INTRODUCTION Using the in-situ measurements of water isotopes as a measure of the accuracy of column measurements is complicated as they are sensitive to different altitudes. The in-situ measurements are obviously sensitive to surface water vapour, while column measurements are sensitive to all the water vapour in a column of the atmosphere. As water vapour is by far most abundant in the PBL, column measurements are most sensitive to water vapour isotope signatures near to the surface. However, at higher altitudes the atmosphere is very depleted in heavy isotopes, leading to a column more depleted than the surface. Complicating the comparison further, the sensitivity of the column to different atmospheric layers may change with the water vapour concentration. Therefore to use surface in-situ measurements for a quantitative assessment of column retrieval accuracy, some assumptions about the profile of water vapour isotopes is required. Here 3 simple isotopic models are used to construct atmospheric water vapour isotope profiles using the in-situ observations, which after convolving the modeled columns with TCCON water isotope averaging kernels will be compared against column measurements to determine the accuracy of TCCON retrievals.
Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. Since low order regularizations are easier to implement numerically, our methods give a theoretical foundation for their use.