air pollutant concentrations. In these cases, the rollback model can be used to provide
a simple method to assess source pollutant emissions required to satisfy air quality
criteria. The rollback model may be considered as belonging to the same family as the
box model (Benarie, 1980), so that it is sometimes called the receptor-oriented symmet
rical counterpart of the box model. A basic physical assumption of the rollback model
is that the pollutant concentrations are directly proportional to emissions according
to some simple relationship. Without consideration of meterological parameters and
local effects, such as the terrain, the emission control requirements can be determined
as proportional to the amount by which the peak pollutant concentration exceeds the
desired air quality standards.
T he sim plest form of th e rollback m odel is of th e type
X = Xb + Re (2.12)
w here \ is th e p o llu tan t concentration due to emissions w ith ra te e, and Xb is a m easure of background pollu tan t concentration. R is th e constant of proportionality which includes all th e effects relevant to the m eteorology and area source d istribution, and can be determ ined as (de Nevers and M orris, 1975)
R = , X m a x - X b , (2.13)
e
where Xmax is th e highest pollu tan t concentration in the region of interest. According to th e selected air quality stan d ard , th e allowable emission ra te for a new Xmax can be obtained from
e(x*ta - Xb)
(2.14)
^max — / \
( X m a x Xb)
where Xsta is th e designated air quality stan d ard specified for th e pollutant being considered. T he required reduction from a peak of p o llu tan t concentration can be obtained from
p __ qq X f i ’i ß .i ' "Xsta Xmax Xb
where P is th e percentage reduction required (Schuck and P a p e tti, 1973).
A generalised form ulation of th e sim ple rollback m odel is given by (Chang and W einstock, 1973, 1974)
X*- = Xb + Ci>ej (2-16)
i = i j= l
where Xi is th e concentration at receptor z, ej is th e emission ra te for source j , and
Rij is th e source-receptor interaction for source j and receptor i. The Cij m ay be calculated from a moving box model or a Gaussian plum e model.
T h e sim ple rollback m odel has been used successfully to exam ine th e m otor vehicle em ission goals for standards governing CO, N 0 X and hydrocarbons (B arth (1970) and Schuck and P a p e tti (1973)). T he technique has also been employed to describe photo chem ical smog effects in term s of th e prim ary p o llu tan t concentrations (H am m ing et al, 1973). de Nevers and M orris (1975) extended th e basic technique to apply to m ul tip le sources, different stack heights, different source-to-receptor distances and wind direction frequencies. Szepesi (1977) specified source-receptor functions as G aussian for point and area sources. Peterson and Moyers (1980) developed a m odel for th e case w here continuous m easurem ents of am bient concentrations and emissions are available and recorded over tim e intervals corresponding to air quality standards. Georgopoulos and Seinfeld (1982) recom m ended th e use of th e m ean values E(xmax) and E(xsta)
instead of Xmax and Xsta in rollback calculations, which has th e advantage of allowing for th e conservation of mass of non-reactive pollutants.
T he nonlinearity of atm ospheric processes lim its th e usefulness of the rollback m odel, as does its lack of spatial resolution. Horie and O verton (1974) noted th a t th e higher th e percentile value of concentration considered as th e desired air quality goal, th e greater th e un certain ty in th e emissions reductions calculated by th e rollback technique. W hen using th e m odel to predict th e ra te of growth of air pollution due to u rb an developm ent, it m ust be assum ed th a t th e d istrib u tio n of sources is unchanging w ith tim e. Therefore, th e rollback m odel m ay be used for regional analysis of areas w ith m any well d istrib u ted sources of various types and as a first step approach or screening m odel to obtain a crude picture of fu tu re trends.
2.3.4
P erform an ce and V alid ation o f D e te r m in istic M od els
From th e preceding discussion above, it is seen th a t determ inistic models vary in de scriptive and com putational complexity. They can be sim ple (e.g. the Hanna-Gifford ATDL m odel), interm ediate (e.g. Gaussian plum e m odel) or complex (e.g. num erical m odels based on K -theory). Num erous investigations have shown th a t, under cer ta in conditions, a simple modelling approach m ay perform quite well when com pared
w ith m ore complex m odels for estim ation of am bient pollutant concentrations resu lt ing from th e dispersion of pollutants in an airshed. Sim pson and H anna (1981) argued th a t th e advection effects of the atm osphere dom inate horizontal diffusion effects for long periods. Therefore, th e vertical diffusion is relatively less im p o rtan t and it can be accom m odated by sim ple Gaussian or box assum ptions. On the other hand, complex m odels such as th e K-diffusion m odel m erely involve different assum ptions to handle w hat is com m only regarded as a highly stochastic problem , and also require a num erical solution which m ay introduce com putational errors.
T he best feature of determ inistic models is th a t they can be used for approxim ation of th e causal link betw een th e variables, such as those describing emissions, m eteoro logical conditions and terrain , and the dependent p o llu tan t concentrations. These m odels have im proved our understanding of the n a tu re of pollu tan t dispersion in the atm ospheric environm ent and describe the physical processes of pollu tan t dispersion. In practice, m ost applicable determ inistic models are useful at best for predicting the m ean of p o llu tan t concentrations (see e.g. Jakem an et ah, 1988). M any determ inis tic m odels can predict long-term m eans of p o llu tan t concentrations for a wide range of physical circum stances w ith reasonable accuracy. Such models retain sensitivity to variations in bo th m ean emission strengths and m eteorological variables, such as wind speed and wind direction. T hus, determ inistic m odels are generally best suited to estim atio n of m ean pollu tan t concentration under m ean conditions.
D eterm inistic m odelling encounters two m ajo r practical difficulties w ith respect to m odel perform ance. F irst, th e determ inistic m odels are not capable of predicting extrem e p o llu tan t levels especially well, and m any air quality standards require this knowledge. Second, by th eir very n atu re, determ inistic models cannot characterize the u n certain ty in model predictions.
It has been found, for exam ple, by Simpson and H anna (1981) in an exam ination of th e predictive ability of various determ inistic m odels, th a t th e values of th e extrem es of th e distrib u tio n are unlikely to be very accurate. Pasquill and Sm ith (1983) argue
th a t it is th e special n atu re of the meteorological conditions and other circum stances which com bine occasionally to form the worst pollution episodes, and it is difficult to m odel such extrem e occurrences. H anna (1982) refers to ‘n atu ral variability’ as the tu rb u len t fluctuations in wind velocity which m ay occur over tim e periods ranging from m icroseconds to years. Obviously, th e existence of n a tu ra l variability lim its strongly th e estim atio n accuracy of air pollu tan t concentrations using determ inistic models. V enkatram ’s (1983, 1984) analysis reveals th a t th e expected deviation of observations from predictions becomes large when th e sam pling tim e is not m uch greater th an the tim e scale controlling diffusion. From th e study of H anna (1982) and th e theoretical analysis of V enkatram (1984), it is often stated th a t th e accuracy of predictions of existing determ inistic models for ensem ble m eans is approxim ately of order 2.
T he accuracy and application of determ inistic m odels is often restricted due to the lack of essential meteorological or topographical inform ation being available, p articu larly for complex models. Enhancing the d a ta collection substantially raises the costs of model developm ent, which m ay be prohibitive in m any circum stances. Therefore, a simple b u t functional determ inistic m odel is norm ally very im p o rtan t in practice for air quality m anagem ent problem s.
In conclusion, a wide range of determ inistic models is available in th e literatu re (and in com puter packages) for im pact assessm ents of air quality. Such models are m ost useful in predicting concentration values around the m ean or m edian of pollutant concentrations over tim e periods no shorter th an one hour. T he highly stochastic n atu re of tu rb u len t diffusion is a problem atic constraint. D eterm inistic models are unlikely to perform well in estim ating th e upper percentiles of th e distrib u tio n of air pollutant concentrations. Different approaches are required to circum vent these problems and to satisfy th e dem ands of air quality m anagem ent.