Eq. 1.1 — General vector RTE
Full Radiative Transfer Equation (monochromatic, plane-parallel)
$$\mu\,\frac{dI_\nu(\tau_\nu,\mu,\phi)}{d\tau_\nu} = I_\nu(\tau_\nu,\mu,\phi) - J_\nu(\tau_\nu,\mu,\phi)$$
Describes how spectral radiance $I_\nu$ changes with optical depth $\tau_\nu$ along a direction
$(\mu,\phi)$. The left side is the net change. The right side balances extinction loss against the
total source function $J_\nu$ — which combines thermal emission and scattering contributions.
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Symbol Definitions
$I_\nu(\tau_\nu,\mu,\phi)$
Spectral radiance [W m⁻² sr⁻¹ Hz⁻¹] at optical depth $\tau_\nu$, cosine of zenith angle $\mu = \cos\theta$, and azimuth $\phi$
$\mu = \cos\theta$
Cosine of polar zenith angle $\theta$. $\mu>0$ = upward hemisphere, $\mu<0$ = downward hemisphere
$\tau_\nu$
Monochromatic optical depth, measured from TOA downward. Dimensionless. $d\tau_\nu = -k_\nu \rho\, dz$ where $z$ increases upward
$J_\nu(\tau,\mu,\phi)$
Total source function [same units as $I_\nu$]. Sum of thermal emission source $J_\nu^{em}$ and scattering source $J_\nu^{sc}$
$\phi$
Azimuth angle [rad]. For azimuthally symmetric problems (nadir view, uniform atmosphere) the $\phi$ dependence drops out
Plane-parallel assumption: atmosphere is horizontally homogeneous; properties vary only with altitude $z$ (or equivalently $\tau$). Valid when the atmosphere is much thinner than the Earth's radius (~100 km vs 6371 km).
Monochromatic: equation holds at a single frequency $\nu$. Real atmosphere requires integrating over instrument bandpass $\Delta\nu$.
Eq. 1.2 — Source function decomposition
Total Source Function $J_\nu$ — All Contributing Terms
$$J_\nu(\tau,\mu,\phi) = \underbrace{(1-\omega_\nu)\,B_\nu(T)}_{\text{thermal emission}} + \underbrace{\frac{\omega_\nu}{4\pi}\int_0^{4\pi} p_\nu(\Omega,\Omega')\,I_\nu(\tau,\Omega')\,d\Omega'}_{\text{multiple scattering}} + \underbrace{\frac{\omega_\nu}{4\pi}\,F_0^\nu\,p_\nu(\Omega,\Omega_0)\,e^{-\tau/\mu_0}}_{\text{direct solar scattering}}$$
The source function $J_\nu$ has three contributions: (1) thermal emission weighted by co-albedo $(1-\omega_\nu)$,
(2) diffuse multiple-scattering from all directions $\Omega'$, and (3) single scattering of the direct solar beam
(attenuated to depth $\tau$ by $e^{-\tau/\mu_0}$).
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Symbol Definitions
$\omega_\nu$
Single scattering albedo. $\omega_\nu = k_s^\nu / k_e^\nu \in [0,1]$. $\omega=0$: pure absorption; $\omega=1$: pure scattering
$B_\nu(T)$
Planck blackbody function at local temperature $T$ [W m⁻² sr⁻¹ Hz⁻¹]
$p_\nu(\Omega,\Omega')$
Phase function — probability of photon from direction $\Omega'$ scattered into $\Omega$. Normalized: $\frac{1}{4\pi}\int p\,d\Omega' = 1$
$\int_0^{4\pi} d\Omega'$
Integration over all $4\pi$ steradians of incoming directions. In plane-parallel: $\int_{-1}^{1}\int_0^{2\pi} d\mu'\,d\phi'$
$F_0^\nu$
Solar spectral irradiance at TOA [W m⁻² Hz⁻¹]. Extraterrestrial solar flux (Kurucz spectrum)
$\mu_0 = \cos\theta_0$
Cosine of solar zenith angle. Determines solar beam attenuation rate with optical depth
$e^{-\tau/\mu_0}$
Attenuation of direct solar beam from TOA to level $\tau$. This is Beer-Lambert applied to the solar path
Eq. 1.3 — Extinction coefficient decomposition
Volume Extinction Coefficient — All Physical Processes
$$k_e^\nu = \underbrace{k_a^\nu}_{\text{gas absorption}} + \underbrace{k_R^\nu}_{\text{Rayleigh scatter}} + \underbrace{k_M^\nu}_{\text{Mie/aerosol}} + \underbrace{k_c^\nu}_{\text{cloud droplets}}$$
The total extinction coefficient $k_e^\nu$ [m⁻¹] governs how fast the beam is attenuated per unit path length.
It decomposes into absorption by gases (CO₂, H₂O, O₃, O₂, CH₄, N₂O), elastic Rayleigh scattering by molecules,
Mie scattering and absorption by aerosol particles, and scattering/absorption by cloud droplets.
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Symbol Definitions
$k_a^\nu$
Gas absorption coefficient [m⁻¹]. $k_a^\nu = \sum_g n_g \sigma_g^\nu$ where $n_g$ is number density [m⁻³] and $\sigma_g^\nu$ is absorption cross-section [m²] for gas $g$
$k_R^\nu$
Rayleigh scattering coefficient. $k_R^\nu = n_{air}\,\sigma_R^\nu$ where $\sigma_R^\nu \propto \nu^4$ (or $\propto \lambda^{-4}$). Purely scattering: contributes to $\omega_\nu$
$k_M^\nu$
Aerosol extinction coefficient [m⁻¹]. Complex: depends on particle size distribution $n(r)$, refractive index $m(\nu)$, and Mie efficiency $Q_{ext}(x,m)$ where $x=2\pi r/\lambda$
$k_c^\nu$
Cloud extinction coefficient [m⁻¹]. Clouds are optically thick ($\tau_c \sim 10$–100) and largely wavelength-independent in VIS/NIR. Strong contaminant for CO₂ retrieval
Gas absorption cross-sections $\sigma_g^\nu$ come from line-by-line databases (HITRAN, GEISA). Each molecular transition has a central frequency $\nu_0$, line strength $S$, and line shape. The cross-section is:
$$\sigma_g^\nu = S \cdot f(\nu - \nu_0; \gamma_L, \gamma_D)$$
where $f$ is the Voigt profile combining pressure-broadened Lorentzian ($\gamma_L \propto p$) and Doppler-broadened Gaussian ($\gamma_D \propto \sqrt{T}$) line shapes.
