Anomalous reflection from a two-layered marine sediment

Acoustic reflection from the seabed can provide a useful tool for recovering the geo-acoustic parameters of a stratified sediment,¹ including the sound speed gradient within the uppermost layer.² In fact, seabed reflection, notably the frequency dependence of the quasi-angle of intromission, was among the techniques used by Holland and colleagues³ to help characterize the fine-grained (mud) sediment at the New England Mud Patch (NEMP), about 95 km south of Martha's Vineyard, MA, during the Seabed Characterization Experiment conducted in 2017 (SBCEX17) and sponsored by the US Office of Naval Research.⁴ 

To aid the interpretation of seabed reflection data from the NEMP and elsewhere, several analytical solutions were formulated recently for the reflection coefficient for three different sediment sound speed structures: linear and inverse-square, each in a half-space;⁵ and a linear layer overlying a homogeneous basement.⁶ In all three cases, the seawater column was taken to be homogeneous and, in order to focus on the effects of the sound speed profile in the sediment, the density in the sediment layers was treated as uniform, an assumption that, as it happens, is consistent with core data from the mud sediment at the NEMP.^7,8

In the analysis of the reflection coefficient of a linear layer over a homogeneous half space,⁶ the exact, closed form solution was formulated in terms of definite integrals, over finite limits, of modified Bessel functions of order zero or unity. These integrals are straightforward to evaluate using a Simpson's rule or similar algorithm.⁹ In addition to grazing angle, the solution was found to be dependent on frequency, the sound speed gradient in the layer, $γ$⁠, as well as the sound speed in the basement, $c 3$⁠, relative to that in the water column, $c 1$⁠.

For a “fast” basement (⁠ $c 3 > c 1$⁠), a frequency-independent critical grazing angle exists, signifying that total reflection occurs at grazing angles below the critical. A “slow” basement (⁠ $c 3 > c 1$⁠), on the other hand, in the limit of low frequency, exhibits an angle of intromission where no energy is reflected. In this low-frequency limit, the top layer is in effect transparent to the incident sound, the seabed acts as a fluid-fluid boundary with seawater above and basement below, and the expression for the reflection coefficient reduces to the familiar Rayleigh¹⁰ formalism.

In the present paper, reflection from a two-layered sediment is again considered under similar assumptions to those of the previous analysis:⁶ the density and sound speed are taken to be uniform in the seawater column and the basement; the density is constant in the top layer; and shear is considered to be negligible, which is reasonable for a top layer consisting of fine-grained sediment (mud). In the interests of clarity, attenuation is neglected in the early discussions but its effects are examined towards the end of the paper. Analyses of three sound speed profiles in the top layer are discussed, in addition to the fluid-fluid case where the depth of the top layer is allowed to go to zero and the reflection is from the interface between the seawater and the basement (case 1). This elementary solution is included since it provides a useful baseline reference against which reflection from the more complex stratifications may be compared.

In order of increasing complexity, the three sound speed profiles to be considered in the top layer are: uniform throughout the layer (case 2); a uniform gradient in the layer (case 3); and an inverse-square profile in the layer (case 4). The basic approach to obtaining the solution for the reflection coefficient in all three cases is to formulate the wave (Helmholtz) equation for each of the three domains (seawater and two sediment layers) and, if the sound speed gradient is non-zero, to convert the Helmholtz equation for the top layer to the modified Bessel equation. Spatial Fourier transforms are then applied as appropriate followed by the inverse Fourier transforms, and the various constants of integration are obtained from the boundary conditions. The resulting integral over horizontal wavenumber for the field in the water column is evaluated by stationary phase, to obtain the specular reflection, which leads to the final expression for the reflection coefficient.

For the linear profile in the top layer, the nature of the solution for the reflection coefficient has already been discussed for low frequencies,⁶ below about 500 Hz. Higher frequencies are examined in this paper, in the region of several kHz, where the solution takes on a significantly different character, exhibiting delta-function-like spikes of total reflection, representing acoustic glint from the seabed. The angular distribution of the spikes depends strongly on the frequency and the depth of the top layer, and mildly on the gradient of the profile. For the case of the inverse-square profile in the top layer, a new solution is presented, and this too exhibits a glint structure but with its own characteristic angular distribution. As exemplified by the linear and inverse-square cases, the glint is present only when the profile in the top layer has a non-zero gradient but otherwise is absent.

