This paper discusses alternate dependence models for long-term reliability assessment in applications like offshore engineering and wind turbine design, focusing on the environmental contour method. It evaluates existing methods, including the Rosenblatt and Nataf transformations, and introduces a copula-based approach to model the dependence among random variables. Ultimately, it demonstrates that the Nataf transformation is a specific case of the Gaussian copula and provides new insights on generating environmental contours using rank-based correlations.

1. Alternate Dependence Models for Long Term
Reliability Assessment
Anshul Goyal
Abstract
In many applications such as offshore engineering, wind turbine design etc there
is a need to define response or load level associated with the target return period
or probability of exceedance. Inverse reliability methods are needed in such
situations —Environmental Contour is one such method that is used extensively.
If the exact joint distribution of the environmental variables is provided, one can
use the Rosenblatt Transformation approach [1]. However, if the information is
limited to the marginal distributions of the variables and the linear correlation
coefficient (ρ), another popular way of NATAF Transformation is suggested in
literature [2][3].
In this paper, we use the same dataset as used by [3] describing the marginal
and joint probability of the random variables. Initially, we re-solve the problem
done earlier by Gonsalvez et al [3]. More accurate set of contours is obtained
and compared with the earlier study. Next, we propose a better measure of
association among the random variables —rank based correlation also known as
Kendall’s Tau, τ. Archimedean Copula family is used to construct the depen-
dence among the random variables and several different contours are plotted.
Eventually, it is shown that NATAF is nothing but a special case when choice
of Copula is Gaussian. The contour results obtained using the Gaussian Copula
and NATAF overlap each other. Thus, this paper is attempt to create alternate
dependence models using the information about the marginal distributions and
rank based correlation among the random variables.
1 Introduction
The environmental contour method was first developed by Winterstein et al
[1]. The method has been extensively used to estimate design loads for fixed
and floating offshore oil platforms. This method has also been used to estimate
design loads for the wind turbines [4]. The contours obtained are specified for
the N − year return period of any structural response quantity. The advantage
is that the contours are independent of the structure and are very handy to
characterize a multi- dimensional environmental hazard at a site. These contours
1

2. can be traced to evaluate the extreme structural response associated with a
target return period.
The conventional use of the environmental contour method requires the joint
distribution of the random variables. Once the joint distribution is available,
Rosenblatt Transformation can be used to map the variables in standard normal
uncorrelated space to the physical space. However, in case when the number
of random variables increase, it becomes difficult to come with the multivari-
ate joint distributions. In such cases, working with marginal distributions is
still manageable along with the information of the correlation among the ran-
dom variables. There are several kinds of correlations one can derive from the
dataset. Most commonly used is the Pearson’s correlation or the linear cor-
relation coefficient, ρ. However, for cases where the dependence is non-linear;
rank based correlation, τ is a better measure of association [5] which takes into
account the number of concordant and discordant pairs in the dataset.
The first half of the paper explains the problem and the relevant method-
ologies one can apply to solve this inverse problem. The dataset and the initial
calculations set up the problem and provide a basis for making justifications in
later sections.In addition to this, the main focus is to resolve the problems us-
ing NATAF Transformation and compare it with the earlier results [3]. Typical
arguments and discussions are made to overcome the approximations made in
their study.
The next half introduces a more generalized method of generating environ-
mental contours using Copulas. Several members of the Archimedean family are
used to describe the dependence. This section typically provides an algorithm
to generate contours using variety of alternate dependence models. The fact
that NATAF is a special case of Gaussian copula is demonstrated with the help
of overlapping contours from both these methodologies. Several advantages of
the Copula based method are also discussed. The paper ends with a flowchart
comparing all the methods which summarizes author’s general approach.
2 Problem Formulation
We are interested to solve the problem where we come up with the set of envi-
ronmental contours associated with a target return period, Tr and the reliability
index β. The random variables considered in the study are the significant wave
height, Hs and the peak period, Tp. Based on the target return period, the
reliability index β is calculated using Eq 1.
β = Φ−1
−
ln(1 − 1/Tr)
λE
(1)
Here λE = 63/41; is the mean annual rate of the occurrence of extreme
natural events which is modelled as the Poisson’s process.
The first step in solving this inverse problem using the environmental contour
method is to sample the points U = [u1 = βsinθ, u2 = βcosθ] lying on a circle
such that u2
1 + u2
2 = β2
. The set of points u1 and u2 are uncorrelated standard
2

