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Geomathematics (also: mathematical geosciences, mathematical geology, mathematical geophysics) is the application of mathematical methods to solve problems in geosciences, including geology and geophysics, and particularly geodynamics and seismology.
Geophysical fluid dynamics develops the theory of fluid dynamics for the atmosphere, ocean and Earth's interior. [1] Applications include geodynamics and the theory of the geodynamo.
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.
Applications include geomagnetism, magnetotellurics and seismology.
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]
Some statistical problems come under the heading of mathematical geophysics, including model validation and quantifying uncertainty.
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. [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.
Many applications of mathematics in geomorphology are related to water. In the soil aspect things like Darcy's law, Stokes' 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. [13] It can be mathematically modeled with Hooke's Law to show the elastic characteristics while using Lamé 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]
Geotechnical engineering, also known as geotechnics, is the branch of civil engineering concerned with the engineering behavior of earth materials. It uses the principles of soil mechanics and rock mechanics to solve its engineering problems. It also relies on knowledge of geology, hydrology, geophysics, and other related sciences.
Geophysics is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists, who usually study geophysics, physics, or one of the Earth sciences at the graduate level, complete investigations across a wide range of scientific disciplines. The term geophysics classically refers to solid earth applications: Earth's shape; its gravitational, magnetic fields, and electromagnetic fields ; its internal structure and composition; its dynamics and their surface expression in plate tectonics, the generation of magmas, volcanism and rock formation. However, modern geophysics organizations and pure scientists use a broader definition that includes the water cycle including snow and ice; fluid dynamics of the oceans and the atmosphere; electricity and magnetism in the ionosphere and magnetosphere and solar-terrestrial physics; and analogous problems associated with the Moon and other planets.
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field. It is called an inverse problem because it starts with the effects and then calculates the causes. It is the inverse of a forward problem, which starts with the causes and then calculates the effects.
Seismic tomography or seismotomography is a technique for imaging the subsurface of the Earth using seismic waves. The properties of seismic waves are modified by the material through which they travel. By comparing the differences in seismic waves recorded at different locations, it is possible to create a model of the subsurface structure. Most commonly, these seismic waves are generated by earthquakes or man-made sources such as explosions. Different types of waves, including P-, S-, Rayleigh, and Love waves can be used for tomographic images, though each comes with their own benefits and downsides and are used depending on the geologic setting, seismometer coverage, distance from nearby earthquakes, and required resolution. The model created by tomographic imaging is almost always a seismic velocity model, and features within this model may be interpreted as structural, thermal, or compositional variations. Geoscientists apply seismic tomography to a wide variety of settings in which the subsurface structure is of interest, ranging in scale from whole-Earth structure to the upper few meters below the surface.
Geologic modelling,geological modelling or geomodelling is the applied science of creating computerized representations of portions of the Earth's crust based on geophysical and geological observations made on and below the Earth surface. A geomodel is the numerical equivalent of a three-dimensional geological map complemented by a description of physical quantities in the domain of interest. Geomodelling is related to the concept of Shared Earth Model; which is a multidisciplinary, interoperable and updatable knowledge base about the subsurface.
Geodynamics is a subfield of geophysics dealing with dynamics of the Earth. It applies physics, chemistry and mathematics to the understanding of how mantle convection leads to plate tectonics and geologic phenomena such as seafloor spreading, mountain building, volcanoes, earthquakes, faulting. It also attempts to probe the internal activity by measuring magnetic fields, gravity, and seismic waves, as well as the mineralogy of rocks and their isotopic composition. Methods of geodynamics are also applied to exploration of other planets.
Geophysical imaging is a minimally destructive geophysical technique that investigates the subsurface of a terrestrial planet. Geophysical imaging is a noninvasive imaging technique with a high parametrical and spatio-temporal resolution. It can be used to model a surface or object understudy in 2D or 3D as well as monitor changes.
Computational geophysics is the field of study that uses any type of numerical computations to generate and analyze models of complex geophysical systems. It can be considered an extension, or sub-field, of both computational physics and geophysics. In recent years, computational power, data availability, and modelling capabilities have all improved exponentially, making computational geophysics a more populated discipline. Due to the large computational size of many geophysical problems, high-performance computing can be required to handle analysis. Modeling applications of computational geophysics include atmospheric modelling, oceanic modelling, general circulation models, and geological modelling. In addition to modelling, some problems in remote sensing fall within the scope of computational geophysics such as tomography, inverse problems, and 3D reconstruction.
