Mathematical geophysics
Encyclopedia
Mathematical geophysics is concerned with developing mathematical methods for use in geophysics
. As such, it has application in many fields in geophysics
, particularly geodynamics
and seismology
.
develops the theory of fluid dynamics
for the atmosphere, ocean and Earth's interior. Applications include geodynamics
and the theory of the geodynamo.
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
.
, 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
.
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
.
Geophysics
Geophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...
. As such, it has application in many fields in geophysics
Geophysics
Geophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...
, particularly geodynamics
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, mountain building, volcanoes, earthquakes, faulting and...
and seismology
Seismology
Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...
.
Geophysical fluid dynamics
Geophysical fluid dynamicsGeophysical fluid dynamics
Geophysical fluid dynamics is the study of naturally occurring, large-scale flows on Earth and other planets. It is applied to the motion of fluids in the ocean and outer core, and to gases in the atmosphere of Earth and other planets...
develops the theory of fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
for the atmosphere, ocean and Earth's interior. Applications include geodynamics
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, mountain building, volcanoes, earthquakes, faulting and...
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
Magnetotellurics is an electromagnetic geophysical method of imaging the earth's subsurface by measuring natural variations of electrical and magnetic fields at the Earth's surface. Investigation depth ranges from 300m below ground by recording higher frequencies down to 10,000m or deeper with...
and seismology
Seismology
Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...
.
Fractals and complexity
Many geophysical data sets have spectra that follow a power lawPower law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
, meaning that the frequency of an observed magnitude varies as some power of the magnitude. An example is the distribution of earthquake
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...
magnitudes; small earthquakes are far more common than large earthquakes. This is often an indicator that the data sets have an underlying fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
geometry. Fractal sets have a number of common features, including structure at many scales, irregularity, and self-similarity
Self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...
(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
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
of the set, which is generally different from the more familiar topological dimension. Fractal phenomena are associated with chaos
Chaos
- Mythology, philosophy, and religion :* Chaos , the concept in classical creation myths** a chasm or abyss, especially in biblical usage, see Abyss** prima materia, the primordial state from which the cosmos originated...
, self-organized criticality
Self-organized criticality
In physics, self-organized criticality is a property of dynamical systems which have a critical point as an attractor. Their macroscopic behaviour thus displays the spatial and/or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune...
and turbulence
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
.
Data assimilation
Data assimilationData assimilation
Applications of data assimilation arise in many fields of geosciences, perhaps most importantly in weather forecasting and hydrology. Data assimilation proceeds by analysis cycles...
combines numerical models of geophysical systems with observations that may be irregular in space and time. Many of the applications involve geophysical fluid dynamics
Geophysical fluid dynamics
Geophysical fluid dynamics is the study of naturally occurring, large-scale flows on Earth and other planets. It is applied to the motion of fluids in the ocean and outer core, and to gases in the atmosphere of Earth and other planets...
. 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
Weather forecasting
Weather forecasting is the application of science and technology to predict the state of the atmosphere for a given location. Human beings have attempted to predict the weather informally for millennia, and formally since the nineteenth century...
.