Estimation of Source Parameters for Hazard Releases

Source Term Estimation for 

Hazardous Releases 

Fukushima Workshop 

NCAR, 22 nd – 23 rd February 2012 

Dstl/DOC61790 

Dr. Gareth Brown

Presentation Outline 

• Dstl’s approach to Source Term Estimation 

• Recent developments 

– Spatially and temporally gridded urban met object 

– Processing of deposition measurements 

• Toy example 

• Application to Fukushima 

© Crown Copyright 2012 

UK UNCLASSIFIED 

Dstl is part of the 

Ministry of Defence

Source Term Estimation Monte Carlo 

Bayesian Data Fusion (MCBDF) 

• MCBDF provides a 

set of likely source 

terms for calculating 

probabilistic hazards 

• Accounts for 

uncertainty in source, 

meteorology, sensor 

performance & 

human reporting 



P. Robins, V. Rapley, N. Green, Realtime 


Sequential Inference of Static Ministry Parameters of Defence with 

Expensive Likelihood Calculations (2009)

Bayes’ Theorem 

p( θ y) ∝ p( yθ) p( θ) 

Posterior 

probability density 

of hypothesis θ 

given the data y 

Likelihood 

probability density 

of the data y 

conditional on the 

hypotheses θ 

Prior 

probability 

density of θ 

based on prior 

knowledge 

• Iterative update of belief in a hypothesis 

• Likelihood calculation particularly costly 

– Necessary to run dispersion model for sensor data 

– Sensor likelihood function complex 





MCBDF: Bayesian data fusion 

• MCBDF uses Bayesian inference over a sample set of 

hypothesized source-terms and Met. variables 

θ 

1 

= 

* x * y L 0 

 

Source−term 

Met 

( xytmdau , , , , , , , u , ,ln z , DP , ) 

( ) 

Model 

• The dimensionality of problem depends on its complexity 

– Complex source terms 

– Gridded meteorology in time and space 

– Urban dispersion 





Physics Modelling 

• Dispersion is a turbulent process 

θ 

( x, t , t ) 

1 

– Impossible to model accurately 

– Use ensemble puff dispersion models 

– SPRINT model optimised for STE 

– Implements SCIPUFF closure relations 

2 

⎛ 

2 

ccc , ′ ⎞ 

⎜ ⎟ 

⎫ dispersion ⎜ 

2 

⎟ 

⎬ ⎯⎯⎯⎯⎯⎯⎯⎯→ p⎜ χχχ , ′ ⎟ 

⎭ ⎜ ⎟ 

2 

⎜χd χd, 

χ′ 

⎟ 

d 

⎝ ⎠ 





MCBDF: Example Prior 





Sensor Likelihood Calculations 

• When CB sensor measurements are passed to MCBDF 

the likelihood is calculated as 

( 

2) ( ) ( 

2 

µσ µσ ) 

py| , = ∫ pyc | pc| , dc 

 

y ≡ 

∞ 

Sensor measurement 

0 measurement concentration 

density density 

µ ≡ Mean mass-concentration from dispersion simulation 

2 

σ ≡ Mass concentration variance from dispersion simulation 

c ≡ Unobserved ground-truth concentration 





MCBDF: Upon Receipt of a Detection 





Estimating the Posterior 

Distribution 

( θ y) 

( θ) ( θ) 

