In this section we describe our approach of generating the geometry-dependent LER. First, we average all input data over a nominal OMI pixel. The input data include MODIS-derived land BRDF kernel coefficients, land–water flags, terrain heights from a digital elevation model, and chlorophyll values and wind speed over water surfaces. Second, we compute the TOA radiance accounting for surface BRDF. Third, we calculate the geometry-dependent LER from the TOA radiance. Then we use this geometry-dependent LER in cloud and NO2 algorithms to replace the climatological LERs.

The BRDF over land is calculated using the Ross-Thick Li-Sparse (RTLS) kernel model : BRDF=aiso+avolkvol+ageokgeo, where the coefficients aiso, avol, and ageo come from MODIS data, the isotropic kernel, aiso, describes the Lambertian part of light reflection from the surface, the volumetric kernel, kvol, describes light reflection from a dense leaf canopy, and the geometric kernel, kgeo, describes light reflection from a sparse ensemble of surface objects casting shadows on the background assumed to be Lambertian. The kernels are the only angle-dependent functions, the expressions of which are given in . The BRDF coefficients are spatially averaged over an actual satellite FOV and used to calculate TOA radiance for its observation geometry.

The BRDF coefficients depend on wavelength. For the present study we selected two wavelengths in the UV and vis: 354 and 466 nm. These wavelengths are relatively free of atmospheric RRS and trace-gas absorption. The BRDF coefficients at 466 nm are directly taken from the MCD43GF product at 470 nm (MODIS Band 3 has a wavelength range from 459 to 479 nm and a center wavelength of 470 nm) that is provided at a spatial resolution of 30 arcsec . Because the MODIS product is not available at 354 nm, we plan to adjust the 470 nm LERs to account for potential spectral dependences. The adjustment applies the spectral ratio of climatological OMI-derived LERs, Rg(354)/Rg(470), similar to the approach of . In the paper we assume that the BRDF coefficients are spectrally independent to focus on the surface BRDF effects only. Using climatological data of Kleipool et al. (2008) we find that this assumption can be valid for some areas; for example, the climatological ratio Rg(354)/Rg(470) is close to unity (within ±5 %) over the eastern part of North America. However, this is not the case for arid and semiarid areas. We plan to release our geometry-dependent LER product computed for wavelengths other than 470 nm using a spectral correction of the BRDF coefficients. This spectral correction will be based on the ratio Rg(354)/Rg(470) derived from a critical analysis of different existing data sets of climatological satellite-derived LERs.

To calculate TOA radiance over water surfaces, we account for both light specularly reflected from a rough water surface and diffuse light backscattered by water bulk and transmitted through the water surface. We neglect contributions from oceanic foam that can be significant for high wind speeds. Reflection from the water surface is described by the Cox–Munk slope distribution function . We use an isotropic form of the Cox–Munk distribution in which the facet–slope variance is independent of wind direction. All computations use a wind speed of 5 ms-1 which is close to the climatological mean.

Diffuse light from the ocean is described by a Case 1 water model that has chlorophyll concentration as a single input parameter . Our Case 1 water model accounts for the anisotropic nature of light backscattered by the ocean . A spatial distribution of chlorophyll concentration is taken from the monthly SeaWiFS climatology. The common Case 1 water model developed for the vis was extended to the UV using data from . To calculate water-leaving radiance, we need to know the downwelling atmospheric transmittance at the surface. The transmittance is obtained by calculating the total atmospheric direct and diffuse downwelling flux at the surface. The diffuse contribution in the transmittance will itself depend on the water-leaving radiance. To calculate the atmospheric transmittance, we introduce in VLIDORT a module for the iterative calculation of the transmittance, in which the first computation is made for a black surface, and this is then used again as input to the water-leaving contribution. This process is repeated until convergence of the transmittance is achieved (two or three iterations are sufficient).

To estimate LER over mixed surface types, we compute an area-weighted radiance for uniform land and water contributions within an OMI FOV. The LER for heterogeneous surface pixels is then calculated from this linear combination of radiances. The high spatial resolution MCD43GF product also supplies an eight-category land water classification map at the same resolution as the BRDF parameters. We convert this map into a binary land–water mask by merging all shorelines and ephemeral water into the land category and classifying all other water subcategories simply as water. We then compute the areal fraction of land and water for each OMI FOV. For specification of the OMI pixel, we used the OMPIXCOR product that provides coordinates of OMI pixel corners (http://disc.sci.gsfc.nasa.gov/uui/datasets/OMPIXCOR_V003/summary). We used an option of overlapping pixels in the along-track direction corresponding to 75 % energy in the along-track FOV. In this option the edges of the FOV are aligned in the cross track direction but overlap in the along-track direction.

