4D-Var using Hessian approximation and backpropagation applied to automatically-differentiable numerical and machine learning models

The paper explicitly formulates 4D‑Var as nonlinear least squares, recalls the Gauss–Newton update x^{k+1}=x^k−(F^T F)^{-1}F^T f, and shows the 4D‑Var Hessian F^T F=B^{-1}+Ĥ^T R̂^{-1}Ĥ, establishing the Gauss–Newton equivalence when linearization is exact. It then defines Backprop‑4DVar with the approximate Hessian F^T F ≈ α^{-1}[B^{-1}+H_0^T R^{-1}H_0] and the update x^{k+1}=x^k−[F^T F]^{-1}∇J, noting that choosing α^{-1}I recovers vanilla gradient descent . The candidate solution reproduces these identities rigorously and adds a standard descent‑lemma proof that the preconditioned gradient step with M=P^{-1} achieves monotone decrease for sufficiently small α under L‑smoothness, which the paper does not attempt. Aside from this added convergence guarantee (which requires extra regularity assumptions), there is no contradiction. Hence both are correct; the paper states the algorithmic equivalences, while the model supplies additional (optional) analysis.