New theoretical bases are proposed to localize and estimated gross process measurement errors (GE) subject to linear constraints, applying the Maximum Likelihood (ML) principle. GE itself is considered as random variable and two families of distribution are proposed as models. The first, more adequate model is the family of Gamma distributed GE-s, the second, less adequate but more practical, is that of the non-zero mean Gaussian GE-s. The concept of GE situations is introduced and the problem of GE identification is formulated as a mixed discrete-continuous ML estimation to find the actual situation. Algorithm and results of simulation experiments will be given in another paper.