Eq. 2.1 — Planck function (frequency)
Spectral Radiance of a Blackbody (Frequency Form)
$$B_\nu(T) = \frac{2h\nu^3}{c^2}\,\frac{1}{\exp\!\left(\dfrac{h\nu}{k_B T}\right) - 1}$$
The Planck function gives the spectral radiance [W m⁻² sr⁻¹ Hz⁻¹] emitted by a perfect blackbody at temperature $T$. This is the thermal emission source function $J^{em}_\nu = B_\nu(T)$ used in the RTE for gas molecules and the Earth's surface.
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Constants & Variables
$h = 6.626\times10^{-34}$ J·s
Planck's constant
$c = 2.998\times10^8$ m/s
Speed of light in vacuum
$k_B = 1.381\times10^{-23}$ J/K
Boltzmann constant
$T$ [K]
Absolute temperature of the emitting body. Sun: 5778 K → peak ~0.5 µm. Earth surface: 288 K → peak ~10 µm
$\nu$ [Hz]
Frequency. Relation to wavelength: $\nu = c/\lambda$. CO₂ 15µm band: $\nu \approx 2\times10^{13}$ Hz
Wavelength form: $B_\lambda(T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/\lambda k_B T}-1}$. Note $B_\nu d\nu = B_\lambda |d\lambda|$ so they are NOT the same function evaluated at the same argument.
Wien's displacement law (peak wavelength): $\lambda_{max} T = 2898\,\mu\text{m}\cdot\text{K}$
Stefan-Boltzmann law (total emission): $E = \sigma T^4$ where $\sigma = 5.67\times10^{-8}$ W m⁻² K⁻⁴
Eq. 2.2 — Brightness temperature
Brightness Temperature $T_B$ — Inverse Planck
$$T_B(\nu) = \frac{h\nu}{k_B}\,\frac{1}{\ln\!\left(1 + \dfrac{2h\nu^3}{c^2\,I_\nu}\right)}$$
TIR satellites (AIRS, IASI, CrIS) report measured radiance as brightness temperature $T_B$ — the temperature a perfect blackbody would need to emit the observed $I_\nu$. CO₂ absorbs and re-emits at 15 µm. More CO₂ at higher (colder) altitudes → lower $T_B$ at 15 µm → satellite sees colder source.
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Physical interpretation
$T_B < T_{surface}$
In CO₂ absorption bands: satellite "sees" emission from mid-troposphere (~220 K) rather than warm surface (~288 K) → colder = more CO₂
$\Delta T_B / \Delta[\text{CO}_2]$
Sensitivity: ~0.5 K per 10 ppm CO₂ change in strong TIR bands. Weaker sensitivity than SWIR reflected solar
Eq. 2.3 — Rayleigh-Jeans & Wien approximations
Limiting Forms of the Planck Function
$$B_\nu(T) \approx \frac{2\nu^2 k_B T}{c^2} \quad (h\nu \ll k_B T,\;\text{microwave / Rayleigh-Jeans})$$
$$B_\nu(T) \approx \frac{2h\nu^3}{c^2}\,e^{-h\nu/k_B T} \quad (h\nu \gg k_B T,\;\text{UV / Wien})$$
The Rayleigh-Jeans limit (microwave, long $\lambda$) gives linear dependence on $T$ — useful for microwave sounders. The Wien limit (short $\lambda$, UV/VIS) gives exponential dependence. SWIR CO₂ bands (~1.6 µm, 12500 cm⁻¹) at 288 K lie in the Wien regime: $h\nu/k_BT \approx 50 \gg 1$, so reflected solar dominates completely over thermal emission.
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Critical for CO₂ satellite mode selection: At 1.6 µm and 288 K, blackbody thermal emission is $\sim 10^{-8}$ times smaller than reflected solar. Hence daytime SWIR instruments measure purely reflected sunlight. At 4.3 µm, thermal emission starts to be significant. At 15 µm, thermal emission from the atmosphere/surface is the only signal.
Eq. 3.1 — Beer-Lambert transmittance
Monochromatic Beam Transmittance (no scattering)
$$\mathcal{T}_\nu(z_1, z_2) = \exp\!\left(-\int_{z_1}^{z_2} k_a^\nu(z)\,\rho(z)\,dz\right) = \exp\!\left(-\int_{z_1}^{z_2} \sum_g n_g(z)\,\sigma_g^\nu(T(z),p(z))\,dz\right)$$
Fraction of monochromatic radiance transmitted between altitudes $z_1$ and $z_2$ (or equivalently any two optical depths). For a slant path at zenith angle $\theta$, replace $dz$ with $dz/\mu = dz/\cos\theta$ (the "air mass factor"). This is the core of CO₂ column retrieval.
