the harmonics and can beisolated by windowing the spectrum S(ω) with a rectangularwindow c(ω) around the first harmonic. This windowing cuts out abandwidth in the frequency domain around the first harmonic

δ ωð Þ = c ωð ÞS ωð Þ; ð12Þ

with c(ω)=1 ifω is in the frequency range of thewindow and c(ω)=0elsewhere. If a narrowwindow is chosen, only the low frequency in thetime base distortion will be isolated, because the high frequencycomponents will be found further away from the harmonic.4. Finally, we are interested in the distortion from the average accretion

rate in the time domain. This can be computed by translating the

estimate δ(ω) to the origin of the spectrum, applying the InverseDiscrete Fourier Transform (IDFT) and taking the phase of this. δ(ω)is the Fourier transform of the estimated time base distortion. Thisresults in the estimationof the sought timebase distortion, δ(n) (for adefinition see Eq. (2))

δ nð Þ = u exp −jnω0Tsð ÞIDFT δ ωð Þ½ �ð Þω0

ð13Þ

with φ(u) the phase of u, j=ffiffiffiffiffiffiffiffi−1

p, ω0 the fundamental radial

frequency (2π/age) and Ts the sample period. The exponential termshifts the frequency by −ω0.

To summarize, transforming the time signal to the frequencydomain enables the reconstruction of the time base distortion (TBD),δ(n), by a relatively easy calculation. Once the TBD is calculatedemploying Eq. (13), every observation can be dated using Eq. (2).

5.2.2. Assumptions1. The proxy time series is periodic and bandwidth limited;2. The accretion rate is bandwidth limited.3. At least two observations are dated, in order to have a time scale or

an age of the record.

5.2.3. Tuning parametersThe only tuning parameter is the width of the frequency domain

window c. Enlarging the window's width will improve the timeresolution of the accretion rate, but at the same time the noiseinfluence will be larger. Decreasing the window width will decreasethe resolution, but at the same time the noise influence decreases.These properties are opposite to those in the time domain.

5.2.4. Performance in the presence of stochastic errorsThe accretion ratewas estimated by using Eq. (13) for the TBDwith a

window of 1 year−1 around the first harmonic. An RMS error of 0.7% isassociated with the estimated accretion rate (see Fig. 6e). This low errorresults from the frequency domain approach.White noise is spread overall frequencies. Since only a (small) part of the spectrum is used tocompute the TBD and subsequently the accretion rate, the noise on all

Fig. 16. Accretion rate estimated by the phase demodulation method (full line), compared to the true constant accretion rate (dotted line), for the test datasets with potential modelerrors: (a) a periodic signal instead of a sinusoidal signal, (b) an amplitude modulated signal, (c) sinusoidal signal+a linear trend and (d) hiatus.



other lines in the spectrum is filtered off and does not influence theresults.

If the noise level increases, thiswill affect the results.More stochasticnoise will be captured in the frequency window and will thereforeinfluence the estimated accretion rate and increase the associateduncertainty. However, themajority of the stochastic noisewill still act onfrequencies outside thewindowand thus its overall effect is still reducedcompared to a similar approach in the time domain.

5.2.5. Performance in the presence of potential model errors

Tested on the simulation examples shown in Fig. 4, the phasedemodulation method works well, because it was not assumed thatthe signal was sinusoidal. Fig. 16 shows the estimated accretion rates(again a spectral window of seven lines was used to separate the timebase distortion): the second harmonicwas not covered by thewindowused to cut out an estimate of the time base distortion. Consequently,no errors are made (Fig. 16a). A very small error is made for theamplitude modulated signal (Fig. 16b), where an RMS error of 2% isfound. Most of the trend appeared at the low frequency componentsof the spectrum and did not overlap with the window, again resultingin a RMS error of 8% (Fig. 16c). The hiatus produced the largest error(12%) because the discontinuity is spread out in the frequency domain(see Fig. 15d), i.e. it is not bandwidth limited, and is consequently onlypartly covered by the window.

