Describing and classifying the statistical structure of topography and bathymetry is of much interest across the geophysical sciences. Oceanographers are interested in the roughness of seafloor bathymetry as a parameter that can be linked to internal-wave generation and mixing of ocean currents. Tectonicists are searching for ways to link the shape and fracturing of the ocean floor to build detailed models of the evolution of the ocean basins in a plate-tectonic context. Geomorphologists are building time-dependent models of the surface that benefit from sparsely parameterized representations whose evolution can be described by differential equations. Geophysicists seek access to parameterized forms for the spectral shape of topographic or bathymetric loading at various (sub)surface interfaces in order to use the joint structure of topography and gravity for inversions for the effective elastic thickness of the lithosphere. A unified geostatistical framework involves the Matérn process, a theoretically well justified parameterized form for the spectral-domain covariance of Gaussian processes. We provide a computationally and statistically efficient method for estimating the parameters of a stochastic covariance model observed on a regular spatial grid in any number of dimensions. Our proposed method makes important corrections to the well-known Whittle likelihood to account for large sources of bias caused by boundary effects and aliasing. We generalise the approach to flexibly allow for significant volumes of missing data including those with lower-dimensional substructure, and for irregular sampling boundaries. We provide detailed implementation guidelines which maintains the computational scalability of Fourier and Whittle-based methods for large data sets.

Satellite observations of the geomagnetic field contain signals originating both from electrical currents in the core and from magnetized rocks in the lithosphere. At short wavelengths the lithospheric signal dominates, obscuring the signal from the core. Here we present details of a method to co-estimate separate models for the core and lithospheric fields, which are allowed to overlap in spherical harmonic degree, that makes use of prior information regarding the sources. Using a maximum entropy method, we estimate a time-dependent model of the core field together with a static model of the lithospheric field that satisfy the constraints provided by satellite observations as well as statistical prior information, but are otherwise maximally non-committal with regard to the distribution of radial magnetic field at the source surfaces. Tests based on synthetic data are encouraging, demonstrating it is possible to retrieve parts of the core field beyond degree 13 and the lithospheric field below degree 13. Results will be presented from our new model of the time-dependent core surface field up to spherical harmonic degree 30 and implications regarding our understanding of the core dynamo discussed.

A fundamental choice for any geophysical inversion is the parametrization of the subsurface. Voxels and coefficients of basis functions (i.e., spherical harmonics) often are a natural choice, especially since they can simplify forward calculations. An alternative approach is to use a finite collection of discrete anomalies, which leads to transdimensional (TD) techniques, when considered through a Bayesian lens. The most popular form of TD inversion uses a variable number of Voronoi cells as spatial representation. The TD approach addresses the issues of non-uniqueness and lack of resolution in a special way: Instead of smoothing or damping the solutions, the spatial structure of the model is controlled by weighing the number of elements against the achieved data fit. This gives it an intrinsic adaptive behaviour, useful for heterogeneous data coverage. Furthermore, geophysical sensitivity often changes with depth, which TD approaches can also adapt to. In a joint inversion context for several properties (like seismic velocities and densities), spatial coupling between different sought parameters is automatically guaranteed.

In this contribution I will present some examples for using TD inversions on global gravity and surface wave data to simultaneously determine the velocity and density structure within the Earth’s mantle.

The intrinsic non-uniqueness of potential-field inversion of surface scanning data can be circumvented by solving for the potential field of known individual source regions. A uniqueness theorem characterizes the mathematical background of the corresponding inversion problem, and determines when a potential-field measurement on a surface uniquely defines the magnetic potentials of the individual source regions. For scanning magnetometers in rock magnetism, this result implies that dipole magnetization vectors of many individual magnetic particles can be reconstructed from surface scans of the magnetic field. It is shown that finite sensor size still retains this conceptual uniqueness. The technique of micromagnetic tomography (MMT) combines X-ray micro computed tomography and scanning magnetometry to invert for the magnetic potential of individual magnetic grains within natural and synthetic samples. This provides a new pathway to study the remanent magnetization that carries information about the ancient geomagnetic field and is the basis of all paleomagnetic studies. MMT infers the magnetic potential of individual grains by numerical inversion of surface magnetic measurements using spherical harmonic expansions. Because the full magnetic potential of the individual particles in principle is uniquely determined by MMT, not only the dipole but also more complex, higher order, multipole moments can be recovered. Even though a full reconstruction of complex magnetization structures inside the source minerals is mathematically impossible, these additional constraints by far-field multipole terms can substantially reduce the number of possible micromagnetic energy minima. For complex particles with many micromagnetic energy minima it is possible to include the far-field constraints into the micromagnetic minimization algorithm.