## 2 Materials and Methods
### Quasi-Invariant Areas for Evaluating Surface Reflectance Retrieval  
To have value for atmospheric correction, ground truth must be collected in time and space over the selected area, and in sufficient quantity across diverse reflectance targets to constitute statistically valid sampling. Ideally, the sample must define not just single points of SR but entire distributions. CMAC R&D revealed that the value from comparison of individual estimates of SR is limited because greater mathematical context is lacking. The selection of a QIA with a range of reflectance values, low to high, provides reflectance distributions that convey a great deal more information than monotypic plots. For example, a single reflectance measurement could be correct while the underlying distribution is wrong, but not vice versa. An impediment that exists for calibrating SR retrieval methods or for evaluating their output is that sets of appropriately scaled ground truth reflectance datasets available for image snapshots during image capture generally do not exist. Further, due to variable antecedent conditions and cloud cover, ground truth sampling seldom works well in real time. Instead, an optimized solution for ground truth is to select targets with known surface reflectance from archived images rather than from a new collection. For CMAC calibration and verification, a concept similar to pseudo-invariant targets (PITs) [16] was employed: quasi-invariant areas (QIAs), defined as an AOI with a variable SR distribution that remains consistent between image acquisitions over time. QIAs allow investigation of reflectance distributions under varying atmospheric conditions and also for comparison of SR retrieval methods. PITs are homogeneous areas containing only one target of known reflectance, while QIAs can be chosen to contain many separate targets with widely varying reflectances, hence providing the desired reflectance distributions for calibration and testing. When using a QIA to evaluate SR retrieval methods, the first step is to define boundaries that exclude cover that may change between image dates. Such surfaces generally consist of vegetation, depressions and bare soil subject to water retention in puddles or wetted soil. Warehouse and industrial districts are permanent facilities appropriate for testing across extended periods because they are engineered for the quick drainage of precipitation and include vegetation management or elimination. Such QIAs were chosen in the urban area of Southern California (SoCal) east of Los Angeles (Figure 4). QIA boundaries were mapped to exclude vacant land potentially containing vegetation that could grow/senesce during the period of study or present surface conditions of variable wetness from antecedent rainfall. Five QIAs were selected within the sidelap region of adjacent L8/9 tiles, in the Landsat World Reference System-2 path/row 40/36 and 41/36. The AOIs containing these QIAs are located in the vicinity of the Ontario, California Airport. Three of these QIAs were also used to evaluate S2 AC reported in paper 1. Twenty-four L8 images acquired between 2019 and 2022 and seven L9 images acquired in 2022, were selected for analysis. A list of these images is provided in the Supplemental Materials. The selected images were displayed and inspected to assure that no gross anthropogenic changes were visible through the period of analysis, and each was verified to be cloud and cloud-shadow free. If haze was visible in an image, it was closely inspected to assure even expression within each QIA. Band 9 was inspected for each image to ensure that no cirrus was present, since cirrus, though correctable by CMAC, is seldom evenly expressed across QIA scales of a square kilometer or greater. The selected SoCal warehouse and industrial areas present mostly rooftops, paved roadways and parking that are well drained and are sufficiently large so that variability in parked or moving vehicles, or other individual sub-targets across multiple image-to-image comparisons is reduced to minor stochastic variation.