Table I
Evaluating Uncertainty of Computing Methods in Estimating Consequences of Major Industrial Hazards. M. E. Soggiu, L. Lauria, and G. Marsili, Istituto Superiore di Sanità, Rome, Italy
The uncertainty affecting individual death risk estimates in Probabilistic Risk Assessment (PRA) is a key variable in regulatory decision-making about major industrial hazards and it must be evaluated Despite the numerous and different sources of uncertainty, some main contributors to the overall uncertainty can be identified In this paper an exercise of accidental liquid chlorine release was conducted and the uncertainty contributions due to the intrinsic variability of input variables, the discharge and dispersion models, and the computational procedures were evaluated Results point out that variability of input variables accounts for most of the overall uncertainty closer to the release source, while the discharge and dispersion model uncertainty accounts for most of the overall uncertainty at greater distances. However, due to the large positive skewness of frequency distribution of individual death risk, attention should always be paid to the computational procedures. In particular, a sensitivity analysis of the numerical integration models, such as the Monte Carlo method, should always be carried out in order to determine the reliability of the reference values selected for the PRA.
Introduction
The estimate of consequences in Probabilistic Risk Assessment
(PRA) of major industrial hazards requires that the uncertainty
of results be evaluated. In a recent study comparing true
uncertainty and variability affecting risk estimates for an
accidental release of liquid chlorine, we found that the overall
uncertainty increases with distances from release source.
Additionally, at increasing distance, true uncertainty accounts
for an ever greater proportion of total uncertainty (Marsili et
al., this meeting).
Risk estimate for an accidental toxic chemical release is
quite complex and requires the consideration of a large number of
input variables as well as the assumption of hypotheses and
approximations in modelling the phenomena involved. Consequently,
uncertainty due to a large number of sources will affect the risk
estimates.
From a theoretical point of view, the overall uncertainty can
be divided into the intrinsic variability of the input variables
and into the true uncertainty due to modelling, computation and
simulation methods.
In this paper we discuss the different kinds of true
uncertainty, from hereon referred to simply as uncertainty, in
order to evaluate the contributions of each of them to the risk
estimates.
Materials and methods
Frequency distributions of individual death risk were
estimated by a model employing:
For this procedure two different sources of uncertainty can
be identified. The first one is due to the Phast package model
and the second one due to the two best-fitting functions.
Uncertainty due to the discharge and dispersion model was
evaluated from field experiments and is represented by a
fractional bias (FB), in this case indicating an underestimation
by the Phast package, and by the normal mean square error (NMSE),
which accounts for casual error (Hanna S.R. et al., 1991).
The two best-fitting functions, identified by a pseudo
Gauss-Newton algorithm (Ralston M.L. et al. 1978) are:
Cd = a+bF+cF2
and
Pr = e+f Ln(Cd)
where a,b,c,e and f are best-fit function parameters
Cd = expected concentration at distance d
F = hole diameter
Pr = probit value by which death probability can be estimated by integrating the normal
distribution (with mean 5 and variance 1) up to Pr.
Uncertainty introduced in the results by these functions was estimated by the standard errors of the relative fits and is reported in table I, along with FB and NMSE.
Table I
Results
Frequency distributions of individual death risk, reported in
figure 1, were computed at distances of 200, 250, 300 and 600
meters from the release source for the three following cases:
Results show that at increasing distance from release source,
variability accounts less and less for the overall uncertainty.
This finding is particularly emphasized in positively skewed
frequency distribution when the right tail of distribution is
considered. In our study, at distances of 200 and 600 m,
variability respectively accounts for 91% and 22% of the overall
uncertainty at the 95th percentile.
Figure 1: Frequency distributions for individual death risk. First point on the left = 5th percentile; second point = 50th percentile; third point = 95th percentile.
Within the uncertainty, the contribution of both the
uncertainties of the model and of the computing procedure can be
evaluated for the same percentile. In particular, at every
distance the model uncertainty is predominant, but it counts ever
less at greater distances. Overall, the results suggest that
considering the 95th percentile of frequency distribution of
individual death risk, the model uncertainty and the computing
procedure uncertainty, respectively, account for 8.9% and 0.1% at
200 m and for 64% and 14% at 600 m from release source.
In this exercise, the input distributions are strongly
positive skewed and the higher percentiles are used as reference
values for the PRA study. Since the tails are invariably less
stable than the central percentiles it is therefore important to
investigate the numerical stability of the output frequency
distribution.
Therefore, a sensitivity analysis of the Monte Carlo method
(Burmaster D.E. et al., 1994) used in the exercise was
implemented for cases A, B, and C above. For each case, a 1000
trial MC simulation was performed and the sensitivity of the
method was tested re-running the simulation 500 times with
different random input numbers. The 95th percentile for each
simulation was taken; when case A and B were compared, the
resulting distributions overlapped. This overlap might hide a
real difference between the two models. Consequently a second
simulation using 10000 trials was used. The same sensitivity
analysis permitted the demonstration of the difference in the
95th percentile distributions of the two cases. In table II, the
5th, 50th and 95th percentile distributions are reported for the
individual death probability at 300 m from the release source for
case C. The variation coefficient of the higher percentile is
equal to 4%, while for the first MC implementation with 1000
trials this value was equal to 13%.
Table II
The MC simulation was implemented on an IBM mainframe which uses a powerful pseudo random number generator. When 10000 trials are used, a good stability of the estimates of risk is assured, and it becomes possible to evaluate the role of the different uncertainties introduced by the various computation procedures.
References
Marsili G., L. Lauria, M. E. Soggiu (1996): Uncertainty and Variability in Probabilistic Risk Assessment of Accidental Release of Toxic Chemical, this meeting.
Hanna S. R., Strimaitis D. G., Chang J. C. (1991): Evaluation of fourteen hazardous gas models with ammonia and hydrogen flouride fiel data, Journal of Hazardous Materials, 26, 127-158.
Burmaster D. E. and Anderson P. D. (1994): principles of good practice for the use of Monte Carlo technique in Human health and ecological Risk Assessment, Risk Analysis, 14, 477-481.
Ralston M. L. and Jennrich R. I. (1978): DUD, a derivative
free algorithm for nonlinear least squares, Technometrics, 20,
7-14.
Acknowledgement: This research was partially supported by a
financial contribution by the Gruppo Nazionale per la difesa
dai rischi chimico industriali ecologici of the National
research Council. Grant N.95.02008.PF37.