The GNSS tomography provides the opportunity to estimate atmospheric wet refractivity, which is important for precise positioning and navigation, as well as for constraining weather and climate models. The GNSS-derived water vapor is often estimated by implementing voxel-based inversions, but this technique is numerically unstable due to the high number of unknown parameters especially in networks covering large areas or when the spatial resolution increases. To mitigate this problem, in this study, we introduce functional based 3 and 4 dimensional (3D and 4D) inversion formulations. Here, the horizontal changes are modelled by the spherical cap harmonic functions. For the vertical component, empirical orthogonal functions derived from ERA5 (Empirical Reanalysis Fifth generation) are applied as background model. The time-dependency is accounted for by applying polynomial spline functions. Numerical results are based on a network of about 190 GPS stations in Germany during 15 days in summer and winter of 2018. Observations from 8 radiosonde stations are applied for comparisons. Our results indicate that the functional tomography is effective and retrieves wet refractivity indices with the mean RMSE (Root Mean Square Error) of about 1.9 ppm. These values are found to be up to 22% smaller than those derived by comparing ERA5 with the radiosonde data.

GNSS 层析成像提供了估算大气湿折射率的机会，这对于精确定位和导航以及约束天气和气候模型非常重要。GNSS 衍生的水蒸气通常通过实施基于体素的反演来估计，但由于未知参数的数量很多，尤其是在覆盖大面积的网络中或空间分辨率增加时，这种技术在数值上不稳定。为了缓解这个问题，在本研究中，我们引入了基于函数的 3 维和 4 维（3D 和 4D）反演公式。在这里，水平变化由球冠谐波函数建模。对于垂直分量，从ERA5（经验再分析第五代）导出的经验正交函数被用作背景模型。通过应用多项式样条函数来考虑时间相关性。数值结果基于 2018 年夏季和冬季的 15 天内德国约 190 个 GPS 站的网络。应用来自 8 个无线电探空站的观测进行比较。我们的结果表明，功能断层扫描是有效的，并且以大约 1.9 ppm 的平均 RMSE（均方根误差）检索湿折射率。发现这些值比通过比较 ERA5 与​​无线电探空仪数据得出的值小 22%。我们的结果表明，功能断层扫描是有效的，并且以大约 1.9 ppm 的平均 RMSE（均方根误差）检索湿折射率。发现这些值比通过比较 ERA5 与​​无线电探空仪数据得出的值小 22%。我们的结果表明，功能断层扫描是有效的，并且以大约 1.9 ppm 的平均 RMSE（均方根误差）检索湿折射率。发现这些值比通过比较 ERA5 与​​无线电探空仪数据得出的值小 22%。