Effective roughness on the sea surface for determining variability characteristics of reflected sound waves

We use the 2D-FDTD simulation model shown in Fig. 1 to evaluate the variability characteristics of reflected sound waves. The analytic area is discretized by the staggered grid. The sound pressure and particle velocity on the grid point are calculated using the central difference scheme. In the FDTD simulation, the Gaussian pulse is transmitted from the sound source, and the surface boundary with zero pressure is generated by Eq. (1). Other boundaries are set perfectly matched layer (PML) to absorb sound waves. ²⁶⁾ Table I shows the basic parameters for the 2D-FDTD simulation model.

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Table I. Basic parameters for the 2D-FDTD simulation model.

Parameters	Values
Sound source	Gaussian pulse
Analysis area	Width: 7 m, Depth: 3 m
Density: $\rho$	1000 kg m−3
Sound speed: $c$	1475 m s−1
Surface boundary condition	Sound pressure is zero
Side and bottom boundary condition	PML
Spatial step	5 mm
Timestep	1.7 us
Surface wavelength: ${\rm{\Lambda }}$	0.32–5.08 m
rms of surface water level: ${\sigma }_{z}$	4.12–37.1 mm

In strictly, the reflected sound waves on the sea surface are affected by the Doppler effect because the sea surface fluctuates constantly. However, the fluctuating velocity of the sea surface is on the order of several seconds and is extremely small compared to the speed of sound in seawater (about 1500 m s⁻¹). Therefore, in the calculation of the FDTD, the sea surface is treated as a stationary boundary with no temporal variation. Since this study aims to evaluate the sea surface reflection characteristics, which is the main factor of the multipath effect, there is no problem in treating the sea surface as a stationary boundary.

Sea surface waves are represented by the superposition of regular waves with different surface wavelengths. In addition, sea surface waves are classified into dispersible waves having different phase velocities depending on the surface wavelength. The Bretschneider-Mitsuyasu spectrum shown in Eq. (1) is well known to give a good approximation of the sea surface spectrum that is observed on the deep offshore coast of Japan. In order to simulate sea surface waves, we used a single summation method ^27,28) shown in Eq. (2) based on the Bretschneider-Mitsuyasu spectrum.

$$\begin{eqnarray}&&S\left(f\right)=0.257{H}_{1/3}^{2}{T}_{1/3}{\left({T}_{1/3}f\right)}^{-5}\exp \left[-1.03{\left({T}_{1/3}f\right)}^{-4}\right]\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\eta \left(x,t\right)=\displaystyle \sum _{n=1}^{{N}_{s}}{a}_{n}\,\cos ({k}_{n}x\,-2\pi {f}_{n}t+{\varepsilon }_{n})\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{a}_{n}=\sqrt{S\left({f}_{n}\right){\rm{\Delta }}{f}_{n}}.\end{eqnarray} \tag{ 3 }$$

Here, $\eta \left(x,t\right)$ is the water level. ${a}_{n},$ ${k}_{n},$ ${f}_{n},$ and ${\varepsilon }_{n}$ represent the amplitude, the wavenumber, the frequency, and the initial phase. ${N}_{s}$ is the sampling number in the frequency domain. $S(f)$ is the ocean wave spectrum, ${H}_{1/3}$ is the significant wave height, and ${T}_{1/3}$ is the significant wave period. In the calculation of $\eta (x,t)$ shown in Eq. (1), ${k}_{n}={\left(2\pi {f}_{n}\right)}^{2}/g\,$ ($g:$ 9.8 ${\rm{m}}\,{{\rm{s}}}^{-2}$) and ${\varepsilon }_{n}$ is randomly given in the range of 0–$2\pi$ rad.

In the simulation using the FDTD method, the water level $\eta \left(x\right)$ at a certain time is set as the surface boundary. In order to evaluate the statistical properties of fluctuation of reflected sound waves, we generate a total of 250 surface boundaries observed every 0.3 s with the same ${H}_{1/3}$ and ${T}_{1/3}$ conditions.

