An Opportunity for Improved Observation of Ocean Currents in the Coastal Zone

Oleg A. Godin
School of Earth and Ocean Sciences, University of Victoria, P.O.Box 1700,
Victoria, British Columbia V8W 2Y2 , Canada
Dmitry Yu. Mikhin
P. P. Shirshov Oceanography Institute of the Russian Academy of Sciences,
Moscow 117851, Russia

Abstract - We present a new low-frequency acoustic technique for three dimensional current monitoring which is based on an innovative use of reciprocal acoustic transmission and matched-field processing. Robustness and accuracy of the inversion in the presence of acoustic ambient noise and uncertainties in knowledge of the ocean bottom topography, sound velocity field in water, and other environmental parameters as well as in positions of acoustic transceivers are addressed via numerical simulations. Amount of the input data required for the successful MNT inversion is evaluated in terms of the number of acoustic transceivers comprising the monitoring system and of their signal's parameters. [1]

I. INTRODUCTION

Real-time, non-invasive measurements of flow velocity field in three dimensions over horizontal scales large compared to the ocean depth would be perhaps the most important contribution of acoustics to coastal oceanography. Such observations are certainly beyond the reach of direct in situ measurements as well as of currently available high-frequency acoustic methods. There is, however, a long list of compelling reasons for such observations. Knowledge of the current field contributes to an understanding of the dynamics of ocean circulation and upwelling and is necessary for estimating vorticity, heat transfer, and other important characteristics of an ocean region. Current measurements can be used to verify general circulation models and the assumptions made in their derivation. There are situations where current remote sensing in the coastal zone can contribute to global climate studies through monitoring of major current systems in narrow, coastal straits. An example is estimating the heat flux associated with the Gulf Stream system by monitoring the flow in the Strait of Florida where the system is confined between Florida and the Bahama Banks. Monitoring the flow further downstream would be logistically more difficult.
There are also a number of societal concerns that can be addressed through coastal current monitoring. Real-time current measurements over horizontal scales large compared to the ocean depth are of value to shipping and fishing interests, in tracking potential pollutants, and in search and rescue operations. They can be used to regulate the discharge of potential pollutants and can support the design of new discharge systems.
Significance of low-frequency acoustics for flow velocity field measurement in shallow water has been recognized long ago. An effort was made to apply travel-time acoustic tomography, developed by Munk and Wunsch [1] for the deep ocean, in the Strait of Florida [2]. The deep-ocean approach proved to be inapplicable to current measurements in shallow water because of an inability to resolve and identify bottom-interacting acoustic rays, which translates into a lack of vertical resolution in flow velocity measurements.
A new full-field acoustic technique for current monitoring was recently put forward [3]. It is based on an innovative use of reciprocal acoustic transmission and matched-field processing. This Matched Non-reciprocity Tomography (MNT) technique does not rely on rays or normal modes but uses nonreciprocity of the full acoustic field, measured as a function of frequency or position, as input data for the current velocity inversion. (Nonreciprocity of an acoustic quantity, by definition, is a difference in the quantity value for sound waves propagating in opposite directions between two given points in space.) Further research has revealed [4, 5] that nonreciprocity of various acoustic quantities (amplitude, phase and complex amplitude of a CW, etc.) possess quite different sensitivity to variations of transceiver positions, flow velocity, sound speed, and other environmental parameters. Acoustic phase non-reciprocity was identified as the most promising quantity to be used as input data for the full-field current inversion.
In this paper, performance of the MNT-based current inversion in the presence of various mismatches is evaluated via numerical simulations for an environmental scenario of particular interest.
II. A PPROACH
Our environmental scenario is chosen to represent conditions met in the Strait of Florida in summer during the shallow-water tomography experiment described in [2]. Flow velocity and sound speed in water are assumed to be range-independent. Vertical profiles of and of current velocity horizontal component in the vertical section containing source and receiver are shown in Fig. 1. These profiles are obtained from data presented in [2, 6]. Note a well-defined near-bottom sound channel which causes downward refraction and multiple bottom interactions of acoustic signals. The ocean bottom was modeled as a homogeneous fluid halfspace. As surface-scattered signals were not observed in the experiment [2], we chose to represent the ocean surface by a pressure-release horizontal boundary.
FIGURE 1
Sound speed and current velocity profiles used for numerical simulations in two cases considered