Given the computed TOA radiance, Icomp, the LER is calculated by inverting Icompλ,θ,θ0,ϕ,Ps,R=I0λ,θ,θ0,ϕ,Ps+RTλ,θ,θ0,Ps1-RSbλ,Ps, where λ is wavelength, θ is the VZA, θ0 is the SZA, ϕ is the relative azimuth angle, R is the LER, I0 is the TOA radiance calculated for a black surface, T is the total (direct + diffuse) solar irradiance reaching the surface converted to the ideal Lambertian reflected radiance (by dividing by π) and then multiplied by the transmittance of the reflected radiation between the surface and TOA in the direction of a satellite instrument, and Sb is the diffuse flux reflectivity of the atmosphere for the case of its isotropic illumination from below (Eq. 200 in ; ). To speed up computations, we created lookup tables of the quantities I0, T, and Sb for selected wavelengths.

Averaging the BRDF coefficients over an OMI pixel may not be equivalent to averaging the high-resolution surface LER over the OMI pixel. We carried out a numerical experiment of calculations of TOA radiances using the high-resolution BRDF coefficients and OMI geometries for the Washington–Baltimore area of the United States (Fig. ). The TOA radiances were converted into LERs using Eq. () and then the LERs were averaged over OMI pixels. The resulting LERs were compared with that calculated from the standard procedure of averaging the BRDF coefficients first. We found that the mean LER difference was equal to 0.75×10-5 with a standard deviation of 4.2×10-4, which is quite acceptable for our purposes.

It should be noted that aerosols are not included in the computation of the geometry-dependent LER. Scattering by aerosols in the atmosphere reduces the BRDF effects . Therefore, the use of the geometry-dependent LER may result in overestimation of the BRDF effects. While non-absorbing aerosols are implicitly accounted for in the cloud algorithms (see Sect. 2.3.1), the aerosols directly affect the AMF, and thus trace-gas retrievals.

Figure  shows a data flow diagram that summarizes the generation of the geometry-dependent LER for satellite instruments. The diagram shows input data, all the steps of processing the data, and outputs. The input data include MODIS-derived land BRDF kernel coefficients and land–water flags, chlorophyll values, terrain heights from a digital elevation model, and wind speed at a height of 10 m. All the input data are averaged over a nominal OMI pixel using the OMI pixel corner information. The averaged input data, OMI pixel geometry, and atmospheric profiles are used to compute the TOA radiance with VLIDORT. The geometry-dependent LER is calculated from the TOA radiance using Eq. () and a pre-computed lookup table of radiative transfer parameters. Values of the geometry-dependent LER for each OMI pixel along with ancillary data are written in an HDF5-EOS output file for every OMI orbit.

Because reflection of incoming direct and diffuse solar light from non-Lambertian surfaces depends on satellite observational geometry, the same area observed at different geometries can have different LERs. Figure  shows the MODIS-based high spatial resolution LER over the Baltimore–Washington area of the United States for 2 consecutive days (17 and 18 January 2005) computed using the OMI observational geometry. The SZA and VZA values are in the similar ranges for both days. However, there is a large difference in the relative azimuth angle, which varies from around 63∘ for 17 January to about 118∘ for 18 January. Since the land tends to have strong backward scattering, this explains the higher LER for 18 January than that for 17 January. The differences, if not accounted for, may produce errors in the trace-gas retrievals.

A comparison of the computed geometry-dependent and climatological LERs at 466 nm is shown in Fig.  for OMI orbit 12414 of 14 November 2006. The climatological LERs (monthly) are derived from OMI observations . In general, the eastern portion of the orbital swath (that has a later Equator crossing time) has higher values of the LERs than the western part. This is an effect of the OMI observational geometry and BRDF increases in the backscattered direction over land. Over the US, the western portion of the orbital swath has higher values of the LER than the eastern portion. This is explained by the dominance of the difference in climatological surface reflectances between the east and the west (see Fig. b) as compared with the BRDF increase due to the OMI observational geometry.

