For example, the observed $CO2$ differential optical depth, $\Delta \tau $, associated with a given $CO2$ spectral feature as illustrated in Fig. 1, is also given by Display Formula
$\Delta \tau =\u222b0psfc\Delta \sigma (\lambda on,\lambda off,T,p)\xb7\eta (T,p)\xb7(1\u2212qg\xb7Mdry)dp,$(1)
where $\Delta \sigma $ is the $CO2$ differential absorption cross section, $\eta $ is the $CO2$ number density, $g$ is the local acceleration due to gravity, $Mdry$ is the molecular mass of dry air ($28.9644e\u22123/NA\u2009\u2009kg/molecule$, where $NA$ is Avogadro’s number), $q$ is the local specific humidity, $psfc$ is the surface pressure, and $\lambda on/\lambda off$ represent the on/off-line wavelengths. $XCO2$ is given by Display Formula$XCO2=\Delta \tau +\Delta \tau other\u222b0psfc\Delta \sigma (\lambda on,\lambda off,T,p)(1\u2212q)dp,$(2)
where $\Delta \tau other$ represents the residual observed differential optical depth due to other species in the region of interest. Optimally, $\Delta \tau other$ approaches zero in the case where the region of interest is void of other absorption features, e.g., water vapor and other trace gases. The measurement error terms are not only driven by the instrument design but also the ancillary meteorological data employed in the retrieval process and the interplay between the two. Both $\Delta \sigma $ and $\eta $ vary as a function of pressure and temperature. As illustrated by these equations, the accuracy of retrieved $XCO2$ values depends not only on the error characteristics of the observed $\Delta \tau $ but also on the ability to accurately characterize the P, T, and WV concentrations along the observed path. In the case of global space-based monitoring systems, retrievals of $XCO2$ typically rely on P/T/WV values derived from meteorological analyses that combine atmospheric general circulation models with assimilation of both ground-based measurements, e.g., rawinsonde observation (RAOB) and hourly surface weather observations, and satellite observations to globally estimate the atmospheric state. This work addresses the impact of uncertainties in atmospheric temperature, moisture, and surface pressure knowledge on the two observed quantities that provide an estimate of $XCO2$ as given in Eq. (2). These parameters are the $CO2$ column mixing ratio and the dry air surface pressure, which can be derived from the observed surface pressure and the water vapor mixing ratio profile. Similar to their passive counterparts, these measurements are also impacted by a number of other atmospheric parameters such as aerosols and clouds. However, the impact of these parameters on the measured quantities is highly dependent on instrument implementation and is beyond the scope of this work.