CVEN 607 Special Topics:Source apportionment

Qi Ying

Zachry Department of Civil Engineering

Texas A&M University

Email: qying@civil.tamu.eduWeb: https://ceprofs.civil.tamu.edu/qying/

Source-Receptor Relationship using Statistical Receptor Models

Chemical Mass Balance

Source-Receptor Relationship using Deterministic CTMs

( ) i=1,2,3

i

i

i i i

u CC CK R S L

t x x x

= − + + + −

Reactive Transport Modeling

Hybrid modeling

• Combine CTM-based and receptor-based source apportionment

4

( ) i=1,2,3

i

i

i i i

u CC CK R S L

t x x x

= − + + + −

Reactive Transport Modeling

Chemical Mass Balance

Receptor-based source apportionment models• The foundation of all receptor-based models for

source apportionment is a simple mass conservation argument:

Measured aerosol concentrations, C, at a receptor will be a linear combination of the contributions (Sj) from p potential sources,

C =

j=1

p

Sj

Receptor-based source apportionment models• Same argument can be made for the concentration

of aerosol components

Ci =

j=1

p

ai,jSj , i = 1,2, … n

m: number of sourcesn: number of aerosol chemical componentsai,j: mass fraction of chemical component i in aerosol

emitted from source j

Chemical Mass Balance

• Tracer element method: it assumes that each aerosol source type has a unique chemical component that does not exist in other source types.

Ci =

j=1

p

ai,jSj = ai,tSt

Thus,

St =Ciai,t

Requirements: (1) ai,𝑡 is well know and invariant

between source and receptor (2) Ci can be measured accurately and

precisely in the ambient sample(3) Concentrations of component i

comes only from source t

Chemical Mass Balance

• Some examples of unique tracers

Cass, Trends in Analytical Chemistry, 17(6), 1998

Chemical Mass Balance

• Molecular markers for secondary organic aerosol

Chemical Mass Balance

• If no source-specific tracers can be applied, the set of chemical mass balance equations can be solved simultaneously to obtain estimations of source contributions. Consider this example:

Measured PM2.5 concentration at the receptor: PM2.5 32 μg m-3, 2.58 μg m-3 of Si and 3.84 μg m-3 of Fe.

Sources: power plant and soil dust. Source profiles:

soil: 200 mg/g of Si, 32 mg/g of Fe power plant: 10 mg/g of Si and 150 mg/g of Fe.

Determine contributions of power plant and soil dust to measured PM2.5 at the receptor.

Chemical Mass Balance

• Solution:Write mass balance equations for both chemical components

2.58 20% 1%

3.84 3.2% 15%

total

total

Si S P

Fe S P

= = +

= = +

Solve the two-equation set：

S: PM2.5 from soilP: PM2.5 from power plant

S=12 μg m-3 and P=18 μg m-3

Chemical Mass Balance

• A unique solution can be obtained if the number of equations (chemical species) equals the number of unknowns (sources).

• There is usually no solution: if the number of equations is more than the number of unknowns (overdetermined system) – This is common.

• There is usually infinite number of solutions: if the number of equations is fewer than the number of unknows (underdetermined system)

Chemical Mass Balance

• How do you obtain source apportionment if there is no solution for the equation set?

1 11 12 1 1 1

2 21 22 2 2 2

1 2

...

...

... ... ... ... ... ... ...

...

i ij j i

m

m

n n n nm m m

c a a a s e

c a a a s e

c a a a s e

= +

= +

c A s e

Measured species concentrations

Source profiles (Speciation profiles) Matrix

Sourcecontributions

Residual

Chemical Mass Balance

• Ordinary linear least-squares (OLS) method.

Define an objective function based on the residual vector e:

φ2 =

i=1

n

ei2 =

i=1

n

Ci −

j=1

p

aijSj

2

The most probable solution of S is the one that minimize the objective function φ2. The solution can be written in matrix form:

𝐒 = 𝐀T𝐀−1𝐀T𝐂

Chemical Mass Balance

• Consider the fact that some species have larger measurement uncertainty than other species, the objective function can be modified to use weighted residuals

φ2 =

i=1

nCi −σj=1

m aijSj2

σi2

The most probable solution of S in OLS sense is:

𝐒 = 𝐀T𝐖𝐀−1𝐀T𝐖𝐂

W is a diagonal matrix with 𝜎𝑖2 on the diagonal.

