  For a smooth surface X over an algebraically closed field of positive
characteristic, we consider the ramification of an Artin-Schreier extension of
X. A ramification at a point of codimension 1 of X is understood by the Swan
conductor. A ramification at a closed point of X is understood by the invariant
r_x defined by Kato [2]. The main theme of this paper is to give a simple
formula to compute r_x' defined in [4], which is equal to r_x for good
Artin-Schreier extension. We also prove Kato's conjecture for upper bound of
r_x.
