This paper presents a finite element model based state estimation algorithm to reconstruct the complete dynamic response of an instrumented structure subject to unmeasured disturbances in the form of realizations of random excitations. The proposed algorithm operates on noise contaminated measurements of dynamic response, a finite element model of the system and a spectral density description of the random excitations and measurement noise. Although this problem typically falls within the category of Kalman filtering; the main contribution of the paper is that it develops an estimator with similar state error characteristics as the Kalman filter, but with the advantage that it can be directly implemented as a modified version of the finite element model of the system. The proposed observer results in a modified version of the model of the system with added dampers and applied forces which are linear combinations of the measurements applied at the sensor locations. The proposed method is successfully illustrated in a 10 degree of freedom spring-mass-dashpot structure with two noise contaminated measurements and subject to realizations of random excitations at all degrees of freedom.