Question 4 of 4: Three-Point Strike and Dip Problem

Question D: Three-Point Strike and Dip Problem (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. Three coal-seam outcrops on a topographic map, scale 1 cm = 100 m: A at elevation 500 m, B at 300 m, C at 400 m (map positions read from the printed map; see table below).

Find. (a) strike and dip of the (assumed planar) coal seam. (b) the outcrop pattern the seam traces across this topography.

Approach. Fit a plane $z=aX+bY+c$ through the three digitized points; the plane's dip is the magnitude of its steepest gradient and the dip direction is the azimuth of that gradient (down-slope); strike is perpendicular to dip direction. Cross-check by the classical structure-contour method: interpolate the point on chord A–B at C's own elevation (400 m) and draw the line from that point to C — this line is a structure contour of the seam and must trend parallel to strike.

1. Plane fit. Solving $z=aX+bY+c$ through A, B, C gives $a=-0.394$, $b=-0.410$ (both per metre) — i.e. elevation decreases toward increasing East and increasing North, so the seam dips toward the NE.
2. Dip magnitude and direction. $\text{dip}=\arctan\sqrt{a^2+b^2}=\arctan(0.569)=\boxed{29.6^{\circ}\approx30^{\circ}}$, in the down-gradient direction $\text{dip dir.}=\operatorname{atan2}(-a,-b)=\boxed{044^{\circ}\ (\text{N44}^{\circ}\text{E})}$.
3. Strike. Perpendicular to dip direction: $\boxed{\text{N46}^{\circ}\text{W (134}^{\circ}/314^{\circ}\text{)}}$.
4. Cross-check (structure-contour method). Interpolating chord A(500)–B(300) at $z=400$ (the same elevation as C) gives a point 50% of the way from A to B; the line from that point to C is, by construction, a 400 m structure contour of the seam — its trend comes out at N46°W, matching the plane-fit strike exactly (independent confirmation the digitized point positions are self-consistent).

(b) Outcrop pattern. With strike and dip fixed, the seam's own structure contours are a family of parallel straight lines trending N46°W, each 100 m higher than its NE neighbour and spaced $100/\tan(29.6^{\circ})\approx176\text{ m}$ apart measured perpendicular to strike (1.76 cm on the map). The seam's outcrop trace is found by overlaying this structure-contour family on the topographic map and marking every point where a structure contour of value $V$ crosses a topographic contour of the same value $V$ — A, B and C are three such points by construction. Because the seam dips NE while the mapped topography rises to a hill (peak >500 m around A) in the NW and falls toward a valley in the SE, the Rule of V's applies: the outcrop trace will V in the dip direction (pointing NE) where it crosses the valley/drainage low south-east of A–B–C, and V against the dip direction (pointing SW, i.e. up-dip) where it wraps around the western hill's nose — an asymmetric, NW–SE-trending sinuous band, narrower where the ground is steep and wider where it is gentle.