INTRODUCTION
Among the topical issues that are actively discussed by specialists in the field of applied ocean acoustics and related issues of acoustic oceanography, a special place is occupied by the issue of assessing the necessary requirements on the volume and quality of information regarding the properties and parameters of the ocean environment to improve the practical capabilities of receiving antenna systems. In essence, this means quantitative assessments of the requirements for the means of operational control of the environment and values of its parameters that affect the efficiency of detection and estimation of signal parameters against the background of the sea noise and interference in real ocean conditions [1, 2]. There is no doubt that correct information about the propagation environment of received signals should be taken into account in their processing algorithms [3–7] and in this sense, “environmental matching of the signal processing” based on an adequate model, as an approach, is not just appropriate, but necessary. However, what kind of information is required for the realization of such an approach, about what parameters of the environment and, in particular, with what allowable error of their estimation in natural conditions, is not obvious in advance and is a subject of discussion. Without analyzing these fundamental aspects, both the approach itself and specifying it with specific algorithms, remain undefined to the extent necessary for effective practical application.
In recent works [8, 9], this issue was considered by numerical modeling of the efficiency of a vertical linear array (VLA) as a spatial filter of narrowband signals from remote sources in a shallow sea channel and received under conditions of a priori inaccurate knowledge of its parameters. This paper develops the results obtained there with a focus on a comparative analysis of substantially different signal processing methods that rely on the same (assumed known) propagation channel model. The difference between the methods lies not only in their different mathematical formulations, but also in the fact that they use a priori information regarding the conditions of reception of a useful signal to different degrees. As will be shown below, such conditions are mainly determined by the possible presence of intense interference and ambient sea noise at the array input.
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We have chosen the shallow sea channel as a medium for signal propagation not only for reasons of practical importance, but also because in such a channel the sound fields are strongly influenced by the underlying seabed [10]. Consequently, geoacoustic channel parameters should be explicitly included among the parameters to be evaluated along with the sound velocity vertical profile (SVVP) and the underwater channel depth. Obviously, the amount of a priori uncertainty of the medium model only increases, and significantly, the uncertainty of data on bottom parameters in natural conditions is usually quite significant. In turn, the VLA as one of the basic configurations of sonar antennas is of particular interest due to the practical possibility of selecting spatial harmonics of received signals (waveguide modes or ray paths) in the vertical plane of the channel. The limitation of the problem formulation associated with the fixed frequency assumption is not fundamental in this context, since spatial processing is usually understood as frequency-dependent filtering of signals using arrays of a certain (not too small) wave size. In the case of receiving signals with finite spectrum width, their array processing involves the use of frequency accumulation of the output signals of separate narrow-band spatial filters [11–13].
Following the works [8, 9], we continue the consideration of one of the realistic scenarios, the reception of a weak useful signal generated at the VLA input by a remote localized source, against the background of wind-generated sea noise and intense interference generated by another localized source (possibly several). The operating frequency is assumed to be low (the range of the first hundreds of Hz), with this in mind we rely on the well-known normal-mode formalism for describing sound fields in an underwater channel [10, 14]. We characterize the efficiency of the processing methods by the array gain performance in terms of the signal/(noise + interference) ratio (SNIR). In fact, this is the main, although heuristic in nature, characteristic of extended array as an effective spatial filter of a useful signal against additive noise and interference; at the same time, its relationship to the likelihood ratio as a statistical criterion for the weak signal detection is well known (e.g., [11, 12]).
With respect to the underwater channel model, we assume that the SVVP, channel depth, and underlying bottom parameters are not known exactly, but with some deviations from those reference values, which are used in the calculation of the array weight vector according to the choice of the spatial processing method. These deviations, set within certain intervals, form an ensemble of “real” values of the environment parameters and thus characterize the level of a priori uncertainty of the reference model. In such a reception scenario, both the reference model itself and the processing methods based on the model become fundamentally mismatched, which is most consistent with the practice of receiving signals from remote sources in real sea conditions. The exact matching is considered as that ideal scenario that is met by the potential of signal processing techniques in a given channel; in this sense its consideration is undoubtedly useful.
Here, we quantitatively compare three essentially different methods of array signal processing: (i) the matched-signal processing (MSP) which originally was called matched-field processing in English-language literature [3, 4], (ii) heuristic method of matched-mode filtering (MMF) for one of the signal-carrying modes (with the condition that this mode is selected according to a certain criterion) [15, 16], and (iii) optimal signal processing (OSP), where optimization is resulted from the criterion of the output SNIR maximum [11]. These known methods are essentially different approaches to spatial filtering of a multimode signal against a background of modal noise and interference. Our main interest is precisely to reveal, based on their specific example, the dependence of the array gain loss for the model-based signal processing on the level of a priori uncertainty of the model and on the “external” conditions of useful signal reception. Such conditions are understood here, first of all, as relative intensity levels and mode composition of the interference and noise fields at the VLA input.
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PROBLEM STATEMENT AND BASIC EQUATIONS
The sound channel of the shallow sea and the position of the VLA in it were set using data from one of the Institute of Applied Physics, Russian Academy of Sciences field experiments with a stationary acoustic path in the Barents Sea [17] using a tone source at 240 Hz (this choice is made only for certainty of the modeling conditions). The depth of the VLA arrangement site is 160 m, the reference SVVP along the signal propagation path (shown as a bold line in Fig. 1) is constructed by linear interpolation of the measurement data at 32 points along the depth, following in 5 m intervals (marked by dots). It is further assumed that, due to its natural variability and measurement error, the real SVVP is known with some arbitrary deviations from the reference profile. The model of its “instantaneous” realizations is chosen in the form of:where \({{c}_{i}}\) are the realizations of sound velocity at each ith measurement horizon; \({{\left( {{{c}_{0}}} \right)}_{i}}\) are the measured values; \({{\xi }_{i}}\) are the statistically independent random variables uniformly distributed over the interval \([ - 1,\;1]\); and \({{\left( {\operatorname{var} \left( c \right)} \right)}_{i}}\) is the amplitude of deviations, which is a modeling variable and monotonically decreases with depth (as index i increases) from maximum value at the surface to zero at the bottom. The narrowing “arm” of the SVVP variations and several of its random realizations are shown in Fig. 1. We proceed here under the realistic assumption that the variations in the SVVP have a depth dependence and are maximal in the upper layer, which is most affected by temperature variations in the atmosphere. We limit the maximum interval of variations in the modeling to 5 or 10 m/s (which corresponds to temperature variations at the level of 1 or 2°C), and it is passed with a small step of 0.1 m/s in calculations shown below.
$${{c}_{i}} = {{\left( {{{c}_{0}}} \right)}_{i}} + {{\left( {\operatorname{var} \left( c \right)} \right)}_{i}}{{\xi }_{i}},\,\,\,\,i = 1,...,32,$$
(1)
Fig. 1.
