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VARIATIONAL INVERSION BY PROBABILISTIC PCA MODELIZATION FOR SHALLOW-WATER ACOUSTIC TOMOGRAPHY: APPLICATION TO SOUTH ELBA ENVIRONMENT

M. Berradaa, F. Badrana,b, M. Crépona, J.-P. Hermandc, S. Thiriaa a

Laboratoire d’Océanographie et du Climat – Expérimentation et Approches Numériques, Université Pierre et Marie Curie, T. 45, 5-ème étage 4, place Jussieu, 75005 Paris, France. b Laboratoire CEDRIC, Conservatoire National des Arts et Métiers – 292, rue Saint Martin, 75003 Paris, France. c Environmental Hydroacoustics lab, Université libre de Bruxelles (U.L.B.), av. Franklin D. Roosevelt 50, CP 194/05, B-1050 Bruxelles, Belgique. M. Berrada, LOCEAN-UPMC, T. 45, 5-ème étage 4, place Jussieu, 75005 Paris, France. Fax: (0033) (0)1 44 27 71 59, [email protected]

Abstract: This paper presents the application of a probabilistic approach for variational inversion in acoustic tomography. The aim is to determine the time-evolving, rangeaveraged, vertical profile of speed of sound c( z , ) in a shallow water environment from the acoustic pressure fields generated by a monochromatic sound source and measured on a sparse vertical hydrophone array. A variational approach that minimizes a cost function which measures the distance between observations and their modelled counterparts is used. As the tomographic inversion is an ill-posed problem a regularization term in the form of a quadratic restoring term to a background is added. To avoid inverting for the variancecovariance matrix associated with the above weighted quadratic background, it is proposed to model the sound speed vector using probabilistic principal component analysis (PPCA). The probabilistic PCA introduces an optimum reduced number of non correlated latent variables  which determine a new control vector and introduce a new regularization term, expressed as  T  . PPCA represents a rigorous formalism for the use of a priori information and allows for an efficient implementation of the variational inverse method. In the present work the probabilistic PCA is applied to an acoustic tomography scenario in the South Elba environment. Keywords: probabilistic principal component analysis, empirical orthogonal function, tomographic inversion, variational method, adjoint method, regularization.

1. INTRODUCTION The application of a probabilistic approach for variational inversion in shallow-water acoustic tomography is presented. The classical approach consists of introducing a cost function J that represents the distance between actual measurements and their prediction computed from the so-called direct model. In this paper, the direct model is a numerical acoustic propagation model based on the wide-angle PE (WAPE) due to Claerbout [1]. Some applications of the variational approach in tomographic inversion is given in [2], where the gradient of the cost function is computed by the adjoint approach using the semi-automatic adjoint code generator YAO via modular graph approach [2,3,4]. In tomographic inversion, two problems have to be considered: first, the observed and computed measurements cannot be identical due to correlated additive noise on the vertical receiver array (VRA) signals and uncertainties in the physical modelling of the shallow water environment and acoustic propagation, and, second, the inverse problem is ill-posed having several different solutions. Different approaches have been proposed to overcome this second problem. One can use a regularization method involving the introduction of a penalty term in the cost function J [2]. Another approach proposes to restrict the search of the control vector, which is here the vertical sound speed profile c  c(z ) , to a subspace of reduced dimension defined by Principal Components Analysis (PCA) [5,6,7]. The drawback is that the dimension of the PCA subspace is provided implicitly by the PCA methodology without any criterion of optimality, and does not explicitly introduces a penalty term into the cost function. The present work proposes to address the issues of dimensional reduction and regularization by using a Probabilistic Principal Component Analysis (PPCA) model to decompose the sound speed profile. This approach allows the intrinsic dimension of the data (here, the sound speed profile) to be determined, and provides the necessary a priori knowledge required for the regularization of the variational inversion solution. The PPCA model associated with Bayesian formalism allows us to define a generalized cost function with a penalty term. This approach is applied to acoustic data synthesized from NCOM oceanic model predictions, obtained during the MREA BP’07 experiment, southeast of Elba, Italy. The paper is organized as follows. Section 2 reviews the variational approach and the chosen cost function together with its background terms. Section 3 introduces the PPCA model and the associated Bayesian formalism. Section 4 demonstrates the adequacy of the PPCA approach using MREA BP’07 ocean prediction data and synthetized acoustic data.

