Over the last four decades there has been considerable research in the improvement of imaging exo-atmospheric objects through air turbulence from ground-based instruments. Whilst such research was initially motivated for military purposes, the benefits to the astronomical community have been significant. A key topic in this research is isoplanatism. The isoplanatic angle is an angular limit that separates two point-source objects, where if independent measurements of wavefront perturbations were obtained from each source, the wavefront distortion would be considered equivalent. In classical adaptive optics, perturbations from a point-source reference, such as a bright, natural guide star, are used to partially negate perturbations distorting an image of a fainter, nearby science object.

Various techniques, such as atmospheric tomography, maximum a posteriori (MAP), and parameterised modelling, have been used to estimate wavefront perturbations when the distortion function is spatially variant, i.e., angular separations exceed the isoplanatic angle, θ₀, where θ₀ ≈ 10 μrad for mild distortion at visual wavelengths. However, the effectiveness of such techniques is also dependent on knowledge a priori of turbulence profiles and configuration data.

This dissertation describes a new method used to estimate the eigenvalues that comprise wavefront perturbations over a wide, spatial field. To help reduce dependency on prior knowledge for specific configurations, machine learning is used with a recurrent neural network trained using a posteriori wavefront ensembles from multiple point-source objects. Using a spatiotemporal framework for prediction, the eigenvalues, in terms of Zernike polynomials, are used to reconstruct the spatially-variant, point spread function (SVPSF) for image restoration. The overall requirement is to counter the adverse effects of atmospheric turbulence on the images of extended astronomical objects. The method outlined in this thesis combines optical wavefront sensing using multiple natural guide stars, with a reservoir-based, artificial neural network. The network is used to predict aberrations caused by atmospheric turbulence that degrade the images of faint science objects.

A modified geometric wavefront sensor was used to simultaneously measure phase perturbations from multiple, point-source reference objects in the pupil. A specialised recurrent neural network (RNN) was used to learn the spatiotemporal effects of phase perturbations measured from several source references. Modal expansions, in terms of Zernike coefficients, were used to build time-series ensembles that defined wavefront maps of point-source reference objects. The ensembles were used to firstly train an RNN by applying a spatiotemporal training algorithm, and secondly, new data ensembles presented to the trained RNN were used to estimate the wavefront map of science objects over a wide field. Both simulations and experiments were used to evaluate this method.

The results of this study showed that by employing three or more source references over an angular separation of 24 μrad from a target, and given mild turbulence with Fried coherence length of 20 cm, the normalised mean squared error of low-order Zernike modes could be estimated to within 0.086.

A key benefit in estimating phase perturbations using a time-series of short exposure point-spread functions (PSFs) is that it is then possible to determine the long exposure PSF. Based on the summation of successive, corrected, short-exposure frames, high resolution images of the science object can be obtained. The method was shown to predict a contiguous series of short exposure aberrations, as a phase screen was moved over a simulated aperture. By qualifying temporal decorrelation of atmospheric turbulence, in terms of Taylor's hypothesis, long exposure estimates of the PSF were obtained.