Lagrangian Data Assimilation by hcj

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									     Lagrangian Data Assimilation:
       Method, Applications, and
Strategy for Optimal Drifter Deployment


             Kayo Ide, UCLA



C.K.R.T. Jones, Guillaume Vernieres,UNC-CH
       Hayder Salman, Cambridge U.
              Liyan Liu, NCEP
        Lagrangian Instruments in the Ocean: Drifters
Observations at sea surface
   T            : Temperature
along (x(2D) )(tk)) at sea surface



                                                                                   Float Package




                                                                                 Temperature Sensor




    Data available from
    http://www.aoml.noaa.gov/phod/dac/dacdata.html   http://www.drifters.doe.gov/design.html
         Lagrangian Instruments in the Ocean: Floats
Observation on the isopyncnal surface
   (T,S )
   (u,v)
along (x(2D) )(tk), p(x(2D) )(tk))




       http://www.dosits.org/gallery/tech/ooct/rafos1.htm   http://www.whoi.edu/instruments/
           Global Ocean Observing System by Drifters




Global observation network by drifters
     1250 drifters to cover at the 5ox5o resolution
Drifters are used as the platform
     Eulerian observations of T (SLP, Wind)

                             http://www.aoml.noaa.gov/phod/dac/gdp.html
Assimilation and Short-Range Forecast for Regional Ocean

  Real-Time Regional ocean off the U.S. West Coast




                                                                           Observations:
                                                                              Remote-sensing
                                                                              In situ




 Model: Regional Ocean Modeling System         Method: Incremental 3D-Var
 (ROMS)                                            Weak constraints by dynamic balance
    One-way nested configuration with              Inhomogeneous / anistropic background error
    increasing resolution for smaller domain       covariance using Kronecker product
    COAMPS forcing
                                                            Li, Chao, McWilliams, Ide (2007a,b)
        Ocean Observations: Remote-Sensing by Satellite
  Sea Surface Temperature (SST)




                                Sea Surface Height (SSH)

Data available at
http://ourocean.jpl.nasa.gov/




                                           http://nereids.jpl.nasa.gov/
  Ocean Observations: In Situ by Stationary Platforms

Mooring
 (T , S, p)
 (u, v)




  Data available at http://ourocean.jpl.nasa.gov   http://www.mbari.org
       Ocean Observations: In Situ by Movable Platforms
Glider
  At the surface: xG(2D)
  In the water:
  (T , S, p)



Ship                                                                 http://www.mbari.org

  (T , S, p)
   (u, v)




                                    Data available at http://ourocean.jpl.nasa.gov
                        Ocean Observations

Currently available observations are inhomogeneous and sparse in space &
sporadic in time. Available observations are mostly
   T and S
   In the upper ocean
                    Routine Observation for ROMS 3D-Var system
    Type                                 Platform            Number / day
    SST            Remote-sensing        Satellite           O(102) - O(104)
    SSH            Remote-sensing        Satellite           0 - O(102)
    T&S            In situ               Mooring, ship,      O(102)
                                         glider, AUV


Ocean observations are precious
   New types of observations: SSS by Satellite, Coastal HF radar
   New technology for cost effectiveness: Lagrangian data
   Ocean Observation: Remote-Sensing by HF Radar
Coastal Oceans Currents Monitoring Program (COCMP)




http://www.cocmp.org/




                                          http://www.cencoos.org/currents
 Lagrangian Dynamics of Drifters




                      QuickTime™ and a
                         decompressor
                are need ed to see this picture.




Data available from http://www.aoml.noaa.gov/phod/dac/dacdata.html
                               Outline

Ocean observation for data assimilation systems

Lagrangian data assimilation (LaDA) method

Application I: Double-gyre circulation. “Proof of concept”

Application II: Gulf of Mexico. “Efficiency”

Design of optimal deployment strategy using dynamical systems
theory

Concluding remarks
   Summary
   Future Directions
Basic Elements of Lagrangian Data Assimilation System

           Eulerian Model:                         Lagrangian Observation:
              State xF                                   Location yD

              M                                                    M      
             u t
                   
              ijl k
                          
                          
                                                             r ( x) t
                                                             D, j k       
                                                                             
     
   xF tk           
             v ijl tk
             
                          
                          
