A sketch of the 2-D system and the entrance optics of the telescope are shown in Fig. 1. The 2-D telescope allows EA scans (-20 to +90∘) at any AA and azimuth scans from -180 to +180∘ at any EA. The telescope is designed in two tiers. The upper tier contains the optics and a motor (Intelligent Motion Systems Inc. MDrive17) to rotate the housing with the prism for the EA mode, and the lower tier contains a stronger motor (MDrive34) for moving the upper tier to address different AA. There are two view ports, one for the measurement of scattered solar photons, and the other to observe the direct solar beam. The active one can be chosen by opening a shutter just behind the sapphire window. The telescope theoretical field of view (FOV) of the scattered light view port was determined to have an opening angle of 0.95∘ (full angle). The direct solar light is collected into an integrating sphere with a diameter of 2.54 cm. This sphere serves to homogenize the light, and correct for the possibility of pointing inaccuracies and atmospheric lens effects; this paper does not use any data from the sphere, which will be the subject of a separate publication (Ortega et al., 2015b). The photons are directed onto an f/4 2.54 cm lens via the same optical axis; the lens focuses the light into the mono-fiber, which is coupled to the fiber bundle (see Sect. 2.3). In order to merge the light from both view ports onto a common optical axis, a hole (0.64 cm diameter) was drilled in the prism which is used for the direct sun beam; the scattered light is reflected by 90∘ along the long side of the prism. The exit port of the integrating sphere is coupled via the hole, and both beams merge past the prism. A custom software package has been developed in LabView to choose between the different scan modes described below.

The capabilities of the CU 2-D-MAX-DOAS instrument include three different modes of measurement: (1) the traditional off-axis (EA) scan at any AA, (2) the AA scan at any EA, and (3) direct sun observations. The different modes of measurement significantly enhance the information that is accessible about trace gases and aerosol properties that can be retrieved simultaneously. A summary of the three different modes of measurement is shown in Table 1. The particular focus in developing the azimuth scanning capabilities is to enable the measurement of chemical gradients surrounding the measurement site in near-surface air (Part 1 – this work). Part 2 is dedicated to measurements of aerosol micro-physical properties in the air column, i.e., effective radius of an aerosol size distribution and complex refractive index.

Pointing accuracy was characterized in some detail, because it plays an important role in the inversion of MAX-DOAS measurements. For example, the exact knowledge of EA is inherent to vertical sensitivity (Hönninger et al., 2004), and the importance of accurate pointing is a prerequisite to ensuring a meaningful comparison/validation of MAX-DOAS instruments (Roscoe et al., 2010). For 2-D systems, knowledge about the pointing further plays an important role for the placement of hot spots, and characterization of horizontal inhomogeneity at any given EA as the AA is varied. The factors that influence knowledge about EA and AA accuracy are mechanical design, angle calibration, and the alignment procedure. In this section we describe the methodology used to ensure maximum accuracy of the telescope pointing.