p ∝ ∏ p y p 

i 

• Calculated as the product of the individual likelihoods 

and the prior 

• Posterior estimated by sampling lots of hypotheses 

• MCBDF uses Differential Evolution Markov Chain 

(DEMC) Monte Carlo to generate new hypotheses 

i 





Calculating Hazard Areas 

• The Hazard Calculator 

obtains the probability of 

exceeding a specified 

threshold dosage 

• A weighted sum of these 

gridded probabilities is 

used to define the 

probable hazard area 

Release location 

Sensor network 

Red: ground truth; green: predicted 









Recent Developments 

• MCBDF initially developed for rapid warning of chemical 

attacks based on input from sensor networks 

– Short effective duration approximates to spatially and temporally 

invariant meteorology 

• However STE for Bio and Rad releases in major cities 

requires 

– Fast dispersion models accounting for urban areas 

– Much longer temporal windows 

– Efficient methods to compute long duration dosage / deposition 

– Spatially varying meteorology for large scale dispersion 





Meteorological Inference 

• Urban capability 

• Gridded urban meteorological model 

defined 

– Wind flow components gridded in 2D 

space and time 

– Surface properties gridded in 2D space 

– Atmospheric stability gridded in time 

• 60 – 400 extra dimensions in 

parameter space 

– MCMC techniques still efficient if 

converged 

– Much more difficult to achieve 

convergence 

f 

Sykes et al, SCIPUFF Tech Doc 

⎡ ⎛ 

( zh , , α ) = exp ⎢−α 

⎜1− 

⎣ ⎝ 

c c c c 

Canopy height 

z 

h 

c 

⎞⎤ 

⎟⎥ 

⎠⎦ 

Canopy flow parameter 





Toy Problem Example 

• Single, short duration release 

• Single isotope 

• Quantitative deposition measurements 

• Spatially gridded, but temporally invariant meteorology 

• No transport due to rain run-off before measurements 

taken 





Deposition data 

• Likelihood model for quantitative measurement of 

deposited material 

– Assume local detection only, alpha, beta, not longrange 

gamma 

– Uncertainty per measurement, not fixed 

• Time taken to collect Poisson count statistics 

• Energy spectrum uncertainty on which isotope is being measured 

• Naturally occurring background (with uncertainty) subtracted 

• Simple, normally distributed measurement error model 

• Lower limit and saturation can be input 





Sample Points 

• 88 points 

• 20km 

exclusion zone 

shown 

• Release point 

known 

• No met data 





Hazard Plot 

• First 

measurement 

above 

background 

• Close to 

source 

• Huge met 

uncertainty 






• First 3 rings of 

data 

• Less 

uncertainty 

close to 

source 






• 4 rings of data 






• 6 rings of data 






• All data 

• More time for 

convergence 





Problems to solve for Fukushima 

• Parameters to infer: 

– Mass released in each hour over the last year 

– Gridded meteorology for each hour over the last year 

• 10km spatial resolution 

• Modelling: 

– Gamma sensors pick up spatially averaged dose 

• Function of range needs integrating 

– 1/r 2 , attenuation through air, scatter build up factors 

– Longer range second order closure dispersion model needed 

• SCIPUFF 

• Other? 

– Further transport of deposited material due to water run-off 





Backup 





The Concentration Sensor 

Likelihood Model 

• Critical to the performance of MCBDF is an accurate 

probabilistic description of the detector’s response 

2 

σ e 

L 

L 

≡ 

≡ 

≡ 

( 

2 

, σ ) e 

⎧ Φ Lc y= 

L 

( ) 

⎪ 

( 

2 

p yc = ⎨ φ yc, 

σ ) e 

L< y< 

L 

⎪ 

( 

2 

⎪1 −Φ Lc, 

σ ) = 

⎩ 

e 

y L 

Measurement error variance 

Sensor saturation point 

Sensor limit of detection 

normal distribution CDF 





Deposition modelling 

• Improved physics improves consistency of sensor data 

likelihood modelling 

– More accurate source term estimation in presence of deposition 

• Amount of deposition inferred from data and/or uncertainty 

correctly passed to hazard calculations 

17 February 2012 





Deposition data 

• Likelihood model for detection of deposited material already in 

place 

– Biological hazards 

– Similar to a probit model 

– Extended to include: 

• Probability of false alarm 

• Probability of false negative 

– Integrate out unobserved true amount of deposited material 






Prelim. deposition results (simulated met. constant) 

Collectors + Identifiers (dosage) 

+ Survey ID (deposition) 

