Evaluation of the fourth-order tesseroid formula and new combination approach to precisely determine gravitational potential

Calculating topographic gravitational potential (GP) is a time-consuming process in terms of efficiency. Prism, mass-point, mass-line, and tesseroid formulas are generally used to calculate the topographic GP effect. In this study, we reformulate the higher-order formula of the tesseroid by Taylor series expansion and then evaluate the fourth-order formula by numerical tests. Different simulation computations show that the fourth-order formula is reliable. Using the conventional approach in numerical calculations, the approximation errors in the areas of the north and south poles are extremely large. Thus, in this study we propose an approach combining the precise numerical formula and tesseroid formulas, which can satisfactorily solve the calculation problem when the computation point is located in the polar areas or areas very near the surface. Furthermore, we suggest a "best matching choice" of new combination approach to calculate the GP precisely by conducting various experiments. Given the computation point at different positions, we may use different strategies. In the low latitude, we use a precise numerical formula, the fourth-order tesseroid formula, the second-order tesseroid formula, and the zero-order formula, in the 1 degrees range (from the computation point), 1 degrees to 15 degrees range, 15 degrees to 40 degrees range, and the range outside 40 degrees, respectively. The accuracy can reach 2 x 10(-5) m(2) s(-2). For the high latitude, we use the precise numerical formula, fourth-order tesseroid, second-order tesseroid, and zero-order tesseroid formulas in the ranges of 0 degrees to 1 degrees, 1 degrees to 10 degrees, 10 degrees to 30 degrees, and the zones outside 30 degrees, respectively. However, if an accuracy level of 2 x 10(-5) m(2) s(-2) is required, the zero-order tesseroid formulas should not be used and the second-order tesseroid formula should be used in the region outside 15 degrees for the low latitude and in the region outside 10 degrees for the high latitude.