Introduction
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.
originally built instantaneous mantle circulation
models to understand the effect of plates on large-scale mantle flow. Since
then, they have been used, among other applications, to understand the
dynamics and evolution of the deep Earth mantle structures
,
to study the evolution of mantle plumes and their relationship to hotspots
, and to infer changes in the Earth's rotation axis
, sea-level or
dynamic topography .
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 seefor a
description of this method. This density field is then
advected backward in time with plate tectonic reconstructions as imposed
boundary condition . This method has a limited
numerical cost and exploits the two most instructive constraints on mantle
circulation: plate tectonic reconstructions and seismic tomography. However,
this technique neglects thermal diffusion, so it is not able to reconstruct
past thermal structures that have completely diffused before present and it
is limited to times and regions for which the effect of diffusion is thought
to be small. This limits reconstructions to the last 50 to 75 Ma
or even to shorter periods if we consider the
uncertainties in tomographic 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
, or with a more sophisticated method, by blending a
convection solution with thermal and kinematic models of plates and slabs
. This approach allows the use of models of
convection with chemical heterogeneities .
Also, it is not anymore the reconstruction method that limits the timespan of
the reconstruction but the availability of plate tectonic reconstructions.
This led to mantle circulation models integrating up to 450 Ma of plate
reconstruction history . However, this method considers
plate tectonic reconstructions as perfect estimates of surface tectonics:
uncertainties affecting the reconstructions are not taken into account
although they are substantial, especially as reconstructions go further into
the past for example, there is almost no information on the state of
the ocean floor before 140 Ma, see e.g.,. This method
also requires the choice of an arbitrary initial temperature field to compute
the evolution.
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 .
The full inverse problem for mantle circulation, as stated by
, would take into account model errors, numerical
approximations, errors in plate reconstructions and on the estimation of the
current tomography-derived temperature field to provide the best fit given
all sources of information. However, solving the full inverse problem of
mantle circulation is still a great challenge given the nonlinearities in
mantle convection dynamics and the computational power required to compute a
realistic forward mantle convection evolution alone
. So far, variational data
assimilation dominates over other methods to estimate mantle circulation
. To
simplify the problem, they minimize the misfit between the final temperature
field of the mantle circulation model and the one deduced from seismic
tomography. These mantle circulation models impose plate tectonic
reconstructions as boundary conditions, as in the first two approaches.
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
.
These models physically link surface tectonics comparable to that of the
Earth to mantle convection processes
. In
, we took advantage of this link to build a
sequential data assimilation algorithm able to integrate plate
reconstructions into a mantle convection code while taking into account the
uncertainties in those plate tectonic reconstructions. This technique
assimilates a time series of surface observations chronologically, by
repeating two stages (analysis and forecast), until all observations are taken
into account. Whenever an observation is available, the analysis evaluates
the most likely state of the mantle at this time, considering a prior guess
(supplied by the forecast) and the new observations at hand. For this
evaluation, we used the classical best linear unbiased estimate
. Then, the forward model of mantle
convection forecasts the evolution of the mantle until the next observation
time. We tested this algorithm on synthetic experiments. It proved to be
efficient in recovering mantle circulation given constraints on the amplitude
of errors affecting observations and the timespan between observations.
Here we extend this work by applying a more advanced sequential data assimilation
method, the ensemble Kalman filter (EnKF, originally described in
, and in its corrected version in
). This method is particularly suited for high-dimensional
nonlinear dynamical models . Instead
of estimating the most likely state of the mantle, the EnKF provides at each
time an approximation of the probability density function of the state of the
system in the form of a finite ensemble of states. During the forecast stage,
each member of the ensemble evolves independently. For the analysis, we use
the second order statistics of the ensemble to correct each ensemble member
with the new observations at hand. We evaluate this method with synthetic
experiments in 2-D-spherical annulus geometry
and compare it to the algorithm developed in .
The EnKF provides more accurate estimations than the former method, and is
even able to reconstruct evolutions that the former method could not.
Moreover, the EnKF also estimates locally the error in
the reconstruction. Both covariance inflation and localization eliminate
spurious correlations arising from the finite size of the ensemble that is
used to compute them.
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 . Section 5 is a discussion on the
choice of the method and the challenges involved in the application of such a
method to a realistic setting.
Presentation of the problem
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.
Mantle convection model
On the timescales and length scales we are interested in (≥ 10 kyr,
≥ 1000 km), the mantle can be modeled as a continuous viscous medium.
To compute mantle circulation, we solve the equations of conservation of mass
(Eq. below), momentum (Eq. below) and energy
(Eq. below) for an isochemical mantle under the Boussinesq
approximation. The system of equations is nondimensionalized to the thermal
diffusion timescale see. Given the high Prandtl
number of the mantle (of the order of 1024), inertia is neglected. With
these assumptions, the equations of conservation of mass and momentum become
diagnostic equations of the form
∇⋅u=0,∇⋅σ-∇p+RaTTer=0,
where σ, u, p and T are the
nondimensional deviatoric stress, velocity, dynamic pressure and
temperature, respectively. RaT is the Rayleigh number based on
the temperature difference between the top and bottom boundaries of the
domain, defined as
RaT=ρ0g0α0ΔTa3μ0κ0
with ρ0 being the density for T=0, g0 the gravitational acceleration,
α0 the thermal expansivity, ΔT the temperature drop, a the
depth of the layer, κ0 the thermal diffusivity and μ0 the dynamic
viscosity of the system. The vertical velocities and shear stress at the
surface and the base of the model are set to zero.
The deformation response of mantle material to stress is implemented as a
linear relationship linking the strain rate tensor
ϵ˙ to the deviatoric stress tensor
σ as
σ=2μeffϵ˙=μeff∇u+∇uT.
The choice of the effective viscosity μeff is crucial for the
development of plate-like tectonics at the surface of the convective system.
For μeff, we choose a composite rheology with a viscous
Newtonian component μn and a pseudo-plastic component, implemented
with an equivalent pseudo-plastic viscosity μy, such that
μeff=min(μn,μy).
Values of the parameters of the forward model.
Symbol
Meaning
Value
RaT
Rayleigh number based on temperature difference
106
RaH
Rayleigh number based on internal heating
2.05 × 107
L
number of grid points in longitude
384
M
number of grid points in radius
48
ra
radius of the top of the domain
2.2
rb
radius of the bottom of the domain
1.2
Ta
temperature at the top of the domain
0
Tb
temperature at the bottom of the domain
0.9
EA
activation energy
23.03
T1
temperature at which μT=1
1
β
factor of viscosity reduction for partial melting
10
Ts0
solidus temperature at r=ra
0.6
∇rTs
radial gradient of the solidus temperature
2
σY
yield stress
104
∇rσY
radial gradient of the yield stress
2 × 105
The Newtonian viscosity μn follows an Arrhenius law
μn=μ0expEAT+T1
with μ0=exp-EA2T1, T1 the
temperature at which the nondimensional μn=1 and EA
the nondimensional activation energy. This law reflects the thermal
activation of crystal deformation, and creates a highly viscous upper
boundary layer (the lithosphere), while the rest of the mantle is less
viscous. We also implement the decrease in viscosity in the asthenosphere
(the layer below the lithosphere) by reducing by a factor of 10 the
viscosity μn when the temperature is above a solidus equation
Ts=Ts0+∇rTs(ra-r) with ra the
surface value of r. The presence of a weak asthenosphere tends to favor
plate-like behavior , and is
compatible with laboratory and observational data
.
The pseudo-plastic part μy is defined by
μy=σyield2ϵ˙II,
where ϵ˙II is the second invariant of the strain
rate tensor and σyield=σY+(ra-r)∇rσY, with σY and
∇rσY the yield stress at the surface and the
depth-dependence of the yield stress, respectively.
