Chapter 2: Line-by-line physics

Every opacity climt uses ultimately traces back to HITRAN: the atlas of spectral lines for each greenhouse gas. A line-by-line (LBL) calculation means evaluating the opacity at enough wavenumbers to resolve every individual line — tens of thousands of points per cm⁻¹, over thousands of cm⁻¹, on a T-p grid. It is expensive.

But you only have to do it once. All of correlated-k is a lossy compression of an LBL calculation.

The Voigt line shape

Each spectral line at rest wavenumber $\nu_0$ with intensity $S$ has a profile that is the convolution of two broadening mechanisms:

Doppler broadening: $\Delta\nu_D \propto \sqrt{T/M}$ — pure Gaussian, dominant in the upper atmosphere and for light gases.
Pressure broadening: $\Delta\nu_L \propto p$ — pure Lorentzian, dominant in the troposphere.

The convolution is the Voigt profile:

\[\phi(\nu) = \frac{y}{\pi}\int_{-\infty}^{\infty} \frac{e^{-t^2}}{y^2 + (x-t)^2}\,dt,\]

with $x = (\nu-\nu_0)/\Delta\nu_D$, $y = \Delta\nu_L/\Delta\nu_D$. The absorption cross-section at wavenumber $\nu$ is the sum over all lines of $S_i\,\phi_i(\nu)$.

What `linepyline` does

linepyline (Rodrigo Caballero, GPL-3) wraps a Voigt evaluator around the HITRAN line database. You build an rtm object, ask it for a wavenumber grid, and evaluate the mass absorption coefficient (m²/kg) for a gas at a given (T, p):

import linepyline as lpl

r = lpl.rtm()                                    # HITRAN2024 + MT_CKD by default
nu = r.get_nu_grid(1000.0, 1200.0, 0.01)         # cm⁻¹, 0.01 cm⁻¹ spacing
kappa = r.get_kappa_hitran("H2O", 1000.0, 1200.0, 0.01,
                           p=1.0e5, T=296.0)      # m²/kg

Figure 1: **Figure 2.1** — H₂O line-by-line mass absorption coefficient, 1000–1200 cm⁻¹, T=296 K, p=1 bar. Every spike is an individual line. The baseline between lines is not zero — there is a weak continuum and a far-wing contribution.

Warning

linepyline is an optional dependency. The chapters that use it guard against its absence with a try/except ImportError block. If you see “linepyline not available” in notebook output, install it separately from HITRAN’s line data.

Why not just ship LBL?

A single LBL column from 10 to 30 000 cm⁻¹ at R = 500 000 is ≈ 10⁷ points. Multiply by a T × p grid (14 × 20) and keep one gas in double precision: 22 GB. Three gases: 66 GB. This is why we compress into k-distributions (Chapter 3).

Try it yourself

Run examples/k_distribution_demo.ipynb cell by cell. The first half uses linepyline to reproduce Figure 2.1, with a pre-baked fallback array when linepyline is absent.

References

Goody, R. M., and Y. L. Yung. 1989. Atmospheric Radiation: Theoretical Basis. 2nd ed. Oxford University Press. https://academic.oup.com/book/40894.

Rothman, L. S. et al. 2013. “The HITRAN2012 Molecular Spectroscopic Database.” Journal of Quantitative Spectroscopy and Radiative Transfer 130: 4–50. https://doi.org/10.1016/j.jqsrt.2013.07.002.

--- title: "Chapter 2: Line-by-line physics" bibliography: ../../references.bib --- Every opacity climt uses ultimately traces back to HITRAN: the atlas of spectral lines for each greenhouse gas. A line-by-line (LBL) calculation means evaluating the opacity at enough wavenumbers to resolve every individual line — tens of thousands of points per cm⁻¹, over thousands of cm⁻¹, on a T-p grid. It is expensive. But you only have to do it once. All of correlated-k is a lossy compression of an LBL calculation. ## The Voigt line shape Each spectral line at rest wavenumber $\nu_0$ with intensity $S$ has a profile that is the convolution of two broadening mechanisms: - **Doppler broadening**: $\Delta\nu_D \propto \sqrt{T/M}$ — pure Gaussian, dominant in the upper atmosphere and for light gases. - **Pressure broadening**: $\Delta\nu_L \propto p$ — pure Lorentzian, dominant in the troposphere. The convolution is the **Voigt profile**: $$\phi(\nu) = \frac{y}{\pi}\int_{-\infty}^{\infty} \frac{e^{-t^2}}{y^2 + (x-t)^2}\,dt,$$ with $x = (\nu-\nu_0)/\Delta\nu_D$, $y = \Delta\nu_L/\Delta\nu_D$. The absorption cross-section at wavenumber $\nu$ is the sum over all lines of $S_i\,\phi_i(\nu)$. ## What `linepyline` does `linepyline` (Rodrigo Caballero, GPL-3) wraps a Voigt evaluator around the HITRAN line database. You build an `rtm` object, ask it for a wavenumber grid, and evaluate the mass absorption coefficient (m²/kg) for a gas at a given (T, p): ```python import linepyline as lpl r = lpl.rtm() # HITRAN2024 + MT_CKD by default nu = r.get_nu_grid(1000.0, 1200.0, 0.01) # cm⁻¹, 0.01 cm⁻¹ spacing kappa = r.get_kappa_hitran("H2O", 1000.0, 1200.0, 0.01, p=1.0e5, T=296.0) # m²/kg ``` ![**Figure 2.1** — H₂O line-by-line mass absorption coefficient, 1000–1200 cm⁻¹, T=296 K, p=1 bar. Every spike is an individual line. The baseline between lines is not zero — there is a weak continuum and a far-wing contribution.](_artifacts/02_lbl_H2O_1000_1200.png){#fig-lbl-spectrum} ::: {.callout-warning} `linepyline` is an optional dependency. The chapters that use it guard against its absence with a `try/except ImportError` block. If you see "linepyline not available" in notebook output, install it separately from HITRAN's line data. ::: ## Why not just ship LBL? A single LBL column from 10 to 30 000 cm⁻¹ at R = 500 000 is ≈ 10⁷ points. Multiply by a T × p grid (14 × 20) and keep one gas in double precision: 22 GB. Three gases: 66 GB. This is why we compress into k-distributions (Chapter 3). ::: {.callout-tip} ## Try it yourself Run `examples/k_distribution_demo.ipynb` cell by cell. The first half uses `linepyline` to reproduce Figure 2.1, with a pre-baked fallback array when `linepyline` is absent. ::: ## Further reading - @rothman2013 — the HITRAN 2012 spectroscopic database. - @goody1989, Chapter 3 — derivation of Voigt line shapes and pressure broadening.

The Voigt line shape

What linepyline does

Why not just ship LBL?

Further reading

References

What `linepyline` does