The coordinate system for azimuth viewing is defined as zero corresponding to true north. In order to align the 2-D telescope along the absolute north, we use a three-step process: (1) the raw absolute north is first determined using a digital/manual compass, correcting for the magnetic declination (coarse alignment). (2) We use the scattered solar port and measure the solar radiance distribution at constant EA while varying the AA from left to right of the solar disk. To avoid saturation, the fluxes cannot be measured pointing directly to the sun but need to be performed 5∘ below and above the sun position. For cloud-free days and angles close to the sun we expect the distribution of radiances to be symmetric with a maximum intensity at the center of the sun's disk. Experimentally, however, a Gaussian fit shows an initial horizontal offset around the sun, which is normally greater than the internal encoder resolution of the AA stepper motor (0.17∘). The CU data acquisition software has an option to (3) account for the offset determined from step 2 by software; then we repeat step 2 until the radiances acquired on the left and right side are similar and with an offset smaller than 0.17∘ (fine alignment). An example of the experimental final alignment is shown in Fig. 2a. In this figure normalized radiances (L, black) are plotted as a function of solar relative azimuth angles (SRAAs). Negative SRAAs represent measurements on the left side (counterclockwise) and positive values are on the right side of the sun (clockwise). A Gaussian fit applied to this example yields the deviation around the center of the sun, which is around 0.06∘. The variation of the AA offset over the course of the day is characterized by means of solar aureole measurements (mode 2). In this mode, solar scattered photons at high time resolution (∼ 1 s) are collected for SRAAs > 5∘ typically in steps of 5∘ up to 180∘ for the left and right side of the solar disk. Using these measurements a Gaussian function is fitted under homogenous and cloud-free conditions. Figure 2b shows the diurnal average azimuth offset obtained in steps of 5∘ solar zenith angles (SZAs). The initial alignment of the EA housing is performed using a spirit level. To characterize the absolute EA along the line of sight as viewed by the optics, we measure radiances as a function of EA across a remote object (e.g., the horizon). The gradients across the target are used to determine the EA offset, which is then accounted for by software. Figure 2c shows the normalized radiances measured during the MAD-CAT campaign (see Sect. 2.2), with the 2-D telescope pointing at an AA of 242∘ towards a hilltop that was chosen as a target because of its well-known geometric EA of 1.1∘ (J. Remmers, personal communication, 2013). The derivative of the radiances (dL / dEA) gives the instantaneous gradients with EAs with a typical Gaussian shape; a Gaussian shaped fit is then applied to the derivative of radiance data to obtain quantitatively the EA offset. The EA offset for the example in Fig. 2c is around 0.35∘. We have made no further attempts to account for atmospheric refraction, which for extraterrestrial objects close to the horizon the effect of refraction would be less than 0.2∘ (depending on temperature, pressure, and wavelength) (Gisi et al., 2011). In order to know the EA offset along different azimuth views, we used a manual digital level. Figure 2d shows the tilt measurements as a function of the AA; here the EA offset is smaller than the resolution of the encoder accuracy of 0.17∘ for most of the AA.

The retrieval of multi-wavelength aerosol extinction profiles is based on normalized radiances at a given wavelength, similar to the inversion described in Prados-Roman et al. (2011) for aircraft MAX-DOAS measurements. The idea behind the sun-normalized radiances retrieval builds on the minimization of the cost function in Eq. (1) via the nonlinear Levenberg–Marquardt approach. yL-FLxL,bSe-12 In this equation, yL are the measured sun-normalized radiances yL=lnLiλLrefλ, FL(x,b) is the sun-normalized radiance simulated with the input parameters b and the aerosol profile xL, and Se is the diagonal covariance measurement error matrix. The full spherical Monte Carlo Radiative Transfer Model (McArtim) has been used for this work. It has been shown in Deutschmann et al. (2011) that McArtim is a suitable algorithm to represent the atmospheric radiative transfer in the UV–Vis–IR wavelength range. The pressure and temperature profiles were taken from the US Standard Atmosphere, and the retrieval grid was chosen to be 300 m thick between 0 and 3.3 km to represent the boundary layer. Typical aerosol optical parameters for urban atmosphere were used in the RTM (Dubovik et al., 2002). Assumptions for all the input parameters b (absorption cross section, aerosol optical properties, atmospheric conditions, etc.) used in the forward model are considered in the error propagation.

The wavelengths chosen to calculate the aerosol extinction profiles are 350, 450, and 560 nm, which are representative for the window intervals where NO2 is retrieved (see Table 3). These wavelengths prevent significant interferences from strong absorbers such as H2O and O3. The normalized radiances are calculated with the 90∘ EA as the reference angle in order to use lower EAs and maximize the information in the boundary layer. The aerosol extinction convergence criteria are achieved when the residuals of the cost function in Eq. (1) fall into a valley under a certain feasible set of extinction profiles calculated with the constrain parameters.

