Recent advances in mantle convection modeling led to the release
of a new generation of convection codes, able to self-consistently generate
plate-like tectonics at their surface. Those models physically link mantle
dynamics to surface tectonics. Combined with plate tectonic reconstructions,
they have the potential to produce a new generation of mantle circulation
models that use data assimilation methods and where uncertainties in plate
tectonic reconstructions are taken into account. We provided a proof of this
concept by applying a suboptimal Kalman filter to the reconstruction of
mantle circulation

Mantle circulation models are estimates of mantle flow history. They combine
two sources of information: observations on the dynamics or 3-D structure of
the Earth's mantle and a numerical model of mantle convection. In their
effort to reconcile both observations and our physical understanding of
mantle dynamics, they serve a wide variety of purposes and disciplines.

The geodynamics community has developed three alternative approaches to the problem of the reconstruction of mantle circulation.

The first approach,
backward advection, consists of starting at present by estimating the current
density field of the mantle from seismic tomography models

The second approach, the semi-empirical sequential method, estimates mantle
circulation by integrating plate tectonic reconstructions chronologically
into a mantle convection model. Plate tectonic reconstructions are either
introduced as velocity boundary conditions, as first described by

The third approach uses data assimilation methods to solve the
mantle circulation problem. Data assimilation methods are inverse methods
dealing with the specific problem of estimating the evolution of a dynamical
system from asynchronous data and a physical model

Here, we take a different view of data assimilation methods for mantle
circulation models by focusing on how to take into account the uncertainties
in plate tectonic reconstructions. For almost a decade, 3-D spherical mantle
convection models have shown the capability to self-consistently produce
plate-like tectonics at their surface

Here we extend this work by applying a more advanced sequential data assimilation
method, the ensemble Kalman filter (EnKF, originally described in

This paper is organized as follows: in Sect. 2, we present our
simplifications on the general mantle circulation reconstruction problem and
the correspondence with the notation in the EnKF algorithm. Then, in Sect. 3,
we detail the EnKF method and justify the variants chosen for the
application to mantle circulation. Section 4 presents the results obtained on
synthetic experiments and compares them to results obtained by the method
described in

We aim at reconstructing mantle circulation for the last hundreds of millions of years by combining a mantle convection model with plate tectonic reconstructions, using an EnKF. To study the behavior of the EnKF on such a problem, we consider a simplified mantle convection model. This section describes the model used to compute a mantle evolution, the dataset assimilated in this evolution, and finally the backbone of ensemble Kalman filtering.

On the timescales and length scales we are interested in (

The deformation response of mantle material to stress is implemented as a
linear relationship linking the strain rate tensor

The choice of the effective viscosity

Values of the parameters of the forward model.

The Newtonian viscosity

The pseudo-plastic part

This composite rheology allows the development of strong plates delimited by
narrow weak zones (i.e., plate boundaries), and is currently the best way to
self-consistently generate plate-like tectonics at the surface of global mantle convection models

The energy conservation equation is the only prognostic equation of the system

These equations are solved using the finite volume, multigrid parallel code
STAGYY

Note that this paper focuses on the methodology of ensemble data assimilation
for a convecting system similar to that of the Earth's mantle. Hence, we
choose a rather simple model that can reproduce plate-like tectonics at the
surface. We rely on simplifications such as a 2-D geometry, incompressible
and isochemical mantle, and a rheology which does not take into account the
history of the material. Although some of the complexities we ignore may play
a fundamental role in the reconstruction of the Earth's mantle evolution, we
choose to focus in this paper on the data assimilation methodology. Moreover, we choose to
keep the same parameters as the test case of

To ease the comparison with Earth's mantle convection, we rescale the
nondimensional time in the evolution,

The dynamics of the convective system we just described depends on the two
dimensionless numbers

The state of the Earth's surface is the time-integrated expression of mantle
circulation. On a global scale, the main source of information for the last

Plate tectonic reconstructions use the geometric theory of plate tectonics to
integrate all these observations. The result is a time series of maps of
seafloor ages, plate layout and kinematics. The continuously closed plate
algorithm

Although we are aware that these plate tectonic reconstruction maps are in
themselves models and not direct observations, we propose to develop an
assimilation method that use them as data to assimilate in our mantle
convection model. This solution is generally chosen in mantle circulation
reconstructions

In this study, we limit ourselves to the test of data assimilation in
synthetic experiments. In the model described in Sect.

