The focus of this paper is on Lambertian surface reflection in combination with clear-sky atmospheric conditions. For these conditions, there exists a simple relationship between the Earth reflectance R and the (Lambertian) surface albedo As : 2R(μ,μ0,ϕ,ϕ0,As)=R0(μ,μ0,ϕ-ϕ0)+AsT(μ,μ0)1-Ass⋆. In Eq. (), the first term on the right is the so-called path reflectance R0. This is the atmospheric contribution to the Earth reflectance for a Rayleigh atmosphere which is bounded below by a non-reflecting surface. The second term in Eq. () is the surface contribution to the Earth reflectance. This term depends on the surface albedo As, on the total transmission T of the atmosphere, and on the spherical albedo s⋆. The property s⋆ is the spherical albedo of the Rayleigh atmosphere for illumination from below. The parameters R0, T, and s⋆ can in principle be calculated using any radiative transfer model see, e.g.,.

From a given measured reflectance Robs, the surface albedo As can now be determined from Eq. (): 3As=Rλobs-Rλ0Tλ(μ,μ0)+sλ⋆(Rλobs-Rλ0). Both parameters Rλobs and Rλ0 depend on μ, μ0, and ϕ-ϕ0 and so, in general, does As. When clear-sky conditions apply, the parameter As is the Lambertian-equivalent reflectivity (LER) of the surface.

Traditional, non-directional surface LER databases are built on the assumption that all surface types act as Lambertian reflectors. That is, one assumes that the amount of light being reflected by the surface does not depend on the direction of incoming and reflected light. This is illustrated in Fig. a. The Lambertian assumption is, unfortunately, in many cases not justified. A more realistic description of the reflective properties of a surface requires a bi-directional reflectance distribution function (BRDF) . The BRDF is a function of the incoming and outgoing directions. A hypothetical surface BRDF is shown in Fig. b. Here, the surface BRDF contains a retroreflection lobe resulting from increased reflection by vegetation in the backscattering direction.

In the retrieval algorithm of the traditional GOME-2 surface LER database , grid cells acts as storage containers in which all observations with fitting geolocation are stored, irrespective of viewing geometry and scene conditions. Statistical methods are then employed to identify the cloud-free scenes. For the directional GOME-2 surface DLER, the grid cell container is split into five sub-containers, each representing a certain viewing angle range (see Fig. b). The traditional retrieval algorithm is then run five times, deriving surface LER for each of the five viewing angle containers. The viewing angle dependence can then be analysed. This procedure is explained in Fig. .

The coloured circles in Fig.  represent the surface LER retrieved by GOME-2 for a grid cell over the Sahara desert, for the five viewing angle containers and for the 26 wavelength bands defined in Sect. . The viewing angle θv presented on the horizontal axis is defined as follows. 4θv=-θ,for the east viewing directionθ,for the west viewing direction The centres of the viewing angle containers are indicated by the dotted vertical lines. Parabolic curves are fitted to the five retrieved surface LER values for all wavelength bands. Labels are provided for most of the wavelength bands.

From Fig.  it can be concluded that there is a clear dependence on θv. The parabolic fits suggest that the dependence may be parameterised as a function of the viewing angle θv in the following way: 5ADLER=ALER+c0+c1⋅θv+c2⋅θv2, where θv is negative on the east side of the orbit swath and positive on the west side of the orbit swath; see Eq. (). The coefficients c0, c1, and c2 are wavelength dependent and are calculated for each grid cell, provided that all five viewing angle segments are sufficiently filled with observations. For water bodies the coefficients c0, c1, and c2 are set to zero because surface DLER and BRDF data cannot be cast into a climatology easily for water surfaces. This is because of the strong dependence on the viewing and solar angles for sun glint conditions and because of the dependence on parameters such as wind speed and chlorophyll concentration. The provided surface reflectance over water is therefore the standard minimum LER and more representative of the diffuse component of the surface reflection .

