A model-based analysis of sound transmission in a deep ice-covered Arctic ocean recorded during the Ice Experiment 2014 is presented. A source of opportunity transmitted mid-frequency (3500 Hz) 5 s duration continuous wave pulses. The source and receiver were omnidirectional, located under ice at a ∼30 m depth at a ∼719 m distance from each other. Recorded acoustic intensity time series showed a clear direct blast signal followed by an about 30 s duration reverberation coda. The model considers several types of arrivals contributing to the received signal at different time intervals. The direct signal, corresponding to a short-range nearly horizontal propagation, is strongly affected by the presence of a weak near-surface (within 50 m depth) acoustic channel. Reverberation coda that follows the direct signal corresponds to medium-range bottom- and ice-bounced arrivals from steep angles which are controlled by reflectivity and scattering strengths of ice and bottom, their physical properties, and acoustical parameters.

Most acoustic experiments in the Arctic are aimed at better understanding the sound interaction with a complicated ice-covered dynamic ocean environment and, based on this understanding, finding a way to remotely assess and monitor the environment at different spatial and temporal scales. In many cases, the emphasis is on low-frequency long-range propagation and large-scale observations (up to across the Arctic Ocean scales); see, e.g., overviews by Howe et al. (2019) and Worcester et al. (2020). Also, in many cases, the presence of ice cover, fine structure of a relatively thin surface duct (within 45–60 m depths), may not be critical as only acoustic paths which are below this duct survive over the large ranges of interest. The same reason allows neglecting paths with multiple bottom interactions. At higher frequencies and shorter ranges, however, the related effects become more important and directly observable; see, e.g., Yang and Hayward (1993) and Jensen et al. (2011). Such observations may be a valuable source of information, although at smaller spatial scales, about ice cover state and characteristics, bottom parameters, and under ice water properties and dynamics.

This paper presents results of midfrequency acoustic observations for transmission of 3.5 kHz, 5 s pulses in a deep ice-covered Arctic ocean made during the Ice Experiment 2014 (ICEX14). Details of the experiment and environmental conditions are described in Williams et al. (2018) and Ivakin and Williams (2020). Recorded acoustic intensity time series showed a clear direct blast signal followed by an about 30 s duration reverberation coda. The main goal of this study is to provide a model-based analysis for each segment of the measured time series, suggest their physical explanation in terms of specific sound propagation and scattering mechanisms, and particularly quantify roles of the Arctic near-surface duct, ocean bottom, and ice cover.

The paper is organized as follows: in Sec. II, the ICEX14 data on acoustic propagation and reverberation are described. In Sec. III, a direct blast signal is considered using combined ray-based and full-field (PE) models for under-ice propagation. In Sec. IV, a model for the reverberation coda is presented, and results of model-data comparison are discussed. Summary and concluding remarks are given in Sec. V.

The dataset on acoustic propagation and reverberation discussed in this paper was obtained in an ice-covered region of the Beaufort Sea in March 2014 during one of the ICEX14 experiments. Water depth was greater than 3.5 km and the ice in the area was ∼2 m thick. A typical depth-dependence of sound speed measured in the ICEX14 area is shown in Fig. 1 and is used in this paper as a representative example, as no such profiles are available for the specific location and time of acoustic measurements of opportunity (discussed here) during their short (about 5 min) duration.

FIG. 1.

A typical sound speed depth-profile measured during the ICEX14.

FIG. 1.

A typical sound speed depth-profile measured during the ICEX14.

Close modal

Midfrequency transmissions from an omnidirectional source of opportunity located under ice at a ∼30 m depth were recorded on a hydrophone located at the same depth at a ∼719 m range from the source. The acoustic measurements discussed here were performed within a short, 5 min time span during which five consecutive 5 s duration CW pulses at 3.5 kHz were transmitted, one per minute. Figures 2(a) and 2(b) show time series of a relative intensity (RI), that represents recorded acoustic intensity level (IL), squared magnitude of received pressure envelope (in dB units), relative to source level (SL). Note that it has the same magnitude as, but opposite sign as, transmission loss (TL), i.e., RI = − TL = IL − SL. All five received time series, aligned relative to each ping's start time, are shown by solid lines, and the average over the five pings is shown by a thick dashed line.

FIG. 2.

