Let's learn about sampling and interpolation. This is now one Tuesday session (Thursday is Interpolation Explorer and Lab 8): it runs from sampling design through splines to kriging, so skip optional slides as time requires. The surface on the right is the end product of everything we do today — a continuous surface built from a scatter of measured points.

stamp:begin

stamp:end

Everything on the right-hand figure is the whole idea in miniature: an unknown value y at position x, estimated as a weighted blend of two known values. Every method in this deck is a more elaborate version of that one picture. That figure comes back several times on purpose.

Give them three or four minutes. Collect several answers and, more importantly, several rules: nearest city, plain average, distance-weighted average, adjust for elevation. Those four answers are Thiessen, a global mean, IDW, and a trend surface. Name them now, then spend the rest of the session making each one precise.

Map: Utah from UGRC's state boundary, cities at the centroids of their UGRC municipal boundaries (tools/week08_interpolation_figures.py). Point out before anyone answers: American Fork is 20 km from Provo, Moab and Cedar City are over 250 km away and 1,000 m lower and higher than each other - the map is why nearest-city and distance-weighted answers win.

Note the phrase "within the region sampled." Predicting outside that region is extrapolation, and every method here degrades badly when you ask it to extrapolate. Fixed the source's "Can not" to "cannot."

The two halves are inseparable. A brilliant interpolator cannot rescue a bad sample, and a beautiful sample is useless until you interpolate it. Spend the first part of class on the left half, the rest on the right half.

The next four slides all use the same made-up scenario: gold is distributed through the subsurface, you cannot dig everywhere, so you choose where to dig. Keep the scenario fixed and change only the sampling design so the comparison is fair.

Left: the true surface, drawn as contours. Right: the cartoon reason we care. The whole point of the next four slides is that you must commit to a sampling design before you know what the surface looks like.

Systematic is the default people reach for. The bias failure mode is worth naming: if the terrain has ridges at roughly your sample spacing, a grid can land on ridge tops every time and you will never see the valleys.

"The location of points may be a problem" is a field-work concern: random points land in rivers, on private land, and halfway up cliffs. Ask the class what they would do with a point they cannot reach — moving it quietly destroys the randomness that justified the design.

Cluster sampling is the design that budgets produce. It buys many observations cheaply, but the observations are not independent: points inside one cluster tell you nearly the same thing.

Adaptive is the best design and the hardest to execute, because it needs prior knowledge of the thing you are going out to measure. In practice it means a coarse first pass followed by targeted infill. Fixed the source's "can not" to "cannot."

TODO(instructor): decide how much of sampling design belongs in CE 414 versus the statistics prerequisite, and whether to add a slide on sample size and spacing.

This repeats the framing from earlier in the session, which is exactly what the source deck did: it restated the sampling/interpolation definition slide as the transition out of sampling designs and into interpolation methods. Everything from here on is the second half.

The last bullet is the honest one and it is where the rest of a career in this material lives: hold points back, interpolate without them, and compare. Everything else on this slide is bookkeeping.

TODO(instructor): decide whether to teach train/test validation explicitly this week — holding out a subset, computing RMSE and bias, and talking about uncertainty in the interpolated surface — or to leave it to the lab. If it stays, it probably needs its own slide rather than one bullet.

Point out that the raster path and the contour path are two renderings of the same estimated surface, not two different analyses. In practice you interpolate to a raster and derive contours from it.

This pair is the reference for the next twenty slides. Left is the truth, which in a real project you never have. Right is everything the interpolator gets to see. Every method that follows is judged by how close it comes to the left panel using only the right panel.

Thiessen is the zero-weighting case: the nearest point gets a weight of one and everything else gets zero. Worth saying out loud, because it makes the later methods feel like refinements rather than unrelated tricks.

Walk the four panels in order. The key geometric fact: every location inside a polygon is closer to that polygon's sample point than to any other. In ArcGIS Pro this is a single tool; students should still be able to draw it by hand on an exam.