2. VARIATIONAL TOMOGRAPHIC INVERSION

The principle of variational inversion consists of minimizing a given distance between the measurements and the outputs of the so-called direct model whose input parameters (the quantities to be retrieved) are adjusted to obtain the best fit between the observed and simulated measurements (here, acoustic pressures on a vertical array). Here, the used direct acoustic model, denoted by G f , is based on the wide-angle parabolic equation (WAPE) model [1]. For further details on G f , see [2]. For a given sound speed profile, at a given time  , c( z , ) , a predicted vector field   G f [c] can be computed at each of the N

elements of a vertical receiver array (VRA) for frequency f , and compared to the measurements (processed acoustic signals s j (t ), j  1,..., N ). The mismatch between the computed and observed values is quantified by using the following cost function, proposed in [8],

 * Rˆ  ˆ J o (c)  trR  * ,

(1)



ˆ is the estimated where * is the Hermitian transpose operator, tr is the trace operator, R ˆ  is the linear Bartlett processor. The spatial correlation matrix at frequency f and  * R ˆ is estimated from the acoustic signals s (t ) [8]. The gradient of J is computed by matrix R j

o

the adjoint approach which is implemented, here, by using the semi-automatic adjoint code generator YAO [4]. For further details on the implementation, see Refs. [2,3]. Because of additive noise on the VRA signals, uncertainties about the environment, and inaccuracies in the acoustic propagation modeling, a priori information about the desired control vector must be introduced. A well-known procedure used in data assimilation is to modify the cost function such as that:

J (c ) 

1 J o (c)  J b (c), T

(2)

where J o (c) is given by (1) and J b (c)  (c  cb )T B1 (c  cb ).

(3)

The vector cb is called the “background”, and B is the covariance matrix of the distance to the background. Equation (3) leads to a local search of the desired control vector c * in the vicinity of cb . In the cost function (2), T is a continuous hyper parameter which determines the weighted compromise between J o (c) and J b (c) . Due to the high dimensionality of the c( z , ) vectors and the strong correlation between their components, it becomes difficult to estimate the matrix B1 , which is often ill conditioned. A possible approach is to introduce an a priori information constraint explicitly by restricting the subspace of possible control vectors and removing the background term. This explicit approach has been introduced for environmental inversion [5,6,7] by using Principal Component Analysis (PCA), also known as Empirical Orthogonal Functions (EOF). Another transformation, which is implicit, consists of using the Probabilistic PCA model (hereinafter called PPCA) and the Bayesian formalism.

3. PPCA APPROACH TO VARIATIONAL INVERSION

In the following, c( z , ) is assumed to be evaluated at M points of the discrete space with respect to depth z . The Probabilistic PCA (PPCA) model [9,10] allows the control vector c to be interpreted probabilistically. It introduces an explicit latent variable   IR q ( q  M )

whose prior distribution N (0, I q ) is isotropic and normal, where I q is the identity matrix of order q . Let us assume that: c  W  cb   ,

(4)

where   N (0,  2 I M ) is a stochastic isotropic and normally distributed process with standard deviation  , I M the identity matrix of order M , W a M  q matrix of range q and cb a vector over IR M . The columns of W define a linear subspace E q of dimension q over IR M , and W  cb represents the associated affine subvariety, which contains the vector cb . The different profiles c can be considered as the sum of a vector belonging to the affine subvariety and a noise  . Under these conditions, it can be shown that the profile c is normally distributed, that its mean is the vector cb , and that its variance-covariance matrix is given by: B  WW T   2 I M .