In this study, the wave height and wavelength of the sea surface are adjusted by controlling ${H}_{1/3}$ and ${T}_{1/3}$ in Eq. (1). In statistically evaluating the variability characteristics of reflected sound waves, the effective value of water level ${\sigma }_{z}$ as overall sea roughness and the wavelength of the sea surface ${\rm{\Lambda }}$ are defined as Eqs. (4) and (5), respectively.

$$\begin{eqnarray}&&{\sigma }_{z}=\sqrt{\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}{\eta }^{2}\left(0,{t}_{i}\right)}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\rm{\Lambda }}=\displaystyle \frac{g}{2\pi }{T}^{2}=1.56{T}^{2}.\end{eqnarray} \tag{ 5 }$$

Here, Eq. (4) is the Rayleigh roughness parameter $2k{\sigma }_{z}\,\cos \,\phi$ and is calculated from the water level directly above the sound source on the sea surface used in the FDTD simulation. Equation (5) shows the relationship between the wavelength ${\rm{\Lambda }}$ and the period $T$ of sea surface waves observed under the condition of deep water waves dominated by gravity waves. Since the Bretschneider-Mitsuyasu spectrum has a frequency bandwidth, the expected value of the period T of sea surface waves is calculated from the Eq. (1), and the average wavelength ${\rm{\Lambda }}$ of sea surface waves is defined. In the previous paper, the relationship ${H}_{1/3}=4.004{\sigma }_{z}$ and ${T}_{1/3}=1.1T$ are derived based on statistical theory. ^29–32)

In this study, the Gaussian pulse used as the sound source is treated as impulse responses with a limited frequency band in order to reduce the numerical dispersion of the FDTD method. The amplitude and phase of reflected sound waves are calculated by convolving arbitrary tone burst waves with impulse responses obtained by the FDTD simulation.

Figure 2 shows the irradiation range of burst waves on the sea surface that contributes to reflected sound waves. The reflection points on the sea surface directly above the sound source contribute to the rising part of burst waves. Then, the contribution from horizontally distant reflection points on the sea surface increase over time. The irradiation range of burst waves $2L$ can be described by Eq. (6). Here, $R$ is the distance from the sound source to the sea surface, sound speed $c,$ and the tone burst wave length ${\rm{\Delta }}t.$

$$\begin{eqnarray}&&\begin{array}{l}L=\sqrt{{\left(R+\displaystyle \frac{c{\rm{\Delta }}t}{2}\right)}^{2}-{R}^{2}}=\sqrt{{R}^{2}+c{\rm{\Delta }}tR+\displaystyle \frac{{c}^{2}{\rm{\Delta }}{t}^{2}}{4}-{R}^{2}}\\ \,=\,\sqrt{c{\rm{\Delta }}tR+\displaystyle \frac{{c}^{2}{\rm{\Delta }}{t}^{2}}{4}}.\end{array}\end{eqnarray} \tag{ 6 }$$

The longer the burst wavelength, the wider the irradiation range of sound waves on the sea surface that contributes to reflected sound waves. Since the distance attenuation becomes large and the angle of incidence of sound waves on the sea surface becomes shallow, the contribution from horizontally distant reflection points gradually decreases, and the amplitude and phase of reflected sound waves converge to a constant value. In this study, we evaluate the variability characteristics of reflected sound waves using the average amplitude $r$ and phase $\theta$ in the 85%–95% section of the latter half of burst length contributed many reflection points.

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When there is no wave on the sea surface, the reflected sound waves can be regarded as sound waves from the mirror image of the sound source. The reflected sound waves are coherent waves with no temporal fluctuation. As the wave height of the sea surface increases, reflection points randomly fluctuating on the sea surface increase. As a result, the coherent component of the reflected sound waves decreases, and the incoherent component gradually increases. The statistical properties of a signal in which coherent and incoherent components coexist can be represented by the Rice distribution. ^33–36)