(Color online) Relative Intensity timeseries (solid lines) and their average (dashed) for all five transmitted pings (1–5). The entire time series are shown with an ellipse that surrounds direct arrivals (a), which are zoomed in for more detail (b).

FIG. 2.

(Color online) Relative Intensity timeseries (solid lines) and their average (dashed) for all five transmitted pings (1–5). The entire time series are shown with an ellipse that surrounds direct arrivals (a), which are zoomed in for more detail (b).

Close modal

A distinctly different behavior of the data are seen in two different time intervals. The short times (∼0.5–5.5 s) correspond to direct arrival of the 5 s duration pulses transmitted from the rather short direct-path distance (∼719 m). The time series zoomed in and shown in Fig. 2(b) for detail, vary approximately 0.9 dB around the average value, which gives an estimate for the transmission loss of the direct arrivals, TL45.9±0.9dB. The direct arrival signal and following reverberation coda [Fig. 2(a)] are above noise level (∼ −120 dB, seen as a background) within a time interval t ∼ 0.5–35 s, which is associated with short-to-medium ranges, ct ∼ 0.75–50 km. Here, c is the sound speed in the water and t is the propagation time counted from beginning of each ping.

The second time interval includes reverberation signal associated with possible bottom bounces. The signal starts at time ∼5 s that corresponds to ∼7 km range (approximately two bottom depths), although it can be seen only after ∼5.7 s because its beginning (at ∼5–5.7 s) is well below the level of direct blast signal. If just a little shorter transmitted pulse was used, e.g., 3 s instead of 5 s duration, the direct arrival would have a much clearer separation from the coda. The shape of the coda shows a somewhat different behavior in the first part (t < 10 s) where it has a plateau at ∼−80 dB level, then a sharp back edge at t ∼ 10 s, and following decay to a noise level at t >35 s.

Note that the source and receiver clocks were not synchronized, so the estimates of arrival time for the beginning of direct blast signal relative to the ping start time seen in Fig. 2(b) (∼0.7 s) and related parameters assume a typical value for sound speed in the water under ice at the source–receiver depth (30 m), c0 = 1435 m/s (see Fig. 1). For some other important estimates, however, only relative times are needed. For instance, the time difference between back edges of the direct arrival and the first part of the coda (clearly seen in Fig. 2 at 5.5 and 10 s, respectively) provides a simple estimate for a specific ocean depth under the drifting ice during the 5 min duration acoustic experiment, ∼3700 m, which will be used later in analysis of the coda.

In this section, the short-time interval of the received signal (at ∼0.5–5.5 s) corresponding to direct arrivals is considered in more detail and analyzed. Numerical simulations of under-ice propagation in the surface duct can be made using elastic full-wave codes, see, e.g., Jensen et al. (2011), and Frank and Ivakin (2018a,b). Although such calculations are time consuming, they could be worthwhile in the case where the sound speed profile (SSP) and source location are known with sufficient accuracy. As it is not the case in our experiment, for simplicity and faster calculations, we use a Range-dependent Acoustic Model (RAM) based on Parabolic Equation (PE) initially developed for fluid ocean environments with free flat surface (Collins, 1999), for brevity called here standard PE, which is slightly modified here to take into account effects of elastic ice cover, either flat or rough, using some simple ray-based considerations. The goal is to understand and quantify mechanisms affecting the measured intensity level and its variation over time of observations (∼5 min).

The SSP (see Fig. 1) clearly shows presence of a near-surface duct (∼45 m thick) above a gradient/transition layer (∼45–55 m). Although this profile is just a representative example, the surface duct is a typical feature of this ice-covered environment, so we will start from this SSP, which defines a set of environmental inputs to an initial scenario (scenario 1, free flat surface, and standard PE model is used). Other inputs are defined by the source and receiver locations, their depths, and range (horizontal separation). Then, we consider other scenarios to assess the effects of possible corrections, variations, and uncertainties in the range- and depth-dependences of the acoustic intensity caused by relatively small changes of SSP. Finally, effects of ice cover will be shown using some ray-based corrections to standard PE solutions. For all the scenarios, a connection between the simulated range–depth dependencies with observed acoustic intensity time series will be discussed in Secs. III B, III C, and III D.

