Deep-image prior (DIP) is a novel approach to solving ill-posed inverse problems whose solution is parameterized with an untrained deep neural network and cascaded with the forward modeling operator. A key component to the success of such a method is represented by the choice of the network architecture, which must act as a natural prior to the inverse problem at hand and provide a strong inductive bias toward the desired solution. Inspired by the close link between neural networks and iterative algorithms in classical optimization, we apply an unrolled version of the gradient descent (GD) algorithm as our DIP network architecture, denoted as the deep-unrolling (DU) architecture. Each layer of the unrolled network is comprised of two parts: the first part corresponds to the GD step of the data-fidelity term, whereas the second part, formed by a six-layer convolutional neural network (CNN), plays the role of a regularizer in the associated objective function. The developed DU architecture is applied to the problem of image-domain least-squares migration (IDLSM) to invert migrated seismic images for their underlying reflectivity and is denoted as DU-IDLSM. As such, the DU architecture parameterizes the reflectivity, and the input of each layer of the unrolled network is the reflectivity at the previous layer. Similar to the classical DIP approach, the parameters of the DU architecture are optimized in an unsupervised fashion by minimizing the data misfit function itself. Through experiments with a part of the Sigsbee2A model and a marine field data set, we test the effectiveness of the DU-IDLSM approach and highlight two key benefits. First, the DU architecture can effectively regularize the inversion process, resulting in reflectivity estimates with fewer artifacts and higher image resolution than those produced by conventional IDLSM approaches. Second, we indicate that by including dropout layers in the CNN architecture, DU-IDLSM can produce a qualitative measure of the uncertainty associated with the least-squares migration process.

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