NUMERICAL SIMULATIONS OF ACOUSTIC TOMOGRAPHY OF OCEAN CURRENTS IN COASTAL REGIONS
Oleg A. Godin and Dmitry Yu. Mikhin
P.P.Shirshov Oceanology Institute of the Russian Academy of Sciences
23 Krasikova St., Moscow 117218, Russia
Ocean Acoustic Tomography (OAT) is widely recognized as a powerful technique for studying the thermal and current structure of the ocean. A number of sea trials have been conducted since the late 1970's to determine the feasibility of OAT (see  for summary). Both temperature tomography and current tomography have been shown to be feasible in the deep ocean for propagation paths up to about 1000 km. This situation is to be contrasted with that of OAT in coastal regions. Field experiments on current tomography in such regions have not been quite successful and clearly demonstrated that the deep-water approach is not suitable for shallow-water conditions. The reasons are the inability to resolve ray paths in time with signal bandwidth available and the intrinsic inability to identify ray paths that have interacted with the ocean bottom [2, 3]. As a result the current velocity inversion possesses very poor or non-existent resolution in the vertical plane.
A new, full-field approach to acoustic monitoring of ocean currents called Matched Non-reciprocity Tomography (MNT) was recently put forward in order to overcome problems with ray resolution and identification in shallow, coastal areas. The method is based on the non-reciprocity of sound propagation in a flow field. Reciprocal propagation is necessary to separate the relatively subtle acoustic effects of currents from the influence of variations in sound speed and bottom topography. However, non-reciprocal, by itself, does not ensure that a full field inversion will be stable with respect to the inevitable uncertainties in environmental conditions and experiment geometry. As originally proposed by Godin and Mikhin [4-6], the approach consists in inverting non-reciprocity in phase of a CW signal measured at a set of points in a given vertical plane. Analytic studies indicate non-reciprocity of phase of a CW signal propagating in opposite directions is sensitive to flow velocity and does not depend, to first order, on fluctuations in sound speed and on uncertainties in transceiver location, that is crucial for MNT to be feasible.
Generally speaking, information about the current field is contained not only in the depth dependence of the phase non-reciprocity but also in its frequency dependence. A change in the current field results in changes both in the depth dependence of the phase non-reciprocity and in its functional dependence on frequency. In this report we address the problem if it is possible to exploit this frequency dependence instead of (or in addition to) the depth dependence when solving the inverse problem.
To find the current velocity through MNT, one should minimize a cost function describing the mismatch between experimentally measured and predicted non-reciprocity in acoustical phase. The predicted value is a function of several parameters used in the model to describe the flow field. The minimum with respect to these parameters gives the solution. In field experiments the input data are deteriorated by uncertainties in environmental conditions and in experiment geometry. To determine the requirements to the experiment design for at-sea MNT trials and to construct the optimum inversion scheme it is necessary to explore by computer modeling the sensitivity of the inversion to different mismatches. Such simulations consist in modeling sound propagation between a moored transceiver and a near-vertical array of transceivers under appropriate hydrographic conditions, including potential sources of errors, and then inverting the synthesized data to determine the flow.
Fig.1. Sound speed (a) and current velocity (b) profiles used in modeling.
The simulations presented in the report were fulfilled for a medium model resembling the propagation conditions in the Straits of Florida in summer months: the vertical profiles of sound speed and flow velocity are shown in Fig.1. The model is based on the description of the field tomographic experiment in this area [7, 8]. In particular, this experiment demonstrated the inappropriateness of the deep-water OAT technique for shallow water conditions. The sound speed decreases rapidly with depth making propagation bottom-limited. The ocean bottom was modeled as a fluid halfspace. The medium was assumed to be range-independent. In the reconstruction of the flow the sound speed profile was supposed to be known (exactly or approximately). In at-sea surveys it may be preliminary found, for example, with the Matched Field Processing technique. This problem has been carefully studied by many authors and is not addressed here.
The current profile sought-for in inversion was a linear combination of a barotropic component and a baroclinic one linearly depending on depth z, their amplitudes being and respectively. The cost function was calculated at nodes of a rectangular grid on the plane with steps of 0.1 along both axes. The current profile corresponding to the node having the minimal cost function was taken to be the solution to the inverse problem.
For a cost function we used the sum of the squared differences between the sines and cosines of simulated phase non-reciprocity and those of the model-predicted non-reciprocity. In the process of averaging over the array transceivers, higher weights were given to those with higher acoustic intensities. A version of this cost function considered earlier [4, 5] used the differences between the simulated and theoretical phase non-reciprocity instead of sine and cosine components. The new cost function gives somewhat better results as it eliminates problems with discontinuity of phase reduced to the interval.
Fig.2. Simulated in degrees vs. signal frequency and transceiver depth.
