In atmospheric chemistry, differential optical absorption spectroscopy (DOAS) is used to measure concentrations of trace gases. When combined with basic optical spectrometers such as prisms or diffraction gratings and automated, ground-based observation platforms, it presents a cheap and powerful means for the measurement of trace gas species such as ozone and nitrogen dioxide. Typical setups allow for detection limits corresponding to optical depths of 0.0001 along lightpaths of up to typically 15 km and thus allow for the detection also of weak absorbers, such as water vapour, nitrous acid, formaldehyde, tetraoxygen, iodine oxide, bromine oxide and chlorine oxide.

DOAS instruments are often divided into two main groups: passive and active ones. The active DOAS system such as longpath(LP)-systems and cavity-enhanced(CE) DOAS systems have their own light-source, whereas passive ones use the sun as their light source, e.g. MAX(Multi-axial)-DOAS. Also the moon can be used for night-time DOAS measurements, but here usually direct light measurements need to be done instead of scattered light measurements as it is the case for passive DOAS systems such as the MAX-DOAS.

The change in intensity of a beam of radiation as it travels through a medium that is not emitting is given by Beers law:

where I is the intensity of the radiation, ${\displaystyle \rho }$ is the density of substance, ${\displaystyle \beta }$ is the absorption and scattering cross section and s is the path. The subscript i denotes different species, assuming that the medium is composed of multiple substances. Several simplifications can be made. The first is to pull the absorption cross section out of the integral by assuming that it does not change significantly with the path—i.e. that it is a constant. Since the DOAS method is used to measure total column density, and not density per se, the second is to take the integral as a single parameter which we call column density:

The new, considerably simplified equation now looks like this:

If that was all there was to it, given any spectrum with sufficient resolution and spectral features, all the species could be solved for by simple algebraic inversion. Active DOAS variants can use the spectrum of the lightsource itself as reference. Unfortunately for passive measurements, where we are measuring from the bottom of the atmosphere and not the top, there is no way to determine the initial intensity, I₀. Rather, what is done is to take the ratio of two measurements with different paths through the atmosphere and so determine the difference in optical depth between the two columns (Alternative a solar atlas can be employed, but this introduces another important error source to the fitting process, the instrument function itself. If the reference spectrum itself is also recorded with the same setup, these effects will eventually cancel out):

A significant component of a measured spectrum is often given by scattering and continuum components that have a smooth variation with respect to wavelength. Since these don't supply much information, the spectrum can be divided into two parts:

where ${\displaystyle \alpha }$ is the continuum component of the spectrum and ${\displaystyle \beta ^{*}}$ is that which remains and we shall call the differential cross-section. Therefore: