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. Scattered height observations of a spatially correlated surface.
Find. What kriging is, the main parameters it estimates, and how they are used.
What kriging is. Kriging is a geostatistical interpolation method that treats elevation as a realization of a spatially correlated random field and estimates the value at an unsampled point as a weighted linear combination of neighbouring observations, $\hat z_0 = \sum_i \lambda_i z_i$, in which the weights are chosen to make the estimator unbiased ($\sum_i \lambda_i = 1$) and of minimum estimation variance. It is therefore the Best Linear Unbiased Estimator (BLUE). Unlike deterministic schemes (IDW, splines) whose weights depend on distance alone, kriging derives its weights from the measured spatial correlation structure of the data, so it adapts to how quickly the terrain de-correlates with distance and accounts for the clustering (redundancy) of samples.
Main parameters estimated. The spatial correlation is captured by the semivariogram $\gamma(h) = \tfrac{1}{2}\,\mathrm{E}\!\left[(z(x) - z(x+h))^2\right]$, computed from the data and fitted with a model (spherical, exponential, Gaussian). Its three parameters are: the nugget $c_0$ — the variance at zero separation, representing measurement error and micro-scale variability; the sill $c_0 + c$ — the plateau variance reached where pairs become spatially independent (the overall data variance); and the range $a$ — the separation distance at which the sill is reached, i.e. the distance beyond which points are no longer correlated.
How they are used. The fitted variogram supplies the covariances that build the kriging system: $\sum_j \lambda_j\,\gamma(x_i,x_j) + \mu = \gamma(x_i,x_0)$ together with $\sum_j \lambda_j = 1$ (with a Lagrange multiplier $\mu$ enforcing unbiasedness). Solving it gives the weights $\lambda_i$ that minimize the estimation variance; substituting into $\hat z_0 = \sum_i \lambda_i z_i$ produces the interpolated height, and the same system returns the kriging variance $\sigma_K^2$ at every point — a map of estimation uncertainty. Thus the range controls how far the influence of a sample extends, the sill scales the variance, and the nugget determines how much the estimate is smoothed toward the mean near the data. This is why kriging is preferred where a statistically optimal surface and a rigorous uncertainty map are needed, at the cost of the variogram-modelling effort.