Once the aerosol extinction profiles have been derived, they are used to constrain the linear inversion of NO2 vertical profiles using the EA scan at standard AA. A set of NO2 dSCDs from the EA scan are related to the trace gas vertical profile x in the following equation: ytg=Kx+ε, where K is the weighting function matrix that expresses the sensitivity of measurement ytg to the true profile x, and ε represents the measurement and radiative transfer errors. The maximum a posteriori solution to the above problem using OE is given by x=xa+KTSε-1K+Sa-1KTSε-1ytg-Kxa, where xa is the a priori profile used to constrain the ill-posed inversion problem (Rodgers, 2000), and Sε and Sa are the measurement error and a priori error covariance matrices, respectively. Sε is built using the square of the DOAS fit error as the diagonal elements, and the non-diagonal elements were set to 0 assuming the errors are uncorrelated. As an a priori profile we use an exponentially decreasing NO2 profile with a fixed scale height of 0.5 km and 5 × 1015 molecules cm-2 vertical column density (VCD) at the ground, similar to the approach used in Hendrick et al. (2014). The Sa matrix was treated as a tuned parameter in order to avoid non-real oscillations in the retrieved profiles (Clémer et al., 2010; Baidar et al., 2013; Hendrick et al., 2014). The diagonal elements of the Sa matrix were set to account for large variations, up to 100 %, of the initial a priori profiles. The non-diagonal elements of the Sa matrix were correlated through the altitudes (z) with an exponential Gaussian decay function (Barret et al., 2003; Clémer et al., 2010): Sa=Sai,iSaj,jexp-ln2zi-zjγ2. In this equation zi and zj are the altitudes of the ith and jth grid layers, respectively, and γ is the correlation parameter, which was set to 0.3 km, similar to the inversion grid height. The wavelengths used to retrieve NO2 vertical profiles are the same as for the aerosol extinction profiles (350, 450, and 560 nm).

We apply the parameterization approach introduced in Sinreich et al. (2013) to the AA scan to obtain the horizontal near-surface mixing ratios of NO2. In short, Sinreich et al. (2013) pointed out that dSCD obtained from MAX-DOAS in a single and low EA can be converted into near-surface box average mixing ratios by means of parameterization of the RTM. In the following, we use terminology consistent with that of Sinreich et al. (2013); however we refer to MLH instead of planetary boundary layer (PBL) to avoid confusion with the meteorological term. Briefly, the differential light path of the EA with respect to the reference is determined by means of O4 dSCD weighted by a correction factor. The correction factor, fc, accounts for the different O4 and NO2 vertical profile shapes and the difference in absorption wavelengths. Sinreich et al. (2013) showed that this method does not depend on the actual aerosol load and that it only weakly depends on the aerosol layer height, if sufficient aerosol is present. This was assessed in Sinreich et al. (2013) by a “collapsing” of the O4 dSCDs in a set of two low EAs, which is however not a prerequisite for this approach to work for very low EAs (see Supplement text in Sinreich et al., 2013; Wang et al., 2014). An important prerequisite for calculating fc is knowledge about the trace gas MLH. As long as this height is higher than 500 m, the method is insensitive to the actual value of the trace gas layer height. We apply this method to multiple wavelengths to average NO2 over different horizontal distances. At each wavelength the near-surface volume mixing ratio of NO2 can be evaluated using the following equation (Sinreich et al., 2013): VMRNO2=1fcdSCDNO2⋅CO4dSCDO41CF, where dSCDNO2 and dSCDO4 are the measured dSCD for NO2 in units of molec× cm-2 and O4 in units of molec2× cm-5, respectively; CF is the conversion factor from the concentration to the VMR; CO4 the concentration of O4 in units of molec× cm-3; and fc (unitless) the correction factor which adjusts for the differences in the NO2 and the O4 profile shapes. The correction factor is calculated by the following equation: fc=dAMFNO2⋅MLHNO2⋅CO4dAMFO4⋅VCDO4, where dAMFNO2 and dAMFO4 (unitless) are the differential air mass factors (AMF) for NO2 and O4, respectively; VCDO4 is a typical O4 VCD in units of molec2× cm-5; and MLHNO2is the NO2 MLH in units of centimeters. The MLHNO2 represents the depth of the lower atmosphere where NO2 is dispersed as a result of turbulent vertical mixing processes. In a first step, the MLHNO2 is estimated using the NO2 vertical profiles (see Sect. 4.1.2), and it is used subsequently to calculate the value of fc.The dAMF is defined as the difference in AMF (light path enhancement in the atmosphere relative to vertical path through the atmosphere) between the measured and reference viewing geometry. Generally, dAMFs are calculated with a RTM at low uncertainty; however the calculation of dAMFs depends on the state of the atmosphere (aerosol load), geometry of the measurements, and wavelength (Wagner et al., 2007). Although the correction factors and the sensitivity were presented in Sinreich et al. (2013), and applied in Wang et al. (2014), these studies explored a limited subset of geometries and wavelength ranges. In this work we expand significantly the number of AAs and wavelength ranges. This requires a more detailed description of the correction factors and their dependency on input parameters to the RTM. In order to know the variability of the correction factors, we change the magnitude of different input parameters such as the aerosol asymmetry parameter (g), single scattering albedo (SSA), surface albedo (SA), and NO2 as well as aerosol load and vertical extent. Table 5 shows the main input parameters to calculate fc and its variability for the three wavelengths. A good starting point for the aerosol optical properties is using the aerosol column properties derived from the co-located AErosol RObotic NETwork (AERONET) sun photometer as shown in Table 5. Barnard et al. (2008) showed that the SSA at UV wavelengths can be significantly lower than in the visible due to light absorption by “brown-carbon” aerosols. The discrete wavelengths used by AERONET do not cover 350 nm; hence we use a lower limit of SSA to calculate the variability in the UV (Table 5). The RTM is parameterized by using the aerosol load determined in Sect. 4.1.1. In contrast, the diurnal geometry of the measurements, i.e., the SZA and SRAA, are known parameters. The magnitude of fc and its variability are shown in Sect. 4.2.

