The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

We have created a digital age grid of the ocean floor with a grid node interval of 6 arc min using a self-consistent set of global isochrons and associated plate reconstruction poles. The age at each grid node was determined by linear interpolation between adjacent isochrons in the direction of spreading. Ages for ocean floor between the oldest identified magnetic anomalies and continental crust were interpolated by estimating the ages of passive continental margin segments from geological data and published plate models. We have constructed an age grid with error estimates for each grid cell as a function of (1) the error of ocean floor ages identified from magnetic anomalies along ship tracks and the age of the corresponding grid cells in our age grid, (2) the distance of a given grid cell to the nearest magnetic anomaly identification, and (3) the gradient of the age grid: i.e., larger errors are associated with high age gradients at fracture zones or other age discontinuities. Future applications of this digital grid include studies of the thermal and elastic structure of the lithosphere, the heat loss of the Earth, ridge-push forces through time, asymmetry of spreading, and providing constraints for seismic tomography and mantle convection models.

Digital isochrons of the world's ocean floor

We present a revised global plate motion model with continuously closing plate boundaries ranging from the Triassic at 230 Ma to the present day, assess differences among alternative absolute plate motion models, and review global tectonic events. Relatively high mean absolute plate motion rates of approximately 9–10 cm yr^(−1) between 140 and 120 Ma may be related to transient plate motion accelerations driven by the successive emplacement of a sequence of large igneous provinces during that time. An event at ∼100 Ma is most clearly expressed in the Indian Ocean and may reflect the initiation of Andean-style subduction along southern continental Eurasia, whereas an acceleration at ∼80 Ma of mean rates from 6 to 8 cm yr^(−1) reflects the initial northward acceleration of India and simultaneous speedups of plates in the Pacific. An event at ∼50 Ma expressed in relative, and some absolute, plate motion changes around the globe and in a reduction of global mean plate speeds from about 6 to 4–5 cm yr^(−1) indicates that an increase in collisional forces (such as the India–Eurasia collision) and ridge subduction events in the Pacific (such as the Izanagi–Pacific Ridge) play a significant role in modulating plate velocities.

Ocean Basin Evolution and Global-Scale Plate Reorganization Events Since Pangea Breakup

This chapter is in part an update of a previous, more abbreviated review ( Hofmann, 1997 ) It covers the subject in greater depth, and it reflects some significant changes in the author's views since the writing of the earlier paper In particular, the spatial range of equilibrium attained during partial melting may be much smaller than previously thought, because of new experimental diffusion data and new results from natural settings Also, the question of ‘layered’ versus ‘whole-mantle’ convection, including the depth of subduction and of the origin of plumes, has to be reassessed in light of the recent breakthroughs achieved by seismic mantle tomography As the spatial resolution of seismic tomography and the pressure range, accuracy, and precision of experimental data on melting relations, phase transformations, and kinetics continue to improve, the interaction between these disciplines and geochemistry sensu stricto will continue to improve our understanding of what is actually going on in the mantle The established views of the mantle being engaged in simple two- or single-layer convection are becoming obsolete In many ways, we are just at the beginning of this new phase of mantle geology, geophysics, and geochemistry

Sampling mantle heterogeneity through oceanic basalts: Isotopes and trace elements

/pdf/revised-definition-of-large-igneous-provinces-lips-3c10x1yfrr.pdf

Revised definition of Large Igneous Provinces (LIPs)

Although plate tectonics has pushed the frontiers of geosciences in the past 50 years, it has legitimate limitations, and among them we focus on both the absence of dynamics in the theory and the difficulty of reconstructing tectonics when data are sparse. In this manuscript, we propose an anticipation experiment, proposing a singular outlook on plate tectonics in the digital era. We hypothesize that mantle convection models producing self-consistently plate-like behavior will capture the essence of the self-organization of plate boundaries. Such models exist today in a preliminary fashion, and we use them here to build a database of mid-ocean ridge and trench configurations. To extract knowledge from it, we develop a machine learning framework based on Generative Adversarial Networks (GANs) that learns the regularities of the self-organization in order to fill gaps of observations when working on reconstructing a plate configuration. The user provides the distribution of known ridges and trenches, the location of the region where observations lack, and our digital architecture proposes a horizontal divergence map from which missing plate boundaries are extracted. Our framework is able to prolongate and interpolate plate boundaries within an unresolved region but fails to retrieve a plate boundary that would be completely contained inside of it. The attempt we make is certainly too early because geodynamic models need improvement and a larger amount of geodynamic model outputs, as independent as possible, is required. However, this work suggests applying such an approach to expand the capabilities of plate tectonics is within reach.

