Functional Data Assimilation with White-Noise Data Error and - - PowerPoint PPT Presentation

Functional Data Assimilation with White-Noise Data Error and Applications to Assimilation of Active Fires Satellite Detection Data

Jan Mandel, University of Colorado Denver James Haley, University of Colorado Denver Ivan Kasanicky, Czech Academy of Sciences Adam K. Kochanski, University of Utah Martin Vejmelka, AVAST Supported by NASA grant NNX13AH59G and NSF grant DMS-1216481 October 1, 2017 The 3rd Annual Meeting of SIAM Central States Section Colorado State University, Fort Collins, CO

Bayes theorem in infinite dimension

Forecast and analysis are probability distributions Bayes theorem: pa (u) ∝ p (d|u) pf (u) No Lebesgue measure, no densities. Integrate over an arbitrary measurable set A instead:

Z

A

pa (u) du ∝

Z

A

p (d|u) pf (u) du µa (A) ∝

Z

A

p (d|u) dµf (u) Data likelihood is the Radon-Nikodym derivative: p (d|u) ∝ dµa dµf

• Normalize: µa (A) =

R

A p(d|u)dµf(u)

R

V p(d|u)dµf(u)

• But how do we know that

R

V

p (d|u) dµf (u) > 0?

Infinite-dimensional data, Gaussian measure error bad

• The simplest example: µf = N (0, Q), H = I, d = 0, R = Q, U = V.

The whole state is observed, data error distribution = state error

• distribution. Come half-way? Wrong.
• p (d|u) = const e−1

2|u|2 R−1 = const e

− 1

2

D

R−1/2u,R−1/2u

E

• data likelihood p (d|u) > 0 if u ∈ R1/2 (V ) = D

⇣

R−1/2⌘

• p (d|u) = e−∞ = 0 if u /

∈ R1/2 (V ) = Q1/2 (V )

• Q1/2 (V ) is the Cameron-Martin space of the measure N (0, Q)
• But µ = N (0, Q) ⇒ µ

⇣

Q1/2 (V )

⌘

= 0. Thus,

Z

V

p (d|u) dµf (u) = 0

Actually, who says that there even has to be an observation

• perator and its value subtracted from the data?

Data likelihood carries the information what the data says about the uncertainty of the state. Data likelihood is just a function of the state u: p(d|u) is the likelihood of data d given u

Examples of positive data likelihood

• White noise: (V, h·, ·i) is a Hilbert space and

p (d|u) = e1

2hdHu,dHui

• Pointwise Gaussian: µf is a random field on domain D ⇢ R2, data is

a function d : D ! R, and p (d|u) = e1

2

R

D|d(x)u(x)|2dx

• Pointwise positive:

p (d|u) = e

R

D f(x,d(x),u(x))dx

(the satellite sensing application will be like that)

Application: Assimilation of Satellite fire detection

• Data assimilation methodology (NSF)
• Wildland fire simulation and forecasting with assimilation of satellite Active Fires

detection data. Running autonomously for any fire anywhere any time in the CONUS, distributed on the web as a fire forecast layer over weather forecast. (NASA)

• A lightweight fire-smoke model over entire CONUS driven by HRRR weather forecast

and satellite sensing autonomously, filling missing data by model (current HRRR- Smoke model at NOAA is statistics based from satellite detections, without a fire model) (Joint Polar Satellite System (JPSS) - Smoke initiative)

• Assimilation of night infrared imagery from Unmanned Aerial Systems (UAS, a.k.a.

drones) (NightFOX project at NOAA)

Satellite Fire Detection - 2010 Fourmile Canyon Fire, Boulder, CO

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The data granules

Wildland Fire Behavior and Risk Forecasting PI: Sher Schranz, CSU/CIRA

As of: March 1, 2016

The model: WRF-SFIRE

WRF is a large open source project headed by NCAR

2013 Patch Springs Fire, UT

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Satellite data and fire simulation

• Level 2 MODIS/VIIRS Active

Fires

• Intersection of the granules

with the domain

• No fire detection is also

information!

• Except when data missing

(e.g., cloud)

• Shown with fire perimeter

and wind field from simulation

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Fire detection with simulated fire perimeter and wind vectors (detail)

Fire detection 0-6 hrs old Low confidence Nominal confidence High confidence Water Ground – no fire Cloud – no detection possible

Occasional imperfect data

Data assimilation: Connecting the data with the model

• Active Fires data resolution 375m-1km is much

coarser than the fuel data (Landfire) and fire models 30m-200m

• False positives, false negatives, geolocation

errors, artifacts possible

• Confidence levels given when fire is detected
• Unfortunately there is no confidence level for

land or water detection (no fire)

