18-Geom-B1 Digital Terrain Modelling · December 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Exams, December 2017 — 3 hours, closed book (one approved Casio or Sharp calculator permitted). The schedule prints TWELVE questions and states that "10 questions constitute a complete paper": Part A (Q1–Q8) is compulsory, Part B requires ONE of Q9–Q10, and Part C requires ONE of Q11–Q12, for a 100-mark paper. All twelve questions are solved below for completeness (a candidate would answer only Q1–Q8 plus one from each of Parts B and C).
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, 2014); Isaaks & Srivastava, An Introduction to Applied Geostatistics (Oxford, 1989). Canadian datums throughout (NAD83(CSRS), CGVD2013).
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. Sampled heights and three named interpolation methods (trend surface, IDW, kriging).
Find. (11.1) the interpolation process; (11.2) a table classifying the three methods by scope, model type and exactness.
11.1 The interpolation process. Surface interpolation estimates the elevation at an unsampled location from nearby measured points, on the assumption that terrain is spatially continuous and autocorrelated (near points resemble each other). The general procedure is: (1) for the target point, select a neighbourhood of surrounding samples (all points for a global method, or the nearest $n$ / those within a search radius for a local method); (2) choose a model of spatial behaviour — a mathematical trend, a distance-decay weighting, or a fitted statistical covariance; (3) compute the estimate, almost always as a weighted combination of the neighbours, $\hat z_0 = \sum_i \lambda_i z_i$, with the weights set by the chosen model; and (4) optionally assess the error (residuals at known points, or, for kriging, an estimation variance). Repeating this at every grid node produces the interpolated (gridded) surface. Methods differ in how the weights $\lambda_i$ are determined and in whether they reproduce the data exactly.
11.2 Classification.
| Method | Scope | Model type | Exactness |
|---|---|---|---|
| Trend surface | Global (one polynomial fits all data) | Deterministic (least-squares polynomial) | Inexact (smooths through, does not honour points) |
| Inverse distance weighting (IDW) | Local (search neighbourhood) | Deterministic (distance-decay weights $d^{-p}$) | Exact (weight → ∞ at a data point) |
| Kriging | Local (usually a moving neighbourhood; can be global) | Stochastic (geostatistical; fitted variogram) | Exact (reproduces sampled values; ordinary kriging) |
Trend surface is global & inexact because a single low-order polynomial is least-squares-fitted to every point and captures only the regional trend. IDW is local, deterministic & exact — weights depend only on distance and a target coincident with a sample returns that sample’s value. Kriging is the only stochastic method: it derives its weights from a fitted model of the data’s spatial correlation (the semivariogram) and, in its ordinary form, is an exact interpolator that also returns an estimation variance.
