P1: OTA/XYZ P2: ABC
JWST061-09 JWST061-Caers April 6, 2011 13:24 Printer Name: Yet to Come
9.2 THE CONCEPT OF DISTANCE 155
simulated on a one million cell grid and what is needed is contaminant concentration at
10 years for a groundwater well located at coordinate (x,y), then a single input model has
dimension of 3 × 10
6
while the target response is a single variable.
This simple but key observation suggests that the uncertainty represented by a set of
Earth models can be represented in a simpler way, if the purpose of these Earth mod-
els is taken into account. Indeed, many factors may affect contaminant concentration at
10 years to a varying degree of importance. If a difference in value of a single input vari-
able (porosity at location (x,y,z)) leads to a considerable difference in the target response,
then that variable is critical to the decision making process. Note that the previous state-
ment contains the notion of a “distance” (a difference of sorts). However, because Earth
models are of large dimension, complex and spatially/time varying, it may not be trivial
to discern variables that are critical to the decision making process easily. To make the
uncertainty puzzle simpler, the concept of a distance is introduced. A distance is a single,
evidently positive value that quantifies the difference between any two “objects.” In our
case the objects are two Earth models. If there are L Earth models, then a table of L × L
distances can be specified. The mathematical literature offers many distances to choose
from; a very common distance that is introduce later is the Euclidean distance (in 2D it
would be a measure of the distance between two geographic locations on a flat plane).
The choice between distances provides an opportunity to choose a distance that is related
to the response differences between models. This will allow “structuring” uncertainty
with a particular response in mind and create better insight into what uncertainty affects
the response most. Earth models can be considered as puzzle pieces: if two puzzle pieces
are deemed similar then they can be grouped and represented by some average puzzle
piece (the sky, the grass, etc.). This, however, requires a definition of similarity. This is
where the distance comes in. In making that distance a function of the desired response,
the grouping becomes effective for the decision problem or response uncertainty question
we are trying to address. For example, if contaminant transport from a source to a specific
location is the target, then a distance measuring the connectivity difference (from source
to well) between any two Earth models would be a suitable distance.
Defined in the next section are some basic concepts related to distances that allow the
uncertainty represented by a large variety of models to be analyzed very rapidly. As an
initial illustration of how a distance renders complex phenomena more simple, consider
the example in Figure 9.1. As discussed in Chapter 8, uncertainty in structural geometry
is complex and attributed to various sources. Figure 9.2 shows a few structural models
from a case study whereby a total of 400 structural Earth models are built. As discussed
in Chapter 8, a structural model consists of horizon surfaces cut by faults. To distinguish
between any two structural models, the joint difference between corresponding surfaces
in a structural model is then used as a distance, termed d
H.
Figure 9.1 explains how this is
done exactly. A surface consists of points x,y with a certain depth z (at least for surfaces
of non-overhang structures). The joint distance between these depth values z for each
surface of a model k and the same surface of a model is then a measure of the difference
in structural model. How this distance is actually calculated is not the point of discussion
for this book. Figure 9.1 shows that such difference depends on the difference in fault
structures as well as the difference in horizons (as was outlined in Chapter 8).