Multi-level surrogate modelling offers the promise of fast approximation to expensive simulation codes for the purposes of uncertainty quantification (UQ). The hope is that a large number of cheap samples from the simulator on coarse grids, can be corrected by a few expensive samples on a fine grid, to build an accurate surrogate. Of the various multi-level approaches, a correction-based method using Gaussian process regression (Kriging) is studied here. In particular, we examine the "additive bridge-function" method, for which-although widely applied-results on theoretical convergence rates and optimal numbers of samples per level are not present in the literature. In this paper, we perform a convergence analysis for the expectation of a quantity of interest (QoI), utilizing convergence results for single-fidelity Kriging, as well as existing multi-level analysis methodology previously applied in context of polynomial-based methods. Rigorous convergence and computational cost analyses are provided. By minimizing the total cost, optimal numbers of sampling points on each grid level are determined. Numerical tests demonstrate the theoretical results for: a 2d Genz function, Darcy flow with random coefficients, and Reynold-Averaged Navier-Stokes (RANS) for the flow over an airfoil with geometric uncertainties. The efficiency and accuracy of this method are compared with standard-and multi-level Monte Carlo. All the test cases show that using our multi-level kriging model significantly reduces cost.