PERFORMANCE BOUNDS ON THE DETECTION AND
LOCALIZATION IN A STOCHASTIC OCEAN
ARTHUR B. BAGGEROER AND HENRIK SCHMIDT
Massachusetts Institute of Technology, Cambridge MA 02090, USA
E-mail: [email protected]
Methods for predicting the performance of a sonar for both and localization with a known
propagation and clutter distribution are well known. Many oceanographic processes
such as internal waves and bottom roughness require a stochastic description since
they are dynamic and/or have a fine scale beyond which measurements are unrealistic.
These and other processes randomize the temporal and spatial structure of signals and
a single degree of freedom model for a replica is no longer applicable; hence, virtually
all the single replica processing is not optimum. For example, one can have a power
flux across and array from a signal with good SNR, yet still not be able to localize
because the phase structure has been randomized too much by the stochastic processes
of the ocean. Here we introduce a stochastic model for propagation from which the
number of degrees of freedom in a signal observed by a sonar array. The objective
is a model which reasonably represents the observed signal as observed by a sonar.
We then apply these models to developing Chernoff bounds for detection and CramerRao for localization. These models also indicate new receiver structures for a passive
sonar which exploits the uncertainty rather than allow it to degrade from the optimal
performance. Generally, we can infer that if the eigenspectrum of the observed signal has
a “red” structure, which is the case in a partially saturated ocean, one can recover some
of the performance compared to a deterministic propagation model. If the eigenspectrum,
however, approaches “white” spectrum, then the performance degrades to simple energy
detection and localization is not possible.
1 Introduction
There are many models for characterizing the random components of the ocean. For
well known examples, there are Pierson-Moscovitz spectra the ocean surface >dH, GarrettMunk spectra for internal waves, >1H, and correlation functions for seafloor roughness >nH.
Similarly, there are very large number of theories for predicting the coherent and incoherent components of scattered fields. Predicting the performance of a sonar system,
however, is difficult because these models do not lead to representations used in detection and estimation analysis. Some theories that do lead to useful representations are
the temporal-spatial correlation for propagation fluctuations through internal waves >;H
and the propagation of second moments by the parabolic equation >xH. The former uses
a ray parameterization and assumes the rays are uncorrelated. which is questionable at
low frequencies. The latter does lead to full field results, however, implementations the
equation for the second moment has not attained the maturity of those for field itself, for
example the sophisticated codes of Collins >EH.
507
N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 507-514.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
508
A.B. BAGGEROER AND H. SCHMIDT
Performance predictions for detection and localization are generally based on likelihood ratios tests (LRT’s) [7]. For Gaussian statistics these lead to a combination of
linear and quadratic operations upon the observed data. This contrasts with some of the
theoretical and observed statistics for the probability density function (PDF’s) at a single
receiver; however, it is not clear that the same data could be well represented by a Rician
distribution where a phase randomized mean is a component of the data as well. Moreover, the authors are not aware of any multivariate PDF’s as needed for a sensor array in
the literature and these are essential for performance predictions for data from an array of
sensors. Consequently, we use a Gaussian vector model for the data which also leads to
Wishart PDF’s for the sample covariance of the observed data. Essentially, our arguments
for this model is that tractable analytic results which incorporate all the propagation and
stochastic model effects and the current literature cannot make a firm case for alternative
models because of approximations employed and/or cannot be distinguished from vector
Gaussian model or generalizations of it such as incorporating a mean or using a Rician
PDF.
Currently, the complexity of range dependent propagation with uncertain perturbation introduced by an environmental model is sufficiently complicated that the statistics
required by a detection or localization performance can only be provided by estimated
moments such as the sample covariance. In general, there are no tractable analytic methods for computing these moments except in the single replica, or one degree of freedom,
for the signal which corresponds to no uncertainty in the propagation. We consequently
use sample covariances obtained by running a range dependent propagation code such
as FEPE (RAM) or range dependent OASIS and accumulating the results for different
samples of the ocean obtained using uncertainty models of the ocean [6, 8].
