us76_u86_4-spectra

The us76_u86_4-spectra data set is an absorption cross section data set for the the us76_u86_4 absorbing gas mixture and computed using the SPECTRA Information System.

Description

Absorber

The us76_u86_4 absorber is a gas mixture defined by the mixing ratios provided in the table below.

Species	N2	O2	CO2	CH4
Mixing ratio [%]	78.084	20.9476	0.0314	0.0002

Each gas species in this gas mixture is an absorber. The us76_u86_4 absorber is named in this way because it corresponds to the gas mixture defined by the us76 atmosphere model [NNU76] in the region of altitudes below 86 km, and restricted to the 4 main molecular species, namely N2, O2, CO2 and CH4.

Data source

The absorption cross sections data in this data set were computed using the SPECTRA Information System, with inputs available here. SPECTRA uses the HITRAN database (2016-edition) [GRH+17].

Structure

The us76_u86_4-spectra data set includes two data variables:

• absorption cross section (xs)

• mixing ratios (mr)

and three dimension coordinates:

• wavenumber (w)

• pressure (p) and

• molecule (m).

The absorption cross section data variable is tabulated with respect to the wavenumber and pressure coordinates. The mixing ration data variable is tabulated with respect to the pressure and molecule coordinates.

Note

The data set does not include a temperature dimension coordinate because the temperature coordinate is mapped onto the pressure coordinate according to the pressure-temperature relationship defined by the us76 thermophysical profile. This is why this data set is tied to the us76 thermophysical profile.

The graph below illustrates the pressure and temperature points used to compute the absorption cross section. There are 64 points (in red). The pressure values are geometrically spaced between \(10^{-6} \mathrm{atm}\) (\(0.101325 \mathrm{Pa}\)) and \(1 \mathrm{atm}\) (\(101325 \mathrm{Pa}\)). The temperature values are defined as the ordinate values when the abscissa values are set to these 64 pressure values, and the pressure-temperature relationship is given by the US76 model (blue curve).

Wavenumber coordinate

The wavenumber coordinate is a linearly spaced mesh between \(4000 \, \mathrm{cm}^{-1}\) (\(2500 \, \mathrm{nm}\)) and \(25711 \, \mathrm{cm}^{-1}\) (\(390 \, \mathrm{nm}\)) with a constant wavenumber step of \(0.01 \, \mathrm{cm}^{-1}\) (corresponding wavelength step varies between \(~ 0.00015 \, \mathrm{nm}\) and \(~ 0.006 \, \mathrm{nm}\)).

Pressure coordinate

The pressure coordinate is a geometrically spaced mesh between \(0.101325 \, \mathrm{Pa}\) (\(10^{-6} \, \mathrm{atm}\)) and \(101325 \, \mathrm{Pa}\) with 64 mesh points.

The minimum pressure value of \(10^{-6} \, \mathrm{atm}\) corresponds to a maximum altitude of approximately 93 km, in the us76 thermophysical profile. This minimum value is a restriction of SPECTRA. For higher altitudes, that is to say for lower pressure values, the absorption cross section value is approximated to 0.

Note

Given that the maximum value of the absorption cross section at 93 km is:

\[\max_{\nu} \sigma_{a} \approx 9 \, 10^{-23} \, \mathrm{cm}^2,\]

which corresponds to a maximal absorption coefficient value of:

\[\max_{\nu} k_{a} \approx 4 \, 10^{-4} \, \mathrm{km}^{-1},\]

this approximation seems reasonable.

Interpolation accuracy

We assess the accuracy of the wavenumber and pressure interpolation of the us76_u86_4-spectra absorption data set.

Method

We down-sample the original data set on the corresponding axis, i.e. either wavenumber or pressure, by selecting data corresponding to even-indices coordinate values. Then, we interpolate the down-sampled data set on the odd-indices coordinate values. We assess the interpolation accuracy by computing the relative errors:

(1)\[\epsilon(p_i, q) = \frac{ \lvert \sigma_{\mathrm{a, interpolated}}(p_i, q) - \sigma_{\mathrm{a, original}}(p_i, q) \rvert}{ \sigma_{\mathrm{a, original}}(p_i, q)}\]

where:

• \(\sigma_{\mathrm{a, original}}\) is the original data set absorption cross section,

• \(\sigma_{\mathrm{a, interpolated}}\) is the interpolated down-sampled data set absorption cross section,

• \(p_i\) is the interpolation coordinate (i.e. wavenumber or pressure) where \(i\) is odd, and

• \(q\) is the other coordinate (i.e. pressure or wavenumber).

Note

Since the assessed interpolation accuracy here is that of the downsampled data set, we can expect that the interpolation accuracy of the original data set to be better.

We discard the cross section data where the following condition is met:

(2)\[k_{\mathrm{a}}(\nu_{i}, p) < 10^{-3} \, \mathrm{cm}^{-1},\]

where \(k_{\mathrm{a}}(\nu_{i}, p)\) is computed with:

\[k_{\mathrm{a}}(\nu_{i}, p) = n(p) \, \sigma_{\mathrm{a}}(\nu_{i}, p)\]

where \(n(p)\) is the number density corresponding to the pressure \(p\) in the us76 thermophysical profile.

We apply (2) because we estimate that these absorption coefficient values are too small to influence radiative transfer computations significantly.

Note

By interpolating the down-sampled absorption cross section data set at the odd-index coordinate values, we maximise the interpolation error with respect to the reference data set because each interpolation point is exactly at the middle of each data interval. This means that the relative errors are computed in the worst case scenarios, hence provide a conservative estimate of the interpolation accuracy.

Results

Below are illustrated some histograms of the values of \(\epsilon\) computed with (1).

Wavenumber interpolation

Below is given the histogram of the relative errors \(\epsilon (\nu_i, p)\) for \(p=101325 \, \mathrm{Pa}\) when interpolating the down-sampled data set on to the even-index wavenumber values of the original data set.

Pressure interpolation

Below is illustrated the histogram of the relative errors \(\epsilon(p_i, \nu)\) for \(p_i = 65349 \, \mathrm{Pa}\) and for all wavenumbers values of the original data set.

Analysis

Wavenumber interpolation

We make the following observations:

• At standard pressure (ground level altitude), the interpolation accuracy is fine – better than 5 % except for some outliers.

• As the pressure decreases (corresponding altitude increases), the interpolation accuracy gets poorer and poorer, although the errors counts decrease simultaneously.

• The counts number decreases with decreasing pressure because lower pressure means lower number density hence fewer absorption cross section data points satisfy (2).

Pressure interpolation

We make the following observations:

• At ~4 km altitude (\(p = 65349 \, \mathrm{Pa}\)), the interpolation accuracy is relatively bad – up to 50 % – except for a few outliers.

• The interpolation accuracy generally improves with increasing altitudes (decreasing pressure).

• For altitudes ~ 21 km and higher (\(p < 4703 \, \mathrm{Pa}\)), the interpolation accuracy is better than 5 %, except for some outliers.

• The counts number decreases with decreasing pressure because lower pressure means lower number density hence fewer absorption cross section data points satisfy (2).

Conclusion

It is not easy to draw any conclusion regarding the interpolation accuracy of the us76_u86_4-spectra absorption data set from the above analysis. Since the interpolation accuracy of the down-sampled data set is not quite satisfactory, there is not guarantee that the interpolation accuracy of the actual data set is. On the other hand, the interpolation accuracy of the down-sampled data set was estimated in the worst case scenarios, so that the average accuracy is in fact better than the reported accuracy. Nevertheless, we plan to generate a new version of the data set with finer wavenumber and pressure meshes to improve the interpolation accuracy.