Role of Spectral Line Physics in CO₂ Retrieval

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Physical Origin of Spectral Lines

CO₂ molecules absorb radiation when photons match the energy difference between molecular quantum states. For vibrational–rotational transitions: $$\Delta E = h \nu_0$$ Thus absorption occurs at discrete frequencies \( \nu_0 \).

In SWIR bands \(\sim 1.6 \mu m, \sim 2.0 \mu m\), these correspond to overtone and combination vibrational bands of CO₂. Each transition produces a spectral line. The observed absorption band is a superposition of thousands of individual lines.

From Microscopic Transitions to Macroscopic Absorption

Microscopically, absorption probability per molecule: $$ \sigma_\nu $$ Macroscopic absorption coefficient: $$ \alpha_\nu = n \sigma_\nu $$ Where:
  • \( n \) = number density
  • \( \sigma_\nu \) = absorption cross-section
Spectral line physics defines \( \sigma_\nu \).

Mathematical Description of a Spectral Line

For a single transition: $$ \sigma_\nu = S(T) f(\nu - \nu_0) $$ Where:
  • \( S(T) \) = line strength
  • \( f(\nu - \nu_0) \) = line shape function
  • \( \nu_0 \) = line center
Line Strength \( S(T) \)
Line strength depends on temperature via Boltzmann population: $$ S(T) = S(T_0) \frac{Q(T_0)}{Q(T)} \exp\left( -\frac{E_l}{k_B} \left( \frac{1}{T} - \frac{1}{T_0} \right) \right) $$ Where:
  • \( Q(T) \) = partition function
  • \( E_l \) = lower state energy
This tells you how many molecules are in the absorbing state. Temperature errors → line strength errors → CO₂ bias.
Line Shape Physics
Real lines are broadened by physical mechanisms.
  1. Doppler Broadening: Due to molecular thermal motion.

    Velocity distribution → frequency shift.

    Width: $$ \alpha_D =\nu_0 \sqrt{\frac{2k_B T}{mc^2}} $$ Line shape: $$ f_D(\nu) = \frac{1}{\alpha_D \sqrt{\pi}} \exp\left( -\frac{(\nu - \nu_0)^2}{\alpha_D^2}\right) $$ Dominates in upper atmosphere (low pressure).
  2. Pressure (Collisional) Broadening Molecular collisions perturb energy levels.

    Lorentzian profile:

    $$ f_L(\nu) = \frac{1}{\pi} \frac{\gamma}{ (\nu - \nu_0)^2 + \gamma^2} $$ Where: $$ \gamma = \gamma_{air} p_{air} + \gamma_{self} p_{CO_2} $$ Dominates in lower troposphere.
  3. Voigt Profile: Real atmosphere → convolution of Doppler and Lorentz: $$ f_V(\nu) = f_D * f_L $$ This profile determines exact absorption line shape.

From Spectral Lines to Optical Depth

Gas optical depth: $$ \tau_\nu^{CO_2} = \int_0^\infty n_{CO_2}(z) \sigma_\nu^{CO_2}(T,p) dz $$ But: $$ \sigma_\nu^{CO_2} = \sum_i S_i(T) f_i(\nu - \nu_i, p, T) $$ Thus: $$ \tau_\nu = \sum_i \int n(z) S_i(T(z)) f_i(\nu - \nu_i, p(z), T(z)) dz $$ This equation shows:
  • Line strengths weighted by temperature
  • Line shapes weighted by pressure
  • Integrated over vertical column
This is the physical core of CO₂ retrieval.

Why Spectral Line Physics Is Critical

Retrieval does not simply measure total absorption. It fits:
  • Line depth
  • Line width
  • Line wings
  • Line asymmetry
These contain information about:
  • CO₂ amount
  • Pressure profile
  • Temperature
  • Path length
Even small errors in line shape modeling distort absorption depth. For 1 ppm precision, spectroscopy must be extremely accurate.

How Retrieval Algorithms Use Spectral Line Physics

Forward model workflow:
  1. Read line parameters from database (e.g. HITRAN)
    • Line center
    • Line strength
    • Air broadening coefficients
    • Temperature exponents
  2. Compute line strength at actual temperature.
  3. Compute Voigt profile at each altitude.
  4. Compute absorption cross-section:
    5. $$ \sigma_\nu(z) = \sum_i S_i(T(z)) f_i(\nu, p(z), T(z)) $$
  5. Compute vertical optical depth: $$ \tau_\nu = \int n_{CO_2}(z) \sigma_\nu(z) dz $$
  6. Insert into forward radiance model: $$ I_\nu = I_0 e^{-\tau_\nu} $$
  7. Convolve with instrument line shape.
This produces simulated spectrum. The algorithm then adjusts CO₂ to match measured line depths.

How Spectroscopy Errors Cause Bias

Suppose line strength has error: $$ S = S_{true} + \delta S $$ Optical depth becomes: $$ \tau_\nu \propto S $$ Measured absorption depth fixed. Thus retrieval compensates by adjusting CO₂: $$ \delta XCO_2 \approx -\frac{\delta S}{S} XCO_2 $$ 1% line strength error → ~1% CO₂ bias (~4 ppm). That is mission-critical.

Subtle Spectroscopic Effects

Advanced retrievals must also include:
  • Line mixing
  • Speed dependence
  • Non-Voigt corrections
  • CIA (collision-induced absorption)
  • Temperature-dependent broadening
Ignoring these can introduce latitude-dependent bias.

Conceptual Summary

Spectral line physics defines: $$\sigma_\nu \rightarrow \tau_\nu \rightarrow I_\nu \Rightarrow XCO_2$$ Thus spectroscopy is the physical bridge between quantum mechanics and climate science.

Reference

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