Issue 
A&A
Volume 645, January 2021



Article Number  A20  
Number of page(s)  7  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/202039040  
Published online  22 December 2020 
Spectral binning of precomputed correlatedk coefficients^{★}
Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geoffroy SaintHilaire, 33615 Pessac, France
email: jeremy.leconte@ubordeaux.fr
Received:
27
July
2020
Accepted:
26
October
2020
With the major increase in the volume of the spectroscopic line lists needed to perform accurate radiative transfer calculations, disseminating accurate radiative data has become almost as much a challenge as computing it. Considering that many planetary science applications are only looking for heating rates or midtolow resolution spectra, any approach enabling such computations in an accurate and flexible way at a fraction of the computing and storage costs is highly valuable. For many of these reasons, the correlatedk approach has become very popular. Its major weakness has been the lack of ways to adapt the spectral grid/resolution of precomputed kcoefficients, making it difficult to distribute a generic database suited for many different applications. Currently, most users still need to have access to a linebyline transfer code with the relevant line lists or highresolution cross sections to compute kcoefficient tables at the desired resolution. In this work, we demonstrate that precomputed kcoefficients can be binned to a lower spectral resolution without any additional assumptions, and show how this can be done in practice. We then show that this binning procedure does not introduce any significant loss in accuracy. Along the way, we quantify how such an approach compares very favorably with the sampled cross section approach. This opens up a new avenue to deliver accurate radiative transfer data by providing midresolution kcoefficient tables to users who can later tailor those tables to their needs on the fly. To help with this final step, we briefly present Exo_k, an openaccess, opensource Python library designed to handle, tailor, and use many different formats of kcoefficient and crosssection tables in an easy and computationally efficient way.
Key words: planets and satellites: general / planets and satellites: atmospheres
© J. Leconte 2020
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Despite the considerable increase in computing power, linebyline radiative transfer is still considered a computationally intensive task in many cases of interest. One of the reasons for this state of affairs is that recent progress in spectroscopy has multiplied the size of molecular line lists by more than three orders of magnitude (Tennyson & Yurchenko 2012). We are at a point at which line list themselves, and a fortiori, highresolution cross sections, represent a data volume that is not trivially handled and exchanged.
To face this while maintaining flexibility, some have opted for sampled or binned down crosssection tables^{1} (Line et al. 2015; Waldmann et al. 2015). But Garland & Irwin (2019) showed that this approach is inaccurate if the resolution used is not sufficiently high, and too computationally expensive if the resolution is high enough (we briefly revisit this question in Sect. 2 whose results are summarized in Figs. 1 and 2). It seems that for typical resolutions below ~1000, methods designed to group absorptions, such as the socalled correlatedk method (Liou 1980; Lacis & Oinas 1991), are more efficient while remaining accurate (Irwin et al. 2008; Amundsen et al. 2017; Garland & Irwin 2019; Zhang et al. 2020). It is especially true for multidimensional models that need a very fast radiative transfer and use only 10–30 bins for the whole spectrum (Showman et al. 2009; Wordsworth et al. 2011; Leconte et al. 2013; Amundsen et al. 2017).
For these reasons, it is interesting to directly distribute reference kcoefficients for each molecule instead of highresolution cross sections. This is currently done for the ExoMol project (Chubb et al. 2020). This approach decreases the data volume necessary to handle while keeping an optimal accuracy. A current drawback with precomputed kcoefficients, however, is that the person computing the tables needs to choose a spectral resolution (or wavenumber grid) without knowing exactly what the table will be used for, while kcoefficients are used at their highest potential when tailored to the needed resolution.
To circumvent this problem we present a way to accurately bin down precomputed kcoefficients to an arbitrary spectral grid without any significant additional loss in accuracy. This method is faster than computing the kcoefficients directly from linebyline data. It therefore opens up a new way of disseminating accurate, reference molecular opacities: spectroscopists only need to compute kcoefficients once with a resolution that is higher than the highest resolution they envision their users need (e.g., the resolution of the instrument you want to compare your model to). Users can then easily bin down the data to their spectral grid of choice, possibly even on the fly.
