The surface distributions resulting from the ensemble computation have signif- icantly different properties than distributions common for ensemble data resulting from simulations or, for example, the probabilistic iso-surfaces presented by Pfaf- felmoser et al. [80, 81] or P¨othkow et al. [87, 88]. Whereas the distributions of the
Figure 4.1:Cut sections of all surfaces of one ensemble rendered into a single slice view with reduced opacity. It is clearly visible that on the left side there is close to no variation. On the right side, the surfaces spread and form several clus- ters of very similar surfaces. This behavior is very different from a Gaussian distribution, which is often assumed by probabilistic approaches. Here, how- ever, a mean surface, which would be the perfect representative for a Gaussian distribution, would lie between the clusters and would not correspond to any desired horizon surface.
latter can usually be modeled as a Gaussian distribution, each parameter setting in our approach can produce a completely different surface. In fact, typically there is very little variation as long as the surfaces tag the same horizon. In uncertain areas, however, it often happens that different parameter settings lead to surfaces tagging different horizons, which results in disconnected clusters of very similar surfaces in each cluster. This behavior is clearly visible in the example shown in Figure 4.1.
Here cut sections of all surfaces from one ensemble are blended on top of a seismic slice. The clustering is clearly visible. Thus, the visualization techniques presented by Pfaffelmoser et al. and P¨othkow et al. [80, 87] are not applicable for our ensem- bles. That is, the mean surface used in these approaches as a representative surface would not correspond to a horizon surface, since it would likely result in a surface in between two possible segmentations, but tag neither one correctly.
Instead of synthesizing a surface, we have decided to extract the maximum likelihood surface for use as the representative of the ensemble, as described in Section 4.2. A comparison of the maximum likelihood surface and a mean surface can be seen in Figure 4.2. Figure 4.2a shows an example of a maximum likelihood surface. Even though there is a somewhat large variance in the ensemble, the max- imum likelihood surface fits the underlying data quite well. In contrast, the mean surface shown in Figure 4.2b is basically a mixture of two large clusters of surfaces and does not fit either one of the tagged horizons for large parts of the surface.
Even though the median surface is typically very similar to the maximum likeli- hood surface, we prefer the maximum likelihood surface for another reason. As this surface is an actual result of the optimization, we can use it as a basis to further edit the surface, e.g. by adding constraints, without recomputing large parts of the surface.
(a)Maximum likelihood surface (b)Mean surface
seed area, clear ridge
uncertain area, optimization tries
some traces stay on wrong upper horizon, mean would be in between
(c)Cut section of the complete ensemble
Figure 4.2:Comparison of the maximum likelihood surface (a) with synthetic mean sur- face (b). The color coding indicates the difference between the amplitude at the volume position passed by the surface and the targeted amplitude. Black means a small difference (better), red a bigger difference (worse). It is clearly visible why the mean surface is not suitable in this case. Whereas the maximum likelihood surface is a good fit for the ridge line for large parts of the surface, the front part of the mean surface shows large difference values. (c) shows cut sections of all surfaces of one ensemble rendered into a single slice view with reduced opacity. The multimodal distribution causing the bad results for the mean surface shown in (b) can be seen clearly.
However, simply displaying the maximum likelihood surface itself without any additional information does not provide much information about the ensem- ble. Therefore, we also depict the results of the statistical analysis described in Sec- tion 4.2. In the standard setting, we compute what we callvirtualsurfaces from the mean, median, and maximum mode from each(x, y)-position. Virtualhere means that we do not render these as surfaces, but only use them in order to compute the distance to the maximum likelihood surface for each(x, y)-position. These dis- tances are then color-coded and used to texture the maximum likelihood surface.
If desired by the user, however, these virtual surfaces can also be rendered directly.
We allow pseudo-coloring the surface with these results, using one of several pre-defined, or user-defined color maps. These properties immediately provide a good idea about how the surface extraction behaves in different areas, i.e. very stable areas are clearly visible throughout all properties, indicated by small values in range, standard deviation, variance, close to zero values in the skewness, or very large values in the kurtosis. In addition, it is possible to automatically animate all surfaces in a pre-defined range. Animating the ensemble gives a nice impression of the parameters that result in similar surfaces, as well as of which areas in the dataset react more or less to changes in the parametrization of the cost function.
Similar surfaces or surface parts in the ensemble will result in little variation in the animation, whereas areas of large variance will show more movement and thus automatically draw the user’s attention. An example for this can be seen in Figure 4.3.
All the described techniques have in common that they can be used to visualize the complete ensemble or any user-defined subset. Using a set of sliders, the user can define a subrange for each parameter, and the statistical analysis is carried out on the fly for this range. This allows an interactive exploration of the parameter
Figure 4.3:Visualization of a horizon ensemble using the representative surface aug- mented with the variance color-coded on the left and the histogram of a se- lected position on the right.
space, which is helpful to define interesting ranges for each parameter. To allow this live exploration of the parameter space, we implemented an efficient, com- pletely GPU-based pipeline for computing the statistics as well as rendering the results, as presented in Section 4.4.
The same pipeline also allows efficient implementation of a number of other features. If desired, the user can choose to render any surface from the ensemble.
This requires no data transfer to or from the GPU, except for the ID of the surface in the ensemble to render. In addition, it is possible to automatically animate all surfaces in a predefined range. In the presented application this can be useful in two ways; As shown by Brown [12], as well as Lundstr¨om et al. [57], animation is a powerful tool for visualizing uncertainty. The user can choose to animate through all members of a single time step to get an impression of the surface distribution.
The described visualization techniques can give a very good impression of the quantitative variation in the data. Detailed information on the surface distribu- tion can be gained by animating through or manually selecting individual surfaces from the ensemble. However, it is hard to compare more than two surfaces this way. We therefore provide an additional view showing the histogram and proba-
bility distribution for a selected position. The position to investigate can be picked directly in the 3D view. To facilitate easy comparison, we color the bin correspond- ing to the current representative surface differently than the remaining bins. An example for the histogram is shown in Figure 4.3 on the right.