18-Geom-B1 Digital Terrain Modelling · December 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Exams, December 2016 — 3 hours, closed book (an approved Casio or Sharp calculator permitted). TWELVE numbered questions constitute a complete paper; each carries the margin value shown and the schedule totals 100 marks. Most answers are required in essay format, so clarity and organization are graded. All twelve questions are solved below for completeness.
Reference texts: Li, Zhu & Gold, Digital Terrain Modeling — Principles and Methodology (CRC Press, 2005); Maune (ed.), Digital Elevation Model Technologies and Applications: The DEM Users Manual (2nd ed., ASPRS, 2007); Wilson & Gallant, Terrain Analysis — Principles and Applications (Wiley, 2000); Wolf, Dewitt & Wilkinson, Elements of Photogrammetry with Applications in GIS (4th ed., McGraw-Hill); Natural Resources Canada High-Resolution DEM (HRDEM) and CDEM product specifications. Canadian datums throughout (NAD83(CSRS), CGVD2013).
Note: this December 2016 paper is the same twelve-question essay set as the December 2014 and December 2015 04-Geom-B1 exams, in a slightly different order (here Q1 asks the primary use of the DTM/DEM/DSM, Q2 the five definitions, Q3 the sampling-interval drivers). The margin marks are internally consistent this year — every sub-part annotation matches its header weight and the schedule sums to 100.
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
Given. Scattered height observations to be interpolated onto unsampled locations.
Find. How Kriging performs the interpolation and why it is used.
How. Kriging is a geostatistical interpolator that treats elevation as a spatially-correlated random field. First the spatial dependence is modelled: the experimental semivariogram $\gamma(h)=\tfrac{1}{2}\,\mathrm{E}\!\left[(z(x)-z(x+h))^2\right]$ is computed from the data and a model (spherical, exponential, Gaussian) with a nugget, sill and range is fitted. The unknown height at a target point is then estimated as a weighted linear combination of the neighbouring observations, $\hat z_0=\sum_i \lambda_i z_i$, where the weights $\lambda_i$ are solved from the Kriging system so the estimator is unbiased ($\sum\lambda_i=1$) and has minimum estimation variance — i.e. it is the Best Linear Unbiased Estimator (BLUE). The same system also returns a Kriging variance at every point.
Why. Kriging is used because, unlike deterministic schemes (IDW, splines), it adapts the weights to the measured spatial structure of the terrain rather than to distance alone, it accounts for clustering/redundancy of samples (declustering), it is statistically optimal (minimum-variance, unbiased), and — uniquely — it supplies a map of estimation uncertainty that flags where more data are needed. That makes it well suited to smoothly varying surfaces and to quality-controlled DEM production, at the cost of variogram modelling effort and heavier computation.