5.2.6. Conclusions on the phase demodulation method

The phase demodulation technique performed well in thepresence of stochastic noise and in the presence of the model errorstested here. However, this is partly due to the subjective choice ofusing a window of seven lines to separate the time base distortion.One could choose a different window width and would then drawdifferent conclusions from the same data. To address this subjectiv-ity, the last method discussed in this overview uses a parametricmodel combined with an automated model selection criterion toselect the most appropriate model complexity according to themeasurements.

5.3. Parametric time base distortion method, coupled to an automatedmodel selection procedure

5.3.1. Method description

This method combines Martinson et al.'s method for theestimation of the time base with a parametric representation ofthe signal, which will make it less vulnerable to model errors (deBrauwere et al., 2008). This is a parametric method, implyingparameters to be optimized. But first they need to be initialized. Thiscan be achieved by, e.g., the phase demodulation method. In addition,this method employs an automated model selection criterion, whichis able to estimate the optimal complexity of the model, i.e. howmany terms/parameters are necessary to model the target andmapping function? Such an automated model selection procedureenables to simultaneously maximize accuracy and precision in anobjective way.

Martinson et al.'s method, as described by Eqs. (1)–(5) can begeneralized in order to handle unknown periodic signals. Therefore, it isassumed that the time series is described by Eq. (1). In contrast withMartinson et al.'s method, the amplitudes Ai (i=0,…,2 h), frequency ω,and number of harmonics h have to be estimated based on theobservations and are not subjectively chosen by the operator. The timevariable, t, is unknown, due to the variations in accretion rate, and can bedecomposed into a constant time step and a distortion of this constantstep, as in Eq. (2). The time base distortion, δ(n), depends on thecoefficientsB and on the number of coefficients b (Eq. (3)), whichwill beoptimized too.

1. First, the investigator has to set the most complex signal and timebase model that will be considered, which are defined by thenumbers hmax and bmax. The number of parameters of this mostcomplex model is 2hmax+bmax+2 (see Eqs. (1)–(3) and (5)) andthis number cannot be higher than the number of observations,otherwise the system is underdetermined and hence not allparameters are (uniquely) identifiable. Most probably, the dataprofile is not that complex and the investigator may decide to limitthe most complex model to a smaller complexity.

2. Secondly, all models with complexity ≤(hmax, bmax) are optimizedindividually. The parameters of a model with complexity (h,b) aregrouped in a vector,

θ = A0;A1; :::;A2h;B1; :::;Bb;ω½ �: ð14Þ

The optimal values of these parameters in a least squares senseare found by

θ = argminθ

K θð Þ; ð15Þ

with K(θ) defined in Eq. (5).3. Thirdly, the optimal signal and TBD model complexities, i.e. h and

b, have to be estimated. This is achieved by assigning a value toeach model which expresses how suitable it is to describe the data.This value cannot simply be a measure of the fit, since this wouldonly assess the accuracy of the model and will generally designatethe most complex model as the best one. Too complex models havethe disadvantage of being sensitive to noise, and hence are notprecise. Instead, an intermediately complex model must bedetermined, which is complex enough to avoid most systematicerrors, but is not so complex that it actually fits the noise in thedata. To achieve this trade-off, the model selection criterionconsists of two factors, one expressing the model fit (namely theresidual cost function), which decreases with increasing modelcomplexity, and one so-called “penalty factor” which penalisesincreasing model complexities. Several expressions have alreadybeen proposed for this penalty factor (Akaike, 1974; Rissanen, 1978,Schwarz, 1978), but its most appropriate form depends upon thesignal-to-noise ratio and the availability of a noise model (Schou-kens et al., 2002; De Ridder et al., 2005; de Brauwere et al., 2005). Inthe examplesworked out in this paper the criterion to beminimizedhas the following expression (De Ridder et al., 2005):

MDLc =K θ� �N

exp pc nθ;Nð Þ½ �with pc nθ;Nð Þ = ln Nð Þ nθ + 1ð ÞN − nθ − 2

; ð16Þ

with N the total number of observations and nθ the number of freemodel parameters. The penalty factor was derived for cases where themeasurement noise variance is not explicitly known, but is normallydistributed. To construct this penalty term, it is assumed that one of themodels from the model set is the ‘true model’. The model with thehighest probability to be that ‘true model’ is selected. If the aimwas toselect the model that would result in the smallest mismatch if themeasurements were repeated, a different penalty term should be used(Akaike,1974). If themeasurement noise covariancematrix is known, amodified penalty term should be used (De Ridder et al., 2005).

