  The lack of low frequency information and a good initial model can seriously
affect the success of full waveform inversion (FWI), due to the inherent cycle
skipping problem. Computational low frequency extrapolation is in principle the
most direct way to address this issue. By considering bandwidth extension as a
regression problem in machine learning, we propose an architecture of
convolutional neural network (CNN) to automatically extrapolate the missing low
frequencies without preprocessing and post-processing steps. The bandlimited
recordings are the inputs of the CNN and, in our numerical experiments, a
neural network trained from enough samples can predict a reasonable
approximation to the seismograms in the unobserved low frequency band, both in
phase and in amplitude. The numerical experiments considered are set up on
simulated P-wave data. In extrapolated FWI (EFWI), the low-wavenumber
components of the model are determined from the extrapolated low frequencies,
before proceeding with a frequency sweep of the bandlimited data. The proposed
deep-learning method of low-frequency extrapolation shows adequate
generalizability for the initialization step of EFWI. Numerical examples show
that the neural network trained on several submodels of the Marmousi model is
able to predict the low frequencies for the BP 2004 benchmark model.
Additionally, the neural network can robustly process seismic data with
uncertainties due to the existence of noise, poorly-known source wavelet, and
different finite-difference scheme in the forward modeling operator. Finally,
this approach is not subject to the structural limitations of other methods for
bandwidth extension, and seems to offer a tantalizing solution to the problem
of properly initializing FWI.
