Appraisal problem in the 3D least squares Fourier seismic data reconstruction

Least-squares Fourier reconstruction is basically a discrete 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 ill-conditioned. For this reason the LS Fourier reconstruction with minimum norm (FRMN) adopts a damped least-squares inversion to retrieve a unique and stable solution. In this work we show how the damping can introduce artefacts on the reconstructed 3D data. To quantitatively describe this issue, we introduce the concept of “extended” model resolution matrix (EMRM) and we formulate the reconstruction problem as an appraisal problem. Through the simultaneous analysis of the EMRM and of the noise term, we discuss the limits of the FRMN reconstruction and we assess the validity of the reconstructed data and the possible bias introduced by the inversion process. Also, we can guide the parametrization of the forward problem to minimize the occurrence of unwanted artefacts. A simple synthetic example and real data from a 3D marine common shot gather are used to discuss our approach and to show the results of FRMN reconstruction.