Acoustic wave reflection from the transition layer of surficial marine sediment

Acoust. Sci. & Tech. 25, 3 (2004) PAPER Acoustic wave reflection from the transition layer of surficial marine sediment Masao Kimura and Takuya Tsurumi School of Marine Science and Technology, Tokai University ( Received 13 March 2003, Accepted for publication 2 October 2003 ) Abstract: It has been recently reported that the physical properties such as the porosity and the density in the surficial marine sediment vary largely with the depth. The characteristics of acoustic wave reflection from such transition layer of the surficial marine sediment seem to be very different from that from homogeneous sediment. In this study, the incident angle dependence of the reflection coefficient for the transition layer of the surficial marine sediment model is calculated using OASES (Biot-Stoll model). The effects of the transition layer and the frequency dependence of the reflection coefficient characteristics are investigated. Next, the characteristics of the incident angle for the reflection coefficient are measured in water tank and in situ. Beach sand (fine sand) are used for water tank measurements, and in situ measurements are done in a very fine sand bottom. The operating frequency is 150 khz. The incident angles are 0, 20, 50, and 60 degrees. The measured results are compared with the calculated results using OASES (Biot-Stoll model), and the effects of the transition layer on the characteristics of the reflection coefficient are investigated. Keywords: Reflection, Marine sediment, Transition layer, Porosity, Velocity PACS number: 43.30.Ma [DOI: 10.1250/ast.25.188] 1. INTRODUCTION To measure the characteristics of surficial marine sediment acoustically, it is required to examine the acoustic wave reflection characteristics from the surface of the surficial marine sediment in detail. The reflection characteristics from the surface of the homogeneous watersaturated sediments have been reported [1,2]. It has been recently reported that the physical properties such as the porosity and the density in the surficial marine sediment vary significantly with the depth [3 5]. The characteristics of the acoustic wave reflection from such transition layer of the surficial marine sediment seem to be very different from that from homogeneous sediments. Few reports are available on the calculated results of the reflection coefficients from the surface of the sediments with the top transition layer using the Biot-Stoll model other than Stern et al. s paper [6]. Acoustical analyses and measurements of surficial marine sediment with the top transition layer are very important to evaluate the coastal environments. In this study, the reflection characteristics from the surface of the sediments with the top transition layer depending on the frequency are obtained using OASES (Biot-Stoll model) [7]. The effects of the transition layer on e-mail: mkimura@scc.u-tokai.ac.jp the reflection coefficients are investigated. Next, the characteristics of the incident angle for the reflection coefficient are measured in water tank and in situ. Beach sand (fine sand) is used for water tank measurements, and in situ measurements are done in a very fine sand bottom. The measured results are compared with the calculated results using OASES (Biot-Stoll model). 2. REFLECTION COEFFICIENTS FOR PLANE ACOUSTIC WAVE Sediments are fluid-saturated porous media which are composed of the frame, which is the aggregate of the grain, and the pore fluid, which fills the pore. The Biot-Stoll model [8 11], which is the acoustic wave propagation model in fluid-saturated porous media, is used for modeling sediments. The wave equations for fluid-saturated porous media are expressed as follows [12] r 2 u þðh Þ½rðr uþš C½rðr wþš ¼ @2 u @t @ 2 w 2 f @t ; ð1þ 2 C½rðr uþš M½rðr wþš @ 2 u ¼ f @t f @ 2 w 2 @t F @w 2 k @t ; ð2þ 188

