^{1,a)}and Nicholas C. Makris

^{1,b)}

### Abstract

A method is derived for instantaneous source-range estimation in a horizontally stratified oceanwaveguide from passive beam-time intensity data obtained after conventional plane-wave beamforming of acoustic array measurements. The method has advantages over existing source localization methods, such as matched field processing or the waveguide invariant. First, no knowledge of the environment is required except that the received field should not be dominated by purely waterborne propagation. Second, range can be estimated in real time with little computational effort beyond plane-wave beamforming. Third, array gain is fully exploited. The method is applied to data from the Main Acoustic Clutter Experiment of 2003 for source ranges between , where it is shown that simple, accurate, and computationally efficient source range estimates can be made.

I. INTRODUCTION

II. DERIVATION OF THE ARRAY INVARIANT

A. Beam-time migration for horizontal arrays in stratified waveguides

B. Array invariant for horizontal arrays in ideal waveguides

C. Array invariant for horizontal arrays in stratified waveguides

D. Array invariant for vertical arrays in stratified waveguides

III. ILLUSTRATIVE EXAMPLES BY SIMULATION

A. Horizontal array

B. Vertical array

C. Environmental invariance

IV. EXPERIMENTAL DEMONSTRATION OF THE ARRAY INVARIANT

A. Source, receiver geometry, and environmental parameters

B. Instantaneous range estimation by the array invariant method

V. COMPARISON OF THE ARRAY INVARIANT METHOD TO OTHER RANGE ESTIMATION TECHNIQUES

VI. CONCLUSION

### Key Topics

- Acoustic waveguides
- 22.0
- Speed of sound
- 19.0
- Acoustical measurements
- 11.0
- Oceans
- 11.0
- Matched field processing
- 9.0

## Figures

The geometry of the coordinate system for a horizontal line array (a), or a vertical line array (b). The horizontal line array is aligned parallel to the -axis. The vertical line array is located along the -axis. A source is located at .

The geometry of the coordinate system for a horizontal line array (a), or a vertical line array (b). The horizontal line array is aligned parallel to the -axis. The vertical line array is located along the -axis. A source is located at .

Definition of the elevation angle and the bearing of plane waves. The angles are defined in the “coming from” direction.

Definition of the elevation angle and the bearing of plane waves. The angles are defined in the “coming from” direction.

Group velocity and modal elevation angle as a function of frequency in an ideal waveguide. The water depth and the sound speed are and , respectively, and the boundaries are assumed to be pressure release. The vertical lines at 30, 40, and will be referred to in Fig. 5.

Group velocity and modal elevation angle as a function of frequency in an ideal waveguide. The water depth and the sound speed are and , respectively, and the boundaries are assumed to be pressure release. The vertical lines at 30, 40, and will be referred to in Fig. 5.

The beam-time migration lines as a function of reduced travel time and array scan angle for various source ranges over the full frequency band shown in Fig. 3. The sound speed is and source bearing is . It can be seen that all the merge to a single beamformer migration line .

The beam-time migration lines as a function of reduced travel time and array scan angle for various source ranges over the full frequency band shown in Fig. 3. The sound speed is and source bearing is . It can be seen that all the merge to a single beamformer migration line .

(a) The beam-time migration lines for modes in the band shown in Fig. 3 as a function of reduced travel time and array scan angle for a source at and . The beam-time migration lines appear as discrete line segments. The beginning and end of each segment is marked by mode number . (b) The same as (a), but for modes in the band. As group velocity and elevation angle of a given mode changes, for that mode migrates to a different location in the beam-time plot. This migration is constrained to occur within the curve given by Eq. (8). (c) The same as (a), but for modes in the band. As the frequency band of the source increases, for the individual modes overlap to form the continuous .

(a) The beam-time migration lines for modes in the band shown in Fig. 3 as a function of reduced travel time and array scan angle for a source at and . The beam-time migration lines appear as discrete line segments. The beginning and end of each segment is marked by mode number . (b) The same as (a), but for modes in the band. As group velocity and elevation angle of a given mode changes, for that mode migrates to a different location in the beam-time plot. This migration is constrained to occur within the curve given by Eq. (8). (c) The same as (a), but for modes in the band. As the frequency band of the source increases, for the individual modes overlap to form the continuous .

