Atmospheric trace gases can be measured with ground-, aircraft- and space-based spectrometers that measure solar irradiance scattered into the line of sight of the instrument (see Fig. ). In the case of aircraft- and space-based observations, a large fraction of the measured photons usually travels along a main path (thick dashed line) representing a single reflection at the surface. In the case of ground-based observations, the measured photons must follow a path with at least a single atmospheric scattering into the line of sight of the instrument (except for direct sun observations). Atmospheric scattering and absorption is determined by the distribution and properties of molecules, aerosols and clouds, and it depends on the wavelength of the radiation. Molecular scattering is particularly important in the UV range of the spectrum. Photons are absorbed by the trace gases along the optical path from the sun to the instrument. For a weak absorber such as NO2, the abundance of the trace gas along the mean optical path can be obtained by fitting an absorption cross section to the measured spectrum. Thereby, the mean optical path is the total length of all individual photon paths divided by the number of photons collected by the instrument. The result of the fit is a SCD, which is defined as 1SCD=∫pathc(l)dl, with trace gas concentration c and optical path l. SCDs are not an intrinsic property of the atmosphere, since they depend on the illumination and viewing geometry. Therefore, for most applications, the main quantity of interest is the VCD. It is defined as 2VCD=∫z0TOAc(z)dz, with surface elevation z0 and top of the atmosphere, TOA. AMFs, defined as 3AMF=SCDVCD, can be computed for a vertically varying atmosphere by dividing the atmosphere in layers with uniform properties (see Fig. a and b). The total AMF is then computed from the individual layer AMFs as 4AMF=∑k=1nzAMFkVCDk∑k=1nzVCDk, with AMFk and VCDk being the AMF and VCD in the kth layer, respectively. The total AMF is thus a function not only of the atmospheric properties in each layer but also of the shape of the vertical profile of the trace gas .

Similarly, the atmosphere can be divided into boxes in all three dimensions (i,j,k) with homogeneous optical properties for each box (see Fig. c and d). The total AMF can be computed from the 3D-box AMFs AMFi,j,k as 5AMF=∑i=1nx∑j=1ny∑k=1nzAMFi,j,kVCDi,j,k∑k=1nzVCDk, where the denominator is a sum over VCDs in k different vertical layers that could, for example, be taken at the location of an instrument or above the ground pixel of an aircraft- or space-based instrument. In this case, the AMF can be interpreted as the instrument sensitivity to the trace gas under investigation for measuring that specific VCD.

The libRadtran RTM (available at http://www.libradtran.org, last access: 12 August 2020) can be used to calculate basic radiative quantities with different numerical solvers . One of its solvers is MYSTIC, which uses the Monte Carlo technique to trace individual photons on their way from the source (e.g., sun) to the target (e.g., measurement instrument). Scattering, absorption and reflection processes are treated as random decisions with respective probability distributions. MYSTIC calculates radiative quantities (irradiance, actinic flux at levels, radiance, absorption, emission, actinic flux, photon's path length and air mass factors) in 1D or 3D domains in spherical geometry or in plane-parallel geometry . The 1D-layer and 3D-box AMFs were implemented following the same methodology as in McArtim, which to our knowledge is the only other existing RTM capable of computing 3D-box AMFs . Note that McArtim is no longer actively developed.

AMFs depend on absorption and scattering processes affecting the light path in the atmosphere. AMFs can be readily calculated from the photon paths simulated by a Monte Carlo radiative transfer model. The Monte Carlo technique traces the paths of individual photons by describing the effects of absorption, scattering and reflection as random events with specific probabilities . To obtain a robust measure of the mean optical path, a large number of photon paths need to be traced.

SCDs, VCDs and AMFs can be computed for the whole atmosphere, for individual vertical layers or for individual 3D boxes. For the general case of an atmospheric box i with constant concentration and optical properties, the AMF can be written as 6AMFi=SCDiVCDi=∫pathcidl∫zizi+1cidz=∫pathdlhi=Lihi, where Li=∫pathdl is the mean optical path within the box of all photons that reach the instrument and hi is the height of the box. Since the 3D-box/1D-layer AMFs are usually simulated for a sensor at a specific location in a three-dimensional model domain, the photons are traced backwards from the sensor towards the sun to increase computational efficiency as described in and . In addition, the commonly used variance reduction method, known as “local estimate”, is applied at each scattering event . The method computes the probability of an individual photon to be scattered into the direction of the sun that is assigned as a weight wn to the photon. The weights of all photons can be summed up to obtain the radiance at the sensor. When a photon is scattered, a weighted photon path length (wn⋅li) is also calculated, where li are the path lengths in each individual box i traversed by the photon before the scattering event. The mean optical path within a box i is then obtained by summing up the weighted photon path lengths of all photons as follows: 7Li=∑nNwnli,n∑nNwn, where N is the total number of photons. Li is then divided by the height of the box/layer to obtain the 3D-box/1D-layer AMF.