• Missing values: clouds, instrument error
• Improve the model in a statistical sense, not

for direct use such as initialization

Comparing Active Fires detection data and fire arrival time from simulation

Measuring how well does the model fit the data

• The fire-spread model state is encoded as the fire arrival

time T

• The fire is very likely to be detected at a location when it

arrives there, and for a few hours after

• The probability of detection quickly decreases afterwards
• The fire arrival time T from the model + sensor properties

allow us to compute the probability of fire detection or non- detection in every pixel

• Plug in the actually observed data => data likelihood
• Data likelihood = the probability of the data values

given the state T

Probability of fire detection

[Based on Morisette et. al 2005; Schroeder et al. 2008, 2014 ]

Actively burning area in the pixel Proxy for radiative heat flux in the pixel. Probability of fire detection Time since fire arrival (h) Heat flux

Data likelihood of active fires detection

Log probability of positive detection

• Fire is likely to be

detected for few hours after arrival at the location.

• The probability of

detection quickly decreases afterwards.

• The tails model the

uncertainty of the data and the model in time and space.

Logarithm of probability is more convenient. Instead of multiplying probabilities, add their logarithms.

Probability of no fire detection

Determined by the probability of detection. probability( yes|T ) + probability( no|T ) = 1

Evaluating the fit of the model state to the data: Data likelihood

• The model state is encoded as the fire arrival time T.
• Data likelihood = probability of the data given the model state T.
• Computing with logarithms is more conventient - add not multiply
• log(data likelihood) =

= confidence level(cell) ⋅ log data likelihood

grid cells

∑

(cell)

granules

∑

= c(cell) f

grid cells

∑

(cell,T granule

granules

∑

−T)

• Cloud mask or missing data in a cell implemented by c(cell) = 0

5 6 40.39 7 8 9 40.38

Hours since simulation start

10 11 12

Lat

40.37 13

• 112.625
• 112.63

40.36

• 112.635

Lon

• 112.64
• 112.645
• 112.65

40.35

• 112.655
• 112.66
• 6
• 5.5
• 5
• 4.5
• 4
• 3.5

#104

Automatic ignition from Active Fires satellite detection data by maximum likelihood

• Find the most

probable fire given the data from the first few MODIS/VIIRS detections

log Pr(data|T) = c(cell) f

grid cells

∑

(cell,T granule

granules

∑

−T,x, y) → max

T

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Penalty by Powers of Laplacian

• Penalty by equivalent to prior assumption that error in T is

a gaussian random field with mean Tf and covariance A

• Here,
• With zero boundary conditions on rectangle, the eigenvectors are of the

form

• Evaluate the action of powers of A by Fast Fourier Transform

(FFT)

! A = − ∂2 ∂2x − ∂2 ∂2 y ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

−p

!!!!!!!!p >1,!!λ jk ∝

jπ a

( )

2

+

kπ b

( )

2

⎛ ⎝ ⎜ ⎞ ⎠ ⎟

−p

→0

! T ∼ !e

−1 2 T−Tf

A−1 2

⇔T =T f + θkλk

1/2 k

∑

Tk ,!θk ∼ N(0,1),!ATk = λkTk

!

T−Tf

A−1 2

!λk →0!fast! ⇒ !random!field!smooth

!Tjk(x, y)∝sin jπx

a sin kπ y b

Data assimilation results 2013 Patch Springs fire, UT

Forecast Analysis

But fire is coupled with the atmosphere

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Atmosphere Heat release Fire propagation Wind Heat flux

• Heat flux from the fire changes the state of the atmosphere
• ver time.
• Then the fire model state changes by data assimilation.
• The atmospheric state is no longer compatible with the fire.
• How to change the state of the atmosphere model in

response data assimilation into the fire model?

• And not break the atmospheric model.
• Re

Replay the fire from given fire arrival time

Spin up the atmospheric model after the fire model state is updated by data assimilation

Fire arrival time changed by data assimilation

Active fire detection

Atmosphere out

• f sync with fire

Forecast fire simulation Coupled atmosphere-fire Replay heat fluxes derived from the changed fire arrival time Rerun atmosphere model from an earlier time Continue coupled fire-atmosphere simulation Atmosphere and fire in sync again

Assimilation Cycles with Atmosphere Spin Up

8/11/13 8/12/2013 8/13/2013 8/14/2013 8/15/2013 8/16/2013 8/17/2013 Cycle 0 Satellite data Spin up Satellite data Spin up Satellite data Spin up Simulation Simulation Cycle 1 Cycle 2 Satellite data Analysis Spin up Analysis Simulation Simulation Cycle 3 Cycle 4 Simulation Analysis Analysis

Conclusion

• A simple and efficient method – implemented by FFT 2-3 iterations

are sufficient to minimize the cost function numerically, 1 iteration already pretty good

• Pixels under a cloud cover do not contribute to the cost function
• Robust for missing and irregular data

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