                                                   
                                               yD, j tk     rD, j tk
                                                            
                                                                (y )
                                                                           
                                                                             
                   
              hijl tk       NF : 10   5-7
                                                             rD,pj ) tk
                                                                (
                                                                             LD  2 [or 3]
                                                                        
              M                                                    M              per drifter




                               Data Assimilation Method
   Data Assimilation Method: Kalman-Filter Approach
   Forecast from tk-1 to tk:                     Observation at tk:
         x a 1  xk 
                     f
                                                           y o 
          a  f                                         o
            k                                                 k

         Pk 1  Pk                                     R k 

             
x k  x k ~ N 0,Pkf
  f     t
                                                            y o  y k ~ N 0,R o
                                                               k
                                                                     t
                                                                               k     
                                                                         t
                                                                        yk        h x k
                                                                                             t
                                                                                             k




                                                                   k 
                                                     xa  xk  Kk yo  yk
                                                      k
                                                           f            f
                                                                                                 
                                                     Pka    I  K H P                 f


      k
            t
                 
    x a  x k ~ N 0,Pka       Analysis at tk:                          k        k       k




                                x k   x k 
                                  f        a                    
                                                       y k  hk x k
                                                         f        f


                                 f   a
                                                             PH                                    
                                                                                                     1
                                Pk  Pk             Kk         R f
                                                                    k
                                                                        T
                                                                        k
                                                                                     o
                                                                                     k
                                                                                          HkPkfHT
                                                                                                 k

                                                                
                                                       Hk        h
                                                               x k         xf
                                                                             k
        Elements of Assimilating Lagrangian Data
Essence of analysis in data assimilation

             
  xa  x f  K yo  y f        
      K PH                            
                                        1
             f
               R T       o
                              H PfHT
      y  h x 
        f            f




Elements in hands
   Forecast flow state             xF as xf
   Lagrangian observation yD as yo




Missing elements
   h that gives yfD from xfF , because nothing in xfF directly relates to yoD
   Pf that gives K for optimal impact of yo on xa
Assimilation of Lagrangian Data: Conventional Method
                                                         Transform from Lagrangian data

                                                                        
                                                               yD  yD t j
                                                                o    o




                                                         to Eulerian (velocity) data

                                                               yo 
                                                                           
                                                                      yD t j  yD t j 1
                                                                       o        o

                                                                V
                                                                           t j  t j 1


                                                         Observation operator

                                                              y V  HV x f
                                                                f




                                                         Hv is linear spatial interpolation.


Feedback the mismatch of observation (innovation) into the model variable
          f
                  
    xF  xF  K V y o  y V
     a
                    V
                          f
                                                             Carter (1989)
                                                              Kamachi, O’Brien (1995)
                              R                    
                                                    1
              K V  P HV
                      f   T        o
                                   V
                                        HVP HV
                                            f   T
                                                              Tomassini, Kelly, Saunders (1999)
Assimilation of Lagrangian Data: Conventional Method
                                                         Transform from Lagrangian data

                                                                        
                                                               yD  yD t j
                                                                o    o




                                                         to Eulerian (velocity) data

                                                               yo 
                                                                           
                                                                      yD t j  yD t j 1
                                                                       o        o

                                                                V
                                                                           t j  t j 1


                                                         Observation operator

                                                              y V  HV x f
                                                                f




                                                         Hv is linear spatial interpolation.


Feedback the mismatch of observation (innovation) into the model variable
          f
                  
    xF  xF  K V y o  y V
     a
                    V
                          f
                                                             Carter (1989)
                                                              Kamachi, O’Brien (1995)
                              R                    
                                                    1
              K V  P HV
                      f   T        o
                                   V
                                        HVP HV
                                            f   T
                                                              Tomassini, Kelly, Saunders (1999)
Assimilation of Lagrangian Data: Conventional Method
                                                         Transform from Lagrangian data

                                                                        
                                                               yD  yD t j
                                                                o    o




                                                         to Eulerian (velocity) data

                                                               yo 
                                                                           
                                                                      yD t j  yD t j 1
                                                                       o        o

                                                                V
                                                                           t j  t j 1


                                                         Observation operator

                                                              y V  HV x f
                                                                f




                                                         Hv is linear spatial interpolation.


