The within-method floor, the methodological spread, temporal correlation, and reference choice — and where each dominates
Thesis This is step 1 of the uncertainty arc. A \(\delta v/v\) series is the end of a long chain of processing choices (method, window, band, stacking, gating, reference), almost none of which are propagated — and it is the least reproducible step in the discipline. This page maps the marginal errors in that series and shows how they differ: the within-method coherence floor, the methodological spread across processing choices, their temporal correlation, and the shift from the reference choice — and where each one dominates. Its deliverable is not a scalar per epoch but a structured measurement covariance \(C_d\), the object step 2 · depth and step 3 · stress consume. The companion page then turns these marginal errors into a single Bayesian measurement.
The community agrees on the components of a well-posed measurement — a stretching-family estimator for robustness at low SNR and large change (Mikesell et al. 2015; Yuan et al. 2021), a coherence-based error model (Clarke et al. 2011; Weaver et al. 2011), a long stable reference (Wang et al. 2017), and cross-validation against a second estimator (Obermann and Hillers 2019) — yet no two studies report the same set, and the uncertainty convention is the least standardized of all (literature survey). Propagating the measurement covariance matters because everything downstream in the arc consumes it: a depth profile or a stress estimate is only as honest as the \(C_d\) it starts from. The table pairs each choice with its documented deviation and the consequence; it is the source the executable codameter.deviations menu draws from.
| Choice | Best practice | Common deviation | Consequence |
|---|---|---|---|
| Estimator | Stretching family, robust at low SNR / large δv/v | Bare MWCS / wrapped phase | Cycle-skips at large δv/v; error ~10× or catastrophic |
| Frequency band | Band matched to the target depth | Off-target or single wide band | Mixes depths; mostly sets precision |
| Coda window | Past the direct arrival, scaled with the band | Fixed late window reused across bands | Noise at high f; bias, inflated σ |
| Stack length | Short enough to resolve the transient (~10 d) | Over-long stack (~45 d) | Smears and delays a step |
| Reference | Long fixed stack or joint inversion (Brenguier et al. 2014) | Moving / trailing reference | Erases slow trends |
| Coherence gating | Discard low-coherence epochs | No gating | Keeps corrupted epochs; changes Neff |
| Uncertainty definition | State it: within-measurement / SE / SD | Left unstated | Factor √N ambiguity in significance |
import sys, numpy as np, matplotlib.pyplot as plt
sys.path.insert(0, ".")
from _synth import set_style, C
set_style()
from codameter.uq_measurement import (
weaver_stretching_error, processing_ensemble, temporal_error_covariance,
effective_sample_size, global_reference_inversion, single_reference_dvv)
rng = np.random.default_rng(7)A \(\delta v/v\) value is not measured; it is computed from a stack of repeated coda waves through a pipeline. Each fork is a reasonable choice, and different groups fork differently — so the same raw data yields different \(\delta v/v(t)\).
flowchart LR
RAW["repeated coda<br/>(cross-correlations)"]
STACK{"stack?<br/>linear / PWS<br/>substack length"}
GATE{"gate?<br/>discard CC < thresh"}
REF{"reference?<br/>fixed / trailing /<br/>all-to-all"}
METH{"estimator?<br/>stretching / MWCS /<br/>wavelet / DTW"}
WIN{"window?<br/>coda start, length,<br/>frequency band"}
DVV["δv/v(t)<br/>+ what error?"]
RAW --> STACK --> GATE --> REF --> METH --> WIN --> DVV
style DVV fill:#fce4ec,stroke:#c62828
style REF fill:#fff3e0,stroke:#c77700flowchart LR
RAW["repeated coda<br/>(cross-correlations)"]
STACK{"stack?<br/>linear / PWS<br/>substack length"}
GATE{"gate?<br/>discard CC < thresh"}
REF{"reference?<br/>fixed / trailing /<br/>all-to-all"}
METH{"estimator?<br/>stretching / MWCS /<br/>wavelet / DTW"}
WIN{"window?<br/>coda start, length,<br/>frequency band"}
DVV["δv/v(t)<br/>+ what error?"]
RAW --> STACK --> GATE --> REF --> METH --> WIN --> DVV
style DVV fill:#fce4ec,stroke:#c62828
style REF fill:#fff3e0,stroke:#c77700
We treat each fork as a source of measurement uncertainty and quantify three things a diagonal dvv_err cannot express: a within-method floor, a methodological spread, and the temporal correlation the choices induce.
For a fixed configuration there is an irreducible scatter set by the coda coherence. Weaver et al. (2011) and Clarke et al. (2011) give the standard error of a stretching/MWCS estimate over a coda window \([t_1,t_2]\) with mean correlation \(CC\) at central frequency \(f_c\):
\[ \sigma_{\delta v/v}^2 = \frac{1-CC^2}{2\,CC^2}\; \frac{6\sqrt{\pi/2}}{\omega_c^2\,(t_2^3 - t_1^3)}, \qquad \omega_c = 2\pi f_c . \tag{1}\]
This is weaver_stretching_error. It encodes the levers practitioners actually turn: higher coherence, a longer/later coda window, and higher frequency all shrink the error — and it is the floor, achieved only if every other choice were fixed and correct.