First, starting with scenario 1, we want to demonstrate that, while the surface waveguide is relatively weak (with only ∼7 m/s for total variations of the sound speed), it happened to be sufficient, for this specific experiment geometry, ranges, and depths, to cause a noticeable enhancement of intensity due to a focusing effect of refraction within the gradient layer that appears, in ray-based terminology, as a caustic near the receiver location (Brekhovskikh and Lysanov, 2003).

Using a standard PE solution for scenario 1 (SSP given in Fig. 1 with free ocean surface) results in Fig. 3 which shows range–depth contours for relative intensity (-TL) and clearly confirms that the receiver is located in the area of enhanced intensity.

FIG. 3.

(Color online) A range–depth image of Relative Intensity in dB units, −TL. It resulted from PE-based calculations assuming free ocean surface and sound speed depth-profile shown in Fig. 1. Locations of source and receiver are shown by crosses. Note an enhancement of intensity near the receiver location caused by focusing effects of refraction in the surface duct.

FIG. 3.

(Color online) A range–depth image of Relative Intensity in dB units, −TL. It resulted from PE-based calculations assuming free ocean surface and sound speed depth-profile shown in Fig. 1. Locations of source and receiver are shown by crosses. Note an enhancement of intensity near the receiver location caused by focusing effects of refraction in the surface duct.

Close modal

For the sake of faster computer simulations, now we assume that, at short times considered here, deep ocean layers have little effect on propagation, and can be ignored, i.e., they will be truncated and replaced by a non-reflecting absorbing bottom that plays here a role of a perfectly matched layer (PML). Note that the initial profile (SSP1) is truncated at the deepest point of available SSP data (500 m), which significantly increases the calculation time, even using a rather fast standard PE model. Examining SSP variations would require multiple calculations and a much larger volume of simulations and associated computer time. To reduce those, the SSP can be truncated “properly,” i.e., without any significant effect on model-data comparisons.

For an illustration, consider two more profiles, 2 and 3, shown in Fig. 4(a), both being significantly different from SSP1 and from each other, and compare the results of simulations for the three profiles, which are presented in Fig. 4(b), for the range- and depth-dependences of acoustic intensity. It is clearly seen that effects of stratification in water columns deeper than ∼60 m are rather small for most of the depths and ranges. There are some locations though with noticeable sensitivity caused by wave interference due to multipaths (using ray terms). At those locations, amplitudes of different rays are comparable that results in more pronounced minima and maxima of interferential patterns.

FIG. 4.

(Color online) Sound speed profiles (SSP 1–3) which differ only at depths below the surface duct (a), and corresponding range- and depth-dependences of Relative Intensity in dB units, −TL (b). Free of ice ocean surface is assumed.

FIG. 4.

(Color online) Sound speed profiles (SSP 1–3) which differ only at depths below the surface duct (a), and corresponding range- and depth-dependences of Relative Intensity in dB units, −TL (b). Free of ice ocean surface is assumed.

Close modal

SSP3 is chosen as a “properly” truncated sound speed profile for further comparisons to show critical importance of SSP variations within the surface channel. The standard PE model with ice-free flat ocean surface is used for simulations and results of comparisons for SSP 3–5 shown in Fig. 5(a) are presented in Fig. 5(b) again for range- and depth-dependencies of acoustic intensity. It is seen that even small SSP variations in the surface duct and its lower boundary, the gradient layer, may result in very large (tens of dB) variations of transmission loss at the receiver.

FIG. 5.

(Color online) Sound speed profiles (a) which differ only within the surface duct (unlike those shown in Fig. 4), and corresponding range- and depth-dependences of Relative Intensity in dB units, −TL (b). Free of ice ocean surface is assumed.

FIG. 5.

(Color online) Sound speed profiles (a) which differ only within the surface duct (unlike those shown in Fig. 4), and corresponding range- and depth-dependences of Relative Intensity in dB units, −TL (b). Free of ice ocean surface is assumed.

Close modal

For further analysis, we choose scenario 4 as it gives results for the predicted transmission loss close to that observed in our experiments. Our goal now is to suggest a way to correct the results for presence of ice cover. To make these corrections, we use a ray-based approach combined with the method of images and consider an additional acoustic field from an imaginary source assuming that its amplitude is taken such that it will (1) compensate the reflection from the free surface, and (2) add the reflection field corresponding to the real surface, the ice cover. It is easy to show that, in case of a homogeneous water layer under the ice, such source amplitude is 1+V, where V is the complex reflection coefficient of the ice layer. In the case of a slightly stratified water under ice, it would have a simple additional phase-correction factor for the imaginary source amplitude. In our case of a very slight stratification at 0–40 m depths, as in scenario 3 [Fig. 5(a)], this factor is not critical and can be omitted.

