Local crustal magnetic field inversion from satellite data

In many cases, planetary magnetic field data are of spatially varying quality, or are only regionally available (e.g. the MESSENGER satellite mission to Mercury). Or, to save computational costs, we may want to calculate a high-resolution crustal magnetic field model only locally instead of solving for the entire planet.

Such local, or regional crustal magnetic field inversions can be computationally efficiently done with our altitude-cognizant gradient vector Slepian functions as Frederik Simons and I demonstrate in our research article.


Vector Slepian functions

The counterpart on the sphere to Fourier analysis are spherical harmonic functions.

Similar to Fourier analysis, in which a (spatial) truncation of a signal leads to an infinitely wide wavenumber spectrum, an infinitely high spherical harmonic degree would be required to describe a signal that has non-zero values only within a target region (e.g. Eurasia).

If we want to calculate Earth's crustal magnetic field or Earth's gravity field locally within a target region on Earth's surface from satellite data we require functions that have a finite bandlimit and which concentrate on the target region as well as possible.

Such functions can be generated in a mathematically optimal way by solving a maximization problem. They are named "Slepian" functions after one of their discoverers, David Slepian.

This poster shows the basic concept. The paper describing the construction of vector Slepian functions in detail is available on my publications page.


Adaptive wavelet electrical resistivity tomography

In electrical resistivity tomography surface based (or borehole) measurements of electrical potential fields generated by actively injected electrical current are used to image the electrical conductivity of the subsurface.

The classical method to calculate the conductivities from the known injected currents and the measured electrical potentials makes use of a fixed parameter grid that needs to be chosen initially.

In our adaptive wavelet method the subsurface conductivity is not parameterized using constant subsurface blocks but as a superposition of three-dimensional Haar wavelets.

The list of Haar wavelets used to describe the subsurface conductivity is automatically adapted depending on a measure for the resolving power of the potential value data set.

The method is described in detail in my PhD Thesis and links to the articles published in the course of my PhD can be found on my publications page.


Background figure

The figure in the background shows the 500th best-suited altitude-cognizant gradient vector Slepian function with maximum spherical-harmonic degree 100 concentrated over North America and optimized for average satellite altitude 300 km.

Alain Plattner
Department of Geological Sciences
University of Alabama
201 7th Avenue
Room 2023 Bevill Building
Tuscaloosa, AL 35487

Email: amplattner (at) ua (dot) edu