commit f61b68b95b5be8ce26c23df2e32b5aa271cfb5a5
parent 0ef95571e3694da55a7604ac0488ae640931cbbb
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Mon, 31 Oct 2016 15:28:40 +0300
Edit ARMA part (#1 pass).
Diffstat:
| phd-diss.org | | | 281 | +++++++++++++++++++++++++++++++++++++++++-------------------------------------- |
1 file changed, 145 insertions(+), 136 deletions(-)
diff --git a/phd-diss.org b/phd-diss.org
@@ -20,7 +20,7 @@
\clearpage
#+end_latex
-* Introduction
+* TODO Introduction
**** Topic relevance.
**** State-of-the-art.
**** Goals and objectives.
@@ -29,139 +29,32 @@
**** Methodology and research methods.
**** Theses for the defense.
**** Results verification and approbation.
-* Problem statement
-* Related work
-* ARMA model for ocean wave simulation
+* TODO Problem statement
+* TODO Related work
+* TODO ARMA model for ocean wave simulation
** The reasons for the search for a new ocean simulation model
ARMA model emerged in response to difficulties encountered by practitioners who
used wave simulation models developed in the framework of linear wave theory.
-The problems they have encountered in their work can be summarised as the
-following.
+The problems they have encountered with Longuet---Higgins model (a model which
+is entirely based on linear wave theory) can be summarised as the following.
1. /Periodicity/. Linear wave theory approximates waves by a sum of harmonics,
so period of the whole wavy surface realisation depends on the number of
harmonics in the model. So, reasonably large realisations require large
- number of coefficients. This in turn results in overall bad performance of
- the model, no matter how optimised the software implementation is.
+ number of coefficients. This in turn results in overall bad performance of a
+ model based on this theory, no matter how optimised the software
+ implementation is.
2. /Linearity/. Linear wave theory gives mathematical definition for ocean waves
which have small amplitudes compared to their lengths. Waves of this type
- occur in the ocean, so waves in shallow water as well as storm waves, for
- which this assumption does not hold, are not perfectly captured by linear
+ occur mostly in the ocean, so waves in shallow water as well as storm waves,
+ for which this assumption does not hold, are not perfectly captured by linear
theory.
-3. /Probabilistic convergence/. Wave phase, which is often generated by pseudo
- random number generator (PRNG), has uniform distribution, and this makes wavy
- surface characteristics (average wave height, wave period, wave length etc.)
- sometimes converge slowly to the desired values. Convergence rate depends on
- the values generated by PRNG, which makes it probabilistic.
+3. /Probabilistic convergence/. Phase of a wave, which is often generated by
+ pseudo random number generator (PRNG), has uniform distribution, and this
+ makes wavy surface characteristics (average wave height, wave period, wave
+ length etc.) sometimes converge slowly to the desired values. Convergence
+ rate depends on the values generated by PRNG, which makes it probabilistic.
-** The shape of ACF for different types of waves
-*** Two methods to find ocean wave's ACF
-**** Analytic method of finding the ACF.
-The simplest way to find auto-covariate function for a particular ocean wave
-profile is to apply Wiener---Khinchin theorem. According to this theorem the
-autocorrelation $K$ of a function $\zeta$ is given by the Fourier transform of
-the absolute square of the function:
-\begin{equation}
- K(t) = \Fourier{\left| \zeta(t) \right|^2}.
- \label{eq:wiener-khinchin}
-\end{equation}
-When $\zeta$ is replaced with actual wave profile, this formula gives you
-analytic expression for the corresponding ACF.
-
-For three-dimensional wave profile (2D in space and 1D in time) analytic
-expression is a polynomial of high order and is best obtained via computer
-algebra software. Then for practical usage it can be approximated by
-superposition of exponentially decaying cosines (which is how ACF of a
-stationary ARMA process looks like cite:box1976time).
-
-**** Empirical method of finding the ACF.
-However, for three-dimensional case there exists simpler empirical method which
-does not require sophisticated software to determine shape of the ACF. It is
-known that ACF represented by exponentially decaying cosines of a wave profile
-satisfies first order Stokes' equations for gravity waves cite:boccotti1983wind.