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Symbol Definitions
$k_a^\nu(z)$ [m² kg⁻¹]
Mass absorption coefficient at altitude $z$, frequency $\nu$. Pressure- and temperature-dependent via line-shape broadening
$\rho(z)$ [kg m⁻³]
Air density at altitude $z$. From hydrostatic equation: $dp/dz = -\rho g$
$n_g(z)$ [m⁻³]
Number density of gas $g$ at altitude $z$. For CO₂: $n_{CO_2}(z) = x_{CO_2}\cdot n_{air}(z)$ where $x_{CO_2}$ is the mole fraction (~420 ppm)
$\sigma_g^\nu(T,p)$ [m²]
Absorption cross-section. Temperature affects Doppler width ($\gamma_D \propto \sqrt{T}$) and line strength ($S \propto \exp(-E''/k_BT)$). Pressure affects Lorentz width ($\gamma_L \propto p$)
$\tau_\nu = -\ln\mathcal{T}_\nu$
Optical depth. $\tau=1$: ~63% absorbed. $\tau=3$: ~95% absorbed. CO₂ 4.3µm band: $\tau > 100$
Air mass factor (AMF): For a satellite looking at zenith angle $\theta$ and solar illumination at $\theta_0$:
$$M = \frac{1}{\mu} + \frac{1}{\mu_0} = \frac{1}{\cos\theta} + \frac{1}{\cos\theta_0}$$
The effective CO₂ optical depth is $\tau_{eff} = M \cdot \tau_{nadir}$. CO₂ column sensitivity $\propto M$ — glint observations (small $\theta$) maximize sensitivity.
Eq. 3.2 — Voigt line profile
Spectral Line Shape — Voigt Profile (Convolution of Lorentz & Gauss)
$$\sigma_g^\nu = S(T)\cdot V(\nu-\nu_0;\,\gamma_D,\gamma_L) = S(T)\cdot \frac{1}{\gamma_D\sqrt{\pi}}\,\mathrm{Re}\!\left[w\!\left(\frac{\nu-\nu_0+i\gamma_L}{\gamma_D}\right)\right]$$
$$\gamma_D = \frac{\nu_0}{c}\sqrt{\frac{2k_BT\ln 2}{m}} \quad\text{(Doppler/Gaussian HWHM)}$$
$$\gamma_L = \gamma_L^{ref}\left(\frac{T_{ref}}{T}\right)^{n_L}\frac{p}{p_{ref}} \quad\text{(Lorentz/collision HWHM)}$$
$w(z)$ is the Faddeeva (complex error) function. The Voigt profile is exact for an isolated spectral line broadened by both thermal motion (Doppler) and pressure collisions (Lorentz). High altitudes: Doppler-dominated. Troposphere: Lorentz-dominated. The ratio $\gamma_L/\gamma_D$ determines the Voigt shape parameter.
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Symbol Definitions
$S(T)$ [cm molecule⁻¹]
Temperature-dependent line strength. $S(T) = S(T_{ref})\frac{Q(T_{ref})}{Q(T)}\exp\!\left[-\frac{hcE''}{k_B}\!\left(\frac{1}{T}-\frac{1}{T_{ref}}\right)\right]\frac{1-e^{-hc\nu_0/k_BT}}{1-e^{-hc\nu_0/k_BT_{ref}}}$
$E''$ [cm⁻¹]
Lower state energy of the transition. Determines how line strength changes with temperature
$Q(T)$
Total internal partition function of the molecule at temperature $T$
$n_L$
Temperature exponent for Lorentz broadening (~0.5–0.75 for CO₂ lines)
$m$
Molecular mass of CO₂ = 44 amu = $7.31\times10^{-26}$ kg
Eq. 3.3 — Band transmittance & k-distribution
Spectrally Averaged Transmittance — k-Distribution Method
$$\bar{\mathcal{T}}(\Delta\nu) = \frac{1}{\Delta\nu}\int_{\Delta\nu} e^{-k_\nu u}\,d\nu = \int_0^\infty f(k)\,e^{-ku}\,dk = \int_0^1 e^{-k(g)\,u}\,dg$$
The k-distribution method rearranges the spectral integration: instead of integrating over frequency $\nu$ (rapidly varying), we integrate over the cumulative distribution $g$ of absorption coefficients $k$ (smooth, monotonic). This reduces computation from thousands of line-by-line points to ~16 Gaussian quadrature points per band, enabling fast forward models in retrieval algorithms.
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Symbol Definitions
$u = \int \rho q\,dz$
Gas column amount (absorber path) [kg m⁻² or molec cm⁻²]. For CO₂: $u_{CO_2} = x_{CO_2}\cdot u_{air}$
$f(k)$
Probability density function of absorption coefficients in band $\Delta\nu$. Smooth function → efficient quadrature
$g = \int_0^k f(k')\,dk'$
Cumulative distribution function (CDF) of $k$. Monotonically increasing from 0 to 1 → smooth integrand
$k(g)$
Inverse CDF — the absorption coefficient at CDF value $g$. Used in radiative transfer models (RRTMG, LIDORT)
Eq. 4.1 — Rayleigh scattering cross-section
Rayleigh Scattering Cross-Section (Molecules)
$$\sigma_R(\lambda) = \frac{8\pi^3}{3}\,\frac{(n^2-1)^2}{N^2\lambda^4}\,\frac{6+3\delta}{6-7\delta}$$
Rayleigh scattering cross-section per molecule scales as $\lambda^{-4}$. The factor $(6+3\delta)/(6-7\delta)$ is the King correction factor accounting for molecular anisotropy ($\delta$ is the depolarization ratio; $\delta\approx0.0279$ for air). This purely-scattering process has single scattering albedo $\omega_R = 1$ (no absorption).
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Symbol Definitions
$n(\lambda)$
Real refractive index of air. $n-1 \approx 2.9\times10^{-4}$ at STP. Wavelength-dependent (dispersion)
$N$ [m⁻³]
Molecular number density. At STP: $N_0 = 2.687\times10^{25}$ m⁻³ (Loschmidt number)
$\delta$
Depolarization ratio (~0.0279 for air). Accounts for non-spherical electron clouds in N₂, O₂
$\lambda^{-4}$ dependence
Blue (0.45µm) scatters $(0.7/0.45)^4 \approx 5.9\times$ more than red (0.7µm). Makes sky blue, sunsets red
Eq. 4.2 — Phase function
Phase Function — Angular Distribution of Scattered Light
$$p(\Theta) = \text{normalized angular distribution},\quad \frac{1}{4\pi}\int_0^{4\pi} p(\Theta)\,d\Omega = 1$$
$$p_R(\Theta) = \frac{3}{4}(1+\cos^2\Theta) \quad\text{(Rayleigh)}$$
$$p_{HG}(\Theta) = \frac{1-g^2}{(1 + g^2 - 2g\cos\Theta)^{3/2}} \quad\text{(Henyey-Greenstein, aerosols)}$$
The Rayleigh phase function $p_R$ is symmetric (equal forward/backward scatter). The Henyey-Greenstein function $p_{HG}$ parameterizes aerosol/cloud scattering by asymmetry factor $g = \langle\cos\Theta\rangle$. For clouds $g\approx0.85$ (strongly forward-scattering). The phase function enters the scattering source integral in the RTE.
