Morphing and wavelet EnKF data assimilation Jan Mandel Based on - - PowerPoint PPT Presentation

Morphing EnKF Spectral 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 Mathematics Department 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

Morphing EnKF Spectral and wavelet EnKF Applications

Outline

1

Morphing EnKF Data assimilation and EnKF Automatic image registration The morphing transformation

2

Spectral and wavelet EnKF State covariance approximation FFT and wavelets Examples

3

Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

Data assimilation – continuous Bayesian view

Model

must support the assimilation cycle: export, modify, and import state the state must have metadata: what, when, where must support meaningful continuous adjustments to the state – no discrete datastructures

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 it is not the purpose to minimize the error

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

A simple reaction-diffusion 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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

Automatic registration procedure

Avoid getting 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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

Grid refinement

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

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

Linear combinations of transformed states are now physically realistic. The first and the last state are actual simulation states, those in between are generated automatically by morphing. Both the position and the amplitudes are interpolated.

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Data assimilation and EnKF Automatic image registration The morphing transformation

Morphing transform makes distribution more 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 (b), and (c) the spatial transformation component in the X-axis. The transformation has made bimodal distribution into unimodal.

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications State covariance approximation FFT and wavelets Examples

Ensemble Kalman Filter (EnKF)

Uses an ensemble of solutions {uk}N

k=1 to estimate model

errors. 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 sample covariance (QN). But, the sample covariance is just an approximation; why not use a different approximation?

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications State covariance approximation FFT and wavelets Examples

Sample covariance

u ← 1 N

N

• k=1

uk QN ← 1 N − 1

N

• k=1

(uk − u) (uk − u)T

Simple formula to estimate covariance from an ensemble Well known convergence behavior Has rank at most N − 1 Usually creates spurious long range correlations, but can estimate the diagonal well

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications State covariance approximation FFT and wavelets Examples

Covariance tapering

Artificially eliminate long range correlations using a tapering function.

Large sample covariance (N = 1000) Small sample covariance (N = 10) Tapered small sample covariance (N = 10)

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications State covariance approximation FFT and wavelets Examples

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 from discontinuities Wide range of waveforms to choose from for different kinds of features Coiflet octave 2 Coiflet octave 4

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications State covariance approximation FFT and wavelets Examples

Spectral covariance estimation

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

˜ uk ← S(uk) diag ˜ QN ← 1 N − 1

N

• k=1

( ˜ uk − ˜ uk)2

˜ QN is the covariance in spectral space. We can convert back to model space for diagnostics.

QN ← S−1(I) ˜ QnS(I)

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

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications State covariance approximation FFT and wavelets Examples

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 sin jx 2π 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.

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications State covariance approximation FFT and wavelets Examples

Comparison of covariance estimates

Large sample (N = 1000) Small sample (N = 10) FFT covariance (N = 10) Wavelet covariance (N = 10) Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications State covariance approximation FFT and wavelets Examples

Spectral EnKF

EnKF using only the diagonals of the covariances in spectral space. For one variable (where C(·, ·) is the diagonal of the sample covariance): ˜ 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 individually All computations are done on diagonal matrices Faster than EnKF with localization similar to tapering

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications State covariance approximation FFT and wavelets Examples

The spectral EnKF—1D model revisited

Comparison using an observation of the first variable. Shown is the first ensemble member for each method. EnKF FFT EnKF Wavelet EnKF v1 w1

Forecast: black dashed line Data: red dotted line Analysis: blue solid line

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

Morphing EnKF for a 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 High fire heat areas in propagating surface fire. Forecast ensemble members are have their fire areas spread over the simulation domain. Analysis brings the ensemble closer to the data and moves the fire areas.

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

WRF coupled with wildfire spread

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

Data assimilation for the coupled WRF and fire model

Data source No assimilation Standard EnKF Morphing EnKF

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

A real coupled WRF and fire run

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

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

Assimilation of precipitation radar data

OPERA radar data given every 15 minutes for 20 hours over Western Europe, processed to be directly comparable with the RAIN field. WRF simulation showing incorrect precipitation. The precipitation field in WRF is diagnostic only. Data assimilation must rely

• n correlation between

RAIN and other fields.

model radar

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

To test morphing on an operational system: Registration used to compute a morphing function that moves the model RAIN field to the radar data. Morphing function applied to standard fields: wind (U, V, W), temperature (T), pressure (PH), humidity (MU). No amplitude changes yet. Restart model from the morphed WRF output file and run next 15 minutes. Repeat.

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 00_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 00_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 00_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 01_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 01_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 01_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 01_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 02_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 02_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 02_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 02_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 03_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 03_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 03_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 03_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 04_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 04_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 04_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 04_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 05_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 05_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 05_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 05_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 06_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 06_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 06_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 06_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 07_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 07_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 07_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 07_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 08_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 08_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 08_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 08_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 09_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 09_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 09_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 09_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 10_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 10_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 10_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 10_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 11_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 11_15 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 11_30 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 11_45 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

no morphing radar small morphing large morphing

Time 12_00 : RAIN at surface Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

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

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

Other possible applications of morphing

Forecasting in geosciences

precipitation, storms, squall lines hurricanes pollution transport location of ocean currents

Forecasting in sociology and political science

spread of epidemics (pilot project funded by NIH) spread of social networks and memes spread of behavior patterns and social strata (criminality, abuse, gentrification)

Anything where movement of features in space is important, especially when the features reinforce themselves instead of dissipating We are looking for applications and collaborators!

Jan Mandel Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

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 Morphing and wavelet EnKF

Morphing EnKF Spectral and wavelet EnKF Applications Wildfire reaction-diffusion model WRF coupled with wildfire spread Assimilation of precipitation radar data to WRF

References

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

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

• J. D. BEEZLEY, J. MANDEL, AND L. COBB, Wavelet ensemble

Kalman 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, Data

assimilation for wildland fires: Ensemble Kalman filters in coupled atmosphere-surface models, IEEE Control Systems Magazine, 29 (2009), pp. 47–65.

• J. MANDEL, J. D. BEEZLEY, AND A. K. KOCHANSKI, Coupled

atmosphere-wildland fire modeling with WRF 3.3 and SFIRE 2011, Geoscientific Model Development, 4 (2011), pp. 591–610.

Jan Mandel Morphing and wavelet EnKF