The reference SVVP (bold line, points on which correspond to sound velocity measurement depth) and five random realizations of SVVP (1) within an interval monotonically narrowing with depth. On the left are the positions of the VLA elements by depth (circles); on the right are the sample values of the depths of the sources of useful signal (S, 80 m) and interference (N, 10 m).
We emphasize that the intervals of a priori uncertainty of the SVVP we set do not exceed the variability of the reference SVVP (~20 m/s) within the depths of 40–60 m, i.e., the seasonal hydrology of the summer-type channel is generally preserved. Consequently, it is precisely those relatively small and short-term variations in the sound speed field that are subject to assessment by operational oceanography. At the same time, the chosen model of SVVP variations does not pretend to be highly consistent with the character of the variability of the real ocean, which is, moreover, very variable for different marine areas and in different seasons. Additional corrections to model (1), such as “smoothing” the realizations by introducing a heuristic constraint on the magnitude of the vertical sound speed gradient or mutual correlations of speed variations at close horizons are more in line with in-situ conditions, but are not the subject of this paper. We note only that such correction is expected to lead to some weakening of the effects demonstrated below, but will not affect their nature (separate calculations with such corrections of model (1) have been made).
Figure 1 also shows: the position of the VLA elements, consisting of 13 elements, located equidistantly after 8.5 m in the depth interval from 44.5 to 146.5 m below the mixed and relatively warmed layer; the particular depths of the point sources of signal (S, 80 m) and interference (N, 10 m). These positions of the array and sources are consistent with most of the illustrations below.
The underlying bottom is simplified as a homogeneous liquid absorbing half-space; the reference values of the parameters are set as follows: density ρb = 1.8 g/cm3, sound velocity cb = 1750 m/s, sound attenuation coefficient βb = 0.07 dB/km Hz or 0.12 dB/λ. Taking into account the significant variation of geoacoustic parameters of bottom rocks of different types, we limit the intervals of assumed deviations from the specified values to rather large values: 50% in either direction for density, –15 to 25% for the sound velocity, from 0.01 to 0.7 dB/km Hz for the attenuation coefficient. Such intervals include values of the relevant parameters for most known sedimentary bedrock types. Since the bottom parameters are inherently stable quantities, we model their a priori uncertainty within the specified intervals with a fixed step: 1% for density and velocity, 0.01 dB/km Hz for attenuation coefficient. The uncertainty of the channel depth is given selectively by individual error values within the relative value of 3%, which is comparable to the errors given by the available World Ocean Atlases. For a given channel, the absolute error of bathymetry in such an estimate from above is 5 m (within the wavelength for a given frequency).
Thus, the a priori uncertainty of the computational model of the sound propagation channel for the whole set of its main physical parameters is modeled by us taking the natural specificity of these parameters and the possibility of obtaining correct information about them in natural conditions into account.
Let us further specify the scheme of calculation of sound fields at the VLA input. The calculation is performed in the framework of the normal-mode formalism using the software developed earlier by Dr. I.A. Shereshevsky and then adapted by Dr. I.P. Smirnov (IAP RAS) for modeling of multimode signal processing methods in MatLab environment in various problem statements (see, for example, [8, 18, 19]). The calculated expression for the components of the useful signal vector at the input of the jth array element has a well-known form of superposition of separate normal modes [10, 14]:
$$s\left( {{{z}_{j}}} \right) = \sum\limits_{m = 1}^M {\frac{{{{u}_{m}}\left( {{{z}_{{\text{S}}}}} \right){{u}_{m}}\left( {{{z}_{j}}} \right)\exp \left\{ {\left( {i{{\kappa }_{m}} - {{\alpha }_{m}}} \right){{r}_{{\text{S}}}} + {{i\pi } \mathord{\left/ {\vphantom {{i\pi } 4}} \right. \kern-0em} 4}} \right\}}}{{\sqrt {{{\kappa }_{m}}{{r}_{{\text{S}}}}} }}} .$$
(2)
Here, \({{u}_{m}}\left( {{{z}_{j}}} \right)\) is the value of the waveguide eigenfunction at the depth of the jth element; \({{\kappa }_{m}}\)and \({{\alpha }_{m}}\) is the propagation constant and attenuation coefficient of the mth mode (real and imaginary part of the longitudinal wave number of the mode, respectively); \({{r}_{{\text{S}}}}\) is the horizontal distance to the signal source; \({{z}_{{\text{S}}}}\) is the depth of the source; and M is the total number of propagating modes of the discrete spectrum. We use a similar expression to calculate the interference signal at the array input by substituting the coordinates \({{r}_{{\text{N}}}},{{z}_{{\text{N}}}}\) for the corresponding source into expression (2). Mode attenuation coefficients \({{\alpha }_{m}}\) are calculated for the given bottom parameters by the following formula [10]:
$${{\alpha }_{m}} = \frac{{{{\rho }_{{\text{w}}}}}}{{{{\rho }_{b}}}}\frac{{k_{b}^{2}u_{m}^{2}\left( H \right)}}{{2{{\kappa }_{m}}\sqrt {\kappa _{m}^{2} - k_{b}^{2}} }}{{\alpha }_{b}}.$$
Here, H is the channel depth (given 160 m); \({{k}_{b}} = {{2\pi f} \mathord{\left/ {\vphantom {{2\pi f} {{{c}_{b}}}}} \right. \kern-0em} {{{c}_{b}}}}\)is the wave number of the sound wave in the bottom for a given frequency (240 Hz); \({{\rho }_{{\text{w}}}}\) is the density of water (g/cm3); and the parameter \({{\alpha }_{b}}\) characterizes the absorption properties of bottom sediments and is related to the attenuation coefficient \({{\beta }_{b}}\) by a relation: \({{\beta }_{b}} = \left( {{{{\text{40}}\pi {\kern 1pt} {\text{log}}{\kern 1pt} e} \mathord{\left/ {\vphantom {{{\text{40}}\pi {\kern 1pt} {\text{log}}{\kern 1pt} e} {{{c}_{b}}}}} \right. \kern-0em} {{{c}_{b}}}}} \right){{\alpha }_{b}}\) (dB/km Hz).
