THE SOLUTION OF INVERSE PROBLEMS IN UNDERWATER GEOACOUSTICS BY OPTIMAL CONTROL METHODS

Abstract- The propagation of acoustic signals in a waveguide can be efficiently modelled by a parabolic approximation of the wave equation. Small and large angle parabolic approximation based on rational Padé approximations are used together with a local and a non local (physically more realistic) impedance boundary condition on the sea floor. The inverse problem consists of computing the coefficients of the impedance condition (in order to reconstruct the acoustic field in the waveguide) from measurements taken along a vertical hydrophone array placed at a given range. The source signal (at range zero) is known, and a zero pressure boundary condition is applied on the sea surface. The optimal control method for this problem is set up as follows. First, define a least-squares cost function (usually the mismatch between the known measurements and the simulated field), constrained by the parabolic wave equation. Take the variation of the cost function with respect to the control function (that appears in the impedance boundary condition). Obtain an explicit representation of the gradient of the cost function by introducing an adjoint boundary value problem. Finally, use this exact gradient in a gradient-based optimization loop. The optimal control, thus computed, solves the inverse problem with high accuracy and is computationally inexpensive when compared to the classical signal theory approaches currently used in acoustics. We present the physical problem, the steps of the mathematical modelling and the op