3D assimilaiton update: en route Lagrangian data

Transcription

1 / 3 3D assimilaiton update: en route Lagrangian data Elaine Spiller Amit Apte, Chris Jones Marquette University TIFR CAM, Bangalore and University of North Carolina January 24, 22

2 Outline 2 / 3 Motivation Test problem Observation operator and preliminary results Looking forward

3 Lagrangian instruments Argo float glider Goal collect below-surface measurements to better understand 3D dynamics and structures Lagrangian instruments collect data en route (temperature, pressure, salinity) Observations depend on unknown drifter paths What to do with that data? 3 / 3

4 Float depth profile 4 / 3 Cartoon: Argo float depth profile Cartoon: Argo float depth profile 2 m 7 days 2 m 7 days 7- day float results in O()-O() km traveled high frequency data in dive/ascent just before surfacing in water column beneath surfacing location low frequency en-route measurements at depth, no latitude/longitude information en-route measurements averaged, not used in assimilation

5 Float depth and overview 5 / 3 Cartoon: Argo float depth profile Cartoon: Overview of surfacing locations 2 m 7 days north east Lagrangian DA can help ascertain velocities w/o averaging

6 Float depth and overview 6 / 3 Cartoon: Argo float depth profile Cartoon: Overview of surfacing locations 2 m 7 days north east Some possible Lagrangian paths

7 Float depth and overview 7 / 3 Cartoon: Argo float depth profile Cartoon: Overview of surfacing locations 2 m 7 days north east need path & speed for subsurface observation locations

8 Float depth and overview 8 / 3 Cartoon: Argo float depth profile Cartoon: Overview of surfacing locations 2 m 7 days north east Can en-route observations help Lagrangian DA?

9 Assimilated 3-D Lagrangian paths are (possibly) useful for 9 / 3 aid in resolving Lagrangian structures assimilating data into high resolution models avoiding averaging via determining en-route data collection locales along paths which cross multiple grid cells

10 Depth profile for gliders / 3 Cartoon: glider flight depth profile m 7 hours (~ 5km) roll, pitch with preprogrammed flight plan paths are semi-lagrangian predict path with estimated velocity field and flight plan

11 Assimilating glider paths is (possibly) useful for / 3 figuring out what happened when glider surfaces far from where predicted improving local velocity estimates for planning next flight describing 3-D transport paths like those theorized to exist in the meridional overturning conveyor belt (Lozier, 2) need Lagrangian paths to help encorporate en-route data

12 Assimilating glider paths is (possibly) useful for 2 / 3 figuring out what happened when glider surfaces far from where predicted improving local velocity estimates for planning next flight describing 3-D transport paths like those theorized to exist in the meridional overturning conveyor belt (Lozier, 2) need Lagrangian paths to help encorporate en-route data Can en-route data help Lagrangian DA?

13 Assimilating glider paths is (possibly) useful for 3 / 3 figuring out what happened when glider surfaces far from where predicted improving local velocity estimates for planning next flight describing 3-D transport paths like those theorized to exist in the meridional overturning conveyor belt (Lozier, 2) need Lagrangian paths to help encorporate en-route data Can en-route data help Lagrangian DA? Possibly if gradients are strong

14 Observations and likelihood Observations will be related to the state variable by some observation function y = H(x). (For LaDA H(x) = x d, the instrument s location.) We can think of observations as random variables distributed as Or, Y j = H(X j )+ noise. Y j (X j = x j ) g(y x j ). g(y x) is the likelihood how likely was an observation given the possible states? With a whole set of observations {Y j } we can write down the likelihood for the time-series of observations n p(y :j x :j ) = g(y k x k ) j= 4 / 3

15 Inference: goal for data assimilation Given a background distribution of initial conditions, µ(x ), and observations, Y :n, we want to infer the distribution of physical states X :n. Prior Likelihood p(x :n ) = µ(x o ) p(y :n x :n ) = n m(x j x j ) j= n g(y = H(x j ) x j ) j= Posterior, obtained by Bayes rule p(x :n y :n ) = p(y :n x :n )p(x :n ) p(y :n ) recall, p(y :n ) = p(y :n x :n )p(x :n )dx :n 5 / 3

16 Breakdown of DA schemes: representation of posterior 6 / 3 Sample posterior: particle filter or MCMC handles nonlinear/nongaussianity naturally doesn t scale well as dimension increases Approx posterior as Gaussian: Kalman filter (family) relies on Gaussian/linear assumptions ENKF samples to estimate covariance Find mode of posterior: variational DA what if posterior is multi-modal w/nearly even masses? For all cases, including en-route data changes observation function, H(x), and hence likelihood

