GNGTS 2024 - Atti del 42° Convegno Nazionale

Session 3.3 GNGTS 2024 NUMERICAL PROBLEMS OF THE UNCERTAINTY ASSESSMENT OF THE SIBSON INTERPOLATION METHOD M. Iurcev 1 , F. Petenat 1 1 Isttuto di Oceanografa e di Geofsica Sperimentale, OGS, Trieste Introducton In 2022 we presented a note (Iurcev et al. 2022) on the assessment of uncertaintes for the Natural Neighbours – hereafer NN - interpolaton method (Sibson 1980, 1981) with bidimensional scalar data. This non-parametric interpolaton method uniquely determines the interpolated data and is therefore classifed as a deterministc method. However, it is important to quantfy the uncertaintes due to the spatal distributon of the dataset to be interpolated. The implementaton of this approach is now reported in Iurcev et al. 2023. This approach is based on a gradient method derived from the bivariate version of the Mean Value Theorem MVT (also known as the Lagrange Theorem) in ℝ 2 , combined with Sibson’s formula for interpolaton (Iurcev et al. 2023). This deterministc method, based on the MVT, raises two major issues. The frst problem concerns gradient estmaton. The second issue is the unknown locaton of the points of MVT, along the line between the interpolaton point and the i-th Natural Neighbour. The purpose of this note is to show how we have tried to solve these two issues. Gradient estmaton In Iurcev et al. 2022, we presented an approach that is widely used in the literature. The approximaton of the gradient using fnite diferences superimposed on a regular grid in which the functon value is known or estmated. However, this method introduces an additonal level of uncertainty as the functon must be interpolated through the grid. The proposed approach is the Local Least Squares plane approximaton of the unknown surface. The OLS (Ordinary Least Squares) approximaton requires a subset of points x i , f( x i ) in the neighbourhood. If there are at least three non-colinear points in ℝ 3 space, the linear regression defnes a plane whose slope is a possible gradient estmator. In this context, two diferent least squares strategies for computng the gradient for bivariate interpolaton of surfaces are investgated by Belward et al. (2008). The two methods are based on the generalizaton of Moving Least Squares (MLS). The frst method is the classical method based on a linear system of equatons in which the gradient is derived by a second order truncated Taylor expansion. In the second method, the gradient is a consequence of the Finite Volume Method (FVM) soluton which is used to solve a difusion equaton. Belward et ξ i