20km 

10km 

10km 

5km 

1 detection, 

8 nulls 






Burn In and Convergence 

• Initial hypotheses may be far from the 

peak of the posterior 

• But rapid answers are required 

– Data continually changing the posterior 

– limited time for new sample weights to 

make the old ones insignificant 

– Limited time for samples to spread out and 

capture true uncertainty 

• Detection of non-convergence delays 

message processing 

Re-weight 

Re-sample 






Hazard Calculation 

• Probit slope model 

– S used to indicate uncertainty in appropriate value for 

χ d 50 

1 ⎛ ⎛ S 1 log 

2 ⎜ ⎜ 

⎝ ⎝ 2 

( χd 

) = ⎜ + erf ⎜ 

10 

p Hazard 

⎛ χd 

⎜ 

⎝χ 

d 50 

⎞⎞⎞ 

⎟ 

⎟ 

⎠ 

⎟ 

⎠⎠ 

∞ 

( , ′ 2 ) ( , ′ 

2 

) ( ) 

p Hazard χ χ = ∫ p χ χ χ p Hazard χ dχ 

d d d d d d d 

0 

• Average over 1000 release/met samples from posterior. 






Deposition data likelihood (clipped normal) 

2 

( χd, 

σe 

) 

2 

( , ) 

2 

( χd 

σe 

) 

⎧ Φ L y = L 

⎪ 

p( y χd) 

= ⎨ φ y χd σe 

L< y< 

L 

⎪ 

⎪1 −Φ L , y = L 

⎩ 

2 2 

( χ , ) unclipped 

( , 

d 

χ ′ 

d 

⎯ ⎯ ⎯ ⎯→ µ 

N 

σN) 

∞ 

⎧ 

2 2 2 

⎪ ( L χd, σ ) ⎡ 

e ( 0 µ 

N, σN) δ ( χd) φ( χ 

d 

µ 

N, 

σ ) ⎤ 

∫ Φ Φ + 

N 

dχd 

y = L 

⎣ ⎦ 

⎪ 0 

∞ 

2 ⎪ 2 2 2 

p( y µ 

N, σN) 

= ⎨ φ( y χd, σ 

e ) ⎡ ( 0 µ 


d 

µ 

N, 

σN) 

⎤ 

∫ 

Φ + dχd 

L< y< 

L 

⎣ ⎦ 

⎪ 0 

⎪∞ 

2 2 2 

⎪ ( 1 ( L χd, σ 

e )) ⎡ ( 0 µ 


d 

µ 

N, 

σN) 

⎤ 

∫ −Φ Φ + dχd 

y = L 

⎣ 

⎦ 

⎪⎩ 0 

L 

⎧ 

2 2 2 

⎪ Φ( L 0, σ 

e ) Φ ( 0 µ 

N, σ 

N) + k( y) ⎡1 ( 0 µ ( y) , σ ( y) 

) ⎤ 

∫ −Φ dy y = L 

⎣ ⎦ 

⎪ 

−∞ 

2 ⎪ 2 2 2 

p( y µ 

N, σN) 

= ⎨ φ( y 0, σ 

e ) Φ ( 0 µ 

N, σ 

N) + k( y) ⎡1 −Φ ( 0 µ ( y) , σ ( y) 

) ⎤ L< y< 

L 

⎣ ⎦ 

⎪ 

∞ 

⎪ ⎡ 

2 2 2 

1−Φ( L 0, σ 

e ) ⎤ Φ ( 0 µ 

N, σ 

N) + k( y) ⎡ 1−Φ ( 0 µ ( y) , σ ( y) 

) ⎤ dy y = L 

⎪⎣ ⎦ ∫ ⎣ ⎦ 

⎩ 

L 

2 2 

( x , ) ≡ ( x , ) 