𝛅𝐒 = 𝐀T𝐖𝐀−1𝟏

Chemical Mass Balance

• Effective variance least-squares method.

The aerosol source compositions are also measured quantities and have uncertainties associated with them.

φ2 =

i=1

nCi −σj=1

m aijSj2

σC,i2 + σj=1

m σa,i,j2 Sj

2 =

i=1

nCi −σj=1

m aijSj2

Veff,i

The most probable solution of S in OLS sense is:

𝐒 = 𝐀T𝐖𝐀−1𝐀T𝐖𝐂

W is a diagonal matrix with Veff,i on the diagonal.

How to solve this?

Chemical Mass Balance

• Effective variance least-squares method - Iterative solution.

Since the effective variance contains the unknown quantities S, it needs to be solved using an iterative approach.

(1) First, it is solved with OLS without considering uncertainties in the source profiles.

(2) The resulted S are used to determine Veff,i, and a new solution of S can be obtained.

(3) Step (2) is repeated until there is little change of S.

Chemical Mass Balance

• Muticollinearity problem

When two sources have nearly identical composition or one source composition is almost identical to a linear combination of other sources, the least-square solution is mathematically unstable, i.e. small errors in the measurements will be magnified into large errors in calculated source contributions.

Chemical Mass Balance

• Example - Multicollinearity

𝐴 =

0.1 0.11 0.40.5 0.48 0.30.2 0.21 0.050.2 0.20 0.25

𝐶 =

1.522.360.771.35

Ignore measurement uncertainties in ambient samples and source profiles,

𝐶′ =

1.502.400.801.35

𝐒 = 𝐀T𝐀−1𝐀T𝐂 =

−9.6031 23.0187 −44.3596 −3.38558.4124 −22.0746 47.4383 3.54212.2549 0.0980 −2.0588 0.6863

1.522.360.771.35

=123

𝐒′ = 𝐀T𝐀−1𝐀T𝑪′ =

−9.6031 23.0187 −44.3596 −3.38558.4124 −22.0746 47.4383 3.54212.2549 0.0980 −2.0588 0.6863

1.502.400.801.35

=0.782.372.90

Less than 2% difference

~20% difference in source apportionment results!

Chemical Mass Balance

• Profile variabilityEven though each set of selection can lead to satisfying results numerically, large variations in source contributions can happen.

Subramanian, et al., AE, 40, 8002-8019, 2006

Software for CMB

• US EPA https://www3.epa.gov/scram001/receptor_cmb.htm

Multivariate models

• Multivariate models deal with a series of measurements of aerosol component during a sampling period. The mass balance equations can be written as

Ci,k =

j=1

m

ai,jSj,kk: sample index

The multivariate models use Ci,k with the objective of estimating the number of sources m, and more ambitiously, determining aij and Sjk

Multivariate models

• The traditional CMB could not incorporate additional information contained in multiple samples measured at different times with varying source emissions.

http://people.clarkson.edu/~phopke/PMF-Guidance.htm

Manual multivariate analysis

Only one source contribute to Fe and Si

Soil?

High K, low Fe --> fire?

High Fe, low K --> soil?

Soil, fire mixture

High S from long range transport

High absorbing fire particles

Principle Component Analysis (PCA)• Eigenvalue analysis of the covariant matrix

𝑐𝑜𝑣( റ𝑥, റ𝑦) =σ𝑛=1𝑛 𝑥𝑖 − ҧ𝑥 𝑦𝑖 − ത𝑦

𝑛 − 1covariance

covariance matrix – covariance is for 2-dimension data. If the dataset is n dimension, a covariance matrix can be formed by selecting data from the ith dimension and the jth dimension for all permutations of (i,j)

For example, for a series of aerosol samples for K, Si, Fe, the following covariance matrix C can be calculated:

𝐶 =

𝑐𝑜𝑣(𝐾, 𝐾) 𝑐𝑜𝑣(𝐾, 𝑆𝑖) 𝑐𝑜𝑣(𝐾, 𝐹𝑒)𝑐𝑜𝑣(𝑆𝑖, 𝐾) 𝑐𝑜𝑣(𝑆𝑖, 𝑆𝑖) 𝑐𝑜𝑣(𝑆𝑖, 𝐹𝑒)𝑐𝑜𝑣(𝐹𝑒, 𝐾) 𝑐𝑜𝑣(𝐹𝑒, 𝑆𝑖) 𝑐𝑜𝑣(𝐹𝑒, 𝐹𝑒)

Principle component analysis (PCA)• Eigenvalues λ and eigenvectors (𝒙)

𝑨𝒙 = λ𝒙Prefix eigen is from German, meaning “proper” or “characteristic”

𝑨 − λ𝑰 𝒙 = 0Non-zero solution of x exist only when det 𝑨 − λ𝑰 = 0

For example: 𝐴 =2 11 2

𝑑𝑒𝑡 𝑨 − λ𝑰 =2 − λ 11 2 − λ

= 3 − 4λ + λ2 =0

𝑥 λ = 1 =1−1

𝑥 λ = 3 =11

Principle component analysis (PCA)• Eigenvectors and the corresponding eigenvalues of

the covariant matrix of the species (mean-subtracted) can be solved using a numerical analysis package such as MATLAB.

• The number of real eigenvalues represents the number of factors needed to explain the observation dataset. The importance of each factor is represented by the numerical value of the eigenvalues.

Principle component analysis (PCA)• Limitation of PCA

• The eigenvectors usually do not carry physical meaning.

• It is good for identifying the number of major factors (i.e. sources) but cannot directly provide source contributions results.

Positive Matrix Factorization

• PMF solves the mass conservation equations by minimizing the a weighted objective function given by:

Q =

i=1

n

𝑘=1

𝑚Ci,𝑘 − σj=1

paijSj,k

2

σi,k2

p: number of sourcesn: number of speciesm: number of measurements

Potential source contribution function (PSCF)• Potential source region is divided into a gridded array of cells

with grid size ∆x, ∆y.

• Calculate a large number (M) of back-trajectories. Each trajectory has n end-points with a time increment of ∆t. The total time interval for one trajectory is n∆t, and the total time of all trajectories is T=M*n*∆t=N ∆t. N is the total number of trajectory end points.

• For each grid cell (i,j), ni,j represents the number of trajectory end points are in the grid cell (i,j).

• The probability Pi,j represents the residence time distribution of air parcels arriving at the receptor at each grid cell (i,j).

𝑃𝑖,𝑗 ≅𝑛𝑖,𝑗∆𝑡

𝑁∆𝑡=𝑛𝑖,𝑗𝑁

(It is easy to see that σ𝑖σ𝑗 𝑃𝑖,𝑗 = 1)

Potential source contribution function (PSCF)

Ashbaugh, et al. 1985, AE, 19, 1263-1270

Potential source contribution function (PSCF)• It is also possible to define the probability Pi,j of a

subset of the trajectories at the receptor site. For example, the subset can be those trajectories with concentration C>Ccutoff at the receptor. The probability is denoted as P[S]i,j

𝑃[𝑆]𝑖,𝑗 ≅𝑚𝑖,𝑗∆𝑡

𝑁∆𝑡=𝑚𝑖,𝑗

𝑁

where m is the number of trajectory end points in grid cell (i,j) for the subset of trajectories

Potential source contribution function (PSCF)

Ashbaugh, et al. 1985, AE, 19, 1263-1270

Potential source contribution function (PSCF)• For both Pij and P[S]ij, maximum density usually

occurs at the receptor site where all trajectories converge. It will be useful to remove this inherent peak, in a way that best defines the regions where air parcels are most likely to entrain air pollutants.