Finally, the results below pertain to a passive sonar model such as represented by
classical beamforming or Matched Field Processing. Similar results can be derived for
active systems, or Matched Field Tomography or Ocean Acoustic Tomography, but they
are not included because of space restrictions imposed on the papers. Moreover, we
concentrate on presenting the theoretical framework with examples to be subsequently
published.
2 Stochastic ocean model
Virtually all sonars are implemented in the frequency domain by transforming a sequence
of L data segments, possibly overlapping, to form a “snapshot” vector for an array with
N sensors given by
! Tl + T2
l
w(t − Tl )r(t)e−j2πft dt
R (f ) =
Tl − T2
where
Rl (f ) is the N x1 vector of the l the “snapshot;”
r(t) is the vector of observations;
w(t) is a window, or taper, function controlling bandwidth and sidelobes;
Tl is the center of the window for l the “snapshot”;
T is the duration of the window.
STOCHASTIC OCEAN PERFORMANCE BOUNDS
509
There are a number of issues in the windowing operation including the spread in group
delays and transit time across an array for the signal, doppler effects due to source/receiver
motion and environmental processes such as internal waves and the overall stationarity
of the field. [9]. These effects lead what is termed “doppler spreading” in detection and
estimation theory and are important issues for the performance of sonar systems; however,
other than noting that source/receiver motion dominates variability except in fixed-fixed
systems, we do not address these issues in uncertainty.
Calculating the performance of detection and localization for a sonar requires the
mean vector and covariance matrix of Rl (f ). If the additional assumption of Gaussianity
is applied, then analytic expressions for the false alarm and detection probabilities as well
as localization accuracy can be determined. For a stochastic ocean we use the following
model
Ndof
Rl (f ) =
"
B̃il (f )Gi (f, a) + Nil (f ),
i
where
Ndof is the number of degrees of freedom for the representation;
Gi (f, a, i = 1, Ndof is a sequence of “Green’s functions;”
B̃il (f ) are complex Gaussian random variables of the snapshots which are identically distributed with zero mean and covariance ΣB (f);
Nil (f ) are additive noise components with spectral covariance Sn (f );
a is a parameter vector which representing both localization (range, bearing, depth)
and environmental parameters (sound speeds, layer thicknesses, random models).
Some comments regarding the model are appropriate here. First, there are numerous
results which argue a different PDF for the marginal, i.e. first order density, of a signal
propagating through a random ocean. The log-normal distribution, for example, is often
cited for the envelope which differs from a Raleigh PDF resulting from a Gaussian. Our
argument is that these results are all for a single path and observation; when one considers
a realistic ocean with many paths and a typical detection or localization process where
there is averaging across a large number “snapshots,” the central limit theorem rapidly
takes over and one cannot distinguish the differences. Moreover, there are no multivariate
distributions consistent with these marginal distributions. Next, for the case Ndof = 1
corresponds to a single, spatially coherent propagation path and reduces to the model used
for classical Matched Field Processing (MFP). Finally, the “Green’s functions,” Gi (f, a
are not truly Green’s functions in the classical usage. Such functions are well defined
only a given propagation environment, not an ensemble of them. They are intended to
represent a set of functions which span the covariance representation for the signal which
is required to incorporates the uncertainty. Examples of this are discussed below.
The ensemble covariances of the signal and noise are the important quantities for
performance prediction algorithms. For this we define the N xNdof Green’s function
matrix a
G(f, a) = [ G1 (f, a) | G1 (f, a) | · · · | Gdof (f, a) ] .
H
aA
indicates the complex transpose of the matrix A.
510
A.B. BAGGEROER AND H. SCHMIDT
This leads to the following for the ensemble covariance of the “snapshots”
Kr (f ) = G(f, a)ΣB (f )G H (f, a) + Kn (f).
We note the ΣB (f) need not, and in most cases, is not diagonal, so the model can include
phenomena such as ray/mode correlation. Furthermore, Sn (f ) can also be an arbitrary,
for example, as derived from a Kuperman-Ingenito noise model [10].
