For space borne applications the DOAS (Differential Optical Absorption Spectroscopy) method consists of two steps.

The DOAS method is used to fit the differential absorption cross-sections to the measured sun-normalized Earth radiance spectrum, to obtain the slant column density.

The slant column density is translated into the vertical column density using the air mass factor. This air mass factor can include a correction for cloud effects, in order to account for the trace gas amount obscured by clouds.

Figure 1: Sketch of satellite radiation measurement and geometry in a plane parallel atmosphere. The blue lines are the optical path relevant to the slant column density (first step of the DOAS algorithm). The red line is relevant to the vertical column density (second step of the DOAS algorithm).

1. Deriving the Slant Column Density

The first step in the DOAS algorithm is to determine the slant column density, which is defined as the amount of the trace gas along an average path taken by photons within a fit window as they travel from the sun, through the atmosphere to the satellite sensor. This path is represented in Figure 1 by the blue lines. The slant column density is determined by fitting a function to the ratio of the measured Earth radiance to the solar irradiance data. This fit is applied to data taken in a certain wavelength range, called the fit window, which needs to be optimised for each trace gas separately. A polynomial function, which serves as a high-pass filter, is applied to account for scattering and absorption that vary gradually with the wavelength, e.g. reflection by the surface and scattering by molecules, aerosols, and clouds. Also, the high-pass filter takes out gradually varying radiometric calibration errors and other instrumental multiplicative effects. The DOAS equation can be written:

Where

• I(?) is the measured earthshine backscattered radiance spectrum,
• E₀(?) is the extraterrestrial solar irradiance spectrum
• µ₀ is the cosine of the solar zenith angle ?₀ (see Figure 1)
• ?_j^diff is the differential absorption cross-section of specie j, it represents the high-pass filtered parts of the absorption cross-sections
• N_s,j is the slant column density of the relevant absorber j
• c_p are the coefficients of the polynomial function.

Note that this equation represents only one of the many possible implementations of DOAS. Other implementations use a slightly different equation.

In order to optimize the fitting procedure, additional structured spectral effects have to be considered carefully such as the Ring effect (Grainger and Ring, 1962 [RD35]). The linearity of the above equation may be broken down by instrumental aspects such as small wavelength shifts between I and E₀, which requires accurate calibration, or possible contamination of measured radiances by spectral stray-light, which requires the introduction of an intensity offset parameter.

The fit parameters N_s,j and c_p are then determined using a non-linear, least-squares fit. Information on the quality of the fit and the fit parameters is derived from the associated covariance matrix.

2. Air Mass Factor and Vertical Column Density Computation.

In DOAS, the Air Mass Factor (AMF) is used to translate the slant column density into a vertical column density. The air mass factor M_j is defined as the ratio of the slant column density N_s,j and the vertical column density N_v,j. M_j depends on the viewing and solar geometry, the fit window used, the surface albedo, the surface pressure, the actual trace gas profile, clouds and aerosols, as they all affect the apparent slant column amount. The air mass factor is derived from radiative transfer calculations. For computing AMFs, the TROPOMI DOAS algorithm makes use of either pre-calculated LUTs or calls to a radiative transfer model, depending of the processor.

M_j = N_v,j  / N_s,j

The AMF calculation accounts for cloud-contaminated pixels. The independent pixel approximation (IPA) is used to express the AMF as a linear combination of a cloudy AMF (M_cl,j) and a clear-sky AMF (M_cr,j).

M_j = w M_cl,j + (1 – w) M_cr,j

Weight w is the fraction of the radiance that is due to the cloudy part of the ground pixel.

The amount of trace gas below the Lambertian cloud is called the "ghost column" N_G,j and is computed by integrating the a priori trace gas profile from the surface to the cloud pressure. Finally, the vertical column density can be computed:

 N_v,j = (N_s,j + w M_cl,j N_G,j) / (w M_cl,j + (1 – w) M_cr,j) for a partially cloud scene.

Or

N_v,j = N_s,j / M_cr,j    for a clear sky scenario (w=0)