Analytical path-summation imaging of seismic diffractions

Introduction

Path-integral formulation provides an efficient imaging method, which does not require subjective and time-consuming velocity picking and produces an image of the subsurface without specifying a velocity model (Landa et al., 2006; Burnett et al., 2011). The concept of path-integral imaging is based on the fact that an image can be obtained from summing (stacking) of wavefields along wavefronts: the image accumulates contributions from only those events that are nearly in phase and which correspond to the true media features (Landa, 2004; Keydar, 2004). In other words, path-integral method constructs the final image by summing a set of images obtained from the range of different velocity distributions without searching for the "optimal" one.

Landa et al. (2006) and Landa (2009) propose to measure the flatness of common image gathers and use it to weight images before stacking them to perform path-integral imaging. These weights are used to enhance coherent summation of images migrated with velocities in the vicinity of the true velocity and to cancel contributions from incorrect velocities. This approach brings path-integral application for seismic imaging purposes in correspondence with path-integral formulation in quantum mechanics (Feynman and Hibbs, 1965). In this paper, we follow Burnett et al. (2011) and employ the inherent property of diffractions' behavior under time-migration transformation - apex stationarity - for time path-summation imaging implementation for diffraction imaging. We use a term "path summation" in order to highlight that no weighting is applied before stacking the images as opposed to path-integral scheme, where weights represent assumed probability of each individual image.

A path-summation image can be calculated through a summation of a set of constant velocity images, generated by velocity continuation (VC), which accounts for both lateral and vertical shifts of the event under velocity perturbation. The VC transformation is continuous and analytical: a velocity step between constant velocity images can be in theory infinitely small (Fomel, 2003b; Claerbout, 1986; Burnett and Fomel, 2011; Decker and Fomel, 2014; Fomel, 2003a,1994). Diffractions are sensitive to velocity perturbation (Novais et al., 2006): hyperbola flanks change their shape under VC transformation whereas their apexes remain stationary. Therefore, stacking constant velocity images can superimpose diffraction apexes constructively, cancel out hyperbola flanks and generate an image (Burnett et al., 2011). If the constant velocity images are weighted before stacking by their corresponding velocities, they produce a double path-summation image, which can be used for velocity estimation (Santos et al., 2016; Schleicher and Costa, 2009).

The forementioned approach for path-summation migration requires a sequence of VC steps to generate a set of constant velocity images and therefore may consume significant computational resources. Moreover, produced images appear to be contaminated by artifacts compared to the images produced by picking velocities (Burnett et al., 2011). The artifacts come from incomplete cancellation of under- and overmigrated flanks of hyperbolas. We refer to these flanks as tails. Tails might overlap with useful signal and generate spurious features in the images. Note that tails are an intrinsic feature of unweighted path-summation images, as opposed to path-integral images, where the weighting describing the probability of each image and corresponding velocity is applied before summation.

In this paper, we propose a direct analytical approach to performing path-summation and double path-summation migration as opposed to stacking of multiple constant-velocity images. Computational efficiency is improved significantly because we do not require multiple velocity continuation steps for calculation of constant velocity images but instead compute the path integral in one step at the cost of only two Fast Fourier Transforms (FFTs). To improve the image quality, we propose to apply a Gaussian weighting scheme to eliminate tails in path-summation migration images. We test the effectiveness and the robustness of the proposed workflow on synthetic and field data examples.

Analytical path-summation imaging of seismic diffractions