Reliability analysis of offshore structures within a time varying environment

Abstract

The analysis and design of offshore structures necessitates the consideration of environmental loads. Realistic modeling of the environmental loads is particularly important to ensure reliable performance of these structures. In this paper, structural reliability analysis of offshore structures subjected to a time varying environment is investigated. In this work, an extreme value statistical model for the wave height is adopted as a basis for the performance assessment of a jacket structure. Due to the changing environment, the model parameters are modeled to be time varying. To deal with this issue, two segmentation algorithms are proposed and applied to the observed data in order to derive piecewise stationary processes for a statistical analysis. The investigation includes the extreme value modeling of the wave height in the characterization of the sea load. The implementation of the segmentation algorithms in the original data eventually leads to approximations of the safety quality of the existing structure within different time interval. The computed result is developed to reflect the time varying effects in the failure probability of structures. The results are compared with the traditional extreme values approach in view of the accuracy and information content. The investigation is also extended to a case where the design of the structure ignores the time varying property.

Appendices

Appendix 1: Testing values of threshold and time span in pot approach for each identified seasonal sectors

To check the adequacy of the Poisson-GPD model for H _S in different discretized sectors, the exceedances are tested in both the Pareto model and Poisson process model with reference to the appropriate threshold and minimum time difference (time span) to reduce serial correlation (Coles 2001; Zhang and Lam 2014).

The Poisson-GPD model assumes the occurrence rate to follow a Poisson’s distribution. A simple test can be performed by transforming the random variable t _i (i = 1,…,n), which is the exact occurrence time of the exceedances, to a uniformly distributed random variable y _i (i = 1,…,n) based on the property of Poisson process (Luceño 2006)

$$ y_{i} = 1 - e^{{ - \int_{{t_{i - 1} }}^{{t_{i} }} {\lambda dt} }} , $$

(12)

where λ is the average occurrence rate for the extremes within the time period of the discretized seasonal sector.

Therefore, by checking the quantile–quantile plot which compares the empirical value y _i and the theoretical value r _i  = (i−1/2)/n for y ₁  < y ₂  < ··· y _n , a suitable time span can be selected in the POT method. Here, time span is tested at a value of 12 h for the applied data. The threshold selected is taken as 1.5 times the mean value for each season which is a reasonable selection from a practical point of view (Boccotti 2000). However, in order to justify the chosen threshold and time span are good enough, the suitability of several other values of threshold and time span is going to be compared in the following. Figure 16 illustrates the comparison between y _i and r _i in each of the identified four seasonal sectors for using the time span at 12, 24 and 36 h. The figure shows that the Poisson process model for adopting time span at 12 h is most adequate as the maximum difference between the empirical CDF and the theoretical CDF is the smallest one among the tested three. This supports the adequate use of time span at the value of 12 h.

Fig. 16

Test of Poisson process in each identified seasonal sector

The appropriateness of the threshold is tested by using the mean residual plot and the L-moment plot. The mean residual life plot, or sometimes called mean excess plot, is used to test the stability of the mean value of the exceedances with the change of threshold. Theoretically, the mean of a GPD for Eq. (1) can be estimated as

$$ E[x] = \mu + \frac{{\tilde{\sigma }}}{1 - \xi },\quad \xi < 1. $$

(13)

Since the estimated scale parameter is a linear function with the threshold, by substituting $\tilde{\sigma } = \sigma + \xi \left( {u - \mu } \right)$ in this function, the estimated mean function also becomes a linear one with a gradient of

$$ k = \frac{\xi }{1 - \xi }. $$

(14)

This consistent linear relationship between the mean excess and the threshold must be maintained for the possible thresholds.

The mean residual plots for all the identified four seasonal sectors are shown in Fig. 17. It is seen that the selected threshold is within the region that has a linear trend appeared in the figures. This supports the hypothesis of GPD model for the data above the chosen threshold.

Fig. 17

Test of threshold (vertical dotted line) by mean residual plot with 95 % confident intervals (green line) for all four seasonal sectors

Another model checking which utilizes the L-moment plot is also used in this study. The L-moment theory is established based on order statistics that could give a measure to the properties of a distribution such as skewness and kurtosis. It parallels the theory of conventional moment, but is more robust in the inference when there are outliers. The adequacy of the GPD fit to a sample data may be assessed through comparing the L-moment ratio (τ ₄/τ ₃) against the fitted model, where the theoretical L-moment ratio of the GPD is approximately (Hosking 1990)

$$ \tau_{4} = \frac{{\tau_{3} \left( {1 + 5\tau_{3} } \right)}}{{5 + \tau_{3} }}, $$

(15)

where τ ₃ is the L-skewness and τ ₄ is the L-kurtosis. By increasing the threshold, the plot will show a trajectory around the theoretical curve. An appropriate threshold is selected when the value of the L-moment ratio is close to the value given by Eq. (15).

