We created a specific GCM–RTM interface inside the COSP module of the EC-Earth climate model, in order to perform the radiative transfer calculations online, passing the instantaneous atmospheric fields on a global scale to the RTM with a time step of 6 h.

In the radiative scheme of IFS, the spectral emissivity of the surface is assumed to be constantly equal to 0.99 outside the atmospheric window region (800–1250 cm-1). Conversely, within this region, the emissivity depends on eight types of surfaces, namely open sea, sea ice, interception layer, low and high vegetation, exposed and shaded snow and bare ground. These emissivity values are interpolated to a regular wavenumber grid with steps of 10 cm-1, in the range from 100 to 3000 cm-1 and supplied to the σ-IASI RTM. The surface pressure and surface temperature are directly supplied to σ-IASI, while the simulated temperature, humidity and gases concentration profiles are first interpolated to the fixed pressure grid used by σ-IASI. Carbon dioxide, methane and nitrous oxide concentrations are horizontally and vertically uniform, depending only on time. The ozone mixing ratio used in the model simulation is a function of pressure, latitude and time, as described in . Finally, concentrations of the other trace gases required by σ-IASI (SO2, CO, HNO3, NH3, OCS, HDO and CF4) are not modelled in the IFS; thus, they are extracted from the U.S. standard atmosphere of the Atmospheric Constituent Profiles dataset .

In order to minimize the huge impact of the radiative code on the GCM computing performance, the look-up tables (LUTs) of optical depth parametrization coefficients are allocated and loaded from file only once at the beginning of the simulation and then stored and deallocated at the end of the process. Moreover, the outgoing radiance is computed only once every four latitude × longitude grid points of the EC-Earth model, for a total of about 6000 simulated spectra every 6 h.

To limit the data storage required, the high-resolution spectrum computed by σ-IASI is convolved with a 10 cm-1 wide box function and sampled every 10 cm-1. Since EC-Earth does not include variables with a spectral dimension, we stored the simulated spectra in new auxiliary 4D variables declared in the IFS grib code scheme, using the dimension corresponding to vertical model levels as the spectral dimension. These simplifications allowed us to strongly reduce the computational cost of the model run, passing from an initial value of 90 000 core hours per simulated year (CHPSY) to 4000 CHPSY, which is comparable to the cost of the other simulators already present in COSP and about 8 times higher than an EC-Earth standard atmosphere-only simulation without COSP (about 500 CHPSY).

In order to better correlate the differences between modelled and observed radiances to model biases, we first studied, for a reference tropical atmosphere, the sensitivity of the radiance, computed with σ-IASI, to the model temperature and trace species concentration.

Figure shows a spectrum of the TOA spectrally resolved radiance simulated by σ-IASI in clear-sky conditions in the spectral range between 50 and 2250 cm-1. The spectral intervals measured by IASI and FORUM are highlighted, together with the approximated spectral ranges of the atmospheric window regions and the main gas absorption bands, which are summarized in Table .

The FIR region (from 100–667 cm-1) is dominated by the signature of the rotational band of water vapour (blue shade), whose study will be consolidated with the help of future FORUM measurements . In anticipation of FORUM measurements, we focus here on part of the MIR region of the spectrum measured by IASI, specifically between 645–2250 cm-1. In this region, the spectrum undergoes a strong absorption between 640–750 cm-1 due to CO2. In more detail, in the core of the CO2 band (660–670 cm-1), the atmosphere appears opaque from space, and the radiance reaching the TOA originates from the stratosphere. On the contrary, in the wings of the CO2 band measured by IASI (700–750 cm-1), the effective emission level is located in the middle to upper troposphere. From 800 to 950 cm-1 and from 1100 to 1250 cm-1 (red shades), the atmosphere is almost transparent, and the radiance reaching the TOA mainly originates from the surface or the atmospheric layers closest to the surface. Other strong absorption bands are located between 980 and 1080 cm-1 (ozone; green shade), between 1200 and 1400 cm-1 (methane; pink shade) and between 1250 and 1350 cm-1 (nitrous oxide; grey shade). Finally, the rotational–vibrational water vapour band, located between 1400 and 1850 cm-1, is highlighted.