To help with the latter, we developed Exo_k^{2}, an opensource Python 3 library that can directly load precomputed kcoefficient and crosssection tables, change their spectral resolution as well as pressuretemperature grids, and save them back on disk. The library can handle many different formats used in various codes: for example, TauREx (AlRefaie et al. 2019; both pickle and HDF5), petitRADTRANS (Mollière et al. 2019), Nemesis (Irwin et al. 2008), ARCIS (Min et al. 2020), Exo transmit (Kempton et al. 2017), LMD Generic global climate model (Wordsworth et al. 2011; Leconte et al. 2013). Several of these formats are provided by the latest ExoMol release (Chubb et al. 2020). Other formats are being implemented and new ones can be added on request. It is also possible to compute kcoefficient tables from highresolution spectra. One can also combine the opacities of several molecules using the random overlap method (Lacis & Oinas 1991) to create a table for a given atmospheric mix, which is very useful for climate models, for example. Finally, the library includes methods to directly interpolate and combine the opacities, and even compute transmission and emission spectra for 1D planetary atmospheres. Leveraging on the Numba library, these operations are performed in a very computationally efficient way. So Exo_k can be imported directly into any radiative transfer code to handle molecular opacities easily and efficiently.
In this article, we first briefly compare the relative accuracy and efficiency of the sampled cross section and correlatedk approaches in Sect. 2. After introducing the necessary concepts and notations, Sect. 3 demonstrates rigorously why and how to bin down precomputed kcoefficients to an arbitrary resolution. Then, in Sect. 4, we validate the numerical algorithm presented in this work and implemented in Exo_k.
2 Sampled cross sections versus correlatedk tables
To accurately compare the efficiency of various radiative transfer methods when a given resolution and precision are needed, we briefly estimate theaccuracy of atmospheric modeling when using the following types of input radiative data: sampled and binned cross section and kcoefficient tables.
We compare this for two different atmospheric configurations and observation geometries:

Transmission spectrum of a hot Jupiter with a 1000 K isothermal H_{2}/H_{2}O atmosphere in front of a Sun.

Emission spectrum of a pure CO_{2} Marslike atmosphere.
The parameters for these two cases are given in Table 1.
Each time, we start by computing a reference spectrum using highresolution monochromatic cross sections (with a typical step of 0.01 cm^{−1}). For our CO_{2} atmosphere, we use spectra from Turbet & Tran (2017). For our hotJupiter case, we use water spectra produced by the Exomol project with H_{2} broadening (K. Chubb, private communication) based on the line list produced by Polyansky et al. (2018). Collisioninducedabsorptions (CIA) are taken from the HITRAN database (Richard et al. 2011).
Then, the highresolution cross sections are sampled at various resolutions (R_{sp} = {1000, 3000, 10 000, 30 000}) and a new set of spectra are computed at each of these resolutions.
The spectra obtained from the sampled cross sections are then binned (preserving area) at selected final resolutions (R_{fin} = {10, 20, 50, 100, 200, 500, 1000}). The transmission spectra computed in the hotJupiter case are shown in the left column of Fig. 2. Finally, these spectra are compared tothe properly binned reference spectrum (right column of Fig. 2). The root mean squared (RMS) error is shown for each (R_{sp}, R_{fin}) in Fig. 1. The emission spectra computed in the Marslike case are shown in Fig. 3.
For the correlatedk approach, kcoefficient tables are computed from the highresolution spectra directly at the final desired resolution (computed with 20 GaussLegendre quadrature points in gspace) and compared to the reference spectrum binned at the same resolution. This gives the black dashed curves in Fig. 2. The RMS error is shown by the black curve in Fig. 1.
For all the computations described above we used a standard algorithm to describe the transmission and emission spectra of a 1D atmosphere that have been implemented in the Exo_k library^{2}.
