Geomathematics
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Geomathematics (also: mathematical geosciences, mathematical geology, mathematical geophysics) is the application of mathematical methods to solve problems in
geosciences Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four spheres ...
, including
geology Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
and
geophysics 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. The term ''geophysics'' som ...
, and particularly geodynamics and
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
.


Applications


Geophysical fluid dynamics

Geophysical fluid dynamics Geophysical fluid dynamics, in its broadest meaning, refers to the fluid dynamics of naturally occurring flows, such as lava flows, oceans, and planetary atmospheres, on Earth and other planets. Two physical features that are common to many of th ...
develops the theory of
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
for the atmosphere, ocean and Earth's interior. Applications include geodynamics and the theory of the geodynamo.


Geophysical inverse theory

Geophysical
inverse theory 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, sound source reconstruction, source reconstruction in ac ...
is concerned with analyzing geophysical data to get model parameters. 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. 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.


Fractals and complexity

Many geophysical data sets have spectra that follow a
power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
, meaning that the frequency of an observed magnitude varies as some power of the magnitude. An example is the distribution of
earthquake An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...
magnitudes; small earthquakes are far more common than large earthquakes. This is often an indicator that the data sets have an underlying
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
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 In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
of the set, which is generally different from the more familiar
topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topological invariant, topologically invariant way. Informal discussion F ...
. Fractal phenomena are associated with chaos, self-organized criticality and
turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
. Fractal Models in the Earth Sciences by
Gabor Korvin Gabor Korvin (born in 1942 in Hungary) 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 th ...
was one of the earlier books on the application of Fractals in the Earth Sciences.


Data assimilation

Data assimilation Data assimilation is a mathematical discipline that seeks to optimally combine theory (usually in the form of a numerical model) with observations. There may be a number of different goals sought – for example, to determine the optimal state es ...
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 In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
. 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 Weather forecasting is the application of science and technology forecasting, to predict the conditions of the Earth's atmosphere, atmosphere for a given location and time. People have attempted to predict the weather informally for millennia a ...
.


Geophysical statistics

Some statistical problems come under the heading of mathematical geophysics, including
model validation In statistics, model validation is the task of evaluating whether a chosen statistical model is appropriate or not. Oftentimes in statistical inference, inferences from models that appear to fit their data may be flukes, resulting in a misunderstan ...
and quantifying uncertainty.


Terrestrial Tomography

An important research area that utilises inverse methods is
seismic tomography Seismic tomography or seismotomography is a technique for imaging the subsurface of the Earth with seismic waves produced by earthquakes or explosions. P-, S-, and surface waves can be used for tomographic models of different resolutions based on ...
, a technique for imaging the subsurface of the Earth using seismic waves. Traditionally seismic waves produced by
earthquakes An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...
or anthropogenic seismic sources (e.g., explosives, marine air guns) were used.


Crystallography

Crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
is one of the traditional areas of
geology Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
that use
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. Crystallographers make use of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
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 An electron microscope is a microscope that uses a beam of accelerated electrons as a source of illumination. As the wavelength of an electron can be up to 100,000 times shorter than that of visible light photons, electron microscopes have a hi ...
to be able to show relationship between light diffraction angles, wavelength, and the d-spacings within a sample.


Geophysics

Geophysics 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. The term ''geophysics'' som ...
is one of the most
math Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
heavy disciplines of
Earth Science Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four spheres ...
. There are many applications which include
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
,
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particle ...
, 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 field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s. 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.


