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Comparing training-image based algorithms using an
analysis of distance
Xiaojin Tan1, Pejman Tahmasebi1, and Jef Caers1 1 Department of Energy Resources Engineering, Stanford University, USA;
Abstract
As additional multiple-point statistical (MPS) algorithms are developed, there is an increased need for scientific ways for comparison beyond the usual visual comparison or simple metrics, such as connectivity measures. In this paper we start from the general observation that any (not just MPS) geostatistical simulation algorithm represents two types of variability: 1) the within-realization variability, namely, that realizations reproduce a spatial continuity model (variogram, Boolean or training-image based), 2) the between-realization variability representing a model of spatial uncertainty. In this paper, we argue that any comparison of algorithms needs, at a minimum, to be based on these two randomizations. In fact, for the MPS algorithms, we illustrate, with examples, that there is often a trade-off: increased pattern reproduction entails reduced spatial uncertainty. In this paper we make the subjective choice that the best algorithm maximizes pattern reproduction while at the same time maximizes spatial uncertainty. In order to render these fundamental principles quantitative, we rely on a distance-based measure for both within-realization variability (pattern reproduction) and between-realization variability (spatial uncertainty). To compare any two (or more) algorithms, we first generate a set of realizations with each algorithm. Each realization is up-gridded into a set of successively coarser grids; a single realization is turned into a multi-resolution pyramid of grids. The same operation is performed on the training image. We calculate for each realization the Jensen-Shannon divergence (a distance) between the pattern histograms of each multi-resolution grid of the realizations generated with both algorithms and the training image. A weighted average of the Jensen-Shannon divergence of all multi-resolution grids represents a single quantitative measure of the within-realization variability (pattern reproduction) of a given algorithm. The same distance, but now calculated and averaged between sets of realizations of both algorithms is then a single quantitative measure of the between-realization variability. We illustrate in this paper that this method is efficient and effective for 2D, 3D, continuous and discrete training images. Multiple-point statistics, spatial uncertainty, training images, pattern modeling
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Introduction
Any presumed novel contribution in science requires evidence on a simple question: has the
proposed methodology improved on the state-of-the-art? Specifically to uncertainty and
stochastic modeling, such proof is neither always evident nor trivial. If we make any
probabilistic assessment of an event certain to happen or not, e.g. it will rain tomorrow or not,
then, there is no trivial objectifiable manner, without additional assumptions, typically relating
to stationarity, in assessing the correctness of that probabilistic assessment. In complex higher-
dimensional stochastic modeling, such as geostatistical simulation, the same question arises.
The geostatistical literature contains many probabilistic (or other) models of spatial uncertainty
through stochastic simulation. In this paper we study algorithms that rely on a training image as
source for specifying spatial continuity. More specifically, we investigate the following question:
given k algorithms plus a training image deemed relevant for the spatial phenomenon to be
modeled: how would one rank these algorithms?
We immediately need to acknowledge that such question cannot be addressed in a fully
objective manner. We will need to decide on some measure of “best” and such decision can be
questioned. However, what we do argue in this paper is that such measure should, at a
minimum, be based on two concepts that are often counteracting each other: 1) the
reproduction of the statistics deemed relevant for the field and 2) the variability between the
generated realizations (also termed the space of uncertainty). Specifically to geostatistics with
training images, the question of performance of the algorithms is important and remains
unaddressed: how do we measure progress in the development of new algorithms now that
many researchers enter this field each with their own approaches? Some papers rely on visual
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comparisons, “it looks better”; others rely on some summary statistics that are checked against
the training image (Soleng et al., 2006; De Iaco & Maggio, 2011). However the summary
statistics addresses one issue only, reproduction of statistics, and ignores the question related
to space of uncertainty.
In this paper, we propose a practical methodology that allows ranking training image-based
geostatistical algorithms regardless of the nature of the training image, namely, whether they
are discrete or continuous, two or three dimensional, have small or large grid dimensions. Our
methodology relies on the following definition of best: an algorithm A is better than an
algorithm B if the training image statistics are reproduced better while at the same time the
space of uncertainty (the variability between realizations) is larger. To make this quantitative
we rely on the concept of distances between realizations and distances between the statistics
of a realization and the target training image statistics. We propose a ratio of these two
counteracting distances as a measure of performance and rank algorithms on that basis. We
provide various examples of comparisons of some MPS algorithms.
Methodology
Why distance and not covariance?
