docs/tutorials/nomenclature.rst
Nomenclature
============
Roman letters
-------------
+----------------------------------+----------------------------------+
| :math:`A_{c}`, :math:`A_{t}` | Zero-crossing wave crest height |
| | and trough excursion. |
+----------------------------------+----------------------------------+
| :math:`a_{i}` | Lower integration limit. |
+----------------------------------+----------------------------------+
| :math:`b_{i}` | Upper integration limit. |
+----------------------------------+----------------------------------+
| :math:`c_{0}` | Truncation parameter of |
| | truncated Weibull distribution. |
+----------------------------------+----------------------------------+
| :math:`\mbox{\sf C}[X,Y]` | Covariance between random |
| | variables :math:`X` and |
| | :math:`Y`. |
+----------------------------------+----------------------------------+
| :math:` | Directional spreading function. |
| D(\ensuremath{\omega },\theta),` | |
| :math:`D(\theta)` | |
+----------------------------------+----------------------------------+
| :math:`dd_{crit}\, d_{crit}\, % | Critical distances used for |
| z_{crit}` | removing outliers and spurious |
| | points. |
+----------------------------------+----------------------------------+
| :math:`\mbox{\sf E}[X]` | Expectation of random variable |
| | :math:`X`. |
+----------------------------------+----------------------------------+
| :math:`E(\ensuremath{\omega | Quadratic transfer function. |
| }_{i},\ensuremath{\omega }_{j})` | |
+----------------------------------+----------------------------------+
| :math:`f` | Wave frequency |
| | :math:`[\textit{Hz}]`. |
+----------------------------------+----------------------------------+
| :math:`f_{p}` | Spectral peak frequency. |
+----------------------------------+----------------------------------+
| :math:`F_{X}(\cdot)`, | Cumulative distribution function |
| :math:`f_{X}(\cdot)` | and probability density function |
| | of variable :math:`X`. |
+----------------------------------+----------------------------------+
| :math:`G(\cdot),\,g(\cdot)` | The transformation and its |
| | inverse. |
+----------------------------------+----------------------------------+
| :math:`g` | Acceleration of gravity. |
+----------------------------------+----------------------------------+
| :math:`H`, :math:`h` | Dimensional and dimensionless |
| | wave height. |
+----------------------------------+----------------------------------+
| :math:`H_{m0}`, :math:`H_s` | Significant wave height, |
| | :math:`4\sqrt{m_{0}}`. |
+----------------------------------+----------------------------------+
| :math:`H_{c}` | Critical wave height. |
+----------------------------------+----------------------------------+
| :math:`H_{d}`, :math:`H_{u}` | Zero-downcrossing and |
| | -upcrossing wave height. |
+----------------------------------+----------------------------------+
| :math:`h` | Water depth. |
+----------------------------------+----------------------------------+
| :math:`h_{\max}` | Maximum interval width for |
| | Simpson method. |
+----------------------------------+----------------------------------+
| :math:`H_{rms}` | Root mean square value for wave |
| | height defined as |
| | :math:`H_{m0}/\sqrt{2}`. |
+----------------------------------+----------------------------------+
| :math:`K_{d}(\cdot)` | Kernel function. |
+----------------------------------+----------------------------------+
| :math:`k` | Wave number |
| | :math:`[\textit{rad/m}]` or |
| | index. |
+----------------------------------+----------------------------------+
| :math:`L_{p}` | Average wave length. |
+----------------------------------+----------------------------------+
| :math:`L_{\max}` | Maximum lag beyond which the |
| | autocovariance is set to zero. |
+----------------------------------+----------------------------------+
| :math:`M,\,M_{k}` | Local maximum. |
+----------------------------------+----------------------------------+
| :math:`M_{k}^{tc}` | Crest maximum for wave no. |
| | :math:`k`. |
+----------------------------------+----------------------------------+
+----------------------------------+----------------------------------+
| :math:`m,\,m_{k}` | Local minimum. |
+----------------------------------+----------------------------------+
| :math | Rainflow minimum no. :math:`k`. |
| :`m_{k}^{{\protect\mbox{\protect | |
| \footnotesize\protect\sc rfc}}}` | |
+----------------------------------+----------------------------------+
| :math:`m_{k}^{tc}` | Trough minimum for wave no. |
| | :math:`k`. |
+----------------------------------+----------------------------------+
| :math:`m_{n}` | n’th spectral moment, |
| | :math:`\int |
| | _{0}^{\infty}\omega^{n} S_{\eta |
| | \eta}^ |
| | {+}(\omega) \,\mathrm{d}\omega`. |
+----------------------------------+----------------------------------+
| :math:`N` | Number of variables or waves. |
+----------------------------------+----------------------------------+
| :math:`N_{c1c2}` | Number of times to apply |
| | regression equation. |
