Sub-block a model from solids¶

The previous tutorial kept vein proportions on 5 m blocks. Many models instead need one domain per cell, for example to estimate grade inside each vein with its own samples. With 10 m parent blocks and veins one to two meters thick, a label per parent block puts most of the vein volume in the wrong place. Sub-blocking splits the parents a vein surface crosses into smaller cells, so the labels follow the contacts. You compare the volume the parents capture and the volume sub-blocks of several sizes capture with the volume of the solids.

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, HIGHLIGHT, save

What you'll learn

How BlockModel.subblock splits parent blocks along solid contacts.
Why a total volume close to the solid's can hide volume in the wrong place, and how to measure it.
How to choose the sub-grid from the thickness of the solids.

Prerequisites: flag solid proportions in a block model and sub-blocks, which introduces subblock, from_meshes and regularize on the stacked lenses.

The data¶

The four veins of the vein-gold grade-control dataset and a parent grid of 10 m blocks around them.

data = bt.datasets.vein_gold_grade_control()
names = ["V1", "V2", "V3", "V4"]
veins = [data[f"vein_{name}"] for name in names]
parents = bt.BlockModel.from_extents(*veins, size=(10, 10, 10), buffer=10, snap=True)
solid_volume = sum(vein.volume for vein in veins)
print(parents)
print(f"veins: {solid_volume:,.0f} m3")

Out:

BlockModel(regular, 140352 of 140352 cells, count [32, 86, 51], size [10.0, 10.0, 10.0], rotation [0.0, 0.0, 0.0])
veins: 1,035,991 m3

Step 1: Sub-block against the solids

subblock takes (mesh, "inside", label) domains in priority order and n, the sub-cells per parent edge. Every parent a vein surface crosses is split into n × n × n sub-cells; each sub-cell takes the label of the first domain holding its center, and neighboring sub-cells with one label merge into larger sub-blocks. Parents no surface crosses stay whole. Listing V4 last gives the space it shares with the other veins to them, the same priority as in the previous tutorial. With n = 1 there is no splitting: each parent takes the label at its center.

Figure 1. A parent block the contact crosses is split into sub-cells (here 4 × 4); each takes the domain at its center, and sub-cells of one domain merge along a row. Parents the contact misses stay whole.

domains = [(vein, "inside", name) for vein, name in zip(veins, names)]
models = {n: parents.subblock(domains, n) for n in (1, 2, 4, 8, 16)}
for n, model in models.items():
    print(f"n = {n:<2}: {len(model):>7,} blocks, smallest {model.volumes.min():g} m3")

Out:

n = 1 :   1,046 blocks, smallest 1000 m3
n = 2 :   6,397 blocks, smallest 125 m3
n = 4 :  33,031 blocks, smallest 15.625 m3
n = 8 : 152,906 blocks, smallest 1.95312 m3
n = 16: 679,610 blocks, smallest 0.244141 m3

Step 2: Measure captured and misplaced volume

The volume labeled as a vein can match the solid while sitting in the wrong place: a block labeled V1 that is mostly outside V1 is offset by vein volume in blocks left unlabeled. The misplaced volume adds the two: labeled volume outside the vein, plus vein volume outside its labeled blocks. Mesh.proportion on the labeled blocks measures both.

def volumes(model):
    labels = model["domain"]
    captured = misplaced = 0.0
    for vein, name in zip(veins, names):
        part = model.filter(labels == name)
        inside = (vein.proportion(part) * part.volumes).sum()
        captured += part.volumes.sum()
        misplaced += part.volumes.sum() - inside + vein.volume - inside
    return captured, misplaced


print(f"{'':7}{'labeled (m3)':>14}{'vs solids':>11}{'misplaced (m3)':>16}{'share':>7}")
for n, model in models.items():
    captured, misplaced = volumes(model)
    print(
        f"n = {n:<2}{captured:14,.0f}{captured / solid_volume - 1:+11.1%}{misplaced:16,.0f}"
        f"{misplaced / solid_volume:7.0%}"
    )