Figure 6 shows the multi-wavelength aerosol extinction diurnal vertical distribution using the standard AA of 50.8∘. The aerosol extinction follows the typical wavelength dependence, where it increases as the wavelength decreases. The integrated extinction profiles over altitude, known as aerosol optical depth (AOD), are compared with the AOD retrieved with the co-located AERONET sensor to partially demonstrate aerosol homogeneity around the site. The AERONET sun photometer uses discrete fixed wavelengths, which are not the same as the wavelengths we applied here; hence the aerosol wavelength dependence (Ängström exponent) is used to interpolate the AOD at the wavelengths of interest. Even though the directionality of AERONET (a solar tracker) and the standard EA scan of the 2-D-MAX-DOAS measurements are different, they show a generally good multi-wavelength AOD agreement with a slope ranging from 0.77 to 0.93 (±0.03) from the visible to the UV. The agreement between the two instruments could be coincidental, or indicate that the aerosol load around the city of Mainz is homogeneous. Furthermore, observations of the 2-D analysis of O4 dSCD (elevation angle of 2∘) do not show a significant difference with respect to aerosol azimuth distribution.

The diurnal variation of NO2 in the boundary layer is shown in Fig. 6 and indicates an increase of NO2 in the early morning followed by a decrease in the evening. The vertical extent of NO2 in the boundary layer for the three wavelengths is very similar: most of the NO2 is located in the layer below 1 km. Small differences in the NO2 vertical distribution are found for the three wavelengths, and these may be related to the smoothing in the retrieval and to some extent the sensitivity range for each wavelength.

