Calculation of Air Mass Factors

Air mass factors (AMFs) relate apparent column densities (ACDs) to vertical column densities, VCDs, through,

$$\delta=\frac{\rm ACD}{\rm VCD}$$ (11.14)

where $\delta$ represents the air mass factor. The effective path enhancement for an atmosphere containing an absorbing layer can be obtained by considering the Beer-Lambert law (Perliski and Solomon, 1993),

$$I=I_o \exp{(-\sigma_l \delta \Delta N_l)}$$ (11.15)

where $\sigma_l$ is the absorption cross-section and $\Delta N_l$ is the vertical column amount of species l. Physically, the air mass factor represents the effective enhancement in optical path through the absorbing layer. An expression for air mass factor can be arrived at by inverting equation (6.15) to obtain (Perliski and Solomon, 1993; McKenzie et al., 1991; Ridley et al., 1984),

$$\delta_l = \frac{-\ln{ \left( \frac{I}{I_o} \right) } } {\sigma_l \cdot \Delta N_l}$$ (11.16)

and in a plane parallel purely absorbing atmosphere, this reduces to $\delta=1/\mu$ for the direct solar beam As multiple scattering is important, air mass factors are calculated with the aid of radiative transfer models. The model is run once using the background atmosphere to determine I_o. Then, an absorber at some level is perturbed by column amount $\Delta N_l$ and the model is re-run to calculate I. An air mass factor of zero indicates that modifying the amount of absorber has no impact on the radiances. However, real air mass factors (which include multiple scattering) must be larger than zero as perturbing the atmosphere at one point will effect the radiances at all others. In general, each species will have a different value for the air mass factor at the same wavelength. However, if the optical depth of both the absorber and background atmosphere are small then air mass factors will only vary with wavelength and geometry. Also, for optically thin absorbers the air mass factor does not depend on the amount by which the layer is perturbed. This is equivalent to saying there is a linear response as doubling $\Delta N_l$ also doubles the numerator in equation (6.16). At large optical depths this is not the case as the majority of the signal originates close to the observation point and hence the effect of the perturbed absorber will not be seen.

In general, air mass factors depend on a number of factors, including: solar zenith angle, height of the layer being perturbed, wavelength, the type and abundance of aerosols, and the altitude and direction of observation. The effects of these variables have been examined in some detail for zenith-sky geometry (Perliski and Solomon, 1993; Slusser et al., 1996) but, to date, their effects on nadir and limb geometry have not been assessed. Air mass factors will also depend on how the absorber profile varies within the perturbed layer itself.

 

Air mass factor calculations made using the model developed for this work compare well with Monte Carlo-calculated AMFs for zenith-sky geometry (Slusser et al., 1996). A brief survey of how solar zenith angle, height of the ER-2, height of the absorbing layer, wavelength, and aerosols impact both limb and nadir geometry is carried out. Nadir air mass factors are considered first. Refer to Figure 6.2 for an illustration of the the nadir geometry.

Figure 6.11 shows nadir AMFs for an albedo of 0.6 and 0.3 using different absorbing layer heights as a function of solar zenith angle. These calculations are for BrO at 350 nm, although the optical depths are small enough that they are applicable for any absorber at this wavelength. For the layers nearer to the surface, the AMF increases slightly with SZA and then decreases. This is due primarily to the increasing pathlength, and hence increasing extinction, of the direct solar beam. At larger SZAs, most of the energy will have been scattered out before reaching these levels. For this same reason, the lower the layer, the smaller the AMF. The higher layers tend to continually increase with SZA due to the increase in overall pathlength. A larger albedo will increase the AMF as a larger fraction of the signal will have passed through the absorbing layer (at least) twice (as long as the atmosphere is not too thick). The dashed lines in Figure 6.11 refers to the optically-thin, plane-parallel, geometric AMF of $1+\sec{\theta_o}$. The 8-20 km layer resembles this but only in a qualitative way.