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
DTDt=∇2T+RaHRaT,
with RaH being the Rayleigh number based on internal heating
RaH=ρ02g0α0Ha5μ0k0κ0
with H being the dimensional heating rate and k0 the thermal conductivity. We
set isothermal top and bottom boundaries with temperatures Ta and
Tb, respectively. The models presented here have 10 % basal heating
and 90 % internal heating.
These equations are solved using the finite volume, multigrid parallel code
STAGYY , on a spherical annulus staggered grid.
This geometry provides results closer to the spherical geometry than
cylindrical geometry . In the following, the
longitudinal coordinate of a point is ϕl, with
l∈{1,2,…,L} and its radial coordinate is rm with
m∈{1,2,…,M}, r varying from rb to ra.
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 in
order to enable direct comparison between the methods. Table
lists the chosen parameter values.
To ease the comparison with Earth's mantle convection, we rescale the
nondimensional time in the evolution, t, by the transit time of the
convective system. By definition, the transit time of the Earth's mantle is
ttE=aE/vrmsE=70 Myrs,
with aE=2890 km, the thickness of the Earth's mantle, and
vrmsE=4.13 cm yr-1, the root mean square of
surface velocities of the Earth, as estimated by plate tectonic
reconstructions . We compute the same quantity for the
model ttm=a/vrms, with a=ra-rb. The
scaled time ts is then
ts=tttEttm.
The dynamics of the convective system we just described depends on the two
dimensionless numbers RaT and RaH. In our model,
RaT=106 and RaH=2.05 × 107. These values are
1 to 2 orders of magnitude lower than the current Earth estimates, but
high enough to ensure chaotic convection with thermal turbulence
. In this regime, the top and
bottom boundary layers develop instabilities that can trigger transient
descending and ascending currents, respectively. This leads to a highly
time-dependent flow, and the exponential growth of perturbations of the
initial state of the system, as studied by ,
in a series of twin experiments in 3-D spherical geometry. We computed the
Lyapunov time corresponding to the time over which initial perturbations grow
exponentially by a factor of e, and found for our models a Lyapunov time of
140 Myr, similar to the times estimated
for their most Earth-like model.
Observations of mantle circulation
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
100 Myr is the database of the localization and identification of magnetic
anomalies on the seafloor, translated into maps of seafloor ages
. This information is complemented
with regional geological studies giving constraints on the timing and
geometry of tectonic events as well as a synthesis of paleontological,
structural geology, stratigraphical, magnetic anomalies, gravity data and
seismic studies. In addition, paleomagnetic data provide constraints on the
paleolatitude of continental blocks .
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 produces plate tectonic reconstruction maps
continuous in space and time .
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 ,
because it provides continuous surface boundary conditions in space and time
for the period of reconstruction. One advantage of the technique we develop
is that it is possible to consider errors in the data that are assimilated,
another is that the reconstructions do not need to be known at all times and
at all points on the surface. Hence it is possible, in principle, to design a
data assimilation scheme using direct observations. However, this would
require further developments both on the database design and on the data
assimilation algorithm. Sequential data assimilation methods for mantle
circulation are still in their infancy, so we opt for a simpler structure of
the data to be assimilated: a time series of maps of surface velocity and
seafloor age, as given by plate tectonic reconstructions.
In this study, we limit ourselves to the test of data assimilation in
synthetic experiments. In the model described in Sect. , the
absence of small-scale convection at the base of the boundary layer makes the
surface heat flux an excellent proxy for the age of the seafloor
. Consequently, we consider surface heat flux and
surface velocity as the data to assimilate.
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. , we choose for the amplitude of the
uncertainties in heat flux and velocity an arbitrary value of 10 % of their respective root mean square values.
We further discuss this choice in Sect. .
Ensemble Kalman filtering framework notations
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
y1o,y2o,…,yKo,
where the subscripts {1,2,…,K} refer to the times at which
observations are available. As seen in the previous section, the data used
for our experiments are surface velocity and surface heat flux. The data
vector at time k is thus defined as
yko=qko(ϕ1),qko(ϕ2),…,qko(ϕL),uϕko(ϕ1),uϕko(ϕ2),…,uϕko(ϕL)T,
where qko(ϕl) and uϕko(ϕl) are the observed values of surface heat flux and surface
horizontal velocity at the kth time step and longitude ϕl,
and (⋅)T means transpose. We model errors in observations by a random
vector of zero mean and covariance matrix Rk (we suppose unbiased
observations). Although Rk is a diagonal matrix of constant value
and size in our experiments, it is not generally the case. Correlations
between errors in observations could be specified in Rk.
Notations and dimensions of data assimilation
variables.
Symbol
Meaning
Size (literal)
Size (value)
x
state
LM+L
18 816
y
data
L+L
768
H
observation matrix operator
(L+L)×(LM+L)
768×18 816
R
observation error covariance matrix
(L+L)×(L+L)
768×768
P
state error covariance matrix
(LM+L)×(LM+L)
18 816 × 18 816
X
ensemble state
(LM+L)×N
18 816 × N,
(N=96, 288 or 768)
The evolution of the state of the system is estimated sequentially during the
period where observations are available. At each time step
k∈{1,2,…,K}, we define two state vectors: the a priori state, or
forecast state xkf, and the analysis state
xka, which is the state corrected after having
assimilated the observations yko. The system of equations
developed in Sect. shows that we can compute velocity,
viscosity and pressure values at each grid point from the sole knowledge of
the temperature field: the temperature field describes completely the state
of the system. However, the relation between surface velocities and the
temperature field is nonlinear. We choose to include in the state the whole
temperature field, but also add the surface velocities, to form an augmented
state vector, following the suggestion of ,
Sect. 4.5. This formulation establishes a linear relationship between the
state and data (see last paragraph of this section), which simplifies the
computations thereafter. This formulation also implies that the analyzed
surface velocities and temperature field do not necessarily satisfy
Eqs. () and (). However, momentum and mass
conservation are ensured at the time step following the analysis, and we did
not observe any instability developing due to this formulation. The state of
the mantle at a time step k∈[1,K] is defined as
xk=Tk(ϕ1,r1),Tk(ϕ1,r2),…,Tk(ϕL,rM),uϕk(ϕ1),uϕk(ϕ2),…,uϕk(ϕL)T,
where Tk(ϕl,rm) is the value of temperature at the kth
time step, longitude ϕl and radius rm, and uϕk(ϕl) is the surface horizontal velocity at the kth time
step and longitude ϕl. The forecast and analyzed
states are uncertain as well; their uncertainties are represented by two
random vectors of zero expected value and covariance matrices
Pkf and Pka, respectively. We
compute explicitly these covariance matrices only for the initialization step
(see Sect. ). Otherwise, the uncertainty in the forecast and
analyzed states is represented by two ensembles of N states
{xknf}n∈[1,N] and
{xkna}n∈[1,N], such that their average equals
xkf and xka, respectively, and their
respective sample covariance matrices approximate Pkf
and Pka. The ensemble of states
{xknf}n∈[1,N] and
{xkna}n∈[1,N] are stored in the matrices
Xkf and Xka, where the nth
column is the state of the nth ensemble member xknf
and xkna, respectively.
Finally, we introduce the observation operator, which maps a given state
vector xkne (e being f or a) to the corresponding data
ykne. The surface heat flux is approximated by a first order
discretization of Fourier's law. The observation operator is then linear,
with its velocity part being simply the identity, and can be represented by
the matrix H such that
∀k∈{1,2,…,K},∀n∈{1,2,…,N},ykne=Hxkne.
Table summarizes the dimensions of the vectors and matrices for our problem.
