Engineering Rock Mechanics

168 Anisotropy and inhomogeneity

STATISTICS

ftx)l A‘ Probability ~

function

X

GEOSTATISTICS

Semi-variogram

I I)

ah

SAMPLED DATA:

13 21 19 31 23 42 68 51 74

Figure 10.3 Methods of quantifying inhomogeneity through statistics and

geostatistics.

(a) lump all the data into one histogram;

@) separate the site into a number of discrete ‘structural regions’ and create

(c) use the techniques of geostatistics, specifically semi-variograms and

........

a histogram for each; and

kriging.

Note that the use of these three different techniques represents three ways

in which the locations of the sampling points are taken into account. When

the data are lumped together, the information on location is suppressed,

except that all samples came from an assumed single universe. In the

second case, the location information is similarly suppressed, except that

different prubability density functions can now be distinguished in the

different sampled areas, and hence statements can be made about

whether there is any variation between the regions of the sampled uni-

verse. In the third case, the distances between the specific sampling points

are explicitly taken into account in the creation of a semi-variogram, as

shown in the top right of Fig. 10.3 and explained below.

In this context of inhomogeneity, we should distinguish between accu-

racy and precision-which are our two main parameters for assessing the

level of inhomogeneity using the lumped histogram approach. Accuracy

is the ability to obtain the correct answer ’on the average’: the sampled

mean is, on the average, the true mean, i.e. there is no bias in the

measurements. Precision, commonly measured by the standard deviation,

is the degree of spread of the measurements, whether or not they are accurate.

Considering the probability density function at the top left of Fig. 10.3, the

material will be inhomogeneous if the spread of results is greater than that

which would result from sampling error alone. In fact, this is our sole

measure when the data are lumped together. However, if different prob-

ability density functions are constructed for the different structural

regions, we can utilize the differences between their means as a measure

of the inter-’structural region’ inhomogeneity and the spread of individual

histograms as a measure of the intra-’structural region’ inhomogeneity.