In multivariate geostatistics, suppose that we relax the usual second-orderstationarity assumptions and assume that the component processes are intrinsic random functions of general orders. In this article, we introduce a generalized crosscovariance function to describe the spatial cross-dependencies in multivariate intrinsic random functions. A nonparametric method is then proposed for its estimation. Based on this class of generalized cross-covariance functions, we give cokriging equations for multivariate intrinsic random functions in the presence of measurement error. A simulation is presented that demonstrates the accuracy of the proposed nonparametric estimation method. Finally, an application is given to a dataset of plutonium and americium concentrations collected from a region of the Nevada Test Site used for atomic-bomb testing. © International Association for Mathematical Geosciences 2009.