\documentclass[twoside]{article}
\usepackage[silent,pdf,hyper]{elxshort}
\usepackage{color}
\usepackage{hyperref}
\journalid{ES}{19}{1}{19}
\citnumber{}{5}
\paperid{2019ES000662}
\papertype{RAR}
\cpright{}{2019}
\lefthead{ermoshkin and kapustin}
\righthead{estimation of the wind-driven wave spectrum} %  using a high spatial %resolution coherent radar
\received{13}{March}{2019}
\accepted{16}{April}{2019}
\published{14}{June}{2019}
\definecolor{brown}{RGB}{157,23,4}

\authaddr{Aleksey V. Ermoshkin,  Institute of Applied Physics RAS, %of the Russian Academy of Sciences,
Nizhniy Novgorod, Russia.
%\qquad \qquad \qquad \qquad \qquad \qquad
\mbox{(al-ermoshkin@yandex.ru)}}


\begin{document}

\title{Estimation of the wind-driven wave spectrum using a high spatial resolution coherent radar}

\author{1}{Ermoshkin}{A. V.}{}{1}{}{}
\author{2}{Kapustin}{I. A.}{}{1}{}{}

\affil{1}{Institute of Applied Physics of the Russian Academy of Sciences}{Nizhniy Novgorod}{}{Russia}

\keywd{Coherent radar}
\keywd{Doppler velocity}
\keywd{wind wave spectra}

%	Estimation of the wind-driven wave spectrum using a high spatial %resolution coherent radar
%    A.V. Ermoshkin, I.A. Kapustin
%     Institute of Applied Physics of the Russian Academy of Sciences, %Nizhniy Novgorod, Russia
%      Cor. author: Aleksey V. Ermoshkin: al-ermoshkin@yandex.ru
 %     Keywords: coherent radar, Doppler velocity, wind wave spectra

 \abstract{The paper deals with experimental determination of the wind-driven wave spectrum by using the remote
method of high-resolution coherent radar sensing of the water surface. The method is applied to the conditions of the
fetch-limited wind wave growth, which is typical for enclosed waters and the sea nearshore, where the dominant
wavelength is of the order of ten meters. The experiments were performed using a coherent X-band panoramic
digital radar operating with the horizontal polarization for transmission and reception. The paper considers an
algorithm used to form velocity images of the water surface from the data of coherent radar sensing on the basis of
the phase angle difference method. The paper shows the possibility to recover wind wave spectra from the data on
the Doppler shift of microwave radio waves. A theoretical justification and an experimental verification of the
method are provided. Appearance of the third harmonic of the wind waves recovered from the radar sensing data of
the spatio-temporal spectra of Doppler velocities is demonstrated for the first time. The functions that relate to the
wave elevation spectra and the Doppler velocity spectra are determined. It is shown that the linear free-surface wave
approximation is valid for the reconstruction of wave spectra.
 %The temporal and omnidirectional spectra of the spatial wave
 The temporal and spatial omnidirectional wave spectra
 obtained in the experiment are described well by the power model functions using the exponent factor
of $-4$ and $-5/2$, respectively. The presented results may be used for studies of interaction between the harmonics of
wind waves with the bottom topography, wave-current interaction, diagnostic of non-homogeneous currents, and
other hydrophysical processes.}


\section{Introduction}

     The idea of using panoramic radar, for example, marine radar, to determine the kinematic
characteristics of wind waves is hugely attractive, primarily due to the simplicity of
measurements, and such evident advantages of radar sensing as all-weather and day-and-night
operation. The methods of extracting the period and direction of the waves from the radar data
are well-developed
[\itc{Young et al.,} \reflink{Youn85}{1985}].
 However, the estimation of the wave spectrum is
difficult due to the complex physical features of the modulation transfer function
 [\itc{Ermakov et al.,} \reflink{Erma14}{2014}].
 The possibility of measuring the wave height based on the data on the backscattering
intensity has been studied in a number of papers, e.g., in
[\itc{Dankert et al.,} \reflink{Dank05}{2005}].
Those papers
emphasize the necessity of experimental determination of the wave height as a function of the
signal-to-noise ratio
[\itc{Ivonin et al.,} \reflink{Ivon11}{2011}].
 In contrast to the backscattering intensity, the Doppler
frequency shift of the reflected radio wave is associated with the radial velocity of the scattering
elements. The variable component of the Doppler shift is determined by the orbital velocities of
the wind waves. This is the principle underlying operation of a radar device used to determine
wave heights
[\itc{Rosenberg et al.,} \reflink{Rose73}{1973}].
Studies of Doppler characteristics of microwave radio
waves scattering were described in a large number of papers, both theoretical and experimental,
starting from the 1960s. Currently, the main model used to explain the features of the Doppler
spectra is the two-scale scattering model
[\itc{Bass and Fuks,} \reflink{Bass72}{1972}].
The possibility of determining
the wave height using coherent radar data at a low grazing angle was studied in
[\itc{Carrasco et al.,} \reflink{Carr17}{2017};
\itc{Lyzenga et al.,} \reflink{Lyze10}{2010}].
 Those works used the approximation of linear free-surface waves to
find a relation between the wave height and its orbital velocity measured with a Doppler radar.
The numerical modeling described in
[\itc{Nwogu and Lyzenga,} \reflink{Nwog10}{2010}]
supports the possible
determination of the wave height based on the radar measurements of the Doppler velocity.

