Estimating fugitive emissions of airborne particulatematter using a Gaussian plume model

Bamdad Hosseini and John M. StockieSimon Fraser University

The 2015 AMMCS-CAIMS congress

June 10, 2015

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Fugitive emissions

Companies can place sensors on well-known sources.

Unknown sources: piles of material, moving trucks, vents andwindows, etc.

Fugitive emissions are difficult to measure directly.

Indirect measurements → source inversion

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Fugitive emissions

Teck Resources Ltd. at Trail, British Columbia.

Integrated lead-zinc smelter.

Different pollutants: zinc, lead, arsenic, SOx etc.

During August 20 to September 19,2014.

Multiple sensors: Dust-Fall jars, Xact, PM10 and TSP.

Horizontal wind velocity and direction at a single post.

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The forward problem

Simplifying assumptions:Ground is flat.Horizontal wind velocity changes with altitude.One type of contaminant with no chemical reactions.Fixed deposition and settling velocities.Only dry deposition (Ignore washout due to rain).All sources in an area are approximated by a single point source.

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The forward problem

Gaussian plume (Ermak’s1) solution for a point source at (0, 0, H),x-axis pointing in the direction of wind:

c(x) =q

2πσyσz |u|exp

(−y2

2σ2y

−Wsetz

2Kz−W2

setσ2z

8K2z

)

×[exp

(−H2

2σ2z

)+ exp

(−

(H + 2z)2

2σ2z

)− exp

(Wo(H + 2z)

Kz+W2

o σ2z

2K2z

)erf

(Woσz√2Kz

+H + 2z√2σ2

z

)].

(1)

c concentration

q rate of emissions

u wind velocity

Wset settling velocity

Wdep deposition velocity

Wo = Wdep − 12Wset

Kz vertical eddy diffusioncoefficient

σy, σz horizontal and verticalplume standard deviation

Linear in the emission rates!

Commonly used in industry standard software.

1Donald L. Ermak. “An analytical model for air pollutant transport and depositionfrom a point source”. In: Atmospheric Environment 11.3 (1977), pp. 231–237.

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The forward problem

Wind data measured at times {τ1, τ2, · · · , τNw}.Piece-wise constant wind data for t ∈ (τi, τi+1].

Piece-wise constant Ermak’s solution for each sensor.

qi(t), rate of emission of source i ∈ {1, 2, · · · , Nq}.Sensor j is actively measuring on intervals {Ij,1, Ij,2, · · · , Ij,Nj},Ij,k := {t ∈ (aj,k, bj,k]}.

Model sensor measurements as dj,k = Bj∫Ij,k

∑Nqi=1 ci,j(t)dt.

Discretize on a grid {t1, t2, · · · , tNT }, over the interval (0, T ].

Concatenate and regroup to get

d = Fq

.

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The inverse problem

Additive, uncorrelated Gaussian noise

d = Fq + εεε, ε ∼ N(0,ΣΣΣ).

Bayes’ rule2

πpost(q|d) ∝ exp

(−1

2

∥∥∥ΣΣΣ−1/2(Fq− d)∥∥∥22

)︸ ︷︷ ︸

likelihood

πprior(q)

Three choices for the prior: constant+positivity, smoothness andsmoothness+positivity

2Jari P. Kaipio and Erkki Somersalo. Statistical and computational inverse problems.Springer, 2005.

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The inverse problem

Constant+positivity:Take p = (p1, · · · , pNq

) and q = BpReduces to Nq dimensional problem.Maximum likelihood estimator (MLE) will suffice

qc = Bpmle, pmle := arg minp�0

∥∥∥ΣΣΣ−1/2(FBp− d)∥∥∥22.

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The inverse problem

Smoothness:Use constant emissions as prior mean.Gaussian smoothness prior, πprior(q) = N(qc,C).C = INq

⊗ L−2, where INqis the Nq ×Nq identity matrix.

L is a discretization of α(I − γ∂tt) with homogeneous Neumann bc.

0 200 400 600 800

−2

0

2

4

6

Time (hr)

mo

l/h

r

Sample 1

q1

q2

q3

q4

q5

q6

q7

0 200 400 600 800

−2

0

2

4

6

Time (hr)

mo

l/h

r

Sample 2

Linear forward map → πpost = N(qs,Cs),

qs := qc + CF∗(ΣΣΣ + FCF∗)−1(d− Fqc),

Cs := C−CF∗(ΣΣΣ + FCF∗)−1FC.

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The inverse problem

Smoothness+positivityIntroduce an auxiliary variable: q = max{v,0}.Pose nonlinear problem for v

π(v|d) ∝ exp

(−1

2

∥∥∥ΣΣΣ−1/2(F(max{v,0})− d)∥∥∥22

)πprior(v).