3. normal variables. The real problem in hand is to map these standard normal
uncorrelated variables to set of physical space containing the variables of interest
(here hs and tp) which are correlated.
T(u1, u2) = [(h∗
s, t∗
p); ρ, τ] (2)
Here T is the transformation which maps back the uncorrelated u1 and u2
to the physical space of hs and tp correlated by ρ or τ.
Section 3 provides the information on joint and the marginal distribution
model of Hs and Tp which has been kept consistent throughout the paper. This
is used for initial calculation of ρ and τ. Based on this dataset we have presented
three methods to solve the inverse problem constrained by the probabilistic
description of the random variables. These can be listed as
• Rosenblatt Transformation: When the joint description of the random
variables, FHs,Tp
(hs, tp) is specified.
• NATAF Transformation: When the marginal distribution FHs
(hs) and
FTp
(tp) is specified along with the correlation coefficient ρ.
• Copula based Transformation When the marginal distribution FHs
(hs)
and FTp
(tp) is specified along with the rank based correlation,τ and ρ in
case of Gaussian Copula.
The flowchart explains the set up of the Inverse problem.
Figure 1: Inverse Problem Setup
3 Dataset& Initial Calculations
3.1 Dataset
The closed form expressions for the probability density functions have been
taken from Gonsalvez et al [3]. The marginal density, fHS
(hs) is a three param-
eter Weibull distribution, Eq 3 with the parameters values specified in Table
3

4. 1. The conditional density fTp|Hs
(tp) is a log-normal distribution, Eq 5 with
conditional mean and variance given by Eq 6. The parameters for conditional
density are given in Table 2 The exact marginal density fTp
(tp) is obtained by
numerical integration of Eq 7. The joint probability density is given by Eq 8.
fHs
(hs) =
ζ1
α
hs − γ
α
ζ1−1
exp −
hs − γ
α
ζ1
(3)
FHs
(hs) = 1 − exp −
x − γ
α
ζ1
(4)
fTp|Hs
(tp) =
1
√
2πσtp
exp −
1
2
lntp − µ
σ
2
(5)
µ = α1 + α2hζ2
s ; σ2
= a1 exp(−b1 hs) + a2 exp(−b2 hs) (6)
fTp
(tp) =
∞
γ
fHs,Tp
(hs, tp)dhs (7)
f(hs, tp) = f(hs)f(tp|hs) (8)
Table 1: Parameters of the Marginal distributions of hs
Distribution Scale Shape Location
Weibull α = 2.154 ζ1 = 1.273 γ = 0.763
Table 2: Parameters of the conditional distribution of tp|hs
α1 α2 ζ2 a1 a2 b1 b2
1.9307 0.1607 0.6898 0.0012 0.1161 0.0 0.2721
3.2 Initial Calculations
• Assume that the full information about the joint distribution is given,
calculate the linear correlation coefficient, ρ and rank based correlation, τ
which will be useful in the later sections.
ρhs,tp
=
E (hs − µhs
)(tp − µtp
)
σhs
σtp
(9)
4

5. τhs,tp
= 4
∞
0
∞
0
πHs,Tp
(hs, tp)
∂2
πHs,Tp
(hs, tp)
∂hs∂tp
dhsdtp − 1 (10)
πHs,Tp
(hs, tp) = 1 − FHs
(hs) − FTp
(tp) + FHs,Tp
(hs, tp) (11)
• These results have been tabulated in Table 3 which contains the values
obtained from closed form probability density/distribution functions as
well simulated set of 1 million data points.
Table 3: Comparison of parameters from closed form expressions and simulated
data
Closed Form Simulation
µ σ µ σ
Hs 2.7611 1.581 2.7637 1.583
Tp 9.7866 2.6297 9.788 2.6282
Hs, Tp
Correlation, ρ
0.4405 0.4421
Kendall Tau, τ
0.2804 0.2876
0 2 4 6 8 10 12 14 16 18
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Hs
Marginal pdf of Hs
Figure 2: Marginal density of Hs
5

6. 0 5 10 15 20 25 30
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
discrete probability density of Tp
Tp
Original Marginal
Figure 3: Marginal density of Tp
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
0.01
0.02
0.03
0.04
0.05
0.06
Hs
Joint density of Hs and Tp
Tp
density
Figure 4: Joint density of Hs and Tp
6

7. 0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
0.2
0.4
0.6
0.8
1
Hs
Cumulative Probability Distribution of F(Hs,Tp)
Tp
cdf
Figure 5: Cumulative Distribution of Hs and Tp
0
5
10
15
20
25
30
0
10
20
30
0
0.2
0.4
0.6
0.8
1
Tp
Complimentary Cumulative Distribution of Hs and Tp
Hs
complimentcdf
Figure 6: πHs,Tp (hs, tp)
7