A critical parameter for the calculation of fc in Eq. (6) is prior knowledge of the MLH (Sinreich et al., 2013). In this work we estimate the MLH using NO2 as a tracer for the vertical extent of mixing. In general, the estimation of the altitude sensitivity in the vertical profile retrieval is well represented by the averaging kernels and eigenvectors which are derived in OE (Rodgers 2000; Frieß et al., 2006). According to the averaging kernel of the NO2 vertical profile the boundary layer is well constrained by the measurements (see Fig. 6). The number of DOF is 2–2.5 for each wavelength, with the highest sensitivity close to the surface (0–200 m layer). NO2 is a good tracer for MLH, because it has a relatively short effective lifetime (limited to few hours by OH + NO2 reactions). The MLH can be described as the depth of the lower atmosphere in which the NO2, emitted mostly as NO and formed within this layer, is dispersed almost uniformly as a result of turbulent vertical mixing processes (Emeis and Schäfer, 2006). Usually NO2 decreases strongly above this mixing height as chemical reactions remove NO2 as a result of photochemistry and hydrolysis of N2O5 at the surface of wet aerosols (Baidar et al., 2013; Ryerson et al., 2013). In this work we approximate the MLHNO2 as the altitude over which the NO2 mixing ratio decreases to 1/(2e) of the near-surface value. The gray circles in the NO2 vertical distribution plot of Fig. 6 represent the MLHNO2 determined by this definition. As can be seen, the MLHNO2 varies as a function of the time of day. There is some sensitivity to a residual layer in the early morning. Around 10:00 local time (LT) the MLHNO2 starts to increase due to the development of the convective boundary layer and efficient mixing. The MLHNO2 variation for the different wavelengths may be caused by the smoothing factor in the NO2 vertical profile retrieval, though a mountain terrain effect may also play a role at the longer wavelengths. For the calculation of fc the time-resolved average MLHNO2 obtained with the three wavelengths is used and the standard deviation is used to estimate its variability. The sensitivity of the height distribution of NO2 on fc was tested with an independent method. This approach assumes a NOx box profile shape, assigns the near-surface NO2 VMR from OE, and assigns an effective mixing height (MLHNO2,eff) based on the measured NO2 VCD. The retrieved MLHNO2,eff was compared with the MLHNO2 calculated before. We found a good correlation (R2=0.81) with a linear fit of MLHNO2,eff=0.93±0.06⋅MLHNO2+0.10±0.04[R2=0.81]. The resulting MLHNO2,eff is within the error near the lower limit of the standard deviation, which is used to calculate the variability associated with the MLHNO2 as mentioned before.

The results of the azimuthal diurnal variation of the correction factors are shown in the form of a polar plot in Fig. 7a. The radii of the polar plots shown in this section represent the local time and the color code the magnitude of the parameters. The values of fc represent the mean values of the sensitivity studies shown in Sect. 3.2.3 and summarized in Table 5. A value of fc=1.00 means that the radiation field is equally sensitive to NO2 and O4; a value of fc < 1 means that the sensitivity increases towards detecting O4 with respect to NO2. A detailed description of the variability and geometry dependency is given in Sinreich et al. (2013) and shown in Fig. S2. In short, fc does not change drastically with the variability of g, SSA, and SA (Sinreich et al., 2013). In contrast, the MLH can have a larger impact. We estimate the variability of fc to be on the order of 5–8 %. This low error is possible since we estimate the MLH and its variation using NO2 vertical profiles. The most surprising effect is the clear SRAA dependence where fc increases for SRAA close to the sun, especially for SZA between 40 and 60∘ (Fig. S2). This behavior is due to low-O4 dAMFs obtained with the RTM. For these cases fc can in fact exceed unity; however the O4 dSCDs do not show a significant SRAA dependence. This behavior in calculated O4 dAMFs is currently not understood. However, we note that it is consistent with earlier observations that measured O4 dSCD under certain conditions can exceed those calculated by RTM (Clémer et al., 2010; Wagner et al., 2011; Irie et al., 2011, Merlaud et al., 2011). Recent testing of measured O4 dSCD from aircraft found agreement within 2–3 % with those from other geometries (Spinei et al., 2015). For lack of a physical explanation, for the cases when fc>1 we set fc to be unity.

Figure 7b shows the diurnal variation of Leff calculated with Eq. (7). The results indicate a strong wavelength dependence, which is exploited in the onion-peeling approach introduced in Sect. 4.3. The 350 nm Leff displays values between 5 and 10 km; the 450 nm one can reach horizontal distances of 10–20 km; and the 560 nm one shows sensitivity to horizontal distances of 18–30 km. In the early morning the Leff is larger, likely due to low-O4 dAMF, and is more evident for the east view due to the low SRAAs. In general, there is not a significant AA dependence, indicating azimuth homogeneity as stated in Sect. 4.1.1.

