COMPUTER SIMULATIONS OF ACOUSTIC TOMOGRAPHY OF OCEAN CURRENTS IN COASTAL REGIONS
Dmitry Yu. Mikhin [1], and Oleg A. Godin [1][2]
An innovative inversion technique called Matched Non-Reciprocity Tomography (MNT) was proposed by the authors for acoustic monitoring of oceanic flows in coastal areas [1]. The method is an extension of Matched Field Processing (MFP) to the problem of current field tomography. The long-term goal of our extensive computer simulations is to establish applicability and robustness of MNT technique under realistic environmental conditions and under the presence of the various uncertainties one faces in field experiments. These uncertainties create mismatches in the input data set for flow reconstruction and degrade the solution. We focus our attention on shallow-water propagation conditions in the Straits of Florida, which is one of the best potential areas for a MNT-based observation system and for an at-sea MNT validity test. In the present paper we study the influence of mismatches in the sound speed field on MNT performance. The novelty in the model as compared to our previous publications [1-3] is primary that the medium is horizontally inhomogeneous .
RECONSTRUCTION OF FLOW IN RANGE-DEPENDENT MEDIA
The scenario of numerical experiments is similar to that used in [2-4]. The normal mode model is applied to simulate reciprocal propagation of multi-tonal signals between a single transceiver and a near-vertical array of transceivers. The acoustic devices are anchored on the sea floor approximately 36 km apart. The flow direction is assumed to make a 45° angle with the vertical plane passing through both moorings. Sound speed and current velocity profiles (Fig. 1a) are superimposed by 2D perturbation and/or placed over a bathymetry linearly varying with range. From the point of view of the inversion, bottom inclination, perturbation , additive acoustic noise, uncertainties in experiment geometry (propagation range) and in bottom geophysical parameters (sound speed, density) are considered as mismatches. They contaminate the "experimental" data and are absent when the "theoretical" non-reciprocity is simulated. Godin and Mikhin [3] thoroughly studied degrading of the inversion results with increase of the last three mismatches. In the present paper we concentrate on the first two. The impact of variations in the bottom depth was also addressed in [4] for a different model of topography uncertainties. In addition, we will consider a limited number of array elements as one more potential source of tomography errors.
The flow velocity field is obtained from the simulated data by broad-band MNT processing [3]. The inversion is range-independent, i.e., the current profile sought-for by the inversion depends on depth z only . The flow is represented as a linear combination of barotropic and baroclinic components, their amplitudes being and respectively. The baroclinic one linearly depends on depth. The forward solutions are constructed on a grid in the plane. Originally the grid steps along both axes are chosen to be 0.1, and then are decreased to 0.025. Matched non-reciprocity processing for CW signal is usually performed by minimizing the weighted sum of differences in sine and cosine components of phase non-reciprocity over the array elements. Absolute phase can be erroneous in experimental measurements. The negative impact of such errors is eliminated by adjusting the mean phase at the array to provide the best correspondence with theoretical data. The CW cost function is
(1)
Here is the cost function to be minimized; and are the weighting factor and the phase non-reciprocity, respectively; are the complex acoustic pressures simulated for downstream and upstream propagation under trial "theoretical" and "experimental" environments; is an argument of a complex number x; angular brackets stand for averaging over array aperture. For multi-tonal signals the output of the CW MNT processor (1) is averaged over frequency. The current profile corresponding to the grid node with the minimal cost function is the solution to the inverse problem. More complicated Maximum Likelihood Method (MLM)-based MNT processor is described in the next section.
To study the impact of mismatch we represent the sound speed field perturbation as a truncated Fourier series with range and depth:
, (2a)
, (2b)
where km and m. The 1.2 factor is introduced to avoid periodicity effects. This perturbation imitates the error of the sound speed tomography and is not related directly to any physical wave phenomenon. We consider two models of range and depth power spectrum. A white spectrum model assumes all sine and cosine components in (2b) have the same relative amplitudes independent of their space scales, whereas an inverse squared spectrum model assumes the relative amplitude of decays as , which tapers small scale variability. A random number generator gives coefficients for a particular perturbation. Then are normalized so that RMS deviation of the sound speed field equals 1 m/s for . Hence, A is the RMS deviation of the true sound speed field from that used in inversion. It is further referred to as the perturbation amplitude. Typical plots of for both models and are presented in Figs. 1a and 1b. We assume and . The inverse problem is sequentially solved for A varying from -5 to +5. The modeling is repeated 32 times for different sets of random coefficients to obtain ensemble-averaged minimal, maximal, and average errors in the reconstructed current velocity profile.
