The change of curvature as an invariant measure for studying height deformation in geodetic control networks

One of the existing problems in geodetic control networks for computing displacement vectors is finding a fixed station which has no deformation within the observation interval. This problem comes from the datum-dependency characteristic of the displacement vector. Accordingly, extraction of invariant parameters of the deformation tensor in geodetic control networks which are independent from datum has an effective role in the accurate interpretation of results in geodynamic studies. In height networks, change of the curvature tensor and its associated invariants have been introduced for assessing vertical deformation. The summation and difference of elongation of change of the curvature tensor are two common invariants of this tensor. Additionally, changes of the Gaussian and mean curvature parameters are two other key invariants with physical interpretations that are used to describe deformation behavior. Although many methods have been proposed to calculate deformation tensor fields on the earth’s surface, few refer to the actual surface of the Earth. Most of these methods formulate the problem on reference surfaces such as projection planes or spheres and, consequently, their results suffer from possible effects of inaccuracy and incompleteness of the mathematical models of projections. In the present study, we used a method of differential geometry that allows deformation analysis of the actual surface of the Earth for a more reliable and accurate estimate of the surface deformation measures. The method takes advantage of the simplicity of 2-dimensional spaces versus 3-dimensional spaces without losing or neglecting information and effect of the third dimension in the final results. Khorasan is a large province in the northeast of Iran where over-extraction of water resources for industrial and agricultural purposes has caused an extensive subsidence in some cities such as Mashhad, Neyshabour and Kashmar. Since the employment of first-order precise leveling network of Iran, which has been utilized twice for over two centuries in these cities, precise leveling has been one of the observations methods for measuring the subsidence. On the other hand, GPS and InSAR observations in these areas revealed the extent and magnitude of the subsidence. In this paper, we estimated the invariant parameters of the tensor of curvature using precise leveling observation to study the height deformation behavior of the province of Khorasan in these subsidence areas. We computed the invariant parameters of change of curvature tensor relative to two distinct datums to evaluate their datum-independency characteristic as well as studying subsidence behavior. Obtained results relative to 2 datums show the unique value and pattern for each parameter and reveal the subsidence area. The computed maximum mean curvature change was approximately, and the maximum Gaussian curvature change was approximately, which confirms the high rate of subsidence in this area. This application reveals the capabilities and strengths of the proposed method and suggests the superseding of these invariant parameters with other geodetic network adjustment results, especially in areas in which a fixed datum is undefined.