10.2 Atmospheric radiative transfer in the mm/submm

Due to space constraints in this book we cannot provide a very detailed overview of all aspects involved in modeling the longwave atmospheric spectrum. The author and co-workers have recently published an in-depth description of their radiative transfer model ATM (Atmospheric Transmission at Microwaves, Pardo et al [2001b]) that can serve as a reference for the information that is missing here. That model has been adopted in this chapter.

10.2.1 Introduction

Accurate modeling of the longwave emission/absorption spectrum of the terrestrial atmosphere is needed in many scientific applications. In the astrophysical domain, it is needed to predict the atmospheric attenuation at a given frequency for ground-based and airborne observatories, to calculate system noise temperatures and to estimate phase delays for interferometry. In remote sensing of the atmosphere and the Earth's surface, obtaining useful data for meteorological and environmental studies relies upon an accurate knowledge of the atmospheric spectrum.

We will see in this chapter how to model the Earth's atmosphere longwave spectrum. For millimeter and submillimeter astronomy applications we need to know for a given path through the atmosphere, the opacity, radiance, phase delay and polarization^10.1.

10.2.2 Unpolarized radiative transfer equation

The unpolarized radiative transfer in non-scattering media is described by a relatively simple differential scalar equation:

$$\frac{dI_{\nu}(\vec{r},\vec{n}}{ds}=\epsilon_{\nu}-\kappa_{\nu} {I_{\nu}(\vec{r},\vec{n})}$$ (10.12)

where $I_{\nu}$ is the radiance (in W$\;$m$^{-2}$ster$^{-1}$cm$^{-1}$), $\epsilon_{\nu} dw d\nu ds d\sigma$ and $\kappa_{\nu} I(\vec{r},\vec{n},\nu) dw d\nu d\sigma ds$ are the amounts of energy emitted and absorbed at frequency $\nu$ in a pencil of solid angle $dw$ in the direction $\vec{r}$ through a cylinder of length $ds$ and cross-section $\vec{d\sigma}$= $d\sigma \vec{n}$. $\epsilon_{\nu}$ and $\kappa_{\nu}$ are the macroscopic absorption and emission coefficients. The absorption coefficient gives the fractional loss of intensity (at a given wavelength) per length through an absorbing medium.

After rearranging equation 10.12 and considering absence of scattering, the radiative transfer problem is unidimensional in the direction of $\vec{r}$. We can formulate it under Local Thermal Equilibrium (LTE) conditions as follows:

$$\frac{dI_{\nu}(s')}{d\tau_{\nu}}=-I_{\nu}(s')+S_{\nu}(T[s'])$$ (10.13)

where $s'$ is a coordinate along the path, $S_{\nu}=\epsilon_{\nu}/\kappa_{\nu}$ is the so-called source function, and $d\tau_{\nu}=\kappa_{\nu}ds$ is the differential opacity.

The solution of this equation can be given in an integral form:

$$I_{\nu}(s)=I_{\nu}(0)e^{-\tau_{\nu}(0,s)}+\int_{0}^{s} S_{\nu}(s')e^{- \tau_{\nu}(s',s)}\kappa_{\nu}(s')ds'$$ (10.14)

In general, the line-by-line integrated opacity corresponding to a path through the terrestrial atmosphere is calculated in a discrete way as follows:

$$\tau_{\nu}(s',s)=\sum_{i(layers)}^{}[\sum_{j(molec.)}^{ } (\sum_{k(resonances)}^{ } \kappa_{\nu_{k}})_{j}]_{i}\cdot \Delta s_{i}$$ (10.15)

where $\Delta s_{i}$ is the path through the homogeneous i-th layer in the path from $s'$ to $s$ and no line coupling between different species is assumed. As pressure increases the calculations have to use thiner layers to follow the opacity distribution.

The absorption coefficient of an electric dipole (E1) resonance in the atmosphere is given in general by the following equation:

$$(\kappa_{\nu})_{lu}=\frac{8\pi^{3}N\nu}{3hcQ}(e^{-E_{l}/KT}- e^{-E_{u}/KT}) \cdot \mid <u\mid\mu\mid l >\mid^{2} f(\nu,\nu_{l\rightarrow u})$$ (10.16)

where $N$ is the number density in the relevant vibrational state the molecule, $E_{n}$ the energy level of the state and g$_{n}$ its degeneracy, $Q$ is the partition function, $\mu$ is the dipole operator of the transition and $\mid u>$, $\mid l>$ are the wavefunctions of the upper and lower states, and, finally, $f(\nu,\nu_{l\rightarrow u})$ is the line shape function. This the basic expression used in the ATM model.