The implementation of the 1D-layer and 3D-box AMF module in MYSTIC was evaluated against the results of different RTMs presented in an extensive RTM comparison study . The simulated scenarios are representative for ground-based Multi-Axis-DOAS (MAX-DOAS) measurements of scattered sunlight spectra for different elevation angles (see Fig. b and d for the case of zenith-sky observations). The nine models included four models using full spherical geometry, four models using spherical geometry only for a subset of interactions and one model using plane-parallel geometry. The 1D-layer AMFs computed by these models agreed very well with differences mostly below 5 %, which could mainly be attributed to the different treatments and approximations of the Earth's sphericity and to model initialization parameters .

For the comparison, we computed 1D-layer and 3D-box AMFs with MYSTIC in plane-parallel geometry as well as 1D-layer AMFs in spherical geometry for all scenarios presented in . The 3D-box AMFs have not yet been implemented with spherical geometry.

1D-layer and 3D-box AMFs were computed for five wavelengths (310, 360, 440, 477, 577 nm), seven elevation angles (1, 2, 3, 6, 10, 20, 90∘) and three aerosol scenarios (aerosol extinction of 0.0, 0.1 and 0.5 km-1). For the aerosol scenarios, an aerosol layer was prescribed between 0 and 2 km with an asymmetry parameter of 0.68 and a single-scattering albedo of 1.0. No aerosols were prescribed above 2 km. For the simulations, 17 vertical layers were used with a thickness of 100 m below 1000 m and a thickness of mostly 1000 m above see Table 1 in. Profiles of temperature, pressure, density and ozone concentration were taken from the US Standard Atmosphere . Ozone cross sections (in cm2) were 9.59×10-20, 6.19×10-23, 1.36×10-22, 5.60×10-22 and 4.87×10-21 at 310, 360, 440, 477 and 577 nm, respectively. Other atmospheric absorbers were ignored. Further details can be found in . For each scenario, we traced 1 million photons, which balances statistical noise expected from a Monte Carlo approach with computation time. The computed 3D-box AMFs were integrated horizontally to obtain 1D-layer AMFs that can be compared with the 1D-layer AMFs from other models. MYSTIC was mainly compared to SCIATRAN (Version 2.2; ). SCIATRAN was chosen because it agrees well with the mean of the models in , and because it is based on the discrete ordinate method to solve the radiative transfer equation, which is fundamentally different from a Monte Carlo solver, and finally because it offers both plane-parallel and spherical solutions. In addition, we compared MYSTIC to the mean of eight of nine RTMs in the comparison study. The PROMSAR/Italy model was not included because of its large deviation from the mean seefor details.

The comparison of 1D-layer AMF profiles calculated with the MYSTIC 1D modules with SCIATRAN for the 67 observation scenarios used in is summarized in Fig.  in the form of a scatter plot. The horizontally integrated AMFs from MYSTIC's 3D module perfectly agree with its 1D module with plane-parallel geometry within the statistical noise of the Monte Carlo approach. When tracing 1 million photons, the difference between 1D and 3D module was smaller than 0.5 %. Therefore, only results from the 1D module were plotted against the SCIATRAN results. The agreement between MYSTIC and SCIATRAN is very good for almost all scenarios with relative differences mostly below 5 %. Overall, 97 % of the compared points are within a relative difference of 5 % for spherical geometry and 92 % for plane-parallel geometry. The mean of the relative differences for spherical geometry is 0.9 % and its standard deviation 2.0 %, and for the plane-parallel geometry the mean is 0.3 % with a standard deviation of 2.7 %.

To illustrate the differences in AMF profiles between the two RTMs, we selected four scenarios with a wavelength of 577 nm, because at this wavelength we observe comparatively large differences between the two models. To illustrate a usual scenario with low difference, we also selected the same scenarios but with a 360 nm wavelength. The upper row of Fig.  (scenario at 577 nm) and Fig.  (scenario at 360 nm) shows MYSTIC 1D-layer AMF profiles for the selected scenarios with a low elevation angle of 3∘ and a high elevation angle of 90∘ (zenith) without and with aerosols, respectively. For comparison, the corresponding profiles computed with SCIATRAN are also shown. The lower row presents the relative differences between MYSTIC and SCIATRAN. Since plane-parallel and spherical modes have different geometrical assumptions, we compare plane-parallel models and spherical models separately.