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Symbol Definitions
$\Theta$
Scattering angle between incident direction $\hat{\Omega}'$ and scattered direction $\hat{\Omega}$. $\cos\Theta = \hat{\Omega}\cdot\hat{\Omega}'$
$g = \langle\cos\Theta\rangle$
Asymmetry parameter [−1,1]. $g=0$: isotropic. $g=1$: complete forward scatter. $g=-1$: complete backscatter. Air (Rayleigh): $g=0$. Aerosols: $g\approx0.6$–$0.7$. Clouds: $g\approx0.85$
Legendre expansion
$p(\cos\Theta) = \sum_{l=0}^{L} (2l+1)\,\chi_l\,P_l(\cos\Theta)$ where $\chi_l$ are expansion coefficients (moments). Used in adding-doubling and discrete ordinates methods
Eq. 4.3 — Mie scattering efficiency
Mie Theory — Extinction & Scattering Efficiencies
$$Q_{ext}(x,m) = \frac{2}{x^2}\sum_{n=1}^\infty (2n+1)\,\mathrm{Re}(a_n + b_n)$$
$$Q_{sca}(x,m) = \frac{2}{x^2}\sum_{n=1}^\infty (2n+1)\,(|a_n|^2 + |b_n|^2)$$
$$x = \frac{2\pi r}{\lambda},\qquad k_M^\nu = \int_0^\infty n(r)\,\pi r^2\,Q_{ext}(x,m)\,dr$$
Mie theory gives exact solution for scattering by a sphere of radius $r$, complex refractive index $m=n_r-in_i$, and size parameter $x$. $a_n,b_n$ are Mie coefficients (Ricatti-Bessel functions). The aerosol extinction coefficient $k_M^\nu$ integrates over the particle size distribution $n(r)$. For CO₂ retrieval, aerosol $\{x_{CO_2},\tau_{aer},r_{eff},m\}$ are all retrieved simultaneously.
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Symbol Definitions
$x = 2\pi r/\lambda$
Size parameter. $x\ll1$: Rayleigh regime. $x\sim1$: resonance (strong Mie effects). $x\gg1$: geometric optics
$m = n_r - in_i$
Complex refractive index. Real part $n_r$: refraction/scattering. Imaginary part $n_i$: absorption. $n_i=0$: non-absorbing aerosol
$n(r)$ [m⁻⁴]
Aerosol particle size distribution. Often lognormal: $n(r) = \frac{N}{\sqrt{2\pi}\ln\sigma_g}\frac{1}{r}\exp\!\left[-\frac{(\ln r - \ln r_g)^2}{2\ln^2\sigma_g}\right]$
$\omega_{aer} = Q_{sca}/Q_{ext}$
Aerosol single scattering albedo. Black carbon: $\omega\sim0.2$ (absorbing). Sulfate: $\omega\sim0.99$ (scattering)
Eq. 5.1 — Schwarzschild equation
RTE for Thermal IR — No-Scattering Limit ($\omega_\nu=0$)
$$\mu\frac{dI_\nu}{d\tau_\nu} = I_\nu - B_\nu(T(\tau_\nu))$$
In the thermal infrared (TIR, 4–100 µm), scattering is negligible compared to absorption/emission for clear-sky conditions. Setting $\omega_\nu = 0$ in the full RTE eliminates the scattering integral. The source function reduces to the Planck function $B_\nu(T)$ at the local temperature — Kirchhoff's law: absorptivity equals emissivity.
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Kirchhoff's law of thermal radiation: In thermodynamic equilibrium (LTE — Local Thermodynamic Equilibrium), the emissivity $\epsilon_\nu$ equals the absorptivity $\alpha_\nu$ at every frequency: $\epsilon_\nu = \alpha_\nu = 1 - \omega_\nu$. LTE holds throughout the troposphere and stratosphere (up to ~80 km). Above that, non-LTE corrections needed (important for 4.3 µm limb sounders).
Eq. 5.2 — Formal solution of Schwarzschild
Upwelling TIR Radiance at Any Level — Formal Integral Solution
$$I_\nu(\tau^*,\mu) = \underbrace{\epsilon_s B_\nu(T_s)\,\mathcal{T}_\nu(\tau^*,0;\mu)}_{\text{surface emission term}} + \underbrace{(1-\epsilon_s)\,I_\nu^{\downarrow}(0)\,\mathcal{T}_\nu(\tau^*,0;\mu)}_{\text{surface reflection of downwelling}} + \underbrace{\int_0^{\tau^*} B_\nu(T(\tau'))\,\frac{\partial\mathcal{T}_\nu(\tau^*,\tau';\mu)}{\partial\tau'}\,d\tau'}_{\text{atmospheric emission}}$$
The upwelling radiance at TOA ($\tau^*$) seen by a nadir-viewing TIR satellite is the sum of: (1) surface thermal emission attenuated to TOA, (2) surface reflection of downwelling atmospheric radiation, and (3) the integral of atmospheric emission from all layers, each weighted by the transmittance from that layer to TOA.