 

Table 6.4: Model-calculated, nadir air mass factors at 350 nm as a function of the placement of the absorbing layer. The basic column refers to calculations made using $z_{\rm ER2}=20$ km, $\theta_o=70^{\circ}$, $\Lambda=0.6$, background stratospheric aerosols, and water droplets and ice aerosols inside the planetary boundary layer (PBL), taken as the lowest 1 km of the atmosphere. Others columns specify what has been changed from the basic calculation. The calculations were made for BrO but apply equally well to any absorber.

Absorbing		$z_{\rm ER2}=$	$z_{\rm ER2}=$	No aerosols	No strat.	Single
Layer (km)	Basic	10 km	5 km	in PBL	aerosols	Scattering
0-1	3.11	3.63	4.32	2.76	3.17	1.67
0-5	3.29	3.81	4.37	3.15	3.33	1.82
5-8	3.81	4.32	3.81	3.78	3.83	2.29
8-20	4.09	3.73	3.37	4.07	4.08	3.07

The impact of changing other variables is summarized in Table 6.4. Lowering the height of the observation tends to increase the AMF as the signal contains a larger fraction of photons which have sampled the absorbing layer. The exception is for the 8-20 km layer where the AMF decreases as less of the absorber is below the observation point. Removing the aerosols from the planetary boundary layer (PBL) acts to decrease the AMF, especially if the absorbing layer overlaps the PBL, due to the reduced scattering. Similarly, without stratospheric aerosols, the AMF may increase if the absorbing layer is low, or decrease if it overlaps the aerosol layer. Comparisons with single scattering AMFs indicates that multiple scattering is important, although less so for 8-20 km layer.

 

Table 6.5: Typical elevation angles used in a limb retrieval.

 

Step	Elevation Angle	Tangent Heighta	Peak in AMF at (km)
	(deg.)	(km)	350 nm	500 nm
0	5.0	-	20-21	20-21
1	3.5	-	20-21	20-21
2	2.0	-	20-21	20-21
3	0.5	-	20-21	20-21
4	-1.0	19.0	19-20	19-20
5	-2.5	13.9	18-19	14-15
6	-4.0	4.4	18-19	18-19
7	-5.5	-	18-19	18-19
8	-7.0	-	18-19	18-19
9	-8.5	-	18-19	18-19
Nadir	-90	-	8-9	1-2

^a Uplooking steps and downlooking steps with a direct surface component do not have a tangent height; does not include refraction.

 

 

Air mass factors are now examined as a function of where the absorbing layer is placed for limb and nadir viewing angles. The limb viewing geometry is illustrated in Figure 6.2. A set of ten limb scan steps, starting at ${\rm EA}=5^{\circ}$, are used which represent a typical CPFM scan. These angles are given in Table 6.5. The thickness of the perturbed layer was 1 km. Figure 6.12 shows the calculations at 500 nm. The first five steps are shown in panel (a). Steps 0 to 3 all have peaks, indicating the height to which they are most sensitive, near 20 km, the altitude of the ER-2. This is due simply to the fact that the largest pathlength enhancements occur for all near 20 km. Step 4 has its peak slightly below 20 km, near its tangent height of 19 km. Panel (b) has the remaining steps, including the nadir. Step 5 also peaks near its tangent height, 13.9 km, but steps 6 to 9 all peak near the 20 km. The reason for this is that while the pathlengths are longer closer to the surface, the optical depths are large enough that the majority of the signal is originating from layers near the observation point. The nadir AMF is nearly constant with a slight maximum just above the surface.

Similar calculations were carried out at 350 nm and are presented in Figure 6.13. The increase in Rayleigh scattering acts to push the heights at which the maximum AMF occurs closer to 20 km. This also leads to an overall decrease in AMF peak values and an increase in width. With the exception of the nadir, which had a peak at 8-9 km, the maximum occurs at all angles between 18-21 km. The heights at which each angle has a maximum in AMF are summarized in Table 6.5.