Corrected Kriging update formulae for batch-sequential data assimilation

Recently, a lot of effort has been paid to the efficient computation of Kriging predictors when observations are assimilated sequentially. In particular, Kriging update formulae enabling significant computational savings were derived in Barnes and Watson (1992), Gao et al. (1996), and Emery (2009). Taking advantage of the previous Kriging mean and variance calculations helps avoiding a costly $(n+1) \times (n+1)$ matrix inversion when adding one observation to the $n$ already available ones. In addition to traditional update formulae taking into account a single new observation, Emery (2009) also proposed formulae for the batch-sequential case, i.e. when $r > 1$ new observations are simultaneously assimilated. However, the Kriging variance and covariance formulae given without proof in Emery (2009) for the batch-sequential case are not correct. In this paper we fix this issue and establish corrected expressions for updated Kriging variances and covariances when assimilating several observations in parallel.