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Matched-Field Performance Prediction with Model Mismatch Yann Le Gall, Francois-Xavier Socheleau, Member IEEE, and Julien Bonnel, Member IEEE

Abstract—Matched-field estimation is known to be sensitive to mismatch between the assumed replica of the acoustic field and the actual field. An interval error-based method (MIE) is proposed to predict the mean-squared error (MSE) performance for multisnapshot and multifrequency maximumlikelihood matched-field estimation under model mismatch. The source signal is assumed deterministic unknown. Global errors are predicted by deriving exact expressions of pairwise error probabilities with model mismatch in conjunction with the use of the Union bound. Local errors are approximated using a Taylor expansion of the MSE. Numerical examples show the accuracy of the method. Index Terms—Matched-field processing, mismatch, method of interval errors, underwater acoustics

I. I NTRODUCTION In underwater acoustics, nonlinear inverse problems are commonly solved using Matched-field processing (MFP) [1]. From data collected with an array of sensors, MFP compares measured acoustic fields with replica fields computed by a propagation model. Estimates of parameters such as source position, geoacoustic and/or oceanographic quantities are obtained by maximizing a cost function that quantifies the match between the measured data and the simulated replicas. Noise in the measured data as well as mismatch between the assumed replica fields and the actual field may result in poor estimation performance. MFP estimation suffers from large ambiguities that lead to non-local errors at low signal-to-noise ratio (SNR) and mismatch has a serious effect on performance, particularly at high SNR. Therefore, it is important to quantify the performance of MFP to know and understand its limitation as well as to determine the operational conditions required to obtain acceptable estimates. The performance prediction of MFP is not a simple problem if we wish to avoid computationally intensive Monte-Carlo simulations. MFP performance analysis is commonly carried out through the computation of limiting bounds on the meansquare error (MSE) [2]–[5] or by using the method of interval errors analysis (MIE) [5]–[8]. MIE does not bound the performance but approximates the MSE by capturing local as well as global errors for a particular estimator [9]–[12]. To the best of This work was funded by the French Government Defense procurement agency (Direction G´en´erale de l’Armement). Yann Le Gall was with ENSTA Bretagne, he is now with Thales Underwater Systems, Route de Sainte Anne du Portzic, 29601 BREST Cedex, France (email: [email protected]). Julien Bonnel is with ENSTA Bretagne, UMR CNRS 6285 Lab-STICC, 2 rue Francois Verny, 29806 Brest Cedex 9, France (e-mail: [email protected]). F.-X. Socheleau is with Institut Mines-Telecom, Telecom Bretagne, UMR CNRS 6285 Lab-STICC, Technople Brest Iroise-CS 83818, 29238 Brest Cedex, France (e-mail: [email protected]).

our knowledge, performance analysis in the presence of model mismatch has never been considered for the full multisnapshot and multifrequency MFP problem under the deterministic signal model (conditional model) where the source signals are assumed deterministic unknown; although this data model is very common for parameter estimation in the maximumlikelihood MFP framework [13], [14]. The aforementioned references taking mismatch into account consider either a monofrequency stochastic model (unconditional model) [7] or a particular case of the monofrequency deterministic model where relative source spectral information is available between snapshots [10]. In this paper, the MIE approach is derived to predict MFP MSE performance in the presence of model mismatch for the full multisnapshot and multifrequency MFP problem under the deterministic signal model. Note that the approach in [7], [12] accounts for an unknown noise covariance matrix whereas this latter is assumed known here. The paper is organized as follows. The observation model and the MFP estimator are described in Section II. Section III presents the MIE analysis corresponding to our MFP context. Numerical examples illustrating the results are shown in Section IV, followed by conclusions in Section V. Notation: Throughout this paper, lowercase boldface letters denote vectors, e.g., x, and uppercase boldface letters denote matrices, e.g., A. The superscripts T and H mean transposition and Hermitian transposition, respectively. IN and 0N,M denote the N ×N identity matrix and the N ×M zero matrix, respectively. Re[·] denotes real part of a complex quantity. Operator k·k designates the standard Euclidean norm. Finally, E {·} denotes expectation and P denotes the probability measure. II. O BSERVATION MODEL

AND MAXIMUM LIKELIHOOD

ESTIMATE

Consider an array of N sensors in the ocean acoustic waveguide. The goal of MFP is to estimate a set of unknown parameters θ from the array measurements. This parameter set may be the source position (range, depth, bearing...) for source localization, and/or ocean environmental parameters (bathymetry, sound speed, density...) for tomography and geoacoustic inversion. For each snapshot l and each frequency fm , the complex array output is modeled by the N × 1 vector yl (fm ) = sl (fm ) · g(fm , θ) + wl (fm ), l = 1, ..., L, m = 1, ..., M

(1)

where, • L is the number of snapshots available for each of the M frequencies.