φ µσ φ µ σ 

σ 

σσ 

2 2 

2 1 2 

= 

2 2 

σ1 + σ2 

2 µ 

1 

µ 

2 

µ σ ⎛ ⎞ 

= ⎜ + σ 

2 2 ⎟ 

⎝ 1 σ 

2 ⎠ 

k 

( x , 2 ) ( x , 2 ) ≡ k ( x , 

2 

) 

1 1 2 2 

φ µ σ φ µ σ φ µ σ 

σ 

= 

σσ 

1 2 

2 

e 

π 

2 2 2 

1⎛µ 1 µ 2 µ ⎞ 

− + − 

2⎜ 

2 2 2 

σ1 σ2 

σ ⎟ 

⎝ 

⎠ 

Romberg (closed and semi-infinite) numerical integration 






Deposition data likelihood (clipped gamma) 

2 

( χd, 

σe 

) 

2 

( , ) 

2 

( χd 

σe 

) 

⎧ Φ L y = L 

⎪ 

p( y χd) 

= ⎨ φ y χd σe 

L< y< 

L 

⎪ 

⎪1 −Φ L , y = L 

⎩ 

2 * 

( χ , ) ( , , 

d 

χ′ d 

⎯ ⎯→ sk λ) 

Romberg (closed and 

semi-infinite) 

numerical integration 

⎧ 

⎡ 

⎛ ⎛χd 

+ λ ⎞⎞ 

⎤ 

* 

⎪ ∞ 

⎢ 

k −1 

exp 

2 χd 

λ ⎜ − ⎜ ⎟ 

s ⎟ 

⎥ 

⎪ 

⎛ + ⎞ ⎝ ⎠ 

Φ ( L χd, σ ) ⎢ 

⎝ ⎠ 

e 

+ (1 − γδχ ) ( ) ⎥ 

* 

d 

dχd 

y = L 

⎪ ∫ 

⎢ 

⎜ ⎟ 

s s ( k ) 

0 

⎝ ⎠ Γ 

⎥ 

⎪ 

⎢ 

⎥ 

⎪ 

⎣ 

⎦ 

⎪ 

⎡ 

χd 

λ 

⎤ 

⎪ 

⎛ ⎛ + ⎞⎞ 

* 

∞ 

⎢ 

k −1 

exp 

2 2 χd 

λ ⎜ − ⎜ ⎟ 

s ⎟ 

⎥ 

⎪ 

⎛ + ⎞ ⎝ ⎠ 

p( y µ 

N, σN) 

= φ( y χd, σ 

⎝ ⎠ 

⎨ 

e ) ⎢ 

(1 γδχ ) ( ) ⎥ 

∫ 

⎜ ⎟ 

+ − 

* 

d 

dχd 

L< y< 

L 

⎪ 

⎢ s s ( k ) 

0 

⎝ ⎠ Γ 

⎥ 

⎪ 

⎢ 

⎥ 

⎣ 

⎦ 

⎪ 

⎪ ⎡ 

⎛ ⎛χd 

+ λ ⎞⎞ 

⎤ 

⎪ 

* 

∞ 

⎢ 

k −1 

exp 

2 ⎛χd 

+ λ ⎜ − ⎜ ⎟ 

⎞ 

s ⎟ 

⎥ 

⎪ ( 1 −Φ( L χd, 

σe 

)) 

⎢ 

⎝ ⎝ ⎠⎠ 

⎜ ⎟ 

+ (1 − γδχ ) ( ) ⎥ 

* 

d 

dχd 

y = L 

⎪∫ 

⎢⎝ 

s 

0 

⎠ sΓ( k ) 

⎥ 

⎢ 

⎥ 

⎪⎩ 

⎣ ⎦ 

⎧ 

⎡ 

⎛ ⎛χd 

+ λ ⎞⎞⎤ 

* 

⎪ 

∞ 

⎢ 

k −1 

exp 

2 2 χd 

λ ⎜ − ⎜ ⎟ 

s ⎟⎥ 

⎪ 

⎛ + ⎞ ⎝ ⎠ 

Φ( L 0, σe )( 1− γ) + Φ( L − χd 0, σ ) ⎢ 

⎝ ⎠⎥ 

e 

dc y = L 

* 

⎪ 

∫ 

⎢ 

⎜ ⎟ 

( ) 

