Time-lapse image registration using the local similarity attribute |

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 , where

is the solution of the least-squares minimization problem

and

is the solution of the least-squares minimization

Analogously, the local similarity is a variable signal defined as the product of two variable signals and that are the solutions of the regularized least-squares problems

where 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 that
matches the monitor image with the
base image by applying the
transformation ? One can define the
base traveltime as an integral in depth, as follows:

where is the monitor depth, and are seismic velocities as functions of time rather than depth, and is the part of the time shift caused by the reflector movement:

In a situation where the change of with can be neglected,
a simple differentiation of the function
detected by the local similarity analysis provides an estimate
of the local ratio of the velocities:

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

Time-lapse image registration using the local similarity attribute |

2013-07-26