scholarly journals Estimating the Long-Term Reliability of Steel and Cast Iron Pipelines Subject to Pitting Corrosion

2021 ◽  
Vol 13 (23) ◽  
pp. 13235
Author(s):  
Robert E. Melchers ◽  
Mukshed Ahammed
Keyword(s):  
Pitting Corrosion ◽  
Asset Management ◽  
Probability Distributions ◽  
Water Injection ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Pit Depth ◽  
Wall Perforation ◽  
System Sustainability

Water-injection, oil production and water-supply pipelines are prone to pitting corrosion that may have a serious effect on their longer-term serviceability and sustainability. Typically, observed pit-depth data are handled for a reliability analysis using an extreme value distribution such as Gumbel. Available data do not always fit such monomodal probability distributions well, particularly in the most extreme pit-depth region, irrespective of the type of pipeline. Examples of this are presented, the reasons for this phenomenon are discussed and a rationale is presented for the otherwise entirely empirical use of the ‘domain of attraction’ in extreme value applications. This permits a more rational estimation of the probability of pipe-wall perforation, which is necessary for asset management and for system-sustainability decisions.

On the Probability Distribution of Mooring Line Tensions in a Directional Environment

2011 ◽  
Author(s):  
Jan Mathisen ◽  
Siril Okkenhaug ◽  
Kjell Larsen
Keyword(s):  
Time Series ◽  
Probability Distributions ◽  
Extreme Value Distribution ◽  
Line Tension ◽  
Extreme Value ◽  
Mooring Line ◽  
Large Set ◽  
Short Term ◽  
Reliability Method ◽  
Line Design

A joint probabilistic model of the metocean environment is assembled, taking account of wind, wave and current and their respective heading angles. Mooring line tensions are computed in the time domain, for a large set of short-term stationary conditions, intended to span the domain of metocean conditions that contribute significantly to the probabilities of high tensions. Weibull probability distributions are fitted to local tension maxima extracted from each time series. Long time series of 30 hours duration are used to reduce statistical uncertainty. Short-term, Gumbel extreme value distributions of line tension are derived from the maxima distributions. A response surface is fitted to the distribution parameters for line tension, to allow interpolation between the metocean conditions that have been explicitly analysed. A second order reliability method is applied to integrate the short-term tension distributions over the probability of the metocean conditions and obtain the annual extreme value distribution of line tension. Results are given for the most heavily loaded mooring line in two mooring systems: a mobile drilling unit and a production platform. The effects of different assumptions concerning the distribution of wave heading angles in simplified analysis for mooring line design are quantified by comparison with the detailed calculations.

Tree-dependent extreme values: the exponential case

1990 ◽  
Vol 27 (01) ◽  
pp. 124-133 ◽  
Author(s):  
Vijay K. Gupta ◽  
Oscar J. Mesa ◽  
E. Waymire
Keyword(s):  
Branching Process ◽  
Extreme Values ◽  
Rooted Tree ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Extinction Time ◽  
Brownian Excursion ◽  
Weighted Trees

The length of the main channel in a river network is viewed as an extreme value statistic L on a randomly weighted binary rooted tree having M sources. Questions of concern for hydrologic applications are formulated as the construction of an extreme value theory for a dependence which poses an interesting contrast to the classical independent theory. Equivalently, the distribution of the extinction time for a binary branching process given a large number of progeny is sought. Our main result is that in the case of exponentially weighted trees, the conditional distribution of n–1/2 L given M = n is asymptotically distributed as the maximum of a Brownian excursion. When taken with an earlier result of Kolchin (1978), this makes the maximum of the Brownian excursion a tree-dependent extreme value distribution whose domain of attraction includes both the exponentially distributed and almost surely constant weights. Moment computations are given for the Brownian excursion which are of independent interest.

Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

1982 ◽  
Vol 14 (04) ◽  
pp. 833-854 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Extreme Value Theory ◽  
Convergence Rates ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I ◽  
Type Ii

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Limit laws for maxima of a sequence of random variables defined on a Markov chain

1970 ◽  
Vol 2 (2) ◽  
pp. 323-343 ◽  
Author(s):  
Sidney I. Resnick ◽  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Distributions ◽  
Limit Laws ◽  
Limiting Behavior ◽  
Normalizing Constants

Consider the bivariate sequence of r.v.'s {(Jn, Xn), n ≧ 0} with X0 = - ∞ a.s. The marginal sequence {Jn} is an irreducible, aperiodic, m-state M.C., m < ∞, and the r.v.'s Xn are conditionally independent given {Jn}. Furthermore P{Jn = j, Xn ≦ x | Jn − 1 = i} = pijHi(x) = Qij(x), where H1(·), · · ·, Hm(·) are c.d.f.'s. Setting Mn = max {X1, · · ·, Xn}, we obtain P{Jn = j, Mn ≦ x | J0 = i} = [Qn(x)]i, j, where Q(x) = {Qij(x)}. The limiting behavior of this probability and the possible limit laws for Mn are characterized.Theorem. Let ρ(x) be the Perron-Frobenius eigenvalue of Q(x) for real x; then:(a)ρ(x) is a c.d.f.;(b) if for a suitable normalization {Qijn(aijnx + bijn)} converges completely to a matrix {Uij(x)} whose entries are non-degenerate distributions then Uij(x) = πjρU(x), where πj = limn → ∞pijn and ρU(x) is an extreme value distribution;(c) the normalizing constants need not depend on i, j;(d) ρn(anx + bn) converges completely to ρU(x);(e) the maximum Mn has a non-trivial limit law ρU(x) iff Qn(x) has a non-trivial limit matrix U(x) = {Uij(x)} = {πjρU(x)} or equivalently iff ρ(x) or the c.d.f. πi = 1mHiπi(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {Mn} are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.

Rate of convergence for the generalized Pareto approximation of the excesses

2003 ◽  
Vol 35 (04) ◽  
pp. 1007-1027 ◽  
Author(s):  
J.-P. Raoult ◽  
R. Worms
Keyword(s):  
Distribution Function ◽  
Uniform Convergence ◽  
Rate Of Convergence ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Generalized Pareto

Let F be a distribution function in the domain of attraction of an extreme-value distribution H γ. If F u is the distribution function of the excesses over u and G γ the distribution function of the generalized Pareto distribution, then it is well known that F u (x) converges to G γ(x/σ(u)) as u tends to the end point of F, where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F̅ u (x)-G̅γ((x+u-α(u))/σ(u)), where α and σ are two appropriate normalizing functions.

Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

1982 ◽  
Vol 14 (4) ◽  
pp. 833-854 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Extreme Value Theory ◽  
Convergence Rates ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I ◽  
Type Ii

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Tail equivalence and its applications

1971 ◽  
Vol 8 (01) ◽  
pp. 136-156 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
Normalizing Constants ◽  
Independent Identically Distributed

If for two c.d.f.'s F(·) and G(·), 1 – F(x)/1 – G(x) → A, 0 &lt;A &lt;∞ , as x → ∞, then for normalizing constants an &gt; 0, bn, n &gt; 1, Fn (anx + bn ) → φ(x), φ(x) non-degenerate, iff Gn (anx + bn )→ φ A−1(x). Conversely, if Fn (anx+bn )→ φ(x), Gn (anx + bn ) → φ'(x), φ(x) and φ'(x) non-degenerate, then there exist constants C &gt;0 and D such that φ'(x) =φ(Cx + D) and limx→∞ 1 — F(x)/1 — G(x) exists and is expressed in terms of C and D, depending on which type of extreme value distribution φ(x) is. These results are used to study domain of attraction questions for products of distribution functions and to reduce the limit law problem for maxima of a sequence of random variables defined on a Markov chain (M.C.) to the independent, identically distributed (i.i.d.) case.

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