1
Chapter 1
Introduction
As far back as the seventeenth century, the air quality in some European cities had
degraded to the point where it became a health hazard. In 1661, Evelyn conducted a
study of the air quality at London, England. From his findings, he suggested that the air
pollution in London could be lessened by using taller chimneys and moving polluting
industries 5 to 6 miles from London to provide for better dispersion of pollutants before
reaching London (DeMarrais, 1979). This was among the first realizations that air quality
is dependent upon both emissions and meteorological conditions.
Today, this association is known as the source-receptor relationship (SRR).
Specifically, the SRR is the quantification of the contribution of a source's emissions
upon a receptor's concentration. The SRR is dependent on meteorological transport and
physico-chemical transformation and removal rate processes. The understanding of the
SRR has become a central issue in air quality management. Only when the identification
of relevant sources and their impact upon the receptor are known, can a rational control
policy be developed that will minimize costs of implementation and attain desired results.
The quantification of the SRR has been an active field of research for many years,
and continues to be developed. Today, the primary processes involved in the relationship
are understood. Based upon this understanding, a number of methods and techniques
have also been developed for the quantification of the absolute and relative impact of a
source on a receptor, and the simulation of receptor concentrations.
An important factor influencing the ambient concentrations is the emission
inventory. Without proper emission inventories, the absolute source contributions cannot
be quantified, and any control policies implemented will most likely not attain the desired
2
results. The importance of the emission inventories has lead to considerable effort in the
creation of reliable inventories. Current emission inventories have generally been created
using source oriented approaches. These approaches include the measurement of source
emission rates (Pierson and Branchaczek, 1983; Dilts et al., 1991; Stephens and
Groblicki, 1991), and estimation techniques via emission models (Pope and Lynch,
1991). The emission models are generally based upon emission factors, material
balances, and activity factors (Husar 1994; Fieber et al. 1991).
The primary problem of many of these emission inventories is their large
uncertainty. There is even concern that emission inventories may be the least accurate of
the major inputs to photochemical models (Seinfeld, 1988). The uncertainty of the
emission inventories is especially a problem for natural emissions, such as biogenic
emissions and soil dust. In one assessment of the uncertainties in a biogenic emission
model, Cheung et al. (1991) found that slight modifications to the model inputs could
result in variations of up to 50% between estimated emissions. Emission estimates from
motor vehicle also have large uncertainties. Studies comparing results from motor
vehicle emission models to tunnel data indicate that the estimated emissions
underestimate measured data by factors of 2 - 4 (Taylor, 1991). The concern of the
uncertainties of current emission inventories was voiced at a recent conference on acid
rain. One of the conclusions from this conference was that better quantification of
anthropogenic emissions were needed, especially of NOX, NH3, base cations such as Ca+,
and VOCs (Rodhe, 1995).
Since the 1970's, there has been extensive monitoring of ambient concentrations
and precipitation chemistry throughout the US. The availability of the receptor data has
spurred the development of techniques for the identification of source types and regions,
and the quantification of emission fields from receptor data. These techniques essentially
infer source emission patterns from the receptor data and known SRR.
3
These receptor oriented approaches have many uses and benefits, including the
verification and improvement of established emission inventories, and the identification
and quantification of unknown source regions and emission rates. Due to these benefits,
there has been considerable development of these receptor oriented approaches as
reviewed below.
1.1 Background
The development of receptor oriented approaches for the identification of source
types and regions, and the quantification of emission fields has followed three paths. One
technique, receptor models, uses only receptor data to perform source apportionment.
The second group of techniques uses receptor data combined with airmass histories to
identify and apportion source contributions to the receptor concentrations. The third
technique employs a dispersion model to perform inverse air pollution modeling for the
estimation of emission fields using receptor data. These three receptor oriented
approaches are further discussed below.
1.1.1 Source Type Identification Using Receptor Models
Receptor models use ambient monitoring data to estimate the contribution of
various source types to receptor concentrations. Although, there are a number of different
receptor modeling techniques, they are all generally based upon the model:
ci =
where ci
sj
fij

