Geomathematics

Geomathematics is the application of mathematical methods to solve complex problems in the geosciences. It is also known as mathematical geosciences, mathematical geology, or mathematical geophysics. Scientists use these tools to study the Earth's physical properties and its many moving parts. This field includes the study of geology, geophysics, geodynamics, and seismology. By using math, researchers can model how the planet functions from its surface to its deep interior.

One central method in this field is geophysical inverse theory. This theory addresses the question of what can be known about the Earth's interior using only surface measurements. Scientists use a forward model to predict what they should see on the surface based on a specific internal distribution. For example, the density of rocks determines the value of gravitational acceleration measured at the surface. Inverse theory works backward to determine the spatial distribution of variables like density or seismic wave velocity. This process is vital for studying geomagnetism, magnetotellurics, and seismology.

Geomathematics also examines patterns through the study of fractals and complexity. Many geophysical data sets follow a power law, where the frequency of a magnitude varies based on a power of that magnitude. A common example is the distribution of earthquake magnitudes, where small earthquakes occur much more frequently than large ones. These data sets often possess an underlying fractal geometry. Fractal sets are characterized by irregularity, structure at many scales, and self-similarity. This means the parts of the set look much like the whole. These phenomena are often linked to turbulence, chaos, and self-organized criticality.

History shows that mathematical geosciences have deep roots in early scientific discovery. A foundational example is the method used by Al-Biruni to determine the Earth's radius. He combined geometry and surveying to solve this geophysical puzzle. By measuring the height of a mountain and the dip angle to the horizon, he used early trigonometry and algebra. This work represents an early application of mathematical tools to understand the scale of our planet.

In the study of fluids, geomathematics utilizes geophysical fluid dynamics. This branch develops theories for the movement of fluids in the atmosphere, the ocean, and the Earth's interior. It is essential for understanding geodynamics and the theory of the geodynamo. To make accurate predictions, scientists use fluid dynamic models governed by partial differential equations. These models require accurate initial conditions to work well. Since initial conditions are often unknown, scientists use data assimilation. This method combines numerical models with irregular observations to improve those initial conditions. This is especially important for modern weather forecasting.

Mathematics is also essential for studying the Earth's surface and its components, such as soil and water. In geomorphology, Darcy's law describes how fluid flows through a uniform, saturated soil. This work is often categorized as hydrogeology. Stokes' law is used to measure how quickly different sized particles settle out of a fluid. This helps scientists perform pipette analysis to find the percentages of sand, silt, and clay in a soil sample. Additionally, stream power is used to calculate a river's ability to incise into its bed. This helps predict if a river will change course or how it is affected by dams.

Glaciology relies heavily on theoretical, experimental, and modeling approaches. Scientists use Hooke's law to model the elastic characteristics of ice. They may also use Lamé constants to describe these properties. Because polycrystalline ice deforms slower than single crystalline ice, the math must account for stress on blocked basal planes. For glaciers with variable thickness, scientists use the shallow-ice approximation. This helps them predict stress and velocity, which are affected by temperature and ice properties. The basal shear-stress formula is another key tool in this area of study.

Finally, the field extends into crystallography and high-level geophysics. Crystallographers use linear algebra and the Metrical Matrix to study the structure of crystals. This matrix uses basis vectors of unit cell dimensions to find bond lengths and the volume of a unit cell. They also use Miller's Index and Bragg's equation to understand light diffraction angles and wavelengths. In broader geophysics, methods like seismic tomography use inverse methods to image the subsurface. Seismic waves from earthquakes or human sources, such as marine air guns, allow for high-resolution imaging. While seismic methods are expensive, they are used frequently due to their accuracy and ability to penetrate the Earth.