10.2.3 Spectroscopic parameters

Both transition probabilities $\mid <u\mid\mu\mid l >\mid^{2}$ and rotational energy levels (from which both resonance frequencies and population factors under LTE are determined) can be obtained from the rotational hamiltonians. The number of rotational constants depends on the type of molecule. The cases to be considered in the atmosphere are diatomic or linear molecules (with no magnetic moment), symmetric rotors in $^{3}\Sigma$ electronic state, and asymmetric rotors.

10.2.3.1 Transition probabilities:

The way we parametrize them is the following:
$\mid<J,\tau\mid\mu\mid J^{\prime}, \tau^{\prime} >\mid^{2}=\mu_{g}^{2}\lambda_{g}(J,\tau,J^{\prime}, \tau^{,})$. $J,J^{\prime}$ represent rotational quantum numbers, $\tau,\tau^{\prime}$ are other quantum numbers, $\mu_{g}$ is the value of the dipole moment (electric or magnetic), and $\lambda_{g}(J,\tau,J^{\prime},\tau^{\prime})$ is a dimensionless parameter called the oscillator strength of the particular transition.

10.2.3.2 Partition function:

The case of symmetric linear or diatomic molecules ($^{16}$O$_{2}$, N$_{2}$O or CO) and the general asymmetric rotors must be treated individually. In the first case the energy levels are $\sim$hBJ(J+1) where B is the rotational constant of the molecule in the considered vibrational state. There are corrections to this simple rigid rotor expression but for CO, for example, the associated parameter D is five orders of magnitude smaller than B and for other important atmospheric molecules the ratio B/D is even larger. In the case of asymmetric rotors (O$_{3}$, SO$_{2}$,...) there are three main rotational constants (A, B, C) in the Hamiltonian related with the three principal axes. There are also corrections due to centrifugal and other effects.

Analytical expressions (see Pardo et al [2001b]) can be used in the atmosphere for both types of molecules within an error not larger than 0.4%.

10.2.3.3 Spectroscopy of H$_2$O, O$_2$, O$_3$

Of the major molecular constituents of the atmosphere (see Table 10.1) only water vapor and ozone, owing to their bent structure, have a non-zero electric dipole moment. Molecular nitrogen, an homonuclear species, and CO$_2$, a linear symmetric species, have no permanent electric or magnetic dipole moment in their lowest energy states. These latter molecules, as is the case for most of gaseous molecules, are singlet states, with electrons arranged two-by-two with opposite spins. O$_2$ has a permanent magnetic dipole owing to two parallel electron spins. It thus presents magnetic dipole transitions of noticeably intensity due to the large abundance of this molecule in the atmosphere.

We present here the details about the spectroscopy of H$_{2}$O, O$_{2}$ and O$_{3}$ because these molecules dominate the longwave atmospheric spectrum as seen from the ground.

10.2.3.3.1 Water vapor

Water vapor is a C$_{2v}$ molecule (degree of symmetry: $\sigma$=2) with a relatively high electric dipole moment: $\mu$=1.88$\cdot$10$^{-19}$esu$\cdot$cm. The first vibrational modes of H$_{2}^{16}$O are at 3693.8 cm$^{-1}$ (1,0,0), 1614.5 cm$^{-1}$ (0,1,0) and 3801.7 cm$^{-1}$ (0,0,1). The nuclear spins (1/2 for H, 0 for $^{16}$O and $^{18}$O, 1/2 for $^{17}$O, and 0 for D) lead to two spectroscopically separated species of water: I=1 (statistical weight g=3) [orto-$^{16,18}$H$_{2}$O], and I=0 (g=1) [para-H $_{2}^{16,18}$O]. HDO and H$_{2}^{17}$O have a hyperfine structure.

For $^{16,18}$H$_{2}$O each level is denoted, as usual for asymmetric top molecules by three number s $J_{K_{-1},K_{+1}}$. $J$, which is a ``good'' quantum number, represents the total angular momentum of the molecule; by analogy with symmetric tops, $K_{-1}$ and $K_{+1}$ stand for the rotational angular momenta around the axis of least and greatest inertia. Allowed radiative transitions obey the selection rules $\Delta{J}=\pm1, \Delta{K}=\pm1, 3,$ with $K_{-1},K_{+1}: odd,odd\leftrightarrow even,even$ or $o,e\leftrightarrow e,o$. The levels with $K_{-1}$ and $K_{+1}$ of the same parity belong to the para species, those of opposite parity belong to orto water.

