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

Abstract

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). \ud 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.\ud 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

Reconstruction of seismic data based on SFISTA and curvelet transform

“…It mainly consists of the sparse transform, measurement matrix, and reconstruction algorithm. The sparse transforms that are often used include the Fourier transform (Zhang et al, 2013;Ciabarri et al, 2014), curvelet transform (Hennenfent et al, 2010;Liu et al, 2015;Zhang et al, 2017;Zhang et al, 2019;Tian and Qin, 2022), contourlet transform (Eslami and Radha, 2006), and seislet transform (Liu W et al, 2016). Because the curvelet transform undergoes multi-scale and multi-direction analysis and can perform the optimal local decomposition of seismic data (Yang et al, 2017), the curvelet transform is employed in this paper as a sparse transform, and the contourlet transform is also used in this paper for comparison analysis.…”

Achieving Robust Compressive Sensing Seismic Acquisition with a Two-Step Sampling Approach

“…This type of noise is usually represented by the surface waves that are suffering the most from the sparse aliasing sampling intervals. On the data processing side, different algorithms can be applied to mitigate the aliasing in the data, for instance, reconstruction methods based on the prediction error filter [5,6]. In addition to a long-standing dilemma of shooting a dense survey or working with aliased seismic gathers, the ideal regular seismic layout can be compromised by field deployment circumstances: obstacles and areas without access, equipment malfunction, zones with poor recording conditions, logistical constraints, and surface topography.…”

“…The field implementation of CS-based acquisition has been pioneered by [9] and further practiced by [10]. It is worth pointing out that, even before the first applications of the CS framework for the problems of seismic data interpolation, a vast amount of techniques for data reconstruction and dealiasing of spatial wavenumbers has been developed [6]. These techniques are being continuously improved [11].…”