Persistence provides a way of grading the importance of homological features in the sublevel sets of a real-valued function. Following the definition given by Edelsbrunner, Morozov and Pascucci, an ε-simplication of a function f is a function g in which the homological noise of persistence less than ε has been removed. In this paper, we give an algorithm for constructing an ε-simplication of a function defined on a triangulated surface in linear time. Our algorithm is very simple, easy to implement and follows directly from the study of the ε-simplication of a function on a tree. We also show that the computation of persistence defined on a graph can be performed in linear time in a RAM model. This gives an overall algorithm in linear time for both computing and simplifying the homological noise of a function f on a surface.