Geomathematics or Mathematical Geophysics is the application of mathematical intuition to solve problems in Geophysics. The most complicated problem in Geophysics is the solution of the three dimensional inverse problem, where observational constraints are used to infer physical properties. The inverse procedure is much more sophisticated than the normal direct computation of what should be observed from a physical system. The estimation procedure is often dubbed the inversion strategy (also called the inverse problem) as the procedure is intended to estimate from a set of observations the circumstances that produced them. The Inverse Process is thus the converse of the classical scientific method.
An important research area that utilises inverse methods is seismic tomography, a technique for imaging the subsurface of the Earth using seismic waves. Traditionally seismic waves produced by earthquakes or anthropogenic seismic sources (e.g., explosives, marine air guns) were used.
Crystallography is one of the traditional areas of geology that use mathematics. Crystallographers make use of linear algebra by using the Metrical Matrix. The Metrical Matrix uses the basis vectors of the unit cell dimensions to find the volume of a unit cell, d-spacings, the angle between two planes, the angle between atoms, and the bond length. Miller's Index is also helpful in the application of the Metrical Matrix. Brag's equation is also useful when using an electron microscope to be able to show relationship between light diffraction angles, wavelength, and the d-spacings within a sample.
Geophysics is one of the most math heavy disciplines of Earth Science. There are many applications which include gravity, magnetic, seismic, electric, electromagnetic, resistivity, radioactivity, induced polarization, and well logging. Gravity and magnetic methods share similar characteristics because they're measuring small changes in the gravitational field based on the density of the rocks in that area. While similar gravity fields tend to be more uniform and smooth compared to magnetic fields. Gravity is used often for oil exploration and seismic can also be used, but it is often significantly more expensive. Seismic is used more than most geophysics techniques because of its ability to penetrate, its resolution, and its accuracy.
Many applications of mathematics in geomorphology are related to water. In the soil aspect things like Darcy's law, Stoke's law, and porosity are used.
Mathematics in Glaciology consists of theoretical, experimental, and modeling. It usually covers glaciers, sea ice, waterflow, and the land under the glacier.
Polycrystalline ice deforms slower than single crystalline ice, due to the stress being on the basal planes that are already blocked by other ice crystals. It can be mathematically modeled with Hooke's Law to show the elastic characteristics while using Lamé constants. Generally the ice has its linear elasticity constants averaged over one dimension of space to simplify the equations while still maintaining accuracy.
Viscoelastic polycrystalline ice is considered to have low amounts of stress usually below one bar. This type of ice system is where one would test for creep or vibrations from the tension on the ice. One of the more important equations to this area of study is called the relaxation function. Where it's a stress-strain relationship independent of time. This area is usually applied to transportation or building onto floating ice.
Shallow-Ice approximation is useful for glaciers that have variable thickness, with a small amount of stress and variable velocity. One of the main goals of the mathematical work is to be able to predict the stress and velocity. Which can be affected by changes in the properties of the ice and temperature. This is an area in which the basal shear-stress formula can be used.