The Pekeris waveguide with sand bottom, where , , and are the sound speed, density, and attenuation of the water column, and , , and are those of the sea-bottom.

The Pekeris waveguide with sand bottom, where , , and are the sound speed, density, and attenuation of the water column, and , , and are those of the sea-bottom.

(a) Beam-time image with true source range and bearing in the Pekeris sand waveguide. The dotted and dashed vertical lines are at and , respectively, where is the scan angle of the array corresponding to the global maximum of the data. The black solid line is the linear least squares fit of peak intensity angle versus time using Eq. (26). (b) The black solid line is the same as shown in (a), and the black dashed line is the linear least squares fit using Eq. (28). The two least squares fits are nearly identical to each other.

(a) Beam-time image with true source range and bearing in the Pekeris sand waveguide. The dotted and dashed vertical lines are at and , respectively, where is the scan angle of the array corresponding to the global maximum of the data. The black solid line is the linear least squares fit of peak intensity angle versus time using Eq. (26). (b) The black solid line is the same as shown in (a), and the black dashed line is the linear least squares fit using Eq. (28). The two least squares fits are nearly identical to each other.

(a) Vertical wavenumber versus frequency for modes in the Pekeris sand waveguide of Fig. 6. Each horizontal line corresponds to a specific mode. Higher-order modes have higher wavenumbers. (b) Frequency derivatives of . This figure shows that is effectively a constant function of frequency so that relation (13) is satisfied for the Pekeris waveguide except near mode cut-off.

(a) Vertical wavenumber versus frequency for modes in the Pekeris sand waveguide of Fig. 6. Each horizontal line corresponds to a specific mode. Higher-order modes have higher wavenumbers. (b) Frequency derivatives of . This figure shows that is effectively a constant function of frequency so that relation (13) is satisfied for the Pekeris waveguide except near mode cut-off.

(a) Beam-time image identical to that in Fig. 7. The black solid line is the exact beam-time migration line given in Eq. (12), for modes up to . The last four modes with mode cut-off in the band, as shown in Fig. 8, are neglected. The gray solid line is the beam-time migration line for non-waterborne modes from Eq. (14). (b) The black and gray solid lines are the detailed shapes of the same and shown in (a). The two least squares fits and in Fig. 7, overlain as gray dashed and dotted lines, show good agreement with the exact beam-time migration line .

(a) Beam-time image identical to that in Fig. 7. The black solid line is the exact beam-time migration line given in Eq. (12), for modes up to . The last four modes with mode cut-off in the band, as shown in Fig. 8, are neglected. The gray solid line is the beam-time migration line for non-waterborne modes from Eq. (14). (b) The black and gray solid lines are the detailed shapes of the same and shown in (a). The two least squares fits and in Fig. 7, overlain as gray dashed and dotted lines, show good agreement with the exact beam-time migration line .

(a) Beam-time image for in the Pekeris sand waveguide. The black solid line is the linear least squares fit of peak intensity versus time calculated using Eq. (30). The gray solid line is the beam-time migration line for non-waterborne modes, , in Eq. (18). (b) The black solid line is the exact beam-time migration line . The gray solid and dashed lines are and in (a), respectively. It can be seen that the exact beam-time migration line can be well approximated by the least squares fit .

(a) Beam-time image for in the Pekeris sand waveguide. The black solid line is the linear least squares fit of peak intensity versus time calculated using Eq. (30). The gray solid line is the beam-time migration line for non-waterborne modes, , in Eq. (18). (b) The black solid line is the exact beam-time migration line . The gray solid and dashed lines are and in (a), respectively. It can be seen that the exact beam-time migration line can be well approximated by the least squares fit .

The same as Fig. 9(a), but for the deep Pekeris sand waveguide. The exact beam-time migration line is plotted for the first 36 modes of the 41 propagating modes. It can be seen by comparison of Fig. 9(a) and Fig. 11 that the exact beam-time migration line in a Pekeris waveguide is invariant over the waveguide depth.