The radiance reaching the TOA originates mainly from upper atmospheric layers in the spectral regions of strong absorption, while in more transparent regions it originates from the lower atmospheric layers . More accurate information on the atmospheric layers contributing to the observed OLR spectrum can be extracted from the analysis of the Jacobians, defined as the partial derivatives of radiance with respect to any atmospheric parameters. In Fig. , we show the Jacobians of the most relevant variables computed with the σ-IASI RTM for a tropical standard atmosphere over ocean at the IASI sampling of 0.25 cm-1 from 10 to 2250 cm-1. For a better readability of the graph, the Jacobian values shown are the absolute values and normalized, separately, to their maximum value for each quantity.

As we can see from Fig. , the entire spectrum is sensitive to the temperature profile (red areas).

The atmospheric window AW1 is more transparent than the atmospheric window AW2, where the radiation is slightly sensitive to the water vapour concentration. In the first case (AW1), the radiation is affected by the temperature of the atmospheric layers between 0 and 3 km, while in the second one (AW2), it is controlled by the temperature of layers at greater heights – up to about 7 km.

Specific features can be noticed for each gas.

The outgoing radiance between 1400 and 1850 cm-1 (WV2) is attenuated by H2O in the upper troposphere (blue area) from approximately 10 to 20 km. Water vapour also reduces the OLR in the FIR region (WV1).

A change in sign of the Jacobian is observed for all gases between the troposphere and stratosphere (see Fig. S1 in the Supplement). In fact, in the troposphere, absorption processes dominate the emission ones (so increased concentration reduces the OLR), while the opposite happens in the stratosphere.

As already mentioned, due to the complication with comparing all-sky measurements at different spatial resolutions, here we focus only on the clear-sky part, leaving the analysis of cloudy sky to a future work. In addition, the low temporal sampling of the model (the RTM is called every 6 h) and the uncertainties in the land surface emissivity do not allow us to perform accurate comparisons between measured and observed radiances over land.

Therefore, we focus on the comparison of daytime measurements and model outputs over ocean.

The selection process of simulated and measured spectra used to build the clear-sky radiance climatologies goes through the following steps. To save computing time, the EC-Earth model simulates spectra in correspondence of only once every four latitude × longitude grid cells (see Fig. S2 in the Supplement). The dimension of the model cells is 0.7×0.7∘. For each of these cells, we compute the monthly average radiance using only the simulated spectra with the local solar time between 6 and 12 h over the ocean and only if the current cloud cover of the model cell is less than 30 %. Generally, the radiative computation in clear-sky conditions in climate models exploits the same all-sky properties (e.g. surface temperature, temperature/humidity profile, surface albedo and aerosol) but with the clouds removed. Since the temperature and humidity profiles of the model are indirectly affected by the presence of clouds, this causes a systematic negative bias when comparing observed and simulated clear-sky radiative fluxes. According to , this difference can reach up to -12 Wm-2 in the convectively active regions at tropical latitudes. In the CERES EBAF 4.1 dataset, a new adjustment factor is introduced to generate the TOA clear-sky fluxes that are more in line with the clear-sky fluxes represented in climate models, as described in and . On the contrary, in the EC-Earth vs. IASI comparison, to mitigate this problem, we selected only the spectra computed over geographical grid points in which the simulated total cloud cover is less than 30 %. This threshold is the result of a trade-off between reducing the impact of this potential source of bias and keeping a significant number of measurements in the analysis. In principle, a lower threshold would be more desirable, but at the same time, this reduces the statistics. We then compute the monthly zonal averages by averaging the monthly mean radiances relating to the model cells within the considered latitude belt . With this procedure, all model cells contribute to the zonal mean with equal weight.

Concerning the measurements, IASI spectra are selected from 2∘×2∘ cells centred on the ECE model cells for which spectra are simulated. On the one hand, the dimension of these cells is large enough to allow the selection of a sufficiently large number of IASI spectra. On the other hand, these cells do not overlap each other; thus, each IASI measurement contributes only once to the statistics. For each of these cells, we compute the monthly average radiance using IASI measured spectra that meet the following conditions:

The radiance is measured in daytime, in the near-nadir geometry over the ocean, and corresponds to clear-sky conditions (cloud mask of AVHRR = 0).