j
fij*sj
(1-1)
Receptor concentration of species i [g/m3]
Source strength. The estimated contribution of source j to the receptor
concentration i measured at the receptor. [g/m3]
Source profile or finger print. The mass fraction of species i from
emission of the jth source as measured at the receptor site. [g/g]
One of the first receptor models was the chemical mass balance, CMB, and was
developed by Winchester and Nifong (1971) and Hidy and Friedlander (1972). This is a
4
simple and straight forward method that can be applied to individual ambient samples. It
assumes that the receptor species concentrations and source profiles are known, and sj is
determined via inversion of fij in Equation 1-1.
Factor analysis is a more flexible apportioning scheme that can be used when
multiple data samples are available (Hwang et al. 1984; Hopke, 1985; National Research
Council, 1993). Instead of having to determine the fij a priori as in CMB, factor analysis
determines them from the data. Thus, it can be used to check and refine source profiles
used in a CMB. The determination of the source profiles are based on the concept that
two aerosol species from the same source should correlate well together. However,
usually the same chemical species will be emitted by multiple sources, and some a priori
knowledge of the source profiles is necessary to obtain the proper fij matrix.
A problem with CMB and factor analysis is that they do not account for secondary
species or aerosol growth due to water. This can lead to a large unaccounted fraction of
the fine mass (White, 1986). However, secondary aerosols can be taken into account by
estimating their total mass from measured data (Friedlander, 1977; Dzubay, 1980). The
source of the gases responsible for the secondary aerosol are not identified.
Linear regression is another receptor modeling technique that has been widely
used. It can take into account secondary aerosols and growth of aerosols due to water
condensation during transport (Belsley et al. 1980; Kleinman et al. 1980). This method
works by estimating coefficients of independent variables, and tracers, in a linear
relationship using multiple data samples.
These receptor models have several drawbacks. First, they rely on tracers to
identify sources and to be scaled up to yield source mass contributions to the receptor
concentrations. To obtain reliable results, these chemical tracers must be stable and
measurable species. This is not always the case in many remote areas such as National
Parks. Also, it is difficult for receptor models to be able to identify specific sources or
source areas.
5
1.1.2 Source Region Identification Using Airmass Histories
Airmass histories trace the pathway or trajectory of an airmass prior to its arrival
at a receptor site. When used in conjunction with receptor concentrations, potential
source areas of air pollutants and their contributions to the receptor concentrations can be
identified.
Various statistical methods to identify source regions using long term
concentrations and their respective airmass histories have been developed. Miller (1981)
used airmass histories to create flow climatologies by classifying the airmass histories
based upon their transport speed and sector of travel. This technique allows the
identification of regions of influences upon a receptor site. White et al. (1994) used this
method in conjunction with fine sulfur aerosol and extinction measurements to identify
the transport over sectors associated with high and low concentration and extinction
values. Moody and Samson (1989) developed a trajectory classification technique using
cluster analysis, where airmass histories with similar pathways and transport speeds were
grouped together. They used this analysis to determine what fraction of chemical
variability in precipitation composition could be related to differences in atmospheric
transport.
Flow climatologies and cluster analysis can only identify broad source regions
influencing the receptor concentrations. Ashbaugh (1983), Ashbaugh et al. (1985), and
Poirot and Wishinski (1986) developed a technique specifically designed to identify
source regions, known as the residence time analysis. In the residence time analysis, a
grid is superimposed over the spatial domain and the residence time of all airmass
histories within each grid cell is accumulated, then normalized by the total residence time.
Combining the residence time analysis with receptor concentrations, conditional
probability densities can be created that identify those source regions which influence
high and low receptor concentrations (Ashbaugh, 1983; Gebhart and Malm, 1991;
Vasconcelos, 1995). This technique is known as conditional frequency analysis (CFA).
6
Vasconcelos (1995) extended the residence time analysis by developing a technique that
calculates statistical significance for the features of the residence time probability
densities.
Seibert et al. (1994) developed a similar method to the residence time analysis.
They calculated weighted logarithmic mean concentrations for each grid cell from
receptor concentrations and airmass histories. The concentration for each grid was equal
to the average of all of the receptor concentrations with airmass histories passing over the
grid cell, and weighted by the residence time of the airmass history in the grid cell.
Therefore, grid cells that have airmass histories associated with high receptor
concentrations passing over them, and having high residence times, will have higher
calculated concentrations than grid cells with airmass histories associated with low
receptor concentrations passing over them having high residence times. The grid cells
with high calculated concentrations are assumed to be the pollutant source regions. Stohl
(1995) refined this technique by introducing an iterative procedure that distributes the
receptor concentrations along the corresponding trajectories in an unequal fashion. The
identified high emission regions have a higher fraction of the receptor concentrations than
the regions of low emissions. Using measured sulfate data, Stohl demonstrated that the
new technique was able to identify many of the known SO2 source regions in Europe with
higher spatial resolution than the method by Seibert et al. (1994).
1.1.3 Source Field Reconstruction using Dispersion Models
Dispersion models include the simulation of the transport of pollutants as well as
kinetics processes. If source emission rates are known, then the model can be used to
simulate receptor concentrations and deposition fields. If the receptor concentrations are
known, then the dispersion model can be used to estimate the emission field. This
essentially entails running the model "in reverse," tracing the path of the pollutants at the
receptor back to their many sources, and reversing the roles of dilution and removal
processes so that they become additions to the polluted airmass.
7
All techniques for the inference of information about pollutant emission fields
from receptor data are inversion processes, and are subjected to possible unbounded error
amplification of measurement and model errors due to the problem being "ill-posed." An
ill-posed problem is a continuous problem without a unique stable solution. For a
discrete problem it is known as ill-conditioned. Jackson (1972), Parker (1977), Allison
(1979), and Tarantola (1987) discuss the theory and solution techniques behind the
general inverse problem and error amplification due to ill-posedness and ill-conditioning.
Enting (1985), Newsam and Enting (1988), and Enting and Newsam (1990) have
examined the mathematical implications of ill-posedness as applied to the retrieval of
emission fields. They have found that the retrieval of surface concentrations is a mildly
ill-posed problem, and is dependent upon the advective and diffusive rates of the
atmosphere and the availability of measurements. The degree of error amplification was
a function of the spatial and temporal resolution of the retrieved emission fields. The
adverse affects of the ill-posedness can be minimized by incorporating a priori
information. Chapter 4 provides an in-depth discussion of the error amplification
encountered from the retrieval of emissions from a discrete integral equation.
The methods for emission estimation from receptor data fall into three categories:
trial and error, mass balance, and optimal estimation methods. Using a trial and error
technique, Fung et al. (1991) reconstructed global methane emission fields. The
reconstruction employed the use of a global three dimensional tracer transport model,
observed data, and current knowledge of the location and source types and sinks of
methane. Using the known information of methane emissions as constraints, candidate
emission fields were constructed. Using the model and candidate emission fields the
observed data were simulated. The emission fields were adjusted to reduce the residuals,
the difference between the simulated and observed data. Six methane emission scenarios
were found that could equally approximate the observed data. The inability to obtain a
unique solution is an illustration of the ill effects of the ill-conditioned problem.
8
The mass balance approach uses a set of differential equations relating the change
of concentrations with time in "reservoirs" to the fluxes between the reservoirs, losses due
to kinetic process, and additions from sources:
ci (t )
  Lij ci (t )  Ri (t )  Si (t )
t
j
where ci(t)
Lij
Ri(t)
Si(t)
(1-2)
is the concentration at the reservoir i
is the flux between reservoir j and reservoir i
is the removal rate of mass in reservoir i due to kinetic processes
is the emission rate of mass into reservoir i
The differential equations are then numerically solved for the sources. This technique has
been applied to the retrieval of emissions on the global scale for long lived pollutants,
such as CO2 (Tans et al. 1989; Enting and Mansbridge, 1989; Enting and Mansbridge,
1991) and chlorofluorocarbons (Fraser et al. 1985).
Most optimal estimation techniques retrieve emission fields via the inversion of
the discrete form of the SRR using methods such as least squares. The discrete form of
the SRR can be written as:
cil = Vil-1
where cil
ejk
Tiljk
time
V

