Correlated-k Method

Line-by-line radiation resolves absorption at very fine spectral spacing. That is accurate but too expensive for most weather and climate simulations. A correlated-k model compresses each spectral band into a small quadrature over sorted absorption strength.

From Wavenumber to g Space

Within a spectral band $b$, define the cumulative distribution of absorption coefficients by

\[g(k) = \frac{1}{\Delta \nu_b} \int_{\nu \in b} \mathbf{1}\left(\kappa_\nu \le k\right)\,d\nu.\]

The band-integrated transmissivity can then be approximated as

\[\bar{T}_b(u) = \frac{1}{\Delta \nu_b}\int_{\nu \in b} e^{-\kappa_\nu u}\,d\nu \approx \sum_{i=1}^{N_g} w_{b,i} \exp\left(-k_{b,i} u\right).\]

The nodes $k_{b,i}$ and weights $w_{b,i}$ are the g points. The "correlated" assumption is that the sorted absorption ordering remains usable as pressure, temperature, and gas composition vary across the column.

How ecCKD Chooses g Points

ecCKD is not an ordinary per-band Gaussian quadrature over the cumulative distribution above. The construction, described by Hogan & Matricardi (2022) and in the tool documentation (doc/ecckd_documentation.tex in the upstream ecCKD repository, pinned here as the ecckd_source artifact), proceeds in distinct steps:

1. Reorder each gas once, on a fixed reference profile. Wavenumbers are ranked within each band on the median present-day profile of the CKDMIP MMM dataset — in the longwave by the height of the peak cooling rate (Hogan, 2010), in the shortwave by the height at which the zenith optical depth reaches 0.25 from the top of atmosphere. The result is a unique mapping from wavenumber to g space (the gpoint_fraction variable in the CKD-definition files), unlike CKD implementations that reorder independently at each pressure level.
2. Partition each gas's g space against an error tolerance. For each gas and band, the ordered spectrum is partitioned into g points such that a penalty function — heating-rate error plus a weighted flux error of a quasi-monochromatic calculation against line-by-line truth — stays below a configured heating-rate tolerance in K day⁻¹. Fewer g points are needed at looser tolerance.
3. Merge gases with the full-spectrum hypercube partition. The single-gas partitions are overlapped using the hypercube-partition method of Hogan (2010), which sets the combined number of g points per band to $1 - n_\mathrm{gas} + \sum_i n_{g,i}$ rather than the product of single-gas counts.

References: Hogan (2010), J. Atmos. Sci., 67, 2086–2100, doi:10.1175/2010JAS3202.1; Hogan & Matricardi (2020), Geosci. Model Dev., 13, 6501–6521, doi:10.5194/gmd-13-6501-2020 (CKDMIP); Hogan & Matricardi (2022), J. Adv. Model. Earth Syst., DOI 10.1029/2022MS003033 (the ecCKD tool).

What ecCKD Stores

An ecCKD CKD-definition file stores tabulated gas absorption and supporting spectral data. NumericalRadiation loads these files into EcCKDTabulatedGasOpticsModel, which holds:

• longwave and shortwave absorption tables,
• pressure and temperature interpolation grids,
• gas names and optional water-vapor dependent tables,
• longwave and shortwave quadrature weights,
• shortwave Rayleigh scattering data when available.

At runtime, optical_properties! interpolates the tables to each layer, multiplies by gas path amounts, and fills LongwaveOptics and ShortwaveOptics.

Accuracy and Cost

Increasing the number of g points improves the quadrature approximation but also increases the cost of optical-property evaluation and column transport. The published ecCKD pairs exposed by this package are:

The validation workflow compares these reduced models against ecRad and RRTMGP-style references, then reports flux and heating errors as a function of g-point count. New reduced models should be judged by the same metrics before being used in prognostic simulations.