On the basis of these assumptions, a systematic comparison has been performed using a dataset that covers the years from 2008 to 2016, for latitudes ranging from 60∘ S to 60∘ N, over the ocean.

Figure  shows the 9-year zonal average of the brightness temperature (BT; defined in Appendix ) differences (model minus observations). Plots describing BT biases allow us to show more clearly the differences over all the selected spectral range. On the contrary, radiance biases in the Planckian tail tend to become too small to be visible. Considering that model SSTs are constrained to be equal to the observed values, we expect small differences between model and IASI spectral radiances in the atmospheric spectral windows (AW1 and AW2). Thus, the limited discrepancies in BTs obtained in the spectral window AW1 in the tropical belt (30∘ S, 30∘ N) confirm the self-consistency of the performed comparison. Instead, as mentioned in Sect. , the atmospheric window AW2 is more sensitive to the presence of water vapour and shows a small positive bias, approximately equal to 0.3 K. At mid-latitudes, however, especially in the Southern Hemisphere, a negative model bias of about 1 K is present in both the atmospheric windows, thus making difficult the comparison of all the frequencies at these latitudes. As mentioned before, this model bias is thought to be linked to the limited number of spectra available at these latitudes and also to the cloud cover representation in the model. This aspect will be further discussed in Sect. .

Significant discrepancies, of about 3.5 K, are present in the CO2 band at all latitudes, which might indicate a warm bias in the model temperature of the upper troposphere and stratosphere. A warm bias of about 1 K is also seen in the rotational–vibrational water vapour band (WV2), but this is limited to the tropical latitudinal belt between 30∘ S and 30∘ N.

In a similar way, the bias visible in the O3 band is strictly dependent on latitude and is characterized by a positive sign at the tropics, while it tends to take negative values at mid-latitudes. As described in Sect. , this spectral band is affected by surface, lower troposphere and stratospheric temperatures.

On the basis of the above considerations, we focus our analysis on the discrepancies found over tropical ocean, where the BT differences in the atmospheric windows are close to zero.

Figure shows the 2008–2016 average of the model-simulated and observed BTs over ocean, at tropical latitudes (30∘ S, 30∘ N; top panels) and their differences (bottom panels). In this case, we see that the model is generally in good agreement with the observations, and the most significant discrepancies are found in the CO2 band, in the O3 band and along the water vapour absorption band (WV2).

It is possible to characterize the height dependence of the model temperature bias by focusing the analysis on specific spectral bands. Therefore, Fig.  shows the 9-year monthly averages of simulated and observed BTs over ocean in four spectral intervals, together with the respective temperature Jacobians, which highlight the atmospheric levels that are most sensitive to the temperature.

The largest sensitivity to temperature in the stratosphere is found in the spectral interval centred at 660 cm-1 (Fig. a), while the channel at 700 cm-1 is sensitive to temperature in the upper troposphere and stratosphere (Fig. b). The maximum sensitivity to temperature in the middle troposphere is reached in the spectral interval centred at 730 cm-1. Finally, as usual, the spectral channels in the atmospheric window, in this case averaged between 845 and 855 cm-1, are a proxy for the lower troposphere and surface temperature. As already mentioned, the spectral intervals centred at 660, 700 and 730 cm-1 are not only sensitive to temperature but also to the CO2 concentration. The model, however, uses CO2 global average concentrations smoothly increasing with time, according to the actual measurements; thus, any uniform warm model bias cannot be attributed to an erroneous carbon dioxide concentration (see Sect. ). The regional and seasonal variabilities in the CO2 concentrations amount at most to a few parts per million and thus could cause only small local, seasonal biases.

