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General second-order covariance of Gaussian maximum likelihood estimates applied to passive source localization in fluctuating waveguides
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10.1121/1.3488303
/content/asa/journal/jasa/128/5/10.1121/1.3488303
http://aip.metastore.ingenta.com/content/asa/journal/jasa/128/5/10.1121/1.3488303

Figures

Image of FIG. 1.
FIG. 1.

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, .

Image of FIG. 2.
FIG. 2.

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).

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

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.

Image of FIG. 6.
FIG. 6.

(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

Image of FIG. 7.
FIG. 7.

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.

Image of FIG. 8.
FIG. 8.

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.

Image of FIG. 9.
FIG. 9.

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.

Image of FIG. 10.
FIG. 10.

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.

Image of FIG. 11.
FIG. 11.

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.

Image of FIG. 12.
FIG. 12.

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.

Image of FIG. 13.
FIG. 13.

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

Generic image for table
TABLE I.

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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/content/asa/journal/jasa/128/5/10.1121/1.3488303
2010-11-24
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: General second-order covariance of Gaussian maximum likelihood estimates applied to passive source localization in fluctuating waveguides
http://aip.metastore.ingenta.com/content/asa/journal/jasa/128/5/10.1121/1.3488303
10.1121/1.3488303
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