WAFO - A MATLAB toolbox for random waves

and loads

Sofia Aberg

Mathematical StatisticsCentre for Mathematical Sciences

Lund University

GoteborgAugust 15-19, 2005

Outline

Introduction

OverviewRandom sea wavesFatigue analysisExtreme value analysis

Example from recent researchEnvironmental standardsDistribution of the maximum in bounded regionsrind

Results

WAFO briefly described

Wave Analysis for Fatigue and Oceanography

◮ Statistical analysis and simulation of random waves

◮ Calculation of theoretical distributions related to characteristic waveparameters

◮ Applications to sea waves and fatigue analysis

Developed by the WAFO-group:P.A. Brodtkorb, M. Frendahl, P. Johannesson, G. Lindgren, I. Rychlik, J.Ryden, E. Sjo + others

WAFO - Philosophy

MAKE SCIENTIFIC COMPUTATIONS REPRODUCIBLE!!!

◮ Available free of charge at the Internet

◮ More than 250 routines organised in modules related to applications

◮ Easy to find routines, easy to add new ones

◮ Help pages in nice html-interface

◮ Tutorial with many examples

Important module: algorithms and code for generation of results inselected articles

www.maths.lth.se/matstat/wafo/

Random sea waves

Modelling of sea waves as stationary transformed Gaussian processes.

◮ Extraction of wave characteristics from data

◮ Estimation of spectrum

◮ Spectral simulation

◮ Calculations of exact distributions for characteristic wave parameters

Example: Wave spectra

Estimation of spectrum from data.

>> Sest = dat2spec(data,200);

0 50 100 150 200 250−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

Sur

face

ele

vatio

n (m

)

>> wspecplot(Sest);

0 1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Spectral density

Frequency [rad/s]

S(w

) [m

2 s /

rad]

fp1 = 1.1 [rad/s]fp2 = 0.58 [rad/s]

Joint distribution of wave characteristics

Definition of crest length and crestamplitude.

Ac

Tc

>> f = spec2thpdf(Sest,...);

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ampl

itude

[m]

period [s]

Joint density of (Tc,Ac)v = 0

Level curves enclosing:10305070909599

Routines related to random loads and fatigue

◮ Extraction of rainflow cycles from data

◮ Calculation of expected rainflow matrix

◮ Switching Markov loads

◮ Visualization of cycle counts etc.

0 10 20 30 40 50 60 70 80−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

Sur

face

ele

vatio

n (m

)

Turning points in data

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

min

max

Rainflow cycle counts

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

min

Max

Smoothed observed rainflow matrix

Extreme value analysis

WAFO contains a module for extreme value analysis. For example onecan

◮ estimate parameters in the Generalized Extreme Value distributionand the Generalized Pareto distribution

◮ simulate from the GEV and GPD

◮ make probability and quantile plots

0 2 4 6 8 10 12 14−2

−1

0

1

2

3

4

5

6

7

8Gumbel Probability Plot

X

−lo

g(−

log(

F))

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(x

)

Empirical and GEV estimated cdf (PWM method)

x

Environmental standard for ozone

Ozone is an airpollutant which may cause severe lung damage. In theU.S. the 1-hour air-quality standard for ozone is 0.12 ppm. If we getconcentrations above 0.18 ppm it is considered to be a serious violationof the clean air act.

Can it be assured that people are not exposed todangerous concentrations even though the standardis attained?

Experiment

Suppose that a measurement is obtained exactly at the standard level.Define a region in the plane such that it consists of all points correlatedat least 0.7 with the monitoring site.

Given the observation, what is then the distribution of the maximum overthis region?

One dimensional counterpart

Let {X (t), t ∈ [0, T ]} be a random process. Then

P( maxt∈[0,T ]

X (t) > u) = P(X (0) > u) + P(X (0) ≤ u, maxt∈[0,T ]

X (t) > u)

= P(X (0) > u) + P(X (0) ≤ u, N+T (u) ≥ 1),

where N+T (u) is the number of up-crossings of level u of the process

X (t) in the interval [0, T ].

τ

u

first up−crossing of u

Rice bounds

Classical Rice upper bound:

P(X (0) ≤ u, N+T (u) > 0) ≤ E (N+

T (u))

=

∫ T

0

E (X ′(t)+|X (t) = u)fX (t)(u)dt

Let τ be the first time the process crosses level u. Then by using thefirst passage density

P(X (0) ≤ u, maxt∈[0,T ]

X (t) > u) = P(τ ∈ [0, T ])

=

∫ T

0

E (X ′(t)+{X (s) < u, ∀s < t}|X (t) = u)fX (t)(u)dt

Two dimensions

Let W (x), x ∈ R2 be a random field and let S be a bounded region in R

2

with boundary ∂S. Then

P(maxx∈S

W (x) > u)

= P(maxx∈∂S

W (x) > u) + P(maxx∈∂S

W (x) ≤ u, maxx∈S

W (x) > u).

Needs an analogue to the up-crossings in one dimension!!

Analogue to up-crossings

Use up-crossings in the x-direction.

These points satisfy

W (x) = u, W01(x) = 0, W02 < 0, W10 > 0

Denote the number of such points by NS(u).

Upper bound in two dimensions

As in one dimension bound the probability that the number of suchpoints is greater than one with an expectation. This gives, after somecalculations :-)

P(maxx∈∂S

W (x) ≤ u, maxx∈S

W (x) > u) = P(maxx∈∂S

W (x) ≤ u,NS(u) ≥ 1)

≤

ZS

E(W02(x)−W +

10{W (s) ≤ u,∀s ∈ ∂S}|W (x) = u, W01(x) = 0)fW (x),W01(x)(u, 0)dx

rind

To compute an upper bound for P(maxx∈S W (x) > u) we need tocompute multivariate normal expectations of the form

E (|Xd(1) · . . . · Xd(Nd)|{ai < Xd(i) < bi , cj < Xt(j) < dj}|Xc = xc) fXc (xc)

The WAFO-function rind is custom made for these type of calculationsunder the Gaussian assumption!! The input is just the mean andcovariance matrix of the variables.

Ozone example revisited

Model and assumptions

◮ ξ(x) square root of true underlying ozone field. Assumed to beGaussian.

◮ observation z(x0) = ξ(x0) + ǫ(x0)

◮ conditional process W (x) = {ξ(x)|ξ(x0) + ǫ(x0) = z(x0)}

◮ E (ξ(x)) = 0.235, Var(ξ(x)) = 0.0642, Var(ǫ(x0)) = 0.0322

◮ Covariance of ξ(x) is of the squared exponential type

Intensities computed by rind

Intensity on the boundary:

0 1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

θ

Inte

nsity

Intensity within the region:

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Distribution of the maximum

Ditribution of the maximum in the 0.7-correlation region on a linear andlog-scale respectively:

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

u (ppm)

P(M

S(W

)>u)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

−12

10−10

10−8

10−6

10−4

10−2

100

u (ppm)P

(MS(W

)>u)

Note that there is a 5% risk of serious violation!!

Reminder!

The homepage of WAFO on the internet

http://www.maths.lth.se/matstat/wafo/

Versions to download are available for Windows and Unix.