We describe a new 4D-Var inversion framework for nitrous oxide (N2O) based on the GEOS-Chem chemical transport model and its adjoint, and apply it in a series of observing system simulation experiments to assess how well N2O sources and sinks can be constrained by the current global observing network. The employed measurement ensemble includes approximately weekly and quasi-continuous N2O measurements (hourly averages used) from several long-term monitoring networks, N2O measurements collected from discrete air samples onboard a commercial aircraft (Civil Aircraft for the Regular Investigation of the atmosphere Based on an Instrument Container; CARIBIC), and quasi-continuous measurements from the airborne HIAPER Pole-to-Pole Observations (HIPPO) campaigns. For a 2-year inversion, we find that the surface and HIPPO observations can accurately resolve a uniform bias in emissions during the first year; CARIBIC data provide a somewhat weaker constraint. Variable emission errors are much more difficult to resolve given the long lifetime of N2O, and major parts of the world lack significant constraints on the seasonal cycle of fluxes. Current observations can largely correct a global bias in the stratospheric sink of N2O if emissions are known, but do not provide information on the temporal and spatial distribution of the sink. However, for the more realistic scenario where source and sink are both uncertain, we find that simultaneously optimizing both would require unrealistically small errors in model transport. Regardless, a bias in the magnitude of the N2O sink would not affect the a posteriori N2O emissions for the 2-year timescale used here, given realistic initial conditions, due to the timescale required for stratosphere-troposphere exchange (STE). The same does not apply to model errors in the rate of STE itself, which we show exerts a larger influence on the tropospheric burden of N2O than does the chemical loss rate over short (< 3 year) timescales. We use a stochastic estimate of the inverse Hessian for the inversion to evaluate the spatial resolution of emission constraints provided by the observations, and find that significant, spatially explicit constraints can be achieved in locations near and immediately upwind of surface measurements and the HIPPO flight tracks; however, these are mostly confined to North America, Europe, and Australia. None of the current observing networks are able to provide significant spatial information on tropical N2O emissions. There, averaging kernels (describing the sensitivity of the inversion to emissions in each grid square) are highly smeared spatially and extend even to the midlatitudes, so that tropical emissions risk being conflated with those elsewhere. For global inversions, therefore, the current lack of constraints on the tropics also places an important limit on our ability to understand extratropical emissions. Based on the error reduction statistics from the inverse Hessian, we characterize the atmospheric distribution of unconstrained N2O, and identify regions in and downwind of South America, central Africa, and Southeast Asia where new surface or profile measurements would have the most value for reducing present uncertainty in the global N2O budget.