We use two independent computer programs for predicting multi-tonal acoustic fields under the "experimental" environmental conditions and experimental geometry. The first code is developed by the authors and accurately accounts for the current's influence on sound propagation. The program solves the eigenvalue problem for layered moving media [5, Sect. 4.4] by applying Pruffer transformation of variables. The exact modal expressions [5, Sect. 4.4] are applied to calculate the acoustic field. No limitations on Mach number of a flow and its vertical gradients are imposed. The main drawback of the code is that horizontal inhomogeneities are treated in adiabatic approximation [5, Sect. 7.3], which unavoidably fails for horizontally inhomogeneous media with large A. The other model is one-way coupled mode implementation of KRAKEN [6]. We apply the Effective Speed of Sound Approximation [7] in order to use KRAKEN in moving media. The model partially accounts for mode coupling, but describes the effects of currents approximately. Although both models used to simulate experimental data are approximate, the approximations are built using different small parameters of the problem that assure confidence in the results. The medium is assumed layered during the inversion and the trial acoustic fields are calculated using our code.

Fig. 1. Environmental conditions of the numerical experiment: a) background sound speed (top) and current velocity (bottom) profiles; b) and c) - typical perturbations of the sound speed field for white (b) and inverse squared (c) power spectra.
First, let us consider the inversion results for a dense transceiver array. The single transceiver is placed at 530 m depth (40 m above the sea floor), and the array consists of 23 elements equally spaced from 560 to 10 m in depth. Their central frequency is taken to be 50 Hz and the frequency bandwidth 10 Hz. These values, relatively low for real-life devices, are chosen for faster simulations. Based on [3], the MNT-based inversion is expected to give good mapping of the current field even for 2-3 transceivers in the array if the data at 11 or more frequencies are available. In the present paper we use 11 frequency bins equally spaced from 40 to 60 Hz.
The errors of the reconstructed in-plane current profile for white-spectrum perturbation are shown in Fig. 2. The solid green, red, and violet lines show the RMS, minimal and maximal values, respectively, of the inversion error (top) and the corresponding cost function value (bottom). They were generated using the adiabatic mode model on a grid , with 0.1 steps along both axes. The dashed cyan lines show RMS values plus and minus one standard deviation. Two more mismatches are included concurrently with : 20 dB additive (pseudo)-random acoustic noise, which is assumed depth and frequency independent for simplicity, and 50 m propagation range error. The reconstruction error remains within 10 cm/s (yellow line on the top plot in Fig. 2) for the relatively wide interval . However, the maximal error and the dispersion are too high, and the lines are not smooth. For comparing adiabatic and couple-mode results, green crosses on the same plot show RMS values , predicted with KRAKEN. Coupled-mode model reveals further increase of for negative A. The maximal error also grows (not shown).

Fig. 2. Ensemble-averaged errors of the reconstructed current velocity profile (top) and the respective values of MNT cost function (1) in the point of minimum (bottom) vs. perturbation amplitude A. Line types and markers correspond to different computer models and parameter grid steps.
The ragged behavior of the plots can be a result of large separation between the grid nodes, and, hence, poor resolution. To verify this assumption, the modeling was repeated with KRAKEN on a finer grid , with 0.025 steps (green and red triangles at Fig. 2 show the obtained RMS and maximal values). The improvement is drastic for large A: in some cases is lower than the original . The detailed grid is used in all subsequent examples. The admissible reconstruction error cm/s is reached for RMS sound speed perturbations m/s (yellow lines on the bottom plot in Fig. 2). Although the improvement has been achieved, it also indicates that the cost function minima at large perturbations are flat and degraded. Having added new points between the nodes of the original grid, we discovered new maxima comparable to the original one. The result suggests we approached the bottom of a stretched canyon on the cost function landscape far from the original point. The same can be concluded from plot at large A. The function reveals saturation after and its value is relatively high. Finer parameter space sampling or an increase in A leads to substantial migration of the minimum position, but the cost function in the minimum remains almost unchanged. On the other hand, detailed sampling hardly affects the minimum position for small A. The landscape depression is sharp with a low cost function value.
