## 1. Introduction

Backscatter autocorrelation is the fundamental measurement in pulse-to-pulse coherent Doppler sonar (Garbini et al. 1982; Lhermitte and Serafin 1984). While velocity is determined from the phase of the complex autocorrelation coefficient, the coefficient magnitude is often used as a measure of data quality. Corresponding to each velocity measurement, many commercially available instruments provide a measure of autocorrelation, for example, as a coefficient between 0% and 100%. Recommended minimum values for the autocorrelation coefficient assist the user in collecting high-quality measurements and diagnosing instrumentation problems when necessary. In addition to qualitative assessment, autocorrelation can also provide quantitative information on the expected magnitude of measurement errors. For example, pulse-to-pulse autocorrelation has been used to identify and replace spurious Doppler velocimeter measurements in the surf zone (Elgar et al. 2005). Also, the relationship between velocity measurement error and autocorrelation has been determined through laboratory testing of a coherent Doppler sonar (Zedel et al. 1996).

*z*} of complex-valued backscatter samples, the autocorrelation at a lag of

_{n}*k*pulse-to-pulse intervals is

*E*denotes expected value, * denotes complex conjugation, and

*Ļ*is the time interval between successive acoustic transmissions. The autocorrelation coefficient is defined as

*Ļ*

^{2}is the variance of the sequence {

*z*}. By definition,

_{n}*Ļ*is a number between zero and one that expresses the degree of pulse-to-pulse autocorrelation.

*z*

_{1}, ā¦ ,

*z*corresponding to an ensemble of

_{M}*M*pulses (ZrniÄ 1977),

*Ļ*:

*z*is therefore well described by a complex Gaussian distribution, that is,

*z*=

*x*+

*iy*, where

*x*and

*y*are independent normally distributed random variables with equal variances. In general, a time series of backscatter samples is a nonstationary random process because Doppler frequency, autocorrelation, and amplitude are functions of time. However, in the analysis of coherent Doppler systems, backscatter samples are frequently modeled as being drawn from a wide-sense stationary (WSS) random process with Gaussian power spectral density (Garbini et al. 1982; Lhermitte and Serafin 1984; Zedel et al. 1996),

*f*is the mean Doppler frequency and

_{D}*Ļ*denotes the spectral width. In this article, the term āGaussian distributionā refers to the probability distribution of a single backscatter sample. The term āGaussian random processā refers to a time series where (i) each sample obeys a Gaussian distribution, and (ii) the power spectrum of the time series is a Gaussian function as in (5).

_{f}*Ļ*is

*Ļ*determines the width of the Doppler spectrum, and hence the variance of velocity measurements.

To compare coherent Doppler sonar observations with expected performance based on a Gaussian random process, it is necessary to examine the relationship between the true autocorrelation coefficient *Ļ* and its estimate *Ļ* may be inverted to obtain unbiased estimates of autocorrelation from biased samples of *Ļ*. Because the effectiveness of pulse-pair averaging depends on the correlation between successive measurements, a sonar designer may wish to know how much averaging is required to sufficiently attenuate measurement errors for a given observed coefficient

This article is organized as follows. In section 2, a new formula is presented for the asymptotic estimator

## 2. Theory

*M*ā ā, the numerator converges to |

*R*(

*Ļ*)| =

*ĻĻ*

^{2}. Assuming that each sample

*z*is described by a Gaussian distribution, the denominator converges to the mean

_{n}*Ī¼*of the product of two dependent Rayleigh random variables |

*z*| and |

_{n}*z*

_{n+1}|. The product |

*z*ā

_{n}*z*

_{n+1}| is described by the probability distribution (Simon 2002, chapter 6)

*r*= |

*z*ā

_{n}*z*

_{n+1}|,

*K*

_{0}is a modified Bessel function of the second kind, and

*I*

_{0}is a modified Bessel function of the first kind. The mean

*Ī¼*is determined from

**E**(

*k*),

*Ļ*, the first-order Taylor series is

*Ļ*near one, the first-order Taylor series is given by

*Ļ*< 1.

