Mid-frequency acoustic propagation in shallow water on the New Jersey shelf. II. Intensity fluctuation

During the Shallow Water 2006 Experiment on 11 August '06, a mid-frequency acoustic propagation experiment was performed in the frequency band $2–10kHz$ (referred to as the mid-frequency band) and at ranges $1–9km$⁠. The mean intensity has been found to be well modeled by a simple model of incoherent sum of modes.¹ The intensity fluctuation over short time periods is found to be significant and causes uncertainty in predictions of sound propagation. In this work the acoustic arrivals are treated as random and the scintillation index is used for the purpose of quantifying intensity fluctuations.

Mid-frequency sound intensity fluctuations in the deep ocean have been measured² and studied by using the method of moments and simulations,³ where it is found that the intensity fluctuation is small and not saturated. Observations of low-frequency (hundreds of Hertz) intensity fluctuation in the deep ocean have been reported. For example, Duda et al.⁴ found strong focusing that makes the scintillation index at $400Hz$ to be greater than 1. There are some efforts devoted to intensity fluctuations in shallow water at low frequency.^5,6 To the authors’ knowledge, research on mid-frequency intensity fluctuation in shallow water is limited.^7–9 

At the experiment site, a typical strong summer thermocline existed, and in addition, warm, salty slope water underlay the colder mid-depth water. Thus, there was a sound speed minimum at approximately $32m$ depth, forming a sound channel. The acoustic propagation measurement was carried out along the $80m$⁠, isobath for $1,2,…,9km$ ranges. At each of these stations, 100 identical acoustic transmissions were made in order to study the statistics of sound intensity on single hydrophones. The signals transmitted and studied in this letter are wideband pulses between 20 and 10 $kHz$⁠. Details of the experiment description can be found in Ref. 1. The letter describes the intensity fluctuation as a function of frequency in terms of scintillation index. Cross-frequency correlations of intensity and the cumulative distribution functions of intensity fluctuations are presented.

On the same day but before the acoustic data at the different range stations were taken, data from a towed conductivity-temperature-depth (CTD) chain over an extended period showed that sound speed close to the sound channel had complex variations,¹ believed to be caused by ambient internal waves and possibly neutrally buoyant thermohaline intrusions. No strongly nonlinear internal waves were observed that day, a special case during the experiment. These ambient internal waves cause intensity fluctuations. These fluctuations are too complicated to be treated deterministically, and necessarily must be dealt with as a random process. The intensity of the 100 pings at selected range stations are shown in Fig. 1, where the ping-to-ping interval is roughly $20s$⁠, and the signal-to-noise ratio is $45dB$ after pulse compression of frequency modulated signals. The $10-ms$-long windows in Fig. 1 contain at least 97% of the energy of all arrivals for all range stations. Due to space limitations, results from four of the representative range stations are shown; however, results for the range stations not shown have characteristics consistent with those presented in this letter. Because the acoustic source had an unmeasured horizontal motion $O(1m)$ relative to the receivers and $0.5m$ uncertainty in depth, the data shown in the figure have been pulse compressed and lined up with the first ping at each range. Correlating integrated intensity between two receivers vertically separated by $0.9m$⁠, the correlation coefficient is found to be 0.95, thus source vertical motion was ruled out as having substantial impact on intensity fluctuations.

At the $1km$ range, measured sound intensity can be separated into individual arrivals; at other ranges, the arrivals are clustered and not separable. The received pressure can have broadband deep fades, such as the ones shown at the $1km$ range in Fig. 1; ping-to-ping fluctuations increase as the range increases. Calculations show that the peak intensity can deviate from its mean value by as much as $12dB$⁠. This letter seeks to quantify the fluctuation.

Intensity fluctuations are often quantified in the scintillation index, SI, defined as $SI=⟨I2⟩∕⟨I⟩2−1$⁠, where $I$ is sound intensity and the angular brackets stand for ensemble average. The intensity, $I$⁠, can be chosen to have different meanings, thereby leading to different scintillation indices. For example, $I$ could be a time integral, a peak value, or a modal intensity. In both theoretical and simulation approaches dealing with wave propagation in random media, $I$ is naturally chosen to be a single-frequency intensity at a single receiver position; this choice is made here, in order to be able to compare with theoretical and modeled results. A broadband SI is expected to be smaller than the single-frequency SI. If $I$ is associated with a nonrandom process, $SI=0$⁠; if $I$ has a number of statistically independent contributions with random phases (uniformly distributed on $[0,2π]$⁠), $SI=1$⁠, and is said to be saturated. In the saturated case, the intensity has an exponential distribution. In order to properly extract the single-frequency SI from data, a suitably small bandwidth needs to be found within which sound intensity is correlated. The appropriate bandwidth can be determined from the cross-frequency intensity correlation. The cross-frequency intensity correlation is a fourth moment of the sound pressure field and was calculated in the following steps. Pressure time series $100ms$ in duration were Fourier transformed and squared, resulting in intensity with $10Hz$ resolution. For the 100 pings at each range station, the intensity at each frequency bin was normalized by its mean, and then the intensity cross-frequency correlation was calculated. For each lag, $δf$⁠, the correlation function was averaged over the mean frequency $f$ for which $f+δf∕2$ and $f0δf∕2$ are within a chosen band. The bands were chosen to be: 2–3.5, 3.5–5, 6.5–8, and 8–10 $kHz$⁠.This averaged cross-frequency correlation function is given in Fig. 2 for four ranges.

