Study on sound-speed dispersion in a sandy sediment at frequency ranges of 0.5–3 kHz and 90–170 kHz

In order to study the properties of sound-speed dispersion in a sandy sediment, the sound speed was measured both at high frequency (90–170 kHz) and low frequency (0.5–3 kHz) in laboratory environments. At high frequency, a sampling measurement was conducted with boiled and uncooked sand samples collected from the bottom of a large water tank. The sound speed was directly obtained through transmission measurement using single source and single hydrophone. At low frequency, an in situ measurement was conducted in the water tank, where the sandy sediment had been homogeneously paved at the bottom for a long time. The sound speed was indirectly inverted according to the traveling time of signals received by three buried hydrophones in the sandy sediment and the geometry in experiment. The results show that the mean sound speed is approximate 1710–1713 m/s with a weak positive gradient in the sand sample after being boiled (as a method to eliminate bubbles as much as possible) at high frequency, which agrees well with the predictions of Biot theory, the effective density fluid model (EDFM) and Buckingham’s theory. However, the sound speed in the uncooked sandy sediment obviously decreases (about 80%) both at high frequency and low frequency due to plenty of bubbles in existence. And the sound-speed dispersion performs a weak negative gradient at high frequency. Finally, a water-unsaturated Biot model is presented for trying to explain the decrease of sound speed in the sandy sediment with plenty of bubbles.


Introduction
Measurements of sound speed in marine sediments are usually conducted with unconsolidated and well-sorted sand. For the unconsolidated sediments, the "sound speed" usually refers to the phase speed of the compressional wave, or the speed of the fast wave in Biot theory (Biot, 1956a(Biot, , 1956b. By all appearances, marine sediments, especially unconsolidated sand, are essentially porous in actual marine environments. Because the motion of fluid and granular phase may be different in response to an acoustic excitation, which will lead to varying of sound speed with the frequency, marine sediment is a kind of dispersion medium. The significance of study on sound-speed dispersion in marine sediments lies in at lest two aspects. On the one hand, if the dispersion is very strong, it should specifically present a method to deal with sound-speed dispersion in acoustic models or geoacoustic data. On the other hand, because the sound speed in sediments determines the critical grazing angle, the sound-speed dispersion is very important for sonar performance in detecting buried objects. Especially for classing the buried objects based on physical scattering models, the mismatch of sound speed may lead to mistake of the detection system in identifying the type of buried objects. It is remarkable that the sound-speed dispersion and the dependence of attenuation on frequency are closely linked together under the causality condition. They should be subjected to the Kramers-Kronig relationships (Horton, 1974(Horton, , 1981Donnell et al., 1981). The level of dispersion implies the relationships between attenuation and frequency, and vice versa. The dispersion depends on the value of attenuation and its relation to the frequency.
Measurements of sound speed in unconsolidated sediments have been carried out since the 1950's. The available data covered a broad range of frequency, from a few hertz to megahertz. To get these data, many kinds of techniques were employed, including in situ measurements (Thorsos et al., 2001;Turgut and Yamamoto, 2008;Zimmer et al., 2010), laboratory measurements (Wingham, 1985;Sessarego et al., 2008) and remote sensing measurements (Holland et al., 2005;Zhou and Zhang, 2009). The key is how to minish the measurement uncertainty as much as possible, because the sound-speed dispersion is very weak in most of sediments. Fluid and elastic (or viscoelastic) theories are both based on a single phase system, which views the sound speed as independent of frequency. Therefore, they are not suitable for predicting sound speed in sediments. The main methods that can be used to predict sound speed in sediments include Hamilton's empirical relationships (Hamilton, 1980), Biot theory (Biot, 1956a(Biot, , 1956b and Buckingham's theory (Buckingham, 1997(Buckingham, , 1998(Buckingham, , 2000(Buckingham, , 2005. To test the applicability of these methods, it is necessary to obtain accurate data of sound speed. However, many kinds of gasses may be mixed into sediments, suffering from various effects of physics and chemistry in actual marine environments. Then the sediments become a three-phase system consisting of gasses, liquids and solids, where water-saturated models of acoustic propagation in sediments are not applicable any more. Furthermore, the influence of bubbles on sound speed is usually not taken into account in in-situ measurements when the bubbles are few, or the bubbles are eliminated as much as possible through boiling the sediment samples in laboratory measurements. Recently, some investigators have concerned on the influence of plenty of bubbles at high frequency. It is found in Li et al. (2015) that the sound speed performs enormous decrease, but the volume percentage of the gas seems to be too large, which may be the result of destruction in collection of sediment samples. While the data in Tao et al. (2010) showed little inference of bubbles on the sound speed. In this paper, much attention is paid to the influence of plenty of bubbles in existence on sound-speed dispersion in sandy sediment both at high and low frequencies.
