GNGTS 2024 - Atti del 42° Convegno Nazionale

Session 3.3 GNGTS 2024 al. (2008) show that "the uniqueness of the gradient estmates, [using both methods], is not a result of the analytcal propertes of the approximaton processes, it is a consequence of the method of linear least squares". We studied some possibilites to compute the gradients by OLS, using the subset of points around a grid point. A method is an "n" estmate (based on NNs) and another using an "r" estmate (using the distance within a fxed radius). Of course, the choice of radius in the r-method is quite arbitrary, whereas the n-method is uniquely defned. If the radius is too small, the subset used for the OLS estmator may be for many interpolaton points. If the radius is too large, the gradient estmate will be very poor. The best choice for the fxed radius depends on the local density of the dataset. As described in De Keyser et al. (2007), the method is only valid if the so-called spatal homogeneity conditon is fulflled. To test the gradient, we used a random dataset of 500 points in the unitary square [0,1] 2 of Franke (1979) functon. The Franke functon is a diferentable functon that is ofen used as a test functon in literature. The main problem for the optmal radius depends strictly on the local spatal density of the dataset, instead. The NN bypasses this problem, but at the same tme is not feasible if we approximate the points ξ i with the relatve NNs, since the vectorial expression becomes zero. The statstcal tests performed so far suggest that OLS gradient estmaton with a fxed radius can provide reasonable estmates. There are many ways to combine gradient estmaton and our equatons. Since the Franke functon is known, it is also possible to obtain a "semi-exact estmator" using its exact gradient. The only problem is the true locaton of points ξ i , which must be approximated by the point of interpolaton x * , the data points x i , the midpoint, or in some other way. Although many interestng questons have been raised, the investgaton is stll ongoing and requires further analysis from both theoretcal and experimental perspectves. References Belward J.A., Turner I.W., Ilić M.; 2008: On derivatve estmaton and the soluton of least squares problems . J. Comp. and Appl. Math., 222, 511–523. De Keyser J., Darrouzet F., Dunlop M.W., Décréau P.M.E.; 2007: Least-squares gradient calculaton from mult-point observatons of scalar and vector felds: methodology and applicatons with Cluster in the plasmasphere . Ann. Geophys., 25, 971–987, www.ann-geophys.net/25/971/2007/ . Franke R.; 1979: A critcal comparison of some methods for interpolaton of scatered data . Naval Postgraduate School, Monterey, CA, USA, Technical Report, NPS-53-79-003, < hdl.handle.net/ 10945/35052>. Iurcev M., Majostorovic M., Petenat F.; 2022: Introducton to Interpolaton Uncertainty of the Natural Neighbors Method (Sibson). 40° Convegno Nazionale Gruppo Nazionale di Geofsica della Terra Solida GNGTS.Online, 27-29 Giugno. 2022, Trieste. Abstract, 504-506.