To our knowledge, the amplitude of the uncertainty in global plate tectonic reconstructions has not yet been assessed.

In the synthetic test we perform in Sect.

Our aim is to assimilate a time series of observations (surface velocities
and heat fluxes) into a mantle convection model to estimate the evolution of
the state of the mantle. We introduce here the general formulation of
ensemble Kalman filtering and link them to our problem. We use the notation
system recommended by

The time series of data is defined as a set of column vectors

Notations and dimensions of data assimilation variables.

The evolution of the state of the system is estimated sequentially during the
period where observations are available. At each time step

Finally, we introduce the observation operator, which maps a given state
vector

Table

The EnKF

To implement the EnKF, we used the software environment Parallel Data
Assimilation Framework

We compute the second order statistics of the background state from a series
of

We assimilate the first set of observations

This method of generating the starting ensemble takes advantage of the
extensive knowledge we have on the background statistics of the model.
Several other methods have been tested to generate a starting ensemble, such
as starting with random decorrelated snapshots of mantle convection
simulations, second order exact sampling from

Between time steps

The forecast state is the average of the ensemble

We correct the forecast ensemble variance with an inflation factor

The inflation factor is

The analyzed state

The Kalman gain is defined as

We also tested the domain localization strategy as described in

We used the software environment PDAF

We modified the STAGYY code following the procedure recommended by PDAF

We test the data assimilation scheme on twin experiments using the model described in Sect.

After describing the setup used for twin experiments, we test the robustness of the EnKF method and compare it to that of method 1. Then, we determine the range of data assimilation parameters which are suitable to conduct an ensemble data assimilation. Finally, we assess the ability of the scheme to actually reconstruct specific geodynamic structures.

Twin experiments are a way to assess the accuracy of a data assimilation procedure in a controlled environment, where the true evolution is perfectly known.

First, we compute a reference state evolution using the forward numerical
model, considered as the true state evolution, from which we extract the set
of true state vectors

Then, we perform ensemble data assimilation for the dataset

Notations and range of values tested for data assimilation parameters

We present here tests with different assimilation parameters, varying the
number of members

We compute four different state evolutions to test the accuracy of the
EnKF for different dynamical cases (the four state
evolutions are described in the next section). Figure

The evolutions of the global errors in the estimated temperature field and
surface horizontal velocity field over the time period

Time evolution of the errors in the estimated
temperature field (panels

We test the EnKF on one evolution, with sizes of the ensemble

We choose the average error in temperature after analysis

The error evolutions for temperature and surface horizontal velocity follow
the analysis–forecast sequence: at each analysis time (every

For the surface horizontal velocity (Fig.

On the contrary, the evolution of the error in the temperature depends on the
parameters of the assimilation. Figure

We compute the error in the estimated temperature by comparing it to the true
temperature field. However, in a realistic case, the true temperature is not
known, and the evaluation of the data assimilation algorithm is based on the
study of the statistics of the innovation vector

Figure

The evolution of the cumulative mean of the innovation

The norm of the instantaneous innovation

Evolution of

We also tested the assimilation algorithm for four different state evolutions,
with the optimal parameters for an ensemble size of

Evolution of the error (

The spread of the ensemble is an estimation of the uncertainty in the state.
We compare the evolution of

Although we computed the four state evolutions using the same forward
modeling code and with the same values of physical parameters (as described
in Table

In the four cases, the errors in the estimated temperature field
systematically decrease during the analysis step for the EnKF algorithm. The
errors stay below the first analysis error for evolutions A, B and C, while
they reach slightly higher values for evolution D. The error of the EnKF is
always lower than that obtained with method 1 for the first

Comparison of temperature field evolutions for
evolution B.

Example of temperature fields of the members of the
ensemble. This example is taken after

For evolution B, method 1 fails to accurately reconstruct the evolution, with
the error reaching values greater than

In Fig.

Rank histograms were first described independently by

Selection of the variable and the verification. We compute rank histograms
for surface heat fluxes (Fig.

Selection of the sampling points. To be able to interpret our rank histograms,
we need to populate them with samples that are independent. To do so, we use
the four evolutions presented in Fig.

