The detection and localization of an acoustic source has long been the motivation of early sonar systems. With the advent of quieter and quieter submarines due to new manufacturing technologies and the recent proliferation of diesel powered vessels, the need for more sophisticated processing techniques has been apparent for quite some time. It has often been contemplated that the incorporation of ocean acoustic propagation models into signal processing schemes can offer more useful information necessary to improve overall processor performance and to achieve the desired enhancement/detection/localization even under the most hostile of conditions. For our problem in ocean acoustics the model-based approach is shown in Figure 1. The underlying physics is represented by an acoustic propagation model depicting how the sound propagates from a source or target to the sensor measurement array of hydrophones. Noise in the form the background or ambient noise, shipping noise, uncertainty in the model parameters etc. is shown in the figure as input to both the propagation and measurement system models. Besides the model parameters and initial conditions, the raw measurement data is input to the model with the output being the filtered or enhanced signal.
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Figure 1. Model-Based Approach: Structure of the model-based processor showing the incorporation of propagation (ocean), measurement (sensor array), and noise (ambient) models. |
Model-based techniques offer high expectations of performance, since a processor based on the predicted physical phenomenology that inherently has generated the measured signal must produce a better (minimum error variance) estimate then one that does not [1,2]. The uncertainty of the ocean medium also motivates the use of stochastic models to capture the random and often nonstationary nature of the phenomena ranging from ambient noise and scattering to distant shipping noise. Therefore, processors that do not take these effects into account are susceptible to large estimation errors. This uncertainty was discussed by Tolstoy [3], in the work of Carey [4] when investigating space-time processing, and in the overviews by Sullivan and Middleton [5] and Baggeroer [6]. However, if the model embedded in this processor is inaccurate or for that matter incorrect, than the model-based processor (MBP) can actually perform worse. Therefore, it is necessary, as part of the MBP design procedure, to estimate/update the model parameters either through separate experiments or jointly (adaptively) while performing the required processing [7].
Incorporating a propagation model into a signal processing scheme was most probably initiated by the work of Hinich [9] in 1973 who applied it to the problem of source depth estimation. However, as early as 1966, Clay [10] suggested matching the modal functions of an acoustic waveguide to estimate source depth. The concept of matched-field processing (MFP), that is, comparing the measured pressure-field to that predicted by a propagation model to estimate source range and depth was introduced by Bucker [11] in 1976. In MFP, the localization problem is solved by exhaustively computing model predictions of the field at the array for various assumed source positions. The final position estimate is the one achieving maximum correlation with the measured field at the array. Many papers have been written exploiting and improving on the MFP and are best summarized in the text of Tolstoy [3] and the special issues of Doolittle [12] and Stergiopoulos [13]. However, matched-field is primarily aimed at the localization problem, indeed most estimators implemented by MFP are focused on seeking an estimation of localization parameters.
In ocean acoustics there are many problems of interest other than localization that are governed by propagation models of varying degrees of sophistication. Our discussion will eventually lead to adaptively adjusting parameters of the propagation model to "fit" the ever- changing ocean environment encompassing our sensor array [14]. In fact, one way to think about this processor is that it passively listens to the ocean environment and "learns" or adapts to its changes. It is clear that the resulting processor will be much more sensitive to changes than one that does not, thereby, providing current information and processing. One recent paper utilizes such a processor as the heart of its model-based localization scheme [15].
With this background in mind, we discuss the development of a "model- based processor," that is, a processor that incorporates a mathematical representation of the ocean acoustic propagation and can be used to perform various signal processing functions ranging from simple filtering or signal enhancement, to adaptively adjusting model parameters, to localization, to tracking, to sound speed estimation or inversion. In all of these applications the heart of the processing lies in the development of the MBP and its variants. Clearly each of the MFP methods described above can be classified as model-based, for instance, the MFP incorporates a fixed (parametrically) propagation model. However, here we will emphasize the recursive, state- space, forward propagation scheme of Candy and Sullivan [7]. We choose to differentiate between the terms model- based processing and matched-field processing, primarily to emphasize the fact that this discussion is based on the existing state-space framework that enables access to all of the statistical properties inherited through this formalism such as the predicted conditional means and covariances [1,2]. This approach also enables us access to the residual or innovation sequence associated with model- based processors (Kalman filter estimator/identifiers) permitting us to monitor the performance of the embedded models in representing the phenomenology (ocean acoustics, noise, etc.) as well as the on-line potential of refining these models adaptively using the innovations [7,8]. The state-space formalism can be considered to be a general framework that already contains the signal processing algorithms and it is the task of the modeler to master the art of embedding his models of interest. Thus, in this sense, the modeler is not practicing signal processing per se, but actually dealing with the problem of representing his models within the state-space framework. Furthermore, this framework is not limited to localization, but because of its flexibility, tomographic reconstructions can be performed to directly attack the mismatch problem that plagues MFP [3,12]. This can be accomplished by constructing an "adaptive" MBP that allows continuous updating of the model parameters and is easily implemented by augmenting them into the current state vector. That is, unlike the conventional view of the inverse problem, where the functional relationship between the measurements and the parameters of interest must be invertible, we simply treat these parameters as quantities to be estimated by augmenting them into the state vector. In MFP, most of the techniques employed to "correct" this mismatch problem usually achieve their result by a desensitization of the algorithm. Adaptive MBP does not sacrifice any potential information available in the model, but actually refines it by adaptively or recursively updating parameters. In this way the original states and the augmented states are updated by the recursive processor in a self-consistent manner. The fact that the relationship between the original states and the parameters of interest may be complicated and/or nonlinear is not an issue here, since only the "forward" problem is explicitly used in each recursion via the measurement relations. Thus, the usual complications of the inverse problem are avoided at the expense of creating a higher dimensional state-space. All that is necessary is the parameters of interest be observable or identifiable in the system theoretic sense [8,15].
