\chapter{Literature Review}

In this chapter, literature relevant to the present study will be reviewed.
Three sections will be detailled: the separation of incident and reflected
components from wave measurements, the modelisation of wave impacts on a
rubble-mound breakwater, and the modelisation of block displacement by wave
impacts.

\section{Separating incident and reflected components from wave buoy data}

\subsection{Introduction}

The separation of incident and reflected waves is a crucial step in numerically
modeling a sea state. Using the raw data from a buoy as the input of a wave
model will lead to incorrect results in the domain as the flow velocity at the
boundary will not be correctly generated.

Several methods exist to extract incident and reflected components in measured
sea states, and they can generally be categorised in two types of methods:
array methods and PUV methods \parencite{inch2016accurate}. Array methods rely
on the use of multiple measurement points of water level to extracted the
incident and reflected waves, while PUV methods use co-located pressure and
velocity measurements to separate incident and reflected components of the
signal.

\subsection{Array methods}

\subsubsection{2-point methods}

Array methods were developped as a way to isolate incident and reflected wave
components using multiple wave records.
\textcite{goda1977estimation,morden1977decomposition} used two wave gauges
located along the wave direction, along with spectral analysis, in order to
extract the incident and reflected wave spectra. Their work is based on the
earlier work of \textcite{thornton1972spectral}. \textcite{goda1977estimation}
analyzed the wave spectrum components using the Fast Fourier Transform, and
suggests that this method is adequate for studies in wave flumes. They noted
that this method provides diverging results for gauge spacings that are
multiples of half of the wave length. \textcite{morden1977decomposition}
applies this technique to a field study, where the sea state is wind generated.
\textcite{morden1977decomposition} showed that, using appropriate spectral
analysis methods along with linear wave theory, the decomposition of the sea
state into incident and reflected waves is accurate. A relation between the
maximum obtainable frequency and the distance between the sensors is provided.
According to \textcite{morden1977decomposition}, the only needed knowledge on
the wave environment is that wave frequencies are not modified by the
reflection process.

\subsubsection{3-point methods}

In order to alleviate the limitations from the 2-point methods,
\textcite{mansard1980measurement} introduced a 3-point method. The addition of
a supplementary measurement point along with the use of a least-squares method
most importantly provided less sensitivity to noise, non-linear interactions,
and probe spacing. The admissible frequency range could also be widened. A
similar method was proposed by \textcite{gaillard1980}. The accuracy of the
method for the estimation of incident and reflected wave components was once
again highlighted, while the importance of adequate positioning of the gauges
was still noted.

\subsubsection{Time-domain method}

\textcite{frigaard1995time} presented a time-domain method for reflected and
incident wave separation. This method, called SIRW method, used discrete
filters to extract the incident component of an irregular wave field. The
results were as accurate as with the method proposed by
\cite{goda1977estimation}, while singularity points are better accounted for.
The main advantage of the SIRW method is that it works in the time-domain,
meaning that real time computations can be performed.
\textcite{frigaard1995time} also mentions the possibility of replacing one of
the wave gauges by a velocity meters to prevent singularities.

This method was improved by \textcite{baldock1999separation} in order to
account for arbitrary bathymetry. Linear theory is used to compute shoaling on
the varying bathymetry. Resulting errors in the computed reflection coefficient
are low for large reflection coefficients, but increase with lower
coefficients. The neglect of shoaling can lead to important error in many
cases. The presented method could also be extended to three-dimensionnal waves
and bathymetry by considering the influence of refraction.

\subsubsection{Further improvements}

Further additions were made to array methods. \textcite{suh2001separation}
developped a method taking constant current into account to separate incident
and reflected waves. This method relies on two or more gauges, using a least
squares method. Results are very accurate in the absence of noise, but a small
amount of error appears when noise is added.