+----------------------------------+----------------------------------+
| ``NIT`` | Order in the integration of wave |
| | characteristic distributions. |
+----------------------------------+----------------------------------+
| :math:`n_{i},` :math:`n` | Sample size. |
+----------------------------------+----------------------------------+
| :math:`\mbox{\sf P}(A)` | Probability of event :math:`A`. |
+----------------------------------+----------------------------------+
| :math:`O(\cdot)` | Order of magnitude. |
+----------------------------------+----------------------------------+
| :math:`Q_p` | Peakedness factor. |
+----------------------------------+----------------------------------+
| :math:`R_{\eta}(\tau)` | Auto covariance function of |
| | :math:`\eta(t)`. |
+----------------------------------+----------------------------------+
| :math:`S_{p}` | Average wave steepness. |
+----------------------------------+----------------------------------+
| :math:`S_s` | Significant wave steepness. |
+----------------------------------+----------------------------------+
| :math:`S_{\eta \eta}^{+}(f), % | One sided spectral density of |
| S_{\eta \eta}^{+}(\omega)` | the surface elevation |
| | :math:`\eta`. |
+----------------------------------+----------------------------------+
| :math: | Directional wave spectrum. |
| `S(\ensuremath{\omega },\theta)` | |
+----------------------------------+----------------------------------+
| :math:`s` | Normalized crest front |
| | steepness. |
+----------------------------------+----------------------------------+
| :math:`s_{c}` | Critical crest front steepness. |
+----------------------------------+----------------------------------+
| :math:`s_{cf}` | Crest front steepness. |
+----------------------------------+----------------------------------+
| :math:`s_N` | Return level for return period |
| | :math:`N`. |
+----------------------------------+----------------------------------+
| :math:`s_{rms}` | Root mean square value for crest |
| | front steepness, |
+----------------------------------+----------------------------------+
| | i.e., |
| | :math:`5/4\,H_{m0}/T_{m02}^{2}`. |
+----------------------------------+----------------------------------+
| :math:`T_{c}`, :math:`T_{cf}`, | Crest, crest front, and crest |
| :math:`T_{cr}` | rear period. |
+----------------------------------+----------------------------------+
| :math:`T_{m(-1)0}` | Energy period. |
+----------------------------------+----------------------------------+
| :math:`T_{m01}` | Mean wave period. |
+----------------------------------+----------------------------------+
| :math:`T_{m02}` | Mean zero-crossing wave period |
| | calculated as |
| | :math:`2\pi\sqrt{m_{0}/m_{2}}`. |
+----------------------------------+----------------------------------+
| :math:`T_{m24}` | Mean wave period between maxima |
| | calculated as |
| | :math:`2\pi\sqrt{m_{2}/m_{4}}`. |
+----------------------------------+----------------------------------+
| :math:`T_{Md}` | Wave period between maximum and |
| | downcrossing. |
+----------------------------------+----------------------------------+
| :math:`T_{Mm}` | Wave period between maximum and |
| | minimum. |
+----------------------------------+----------------------------------+
| :math:`T_{p}` | Spectral peak period. |
+----------------------------------+----------------------------------+
| :math:`T_{z}` | Mean zero-crossing wave period |
| | estimated directly from time |
| | series. |
+----------------------------------+----------------------------------+
| :math:`T` | Wave period. |
+----------------------------------+----------------------------------+
| :math:`U_{10}` | 10 min average of windspeed |
| | :math:`10 [m]` above the |
| | watersurface. |
+----------------------------------+----------------------------------+
| :math:`U_{i}` | Uniformly distributed number |
| | between zero and one. |
+----------------------------------+----------------------------------+
| :math:`V`, :math:`v` | Dimensional and dimensionless |
| | velocity. |
+----------------------------------+----------------------------------+
| :math:`\mbox{\sf V}[X]` | Variance of random variable |
| | :math:`X`. |
+----------------------------------+----------------------------------+
| :math:`V_{cf}`, :math:`V_{cr}` | Crest front and crest rear |
| | velocity. |
+----------------------------------+----------------------------------+
| :math:`V_{rms}` | Root mean square value for |
| | velocity defined as |
| | :math:`2 H_{m0}/T_{m02}`. |
+----------------------------------+----------------------------------+
| :math:`W_{age}` | Wave age. |
+----------------------------------+----------------------------------+
| :math:`W(x,t)` | Random Gassian field. |
+----------------------------------+----------------------------------+
| :math:`X(t)` | Time series. |
+----------------------------------+----------------------------------+
| :math:`X_{i}`, :math:`Y_i`, | Random variables. |
| :math:`Z_{i}` | |
+----------------------------------+----------------------------------+
