A variational formulation for interpolation of seismic traces with derivative information

Abstract

We construct a variational formulation for the problem of interpolating seismic data in the case of missing traces. We assume that we have derivative information available at the traces. The variational problem is essentially the minimization of the integral over the smallest eigenvector of the structure tensor associated with the interpolated data. This has the physical meaning of penalizing the local presence of more than one direction in the interpolation. We show that the solution to the variational problem also satisfies an elliptic partial di↵erential equation. Moreover, this can be obtained by considering the steady state solution of a non-standard anisotropic di↵usion problem. We show existence and uniqueness for this type of anisotropic di↵usion. In particular, the uniqueness property is important as it guarantees that the unique solution to the variational problem that we constructed can be obtained by the numerical schemes we propose.

Increasing the Lateral Resolution of 3D-GPR Datasets through 2D-FFT Interpolation with Application to a Case Study of the Roman Villa of Horta da Torre (Fronteira, Portugal)

“…The use of mathematical transforms [8] and predictive filters [21] in the f-x domain [22] is one of the interpolation strategies for estimating missing traces in seismic data. We can perform the data estimation using predictive filters as autoregressive operators since linear events overlap in the f-x domain.…”

Increasing the Lateral Resolution of 3D-GPR Datasets Through 2D-FFT Interpolation. Application to the Case Study of Roman Villa of Horta Da Torre (Fronteira, Portugal)

“…The use of mathematical transforms [21] and predictive filters [22] in the f-x domain [23] is one of the interpolation strategies for recovering missing traces in seismic data. We can perform the data recovery using predictive filters as autoregressive operators since linear events overlap in the f-x domain.…”

Pwd-pinn: Slope-assisted seismic interpolation with physics-informed neural networks

“…Moreover, Fomel and Guitton (2006) introduced the so-called plane wave construction filters (i.e., inverse of PWD) to guide the solution of geophysical inverse problems to be consistent with a-priori local slopes. Similar ideas, although based on other ways of estimating slopes, have been also proposed in the literature by Hellman and Boyer (2016), Andersson et al (2015), Ramirez et al (2015) and Andersson et al (2016). In this work, we propose to tackle the problem of seismic interpolation guided by local slopes using a recent development in the field of machine learning, the so-called physics-informed neural networks (PINNs - Raissi et al 2019).…”