j
k
Tiljk *ejk
(1-3)
Receptor concentration of species i at time l. [g/m3]
Emitted mass from source j at time k. [kg]
Transfer matrix: the probability of emissions from source j released at
k impacting the receptor i at time l.
Volume of receptor i at time l. [km3]
In one study, Yamartino and Lamich (1979) developed a source finding algorithm based
upon least squares principles. They applied the algorithm for the retrieval of temporally
constant CO emissions at the Williams Air Force base. They used CO monitoring data
over a 13 month period, and calculated the transfer matrix from a non kinetic Gaussian
plume model. They were able to identify the locations and strengths of the sources with
the largest contributions to the monitoring sites.
9
Prahm et al. (1980) developed a technique for the quantification of emissions on a
regional scale. This technique used a two dimensional receptor oriented Lagrangian
model for the calculation of the transfer matrix, which was inverted using least squares.
The technique was applied for the retrieval of temporally constant SO2 emissions from 37
source regions over Western Europe. The input data consisted of more than two years of
SO2 and SO42- concentrations at 45 receptors over Western Europe. The results compared
favorably with European emission inventories. This work is significant, because it is the
only study conducted on the regional scale, and it used a secondary species in order to
retrieve the emission field.
Using a global two dimensional chemical transport model, Brown (1993)
retrieved estimates of CFC11, methylchloroform, and methane emissions from their
ambient concentrations. Instead of using Equation 1-3, a set of linear equations were
created by relating the change in receptor concentrations due to changes in emission rates
by a Jacobian matrix containing the variation at each receptor for an increment in each
source. The Jacobian matrix and the change in receptor concentrations due to a fixed
change in the emission rates were calculated from the model. The system was inverted
with equal numbers of knowns and unknowns for the determination of the change in the
emissions that would fit the observed concentrations. Repeating the inversion over
different time periods allowed for the retrieval of temporal variations in the emission
field.
The Kalman filter has also been used for the retrieval of emissions. The Kalman
filter is a recursive least squares approach that allows for the carrying over a portion of
the information from one inversion to another. The rates of variation of the unknown
parameters and measures of the goodness of fit can be controlled. This is particularly
useful where parameters are slowly varying in time, as is the usual case with pollutant
emissions. Mulholland (1989) estimated SO2 emissions at nine power stations from the
time series of SO2 concentrations at eight receptors using a Kalman filter to invert
Equation 1-3. Hartly and Prinn (1993), using a similar set of linear equations as Brown,
10
employed the Kalman filter to retrieve global emissions of CFCL3 from surface
concentrations. They assumed that the emission rates were constant over a three month
period of time and used the Kalman filter in an iterative process seeking convergence of
simulated and measured concentrations as the emissions were continually updated with
each iteration of the Kalman filter.
In a recent study by Mulholland and Seinfeld, a Kalman filtering technique was
developed to find spatial and temporal adjustment factors to an existing CO emission
inventory in the South Coast Air Basin of California. The use of an existing emission
inventory greatly reduced the error amplification due to ill-conditioning of the problem.
This allowed for the retrieval of higher spatial and temporal resolution of the emission
fields.
1.2 Scope of Research
The primary goal of this project is the development of a method for the
reconstruction of emission fields from ambient monitoring data and a dispersion model.
In attainment of this goal, the following objectives will be accomplished:

Formulate an expression relating source emissions to receptor concentrations, i.e., the
source receptor relationship (SRR).

Employ a dispersion model for the calculation of transport and kinetic parameters in
the SRR.

Develop and test an optimal estimation technique for the retrieval of emission fields
from the SRR using the known ambient monitoring data.

Validate the procedure by retrieving North American seasonal SO2 emission fields
from SO42- concentration and wet deposition data.

Apply the retrieval process for the reconstruction of North American seasonal NH3,
and NO2 emission fields from NH4+ and NO3- wet deposition data.
11
Chapter 2 presents the mathematical development of the Lagrangian formulation
of the SRR. Chapter 3 presents the CAPITA Monte Carlo model. This is a Lagrangian
dispersion model for the simulation of regional scale transport, transformation, and
removal of atmospheric pollutants for the calculation of concentration and deposition
fields. The model is also used to calculate the transport and kinetic parameters in the
SRR. In Chapter 4, the retrieval process is developed, tested, and used to retrieve the
SO2, NH3, and NO2 emission fields. Results from this study are summarized in Chapter
5.
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Chapter 2
Physical Principles of the General Source
Receptor Relationship
The formulation of the source receptor relationship (SRR) can be derived from
either an Eulerian or Lagrangian point of view. The Eulerian viewpoint defines a
coordinate system that is fixed with respect to the ground. The conservation equations
describing the behavior of species are then applied to fixed control volumes. In the
Lagrangian viewpoint, the coordinate system is not fixed, but attached to moving fluid
particles.
The formulation of the SRR in the Eulerian framework is favorable because it can
incorporate all of the physical and chemical rate processes in the control volumes, i.e.
species transformation and removal, and between control volumes, i.e. transport and
diffusion. The chemical reactions within a control volume may involve "nonlinear"
interactions of species from multiple sources. Consequently, its solution can yield the
spatial temporal pattern of dispersing and chemically reacting pollutants. The primary
drawback of the Eulerian models is that source apportionment cannot conveniently be
done, and the specific roles of transport, transformation, and removal in the SRR can not
be explicitly computed. Also, incorporating high resolution meteorology and chemistry,
the computer power required for the solution makes them suitable only for research type
applications that are limited in spatial and temporal scope (e.g. a few days or weeks per
year).
In the Lagrangian formulation, all of the transport and kinetic processes are
applied to the individual fluid particles. The primary problem with this approach is that it
does not easily lend itself to problems involving nonlinear chemical reactions. However,
performing source apportionment, and determining the specific roles of the transport
13
transformation, and the removal processes in the SRR are easily accomplished. Because
of these unique abilities, the Lagrangian approach was used in this study. The next
section develops the concepts and equations of the SRR from the Lagrangian viewpoint.
2.1 Lagrangian Formulation of the Source Receptor
Relationship
The development of the Lagrangian formulation can be seen from the standpoint
of a single particle in the turbulent atmosphere. In the turbulent atmosphere, the exact
movement of the particle is not known. Hence, if a particle is at the location X' at time t',
the position of the particle X at some latter time, t can not be predicted exactly. This is
demonstrated in Figure 2-1, where a single particle released from the Grand Canyon in
the turbulent atmospheric boundary layer is tracked for 36 hours. The position of the
particle is marked at several times after its release from the source. This process was
repeated 50 times. Each time the particle followed a different path from the source, but
was transported in the same general direction, north of the Grand Canyon. Thirty six
hours after impacting the receptor, the particle could be located anywhere in a region
stretching from Washington to Montana, well into Canada.
In Figure 2-1, it is evident that while the exact transport of a particle in turbulent
flow is not known, there are higher probabilities that the particle will reside in one region
as opposed to another. For example, there is a high probability that the particle will
reside in Utah twelve hours after its release, but a low probability that it will be in Texas.
This probability of particle transport can be described by the transit conditional
probability density function, Pt(X,t | X',t'), which has units of m-3. This function gives the
probability that a particle located at position X' and time t' will be transported to the
position X at time t. Pt is a probability density, because the larger a volume of fluid is,
the higher the probability that the particle will be located somewhere in that volume at
14
time t. If the volume is infinite in size, the probability that the particle is within this
volume is 1:
  