(5)

The matrix B can be determined by estimating the model parameters ( W, cb ,  2 ) based on an observation data set A . In the following, A is a subset of sound speed profiles and locally represents the control vector. Through Maximum Likelihood (ML) estimation [9,10], the three parameters of the PPCA model can be determined. A possible optimal solution is such that: 

cb is the mean of the data set A ;



W  U (L   2 I q )1 / 2 where

U  (u1 , u 2 ,..., u q ) is comprised of the first q eigenvectors of the empirical variance-covariance matrix of A . L is a diagonal matrix ( q  q ), whose elements are the corresponding eigenvalues i ; and finally,



(6)

M 1    i . M  q i  q 1 2

The sum



M i  q 1

i represents the residual variance of the data, not taken into account by the

first q principal axes. Therefore,  2 is the average of the residual variance of the remaining ( M  q ) principal axes. If one assumes that the data subset A is statistically representative of the control vector c and can be generated by the PPCA model (4), the residual variance must be evenly distributed among the M  q remaining principal axes. Thus, the number of axes q must be chosen in such a way that this property is verified. On the other hand, if



M

i  q 1

i

is small enough and the number M  q of remaining axes is large enough, the value of  2 found when estimating the residual variance (6) is small. The choice of q is not as critical as for the explicit approach of PCA. Only a few values of q will be sufficient and this choice will lead to a rigorous inverse methodology. In this case, the parameter  in (4) turns out to

be a normally distributed noise, and all of the useful information in A is contained in the principal affine linear subspace of dimension q . As a consequence, c  W  cb , where  is the latent variable associated with c , and is normally distributed since   N (0, I q ) . The affine subvariety E q  cb of dimension q and the density function N (0, I q ) totally describe the control vector c . The solutions of the variational inversion must be found in this subvariety. The latent variable becomes the control vector and the first term of the total cost function (2) can be rewritten as a function of  as: J o (c)  J o ( W  cb )   o ( ).

(7)

Using Bayesian formalism, the following total cost function is found: J ( ) 

1  o ( )   T  , T

(8)

since the a priori information on the new control vector  is known. The inverse problem is solved by minimizing (8) with respect to  . The main advantages of the PPCA reformulation (8) are that the difficulty of estimating B 1 is circumvented, and the control vector becomes  , which has a smaller dimension ( q  M ), and whose components are not correlated. This provides improved preconditioning for the minimization process. Finally, this approach selects q according to its ability to fully reproduce the process without any loss of useful information. The minimization process thus leads to a realistic solution, which follows the a priori distribution of the data. In the following, one illustrates this methodology with a realistic test of acoustic tomography in a shallow water environment.

4. NUMERICAL VALIDATION USING MREA BP’07 DATA During the MREA BP’07 experiments, southeast of Elba Island in the Mediterranean Sea, a large set of in-situ acoustic and environmental data were collected for developing geoacoustic inversion [11] and acoustic tomography [12] methods. In the present work, 4-day prediction results of the temperature and salinity fields, obtained with NCOM model [12], were used to create a set of sound speed profiles (SSP). The SSPs were predicted at 1-h interval given rise to 96 profiles (Fig. 1). See [11,12] for details on the experimental data and ocean modelling. The MREA BP’07 profiles represent the evolution of the range-averaged SSP along a 15km transect (A-B in Fig. 1 of Ref. [12]) for a duration of four days, and constitute the data set A . The PPCA provides a model of this behaviour. Figure 2 shows the Cumulative Percentage of Total Variability (CPTV) with respect to total energy, for each of the first 15 PPCA axes. For q  8 axes, the CPTV is approximately 99%. Following the discussion about the choice of q in the previous section, and the results shown in Fig. 2, values of q greater than 8 provide an appropriate trade-off.

