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In several geophysical applications, such as full waveform inversion and datamodelling, we are facing the solution of inhomogeneous Helmholtz equation. Thedifficulties of solving the Helmholtz equa- tion are two fold. Firstly, in thecase of large scale problems we cannot calculate the inverse of the Helmholtzoperator directly. Hence, iterative algorithms should be implemented. Secondly,the Helmholtz operator is non-unitary and non-diagonalizable which in turndeteriorates the performances of the iterative algorithms (especially for highwavenumbers). To overcome this issue, we need to im- plement properpreconditioners for a Krylov subspace method to solve the problem efficiently.In this paper we incorporated shifted-Laplace operators to precondition thesystem of equations and then generalized minimal residual (GMRES) method usedto solve the problem iteratively. The numerical results show the performance ofthe preconditioning operator in improving the convergence rate of the GMRESalgorithm for data modelling case. In the companion paper we discussed theapplication of preconditioned data modelling algorithm in the context offrequency domain full waveform inversion. However, the analysis of the degreeof suitability of the preconditioners in the solution of Helmholtz equation isan ongoing field of study.

Application of shifted-Laplace preconditioners for heterogenous Helmholtz equation- part 1: Data modelling