commit ac0a4ffbef95fd97ca086717d2d06bed4ee1a313
parent e2a1c2f1180ac53c44caf6a4911ded3e0c9a8d7d
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Sun, 28 May 2017 09:44:16 +0300
Add related work and governing equations.
Diffstat:
| arma.org | | | 396 | ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++- |
| preamble.tex | | | 30 | ++++++++++++++++++++++++++++-- |
| refs.bib | | | 82 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++-- |
3 files changed, 501 insertions(+), 7 deletions(-)
diff --git a/arma.org b/arma.org
@@ -4,7 +4,7 @@
#+LATEX_CLASS: scrartcl
#+LATEX_CLASS_OPTIONS:
#+LATEX_HEADER_EXTRA: \input{preamble}
-#+OPTIONS: H:2 num:0 todo:nil toc:nil
+#+OPTIONS: H:5 num:0 todo:nil toc:nil
#+begin_abstract
Simulation of sea waves is a problem appearing in the framework of developing
@@ -73,7 +73,7 @@ isotropy within the marine environment. In this case eq.\nbsp{}eqref:eq-continui
reduces to
\begin{equation}
\label{eq-continuity-2}
- \mathoperator{div}\vec{V} =
+ \text{div}\vec{V} =
\frac{\partial{V_x}}{\partial{x}} +
\frac{\partial{V_y}}{\partial{y}} +
\frac{\partial{V_z}}{\partial{z}} = 0
@@ -145,13 +145,403 @@ becomes an even more significant issue in a nonlinear computation where the wave
model is even more complex. Thus identifying a significantly less time intensive
method for modelling the ambient ocean-wave environment has the potential for
significantly speeding the total simulation process.
-
* Related work
+In\nbsp{}cite:spanos1982arma ARMA model is used to generate time series spectrum of
+which is compatible with Pierson---Moskowitz (PM) approximation of ocean wave
+spectrum. The authors carry out experiments for one-dimensional AR, MA and ARMA
+models. They mention excellent agreement between target and initial spectra and
+higher performance of ARMA model compared to models based on summing large
+number of harmonic components with random phases. The also mention that in order
+to reach agreement between target and initial spectrum MA model require lesser
+number of coefficients than AR model. In\nbsp{}cite:spanos1996efficient the authors
+generalise ARMA model coefficients determination formulae for multi-variate
+(vector) case.
+
+One thing that distinguishes present work with respect to afore-mentioned ones
+is the study of three-dimensional (2D in space and 1D in time) ARMA model, which
+is mostly a different problem.
+1. Yule---Walker system of equations, which are used to determine AR
+ coefficients, has complex block-block structure.
+2. Optimal model order (in a sense that target spectrum agrees with initial) is
+ determined manually.
+3. Instead of PM spectrum, analytic formulae for standing and propagating
+ waves ACF are used as the model input.
+4. Three-dimensional wavy surface should be compatible with real ocean surface
+ not only in terms of spectral characteristics, but also in the shape of wave
+ profiles. So, model verification includes distributions of various parameters
+ of generated waves (lengths, heights, periods etc.).
+Multi-dimensionality of investigated model not only complexifies the task, but
+also allows to carry out visual validation of generated wavy surface. It is the
+opportunity to visualise output of the programme that allowed to ensure that
+generated surface is compatible with real ocean surface, and is not abstract
+multi-dimensional stochastic process that is real only statistically.
