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: p a ( u ) ∝ p ( d | u ) p f ( u ) No Lebesgue measure, no densities. Integrate over an arbitrary measurable set A instead: Z Z p a ( u ) du ∝ p ( d | u ) p f ( u ) du A A Z µ a ( A ) ∝ p ( d | u ) dµ f ( u ) A Data likelihood is the Radon-Nikodym derivative: p ( d | u ) ∝ dµ a dµ f A p ( d | u ) dµ f ( u ) R • Normalize: µ a ( A ) = V p ( d | u ) dµ f ( u ) R p ( d | u ) dµ f ( u ) > 0? • But how do we know that R V
Infinite-dimensional data, Gaussian measure error bad • The simplest example: µ f = 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. D E − 1 R − 1 / 2 u,R − 1 / 2 u • p ( d | u ) = const e − 1 2 | u | 2 R − 1 = const e 2 • data likelihood p ( d | u ) > 0 if u ∈ R 1 / 2 ( V ) = D ⇣ R − 1 / 2 ⌘ • p ( d | u ) = e −∞ = 0 if u / ∈ R 1 / 2 ( V ) = Q 1 / 2 ( V ) • Q 1 / 2 ( V ) is the Cameron-Martin space of the measure N (0 , Q ) Z Q 1 / 2 ( V ) p ( d | u ) dµ f ( u ) = 0 ⇣ ⌘ • But µ = N (0 , Q ) ⇒ µ = 0. Thus, V
Actually, who says that there even has to be an observation operator 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 ) = e � 1 2 h d � Hu,d � Hu i • Pointwise Gaussian: µ f is a random field on domain D ⇢ R 2 , data is a function d : D ! R , and D | d ( x ) � u ( x ) | 2 dx p ( d | u ) = e � 1 R 2 • Pointwise positive: R D f ( x,d ( x ) ,u ( x )) dx p ( d | u ) = e (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 9
The data granules
Wildland Fire Behavior and Risk Forecasting As of: The model: WRF-SFIRE March PI: Sher Schranz, CSU/CIRA 1, 2016 WRF is a large open source project headed by NCAR 2013 Patch Springs Fire, UT
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 | 12
Fire detection with simulated fire perimeter and wind vectors (detail) Water Ground – no fire Cloud – no detection possible Fire detection 0-6 hrs old Low confidence Nominal confidence High confidence 13
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 ] Probability of fire detection Actively burning area in the pixel Proxy for radiative heat flux in the pixel. Heat flux Time since fire arrival (h)
Data likelihood of active fires detection Logarithm of probability is more convenient. Instead of multiplying probabilities, add their logarithms. 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.
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 = (cell) g ranules grid cells ∑ ∑ = − T ) (cell, T granule c (cell) f g ranules grid cells • Cloud mask or missing data in a cell implemented by c (cell) = 0
Automatic ignition from Active Fires satellite detection data by maximum likelihood ∑ ∑ log Pr(data| T ) = − T , x , y ) → max (cell, T granule c (cell) f T g ranules grid cells • Find the most 13 -3.5 probable fire 12 11 Hours since simulation start given the data -4 10 from the first few 9 8 -4.5 MODIS/VIIRS 7 6 detections -5 5 40.39 40.38 -5.5 40.37 -112.625 40.36 -112.63 -112.635 -6 -112.64 Lat -112.645 -112.65 40.35 -112.655 # 10 4 -112.66 Lon
Penalty by Powers of Laplacian 2 T − T f ! A − 1 • Penalty by equivalent to prior assumption that error in T is a gaussian random field with mean T f and covariance A − 1 2 ⇔ T = T f + 2 T − T f ∑ θ k λ k 1/2 T k ,! θ k ∼ N (0,1),! AT k = λ k T k T ∼ ! e A − 1 ! k ! λ k → 0!fast! ⇒ !random!field!smooth • − p ⎛ ⎞ − p ( ) A = − ∂ 2 − ∂ 2 ( ) ⎛ ⎞ • Here, 2 2 !!!!!!!! p > 1,!! λ jk ∝ j π + → 0 k π ⎜ ⎟ ⎜ ⎟ ∂ 2 y ⎝ ⎠ a b ∂ 2 x ⎝ ⎠ ! • With zero boundary conditions on rectangle, the eigenvectors are of the ! T jk ( x , y ) ∝ sin j π x a sin k π y form b • Evaluate the action of powers of A by Fast Fourier Transform (FFT) 23
Data assimilation results 2013 Patch Springs fire, UT Analysis Forecast
But fire is coupled with the atmosphere Atmosphere Heat flux Wind Fire Heat propagation release Heat flux from the fire changes the state of the atmosphere • over 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 • 25
Spin up the atmospheric model after the fire model state is updated by data assimilation Rerun atmosphere Atmosphere and Atmosphere out Coupled model from an of sync with fire fire in sync again atmosphere-fire earlier time Continue coupled Replay heat fluxes Forecast fire Fire arrival time fire-atmosphere derived from the simulation changed by simulation changed data assimilation fire arrival time Active fire detection
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 Simulation Satellite data Cycle 1 Analysis Spin up Simulation Satellite data Cycle 2 Analysis Spin up Simulation Satellite data Cycle 3 Analysis Spin up Simulation Satellite data Cycle 4 Analysis Spin up Simulation
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 28
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