Data assimilation combines models with observations to estimate system state. Linear methods assume Gaussian statistics — fast but wrong when the distribution is multimodal, skewed, or heavy-tailed. Fully nonlinear methods like particle filters handle any distribution but are computationally prohibitive in high dimensions. The gap between “fast but wrong” and “right but slow” is where most real atmospheric and geophysical problems live.
This paper fills the gap with adaptive triangular transport maps using P-spline basis functions. The maps transform the non-Gaussian forecast distribution into something the analysis step can handle. Information criteria automatically select the map complexity — how nonlinear, how many basis functions — based on the actual statistics and available ensemble size. No hyperparameter tuning required.
The adaptation is the key: the method adjusts its own complexity to match the problem. When the distribution is nearly Gaussian, the map is nearly linear and the method recovers Kalman-like efficiency. When the distribution is strongly non-Gaussian, the map becomes more expressive. The cost scales with actual non-Gaussianity, not worst-case non-Gaussianity. Tests on a high-dimensional groundwater model show it works where ensemble Kalman filters fail and particle filters are too expensive.