The multiquadric (MQ) method was discovered by Rolland L. Hardy in the late 1960s. It is a true scattered data, grid-free scheme for mapping of surfaces and volumes. It is a physically deterministic process and is related to concepts of the potential. With proper constraints it satisfies equilibrium and minimum energy conditions. The geostatistical kriging (K) method, developed by matreron in the 1960s, is a stochastic process procedure based on the theory of regionalized variables. It is claimed to be different from classical Statistics and Probability;;This dissertation consists of two major research areas: a comparative study of both the MQ and K theories, and a comparative study of the application of both methods to directly or indirectly related mineral resources problems. The results of the research shows that the MQ method yields equal or better accuracy than the K method. The MQ method avoids the costly pre-processing steps which are required by the K method. The MQ variation varies with location but the K variance does not. Both methods were able to produce representations of isarithmic contours or surfaces for the variable under investigation (temperature, wind velocity, iron grade, manganese percent, simulated data, and Iowa coal thickness). However, the K method could not rigorously produce three dimensional maps for the cases in this research which involved non-simulated data;The K stochastic error prediction formula yields an overly optimistic result. This formula can be used for comparing relative accuracy among various theoretical variograms, but for error of prediction purposes cross validation should be used as with the MQ method;Research produced the following outcomes: the MQ system is found to yield an unbiased estimation, a formula for the equal weighting of data for semivariograms in the K method, a proof of positive definiteness of the Q[superscript] TQ matrix, methods of preventing ill conditioning in coefficient matrices, methods of preventing underflow and overflow in computation, vector data analysis, and multiplane plotting. This research was supported (in part) by the Iowa State Mining and Mineral Resource Institute through the Department of the Interior's Mineral Institute program administered by the U.S. Bureau of Mines under allotment Grant G1164119 and G1174119.