In the upper atmosphere, the 1D-layer AMFs decrease with altitude in all scenarios (Figs.  and ), because the atmospheric density is decreasing, which lowers the amount of scattering and, correspondingly, the mean photon path length. In the lowest layers, however, the profile shapes are different for the two elevation angles with a rapid decrease with altitude in the low elevation angle scenarios and a local maximum between 2 and 5 km in the high elevation angle scenarios. This local maximum is caused by multiple scattering, which contributes to the horizontal light paths in those layers. The reduction towards the surface in the latter scenarios is due to the low surface albedo. For an elevation angle of 3∘, AMFs are high close to the ground because of the long light path in the layers due to the low elevation angle. Since aerosols increase scattering, photon path lengths and correspondingly 1D-layer AMFs are low in the lowest 2 km, when an aerosol layer is present.

1D-layer AMFs computed with spherical and plane-parallel geometry show noticeable differences for long wavelengths and low aerosol extinction, especially at altitudes above 5 km where extinction coefficients are small (see upper- and lower-left part in Fig. ). In plane-parallel geometry, if one of these photons is traveling horizontally, it will strongly contribute to increase the mean photon path in that specific layer. In spherical mode, the same photon would change layer because of the curved atmospheric layers, and therefore its contribution to the mean photon path will be divided between the crossed layers. Furthermore, in a curved atmosphere, the zenith angle of the photon, which was initially traveling horizontally, will increase. At low altitude, these effects are smaller, and, conversely, 1D-layer AMFs computed with spherical and plane-parallel geometry agree better (mostly <5 %).

AMF profiles calculated with MYSTIC generally agree very well with those calculated with SCIATRAN with relative differences mostly smaller than 5 %. However, significant differences (up to 23 % relative difference) are seen between the plane-parallel solutions of the two models above 5 km for the scenarios without aerosols at 577 nm (Fig. ). In contrast to the plane-parallel case, the spherical solution of MYSTIC is in good agreement with the spherical solution of SCIATRAN. The difference between SCIATRAN plane parallel and MYSTIC plane parallel is attributed to the different solution methods of the radiative transfer equation. A possible explanation is the following: in discrete ordinate methods, the directions of the radiation field are discretized and do not include the exact horizontal direction, for which in plane-parallel geometry the photon path length becomes extremely large in an optically thin medium like the higher atmosphere. In a Monte Carlo model, this horizontal direction is included; therefore the 1D-layer AMF might be larger. This hypothesis could be tested by including more streams (discrete directions) in SCIATRAN and verifying if the solution approaches the higher AMFs from the MYSTIC solution.

The simulations for the same scenarios but with 360 nm wavelength agree very well with SCIATRAN for both spherical and plane-parallel geometries (relative difference <5 %). The differences mentioned above are much smaller at this wavelength because atmospheric scattering events increase with lower wavelength and thus, prevent those very long photon paths. We also investigated a scenario with a wavelength of 440 nm, which is a typical wavelength of the window used for NO2 fitting (see Fig. S4 in the Supplement), for which MYSTIC and SCIATRAN also agree very well (<5 % relative difference), but as for simulations at 577 nm discussed above, the simulations at 440 nm show significant differences between plane-parallel and spherical geometry for layers above 5 km. These differences are, however, smaller than at 577 nm because the optical thickness of Rayleigh scattering is higher at 440 nm.

Overall, MYSTIC agrees very well with SCIATRAN with differences mainly smaller than 5 %. An exception is the high elevation scenario without aerosols, where the plane-parallel solutions of MYSTIC and SCIATRAN differ by up to 23 % for a wavelength of 577 nm at altitudes above 5 km. It should be noted that for these cases the 1D-layer AMFs are very small, and therefore the absolute differences, which are relevant for most applications, are also small. The 1D-layer AMFs computed with MYSTIC also agree very well with the other models presented in . Differences larger than 5 % are mainly attributable to differences between plane-parallel and spherical solutions (see the Supplement). When comparing MYSTIC with the mean of the models, 88.3 % of the compared points are within a relative difference of 5 % for spherical geometry, 81.5 % for plane-parallel geometry, and 97.5 % for the mean of plane-parallel and spherical geometry. The mean of spherical and plane-parallel geometry agrees best because the models in represents a mixture of spherical and plane-parallel solutions.