   P  X , t X' , t ' dX  1
t
(2-1)
  
Figure 2-1. The position of 50 particles 6, 12, 24, and 36 hours after being released from a surface level
source at the Grand Canyon on 1/15/1992 00:00.
If the exact position of a particle is not known, but there is a probability density
function (X', t') for finding the particle at position X' and time t' then the probability of
finding a particle at X at t can be described by:
 X , t 
  
   P  X , t X ' , t '  X ' , t ' dX '
t
(2-2)
  
The function (X,t) describes the probable location of one particle. If however, n
particles exist, then the position of the ith particle can be described by the function i (X,
t). It has been shown by Lamb (1971) that the ensemble mean concentration at the point
X can then be defined by:
n
c X , t   i  X , t 
i 1
(2-3)
15
If each (X, t) in Equation 2-3 is replaced by Equation 2-2, and the (X',t') is replaced
with an initial particle distribution at Xo and to and particle sources, with units of
particles/volume/time, then Equation 2-3 becomes:
c X , t  
  
   P X,t X
t
o
, t o  c X o , t o  dX o
  
   t
+
    P  X , t X ', t ' S  X' , t ' dt ' dX'
t
(2-4)
   to
where c
Pt
S
X
X'
t
t'
receptor concentration
transit probability density function
source emission rate
receptor location
source location
receptor time
source release time
[particles/m3]
[1/m3]
[particles /m3/sec]
[sec]
[sec]
This equation represents the source receptor relationship for a conservative
species. The first term on the right represents those particles present at time to. The
second term accounts for particles added from to to t by the sources at X'. A similar
development was first put forth by Lamb and Neiburger (1971) and Lamb and Seinfeld
(1973).
This equation can be better understood by examining it under restrictive
conditions. The simplest form of Equation 2-4 can be seen by considering a single source
at X', that releases a puff of pollution over the time t' + t, such that S(X', t') = M/t/V,
where M is the pollutant mass released. Then the impact of the source at every point X
and time t is:
c X , t   Pt  X, t X' , t ' M
(2-5)
If the receptor encompasses a finite volume X then the concentration is found by
averaging Equation 2-5 over the volume:
 M
c  X, t   
 X

X

Pt  X, t X' , t ' dX 

(2-6)
16
If the source has continuous emissions, such that S(X', t') = q /V where q is the
source strength [g/sec], then the concentration at every point X and time t can be defined
by:
t
c X , t    Pt  X, t X' , t ' qdt '
(2-7)
to
where to is the initial release time of the source. At any receptor volume X and time t
there is a probability that emissions that were released from the same source at different
times can impact it. This equation demonstrates the inherent time lag between the source
emissions and receptor impact. If multiple sources are added to the scenario, then the
resulting equation would be equivalent to Equation 2-4 without an initial particle
distribution.
2.1.1 Source Receptor Relationship from the Source and Receptor Viewpoint
In the SRR there are two time dimensions, the source time, t' and receptor time, t.
These time dimensions can be related to each other via the particle age . Using , the
SRR can be defined from a source viewpoint, where t is replaced by t' +  in Equation 24, or a receptor viewpoint, where t' is replaced by t -  in Equation 2-4. When inverting
the SRR for the retrieval of emission fields it is often more convenient to view the SRR
from the receptor viewpoint. For this reason, further development of the SRR will be
conducted from the receptor viewpoint. The SRR from the receptor viewpoint is:
c X , t  
  
   Pt  X , t X o , t   o  c X o , t   o 

   0
+
dX o
    Pt  X , t X ', t    S  X' , t    ddX'
(2-8)
  o
2.1.2 Kinetic Species
The source receptor relationship, as defined by Equation 2-8, does not account for
multiple species or the kinetic processes affecting the species. Multiple species can be
17
included in the equation by adding a species index p to the receptor concentration term
and a species index p' to the source term. The indexes p and p' each refer to the same set
of species. Each fluid particle then contains the mass of multiple species, where the
initial mass of each species is determined by the source's emission rate.
The kinetics responsible for dry and wet deposition and the transformation
between the species, can be incorporated into the equation via the kinetic probability,
Pk(p,X,t |p',X',t-). The kinetic probability determines the likelihood of the initial mass of
species p', when released from the source, as remaining as species p', being transformed
to a new species p or being dry or wet deposited when the particle has age . The kinetic
probability is described by a system of pseudo first order kinetic rate equations applied to
each particle. In the SRR, the transit conditional pdf is multiplied by the kinetic
probability.
The mass of species p at the receptor is a combination of primary and secondary
contributions. The primary contribution is the product of the initial particle mass of
species p at the source times the probability that it was not removed during transport to
the receptor. The contribution of a secondary species is the product of the initial mass of
the precursor species p' times the probability that p' was transformed to p and not
removed during the transport from the source to the receptor, and times the ratio of the
molecular weight of p to p'. If there are multiple reactants then the total secondary
contribution is the sum of the secondary contribution from each species p'. In the
notation of the kinetic probability, the probability of primary mass contribution is defined
by p = p', and the probability of secondary mass contributions is defined by p  p'. The
mass of the particle in the form of species p, assuming equimolar conversions, is then:
18
Primary contribution to species p



m p, X , t   Pk  p, X , t p , X' , t    m p, X' , t   o 
 Pk  p, X , t p'1 , X' , t    m p'1 , X' , t   o 
Mp
M p'1
Mp
 Pk  p, X , t p'2 , X' , t    m p'2 , X' , t   o 
......
M p' 2


