Method

In geophysical estimation problems, regularization is used to solve ill-posed problems by providing additional constraints on the estimated model. Shaping regularization (Fomel, 2008,2007) implies a mapping of the input model $\mathbf{m}$ to the space of acceptable functions. The mapping is controlled by the shaping operator $\mathbf{S_m}$. In the linear case, the solution of the estimation problem using shaping regularization is defined as:

$$\mathbf{\hat{m}=[I+S_m(BF-I)]^{-1}S_mBd},$$ (1)

where $\mathbf{F}$ is the forward operator, $\mathbf{B}$ is the backward operator, and $\mathbf{d}$ is the data. We implement the Generalized Minimum Residual (GMRES) algorithm (Saad and Schultz, 1986) to perform the linear inversion in equation 1.

Shaping regularization in the application to NMO stack utilizes signal from different offsets to reconstruct a high resolution stack. The model $\mathbf{m}$ is in this case a seismic trace at zero-offset and the data $\mathbf{d}$ is a CMP gather. We define the linear operators used in the shaping regularization scheme as:

• $\mathbf{F}$ (forward operator) applies predictive painting (Fomel, 2010) to spread information using a known dip field and then subsamples in time.
• $\mathbf{B}$ (backward operator) interpolates the data in time to a denser grid and stacks in a recursive fashion using local slopes.
• $\mathbf{S_m}$ (shaping operator) is a bandpass filter that controls frequency content.

We implement PWC stacking in the backward operator of shaping regularization which follows local slopes of a CMP gather. The key idea of this stacking procedure is to start at the farthest offset trace of the gather and make a local slope prediction of the preceding trace using PWC (Figure 1a). The partially corrected trace is then stacked with the uncorrected neighbor, which is the input for the next local prediction (Figure 1b). The process is repeated in the offset direction until the zero offset trace is reconstructed (Figure 1c). This recursive stacking approach results in higher resolution stacks compared to conventional NMO and stack. The procedure is equivalent to computing the zero scale of the seislet transform (Fomel and Liu, 2010). Advantages of PWC stacking include eliminating the effects of “NMO stretch" as well as the problem of non-hyperbolic moveout. The approximate inverse of PWC stacking is defined by predictive painting (Fomel, 2010). This algorithm is comprised of two main steps, namely estimating local slopes of seismic events using plane-wave destruction (PWD) (Fomel, 2002) and spreading information from a seed trace inside a volume. In this application, we use the updated model to spread information across the CMP gather using the estimated dip field.

schem5
Figure 1. Schematic of the PWC stacking algorithm. (a) Stack far offset trace T$_2$ with neighboring trace T$_1$, (b) stack updated trace T'$_1$ with neighboring trace T$_0$, (c) final accumulated stack.

In PWC stacking, each seismic trace is predicted from its neighbors that are shifted along the event slopes. Slopes are estimated by PWD, which minimizes the prediction error to estimate optimal slopes. PWD can be sensitive to conflicting slopes at far offsets of a CMP gather when the dip is large and cause PWC stacking to fail in characterizing an optimal stack. To account for this, we first apply a constant velocity NMO correction to the CMP gather, which results in smoothly varying slopes without crossing events. We then estimate the moveout $t(x)$ of the corrected seismic events at offset $x$ as follows:

$$t(x) = \sqrt{t_{0}^{2}+\frac{x^{2}}{v_{0}^{2}} + x^{2} \left(\frac{1}{v^{2}} - \frac{1}{v_{0}^{2}}\right)},$$ (2)

where $t_0$ is the zero offset travel-time, $v$ is the NMO velocity estimated by a conventional method and $v_0$ is a constant velocity. Adding the correction factor due to the constant velocity NMO correction from equation 2, the dip field becomes:

$$p={\frac{x}{t}}\left(\frac{1}{v^{2}}-\frac{1}{v_{0}^{2}}\right).$$ (3)

We use this estimated dip as the initial model for PWD. This dip estimation scheme follows the velocity-dependent formulation of the seislet transform (Liu et al., 2015) and provides us with a better estimation of the dip field for CMP gathers with large dipping events and conflicting slopes at far offsets. We implement this dip estimation method to compute the PWC stack in an iterative fashion while using shaping regularization (equation 1) to yield a high resolution stack.