3
Green’s function representations
There are several possible ways to develop the above representation for the covariance of
a signal propagating in a stochastic ocean. The tradeoff depends upon implementation of
a numerical model for estimating the sample covariance as well as the covariance matrix
ΣB (f). Ideally, one wants a representation with this covariance to be very close to a
diagonal matrix. (An exact diagonal matrix corresponds to adiabatic propagation.) In the
this sections we indicate possible choices for the representation of G1 (f, a).
For a range independent environment the Green’s function can be represented as a
sum of normal modes. Extensions to range dependent propagation lead to coupled mode
codes. The basic concept is to expand the output in terms of the normal modes at the
array receiver. The weightings of the modes form a random vector with a covariance
which is a transformation on ΣB (f ). If there is no modal coupling, i.e. this matrix has
rank one whereas if the propagation saturates this matrix has full effective rank of Ndof .
A second alternative is to apply a singular decomposition to the array receiver output.
In this approach the eigenvectors are orthonormal and ΣB (f ) is diagonal. The eigenvalue
distibution represents the effective number of degrees of freedom in the signal. For
example, if there is no modal coupling, the there is only one non zero eigenvalue and the
eigenvector corresponds to the adiabatic solution to wave equation [11]. In general, this
leads to the above representation where the eigenvectors are the Gi (f, a) and a diagonal
covariance matrix for ΣB (f ).
Finally, the simplest and most practical representation for the array receiver is to
form a set of preformed beams which form a complete orthogonal basis at the array
receiver. In this case the representation is in terms of the beam steering vector and
the covariance matrix represents the intrabeam correlation. For example, phase coupled
multipath as occurs in a deterministic ocean leads to full coherence among the respective
beams whereas uncorrelated multipath corresponding to saturation leads to zero coherence.
4
Detection performance and the Chernoff bound for a stochastic ocean
The fundamental question here is how to employ the above models to make performance
predictions for a sonar system. As posed, this is a spatial extension to the so called
detection and estimation of a random signal embedded in additive noise problem. This
has been examined in the signal and array processing literature [7], so we extend the
results in this literature.
Optimal detection uses a likelihood ratio test to call the presence or absence of a
target. The key performance parameters are the detection probability, PD and false alarm
probability, PF . Except in the case of a diagonal ΣB , which leads to ξ 2 probability
511
STOCHASTIC OCEAN PERFORMANCE BOUNDS
densities, PD and PF can only be approximated albeit as closely as one needs with
Chernoff bounds. In the interest of space we omit derivations and give just results.
For the binary detection case with a given value of a we have two hypotheses:
H1 : KR ((f )|H1 ) = G(f, a)ΣB (f )G H (f, a) + Kn (f ) signal plus noise
H0 : KR ((f )|H0 ) = Kn (f )
noise only
In M-ary formulation where one formulates a hypothesis test over all the parameters a
such as typically done for MFP ambiguity surfaces. For the Chernoff bound and approximation for PD and PF we define the semi-invariant of the likelihood ratio conditioned
on hypothesis H0 . This is given by
!
µ(s) = L
ln(|K1 |s−1 |K0 |−s + |sK0 + (1 − s)K1 |df
W
where
W is the bandwidth of the signal;
s is a free parameter with 0 ≤ s ≤ 1 which can be used to optimize the bounds
(In the low SNR case typical of sonar detection, s = 1/2.);
K1 = KR ((f )|H1 ) is a shortened notation for the covariance on the signal
present hypothesis;
K0 = KR ((f )|H0 ) is a shortened notation for the covariance on the null
hypothesis.
All the quantities involved can be determined from the problem formulation depending
upon the Green’s function representation. One can readily derive the following expressions for PD and PF [7]
1
ú
eµ(s)−sµ(s)
2πs2 µ̈(s)
and
PF = #
1
ú
eµ(s)+(1−s)µ(s)
PD = 1 − #
2π(1 − s)2 µ̈(s)
By sweeping s over its range [0, 1], one can sweep out the entire Receiver Operating
Characteristic (ROC) for the problem.