Figure  shows significant differences between the geometry-dependent and climatological LERs for both land and ocean. Over land, the climatological LERs are mostly higher than the geometry-dependent LERs. This is presumably because the geometry-dependent LERs are derived from atmospherically corrected MODIS radiances while the climatological LERs are affected by residual aerosols. Moreover, climatological LERs are inherently contaminated by clouds due to substantially larger sizes of OMI pixels as compared with those of MODIS. This is particularly true for the Amazonian region where clouds are persistent.

Over ocean, the geometry-dependent LERs are systematically higher than the climatological LERs in areas affected by sun glint and at large VZAs. This is because the climatological LERs are based on the mode of LERs from a long time series of observations over a given area; this minimizes the impact of observations affected by sun glint and high values that occur at large VZAs. The total ocean reflectance is comprised of three components: direct and diffuse solar light reflected from the ocean surface and water-leaving light. The fraction of each component strongly depends on geometry. Reflection of direct solar light dominates in the sun-glint area. At the edges of the swath the relative contribution of reflected diffuse light increases because the sky radiance increases to the horizons and the reflection angle increases thus the Fresnel reflection increases. The higher values of LER nearer to the eastern part of the swath than at the western part are mostly due to sky light reflected from the ocean surface. The angular distribution of the sky radiance is not symmetric in the plane of satellite observations because the sun is in the western part of the swath. The sky radiance is higher in the eastern part of the swath, and it is reflected at higher angles than the light from the western part. Additionally, the higher reflection angle results in higher Fresnel reflection in the eastern part of the swath. This is confirmed by our calculations of the view angle dependence of the reflected light only, i.e., no water-leaving radiance included.

Figure  shows the geometry-dependent LERs computed at 466 and 354 nm and their differences for the same OMI orbit 12414 of 14 November 2006. Here, we assume that the BRDF coefficients over land are spectrally independent. The LER differences over land are thus solely due to the smoothing effect of enhanced Rayleigh scattering in UV that increases the diffuse to direct incident irradiance ratio as compared with 466 nm. Over land, LER(354) < LER(466), but the differences are relatively small (<0.015).

Over the ocean, the LER differences additionally result from the spectral dependence of water-leaving radiance. Over the sunglint areas, the solar light reflected from the ocean surface is significantly brighter at 466 nm than at 354 nm, thus leading to higher LERs. Over areas less affected by sunglint, LER(354) > LER(466) in general due to higher amounts of water-leaving radiance.

It is interesting to note that the patterns of rivers and their tributaries are evident in the LER maps of Fig.  for both 354 and 466 nm. This effect is most pronounced when rivers are viewed from the OMI measurement geometry that registers the reflectance signal of Fresnel reflection from smooth river surfaces. It may be somewhat surprising that this appears at OMI spatial resolution; we can explain the effect by considering that while the LER from FOVs comprised of river areas and surrounding land is weighted linearly by the areal fraction of each, reflectance from the river surface is disproportionally high due to the Fresnel reflection in sun-glint geometry. Outside of the regions where OMI observes glint, the LER in the Amazon basin may still be higher than expected due to the turbidity of some rivers in the Amazon floodplain that varies seasonally.

We consider the BRDF effect on the NO2 AMFs only, because the retrieved NO2 amount is inversely proportional to the AMF. The geometry-dependent LER approach provides an exact match of TOA radiances with the full BRDF approach but not of the photon path lengths. This simplification can lead to some biases in the calculation of AMFs and thus to biases in the retrieved NO2 vertical columns. have estimated the biases. They compared the box AMFs and NO2 vertical columns calculated with the full BRDF with that calculated with black sky albedo and white sky albedo. According to their data, maximum differences in the box AMFs are up to 10 % at the surface and differences in the NO2 vertical columns are smaller than 12 %. We carried out calculations of NO2 scattering weights and AMFs with full BRDF treatment and compared them with that calculated with the corresponding geometry-dependent LER. Figure a shows an example of the altitude dependence of scattering weights calculated with the full BRDF treatment and the geometry-dependent LER. It can be seen that the difference between the scattering weights is small. An AMF difference for this case is 5.6 %. Figure b shows a scatter plot of the full BRDF AMFs vs. the geometry-dependent LER AMFs calculated for OMI measurements over the eastern US for orbit 12414 of 14 November 2006. Differences in AMFs due to different treatment of the surface are within ±6 % (at 95 % confidence interval) and always less than 10 %.