The proposed methodology heavily relies on distances as a concept for variation, not on
variance or covariance, hence this choice, often advocated by the authors (see Suzuki and
Caers, 2008; Scheidt and Caers, 2009; Caers, 2011) requires some motivation. Given a set of L
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random vectors x, for example a set/ensemble of L geostatistical realizations of size N, typically
L<<N, gathered in a data matrix X of size N×L, then the centered ensemble covariance is simply
1 1
,T TC HX HX H I
L L 11 C is of size N×N, H is a centering matrix (1)
We will use capital italic for matrix and bold for vectors. The dimensions of these covariances
grow rapidly (N), hence it is often preferable to work with dot-products (Shawe-Taylor and
Cristianini, 2004), calculated from the same information as
( )TD HX HX of size L×L (2)
Clearly for all cases of practical interest in geostatistics we have: L<<N. More importantly, the
dot-product is related to the Euclidean distance, namely, each entry dij in D can be calculated as
function of a Euclidean distance (Borg & Lingoes, J. 1987)
2 2 21
2ij ki kj ijd e e e (3)
with eij the Euclidean distance between two realizations xi and xj; ekj is then the Euclidean
distance between realization xk and xj . Dot-products and Euclidean distances have simple
extensions to non-Euclidean spaces such as the Manhattan distance, Minkovski distance,
Hausdorff distance and many others. Kernels (Schölkopf & Smola, 2002; Shawe-Taylor and
Cristianini, 2004) are used to transform dot-products into new dot-products that separate the
space of X better. This is termed the “kernel trick” in the computer science literature and
commonly used for non-linear regression and classification problems. There are no trivial non-
Euclidean extensions of covariance or variance, or, such extensions do not address the
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dimensionality problem (size of N) of the covariance matrix, which is still a measure of linear
relationship/variation. Higher order cumulants (Dimitrakopoulos et al., 2010) may capture more
complex variation, but their dimensionality explodes even faster than covariances. The same
observation can be made for “cliques” in a Markov random field setting (Tjelmeland & Besag,
2001).
Hence, while the subsequent methodology could be presented in terms of variances and
covariances (typically termed “analysis of variance” or ANOVA), it would neither have the
practicality nor the effectiveness of the same methodology presented using an “analysis of
distance” (which we will term: ANODI).
A multi-resolution pyramid of realizations and training image
In this paper we consider that some target statistics are presented in an exhaustive training
image, which is common to most multiple-point geostatistical modeling. We also recognize that
statistical variation occurs at multiple scales; hence, any comparison methodology should
include differences in such variation. For this purpose, we create, from any single realization a
pyramid of multiple resolution views (similar to Heeger & Bergen, 1999) of the same realization,
see for example in Figure 1.
The typical approach in MPS is to use a multi-grid approach by subsampling finer grids into a
coarser grid which was introduced in the snesim algorithm and needed for practical
applications (Strebelle, 2002; Caers et al., 2003). However, in doing so, finer scale variability is
simply lost at coarses scales. Instead, we rely on a multi-resolution method first proposed in
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MPS by Honarkhah (2011). Figure 1 shows a result for a simple training image. Instead of
subsampling the image, we interpolate the coarse grid values from the finest scale resolution.
This interpolation relies on bi-cubic interpolation in 2D and tri-cubic interpolation in 3D (Lekien
& Marsden, 2005). In a general context, tri-cubic interpolation is used to obtain values at
arbitrary locations from a gridded data set. In 3D, the interpolated value v at any arbitrary
location (x,y,z) is obtained from the 64 nearest neighbor gridded values at locations( , , )i j kx y zu u u
as follows
3 3 3
0 0 0
( , , )i j kijk x y z
i j k
v x y z a u u u
The coefficients are obtained by solving a set of 64 equations (see Lekien & Marsden, 2005 for
details). In case of binary variables, the interpolated value is no longer binary. In that case we
use Otsu’s thresholding method (Otsu, 1979) to turn the continuous valued tri-cubic
interpolations back to binary variables (categorical values are basically a vector of binary
indictor variables).
In terms of mathematical notation this operation is summarized as follows. We consider a
training image as ti that expands into a set of multi-resolution grids g=1,…,G as
expand1 2{ , , , }G ti ti ti ti ti
(4)
Each realization =1,…,L generated with algorithm k can be expanded likewise
expand( ) ( ) ( ) ( ) ( ),1 ,2 ,{ , , , }k k k k k G re re re re re
(5)
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Should realization and training image be created on different grids then G is the maximum
attainable coarse resolution common to both grids.
Summarizing pattern frequencies by means of clustering
We now consider the multi-resolution view of a training image or geostatistical realization and
attempt to summarize its statistics. We propose two ideas: 1) relying on multiple-point
histograms (MPH, see for example Deutsch and Gringarten, 2000; Lange et al., 2012), which
only works for small cases and binary variables and 2) relying on grouping patterns into clusters
and therefrom deducing a cluster-based histograms of patterns (CHP).
A multiple-point histogram (MPH) records for binary realizations, within a given fixed template,
every possible pattern configuration and creates a frequency table of these patterns (see for
example Lange et al., 2012). To create a meaningful table, one has to use a small template (for
example 4 × 4 and for 3D to 3 × 3 × 2), otherwise each bin would contain either one or zero
patterns. One then calculates an MPH for each grid in the pyramid. In terms of mathematical
notation this operation can be summarized as follows
summarize1 2 1 2{ , , , } { ( ), ( ), , )}G Gti ti ti MPH ti MPH ti MPH(ti
(6)
where each MPH vector consists of counts MPHi; hence for the training image at multi-
resolution g we have the notation,
,1( ) { ( ), , ( )}pat gg g n gMPH MPHMPH ti ti ti
(7)
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with npat,g the number of different patterns for that resolution g. MPHs only apply to small cases
(usually 2D) and binary variables because the frequency table explodes when going to 3D or
dealing with multi-category and continuous variables. We will use it in this paper therefore as a
reference of comparison with the methodology presented next, which is applicable to any case.