-So, if the shape of the wave profile is the only concern, then one can simply
-multiply it by a decaying exponent to get appropriate ACF. This ACF will not
-reflect other wave profile parameters such as wave height and period, but opens
-possibility to simulate waves of a particular non-analytic shape by "drawing"
-their profile, then multiplying it by an exponent and using the resulting
-function as ACF. So, this empirical method is imprecise but offers simpler
-alternative to Wiener---Khinchin theorem; it is mainly useful to test ARMA
-model.
-
-*** Examples of ACFs for various types of wave profiles
-**** Standing wave.
-For three-dimensional standing wave the profile is approximated by
-\begin{equation}
- \zeta(t, x, y) = A \sin (k_x x + k_y y) \sin (\sigma t).
- \label{eq:standing-wave}
-\end{equation}
-In order to get ACF via analytic method one needs to multiply this expression by
-a decaying exponent, because Fourier transform is defined for a function $f$ that
-$f \underset{x \rightarrow \pm \infty}{\longrightarrow} 0$. The formula of the
-profile then transforms to
-\begin{equation}
- \zeta(t, x, y) =
- A
- \exp\left[-\alpha (|t|+|x|+|y|) \right]
- \sin (k_x x + k_y y) \sin (\sigma t).
- \label{eq:decaying-standing-wave}
-\end{equation}
-Them, if one takes 3D Fourier transform of this expression via any capable
-computer algebra software, the resulting polynomial may be fitted to the
-following ACF approximation.
-\begin{equation}
- K(t,x,y) =
- \gamma
- \exp\left[-\alpha (|t|+|x|+|y|) \right]
- \cos \beta t
- \cos \left[ \beta x + \beta y \right].
- \label{eq:standing-wave-acf}
-\end{equation}
-So, after applying Wiener---Khinchin theorem we get the same formula but with
-sines replaced with cosines. This replacement is important because the value of
-ACF at $(0,0,0)$ equals to the variance of wave elevation, and if one used sines
-the value would be wrong.
-
-If one tries to replicate the same formula via empirical method, the usual way
-is to adapt eqref:eq:decaying-standing-wave to match eqref:eq:standing-wave-acf.
-This can be done by changing the phase of the sine, or by replacing sine with
-cosine to move the maximum of the function to $(0,0,0)$.
-
-**** Propagating wave.
-Three-dimensional profile of this type of wave is approximated by
-\begin{equation}
- \zeta(t, x, y) = A \cos (\sigma t + k_x x + k_y y).
- \label{eq:propagating-wave}
-\end{equation}
-For the analytic method one may repeat steps from the previous two paragraphs
-with ACF approximated by
-\begin{equation}
- K(t,x,y) =
- \gamma
- \exp\left[-\alpha (|t|+|x|+|y|) \right]
- \cos\left[\beta (t+x+y) \right].
- \label{eq:propagating-wave-acf}
-\end{equation}
-For the empirical method propagating wave profile is simply multiplied by
-a decaying exponent without need to adapt the maximum value of ACF.
-
-*** Comparison of studied methods
-To summarise, the analytic method of finding ocean wave's ACF reduces to the
-following steps:
-- Make wave profile decay when approach $\pm \infty$ by multiplying it by
- a decaying exponent.
-- Take Fourier transform of absolute square of the decaying wave profile using
- computer algebra software.
-- Fit the resulting polynomial to the appropriate ACF approximation.
-
-Two examples in this section showed that in case of standing and propagating
-waves their decaying profiles resemble the corresponding ACFs with the exception
-that the origin should be moved to the function's maximal value for the ACF to
-be useful in ARMA model simulations. So, using the empirical method the ACF is
-found in the following steps:
-- Make wave profile decay when approach $\pm \infty$ by multiplying it by
- a decaying exponent.
-- Move maximum value to the origin by adjusting phases or using trigonometric
- identities to shift the phase of the resulting function.
+*** TODO Insert a paragraph about ARMA model adavntages
** Governing equations for 3-dimensional ARMA process
*** Three possible processes
@@ -312,12 +205,12 @@ MA process exclusively.