At the $1km$ range, the intensity correlation shows regular oscillations with the same spacing in frequency difference in each frequency band. This oscillation can be qualitatively understood. Note that in Fig. 1, there are two major arrivals separated by $6ms$⁠. The first is the direct arrival that stays in the water column, and the second is a bottom bounce. Assume that the two arrivals have independent scintillation, with scintillation indices $S1$ and $S2$⁠, both independent of frequency. Also assume that the time spacing $T$ between the arrivals is a constant. The total arrival is written as $a=a1exp(−i2πfT1)+a2exp(−i2πfT2)$⁠, where $T=T2−T1$⁠, then to the lowest nontrivial order the cross-frequency intensity correlation for frequencies $fa$ and $fb$ can be found to be

where $F(R,f)=[1+Rcos(2πfT)∕1+2Rcos(2πfT)+R2]$⁠, $R$ is the amplitude ratio of the two arrivals, and $δ(f)=I(f)−⟨I(f)⟩$⁠. Because the above-presented expression is symmetric with respect to the two frequencies and the dependence of $T$ is periodic, the cross-frequency correlation in the two-dimensional space has a “chessboard” pattern with periodicity of $1∕T$⁠. After averaging correlation over the mean frequency within a chosen band, this chessboard pattern becomes a regular oscillation as shown in the $1km$ case in Fig. 2, where the periodicity is seen to be $1∕T$⁠, as predicted. If the first arrivals were isolated and a correlation was separately calculated, the oscillation disappears and the correlation improves as seen in Fig. 2 given as a dashed curve.

At all ranges longer than $1km$⁠, the arrivals are not separable and are treated as a single arrival. At all frequencies, the intensity cross-correlation function has a sharp peak with a half-power width of 50–200 $Hz$ with no clear range dependency, as shown in Fig. 2. The origin of the narrowness of correlation is yet to be investigated. From the estimated width of the intensity cross-frequency correlation, it is safe to assume that the $10Hz$ bin is narrow enough to assure that within the bin the intensity is fully correlated; hence a single-frequency scintillation index can be estimated within each bin. All the intensity values from 2–10 $kHz$ with $10Hz$ bin width were divided into the same four frequency bands as when estimating the cross-frequency correlations. In each of the bands, the intensity cumulative distribution function (CDF) $C(I)$ is estimated for each of the nine ranges and is shown in Fig. 3. Here $C(I)$ is defined as the probability that the intensity is less than $I$⁠. In Fig. 3 we plotted on a logarithmic scale the complementary function, 10 $C(I)$⁠, for easier comparison to the exponential distribution function, which is a straight line on the logarithmic scale. We also plotted numerically determined 95% confidence bounds for the exponential distribution in Fig. 3 by faint dashed curves. If the intensity follows the exponential distribution, data would fall within the bounds. As shown in Fig. 3, the small intensity regime of $C(I)$ for all frequencies and ranges follows the exponential distribution. The probability distribution function (PDF) is the derivative of $C(I)$⁠, and, for an exponential distribution, is largest at zero, thus the intensity PDF peaks at zero. This intensity fluctuation can also be observed in the ping-to-ping variations in intensity as in Fig. 1, where the first arrivals at the $1km$ range show deep fades at several places. As there were no strongly nonlinear internal waves on the day of the measurements, one would expect the intensity PDF to peak around the mean intensity and diminish at small intensities. The observation that the intensity at small values follows the exponential distribution for a wide band of frequencies is to us a surprising result. Saturation happens when a large number of independent arrivals contribute additively to the signals. However, our observations show that even at the $1km$ range, the small intensities follow the exponential distribution, even though at such short range it is not expected that the arrivals are saturated. The large intensity values are understood to be caused by focusing effects. Clearly they do not follow the exponential distribution but they constitute low probability. Therefore, we conclude that the intensity fluctuation is not saturated, and the reason needs to be investigated.

Mid-frequency $(2–10kHz)$ sound intensity fluctuations at $1–9km$ range stations were investigated through the single-frequency scintillation index. The intensity fluctuations are due to water column sound speed variability from ambient internal waves; the data were collected on a day when there were no strongly nonlinear internal waves present. It is highly unlikely that fish scattering is the cause of the fluctuation because the observed large fluctuation requires high fish density. Also, such fish presence would result in noticeable attenuation in the mean intensity versus range, which is not observed.¹ 

Two findings were not anticipated before the experiment: intensity is only correlated over a narrow frequency band (50–200 $Hz$⁠) and the bandwidth is independent of center frequency and range; the intensity PDF peaks at zero for all frequencies, and follows an exponential distribution at small values. What causes the observed deep fades in intensity? Such deep fades manifest as uncertainty in predicting mid-frequency sound propagation in shallow water. To understand the intensity fluctuation in order to estimate uncertainty, two potentially profitable approaches are numerical simulations based on the parabolic equation approximation and transport theory such as the one proposed by Dozier and Tappert,¹⁰ where the validity of some key assumptions needs to be investigated.

This work was supported by the Office of Naval Research.