A Bayesian Approach to Lagrangian Data Assimilation Chris Jones, UNC - - PDF document

SLIDE 1

A Bayesian Approach to Lagrangian Data Assimilation

Chris Jones, UNC‐CH and University of Warwick Andrew Stuart, Jochen Voss, University of Warwick Amit Apte, TIFR Bangalore Supported by the Office of Naval Research and the National Science Foundation

f f f 1 1 2 1 1

Model forecast: ( ), ( ), ( ) x t x t t P

t t 1 1 2 1

( ), ( ) x t x t

t t =

t t 1 2

( ), ( ) x t x t

f f f 1 2

Initial conditions: ( ), ( ), ( ) x t x t t P

1

t t =

• t

1 1 1 1

Measurement: ( ) ( ) y t x t ε = +

a a a 1 1 2 1 1

State estimate: ( ), ( ), ( ) x t x t t P

Gain Matrix

( )

posterior

• bs

prior

( ) ( ) P x y P y x P x =

Bayes

SLIDE 2

Framework for DA Approach

( ) dx f x dt =

n

x R ∈

(0) x x ζ = ฀ ( ) is the pdf of initial conditions p x

ζ 1

• bs:

,..., K t t t = ( ( ))

k k k

y h x t η = +

Bayesian formulation

Prior distribution:

( )

prior

P x ) | ( x y Pobs

( )

2 2

1 ( | ) exp ( ) 2

• bs

P y x y h x σ ⎡ ⎤ ∝ − − ⎢ ⎥ ⎣ ⎦ ) ( ) | ( ) | ( x P x y P y x P

prior

• bs

posterior

∝

from initial “initial condition” Observational likelihood: from (Lagrangian) data Bayes rule: Ultimate Goal: Obtain the posterior distribution of initial conditions Assumption: Perfect model, but in principle unnecessary

SLIDE 3

State Estimation

Model runs + observations state estimate

x

t

1

t

2

t

N

t

3

t

( )

N

x t

model model model

( ) y t

1

( ) y t

2

( ) y t

3

( ) y t ( )

N

y t

OBS

+

Bayes:

( )

posterior

• bs

prior

( ) ( ) P x y P y x P x =

M

}

sample posterior

Model runs

Langevin Sampling

ds dW Z L ds dZ 2 ) ( + =

W Z L Z Z

s s k k k

2 ) (

1

∆ + ∆ + =

+

) ( log ) ( Z Z L

z

ρ −∇ =

Langevin dynamics : in discretized version generates N samples from the distribution with

N k k

Z

1

} {

=

( ) Z ρ

Basis: Invariant distribution of the Langevin dynamics is It works provided the Langevin equation is ergodic.

) (Z ρ

SLIDE 4

Deterministic model: 3

x x x = − &

1

1 −

Estimation problem: Initial condition

x

Observations:

( )

i i

y x i η = ∆ + 1, , ; i N N T = ∆ = L

) , ( ~

2

R N

i

η

Test case

Comparing EnKF and Langevin sampling

−1.5 −1 −0.5 0.5 1 1.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Exact posterior at t=T

• Approx. using EnKF
• Approx. using Langevin SDE

−1.5 −1 −0.5 0.5 1 1.5 0.5 1 1.5 Exact posterior at t=T

• Approx. using EnKF
• Approx. using Langevin SDE

EKF posterior −1.5 −1 −0.5 0.5 1 1.5 0.5 1 1.5 2 Exact posterior at t=T

• Approx. using EnKF
• Approx. using Langevin SDE

EKF posterior −1.5 −1 −0.5 0.5 1 1.5 0.5 1 1.5 2 Exact posterior at t=T

• Approx. using EnKF
• Approx. using Langevin SDE

EKF posterior

. 3

x x x = −

SLIDE 5

Lagrangian DA

• In ocean, subsurface info is often (quasi‐) Lagrangian
• State estimation (as opposed to forecasting) is of interest in
• cean
• Particularly appropriate for float data
• Natural to use an augmented state‐space approach
• Obs are in a clearly defined low‐dimensional subspace but

encode key aspects of full dynamics

• Can potentially capture large‐scale coherent features

Lagrangian DA and State Estimation

( ) ( )

t x x M dt dx t x M dt dx

F D D D F F F

, , ; , = =

Augmented model : ) , ( ~ ; ) (

2

R N t x y

i i i D i

ξ ξ + = Observations : at times (t1,t2, … tm ) Goal : Estimate initial conditions x(0) using the observations Idea : Use Bayesian formulation and Langevin sampling with initial conditions

( )

(0), (0)

F D

x x x ≡

SLIDE 6

Model Problem for Lagrangian DA

Linearized shallow water model:

2‐mode approximation:

1 k l m = = =

Geostrophic mode with amplitude Inertial‐gravity mode that is time periodic

u

Augmented System

• bs at:

with Gaussian errors uncorrelated and independent of each other

1, ...,

k

t k k N δ = =

SLIDE 7

Experiments and Methods

• 1. Short trajectory
• 2. Long trajectory staying in cell
• 3. Trajectory crossing cell boundaries
• A. Langevin Stochastic DE
• B. Metropolis Adjusted Langevin Algorithm
• C. Random Walk Metropolis Hastings
• D. EnKF

Short Trajectory

SLIDE 8

Long Trajectory in Cell

3 observations sets, # of obs: Obs set 1: 100 Obs set 2: 20 Obs set 3: 6

Comparison

MALA improves with increased number of observations (frequency kept same) but EnKF does not.

Obs set 4, has same fequency as 3, but extends trajectory and makes 20 obs

SLIDE 9

Scatter Plots

EnKF is handicapped by trying to effectively approximate by a Gaussian and thus not accounting for nonlinear effects

Trajectory crossing cell boundary

SLIDE 10

Scatter Plots Conclusions

• In model problem, a modified Langevin

sampling does particularly well.

• Going to higher dimensions is obviously a
• challenge. Salman (2007) has a hybrid

method that appears to work well

• Nonlinearity is well addressed, but saddle issue is

not resolved

• Filtering vs. smoothing is a serious debate because
• f chaotic effects