The same as Fig. 9(a), but for the deep Pekeris sand waveguide. The exact beam-time migration line is plotted for the first 36 modes of the 41 propagating modes. It can be seen by comparison of Fig. 9(a) and Fig. 11 that the exact beam-time migration line in a Pekeris waveguide is invariant over the waveguide depth.

Horizontally stratified waveguide with linear sound speed gradient. The sound speed is constant up to depth, and linearly decreases to at depth. The density and attenuation of the water column are the same as those in Fig. 6, but the geoacoustic parameters of the sea-bottom are assumed to be different.

Horizontally stratified waveguide with linear sound speed gradient. The sound speed is constant up to depth, and linearly decreases to at depth. The density and attenuation of the water column are the same as those in Fig. 6, but the geoacoustic parameters of the sea-bottom are assumed to be different.

Beam-time image for and in the environment shown in Fig. 12. The exact beam-time migration line is plotted for the first 27 modes of the 31 propagating modes. The exact beam-time migration line is nearly identical to that of the Pekeris waveguide shown in Fig. 9, and it effectively spans the entire line.

Beam-time image for and in the environment shown in Fig. 12. The exact beam-time migration line is plotted for the first 27 modes of the 31 propagating modes. The exact beam-time migration line is nearly identical to that of the Pekeris waveguide shown in Fig. 9, and it effectively spans the entire line.

Mode shape of the first ten modes at 390 and , for the environment shown in Fig. 12. Only the first three modes are waterborne since they are trapped in the refract-bottom-reflect sound speed channel between and shown in Fig. 12.

Mode shape of the first ten modes at 390 and , for the environment shown in Fig. 12. Only the first three modes are waterborne since they are trapped in the refract-bottom-reflect sound speed channel between and shown in Fig. 12.

Vertical wavenumber of the first ten modes in the environment shown in Fig. 12. The solid lines represent , and the dashed lines represent . Only the first three modes are waterborne, and exhibit rapid change of versus frequency.

Vertical wavenumber of the first ten modes in the environment shown in Fig. 12. The solid lines represent , and the dashed lines represent . Only the first three modes are waterborne, and exhibit rapid change of versus frequency.

The source position and the two receiver ship tracks on May 7, 2003. The source to receiver distance varied from to . The depth contour of the sea-bottom in meters is also shown in the figure. The arrows show the heading of the receiver ship along the tracks. The origin of the coordinates in Figs. 16 and 17 is at 38.955°N and 73.154°W.

The source position and the two receiver ship tracks on May 7, 2003. The source to receiver distance varied from to . The depth contour of the sea-bottom in meters is also shown in the figure. The arrows show the heading of the receiver ship along the tracks. The origin of the coordinates in Figs. 16 and 17 is at 38.955°N and 73.154°W.

The source position and the two receiver ship tracks on May 1, 2003. The source to receiver distance varied from to .

The source position and the two receiver ship tracks on May 1, 2003. The source to receiver distance varied from to .

Sound speed profiles measured by XBT’s during the MAE 2003. Two XBTs were deployed for tracks 141a̱1 and 141ḏ1 [(a) and (b)], and three XBTs were deployed for tracks 84̱1 and 85̱4 [(c) and (d)]. The Greenwich Mean Time of the deployment are shown in the parentheses.

Sound speed profiles measured by XBT’s during the MAE 2003. Two XBTs were deployed for tracks 141a̱1 and 141ḏ1 [(a) and (b)], and three XBTs were deployed for tracks 84̱1 and 85̱4 [(c) and (d)]. The Greenwich Mean Time of the deployment are shown in the parentheses.

(a) The beam-time sound pressure level image measured during the MAE 2003. The dotted vertical line is at , and the dashed vertical line is at , where and . The slanted line is the linear least squares fit of peak beam-time migration. The receiver depth is . (b) Simulation of the measurement shown in (a) using the sound speed profile in Fig. 18(b) XBT3. The positions of and are nearly identical. The slant line is up to the 20th mode.

(a) The beam-time sound pressure level image measured during the MAE 2003. The dotted vertical line is at , and the dashed vertical line is at , where and . The slanted line is the linear least squares fit of peak beam-time migration. The receiver depth is . (b) Simulation of the measurement shown in (a) using the sound speed profile in Fig. 18(b) XBT3. The positions of and are nearly identical. The slant line is up to the 20th mode.