*** Process selection criteria for different wave profiles
The main problem of ARMA model application to ocean wave generation is that for
-different types of wave profiles different processes must be used: standing
+different types of wave profiles different processes /must/ be used: standing
waves can only be generated by AR process, and propagating waves can only be
generated by MA process. This statement comes from practice: if one tries to use
the processes the other way round, the resulting realisation either diverges or
does not look like real ocean waves. (The latter happens for non-invertible MA
-process, and sometimes for AR process for unknown reason.) So, the best way to
+process, and sometimes for AR process for an unknown reason.) So, the best way to
apply ARMA model to ocean wave generation is to use AR process for standing
waves and MA process for progressive waves.
@@ -330,16 +223,22 @@ The final problem is how to mix 3-dimensional AR and MA processes. Again the new
formula to recompute ACF needs to be derived in case of three dimensions.
It appears that some statements made for AR and MA processes in cite:box1976time
-should be flipped for three-dimensional case. For example, the authors say that
-ACF of MA process is cut at $(q_1,q_2,q_3)$ and ACF of AR process decays to
-nought infinitely, but in practice making ACF of 3-dimensional MA process not
-decay results in it being non-invertible and producing realisation that does not
-look like real ocean waves, whereas doing the same for ACF of AR process does
-not have severe consequences. The second example: the authors say that one
-should allocate the first $q$ points of ACF to MA process (as it often needed to
-describe the peaks in ACF) and leave the rest points to AR process, but in
-practice in case of ACF of propagating wave AR process is stationary only for
-the first time slice of the ACF, and the rest is left to MA process.
+(a book from creators of ARMA process) should be flipped for three-dimensional
+case. For example, the authors say that ACF of MA process is cut at
+$(q_1,q_2,q_3)$ and ACF of AR process decays to nought infinitely, but in
+practice making ACF of 3-dimensional MA process not decay results in it being
+non-invertible and producing realisation that does not look like real ocean
+waves, whereas doing the same for ACF of AR process does not have severe
+consequences. The second example: the authors say that one should allocate the
+first $q$ points of ACF to MA process (as it often needed to describe the peaks
+in ACF) and leave the rest points to AR process, but in practice in case of ACF
+of a propagating wave AR process is stationary only for the first time slice of
+the ACF, and the rest is left to MA process.
+
+So, for now the only safe scenario is to use AR process for standing waves and
+MA process for propagating waves. With new formulae for 3 dimensions a single
+mixed ARMA process might be a better choice, but this is the objective of the
+future research.
** Verification of wavy surface integral characteristics
Research shows that several ocean wave characteristics (e.g. wave height, wave
@@ -441,6 +340,116 @@ exit
| \includegraphics{propagating-elevation} | \includegraphics{propagating-wave-height-x} |
| \includegraphics{propagating-wave-length-x} | \includegraphics{propagating-wave-period} |
+** The shape of ACF for different types of waves
+*** Two methods to find ocean wave's ACF
+**** Analytic method of finding the ACF.
+The simplest way to find auto-covariate function for a particular ocean wave
+profile is to apply Wiener---Khinchin theorem. According to this theorem the
+autocorrelation $K$ of a function $\zeta$ is given by the Fourier transform of
+the absolute square of the function:
+\begin{equation}
+ K(t) = \Fourier{\left| \zeta(t) \right|^2}.
+ \label{eq:wiener-khinchin}
+\end{equation}
+When $\zeta$ is replaced with actual wave profile, this formula gives you
+analytic expression for the corresponding ACF.
+
+For three-dimensional wave profile (2D in space and 1D in time) analytic
+expression is a polynomial of high order and is best obtained via computer
+algebra software. Then for practical usage it can be approximated by
+superposition of exponentially decaying cosines (which is how ACF of a
+stationary ARMA process looks like cite:box1976time).
+
+**** Empirical method of finding the ACF.
+However, for three-dimensional case there exists simpler empirical method which
+does not require sophisticated software to determine shape of the ACF. It is
+known that ACF represented by exponentially decaying cosines of a wave profile
+satisfies first order Stokes' equations for gravity waves cite:boccotti1983wind.
