  We propose a new numerical method to solve the linearized problem of travel
time tomography with incomplete data. Our method is based on the technique of
the truncation of the Fourier series with respect to a special basis of L2.
This way we derive a boundary value problem for a system of coupled partial
diffeerential equations (PDEs) of the first order. This system is solved by the
quasi-reversibility method. Hence, the spatially dependent Fourier coefficients
of the solution to the linearized Eikonal equation are obtained. The
convergence of this method is established. Numerical results for highly noisy
data are presented.