Secondary contributions to species p
or in compact form:
m p, X , t    Pk  p, X , t p' , X' , t   m p', X' , t   o 
p'
Mp
M p'
(2-9)
where m(p,X,t)
mass of species p in the particle at location X and time t
Mp, Mp'1, etc. molecular masses of species p, p'1, etc.
Pk
Kinetic probability
While this equation assumes equimolar conversion of reactant p' to product p, it can
easily be generalized for an arbitrary reaction stoichiometry.
As an example, consider the system where at a particle's release from a source, it
contains only sulfur where 90% of the sulfur is in the form of SO2 and 10% of the sulfur
is in the form of SO42-. The pseudo first order kinetic rate equations defining the fraction
of the initial sulfur in the particle in the form of SO2, fSO2 and SO42-, fSO4 at age  are:
df SO2
  kt  kr 2  f SO2,
dt
df SO4
 kt f SO2  kr 4 f SO4
dt
where f SO2 & f SO4
kt
kr2
kr4
t
t-o
f SO2  f SO2 o = 0.9 @ t   o
(2-10)
f SO4  f SO4o = 0.1 @ t   o
(2-11)
fraction of initial sulfur in the form of SO2 and SO42respectively
transformation rate of SO2 to SO4
[1/sec]
deposition rate of SO2
[1/sec]
2deposition rate of SO4
[1/sec]
receptor time
[sec]
2source release time of SO2 the SO4
[sec]
19
The solutions are:

Pk SO 2 , X , t SO 2 , X ', t 
f SO2 ( t )  f SO2o
f SO 4 ( t )  f SO 2o

 k  k  
e  t r2 
 kt 

(2-12)
e 

 k r 4   

 e  t r2 
 f SO 4 o
 kr 2  kr 4
kt


 k k
Pk SO 4 , X , t SO 2 , X ', t 

 
e 
 kr 4  

(2-13)
Pk SO 4 , X , t SO 4 , X ', t 

As can be seen, there are no secondary contributions of SO2, so that the fraction of sulfur
in the particle as SO2 is dependent upon only primary contributions. The fraction of
sulfur in the particle as SO42- has contributions from both primary and secondary species
and the total mass of SO42- at time t is found through summing these contributions
together. Figure 2-2, presents the change of the three kinetic probabilities with the age of
the particle, and the corresponding sulfur budget.
The addition of multiple species and kinetic processes to the SRR does not affect
the transit pdf since all chemical species undergo the same transport within the fluid.
Introducing the kinetic probability into Equation 2-8 and generalizing for multiple
species, the mean concentration of species p at a receptor X and time t is:
c p, X , t  
  
   Pt  X , t X o , t   o  Pk  p , X , t p' , X o , t   o  c p' , X o , t   o  M p dX o
M

  
+
0
 
  o
p'
Pt  X , t X' , t     Pk  p, X , t p' , X' , t    S  p', X' , t   
where Pk
p
p'
Mp, Mp'
p'
p'
Mp
M p'
ddX'
(2-14)
kinetic probability
receptor species indexes
source species indexes
molecular masses of species p and p' respectively
This is similar to the Lagrangian formulation of the SRR by Lamb and Neiburger (1971),
Lamb (1971), Lamb and Seinfeld (1973), and Cass (1981); however, their development of
the kinetic probability is slightly different.
20
Primary SO2
1
Primary SO4
1
SO42- Pk(SO42-,X,t | SO2, X',t-age)
0.8
Kinetic Probability
Kinetic Probability
0.8
0.6
Primary SO2 & Secondary SO42Deposited
0.4
0.2
2-
Primary SO4 Deposited
0.6
0.4
Pk(SO4,X,t | SO4, X',t-age)
0.2
Pk(SO2,X,t | SO2, X',t-age)
0
0
0
1
2
3
4
Quantum Age, Days
0
1
2
3
4
Quantum Age, Days
Total Sulfur Budget
1
2-
Fraction of Total Sulfur
SO4
0.5
SO2 & SO42- Deposited
SO2
0
0
1
2
3
4
Quantum Age, Days
Figure 2-2. Kinetic probabilities and sulfur budget for a fluid particle that was released from a source with
90% of its sulfur as SO2 and 10% of its sulfur as SO42-. The rate coefficients used to create these figures are
kt = 1 %/hr, kr2 = 3%/hr and kr4 = 1.5 %/hr
Deposition Rates. The SRR in Equation 2-14 relates the impact of the emissions
of multiple chemically active species to ambient receptor concentrations. However, over
any period of time, t, a fraction of the matter comprising a particle can be dry and/or wet
deposited, creating deposition rates. The calculation of deposition rates can be included
into Equation 2-14, by incorporating a species states index q. This index identifies the
probability of the mass of species p remaining in the particle or the probability that the
mass has been dry and/or wet deposited at the surface position X during the time t+t.
Both the transit conditional pdf and kinetic probability are dependent upon the species
index. The transit conditional pdf is dependent upon a surface area as opposed to a
21
volume for deposition calculations, while the kinetic probability is dependent upon the
dry and wet deposition rate equations. The addition of the species state to the SRR
results in:
c  p, q, X , t  
  
   Pt  q, X , t X o , t   o   Pk  p , q, X , t p' , X o , t   o  c p' , X o , t   o  M p dX o
M

  
+
0
 
  o
p'
p'
Pt  q , X , t X' , t     Pk  p, q, X , t p' , X' , t    S  p', X' , t   
p'
Mp
M p'
ddX'
(2-15)
where p
p'
q
c
Pt
Pk
S
X
X'
t

t -
co
Mp, Mp'
receptor species index
source species index
species state. q identifies the fluid particles mass as being ambient,
dry deposited, or wet deposited.
receptor concentration for q = ambient species mass
[g/m3]
receptor deposition rate * t for q = deposited species mass
[g/m2]
transit probability density function
q = ambient species mass
[1/m3]
q = deposited species mass
[1/m2]
kinetic probability
source emission rate
[g/m3/sec]
receptor location
source location
receptor time
age
source release time
initial concentration field
[g/m3]
molecular masses of species p and p' respectively
2.1.3 Discretisation of the SRR
It is often difficult to obtain a functional forms of the SRR. Consequently, models
are employed to estimate the transit, Pt, and kinetic, Pk, probabilities of the SRR (see
Chapter 3). In order to estimate the SRR using a model, it is necessary to work with a
discrete form of Equation 2-15.
22
In the discrete form of the SRR, the infinitesimal source and receptor volumes
become volumes of finite size, the continuous time is treated as discrete steps in time, and
the integrals become summations. It is also necessary to average the SRR over the
receptor concentration volumes or deposition areas and time increments. The averaging
converts Pt from a conditional pdf to a conditional probability function and is dependent
upon the size of the receptor volume/area used in the averaging. The discretised SRR,
ignoring initial concentrations can be written as:
c p ,q , i , l
1