2

• •

g(fm , θ) is a complex N × 1 vector representing the transfer function of the medium at frequency fm for the propagation from the source to each of the N receivers. It is usually termed Green’s function. sl (fm ) is an unknown deterministic complex scalar representing the source amplitude and phase. wl (fm ) is a complex N × 1 vector representing a circularly-symmetric, zero mean Gaussian noise that is independent of the source signal. It is assumed to be white in frequency and time with positive definite spatial covari2 ance matrix Σw (fm ) = σw (fm )Cw (fm ). This matrix is assumed known and to simplify subsequent derivations and analysis we will work in whitened coordinates and 2 assume that Σw (fm ) = σw (fm )IN .

The set of observations is collected in the following vector T T y = [y1T (f1 ) . . . yL (f1 ) . . . y1T (fM ) . . . yL (fM )]T . (2)

To estimate θ, matched-field methods compare the data measured on the array with replicas of the acoustic field derived from the wave equation. Model mismatch arises when these replicas differ from the actual field. This occurs for instance when the assumed oceanic waveguide differs from the true one or when system parameters are erroneous (inaccurate receivers’ positions...). Formally, the assumed Green’s functions gǫ (fm , θ) used to estimate θ differs from the true Green’s function g(fm , θ). When sl (fm ) is assumed unknown and Σw (fm ) is known, the ML estimate of the parameter set θ under model mismatch is [14] θˆ = arg max C(θ),

(3)

θ

where C(θ) =

M X L X g

m=1 l=1

ǫ

H

(fm , θ)Rl,m gǫ (fm , θ) , 2 (f ) σw m

(4)

m ,θ) Rl,m = yl (fm )ylH (fm ) and gǫ (fm , θ) = kggǫǫ (f (fm ,θ)k . If the true set of parameters is denoted as θ0 , the search function C(θ) with a noise-free observation is called the ambiguity function (AF) ψ(θ) and satisfies



the amplitude ratio between the mainlobe and the sidelobes is smaller than in the no-mismatch case: the “threshold region” where global errors dominate will start at larger SNR. 1

without mismatch with mismatch

0.9 0.8

Ambiguity function



0.7 0.6 0.5 0.4 0.3

θmis

0.2

θ0

0.1 0 1000

2000

3000

4000

5000

6000

7000

8000

9000

=rs

Fig. 1. Example of ambiguity functions with and without model mismatch, γ(l, fm ) = 1. θ0 is the true parameter set and θ mis denotes the parameter set that maximizes the AF with mismatch.

Note that these observations may not hold true in all cases, they are scenario-dependent but typically occurred for the simulated cases considered by the authors that focused on shallow water environments (pekeris waveguides, summer water sound speed profiles, sediment layers over a basement). III. M ETHOD OF INTERVAL ERRORS MIE provides an approximation of the mean square error (MSE) for a given estimator (see the tutorial treatment by Van Trees and Bell in [16]). As shown in Fig. 1, the AF usually exhibits a typical mainlobe/sidelobes behavior. MIE builds upon the decomposition of the MSE into two terms: the local errors that concentrate around the mainlobe peak and outliers that concentrate around the sidelobe peaks. Consider the true parameter θ0 and a discrete set of No parameter points {θ1 , θ2 . . . θNo } sampled at the sidelobe maxima of the AF with mismatch. The conditional MSE of the ML estimator can then be approximated as [8]–[12]: o n ˆ − θ 0 )(θ ˆ − θ0 )T ≈ Ey (θ ! No X 1− Pe (θ k |θmis ) × MSE(local) (θ 0 ) k=1

ψ(θ) =

M X L X

γ(l, fm )|gǫ H (fm , θ)g(fm , θ 0 )|2 ,

(5)

m=1 l=1

where γ(l, fm ) = g(fm ,θ0 ) kg(fm ,θ0 )k .