EnKF with localization and inflation
The EnKF
is a sequential data assimilation algorithm using the same equations as the
Kalman filter for the analysis step, but Monte Carlo methods to forecast the
error statistics of the state. We explain here how we adapt the EnKF
to our problem and justify the choice of the starting ensemble.
To implement the EnKF, we used the software environment Parallel Data
Assimilation Framework
PDAF,.
Initialization: first analysis and generation of the starting ensemble
We compute the second order statistics of the background state from a series
of 400 decorrelated snapshots of convection simulations by following the
procedure detailed in , Sect. 4.1. The model
setup is spherically symmetric, so the expected value and covariance of the
background temperatures and surface velocities must satisfy
∀(ϕ,r),〈T(ϕ,r)〉=〈T(0,r)〉,
∀(ϕ1,ϕ2,r1,r2),Cov(T(ϕ1,r1),T(ϕ2,r2))=Cov(T(0,r1),T(ϕ1-ϕ2,r2)),
=Cov(T(0,r1),T(ϕ2-ϕ1,r2)),
where 〈⋅〉 stands for the expectation operator and
Cov(⋅,⋅) stands for the covariance operator. Likewise, we
have
∀ϕ,〈uϕ(ϕ)〉=〈uϕ(0)〉,
∀(ϕ1,ϕ2),Cov(uϕ(ϕ1),uϕ(ϕ2))=Cov(uϕ(0),uϕ(ϕ1-ϕ2)),∀(ϕ1,ϕ2,r1),Cov(T(ϕ1,r1),uϕ(ϕ2))=Cov(T(0,r1),uϕ(ϕ2-ϕ1)),
=-Cov(T(0,r1),uϕ(ϕ1-ϕ2)).
We use these symmetries to compute
〈T(0,rm)〉,withm∈{1,…,M}
Cov(T(0,rm),T(ϕl′,rm′)),withm∈{1,…,M},l′∈{1,…,L/2},andm′∈{1,…,M}
〈uϕ(0)〉,Cov(uϕ(0),uϕ(ϕl)),withl∈{1,…,L/2}
Cov(uϕ(0),T(ϕl,rm)),withl∈{1,…,L/2},andm∈{1,…,M}
and build with these values the first forecast state of expected value
x1f and associated covariance matrix P1f. For the
model used in this study (see Table ), the covariance matrix
P1f has (LM+L)2=188162=354041856 components. By
using the symmetries in the system, we are able to reduce the number of
independent components in the covariance matrix to L/2(M+1)2=3557400.
P1f is eigendecomposed and rank reduced into
P1rf=VΛVT, with
Λ a diagonal matrix containing the nr=1928 largest
eigenvalues of P1f (which accounts for 99.98 % of its
cumulative variance) and V the corresponding matrix of
eigenvectors.
We assimilate the first set of observations y1o using the
classical best linear unbiased
estimator equations (see , for example). When the
forecast covariance matrix is eigendecomposed and rank reduced, these
equations can take the form
x1a=x1f+VAVTHTR-1(y1o-Hx1f),P1a=VAVT,
with
A=Λ-1+VTHTR-1HV-1.
After the first analysis, we generate an ensemble of N initial states from
the first analyzed state average x1a and associated
covariance matrix P1a. To do so, we follow the second order
exact sampling method . First,
A is eigendecomposed
A=VaΛaVaT.
The ensemble members are then computed following
X1a=||x11a…x1Na||=||x1a…x1a||+N-1VVaΛa1/2ΩN×(N-1)T0(nr-N)×N,
where ΩN×(N-1) is a random matrix whose
columns are vectors forming an orthonormal basis and each of them is
orthogonal to 1N, the column vector of dimension N full
of 1, 1N=[1,…,1]T.
0(nr-N)×N is a (nr-N)×N matrix full of
0. ΩN×(N-1) is generated through the
algorithm described in the appendix of . The
matrix ΩN×(N-1) is designed so that the sample
mean of the starting ensemble is equal to x1a and its
sample covariance matrix is equal to matrix P1a reduced to its
N largest eigenvalues.
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 x1f and
P1f, and several assimilations of the first
observations y1o. These alternative solutions resulted in
reconstructions with larger initial errors and slower error decrease
throughout the assimilation window, if any.
Forecast
Between time steps k-1 and k, the forward numerical code STAGYY computes
independently the evolution of each of the analyzed states
{xk-1,na}n∈[1,N] to produce a forecast ensemble
{xk,nf}n∈[1,N].
The forecast state is the average of the ensemble
xkf=1NXkf1N.
The forecast error covariance matrix
is given by the sample covariance matrix of the ensemble of forecast states
Pkf=1N-1XkfIN-1N1N1NTIN-1N1N1NTTXkfT,
where IN is the identity matrix of dimension N×N. After
several assimilation cycles, the finite size of the ensemble induces the
underestimation of the error variance , and can lead to
filter divergence. We observed this behavior in our case; thus, to stabilize
the filter, we apply covariance inflation, as suggested in
and .
We correct the forecast ensemble variance with an inflation factor γ
according to
Xkf←1NXkf1N1NT+XkfIN-1N1N1NTγ,
where ← means that we replace the matrix on the left-hand side by
the term on the right-hand side. γ is computed following the same
principles as in the suboptimal Kalman filter developed in
, i.e., by comparing the error in observations
and the standard deviation of the innovation dk defined as
dk=yko-1NHXkf1N.
The inflation factor is
γ=Vd-VoVf,
with
Vd=TrdkdkT,Vo=Tr(Rk),Vf=TrHXkfIN-1N1N1NTIN-1N1N1NTTXkfTHT,
where Tr(⋅) means the trace. The inflation factor is then truncated
between a minimum value of 1 (to prevent further contraction of the
ensemble spread) and a maximum value of γ+=1.25 (to prevent
overspread). Several values of maximum inflation factor have been tested,
from γ+=1.1 to γ+=2, and showed little impact on the
efficiency of the assimilation. A constant inflation factor was also tested,
but the results with an adaptive inflation factor were substantially more
accurate, especially for the first assimilation times.
Analysis
The analyzed state xkna of the nth member of the ensemble is
xkna=xknf+Kkykno-Hxknf,
where Kk is the Kalman gain. ykno is the observed data vector yko to
which a random perturbation of zero expected value and covariance matrix
Rk is added, as is recommended in
.
The Kalman gain is defined as
Kk=(Pkf∘C)HTH(Pkf∘C)HT+Rk-1,
where the matrix Pkf is the sample covariance matrix of
the ensemble of forecast states {xknf}n∈[1,N]. We
use a limited ensemble size (maximum 768) to estimate
Pkf. Spurious correlations ensue, especially between
distant points. To mitigate this effect, we implement localization directly
on the forecast error covariance matrix by Schur multiplying (symbol
∘) Pkf by the localization matrix
C, as introduced by and
. The matrix C is
itself the Schur product of a vertical localization matrix
Cv and a horizontal localization matrix
Ch. The value of
Cv(i,j) depends on the absolute radius
difference between the ith and the jth components of the state vector and
on the vertical correlation length ℓv. The value of
Ch(i,j) depends on the absolute angle
difference of the ith and the jth components of the state vector and on
the vertical correlation length ℓh. Both values follow a
Gaspari–Cohn compactly supported fifth-order piecewise rational function
similar to a Gaussian but with a compact support; Eq. 4.10
of.
We also tested the domain localization strategy as described in
, since it is in some cases computationally more
efficient and already implemented in PDAF. However, it led to a systematic
failure of the assimilation. This is due to the nature of our problem: all
the observations are located at the surface of the model and we aim at
estimating the temperature field over the whole depth of the mantle. A
vertical localization is as necessary as a horizontal localization, hence the
localization has to be done directly in the forecast error covariance matrix
and not only in the data space.