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Symbol Definitions
$\epsilon_s$
Surface emissivity in TIR. Ocean: $\epsilon_s\approx0.98$–0.99. Land: 0.90–0.98. Different for each frequency band
$T_s$
Surface skin temperature [K]. The temperature at the air-surface interface, not the 2 m air temperature
$\mathcal{T}_\nu(\tau^*,\tau';\mu)$
Transmittance from layer $\tau'$ to TOA $\tau^*$ in direction $\mu$. $= \exp\!\left(-(\tau^*-\tau')/\mu\right)$ for uniform atmosphere
$\partial\mathcal{T}/\partial\tau'$
Weighting function for layer $\tau'$ — how much that layer contributes to TOA radiance. Peaks where transmittance changes fastest (see Tab 7)
$I_\nu^{\downarrow}(0)$
Downwelling atmospheric radiance at the surface. Significant in TIR bands; negligible in SWIR (except cloud reflection)
Eq. 6.1 — Complete SWIR TOA radiance
Top-of-Atmosphere Upwelling Radiance — All Terms (OCO-2 / GOSAT context)
$$I_\nu^{TOA}(\mu,\phi) = \underbrace{\frac{\mu_0 F_0^\nu}{\pi}\,a_s(\mu,\mu_0,\phi)\,\mathcal{T}_\nu^2(0,z_s;\mu,\mu_0)}_{\text{(A) surface-reflected solar}} + \underbrace{\int_0^\infty J_\nu^{sc}(z)\,\frac{\partial\mathcal{T}_\nu(z,\infty;\mu)}{\partial z}\,dz}_{\text{(B) atmospheric scattering}} + \underbrace{\text{(TIR emission — negligible at 1.6 µm)}}_{\approx 0}$$
In the SWIR at 1.6 µm and 2.06 µm, term (A) — solar photons reflected off the surface traversing the atmosphere twice — completely dominates. Term (B) is scattered solar light from the atmosphere (aerosol/Rayleigh path radiance). The $\mathcal{T}_\nu^2$ factor shows the two-way absorption: down path × up path.
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Symbol Definitions
$a_s(\mu,\mu_0,\phi)$
Bidirectional reflectance distribution function (BRDF) of the surface [sr⁻¹]. For Lambertian surface: $a_s = A_s/\pi$ where $A_s$ is hemispherical albedo. Ocean glint: sharply peaked BRDF
$\mathcal{T}_\nu^2$
Two-way transmittance = $\mathcal{T}_\nu(\text{TOA}{\to}z_s;\mu_0)\times\mathcal{T}_\nu(z_s{\to}\text{TOA};\mu)$. For nadir view + vertical sun: $= e^{-2\tau_\nu}$. Encodes the entire CO₂ column absorption
$J_\nu^{sc}(z)$
Scattering source function at altitude $z$ (aerosol + Rayleigh scattered solar). This "path radiance" contaminates the CO₂ signal by providing photons that bypassed the surface
$F_0^\nu/\pi$
Converts solar irradiance [W m⁻² Hz⁻¹] to equivalent isotropic radiance [W m⁻² sr⁻¹ Hz⁻¹]. Factor $\mu_0$ gives the projection onto horizontal surface
Decoupled approximation used in fast forward models (e.g. XCO2 retrieval in OCO-2 ACOS algorithm): the two-way transmittance separates into sun-path and view-path. This holds when multiple scattering is small. Full vector (polarized) multiple-scattering RT (VLIDORT, LIDORT) is used for accurate aerosol correction.
Eq. 6.2 — O₂ A-band normalization
Ratio Method — Eliminating Geometric & Albedo Uncertainties
$$\frac{I_{CO_2}^{obs}(\nu_{CO_2})}{I_{O_2}^{obs}(\nu_{O_2})} = \frac{\mathcal{T}_{CO_2}(\nu_{CO_2})\cdot\mathcal{T}_{H_2O}(\nu_{CO_2})\cdot\mathcal{T}_{aer}(\nu_{CO_2})}{\mathcal{T}_{O_2}(\nu_{O_2})\cdot\mathcal{T}_{aer}(\nu_{O_2})} \cdot f(\text{albedo, geometry})$$
$$N_{CO_2} \propto -\mu\mu_0\,\ln\left(\frac{I_{CO_2}^{obs}}{I_{CO_2}^{cont}}\right) \cdot \frac{1}{\sigma_{CO_2}^{eff}},\quad p_s \propto -\mu\mu_0\,\ln\left(\frac{I_{O_2}^{obs}}{I_{O_2}^{cont}}\right)\cdot\frac{1}{\sigma_{O_2}^{eff}}$$
O₂ is 20.946% of dry air — fixed and well-known. Its column amount $N_{O_2} \propto p_s$ (surface pressure). Dividing CO₂ absorption by O₂ absorption cancels: (a) viewing geometry uncertainty, (b) surface albedo, (c) instrumental gain. What remains is the ratio of CO₂ to O₂ column → $x_{CO_2}$ (dry-air mole fraction = XCO₂).
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Symbol Definitions
$I^{cont}$
Continuum radiance — interpolated radiance at a non-absorbing "window" wavelength adjacent to the band. Provides the baseline for computing absorption depth
$\sigma^{eff}$
Effective cross-section averaged over the instrument spectral response function. Depends on CO₂ amount itself (non-linear: strong lines saturate first)
$\text{XCO}_2$
$= N_{CO_2} / N_{dry-air}$ in parts per million (ppm). Column-averaged dry-air mole fraction. ~420 ppm globally. OCO-2 target precision: 0.3–1 ppm (0.07–0.25%)
Eq. 7.1 — Two-stream equations
Coupled Upward/Downward Flux Equations
$$\frac{dF^+}{d\tau} = \gamma_1 F^+ - \gamma_2 F^- - \gamma_3\,\mu_0 F_\odot\,e^{-\tau/\mu_0} - (1-\omega)\,\pi B$$
$$\frac{dF^-}{d\tau} = \gamma_2 F^+ - \gamma_1 F^- + \gamma_4\,\mu_0 F_\odot\,e^{-\tau/\mu_0} + (1-\omega)\,\pi B$$
The two-stream approximation collapses the full angular dependence of $I(\mu)$ into two hemispheric fluxes: $F^+$ (upwelling) and $F^-$ (downwelling). The coefficients $\gamma_{1..4}$ depend on the closure scheme. The solar forcing appears as an inhomogeneous source term. This is the basis of climate model radiation schemes (e.g. RRTMG).
