Microcanonical ensemble

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https://en.wikipedia.org/wiki/Microcanonical_ensemble

In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it cannot exchange energy or particles with its environment, so that (by conservation of energy) the energy of the system does not change with time.

The primary macroscopic variables of the microcanonical ensemble are the total number of particles in the system (symbol: N), the system's volume (symbol: V), as well as the total energy in the system (symbol: E). Each of these is assumed to be constant in the ensemble. For this reason, the microcanonical ensemble is sometimes called the NVE ensemble.

In simple terms, the microcanonical ensemble is defined by assigning an equal probability to every microstate whose energy falls within a range centered at E. All other microstates are given a probability of zero. Since the probabilities must add up to 1, the probability P is the inverse of the number of microstates W within the range of energy,

${\displaystyle P=1/W,}$

The range of energy is then reduced in width until it is infinitesimally narrow, still centered at E. In the limit of this process, the microcanonical ensemble is obtained.

Applicability

Because of its connection with the elementary assumptions of equilibrium statistical mechanics (particularly the postulate of a priori equal probabilities), the microcanonical ensemble is an important conceptual building block in the theory. It is sometimes considered to be the fundamental distribution of equilibrium statistical mechanics. It is also useful in some numerical applications, such as molecular dynamics. On the other hand, most nontrivial systems are mathematically cumbersome to describe in the microcanonical ensemble, and there are also ambiguities regarding the definitions of entropy and temperature. For these reasons, other ensembles are often preferred for theoretical calculations.

The applicability of the microcanonical ensemble to real-world systems depends on the importance of energy fluctuations, which may result from interactions between the system and its environment as well as uncontrolled factors in preparing the system. Generally, fluctuations are negligible if a system is macroscopically large, or if it is manufactured with precisely known energy and thereafter maintained in near isolation from its environment. In such cases the microcanonical ensemble is applicable. Otherwise, different ensembles are more appropriate—such as the canonical ensemble (fluctuating energy) or the grand canonical ensemble (fluctuating energy and particle number).

Properties

Thermodynamic quantities

The fundamental thermodynamic potential of the microcanonical ensemble is entropy. There are at least three possible definitions, each given in terms of the phase volume function v(E), which counts the total number of states with energy less than E (see the Precise expressions section for the mathematical definition of v):

• the Boltzmann entropy

  ${\displaystyle S_{\text{B}}=k\log W=k\log \left(\omega \frac{dv}{dE}\right),}$

• the 'volume entropy'
  ${\displaystyle S_v=k\log v,}$
• the 'surface entropy'
  ${\displaystyle S_s=k\log \frac{dv}{dE}=S_{\text{B}}-k\log \omega .}$

In the microcanonical ensemble, the temperature is a derived quantity rather than an external control parameter. It is defined as the derivative of the chosen entropy with respect to energy. For example, one can define the "temperatures" T_v and T_s as follows:

${\displaystyle 1/T_v=d{S}_v/dE,}$

${\displaystyle 1/T_s=d{S}_s/dE=d{S}_{\text{B}}/dE.}$

Like entropy, there are multiple ways to understand temperature in the microcanonical ensemble. More generally, the correspondence between these ensemble-based definitions and their thermodynamic counterparts is not perfect, particularly for finite systems.

The microcanonical pressure and chemical potential are given by:

${\displaystyle \frac{p}{T}=\frac{\mathrm{\partial }S}{\mathrm{\partial }V};\frac{\mu }{T}=-\frac{\mathrm{\partial }S}{\mathrm{\partial }N}}$

Phase transitions

Under their strict definition, phase transitions correspond to nonanalytic behavior in the thermodynamic potential or its derivatives. Using this definition, phase transitions in the microcanonical ensemble can occur in systems of any size. This contrasts with the canonical and grand canonical ensembles, for which phase transitions can occur only in the thermodynamic limit—i.e., in systems with infinitely many degrees of freedom. Roughly speaking, the reservoirs defining the canonical or grand canonical ensembles introduce fluctuations that "smooth out" any nonanalytic behavior in the free energy of finite systems. This smoothing effect is usually negligible in macroscopic systems, which are sufficiently large that the free energy can approximate nonanalytic behavior exceedingly well. However, the technical difference in ensembles may be important in the theoretical analysis of small systems.

