Lagrangian Observations in the Atmosphere Balloons Horizontal - - PowerPoint PPT Presentation

Lagrangian Observations in the Atmosphere

Balloons Horizontal position, pressure, temperature

Hertzog, Basdevant, Viall, Mechoso (2004) http://wwwsouthpoledude.com/mcmurdo0506/

T (K) Days in 2002

40 80 220 190

SPB 6: Obs; ECMWF; NCEP

SuperPressure Balloon (SPB) observations and comparison with reanalysis (ECMWF and NCEP)

SPB6: Orange

Cloud features Horizontal position

Following up on the previous talk by Andy Tangborn

Lagrangian Data Assimilation (LaDA): Method and Mathematical Challenges

Kayo Ide, UCLA Guillaume Vernieres & Chris Jones, UNC-CH Hayder Salman, Cambridge Unversity

http://www.drifters.doe.gov/design.html Data available from http://www.aoml.noaa.gov/phod/dac/dacdata.html

Two Mathematical Issues in LaDA

• 1. High nonlinear drifter dynamics  Handing of chaotic data
• 2. Limited number of observations  Deployment strategy

Basic Elements of Lagrangian Data Assimilation System

xF tk

( ) =

 uijl tk

( )

vijl tk

( )

hijl tk

( )

 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ NF  10

5-7

yD,j tk

( ) =

 rD,j

(x) tk

( )

rD,j

(y ) tk

( )

rD,j

(p) tk

( )

⎡ ⎣ ⎤ ⎦  ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ LD = 2 [or 3] per drifter

Application to Gulf of Mexico: Next Presentation by Guillaume Vernieres

x

a =

xF

a

xD

a

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = xF

f

xD

f

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + P

FD f

P

DD f

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ P

DD f + RDD

• (

)

−1 yD

• − xD

f

( )

Lagrangian Data Assimilation (LaDA) Method

Eulerian Models xF Lagrangian Data yD Lagrangian Data Assimilation (LaDA) x ≡ xF xD ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

yD

• = Hx

t + εD = xD t + εD

Augmented model state x Partial observation y of x Direct assimilation of yD into xF P ≡ P

FF

P

FD

P

DF

P

DD

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ H = 0 I

( ) εD  N 0

RDD

• (

)

EKF: Ide, Jones, Kuznetsov (2002); EnKF: Salman,Kuznetsov, Jones, Ide (2005)

xF tk

( )

xD tk

( )

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = mF xF tk−1

( )

( )

mD xF tk−1

( ),xD tk−1 ( )

( )

⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟

Explicit computation of flow-dependent P is required

Proof of Concept: Application to Mid-latitude Ocean Circulation

Ocean circulation

1-layer shallow-water model Wind-driven: τ=0.05 Ns-2 Domain size: 2000km x 2000km

Perfect model scenario

Model spin-up for 12yrs Nature run with H0=500m Ensemble with (Hmean, Hstd)=(550m,50m) Drifters Released at the beginning of 13yr Observed every day with σstd=200m

Case: ν=500m2s-1, (∆T, LD )=(1day, 1)

T=0 days T=90 days Truth Without DA Ensemble Mean With LaDA Analysis

Weakly chaotic ocean dynamics by basin-scale Rossby wave Background color contour: instantaneous h

Trajectory for 90days

Case: ν=500m2s-1, (∆T, LD )=(1day, 1)

T=0 days T=90 days Truth Without DA Ensemble Mean With LaDA Analysis

Issue 1: Handling of Chaotic Data for xD

Spatially-Temporally chaotic ocean dynamics in truth: ν=400 m2s-1 Triangles in color: Drifters motion in truth Curves in purple: Material boundaries (manifolds for xD) Background contour: Instantaneous h in truth

Sudden Filter Divergence by the Hyperbolic Effect for xD

Large innovation d can occur near a hyperbolic trajectory(HT) Spread of Pf

DD brings xa D properly close to yo

Unreastically large Δxf

D can occur due to poor representation of Pf FD

Ensemble spread of xD (ΔT=20) Ensemble mean (forecast/analysis)

x

D f

x

D a

yD

• d = y
• − x

D f

ΔxF

a

ΔxD

a

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = P

FD f

P

DD f

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ P

DD f + RD

• (

)

−1d

xD(t=0) xD(t=20) xD(t=40) Hyperbolic Trajectory (HT)

Two potential sources for large d=yo – xD

t

1. Large observation error: yo – xD

t  No update (outlier yo)

2. Fast dispersion in: xD

f – xD t  Update at least xD & check for xF

Outline of the quality control scheme

a) Detection of hyperbolic effect (HE) in xD

f using 2 tests

1. Against Rd

• :

2. Against (PDD

f+RD

• ):

b) Sanity check for ΔxF

a

• 1. Against PFF

f:

Update with QC Scheme

Quality Control Scheme to Handle Chaotic Data for xD

More sophisticated approach (Particle filter): Spiller, Budhiraja, Ide, Jones (2008)

cD

• = d

T RD

• (

)

−1d

cD

• f = d

T P DD f + RD

• (

)

−1d

cF

a = ΔxF aT P FF f

( )

−1ΔxF a

CD

• f≥βo

CD

• f<βo

CD

• <βo

xF

& xD

xF

& xD

CD

• ≥βo

No Update HE detected CF

a<βa

CF

a≥βa

xF

& xD

xD only

Issue 2: Towards Observing System Design

Observing system:

Trajectories significantly differ depending on deployment locations/times Once Lagrangian instruments are released, they go with the flow. Number of drifters and resources to deploy instruments are limited.