10.2.3.3.2 Molecular oxygen

Molecular oxygen, although homonuclear, hence with zero electric dipole moment, has a triplet electronic ground state, with two electrons paired with parallel spins. The resulting electronic spin couples efficiently with the magnetic fields caused by the end-over-end rotation of the molecule, yielding a magnetic dipole moment of two Böhr magnetons, $\mu_{mag}$=2 $\mu_{B\ddot{o}hr}$=1.854$\cdot$10$^{-2}$Debyes. The magnetic dipole transitions of O$_2$ have intrinsic strengths $\sim10^{2-3}$ times weaker than the water transitions. O$_2$, however, is 10$^{2-3}$ times more abundant than H$_2$O, so that the atmospheric lines of the two species have comparable intensities.

The spin of 1 makes of the ground electronic state of O$_2$ a triplet state ($^3\Sigma$). N, the rotational angular momentum couples with S, the electronic spin, to give J the total angular momentum: N +S = J. The N$\cdot$S interaction (and the electronic angular momentum-electronic spin interaction L$\cdot$S) split each rotational level of rotational quantum number $N\geq1$ into three sublevels with total quantum numbers

$$J= N+1, J=N {\rm     and    } J=N-1$$

the $J= N+1$ and $J=N-1$ sublevels lying below the $J= N$ sublevel by approximately $119 (N+1)/(2N+3)$GHz and $119/(2N-1)$GHz, respectively. Note that the two identical $^{16}$O nuclei have spins equal to zero and obey the Bose-Einstein statistics; there are only odd $N$ rotational levels in such a molecule.

The magnetic dipole transitions obey the rules $\Delta{N}= 0, \pm 2$ and $\Delta{J}= 0, \pm 1$. Transitions within the fine structure sublevels of a rotational level (i.e. $\Delta{N}= 0$) are thus allowed. The first such transition is the $(J,N)= 1,1\leftarrow0,1$ transition, which has a frequency of 118.75 GHz. The second, the $1,1\leftarrow2,1$ transition, has a frequency of 56.26 GHz. It is surrounded by a forest of other fine structure transitions with frequencies ranging from 53 GHz to 66 GHz. The first "true" rotational transition, the $N= 3\leftarrow 1$ transitions, have frequencies above 368 GHz (368.5, 424.8, and 487.3 GHz). In addition, the permanent magnetic dipole of this molecule can interact with an external magnetic field, leading to a Zeeman splitting of the energy levels. In our atmosphere this splitting is of the order of 1-2 MHz.

The rare isotopomer $^{18}$O$^{16}$O is not homonuclear, hence has odd $N$ levels and a non-zero electric dipole moment. This latter, however, is vanishingly small (10$^{-5}$D).

10.2.3.3.3 Ozone

The quantum numbers are as in other asymmetric top molecules, such as H$_{2}$O. As noted above, ozone is mostly concentrated between 11 and 40 km altitude; it shows large seasonal and, mostly, latitude variation. Because of its high altitude location, its lines are narrow: at 25 km, $\rho_a$, hence $\Delta \nu$, is reduced by a factor of 20 with respect to see level; moreover, the dipole moment of ozone ($\mu$= 0.53 Debyes), 3.5 times smaller than that of H$_2$O, further reduces the ozone line widths. Because of their small widths and despite the small ozone abundance, ozone lines have significant peak opacities, especially as frequency increases. This fact can be seen in the high resolution FTS measurements presented in this chapter.

10.2.4 Line shapes

10.2.4.1 Absorption

In the lower atmospheric layers (up to $\sim$ 50 km, depending on the molecule and the criteria) the collisional broadening mechanism (also called pressure-broadening) dominates the line shape. One approximation to the problem considers that the time between collisions, $\tau_{col}$ ($\propto$ $p^{-1}$), is much shorter than the time for spontaneous emission, $\tau_{rad}$, which is, in the case of a two level system, 1/A $_{u\rightarrow l}$ where A $_{u\rightarrow l}$ is the Einstein's coefficient for spontaneous emission. This approximation leads to the so-called Van Vleck & Weisskopf profile, normalized as follows to be included in equation 10.16:

$$f_{VVW}(\nu,\nu_{l\rightarrow u})= \frac{\nu \Delta \nu}{\pi \nu_... ...rightarrow u})^{2}} +\frac{1}{(\Delta \nu)^{2}+(\nu+\nu_{l\rightarrow u})^{2}})$$ (10.17)

where $\Delta \nu$ is the collision broadening parameter.