Same points, now partitioned. Note the dense cluster on the right of the map: where samples crowd together the polygons become tiny slivers, and where samples are sparse a single polygon can cover a large area with one measured value.

The 3D view makes the defining property obvious: the surface is flat inside each polygon and jumps discontinuously at every boundary. That is fine for a categorical variable such as soil type, and wrong for elevation or temperature. The source slide title misspelled this as "Theissen" — corrected to "Thiessen."

Tie the disadvantages back to the 3D view they just saw. The odd-shaped boundaries and the step discontinuities are the same fact seen twice.

Combined the source's text slide with the figure that followed it. The failure mode to demonstrate: shrink the radius until empty circles appear as holes in the output, grow it until the surface flattens into the global mean. Fixed the source's "many contain no points" to "may contain no points."

Read the diagram top to bottom. The input sample layer floats above the output raster; each output cell takes the average of whatever samples fall inside its circle. Point at the cell that got three samples and the cell that got one, and ask which value they trust more. Nothing in the method distinguishes them — the output carries no record of how many points went into each cell.

Original surface on the left, fixed-radius result on the right. The interpolated surface is visibly blockier and the fine drainage detail on the left is gone. It also does not pass through the measured values — this is a non-exact method, which comes back at the end of the session.

IDW is the first method that uses more than one point and treats them unequally. Corrected the source's "nearby know points" to "known points."

Read the equation as a weighted average whose weights are 1/d^n, normalized by their own sum so they add to one. That normalization is the denominator. If n = 0 every weight is equal and IDW collapses to a plain local mean.

Walk the arithmetic on the figure: three known points at distances 4, 2 and 6 with values 50, 30 and 52, weights of 1/4, 1/2 and 1/6, and an estimate of 34. This is a linear IDW, exponent one. Have the class redo it with the exponent squared and watch the answer move toward the nearest point.

These are the only two knobs students will turn in the lab, so make them concrete. As n grows without bound, IDW converges on the Thiessen result: the nearest point takes all the weight.

Original surface on the left, IDW with exponent one and the six nearest points on the right. Point out the small closed contour rings around isolated sample points — the "bull's-eyes" that are the signature artifact of IDW.

Both panels use exponent two; the left uses six neighbors, the right uses twelve. More neighbors smooths the surface without removing the bull's-eyes, because the exponent, not the neighbor count, is what concentrates weight on the nearest point.

Compare this against the Thiessen surface from earlier. IDW is continuous — no cliffs at polygon edges — but it is not smooth: the surface has a kink at every sample point, and each sample sits at a local peak or pit.

Trend surface is regression, not local interpolation: every sample point influences every part of the surface. That is why the result on the right looks nothing like the original — it captures the regional slope and discards local detail on purpose.

Combined the source's text slide with the figure that followed it. Splines give the smoothest-looking result of anything in this deck, which is not the same as the most accurate: the surface can overshoot well past the range of the measured values between widely spaced points. Corrected "are use to interpolate" to "are used to interpolate."

The source deck labeled this "Day 2: Kriging." Everything before this point weights points by a rule you choose; kriging derives the weights from the data itself.

Corrected the source's run-together "dependson" to "depends on." The three-component split is what makes kriging different in kind from IDW: it models the structure of the data before it interpolates.

The figure splits a real surface (Little Cottonwood Canyon, 30 m, from the Lab 10 DEM) the way the bullets describe: a least-squares 2nd-order trend (std 97 m), what is left smoothed over 210 m - the autocorrelated part, ridges and canyons (std 210 m) - and the rest, the random part, which here is mostly channels and crests narrower than 210 m (std 16 m). The split is illustrative: a different smoothing width moves the line between 'autocorrelated' and 'random', which is exactly the judgment a variogram formalizes. tools/week08_interpolation_figures.py. If the slide is crowded, drop the bullets' examples and talk to the figure.

Corrected the source's "is know as" to "is known as." The red numbers on the figure are the lag distances for each pair. With four points there are six pairs; with a hundred points there are 4,950, which is why the next step is to bin them.

