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

Get Complete Project Material File(s) Now! »

Rheology and Self-Consistent Plate Generation

Mantle material is polycrystalline. Its response to stress depends on various mecha-nisms of deformation at the crystal scale (Karato, 2013; Kohlstedt, 2015). The combi-nation of all these mechanisms results in a complex and nonlinear rheology which is key to understanding mantle dynamics (Turcotte & Oxburgh, 1972; Davies & Richards, 1992; Bercovici, 2015).
In section 1.1, we supposed a relationship between the deviatoric stress tensor τ and the strain rate tensor ǫ˙ of the form τ = 2µǫ˙, (1.24).
with µ the viscosity and ǫ˙ = ∇u + [∇u]T /2. This formulation can take into account a wide range of deformation mechanisms by considering a composite rheology, where the effective viscosity µ is 1 = 1 + 1 + … (1.25).
with µ1, µ2… the effective viscosities of each mechanism, potentially varying with tem-perature, pressure, strain rate, grain size and composition.

A Data Assimilation Framework for Mantle Circulation Problems

Data assimilation methods have originally been developed as a way to estimate the state of the atmosphere at present, so as to obtain an initial state for numerical weather predictions (see for example Ghil, 1989, for a short account of early data assimilation developments in weather forecasting). The improvement of both numerical weather models and the observational network, along with the increase in computational power, led to the evolution from subjective analysis of sparse observations, to objective anal-ysis of a denser network of observations and finally to the assimilation of data not only spatially, but also through time, i.e. 4-D data assimilation (Charney et al., 1969; Smagorinsky et al., 1970; Rutherford, 1972). Atmospheric data assimilation borrows many concepts from control and estimation theory (see for example Ghil et al., 1981; Le Dimet & Talagrand, 1986). The main difference is in the larger size of the prob-lem and dataset used in atmospheric data assimilation, which pushed the develop-ment of specific techniques. The application of data assimilation methods has been progressively opening to various geophysical systems, following the development of observational networks and numerical models in the corresponding disciplines. Cur-rent applications include oceanography (Chassignet & Verron, 2006), geomagnetism (Fournier et al., 2010), oil and gas reservoir (Evensen, 2009a, Chapter 17), and glaciol-ogy (Bonan et al., 2014), for example. In parallel, the goals of data assimilation methods have diversified: the original aim was to estimate the current state of a system to fore-cast future evolutions, but other applications have emerged, such as reanalysis of past data (hindcasting), sensitivity analysis of models to different parameters and initial and boundary conditions, assessment of the numerical model, and parameter estima-tion. Concerning mantle convection, data assimilation techniques provide powerful tools to investigate fundamental questions such as:
• what is the predictive power of state of the art mantle convection models, or how well do we model mantle dynamics?
• what was the circulation of the mantle in the past?
• how sensitive are mantle circulation models to the variation of key parameters, such as rheological ones?
The integration of data in dynamical models is a longstanding problem in global man-tle convection studies (Hager & O’Connell, 1979) and in geodynamics in general (see for example the introductory book by Ismail-Zadeh et al., 2016). One of the ways to solve this problem has been to apply data assimilation methods to mantle circulation reconstructions. This approach was pioneered by Bunge et al. (2003) and Ismail-Zadeh et al. (2004) who developed variational data assimilation methods for mantle circula-tion reconstructions. In these articles, they present the data assimilation problem of mantle circulation reconstruction in its continuous form (using a set of partial differ-ential equations similar to the one described in Section 1.2 and associated continuous fields). Here, we choose a different approach and formulate the data assimilation prob-lem directly on the discretized version of mantle flow equations. We use the standard data assimilation formulation advised by Ide et al. (1997) and define the unknown (the state of the mantle), the sources of information (observations, background state and dynamical model) and how each of these components is related to the others.

