3D fourier reconstruction of irregularly sampled seismic Gathers : analysis of the extended model resolution matrix

Fourier reconstruction is basically a linear inverse problem that attempts to recover the Fourier spectrum of the seismic wave-field from irregularly sampled data along the spatial coordinates. The estimated Fourier coefficients are then used to reconstruct the data in a regular grid via a standard inverse Fourier transform (IDFT or IFFT). Unfortunately, this kind of inverse problem is usually under-determined and illconditioned. For this reason the Fourier reconstruction with minimum norm (FRMN) adopts a weighted damped least-squares inversion to retrieve a unique and stable solution. In this work we show how damping can introduce deleterious artifacts on the reconstructed 3D data. To quantitatively describe this issue, we introduce the concept of extended resolution matrix (ERM) and we formulate the reconstruction problem as an appraisal problem. Through the simultaneous analysis of the ERM and of the noise term, we can assess the validity of the reconstructed data and verify the possible bias introduced by the inversion process. Also, we can guide the parametrization of the forward problem to minimize the occurrence of unwanted artifacts. Real data from a 3D marine common shot gather are used to discuss our approach and to show the results of FMNR reconstruction.