## What is WAFO

WAFO is a toolbox of Matlab routines for statistical analysis and simulation of random waves and random loads. WAFO is freely redistributable software, see WAFO licence and contain tools for:

**Fatigue** **Analysis**

- Fatigue life prediction for random loads
- Theoretical density of rainflow cycles

**Sea** **modelling**

- Simulation of linear and non-linear Gaussian waves
- Estimation of frequency spectra and directional wave spectra
- Model spectra (JONSWAP, Torsethaugen, Bretschneider, Pierson-Moskowitz,...)
- Joint wave height, wave steepness, wave period distributions

**Statistics**

- Extreme value analysis
- Multivariate Gaussian probabilities and expectations
- Profile log-likelihood analysis (Available in next release)
- Regression analysis (Available in next release)
- Hypothesis tests (one-sample and two sample T-test, Wilcoxon signed rank and rank-sum test,....)
- Kernel density estimation (Fast data-gridders with various bandwidth selectors)
- Hidden markov models
- Probability distributions and random number generators (Beta, Chi-2, Exponential, Gamma, GPD, GEV, Gumbel, Inverse Gaussian, Lognormal, Normal, Rayleigh, Students-T, Weibull,...)
- Bootstrap (Available in next release)
- Design of Experiments

**Numerics**

- Derivatives (gradient, hessian, jacobian) (Available in next release)
- Integration (Gaussian quadratures (1D and 2D), simpson, Clenshaw-Curtis)
- Smoothing spline

The toolbox also contains scripts for generation of plots in some publications and scripts for comparison with results in the literature. Data sets from measurements of the sea surface are included.

WAFO replaces the two toolboxes Wave Analysis Toolbox (WAT) and Fatigue Analysis Toolbox (FAT). The development of numerical methods for calculating distributions of random waves dates back to the 70's, which is detailed in the next section.

WAFO is developed for use with Matlab version 5.x/6.x/7.x./8.x Some routines call programs written in Fortran. Versions for PC/Windows and Unix environments are available.

We encourage you to submit new code for possible inclusion in future versions of WAFO.

## Users of WAFO around the world

Forsvarets
forskningsinstitutt (FFI), Horten, Norway

Metocean group
at IFREMER, Brest, France

Volvo Trucks,
Göteborg, Sweden

Volvo Cars,
Göteborg, Sweden

Scania, Södertälje,
Sweden

Trafikverket
(Swedish Transport Administration), Borlänge

PSA, Division of
Automotive Research and Innovations, Paris

Ford Motor Company,
Analytical Road Loads VEV, USA

Dpt. of Civil and
Offshore Engineering, Heriot-Watt University, Edinburgh, UK

## History

At the department of mathematical statistics, research related to level crossings in stochastic processes has been performed starting in the early 70's by Georg Lindgren [1]. Much work has been devoted to the problem of how to find statistical distributions for wave length and wave period for Gaussian processes, with obvious applications in oceanography and statistical analysis of fatigue of materials. Routines were originally written i Fortran, and in 1978 a computer program for wave-length and amplitude analysis of Gaussian waves, WAMP, was presented by Lindgren and Kenneth Holmström.

Further progress was made in the 80's, when a so-called regression
approximation was developed by Igor Rychlik [2]. This made it possible to
calculate distributions for characteristic wave parameters for a broader class
of processes. A first collection of Matlab routines appeared in 1993, called *
Fatigue Analysis Toolbox* (FAT). A co-worker at that time was Mats Frendahl.
Some years later (1995), a Wave Analysis Toolbox
(WAT) was developed. That toolbox focused on problems occuring in oceanography
while still keeping routines from FAT for rainflow analysis of loads. Finn
Lindgren made important contributions to WAT.

A fast development took place in 1999. Pär Johannesson, a PhD student in Lund, had written a Matlab toolbox for switching Markov chains. Further, Per A Brodtkorb (NTNU, Trondheim) was a visiting researcher in Lund. Many new routines were developed, with emphasis on applications from oceanography and marine technology, and old ones were made more efficient. The idea of updating WAT was born. Two further PhD students from Lund working with applications of random waves (Eva Sjö, and Jesper Rydén) were involved in the updating work, contributing with new routines and reorganising the stuff [3].

By the end of January 2000, a first version of WAFO was released on the Internet.An updated version for Unix was released by the end of May. Next version (2.0.1) was released at July 5, with an accompanying tutorial appearing July 6.

(For a complete list of references, see the Tutorial.)

- Lindgren, G. (1972). Wave-length and amplitude in Gaussian noise.
*Adv Appl Prob***4**, pp. 81-108. - Rychlik, I. (1987). Regression approximation of wave length and amplitude
distributions.
*Adv Appl Prob***19**, pp. 396-430. - Brodtkorb, P.A., Johannesson, P., Lindgren, G., Rychlik, I., Rydén, J. and
Sjö, E. (2000). WAFO - a Matlab toolbox for analysis of random waves and
loads.
*Proceedings of the 10th International Offshore and Polar Engineering conference*, Seattle, Vol III, pp. 343-350.

## Cite WAFO

We have invested a lot of time and effort in creating WAFO, please cite it when using it for data analysis. To cite WAFO in publications use:

Brodtkorb, P.A., Johannesson, P., Lindgren, G., Rychlik, I.,
Rydén, J. and Sjö, E. (2000). "WAFO - a Matlab toolbox for analysis of random
waves and loads", *Proc. 10th Int. Offshore and Polar Eng. Conf.*, Seattle,
USA, Vol III, pp. 343-350.

or

WAFO-group (2000). "WAFO - A Matlab Toolbox for Analysis of Random Waves and Loads - A Tutorial" Math. Stat., Center for Math. Sci., Lund Univ., Lund, Sweden. ISBN XXXX, URL http://www.maths.lth.se/matstat/wafo.

A BibTeX entry for LaTeX users is

@InProceedings{BrodtkorbEtal2000Wafo,

author = {Brodtkorb, {P.A.} and Johannesson, P. and Lindgren, G. and Rychlik, I. and Ryd{\'e}n, J. and Sj{\"o}, E.},

title = {{WAFO} - a {M}atlab Toolbox for the Analysis of Random Waves and Loads},

booktitle = {Proc. 10'th Int. Offshore and Polar Eng. Conf., ISOPE, Seattle, USA},

pages = {343--350},

year = {2000},

volume = {3},

}

or

@Manual{WAFO-group2000Wafo,

author = {WAFO-group},

title = {{WAFO} - A Matlab Toolbox for Analysis of Random Waves and Loads - A Tutorial},

year = {2000},

organization = {Math. Stat., Center for Math. Sci., Lund Univ.},

address = {Lund, Sweden}},

note = {{ISBN} XXXX},

url = {http://www.maths.lth.se/matstat/wafo}

}