ENVIRONMENTAL UNCERTAINTY IN ACOUSTIC INVERSION
STAN E. DOSSO AND MICHAEL J. WILMUT
School of Earth and Ocean Sciences, University of Victoria, Victoria B.C. Canada
E-mail: [email protected]
Acoustic processing for source localization is often limited by uncertainties in the physical properties of the ocean environment, such as the seabed geoacoustic parameters
and water-column sound-speed structure. Uncertainties in environmental parameters
can be the result of measurement uncertainties and of spatial and temporal variability. Quantifying environmental uncertainties and examining how they are transfered
into uncertainties for source localization represents an important and challenging problem. In this paper, Bayesian inference theory is applied to estimate the uncertainties of
geoacoustic inversion in the form of marginal probability distributions for the seabed
parameters. In addition, a Bayesian form of focalization is developed which incorporates uncertain environmental parameters into the source localization, and quantifies the
resulting localization uncertainty in terms of the 2-D marginal probability distribution
for source range and depth (i.e., a probability ambiguity surface). Localization uncertainties are examined as a function of the uncertainties in geoacoustic parameters and
ocean sound-speed profile.
1 Introduction
Acoustic fields propagating in a shallow-water ocean environment are affected by the location of the acoustic source and by physical properties of the environment (water column
and seabed). Environmental properties are often not well known, due to a lack of data,
measurement uncertainties, and spatial and temporal variability. Considerable effort has
been applied in recent years to invert ocean acoustic fields for source location and for environmental properties, particularly geoacoustic parameters. Although these two inverse
problems are often treated independently, in practical applications uncertainty always exists in environmental parameters and degrades the ability to perform localization. The
problems are further linked as geoacoustic inversion is often applied to provide improved
environmental information for subsequent source localizations. The goals of this paper
are to quantify the uncertainties of geoacoustic inversion and source localization, and to
examine localization uncertainties as a function of the uncertainties in geoacoustic parameters and ocean sound-speed profile (SSP). As acoustic inverse problems are strongly
nonlinear, the analysis is carried out using Bayesian inference theory.
In Bayesian inversion [1], the solution is characterized by its posterior probability
density (PPD). The PPD combines prior information for the unknown model parameters
with the information from an observed data set expressed in terms of a likelihood function.
To interpret the multi-dimensional PPD, the state of information about model parameters
is typically quantified in terms of moments of the PPD, such as marginal probability
distributions. Estimating these moments involves computing multi-dimensional integrals
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Sonar Performance, 171-178.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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of the PPD, which is usually carried out using sampling methods. Obtaining efficient,
unbiased sampling and verifying that the integral estimates have converged are important
issues in Bayesian inversion. For instance, Monte Carlo integration is based on sampling
at random from a uniform distribution over the parameter space. However, if the integrand is concentrated in localized regions of the space, many of these models will not
contribute significantly to the integral. The method of importance sampling draws samples from regions that contribute most to the integral, providing more efficient sampling.
In particular, if the sample of models can be drawn from the PPD itself, the integral evaluation is straightforward and efficient. This can be accomplished using Gibbs sampling
(GS) methods, such as Metropolis and heat-bath sampling, which also form the basis for
simulated annealing inversion.
A Bayesian formalism was applied to geoacoustic inversion by Gerstoft and Mecklenbräuker [2], who used a genetic-algorithms based approach to sample the PPD. Dosso
[3] developed a fast Gibbs sampler (FGS) approach to Bayesian inversion, which provides
an efficient, unbiased sampling of the PPD with a rigorous convergence criterion. The
FGS algorithm was compared to standard GS and Monte Carlo integration for benchmark
test cases and found to produce identical results in orders of magnitude less computation
time. FGS analysis has been applied to a broadband, shallow-water geoacoustic survey
[4]. The marginal distributions obtained for measured data agreed well with those of
synthetic test cases, illustrating that simulations can provide a meaningful evaluation of
practical cases. The marginals were found to have simple, smooth forms that facilitate
straightforward comparisons of the information content for different cases.
To address environmental uncertainty in source localization, Collins and Kuperman
[5] included environmental parameters as additional unknowns in a simulated-annealing
inversion for the optimal source location, an approach known as focalization. Shorey et
al. [6] applied a Bayesian formulation to this problem, referred to as the optimal uncertain
field processor (OUFP), which estimates the source location by integrating over uncertain
environmental parameters. As their goal was to estimate the maximum-probability source
location rather than characterize the localization uncertainty, intensive sampling was not
required, and Monte Carlo integration for a fixed number of samples was applied.
