Morphing ensemble Kalman filter

Transcription

1 Morphing ensemble Kalman filter and applications Center for Computational Mathematics Department of Mathematical and Statistical Sciences University of Colorado Denver Supported by NSF grants CNS and ATM Institute of Computer Science Czech Academy of Sciences Prague, June 3, 2009

2 Outline Data assimilation 1 Data assimilation 2 Wildfire Epidemic spread Other applications 3 structure Available software Future developments

3 Data assimilation Model must support the assimilation cycle: export, modify, and import state the state must be described: what, when, where changes to the state must be meaningful: no discrete datastructures (such as tracers) Data must have error estimate must have metadata: what, when, where Observation function connects the data and the model creates synthetic data from model state to compare Data assimilation algorithm adjusts the state to match the data balances the uncertainty in the data and in the state

4 The Ensemble Kalman Filter () uses an ensemble of simulations to estimate model uncertainty by sample covariance converges to Kalman Filter (optimal filter) in large ensemble limit and the Gaussian case uses the model as a black box adjusts the state by making linear combinations of ensemble members (OK, locally in local versions of the filter, but still only linear combinations) if it cannot match the data by making the linear combinations, it cannot track the data probability distributions close to normal needed for proper operation

5 The Ensemble Kalman Filter () X a = X f + K ( Y HX f ), K = P f H T (HP f H T +R) 1 X a : Analysis/Posterior ensemble X f : Forecast/Prior ensemble Y : Data K : Kalman gain H: Observation function P f : Forecast sample covariance R: Data covariance Basic assumptions: Model and observation function are linear Forecast and data distributions are independent and Gaussian (if not, routinely used anyway)

6 A simple wildfire model Temperature (K) X (m) 1D temperature profile 2D temperature profile Solutions produce non-linear traveling waves and thin reaction fronts.

7 An example in 2D: non-physical results Forecast ensemble Data Analysis ensemble Forecast ensemble generated by random spatial perturbations of the displayed image Analysis ensemble displayed as a superposition of semi-transparent images of each ensemble member Identity observation function, H = I Data variance, 100 K

8 What went wrong? Data assimilation The Kalman update formula can be expressed as X a = A(X f ) T, so Xi a span{x f }, where the analysis ensemble is made of linear combinations of the forecast. Probability density Temperature (K) Non-Gaussian distribution: Spatial perturbations yield forecast distributions with two modes centered around burning and non-burning regions.

9 Solution: morphing (picture Gao & Sederberg 1992) Need correction of location, not just amplitude Solution: Use morphs instead of linear combinations Define morphing transform, carries explicit position information In the morphing space, probability distributions are much closer to Gaussian, standard succesfull Initial ensemble: smooth random perturbation of amplitude and location Applicable to any problem with moving features (error in speed causes error in location), not necessarily sharp

10 Image morphing Data assimilation A morphing function, T : Ω Ω defines a spatial perturbation of an image, u. It is invertible when (I + T ) 1 exists. An image u morphed by T is defined as ũ = u(x + Tx) = u (I + T )(x). u I + T = ũ

11 Automatic image registration Goal: Given two images u and v, find an invertible morphing function, T, which makes u (I + T ) v, while ensuring that T is small as possible. Image registration problem J u v (T ) = u (I + T ) v R + T T min T r R = c R r 2 T T = c T T 2 + c T 2 c R, c T, and c are treated as optimization parameters

12 Automatic registration procedure Avoid trapped in local minima! Multilevel method Start from the coarsest grid and go up On coarse levels, look for an approximate global match, then refine Smoothing by a Gaussian kernel first to avoid locking the solution in when some fine features match by an accident while the global match is still poor On all levels map out the solutions space by sampling iterate by steepest descent from the best match

13 Minimization by sampling Probe the solution space by moving the center to sample points and evaluating the objective function and taking the minimum. Morphing function on grid points determined by some sort of interpolation. Refine the grid and repeat until desired accuracy is reached. When using bilinear interpolation, invertibility is guaranteed when all grid quadrilaterals are convex. Smoother interpolation... invertibility more complicated

14 Grid refinement Data assimilation The objective function need only be calculated locally, within the subgrid, allowing acceptable computational complexity, O(n log n).