We summarize the approach: 1) A model which allows more than one spatial degree
of freedom to incorporate the uncertainty in the propagation. 2) We suggested several
characterizations -normal modes, SVD’s or preformed beams. Generally, these need
to estimated by simulation since the theory for propagation in a stochastic cannot now
provide the needed covariances. 3) Once these covariances are known, well established
bounds from detection theory can be applied. The novel item is the model to represent
a stochastic ocean in a format that sonar predictions can be done. All can be addresses
directly by numerical methods.
512
A.B. BAGGEROER AND H. SCHMIDT
5 Bounds on localization and tomography for a stochastic ocean
Determining the obtainable accuracy of parameter estimates with MFP has been extensively studied. There have been two approaches -extensive simulations [12] and analytic
approaches using bounds [13–16]. While simulations are useful, much depends on the
algorithm and its implementation; in addition, there is a complicated interplay among the
parameters which is hard to extract from simulations.
The bounds performance of a parameter estimation with a single degree of freedom
with no parameter model mismatch have three regions of interest as indicated in figure
below (from [16]). At SNR" s > SNR1 the performance is well predicted by a linearized
2
e (dB)
3 dB
No Information
Region
Asymptotic
Region
3 dB
Transition
Region
SNR2
SNR1
SNR (dB)
Figure 1. Performance realms in parameter estimation
analysis as given by the Cramer-Rao bound [13, 14]. At medium levels SNR2 < SN R <
SNR1 the performance enters an “asymptotic region” where sidelobes play the dominant
role [15, 16]. In this region nonlinear bounds such as the Weiss-Weinstein, Ziv-Zakai or
Barankin bounds have been used successfully. Finally, for SN R < SNR2 there is no
information in the observed signal and one has only a priori knowledge.
In this section we present generalizations of the results of Baggeroer and Schmidt [13]
to the model for a stochastic ocean indicated above. For this we define the following
quantities b :
d2 (f, a) = G † (f, a)S−1
n (f)G(f, a),
∂G(f,a)
li (f, a) = G † (f, a)S−1
n (f) ∂ai ,
li,j (f, a) =
∂G † (f,a) −1
Sn (f )∂G(f, a)∂aj ,
∂ai
$
%−1
Γ(f, a) = 2 I + d2 (f, a)Sb (f )
.
b The several spectral quantities scale to the covariance by multiplying by T , the window duration,
i.e. Kx = T Sx .
STOCHASTIC OCEAN PERFORMANCE BOUNDS
513
The Cramer-Rao bound is expressed in terms of the Fisher Information Matrix, J.
Asymptotically, the performance of any parameter estimation algorithm is given the variance being lower bounded as follows:
σa2i,i ≥ [J−1 ]i,i
where
σa2i,i is the variance, or the inverse of the output SNR for the i the parameter;
[J−1 ]i,i is the i the diagonal matrix of the inverse of the Fisher information matrix.
Furthermore, one can demonstrate the the cross terms J−1
i,j indicate the coupling among
the parameters. This is important because it permits one to search for a minimal parameterization of the models which improves the accuracy of most estimation algorithms.
The only remaining issue is to determine an expression for the elements of the Fisher
Information Matrix. One can derive the following result for this [17]
Ji.j =
!
T
∆W
&
&&
'
T r Sb (f )Γ(f, a) Ev(li,j (f, a))Sb (f)d2 (f, a) − Ev(l†i (f, a)Sb (f )lj (f, a))
+ (Ev(li (f, a))Sb (f )Γ(f, a)Ev(lj (f, a)))]] df.
c
Again, all these quantities can be directly computed numerically. One can readily verify
that this reduces to the earlier derived single replica results for Ndof = 1. These sections
of the above can be interpreted fairly easily. The first line indicates the “convexity” of
the ambiguity normalized to an SNR. The second line indicates the effect of increasing
the observed power as a result of changing a parameter. Both lines are scaled by the a
SNR factor which increases with SNR for SNR < 1 while at high SNR’s it approaches
unity. This is a common characteristic of Gaussian detection and estimation. Overall,
we have a method for quantifying the obtainable performance for parameter estimation
in a stochastic ocean. Bounds for the performance in the asymptotic region can also be
derived, however, they are computationally extensive at the current time.
