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18.3 The Variance Covariance Matrix
When accumulating the matrix, care has to be taken to ensure that errors that correlate from one survey station to the next are added to the delta values directly before summing but errors that can vary from one survey to the next are summed in their product form. This ensures that random and systematic effects are propagated correctly. Some errors also continue from one survey leg to another and some even from one well to another such as the error in magnetic north.
Once the matrix is complete a technique is required to derive the size and orientation of the three main axes.
18.4 Eigen Values and Eigen Vectors
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 h o nc M xdscb h “ ud cos”ofh h xs ndh Egn lus llush lngh. h mathematics of converting a 3 x 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 ‘ w ng’ c o s h l h main dimensions of the ellipse on the leading diagonal with zeros for all the covariances when seen from these axes.
y ccumul ng h s ll pso ds long h w llbo , w c
n“unc ny n lop ”lk funn l ound h
wellbore which we can then use to ensure a safe passage when drilling close to other wells.
18.5 Collision Risk
It would be wrong to think that if two uncertainty envelopes were touching there was always a high risk of collision. We often use between 2 and 3 standard deviations when determining the size of our ellipses which means that probability of being outside the ellipse is very small.
Figure 82: Calculating and plotting the collision risk.
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