## 1 Introduction  
Satellite retrievals of the atmospheric dry-air column-average mole fraction of CO\({}_{2}\) (XCO\({}_{2}\)) based on hyperspectral measurements in appropriate near (NIR) and short wave infrared (SWIR) O\({}_{2}\) and CO\({}_{2}\) absorption bands can help to answer pressing questions about the carbon cycle [1]. However, the precision and even more the accuracy requirements for applications like surface flux inversion or emission monitoring are demanding [2; 3; 4]. As an example, large scale biases of a few tenths of a ppm can already hamper an inversion with mass-conserving global inversion models [2; 3].  
Several theoretical studies suggest that multiple scattering at aerosols and clouds are a significant error source for XCO\({}_{2}\) retrievals [5; 6; 7]. Therefore, so called full physics retrievals were set up aiming to minimize these scattering related errors by explicitly fitting scattering related properties such as cloud water/ice content, aerosol optical thickness, cloud height, etc. [8; 9; 10; 11; 12].  
However, due to limited information content in the used absorption bands [13; 14; 15], only few scattering parameters can simultaneously be retrieved, i.e., many other properties (e.g., scattering phase function (SPF), number of cloud layers, etc.) rely on empirical estimates. Additionally, the needed RT calculations with multiple scattering (especially with polarization) can produce computational costs which are several orders of magnitude larger than for absorption only models. This is true even when making use of short cuts and approximations such as the low streams interpolation method [12], correlated-k method [16], or neglecting RT effects like polarization [17]. Much larger speedups can be achieved by, e.g., tabulating the RT [18] or with the photon path-length distribution function (PPDF) method [19; 20].  
A recent study comparing OCO-2 Orbiting Carbon Observatory-2 [21; 22] and GOSAT Greenhouse Gases Observing Satellite [23] XCO\({}_{2}\) retrievals with and without any explicit consideration of scattering at clouds and aerosols suggests that the introduced errors in measured data are small after appropriate filtering [24]. Additionally, [PERSON] et al. showed with simulated and measured GOSAT data that considering scattering at only one layer can be sufficient to obtain results with state of the art precision and accuracy [20].  
Within the next section, we propose a scalar RT model which approximates multiple scattering effects at an optically thin isotropic scattering layer with only little extra computational costs compared to an absorption only RT model. This model is the heart of the Fast atmOspheric traCe gAs retrievaL FOCAL which is introduced in Section 3. In this section, we also introduce various FOCAL setups and perform retrieval experiments for an OCO-2 like instrument. These experiments are not aiming to comprehensively cover the majority of potential geophysical scenarios, because the final quality depends on the full retrieval scheme including, e.g., potential instrument and forward model errors and different post-filtering capabilities. The aim is rather to identify a promising candidate retrieval setup serving as starting point for the development of a full retrieval scheme and its application to actually measured OCO-2 data. Whilst part 1 of this publication is on theoretical aspects of the RT and the retrieval, part 2 [25] deals with the application to measured OCO-2 data including noise model, zero level offset, pre- and post-filtering, bias correction, and validation.