The PDF of amplitude variability in the Rice distribution can be shown as

$$\begin{eqnarray}&&\,p\left(x\right)=2r\left(1+\gamma \right)\exp \left\{-\left[\left(1+\gamma \right){r}^{2}+\gamma \right]\right\}{I}_{0}\left[\left[2r\sqrt{\gamma \left(1+\gamma \right)}\right]\right]\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\gamma ={s}^{2}/\left\langle {n}^{2}\right\rangle ,\end{eqnarray} \tag{ 8 }$$

where $p,$ $r,$ and ${I}_{0}$ are the probability density, the amplitude of the sound wave, and the modified Bessel function, respectively. $\gamma$ is the energy ratio of the coherent and incoherent components, which is expressed by using an rms of sinewave $s$ and a rms of noise $\left\langle {n}^{2}\right\rangle .$ In the PDF of the amplitude variability of the Rice distribution, the Rayleigh distribution is obtained when $\gamma =0$ and the Gaussian distribution when $\gamma$ is large. Figure 3 shows the Rice distribution expressed using complex vectors. As shown in Fig. 3, the Rice distribution follows the normal distribution $N\left(s,\left\langle {n}^{2}\right\rangle \right)$ for the $x$-axis and the normal distribution $N\left(0,\left\langle {n}^{2}\right\rangle \right)$ for the $y$-axis in the complex plane. Assuming that the sea surface is a sufficiently random rough surface, previous studies ^12–17) have shown that the variability characteristic of reflected sound waves can be described using the Rayleigh roughness parameter $2k{\sigma }_{z}$ and the Rice distribution.

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However, we have reported that the variability characteristics of reflected sound waves change depending on the surface wavelength ${\rm{\Lambda }}$ even with the same Rayleigh roughness parameter $2k{\sigma }_{z}.$ ²⁴⁾ In Ref. 24, we show that as ${\rm{\Lambda }}$ becomes smaller, the variability characteristics displayed on the complex plane approach the Rice distribution shown in Fig. 3. Figure 4 shows the variability of reflected sound waves on the complex plane obtained under the condition that the Rayleigh roughness parameter $2k{\sigma }_{z}$ is fixed and surface wavelength ${\rm{\Lambda }}$ are changed in the FDTD simulation. Here, the data are normalized by the amplitude ($r=1$) and phase ($\theta =1$) observed under the condition that there is no wave on the sea surface. In addition, Fig. 4 expresses the effect of surface wavelength ${\rm{\Lambda }}$ on variability characteristics using complex vectors. When ${\rm{\Lambda }}$ is very large, and the sea surface can be regarded as a flat surface within the burst wave irradiation range, the sea surface acts as a specular reflection boundary. In this condition, only phase fluctuation according to the Rayleigh roughness parameter $2k{\sigma }_{z}$ is observed without the amplitude fluctuation. This characteristic can be expressed by complex vectors in which the phase of coherent component s of reflected sound waves fluctuates according to $2k{\sigma }_{z}.$ As ${\rm{\Lambda }}$ becomes smaller, irregularly fluctuating reflection points increase within the irradiation range of the burst wave on the sea surface. Therefore, the coherent component s of a reflected sound wave is gradually lost, and the incoherent component n increase. Under the condition that ${\rm{\Lambda }}$ is small enough, the reflected sound waves from multiple randomly fluctuating reflection points interfere with each other with a phase difference according to $2k{\sigma }_{z}.$ As a result, the sea surface approaches a sufficiently random rough surface, and the variability characteristics of reflected sound waves converge to the Rice distribution described in previous studies. ^12–17)

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In our previous study, ²⁴⁾ we could not quantitatively evaluate the variability characteristics of reflected sound waves on a relatively smooth surface with a large surface wavelength ${\rm{\Lambda }}.$ In the FDTD simulation of sea surface wavelength, we found that the variability characteristics of reflected sound waves are different from the Rice distribution shown in Fig. 4 on a relatively smooth sea surface where ${\rm{\Lambda }}$ is large. In this condition, it is considered that the geometrical similarity between the shape of the sea surface and sound wave directly contributes to the variability characteristics of reflected sound waves. For the variability characteristics of reflected sound waves to be the same on a relatively smooth surface, a geometric similarity relationship must be established between the shape of the sea surface and sound waves, as shown in Fig. 5 and Eq. (9). Here, $R$ is the distance between the sound source, receiver, and sea surface. $\lambda$ is the wavelength of the sound wave.