The Unknown: the Evolution of the True State of the Mantle

The true state gathers all the variables that are necessary and sufficient to compute the 3-D flow of the Earth’s mantle at a given time.
In the continuum approximation (see section 1.1), the evolution of mantle flow is de-scribed by a set of fields varying continuously in space and time. However, the com-putation of a dynamical evolution of those fields requires their discretization (see sec-tion 1.5). So, in practice, we aim at estimating the evolution of the discretized state of the mantle. In this case, the true state can take the form of a vector containing the values of the discretized fields at a given time step k. This vector is named the true state vector and noted xtk. Then, we write the evolution of the true state as a time series of true state vectors x1t, x2t, …, xKt , (2.1).
The type and number of discretized fields included in the true state depend on the model chosen for mantle convection. For the model described in Chapter 1, the 3-D flow of the mantle at a given time is entirely defined from the sole knowledge of the temperature field. In this case, the true state vector contains the temperature values of the mantle at points defined on the numerical grid. For a model taking into account the variations of composition in the mantle, we would have to add the discretized compositional fields, for example.

The Background State of the Mantle

The background state of the mantle corresponds to the estimation of the state of the mantle before the observed data have been integrated. This a priori state comes from our knowledge of the physics of mantle convection, or the integration of another set of observed data, for example. The background state is linked to the first true state vector through xb = x1t + ǫb, (2.6).
where ǫb is the error vector of the background state vector. In this work, we used the results of a very long evolution computed with the dynam-ical model M to infer the average temperature field in the mantle and evaluate the amplitude of the background error vector (see section 3.4.1 for more details).

Surface Kinematics History of the Mantle

Observations on the global horizontal motions at the surface of the Earth are generally sparse and indirect and, for a significant part, are of qualitative nature. The concepts and methods associated to plate tectonics theory enable us to produce a simple time series of quantitative data that can be integrated in a data assimilation framework as described in Section 2.1.
The modern formulation of plate tectonics theory is the result of a collective effort of the geoscience community to integrate at a global scale observations on the horizon-tal motions of the Earth’s surface from a wide range of disciplines (see for example Oreskes, 2003, for a historical description of the origins of plate tectonics theory and an account of its development by some of the researchers who took part in this scientific revolution). Here, we summarize the evolution of ideas and highlight key observa-tions that led to the formulation of plate tectonics theory in the late 60’s by McKenzie
& Parker (1967), Morgan (1968) and Le Pichon (1968). We then describe the current models for past surface motions, which are to a large extent still based on these princi-ples and observations. In the beginning of the 20th century, Alfred Wegener (1880-1930) proposed the conti-nental drift hypothesis: the continents once formed a single landmass (called Pangaea) and have been drifting apart since then. Throughout his career, he collected obser-vations supporting his theory and documented them in the successive editions of his book, « Die Entstehung der Kontinente und Ozeane » (see for example Wegener, 1924, the English translation of the third edition). Wegener bases his theory on the comple-mentary shape of continents and the continuity of geological formations, paleoclimates and fossil repartition once continents are reunited.
Although the continental drift theory was rejected by a large part of the community (Oreskes, 1999), it gained a few early supporters. For example, Du Toit (1937) collected further geological observations supporting the theory of continental drift. Holmes (1931, 1944) proposed mantle convection as a viable mechanism for continental drift. Geodesist Felix Andries Vening-Meinesz pioneered the oceanic geodetic studies and interpreted the negative gravity anomalies at oceanic trenches (the deepest regions of the ocean floor) as the locations of downwelling convective currents (see Vening-Meinesz, 1948, for a synthesis of his gravity expeditions between 1923 and 1938). Griggs (1939b) designed experimental models of continental drift driven by convection cur-rents.
The 1960 − 1970 decade saw the progressive formulation and acceptance of the plate tectonics theory, starting with the sea-floor spreading theory (Dietz, 1961; Hess, 1962) and new evidence of large scale lateral motion of continents (Runcorn, 1961) and fin-ishing with the « new global tectonics » (i.e. plate tectonics) (Morgan, 1968; Le Pichon, 1968).
The intense marine geophysical exploration of the previous decades (started for mili-tary purposes during WWII and continued from then on) had produced an extensive dataset of the bathymetry, heat flow, seismic properties, gravity and magnetic anoma-lies of the seafloor, giving a global view of the oceanic domain and requiring new interpretations. Parallelly, the progresses in seismology made possible the detection and precise localization of ever smaller and distant earthquakes, as well as a better de-termination of earthquake properties such as the focal mechanism, and the associated slip (see e.g. Isacks et al., 1968, for a synthesis of data interpreted in the light of plate tectonics theory).

