Seeking the optimal operating policy by an off-line controller for pipelines carrying natural gas has an inherent state estimation problem associated with deviations from demand forecast. This paper presents a Kalman-filter-based observer for the real-time estimation of deviations from the states previously obtained by an off-line controller optimally, around an expected demand function. The observer is based on the linearized form of the non-linear partial differential equations which are the state space representation of isothermal and unidirectional gas flow through a pipeline. Data for the observer are produced by a dynamic simulator. The simulator and linearized observer equations are solved using an implicit finite element method. The observer has been tested on a pipeline subject to certain deviations from demand forecast. It converges in a short span of time.