^{1,a)}, B. G. Kinda

^{2}, J. Bonnel

^{2}, Y. Stéphan

^{3}and S. Vallez

^{4}

### Abstract

An inversion scheme is proposed, relying upon the inversion of the noise of a moving ship measured on a single distant hydrophone. The spectrogram of the measurements exhibits striations which depend on waveguide parameters. The periodic behavior of striations versus range are used to estimate the differences of radial wavenumber between couples of propagative modes at a given frequency. These wavenumber differences are stacked for several frequencies to form the relative dispersion curves. Such relative dispersion curves can be synthesized using a propagation model feeded with a bottom geoacoustic model. Inversion is performed by looking for the bottom properties that optimize the fit between measured and predicted relative dispersion curves. The inversion scheme is tested on simulated data. The conclusions are twofold: (1) a minimum 6 dB signal to noise ratio is required to obtained an unbiased estimate of compressional sound speed in the bottom with a 3 m s^{−1} standard deviation; however, even with low signal to noise ratio, the estimation error remains bounded and (2) in the case of a multi-layer bottom, the scheme produces a single depth-average compressional sound speed. The inversion scheme is applied on experimental data. The results are fully consistent with a core sample measured around the receiving hydrophone.

I. INTRODUCTION

II. THE MOVEBOAT2006 EXPERIMENTAL SETUP

III. FORWARD MODELING

A. Propagation and source movement

B. Channel propagation features

IV. INVERSION METHOD

V. APPLICATION IN A VERY SHALLOW WATER ENVIRONMENT

A. Signal to noise ratio estimation

B. Simulations

1. Study 1: Impact of noise on estimation accuracy

2. Study 2: Impact of a structure mismatch between guessed waveguide and true waveguide

C. Experimental data

VI. DISCUSSION

A. Losses and gains of our passive inversion scheme

B. Sensitivity to the hypothesis of a range-independent waveguide

C. Hamilton parameterization of the sediment

VII. CONCLUSION

### Key Topics

- Marine vessels
- 31.0
- Marine vehicle noise
- 27.0
- Acoustic noise
- 23.0
- Speed of sound
- 23.0
- Microphones
- 15.0

## Figures

MOVEBOAT2006 chart presenting Vilanova i la Geltru harbor, isobath line, hydrophone’s position, fish farm and typical tracks (parallel to the shoreline above the 15 meter isobath) of cooperative DOMIGO trawler.

MOVEBOAT2006 chart presenting Vilanova i la Geltru harbor, isobath line, hydrophone’s position, fish farm and typical tracks (parallel to the shoreline above the 15 meter isobath) of cooperative DOMIGO trawler.

Schematic RDC curves in the (*k*, *f*) plane. For frequencies lower than *f* _{2} only mode 1 propagates, no interference exists. For frequencies in [*f* _{2}, *f* _{3}] modes 1 and 2 propagate and interfere together to create a single curve between *f* _{1} and *f* _{2} along *k* _{2}(*f*) − *k* _{1}(*f*). For frequencies higher than *f* _{3}, modes 1, 2, and 3 propagate ant interfere together to created three relative dispersion curves located along *k* _{2}(*f*) − *k* _{1}(*f*), *k* _{3}(*f*) − *k* _{1}(*f*), and *k* _{3}(*f*) *− k* _{2}(*f*).

Schematic RDC curves in the (*k*, *f*) plane. For frequencies lower than *f* _{2} only mode 1 propagates, no interference exists. For frequencies in [*f* _{2}, *f* _{3}] modes 1 and 2 propagate and interfere together to create a single curve between *f* _{1} and *f* _{2} along *k* _{2}(*f*) − *k* _{1}(*f*). For frequencies higher than *f* _{3}, modes 1, 2, and 3 propagate ant interfere together to created three relative dispersion curves located along *k* _{2}(*f*) − *k* _{1}(*f*), *k* _{3}(*f*) − *k* _{1}(*f*), and *k* _{3}(*f*) *− k* _{2}(*f*).

