The Discontinuous Nature of Kriging Interpolation for Digital Terrain Modeling

Kriging is a widely employed method for interpolating and estimating elevations from digital elevation data. Its place of prominence is due to its elegant theoretical foundation and its convenient practical implementation. From an interpolation point of view, kriging is equivalent to a thin-plate spline and is one species among the many in the genus of weighted inverse distance methods, albeit with attractive properties. However, from a statistical point of view, kriging is a best linear unbiased estimator and, consequently, has a place of distinction among all spatial estimators because any other linear estimator that performs as well as kriging (in the least squares sense) must be equivalent to kriging, assuming that the parameters of the semivariogram are known. Therefore, kriging is often held to be the gold standard of digital terrain model elevation estimation. However, I prove that, when used with local support, kriging creates discontinuous digital terrain models, which is to say, surfaces with “rips” and “tears” throughout them. This result is general; it is true for ordinary kriging, kriging with a trend, and other forms. A U.S. Geological Survey (USGS) digital elevation model was analyzed to characterize the distribution of the discontinuities. I show that the magnitude of the discontinuity does not depend on surface gradient but is strongly dependent on the size of the kriging neighborhood.

Fast Algorithms Using Minimal Data Structures for Common Topological Relationships in Large, Irregularly-spaced Topographic Data Sets

Digital terrain models (DTM) typically contain large numbers of postings, from hundreds of thousands to billions. Many algorithms that run on DTMs require topological knowledge of the postings, such as finding nearest neighbors, finding the posting closest to a chosen location, etc. If the postings are arranged irregu- larly, topological information is costly to compute and to store. This paper offers a practical approach to organizing and searching irregularly-space data sets by presenting a collection of efficient algorithms (O(N),O(lgN)) that compute important topological relationships with only a simple supporting data structure. These relationships include finding the postings within a window, locating the posting nearest a point of interest, finding the neighborhood of postings nearest a point of interest, and ordering the neighborhood counter-clockwise. These algorithms depend only on two sorted arrays of two-element tuples, holding a planimetric coordinate and an integer identification number indicating which posting the coordinate belongs to. There is one array for each planimetric coordinate (eastings and northings). These two arrays cost minimal overhead to create and store but permit the data to remain arranged irregularly.

Creating Useful Products From Connecticut's 2000 LIDAR Data Set JHR 08-314 Project 07-2

The State of Connecticut owns a LIght Detection and Ranging (LIDAR) data set that was collected in 2000 as part of the State’s periodic aerial reconnaissance missions. Although collected eight years ago, these data are just now becoming ready to be made available to the public. These data constitute a massive “point cloud”, being a long list of east-north-up triplets in the State Plane Coordinate System Zone 0600 (SPCS83 0600), orthometric heights (NAVD 88) in US Survey feet. Unfortunately, point clouds have no structure or organization, and consequently they are not as useful as Triangulated Irregular Networks (TINs), digital elevation models (DEMs), contour maps, slope and aspect layers, curvature layers, among others. The goal of this project was to provide the computational infrastructure to create a first cut of these products and to serve them to the public via the World Wide Web. The products are available at http://clear.uconn.edu/data/ct_lidar/index.htm.

Validation of a Spatially Continuous EDEN Water-Surface Model for the Everglades, Florida

The Everglades Depth Estimation Network (EDEN) is an integrated network of realtime water-level monitoring, ground-elevation modeling, and water-surface modeling that provides scientists and managers with current (2000-present), online water-stage and water-depth information for the entire freshwater portion of the Greater Everglades. Continuous daily spatial interpolations of the EDEN network stage data are presented on grid with 400-square-meter spacing. EDEN offers a consistent and documented dataset that can be used by scientists and managers to: (1) guide large-scale field operations, (2) integrate hydrologic and ecological responses, and (3) support biological and ecological assessments that measure ecosystem responses to the implementation of the Comprehensive Everglades Restoration Plan (CERP) (U.S. Army Corps of Engineers, 1999). The target users are biologists and ecologists examining trophic level responses to hydrodynamic changes in the Everglades. The first objective of this report is to validate the spatially continuous EDEN water-surface model for the Everglades, Florida developed by Pearlstine et al. (2007) by using an independent field-measured data-set. The second objective is to demonstrate two applications of the EDEN water-surface model: to estimate site-specific ground elevation by using the validated EDEN water-surface model and observed water depth data; and to create water-depth hydrographs for tree islands. We found that there are no statistically significant differences between model-predicted and field-observed water-stage data in both southern Water Conservation Area (WCA) 3A and WCA 3B. Tree island elevations were derived by subtracting field water-depth measurements from the predicted EDEN water-surface. Water-depth hydrographs were then computed by subtracting tree island elevations from the EDEN water stage. Overall, the model is reliable by a root mean square error (RMSE) of 3.31 cm. By region, the RMSE is 2.49 cm and 7.77 cm in WCA 3A and 3B, respectively. This new landscape-scale hydrological model has wide applications for ongoing research and management efforts that are vital to restoration of the Florida Everglades. The accurate, high-resolution hydrological data, generated over broad spatial and temporal scales by the EDEN model, provides a previously missing key to understanding the habitat requirements and linkages among native and invasive populations, including fish, wildlife, wading birds, and plants. The EDEN model is a powerful tool that could be adapted for other ecosystem-scale restoration and management programs worldwide.