## 3. Numerical simulation

### a. Gaussian random process

*Ī¶*

_{1}, ā¦ ,

*Ī¶*denote an ensemble of

_{M}*M*independent identically distributed samples from a complex Gaussian distribution with zero mean and unit variance. For a Gaussian power spectrum, the backscatter autocorrelation sequence

*R*is given by (6) and (7),

_{k}**z**denote samples

*z*

_{1}, ā¦ ,

*z*arranged as a vector. The corresponding covariance matrix

_{M}**of independent samples**

*Ī¶**Ī¶*

_{1}, ā¦ ,

*Ī¶*has covariance matrix

_{M}**z**has covariance matrix given by

Samples of the autocorrelation estimate ^{7} simulated pings for ensemble lengths of *M* = 10, 20, 40, and 100, and true autocorrelation coefficients ranging from 0.2 to 0.98 in increments of 0.02. For each pair (*Ļ*, *M*), the mean autocorrelation estimate was calculated as an approximation to the expected value *M* ā ā, simulations for larger values of *M* indicated that in the interval of *Ļ* ā„ 0.2, the autocorrelation ratio is within 1% of the asymptotic ratio when *M* is greater than or equal to 600.

Estimated autocorrelation coefficients from simulation of a Gaussian random process. Each curve represents the ratio *Ļ* for a fixed ensemble length *M*. The dashed line is the asymptotic ratio

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

Estimated autocorrelation coefficients from simulation of a Gaussian random process. Each curve represents the ratio *Ļ* for a fixed ensemble length *M*. The dashed line is the asymptotic ratio

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

Estimated autocorrelation coefficients from simulation of a Gaussian random process. Each curve represents the ratio *Ļ* for a fixed ensemble length *M*. The dashed line is the asymptotic ratio

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

### b. Coherent Doppler sonar model

Numerical simulation of steady flow was also performed with the coherent Doppler sonar model described in Zedel (2008). The model simulates pulse-to-pulse coherent scattering from a cloud of moving particles for arbitrary multistatic sonar geometries. Physical effects such as spherical spreading, acoustic absorption, frequency-dependent beam patterns, transducer frequency response, and receiver noise are included in the model. The model supports simulation of arbitrary pulse shapes, including the use of multiple carrier frequencies.

Simulations were performed for a monostatic sonar measuring horizontal velocities of 0.5, 1.5, 3.0, and 4.5 m s^{ā1}. In the model, the sonar was tilted 5Ā° from vertical to reproduce the geometry of the towing tank experiment described in section 4. Parameters for the coherent Doppler sonar simulation are listed in Tables 1 and 2.

Coherent Doppler sonar parameters.

Parameters for the coherent Doppler sonar simulation.

*Ļ*

^{2}is the variance of the backscatter sequence {

*z*} and

_{n}*N*= 2 Ć 10

^{5}is the total number of simulated pings. The mean autocorrelation estimate

*M*= 10, 20, 40, 100, and 1000. Figure 3 shows the ratio

*M*is increased, the ratio converges toward the asymptotic ratio

Estimated autocorrelation coefficients from the coherent Doppler sonar simulation. Each circle represents the ratio *Ļ* for ensemble lengths of *M* = 10, 20, 40, 100, and 1000. The asymptotic ratio *M*, circles converge downward to the dashed line.

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

Estimated autocorrelation coefficients from the coherent Doppler sonar simulation. Each circle represents the ratio *Ļ* for ensemble lengths of *M* = 10, 20, 40, 100, and 1000. The asymptotic ratio *M*, circles converge downward to the dashed line.

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

Estimated autocorrelation coefficients from the coherent Doppler sonar simulation. Each circle represents the ratio *Ļ* for ensemble lengths of *M* = 10, 20, 40, 100, and 1000. The asymptotic ratio *M*, circles converge downward to the dashed line.

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

## 4. Apparatus

A towing tank experiment was performed using the multifrequency coherent Doppler sonar described in Hay et al. (2008). Each circular piezocomposite transducer has a diameter of 2 cm, a nominal center frequency of 1.7 MHz, and a bandwidth of approximately 1 MHz. Carrier frequencies, profiling range, range resolution, pulse length, pulse-to-pulse interval, and ensemble length are configurable in software. The dimensions of each sample volume are determined by the beam pattern, carrier frequency, and range resolution. Nominally, each sample point has a diameter of 2 cm and a height of 3 mm. The parameters in Table 1 also apply for the sonar used in the towing tank experiment.