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Lambda }}^{\prime} }{{\rm{\Lambda }}}=\displaystyle \frac{{\sigma }_{z}^{\prime} }{{\sigma }_{z}}=\displaystyle \frac{R^{\prime} }{R}=\displaystyle \frac{\lambda ^{\prime} }{\lambda }\end{eqnarray} \tag{ 9 }$$

${\rm{\Lambda }}$ and $R$ are normalized by the frequency of sound waves, such as $k{\rm{\Lambda }}$ and $kR,$ respectively. The unit of $k{\rm{\Lambda }}$ and $kR$ is radians, which is the same as the Rayleigh roughness parameter $2k{\sigma }_{z}.$

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In this section, we adjusted ${\rm{\Lambda }},$ $R,$ and $f$ so that $k{\rm{\Lambda }}$ and $kR$ are fixed, and we evaluated the effect of the geometrical similarity between the shape of the sea surface and sound waves on the variability characteristics of reflected sound waves. Table II shows the FDTD simulation conditions. Figure 6 shows the energy ratio $\gamma$ of the Rice distribution for amplitude variability and the standard deviation ${\sigma }_{\theta }$ for phase variability when the Rayleigh roughness parameter $2k{\sigma }_{z}$ are changed under the fixed condition of $k{\rm{\Lambda }}$ and $kR.$ In Fig. 6, $\gamma$ and ${\sigma }_{\theta }$ are estimated by the maximum likelihood method using the amplitude and phase of reflected sound waves. When the sea surface is a sufficiently random rough surface, $\gamma$ is uniquely determined by $2k{\sigma }_{z},$ so that $\gamma$ is displayed as ${\gamma }_{0}$ in Fig. 6. Figure 6 shows that γ is determined by $2k{\sigma }_{z}$ regardless of frequency, $\gamma$ takes a different value depending on $k{\rm{\Lambda }}.$ These results indicate that on a relatively smooth sea surface with large $k{\rm{\Lambda }},$ $\gamma$ is larger than ${\gamma }_{0};$ that is, the coherent component of reflected sound waves is large. Therefore, it is considered that the roughness of the sea surface, which is smaller than the overall sea roughness ${\sigma }_{z},$ contributes to $\gamma$ for amplitude variability of reflected sound waves. On the other hand, ${\sigma }_{\theta }$ equals to $2k{\sigma }_{z},$ regardless of $k{\rm{\Lambda }},$ which indicates ${\sigma }_{z}$ contributes to ${\sigma }_{\theta }$ for the phase variability.

Table II. FDTD simulation conditions evaluating the effect of geometrical similarity.

$k{\rm{\Lambda }}\,$[rad]	$kR\,$[rad]	${\rm{\Lambda }}\,$[m]	$f\,$[Hz]	$R\,$[m]
8.95	53.25	0.32	6613	1.89
		0.38	5465	2.29
		0.42	5000	2.5
13.26	53.25	0.38	8099	1.54
		0.42	7410	1.69
		0.46	6806	1.84
		0.50	6272	1.99
		0.62	5000	2.5
20.72	53.25	0.62	7813	1.6
		0.71	6806	1.84
		0.81	5981	2.09
		0.97	5000	2.5
27.07	53.25	0.81	7812	1.6
		0.97	6531	1.91
		1.09	5844	2.14
		1.27	5000	2.5
42.29	53.25	1.27	7812	1.6
		1.99	5000	2.5
60.90	53.25	1.99	7200	1.74
		2.86	5000	2.5
82.89	53.25	2.86	6806	1.84
		3.89	5000	2.5
108.26	53.25	2.86	8889	1.41
		3.89	6531	1.91
		5.08	5000	2.5

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From the results in the above section, it is considered that as the sea surface wavelength ${\rm{\Lambda }}$ increases, the sea surface roughness that contributes to reflected sound fluctuations becomes smaller than the overall sea surface roughness ${\sigma }_{z}.$

We determine the range of the sea surface, $2{\rm{\Delta }}x,$ that contributes to the amplitude variability characteristics as shown in Fig. 7. The effective roughness, ${\sigma }_{z0},$ is defined as the roughness within the contribution range of this sea surface, $2{\rm{\Delta }}x.$ The effective roughness is smaller than the roughness of the entire sea surface, ${\sigma }_{z}$ when the sea surface wavelength, ${\rm{\Lambda }},$ is large. The effective roughness increases as the sea surface wavelength, ${\rm{\Lambda }},$ decreases. When ${\rm{\Lambda }}$ is sufficiently small, and the sea surface approaches the random rough surface, the effective roughness becomes ${\sigma }_{z}.$

