4DVAR

Four-dimensional variational data assimilation, or 4D-Var, is a method for estimating the initial state of a dynamical system by minimizing a cost function over a time window that integrates an forecast model with observations collected within that window. The goal is to produce a trajectory that best fits the observations while staying close to a prior estimate of the state.

In 4D-Var, the state at the initial time is denoted x0, with xb representing the background (or prior) state and B its error covariance. Observations yk are available at times tk within the window, with Hk as the observation operator mapping model states to observed quantities, and Rk their error covariance. The forward model M advances the state from the initial time to later times, so xk = M(t0, tk; x0). A common form of the cost function is J(x0) = 1/2 (x0 - xb)^T B^{-1} (x0 - xb) + 1/2 sum over k of (Hk xk - yk)^T Rk^{-1} (Hk xk - yk). The 4D-Var solution minimizes J with respect to x0, typically using iterative optimization and gradients computed via the adjoint of the forecast model.

Variants include strong-constraint 4D-Var, which assumes a perfect forecast model and excludes model error, and weak-constraint 4D-Var, which allows for model error by adding a term for model dynamics errors. Another common variant is incremental 4D-Var, which solves a sequence of linearized problems to improve numerical efficiency.

4D-Var is widely used in numerical weather prediction and other geoscience data assimilation tasks, requiring a differentiable forecast model and adjoint capabilities. It is computationally intensive, often implemented on high-performance computing platforms, and is frequently complemented by ensemble methods in hybrid schemes.