A probabilistic approach to seismic diffraction imaging

Introduction

Though less popular than reflection imaging for characterizing the subsurface, diffraction imaging has been gaining increasing attention (Landa, 2012). Seismic diffractions occur when a seismic wave interacts with an object on the order of its wavelength, such as a fault, fracture, or void (Decker et al., 2015; Schwarz, 2019; Moser and How-ard, 2008; Popovici et al., 2015; Klokov and Fomel, 2012; Harlan et al., 1984; Fomel et al., 2007), and may even be able to resolve objects beyond the seismic wavelength (Khaidukov et al., 2004).

Seismic diffractions are significantly weaker than reflections (Klem-Mus-atov, 1994), requiring their quarantining from the stronger reflection signal to be usable. Numerous methods exist for separating diffraction signal from reflection, in both the data and image domains (Kozlov et al., 2004; Moser and How-ard, 2008; Berkovitch et al., 2009; Fomel et al., 2007; Harlan et al., 1984). Attributes correlated to a diffraction occurring, like focusing (Khaidukov et al., 2004) or angle-gather flatness (Landa et al., 2008; Reshef and Landa, 2009) only become available after migration, so these methods typically function by predicting and removing reflections, although recent excitement in applying machine learning techniques to problems related to seismic imaging and interpretation (Kaur et al., 2019; Wu et al., 2020; Pham et al., 2019) has extended into the use of pattern recognition (de Figueiredo et al., 2013 and deep learning (Tschannen et al., 2020) for diffraction detection. Reflection removal leaves the diffractions but also the already present noise. Diffraction images often feature poor signal to noise ratios and situations where it is difficult to distinguish what features are diffraction, noise, or migration artifacts (Decker et al., 2013; Harlan et al., 1984; Fomel et al., 2007). This difficulty motivated Decker et al. (2017b) to treat the semblance of diffraction image angle gathers as a proxy for diffraction likelihood at different locations in space, and use that semblance as a model weight for least-squares Kirchhoff migration (Nemeth et al., 1999). However, in practice this involved thresholding semblance at a selected level, making the process both arbitrary and non-linear. To overcome this issue we turn to path-integral seismic imaging.

Path-integral seismic imaging (Landa et al., 2006) applies the stationarity of Feynman path integrals to the problem of seismic time imaging to create images without knowing a velocity model. Integrating (stacking) over all possible seismic signal paths, which are dependent on velocity, with an appropriate weighting function produces a seismic image based on the concept that the seismic image is stationary at the correct velocity. Thus, paths with incorrect velocity interfere destructively during summation whereas correct velocity signal interferes constructively. Schleicher and Costa (2009) utilized this concept to determine seismic migration velocity, while Burnett et al. (2011), Merzlikin and Fomel (2017), Merzlikin et al. (2019), and Merzlikin et al. (2020) applied the technique to seismic diffraction imaging.

We observe that if the weight functions used in the path-integral imaging equations are treated as probability distributions, the imaging and velocity analysis techniques become equivalent to calculating expectation values for the time image and time migration velocity respectively (Fomel and Landa, 2014). This is immediately clear when examining the imaging condition of Landa et al. (2006), which treats the imaging process as a weighted summation of images within a set corresponding to the probability of each. This observation inspires us to use attributes corresponding to diffraction probability as weight functions. In order to utilize such a framework we first need a suite of seismic images over different migration velocities and their angle gathers, which are provided by oriented velocity continuation (OVC). OVC involves the continuous propagation of slope-decomposed seismic images along their characteristics over different time migration velocities (Decker et al., 2017a; Decker and Fomel, 2014). Other continuation operators exist, for example Burnett and Fomel (2011) proposed a method for applying azimuthally anisotropic velocity continuation to zero-offset data. The slope-decomposed partial images produced by OVC are equivalent to dip-angle gathers (Decker and Klokov, 2014), enabling us to compute gather semblance and other path-integration weights. OVC provides us with an ensemble of slope-decomposed seismic images over a range of different time migration velocities. We propose to use these images to construct the weight functions for path integration.

This paper demonstrates the principles of our probabilistic imaging process and applications to demonstrate its utility. In the next section we outline how we may construct weight functions for path-integral imaging correlated to the probability of a correctly migrated diffraction occurring at a location in space for a given velocity. We illustrate our methodology on a toy model example and then apply it to a synthetic data set to illustrate its robustness to noise relative to deterministic methods for diffraction imaging and equal weight path-integral imaging. The method is applied to a field data set of the Nankai trough to show how it is able to both generate a diffraction image with suppressed noise and output a plausible velocity field that is able to highlight geologically interesting features like a velocity inversion. Finally, we end the paper with conclusions and outline some promising avenues for future work.

A probabilistic approach to seismic diffraction imaging