Feedback the mismatch of observation (innovation) into the model variable
          f
                  
    xF  xF  K V y o  y V
     a
                    V
                          f
                                                             Carter (1989)
                                                              Kamachi, O’Brien (1995)
                              R                    
                                                    1
              K V  P HV
                      f   T        o
                                   V
                                        HVP HV
                                            f   T
                                                              Tomassini, Kelly, Saunders (1999)
Assimilation of Lagrangian Data: Conventional Method
                                                         Transform from Lagrangian data

                                                                        
                                                               yD  yD t j
                                                                o    o




                                                         to Eulerian (velocity) data

                                                               yo 
                                                                           
                                                                      yD t j  yD t j 1
                                                                       o        o

                                                                V
                                                                           t j  t j 1


                                                         Observation operator

                                                              y V  HV x f
                                                                f




                                                         Hv is linear spatial interpolation.


Feedback the mismatch of observation (innovation) into the model variable
          f
                  
    xF  xF  K V y o  y V
     a
                    V
                          f
                                                             Carter (1989)
                                                              Kamachi, O’Brien (1995)
                              R                    
                                                    1
              K V  P HV
                      f   T        o
                                   V
                                        HVP HV
                                            f   T
                                                              Tomassini, Kelly, Saunders (1999)
      Lagrangian Data Assimilation (LaDA) Method
Elements in hands
                                                                               xF
   Augmented state x and model m
       Flow state xF and model mF
       Drifter state xD and model mF
             x                                         xD
         x   F
             x  D



                        
                mF xF tk 1    
      
  m x tk 1                   
                          
                mD xF tk 1 ,xD tk 1
                               

  Observation yD and operator h that relates yD to x
                    x 
          
      h x      0 I  F
                     xD 
                                                       Ide, Jones, Kuznetsov (2002)
                                                       [Ide and Ghil (1997)]

Missing elements
   Pf that gives K for optimal impact of yo on xa
      Lagrangian Data Assimilation (LaDA) Method
Elements in hands
   Augmented state x and model m
       Flow state xF and model mF
       Drifter state xD and model mF
             x 
         x   F
              xD 
                                                                   
                                                       dxD, j  uF xD, j dt


                      
                mF xF tk 1  
      
  m x tk 1                 
                        
                mD xF tk 1 ,xD tk 1
                             

  Observation yD and operator h that relates yD to x
                    x 
          
      h x      0 I  F
                     xD 
                                                       Ide, Jones, Kuznetsov (2002)
                                                       [Ide and Ghil (1997)]

Missing elements
   Pf that gives K for optimal impact of yo on xa
      Lagrangian Data Assimilation (LaDA) Method
Elements in hands
   Augmented state x and model m
       Flow state xF and model mF
       Drifter state xD and model mF
             x 
         x   F
              xD 

                      
                mF xF tk 1  
      
  m x tk 1                 
                        
                mD xF tk 1 ,xD tk 1
                             

  Observation yD and operator h that relates yD to x
                    x 
          
      h x      0 I  F
                     xD 
                                                       Ide, Jones, Kuznetsov (2002)
                                                       [Ide and Ghil (1997)]

Missing elements
   Pf that gives K for optimal impact of yo on xa
              Ensemble-Based Data Assimilation
Use of ensemble X   ,...,x  to represent the uncertainty of x
                       x        1   Ne


in particular, mean and covariance

  mean
       xF  1 Ne  xF,n 
    x        
       xD  Ne n1  xD,n 


  covariance

         P    PFD                                                       rx 
     P   FF
          PDF PDD 
                                                                   xD   
                                                                          ry 
                          
                     x x x x                           
                                             xF,n  x xD,n  x 
                                         T                    T
               Ne
           1
                                                             
                       F,n   F,n


                          
         Ne  1 n1  x  x x  x
                                                          
                                             xD,n  x xD,n  x 
                                         T                    T
                     D,n    F,n                                
                              Ensemble-Based LaDA
                                                                    xD
1. Ensemble forecast from tk-1 to tk
       f
              
      xF,n tk  mF xF,n ,
                    a
                                           tk 1 
       f
      xD,n   t  m x
              k     D
                          a
                          F,n
                                   a
                                , xD,n , tk 1   
    for n = 1,…, Ne


