Morphing ensemble Kalman filter and applications Jan Mandel and - - PowerPoint PPT Presentation

Data assimilation Applications Software

Morphing ensemble Kalman filter

and applications Jan Mandel and Jonathan D. Beezley

Center for Computational Mathematics Department of Mathematical and Statistical Sciences University of Colorado Denver Supported by NSF grants CNS-0623983 and ATM-0719641 Institute of Computer Science Czech Academy of Sciences

Prague, June 3, 2009

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software

Outline

1

Data assimilation EnKF Registration and morphing

2

Applications Wildfire Epidemic spread Other applications

3

Software Software structure Available software Future developments

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

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

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

The Ensemble Kalman Filter (EnKF)

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

• peration

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

The Ensemble Kalman Filter (EnKF) X a = X f + K

• Y − HX f

, K = PfHT(HPfHT +R)−1 X a: Analysis/Posterior ensemble X f: Forecast/Prior ensemble Y: Data K: Kalman gain H: Observation function Pf: Forecast sample covariance R: Data covariance Basic assumptions: Model and observation function are linear Forecast and data distributions are independent and Gaussian (if not, EnKF routinely used anyway)

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

A simple wildfire model

200 400 600 800 1000 300 400 500 600 700 800 900 1000 1100 1200 Temperature (K) X (m)

1D temperature profile 2D temperature profile Solutions produce non-linear traveling waves and thin reaction fronts.

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

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

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

What went wrong?

The Kalman update formula can be expressed as X a = A(X f)T, so X a

i ∈ span{X f}, where the

analysis ensemble is made of linear combinations of the forecast.

500 1000 1500 Temperature (K) Probability density

Non-Gaussian distribution: Spatial perturbations yield forecast distributions with two modes centered around burning and non-burning regions.

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

Solution: morphing EnKF

(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 EnKF 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

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

Image morphing

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 ˜ u = u(x + Tx) = u ◦ (I + T)(x). u

• I + T

= ˜ u

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

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 Ju→v(T) = ||u ◦ (I + T) − v||R + ||T||T → min

T

||r||R = cR||r||2 ||T||T = cT ||T||2 + c∇||∇T||2 cR, cT , and c∇ are treated as optimization parameters

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

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

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

Minimization by sampling

Probe the solution space by moving the center to sample points and evaluating the

• bjective 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

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

Grid refinement

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

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

Image smoothing

A smoothed temperature profile (in blue) with bandwidth 200 m. Gaussian kernel with bandwidth h Gh(x) = ch exp

• −xTx

2h

• Smoothing by convolution with Gh(x)

improves performance of steepest descent methods applied to Ju→v(T).

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

The morphing transformation

Augment the state by an explicit information about space deformation: Morphing transformation Given a reference state u0 Mu0ui = Ti The registration map ri = ui ◦ (I + Ti)−1 − u0 Residual (of amplitude) M−1

u0 [Ti, ri] = ui = (u0 + ri) ◦ (I + Ti)

The inverse transform ui,λ = (u0 + λri) ◦ (I + λTi) intermediate states for 0 < λ < 1 Linear combinations of [ri, Ti] give intermediate states. Apply Mu0 to the ensemble and the data, run the EnKF on the transformed variables, and apply the inverse transformation to get the analysis ensemble.

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

Linear combinations of transformed states are now physically realistic.

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software EnKF Registration and morphing

Morphing Transform Makes Distribution Closer to Gaussian

500 1000 1500 Temperature (K) Probability density −400 −200 200 400 Temperature (K) −150 −100 −50 50 100 150 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.

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software Wildfire Epidemic spread Other applications

Reaction-diffusion PDE fire model

X (m) Y (m) 100 200 300 400 500 100 200 300 400 500 X (m) Y (m) 100 200 300 400 500 100 200 300 400 500 X (m) Y (m) 100 200 300 400 500 100 200 300 400 500

Data Forecast Analysis

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software Wildfire Epidemic spread Other applications

WRF-Fire: fireline propagation coupled with weather

Data source No assimilation Standard EnKF Morphing EnKF

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software 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

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software 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!

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software Software structure Available software Future developments

Software architecture

Separate executables communicate by NETCDF files

model

• bservation function

EnKF 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

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software Software structure Available software Future developments

Parallel software structure

CPU Real data pool CPU

…

State Synthetic data Fire - atmosphere model Observation function State State

… Ensemble member 1

Morphing Parallel linear algebra

Advancing the ensemble in time Data assimilation

State Synthetic data Fire - atmosphere model Observation function State State

Ensemble member N Ensemble Kalman filter

CPU CPU Morphing State State

…

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software Software structure Available software Future developments

Available software

Morphing - available now

Automatic registration Morphing transform Smooth random perturbation to general an initial ensemble

EnKF - 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

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software Software structure Available software Future developments

Assimilation of point data (near future)

Our current morphing software is limited to raster data over all

• r a big part of the domain, such as images. Extension to point
• bservations by matching lines in timespace - assimilate into

many time levels at once. Also to handle delayed observations. Spacetime morping EnKF 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).

Jan Mandel and Jonathan D. Beezley Morphing EnKF

Data assimilation Applications Software Software structure Available software Future developments

References

• J. D. BEEZLEY, High-dimensional data assimilation and

morphing ensemble Kalman filters with applications in wildfire modeling. Ph.D. Thesis, 2009.

• J. D. BEEZLEY AND J. MANDEL, Morphing ensemble

Kalman filters, Tellus, 60A (2008), pp. 131–140.

• 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. 47–65.

• J. MANDEL, L. COBB, AND J. D. BEEZLEY, On the

convergence of the ensemble Kalman filter. arxiv:0901.2951, 2009.

Jan Mandel and Jonathan D. Beezley Morphing EnKF