In this paper, Section 2 provides the detail of sound speed measurements with a boiled and an uncooked sand sample at high frequency. The low-frequency measurements are in Section 3, including an inverse method, a direct measurement and a sampling measurement with a boiled sample. Data analysis and discussions are presented in Section 4. Finally, conclusions and suggestions are given in Section 5.

Measurements of sound-speed dispersion at high frequency
In this section, the sound speed is obtained through comparing the received signals with and without a sand sample (collected from the bottom of the large water tank shown in Section 3) filled in a glass container. 2.1 With a boiled sand sample As shown in Fig. 1, the whole equipment was posi-tioned in a glass tank with the size of 1.2 m×1.0 m×1.0 m, where the depth of pure water was 0.82 m. Two glass containers with the same horizontal dimension but different heights were secured on a horizontal supporting frame. The glass containers had an overall length and width of 0.34 m and a wall thickness of 4.6 mm, giving an effective horizontal dimension of 0.30 m×0.30 m relative to the frame. The inner heights of the two glass containers were 99.8 mm and 199.7 mm, respectively. The higher one was filled with a boiled sand sample to eliminate the bubbles as much as possible, and allowed to be sit for 22 days. By this time, the observed sound speed was basically steady and subsequently the normal measurements were carried out. The surface of the sand sample was scraped even with the glass container to keep the thickness of the sand sample and the inner height of the glass container approximately as the same. The choice of thickness of the sand sample was based on two considerations. First, the thickness should be several times the acoustic wavelength, so that a steady transmission of sound wave can be established and the sound-speed dispersion can be clearly exhibited. Second, the direct wave and reflective wave from the lower and upper surface of the sand sample should not overlap in time domain. The source and hydrophone (B&K8105) were fixed on a slip frame face to face with a distance of 491 mm. The source was 80 mm above the sand surface. And the hydrophone was located 195 mm above the upper surface of the slip frame, 300 mm above the bottom of the glass tank and 210 mm below the bottom of the glass containers. With this configuration, the pulse length of the transmitted signal must be selected carefully to make sure as far as possible that the direct wave and multipaths do not overlap in time domain. The main multipaths that need to be considered include reflective waves from the active face of the source, the bottom of glass containers and the upper surface of the slip frame. Fortunately, the -3 dB beam width of the source is about 10°, which allows the sound wave to avoid ensonifying the wall of glass containers and reduces the disturbance of multipaths. A series of CW pulses with 5 cycles weighted by Blackman window were transmitted. The central frequency varied from 90 kHz to 170 kHz with a step of 10 kHz, and the repetition rate was 1 pulse/s. Synchronously, the received signals were filtered with a pass band of 10-230 kHz, collected by a PCI data acquisition card and stored in a PC. The sampling rate was 25 MHz.
Firstly, the received signals through only the glass container (without the sand sample) were recorded in advance. Then the source and hydrophone were moved to the side of the sand sample, and the corresponding signals were recorded. For the sake of making a statistical analysis of the sound speed, the received signals were collected 30 times for every condition.
The received signals without and with the sand sample for each frequency are shown in Fig. 2. It is clear that the direct wave and other multipaths are basically separated in time domain. Fig. 2 also indicates that the sound speed in the boiled sand sample is faster than that in water.