In an alternative approach, we cluster patterns into groups using the methodology described in
Honarkhah and Caers (2010); this methodology works for 3D and continuous values. For each
cluster, we record the number of patterns and calculate the prototype, which is a
representative of each cluster; the prototype could be a mean of the patterns within that
cluster or a medoid pattern. Unlike in the MPH approach, the template size for each grid g is
determined automatically using a so-called elbow plot (see Honarkhah and Caers, 2010), which
determines a template size relevant to the pattern variation in the image. We repeat this
exercise for all multi-resolution grids in the pyramid for which this is possible. We do the same
for all realizations. However, for the realizations, the clustering is slightly different. Since we
want to make sure that the clustering results between training image and realization are
comparable for each multi-resolution grid, we cluster the realization patterns by calculating
distances with the training image prototypes and simply assign the pattern to the closest
prototype. In this fashion, we obtain the same number of groups/clusters with the same
prototypes for realizations and training image. In the example section, we show that such
clustering leads to similar results as using the full multiple-point histogram described above in
case of a simple 2D binary example.
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In terms of mathematical notation the above operations can be summarized as follows. Each
pyramid is now summarized as a list of what we will term cluster-based histograms of patterns
CHP, itself a list of frequencies:
summarize1 2 1 2{ , , , } { ( ), ( ), , ( )}G Gti ti ti CHP ti CHP ti CHP ti (8)
and similarly for
summarize( ) ( ) ( ) ( ) ( ) ( ),1 ,2 , ,1 ,2 ,{ , , , } { ( ), ( ), , ( )}k k k G k k k Gre re re CHP re CHP re CHP re
(9)
Each CHP vector consists of frequencies; hence for the training image at multi-resolution g we
have the notation,
1( ) { ( ), , ( ), , ( )}g gg g c g C gCHP CHP CHPCHP ti ti ti ti
(10)
with Cg the number of clusters for the multi-resolution grid g.
Calculating distances
As stated in the introduction, two distances need to be calculated 1) a distance summarizing
pattern reproduction and 2) a distance summarizing space of uncertainty. We consider the
former distance first. The distance will need to summarize the total difference between CHPs
of realizations generated with algorithm k and training image. To do so, we propose a statistical
measure of distance termed the Jensen-Shannon divergence (Cover and Thomas, 1991; Endres
and Schindelin, 2003) which for two frequency distributions pi and qi is the average of two
Kullback-Leibler divergences
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1 1( , ) log log
2 2i i
JS i ii ii i
p qd p q
q p
p q
(11)
Other statistical measures of difference that have been used in this context (Lange et al., 2012)
are the chi-squared distance, but such distance is not robust in the context of CHPs whose
frequencies vary widely. If we use the CHPs, then in terms of a single multi-resolution grid g,
the JS distance between the th realization and training image is
( ),( ) ( )
, , ( )1 1 ,
( ) ( )1 1( , ) ( )log ( )log
2 ( ) 2 ( )
g g
g g
g g
g gg g
C Cc k g c g
JS g k c k g c gc cc g c k g
CHP CHPd CHP CHP
CHP CHP
re ti
re ti re titi re
(12)
cg is a counter for the number of clusters Cg obtained for each multi-resolution grid g. The same
formula is applied to calculate differences between realizations and ' for each multi-
resolution grid g, in notation, the distances ( ) ( '), ( , )JS g k kd re re . The same Eq. (11) can be used to
calculate differences in MPH.
For each multi-resolution grid g, given L realizations of each algorithm k and a training image,
we can calculate k distance tables of (L+1)×(L+1) (k is the number of algorithms for
comparison). Multi-dimensional scaling (Borg & Groenen, 1997; Caers, 2011) can be used to
visualize the various differences. This visual appreciation may already reveal significant
differences between the algorithms and should be used as supporting visual evidence. It may
also reveal how algorithms compare as the resolutions change.
The between realization variability (space of uncertainty) is now calculated as an average over
all realizations per each multi-resolution grid g:
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, ,
1 '
'
1
1( , )
( 1)
L Lbetweeng k JS g k kd d
L L
re re
(13)
The within-realization variability, namely how well it reproduces the target statistics, has then
the following average per each multi-resolution grid g:
( ), ,
1
1( , )
Lwithing k JS g kd d
L
re ti (14)
Ranking algorithm performance with ANODI
As stated in the introduction, we would like to obtain a ranking of the algorithms based on the
distances calculated above. We emphasize the idea of ranking, meaning that only a relative
ordering is needed, and no absolute quantification of performance is required. This motivates
our decision to use a ratio of distances between two algorithms k and m, namely, for the
between-realization variability we use the ratio,
,
,
betweeng k
betweeng m
d
d (15)
The final value (as a ratio) that quantifies the between-realization differences between two
algorithms k and m is a weighted average per multi-resolution grid as follows:
,
,1 ,
betweenGg kbetween
k m g betweeng g m
dr w
d
(16)
We decided for the following fixed weights:
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1
2g g
w (17)
meaning that higher resolution grids get more weight than lower resolution ones. This choice is
based on two principles, namely 1) lower resolution grids contain less information and have less
variability than higher resolution grids in the same pyramid, and 2) shorter scale patterns are
more critical than larger scale patterns. For example, the channel disconnection seen in the
bottom right of Figure 1 represents a short scale pattern.