We show that the correlatedk approach reaches an accuracy that is one to two orders of magnitude better than sampled cross sections, especially for resolutions bigger than 100. The relative drop in accuracy of the correlatedk method at R_{fin} = 10 is due to the low sampling of the CIA absorption, which is treated as constant in each spectral bin. This could be alleviated by applying the random overlap method to the CIA as well. To compare numerical efficiency, we focus on the R_{fin} = 10 case for which the R_{sp} = 30 000 roughly manages to reach an accuracy equivalent to the correlatedk method. Then, considering that our kcoefficients were computed using 20 quadrature points, the computation is still 150 times faster with the correlatedk method compared to the sampled cross sections.
We conclude this comparison by mentioning binneddown cross sections – in which the finely sampled cross sections are integrated over a coarser grid, thereby preserving area. Although it seems the most natural way to bin cross sections, this method performs even worse than the sampled cross sections (by about one order of magnitude; Fig. 4). This rather counterintuitive result stems from the nonlinear nature of radiative transfer: the opacity dilution due to the binning procedure is not sufficient to increase the transparency of the gas near optically thick lines, while decreasing it in the line wings. Overall, this systematically over estimates the opacity of the atmosphere and creates a systematic bias – on the order of 100 ppm in our hotJupiter test case.
Standard parameters used in our two fiducial cases.
Fig. 1
Root mean square error on the transit spectrum of our fiducial hot Jupiter over the 1–2 μm window as a function of the final observed resolution. Dashed curves represent the error obtained with sampled crosssection tables at various sampling resolution (from top to bottom: R_{sp} = {1000, 3000, 10 000, 30 000}). The spectra are shown in Fig. 2. The black curve with dots represents the error using kcoefficients computed from highresolution spectra directly at the final resolution. The gray curves show results when using binned down kcoefficients using natural (solid with stars) and noninteger (dotted) binning (see Sect. 3.4 for details). 
Fig. 2
Effect of sampling resolution on the transit spectrum of a hot Jupiter. The various rows show the effect for different final observing resolutions (R_{fin}, specified in the top left corner of the panels). Left column: transit depth in the 1–2 μm window (with an offset of 11500 ppm). The various colors correspond to the four sampling resolutions (i.e., the resolution used in radiative transfer calculation before binning; R_{sp}) shown in the legend. The curve with dots shows the reference case computed from our very highresolution cross sections (Δσ ≈ 0.01 cm^{−1}). The highresolution reference spectrum before binning is shown as the shading in the top left panel. Right column: difference (in ppm) between the spectra of the left panel and the reference case. The difference between the calculation with kcoefficients and the reference case is shown with the black dashed curve; the two spectra would be indistinguishable in the left panels. The RMS standard deviation over the spectral region is shown in Fig. 1. 
Fig. 3
Effect of sampling resolution on the emission spectrum of our Marslike case around the 15 μm CO_{2} band. The rows show the effect for different final observing resolutions (R_{fin}). Left column: flux. The various colors correspond to the four sampling resolutions (i.e., the resolution used in radiative transfer calculation before binning; R_{sp}) shown inthe legend. The reference case is computed from our very highresolution cross sections (Δσ ≈ 0.01 cm^{−1}). The highresolution reference spectrum before binning is shown as the shading in the top left panel. Right column: relative error between the spectra of the left panel and the reference case. The error between the calculation with kcoefficients and the reference case is shown with the black dashed curve; the two spectra would be indistinguishable in the left panels. 
Fig. 4
Effect of sampling resolution on the transit spectrum of a hot Jupiter using cross sections that have been binned by preserving the total area. The panels can be compared to the results for sampled cross sections in Fig. 2 (see the caption for details). The binning systematically overestimates the opacity. 
3 Method to bin down kcoefficients
3.1 Correlatedk formalism
Before demonstrating how to bin down kcoefficients, we briefly introduce the basic concepts we need. In the correlatedk method (see Liou 1980 for details), the wavenumber space, where ν is the wavenumber, is divided into N_{b} spectral bands of width . In each band, the transmission through a slab of matter can be written as (1)
where N_{g} is the number of points (or abscissas) used to discretize the gspace and w^{b} are the associated set of weights for each band. The symbol denotes the inverse of the cumulative density function of the opacity within band b – the socalled kdistribution – and should not be confused with k^{ν}, the absorption coefficient, which is a function of the wavenumber. The socalled correlatedk coefficients (or simply kcoefficients) are the , which are the discretized version of the kdistribution ().
