ES&T
FEATURE Spatial concentration distributions
These can be used to describe concentration variations in actual environmental data and in evaluative models of chemical fate in the environment
Donald Mackay Sally Paterson Department of Chemical Engineering and Applied Chemistry University of Toronto Toronto, Ontario M5S I A4, Canada When one assesses the environmental behavior of new and existing chemicals, it is instructive to calculate the partitioning, reaction, and interphase transport characteristics of the chemical in a hypothetical or evaluative environment or "unit world." This concept, originally suggested by Baughman and Lassiter (/), has been used by others, notably Mackay (2), Mackay and Paterson (3, 4), Neely (J), Neely and Mackay (6), Klöpffer et al. (7), and Hushon et al. (8). A common assumption in such assessments is that each environmental compartment, medium, or phase, such as water, air, or soil, is homogeneous. Some heterogeneity can be in0013-936X/84/0916-207A$01.50/0
troduced, however, by considering multiple compartments of similar material, as for example the troposphere and stratosphere, or the epilimnion and hypolimnion. In this article, we address the issue of characterizing the heterogeneous spatial distribution of a chemical within a compartment using a probability density function. The advantage of this approach is that rather than merely stating that "the mean chemical concentration in the water (or fish, or air) is 1.0 mol/m 3 ," it is possible to add that "approximately 10% of the volume may experience a concentration in excess of 10 mol/m 3 and 1% may experience a concentration in excess of 50 mol/m 3 ." Although a mean concentration may be judged to be "acceptable," it is likely that a small fraction of the environment experiences an "unacceptable" concentration. It is useful to know if this small fraction is 1% or 0.0001%. A chemical in the environment also varies in concentration temporally,
© 1984 American Chemical Society
especially in the atmosphere; thus there is a dual variability. In this discussion, however, we assume a steady-state condition—that is, concentrations do not change with time, or at least the spatial distribution is constant, although the locations subject to a given concentration range may change with time. This steadystate assumption is most valid for relatively immobile compartments such as sediments or soils. The issue of the spatial statistical distribution of pollutant concentrations has many features in common with the issue of the temporal distribution, which has been reviewed recently for air by Georgopoulos and Seinfeld (9) and for water by Dean (70). In evaluative environmental calculations, the information provided is usually the compartment volume and the amount of chemical present, often in the form of the mean concentration or fugacity. The task that we address here is to convert this data into a distribution of concentrations, which Environ. Sei. Technol., Vol. 18, No. 7, 1984
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FIGURE 1
Distribution and cumulative functions
Concentration
Concentration
Concentration
Concentration
when integrated, corresponds to the same total amount of chemical. Distribution functions We assume that an environmental c o m p a r t m e n t has a total volume VT m 3 and contains M mol of chemical; thus the mean concentration CM is M/VT mol/m 3 . If representative 208A
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samples of the entire volume could be taken and analyzed, the data could be categorized according to the concentration range in which each sample lies, and a histogram could be prepared of volume vs. concentration as shown in Figure la. The total volume—that is, the total heights of the r e c t a n g l e s — m u s t equal VT. T h e
number of moles in each rectangle m, can be calculated as C/K, (where C, is the mean concentration in the rectangle), and when totaled, £ m , must yield M. It is also possible to present these data in cumulative distribution form in which the total volume experiencing a concentration less than Cis plot-
ted against C. This is shown in Figure lb, which is obtained simply by adding each rectangle to all those of lower concentrations. The cumulative curve shows immediately the fraction of the volume (F) experiencing concentrations of less than or greater than a given value. In the example given, half the volume has a concentration above 3.8 mol/m 3 , and 10% is in excess of 5.5 mol/m 3 . It is useful and instructive to fit equations to these distributions so that changes with time or space can be ascertained rigorously. The selection of the equation ideally should be based on an understanding of the fundamental causes of the variation, but at the present state of the art we are unable to justify any particular equation. Our present selection is thus based entirely on trial and error and convenience, but it is hoped that future selections will be more soundly based. This "curve-fitting" procedure can be viewed as fitting an equation to the distribution or cumulative distribution curves, or (equivalently) redrawing the axes to linearize the cumulative distribution curve. T h e curve obtained is in principle a continuous distribution function or probability density function as reviewed in the texts by Aitchison and Brown (77), Hahn and Shapiro (72), Johnson and Kotz (13), and Elderton and Johnson (14). Each function has two forms: the bell-shaped distribution function corresponding to the continuous line in Figure 1 c in which the ordinate is designated y, and the cumulative function (Figure Id) with ordinate F. The
