The increasing availability of satellite-based observations of atmospheric greenhouse gas (GHG) (e.g., GOSAT, OCO-2) offers an unprecedented opportunity to better monitor and understand the global carbon cycle. Current atmospheric source inversion systems using large satellite datasets are often limited by the computational cost of performing long-term inversions as well as quantifying the associated uncertainties. Previous approaches have relied on various dimension reduction techniques to make those computations more tractable, although they have often fallen short of combining both theoretical optimality and computational scalability. To address those limitations, we introduce a new approach for high-dimensional GHG source inversions, which combines cutting-edge randomization methods for large matrices decomposition and optimal dimension reduction techniques that maximize observational constraints. The resulting algorithm dramatically improves the computational scalability of the inversions. An additional and equally useful feature of this method is its ability to provide as a by-product of the optimization the spatiotemporal flux patterns that are independently and most constrained by the observations along with their associated posterior errors. Such diagnostics are crucial to better understand the information content of satellite-based observations used in current GHG source inversion systems and enable to answer critical questions such as: can current satellite observations provide seasonal to sub-seasonal constraints on the carbon cycle? At which spatial resolutions can those observations add significant information to the prior (bottom-up) estimates? Those aspects are illustrated with a pseudo-experiment based on a monthly methane source inversion over North America using GOSAT XCH4 column observations.