The advantage of introducing a model selection criterion elim-inates all interferences from the user regarding the choice of themodel complexity, which makes the proposed method more objectiveand user independent. Of course, the choice of a periodic signal modelis still made by the user. Practically, this means that only a model sethas to be defined, i.e. the set of all periodicmodels with complexities hand b≤predefined maximum values hmax and bmax. Next, all modelsof this set are used to fit the observations, and Eq. (16) can be used toselect the best model (and thus best estimated time base and timeseries) within this set.



5.3.2. Assumptions1. The signal is periodic and bandwidth limited.2. At least two observations are dated, in order to establish a time

scale.3. The noise is Gaussian (assumption underlying the MDLc criterion

for model selection).

The first assumption can in principle be replaced by anotherproperty of the signal, like a periodic signal with trend, or apolynomial signal, a signal with multiple non-harmonically relatedperiodicities, etc., as long as it can be expressed in a mathematicalexpression of reasonable complexity w.r.t. the data. The mainlimitation is that in some cases it may be difficult to find good initialvalues, because the phase demodulation method cannot be used fornon-periodic signals.

5.3.3. Tuning parametersNo tuning parameters are used in this method. The maximal

complexity of the time base distortion model (bmax) and of the signalmodel (hmax) can be set by the user in order to limit the number ofmodels in the model set to be considered for model selection. But ifcalculation time is no constraint the maximal number of parameterscan be chosen (bmax+2hmax+2=the number of observations) andthe model selection criterion will select the best model from thisgeneral model set.

5.3.4. Performance in the presence of stochastic errorsAssuming white noise, the method proposed above is able to

detect that no variations in accretion rate occurred, i.e. the modelselection criterion rejected all time base distortion parameters andthus selected the model with b=0. Consequently, the RMS error andthe uncertainty on the time base were zero.

Increasing the noise level will not influence the results much. Thishas two reasons: the optimization filters off most of the noise and theerror-propagation into the parameters is minimized by tuning thenumber of parameters. However, note that some featuresmay no longerbe significant if the noise level increases which will result in the modelselection criterion rejecting that part (or complexity) of the model.

5.3.5. Performance in the presence of potential model errorsTested on the simulation examples with model errors, shown in

Fig. 4, the combination of a parametric signal and time base distortionmodel, with amodel selection criterionperformedwell. For the periodicsignal (Fig. 17a), the amplitude modulated signal (Fig. 17b) and thetrended signal (Fig. 17c) the method was able to reconstruct the correctaccretion rate. Only for the hiatus this parametric time base distortionapproach produced a quite large error (RMS error of 19%—Fig. 17d). Thereason is the sameas for thephase demodulationmethod: the hiatus is adiscontinuity in the time domain,which is spread over all frequencies inthe frequency domain. The model selection criterion rejects the morecomplex models, which results in a simple time base distortion modelwith a large error. This error can, at least partly, be avoided if theindividual uncertainties on the observations are known. In that case adifferent penalty factor can be used which does not penalize complexmodels so strongly (De Ridder et al., 2005).

5.3.6. Conclusion on the parametric time base distortion methodThe parametric time base distortion approach performs well in the

presence of stochastic noise as well as in the presence of most modelerrors tested here. This is partly a result of the model selectioncriterion, used to tune the complexity such that the stochastic noise isfiltered off, without introducing systematic errors.

