Volumes with Rasters

How Much Rock Is in a Mountain? — and Lab 9

CE 414 Engineering Applications of GIS
Civil & Construction Engineering
Brigham Young University
Dr. Dan Ames

CE 414 · Week 10 — Volumes with RastersLast Updated: 2026-10-09© 2026 Daniel P. Ames · CC BY 4.0

Today's Goals

By the end of class you should be able to:

  • Write any raster volume as height × cell area, summed
  • Compute it in ArcGIS Pro two ways: Raster Calculator + Zonal Statistics, and Surface Volume
  • Explain why the base surface is the biggest choice in the answer
  • Say what Lab 9 asks for, and how you will know you are right
CE 414 · Week 10 — Volumes with Rasters

Part 1 — A Volume Is a Sum

CE 414 · Week 10 — Volumes with Rasters

Height × Area, Summed

  • Every cell: height above the base × cell area
  • 10 m cells: each meter of height is 100 m³
  • Sum the cells: m³; ÷ 1,000³ for km³
  • Lake (Week 7), pad (Tuesday), mountain (today): the same sum
CE 414 · Week 10 — Volumes with Rasters

In ArcGIS Pro, Two Ways

  • Raster Calculator for height, × 100 ÷ 1,000³ per cell, then Zonal Statistics, statistic Sum: any base, even a curved one (Lab 9)
  • Surface Volume (3D Analyst): volume above or below a flat plane, written to a text file
  • Above 1,572.2 m, the butte holds 4.99 km³ over 26.2 km²
CE 414 · Week 10 — Volumes with Rasters

Part 2 — The Base Is the Answer

CE 414 · Week 10 — Volumes with Rasters

Same DEM, Four Bases

  • Three flat bases from the outline's edge: 5.46, 4.98, 4.51 km³
  • Lab 9's interpolated plain: 5.14 km³
  • The base alone moves the answer by about a fifth
CE 414 · Week 10 — Volumes with Rasters

Why a Flat Base Is Wrong Here

  • The plain is not level: it falls toward the north by tens of meters across the butte
  • A flat base buries the butte's low side and lifts its high side
  • An interpolated base follows the plain, from points sampled all around
CE 414 · Week 10 — Volumes with Rasters

Where Does the Butte End?

  • The outline is a choice: Lab 9's was drawn by a rule from the DEM
  • Lava flows of the plain may lap against the dome: that part is buried
  • A volume above an interpolated surface is what the DEM can see
CE 414 · Week 10 — Volumes with Rasters

Part 3 — Lab 9, Big Southern Butte

CE 414 · Week 10 — Volumes with Rasters

Lab 9 at a Glance

  • The data: a 10 m DEM of the butte and a reference outline, prepared for you
  • The model: random points on the plain → IDW → height above the plain → Zonal Statistics Sum
  • Then: four runs that change the points, the outline, and the method

Lab 9 — Big Southern Butte

CE 414 · Week 10 — Volumes with Rasters

How You Will Know You Are Right

  • Volume: 5.145 km³
  • Tallest cell: 729.1 m above the plain; mean 183.5 m
  • Sanity check: 28.03 km² × 183.5 m ≈ 5.14 km³
CE 414 · Week 10 — Volumes with Rasters

Final Thoughts

Quiz: The Base Is the Answer. Five questions on raster volumes, units, Surface Volume and choosing a base. Scan the code, or open the link under it.

  • Not graded, nothing recorded: a quick check that today's material made sense
  • Note the units question; units are a common mistake in Lab 9

Before next class

  • Lab 9 — Big Southern Butte — due Saturday 11:59 pm
  • Quiz 8 (Volumes and Spatial Analysis, Chapter 9) — due Saturday 11:59 pm
  • Next week: coordinate systems and GPS refreshers, and Lab 10 — Avalanche Hazard
  • Office hours: calendly.com/dan-ames/office-hours

center

byu-hydroinformatics.github.io/ce414-gis-applications/quizzes/volumes/

CE 414 · Week 10 — Volumes with Rasters

Thursday of Week 10. Tuesday's cut and fill was a difference of two surfaces that both exist: the ground and a design. Today the bottom surface is one nobody can see: the plain under Big Southern Butte, which Lab 9 rebuilds by interpolation (Week 9's methods) before it measures the dome. The title image is a profile through the summit from the Lab 9 DEM.

stamp:begin

stamp:end

The only things that change between Week 7's lake, Tuesday's pad and Lab 9's butte are the two surfaces: lid and bottom. For a lake the lid is the water level and the bottom the lake bed; for the pad, ground and design; for the butte, the DEM and a rebuilt plain. Units again: the DEM must be projected so the cell size is in meters; a geographic DEM's "cell size" is in degrees.

Surface Volume run in ArcGIS Pro 3.7.1's arcpy on the Lab 9 DEM (projected to UTM 12N, 10 m) clipped to the reference outline: plane 1,572.2 m (the mean of the outline's edge cells), ABOVE, 4.988 km³, 2-D area above the plane 26.17 km². The same number came from summing the cells directly. It only takes a flat plane, which is why Lab 9 uses the Raster Calculator route. The figure is the finished Lab 9 model.

Flat bases: the lowest edge cell (1,554.8 m), the mean of the edge (1,572.2 m) and the highest edge cell (1,588.9 m), each summed signed over the 280,311 cells of the outline (tools/week10_volume_figures.py). The interpolated value is Lab 9's published Step 8 check (5.145 km³); none of Lab 9's Step 9 sensitivity results are shown here. Ask which one is right before showing the next slide.

Read the profile: the plain at the north end sits well below the plain at the south end. A single elevation cannot fit both. Lab 9 samples random points on a ring of plain around the outline and interpolates across the gap, which is the Week 9 lesson put to work.

The example map is the Lab 9 page's baseline sheet (height above the rebuilt plain). The reference outline is every cell more than 10 m above a plane fitted to the plain (READ-ME-FIRST.txt in the Lab 9 package). Lab 9 Step 9 has each student digitize their own outline; the answers are theirs to find.

Lab 9 is due Saturday of this week. Remind them: the Random Number Generator seed in Step 0 is what makes their baseline match the page, and a run from the tool dialog deletes everything that is not a model parameter.

All three are TIP check values on the Lab 9 page (Steps 7 and 8). The sanity check is the one to teach: area times mean height must equal the sum, or the units are wrong somewhere.

Four minutes, in pairs. If the room has no signal, put the URL on the board.

Deck written October 9, 2026. Figures: tools/week10_volume_figures.py; the model diagram and example map are copied from the Lab 9 page.