2. Ensemble update at tk to incorporate
        o
      y D tk   and      o
                        R DD tk   
                                                                                                       xF
    Analysis (dropping tk )
             xF,n 
               a

       xa   a 
        n
             xD,n 
                xF,n   PFD  f
                  f

                                                      y                      
                            f

                                    
                                                     1
               f    f  PDD  R DD
                                     o                      o
                                                                D,n  xD,n
                                                                  o      f

                xD,n   PDD 
                                                            D


                                                                           Salman, Kuznetsov,Jones, Ide (2006)
                                                                           Salman, Ide, Jones (2007)
                              Ensemble-Based LaDA
                                                                    xD
1. Ensemble forecast from tk-1 to tk
       f
              
      xF,n tk  mF xF,n ,
                    a
                                           tk 1 
       f
      xD,n   t  m x
              k     D
                          a
                          F,n
                                   a
                                , xD,n , tk 1   
    for n = 1,…, Ne


2. Ensemble update at tk to incorporate
        o
      y D tk   and      o
                        R DD tk   
                                                                                   xF
    Analysis (dropping tk )
             xF,n 
               a

       xa   a 
        n
             xD,n 
                xF,n   PFD  f
                  f

                                                      y                      
                            f

                                    
                                                     1
               f    f  PDD  R DD
                                     o                      o
                                                                D,n  xD,n
                                                                  o      f

                xD,n   PDD 
                                                            D
                              Ensemble-Based LaDA
                                                                    xD
1. Ensemble forecast from tk-1 to tk
       f
              
      xF,n tk  mF xF,n ,
                    a
                                           tk 1 
       f
      xD,n   t  m x
              k     D
                          a
                          F,n
                                   a
                                , xD,n , tk 1                   yD
    for n = 1,…, Ne


2. Ensemble update at tk to incorporate
        o
      y D tk   and      o
                        R DD tk   
                                                                                   xF
    Analysis (dropping tk )
             xF,n 
               a

       xa   a 
        n
             xD,n 
                xF,n   PFD  f
                  f

                                                      y                      
                            f

                                    
                                                     1
               f    f  PDD  R DD
                                     o                      o
                                                                D,n  xD,n
                                                                  o      f

                xD,n   PDD 
                                                            D
                              Ensemble-Based LaDA
                                                                    xD
1. Ensemble forecast from tk-1 to tk
       f
              
      xF,n tk  mF xF,n ,
                    a
                                           tk 1 
       f
      xD,n   t  m x
              k     D
                          a
                          F,n
                                   a
                                , xD,n , tk 1                   yD
    for n = 1,…, Ne


2. Ensemble update at tk to incorporate
        o
      y D tk   and      o
                        R DD tk   
                                                                                   xF
    Analysis (dropping tk )
             xF,n 
               a

       xa   a 
        n
             xD,n 
                xF,n   PFD  f
                  f

                                                      y                      
                            f

                                    
                                                     1
               f    f  PDD  R DD
                                     o                      o
                                                                D,n  xD,n
                                                                  o      f

                xD,n   PDD 
                                                            D
                              Ensemble-Based LaDA
                                                                    xD
1. Ensemble forecast from tk-1 to tk
       f
              
      xF,n tk  mF xF,n ,
                    a
                                           tk 1 
       f
      xD,n   t  m x
              k     D
                          a
                          F,n
                                   a
                                , xD,n , tk 1   
    for n = 1,…, Ne


2. Ensemble update at tk to incorporate
        o
      y D tk   and      o
                        R DD tk   
                                                                                   xF
    Analysis (dropping tk )
             xF,n 
               a

       xa   a 
        n
             xD,n 
                xF,n   PFD  f
                  f

                                                      y                      
                            f

                                    
                                                     1
               f    f  PDD  R DD
                                     o                      o
                                                                D,n  xD,n
                                                                  o      f

                xD,n   PDD 
                                                            D
        Mechanisms of Lagrangian Data Assimilation (LaDA)
          Forecast from tk-1 to tk:                                              Observation at tk:
                  x a 1  xk 
                              f
                                                                                        y o 
                   a  f                                                             o
                     k                                                                     k

                  Pk 1  Pk                                                         R k 
         xF 
            f      mF xF 
                          a
                                                                                     o
                                                                                                      o
                                                                                                             
         x f    m xa , xa 
                                                                                      yD tk      with R DD tk
                       