Consider now the reflection coefficient from an elastic ice cover. Figure 6 shows its real and imaginary parts as functions of incidence angle for several cases. Two curves on each subplot show results for two cases of flat ice; one is for 2 m ice layer thickness, and another is for a half-space. Ice parameters (see Table I) are representative but typical for use in various elastic models and computer simulations (see, e.g., McCammon and McDaniel, 1985; Williams and Francois, 1992). Important is that at near-grazing incidence, in the 70°–90° interval, the results in both cases are coincident and correspond to the limit case of the infinite layer thickness, or half-space. This is because, at these angles, both compressional and shear waves in ice are evanescent and their penetration to ice is smaller than the ice layer thickness. At the ranges of interest here, ∼720 m, all possible incidence angles are well within the mentioned interval of near-grazing incidence angles. For source and receiver depths ∼30 m, the specular reflection (incidence) angle θ is ∼85°, and, according to Fig. 6, one obtains W=1+V ∼ 0.2 + 0.4i, the needed input value for flat ice.

FIG. 6.

(Color online) Real (a) and imaginary (b) parts of the reflection coefficient from an ice cover represented by either flat or rough elastic half-space and 2 m-thick layer.

FIG. 6.

(Color online) Real (a) and imaginary (b) parts of the reflection coefficient from an ice cover represented by either flat or rough elastic half-space and 2 m-thick layer.

Close modal
TABLE I.

Parameters used in the models of ice cover and bottom.

Model parametersIceBottom
Density, g/cm3 0.9 2.7 
Compressional wave speed, m/s 3800/(1 + 0.01i) 4800/(1 + 0.01i) 
Shear wave speed, m/s 1600/(1 + 0.03i) 2000/(1 + 0.03i) 
Thickness, m 2.0 ∞ 
RMS roughness, cm 0/20/50 0/3.5/5 
Backscattering strength at nadir, M 0.015 0.01 
Power exponent, γ 2.0 2.5 
Model parametersIceBottom
Density, g/cm3 0.9 2.7 
Compressional wave speed, m/s 3800/(1 + 0.01i) 4800/(1 + 0.01i) 
Shear wave speed, m/s 1600/(1 + 0.03i) 2000/(1 + 0.03i) 
Thickness, m 2.0 ∞ 
RMS roughness, cm 0/20/50 0/3.5/5 
Backscattering strength at nadir, M 0.015 0.01 
Power exponent, γ 2.0 2.5 

Two more curves on each of subplots in Fig. 6 show results corrected for the ice roughness using corresponding coherence-loss factors in reflection coefficients, Vrough=VflatexpP2/2, where P=2kσcosθ is the Rayleigh parameter, k is wave number in water, and σ is the ice RMS (root-mean-square) roughness. Two values for the roughness parameter are taken, σ= 0.2 m (moderately rough ice) and σ= 0.5 m (very rough ice). At θ= 85°, in the case of a very rough ice, one obtains W=1+Vrough 0.6 + 0.2i, another needed input value to show the effect of ice roughness. Results comparing the cases of free surface, flat ice, and very rough ice, are shown in Fig. 7, for depth- and range-dependences of transmission loss in the vicinity of the receiver location.

FIG. 7.

(Color online) Depth- (a) and (b) range-dependences of −TL derived using combined PE–ray-based approach that accounts for presence of elastic ice cover, either flat or rough, and allows comparisons to the ice-free case. Results are presented for SSP4 shown in Fig. 5(a) and zoomed in for more detail in the vicinity of the receiver location.

FIG. 7.

(Color online) Depth- (a) and (b) range-dependences of −TL derived using combined PE–ray-based approach that accounts for presence of elastic ice cover, either flat or rough, and allows comparisons to the ice-free case. Results are presented for SSP4 shown in Fig. 5(a) and zoomed in for more detail in the vicinity of the receiver location.

Close modal

The curves for the relative intensity level (-TL) as function of range and depth shown in Fig. 7, as expected, are oscillating and give TL values very sensitive to the receiver location (and the source location as well, due to the reciprocity principle).