+
* Three-dimensional ARMA process as a sea wave simulation model
+
+Another approach to modelling wind waves is possible in terms of the
+representation of a stochastic moving surface as a linear transformation of
+white noise with memory. These methods are one of the most popular ways of
+modelling stationary ergodic Gaussian random processes with given correlation
+characteristics (Box, et al., 2008). However, these methods have were not used
+to simulate wind waves for a long time. The first attempts to model
+two-dimensional disturbances were undertaken in the early 70's (cf. Kostecki,
+1972), and the impetus for this was the development of the resonance theory of
+waves in wind. However, the formal mathematical framework was developed by
+Gurgenidze & Trapeznikov (1988) and Rozhkov & Trapeznikov (1990). They built a
+one-dimensional model of ocean waves \(\zeta(t)\), on the basis of an
+autoregressive-moving average (ARMA) model:
+\begin{equation}
+ \zeta_{\vec i}
+ =
+ \sum\limits_{\vec j = \vec 0}^{\vec N}
+ \Phi_{\vec j} \zeta_{\vec i - \vec j}
+ +
+ \sum\limits_{\vec j = \vec 0}^{\vec M}
+ \Theta_{\vec j} \epsilon_{\vec i - \vec j}
+ ,
+ \label{eq-arma-process}
+\end{equation}
+where \(\zeta\)\nbsp{}--- wave elevation, \(\Phi\)\nbsp{}--- AR process
+coefficients, \(\Theta\)\nbsp{}--- MA process coefficients,
+\(\epsilon\)\nbsp{}--- white noise with Gaussian distribution,
+\(\vec{N}\)\nbsp{}--- AR process order, \(\vec{M}\)\nbsp{}--- MA process order,
+and \(\Phi_{\vec{0}}\equiv{0}\), \(\Theta_{\vec{0}}\equiv{0}\). Here arrows
+denote multi-component indices with a component for each dimension. In general,
+any scalar quantity can be a component (temperature, salinity, concentration of
+some substance in water etc.). Equation parameters are AR and MA process
+coefficients and order.
+
+Any ARMA process can be uniquely represented as a process moving average and
+autoregression process of general infinite order (Gurgenidze & Trapeznikov,
+1988), and the parameters of the spectral representation are defined by the rule
+of division of power series (in a rational factorized form, Rozhkov &
+Trapeznikov, 1990):
+\begin{equation*}
+ S(\omega) =
+ \frac{\Delta\sigma^2}{\pi}
+ \frac{\prod\limits_m (1 - z_m e^{-im\omega\delta})(1 - z_m e^{im\omega\delta})}
+ {\prod\limits_n (1 - p_n e^{-in\omega\delta})(1 - p_n e^{in\omega\delta})},
+\end{equation*}
+where \(z_m\) and \(p_n\) are the zeros of numerator (MA), and denominator (AR),
+respectively, which form a pair of mutually conjugate numbers. If some of the
+zeros are located near the unit circle, then the spectral density will have
+pronounced dips.
+
** Autoregressive (AR) process
+AR process is ARMA process with only one random impulse instead of theirs
+weighted sum:
+\begin{equation}
+ \zeta_{\vec i}
+ =
+ \sum\limits_{\vec j = \vec 0}^{\vec N}
+ \Phi_{\vec j} \zeta_{\vec i - \vec j}
+ +
+ \epsilon_{i,j,k}
+ .
+ \label{eq-ar-process}
+\end{equation}
+The coefficients \(\Phi\) are calculated from ACF via three-dimensional
+Yule---Walker equations, which are obtained after multiplying both parts of the
+previous equation by \(\zeta_{\vec{i}-\vec{k}}\) and computing the expected value.
