Imagine looking at an almond which you are holding in your hand. You can choose to look at it from above, in front or any other direction in space. However, no matter how the almond is oriented there will always be three orthogonal viewing vectors that let you see the ellipses of the almond at their full size. In other words if you line up your eye on one axis, you will see the ellipse formed by the other two with no foreshortening effects.
Eigen vectors provide a way to derive the best viewing angles to see the 3 uncertainty ellipses. The Eigen vectors of the Covariance Matrix describe the “attitude vectors” of the three axes and the Eigen values tell us their length. The mathematics of converting a 3 × 3 Covariance Matrix to its Eigen vectors and Eigen values is very straightforward and provides us with the dimensions and orientation of our uncertainty ellipsoid. It is a matrix manipulation that finds the ‘viewing’ vectors that leave the main dimensions of the ellipse on the leading diagonal with zeros for all the covariances when seen from these axes.
By accumulating these ellipsoids along the wellbore, we create an “uncertainty envelope” like a funnel around the wellbore which we can then use to ensure a safe passage when drilling close to other wells.
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