Conjugate guided gradient (CGG) method for robust inversion and its application to velocity-stack inversion

Application of the CGG method in Velocity-stack inversion

The CGG method described in the previous section can be used to any inversion problem whose required properties are the robustness to spiky noise and the parsimony of the model. In this section, the CGG method is tested on a velocity-stack inversion which is useful not only for velocity analysis but also for various data processing applications. The conventional velocity-stack is performed by summing or estimating semblance Taner and Koehler (1969) along the various hyperbolas in a CMP gather, resulting in a velocity-stack panel. Ideally a hyperbola in a CMP gather should be mapped onto a point in a velocity-stack panel. Summation along a hyperbola, or hyperbolic Radon transform (HRT), does not give such resolution. To obtain a velocity-stack panel with better resolution, Thorson and Claerbout (1985) and Hampson (1986) formulated it as an inverse problem in which the velocity domain is the unknown space. If we find an operator $\mathbf H$ that transforms a point in a model space (velocity-stack panel) $\mathbf m$ into a hyperbola in data space (CMP gather) $\mathbf d$,

$$\mathbf d = \mathbf H \mathbf m ,$$ (11)

and also find its adjoint operator $\mathbf H^T$, we can pose the velocity-stack problem as an inverse problem. The adjoint operator $\mathbf H^T$ corresponds to the velocity stacking operator for a given range of velocities(or slownesses), which generates a velocity-stack panel and can be described as

$${\mathbf m}(s,\tau) = \sum_{h=h_{min}}^{h_{max}} {\mathbf d}(h,t=\sqrt{\tau^2+h^2 s^2})$$ (12)

where ${\mathbf d(h,t)}$ denotes the common-midpoint (CMP) gather and ${\mathbf m}(s,\tau)$ denotes velocity-stack panel. The slowness-time pair $(s,\tau)$ are the coordinate axes of the velocity-stack and the offset-time pair $(h,t)$ are the coordinate axes of the CMP gather. A straightforward definition for the forward operator $\mathbf H$ is the adjoint of the operator $\mathbf H^T$ defined by Equation (12). Through the suitable definition of the inner product, $\mathbf H$ turns out to be simply the process of reverse NMO and stacking (Thorson and Claerbout, 1985):

$${\mathbf d}(h,t) = \sum_{s=s_{min}}^{s_{max}} {\mathbf m}(s,\tau = \sqrt{t^2-h^2 s^2}).$$ (13)

Inverse theory helps us to find an optimal velocity-stack panel which synthesizes a given CMP gather via the operator $\mathbf H$. The usual process is to implement the inverse as the minimization of a least-squares problem and calculate the solution by solving the normal equation:

$$\mathbf H^T \mathbf H \mathbf m = \mathbf H^T \mathbf d.$$ (14)

Since the number of equations and unknowns may be large, an iterative least-squares solver such as CG is usually preferred to solving the normal equation directly.

The least-squares solution has some attributes that may be undesirable. If the model space is overdetermined and has busty noise in data, the least-squares solutions usually will be spread over all the possible solutions. Other methods may be more useful if we desire a parsimonious representation of the solution. To obtain a more robust solution, Nichols (1994) and Trad et al. (2003) used the IRLS method for $\ell ^1$-norm minimization, and Guiiton and Symes (2003) used a quasi-Newton method called limited-memory BFGS  Goldfarb (1970); Nocedal (1980); Broyden (1969); Fletcher (1970); Shanno (1970) for Huber-norm minimization. Another possibility is the CGG method proposed in the preceding section. In the next subsections the results of the CGG method for the velocity-stack inversion are compared with the results of conventional LS method and $\ell ^1$-norm IRLS method.

Conjugate guided gradient (CGG) method for robust inversion and its application to velocity-stack inversion