Inversion scheme diagram.

Inversion scheme diagram.

Flowchart to define and optimize the objective function J. Measurement are processed in box *B* _{1} to compute *I* _{mes}(*k, f*). In the box *B _{2},* the map

*I*

_{mes}(

*k*,

*f*) is matched to a synthetic binary map

*M*(

*k*,

*f*,

*θ*) obtained from simulation. Applied on

*I*

_{mes}(

*k*,

*f*) the goal of

*M*(

*k*,

*fθ*) is to extract the power contained by the measurement around some simulated RDC. To do so, for a given

*θ*the simulated RDC are computed in box

*B*

_{3}, RDC appear to have an ideal infinite resolution in the (

*k, f*) plane. To account for the bounded range

*R*

_{max}

*− R*

_{min}of the measurements, each RDC is broaden by the expected resolution of

*I*(

_{m}*k, f*) [i.e., 2

*π/*(

*R*

_{max}

*− R*

_{min})] to form a binary masking map in the (

*k*,

*f*) plane with one around the simulated RDC and zero elsewhere. An optimization procedure is applied on

*θ*to optimize the amount of power of

*I*(

_{m}*k, f*) contained in the mask

*M*(

*k, f*,

*θ*).

Flowchart to define and optimize the objective function J. Measurement are processed in box *B* _{1} to compute *I* _{mes}(*k, f*). In the box *B _{2},* the map

*I*

_{mes}(

*k*,

*f*) is matched to a synthetic binary map

*M*(

*k*,

*f*,

*θ*) obtained from simulation. Applied on

*I*

_{mes}(

*k*,

*f*) the goal of

*M*(

*k*,

*fθ*) is to extract the power contained by the measurement around some simulated RDC. To do so, for a given

*θ*the simulated RDC are computed in box

*B*

_{3}, RDC appear to have an ideal infinite resolution in the (

*k, f*) plane. To account for the bounded range

*R*

_{max}

*− R*

_{min}of the measurements, each RDC is broaden by the expected resolution of

*I*(

_{m}*k, f*) [i.e., 2

*π/*(

*R*

_{max}

*− R*

_{min})] to form a binary masking map in the (

*k*,

*f*) plane with one around the simulated RDC and zero elsewhere. An optimization procedure is applied on

*θ*to optimize the amount of power of

*I*(

_{m}*k, f*) contained in the mask

*M*(

*k, f*,

*θ*).

Accuracy of our inversion scheme versus SNR, mean value and standard deviation of compressional sound speed of the bottom estimates (*N* = 100 runs of independent simulations for each SNR value). For SNR < 6 dB, estimates are biased, for SNR > 6 dB estimates are unbiased and have a standard deviation less than 3 m s^{−1}.

Accuracy of our inversion scheme versus SNR, mean value and standard deviation of compressional sound speed of the bottom estimates (*N* = 100 runs of independent simulations for each SNR value). For SNR < 6 dB, estimates are biased, for SNR > 6 dB estimates are unbiased and have a standard deviation less than 3 m s^{−1}.

(a) Shape of objective function J: SNR = 2.5 dB (dots); SNR = 5.58 dB (triangles); SNR = 10,70 dB (crosses); SNR = 23.6 dB (circle); SNR = 63.5 dB (diamond); (b) comparison of normalized criterium shape depending on SNR values. The highest the SNR is, the greater the maximum value of the objective function (a) and the narrower the peak around the maximum (b).

(a) Shape of objective function J: SNR = 2.5 dB (dots); SNR = 5.58 dB (triangles); SNR = 10,70 dB (crosses); SNR = 23.6 dB (circle); SNR = 63.5 dB (diamond); (b) comparison of normalized criterium shape depending on SNR values. The highest the SNR is, the greater the maximum value of the objective function (a) and the narrower the peak around the maximum (b).

