In recent years, aircraft and satellite observations of atmospheric trace gases have been used to infer surface and tropospheric sources and sinks of those gases using Chemistry Transport Models (CTMs). While the data assimilation techniques used in these inversion studies are well studied and understood, there are several technical issues that complicate the analysis of the estimated sources. In particular, current assimilation techniques such as 4D-Var tacitly assume the employed CTM to be perfect. Due to this, model biases can lead to spurious corrections to a priori source estimates which can greatly complicate their subsequent analysis. In this talk, I will present a "weak-constraint" version of 4D-Var which allows for the simultaneous estimation of trace gas sources as well as potential transport errors. I will further present results from an application of this weak-constraint 4D-Var method to the estimation of CO sources using MOPITT CO observations, alongside a discussion of benefits and problems of this method.