A variational approach for picking optimal surfaces from semblance-like panels

Introduction

The concept of picking normal moveout (NMO) velocity from the dominant semblance trend in velocity spectral display panels was pioneered by Taner and Koehler (1969). These panels measure how well applying a parameterized operation, like the NMO correction, over a parameter sweep transform data in some metric. In the case of Taner and Koehler (1969), this meant NMO gather flatness as measured by semblance. Numerous other geophysical applications exist including dip moveout (DMO) velocity analysis (Deregowski, 1986; Yilmaz, 2001; Hale, 1984); migration velocity analysis (Decker and Fomel, 2021; Deregowski, 1990; Decker et al., 2017; Fowler, 1988; Fomel, 2003); image registration with different time shifts (Hale, 2013); data alignment by local similarity scan and similar applications (Fomel, 2009; Bader et al., 2019; Fomel, 2007a); deconvolution with dynamic frequency wavelets (Decker and Fomel, 2018); picking geobodies, faults, and seismic horizons in automatic interpretation (Yan and Wu, 2021; Wu and Fomel, 2018a,b); and determining the principal anisotropic axis in image gathers (Decker and Zhang, 2020).

Because manually selecting such trends can be a labor-intensive process, a rich tradition of research has focused on increasing the ability of computers to learn the dominant trends in semblance-like panels with reduced need for human intervention. This has involved overcoming difficulties in the picking process where artificially or anomalously high semblance values do not reflect a high-quality fit. These artificially high values are often caused by artifacts related to lateral velocity variations (Hubral and Krey, 1980) and care must be taken to avoid outputting an unphysical velocity model. Early efforts, like those of Sherwood and Poe (1972), involved interpolating a line between the largest semblance values detected, but this required interpreter supervision to avoid artifacts and unphysical velocity trends. de Bazelaire (1988) used the theory of geometric optics (Born and Wolf, 1959) to devise an ``optical stack'' method that automatically output velocity information. Doicin et al. (1994) modified the optical stack approach by applying constraints which force the method to avoid unphysical velocity selections. Alder and Brandwood (1999) expanded on the method of Doicin et al. (1994) by using three-dimensional interpolation and locally scaled regression to determine a dense three-dimensional velocity field. This work was further extended by Siliqi et al. (2003) to solve for two parameters simultaneously: velocity and an annellipticity term which can help account for the effects of dipping reflectors or seismic anisotropy. Arnaud et al. (2004) explored how these picking algorithms may be implemented in situations where the phase or amplitude of a seismic reflection changes with offset. Larner and Celis (2007) demonstrated how selective-correlation velocity analysis could be used to improve the resolution of velocity spectra and lead to more accurate picks. Research has progressed on methods for automatic multiparameter scanning (Tao et al., 2012), including the efficient application of the three-dimensional version of the Fourier integral operator butterfly algorithm (Candès et al., 2009) to the problem by Hu et al. (2015).

The approach outlined here follows a branch of inquiry originated by Toldi (1989), who introduced a method for iteratively finding the best path through semblance using a conjugate-gradient solver given a priori knowledge of the likely velocity field and enforcing penalties for large changes in velocity. Symes and Carazzone (1991) proposed applying a variational principle to invert for velocity models from seismic reflection data using differential semblance. Differential semblance is advantageous because, in the case of noiseless data, it possesses a single minima and exhibits convex behavior near that minimum, but with a reduction in velocity resolution compared to regular semblance (Symes, 1998; Mulder and ten Kroode, 2002; Li and Symes, 2007; Symes, 1999). Harlan (2001) simplified the method of Toldi (1989), replacing the a priori constraints of that method with a stiffness penalty and explicitly casting the ideal path through the semblance panel as the maximization of the variational integral:

$$\max_{v(\mathbf{x})} \int \alpha \left[v(\mathbf{x}),\mathbf{x} \right] d\mathbf{x},$$ (1)

where $v(\mathbf{x})$ is a smooth surface defining the velocity and $\alpha$ is semblance. This maximization was performed using a Gauss-Newton algorithm (Luenberger and Ye, 1984). Although Harlan (2001) proposed a framework for solving for $v(\mathbf{x})$ , a multidimensional surface, no examples were provided.