For the within-realization variability, we use the same idea namely
,
,1 ,
withinGg kwithin
k m g withing g m
dr w
d
(18)
The ratio of both
,,
,
betweenk mtotal
k m withink m
rr
r
(19)
can then be used to rank algorithms; the best algorithm has the largest ratio compared to all
other algorithms.
Application of the methodology
A 2D binary example
We first illustrate various elements of the methodology in considerable detail using a simple 2D
channelized training image (Strebelle’s training image, Strebelle (2002)) of size 101×101 (Figure
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2). Three algorithms are considered: dispat (a distance-based pattern simulation, Honarkhah
and Caers, 2010), ccsim (cross-correlation-based pattern simulation, Tahmasebi et al. 2012;
Tahmasebi and Caers, this issue) and also sisim (sequential indicator simulation implemented in
SGEMS, Remy et al., 2009) where the variogram is calculated from the training image, see
Figure 2. The latter case is included to illustrate the difference between multi-point and two-
point algorithms in terms of space of uncertainty. 50 unconditional realizations are generated
with each algorithm. ccsim is a pattern-based method that is an improvement on simpat,
although only qualitative visual appreciations were made in Tahmasebi et al (2012). Hence, in
this paper we aim to demonstrate this improvement quantitatively. An example of a multi-
resolution pyramid of a single realization is shown in Figure 1. These pyramids are constructed
for all realizations as well as the training image. We will present the details of the results for
both the MPH, which is feasible for this simple case, and CHP approach.
Using a 4 × 4 template we create the MPHs of Eq. (6) for each multi-resolution grid up to g=10.
Since we have 50 realizations of each algorithm and 1 training image, we can construct a table
of 151 × 151 JS divergences. Multi-dimensional scaling (MDS) is used to visualize these
distances (and hence realizations vs training image), see Figure 3. The 151 × 151 Euclidean
distances between the points in Figure 3 approximate the 151 × 151 JS divergence distances.
MDS relies on eigenvalue decomposition; the x-axis in Figure 3 refers to the largest eigenvalue,
the y-axis to the second largest eigenvalue. The units on these axes are not important, what
matters are relative distances between these points. However, we do plot the percentage of
contribution of the each eigenvalue to the total sum of all eigenvalues. A large contribution in
these first two eigenvalues means that higher dimensions (3, 4, 5,…) contain only a small
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percentage of the total variance (or distance). Note that these plots are only used for
visualization, we always use the actual JS-divergences in the calculations of ration in Eq. (19).
From this visual appreciation, we observe that dispat and ccsim reproduce the statistics of the
training image better than sisim (as expected) and that among the MPS algorithms ccsim seems
to outperform dispat. We also notice that the cloud of points for sisim is larger than for dispat
and ccsim, suggesting a larger space of uncertainty. Note however that this MDS plot is just for
g=1. If for example the same plot is made for multi-resolution g=6 than the differences are
much less pronounced, meaning that at least for that coarser resolution, all three algorithms
seem to have similar characteristics, see Figure 4.
Consider now repeating the same analysis, but using CHPs instead of MPHs. In such case, a CHP
is created for each multi-resolution grid g for each realization as well as the training image. To
create such CHP we use a template (the template size obtained is 20 × 20) to scan each
resolution of the training image for patterns. These patterns are then clustered in groups using
a kernel k-means clustering (see Honarkhah and Caers, 2010) for details. A class representative,
termed prototype is calculated. These prototypes are then used to cluster the patterns of the
realizations of the various algorithms, simply by assigning each pattern to the closest prototype.
In Figure 5, 48 clusters are obtained for the first resolution g=1 of the training image. Above
each prototype are the different counts of patterns within that cluster for the training image, a
single dispat, ccsim and sisim realization.
We then calculate JS-divergence distances given these CHPs. Consider for example multi-
resolution g=1, i.e. the original realizations and training image. We again create a table of
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151×151 JS-divergence distances and MDS is used to visualize the distances, see Figure 6. Figure
7 provides a plot for g=3. In the CHP approach, we use less multi-resolution grids since the
template determined by the elbow plot is large (20 × 20) as compared to the fixed template
used by MPH (4 × 4).
Eq. (13) now calls for the calculation of an average of between-realization distances for each
algorithm k for each multiple grid g. We again consider using the MPH first. The result for MPH
is summarized in Table 1. From these distances we calculate the ratios in Eq. (15). These ratios
are then weighted according to Eq. (16). Table 2A presents the matrix representing Eq (16).
Note that these ratios compare spaces of uncertainty. We confirm what we suspected based on
the MDS plots: sisim has the largest space of uncertainty, while dispat has a slightly larger space
of uncertainty than ccsim. We now turn to the within-realization variability, meaning the
reproduction of statistics. Table 3 summarized the distances of Eq. (14). Table 2B summarizes
the ratios. Note that the algorithm having the best pattern reproduction is ccsim, significantly
better than dispat, confirming our visual appreciation from the MDS plots as well as
realizations. Table 2C compares the final ratio of ratios, namely Eq. (19), from which it can be
deduced that ccsim is the overly better algorithm with a quantitative rank of
1 : 0.70 : 0.46 (ccsim : dispat : sisim),
if we use “1” for the best algorithm. We will use this notation from now for all further
comparisons.