Implementation of the EnKF
We used the software environment PDAF
in combination with the mantle
convection code STAGYY to develop an EnKF code for
mantle convection. PDAF provides a set of core
routines computing in parallel the analysis steps for a range of ensemble-based
data assimilation techniques. It provides as well a set of standard
routines to adapt the parallelization of a preexisting parallel forward
numerical model and integrate the data assimilation routines. The final
product is a highly scalable ensemble data assimilation code running both
forecasts and analyses in parallel.
We modified the STAGYY code following the procedure recommended by PDAF
see the online documentation wiki in. We also made a few
modifications to PDAF routines to allow for localization directly in the
forecast error covariance with the EnKF. Additionally, we
designed a basic observation database so as to load, in a single step, all the
observations used in the data assimilation procedure.
A posteriori evaluation of the EnKF method
We test the data assimilation scheme on twin experiments using the model described in Sect. .
Throughout this section, we compare the results of the EnKF for mantle
circulation reconstructions to the results computed using the method
developed in , hereafter referred to as method 1.
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 experiment setup
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 {xkt}k∈[1,K]. Here, the timespan of
the state evolution is 150 Myr and we sample true state vectors every
10 Myr. From these state vectors, we compute a time series of surface heat
fluxes and surface velocities, following Eq. (). We add to these
observations a random Gaussian noise of standard deviation 10% of the root
mean square of surface heat flux qrms and surface
velocities vrms (we compute
qrms and vrms from a free run of the dynamical model,
they represent long-term averages and are characteristic of the system
dynamics). We obtain the time series of observations to assimilate
{yko}k∈[1,K]. It follows that the observation
error covariance matrix R is diagonal and time independent.
Then, we perform ensemble data assimilation for the dataset
{yko}k∈[1,K], with the observation error
covariance matrix R. We did not consider any model error in the
filter we describe, so the parameters of the model used in the data
assimilation realizations are the same as those of the reference model.
Notations and range of values tested for data
assimilation parameters
Symbol
Meaning
Value
N
number of ensemble members
96 to 768
K
number of observation times
16
γ+
maximum inflation factor
1.25
ℓv
vertical correlation length
0.3 to 1
ℓh
horizontal correlation angle
π/10 to π/2
We present here tests with different assimilation parameters, varying the
number of members N, the vertical correlation length ℓv and the
horizontal correlation angle ℓh. Table details the range
of parameters tested.
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 shows the
initial and final states of these evolutions, together with the result of
global error evolution, and will be discussed in the next section.
Robustness of the assimilation algorithm
The evolutions of the global errors in the estimated temperature field and
surface horizontal velocity field over the time period {1,…,K} are
ϵTf(1),ϵTa(1),ϵTf(2),…,ϵTf(K),ϵTa(K)andϵuϕf(1),ϵuϕa(1),ϵuϕf(2),…,ϵuϕf(K),ϵuϕa(K),
respectively, where ϵTe(k) and ϵuϕe(k),
e standing for a (analysis) or f (forecast), are
ϵTe(k)=∑l=1L∑m=1MT‾ke(ϕl,rm)-Tkt(ϕl,rm)2V(ϕl,rm)∑l=1L∑m=1MV(ϕl,rm)andϵuϕe(k)=∑l=1Lu‾ϕke(ϕl)-uϕkt(ϕl)2V(ϕl,ra)∑l=1LV(ϕl,ra)
with V(ϕl,rm) being the volume of the grid cell at
longitude ϕl and radius rm,
T‾ke(ϕl,rm) the average temperature and
u‾ϕke(ϕl) the average horizontal velocity of
the estimated ensemble (either forecast or analysis) at longitude
ϕl and radius rm and ra, and where the superscript t
still refers to the true state.
Time evolution of the errors in the estimated
temperature field (panels a to c) and the estimated surface
velocities (panels from d to f) obtained from data
assimilations with the same 150 Myr observation dataset, but different
assimilation parameters. The size of the ensemble is N=96 for (a)
and (d), N=288 for (b) and (e) and N=768 for
(c) and (f). The assimilations are computed for any
combination of data assimilation parameters: γ+=1.25, ℓv=0.3,
0.5, 0.7 and 1 and ℓh=π/10, π/8, π/6, π/4 and
π/2. The black lines represent the evolution of the error for the
assimilation with the minimum average error in the analyzed temperature
field: N=96, ℓz=0.5, ℓh=π/6 and γ+=1.25 for
(a) and (d), N=288, ℓz=0.7, ℓh=π/10 and
γ+=1.25 for (b) and (e), N=768, ℓz=0.5,
ℓh=π/4 and γ+=1.25 for (c) and (f). The
gray area is delimited by the maximum and minimum values of errors at each
time, for all data assimilations. The background error is represented in red
for reference.
We test the EnKF on one evolution, with sizes of the ensemble N=96, 288
and 768 and for each combination of the following values of the data
assimilation parameters: vertical correlation length ℓv=0.3,
0.5, 0.7 and 1 and horizontal correlation angle
ℓh=π/10, π/8, π/6, π/4 and π/2. We show in
Fig. , for each ensemble size, the maximum and minimum values of
errors in temperature (Fig. a–c) and in surface horizontal
velocity (Fig. d–f), obtained for all
these parameters, as a function of time. We also represent the background
error in temperature ϵTb(k) and on surface horizontal
velocity ϵuϕb(k)
ϵTb(k)=∑l=1L∑m=1MTb(rm)-Tkt(ϕl,rm)2V(ϕl,rm)∑l=1L∑m=1MV(ϕl,rm)andϵuϕb(k)=∑l=1Luϕb(ra)-uϕkt(ϕl,ra)2V(ϕl,ra)∑l=1LV(ϕl,ra),
where Tb and uϕb are 1-D profiles corresponding to the average
temperature and horizontal velocity, respectively, computed from a long run.
We choose the average error in temperature after analysis
ϵ‾Ta=1K∑k=1KϵTa(k)
as the global measure for the quality of the assimilation. For each ensemble
size, the error evolution of the best assimilation (in the sense of minimum
ϵ‾Ta) is also shown in Fig. .
The error evolutions for temperature and surface horizontal velocity follow
the analysis–forecast sequence: at each analysis time (every
10 Myr), the
error decreases abruptly, and during the forecast phases, the error
increases.
For the surface horizontal velocity (Fig. d–f), the error
evolutions are very similar regardless of the data assimilation parameters:
the error decreases drastically during the analysis to a value of 25 to
50, while the amplitude of the error growth during the forecast phase
evolves from around 200 for the first forecasts to around 100 at the end
of the assimilation.
On the contrary, the evolution of the error in the temperature depends on the
parameters of the assimilation. Figure a–c shows that, for any size
of the ensemble, it is possible to find a set of parameters leading to a
drastic reduction of the global error in the temperature field after a few
analyses. This first phase, when errors decrease quickly, lasts approximately
70 Myr, which corresponds to the transit time of the dynamic system. After
this phase, the error in temperature slowly increases with time, while
remaining well below the errors measured for the first analyses. We can see
that for N=288 and N=768, any combination of vertical and horizontal
correlation lengths leads to errors lower than the first analysis. However,
the difference between the maximum and the minimum errors obtained is greater
than 0.01, which is large considering the background error is only around
0.1. The best error evolutions for N=288 and N=768 are very similar,
with a minimum error of 0.0318 and 0.0302 after 90 Myr, and an average
global error after analysis of 0.0391 and 0.0378, respectively. During
the assimilation of a dataset, most of the computational time is dedicated to
the forecast step, so the data assimilation with 768 members is 2.7 times
more expensive (computationally speaking) than the assimilation with 288
members. Since we obtain very similar results for N=288 and N=768, we
favor the assimilation with 288 members.