The time difference of received signals with and without the sand sample is denoted by t and the thickness of the sand sample is denoted by d, then the sound speed can be calculated as: represents the sound speed in water. The sound 0 ± C < T < 95 ± C speed of pure water can be calculated using the following formula with the temperature range of (Grosso, 1972) c w (T) = 1402:388+5:03711T¡0:0580852T 2 +3:342£ The uncertainty of sound speed determined by Eq.
(2) is ±0.015 m/s. During the experiment, the temperature was 23.6°C, so the corresponding sound speed in water was 1492.87 m/s.
Because the sound-speed dispersion over the measurement frequency band is not strong, the distortion of signal waveform arising from the dispersion properties of the sandy sediment is not prominent. The time delay can be determined according to the position of correlation function peak. The measured sound speed is shown in Fig. 3. The mean values vary approximately from 1710 m/s to 1713 m/s with an uncertainty of about 4.5 m/s. It is found that the sound-speed dispersion in the sand sample is very weak over the measurement frequency band, which is smaller than the measurement uncertainty. Here, the uncertainty of the measured sound speed derives from the uncertainty of the measured thickness of the sand sample and the uncertainty in estimation of the time difference. According to the transfer equation of uncertainty The uncertainty of measured sound speed can be obtained from Eq. (1) and Eq. (3) as follows: Eq. (4) implies the assumption that the uncertainty of time difference and thickness are independent. In view of the slight extrusion by scraping the surface of the sand sample and further deposition after scraping, it is believed that the uncertainty of the measured thickness of the sand sample is about 0.5 mm. The uncertainty of time difference includes the uncertainty of the time-delay estimation derived from the position of correlation function peak, denoted by , and the statistical uncertainty (standard variance of processing results for 30 pings). The total uncertainty of the time difference is ¢t = q (¢t 1 ) 2 + (¢t 2 ) 2 ; (5) ¢t 1 where is determined by the calibration experiment conducted in the water. It is found that the uncertainty of timedelay estimation is about 0.3 μs.

With an uncooked sand sample
Because the collection was not conducted in water environments, which leaded to be gas-bearing sediment, plenty of bubbles were formed in the sand sample. In order to demonstrate the influence of plenty of bubbles on the sound speed at high frequency, another sand sample (wet but not water-saturated) was measured immediately without being cooked. The measurement system and method were the same as in Section 2.1. The thickness of the sand sample was 32.2 mm. The temperature during the experiment was 24.4°C, and the sound speed in water was 1495.07 m/s obtained from Eq. (2). The received signals without and with the sand sample for each frequency are shown in Fig. 4. It is found that the direct wave through the sand sample is very weak, which implies that the attenuation of the uncooked sand sample is enormous. And the sound speed is much lower than that in water. The measured sound speed according to the received signals is shown in Fig. 5, which indicates that the sound speed in the sand sample obviously decreases at high frequency. Furthermore, the sound-speed dispersion performs a weak negative gradient with the frequency.

Measurements of the sound-speed dispersion at low frequency
In this section, three different measurements of the sound speed in the sandy sediment at low frequency are presented, including an inverse method, a direct measurement and a sampling measurement with a boiled sample.
The configurations in different measurements are shown in  p where y 1 , y 2 , y 3 and c 1 are unknown variables that can be obtained from optimization algorithm. The measurement frequency band was selected in the range of 0.5-3 kHz according to the frequency response of the source. CW pulses with a length of 2 ms were adopted as the transmitted signals. Different numbers of cycles were filled for different central frequencies to guarantee the same pulse length and signal bandwidth. Over the measurement frequency band, the sound-speed dispersion is relatively prominent from the view of model predictions. So the form of the transmitted signal is selected for the sake of obtaining narrow signal bandwidth as much as possible to make sure that the measured sound speed would approximately represent the value at its center frequency. The pulses were transmitted each second and the data were collected synchronously. The sampling rate was 2 MHz.