This lineshape describes quite well the resonant absorption in the lower atmospheric layers except for very large shifts from the line centers. For example, all the mm/submm resonances of H$_{2}$O and other molecules up to 1.2 THz are well reproduced using this approximation within that frequency range. Some properties of the collision broadening parameters in the atmosphere are:

$$\Delta\nu(p,T)=\Delta\nu(p_{0},T_{0}) \frac{p}{p_{0}}(\frac{T_{0}}{T})^{\gamma}$$ (10.18)

$$\Delta\nu(M-dry\;air)=X_{N_{2}}\Delta\nu(M- N_{2})+X_{O_{2}}\Delta\nu(M-O_{2})$$ (10.19)

where $X$ are the volume mixing ratios. Laboratory measurements for individual lines are the only source of precise information about the parameters $\gamma$ and $\Delta\nu(M-N_{2},O_{2})$ for the different atmospheric trace gases $M$. The exponent $\gamma$ has been found in most cases to be in the range 0.6 and 1.0. For O$_{2}$ and H$_{2}$O self-collisions have to be considered.

When the pressure gets very low the Doppler effect due to the random thermal molecular motion dominates the line broadening:

$${\cal{F}}_{D}(\nu,\nu_{l\rightarrow u})=\frac{1}{\Delta \nu_{D}}(... ...2}{\pi})^{1/2} exp[-(\frac{\nu-\nu_{l\rightarrow u}}{\Delta \nu_{D}})^{2}\ln 2]$$ (10.20)

where the halfwidth parameter is given by:

$$\Delta \nu_{D}=\frac{\nu_{l\rightarrow u}}{c}\sqrt{\frac{2\ln 2 kT}{m}} = 3.58\cdot 10^{-7} \nu_{l\rightarrow u} \sqrt{\frac{T}{M}}$$ (10.21)

M being the molecular weight of the species in g/mol.

If the collisional and thermal broadening mechanisms are comparable the resulting line-shape is the convolution of a Lorentzian (collisional line shape at low pressures) with a Gaussian: a Voigt profile.

10.2.4.2 Phase delay

Besides absorption, the propagation through the atmosphere also introduces a phase delay. This phase delay increases as the wavelength approaches a molecular resonance, with a sign change across the resonance. The process can be understood as a forward scattering by the molecular medium in which the phase of the radiation changes.

Both the absorption coefficient and the phase delay can be treated in a unified way for any system since both parameters are derived from a more fundamental property, the complex dielectric constant, by means of the Kramers-Krönig relations. A generalized (complex) expression of the VVW profile, which accounts for both the Kramers-Krönig dispersion theory as well as line overlapping effects (parameter $\delta$) is the following ( $\nu_{l\rightarrow u}\equiv\nu_{lu}$):

$${\cal{F}}_{VVW}(\nu,\nu_{lu})=\frac{\nu}{\pi \nu_{lu}} [\frac{1-i\delta}{\nu_{lu}-\nu-i\Delta \nu} +\frac{1+i\delta}{\nu_{lu}+\nu+i\Delta \nu}]$$ (10.22)

the imaginary part of which reduces to equation 10.17 when $\delta$=0.

10.2.5 Non-resonant absorption

10.2.5.1 H$_{2}$O pseudocontinuum

Lines with resonant frequencies up to a given frequency (10 THz in our case) are included in line-by-line calculations. Since the true lineshape is not known accurately beyond a few times the halfwidth from the line centers (this may be due to the finite collision time, the complexity of the calculations and the lack of precise laboratory data), a broadband ``continuum''-like absorption term needs to be included for accurate results in the longwave domain.

10.2.5.2 Dry continuum-like absorption

The non resonant absorption of the dry atmosphere is made up of two components: collision induced absorption due to transient electric dipole moments generated in binary interactions of symmetric molecules with electric quadrupole moments such as N$_{2}$ and O$_{2}$, and the relaxation (Debye) absorption of O$_{2}$.