The discrepancy of the results obtained by the two models indicates that the adiabatic theory fails for strong perturbations at least for negative A. KRAKEN solution is free from ESSA-induced errors in the absence of flows. Comparison of one-way acoustic fields predicted for media at rest with the same sound speed profile and perturbations revealed the adiabatic theory is applicable up to for inverse squared spectrum and up to only for white spectrum.

Fig. 3. Ensemble-averaged errors of the reconstructed current velocity profile (top) and the respective values of MNT cost function (1) in the point of minimum (bottom) vs. perturbation amplitude A for different number of transceivers in the array. KRAKEN coupled-mode model is used.
So far, we assumed the transceiver array to be dense. This is hardly the case with real experimental facilities. Decreasing the number of array elements, each one an expensive source-receiver pair, makes an MNT-based observation system more affordable. For a concept validity test one may use a few commercial transceivers attached to a single mooring instead of a specialized transceiver array. We study the performance of our system for a moderate number of array cells. The ensemble-averaged errors in the reconstructed current velocity field and MNT cost functions are calculated for , white perturbation spectrum, and for , inverse squared spectrum (Fig. 3). These few-element arrays are synthesized using the data from each 6th and each 4th transceiver of the original aperture. The four-element array has cells at 560, 410, 260 and 110 m, and the six-element one at 560, 460, 360, 260, 160 and 60 m. The perturbations are specified by the same sets of coefficients as earlier. An additional 50 m propagation range mismatch and dB acoustic noise are added. The current reconstruction for , white spectrum, is now considered as an reference case. The corresponding and are shown in Fig. 3, where solid green, red, violet, and dashed cyan lines stand for RMS, maximal, minimal and RMS plus and minus standard deviation values. The crosses and circles show for six- and four-element arrays. The additional errors in due to finite number of transceivers proved to be considerably smaller than standard deviation in the reference case. The maximal and minimal errors for the few-element arrays are shown by dashed and dotted red and violet lines. They are close to the corresponding plots for . The MNT-based tomography technique provides an accurate solution for the inverse problem although the input data set is limited and contaminated by the mismatches. The inversion reveals no obvious dependence on the perturbation spectrum, and, hence, on the typical horizontal and vertical scales of sound speed inhomogeneities within the considered limits.
ALTERNATIVE INVERSION METHODS AND ADVANCED BEAMFORMERS IN CURRENT TOMOGRAPHY
The described MNT inversion is based on a relatively simple RMS matching of theoretical and experimental data originated from the least squares method. It is widely recognized that more advanced methods (often called beamformers) can dramatically outperform simple least squares in MFP problems. Bartlett and MLM beamformers can serve as good examples [8]. However, caution should be warranted when the existing methods are extended to use in current tomography. The physics of such a problem is quite distinct, and the optimal beamformer might differ as well.
The paper by Elisseeff and Schmidt [9] serves as an illustration of common misconceptions in current tomography. It considers reconstruction of current velocity profile in a stratified environment. The flow was assumed uniform in the upper 40 m layer, followed by 10 m transition zone and 175 m motionless water column down to the bottom at 225 m depth. For inversion purposes the current was described with two parameters: the velocity of the uniform layer and the transition zone depth. Multi-tonal input acoustic data at 8 frequencies ranging from 200 to 250 Hz were obtained with two sources and two 32-element receiving arrays moored 2 km apart. This scheme of measurements is the first problem in the Elisseeff's approach. Ocean current tomography is normally based on the acoustic reciprocity principle, which guarantees that in the absence of currents, the sound pressure at point created by a monopole source [3] at point equals exactly to pressure at point emitted by the identical source at [10]. The only way to separate relatively subtle effects of flows from those of the sound speed field variations is to use the difference of some acoustical quantities, e.g., propagation times [11], measured in reciprocal transmission scheme. Obviously, this is not the case in [9]. The reciprocity of their data holds only due to the symmetrical experiment geometry in a stratified environment. Even in a medium at rest, mismatches in the depths of devices or the medium range-dependence breaks the symmetry and will be interpreted as a flow presence!

Fig. 4. Current reconstruction over a sloping bottom using true non-reciprocity (green) and a difference of acoustic fields obtained independently by 2 sources and 2 receiving arrays (blue). The plots show the mean error in the reconstructed flow (top left), the corresponding cost function (bottom left), and the mean errors in barotropic (bottom right) and baroclinic (top right) components of the current.
