Radiative Transfer Equation - Atmospheric Absorption Models for the Millimeter Wave Range

Considering the radiative transfer equation (RTE) of a non-scattering atmosphere, one is confronted with the three physical effects of absorption, induced emission, and spontaneous

1The source code of Arts and its user guide can be downloaded via the Arts home page http://www.sat.uni-bremen.de/arts/. The whole package is free for use under the GNU license agreement.

21 E

Elow up

I

ⁱⁿ

I

^out

B

¹²

B

²¹

A

molecular system

Figure 2.1: Einstein coefficients for the induced emission (B21) and absorption (B12) as well as for the spontaneous emission (A21). E_low and Eup denote the energy levels of the transition.

emission. All three effects are related with the appropriate Einstein coefficients which are shown in Figure 2.1. The Einstein coefficientB₁₂ has the meaning of a probability per unit time per unit energy density for absorption, B21 is the probability per unit time per unit energy density for induced emission, andA₂₁is the probability per unit time for spontaneous emission. The link between the three Einstein coefficients is given by (Weissbluth, 1978):

g₁B₁₂=g₂B₂₁, A₂₁= 2h ν³

c² B₂₁ (2.1)

where g_1,2 are the statistical weights of the lower and upper energy states, h is the Planck constant, ν the transition frequency, and c is the speed of light. For the deduction of the radiative transfer equation two-state molecules are assumed for simplicity. The generalization to multi-state molecules is then rather simple and bears no additional complication.

To deduce the differential form of the RTE we consider an ideally infinitesimally small volume dV = dAdsaround a pointP within the line of sight (see Figure 2.2). The difference of radiation intensity entering and leaving the volume dV (in units of energy per time per area per solid angle per frequency) is expressed in terms of the Einstein coefficients as (Read, 1980;Rybicki and Lightman, 1976; Armstrong and Nicholls, 1972)

dIν(P) = −h ν (N1B12−N2B21) φ(ν)Iν(P) ds/ c

| {z }

absorption and induced emission

+ h ν N₂A₂₁φ(ν) ds/ c

| {z }

spontaneous emission

(2.2) whereN₁ andN₂ are the number densities of molecules (molecules per volume) in state 1 and 2, respectively. The function φ(ν) is introduced to describe the ability of the molecules to emit or absorb radiation of frequencyν, which are well separated from the nominal transition frequency of an unperturbed molecule². The energy states involved in transitions 1*)2 are

2The lifetime of completely undisturbed molecule in excited state 2 is inversely proportional toA21. The natural line width for rotational transitions of undisturbed molecules is of the order of 1-100µHz (Hollas, 1998; Townes and Schawlow, 1955). Hence the function φ(ν) is under very low pressure conditions well approximated by Dirac’s delta function,δ(E2−E1−h ν). For higher pressure conditions the intermolecular interaction disturbs the molecular energy levels noticeably and thereforeφ(ν) deviates in this case from the delta function significantly.

neither infinitely sharp nor entirely fixed. Molecules react on perturbations, like for example interactions with other molecules, with the readjustment of the energy levels. This enables the molecules to absorb or emit radiation in a frequency band around the undisturbed transition frequency.

In the formulation of Equation (2.2) it is implicitly assumed that the line shape of emission and absorption is the same implying that the molecular system absorbs as much as it emits in any frequency range. This assumption is known as the law of detailed balance (Rybicki and Lightman, 1976; Breene, 1957).

With Equation (2.2) an effective volume absorption coefficient can be defined in terms of the Einstein coefficients (Rybicki and Lightman, 1976):

αν def

= h ν (N1B12−N2B21)φ(ν)/ c

= h ν N1B12

1−g1N2

g₂N₁

φ(ν)/ c (2.3)

where the absorption coefficient, α_ν, has the unit of inverse length. It is furthermore conve-nient to define the following source function,S_ν(s), by

S_ν(P) def

= N₂A₂₁

N₁B₁₂−N₂B₂₁ = 2h ν³ c²

g₂N₁ g₁N₂ −1

−1

. (2.4)

With these definitions in hand one gets a more compact expression for Equation (2.2) of the form

dI_ν(P) = α_ν(P) [S_ν(P)−I_ν(P)] ds (2.5)

The intensity difference thus consists of one term which reduces the intensity and one term which increases the intensity along the element dsof the line of sight.

The energy of rotational transitions in the STHz range is much smaller that the transla-tional energy of the molecules, hν kBT. It is thus well justified to assume that the thermalization process will result in a Maxwell–Boltzmann distribution for the occupation probabilities of the rotational energy levels in the lower atmosphere (Goody, 1995). The validity of Maxwell–Boltzmann statistics requires local thermodynamic equilibrium (LTE).

Assuming LTE conditions in the volume dV yields for the number densities of the two-level molecules

g1N2

g₂N₁ = e^−(E²^−E¹^)/k^B^T = e^−hν/k^B^T (2.6) Inserting this relation into the expression for the absorption coefficient gives

α^LTE_ν = h ν g₁N₁B₁₂ ^h1−e^−hν/k^B^Tⁱφ(ν)/ c (2.7)

= h ν g₂N₂B₂₁ ^he^hν/k^B^T −1ⁱφ(ν)/ c (2.8) The absorption coefficient is implicitly a function ofP on the line of sight since the thermo-dynamic properties like temperature and number density can vary as well along the line of sight.

For LTE conditions the source function is equal to the Planck function³, B_ν(T), which has the unit of power per unit area per units solid angle per unit frequency interval aroundν at pointP along the line of sight (W/m²/steradian/Hz) (Andrews, 2000)

S_ν^LTE(P) = B_ν(T(P)) = 2·h·ν³

c² · 1

e^hν/k^B^T −1 (2.9)

3Bν(T) equals (c/4π)·u(T, ν) whereu(T, ν) is the blackbody spectral energy density.

LOS

P s

P dA ds ds

sensor

Figure 2.2: Radiative transfer along the line of sight (LOS) of the sensor. The Schwarzschild equation considers the radiation budget of a small volume dV = dA·dsat a pointP on the line of sight.

With Equation (2.9) one gets the final version of the differentialRTEunder LTE conditions, which is also known as the Schwarzschild equation:

dIν(P)|_LTE = α^LTE_ν (P)·ds·[Bν(T(P))−Iν(P) ] (2.10)

= dτ_ν(P)·[B_ν(T(P))−I_ν(P) ]

Since in all the subsequent calculations LTE is assumed, the superscript LTE is further omitted. In the second expression of the Schwarzschild equation the differential optical thickness dτν(P) is introduced which is the product of the absorption coefficient and the line of sight element ds.

Qualitatively, the Schwarzschild equation describes the net difference of radiation intensity due to absorption and emission in a small volume dV around point P on the line of sight (see Figure 2.2).

To determine the effective intensity which can be seen by a passive instrument (e.g., a radiometer), one has to integrate Equation (2.10) along the line of sight from the outer limit of the atmosphere to the instrument (Sb → So). The background intensity, Iν(Sb), at Sb depends on the viewing geometry. In the zenith and limb viewing case this input is equal to the cosmic microwave background (CMB) radiation (White and Cohn, 2002), Iν(Sb) = ICMB. Nadir measurements in the MW and MMW frequency range are on the other hand sensitive to the Earth surface emission,I_ν(Sb) =_νB_ν(Ts), with_ν as the surface emissivity andTs as the surface temperature. If the instrument is located at point So, the integration of the Schwarzschild equation leads to the radiative transfer equation

I_ν(So) = I_ν(Sb)·exp − Z So

dτ_ν(s)

(2.11) +

Z So

S_b

α_ν(s)·B_ν(T(s))·exp − Z So

dτ_ν(s⁰)

! ds.

Figure 2.3: Example of an integrated intensity calculation for a cloud-free mid-latitude sum-mer atmosphere, consisting of oxygen, nitrogen, and water vapor. A nadir viewing geometry with a platform altitude of 830 km is assumed. The calculation is performed for different frequencies up to 400 GHz. The intensity unit is thermodynamic brightness temperatureTB with the definitionI_ν(Sb) =B_ν(TB) (see Equation (2.12)).