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The energy ratio $\gamma ,$ which is a parameter of the Rice distribution that describes the amplitude variability characteristics of the reflected sound waves, can be determined by the new Rayleigh roughness parameter $2k{\sigma }_{z0}$ using the effective roughness of the sea surface ${\sigma }_{z0}$ as shown in Fig. 8. The Rayleigh roughness parameter, $2k{\sigma }_{z}$ represents the phase difference between the reflection from the average water level and the reflection from crests and troughs of the entire sea surface waves. On a relatively smooth sea surface with a large ${\rm{\Lambda }},$ the sea level range that contributes to the reflected sound waves is limited. The water level difference in this range is smaller than the overall water level difference, so the phase difference, $2k{\sigma }_{z0}$ is smaller than $2k{\sigma }_{z}.$

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The effective roughness of the sea surface ${\sigma }_{z0}$ is estimated from the relationship between the energy ratio $\gamma$ and the Rayleigh roughness parameter $2k{\sigma }_{z}$ obtained in Fig. 6. We assume that the energy ratio ${\gamma }_{0}$ obtained on a sufficiently random rough surface can be described uniformly by the Rayleigh roughness parameter $2k{\sigma }_{z0}$ expressed by the effective roughness ${\sigma }_{z0}$ of the sea surface. From the relationship between $\gamma$ and $2k{\sigma }_{z}$ obtained by the FDTD simulation and the relationship between ${\gamma }_{0}$ and $2k{\sigma }_{z0}$ on the sufficiently random rough surface, ${\sigma }_{z0}/{\sigma }_{z}$ is given by Eq. (10).

$$\begin{eqnarray}&&\displaystyle \frac{2k{\sigma }_{z0}}{2k{\sigma }_{z}}=\displaystyle \frac{{\sigma }_{z0}}{{\sigma }_{z}}.\end{eqnarray} \tag{ 10 }$$

Figure 9 shows ${\sigma }_{z0}/{\sigma }_{z}$ obtained from the simulation results in Fig. 6. If $k{\rm{\Lambda }}$ is fixed, ${\sigma }_{z}$ and ${\sigma }_{z0}$ are in a proportional relationship, so ${\sigma }_{z0}/{\sigma }_{z}$ is an almost constant value. ${\sigma }_{z0}/{\sigma }_{z}$ approaches 1 when $k{\rm{\Lambda }}$ is sufficiently small and decreases as $k{\rm{\Lambda }}$ increases. The range of the sea surface that contributes to the amplitude variability $2{\rm{\Delta }}x$ is calculated from ${\sigma }_{z0}/{\sigma }_{z}$ obtained by the FDTD simulation. From the shape of the sea surface used in the FDTD simulation, the spatial effective roughness ${\sigma }_{z0}(x)/{\sigma }_{z}$ is calculated for each ${\rm{\Lambda }}.$ $2{\rm{\Delta }}x$ can be calculated from ${\sigma }_{z0}(x)/{\sigma }_{z}$ corresponding ${\sigma }_{z0}/{\sigma }_{z}$ obtained by the FDTD simulation. Since $2{\rm{\Delta }}x$ changes with $R,$ we normalized $2{\rm{\Delta }}x$ using the sound wave propagation time shown in Fig. 7. As shown in Fig. 7, the propagation time from the sound source to the receiver is shown in $2R/c,$ where $c$ is the speed of sound. Here, $2{\rm{\Delta }}x$ that contributes to the amplitude variability can be replaced with the range irradiated during the sound wave propagation time of $2R/c\,\sim \,2\sqrt{{R}^{2}+{\rm{\Delta }}{x}^{2}}/c.$ Therefore, the propagation time that contributes to amplitude variability is expressed by Eq. (11) using $T=2R/c.$ Here, the coefficient of sound wave propagation time, $\alpha ,$ is defined by Eq. (12). Table III shows the $2{\rm{\Delta }}x$ and $\alpha$ obtained by the FDTD simulation.