For a given frequency in the channel with the specified reference parameter values, the number of modes is \(M = 28\), i.e., the modeled sound propagation channel is really a multimode waveguide. It is important that in such a waveguide qualitatively different situations can be realized in terms of the mutual location of the intensity spectra of the useful signal, interference, and sea noise by mode numbers. It is natural to expect that the effectiveness of our (or any other) chosen processing methods will strongly depend on this, both in the case of full certainty of the channel model (exact matching) and under its a priori uncertainty.
As in [8, 9], the channel is assumed to be horizontally homogeneous and deterministic, hence, the effects of mode interaction and scattering on random inhomogeneities of the channel accumulated with distance are not taken into account. Physically, this limits the specified distances to the corresponding sources at the level of the first tens of kilometers from above (for the selected low frequency range). Specifically, the source of the useful signal is set in the interval of average depths (selectively at a depth of 40, 80, or 120 m), the source of interference, under the surface (at a depth of 5 or 10 m), the distances to both sources are fixed the same (15 km), the angular positions of the sources in the horizontal plane do not play a role and therefore are not set. The ratio of the total (by elements) intensities of the signal to the interference at the VLA input is also an important parameter of the problem and is set low at the level of –10 or –20 dB.
The noise background is assumed here to consist of two independent components: spatially white noise, and the ambient sea noise, which is of wind origin and is generated in the near-surface layer of the channel; we use one of the most cited models [20] for its consideration. Such noise is obviously not fully coherent interference at the array, in qualitative difference from the noise interference from a localized source. Crucially, the sea noise is anisotropic in the vertical plane of the channel due to a certain mode composition shifted to high-order modes effectively excited from the sea surface. We also note that the adequate choice of a spatially distributed environmental noise model depends on the chosen frequency range. For example, for lower frequencies (~100 Hz and below) it is necessary to take the noise of distant shipping into account, which becomes its main source. Therefore, and with decreasing frequency and the corresponding increase in the size of the “noise-forming” water area, the transformation of the noise mode composition into the region of low mode numbers and the corresponding smoothing of vertical anisotropy becomes typical. Most importantly, both supposed sources of intense interference at the array input, the source localized in the waveguide and the ambient noise, together occupy a certain part of the mode spectrum of the waveguide (even a significant part of it) and thus sharply limit the possibilities of noise-resistant reception of the signal field modes.
Thus, we focus on a practically demanded scenario of VLA reception of a weak and spatially correlated (coherent) signal of a remote underwater object “masked” by a powerful coherent interference of a surface vessel and partially coherent anisotropic sea noise.
The computational expressions for finding the antenna gain under such a scenario are well known, they are of the form:
$$\begin{gathered} G = \frac{{{\text{SNI}}{{{\text{R}}}_{{{\text{out}}}}}}}{{{\text{SNI}}{{{\text{R}}}_{{{\text{in}}}}}}},\,\,\,\,{\text{SNI}}{{{\text{R}}}_{{{\text{in}}}}} = \frac{{\sigma _{{\text{S}}}^{2}}}{{K + \sigma _{{\text{N}}}^{2} + \sigma _{{{\text{mn}}}}^{2}}}, \\ {\text{SNI}}{{{\text{R}}}_{{{\text{out}}}}} = \frac{{{{{\mathbf{w}}}^{ + }}{{{{\mathbf{\tilde {R}}}}}_{{\text{S}}}}{\mathbf{w}}}}{{{{{\mathbf{w}}}^{ + }}\left( {{\mathbf{E}} + {{{{\mathbf{\tilde {R}}}}}_{{\text{N}}}} + {{{{\mathbf{\tilde {R}}}}}_{{{\text{mn}}}}}} \right){\mathbf{w}}}}, \\ \end{gathered} $$
(3)
$$\begin{gathered} {{{{\mathbf{\tilde {R}}}}}_{{\text{S}}}} = \sigma _{{\text{S}}}^{2}{\mathbf{\tilde {s}}}{\text{*}}{{{{\mathbf{\tilde {s}}}}}^{{\text{T}}}},\,\,\,\,{{{{\mathbf{\tilde {R}}}}}_{{\text{N}}}} = \sigma _{{\text{N}}}^{2}{\mathbf{\tilde {n}}}{\text{*}}{{{{\mathbf{\tilde {n}}}}}^{{\text{T}}}}, \\ {{{\tilde {R}}}_{{{\text{mn}}}}}\left( {{{z}_{i}},{{z}_{j}}} \right) = \sigma _{{{\text{mn}}}}^{2}{{{\tilde {r}}}_{{{\text{mn}}}}}\left( {{{z}_{i}},{{z}_{j}}} \right), \\ \end{gathered} $$
(4)
$${{\tilde {r}}_{{{\text{mn}}}}}\left( {{{z}_{i}},{{z}_{j}}} \right)\sim \sum\limits_{m = 1}^M {\frac{{{{{\tilde {u}}}_{m}}\left( {{{z}_{i}}} \right){{{\tilde {u}}}_{m}}\left( {{{z}_{j}}} \right)\tilde {u}_{m}^{2}\left( {{{z}^{/}}} \right)}}{{{{{\tilde {\alpha }}}_{m}}{{{\tilde {\kappa }}}_{m}}}}} .$$
(5)
Here, G is the array gain, which is usually defined as the relation of the SNIR at the array output (\({\text{SNI}}{{{\text{R}}}_{{{\text{out}}}}}\)) to the input SNIR averaged over the array elements (\({\text{SNI}}{{{\text{R}}}_{{{\text{in}}}}}\)); K is the number of array elements; \(\sigma _{{\text{S}}}^{2},\) \(\sigma _{{\text{N}}}^{2},\) \(\sigma _{{{\text{mn}}}}^{2}\) are the total input intensities of the useful signal (lower index S), interference (index N) and modal noise (index mn), which are normalized to the white noise intensity \({{\sigma }^{2}}\) (hereinafter \({{\sigma }^{2}} = 1\)) and set as dimensionless values; matrices \({{{\mathbf{R}}}_{{\text{S}}}}\) and \({{{\mathbf{R}}}_{{\text{N}}}}\) of dimension \(\left( {K \times K} \right)\) are the normalized correlation matrices of the useful signal and interference, respectively, which are dyadic matrices of unit rank due to the initial assumption of complete coherence of the fields of localized sources at the array input (the normalization \(\left\| {{{{\mathbf{R}}}_{{\text{S}}}}} \right\| = \operatorname{Sp} \left( {{{{\mathbf{R}}}_{{\text{S}}}}} \right) = \sigma _{{\text{S}}}^{2},\) \(\left\| {{{{\mathbf{R}}}_{{\text{N}}}}} \right\| = \operatorname{Sp} \left( {{{{\mathbf{R}}}_{{\text{N}}}}} \right) = \sigma _{{\text{N}}}^{2}\) is used); \({{{\mathbf{R}}}_{{{\text{mn}}}}}\) is the similarly normalized correlation matrix of sea noise, the elements of which (5) are calculated by the formulas of [20] assuming that the statistical ensemble of noise sources is isotropically distributed in the horizontal plane of the channel at a fixed depth of \({{z}^{/}} = 1\) m; E is the unit correlation matrix of white noise; the vectors \({\mathbf{\tilde {s}}},\;{\mathbf{\tilde {n}}}\) are the normalized column vectors of signal and interference replicas at the array input, respectively (normalized by \({{\left| {{\mathbf{\tilde {s}}}} \right|}^{2}} = \;{{\left| {{\mathbf{\tilde {n}}}} \right|}^{2}} = 1\)); w is the array weight vector of ; and the upper indices “*”, “T”, “+” are the signs of complex conjugation, transpose and Hermite conjugation of matrices, respectively. The upper tilde sign in expressions (3–5) means that the corresponding vectors and matrices are calculated for the model parameters varying with respect to their reference values, i.e., for the assumed conditions of mismatch between the reference model of the channel and the real channel.
We emphasize that all fields (signal, interference, and noise) are calculated as reduced to the VLA input so as to operate with the dimensionless ratio of their normalized intensities of the form \({{\sigma }^{2}}:\sigma _{{\text{S}}}^{2}:\sigma _{{\text{N}}}^{2}:\sigma _{{{\text{mn}}}}^{2}\). This allows us not to specify the actual capacities of the sources themselves, treating them as free parameters. It is clear that taking the total field attenuation of a single source along the propagation path, including the cylindrical divergence and mode attenuation into account, these powers become dependent on the distance to the source. For the purposes of our consideration such calculations do not play a role. The specified hierarchy of signal intensities at the array input is practically important; its principal influence will be demonstrated below.
Within the given scenario of signal reception the array weight vectors for MSP, MMF and OSP have the following forms:
$${{{\mathbf{w}}}_{{{\text{mf}}}}} = {\mathbf{s}}^{*},\,\,\,\,{{{\mathbf{w}}}_{q}} = {{{\mathbf{u}}}_{q}},\,\,\,\,{{{\mathbf{w}}}_{{{\text{opt}}}}} = {{\left( {{\mathbf{E}} + {{{\mathbf{R}}}_{{\text{N}}}} + {{{\mathbf{R}}}_{{{\text{mn}}}}}} \right)}^{{ - 1}}}{\mathbf{s}}{\text{*}}.$$
(6)
Here, \({{{\mathbf{u}}}_{q}}\)is the normalized vector of the modal shape with number q over the VLA elements (modal vector), which is valid for the vertical array (the sign of complex conjugation is omitted here).
Thus, according to (6), each array processor is assumed to be “tuned” to those replicas of the input signals that correspond to the reference model of the channel, while the array input contains “real” signals appearing in expressions (3)–(5) and they differ from their reference replicas due to variations of the model parameters in the given intervals. In essence, this is our approach to modeling processing methods under a priori uncertainty of the medium model in order to calculate their efficiency loss, defined by the antenna gain G (3). On this basis, we will obtain quantitative estimates of the requirements for the acceptable level of model uncertainty in terms of the set of its physical parameters.
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MODELING RESULTS AND DISCUSSION
We begin our discussion of the simulation results by showing which mode number distributions correspond to the signals at the VLA input within the reference channel model. Figure 2 shows the moduli of modal amplitudes (i.e. mode excitation coefficients) in the field of the useful signal (curve 1) and interference (curve 2) at different depths of both sources equidistant from the VLA at a distance of 15 km; curve 3 in Fig. 2c corresponds to the ambient sea (modal) noise (the dots everywhere mark discrete values for each mode number; the connecting lines are drawn for illustration only). For clarity of comparison, the spectra are given here on the same scale, while in reality they are very different and proportional to the given values of the input intensities of the corresponding fields.
Fig. 2.
The normal mode spectra of the signal field (dashed curve 1) and interference (solid curve 2) for different depth positions of the sources: (a), 40 m (signal) and 5 m (interference); (b), 120 m (signal) and 10 m (interference); (c), 80 m (signal) and 10 m (interference) with modal noise added (solid curve 3).
The mode spectrum of the useful signal was expectedly broader and strongly “chopped,” since its source is located in the region of the middle depths of the channel. It is well seen that only a small fraction of the signal field modes appear relatively “free” from the interference, namely, a few low-number modes in Fig. 2a and, in addition to them, a few medium-number modes in Fig. 2b (in the region of the deep dip in the interference spectrum). However, taking into account the sea modal noise, which, as well as the interference, “starts” from about the sixth mode, the values of the input SNIR over the middle and high modes are additionally and strongly reduced. This leads, as will be shown below, to an even greater difference between the gains of the matched signal filter, which does not take into account the interference and modal noise, and the optimal filter, which performs their effective suppression by inverting the total correlation matrix of the interference and noise according to expression (6).
Note that taking the relative smoothness of the mode spectrum of the “subsurface” interference field into account, it can be qualitatively interpreted in a broader sense as the spectrum of not one, but a set of interference sources localized in some depth interval that together form an intense interference at VLA inputwith a certain mode composition.