0 20

Depth (m)

40 60 80 100 Water Column Sediment Layer Bottom

120 1508

1510

1512 1514 sound speed (m/s)

1516

1518

Figure 1. MREA BP’07 sound speed profiles (blue) and their ensemble average (cyan). The sediment layer and bottom geoacoustic properties are given in Ref. [8]. 100 90 80 70

%

60 50 40 30 20 10 0

1

2

3

4

5

6

7

8 9 #axes

10 11 12 13 14 15

Figure 2. Energy of the first 15 axes. The bar graph represents the percentage of total variability of each PCA axis. The cumulative percentages of total variability (CPTV) are indicated by circle. The first eight axes include almost 99% of the total energy. Under twin experiments, a monochromatic acoustic source of frequency of 500 Hz is positioned at z s  69.2 m depth and the seafloor depth is z l  113.1 m . The synthesized acoustic signals are sampled on a vertical array (VRA) of 32 hydrophones, spaced at 2 m intervals, between the depths of 37.2 m and 99.2 m . To simulate a realistic scenario, we added a normally distributed noise of amplitude 0.01 to the acoustic signals. This gives a signal-to-noise ratio comparable to the levels observed during the Yellow Shark experiment ˆ used to [8]. These simulated measurements were later used to compute the matrix R determine the observation error (1). In this article, we consider a snapshot in the tracking scenario developed in Ref. [14]. The profile c * farthest from cb in A , based on the Euclidian distance, is selected as the “actual profile” to be retrieved. In other words, the synthesized acoustic signals were generated using c * as input to the WAPE model G f . The initial values at the start of the inversion were given as   0 . It corresponds to mean profile cb .

For the dimension of the reduced subspace we choose q  8 as the largest possible value allowed by the PPCA model. We thus minimized the cost function (8) with respect to , which is a 8-dimensional normally distributed variable [  N (0, I 8 ) ]. The “L-curve” method [13] is used to estimate the value of T in (2), which is here TLC  2.5 104 . Figure 3 shows the true centred profile ( cz true  c *  cb ), together with that estimated using TLC as regularization parameter with 8 axes ( cz est1  c est1  cb ). Also shown is the estimated centred profile ( cz est 2  c est 2  cb ), with the background term removed, in order to illustrate the usefulness of this term. 0

20

Depth (m)

40

60

80

100

cztrue czest1 czest2

120 −2

−1.5

−1 −0.5 Sound speed (m/s)

0

0.5

Figure 3. Acoustic tomography results using 8 PCA axes. Starting from   0 (corresponding to the mean profile) the true centred profile is shown by the blue solid line ( cz true ). The profile that was estimated using TLC  2.5 104 as the regularization parameter is shown by the red dashed line ( cz est1 ), and the one estimated with the background term removed is shown by the black dotted line ( cz est 2 ). Clearly, the control vector cest1 estimated according to inverse PPCA methodology is close to the “true” profile c * to be retrieved. We can also see from the estimated profile cest 2 that the performances are improved when the background term is used. For the present twin experiment we have used a monochromatic acoustic source but the methodology can be easily extended to variational inversion across multiple frequencies, like in Refs. [2,3,14].

5. CONCLUSION In this paper we have presented a variational approach to solve the inverse problem of acoustic tomography in a shallow water environment, based on the probabilistic principal component analysis (PPCA). We have shown that this method can provide an appropriate representation of the regularization term in the cost function, and can significantly reduce the number of control parameters. In the context of twin experiments with additive measurement noise, the methodology proposed here leads to satisfactory and robust results.

6. ACKNOWLEDGEMENTS This work was supported by the Service Hydrographique et Océanographique de la Marine Française (Hydrographic and Oceanographic Department of the French Navy). The support of all individuals and institutions involved in the MREA BP'07 Joint Research Program is highly appreciated. Especially, the contributions of the Royal Netherlands Navy and NATO Undersea Research Centre in the observational program are hereby deeply acknowledged.

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