^{1}, Michele Zanolin

^{2}, Purnima Ratilal

^{3}, Tianrun Chen

^{4}and Nicholas C. Makris

^{5,a)}

### Abstract

A method is provided for determining necessary conditions on sample size or signal to noise ratio (SNR) to obtain accurate parameter estimates from remote sensing measurements in fluctuating environments. These conditions are derived by expanding the bias and covariance of maximum likelihood estimates (MLEs) in inverse orders of sample size or SNR, where the first-order covariance term is the Cramer-Rao lower bound (CRLB). Necessary sample sizes or SNRs are determined by requiring that (i) the first-order bias and the second-order covariance are much smaller than the true parameter value and the CRLB, respectively, and (ii) the CRLB falls within desired error thresholds. An analytical expression is provided for the second-order covariance of MLEs obtained from general complex Gaussian data vectors, which can be used in many practical problems since (i) data distributions can often be assumed to be Gaussian by virtue of the central limit theorem, and (ii) it allows for *both* the mean and variance of the measurement to be functions of the estimation parameters. Here, conditions are derived to obtain accurate source localization estimates in a fluctuating oceanwaveguide containing random internal waves, and the consequences of the loss of coherence on their accuracy are quantified.

I. INTRODUCTION

II. GENERAL ASYMPTOTIC EXPANSIONS FOR THE BIAS AND COVARIANCE OF THE MLE

A. Asymptotic statistics of the MLE

B. General multivariate Gaussian data

C. Mean and variance of the measured field

III. ILLUSTRATIVE EXAMPLES

A. Undisturbed waveguide

B. Waveguide containing internal waves

1. Importance of the joint-moment terms in calculating the second-order covariance

C. Discussion

IV. CONCLUSIONS

### Key Topics

- Acoustic waveguides
- 27.0
- Internal waves
- 23.0
- Acoustic source localization
- 12.0
- Coherence
- 10.0
- Intense sound sources
- 10.0

## Figures

Geometry of an ocean waveguide environment with two-layer water column of total depth , and upper layer depth of . The density and sound speed in the upper layer are and , respectively. The density and sound speed in the lower layer are and , respectively. The bottom sediment half-space is composed of sand with density and sound speed . The attenuations in the water column and bottom are and , respectively. The internal wave disturbances have coherence length scales and in the and directions, respectively, and are measured with positive height measured downward from the interface between the upper and lower water layers. The internal wave disturbances, when present, are assumed to have a height standard deviation of . In the case of a deterministic waveguide with no internal waves, .

Geometry of an ocean waveguide environment with two-layer water column of total depth , and upper layer depth of . The density and sound speed in the upper layer are and , respectively. The density and sound speed in the lower layer are and , respectively. The bottom sediment half-space is composed of sand with density and sound speed . The attenuations in the water column and bottom are and , respectively. The internal wave disturbances have coherence length scales and in the and directions, respectively, and are measured with positive height measured downward from the interface between the upper and lower water layers. The internal wave disturbances, when present, are assumed to have a height standard deviation of . In the case of a deterministic waveguide with no internal waves, .

Signal to additive noise ratio (SANR) at 415 Hz in an undisturbed waveguide with no internal waves. The SANR received at the 10-element vertical array described in Sec. III is plotted as a function of source range and depth . The observed range-depth pattern is due to the underlying modal coherence structure of the total acoustic field intensity. The receiver array is centered at and . The source level is fixed as a constant over range so that is 0 dB at 1 km source range at all source depths. For the undisturbed waveguide, SANR is equivalent to signal to noise ratio (SNR).

Signal to additive noise ratio (SANR) at 415 Hz in an undisturbed waveguide with no internal waves. The SANR received at the 10-element vertical array described in Sec. III is plotted as a function of source range and depth . The observed range-depth pattern is due to the underlying modal coherence structure of the total acoustic field intensity. The receiver array is centered at and . The source level is fixed as a constant over range so that is 0 dB at 1 km source range at all source depths. For the undisturbed waveguide, SANR is equivalent to signal to noise ratio (SNR).