Information entropy

For a given mechanical system (fixed N, V) and a given range of energy, the uniform distribution of probability P over microstates (as in the microcanonical ensemble) maximizes the ensemble average −⟨log P⟩.

Thermodynamic analogies

Early work in statistical mechanics by Ludwig Boltzmann led to his eponymous entropy equation for a system of a given total energy, S = k log W, where W is the number of distinct states accessible by the system at that energy. Boltzmann did not elaborate too deeply on what exactly constitutes the set of distinct states of a system, besides the special case of an ideal gas. This topic was investigated to completion by Josiah Willard Gibbs who developed the generalized statistical mechanics for arbitrary mechanical systems, and defined the microcanonical ensemble described in this article. Gibbs investigated carefully the analogies between the microcanonical ensemble and thermodynamics, especially how they break down in the case of systems of few degrees of freedom. He introduced two further definitions of microcanonical entropy that do not depend on ω - the volume and surface entropy described above. (Note that the surface entropy differs from the Boltzmann entropy only by an ω-dependent offset.)

The volume entropy S_v and associated T_v form a close analogy to thermodynamic entropy and temperature. It is possible to show exactly that

${\displaystyle dE=T_{\mathrm{v}}d{S}_{\mathrm{v}}-⟨P⟩dV,}$

(⟨P⟩ is the ensemble average pressure) as expected for the first law of thermodynamics. A similar equation can be found for the surface (Boltzmann) entropy and its associated T_s, however the "pressure" in this equation is a complicated quantity unrelated to the average pressure.

The microcanonical T_v and T_s are not entirely satisfactory in their analogy to temperature. Outside of the thermodynamic limit, a number of artefacts occur.

• Nontrivial result of combining two systems: Two systems, each described by an independent microcanonical ensemble, can be brought into thermal contact and be allowed to equilibriate into a combined system also described by a microcanonical ensemble. Unfortunately, the energy flow between the two systems cannot be predicted based on the initial T's. Even when the initial T's are equal, there may be energy transferred. Moreover, the T of the combination is different from the initial values. This contradicts the intuition that temperature should be an intensive quantity, and that two equal-temperature systems should be unaffected by being brought into thermal contact.
• Strange behavior for few-particle systems: Many results such as the microcanonical Equipartition theorem acquire a one- or two-degree of freedom offset when written in terms of T_s. For a small systems this offset is significant, and so if we make S_s the analogue of entropy, several exceptions need to be made for systems with only one or two degrees of freedom.
• Spurious negative temperatures: A negative T_s occurs whenever the density of states is decreasing with energy. In some systems the density of states is not monotonic in energy, and so T_s can change sign multiple times as the energy is increased.

The preferred solution to these problems is avoid use of the microcanonical ensemble. In many realistic cases a system is thermostatted to a heat bath so that the energy is not precisely known. Then, a more accurate description is the canonical ensemble or grand canonical ensemble, both of which have complete correspondence to thermodynamics.

Precise expressions for the ensemble

The precise mathematical expression for a statistical ensemble depends on the kind of mechanics under consideration—quantum or classical—since the notion of a "microstate" is considerably different in these two cases. In quantum mechanics, diagonalization provides a discrete set of microstates with specific energies. The classical mechanical case involves instead an integral over canonical phase space, and the size of microstates in phase space can be chosen somewhat arbitrarily.

