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Literature Cited (1) Oliver, R. M. Enuiron. Sci. Technol. 1985, 19, 225-231. (2) Peterson, T. W.; Moyers, J. L. Atmos. Enuiron. 1980,14, 1439-1444. (3) Larsen, R. I. Environmental Protection Agency, North Carolina, Research Triangle Park, NC, 1971, Report AP-89. (4) Larsen, R. I. J. Air Pollut. Control Assoc. 1977,27,454-459. ( 5 ) Mage, D. T. Atmos. Enuiron. 1982, 16, 1273-1274. (6) Mage, D. T.; Ott, W. R. J. Air Pollut. Control Assoc. 1978, 28, 796-798. (7) Peterson, T. W.; Moyers, J. L. Atmos. Environ. 1982, 16, 1274. (8) Larsen, R. I. J. Air Pollut. Control Assoc. 1978,28, 798. (9) Scorer, R. S. “Air Pollution”;Pergamon Press: London, 1968. (10) Larsen, R. I. J. Air Pollut. Control Assoc. 1961,11, 71-76.
‘This article has not been subjected to Agency review and does not necessarily reflect the view of the Agency. David T. Mage Environmental Monitoring Systems Laboratory U.S. Environmental Protection Agency, MD-56+ Research Triangle Park, North Carolina 2771 1
SIR. Dr. Mage in ref 1raises four issues that might be briefly stated as follows: (1)the misuse of mass conservation; (2) the invalidity of a rollback model whose illustration is based on the use of log-normally distributed concentrations; (3) the estimation of diffusion or meteorological parameters; (4)the absence of an important list of references. I am grateful for the list of references, apologize for their omission, and am happy to include them at this late date even though I still do not feel all of the references are particularly relevant to my paper (2). The model that I proposed begins with eq l a (2) and the additional assumption that ECy) is deterministic and that randomness in receptor concentrations #(z,y,t) is obtained through the random error terms t ( z , t ) . Note that in my paper a(z,y) is neither emission nor time dependent though I certainly believe there are real-life situations where such dependencies must and can be easily included. The similarity (linear dependence) to eq 13 (3) is obvious. What is assumed is that the expectation of errors is zero and that error and concentration distributions are stationary, Le., invariant to time shifts, although errors may be correlated. If the reader is dissatisfied with those assumptions, then he should not proceed or at least he should proceed long enough to decide whether some of the results make sense and whether some of the assumptions can be relaxed in the interest of more useful models. In particular, relaxing the assumption of emission independence or strict stationarity may be quite simple. Of course, my model is different from the one proposed by Peterson and Moyers ( 4 ) in which both D and E are random with random concentration C written as their product. I did not then nor would I now claim that either the multiplicative or the additive formulation is the “right” model. I do not believe that one correct model exists any more than I believe there is a unique pollution density distribution that once and for all lays to rest the global behavior of random concentrations. What I did want to study were the specific assumptions that lead to emission rollback models and the insights that the use of simple long-term emission control policies will have upon the counting distributions of random exceedances in a given interval of time. Further, I wanted to find the procedure 0013-936X/85/0919-0867$01.50/0
Figure 1. Flows throughout a region.
one might follow to determine how emissions could be reduced to ensure a given probability of violation. Let me emphasize that I believe in model building as a step in developing insights, not as conclusive proof that the world and model behave as one. Stationarity and Conservation of Expected Flows
Mass conservation as I use it is meant to apply in expectation, not deterministically on a differential area in some hypothetical surface. I, also, reject the notion that deterministically the “output” will exactly equal the “input” since in any finite period the total pollution mass introduced to a region may differ from the total amount of pollutant leaving the region simply because inventories of pollutants within the given region may increase or decrease. Consider the graph in Figure 1 as an idealized representation of a region in which source emissions leave node 0 and flow to nodes 1 , 2 , and eventually j . The nodes are meant to be an idealization, depicting the collection of flows in one region just as an arc between nodes depicts a single flow path as an idealization of the many complicated real-world pollutant trajectories. The directions of flow are indicated by the arrowheads on the arcs connecting a pair of nodes. There exist flow paths between regions such as the arc connecting node 1to node 2. We note that pollutants can also leave nodes 1, 2, and j for other nodes in the network not depicted in Figure 1. Denote pollutant inventory level at time t by I ( t ) with an appropriate node subscript. The inventory level of node j at time period t + 1 is related to the level at time period t by the equation I;(t
+ 1) = I;(t) + Cfij(t + 1) - Cfji(t+ 1) 1
1
j = 0, 1, 2,
... IJ( (1)
where fv(t)denotes the mass flow from node i to j in period t. Clearly, in any time period, the total flow, Cifij(t),into j (or any subset of J) may differ substantially from the total outbound flow Cifji(t).Let us assume that inventory levels, as well as flows on arcs, are random variables whose distributions are stationary over time. Then E [ I ; ( t ) ]= E [ l j ( s ) for ] all s, t , and expectation of both sides of eq 1 yields the result that expected flows into a node (region) must equal expected outflows from the same node. Another way to say this is that there is conservation of expected input and output flow or expected net flow into a region equals zero. This relationship not only holds for a single node but also holds for any subset of nodes including the entire network. Thus, under stationarity of inventory levels both local and global conservation in expectation
External pollutant flow rates (due to emissions) such as a source E at node (0) can be easily included in eq 1 and
