A new Monte Carlo method for environmental contour estimation

Arne B. HusebyUniversity of Oslo, Norway

Erik VanemDNV - GL, Norway

Bent NatvigUniversity of Oslo, Norway

ABSTRACT: Environmental contour estimation is an efficient and widely used method for identifying extreme condi-tions as a basis for e.g., ship design. Monte Carlo simulation is a flexible method for estimating such contours. A mainchallenge with this approach, however, is that extreme conditions typically correspond to events with low probabilities.Thus, in order to obtain satisfactory estimates, large numbers of simulations are needed. While these simulations can becarried out very fast, the analysis of the resulting data can be very time-consuming. In the present paper we propose a newMonte Carlo method where only the extreme simulation results are stored and analyzed. This method utilizes the fact thatan unbiased estimate of an environmental contour does not depend on the exact values of the non-extreme results. It issufficient to know the number of such results. Probabilistic structural reliability analysis is performed to ensure that me-chanical structures can withstand certain design loads. Obtaining precise environmental contours has become an importantpart of this analysis. The proposed method improves precision and speeds up calculations.

1 INTRODUCTION

Probabilistic structural reliability analysis is performed toensure that a structure is able to withstand the requireddesign loads. A realistic description of the environmentalloads and structural response is a crucial prerequisite forstructural reliability analysis of structures exposed to envi-ronmental forces. The concept of environmental contoursis an efficient method of estimating extreme conditions asbasis for design. See (Winterstein et al. 1993) and (Haver& Winterstein 2009). It is widely used in marine structuraldesign. See e.g., (Baarholm et al. 2010), (Fontaine et al.2013), (Jonathan et al. 2011), (Moan 2009) and (Ditlevsen2002). The traditional approach is to use the well-knownRosenblatt transformationintroduced in (Rosenblatt 1952)to transform the environmental variables into independentstandard normal variables and identify a sphere with de-sired radius in the transformed space. Environmental con-tours are then found by re-transforming the sphere back tothe original space. This approach is closely related to theFORM-approximation (First Order Reliability Method),where the failure boundary in the transformed space is ap-proximated by a hyperplane at the design point.

However, a supporting hyperplane in the transformedspace will generally not correspond to a supporting hyper-plane in the original space, and this may introduce biasesin the failure probability estimates in either direction. See(Huseby et al. 2013). To avoid such biases, contours in theoriginal space can be constructed by using Monte Carlo

simulations on the joint environmental model. The result-ing environmental contours would then have supportinghyperplanes with the desired failure probability. This yieldsa more straightforward interpretation of the contours. An-other advantage of this approach is a more flexible frame-work for establishing environmental contours, which forexample simplifies the inclusion of effects such as futureprojections of the wave climate related to climatic change.See (Vanem & Bitner-Gregersen 2012).

In Section 2 we review the basic definitions of envi-ronmental contours and the fundamental existence theo-rem proved in (Huseby et al. 2014). Moreover, we showhow this contour can be expressed in terms of a certainpercentile function. In Section 3 we explain briefly howto estimate the contour using direct Monte Carlo simu-lations. More details on this are given in (Huseby et al.2013) and (Huseby et al. 2014). We then propose a newMonte Carlo method where only the extreme simulationresults are stored and analyzed. This method utilizes thefact that an unbiased estimate of an environmental contourdoes not depend on the exact values of the non-extremeresults. Knowing the number of such results is sufficient.An important part of this approach is how to choose the setof non-extreme results which can be discarded. This isssueis discussed in Section 4. Finally the suggested method isdemonstrated on a few numerical examples in Section 5.

2 BASIC CONCEPTS AND RESULTSEnvironmental contours are defined in various ways in theliterature. See e.g., (Leira 2008), (Moan et al. 2005) and(Haver 1987). In this paper, however, we apply the samedefinition of environmental contours as used in (Husebyet al. 2013). Let X be a vector of environmental variableswith possible values in the set X ⊆ Rn and assume that thedistribution of X is absolutely continuous with respect tothe Lebesgues measure in Rn. Moreover, let Pe ∈ (0,0.5)be a given exceedance probability. The objective is to iden-tify a convex set B ⊂ X such that for every supporting hy-perplane1 Π of B, we have P [X ∈ Π+] = Pe, where Π+

denotes the halfspace bounded by the hyperplane Π andnot containing B. We also introduce Π− which denotes thehalfspace complementary to Π+. Thus, B ⊆ Π−. The re-sulting environmental contour is the boundary of the setB and denoted ∂B. Whenever such a set B can be found,safely designed structures can easily be identified by con-sidering the points in ∂B.

