Note

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downhole nugget

along a hole, samples sit one sample length apart, far closer than any two holes, so the downhole variogram shows the nugget best. the data: Ni of a nickel laterite, sampled at 1 m down 448 vertical holes, horizon by horizon.

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
from common import ACCENT, GRAY, HIGHLIGHT, save

tables = bt.datasets.nickel_laterite_profile()
intervals = bt.merge_intervals(tables["assays"], tables["horizons"])
drillholes = bt.Drillholes(tables["collars"], tables["surveys"], intervals)
runs = drillholes.composite(None, ["NI_PCT"], domain="HORIZON")
horizon, thickness = runs["HORIZON"], runs["length"]
for name in ["FERR", "LIM", "SAP", "BRK"]:
    print(f"{name:5} mean thickness {thickness[horizon == name].mean():5.1f} m")

Out:

FERR  mean thickness   2.3 m
LIM   mean thickness   7.4 m
SAP   mean thickness  10.9 m
BRK   mean thickness   3.5 m

a downhole variogram needs several lags inside one horizon, and only the limonite LIM and the saprolite SAP are thick enough. holes= keeps only pairs down the same hole; lag k gathers pairs about k composite lengths apart. nugget() extrapolates a line through the first three lags to zero. composites never cross a horizon, and each horizon gets its own variogram, standardized by its variance, at three composite lengths.

horizons, lengths = ["LIM", "SAP"], [1.0, 1.5, 2.0]
nuggets, downhole = {}, {}
for length in lengths:
    composites = drillholes.composite(length, ["NI_PCT"], domain="HORIZON", residual="merge")
    for name in horizons:
        sub = composites.filter(composites["HORIZON"] == name).drop_null("NI_PCT")
        exp = bt.experimental_variogram(sub, "NI_PCT", length, 6 * length, standardize=True, holes="HOLE_ID")
        nuggets[name, length], downhole[name, length] = exp.nugget(), exp
print("nugget / variance")
print("      " + "".join(f"{h:>7}" for h in horizons))
for length in lengths:
    print(f"{length:3.1f} m " + "".join(f"{nuggets[h, length]:7.2f}" for h in horizons))

Out:

nugget / variance
          LIM    SAP
1.0 m    0.20   0.14
1.5 m    0.19   0.15
2.0 m    0.35   0.20

at 1 m the limonite carries a fifth of its Ni variance in the nugget, the saprolite about a seventh; 1.5 m composites give nearly the same. at 2 m the first three lags span 6 m, most of the limonite's 7.4 m. the line then reaches zero across the bend of the variogram, and the nugget comes out larger in both horizons. carry the nugget of the shortest composites into the fits between holes, whose closest spacing is 25 m.

fig, (a, b) = plt.subplots(1, 2, figsize=(9.2, 3.4), layout="constrained")
for name, color in (("LIM", ACCENT), ("SAP", GRAY)):
    exp = downhole[name, 1.0]
    a.plot(exp.lags, exp.gammas, "o-", color=color, ms=3, label=name)
    a.plot([0, exp.lags[0]], [exp.nugget(), exp.gammas[0]], ":", color=color)
    a.plot(0, exp.nugget(), "s", color=HIGHLIGHT, ms=5, clip_on=False)
a.set(xlim=(0, 6), ylim=(0, None), xlabel="Downhole lag (m)", ylabel="standardized γ(h)")
a.set_title("Downhole Ni variograms, 1 m composites")
a.legend(loc="lower right")
for name, color in zip(horizons, (ACCENT, GRAY)):
    b.plot(lengths, [nuggets[name, length] for length in lengths], "o-", color=color, ms=4, label=name)
b.set(xticks=lengths, ylim=(0, None), xlabel="Composite length (m)", ylabel="nugget / variance")
b.set_title("Extrapolated nugget by horizon")
b.legend(loc="lower right", ncol=2)
save(fig, "downhole")

Total running time of the script: ( 0 minutes 0.179 seconds)

Download Python source code: example_05_05.py

Download Jupyter notebook: example_05_05.ipynb

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