An important new feature is that presence of ice cover, either flat or rough, results in additional range dependent oscillations [O(10m)] of TL, which are much smaller than those in the free surface case, and which also may significantly increase the slope of the TL curve up to ∼1 dB/meter. Assume, for instance, that source–receiver exact locations allow some dynamical changes around nominal range, say ± a few meters horizontally, which for the source of opportunity and an ice-tethered receiver in this experiment is a reasonable estimate, considering possible relative ice–water movement caused, e.g., by currents and winds (Williams et al., 2018). Then, using the variations seen in Fig. 7(b) at 719 m over a few meters, scenarios with ice cover, both flat and rough, one obtains approximate bounds for transmission loss, −TL −48 ± 2 dB.

Although this estimate is within a reasonably good fit to rough bounds of measured transmission loss, −TL 46±1 dB [see Fig. 2(a)], the difference (∼2 dB) can still be informative. For instance, Fig. 7(a) shows a significant sensitivity to rather small vertical variations of source and receiver locations. Changing either source or receiver depth from 30 to 28 m, according to Fig. 7(a), would enhance the acoustic intensity to the observed values.

Another practical way to use this difference between observed and predicted transmission loss may be adjusting the gradient layer depth and/or its shape for which the slight evolution within the time of acoustic observations (here, the 5 min time span that includes all the five pings) may be a reasonable assumption. Assume, for instance, that the gradient layer slightly changes its depth, say, for a few tens of centimeters from its typical depth (∼50 m) over a few minutes, which can be caused, for instance, by internal waves. Then, according to Fig. 5, corresponding variations of acoustic intensity at the receiver could be on the order or greater than observed.

Both types of dynamics, horizontal changes in source-to-receiver distance, and vertical changes in the gradient layer depth, can be important as a potential tool for monitoring ocean water dynamics, which can be caused, for instance, by local currents, internal waves, or other mechanisms of SSP evolution. Recently, a somewhat similar approach based on measuring reflection from a gradient layer in the water column was used at much lower frequencies using seismo-acoustic instruments to observe dynamics of the varying gradient layer parameters, depth, and structure (Zou and Zhang, 2021).

Note, however, that such monitoring would require a sufficiently high stability of transmitted sound level and well controlled source–receiver locations. Here, to the authors' knowledge, the stability of the source location and sound level were high enough to somewhat narrow down the number of possible mechanisms and to believe that observed temporal variations of the received intensity and transmission loss [Fig. 2(b)] can be explained by either some variations in the receiver location (around the nominal location, here 30 m depth and 719 m range) or slight evolution of the sound speed profile during the experiment (within ∼5 min time span). Both can be caused by hydrodynamic processes, for instance, internal wave or currents/winds driven relative ice-water movement, which can change the range between source and receiver over the timescale ∼1 min between the pings.

For more insight into effects of the surface duct, and particularly the role of transition layer and its focusing effects, consider a ray model for a simple profile shown in Fig. 5(a) (profile 5), which corresponds to a linear gradient layer between two homogeneous half-spaces, z<z1 and z>z2, with sound speeds c1 and c2, respectively, assuming that the source and receiver are located above the gradient layer at depths zS and zR. In this case, the intensity of the refracted path is increased by a focusing effect of the gradient layer, and for the focusing factor's range–depth dependence one obtains F(r,z)=1/1q, where q(r,z)=2H(z)/ar2,H=z1+z2zSz, and a=(1/c1)dc/dz=|c2c1|/(c1|z2z1|). The focusing factor is constant in the range–depth space along the curves where qr,z is fixed, with qr,z=1 being equation of the caustic. This also explains connection between effects of changes in the thickness and location of gradient layer, and source–receiver geometry. For illustration, consider a numerical example. Given the fixed source–receiver depths and range (zR = zS = 30 m, r= 719 m), the relative change of sound speed over the gradient layer (its “strength”) and its location (|c2c1|/c1=0.005, z1 = 47 m), one founds that caustic will appear near the receiver location if the gradient layer thickness increases from ∼7 to ∼20 m.