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Now consider the same calculations but based on the CHP. Tables 4, 5 and 6 summarize the
distances and ratios for the CHP approach. In the case of using CHPs, we find the following
qualitative ranking
1 : 0.67 : 0.35 (ccsim : dispat : sisim),
In other words, for this example, the clustering did not affect the ranking, only how much
better or worse each algorithm faired in our comparison.
Trading uncertainty with pattern reproduction
In this example, we keep the same Strebelle training image, but study quantitatively the impact
of parameter choices for one single algorithm, namely snesim (single normal equation
simulation, Strebelle, 2002). A key parameter in this algorithm is the size of the search
template, expressed in one parameter ns. A larger ns will lead to searching for a larger data-
event near the location being simulated, and hence a more complete multiple-point statistics
than smaller data events. However, at some point, due to the limited size of the training image
(and hence the limited amount of unique multiple-point statistics) any increase in this
parameter will not improve the results any further. We use the snesim implementation in S-
GEMS (Remy et al., 2009, parameter is termed “# nodes in search template”) and run the
snesim algorithm for three choices of ns, namely 10, 50 and 200, generating 50 realizations
each.
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Two realizations with each value of ns are shown in Figure 8. Figure 9 shows the MDS plot for
g=1. The ratios for pattern reproduction, space of and uncertainty as well as total ratio are
Space of uncertainty (“between”): 1.65 : 1 : 1 (ns=10 : ns=50 : ns=200)
Pattern reproduction (“within”): 2.75 : 1 : 1 (ns=10 : ns=50 : ns=200)
Total (“between/within”): 0.60 : 1 : 1 (ns=10 : ns=50 : ns=200)
Clearly going from ns=50 to ns=200 does not constitute an improvement. It is also clear there is
a trade-off between space of uncertainty and pattern reproduction. Using the same (limited) TI,
a better pattern reproduction leads to a smaller variation between the realizations (going from
ns=10 to ns=50). We therefore conjecture that any post-processing of snesim realizations
(Strebelle & Remy, 2005) would lead to a further reduction in uncertainty represented by the
post-processed realizations.
Continuous-valued 2D cases
A continuous training image, shown in Figure 10 of size 130 x 100 is used. This training image is
sampled from a multi-Gaussian model with an isotropic Gaussian variogram with range equal to
20. The aim here is to test how the methodology can be used to assess cases with continuous
training images. In this section we compare two algorithms: ccsim and sgsim.
50 realizations were generated with each algorithm. For sgsim, we model the variogram from
the training image. Some realizations are shown in Figure 10. QQ-plots (not shown) show
excellent reproduction of the marginal Gaussian distribution, while the ergodic fluctuations of
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the variograms are quite similar for both methods, see Figure 12. Figure 11 shows an MDS plot
based on the JS-divergence distances. We notice a wider spread for the sgsim realizations, at
least for multi-resolutions g=1. One also notices that the ccsim realizations look visually
different from the sgsim realizations yet have very similar variogram reproduction, see Figure
12.
In terms of ratios we obtain the following result
Space of uncertainty (“between”): 1.48 : 1 (sgsim : ccsim)
Pattern reproduction (“within”): 1.10 : 1 (sgsim : ccsim)
Total (“between/within”): 1.24 : 1 (sgsim : ccsim)
Clearly sgsim generates a larger space of uncertainty than ccsim but their performance in
reproducing patterns of the given training image is similar. It should be understood that the aim
of both algorithms is very different: ccsim tries to reproduce the patterns of a limited size
training image, while sgsim attempts to sample a multi-Gaussian model with a given
covariance. The limited size of the training image results in a small pattern database resulting in
a smaller space of uncertainty.
3D binary case
Finally, we demonstrate a case with a 3D binary training image, see Figure 13. The training
image is a binary sand/shale conceptual model with the dimensions of 69×69×39. Two MPS
algorithms, ccsim and dispat, are used for comparison. Figure 13 depicts the projection of the
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constructed metric space by means of multi-dimensional scaling using JS divergence when
multi-resolution g=1. The comparison in terms of ratios confirms the above findings on the
relative ranking between ccsim and dispat
Space of uncertainty (“between”): 1.14 : 1 (dispat : ccsim)
Pattern reproduction (“within”): 1.90 : 1 (dispat : ccsim)
Total (“between/within”): 0.60 : 1 (dispat : ccsim)
Conclusions
In this paper we provide a methodology for comparing the performance of training image-
based geostatistical algorithms in the case of unconditional simulations. Our methodology is
based on the observation that in generating geostatistical realizations two types of variability
are created: variability within the realizations and variability between the realizations. We
establish a distance-based approach to measure the trade-off between these two variabilities.
While one can debate the particular subjective choices made in the presented comparison
methodology, the variety of examples illustrate that the results obtained from this
methodology are consistent with expectations.
