UNCERTAINTIES IN LCA

Analytical uncertainty propagation in life cycle inventory and impact assessment: application to an automobile front panel

Jinglan Hong • Shanna Shaked • Ralph K. Rosenbaum • Olivier Jolliet

Received: 25 March 2009 / Accepted: 24 February 2010

© Springer-Verlag 2010

Responsible editor: Andreas Ciroth

Jinglan Hong (*)

School of Environmental Science and Engineering, Shandong University, No. 27 Shanda Nan Road, 250100 Jinan, China

Jinglan Hong • Shanna Shaked • Olivier Jolliet (*)

School of Public Health, University of Michigan, Ann Arbor, MI, USA

Ralph K. Rosenbaum

CIRAIG, École Polytechnique de Montréal, Department of Chemical Engineering, 2900 Édouard-Montpetit, P.O. Box 6079, Stn. Centre-ville, Montréal (Québec) H3C 3A7, Canada

Supporting information

S1. Characteristics of a lognormal distribution

a) Single scenario

We providehere a condensed introduction to the basic concepts of lognormal distributions.Deeper insights intouncertainty representation in the context of life cycle assessment were published by Heijungs and Frischknecht (2005). For lognormally distributed parameters, Strom and Stansbury (2000) discuss the determination and role of these parameters.

In the context of environmental multimedia modelling, a broadly used uncertainty range is the 95% confidence interval for a normal distribution. For a large number of pointsxi, this approximately corresponds to the integration over the range from to around the mean . The limits of this interval are the 2.5th percentile as the lower bound at and the 97.5th percentile as the upper bound at .


Figure S1.Lognormal distributions of steel scenario with geometric mean, mean and 95% confidenceinterval.

For a lognormally distributed parameter, which isoftenused in environmental modelling, the corresponding distribution and the 95% confidence interval are depicted in Fig.S1. This 95% confidence interval corresponds to the integration over the range to ,where σ is the standard deviation of the natural logarithm of x and μthe geometric mean of x. This factor is also defined as the squared geometric standard deviation (GSD2):

/ (S1)

The squared geometric standard deviation GSD2therefore defines the 2.5th and 97.5th percentiles, i.e. the 95% confidence interval bounds, of a probability distribution around the geometric meanμ:

/ (S2)

The exponent 2 is in fact the rounded value of 1.96, the two-sided critical value at significance level 0.95 from a table of the lognormal distribution.

The geometric standard deviation of a set of points xkcan be calculated as follows:

(S3)

where µ is the geometric mean of the lognormal distributionand n is the number of points sampled.

The geometric mean can be calculatedas a function ofthe mean of the variable's logarithmor of the mean, as follows:

(S4a)

and the corresponding probability distribution is given by

(S4b)

b) Comparison between two scenarios

Figure S2.Comparison ofthe lognormal distributions of Steel and Aluminum scenarios (with high uncertainty on the independent parameter gasoline consumption GSD2independent=1.77)

When comparing two normally, or here lognormally, distributed scenarios A and B, one might think that the probability that scenario A is higher than scenario B is the fractional area of overlapbetween the two distributions (Fig.S2). However, the impact scores of two LCA scenarios are generally not independent since they are based on common LCI processes and LCIA characterization factors. Therefore when one set of parameters (p1,p2,p3) yields a high result in scenario A, it is likely to also yield a high result in scenario B. In a Monte Carlo simulation, it is therefore necessary to run the analysis for A and B in parallel, as follows: a) randomly select one set of parameters (p1,p2,p3), b) calculatethe impact score for both scenarios A and B based on this same set of parameters, and c) repeat the operation a large number of times to determine the probabilities P(A-B>0) or P(A/B>1).

S2. Derivation of the Taylor series expansion applied to lognormal distributions

According to Morgan and Henrion (1990), the variance associated with the output y* of a model can be approximated as a function of the input variables xi*:

(S5a)

In our case, the slope is evaluated at X0 equal to the mean.Transforming variable y* into y*=ln y and xi*=ln xi, one gets the following general equation:

(S5b)

In the particular caseof lognormal distributions for the output dependent variable and for each of the input variables, . Defining the relative sensitivity to parameter i as
, one gets:

(S5c)

The sensitivity to input parameter i describes the relative change in the model output (Δy/y) due to the relative change in this input parameter i (Δxi/xi). For practical LCA applications where all input variables arelognormally distributed, the hypothesis that the output variable islognormally distributed is mostly satisfied except in the case of dominant avoided processes for which the distribution can cross the zero.

