In this paper we use the Laest Squares Collocation (LSC) method for the "Geoid Determination" and the "Earth Gravity Field Modeling" in the Coastal Pars region in southern Iran. The LSC is one of the Earth Gravity Field Modeling methods which does not need regularization, opposite to the Geodetic Boundary Value Problem (GBVP) solutions, such as Stokes. Also, unlike statistical methods, the LSC has the ability to account for the systematic effects in the data (trend), it predicts quantities between the data points (interpolation), and estimates the quantity at the data point (filtering). The main advantage of LCS methods is their capability of incorporating heterogeneous data, for which gravimetric or geometric data can be used as inputs of the target function.
In the first Section, we briefly introduce the LSC. In Section 2, we describe fundamentals of the LSC in a geometric space, and the way it connects the statistical concept of the covariance function and error least squares constrain in order to reproduce the kernel function in the Hilbert space which leads to the "Least Squares Collocation". Furthermore, the Wiener-Kolmogrov formula (Equation 7) is introduced as a solution for the LSC. Further in Section 2, we explain our approach to use the LSC with random errors to adapt its theory to the noisy data (Equation 28).
In Section 3, the concept of "True Covariance Function"(Equation 12), and the procedure of estimating its "Empirical Covariance Function"(Equation 34) based on two essential assumptions: "Non Stationarity" and "Ergodicity", are described. We divide the covariance function into global and local subclasses and individually explain their structures. Also, we describe the covariance function modeling in the LSC by fitting an analytical covariance model (derived from a true covariance function) to an empirical covariance function (obtained from local gravity data) (Equation 63). We demonstrate that an analytical covariance model can be generated by determining the covariance model parameters using the least squares inverse (Equation 65).
In Section 4, we use gravity anomaly data for determining Geoid by applying the LSC. Tscherning's algorithm (Figure 5) is used for the purpose of implementing the LSC theory. As in the collocation theory, the function that was used in the Hilbert space must be harmonic, In our observational space (a sphere that represents the Earth), we assume that there is no mass above the Geoid surface. In order to guarantee this, the "Remove-Compute-Restore" method is used. Based on the field operation conducted the Department of Geophysics, (Exploration Directorate of National Iranian Oil Company, 2004) in our case study, the value of the topographic density has estimated about 2.3 .
Finally in the Section 5, we evaluate the results with 15 GPS/Leveling control points in the region and the root mean squared (RMS) value of 0.052544 meters is achieved. In another experiment we use the LSC for determination of the geoid, using the same data, but having topographic density of 2.67 . The achieved RMS in this experiment is 0.06695 meters. Comparing these two experiments indicates that, in the Coastal Pars region, the topographic density value (2.3 ) determined by the Department of Geophysics, (Exploration Directorate of National Iranian Oil Company, 2004), provides a better estimation compared to the global value (2.67 ). The Section is wrapped by further analysis between the Geoid results of the LSC and Geoid derived from the Earth Gravity Model released (EGM1) 1996 and the EGM 2008 Geopotential models in the region. Our analysis demonstrates that the Geoids obtained from the EGM's models have about 20 centimeters shift compared to those obtained by the LSC.