8. 4 Dependence Models
4.1 Rossenblatt Transformation
This is the conventional technique when the complete information about the
joint density of Hs and Tp is available. For this sub-section we assume that this
information is available and is given by Eq 3- Eq 8.
• Generate u1 and u2 given by Eq 12 such that u2
1 + u2
2 = β2
; θ lies between
[0, 2π].
u1 = βcosθ u2 = βsinθ (12)
• The mapping from standard normal uncorrelated space ,U to the physical
space (Hs&Tp) is done by Eq 13
h∗
s = F−1
Hs
(Φ(u1)); t∗
p = F−1
Tp|Hs
(Φ(u2)) (13)
0 1 2 3 4 5 6 7 8
4
6
8
10
12
14
16
18
Hs
Tp
50 yrs contour using Rosenblatt Transformation
Figure 7: Rosenblatt contours for Tr = 50yrs
4.2 NATAF Transformation
In this section, the information is limited to FHs
(hs), FTp
(tp) and ρhs,tp
. Our
approach is different from Gonsalvez et al since we are using the ρhs,tp
calculated
using Eq 9 unlike using the simulated data. This way the calculation of ρhs,tp
is more accurate. Moreover, the marginal density,fTp
(tp) is obtained using Eq
7. This is again more accurate than the fitted log normal density of tp used by
Gonsalvez et al. Fig 8 shows the comparison of fTp
(tp).
8

9. 0 5 10 15 20 25 30
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
discrete probability density of Tp
Tp
Original Marginal
Fitted Lognormal Marginal
Figure 8: Comparison fTp (tp) in both the studies.
There are several steps involved in the NATAF Transformation which have
been listed
• Transform the variables hs and tp into normal variates y1 and y2 using
Eq14
y1 = Φ−1
(FHs
(hs)) y2 = Φ−1
FTp
(tp) (14)
• Here y1 and y2 are jointly normal with the correlation ρ′
, Eq15. This is
obtained by recursively solving Eq 16 until a value of ρ′
gives the exact
value of ρ. The value of ρ′
for this case is 0.4608.
φ2(y1, y2, ρy1,y2
) =
1
2π (1 − ρ2
y1,y2
)
exp −
1
2(1 − ρ2
y1,y2
)
y2
1+y2
2−2ρy1,y2
y1y2
(15)
ρhs,tp
=
∞
0
∞
0
hs − µhs
σhs
tp − µtp
σtp
φ2(y1, y2, ρ′
)dy1dy2 (16)
• The next step is to relate the uncorrelated standard normal variables u1
and u2 to the correlated standard normal variates y1 and y2. Using rules
of probability transformation, Eq 17 shows the relation between the two
variables. This transformation ensures that E[y1] = E[y2] = 0 and σy1
=
σy2
= 1. Also ρy1,y2
= ρ′
.
y1 = u1 y2 = ρ′
y1 + (1 − ρ′2
)u2 (17)
9

10. • The final step is to map the variables from Y space to the physical space
containing h∗
s and t∗
p. This is done using Eq 18. Thus NATAF essentially
involves two transformation —the first one from uncorrelated U space to
correlated Y space and the final one from the ρ′
correlated Y space to the
ρ correlated physical space.
h∗
s = F−1
Hs
(Φ(y1)) t∗
p = F−1
Tp
(Φ(y2)) (18)
0 1 2 3 4 5 6 7 8
4
6
8
10
12
14
16
18
Hs
Tp
50 yrs contour using NATAF Transformation
Figure 9: NATAF based contours for Tr = 50yrs
0 1 2 3 4 5 6 7 8
4
6
8
10
12
14
16
18
Hs
Tp
50 yrs contour comparison
Gonsalvez et al
Corrected Contours
Figure 10: Comparison of NATAF based contours for Tr = 50yrs
4.3 Copula
Copula is a function which joins or combines the multivariate joint distribution
function to the marginal distribution functions of the associated random vari-
ables. Fig 11 explains copulas as a mapping which takes the information from
the marginal distributions as the input and map them to give multivariate joint
10

11. distribution. The information about the dependent structures is encoded in the
copula family of choice. There are wide variety of copula families available for
constructing the dependence among the random variables. Hence, Copulas of-
fer a very flexible way to model the dependence which allows us to solve this
Inverse problem using multiple number of ways. Here we are concerned about
the Archimedean Copula family much because it is much easier to construct.
For the purpose of comparison with the NATAF based contours, we use the
Gaussian Copula which is the member of elliptical copula family.
Figure 11: Copula as mapping
There are sequence of steps which are listed to generate contours using Cop-
ulas
• Similar to Eq 9 we now instead calculate Kendall tau, τ analytically Eq
10 & Eq11. The expression is solved using numerical integration and the
value obtained is 0.2876. This is also verified by calculating τ from the
simulation data shown in Table 3.
• Define z1 and z2 using Eq 19
z1 = FHs
(hs) z2 = FTp
(tp) (19)
• In this problem we are interested to map all the points in U space to the
physical space of the random variables. Hence we introduce ∗ notation
on the Z space which are related to the values in U space.Define z∗
1 =
FHs
(h∗
s) = Φ(u1).
11

