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Geomathematics (also:mathematical geosciences,mathematical geology,mathematical geophysics) is the application ofmathematical methods to solve problems ingeosciences, includinggeology andgeophysics, and particularlygeodynamics andseismology.
Geophysical fluid dynamics develops the theory offluid dynamics for the atmosphere, ocean and Earth's interior.[1] Applications include geodynamics and the theory of thegeodynamo.
Geophysicalinverse theory is concerned with analyzing geophysical data to get model parameters.[2][3] 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.[4]
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 aforward model predicting the surface observations given the distribution of this variable.
Applications includegeomagnetism,magnetotellurics and seismology.
Many geophysical data sets have spectra that follow apower law, meaning that the frequency of an observed magnitude varies as some power of the magnitude. An example is the distribution ofearthquake magnitudes; small earthquakes are far more common than large earthquakes. This is often an indicator that the data sets have an underlyingfractal geometry. Fractal sets have a number of common features, including structure at many scales, irregularity, andself-similarity (they can be split into parts that look much like the whole). The manner in which these sets can be divided determine theHausdorff dimension of the set, which is generally different from the more familiartopological dimension. Fractal phenomena are associated withchaos,self-organized criticality andturbulence.[5] Fractal Models in the Earth Sciences byGabor Korvin was one of the earlier books on the application ofFractals in theEarth Sciences.[6]
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 ofpartial 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 inweather forecasting.[7]
Some statistical problems come under the heading of mathematical geophysics, includingmodel validation and quantifying uncertainty.
An important research area that utilises inverse methods isseismic tomography, a technique for imaging the subsurface of the Earth usingseismic waves. Traditionally seismic waves produced byearthquakes or anthropogenic seismic sources (e.g., explosives, marine air guns) were used.
Crystallography is one of the traditional areas ofgeology that usemathematics. Crystallographers make use oflinear algebra by using theMetrical Matrix. TheMetrical Matrix uses the basis vectors of theunit 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.[8] Miller's Index is also helpful in the application of theMetrical Matrix.Brag's equation is also useful when using anelectron microscope to be able to show relationship between light diffraction angles, wavelength, and the d-spacings within a sample.[8]
Geophysics is one of the mostmath heavy disciplines ofEarth Science. There are many applications which includegravity,magnetic,seismic,electric,electromagnetic,resistivity, radioactivity, induced polarization, andwell logging.[9] 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.[9] While similargravity fields tend to be more uniform and smooth compared tomagnetic fields. Gravity is used often foroil exploration and seismic can also be used, but it is often significantly more expensive.[9] Seismic is used more than most geophysics techniques because of its ability to penetrate, its resolution, and its accuracy.
Many applications ofmathematics ingeomorphology are related to water. In thesoil aspect things likeDarcy's law,Stokes' law, andporosity are used.
Mathematics inGlaciology consists of theoretical, experimental, and modeling. It usually coversglaciers,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.[13] It can bemathematically modeled withHooke's law to show the elastic characteristics while usingLamé constants.[13] Generally the ice has its linearelasticity constants averaged over one dimension of space to simplify the equations while still maintaining accuracy.[13]
Viscoelasticpolycrystalline ice is considered to have low amounts ofstress usually below onebar.[13] This type of ice system is where one would test forcreep orvibrations from thetension on the ice. One of the more important equations to this area of study is called the relaxation function.[13] Where it's astress-strain relationship independent of time.[13] This area is usually applied to transportation or building onto floating ice.[13]
Shallow-Ice approximation is useful forglaciers that have variable thickness, with a small amount of stress and variable velocity.[13] 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.[13]
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