The estimation of states in noisy dynamical systems is a problem whose solution is of significant importance in various scientific disciplines. Algorithms for filtering, smoothing and prediction estimates of lumped parameter system states have been derived by Kalman and Bucy [5], Bryson and Frazier [l], Cox [3], and Detchmendy and Sridhar [4]. The techniques utilized for generating these algorithms include orthogonal projection theory [5], maximum likelihood estimate [3], and the classical least squares error criterion combined with an invariant embedding technique [4]. The class of problems considered in these investigations were dynamical systems described by ordinary differential equations with additive disturbances. Also, statistical assumptions concerning the characteristics of the input disturbances were an integral part of the algorithm derivations in most cases. Many dynamical systems are distributed in space as well as time and are defined as distributed parameter systems. A general class of such systems are those defined by partial differential equations. Estimation algorithms for partial differential systems have been generated by Tzafestas and Nightingale [ll], Thau [lo], and Seinfeld [9]. Seinfeld [9] utilized the least squares estimation criterion and an approximation to the minimum value of the error criterion to generate an approximation to the optimal estimate for nonlinear partial differential equation systems.