2
Theory
This section briefly summarizes the GS approach to Bayesian acoustic inversion; a more
complete treatment of Bayesian theory is given in [1]. Let m and d represent model and
data vectors, respectively, with elements mi and di considered to be random variables.
Bayes’ rule for conditional probabilities leads to
P (m|d) ∝ L(d|m) P (m),
(1)
where P (m|d) represents the PPD, L(d|m) is the likelihood function, and P (m) is
the prior distribution. In this paper, the prior is taken to be a uniform distribution over
pre-defined lower and upper bounds for each parameter. The likelihood function is determined by the form of the data and errors: for Gaussian errors L(d|m) ∝ exp [−E(m)],
representing a Gibbs distribution, where E(m) is the appropriate error function. For
multi-frequency acoustic data d= {df , f = 1, F } due to a source with unknown spectrum
and random errors assumed to be uncorrelated over frequency and space with standard
ENVIRONMENTAL UNCERTAINTY IN INVERSION
173
deviation σf at the fth frequency, the error function can be written [3]
E(m) =
F
!
f=1
(1 − Bf (m)) |df |2 /σf2 .
(2)
In (2), Bf (m) represents the (normalized) Bartlett processor defined
Bf (m) =
|d†f df (m)|2
|df |2 |df (m)|2
,
(3)
where df (m) is the replica acoustic field computed for model m. The normalized PPD
can thus be written
exp [−E(m)] P (m)
P (m|d) = "
,
(4)
exp
[−E(m! )] P (m! ) dm!
m!
where the integration spans the model space.
To interpret the multi-dimensional PPD requires computation of its integral properties.
Here, parameter uncertainties are considered in terms of marginal probability distributions.
The 1-D marginal distribution for parameter mi is defined
#
P (mi |d) =
δ(m!i −mi ) P (m! |d) dm! ,
(5)
m!
where δ is the Dirac delta function; higher-dimensional marginal distributions are defined
in a similar manner. Multi-dimensional PPD integrals such as (5) are typically evaluated
using sampling methods. This paper applies the GS methods of Metropolis and heat-bath
sampling which asymptotically sample from the PPD itself, providing a straightforward
and efficient evaluation of integrals such as (5).
In Metropolis sampling, each parameter of the model is perturbed in turn, with perturbations accepted if a uniform random number ξ drawn from the interval [0, 1] satisfies
ξ ≤ exp [−∆E/T ] ,
(6)
where T represents a control parameter referred to as temperature. In heat-bath sampling,
a discretized 1-D distribution is formed for each parameter mi in turn by sampling
exp [−E(m)/T ] across the parameter bounds while keeping all other parameters fixed.
A new value for mi is then drawn at random from this distribution. Markov-chain analysis
[1] indicates that, in the limit of a large number of perturbations, both the Metropolis and
heat-bath methods provide an unbiased sampling of the Gibbs distribution
exp [−E(m)/T ]
,
!
m! exp [−E(m )/T ]
PG (m) = $
(7)
which is proportional to the PPD (4) for uniform priors. In simulated annealing inversion,
T is reduced during sampling to ultimately obtain the model that minimizes E and
hence maximizes the PPD. In Bayesian analysis, intensive sampling is carried out at the
fixed temperature T = 1 to sample from the PPD (4) and evaluate integrals such as (5).
However, cooling from a high temperature to T = 1 prior to accumulating the sample is
recommended to initiate the sampling in a high-probability region of the space.
A fast Gibbs sampling (FGS) algorithm for geoacoustic inversion based on Metropolis
sampling has recently been developed [3]. In FGS, the efficiency of standard Metropolis
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S.E. DOSSO AND M.J. WILMUT
GS is improved by sampling in a transformed parameter space rotated to minimize interparameter correlations, and by adaptively determining appropriate perturbation sizes for
sampling individual parameters. This approach is applied here for geoacoustic inversion.
While Metropolis sampling has proved effective for geoacoustic inversion, for acoustic
localization with uncertain environmental parameters (focalization) the heat bath method
can provide important advantages. The major challenge in localization involves the potentially large number of isolated, locally-optimal solutions dispersed over source range
and depth (r and z), referred to as side-lobes on an ambiguity surface. As a result,
independent 1-D sampling in r and in z can be very inefficient since the probability of
jumping between locally-optimal solutions not aligned with the parameter axes is small.