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Closure coefficients (Eddington scheme)
$\gamma_1 = \frac{7-\omega(4+3g)}{4}$
Loss from upwelling: extinction − backscattering
$\gamma_2 = \frac{-(1-\omega(4-3g))}{4}$
Coupling: downwelling backscatter into upwelling
$\gamma_3 = \frac{2-3g\mu_0}{4}$
Direct solar beam fraction forward-scattered upward
$\gamma_4 = 1-\gamma_3$
Direct solar beam fraction forward-scattered downward
$g$ [−1,1]
Asymmetry parameter of the phase function
Net flux and heating rate:
$$Q(z) = -\frac{\partial(F^+-F^-)}{\partial z} = \rho c_p \frac{\partial T}{\partial t}\bigg|_{rad}$$
The atmospheric heating/cooling rate due to CO₂ at 15 µm drives the greenhouse effect: more CO₂ → increased downwelling $F^-$ at surface → warming.
Eq. 8.1 — Contribution function (TIR)
TIR Weighting Function — Where the Satellite "Sees" in the Atmosphere
$$W_\nu(z) = \frac{\partial I_\nu^{TOA}}{\partial T(z)} = B_\nu'(T(z))\,\frac{\partial\mathcal{T}_\nu(z,\infty)}{\partial z} = B_\nu'(T(z))\cdot(-k_\nu(z)\,\rho(z)/\mu)\cdot\mathcal{T}_\nu(z,\infty)$$
The TIR weighting function $W_\nu(z)$ tells us which atmospheric level contributes most to the observed TOA radiance at frequency $\nu$. It is the product of the Planck derivative (sensitivity to temperature) and the transmittance gradient (opacity structure). Peak of $W_\nu$ determines the "sounding altitude" of that channel.
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Symbol Definitions
$B_\nu'(T) = \partial B_\nu/\partial T$
Derivative of Planck function with respect to temperature. Maximum in TIR where both $B_\nu$ and its sensitivity to $T$ are significant
$\partial\mathcal{T}/\partial z$
Transmittance gradient — peaks where opacity transitions from transparent to opaque. For CO₂ at 15µm: peaks at ~10–15 km. Wings of band: peaks lower (more transparent)
$\int_0^\infty W_\nu(z)\,dz = 1-\mathcal{T}_s$
Normalization: integral of weighting functions equals total atmospheric opacity (1 − surface transmittance)
Information content (Shannon entropy):
$$H = \frac{1}{2}\ln\det(\mathbf{I} + \mathbf{S}_a^{1/2}\mathbf{K}^T\mathbf{S}_e^{-1}\mathbf{K}\mathbf{S}_a^{1/2})$$
Number of independent pieces of information = $\text{tr}(\mathbf{A})$ where $\mathbf{A}$ is the averaging kernel matrix (see Eq. 9.3). TIR sounders typically retrieve 2–5 independent CO₂ layers.
Eq. 8.2 — Jacobian (SWIR)
SWIR Jacobian — Radiance Sensitivity to CO₂ Profile
$$K_\nu^{(l)} = \frac{\partial I_\nu^{TOA}}{\partial N_{CO_2}^{(l)}} = -\frac{\mu_0 F_0^\nu}{\pi}\,a_s\,e^{-\tau_\nu^{total}/\mu_0}\,e^{-\tau_\nu^{total}/\mu}\cdot\left(\frac{\sigma_{CO_2}^\nu}{\mu_0} + \frac{\sigma_{CO_2}^\nu}{\mu}\right)$$
The SWIR Jacobian $\mathbf{K}$ is the matrix of partial derivatives of the measured spectrum with respect to each atmospheric state variable (CO₂ in each layer, surface albedo, aerosol optical depth, etc.). It linearizes the forward model around a prior state and is the key quantity in the retrieval inverse problem. Each row is a different wavelength; each column is a different state variable.
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Symbol Definitions
$N_{CO_2}^{(l)}$ [mol/m²]
CO₂ column amount in atmospheric layer $l$. The state vector $\mathbf{x}$ contains CO₂ in all $L$ layers plus nuisance parameters
$\sigma_{CO_2}^\nu/\mu_0 + \sigma_{CO_2}^\nu/\mu$
Two-way path enhancement — CO₂ in any layer absorbs once on the way down ($1/\mu_0$) and once on the way up ($1/\mu$). Greater sensitivity at oblique angles
$e^{-\tau^{total}}$
Attenuation by all gas + aerosol above layer $l$. Lower layers contribute less (more attenuation above). SWIR weighting functions nearly uniform with altitude — good column sensitivity
Eq. 9.1 — Cost function
Optimal Estimation Cost Function — Maximum A Posteriori Solution
$$\mathcal{J}(\mathbf{x}) = \underbrace{(\mathbf{x}-\mathbf{x}_a)^T \mathbf{S}_a^{-1} (\mathbf{x}-\mathbf{x}_a)}_{\text{prior constraint}} + \underbrace{(\mathbf{y}-\mathbf{F}(\mathbf{x}))^T \mathbf{S}_e^{-1} (\mathbf{y}-\mathbf{F}(\mathbf{x}))}_{\text{measurement fit}} \rightarrow \text{minimise}$$
The retrieved state $\hat{\mathbf{x}}$ minimizes $\mathcal{J}$: a weighted compromise between matching the measurement $\mathbf{y}$ (observed spectrum) via the forward model $\mathbf{F}(\mathbf{x})$ and staying close to the prior $\mathbf{x}_a$. The prior constrains underdetermined aspects (e.g. CO₂ in individual layers when only the column is well-measured).