|sl (fm )|2 2 2 (f ) kg(fm , θ 0 )k σw m



Pe (θk |θmis ) × (θ k − θ0 )(θ k − θ0 )T , (6)

k=1

and g(fm , θ0 ) =

The study of this AF helps to understand the impact of model mismatch on the estimation performance. An example is shown in Fig. 1 for a range estimation problem. It is observed that: •

+

No X

the maximum of the AF with mismatch is at a value θmis different from the true value θ0 : the estimator will be biased, the mainlobe is wider than in the no-mismatch case: the local error around θmis will be larger than the Cram´er-Rao bound (CRB) [15],

where Pe (θk |θmis ) is the pairwise error probability of the ML estimator under model mismatch (3), i.e. the probability of deciding in favor of the parameter θk in the binary hypothesis test {θk , θmis }. The pairwise error probability Pe (θ k |θ mis ) is used as an approximation of the probability that the estimate PNo Pe (θ k |θmis )) is used falls on the sidelobe k and (1 − k=1 as an approximation of the probability that the estimate falls on the mainlobe (i.e the probability of local errors). MSE(local) (θ 0 ) is the asymptotic MSE of the ML estimator. In the absence of mismatch, the CRB is usually a good predictor of the performance in this region [5]. Since mismatch is possible, these local errors must be approximated by other means as described in III-B.

3

Note that the MIE approximation begins to over predict the MSE at low SNR. Thus, it is hard limited not to exceed the variance of a uniform distribution in the search interval of the ML estimator. A. Pairwise error probability The pairwise error probability Pe (θ k |θmis ) of the ML estimator under model mismatch (3) is given by Pe (θ k |θ mis ) = P

M X L X |gǫ (fm , θk )yl (fm )|2 2 (f ) σw m m=1 l=1

! M X L X |gǫ (fm , θ mis )yl (fm )|2 ≥ 0 . (7) − 2 (f ) σw m m=1 l=1

Following the same approach as in [5, Sec. V-C], this pairwise error probability can be expressed as Z ∞ 1 1 × Pe (θ k |θmis ) = 1 − 2π −∞ jω + β (8) e−c dω, QM L L m=1 (1 + λ1m (jω + β)) (1 + λ2m (jω + β)) M X L X λ1m |µ1m,l |2 (jω + β)

λ2m |µ2m,l |2 (jω + β) , (1 + λ1m (jω + β)) (1 + λ2m (jω + β)) m=1 l=1 (9) for some β > 0 such that 1 + βλ1m > 0 and 1 + βλ2m > 0. λ1m ,2m are the non-zero eigenvalues of c=

+

gǫ (fm , θ k )gǫ H (fm , θk ) − gǫ (fm , θmis )gǫ H (fm , θmis ) (10) and satisfy λ1m ,2m = ±

q 1 − |gǫ H (fm , θk )gǫ (fm , θmis )|.

(11)

The values µ1m,l , µ2m,l can be expressed as a function of the SNR γ(l, fm ). In the general case with possible mismatch they satisfy s γ(l, fm )(1 − λ1m ,2m ) × µ1m,l ,2m,l = 2λ21m ,2m  1 × gǫ H (fm , θmis )g(fm , θ0 ) − 1 − λ1m ,2m  H H gǫ (fm , θmis )gǫ (fm , θ k )gǫ (fm , θk )g(fm , θ0 ) . (12) B. Asymptotic local MSE When expanded, the MSE can be expressed as  T   ˆ ˆ = θ − θ0 θ − θ 0 Ey n o o n ˆ θˆT + Ey ∆θˆ bT (θ 0 ) Ey ∆θ∆ n oT + b(θ0 )Ey ∆θˆ + b(θ 0 )bT (θ0 ),

