Geomathematics (also: mathematical geosciences, mathematical geology, mathematical geophysics) is the application of

mathematical 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 ...

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 spher ...

, 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 Ea ...

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 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, ...

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) ...

for the atmosphere, ocean and Earth's interior. Applications include geodynamics and the theory of the geodynamo.

Geophysical inverse theory

Geophysical inverse theory 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 Magnetotellurics (MT) is an electromagnetic geophysical method for inferring the earth's subsurface electrical conductivity from measurements of natural geomagnetic and geoelectric field variation at the Earth's surface. Investigation depth ...

and seismology.

Fractals and complexity

Many geophysical data sets have spectra that follow a

power law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one qua ...

, 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 il ...

geometry. Fractal sets have a number of common features, including structure at many scales, irregularity, and

self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically s ...

(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, o ...

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 topologically invariant way. Informal discussion For ordinary Euclidean ...

. 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 was one of the earlier books on the application of

Fractals 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 ill ...

in the

Earth Sciences 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 spher ...

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 to predict the conditions of the atmosphere for a given location and time. People have attempted to predict the weather informally for millennia and formally since the 19th centu ...

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 misundersta ...

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, fr ...

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 wo ...

is one of the traditional areas of

that use 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 matric ...

by using the Metrical Matrix. The Metrical Matrix uses the basis vectors of the

unit cell In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessari ...

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 ...

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 spher ...

. 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 str ...

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 particles ...

seismic 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 ...

electric Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described b ...

electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...

, radioactivity, induced polarization, and

well logging Well logging, also known as borehole logging is the practice of making a detailed record (a ''well log'') of the geologic formations penetrated by a borehole. The log may be based either on visual inspection of samples brought to the surface ...

. 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 t ...

s. Gravity is used often for

oil exploration Hydrocarbon exploration (or oil and gas exploration) is the search by petroleum geologists and geophysicists for deposits of hydrocarbons, particularly petroleum and natural gas, in the Earth Earth is the third planet from the Sun ...

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 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 ...

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 ...

aspect things like

Darcy's law Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of ...

, 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. *

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 rock (geology), rocks of the Earth's crust (ge ...

. * 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 accelera ...

is used to find the ability of a river to incise 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 ...

. 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, a ...

s can be used in multiple areas of

including: The exponential growth equation, distribution of sedimentary rocks, diffusion of gas through rocks, and crenulation cleavages.

Glaciology

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, c ...

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 over many years, often centuries. It acquires distinguishing features, such a ...

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 o ...

, waterflow, and the land under the glacier.

Polycrystalline A crystallite is a small or even microscopic crystal which forms, for example, during the cooling of many materials. Crystallites are also referred to as grains. Bacillite is a type of crystallite. It is rodlike with parallel longulites. Stru ...

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 by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...

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 kno ...

. Generally the ice has its linear elasticity constants averaged over one dimension of space to simplify the equations while still maintaining accuracy.

Viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linear ...

polycrystalline A crystallite is a small or even microscopic crystal which forms, for example, during the cooling of many materials. Crystallites are also referred to as grains. Bacillite is a type of crystallite. It is rodlike with parallel longulites. Stru ...

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 (un ...

. 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 ...

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, such as the motion of a pendulum—or random, suc ...

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

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), 2011 * ''International'' (New Order album), 2002 * ''International'' (The T ...

'' *''

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 ...