The properties of the set B depends on the probabilitydistribution of the vector X. If B has the property men-tioned above, we say that X admits a Pe-contour. Fromthe above definition we see that the construction of B isstrongly linked to hyperplanes Π with the property thatP [X ∈ Π+] = Pe. We will refer to such hyperplanes asPe-exceedance hyperplanes, and we denote by P(Pe) thefamily of all Pe-exceedance hyperplanes. In (Huseby et al.2014) necessary and sufficient conditions for the existenceof Pe-contours are given. In particular it is shown that inthe general case B can be defined as follows:

B =⋂

Π∈P(Pe)

Π−. (1)

Note that all the halfspaces Π− ∈ P(Pe) are convex sets.Hence, the definition of B given in (1) implies that the setB is indeed a convex set as well since it is an intersectionof convex sets.

In the remaining part of the paper we consider the two-dimensional case where X = (T,H). In this case it is con-venient to introduce a functionC(θ) defined for θ ∈ [0,2π)as:

C(θ) = inf{C : P [T cos(θ) +H sin(θ) > C] = Pe}.(2)

Thus, C(θ) is the (1− Pe)-percentile in the distribution ofthe random variable Y (θ) = T cos(θ) +H sin(θ). We alsointroduce:

Π(θ) = {(t, h) : t cos(θ) + h sin(θ) = C(θ)}

Π+(θ) = {(t, h) : t cos(θ) + h sin(θ) > C(θ)},

Π−(θ) = {(t, h) : t cos(θ) + h sin(θ) ≤ C(θ)}.

1A hyperplane Π is a supporting hyperplane of a convex set B if Bis entirely contained in one of the two closed half-spaces determined byΠ and B has at least one boundary-point on Π.

In (Huseby et al. 2014) it is shown that the expression forB given in (1) can be simplified to:

B =⋂

θ∈[0,2π)

Π(θ)−. (3)

Existence of Pe-contours can then be expressed as a condi-tion on the function C:

Theorem 2.1 X admits a Pe-contour if and only if for anyθ ∈ [0,2π) and δ ∈ (0, π/2), we have:

1

2[C(θ− δ) +C(θ+ δ)] > cos(δ)C(θ). (4)

We observe that if C is convex in [θ − δ, θ + δ], thenthe condition (4) holds for this particular θ and δ. If Cis everywhere convex, then (4) will be satisfied for everyθ ∈ [0,2π) and δ > 0. However, since C typically willhave the property that limθ→2πC(θ) = C(0), C cannotbe strictly convex everywhere. Thus, unless C is constant,this function will have parts where it is concave as well.Intuitively, Theorem 2.1 states that C in its concave partsshould behave locally like the cosine function around zero.On the other hand, if C has parts with a more dramaticallyconcave shape, the condition (4) may not hold, and in suchcases X does not admit a Pe-contour.

The function C(θ) can be used to identify the boundaryof the set B. In order to show this, we assume that C(θ)is differentiable. For a given angle θ ∈ [0,2π) and a smallnumber δ > 0 we consider the intersection between the twoPe-exceedance hyperplanes Π(θ) and Π(θ+ δ). This pointcan be found by solving the following linear equations:

t cos (θ) + h sin (θ) = C(θ), (5)

t cos (θ+ δ) + h sin (θ+ δ) = C(θ+ δ),

with the solution:

t =sin (θ+ δ)C(θ)− sin (θ)C(θ+ δ)

sin(δ)(6)

h =− cos (θ+ δ)C(θ) + cos (θ)C(θ+ δ)

sin(δ)

As δ → 0 the intersection point (t, h) will converge toa point in Π(θ) which we denote by (t(θ), h(θ)). Usingl’Hopital’s rule it is easy to see that (t(θ), h(θ)) is givenby:(

t(θ)h(θ)

)=

[C(θ) −C ′(θ)C ′(θ) C(θ)

]·(

cos (θ)sin (θ)

), (7)

where C ′(θ) denotes the derivative of C(θ).As θ runs through all angles in [0,2π), the point

(t(θ), h(θ)) will move along the boundary of the set B.Thus, the environmental contour can be expressed as:

∂B = {(t(θ), h(θ)) : θ ∈ [0,2π)}. (8)

In the next section we shall see how the function C andhence also the environmental contour ∂B can be estimatedusing Monte Carlo simulation.