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 dk at forecast
number k
dk=yko-Hxkf.
After each forecast and just before analysis, we compute the Euclidean norm
of the instantaneous innovation dki and the Euclidean norm of the
cumulative mean innovation dkc
dki=∥dk∥anddkc=1k∑i=1kdi.
Before computing these norms, we normalize the part of the innovation
corresponding to surface heat flux and velocities by their respective root
mean square values qrms and vrms (corresponding to
time averages, characteristic of the dynamic system we are studying).
Figure shows the evolution of dki and dkc as a
function of the number of forecasts for data assimilations with different
sizes of ensemble and their respective optimum vertical and horizontal
correlation lengths.
The evolution of the cumulative mean of the innovation dkc
allows us to check some aspects of the consistency of the data assimilation
algorithm. Indeed, the derivation of the EnKF equations assumes that the
error in observations yo and the error in the forecast data
Hxf are unbiased. Such hypotheses imply that the
statistically expected value of d is zero, which means that the norm
of the cumulative innovation should converge to zero as the number of
forecasts increases. Figure a shows the cumulative innovation
constantly decreasing throughout the assimilation, with comparable values for
N=288 and N=768, and slightly higher values for N=96.
The norm of the instantaneous innovation dki measures the distance
between the forecast data and the observation, and therefore allows us to
monitor the success (or failure) of the assimilation. In Fig. b,
we can see that the norm of the instantaneous innovation decreases during the
first eight forecasts, i.e., 70 Myr, and then oscillates for the rest of the
assimilation. The comparison of Figs. and reveals
one important pitfall of the application of data assimilation to our problem.
After the 10th assimilation, the instantaneous innovation
(Fig. b) is almost the same for N=96, N=288 and N=768,
while the global error in the estimated temperature field (Fig. )
is clearly higher for N=96 than for N=288 or 768. This is because the
instantaneous innovation measures the distance between observed and forecast
data at the surface, while the error measures the distance between the
estimated and true temperature field, not only at the surface but also at
depth. This means that for a same innovation at the surface, the error in the
temperature field at depth can vary substantially. In other words, the
instantaneous innovation does not necessarily vary the same way the true
error in the temperature field does.
Evolution of (a) the cumulative mean
innovation and (b) the norm of the instantaneous innovation as a
function of the number of forecasts performed, and for different ensemble
sizes. For each size of the ensemble, the evolutions correspond to the best
combinations of correlation length parameters: N=96, ℓz=0.5,
ℓh=π/6 and γ+=1.25; N=288, ℓz=0.7, ℓh=π/10
and γ+=1.25, and N=768, ℓz=0.5, ℓh=π/4 and
γ+=1.25.
We also tested the assimilation algorithm for four different state evolutions,
with the optimal parameters for an ensemble size of N=288 members
(ℓv=0.7 and ℓh=π/10). Figure shows the initial and
final temperature fields of the evolutions, together with the evolution of
the global error, the spread of the ensemble, the background error and the
error evolution using method 1.
Evolution of the error (ϵTe, red) as
a function of time for four different evolutions with N=288,
γ+=1.25, ℓv=0.7 and ℓh=π/10,
compared to the evolution of the spread of the ensemble
(σTe, blue), the evolution of the error with the
technique of (ϵTe method 1,
yellow) and the background error (ϵTb, purple). The
initial and final states of the true evolutions are represented on the left
of each corresponding graph.
The spread of the ensemble is an estimation of the uncertainty in the state.
We compare the evolution of ϵTe to the global standard
deviation of the temperature field of the ensemble:
σTf(1),σTa(1),σTf(2),…,σTf(K),σTa(K)
with σTe(k) defined as
σTe(k)=∑n=1N∑l=1L∑m=1MTkne(ϕl,rm)-T‾ke(ϕl,rm)2V(ϕl,rm)(N-1)∑l=1L∑m=1MV(ϕl,rm).
We compute the error for an estimated state evolution with method 1 using Eq. ().
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 ), they show different geodynamic configurations:
evolution A has a shorter wavelength of convection, with the persistence of
four subductions, three ridges and five upwellings, the death of one ridge and the creation
of two. Evolutions B, C and D have longer wavelengths of convection, with two
major downwellings, stable throughout the evolutions. In evolution B, one of
these downwellings has a very large negative temperature anomaly at the
bottom of the domain. In evolution C, the remnant of a subduction merges with
a larger subduction into a single downwelling.
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
50 Myr.
The average error is lower for the EnKF than for method 1 in three out of
four cases. The average standard deviation of the ensemble (ensemble spread)
is of the same order of magnitude as the true error. However, its evolution
is not the same as the true error, with, for example, differences between
both of more than 0.02 for some part of evolution C. Moreover, in three out
of four cases (evolutions B, C and D), the spread of the ensemble is much
lower than the true error. For evolutions C and D, the results of the two
methods are comparable, whereas the assimilation with EnKF performs better
than method 1 for evolutions A and B.
Comparison of temperature field evolutions for
evolution B. (a) Depicts the evolution of the analyzed temperature
field with method 1, (b) is the true evolution of the temperature
field and (c) is the evolution of the analyzed temperature field
with the ensemble Kalman filter (EnKF) N=288, ℓv=0.7 and
ℓh=π/10.
Example of temperature fields of the members of the
ensemble. This example is taken after 80 Myr for the assimilation of
evolution B, with EnKF N=288, ℓv=0.7 and
ℓh=π/10.
For evolution B, method 1 fails to accurately reconstruct the evolution, with
the error reaching values greater than 0.08 at the end of the assimilation.
This case is further investigated in Figs. and .
Figure compares the true temperature field evolution with the
analyzed temperature field of method 1 and of the EnKF with
N=288, ℓv=0.7 and ℓh=π/10. The sudden increase in the error
of the estimated temperature field for method 1 seen in Fig. b
happens after around 80 Myr of assimilation, when the direction of bending
at the bottom of the domain changes for the downwelling on the left side (see
Fig. b). The analyzed temperature field of method 1 does not
predict this change of direction (see Fig. a), while the
analyzed temperature field of the EnKF predicts it (see Fig. c).
Method 1 computes only the evolution of the best estimate of the system. The
computation of only one estimate ignores that, in this case, a slight
perturbation of the estimated state could lead to totally different
dynamics. On the contrary, the EnKF method computes the evolution of an
ensemble of perturbed solutions and thus takes into account the nonlinearity
of the solution, at least for the forecast stage. Figure shows
examples of the analyzed temperature fields of different ensemble members for
evolution B, after 80 Myr of assimilation. Although the average
temperature fields display a downwelling bending to the right, the ensemble
members show a wide variety of downwelling geometries.
Reliability of the ensemble
In Fig. , the standard deviation of the temperature of the ensemble, σTe, is
lower than the error in temperature, ϵTe, for some state
evolutions. This indicates that we cannot rely on the spread of the ensemble
to estimate accurately the evolution of the global error in the temperature
field. To investigate the reliability of the ensemble in more detail, we
compute rank histograms for surface heat flux and velocity
(Fig. ), and for the temperature at the surface, mid-domain and
at the bottom (Fig. ).
Rank histograms were first described independently by
,
and Talagrand (; ). They are a tool to
diagnose systematic biases and misestimations of the uncertainty in an
ensemble of forecasts . To obtain the rank
histograms of Figs. and , we proceed as follows:
Selection of the variable and the verification. We compute rank histograms
for surface heat fluxes (Fig. a, c, e), surface velocities
(Fig. b, d, f), and surface, mid-mantle and bottom temperature
(Fig. ). For Figs. a, b and , the
ensemble is checked against the true value, while for Fig. c–f,
it is checked against the observed value. In this context, the true
values are the verification for Figs. a, b and
, and the observed values are the verification for
Fig. c–f, respectively.