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Symbol Definitions
$\mathbf{x} \in \mathbb{R}^n$
State vector. Contains: CO₂ profile on $L$ levels, H₂O profile, surface pressure $p_s$, surface albedo $A_s$, aerosol optical depth $\tau_{aer}$, aerosol size parameters, temperature profile, SIF (solar-induced fluorescence), instrument zero level offset. Typically $n \sim 20$–60
$\mathbf{y} \in \mathbb{R}^m$
Measurement vector: observed spectral radiances at $m$ frequencies across O₂ A-band, 1.6 µm CO₂ band, 2.06 µm CO₂ band. OCO-2: $m\approx3\times1016 = 3048$ spectral points
$\mathbf{F}(\mathbf{x}) \in \mathbb{R}^m$
Forward model: computes the radiance spectrum that the satellite WOULD see given state $\mathbf{x}$. Calls radiative transfer code (e.g. LIDORT) + instrument model (ILS, noise)
$\mathbf{S}_a \in \mathbb{R}^{n\times n}$
Prior (a priori) error covariance. Encodes expected variance and correlations in the state variables before measurement. CO₂: $S_a^{CO_2} \sim (3\text{ ppm})^2$ vertically correlated
$\mathbf{S}_e \in \mathbb{R}^{m\times m}$
Measurement error covariance. Diagonal terms: instrument noise ($\text{SNR}^{-2}$). Off-diagonal: correlated errors from forward model approximations ("model error" $\mathbf{S}_b$). Full: $\mathbf{S}_e = \mathbf{S}_n + \mathbf{K}_b\mathbf{S}_b\mathbf{K}_b^T$
Eq. 9.2 — Gauss-Newton iteration
Iterative Solution — Gauss-Newton / Levenberg-Marquardt Update
$$\mathbf{x}_{i+1} = \mathbf{x}_a + \mathbf{G}_i\left[\mathbf{y} - \mathbf{F}(\mathbf{x}_i) + \mathbf{K}_i(\mathbf{x}_i - \mathbf{x}_a)\right]$$
$$\mathbf{G}_i = (\mathbf{K}_i^T\mathbf{S}_e^{-1}\mathbf{K}_i + \mathbf{S}_a^{-1})^{-1}\mathbf{K}_i^T\mathbf{S}_e^{-1} \quad\text{(gain matrix)}$$
$$\hat{\mathbf{S}} = (\mathbf{K}^T\mathbf{S}_e^{-1}\mathbf{K} + \mathbf{S}_a^{-1})^{-1} \quad\text{(posterior covariance)}$$
At each iteration, the Jacobian $\mathbf{K}_i = \partial\mathbf{F}/\partial\mathbf{x}|_{\mathbf{x}_i}$ is computed (analytically or via finite difference). The gain matrix $\mathbf{G}$ maps measurement residuals into state updates. Convergence criterion: $(\mathbf{x}_{i+1}-\mathbf{x}_i)^T\hat{\mathbf{S}}^{-1}(\mathbf{x}_{i+1}-\mathbf{x}_i) < \epsilon n$. Typically 3–10 iterations.
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Symbol Definitions
$\mathbf{K}_i = \partial\mathbf{F}/\partial\mathbf{x}$
Jacobian matrix [m×n] evaluated at current iterate $\mathbf{x}_i$. Row $j$, column $k$: sensitivity of radiance at frequency $\nu_j$ to state variable $x_k$
$\mathbf{G}$ [n×m]
Gain (contribution) matrix. $\hat{\mathbf{x}} = \mathbf{x}_a + \mathbf{G}(\mathbf{y}-\mathbf{F}(\mathbf{x}_a))$ at convergence. Rows show which spectral channels drive each state variable
$\hat{\mathbf{S}}$ [n×n]
Posterior error covariance. $\hat{\sigma}_{XCO_2} = \sqrt{\mathbf{c}^T\hat{\mathbf{S}}\mathbf{c}}$ where $\mathbf{c}$ is the column operator. Target: $\hat{\sigma}_{XCO_2} < 1$ ppm
Levenberg-Marquardt damping: Replace $\mathbf{K}^T\mathbf{S}_e^{-1}\mathbf{K}$ with $\mathbf{K}^T\mathbf{S}_e^{-1}\mathbf{K} + \gamma\mathbf{S}_a^{-1}$ where $\gamma>0$ damps oscillation for highly nonlinear forward models (aerosol retrieval). $\gamma\to0$: pure Gauss-Newton. $\gamma\to\infty$: steepest descent direction.
Eq. 9.3 — Averaging kernel
Averaging Kernel Matrix $\mathbf{A}$ — What the Retrieval Actually Measures
$$\mathbf{A} = \mathbf{G}\mathbf{K} = \hat{\mathbf{S}}\mathbf{K}^T\mathbf{S}_e^{-1}\mathbf{K} = \mathbf{I} - \hat{\mathbf{S}}\mathbf{S}_a^{-1}$$
$$\hat{\mathbf{x}} - \mathbf{x}_a = \mathbf{A}(\mathbf{x}_{true}-\mathbf{x}_a) + \mathbf{G}\boldsymbol{\epsilon}$$
The averaging kernel $\mathbf{A}$ [n×n] tells us: if the true atmosphere deviates from the prior by $(\mathbf{x}_{true}-\mathbf{x}_a)$, how much of that deviation does the retrieval actually see? Row $i$ of $\mathbf{A}$ gives the sensitivity of retrieved $\hat{x}_i$ to the true profile. For XCO₂, the column-averaging kernel (1D) should be ≈1 uniformly from surface to TOA for unbiased column retrieval.