Firstly, a series of reference signals were recorded in advance. The source and hydrophone #4 (B&K8105) were located at the center of water column in depth and width direction, keeping face to face. The distance between them was 0.7 m. The received signals at 0.5 kHz, 1 kHz, 2 kHz and 3 kHz were selected as the reference signals. The received signals of the three buried hydrophones are shown in Fig. 8. It is found that the received signals at 0.5 kHz, 1 kHz, 2 kHz and 3 kHz can be observed. Therefore, the received signals at above four frequency points were only used to  YU Sheng-qi et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 103-113 calculate the sound speed at corresponding frequencies. The arrival time of the received signals is obtained from the time difference with the reference signals and their arrival time to eliminate the electronic delay of the equipments as far as possible. Because the time-delay resolution of signals is relatively small due to the relatively narrow bandwidth, the calculation of time difference is based on the zero crossing detection rather than the correlation estimation.
According to the arrival time of the received signals in Fig. 8 and the predicted sound speed at low frequency (derived from fitting the measured data at high frequency in Fig. 3 based on Biot theory) and the geometry in experiment, a large time lag between the actual arrival time and the theoretical value can be found. The main probable reasons for this phenomenon lie in: (1) the effect of wave guide is remarkable in the experiment environment, and the measured sound speed corresponds to the group speed of the normal mode rather than the intrinsic sound speed of the medium.
(2) The sound speed in the sandy sediment at the bottom of the water tank is much lower than the theoretical predictions.
In order to estimate the effect of wave guide in the experimental environment, the waveforms of the three buried hydrophones are predicted at 2 kHz based on the fast field method according to the geometry and the predicted sound speed at low frequency. These predicted waveforms in time domain are presented in Fig. 9. Intercept partial reference signals (Make the time delay zero and abandon the part of overlap multipaths.) as the transmitted signals in predicting waveforms to guarantee the accuracy. Besides the geometric parameters, the values of sound speed in the water and sandy sediment are set to 1450 m/s and 1580 m/s, and their densities are 1000 kg/m 3 and 1800 kg/m 3 , respectively. The simulation results indicate that the waveforms of the measured signals perform somewhat widening, while the arrival time has a large lag compared with the predicted waveforms as well. Therefore, the effect of wave guide is not the main cause to bring the remarkable time delay of the received signals, and the sound speed in the sandy sediment at low frequency may be also much lower than the theoretical values.

Direct measurement
To further confirm that the sound speed in the sandy sediment at the bottom of the water tank is much lower than the theoretical values, a direct measurement is conducted as follows. The source was suspended vertically to make its radiation surface face the bottom. Then the source was moved over hydrophone #1, and hydrophone #4 was placed on the sand surface over hydrophone #1. The distance between the source and hydrophone #4 was about 0.3 m. This configuration is shown in Fig. 6b. The temperature during the experiment was 14.3°C, and the sound speed in water was 1463.46 m/s obtained from Eq. (2). The received signals of the two hydrophones at 2 kHz is shown in Fig. 10a, where the gain of the received signals of hydrophone #1 is ten times that of hydrophone #4, and other conditions are the same. The sound speed was obtained from the distance and the time difference of the received signals of hydrophone #1 and hydrophone #4.

Sampling measurement with the boiled sample
In the last case, the measurement of the sound speed in a boiled sand sample at low frequency was conducted as follows. A certain volume of boiled sand sample was placed in a cylindric container with a diameter of 0.34 m. The container has been located on the bottom of the water tank for several days to make sure that the sand sample was steady. Hydrophone #4 was placed at the bottom of the container in  advance, and hydrophone #5 was placed on the surface of the sand sample with a distance of 0.125 m above hydrophone #4. The radiation surface of the source faced the two hydrophones with a distance of about 0.3 m above hydrophone #5. This configuration is shown in Fig. 6c. The temperature during the experiment was 15.7°C, and the sound speed in water was 1468.36 m/s obtained from Eq. (2). The received signals of the two hydrophones at 2 kHz is shown in Fig. 10b, where the gain of the received signals of hydrophone #1 is ten times that of hydrophone #4, and other conditions are the same. Furthermore, the received signals can still be observed at higher frequencies as shown in Fig. 11. The sound speed was obtained from the distance and the time difference of the received signals of hydrophone #4 and hydrophone #5.