READ  Diversity of Citrus tristeza virus populations in commercial Star Ruby orchards

Dynamical Mantle Circulation Reconstructions

Three alternative strategies have been proposed to reconstruct the history of mantle circulation: backward advection, semi-empirical sequential methods and variational data assimilation. In the present dissertation, we propose a fourth one: sequential data assimilation, that we develop in Chapters 3 and 4. Backward advection and empiri-cal sequential methods have already produced numerous estimations of global mantle circulation evolutions in the last decades. The variational data assimilation methods are more recent, and have so far been applied to regional reconstructions (Liu & Gur-nis, 2008; Spasojevic et al., 2009) and just recently to global mantle circulation models (Glišovi´c & Forte, 2014; Horbach et al., 2014; Glišovi´c & Forte, 2016). Backward advection and empirical sequential methods are direct methods: they in-tegrate data in mantle convection models as boundary or initial conditions, without taking into account the uncertainties on the data and/or the model. We present these methods in the first part of this section. On the contrary, variational and sequential data assimilation consider mantle circulation reconstruction as an inverse problem, and explore (at least partially) the mantle state space to determine the evolution that best fits the observed data, given uncertainties on both data and the convection model. We describe these techniques in the second part of this section.

Convection Model with Plate-Like Behaviour

The forward model M is our source of prior information. It solves the equations of conservation of mass, momentum and energy with classical simplifications for man-tle convection: infinite Prandtl number and Boussinesq approximation. We further assume an isochemical mantle, and non-dimensionalize the equations to thermal dif-fusion scales (for a full development of the equations, see Ricard (2015) for example). We obtain:
∇ · u = 0, (3.13).
∇ · µ ∇u + (∇u)T − ∇p + RaT er = 0, (3.14).
=∇2T +H, (3.15).
where u, p, T and t are the non-dimensional velocity, dynamic pressure, tempera-ture and time, respectively. We work in spherical coordinates (r, θ, φ) of unit vectors (er, eθ, eφ). Ra is the Rayleigh number and H is the non-dimensional internal heating rate. The models presented here have 10% basal heating and 90% internal heating.
The temperatures at the top and bottom boundaries are set to Ta and Tb. The surface and the base of the model are shear-stress free. The dynamic viscosity µ varies with temperature and stress following the equation µ = µT−1 + µy−1 −1 , (3.16).
µT decreasing exponentially with temperature (according to Arrhenius law), and di-vided by β when reaching the solidus: µT = exp EA EA if T < Ts, (3.17) T +T1 2T1 µT =β − 1 exp EA − EA if T > Ts. (3.18)  +T1 2T1.
T1 is the temperature for which µT = 1, EA is the activation energy and Ts = Ts0 + ∇rTs(ra − r) with ra the surface value of r. Ts models the variation of solidus with depth and is tuned so that the viscosity drop is located at the base of the top bound-ary layer. This results in a weaker asthenosphere and favours plate-like behaviour (Richards et al., 2001; Tackley, 2000b). µy is defined by σyield = σY + (ra − r).∇rσY , µy = σyield , (3.19).
with σY , ∇rσY and ǫ˙ being the yield stress at the surface, the depth-dependence of the yield stress and the second invariant of the strain rate tensor respectively. The strain rate tensor is linked to the velocity by ǫ˙ = 1 ∇u + (∇u)T . (3.20).
Solutions are computed using StagYY (Tackley et al., 1993), a finite-volume, multigrid convection code. We use a spherical annulus grid which provides results closer to the spherical grid than a cylindrical geometry (Hernlund & Tackley, 2008). The grid is refined in the radial direction near the upper boundary of the model. In the following, the longitudinal coordinate of a point is written φm, with m ∈ {1, 2, …, M } and its radial coordinate is written rn with n ∈ {1, 2, …, N }, r varying from rb to ra. The value of the parameters used for this work are given in Table 3.1.

