Lecture 5: Thickness using linear algebra

Summary

The lecture derives a general method for calculating the thickness of a sedimentary layer using linear algebra. Points on the layer’s top and base are expressed in a coordinate system aligned with the layer, and their separation is measured along the layer’s pole axis.

Method

1. Start with East–North–Up (ENU) coordinates. The locations of points on the layer’s top and base are given by their east, north, and up coordinates.

2. Define a layer-aligned coordinate system. The new axes are:

   • S: strike direction
   • D: dip direction
   • P: pole to the layer

3. Build a transformation matrix. A (3 \times 3) matrix (A) transforms coordinates from the ENU system to the strike–dip–pole system. Its components are direction cosines: the cosines of the angles between the new axes and the original ENU axes. The matrix depends on the layer’s strike and dip.

4. Transform each point’s position vector. Apply the matrix as [ \mathbf{V}’ = A\mathbf{V}. ] In index notation, this is [ V’i = AV_j. ]

5. Interpret the transformed coordinates. The third transformed coordinate, (V’_3), gives the point’s position along the pole axis (P).

6. Calculate thickness. Transform points on the top and base, then subtract their pole-axis coordinates. The thickness is the difference between the top and base (P) coordinates.

The lecture also explains the Einstein summation convention used in the index notation: a repeated index, such as (j), indicates summation, while the free index, such as (i), identifies the separate equations. Expanding the notation gives three equations, each with three terms.

The class exercise applies this procedure to points at unit contacts, assuming the units share the same strike and dip.

Speakers and Sources

• Speaker: An unnamed lecturer.
• Sources and materials mentioned: A class exercise and sections from a “Computational Geosciences” resource. The lecture also refers to formulas for layer thickness from the previous lecture.