To construct the microcanonical ensemble, it is necessary in both types of mechanics to first specify a range of energy. In the expressions below the function ${\displaystyle f\left(\frac{H-E}{\omega }\right)}$ (a function of H, peaking at E with width ω) will be used to represent the range of energy in which to include states. An example of this function would be

${\displaystyle f(x)=\{\begin{array}{ll}1,& \mathrm{i}\mathrm{f}|x|<\frac{1}{2},\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}}$

or, more smoothly,

${\displaystyle f(x)=e^{-\pi x^2}.}$

Quantum mechanical

Example of microcanonical ensemble for a quantum system consisting of one particle in a potential well.

 

Plot of all possible states of this system. The available stationary states displayed as horizontal bars of varying darkness according to |ψ_i(x)|².

 

An ensemble containing only those states within a narrow interval of energy. As the energy width is taken to zero, a microcanonical ensemble is obtained (provided the interval contains at least one state).

The particle's Hamiltonian is Schrödinger-type, Ĥ = U(x) + p²/2m (the potential U(x) is plotted as a red curve). Each panel shows an energy-position plot with the various stationary states, along with a side plot showing the distribution of states in energy.

 

Further information on the representation of ensembles in quantum mechanics: Statistical ensemble (mathematical physics)

A statistical ensemble in quantum mechanics is represented by a density matrix, denoted by ${\displaystyle \hat{\rho }}$. The microcanonical ensemble can be written using bra–ket notation, in terms of the system's energy eigenstates and energy eigenvalues. Given a complete basis of energy eigenstates |ψ_i⟩, indexed by i, the microcanonical ensemble is

${\displaystyle \hat{\rho }=\frac{1}{W}\sum _{i}f\left(\frac{H_i-E}{\omega }\right)|{\psi }_i⟩⟨{\psi }_i|,}$

where the H_i are the energy eigenvalues determined by ${\displaystyle \hat{H}|{\psi }_i⟩=H_i|{\psi }_i⟩}$ (here Ĥ is the system's total energy operator, i. e., Hamiltonian operator). The value of W is determined by demanding that ${\displaystyle \hat{\rho }}$ is a normalized density matrix, and so

${\displaystyle W=\sum _{i}f\left(\frac{H_i-E}{\omega }\right).}$

The state volume function (used to calculate entropy) is given by

${\displaystyle v(E)=\sum _{H_i<E}1.}$

The microcanonical ensemble is defined by taking the limit of the density matrix as the energy width goes to zero, however a problematic situation occurs once the energy width becomes smaller than the spacing between energy levels. For very small energy width, the ensemble does not exist at all for most values of E, since no states fall within the range. When the ensemble does exist, it typically only contains one (or two) states, since in a complex system the energy levels are only ever equal by accident (see random matrix theory for more discussion on this point). Moreover, the state-volume function also increases only in discrete increments, and so its derivative is only ever infinite or zero, making it difficult to define the density of states. This problem can be solved by not taking the energy range completely to zero and smoothing the state-volume function, however this makes the definition of the ensemble more complicated, since it becomes then necessary to specify the energy range in addition to other variables (together, an NVEω ensemble).

Classical mechanical

Example of microcanonical ensemble for a classical system consisting of one particle in a potential well.

 

Plot of all possible states of this system. The available physical states are evenly distributed in phase space, but with an uneven distribution in energy; the side-plot displays dv/dE.

 

An ensemble restricted to only those states within a narrow interval of energy. This ensemble appears as a thin shell in phase space. As the energy width is taken to zero, a microcanonical ensemble is obtained.

Each panel shows phase space (upper graph) and energy-position space (lower graph). The particle's Hamiltonian is H = U(x) + p²/2m, with the potential U(x) shown as a red curve. The side plot shows the distribution of states in energy.

 

Further information on the representation of ensembles in classical mechanics: Statistical ensemble (mathematical physics)

In classical mechanics, an ensemble is represented by a joint probability density function ρ(p₁, … p_n, q₁, … q_n) defined over the system's phase space. The phase space has n generalized coordinates called q₁, … q_n, and n associated canonical momenta called p₁, … p_n.

