  Detecting a specific horizon in seismic images is a valuable tool for
geological interpretation. Because hand-picking the locations of the horizon is
a time-consuming process, automated computational methods were developed
starting three decades ago. Older techniques for such picking include
interpolation of control points however, in recent years neural networks have
been used for this task. Until now, most networks trained on small patches from
larger images. This limits the networks ability to learn from large-scale
geologic structures. Moreover, currently available networks and training
strategies require label patches that have full and continuous annotations,
which are also time-consuming to generate.
  We propose a projected loss-function for training convolutional networks with
a multi-resolution structure, including variants of the U-net. Our networks
learn from a small number of large seismic images without creating patches. The
projected loss-function enables training on labels with just a few annotated
pixels and has no issue with the other unknown label pixels. Training uses all
data without reserving some for validation. Only the labels are split into
training/testing. Contrary to other work on horizon tracking, we train the
network to perform non-linear regression, and not classification. As such, we
propose labels as the convolution of a Gaussian kernel and the known horizon
locations that indicate uncertainty in the labels. The network output is the
probability of the horizon location. We demonstrate the proposed computational
ingredients on two different datasets, for horizon extrapolation and
interpolation. We show that the predictions of our methodology are accurate
even in areas far from known horizon locations because our learning strategy
exploits all data in large seismic images.