The MODIS Ross–Li surface BRDF model is a linear kernel-based BRDF model used to describe the surface reflectance of land surfaces. The surface anisotropy is described by two geometry-dependent kernels which have to be combined with the provided kernel coefficients if one wants to calculate the BRDF. The Li–Sparse kernel Kgeo is the geometric kernel which describes the contribution of sunlit and shaded parts of a scene due to the presence of three-dimensional objects, typically trees. The Ross–Thick kernel Kvol is the volumetric kernel which describes the smaller-scale variation of the leaf canopy, i.e. the orientation of the leaves themselves.

The geometric and volumetric kernels are independent of wavelength. The wavelength dependence of the BRDF is contained entirely in the kernel coefficients. The expression for the surface reflectivity is as follows . Ag(λ,θ,θ0,ϕ-ϕ0)=fiso(λ)+fvol(λ)⋅Kvol(θ,θ0,ϕ-ϕ0)6+fgeo(λ)⋅Kgeo(θ,θ0,ϕ-ϕ0) The exact expressions needed to calculate Kvol and Kgeo are provided in Appendix . The coefficients fiso, fvol, and fgeo are the kernel coefficients of the isotropic, volumetric, and geometric contributions.

For our model calculations we make use of the “Doubling-Adding KNMI” (DAK) radiative transfer code which will be described extensively in Sect. . Here we make use of surface reflection defined by a BRDF instead of Lambertian surface reflection as described by . We thereto provide DAK the three kernel coefficients (fiso, fvol, and fgeo) as defined in the MODIS ATBD . Using this set-up, the TOA reflectance is calculated at a number of wavelengths, for the VZA and SZA nodes θ and θ0 that are also part of the look-up tables (LUTs) described in Sect. , and for 360 equidistant values of the relative azimuth angle ϕ-ϕ0. The surface elevation is set to zero (sea level) and the ozone column to 350 DU. As before, cloud and aerosols are not included. The calculations are performed monochromatically.

Next, the simulated surface DLER is retrieved from the simulated TOA reflectances using a similar set-up as the one described in Sect. . The only difference here is that the input reflectances are not measured but simulated by DAK, i.e., they are based on surface reflection described by the BRDF kernel coefficients (fiso, fvol, and fgeo). The differences between BRDF and DLER for all angles θ, θ0, and ϕ-ϕ0 are then analysed as a function of wavelength.

The results for 772 nm are presented in Fig.  in the form of polar plots. The solar zenith angle was set to 32∘. Figure a presents the BRDF, characterised by the kernel coefficients (fiso,fvol,fgeo)=(0.36,0.24,0.03). These kernel coefficients are representative of vegetated surfaces such as forests . Figure b shows the TOA reflectance calculated by the DAK radiative transfer model (RTM). Note that the reflectance is similar to the BRDF. Figure c presents the retrieved DLER. The differences between BRDF and DLER appear to be small. This is confirmed by Fig. d, which presents the BRDF (red curve) and DLER (dotted blue curve) inside the principal plane (ϕ-ϕ0=0∘ or 180∘). Differences are found, but they are small even for large viewing zenith angles.

These results are not unexpected because at 772 nm the Rayleigh optical thickness is quite low (about 0.02), so scattering in the atmosphere is relatively weak. This means that (single) surface reflection dominates and that the reflectance at the TOA is similar to the BRDF of the surface. Moreover, in this situation the retrieved surface DLER and BRDF are similar. Note that the behaviour of the BRDF for extreme viewing zenith angles in the forward scattering direction is suspicious because the BRDF becomes negative for viewing zenith angles close to 90∘. In the DAK RTM, the surface BRDF is therefore not allowed to become negative.

