We consider modeling of wave propagation phenomena
in some noisy and cluttered environments. We then show how
the noisy environment may have an effect when trying
to use wave reflections for imaging purposes. In particular
we discuss the so called parabolic approximation regime
corresponding to long range propagation.

Adding a column of numbers produces `carries' along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to answer natural questions: How many carries are typical? Where are they located? (Many further examples, from combinatorics, algebra and group theory, have essentially the same neat formulae.) The examples give a gentle introduction to the emerging fields of one-dependent and determinantal point processes. This work is joint with Alexei Borodin and Persi Diaconis.