The tropospheric NO2 AMF, AMFtrop, is calculated using the MLER model with input cloud parameters from the O2–O2 algorithm assuming a priori NO2 vertical profile shapes (see Fig. ): AMFtrop=AMFg(Ps,Rg)(1-fr)+AMFc(OCP,Rc)fr. The effect of a surface reflectivity change, ΔRg, of 0.01 on AMFg is shown as a function of Rg in Fig. . The Jacobian, J=ΔAMFg/ΔRg, is always positive because larger surface reflectances increase satellite sensitivity to NO2 absorption in the lowest atmosphere. J decreases with increasing Rg and for unpolluted NO2 mixing ratio profiles (Fig. ).

An effect of replacing the climatological LERs with geometry-dependent LERs on AMFg for OMI observational geometries and ground resolution can be estimated from Figs.  and  using ΔRg=LER(BRDF)-LER. The effect is largest over polluted regions in the eastern US, where ΔRg is negative with values from -0.03 to -0.02 (Fig. ), LER ∼ 0.05, and ΔAMFg∼ -20 to -30 %. The BRDF effect reverses over water for glint geometries and large viewing angles, but Rg is large here and the effect on AMFg is reduced (i.e., small Jacobian).

To estimate the BRDF effect on AMFtrop we need to account for the fr change as well. By differentiating Eq. () and assuming that AMFg and fr are independent variables and both depend on ΔRg, we get ΔAMFtrop=ΔAMFg(Rg)(1-fr)+ΔfrAMFc(OCP,Rc)-AMFg(Rg). The cloud AMF strongly depends on the OCP, since high clouds (low OCP) have a shielding effect and low clouds (high OCP), aerosols, and fog can enhance AMFc. Assuming a negligible NO2 mixing ratio above the cloud OCP, we can neglect AMFc and Eq. () simplifies to ΔAMFtrop=ΔAMFg(Rg)(1-fr)-ΔfrAMFg(Rg). Over land, replacing the climatological LERs with the geometry-dependent LERs reduces the surface LER on average (i.e., ΔRg<0), leading to smaller values of AMFg (Fig. ). At the same time, the mean ECF increases by 0.02 (Fig. ) and this produces even larger increases in fr (Δfr∼0.04). Therefore both terms in the above equation are negative meaning that switching to a geometry-dependent LER reduces AMFtrop even more over land. The effect is mixed over water, since both ΔRg or Δfr can change signs for certain geometries. It should be noted that we derived Eqs. () and () to qualitatively illustrate the effect of changing surface reflectance on AMF in cloudy conditions. The equations are not used to produce data in the figures of Sect. . The data in the figures are obtained numerically using Eq. ().

Figure  shows that the calculated impact on AMFtrop arising from replacing the climatological OMI-based LERs with geometry-dependent LERs exhibits a strong spatial variation with smaller effects over ocean, unpolluted, or cloudy areas. Over land, where the geometry-dependent LER is generally lower than the climatological LER, use of the BRDF data results in lower AMFs and higher tropospheric NO2 VCDs. The effect is enhanced over polluted areas such as eastern US, where the changes in AMF can reach up to 50 %. The effect is reduced for unpolluted and overcast conditions and mixed over oceans, because Rg increases for sunglint and large VZA directions but decreases for other directions.

Figure  compares the clear-sky AMFtrop calculated using climatological and geometry-dependent LERs for OMI orbit 12414. The use of geometry-dependent LERs generally leads to lower AMFtrop by up to 29 % over land and 15 % over ocean. Differences in O2–O2 cloud parameters resulting from the use of geometry-dependent LERs add additional scatter, changing AMFtrop by -42 to 5 % over land and -22 to 13 % over ocean. AMFtrop differences are large for low AMFs, driven by enhanced differences in LER, OCP, or fr. To make the numbers characterizing the AMF differences be more representative, we calculated the tropospheric NO2 AMFs using the geometry-dependent LER and compared them with those calculated with the climatological LER for 2 days: 14 November and 14 July 2006. The AMF differences arising from both replacing the climatological LERs with the geometry-dependent LERs and changing the cloud parameters exhibit strong spatial variations with smaller effects over the ocean, unpolluted, or cloudy areas similar to Fig . A global analysis of the AMF differences shows that the AMF differences for OMI orbit 12414 are consistent with those for both days.

Figure  illustrates how the use of geometry-dependent LER changes NO2 retrievals over clean and polluted areas. Consistent with previous studies by , AMFs are considerably lower with geometry-dependent LERs. This suggests that the current operational NO2 products based on climatological LERs could be underestimated by up to 48 % over China. The eastern US exhibits similar but somewhat smaller differences. Minor changes are expected over unpolluted and overcast areas.