However, 2-D sampling in the r-z plane precludes this difficulty. To accomplish this
within the heat bath method, a range-depth probability surface is generated and taken
to be equivalent to the results of intensive 2-D sampling of these parameters. Hence,
2-D marginal probability distributions can be formed by summing the r-z probability distributions for a large number of realizations of the environmental parameters, generated
using GS. Note that this is equivalent to averaging r-z probability surfaces over environmental parameters sampled from the PPD, while the OUFP method [6] averages over
parameters sampled from the prior distribution. For range-independent or adiabatically
range-dependent problems, r-z probability surfaces can be computed efficiently since the
modal properties need be evaluated only once.
The final component required in GS is a rigorous convergence criterion. Convergence
is judged here in terms of the difference between two two independent samples of models collected in parallel with the GS algorithm. For geoacoustic inversion, the criterion
adopted requires the maximum difference between the 1-D cumulative marginal distributions for all parameters be suitably small (less than 0.1). For source localization in an
uncertain environment, an additional requirement is included that the difference between
the two 2-D cumulative marginal distribution in r and z be less than 0.02. In each case,
the final sample for integration is taken to be the union of the two independent samples.
3
Inversion examples
This section illustrates the Bayesian analysis of geoacoustic inversion and source localization in an uncertain environment with realistic synthetic examples. The range-independent
environment considered in both cases is given in Fig. 1, which shows the environmental parameters consisting of the ocean SSP c1 (at surface), c2 (at 10-m depth) and c3
(at seabed, 100-m depth), sediment and basement sound speeds, cs and cb , sediment
thickness, h, and seabed density and attenuation, ρ and α (constant over sediment and
basement). Acoustic fields due to a source at range r and depth z are measured at a
19-sensor vertical line array (VLA) which extends from 0–90 m depth.
The first example consists of inversion for the seabed geoacoustic parameters with the
SSP known exactly and source range and depth known to within small corrections (typical
assumptions for geoacoustic inversion). The data consist of acoustic-field measurements
at 50, 100, and 200 Hz, with complex Gaussian errors added to the data to achieve
a signal-to-noise ratio of SNR = 10 dB at each frequency. The marginal probability
distributions for all parameters were computed using the FGS algorithm and are shown
in Fig. 2(a) (top set of distributions). This figure also indicates the true values and search
ENVIRONMENTAL UNCERTAINTY IN INVERSION
*
Source
(r, z)
175
c1
c2
Water
VLA
c3
h
cs
Sediment
cb
Basement
Figure 1. Schematic diagram of ocean environment for geoacoustic inversion and source localization
examples. Water depth is 100 m. Parameters are described in text.
bounds for each parameter. Wide bounds are applied for the geoacoustic parameters so
that the data (not the prior information) constrain the solution. The marginal probability
distributions are unimodal and approximately symmetric, with smooth, simple forms.
Figure 2(a) shows that the sediment properties, h and cs , are well determined, while cb ,
ρ, and α are less well determined but still constrained by the data. The source range
r is poorly determined within the small correction bounds, while the source depth z is
reasonably well determined. Figure 2(b) shows the marginal distributions determined
when the SSP parameters are considered unknown within a 10 m/s interval. The SSP
parameters themselves are almost completely undetermined via inversion; however, the
effect on the other parameters is minimal (distributions for cs and z are slightly wider).
This indicates that uncertainty in the SSP, possibly due to spatial or temporal variability,
does not preclude good results for geoacoustic inversion, provided this uncertainty is
a
b
a
b
Figure 2. Marginal probability distributions from geoacoustic inversion. Upper set of distributions
(a) shows results with known SSP; lower set (b) shows results with unknown SSP. Dotted lines
indicate true parameter values, and the range of abscissa values indicates the parameter search
bounds.
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S.E. DOSSO AND M.J. WILMUT
a
b
Figure 3. (a) Ambiguity surface for Bartlett match. (b) Probability ambiguity surface. Environmental parameters are set to their true values and SNR = 5 dB. Distributions are normalized to a
common maximum value.
accounted for in the inversion procedure. It is important to note that this result does not
imply that a good knowledge of the SSP is irrelevant in geoacoustic inversion. Holding
the water-column parameters fixed at incorrect values within the prior bounds can lead
to poor inversion results. This point is examined later in the paper.
