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,

mariebocher@gmail.com2 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.