Parameter selection and covariance updating

• Simple expression for covariance updating without the need for forward propagation.
• Equivalence of previously published methods within the assumption of small perturbation.
• Parameter selection based on decomposition of the covariance matrix.

A simple expression is developed for covariance-matrix correction in stochastic model updating. The need for expensive forward propagation of uncertainty through the model is obviated by application of a formula based only on the sensitivity of the outputs at the end of a deterministic updating process carried out on the means of the parameters. Two previously published techniques are show to reduce to the same simple formula within the assumption of small perturbation about the mean. It is shown, using a simple numerical example, that deterministic updating of the parameter means can result in correct reconstruction of the output means even when the updating parameters are wrongly chosen. If the parameters are correctly chosen, then the covariance matrix of the outputs is correctly reconstructed, but when the parameters are wrongly chosen is found that the output covariance is generally not reconstructed accurately. Therefore, the selection of updating parameters on the basis of reconstructing the output means is not sufficient to ensure that the output covariances will be well reconstructed. Further theory is then developed by assessing the contribution of each candidate parameter to the output covariance matrix, thereby enabling the selection of updating parameters to ensure that both the output means and covariances are reconstructed by the updated model. This latter theory is supported by further numerical examples.