Out:

         labeled (m3)  vs solids  misplaced (m3)  share
n = 1      1,046,000      +1.0%       1,625,804   157%
n = 2      1,038,125      +0.2%       1,222,097   118%
n = 4      1,033,672      -0.2%         776,380    75%
n = 8      1,033,709      -0.2%         412,907    40%
n = 16     1,033,922      -0.2%         172,732    17%

Every sub-grid labels within about 1 % of the solids' total volume, the whole parents included. The misplaced volume tells them apart. Labeled at their centers, 10 m parents misplace 157 % of the vein volume: more than the veins hold. Each halving of the sub-cell cuts the misplaced volume, to 40 % at n = 8 (1.25 m sub-cells) and 17 % at n = 16 (0.625 m). What remains is the sliver along each contact thinner than a sub-cell, and a vein 1 m thick is mostly such slivers.

Pitfall

A total that matches the solid does not show that the labels sit in the right place. Centered labels err on both sides of a contact, and the errors cancel in the total. Measure the misplaced volume, or compare a section with the solid outlines.

Step 3: Look at a section

A vertical east–west section across V1 and V4 at northing 15 000.3 m, with the vein outlines. The northing falls between sub-cell faces, so the section cuts each cell once.

plane = ((0, 15000.3, 0), 90, 90)
scheme = bt.Categories(names, colors=[ACCENT, "#6f9fc9", HIGHLIGHT, GRAY])
fig, axes = plt.subplots(1, 3, figsize=(12, 5.2), sharey=True, layout="constrained")
for ax, n in zip(axes, (1, 4, 16)):
    model = models[n]
    codes = scheme.encode(model["domain"])
    edges = {"edgecolor": "white", "linewidths": 0.3 if n < 16 else 0}
    bt.plot.section(model, codes, plane=plane, scheme=scheme, colorbar=False, ax=ax, **edges)
    bt.plot.slab(np.empty((0, 3)), plane=plane, thickness=1, meshes=veins, ax=ax)
    title = "10 m parents, label at center" if n == 1 else f"n = {n}: {10 / n:g} m sub-cells"
    ax.set(title=title, xlim=(8040, 8090), ylim=(660, 720))
    if ax is not axes[0]:
        ax.set_ylabel("")
bt.plot.category_legend(scheme, ax=axes[2], loc="lower right")
save(fig, "section")

10 m parents, label at center, n = 4: 2.5 m sub-cells, n = 16: 0.625 m sub-cells

Check before you move on

Sub-blocks must tile their parents: no parent holds more sub-block volume than its own. Each vein's labeled volume at n = 16 should match its solid to within a percent.

fine = models[16]
per_parent = np.bincount(fine.index.astype(np.int64), weights=fine.volumes)
print(f"largest volume in one parent: {per_parent.max():g} m3 of {np.prod(parents.size):g} m3")
assert per_parent.max() <= np.prod(parents.size) + 1e-6
labels = fine["domain"]
for vein, name in zip(veins, names):
    labeled = fine.volumes[labels == name].sum()
    print(
        f"{name}: sub-blocks {labeled:9,.0f} m3, solid {vein.volume:9,.0f} m3 ({labeled / vein.volume - 1:+.2%})"
    )

Out:

largest volume in one parent: 648.438 m3 of 1000 m3
V1: sub-blocks   658,560 m3, solid   658,570 m3 (-0.00%)
V2: sub-blocks   142,107 m3, solid   142,085 m3 (+0.02%)
V3: sub-blocks   179,841 m3, solid   179,815 m3 (+0.01%)
V4: sub-blocks    53,415 m3, solid    55,521 m3 (-3.79%)

The largest parent holds 648 m³ of vein sub-blocks, under its 1000 m³. V1 to V3 match their solids to within 0.02 %. V4 falls 3.79 % short, the space it gave to the veins it crosses, as in the previous tutorial.

The decision¶

Size the sub-cell from the thinnest solid. The veins are 0.7 to 2 m thick, so n = 16 gives 0.625 m sub-cells, about half the thickness of V2 and V3. It needs 679,610 blocks against 152,906 at n = 8, and it misplaces 17 % of the vein volume against 40 %. The labels then follow the contacts close enough to select samples and estimate each vein apart. For tonnage, also store each sub-block's proportion inside its vein, as in the previous tutorial, so the volume balances without the slivers.