     The purpose of this work is to study the possibility of remote radar measurement of spatio-temporal spectra of wind waves for the meter and decameter ranges. In well-known papers
[\itc{Lyzenga et al.,} \reflink{Lyze10}{2010}]
that used the data from pulse radar, the spatial resolution was 10 m, while
in
[\itc{Plant and Farquharson,} \reflink{Plan12}{2012}],
it was 7.5~m. Thus, the spatial spectra of radar images were
limited to a wave number of 0.3--0.4~rad/m. This is not sufficient for analysis of the wave spectra
in the fetch-limited conditions, e.g., in the coastal zone and in enclosed water bodies. A
significant extension in the range of the measured wavelengths became possible due to a high
spatial resolution of the radar used (less than 1~m) and coherence of the radiation. In this work,
field experiments were performed on a Stationary Oceanographic Platform (SOP) of the Black
Sea Hydrophysical Test Site of the Russian Academy of Sciences in the coastal area of the Black
Sea near Katsiveli. During the experiments, velocity radar images of wavy water surface were
obtained and used for spectrum processing. Comparative analysis of the temporal and spatial
spectra of wind waves, which were determined based on the data obtained with high spatial
resolution coherent radar and a contact wave gauge (string wave recorder), was carried out. It has
been shown that the wave spectra reconstructed from velocity radar images correlate with the
data of roughness contact measurements, both qualitatively and quantitatively.

\subsection{Theoretical Background}

\enlargethispage{-.9pc}

\begin{sloppypar}
     Let us consider the problem of propagation of linear free-surface waves and obtain an
expression that relates the spectra of the measured Doppler velocities to the spectra of the wave
height
%[\itc{Laboratory Manual in Ocean Dynamics,} \reflink{Labo92}{1992}].
[\itc{Nekrasov and Pelinovskiy (eds.),} \reflink{Nekr92}{1992}].
For linear deep-water waves propagating
along the $x$ axis ($kh \gg 1$), the potential may be expressed as
\end{sloppypar}

 \begin{eqnarray*}
\varphi (x,t)   = \frac{g}{\omega} \eta_0 \sin (\omega t - kx + \alpha)
\end{eqnarray*}
 where $\eta_0$  is the wave amplitude, and $\omega$  and $k$ are the frequency and wave number, respectively.

Then, the expression for the horizontal component of velocity has the form

\begin{eqnarray*}
u (x,t)   = \frac{\partial\varphi }{\partial x}=
\frac{g k}{\omega}
 \eta_0 \cos (\omega t - kx + \alpha)
\end{eqnarray*}
 while that for the wave amplitude is

 \begin{eqnarray*}
\eta (x,t)   = \frac{1 }{g}\frac{\partial\varphi }{\partial t}=
 \eta_0 \cos (\omega t - kx + \alpha)
\end{eqnarray*}
 The relation between the wave elevation spectrum
and the horizontal velocity spectrum is determined by the following expression:
\begin{equation}\label{1}
S_{\eta} (k,\omega) =
\Bigl(
\frac{\omega}{g k} \Bigr)^2 S_u (k,\omega)
\end{equation}   	 			% (1)


\begin{figure*}[t]                     % Fig 1
\figurewidth{35pc}
\setimage{}{}{41pc}{}{2019es000662-f01}
\caption{The experimental setup. Circles correspond to ranges (grazing angles) 100~m
(80.8\deg), 200~m (85.4\deg), 300~m (87\deg), 400~m (87.7\deg) and 500~m (88.2\deg). 1 -- coherent radar, 2 -- weather station, 3 -- acoustic Doppler current profiler, 4 -- string wave gauge.}
\end{figure*}