By analyzing the spectra of energy-carrying modes of the received fields and the resulting distribution by mode numbers of the SNIR value (modal SNIRs), it is possible to correctly select the most “useful” mode of the signal field, for which this value will be the largest, and to implement the MMF processor for this single mode. For example, for the spectra shown in Fig. 2, the numbers of such modes are as follows: 5, 1, and 2, respectively. As will be seen in the following, such a mode filter can provide close to an optimal filter in terms of antenna gain, but also turns out to be much simpler to implement and more robust to environmental mismatch. We emphasize that such a heuristic mode-filtering procedure is inherently adaptive, since the choice of the specified mode cannot be made a priori, but only on the basis of empirical estimation of the modal SNIRs. Importantly, however, this choice is determined not only by the positions of the relevant sources, which is obvious, but also by the position of the receiving VLA. It was previously shown [16, 21] that the positions of individual VLA elements and the VLA as a whole in depth are a factor that significantly affects the antenna gain in various methods of processing multimode signals in the background of interference, including the method of matched-mode filtering. This indicates additional possibilities for improving processing efficiency associated with a special selection of the VLA arrangement in a channel (we will return to this aspect below).
Before proceeding to further discussion, let us give a useful illustration, which clearly demonstrates the qualitative difference between the processing methods considered here. In [22, 23], a characteristic of an array that receives multimode signals was introduced, which is a direct physical analog of the conventional plane-wave array beampattern, the beampattern in mode domain, or modal pattern. This characteristic, like the modal spectra of received signals, is a discrete function of mode numbers, i.e., a vector of dimension M, each element of which is the response of the array with some weight vector to a separate waveguide mode (mode response): \({{F}_{q}} = {\mathbf{u}}_{q}^{T}{\mathbf{w}}.\) The modal pattern is then defined in a general form as \({\mathbf{F}} = {\mathbf{U}}_{{}}^{T}{\mathbf{w}},\) where \({\mathbf{U}}\) is a matrix of dimension \(\left( {K \times M} \right),\) which is the matrix of the mode structure at the array input (consisting of column vectors \({\mathbf{u}}_{q}^{{}}\), \(q = 1,\;2,\;...,\;M\)). The modal pattern explicitly defines the “directivity” of the VLA in mode domain and is therefore a convenient tool for analyzing methods of array signal processing in multimode transmission channels (not only for vertical arrays).
For example, for the scenario of receiving a useful signal against a background of interference and mode noise (Fig. 2c), provided that the normalized field intensities at the VLA input have the ratio 1 : 10 : 100 : 100 (in the above order), the array modal patterns with weight vectors (6) are shown in Fig. 3. It is clearly seen that the optimal processing (curve 3) provides spatial filtering of only a small group of the very first modes, “free” from the intensive interference and noise. This is a physical reason to receive the useful signal against their background with the highest gain. In turn, the MSP (curve 1) has high “side lobes” in the middle and high number regions, which leads to a strong weakening of its efficiency in this scenario. The MMF for the second mode (curve 2) has a close to optimal modal pattern and, accordingly, also provides effective suppression of interference and modal noise. However, at the same time it suppresses a part of interference-resistant signal modes, because the “main lobe” of its modal pattern turns out to be too narrow (the mode number here is chosen \(q = 2\) according to the above estimates).
Fig. 3.
VLA modal patterns for the MSP (1, dashed curve), MMF of the second mode (2, dashed-dotted curve) and OSP (3, solid curve) in the case of signal reception on the background of the interference and sea modal noise. The mode spectra of the received fields correspond to Fig. 2c when the ratio of normalized intensities at the array input 1 : 10 : 100 : 100.
1. The Influence of Variations in SVVP
The influence of random variations of the SVVP on the VLA gain (so far without taking into account variations of the bottom parameters) is demonstrated in two qualitatively different situations of signal reception, in the absence of intense modal noise (Fig. 4) and against its background (Fig. 5). For a better comparison of processing methods, the dependencies of interest are shown here in a relatively large interval of variations up to 10 m/s. Minor curve breaks are associated with a finite number of velocity profiles, which was chosen to be 100 for each value of variation amplitude from zero to maximum with a step of 0.1 m/s.
Fig. 4.
The dependences of the VLA gain on the amplitude of the SVVP variations for matched signal filtering (1, dashed curve), matched filtering of the second mode (2, dashed-dotted curve) and optimal processing (3, solid curve) in the case of signal reception on the background of the interference without taking into account the modal noise: normalized field intensities at the array input are equal to (a), 1 : 10 : 100 and (b), 1 : 10 : 1000. Source locations are: (15 km, 80 m) and (15 km, 10 m) for signal and interference, respectively (also in the following figures).
Fig. 5.
Similar Fig. 4 dependences in the case of signal reception on the background of interference and modal noise: (а) 1 : 10 : 100 : 100 and (b) 1 : 10 : 100 : 1000.
Figure 4 shows the array gain loss at two different values of the input signal-to-interference ratio (SIR): –10 dB (Fig. 4a) and –20 dB (Fig. 4b); Fig. 5, at fixed SIR at –10 dB and different intensity of modal noise (10 dB and 20 dB to the signal level, Fig. 5a and 5b, respectively). The horizontal dashed line in both figures marks the gain level G = K, which, as is well known, corresponds to the matched-signal spatial processing of a fully coherent signal against a background of spatially-white noise. It is physically obvious that in the situation of intense and significantly anisotropic interference this level can be significantly exceeded, since even matched signal filtering can partially (and even significantly) suppress the interference provided that there are noticeable differences in the modal spectra while the optimal processor provides such suppression in the best possible way. Such “anomalies” of the VLA gain at reception of multimode signals have been repeatedly demonstrated earlier [15, 21, 24].