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. , and, for each evolution,
we select points that are spaced from each other by the correlation angle
ℓh=π/10, and taken after 10, 80 and 150 Myr of assimilation.
We obtain 120 sampling points per histogram.
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 17 ranks as the bin width (289=172).
If the ensemble statistics are reliable, then the true value of a given
variable and the values of the ensemble of forecasts can be considered as
random draws from the same distribution. In this hypothesis, the rank of the
true value follows a uniform law, and the rank histogram should be flat. We
represent the expected rank counts for a flat histogram as a dashed line in
Figs. and . If this is not the case, the shape of
the rank histogram provides indications of the existence of biases and under-
or over-dispersion of the ensemble even though the shape of a rank
histogram can also be affected by other factors; see
e.g.,.
To guide our interpretations, we perform the χ2 goodness-of-fit test
see e.g.,Sects. 5.2.5 and 7.7.2 to test if our
rank histograms are significantly non-uniform. We compute the value
χ2=∑i=117#oi-#ei2#ei,
where #oi is the bin count in the ith bin and #ei is the expected
count for a uniform distribution 120/17≈7.06. The values of χ2
are written on each histogram of Figs. and . If the
ranks we sampled come from a uniform distribution, then χ2 follows a
chi-square probability law with 17-1-1=15 degrees of freedom. In this hypothesis, the probability to obtain a
χ2≥χc2=24.996 is 0.05. We take this value of
χc2 as the critical value over which we consider that the
rank histogram is significantly non-uniform.
Rank histograms of the surface true heat
flux (a) and velocity (b) as well as the surface observed
heat flux (c) and velocity (d), computed from the four
evolutions of Fig. . The dashed lines represent the count for each
bin if the rank histograms were flat.
The left column of Fig. represents the rank histograms of the
true surface heat flux (Fig. a) and velocity
(Fig. b). Figure a shows a slightly higher
occurrence of the true heat flux in higher ranks within the ensemble. This
would suggest that the ensemble estimation of surface heat flux is biased
towards the lower values. However, the χ2 value for Fig. a
is well below the critical value χc2, so that we
cannot say that the rank histogram is significantly non-uniform. On the
contrary, the rank histogram of surface velocities (Fig. b)
has a χ2=26.43>χc2: it is significantly non-uniform. This
histogram is more populated in the bins corresponding to the lowest and
highest ranks (1–17 and 255–289). This would suggest ensemble
under-dispersion, even though the shape of the rank histogram is more complex
than the classical U shape associated with ensemble under-dispersion (in
particular, the middle ranks, 137–153, are also highly populated).
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. a, b? The middle column of Fig.
represents the rank histograms of the observed heat flux in
Fig. c and observed velocity in Fig. d. Both
Fig. c, d have a distinct U shape, with χ2 of
310.14 and 26.14, respectively. For the surface heat flux, the difference
between rank histograms of the truth and the observation is dramatic (the
χ2 value jumps from 12.14 to 310.14). For the surface velocity, the
difference is less striking, even though the U shape is much clearer on the
rank histogram of observed velocities. The more pronounced U shape for the
rank histograms of both observed heat flux and velocity indicates that the
ensemble is not as under-dispersed around the truth as what could be deduced
from the rank histograms with noised observations. In other words, noise in
the observation has a major effect on the shape of the rank histogram, so
that we cannot interpret the reliability of the ensemble by looking directly
at the rank histograms of the observations.
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 and
. The noise we add to each ensemble member
has the same standard deviation as the noise affecting the observed data. The
right column of Fig. represents such rank histograms for heat
flux in Fig. e and velocity in Fig. f. Both
χ2 scores are well below χc2 thus, we cannot say that the rank
histograms are significantly non-uniform. It is not possible to detect the
under-dispersion of the ensemble for surface velocity using only observed
data.
Rank histograms for temperature at the surface
(a), mid-mantle (b) and at the bottom of the model
(c), computed from the four evolutions of Fig. . The dashed
lines represent the count for each bin if the rank histograms were flat.
Figure shows the rank histograms for temperature at different
depth. At the surface (Fig. a), the rank histogram is the same
as the rank histogram for the true heat flux Fig. a, since there
is a linear relationship between surface temperature and surface heat flux.
It follows that the rank histogram of surface temperature is not
significantly non-uniform. At mid-mantle (Fig. b), the rank
histogram of temperature is significantly non-uniform, with a
χ2=36.43≥χc2. It is more populated towards the higher
values, which could indicate that the temperature ensemble in the mid-mantle
is biased towards the lower values. At the bottom (Fig. c), the
rank histogram of temperature is also significantly non-uniform, with a
χ2=37.29≥χc2. The first bin of the rank histogram is highly
populated, while the rest of the histogram is roughly flat. This suggests
that the ensemble is biased towards the hotter temperatures at the bottom of
the model.
In conclusion, Fig. shows that the ensemble is reliable at the
surface for temperature, but becomes unreliable at depth. The lower value of
the standard deviation σTe compared to the true error
ϵTe observed in Fig. in three out of four cases is
due to a misestimation of the error in the temperature at depth by the
ensemble. We discuss this point in more detail in Sect. .
Effect of the data assimilation parameters on the quality of the estimation
As shown in Fig. , the choice of N, ℓv and ℓh is
critical to minimize errors in the assimilation, with errors in the estimated
temperature field varying from 0.03 to more than 0.1 depending on the
choice of parameters. We further investigate the effect of these parameters
by comparing the average global errors after analyses,
ϵ‾Ta, for different combinations of N, ℓv and
ℓh. Figure displays the values of ϵ‾Ta
for sizes of ensemble N=96, 288 and 768 (Fig. a–c, respectively) with ℓv varying between 0.3 and 1, and ℓh
between π/10 and π/2. As in Fig. , we observe a dichotomy
between assimilations with N=96 members, with higher errors, and
assimilations with N=288 and 768, with lower errors.
Values of the average analyzed error for assimilations
performed using the dataset generated by evolution A of Fig. , with
different sizes of the ensemble, and vertical and horizontal correlation
lengths. (a) For 96 ensemble members, (b) for 288
ensemble members and (c) for 768 ensemble members. The dashed lines
delimit the zones for which errors are less than
ϵ‾Tmina(N)+0.002.
For each size of ensemble N we identify the pair (ℓv,ℓh) that
leads to the assimilation with the lowest error
ϵ‾Tmina(N). From this minimum value
ϵ‾Tmina(N), we select all the pairs
(ℓv,ℓh) that lead to data assimilation with global errors less than
ϵ‾Tmina(N) + 0.002. As the size of the ensemble
increases, the optimal lengths of correlations (ℓv,ℓh) tend to
increase. This is a classical effect , observed in
EnKFs for various dynamical systems. As N increases, the
amplitude of noise in the sample correlation matrix Pf decreases,
and small, yet real, correlations between distant points can be taken into
account . Between ensemble sizes of N=96 and
N=288, the zone of optimal correlations is displaced towards the greater
vertical correlation lengths. When we increase the size of the ensemble from
N=288 to N=768, the zone of optimal correlations is displaced towards
greater horizontal correlation angles. So the accurate estimation of
correlations between points on the same vertical level needs less samples
than between points on the same horizontal level. This is due to the
specifics of mantle convection dynamics. The highly nonlinear rheology
produces plates at the surface with values of velocity and temperature that
may vary substantially (by 1 or 2 orders of magnitude) on short distances
in the horizontal direction, especially because of pseudoplasticity. On the
contrary, highly viscous cold downwellings establish a strong continuity in
the vertical direction. Given that a small perturbation can trigger the
formation of a new plate boundary (see Sect. ), those scales
of variability reverberate through the ensemble covariance matrix.
