Modal depth function estimation using time-frequency analysis
Spectrograms of the recorded field for an impulsive source (arbitrary linear scale) and corresponding theoretical time-frequency dispersion curves (black curves) in a Pekeris waveguide. Water depth is D = 130 m and water sound speed is ms−1. Bottom sound speed and density are ms−1 and kg m−3, respectively. Source and receiver depths are both m, so that all of the modes are equally excited. Three source−receiver ranges are considered: (a) 1 km, (b) 5 km, and (c) 60 km.
Modal separability in a Pekeris waveguide using TF analysis (the waveguide parameters are the same as in Fig. 1). In the light gray area, the modes are naturally separated in the TF domain, whereas in the white area, the modes are not separated in the TF domain. The dark gray area corresponds to the domain where only one mode is propagating. The arrow corresponds to the frequency bandwidth of the source presented in Sec. ???, for which the modal separation is improved using warping. The crosses refer to the two operating points where the modes are extracted (see Fig. 5).
Principle of modal filtering using warping operator.
Warped version of the spectrograms presented in Fig. 1 (arbitrary linear scale). Source–receiver ranges are: (a) 1 km, (b) 5 km, and (c) 60 km.
Extracted modes along the VLA (source–receiver range r = 5 km) at frequencies f = 45 Hz (stars) and f = 75 Hz (circles), and the corresponding theoretical mode (continuous lines 45 Hz; dotted lines 75 Hz).
Modal amplitude extraction accuracy according to the SNR (results for f = 50 Hz).
Extracted modes along the VLA (source–receiver range r = 1 km) at frequencies f = 45 Hz (stars) and f = 75 Hz (circles), and the corresponding theoretical mode (continuous lines 45 Hz; dotted lines 75 Hz). A sign estimation error occurs at depth z = 70 m for mode 2 at f = 75 Hz. This error is propagated along the mode for depths m, but the amplitude estimation is not affected.
Nonisovelocity celerity profiles used to benchmark the estimation: (a) summer profile and (b) polar profile.
Nonisovelocity simulations: extracted modes along the VLA (source–receiver range r = 5 km) at frequencies f = 50 Hz, and corresponding theoretical mode. Continuous lines and stars correspond to the summer profile while dotted lines and circles correspond to the polar profile.
Spectrograms (arbitrary linear scale) of the received signal (a) and the corresponding warped signal (b). Both source and receiver are at depth mm. Note that the frequency scales are different in (a) and (b).
Extracted modes along the VLA (stars) and the corresponding theoretical modes (continuous curves) for the experimental data.
Modal separability in a Pekeris waveguide using TF analysis. The water sound speed is ms − 1, and the bottom sound speed and density are ms−1 and kg m−3, respectively. Each curve corresponds to the modal separability limit at a given depth (for a given depth, modes are resolvable below the curve, whereas they are unresolvable above it). The depths illustrated are for 50, 70, 90, 110, 130, and 150 m.
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