Morphing and wavelet EnKF data assimilation - UC Denverjmandel/slides/psu2011_jm.pdf · Morphing EnKF Spectral and wavelet EnKF Applications Morphing and wavelet EnKF data assimilation

Morphing EnKFSpectral and wavelet EnKF

Applications

Morphing and wavelet EnKF data assimilation

Jan Mandel

Based on joint work with J. D. Beezley, L. Cobb, A. Krishnamurthy, A. K.Kochanski, K. Eben, P. Jurus, and J. Resler

Center for Computational MathematicsDepartment of Mathematical and Statistical Sciences

University of Colorado Denver

Supported by NSF grant AGS-0835579

Penn State, November 3, 2011

Jan Mandel Morphing and wavelet EnKF


Applications

Outline

1 Morphing EnKFData assimilation and EnKFAutomatic image registrationThe morphing transformation

2 Spectral and wavelet EnKFState covariance approximationFFT and waveletsExamples

3 ApplicationsWildfire reaction-diffusion modelWRF coupled with wildfire spreadAssimilation of precipitation radar data to WRF



Applications

Data assimilation and EnKFAutomatic image registrationThe morphing transformation

Data assimilation – continuous Bayesian view

Modelmust support the assimilation cycle: export, modify, andimport statethe state must have metadata: what, when, wheremust support meaningful continuous adjustments to thestate – no discrete datastructures

Datamust have error estimatemust have metadata: what, when, where

Observation functionconnects the data and the modelcreates synthetic data from model state to compare

Data assimilation algorithmadjusts the state to match the databalances the uncertainty in the data and in the stateit is not the purpose to minimize the error



Applications


The Ensemble Kalman Filter (EnKF)

uses an ensemble of simulations to estimate modeluncertainty by sample covarianceconverges to Kalman Filter (optimal filter) in largeensemble limit and the Gaussian caseuses the model as a black boxadjusts the state by making linear combinations ofensemble members (OK, locally in local versions of thefilter, but still only linear combinations)if it cannot match the data by making the linearcombinations, it cannot track the dataprobability distributions close to normal needed for properoperation



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The Ensemble Kalman Filter (EnKF)

X a = X f + K(Y − HX f ), K = P f HT (HP f HT +R)−1

X a: Analysis/PosteriorensembleX f : Forecast/PriorensembleY : Data

K : Kalman gainH: Observation functionP f : Forecast samplecovarianceR: Data covariance

Basic assumptions:

Model and observation function are linearForecast and data distributions are independent andGaussian (if not, EnKF routinely used anyway)



Applications


A simple reaction-diffusion wildfire model

0 200 400 600 800 1000300

400

500

600

700

800

900

1000

1100

1200

Tem

pera

ture

(K)

X (m)

1D temperature profile 2D temperature profile

Solutions produce non-linear traveling waves and thin reactionfronts.



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An example in 2D: non-physical results

Forecast ensemble Data Analysis ensemble

Forecast ensemble generated by random spatialperturbations of the displayed imageAnalysis ensemble displayed as a superposition ofsemi-transparent images of each ensemble memberIdentity observation function, H = IData variance, 100 K



Applications


What went wrong?

The Kalman update formulacan be expressed as

X a = A(X f )T ,

so X ai ∈ spanX f, where the

analysis ensemble is made oflinear combinations of theforecast.

0 500 1000 1500Temperature (K)

Prob

abilit

y de

nsity

Non-Gaussian distribution:Spatial perturbations yieldforecast distributions with twomodes centered aroundburning and non-burningregions.



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Solution: morphing EnKF

(picture Gao & Sederberg 1992)

Need correction of location, not justamplitudeSolution:

Use morphs instead of linearcombinationsDefine morphing transform, carriesexplicit position informationIn the morphing space, probabilitydistributions are much closer toGaussian, standard EnKF succesfullInitial ensemble: smooth randomperturbation of amplitude and location

Applicable to any problem with movingfeatures (error in speed causes error inlocation), not necessarily sharp



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Image morphing

A morphing function, T : Ω→ Ω defines a spatialperturbation of an image, u.It is invertible when (I + T )−1 exists.An image u “morphed” by T is defined asu = u(x + Tx) = u (I + T )(x).

u I + T = u



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Automatic image registration

Goal: Given two images u and v , find an invertible morphingfunction, T , which makes u (I + T ) ≈ v , while ensuring that Tis “small” as possible.