Much of the formalism for this model-based signal processing has been worked out [7]. It is concerned with the incorporation of environmental (propagation, seabed, sound speed, etc.), measurement (sensor arrays) and noise (ambient, shipping, surface, etc.) models along with measured data into a sophisticated processing algorithm capable of detecting, filtering (enhancing) and localizing an acoustic source(target) in the complex ocean environment. This technique offers a well-founded statistical approach for comparing propagation/noise models to measured data and is not constrained to a stationary environment which is essential in the hostile ocean. Not only does the processor offer a means of estimating various quantities of high interest (modes, pressure-field, sound speed, etc.), but it also provides a methodology to statistically evaluate its performance on-line. Although model- based techniques have been around for quite a while, [1,2] they have just recently found their way into ocean acoustics. Some of the major advantages of model-based processors are that they are:
First let us examine the inherent structure of the model-based processor. Model-based processing is a direct approach that uses in-situ measurements. More specifically, the acoustic measurements are combined with a set of preliminary sound speed and other model parameters usually obtained from a priori information or a sophisticated simulator (such as SNAP [16], the SACLANT normal-mode propagation model) that solves the underlying boundary value problem to extract the initial parameters/states in order to construct the forward propagator and initialize the algorithm. The algorithm then uses the incoming data to adaptively update the parameter set jointly with the acoustic signal processing task (detection, enhancement, and localization). In principle, any propagation model can be included in this method. There are basically three advantages to this approach. First, it is recursive and, therefore, can continuously update the estimates of the sonar and environmental parameters. Second, it can include the system and measurement noise in a self consistent manner. By noise, is meant not only acoustic noise, but also errors in the input parameters of the model. Third, one of the outputs of the MBP is the so-called innovation sequence, which provides an on-line test of the "goodness of fit" of the model to the data. This innovation sequence plays a major role in the recursive nature of this processor by providing information that can be used to adaptively correct the processor and the propagation model itself, as well as provide the input to a sequential detector [7]. Along with the ability of this processing scheme to self consistently estimate parameters of interest along with the signal processing task, stand-alone estimators can also be used to provide refined inputs to the model. Further, estimates can be refined as demonstrated in a new towed array processing scheme [17].
Next we discuss the basics of the model-based approach to signal processing. Formally, the model-based approach is simply "incorporating mathematical models of both physical phenomenology and the measurement process including noise into the processor to extract the desired information." This approach provides a mechanism to incorporate knowledge of the underlying physics or dynamics in the form of mathematical propagation models along with measurement system models and accompanying uncertainties such as instrumentation noise or ambient noise as well as model uncertainties directly into the resulting processor. In this way the model-based processor enables the interpretation of results directly in terms of the physics. The model-based processor is really a modeler's tool enabling the incorporation of any a-priori information about the problem to extract the desired information. The fidelity of the model incorporated into the processor determines the complexity of the model-based processor. These models can range from simple implied non-physical representations of the measurement data such as the Fourier or wavelet transforms to parametric black-box models used for data prediction to lumped mathematical physical representations usually characterized by ordinary differential equations to full physical partial differential equation models capturing the critical details of wave propagation in a complex medium. The dominating factor of which approach is the most appropriate is usually determined by how severe the measurements are contaminated with noise and underlying uncertainties. If the signal-to-noise ratio (SNR) of the measurements is high, then simple non- physical techniques can be used to extract the desired information. This approach of selecting the appropriate model is depicted in Figure 2 where we note that as we progress up the "modeling" steps to increase the SNR, the complexity of the model increases to achieve the desired results.
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Figure 2. Fidelity of the embedded model determines the complexity of the resulting model-based processor required to achieve the desired SNR: (a) Simple implied model (Fourier, Wavelet, etc.). (b) Black-box model (data prediction model). (c) Gray-box model (implied physical model). (d) Lumped physical model (differential equations). (e) Full physical model (partial differential equations). |
Consider the following example taken from ocean acoustics to motivate the approach.
Suppose we have a plane wave signal characterized by
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where sk(t) is the space-time signal measured by the kth sensor, a is the plane wave amplitude factor b,q,wo, with the respective wavenumber, bearing, and temporal frequency parameters. Let us further assume that the signal is measured by a horizontal array. A simple but important example in ocean acoustics is that of a 50 Hz plane wave source (target) at a bearing of 45° impinging on a 2-element array at a 10 dB SNR (see Figure 3 below).