The experiment was performed in the Marine Craft Model Towing Tank at Dalhousie University. The tank has horizontal dimensions of 30 m Ć 1 m and a depth of 1 m. An instrumented carriage is propelled by an electric motor along rails mounted above the tank. Carriage speed is computer controlled and programmable over a range from 0 to 3.0 m s^{ā1}. Constant speed is sustained over a rail length of approximately 25 m. The towing carriage and instrumentation are shown schematically in Fig. 4. The sonar was rotated to point 5Ā° aft (i.e., counterclockwise in Fig. 4) to avoid receiving multiple reflections from the tank bottom. The sonar was located on the tank center line with the center transducer 56 cm above the bottom. Water in the tank was seeded with agricultural lime. Prior to each run, approximately 0.5 kg of lime was added to replace scatterers lost to settling. A rough estimate of sediment concentration was 1 g L^{ā1}.

Side view schematic of the towing tank showing the multifrequency coherent Doppler sonar. Instrumentation was attached to a carriage that moved along rails mounted above the water. Transducer beam patterns are indicated (dashed lines).

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

Side view schematic of the towing tank showing the multifrequency coherent Doppler sonar. Instrumentation was attached to a carriage that moved along rails mounted above the water. Transducer beam patterns are indicated (dashed lines).

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

Side view schematic of the towing tank showing the multifrequency coherent Doppler sonar. Instrumentation was attached to a carriage that moved along rails mounted above the water. Transducer beam patterns are indicated (dashed lines).

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

## 5. Experimental procedure

Carriage speed was varied from 0.05 to 3.0 m s^{ā1} by programming the desired speed into the towing tank control system. Results are presented in section 6 for velocities of 0.5, 1.5, and 3.0 m s^{ā1}. The control system software automatically calculated an acceleration and deceleration profile to maximize the time at constant speed subject to the tank length constraint. Two runs were performed for each speed with a duration of 55 s, or the time elapsed in traversing the entire tank length, whichever was less. Carriage speed was recorded by the control system.

Autocorrelation coefficients were recorded by the sonar data acquisition system using a fixed ensemble length of *M* = 10. Because it was not possible to simultaneously record data with multiple ensemble lengths, an indirect approach was taken to assess the validity of simulations in section 3. For each carriage speed, the *M* = 10 curve from Fig. 2 was used to infer the true autocorrelation coefficient *Ļ* from the mean of the observed estimates

## 6. Results

In Table 3, the mean autocorrelation estimate ^{ā1} from the 41-cm range bin of the center transducer receiver channel. Here, *M* = 10 curve in Fig. 2. These values of *Ļ* were used to generate autocorrelation coefficients from a Gaussian random process, as described in section 3.

Estimated autocorrelation coefficients from towing tank data.

Distributions of towing tank autocorrelation coefficients are shown in Fig. 5 for carriage speeds of 0.5, 1.5, and 3.0 m s^{ā1}. Values of *M* = 10 and *Ļ* as listed in Table 3. Dashed lines in Fig. 5 represent distributions of ^{7} simulated pings, with autocorrelation coefficients grouped in 200 equally spaced bins.

Distributions of measured and simulated autocorrelation coefficients. A histogram of *M* = 10 curve in Fig. 2 are also shown (dotted lines). Simulated distributions with no bias correction are represented (dashed lines). Carriage speed is (top) 0.5 and (bottom) 3.0 m s^{ā1}.

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

Distributions of measured and simulated autocorrelation coefficients. A histogram of *M* = 10 curve in Fig. 2 are also shown (dotted lines). Simulated distributions with no bias correction are represented (dashed lines). Carriage speed is (top) 0.5 and (bottom) 3.0 m s^{ā1}.

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

Distributions of measured and simulated autocorrelation coefficients. A histogram of *M* = 10 curve in Fig. 2 are also shown (dotted lines). Simulated distributions with no bias correction are represented (dashed lines). Carriage speed is (top) 0.5 and (bottom) 3.0 m s^{ā1}.