Ocean acoustic localization MLE behavior given a single sample for (a) range estimation and (b) depth estimation for a 415 Hz source placed at 50 m depth in an undisturbed waveguide with no internal waves. The MLE first-order bias magnitude (solid line), square root of the CRLB (circle marks) and square root of the second-order variance (cross marks), as well as the measured signal to additive noise ratio (SANR, dashed line) are plotted as functions of source range. Given the necessary sample size conditions in Eq. (6), whenever the first-order bias and the second-order variance attain roughly 10% of the true parameter value and the CRLB, respectively, more than a single sample will be needed to obtain unbiased, minimum variance MLEs. The source level is fixed as a constant over range so that is 0 dB at 1 km source range.

Ocean acoustic localization MLE behavior given a single sample for (a) range estimation and (b) depth estimation for a 415 Hz source placed at 50 m depth in an undisturbed waveguide with no internal waves. The MLE first-order bias magnitude (solid line), square root of the CRLB (circle marks) and square root of the second-order variance (cross marks), as well as the measured signal to additive noise ratio (SANR, dashed line) are plotted as functions of source range. Given the necessary sample size conditions in Eq. (6), whenever the first-order bias and the second-order variance attain roughly 10% of the true parameter value and the CRLB, respectively, more than a single sample will be needed to obtain unbiased, minimum variance MLEs. The source level is fixed as a constant over range so that is 0 dB at 1 km source range.

Undisturbed waveguide. , where is the sample size necessary to obtain an unbiased source range MLE whose MSE attains the CRLB and has , where , , are calculated using Eqs. (6) and (7), given a 415 Hz source at 50 m depth. Source level is fixed as a constant over range so that is 0 dB at 1, 10, 20, and 30 km source range (black circles), respectively, for the four curves shown.

Undisturbed waveguide. , where is the sample size necessary to obtain an unbiased source range MLE whose MSE attains the CRLB and has , where , , are calculated using Eqs. (6) and (7), given a 415 Hz source at 50 m depth. Source level is fixed as a constant over range so that is 0 dB at 1, 10, 20, and 30 km source range (black circles), respectively, for the four curves shown.

Undisturbed waveguide. of the square root of the CRLB for (a) source range , (b) source depth MLEs given a single sample. , the sample sizes or SNRs necessary to obtain (c) source range, (d) source depth MLEs that become unbiased and have MSEs that attain the CRLB. Given any design error threshold, the sample size necessary to obtain an accurate source range or depth MLE is then equal to , where . The source level is fixed as a constant over range so that is 0 dB at 1 km source range at all source depths.

Undisturbed waveguide. of the square root of the CRLB for (a) source range , (b) source depth MLEs given a single sample. , the sample sizes or SNRs necessary to obtain (c) source range, (d) source depth MLEs that become unbiased and have MSEs that attain the CRLB. Given any design error threshold, the sample size necessary to obtain an accurate source range or depth MLE is then equal to , where . The source level is fixed as a constant over range so that is 0 dB at 1 km source range at all source depths.

(a) Signal to additive noise ratio (SANR), (b) signal to noise ratio (SNR), and (c) the ratio of coherent to incoherent intensity at 415 Hz in a waveguide containing random internal waves. The internal wave disturbances have a height standard deviation of and coherence lengths of . This medium is highly random so that incoherent intensity dominates at all depths beyond about 20 km. The total received intensity, given by the numerator of SANR in Eq. (16) follows a decaying trend with local oscillations over range. All quantities are plotted as functions of source range and depth received at the 10-element vertical array described in Sec. III. The receiver array is centered at and . The source level is fixed as a constant over range so that is 0 dB at 1 km source range at all source depths

