data spacing¶

the equivalent data spacing (cabral pinto and deutsch, 2017) is the spacing of a square grid of holes that would put as much data around a location as the real drilling does. it reads in meters whatever the layout: a cell inside a square grid of 700 m scores about 700 m, and a cell among irregular holes scores the square grid they are worth. it is the usual basis for resource classification (classification).

import sys
from pathlib import Path

HERE = Path(__file__).parent if "__file__" in globals() else Path.cwd()
sys.path.insert(0, str(HERE.parents[1]))

import boitata as bt
import matplotlib.pyplot as plt
import numpy as np
from common import ACCENT, GRAY, INK, map_axes, save

data = bt.datasets.coal_seam_thickness()
holes, lease, grid = data["boreholes"], data["boundary"], data["grid"]
print(f"{len(holes)} holes, {grid['INSIDE'].sum():.0f} cells of 100 m inside the lease")

Out:

295 holes, 7162 cells of 100 m inside the lease

in plan¶

a coal seam drilled on a regular mesh, then infilled where the seam is thick. with search=None, data_spacing takes the plan distances d from a target to its nearest holes. for each n it averages the n-th and n + 1-th, r = (d_n + d_n+1) / 2, and turns the circle of radius r, which holds n holes, into the area per hole: sqrt(pi r² / n). the result is the mean over n from 4 to 10. hull=True leaves cells outside the convex hull of the holes blank, where any spacing would be an extrapolation.

inside = grid["INSIDE"] == 1
spacing = bt.data_spacing(grid, holes, None, hull=True)
shown = np.where(inside, spacing, np.nan)
p10, p50, p90 = np.nanpercentile(shown, [10, 50, 90])
print(f"spacing in the lease: median {p50:.0f} m, P10 {p10:.0f} m, P90 {p90:.0f} m")
print(f"{np.mean(np.isnan(spacing[inside])):.1%} of the lease lies outside the hull of the holes")
fig, (a, b) = plt.subplots(1, 2, figsize=(11, 4), layout="constrained", width_ratios=[1.5, 1])
bt.plot.section(grid, shown, axis="z", index=0, colorbar=False, ax=a)
a.plot(*lease.coords[:, :2].T, color=GRAY, lw=0.8)
a.scatter(*holes.coords[:, :2].T, s=2, color=INK)
fig.colorbar(a.collections[0], ax=a, shrink=0.8, label="Spacing (m)")
map_axes(a, "Equivalent spacing in plan")
bt.plot.histogram(shown[~np.isnan(shown)], bins=np.arange(0, 1100, 50), stats=True, ax=b, color=ACCENT)
b.set(title="Cells of the lease", xlabel="Spacing (m)")
save(fig, "plan")

Equivalent spacing in plan, Cells of the lease

Out:

spacing in the lease: median 535 m, P10 391 m, P90 655 m
9.5% of the lease lies outside the hull of the holes

the infill clusters read 200 to 400 m and the regional mesh 550 to 700 m. the measure averages over the 4 to 11 nearest holes, so it changes smoothly across the edge of the infill rather than jumping at it.

in 3D¶

an iron formation cut by 187 diamond holes, about 100 m apart. with a search, the spacing comes from the samples inside its ellipsoid: sqrt(V / (c n)), with V the volume of the ellipsoid, n the samples inside it around the block and c their length. n c is the length of hole inside V, so V / (n c) is the plan area per hole for vertical holes. a Drillholes stands for its interval midpoints, and c defaults to the median interval length; for composites in a PointSet, pass composite_length=. the ellipsoid here reaches 150 m in plan and 30 m vertically, so it follows the flat formation.

plateau = bt.datasets.iron_formation_plateau()
drillholes = bt.Drillholes.from_tables(plateau)
model = plateau["block_model"]
formation = model.filter(np.isin(model["LITH"], ["IC", "HC", "HF", "IF"]))
search = bt.Search(150.0, ratios=(1.0, 0.2))
volume = bt.data_spacing(formation, drillholes, search, hull=True)
plan = bt.data_spacing(formation, drillholes, None, holes="HOLE_ID", hull=True)
for name, values in (("3D", volume), ("plan", plan)):
    p10, p50, p90 = np.nanpercentile(values, [10, 50, 90])
    print(
        f"{name:>4}: median {p50:.0f} m, P10 {p10:.0f} m, P90 {p90:.0f} m, blank {np.mean(np.isnan(values)):.1%}"
    )

Out:

  3D: median 111 m, P10 93 m, P90 153 m, blank 3.1%
plan: median 101 m, P10 90 m, P90 112 m, blank 0.7%

the plan form also works on drill holes: holes= counts each hole once, at its nearest sample, so a hole with a hundred samples down its length is one hole, and every block in a column gets the same value. the 3D form sees depth. both agree near 100 m where the holes cross the formation, but blocks below the ends of the holes have fewer samples around them and read wider, up to blank where no sample is in reach. when most blocks have fewer than 4 samples inside the ellipsoid, data_spacing warns: counts that small are unstable, and the ellipsoid should grow.

fig, axes = plt.subplots(1, 2, figsize=(11, 3), layout="constrained", sharey=True)
for ax, values, title in (
    (axes[0], plan, "Plan, holes counted once"),
    (axes[1], volume, "3D, ellipsoid 150 × 30 m"),
):
    bt.plot.section(
        formation, values, axis="y", index=40, colorbar=False, vmin=50, vmax=250, ax=ax, holes=drillholes
    )
    ax.set(title=title, xlabel="Easting (m)")
axes[1].set_ylabel("")
fig.colorbar(axes[1].collections[0], ax=axes, shrink=0.9, extend="max", label="Spacing (m)")
save(fig, "section")