M. KIMURA and T. TSURUMI: ACOUSTIC WAVE REFLECTION FROM MARINE SEDIMENT s ¼ A 1 exp½jð!t k 1p cos 1 z k w sin xþš expðk 1a zþ þ A 2 exp½jð!t k 2p cos 2 z k w sin xþš expðk 2a zþ; ð8þ f ¼ B 1 exp½jð!t k 1p cos 1 z k w sin xþš expðk 1a zþ þ B 2 exp½jð!t k 2p cos 2 z k w sin xþš expðk 2a zþ; ð9þ s ¼ C exp½jð!t k sp cos s z k w sin xþš expðk sa zþ; ð10þ f ¼ D exp½jð!t k sp cos s z k w sin xþš expðk sa zþ; ð11þ Fig. 1 Incident and reflected potentials in seawater and refracted potentials in marine sediment. w ¼ ðu UÞ; where, u is the displacement of the frame, U is the displacement of the fluid, w is the relative displacement of the frame to the fluid, is the shear modulus, is the porosity, is the density of the sediment, f is the density of the fluid, k is the permeability, is the structure factor, is the viscosity, F is the viscous correction factor, and H, C, M are the constants determined by the moduli of the grain, the fluid and the frame. As shown in Fig. 1, we consider a plane longitudinal wave incident at an angle to a sediment half-space at z ¼ 0. There exist incident and reflected waves in the seawater, and three refracted waves the first kind of longitudinal wave (fast wave), the second kind of longitudinal wave (slow wave), and the shear wave, in the sediment. The incident and reflected waves in the seawater will have the displacement potentials, i ¼ A i exp½jð!t k w cos z k w sin xþš; r ¼ A r exp½jð!t þ k w cos z k w sin xþš; where k w ¼!=c w.! is the angular frequency, and c w is the sound velocity in the seawater. Two displacements u and w can be represented by scalar potentials s and f, and vector potentials s, f, u ¼r s þ rot s ; ð3þ ð4þ ð5þ ð6þ where k 1p and k 2p are the equal phase wave numbers for the first and second kinds of longitudinal waves and k 1a and k 2a are the equal amplitude wave numbers. k sp and k sa are the equal phase and equal amplitude wave numbers for the shear wave. The relationships between the complex amplitudes A 1, A 2, B 1, B 2, C, and D are Hk l1 2 B 1! 1 ¼ H 2 2 A 1 CHk l1 ; ð12þ fh! 2 Hk l2 2 B 2! 1 ¼ H 2 2 A 2 CHk l2 fh! 2 D C ¼ f 1 k s 2! 2 ; ð13þ ; ð14þ where k l1, k l2, and k s are the complex wave numbers for the first and second kinds of longitudinal waves and the shear wave, and H ¼ ðk r K b Þ 2 þ K b þ 4 ; ð15þ D K b 3 C ¼ K rðk r K b Þ ; ð16þ D K b D ¼ K r 1 þ K r 1 ; ð17þ K f where K f, K r, K b are bulk moduli of the pore fluid, the grain and the frame, respectively. The following four boundary conditions are required at a seawater-sediment interface. (1) For continuity of fluid movement w ¼r f þ rot f : In the sediment, the scalar and the vector potentials defined in Eqs. (6) and (7) are ð7þ @ i @z þ @ r @z ¼ @ s @z þ @ s @x @ f @z @ f @x : (2) For equilibrium of normal traction ð18þ 189

Acoust. Sci. & Tech. 25, 3 (2004) H @2 s @x þ @2 s 2 @2 s 2 @z 2 @x 2 C @2 f @x þ @2 f ¼ 2 @z 2 f @2 s @x@z @ 2 i @t 2 (3) For equilibrium of fluid pressure where M @2 f @x þ @2 f C @2 s 2 @z 2 @x 2 @ 2 i ¼ f @t þ @2 r ; 2 @t 2 þ @2 r @t 2 þ @2 s @z 2 : ð19þ ð20þ M ¼ K r 2 : ð21þ D K b (4) For equilibrium of tangential traction 2 @2 s @x@z @2 s @z @2 s ¼ 0: ð22þ 2 @x 2 By combining Eqs. (4), (5) and (8) (22), the following four linear complex equations can be obtained. 0 10 1 0 1 c 11 c 12 c 13 c 14 A r Y 1 c 21 c 22 c 23 c 24 A 1 B CB C @ c 31 c 32 c 33 c 34 A@ A 2 A ¼ Y 2 B C @ Y 3 A : ð23þ c 41 c 42 c 43 c 44 C Y 4 In Eq. (23), the components of fcg and fyg are given by the physical parameters of seawater and sediment. A r is the complex amplitude of the reflected wave. Therefore, the reflection coefficient can be obtained by solving Eq. (23), once A i is specified. 3. CALCULATIONS OF REFLECTION COEFFICIENTS FOR PLANE ACOUSTIC WAVE The physical parameters of the water-saturated sediment whose upper layer is seawater are shown in Table 1. In these calculations, the sound velocity in the seawater is assumed to be 1,500 m/s. Recently, it has been reported that the physical properties such as the porosity and the density in the surficial marine sediment vary largely with the depth [3 5]. The characteristics of the acoustic wave reflection from such a transition layer of the surficial marine sediment seem to be very different from that from homogeneous sediment. Thus, the reflection characteristics from the surface of the sediments with the top transition layer depending on