(a) Signal to additive noise ratio (SANR), (b) signal to noise ratio (SNR), and (c) the ratio of coherent to incoherent intensity at 415 Hz in a waveguide containing random internal waves. The internal wave disturbances have a height standard deviation of and coherence lengths of . This medium is highly random so that incoherent intensity dominates at all depths beyond about 20 km. The total received intensity, given by the numerator of SANR in Eq. (16) follows a decaying trend with local oscillations over range. All quantities are plotted as functions of source range and depth received at the 10-element vertical array described in Sec. III. The receiver array is centered at and . The source level is fixed as a constant over range so that is 0 dB at 1 km source range at all source depths

Ocean acoustic localization MLE behavior given a single sample for (a) range estimation and (b) depth estimation for a 415 Hz source placed at 50 m depth in a waveguide containing random internal waves. The internal wave disturbances have a height standard deviation of and coherence lengths of . The MLE first-order bias magnitude (solid line), square root of the CRLB (circle marks) and square root of the second- order variance (cross marks), as well as the signal to additive noise ratio (SANR, dashed line) and signal to noise ratio (SNR, dash-dotted line) are plotted as functions of source range. Other than the first-order bias and CRLB of the range MLE, the remaining quantities have increased by at least an order of magnitude when compared to the static waveguide scenario in Fig. 3. Given the necessary sample size conditions in Eq. (6), whenever the first-order bias and the second-order variance attain roughly 10% of the true parameter value and the CRLB, respectively, more than a single sample will be needed to obtain unbiased, minimum variance MLEs. The source level is fixed as a constant over range so that is 0 dB at 1 km source range.

Ocean acoustic localization MLE behavior given a single sample for (a) range estimation and (b) depth estimation for a 415 Hz source placed at 50 m depth in a waveguide containing random internal waves. The internal wave disturbances have a height standard deviation of and coherence lengths of . The MLE first-order bias magnitude (solid line), square root of the CRLB (circle marks) and square root of the second- order variance (cross marks), as well as the signal to additive noise ratio (SANR, dashed line) and signal to noise ratio (SNR, dash-dotted line) are plotted as functions of source range. Other than the first-order bias and CRLB of the range MLE, the remaining quantities have increased by at least an order of magnitude when compared to the static waveguide scenario in Fig. 3. Given the necessary sample size conditions in Eq. (6), whenever the first-order bias and the second-order variance attain roughly 10% of the true parameter value and the CRLB, respectively, more than a single sample will be needed to obtain unbiased, minimum variance MLEs. The source level is fixed as a constant over range so that is 0 dB at 1 km source range.

Fluctuating waveguide containing internal waves. , where is the sample size necessary to obtain an unbiased source range MLE whose MSE attains the CRLB and has , and , , are calculated using Eqs. (6) and (7), given a 415 Hz source placed at 50 m depth. Source level is fixed as a constant over range so that is 0 dB at 1, 10, 20, and 30 km source range (black circles), respectively, for the four curves shown.

Fluctuating waveguide containing internal waves. , where is the sample size necessary to obtain an unbiased source range MLE whose MSE attains the CRLB and has , and , , are calculated using Eqs. (6) and (7), given a 415 Hz source placed at 50 m depth. Source level is fixed as a constant over range so that is 0 dB at 1, 10, 20, and 30 km source range (black circles), respectively, for the four curves shown.

Fluctuating waveguide containing internal waves. of the square root of the CRLB for (a) source range , (b) source depth MLEs given a single sample. , the sample sizes or SNRs necessary to obtain (c) source range, (d) source depth MLEs that become unbiased and have MSEs that attain the CRLB. Given any design error threshold, the sample size necessary to obtain an accurate source range or depth MLE is then equal to , where . The internal wave disturbances have a height standard deviation of and coherence lengths of . The source level is fixed as a constant over range so that is 0 dB at 1 km source range at all source depths.

Fluctuating waveguide containing internal waves. of the square root of the CRLB for (a) source range , (b) source depth MLEs given a single sample. , the sample sizes or SNRs necessary to obtain (c) source range, (d) source depth MLEs that become unbiased and have MSEs that attain the CRLB. Given any design error threshold, the sample size necessary to obtain an accurate source range or depth MLE is then equal to , where . The internal wave disturbances have a height standard deviation of and coherence lengths of . The source level is fixed as a constant over range so that is 0 dB at 1 km source range at all source depths.

