Hierarchical Dynamic Time Warping methodology for aggregating multiple geological time series
Introduction
Past climate (paleoclimate) reconstructions rely primarily on the investigation and inter-comparison of diverse sets of environmental proxy time-series. These comparisons aim to date, and tune age models of records using external forcing (e.g. Hays et al., 1976), test for coherence between climate signal in various spatial and temporal time scales (e.g. Cheng et al., 2015; Deininger et al., 2017; Mayewski et al., 2004) or compound high definition datasets to reduce noise or remove locality (or outlier) effects (Nagra et al., 2017; Orland et al., 2014; Treble et al., 2005). While there is no shortage of paleoclimate datasets, the inherent age uncertainties in proxy time-series often impede matching or synchronisation between signals. These uncertainties arise from the nature of deposition of different proxy recorders, the accuracy and precision of applied dating techniques and use of different sampling methodologies for various proxies. Many different signal matching methodologies are commonly applied to address the issue of signal feature alignment in geological records (i.e. cycles, peaks, troughs). These are either done by manual matching processes (e.g. Mayewski et al., 2004; Prell et al., 1986), or by automated correlation algorithms (e.g. Lisiecki and Lisiecki, 2002; Riechelmann et al., 2019). The former methodologies are time-consuming and bias prone, while the latter are susceptible to over or under -fitting, parameterisation problems and lastly, the always present human bias.
Dynamic Time Warping (DTW) is a robust similarity measurement algorithm between two time-series which may vary in speed, length or are unsynchronised. While DTW was initially developed for speech recognition (Sakoe and Chiba, 1978), today we can find DTW applied to temporal sequences of video, audio, movement, graphics data, geological data or any data which can be turned into a linear sequence (e.g. Egglestone and Peirce, 1995; Kim et al., 2001; Lisiecki and Lisiecki, 2002; Ten Holt et al., 2007). Fig. 1 shows a representative DTW process performed on a section of two adjacent Laser Ablation (LA) – ICP-MS traverses of Barium concentrations (Ba [ppm]) measured in a speleothem from Moondyne Cave in southwestern Australia (Treble et al., 2003). Previous geological applications of DTW included determining effective porosity and tortuosity of different soils (Egglestone and Peirce, 1995), correlating geologic strata and well logs (Lallier et al., 2016; Silversides and Melkumyan, 2016; Waterman and Raymond, 1987), seismic patterns in volcano monitoring (Orozco-Alzate et al., 2015), stream water-level travel time (Claure et al., 2018) and synchronisation and dating of paleoclimate records (Ajayi et al., 2020; Hay et al., 2019; Lisiecki and Lisiecki, 2002). Lisiecki and Lisiecki (2002) experimented with the alignment of several millennial-scale records: SPECMAP data, artificially distorted SPECMAP dataset, ice oxygen isotope curves from two different Greenland locations (GRIP and GISP2) and even two different proxies from the same Greenland ice core (ice δ18O and δ15N in GRIP). While showing excellent results in all experiments, the work of Lisiecki and Lisiecki (2002) does not attempt matching of more than two signals per experiment. A more recent study stemming from their work uses a probabilistic stratigraphic alignment algorithm (HMM-Match) to constrain age uncertainties between 35 different marine core records by aligning each to a single “master” record (see Lin et al., 2014). In all the implementations mentioned above for multiple signals, the matching between two “slave” records is performed through the alignment vector with the master record.
In sections 2 The hierarchical Dynamic Time Warping methodology, 3 Testing HDTW properties, we present a Hierarchical aggregation approach to DTW outputs (HDTW) which allows for the alignment of 2n signals with each pair given equivalent weight at each level of the hierarchy (no master record). The HDTW process indexes the records on a unified “path matrix” that, in turn, serves to highlight non-local features (outliers). One potential advantage for implementing HDTW in geosciences is the alignment of several parallel micron resolution analysis traverses measured in a single sample. In section 4, we explore a specific use case for HDTW of a sub-annual calibration of a paleoclimate time-series measured in micron-scale resolution on speleothems.
Section snippets
Why implement a (hierarchical) aggregation layer?
Vaughan and Gabrys (2016) describe a method to aggregate 2n signals (for n > =1) by applying a hierarchical aggregation layer to the DTW process. This layer produces a single combined trajectory. This trajectory is then used to analyse multivariate data from virtual reality training simulators. In their method, simplified in Fig. 2, every two traverses are compared (DTW-wise), and a single mean signal is extrapolated and propagated to the next level input (e.g. level 0 to level 1 in Fig. 2). On
Warping of multiple signals of different resolution
One of the strengths of DTW is the ability to warp signals of different lengths and speeds (Lisiecki and Lisiecki, 2002). Here, we test if the HDTW process yields a similar output for multiple signals using a speleothem sample (SO-38, Appendix B.1). Sample SO-38 is a flowstone from the Soreq Cave. The sample was removed in 2005 and covers the last two centuries approximately. The measured low-resolution stable oxygen isotope ratio (δ18O) and stable carbon isotope ratio (δ13C) time series of the
Application: calibrating sub-annual time series using HDTW aligned features
The interpretation of paleo-records requires working with multiple proxy signals (e.g. Almogi-Labin et al., 2009; Mayewski et al., 2004), often from several coeval samples of the same repository (e.g. Cheng et al., 2015) or, specifically in micron-scale analyses, parallel sampling within a single sample (Langner and Mulitza, 2019; Treble et al., 2005). For the most part, such signals are not synchronised among themselves due to the different nature of the repository (deposition or growth rate),
Conclusions
This paper presents a new Hierarchical aggregation methodology for DTW, or HDTW, that allows the alignment of multiple (2n) input signals. HDTW offers downward indexing process that serves to highlight local and non-local features.
A sub-annual age-model utilising HDTW for signals and features alignment is successfully applied to produce an annual calibration for a fluorescence banding image and LA-ICP-MS trace element traverse. The output provides comparable results to previous age models with
Code and data availability
The HDTW module & example scripts (inc. data) are available at https://github.com/AsafGazit/HDTW, GitHub an open-source online data repository. Code was developed by Asaf Gazit and Yuval Burstyn. Contact Email: [email protected]. Year first available: 2020. The HDTW script can run on any operating system supporting 3.6 Python environment and dependencies required (list of dependencies is in section 2, GitHub repository and Appendix A). Code size is 36 kb. Example scripts are 36 kb. The code
Author contributions
YB designed the concept of this study, drafted the manuscript with support of all co-authors and supervised the analysis of sample SO-38. AG designed the concept of this study, drafted the manuscript with support of all co-authors and developed the HDTW code. OD supervised the analysis of sample SO-38 and development of LA-ICPMS analysis methodology and assisted in reviewing and editing the manuscript for publication. AG and YB developed the peak counting algorithm. AG created the python module
Funding
The authors wish to acknowledge the support in sample analysis of Microbeam Laboratory at The Fredy & Nadine Herrmann Institute of Earth Sciences, The Hebrew University of Jerusalem.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The Authors are grateful to Dr. Miryam Bar-Matthews and Dr. Avner Ayalon for making sample SO-38 available for testing the methodology. We wish to thank Prof. Grégoire Mariethoz and Dr. Leor Katz for their support and suggestions on improving this manuscript. YB wishes to thank Edna Rintel and Tzipora Burstyn for additional financial support. AG wishes to thank his parents, Elisheva and Shlomo, and the rest of the Gazit family for their endless support. AG wishes to express his gratitude and
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