To illustrate Equation (2.11) Figure 2.3 shows anArtscalculation of the integrated intensity, I_ν(So), of a nadir looking space borne passive instrument for a mid-latitude summer atmo-sphere (Kneizys et al., 1996). The unit for the intensity is chosen asthermodynamic brightness temperature⁴ (TB). The thermodynamic brightness temperature is equal to the temperature of a blackbody whose emitted radiation intensity is equal to the measured intensity. Hence TB is determined by inverting the Planck spectral radiance function (see Equation (2.9))

B_ν(TB) = I_ν(So) (2.12)

TB = h·ν kB

. ln 1 + 2·h·ν³ I_ν(So)·c²

The low brightness temperatures in Figure 2.3 coincide with strong absorption of oxygen and water vapor spectral lines. At these frequencies the atmosphere is partially optically thick and hence only radiation from the higher atmospheric layers can reach the top of the atmosphere. For example in the mid-latitude summer atmosphere used in Figure 2.3, a TB of 220 K corresponds to physical temperatures around the upper troposphere and a TB of 240 K to a physical temperature around 9 km.

With the optical thickness introduced in Equation (2.10) one can reformulate the integral form of the radiative transfer equation in terms of the frequency dependent transmission of the atmosphere. The transmission , T_ν, between two points along the line of sight (P,So) and the corresponding optical thickness,τ_ν, are defined as

T_ν(P,So) = exp [−τ_ν(P,So)] (2.13)

4The thermodynamic brightness temperature is also called Planck brightness temperature

τν(P,So) = Z P

dτν(s) = Z P

αν(s) ds (2.14)

Introducing the transmissionT_ν into Equation (2.11) leads to the expression I_ν(So) =I_ν(Sb)· T_ν(Sb,So) +

Z So

α_ν(s)·B_ν(T(s))· T_ν(s,So) ds (2.15) From Equation (2.15) we can distinguish between optically thin (T_ν →1) and optically thick (T_ν →0) conditions. In the optically thin case,I_ν(So) is the sum of the background radiation and the atmospheric contribution along the whole line of sight fromSb up to So:

Iν(So)≈Iν(Sb) + Z So

αν(s)·Bν(T(s)) ds (2.16)

To illustrate the case of a partially optically thick atmosphere, we simplify the situation by dividing the transmission into two parts

T_ν =

( 1 : So ≤ s ≤ P 0 : P < s ≤ Sb

(2.17) where the atmosphere from the instrument to pointP is totally transparent and behind P optically thick. Introducing this step function into Equation (2.15) leads immediately to the result that only the second term with the integral from P to So will contribute to I_ν(So).

Hence the instrument would not be sensitive to the region behindP: Iν(So) =

Z So

αν(s)·Bν(T(s)) ds+Iν(P) (2.18)

An illustration of Equation (2.16) is shown in Figure 2.4, where anArts transmission cal-culation for a mid-latitude summer atmosphere is shown. Plotted are the transmissions for a nadir viewing space borne instrument at frequencies up to 400 GHz. The distinct features where the transmission is very low are caused by the strong absorption lines of water vapor and oxygen (60 GHz). Comparing the brightness temperatures and transmissions in Fig-ure 2.3 and FigFig-ure 2.4 illustrates the correspondence between lowTB and low transmission (optically thick atmosphere) and the reverse case, i.e. highTB and high transmission of an optically thin atmosphere.

Clearly, the integral equation is unique only in the forward direction which means by defining an atmosphere completely, Equation (2.11) gives a single solution for Iν(So). The inverse is not true, a measured intensity Iν(So) does not determine the atmosphere along the line of sight uniquely. Many different atmospheric situations can produce the same intensity at the detector. This makes it necessary to operate multi-frequency instruments in order to retrieve vertical profiles of temperature or trace gas concentration. The different transmission at different frequencies permits a multi-frequency instrument to detect radiation from different layers of the atmosphere.

Im Dokument Atmospheric Absorption Models for the Millimeter Wave Range (Seite 52-57)