Lagrangian Data Assimilation: Optimal Drifter Deployment Strategy

1. Lagrangian Data Assimilation:Method, Applications, andStrategy for Optimal Drifter Deployment Kayo Ide, UCLA C.K.R.T. Jones, Guillaume Vernieres,UNC-CH Hayder Salman, Cambridge U. Liyan Liu, NCEP

2. 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

3. 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.whoi.edu/instruments/ http://www.dosits.org/gallery/tech/ooct/rafos1.htm

4. 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

5. 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 (ROMS) • One-way nested configuration with increasing resolution for smaller domain • COAMPS forcing • Method: Incremental 3D-Var • Weak constraints by dynamic balance • Inhomogeneous / anistropic background error covariance using Kronecker product Li, Chao, McWilliams, Ide (2007a,b)

6. 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/

7. 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

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

9. 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 • Ocean observations are precious • New types of observations: SSS by Satellite, Coastal HF radar • New technology for cost effectiveness: Lagrangian data

10. Ocean Observation: Remote-Sensing by HF Radar Coastal Oceans Currents Monitoring Program (COCMP) http://www.cocmp.org/ http://www.cencoos.org/currents

11. Lagrangian Dynamics of Drifters Data available from http://www.aoml.noaa.gov/phod/dac/dacdata.html

12. 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

13. Basic Elements of Lagrangian Data Assimilation System Eulerian Model: State xF Lagrangian Observation: Location yD Data Assimilation Method

14. Data Assimilation Method: Kalman-Filter Approach Forecast from tk-1 to tk: Observation at tk: Analysis at tk:

15. Elements of Assimilating Lagrangian Data Essence of analysis in data assimilation • Elements in hands • Forecast flow statexF asxf • Lagrangian observation yD as yo • Missing elements • h that gives yfD from xfF , because nothing inxfF directly relates toyoD • Pfthat gives K for optimal impact of yoon xa

16. Assimilation of Lagrangian Data: Conventional Method Transform from Lagrangian data to Eulerian (velocity) data Observation operator Hv is linear spatial interpolation. Feedback the mismatch of observation (innovation) into the model variable Carter (1989) Kamachi, O’Brien (1995) Tomassini, Kelly, Saunders (1999)

17. Assimilation of Lagrangian Data: Conventional Method Transform from Lagrangian data to Eulerian (velocity) data Observation operator Hv is linear spatial interpolation. Feedback the mismatch of observation (innovation) into the model variable Carter (1989) Kamachi, O’Brien (1995) Tomassini, Kelly, Saunders (1999)

18. Assimilation of Lagrangian Data: Conventional Method Transform from Lagrangian data to Eulerian (velocity) data Observation operator Hv is linear spatial interpolation. Feedback the mismatch of observation (innovation) into the model variable Carter (1989) Kamachi, O’Brien (1995) Tomassini, Kelly, Saunders (1999)

19. Assimilation of Lagrangian Data: Conventional Method Transform from Lagrangian data to Eulerian (velocity) data Observation operator Hv is linear spatial interpolation. Feedback the mismatch of observation (innovation) into the model variable Carter (1989) Kamachi, O’Brien (1995) Tomassini, Kelly, Saunders (1999)

20. Lagrangian Data Assimilation (LaDA) Method • Elements in hands • Augmented state x and model m • Flow statexF and model mF • Drifter statexD and model mF xF xD • Observation yD and operator h that relates yD to x Ide, Jones, Kuznetsov (2002) [Ide and Ghil (1997)] • Missing elements • Pfthat gives K for optimal impact of yoon xa

21. Lagrangian Data Assimilation (LaDA) Method • Elements in hands • Augmented state x and model m • Flow statexF and model mF • Drifter statexD and model mF • Observation yD and operator h that relates yD to x Ide, Jones, Kuznetsov (2002) [Ide and Ghil (1997)] • Missing elements • Pfthat gives K for optimal impact of yoon xa

22. Lagrangian Data Assimilation (LaDA) Method • Elements in hands • Augmented state x and model m • Flow statexF and model mF • Drifter statexD and model mF • Observation yD and operator h that relates yD to x Ide, Jones, Kuznetsov (2002) [Ide and Ghil (1997)] • Missing elements • Pfthat gives K for optimal impact of yoon xa

