Abstract. We consider a stochastic differential equation modelfor Earth's axial magnetic dipole field.Our goal is to estimate the model's parametersusing diverse and independent data sources that had previously been treated separately,so that the model is a valid representation of an expanded paleomagnetic recordon kyr to Myr timescales.We formulate the estimation problem within the Bayesian frameworkand define a feature-based posterior distributionthat describes probabilities of model parameters givena set of features derived from the data.Numerically, we use Markov chain Monte Carlo (MCMC)to obtain a sample-based representation of the posterior distribution.The Bayesian problem formulation and its MCMC solutionallow us to study the model's limitations and remaining posterior uncertainties.Another important aspect of our overall approach is thatit reveals inconsistencies between model and data or within the various data sets.Identifying these shortcomings is a first and necessary step towards building more sophisticated models or towards resolving inconsistencies within the data.The stochastic model we derive representsselected aspects of the long-term behavior of the geomagnetic dipole fieldwith limitations and errors that are well defined.We believe that such a model is useful (besides its limitations) for hypothesis testing and give a few examples of how the model can be used in this context.
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A comprehensive model for the kyr and Myr time scales of Earth's axial magnetic dipole field
We consider a stochastic differential equation model for Earth's axial magnetic dipole field. The model's parameters are estimated using diverse and independent data sources that had previously been treated separately. The result is a numerical model that is informed by the full paleomagnetic record on kyr to Myr time scales and whose outputs match data of Earth's dipole in a precisely defined feature-based sense. Specifically, we compute model parameters and associated uncertainties that lead to model outputs that match spectral data of Earth's axial magnetic dipole field but our approach also reveals difficulties with simultaneously matching spectral data and reversal rates. This could be due to model deficiencies or inaccuracies in the limited amount of data. More generally, the approach we describe can be seen as an example of an effective strategy for combining diverse data sets that is particularly useful when the amount of data is limited.
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- Award ID(s):
- 1740858
- NSF-PAR ID:
- 10111676
- Date Published:
- Journal Name:
- Nonlinear Processes in Geophysics Discussions
- ISSN:
- 2198-5634
- Page Range / eLocation ID:
- 1 to 24
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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SUMMARY It is well known that the axial dipole part of Earth’s magnetic field reverses polarity, so that the magnetic North Pole becomes the South Pole and vice versa. The timing of reversals is well documented for the past 160 Myr, but the conditions that lead to a reversal are still not well understood. It is not known if there are reliable ‘precursors’ of reversals (events that indicate that a reversal is upcoming) or what they might be. We investigate if machine learning (ML) techniques can reliably identify precursors of reversals based on time-series of the axial magnetic dipole field. The basic idea is to train a classifier using segments of time-series of the axial magnetic dipole. This training step requires modification of standard ML techniques to account for the fact that we are interested in rare events—a reversal is unusual, while a non-reversing field is the norm. Without our tweak, the ML classifiers lead to useless predictions. Perhaps even more importantly, the usable observational record is limited to 0–2 Ma and contains only five reversals, necessitating that we determine if the data are even sufficient to reliably train and validate an ML algorithm. To answer these questions we use several ML classifiers (linear/non-linear support vector machines and long short-term memory networks), invoke a hierarchy of numerical models (from simplified models to 3-D geodynamo simulations), and two palaeomagnetic reconstructions (PADM2M and Sint-2000). The performance of the ML classifiers varies across the models and the observational record and we provide evidence that this is not an artefact of the numerics, but rather reflects how ‘predictable’ a model or observational record is. Studying models of Earth’s magnetic field via ML classifiers thus can help with identifying shortcomings or advantages of the various models. For Earth’s magnetic field, we conclude that the ability of ML to identify precursors of reversals is limited, largely due to the small amount and low frequency resolution of data, which makes training and subsequent validation nearly impossible. Put simply: the ML techniques we tried are not currently capable of reliably identifying an axial dipole moment (ADM) precursor for geomagnetic reversals. This does not necessarily imply that such a precursor does not exist, and improvements in temporal resolution and length of ADM records may well offer better prospects in the future.
