Post-flattening inversion

There are two general approaches to calculating moveout parameters using the flattening methodology. The first approach is to perform parameter estimation in two phases. First, solve for the non-linear $\boldsymbol \tau$ field, then construct a linear problem to find the moveout parameters that best fit the $\boldsymbol \tau$ field.

The flattening algorithm provides a time-shift $\bf \boldsymbol \tau$ field that is function of depth z, offset h, and CRP $\bf x$. As a first test we want to estimate moveout of a volume migrated using downward continuation migration. Biondi and Symes (2003) demonstrated that residual moveout $\Delta z$can be approximated (assuming zero geologic dip) as a function of angle $\theta$ and depth z through

$$\Delta z = z \rho {\rm tan}(\theta)^2 ,$$ (7)

where $\rho$ is the moveout paremeter. We can estimate $\rho(z,\bf z)$ as a global inverse problem. Defining the above moveout equation above as $\bf B$mwe obtain the objective function Q,

$$Q(\boldsymbol \rho) = \vert\boldsymbol \tau - \bf B \boldsymbol \rho\vert^2.$$ (8)

We can ensure spatial smoothness by introducing a roughener $\bf A$to the objective function to obtain,  

$$Q(\boldsymbol \rho) = \vert\boldsymbol \tau - \bf B \boldsymbol \rho\vert^2 + \epsilon^2 \vert\bf A \boldsymbol \rho\vert^2 ,$$ (9)

where $\epsilon$ is scaling parameter.

To test the methodology I migrated a line from a 3-D North Sea dataset. Figure  displays two cross-sections of the migrated data (left) and the $\boldsymbol \tau$ field (right) calculated from the volume. A moveout field $\boldsymbol \rho$ is then calculated from the $\boldsymbol \tau$ field using a conjugate gradient algorithm to minimize equation 9. Figure  shows the resulting moveout field. The inversion approach has an additional advantage, it easy to assess where the moveout parameterization effectively described the time shifts and where it failed. Figure  shows the result of stacking the absolute value of the residual over the offset plane. Areas of high amplitude represent areas where a single parameter did not accurately describe $\boldsymbol \tau$.

 

data
Figure 1 The left panel shows three cross-sections of the migrated image (depth, inline, angle). The right panel shows the time shifts calculated from the volume.

 

rho1
Figure 2 The result of inverting for the moveout parameter $\boldsymbol \rho$ from the time shifts shown in the right panel of Figure .

 

resid1
Figure 3 The spatial error fitting error associated with the time shifts shown in Figure  and the moveout parameter shown in Figure .

Rather than solving for a single moveout parameter at each location, we can solve for multiple moveout parameters simultaneously. To test this approach I introduced a new operator $\bf C$ that estimates the moveout parameter $\boldsymbol \mu$ by searching for higher order moveout anomalies. For $\bf C$I chose an arbitrary moveout function,

$$\Delta z = \mu z {\rm tan}(\theta)^4$$ (10)

that attempts to see if a higher polynomial of the same form as $\bf C$ to help to describe the moveout. The optimization goal of equation ($\ref{bob3/eq:single}$) becomes  

$$Q(\boldsymbol \rho,\boldsymbol \mu) = \vert\boldsymbol \tau ... ...mbol \rho\vert^2 +\epsilon^2 \vert\bf B \boldsymbol \mu\vert^2.$$ (11)

Figure  shows the resulting $\boldsymbol \rho$ (left) and $\boldsymbol \mu$ (right) fields. Note how similar the $\boldsymbol \rho$ field is to the one in Figure , indicating that a two-stage estimation approach would have yielded a similar result. Figure  shows the resulting residual. Note the decrease in some areas compared to Figure , but still showing areas where the moveout is significantly more complex.

 

rho2
Figure 4 The result of inverting for both $\rho$ (left panel) and $\mu$ (right panel). Note the similarity to the single parameter estimation shown in Figure .

 

resid2 Figure 5 The fitting error associated with the two parameter fitting shown in Figure .

The methodology of this section assumed that the $\boldsymbol \tau$ field was accurate. The non-linear nature means this assumption is problematic, particularly when we are far from the correct solution. In the context of the moveout problem, this means we are far from flat the defacto starting guess.