0 

⎝ s ⎠ sΓ 

k ⎥ 

⎪ 

⎢ 

⎥ 

⎪ 

⎣ 

⎦ 

⎪ 

⎡ 

χd 

λ ⎤ 

⎪ 

⎛ ⎛ + ⎞⎞ 

* 

∞ 

⎢ 

k −1 

exp 

2 2 2 χd 

λ ⎜ − ⎜ ⎟ 

s ⎟⎥ 

⎪ 

⎛ + ⎞ ⎝ ⎠ 

p ( y µ 

N, σN) 

= φ( y 0, σe )( 1 γ) φ( y χd, 

σ 

⎝ ⎠ 

⎨ 

− + 

e ) ⎢ 

⎥ 

∫ 

⎜ ⎟ 

dc L < y < L 

* 

⎪ 

⎢ s s ( k ) 

0 

⎝ ⎠ Γ ⎥ 

⎪ 

⎢ 

⎥ 

⎣ 

⎦ 

⎪ 

⎪ ⎡ 

⎛ ⎛χd 

+ λ ⎞⎞⎤ 

⎪ ∞ 

⎢ 

k 

2 2 

( 1−Φ( L 0, σe ))( 1− γ) + ( 1−Φ( L−χd 0, σe 

)) 

* −1 

exp 

χd 

λ ⎜ − ⎜ ⎟ 

s ⎟⎥ 

⎪ ⎢⎛ + ⎞ ⎝ ⎝ ⎠⎠ 

⎜ ⎟ 

⎥dc x = L 

* 

⎪ 

∫ 

⎢ s sΓ( k ) ⎥ 

0 

⎝ ⎠ 

⎢ 

⎥ 

⎪⎩ 

⎣ ⎦ 






Urban meteorology 

• Displacement height 

• Canopy blending 

• Mean wind vector 

• Shifted wind vector 

z ( h, α ) = 0.7 hF( α ) 

d c c c c c 

( α ) 2 

c 

⎛ 

⎞ 

Fc( αc) = 1−exp⎜− 

⎟ 

⎜ 0.25 + 0.5α 

c ⎟ 

⎝ 

⎠ 

u 

' 

( xyzt , , , ) = gxyz ( , , ) u( xyzt , , , ) 

' 

' fsl 

( z , z0, L) 

ux(, xyzt , ,) = 

u* 

x(, xyt ,) 

k 

' 

' fsl 

( z , z0, L) 

uy(, xyzt , ,) = 

u* 

y(, xyt ,) 

k 

u(, xyzt , ,) = 0 

' 

z 

⎧( hc < zs + zd) and ( z < hc) 

⎨ 

⎧h ( 

c s d) 

c 

− zd 

if ⎩ or h ≥ z + z 

⎪ 

⎪ 

⎪ z − zd 

if ( hc < zs + zd) and ( hc ≤ z < zs + zd) 

( , , ) = ⎨ 

⎪⎪ 

zs if ( hc < zs + zd) and ( z ≥ zs + zd) 

⎪ 

⎪⎩ 

' 

z xyz 

• Surface layer profile 

• Canopy layer profile 

• Blending function 

⎛ z ⎞ z 

fsl 

( zz , 

0, L) = ln ⎜ + 1 ⎟−Ψm( ) 

⎝ z0 

⎠ L 

⎡ ⎛ z ⎞⎤ 

fc( zh , 

c, αc) = exp ⎢−αc⎜1 

− ⎟⎥ 

⎣ ⎝ hc 

⎠⎦ 

gxyz ( , , ) = F( α ) f( zh , , α ) + (1 −F( α )) 

c c c c c c c 

fsl 

(, zz0, L) 

f ( h , z , L) 

sl 

c 

0 






Wind vector spatial derivatives 

• Calculated by chain rule, e.g.: 