The numerical simulations consist of modeling sound propagation in a vertical plane between a moored transceiver and a vertical transceiver array. The propagation model [3] is based on a finite-difference solution of the parabolic wave equation for sound in a moving fluid. The assumed geometry of the numerical experiment provides for remote sensing of a single component of the current velocity, namely, . Monitoring of both horizontal components of the current, not considered in this paper, requires an additional point transceiver. As with other tomography techniques [1, Sec. 6.9], observations in many vertical planes crossing the ocean region under study are necessary to get resolution in the horizontal plane. Collecting such a data set for the MNT inversion would require two or three arrays and a number of point transceivers. Hence, resources for a three-dimensional current velocity MNT inversion are not dramatically different from those for the inversion in a single vertical plane.
To find the current velocity through MNT, one should minimize a cost function, which is a measure of similarity between measured and predicted non-reciprocity in acoustic phase. We used one of two cost functions, either the RMS difference between the simulated phase non-reciprocity for a "true" environment and that for a "trial" environment, or the averaged sum of discrepancies squared of sines and of cosines of the two phase non-reciprocities. During averaging over the array transceivers, higher weights were given to those with higher acoustic intensities. In the process of inversion, experimental geometry and environmental parameters other than current velocity field were assumed to be known (exactly or approximately). Unless stated to the contrary, we looked for the solution to the inverse problem in a two-dimensional parametric space defined by the magnitude of the barotropic current component and the magnitude of the baroclinic current component with linear depth dependence. This was done by calculating a cost function at grid nodes in this space. The current profile corresponding to the node having the minimal cost function was taken to be the solution to the inverse problem.
Ambient noise and inevitable uncertainties in our knowledge of system and environmental parameters contribute to discrepancies between measured and simulated acoustic fields and affect the inversion results. The mismatches we considered include discrepancies in horizontal transceiver separation, array tilt, bottom geoacoustic parameters, bathymetry, as well as acoustic noise and random and systematic errors in the sound speed. In addition, the effect of number of transceivers in the array on quality of the inversion was considered.
III. N UMERICAL RESULTS
A. Monochromatic Sound, Range-Independent Environment
Consider first a range-independent problem. Let a moored transceiver be situated at the depth of 360 m and 40 m above the ocean bottom. Horizontal separation between the moored transceiver and a vertical transceiver array is 20 km, sound frequency Hz. Relevant sound speed and current velocity profiles are shown at Fig. 1a. There exist 10 normal modes with phase velocity below sound speed at the ocean surface. With a grid in parametric space chosen, two adjacent nodes correspond to 10 cm/s difference in the barotropic component of the current velocity or 5.77 cm/s RMS difference in the baroclinic component. In an ideal case, when there no mismatches and the array consists of equidistant transceivers spanning the whole water column, the cost function has a sharp global minimum (Fig. 2a). The minimum occurs in the node which is the closest one to the point in the parametric space corresponding to the best fit (in the RMS sense) of the given profile of current velocity by a linear function of depth.
FIGURE 2
Inverse MNT cost function vs. the model parameters in no-mismatch case (a) and under the presence of 6 different mismatches (b) (all details are given in the text)

