Albert Boulanger

Lamont Doherty Earth Observatory

December 1996

*Last Revised 1/15/97*

We document our recent study of the statistical
significance of our use of region growing to extract volumetric regions
that represent reservoirs before differencing. |

Recently we have undertaken a study justifying our use of region growing from a statistical point of view. Our approach is to look at the spatial and local statistics of seismic amplitude data inside and outside the regions. To do this, we have written two new modules:

does correlograms and variograms based on GSLIB methodology. | |

does local statistics computations based on a box size. Does a generalized coherence calculation and a variance calculation. |

The equations for the variogram and the correlogram are as follows:
and the head mean is: |

Here is an
example network for the typical use of the variogram module:

Here is an example
network for the typical use of the local stats module:

We first look at the variograms and correlograms for the runsum seismic
data from a turbidite field in the GOM. First, we compute the spatial statistics
of the regions and then for all the data outside the regions. This is for
the 94 dataset:

Correlogram for W94 |
Variogram for W94 |

This analysis shows better correlations inside the regions than outside
although at very short lags the outside is very slightly higher. Below
is the correlogram and variogram for the 88 data. Here the outside is higher
till a lag of about 4. The inside Z direction has very few data points
at about a lag of 8 which contributes to its behavior there.

Correlogram for T88 |
Variogram for T88 |

Note: We use the X direction only for the rest of this study
when presenting variograms and correlograms. |

We became interested in how the variance was distributed in the data. We developed a local statistics module that generated local statistical estimates throughout the seismic volume suitable for visualization. The following shows these results.

This composite image shows the regions, a cohrence image, and a variance image. We see that the regions are surrounded by areas of high variance and low coherence. The variablity of the seismic data is highest around reservoirs. (At least in bright spot country.) Below are two animations. One has the variance fading-in on a line slice of the data. The other is a 3D perspective of the 1988 data and slices of the variance in the crossline direction. We see the high degree of correlation between areas of high variance and the boundaries of the reservoirs determined by our region growing algorithm.

From this we see that there are areas just outside the regions that should be accounted for when comparing the statistics inside and outside the regions. In order to treat this, we use an 3D erosion operation to eat away some of the volume before computing any statistical measure of variabilty outside the regions.

Correlelogram with Erosion |
Number of Points |

From the correlelogram, we see that erosion does improve the correlation of the data outside the region a little. However a drawback or erosion, is that it reduces the population outside the region as the number of points graph indicates.

Next we turn to the spatial analysis of seismic differences inside and outside our regions. The way that we use the two regions for difference analysis is we take the union of their volumes as the total area of interest. Thus, there may be areas included in this union that may not be in on or the other survey for a variety of reasons, but often due to the shrinkage of the reservoir due to production. We call this difference Union Difference. Unlike the single survey case, the differences inside the region is less correlated with higher variance than outside (using an erosion of 4 to compensate for the high variance at the region boundaries). Since the regions are areas were there are indeed seismic differences, this result is not unsurprising.

Correlogram Union Difference |
Variogram Union Difference |

Separating the Areas of Negative, Zero and Positive Change within Regions

Lamont 4D Software also identifies areas ** within regions **that
have either negative, zero or positive change. This is used to interpret
seismic differences and we wish to analyze using spatial statistics these
identified areas of change inside and outside the grown regions. Before
we apply this analysis, we need to discuss the methodology used to identify
those subparts of the regions.

Before we use union difference on the regions, we first use a straight difference of the regions with zero values set outside the regions. The purpose of this is to understand what populations inside the regions represent negative, zero and positive change. The figure below illustrates the isolation of the areas within the regions of negative, no, and positive change. The significant separations of the histograms are due to simple differencing of the two volumes after region growing. This emphasizes where a region has shrunk or grown to, since values of amplitude or impedance in locations outside the two regions have been replaced with zero values. This expands the histogram of amplitudes or impedances within the volume into three populations:

- Those voxels with high and sustained amplitudes or low and sustained impedances. We select green color for these voxels to imply bypassed amplitudes or impedances.
- Those within a region at time 1, but outside the region at time 2. These are areas of shrinkage, and we color them blue to denote dimming of amplitudes or increase of impedances.
- Those within a region at time 2, but not at time 1. These are areas of growth in the regions, and we color them red to denote brightening of amplitudes or decrease in impedances.

This histogram is then superimposed with the histogram based on union differencing, which has a single peak with the negative and positive change populations pulled in. Below is a blow up of the central peaks of these two histograms:

Animation of the subparts of the regions identified as negative (blue), and no (green) change. |

The difference in the negative parts of the central peaks represent the
the negative change population, and the threshold for separating this population
is set by eyballing. Likewise for the positive. In the case for this turbidite
field, most of the change is dimming as significant change of the negative
part of the central peaks indicate.

Our approach is to use the methodology described above for selecting the threshold values for selecting the three populations of interest within the regions to both outside and inside the regions and to use spatial statistics to compare the outside population with the inside population. For this study the cutoffs selected were -48 and 58.

Below are the correlogram, number of points, and variogram for the zero change population both inside and outside the regions. The correlation of the areas of zero change is higher outside the regions if we erode than inside. However, notice that the number of points is decreased dramatically if erosion is used. This will be a problem for the negative and positive populations as well.

Correlogram of Zeroes |
Number of Points for Zeroes |

Variogram of Zeroes |

Below are the correlogram, number of points, and variogram for the negative change population. Notice that the inside correlation is better than the outside -- eroded or not. However notice that the number of points for erosion is very small.

Correlogram of Negatives |
Number of Negative Points |

Variogram of Negatives |

Spatial Distribution of Negatives Outside of Regions |

The decrease in the negative population led to the investigation of how the negative population outside the region was distributed. The figure to the right has two panels that show this distribution. The lower panel is the grey scale amplitudes with the regions masked out and set to zero. The darker areas are areas of negative change. The panel above this is a map of the amplitudes identified as negative change. Notice that these areas tend to be concentrated near the grown regions. This is probably due to shrinkage of the reservoir where the initial extent of the reservoir was underestimated by region growing. It also could be due to left-over mis-registration, but since these areas seem symmetric these negative values are most probably reservoir shrinkage from water encroachement. (The distribution of zero changes is very even on the outside. Positive changes were concentrated but had very little data.)

Our final set of images cover the positive change population. Again the correlation inside the regions of the positives are better than outside, although there are so few datapoints of positive change within this turbidite field. We suspect that the humped correlogram and the recovery in the variogram is due to lack of points. It is known that this field has more decrease in amplitude overall with little gas production.

Correlogram of Positives |
Number of Positive Points |

Variogram of Positives |

This is our beginning on characterizing statistically how our 4D methodology
works. For me, it was a learning exercise in spatial statistics. Several
questions need to be cleared up such as the better correlations at very
short lags for the outside data in the 88 data. We have shown how high
variability surrounds our grown regions and that one needs to understand
this spatial variability before modeling the noise. We had begun this investigation
looking for a nugget effect which would give a handle on noise in the data.
There were no nuggets however. Our next planned approach is to look at
region specific SNR and Contrast to Noise (CNR) measures as outlined in
the paper, *A Comparative Analysis of Several Transformations for Enhancement
and Segmentation of Magnetic Resonance Images*, Soltanian-Zadeh, et
al. 1994 to model the noise in 4D differencing. We also will enhance the
variogram module to do cross-semivariograms and covariance in order to
look at the cross variance between the surveys.

If you have comments or suggestions, email me at *aboulanger@ldeo.columbia.edu*