The NO2 plume was computed with the Graz Lagrangian dispersion Model (GRAL) for a 262.5 m tall stack located at x=1.9 km and y=1.3 km. NO2 molecules were released at this altitude at a constant rate of 40 kgh-1. NOx chemistry was ignored for simplicity. The model domain had a size of 4 km × 4 km and extended from the surface to 21 km altitude. The simulated NO2 was sampled on an output grid with a 100 m horizontal resolution and 20 vertical levels with 25 m resolution from 0 to 500 m. For the simulation we assumed neutral atmospheric stability and southerly wind with a speed of 5 ms-1 at 12 m above ground. The full vertical wind profile is generated within the model based on similarity theory. The NO2 background from the US Standard Atmosphere was added to the simulated NO2 field, which was extended to 21 km altitude (see vertical resolution profile in the Supplement). The resulting NO2 VCDs are shown in Fig. a. In the following, the simulated NO2 concentration field and the corresponding NO2 VCDs are referred to as the true NO2 field and as the true total VCD, respectively. The true VCD will be used as a reference to demonstrate the limitations of 1D radiative transfer calculations.

Using MYSTIC, we computed the SCDs that would be observed from an airborne push-broom spectrometer flying parallel to the plume axis from south to north at an altitude of 6 km. The field of view in the across-track direction of the instrument covers the full x-direction of the model domain. The SCDs were obtained by computing 3D-box AMFs for each single observation (i.e., for each ground pixel) and multiplying these AMFs with the 3D NO2 field from the simulation (which corresponds to the numerator in Eq. ).

As an example, Fig.  illustrates the 3D-box AMFs for an instrument pointing downwards at a zenith angle of 4.8∘ and an azimuth angle of 90∘. The sun is placed in the west (SAA=90∘) at a SZA of 20∘ – i.e., the instrument is facing the sun. The figure shows the 2D cross section of 3D-box AMFs in the principal plane of the observations, which aligns with the x–z plane in this geometry. Figure b and c show the horizontally and vertically integrated 3D-box AMFs, i.e., layer and column AMFs, respectively. The layer AMFs are identical to 1D-layer AMFs. The 3D-box AMFs are high along the line of sight of the instrument and largest just below the aircraft. Most photons travel directly along the geometric path from the sun to the ground pixel and then to the instrument. Although 3D-box AMFs are highest along the geometric path due to the relatively bright surface, a non-negligible fraction of photons is scattered into the line of sight without reaching the surface, leading to an increase in 3D-box AMFs within a parallelogram bounded by the line of sight and the position of the sun.

The column AMFs (Fig. c) are highest close to the instrument and decrease with distance to the instrument in the -x direction due to atmospheric scattering. After the “reflection point”, values continue to decrease with distance to the instrument but at a lower rate. Due to periodic boundaries this decrease continues on the right of the instrument (x≥3.9 km). Layer AMFs (i.e., 1D-layer AMFs) (Fig. b) are highest directly below the instrument. They change by a factor of 2 at the altitude of the aircraft because layers below are crossed (at least) twice by the photons, while layers above are only crossed once.

3D-box AMFs and corresponding SCDs were computed for four different solar zenith angles and four different relative azimuth angles between the sun and the plume axis (and flight direction). We used a default aerosol scenario with a rural-type aerosol representative of spring–summer conditions in the PBL (0–2 km) and a background aerosol above 2 km (visibility of 50 km in the PBL). The parameters used for the AMF calculation are summarized in Table . Note that with perfect knowledge of the relative NO2 distribution, the true total VCD could be reproduced exactly from the SCDs using 3D radiative transfer calculations. The computational cost of calculating 3D-box AMFs is considerably larger than for 1D-layer AMFs. The computational time for calculating 3D-box AMFs for the scenarios here (see Table  with SZA=20∘, SAA=90∘, VAA=90∘ and VZA=2∘) is around 218 s with 1 million photons using a single core of our local machine (Intel Xeon W-2175 CPU @ 2.5 GHz). The computational time for the corresponding 1D-layer AMFs is only about 4 s with 1 million photons. Note, however, that even less photons would be sufficient to obtain a similar noise level as for the 3D-box AMFs.

