# Validation & Equation Reference

This tool is a browser re-implementation of the **PNNL / U.S. DOE Sky Glow Comparison
Tool** (PNNL-SA-138348), based on the Kocifaj radiative-transfer model. Its equations and
outputs were verified, line by line, against the authoritative sources below.

**Reference sources**

1. **PNNL spreadsheet** — `Sky Glow Comparison Tool for Public Release V1.xlsm`
   (worksheet formulas, the embedded scattering-function tables, and the VBA modules
   `Calculate.bas`, `AddSPD.bas`, and `CCT_FromUV_Function.bas`).
2. **Independent field check** — a manual run of *DOE Sky Glow Calculator v1* recorded by
   the user in 2022 (scotopic weighting, clear/low-particulate, FAR and NEAR observers).

All scattering-function tables (48 of them), the 12 example SPDs, the CIE 1931 2° colour-
matching functions, the scotopic V′(λ) curve, and the photopic V(λ) curve embedded in this
app are taken directly from that spreadsheet.

---

## 1. Equation comparison

### 1a. Sky-glow model — *equation-for-equation identical*

| Step | PNNL spreadsheet (`Calculate.bas` / `AddSPD.bas`) | This app (`index.html`) | Status |
|---|---|---|---|
| Chromaticity *x, y* | `SUMPRODUCT(SPD, CIE-1931 CMF) / (X+Y+Z)` | `Σ spd·cmf / (X+Y+Z)`, same CMFs | ✅ identical |
| SPD normalization | Each SPD pre-scaled to **1000 lm**: `Φ = Σ 683·SPD·V(λ)·Δλ`, `SPD′ = SPD·1000/Φ` | Applied inline: **`÷ Σ SPD·V(λ)`** | ✅ equivalent — the constants (683, Δλ, 1000) cancel in the baseline ratio |
| Sky glow per source | `Σ (SPD′ · SF)` | `Σ (SPD · SF)` | ✅ identical |
| Scattering-function selection | match `atm` + `obs` + `uplight` + `weighting` across 48 tables | `getSF()` — same match over the same 48 tables | ✅ identical |
| Spectral integration | Trapezoidal rule, `5·(f[n+1]+f[n])/2` summed over 380–770 nm | Rectangular sum `Σ f[n]` | ✅ equivalent for the ratio (endpoints ≈ 0; the ×5 cancels) |
| Lumen output % | `SPDtest × (lmout/100)` | `× lumenPct` | ✅ identical |
| **Relative sky glow** | `SGtest / SGbase` | `sg(test) / sg(base)` | ✅ identical |

The only implementation differences (trapezoidal vs. rectangular integration; lumen
normalization applied at input vs. inline) are mathematically equivalent for the reported
relative-sky-glow ratio and account for the ≈0.3 % residual seen in §2b.

### 1b. Correlated colour temperature (CCT) — *exact port*

| Aspect | PNNL spreadsheet | This app | Status |
|---|---|---|---|
| Colour-matching functions | CIE 1931 2°, *x,y* → CIE 1960 UCS *u,v* | same | ✅ identical |
| *u, v* transform | `u = 4x/(−2x+12y+3)`, `v = 6y/(−2x+12y+3)` | same | ✅ identical |
| CCT algorithm | Robertson (1968) / Mori isotemperature-line interpolation — `CCT(u,v)` VBA + `BBR-Isotemp` table | Faithful port of the same algorithm + the same 32-row isotemperature table | ✅ identical |

> **Note:** the original artifact used McCamy's 1992 cubic approximation, which agreed with
> the spreadsheet to within ~1 %. It has been replaced with a direct port of the
> spreadsheet's Robertson method (including the "below 1667 K → not reported" guard) so CCT
> now matches the spreadsheet exactly.

---

## 2. Validation results

### 2a. CCT — every source matches the spreadsheet exactly

| Source | This app (Robertson) | PNNL spreadsheet |
|---|---|---|
| HPS Example | 2040 K | 2040 K |
| LPS Example* | 1718 K | 1718 K |
| Metal Halide Example | 3925 K | 3925 K |
| PC Amber LED Example | 1873 K | 1873 K |
| LED Example 1 | 2703 K | 2703 K |
| LED Example 2 | 2978 K | 2978 K |
| LED Example 3 | 3940 K | 3940 K |
| LED Example 4 | 4098 K | 4098 K |
| LED Example 5 | 5196 K | 5196 K |
| LED Example 6 | 6103 K | 6103 K |
| MH 2 (4041 K) | 4041 K | 4041 K |
| MV H38JA-100/DX | 3430 K | 3430 K |

\* Low-pressure sodium is effectively monochromatic; its CCT is mathematically ill-defined
and shown only for completeness.

