Precise estimation of covariance parameters in least-squares collocation by restricted maximum likelihood

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Precise estimates of the covariance parameters are essential in least-squares collocation (LSC) in the case of increased accuracy requirements. This paper implements restricted maximum likelihood (REML) method for the estimation of three covariance parameters in LSC with the Gauss-Markov second-order function (GM2), which is often used in interpolation of gravity anomalies. The estimates are then validated with the use of an independent technique, which has been often omitted in the previous works that are confined to covariance parameters errors based on the information matrix. The crossvalidation of REML estimates with the use of hold-out method (HO) helps in understanding of REML estimation errors. We analyzed in detail the global minimum of negative log-likelihood function (NLLF) in the estimation of covariance parameters, as well, as the accuracy of the estimates. We found that the correlation between covariance parameters may critically contribute to the errors of their estimation. It was also found that knowing some intrinsic properties of the covariance function may help in the scoring process.