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Symbol Definitions
$\mathbf{A} = \mathbf{I}$
Ideal: perfect retrieval. $A_{ij}=\delta_{ij}$ means retrieved $x_i$ responds only to true $x_i$
$\mathbf{A} = \mathbf{0}$
No information from measurement — retrieval equals prior. Degrees of freedom for signal: $d_s = \text{tr}(\mathbf{A})$
$\mathbf{c}^T\mathbf{A}$
Column-averaging kernel for XCO₂. $\mathbf{c}$ is pressure-weighting vector. Should be ~1 at all altitudes for unbiased XCO₂. Deviates near surface (low surface albedo) and above tropopause
$\mathbf{G}\boldsymbol{\epsilon}$
Noise-induced retrieval error. $\boldsymbol{\epsilon}$ is measurement noise vector. $\hat{\mathbf{S}}_n = \mathbf{G}\mathbf{S}_e\mathbf{G}^T$ is the noise error covariance
Eq. 10.1 — Column-averaged XCO₂
Pressure-Weighted Column-Averaged Dry-Air Mole Fraction
$$X_{CO_2} = \frac{\int_0^{p_s} x_{CO_2}(p)\,dp}{\int_0^{p_s} dp} = \frac{1}{p_s}\int_0^{p_s} x_{CO_2}(p)\,dp \approx \sum_{l=1}^{L} w_l\,x_{CO_2}^{(l)}$$
$$w_l = \frac{\Delta p_l}{p_s},\quad \sum_{l=1}^L w_l = 1$$
XCO₂ is the pressure-weighted vertical average of the CO₂ mole fraction profile $x_{CO_2}(p)$. The weights $w_l = \Delta p_l / p_s$ are proportional to the pressure thickness of each layer. This definition excludes water vapor (dry-air mole fraction). The actual retrieved quantity is the prior profile adjusted by the retrieval: $\hat{x}_{CO_2}(p) = x_a(p) + [\hat{\mathbf{x}}-\mathbf{x}_a]_{CO_2}$.
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Symbol Definitions
$x_{CO_2}(p)$
CO₂ dry-air mole fraction [mol/mol] at pressure level $p$. Ranges from ~420 ppm near surface (NH summer) to ~415 ppm at tropopause, with seasonal ±10 ppm cycle
$p_s$ [Pa]
Surface pressure. Retrieved from O₂ A-band depth. Standard: ~101325 Pa. Varies ±5000 Pa with terrain and weather. Uncertainty in $p_s$ → XCO₂ bias at ~0.25 ppm/hPa
$w_l = \Delta p_l/p_s$
Layer pressure weight. Lower atmosphere has larger $\Delta p$ → weighted toward surface CO₂. Important because surface fluxes (fossil fuels, biosphere) affect surface layers first
Eq. 10.2 — Atmospheric transport & flux inversion
CO₂ Continuity Equation & Bayesian Flux Inversion
$$\frac{\partial c}{\partial t} + \nabla\cdot(\mathbf{u}c) = \nabla\cdot(D\nabla c) + S(\mathbf{x},t)$$
$$\mathbf{y}_{XCO_2} = \mathbf{H}\,\mathbf{s} + \boldsymbol{\epsilon},\qquad \hat{\mathbf{s}} = \mathbf{s}_b + (\mathbf{H}^T\mathbf{R}^{-1}\mathbf{H}+\mathbf{B}^{-1})^{-1}\mathbf{H}^T\mathbf{R}^{-1}(\mathbf{y}-\mathbf{H}\mathbf{s}_b)$$
Satellite XCO₂ observations are used in atmospheric inverse modeling to infer surface CO₂ fluxes $\mathbf{s}$ (fossil fuel emissions, biosphere uptake, ocean exchange). The transport operator $\mathbf{H}$ maps surface fluxes to XCO₂ observations via an atmospheric transport model (GEOS-Chem, TM5, etc.). The Bayesian inversion produces optimized fluxes $\hat{\mathbf{s}}$ with uncertainty.
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Symbol Definitions
$c$ [ppm or mol/m³]
CO₂ concentration field in 3D atmosphere + time
$\mathbf{u}$ [m/s]
Wind velocity field from meteorological reanalysis (ERA5, MERRA-2). Advects CO₂ from source regions
$S(\mathbf{x},t)$ [mol m⁻³ s⁻¹]
Source/sink term. Positive: fossil fuels, respiration, fires. Negative: photosynthesis, ocean uptake. What we want to infer
$\mathbf{H}$ [obs×flux]
Transport/observation operator. Maps surface fluxes → XCO₂ at satellite overpass locations. Computed by adjoint of atmospheric transport model
$\mathbf{B}$ [flux×flux]
Background flux error covariance. Encodes spatial correlations between flux regions (ecosystems, countries)
$\mathbf{R}$ [obs×obs]
Observation error covariance. Includes retrieval noise + transport model error + representativeness error
Global carbon budget check: The global CO₂ growth rate = fossil emissions + land use change − terrestrial uptake − ocean uptake. Satellites constrain the spatial distribution of uptake/emission, resolving where the "missing sink" is.
Eq. 10.3 — Solar-induced fluorescence (SIF)
SIF as a Proxy for Gross Primary Production
$$I^{TOA}_{SIF}(\nu) = \frac{1}{\pi}\,\Phi_F(\nu)\cdot\mathbf{F}^{PAR}\cdot f_{green} + \text{surface reflection} + \text{atmospheric scattering}$$
$$\text{GPP} \approx \text{LUE} \times \text{APAR} = \text{LUE}\times f_{PAR}\times F^{PAR}$$
Chlorophyll re-emits a fraction of absorbed sunlight as fluorescence at 0.69 and 0.74 µm, overlapping CO₂ retrieval windows. Modern CO₂ satellites (OCO-2, GOSAT, TROPOMI) simultaneously retrieve SIF, which correlates with photosynthesis (GPP). SIF fills Fraunhofer lines in the solar spectrum — detectable as "infilling." This links CO₂ column measurements to the land biosphere carbon sink.
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Symbol Definitions
$\Phi_F(\nu)$
Fluorescence yield spectrum [sr⁻¹]. Ratio of emitted fluorescence photons to absorbed PAR photons. Depends on vegetation type and stress state
$F^{PAR}$
Photosynthetically active radiation (0.4–0.7 µm) flux [W/m²]
$f_{green}$
Fraction of pixel covered by photosynthetically active vegetation
LUE
Light use efficiency [mol CO₂ / mol photon]. Links SIF signal to actual CO₂ uptake rate