Plane-wave and cylindrical-wave acoustic reflection from a marine sediment with layering representative of the New England Mud Patch

This paper is one in a series on the reflection coefficient of marine sediments. The first of the previous papers on the topic addresses semi-infinite sediments exhibiting an upward refracting sound speed profile, either linear or inverse-square.¹ This was followed by an analysis of plane-wave reflection from a two-layer sediment, the top layer with an upward-refracting (linear or inverse-square) sound speed profile above a homogeneous basement.^2,3 An unexpected phenomenon that emerged from that particular two-layer analysis was the presence of acoustic “glint,” whereby, in the kHz frequency range very narrow spikes of total reflection occurred at several discrete grazing angles. The distribution across grazing angles of the glint is heavily dependent on the depth of the top layer, the frequency, and to a lesser extent the gradient or shape of the profile in the layer.

Theoretical investigations of sound waves in stratified fluids have been pursued by previous authors, notably Tolstoy,^4,5 Rutherford and Hawker,⁶ and Robins,⁷ who considered reflection from a fluid layer of varying density and sound speed sandwiched between two homogeneous fluids.

In 2017, the Office of Naval Research sponsored a research program to investigate the geo-acoustic and geo-physical properties of fine-grained sediments. In support of this effort, the international, multi-institutional Seabed Characterization Experiment⁸ (SBCEX17) was conducted on the New England Mud Patch (NEMP), located about 95 km south of Martha's Vineyard, MA. The sediment at the NEMP^9–12 consists of a near-homogeneous layer of fine-grained material (mud), which increases in thickness from roughly 2 m to 12 m along a northwest to southeast transect approximately 15 km long. Beneath the mud, the basement consists of an initial admixture of sand, which builds in concentration steadily with increasing depth, eventually becoming a medium-to-coarse sand layer¹³ with little if any mud present. This description of the sediment stratification at the NEMP is, it should be noted, a considerably simplified version of the detailed surveys presented by Twitchell et al.,¹² Goff et al.,⁹ and Chaytor et al.¹³ 

At site SC2 in the southeast of the NEMP, where the near-homogeneous mud layer is relatively thick, extending to approximately 10.5 m beneath the seabed, Jiang et al.¹⁴ conducted acoustic reflection experiments, which returned data that they inverted to recover the sediment geo-acoustic properties: the sound speed, density, and attenuation as functions of depth. From their Table III,¹⁴ the results are as depicted by the solid blue circles in Fig. 1. Their values for the sound speed and attenuation in Figs. 1(a) and 1(c) are for a frequency of 1175 Hz. According to Table I in Jiang et al.,¹⁴ the sound speed at the base of the water column was 1472.3 m/s, which is substantially less than 1969 m/s, the deepest data point shown in Fig. 1(a), suggesting that the plane-wave reflection coefficient might be expected to show a critical grazing angle somewhere in the region of 41.6°.

In passing, it is worth noting that Jiang et al.,¹⁴ in their Figs. 6 and 7, show a sediment stratification at the SC2 site with interfaces at 11.7 m (mud base), 12.8 m (sand base), and 14.4 m (deep base). Above the mud base is a “geo-acoustic transition layer” between depths of 10.8 and 11.7 m. Although inferred from inversions yielding marginal posterior probability profiles, none of these boundaries is visible in the sediment sound speed data in Fig. 1(a), where at depths greater than 10.5 m, it is evident that the monotonic-increasing inverse-square profile displays a conspicuously good match to the data points.

The purpose of this paper is to develop an analysis of the plane wave and cylindrical-wave reflection coefficients of a two-layer sediment representative of the NEMP, based upon the idealized profiles depicted by the red lines in Fig. 1. The theoretical approach to be followed is similar to that in the previous analyses,^1–3 but with obvious differences in the geo-acoustic properties of the layers. It is assumed that the ocean and upper sediment (mud) layers are homogeneous and that the lower (basement) layer supports an inverse-square sound speed profile [Fig. 1(a)] but a depth-independent density [Fig. 1(b)]. Shear is considered to be negligible in the fine-grained material at the NEMP. Attenuation in the ocean is treated as finite but vanishingly small, and initially, to clarify the discussion of the plane wave reflection coefficient, attenuation in both sediment layers is neglected. A little later, a realistic level of attenuation is introduced into the mud and basement layers, as illustrated by the red line in Fig. 1(c), which has a significant effect on both reflection coefficients.

