† Corresponding author. E-mail:
Project supported by the Foundation of Key Laboratory of Marine Intelligent Equipment and System of Ministry of Education, China (Grant No. SJTU-MIES1908) and the National Natural Science Foundation of China (Grant No. 41775027).
Traditional geoacoustic inversions are generally solved by matched-field processing in combination with meta-heuristic global searching algorithms which usually need massive computations. This paper proposes a new physical framework for geoacoustic retrievals. A parabolic approximation of wave equation with non-local boundary condition is used as the forward propagation model. The expressions of the corresponding tangent linear model and the adjoint operator are derived, respectively, by variational method. The analytical expressions for the gradient of the cost function with respect to the control variables can be formulated by the adjoint operator, which in turn can be used for optimization by the gradient-based method.
Sound wave is the only information carrier that can travel long distances in the ocean. The exploration and development of marine resources, underwater target detection and identification, and environmental monitoring are all dependent on underwater acoustic detection technology.[1–3] Accurate acquisition of geoacoustic parameters is a prerequisite for the effective implementation of this technology. Due to the complexity of the marine environment, it is difficult for traditional in situ methods to obtain large-scale geoacoustic parameters in real time. With the advancement of sonar technology and inverse problem theory, the use of acoustic signals to invert the parameters of underwater acoustic environments has received more and more attention.[4–6]
Most previous geoacoustic studies have focused on using matched-field processing methods in combination with heuristic searching methods, such as genetic algorithm,[7] simulated annealing,[8] and sequential Monte Carlo sampling,[9] which always need huge computing resources because of too many forward propagation model runs. The adjoint method coming from optimal control theory is known to give accurate and efficient data assimilation processes in oceanography and meteorology.[10,11] However, it has rarely been applied in underwater acoustics for inversion purposes.[12]
For underwater acoustics modelling, a downgoing radiation condition must be imposed on the transmitted component of the field since the seabed is penetrable to sound waves, especially at low frequencies. Although many acoustic propagation tools are now available, the parabolic equation (PE) model has the advantages of dealing with the range-dependent environments efficiently and obtaining the full wave solution of the sound field. In the PE models, a downgoing radiation condition can be done by appending an absorbing layer[13] or alternatively by applying a non-local boundary condition (NLBC).[14] The latter is more attractive for limiting the computations in the interested region and time saving.
The remainder of this paper is organized as follows: The forward propagation model, i.e., a non-local boundary condition for the finite-difference PE solver, is briefly introduced in Section
With the far-field assumption and paraxial approximation, the standard PE can be written as[13]
To obtain a well-posed initial boundary value problem of Eq. (
Note the computation domain be
Set perturbations to c(z),
Define inner product as
Combining the above four equations and considering the lower boundary layer of the tangent linear model,
Now return to the cost function, the Gâteaux differential of
Combining Eq. (
Owing to
With these gradients, optimization could be generally accomplished through using the iterative gradient-based methods,
The flowchart of the whole iteration process is shown in Fig.
The analytical theory framework for geoacoustic inversion by the adjoint PE method is introduced above. It should be noted that parameter estimation problems have demonstrated to be ill-posed, especially when the problems are severely nonlinear and the observed data are contaminated by noise.[15] These problems usually have multiple local optimal solutions. To deal with the ill-posedness of the inversion, an efficient method is to carry out regularization that can make the cost function be convex. Detailed regularization techniques for inverse problem can refer to Ref. [16].
On the other hand, the theoretical derivations are based on small perturbation approach. Therefore, the adjoint process is a local optimization method and the inversion accuracy is dependent on the initial guess,[17] especially for severely nonlinear inverse problems. If the initial guess is deviated far from the true solution, the iterations will be always trapped to a local optimal solution. In practical operations, a good initial guess can be obtained from historical observations and/or empirical values.
Geoacoustic parameters are very important for underwater acoustic detection. How to acquire these parameters accurately from the perspective of adjoint PE method needs systematical research. This paper only gives the basic theoretical framework, and future work will be focused on discussing the uniqueness and stability of the solution, as well as collecting the experimental data to validate this theoretical method.
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