Magnitude 7

[Nguyet Edmondson]

Inmathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering (or ranking) of the class of objects to which it belongs. Magnitude as a concept dates to Ancient Greece and has been applied as a measure of distance from one object to another. For numbers, the absolute value of a number is commonly applied as the measure of units between a number and zero.

In vector spaces, the Euclidean norm is a measure of magnitude used to define a distance between two points in space. In physics, magnitude can be defined as quantity or distance. An order of magnitude is typically defined as a unit of distance between one number and another's numerical places on the decimal scale.

They proved that the first two could not be the same, or even isomorphic systems of magnitude.[2] They did not consider negative magnitudes to be meaningful, and magnitude is still primarily used in contexts in which zero is either the smallest size or less than all possible sizes.

A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space.[8] The norm of a vector v in a normed vector space can be considered to be the magnitude of v.

When comparing magnitudes, a logarithmic scale is often used. Examples include the loudness of a sound (measured in decibels), the brightness of a star, and the Richter scale of earthquake intensity. Logarithmic magnitudes can be negative. In the natural sciences, a logarithmic magnitude is typically referred to as a level.

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.

The study of earthquakes is challenging as the source events cannot be observed directly, and it took many years to develop the mathematics for understanding what the seismic waves from an earthquake can tell about the source event. An early step was to determine how different systems of forces might generate seismic waves equivalent to those observed from earthquakes.[15]

The simplest force system is a single force acting on an object. If it has sufficient strength to overcome any resistance it will cause the object to move ("translate"). A pair of forces, acting on the same "line of action" but in opposite directions, will cancel; if they cancel (balance) exactly there will be no net translation, though the object will experience stress, either tension or compression. If the pair of forces are offset, acting along parallel but separate lines of action, the object experiences a rotational force, or torque. In mechanics (the branch of physics concerned with the interactions of forces) this model is called a couple, also simple couple or single couple. If a second couple of equal and opposite magnitude is applied their torques cancel; this is called a double couple.[16] A double couple can be viewed as "equivalent to a pressure and tension acting simultaneously at right angles".[17]

The single couple and double couple models are important in seismology because each can be used to derive how the seismic waves generated by an earthquake event should appear in the "far field" (that is, at distance). Once that relation is understood it can be inverted to use the earthquake's observed seismic waves to determine its other characteristics, including fault geometry and seismic moment.[citation needed]

The debate ended when Maruyama (1963), Haskell (1964), and Burridge and Knopoff (1964) showed that if earthquake ruptures are modeled as dislocations the pattern of seismic radiation can always be matched with an equivalent pattern derived from a double couple,[citation needed] but not from a single couple.[22] This was confirmed as better and more plentiful data coming from the World-Wide Standard Seismograph Network (WWSSN) permitted closer analysis of seismic waves. Notably, in 1966 Keiiti Aki showed that the seismic moment of the 1964 Niigata earthquake as calculated from the seismic waves on the basis of a double couple was in reasonable agreement with the seismic moment calculated from the observed physical dislocation.[23]

A double couple model suffices to explain an earthquake's far-field pattern of seismic radiation, but tells us very little about the nature of an earthquake's source mechanism or its physical features.[24] While slippage along a fault was theorized as the cause of earthquakes (other theories included movement of magma, or sudden changes of volume due to phase changes[25]), observing this at depth was not possible, and understanding what could be learned about the source mechanism from the seismic waves requires an understanding of the source mechanism.[5]

In a pair of papers in 1958, J. A. Steketee worked out how to relate dislocation theory to geophysical features.[30] Numerous other researchers worked out other details,[31] culminating in a general solution in 1964 by Burridge and Knopoff, which established the relationship between double couples and the theory of elastic rebound, and provided the basis for relating an earthquake's physical features to seismic moment.[32]

The first calculation of an earthquake's seismic moment from its seismic waves was by Keiiti Aki for the 1964 Niigata earthquake.[36] He did this two ways. First, he used data from distant stations of the WWSSN to analyze long-period (200 second) seismic waves (wavelength of about 1,000 kilometers) to determine the magnitude of the earthquake's equivalent double couple.[37] Second, he drew upon the work of Burridge and Knopoff on dislocation to determine the amount of slip, the energy released, and the stress drop (essentially how much of the potential energy was released).[38] In particular, he derived an equation that relates an earthquake's seismic moment to its physical parameters:

with μ being the rigidity (or resistance to moving) of a fault with a surface area of S over an average dislocation (distance) of ū. (Modern formulations replace ūS with the equivalent D̄A, known as the "geometric moment" or "potency".[39]) By this equation the moment determined from the double couple of the seismic waves can be related to the moment calculated from knowledge of the surface area of fault slippage and the amount of slip. In the case of the Niigata earthquake the dislocation estimated from the seismic moment reasonably approximated the observed dislocation.[40]

Seismic moment is a measure of the work (more precisely, the torque) that results in inelastic (permanent) displacement or distortion of the Earth's crust.[41] It is related to the total energy released by an earthquake. However, the power or potential destructiveness of an earthquake depends (among other factors) on how much of the total energy is converted into seismic waves.[42] This is typically 10% or less of the total energy, the rest being expended in fracturing rock or overcoming friction (generating heat).[43]

Nonetheless, seismic moment is regarded as the fundamental measure of earthquake size,[44] representing more directly than other parameters the physical size of an earthquake.[45] As early as 1975 it was considered "one of the most reliably determined instrumental earthquake source parameters".[46]

Most earthquake magnitude scales suffered from the fact that they only provided a comparison of the amplitude of waves produced at a standard distance and frequency band; it was difficult to relate these magnitudes to a physical property of the earthquake. Gutenberg and Richter suggested that radiated energy Es could be estimated as

Seismic moment is not a direct measure of energy changes during an earthquake. The relations between seismic moment and the energies involved in an earthquake depend on parameters that have large uncertainties and that may vary between earthquakes. Potential energy is stored in the crust in the form of elastic energy due to built-up stress and gravitational energy.[53] During an earthquake, a portion Δ W \displaystyle \Delta W of this stored energy is transformed into

To make the significance of the magnitude value plausible, the seismic energy released during the earthquake is sometimes compared to the effect of the conventional chemical explosive TNT.The seismic energy E S \displaystyle E_\mathrm S results from the above-mentioned formula according to Gutenberg and Richter to

For comparison of seismic energy (in joules) with the corresponding explosion energy, a value of 4.2 x 109 joules per ton of TNT applies.The table[55] illustrates the relationship between seismic energy and moment magnitude.

Ocean drilling expeditions have repeatedly probed those lightless depths and uncovered cells that survive almost in suspended animation, consuming orders of magnitude less energy than their neighbors at the surface.

Earthquake magnitude, energy release, and shaking intensity are all related measurements of an earthquake that are often confused with one another. Their dependencies and relationships can be complicated, and even one of these concepts alone can be confusing.

The time, location, and magnitude of an earthquake can be determined from the data recorded by seismometer. Seismometers record the vibrations from earthquakes that travel through the Earth. Each seismometer records the shaking of the ground directly beneath it. Sensitive instruments, which greatly magnify these ground motions, can detect strong earthquakes from sources anywhere in the world. Modern systems precisely amplify and record ground motion (typically at periods of between 0.1 and 100 seconds) as a function of time.

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