To illustrate the importance of truly reciprocal input data we modified the MNT scheme (1) according to Elisseeff's design with 2 sources and 2 receiving arrays. The field emitted by an element of the transceiver array and recorded by the stand-alone transceiver at was replaced with a field from an additional stand-alone source at recorded at a element of an additional array. The performance of the original and modified methods is compared in a motionless medium with bottom depth linearly varying in range. The range-mean ocean depth of 570 m was held constant over modeling. The experimental data are simulated with KRAKEN over a sloping bottom and the inversion is done in a stratified environment with the same sound speed profile. Both receiving arrays have 23 hydrophones located equidistantly from 560 to 10 m in depth. Both sources are at 530 m depth horizon. As earlier, a -20 dB acoustic noise and a propagation range mismatch of 50 m are added. The reconstruction results are shown in Fig. 4, where the abscissa axis is the difference of the bottom depths at the mooring sites. The symmetry, and hence, the modified solution, breaks completely for m at 36 km, which corresponds to the bottom inclination as small as 0.04°. The true medium is motionless, while the tomography based on the wrong measurement scheme predicts a noticeable flow up to 0.7 m/s at the surface and 0.1 m/s at the bottom. It is important that the observed error is not due to a particular inversion method, but occurs because the given data are close to the exact acoustic field non-reciprocity in some moving medium. Truly reciprocal input data give us perfect results for all up to the maximal considered 80 m drop.
One more problem of applying matched-field methods in current tomography is the choice of a physical quantity taken as an input data for inversion. The analysis, first presented in the pioneering paper on MNT [1], revealed that a straightforward approach of matching complex non-reciprocities (defined as differences of complex acoustic fields down and up the flow) shows very poor results under the presence of relatively subtle mismatches. To understand such a behavior let us assume that at long ranges both down- and up-stream fields consist of single modes with eigenvalues and eigenfunctions , respectively. Let us also assume that all propagation conditions, hence and , are known exactly and so is the experiment geometry, except for the propagation range, which has mismatch. The non-reciprocity of the complex field is
, (3)
where , , and . Hence, the complex non-reciprocity as well as the complex fields themselves are locally periodic with range, the period being about a wavelength . On the other hand, the differences of the phases and of the amplitudes of the same fields have much larger typical scale of horizontal variations about , where M is a typical Mach number. The presented derivation can be directly extended to the case of multi-modal propagation, and even to range-dependent media, if we also assume that the considered propagation range mismatches are much less than the spatial interval of modal interactions and interference. Similarly, one can prove instability of complex non-reciprocity with respect to variations of the sound speed in water and in the bottom.
The possibility of matching complex non-reciprocities was also addressed by Elisseeff and Schmidt, who used an advanced inversion technique based on a MLM beamformer and utilized multi-frequency data by averaging the output in dB of CW beamformer over frequency. Their claim that the sophisticated multi-tonal algorithm managed to reconstruct the flow under the presence of acoustic noise [4] and propagation range mismatch contradicts the analysis presented here. Certainly, the local period of spatial variations of complex non-reciprocity depends on frequency and averaging over f helps to diminish the negative influence of propagation range mismatch, but cannot eliminate it completely. The considered mismatch of m is just too small to deteriorate the solution. However, the ambiguity surface for such a tiny mismatch [9] is obviously degraded compared to the no-mismatch case. More importantly, the conclusion of [9] refers to a completely unrealistic case of layered medium with precisely known sound speed field in water and in the bottom.
To study performance of advanced beamformers under the presence of concurrent and strong mismatches we implemented two versions of the MLM algorithm. Our approach is similar to that of [9] to facilitate comparison of the results. The first method is based on matching the non-reciprocity of complex fields , as Elisseeff and Schmidt did, except for we use true non-reciprocity measured by a transceiver array. As far as we could conclude from [9}, for layered media this beamformer is exactly the same as in [9]. It is further referred to as Elisseeff's one. In the second method the value to be matched is , where the asterisk denotes complex conjugate. The complex product depends on the difference of phases in down- and up-stream propagation. Hence, it is stable to the mismatches considered in the previous paragraph. Note that terms equivalent to enter I and in the cost function (1), which is highly stable to the mismatches. Of course, the presented considerations do not prove that our MLM beamformer based on is the optimal in a strict statistical sense. However, they suggest that this method is a good initial choice for studying the performance of advanced beamformers in flow tomography.