It should be understood that we are not researching the following question: which are the best
statistics to reproduce? In other words, we do not investigate the choice of training image (or
training images). Such choice is important and in making such choice, inconsistencies with
respect to local data may be present. We also limit ourselves to unconditional simulation. In the
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case of conditional simulation, the space of uncertainty will evidently be affected by
conditioning data. Hence, one would need to investigate the “conditioning performance” of the
algorithm. For example, in Caers (2012) it is argued that such performance needs to be
assessed relative to a rejection sampler. Future research will focus on integrating this additional
“conditioning” component into the framework proposed in this paper, possibly by adapting a
weighting in Eq. (17) that favors short scale pattern variability.
Acknowledgements
We appreciate the discussions on distances derived from frequency tables with Katrine Lange
of the Technical University of Denmark during her visit at Stanford University.
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References
Borg, I., Lingoes, J. 1987. Multidimensional Similarity Structure Analysis. Springer-Verlag, New York. Borg, I., Groenen, P. 1997. Modern multidimensional scaling: theory and applications. New-York, Springer. Caers J., Strebelle S., Payrazyan K., 2003. Stochastic integration of seismic data and geologic scenarios: A West Africa submarine channel saga. The Leading Edge (Tulsa, OK), 22 (3) , pp. 192-196. Caers, J., 2011. Modeling Uncertainty in the Earth Sciences. Wiley-Blackwell, 246p. Caers, J., 2012. On internal consistency, conditioning and models of uncertainty. In: Ninth International Geostatistics Congress, Oslo, Norway June 11 – 15, 2012. Chiles , J.P. and Delfiner P., 1999. Geostatistics: Modeling Spatial Uncertainty (Wiley Series in Probability and Statistics). Cover, T. M., Thomas, J. A., 1991. Elements of information theory. Wiley-Interscience, New York, NY, USA. Deutsch, C., Gringarten, E., 2000. Accounting for multiple-point continuity in geostatistical modeling. In: of Southern Africa, G. A. (Ed.), 6th International Geostatistics Congress. Vol. 1. Cape Town, South Africa, pp. 156-165. Dimitrakopoulos, R., Mustapha, H., & Gloaguen, E., 2010. High-order statistics of spatial random fields: exploring spatial cumulants for modeling complex non-Gaussian and non-linear phenomena. Mathematical Geosciences, 42(1), 65-99. Endres, D. M. & Schindelin, J. E., 2003. A new metric for probability distributions. IEEE Transactions on Information Theory 49 (7), 1858-1860. Goovaerts, P., 1997. Geostatistics for Natural Resources Evaluation (Applied Geostatistics Series), Cambridge University Press. Heeger, D. J., & Bergen, J. R. (1995, September). Pyramid-based texture analysis/synthesis. In Proceedings of the 22nd annual conference on Computer graphics and interactive techniques (pp. 229-238). ACM. Honarkhah, M. & Caers, J. 2010. Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling, Mathematical Geosciences, 42: 487-517.
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Honarkhah, M., 2011. Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling, PhD dissertation, Stanford University, USA. (https://pangea.stanford.edu/ERE/pdf/pereports/PhD/Honarkhah2011.pdf) Honarkhah, M. & Caers, J, 2012. Direct Pattern-Based Simulation of Non-stationary Geostatistical Models. Math Geosci (2012) 44:651–672. De Iaco, S., & Maggio, S. (2011). Validation techniques for geological patterns simulations based on variogram and multiple-point statistics. Mathematical Geosciences, 43(4), 483-500. Lange, K., Frydendall, J., Cordua, K. S., Hansen, T. M., Melnikova, Y., & Mosegaard, K., 2012. A Frequency Matching Method: Solving Inverse Problems by Use of Geologically Realistic Prior Information. Mathematical Geosciences, available online. Lekien, F., & Marsden, J. (2005). Tricubic interpolation in three dimensions. International Journal for Numerical Methods in Engineering, 63(3), 455-471. Otsu, N., 1979. A threshold selection method from gray level histograms. IEEE Trans. Systems, Man and Cybernetics 9, 62-66. Remy, N., Boucher, A., and Wu, J., 2009. Applied Geostatistics with SGEMS: A User's Guide, Cambridge University Press: New York, NY. Shawe-Taylor, J. & Cristianini, N., 2004. Kernel Methods for Pattern Analysis, Cambridge University Press, 2004. Scheidt, C., & Caers, J., 2009. Representing spatial uncertainty using distances and kernels. Mathematical Geosciences, 41(4), 397-419. Schölkopf, B. and Smola, A.J., 2002. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press. Soleng, H. H. Syversveen, H.H. and Kolbjørnsen, O., 2006. Comparing Facies Realizations: defining metrices on realization space. In Ecmor X. Proceedings of the 10th European Conference in the Mathematics of Oil Recovery, pages A014. European Association of Geoscientists & Engineers. 2006 Strebelle, S., 2002. Conditional simulation of complex geological structures using multiple-point statistics. Mathematical Geology 34 (1), 1-22. Strebelle S. & Remy, N., 2005. Post-processing of Multiple-point Geostatistical Models to Improve Reproduction of Training Patterns Geostatistics Banff 2004. Quantitative Geology and Geostatistics, Volume 14, 5, 979-988.
