  Incorporating prior knowledge on model unknowns of interest is essential when
dealing with ill-posed inverse problems due to the nonuniqueness of the
solution and data noise. Unfortunately, it is not trivial to fully describe our
priors in a convenient and analytical way. Parameterizing the unknowns with a
convolutional neural network (CNN), and assuming an uninformative Gaussian
prior on its weights, leads to a variational prior on the output space that
favors "natural" images and excludes noisy artifacts, as long as overfitting is
prevented. This is the so-called deep-prior approach. In seismic imaging,
however, evaluating the forward operator is computationally expensive, and
training a randomly initialized CNN becomes infeasible. We propose, instead, a
weak version of deep priors, which consists of relaxing the requirement that
reflectivity models must lie in the network range, and letting the unknowns
deviate from the network output according to a Gaussian distribution. Finally,
we jointly solve for the reflectivity model and CNN weights. The chief
advantage of this approach is that the updates for the CNN weights do not
involve the modeling operator, and become relatively cheap. Our synthetic
numerical experiments demonstrate that the weak deep prior is more robust with
respect to noise than conventional least-squares imaging approaches, with
roughly twice the computational cost of reverse-time migration, which is the
affordable computational budget in large-scale imaging problems.