10.2.5.3 Pseudocontinua in ATM

In the ATM model model, we introduce collisionally-induced dry absorption and longwave (foreign) pseudocontinuum water vapor absorption derived from our previous FTS measurements performed on top of Mauna Kea, Hawaii. For both terms we use $\nu^{2}$ frequency power laws, with the coefficients as determined by Pardo et al [2001a].

$$\kappa_{c,H_{2}O}=0.031\cdot\Bigl(\frac{\nu}{225}\Bigr)^{2} \cdot... ...1013}\cdot\frac{p-e}{1013}\Bigr]\cdot \Bigl(\frac{300}{T}\Bigr)^{3}\;\;\;m^{-1}$$ (10.23)

$${\Large\kappa}_{c,dry}=0.0114 \Bigl(\frac{p-e}{1013}\Bigr)^{2}\Bigl(\frac{300}{T}\Bigr)^{3.5} \Bigl(\frac{\nu}{225}\Bigr)^{2}\;\;\;m^{-1}$$ (10.24)

The validity of these expressions is restricted to frequencies $\leq$1.1 THz (the upper limit of our current data), although there are indications that they can be extended to 2 THz with no important loss of accuracy.

10.2.6 Radiative transfer through atmospheric hydrometeors

The equation describing the transfer of radiation through an atmosphere that contains scatterers is as follows:

$$\mu\frac{dI(z,\mu)}{dz}={K} (z,\mu)I(z,\mu) -2\pi\int_{-1}^{1}M(z,\mu,\mu') I(z,\mu')d\mu'-\sigma(z,\mu)B[T(z)]$$ (10.25)

where:
$\bullet$ $I$=(I,Q,U,V)$^{T}$ is the Stokes vector describing the propagating electromagnetic field.
$\bullet$ $K$ is the extinction matrix.
$\bullet$ $M$ is the zeroth Fourier component of the phase matrix.
$\bullet$ $\sigma$ is the emission vector.
$\bullet$ $B(T)$ is the blackbody spectrum at temperature T.
$\bullet$ $\mu$=cos($\theta$) ($\theta$=zenith angle, 0 $\Rightarrow$downwards). The frequency dependence in several of these quantities is implicit. Finally, z is the vertical coordinate in a plane-parallel atmosphere.

Figure 10.1: Simulations of the effect of clouds in the brightness temperature of the atmosphere at zenith. The liquid water and ice layers have been placed between 6.0 and 6.5 km, the equivalent water path is 0.1 km/m$^{2}$ for both but the size of the spherical particles is 40 $\mu$m for liquid water and 100 $\mu$m for ice. The considered atmosphere contains 1 mm of water vapor column.

$K$, $M$ and $\sigma$ are related according to:

$$K(\mu)=2\pi\int_{-1}^{+1}M(z,\mu,\mu^{,})d\mu^{,}+\sigma(\mu)$$ (10.26)

which is a consequence of the detailed energy balance.

The frequency independent (excluding Raman effect or fluorescence) far-field scattering by single non-spherical particles has been incorporated to the ATM model. It is computed using state-of-the-art T-Matrix routines developed by Mishchenko [2000]. For the integration over all possible orientations to be possible, it is necessary to be in the single scattering regime. Then, each particle is in the far-field zone of the others. This implies that the average distance between particle centers is larger than 4 times their radii. This requirement is usually satisfied by cloud and precipitation particles. The scattered fields are then incoherent and their Stokes parameters can simply be added.

For estimates concerning the effect of clouds to ground-based observations a plane parallel geometry is assumed, as a first approach, with thermal emission as the only source of radiation. The hydrometeors can be either totally randomly-oriented or at least azimuthally randomly oriented. In both cases, the radiation field is azimuthally symmetric leading the Stokes parameters U and V to vanish so that the dimension of tensorial equation (10.25) reduces to be 2. This radiative transfer equation can then be integrated using the quite standard method called doubling and adding, introduced in the ATM model according to Evans & Stephens [1991].

To illustrate the effect of two different types of clouds, we have performed first a simulation where we add to a clear atmosphere containing 1 mm of water vapor a layer between 6 and 6.5 km that contains spherical water droplets with a radius of 40 $\mu$m and a liquid water path of 0.1 km/m$^{2}$. The second simulation replaces the water droplets for spherical ice particles with a radius of 100 $\mu$m. The effect of liquid water is quite large since it is a quite effective absorber. The effect of ice is much more related with its scattering properties and becomes more important at shorter wavelengths.