  In current seismic acquisition practice, there is an increasing drive for
sparsely (in space) acquired data, often in irregular geometry. These surveys
can trade off subsurface information for efficiency/cost - creating a problem
of "missing seismic data" that can greatly hinder subsequent processing and
interpretation. Reconstruction of regularly sampled dense data from highly
sparse, irregular data can therefore aid in processing and interpretation of
these far sparser, more efficient seismic surveys. Here, two methods are
compared to solve the reconstruction problem in both space-time and
wavenumber-frequency domain. This requires an operator that maps sparse to
dense data: the operator is generally unknown, being the inverse of a known
data sampling operator. As such, here the deterministic inversion is
efficiently solved by least squares optimisation using a numerically efficient
Python-based linear operator representation. An alternative approach is
probabilistic and uses deep learning. Here, two deep learning architectures are
benchmarked against each other and the deterministic approach; a Recurrent
Inference Machine (RIM), which is designed specifically to solve inverse
problems given known forward operators, and the well-known U-Net. The trained
deep learning networks are capable of successfully mapping sparse to dense
seismic data for a range of different datasets and decimation percentages,
thereby significantly reducing spatial aliasing in the wavenumber-frequency
domain. The deterministic inversion on the contrary, could not reconstruct the
missing data and thus did not reduce the undesired spatial aliasing. Results
show that the application of Deep Learning for seismic reconstruction is
promising, but the treatment of large-volume, multi-component seismic datasets
will require dedicated learning architectures not yet realisable with existing
tools.
