• Coherence extrapolation for underwater ambient noise

Table I shows the parameters of two environments with a seabed consisting of three and five sediment layers (including a bottom half-space). For both environments, the water depth is 130 m. These parameters are input to Eq. (1) to obtain the coherence function s(z) at f = 750 Hz with constant water sound speed c_r = 1500 m/s. Figures 1(a)–1(c) show the extrapolation of s(z) for the three-layer environment for increasing P. The simulated coherence (shown as solid lines) represents what would be “measured” with a 100-element array with inter-element spacing of 5 cm (4.95 m aperture). The real (and even) part of the coherence is shown for z > 0, while the imaginary (and odd) part is shown for z < 0. Notice that the coherence functions are normalized to unit maximum for display purpose. Because the extrapolated results are better for the imaginary part than for the real part of the coherence, zoomed-in views of the extrapolation are provided only for the latter. This convention will be followed throughout this paper. The extrapolation results are shown as dashed lines: Figs. 1(a)–1(c) correspond to P = 0, 9, and 11 in Eq. (7). For P = 0, the real part is coarsely approximated by ψ₀(z), and the representation of the true covariance is inaccurate even within the aperture. For P = 9 (which includes the PSFs with μ_p > 0.5), the extrapolated coherence matches well within the aperture, and it predicts the oscillatory behavior of the covariance for $|z|>Z/2$. Adding two more PSFs (which corresponds to including all PSFs with μ_p > 0.06) to the summation gives even more accurate results shown in Fig. 1(c). In this paper, selection of the number of terms P was done based on minimizing the difference between the extrapolated coherence and the coherence measured at the aperture. Similar results can be obtained by plotting $\sqrt{{{\sum }_{p=1}^P|a_p|}^2}$ in a loglog scale against the L2 norm of the error within the aperture given by $\sqrt{{{\sum }_{t=1}^N|s_{1t}^N(P)-s_{1t}|}^2}$ [notice that $s_{1t}^N(P)$ emphasizes the dependence of the extrapolated coherence on P]. This plot results in an L-shaped trace that provides a graphical tool to select the value P_elbow corresponding to the L-shape elbow. The rationale of the approach inspired in the L-curve method is that even though the coefficients $|a_p|$ tend to grow (due to normalization by decreasing eigenvalues μ_p), their contribution to minimizing the errors $|s_{1t}^N(P)-s_{1t}|$ is not significant for P > P_elbow. In addition, as pointed out previously, it is a good practice to limit P because problems of numerical instability can occur by adding more terms to Eq. (7) due to inaccuracies in determining the projection coefficients a_p.

Figure 1(d) shows extrapolation results for a more challenging signal s(z), obtained from the coherence function at f = 750 Hz of the five-layer sediment profile described in Table I. In this case, the real part of the coherence exhibits more structure in its tail, as opposed to the decaying-envelope behavior of s(z) in Figs. 1(a)–1(c). Even though the extrapolation results are less accurate, the trend of oscillations is estimated correctly. Similar extrapolation results were obtained for simulated data corresponding to more sparse VLAs with Δ = 0.5 m inter-element spacing.

Throughout this paper, the reason for obtaining better extrapolation results for the imaginary part of the coherence can be seen from Eqs. (1) and (2): Consider that the Fourier transform of the coherence, $\mathcal{F}\left({\mathbf{s}}_{\mathrm{ab}}\left(z\right)\right)$, is real⁹9. M. Siderius, L. Muzi, C. H. Harrison, and P. L. Nielsen, “Synthetic array processing of ocean ambient noise for higher resolution seabed bottom loss estimation,” J. Acoust. Soc. Am. 133, EL149–EL155 (2013). https://doi.org/10.1121/1.4774074 and can be expressed as a summation of even and odd functions. Because $\tilde{G}(k_z)=\tilde{G}(-k_z)$ and $\tilde{R}(k_z)=\tilde{R}(-k_z)$, the even function [responsible for the real part of s_ab(z)] is strongly dependent on the seabed reflection coefficient, which introduces complicated features, while the effect of the seabed in the odd function [which gives the imaginary part of s_ab(z)] tends to cancel out due to symmetry, giving a smoother (easier to extrapolate) function.

The coherence extrapolation was applied to measured data from the BOUNDARY2003 experiment,¹²12. C. H. Harrison, “Performance and limitations of spectral factorization for ambient noise sub-bottom profiling,” J. Acoust. Soc. Am. 118, 2913–2923 (2005). https://doi.org/10.1121/1.2048967 collected by the NATO-STO Centre for Maritime Research and Experimentation (CMRE formerly NATO Undersea Research Centre). The aperture used in the experiment consists of a 32-element drifting array of omni-directional pressure sensors with Δ = 0.18 m inter-element spacing. The data were collected in a region with water depth of approximately 131 m at a sampling rate of 12 kHz, and the frequency-dependent data covariance was computed from snapshots recorded over a 5-min interval. The 32 × 32 element covariance matrix was made Toeplitz by taking the mean along diagonals, and this is taken as a reference (i.e., ground truth) in this section on the understanding that it differs from the true covariance, which could only be measured accurately with a larger (and denser) aperture array.

Figure 2 shows the extrapolation for N = 12 at frequencies f = 703, 797, 891, 1008, 1102, and 1195 Hz. The extrapolated covariance with N_e = 32 is in agreement with the N_e-element reference covariance, and they exhibit similar behavior as the simulated results obtained in Sec. IV A. Similar to Figs. 1(c) and 1(d), the extrapolated imaginary part (z < 0) closely matches the reference, and the extrapolated real part (z > 0) follows the oscillatory pattern of the reference but is less accurate. In addition, the reconstructed coherence within the observation aperture ($|z|<Z/2$) closely matches the reference.