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2. If the reader cares to do so, he can carry the same analysis into differential areas rather than the discrete network formulation described above. With mild restrictions the results are the same. The stationarity assumption is not as restrictive as might first appear. Focus attention on a single node. Suppose that in summer the wind is dominantly from source to receptor but that in winter the wind is dominantly from receptor to source. As a result, summer concentrations are primarily background and source induced, while winter concentrations are due to the background. What we have here are four parameters a,, a,, b,, and b,, with subscripts s for summer and w for winter, not a single model in a global setting to handle all time periods. In the winter model the parameter a,(z,y) is zero. Thus, in a very real sense it is possible to include time-dependent or seasonal effects as long as one is not primarily concerned with the transient problems that surely arise in the transition between the summer and winter periods. Emission dependence can be handled in a similar way as would be the case for a model where subscripts s refer to strong and w to weak source emissions. Although my paper does not deal with statistical estimation per se (see a later section of this response), estimation of a(z,y) and b(z) (with or without subscripts) must of course be made with data obtained from periods of time consistent with the theoretical assumptions. Common sense must be used to estimate parameters of the model under consideration. In the above example one should not use winter data to estimate the meteorological parameter for summers any more than one should estimate a,(z,y) by dividing a realized background observations b,(z) by the emission level!
Log- Normal us. LN3C Distribution Many of Dr. Mage’s comments are related to the issue of how poorly the log-normal distribution used by this author fits actual air pollution concentration data; the implication is that the act of using the log normal somehow invalidates the theoretical model. Log-normality assumptions are not used to derive the rollback eq 5a-d. To be more to the point, eq 7 (2)is valid for any distribution function, not just the lognormal. The most important insights one obtains from the use of the log-normal density in the rollback model is that it reveals how important the shape of its tail is in describing the frequency, magnitude, and statistics of Poisson exceedances. While the use of the log-normal density was meant as an illustration, let me hasten to point out that the use of the so-called “correct” modified log-normal distribution leads to an identical procedure for predicting the future exceedance counts. This is not an accident. It is due to the fact that the tail of either the log normal or the LN3C have identical shape for large arguments (see Figure 1 of ref 5). With rare event exceedances it really does not matter what the density function is called or what it looks like at the origin. I may have failed to make this clear, but I did not intend in my earlier paper nor do I intend in this discussion to defend or promote a particular probability density function as “the right one” to use. I made no mention of fitting or sampling methodology which is a particularly important and difficult subject and one in which Dr. Mage has considerable experience. In my own limited experience the distribution that best fits air pollution data in a particular situation, as distinct from a global setting where one’s interest is in estimating a universal distribution function, 868
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is not in a family of one-, two-, or even three-parameter models. Furthermore, it is not at all obvious that identification of a named density is always an important problem since, in a given situation, one might as well use sample data to directly estimate the underlying distribution function. I very much appreciate the careful statistical work that Mage and Ott have provided in ref 5, but surely it is not the case that all strategies for emission controls must now and forever be analyzed in the context of an LN3C density function, rather than a log normal! Fitting and Parameter Estimation
The matter of how one makes assumptions that yield mathematical models and obtains insights from these models is often confused with the discussion of how one estimates model parameters. It was precisely this feature that kept me from referencing the discussing by Mage (3), which addresses certain estimation questions that arise in connection with the model of Peterson and Moyers (6). Estimates of the parameters a and b in eq l a (2) are obviously important. To simplify the discussion immediately following, let us suppose that backgrounds are negligible and one has realizations of emission-concentration pairs. In such situations one way to estimate a(z,y) (assuming a constant error variance in eq la) is the well-known proportionality to the sum of products of concentration-emission pairs, i.e.
CEi(U)+i(z) d(Z,Y) =
i
CE3.Y)
(3)
i
where subscript i now refers to the distinct realizations in an appropriate sampling period. If, on the other hand, it is suspected for theoretical or practical reasons, that the error variance is emission dependent, then there exist straightforward ways of estimating a(z,y). For example, if the standard deviation of &,t) is proportional to emission level, the weighted least-squares estimate of a(z,y) is proportional to the sum of ratios (not products) of realized concentration-emission pairs. Clearly, estimation must proceed in conjunction with the particular circumstances at hand and the theoretical structure of the underlying model. I simply do not understand Mage’s statement that “when the wind is from the monitor to the stack the authors would also have the parameters a(z,y,t)and D equal b / E . . Subtracting off the background value from the measured C .. . will lead to values of a(z,y,t) and D equal to zero . . .”. The suggestion to estimate a(z,y) in such a manner is not mine, and I wholeheartedly agree it is not a good thing to do. One realization does not an estimate make! If a, is zero in the summer-winter model described earlier, then the winter residual t,(t) would be the difference between a realized winter concentration measured at the receptor and the forecast b,. Clearly, one will get a large negative residual in the summer when an unlikely realization results in a wind from receptor to source and the linear model (0ver)prediets the receptor concentration as a,E b,; similarly, in winter with an unlikely wind from source to receptor, a very large positive residual will result when the model (under)predicts the receptor concentration with b,. Indeed, this is precisely why one constructs residuals and uses them to estimate parameters and goodness of fit.
