In this work, various methods for the estimation of the parameter uncertainty and the covariance between the parameters and the state variables are investigated using the local ensemble transform Kalman filter (LETKF). Two methods are compared for the estimation of the covariances between the state variables and the parameters: one using a single ensemble for the simultaneous estimation of model state and parameters, and the other using two separate ensembles; for the initial conditions and for the parameters. It is found that the method which uses two ensembles produces a more accurate representation of the covariances between observed variables and parameters, although this does not produce an improvement of the parameter or state estimation. The experiments show that the former method with a single ensemble is more efficient and produces results as accurate as the ones obtained with the two separate ensembles method. The impact of parameter ensemble spread upon the parameter estimation and its associated analysis is also investigated. A new approach to the optimization of the estimated parameter ensemble spread (EPES) is proposed in this work. This approach preserves the structure of the analysis error covariance matrix of the augmented state vector. Results indicate that the new approach determines the value of the parameter ensemble spread that produces the lowest errors in the analysis and in the estimated parameters. A simple low-resolution atmospheric general circulation model known as SPEEDY is used for the evaluation of the different parameter estimation techniques.