Optimal Inference of Velocity Fields

Transcription

1 Optimal Inference of Velocity Fields NJIT EXTREEMS- Mentor: Richard Moore January 30, 2014

2 What is data assimilation? Data assimilation: The incorporation of observational data into a physical model of a system to provide optimal estimates of the system s state and uncertainty. MERRA reanalysis, NASA GMAO

3 Why is necessary? We blend models with data to filter, smooth, or predict the state of a system that is too complex to get a handle on using just the data or just the model, because Model runs at required resolution/timescales too expensive Model parameters only known in sense of distribution Observational measurements have error and are not at locations/times needed Observational measurements may not be of QOIs Examples of : Numerical weather prediction Climate models (e.g., NASA GISS ModelE, NCAR CESM) Management of species and natural resources Navigation/control (e.g., pursuit) Economic models

4 Techniques for Variational methods Minimize an objective (cost) function (e.g., least squares) Hard or soft constraints reflect assumptions about statistical distributions, model behavior, etc. Probabilistic methods Bayes rule Maximum (a posteriori) likelihood estimator Ensemble methods Several random realizations of state generated to compute mean/variance of Gaussian distribution Particle filters to compute moments of nonparametric distribution

5 Using gliders to infer ocean state Underwater autonomous vehicles (gliders) are being used to measure various ocean parameters, including: temperature, pressure, salinity biological species markers (e.g., chlorophyll fluorescence) flow velocity (directly or indirectly) The Economist, 2012

6 How gliders work These gliders can: communicate with shore station when at surface steer (by banking or using rudder) propel forward (0.5 m/s) using buoyancy-driven locomotion NOAA IOOS

7 (a) Spray gliders (b) Slocum gliders Figure ASAP Experiment, Monterey Bay, CA Location of glider profiles collected during the 2006 ASAP field experiment. (a) Spray gliders collected profiles primarily on the boundary of the ASAP domain. Profiles north or west of the domain were collected during large, current-induced deviations from the desired track. Profiles collected along the modified domain boundary are contained in a gray ellipse marked with an arrow. (b) Slocum gliders collected profiles inside the mapping domain and on its boundary. Profiles collected over the canyon head are contained in a gray ellipse marked with an arrow m/s m/s Lat (deg) Lat (deg) Lon (deg) Lon (deg) (a) 17:00 GMT August 8 (b) 00:00 GMT August 11 Figure6 9. Slocum Snapshots of ocean gliders, flow as computed 4 Spray from glider depth-averaged gliders flow estimates using OA. (a) Flow transition from relaxation to upwelling advected gliders out of the mapping domain; (b) equatorward flow indicative of an upwelling. (Mostly) autonomously coordinated for 24 days in the western corner, or how to direct the gliders around the offending current, was the subject of discussions among all the team members, but no solution was found until the current weakened Results of Slocum Glider Operations Strong and highly variable flow conditions such as the ones shown in Figure 9 presented a major challenge to steering the gliders along their assigned tracks with the Control formulated to minimize objective function Journal of Field Robotics DOI /rob Leonard et al., J. Field Robot. 2010

