# 3.4.2: Analysis of the time-series

- Page ID
- 16290

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The mean and the standard deviation are important statistical properties that can be derived from an arbitrary time-series. For a stationary signal the mean should be constant (zero for purely oscillatory signal), whereas the standard deviation \(\sigma\) is a vertical measure that can be related to the wave height as we will see later. The variance of the demeaned surface elevation \(\sigma^2\) is an important quantity in the statistical description of waves since it is related to the mean wave energy per unit area \(E\) which is the sum of the potential and kinetic energy (\(E = E_p + E_k\)). The potential energy is \(E_p = \tfrac{1}{2} \rho g \sigma^2\). In the linear approximation (valid for small amplitude waves compared to the wavelength and water depth) the potential and kinetic energy are the same (or \(E_p = E_k = \tfrac{1}{2} E\)) and hence:

\[E = \rho g \sigma^2\]

The difference between variance and energy therefore is just a factor \(\rho g\).

Alternatively, when the short-term time record (order 20 minutes) is considered as a series of individual waves with their own wave height and period average parameters can be taken of the series of wave heights and periods in order to characterise the record. Remember that useful averages require a stationary record.

Before starting the wave-by-wave analysis the mean water level should be subtracted from the record. We then have a purely oscillatory surface elevation signal about the mean. In the case of for instance a tidal variation it is possible that the record is not entirely stationary but that there is a slight variation in the mean water level. For a short record length this trend will be approximately linear and can be determined by regression analysis and then subtracted from the signal so that the signal is stationary again.

When the signal is demeaned, individual waves can be defined as having a wave height equal to the difference in elevation between the crest and the trough. The wavelength and wave period are the distance and period respectively between two subsequent downward or upward zero-crossings. An advantage of using downward crossings is that it relates more directly to visual observations of wave heights. The reason is that observers tend to define a wave as starting with the trough, such that the wave height is taken to be the height of the crest relative to the preceding trough.

Various average parameters can now be derived of which the most obvious probably is the mean wave height. Nevertheless, the mean wave height is not used that often. Of more practical use is the significant wave height \(H_s\) or \(H_{\tfrac{1}{3}}\). The significant wave height is defined as the average height of the highest one third of the waves:

\[H_{1/3} = \dfrac{1}{N/3} \sum_{j = 1}^{N/3} H_j\]

where \(H_j\) is the \(j\)-th wave (with \(j = 1\) the largest wave, \(j = 2\) the second largest etc.) and \(N\) is the total number of waves.

It is called significant wave height because it approximately corresponds to visual estimates of experienced observers at sea of a representative wave height. Apparently observers tend to bias their estimates to the higher waves in the record. Its correspondence with visual estimates and therefore easy quantification from ships and large databases makes it a useful measure for coastal engineers.

Another often used parameter is \(H_{rms}\), the root-mean-square wave height, which is obtained by taking the square root of the mean of the wave heights squared:

\[H_{rms} = \sqrt{\dfrac{1}{N} \sum_{i = 1}^{N} H_i^2}\]

It can be seen as a wave energy measure since the wave energy is related to the wave height squared (see also Sect. 3.4.4).

Other measures like \(H_{1/10}\) and \(H_{1/100}\) are also used and they are defined analogous to \(H_{1/3}\) as the average of the highest 1/10 and 1/100 of the waves respectively.

The mean of all wave periods is called the mean wave period or zero-crossing wave period:

\[\overline{T_0} = \dfrac{1}{N} \sum_{i = 1}^{N} T_i\]

Similar to \(H_{1/3}\) the significant wave period is defined as the average wave period of the highest one-third of the waves. The significant wave period is not correlated to visual estimates and therefore has less physical meaning:

\[T_{1/3} = \dfrac{1}{N/3} \sum_{j = 1}^{N/3} T_j\]

What values would typically be found at for instance the North Sea? For sea conditions significant wave heights range from order 1 m during quiet periods (with mean wave periods of order 5 s) to 10 m and more during storms (with mean periods of order 10 s). North Sea swell has wave heights of 0.5 m to 1 m and wave periods of around 10 s.