Determination of the rank of the verification. At each sampling point, we determine the rank of the verification in a vector composed of all the values taken by the ensemble plus the verification, in ascending order.

Computation of the rank histogram. In order to have bins of constant
width, we choose

To guide our interpretations, we perform the

Rank histograms of the surface true heat
flux

The left column of Fig.

In an assimilation with Earth data, the truth is not known, and we would have
to draw rank histograms using observed data. The question is, would we come
to the same conclusion about the reliability of the ensemble as with
Fig.

Since the noise in the observations largely affects the shape of the rank
histograms, we need to add noise to the ensemble members before computing the
rank histograms, as explained in

Rank histograms for temperature at the surface

Figure

In conclusion, Fig.

As shown in Fig.

Values of the average analyzed error for assimilations
performed using the dataset generated by evolution A of Fig.

For each size of ensemble

For the ensemble size

Values of

Figure

Overall, the average ensemble spread

The average norm of instantaneous innovations and the norm of the cumulative innovations display the same behavior: they decrease with increasing vertical and horizontal correlation lengths. The longer the correlation lengths, the closer the forecast data are to the observations, and the less biased the assimilation. This means that a better fit to the observations does not necessarily imply a better fit to the true temperature field. In a realistic context, the result of the assimilation should be checked against independent data to evaluate its accuracy. In the case of the Earth's mantle, independent data could be, for example, the geoid or tomographic models.

We focus on three key flow structures: (1) downwelling slabs (subduction), (2) ridges, i.e., shallow structures resulting from divergent plates at the surface, and (3) plumes, hot upwellings rising from the base of the model.

Figure

Comparison of estimated states after

Globally, the EnKF288 and EnKF96 solutions for the temperature field are smoother than the solution of method 1. We observe this difference especially in the asthenosphere, the part of the mantle below the top boundary layer. For method 1, the asthenosphere shows short wavelength temperature variations. These variations are absent from the true temperature field and are inconsistent with convection solutions with the chosen parameters. They stem from the amplification of the noise in the observations during the analysis. Moreover, the asthenosphere of the analyzed temperature field of method 1 is hotter than the true temperature.

Both EnKF96 and EnKF288 reconstruct successfully the ridges locations and
structures, as testified by their error fields. On the contrary, method 1
fails to reconstruct the ridge on the top right of the domain. It also
predicts a ridge that does not exist in the true state (in the top left
quadrant). On the right of the domain, another ridge is associated with a
vertical positive temperature anomaly underneath. This pattern is found
regularly under ridges when applying method 1. This is due to the use of a
constant forecast error covariance matrix,

All three assimilations reconstruct the subductions and accurately predict the bending direction of slabs at the base of the model. Method 1 tends to underestimate the amplitude of the negative temperature anomalies, whereas both EnKF assimilations overestimate them. This is especially noteworthy for the bottom left subduction. Moreover, the estimated slabs are wider than the true slabs. However, we note two arguments in favor of the EnKF: first, the estimation of the slab improves when the size of the ensemble increases; second, the local standard deviations of the ensemble indicate that the estimation in this part of the domain is less accurate.

Both EnKF288 and EnKF96 solutions do not show any plume at the base of the
mantle. However, the ensemble spread shows a greater uncertainty in the
places where plumes occur. Method 1 predicts the approximate location of all
plumes, but their geometry is not accurate. Method 1 provides only one
estimate of the temperature field. In this evolution, the plumes are allowed
to develop. EnKF96 and EnKF288 provide an ensemble of states. Each state
develops plumes at different locations and their averages show only a
slightly hotter anomaly over a wide area of possible location for the plumes,
as we showed earlier in Fig.

To illustrate how different flow structures are reconstructed, we plot in
Fig.

Detailed results of the assimilation depicted in
Fig.