3 ESTIMATING ENVIRONMENTAL CONTOURSUSING MONTE CARLO SIMULATION

In this section we show how to estimate environmentalcontours using Monte Carlo simulation. More specifically,we use Monte Carlo simulations in order to estimate theset B introduced in the previous section, and then use itsboundary ∂B as the desired contour. As in the previous sec-tion a two-dimensional environmental space is assumed,and we let T and H denote the two environmental vari-ables. Moreover, for any θ ∈ [0,2π) we introduce the ran-dom variable Y (θ) = T cos(θ) +H sin(θ) and its cumula-tive distribution function Fθ(y) = P (Y (θ) ≤ y).

We start out by performing a Monte Carlo simulation onthe joint environmental model (or alternatively on an em-pirical distribution) producing a total of n sample points:

(T1,H1), . . . , (Tn,Hn) (9)

For a given angle θ ∈ [0,2π) we calculate the projectionsof these points onto the unit vector (cos(θ), sin(θ)), i.e.:

Yi(θ) = Ti cos(θ) +Hi sin(θ), i = 1, . . . , n (10)

These projections are then sorted in ascending order:

Y(1)(θ) ≤ Y(2)(θ) ≤ · · · ≤ Y(n)(θ). (11)

Using the sampled values Y1(θ), . . . , Yn(θ) the cumula-tive distribution function Fθ can be estimated by the well-known empirical cumulative distribution function:

Fθ(y) =1

n

n∑i=1

I(Yi(θ) ≤ y). (12)

Now, for a given exceedance probability Pe, we recall thatC(θ) is the (1−Pe)-percentile in the distribution of Y (θ).Thus, in order to estimate C(θ), we look for a value y suchthat Fθ(y) = (1− Pe). We then note that for k = 1, . . . , n

we have Fθ(Y(k)(θ)) = k/n. Hence, we proceed by identi-fying an integer k such that:

k

n≈ 1− Pe. (13)

That is2, k ≈ n(1−Pe). Having determined k, an unbiasedestimate for C(θ) is:

C(θ) = Y(k)(θ). (14)

Furthermore, using standard numerical methods we esti-mateC ′(θ) by C ′(θ). Using these estimates in combinationwith (7) we obtain the contour estimate. Further details anda couple of alternative methods are given in (Huseby et al.2013).

For moderately small exceedance probabilities thismethod works very well. However, in many typical appli-cations Pe can be very small, i.e., less than 0.1%. In suchcases a large number of simulations are needed in order to

2Assuming that Pe ∈ (0,1) is a rational number, we can in factensure that k = n(1− Pe) by choosing n sufficiently large.

obtain stable estimates. At the same time most of the sim-ulations yield results close to the central area of the jointdistribution, and thus very few results provide informationabout the contour area.

While Monte Carlo simulation on a bivariate distributioncan be done extremely fast on modern computers, process-ing the results in order to obtain the contours can be muchmore time consuming. In fact, even storing a large numberof simulation results in the computer memory can representa challenge.

In order to address these issues, we propose an alterna-tive sampling scheme. Assume that we can find a subsetE of the set B. Although the set B is yet to be estimated,finding such a subset is usually not difficult using e.g., acrude preliminary estimate of the set B. We will return tothe problem of choosing a suitable set E later. Since E is asubset of B, it follows that for any supporting hyperplaneΠ(θ) of B we will have that E ⊂ B ⊂ Π−(θ). See Figure 1.

E ∂B

Π(θ)

Figure 1: The set E , part of the environmental contour ∂B andthe supporting hyperplane Π(θ).

