Presently, the scattering model of Jackson et al. (1996) (Williams and Jackson, 1998) has been adopted extensively for the calculation of a bottom scattering strength at high frequency (10–100 kHz). In this mode, only the effect of surface layer of the sediment was taken into account. Bottom scattering was attributed to the seafloor roughness and the sediment heterogeneity, and the corresponding scattering strength was calculated by a composite-roughness scattering model and a volume scattering model considering refraction and attenuation in the sediment, respectively. The predicted bottom scattering strength agreed well with the measured data of several experimental sites with different sediment types.

Some more complex effects below the water-sediment interface, such as the sound speed gradient in the sediment and the seafloor stratification (Li et al., 1987), have been further considered in bottom scattering models since the 1990s, which are available to predict the bottom scattering strength at low frequency (100 Hz–10 kHz). In an early scattering model for an arbitrarily stratified seafloor (Ivakin, 1986), the effect of sediment volume scattering was merely taken into account. Specialized profiles of sound speed and density were employed to describe the continuous stratification. An analytic solution for acoustic field in the sediment was obtained, and the influence of internal boundaries on the frequency-angular dependence of the bottom scattering strength was demonstrated. In addition, the effect of continuous stratification was considered and some general properties of the backscattering strength were established by the WKB approximation. It was shown that the Lambert-like angular dependence could sometimes arise from the influence of a surface gradient on sediment volume scattering. In Efimov and Ivakin (1987), staircase functions were employed to approximately characterize arbitrary profiles, corresponding limitations in computation algorithms were discussed, and model-data comparisons were provided.

With the development of research, scattering caused by rough interfaces in layered sediments was further considered in bottom scattering models. In Ivakin (1994b), a first-order solution was presented for stratified fluid sediments with arbitrary number of interfaces, and the bottom bistatic scattering strength was expressed through the auto-spectra and cross-spectra of the roughness of different interfaces. In Ivakin (1994a), both roughness and volume scattering mechanisms were considered and compared. It was shown that the roughness and the volume scattering amplitudes for layered sediments can be denoted in a similar form. A somewhat different approach was presented in Essen (1994) and Moe and Jackson (1994), where the rough water-sediment interface was viewed as the only scattering mechanism, but the effect of internal reflections due to stratification was contained. Some simplifications were introduced in Lyons et al. (1994), where the roughness and volume scattering in a two-layer system were processed. A treatment of the roughness scattering from a multilayered seafloor was demonstrated in Tang (1996), where the limit of close stacking of weakly scattering interfaces was shown to be indistinguishable from the volume scattering. In Ivakin (1998a), a unified perturbation approach to deal with the roughness and the volume scattering was presented, and generalized to the case that an arbitrary scattering basement was covered by stratified fluid sediment. The basement was assumed to be fluid (rough and/or heterogeneous) in numerical simulations of Efimov and Ivakin (1987) and Ivakin(1994b, 1998b), that is, the scattering from basement was considered, but ignored its elasticity. A numerical example of the scattering from a rough and heterogeneous elastic basement was presented in Ivakin (1997), which involved a first-order solution for the scattering amplitude of an elastic half-space medium (Ivakin, 1990; Ivakin and Jackson, 1998; Jackson and Ivakin, 1998). Finally, a geoacoustic bottom interaction model (GABIM) was established in Jackson et al. (2010), and corresponding calculation software was developed as well. Both roughness and volume scattering were considered in the GABIM, where the zeroth-order field solution required in the scattering kernels was calculated through the ocean acoustics and seismic exploration synthesis (OASES), and the volume scattering from the sediment with a complicated structure was expressed through an equivalent interface scattering cross section. The GABIM applies to the seafloor with arbitrary stratification, where the water-sediment interface is roughness, but allows only one internal roughness boundary (i.e., the sediment-basement interface). The sediment layers and the basement can all be heterogeneous responsible for the volume scattering, but the shear effect of the basement is only partly taken into account.

A simple and practical seafloor model is adopted in this paper. The seafloor is assumed to be made up of a fluid sediment layer with a finite thickness and an elastic half-space basement, and the shear effect of the basement is completely considered. On the basis of the seafloor model, a backscattering model for the stratified seafloor is established. The predicted bottom backscattering strength includes contributions of four kinds of scattering mechanisms, i.e., the roughness scattering from the water-sediment interface, the volume scattering from the sediment layer, the roughness scattering from the sediment-basement interface, and the volume scattering from the basement.