X q,i,l

X q,i,l
1
1
c p, q, X , t  dtdX 

X q,i,l
t t


X q,i,l

* Pk p , q , X , t p', X' , t   S  p', X' , t   
p'
c p , q , i, l
1

X q,i,l
where c
Pt
Pk
e
S
X a s ,i,l
X'
t'
dX q
q
p
p'
i
j
l
k
J
1    0
     P  q, X , t X' , t   
t t      o t
Mp
M p'
0
Mp
j 1 k  K p'
M p'
   Pt q , i, l , j, l  k Pk p, q , i, l p' j, l  k
ddX' dtdX q
e p' , j,l  k
receptor concentration for q = ambient species mass
receptor deposition rate * t for q = deposited species mass
transit probability
kinetic probability
= S  j, l  k  X' , source mass emissions
source emissions rate
finite volume when q = ambient species mass
finite surface area when q = deposited species mass
finite source volume
source release time increment
infinitesimal receptor volume at q = ambient species mass
infinitesimal receptor area at q = deposited species mass
identifies the species state, i.e. ambient concentration,
dry or wet deposited mass
identifies the receptor species indexes
identifies the source species index
identifies the discrete receptor location
identifies the discrete source location
identifies the discrete receptor time increment
identifies the discrete particle age increment
(2-16)
(2-17)
[g/m3]
[g/m2]
[g]
[g/m3/sec]
[m3]
[1/m2]
[m3]
[sec]
[m3]
[m2]
23
K
identifies the maximum number of age increments back in time
J
identifies the number of sources
Mp, Mp' molecular masses of species p and p' respectively
2.1.4 Transfer Matrix
In Equation 2-17, the source emissions are related to the receptor concentrations
and deposition rates by Pt and Pk. In some circumstances, it is not necessary to maintain
the separation of Pt and Pk, and Equation 2-17 can be simplified by introducing the
transfer matrix, the product of the transit and kinetic probability. Thus the SRR becomes
(National Research Council, 1993):
c p , q ,i ,l 
1
X q ,i,l
where Tp,q,i,l p', j ,l k = Pt
J
0
Mp
j 1 k  K p'
M p'
   Tp,q,i,l p', j,l k
P
q , i , l , j , l  k k p, q , i , l p' j , l  k
e p' , j ,l  k
(2-18)
is the transfer matrix.
Each element of the transfer matrix is the probable contribution of species p in state q at
the receptor i and time l with age k released from the source j as species p'.
2.2 Matrix Representation of the SRR
The source receptor relationship defined in Equations 2-17 and 2-18 operates in a
seven dimensional vector space with the dimensions: receptor volume or area, receptor
time, receptor species, species state, source volume, particle age, and source species.
Insights into the meaning of this relationship and its solution can be obtained by
redefining the SRR in terms of a receptor and source vector subspaces, where the receptor
space has the dimensions receptor volume, time, species and species state, and the source
vector space has the dimensions source volume, particle age, and source species. Taking
the molecular weight ratio as a part of Pk, the SRR then becomes:
c =
1
X 
 ( Pt Pk ) e

(2-19)
24
where 

is the receptor index
is the source index.
The receptor index, , identifies the receptor values, i.e. the receptor concentrations and
deposition rates in Equation 2-17, and the source index,, identifies the emission values,
i.e. the source mass emissions in Equation 2-17.
A graphical depiction of Equation 2-19 and how the grouping of the seven
dimension into the receptor and source space is accomplished is presented in Figure 2-3.
As shown, the number of receptor values, m, in the receptor vector, c, is equal to the
product of the number of indices of the receptor volume or area, I, receptor time, L,
receptor species, P, and species state, Q: m = ILPQ. Each row of the transfer matrix
corresponds to one of the receptor values and defines the relative contribution of every
emission value to this receptor value. Similarly, the number of emission values, n, in the
emission vector , e, is equal to the multiple of the number of sources, J, source species,
P', and the number of receptor times plus age, L+K: m = JP'(L+K). Each column of the
transfer matrix corresponds to one of the emission values, and represents the relative
contribution of that emission value to all receptor values.
In Equation 2-19, the transfer matrix linearly maps the emission values from the
source space to the receptor values in the receptor space. Consequently, every
concentration and deposition rate is equal to a linear combination of its row in the transfer
matrix and the emission values in the source space. The calculation of the receptor value
is independent of all other receptor values. This independence allows for the addition and
removal of receptor values and corresponding rows of the transfer matrix without
affecting other receptor values. However, each receptor value is potentially dependent
upon all emission values. Therefore, emission values cannot be indiscriminately added
and removed from the system.
25
Figure 2-3. A graphical representation of the matrix form of the SRR, Equation 2- 19. The vector on the left hand side contains the receptor values. Also shown are how
the four dimension of the receptor subspace are group together for each index in the receptor space. The vector on the right hand side contains the emission values. The
grouping of the three dimensions of the source space is also shown. The matrix is the transfer matrix that maps the emission values from the source space to the receptor
space
26
27
2.3 Reducing the SRR Resolution
Often the desired resolution of the dimensions in the SRR are coarser than the
resolution of the pre-calculated probabilities, Pt and Pk. In these cases, it is necessary to
integrate and average over the dimensions of the pre-calculated probabilities to match the
desired resolution. The reduction in the resolution simplifies the solution of the SRR by
reducing the number of receptor and emission values. It is especially important to reduce
the number of emission values when solving for the emission values, because inversion
problems can become intractable as the number of unknowns increases (see Chapter 4).
The dimensions of the SRR that can be reduced are the receptor volume or area,
source volume, receptor time, and particle age. This section will present the methodology
for performing the reduction in the resolution of each of these four dimensions, and the
assumptions and limitations resulting from the process. All examples will be conducted
using a SRR applicable to a situation where the ambient concentrations of a single
primary species being emitted from multiple sources is calculated. Therefore, the species
indexes, p and p', and the species state index q, will be ignored. Also, the initial
concentration field is also ignored. All discussions concerning this simplified form of the
SRR are applicable to the more general form.
2.3.1 Reducing the Receptor Volume Resolution
To reduce the resolution of the receptor volume, the SRR is averaged over the
receptor volumes that are to be grouped together. The aggregated receptor volume index
is represented by the index i', to differentiate it from the original receptor volume index, i.
With the reduction of the receptor volume resolution by grouping p receptor volumes
together Equation 2-17 becomes:
c' ( i' , l ) 
1 i p
1 i p 1
c( i, l )  

