Sea Ice Data Assimilation Challenges

Section 0
Sea ice, Unstable subspace, Model error:
three topics in data assimilation
Amit Apte
International Centre for Theoretical Sciences (ICTS-TIFR)
Bangalore, India
Data assimilation working group, and in particular,
Anugu Sumith Reddy, ICTS-TIFR (unstable subspace);
Colin Guider, UNC (Sea ice)
Amit Apte (ICTS-TIFR, Bangalore) Data Assimilation working group ( apte@icts.res.in ) p. 1 / 32

Section 0
Sea ice, Unstable subspace, Model error:
three topics in data assimilation
Amit Apte
International Centre for Theoretical Sciences (ICTS-TIFR)
Bangalore, India
Data assimilation working group, and in particular,
Anugu Sumith Reddy, ICTS-TIFR (unstable subspace);
Colin Guider, UNC (Sea ice)

Section 1 Data assimilation: 3 minute introduction
What is data assimilation?
The art of optimally incorporating
partial and noisy observational data of a
chaotic and complex dynamical system with an
imperfect model of the data noise and the system dynamics to get a
prediction and the associated uncertainty for the system state
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Nonlinear ﬁltering ≡ determination of posterior i.e.
conditional distribution given the observations
Consider a Markov chain (or dynamics or plant)
xt+1 = m(xt) + ζt with x0 ∈ Rd
unknown
We will assume a probability density pa(x0) for the initial condition.
We will consider the problem of estimating the state xt at some time
t given observations at times t1, t2, . . . , tN.
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –
Observations yt ∈ Rp (typically p ≤ d) are local in time.
yt = h(xt) + ηt t = 1, . . . , N
h is called the observation operator. ηt is observational noise. Eventually
we will assume independence between ηt and ζt.
Main object of interest: the posterior distribution of the state conditioned
on the observations

Nonlinear ﬁltering ≡ determination of posterior i.e.
conditional distribution given the observations
Consider a Markov chain (or dynamics or plant)
xt+1 = m(xt) + ζt with x0 ∈ Rd
unknown
We will assume a probability density pa(x0) for the initial condition.
We will consider the problem of estimating the state xt at some time
t given observations at times t1, t2, . . . , tN.
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Filtering density satisﬁes a recursion relation
A notation: y1:t = {y1, y2, . . . , yt} and x1:t = {x1, x2, . . . , xt}
“prediction” given by
pf
(x1:t, xt+1|y1:t) = pa
(x1:t|y1:t) · pm
(xt+1|xt)
“update” given by Bayes theorem
pa
(x1:t+1|y1:t, yt+1) ∝ pf
(x1:t+1|y1:t) · pη(yt+1|xt+1)
we obtain the following recursive relation for the posterior distribution
pa
(x1:t+1|y1:t+1) ∝ pa
(x1:t|y1:t) · pm
(xt+1|xt) · pη(yt+1|xt+1)
where pη(yt+1|xt+1) is the observational noise and pm(xt+1|xt) is the
Markov transition Kernel for the dynamical model.
In continuous time setup: Zakai or Kushner-Stratonovich equation for the
conditional distribution

Kalman ﬁlter in two moments
Linear Gaussian case:
Linear model: m(xt) = Mtxt
Linear observation operator: h(xt) = Htxt
Gaussian noise: ζt ∼ N(0, Qt) and ηt ∼ N(0, Rt)
Gaussian initial condition: x0 ∼ N(xa
0 , ∆0)
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

0 , ∆0)
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –
Kalman ﬁlter gives a recursion relation for the mean and covariance:
“Update step” given by (with Kt = ΣtHT
t (HtΣtHT
t + Rt)−1
)
xa
t = xf
t + Kt(yt − Htxf
t ) and ∆t = (I − KtHt)Σt
“Prediction step” gives forecast error covariance Σt:
xf
t+1 = Mtxa
t and Σt+1 = Mt∆tMT
t + Qt
The covariance forecast or update do not depend on the mean.

Ensemble Kalman ﬁlter
Suppose we have a sample {xi
t }, i = 1, . . . , N from pa(xt|y1:t), i.e.,
we approximate pa(xt|y1:t) ≈ (1/N) N
i=1 δ(xt − xi
t )
Ensemble Kalman ﬁlter prediction uses the nonlinear model and
update uses sample statistics.
“Prediction step” uses the full nonlinear model:
xf ,i
t = m(xa,i
t−1)
“Update step” given by
xa,i
t = xf ,i
t + K(yi
t − Hxf ,i
t )
Here K = ˆPf
t HT
(H ˆPf
t HT
+ R)−1
is calculated using the sample
covariance ˆPf
t .

Section 2 Challenges in sea ice data assimilation
Outline
Data assimilation: 3 minute introduction
Challenges in sea ice data assimilation
Role of unstable subspace in data assimilation
Role of model error in data assimilation
Thanks to Colin Guider for the slides in this section

neXtSIM: a new Lagrangian sea ice model
neXtSIM simulation video
Lagrangian model
running on triangular,
unstructured mesh
When mesh becomes too
distorted, a remeshing
process occurs
Variables: Ice and snow
thickness; Ice
concentration; Ice
damage; Internal stress
tensor; Sea ice velocity
https://www.youtube.com/watch?v=cFfQsu6xWZ4

The number of edges and vertices changes in time
=⇒ the state space dimension changes in time!