6. Comparison of the methods

The methods described above are compared on three criteria:robustness to stochastic noise (Section 6.1), the use of tuningparameters (Section 6.2) and the sensitivity to model errors andassumptions used (Section 6.3).

6.1. Robustness to stochastic noise

The most important property of an accretion rate reconstructionmethod is that it must be relatively insensitive to stochastic noise,because every record isdisturbedby it. If this noise ispropagated into thereconstructed time base, the time series becomes unreliable too. In

Fig. 17. Accretion rate estimated by the parametric time base distortion approach (full line), compared to the true constant accretion rate (dotted line), for the test datasets withpotential model errors: (a) a periodic signal instead of a sinusoidal signal, (b) an amplitude modulated signal, (c) sinusoidal signal+a linear trend and (d) a hiatus.



general, non-parametric methods are more sensitive to stochastic noisethanparametricmethods, as is illustrated inTable 2. Parametricmethodsreduce the effect of noise in themeasurements by the use of a relativelysimple parametric model for the observations. This “simplification”enables, if it is well chosen, to suppress the stochastic noise and onlykeep the relevant signal. The reason why the Parametric time basedistortion (TBD) method works best is because it is able to accuratelychoose the complexity of the parametric model. The other parametricmethods could be further improved if they were also combined withsuch a model selection procedure. Apart from the parametric methods,the non-parametric methods by Wilkinson and Ivany (2002) and DeRidder et al. (2004) also perform well. This is a consequence of theaveraging (sliding window) process present in both methods, whichactually filters off the high frequency components being mostly noise.

6.2. Use of tuning parameters

Tuning parameters must be defined by the investigator. Conse-quently, the same record can be converted to a different time seriesusing the samemethod butwith different tuning parameters. Therefore,it is better to eliminate tuning parameters whenever possible. Only thecorrelation maximization proposed by Yu and Ding (1998), and theparametric TBD approach (de Brauwere et al., 2008) do not use tuningparameters (Table 2).

6.3. Assumptions and model errors

As was already repeatedly mentioned, it is impossible to transforma proxy record into a time series without making additionalassumptions. Depending upon the nature of the record, the assump-tions made by one method may be better than those of an alternativemethod. For example, if the record does not showany pattern, the onlymethod that can be used is probably the anchor point method,because all the others somehowmake assumptions about the shape ofthe time series. On the other hand, if the record seems to be sinusoidal,

mapping methods assuming a sinusoidal target function can be used.In the real world, the investigator does not know which assumptionsare correct and what is worse: the result of the violation of one orseveral assumptions is hard to predict. We have tried to test themethods on some possible violations of the assumptions. Table 2 alsosummarizes the performances of the methods in the presence of thetested model errors. Note that the list of possible model errors is farfrom complete and that the fact that a method performs well in thepresence of one particular model error does not guarantee that it willwork well on all real world records.

The worst method appears to be the correlation maximization by Yuand Ding (1998), because the accretion rate is not bounded by anyexpression or tuning parameter. Consequently it performs well in theabsence of any errors, but in all other cases, these errors are fullypropagated into the accretion rate. The most reliable method is theParametric TBDmethod, although it is not copingwell when reconstruct-ing a signal with hiatus. This kind of signal appears to be a problem formost methods, because a hiatus cannot be described by the assumedtarget functions. Only methods using individually tuned anchor points(anchor pointmethod andmethod by Yu andDing (1998)) can copewitha growth stop, if they are well tuned (possibly by hand).

7. Real world example

7.1. Dataset description

So far, the different methods have been compared on syntheticdata. In this paragraph a real world example is described. The δ18O-signal measured in a Saxidomus giganteus is used (see Fig. 18). Thespecimen was sampled in Washington state along the West Coast ofthe U.S.A. (Gillikin et al., 2005b). The large winter–summer variationsare reflected in these signals and thus it seems appropriate to usemethods assuming a periodic signal.