15 Image smoothing Data assimilation Gaussian kernel with bandwidth h { G h (x) = c h exp x T } x 2h A smoothed temperature profile (in blue) with bandwidth 200 m. Smoothing by convolution with G h (x) improves performance of steepest descent methods applied to J u v (T ).

16 The morphing transformation Augment the state by an explicit information about space deformation: Morphing transformation Given a reference state u 0 { Ti The registration map M u0 u i = r i = u i (I + T i ) 1 u 0 Residual (of amplitude) u 0 [T i, r i ] = u i = (u 0 + r i ) (I + T i ) The inverse transform u i,λ = (u 0 + λr i ) (I + λt i ) intermediate states for 0 < λ < 1 M 1 Linear combinations of [r i, T i ] give intermediate states. Apply M u0 to the ensemble and the data, run the on the transformed variables, and apply the inverse transformation to get the analysis ensemble.

17 Linear combinations of transformed states are now physically realistic.

18 Morphing Transform Makes Distribution Closer to Gaussian Probability density Temperature (K) Temperature (K) Perturbation in X axis (m) (a) (b) (c) Typical pointwise densities near the reaction area of the original temperature (a), the residual component after the morphing transform, and (c) the spatial transformation component in the X-axis. The transformation has made bimodal distribution into unimodal.

19 Wildfire Epidemic spread Other applications Reaction-diffusion PDE fire model Y (m) Y (m) Y (m) X (m) X (m) X (m) Data Forecast Analysis

20 Data assimilation Wildfire Epidemic spread Other applications WRF-Fire: fireline propagation coupled with weather Data source No assimilation Standard

21 Wildfire Epidemic spread Other applications Epidemic spreads in waves similar to wildfire Proposal with Loren Cobb to National Institute of Health just before the swine flu epidemic, publicity on ABC TV news

22 Wildfire Epidemic spread Other applications Other possible applications in future Forecasting in geosciences precipitation, storms, squall lines position of hurricane vortex pollution transport location of ocean currents Forecasting in sociology and political science spread of social networks and memes improve accuracy of election polls Anything where movement of features in space is important We are looking for applications and collaborators!

23 architecture Data assimilation structure Available software Future developments Separate executables communicate by NETCDF files model observation function morphing transform Avoid conflicts of software requirements when building the executables NETCDF files contain metadata: names of variables, units, dimensions, descriptions,... Arrays to operate on are selected by text files (namelists) Run from scripts without recompilation for different problems, create the namelists on the fly

24 Parallel software structure structure Available software Future developments CPU CPU State State Ensemble member 1 Fire - atmosphere model State Observation function Synthetic data Morphing Ensemble Kalman filter State CPU CPU State State Ensemble member N Fire - atmosphere model State Observation function Synthetic data Parallel linear algebra Morphing State Advancing the ensemble in time Data assimilation Real data pool

25 Available software Data assimilation structure Available software Future developments Morphing - available now Automatic registration Morphing transform Smooth random perturbation to general an initial ensemble - coming soon Based on massively parallel scalable linear algebra Observation and model interface DART compatible Built on existing packages as much as possible - ScaLAPACK, FFTW,... Contemporary sofware engineering for flexibility and easy maintenance Free open source licensing

26 structure Available software Future developments Assimilation of point data (near future) Our current morphing software is limited to raster data over all or a big part of the domain, such as images. Extension to point observations by matching lines in timespace - assimilate into many time levels at once. Also to handle delayed observations. Spacetime morping will match the dotted line - a time series of observations at a fixed location - by a deformation of the space at the analysis time (upper left edge).

27 References Data assimilation structure Available software Future developments J. D. BEEZLEY, High-dimensional data assimilation and morphing ensemble Kalman filters with applications in wildfire modeling. Ph.D. Thesis, J. D. BEEZLEY AND J. MANDEL, Morphing ensemble Kalman filters, Tellus, 60A (2008), pp J. MANDEL, J. D. BEEZLEY, J. L. COEN, AND M. KIM, Data assimilation for wildland fires: Ensemble Kalman filters in coupled atmosphere-surface models, IEEE Control Systems Magazine, (2009), pp J. MANDEL, L. COBB, AND J. D. BEEZLEY, On the convergence of the ensemble Kalman filter. arxiv: , 2009.