The paper is organized as follows. The stratified seafloor model and its parameterization are presented in Section 2. The acoustic filed solution in each layer is formulated as well. The backscattering model is established in Section 3. Section 4 demonstrates the simulation results, including the dependence of the backscattering strength on the acoustic frequency and the sediment-layer thickness, and the influence of shear effect of the basement on the backscattering strength. Finally, some conclusions are summarized in Section 5.

The bottom scattering model proposed in this paper applies to the prediction of the backscattering strength for stratified seafloors at low frequency. Because a transmission depth is relative deep in this frequency range, the influence of the seafloor stratification on the acoustic scattering becomes more significant. When a shallow bedrock exists in the seafloor, or the acoustic frequency is low enough that the sound wave interacts with the basement, a bottom scattering model based on the seafloor model shown in Fig. 1 is more reasonable. The seafloor is made up of a sediment layer with thickness of d and a half-space basement, and either layer possesses weak heterogeneity responsible for the volume scattering. Because there is a certain contrast of acoustic impedance between the unconsolidated sediment layer and the basement, the roughness scattering from the sediment-basement interface is further taken into account. Then the scattering mechanisms include the roughness scattering from the water-sediment interface, the volume scattering from the sediment layer, the roughness scattering from the sediment-basement interface, and the volume scattering from the basement.

In the parameterization of the seafloor, the sediment layer is viewed as fluid, which is parameterized by compressional wave speed and attenuation and mass density. The basement is viewed as an elastic medium, whose shear effect is parameterized by shear wave speed and attenuation. It is further supposed that all the parameters are independent of depth in each layer with the exception of small random fluctuation responsible for the volume scattering. The speed and the density are expressed in the form of dimensionless ratios of corresponding parameters to that in the water as follows:

where the subscript n (=1, 2) represents the sediment layer and the basement, respectively; p denotes the compressional wave; t denotes the shear wave; c_w is the sound speed in the water (the acoustic absorption is ignored); ρ_n stands for the density of the sediment layer or the basement; and ρ_w is the density of the water. The attenuation of the sediment layer and the basement is given by loss parameters ${\delta _{{{\rm p}},n}}$ and ${\delta _{{\rm t},2}}$, which are related to the attenuation coefficient ${\alpha _{{{\rm p}},n}}$ and ${\alpha _{{{\rm t}},2}}$ in dB/m as

where f is the acoustic frequency.

The seafloor roughness is usually characterized by the roughness spectrum in scattering models. One of the simplest and most often used isotropic spectral forms is the “power law”:

where the parameter w₂ is called the spectral strength; and the parameter γ₂ is called the spectral exponent. Because the contribution of all scale roughness is considered in Eq. (6), this form of the roughness spectrum can be extended to low frequency in the case of being invariable of w₂ and γ₂.

The effect of acoustic refraction and transmission through the interface is considered in modeling of the volume scattering. The volume scattering is attributed to independent and isotropic point scatters in the sediment layer and the basement, and a dimensionless parameter is employed to quantify the volume scattering as follows:

where the commonly used approximation of single scattering is implied. The parameter ${\sigma _{2,n}}$ is referred to as the volume parameter; and the parameter ${\sigma _{{\rm{v}},n}}$ is the volume scattering cross section, which is assumed to be independent of depth in each layer. It is remarkable that besides the contribution of continuous heterogeneity to the volume scattering, possible discrete scatters or strong scattering layers are not considered in the volume scattering model.

When calculating the scattering strength of the stratified seafloor, it involves the pressure field in each layer derived from the incident plane wave in the water with unit strength. Because the seafloor model contains an elastic basement, it is more convenient to firstly solve the potential function of the displacement in stead of the pressure. According to the field theory, the displacement filed can be divided into

where $\phi $ is the scalar potential function of the displacement; and ${\vec \psi } $ is the vector potential function of the displacement. The former is used to describe the compressional wave, and the latter is for the shear wave. ${\vec r} $ is the position vector of an arbitrary space point. For a uniform medium, $\phi $ and ${\vec \psi} $ are subject to the following two uncoupled Helmholtz equations, respectively:

where c_p and c_t denote the compressional wave speed and the shear wave speed, respectively; and the parameter ω is the angular frequency.

It is supposed that the coordinate origin lies on the water-sediment interface and the positive direction is downwards. The formal solution of the scalar potential function of the displacement in the water can be written as

where the coefficient A₀ indicates the complex amplitude of the reflected wave in the water.

The sediment is viewed as fluid, and the formal solution of the scalar potential function of the displacement in the sediment layer is

where the coefficients A₁ and B₁ indicate the complex amplitude of the up-going and down-going waves in the sediment layer, respectively.