The probability density function for the microcanonical ensemble is:

${\displaystyle \rho =\frac{1}{h^{n}C}\frac{1}{W}f\left(\frac{H-E}{\omega }\right),}$

where

• H is the total energy (Hamiltonian) of the system, a function of the phase (p₁, … q_n),
• h is an arbitrary but predetermined constant with the units of energy×time, setting the extent of one microstate and providing correct dimensions to ρ.
• C is an overcounting correction factor, often used for particle systems where identical particles are able to change place with each other.

Again, the value of W is determined by demanding that ρ is a normalized probability density function:

${\displaystyle W=\int \dots \int \frac{1}{h^{n}C}f\left(\frac{H-E}{\omega }\right)d{p}_1\dots d{q}_n}$

This integral is taken over the entire phase space. The state volume function (used to calculate entropy) is defined by

${\displaystyle v(E)=\int \dots {\int }_{H<E}\frac{1}{h^{n}C}d{p}_1\dots d{q}_n.}$

As the energy width ω is taken to zero, the value of W decreases in proportion to ω as W = ω (dv/dE).

Based on the above definition, the microcanonical ensemble can be visualized as an infinitesimally thin shell in phase space, centered on a constant-energy surface. Although the microcanonical ensemble is confined to this surface, it is not necessarily uniformly distributed over that surface: if the gradient of energy in phase space varies, then the microcanonical ensemble is "thicker" (more concentrated) in some parts of the surface than others. This feature is an unavoidable consequence of requiring that the microcanonical ensemble is a steady-state ensemble.

Examples

Ideal gas

The fundamental quantity in the microcanonical ensemble is ${\displaystyle W(E,V,N)}$, which is equal to the phase space volume compatible with given ${\displaystyle (E,V,N)}$. From ${\displaystyle W}$, all thermodynamic quantities can be calculated. For an ideal gas, the energy is independent of the particle positions, which therefore contribute a factor of ${\displaystyle V^N}$ to ${\displaystyle W}$. The momenta, by contrast, are constrained to a ${\displaystyle 3N}$-dimensional (hyper-)spherical shell of radius ${\displaystyle \sqrt{2mE}}$; their contribution is equal to the surface volume of this shell. The resulting expression for ${\displaystyle W}$ is:

${\displaystyle W=\frac{V^N}{N!}\frac{2{\pi }^{3N/2}}{\mathrm{\Gamma }(3N/2)}{\left(2mE\right)}^{(3N-1)/2}}$

where ${\displaystyle \mathrm{\Gamma }(.)}$ is the gamma function, and the factor ${\displaystyle N!}$ has been included to account for the indistinguishability of particles (see Gibbs paradox). In the large ${\displaystyle N}$ limit, the Boltzmann entropy ${\displaystyle S=k_{\mathrm{B}}\log W}$ is

${\displaystyle S=k_{\mathrm{B}}N\log \left[\frac{V}{N}{\left(\frac{4\pi m}{3}\frac{E}{N}\right)}^{3/2}\right]+\frac{5}{2}{k}_{\mathrm{B}}N+O\left(\log N\right)}$

This is also known as the Sackur–Tetrode equation.

The temperature is given by

${\displaystyle \frac{1}{T}\equiv \frac{\mathrm{\partial }S}{\mathrm{\partial }E}=\frac{3}{2}\frac{N{k}_{\mathrm{B}}}{E}}$

which agrees with analogous result from the kinetic theory of gases. Calculating the pressure gives the ideal gas law:

${\displaystyle \frac{p}{T}\equiv \frac{\mathrm{\partial }S}{\mathrm{\partial }V}=\frac{N{k}_{\mathrm{B}}}{V}\to pV=N{k}_{\mathrm{B}}T}$

Finally, the chemical potential ${\displaystyle \mu }$ is

${\displaystyle \mu \equiv -T\frac{\mathrm{\partial }S}{\mathrm{\partial }N}=k_{\mathrm{B}}T\log \left[\frac{V}{N}{\left(\frac{4\pi mE}{3N}\right)}^{3/2}\right]}$

Ideal gas in a uniform gravitational field

The microcanonical phase volume can also be calculated explicitly for an ideal gas in a uniform gravitational field.

