Question 13 of 23: Variogram Trends – Tolerance, Bandwidth and Anisotropy

Question 3.3: Variogram Trends – Tolerance, Bandwidth and Anisotropy (5 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.

Computing directional experimental variograms — along strike, down dip, and across a deposit — and comparing how quickly each direction's γ(h) rises reveals TRENDS in continuity: a direction whose variogram rises slowly (long range, or low sill relative to others) is a direction of greater geological continuity (e.g. along a vein's strike or a stratigraphic horizon), while a fast-rising, short-range direction cuts across continuity (e.g. across a vein's true width). Mapping range as a function of direction (a directional variogram “rose”) directly recovers the deposit's own preferred orientation without relying solely on geological interpretation.

3.3.1 Tolerance (included angle). Because real samples rarely lie exactly along the nominal search direction, an angular tolerance (e.g. ±22.5°) is allowed around the target azimuth/dip so that enough pairs fall inside the search cone to compute a stable experimental point; too tight a tolerance starves the calculation of pairs, too loose blurs directions together.

3.3.2 Band width. A maximum perpendicular (off-axis) distance capping how far a candidate pair may stray from the search line even while satisfying the angular tolerance — without it, the angular cone widens without limit at long lag distances and starts admitting pairs that are geologically irrelevant to the intended direction.

3.3.3 Anisotropy. The property of a variogram whose range and/or sill genuinely differ with direction, reflecting real directional continuity in the ore body (e.g. longer range along strike than across true width); GEOMETRIC anisotropy (same sill, different range by direction, related by a simple axis-scaling/rotation) is the most common and easiest to correct for in kriging by rescaling distance directionally before applying an isotropic model.

3.3.4 Azimuth 90/dip 0 vs. azimuth 270/dip 0. These two horizontal directions point exactly opposite along the SAME physical line (090° and 270° are 180° apart, i.e. collinear, both dip 0°); a variogram computed along a straight line is inherently symmetric about the origin (pairs are counted regardless of which sample is treated as “first”), so the two headings sample the identical set of point pairs and must give an identical γ(h).

3.3.5 Azimuth 90/dip 45 vs. azimuth 270/dip 45. With a non-zero dip, azimuth 90°/dip 45° and azimuth 270°/dip 45° are NOT collinear opposites — one plunges down-and-toward-090° while the other plunges down-and-toward-270°, tracing two different lines through 3-D space (mirror images through the vertical, not the same line). If the deposit itself has an inclined structural fabric (e.g. a dipping vein or a plunging fold axis), continuity down-dip in one compass sense can genuinely differ from continuity down-dip in the opposite compass sense, so the two variograms need not match — only a horizontal (dip 0) pair is guaranteed to be identical by the simple collinearity argument used in 3.3.4.