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SUMMARY Earth’s internal magnetic field is dominated by the contribution of the axial dipole whose temporal variations are wide ranging and reflect characteristic timescales associated with geomagnetic reversals and large scale palaeosecular variation, ranging down to decadal and subannual field changes inferred from direct observations. We present a new empirical power spectrum for the axial dipole moment based on composite magnetic records of temporal variations in the axial dipole field that span the frequency range 0.1 to 5 × 105 Myr–1 (periods from 10 million to 2 yr). The new spectrum is used to build a stochastic representation for these time variations, based on an order 3 autoregressive (AR) process and placed in the context of earlier stochastic modelling studies. The AR parameter estimates depend on the frequency of transitions in the spectral regime and may be influenced by Ohmic diffusion, advection and torsional oscillations in Earth’s core. In several frequency ranges across the interval 200–5000 Myr–1(5000 to 200 yr periods) the empirical power spectrum lies above the AR3 model and may be influenced by Magneto–Coriolis (MC) waves in Earth’s core. The spectral shape and parameter estimates provide a potentially useful guide for developing assessments of whether numerical dynamo simulations meet criteria for being considered Earth like.more » « less
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Abstract. Many applications in science require that computational models and data becombined. In a Bayesian framework, this is usually done by defininglikelihoods based on the mismatch of model outputs and data. However,matching model outputs and data in this way can be unnecessary or impossible.For example, using large amounts of steady state data is unnecessary becausethese data are redundant. It is numerically difficult to assimilate data inchaotic systems. It is often impossible to assimilate data of a complexsystem into a low-dimensional model. As a specific example, consider alow-dimensional stochastic model for the dipole of the Earth's magneticfield, while other field components are ignored in the model. The aboveissues can be addressed by selecting features of the data, and defininglikelihoods based on the features, rather than by the usual mismatch of modeloutput and data. Our goal is to contribute to a fundamental understanding ofsuch a feature-based approach that allows us to assimilate selected aspectsof data into models. We also explain how the feature-based approach can beinterpreted as a method for reducing an effective dimension and derive newnoise models, based on perturbed observations, that lead to computationallyefficient solutions. Numerical implementations of our ideas are illustratedin four examples.
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Abstract A foundational assumption in paleomagnetism is that the Earth's magnetic field behaves as a geocentric axial dipole (GAD) when averaged over sufficient timescales. Compilations of directional data averaged over the past 5 Ma yield a distribution largely compatible with GAD, but the distribution of paleointensity data over this timescale is incompatible. Reasons for the failure of GAD include: (a) Arbitrary “selection criteria” to eliminate “unreliable” data vary among studies, so the paleointensity database may include biased results. (b) The age distribution of existing paleointensity data varies with latitude, so different latitudinal averages represent different time periods. (c) The time‐averaged field could be truly non‐dipolar. Here, we present a consistent methodology for analyzing paleointensity results and comparing time‐averaged paleointensities from different studies. We apply it to data from Plio/Pleistocene Hawai'ian igneous rocks, sampled from fine‐grained, quickly cooled material (lava flow tops, dike margins and scoria cones) and subjected to the IZZI‐Thellier technique; the data were analyzed using the Bias Corrected Estimation of Paleointensity method of Cych et al. (2021,
https://doi.org/10.1029/2021GC009755 ), which produces accurate paleointensity estimates without arbitrarily excluding specimens from the analysis. We constructed a paleointensity curve for Hawai'i over the Plio/Pleistocene using the method of Livermore et al. (2018,https://doi.org/10.1093/gji/ggy383 ), which accounts for the age distribution of data. We demonstrate that even with the large uncertainties associated with obtaining a mean field from temporally sparse data, our average paleointensities obtained from Hawai'i and Antarctica (reanalyzed from Asefaw et al., 2021,https://doi.org/10.1029/2020JB020834 ) are not GAD‐like from 0 to 1.5 Ma but may be prior to that.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.5194/npg-2018-42