The Data: Surface Heat Flux and Surface velocities

As a first approach, the data yo we use are not direct measurements per se, but plate reconstruction models. For instance, Seton et al. (2012) or Shephard et al. (2013) pro-posed plate tectonics reconstructions for the last 200 My, using the continuously clos-ing plates methodology (Gurnis et al., 2012) so that a reconstruction can be numerically computed for any time between a given ti and the subsequent ti+1. These reconstruc-tions integrate paleomagnetic, paleobiological and geological data to provide continu-ous maps of surface velocity and seafloor age as well as the position and geometry of continents. It is this type of data that is used today in convection reconstructions with imposed boundary conditions. One fundamental difference between these methods and our sequential assimilation method is that the latter naturally takes into account uncertainties in the reconstructions. A second difference is that we do not need the surface data to be known at all times.
Plate reconstructions provide estimates of the velocity at any location on the surface of the Earth in the approximation of the plate tectonics theory, as well as the age of the seafloor. In the model we use in this manuscript, surface heat flux is an excellent proxy for the age of the seafloor (Coltice et al., 2012). Consequently, we propose to consider surface heat flux and surface velocity as the data to assimilate. However, with more sophisticated models, small scale convection would require an explicit computation of the age of the seafloor.

Quality of the data assimilation estimate

We evaluate the quality of the data assimilation scheme on its ability to retrieve the true temperature fields and to match surface data.
Fig. 3.6 shows examples of the final forecast state (second column) for evolutions with different parameters, after 300 My of data assimilation: t10γ10 (31 observation times, model 1), t50γ10 (7 observation times, model 1), t10γ10Q40 (31 observation times, model 2) and t10γ10Ra7 (31 observation times, model 3). The two first cases are done using the same model parameters, so we display the data assimilation results of the same evolution for better comparison. The true temperature fields for each case are displayed on the first column. The local error on the third column of Fig. 3.6 is the absolute value of the error at each coordinate (φm, rn)
ǫTf (φm, rn) = |T f (φm, rn) − T t(φm, rn)|. (3.62)

Table of contents :

1 The Mantle Circulation Forward Problem 
1.1 Fluid Mechanics for Mantle Convection
1.2 Conservation Equations
1.3 Initial and Boundary Conditions
1.4 Rheology and Self-Consistent Plate Generation
1.5 Numerical Approximation
2 Data and the Mantle Circulation Inverse Problem 
2.1 A Data Assimilation Framework for Mantle Circulation Problems
2.1.1 The Unknown: the Evolution of the True State of the Mantle
2.1.2 Observed Data on Mantle Circulation
2.1.3 The Dynamical Model
2.1.4 The Background State of the Mantle
2.2 Data on Mantle Circulation
2.2.1 Mantle Temperature Field at Present
2.2.2 Surface Kinematics History of the Mantle
2.3 Dynamical Mantle Circulation Reconstructions
2.3.1 Direct Methods
2.3.2 Data Assimilation Methods
3 A sequential data assimilation approach for the joint reconstruction of mantle convection and surface tectonics 
3.1 Introduction
3.2 The Extended Kalman Filter
3.2.1 Initialization
3.2.2 Analysis and forecast sequence
3.3 Convection model, Mantle State Vector and Tectonic Data
3.3.1 Convection Model with Plate-Like Behaviour
3.3.2 The State of the mantle
3.3.3 The Data: Surface Heat Flux and Surface velocities
3.3.4 The Observation Operator and the Augmented State
3.4 Sequential Data Assimilation Algorithm for Mantle Convection
3.4.1 Initialization
3.4.2 Analysis
3.4.3 Forecast
3.5 Synthetic Experiments
3.5.1 Setup of the Experiments
3.5.2 Quality of the data assimilation estimate
3.6 Discussion
3.7 Conclusion
4 Ensemble Data Assimilation For Mantle Circulation 
4.1 Introduction
4.2 Presentation of the Problem
4.2.1 Mantle Convection Model
4.2.2 Observations of Mantle Circulation
4.2.3 Ensemble Kalman Filtering Framework: Notations
4.3 Ensemble Kalman Filter with Localization and Inflation
4.3.1 Initialization: First Analysis and Generation of the starting ensemble
4.3.2 Forecast
4.3.3 Analysis
4.3.4 Implementation of the Ensemble Kalman Filter
4.4 A posteriori Evaluation of the Ensemble Kalman Filter Method
4.4.1 Twin Experiment Setup
4.4.2 Robustness of the Assimilation Algorithm
4.4.3 Effect of the data assimilation parameters on the quality of the estimation .
4.4.4 Accuracy of the Reconstruction of Geodynamic Structures
4.5 Discussion
4.6 Conclusion
5 Discussion and Conclusion 
Bibliography 

GET THE COMPLETE PROJECT

Related Posts