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Suzuki, S., & Caers, J., 2008. A distance-based prior model parameterization for constraining solutions of spatial inverse problems. Mathematical Geosciences, 40(4), 445-469. Tahmasebi, P., Hezarkhani, A., & Sahimi, M., 2012. Multiple-point geostatistical modeling based on the cross-correlation functions. Computational Geosciences, 16:779–797. Tahmasebi, P. & Caers, J. (this issue). Fast stochastic pattern simulation using multi-scale search. Mathematical Geosciences, submitted. Tjelmeland, H., & Besag, J., 2001. Markov Random Fields with Higher‐order Interactions. Scandinavian Journal of Statistics, 25(3), 415-433.
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Figure & Table captions
Figure 1: a single binary realization is decomposed into a pyramid of realizations at different resolution. Figure 2: 2D binary case: Training image, two realizations generated with dispat, ccsim and sisim. The variogram is calculated from the training image (x and y direction variogram shown here). Figure 3: MPH approach: 2D projection of realizations using multi-dimensional scaling (MDS). The distance is the JS-divergence for multi-resolution g=1 using MPHs. The axis are omitted (as typical for MDS plots, see Caers, 2011) since only the relative scatter of points is relevant. We also report the contribution, in %, of each eigenvalue to the total sum of all eigenvalues. Figure 4: MPH approach: multi-resolution g=6: selected realizations and MDS plot. Figure 5: list of 48 prototypes. Above each prototype is listed the number of patterns for the cluster for which the prototype is representative in the following order: training image – ccsim realization – a dispat realization – sisim realization Figure 6: CHP approach: 2D projection of realizations using multi-dimensional scaling (MDS). The distance is the JS-divergence for multi-resolution g=1 based on CHPs. The axis are omitted (as typical for MDS plots, see Caers, 2011) since only the relative scatter of points is relevant. Figure 7: CHP approach: multi-resolution g=3: selected realizations and MDS plot. Figure 8: snesim realization for different number of nodes in the search template. Figure 9: MDS plot of snesim realizations for multi-resolution g=1. Figure 10: 2D continuous case: training image, ccsim and sgsim realizations. Figure 11: MDS plot for multi-resolution g=1 Figure 12: variogram model (red) versus experimental variograms (blue) Figure 13: 3D binary case: training image, ccsim and dispat realization and MDS plot for multi-resolution g=1. Realizations are shown in two ways, with and without the background (blue) category removed. Table 1: 2D binary case: table with averaged “between”-realization JS divergences for the MPH approach
25
Table 2: ratios of the within and between realization variability and total ratio for the MPH approach Table 3: 2D binary case: table with averaged “within”-realization JS divergences for the MPH approach Table 4: 2D binary case: table with averaged “between”-realization JS divergences for the CHP approach Table 5: ratios of the within and between realization variability and total ratio for the CHP approach Table 6: 2D binary case: table with averaged “within”-realization JS divergences for the CHP approach
Multi-resolution 1
(101 x 101)
Multi-resolution 2
(51 x 51)
Multi-resolution 3
(26 x 26)
Figure 1
Pyramid of one single image
Training image
Figure 2
Variogram of Training image
Lag Distance
0 10 20 30 40 50
0
0
.1
0.2
0.3
dispat
ccsim
sisim
closest sisim
50th closest
closest dispat model 20th closest 30th closest 50th closest
Training image
closest ccsim model
40th closest
50th closest
Figure 3
ccsim
sisim
dispat
training image
20%
10%
Figure 4
Training image dispat ccsim sisim
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6x-y Plot & Multi Scale = 6
DISPAT
CCSIM
SISIM
Training Image
14%
20%
72-42-117-49 39-34-28-21 45-43-44-47 23-24-22-19 57-65-88-65 19-65-32-19 82-36-91-32
31-52-16-22 16-18-2-16 57-29-59-25 23-7-16-23 16-16-12-12 52-14-45-13 18-17-22-40
24-5-5-10 51-98-55-50 9-2-4-5 53-50-62-153 89-73-145-150 47-38-66-70 30-34-18-17
23-25-9-26 24-59-39-47 11-23-3-4 46-7-66-9 12-7-0-4 10-8-1-0 35-65-32-21
16-11-1-3 16-13-0-22 47-72-74-177 51-31-46-34 51-96-53-21 35-40-22-16 13-37-1-40
