Deligne and Kato proved a formula computing the dimension of the nearby cycles complex of an $\ell$-adic sheaf on a relative curve over an excellent strictly henselian trait. In this article, we reprove this formula using Abbes-Saito's ramification theory.
"Ramification and nearby cycles for $\ell$-adic sheaves on relative curves." Tohoku Math. J. (2) 67 (2) 153 - 194, 2015. https://doi.org/10.2748/tmj/1435237040