The situation changes quite a bit at 463 nm (see Fig. ). Figure  shows that at this wavelength the typical value of the BRDF is much lower than at 772 nm. Because of increased Rayleigh scattering in the atmosphere, the TOA reflectance is now very different from the surface BRDF. The retrieved DLER shows quite some differences compared to the BRDF. This is caused by an increased occurrence of scattering in the atmosphere (Rayleigh optical thickness ∼0.2 at 463 nm) and multiple scattering via the surface. The effects may seem to be modest in an absolute sense, but relative to the typical value of the BRDF (∼0.03) they can be quite large, depending on the viewing and solar angles that are involved. For instance, inside the “hot spot” the differences are 0.003, of which the BRDF is 0.045 and the DLER is 0.042. The differences therefore can go up to 6 % at this wavelength.

Based on the results we can distinguish three wavelengths regimes. For λ>1000 nm the functional behaviour of DLER and BRDF is nearly identical and interchangeable. For 500<λ<1000 nm the DLER and BRDF values are similar, and DLER and BRDF can be interchanged in most practical situations. For example, at 555 nm the difference in the hot spot region is less than 3 % for θ0=32∘ and less than 8 % for θ0=60∘. For λ<500 nm, however, DLER and BRDF differ by too much for them to take over each other's role. For example, in the UV at 380 nm the differences go up to 12 % for θ0=32∘ and even 30 % for θ0=60∘. It should be noted that the absolute differences in these cases are small (maximum of the order of 0.01). Please note that vegetated surfaces have a relatively strong surface anisotropy compared to most other surface types such as desert. The provided numbers therefore represent worst-case situations.

Depending on the application, using BRDF instead of DLER, or DLER instead of BRDF, can be acceptable even below 500 nm. For the validation study presented in Sect.  we will, however, restrict ourselves to λ>500 nm.

Table  provides a list of the chosen wavelength bands, as well as their central wavelength and bandwidth. Note that the wavelength bands at 328, 585, 685, 697, and 712 nm were not present in the previous version of the GOME-2 surface LER database their Table 2. Most of the wavelength bands are 1 nm wide. The wavelength bands are therefore narrow enough to be considered monochromatic but also wide enough to effectively minimise the impact of the Ring effect . The reflectances for the wavelength bands are calculated from the reflectances measured by the individual detector pixels that fall within the wavelength band. A boxcar weighting function w is applied to each detector pixel reflectance. This weighting function is defined as follows. 7wij=1,for|λi-λjc|≤ωj0,for|λi-λjc|>ωj In this equation, λi is the wavelength of detector pixel i, λjc is the central wavelength of wavelength band j, and 2ωj is the width of wavelength band j. Normalisation of the resulting band reflectance is performed by dividing the result with the number of participating detector pixels, denoted by Nj.

Most of the wavelength bands are positioned in the continuum parts of the spectrum, avoiding absorption bands as much as possible. This is essential because having to take absorption by atmospheric species into account complicates the radiative transfer calculations considerably. For wavelength bands located in the continuum monochromatic simulations are sufficient. This is not the case for a number of wavelength bands which are affected too much from absorption by trace gases. These wavelength bands (see Table ) require a spectral handling of the radiative transfer calculations. This approach is described in Sect. .

Three examples of the impact of absorption by oxygen, water vapour, and ozone are given in Fig. . Figure a shows the situation for the wavelength band at 758 nm, positioned just in front of the O2-A band, while Fig. b shows the situation for the wavelength band near 697 nm, which is spectrally surrounded by water vapour absorption lines. Figure c presents the situation for the wavelength band at 328 nm, where ozone absorption is quite variable over the extent of the wavelength band. The black curves represent the simulated reflectance spectra. These spectra were calculated for clear-sky conditions, for a surface albedo As=0.5 at sea level, for nadir view and local noon (θ=θ0=0∘), for an ozone column of 350 DU, and for a water vapour column of 4.0 gcm-2. For comparison, the horizontal green curves represent the reflectance spectra without taking absorption by oxygen and/or water vapour into account in the radiative transfer calculations.

