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18-Geom-B1 Digital Terrain Modelling · December 2014

Question 11 of 12: DEMs for Orthophotos and Volume Computation

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: National Exams, December 2014 — 3 hours, closed book (any non-communicating calculator permitted). TWELVE numbered questions constitute a complete paper; each is of varying value 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).

Check / source note. The Question 1 header carries 9 marks but its annotation reads “(3 × 2 marks)” = 6. The header value is authoritative because it is the figure that makes the printed schedule sum to the stated 100 Total marks; the “3 × 2” is a typographic error for 3 × 3. Answers are graded on all three parts of Q1 equally.

Question 11: DEMs for Orthophotos and Volume Computation (10 marks)

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. A DEM together with (a) an aerial/satellite image and its exterior orientation, and (b) two surfaces whose enclosed volume is wanted.

Find. The role of the DEM in each task.

(a) Orthophoto generation. A raw perspective photograph has relief displacement: points off the datum are shifted radially by their height, so scale varies across the image. Orthorectification removes this using the DEM: the process is differential rectification. For every output orthophoto pixel of known ground position $(X,Y)$, the DEM supplies the ground height $Z$; the collinearity equations, with the camera's interior and exterior orientation, then project that 3-D ground point $(X,Y,Z)$ back into the original image to find the pixel that images it, and its grey value is resampled into the orthophoto. Doing this cell-by-cell converts the perspective image into an orthographic, map-accurate image at uniform scale. The DEM's accuracy directly limits the horizontal accuracy of the orthophoto (a height error $\Delta Z$ produces a horizontal shift $\approx \Delta Z\cdot r/H$, the radial-distance-over-flying-height ratio).

(b) Volume computation. Volumes are computed as the space between two surfaces—the DEM and a reference (a design grade, a datum plane, or an earlier DEM). Grid method: at each cell of area $A$ take the height difference $\Delta z$ between surfaces and sum, $V=\sum A\,\Delta z$ (rectangular/average-end-area prisms). TIN method: for each triangular prism between the two triangulated surfaces, $V=\bar{h}\cdot A_{\triangle}$ with the mean of the three corner differences, summed over all facets. Comparing a “before” and “after” DEM gives cut ($\Delta z<0$) and fill ($\Delta z>0$) volumes for earthworks, stockpiles or reservoir capacity. Finer sampling and honoured breaklines improve the estimate; the prismoidal (Simpson) rule refines it where curvature matters.