8 2006 ASAP Experiment, Monterey Bay, CA 726 Journal of Field Robotics 2010 (a) GCT 2 (b) GCCS planner panel, July 30, 2006, at 23:10 GMT Used Figure objective 5. (a) GCT 2 defines analysis a coordinated pattern for for. the four Slocum Global gliders, withoptimization the pair we08 and we10 to move over on opposites of the north track, the pair we09 and we12 on opposites of the middle track, and the two pairs synchronized on their respective tracks. Glider we07 should move independently around the south track (the sixth glider had not yet been deployed). The dashed lines show the superelliptical tracks, the circles show a snapshot of the glider positions, and the color coding defines each glider s track assignment. The thin gray lines show the feedback interconnection topology for coordination (all but we07 respond to each other), and the arrows show the prescribed direction of rotation for the gliders. (b) Several real-time status and assessment figures, for scalar field f (x, t): movies, and logs were updated regularly on the Glider Planner and Status page (Princeton University, 2006a). Shown here is a snapshot of one of the panels, which was updated every minute. It presents, for each glider, surfacings over the previous 12 h (squares), waypoints expected to be reached before the next surfacing (gray triangle), next predicted surfacing (gray circle with red fill), new waypoints over the next 6 h (blue tj+1 triangles), and planned position in 24 h (hollow red circle). Each glider is identified with a label at the planned position in 24 h. restricted set of trajectories to minimize forecast mapping error 1 B Design and Local Optimization of GCTs A desired motion pattern for the fleet of gliders under GCCS control is specified as a set of glider coordinated trajectories (GCT). A GCT has three main components, all contained in an XML file and used as input to the GCCS (Princeton University, 2006c). The first component is the operatinget domain, al., J. which Fieldspecifies Robot. the shape, 2010 location, size, Leonard and orientation of the region where the gliders operate. B dx t j dt E[(f (x, t) ˆf (x, t)) 2 ] Adaptations to sampling plans were implemented by switching to a new GCT. In the case of a switch of GCT, the GCCS would be manually interrupted, the new GCT file swapped for the old one, and then the GCCS restarted. The Princeton Glider Planner and Status page (Princeton University, 2006a), linked to the VCR, was consulted for determining adaptations as it maintained up-to-date maps of glider positions and GCCS planning, OA predicted cur-

9 2006 ASAP Experiment, Monterey Bay, CA More details: Used 3d circulation models for and optimization After computing optimal trajectories, gliders were controlled to their trajectories using feedback Additional feedback terms were used to keep gliders from bunching together Optimal trajectories switched manually every 2 days, gliders communicated (i.e., feedback applied) upon surfacing every 3 hours Spray gliders measured velocity field directly using acoustic Doppler profilers Slocum gliders provided velocity estimates indirectly by comparing measured displacement to dead reckoning estimate

10 Alternative approach Rather than globally optimizing the mapping error over a limited set of trajectories, we can consider locally optimal minimizers obtained using the calculus of variations. This might be: quicker less restrictive (i.e., more optimal) less prone to getting stuck in regions where the flow is stronger than the propulsion We will explore simple inference problems (of my choosing) using an unknown scalar field and a known velocity field, to develop heuristics for controlling autonomous observers toward regions of high uncertainty. For simplicity, we will use the Kalman filter as a basis for.

11 Setup Goal: to infer unknown scalar field f (x) on periodic domain. Observations: K gliders, each taking observations at times t j, j = 1,..., J: y k j = f (x k (t j )) + η k j, k = 1,..., K where gliders evolve between observations according to dx k dt = v(x k ) + u k (t) and η is uncorrelated Gaussian noise, with E[η k j ηm l ] = σ 2 δ jl δ km. have maximum relative speed, u k (t) u max. How to choose u k to learn f quickly?

12 1d Example: v(x) = sin 2x, f (x) = x sin 2 3x

13 Objectives of this project We will learn about: 1 The Kalman filter 2 The calculus of variations and optimal control 3 Monte Carlo simulations (maybe) And we will use these to: 1 Formulate the inference problem for simple 1d flows 2 Formulate the optimal control problem for inferring scalar fields with minimal variance 3 Solve or approximate the controlled problem 4 Test our ideas in MATLAB And if we re lucky/good we will also: 1 Incorporate time-evolution in the flow model 2 Parallelize our code using MATLAB

14 Useful resources (will be posted on dropbox) A. E. Bryson, Applied Optimal Control: Optimization, Estimation and Control (CRC Press, 1975). W. CHEN, E. LUNASIN, REU report, U. Michigan (2013). A. H. Jazwinski, Stochastic Processes and Filtering Theory (Courier Dover Publications, 2007). N. E. Leonard, et al., Proceedings of the IEEE 95, 48 (2007). P. F. Lermusiaux, Physica D: Nonlinear Phenomena 230, 172 (2007). S. K. Park, L. Xu, Data Assimilation for Atmospheric, Oceanic and Hydrological Applications (Vol. II) (Springer, Dordrecht, 2013), first edn. Glider observing assets map. URL asset_map.html. SCCOOS spray gliders. URL J. Anderson, et al., Bulletin of the American Meteorological Society 90, 1283 (2009).