Gaussian random field has been widely applied to quantify high-dimensional uncertainties in the spatial or temporal domain. A common practice in Gaussian random field modeling is to use the exponential function to represent the covariance matrix. However, the exponential function-based covariance formulation does not allow negative values, thus it cannot capture the negative correlation between two locations in the input domain. To resolve this issue, this work reports new formulations of the covariance matrix based on oscillating functions, and a process of reconstructing Gaussian random field models from observation data. The proposed covariance functions are compared with the traditional exponential covariance functions on two test cases, where the datasets show negative correlations. The results of comparative studies demonstrate that the proposed formulations improve the accuracy of Gaussian random field models effectively.