The results are stated below for a 3-dimensional ideal gas of ${\displaystyle N}$ particles, each with mass ${\displaystyle m}$, confined in a thermally isolated container that is infinitely long in the z-direction and has constant cross-sectional area ${\displaystyle A}$. The gravitational field is assumed to act in the minus z direction with strength ${\displaystyle g}$. The phase volume ${\displaystyle W(E,N)}$ is

${\displaystyle W(E,N)=\frac{(2\pi {)}^{3N/2}{A}^N{m}^{N/2}}{g^N\mathrm{\Gamma }(5N/2)}{E}^{\frac{5N}{2}-1}}$

where ${\displaystyle E}$ is the total energy, kinetic plus gravitational.

The gas density ${\displaystyle \rho (z)}$ as a function of height ${\displaystyle z}$ can be obtained by integrating over the phase volume coordinates. The result is:

${\displaystyle \rho (z)=\left(\frac{5N}{2}-1\right)\frac{mg}{E}{\left(1-\frac{mgz}{E}\right)}^{\frac{5N}{2}-2}}$

Similarly, the distribution of the velocity magnitude ${\displaystyle |\overrightarrow{v}|}$ (averaged over all heights) is

${\displaystyle f(|\overrightarrow{v}|)=\frac{\mathrm{\Gamma }(5N/2)}{\mathrm{\Gamma }(3/2)\mathrm{\Gamma }(5N/2-3/2)}\times \frac{m^{3/2}|\overrightarrow{v}{|}^2}{2^{1/2}{E}^{3/2}}\times {\left(1-\frac{m|\overrightarrow{v}{|}^2}{2E}\right)}^{\frac{5(N-1)}{2}}}$

The analogues of these equations in the canonical ensemble are the barometric formula and the Maxwell–Boltzmann distribution, respectively. In the limit ${\displaystyle N\to \mathrm{\infty }}$, the microcanonical and canonical expressions coincide; however, they differ for finite ${\displaystyle N}$. In particular, in the microcanonical ensemble, the positions and velocities are not statistically independent. As a result, the kinetic temperature, defined as the average kinetic energy in a given volume ${\displaystyle Adz}$, is nonuniform throughout the container:

${\displaystyle T_{\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c}}=\frac{3E}{5N-2}\left(1-\frac{mgz}{E}\right)}$

By contrast, the temperature is uniform in the canonical ensemble, for any ${\displaystyle N}$.

Satellite temperature measurements

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Satellite_temperature_measurements

Comparison of ground-based measurements of near-surface temperature (blue) and satellite based records of mid-tropospheric temperature (red: UAH; green: RSS) from 1979 to 2010. Trends plotted 1982-2010.

 

Atmospheric temperature trends from 1979-2016 based on satellite measurements; troposphere above, stratosphere below.

 

For broader coverage of this topic, see Temperature measurement.

Satellite temperature measurements are inferences of the temperature of the atmosphere at various altitudes as well as sea and land surface temperatures obtained from radiometric measurements by satellites. These measurements can be used to locate weather fronts, monitor the El Niño-Southern Oscillation, determine the strength of tropical cyclones, study urban heat islands and monitor the global climate. Wildfires, volcanos, and industrial hot spots can also be found via thermal imaging from weather satellites.

Weather satellites do not measure temperature directly. They measure radiances in various wavelength bands. Since 1978 microwave sounding units (MSUs) on National Oceanic and Atmospheric Administration polar orbiting satellites have measured the intensity of upwelling microwave radiation from atmospheric oxygen, which is related to the temperature of broad vertical layers of the atmosphere. Measurements of infrared radiation pertaining to sea surface temperature have been collected since 1967.