12. • The above definition can easily map the first variable u1 to the h∗
s. How-
ever, to map the other variable u2 to t∗
p we need to construct the condition
distribution of FTp|Hs
(tp). To do so it is necessary to first construct the
joint distribution, FHs,Tp
(hs, tp) using Copulas.
• From the statement of the Sklar Theorem, Eq 20, the joint distribution
is constructed using a copula family, C(z1, z2, τ), where τ depends of the
value of the parameter θ which describes the dependence. There are many
copula choices available to model the dependence listed in 4. Each copula
choice a generator Ψ and a parameter θ which encodes the dependence
and is related to correlation among the random variables. Table 5 lists
the generator Ψ, parameter formulae, θ and calculated values in our study.
FHs,Tp
(hs, tp) = C(FHs
(hs), FTp
(tp)) = C(z1, z2; τ); τ = f(θ) (20)
• Once the joint distribution is known, we can easily obtain the joint den-
sity using Eq 21 where the copula density c(z1, z2) is related to copula
distribution C(z1, z2) using Eq 22
fHs,Tp
= c(z1, z2; τ)fHs
(hs)fTp
(tp) (21)
c(z1, z2; τ) =
∂2
C(z1, z2; τ)
∂z1∂z2
(22)
• The conditional density fTp|Hs
(tp) can be obtained using Eq 23. However
for the inverse problem we are interested for the value of h∗
s and t∗
p cor-
responding to u1 and u2 lying on the circle. Thus Eq 23 becomes more
specific by Eq 24
fTp|Hs
(tp) = c(z1, z2; τ)fTp
(tp) (23)
fTp|H∗
s
(tp) = c(z∗
1 , z2; τ)fTp
(tp) (24)
• Obtain the conditional probability distribution, FTp|H∗
s
(tp) using Eq 25
FTp|H∗
s
(tp) =
tp
0
fTp|H∗
s
(ζ)dζ (25)
• With the above information known, the last step is to obtain h∗
s and t∗
p
using Eq 26
H∗
s = F−1
Hs
(Φ(u1)) T∗
p = F−1
Tp|H∗
s
(Φ(u2)) (26)
12

13. Table 4: Elliptical and Archimedean Copulas
Class Name C(z1, z2)
Elliptical
Normal
Φ−1
(z1)
−∞
Φ−1
(z2)
−∞
1
2π
√
1−ρ2
exp − s2
−2ρst+t2
2(1−ρ2) dsdt
Student
T −1
(z1)
−∞
T −1
(z2)
−∞
1
2π
√
1−ρ2
1 + s2
−2ρst+t2
ν(1−ρ2)
− ν+2
2
dsdt
Archimedean
Frank −1
θ log 1 + (e−θz1 −1)(e−θz2 −1)
e−θ−1
Clayton z−θ
1 + z−θ
2 − 1
−1/θ
Gumbel exp − (−log(z1))θ
+ (−log(z2))θ 1/θ
Table 5: Generator function ψ and parameters θ of Archimedean Copulas
Name ψ θ τ θcal
Frank −log exp(−θt)−1
exp(−θ)−1 (−∞, ∞)0 1 − 4
θ (D1(−θ) − 1) 2.6988
Clayton 1
θ (t−θ
− 1) [−1, ∞)0 θ
θ+2 0.7793
Gumbel (−log(t))−θ
[1, ∞] θ−1
θ 1.3897
0 1 2 3 4 5 6 7 8
4
6
8
10
12
14
16
18
Hs
Tp
50 yrs contour using Gaussian Copula
Gumbel Copula
Rosenblatt
NATAF
Figure 12: Comparison of Rosenblatt, NATAF and Gaussian Copula based contours
for Tr = 50yrs
13

14. 0 1 2 3 4 5 6 7 8
4
6
8
10
12
14
16
18
Hs
Tp
50 yrs contour using Clayton Copula
Clayton Copula
Rosenblatt
Figure 13: Comparison of Clayton and Rosenblatt based contours for Tr = 50yrs
0 1 2 3 4 5 6 7 8
4
6
8
10
12
14
16
18
Hs
Tp
50 yrs contour using Frank Copula
Frank Copula
Rosenblatt
Figure 14: Comparison of Frank and Rosenblatt based contours for Tr = 50yrs
0 1 2 3 4 5 6 7 8
4
6
8
10
12
14
16
18
Hs
Tp
50 yrs contour using Gumbel Copula
Gumbel Copula
Rosenblatt
Figure 15: Comparison of Gumbel and Rosenblatt based contours for Tr = 50yrs
14

15. Figure 16: Algorithm for solving the Inverse problem using all three methods based
on the available information
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15