For the wavelength band at 758 nm the impact of oxygen absorption is obviously very small. On the other hand, for the wavelength band near 697 nm the impact of water vapour absorption is much larger. A monochromatic calculation is clearly not sufficient in this case. To proceed, we first define Gj(λ), the spectral response function of wavelength band j, as a weighted superposition of the slit functions of the individual detector pixels by using the boxcar weighting defined in Eq. (). That is, for the response function Gj we have 8Gj(λ)=1Nj∑i=1wij⋅Si(λ), where Si is the normalised slit function of detector pixel i from the appropriate spectral band and Nj the number of detector pixels that make up wavelength band j. The resulting response functions G758, G697, and G328 are presented in Fig.  as blue curves, in arbitrary units. The vertical green lines indicate the wavelengths λi of the detector pixels that contribute to the reflectance of the wavelength band. Next, we calculate the simulated band reflectance. For this we first need to simulate the spectrum surrounding the wavelength band at a high spectral resolution. We use a spectral resolution of 0.01 nm. The spectral sampling is then increased by a factor of 100 using Akima interpolation . This allows for accurate numerical integration, and the resulting expression for the simulated band reflectance is 9Rjsim=∑kΔλ⋅Gj(λk)⋅Rsim(λk), where the summation over k involves a summation over the wavelengths λk and Δλ=10-4 nm.

The impact of neglecting absorption by oxygen and/or water vapour and using monochromatic calculations can now be calculated. For the 758 nm case given in Fig.  this effect is only 0.003 on the reflectance and about the same for the surface LER. This wavelength band could therefore be treated monochromatically. For the 697 nm case, however, the effect is 0.018, which is too high to justify monochromatic calculations. For the 328 nm wavelength band, adopting monochromatic calculation would lead to an error of -0.027. For the wavelength bands at 328, 697, 712, 758, and 772 nm we use Eq. () to calculate the simulated band reflectance. In Table  this is indicated by the label “S” in the fifth row. For the other wavelength bands we use monochromatic calculations, indicated by “M” in the fifth row of Table . For the 758 and 772 nm wavelength bands a monochromatic calculation would have sufficed, but because of their strong importance to cloud and aerosol retrieval using the O2-A band, we decided to go further than necessary by adopting spectral calculations.

For the radiative transfer calculations we make use of the radiative transfer code “Doubling-Adding KNMI” (DAK) . The DAK code is able to calculate all four components of the Stokes vector , and in its minimal set-up it features molecular scattering and Lambertian surface reflection, but the user can decide to include many other features such as scattering by clouds and/or aerosols, absorption by various trace gases, and surface reflection defined by a BRDF. The extension to BRDF was described by . In the calculations we did not include clouds and aerosols and adopted Lambertian surface reflection. Polarisation is included in the calculations. We used a standard mid-latitude summer (MLS) atmosphere for the atmospheric profiles and included absorption by ozone, NO2, and O2–O2 for all wavelength bands. For some of the wavelength bands we also included absorption by oxygen and water vapour (see Table ).

For all 26 wavelength bands look-up tables (LUTs) were created. The LUTs were made for 7 ozone column values (50, 200, 300, 350, 400, 500, and 650 DU), for 10 surface heights (ranging from 0 to 9 km in steps of 1 km), for water vapour columns of 0 and 4 gcm-2, and for 42 non-equidistant values of μ and μ0. The dependence on the relative azimuth angle ϕ-ϕ0 can be treated analytically. To explain, because the simulations represent clear-sky Rayleigh atmospheres, the Fourier expansion of the reflectance in terms of the relative azimuth angle ϕ-ϕ0 ends after only three terms. More specifically, we have 10R0=a0(μ,μ0)+∑i=122ai(μ,μ0)cos⁡i(ϕ-ϕ0). We therefore do not store the reflectances R0 in the LUTs but instead store the Fourier coefficients a0, a1, and a2. The reflectance R0 can be calculated from these. The LUTs contain the parameters a0, a1, a2, T, and s⋆.

