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. Irregularly distributed elevation observations to be converted to a regular grid.
Find. (8.1) a definition of gridding; (8.2) the moving-average gridding method.
8.1 Gridding. Gridding is the process of interpolating irregularly (or differently) spaced elevation data onto the regular nodes of a grid — converting scattered mass points, contour strings or an arbitrary point cloud into a raster DEM with uniform spacing. For each grid node whose height is unknown, an interpolation rule estimates $Z$ from the surrounding observations. Gridding is needed because most acquisition methods produce irregular samples, whereas grid storage and raster analysis (slope, hydrology, map algebra, differencing) require regularly posted heights. The choice of interpolation method (nearest neighbour, inverse-distance weighting, moving average, splines, kriging), the search neighbourhood and the grid spacing together govern the accuracy and smoothness of the resulting surface.
8.2 Moving-average gridding. Moving-average gridding assigns each grid node the weighted average of the observations that fall within a search window (a circle or ellipse) centred on that node. As the window “moves” from node to node across the grid, the estimate is $\hat z_0 = \dfrac{\sum_i w_i\,z_i}{\sum_i w_i}$, where the weights $w_i$ are commonly inverse functions of distance (e.g. $w_i = 1/d_i^{\,p}$, giving inverse-distance weighting as a special case) or simply equal (a plain local mean). Nodes with no data in the window are left blank or the radius is enlarged. It is fast, simple and robust to noise (the averaging smooths random error), which makes it a common default; its drawbacks are that it is a smoothing (inexact) estimator that does not honour the data exactly, it flattens peaks and pits and cannot extrapolate beyond the data range, and results depend on the window size — too small leaves gaps, too large over-smooths real relief.