The effect of the strong initial divergence between the curves that we expected, especially for the optimal and matched-signal processing, which exceeds the value of 10 dB in Fig. 4a and 20 dB in Fig. 4b. At the estimation level, the incremental gain of the OSP over the MSP is equal to the inverse of the SIR, as evidenced by these values. However, as the variation of the SVVP increases, the optimal processing degrades rapidly; this is also an expected effect due to the presence of intense interference [8]. In fact, under such conditions, optimal processing not only partially “loses” the useful signal (as does matched-signal processing), but also suppresses the interference less, with the result that the OSP gain loss grows faster, and the more intense the interference is, the more severe they are. In its turn, the matched mode filtering of the most effective 2nd mode, occupying at first an intermediate position (its initial loss to the optimal processing is quite significant, about 8 dB), turns out to be much less sensitive to the variations of the SVVP. As a result, this method provides the largest antenna gain already at moderate SVVP variations of ~2 m/s, and then its gain over the other two methods only grows. The reason for the high stability of the MMF processor is physically quite clear and is related to the fact that the modes of the first numbers are the least sensitive to variations of the SVVP, especially if these variations appear to be the strongest only in the upper layer of the channel (Fig. 1). Thus, if we set the value of acceptable array gain loss equal for all processing methods as a requirement for the accuracy of operational control of SVVP variability, these requirements will turn out to be radically different. For example, for a given loss level of –3 dB, we have the estimates: ~0.5 m/s for optimal processing, ~2 m/s for matched-signal processing, and ~5 m/s for optimal (in the above sense) matched-mode filtering.
Figure 4 illustrates also another important effect, the possibility of nonmonotonic gain loss dependence for the MSP technique. This was also pointed out in [8], and this positive effect, which partly “saves” the MSP under mismatch conditions, is universal in the sense of its manifestation, and even better, in the case of variations not only of the SVVP, but also of the bottom parameters of the model (we will dwell on this below). This feature, at first glance somewhat paradoxical, is also explained by the presence of an intense modal interference, partially “covering” the signal modes, and in its absence it disappears.
From the comparison of Figs. 4 and 5 we can clearly see the significant influence of intensive modal noise. For the case of modal spectra considered (Fig. 2c), the modal noise has a “stabilizing” effect on the efficiency of optimal processing with a multiple increase in the permissible variations of sound speed (already up to ~5–10 m/s) and, on the contrary, leads to additive loss of the MSP gain. Physically, this is due to the fact that the optimal processor is in this case a mode filter of a narrow group of the very first modes, practically “free” from interference and noise (Fig. 3), but it is these modes that are the least sensitive to variations in the channel parameters and provide an increase in the OSP robustness. Matched-signal processing is “tuned,” by definition, to all signal-carrying modes, including the middle and high number modes, which are completely “closed” by the intense noise, and therefore acquires in this case additional gain loss. The VLA gain for the MMF method still weakly depends on the environmental variations. Therefore, we can classify the MMF technique, provided that it is correctly “tuned” to one of the first modes, as a method that is robust not only to the channel model mismatch, but also to the modal noise level (within the framework of the used model, to the wind wave intensity in the array site).
We emphasize, however, that the qualitatively different influence of modal noise on array gain loss for different signal processing methods noted here should not be considered as a general conclusion, since it is caused by specific modal spectra of the corresponding fields (Fig. 2c). The general conclusion is rather a different statement, the influence of a priori uncertainty of the channel model (in this case, on the SVVP in the water column) on the efficiency of different processing methods and in different conditions of signal reception is highly variable and even not always unambiguous, if we keep in mind the possibility of a nonmonotonic dependence of gain loss with increasing mismatch.
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2. The Influence of Variations in Bottom Parameters
Let us now consider an aspect specific to shallow sea channels, concerning the influence of significant uncertainty in the underlying bottom parameters, sound speed, density and attenuation coefficient. Intuitively, it seems clear that the first of these should be much more “influential,” since the longitudinal wave numbers of the modes and hence the mutual phase shifts of the modes at the VLA input for each of the coherent signals of the localized sources depend on the sound velocity in the bottom (as well as in the water column). Since the computational model of signal propagation used by us does not take phase fluctuations of the modes into account, the phase detuning of the modes in the fields of both the useful signal and interference as a result of sound velocity deviations becomes a critical effect. In addition, not only the wave numbers but also the total number of modes of the discrete spectrum and, consequently, the modal spectra of the received signals depend on the sound speed in the bottom, while the number of modes depends weakly on the bottom density. For example, at a relative decrease of the sound speed from the reference value by 10% the number of modes decreases noticeably to \(M = 19\), and at the same relative increase, increases to \(M = 34\), while varying the density in the whole given interval does not practically affect the number of modes. We note that in the asymptotics of the infinitely large sound speed corresponding to an absolutely rigid bottom, \(M = 53\), which is much higher than the typical values for our statement.
The following two figures demonstrate the joint effect of density and sound velocity variations in the bottom (now without taking into account SVVP variations) in the case of intense modal noise: as three-dimensional dependencies (Fig. 6) and as equal level lines on the plane (Fig. 7). First of all, let us note the different scales of the values of interest on the vertical scale in these graphs. As in the previous figures, here we demonstrate significant differences in the initial VLA gains for different processing methods (when the reference model is accurately matched to the real channel) and qualitatively different nature of gain loss under conditions of increasing mismatch. The gain level isolines in Fig. 7 are plotted with the same 3 dB interval for all three methods, which makes it easy to estimate the width of the “stability zone” for both varying parameters for each of them and make corresponding comparative conclusions.
Fig. 6.
Dependences of VLA gain on amplitudes of relative variations of sound velocity in the bottom and bottom density for (a) matched-signal processing, (b) matched-mode filtering of the second mode and (c) optimal processing and in the case of signal reception on the background of interference and modal noise: 1 : 10 : 100 : 100. The circles mark the gain values in the absence of variations (at the origin of coordinates); the triangles mark the main maxima of dependences for each method.
Fig. 7.
The same dependencies as in Fig. 6, in the form of lines of equal gain level plotted in 3 dB intervals.