\begin{sloppypar}
     The horizontal component of velocity at the sea surface may be determined by measuring
the Doppler velocity of the reflecting elements, which may be written as
 \begin{eqnarray*}
u_d(x, t, \vartheta ) =
\end{eqnarray*}
 \begin{eqnarray*}
 (u^x_{br} (x) + u (x,t) + u_c^x (x)) \cos \vartheta \sin \theta
\end{eqnarray*}
 where $\vartheta$  is the azimuth angle of sensing with
respect to the wave propagation direction, coinciding with the $x$ axis,   $u_c^x$ is the component of the
current velocity in the direction of the wave propagation,
 $u_{br}^x$ is the phase velocity of the
scattering (Bragg) wave
\end{sloppypar}


 \begin{eqnarray*}
u_{br}^x = \sqrt{ \frac{ g \lambda}{4 \pi \sin \theta} + \frac{2 \pi \sigma}{\rho\lambda} \sin \theta}
\end{eqnarray*}
$\lambda$  is the wavelength of the radar,
$\theta$ is the
incidence angle,
 $\sigma$ is the surface tension coefficient, and
  $\rho$ is the water density. The time-averaged Doppler velocity is

\begin{eqnarray*}
\overline{u_d (x, \vartheta)}^T =
\end{eqnarray*}
\begin{eqnarray*}
\overline{(u_{br}^x (x) + u(x,t) +  u_{c}^x (x))}^T \cos \vartheta \sin \theta \approx
\end{eqnarray*}
\begin{eqnarray*}
(u_{br}^x (x) +  u_{c}^x) \cos \vartheta \sin \theta
\end{eqnarray*}
 because
 $u_{br}^x$  and  $u_{c}^x$ are
time-independent values and
$\overline{u (x, t)}^T \approx 0$. Then, it is possible to use
\eqref{1}           %(1)
to relate the spectrum of
the measured Doppler velocities and the wave elevation spectrum with the following linear
equation:
\begin{equation}\label{2}
S_{\eta} (k,\omega) =
\Bigl(
\frac{\omega}{g k \cos \vartheta \sin \theta} \Bigr)^2
% S_{\vartheta/g} (k,\omega)
%S_{u^0/d^0} (k,\omega)
S_{\tilde{u}_d} (k,\omega)
\end{equation}   	 			% 		(2)
     where
\begin{eqnarray*}
% \vartheta/g(x, t, \vartheta) = u_d (x, t, \vartheta) -
\tilde{u}_d (x, t, \vartheta) = u_d (x, t, \vartheta) -
\overline{u_d(x, \vartheta)}^T
\end{eqnarray*}


\subsection{Experiment and Data Analysis}

     To check the theoretical conclusions for the fetch-limited conditions, a field experiment
was carried out. A coherent X-band radar system operating with the HH polarization mode was
used as a source of the radar data. The radar was installed at a height of 16 m above sea level,
and the range of the incident angles $\theta$  was from 40\deg \ to 88.2\deg
(\figref{1}).           % (Figure 1).
 The wind velocity, air
temperature, pressure, humidity of the near-surface atmospheric layer were measured using a
Vantage Pro2 Weather Station (Davis Instruments$^{\mathrm TM}$) installed at a height of 26 m (2 in
\figref{1}).           % Figure 1).
The marine current velocity at depths from 1 to 25 m with a step of 0.5 m were obtained
according to the data of the acoustic Doppler current profiler (ADCP WorkHorse Monitor 1200~kHz, RDI$^{\mathrm TM}$), which was hung out on the cables from the middle deck (3 in
\figref{1}).           % Figure 1).
 The
ADCP was oriented vertically down and was sunk to a depth of 0.3 m. The surface elevation was
measured with a string wave gauge installed at the SOP (4 in
\figref{1}).           % Figure 1).