Use smoothness prior on v

πprior(v) = N(qc,C)

Use Markov Chain Monte Carlo to compute vCM (posterior mean),qsp := max{vCM,0}, standard deviations, etc.

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Application: setup

During August 20 to September 19, 2014.

PbO particulates.

Density [kgm−3] Diameter [m] Deposition vel. [ms−1] Settling vel. [ms−1]

ρ d Wdep Wset

9530 5× 10−6 0.005 0.0026

Wind data at 10 minute intervals.

0 5 10 15 20 25 30 35−5

0

5

10

time (days)

win

d v

elo

city

(m

/s)

raw data

regularization

0 5 10 15 20 25 30 350

100

200

300

400

time (days)

win

d d

irec

tion (

deg

)

raw data

regularization

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Application: setup

Type Dust-fall jar TSP PM10 Xact

Sched. entire month every two days every week every hour

SNR 11 100 100 100

# meas. 1 16 6 761

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Application: synthetic data set

0 200 400 600 800

0

1

2

3

4

5

6

Time (hr)

mol/

hr

Artificial

q1

q2

q3

q4

q5

q6

q7

0 200 400 600 800

0

1

2

3

4

5

6

Time (hr)

mol/

hr

qs

q1

q2

q3

q4

q5

q6

q7

0 200 400 600 800

0

1

2

3

4

5

6

Time (hr)

mol/

hr

qsp

q1

q2

q3

q4

q5

q6

q7

q1 q2 q3 q4 q5 q6 q70

1

2

3

4

5Average emissions

mol/

hr

Artificialq

c

qs

qsp

0 200 400 600 8000

0.5

1

1.5

2

qs standard deviation

Time (hr)

mol/

hr

0 200 400 600 8000

0.5

1

1.5

qsp

standard deviation

Time (hr)

mol/

hr

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Application: actual data

[tn/yr]

qc 54.0

qs 57.5± 5

qsp 63.0± 5

Prev. study 25− 40

Indep. 15− 500 200 400 600 800

0

5

10

15

20

Time (hr)m

ol/

hr

qs

q1

q2

q3

q4

q5

q6

q7

0 200 400 600 8000

5

10

15

20

Time (hr)

mo

l/h

r

qsp

q1

q2

q3

q4

q5

q6

q7

q1 q2 q3 q4 q5 q6 q70

5

10

15Average emissions

mo

l/h

r

q

c

qs

qsp

0 200 400 600 8000

0.5

1

1.5

2

2.5

3

qs standard deviation

Time (hr)

mo

l/h

r

0 200 400 600 8000

0.5

1

1.5

2

2.5

3

qsp

standard deviation

Time (hr)

mo

l/h

r

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Application: impact assessment

0.05

0.05

0.05

0.0

5

0.05

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.5

0.5

0.5

0.5

0.5

1

1

1

1

2

0.5

1

12

2 2

2

2

2

2

x [m]

y [

m]

Monthly total deposition at ground level [mg/m2]

1000 1500 2000 2500

0

500

1000

1500

2000

0.5

1

1.5

2

2.5

3

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Application: uncertainty propagation

Approximate posterior by N(qsp, C̃sp).

Forward problem is linear

πdeposition = N(F̃qsp, F̃C̃spF̃∗).

0.05

0.05

0.05

0.0

50.05

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.5

0.5

0.5

0.5

0.5

1

1

1

1

2

0.5

1

12

2 2

2

2

2

2

x [m]

y [

m]

Monthly total deposition at ground level [mg/m2]

1000 1500 2000 2500

0

500

1000

1500

2000

0.5

1

1.5

2

2.5

3

0.01

0.01

0.0

1

0.0

1

0.01

0.01

0.0

1

0.020.02

0.0

2

0.0

2

0.02

0.02

0.05

0.05

0.05

0.05

0.1

0.1

0.1

0.05

0.05

0.1

0.1

0.02

0.1

0.2

x [m]

y [

m]

Monthly total deposition std [mg/m2]

1000 1500 2000 2500

0

500

1000

1500

2000

0

0.05

0.1

0.15

0.2

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Closing remarks

Estimating fugitive emissions is difficult.

We solve three inverse problems.

Constant and positive emissions.Smoothness.Smoothness and positivitiy.

Insufficient data.

Room for improvement:

Better prior.Faster algorithms.Improve uncertainty propagation.Include wind uncertainty.Design of experiments.

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Thank you!

Ermak, Donald L. “An analytical model for air pollutant transport and depositionfrom a point source”. In: Atmospheric Environment 11.3 (1977), pp. 231–237.Kaipio, Jari P. and Erkki Somersalo. Statistical and computational inverseproblems. Springer, 2005.

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