The SCDs computed for the scenario with the sun illuminating the scene from the west at a solar zenith angle of 40∘ are presented in Fig. b. The SCDs are larger than the VCDs (panel a) because the AMFs (panel c) are generally larger than 1. The SCD plume is wider and shifted towards the east compared to the VCD plume. The widening is due to both geometric effects and atmospheric scattering. Geometric effects are caused by the fact that photons following the main geometric path from the sun to the surface and to the instrument may traverse the plume either on the way from the sun to the surface or from the surface to the instrument (or both). These two pathways are separated horizontally. For high solar zenith angles (here SZA=40∘) this leads to two SCD maxima close to the source as seen in Fig. b. The westerly maximum corresponds to the direct observation of the plume (photons reflected by the surface pass the plume on the direct way to the aircraft), whereas the easterly maximum corresponds to its mirror image (photons first travel through the plume before they get reflected at the surface and reflected to the aircraft). This is further illustrated in Fig. , where two of the three illustrated direct paths (i.e., three viewing zenith angles) cross the NO2 maximum – main photon path (1) and (3) in Fig. . The main photon path for the observation angle (2) in Fig.  misses the plume maximum, which is why total SCD is lower for this observation. Atmospheric scattering leads to an additional horizontal smoothing of the plume, but in the case of a medium-high surface albedo of 0.2, the geometric effects dominate.

For each scenario, total AMFs were also computed from 1D-layer AMFs, which requires a NO2 profile (Eq. ). The most obvious approach is to use the true NO2 profile above the ground pixel the instrument is pointing towards, which is based on the idea that the AMF is used to convert an SCD to a VCD above a ground pixel (Fig. d, e, f). Alternatively, a NO2 background profile from the US Standard Atmosphere was used for each ground pixel, which assumes that no information on the spatial variability in NO2 is available (Fig. g, h, i).

Figure f and i show the total AMFs computed with the true and background NO2 profile, respectively. In both cases, AMFs increase with distance from the aircraft due to the increasing viewing zenith angle. For the true NO2 profiles, AMFs are higher inside the plume. This can be explained by the fact that the measurements are more sensitive to NO2 inside the plume than to the background NO2 outside because the plume is located at an altitude where the 1D-layer AMFs are higher.

Figure d and g show the VCDs obtained by dividing the true SCDs in Fig. b by the 1D-layer AMFs in Fig. f and i, respectively. Since geometric distortions and horizontal smoothing due to scattering cannot be corrected for when using a 1D radiative transfer model, all structures seen in the SCDs are essentially preserved in the VCDs, including the double peak structure, the widening of the plume and the horizontal displacement. Figure e and h show the differences in these VCDs from the true VCDs. In both cases, the location of the plume is shifted towards the aircraft relative to the true position. Within the maximum of the plume, this displacement leads to an underestimation of the true VCDs by -60.8 µmol m-2 when using the NO2 profile (Fig. e) above the ground pixel and by -54.6 µmol m-2 when using the constant NO2 profile (Fig. h).

The displacement of the calculated VCD plume and the magnitude of the bias depend on the position of the sun as demonstrated in Figs.  and . The shift increases with increasing SZA due to the geometric effects explained earlier. The relative azimuth angle between the viewing direction and the sun also plays a critical role. The displacement is smaller when the aircraft is flying directly away from the sun (SAA=0∘) or towards the sun (SAA=180∘) and the sun illuminates the scene along the plume axis, but even in these cases it is not negligible. Biases are typically larger when the spatial displacement is large.

A possible application of airborne imaging spectroscopy is the estimation of NO2 emissions from point sources. Measurements from airborne spectrometers have been used, for example, to estimate CO2 emissions from power plants or CH4 emissions from coal mine ventilation shafts . The emissions can be estimated using a mass-balance approach by integrating the NO2 VCD enhancement above the background across the plume and multiplying this integral (referred to as line density in the following) with a mean wind speed to obtain a flux. The flux is equivalent to the source strength under the assumption of steady-state conditions.

We computed line densities 300 m downstream of the source for the true VCD field and for fields computed with 1D-layer AMFs for different solar zenith and azimuth angles. The VCD cross sections are shown in Fig. . The line densities were multiplied with a wind speed of 9.1 m s-1, which is the wind speed at the stack height of 262.5 m in the GRAL simulation.

Table  summarizes the computed line densities and fluxes for the different scenarios. In all scenarios, emissions were significantly underestimated by 9 %–37 % (relative to the true VCD) depending on the solar azimuth and zenith angle. Note that the emission estimation for the true VCD is slightly higher than the emission input for the dispersion model due to simplification of the mass-balance approach, which does not account for the vertical variability in wind speeds across the plume. The bias in the plume emission estimation using 1D-layer AMFs generally increases with solar zenith angle. This bias also depends on the solar azimuth angle. The largest bias occurs, when the SAA is 0 or 180∘, i.e., the instrument is flying towards and away from the sun.