In the theoretical development, the wave (Helmholtz) equations are set up for the ocean and the two sediment layers, and spatial Fourier transforms with respect to horizontal distance are applied over all three domains. A further spatial Fourier transform is performed over elevation in the (semi-infinite) ocean, followed by the corresponding inverse transform; and the partial differential equation for the basement, which supports the inverse-square profile, is transformed into the modified Bessel equation¹⁵ by an appropriate mapping of the depth coordinate. After evaluating the various constants of integration that are involved with the aid of the boundary conditions, a Sommerfeld–Weyl^16,17 type of horizontal wavenumber integral for the field in the water column is obtained. A numerical evaluation of this integral yields the cylindrical-wave reflection coefficient, which incorporates the curvature of the wavefronts due to the proximity of the source and receiver to the seabed.

By setting the source and receiver far above the seabed and then applying a stationary phase analysis to the wavenumber integral, a closed-form expression emerges for the plane-wave reflection coefficient, which, with no losses present, exhibits a critical grazing angle, α_c, when evaluated under the geo-acoustic conditions of site SC2 at the NEMP. An apparent, or quasi, critical grazing angle,¹⁸  α_q, is predicted by the numerical integration for the cylindrical-wave reflection coefficient, which is less than α_c by an amount that depends on the source-plus-receiver elevation above the seafloor. Unlike the case of a top layer with an inverse-square profile,³ no glint is predicted when a similar profile is present in the basement layer.

As in the previous analyses,^1–3 two coordinate systems are used to characterize the acoustic field in the vertical: z, in the downward direction with its origin at a distance, h, above the mud-basement interface; and $z¯$ directed upwards with its origin at a distance, d, beneath the seabed at the bottom of the mud layer. These vertical coordinate systems, along with the sound speeds in the three domains, are illustrated in the schematic of Fig. 2.

The second wavenumber integral in Eq. (15), taken over the real axis, includes all contributions, notably the normal modes in the seawater-mud-basement waveguide, to the reflected field in the water column. The characteristic equation for the modal eigenvalues may be obtained from Eq. (16) by setting the denominator to zero, which yields a transcendental equation whose solutions are the poles of R₀(p) in the complex p-plane. Also accommodated by the second integral in Eq. (15) is the curvature of rays due to the upward refracting, inverse-square profile in the basement, including the effects of any caustic that may appear in the water column as a result of such curvature.¹⁸ 

The function R₀(p_s) in Eq. (22) is the plane wave reflection coefficient of a sediment with idealized layering, as illustrated in Figs. 1 and 2. As a simple check, it is a straightforward matter to show that in the limit as d goes to zero, Eq. (22) reduces to the correct form for a semi-infinite basement with an inverse-square profile and no overlying homogeneous mud layer.¹ 

One issue remains to be addressed before the expressions for the reflection coefficients in Eqs. (20) and (22) can be evaluated. It concerns the function N in Eq. (18), which involves Macdonald's function and its derivative with respect to the argument.

These two integrals may be computed numerically by a Simpson's rule or similar algorithm,²¹ taking care with the choice of the upper limit and the sampling rate, both of which may be determined from a visual inspection of the integrands. The function N in Eq. (18) may then be evaluated, along with M₁ and M₂ in Eqs. (17a) and (17b), respectively, thereby facilitating the determination of the cylindrical-wave [Eq. (20)] and plane-wave [Eq. (22)] reflection coefficients, R and R₀, respectively.

Site SC2 is located in the southeast of the NEMP, where the mud layer is approximately 10.5 m thick. Figure 1 shows data on the sound speed structure, density, and attenuation in the mud layer and the basement, as reported by Jiang et al.¹⁴ in their Table III, along with the associated idealized profiles used here to evaluate the cylindrical-wave and plane-wave reflection coefficients in Eqs. (20) and (22), respectively. As an adjunct to Fig. 1, Table I summarizes the numerical values of the geo-acoustic parameters for the site, as specified in Jiang et al.;¹⁴ and Table II lists the inverse-square parameter values used here to characterize the sound speed and attenuation profiles in the basement, as illustrated by the red curves in Figs. 1(a) and 1(c). With the aid of the numerical values in Tables I and II, the expressions for the reflection coefficients, R and R₀, in Eqs. (20) and (22), respectively, may be evaluated as functions of grazing angle, α, with the thickness of the mud layer, d, the frequency, f, and the source-plus-receiver elevations above the seabed, W, treated as parameters.

The region of total reflection is illustrated in Fig. 3, which shows |R₀| from Eq. (22) for site SC2, under lossless conditions, plotted for two frequencies, 1175 and 2975 Hz (corresponding to the lowest and highest frequencies, respectively, used by Jiang et al.¹⁴ in the inversion of their SC2 data.) Beyond α_c, the red and blue curves represent, respectively, the numerical evaluation of the integrals representing the modified Bessel functions in Eqs. (26) and their Debye asymptotic approximations in Eqs. (29) and (30).

