^{1,a)}, C. Gervaise

^{1}, P. Roux

^{2}, B. Nicolas

^{3}and J. I. Mars

^{3}

### Abstract

Acoustic propagation in shallow water is characterized by a set of depth-dependent modes, the modal depth functions, which propagate in range according to their horizontal wavenumbers. For inversion purposes, modal depth function estimation in shallow water is an issue when the environment is not known. Classical methods that provide blind mode estimation rely on the singular value decomposition of the received field at different frequencies over a vertical array of transducers. These methods require that the vertical array spans the full water column. This is obviously a strong limitation for the application of such methods in an operational context. To overcome these shortcomings, this study proposes to replace the spatial diversity constraint by a frequency diversity condition, and thus considers the case of a field emanating from an impulsive source. Indeed, because of the discrete nature of the wavenumber spectrum and due to their dispersive behavior, the modes are separated in the time-frequency domain. This phenomenon enables the design of a modal filtering scheme for signals received on a single receiver. In the case of a vertical receiver array, the modal contributions can be isolated for each receiver even when using a partial water column spanning array. This method thus eliminates the receiving constraints of classical methods of modal depth function estimation, although it imposes the use of an impulsive source. The developed algorithm is benchmarked on numerical simulations and validated on laboratory experimental data recorded in an ultrasonicwaveguide. Practical applications are also discussed.

I. INTRODUCTION

II. MODAL PROPAGATION

A. Normal mode theory and VLA

B. Time-frequency representation of the field recorded on a single hydrophone after an impulsive emission

III. MODAL EXTRACTION SCHEME

A. Modal filtering on a single receiver

1. Natural modal separability in the time-frequency domain

2. Filtering scheme for unresolved modes

B. Modal shape estimation along a VLA

1. Mode amplitude estimation

2. Mode sign estimation

IV. APPLICATIONS

A. Simulated data

1. Mode extraction depends on frequency

2. Mode extraction depends on the SNR

3. Mode extraction under short-range conditions

4. Mode extraction under nonisovelocity conditions

B. Experimental data

1. Data set description

5. Mode extraction results

VI. CONCLUSIONS

### Key Topics

- Fourier transforms
- 19.0
- Acoustic waveguides
- 7.0
- Microphones
- 7.0
- Real functions
- 7.0
- Signal processing
- 6.0

## Figures

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.

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).

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.

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.

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).

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).

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.

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 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.

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).

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.

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.

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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