Note that the combined ray-PE model, used here to describe the direct arrival, is a small-angle approximation of the full-wave solution, i.e., it includes only possible nearly horizontal short-range paths with ice reflections and in-water refractions, but excludes nearly vertical bottom and ice reflections which arrive later along medium-range paths as a part of reverberation coda. These steep-angle paths are practically not affected by water stratification, and associated arrivals' intensities can be estimated following, in general, an approach used in Ainslie et al. (2016) and a semi-coherent multiple specular reflections (MSR) model described therein. However, some modifications are needed to adjust this model to a medium-range ice-covered deep-ocean reverberation scenario the typical details of which are described, e.g., in Yang and Hayward (1993) and Jensen et al. (2011).

One such modification is that now the model replaces a free water surface by an ice cover whose effect is accounted in terms of an angular dependence of coherent reflection coefficient (see Fig. 6). Correspondingly, the propagation loss factors are corrected to account for additional loss due to reflections from ice. A second modification exploits conditions, rather common for deep water scenarios where source and receiver depths are much smaller than the ocean depth, which allows a simplified treatment of the so-called “4-paths semi-coherent effect.” In general, this is similar to an effective directivity pattern effect discussed in Jensen et al. (2011). In the case of ice cover, combined source and receiver Lloyd-mirror–like patterns form an “effective directivity” defined in terms of reflection coefficients of ice at outgoing (from the source) and arriving (to the receiver) angles from nadir. Then the “4-paths effect” is automatically included for all propagation paths between ice and bottom. However, this effect can be significantly reduced in the case of rough ice because of the coherence-loss correction factor, as was discussed earlier in Sec. III C. This means that in the case of rough ice, at these steep angles, the paths that include ice reflections can be neglected.

Note also that the approach used in Ainslie et al. (2016) originally assumes monostatic scenarios with no horizontal offset of the receiver from the source. In our experiment, the source–receiver separation in the horizontal plane is considerable. Strictly speaking, a bistatic scenario should be considered, see, e.g., Ellis and Crowe (1991) and Williams and Jackson (1998). It is known, however, that bistatic effects in reverberation are negligible at large enough times much greater than the time of direct propagation between source and receiver; see, e.g., Urick (1960, 1970). This is the case for the geometry of our midrange reverberation measurements where ranges are large (∼7 km and larger) compared to the distance between source and receiver (719 m).

Coherent part of reverberation corresponds to the model where the ocean boundaries are either flat or defined by coherent components of reflection coefficients. Consider first a case of flat boundaries with high reflectivity. In this case, reverberation coda is a sequence of multiple specular reflections of the emitted pulse from the boundaries with correspondingly increasing time delays and decreasing amplitudes due to the spherical spread and multiple reflection losses. An example of such coda is shown in Fig. 8(a), and some associated paths including specular reflections from ice and bottom at steep (near vertical) incidence (with one and two bottom bounces) are shown in Fig. 8(b). The contribution of direct arrivals, for simplicity, is shown by a PE arrow that replaces three possible types of short-range paths of propagation within the surface channel, a direct path, and two nearly horizontal ones, upward refracted by the gradient layer and the one reflected from ice cover at ∼85° incidence angle.

FIG. 8.

(Color online) (a) ICEX14 data compared to a model that includes, in addition to direct arrivals, only coherent component of reverberation caused by specular reflections from bottom and ice given their reflection coefficients at steep (near vertical) incidence, with associated paths (b). Bottom is a flat elastic basalt half-space, while ice cover is a flat 2 m-thick elastic layer, either flat or rough.

FIG. 8.

(Color online) (a) ICEX14 data compared to a model that includes, in addition to direct arrivals, only coherent component of reverberation caused by specular reflections from bottom and ice given their reflection coefficients at steep (near vertical) incidence, with associated paths (b). Bottom is a flat elastic basalt half-space, while ice cover is a flat 2 m-thick elastic layer, either flat or rough.

Close modal

It is assumed that ice cover is a flat 2 m-thick elastic layer with a complex reflection coefficient given in Fig. 6. Also assumed is that bottom can be represented as a flat elastic basalt half-space. In this case, the reflectivity at the relevant near-vertical directions for both boundaries are high enough: ∼0.8 for bottom and ∼0.87 for ice. This, along with additional enhancement caused by the “4-path effect” for source and receiver located not far from the ice, results in a high reverberation level seen in Fig. 8(a) until times and ranges including up to six bottom bounces (∼50 km range), i.e., for all ranges and times where the ICEX14 reverberation coda was observed [taken from Fig. 2(a) for the average relative intensity and shown in Fig. 8(a) for comparison).