An intuitive way to understand that is to say that now, in band b, the opacity can be described by N_{g} representative values () for each of the N_{g} points in gspace (g_{i}), each of these values being affected a weight (wb_{i}). Then any radiative transfer calculation can be computed separately in the N_{g} bins to be later summed up with the proper weights.
An important, although often forgotten assumption of the correlatedk formalism is that both the spectral incoming flux impinging on our medium () and the source function of the latter should be nearly constant within each spectral band (Lacis & Oinas 1991) so that it can be written , where is the total flux within the band. Simply put in the context of a purely absorbing medium, which can be generalized, this ensures that the outgoing flux after a path u is given by (2)
This assumption is important in our context because it is the only one that we need to make to be able to combine the kcoefficients of various bands. Finally, the bolometric flux can be obtained with (3)
3.2 Combining kcoefficients of various bands
To reduce the resolution of precomputed kcoefficients, we essentially need to answer the following question: Can we determine the kcoefficients of a “superband” that is the union of two smaller, nonoverlapping “subbands”^{3}, b_{1} and b_{2}, for which we know the kcoefficients, and ?
In this form, the answer is not trivial. However, when computing kcoefficients, the crucial mathematical object is notthe kdistribution () but its inverse, the cumulative density function (CDF) of the opacity within the band (hereafter called the gdistribution, ĝ^{b} (k)) defined as (4)
where H(x) is equal to 0 where x is positive and 1 elsewhere.
The question can thus be reframed to ask whether we can compute the gdistribution of the superband, , if we know the gdistribution within the two subbands (ĝ^{b1} and ĝ^{b2}). This is rather straightforward, because as long as the two smaller bands do not overlap, we have and (5)
Therefore the gdistribution in the superband is just the sum of the gdistributions in the subbands weighted by their spectral extent.
This can be understood more intuitively when discussing in terms of probability. We want to know if , that is, the probability that a monochromatic ray of light falls in a spectral region where the opacity k^{ν} is lower than k. To compute this, we have to sum, for each band, the probability of the ray falling in the given subband; this probability is , as long as the incoming spectral flux is constant over the superband, multiplied by the probability that k^{ν} < k within this subband, which is ĝ^{b}(k). So we recover Eq. (5) as long as we keep the essential assumption of the correlatedk method that the incoming flux and sources functions do not significantly vary over the wavelength range that we wish to describe by a single distribution.
3.3 Numerical algorithm
As shown above, combining the gdistributions of various bands in a single gdistribution for a large band is straightforward. However, to use this in a radiative transfer code, we usually need the kcoefficients () that are tabulated for fixed gpoints (g_{i}) with the associated weights (wb_{i}). The difficulty is that although g_{i} can be seen as the values of ĝ^{b} in band b for the points located at in opacity space, the distribution ĝ^{b} is sampled at different locations in opacity space in each subband.
To circumvent this problem, we recommend the following approach, which is the one implemented in Exo_k. The whole process is illustrated in Fig. 5 with only two bands, but can be carried out for any numberof bands. For the moment, however, we restrict ourselves to the case in which the superband is composed ofan integer number of subbands (hereafter, natural binning). Once these subbands have been identified, we determine (6)
and compute an evenly logdistributed grid of N_{k} points () between these two values. Then, for each subband, we interpolate on this grid, which yields N_{k} values per band, , representingall the ĝ^{b} on the same grid for all bands. It is thus now easy to compute the global gdistribution which is given by (7)
This generally yields an oversampled version of the distribution because we choose N_{k} > N_{g} (the black curve in the middle panel of Fig. 5). Recomputing the kcoefficients is thus now simply a matter of resampling onto our original ggrid (g_{i}) – going from the black to the gray curve in the right panel of Fig. 5 – as discussed in Lacis & Oinas (1991) and Amundsen et al. (2017). However, unlike those authors, we think that, to be consistent with the usual quadrature rules, the finely sampled kdistribution should not be weightaveraged over the final gbins, but sampled at the precise abscissa of the gpoint determined by the quadrature rule used, following Irwin et al. (2008).
