Engineering Rock Mechanics

1 32 Discontinuities

than or equal to 50 MPa, less than or equal to 25 MPa, greater than 25 MPa,
or greater than 25 MPa and less than or equal to 30 MPa?
We can also establish, for example, how long a scanline should be in
order to determine the discontinuity frequency, or mean spacing, to within
specified tolerances. For this we utilize the central limit theorem, which
states that the means of random samples of size N taken from a
population of any distribution that has a mean X and a standard deviation
CT will tend to be normally distributed with a mean x and a standard
deviation of o/Nln. However, in the case of a negative exponential
distribution, the standard deviation is equal to the mean: they are both 2. Hence,
utilizing the concept of confidence intervals in the normal distribution, we
can find the appropriate scanline length for any desired confidence level.
As an example, let us say that we wish to determine the mean spacing
X with an 80% confidence that the error will be less than +20%. The 80%
confidence implies that we require our sampled mean to be located within
the zone which is 80% of the total area under the standardized normal
probability density function, The half bandwidth of this zone for the
standardized normal distribution is given by the standardized normal
variable, z = 1.282 (found from statistical tables), so the half bandwidth for
our parameters can then be calculated from zo/Nm. This half bandwidth
is equated to the allowable proportional error, E, which in this example is
0.2, i.e. 20% from the true value of the mean. Then we calculate N from the
equality

zo/N112 = EX.

As CT = X for the negative exponential distribution, we have

In the example, N = 1.2822/0.22 = 41.

Alternatively, for 90% confidence that we will be within a +lo% error

bandwidth, we find that N = 1.645210.12 = 271. These two examples are

highlighted by the dashed lines in Fig. 7.18.

Sampling bias and mean orientation. From Section 7.2.3, we know that the

discontinuity frequency, &, along a line subtending an angle 8 to the

normal of a discontinuity set with frequency A is & = A cos 8. As 8 tends

to 90°, so A, tends to zero. Consequently, as the scanline is rotated to become

nearly parallel to discontinuities, so the number of sampled discon-

tinuities per unit length of scanline reduces. This clearly introduces a

sampling bias when A is being estimated, resulting from the relative

directions of the discontinuities and the scanline. The bias can be removed

by using a weighting factor: this weights the number of intersected

discontinuities associated with each set to give an effective number of

intersected discontinuities and hence removes the bias. The weighting

factor, w, is calculated from the expression llcos 8, where 8 is the angle

between the normal to the discontinuity and the scanline for

each set.