Top row: a spatially autocorrelated layer, and its scatter of nearby pairs falls on a line. Bottom row: an uncorrelated layer, and the same scatter is a cloud. That contrast is the entire justification for interpolating at all — if the bottom panel were your data, no method in this deck would beat the global mean. Fixed the source's "Higher autocorrelations indicates" to "Higher autocorrelation means."

Semi-variance is a dissimilarity measure: it goes up as points get less alike. The "semi" is the factor of one half in front of the sum.

Merged the source's tolerance slide with the variogram slide that immediately followed it. The jagged line is the semi-variance actually calculated from the data; the smooth line is the model fitted to it. Only the smooth line is used to interpolate.

Definitions follow Esri's "Understanding a semivariogram: the range, sill, and nugget" (ArcGIS Pro help, checked October 2, 2026); the partial sill is the sill minus the nugget. The figure is a real one: 150 random elevation samples from Little Cottonwood Canyon, 11,175 pairs binned every 300 m, a spherical model fitted - range about 3,360 m, sill about 63,000 m2, nugget 0 (elevation is measured precisely, and the samples resolve the terrain). Ask: what would put a nugget here? (GPS elevations with a few meters of error, or samples much farther apart than the gullies.) The source deck's idealized Figure 9-14 with hand-drawn red marks is no longer used on this slide.

Combined two consecutive text-only slides from the source. The comparison in the second half is the sentence worth remembering: IDW's weights come from a rule you picked, kriging's weights come from the data's own spatial structure.

Left: the semivariogram cloud, every pair of the 150 samples. Right: the pairs binned every 300 m and a spherical model fitted to the bins. The model, not the cloud, sets kriging's weights.

Original surface on the left, kriged interpolation on the right. Compare it against the IDW panels from earlier: the bull's-eyes are gone, because the weights now reflect how quickly the variable actually decorrelates with distance instead of a fixed 1/d^n rule.

The 3D view of the same result. Set this beside the Thiessen and IDW surfaces from earlier in the deck: stepped, kinked, and smooth. Same points, three surfaces.

Optional slide; it answers "how would you judge which one to trust" with a test students can run. 150 random points of a real 30 m DEM (Little Cottonwood Canyon, 2,180-3,490 m) interpolated by six ArcGIS Pro tools, and each surface checked at 60 hold-out points the methods never saw. Root-mean-square error at the hold-out points: kriging with a 300 m lag 87 m, natural neighbor 98, IDW 101, spline 108, Thiessen 114, kriging with the default lag 142, 2nd-order trend 228 - and just using the mean of the samples 218. Two lessons: hold out some points and measure, don't eyeball; and the same method with a careless setting (kriging's default lag here) can do worse than a much simpler one. tools/week08_interpolation_figures.py; numbers in tools/week08_interpolation_numbers.json. This addresses the earlier TODO about teaching train/test validation; Dan to decide whether it stays.

Optional, with the slide before it. The orange bar is the lowest error. Kriging with a sensible lag wins here; kriging with the tool's default lag loses to every method but the trend. The dotted line is the error of ignoring location entirely and guessing the mean of the 150 samples (218 m).

The test, run in ArcGIS Pro on 150 samples snapped to 30 m cell centers (tools/week08_interpolation_figures.py): Thiessen, Natural Neighbor, IDW and kriging with a 300 m lag give the sample value back exactly; the regularized spline is off by 0.01 m on average (it is built to pass through the points, the regularization relaxes that slightly); the 2nd-order trend misses by 166 m on average; and Kriging with the tool's default lag fitted a model that is almost all nugget (nugget 39,666 m2 of a 42,459 m2 sill, range 30 m) and misses by 95 m. So "is kriging exact?" has the honest answer "it depends on the nugget" - which is why this slide replaced the source deck's list that called spline and kriging non-exact. Fixed-radius averaging (earlier in the deck) is also not exact: it averages several samples near each point.

Vote on all four before discussing any of them. The interesting argument is between adaptive and random: adaptive puts points where the surface changes fastest, which is where interpolation errors are largest, but it needs prior knowledge that a real project usually does not have. There is no answer key here on purpose.