For the ensemble size N=288 and all the values of (ℓv,ℓh),
we additionally evaluate the average global ensemble spread
σ‾Ta=1K∑k=1KσTa(k),
the average norm of the instantaneous innovation
di‾=1K∑k=1Kyko-Hxkf
and the cumulative mean innovation after K forecasts
dKc=1K∑k=1Kyko-Hxkf.
These three values are indicators of the accuracy of the assimilation and can be computed in the
case of an assimilation with Earth data, unlike ϵ‾Ta.
Values of (a) mean analyzed error,
(b) mean ensemble spread, (c) norm of instantaneous
innovation and (d) norm of cumulative innovation after K=16 forecasts
for N=288, and different vertical and horizontal correlation lengths. The
dashed line delimits the zone for which errors are less than
ϵ‾Tmina(288)+0.002.
Figure represents these results along with the true error
ϵ‾Ta. The ensemble of optimal data assimilation
parameters is also outlined
(ϵ‾Ta<ϵ‾Tmina(N)+0.002).
Overall, the average ensemble spread σ‾Ta
(Fig. b) decreases when ℓh and ℓv increase, with a
minimum for ℓh=π/2 and ℓv=1. The higher the correlation
lengths, the more covariances will be taken into account in the analysis, and
the analyzed members will be closer to each other and
σ‾Ta lower. The average ensemble spread
σ‾Ta is of the same order of magnitude as the true error
ϵ‾Ta. Moreover, there is a local minimum of
σ‾Ta at ℓv=0.7 and ℓh=π/10. These parameters
correspond to the minimum true error ϵ‾Ta.
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.
Accuracy of the reconstruction of geodynamic structures
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 shows the final state of the assimilation after
150 Myr for the evolution A of Fig. . We selected three assimilations: EnKF96,
an EnKF with N=96, ℓv=0.5 and ℓh=π/6
(Fig. a); EnKF288 an EnKF with N=288, ℓv=0.7 and
ℓh=π/10 (Fig. b) and the assimilation with method 1
(Fig. c). We do not show the EnKF with 768
members since the resulting temperature field is almost indistinguishable
from that of EnKF288. The first column represents the true temperature field,
which is the same for all assimilations. The second column is the analyzed
temperature field, i.e., the average of the temperature fields of the analyzed
ensemble members. The third column is the absolute temperature error, and the
fourth column is the standard deviation of the ensemble spread, which is an
estimate of the error in the analyzed temperature field.
Comparison of estimated states after 150 Myr for evolution A of Fig. . First row: ensemble Kalman filter (EnKF) with
N=96, ℓv=0.5 and ℓh=π/6; second row: N=288, ℓv=0.7
and ℓh=π/10 (these localization length correspond to the optimal
parameters determined previously.); third row for method 1 (third row). The
first column represents the true temperature field at 150 Myr, the second
column is the analyzed temperature field, the third column is the absolute error in
temperature value and the fourth column is the estimated error in the analyzed field
(spread of the ensemble). In the true temperature field of EnKF288, we framed
the location of the subduction (a), plume (b), ridge
initiation (c) and stable ridge (d) studied in
Fig. .
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, P0f for the analysis.
This constant matrix does not take into account the specifics of the dynamics
under a ridge, where the positive anomaly is generally shallow. We do not
observe this detrimental effect in the EnKF assimilations, where we compute
the forecast error covariance matrix Pkf at each analysis time
from the forecast ensemble.
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. for another assimilation.
To illustrate how different flow structures are reconstructed, we plot in
Fig. the time evolution of the EnKF288 ensemble surface,
mid-domain and bottom temperature at the longitude of (a) a subduction, (b) a
plume, (c) a ridge initiation and (d) a stable ridge. Figure
shows the location of these geodynamical features on the true temperature
field. We plot the temperature evolutions at the surface, mid-mantle and at
the bottom of the domain. Note that the surface and bottom values of
temperature actually correspond to the values of the first points below the
surface and above the bottom of the domain, respectively.
Detailed results of the assimilation depicted in
Fig. second row, and Fig. a. Each graph represents the
time evolution of the temperature value at points on four profiles,
corresponding to different geodynamic contexts: (a) a subduction,
(b) an upwelling, (c) a ridge initiation and (d) a
stable ridge. The first row corresponds to points at the surface of the
domain, the second row to points in the mid-mantle and the third row to
points at the bottom of the domain. The lateral coordinates of the points are
shown in Fig. (second row first column). The red line is the
true temperature, the black line is the average of the ensemble, the dark
grey area represent the average plus or minus the standard deviation of the
ensemble and the light gray area is the area spanned by the minimum and the
maximum value taken by the ensemble of 288 members.
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 0.01. The correction associated with the
analysis gradually decreases with depth due to both covariance localization
and the dynamics of the system.
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 40 Myr, and at the bottom of the model after
70 Myr, which is the value of the transit time. At the surface, the spread
of the ensemble decreases as more data are assimilated. On the contrary, the
spread of the ensemble remains steady for mid-mantle depths and at the bottom
of the domain. For these depths, only the average temperature varies.
At the surface for the plume, the spread of the ensemble is very low except
for a peak at 40 Myr, which corresponds to an instability, corrected after
one analysis. We note that this instability greatly affects method 1 since it
leads to the false prediction of the ridge seen in Fig. . At
mid-mantle, the ensemble average is slowly converging to the true
temperature. At the bottom, the estimated temperature is lower than the true
temperature, although it slightly increases throughout the assimilation.
The ridge initiation shows how new observations affect the spread of the
ensemble. At the surface, the spread of the ensemble remains low until
100 Myr, the time of initiation of the ridge. From then on, the estimated
temperature increases and the ensemble members follow the cycle of increasing
spread during forecast and dramatic decrease in spread during analysis. The
temperature in the mid-mantle is estimated with a very good accuracy after
50 Myr. On the contrary, the assimilation does not predict the evolution of
the temperature at the bottom of the domain, although the true temperature
falls within the zone defined by the standard deviation of the ensemble after
50 Myr.
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 100 Myr. At
the bottom of the domain the temperature is underestimated although it
follows the variations in the true temperature: increase of temperature at
the beginning of the assimilation and slight decrease at the end of the
assimilation.
Discussion
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 seefor a comparison
of EnKF and 4-D variational methods. For the mantle circulation
problem, this results in a series of derivation of the adjoint model
considering different approximations (;
; ;
). The ability of a data assimilation scheme to
adapt to different numerical codes is a particularly important issue for
mantle convection since models are in constant evolution, with current
developments including the implementation of chemistry, nonlinear rheologies,
elasticity, phase transitions and compressibility see e.g.,for a
review of recent developments in mantle convection
codes. In particular, this ease of implementation allows
us to work on models self-consistently producing plate-like tectonics at
their surface, and hence to obtain forecasts whose data can be ultimately
compared with plate reconstructions.
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 . The main
difference between the two filters is that the EnKF evaluates the state
covariance matrix with an ensemble of members. This ensemble approach allows
the nonlinear evolution of errors during the forecast stage. This leads to a
higher precision in the reconstruction, but also to a more robust scheme,
able to reconstruct evolutions which could not be reconstructed with the
former method (as illustrated by Figs. and ).
Moreover, the ensemble assimilation provides an estimate of the uncertainty in
the reconstruction at each point of the domain. The estimation of
uncertainties could be valuable information for plate tectonic
reconstructions, especially for regions and times where data are scarce,
because they show the different possible scenarios supported by the ensemble.
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 in combination with the parallel code
STAGYY produces a highly parallel ensemble filter, able to perform the
assimilations on 768 cores in 20 min for an ensemble of 96 members and
3 h for an ensemble of 768 members.