Fig. 5. Ambiguity surfaces generated by Elisseeff's MLM beamformer (left) and phase-based MLM beamformer (right) in no-mismatch case (top) and under the presence of 5 m/s mismatch in the bottom sound speed (bottom).
We begin with comparing the performance of two MLM beamformers in a simple problem when the only mismatch is an error in the bottom sound speed . The medium is layered with sound speed and current velocity profiles shown in Fig. 1a. The data are obtained on a 23-element array under the same geometry and environmental conditions as in our previous models. SNR in both beamformers is taken to be 50 dB, but the conclusions presented below proved to be independent of this choice. The color plots of MLM outputs for no-mismatch case and for m/s are shown in Fig. 5. In the absence of errors both beamformers show excellent performance. However, when a moderate mismatch is introduced, the first of them fails. Its main extremum stretches and shifts from the correct location. Moreover, the sidelobes become comparable to the main lobe. An error m/s destroys the solution completely. The same happens when a 90 m (i.e., ~3) error in propagation range is introduced. The non-reciprocity of complex fields proves to be highly sensitive to errors in the reciprocal parameters, as Eq.(3) implies. The second MLM beamformer, which mainly uses the phase information, is not affected by both mismatches. For its sidelobes are 13 dB below the main peak.

Fig. 6. The results of current velocity reconstruction by Elisseeff's MLM beamformer (blue) and phase-based MLM beamformer (green) under the presence of random uncertainties in the sound speed field with RMS amplitude A. Solid lines with markers stand for RMS values, dashed lines - for minimal and maximal values of corresponding parameters.
In the final simulation the same MLM beamformers are compared with each other as well as with the cost function (1) under the presence of random mismatch. The modeling conditions are similar to the reference case (Fig. 3), except for no additional mismatches are given for MLM inversions. The acoustic noise is introduced as a diagonal contribution to the covariance matrix and not by adding random values to , as done in the previous section. The same sound speed field perturbations with white power spectrum are used. The experimental data for 23 array elements are modeled by coupled-mode KRAKEN code and inversion is performed over a fine grid in parameter space.
The results of current tomography by two MLM beamformers are presented in Fig. 6. In addition to errors in the flow profile (top left plot) and MLM output in the points of maxima (bottom left), Fig. 6 shows separately the reconstructed amplitudes of barotropic (bottom right) and baroclinic (top right) eigenfunctions. The heavy horizontal lines on the right plots stand for the optimal amplitudes of the current components. White noise level of -20 dB is assumed. As expected, the phase-based beamformer outperforms its complex-field based rival. Actually, the plot even underrates the error of Elisseeff's beamformer. Its inaccuracy in reconstructing the baroclinic current component is so high, that is almost always on the lower limit of our grid. Had we used wider grid limits, the mean amplitude of would further decrease causing higher RMS error in current velocity profile. Even the barotropic component, i.e., the mean current averaged over depth and distance, is reconstructed with large errors. On the opposite, our MLM beamformer works very well, especially for measuring the mean flow. One might question the reliability of ensemble-averaged predicted by our method, as its minimal and maximal values are sometimes on the lower or upper boundaries of the grid. Fig. 7 presents histograms of predicted for m/s. For the first two values the plots show obvious points of maxima, which makes us confident in the results. For large A the level of the main lobe reveals saturation, similar to the cost function (1) (Fig. 3). As explained above, this indicates that the main peak becomes flat and comparable to the sidelobes. Based on this criterion and cm/s condition, we estimate the limit of MLM robustness as m/s. The solution is still accurate for stronger perturbations, but can become unstable if more mismatches are introduced.

Fig. 7. Histograms of found by phase-based MLM beamformer for A=2, 4 and 5 m/s (blue, green and violet).
Finally, we compare the performance of phase-based MLM beamformer and the cost function (1) for the same perturbation field . The errors in the predicted amplitudes of barotropic and baroclinic components are shown in Fig. 8 (bottom and top). White noise level for MLM solution is taken at -20 dB as in the previous section. The average deviations of amplitudes of the barotropic and baroclinic components from their optimal values (heavy horizontal lines in Fig. 8) are smaller for MLM method. On the opposite, the least-squares method (1) has lower uncertainty of the solution. The presented modeling suggests that the proper selection of the acoustical quantity to be measured in the experiment is much more important for successful flow reconstruction than the particular choice of beamformer.
