3DVar

3DVar, short for three-dimensional variational data assimilation, is a method used to estimate the state of a geophysical system, such as the atmosphere or ocean, by combining a background forecast with observations at a single analysis time. It aims to produce the most probable state given the prior information and new data.

The method minimizes a cost function that measures the mismatch between the analysis and both the background and the observations. A common form is J(x) = (x - x_b)^T B^{-1} (x - x_b) + (H(x) - y)^T R^{-1} (H(x) - y), where x is the state vector, x_b is the background (forecast) state, B is the background error covariance, y are the observations, R is the observation error covariance, and H maps the model state to observation space. The analysis x^a that minimizes J(x) is the best estimate of the true state under the Gaussian error assumptions.

Implementation often uses linearization around the background and an incremental approach, solving a sequence of linear problems with an adjoint model to compute gradients. 3DVar is typically applied within a single assimilation window and produces one analysis at the center time of that window.

Compared with four-dimensional variational data assimilation (4DVar) or ensemble methods, 3DVar is simpler and computationally cheaper, since it does not explicitly exploit temporal evolution over a time window. Variants include weak-constraint 3DVar, which allows some model error, and incremental 3DVar, which improves convergence for nonlinear problems. Limitations include reliance on accurate B, R, and H, the assumption of Gaussian errors, and potential nonlinearity effects.