the frequency are calculated [13,14]. The equation for porosity variation with the depth is assumed by referring Carbo s paper [15] as follows, ¼ min þð1 min Þ expð z 0:75 Þ; ð24þ where, min is the minimum value of the porosity, and the values of are assumed to be 1.0 (for model 1) and 2.0 (for model 2). These values of is reasonable values for the sediment types from sand to silt and clay [3,4]. The characteristics of porosity variation with depth are shown in Figs. 2 3. In these figures, the porosity at a depth below 20 cm is constant, and the value is 0.62 for silt model and 0.35 for medium sand model. According to the change of the porosity with the depth, the bulk and the shear moduli of the frame [16], the permeability and pore size parameter are changed using the following equations. Table 1 Physical parameters of marine sediment models. Physical parameters Silt Medium sand Grain Diameter 5.5 1.5 d (mm) 0.0221 0.354 Density r (kg/m 3 ) 2,650 2,650 Bulk modulus K r (Pa) 3:60 10 10 3:60 10 10 Pore fluid Density f (kg/m 3 ) 1,000 1,000 Bulk modulus K f (Pa) 2:25 10 9 2:25 10 9 Viscosity (Pas) 1:00 10 3 1:00 10 3 Frame Porosity 0.62 0.35 Permeability k (m 2 ) 4:48 10 12 7:06 10 11 Pore size a (m) 1:20 10 5 6:35 10 5 Structure factor 1.25 1.25 Bulk modulus K br (Pa) 1:43 10 7 5:62 10 7 Bulk logarithmic 0.15 0.15 decrement Shear modulus r (Pa) 4:34 10 6 8:46 10 6 Shear logarithmic s 0.15 0.15 decrement 190

M. KIMURA and T. TSURUMI: ACOUSTIC WAVE REFLECTION FROM MARINE SEDIMENT Fig. 2 Porosity versus depth for transition layer of silt model. is for medium sand model. The above frequencies are selected by considering the interrelation between the wavelength of the acoustic wave and the thickness of the transition layer. The reflection coefficients and the phase angles for the homogeneous and for the transitional sediments at the frequency of 1.5 khz are almost the same. However, the reflection coefficients for the transitional sediment at the frequencies of 15 khz and 150 khz are smaller than that for the homogeneous sediment and the phase angles are different each other, for both silt and medium sand models. This seems to be due to the fact that the effective depth to the reflection coefficient decreases as the frequency increases, and the porosity increases as the depth decreases. The vibrated characteristics in Fig. 4(b) and Fig. 5(a), (b) at the frequency of 150 khz is seems to be due to the insufficient numbers for layers. From these calculated results, it is expected to be possible to estimate the types of sediments and the porosity profile with the depth from the measured results of the angle of incidence characteristics of the reflection coefficients. Fig. 3 Porosity versus depth for transition layer of medium sand model. log K br ¼ 2:20 þ 8:52; ð25þ log r ¼ 0:265 þ 7:07; 1 ð26þ k ¼ d2 3 36k 0 ð1 Þ 2 ðk 0 ¼ 5Þ; ð27þ a ¼ d 3 1 : ð28þ It is difficult to calculate the reflection coefficients for the sediment whose porosity continuously varies with the depth such as the transition layer [6]. Thus, these calculation are performed by separating the layer into thin layers and forming numerous layers using OASES (Biot- Stoll model) [7]. The calculated results using OASES are ja r =A i j for the transitional sediment. In this calculation, the transition layer whose thickness is 20 cm, is divided into 61 layers. Below the depth of 20 cm, the porosity is constant, that is the medium below the depth consists of semi-infinite sediment. The calculated results for the incidence angle characteristics of the reflection coefficients and the phase angles at the frequencies of 1.5 khz, 15 khz and 150 khz are shown in Figs. 4 5. Figure 4 is for silt model, and Fig. 5 4. EXPERIMENTS FOR ACOUSTIC WAVE REFLECTION 4.1. Tank Experiments Beach sand with the diameter of 0.22 mm is used as a medium. This sand is set in the bottom of the tank whose size is 2 3 2 m, as shown in Fig. 6. The thickness of the sand is 42 cm. The measuring system is set on the sand. Four transmitting and four receiving transducers are fixed in the circular frame. The transducers are piezoelectric transducers whose resonant