Inspired by the work of Deschamps and Cohen (2001) in virtual endoscopy, Fomel (2009) continued with the variational approach of Harlan (2001). Fomel (2009) noted that in the one-dimensional case of a velocity path through a single gather, Equation 1 could be reformulated to appear analogous to a ray-tracing equation solving for the minimal travel time of the following integral,

$$\min_{v(t)} \int_{t_{o}}^{t_f} \exp \left(- \alpha \left[v(t),t \right] \right)\sqrt{\lambda^2 + \left( \frac{dv}{dt} \right)^2}dt,$$ (2)

where $\lambda$ is a parameter that modifies the cost of changing position in $t$ relative to changing position in $v$ . Using variational methods (Gelfand and Fomin, 2000; Lanczos, 1966; Greenberg, 1978), the optimal $v(t)$ in Equation 2 is determined by solving the eikonal equation (Babich and Buldyrev, 1972; Yilmaz, 2001) for $U(t,v)$ ,

$$\left(\frac{\partial}{\partial v} U(t,v)\right)^2 + \frac{1}{\lam... ...partial t} U(t,v)\right)^2 = \exp \left(-2 \alpha \left[v(t),t \right] \right),$$ (3)

which, given a $v(t_o)$ , may be done using a finite-difference algorithm (Iserles, 1996). After determining $U(v,t)$ , the method finds $v(t)$ by tracking backward along $\nabla U$ from the $v(t_f)$ that minimizes $U(v,t_f)$ . Oscillations in $v(t)$ are dampened using shaping regularization (Fomel, 2007b).

We propose to expand the approach of Fomel (2009) to picking a multidimensional surface by minimizing a functional resembling semblance-weighted total variation regularization. This formulation enables direct use of information from spatially adjacent semblance panels to determine a continuous velocity field without explicitly enforcing smoothing. It improves upon the existing one-dimensional, gather-by-gather approach of Fomel (2009) by incorporating spatially adjacent information into the picking algorithm. The proposed approach is equivalent to a nonlinear elliptic partial differential equation, which can be challenging to solve directly. Instead, we propose finding minimizing surfaces iteratively. Because common iterative methods seek out the nearest minimizer for the velocity functional, which can be highly multimodal, successful implementation may require an accurate starting model. Additionally, gradient or steepest descent methods, which can be used for iteratively finding a minima for a functional, may require many iterations to converge.

Continuation, or graduated optimization (Chapelle et al., 2006; Hazan et al., 2016; Blake and Zisserman, 1987; Mobahi and Fisher, 2015; Xue et al., 2016; Chaudhuri and Solar-Lezama, 2011), is a method of non-convex optimization. Using this approach, local minima may be avoided by solving a series of successively more challenging, or less convex, approximations to an optimization problem. The solution of a smoother, or more convex, problem is used as the starting model for more rugose one. Increasing the convexity of a minimization problem is frequently accomplished by convolving the objective function with a Gaussian kernel (Wu, 1996). Gaussian convolution may be prohibitively expensive, but may be efficiently approximated by triangle smoothing (Claerbout, 1993).

The limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm ($\ell$ -BFGS) refers to a class of quasi-Newton schemes for accelerating the convergence of an iterative minimizer without the need for a large amount of computer memory (Li and Fukushima, 2001; Nocedal, 1980; Liu and Nocedal, 1989). $\ell$ -BFGS works by using several gradient computations to build an approximation for the Hessian of a system. Such memory considerations become essential for large problems, as the Hessian is on the order of the number of samples in the input vector squared. Using the approximate Hessian enables the algorithm to draw on information about the curvature of cost function level sets, and thus provide a step direction leading more directly to a minimum.

In the following sections, we propose an extension of Fomel (2009) for picking velocity surfaces from semblance volumes. The problem is then discretized, and a $\ell$ -BFGS algorithm is used to accelerate convergence. We then demonstrate how continuation may be applied to the picking algorithm to bypass local minima and make the algorithm behave more like a global optimizer. Finally, the automatic picking algorithm is applied to an automatic interpretation problem to illustrate its versatility and applicability beyond picking velocity surfaces from semblance scans.

A variational approach for picking optimal surfaces from semblance-like panels