Q/kAir Pollution Damages and Costs: An Analysis of Uncertainties. A. Rabl, Ecole des Mines, 60 boul. St.-Michel, F-75272 Paris CEDEX 06, e-mail: RABL@CENERG.ENSMP.FR
We perform an uncertainty analysis of the impact pathway
methodology which traces the fate of each pollutant or other
burden, from source to receptors, using dose-response functions
to evaluate the damage [EC 1995, ORNL/RFF 1994, Ontario Hydro
1993, Rowe et al 1995]. We extend recent work by Slob [1994] who
points out that a lognormal approximation can be used for
multiplicative processes, thanks to the central limit theorem,
thus bypassing the need for a detailed and tedious Monte Carlo
calculation. Of particular importance for policy applications is
the expectation value of the total damage (as opposed to damage
to individuals or damage from episodes); this is addressed in the
present paper. Recently we have shown [Curtiss and Rabl 1996]
that, under certain uniformity assumptions, the total damage can
be calculated in closed form with a simple multiplicative formula
(Eq. 1 below), and that this simple formula is correct within an
order of magnitude under a wide variety of typical conditions.
Therefore we use this formula as starting point for the
uncertainty analysis. From a survey of the literature we offer
estimates of the uncertainties of the factors. As an application
we show results for health impacts from air pollution: the
geometric standard deviation is on the order of 3, and it may be
as large as 5 when economic valuation is included. To the extent
that the distribution of the result is lognormal, the geometric
mean equals the median and the geometric standard deviation has a
simple interpretation in terms of multiplicative confidence
intervals around the median.
Damage and cost are calculated by the impact pathway
methodology whose principal steps are:
The numbers are summed over all receptors (population, crops,
buildings, ...) of concern.
In this paper we focus on the case of linear dose-response
functions because they seem relevant to radionuclides, to many
carcinogens, and to most health impacts of the classical air
pollutants [see e.g. the discussion of health effects in EC
1995]; even if the dose-response functions were really shaped
like hockey sticks, their thresholds appear to, be below typical
ambient concentrations and the impact is the same as if there
were no threshold at all.
When summing over all receptors, one can find significant
cancellations because the law of conservation of matter helps
reduce the errors from the dispersion model. This can be
understood by considering a simplified example of an impact due
to the deposition of a pollutant if the dose-response function is
linear and the density of receptors uniform. In this limiting
case the error due to the dispersion model is zero because
overprediction at one site is exactly compensated by
underprediction elsewhere, assuming that the analysis covers the
entire geographic range over which the pollutant is deposited.
Thus the net error due to dispersion models can be much smaller
than commonly believed.
Curtiss and Rabl [1996] have formalized this argument and
shown that, in a uniform world (in the sense specified at the end
of this paragraph) the equation for the total damage can be
integrated in closed form to yield a very simple formula for the
total damage D
| D = d Q/k |
(1) |
where =
receptor density,
Q = emission rate of pollutant,
d = dose-response function slope, and
k = removal velocity. The latter is defined as ratio F(x)/c(x)
of surface concentration c and total removal flux F (due to dry
deposition, wet deposition or decay) at a point x. This
formula is exact in a uniform world (linear dose-response
function, uniform receptor density and uniform atmospheric
removal rate independent of x). The generalization to
secondary pollutants is straightforward.
By detailed numerical evaluations, using real data for
atmospheric dispersion and geographic receptor distribution,
Curtiss and Rabl [1996] have demonstrated that this simple
formula is remarkably relevant: it approximates the exact result
within a factor of three for a wide range of situations (from
rural to extremely urban). This insensitivity to geographical
detail implies that for the common problem of dispersed sources
the damage can be estimated by using Eq. 1 with the average
receptor density in the region(s) where the emission(s) occur.
To determine the uncertainty of the results, one needs to
1) determine the component uncertainties (of each step of impact pathway analysis)
2) combine the component uncertainties.
It is appropriate to group the main contributions to the
uncertainty in five qualitatively different categories1:
The best way of dealing with the second and third of these
categories is to indicate how the results depend on these choices
and present numbers for different scenarios if the effect on the
result is not obvious.
Even though the complete characterization of uncertainty
requires an entire probability distribution rather than just a
single number or interval, one can often assume that the
distributions are approximately lognormal (for multiplicative
effects); thus it is indeed sufficient to specify just two
numbers:
the geometric mean µg (
median) and geometric
standard deviation 
g.
By definition a variable x has a lognormal distribution if
ln(x) is normal.
The exact probability distributions of the factors do not
matter; even if they are not lognormal, lognormality is likely to
be a good approximation the distribution of the product, thanks
to the central limit theorem. The geometric standard deviation of
the product is obtained from the gi of the n
factors by using the formula
| [ln(g,tot)]2
= [ln(g1)]2
+ [ln(g2)]
2 + ... + [ln(gn)] 2
. |
(2) |
With the assumption of lognormality &mircog
and g
have a simple interpretation in terms of approximate
multiplicative confidence intervals about the geometric mean µg
(median), e.g., [µg/g, µg
· g] for
68% and [µg/g2,
µg · g2]
for 95% confidence.
As an example consider mortality due to particulates; this is
the dominant impact for coal fired power plants, apart from
global warming [ORNL/RFF 1994, EC 1995, Rowe et al 1995, Curtiss
et al 1995]. Emission rates of the major air pollutants are quite
well determined, and we estimate a geometric standard deviation
around 1.1. A survey of deposition velocity data [Seinfeld 1986,
Sehmel 1980] suggests that a lognormal distribution may be
reasonable, with a geometric standard deviation around 2.5 or
perhaps less. For the dose-response function we estimate a
geometric standard deviation around 1.5, based on the confidence
intervals in the literature [e.g. EC 1995]. For the economic
valuation we cite a survey by Ives, Kemp and Thieme [1993] of 78
Value of Statistical Life studies; if one plots these values as a
histogram one finds a distribution that is very close to
lognormal, with geometric standard deviation of 3.4.
As shown in Table 1, one finds from Eq.2 that the geometric
standard deviation of the physical damage is g = 2.7. If
the median damage has been found to be µg = 2
deaths/yr, the one g
interval is 2/2.7 = 0.74 to 2*2.7 = 5.40 deaths/yr. This implies
that air pollution damage can be estimated to about an order of
magnitude, with the present state of knowledge. The true error
might, however, be larger because of effects not taken into
account, such as the composition of the particulates.
g
for acute mortality due to particulates. median), g, µg
· g] for
68% and [µg/g2,
µg · g2]
for 95% confidence
This work has been supported in part by a grant from the
European Commission, DG XII, under contract JOUL2-CT-95-0067 in
the ExternE Program of JOULE.
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1
Although they may overlap in practice; for instance the intergenerational discount rate involves both ethical choice and scenarios [Rabl 1996].