Data analyses and discussions
This section provides the comparison of measured sound speed with predictions of various theories and achieves some valuable conclusions.
Firstly, the measured sound speed in the boiled sand sample at high frequency is compared with the predictions of Biot theory, the EDFM and Buckingham's theory. The parameters of pore water involved in various theories refer to the values of pure water in Hefner and Williams (2006) as known. And other unknown parameters are obtained   YU Sheng-qi et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 103-113 through fitting to the measured sound speed. The fitted results are shown in Fig. 12. The sound speed is given in the form of the speed ratio of sediment/water. Thus, the sound speed in sediments under different temperatures can be calculated according to the corresponding sound speed in water. The search ranges and the values for best fitting to the measured sound speed are listed in Table 1. For Biot theory, the permeability, the pore size parameter and the tortuosity are expressed by the porosity and the mean grain size according to Schock (2004) in the process of searching. The porosity, the mean grain size, the mass density of grains and the bulk density of sediments were measured. The measured confidence intervals (mean value ± standard variance) are taken as the search ranges. Unmeasured parameters, including the bulk modulus of grains, the bulk modulus of frame and the shear modulus of frame, are set to larger search ranges referring to Williams et al. (2002). In the ED-FM, the bulk modulus of frame and the shear modulus of frame are set to zero (Williams, 2001). For Buckingham's theory, the strain-hardening index, the relaxation coefficient, the compressional wave speed given by Wood's equation and the bulk density of sediments are viewed as four independent parameters to search. Except for the bulk density of sediments, the other three parameters are set to larger search ranges referring to Williams et al. (2002). The fitted results show that the EDFM is a reasonable simplification and approximation to Biot theory, because the model predictions are basically the same. With the best values, the predictions of Biot theory, the EDFM, and Buckingham's theory agree well with the measured sound speed in the boiled sand sample at high frequency.
Then the measured sound speed at low frequency is shown in Fig. 13. Because the temperature in low-frequency and high-frequency measurement is different, the measured results are also presented in the form of the speed ratio of sediment/water to reduce the influence of temperature. The results of the inverse method and direct measurements are shown as the circular data point with error bars and the square data point in Fig. 13, respectively. For the inverse method (as an in situ indirect measurement), the measured sound speed are presented in the form of statistical results of 20 estimations of the optimum value (mean ± standard variance), according to the arrival time of the three buried hydrophones and the geometry on the basis of Eqs. (6)-(11). Compared with the results of the sampling measurement at high frequency, the measurement uncertainty of sound speed with the inverse method is larger, which mainly derives from the uncertainty of arrival time estimation of the received signals and the uncertainty of optimization algorithm in optimizing process. One can see that the results of direct and indirect measurements are coincident, and the sound speed in the sandy sediment at low frequency is indeed much lower than predictions of various theories.
In the following, the question is why the sound speed obviously decreases at low frequency. The answer is the existence of plenty of bubbles in the sandy sediment. For the process of burying the hydrophones is not conducted in water-saturated environment, it is difficult to avoid mixing air into the sediment. Then plenty of bubbles were formed. As illustrated in Fig. 8, there is a prominent decrease of the amplitude of the received signals with the increase of frequency. However, the attenuation in the sandy sediment at low frequency is rather small in theory. This contradiction is just the evidence of the existence of plenty of bubbles in the sandy sediment. Plenty of bubbles make the attenuation in sediments enormously increase, and it is further improved with the increase of frequency. Especially for high-frequency sound wave, the existence of bubbles will enormously reduce the transmission depth of sound wave, which is also the reason why the sand sample should be boiled before being measured at high frequency.