### 2b. Relative sky glow vs. the spreadsheet

Scenario: **near** observer, **clear/low-particulate** atmosphere, **unweighted**, baseline
and comparison **uplight 0 %**, **100 % lumen output**, baseline = HPS — matching the
spreadsheet's saved state.

| Comparison source | This app | PNNL spreadsheet | Δ |
|---|---|---|---|
| MH 2 (4041 K) | 1.789 | 1.794 | 0.3 % |
| MV H38JA-100/DX | 1.733 | 1.739 | 0.3 % |
| LED 2719 K | 1.292 | 1.295 | 0.3 % |
| LED V Pump 2724 K | 1.571 | 1.576 | 0.3 % |
| LED 285 (4075 K) | 1.407 | 1.411 | 0.3 % |
| LED 231 (4224 K) | 1.462 | 1.467 | 0.3 % |
| LED Example 5 | 1.490 | 1.495 | 0.4 % |

Residual ≤ 0.4 % is the trapezoidal-vs-rectangular integration difference plus the 5 nm
vs. 1 nm V(λ) grid.

### 2c. Independent field check — DOE Sky Glow Calculator v1 (2022 manual run)

Scotopic weighting, clear/low particulate. All values relative to **HPS @ 2 % uplight,
100 % output = 1.00**. Format: *manual run / **this app***.

**FAR observer**

| Source | 2 % · 100 % | 2 % · 50 % | 0 % · 100 % | 0 % · 50 % |
|---|---|---|---|---|
| HPS | 1.00 / **1.00** | 0.50 / **0.50** | 0.05 / **0.05** | 0.02 / **0.02** |
| 4075 K | 2.45 / **2.45** | 1.23 / **1.22** | 0.11 / **0.11** | 0.06 / **0.06** |
| 2719 K | 1.96 / **1.95** | 0.98 / **0.98** | 0.09 / **0.09** | 0.05 / **0.05** |

**NEAR observer**

| Source | 2 % · 100 % | 2 % · 50 % | 0 % · 100 % | 0 % · 50 % |
|---|---|---|---|---|
| HPS | 1.00 / **1.00** | 0.50 / **0.50** | 0.80 / **0.80** | 0.40 / **0.40** |
| 4075 K | 2.53 / **2.53** | 1.27 / **1.27** | 2.04 / **2.04** | 1.02 / **1.02** |
| 2719 K | 1.99 / **1.99** | 0.99 / **0.99** | 1.60 / **1.60** | 0.80 / **0.80** |

14 of 16 cells match exactly; the two others differ by 0.01 in the last digit (rounding).
This independently confirms the model's signature behaviour — strong uplight sensitivity for
a distant observer (HPS 1.00 → 0.05 from 2 % to 0 % uplight) versus weak sensitivity nearby
(1.00 → 0.80) — and that lumen output scales the result linearly.

---

## Notes on the Sky Glow Index column

The "Sky Glow Index" shown in the comparison table is the modeled sky glow **per lumen**:
`Σ(SPD·SF) ÷ Σ(SPD·V(λ)) × lumen%`. It is in arbitrary model units (the native scale of the
scattering-function tables, ~10⁻⁷) and is meaningful **only relative to the baseline** — it is
not cd/m² or mag/arcsec². The "vs Baseline" column is the interpretable result.

_Last validated: 2026-06 against `Sky Glow Comparison Tool for Public Release V1.xlsm`._

---

## 3. Cross-check vs the strategy white paper, Table 4-1 (added 2026-07-25)

Table 4-1 of the DarkSky "Holistic Strategy to Reduce Light Pollution" draft
(built years earlier with the DOE/Excel Sky Glow Calculator v1) was recomputed
with this app: baseline HPS 2% uplight / 100% output, scotopic weighting,
clear low-particulate, both observers, 12 cells per observer.

**Result: 16 of 24 cells reproduce exactly at the paper's displayed precision.**

| row | verdict |
|---|---|
| HPS (both observers, all 8 cells) | identical (1.00/0.50/0.05/0.02 FAR; 1.00/0.50/0.80/0.40 NEAR) |
| "2719 K" LED (8 cells) | identical to our `LED - Example 1 (2703 K)` within ±0.01 — same SPD, different CCT algorithm label |
| "4075 K" LED (8 cells) | NOT one of the 12 embedded example SPDs. Paper: FAR 2.45, NEAR 2.53. Ours: Example 3 (3940 K) 2.24/2.28, Example 4 (4098 K) 2.86/2.98. The offset is a constant ratio across all 8 cells → identical scattering model, different input spectrum. The paper's SPD is evidently the specific street-light LED from the PNNL 2017 report / DOE calculator v1, intermediate in blue content between our Examples 3 and 4. |

Follow-up: obtain the 4075 K SPD from the DOE v1 workbook and add it to the
library (the app accepts pasted SPDs); it should then reproduce the remaining
8 cells exactly.

**Resolution (2026-07-25, same day):** the paper's exact spectra were located in
the PNNL workbook's *Saved SPDs* sheet — `LED 2719 K` (col M) and `LED 285
4075K` (col AX) — and added to the shared spectral library (v1.2.0). With
them, all **24/24 cells of Table 4-1 reproduce** at the paper's displayed
precision (two cells sit on an exact .xx5 and round oppositely: 1.23→1.22,
1.96→1.95). The white paper's table is now reproducible in this app by
selecting those two sources.