Multigrid/Multiresolution Interpolation: Reducing Oversmoothing and Other Sampling Effects

Abstract

Traditional interpolation methods, such as IDW, kriging, radial basis functions, and regularized splines, are commonly used to generate digital elevation models (DEM). All of these methods have strong statistical and analytical foundations (such as the assumption of randomly distributed data points from a gaussian correlated stochastic surface); however, when data are acquired non-homogeneously (e.g., along transects) all of them show over/under-smoothing of the interpolated surface depending on local point density. As a result, actual information is lost in high point density areas (caused by over-smoothing) or artifacts appear around uneven density areas (“pimple” or “transect” effects). In this paper, we introduce a simple but robust multigrid/multiresolution interpolation (MMI) method which adapts to the spatial resolution available, being an exact interpolator where data exist and a smoothing generalizer where data are missing, but always fulfilling the statistical requirement that surface height mathematical expectation at the proper working resolution equals the mean height of the data at that same scale. The MMI is efficient enough to use K-fold cross-validation to estimate local errors. We also introduce a fractal extrapolation that simulates the elevation in data-depleted areas (rendering a visually realistic surface and also realistic error estimations). In this work, MMI is applied to reconstruct a real DEM, thus testing its accuracy and local error estimation capabilities under different sampling strategies (random points and transects). It is also applied to compute the bathymetry of Gulf of San Jorge (Argentina) from multisource data of different origins and sampling qualities. The results show visually realistic surfaces with estimated local validation errors that are within the bounds of direct DEM comparison, in the case of the simulation, and within the 10% of the bathymetric surface typical deviation in the real calculation.

Using Landsat Image Series to Identify and Characterize Persistent Oceanographic Structures in a Dynamic Marine Protected Area (North of San Jorge Gulf, Argentinian Patagonia)

“…The study area corresponds to the northern coastal zone of San Jorge Gulf, the largest gulf of the Argentinian Patagonian shelf. This gulf is a semi-open basin with an extension of 39,340 km 2 , located between 45 • S (Dos Bahías Cape) and 47 • S (Tres Puntas Cape) (Figure 1) and with a mean depth of 81 m with a maximum depth of around 100 m, mainly covered by silt with coarse granulometric fractions in the north and south ends of the gulf [52,53]. Circulation within the gulf is seasonally variable, in summer a cyclonic gyre develops in the south of the gulf and a weak anticyclonic gyre in the north; in winter a large anticyclonic gyre is present in the southwestern sect.…”