Page 123 - Reliability and Maintainability of In service Pipelines

112 Reliability and Maintainability of In-Service Pipelines

where the function ψ xðÞ is the derivative of the logarithm of the gamma function:

ΓðxÞ @logΓðxÞ
ψ xðÞ 5 5 ð4:19Þ
ΓðxÞ @x

4.3.2.2 Method of Moments

In statistics, the method of moments is a method of estimation of population para-
meters such as mean and variance by equating sample moments with unobserv-
able population moments and then solving those equations for the quantities to
be estimated. Assuming transformed times between inspections as
b
b
w i 5 t 2 t i21 ; i 5 1; ... ; n, the method-of-moments estimates c and λ from (van
i
Noortwijk and Pandey, 2003):
P n δ
^ i51 i d n 5 δ
^ cλ 5 P n 5 b ð4:20Þ
i51 w i t n
!
P n 2 n
^ i51 w i 5 X 2
n
2 δ i 2δw i ð4:21Þ
d n λ 1 2 P
i51 w i i51
The first equation from both methods (i.e., Eqs. 4.17 and 4.20) are the same
and the second equation in the method of moments is simpler since it does not
necessarily require iterations to find the unknown parameter (^ c).
The flowchart in Fig. 4.3 illustrates the gamma distributed degradation model
in the case of availability of corrosion measurements. To use this procedure, at
least two measures of corrosion depth should be available for calculation of δ i in
Eqs. (4.18 and 4.21).

4.3.3 DEVELOPING GAMMA DISTRIBUTED DEGRADATION MODEL IN CASE OF
UNAVAILABILITY OF CORROSION DEPTH DATA

In practice, most of the time for reliability analysis of corrosion affected pipes,
data such as corrosion depth are not available. Therefore, a method should be
developed for such cases of using the gamma distributed degradation model. As
was mentioned in Section 4.4.1, in order to calculate the probability of failure
over elapsed time (Eq. 4.14), the parameters corresponding to shape and scale
parameters (α and λ) should be estimated. The steps for this purpose are:
a. Determining the approximate moments (mean and variance)
b. Estimating values for α and λ by using Eqs. (4.12 and 4.13)
Assuming X 1 ; X 2 ; ... ; X n as basic random variables, moment approximation
(i.e., step (a)) can be carried out by expanding the function Y 5 YðX 1 ; X 2 ; ... ; X n Þ