Prediction error filters to enhance differences

Histogram normalization

The previous section discussed a covariance-based approach to compare two volumes. In this section I will discuss a more tradition amplitude-based approach. Strict differences between the volumes is an option in some instances but often approaches have significantly different amplitude profiles. One solution is called histogram equalization.

My implementation of this approach is to first calculate the amplitude in volume `a' at several different quantiles

$$m_a(i)=Q(a,i),$$ (6)

where $m_a$ is the amplitude at a given percentile $i$ of volume $a$ using the quantile function $Q$. The vector $m_a$ is basically a discrete version of the data's cumulative distribution function (CDF). I then found the amplitude in volume `b' at the same quantiles producing the amplitude map $m_b$. Figure 5 shows the cumulative distribution function for the one and eight velocity PSPI migrations shown in the previous section. Note how the two curves are similar, diverging only at their edges. Finally I looped through volume `b', for each sample I found its approximate quantile by finding the samples of $m_b$ that contained the value and performing linear interpolation. I was able to remap into the amplitude profile of `a' using $m_a$. Figure 6 shows the difference between the one and eight reference velocity PSPI migration after histogram normalization. Note the image seems to emphasize the major reflectors of the image rather than the differences.

cdf Figure 5. The solid curve is the CDF for the eight velocity PSPI migration shown in Figure 1, the dashed curve shows the CDF using one velocity (shown in Figure 3). Note how the two curves are similar diverging only at their edges.

onevel-diff Figure 6. The difference between the one and eight reference velocity PSPI migration after histogram normalization.

Prediction error filters to enhance differences