         D k  D F D           
                                k 1

 PFF
    f
        PFD 
          f
                  PFF
                    a
                              PFD 
                               a


 Pf           M a
  DF
        Pf 
         DD  k
                 P      DF
                              Pa 
                                  DD   k 1
                                                   Analysis at tk:

                                                                                       Other Methods
                                                    x k   x k 
                                                         f           a                  OI:     Molcard et al (2003)
                                                     f   a                         4D-Var: Nodet (2006)
                                                    Pk  Pk 


                              
            xF   PFD PDD  R DD             y
                                              1
   xF                                                                   PFF    PFD        PFF PFD 
     a         f        f f     o

                                                               
                                                                            a      a             f   f

   xa    x f    f                                  xD                              F f
                                                     o       f
                                                                          Pa     PDD  k     PDF PDD  k
   D  k  D   P Pf  R o                 
                                              1
                                                                                                     
                                                     D                             a                f
                      DD DD    DD                              k
                                                                            DF
       Application I. Mid-latitude Ocean Circulation:
                     “Proof of Concept”

      nature run (simulated truth)
             ht=500m
        x1000km




                                x1000km




Ocean circulation                         Perfect model scenario
  1-layer shallow-water model               Model spin-up for 12 yrs
  Domain size: 2000km x 2000km               - Nature run (truth) with H0=500m;
  Wind-driven: =0.05 Ns-2                   - Ensemble with (Hmean, Hstd)=(550m,50m)
                                            Drifter released at the beginning of 13 yrs
 Salman, Kuznetsov,Jones, Ide (2006)        observed every day
      Ex.1: ν=500m2s-1, (∆T, LD )=(1day, 1), (Ne, rloc)=(80, ∞)
        Truth                With LaDA            Without DA

T=0




T=90
 days
      Ex.1: ν=500m2s-1, (∆T, LD )=(1day, 1), (Ne, rloc)=(80, ∞)
        Truth                With LaDA            Without DA

T=0




T=90
 days
                       Application II. Gulf of Mexico
                       “Why is the LaDA Efficient?”
Ocean circulation:                        Data assimilation system
Loop-current eddy                           Perfect model scenario
   3 layer shallow-water model with the          Ne =32-1028
   structured curvilinear grid                   LD =2-6
   Horizontal resolution: 5-13km            Initial perturbation in layer depth
   (average 8.3km)
   Vertical resolution: 2 layers            only (velocity determined by
   at 200m, 800m, 2800m                     geostrophic balance)
   Current forcing at 22.4Sv




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Vernieres, Ide, Jones, work in progress
         Motivation for Eddy Tracking


Aug 28                       Aug 28                             Aug 31




           NOAA GOM surface dynamics report for Katrina

         http://www.aoml.noaa.gov/phod/altimetry/katrina1.pdf
Benchmark Case: (Ne, LD)=(1028, 6)

    Analysis          Control



                                Time=0




                                Time=30




                                Time=50 days
Effect of Number of Drifters: LD (Ne=384)




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                Analysis Mechanism: Representer
Analysis equation:
    xF   xF   PFD  f
      a      f        f

                                                  y            
                                                   1
           f    f  PDD  RDD
    xa   x   P 
                                o                        o
                                                            xD
                                                              f

    D
                                                         D
             D       DD



Representer          1 Ne f                                            
                                                                   
                                                                       T
                              xF,n  xF xD,n  xD
                                       f  f      f

             PFD   Ne  1 n1
                f                                                       
    P fHT   f                                                      
             PDD   1 Ne f                                           T

                     N  1 n1
                     e
                                       f  f
                                           
                              xF,n  xF xD,n  xD
                                                 f
                                                                     
                                                                        

                                                       u f       uF,n  
                                                                       f

                                                  Ne 
                                                                    f   f
                                                          F,n
                                                         f 
       f
        
     PFD (u,v,h), (xD ,y D )   
                               ijk , l
                                         
                                             1
                                                    vF,n    vF,n   (xD,n ,y D,n ) l  (xD ,y D )l 
                                           Ne  1 n1  f                      
                                                                                     f           f    f
                                                                                                           
