Mohamed Berrada

Adjoint-based acoustic inversion for the physical characterization of a shallow water environmenta)

Recently the concept of adjoint modeling has been introduced in shallow water acoustics for solving inverse problems. Analytical adjoints have been derived for normal modes and for both the standard parabolic equation and Claerbout’s wide-angle approximation (WAPE). This paper proposes the application of a semiautomatic adjoint approach that has been successfully applied in the past for multidimensional variational data assimilation in meteorological and climate modeling. Starting from a modular graph representation of the underlying forward model, a programming tool facilitates the generation and coding of both the tangent linear and the adjoint models. The potential of this numerical adjoint approach for the physical characterization of a shallow water environment is illustrated with two applications for geoacoustic inversion and ocean acoustic tomography using Claerbout’s WAPE in combination with nonlocal boundary conditions. Furthermore, the adjoint optimization is extended to multiple frequencies and...

SEMI-AUTOMATIC ADJOINT PE MODELING FOR GEOACOUSTIC INVERSION ∗

Recently, an analytic adjoint-based method of optimal nonlocal boundary control has been proposed for inversion of a waveguide acoustic field using the wide-angle parabolic equation [Meyer and Hermand, J. Acoust. Soc. Am. 117, 2937-2948 (2005)]. In this paper a numerical extension of this approach is presented that allows the direct inversion for the geoacoustic parameters which are embedded in a discrete representation of the nonlocal boundary condition. The adjoint model is generated numerically and the inversion is carried out jointly across multiple frequencies. To demonstrate the effectiveness of the implemented numerical adjoint, an illustrative example is presented for the geoacoustic characterization of a Mediterranean shallow water environment using realistic experimental conditions.

A global optimization method using a random walk on a topological map and local variational inversions

This paper presents an improved variational method suitable for inverting a problem associated with integral constrains. The method allows a global minimization. We minimized a cost function representing the mismatch between the measurements and the output of a numerical model, to which we added a restoring term to a background. A way to process the covariance matrix associated with the above-weighted quadratic background is to model the control vectors using probabilistic principal component analysis (PPCA). The use of PPCA presents difficulties in the case of a large dataset representing the overall variability of the control space. We therefore developed a method based on a topological map model, which allows partition of the dataset into subsets more suited to the PPCA approach and thus leading to a local inversion by the variational method. A random walk based on a Markov chain was used to find the most appropriate subsets of the topological map by taking into account a priori information on the unknown vector. This random walk on a topological map allows us to reduce the number of subsets able to give the optimal solution and thus to achieve a better performance at a lower cost. An example of the application of this method to the shallow water acoustic tomography inverse problem, showing its effectiveness, is presented.

A robust probabilistic approach for variational inversion in shallow water acoustic tomography

This paper presents a variational methodology for inverting shallow water acoustic tomography (SWAT) measurements. The aim is to determine the vertical profile of the speed of sound c(z), knowing the acoustic pressures generated by a frequency source and collected by a sparse vertical hydrophone array (VRA). A variational approach that minimizes a cost function measuring the distance between observations and their modeled equivalents is used. A regularization term in the form of a quadratic restoring term to a background is also added. To avoid inverting the variance?covariance matrix associated with the above-weighted quadratic background, this work proposes to model the sound speed vector using probabilistic principal component analysis (PPCA). The PPCA introduces an optimum reduced number of non-correlated latent variables ?, which determine a new control vector and a new regularization term, expressed as ?T?. The PPCA represents a rigorous formalism for the use of a priori information and allows an efficient implementation of the variational inverse method.

Adjoint-based shallow water tomography with partially known bottom geoacoustic properties

Recently, a numerical extension of the wide-angle PE (WAPE) adjoint-based inversion method has been presented that allows direct inversion of the geoacoustic parameters that are embedded in a discrete representation of nonlocal boundary conditions at the water-sediment interface [Hermand et al., ICTCA 2005]. In contrast to conventional ocean acoustic tomography (OAT), correct modeling of sound interaction with the bottom is particularly important for shallow water acoustic tomography (SWAT). Uncertainty or partial knowledge of the bottom acoustic parameters can significantly degrade SWAT performance. In this context the paper discusses an adjoint-based tomography approach that includes the geoacoustic parameters as additional unknowns in the parameter space. A priori information about the bottom characteristics, e.g., the approximate thickness and composition of sediment layers and hard rock basement are often available from site surveys with sub-bottom profiling systems or in the form of previously archived in situ data. This information can be used as an initial solution to start the inversion process or be included in an augmented cost function through the use of regularization. By means of adjoint modeling exact gradient information can be obtained to determine the sound speed profile of the water column and concomitantly adjust the partially known sediment and bottom halfspace parameters. A case study based on environmental data obtained in Mediterranean shallow waters is used to present first results.

Probabilistic PCA and Ocean Acoustic Tomography Inversion with an Adjoint Method

We present an Ocean Acoustic Tomography (OAT) inversion in a shallow water environment. The idea is to determine the celerity c(z), z is depth, knowing the acoustic pressures caused by a multiple frequencies source and collected by a sparse receiver array. The variational approach minimizes a cost function which measures the adequacy between the measurements and their forward model equivalent. This method introduces also a regularisation term in the form (c(z)‐cb)tB‐1(c(z)‐cb), which supposes that c(z) follows an a priori normal law. To circumvent the problem of estimating B‐1, we propose to model the celerity vectors by a probabilistic PCA. In contrast to the methods which use PCA as a regularization method and filter the useful information, we take a sufficient number of axes which allow the modelization of useful information and filter only the noise. The probabilistic PCA introduces a reduced number of non correlated latent variables η which act as new control parameters introduced in the cost functio...

Adjoint approach to the physical characterization of a shallow‐water environment

In underwater acoustics a variety of different applications of adjoint models has been proposed in recent years. Adjoints have been derived for normal modes and for both the standard parabolic equation and Claerbout’s wide‐angle approximation. This paper reviews the analytic nonlocal boundary control approach proposed in an earlier paper by the authors [Meyer & Hermand, ‘‘Optimal nonlocal boundary control of the wide‐angle parabolic equation for inversion of a waveguide acoustic field,’’ J. Acoust. Soc. Am. 117, 2937–2948 (2005)] and presents a numerical extension that allows direct inversion of the geoacoustic parameters that are embedded in a discrete representation of the nonlocal boundary condition at the water‐sediment interface. The effectiveness of this numerical adjoint approach for the physical characterization of a shallow‐water environment is illustrated with applications for geoacoustic inversion and ocean acoustic tomography. In particular, it is shown how a joint inversion across multiple fr...

A numerical adjoint parabolic equation (PE) method for tomography and geoacoustic inversion in shallow water

Recently, an analytic adjoint‐based method of optimal nonlocal boundary control has been proposed for inversion of a waveguide acoustic field using the wide‐angle parabolic equation [Meyer and Hermand, J. Acoust. Soc. Am. 117, 2937–2948 (2005)]. In this paper a numerical extension of this approach is presented that allows the direct inversion for the geoacoustic parameters which are embedded in a spectral integral representation of the nonlocal boundary condition. The adjoint model is generated numerically and the inversion is carried out jointly across multiple frequencies. The paper further discusses the application of the numerical adjoint PE method for ocean acoustic tomography. To show the effectiveness of the implemented numerical adjoint, preliminary inversion results of water sound‐speed profile and bottom acoustic properties will be shown for the YELLOW SHARK ’94 experimental conditions.