The next example involves source localization in an uncertain environment. The
environmental parameters are as given in Fig. 1, and the data consist of acoustic fields
due to a source at (r, z) = (3 km, 50 m) at a frequency of 100 Hz with SNR = 5 dB. The
inversion results are considered in terms of the 2-D marginal probability distribution over
r and z, referred to as a probability ambiguity surface (PAS). The PAS is compared to
a standard ambiguity surface (AS) for the case of exact environmental parameters (i.e.,
no integration required) in Fig. 3. The AS consists of a plot of the Bartlett processor
B(m), defined by (3), at all points of a grid from 1–6 km range and 0–100 m depth.
The PAS in this case consists of a plot of P (m|d) ∝ exp[B(m) |d|2 /σ 2 ] over the same
grid. The AS and PAS have maxima at the same point; however, the exponentiation of
the PAS stretches the surface by an amount determined by the ratio of the squared data
magnitude to its variance. The AS indicates the fit to the data vs. r and z; the PAS
indicates the probability that particular (r, z) points correspond to the source position.
The probability that the source is within a particular r-z region can be determined by
integrating (summing) P over the region.
Figure 4 shows PAS, computed using heat-bath sampling, for a variety of states of
enviromental information including wide and narrow geoacoustic bounds and known and
unknown SSP (10 m/s uncertainty), as described in the figure caption. The wide geoacoustic bounds correspond to the parameter search bounds indicated in Fig. 2, i.e., the
available information prior to geoacoustic inversion. The narrow geoacoustic bounds correspond to 95% maximum-probability intervals for the parameters determined from the
geoacoustic inversion results (i.e., the marginal distributions in Fig. 2). For comparison,
Fig. 4(a) shows the PAS for the exact environment. The integrated probability for an
acceptable region defined to be within 250-m range and 5-m depth of the true source
location is P̄ = 0.88. Figure 4(b) shows that poor geoacoustic/SSP information significantly degrades the PAS, with a wider main peak and substantial sidelobes. The highest
peak is at the correct source location, but the integrated probability for the acceptable
region is just P̄ = 0.23. Applying narrow geoacoustic bounds in Fig. 4(c) improves the
ENVIRONMENTAL UNCERTAINTY IN INVERSION
177
b
a
d
c
f
e
Figure 4. PAS for source localization example. Panel (a) shows PAS for exact enviromental parameters. The remaining panels show PAS for: (b) wide geoacoustic bounds and unknown SSP;
(c) narrow geoacoustic bounds and unknown SSP; (d) wide geoacoustic bounds and known SSP;
(e) narrow geoacoustic bounds and known SSP; and (f) exact geacoustic parameters and unknown
SSP.
PAS substantially, with P̄ = 0.47. Figure 4(d) shows similar resuts for wide geoacoustic
bounds and known SSP, with P̄ = 0.45. Figure 4(e) and (f) show good results for narrow
geoacoustic bounds and known SSP and for exact geoacoustics and unknown SSP, with
P̄ = 0.70 and 0.65, respectively.
The PAS shown in Fig. 4 quantify the state of available information for source localization under differing states of environmental information. The PAS indicate the
probability that the source is located in a particular r-z interval, providing meaningful
uncertainties for source localization. The reasonably good localization results obtained
for most cases in Fig. 4 does not imply that environmental uncertainties are unimportant
or can be be ignored. Rather, the results indicate how Bayesian analysis takes these
uncertainties into account. To illustrate this, Fig. 5 shows PAS computed for geoacoustic
parameters drawn at random from the wide prior bounds (SSP is exactly known). Very
poor localization results in these cases. Similar results were obtained for the other cases
in Fig. 4(b)–(d).
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S.E. DOSSO AND M.J. WILMUT
a
b
c
d
Figure 5. Panel (a) shows the PAS for exact environment; panels (b)–(d) show PAS for geoacoustic
parameters drawn at random from wide bounds (exact SSP).
4
Summary
This paper applied Bayesian analysis to examine the available information content in
ocean acoustic inverse problems. The dependence of geoacoustic inversion on knowledge
of the ocean SSP and of source localization on SSP and geoacoustic parameters was
considered. The computation of 2-D marginal probability distributions in range and
depth (PAS) quantifies uncertainty in localization, and effectively addresses the effects
of environmental uncertainty. This analysis was carried out efficiently using a variant of
heat-bath GS.
Acknowledgements
We thank Neil Frazer for suggesting the heat bath method for this problem.
References
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