However, introducing ice roughness changes the shape and reduces the level of the coherent component of the coda significantly, which is shown in Fig. 8(a) by a “flat bottom & rough ice” curve. This is due to reduction of the coherent reflection coefficient shown in Fig. 6 at near-normal incidence. As a result, only the path with no reflection from ice and a single bounce from bottom, even if it is highly reflective (as basalt), contributes to the coherent component of the reverberation coda. In the case of rough bottom, reduction of this component will be even more significant, according to additional coherent reflection loss. The coherent reflection coefficient for basalt half-space with a variable RMS roughness parameter is shown in Fig. 9 as a function of incidence angle. For other input parameters needed in the calculations, density, compressional and shear speeds, typical values were used which are given, for both ice and basalt, in Table I.

FIG. 9.

(Color online) The reflection coefficient as functions of the incidence angle for bottom basalt half-space which is assumed either flat or rough.

FIG. 9.

(Color online) The reflection coefficient as functions of the incidence angle for bottom basalt half-space which is assumed either flat or rough.

Close modal

The comparison in Fig. 8 leads to several conclusions. First, the model with highly reflective boundaries, which allows only specular reflections at near-normal incidence, can provide an acceptable fit to the reverberation coda observed at medium ranges (and corresponding times). However, introducing the ice roughness, which is necessary for the relatively thick ice conditions, makes the unfit significant even at rather short times and excludes any ice bounces. This means that those bounces should result in mostly non-specular incoherent scattering from ice in directions which are far from near normal, particularly (as discussed later, in Sec. IV C) to backscattering in slant directions, and, therefore, considering incoherent reverberation is necessary.

Also, a good fit at short time corresponding to a single bottom bounce [∼5–10 s; see Fig. 8(a)] is possible only with assumption of perfectly flat highly reflective bottom (as a flat water-basalt interface). Figure 9, however, shows that even a small RMS roughness (∼5 cm) would result in a noticeable reduction of the bottom bounce amplitude, and would worsen the fit of coherent reverb curve [Fig. 8(a)] to the observed level at short times (5–10 s) corresponding to the first bottom bounce at near-normal incidence, and, therefore, a proper compensation from incoherent component of bottom scattering is needed. Also, this requires examining angular dependencies of reflection and scattering coefficients for both bottom and ice at slant directions, as well as accounting for acoustic absorption in water column that can be important for slant paths and correspondingly longer ranges.

In the case of non-flat boundaries, the reverberation time series were calculated using a single scattering approximation that considers arrivals along slant paths which include only one non-specular scattering from bottom or ice, while all others are neglected. The results are shown in Fig. 10(a), and the arrivals, the earliest and strongest ones, are shown in Fig. 10(b), in addition to horizontal and nearly vertical paths discussed earlier and shown in Fig. 8.

FIG. 10.

(Color online) ICEX14 reverberation data-model comparison (a) including coherent bottom reflection and incoherent ice and bottom backscatter arrivals, and associated paths (b) where thick horizontal and vertical arrows replace multipath components shown in Fig. 8.

FIG. 10.

(Color online) ICEX14 reverberation data-model comparison (a) including coherent bottom reflection and incoherent ice and bottom backscatter arrivals, and associated paths (b) where thick horizontal and vertical arrows replace multipath components shown in Fig. 8.

Close modal

Model simulations of reverberation were performed taking into account acoustic absorption coefficient in sea water, 0.2 dB/km at 3.5 kHz, which is a typical value (see Jensen et al., 2011), and corresponds to the loss parameter of water δkw1.57×106. Dependences of backscattering coefficients on incidence angle for bottom and ice are taken in the form mSθ=Mcosθγ, assuming unknown strengths M and power exponents γ. The values of these parameters resulted from model-data comparisons are given in Table I. Interestingly, for the ice scattering strength, the best fit is at γ=2 which corresponds to well-known Lambert Law.

The reverberation data and model prediction that provides best fit to the data are shown in Fig. 10(a) by solid gray and dotted red lines, respectively. The model result is obtained from a summation of four contributions: near-vertical bottom reflection (dash blue), a −120 dB noise, and two slant backscatter mechanisms shown for the bottom and ice scatter contributions by solid blue and dash-dot red curves, respectively. The near-vertical and slant specular reflections are associated with the paths shown in Fig. 10(b). Corresponding reflection coefficients are taken from Fig. 9 (see the curve for basalt with 3.5 cm RMS roughness). The near-vertical coherent bottom reflection is reduced correspondingly [compared to that shown for flat bottom at 5–10 s time in Fig. 8(a)]. However, this reduction is compensated by the bottom scatter incoherent component which also became the dominant mechanism of reverberation (compared to ice scatter) at intermediate times (10–20 s).