Image registration problem

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

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



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Automatic registration procedure

Avoid getting trapped in local minimaMultilevel method

Start from the coarsest grid and go upOn coarse levels, look for an approximate global match,then refineSmoothing by a Gaussian kernel first to avoid locking thesolution in when some fine features match by an accidentwhile the global match is still poor

On all levelsmap out the solutions space by samplingiterate by steepest descent from the best match



Applications


Minimization by sampling

Probe the solution space by moving thecenter to sample points and evaluating theobjective function and taking the minimum.Morphing function on grid pointsdetermined by some sort of interpolation.Refine the grid and repeat until desiredaccuracy is reached.When using bilinear interpolation,invertibility is guaranteed when all gridquadrilaterals are convex.Smoother interpolation... invertibility morecomplicated



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Grid refinement

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



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Image smoothing

A smoothed temperatureprofile (in blue) withbandwidth 200 m.

Gaussian kernel with bandwidth h

Gh(x) = ch exp−xT x

2h

Smoothing by convolution with Gh(x)improves performance of steepestdescent methods applied to Ju→v (T ).



Applications


The morphing transformation

Augment the state by an explicit information about spacedeformation:

Morphing transformation

Given a reference state u0

Mu0ui =

Ti The registration mapri = ui (I + Ti)

−1 − u0 Residual (of amplitude)

M−1u0

[Ti , ri ] = ui = (u0 + ri) (I + Ti) The inverse transformui,λ = (u0 + λri) (I + λTi) intermediate states for 0 < λ < 1

Linear combinations of [ri ,Ti ] give intermediate states. ApplyMu0 to the ensemble and the data, run the EnKF on thetransformed variables, and apply the inverse transformation toget the analysis ensemble.



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Linear combinations of transformed states are now physicallyrealistic.

The first and the last state are actual simulation states, those inbetween are generated automatically by morphing. Both theposition and the amplitudes are interpolated.



Applications


Morphing transform makes distribution more Gaussian

0 500 1000 1500Temperature (K)

Prob

abilit

y de

nsity

−400 −200 0 200 400Temperature (K)

−150 −100 −50 0 50 100 150Perturbation in X−axis (m)

(a) (b) (c)

Typical pointwise densities near the reaction area of the originaltemperature (a), the residual component after the morphingtransform (b), and (c) the spatial transformation component inthe X-axis. The transformation has made bimodal distributioninto unimodal.



Applications

State covariance approximationFFT and waveletsExamples

Ensemble Kalman Filter (EnKF)

Uses an ensemble of solutions ukNk=1 to estimate modelerrors.

Forecast step:

uk ← M(uk )

Analysis step:

KN ← QNHT (HQNHT + R)−1

uk ← uk + KN (d + ek − Huk ) , ek ∼ N (0,R)

The forecast covariance (Q) is approximated by the samplecovariance (QN ). But, the sample covariance is just anapproximation; why not use a different approximation?



Applications


Sample covariance

u ← 1N

N∑k=1

uk

QN ← 1N − 1

N∑k=1

(uk − u) (uk − u)T

Simple formula to estimate covariance from an ensembleWell known convergence behaviorHas rank at most N − 1Usually creates spurious long range correlations, but canestimate the diagonal well



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Covariance tapering

Artificially eliminate long range correlations using a taperingfunction.

Large samplecovariance (N = 1000)

Small samplecovariance (N = 10)

Tapered small samplecovariance (N = 10)



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Fast Wavelet Transforms

An orthogonal change of basis like the fast Fourier transform,but uses localized waveforms.

Advantages over FFT:

Fast (O(n) rather than O(n lg n))

Can handle localized features

Little to no Gibbs effect fromdiscontinuities

Wide range of waveforms tochoose from for different kinds offeatures

Coiflet octave 2

Coiflet octave 4



Applications


Spectral covariance estimation

Basic idea: Do the sample covariance in spectral space andthrow away off diagonal terms.

uk ← S(uk )

diag QN ← 1N − 1

N∑k=1

(uk − uk )2

QN is the covariance in spectral space. We can convert backto model space for diagnostics.

QN ← S−1(I)QnS(I)

We can handle covariances between multiple variables bycomputing blocks of the covariance independently.



Applications


A simple 1D model

Consider a two variable model with solutions generated by

vk ← hk exp(− (x − ck )2 /w2

k )

hk ∼ N (1.0, 0.12)

ck ∼ N (0.3, 0.12)

wk ∼ N (0.1, 0.012)

wk ← 0.3vk +n∑

j=1

λjk

j2 sinjx2π

λjk ∼ N (0, 1).

vk is a Gaussian bump with random center, width, and height.wk is a smooth field with a small correlation to vk .



Applications


Comparison of covariance estimates

Large sample (N = 1000) Small sample (N = 10)

FFT covariance (N = 10) Wavelet covariance (N = 10)



Applications


Spectral EnKF

EnKF using only the diagonals of the covariances in spectralspace.