The diurnal variation of near-surface mixing ratios of NO2 is presented in Fig. 7c. The azimuth inhomogeneity is apparent from this figure, wherein most of the NO2 is located in the west, especially in the early morning (∼ 07:00–08:00) and when the traffic increases and NO2 accumulates (∼ 09:00–11:00). This is surprising, considering that the industrial complex and cities close to the site are located north and northeast, and may indicate that mobile sources continue to be significant contributors to ambient NOx during morning rush hour. The wind direction and speed can provide further information about transport of the air masses. The daytime values, provided by the MAD-CAT team (https://www.blogs.uni-mainz.de/fb08-ipa/wetter/), are shown in Fig. 8. The main wind direction is northeast with typical wind speed around 2–4 m s-1, hence possible transport to the southwest. Another remarkable observation to emerge from the data is that the average NO2 near-surface mixing ratios at 350 nm are greater than the 450 and 560 nm ones, which is exploited in the onion-peeling approach (Sect. 4.3).

To quantitatively assess azimuth inhomogeneity, the relative azimuth asymmetry (RAA) is introduced here. The RAA is defined as (1-[NO2,min / NO2,max]), where NO2,min and NO2,max are the minimum and maximum mixing ratios of NO2 obtained within a period of time. The RAA calculated for the early morning, midday, afternoon, and evening are 68 ± 7, 34 ± 4, 63 ± 4, and 28 ± 6 %, respectively. This variability is independent of the wavelength and reflects that similar changes are probed by the three different effective path lengths, most likely due to changes in NO2 emissions close to the city of Mainz. Similarly, the relative longitudinal asymmetry (RLA) is defined as (1-[NO2,min / NO2,max]), but with the NO2,min and NO2,max taken with a constant AA and for a period of time of 1.5–2 h. For example, the RLA for the standard AA is 41 % (360 nm), 13 % (450 nm), and 11 % (560 nm) during the early morning (07:30–09:00). In this case there is a strong wavelength dependence confirming that major emissions are released close to the city of Mainz and transported to the southwest. In contrast to the early morning, midday values of RLA are lower than 7 % for the three wavelengths, hence are more homogeneous air mass along the same line of sight. Although the RLA may vary slightly for different AAs, the same behavior described before was found for all of the AAs for the same period of time.

Table 6 shows a summary of the errors for both OE and parameterization. In the following, the errors of the vertical profile retrievals by OE are discussed briefly. For a detailed and more formal description of the error associated in the retrieval of vertical profile using OE we refer to the excellent studies by Rodgers (1990, 2000) and Steck (2002). We have considered three main errors: (1) retrieval noise error associated with the uncertainty in the dSCD; (2) the smoothing error, which represents how in average the constrained retrieval differs from the true state due to the vertical smoothing; and (3) systematic errors from the forward model parameters. The retrieval noise is so far the easiest error to be determined since it is retrieved directly from the gain matrix (Gy) and the Sε covariance matrix error as Sn=GySεGyT. As mentioned before Sε is built using the square of the DOAS fit error, which represents a final error of less than 1 %; however the final noise error considers uncertainties in the NO2 absorption cross section (∼ 5 %). The error due to smoothing and the forward model covariance matrices listed in Table 6 were taken from sensitivity studies. The smoothing error is determined based on the inversion with simulated NO2 dSCDs. First, simulated NO2 dSCDs under pre-defined scenarios were used to retrieve the NO2 vertical profiles under known conditions. The difference between the “real” and the retrieved NO2 profiles was taken as an estimate of this error. On the other hand the forward model parameters are estimated by changing the most important input parameters such as aerosol optical properties in an expected range. In general, lower error was found for layers close to the surface (below 0.6 km). At higher altitudes, the smoothing error is higher; hence the total error increases due to lack of sensitivity and uncertainties in the aerosol profiles.