Data assimilation system:

Regions where Lagrangian observations effectively correct the model state are restricted around the trajectories.

Observing system design = targeting (without metric) by taking into account of

Lagrangian flow structures evolving in {xF(t)} Freely moving instruments {xD(t)} in {xF(t)}

Directed Deployment of Drifters for Targeting in xF for xD

Eddy tracking

Deploy so that drifters will stay in the eddy

Survey of largest possible area

Deploy where drifters will spread out quickly visit various parts of the flow

Balanced performance

Use combination

If no information about Lagrangian flow dynamics

Deploy uniformly

• r based on educated guesses

(and cross your fingers)

Lagrangian Flow Template by Dynamical Systems Theory

Stable and unstable manifolds from the hyperbolic trajectories (HTs) = “material boundaries” of distinct Lagrangian flow regions Dynamical systems theory: A tool analyze Lagrangian dynamics given a time sequence of (Eulerian) flow field

Eulerian Model Field {xF(t), t0-T≤t≤t0-T} Lagrangian flow template for {xD(t0)} How to obtain Lagrangian flow template? How to detect manifolds?

Dynamical Systems Theory

Lagrangian Flow Template for xD in Unsteady Flow xF

Direct Lyapunov Exponents (DLE): divergence of the nearby trajectories

Application to transport: Haller (2001, 2002) and Others. Application to DA: Salman, Ide, Jones (2007) Reconstruction of velocity field: Poje, Toner, Kirwan, Jones (2002)

DLE :

xD t0 +T ;x0 + δx0,t0

( ) ≈ xD t0 +T ;x0,t0 ( ) + δxD t0 +T ;x0,t0 ( )

maxδx0 δxD t0 +T ;x0,t0

( ) = exp σ t;x0,t0 ( )

{ } δx0

σ t;x0,t0

( )

High divergence: forward DLE (T>0) → stable manifolds backward DLE (T<0) → unstable manifolds

Unstable manifold stable manifold

Background contour: instantaneous h

High values of DLE (gray: T<0) and (red: T<0) at t0=0 with T=10days

Targeting based on Lagrangian Flow Template (Manifolds)

Local targeting: Eddy tracking ⇒Deploy inside of eddy, surrounded by the stable & unstable manifolds Global targeting: Survey of the largest possible area ⇒Deploy both side of the stable manifolds for fastest divergence along the unstable manifolds ⇒Deploy into the jet/current defined by the manifolds Mixed strategy: Balanced performance ⇒Deploy using local & global targeting

Lagrangian flow template Mixed strategy

Targeting

Observing System Design for Mid-Latitude Ocean

Local Targeting (3 eddies x 3) Global Targeting (3 HT x 3) Uniform Deployment (3 x 3) Mixed Strategy (3 eddies x 1 2 HT x 3)

Perfect model scenario

Model spin-up for 12yrs Nature run with h0=500m Ensemble with (hmean, hstd)=(550m,50m) Drifters Released at the beginning of 13yr Observed every day with σstd=200m

Test strategies using 9 drifters

Local Targeting Strategy

Truth Local targeting T=25days T=300days T=100days

Background color contour: instantaneous h

Initial error in h: Global Failure to capture new eddy generation

Global Targeting Strategy

Truth Global targeting T=25days T=300days T=100days

Mixed Strategy = Global + Local

Truth Mixed T=25days T=300days T=100days

Spatial Distribution of RMSE in h

Local Targeting T=25days T=300days T=100days Global Targeting

Failure to capture new eddy generation Effective basin-scale estimation by fast divergence

Spatial Distribution of RMSE in h

T=25days T=300days T=100days Mixed Strategy Uniform Deployment

Spatial Distribution of RMSE in KE

Failure to capture new eddy generation

Local Targeting T=25days T=300days T=100days Global Targeting

Initial KE error distribution: local

Spatial Distribution of RMSE in KE

Mixed strategy T=25days T=300days T=100days Uniform Deployment

Remarks on Deployment Strategy

Deployment strategy

It is targeting for xD(t0), using Lagrangian flow template obtained by {xF(t), t0-T≤t≤t0+T} It is naturally built on dynamical systems theory

Eulerian Model Field {xF(t), t0-T≤t≤t0-T} Directed Deployment (e.g., Mixed strategy) Lagrangian flow template xD(t0) Targeting DLE / dynamical systems theory

Summary

Two mathematical issues in LaDA

• 1. Chaotic nature of Lagrangian dynamics
• 2. Deployment strategy that requires Lagrangian (non-instantaenous)

information of the flow

REAL challenges in deployment strategy for LaDA

Drifters are to be released in the real ocean {xt

F(t)}

while Lagrangian flow template (DLE) is obtained by {xm

F(t), t0-T≤t≤t0}.

DLE computation requires {xm

F(t), t0-T≤t≤t0+T} in the past and future

Deployment strategy is doubly penalized by

Predictability limit & uncertainties of Lagrangian dynamics {xD(t)} Predictability limit & uncertainties of flow field {xF(t)}

Deployment strategy can benefit from chaotic nature of Lagrangian dynamics

Lagrangian time scale of {xD(t)} in {xF(t)} is much shorter than Eulerian (evolution) time scale of {xF(t)}