p i
p i X i,l
J
0
  P  i, l j, l  k  P  i, l j, l  k  e j, l  k 
j 1 k  K
t
k
where c'(i',l) is the average concentration over the p receptor volumes.
(2-20)
28
The summation over p on the right hand side of the equation can be removed by
creating aggregate transit probability and kinetic probability, Pt'(i', l | j, l-k) and Pk'(i', l | j,
l-k). The Pt is independent of Pk but dependent upon Xi,l, so that Pt' is defined as:
i p
Pt '  i' , l j, l  k   X'i' ,l 
i
Pt  i, l j, l  k 
X i,l
(2-21)
i p
where X'i' ,l   X i,l the sum of the receptor volumes.
i
Pk is dependent upon Pt, so that the calculation of Pk' requires the transfer matrix. The
aggregate transfer matrix T'(i', l | j, l-k) is:
i p
T '  i', l j, l  k   X'i' ,l 
i
T  i, l j, l  k 
X i ,l
(2-22)
Pk' is then equal to:
Pk '  i', l j, l  k  
T '  i', l j, l  k 
Pt '  i', l j, l  k 
(2-23)
Implementing Pt' and Pk' in to Equation 2-20, the aggregated SRR over the receptor
volume become:
1
c' ( i' , l ) 
X'i ' ,l
J
0
  P '  i' , l j, l  k  P '  i' , l j, l  k  e j, l  k 
j 1 k  K
t
k
(2-24)
The average concentration for the receptor volume i' over the time interval l
calculated by this equation is equivalent to the average concentration calculated from the
integral form of the SRR averaged over the same receptor volume, i', and time interval, l.
In the matrix representation of the SRR, Equation 2-19, the reduction in the
receptor volume resolution is conducted by averaging the grouped receptor values
together and their respective rows in the transfer matrix reducing the number of values in
the receptor space and rows in the transfer matrix by a factor of p.
29
2.3.2 Reducing the Source Volume Resolution
The source resolution is reduced by grouping p source volumes together. The
aggregated source volume is represented by the index j'. This grouping is performed by
summing the emissions from all grouped sources together. The aggregated transit
probability Pt' is equal to the average of the Pt for the grouped sources:
1 j p 1
Pt '  i, l j' , l  k   
P  i, l j, l  k 
p j X i ,l t
(2-25)
Similarly for the transfer matrix:
T '  i, l j' , l  k  
1 j p 1
T  i, l j, l  k 