The simulations showed that phase non-reciprocity is a distinct and non-degenerate function of frequency f. Multi-frequency phase non-reciprocity measurements would be clearly of limited use if was nearly a linear function of f. It is not. This is illustrated by Fig.2 where is plotted as a function of f and of depth of the transceiver in the array. The propagation range and the ocean depth were taken to be 36 km and 570 m. The single transceiver was located at 530 m depth. The flow was assumed to make an angle of 45° with the vertical plane containing the moorings. All simulations were performed with a normal-mode model that explicitly takes into account effects of both in-plane and out-of-plane flow components on the sound field in a vertical plane considered. Moreover, the model is not limited to acoustical effects linear in the current's Mach number.
A number of numerical simulations of the inversion process were carried out to determine if the frequency dependence of the phase non-reciprocity could be used as a replacement for the depth dependence that would be measured with a vertical transceiver array. The possibility of using the frequency dependence together with limited knowledge of the depth dependence was also addressed. Four frequency spectra were considered: a CW sound field with f=50 Hz, and fields at 3, 11, and 41 frequencies, equally spaced between 40 and 60 Hz. It should be particularly emphasized that the Flow Reversal Theorem allows to find the acoustic field non-reciprocity for all source-receiver pairs having calculated the acoustic field only two times for each frequency [4, 5].
Fig.3. The RMS error in the current velocity inversion as a function of SNR for various numbers of elements in the array. All details are given in the text.
Three types of mismatches in the inverse problem were considered: discrepancies in the horizontal separation of the transceivers, discrepancies in the bottom parameters used to calculate the acoustic fields, and additive acoustic noise. Sound speed profile was assumed to be known.
The simulations indicate that the inversion results are insensitive to mismatches in horizontal separation up to 100 m and in the bottom density. A 25% change in the density did not lead to any appreciable change in the problem solution. The effect of a mismatch in the bottom sound speed is greater because of variation in the number of propagating normal modes. The dependence of the inversion error on the mismatch amplitude is shown in Figs.3 and 4 for 11 frequencies. In these figures the RMS difference between the true in-plane current profile and the inversion result is shown as a function of SNR (Fig.3) and the mismatch in the bottom sound speed (Fig.4) for various numbers of elements in the transceiver array: 1 (solid line), 2 (dashed line), 3 (dotted line), 4 (squares), 23 (crosses). The depths of the transceivers in the five cases were, respectively, 510 m; 360 and 560 m; 260, 410, and 560 m; 110 to 560 m with 150 m step; and 10 to 560 m with 25 m step.
Fig.4. The RMS error of the current inversion for different mismatches in the sound speed in the bottom for various numbers of transceivers in the array. The true value of the bottom sound speed is 1550 m/s. All details are given in the text.
The simulated input data for the inversion included following additional mismatches: 9% difference in the bottom density, 50 m error in horizontal separation of the transceivers, a 10 m/s difference between true sound speed in the bottom and its value in the model (only for Fig.3), additive acoustic noise resulting in 20 dB SNR (only for Fig.4). The SNR values were defined as the ratio of the sum of signal intensities over array elements to the sum of noise intensities at the elements for a given frequency. Hence, the SNR used here included processing gain. The noise levels were taken as frequency- and depth-independent for simplicity.
The presented figures demonstrate that because of the intrinsic separation of reciprocal and non-reciprocal acoustic effects, the inversion can be successfully carried out in spite of significant uncertainty in knowledge of the environmental parameters and at considerable noise levels. Then, not unexpectedly, the quality of the inversion increases with the amount of data available (more transceivers in this case). However, with multi-frequency data the improvement saturates rather rapidly. The results suggest that one can achieve a 5% RMS accuracy in the measurement of the surface current with only a few transceivers. Just as in single-frequency case, the three-frequency case required detailed information on the depth-dependence for a robust inversion. With data at 11 or 41 frequencies, the inversion results are good for any number of transceivers in the array. However, for an array consisting of a single transceiver, inversion is not stable and is expected to fail when additional mismatches are taken into account. Inversion with 2 and 3 elements in the array are robust. The degree of robustness depends on the choice of the element depths.
The numerical simulations demonstrate that data on the depth- and frequency-dependence of phase non-reciprocity can be efficiently combined for current inversions with the MNT method. The developed technique is a promising tool for acoustic monitoring of ocean currents in coastal zones. The present investigation is a part of a preparatory theoretical and numerical studies for the planned at-sea MNT validity test in the Straits of Florida.
The authors are very grateful to Dr. D. R. Palmer, NOAA/AOML, for fruitful discussions of the MNT technique and its potential applications. Work supported in part by RFBR, project 95-05-14616, and INTAS, project 93-0557.
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