Satellite datasets show that over the past four decades the troposphere has warmed and the stratosphere has cooled. Both of these trends are consistent with the influence of increasing atmospheric concentrations of greenhouse gases.

Measurements

Satellites do not measure temperature directly. They measure radiances in various wavelength bands, which must then be mathematically inverted to obtain indirect inferences of temperature. The resulting temperature profiles depend on details of the methods that are used to obtain temperatures from radiances. As a result, different groups that have analyzed the satellite data have produced differing temperature datasets.

The satellite time series is not homogeneous. It is constructed from a series of satellites with similar but not identical sensors. The sensors also deteriorate over time, and corrections are necessary for orbital drift and decay. Particularly large differences between reconstructed temperature series occur at the few times when there is little temporal overlap between successive satellites, making intercalibration difficult.

Infrared measurements

Surface measurements

See also: Sea surface temperature § Weather satellites

 

Land surface temperature anomalies for a given month compared to the long-term average temperature of that month between 2000-2008.

 

Sea surface temperature anomalies for a given month compared to the long-term average temperature of that month from 1985 through 1997.

Infrared radiation can be used to measure both the temperature of the surface (using "window" wavelengths to which the atmosphere is transparent), and the temperature of the atmosphere (using wavelengths for which the atmosphere is not transparent, or measuring cloud top temperatures in infrared windows).

Satellites used to retrieve surface temperatures via measurement of thermal infrared in general require cloud-free conditions. Some of the instruments include the Advanced Very High Resolution Radiometer (AVHRR), Along Track Scanning Radiometers (AASTR), Visible Infrared Imaging Radiometer Suite (VIIRS), the Atmospheric Infrared Sounder (AIRS), and the ACE Fourier Transform Spectrometer (ACE‐FTS) on the Canadian SCISAT-1 satellite.

Weather satellites have been available to infer sea surface temperature (SST) information since 1967, with the first global composites occurring during 1970. Since 1982, satellites have been increasingly utilized to measure SST and have allowed its spatial and temporal variation to be viewed more fully. For example, changes in SST monitored via satellite have been used to document the progression of the El Niño-Southern Oscillation since the 1970s.

Over land the retrieval of temperature from radiances is harder, because of inhomogeneities in the surface. Studies have been conducted on the urban heat island effect via satellite imagery. By using the fractal technique, Weng, Q. et al. characterized the spatial pattern of urban heat island. Use of advanced very high resolution infrared satellite imagery can be used, in the absence of cloudiness, to detect density discontinuities (weather fronts) such as cold fronts at ground level. Using the Dvorak technique, infrared satellite imagery can be used to determine the temperature difference between the eye and the cloud top temperature of the central dense overcast of mature tropical cyclones to estimate their maximum sustained winds and their minimum central pressures.

Along Track Scanning Radiometers aboard weather satellites are able to detect wildfires, which show up at night as pixels with a greater temperature than 308 K (35 °C; 95 °F). The Moderate-Resolution Imaging Spectroradiometer aboard the Terra satellite can detect thermal hot spots associated with wildfires, volcanoes, and industrial hot spots.

The Atmospheric Infrared Sounder on the Aqua satellite, launched in 2002, uses infrared detection to measure near-surface temperature.

Stratosphere measurements

Stratospheric temperature measurements are made from the Stratospheric Sounding Unit (SSU) instruments, which are three-channel infrared (IR) radiometers. Since this measures infrared emission from carbon dioxide, the atmospheric opacity is higher and hence the temperature is measured at a higher altitude (stratosphere) than microwave measurements.

Since 1979 the Stratospheric sounding units (SSUs) on the NOAA operational satellites have provided near global stratospheric temperature data above the lower stratosphere. The SSU is a far-infrared spectrometer employing a pressure modulation technique to make measurement in three channels in the 15 μm carbon dioxide absorption band. The three channels use the same frequency but different carbon dioxide cell pressure, the corresponding weighting functions peaks at 29 km for channel 1, 37 km for channel 2 and 45 km for channel 3.