$$\begin{eqnarray}&&T\,\sim \,\left(\displaystyle \frac{\sqrt{{R}^{2}+{\rm{\Delta }}{x}^{2}}}{R}\right)T\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\alpha \equiv \,\left(\displaystyle \frac{\sqrt{{R}^{2}+{\rm{\Delta }}{x}^{2}}}{R}\right).\end{eqnarray} \tag{ 12 }$$

The coefficient of sound wave propagation time, $\alpha ,$ is almost the same in any $k{\rm{\Lambda }}.$ There is a slight difference in $\alpha$ due to the difference in $k{\rm{\Lambda }},$ but it is considered that the backscattering component from the reflection point horizontally away from the reflection point directly above the sound source has changed due to the difference in $k{\rm{\Lambda }}.$ Under the simulation conditions of this paper, the average value of $\alpha$ is calculated as $\alpha$ = 1.0052. The relationship between $R$ and $2{\rm{\Delta }}x$ is given by Eq. (13). Equation (13) shows that the sea surface range $2{\rm{\Delta }}x$ corresponding to about 1/5 of $R$ contributes to the amplitude variability. Equation (13) is estimated from the shape of the sea surface based on the Bretschneider-Mituyasu spectrum, and the relationship changes depending on the wave spectrum

$$\begin{eqnarray}&&2{\rm{\Delta }}x\cong 0.204\,R.\end{eqnarray} \tag{ 13 }$$

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Table III. Estimating the coefficient of sound propagation time $\alpha$ from the result of FDTD simulation in Fig. 6.

$k{\rm{\Lambda }}\,$[rad]	${\sigma }_{z0}/{\sigma }_{z}$	${\rm{\Lambda }}\,$[m]	$f\,$[Hz]	$R\,$[m]	${\rm{\Delta }}x\,$[m]	$\alpha$
42.29	0.553	1.27	7812	1.6	0.195	1.0074
		1.99	5000	2.5	0.32	1.0082
60.90	0.373	1.99	7200	1.74	0.175	1.0051
		2.86	5000	2.5	0.26	1.0054
82.89	0.269	2.86	6806	1.84	0.17	1.0043
		3.89	5000	2.5	0.235	1.0044
108.26	0.210	2.86	8889	1.41	0.125	1.0039
		3.89	6531	1.91	0.175	1.0042
		5.08	5000	2.5	0.23	1.0042

In this section, we verified the effect of the effective roughness ${\sigma }_{z0}$ on the variability characteristics of reflected sound waves. Table IV shows the test parameters for the FDTD simulation. In the simulation, the distance between the sound source, receiver, and sea surface $R$ is fixed at 2.5 m, and the sea surface range $2{\rm{\Delta }}x$ that contributes to the amplitude variability is calculated to be 0.51 m from Eq. (13). Figure 10 shows the relationship between the effective roughness ${\sigma }_{z0}/{\sigma }_{z}$ and ${\rm{\Lambda }}$ under the condition of $2{\rm{\Delta }}x$ = 0.51 m obtained from the shape of the sea surface used in the FDTD simulation.

Table IV. Test parameters for FDTD simulation verifying the effective roughness of sea surface.

Parameter	Value
Frequency: $f$	2.5, 5, 7.5, 10 kHz
Tone burst length: ${\rm{\Delta }}t$	2.4 ms
Tone burst interval	0.3 s
Number of data	250
Distance: $R$	2.5 m
Incident angle of sound：$\phi$	0 deg
Sound speed: $c$	1475 m s−1
Wavelength of sea surface wave: ${\rm{\Lambda }}$	0.32–5.08 m
rms of water level: ${\sigma }_{z}$	4.12–37.1 mm