In both figures, the nonmonotonic character of the dependence for matched-signal processing (Figs. 6a, 7a), which we already noted, is clearly visible, but here this effect is more sharply manifested, there are additional and significant gain maxima and deep dips between them. The high level of such “side lobe” gains indicates that MSP may be rather effective even when the model is significantly mismatched in terms of bottom parameters. Such “returns” to the maximum (for this method) gain values are characteristic and quite predictable (can be calculated on the basis of the model); however, they are highly variable in all parameters of the problem and thus have a quasi-random nature. This feature does not, in our opinion, allow us to give such additional maxima much practical significance, but at the same time, it seems to be essential. Let us specify that the main maximum (13.6 dB) of the MSP gain is achieved in this case at a small density deviation of +3%, although it practically does not differ in magnitude from the initial point of zero deviations for both parameters (these values are here within the “main lobe” of the dependence). It can also be seen that the optimal processing (Figs. 6c, 7c), taking into account the above-mentioned specificity of the joint influence of the intensive interference and modal noise, even here (as in Fig. 5) shows itself to be rather stable in a relatively wide range of uncertainty of the bottom parameters. For the sound velocity, however, this feature is characteristic only at deviations downward from the reference value, i.e., at reduction of the total number of modes of the discrete spectrum. The asymmetrical nature of the dependence clearly demonstrated in this example indicates that the reference value of sound velocity in the bottom should be chosen with a certain “reserve” upwards, the gain loss in the conditions of mismatch may be less significant, and this feature of optimal processing seems to be important in terms of practical recommendations. The MMF for the second mode (Figs. 6b, 7b) also proved to be quite robust to variations in bottom parameters, especially density variations, in this receiving scenario. However, a shift of the main maximum is also visible for it: 22.8 dB for sound speed deviations of –3% and density deviations of –49% versus 20.8 dB at the initial point.
If we now “turn off” the wind waves and with it the intense noise in the middle- and high-order modes, the gain loss expectedly becomes much sharper, especially for the MSP and OSP methods (Figs. 8a, 8c). It can be seen that the area of the central “lobe” narrows sharply and does not exceed the first units of percent for sound speed and ~10% for density. Moreover, the main maximum of the array gain for MSP (28 dB) is now not only noticeably shifted away from the position of the center maximum, but also significantly surpasses it (by about 8 dB). The MMF method (Fig. 8b) consistently demonstrates relatively high stability, as a result of which it turns out to be more effective with increasing model mismatch in terms of bottom parameters. At the same time, and for this method, we can see a shift of the global maximum of the gain (26.2 dB) from the initial value (23.4 dB), i.e., by about the same amount (less than 3 dB, as in Figs. 6, 7), although the shift in the sound speed parameter itself turned out to be quite different. In comparison, the optimal processing provides a maximum gain of about 31 dB, but, as in Fig. 4, the gain loss increases rapidly with increasing sound speed mismatch. Therefore, this method is also characterized by a gully-like dependence, although it is less sharp in comparison with the MSP method.
Fig. 8.
Similar to Fig. 7 dependences in the absence of intense modal noise: 1 : 10 : 100 : 0.
In general, the “inversion” of processing methods with increasing model mismatch noted here and above (in Fig. 4) is quite clear and indicates only that the very term “optimal processing” refers only to that ideal reception scenario when the model of received signals is known exactly and is stationary, i.e., in the absence of any (random or deterministic) deviations of its parameter values from those reference values for which optimization is performed. Essentially the same is true for the term “matched-signal processing,” which in the reception scenario under the uncertainty conditions is obviously “mismatched-signal processing.”
For convenience in comparing the dependences shown in Figs. 4–8, we note that the gain values at the point of zero variations of the parameters in Fig. 5a coincide with the initial values of the corresponding dependences in Figs. 6, 7, and in Figs. 4a, they coincide with the initial values in Fig. 8.
On a par with sound speed and density, the sound attenuation coefficient in sea-bottom sediments can also have significant uncertainty. Without dwelling on the issue of the influence of its uncertainty in detail, we only note that it is even weaker than the influence of uncertainty on the bottom density, and becomes noticeable at the corresponding “errors” not by tens of percent, but by many times. For example, when the difference of the real sound absorption in the bottom from the reference value is three times (0.2 dB/km Hz vs. 0.07 dB/km Hz), the array gain loss for the matched-signal and optimal processing methods does not exceed 2 dB, and is practically absent for the matched-mode processing. A significant influence (more than 3 dB) is shown only at differences up to ~10 times (up to ~0.7 dB/km Hz), but such a strong discrepancy with the reference model actually means the absence of any reasonable information on the bottom sediments. However, the model uncertainty in this parameter should be more significant with increasing distances to the signal and interference sources. Intuitively, it seems clear that in this case errors in the absorption estimation will lead to a more noticeable distortion of the modal spectra of the received signals in comparison with the reference model (first of all, in the region of middle and high modes), and, together with them, to an additional growth of the mismatch effect (see also Fig. 4 in [8]).
Besides, we briefly discuss the impact of the uncertainty of the reference model on the channel depth. Within the limits of relatively small bathymetry errors (up to 3%, which does not exceed the absolute wavelength in our case), this influence can also be expected to be rather weak due to the weak sensitivity of the channel mode structure to variations in the water layer depth at this small level. Such intuitive expectations are confirmed by calculations: for all processing methods the VLA gain loss does not exceed ~1 dB.
Thus, in the situation of multimode signal reception against the background of intensive modal interference and modal sea noise, the VLA gain loss can have a very complex “multi-lobe” dependence on the level of bottom-induced model mismatch. Despite the fact that the variations of these parameters were set noticeably stronger than the variations of SVVP in relative units (units and tens of percents versus tenths of a percent), nevertheless, the corresponding values of the acceptable mismatch impose very strict requirements to the quality of estimation of these parameters in real marine conditions. The method of matched filtering of the most noise-immune signal mode can be close enough in array gain performance to the optimal processing under the conditions of pure matching and at the same time retains its relatively high efficiency even under conditions of significant model uncertainty on the bottom parameters.
The effects of mismatch on individual model parameters (SVVP, bottom parameters, and channel depth) allowed us to show the essentially different nature of their influence on the array gain loss. We now give an illustration of the joint influence of variations in sound velocity in the water column and in the bottom as the two physical parameters of the propagation channel, the uncertainty in the estimation of which has the greatest influence on the gain loss. The corresponding dependences are shown in Fig. 9 for the case of signal reception on the background of intensive modal noise, which we consider as more realistic. All the effects we noted above are clearly visible here: high and narrow “side lobes” of the dependence for MSP (Fig. 9a), leading to an ambiguous influence of mismatch on its efficiency; a more stable, but clearly asymmetric dependence for OSP (Fig. 9c); and a close dependence for the MMF (Fig. 9b) and its dominance at significant deviations of both velocity parameters.
