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The correlation coefficient between two data sequences $a_t$ and $b_t$ is defined as
{c} = {\frac{\displaystyle \sum_t a_t\,b_t}{\displaystyle \sqrt{ \sum_t a_t^2\,\sum_t b_t^2}}}
\end{displaymath} (1)

and ranges between 1 (perfect correlation) and -1 (perfect correlation of signals with different polarity). The definition of the local similarity attribute (Fomel, 2007a) starts with the observation that the squared correlation coefficient can be represented as the product of two quantities $c^2 = p\,q$, where

p=\frac{\displaystyle \sum_t a_t\,b_t}{\displaystyle \sum_t b_t^2}

is the solution of the least-squares minimization problem
\min_p \sum_t \left(a_t - p\,b_t\right)^2\;,
\end{displaymath} (2)


q = \frac{\displaystyle \sum_t a_t\,b_t}{\displaystyle \sum_t a_t^2}

is the solution of the least-squares minimization
\min_q \sum_t \left(b_t - q\,a_t\right)^2\;.
\end{displaymath} (3)

Analogously, the local similarity $\gamma_t$ is a variable signal defined as the product of two variable signals $p_t$ and $q_t$ that are the solutions of the regularized least-squares problems
$\displaystyle \min_{p_t}
\left(\sum\nolimits_t \left(a_t - p_t\,b_t\right)^2 + R\left[p_t\right]\right)\;,$     (4)
$\displaystyle \min_{q_t}
\left(\sum\nolimits_t \left(b_t - q_t\,a_t\right)^2 + R\left[q_t\right]\right)\;,$     (5)

where $R$ is a regularization operator designed to enforce a desired behavior such as smoothness. Shaping regularization (Fomel, 2007b) provides a particularly convenient method of enforcing smoothness in iterative optimization schemes. If shaping regularization is applied iteratively with Gaussian smoothing as a shaping operator, its first iteration is equivalent to the fast local cross-correlation method of Hale (2006). Further iterations introduce relative amplitude normalization and compensate for amplitude effects on the local image similarity. Choosing the amount of regularization (smoothness of the shaping operator) affects the results. In practice, we start with strong smoothing and decrease it when the results stop changing and before they become unstable.

The application of local similarity to the time-lapse image registration problem consists of squeezing and stretching the monitor image with respect to the base image while computing the local similarity attribute. Next, we pick the strongest similarity trend from the attribute panel and apply the corresponding shift to the image.

In addition to its use for image registration, the estimated local time shift is a useful attribute by itself. Time shift analysis has been widely applied to infer reservoir compaction (Hatchell and Bourne, 2005; Janssen et al., 2006; Tura et al., 2005; Rickett et al., 2007). Since the time shift has a cumulative effect, it is helpful to compute the derivative of time shift, which can relate the time shift change to the corresponding reservoir layer. Rickett et al. (2007) define the derivative of time shift as time strain and find it to be an intuitive attribute for studying reservoir compaction.

What is the exact physical meaning of the warping function $w(t)$ that matches the monitor image $I_1(t)$ with the base image $I_0(t)$ by applying the transformation $I_1[w(t)]$? One can define the base traveltime as an integral in depth, as follows:

t = 2\,\int\limits_0^{H_0} \frac{dz}{v_0(z)}\;,
\end{displaymath} (6)

where $v_0(z)$ is the base velocity, and $H_0$ is the base depth. A similar event in the monitor image appears at time
w(t) = 2\,\int\limits_0^{H_1} \frac{dz}{v_1(z)} = \int\li...
...t+\Delta t} \frac{\hat{v}_0(\tau)}{\hat{v}_1(\tau)}\,d\tau\;,
\end{displaymath} (7)

where $H_1$ is the monitor depth, $\hat{v}_0(t)$ and $\hat{v}_1(t)$ are seismic velocities as functions of time rather than depth, and $\Delta t$ is the part of the time shift caused by the reflector movement:
\Delta t = 2\,\int\limits_{H_0}^{H_1} \frac{dz}{v_1(z)}
\end{displaymath} (8)

In a situation where the change of $\Delta t$ with $t$ can be neglected, a simple differentiation of the function $w(t)$ detected by the local similarity analysis provides an estimate of the local ratio of the velocities:

{\frac{d w}{d t}} \approx {\frac{\hat{v}_0(t)}{\hat{v}_1(t)}}\;.
\end{displaymath} (9)

If the registration is correct, the estimated velocity ratio outside of the reservoir should be close to one. One can connect the local velocity ratio to other physical attributes that are related to changes in saturation, pore pressure, or compaction.

We demonstrate the proposed procedure in the next section using several examples.

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Next: Examples Up: Fomel & Jin: Time-lapse Previous: Introduction