Citation: Journal of Atmospheric and Oceanic Technology 28, 7; 10.1175/JTECH-D-10-05002.1

## 7. Discussion

The derivation of the asymptotic autocorrelation coefficient assumed a complex Gaussian probability distribution for each backscatter sample. However, it was not necessary to assume a Gaussian power spectrum for the time series because the asymptotic formula depends only on the expected autocorrelation at a lag of one pulse-to-pulse interval. The formula was presented in terms of an elliptic integral. Although

Numerical simulation of a Gaussian random process showed that the bias of the autocorrelation coefficient increases for short ensemble lengths. For example, the degenerate case of a single pulse pair results in a coefficient of one regardless of the actual pulse-to-pulse autocorrelation. A longer ensemble length is necessary to obtain meaningful autocorrelation estimates. As shown in Fig. 2, a bias persists for all of the ensemble lengths, with convergence to the asymptotic formula occurring for *M* approximately equal to 600. The bias is more significant for small values of the autocorrelation coefficient. For practical applications where reasonably high-quality data are obtained (say *Ļ* ā„ 0.7), there is negligible variation in the ratio *M* is varied. However, a bias is still present for *Ļ* ā„ 0.7, and in this case the bias is well described by the asymptotic Eq. (12).

The coherent Doppler sonar model in Zedel (2008) does not require any Gaussian assumption about the backscatter probability distribution or the time series power spectrum. The model describes the physics of coherent scattering and accounts for the sonar geometry and operating parameters, unlike the simulations of a Gaussian random process in section 3. Simulations of steady flow confirmed that the autocorrelation coefficient converges to the asymptotic formula as ensemble length is increased. However, similarity between Figs. 2 and 3 shows that simulation of a Gaussian random process is sufficient to predict the bias of the autocorrelation coefficient.

In Table 3, the mean observed autocorrelation estimates from the towing tank satisfy *Ļ* for the *M* = 10 curve in Fig. 2. When it is assumed that

It would be interesting to repeat the towing tank experiment with additional runs for each carriage speed while recording the result from each ping. Autocorrelation coefficients could be calculated for a range of ensemble lengths to demonstrate convergence to the asymptotic formula, as in section 3, for the coherent Doppler sonar simulation. Reproduction of Fig. 3 with experimental measurements would require approximately 300 s of data for each speed. At 3.0 m s^{ā1}, the carriage would need to travel 900 m, requiring 36 runs in the Dalhousie University towing tank. For such an endeavor, a longer tank or a continuously operated flume would be more suitable.

Finally, we remark that the definition of the autocorrelation coefficient is not unique. The coefficient considered in this article is an appropriate choice because

## 8. Conclusions

A new formula has been presented for the asymptotic form of an autocorrelation coefficient for coherent Doppler sonar. The derivation showed that the autocorrelation coefficient is a biased estimator in the limit of infinite ensemble length. Numerical simulation of a Gaussian random process indicated that the bias persists for finite pulse-pair averages. Furthermore, the bias increases for shorter ensemble lengths. Validity of the Gaussian random process was confirmed with numerical simulation using a high-fidelity coherent Doppler sonar model, and from sonar measurements in a towing tank where the towing carriage traveled at constant speed. The experiment showed that the distribution of observed autocorrelation coefficients is well predicted by a Gaussian random process once the autocorrelation bias has been removed. Although other autocorrelation coefficients may be defined, the analysis and numerical methods developed in this article could be applied to derive their asymptotic behavior.

## Acknowledgments

We thank Richard Cheel for experimental support and Robert Craig for data acquisition software. Financial support for J. Dillon was provided by the Link Foundation and the Natural Sciences and Engineering Research Council of Canada.

## APPENDIX

### Derivation of the Asymptotic Coefficient

*Ī¼*is given by

*F*is the hypergeometric function (Ahlfors 1966, chapter 8). Equation (A3) is valid when

*a*>

*b*. The gamma function satisfies Ī(1) = 1 and

*b*/

*a*=

*Ļ*implies that

*a*>

*b*is satisfied when

*Ļ*< 1. Equation (A4) becomes

*c*>

*d*> 0 and 0 ā¤

*x*ā¤

*Ļ*. To apply (A10) to (A9), let

*c*>

*d*> 0 is satisfied when

*Ļ*< 1. The parameter

*r*is given by

*x*ā¤

*Ļ*, (A10) may be used to obtain

*Ī“*(0) = 0,

*Ī“*(

*Ļ*) =

*Ļ*/2, and

*É*(0,

*k*) = 0, (A18) reduces to

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