Geostatistical Simulation and Deep Learning in Geomagnetism

The major sources of Earth’s magnetic field that originate within its interior are electrical currents generated by dynamo action in the outer core and the magnetization of the lithosphere. The spatial spectra of these sources overlap and our inability to separate them has hindered progress in understanding the underlying sources, despite the availability of high quality satellite data providing global coverage. In this thesis I develop and implement three new geomagnetic field modelling schemes designed to utilize prior information to enable estimation of separate models for the core and lithosphere magnetic fields. As prior information on the core field I use an ensemble of magnetic fields at the core-mantle boundary obtained from numerical simulations of the geodynamo, while for the lithosphere I use a prior ensemble generated from forward models of lithospheric magnetization based on geological maps.

In a first investigation I derive a baseline model via a classical spherical harmonic inversion by implementing maximum entropy regularization at the Earth’s surface and the core-mantle boundary in order to co-estimate models of the lithospheric field and time-dependent core field. I explore the use of a latent space transform based on a-priori covariances for each source which simplifies the choice of the regularization parameters. A temporal regularization term that minimizes the entropy change over time for the core field is also introduced and explored. Using this approach I am able to derive separate models of the time-dependent core and static lithospheric field minimizing the specified entropy norms and fitting well the satellite data.

I next present and demonstrate a probabilistic inversion scheme via Spherical Direct Sequential Simulation (SDSSIM) making use of the prior information in the form of prior histograms of the radial field on the source surfaces and covariance functions. This is implemented in physical space based on external Green’s functions for Laplace’s equation with Neumann boundary conditions. The method is validated with synthetic test cases demonstrating that it produces realistic
posterior realizations consistent with the known solution while reproducing the prior histogram and covariance functions. An application to real satellite observations is presented that gives the posterior probability distribution for the geomagnetic field at the core-mantle boundary when a lithospheric field has been first removed. A synthetic geomagnetic test case estimating the lithospheric field is also demonstrated, with separate covariance functions applied for the oceans and continents. The implementation presented here is a step toward probabilistic co-estimation of the core and lithospheric field and I describe a possible extension that may enable this to be accomplished.

In the third approach I attempt to learn the many layers of detail present in the prior information by designing two deep learning networks, CLiP_net (Core-Lithosphere Partitioning Network) and Core_net, that employ spherical graph convolutions in a U-net architecture; each take as input a spherical harmonic model and learn features of interest in the radial geomagnetic field in an a-priori latent space. A transformation to physical space is carried out and through spherical harmonic analysis of the output, separate models for the core and lithospheric fields are provided. Training of the networks estimation capabilities are carried out based on information regarding the core and lithospheric fields obtained from the prior ensembles.  CLiP_net provides co-estimated core and lithospheric field models based on a previously estimated internal field model from learned features of the lithospheric field at Earth’s surface and the core field at the core-mantle boundary. Core_net takes as input a model of the core field with little power at high spherical harmonic degree and provides as output an estimate of the core field with power at small scales restored according to learned features from the core field at the core-mantle boundary. Both networks are validated using known models from parts of the prior ensemblenot included during training. CLiP_net successfully reproduces important details of the small scale core field and the large scale lithosphere fields as seen in the validation set. Preliminary results of an application to a time dependent internal field model based on real satellite observations are also presented. Core_net provides estimates of the core field that improve the input models and better resolve the small scale core field. 

The work presented in this thesis together provides a step toward an improved incorporation of The major sources of Earth’s magnetic field that originate within its interior are electrical currents generated by dynamo action in the outer core and the magnetization of the lithosphere. The spatial spectra of these sources overlap and our inability to separate them has hindered progress in understanding the underlying sources, despite the availability of high quality satellite data providing global coverage. In this thesis I develop and implement three new geomagnetic field modelling schemes designed to utilize prior information to enable estimation of separate models for the core and lithosphere magnetic fields. As prior information on the core field I use an ensemble of magnetic fields at the core-mantle boundary obtained from numerical simulations of the geodynamo, while for the lithosphere I use a prior ensemble generated from forward models of lithospheric magnetization based on geological maps. In a first investigation I derive a baseline model via a classical spherical harmonic inversion by implementing maximum entropy regularization at the Earth’s surface and the core-mantle boundary in order to co-estimate models of the lithospheric field and time-dependent core field. I explore the use of a latent space transform based on a-priori covariances for each source which simplifies the choice of the regularization parameters. A temporal regularization term that minimizes the entropy change over time for the core field is also introduced and explored. Using this approach I am able to derive separate models of the time-dependent core and static lithospheric field minimizing the specified entropy norms and fitting well the satellite data. I next present and demonstrate a probabilistic inversion scheme via Spherical Direct Sequential Simulation (SDSSIM) making use of the prior information in the form of prior histograms of the radial field on the source surfaces and covariance functions. This is implemented in physical space based on external Green’s functions for Laplace’s equation with Neumann boundary conditions. The method is validated with synthetic test cases demonstrating that it produces realistic posterior realizations consistent with the known solution while reproducing the prior histogram and covariance functions. An application to real satellite observations is presented that gives the posterior probability distribution for the geomagnetic field at the core-mantle boundary when a lithospheric field has been first removed. A synthetic geomagnetic test case estimating the lithospheric field is also demonstrated, with separate covariance functions applied for the oceans and continents. The implementation presented here is a step toward probabilistic co-estimation of the core and lithospheric field and I describe a possible extension that may enable this to be accomplished. In the third approach I attempt to learn the many layers of detail present in the prior information by designing two deep learning networks, CLiP_net (Core-Lithosphere Partitioning Network) and Core_net, that employ spherical graph convolutions in a U-net architecture; each take as input a spherical harmonic model and learn features of interest in the radial geomagnetic field in an a-priori latent space. A transformation to physical space is carried out and through spherical harmonic analysis of the output, separate models for the core and lithospheric fields are provided. Training of the networks estimation capabilities are carried out based on information regarding the core and lithospheric fields obtained from the prior ensembles. CLiP_net provides co-estimated core and lithospheric field models based on a previously  estimated internal field model from learned features of the lithospheric field at Earth’s surface and the core field at the core-mantle boundary. Core_net takes as input a model of the core field with little power at high spherical harmonic degree and provides as output an estimate of the core field with power at small scales restored according to learned features from the core field at the core-mantle prior information in geomagnetic field modelling. In each case I found that further improvements in the results depend on improving the quality of the prior information and incorporating more of its details. Especially using more detailed information on spatial variation of the priors was found to be a natural next development step across the methods. As the quality of prior information improves in the future, there will be a clear need to move beyond two-point statistics and covariances if we are to better separate and characterize the small scale field in the core.