Hence, for any sampling point (Ti,Hi) ∈ E we have thatYi(θ) ≤ C(θ). Thus, in order to estimate C(θ), we do notneed to know the exact values of the sampling points inE . It is sufficient to know the number of sampling pointsinside this set.

In order to explain this idea further, we assume that wehave performed another Monte Carlo simulation on thejoint environmental model producing a total of n points.However, in this case only the sample points outside of theset E are stored while the sample points inside the set E arediscarded. We assume that the number of stored points is d,while the number of discarded points is e, where d+ e= n.The stored points are denoted:

(T1,H1), . . . , (Td,Hd). (15)

As before we also calculate the projections of these pointsonto the vector (cos(θ), sin(θ)):

Yi(θ) = Ti cos(θ) +Hi sin(θ), i = 1, . . . , d, (16)

and sort the projections in ascending order:

Y(1)(θ) ≤ Y(2)(θ) ≤ · · · ≤ Y(d)(θ). (17)

Assuming that y is greater than the projections of all thediscarded points, we can still estimate the cumulative dis-tribution function Fθ(y) by Fθ(y) which in this case can bewritten as:

Fθ(y) =1

n

d∑i=1

I(Yi(θ) ≤ y) +e

n. (18)

Again we look for a value y satisfying the following condi-tion Fθ(y) = (1−Pe). Using (18) it is easy to see that thiscondition can be expressed as:

Fθ(y) =1

d

d∑i=1

I(Yi(θ) ≤ y) =n(1− Pe)− e

d, (19)

where we have introduced Fθ(y) denoting the empiricaldistribution function based on the stored sample pointsonly. Noting that for k = 1, . . . , d we have Fθ(Y(k)(θ)) =k/d, we proceed by identifying an integer k such that:

k

d≈ n(1− Pe)− e

d. (20)

That is, k ≈ n(1− Pe)− e, and an unbiased estimate forC(θ) is:

C(θ) = Y(k)(θ). (21)

By using the right-hand side of (20) we introduce the fol-lowing quantity:

P ′e = 1− n(1− Pe)− ed

. (22)

This quantity may be interpreted as an adjusted exceedanceprobability which applies to the reduced sample. Whene > 0 it is easy to see that we always have P ′e > Pe. A con-sequence of this is that the percentile estimates becomesmore stable as the exceedance events occur much morefrequent within the reduced sample. Working with this ad-justed exceedance probability also has the technical advan-tage that we may use exactly the same algorithm for thereduced sample as we use for the full sample except thatwe replace Pe by P ′e as input to the calculations.

Note that if C(θ) and C(θ) are computed using the samen sample points, the two estimates will in fact be equal.The advantage of C(θ), however, is that this estimate canbe calculated using only a small fraction of the full sample.Hence, we can easily increase the number n substantiallywithout running into memory problems or long processingtime.

4 CHOOSING THE SET EIn this section we return to the problem of choosing theset E . There are obviously many ways of doing this. In or-der to explain our approach we start out by considering avery simple example where the environmental contour canbe calculated analytically. This is the case when the en-vironmental variables are independent standard normally

distributed. This joint distribution is rotationally symmet-ric, and the exact environmental contour is a circle cen-tered in the origin. See Figure 2. The radius of this circlecan easily be calculated e.g., by considering the supportinghyperplane Π(0). In this case the projection of (T,H) issimply:

Y (0) = T cos(0) +H sin(0) = T. (23)

Thus, Y (0) is standard normally distributed, and C(0) issimply the (1− Pe)-percentile of this distribution. In fact,due to the rotational symmetry of the joint distribution,Y (θ) is standard normally distributed for all θ ∈ [0,2π).Hence, C(θ) is constant, which implies that the environ-mental contour is indeed a circle with radius R equal tothe (1 − Pe)-percentile. Thus, if e.g., Pe = 0.001, thenR = 3.09.

E

∂B

Π(0)

r

R

Figure 2: The environmental contour, ∂B, in the case where thevariables are independent standard normally distributed.