S3. Derivation of the A/B distributions

  1. Derivation of equation(4) in the main text:

)(S6a)

  1. Derivation of equation (6) in the main text:

By definition, the covariance is bounded by the following inequality:

(S6b)

whereσXA andσXB are the standard deviation of the natural logarithms of scenarios A and B, respectively.

Accounting for equation S1, one gets:

Thus

Mathematically,

(S7)

  1. Derivation of equation (8) in the main text:

The overall output uncertainty is represented by theoverall squared geometric standard deviation, (GSDy2), which is calculated as (MacLeod et al., 2002):

/ (2)

Mathematically, equation (2) can be rewritten

thus

and therefore

(S8)

Equation (7) defines that

thus

(S9)

When there are common and independentprocesses in both scenarios, we then obtain:

(S10)

where , , and are the sensitivities and the geometric standard deviations of uncommon processes for scenarios A and B, respectively; , and are the sensitivities and the geometric standard deviation of common processes for both scenarios, respectively. This equation remains valid for negatively correlated scenarios (e.g., those with an avoided burden). In caseswitha positive correlation, SA and SB have the same signand SA/B= SA-SB =|SA|-|SB|. In cases with a negative correlation, SA and SB have opposite signs and SA/B= SA-SB =|SA|+|SB|.

  1. Error function (9)

erf is the error function as defined by the following equation (Abramowitz and Stegun, 1972):

(S11)

S4: Adding the uncertainty for Life Cycle Impact Assessment (LCIA)

The LCIA is carried out by multiplying the inventory results for individual inventory results i(calledelementary flows in ISO 14044) by the characterization factor for this given flow i (Cfi). For example, the characterization factor for the climate change of methane is its Global Warming Potential: CfCH4 = 7 kgCO2equ/kgCH4.

The overall output uncertainty is represented by its overall squared geometric standard deviation (GSDy2), whichis calculated as the sum of the uncertainties due to the ninventory parameters plus the uncertainties due to the mcharacterization factors (Cf1 to Cfm):

(S12)
where is the sensitivity of the output to the characterization factor for substance k. By definition this sensitivity is equal to the relative contribution of substance k to the overall impact score, that is: ,
where Itot .is the total impact score and is the impact score due to substance k, which is the product of total emission of substance k (mk) and the characterization factor for substance k.Therefore:
(S13)

S5.Calculation of the squared geometric standard deviation for the steel scenario using the Taylor series expansion method (equation (2))

Table S1. Squared geometric standard deviation on the climate change score for the steel scenario using the Taylor series expansion method (equation (2))

S6:Calculation of the squared geometric standard deviation for the steel/aluminum scenario using the Taylor series expansion method (equation (8))

Table S2. Squared geometric standard deviation on the climate change score for the steel /aluminum scenario using the Taylor series expansion method (equation (8))

S7. Normality test for the output of the steel scenario

QQ-plot assuming a normal distribution (y)
a) Base scenario
Shapiro-Wilk normality test: W = 0.9961, p-value = 2.7e-10 / QQ-plot assuming a lognormal distribution (log(y)
a) Base scenario
Shapiro-Wilk normality test: W = 0.9988, p-value = 0.001
b) High GSD2independent=1.77
Shapiro-Wilk normality test: W = 0.931, p-value < 2.2e-16 / b) High GSD2independent=1.77
Shapiro-Wilk normality test: W = 0.9929, p-value = 4.9e-15
c) High GSD2common=2.0
Shapiro-Wilk normality test: W = 0.932, p-value < 2.2e-16 / c) High GSD2common=2.0
Shapiro-Wilk normality test: W = 0.9915, p-value < 2.2e-16
d) High GSD2independent and high GSD2common
Shapiro-Wilk normality test: W = 0.919, p-value < 2.2e-16 / d) High GSD2independent and high GSD2common
Shapiro-Wilk normality test: W = 0.9959, p-value = 1.6e-10

Figure S3. QQ plots testing the normality and lognormality of the climate change score distribution established by 10000 Monte Carlo simulations for the different steel scenarios. If the normality assumption is valid, all points should be on the diagonal. a) base scenario; b) High GSD2independent=1.77;c) High GSD2common=2.0; d) High GSD2independent and high GSD2common.