     This coherent radar (Micran$^{\mathrm TM}$) operated in a chirp modulation mode with a modulation
bandwidth of $\Delta f = 191$~MHz and an emitted radio power of 1~W. The range resolution was
\begin{eqnarray*}
\Delta_r= \frac{c}{2\Delta f} = 0.79 \;{\mathrm m},
\end{eqnarray*}
the azimuth resolution was
\begin{eqnarray*}
\Delta_\vartheta= \frac{a \pi R}{360}
\end{eqnarray*}
 where $a = 1\deg$ is the angular
resolution (the antenna length was 2~m), and $R$ was the distance from the resolution element on
the sea surface. Thus, for virtually all conditions of sea disturbance, the condition $\Lambda > 2 \Delta_r$
holds, where $\Lambda$   is the length of the dominant wave, but the condition  $\Lambda > 2 \Delta_\vartheta$ holds for $R$ being
less than 500 meters. The received beat signal (the difference between the reflected and emitted
chirp signals) was subjected to a Fourier transform, which resulted in the vector $Q(r,\vartheta)$
consisting of the amplitude $A$ and phase  $\varphi$ of the reflected radar signal at distances of up to 500~m with a step of 0.79~m. Two spatial-scanning modes were used: lateral and circular. When the
radar operated in the lateral mode, a sequential set of vectors  $Q(r,\vartheta_0, t) = A (r,\vartheta_0, t) e^{i\varphi (r,\vartheta_0, t)}$ was
obtained with the time interval $\tau=3.5$~ms between them at a fixed position of the antenna
azimuth angle $\vartheta_0$. By fixing the distance  $r=r_0$ and applying the Fourier transform to $N$
elements
$Q(r_0, \tau:\tau: N\tau, \vartheta_0)$, we may obtain the Doppler spectra   $D(f)$
of the reflected radar
signal at the distance $r_0$. Numerical simulation of the radar sensing for such a problem statement
was given in
[\itc{Toporkov and Sletten,} \reflink{Topo17}{2017}].
 For a rotating antenna, there is a periodic scanning of
the environment
 $Q(r,\vartheta,n_r T_r)$
 with the antenna rotation period $T_r$, where $n_r$  is the number of
rotations.


     Let us consider the algorithm used to determine Doppler velocities based on the data of
coherent radar sensing as described in
[\itc{Pereslegin and Sinitsyn,} \reflink{Pere11}{2011}].
 For the $k$th vector, we
have the phase value
\begin{eqnarray*}
\varphi_k (r, \vartheta_0)=
\varphi_0 (r, \vartheta_0) - 2 \pi f_d^\vartheta (r, \vartheta_0) \tau k.
\end{eqnarray*}
 Then, on the condition of sensing
the same area, the difference between the two phases of the neighboring vectors is calculated as

\begin{eqnarray*}
\Delta\varphi = \mathrm{arg}(Q(k) \times Q^* (k+1) =
\end{eqnarray*}
\begin{eqnarray*}
\mathrm{arg}\bigl( A^2(r, \vartheta_0) e^{ -i 2 \pi f_d^\vartheta (r, \vartheta_0) \tau} \bigr) =
2 \pi f_d^\vartheta (r, \vartheta_0) \tau.
\end{eqnarray*}
 Using the ratio
 \begin{eqnarray*}
 f_d^\vartheta= \frac{2 u_d}{\lambda} \sin \theta
 \end{eqnarray*}
  and the equation for the phase difference $\Delta\varphi$, the Doppler velocity may be
written as
  \begin{eqnarray*}
 u_d = \frac{\Delta\varphi\lambda}{4 \pi \tau  \sin \theta}.
 \end{eqnarray*}

 According to the method described in
[\itc{Pereslegin and Sinitsyn,} \reflink{Pere11}{2011}],
the sensitivity to velocity fluctuations is determined as

 \begin{eqnarray*}
( u_d)_{\mathrm{sens}} =
\frac{q_0}{\frac{\partial (\Delta\varphi)}{\partial(u_d)} \sqrt{N}},
 \end{eqnarray*}
 where $q_0 >1$
  is the threshold coefficient, $N$ is the number of independent samples of the signal at a
given site, and
$[\partial (\Delta\varphi)]/[\partial(u_d)]$  is the steepness of the characteristic, which is equal to
\begin{eqnarray*}
\frac{\partial (\Delta\varphi)}{\partial(u_d)} =
\frac{4 \pi \tau \sin \theta}{\lambda} \approx  1.4 \: \mathrm{rad \;s/m}
 \end{eqnarray*}
 for the parameters of the radar used.