• One common approach is to normalize P[S] with P,

PSCF𝑖,𝑗 =𝑃[𝑆]𝑖,𝑗

𝑃𝑖,𝑗=𝑚𝑖,𝑗

𝑛𝑖,𝑗

This conditional probability represents the probability of an air parcel will arrive at the receptor with high concentration.

Potential source contribution function (PSCF)

Ashbaugh, et al. 1985, AE, 19, 1263-1270

Grid cells with emission sources would be identified with conditional probability close to 1 if trajectories that have crossed the cells effectively transport the pollutants to the receptor site.

Potential source contribution function (PSCF)• Software for back-trajectory, PSCF and other types

of back trajectory analysis• HYSPLIT: https://ready.arl.noaa.gov/HYSPLIT.php

• TrajStat (Wang et al., 2009): http://www.meteothinker.com/products/trajstat_features.html

• MetCor (Sofowote et al., 2010) https://www.chemistry.mcmaster.ca/faculty/mccarry/MC_downloads.html

Constraining the emission factors using observations

Ying et al., Science of the Total Environment, 624, 355-365.

Robust regression to solve emission factors

http://www.mathworks.com/help/stats/robustdemo.html

m s

, ,

N N 2

1 1 ,( )m jm m j j i m iQ w S a c= =

= −

Observation-model contrainedemission factors

SpeciesDust Residential Transportation Power Industryµ σ µ σ µ σ µ σ µ σ

POC 8.92 10.81 51.03 4.76 22.28 14.39 20.56 17.20 19.78 10.75EC 6.06 4.44 14.84 1.79 13.36 5.55 9.49 7.12 13.98 4.18Na+ 3.37 0.73 1.38 0.24 1.17 0.71 1.67 0.83 1.20 0.53Al 3.17 0.54 0.04 0.07 0.77 0.31 0.71 0.38 0.82 0.23Si 8.74 1.67 0.02 0.23 1.63 0.88 1.46 1.10 1.73 0.64Cl- 21.08 9.24 11.27 2.01 5.42 6.78 20.56 9.17 5.67 4.87K+ 6.34 1.83 3.21 0.69 4.26 2.29 7.78 2.76 5.63 1.76Ca 7.33 0.80 0.00 0.10 0.95 0.45 1.33 0.63 1.13 0.33Ti 0.37 0.05 0.01 0.01 0.07 0.03 0.08 0.04 0.08 0.02Cr 0.04 0.01 0.00 0.00 0.02 0.01 0.02 0.01 0.02 0.01Mn 0.39 0.08 0.08 0.02 0.16 0.07 0.22 0.09 0.18 0.05Fe 4.16 0.89 0.56 0.23 1.99 0.84 2.91 1.08 2.72 0.63Cu 0.08 0.05 0.06 0.01 0.10 0.04 0.13 0.05 0.13 0.03Zn 1.83 0.69 0.60 0.13 1.04 0.53 1.84 0.68 1.55 0.40As 0.25 0.08 0.03 0.02 0.07 0.06 0.10 0.08 0.07 0.05Ba 0.39 0.10 0.02 0.02 0.12 0.07 0.08 0.08 0.13 0.05Pb 0.15 0.26 0.56 0.09 0.49 0.29 1.02 0.37 0.56 0.21

gram per 100 gram of primary PM emission

Results: Predicted vs. Observed Trace Elements

Beihang Site, annual averageMean fractional bias and error based on daily concentrations

Improvement in trace element predictions

Beihang Site, annual averageMean fractional bias and error based on daily concentrations

Regional source contributions to Fe

Co

ncen

tration

(µg m

-3)

➢ Predicted Fe concentrations are high in the northwest dust source regions and in central and north China.

➢ Industrial, transportation and power sectors are all important contributors to Fe.

➢ The high concentrations of Fe may have a significant impact on secondary sulfate formation.

Regional source contributions to Pb

➢ Residential, power and transportation sectors are predicted to be the major source of Pb.

➢ Since a portion of the residential emissions is also due to coal combustion, coal combustion is expected to be the dominant source of Pb in suburban and rural areas.

➢ In urban areas, industrial and vehicle sources are also important.

➢ May have significant health implications