Examples of the magnitude of the DLER surface anisotropy in GOME-2 data are provided in Fig. . In the left column the traditional non-directional GOME-2 surface LER is shown. The parameter presented in the right column is the surface anisotropy parameter, defined here as the difference between the GOME-2 surface DLER at viewing angles θv of +45∘ (west viewing direction) and -45∘ (east viewing direction). For the GOME-2 orbit this parameter is a good indicator of the magnitude and range of the surface anisotropy in the GOME-2 orbit swath. The results in Fig.  are presented for calendar month March and for the wavelength bands at 772, 670, and 555 nm. At 772 nm the surface anisotropy parameter can be as large as 0.2 for vegetated areas. For the typical desert areas the differences are much smaller (0.05–0.10) because non-vegetated surfaces are usually more isotropic than vegetated areas. The surface anisotropy parameter for snow/ice surfaces has the opposite sign. The values over the vegetated areas correspond to percentages of 50 %–125 %, in agreement with what was found already by . The magnitude of the surface anisotropy which is present in the GOME-2 surface DLER is therefore quite substantial.

At 670 and 555 nm the surface anisotropy parameter over vegetation is much lower than at 772 nm. However, this is mainly caused by the fact that the surface reflectance at these wavelength bands is also much lower than at 772 nm. The percentages are more or less the same, in the range of 50 %–125 %. For desert areas the anisotropy parameter is slightly smaller at 670 and 555 nm than at 772 nm. However, the percentages are similar for all three wavelength bands, about 10 %–20 %. For snow/ice surfaces the anisotropy parameter is not depending much on the wavelength. For 555 nm the values are slightly smaller. This is probably because of increased Rayleigh scattering, which results in a more diffuse illumination of the surface. The surface anisotropy parameter varies mostly between -0.1 and -0.3 for snow/ice surfaces.

The directional dependence of the surface DLER was studied for the nine surface types defined in Table . All nine surface types represent land surfaces. Constraints were set on latitude and longitude and, more importantly, on surface type using the Matthews land usage database . Furthermore, coastal areas were excluded and so were all grid cells that contained snow/ice, except for the “Antarctica” and “Greenland” surface types. The results are summarised in Fig. , which presents the GOME-2 surface DLER as a function of the viewing angle θv defined in Eq. (). The solid coloured curves represent the surface DLER at 772 nm, averaged over the grid cells of the surface type region as defined in Table . The legend in the “Antarctica” window explains to which months the curves belong. The grey curves in Fig.  are there to provide an indication of the spread in the surface DLER. The spread is defined as 2.35 times the standard deviation in the data. The legend in the “Greenland” window explains to which months the grey curves belong.

The three desert surface types (“Sahara desert”, “Arabian Peninsula”, and “Australian desert”) show a very similar dependence on the viewing angle. The overall dependence agrees well with that of the single grid cell shown in Fig. . The Australian desert deviates slightly from the other two desert regions because of the much lower values and the larger variability with respect to the calendar month. The latter observation may be partly explained by the different solar zenith angles, but more likely it is caused by the fact that the Australian desert contains more vegetation than the other two desert surface types. The snow/ice surface types (“Antarctica” and “Greenland”) show that the highest value for the surface DLER is reached for the eastward looking direction and not for the westward looking direction as for the other surface types presented in Fig. . There is, at least for Antarctica, a mild dependence on the calendar month. Note that for certain months the surface DLER results are not plotted for these regions. This is because the regions are covered in polar night during these months. The surface DLER is available for these months, but it is a replacement based on other months, which is why the results are not plotted in Fig. .

The four remaining surface type regions (“Amazonian tropical rainforests”, “Asian (sub-)tropical forests”, “Deciduous forests”, and “Grasslands”) show a large dependence on viewing angle. For example, for the month of May (brown curve) the average surface DLER in the Amazonian region varies from 0.21 for θv of -55∘ to 0.36 for θv of +55∘. For the month of November (blue curve) the increase is from 0.22 to 0.49, which is more than a factor of 2. The variability in time is the largest at the westward looking viewing direction.