Having found the environmental contour, there is ofcourse no need to perform any simulations in this case.However, it may still be of interest to explore possiblechoices for the set E . In Figure 2 we have chosen thisset to be a circle inside the set B with radius r < R andcentered in the origin. With this choice it is easy to calcu-late the probability that a sample point is discarded, i.e.,P ((T,H) ∈ E). This probability can be expressed as:

P ((T,H) ∈ E) = P (T 2 +H2 ≤ r2). (24)

Furthermore, since T and H are independent standard nor-mally distributed, it follows that T 2 +H2 is χ2-distributedwith 2 degrees of freedom. Hence, it is easy to see that:

P ((T,H) ∈ E) = 1− exp(−r2/2). (25)

Thus, if e.g., r = 3.0 we get that P ((T,H) ∈ E) =0.988891. Hence, if we e.g., perform n = 1000000 sim-ulations, then the expected number of stored points is d =11109, while the expected number of discarded points ise = 988891. If we plug these numbers along with the ex-ceedance probability Pe = 0.001 into (22) we get the fol-lowing adjusted exceedance probability:

P ′e = 1− 1000000 · (1− 0.001)− 988891

11109= 0.09. (26)

We observe that in this case P ′e is as much as 90 timesgreater than Pe.

We now proceed by extending the methodology to gen-eral joint distributions. As mentioned in the introduction itis always possible to transform any pair of random vari-ables into a pair of independent standard normal variablesusing the Rosenblatt transformation. As before we de-note the environmental variables by T and H . The Rosen-blatt transformation is a transformation Ψ such that if(T ′,H ′) = Ψ(T,H), then T ′ and H ′ are independent stan-dard normal variables. As explained above for these trans-formed variables we can easily find the environmental con-tour, which we denote ∂B′ and a suitable circular set withinthe contour, which we denote by E ′. Unless the Rosenblatttransformation is linear, which happens only when (T,H)has a bivariate normal distribution, we cannot simply ob-tain the environmental contour for T and H by using theinverse Rosenblatt transformation. That is, except for thelinear case, we typically have that Ψ−1(∂B′) 6= ∂B. See(Huseby et al. 2013). However, provided that E ′ is not tooclose to ∂B′, we may still have that Ψ−1(E ′) ⊂ B. Thus,we propose to simply choose E as Ψ−1(E ′). See Figure 3.That is, we let:

E = {(t, h) : Ψ(t, h) ∈ E ′}. (27)

E = Ψ-1(E' )E'

Figure 3: Obtaining the set E by using the inverse Rosenblatttransformation Ψ−1

5 NUMERICAL EXAMPLESIn this section we illustrate the proposed method with twonumerical examples. In both cases we let Pe = 0.001.Moreover, in every plot, the variables T and H are rep-resented by respectively the horizontal and vertical axes.

In the first example we assume that T and H are bivari-ate lognormally distributed with:

E[ln(T )] = E[ln(H)] = 1.0,

SD[ln(T )] = SD[ln(H)] = 1.0,

Cov[ln(T ), ln(H)] = 0.7.

We start out by running n= 50000 simulations where allresults are used in the estimation of the environmental con-tour. The resulting contour is shown in Figure 4. While theoverall shape of the contour is fairly well captured, we seesome noticable numerical instabilities in the form of loops

Figure 4: Environmental contour plot obtained using all 50000points from a simulation run

along the curve. This is a typical phenomenon indicatingthat the data is insufficient for the exceedance probability.

In Figure 5 the same contour curve is shown along witha scatter representing the simulation results. We see thatthere are only a few points in the area of the contour. Thisfact accounts for both numerical and statistical instabilities.

Figure 5: Environmental contour plot and scatter obtained usingall 50000 points from a simulation run

We then proceed by running simulations where a point(t, h) is discarded if Ψ(t, h) ∈ E ′, where the set E ′ is acircle with radius 3. This is less than 3.09, which is the(1− Pe)-percentile of the normal distribution.

The simulation is run until we end up with d = 50000stored points. The number of discarded points in this caseis e = 4452434. Thus, the total number of simulations inthis case is n= 4502434. Since this number is much largerthan the total number of simulations in the case where allresults where used, the latter simulation run takes a littlebit longer time to finish. However, on a standard PC bothruns can be completed in a few seconds anyway. InsertingPe, n, d and e into (22) we obtain the adjusted exceedanceprobability P ′e = 0.09. Using this as input to the contourplot algorithm we obtain the environmental contour shownin Figure 6.