The nomenclature used in the following analyses is much the same as in the previous discussion of low-frequency (<500 Hz) reflection from a surficial layer with a linear sound speed profile overlying an iso-speed basement.⁶ The three domains considered, from top to bottom, are seawater, top (mud) layer, and basement, designated by indices 1, 2, and 3, which are used as subscripts to identify associated parameters. For example, $ρ 1$ is the density in the water column, domain 1. The constants used in the analyses and their numerical values may be found in Table I and the legends of the relevant figures.

Using the parameter values in Table I, the modulus of the reflection coefficient from Eq. (1) is plotted as a function of the grazing angle in Fig. 1. A prominent feature of the curve is the grazing angle of intromission at 7.4°, depicted by the vertical dashed black line, which will provide a useful baseline reference in the ensuing discussions of $| R 1 |$ for cases $i = 2 , 3$⁠, and 4.

The modulus of $R 2$ in Eq. (3) is plotted in Fig. 2(a) as a function of grazing angle for a frequency of 4 kHz, and again in Fig. 2(b) but with the Rayleigh expression for $| R 1 |$ from Eq. (1) superimposed for comparison. It is evident in Fig. 2 that a minimum, representing a quasi-grazing angle of intromission, is exhibited by $R 2$ at approximately $10.5 °$⁠, some $3.1 °$ higher than the true grazing angle of intromission, $β 13 = 7.4 °$⁠, in $R 1$⁠. It is interesting to note that the upper envelope of the undulations in $| R 2 |$⁠, which are associated with resonances and anti-resonances in the top layer of the sediment, is accurately tracked by the Rayleigh expression for $| R 1 |$⁠.

Figure 4 shows $| R 3 |$ as a function of grazing angle for a frequency of 4 kHz. To facilitate the discussion, the curve may be split into two components: a background or continuous curve (blue), which is very similar to but not identical with Fig. 2; and a sequence of very narrow, delta-function-like spikes of total reflection (red), representing acoustic glint from the seabed.

The distribution of the spikes of total reflection across grazing angle is highly sensitive to the frequency and the depth of the top layer, and mildly so to the gradient of the sound speed profile in the layer. As examples of the sensitivities to frequency, layer depth, and gradient, $| R 3 |$ is shown as a function of grazing angle for a frequency of 3 kHz in Fig. 6(a), a layer depth of 1.4 m in Fig. 6(b), and a gradient of $20 s − 1$ in Fig. 6(c), in each case with all else the same as in Fig. 4 and Table I. On comparing the plots in Figs. 4, 6(a), 6(b), and 6(c), it is evident that the angular distribution of the red spikes provides a characteristic “fingerprint,” which could form the basis of a precision methodology for recovering the geo-acoustic parameters of the upper reaches of the sediment. By way of contrast, the continuous, blue portion of the curve is relatively insensitive to the frequency, the layer depth, and the gradient.

The glint phenomenon is not confined to the case of a slow basement. When the sound speed in the basement exceeds that in the water column, representing the fast basement condition, $η 3$ from Eq. (6) is imaginary for grazing angles below the critical, the numerator and denominator in Eq. (7) are complex conjugates, and the modulus of $R 3$ is unity. In this angular regime of total reflection, no glint spikes would be visible since the spikes are themselves angular regions, albeit very narrow, of total reflection. Above the critical grazing angle, a distribution of glint spikes appears, as shown in Fig. 6(d), where the only difference from Fig. 4 is the higher sound speed in the basement. On comparing Figs. 4 and 6(d), it is apparent that the angular distribution, above the critical, of the red glint spikes is identical in the two cases. This could have been anticipated, since the function B in Eq. (8), and hence all the zeros of $B$ are independent of the basement geo-acoustic properties, notably the sound speed c₃. The continuous blue curve in Fig. 6(d), however, beyond the region of total reflection, is substantially different from that in Fig. 4, signifying a moderate sensitivity to the basement sound speed.

The glint is a characteristic feature of the reflection coefficient that arises directly from the gradient of the profile in the top layer. As the gradient goes to zero, the glint vanishes and the expression for $R 3$ in Eq. (7) should reduce to $R 2$ in Eq. (3) for reflection from a layer with a uniform sound speed profile.