is then n oan asymptotic approximation of o by computing n derived ˆ θˆT and Ey ∆θˆ using the same strategy as in Ey ∆θ∆ [17] and [18], except that asymptotic refers to the SNR and not to the number of snapshots (this is more convenient for the deterministic signal model and is also justified by the fact that few snapshots are usually available in MFP applications). More precisely, let θi denote the i-th entry of θ, ∆θˆ is ∂C approximated using a Taylor expansion of ∂θ at infinite SNR, i ∞ 2 H i.e., at θmis and Rl,m = |sl,m | g(fm , θ 0 )g (fm , θ0 ):  ∂C ∂2C ∂C  ˆ ˆ (θmis , R∞ ) + θ, Rl,m ≈ (θ mis , R∞ l,m l,m )∆θ ∂θi ∂θi ∂θT ∂θi  T  M X L X ∂2C ∞ + tr (θ mis , Rl,m ) ∆Rl,m , ∂Rl,m ∂θi m=1 l=1 (14) ˆ where ∆Rl,m = Rl,m −R∞ l,m . The ML estimate θ is obtained with the observed data Rl,m , and the value θmis corresponds to the output of the ML estimator with infinite SNR therefore1 ∂C ˆ (θ, Rl,m ) = 0N ×1 , (15) ∂θ ∂C (θmis , R∞ (16) l,m ) = 0N ×1 . ∂θ Since ∆Rl,m is Hermitian, it can be shown that  T  ∂2C tr (θmis , R∞ ) = ∆R l,m l,m ∂Rl,m ∂θi  H   ∆Rl,m ∂gǫ (fm , θ mis ) g (f , θmis ) . (17) Re 2 2 (f ) ǫ m σw ∂θT m Therefore, from (14), for i = 1 · · · N ,  2 −1 ∂ C ∞ ∆θˆ ≈ − (θ , R ) mis l,m ∂θT ∂θ  H  X  M X L ∆Rl,m ∂gǫ (fm , θmis ) g (f , θ mis ) . × Re 2 2 (f ) ǫ m σw ∂θT m m=1 l=1 (18) n o o n ˆ θˆT and Ey ∆θˆ can After some tedious algebra, Ey ∆θ∆ be obtained from (18). Define am = gǫ (fm , θmis ), cm = g(fm , θ 0 ), Dm = and the N × N matrix [F]i,j =

(13)

where ∆θˆ = θˆ − θmis and b(θ 0 ) = θmis − θ0 . The value θmis as well as the bias b(θ0 ) can easily be obtained by analyzing the AF with mismatch. The asymptotic local MSE

∂gǫ (fm , θ mis ) , ∂θT (19)

H  2 M X L X ∂ am |sl,m |2 Re cm cH m am 2 (f ) σ ∂θ ∂θ j i m=1 l=1 w m   H  ∂am ∂am H + . cm cm ∂θi ∂θj (20)

We then obtain " M L # n o XX −1 H Ey ∆θˆ = −F Re Dm am ,

(21)

m=1 l=1

1 The estimates are local maxima (except in the particular case where they are on the edge of the parameter search space, this case is not considered).

4

n o ˆ θˆT = Ey ∆θ∆ X M X L  1 −1 H T ∗ F Re DH m Dm + Dm am am Dm 2 m=1

air r water Source (rs, zs)

cw VLA

w

l=1

|sl,m |2 H H H T ∗ (22) DH m cm cm Dm + Dm cm cm am am Dm 2 (f ) σw m   −1 H H H T ∗ H + DH D a c c a + D a c D c a m m m m m m m m m m m m F

+

n oT n o + Ey ∆θˆ Ey ∆θˆ .

Finally, an approximation of the asymptotic local MSE is obtained by injecting (21) and (22) in (13). As shown in the next section, this approximation turns out to be very accurate. IV. N UMERICAL EXAMPLES The theoretical results are illustrated for MFP source range estimation in a shallow water environment. The MFP configuration is presented in Fig. 2. The waveguide is a Pekeris waveguide [19] which consists in an isospeed water layer overlaying a semi-infinite fluid basement. This waveguide is classically used to model shallow water environments. The ML estimator (3) assumes that the Pekeris waveguide has the parameters given by the assumed environment in table I. The source depth is assumed known and is fixed to zs = 30 m, whereas the range is unknown in the interval rs = [4, 6] km. The receiving array is a vertical array of N = 12 elements linearly spaced between z1 = 5 m and zN = 95 m. We assume that we observe L = 5 snapshots of the source signal as well as M = 8 frequencies logarithmically spaced from 50 − 500 Hz. The Green’s functions are computed using normal mode theory [20]. Two mismatch scenarios are considered. In the first one, the source range is θ0 = 5000 m and the actual environment is given by scenario 1 in table I. In the second one, the source range is θ0 = 5100 m and the actual environment is given by scenario 2 in table I. Monte-Carlo simulations are performed with 5000 iterations for each SNR. Since the asymptotic MSE can be dominated by the bias b(θ 0 ), results presented n are alsoT o ˆ θˆ in (22). for the bias-free MSE as expressed by Ey ∆θ∆ The results are shown in Fig. 3 and 4 for the first and second scenario, respectively. As a reference, the CRB is also plotted. In both cases, MIE accurately predicts the threshold region as well as the asymptotic performance of the ML estimator (3) with model mismatch. The second scenario is quite unusual: due to mismatch, the mainlobe gets split into two lobes with close amplitudes which explains the notable increase of the bias-free MSE for SNRs lower than 15 dB. In both scenarios, the impact of mismatch is significant: the bias limits the high SNR performance and, because mismatch widens the mainlobe, the bias-free MSE remains much larger than the Cram´er-Rao bound.2 V. C ONCLUSION An interval error-based method has been developed to predict the mean-squared error performance of the full multisnapshot and multifrequency MFP problem with possible 2 Note that in the absence of mismatch, the asymptotic MSE as computed in (21) and (22) coincides with the CRB (not shown here due to lack of space).