                                                                                                           
                                                        h        h  f 
                                                       F,n  ijk  F,n  ijk 
                                                                              


   rFD (u,v,h), (xD ,y D )ijk , l  normalized PFD (u,v,h), (xD ,y D )ijk , l
     f                                             f
         Convergence of rfFD (h1, xD) vs Ne
Ne=32                        Ne=64




Ne=128                        Ne=256




                             At Day 5, LD =2, No localization
         Convergence of rfFD (h1, xD) vs Ne
Ne=384                       Ne=512




Ne=640                        Ne=1024
Convergence of rfFD (h1, xD) vs Ne : RMS over GoM
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Vertical Impact: (Ne, LD)=(384, 2-4)
      Volume of Influence: Lagrangian vs Eulerian




Green: ∂VL={(i,j,k) | rfFD ((u,v,h), (xD  yD))|ijk,l=1 =0.3}
Red: ∂VE={(i,j,k) | rfFE ((u,v,h), SSH)|ijk,l=1 =0.3}
Volume of Influence: Time Evolution




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Volume of Influence: Vertical Structure
         Remarks for Eddy Tracking in the GOM

LaDA can track the detaching eddy quite effectively
Efficiency can be explored using the representer
  Lagrangian observation has large volume of influence than Eulerian
  observation, both horizontally and vertically
  Maximum impact may not necessarily at the location of the observation




For eddy tracking
  Implicit hypothesis:
                                                       QuickTime™ and a
  observations should be for the                          decompressor
                                                 are need ed to see this picture.

  drifters in the eddy
  Implicit action: deployment of
  the drifters in the eddy
  Elements of Drifter Deployment: Lagrangian Tracers
Lagrangian coherent structures        Drifters (microscopic)
i.e., ocean eddies (macroscopic)
  Collection of tracers that evolve      Individually, tracers can be
  and stay together much longer         entrained into or detrained from
  than the Lagrangian                   the coherent structures across
  autocorrelation time scale            the boundaries
   Working Hypotheses for Optimal Drifter Deployment
Optimal deployment strategy should take into account of
  Evolving Lagrangian coherent structures       (macroscopic view)
  Moving observations by drifters {yoD,l(tk)}   (microscopic view)

Working hypotheses
O For eddy tracking
   Deploy drifters in the eddy
O For estimation of the large-scale flow
   Deploy drifters that spread quickly
     and visit various regions of the
     large-scale flow
O For balanced performance
   Use combination
 Without knowledge of the flow field
   Deploy drifters uniformly or
     based on some intelligent guess,
     and hope for the best
      An Immediate Difficulty for Directed Deployment
Use of these hypotheses requires the evolving Lagrangian info.
How to obtain such information?

  We have the data set of instantaneous Eulerian fields {xF(t)}
  but Lagrangian trajectories don’t follow the instantaneous streamlines


  We can simulate a bunch of drifter
  trajectories {xD(t)}
  but the spaghetti diagram does not
  give cohesive information


  We have the drifter observations {yD(tk)}
  but they are too sparse to give the
  complete Lagrangian flow information and
  give no information for the future
   Drifter Deployment Design: Dynamical Systems Theory
                        “Concept”
  Dynamical systems theory: A tool to analyze Lagrangian dynamics
  given a time sequence of the Eulerian flow fields
      Stable and unstable manifolds = “material boundaries” of the distinct
                                       Lagrangian flow regions

  Instantaneous (Eulerian) field                                        Lagrangian flow template




                                        Dynamical
                                         Systems
                                          Theory


                                   Poje, Haller (1999)
                                   Ide, Small, Wiggins (2002)
                                   Mancho, Small, Wiggins, Ide (2003)
                                   ….