The same as Fig. 7, but here the covariance of the measurement is assumed parameter independent so that its derivatives in Eqs. (12) and (13) are set to zero. The asymptotic biases and variances of source range and depth MLEs are typically underestimated, as seen by comparing with Fig. 7. This scenario is equivalent to incorrectly assuming the received measurement is a deterministic signal embedded in purely additive white noise, in which case the SANR and SNR of the measurement are equal and the two curves coincide.

The same as Fig. 7, but here the covariance of the measurement is assumed parameter independent so that its derivatives in Eqs. (12) and (13) are set to zero. The asymptotic biases and variances of source range and depth MLEs are typically underestimated, as seen by comparing with Fig. 7. This scenario is equivalent to incorrectly assuming the received measurement is a deterministic signal embedded in purely additive white noise, in which case the SANR and SNR of the measurement are equal and the two curves coincide.

The same as Fig. 7, but here the mean of the measurement is assumed parameter independent so that its derivatives in Eqs. (12) and (13) are set to zero. The asymptotic biases and variances of source range and depth MLEs may be significantly overestimated, as seen by comparing with Fig. 7. This scenario is equivalent to incorrectly assuming the received measurement is purely random with zero mean, embedded in additive white noise.

The same as Fig. 7, but here the mean of the measurement is assumed parameter independent so that its derivatives in Eqs. (12) and (13) are set to zero. The asymptotic biases and variances of source range and depth MLEs may be significantly overestimated, as seen by comparing with Fig. 7. This scenario is equivalent to incorrectly assuming the received measurement is purely random with zero mean, embedded in additive white noise.

The same as Fig. 9, but here the covariance of the measurement is assumed parameter independent so that its derivatives in Eqs. (12) and (13) are set to zero. The CRLB and the sample sizes necessary to attain it are underestimated when compared with Fig. 9. This scenario is equivalent to incorrectly assuming the received measurement is a deterministic signal embedded in purely additive white noise.

The same as Fig. 9, but here the covariance of the measurement is assumed parameter independent so that its derivatives in Eqs. (12) and (13) are set to zero. The CRLB and the sample sizes necessary to attain it are underestimated when compared with Fig. 9. This scenario is equivalent to incorrectly assuming the received measurement is a deterministic signal embedded in purely additive white noise.

The same as Fig. 9, but here the mean of the measurement is assumed parameter independent so that its derivatives in Eqs. (12) and (13) are set to zero. The CRLB and the sample sizes necessary to attain it are overestimated when compared with Fig. 9. This scenario is equivalent to incorrectly assuming the received measurement is purely random with zero mean, embedded in additive white noise.

The same as Fig. 9, but here the mean of the measurement is assumed parameter independent so that its derivatives in Eqs. (12) and (13) are set to zero. The CRLB and the sample sizes necessary to attain it are overestimated when compared with Fig. 9. This scenario is equivalent to incorrectly assuming the received measurement is purely random with zero mean, embedded in additive white noise.

## Tables

Definitions of the shorthand notations used in Equations B2b-s. perm(a,b,c) is a shorthand for sum of terms obtained by all permutations of the indices a,b, and c. rot(a,b,c) is a shorthand for the sum of terms obtained by rotating the indices a,b, and c. (a↔b) is a shorthand for the sum of terms with indices a and b interchanged. These shorthand notations are used to write Equations B2b-s in a compact manner.

Definitions of the shorthand notations used in Equations B2b-s. perm(a,b,c) is a shorthand for sum of terms obtained by all permutations of the indices a,b, and c. rot(a,b,c) is a shorthand for the sum of terms obtained by rotating the indices a,b, and c. (a↔b) is a shorthand for the sum of terms with indices a and b interchanged. These shorthand notations are used to write Equations B2b-s in a compact manner.

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