INTERPOLATION AS A MATRIX

Here we see how general principles of linear operators are exemplified by linear interpolation. Because the subject matter is so simple and intuitive, it is ideal to exemplify abstract mathematical concepts that apply to all linear operators.

Let an integer k range along a survey line, and let data values x_k be packed into a vector $\bold x$.(Each data point x_k could also be a seismogram.) Next we resample the data more densely, say from 4 to 6 points. For illustration, I follow a crude nearest-neighbor interpolation

scheme by sprinkling ones along the diagonal of a rectangular matrix that is  

$$\bold y \eq \bold B \, \bold x$$ (1)

where  

$$\left[ \begin{array} {c} y_1 \\ y_2 \\ y_3 \\ y_4 \\... ...{array} {c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array} \right]$$ (2)

The interpolated data is simply $\bold y = (x_1, x_2,x_2,x_3, x_4,x_4)$.The matrix multiplication (2) would not be done in practice. Instead there would be a loop running over the space of the outputs $\bold y$ that picked up values from the input.