Therefore, Fig. a confirms the presence of a strong stratospheric warm bias in the model. The more we move towards the lower layers of the atmosphere (Fig. a and b), the more the bias is reduced, until its sign reverses in the spectral band centred at 730 cm-1 (see Fig. c), which is sensitive to the middle troposphere temperature. Finally, as expected, the bias is very small in the atmospheric window over ocean (Fig. d). A more peaked seasonality is present in the ECE curve, which is, however, within the standard deviation of the two curves. The BT differences between ECE and IASI in the spectral intervals of Fig.  are also shown on a global scale in Figs. S5, S6, S7 and S8 in the Supplement. From these plots, it is evident that the biases are homogeneous over the tropical and subtropical latitudes, where we are comparing the simulated and observed BT. Some compensating biases are only present at 850 cm-1 in the atmospheric window. However, these differences are generally very small and always within 1 K.

We now exploit the intervals 725–735 and 1395–1405 cm-1 to explore the accuracy of the representation of the water vapour concentration in the model. In fact, as illustrated by the integrated Jacobians reported in Fig. a, the spectral band at 1400 cm-1 (WV2) is sensitive both to the tropospheric temperature and to the upper tropospheric water vapour concentration. In both spectral intervals, the maximum sensitivity to temperature occurs between 3 and 10 km (green and pink lines in Fig. b). Since the previous analysis (Fig. c) has shown a small negative BT bias at 730 cm-1 assigned to a cold bias of the middle troposphere temperature in the model, if the water vapour concentration were well represented, then we would see a negative BT bias in the spectral band 1395–1405 cm-1. However, since the model BT in the spectral interval centred at 1400 cm-1 shows a slightly positive bias (Fig. c), we conclude that the negative temperature bias of the model seems to be overcompensated for by a dry bias of the water vapour profile in the 7–15 km range. In fact, a too dry upper troposphere in the model allows more radiant energy to reach the TOA, as also witnessed by the negative sign of the water Jacobian shown in Fig. a. BT differences between ECE and IASI in the spectral band 1395–1405 cm-1 are also shown on a global scale in Fig. S9 in the Supplement. Also, in this case, the positive bias is fairly uniform in the latitudinal band between 30∘ S and 30∘ N.

Finally, as in the CO2 case, EC-Earth uses local climatological monthly means of methane and ozone concentrations; thus, the discrepancies occurring in the ozone and methane absorption bands (not shown) are most likely due to biases in the simulated temperature.

For the same period, the average clear-sky OLR flux computed by EC-Earth over ocean, between the latitudes 30∘ S and 30∘ N, is equal to 288.47±0.34Wm-2. This is slightly overestimated compared to the analogous average clear-sky flux obtained from CERES observations that is equal to 287.36±0.32Wm-2. From the Stefan–Boltzmann law, considering the power radiated by a blackbody at the temperature of 295 K (about the average surface temperature of tropical ocean), a difference of 1Wm-2 corresponds to a BT difference of about 0.2 K, i.e. smaller than the biases localized in specific wavenumber ranges that we found from the spectral analysis.

To date, systematic FIR spectral radiance measurements from space are not yet available; thus, we are not able to characterize the discrepancies between the model and observations in the whole OLR spectral range. Despite that, the analysis presented shows clearly that a good agreement between simulated and observed total OLR fluxes may be obtained from the cancellation of opposite-in-sign systematic errors localized in specific spectral ranges. In conclusion, observations of spectrally resolved OLR fluxes from space are needed for a proper tuning of model parameters.

We have seen that a perfect spatial and temporal matching of measurements and simulations is very difficult to actualize; therefore, systematic biases could also arise from the strategy adopted to sample the data.

In order to evaluate the impact of the data sampling strategy, we carried out the following test. We interpolated the EC-Earth model cloud fraction and the measured CERES cloud fraction to a regular space grid of 1∘×1∘ and a time step of 6 h. Then, assuming, alternatively, the interpolated CERES and EC-Earth cloud fractions, we built the statistical distributions for the year 2008 of the same observed SSTs for the grid points with cloud fractions less than 30 %. Figure  shows the SST statistical distributions obtained for EC-Earth and CERES cloud fractions at tropical (Fig. a) and mid-latitudes (Fig. b). At tropical latitudes, the SST distributions obtained with the model (red boxes) and CERES (grey boxes) cloud fractions are quite similar; the average values differ by 0.4 K, and the standard deviations (≈ 4.5 K) differ by less than 0.1 K. On the other hand, at southern mid-latitudes (-60∘ S, -45∘ S; see Fig. b), the offset between the two distributions amounts to 0.9 K.