The process of deriving trends from SSUs measurement has proved particularly difficult because of satellite drift, inter-calibration between different satellites with scant overlap and gas leaks in the instrument carbon dioxide pressure cells. Furthermore since the radiances measured by SSUs are due to emission by carbon dioxide the weighting functions move to higher altitudes as the carbon dioxide concentration in the stratosphere increase. Mid to upper stratosphere temperatures shows a strong negative trend interspersed by transient volcanic warming after the explosive volcanic eruptions of El Chichón and Mount Pinatubo, little temperature trend has been observed since 1995. The greatest cooling occurred in the tropical stratosphere consistent with enhanced Brewer-Dobson circulation under greenhouse gas concentrations increase.

Lower stratospheric cooling is mainly caused by the effects of ozone depletion with a possible contribution from increased stratospheric water vapor and greenhouse gases increase. There has been a decline in stratospheric temperatures, interspersed by warmings related to volcanic eruptions. Global Warming theory suggests that the stratosphere should cool while the troposphere warms.

Top of the stratosphere (TTS) 1979–2006 temperature trend.

The long term cooling in the lower stratosphere occurred in two downward steps in temperature both after the transient warming related to explosive volcanic eruptions of El Chichón and Mount Pinatubo, this behavior of the global stratospheric temperature has been attributed to global ozone concentration variation in the two years following volcanic eruptions.

Since 1996 the trend is slightly positive due to ozone recovery juxtaposed to a cooling trend of 0.1K/decade that is consistent with the predicted impact of increased greenhouse gases.

The table below shows the stratospheric temperature trend from the SSU measurements in the three different bands, where negative trend indicated cooling.

Channel	Start	End Date	STAR v3.0 Global Trend (K/decade)
TMS	1978-11	2017-01	−0.583
TUS	1978-11	2017-01	−0.649
TTS	1979-07	2017-01	−0.728

Microwave (tropospheric and stratospheric) measurements

Microwave Sounding Unit (MSU) measurements

Main article: Microwave Sounding Unit temperature measurements

MSU weighting functions based upon the U.S. Standard Atmosphere.

From 1979 to 2005 the microwave sounding units (MSUs) and since 1998 the Advanced Microwave Sounding Units on NOAA polar orbiting weather satellites have measured the intensity of upwelling microwave radiation from atmospheric oxygen. The intensity is proportional to the temperature of broad vertical layers of the atmosphere. Upwelling radiance is measured at different frequencies; these different frequency bands sample a different weighted range of the atmosphere.

Figure 3 (right) shows the atmospheric levels sampled by different wavelength reconstructions from the satellite measurements, where TLS, TTS, and TTT represent three different wavelengths.

Other microwave measurements

A different technique is used by the Aura spacecraft, the Microwave Limb Sounder, which measure microwave emission horizontally, rather than aiming at the nadir.

Temperature measurements are also made by occultation of GPS signals. This technique measures the refraction of the radio signals from GPS satellites by the Earth's atmosphere, thus allowing vertical temperature and moisture profiles to be measured.

Temperature measurements on other planets

Planetary science missions also make temperature measurements on other planets and moons of the solar system, using both infrared techniques (typical of orbiter and flyby missions of planets with solid surfaces) and microwave techniques (more often used for planets with atmospheres). Infrared temperature measurement instruments used in planetary missions include surface temperature measurements taken by the Thermal Emission Spectrometer (TES) instrument on Mars Global Surveyor and the Diviner instrument on the Lunar Reconnaissance Orbiter; and atmospheric temperature measurements taken by the composite infrared spectrometer instrument on the NASA Cassini spacecraft.

Microwave atmospheric temperature measurement instruments include the Microwave Radiometer on the Juno mission to Jupiter.