⎧ ∂gxyz 

( , , ) ∂αc 

∂gxyz 

( , , ) ∂hc 

⎪ 

+ 

⎪ ∂αc ∂xi ∂hc ∂xi 

if z < h 

∂gxyz ( , , ) ∂gxyz ( , , ) ∂gxyz ( , , ) ∂z 

= + + 

∂xi ⎪ ∂z0 

∂xi ∂z ∂xi 

⎪ 

⎪ 

if z ≥ h 

⎪⎩ 

0 

hc < zs + z 

⎪ 

∂z0 

d 

⎨ 

hc ≥ zs + zd 

∂gxyz 

( , , ) 

= 

∂x 

i 

⎧ 

⎪ 

⎪ 

⎪∂gxyz 

( , , ) ∂αc 

∂gxyz 

( , , ) ∂hc 

+ 

∂α 

∂x ∂h ∂x 

⎪ 

⎪ 

⎪ 

⎪ 

⎪ 

c i c i 

∂gxyz ( , , ) ∂z 

∂gxyz ( , , ) ∂z 

0 

+ + < 

s 

+ 

∂z0 

∂xi 

∂z ∂xi 

c 

c 

if z z z 

⎪ 

⎨ 

⎪ 

⎪ ∂ gxyz ( , , 

s 

+ zd) ∂αc ∂ gxyz ( , , 

s 

+ zd) 

∂hc 

⎪ 

+ 

⎪ ∂αc ∂xi ∂hc ∂xi 

⎪ ∂ gxyz ( , , 

s 

+ zd) 

∂z0 

⎪ + if z ≥ zs 

+ zd 

⎪ ∂z0 

∂xi 

⎪ 

⎪ 

∂ gxyz ( , , 

s 

+ zd) 

⎛ ∂zd ∂αc 

∂zd 

∂h 

⎞ 

c 

+ 

⎜ + ⎟ 

⎪⎩ 

∂z 

⎝∂αc 

∂xi 

∂hc 

∂xi 

⎠ 

d 






Hypotheses 

• Temporal gridding 

• Spatial gridding 

θ 

1 

( ) ( d) x y L ( ) ( c) ( αc) 

= 

1 2 

* * 

0 

Source-term 

Meteorology 

( l , l , t, m,ln d ,ln v , a, u , u , ,ln z ,ln h ,ln , DP ) 

Model 

• Location 

• Time 

• Mass 

• Duration 

• Deposition velocity 

• Agent 

• Friction velocity components 

• Reciprocal Monin Obukhov length 

• Surface roughness 

• Canopy Height 

• Canopy Flow 

• Dispersion model 

• Dispersion model output PDF 

• Floating point values used 

to index discrete values 






Accuracy Compared to ATP45 

• Concentration 

sensor network 

• Meteorology sensor 

• Shear LIDAR 

• SODAR 

• Multiple 

anenometers 

• Probability of hazard 

effect 

• Ground Truth 

• Inferred 






Uncertainty Compared to ATP45 

• Single prompt 

alarm 

• Forecast 

meteorology 

• Probability of 

hazard effect 

• Ground 

Truth 

• Inferred 






Processing dynamic sensor data 

• MCBDF performs sourceterm 

estimation in realtime 

– A time window is 

maintained - typically 30 

minutes into the past for 

chem, 2 days for bio 

• On receipt of new data, 

old hypotheses and old 

data will become obsolete 

and are removed as they 

exit the current time 

window 

– Total likelihood 

housekeeping 

• Remaining hypothesis 

weights are modified by 

the likelihood of new data 






Exam Question 

• Source term – means to an end 

• How much material is in the soil 

– Can the land be used for habitation or farming. 






Summary 

• Deposition survey data as a data source appears 

effective 

• Modelling/inference extensions required

Estimation of Source Parameters for Hazard Releases

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