A synthetic seismogram is the result of forward modelling the seismic response of an input earth model, which is defined in terms of 1D, 2D or 3D variations in physical properties. In hydrocarbon exploration this is used to provide a 'tie' between changes in rock properties in a borehole and seismic reflection data at the same location. It can also be used either to test possible interpretation models for 2D and 3D seismic data or to model the response of the predicted geology as an aid to planning a seismic reflection survey. In the processing of wide-angle reflection and refraction (WARR) data, synthetic seismograms are used to further constrain the results of seismic tomography. In earthquake seismology, synthetic seismograms are used either to match the predicted effects of a particular earthquake source fault model with observed seismometer records or to help constrain the Earth's velocity structure. Synthetic seismograms are generated using specialized geophysical software.
Near-surface geophysics is the use of geophysical methods to investigate small-scale features in the shallow subsurface. It is closely related to applied geophysics or exploration geophysics. Methods used include seismic refraction and reflection, gravity, magnetic, electric, and electromagnetic methods. Many of these methods were developed for oil and mineral exploration but are now used for a great variety of applications, including archaeology, environmental science, forensic science, military intelligence, geotechnical investigation, treasure hunting, and hydrogeology. In addition to the practical applications, near-surface geophysics includes the study of biogeochemical cycles.
The following outline is provided as an overview of and topical guide to geophysics:
Inverse modeling is a mathematical technique where the objective is to determine the physical properties of the subsurface of an earth region that has produced a given seismogram. Cooke and Schneider (1983) defined it as calculation of the earth's structure and physical parameters from some set of observed seismic data. The underlying assumption in this method is that the collected seismic data are from an earth structure that matches the cross-section computed from the inversion algorithm. Some common earth properties that are inverted for include acoustic velocity, formation and fluid densities, acoustic impedance, Poisson's ratio, formation compressibility, shear rigidity, porosity, and fluid saturation.
Vijay Prasad Dimri is an Indian geophysical scientist, known for his contributions in opening up a new research area in Earth sciences by establishing a parallelism between deconvolution and inversion, the two vital geophysical signal processing tools deployed in minerals and oil and gas exploration. In 2010, the Government of India awarded him with the Padma Shri, India's fourth highest civilian award, for his contributions to the fields of science and technology.
Michael Ghil is an American and European mathematician and physicist, focusing on the climate sciences and their interdisciplinary aspects. He is a founder of theoretical climate dynamics, as well as of advanced data assimilation methodology. He has systematically applied dynamical systems theory to planetary-scale flows, both atmospheric and oceanic. Ghil has used these methods to proceed from simple flows with high temporal regularity and spatial symmetry to the observed flows, with their complex behavior in space and time. His studies of climate variability on many time scales have used a full hierarchy of models, from the simplest ‘toy’ models all the way to atmospheric, oceanic and coupled general circulation models. Recently, Ghil has also worked on modeling and data analysis in population dynamics, macroeconomics, and the climate–economy–biosphere system.
In geology, numerical modeling is a widely applied technique to tackle complex geological problems by computational simulation of geological scenarios.
Alik Ismail-Zadeh is a mathematical geophysicist known for his contribution to computational geodynamics and natural hazard studies, pioneering work on data assimilation in geodynamics as well as for outstanding service to the Earth and space science community. He is Senior Research Fellow at the Karlsruhe Institute of Technology in Germany.
Gabor Korvin is a Hungarian Mathematician. He served as a professor at the Department of Earth Sciences, King Fahd University of Petroleum and Minerals. His main areas of research interest include fractal geometry in the earth sciences, statistical rock physics and mathematical geophysics. He is a well-known Applied Mathematician, Geophysicist, Petrophysicist, Historian. At KFUPM he was Coordinator of the Reservoir Characterization Research Group. As Professor, he taught Reservoir Characterization, Seismic Stratigraphy, Petrophysics & Well logging, Solid Earth Geophysics, Geoelectric Exploration, Reflection Seismology, Inverse Problems, Geostatistics and Reservoir Characterization. Fractal Models in the Earth Sciences by Gabor Korvin was one of the earlier books on the application of Fractals in the Earth Sciences.
Susan Marian Ellis is a geophysicist based in New Zealand, who specialises in modelling the geodynamics of the Earth's crust deformation, at different scales. Ellis is a principal scientist at GNS Science and her main interests are in subduction, seismology, tectonics, crust and petrology. Ellis's current work focuses on the influence of faulting on stresses in the crust, and how this is related to geological hazard and the tectonic settings in New Zealand.
Michael Semenovich Zhdanov is a geophysicist, academic and author. He is a Distinguished Professor in the Department of Geology and Geophysics at the University of Utah, Director of the Consortium for Electromagnetic Modeling and Inversion (CEMI), as well as the Founder, chairman and CEO of TechnoImaging.
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