At the surface, the temperature is corrected accurately at each analysis,
with a difference between the true temperature and the analyzed temperature
of less than

For the subduction, the correction is first done on the surface, and then
propagates gradually with depth. The reconstruction of mid-mantle temperature
becomes accurate after

At the surface for the plume, the spread of the ensemble is very low except
for a peak at

The ridge initiation shows how new observations affect the spread of the
ensemble. At the surface, the spread of the ensemble remains low until

For the stable ridge, the spread of the ensemble at the surface is increasing
during forecast and decreasing dramatically during the analysis. At
mid-mantle, the estimated temperature becomes accurate after

We chose the EnKF method for its ease of implementation and flexibility to
adapt to different forward numerical models. Indeed, as long as the nature of
the state and observations does not change, the computation of the analysis
step remains the same regardless of the convection code used. On the
contrary, the alternative method, variational data assimilation, requires the
development of an adjoint code that needs further development for each
additional complexity added to the forward model

The application of the EnKF to the mantle circulation
problem is the continuation of the simpler sequential filter that we
developed in an earlier work

This gain in information and quality for reconstructions comes with a
computational price. While we could perform the former assimilation method in
one core hour, the method developed here requires several hundreds to several
thousands of core hours. However, an efficient parallelization using the PDAF
software

The important computational cost of the EnKF limited us in the number of
assimilations we could test. After checking the stability of the assimilation
results on four different evolutions, we chose to focus on studying the
effect of the parameters of the ensemble data assimilation: the size of the
ensemble and the vertical and horizontal correlation lengths. We found that
the best compromise between the accuracy of the assimilation and the
computational cost was an ensemble of

During these tests, we also evaluated how accurate the estimation of
uncertainties (i.e., the spread of the ensemble) is with respect to the true
error, and, more generally, how reliable the forecast (i.e., the ensemble) is.
If we consider the four assimilations with different data time series
presented in Fig.

Another important question for future applications with Earth data is, how
well can we assess the quality of an assimilation when only observed data are
available, i.e., without any knowledge on the true state? To answer this
question, we investigated the statistics of the cumulative innovation and of
the instantaneous innovation for different ensemble sizes and correlation
lengths. The variation in both cumulative innovation and instantaneous
innovation as a function of ensemble size show the same tendency as the
global average error in the temperature field: the larger the ensemble, the
lower the instantaneous and cumulative innovations, and results for

By construction, sequential data assimilation methods do not propagate new
information back in time. In the case of the reconstruction of mantle
circulation, this is a clear disadvantage since the information on the
Earth's surface tectonics tends to become more reliable as we get closer to
present-day. Consequently, a natural extension of the present work would be
to implement an ensemble Kalman smoother

As a first approach to test the EnKF for mantle circulation reconstructions,
we chose a fairly simple convection model. As already discussed in
Sect.

In the synthetic experiments of Sect.

The choice of the synthetic experiments assimilation window of

First, we set a time series of data covering the whole surface of the domain
and regularly available, every

Second, the observations were perturbed independently with a Gaussian noise
of 10 % of the respective root mean square value of surface heat flux and surface velocities. The estimation of
uncertainties in absolute plate motion models involves estimation of both
uncertainties in relative plate motion and on the absolute reference frame

We applied the EnKF algorithm to the reconstruction of mantle circulation through time. We chose a formulation with covariance inflation and localization to minimize the effect of sampling errors in the estimation of the forecast error covariance matrix. Synthetic twin experiments with different evolutions and for different parameters allowed us to assess the efficiency of the algorithm and to determine the optimal parameters for the assimilation.

This work builds on the developments of a first approach to sequential data
assimilation for mantle circulation made in

We evaluate the accuracy of the EnKF as a function of three main parameters:
the size of the ensemble and two covariance localization parameters, namely
the vertical correlation length and horizontal correlation angle. We find
that a size of the ensemble of the order of

The EnKF was implemented using the parallel data assimilation framework PDAF in a preexisting mantle convection code, STAGYY. The resulting code is highly scalable, which means that the application of the EnKF to realistic data assimilation with plate reconstructions and a 3-D spherical mantle model is within reach in the foreseeable future.

No data sets were used in the article.

The authors declare that they have no conflict of interest.

We thank Jeffrey Anderson and an anonymous reviewer, as well as the editor, Olivier Talagrand, for their very useful comments and suggestions, which greatly helped improve the manuscript. The research leading to these results has received funding from the European Research Council within the framework of the SP2-Ideas Program ERC-2013-CoG, under ERC grant agreement 617588. Calculations were performed using HPC resources from GENCI-IDRIS (grant 2016-047243). The contribution of Alexandre Fournier is IPGP contribution number 3910. Edited by: Olivier Talagrand Reviewed by: Jeffrey Anderson and one anonymous referee