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Focusing efforts in the right direction (eigenvectors) – Part 1

By 7 October 2020January 31st, 2024Space geodesy

Which direction will you take?


In the previous article, we gave a broad overview about how to produce maps of Equivalent Water Heights from gravimetric satellite missions. Let’s make a quantum leap forward and dive fast into into least squares and singular value decomposition.

A geometry problem

Both GRACE and GRACE-FO missions have the same geometry problem. Since the twin satellites are on a polar orbit, they always follow each other in the North-South direction. There are never East-West measurements of the intersatellite distance. This configuration creates an observational weakness which leads to the appearance of North-South stripes in the maps of gravity field. Those stripes are not signal, they are noise. We want to get rid of them. You want to get rid of them Everybody wants to get rid of them. The traditional approach is to invert the full normal equation by the Cholesky method, and then later filter the solution with some advanced type of Gaussian filter called the DDK filter. But we, have a different idea.

Least squares minimum

Let’s suppose we wish to minimize a function of the parameters, such as the sum of the squares of the residuals. Let’s assume the dependency of the residuals to the parameters is linear (we can always linearize, can’t we), then the dependency of the function to the parameters is simply quadratic. If there is only one parameter, the function will be a simple parabola, if two, a paraboloid. If n parameters, then we should imagine some sort of hyper-paraboloid… We can easily plot this function, with its value on the z-axis, and the parameter vector space in the horizontal plane. Of course, we need to be able to imagine that a vector space of dimension n can be represented in a horizontal plane, but readers of this blog don’t lack imagination, do we? In addition, we know that this paraboloid has a unique minimum, which is is very easy to find. That’s convenient. So finding our solution should be quite straightforward.

2601 parameters, standard guess

Now, let’s suppose that our vector space is of dimension 6561. For example, the vector space of gravity field models expanded in spherical harmonics up to degree and order 80. That makes a dimension of 81 x 81 = 6561. Let’s also suppose we don’t want to calculate a minimum of least squares on such a vast vector space, and that we prefer to calculate the minimum on a subspace of a smaller dimension, let’s say 2601. We could take the first subspace that comes to mind, for example the vector space of gravity field coefficients up to degree and order 50, which has the desired dimension. We then could find the minimum in this subspace, and it would provide our gravity solution. This would lead us to the following image:

Inversion of the first 2601 parameters in the canonical basis, i.e. spherical harmonics up to degree and order 50

Boom! We have a solution. However, there seems to be a little problem. This image inevitably makes us think again… How on Earth could we get rid of these really annoying stripes all across the picture?

2601 parameters, with feeling

Let’s put more feeling into it. We could also try to consider the subspace generated by the 2601 steepest principal axes of the paraboloid. After all, this vector space has the same dimension, so it should lead to comparable results. This method has a very seducing virtue, which is that reaching a minimum along the steepest axes is very efficient. You will go down very quickly, while not needing to move too much horizontally. As a bonus, you will also not get too far away from this linearity zone for which the paraboloid does make some sense in the first place. With this attempt, you would find the following gravity solution:

Inversion of 2601 wisely chosen parameters in a wider subspace, i.e. spherical harmonics up to degree and order 80

Boom! Another solution, but no stripes! How the hell did we do it?

Do more with less

In both cases, we used the same number of parameters : 2601. For a same number of parameters, the difference is quite eloquent. Not only did we get rid of the annoying stripes, but also did we obtain a very nice spatial resolution. With such a remarkable and promising difference, the process definitely deserves some more research and exploration (which we will do in the next articles). What is the significance of this result? Do more with less. As long as you make the effort to think astutely. For anyone who doubted it, it emphasizes the importance of focusing efforts in the right direction (the direction of eigenvectors!).