function should have the properties that when C-^°°, F-*\ .0, and y^O, and when C = 0 , F= 0
(1)
It is worth considering the significance of F and y in the present context. F is the fraction of the total volume (V/VT) experiencing less than a given concentration and is dimensionless, ranging from zero to unity. The derivative of F with respect to C is y and is thus the incremental change in volume fraction per concentration increment. It is the slope of the F curve and has units of reciprocal concentration. The volume V, which has a concentration between, for example, 3.0 and 3.5 m o l / m 3 (i.e., AC), can thus be determined either from the F curve as VjAF m 3 or from the y curve as VTyAC. As illustrated in Figures lc and Id, t h e y value is read at a Cvalue of 3.25 mol/m 3 , and AC is 0.5 mol/ m 3 . The amount of chemical in this volume is thus 3.25 VTAF or 3.25 VjyACmoX or 65 mol, as calculated in the figures. T h e total amount of chemical in Vj is M, which is thus J^CVJAFOT Y.CVTyAC, or in differential form
Yr fj (Cy)dC = VTCM
(2)
where CM is the mean concentration. This relationship generally determines one parameter in the empirical equation relating y to C. A second parameter usually determines the degree of spread on either side of CMWhen a third parameter is introduced, it may provide a constraint on the minimum value of C.
Of the numerous functions that can satisfy these constraints, we consider here the normal, lognormal, and Weibull functions and suggest that the Weibull function is particularly suitable for this situation because of its versatility. Indeed, it can closely reproduce both normal and lognormal distributions. Figure 2 lists the three functions and some of their properties. Georgopoulos and Seinfeld have listed and discussed others (9). Data treatment Two data treatment or calculation directions may be encountered. First is the usual "forward" direction in which we process environmental concentration data obtained from a monitoring program for a known VT to obtain M, CM, and an equation with fitted parameters. Second is the "reverse" process in which VT, M, and CM are obtained from "evaluative" calculations and the aim is to devise a suitable distribution equation and its p a r a m e t e r s . T h e s e processes a r e shown schematically in Figure 3. The reverse process requires some additional information in the form of, for example, the likely "spread" of the data, which can be obtained only from environmental measurements for the compound of interest or similar compounds in the environment of interest or from measurements in similar environments into which the compound was similarly introduced. If we can build up, from experience, information about the likely values of the other parameters and the factors that control these values, we may be able to make useful statements about
FIGURE 2
Distribution functions and their properties
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FIGURE 3
Forward and reverse processes for treatment of real and evaluative concentration data Forward process Actual concentrations Density function Cumulative function
Real environment
Concentration known Plot and obtain
Concentration
Concentration
Concentration
Reverse process Predicted concentrations Density function
Concentration
exceedences—that is, the fractions of the environment that have "excessive" concentrations. This will greatly enhance the usefulness of evaluative environmental calculations. It could provide a method by which monitoring data can be examined to check consistency with evaluative estimates. We will examine first the mathematics of the forward and reverse processes and then illustrate the overall process for mirex in the sediments of Lake Ontario. It is suggested that a graphical technique is the best way to treat data because it provides a direct, visual impression of the goodness of fit. Other more rigorous techniques can be used, but at the present state of the art their accuracy is rarely justified. N o r m a l distribution
This function has two adjustable parameters, the mean CM and the spread parameter or standard deviation a. Unfortunately the cumulative function cannot be expressed ana210A
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Cumulative function
Evaluative environment
Concentration
lytically. Thus, error function tables must be used or values must be computed by numerical integration. In the normal distribution function, the concentrations are symmetrically distributed about the mean, CM, which is also the median. That is, at F = 0.5, C is CM and the mode or concentration at which the maximum value of y occurs. Sixty-eight percent of the concentrations lie between (CM - a) and (CM + a) and 95% lie between (CM — 2