CONCLUSIONS
Matched Non-Reciprocity Tomography is an advanced full-field technique for acoustic monitoring of oceanic flows in coastal areas. This paper presents a part of extensive computer simulations aimed to study MNT robustness for realistic environmental conditions and under the presence of uncertainties in experimental geometry and environmental conditions, that degrade the flow reconstruction. The main novelty compared to our previous results [1-4] is that the sound speed is no longer assumed stratified. We study the influence of range-dependent deviations of the true sound speed field and bathymetry from those assumed when solving the inverse problem. The sound speed field perturbations are chosen to be random. They model the remaining error of one-way temperature tomography. The ocean depth is taken to vary linearly with range to imitate smooth decline or raise of the bottom. Two independent normal mode computer codes are applied to simulate multi-tonal acoustic fields propagating in opposite directions between a single moored transceiver and a transceiver array ~36 km apart. The data at 11 frequencies equally spaced from 40 to 60 Hz are used for inversion.

Fig. 8. Comparison of current inversion based on least-squares cost function (1) (blue) and MLM beamformer (green). Solid and dashed lines stand for RMS, minimal and maximal values.
The modeling demonstrates that MNT provides an accurate reconstruction of the flow field for sound speed uncertainties up to 2 m/s RMS concurrently with mismatch in propagation range and -20 dB acoustic noise. This validity limit reveals little or no dependence on the spatial scale of random perturbations. The inversion is robust with as few as 4 transceivers in the array, suggesting the use of several commercial transceivers attached to the same mooring instead of a specialized multi-element array.
An alternative approach to current tomography proposed by Elisseeff and Schmidt [9] is analyzed and compared to the MNT scheme in terms of its efficiency under the presence of experimental mismatches. Reciprocal transmissions between two sources and two receiving arrays are used in [9]. The data obtained in such a scheme of measurements are shown not to be truly reciprocal. The scheme simply relies on symmetry of the problem and can work in stratified media only. We reproduced the Elisseeff's inversion algorithm as close as possible and proved that the inversion indicates strong currents in a motionless medium when the bottom slope is as small as 0.04°. The MNT inversion is not affected by 0.13° decline, i.e., by 80 m bottom drop between the moorings, the maximal value reached in our simulations. The errors of in-situ depth measurements certainly satisfy this limitation.
Finally, we address application of MLM-based methods for solving the inverse problem. Two kinds of MLM beamformers are considered. The first proposed in [9] is aimed at a best correspondence of non-reciprocities of complex acoustic fields down and up the flow. Analytical estimates imply that this method should be unstable with respect to errors in reciprocal parameters such as propagation range or sound speed. The second beamformer is an extension of our old method and matches non-reciprocities of signal phases. In accordance with theoretical analysis, the presented modeling confirm the phase-based beamformer dramatically outperforms its competitor when the input data are contaminated by mismatches in sound speed field (in water or in the bottom) and/or in distance between the moorings. Comparison of phase-based MLM beamformer with our previous least-squares inversion shows that the advanced method has smaller error in the reconstructed flow at the expense of larger uncertainty of the solution.
ACKNOWLEDGMENTS:
The authors are grateful to Acad. Prof. L. M. Brekhovskikh, OI RAS, for support of their work, and to Dr. D. R. Palmer, NOAA/AOML, for fruitful discussions of the MNT technique, its potential applications and requirements for at-sea measurements. The material is based upon work supported in part by Natural Sciences and Engineering Research Council of Canada, the Russian Foundation for Basic Researches under project 96-02-18538, and by the U.S. Civilian Research and Development Foundation under Award No. RG1-190.
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[1] P. P. Shirshov Oceanology Institute of the Russian Academy of Sciences
36 Nakhimovskii Prospect, Moscow 117218, Russia
[2] School of Earth and Ocean Sciences, University of Victoria,
P.O. Box 1700, Victoria, B.C., Canada V8W 2Y2.
[3] We omit more complex sources for simplicity only. For details on the analytic form of reciprocity principle in more general cases refer to [11].
[4] The impact of additive noise is quite different from that of distance and sound speed mismatches, and is not described analytically in the same way as (3) (the numerical examples in this paper and [5, 6] include random noise).