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(6) Peterson, T. W.; Moyers, J. L. Atmos. Enuiron. 1982,16, 1274.
Independence of D and E
Since I concur with the Peterson-Moyers responses (6), to Mage’s critique (3) regarding the assumed independence of D and E, and the boundedness of D, I need not add any more comment to their statements except for the discussion on sample averages. The correctly stated result for expectations in eq 4 (6) is due to the assumed independ, has nothing ence of D and E , i.e., E [ C ]= E [ D ] E [ E ]and to do with rollback or emission control. However, if the realizations of C and E (and derived D ) are those given by Mage (3) and the sample averages are = 5/2, E2 = 3/2, and D, = 3/2 (sample size is denoted by subscript n = 2), then it is true in general that the sample average for C does not equal the product of the sample averages for D and E, i.e.
Robert M. Ollver
Department of Industrial Engineering and Operations Research University of California Berkeley, California 94720
c2
e, z D,E,
finite n
(4)
even when D and E are independent and stationary. With independence the product relation is satisfied for expectation of sample averages, i.e.
E[c,] = E[D,]E[E,] = E [ D ] E [ E = ] E[C]
Comment on “Equilibrium Adsorption of Polycyclic Aromatic Hydrocarbons from Water onto Activated Carbon”
all n
(5)
SIR: In the June 1984 issue of ES&T, Walters and Luthy (1)describe their studies on the adsorption of polycyclic aromatic hydrocarbons from water onto activated carbon. The authors present two linearized forms of the Langmuir adsorption isotherm
and also, in the limit of large sample sizes, the equality holds in eq 4. I simply do not understand the confusion in ref 3 and 6 with regard to sample averages or the claims that one or two realizations of D yield an estimate p ( D ) of the theoretical probability density function, P(D). Summary
and
I appreciate the opportunity to respond to Dr. Mage’s critique as I am naturally interested in the reaction of a practicing professional on these scientific problems. On the basis of his critique I want to give more throught to the implications of mass-transfer rates Qf the pollutants and the underlying carrier (air) as well as some of the unsolved estimation questions which are truly necessary to validate any theory. I respectfully disagree with his claim of the inapplicability of conservation of expected flows or that the use of the log-normal distribution for receptor concentrations invalidates the derivation, application, or implications of a theoretical rollback model based on stationarity and conservation of expected flows. Naturally, where stationarity does not apply one should not force it. On the other hand, I would always encourage its use as a first step in model building in order to better understand order of magnitude effects of emission control policies. Finally, the implication that volume-transfer rates (V) a t a receptor must somehow be less than the volume-transfer rate at an emission source simply does not allow for the possibility of intermediate (nonsource) storage. Such an assumption is overly restrictive for purposes of studying the source-receptor relationship. It is not a question of the second law of thermodynamics; again, it hinges on the proper use of conservation of expected flow and volume-transfer rates. Literature Cited (1) Mage, D. T. Enuiron. Sci. Technol.,preceding paper in this issue. (2) Oliver, R. M. Enuiron. Sci. Technol. 1985, 19, 225-231. (3) Mage, D. T. Atmos. Enuiron. 1982, 16, 1273-1274. (4) Peterson, T. W.; Moyers, J. L. Atmos. Enuiron. 1980,14, 1439-1444. (5) Mage, D. T.; Ott, W. R. J . Air Pollut. Control Assoc. 1978, 28, 796-798. 00 13-936X/85/09 19-0869$01.50/0
(3) The authors make muchado as to the fact that one form fits their data better at low concentrations while the other form fits their data better at high concentrations. In reality, both equations, if properly treated statistically, should give identical estimates of q, and b. Equation 3 is obtained from eq 2 by divided by C,. These equations correspond to the linearizations of the Michaelis-Menten enzyme kinetics equation (4)
attributed to Woolf and to Lineweaver and Burk, respectively, i.e.
(5)
: (e)( A) v,,, 1
=
+
where the variables u and [SI and parameters Kmand Vm, in eq 4 can be identified with the variables qe and C, and parameters ( l / b ) and go in eq 1. Statisticians have demonstrated repeatedly that differences in parameter estimates obtained by using eq 5 and 6 to linearize eq 4 result from incorrect weighting of the experimental data. The same should hold true when eq 2 and 3 are used to linearize eq 1. I submit that the conclusions of Walters and Luthy concerning the relative merits of eq 2 and 3 are based on improper statistical analysis. All of the equations used by these authors, with the exception of Henry’s law, are nonlinear in their pa-
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