Introduction

Spreading some specific information following the local structures accurately and efficiently is important in many geophysical applications, such as seismic flattening (Wu and Hale, 2015a,b; Parks, 2010; Lomask, 2003) and horizon picking in seismic interpretation, structure oriented interpolation (Swindeman and Fomel, 2015), smoothing and denoising. The predictive painting method, proposed by Fomel (2010), is a numerical algorithm that can spread information from a seed trace to its neighbors recursively following local dips with superior computational performance. Predictive painting has been used in different applications. For examples, Fomel (2010) used it to flatten seismic common-midpoint gathers, and to pick horizons in 3D image volumes. Liu et al. (2010) applied it to generate an extended dimension of seismic images to realize structure-oriented smoothing operator for removing non-conforming noise. Casasanta and Fomel (2011) used it for CMP $\tau-p$ moveout correction and for the estimation of interval VTI parameters. Karimi and Fomel (2015), Zhang and Fomel (2016), and Shi et al. (2017) used it for image-guided well log interpolation.

All these applications are based on the assumption that the local dip estimation required by predictive painting is correct. However, when the spreading space contains faults, accurate dip estimation can be challenging due to the existence of conflicting dips. Even when the estimated dip on both sides of the fault is correct, it may not characterize the correct displacement across the fault. We propose to incorporate fault slip information to predictive painting to make it spread information across faults correctly. Fault slip can be estimated by correlating seismic reflectors on the opposite sides of a fault. Aurnhammer and Tonnies (2005) and Liang et al. (2010) proposed windowed cross-correlation methods. Hale (2013) and Wu et al. (2016) used a dynamic warping method that obviates correlation windows. In this paper, the fault slip is estimated by using a local similarity scan (Fomel, 2007a; Fomel and Jin, 2009). Local similarity scan method can estimate a relative time (or depth) shift map between two images, which is similar to that of dynamic warping method and is more accurate than that of windowed crosscorrelation method when the local shifts vary rapidly (Hale, 2013). Compared with dynamic warping, local similarity scan has the advantage of using normalized amplitudes and picking a regularized path with sub-pixel accuracy.

We propose three methods of utilizing the fault slip information in predictive painting: area partition, fault-zone replacement and unfaulting methods. The general idea of unfaulting is not new (Wu et al., 2016; Wei and Maset, 2005; Luo and Hale, 2013). We implement unfaulting of the seismic image by solving a regularized inverse problem using shaping regularization, which can help to get the desired result faster (Fomel, 2007b). In the following, we first briefly review the theory of predictive painting and describe the proposed methods. Then we apply predictive painting to horizon picking, and use several 2D benchmark examples to test the performance of these methods.