D cb b

basement

z

Fig. 2.

MFP configuration.

Environmental parameters Assumed environment Scenario 1 Scenario 2

D (m) 100 99.86 99

cw (m.s−1 ) 1500 1500 1510

ρw (kg.m−3 ) 1000 1000 1000

cb (m.s−1 ) 1800 1750 1750

ρb (kg.m−3 ) 2000 1900 1800

TABLE I E NVIRONMENTAL PARAMETERS OF THE ASSUMED ENVIRONMENT AND OF THE TWO ACTUAL ENVIRONMENTS OF SCENARIOS 1 AND 2.

Fig. 3.

MSE performance for source range estimation in scenario 1.

Fig. 4.

MSE performance for source range estimation in scenario 2.

model mismatch. The deterministic signal model has been considered. Unlike the stochastic model which considers that the source signals follow a Gaussian process, the deterministic model considers that the source signals are deterministic but unknown and makes no assumption on the source signal distribution. It is therefore appropriate for tonal components that may arise from submarines, ships and deployed tomographic sources or for marine mammals’ vocalizations [21]– [25]. Hence, the results provided in this article offer a good framework to analyze MFP performance in various configurations. Furthermore, these results could also be used for the matched-mode processing (MMP) problem [26]–[28] or for coherent MFP [14].

