Assimila'on of func'onal data and applica'on to ac've fires detec'on from satellites Jan Mandel University of Colorado Denver Ins5tute of Informa5cs, Czech Academy of Sciences With Ivan Kasanický, Adam K. Kochanski, Sher Schranz, and Mar5n Vejmelka Computa5onal Mathema5cs in Science and Engineering (CMSE 16) Conference In honor of the 70 th birthday of Professor Zdeněk Dostál Rožnov pod Radhoštěm, Czech Republic May 26, 2016 Par5ally supported by NASA grant NNX13AH59G, NSF grant DMS-1216481, and Czech Science Founda5on grant GA13-34856S .
Probability densi5es are with respect to the Lebesgue measure. But there is no Lebesgue measure in an infinite dimensional space. That is, there is no measure such that All open balls have posi5ve finite measure • It is transla5on or rota5on invariant •
Actually, who says that there even has to be an observa5on operator and its value subtracted from the data? Data likelihood carries the informa5on what the data says about the uncertainty of the state. Data likelihood is just a func'on of the state u : p ( d | u ) is the likelihood of data d given u
Applica5on: Assimila5on of Satellite fire detec5on Funded projects/Goals: Wildland fire simula5on and forecas5ng with assimila5on of satellite • Ac5ve Fires detec5on data. Running autonomously for any fire anywhere any 5me in the US, distributed on the web as a fire forecast layer over weather forecast. (NASA) Wildfire danger and simula5on system for Israel (opera5onal) (Israel • Homeland Security) Op5mal placement of sensors for a large experimental burns in 2017 to • improve smoke modeling (U.S. Joint Fire Science Program) NSF: GEO, CS, DMS – data assimila5on and fire modeling methodology, • computer science infrastructure Czech Grant Agency – data assimila5on methodology •
Measuring how well does the model fit the data • The fire spread model state is encoded as the fire arrival 5me T • The fire is very likely to be detected at a loca5on when it arrives there, and for a few hours aper • The probability of detec5on quickly decreases aperwards • The fire arrival 5me T from the model + sensor proper5es allow us to compute the probability of fire detec5on or non-detec5on in every pixel • Plug in the actually observed data => data likelihood • Data likelihood = the probability of the data values given the state T
Data likelihood of ac5ve fires detec5on Logarithm of probability is more convenient. Instead of mul5plying probabili5es, add their logarithms. Log probability of posi've detec5on • Fire is likely to be detected for few hours aper arrival at the loca5on. • The probability of detec5on quickly decreases aperwards. • The tails model the uncertainty of the data and the model in 5me and space.
Probability of no fire detec5on Determined by the probability of detec5on. probability( yes| T ) + probability( no| T ) = 1
Evalua5ng 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
Automa5c igni5on from Ac5ve Fires data by maximum likelihood • Find the most probable fire given the data. (Much like what a human expert might do.) ∑ ∑ log Pr(data| T ) = − T , x , y ) → max (cell, T granule c (cell) f T g ranules grid cells • Using for igni5on from the first few MODIS/ VIIRS detec5ons (James Haley, PhD student, in progress) • But this ignores any previous modeling – not data assimila5on yet
Data Assimila5on • A basic data assimila5on cycle to ingest data: – Stop the simula5on – Modify the model state (called forecast , T f ) to reconcile the model with new data that arrived since the last cycle, get new es5mated state (called analysis , T ) – Con5nue the simula5on from the analysis T • We do not want to follow the data or the model exactly. Rather, split the difference taking into account their rela5ve uncertain5es. • We want the data likelihood large and the difference between the analysis and the forecast small • This is how weather forecas@ng works nowadays – every few hours (usually 1 to 6), new data are ingested like this.
Assimila'on of MODIS/VIIRS Ac've Fire detec'on: maximum probability es'mate • We want the data likelihood Pr(data , T ) large and the difference between the analysis and the forecast T - T f • Penalize the difference T - T f in a suitable norm 2 → max log Pr(data| T ) − T − T f T • Choose the norm to penalize non-smooth changes – measuring the values of deriva5ves, − p ⎛ ⎞ 2 ≈ u − ∂ 2 ∂ 2 x − ∂ 2 ∫ , a > 1 u udxdy ⎜ ⎟ ∂ 2 y ⎝ ⎠ • Efficient precondi5oned gradient method, easily running on a laptop (not a supercomputer), only one of two itera5ons needed 35
Penalty by powers of Laplacian 2 T − T f • Penalty by equivalent to prior assumption that error in T is ! A − 1 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 • Here, ⎛ ⎞ − p ( ) A = − ∂ 2 − ∂ 2 ( ) ⎛ ⎞ 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) 36
Minimiza5on by precondi5oned steepest descent 2 − J ( T ) = α S − T ( x k , y k ) ∑ 2 T − T f ) → c k f k ( T k min A − 1 C ( T − T f ) = 0 k S − T ( x k , y k ) ∇ J ( T ) = α A ( T − T f ) − F ( T ), F ( T ) = ∑ d c k dt f k ( T k ) k But ∇ J ( T ) is a terrible descent direction, A ill conditioned - no progress at all! Better: preconditioned descent direction A ∇ J ( T ) = α ( T − T f ) − AF ( T ) − p ⎛ ⎞ AF ( T ) = − ∂ 2 ∂ 2 x − ∂ 2 S − T ( x k , y k ) ∑ d c k dt f k ( T k ) ⎜ ⎟ ∂ 2 y ⎝ ⎠ k p > 1:spatial smoothing of the forcing by log likelihood maximization T at ignition point does not change ⇒ descent direction δ from the saddle point problem A δ + C λ = ∂ ∂ t f ( T S − T , x , y ), C T δ = 0 Now one descent iteration is enough. 37
Data assimila5on results 2013 Patch Springs fire, UT Analysis Forecast
Conclusion • A simple and efficient method – implemented by FFT 2-3 itera5ons are sufficient to minimize the cost func5on numerically, 1 itera5on already prewy good • Pixels under a cloud cover do not contribute to the cost func5on • Robust for missing and irregular data • Standard bayesian data assimila5on framework Forecast + data = analysis • Future: – Automa5c igni5on from detec5on by maximum likelihood – Na5ve swath coordinates, data likelihood computed for na5ve pixels, not the cells of the computa5onal mest 45
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