Thursday of Week 9. Tuesday's interpolation lecture built the surfaces: Thiessen, IDW, splines, kriging. Today is the question it ended on: how would you know which one to trust? First with the truth in hand, then without it. Then Lab 8, which does exactly this on Y Mountain. The title image is the Lab 8 example baseline sheet: the true DEM and three rebuilds from the same 2,500 points, with their error maps underneath.

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The profile on the right is one row of cells across the Lab 8 study area: the truth and the three rebuilds from 250 points. Thiessen steps, IDW sags, kriging smooths the peak off. Every idea today is a way of putting a number on the gaps between those lines.

The capture is the Lab 8 Step 6 Raster Calculator, inside ModelBuilder: "%True_DEM%" - "%Thiessen_Surface%". The order matters only for the sign, but say it out loud and keep it fixed: if one student subtracts the other way, every color on their map flips and their "too high" ridges become "too low". Lab 8 uses truth minus surface throughout.

The three error maps from the Lab 8 example baseline (2,500 points, default parameters). Red is where the surface is too high, blue too low. Ask the room before saying it: why is the valley floor nearly white? Because it is flat: any method that returns a nearby sample's value is close when the ground does not change. The error is a map of where the ground changes faster than the samples are spaced. Lab 8 asks each student to find their best method's single worst cell and explain the ground there.

z is the truth, z-hat the surface, n the number of cells. Squaring does two things: it stops positive and negative errors canceling, and it weights big errors heavily, so a surface with a few 100 m misses scores worse than one with many 5 m misses. The capture is Lab 8's second Raster Calculator, Square. Common slip: the root of the mean, not the mean of the roots, and not the root of the sum.

The table is Lab 8 Step 7's output for one method: COUNT 67,337 cells and AREA 60,603,300 square meters, both on the lab page as check values. If COUNT comes out smaller, the surfaces did not cover the whole study area (the Extent environment is the usual cause), and the RMSE is quietly computed on fewer cells. Zonal Statistics as Table opens with the statistic set to All; Mean is the one the RMSE needs.

So report the RMSE, but never alone. The mean error (no squaring) shows bias: near zero means highs and lows balance. The minimum and maximum of the error raster show the worst misses. And the true DEM has its own error, from the lidar and from the 30 m resampling, which never shows up in the RMSE because it is the thing we compare against. The Lab 8 metadata step is there for that reason.

This is the slide from the last interpolation session, on purpose: the six-methods test on Little Cottonwood Canyon held back 60 points. In Lab 8 the Checkpoints feature class is 200 such points, drawn separately from the sample points. Extract Multi Values to Points reads the truth and all three surfaces at each checkpoint in one tool.

The Lab 8 baseline surfaces (2,500 points, defaults), read at 2,000 different random sets of 200 cells, not the lab's own Checkpoints. 90 % of the draws fall within: Thiessen 24.0 to 32.6 m, IDW 17.6 to 24.0, kriging 11.5 to 17.6, around the all-cell values 28.29, 20.71 and 14.46. Yet in all 2,000 draws kriging beat IDW and IDW beat Thiessen, because the same checkpoints judge all three surfaces: a set of points that happens to land on hard ground makes every method look worse together. Lesson: 200 checkpoints are good for choosing a method and loose for quoting its accuracy to a client. Lab 8's third question asks exactly this of each student's own checkpoints. Numbers: tools/week09_interpolation_numbers.json, from tools/week09_interpolation_figures.py.

The capture is the IDW tool inside the Lab 8 model, with Power exposed as a model parameter. Do not give the class the sensitivity numbers: Lab 8 Step 8 is where they find them. Ask for predictions instead, and write them on the board: if 2,500 points gives these RMSEs, what will 10,000 give? Ten times fewer? Half? Does every method gain the same amount? Come back to the board when the labs are in.

Votes only; no answers today. These are the Lab 8 Step 8 questions in advance, so each student goes in with a prediction to test, which is the habit the lab is built around. The three baseline RMSEs are on the lab page as check values, so it is fine to show them.

The figure is the finished Lab 8 model, exported from ModelBuilder. Sample Points, IDW Power and the Semivariogram are its input parameters; the three error rasters and three RMSE tables are its outputs. Two things to warn about: set the Extent environment to True_DEM in Step 0, or IDW and kriging cover less ground at 250 points; and finish Map 1 and the checkpoints before the first run from the tool dialog, because a dialog run deletes everything that is not a parameter, including the surfaces.

Lab 8 Step 1, captured in ArcGIS Pro 3.7.1. The service arrives as a hillshade: 1 m cells, 8-bit unsigned pixels, Web Mercator, and the pop-up at the summit reads 154, where the DEM reads about 2,897 m. That is the service's default raster function at work: the server renders the elevations into a hillshade (its first listed function, Hillshade Gray) before sending them; the service itself stores Float32 meters. Let the class guess why first. The lab's NOTE makes the engineering point: an analysis others must check needs a fixed, dated copy, which is why the course hosts one.

The capture is the model as a tool, set up for run 5 with the example ID ending in 89. Everyone's other runs use the same hosted points and will match a classmate's to the centimeter; run 5 will not. The lab page gives the example's answer (IDW RMSE 20.29 m at power 3.225) so students can check their setup before they use their own digits. Every power falls between 1 and 3.475.

All four are TIP boxes on the lab page. A COUNT below 67,337 means a surface does not cover the study area; an RMSE that matches to the meter but not the centimeter usually means a different neighbor count or a missed environment setting.

The figure is the lab page's example scenario sheet. It is one way to do Map 2, not a template to copy: submissions whose layouts, model labels or symbology match another student's too closely are flagged for follow-up. Say it plainly once, here.

Four minutes, in pairs, then a show of hands on the checkpoint item: most of the room expects 200 points to be too few to rank the methods. If the room has no signal, put the URL on the board; the items read aloud just as well.

Week 9 due items from the DUE table in tools/build_schedule.py: Lab 8, Quiz 7 and Chapter 12 this Saturday. Deck written October 8, 2026. Figures: tools/week09_interpolation_figures.py (error-row crop and the checkpoint test); the ArcGIS Pro captures are from the Lab 8 GUI build (docs/assignments/lab-08/images), copied into this folder.