In most applications of inverse analysis, the measurement errors blurring a dependence between an observed response and material model parameters are assumed to be independent identically distributed random variables. However, when multiple measurements are observed at consecutive time steps on the same sample using the same equipment, it seems more reasonable to assume that these errors form a time series with a positive correlation for measurement errors being close to each other. Moreover, there is no reason to assume that the measurement errors form stationary series because their variability and correlation structure may change with the ongoing measurement process. Here we estimate the sample covariance structure of the measurement errors from several realizations of the observed response vector. Unfortunately, if the number of realizations is small in comparison with the length of the response vector the matrix of sample covariances cannot be considered a good estimate of a true covariance matrix because it will not be positively definite. Several procedures were proposed to estimate the true covariance matrix of a vector with many coordinates when only a few vector realizations are available, see e.g. \cite{Ledoit} or \cite{Musolas}. In our contribution, we consider a damage-plastic model for confined concrete used to simulate the evolution of axial stress and lateral strain within a uniaxial compression test~\cite{Kucerova}. The material properties such as Young modulus, Poisson ratio etc. are parameters of interest. The goal of our study is to learn to what extent a choice of error covariance matrix estimate affects the behaviour of estimated parameters when the Bayesian approach together with polynomial chaos expansion is applied.