The error of the parameterization approach to derive near-surface NO2 VMR is estimated by an error propagation of the inputs to Eq. (5). There are two main factors contributing to the final error: (1) error in the analysis of NO2 and O4 dSCD, and (2) error in the correction factor fc. The typical root mean square (RMS) residual achieved in the fit of NO2 and O4 is on the order of 4.5 × 10-4, 3.5 × 10-4, and 3 × 10-4 for the 360, 450, and 560 nm fits, respectively; lower RMS was observed on other days when the NO2 concentrations were lower. These noise levels correspond roughly to 2–3 times the fit error in the DOAS analysis. We use a 2-sigma fit error for the error propagation in the dSCD of NO2 and O4 in Eq. (5). As mentioned in Sect. 3.2.3 the error in the correction factor takes into account the variations due to uncertainties in the MLHNO2 and aerosol optical properties. Table 5 shows the main input parameters that are varied in order to estimate the variability of the correction factor. Figure S2 shows the O4 dSCD, O4 dAMF, and fc binned by SRAA for different SZA. In this figure the error bars represent the uncertainty estimated using the overall range of variability derived from the above sensitivity studies. The uncertainty of the correction factors is below 10 % most of the time, and for the three wavelengths there is a systematic pattern visible in that the strongest SRAA effects occur at SZAs between 40 and 70∘. Similar findings were observed in Sinreich et al. (2013). The error is normally greater in the 360 nm data than in the data for visible wavelengths. The final error of the VMR is between 12 and 15 %.

The multi-wavelength NO2 near-surface VMRs retrieved with the parameterization approach are compared with those derived from OE for the standard azimuth view in Fig. S5. Note that the two approaches are independent and in particular do not use identical NO2 dSCDs. As pointed out in Sects. 3.2.3 and 3.3 the NO2 derived with the parameterization approach uses only data from the lowest EA, and the mixing ratio is characteristic only over the differential effective path length and height in the lower portion of the MLH. By contrast, the OE uses NO2 dSCDs from a full set of EAs. In reality, the comparison of MLHNO2 in Sect. 4.1.2 shows that there is some variability in NO2 with altitude. The near-surface VMR is subject to smoothness and assumptions about a priori profile shape/magnitude and error covariance matrices, and thus differences in Fig. S5 may represent actual gradients in the chemical state of the atmosphere and should not be interpreted exclusively as an “error” of the retrieval strategy. In order to account for the different horizontal range sensitivity we have averaged the NO2 profiles from Fig. 6 to a similar representative height of the parameterization approach for each wavelength. In general, both methods capture the NO2 variability within the spatial scale for each wavelength. The slopes of linear regressions (parameterization versus OE) varied from 0.83 to 0.88, and intercepts from 0.11 to 1.76 ppbv NO2 (see insert on Fig. S5). There is further generally good correlation, with the R2 ranging between 0.55 and 0.78, for each wavelength considering the difference in sensitivity of air masses probed by the two methods.

The gradients in NO2 identified in this work pose the question of how 3-D measurements can be best validated. Figure 11 emphasizes the relevance of this question in the context of comparable instrument footprint and satellite ground pixel size. Our measurements show that NO2 near-surface mixing ratios vary by a factor of larger than 3 in different azimuth directions, and by up to a factor of 1.5–2 at fixed AA (Sect. 4.2.1). The true NO2 variability is likely higher, since our measurements inherently average over between 5 and 30 km horizontal distance (Sect. 4.2).