frequency is 150 khz, and the diameter is 80 mm. The propagation distance is 1 m. The incident angles are 0,20,50,60. The temperature in water is 22 C. AC pulse whose frequency is 150 khz and whose duration is 0.2 ms, is applied to the transmitting transducer. The received signal by the receiving transducer is inputted to a digital oscilloscope, averaged 64 times, and measured the maximum amplitude of the signal. The amplitude is equivalent to that for the sinusoidal wave. The characteristics of the porosity versus the depth are shown in Fig. 7. In Fig. 7, the measured result and the regressive curve are shown. The porosity was measured for each 4 cm-long sample from the top of the sediment, after taking sample using a GS-type corer. The regressive equation is as follows, ¼ 0:380 þð1 0:380Þ expð 2:79z 0:262 Þ; ð29þ where, z (cm) is the depth from the surface of the sediment. The measured and the calculated results of the incident angle characteristics of the reflection coefficient are shown in Fig. 8. In Fig. 8, there are 3 measured data for the different position of transmitter, for each angle of incidence. The physical parameters of beach sand are shown in 191

Acoust. Sci. & Tech. 25, 3 (2004) Fig. 4 Reflection coefficient versus angle of incidence (a) and phase angle versus angle of incidence (b), for silt model. Table 2. From Fig. 8, it is seen that the measured result of the reflection coefficients is almost the same as the calculated result for the transitional sediment. 4.2. In situ Experiments In situ experiments were performed in the Shimizu harbor with the water depth is about 10 m. The surficial marine sediment is very fine sand whose mean diameter is 0.087 mm. The measuring system same as the system for tank experiments was taken dowm from a small ship, and set on the surface of the bottom. The incident angles are 20,50, and 60. The characteristics of the porosity versus the depth are shown in Fig. 9. The porosity was measured for each 4 cmlong sample from the top of the sediment, after taking sample using a GS-type corer. The regressive equation is as follows, ¼ 0:562 þð1 0:562Þ expð 0:874z 0:456 Þ: ð30þ The measured and the calculated results of the incident angle characteristics of the reflection coefficient are shown in Fig. 10. The physical parameters are shown in Table 2. From Fig. 10, it is seen that the measured result of the reflection coefficients almost agree with the calculated result for the transitional sediment. 5. CONCLUSIONS The reflection coefficient from the transition layer of surficial marine sediment was calculated using OASES (Biot-Stoll model). It is seen that the reflection coefficients for the homogeneous sediment and for the transitional sediment at the frequency of 1.5 khz are almost the same. However the reflection coefficients for the transitional sediment at the frequencies of 15 khz and 150 khz are smaller than that for the homogeneous sediment. The 192

M. KIMURA and T. TSURUMI: ACOUSTIC WAVE REFLECTION FROM MARINE SEDIMENT Fig. 5 Reflection coefficient versus angle of incidence (a) and phase angle versus angle of incidence (b), for medium sand model. Fig. 6 Experimental setup. measured results of the reflection coefficients from the beach sand in tank experiments and very fine sand in in situ experiments were almost agreed with the calculated results from the sediment with the transitional sediments. Fig. 7 Porosity versus depth for transition layer of beach sand (fine sand) in tank. 193

Acoust. Sci. & Tech. 25, 3 (2004) Table 2 Physical parameters of beach sand and seabed sand. Physical parameters Beach sand Seabed sand Grain Diameter 2.2 3.5 d (mm) 0.22 0.087 Density r (kg/m 3 ) 2,558 2,560 Bulk modulus K r (Pa) 3:60 10 10 3:60 10 10 Pore fluid Density f (kg/m 3 ) 1,000 1,000 Bulk modulus K f (Pa) 2:25 10 9 2:25 10 9 Viscosity (Pas) 1:00 10 3 1:00 10 3 Frame Porosity 0.380 0.562 Permeability k (m 2 ) 3:94 10 11 3:89 10 11 Pore size a (m) 4:56 10 5 3:72 10 5 Structure factor 1.25 1.25 Bulk modulus K br (Pa) 4:83 10 7 1:92 10 7 Bulk logarithmic 0.15 0.15 decrement Shear modulus r (Pa) 8:08 10 6 5:37 10 6 Shear logarithmic s 0.15 