\begin{sloppypar}
     In the lateral scanning mode, brightness- and velocity-based
    % $(u_d(r, \vartheta_0, t)$
     $(A(r, \vartheta_0, t)$
and
$u_d(r, \vartheta_0, t)$, respectively)
radar images of the water surface may be obtained in a distance-time system of coordinates (DTI, distance-time image)
(see
\figref{2}).           % Figure 2).
These images were
obtained when scanning upwind in an experiment conducted at a wind velocity of 10~m/s. A
wave structure of the image shows a variation in the slope of the stripes relative to the $x$~axis,
which is explained by various propagation velocities of the wind-driven wave harmonics.
\end{sloppypar}

\def\topfraction{0.98}
\def\textfraction{.02}
\begin{figure}[t]                     % Fig 2
\figurewidth{20pc}
\setimage{}{}{20pc}{}{2019es000662-f02}
\caption{Brightness (a) and velocity (b) radar images of the water surface, obtained in the
upwind lateral scanning mode.}
\end{figure}


     							
     Statistics of the Doppler velocity distribution
      $u_d(r, \vartheta_0, t)$
      should correspond to the wind
wave height distribution function, which is a Gaussian function in the linear approximation
under consideration
%[\itc{Laboratory Manual in Ocean Dynamics,} \reflink{Labo92}{1992}].
[\itc{Nekrasov and Pelinovskiy (eds.),} \reflink{Nekr92}{1992}].
\figref{3}           % Figure 3
 shows the Doppler
velocity distribution functions for various distances (incident angles) shown in
\figref{2}b.           % Figure 2b.
There
is an increase in the average value of the distribution function and a broadening with a growing
incident angle, which is explained by shadowing of some areas of the water surface with crests
of the waves at low grazing angles. However, this trend may be eliminated by subtracting the
time average for each distance. The dashed line in
\figref{3}           % Figure 3
shows an approximation of the
distribution functions with a Gaussian. For incident angles being not too close to the grazing one
(curve 1 in
\figref{3}),           % Figure 3),
we may conclude that the statistics of distribution of the Doppler velocities
is normal.

     Spectral processing of the velocity DTI after subtracting the time-averaged Doppler
velocity for each distance gives a spatio-temporal wave spectrum
\eqref{2}           %(2).
The boundary value of the
wave number is determined as
%\begin{eqnarray*}
%k_N =\frac{\pi}{\Delta_r} \approx 4 \:\mathrm{rad/m},
 %\end{eqnarray*}
 $k_N =\pi/\Delta_r \approx 4$~rad/m,
 which corresponds to the wave frequency in
deep water
$ - \omega(k_N) = \sqrt{g k_N} \approx 6.2 \:\mathrm{rad/s}$.
\figref{4}           % Figure 4
shows the result of the spatio-temporal wave
spectrum reconstruction using this approach and employing the data on the Doppler velocities
given in
\figref{2}b.           % Figure 2b.


\begin{figure}[t]                     % Fig 3
\figurewidth{20pc}
\setimage{}{}{20pc}{}{2019es000662-f03}
\caption{Probability distribution function of the Doppler velocities (solid lines) and its
approximation with a Gaussian (dashed line) for three incident angles: 1 -- 75\deg, 2 -- 85.8\deg, 3 -- 87.6\deg.}
\end{figure}

     					
     The areas of energy localization are prominent in the reconstructed wave spectrum. The
main share of the energy is concentrated near the dispersion curve
 $  \omega = \sqrt{g k} + k u_c$, where $u_c$  is
the velocity of the current at the surface being equal to 0.3~m/s (dashed line 1 in
 \figref{4}b),           % Figure 4b),
 and
the $n$-fold harmonics  $nk, \;n\omega$, where $n=2$ (dashed line 2 in
\figref{4}b)           % Figure 4b)
and $n=3$ (dashed line~3 in
\figref{4}b)           % Figure 4b)
[\itc{Slunyaev and Sergeeva,} \reflink{Slun12}{2012}].
 There are also prominent oncoming waves
 $  \omega = \sqrt{g k} - k u_c$
  (dashed line $-1$ in
\figref{4}b)           %   Figure 4b)
  and the energy area corresponding to the group
structure of the wind waves (0 in
\figref{4}b).           %  Figure 4b).


\begin{figure}[t]                     % Fig 4
\figurewidth{20pc}
\setimage{}{}{20pc}{}{2019es000662-f04}
\caption{Reconstructed spatio-temporal wave spectrum (a) and theoretical dispersion
curves superimposed on the recovered wave spectrum~(b).}
\end{figure}

