^{1}, Mark Andrews

^{1}and Purnima Ratilal

^{1,a)}

### Abstract

The probability distribution of ocean-acoustic broadband signal energy after saturated multipath propagation is derived using coherence theory. The frequency components obtained from Fourier decomposition of a broadband signal are *each* assumed to be fully saturated with energy spectral density that obey the exponential distribution with 5.6 dB standard deviation and unity scintillation index. When the signal bandwidth and measurement time are larger than the correlation bandwidth and correlation time, respectively, of its energy spectral density components, the broadband signal energy obtained by integrating the energy spectral density across the signal bandwidth then follows the Gamma distribution with a standard deviation smaller than 5.6 dB and a scintillation index less than unity. The theory is verified with broadband transmissions in the Gulf of Maine shallow water waveguide in the 300 to 1200 Hz frequency range. The standard deviations of received broadband signal energies range from 2.7 to 4.6 dB for effective bandwidths up to 42 Hz, while the standard deviations of individual energy spectral density components are roughly 5.6 dB. The energy spectral density correlation bandwidths of the received broadband signals are found to be larger for signals with higher center frequencies and are roughly 10% of each center frequency.

This research was funded by the Office of Naval Research Ocean Acoustics Program.

I. INTRODUCTION

II. PROBABILITY DISTRIBUTION FOR SATURATED BROADBAND SIGNAL ENERGY AS A FUNCTION OF SIGNAL BANDWIDTH, MEASUREMENT TIME, FREQUENCY CORRELATION, AND TEMPORAL CORRELATION

A. Probability distribution for the energy of a broadband signal with measurement time smaller than energy spectral density correlation time

B. Probability distribution of broadband signal energy with a measurement time larger than the energy spectral density correlation time

III. BROADBAND TRANSMISSION SCINTILLATION STATISTICS FROM THE 2006 GULF OF MAINE EXPERIMENT

A. Experimental data collection and processing

B. Probability distribution of log-transformed bandwidth-averaged energy spectral densities

C. Mean and standard deviation of energy spectral density across signal bandwidth

D. Dependence of signal energy standard deviation and scintillation index on bandwidth and center frequency

E. Energy spectral density correlation bandwidth from measured broadband data

F. Temporal averaging and broadband spectrum reconstruction in a random multi-modal range-dependent ocean waveguide

G. Discussion and comparison to other shallow water measurements

IV. CONCLUSION

### Key Topics

- Time measurement
- 25.0
- Intracellular signaling
- 20.0
- Rotational correlation time
- 17.0
- Probability theory
- 14.0
- Statistical analysis
- 11.0

##### G01H

## Figures

(A) Locations of source and receiver during GOME'06 (Refs. 5 and 6). Isobath contours have the unit of *m*. The sound speed profiles shown in Fig. 2(A) were collected roughly at the beginning, middle, and end of each receiver track and at the source locations. (B) Normalized histogram of the number of transmissions as a function of source and receiver separation or range.

(A) Locations of source and receiver during GOME'06 (Refs. 5 and 6). Isobath contours have the unit of *m*. The sound speed profiles shown in Fig. 2(A) were collected roughly at the beginning, middle, and end of each receiver track and at the source locations. (B) Normalized histogram of the number of transmissions as a function of source and receiver separation or range.

(A) Sound speed profiles and (B) buoyancy frequency profiles obtained from XBT and CTD measurements at the experimental site. A total of 185 sound speed profiles and 35 buoyancy frequency profiles are shown.

(A) Sound speed profiles and (B) buoyancy frequency profiles obtained from XBT and CTD measurements at the experimental site. A total of 185 sound speed profiles and 35 buoyancy frequency profiles are shown.

Example of source waveform and received broadband signal at a range of 7.6 km for the Tukey-windowed linear frequency modulated pulse centered at = 415 Hz. (A) Source waveform, (B) source spectrum, (C) received signal waveform, and (D) received signal spectrum. The results are normalized for a 0 dB re 1 *μ*Pa at 1 m source level.

Example of source waveform and received broadband signal at a range of 7.6 km for the Tukey-windowed linear frequency modulated pulse centered at = 415 Hz. (A) Source waveform, (B) source spectrum, (C) received signal waveform, and (D) received signal spectrum. The results are normalized for a 0 dB re 1 *μ*Pa at 1 m source level.

Modeled transmission loss for 50-Hz bandwidth Tukey-windowed broadband signals centered at (A) = 415 Hz and (B) = 1125 Hz, calculated using the range-dependent parabolic equation model (Ref. 34). Transmission losses are obtained by averaging over 20 independent Monte Carlo realizations of the broadband signal in the Gulf of Maine environment randomized by internal waves.

Modeled transmission loss for 50-Hz bandwidth Tukey-windowed broadband signals centered at (A) = 415 Hz and (B) = 1125 Hz, calculated using the range-dependent parabolic equation model (Ref. 34). Transmission losses are obtained by averaging over 20 independent Monte Carlo realizations of the broadband signal in the Gulf of Maine environment randomized by internal waves.