+Generic form of YW equations is
+\begin{equation}
+ \label{eq-yule-walker}
+ \gamma_{\vec k}
+ =
+ \sum\limits_{\vec j = \vec 0}^{\vec N}
+ \Phi_{\vec j}
+ \text{ }\gamma_{\vec{k}-\vec{j}}
+ +
+ \Var{\epsilon} \delta_{\vec{k}},
+ \qquad
+ \delta_{\vec{k}} =
+ \begin{cases}
+ 1, \quad \text{if } \vec{k}=0 \\
+ 0, \quad \text{if } \vec{k}\neq0,
+ \end{cases}
+\end{equation}
+where \(\gamma\)\nbsp{}--- ACF of process \(\zeta\), \(\Var{\epsilon}\)\nbsp{}--- white noise
+variance. Matrix form of three-dimensional YW equations, which is used in the
+present work, is
+\begin{equation*}
+ \Gamma
+ \left[
+ \begin{array}{l}
+ \Phi_{\vec 0}\\
+ \Phi_{0,0,1}\\
+ \vdotswithin{\Phi_{\vec 0}}\\
+ \Phi_{\vec N}
+ \end{array}
+ \right]
+ =
+ \left[
+ \begin{array}{l}
+ \gamma_{0,0,0}-\Var{\epsilon}\\
+ \gamma_{0,0,1}\\
+ \vdotswithin{\gamma_{\vec 0}}\\
+ \gamma_{\vec N}
+ \end{array}
+ \right],
+ \qquad
+ \Gamma=
+ \left[
+ \begin{array}{llll}
+ \Gamma_0 & \Gamma_1 & \cdots & \Gamma_{N_1} \\
+ \Gamma_1 & \Gamma_0 & \ddots & \vdotswithin{\Gamma_0} \\
+ \vdotswithin{\Gamma_0} & \ddots & \ddots & \Gamma_1 \\
+ \Gamma_{N_1} & \cdots & \Gamma_1 & \Gamma_0
+ \end{array}
+ \right],
+\end{equation*}
+where \(\vec N = \left( p_1, p_2, p_3 \right)\) and
+\begin{equation*}
+ \Gamma_i =
+ \left[
+ \begin{array}{llll}
+ \Gamma^0_i & \Gamma^1_i & \cdots & \Gamma^{N_2}_i \\
+ \Gamma^1_i & \Gamma^0_i & \ddots & \vdotswithin{\Gamma^0_i} \\
+ \vdotswithin{\Gamma^0_i} & \ddots & \ddots & \Gamma^1_i \\
+ \Gamma^{N_2}_i & \cdots & \Gamma^1_i & \Gamma^0_i
+ \end{array}
+ \right]
+ \qquad
+ \Gamma_i^j=
+ \left[
+ \begin{array}{llll}
+ \gamma_{i,j,0} & \gamma_{i,j,1} & \cdots & \gamma_{i,j,N_3} \\
+ \gamma_{i,j,1} & \gamma_{i,j,0} & \ddots &x \vdotswithin{\gamma_{i,j,0}} \\
+ \vdotswithin{\gamma_{i,j,0}} & \ddots & \ddots & \gamma_{i,j,1} \\
+ \gamma_{i,j,N_3} & \cdots & \gamma_{i,j,1} & \gamma_{i,j,0}
+ \end{array}
+ \right],
+\end{equation*}
+Since \(\Phi_{\vec 0}\equiv0\), the first row and column of \(\Gamma\) can be
+eliminated. Matrix \(\Gamma\) is block-toeplitz, positive definite and symmetric,
+hence the system is efficiently solved by Cholesky decomposition, which is
+particularly suitable for these types of matrices.
+
+After solving this system of equations white noise variance is estimated from
+eqref:eq-yule-walker by plugging \(\vec k = \vec 0\):
+\begin{equation*}
+ \Var{\epsilon} =
+ \Var{\zeta}
+ -
+ \sum\limits_{\vec j = \vec 0}^{\vec N}
+ \Phi_{\vec j}
+ \text{ }\gamma_{\vec{j}}.
+\end{equation*}
** Moving average (MA) process
+MA process is ARMA process with \(\Phi\equiv0\):
+\begin{equation}
+ \zeta_{\vec i}
+ =
+ \sum\limits_{\vec j = \vec 0}^{\vec M}
+ \Theta_{\vec j} \epsilon_{\vec i - \vec j}
+ .
+ \label{eq-ma-process}
+\end{equation}
+MA coefficients \(\Theta\) are defined implicitly via the following non-linear
+system of equations:
+\begin{equation*}
+ \gamma_{\vec i} =
+ \left[
+ \displaystyle
+ \sum\limits_{\vec j = \vec i}^{\vec M}
+ \Theta_{\vec j}\Theta_{\vec j - \vec i}
+ \right]
+ \Var{\epsilon}.