Given that the structure of the solution for $R 4$ in Eq. (22) is the same as that for $R 3$ in Eq. (7), it might be expected that $R 4$ will likewise exhibit the glint phenomenon, and indeed it does. At the zeros of B in Eq. (23) the expression for $R 4$ reduces to $− 1$⁠, representing total reflection. This is illustrated in Fig. 8, which shows $| R 4 |$ as a function of grazing angle for a slow basement and a frequency of 4 kHz. In this particular example, the grazing angles at which the glint occurs are similar to those in Fig. 4 for the linear profile, not surprisingly perhaps since the curvature in the inverse-square profile is extremely subtle, so much so that the two profiles, linear and inverse-square, have almost the same gradient over the depth of the top layer. In the limit, as the curvature of the inverse square profile goes to zero, that is, as $c ∞ → ∞$⁠, it is evident from Eqs. (23)–(26) that $R 4$ in Eq. (22) reduces identically to the solution for a uniform gradient, $R 3$ in Eq. (7); and in turn, as the gradient approaches zero, the glint vanishes and $R 3$ reduces to the solution for a uniform sound speed profile, $R 2$ in Eq. (3), as has already been established.

As the frequency decreases, the zeros in B, or equivalently, the spikes of glint in the reflection coefficient, migrate to higher grazing angles until each vanishes as it passes through normal incidence. This phenomenon is illustrated in Fig. 9, which shows B as a function of grazing angle at four frequencies, 2, 1.5, 1, and 0.5 kHz, in this case for a linear profile under the environmental conditions of Fig. 4 and Table I. The asterisks depict the zeros of B.

In the example shown in Fig. 9, only the first zero remains present at a frequency of 500 Hz, and even that vanishes at approximately 420 Hz. Of course, the vanishing frequencies, that is the frequencies at which the zeros are at normal incidence, depend on the local conditions, notably the shape or gradient of the profile and the depth of the layer. For realistic choices of these environmental conditions the vanishing frequency of the first zero, the last to disappear, is typically in the vicinity of 500 Hz, below which glint no longer appears in the reflection coefficient.

Figure 10 shows the real and imaginary parts of the function B for four values of the loss tangent: $β = 0 ( 0 )$⁠, $β = 0.000 325 ( 0.013 )$⁠, $β = 0.000 650 ( 0.026 )$⁠, and $β = 0.000 975 ( 0.039 )$⁠, where the numbers in brackets represent the attenuation, equivalent to $β$⁠, but expressed in dB/m/kHz. According to Table I in Bowles,¹⁸ the range 0–0.04 dB/m/kHz is representative of the attenuation in many fine-grained marine sediments. The real part of B is insensitive to the attenuation, so much so that for all four values of $β$⁠, the curves in Fig. 10(a) overlap one another precisely, making them visually indistinguishable. The imaginary part of B shows a weak but, as it turns out, a significant dependence on $β$⁠, which introduces a partial suppression of some of the spikes of glint in the reflection coefficient.

This effect is illustrated in Fig. 11, which shows the modulus of the reflection coefficient, |R₃|, for a linear profile under the influence of four levels of attenuation, from zero up to $β = 0.000975 ( 0.039 )$⁠. The black curves in Fig. 11 represent the continuous component of the reflection coefficient, which is unaffected by the attenuation, whilst the red spikes depict the glint. It is evident in Fig. 11 that as the attenuation increases most of the spikes suffer a certain reduction in intensity (with the exception at 22°), to the extent that in Fig. 11(d), the glint event occurring at a grazing angle of 38° is inverted, more aptly being described as an extinction.

One notable feature of the reflection coefficient in the presence of attenuation becomes evident on comparing the unattenuated $| R 3 |$ in Fig. 11(a) with the attenuated versions in the remaining three panels: the increasing attenuation has no appreciable effect on the angular positions of the glint, that is to say, the “fingerprint” of the glint remains unchanged by the action of attenuation. In the examples shown in Fig. 11, the angular positions of the four spikes of glint in each of the four panels fall approximately at 5°, 22°, 38°, and 58°. The fact that the angular positions of the glint are invariant under changes of attenuation could be important if the glint were to be used as the basis of an inversion technique for recovering the geo-acoustic parameters of the seabed.

One might perhaps wonder whether the glint could be observed under controlled conditions in a laboratory experiment or, indeed, with a real-world reflection set-up in the ocean. As with optical (sunlight) glint off a choppy sea surface, it may seem that the positions of the acoustic source and receiver would have to be set precisely to achieve a grazing angle matching that of a particular spike of glint. Achieving such precision would represent a significant challenge. On the other hand, the sensitivity of the angular distribution of the zeros in B to the environmental conditions suggests that, if the glint were detectable, it could provide the basis of a high-precision inversion procedure for recovering the geo-acoustic properties of the seabed.

It may be possible to avoid the requirement for precision placement of the source and receiver by exploiting the fact that the glint moves rapidly across grazing angle with frequency. With this in mind, a source and receiver could be arranged in a static configuration such that the specular angle is somewhere towards the high end of the range, say in the region of 60°. By sweeping the source frequency, a succession of spikes of glint could then be made to pass across the receiver. Of course, rather than use an omni-directional sensor, the receiver could be a vertical line array with a beam steered downward at the specular angle, which would lead to an enhanced signal/noise ratio and an improved probability of detection.