\textcite{inch2016accurate} noticed that the presence of noise lead to
overestimation of reflection coefficient. The creation of bias lookup tables is
proposed in order to account for noise-induced error in reflection coefficient
estimations.

\textcite{andersen2017estimation,roge2019estimation} later proposed
improvements to account for highly non-linear regular and irregular waves
respectively. The improved method provides very accurate results for highly
non-linear waves, but are expected to be unreliable in the case of steep
seabeds, as shoaling is not part of the underlying model.

\subsubsection{Conclusion}

Array methods have been developped enough to provide accurate results in a wide
range of situations. Sensibility to noise has been reduced, and the influence
of shoaling has been considered. Those methods can also be applied to irregular
non-linear waves.

However, they require at least two wave gauges to be used. That means that in
some situations such as the Saint-Jean-de-Luz event of 2017, other methods are
needed since only one field measurement location is available.

\subsection{PUV methods}

The goal of PUV methods is to decompose the wave field into incident and
reflected waves using co-located wave elevation and flow velocity measurements
\parencite{tatavarti1989incoming}. \textcite{tatavarti1989incoming} presented a
detailled analysis of separation of incoming and outging waves using co-located
velocity and wave height sensors. Their method allows to obtain the reflection
coefficient relative to frequency, as well as to separate incident and
reflected wave components. Compared to array methods, this method also strongly
reduces the influence of noise.

\textcite{kubota1990} studied the influence of the considered wave theory on
incident and reflected wave separation. Three methods, based on linear
long-wave theory, small-amplitude wave theory and quasi-nonlinear long-wave
theory respectiveley were developped and compared. The results show that the
quasi-nonlinear approach gave the most accurate results.

%\textcite{walton1992} applied a separation method based on co-located pressure
%and velocity measurements on field, studying two natural beaches. This study
%showed that reflection is not significant on natural beaches. Additionnaly,
%the method that is used allowed for larger reflected energy than incident
%energy.

Research by \textcite{hughes1993} showed how co-located horizontal velocity and
vertical velocity (or pressure) sensors can be used to extract incident and
reflected wave spectra. Their method is based on frequency domain linear
theory, and provided accurate results for full reflection of irregular
non-breaking waves. Low-reflection scenarii were evaluated against the results
from \textcite{goda1977estimation}, and showed good agreement between both
methods. \textcite{hughes1993} also highlights that reflection estimates are
unreliable for higher frequency, where coherency between the two measured
series is lower.

Following the work of \textcite{tatavarti1989incoming},
\textcite{huntley1999use} showed how principal component analysis can alleviate
noise-induced bias in reflection coefficient calculations compared to
time-domain analysis. They also stuied the influence of imperfect collocation
of the sensors, showing that the time delay between sensors leads to a peak in
the reflection coefficient at a frequency related to this time delta.

%%% TODO? %%%
%\begin{itemize} \item \cite{sheremet2002observations}: \end{itemize}

\subsection{Conclusion}

Numerous methods have been developped in order to separate incident and
reflected components from wave measurements. Array methods rely on the use of
multiple, generally aligned, wave gauges, while PUV methods rely on the use of
co-located sensors, generally a wave height sensor and a horizontal velocity
sensor. Array methods generally have the advantage of being more cost-effective
to implement, as the cost of reliable velocity measurement devices can be
important \parencite{hughes1993}. Nevertheless, PUV methods are generally more
accurate regarding noise, varying bathymetry, and can be setup closer to
reflective surfaces \parencite{hughes1993,inch2016accurate}.

In the case of the 2017 event on the Artha breakwater, the results from a
single wave gauge are available, which means that the array methods are not
applicable. A PUV method \parencite{tatavarti1989incoming,huntley1999use}
should then be used to evaluate the reflection coefficient of the Artha
breakwater and to separate the incident and reflected wave components from the
measured data.

\section{Modeling wave impact on a breakwater}

\subsection{SPH models}

\subsection{VARANS models}

\section{Modeling block displacement}