For one variable (where C(·, ·) is the diagonal of the samplecovariance):

uk ← S(uk )

Huk ← S(Huk )

K ← C(uk , Huk )(C(Huk , Huk ) + R)−1

uk ← uk + K (S(d) + ek − Huk ), ek ∼ N (0,R)

uk ← S−1(uk )

Each variable in the model is updated individuallyAll computations are done on diagonal matricesFaster than EnKF with localization similar to tapering



Applications


The spectral EnKF—1D model revisited

Comparison using an observation of the first variable. Shown isthe first ensemble member for each method.

EnKF FFT EnKF Wavelet EnKF

v1

w1

Forecast: black dashed line Data: red dotted line Analysis: blue solid lineJan Mandel Morphing and wavelet EnKF


Applications

Wildfire reaction-diffusion modelWRF coupled with wildfire spreadAssimilation of precipitation radar data to WRF

Morphing EnKF for a reaction-diffusion PDE fire model

X (m)

Y (m

)

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X (m)

Y (m

)

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X (m)

Y (m

)

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Data Forecast Analysis

High fire heat areas in propagating surface fire. Forecastensemble members are have their fire areas spread over thesimulation domain. Analysis brings the ensemble closer to thedata and moves the fire areas.



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WRF coupled with wildfire spread



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Data assimilation for the coupled WRF and fire model

Data source No assimilation

Standard EnKF Morphing EnKF



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A real coupled WRF and fire run

Surface fire heat flux shown in Google Earth. Slower fuels keepburning behind the fireline. Data assimilation coming soon.



Applications


Assimilation of precipitation radar data

OPERA radar data givenevery 15 minutes for 20hours over WesternEurope, processed to bedirectly comparable withthe RAIN field.WRF simulation showingincorrect precipitation.The precipitation field inWRF is diagnostic only.Data assimilation must relyon correlation betweenRAIN and other fields.

model

radar



Applications


To test morphing on an operational system:Registration used to compute a morphing function thatmoves the model RAIN field to the radar data.Morphing function applied to standard fields: wind (U, V ,W ), temperature (T ), pressure (PH), humidity (MU). Noamplitude changes yet.Restart model from the morphed WRF output file and runnext 15 minutes.Repeat.



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no morphing radar

small morphing large morphing

Time 00_15 : RAIN at surface



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Conclusions from morphing test:Morphing standard fields over land causes artifacts in themodel due to conservation laws, esp. the large morphing.With this setup, very small modifications are necessary, butit is unclear if it actually helps.Our collaborators will try different combinations of morphedvariables.Need to look at WRF-Var’s change of variables (streamfunctions, etc).Need to look at HWRF’s use of a simple linear map tomove the vortex and change the hurricane size, with aconservation adjustment.



Applications


Other possible applications of morphing

Forecasting in geosciencesprecipitation, storms, squall lineshurricanespollution transportlocation of ocean currents

Forecasting in sociology and political sciencespread of epidemics (pilot project funded by NIH)spread of social networks and memesspread of behavior patterns and social strata (criminality,abuse, gentrification)

Anything where movement of features in space isimportant, especially when the features reinforcethemselves instead of dissipatingWe are looking for applications and collaborators!



Applications


Assimilation of point data (near future)

Our current morphing software is limited to raster data over allor a big part of the domain, such as images. Extension to pointobservations by matching lines in timespace - assimilate intomany time levels at once. Also to handle delayed observations.

Spacetime morping EnKF will match the dotted line - a timeseries of observations at a fixed location - by a deformation of

the space at the analysis time (upper left edge).



Applications


References

J. D. BEEZLEY AND J. MANDEL, Morphing ensemble Kalmanfilters, Tellus, 60A (2008), pp. 131–140.

J. D. BEEZLEY, J. MANDEL, AND L. COBB, Wavelet ensembleKalman filters, in Proceedings of IEEE IDAACS’2011, Prague,September 2011, vol. 2, IEEE, 2011, pp. 514–518.

J. MANDEL, J. D. BEEZLEY, J. L. COEN, AND M. KIM, Dataassimilation for wildland fires: Ensemble Kalman filters incoupled atmosphere-surface models, IEEE Control SystemsMagazine, 29 (2009), pp. 47–65.

J. MANDEL, J. D. BEEZLEY, AND A. K. KOCHANSKI, Coupledatmosphere-wildland fire modeling with WRF 3.3 and SFIRE2011, Geoscientific Model Development, 4 (2011), pp. 591–610.


Morphing and wavelet EnKF data assimilation - UC Denverjmandel/slides/psu2011_jm.pdf · Morphing EnKF Spectral and wavelet EnKF Applications Morphing and wavelet EnKF data assimilation

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