23. Ensemble-Based Data Assimilation • Use of ensemble to represent the uncertainty of x in particular, mean and covariance • mean • covariance

24. Ensemble-Based LaDA xD • Ensemble forecast from tk-1to tk for n = 1,…, Ne 2. Ensemble update at tkto incorporate and xF Analysis (dropping tk) Salman, Kuznetsov,Jones, Ide (2006) Salman, Ide, Jones (2007)

25. Ensemble-Based LaDA xD • Ensemble forecast from tk-1to tk for n = 1,…, Ne 2. Ensemble update at tkto incorporate and xF Analysis (dropping tk)

26. Ensemble-Based LaDA xD • Ensemble forecast from tk-1to tk yD for n = 1,…, Ne 2. Ensemble update at tkto incorporate and xF Analysis (dropping tk)

27. Ensemble-Based LaDA xD • Ensemble forecast from tk-1to tk yD for n = 1,…, Ne 2. Ensemble update at tkto incorporate and xF Analysis (dropping tk)

28. Ensemble-Based LaDA xD • Ensemble forecast from tk-1to tk for n = 1,…, Ne 2. Ensemble update at tkto incorporate and xF Analysis (dropping tk)

29. Mechanisms of Lagrangian Data Assimilation (LaDA) Forecast from tk-1 to tk: Observation at tk: Analysis at tk: • Other Methods • OI: Molcard et al (2003) • 4D-Var: Nodet (2006)

30. Application I. Mid-latitude Ocean Circulation: “Proof of Concept” • Ocean circulation • 1-layer shallow-water model • Domain size: 2000km x 2000km • Wind-driven: =0.05 Ns-2 • nature run(simulated truth) • ht=500m x1000km x1000km • Perfect model scenario • Model spin-up for 12 yrs • - Nature run (truth) with H0=500m; • Ensemble with (Hmean, Hstd)=(550m,50m) • Drifter released at the beginning of 13 yrs observed every day Salman, Kuznetsov,Jones, Ide (2006)

31. Ex.1: ν=500m2s-1, (∆T, LD )=(1day, 1), (Ne, rloc)=(80, ∞) Truth With LaDA Without DA T=0 T=90 days

32. Ex.1: ν=500m2s-1, (∆T, LD )=(1day, 1), (Ne, rloc)=(80, ∞) Truth With LaDA Without DA T=0 T=90 days

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

34. 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

35. Benchmark Case: (Ne, LD)=(1028, 6) Analysis Control Time=0 Time=30 Time=50 days

36. Effect of Number of Drifters: LD (Ne=384)

37. Analysis Mechanism: Representer • Analysis equation: Representer

38. Convergence of rfFD (h1, xD) vs Ne Ne=32 Ne=64 Ne=128 Ne=256 AtDay 5, LD =2, No localization

39. Convergence of rfFD (h1, xD) vs Ne Ne=384 Ne=512 Ne=640 Ne=1024

40. Convergence of rfFD (h1, xD) vs Ne : RMS over GoM

41. Vertical Impact: (Ne, LD)=(384, 2-4)

42. 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}

43. Volume of Influence: Time Evolution

44. Volume of Influence: Vertical Structure

45. 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: observations should be for the drifters in the eddy • Implicit action: deployment of the drifters in the eddy

46. Elements of Drifter Deployment: Lagrangian Tracers • Drifters (microscopic) • Individually, tracers can be entrained into or detrained from the coherent structures across the boundaries • Lagrangian coherent structures i.e., ocean eddies (macroscopic) • Collection of tracers that evolve and stay together much longer than the Lagrangian autocorrelation time scale

47. 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 • For eddy tracking  Deploy drifters in the eddy • For estimation of the large-scale flow  Deploy drifters that spread quickly and visit various regions of the large-scale flow • 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 O O O 

48. 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)} • butLagrangian 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

49. 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

50. Dynamical Systems Theory for Lagrangian Flow Template: “Method for Detecting Manifolds” • Direct Lyapunov Exponents (Finite Time Lyapunov Exponents: FTLE) • Divergence of the nearby trajectory • FTLE Day 0 Day 60 Day 110 • Theory: Haller (2001, 2002), … • Application to DA: Salman, Ide, Jones (2007)