User Notes

Thanks to helpful feedback from users, here we list a few notes for infrasound array processing with least trimmed squares (LTS).

A note on \({\alpha}\)

To completely remove one element during LTS processing, set \({\alpha}\) = 1 - 2/n.

3 Element Arrays

For 3 element arrays, least trimmed squares cannot be used. This is because we are trying to fit a 2D plane with 3 choose 2 = 3 elements. Ordinary least squares (\({\alpha}\) = 1.0) should be used in this case.

4 Element Arrays

For 4 element arrays, least trimmed squares can be used, but its effectiveness is limited. This is because there are 4 choose 2 = 6 data points used in the regression, but each element is involved in 3 cross correlations (no autocorrelations). Maximum trimming , \({\alpha}\) = 0.5, here actually chooses 4 elements, so the data will still be contaminated. For this reason, we have added an option to remove an element prior to processing. If a four element array is suspected to have an issue with an element, we recommend the user remove the element and process the array as a 3 element array.

5+ Element Arrays

LTS is the most effective for processing arrays with at least five elements.

Uncertainty Quantification

The code now automatically calculates uncertainty estimates using the slowness ellipse method of Szuberla and Olson (2004). The default is currently set at 90% confidence, but this value can be changed in the LsBeam class in lts_classes.py. Note, the values here are 1/2 the extremal values described in Szuberla and Olson (2004) and are meant to approximate confidence intervals, i.e. value +/- uncertainty estimate, not the area of the coverage ellipse. The \({\sigma_\tau}\) value is now calculated automatically by both the ordinary least squares and the least trimmed squares routines.