To study the effect of mismatch in horizontal separation of transceivers, "experimental" data were calculated for various values of the separation, whereas in the process of inversion the separation was fixed at a nominal value of 20 km. With the mismatch increasing, the minimum value of the cost function increases and position of the minimum shifts. We consider inversion successful if shift of the global minimum is not greater than to an adjacent node in the parametric space and there are no qualitative changes in the cost function relief in the vicinity of the minimum. This analysis show that mismatches up to 100 m in the horizontal separation are admissible, which are well above routinely achievable navigation accuracy. When the system mismatch consists of an unknown array tilt, no significant variations of the cost function have been found, provided the tilt is within 20 dg. of vertical.
When a random, uncorrelated acoustic noise is added to the "experimental" value of pressure at each hydrophone, a SNR of 10 to 12 dB has been found necessary for a successful inversion. For a narrow-band signal such a limitation does not seem restrictive.
The relation between amount of acoustic data available and the quality of the MNT inversion has been considered by inverting data from arrays of various length, position, and with various numbers of equidistant elements. Not surprisingly, it turns out that robust inversion requires a sufficient number of array elements located in the lower part of water column where in case of a near-bottom sound source the major part of acoustic energy propagates. This requirement being met, 16 element arrays spanning 30 to 40 per cent of water column are found sufficient for a successful inversion in the case considered. We therefore believe that MNT technique requirements for array positioning and number of elements are not as demanding as those of normal mode approaches requiring mode-resolving arrays.
Acoustic phase at one-way propagation is quite sensitive to sound speed variations. To simulate the effect of uncertainty in knowledge of the sound speed on the MNT inversion, "experimental" data have been simulated in a medium with systematic or quasi-random range-independent variation from the reference state assumed for calculation of the acoustic field replicas. Systematic error in is taken as varying linearly from its maximum value at the ocean surface to zero at the bottom. Amplitude of the random error has its maximum values at 9 prescribed horizons, sign of the error being opposite at any two adjacent horizons, and varies linearly between them. The horizons chosen are those at which values of the unperturbed sound speed are given. It is found that the current velocity inversion is successful when amplitudes of the sound speed variations are up to 0.5-0.7 m/s with systematic variations having more pronounced effects that random ones. It is expected that for range-dependent environments the above limit represents restriction of range-averaged uncertainty in the sound speed.
Figure 2b illustrates the cumulative effect on the inversion of all the mismatches considered above. In this case the vertical array consists of 16 equidistant elements at depths from 250 to 400 m. Mismatch in horizontal separation of the transceivers is 50 m, SNR equals 10 dB, the array tilt is 10 dg., and the amplitudes of random and systematic errors in the sound speed are 0.5 and 0.25 m/s, respectively. In spite of all the mismatches and the limited amount of data available, the cost function still has a single minimum in the domain of the parametric space considered. Although the minimum is not as deep as in the ideal case of Fig. 2a and somewhat smeared, it is sufficiently well-defined and amplitudes of both barotropic and baroclinic components of current velocity are determined correctly by the inversion within a spacing of the computational grid. This translates into depth-averaged RMS accuracy of the current velocity inversion within 10 cm/s.
B. E ffect of Bathymetric Variations
Under conditions of downward refraction and multiple interaction of acoustic energy with the ocean bottom, uncertainty in the knowledge of bathymetry is likely to be the factor of main concern as to applicability of full-field inversions, including the MNT technique. To obtain a quantitative estimate of bathymetric accuracy required for the MNT-based current velocity inversion, "experimental" input data were simulated in a number of range-dependent environments with piece-wise linear bathymetric variations of two kinds using the sound speed and current velocity profiles shown at Fig. 1b, whereas replica fields were calculated assuming a flat bottom at the nominal depth of 570 m. Bathymetric variations of the first kind are periodic functions of range with zero mean characterized by their spatial period ranging from 0.6 to 5.0 km and amplitude ranging from 0 to 30 m. Bathymetric variations of the second kind represent a single bathymetric hill (or, at negative values of its height , an ocean bottom depression). In the numerical simulations varied from -20 m to 20 m. It is assumed that in the vertical plane considered, the bathymetry consists of 3 horizontal segments and two sloping ones. Horizontal separation between a point transceiver and the array equals 36 km. The horizontal segments occupy ranges from 0 to 3 km, 6 km to 30 km, and 33 km to 36 km, respectively. The two short horizontal segments have a nominal depth, whereas that of the long segment differs by . All other mismatches are neglected. In particular, the transceiver array consists of 114 elements equally spaced between the surface and the bottom.
The bathymetric variations considered caused sizable perturbations of acoustic fields propagating up-stream and down-stream. Although suppressed in phase non-reciprocity, the perturbations are still significant for the MNT inversion. Influence of the unknown bathymetry on solution of the inverse problem is illustrated by Fig. 3, where an error in the inversion for the amplitude of the baroclinic current component is shown as a function of the bathymetric hill height and the depression depth. In these simulations, the barotropic component of the current velocity was assumed to be known. The simulation results show that the inversion error depends strongly on acoustic wave frequency and depth of the point transceiver. Rate of the error increase with is different for the hill and the depression. In most cases considered, to determine the current velocity from the MNT-based inversion with accuracy of 10 cm/s RMS one has to know bathymetry within 5 to 15 m, depending on sound frequency.
Effects of periodic variations in ocean depth on the ocean current inversion turn out to be as pronounced as those of a single hill (depression), but more complicated because of their non monotonous dependence on and as well as on frequency. This is likely to be related to resonance phenomena in acoustic normal mode coupling. In most cases considered, for the current velocity to be determined within 10 cm/s RMS, the amplitude of the depth variation has to be less than 2.5 - 5 m. Hence, for both kinds of bathymetric variations 0.5 to 3 per cent accuracy in ocean depth measurements is required. This accuracy is well within reach of commercially available bathymetric mapping systems.
C. Parametric space of high dimensionality
As true current velocity profiles in the examples considered above are not linear functions of depth, there is a discrepancy between the true profile and any inversion result obtained using the two-dimensional parametric space we have assumed so far. With a sufficient amount of input data available, accuracy of the MNT inversion may be increased by using a parametric space of high dimension. We have verified this via numerical simulations involving current velocity inversions in three- and four-dimensional parametric spaces. Second and third order polynomials of depth were used as additional basis functions to represent current velocity in acoustic field replica calculations. It was found that simultaneously with improved inversion accuracy, parametric spaces of high dimension give rise to progressively more complicated landscapes of the cost function. Efficient navigating of such landscapes possessing numerous local extrema requires application of advanced global search strategies, as are needed in other applications of matched-field processing [7; 1, Sects. 6.8, 6.10].
D. Multi-frequency probing signals
FIGURE 3


The error in the reconstructed amplitude of the baroclinic current component vs. the hill height (depression depth)


a) CW sound source is at 530 m depth. The signal frequencies are 100 Hz (solid line with crosses), 50 Hz (long dashes with squares) and 25 Hz (short dashes with triangles). The other details of modeling are given in the text.

b) The same as (a) for two frequencies (50 and 25 Hz) and the source depth of 490 m.