     The spatio-temporal wave spectrum from the coherent radar data may also be determined
in the case of all-round observation of the environment
 [\itc{Trizna,} \reflink{Triz11}{2011}].
  The ratio between the
antenna rotation period $T_r$  and the pulse repetition period $\tau$  determines $N$, which is a number of
the necessary averaged independent measurements of the Doppler velocity in the resolution
element on the wavy surface. Under circular scanning conditions, the amplitude $A(r,\vartheta, n_rT_r)$
and velocity
  $u_d(r,\vartheta, n_rT_r)$ radar images are obtained. As a result of spectral processing of
  $u_d(r,\vartheta, n_rT_r)$, we get the spatio-temporal wave spectrum
 $S_\eta (k, \omega,\vartheta)$  with a boundary wave
number,
 $k_N %= \pi/\Delta_r
 \approx 4$~rad/m, while the limiting frequency for the maximum rotation speed of the
antenna is $T_r\approx 2.5$~s,
 $\omega_N = \pi/T_r \approx 1.2$~rad/s which is significantly smaller than the frequency value
for gravity waves on the water surface at $k_N$. This processing, however, is justified for long
waves. In
[\itc{Nwogu and Lyzenga,} \reflink{Nwog10}{2010}],
 the authors reconstruct the elevation of the sea surface
from measurements of the Doppler velocities with a rotating antenna in the circular scan mode.
In this paper, we consider spectral processing of the velocity image for a single rotation of the
antenna. The field of the Doppler velocities $u_d(x,y)$, interpolated to a uniform grid is shown in
\figref{5}a.           % Figure 5a.
The wind speed in this experiment was 12~m/s. The reconstructed two-dimensional
wave spectrum determined as

\begin{eqnarray*}
S_\eta (k_x, k_y) = \frac{1}{g \sqrt{k_x^2 + k_y^2}} S_{u_d} (k_x, k_y)
 \end{eqnarray*}
 is shown in
\figref{5}b.           % Figure 5b.


\begin{figure}[t]                     % Fig 5
\figurewidth{20pc}
\setimage{}{}{20pc}{}{2019es000662-f05}
\caption{Field of Doppler velocities (a) and reconstructed two-dimensional wave spectrum~(b).}
\end{figure}

    	
     The surface elevation was measured with a string wave gauge installed at the SOP (4 in
\figref{1}),           % Figure 1),
in parallel with the radar observations. The resistive string wave gauge consisted of a
three-channel, analogue processing scheme and an analog-to-digital converter connected to a
computer. Each channel of the wave gauge included a pair of strings (the string material is cantal
A1, string thickness is 0.4~mm) spread by 5~mm, separate channels of the transmitter were spread
to the corners of an equilateral triangle with a side of 20~cm. The wave gauge was calibrated by
raising and lowering the gauge to a predefined height and plotting the voltage-height
transmission curve. The calibration curve was used for recalculation of the measured voltages to
wave heights with subsequent plotting of the wind wave spectra. The quantitative comparison
between the height spectra of the waves obtained by the contact method and those obtained
remotely is given in
\figref{6}.           % Figure 6.
 The frequency spectrum of wind waves was determined based on
the coherent radar data as the average in the Doppler velocities ensemble for the angles being not
too close to the grazing one and corresponding to curve~1 in
\figref{3}:           % Figure 3:

\begin{eqnarray*}
S_\eta (\omega) =
\frac{1}{N} \sum_{i=1}^{N} \Bigl( \frac{1}{\omega \cos \vartheta_0 \sin \theta_i} \Bigr)^2
%S_{u^0/d^0}
S_{\tilde{u}_d}
(\omega,\theta_i)
 \end{eqnarray*}
  (solid line in
  \figref{6}a).           % Figure 6a).
    In the same figure, there is
also a height spectrum of the wind wave obtained basing on the wave gauge data (dotted line)
and the JONSWAP model spectra with  $f^{-5}$ (dot-dash line) and  $f^{-4}$ (points) power dependences
from paper
[\itc{Battjes et al.,} \reflink{Batt87}{1987}].
There is a nice agreement of the remotely measured spectra
with the JONSWAP model spectrum  $f^{-4}$. A comparison of the omnidirectional wave spectrum
obtained from the spectrum of Doppler velocities is shown in
\figref{6}b.           % Figure 6b.
Then, the wave
spectrum recovered from the coherent radar data (solid line in
\figref{6}b)           %  Figure 6b)
 will be determined as

