We use Gaussian random fields (GRFs) that we call generalized integrated Brownian fields (GIBFs), whose covariance functions have been studied in the context of reproducing kernels, for Gaussian process modeling. We introduce GIBFs into the fields of deterministic and stochastic simulation metamodeling, and give a probabilistic representation of GIBFs that is not given in the literature on reproducing kernels. These GIBFs have differentiability that can be controlled in each coordinate, and are built from GRFs which have the Markov property. Furthermore, we introduce a new parameterization of GIBFs which allows them to be used in higher-dimensional metamodeling problems. We also show how to implement stochastic kriging with GIBFs, covering trend modeling and fitting. Lastly, we use tractable examples to demonstrate superior prediction ability as compared to the GRF corresponding to the Gaussian covariance function.