7.2. Tuning parameters for all methods

For the anchor point method the annual minima are taken asanchor points. For Lisiecki and Lisiecki's method the observations aredivided in ten boxes and the dates of their borders are optimized. ForMartinson et al.'s method one basis function per year was used, i.e.b=9. For Wilkinson and Ivany's method a window of 50 observationsis used, which spans approximately 3.4 years. Smaller windows gaveunrealistic accretion rates, probably due to too large stochastic noiseinfluences. For the phase demodulation method a window width of1 year−1 was chosen. For the parametric approach, the maximumcomplexity consisted of bmax=20 basis functions and hmax=4harmonics. The automated model selection procedure selected anoptimal complexity of b=9 and h=1. The method by Yu and Ding(1998) is not applied because the assessment above has shown it isunreliable as soon as noisy data are involved.

Table 2Table summarizing the characteristics and performance of the compared methods.

Method Parametricmodel

Tuningparameters

Target function mustbe known

RMS error for noisysimulation

RMS error for simulations with model error

Periodic Amplitude modulated Trend Hiatus

Anchor points × ×1 9% 5% 7% 7% 13%Yu and Ding (1998) × 35% 107% 206% 65% 0.35%Lisiecki and Lisiecki (2002) × × × 1% 19% 1% 0 16%Wilkinson and Ivany (2002) × × 0.9% 8% 9% 7% 14%Phase demodulation × 0.7% 0 2% 8% 12%Martinson et al. (1982a, 1987) × × × 0.9% 3% 4% 6% 18%Parametric TBD × 0 0 0 0 19%

1The anchor point method does not really require a target function but only the dates of the anchor points. These dates can be derived from independent datings (e.g. special events insediment or ice cores or inflicted stains/wounds in some shells or trees). However, in many cases these dates are assigned based on an assumed target function (e.g. seasonalperiodicity).

Fig. 18. Real world example: δ18O-signal measured in the skeleton of a Saxidomusgiganteus.



7.3. Results

Fig. 19 shows the accretion rates estimated with the differentmethods. The decreasing growth rate is recovered byallmethods, whichis a typical trend for bivalveswhich grow slower as they becomeolder. Asecond common feature in Fig. 19 is the large peak found between thefirst and the second year. The precise date of this peak varies frommethod tomethod, because the dates of the observations are estimatedas well and consequently are subject to errors. Most methods alsorecover two other peaks: one after three years and one after five years.No seasonal cycle is observed in the growth rates. This cycle may beoccurring in reality; these results only show that there is not enoughinformation in the data to recover this high frequency variation.

Looking at the individual reconstructions, one can notice that theanchor point method's reconstruction is quite rough, because onlyannual means are computed. Obviously, the growth rate evolutionwillshow more detail if more anchor points are used but this would alsorequiremakingmore assumptions (about the dates of these additionalanchor points). The anchor point method's result is similar to the onereported in the original study (Fig. 6 in Gillikin et al., 2005b), althoughthe peaks are slightly shifted due to the use of different anchor points(but the same number): while we used the annual δ18O minima, theyused the maxima. Both choices are justifiable and illustrate theimportance of the user's input for the anchor point method. Lisieckiand Lisiecki's method is already an improvement, but the accretionrate remains stepwise constant. Wilkinson and Ivany's methodappears more fitful compared to the others methods. The threeremaining approaches show more or less the same features. Onereason for this is that the automated model selection procedureselected the same complexity for the time base and signal models asthose we as user chose for Martinson et al.'s method. In addition the

time base parameters of both methods are initialized employing thephase demodulationmethod. The small difference betweenMartinsonet al.'s method and the parametric approach (both using a model withthe same complexity), is caused by the optimization routine: the signalmodel remains frozen for Martinson et al.'s method, while in the otherit is optimized simultaneously with the time base. For the phasedemodulation method a window of 1 year−1 was chosen. Thiscorresponds to 9 lines in the discrete Fourier transform, which iscomparable to the 9 timebase distortionparameters used inMartinsonet al.'s method or the parametric approach. Consequently, thebandwidth of these three methods is comparable. The different resultof the phase demodulation is mainly due to the lack of optimizationand thus to a larger noise perturbation in the accretion rate.