Histograms showing distribution of measured log-transformed bandwidth-averaged energy spectral densities received in the 7 to 9 km range for two center frequencies = 415 Hz (left) and = 1125 Hz (right) with (A) and (B) 0.5 Hz bandwidth (nearly monochromatic components), and (C) and (D) 50 Hz bandwidth Tukey windowed signals with an effective bandwidth of 42 Hz. The histograms are overlain with thetheoretical exponential-Gamma distribution modeled using Eq. (8) (black curve), with the number of frequency correlation cells determined from the data mean and standard deviation. The exponential-Gamma distribution corresponding to assumed is also shown for comparison.

Histograms showing distribution of measured log-transformed bandwidth-averaged energy spectral densities received in the 7 to 9 km range for two center frequencies = 415 Hz (left) and = 1125 Hz (right) with (A) and (B) 0.5 Hz bandwidth (nearly monochromatic components), and (C) and (D) 50 Hz bandwidth Tukey windowed signals with an effective bandwidth of 42 Hz. The histograms are overlain with thetheoretical exponential-Gamma distribution modeled using Eq. (8) (black curve), with the number of frequency correlation cells determined from the data mean and standard deviation. The exponential-Gamma distribution corresponding to assumed is also shown for comparison.

Mean and standard deviation of the log-transformed energy spectral density *L _{ε} * as a function of frequency for broadband signals received between the 7 and 9 km range, centered at (A) = 415 Hz and (B) = 1125 Hz.

Mean and standard deviation of the log-transformed energy spectral density *L _{ε} * as a function of frequency for broadband signals received between the 7 and 9 km range, centered at (A) = 415 Hz and (B) = 1125 Hz.

(A) Empirically measured standard deviations of the log-transformed bandwidth-averaged energy spectral densities obtained from broadband transmissions in the Gulf of Maine shown as points. (B) The number of frequency correlation cells are obtained from the measured signal standard deviations via Eq. (10). The dotted curves in (A) and (B) are obtained from the minimum mean-squared error fit to the data points using the equation and coefficients in Table II. The error bar shown applies to all data points.

(A) Empirically measured standard deviations of the log-transformed bandwidth-averaged energy spectral densities obtained from broadband transmissions in the Gulf of Maine shown as points. (B) The number of frequency correlation cells are obtained from the measured signal standard deviations via Eq. (10). The dotted curves in (A) and (B) are obtained from the minimum mean-squared error fit to the data points using the equation and coefficients in Table II. The error bar shown applies to all data points.

Measured SIs in the Gulf of Maine for all four center frequencies as a function of relative bandwidth. The error bar shown applies to all data points.

Measured SIs in the Gulf of Maine for all four center frequencies as a function of relative bandwidth. The error bar shown applies to all data points.

Average energy spectral density correlation coefficient calculated from received broadband signals at the four center frequencies shown as a function of frequency shift within the signal bandwidth.

Average energy spectral density correlation coefficient calculated from received broadband signals at the four center frequencies shown as a function of frequency shift within the signal bandwidth.

Log-transformed time-averaged energy spectral density calculated from received broadband signals in the 7 to 9 km range for thewaveforms centered at (A) = 415 Hz and (B) = 1125 Hz.

Log-transformed time-averaged energy spectral density calculated from received broadband signals in the 7 to 9 km range for thewaveforms centered at (A) = 415 Hz and (B) = 1125 Hz.

## Tables

Two-sided chi-squared test results to verify the distributions of the log-transformed bandwidth-averaged energy spectral density for the four scenarios shown in Fig. 5. A significance level of *α* = 0.05 gives *χ* ^{2} within the range from lower-tail to upper-tail critical values for both the Gamma and exponential distributions.

Two-sided chi-squared test results to verify the distributions of the log-transformed bandwidth-averaged energy spectral density for the four scenarios shown in Fig. 5. A significance level of *α* = 0.05 gives *χ* ^{2} within the range from lower-tail to upper-tail critical values for both the Gamma and exponential distributions.

Empirically determined number of frequency correlation cells is related to relative bandwidth by the “inverted exponential decay” relationship *,* with coefficients *A* and *k* determined by curve fitting as shown in Fig. 7. The case corresponds to one unique independent fluctuation. When *B* becomes very large, tends to *A*, its upper saturation value for each center frequency, which is 3 for the lowest frequency = 415 Hz and 1.6 for the highest frequency = 1125 Hz.

Empirically determined number of frequency correlation cells is related to relative bandwidth by the “inverted exponential decay” relationship *,* with coefficients *A* and *k* determined by curve fitting as shown in Fig. 7. The case corresponds to one unique independent fluctuation. When *B* becomes very large, tends to *A*, its upper saturation value for each center frequency, which is 3 for the lowest frequency = 415 Hz and 1.6 for the highest frequency = 1125 Hz.

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