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)

With the progress of mantle convection modelling over the last decade, it now becomes possible to solve for the dynamics of the interior flow and the surface tectonics to first order. We show here that tectonic data (like surface kinematics and seafloor age distribution) and mantle convection models with plate-like behaviour can in principle be combined to reconstruct mantle convection. We present a sequential data assimilation method, based on suboptimal schemes derived from the Kalman filter, where surface velocities and seafloor age maps are not used as boundary conditions for the flow, but as data to assimilate. Two stages (a forecast followed by an analysis) are repeated sequentially to take into account data observed at different times. Whenever observations are available, an analysis infers the most probable state of the mantle at this time, considering a prior guess (supplied by the forecast) and the new observations at hand, using the classical best linear unbiased estimate. Between two observation times, the evolution of the mantle is governed by the forward model of mantle convection. This method is applied to synthetic 2-D spherical annulus mantle cases to evaluate its efficiency. We compare the reference evolutions to the estimations obtained by data assimilation. Two parameters control the behaviour of the scheme: the time between two analyses, and the amplitude of noise in the synthetic observations. Our technique proves to be efficient in retrieving temperature field evolutions provided the time between two analyses is 10 Myr. If the amplitude of the a priori error on the observations is large (30 per cent), our method provides a better estimate of surface tectonics than the observations, taking advantage of the information within the physics of convection.

/pdf/a-sequential-data-assimilation-approach-for-the-joint-3l9lo7frb4.pdf

A sequential data assimilation approach for the joint reconstruction of mantle convection and surface tectonics

Plate tectonics and mantle controls on plume dynamics

Where does subduction initiate and cease? A global scale perspective

Seismoacoustic Coupling Induced by the Breakup of the 15 February 2013 Chelyabinsk Meteor

http://www.crpg.cnrs-nancy.fr/spip.php?action=acceder_document&arg=180&cle=43039a54ade1d187e6e4fdb4cc13ab6e5bc8bb1d&file=pdf%2F2014.pdf

Nicolas Coltice

Papers

A sequential data assimilation approach for the joint reconstruction of mantle convection and surface tectonics

Plate tectonics and mantle controls on plume dynamics

Where does subduction initiate and cease? A global scale perspective

Seismoacoustic Coupling Induced by the Breakup of the 15 February 2013 Chelyabinsk Meteor

Xenon isotope constraints on the thermal evolution of the early Earth