The basement is viewed as an elastic medium, and the formal solution of the scalar and vector potential functions of the displacement (only with polarization of y direction) in the basement are

where the coefficient B₂ indicates the complex amplitude of the down-going compressional wave in the basement; the coefficient C₂ indicates the complex amplitude of the down-going shear wave in the basement; ${k_{0,z}}$, ${k_{1,z}}$ and ${k_{2,{\rm{p}},z}}$ are the vertical components of wave vector of the compressional wave in each layer; ${k_{2,{\rm{t}},z}}$ is the vertical component of wave vector of the shear wave in the basement; and ${k_{0,x}}$, ${k_{1,x}}$, ${k_{2,{\rm{p}},x}}$ and ${k_{2,{\rm{t}},x}}$ are the corresponding horizontal component of the wave vector, respectively.

According to the continuity of the normal displacement and stress at the water-sediment boundary (z=0), one can obtain

According to continuity of the normal displacement and stress and zero of the shear stress at the sediment-basement boundary (z=d), one can obtain

Substitute the corresponding formal solutions of scalar and vector potential functions of the displacement into above equations, one can obtain

Subsequently, coefficients A₀, A₁, B₀, B₂ and C₂ can be solved according to Eqs (21)–(25). The bottom reflection coefficient is

where the argument θ denotes the grazing angle of the incident wave in the water.

When calculating the roughness scattering from the water-sediment interface, the reflection coefficient ${R_1}(\theta )$ will be used. Coefficients A₁, B₁, B₂ and C₂ will be involved in the calculation of the volume scattering from the sediment layer and the basement. It is worth noting that these coefficients need to be multiplied by the density of corresponding layers to acquire the pressure.

When the seafloor model shown in Fig. 1 is adopted, the bottom scattering is sum of four kinds of mechanisms: the roughness scattering from the water-sediment interface, the volume scattering from the sediment layer, the roughness scattering from the sediment-basement interface, and the volume scattering from the basement. The bottom backscattering strength in dB is defined as

where σ_sr and σ_br denote the roughness scattering cross section of the water-sediment and sediment-basement interfaces, respectively; and σ_sv and σ_bv are the equivalent interface scattering cross section of the volume scattering from the sediment layer and the basement, respectively.

The stratification of the seafloor will also affect the bottom loss in dB, which is defined as

where ${R_1}(\theta )$ is the bottom reflection coefficient at the grazing angle θ.

For the stratified seafloor shown in Fig. 1, when the upper layer is a sandy sediment and the downer one is a rocky sediment, the parameters used for numerical simulations in this section are listed in Table 1. These parameters refer to Jackson and Ivakin (1998), Williams et al. (2002) and Jackson et al. (2010). The sound speed in the water is supposed to be 1 500 m/s.

A backscattering model for a stratified seafloor applying to low frequency (0.1–10 kHz) is established in this paper. In this model, the seafloor is viewed as an elastic half-space basement coved by a fluid sediment layer with a finite thickness. It is believed that the bottom scattering arises from four kinds of mechanisms, including the roughness scattering from the water-sediment interface, the volume scattering from the sediment layer, the roughness scattering from the sediment-basement interface, and the volume scattering from the basement. The numerical simulations of the influence of the acoustic frequency, the sediment-layer thickness, and the shear effect of the basement on the bottom backscattering strength as a function of the grazing angle are presented. The research shows that the roughness scattering from the sediment-basement interface and the volume scattering from the basement are more prominent at relative low frequency (below 1.0 kHz). The contribution of them to the total bottom scattering gradually becomes weak and declines faster at subcritical grazing angles with the increase of frequency. Then the prediction of the model ultimately approaches to that of the high-frequency bottom scattering model of Jackson et al. (1996). The variation regularity of the bottom backscattering strength with the increase of the sediment-layer thickness is similar to that of the frequency dependence. The backscattering strength as a function of the grazing angle gradually approaches to the prediction of the high-frequency bottom scattering model of Jackson et al. (1996) with the increase of the acoustic frequency or the sediment-layer thickness, which validates the backscattering model for the stratified seafloor in terms of the dependence of the acoustic frequency and the sediment-layer thickness. When the sound speed and attenuation of the shear wave in the basement gradually decreases, the prediction tends to that of the full fluid model, which provides another proof. Unfortunately, the model-data comparison is not presented in this paper, because some model parameters are hard or even impossible to measure in situ, such as the roughness under the sediment layer and the heterogeneity in the basement. At present, a try in a large water tank is in preparation. Measurements of the bottom backscattering strength and the model parameters will be synchronously carried out to validate the model through experiments in the near future.