+\end{equation*}
+The system is solved numerically by fixed-point iteration method via the
+following formulae
+\begin{equation*}
+ \Theta_{\vec i} =
+ -\frac{\gamma_{\vec 0}}{\Var{\epsilon}}
+ +
+ \sum\limits_{\vec j = \vec i}^{\vec M}
+ \Theta_{\vec j} \Theta_{\vec j - \vec i}.
+\end{equation*}
+Here coefficients \(\Theta\) are calculated from back to front: from
+\(\vec{i}=\vec{M}\) to \(\vec{i}=\vec{0}\). White noise variance is estimated by
+\begin{equation*}
+ \Var{\epsilon} = \frac{\gamma_{\vec 0}}{
+ 1
+ +
+ \sum\limits_{\vec j = \vec 0}^{\vec M}
+ \Theta_{\vec j}^2
+ }.
+\end{equation*}
+Authors of\nbsp{}cite:box1976time suggest using Newton---Raphson method to solve this
+equation with higher precision, however, this method does not work in three
+dimensions. Using slower method does not have dramatic effect on the overall
+programme performance, because the number of coefficients is small and most of
+the time is spent generating wavy surface.
** Mixed autoregressive moving average (ARMA) process
+:PROPERTIES:
+:CUSTOM_ID: sec:how-to-mix-ARMA
+:END:
+Generally speaking, ARMA process is obtained by plugging MA generated wavy
+surface as random impulse to AR process, however, in order to get the process
+with desired ACF one should re-compute AR coefficients before plugging. There
+are several approaches to "mix" AR and MA processes.
+- The approach proposed in\nbsp{}cite:box1976time which involves dividing ACF into MA
+ and AR part along each dimension is not applicable here, because in three
+ dimensions such division is not possible: there always be parts of the ACF
+ that are not taken into account by AR and MA process.
+- The alternative approach is to use the same (undivided) ACF for both AR and MA
+ processes but use different process order, however, then realisation
+ characteristics (mean, variance etc.) become skewed: these are characteristics
+ of the two overlapped processes.
+For the first approach there is a formula to re-compute ACF for AR process, but
+there is no such formula for the second approach. So, the best solution for now
+is to simply use AR and MA process exclusively.
+
+** Process selection criteria for different wave profiles
+:PROPERTIES:
+:CUSTOM_ID: sec-process-selection
+:END:
+One problem of ARMA model application to ocean wave generation is that for
+different types of wave profiles different processes /must/ be used: standing
+waves are modelled by AR process, and propagating waves 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 correspond to real
+ocean waves. (The latter happens for non-invertible MA process, as it is always
+stationary.) 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.
+
+The other problem is inability to automatically determine optimal number of
+coefficients for three-dimensional AR and MA processes. For one-dimensional
+processes this can be achieved via iterative methods\nbsp{}cite:box1976time, but they
+diverge in three-dimensional case.
+
+The final problem, which is discussed in [[#sec:how-to-mix-ARMA]], is inability to
+"mix" AR and MA process in three dimensions.
+
+In practice some statements made for AR and MA processes in\nbsp{}cite:box1976time
+should be flipped for three-dimensional case. For example, the authors say that
+ACF of MA process cuts at \(q\) 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 results in stationary
+process and adequate realisation. Also, 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.
+
+To summarise, the only established scenario of applying ARMA model to ocean wave
+generation 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 increase model precision, which is one of the objectives of the
+future research.
+
** The shape of ACF for different types of waves
+**** Analytic method of finding the ACF.
+The straightforward way to find ACF for a given 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 formula for the corresponding ACF.
