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

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Revised definition of Large Igneous Provinces (LIPs)

. The core-mantle boundary (CMB) heat flux is an important variable of Earth’s thermal evolution and dynamics. Seismic tomography enables access to seismic heterogeneities in the lower mantle, which can be related to present-day thermal heterogeneities. Alternatively, mantle convection models can be used to either infer the past CMB heat flux or to produce statistically realistic CMB heat flux distributions in self-consistent models. Mantle dynamics modifies the inertia tensor of the Earth, which implies a rotation of the Earth with respect to its rotation axis called True Polar Wander (TPW). This rotation 5 has to be taken into account if mantle dynamics is to be linked to core dynamics. In this study, we explore the TPW and the CMB heat flux produced by a self-consistent mantle convection model. The geoid is also computed and investigated in order to determine the driving mechanism of TPW. This model includes continents, dense chemical piles at the bottom of the shell and plate-like behavior, providing the possibility to link TPW and the CMB heat flux with plate tectonics. A principal component analysis (PCA) of the CMB heat flux is computed to obtain the dominant heat flux patterns. The model shows a 10 geoid dominated by upper mantle structures. Subduction zones and continents are correlated with positive geoid anomalies, about 20 times larger than the observed geoid anomalies. Chemical piles are mostly correlated with negative geoid anomalies because of the anti-correlation between the positions of subducting slabs and the piles. TPW thus tends to lock continents and subduction zones close to the equator, while chemical piles are shifted towards higher latitudes. The positive CMB heat flux anomalies are mostly located at low

True polar wander and heat flux patterns at the core-mantle boundary in a mantle convection simulation

&amp;lt;p&amp;gt;Every planet is singular, with scars and bumps at their surface. One planet, one history. But the physics at play is common to them, connecting planetary bodies together. Tectonics is a common theme of what we can observe on planets of the solar system, and a central question for explanets. More than 20 years of geodynamic modelling has resulted in &amp;amp;#160;identifying a diversity of tectonic regimes for mantle convection, from very active, like heat-pipe (Monnereau and Dubuffet, 2002 among others) and squishy lid (Louren&amp;amp;#231;o et al., 2020) to almost inert, like stagnant lid (Schmeling and Jacoby, 1982). Tectonics is an emergent property deriving from the intimate structure and composition of a planet. It is also a fundamental piece shaping the surface environment. This presentation will attempt to give an overview of tectonic regimes of planets and propose typical evolutional scenari, connecting structural and compositional histories from the depth to the surface.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;&amp;amp;#160;&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;&amp;lt;strong&amp;gt;References&amp;lt;/strong&amp;gt;&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;Louren&amp;amp;#231;o, D. L., Rozel, A. B., Ballmer, M. D., &amp;amp; Tackley, P. J. (2020). Plutonic&amp;amp;#8208;squishy lid: A new global tectonic regime generated by intrusive magmatism on earth&amp;amp;#8208;like planets. Geochemistry, Geophysics, Geosystems, 21, e2019GC008756.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;Monnereau, M., &amp;amp; Dubuffet, F. (2002). Is Io's mantle really molten?. Icarus, 158, 450-459.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;Schmeling, H., &amp;amp; Jacoby, W. R. (1982). On modelling the lithosphere in mantle convection with non-linear rheology. Journal of Geophysics, 50, 89-100.&amp;lt;/p&amp;gt; 

Nicolas Coltice

Papers

True polar wander and heat flux patterns at the core-mantle boundary in a mantle convection simulation

An overview of modeled dynamic histories of rocky planets

Ensemble Assimilation of Geodynamic Observations Into Mantle Circulation Models