Appendix 4

Calculation of the Variances of ν_φ, ν_η. ν_θ

A4.1. Variance of the ν_φ and ν_η, variables

From relationship [3.50] in section 3.3.2 of Chapter 3 we know that the azimuth angle φ_i of the waves with random trials is linked to the ν_φ variable by the following expression:

[A4.1]

The computation of the pdf established in [3.52] enables us to formulate the sought after variance, i.e.:

[A4.2]

is the equation in which D is the variation domain of ν_ø, i.e. the interval [-1 +1]. Integral [A4.2] can then take the detailed form below:

[A4.3]

By using expression [A4.1], we carry out the change of variable leading to integral [A4.4]:

[A4.4]

Analytical solution of this integral allocate the value ½ to the variance:

[A4.5]

Regarding υ_η, a similar calculation leads to the determination of the integral [A4.6]:

[A4.6]

A4.2. Variance of the υ_θ variable

We recall that the definition of the υ_θ variable as found in equation [3.50]:

[A4.7]

Knowing that θ_i covers the [0 +π] interval, amounts to saying that we can divide the domain to conform to the partition suggested in [A4.8]:

[A4.8]

The calculation of the moment of the square of υ_θ is consequently similar to the integral of υ_θ on the two joined intervals [-1 0], i.e.:

[A4.9]

After insertion of the pdf found in [3.54], the variance can be calculated by the integral below:

[A4.10]

Solving this integral analytically leads to the following results:

[A4.11]

Coming back to equation [A4.10] finally gives the value of the variance:

[A4.12]