8. Conclusion

With this study we wanted to focus attention on possible pitfallsand errors inducedwhen transforming a proxy recordmeasured alonga distance axis into a time series. Indeed, this transformation is mostoften not straightforward and the investigator has to make someassumptions about the time series. The first step towards a soundscientific solution is to be aware of the assumptions made. The secondstep is to appreciate the consequences of these assumptions. Indeed,depending upon these assumptions the constructed time series andaccretion rate evolutionwill be more or less sensitive to stochastic andmodel errors. In this study, several methods to establish a time basefor environmental proxy records are classified, described, applied on anumber of test cases and compared, with special attention towardstheir different assumptions. Our hope is that this effort brings somestructure in this rarely discussed topic.

Fig. 19. Accretion rate for the real dataset, estimated by the six methods. Except for (d) all figures have the same axes.



Users can check the performance of their method on theinvestigated test cases, and refresh their knowledge on the underlyingassumptions by reading the respective sections. The main results canbe summarized as follows:

1. The methods can be classified in mapping methods and signalmodel methods. The first class needs the target function to be fullydefined by the user. Especially for those cases where the time series(which is closely related to the target function) is of actual interest,this assumption is to some extent intellectually dishonest, since itwill result in a reconstructed time series equal to what one haspostulated in the first place.

2. The anchor point method's performance depends on the chosenanchor points (a hypothesis which cannot be checked!). If they arechosenwell, the resulting time base is reasonable. However, even inthat case a discretization error is unavoidable.

3. Correlationmaximizationmethods (inparticular themethodproposedby Lisiecki and Lisiecki (2002)) perform better in the presence ofstochastic noise, but are more vulnerable to a misidentification of thetarget function. In addition, the assumption of a constant accretion rateremains a shortcoming.

4. Martinson et al.'s method does not need to assume a constantaccretion rate, but is still sensitive to the choice of the target function.However, this method performs well in the presence of stochasticnoise and reasonably in the presence of the tested model errors.

5. The time domain method developed by Wilkinson and Ivanyperforms slightly worse on all test datasets (noise and modelerrors). The main reason is that the tuning parameter (width of thewindow) has to be carefully balanced: a too large window smoothsall variations in accretion rate, while a too small window deliversresults which are too sensitive to stochastic perturbations.

6. The frequency domain method developed by De Ridder et al.performs better. The main reason is that advantage is taken of thefrequency domain to filter off stochastic noise and certain types ofmodel errors. However, this method also needs the width of aspectral window to be specified by the user.

7. The main advantage of the two previous non-parametric methodsis that they can provide initial values for parametric methods, likeproposed by Martinson et al. (1982a) or by de Brauwere et al.(2008). The latter method is a generalization of Martinson et al.'smethod, where the target function is optimized simultaneouslywith the mapping function. In addition an information criterion isused to automatically optimize the model complexity. Therefore,from all considered methods, this last method implies the leastassumptions on the target function and needs the least subjectiveinput from the user. As such the method can be interpreted as asynthesis of all previous methods, keeping their attractive featureswhile avoiding their weaker points. The price to pay is a morecomplicated technical implementation.

Acknowledgements

Anouk de Brauwere is a post-doctoral researcher of the ResearchFoundation Flanders (FWO-Vlaanderen). This work was also supportedby the Federal Science Policy, Brussels (contract IUAP VI/4 and SSDprojects SD/CS/02 A & B), the Flemish Government and the VrijeUniversiteit Brussel (GOA22/DSWER4, GOA23-ILiNoS, and HOA9), andthe PaleoSalt project (ESF-Euroclimate and FWO G.0642.05). We aregrateful to David Paul Gillikin for providing data about the Saxidomusgiganteus clam.Finally,wewould like to thankGuyMunhovenandDidierPaillard for their constructive comments and criticisms on this work.

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Author's personal copy · Author's personal copy A comparative study of methods to reconstruct a periodic time series from an environmental proxy record Anouk de Brauwerea,, Fjo De