The changing state space dimension poses a challenge

1 dimensional model problem: Burgers’ equation

Adaptive mesh is coupled to the evolution of the dynamical
ﬁeld u

The minimum mesh size deﬁnes a ﬁxed dimensional state
space: one point per interval of size δ1
Numerical experiments using this approach,
with EnKF as the data assimilation scheme

EnKF with ﬁxed dimensional state space deﬁned using
supermesh

Various future directions
A different choice for the reference mesh, using the maximum cell
size, leading to multiple grid points in a cell
Updating the mesh locations during the assimilation step
Moving reference mesh, evolving according to some specified
dynamics
Applications to neXtSIM: definition of 2-dimensional supermesh
Thanks to Colin Guider for the slides and the results in this section

Section 3 Role of unstable subspace in data assimilation
Outline
Thanks to Anugu Sumith Reddy for the slides in this section

Let us recall Kalman ﬁlter
0 , ∆0)
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –
The Kalman ﬁlter algorithm is summarized as
(xf
0 , Σ0) → (xa
0 , ∆0) → · · · (xa
t−1, ∆t−1)
forecast
−→ (xf
t , Σ1)
update
−→ (xa
t , ∆t) · ·
Main questions we ask (and answer1): does the following sequence
converge, and what are the asymptotic properties?
Σ0 → ∆0 · · · → Σt → ∆t · · ·
t→∞
−→ Σ → ∆
1
Gurumoorthy et al. doi:10.1137/15M102857395 and Bocquet et al.
doi:10.1137/16M10677128

Null space of Σt contains the stable Lyapnuov subspace,
rate of convergence ∼ Lyapunov exponents
in the case Qt = 0 for all t (deterministic dynamics): The stable
backward Lyapunov subspace2 of the dynamics is contained within
the null space of Σt asymptotically in time (as t → ∞):
if ut
k is left singular vector of Mt:0 with λt
k < 0, then
lim
t→∞
Σtut
k = 0
Thus,
lim
t→∞
rank(Σt) = min {n0, rank(Σ0)}
where n0 is the dimension of unstable-neutral space = number of
non-negative Lyapyunov exponents.
2
in case it is well deﬁned

Current work: similar results for continuous time
Kalman-Bucy ﬁlter
dxt = Atxtdt , dyt = Ctxtdt + R
1
2
t dWt , x0 ∼ N(m, P)

Stability and projection onto unstable subspace
If ξi are the Lyapunov exponents for ΦT (t, t0), we show that
and hence, projection of Pt onto the “stable subspace” goes to zero
exponentially in time. Futher,

Work in progress and future directions
The above direction of research is motivated by the work of Trevisan,
Carrassi, et al. on the set of methods called “assimilation in unstable
subspace” (AUS)
Relation of the stability result to integral separation and exponential
dichotomy (with Erik Van Vleck)
Small noise limits of linear ﬁltering problem and large deviations
Stability results for linear case with non-Gaussian initial conditions
Eventually stability results for nonlinear ﬁlters for deterministic
dynamical systems
Thanks to Anugu Sumith Reddy for the slides and the results in this
section

Section 4 Role of model error in data assimilation
Outline
Time
true
trajectory
observations
h: obs.
function
obs. error
ensemble
forecast
updated
ensemble
arrows indicate data
assimilation process obs.
space
state
space

Outline
Time
true
trajectory
observations
h: obs.
function
obs. error
ensemble
forecast
updated
ensemble
arrows indicate data
assimilation process obs.
space
state
space

Statistics for representativeness error
True trajectories are solutions of ˙w = g(w)
Model trajectories are solutions of ˙wd = f (wd )
Choose observation space and the observation operators y = h(π(w))
and y = hc(wd )
Fix τ. Solve the above two on time interval (0, τ), starting with the
same initial condition w0. What is the probability distribution of
h(π(w(τ))) − hc(wd (τ))?
Use that probability distribution as (state dependent) observational
noise
Compare with choosing state independent observational noise
(relation to inﬂation??)

Possible setup for implementing the above strategy
Lagrangian data assimilation
Truth: ¨x + ˙x = u(x, t)
Model: ˙x = u(x, t)
Lorenz-96: a well-studied chaotic system with attractor
Truth:
˙xi = (xi+1 − xi−2)xi−1 − xi + F − α
J
j=1
yj,k
˙yj,k = −γ(yj+2,k − yj−1,k)yj+1,k − βyj,k + αxk
Model:
˙xi = (xi+1 − xi−2)xi−1 − xi + F

Some preliminary results on 2-dim Henon map
Truth: xt+1 = 1 − ax2
t + yt , yt+1 = bxt with (a, b) = (1.4, 0.3)
Model: Same map with (a , b ) = (a, b) + (0.001, 0.001)
0.00000.00050.00100.0015
0
50
100
150
200
250
k = 1
0.0000 0.0025 0.0050
0
50
100
150
200
250
k = 2
0.00 0.01
0
100
200
300
400
k = 3
0.00 0.02
0
100
200
300
400 k = 4
0.025 0.000 0.025 0.050
0
100
200
300
400 k = 5
0.1 0.0 0.1 0.2
0
100
200
300
400 k = 8
0.5 0.0 0.5
0
100
200
300
400
500
600
k = 10
2 0 2
0
100
200
300
400
500
600
700
k = 12
2 0 2
0
100
200
300
400
k = 15
2 0 2
0
50
100
150
200
global
Representativeness error depends on time between observations (denoted
by k here), and may be highly non-Gaussian.

Various future directions
Find a better name for representativeness error
State dependence of representativeness error
Develop a framework for using representativeness error in data
assimilation
Thanks to SAMSI (not for the slides in this section but) for providing an
excellent opportunity for the members of the data assimilation working
group to discuss these and other ideas

Sea Ice Data Assimilation Challenges

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