5

R EFERENCES [1] N.R. Chapman, “Inverse problems in underwater acoustics,” in Handbook of Signal Processing in Acoustics, D. Havelock, S. Kuwano, and M. Vorl¨ander, Eds., pp. 1723–1735. Springer Science & Business Media, 2008. [2] A.B. Baggeroer and H. Schmidt, “Cramer-rao bounds for matched field tomography and ocean acoustic tomography,” in Proc. ICASSP. IEEE, 1995, vol. 5, pp. 2763–2766. [3] J. Tabrikian and J.L. Krolik, “Barankin bounds for source localization in an uncertain ocean environment,” IEEE Trans. Signal Process., vol. 47, no. 11, pp. 2917–2927, 1999. [4] W. Xu, A.B. Baggeroer, and C.D. Richmond, “Bayesian bounds for matched-field parameter estimation,” IEEE Trans. Signal Process., vol. 52, no. 12, pp. 3293–3305, 2004. [5] Y. Le Gall, F.-X. Socheleau, and J. Bonnel, “Matched-field processing performance under the stochastic and deterministic signal models,” IEEE Trans. Signal Process., vol. 62, no. 22, pp. 5825–5838, 2014. [6] N. Lee and C. D. Richmond, “Threshold region performance prediction for adaptive matched field processing localization,” in Proc. 12th Adaptive Sensor Array Processing Workshop, 2004. [7] N. Lee, C. D. Richmond, and V. Kmelnitsky, “Mean squared error performance of adaptive matched field localization under environmental uncertainty,” in Proc. Statistical Signal Processing Workshop. IEEE, 2012, pp. 812–815. [8] W. Xu, Z. Xiao, and L. Yu, “Performance analysis of matched-field source localization under spatially correlated noise field,” IEEE J. Ocean. Eng., vol. 36, no. 2, pp. 273–284, 2011. [9] C.D. Richmond, “Capon algorithm mean-squared error threshold SNR prediction and probability of resolution,” IEEE Trans. Signal Process., vol. 53, no. 8, pp. 2748–2764, 2005. [10] C.D. Richmond, “On the threshold region mean-squared error performance of maximum-likelihood direction-of arrival estimation in the presence of signal model mismatch,” in Proc. Fourth IEEE Workshop on Sensor Array and Multichannel Processing. IEEE, 2006, pp. 268–272. [11] F. Athley, “Threshold region performance of maximum likelihood direction of arrival estimators,” IEEE Trans. Signal Process., vol. 53, no. 4, pp. 1359–1373, 2005. [12] C.D. Richmond, “Mean-squared error and threshold SNR prediction of maximum-likelihood signal parameter estimation with estimated colored noise covariances,” IEEE Trans. Inf. Theory, vol. 52, no. 5, pp. 2146– 2164, 2006. [13] C.F. Mecklenbr¨auker and P. Gerstoft, “Objective functions for ocean acoustic inversion derived by likelihood methods,” Journal of Computational Acoustics, vol. 8, no. 02, pp. 259–270, 2000. [14] S.E. Dosso and M.J. Wilmut, “Maximum-likelihood and other processors for incoherent and coherent matched-field localization,” The Journal of the Acoustical Society of America, vol. 132, pp. 2273–2285, 2012. [15] S.M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall, 1993. [16] H.L. Van Trees and K.L. Bell, Bayesian bounds for parameter estimation and nonlinear filtering/tracking, Wiley-IEEE Press, 2007. [17] C. Vaidyanathan and K.M. Buckley, “Performance analysis of the MVDR spatial spectrum estimator,” IEEE Trans. Signal Process., vol. 43, no. 6, pp. 1427–1437, 1995. [18] M. Hawkes and A. Nehorai, “Acoustic vector-sensor beamforming and Capon direction estimation,” IEEE Trans. Signal Process., vol. 46, no. 9, pp. 2291–2304, 1998. [19] C.L. Pekeris, Theory of propagation of explosive sound in shallow water, vol. 27, Geological Society of America Memoirs, 1948. [20] F.B. Jensen, W.A. Kuperman, M.B. Porter, and H. Schmidt, Computational ocean accoustics, 2nd ed, Springer Science & Business Media, 2011. [21] V. Bush, J.B. Conant, and J.T. Tate, Principles and applications of underwater sound, Defense Technical Information Center, 1946. [22] D. Tollefsen and S.E. Dosso, “Bayesian geoacoustic inversion of ship noise on a horizontal array,” The Journal of the Acoustical Society of America, vol. 124, pp. 788–795, 2008. [23] A.V. van Leijen, J.P. Hermand, and M. Meyer, “Geoacoustic inversion in the north-eastern caribbean using a hydrographic survey vessel as a sound source of opportunity,” Journal of Marine Systems, vol. 78, pp. 333–338, 2009. [24] C.-F. Huang and W.S. Hodgkiss, “Matched-field geoacoustic inversion of low-frequency source tow data from the asiaex east china sea experiment,” IEEE J. Ocean. Eng., vol. 29, no. 4, pp. 952–963, 2004.

[25] A.M. Thode, G.L. D’Spain, and W.A. Kuperman, “Matched-field processing, geoacoustic inversion, and source signature recovery of blue whale vocalizations,” The Journal of the Acoustical Society of America, vol. 107, pp. 1286–1300, 2000. [26] G. Le Touz´e, J. Torras, B. Nicolas, and J. Mars, “Source localization on a single hydrophone,” in Proc. OCEANS 2008. IEEE, 2008, pp. 1–6. [27] J. Bonnel, G. Le Touz´e, B. Nicolas, and J. Mars, “Physics-based timefrequency representations for underwater acoustics: Power class utilization with waveguide-invariant approximation,” IEEE Signal Process. Mag., vol. 30, no. 6, pp. 120–129, 2013. [28] Y. Le Gall, F-X. Socheleau, and J. Bonnel, “Performance analysis of single receiver matched-mode processing for source localization,” in Proc. 2nd international conference and exhibition on Underwater Acoustics (UA2014), 2014.

Matched-Field Performance Prediction with Model ...

Julien Bonnel is with ENSTA Bretagne, UMR CNRS 6285 Lab-STICC, 2 rue. Francois ... L(fM )]T . (2). To estimate θ, matched-field methods compare the data measured on the array with replicas of the acoustic field derived from the wave equation. ..... [26] G. Le Touzé, J. Torras, B. Nicolas, and J. Mars, “Source localization.

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Internet Appendix for Pricing Model Performance and ...
Please note: Wiley-Blackwell is not responsible for the content or ... †University of Toronto, Joseph L. Rotman School of Management, 105 St. George Street, Toronto, Ontario,. Canada M5S 3E6 ...... 8In the empirical application in the paper, we use