DATA ASSIMILATION FOR THE TIME-DEPENDENTRECONSTRUCTION OF CONTINENTS.
M. Bocher1, M. G. Tetley21Seismology and wave physics group, Intitute for geophysics, Earth Sciences, ETH, Zürich, Switzerland,
[email protected] AUGURY, Laboratoire de GÃl’ologie de Lyon, UniversitÃl’ Claude Bernard Lyon 1, Lyon, France
AIMAssimilate Paleomagnetic data to reconstruct the motion ofcontinents over the last hundreds of Myr, while preservingbasic geodynamical principles.
PRELIMINARY TESTSSynthetic experiment
130 Myrs0 Myrs 10 Myrs 20 Myrs 60 Myrs 100 Myrs
0 50 100 130
-10
-5
0
69
69.5
68.5
time Myrs
Incl
ina
tion
Dec
lina
tion
Apparent polar Wander
−180˚
−90˚
0˚
90˚
time (Myrs)
130
100500
−180˚
−90˚
0˚
90˚
TrueEstimated with
10 000 particules
−180˚
−90˚
0˚
90˚
−180˚
−90˚
0˚
90˚
Estimated with1000 particules
Estimated with100 particules
ObservationsMOTIVATION• Using plate tectonic theory, we can integrate a wide range of
geological and geophysical observations to produce kinematicplate tectonic reconstructions. These reconstructions are builtvia a largely manual process of integrating many individual time-dependent regional tectonic histories into a geometrically self-consistent global model, making the quantitative estimationof uncertainties very complex.
• The particle filter provides a statistically consistent frameworkwithin which one can assimilate data of variable nature andsource within a dynamical model, providing quantitative uncer-tainties on the estimated trajectory of the system.
Here, we demonstrate a first step to building a data as-similation framework for plate tectonics reconstructions:we apply a particle filter to reconstruct time-dependentcontinental configurations and motions.
THE FORWARD MODELContinents motions are solid body rotations:At Each timestep, we compute the rotation of each continentduring the time δt (here 1 Myrs). This rotation is determinedby 3 parameters:
N
C
C�
P
�
VCδt
Dp
• Dp, the distance from continent centroidto the Euler pole (pole of rotation)
• VC , the velocity of the continent’s cen-troid
• Ts, the fraction of the current rotation tobe kept for the next rotation
Computation of the Random drift for each continent:DP , VC and Ts are random variables. Each of them follows abeta function. We choose the parameters (a, b) of those betafunctions to fit the following geodynamical constraints:
0
0.1
0.2
prob
abili
tyde
nsity
0 5 10 15 18Centroid velocity, cm/yr
max(VC) = 18 cm/yr maximum,mode ∈ [2, 3] cm/yr
0 5000 10000 15000 20000Distance Pole-Centroid (km)
0
2
4
6
8
10
prob
abili
tyde
nsity
×10−5
Dp is related to spin vs translation mo-tions for continents: mode at 10000 km.
0.0 0.2 0.4 0.6 0.8 1.0Ts
0
2
4
6
8
10
prob
abili
tyde
nsity
Ts: how often the continents changedrastically their trajectory.
Collision rules:If during a timestep, two continents overlap each other, thenthey form a cohesive block and are rotated together.
EXAMPLE RANDOM DRIFT SCENARIO
20 Myrs
Present
40 Myrs 60 Myrs
OBSERVATIONS
N
E
z
D
I
H
paleo
pole
paleo N
paleo S
I
Database of incli-nation and declina-tion of the mag-netic field fossilizedin rocks
Uncertainties modelled withFisher statistics:
p(hp)(ho,κ) = C(κ) exp(κ[ho]Thp)
with C(κ) =κ
2π(eκ − e−κ)0 π/2 π
0
1.0
2.0 κ = 5κ = 1
κ = 10κ = 20
prob
abili
tyde
nsity
Angular distance toobserved orientation
North America paleopoles com-puted from the inclination decli-nation database used in Tetley[2018], dated from 0.5 to 550 Ma
DATA ASSIMILATION METHOD
Forecast
Weig
ht
Computation
Resampling
Present t₁ t₂
1/N_p
Forecast
Weig
ht
Computation
Resampling
We use a particle filter [van Leeuwen et al., 2018]:
Initial setup
Np particles {xnp0 }np∈[1,Np] with iden-
tical continental blocks, but different random rota-tions.weight: {ωnp
0 = 1/Np}np∈[1,Np]
pdf: p(x0) =
Np�np=1
ωnp0 δ(x0 − x
np0 ),
with δ the dirac function.
loop over observations:
• Forecast: see forward model.• weight computing: ∀np ∈ [1, Np],
ωnpk
=p(y|xnp
k)
Np�j=1
p(y|xjk)
with p(y|xk) a multivariate Fis-
cher distribution with each component independant of the other.• Stochastic universal resampling.
CONCLUSIONS• We have developed a data assimilation framework for paleomagnetic data:
– based on the Particle Filter,– with a continental drift model consistent with basic geodynamics rules,– where the uncertainties on observations are taken into account.
• For single continents synthetic experiments, a number of particles of ca. 10000 allows us to estimate the trajectory of continentsfor at least 130 Myrs.
THE WAY FORWARD• Perform synthetic tests with data at multiple sites and on different continents• Optimize forward code to allow for more particles• test different resampling techniques, while conserving geodynamical constraints.
REFERENCESMichael G. Tetley. Constraining Earth’s plate tectonic evolution through data mining and knowledge discovery. University of
Sydney, 2018.Peter Jan van Leeuwen, Hans R Künsch, Lars Nerger, Roland Potthast, and Sebastian Reich. Particle filters for applications in
geosciences. arXiv preprint arXiv:1807.10434, 2018.