It is evident that the bubbles can be well eliminated through boiling and string the sand sample. For the boiled sand sample at low frequency, the amplitude of the received signals obviously increases, and the measured sound speed (shown as the triangular data point in Fig. 13) agrees well with the predictions of Biot theory and the EDFM, but lower than the prediction of Buckingham's theory. Here, be-  sides the data error, (including the theory error, the estimation error of arrival time of the received signals and the measurement error of geometric parameters in experiment), the bias between the measured data at low frequency and the predictions of various theories also probably derives from the difference of measurement environment, including the deposition state of the sandy sediment, the temperature in experiment and so on.
The sound speed in the sandy sediment with plenty of bubbles is much lower than the predictions of various theories both at high frequency (shown as the green data point in Fig. 13) and low frequency. Subsequently, we try to explain the phenomenon in theory by a simple way. Biot theory, the EDFM and Buckingham's theory are all based on single or double phase system, but the sediments are also probable a three-phase system made up of gasses, liquids and solids in the actual marine environments, where these models of acoustic propagation in sediments are not applicable any more. If we simply consider that the bubbles, the pore water and the solid grains are uniformly mixed together, and the gas in bubbles and the pore water are equivalent to one kind of fluid, the equivalent density and the equivalent bulk modulus can be obtained as (Berryman and Thigpen, 1985): When the density of gas in bubbles is ρ a =1.20 kg/m 3 , the bulk modulus is K a =8.92×10 4 Pa, the volume percentage of the gas β a is set to different values and and are used in Biot theory (we call it "water-unsaturated Biot model"), the prediction results are shown in Fig. 14. It indicates that the existence of bubbles can lead to a prominent decrease of sound speed in sediments, which is lower than that in water.With the increase of bubbles volume, the sound speed decreases more obviously and its dispersion gets weak gradually. According to the measured sound speed in the sandy sediment with plenty of bubbles both at high frequency and low frequency and the parameters listed in Table 1(a), the parameters related to the bubbles can be obtained through fitting the measured data with the water-unsaturated Biot model. They are ρ a =1.286 kg/m 3 , K a =5.07×10 4 Pa and β a =0.052%. The fitted result is shown in Fig. 13 as the solid line of red color, where the sound speed in water is 1490 m/s. Finally, it needs to point out that the sound speed predicted by various theories is the phase speed. As the contrast between the phase speed and the group speed is relatively small and the accuracy of low-frequency measurement is not high, this contrast is neglected in the case without special instruction.

Conclusions
The sound speed in a kind of sandy sediment was measured both at high frequency (90-170 kHz) and low frequency (0.5-3 kHz) in laboratory environments. The trend of sound-speed dispersion over the frequency range can be obtained according to the measured data. At high frequency, the sound speed was directly obtained through transmission measurement using single source and single hydrophone. At low frequency, the sound speed was indirectly inverted according to the traveling time of signals received by three buried hydrophones in the sandy sediment and the geometry in experiment. A direct measurement was further conducted to confirm the indirect measured data. In the boiled sand sample, the mean sound speed is approximately in the range of 1710-1713 m/s with a weak positive gradient at high frequency, which agrees well with the prediction of Biot theory, the EDFM and Buckingham's theory. The sound speed becomes a little lower at low frequency, which is consistent with the predictions of Biot theory and the EDFM but lower than the prediction of Buckingham's theory. However, the sound speed in the uncooked sandy sediment decreases obviously (about 80%) due to plenty of bubbles in existence both at high frequency and low frequency. It can be interpreted by a water-unsaturated Biot model with the equivalent density and the equivalent bulk modulus of pore fluid instead. That is, the prominent decrease of sound speed is mainly attributed to the decrease of the bulk modulus of pore fluid. The sound-speed dispersion performs a weak negative gradient in the uncooked sand sample at high frequency. To predict the sound speed in sediments with bubbles more accurate, the mechanism of acoustic propagation in water-unsaturated sediments should be further investigated and a more universal model of acoustic propagation in sediments needs to be established. And the accuracy of low-frequency measurement should be further improved with a larger sand sample.