0.15 decrement Fig. 8 Reflection coefficient versus angle of incidence for beach sand (fine sand) in tank. Fig. 10 Reflection coefficient versus angle of incidence for seabed sand (very fine sand). the different kinds of sediments with the transition layer are analyzed in detail. In this study, the amplitude characteristics of the reflection coefficients are analyzed. The phase characteristics of the reflection coefficients are also needed to investigate. REFERENCES Fig. 9 Porosity versus depth for transition layer of seabed sand (very fine sand). The depth dependence of the porosity is seems to be different from each other in the types of the sediment. So, it is required that the incident angle characteristics and the frequency characteristics of the reflection coefficient from [1] R. D. Stoll and T. K. Kan, Reflection of acoustic waves at a water sediment interface, J. Acoust. Soc. Am., 70, 149 156 (1981). [2] M. Kimura, Reflection of plane acoustic wave at a seawatermarine sediment interface, Jpn. J. Appl. Phys., 35, 2948 2951 (1996). [3] K. P. Stephens, P. Fleisher, D. Lavoie and C. Brunner, Scaledependent physical and geoacoustic property variability of shallow-water carbonate sediments from the Dry Tortugas, Florida, Geo-Mar. Lett., 17, 299 305 (1997). [4] K. R. Briggs and M. D. Richardson, Small-scale fluctuations in acoustic and physical properties in surficial carbonate sediments and their relationship to bioturbation, Geo-Mar. Lett., 17, 306 315 (1997). [5] M. Kimura, R. Shimizu, T. Tsurumi and K. Ishida, Measure- 194

M. KIMURA and T. TSURUMI: ACOUSTIC WAVE REFLECTION FROM MARINE SEDIMENT ments of acoustic and physical characteristics of surficial marine sediment, Proc. Autumn Meet. Acoust. Soc. Jpn., pp. 1221 1222 (2001). [6] M. Stern, A. Bedford and H. R. Millwater, Wave reflection from a sediment layer with depth-dependent properties, J. Acoust. Soc. Am., 77, 1781 1788 (1985). [7] H. Schmidt, OASES, Ver. 2.2, User Guide and Reference Manual (1999). [8] M. A. Biot, Theory of elastic waves in a fluid-saturated porous solid. 1. Low frequency range, J. Acoust. Soc. Am., 28, 168 178 (1956). [9] M. A. Biot, Theory of elastic waves in a fluid-saturated porous solid. 2. Higher frequency range, J. Acoust. Soc. Am., 28, 179 191 (1956). [10] R. D. Stoll, Wave attenuation in saturated sediments, J. Acoust. Soc. Am., 47, 1440 1447 (1970). [11] R. D. Stoll, Sediment Acoustics (Springer-Verlag, Berlin, 1989), pp. 5 36. [12] N. P. Chotiros, Biot model of sound propagation in watersaturated sand, J. Acoust. Soc. Am., 97, 199 214 (1995). [13] M. Kimura, Acoustic wave reflection from water-saturated and air-saturated sediments, Jpn. J. Appl. Phys., 41, 3513 3518 (2002). [14] M. Kimura and T. Tsurumi, Characteristics of acoustic wave reflection from the transition layer of surficial marine sediment, Proc. Underwater Technology 2002, Tokyo, pp. 225 230 (2002). [15] R. Carbo, Wave reflection from a transitional layer between the seawater and the bottom, J. Acoust. Soc. Am., 101, 227 232 (1997). [16] M. Kimura and S. Kawashima, Study on physical parameters of the Biot-Stoll marine sediment model, J. Mar. Acoust. Soc. Jpn., 22, 54 63 (1995). Masao Kimura was born in 1947. He graduated from Tohoku University in 1972. He received M.E. degree from Tokai University in 1978, and Dr. of Eng. degree from Tohoku University in 1989. He is currently a professor of the School of Marine Science and Technology, Tokai University. His research interests include acoustic wave propagation in water-saturated porous marine sediments and measuring methods for marine sediments. He is a member of The Acoustical Society of Japan, The Marine Acoustics Society of Japan, and The Acoustical Society of America. Takuya Tsurumi was born in Tochigi, Japan on August 6, 1977. He graduated from Tokai University in 2001 and received M.E. degree in Marine Science and Technology from Tokai University in 2003. He is currently a trainee of Tokai University. His research interests are reflection characteristics from marine sediments. He is a member of The Acoustical Society of Japan, and The Marine Acoustics Society of Japan. 195