As longas we resample on the original grid, we keep the original quadrature with N_{g} points, and the weights are left unchanged. However, it is still possible at this point to use different gpoints, especially if we need to use less gpoints for computational reasons (as is often the case with 3D climate models). For this, we can simply resample on a different ggrid and use the relevant weights, which is an option of Exo_k. In general, Exo_k works for any choice of gspace sampling specified by the user.
Fig. 5
Example of the binning process for two bands. Left: kcoefficients on N_{g} = 20 GaussLegendre quadrature points for two bands ( and ). Middle: gdistribution for the two bands resampled on , a grid of N_{k} = 100 points between k_{min} and k_{max} ( and ). The black line indicates the weighted sum of these two gdistributions, i.e., the gdistribution of the superband (). Right: colored lines are shown as in the left panel. The black line indicates the finely sampled kdistribution (). The gray line represents the kdistribution of the superband resampled on the original ggrid with N_{g} = 20, i.e., the final kcoefficients (). 
3.4 Noninteger binning
Up to now, we only considered the case in which the final superband encompasses an integer number of subbands. This is required for our demonstration to be exact without any further assumptions.
In practice, it is still possible to bin kcoefficients on a spectral grid that does not exactly respect the limits of the subbands (hereafter, noninteger binning). To do that, we replace the spectral extent, Δν^{b}, of the two bands at the boundaries of the superband in Eq. (7) by their spectral extent inside the superband. This is equivalent to binning a highresolution spectrum to a lowresolution grid using a weighted average. In theory, this would remain perfectly accurate if the statistical distribution of the absorption coefficient were constant within the subband. As this is not true in general, we test the accuracy of this approach in the next section.
4 Validation
To validate our approach, we compute spectra in the two cases detailed in Sect. 2. As we do not want to test the accuracy of the correlatedk approach itself, which has already been done before, but whether the binning introduces additional errors, the comparison proceeds as follows:

Starting from very highresolution spectra, we compute a first reference set of tables of kcoefficients with resolutions ranging from R = 1000 to R = 5 and a gspace sampled using 20 GaussLegendre quadrature points. The exact R values used do not matter. However, we make sure that the wavenumber points in the lowresolution grids can be found in the highresolution grid to use only natural binning.

Starting from the R = 1000 table of kcoefficients^{4} we just computed, we use the binning approach described above to compute a new set of ten tables of kcoefficients with the exact same resolutions and wavenumber grids as the reference set.

The emitted or transmitted flux of our fiducial atmosphere is computed at each resolution with both the reference and the binned kcoefficients and the results are compared two by two.
The results of the comparison are shown in Fig. 6. For the Mars emission case, we compare the RMS standard deviation of the relative difference between the two spectra in the 1–200 μm range. For the Hotjupiter in transmission test case, the comparison metric used is the RMS standard deviation of the relative difference between the two spectra in the 1–2 μm window.
All the kcoefficients computed in this work use 20 GaussLegendre quadrature points in gspace. Because the number of intermediate points used to resample the gdistribution in opacity space (N_{k}; see Sect. 3.3) is a free parameter, we tried different values of this parameter. We see that an insufficient intermediate sampling results in significant numerical errors, as could be anticipated. However, when N_{k} is a factor of a few larger than the initial number of gpoints, the accuracy of the method asymptotes below 10^{−3}−10^{−4}, which is itself much lower than the error introduced by the correlatedk method in the first place (see for example Amundsen et al. 2017). We note that although choosing a larger N_{k} may slightly increase the binning time, this does not affect the efficiency of the subsequent radiative transfer calculations in any way. We thus used results with N_{k} = 5N_{g} as a baseline. We also found that using initial data with 40 GaussLegendre points did not significantly affect the performances of the binning procedure. Our algorithm is thus robust to the initial choice of gspace sampling.