The Asian (sub-)tropical forests and the Deciduous forests on the other hand show a temporal variability which is similar for the entire viewing angle range. Grasslands show a low temporal variability. For all four vegetated surfaces the anisotropy of the surface reflection is large. For these surface types the advantage of using DLER instead of LER is therefore substantial, but also for desert and snow/ice surfaces there is a significant improvement. Results for other wavelength bands can be found in Figs. S1–S3 in the Supplement.

In Fig.  we present the results from a comparison between GOME-2 surface DLER and MODIS surface BRDF for three surface type cases. The three reference sites (Amazonian rainforest; equatorial Africa; Libyan desert) were selected primarily on the basis of their homogeneity. Homogeneity is important because the MODIS MCD43C2 product is provided at a 5×5 times higher spatial resolution than the GOME-2 surface DLER database. In all three cases we selected a one-by-one degree latitude and longitude box, containing 16 grid cells from the GOME-2 surface LER database and 400 grid cells from the MODIS MCD43C2 database. The selected grid cells supply all the surface reflectivity parameters that are needed to calculate DLER and BRDF from the equations that were introduced earlier.

We then feed the geometry-dependent DLER and BRDF equations with artificial but realistic GOME-2 viewing and solar angles. The viewing angle θv is varied between -55∘ and +55∘ to simulate the scanning motion of the GOME-2 instrument. This is already sufficient information to calculate the DLER for the 16 selected DLER grid cells (see Eq. ). For the BRDF, the viewing zenith angle is then automatically known (θ=|θv|; see Eq. ), but the solar zenith angle θ0 and relative azimuth angle ϕ-ϕ0 also need to be known. The solar angles θ0 and ϕ0 can be determined from solar position calculations by specifying an hour angle determined from the GOME-2 Equator overpass time of 09:30 LT, taking into account the change in hour angle because of the displacement in the latitude direction (i.e., caused by the rotation of the Earth) and in longitude direction (because of the scanning motion from east to west). The viewing azimuth angle ϕ is quite a constant factor (apart from a 180∘ jump when GOME-2 scans past the exact nadir direction) and was determined from GOME-2 data. With all artificial angles known, the MODIS kernels can be calculated, and subsequently the surface BRDF can be calculated using Eq. () for all 400 MODIS MCD43C2 grid cells.

In Fig. a the scene that is studied is located over the Amazonian rainforest. The blue curves represent the surface BRDF from MODIS band 1 from the 400 MODIS BRDF grid cells. Note that even for what we consider homogeneous scenes there is already quite some variability between the grid cells. The GOME-2 surface DLER at 640 nm is presented in black. The agreement is good, both qualitatively in terms of the directional dependence and in absolute sense. Figure b presents the case of a scene in equatorial Africa. Here the agreement is again good with the correct dependence on the viewing angle. For the east viewing directions (θv<-50∘), however, the agreement seems to be a little less good. It should be noted that there is considerable variability in the 400 blue curves and that some of the blue curves also show the same upward bend for θv<-50∘ as the black curves.

Finally, in Fig. c the scene studied is over the Libyan desert. The Libyan desert is known to be rather stable and is often used as a calibration reference site e.g.. The variability in the approximately 100 km by 100 km large box is low. The agreement between DLER and BRDF is good, with small 3 %–4 % differences for the east viewing directions (θv<-50∘). The lower performance at the most extreme viewing angles could be a result of the fact that in the parabolic fitting procedure explained in Sect.  inaccuracies are not corrected at the swath ends, but they are in the centre of the swath. Also, the parameterisation used for the DLER is a second-order polynomial, so higher-order dependencies are not described well. Another explanation could be background aerosol scattering. There is no explicit filtering or correction for aerosol scattering in the retrieval code. Aerosol scattering would have the largest impact at the extreme viewing angles.

We conclude that the DLER follows the correct directional behaviour. Results for other wavelength bands can be found in Figs. S4 and S5 in the Supplement. In the next section global comparison are performed to be able to draw more quantitative conclusions.