+
+For three-dimensional wave profile (2D in space and 1D in time) analytic formula
+is a polynomial of high order and is best obtained via symbolic computation
+programme. 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\nbsp{}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 satisfies first
+order Stokes' equations for gravity waves\nbsp{}cite:boccotti1983wind. So, if the
+shape of the wave profile is the only concern in the simulation, then one can
+simply multiply it by a decaying exponent to get appropriate ACF. This ACF does
+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 approach; it is mainly useful
+to test ARMA model.
+
+**** Standing wave ACF.
+For three-dimensional plain standing wave the profile is given by
+\begin{equation}
+ \zeta(t, x, y) = A \sin (k_x x + k_y y) \sin (\sigma t).
+ \label{eq-standing-wave}
+\end{equation}
+Find ACF via analytic method. Multiplying the formula by a decaying exponent
+(because Fourier transform is defined for a function \(f\) that
+\(f\underset{x\rightarrow\pm\infty}{\longrightarrow}0\)) yields
+\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}
+Then, apply 3D Fourier transform to both sides of the equation via symbolic
+computation programme, fit the resulting polynomial to the following
+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 initial formula but with
+cosines instead of sines. This difference is important because the value of ACF
+at \((0,0,0)\) equals to the ARMA process variance, 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 either by changing the phase of the sine, or by substituting
+sine with cosine to move the maximum of the function to the origin of
+coordinates.
+
+**** Propagating wave ACF.
+Three-dimensional profile of plain propagating wave is given 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 repeating steps from the previous two paragraphs yields
+\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 the wave profile is simply multiplied by a decaying
+exponent without need to adapt the maximum value of ACF (as it is required for
+standing wave).
+
+**** 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 approaching \(\pm\infty\) by multiplying it by
+ a decaying exponent.
+- Apply Fourier transform to the absolute square of the resulting equation using
+ symbolic computation programme.
+- 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 ACF's maximum should be moved to the origin to preserve simulated
+process variance. Empirical method of finding ACF reduces to the following
+steps.
+- Make wave profile decay when approaching \(\pm\infty\) by multiplying it by
+ a decaying exponent.
+- Move maximum value of the resulting function to the origin by using
+ trigonometric identities to shift the phase.
+
** Evaluation
** Discussion
* Determining wave pressures for discretely given wavy surface
diff --git a/preamble.tex b/preamble.tex
@@ -1,2 +1,28 @@
\usepackage{amsmath}
-\usepackage{cite}-
\ No newline at end of file
+\usepackage{cite}
+\usepackage{latexsym} % \Box macro
+\usepackage{mathtools} % fancy dots in matrices
+
+% custom mathematical expressions
+\newcommand{\Var}[1]{\sigma_{#1}^2}
+\newcommand{\Fourier}[1]{\mathcal{F}\left\{#1\right\}}
+\newcommand{\InverseFourier}[1]{\mathcal{F}^{-1}\left\{#1\right\}}
+\newcommand{\Fun}[1]{\mathcal{D}_1\left(x,#1\right)}
+\newcommand{\FunSecond}[1]{\mathcal{D}_2\left(x,#1\right)}
+\newcommand{\FunThird}[1]{\mathcal{D}_2\left(x,#1\right)}
+\newcommand{\FunThreeD}[1]{\mathcal{D}_3\left(x,y,#1\right)}
+\newcommand{\Sinh}[1]{\cosh\left(#1\right)}