An analysis of specular reflection from a two-layered marine sediment in which the upper layer supports an upward refracting linear or inverse-square sound speed profile leads to exact, closed-form expressions for the reflection coefficient in the two cases. Examination of these solutions reveals that, in the absence of attenuation, total reflection is predicted to occur at certain discrete grazing angles. The angular distribution of these very narrow spikes, designated glint, is very sensitive to the frequency and depth of the upper layer, and mildly so to the gradient or curvature of the profile within the layer. In addition to the discrete spikes, the modulus of the reflection coefficient exhibits a continuous (over grazing angle) background component, which is insensitive to the frequency and to the shape or gradient of the profile. When the gradient is allowed to go to zero, with all else the same, the glint vanishes whilst the continuous component remains essentially unchanged.

Several numerical ocean-acoustic propagation models^19–21 are available for computing the acoustic field in a horizontally stratified ocean. Holland²² developed one such model in which a linear sound speed profile is approximated as a stair-step, that is, as a stack of thin layers each with a uniform sound speed.²³ On running the model under the same environmental conditions as those used to generate Fig. 4, Holland²² found that the modulus of the reflection coefficient was visually indistinguishable from the continuous component, shown in blue in Fig. 4, but the discrete, red spikes of glint were absent.²² 

The fact that Holland's numerical model²² accurately represents the continuous component of the reflection coefficient over the full angular range, from 0° to 90°, supports the idea that the expression for $R 3$ in Eq. (7) along with the function B in Eq. (8) are both correctly formulated. This raises the question as to why the glint is absent from the numerical result. The answer could lie with the stair-step representation of the linear sound speed profile. As established in Sec. IV, the sound speed gradient in a single layer must be non-zero for glint to exist. When the gradient of the sound speed in the layer is zero, glint is absent from the reflection coefficient, and a stack of such zero-gradient layers similarly lacks the means to produce glint.

Taking a mathematical standpoint, under the positive gradient condition in the layer, it is necessary, in order to obtain an analytical solution, to transform the Helmholtz equation into the modified Bessel equation,⁶ a transformation which gives rise to the solutions for $R 3$ and $R 4$ in Eqs. (7) and (22), respectively, both of which exhibit glint. With a gradient of zero, no such Helmholtz-to-Bessel transformation is involved and, as exemplified by $R 2$ in Eq. (3) for the layer with a uniform sound speed profile, glint is absent from the solution for the reflection coefficient. Any model based on the stair-step approximation will be unable to reproduce the discrete spikes of total reflection that are predicted by the analytical solutions for $R 3$ and $R 4$ in Eqs. (7) and (22), respectively, since the glint derives directly from the Helmholtz-to-Bessel transformation.

When attenuation in the top (mud) layer is included, the analysis becomes intractable except for certain special cases, notably for a sound speed and attenuation, respectively, that vary linearly and inversely with depth in the layer. Under such a condition, the continuous component of the reflection coefficient is essentially unaffected by the attenuation whilst the spikes of glint suffer varying degrees of attenuation. The angular positions of these attenuated spikes of glint, however, are unshifted from the locations of their unattenuated counterparts.

Towards the end of the paper, a technique is proposed for detecting the glint, provided of course that the sediment exhibits an upper layer with an upward refracting sound speed profile. It relies on the strong frequency dependence of the angular positions of the spikes of glint. A source and receiver could be set up such that the specular grazing angle is in the region of 60°. By sweeping the frequency, a succession of spikes of glint would pass through the receiver, which could take the form of a vertical line array with downward inclined beam to enhance the probability of detection.

Assuming that the glint could be detected, its angular distribution has potential as the basis of a high-precision inversion technique for recovering the geo-acoustic properties of the seabed. The distribution of the spikes of glint across grazing angle resembles a frequency-dependent “fingerprint” that is sensitive both to the shape or gradient of the sound speed profile in the upper layer and to the depth of the layer. Once available, this information on the sound speed in the layer, in combination with the viscous grain shearing (VGS) theory,^24–26 could then be used to infer the porosity, density, and grain size of the upper reaches of the sediment, along with the intrinsic attenuation in the granular medium.

This research was supported by the Office of Naval Research, Ocean Acoustics Code 322OA, under Grant No. N00014-22-1-2598.

The author has no conflicts of interest to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.