17 Test problem: Inviscid linearized Shallow Water Eqns 7 / 3 Non-dimensional velocity fields u t v t h t = v h x = u h y = u x v y Lagrangian trajectories ẋ(t) = u[x(t), y(t), t] ẏ(t) = v[x(t), y(t), t] Decomposition into Fourier Modes u(x, y, t) = 2π sin(2πx) cos(2πy)u o + cos(2πy)u (t) v(x, y, t) = 2π cos(2πx) sin(2πy)u o + cos(2πy)v (t) h(x, y, t) = sin(2πx) sin(2πy)u o + sin(2πy)h (t)

18 Cellular flow field If u = v = h =, flow field is constant & tracers stay w/in cells y x Otherwise, u o =, v = u 2πh, u = v, & ḣ = 2πv with initial conditions [u o (), u (), v (), h ()] 2 u h v time 8 / 3

19 A few trajectories y y y 9 / 3 Left: u () = v () = h () =, x() =.2, y() =.3 Middle: u () = v () = h () =, x() =.2, y() =.3 Right: u () =.2, v () =.3, h () =.4, x() =.5, y() = x x x

20 Particle filter for standard LADA Test problem: u () = v () = h () =, x() =.2, y () =.3 broad priors on (u, v, h ), tight on (x, y ) at t = run to t = T ( period of coefficients) 5 noisy observations of drifter 2 / 3

21 Particle filter for standard LADA Test problem: u () = v () = h () =, x() =.2, y () =.3 broad priors on (u, v, h ), tight on (x, y ) at t = run to t = T ( period of coefficients) 5 noisy observations of drifter Goal: learn about u (), v (), h () from Lagrangian observations 2 / 3

22 Particle filter for standard LADA Test problem: u () = v () = h () =, x() =.2, y () =.3 broad priors on (u, v, h ), tight on (x, y ) at t = run to t = T ( period of coefficients) 5 noisy observations of drifter Goal: learn about u (), v (), h () from Lagrangian observations y v.23 v x.34 u.5 h.5 22 / 3

23 En route Lagrangian data a test problem 23 / 3 Idea treat height, h(x, y, u, v, h ), as proxy for salinity typical quantity measured en route

24 En route Lagrangian data a test problem 24 / 3 Idea treat height, h(x, y, u, v, h ), as proxy for salinity typical quantity measured en route Sample height, ĥ(t) = h(x d (t), y d (t), t) + noise between surfacings, e.g. traditional observation instants t j

25 En route Lagrangian data a test problem 25 / 3 Idea treat height, h(x, y, u, v, h ), as proxy for salinity typical quantity measured en route Sample height, ĥ(t) = h(x d (t), y d (t), t) + noise between surfacings, e.g. traditional observation instants t j Changes the observation space, so now (z = {x d, y d, u, v, h } whole state) { (x d (t), y d (t)) for t = jt obs H(z) = ĥ(t) for t = t k, (j )T obs < t k < jt obs

26 En route Lagrangian data a test problem 26 / 3 Idea treat height, h(x, y, u, v, h ), as proxy for salinity typical quantity measured en route Sample height, ĥ(t) = h(x d (t), y d (t), t) + noise between surfacings, e.g. traditional observation instants t j Changes the observation space, so now (z = {x d, y d, u, v, h } whole state) H(z) = { (x d (t), y d (t)) for t = jt obs ĥ(t) for t = t k, (j )T obs < t k < jt obs Update Likelihood at surfacing time t j with data {xj o, yj o, ĥo k=...n h }

27 En route Lagrangian data a test problem Idea treat height, h(x, y, u, v, h ), as proxy for salinity typical quantity measured en route Sample height, ĥ(t) = h(x d (t), y d (t), t) + noise between surfacings, e.g. traditional observation instants t j Changes the observation space, so now (z = {x d, y d, u, v, h } whole state) H(z) = { (x d (t), y d (t)) for t = jt obs ĥ(t) for t = t k, (j )T obs < t k < jt obs Update Likelihood at surfacing time t j with data {xj o, yj o, ĥo k=...n h } log(g) = (x d x o ) 2 + (y d y o ) 2 2σd 2 + (h(z k N ) ĥo k )2 /2σh 2 h N h 27 / 3

28 Particle filter w/en route observations traditional LADA: y v.23 v x.34 u.5 h.5 28 / 3

29 Particle filter w/en route observations traditional LADA: y v.23 v x.34 u.5 h.5 en route LADA: y x / 3

30 Particle filter w/en route observations traditional LADA: y v.23 v x.34 u.5 h h.5 en route LADA: v y v x.34 u / 3

31 .5.5 v v Improvement with en-route observations u.5 h.5 Characterizing improvement: compare covariance matrices of prior and posterior distribution ratio of traces ratio determinants standard LaDA.46.9 = = w/height obs.5.9 = = robust over numerical experiments similar improvement for saddle case 3 / 3

32 Looking forward 32 / 3 Future directions: 3D model problem, depth one of observed en-route variables include flight plan control in glider problem assimilate likely paths between surfacing locations endpoints pinned assimilate for most likely paths w/brownian bridge suggestions welcome