The spectra obtained with this set of binneddown kcoefficients are also compared directly to the reference highresolution spectrum as described in Sect. 2 to give the absolute difference shown in gray with stars in Fig. 1. Although not strictly equal because they are computed using a slightly different reference spectrum, the absolute difference shown in Fig. 1 and the relative difference from Fig. 6 can be shown to be consistent by multiplying the latter by the average transit depth as shown on the right axis of Fig. 6. In this particular case, we further wanted to test the impact of noninteger binning. For this a new set of kcoefficients were computed in the exact same way, but with a spectral grid that is slightly shifted in wavenumber to force noninteger binning. As shown by the dotted gray curve in Fig. 1, the accuracy is lower, although this method remains more accurate than sampled cross sections. In addition, the error decreases with the binning factor because the errors at the boundaries of our spectral bins are more diluted.
We therefore conclude that our binning method does not introduce any significant additional errors.
Fig. 6
Numerical relative error introduced by the binning process as a function of the binning factor (ratio of the initial resolution, R_{ini}, to the final one, R_{fin}) for four different values of the number of intermediate sampling points N_{k} (see Sect. 3.3; dotted: N_{k} = N_{g}; dashed: N_{k} = 1.5N_{g}; dotdash: N_{k} = 2N_{g}; solid: N_{k} = 5N_{g}, where N_{g} = 20). Top: RMSrelative topofatmosphere flux error in our Marslike case. Bottom: RMS relative transit depth error in our hotJupiter case. The right axis gives the absolute error to be compared to the results of Figs. 1 and 2. With a sufficient sampling, the errors are under the 10^{−3} − 10^{−4} level depending on the geometry, which is below the error introduced by the correlatedk method in the first place. 
5 Conclusions
We have demonstrated mathematically how the kdistribution of the opacity in a large band can be obtained from the kdistributions inside each of the smaller bands that it encompasses. Then we have presented a numerical algorithm to use this property to bin down precomputed tables of kcoefficients to any arbitrary resolution. Finally, we showed, in two concrete cases, that the numerical error added by this procedure is relatively small compared to the errors due to the more general use of the correlatedk approach.
To facilitate the implementation of this new method, we developed Exo_k, an opensource Python library that has been designed to handle many different sources of radiative data (including, of course, kcoefficient tables, but also cross section and collisioninducedabsorption tables). Because this library is still rapidly evolving, see our extensive online documentation for tutorials, example notebooks, and a complete description of the library’s features^{2}.
We think the flexibility that this approach brings to the correlatedk method opens up a completely new way of disseminating reference radiative data for use in atmospheric models: precomputed kcoefficients can now be directly distributed at a fraction of the data volume while providing fast and accurate results for any user that does not need a higher resolution. For example, R = 1000 kcoefficients computed on 20 gauss points such as that distributed by the ExoMol project take about 50 times less space on disk than the highresolution spectra used to produce them. This is a considerable compression factor, and that does not include the hundreds of CPU hours that would be needed to compute the spectra.
This of course makes us wonder what would be the optimum resolution and quadrature for a flexible and versatile correlatedk database. Our test show that R = 1000 kcoefficients computed on 20 gauss points, when combined with the flexibility brought by a tool like Exo_k, seem to be sufficient for most 1D3D atmospheric model applications^{5} as well as modeling synthetic spectroscopic observations of planetary atmospheres up to that resolution. This method can thus be used to model HST, Spitzer, Ariel (Tinetti et al. 2018), and lowresolution JWST data. For highresolution JWST observation modes, higher resolution radiative data will be needed.
Acknowledgements
I thank F. Selsis for being, as always, an outstanding scientific sparring partner and the TauREx team at UCL for pointing out many important Python tricks and libraries. I am also indebted to K. Chubb and M. Turbet for making some radiative data available before their publication. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement n° 679030/WHIPLASH).
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Here, sampled cross sections specifically refer to cross sections that are computed monochromatically but at a resolution that does not necessarily resolve the lines or their shape. Binneddown cross sections are computed on a finely sampled spectral grid and then integrated over a coarser grid, thereby preserving area.
