Page 83 - Statistics and Data Analysis in Geology

Matrix Algebra

where vh, V,, and Vsh are the proportions of halite, anhydrite, and shale. However,
we want to reverse these equations, and for given values of Pb and At that we
read from the well logs, estimate the proportions of the three constituents of the
rock. Since three unknowns must be estimated, it seems we will require three
equations and, hence, measurements of three log properties. However, because
the proportions of halite, anhydrite, and shale must sum to one, we can use this
constraint to provide the necessary third equation.

The three equations can be set into matrix form as
[:I=[ 2.03 2.98 2.43 1;3] [va
L = cv
7
vh

Vsh

However, what we really want to do is solve for V, given values of L taken from the
well logs. This means that C must be moved to the other side of the equal sign,
which we can do by multiplying both sides of the equation by its inverse, C-l. Then,
[ 2.03 2.98 2.43 y1-l [;I]-[ 21
y
y

Vsh
Perform the necessary matrix inversion and multiplications to determine the
proportions of halite, anhydrite, and shale in the 50-ft interval of the Hutchinson
Salt. Plot the record of lithologic compositions in the form of a lithologic strip log.
Ten of these estimates have been used in Chapter 2 (Table 2.9) to demonstrate the
effects of closure on the calculation of correlations among closed variables.
[Hint: L, as given in file KANSALT.TXT, is a 2 x 50 matrix of Pb and At log responses.
It must be converted to a 3 x 50 matrix by adding a column of 1’s in order for the
dimensions of the matrix multiplication to be correct. What does this column of
1’s represent?]

Exercise 3.6

The state of stress in the subsurface can be represented in a 3 x 3 matrix, Z, whose
diagonal elements represent normal stresses and whose off-diagonal elements rep-
resent shear stresses. The meanings of the nine elements of the stress matrix can
be seen by imagining a cube in a Cartesian coordinate system in which the X-axis
points to the east, the Y-axis points to the north, and the Z-axis points up. The
symbol axx represents the normal stress directed onto the east or west face of the
cube; it will be a positive value if the stress is compressional and a negative value
if the stress is tensional. There is a similar meaning for ayy and aZz. The symbol
a,, represents the shear stress on the east or west face of the cube, acting parallel
to the Y-axis. A shear stress is positive if the compressional or tensional compo-
nent agrees in sign with the direction of force. That is, both components of shear

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