Mathematical geophysics

Mathematical geophysics is concerned with developing mathematical methods for use in geophysics. As such, it has application in many fields in geophysics, particularly geodynamics and seismology.
Areas of mathematical geophysics.
Geophysical fluid dynamics.
Geophysical fluid dynamics develops the theory of fluid dynamics for the atmosphere, ocean and Earth's interior. Applications include geodynamics and the theory of the geodynamo.
Geophysical inverse theory.
Geophysical inverse theory is concerned with analyzing geophysical data to get model parameters. It is concerned with the question: What can be known about the Earth's interior from measurements on the surface? Generally there are limits on what can be known even in the ideal limit of exact data.
The goal of inverse theory is to determine the spatial distribution of some variable (for example, density or seismic wave velocity). The distribution determines the values of an observable at the surface (for example, gravitational acceleration for density). There must be a "forward model" predicting the surface observations given the distribution of this variable.
Applications include geomagnetism, magnetotellurics and seismology.
Fractals and complexity.
Many geophysical data sets have spectra that follow a power law, meaning that the frequency of an observed magnitude varies as some power of the magnitude. An example is the distribution of earthquake magnitudes; small earthquakes are far more common than large earthquakes. This is often an indicator that the data sets have an underlying fractal geometry. Fractal sets have a number of common features, including structure at many scales, irregularity, and self-similarity (they can be split into parts that look much like the whole). The manner in which these sets can be divided determine the Hausdorff dimension of the set, which is generally different from the more familiar topological dimension. Fractal phenomena are associated with chaos, self-organized criticality and turbulence.
Data assimilation.
Data assimilation combines numerical models of geophysical systems with observations that may be irregular in space and time. Many of the applications involve geophysical fluid dynamics. Fluid dynamic models are governed by a set of partial differential equations. For these equations to make good predictions, accurate initial conditions are needed. However, often the initial conditions are not very well known. Data assimilation methods allow the models to incorporate later observations to improve the initial conditions. Data assimilation plays an increasingly important role in weather forecasting.
Geophysical statistics.
Some statistical problems come under the heading of mathematical geophysics, including model validation and quantifying uncertainty.