Fig. 9.
The dependences of VLA gain on variations of SVVP and bottom sound velocity for (a) matched-signal processing, (b) matched-mode fintering for the second mode, and (c) optimal processing in case of signal reception on the background of interference and modal noise: 1 : 10 : 100 : 100.
Finally, we will dwell in more detail on the MMF technique. In the case where the most noise-immune modes of the received signal are low-number modes, this heuristic method provides, as shown above, sufficiently high values of VLA gain and at the same time has relatively high stability under the conditions of both environmental mismatch and significant variations in the sea noise level in the part of medium- and high-number modes. Consequently, it can be considered as an effective suboptimal processing method, and then the question of correct choice of the mode that is defined as a reference mode for its realization becomes practically important. We noted above, relying on the results of [16, 21], that the choice of such a mode depends not only on the spectrum of signal-carrying modes (Fig. 2), but also on the position of the VLA elements in the channel, in fact, on how these modes are “projected” onto the array and what mutual orthogonal properties they possess.
To illustrate this conclusion specific for VLAs, Fig. 10 shows the dependence of the MMF gain on the variations of the SVVP at different choices of the reference mode for three variants of an equidistant VLA with a fixed number of elements \(K = 13\). Figure 10a corresponds to the basic VLA position (Fig. 1), for which all previous dependences are shown; in Fig. 10b is the same VLA placed in the upper part of the channel (the depth of the first element is 13 m, the last one, 115 m); in Fig. 10c is a sparse VLA with the inter-element distance increased to double the wavelength (the depth of the first element is 8 m, the last one, 152 m). It is clearly seen that not only the choice of the reference mode for this processing method and the VLA gain achieved, but also the level of permissible variations of the SVVP have a clear and significant dependence on the array arrangement. This does indicate that the VLA placement in the channel cross-section is another factor to be taken into account when considering the robustness of spatial processing techniques to environmental mismatch.
Fig. 10.
The dependences of the VLA gain for the matched-mode filtering method on the amplitude of the SVVP variations for different numbers of the reference mode (the numbers of the curves correspond to its number) and three VLA positions: (a) baseline position (Fig. 1), (b) at the top of the channel, (c) sparse VLA.
Thus, based on the proposed approach to modeling the problem and the dependencies obtained within its framework, it becomes possible to quantitatively estimate those ranges of acceptable mismatch of the channel model by its main physical parameters, within which the VLA gain loss does not exceed 3 dB (or other specified level) for each of the signal processing methods. All given graphs demonstrate that these ranges, and with them the requirements to the operational parameter estimation in situ, are radically different for different parameters and for different conditions of signal reception against the background of intense interference and sea noise. They are also different (up to an order of magnitude) for different methods of array signal processing; the position of the receiving array also significantly affects the achieved values of gain under mismatch conditions.
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CONCLUSIONS
Let us summarize and formulate conclusions of a generalizing nature, supplementing the conclusions of [8, 9].
1. In the practically important scenario of reception of weak signals against the background of ambient sea noise and interference, jointly “masking” the spatial harmonics of the useful signal (modes, rays, or plane waves), the matched-signal processor noticeably loses to the optimal processing by the value of the gain in the situation of accurate knowledge of all parameters of the propagation channel. It is characteristic that this loss grows not only with a decrease in the relative level of the useful signal at the array input, but also at such a change in the spatial spectra of the signal and interference (for example, at a change in the coordinates of the corresponding sources), at which the “overlap” of their spectra increases. In situations of significant “overlap” (as shown in Fig. 2), a suboptimal processing method is the heuristic method of spatial filtering of one of the modes (actually, a small group of modes of close numbers, taking into account the partial spatial resolution of the modal structure). The necessary condition for its high efficiency is the adaptive selection of the mode (group of modes), which has the highest value of the input modal SNIR.
2. Array processors have different stability to variations of environment parameters, caused by their natural variability and errors of operational control of their values. The levels of acceptable a priori uncertainty of the reference model parameters, which are determined by a given value of array gain loss, for different parameters are not just very different, but depend on what parameters of the model are different from the real environment and what the background interference conditions are really correspond to the signal reception. This means that there are no universal quantitative assessments of the acceptable (in this sense) quality of a multi-parameter model of the ocean environment, and with them the requirements for an operational oceanography system.
3. At the same time, such estimates can and should be obtained in situ, for a given sea area (if it is correctly specified by a set of parameters), depth intervals of signal sources and receiving array position in the channel and frequency range. All these factors strongly but predictably affect the result of signal processing “through” their direct influence on the spatial spectra of the received signals, and therefore should be taken into account when formulating requirements for operational oceanography facilities. Model-based calculations make it possible to give a quantitative forecast of the “corridor of possibilities” of various signal processing methods under realistic mismatch conditions.
4. The term “environmentally-matched signal processing”, or matched-field processing, as applied to the real ocean, especially to complex and multi-parameter (from the point of view of sound propagation) shallow water areas, has no practical meaning if the error levels in the environmental parameters estimation for specific signal processing methods are not quantitatively determined. In our opinion, the main difficulty of the problem lies not in the fact that the very idea does not work (does not work well) [4], but in the fact that its applicability is fundamentally limited by the requirements to the operational information about the real underwater environment, and these requirements are variable.
Finally, we note that the analysis of the influence of the a priori environmental uncertainty in this paper is limited to the consideration of the efficiency of VLA signal processing solely by the criterion of the output SNIR and the array SNIR gain. At the same time, such issue is practically important in another formulation, when the criterion of processing efficiency is determined by the quality of the signal source parameters estimation, first of all, estimation of the source coordinates in a channel [3–7]. In essence, these two criteria do not contradict each other because it is impossible to estimate the parameters of a weak signal if it is not detected against the background of interference and noise with sufficient array gain. However, the estimates of the corresponding requirements for a priori information about the environmental conditions will be different, so this issue will be the subject of separate consideration.
ACKNOWLEDGMENTS
The work is dedicated to the bright memory of our colleagues who recently passed away, Prof. Alexander Sazontov and Dr. Ivan Smirnov. Collaborating and communicating with them was always stimulating, productive, and comfortable.
CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest.
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