Geophysics Math
Bonny Battaglino
Geophysical inverse 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 a forward model predicting the surface observations given the distribution of this variable.
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.[5] Fractal Models in the Earth Sciences by Gabor Korvin was one of the earlier books on the application of Fractals in the Earth 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 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.[7]
An important research area that utilises inverse methods isseismic 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.[8] 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.[8]
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.[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 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.[9] Seismic is used more than most geophysics techniques because of its ability to penetrate, its resolution, and its accuracy.
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 be mathematically modeled with Hooke's Law to show the elastic characteristics while using Lam constants.[13] Generally the ice has its linear elasticity constants averaged over one dimension of space to simplify the equations while still maintaining accuracy.[13]
Viscoelastic polycrystalline ice is considered to have low amounts of stress usually below one bar.[13] 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.[13] Where it's a stress-strain relationship independent of time.[13] This area is usually applied to transportation or building onto floating ice.[13]
Shallow-Ice approximation is useful for glaciers 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]
A Geophysics major typically requires a significant amount of math, including courses in calculus, differential equations, linear algebra, and statistics. These courses are essential for understanding geophysical concepts and data analysis.
While a strong foundation in math is important for success in a Geophysics major, it is not the only factor. Other skills, such as critical thinking, problem-solving, and data analysis, are also crucial. If you struggle with math, you may need to put in extra effort, but it is still possible to succeed in a Geophysics major.
The exact math courses required for a Geophysics major may vary depending on the university or program. However, most programs will require courses in calculus, differential equations, linear algebra, and statistics. Some may also require courses in computer programming and geostatistics.
While it is helpful to have a strong foundation in math before starting a Geophysics major, it is not always necessary. Many programs offer introductory math courses to help students build the necessary skills for more advanced math courses. However, it is important to have a willingness to learn and put in the effort to succeed in math courses.
If you are considering a Geophysics major and are concerned about the math requirements, it is helpful to brush up on your math skills before starting the program. You can also seek out resources such as online tutorials, practice problems, or study groups to help you improve your math skills. Additionally, having a positive attitude and a growth mindset can go a long way in helping you succeed in math courses.
The earliest incarnation of the present CMG Committee was as the Working Group on Geophysical Theory and Computers (WGGTC), which was founded by Vladimir Keilis-Borok. The first meeting of the WGGTC was held in Moscow and Leningrad in 1964 and the last in Moscow in 1971 with intervening meetings held once yearly.
Subsequent to 1971 the group was re-structured as the present Committee on Mathematical Geophysics, which has met on a semi-annual basis since that time, beginning with a meeting in Banff (Canada) in 1972.
The Vladimir Keilis-Borok Medal of the IUGG Commission on Mathematical Geophysics (CMG) has been established by the IUGG Bureau in 2021, and recognizes middle career scientists who made important contributions to the field of mathematical geophysics.
The Union symposium was held during the IUGG General Assembly in Berlin in July 2023 (IUGG2023) The symposium highlighted how fundamental scientific understanding, monitoring, and modelling the Earth System contribute to sustainability both globally, in response to planetary pressures, as well as locally, in response to specific localized developmental constraints. This symposium contributed to the U.N. International Year of Basic Science for Sustainable Development (2022) and the U.N. Decade of Ocean Science for Sustainable Development (2021-2030). Leading experts presented how geosciences and mathematics contribute to the sustainability of our planet and societies: Isabelle Ansorge (South Africa), Anny Cazenave (France), Regina Hock (Norway), Toshio Koike (Japan), Sang-Mook Lee (Rep. of Korea), Valerio Lucarini (UK), Roger Pulwarty (USA), Magdalena Scheck-Wenderoth (Germany), and Jun Xia (China). The symposium was convened by Keith Alverson (Canada) and Alik Ismail-Zadeh (Germany).
The 2024 Vladimir Keilis-Borok Medal is awarded to Valerio Lucarini (University of Leicester, UK) for his outstanding contribution to mathematical geophysics and development of cutting-edge mathematical theories for understanding the climate system fluctuations and its response to forcing. The Medal will be presented during the 34th IUGG Conference on Mathematical Geophysics, 2-7 June 2024, in Mumbai, India, where Professor Lucarini will deliver the Medal Lecture. Congratulations to Valerio Lucarini!
This international publication is the official journal of the IAMG. Mathematical Geosciences is an essential reference for researchers and practitioners of geomathematics, related models, algorithms and computing, who develop and apply data science methods and quantitative models to earth science and geo-engineering problems.
The Geological and Environmental Sciences and Physics Departments offer a program of study leading to a major in Geophysics. Students choosing this program of study are also required to take mathematics courses, which correspond to a minor in mathematics. Students contemplating a Geophysics major should contact these two departments as early as possible for advising. Students need to meet with advisors in both departments. 41-44 hours are required for this major.
The Geophysics option provides the necessary preparation in mathematics and physics to succeed in geoscience applications that require strong backgrounds in these associated disciplines. This option well prepares students for technical careers in geophysics or for advanced studies in graduate school. The B.S. Honors Program is available for this option.
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