Although current velocity inversion is possible with CW sound, one can increase the amount of input data available by using broad-band or multi-frequency signals. We recently considered [8] the possibility of exploiting frequency dependence of phase non-reciprocity instead of or in addition to its depth dependence. Numerical simulations show that phase non-reciprocity is a distinct and non-degenerate function of frequency. An order-of-magnitude estimate of the rate of variation of the phase non-reciprocity with frequency is where is the range, is the average Mach number, and is the average sound speed. This gives 5-10 degrees per Hertz for the environmental parameters we are using. This rate implies that phase non-reciprocity can be resolved at several dozen frequencies with existing transceivers for moderate values for the SNR.
Numerical experiments were carried out [8] under environmental conditions similar to that described above and for four types of probing signals: a CW sound field with frequency of 50 Hz, and fields at 3, 11, and 41 frequencies, equally spaced between 40 and 60 Hz. In the process of inversion, input data at different frequencies were processed incoherently by averaging a cost function over frequency. It was shown, that data on depth- and frequency-dependence of phase non-reciprocity can be efficiently combined for current inversions with the MNT method. Just as in the 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 were accurate and robust with as little as 2 or 3 transceivers in the array. At a SNR of 20 dB, with mismatches of 50 m in horizontal separation of transceivers, 25 per cent and 30 m/s mismatches in the ocean bottom density and sound speed, respectively, no significant effect on the inversion was found.
IV. D ISCUSSION
The new approach to acoustic remote sensing of ocean currents in coastal ocean, MNT, considered in this paper, is based on a remarkable property of long-range underwater sound propagation, namely, that non-reciprocity of acoustic phase does not depend, to first order, on variations of sound speed, ocean bottom geoacoustic parameters, bathymetry or horizontal transceiver separation. As previously discussed [4, 5, 7], the phase non-reciprocity possesses a distinct dependence on current velocity profile as well as on transceiver depth and sound frequency. Therefore, as demonstrated in the present research through numerical simulations under environmental conditions representative of that in the Straits of Florida, the MNT-based inversion of phase-non-reciprocity depth dependence gives rise to good results in spite of significant uncertainty in knowledge of system and environmental parameters and at considerable noise levels. The benign restrictions on accuracy of the array element positioning for the MNT inversions suggest the possibility of using a synthesized aperture instead of a true transceiver array for short-term experiments. The synthesized aperture could be created with a vessel-suspended transceiver or with an underwater winch, by varying the depth during the measurement cycle.
There are a number of ways to further improve quality of MNT-based inversion. For the sake of simplicity, inversion results described above were obtained with a primitive algorithm to search for position of the cost function minima in a low-dimensional parametric space. Application of an advanced search algorithm and/or flow-velocity model of higher dimension, although more computationally demanding, would result in better resolution of the inversion. Simultaneous tomographic inversion of the non-reciprocal component of acoustic field for current velocity and of the reciprocal field component for transceiver positions and/or the ocean bottom geoacoustic parameters, sound speed in water, etc., can further increase the range of admissible system and environmental mismatches, when necessary. Improvements in both resolution and robustness may be achieved by using oceanographically meaningful basis functions to model the flow velocity field. Such basis functions could be obtained from a hydrodynamic model describing ocean circulation in the region under consideration. Using such a basis is particularly valuable to enhance resolution in the horizontal plane.
Theoretical studies suggest that in a coastal ocean, including straits, MNT has a number of important advantages over the traditional ray-travel-time tomography and normal mode tomography, including applicability to a much wider range of propagation conditions and higher robustness with respect to system and environmental mismatches. If the theoretically predicted properties of the MNT technique are confirmed experimentally, this approach would provide a basis for long-term monitoring of ocean dynamics in coastal regions. It seems essential and timely to have the MNT concept validated in an at-sea experiment.
ACKNOWLEDGMENT
The authors gratefully acknowledge many helpful discussions with Dr. D. R. Palmer, NOAA/AOML, of the MNT technique and its possible role in remote sensing of the coastal ocean.
REFERENCES
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[W]ork supported in part by RFBR, project 96-02-18538a, and NSERC Research Partnerships program grant at the University of Victoria, Victoria, B. C., Canada.