See the online documentation for tutorials, example scripts and notebooks, and an extensive and uptodate description of the (evolving) features of the library: http://perso.astrophy.ubordeaux.fr/~jleconte/exo_kdoc/index.html
This choice is motivated by the resolution chosen by the ExoMol project for their kcoefficient tables (Chubb et al. 2020).
All Tables
All Figures
Fig. 1
Root mean square error on the transit spectrum of our fiducial hot Jupiter over the 1–2 μm window as a function of the final observed resolution. Dashed curves represent the error obtained with sampled crosssection tables at various sampling resolution (from top to bottom: R_{sp} = {1000, 3000, 10 000, 30 000}). The spectra are shown in Fig. 2. The black curve with dots represents the error using kcoefficients computed from highresolution spectra directly at the final resolution. The gray curves show results when using binned down kcoefficients using natural (solid with stars) and noninteger (dotted) binning (see Sect. 3.4 for details). 

In the text 
Fig. 2
Effect of sampling resolution on the transit spectrum of a hot Jupiter. The various rows show the effect for different final observing resolutions (R_{fin}, specified in the top left corner of the panels). Left column: transit depth in the 1–2 μm window (with an offset of 11500 ppm). The various colors correspond to the four sampling resolutions (i.e., the resolution used in radiative transfer calculation before binning; R_{sp}) shown in the legend. The curve with dots shows the reference case computed from our very highresolution cross sections (Δσ ≈ 0.01 cm^{−1}). The highresolution reference spectrum before binning is shown as the shading in the top left panel. Right column: difference (in ppm) between the spectra of the left panel and the reference case. The difference between the calculation with kcoefficients and the reference case is shown with the black dashed curve; the two spectra would be indistinguishable in the left panels. The RMS standard deviation over the spectral region is shown in Fig. 1. 

In the text 
Fig. 3
Effect of sampling resolution on the emission spectrum of our Marslike case around the 15 μm CO_{2} band. The rows show the effect for different final observing resolutions (R_{fin}). Left column: flux. The various colors correspond to the four sampling resolutions (i.e., the resolution used in radiative transfer calculation before binning; R_{sp}) shown inthe legend. The reference case is computed from our very highresolution cross sections (Δσ ≈ 0.01 cm^{−1}). The highresolution reference spectrum before binning is shown as the shading in the top left panel. Right column: relative error between the spectra of the left panel and the reference case. The error between the calculation with kcoefficients and the reference case is shown with the black dashed curve; the two spectra would be indistinguishable in the left panels. 

In the text 
Fig. 4
Effect of sampling resolution on the transit spectrum of a hot Jupiter using cross sections that have been binned by preserving the total area. The panels can be compared to the results for sampled cross sections in Fig. 2 (see the caption for details). The binning systematically overestimates the opacity. 

In the text 
Fig. 5
Example of the binning process for two bands. Left: kcoefficients on N_{g} = 20 GaussLegendre quadrature points for two bands ( and ). Middle: gdistribution for the two bands resampled on , a grid of N_{k} = 100 points between k_{min} and k_{max} ( and ). The black line indicates the weighted sum of these two gdistributions, i.e., the gdistribution of the superband (). Right: colored lines are shown as in the left panel. The black line indicates the finely sampled kdistribution (). The gray line represents the kdistribution of the superband resampled on the original ggrid with N_{g} = 20, i.e., the final kcoefficients (). 

In the text 
Fig. 6
Numerical relative error introduced by the binning process as a function of the binning factor (ratio of the initial resolution, R_{ini}, to the final one, R_{fin}) for four different values of the number of intermediate sampling points N_{k} (see Sect. 3.3; dotted: N_{k} = N_{g}; dashed: N_{k} = 1.5N_{g}; dotdash: N_{k} = 2N_{g}; solid: N_{k} = 5N_{g}, where N_{g} = 20). Top: RMSrelative topofatmosphere flux error in our Marslike case. Bottom: RMS relative transit depth error in our hotJupiter case. The right axis gives the absolute error to be compared to the results of Figs. 1 and 2. With a sufficient sampling, the errors are under the 10^{−3} − 10^{−4} level depending on the geometry, which is below the error introduced by the correlatedk method in the first place. 

In the text 
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