Smooth Surface Approximation for Computing Pit Excavation Volume

Abstract

In this paper, a method of volume computation that combines the advantages and avoids the drawbacks of current methods is presented. Earlier methods of computing pit excavation volume were based on the trapezoidal or Simpson-type rules. These methods provide direct formulas for volume computation, but their disadvantage is that they approximate the ground surface by piecewise surfaces that have sharp connections. The existing cubic spline method is based on a smooth ground surface, but its disadvantage is that it requires solving a system of linear equations and integrating a function many times. The proposed method provides a direct formula for volume computation and is based on a smooth approximating surface. The method employs a cubic Hermite polynomial for individual intervals in both directions of the grid, the intervals of which may be unequal. A cubic Hermite volume formula is developed by integrating the surface function in both coordinate directions. The proposed method, which provided comparable accuracy to that of the more elaborate cubic spline method, should be of interest to surveyors concerned with both accuracy and ease of volume computations.