Note again that the values of physical and acoustical parameters of ice and bottom used for calculations of propagation and reverberation in this paper are simply representative to outline the required set of the used model inputs. A comprehensive discussion of these inputs for different types of the ocean bottom can be found, e.g., in Jackson and Richardson (2007), Jackson et al. (2010), and references therein. In the model of ice cover, we considered only the case of a single homogeneous elastic layer. The models used here to specify the reflectivity of bottom and ice can be generalized to also include effects of poroelasticity and layering in a more consistent way following, e.g., Brekhovskikh (1980), McCammon and McDaniel (1985), Williams (2001). Hope et al. (2017), and Chotiros et al. (2021). Such generalizations, however, are beyond the scope of this paper.

In this paper, we present a model-based analysis of an experiment on midfrequency sound transmission in a deep ice-covered Arctic ocean. Recorded acoustic intensity time series showed a clear direct blast signal (having the shape of ∼5 s duration emitted pulse) followed by an about 30 s duration long reverberation coda. For consistent model-data comparison, considering a combined effect of several mechanisms is needed, each dominating within its own time interval. The main result is that each segment of the recorded time series was interpreted by a specific sound propagation or scattering mechanism(s), which was accompanied by an explanation, both qualitative and quantitative.

Figure 11 gives a final comparison of the ICEX14 data with modeling results which include summation of short- and medium-range mechanisms considered in this paper. Short-range transmission loss is controlled by nearly horizontal propagation in the surface channel, including refraction effects in the water gradient layer and reflection from the ice cover.

FIG. 11.

(Color online) The full time series comparison between measured ICEX14 data and model results.

FIG. 11.

(Color online) The full time series comparison between measured ICEX14 data and model results.

Close modal

The direct signal has a level which is slightly varying (within ∼2 dB bounds). This paper suggests a modeling approach that allows quantifying mechanisms affecting the measured intensity level and its variation over time of observations (∼5 min). We show that the level is strongly affected by the structure of a surface duct (within 50 m depth) and, particularly, variations in the depth of a thin gradient layer beneath (typically at ∼50–60 m depths), reflections from ice, refraction in water, and their interference. An important finding is that the presence of ice cover, either flat or rough, results in additional range dependence [O(10 m)] of transmission loss with a slope up to ∼1 dB/m. Potentially, analysis of the direct arrival at ranges where focusing effects of upward refraction become noticeable may provide a base for estimating parameters of the surface duct and its boundaries, the ice cover, and the gradient layer.

Reverberation coda that follows the direct signal and corresponds to medium-range arrivals from steep angles is shown to be strongly affected by reflectivity and scattering strength of the bottom and the ice and controlled by several mechanisms dominating within different time intervals. In the beginning–by first bottom reflection, at intermediate times–by backscatter from bottom, and at the end of the coda–by backscatter from ice. Also shown is that parameters of the bottom significantly affect the level and shape of reverberation signal in all the three time intervals. For instance, the angular-dependent effect of bottom roughness appears important for providing a good fit to reverberation data for two reasons. First, it reduces near-vertical reflection. Second, it also retains the high level for slant reflections needed for understanding why the ice Lambert backscatter contribution is still above noise level that comes to the receiver after two bounces from the bottom.

Generally, we believe that a more comprehensive environmental model could lead to a similar or better results of model-data comparison. Our focus here, however, was to suggest and discuss a simple enough forward model and to show its capability to fit the data using a minimal number of critical environmental parameters. Development of improved inversion model for better inferring environmental properties would be a next step and is a goal for future work. In this paper, the values of physical and acoustical parameters of ice and bottom used for calculations of propagation and reverberation are simply representative so as to outline the required set of model inputs.

This work was supported by ONR Grant Nos. N00014-17-1-2196 and N00014-21-1-2421 to A.I. The ICEX14 data were obtained under the Defense Advanced Research Projects Agency support to K.W. The authors appreciate the thoughtful and constructive reviews of the manuscript.

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