p j X i ,l
(2-26)
Pk' is then found via:
T '  i, l j', l  k 
P '  i, l j', l  k  
k
P '  i, l j', l  k 
t
(2-27)
In performing this aggregation, it is assumed that the emission rates are constant over the
grouped source volumes. Replacing Pt , Pk and e by Pt' , Pk' , and the summed emission
values, e'(j',l-k), Equation 2-17 becomes:
c' ( i, l ' ) 
1
X i ,l
J'
0
  P '  i, l j' , l  k  P '  i, l j' , l  k  e'  j' , l  k 
j '1 k  K
t
k
(2-28)
If the grouped sources have the same emission rate, then the aggregated SRR will
produce the same receptor concentrations and deposition rates as the non-aggregated
SRR. However, if the grouped sources have variable emission rates, the reduction in the
source volume spatially smoothes the emission inventory causing a redistribution of the
emission mass. The spatially redistributed emissions will impose smoothing on the
calculated receptor concentration and deposition fields effectively reducing their spatial
resolution.
In the matrix representation of the SRR, the reduction in the source volume
resolution is accomplished by summing together the grouped emission values and
30
averaging the corresponding columns of the transfer matrix, reducing the number of
emission values and corresponding columns of the transfer matrix by a factor of p.
2.3.3 Reducing the Particle Age Resolution
The particle age resolution is reduced by grouping p age increments together. The
aggregated age index is represented by the index k'. The reduction of the particle age
resolution is performed similarly to the reduction of the source volume. That is, the
emission values are summed together over the time interval (l-k) to (l-k+p), and Pt' and T'
are equal to the average of Pt and T over the time interval (l-k) to (l-k+p) respectively.
The aggregated kinetic probability, Pk', is then ratio of Pt' / T'. The SRR can then be
written as:
c' ( i, l ) 
1
X i ,l
J
0
  P '  i, l j, l  k ' P '  i, l j, l  k ' e'  j' , l  k '
j 1 k ' K '
t
k
(2-29)
If the emission values are constant over the time interval (l-k) to (l-k+p), then the
reduction in age resolution has no affect upon the calculated receptor concentration and
deposition fields. If the emissions are not constant, then the aggregation reduces the
temporal resolution of the emission field which will reduce the resolution of the
calculated receptor concentration and deposition fields.
In the matrix representation of the SRR, the aggregation is equivalent to averaging
the columns of the transfer matrix in Equation 2-19. Time in the source space is
dependent upon the receptor time and age. Therefore, aggregating over the age does not
necessarily reducing the number of emission values and columns of the transfer matrix by
a factor of p. If the receptor time step is less than or equal to the original age time step, k,
then the number of emission values are reduced by only p values. If the receptor time
step is greater than or equal to the aggregated age k', then the emission values will be
reduced by a factor of p.
31
2.3.4 Reducing the Receptor Time Resolution
The resolution of the receptor time can be decreased by averaging over p receptor
time indices, creating a new receptor time index l'. The aggregation is performed
similarly as for the reduction in the receptor space, Equations 2-21 - 2-23, but the
aggregation is performed over the receptor time period l to l+p, and the volumes are
averaged as opposed to summed. The resulting SRR is:
c( i, l ' ) 
where X'i' ,l 
1
X'i ,l '
J
0
  P '  i, l' j, l  k  P '  i, l ' j, l  k  e j, l  k 
j 1 k  K
t
k
(2-30)
1 i p
 X i,l the average of the receptor volumes. In the above aggregation,
p i
the source release time l-k was not aggregated. If the receptor time in the source release
time is to also be aggregated, then the procedure described for the reduction of the age
dimension resolution would be followed.
In the matrix form of the SRR, the reduction in the receptor time resolution by p
indices is accomplished by averaging together the p receptor values and rows of the
transfer matrix for each receptor time index, k'. This reduces the number of receptor
values and rows by a factor of p.
2.3.5 Reducing the Number of Emission Values in the Matrix SRR
The matrix form of the SRR, Equation 2-19, can be simplified by removing
redundant emission values, that is, emission values with the same magnitude. This is
accomplished by summing the rows of the transfer matrix with the redundant emission
values. This can greatly reduce the complexity of the SRR. For example, if the emission
rates are constant, then the number of emission values and columns in the transfer matrix
and is equal to the multiple of the number source volumes and pollutant species, m =
P*J, as opposed to the multiple of the number of source volumes, pollutant species, and
receptor times plus age, m = P*J*(L+K).
32
The reduction in the number of emission values is most important when retrieving
emission fields from known concentration and deposition fields (see section 2.4 and
Chapter 4). This retrieval process is an inversion, and generally in an inversion the fewer
the unknowns the more stable the solution. Therefore, retrieving fewer emission values
allows for a more tractable problem.
2.4 Source and Receptor Oriented Expressions of the SRR
The generalized Lagrangian source receptor relationship is suitable for both
source and receptor oriented analysis. If the source mass emission (e), transit probability,
Pt, and kinetic probability, Pk are known, and for simplicity assume the molecular mass
ratio is 1, then the approach yields the concentration/deposition rates at the receptor:
c=Te
where
(2-31)
T =PtPk/X the transfer matrix normalized by the receptor volume
c
the vector of receptor concentration and deposition rates solved for
e
is the vector of known source mass emissions
This is the standard forward procedure for source oriented dispersion modeling.
The SRR equation can also be written so that the unknown is the emitted mass
vector:
e = (T)-1 c
(2-32)
In this form of analysis, the receptor concentrations/deposition rates as well as the
transit and kinetic probabilities must be known. This equation is the basis for the
retrieval of emissions from measured data (see chapter 4).
A third variation assumes that both the source and receptor concentrations are
known as well as meteorological transport, so that the physical-chemical rate processes
that Pk is dependent upon can be estimated. Chapter 3 presents a methodology for the
estimation of the kinetic rate coefficients for the simulation of the SO2-SO42- system
given measured receptor concentrations/deposition rates, SO2 emission rates, the transit
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probability and the relevant kinetic rate equations. Similarly, characteristics of the transit
probability can be calculated given the proper information. For example, Omatu et al.
(1988) calculated eddy diffusivity coefficients from known mean wind fields, NO2
concentrations, and the physical-chemical rate processes, and Cunnold et al. (1983)
estimated the atmospheric lifetime of CFCl3 from known concentration and a two
dimensional box model of the global atmosphere.
2.5 Input Parameters to the SRR
The source receptor relationship requires four types of variables: emission rates,
receptor concentrations, transit probability density function, and kinetic probability.
Given any three of the variables, the fourth can be extracted using a suitable numeric
technique.
The emission input data can be obtained from emission inventory databases, or
emission inventory models. Most commonly, these are available as gridded data sets, or
as discrete point sources. The emission data usually have poor temporal resolution, either
seasonally or yearly.
The receptor ambient concentrations are available as databases for individual
monitoring sites. Generally the sites are placed randomly in the vicinity of areas of high
receptor sensitivity (e.g. population centers, sensitive lakes, National Parks, and major
source areas). The monitoring data have usually high temporal resolution (hourly or
daily).
The transit conditional pdf, Pt can be calculated via the derivation of analytical
relationships, experimentally, and numerically. The analytical relationships, such as the
Gaussian plume equation, are applicable only under simple circumstances limiting their
usefulness. Experimentally derived Pt, are obtained from unique tracer releases, followed
by extensive ambient monitoring. These experiments are infrequently conducted leading
to a paucity of data to generate the Pt. Numeric simulations of atmospheric transport do
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not suffer from either of these limitations. In this study, numerical solution are conducted
using the CAPITA Monte Carlo Model (see Chapter 3). The CAPITA Monte Carlo
Model is a Lagrangian model that simulates the advection and diffusion processes by
tracking the movement of particles. Pollutant emissions are simulated by releasing an
ensemble of particles at each source release time. The computation of Pt is simply the
fraction of particles that have been released from a source at a specific time and are in the
receptor volume at the time of interest. For example, if the SO2 emissions from the
MOHAVE power plant released on May 24 at 12:00 noon are simulated by 10 particles,
and 1 day later 2 particles are within a receptor volume at Hopi Point, then Pt = 2/10. In
general, Pt for q = ambient concentrations:
Pt
q , i, l , j , l  k
=
# Part. released from source j at time l - k in recepor volume i at time l
Total # Partilces released from source j at time l - k
Similarly for q = deposited mass, but the criteria is over the receptor area as opposed to
within the receptor volume.
The kinetic probability, Pk, is also calculated using the Monte Carlo Model. This
is accomplished by integrating the chemical rate equations using suitable rate constants
along each particle as it ages. The probabilities are calculated directly from the rate
equations as shown in section 2.1.2. The Pk for all particles that have been released from
the same source at the same time and are within the same receptor volume are averaged
together.
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2.6 Effects of Nonlinear Kinetics on the SRR
In the SRR, defined in Equation 2-18, the receptor concentrations are linearly
dependent upon the source emissions via the transfer matrix. However, hidden in the
transfer matrix are possible nonlinear relationships between the receptor concentrations
and the source emissions due to nonlinear transformation and removal rate processes.
These nonlinear relationships can greatly reduce the applicability of the SRR.
A typical use of transfer matrices is to calculate them from known emission fields
and rate equations, then apply them to different emission scenarios to determine the
influence of changes in the emission fields on the receptor concentrations. This is valid
only if the receptor concentrations are linearly dependent on the emission rates. If this is
not the case, then changes in the emission rates will cause erroneous estimates of the
receptor concentrations. For example, the reactions of organic compounds with NOX for
the formation of ozone are highly nonlinear. The reduction of either NOX or organic
concentrations can cause the amount of ozone formed to increase. If a set of transfer
matrices are created that relate emissions of NOX to ozone concentrations for a specific
emission rate, then the application of these transfer matrices to a new NOX emission
inventory with lower emission rates would predict lower concentrations. However, in
actuality the concentrations of ozone may have increased.
Ozone is an extreme example. It can be assumed for many atmospheric
constituents that the kinetic rate processes are approximately linear with small changes in
the atmospheric concentrations. For example, it has been found that for long range
transport of SO2 and NOX, the transformation and removal kinetics are only weakly
nonlinear and can be approximated by pseudo first order parmeterizations (National
Research Council, 1983; Spicer 1983; EPA, 1984). Thus, "what if" analysis’s can be
performed using a set of transfer matrices for SO2 and NOX where different emission
scenarios are employed. However, as the ambient concentrations change significantly
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from those used in the creation of the transfer matrices the resulting concentrations fields
become suspect.
Transfer matrices dependent on nonlinear kinetics can be a particular problem
when retrieving emission fields from receptor data. In these cases, the emission fields are
unknown, and the actual transfer matrices can not be determined.
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