Same points, six methods, one truth to compare against. Push the class past "which is prettiest" to "which one reproduces the drainage pattern on the left." Then note that in a real project you cannot run this comparison, because you never have the left panel — which is why withholding sample points matters.

Every tool in this table produced a surface in tools/week08_interpolation_figures.py (ArcGIS Pro 3.7.1, arcpy, October 2, 2026); the parameters named are the ones the six-methods slide varies. Geostatistical Analyst offers the same methods with more control (and cross-validation built in); not needed for Lab 8. Toolbox paths are written from the arcpy modules used (arcpy.analysis.CreateThiessenPolygons, arcpy.sa.*) - VERIFY the Geoprocessing pane grouping (Spatial Analyst Tools > Interpolation) in the GUI before class.

Five minutes, in pairs, then a show of hands on the one that splits the room: the item asking which method cannot return the measured values unchanged. Most of the room expects IDW to be the non-exact one because its surfaces look averaged, but at a sample location the nearest point is that sample itself, so IDW gives the measurement back; fixed-radius averaging is the one that blends it away. If the room has no signal, put the URL on the board; the items read aloud just as well, and the two class-vote slides are still on screen behind you.

Point them at Lab 8 specifically: it is where the exponent and neighbor-count knobs from the IDW slides get turned by hand.

The method-to-tool slide asked for here now precedes this one.

TODO(instructor): the plan notes a Big Southern Butte example. Decide where it belongs in the sequence — the note is that it should be taught only after the methods above are established — and whether it replaces or supplements the class-vote slides.

Conversion notes (2026-09-03): source "CE 414 Week 8 - Interpolation.pptx", 48 slides, 4:3, converted to 46 Marp slides. No hidden slides in the source. No slides dropped outright; five pairs of source slides were merged because the second was a figure or a continuation of the first: 19+20 (fixed-radius text and figure), 30+31 (trend-surface text/equation and result figure), 32+33 (splines text and figure), 39+40 (tolerance and variogram), and 42+43 (variogram model and the IDW comparison). Source slide 10 is an exact repeat of source slide 4; it is kept, as the source intended, but rendered as a section-divider rather than a duplicate content slide. Source slide 17's title "Theissen" corrected to "Thiessen" — that was the only occurrence of the misspelling; slides 14, 18, 46 and 48 already spelled it correctly. Other objective fixes: "Can not"/"can not" to "cannot" (slides 3, 9); "Some circles many contain no points" to "may contain" (19); "nearby know points" to "known points" (23); "are use to interpolate" to "are used to" (32); "dependson" to "depends on" (35); "is know as the lag distance" to "is known as" (36); "Higher autocorrelations indicates points near each other are alike" to "Higher autocorrelation means..." (37). No ArcGIS 9 / ArcMap / ArcCatalog / ArcToolbox wording appears anywhere in the source deck, so no software-version substitutions were needed; the deck names no GIS software at all, which is itself flagged as a TODO(instructor) above. No screenshots of any GIS user interface exist in this deck, so there are no stale ArcMap-era captures to re-shoot — every figure is a textbook diagram, a scanned figure, a rendered surface, or a hand-drawn equation. The recurring handwritten linear-interpolation figure is kept once on each slide where the source used it (source slides 4, 11, 12, 14, 18, 23, 26) plus the goals and closing slides; it is a deliberate motif, not a duplicate. Five slides that were built from PowerPoint shapes were captured as 200 dpi renders and cropped: the Thiessen construction panels (15), the annotated lag-distance figure (36), the annotated idealized variogram (41), and the two class-vote comparison grids (47, 48). Kriging terminology on the nugget/sill/range slide and the exact/non-exact list is carried over verbatim and flagged with VERIFY comments rather than rewritten. Four slides carry TODO(graphic) markers where no source image exists (the Provo activity, the kriging-components slide, the variogram-to-surface slide, and the exact/non-exact slide); per this pass's instructions no AI images were generated.