  In this article we develop further an algorithm for data assimilation based
upon a shadowing refinement technique [de Leeuw et al., SIAM J. Appl. Dyn.
Sys., 17 (2018)] to take partial observations into account. Our method is based
on regularized Gauss-Newton method. We prove local convergence to the solution
manifold and provide a lower bound on the algorithmic time step. We use
numerical experiments with the Lorenz 63 and Lorenz 96 models to illustrate
convergence of the algorithm and show that the results compare favourably with
a variational technique (weak-constraint four-dimensional variational method)
and a shadowing technique (pseudo-orbit data assimilation). Numerical
experiments show that a preconditioner chosen based on a cost function allows
the algorithm to find an orbit of the dynamical system in the vicinity of the
true solution.