Figure 18 shows the L-moment plot comparing the ratio of L-skewness and L-kurtosis between the empirical value and the theoretical value. In order to compare with other choices of thresholds, the figure also plots out the result for using threshold equals to 0.5, 1.0 and 2.0 times of sample mean. The selected threshold appears to be the most compatible with the GPD model since it generally gives the overall least deviations from the theoretical line for all the four seasons. This confirms that a threshold of 1.5 times the mean is suitable and it agrees well with the previous inference.

Fig. 18

L-moment plot for exceedances over the selected threshold (circle dot) with the theoretical GPD curve (grey line) for all four seasonal sectors

Appendix 2: Testing the number of frequencies in the Fourier expansion

The 2-D Fourier transform is adopted in order to better improve the characterization of parameter changes over different sectors. To avoid over parameterization problem, the model’s quality is being assessed by the Akaike Information Criterion (Akaike 1973). The function is expressed as

$$ {\text{AIC}} = - 2l\left( \theta \right) + 2p, $$

(16)

where l(θ) is the maximum log-likelihood calculated for each candidate model and p is the number of parameters in each referred model. The minimum value in the AIC function indicates the simplest model that could give a good fit to all the collected data.

One should note that the calculation of the likelihood function requires a slight modification in the probability density function for the derived model. If the varying properties of the parameters are taken into account, the probability density function for each of the observed exceedances {x _j } ⁿ _j=1 (j indicate a specific discretized sector; n indicates the number of data within this sector) following a Pareto distribution is expressed as

$$ f_{{x_{j} |\xi_{j} ,\sigma_{j} }} \left( x \right) = \frac{1}{{\sigma_{j} \left( {t_{j} ,\theta_{j} } \right)}}\left( {1 + \xi_{j} \left( {t_{j} ,\theta_{j} } \right)\left( {\frac{{x_{j} - u_{j} }}{{\sigma_{j} \left( {t_{j} ,\theta_{j} } \right)}}} \right)} \right)_{ + }^{{ - \frac{1}{{\xi_{j} \left( {t_{j} ,\theta_{j} } \right)}} - 1}} , $$

(17)

where t _j and θ _j are the seasonality and directionality of the observed exceedance x _j ; and σ(.) and ξ(.) are the scale and shape parameters for the exceedance which can be calculated through the Fourier characterization depending on the observed values of t _j and θ _j . Hence, the likelihood function for the whole set of observed exceedances can be expressed as

$$ \begin{aligned} L\left( {\left\{ {X_{j} } \right\}_{j = 1}^{n} ;\varTheta } \right) &= \prod\limits_{j = 1}^{n} {f\left( {x_{i} ;\varTheta } \right)} \hfill \\ &= \prod\limits_{j = 1}^{n} {\frac{1}{{\sigma_{j} \left( {t_{j} ,\theta_{j} } \right)}}\left( {1 + \xi_{j} \left( {t_{j} ,\theta_{j} } \right)\left( {\frac{{x_{j} - u_{j} }}{{\sigma_{j} \left( {t_{j} ,\theta_{j} } \right)}}} \right)} \right)_{ + }^{{ - \frac{1}{{\xi_{j} \left( {t_{j} ,\theta_{j} } \right)}} - 1}} } . \hfill \\ \end{aligned} $$

(18)

Following that, the log-likelihood function of the discrete model can then be described as

$$ l\left( \varTheta \right) = \log L\left( \varTheta \right) = - \sum\limits_{j = 1}^{n} {\log \left( {\sigma_{j} \left( {t_{j} ,\theta_{j} } \right)} \right) - \sum\limits_{j = 1}^{n} {\left( {\frac{1}{{\xi_{j} \left( {t_{j} ,\theta_{j} } \right)}} + 1} \right)\log \left( {1 + \xi_{j} \left( {t_{j} ,\theta_{j} } \right)\left( {\frac{{x_{j} - u_{j} }}{{\sigma_{j} \left( {t_{j} ,\theta_{j} } \right)}}} \right)} \right)_{ + } } } , $$

(19)

where l(Θ) can be directly put into the calculation of AIC as given in Eq. (16).

In this study, we restrict our attention to several candidate Fourier models which have different frequency terms in representing the seasonality and directionality. That is, we test whether the model should include frequency terms ω ^t = 1, 2, 3, 4 for representing the seasonality and ω ^θ = 1, 2, 3, 4, 5 for representing the directionality. To make it more clearly, different combinations between ω ^t and ω ^θ have been tested in this study. The results are recorded in Table 7.

Table 7 Comparison of AIC between different candidate models

It could be seen that the model having four frequency terms in seasonality and five terms in the directionality has the minimum AIC value. Thus there is an advantage of using this model compared to other candidate models when smooth characterization is employed.