The larger bias of 0.9 K at the southern mid-latitudes is likely contributing to the observed negative model BT bias found in the atmospheric window at the southern mid-latitudes in Fig. . The good agreement between the two SST distributions found in the tropical latitude belt strengthens our confidence in the previous analyses we presented for tropical latitudes. At these latitudes, the choice of comparing model and measured climatologies corresponding to cloud fractions smaller than 30 % ensures that the biases introduced by the data sampling strategy smaller than ≈0.5 K, i.e. also smaller than most of the model biases inferred from Fig. .

We further tested the results of the spectral analysis by comparing the temperature and humidity obtained from the climate model outputs with data provided by ERA5, the latest climate reanalysis from ECMWF. The reanalysis combines available data from different instruments (satellites, ships, weather stations, etc.) with models to generate a complete and continuous global coverage of the main geophysical variables . Figure a shows the temperature differences between EC-Earth and ERA5 reanalysis averaged over 15 years (2000–2014). The strong warm bias in the stratosphere confirms the discrepancy found in our spectral analysis in the region between 655 and 665 cm-1. On the other hand, our spectral analysis did not directly detect the cold model bias visible in Fig.  at the tropopause. In fact, as shown in Fig. , the OLR spectrum is not very sensitive to temperature and gas concentrations at these heights. In addition, the spectral band centred at 700 cm-1 (Fig. b) is partially affected by the positive stratospheric temperature bias, which can easily mask the underlying negative bias at the tropopause. We also note that the small negative model temperature bias present in the troposphere is consistent with the difference in BTs found in the spectral band at 730 cm-1 (Fig. c).

Figure b shows the percentage differences between EC-Earth and ERA5 specific humidity averaged over the period 2000–2014. The absolute values of these differences are visible in Fig. S10 in the Supplement. The negative EC-Earth humidity bias found in the upper troposphere and stratosphere is consistent with the positive bias detected in the model-simulated BTs in the spectral interval between 1395 and 1405 cm-1. Indeed, the drier atmosphere of the model at these heights implies a greater amount of radiation reaching the TOA in the water vapour band.

We also provide an estimation of the BT bias inferred from the temperature and water vapour biases of the EC-Earth model with respect to ERA5. First, we computed the mean profile of these temperature and water vapour biases along the tropical latitudes between 30∘ S and 30∘ N. Then, we performed the scalar product of these profiles with the respective Jacobians (Figs.  and ), which was computed assuming the profiles of a standard tropical atmosphere. The result of this estimation is summarized in Table .

The clear-sky BT biases found in the comparison with IASI (Sect. ) are generally consistent with the estimates inferred here from the comparison with ERA5. Remarkably, the sign is consistent in all cases, although some differences are found in the magnitude. The spectral analysis indicates a stronger positive bias in the stratosphere (660 cm-1) than inferred from ERA5. This is also seen in the spectral band at 700 cm-1. In addition, we also have a more pronounced negative bias at 730 cm-1, possibly produced by a larger negative temperature bias of the model in the middle troposphere. Finally, the BT biases for the spectral bands at 850 and 1400 cm-1 show a very good agreement with the estimates obtained from the ERA5 reanalysis. The positive BT bias at 1400 cm-1 is caused by the negative bias of the model water vapour in the upper troposphere.

It is important to note that the discrepancies between the BT biases, described in the last two rows of Table , may arise from different causes. First, the comparison of temperature and water vapour concentration profiles of Fig.  are performed in all-sky conditions, while the spectral analysis reflects the differences observed in clear-sky conditions. Furthermore, the Jacobian used in the estimation of Table  is referred to a standard tropical atmosphere, on the basis of the dataset of , which may differ from the real one.