Geomorphology

Many applications of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
in
geomorphology Geomorphology (from Ancient Greek: , ', "earth"; , ', "form"; and , ', "study") is the scientific study of the origin and evolution of topographic and bathymetric features created by physical, chemical or biological processes operating at or n ...
are related to water. In the
soil Soil, also commonly referred to as earth or dirt, is a mixture of organic matter, minerals, gases, liquids, and organisms that together support life. Some scientific definitions distinguish ''dirt'' from ''soil'' by restricting the former te ...
aspect things like Darcy's law, Stoke's law, and
porosity Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure ...
are used. * Darcy's law is used when one has a saturated soil that is uniform to describe how
fluid flows In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
through that medium. This type of work would fall under
hydrogeology Hydrogeology (''hydro-'' meaning water, and ''-geology'' meaning the study of the Earth) is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust (commonly in aquif ...
. * Stoke's law measures how quickly different sized particles will settle out of a fluid. This is used when doing pipette analysis of soils to find the percentage sand vs silt vs clay. A potential error is it assumes perfectly spherical particles which don't exist. *
Stream power Stream power originally derived by R. A. Bagnold in the 1960s is the amount of energy the water in a river or stream is exerting on the sides and bottom of the river. Stream power is the result of multiplying the density of the water, the acceler ...
is used to find the ability of a river to
incise Incision may refer to: * Cutting, the separation of an object, into two or more portions, through the application of an acutely directed force * A type of open wound caused by a clean, sharp-edged object such as a knife, razor, or glass splinter * ...
into the
river bed A stream bed or streambed is the bottom of a stream or river (bathymetry) or the physical confine of the normal water flow (channel). The lateral confines or channel margins are known as the stream banks or river banks, during all but flood st ...
. This is applicable to see where a river is likely to fail and change course or when looking at the damage of losing stream sediments on a river system (like downstream of a dam). *
Differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s can be used in multiple areas of
geomorphology Geomorphology (from Ancient Greek: , ', "earth"; , ', "form"; and , ', "study") is the scientific study of the origin and evolution of topographic and bathymetric features created by physical, chemical or biological processes operating at or n ...
including: The exponential growth equation, distribution of sedimentary rocks, diffusion of gas through rocks, and crenulation cleavages.


Glaciology

Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
in
Glaciology Glaciology (; ) is the scientific study of glaciers, or more generally ice and natural phenomena that involve ice. Glaciology is an interdisciplinary Earth science that integrates geophysics, geology, physical geography, geomorphology, climato ...
consists of theoretical, experimental, and modeling. It usually covers
glacier A glacier (; ) is a persistent body of dense ice that is constantly moving under its own weight. A glacier forms where the accumulation of snow exceeds its Ablation#Glaciology, ablation over many years, often Century, centuries. It acquires dis ...
s,
sea ice Sea ice arises as seawater freezes. Because ice is less dense than water, it floats on the ocean's surface (as does fresh water ice, which has an even lower density). Sea ice covers about 7% of the Earth's surface and about 12% of the world's oce ...
, 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 In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
to show the elastic characteristics while using
Lamé constants Lamé may refer to: *Lamé (fabric), a clothing fabric with metallic strands *Lamé (fencing), a jacket used for detecting hits * Lamé (crater) on the Moon * Ngeté-Herdé language, also known as Lamé, spoken in Chad *Peve language, also known ...
. Generally the ice has its linear
elasticity Elasticity often refers to: *Elasticity (physics), continuum mechanics of bodies that deform reversibly under stress Elasticity may also refer to: Information technology * Elasticity (data store), the flexibility of the data model and the cl ...
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 Bar or BAR may refer to: Food and drink * Bar (establishment), selling alcoholic beverages * Candy bar * Chocolate bar Science and technology * Bar (river morphology), a deposit of sediment * Bar (tropical cyclone), a layer of cloud * Bar (u ...
. This type of ice system is where one would test for
creep Creep, Creeps or CREEP may refer to: People * Creep, a creepy person Politics * Committee for the Re-Election of the President (CRP), mockingly abbreviated as CREEP, an fundraising organization for Richard Nixon's 1972 re-election campaign Art ...
or
vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
s 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
glacier A glacier (; ) is a persistent body of dense ice that is constantly moving under its own weight. A glacier forms where the accumulation of snow exceeds its Ablation#Glaciology, ablation over many years, often Century, centuries. It acquires dis ...
s 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.


Academic journals

*''
International Journal on Geomathematics International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' (Kevin Michael album), 2011 * International (New Order album), ' ...
'' *''
Mathematical Geosciences ''Mathematical Geosciences'' (formerly ''Mathematical Geology'') is a scientific journal published semi-quarterly by Springer Science+Business Media on behalf of the International Association for Mathematical Geosciences. It contains original pape ...
''


See also

* Geocomputation * Geoinformatics *
International Association for Mathematical Geosciences The International Association for Mathematical Geosciences (IAMG) is a nonprofit organization of geoscientists. It aims to promote international cooperation in the application and use of mathematics in geological research and technology. IAMG's act ...
(IAMG)


References


Further reading

*
Development, significance, and influence of geomathematics: Observations of one geologist
Daniel F. Merriam Daniel Francis Merriam (February 2, 1927 – April 26, 2017) was an American geologist best known for fostering the development of quantitative modeling in geology after the advent of digital computers. He first joined the Kansas Geological Sur ...
, ''Mathematical Geology'', Volume 14, Number 1 / February, 1982 * * * * * * * {{Geophysics navbox Applied mathematics Earth sciences Geology Geophysics