The red and blue curves in Fig. 3 are extremely well matched across the range of grazing angles from the critical up to normal incidence, with the blue curve almost completely masking the red curve. As the frequency increases, the rate of the oscillations rises but the peak-to-trough height across grazing angle remains essentially the same.

Figure 4 shows the plane-wave reflection coefficient, |R₀|, from Eq. (22), as a function of grazing angle for the two frequencies 1175 and 2975 Hz, computed using the geo-acoustic and inverse-square parameter values in Tables I and II, respectively. It is evident in Fig. 4 that the well-defined region of total reflection of the lossless case is no longer featured at kHz frequencies when attenuation is present. Instead, over the range of grazing angles up to the critical (i.e., the critical grazing angle that would be present in the absence of attenuation), the modulus of the reflection coefficient (red) exhibits a set of contiguous peaks of approximately uniform width, with peak values below unity. The width and level of the peaks both depend on frequency. Above the critical grazing angle, the numerical evaluation of the integrals in Eqs. (26) yields the red curve, which is masked by the almost identical blue curve from the Debye asymptotic approximation in Eq. (29).

At site SC2 on the NEMP the observed reflection coefficient, reported in Fig. 3 of Jiang et al.,¹⁴ exhibits a quasi-critical grazing angle, α_q, of approximately 21°, where the character of the reflectivity changes abruptly. Similarly, at two other sites on the NEMP, designated SWAMI (d ≈ 10 m) and VC31-2 (d ≈ 2 m), quasi-critical grazing angles of approximately 25° and 30°, respectively, have been observed.^27,28 These quasi-critical grazing angles show only a weak, if any, dependence on frequency, and in all three cases, α_q is significantly less than the genuine critical grazing angle, α_c = 44.09°, featured in the lossless, plane-wave reflection coefficient for SC2 (see Fig. 3).

The discrepancy between α_c, as predicted by the plane-wave analysis, and the quasi-critical grazing angles is consistent with the idea, originally proposed by Stickler,¹⁸ that wave-front curvature, arising from the proximity of the source and receiver to the seabed, is responsible for the presence of an apparent critical grazing angle, which appears below and instead of α_c. Under such circumstances, when the source and receiver are close to the seabed, the plane-wave condition W ➛ ∞ underpinning the stationary phase evaluation of the second wavenumber integral in Eq. (15) ceases to apply, in which case the expression for the plane-wave reflection coefficient in Eq. (22) no longer holds. It is then necessary to perform a numerical integration in order to evaluate the cylindrical-wave reflection coefficient R in Eq. (20) using a Simpson's rule²¹ or similar algorithm. As with the integrals in Eqs. (26) for Macdonald's function and its derivative, the upper limit and sampling rate must be chosen with care, in the present case based on a visual inspection of the integrand. Holland and colleagues^28,29 have embedded an integral, analogous to that on the right of Eq. (20), into their inversions for the geo-acoustic parameters of the fine-grained sediments at two sites, the NEMP and the Malta Plateau south of Sicily in the Mediterranean Sea.

The effect of wave-front curvature on the cylindrical reflection coefficient in Eq. (20) is illustrated in Fig. 5 for four source-plus-receiver elevations, W, above the seabed, under the environmental conditions of site SC2. As W rises, the quasi-critical grazing angle increases from 10° in panel (a), reaching 39° in panel (d). It is clear in Fig. 5 that, with increasing W, the quasi-critical grazing angle, α_q, approaches the true value, α_c. Moreover, the cylindrical reflection coefficient in panel (d) is starting to resemble closely its plane-wave counterpart in Fig. 4(a).

As with the observations of Jiang et al.¹⁴ at site SC2, the cylindrical reflection coefficient in Eq. (20) hardly varies with frequency, at least in the low-to-medium kHz range. This is illustrated in Fig. 6, which shows |R| for the two frequencies 1175 and 2975 Hz, with W set equal to 36 m, corresponding to the experimental arrangement reported by Jiang et al.¹⁴ The two theoretical values of α_q, marked by the vertical dashed lines in Fig. 6, as identified by visual inspection, are almost indistinguishable at 21.7°. This frequency independent quasi-critical grazing angle closely matches that observed experimentally by Jiang et al.,¹⁴ as shown in their Fig. 3.