Estimated compressional sound speed of half space bottom for different thick nesses of the sediment layer (crosses); rock basement compressional sound speed (circles); compressional sound speed of the sediment layer (triangles). For a thin sediment layer (area 1), the estimated compressional sound speed is similar to the basement one; for thick sediment layer (area 3), the estimated compressional sound speed in similar to the sediment layer ones, whereas for a middle thickness (area 2), the estimated compressional sound speed is a depth average between sediment and basement ones.

Estimated compressional sound speed of half space bottom for different thick nesses of the sediment layer (crosses); rock basement compressional sound speed (circles); compressional sound speed of the sediment layer (triangles). For a thin sediment layer (area 1), the estimated compressional sound speed is similar to the basement one; for thick sediment layer (area 3), the estimated compressional sound speed in similar to the sediment layer ones, whereas for a middle thickness (area 2), the estimated compressional sound speed is a depth average between sediment and basement ones.

(a) Shape of the cost function J depending on the upper layer thickness versus relative compressional sound speed of the guessed sediment layer (i.e., value - value which optimizes J): 10 m thick sediment layer (circle); 5 m intermediate sediment layer (crosses); 0.25 m thin sediment layer (points). (b) Normalized criterion depending on the layer thickness with the same meaning as below; a middle thickness of sediment layer (curves with crosses) creates a small decrease in the maximum value of J [(a) from 0.6 to 0.56], and a small widening of the peak around the maximum (b).

(a) Shape of the cost function J depending on the upper layer thickness versus relative compressional sound speed of the guessed sediment layer (i.e., value - value which optimizes J): 10 m thick sediment layer (circle); 5 m intermediate sediment layer (crosses); 0.25 m thin sediment layer (points). (b) Normalized criterion depending on the layer thickness with the same meaning as below; a middle thickness of sediment layer (curves with crosses) creates a small decrease in the maximum value of J [(a) from 0.6 to 0.56], and a small widening of the peak around the maximum (b).

(a) and (b) *I* _{mes}(*t*, *f*) for tracks 1 and 2, white boxes identify the data used to compute *I* _{mes}(*k*, *f*), the ship’s range in these boxes is approximately 1500 m between 300 m and 1800 m, striations are clearly visible on *I* _{mes}(*t, f*). (c) and (d) *I* _{mes}(*k*, *f*) for tracks 1 and 2, and corresponding inverted RDC curves (in black) a good match between local maxima of *I _{m} *(

*k*,

*f*) and optimal theoretical RDC is visible. (e) Objective functions for track 1: real data SNR = 12.5 dB (continuous line); simulated data with ship’s range = 1500 m and SNR = 12.5 dB (crosses); simulated data with ship’s range = 750 m and SNR = 12.5 dB (triangles). (f) Objective functions for track 2: real data SNR = 9.8 dB (continuous line); simulated data with ship’s range = 1500 m and SNR = 9.8 dB (crosses); simulated data with ship’s range = 750 m and SNR = 9.8 dB (triangles).

(a) and (b) *I* _{mes}(*t*, *f*) for tracks 1 and 2, white boxes identify the data used to compute *I* _{mes}(*k*, *f*), the ship’s range in these boxes is approximately 1500 m between 300 m and 1800 m, striations are clearly visible on *I* _{mes}(*t, f*). (c) and (d) *I* _{mes}(*k*, *f*) for tracks 1 and 2, and corresponding inverted RDC curves (in black) a good match between local maxima of *I _{m} *(

*k*,

*f*) and optimal theoretical RDC is visible. (e) Objective functions for track 1: real data SNR = 12.5 dB (continuous line); simulated data with ship’s range = 1500 m and SNR = 12.5 dB (crosses); simulated data with ship’s range = 750 m and SNR = 12.5 dB (triangles). (f) Objective functions for track 2: real data SNR = 9.8 dB (continuous line); simulated data with ship’s range = 1500 m and SNR = 9.8 dB (crosses); simulated data with ship’s range = 750 m and SNR = 9.8 dB (triangles).

## Tables

Recorded data during the MOVEBOAT2006 experiment.

Recorded data during the MOVEBOAT2006 experiment.

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