The regional air quality network in Mainz provides in situ NO2 measurements. A comparison with our 3-D distributions is not straightforward and has not been attempted for at least two reasons: (1) previous comparison of NO2 and HCHO in situ observations and open-path spectroscopic measurements show a rather fair level of correlation (0.32 < R2 < 0.77) in ambient air (Harder et al., 1997; Cárdenas et al., 2000; Jimenez et al., 2000; Kim et al., 2001; Thornton et al., 2003; Dunlea et al., 2007), while comparisons under controlled conditions show excellent correlation (also for ambient air) if the air is well mixed (R2 > 0.98) (Thalman et al., 2015). The comparison with network sensors thus is unlikely to carry much meaning in light of the different air masses probed. The inhomogeneity along a constant AA is 50–100 %, and a lower limit for the actual inhomogeneity given the averaging nature of 2-D-MAX-DOAS measurements. Any better agreement would be coincidental. Moreover, this fundamental sampling problem is not unique to a comparison with in situ sensors. For example, car MAX-DOAS measurements have the benefit of integrating vertically but are conducted in air that is offset horizontally from the measurement site of the 2-D-MAX-DOAS and is also subject to the inhomogeneity along a constant AA. (2) Additional artifacts can arise with in situ sensors that measure NO2 indirectly; e.g., molybdenum converters (as used in the Mainz city network) suffer from interferences of NOy species that high-bias the sensor signal attributed to NO2 (Dunlea et al., 2007; Villena et al., 2012). For example, Villena et al. (2012) found that values of NO2 can be overestimated by up to a factor of 4 using molybdenum converters. On the other hand, photolytic converters can show even stronger negative interferences than the molybdenum converters, especially under high pollution levels (high NOx concentrations) (Villena et al., 2012). The sensors used to validate 3-D NO2 distributions should be chosen to actively avoid such potential for bias. In order to validate azimuth distributions of NO2, a first step consists in comparing the NO2 SCDs from multiple 2-D-MAX-DOAS instruments during the MAD-CAT campaign. An according manuscript is currently in preparation for azimuthal distribution of NO2 (Remmers et al., 2015). Additionally, upcoming studies regarding comparison of HONO and CHOCHO dSCD from various instruments have been projected (Ortega et al., 2015a; Wang et al., 2015). The subsequent validation of 3-D distributions of NO2 is non-trivial. In particular, the heterogeneity of NO2 poses the following challenges to designing a sampling strategy for retrieval validation.

The 3-D mapping of the true NO2 variability in an air volume of ∼ 1000 km3 that is probed by the CU 2-D-MAX-DOAS instrument near instantaneously by means of in situ sensors on aircraft would require a fast and flexible airplane. A challenge exists in the fact that the atmospheric state of NO2 is likely changing on the timescales needed to map such an air volume with in situ sensors on aircraft. At a typical airspeed of small aircraft, it would take 3–4 h to map a box of dimensions 32 × 32 × 1 km (∼ 1000 km3) by traverses at distances that are 1 km apart and are flown at a single altitude; however, the variations in the atmospheric state are likely significant as times exceed 30 min (M. Trainer, personal communication, 2010). It would therefore be desirable to have multiple aircraft. Multiple aircraft would have the added benefit that independent vertical profile information could be obtained effectively. The complementary validation by highly time-resolved NO2 column observations from cars (Shaiganfar et al., 2011) and aircraft (Oetjen et al., 2013) provide cost-effective means to access larger air volumes. The retrievals are reasonably straightforward and robust, and have been successfully employed for satellite validation (Oetjen et al., 2013). Imaging spectrometers on the ground (Lohberger et al., 2004) and push broom/whisk broom scanner techniques on aircraft (Kowalewski and Janz, 2009; Heue et al., 2008; General et al., 2014) hold great promise to provide information on finer scales. While imaging techniques greatly increase the amount of information that can be obtained from a single instrument/platform, the quantification of these data is not free of challenges that warrant independent assessments. Ideally, future attempts to validate 3-D distributions will use a combination of multiple aircraft and a combination of active and passive remote sensing of NO2, including networks of ground-based remote-sensing and in situ sensors (Ryerson et al., 2013), and will be further coordinated with measurements from mobile platforms, including in situ measurements from vans and aircraft, car MAX-DOAS, nadir mapping and push-broom imaging from aircraft, and NO2 lidars (Volten et al., 2009; Vlemmix et al., 2011). Such efforts should be supported by atmospheric models to inform the best sampling strategy and bridge between the various spatial scales probed by these diverse available measurements to probe 3-D distributions of NO2. State-of-the-art atmospheric models currently predict NO2 with a resolution of 4 km × 4 km, which is sufficient to represent broad spatial features such as those measured by 2-D-MAX-DOAS. However, this resolution would be insufficient to resolve the spatial gradients in NO2 that can be expected near the edges of street canyons or highways, which can pose challenges if the complementary information derived from in situ sensors and column observations is to be integrated by models. The location for future attempts to validate 3-D distributions of NO2 should include homogeneous yet variable NO2 for best results.