 \begin{eqnarray*}
S_\eta (k) = \frac{1}{k} \frac{1}{g \sqrt{k_x^2 + k_y^2}} S_{u_d} (k_x, k_y),
 \end{eqnarray*}
 where
 \begin{eqnarray*}
S_{u_d} (k, \vartheta) = \frac{1}{k} S_{u_d} (k_x, k_y).
 \end{eqnarray*}
  The spatial spectrum
 \begin{eqnarray*}
S (k) =  S(\omega) \frac{d \omega}{d k} \approx \frac{1}{2} S (\omega)
\sqrt{\frac{g}{k}}
 \end{eqnarray*}
  (dotted line in
  \figref{6}b)           % Figure 6b)
    was obtained based on the wave gauge
data
 [\itc{Hwang et al.,} \reflink{Hwan10}{2010}].
     						


\begin{figure}[t]                     % Fig 6
\figurewidth{20pc}
\setimage{}{}{20pc}{}{2019es000662-f06}
\caption{Temporal (a) and spatial (b) spectra of the wind-driven wave. The solid line
represents the wave reconstructed from Doppler velocities data obtained with the coherent radar;
the dotted line represents the wave gauge data; the dot-dash line represents JONSWAP~$f^{-5}$
spectrum, prime mark is JONSWAP~$f^{-4}$  spectrum~(a) and \mbox{JONSWAP~$k^{-5/2}$}   spectrum~(b).}
\end{figure}


     As it is evident from
     \figref{6}b,           % Figure 6b,
     there is sufficiently good qualitative and quantitative
agreement between the wave spectra obtained with different methods and the model JONSWAP-$k^{-5/2}$ spectrum (dotted), obtained by calculation from JONSWAP-$f^{-4}$. Note the difference
between the spectral peaks of the spatial wave spectra
(\figref{6}b).           % (Figure 6b).
This can be explained by the
influence of the current which was not taken into account during the spatial spectrum calculation
(the dotted line in
\figref{6}b).           % Figure 6b).

\section{Conclusions}

     The temporal spectrum of wind waves is easy to measure according to a wave gauge or
buoy. However, the use of Doppler velocity DTI of the water surface makes it possible to
calculate not only the temporal spectrum, but also the spatio-temporal ($\omega - k$) wave spectrum.
This method provides more detailed information about the wave field than the one-point
measurements. The contact measurements of the spatial spectrum of the wind waves require the
use of a system of sensors, which are synchronized in time and are separated in space and which
measure the elevation of the rough surface. A relatively long recording time is required to collect
the data for spectrum calculation. The spatial panorama of the Doppler velocities of the water
surface is reconstructed in one turn of the coherent radar antenna and can be used to calculate the
wave elevation spectrum. The results of this work show the possibility of reconstructing of the
wave spectra based on the coherent radar measurements.

     Let us briefly formulate the main results of this paper:
  \begin{itemize}
 \item
\begin{sloppypar}
     a linear theoretical relation between the Doppler velocity derived from the coherent radar
measurements and the wave elevation spectrum was obtained for the meter/decameter wave
range;
\end{sloppypar}
\item
\begin{sloppypar}
 a possibility of reconstructing the spatio-temporal wave spectra from radar data on the
Doppler velocities was demonstrated in a field experiment. A comparison was made of the
recovered spectra against those measured by a wave gauge and those obtained by a model
calculation within the JONSWAP; a good agreement between experimental and theoretical data
is shown;
\end{sloppypar}
\item
      limitations of the proposed approach have been defined; they are related to shadowing of
some areas of the wavy water surface by crests of wind waves, which are especially pronounced,
when the incident angles are $\theta > 85$\deg. It should also be noted that breaking of the crest leads to a
higher Doppler velocity as compared with linear theory predictions. Thus, it might lead to
overestimation of wave amplitude values under high wind speed conditions. These limitations
may be partially overcome by using a vertically-polarized radar.
\end{itemize}

     The obtained results may be useful for remote measurements of not only wind waves and
swells (if the wind speed is higher than a certain boundary value). The results can be also useful
for solution of scientific and practical problems about the wave spectrum transformation at the
sea shelf, detection of non-homogeneous currents and other near-surface processes. Spatio-temporal spectra may be used in the evaluation of the wind wave energy distribution and in the
studies of harmonic interactions.

\acknowledgment{This work was done by support of the Russian Science Foundation (Project RSF 18-77-00072).}


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\end{document}