This section presents results from global comparisons between GOME-2 surface DLER and MODIS surface BRDF. This time, the scattering geometry defined by θ, θ0, and ϕ-ϕ0 is prescribed by real GOME-2 observations. For these GOME-2 observations the surface DLER is calculated from the closest grid cell of the GOME-2 surface DLER database for the appropriate viewing angle θv (see Sect. ). The MODIS surface BRDF is taken from the MODIS MCD43C2 snow-free product and based on the exact 25 grid cells that coincide with the grid cell of the GOME-2 surface DLER database. The MODIS surface BRDF is calculated for the prescribed θ, θ0, and ϕ-ϕ0 and averaged over the 25 grid cells. Only observations from the descending forward scan and over land are accepted. Scenes over ocean or scenes containing snow or ice are skipped. Observations with absolute latitudes above 60∘ are also skipped.

A typical outcome for 640 nm is presented in Fig. . The result was obtained for 10 d of GOME-2B observation geometries (10–19 May 2019). In Fig. a and b the traditional, non-directional GOME-2 surface LER is presented against the MODIS surface BRDF for the eastern side of the orbit swath (IndexInScan 1–8; θv<-23∘; Fig. a) and the western side of the orbit swath (IndexInScan 17–24; θv>+23∘; Fig. b). For the eastern side of the orbit swath the correlation between surface LER and MODIS BRDF is quite fair. The linear fit has a slope of 0.845, which admittedly deviates from 1, but Pearson's correlation coefficient r is 0.955, indicating good correlation. The standard deviation of the data points with respect to the linear fit, σ, amounts to 0.027. This value of σ is in line with uncertainties of ∼0.02 that were reported for the GOME-2 surface LER database and for a few other surface LER databases .

For the western side of the orbit swath, Pearson's r still suggests a reasonable correlation between the two datasets, but the linear fit deviates quite a bit more from the one-to-one relationship. Also, the scatter plot suggests that there is no pure linear relationship between the two databases. The explanation for this behaviour is the bias in the traditional LER databases towards the geometries with the lowest surface LER values. For GOME-2 this affects mostly the western side of the orbit swath, as reported earlier by .

In Fig. c and d the results are presented for the comparison between the directional GOME-2 surface DLER and MODIS surface BRDF. For the eastern side of the orbit swath (Fig. c) there are no clear changes compared to the situation in Fig. a. The slope and intercept have improved only marginally. For the western side of the orbit swath (Fig. d), however, the correlation has improved considerably. Both slope and intercept have improved, Pearson's r indicates higher correlation, and σ went down mildly. More importantly, the eastern and western side of the orbit swath now appear to perform equally well.

The analysis presented in Fig.  was repeated for each calendar month for the period 2012–2019 to search for time dependencies in the results. Clear time dependencies, either seasonal or annual, were not found. We also performed the analysis using the SCIAMACHY surface LER database instead of the GOME-2 surface DLER database. The SCIAMACHY surface LER database is by definition a non-directional database, so we could only produce results as in the top row of Fig. . The resulting scatter plots were very comparable. For instance, the linear fit to the data had a slope of 0.860 and an intercept of 0.005, which agrees with the numbers in Fig.  (0.845 and 0.005, respectively). This may suggest that the deviation of the slope from 1 is not just specific for GOME-2 but specific for the differences between the LER databases in general and MODIS BRDF.

In the past, small radiometric calibration errors in the GOME-2 level-1 data have been reported . However, these were mainly found for the UV wavelength range and not so much for the particular wavelength that was studied here (640 nm). Also, other studies have reported good agreement with Advanced Very High Resolution Radiometer (AVHHR) and Advanced Along-Track Scanning Radiometer (AATSR) for the wavelength range 630–670 nm . This suggests that the results and conclusions of this section were not significantly influenced by calibration errors in the GOME-2 data.

Results for other wavelength bands can be found in Figs. S6 and S7 in the Supplement.