6
Summary
We have two important results in this paper: i) We have a robust model for a stochastic
ocean and is coupled to the propagation physics; however, quantities for it now need to be
found by simulation using one of the range dependent codes available. ii) We have given
methods for analyzing detection, i.e. the ROC curve, and for bounding the performance
of any estimation algorithm in the high SNR region and where there is no unmodeled
mismatch.
c The
“even” part of a matrix is given by:
Ev(X) =
X + X†
.
2
514
A.B. BAGGEROER AND H. SCHMIDT
Acknowledgments
This work was supported by the Ocean Acoustic and Underwater Signal Processing Codes
at ONR. In addition, the first author was supported by his SECNAV/CNO Chair for Ocean
Science funded by ONR.
References
1. Neumann, G. and Pierson, W.J., Jr., Principles of Physical Oceanography (Prentice-Hall Inc.,
Englewood Cliffs, NJ, 1966).
2. Garrett, C. and Munk, W.H, Space-time scales of internal waves, Geophys. Fluid Dynamics
3, 225–264 (1972).
3. Ogilvy, J.A., Theory of Wave Scattering from Rough, Random Surfaces (1997).
4. Flatté, S.M., Dashen, R., Munk, W.H., Watson, K.M., and Zachariasen, F., Sound Transmission
Through a Fluctuating Ocean (Cambridge University Press, Cambridge, U.K., 1979).
5. Ishimaru, A., Wave Propagation and Scattering in Random Media (IEEE Press, New York,
1997) (reprint of 1978 original, Oxford University Press).
6. Collins, M. and Westwood, E., A higher order energy-conserving parabolic equation for rangedependent ocean depth, sound speed and density, J. Acoust. Soc. Am. 89, 1068–1075
(1991).
7. Van Trees, H.L., Detection, Estimation and Modulation Theory, Parts I and III (John Wiley
and Sons, 1970) (reprinted 2002).
8. Schmidt, H., Seong, W. and Goh, J.T., Spectral super-element approach to range-dependent
ocean acoustic modeling, J. Acoust. Soc. Am. 98, 465–472 (1995).
9. Baggeroer, A.B. and Cox, H., Passive sonar limits upon nulling multiple moving ships with
large aperture arrays. In Proc. 1999 Asilomar Conference on Signals and Systems (IEEE
Press, New York, 1999).
10. Kuperman, W.A. and Ingenito, F., Spatial correlation of surface generated noise in a stratified
ocean, J. Acoust. Soc. Am. 67, 1988–1996 (1980).
11. Jensen, F.B., Kuperman, W.A., Porter, M.B. and Schmidt, H., Computational Ocean Acoustics
(AIP Press, New York, 1994).
12. Tolstoy, A., Matched Field Processing in Underwater Acoustics (World Scientific Pub. Inc,
Singapore, 1993).
13. Baggeroer, A.B. and Schmidt, H., Parameter estimation theory bounds and the accuracy of full
field inversions. In Full Field Inversion Methods in Ocean and Seismo-Acoustics, edited
by O. Diachok, A. Caiti, P. Gerstoft and H. Schmidt (Kluwer Academic, The Netherlands,
1995) pp. 79–84.
14. Krolik, J. and Narasimhan, Performance bounds on acoustic thermometry of ocean climate in
the presence of mesoscale sound variability, J. Acoust. Soc. Am. 99, 254–265 (1996).
15. Bell, K.L., Ephraim, Y. and Van Trees, H.L., Explicit Ziv-Zakai lower bounds for bearing
estimation, IEEE Trans. Signal Processing 44, 2810–2824 (1996).
16. Xu, W., Performance bounds on matched field methods for source localization and estimation
of environmental parameters. Ph.D. Thesis, Mass. Inst. of Tech. and Woods Hole
Oceanographic Inst. (June 2001).
17. Baggeroer, A.B. and Schmidt, H. Parameter coupling and resolution bounds for source localization and ocean acoustic tomography. MIT Ocean Acoustics Group Internal Memorandum
(to be submitted to J. Acoust. Soc. Am.).