As mentioned earlier, the depth of the mud layer increases from about 2 m to 12 m along a transect from northwest to southeast at the NEMP. Although the quasi-critical grazing angle is sensitive to the depth of the mud layer, its effect is mild, as shown in Figs. 7(a) and 7(b), where the difference in the depths is 7.5 m, giving rise to a shift in a_q of about 5.4°. A more obvious difference between Figs. 7(a) and 7(b) is in the rate of the oscillations, which is significantly slower across all grazing angles in the case of the shallower mud layer. It is worth noting that in Fig. 7, several of the peaks exceed unity, resulting from refraction in the inverse-square basement, giving rise to the formation of one or more caustics,¹⁸ but this does not violate the principle of conservation of energy.

The fine-grained sediment at the NEMP, located on the continental shelf off the East Coast of the United States, consists of a near-homogeneous mud layer overlying a sand-mud basement. The proportion of sand increases with depth in the basement, giving rise to a monotonic increasing sound speed, which is well characterized by an upward-refracting, inverse-square profile. Along a 15 km transect from northwest to southeast on the NEMP, the thickness of the mud layer increases from approximately 2 m to 12 m.^9,12,13

An analysis of acoustic reflection from such a two-layer sediment is developed in this paper, based on a sequence of integral transforms as applied to each of the three layers (homogeneous) seawater, mud, and sand-mud basement. With aid of the boundary conditions, continuity of pressure and continuity of the normal component of particle velocity, an expression is developed in the form of a Sommerfeld–Weyl^16,17 wavenumber inversion integral for the cylindrical reflection coefficient, R, of the sediment [(Eq. (20)]. Under the condition where the source and/or receiver are very far above the seabed, a stationary phase evaluation of the wavenumber integral leads to the analytical, closed-form expression for the plane wave reflection coefficient, R₀, in Eq. (22). The expressions for both R and R₀ involve a modified Bessel function of the third kind, otherwise known as Macdonald's function, which is evaluated in two ways: numerically from its integral representation and in terms of the Debye asymptotic approximation.²² It turns out that these two approaches yield results for |R₀| that are visually almost indistinguishable over the range of grazing angles where the Debye approximation holds.

With values for the geo-acoustic parameters that are representative of site SC2 at the NEMP, and in the absence of attenuation in the sediment, the plane-wave reflection coefficient, R₀, exhibits total reflection at grazing angles up to the critical, α_c ≈ 44.09°, beyond which |R₀| shows oscillations with peak values of approximately 0.4. When realistic levels of attenuation are introduced, the region of total reflection is replaced by a succession of contiguous peaks, the widths of which are very sensitive to the frequency and to the depth of the mud layer. It is hypothesized that these peaks are directly associated with the normal modes in the waveguide formed by the seawater-mud-basement system.

With the source and receiver closer to the seabed but otherwise under the same SC2 conditions, the cylindrical-wave reflection coefficient, R, exhibits a quasi-critical grazing angle that is less than the genuine α_c by an amount that depends on the source-plus-receiver height above the seabed, W, and the depth of the mud layer, d. For example, with W = 36 m and d = 10.5 m, the predicted quasi-critical grazing angle is 21.7°, which is in excellent agreement with its experimentally determined counterpart, as reported by Jiang et al.¹⁴ in their Fig. 3. Moreover, the theoretical quasi-critical grazing angle is found to be essentially independent of frequency, at least over the low-to-mid kHz range, which is again in accord with experimental observations.¹⁴ 

To conclude, a few comments on density and attenuation profiles are in order. The density data in the basement [Fig. 1(b)], although treated as uniform in the present analysis, are well matched by an inverse-square profile. Such a density profile, however, appears to be intractable, prohibiting the development of an analytical solution for the reflection coefficient. In a discussion of density profiles, Robins³⁰ has introduced various coordinate transformations, which lead to solutions for specific cases, but not including the inverse-square. It seems that an analytical solution for the reflection coefficient associated with an inverse-square density profile is a problem that must be left for another time.

With regard to attenuation, the basement data shown in Fig. 1(c) are widely spread, over 2 orders of magnitude from 0.01 to 1 dB/m/kHz, presumably because they represent the effective attenuation due to scattering and other such mechanisms and not the irreversible intrinsic attenuation, in which acoustic energy is converted into heat.²⁴ Be that as it may, it is evident that the data could be equally poorly fitted by an increasing, a constant, or a decreasing profile (the author is indebted to an anonymous reviewer for this choice of words). The decreasing inverse-square profile shown in Fig. 1(c) was selected as a matter of expediency since it makes the analysis of the reflection coefficient tractable.

This research was supported by the Office of Naval Research, Ocean Acoustics Code 322OA, under Grant No. N00014-22-1-2598. The author would like to thank Dr. Charles Holland for several constructive discussions concerning the reflection coefficient of fine-grained sediments.

The author has no conflicts to disclose.

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