A variational approach to reconstruction of an initial tsunami source perturbation

Tsunamis are gravitational, i.e. gravity-controlled waves generated by a given motion of the bottom. There are different natural phenomena, such as submarine slumps, slides, volcanic explosions, earthquakes, etc. that can lead to a tsunami. This paper deals with the case where the tsunami source is an earthquake. The mathematical model studied here is based on shallow water theory, which is used extensively in tsunami modeling. The inverse problem consists of determining an unknown initial tsunami source q ( x , y ) by using measurements f m ( t ) of the height of a passing tsunami wave at a finite number of given points ( x m , y m ) , m = 1 , 2 , … , M , of the coastal area. The proposed approach is based on the weak solution theory for hyperbolic PDEs and adjoint problem method for minimization of the corresponding cost functional. The adjoint problem is defined to obtain an explicit gradient formula for the cost functional J ( q ) = ‖ A q − F ‖ 2 , F = ( f 1 , … , f M ) . Numerical algorithms are proposed for the direct as well as adjoint problems. Conjugate gradient algorithm based on explicit gradient formula is used for numerical solution of the inverse problem. Results of computational experiments presented for the synthetic noise free and random noisy data in real scale illustrate bounds of applicability of the proposed approach, also its efficiency and accuracy.