PA-PRISM Influence Lab — Interactive Intuition Guide

1. The central idea: perturb the station network and watch the field move

Suppose LT and AN use nearly the same gridding machinery, but one station disappears from one product. The station removal is an intervention. The surface response is evidence about the hidden interpolation system.

ΔP(x) = P_AN(x) − P_LT(x)

Then compare that difference against distance from the removed station. If disagreement is largest near the removed station and decays outward, the station has a local influence footprint.

common station LT-only / removed station target grid cell

What is observed vs inferred?

Observed in the experiment: product disagreement is larger near LT-only stations than near common stations, with the two distance curves crossing near ~18 km in the supplied composite.

Inferred, not proved: the shape is consistent with a local weighted interpolation system. A flat-looking inner composite does not by itself prove a literal flat-radius kernel.

Complications: overlapping removed stations, terrain/facet weighting, product lineage differences, and post-processing can broaden or distort the apparent kernel.

Three separate goals

Goal	Question
Reconstruction fidelity	Which parameter set best reproduces PRISM?
Meteorological skill	Which parameter set best predicts withheld gauges?
Perturbation fidelity	Which parameter set reproduces the observed response when stations are removed?

2. Distance-kernel sandbox

Move the sliders. The synthetic map recomputes station weights and the radial curve. This uses a TIER-like adaptive Barnes form for intuition, not a claim that PRISM uses this exact code.

I(d) = exp(−d^y/s), s = S₀ × meanDist/maxDist

illustrative effective distance term ≈ I(d)²

Base scale S₀

4k30k

Distance exponent y

Mean station distance

5 km60 km

Max search distance

50 km300 km

Adaptive scale s—

Half-weight radius—

Weight at 10 km—

Weight at 40 km—

Station-density intuition

With the adaptive scale, sparse neighborhoods broaden the kernel. Dense neighborhoods tighten it. That means “local” does not imply one fixed radius everywhere.

Station density

sparsedense

Max stations used

320

Synthetic positions are deterministic, so changing sliders changes the conceptual experiment without pretending these are real PA stations.

3. Station-removal map

Click anywhere on the map to move the removed station. The difference field is a made-up sum of distance influence × terrain compatibility. It is designed to show why one removed station can create a broad, non-circular footprint.

Terrain influence

nonestrong

Facet mismatch penalty

nonestrong

removed station common stations brightness = larger synthetic |ΔP|

Why the footprint need not be a perfect circle

A local regression can use several gates at once:

W_j(x) ∝ K_d(distance) × K_e(elevation mismatch) × K_f(facet compatibility) × …

Even with a circular distance kernel, terrain can stretch, clip, or redirect the practical influence.

Reverse-engineering trick: fit all removed stations jointly, not one nearest station at a time:

ΔP(x) ≈ Σ_j a_jK(d_xj, terrain_xj, θ) + ε(x)

This reduces contamination when multiple dropped stations overlap.

4. Empirical PA composite from the supplied research result

These are the reported RMS values of relative difference, binned by distance. This is the one panel here that is not synthetic.

How to read the crossover

Near LT-only, 0–3 km0.1644

Near common, 0–3 km0.0854

Approx. crossover~18 km

Interpretationlocal influence signal

The important part is not “18 km equals a secret PRISM radius.” The important part is that station class contains spatial information about product disagreement.

Do not over-read the flat inner bins. A flat-looking 0–11 km composite can come from a true plateau, smoothing, overlapping removals, coarse bins, or adaptive weighting.

Reported values

Distance km	LT-only RMS	Common RMS
0–3	0.1644	0.0854
3–5	0.1550	0.0855
5–7	0.1504	0.0889
7–9	0.1608	0.0963
9–11	0.1665	0.1007
11–14	0.1464	0.1068
17–20	0.1325	0.1291
25–30	0.1248	0.1658
35–40	0.1178	0.1768

5. Terrain smoothing is not precipitation smoothing

These are different objects. Use the slider to visualize repeated Daly-style terrain filtering on a synthetic ridge.

SE_i,j = ½E_i,j + ⅛(E_i+1,j + E_i−1,j + E_i,j+1 + E_i,j−1)

Filter passes

016

Keep these scales separate

Scale	Meaning
DEM cell spacing	How finely elevation is sampled.
Terrain filter scale	Which topographic wavelengths survive into the predictor.
Station search scale	How far the algorithm looks for observations.
Distance-weight scale	How rapidly a chosen station loses influence.
Output grid support	What spatial support the final precipitation value represents.
Inter-cell smoothing	Post-processing that changes the published precipitation field.

An 800 m precipitation grid does not mean the atmosphere was independently observed every 800 m. Fine-scale structure can be inherited from DEM predictors, regression, climatology, and smoothing.

6. Slope bounds and default-state logic

Use the slider to see how a fitted normalized precipitation slope can be accepted, clipped/repaired, or replaced by a default depending on the implementation.

Raw fitted normalized slope

-0.55

Illustrative target elevation difference

-1500 m1500 m

minSlope0.25

defaultSlope1.3

maxInitial4.25

maxFinal3.0

Why final slope ≠ raw regression slope

A published field may combine several states:

raw fit → validity test → station-removal/refit or fallback → default slope → spatial slope filtering → feathering → final field

If many PA raw fits live near the minimum threshold, the final spatial slope surface may reflect branch logic more than unconstrained regression.

Reverse-engineering consequence: infer branch/state before interpreting coefficients. A low final slope does not uniquely reveal the raw fitted slope that existed earlier in the pipeline.

7. TIER-like normals vs CAI time series

TIER-like climatology model

P(x) ~ elevation + terrain-conditioned station weighting

PRISM CAI-style monthly/daily logic

Climatologically aided interpolation

P_time(x) ~ climatological background predictor + current observations

What to fit against what

Question	Best target
Can TIER-family terrain regression emulate PRISM?	PRISM 800 m normals / climatology
Can our model predict real climate?	Withheld independent gauges / dense networks / hydrologic tests
Can our model reproduce hidden station influence?	LT→AN station-removal difference fields
Can TIER reproduce modern CAI/RAI directly?	No, not without adding those later predictor architectures

8. Build the next experiment

Toggle components to see how the research ladder should work. Complexity earns admission only if it improves a predefined objective.

Three-score dashboard

These scores are synthetic teaching metrics, not results. They show why a parameter set can win one objective and lose another.

PRISM reconstruction—

Withheld-gauge skill—

Removal-response fit—

Complexity—

Desired scientific outcome: preserve all three scores separately. The parameterization that best emulates PRISM is not automatically the one that best predicts precipitation.