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 288 members (among the ensemble sizes
we tested, i.e., 96, 288 and 768). Indeed, between the assimilations
with 288 and 768 members, the global average error in the temperature
field (as defined in Eq. ) decreases by 0.0013, while the size
of the ensemble (and hence the computational cost) is multiplied by 2.7. On
the contrary, dividing the size of the ensemble by 3 (from N=288 to 96)
leads to an increase in the global average error of 0.0086. These
differences in errors appear to be small; however, they affect the quality of
the reconstruction of thermal structures. We can see this in
Fig. for example: the global errors in temperature (as defined
in Eq. ) range between 0.0367 and 0.0461, so the difference in
global errors is at most 0.0094. Locally, this translates into the presence
(or absence) of artifactual geodynamic structures (like ridges and
upwellings, visible in the second column of Fig. ). Covariance
localization proved to be important to minimize the error in the
reconstruction of mantle structure: as shown in Fig , for
288 members, the difference in the average error is of 0.0065 between the
optimal correlation length and the least favorable one.
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. , the true global error in temperature is higher
than the ensemble spread in three cases. This would indicate that we are on
average overconfident in our forecasts. To test the reliability of the ensemble
in more detail, we produced rank histograms for temperature at
the surface, mid-depth and at the bottom of the domain (Fig. ).
The rank histogram corresponding to surface temperature does not detect any
biases or over/underspread in the ensemble. On the contrary, the rank
histograms are significantly non-uniform at depth. Our interpretation is that
this tendency is linked to the configuration of the data assimilation
problem, combined with the simple scheme used for covariance inflation
(Sect. ). Indeed, the inflation factor which we propose is
directly linked to the innovation statistics, and it is spatially uniform. It
follows that the inflation factor will adequately correct the spread of the
ensemble at the surface, where the data are located, but not necessarily at
depth, where no observation is available. To improve the reliability of the
ensemble at depth, a solution could be to implement a more complex algorithm
for the inflation factor, especially spatially varying inflation as proposed
by and for example.
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 N=288
and 768 are very close (see Fig. ). On the contrary, the
correlation lengths minimizing the norm of the cumulative innovation and the
instantaneous innovation were different from the ones minimizing the error in
the temperature field. This shows the limits of these indicators to determine
the optimal parameters for the assimilation. In a realistic case, rigorous a
posteriori evaluation of a data assimilation result would require comparison
of the prediction made with independent observations
. For mantle circulation, seismic tomography,
topography, true polar wander or the geoid could play this role.
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
. In the same way as the EnKF uses
sample spatial correlations of the ensemble to update the state of the system
with new observations, the ensemble Kalman smoother uses sample time and
space correlations of successive ensembles to update former states with the
new observations. shows how an ensemble Kalman
smoother can be implemented with a minimal computational cost alongside a
preexisting EnKF. Moreover, shows that such
an algorithm is efficient for nonlinear models, and that in their test case,
optimal localization parameters for the ensemble Kalman smoother coincide
with optimal localization parameters for the EnKF.
As a first approach to test the EnKF for mantle circulation reconstructions,
we chose a fairly simple convection model. As already discussed in
Sect. , a more realistic mantle model would have, among other
things, a 3-D-spherical shell geometry and a higher Rayleigh number. This
would substantially increase the size of the data assimilation problem.
However, we followed the procedure as described in
to implement the EnKF. This results in a highly scalable filter, enabling the
computation of the EnKF assimilation in a reasonable time. An increase in the
Rayleigh number also implies thinner boundary layers, slabs and plumes. This
could translate into lower optimum correlation lengths for the EnKF. A more
realistic model would additionally include a viscosity increase in the lower
mantle , and the presence of continents. This
would tend to lengthen the wavelength of convection in the lower mantle and
therefore might ease the mantle circulation reconstruction see for
exampleSects. 7.02.6.3.2 and 7.02.6.7 for a discussion of both effects
on mantle convection.
In the synthetic experiments of Sect. , the convection model
used to produce the series of data is the same as the forward model used
during the assimilation. For an application with Earth data, this will not be
the case. The equations solved in models of mantle convection still hold some
shortcomings . Moreover, theories, observations and
experiments do not yet fully constrain parameters, especially rheological
ones , and variations in rheology affect the
reconstructions of mantle circulation . Hence it
could be fundamental to take into account model errors. A first order
solution is to increase the inflation parameter γ in Eq. ():
this would overall increase the a priori uncertainty in the mantle
estimation. Performing experiments where the model used to compute the
observation is different from the model used for the assimilation would
provide us with more information on how to implement model errors. Another
solution would be to consider the joint assimilation of the state and model
parameters. Although it is in principle possible for the EnKF
, it could be computationally not tractable.
Indeed, the response of mantle dynamics to different rheological
parametrization is highly nonlinear, and their inversion calls for the
development of techniques focusing on rheology, such as adjoint-based
inversions of rheological parameters
or further applications of
the recently developed pattern recognition techniques for mantle convection
.
The choice of the synthetic experiments assimilation window of 150 Myr is
a compromise between having the possibility to compute assimilations for
various cases and having an assimilation window covering most parts of the
timespan of plate tectonic reconstructions
. However, the
structure of the dataset used for the synthetic experiments is a very
idealized version of the actual plate reconstruction models. We already
discuss this issue in . In the following, we
supplement and update this discussion in the light of research that has
recently come to the fore.
First, we set a time series of data covering the whole surface of the domain
and regularly available, every 10 Myr. Plate tectonic reconstruction data are more complex.
They are based on the estimation of finite relative rotations between
individual plates, structured into a hierarchy describing global relative
motions and anchored in an absolute reference frame. The span of each finite
relative rotation is determined depending on the amount and quality of
information available for a specific context and therefore varies depending
on plate pairs and times. The average span of finite rotations of recent
plate models is of the order of 10 Myr to
5 Myr , but varies over time with, for example,
1 Myr resolution for the last 20 Myr in some regions
, or some gaps in the data such as during the
Cretaceous superchron from 121 to 83 Myr ago . The
continuously closed plate algorithm produces plate
tectonic reconstruction maps continuous in space and time which allows the
creation of a series of global plate reconstructions at regular intervals.
Nonetheless, creating such a regularized time series of reconstruction might
miss tectonic events. Instead, we could adapt the frequency of analyses to
the varying plate reconstruction resolution. Additional synthetic experiments
with a time series whose frequency evolves through time are necessary to
explore the limits of such a method.
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
. The main source of information on the motion of
plates comes from the map of seafloor magnetic anomalies.
developed a method to compute relative
motion of plates and associated uncertainties inferred from magnetic anomaly
identifications. Recently, built an open-source
community database. This database gathers both the coordinates of the seafloor
magnetic anomaly lineations, and their associated plate relative motion and uncertainties,
computed using the Hellinger method . This database could be
used in the future as a basis to automatically produce global plate motion
histories and assess their uncertainties. To our knowledge, this has not been
done so far on a global scale. On a regional scale and for recent time (5
to 20 Myr), applied the
trans-dimensional hierarchical Bayesian method to reduce noise in finite
rotation data and produce time series of high-resolution plate-relative
motions. More recently, applied this technique to
the relative motion of the pacific plate with North America for the last
75 Myr. The uncertainties in relative plate velocities range from 5 to
40 % of the root mean square surface velocity. As we go further back in
time, the quantification of relative plate motion uncertainties becomes
hazardous: most of the seafloor created before 150 Myr has been destroyed
by subduction. These plate tectonic reconstructions involve interpretation of
different types of data, with a limited spatial coverage and relies heavily
on human expertise. For these epochs, maintaining very high uncertainties in
the regions where few data supports the reconstructions would be a solution.