| :math:`x_{c}`, :math:`y_c`, | Truncation parameters. |
| :math:`z_{c}` | |
+----------------------------------+----------------------------------+
Greek letters
-------------
+----------------------------------+----------------------------------+
| :math:`\alpha` | Rayleigh scale parameter or |
| | JONSWAP normalization constant. |
+----------------------------------+----------------------------------+
| :math:`\alpha` | Irregularity factor; spectral |
| | width measure. |
+----------------------------------+----------------------------------+
| :math:`\alpha(h)`, | Weibull or Gamma parameters for |
| :math:`\beta(h)` | scale and shape. |
+----------------------------------+----------------------------------+
| :math:`\alpha_{i}` | Product correlation coefficient. |
+----------------------------------+----------------------------------+
| :math:`\Delta` | Forward difference operator. |
+----------------------------------+----------------------------------+
| :math:`\delta_{i|1}` | Residual process. |
+----------------------------------+----------------------------------+
| :math:`\epsilon_{2}` | Narrowness parameter defined as |
| | :math: |
| | `\sqrt{m_{0}m_{2}/m_{1}^{2}-1}`. |
+----------------------------------+----------------------------------+
| :math:`\epsilon_{4}` | Broadness factor defined as |
| | :math:`\s |
| | qrt{1-m_{2}^{2}/(m_{0} m_{4})}`. |
+----------------------------------+----------------------------------+
| :math:`\epsilon` | Requested error tolerance for |
| | integration. |
+----------------------------------+----------------------------------+
| :math:`\epsilon_{c}` | Requested error tolerance for |
| | Cholesky factorization. |
+----------------------------------+----------------------------------+
| :math:`\eta(\cdot)` | Surface elevation. |
+----------------------------------+----------------------------------+
| :math:`\Gamma` | Gamma function. |
+----------------------------------+----------------------------------+
| :math:`\gamma` | JONSWAP peakedness factor or |
| | Weibull location parameter. |
+----------------------------------+----------------------------------+
| :math:`\lambda_{i}` | Eigenvalues or shape parameter |
| | of Ochi-Hubble spectrum. |
+----------------------------------+----------------------------------+
| :math:`\mu_{X}(v)` | Crossing intensity of level |
| | :math:`v` for time series |
| | :math:`X(t)`. |
+----------------------------------+----------------------------------+
| :math:`\mu_{X}^+(v)` | Upcrossing intensity of level |
| | :math:`v` for time series |
| | :math:`X(t)`. |
+----------------------------------+----------------------------------+
| :math:`\Phi(\cdot),` | CDF and PDF of a standard normal |
| :math:`\phi(\cdot)` | variable. |
+----------------------------------+----------------------------------+
| :math:`\Theta_{n}` | Phase function. |
+----------------------------------+----------------------------------+
| :math:`\rho_{3}`, | Normalized cumulants, i.e., |
| :math:`\rho_{4}` | skewness and excess, |
| | respectively. |
+----------------------------------+----------------------------------+
| :math:`\rho_{ij}` | Correlation between random |
| | variables :math:`X_{i}` and |
| | :math:`X_{j}`. |
+----------------------------------+----------------------------------+
| :math | Covariance matrix. |
| :`\ensuremath{\mathbf{\Sigma} }` | |
+----------------------------------+----------------------------------+
| :math:`\sigma_{X}^{2}` | Variance of random variable |
| | :math:`X`. |
+----------------------------------+----------------------------------+
| :math:`\tau` | Shift variable of time. |
+----------------------------------+----------------------------------+
| :math:`\tau_{i}` | Parameters defining the |
| | eigenvalues of |
| | :math:`\en |
| | suremath{\boldsymbol{\Sigma} }`. |
+----------------------------------+----------------------------------+
| :math:`\omega` | Wave angular frequency |
| | :math:`[rad/s]`. |
+----------------------------------+----------------------------------+
| :math:`\omega_{p}` | Wave angular peak frequency |
| | :math:`[rad/s]`. |
+----------------------------------+----------------------------------+
Abbreviations
-------------
===== ===========================================
AMISE Asymptotic mean integrated square error.
CDF Cumulative distribution function.
FFT Fast Fourier Transform.
GEV Generalized extreme value.
GPD Generalized Pareto distribution.
HF High frequency.
ISSC International ship structures congress.
ITTC International towing tank conference.
IQR Interquartile range.
KDE Kernel density estimate.
LS Linear simulation.
MC Markov chain.
MCTP Markov chain of turning points.
ML Maximum likelihood.
NLS Non-linear simulation.
MISE Mean integrated square error.
MWL Mean water line.
PDF Probability density function.
PSD Power spectral density.
QTF Quadratic transfer function.
SCIS Sequential conditioned importance sampling.
TLP Tension-leg platform.
TP Turning points.
WAFO Wave analysis for fatigue and oceanography.
===== ===========================================