+So, if the shape of the wave profile is the only concern, then one can simply
+multiply it by a decaying exponent to get appropriate ACF. This ACF will not
+reflect other wave profile parameters such as wave height and period, but opens
+possibility to simulate waves of a particular non-analytic shape by "drawing"
+their profile, then multiplying it by an exponent and using the resulting
+function as ACF. So, this empirical method is imprecise but offers simpler
+alternative to Wiener---Khinchin theorem; it is mainly useful to test ARMA
+model.
+
+*** Examples of ACFs for various types of wave profiles
+**** Standing wave.
+For three-dimensional standing wave the profile is approximated by
+\begin{equation}
+ \zeta(t, x, y) = A \sin (k_x x + k_y y) \sin (\sigma t).
+ \label{eq:standing-wave}
+\end{equation}
+In order to get ACF via analytic method one needs to multiply this expression by
+a decaying exponent, because Fourier transform is defined for a function $f$ that
+$f \underset{x \rightarrow \pm \infty}{\longrightarrow} 0$. The formula of the
+profile then transforms to
+\begin{equation}
+ \zeta(t, x, y) =
+ A
+ \exp\left[-\alpha (|t|+|x|+|y|) \right]
+ \sin (k_x x + k_y y) \sin (\sigma t).
+ \label{eq:decaying-standing-wave}
+\end{equation}
+Them, if one takes 3D Fourier transform of this expression via any capable
+computer algebra software, the resulting polynomial may be fitted to the
+following ACF approximation.
+\begin{equation}
+ K(t,x,y) =
+ \gamma
+ \exp\left[-\alpha (|t|+|x|+|y|) \right]
+ \cos \beta t
+ \cos \left[ \beta x + \beta y \right].
+ \label{eq:standing-wave-acf}
+\end{equation}
+So, after applying Wiener---Khinchin theorem we get the same formula but with
+sines replaced with cosines. This replacement is important because the value of
+ACF at $(0,0,0)$ equals to the variance of wave elevation, and if one used sines
+the value would be wrong.
+
+If one tries to replicate the same formula via empirical method, the usual way
+is to adapt eqref:eq:decaying-standing-wave to match eqref:eq:standing-wave-acf.
+This can be done by changing the phase of the sine, or by replacing sine with
+cosine to move the maximum of the function to $(0,0,0)$.
+
+**** Propagating wave.
+Three-dimensional profile of this type of wave is approximated by
+\begin{equation}
+ \zeta(t, x, y) = A \cos (\sigma t + k_x x + k_y y).
+ \label{eq:propagating-wave}
+\end{equation}
+For the analytic method one may repeat steps from the previous two paragraphs
+with ACF approximated by
+\begin{equation}
+ K(t,x,y) =
+ \gamma
+ \exp\left[-\alpha (|t|+|x|+|y|) \right]
+ \cos\left[\beta (t+x+y) \right].
+ \label{eq:propagating-wave-acf}
+\end{equation}
+For the empirical method propagating wave profile is simply multiplied by
+a decaying exponent without need to adapt the maximum value of ACF.
+
+*** Comparison of studied methods
+To summarise, the analytic method of finding ocean wave's ACF reduces to the
+following steps:
+- Make wave profile decay when approach $\pm \infty$ by multiplying it by
+ a decaying exponent.
+- Take Fourier transform of absolute square of the decaying wave profile using
+ computer algebra software.
+- Fit the resulting polynomial to the appropriate ACF approximation.
+
+Two examples in this section showed that in case of standing and propagating
+waves their decaying profiles resemble the corresponding ACFs with the exception
+that the origin should be moved to the function's maximal value for the ACF to
+be useful in ARMA model simulations. So, using the empirical method the ACF is
+found in the following steps:
+- Make wave profile decay when approach $\pm \infty$ by multiplying it by
+ a decaying exponent.
+- Move maximum value to the origin by adjusting phases or using trigonometric
+ identities to shift the phase of the resulting function.
+
** Modeling non-linearity of ocean waves
** Non-physical nature of ARMA model
ARMA model, owing to its non-physical nature, does not have the notion of ocean