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Figure 11 shows the variability characteristics of reflected sound waves obtained when the sea surface wavelength ${\rm{\Lambda }}$ and the frequency $f$ of the sound wave are changed. The energy ratio $\gamma$ indicating the amplitude variability can be uniquely described by the new Rayleigh roughness parameter $2k{\sigma }_{z0}$ expressed by the effective roughness ${\sigma }_{z0},$ and the value converges to the energy ratio ${\gamma }_{0}$ of the Rice distribution. The phase variability ${\sigma }_{\theta }$ can be uniquely described by the Rayleigh roughness parameter $2k{\sigma }_{z}$ expressed by the overall sea roughness ${\sigma }_{z}.$ This means that the long-term vertical movement of sea surface corresponding to $2k{\sigma }_{z}$ appears as phase fluctuation. Figure 11 shows that the amplitude variability characteristic is determined by the local fluctuation in the shape of the sea surface roughness observed on a short time scale, and the phase variability characteristic is determined by the global fluctuation in the overall sea surface roughness observed on a long-time scale. Previous studies ^12–17) have been limited to discussing the variability characteristics of reflected sound waves on a sufficiently random rough surface. We have clarified that the variability characteristics of reflected sound waves on a relatively smooth surface with a large surface wavelength ${\rm{\Lambda }}$ can be expressed using effective roughness ${\sigma }_{z0}.$

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This section report the validity verification of the effective roughness of the sea surface by the water tank experiment. Figure 12 shows the configuration of acoustic devices in a water tank. We controlled the hydraulic drive plungers and generated surface waves according to Eq. (1) spectrum. Omnidirectional hydrophones were used for the sound source and the receiver. The size of a water tank is depth 5 m, width 10 m, and length 210 m. The transducer and hydrophone are set at a depth of 2 m. The sound source transmitted tone burst waves repeatedly from the function generator through a power amplifier, and the received signals were recorded using a data logger through a charge amplifier.

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Table V shows the test parameters for the water tank test. The tone burst length was set to 1.0 msec to separate the reflected sound waves from the sea surface and the wall of the water tank. In order to suppress the effect of reverberation of the tank wall, the tone burst interval was set to a long cycle of 0.3 s. The effective value of water level ${\sigma }_{z}$ was calculated from the data of the wave height meter. Figure 13(a) shows a result of the measurement of water level obtained ${\sigma }_{z}=32.2$ mm condition. From the expected value of wave period $T$ observed by the wave height meter and Eq. (5), the surface wavelength ${\rm{\Lambda }}$ in the water tank test is calculated to be 4.1 m. In the water tank test, the surface wavelength ${\rm{\Lambda }}$ was fixed, and we obtained the variability characteristics data when the water level was changed. Figure 13(b) shows the spectrum of the surface waves obtained in the water tank test.

Table V. Test parameters of water tank test.

Parameter	Value
Frequency: $f$	5, 10 kHz
Tone burst length: ${\rm{\Delta }}t$	1.0 ms
Tone burst interval	0.3 s
Number of data	250
Distance: $R$	2.0 m
Incident angle of sound: $\phi$	0 deg
Sound speed: $c$	1475 m s−1
Wavelength of sea surface wave: ${\rm{\Lambda }}$	4.1 m
rms of water level: ${\sigma }_{z}$	3.5–32.2 mm

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Figure 14 shows the variability characteristics of reflected sound waves of 10 kHz on the complex plane obtained by the water tank test. The results of the water tank test show the variability characteristics on a relatively smooth surface, where the phase fluctuation is larger than the amplitude fluctuation. In the previous section, we indicate that the amplitude variability is described by the new Rayleigh roughness parameter $2k{\sigma }_{z0}$ based on the effective roughness ${\sigma }_{z0}$ on a relatively smooth surface. The sea surface range $2{\rm{\Delta }}x$ in the water tank test that contributes to the amplitude variability is calculated to be 0.41 m from Eq. (13). As a result, the effective roughness ${\sigma }_{z0}/{\sigma }_{z}$ is calculated to be 0.206 from $2{\rm{\Delta }}x,$ Eqs. (1), and (2).

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Figure 15 shows the amplitude and phase variability characteristics of reflected sound waves obtained in the water tank test. The energy ratio $\gamma$ for the amplitude variability converges to ${\gamma }_{0}$ and can be uniquely described by the new Rayleigh roughness parameter $2k{\sigma }_{z0}.$ The phase variability ${\sigma }_{\theta }$ can be uniquely described by the Rayleigh roughness parameter $2k{\sigma }_{z}.$ We confirm the validity of the effective roughness of sea surface on evaluation of amplitude variability characteristic by the water tank test.

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