In this case we get a nice and smooth stable contour.In Figure 7 the same contour is shown along with a scat-

Figure 6: Environmental contour plot obtained using a reducedset of 50000 points from a simulation run

ter representing the simulation results. The area around thecontour is now surrounded by a large number of samplepoints. Thus, with this sampling scheme we get a numeri-cally and statistically stable estimate of the environmentalcontour.

Figure 7: Environmental contour plot and scatter obtained usinga reduced set of 50000 points from a simulation run

Figure 8: The environmental contour obtained using MonteCarlo simulation (green curve) compared to the contour obtainedby the transformation method (red curve)

It is of interest to compare this contour with the corre-sponding contour obtained by the transformation method.Denoting as before the contour in the transformed spaceby ∂B′, the transformed contour is obtained by using theinverse Rosenblatt transformation: Ψ−1(∂B′). In Figure 8we have plotted the two curves in the same diagram. Wesee that the two contours deviate from each other quite not-icable due to the bias resulting from the non-linear trans-formation Ψ−1. In particular in the area around the point(50,50), the transformation method produces a much moreoptimistic contour than the Monte Carlo method, Thus, inthis area the exceedance probability is underestimated ifthe transformation method is used.

In the second example we again assume that T and Hare bivariate lognormally distributed with the same loga-rithmic means and standard deviations. However, in thiscase we let Cov[ln(T ), ln(H)] = 0.0.

As in the previous example we start out by runningn = 50000 simulations where all results are used in the es-timation of the environmental contour. The resulting con-tour is shown in Figure 9. We observe that there are severalnumerical instabilities in the form of loops along the curvein this case as well.

Figure 9: Environmental contour plot obtained using all 50000points from a simulation run

Figure 10: Environmental contour plot and scatter obtained usingall 50000 points from a simulation run

In Figure 10 the same contour curve is shown alongwith a scatter representing the simulation results. We seethat there are only a few points in the area of the contour.Hence, both numerical and statistical instabilities are to beexpected.

As before, we proceed by running simulations where apoint (t, h) is discarded if Ψ(t, h) ∈ E ′, where the set E ′is a circle with radius 3. The resulting contour is shown inFigure 11 and with a scatter in Figure 12.

Figure 11: Environmental contour plot obtained using a reducedset of 50000 points from a simulation run

Figure 12: Environmental contour plot and scatter obtained usinga reduced set of 50000 points from a simulation run

In this case we also get a very smooth contour. We ob-serve that there is a part of the contour where the scatter isless dense compared to the scatter in the previous example.Still there are more than enough points to obtain satisfac-tory results from a statistical perspective. It should also bepointed out that the scatter in fact reaches far outside thearea included in the figures. Thus, there are in fact a lot ofpoints being used in the calculations even though they dono show up in the scatter.

We close this section by comparing the environmentalcurve obtained by Monte Carlo estimation to the corre-sponding contour obtained by the transformation method.Both contours are shown in Figure 13. For this somewhatmore extreme joint distribution we observe that the trans-

Figure 13: The environmental contour obtained using MonteCarlo simulation (green curve) compared to the contour obtainedby the transformation method (red curve)

formation method does not even produce a convex contour.Thus, this contour does not have the required properties.Moreover, most of the contour is biased in the sense thatthe exceedance probability is underestimated.

6 CONCLUSIONSIn the present paper we have presented a new MonteCarlo method where only the extreme simulation resultsare stored and analyzed. In the numerical examples the ex-ceedance probability was chosen to be 0.001, and the con-dition for discarding sampled points were chosen accord-ingly. For even smaller exceedance probabilities this con-dition can easily be adjusted so that we always have suffi-cient data in the area of the contour. Using this new methodit becomes very easy to obtain numerically as well as sta-tistically stable results even for very low exceedance prob-abilities without having to store and analyze millions ofsimulation results. Contrary to the transformation methodthe Monte Carlo method avoids bias resulting from non-linear transformations. As seen from the examples this biascan be substantial.

AcknowledgementsThis paper has been written with support from the ResearchCouncil of Norway (RCN) through the project ExWaCliExtreme Waves and Climate Change.

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