75-37-81-28 10-3-1-6 42-22-43-37 45-69-28-16 25-13-13-17 64-168-91-59 15-2-5-5
24-10-14-31 66-27-95-34 16-13-0-1 11-15-0-31 18-26-10-16 27-2-4-18
Figure 5
ccsim
sisim
dispat
training image
Training image closest ccsim model 40th closest 50th closest
closest sisim
50th closest
closest dispat model
20th closest
30th closest
50th closest
Figure 6
45%
20%
dispat ccsim Training image sisim
Figure 7
20%
14%
Figure 8
Training image
snesim ns=10
snesim ns=200
snesim ns=50
Figure 9
16%
8%
Training image
ccsim
sgsim
Figure 10
Ti, (0, 0), SGSIM, MR=1 closest to Ti 20th closest to Ti
30th closest to Ti 40th closest to Ti 50th closest to TiTi, (0, 0), SGSIM, MR=1 closest to Ti 20th closest to Ti
30th closest to Ti 40th closest to Ti 50th closest to Ti
Ti, (0, 0), CCSIM, MR=1 closest to Ti 20th closest to Ti
30th closest to Ti 40th closest to Ti 50th closest to Ti
Ti, (0, 0), CCSIM, MR=1 closest to Ti 20th closest to Ti
30th closest to Ti 40th closest to Ti 50th closest to Ti
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2x-y Plot & Multi Scale = 1
CCSIM
SGSIM
training image
Training image Ti, (0, 0), SGSIM, MR=1 closest to Ti 20th closest to Ti
30th closest to Ti 40th closest to Ti 50th closest to Ti
Ti, (0, 0), SGSIM, MR=1 closest to Ti 20th closest to Ti
30th closest to Ti 40th closest to Ti 50th closest to Ti
Ti, (0, 0), SGSIM, MR=1 closest to Ti 20th closest to Ti
30th closest to Ti 40th closest to Ti 50th closest to TiClosest SGSIM 30th Closest 50th Closest
Figure 11
Closest CCSIM model
30th Closest
50th Closest
Ti, (0, 0), CCSIM, MR=1 closest to Ti 20th closest to Ti
30th closest to Ti 40th closest to Ti 50th closest to Ti
Ti, (0, 0), CCSIM, MR=1 closest to Ti 20th closest to Ti
30th closest to Ti 40th closest to Ti 50th closest to Ti
Ti, (0, 0), CCSIM, MR=1 closest to Ti 20th closest to Ti
30th closest to Ti 40th closest to Ti 50th closest to Ti
56%
10%
Variograms of ccsim models
Variograms of sgsim models
Figure 12
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
No
rmal
sco
res
No
rmal
sco
res
0 10 20 30 40 50 60 Lag Distance
0 10 20 30 40 50 60 Lag Distance
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Lag Distance
Norm
al S
core
s
Variograms of SGSIM models
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Lag Distance
Norm
al S
core
s
Variograms of CCSIM models
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015x-y Plot & Multi Scale = 1
Dispat
CCSIM
training image
Training image
a dispat model
a ccsim model
46%
19%
Figure 13
multi-resolution dispat ccsim sisim
g=1 0.0455 0.0319 0.1384
g=2 0.1776 0.1915 0.4149
g=3 0.3883 0.4882 0.6092
g=4 0.6444 0.7741 0.7893
g=5 0.8281 0.8968 0.8889
g=6 0.8871 0.9451 0.9470
g=7 0.9523 0.9772 0.9729
g=8 0.9835 0.9860 0.9856
g=9 0.9875 0.9945 0.9892
g=10 0.9339 0.9922 0.9908
Table 1
, 2
' 1
'
,1
1( , )
L Lbetweeng k JS g k kd d
L L
re re
MPH approach
Table 2
dispat ccsim sisim
dispat 1 1.15 0.38
ccsim * 1 0.33
sisim * * 1
alg
o k
algo m
,between
k mr
dispat ccsim sisim
dispat 1 1.63 0.24
ccsim * 1 0.15
sisim * * 1
algo m
,within
k mr
dispat ccsim sisim
dispat 1 0.70 1.58
ccsim * 1 2.20
sisim * * 1
algo m
,,
,
betweenk mtotal
k m withink m
rr
r
(A) (B) (C)
MPH approach
Table 3
multi-resolution dispat ccsim sisim
g=1 0.0496 0.0222 0.2529
g=2 0.1908 0.1672 0.5071
g=3 0.4140 0.4454 0.7201
g=4 0.6859 0.7430 0.9076
g=5 0.8441 0.8910 0.9778
g=6 0.9193 0.9452 0.9921
g=7 0.9566 0.9726 0.9941
g=8 0.9873 0.9824 0.9978
g=9 0.9923 0.9931 0.9990
g=10 0.9702 0.9904 0.9980
( ), ,
1
1( , )
Lwithing k JS g kd d
L
re ti
MPH approach
multi-resolution dispat ccsim sisim
g=1 0.0358 0.0408 0.0416
g=2 0.1166 0.1124 0.1249
g=3 0.2515 0.2269 0.2198
Table 4
, 2
' 1
'
,1
1( , )
L Lbetweeng k JS g k kd d
L L
re re
CHP approach
Table 5
dispat ccsim sisim
dispat 1 0.88 0.86
ccsim * 1 0.98
sisim * * 1
alg
o k
algo m
,between
k mr
dispat ccsim sisim
dispat 1 1.31 0.43
ccsim * 1 0.33
sisim * * 1
algo m
,within
k mr
dispat ccsim sisim
dispat 1 0.67 2.00
ccsim * 1 2.90
sisim * * 1
algo m
,,
,
betweenk mtotal
k m withink m
rr
r
(A) (B) (C)
CHP approach
Table 6
multi-resolution dispat ccsim sisim
g=1 0.0533 0.0405 0.1299
g=2 0.2346 0.2127 0.2620
g=3 0.3106 0.3015 0.3120
( ), ,
1
1( , )
Lwithing k JS g kd d
L
re ti
CHP approach
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