/pdf/an-anticipation-experiment-for-plate-tectonics-3d4xu7pyok.pdf

An Anticipation Experiment for Plate Tectonics

Although more and more robust evidence of whole mantle convection comes from seismic tomography and geoid modeling, the rare gases and other isotopic or trace element signatures of ridge and hotspot basalts indicate the presence of various isolated geochemic al reservoirs in the mantle. We discuss this discrepancy between the fluid dynamic views of mantle convection and the chemical observations. We compare the standard model of geodynamicists where the mantle behaves as a fluid mostly heated from within with the findings of seismic tomography. We suggest that a significant part of the subducted oceanic crust transformed into dense eclogitic assemblages, partially segregates to form a D'' layer growing with time. We show how a two component marble cake mantle filling the whole mantle except D'' can account for the variability of OIBs and MORBs in rare gases. We then present the state of the art in thermochemical convection of the mantle and emphasize the numerical and conceptual progress that must be made to quantitatively test the geochemical hypothesis.

/pdf/geophysical-and-geochemical-models-of-mantle-convection-1sfm6ofjvh.pdf

Geophysical and Geochemical Models of Mantle Convection: Successes and Future Challenges

The interconnectedness of life, water, and plate tectonics is strikingly apparent along mid-ocean ridges (MOR) where communities of organisms flourish off the disequilibrium of chemical potentials created by circulation of hydrothermal fluids driven by Earth's heat [Nisbet and Sleep, 2001; Staudigel et al., 2004]. Moreover, submarine hydrothermal environments may be critical for the emergence of life on Earth [Nisbet and Sleep, 2001]. Oceans were likely present in the Hadean [Valley et al., 2002; Harrison, 2009] but questions remain about early Earth's global tectonics [Van Hunen and Moyen, 2012], including when seafloor spreading began and whether mid-oceanic ridges were deep enough for maximum hydrothermal activities [Kasting et al., 2006]. For example, plate tectonics influences global sea level by driving secular variations in the volume of ocean basins due to continental growth [Flament et al., 2008]. Similarly, variations in the distribution of seafloor age and associated subsidence [Flament et al., 2008], due to assembly and dispersal of supercontinents [Coltice et al., 2012], explains the largest sea level variation over the past 140 Myr [Muller et al., 2008]. Using synthetic plate configurations derived from numerical models of mantle convection [Coltice et al., 2012, 2014] appropriate for early Earth, we show that MOR has remained submerged and its depths potentially constant over geologic time. Thus, conditions in the early Earth existed for hydrothermal vents at similar depths as today, providing environments conducive for the development of life and allowing for processes such as hydrothermal alteration of oceanic crust to influence the mantle's geochemical evolution This article is protected by copyright. All rights reserved.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2016GC006629

Influence of continental growth on mid-ocean ridge depth

Mantle convection models with plate-like behavior produce surface structures comparable to Earth's plate boundaries. However, analyzing those structures is a difficult task, since convection models produce, as on Earth, diffuse deformation and elusive plate boundaries. Therefore we present here and share a quantitative tool to identify plate boundaries and produce plate polygon layouts from results of numerical models of convection: Automatic Detection Of Plate Tectonics (ADOPT). This digital tool operates within the free open-source visualization software Paraview. It is based on image segmentation techniques to detect objects. The fundamental algorithm used in ADOPT is the watershed transform. We transform the output of convection models into a topographic map, the crest lines being the regions of deformation (plate boundaries) and the catchment basins being the plate interiors. We propose two generic protocols (the field and the distance methods) that we test against an independent visual detection of plate polygons. We show that ADOPT is effective to identify the smaller plates and to close plate polygons in areas where boundaries are diffuse or elusive. ADOPT allows the export of plate polygons in the standard OGR-GMT format for visualization, modification, and analysis under generic softwares like GMT or GPlates.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2017GC007030

ADOPT: A tool for automatic detection of tectonic plates at the surface of convection models

The thermal regime of the Earth's interior during the Hadean (the first 700 My after the birth of the solar system) is subject to debate. Evidence for a hotter mantle stems from the abundance of magnesian lavas (komatiites) in the Archean, although their generation might have also resulted from different (hydrous) melting conditions. In this contribution, the present-day mantle abundances of xenon isotopes contributed by extinct and extant radioactivities are used to constrain thermal and magmatic evolution models of the early Earth. Results show that, in the Hadean, heat could escape at a rate much faster than Today. Heat loss from the mantle was driven by magmatism rather than by conduction through the lithospheric lid, precluding modern style plate tectonics. Around the Hadean–Archean transition, a drastic change in the thermal regime led to a secular cooling rate comparable to the modern one, in probable relation to the onset of plate tectonics. Our model also suggests that solid-state convection started later than 50 My after the formation of the solar system, a view consistent with proposed ages for the Moon-forming impact.

Nicolas Coltice

Papers

An Anticipation Experiment for Plate Tectonics

Geophysical and Geochemical Models of Mantle Convection: Successes and Future Challenges

Influence of continental growth on mid-ocean ridge depth

ADOPT: A tool for automatic detection of tectonic plates at the surface of convection models

Xenon isotope constraints on the thermal evolution of the early Earth