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Figure 3. Plane Wave Propagation: (a) Problem geometry. (b) Synthesized 50 Hz, 40o, plane wave impinging on a 2-element sensor array at 10dB SNR. |
We would we like to solve the following basic ocean acoustic processing problem of extracting the source bearing, qo, and temporal frequency, wo parameters. The bearing/ frequency estimation or equivalently localization problem can be considered a problem of estimating a set of parameters, {qo, wo}from noisy array measurements, {pk(t)}. We start with the following models:
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The classical approach to this problem is to first take a sensor channel and perform spectral analysis on the filtered time series to estimate the temporal frequency, wo. The bearing can be estimated independently by performing classical beamforming on the array data. A beamformer can be considered a spatial spectral estimator which is scanned over bearing angle indicating the true source location at maximum power. The results of applying this approach to our problem is shown in Figure 4a demonstrating the outputs of both spectral estimators peaking at the correct frequency and angle parameters.
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Figure 4. Plane Wave Impinging on a 2-element Sensor Array–Frequency and Bearing Estimation Problem: (a) Classical spectral (temporal and spatial) estimation approach. (b) Model-based approach using parametric adaptive (nonlinear) processor to estimate bearing angle, temporal frequency, and the corrsponding residual or innovations sequence. |
The MBP is implemented by incorporating the plane wave propagation, hydrophone array, and noise models (as in Figure 1). However, the temporal frequency and bearing angle parameters are unknown and must be estimated. The solution to this problem is performed by augmenting the unknown parameters into the MBP structure and solving the so-called joint estimation problem [1,2]. This is the parameter adaptive form of the MBP used in most ocean acoustic applications. Here the problem becomes nonlinear due to the augmentation and is more compu- tationally intensive; however, the results are appealing as shown in Figure 4b. Here we see the bearing angle and temporal frequency estimates as a function of time eventually converging to the true values (wo=50Hz,qo=45°). The MBP also produces the "residual or innovations" sequence, (shown in the figure) which is used in determining its overall performance, i.e., it must be statistically zero-mean and white for optimal performance [2].
We summarize the classical and model-based solutions to the frequency and bearing angle estimation problem. The classical approach simply performs spectral analysis temporally and spatially (beamforming) to extract the parameters from noisy data, while the model-based approach embeds the unknown parameters into its propagation, measurement, and noise models through augmentation enabling a solution to the joint estimation problem. The MBP also enables a monitoring of its performance by analyzing the statistics of its residual or innovations sequence. It is this sequence the indicates the optimality of the MBP outputs. This completes the example.
We have discussed the model-based approach to ocean acoustic signal processing and developed a simple example to demonstrate its application. When the underlying SNR of the measured data is low, a MBP can provide a physics- based solution to the problem.
It should also be noted that there is MBP software available SSPACK_PC, which is a toolbox in MATLAB (see web and 3rd party site) [18,19]. Also there is an upcoming short course/tutorial entitled, "Model-Based Ocean Acoustic Signal Processing," at the OCEANS 2000 Conference in Providence, RI. For those interested in learning more about this approach and its applications be sure to sign up.
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James V. Candy is the Chief Scientist for Engineering and Director of the Center for Advanced Signal & Image Sciences at the University of California, Lawrence Livermore National Laboratory. Dr. Candy received a commission in the USAF in 1967 and was a Systems Engineer/Test Director from 1967 to 1971. He has been a Researcher at the Lawrence Livermore National Laboratory since 1976 holding various positions including that of Project Engineer for Signal Processing and Thrust Area Leader for Signal and Control Engineering. Educationally, he received his B.S.E.E. degree from the University of Cincinnati and his M.S.E. and Ph.D. degrees in Electrical Engineering from the University of Florida, Gainesville. He received a commission in the USAF in 1967 and was a Test Director from 1967 to 1971. He is a registered Control System Engineer in the state of California. He has been an Adjunct Professor at San Francisco State University, University of Santa Clara, and UC Berkeley, Extension teaching graduate courses in signal and image processing. Dr. Candy is a Fellow of the IEEE and a Fellow of the Acoustical Society of America (ASA) as well as a member of Eta Kappa Nu and Phi Kappa Phi honorary societies. He was just elected as a "distinguished alumnus" by the University of Cincinnati and will be honored as such at a banquet in June, 2000. He has published over 125 journal articles, book chapters, and technical reports as well as written two texts in signal processing, "Signal Processing: the Model-Based Approach," (McGraw-Hill, 1986) and "Signal Processing: the Modern Approach," (McGraw-Hill, 1988). He has presented short courses sponsored by the IEEE in Applied Signal Processing and more recently Model- Based Ocean Acoustic Signal Processing for IEEE Oceans Society. He is currently the IEEE Chair of the Administrative Committee on "Sonar Signal Processing" and also the Chair of the ASA Interdisciplinary Technical Group on "Signal Processing in Acoustics". as well as being an Associate Editor for Signal Processing of ASA (on- line). His research interests include estimation, identification, spatial estimation, signal and image processing, array signal processing, tomography and biomedical applications.