Hypothesis test 11 — "b-Values Shift With Depth"
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
report.md
# Hypothesis test 11 — "b-Values Shift With Depth" ## Source (verbatim, from the theory HTML) File: `[local provenance path] Theories Vol XXXII - 2026-10-30/theories/bvalue-depth.html` (Theory 08 of 89, Vol. XXXII) - **Claim:** "The earthquake magnitude-frequency b-value declines with depth, so deep and shallow sequences cannot share a statistic." - **Mechanism:** "Higher confining pressure and temperature raise the stress drop for small events." - **Prediction:** "b should fall with depth." - **Refutation:** "Constant b would remove the pressure effect." ## Why probable Confining pressure and temperature genuinely change fault mechanics with depth, and several regional slab studies report lower b at greater depth, so a depth-dependent b is physically plausible. However, b is also set by material heterogeneity, fluids, and catalog completeness, so a *monotonic global* decline is not guaranteed a priori. ## Method - **Data:** USGS ComCat FDSN API (`https://earthquake.usgs.gov/fdsnws/event/1/query?format=csv`), worldwide, 2010-01-01 → 2026-10-30, `minmagnitude=2.5`, `orderby=time`. Downloaded in ~3-month windows (auto-split whenever a query hit the 20,000-row API cap — one 4-month window did). **455,119 rows** downloaded; **452,777 events** after cleaning (valid time/depth/mag, depth ≥ 0, duplicate-id removal). 81 MB of CSV under `data/` (cap was 300 MB). - **Estimator:** maximum-likelihood b, `b = log10(e) / (mean(M) − (M_c − ΔM/2))` with ΔM = 0.1 (Aki 1965, Utsu 1965 bin-width correction). - **Completeness M_c:** primary = **fixed M_c = 4.0 in every bin** (state: fixed cutoff, as permitted); sensitivity = per-bin max-curvature M_c (mode of the FMD + 0.2; Woessner & Wiemer 2005) and fixed M_c = 4.5 and 5.0. - **Depth bins (km):** 0–30, 30–70, 70–150, 150–300, 300–800. - **Uncertainty:** Shi & Bolt `σ = b/√N` **and** delete-one jackknife SE; trend fits are weighted least squares (weights = 1/SE²) of b on the *median* depth in each bin. - **Subduction checks:** (a) all events with depth > 70 km (slab proxy) in finer bins; (b) Tonga–Fiji–Kermadec slab box (lat −45…−15, lon 170…200). - **Robustness:** time halves 2010–2017 vs 2018–2026. ## Results ### Primary: worldwide catalog, fixed M_c = 4.0 | Depth bin (km) | N (mag≥2.5) | N (mag≥4.0) | b (MLE) | SE Shi–Bolt | SE jackknife | max-curv. M_c | b at max-curv. M_c | |---|---|---|---|---|---|---|---| | 0–30 | 268,093 | 124,495 | 0.691 | 0.0020 | 0.0013 | 2.95 | 0.336 (invalid, see below) | | 30–70 | 94,463 | 59,783 | 0.748 | 0.0031 | 0.0020 | 4.55 | 1.126 | | 70–150 | 50,587 | 32,756 | 0.837 | 0.0046 | 0.0033 | 4.45 | 1.161 | | 150–300 | 21,908 | 18,667 | 0.946 | 0.0069 | 0.0053 | 4.45 | 1.205 | | 300–800 | 17,726 | 17,578 | 0.943 | 0.0071 | 0.0059 | 4.45 | 1.103 | **b *rises* with depth** (0.69 → 0.94), i.e., the **opposite** of the prediction "b should fall with depth." Weighted regression of b on median depth: slope **+0.00063 ± 0.00024 per km, p = 0.081** (5 bins). The monotone increase has tiny standard errors (N ≥ 17.5k per bin). The same rise is reproduced in both time halves (2010–2017: 0.688 / 0.745 / 0.814 / 0.894 / 0.846; 2018–2026: 0.693 / 0.751 / 0.856 / 0.988 / 1.012), so it is not an artifact of one time period. ### Sensitivity to completeness choice (worldwide) | Depth bin (km) | b, M_c=4.0 (N) | b, M_c=4.5 (N) | b, M_c=5.0 (N) | |---|---|---|---| | 0–30 | 0.691 (124,495) | 1.124 (70,154) | 1.122 (18,768) | | 30–70 | 0.748 (59,783) | 1.260 (31,177) | 1.291 (7,064) | | 70–150 | 0.837 (32,756) | 1.340 (14,288) | 1.208 (2,891) | | 150–300 | 0.946 (18,667) | 1.400 (6,398) | 1.023 (1,166) | | 300–800 | 0.943 (17,578) | 1.263 (5,692) | 0.760 (1,100) | - At M_c = 4.5 the **rise with depth persists** (1.12 → 1.40, peak at 150–300 km). - Only at M_c = 5.0 does a decline in the claimed direction appear, and only inside the deep part (30–70 km: 1.29 → 300–800 km: 0.76); the full profile is non-monotonic (0–30 km = 1.12 < 30–70 km = 1.29) and N shrinks to ~1,100 in the deepest bin. - Per-bin max-curvature M_c: deeper bins are statistically **flat** (1.13 / 1.16 / 1.21 / 1.10); the shallow bin is unusable because the worldwide M3–4 catalog is incomplete and the FMD is truncated at mag 2.5 (max-curvature M_c = 2.95, b = 0.34 is not physical). ### Subduction-zone subsets (fixed M_c = 4.0 unless noted) - **Depth > 70 km (slab proxy), finer bins:** 70–100: b = 0.801 ± 0.007; 100–150: 0.865 ± 0.006; 150–200: 0.925 ± 0.009; 200–300: 0.977 ± 0.011; 300–400: 1.053 ± 0.020; 400–800: 0.926 ± 0.008. WLS slope +0.00024 ± 0.00015 per km (p = 0.19) — rises then falls; not a decline. With per-bin max-curvature M_c the slope is −0.00016 ± 0.00010 per km (p = 0.19), i.e., a non-significant hint in the claimed direction. - **Tonga–Fiji–Kermadec box:** 0–50: 0.679 ± 0.010 (N=4,559); 50–150: 0.759 ± 0.029 (707); 150–300: 1.024 ± 0.039 (690); 300–450: 1.080 ± 0.085 (163); 450–800: 0.812 ± 0.019 (1,873). Non-monotonic; WLS slope +0.00029 ± 0.00017 per km (p = 0.19). At M_c = 4.5 the Tonga profile is flat (1.08 / 1.31 / 1.27 / 1.59 / 1.26), so the deep-bin dip is not robust to the completeness choice. ### Figure `figures/bvalue_vs_depth.png` (1650×690): left panel — worldwide depth bins at M_c = 4.0 with jackknife error bars and the WLS line (slope +0.00063 ± 0.00024 per km); right panel — slab (depth > 70 km) and Tonga-box subsets with their WLS fits. The claimed downward trend is visually absent; the fixed-M_c points climb with depth. ## Verdict **NOT SUPPORTED** No systematic decline of b with depth in the claimed direction is present. The large-N fixed-M_c analyses show b *increasing* with depth (0.69 at 0–30 km → 0.94 at 300–800 km for M_c = 4.0; 1.12 → 1.40 for M_c = 4.5), stable across both time halves — the opposite of "b should fall with depth." Per-bin completeness gives flat deep bins (≈1.1–1.2). The claimed decline appears only in the small-sample M≥5.0 subset of the deep bins and in one non-robust Tonga deep-bin comparison, so the hypothesis as stated is refuted by this test rather than weakly supported. ## Limitations (honest) 1. **Completeness is depth-dependent.** Shallow offshore events (ridges, trenches) at M4.0–4.5 are under-detected worldwide; deep events lie in well-instrumented slabs. Shallow b jumps from 0.69 (M_c=4.0) to 1.12 (M_c=4.5), showing how much the fixed-M_c=4.0 shallow value is depressed by missing small events. No single global M_c makes shallow and deep bins perfectly comparable. 2. **The answer depends on M_c.** Rise at M_c 4.0/4.5, flat at per-bin max-curvature, deep-only decline at M_c 5.0. We report all of them rather than hiding the dependence. 3. **Tectonic mixing.** Depth bins mix crustal, ridge, transform, and slab regimes with genuinely different b; the confining-pressure mechanism is not isolated by depth alone. The depth > 70 km proxy is not a strict subduction filter, and the Tonga box is only one region. 4. **No declustering.** Aftershock sequences (e.g., 2011 Tohoku) remain in the catalog and can bias b. 5. **Max-curvature M_c fails for the shallow bin** because the catalog is truncated at mag 2.5 and incomplete at M3–4; that bin's b (0.34) is not a real seismicity parameter. 6. **Bin artifacts.** The 300–800 km bin mixes 300–400 and 400–700 km populations with different FMDs; N for M≥5.0 deep events is only ~1,100. 7. The mechanism statement (stress drop vs depth) is not directly testable from an earthquake catalog alone. ## Files - `report.md` — this report - `code/bvalue_test.py` — runnable downloader + analyzer (`python code/bvalue_test.py download|analyze`) - `results.json` — full per-bin and regression results - `data/usgs_2010.csv` … `data/usgs_2026.csv` — 17 year files, 455,119 rows, 81 MB; `data/manifest.json` — download resume manifest - `figures/bvalue_vs_depth.png` — b vs depth with error bars and trend fits Working report — not peer reviewed. Educational information, not medical advice. AI-assisted test: DeepSeek.