Immediate difficulty”                                             Intermediate difficulty:
  How to get Lagrangian flow template                                How to detect manifolds
Dynamical Systems Theory for Lagrangian Flow Template:
            “Method for Detecting Manifolds”
 Direct Lyapunov Exponents (Finite Time Lyapunov Exponents: FTLE)
   Divergence of the nearby trajectory
                                                            
          x t0  T ; x 0   x 0 ,t0  x t0  T ; x 0 ,t0   x t0  T ; x 0 ,t0   
           x  00      0 0        0 0 0
   FTLE max  x t T ; x ,t  exp  t ; x ,t  x
                                                               
       Day 0                                Day 60                                     Day 110




                                                     Theory:            Haller (2001, 2002), …
                                                     Application to DA: Salman, Ide, Jones (2007)
Lagrangian Flow Template of the Double-Gyre Circulation




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Hypothesis Testing Using the Lagrangian Flow Template
Goal: Given LD, design the “optimal” deployment strategy
Perfect model scenario using the shallow-water model
  Nature run 12yr spin-up with H0=500m; Drifter released at year 13
  Ensemble members with (Hmean, Hstd)=(550m,50m)
EnKF Parameters: (Ne, rloc)=(80, 600km)
LaDA Parameters: (∆T, LD)=(1day, 9)
Deployment strategies:


(a) Uniform   (3x3)
(b) Saddle    (3 saddles: 3 each)
(c) Center    (3 centers: 3 each)
(d) Mixed     (3 centers: 1 each;
               2 saddle: 3 each)



 Salman, Ide, Jones (2007) submitted
Distinctive Drifter Motion by the Deployment Strategies




                            Directed deployment
Convergence of the Basin-Scale Error Norms
          Flow Estimation: Uniform Deployment
          Day 25           Day 100          Day 300

Truth




Uniform
     RMSE Spatial Pattern: Uniform Deployment
       Day 25            Day 100          Day 300

h




KE
         Flow Estimation: Center Deployment
         Day 25           Day 100             Day 300

Truth




Center
     RMSE Spatial Pattern: Center Deployment
       Day 25           Day 100          Day 300

h




KE
         Flow Estimation: Saddle Deployment
         Day 25           Day 100             Day 300

Truth




Saddle
     RMSE Spatial Pattern: Saddle Deployment
       Day 25           Day 100          Day 300

h




KE
         Flow Estimation: Mixed Deployment
        Day 25           Day 100             Day 300

Truth




Mixed
     RMSE Spatial Pattern: Mixed Deployment
      Day 25            Day 100          Day 300

h




KE
               Remarks on Deployment Strategy

Deployment strategy
   It is “targeting” in the Lagrangian flow template hidden in a time sequence
   of Eulerian flow field
   It should most naturally be built on dynamical systems theory


Real Difficulty
   Drifters are to be released in the real ocean {xtF (t)}, while the template is
   build for the model flow field {xfF (t)}
   FTLE computation requires i{xfF (t)} in the past and future, thus
   predictability of both the Eulerian flow and Lagrangian dynamics must be
   taken into account.

Predictability of drifters is doubly-penalized by
   Uncertainty of the Lagrangian dynamics
   Uncertainty of the Eulerian flow field
BUT detection of Lagrangian coherent structures is a robust procedure
                         Summary of LaDA

The Lagrangian data assimilation (LaDA) a natural and effective
method for the direct assimilation of Lagrangian observations such
as drifters

Advantage and efficacy of the LaDA are due to
   Large volume of influence horizontally and vertically
   Mobility

Optimal deployment strategy is intimately related to

   Two aspects of Lagrangian tracers: macroscopic (evolution of fluid body
   as Lagrangian coherent structures) and microscopic (dynamics of
   individual tracers)

   Dynamical systems theory, which offers an ideal vehicle for the optimal
   deployment strategy = targeting in the LaDA
 Future Direction I. Further Development LaDA Method

More realistic applications / situations.

Advancement of the optimal deployment strategy
+ building of Lagrangian analysis and forecasting system

Assimilation of float data (3D Lagrangian observations)

Assimilation of quasi-Lagrangian observation instruments, such as
gliders and Autonomous underwater vehicles (AUVs).

   Deployment strategy: estimation/prediction problems  control problem
Theoretical study of observability of Lagrangian observation vs
Eulerian observation.
  Future Direction II. Atmospheric Applications of LaDA
Vorecore balloons
   http://www.southpoledudes.com/mcmurdo0506/

                                                          T




                                                          z




                                                          u

Hertzog, Basdevant, Viall,
Mechoso (2004)



Cloud feature tracking???
                                                          v
Hurricane tracking???
Future Direction III. Development of ROMS LETKF System



Model:
Regional Ocean Modeling
System (ROMS)




                                Method:
                                Local Ensemble
                                Transform Kalman Filter
                                (LETKF)

								
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