+\newcommand{\SinhX}[1]{\sinh\left(#1\right)}
+
+\newcommand{\FourierY}[2]{\mathcal{F}_{#2}\!\left\{#1\right\}}
+\newcommand{\InverseFourierY}[2]{\mathcal{F}^{-1}_{#2}\!\left\{#1\right\}}
+
+\newcommand{\FourierX}[3]{\mathcal{F}_{#2}\!\left\{#1\right\}\!\left(#3\right)}
+\newcommand{\InverseFourierX}[3]{\mathcal{F}^{-1}_{#2}\!\left\{#1\right\}\!\left(#3\right)}
+
+% properly aligned version of sqrt for \zeta_y^2
+\newcommand{\SqrtZeta}[1]{\sqrt{\vphantom{\zeta_x^2}\smash[b]{#1}}}
+
+% wave vector
+\newcommand{\Kvec}{\vec{k}}
+\newcommand{\Kveclen}{\lvert\smash[b]{\Kvec}\rvert}+
\ No newline at end of file
diff --git a/refs.bib b/refs.bib
@@ -34,17 +34,94 @@
@book{kochin1964theoretical,
title={Theoretical hydromechanics},
- author={Kochin, Nikola{\u\i}} and Iliia, A Kibel and Roze, Nikola{\u\i}}},
+ author={Kochin, Nikolai and Iliia, A Kibel and Roze, Nikolai},
year={1964},
publisher={Interscience}
}
@article{beck2001modern,
title={Modern computational methods for ships in a seaway},
- author={BECK, Robert F and REED, Arthur M and SCLAVOUNOS, Paul D and HUTCHISON, Bruce L},
+ author={Beck, Robert F and Reed, Arthur M and Sclavounos, Paul D and Hutchison, Bruce L},
journal={Transactions-Society of Naval Architects and Marine Engineers},
volume={109},
pages={1--51},
year={2001},
publisher={Society of Naval Architects and Marine Engineers}
}
+
+@article{spanos1983arma,
+ title={ARMA algorithms for ocean wave modeling},
+ author={Spanos, PT D},
+ journal={Journal of Energy Resources Technology},
+ volume={105},
+ number={3},
+ pages={300--309},
+ year={1983},
+ publisher={American Society of Mechanical Engineers}
+}
+
+@book{spanos1982arma,
+ title={ARMA Algorithms for Ocean Spectral Analysis},
+ author={Spanos, Pol D},
+ year={1982},
+ publisher={University of Texas at Austin. Engineering Mechanics Research Laboratory}
+}
+
+@article{mignolet1992simulation,
+ title={Simulation of homogeneous two-dimensional random fields: Part I—AR and ARMA models},
+ author={Mignolet, Marc P and Spanos, Pol D},
+ journal={Journal of applied mechanics},
+ volume={59},
+ number={2S},
+ pages={S260--S269},
+ year={1992},
+ publisher={American Society of Mechanical Engineers}
+}
+
+@article{spanos1992simulation,
+ title={Simulation of homogeneous two-dimensional random fields: Part II—MA and ARMA Models},
+ author={Spanos, Pol D and Mignolet, Marc P},
+ journal={Journal of applied mechanics},
+ volume={59},
+ number={2S},
+ pages={S270--S277},
+ year={1992},
+ publisher={American Society of Mechanical Engineers}
+}
+
+@article{spanos1996efficient,
+ title={Efficient iterative ARMA approximation of multivariate random processes for structural dynamics applications},
+ author={Spanos, Pol D and Zeldin, BA},
+ journal={Earthquake engineering \& structural dynamics},
+ volume={25},
+ number={5},
+ pages={497--507},
+ year={1996},
+ publisher={Wiley Online Library}
+}
+
+@phdthesis{zeldin1996representation,
+ title={Representation and synthesis of random fields: ARMA, Galerkin, and wavelet procedures},
+ author={Zeldin, Boris A},
+ year={1996},
+ school={Rice University}
+}
+
+@Book{ box1976time,
+ title = {Time series analysis: forecasting and control, revised
+ ed},
+ author = {Box, George EP and Jenkins, Gwilym M},
+ year = {1976},
+ publisher = {Holden-Day}
+}
+
+@Article{ boccotti1983wind,
+ title = {On wind wave kinematics},
+ author = {Boccotti, Paolo},
+ journal = {Meccanica},
+ volume = {18},
+ number = {4},
+ pages = {205--216},
+ year = {1983},
+ publisher = {Springer}
+}+
\ No newline at end of file
