  Full-waveform inversion (FWI) is a widely used technique in seismic
processing to produce high resolution Earth models that fully explain the
recorded seismic data. FWI is a local optimisation problem which aims to
minimise in a least-squares sense the misfit between recorded and modelled
data. The inversion process begins with a best-guess initial model which is
iteratively improved using a sequence of linearised local inversions to solve a
fully non-linear problem. Deep learning has gained widespread popularity in the
new millennium. At the core of these tools are Neural Networks (NN), in
particular Deep Neural Networks (DNN) are variants of these original NN
algorithms with significantly more hidden layers, resulting in efficient
learning of a non-linear functional between input and output pairs. The
learning process within DNN involves iteratively updating network neuron
weights to best approximate input-to-output mappings. There is clearly
similarity between FWI and DNN. Both approaches attempt to solve for a
non-linear mapping in an iterative sense, however they are fundamentally
different in that the former is knowledge-driven whereas the latter is
data-driven. This article proposes a novel approach which learns
pseudo-spectral data-driven FWI. We test this methodology by training a DNN on
1D multi-layer, horizontally-isotropic data and then apply this to previously
unseen data to infer the surface velocity. Results are compared against a
synthetic model and successfulness and failures of this approach are hence
identified.
