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Focusing efforts in the right direction (eigenvectors) – Part 2

By 4 November 2020January 29th, 2024Space geodesy

Are you on the right track?

Focus your efforts in the right direction

In our previous article, we talked about why it was important to focus efforts in the right direction. That might sound philosophical, but no… it is purely mathematical. Strictly business (i.e. linear algebra). By directions of course, we were referring to the vectors that constitute the basis of the parameters space (the space of the spherical harmonics coefficients). We illustrated how, if given a chosen dimension, we could improve the gravity solution by astutely selecting a better set of vectors, and design a solution within the subspace generated those vectors. In our example, the first subspace was generated by the spherical harmonic functions up to degree and order 50. If you count them on your fingers (and toes), and don’t forget any, you get a dimension of n=2601. These are the first 2601 vectors of the canonical basis. The second subspace (the astute one), on the opposite, was made of the first 2601 eigenvectors of the gravity normal matrix. This is the same number, but they are not the same vectors.

The solution in the first subspace was this:

Solution in the 2601 first dimensions of the canonical basis

While the solution in the second subspace was this:

Solution in the dimensions of the first 2601 eigenvectors
(linear combinations of spherical harmonics up to degree and order 80)

Do you remember? Although the two subspaces are of the same size, the nature of the result is quite different. Indeed, we can get rid of the stripes without any post-inversion filtering, which is very appealing.

What’s the trick?

We rightfully concluded that directions were important. What we didn’t show, however, is what those actual directions « looked like ». Are you curious?

Subspace #1 – The canonical basis

These are the vectors of the canonical basis (spherical harmonics up to degree 50). Each vector represents a dimension (or direction).

Spherical harmonics coefficients C(2,0) and C(2,2)

Spherical harmonics coefficients C(10,1) and C(10,6)

They are perfectly orthogonal, arranged neat and clean, as they should.

Subspace #2 – The eigenvector basis

These are vectors of the alternative basis. They are the eigenvectors of the normal matrix (linear combinations of vectors from the canonical basis up to order 80). They are also orthogonal by the way. They are ranked by descending order of their associated eigenvalue.

Eigenvectors of rank 1 and 10

Eigenvectors of rank 100 and 500

Eigenvectors of rank 6000 and 6200


What to keep and what to leave out

Are you struck by lightning yet? The vectors responsible for the stripes are clearly the ones associated with the lower eigenvalues. They are precisely the ones we left out of our solution, because that’s what the truncation does. The diagonalization of the matrix automatically separates the noise structure from the signal structure. Every direction (dimension) associated with a high eigenvalue is signal. Every direction (dimension) associated with a low eigenvalue is noise. Discarding them from our gravity subspace guarantees that they will not pollute our solution. This explains how we could obtain the results presented at the start of the article. Fantastic!

What next?

In the next article of the “Focusing efforts in the right direction” series, we will show you exactly what happens in the least squares process when we diagonalize the matrix and truncate the eigenvalues. Stay tuned for more!