Vertical wavenumbers at calculated using the sound speed profile in Fig. 18(b) XBT3. This figure shows that relation (13) is satisfied for the MAE waveguide so that the array invariant method should be applicable. This is because the vertical wavenumber is effectively a constant function of frequency.

Vertical wavenumbers at calculated using the sound speed profile in Fig. 18(b) XBT3. This figure shows that relation (13) is satisfied for the MAE waveguide so that the array invariant method should be applicable. This is because the vertical wavenumber is effectively a constant function of frequency.

Experimental range estimates using the array invariant method. The solid lines show measured by GPS. The cross marks show estimated by the array invariant method. (a) Track 141a̱1: 66 range estimates are shown, and 3 noise-corrupted data are ignored. The rms error is . (b) Track 141ḏ1:58 range estimates are shown, and 4 noise-corrupted data is ignored. The rms error is . (c) Track 84̱1: 61 range estimates are shown, and 8 noise-corrupted data are ignored. The rms error is . (d) Track 85̱4: 56 range estimates are shown, and 6 noise-corrupted data are ignored. The rms error is .

Experimental range estimates using the array invariant method. The solid lines show measured by GPS. The cross marks show estimated by the array invariant method. (a) Track 141a̱1: 66 range estimates are shown, and 3 noise-corrupted data are ignored. The rms error is . (b) Track 141ḏ1:58 range estimates are shown, and 4 noise-corrupted data is ignored. The rms error is . (c) Track 84̱1: 61 range estimates are shown, and 8 noise-corrupted data are ignored. The rms error is . (d) Track 85̱4: 56 range estimates are shown, and 6 noise-corrupted data are ignored. The rms error is .

Experimental range estimates using the array invariant method. The range estimates versus GPS measured ranges for tracks 141a̱1, 141ḏ1, 84̱1, and 85̱4 plotted in logarithmic scale. The solid line is the linear regression , where the regression coefficient and the intercept . The correlation coefficient is 0.835.

Experimental range estimates using the array invariant method. The range estimates versus GPS measured ranges for tracks 141a̱1, 141ḏ1, 84̱1, and 85̱4 plotted in logarithmic scale. The solid line is the linear regression , where the regression coefficient and the intercept . The correlation coefficient is 0.835.

Incoherent acoustic intensity measured over the array aperture during the MAE 2003. (a) is one of the measurements from Track 141a̱1, and (b) is the incoherent intensity of the same data shown in Fig. 19 from Track 141ḏ1. The receiver array has 64 channels, the number of which are shown on top of the figures. The range from each channel to the source is shown at the bottom of the figures. The black lines are the interference patterns for (—) (---), (-∙-∙), and (⋯), respectively, calculated using Eq. (31). Variation of from 1 by more than a factor of 2 can be observed.

Incoherent acoustic intensity measured over the array aperture during the MAE 2003. (a) is one of the measurements from Track 141a̱1, and (b) is the incoherent intensity of the same data shown in Fig. 19 from Track 141ḏ1. The receiver array has 64 channels, the number of which are shown on top of the figures. The range from each channel to the source is shown at the bottom of the figures. The black lines are the interference patterns for (—) (---), (-∙-∙), and (⋯), respectively, calculated using Eq. (31). Variation of from 1 by more than a factor of 2 can be observed.

(a) Track 141a̱1: The waveguide invariant parameters calculated using the sound speed profile in Fig. 18(a) XBT2 at . (b) Track 141ḏ1: The waveguide invariant parameters calculated using the sound speed profile in Fig. 18(b) XBT3 at . It can be seen that roughly a factor of 2 change in has occurred in less than 2 hours.

(a) Track 141a̱1: The waveguide invariant parameters calculated using the sound speed profile in Fig. 18(a) XBT2 at . (b) Track 141ḏ1: The waveguide invariant parameters calculated using the sound speed profile in Fig. 18(b) XBT3 at . It can be seen that roughly a factor of 2 change in has occurred in less than 2 hours.

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