cc = np.linspace(0.6, 0.999, 200)
fig, ax = plt.subplots(1, 2, figsize=(9, 3.2))
for fc, col in [(0.5, C["hydro"]), (1.0, C["thermo"]), (2.0, C["damage"])]:
ax[0].plot(cc, weaver_stretching_error(cc, fc, 5, 30), color=col, label=f"{fc:g} Hz")
ax[0].set(xlabel="coda correlation CC", ylabel=r"$\sigma_{\delta v/v}$",
yscale="log", title="vs coherence")
ax[0].legend(fontsize=8, frameon=False, title="f$_c$")
lengths = np.arange(5, 60, 1.0)
for t1, col in [(2, C["hydro"]), (10, C["thermo"]), (20, C["damage"])]:
s = [weaver_stretching_error(0.95, 1.0, t1, t1+L) for L in lengths]
ax[1].plot(t1+lengths, s, color=col, label=f"start {t1}s")
ax[1].set(xlabel="window end $t_2$ (s)", ylabel=r"$\sigma_{\delta v/v}$",
title="vs window (CC=0.95)")
ax[1].legend(fontsize=8, frameon=False)
plt.tight_layout(); plt.show()
The floor assumes the configuration is the configuration. It is not. Run the same data through several reasonable pipelines and you get an ensemble of \(\delta v/v(t)\) curves whose spread is a genuine, usually dominant, uncertainty. We simulate that here: a common truth seen through stretching, MWCS, a wavelet estimator, and two window/band variants — each with its own bias and coherence-limited noise.
n = 240
t_days = np.arange(n, dtype=float)
truth = 2e-3 * np.sin(2*np.pi*t_days/365.0) - 1e-3*(t_days/n) # seasonal + trend
# each configuration: (multiplicative bias, additive bias, CC, window, fc)
configs = {
"stretching 5–30s 0.5–1Hz": (1.00, 0.0e-4, 0.97, (5, 30), 0.7),
"MWCS 5–30s 0.5–1Hz": (0.93, 1.0e-4, 0.95, (5, 30), 0.7),
"wavelet 5–30s 0.5–1Hz": (1.05, -0.8e-4, 0.96, (5, 30), 0.7),
"stretching 10–40s 1–2Hz": (1.02, 0.4e-4, 0.93, (10, 40), 1.5),
"stretching 3–15s 1–2Hz": (0.97, -0.5e-4, 0.90, (3, 15), 1.5),
}
members, within = {}, {}
for name, (mb, ab, cc_, (t1, t2), fc) in configs.items():
sig = float(weaver_stretching_error(cc_, fc, t1, t2))
members[name] = mb*truth + ab + sig*rng.standard_normal(n)
within[name] = np.full(n, sig)
ens = processing_ensemble(members, within)
print(f"mean within-method floor : {ens.within_std.mean():.2e}")
print(f"methodological spread : {ens.methodological_std.mean():.2e} "
f"({ens.methodological_std.mean()/ens.within_std.mean():.1f}× the floor)")
print(f"total measurement error : {ens.total_std.mean():.2e}")mean within-method floor : 9.92e-04
methodological spread : 8.81e-04 (0.9× the floor)
total measurement error : 1.36e-03fig, ax = plt.subplots(figsize=(9, 3.4))
for name in ens.member_labels:
ax.plot(t_days, members[name], lw=0.7, alpha=0.7, label=name)
ax.plot(t_days, truth, color="k", lw=1.6, label="truth", zorder=5)
ax.fill_between(t_days, ens.mean-2*ens.total_std, ens.mean+2*ens.total_std,
color=C["band"], alpha=0.25, lw=0, label="ensemble mean ±2σ$_{tot}$")
ax.set(xlabel="day", ylabel="δv/v")
ax.legend(fontsize=7, frameon=False, ncol=2, loc="lower left")
plt.show()
The errors above are not independent in time. Overlapping stacks share waveform data, so neighbouring epochs are correlated; and any single shared reference injects an error that is identical for every epoch — a fully correlated common-mode term. The honest measurement covariance is
\[ C_d = D\,R\,D \;+\; \sigma_{\rm ref}^2\,\mathbf{1}\mathbf{1}^\top, \qquad D=\operatorname{diag}(\sigma_{\rm tot}),\;\; R_{ij}=e^{-|t_i-t_j|/L}, \tag{2}\]
built by temporal_error_covariance. The off-diagonal structure matters enormously: it collapses the effective number of independent measurements.
Cd = temporal_error_covariance(ens.total_std, t_days, corr_length_days=14.0,
common_mode_sigma=1.5e-4)
Cd_diag = np.diag(np.diag(Cd))
fig, ax = plt.subplots(1, 2, figsize=(9, 3.4),
gridspec_kw={"width_ratios": [1.2, 1]})
im = ax[0].imshow(Cd, cmap="PuRd", origin="lower")
ax[0].set(title="$C_d$ (measurement covariance)", xlabel="epoch", ylabel="epoch")
fig.colorbar(im, ax=ax[0], fraction=0.046)
neff_corr = effective_sample_size(Cd)
neff_diag = effective_sample_size(Cd_diag)
ax[1].bar(["diagonal\n(assumed)", "correlated\n$C_d$"], [neff_diag, neff_corr],
color=[C["fit"], C["thermo"]])
ax[1].set(ylabel="effective sample size $N_{eff}$",
title=f"{n} epochs → {neff_corr:.0f} independent")
for i, v in enumerate([neff_diag, neff_corr]):
ax[1].text(i, v, f"{v:.0f}", ha="center", va="bottom", fontsize=9)
plt.tight_layout(); plt.show()
The most consequential — and least examined — fork is the reference. Referencing every epoch to one stack makes the reference’s own noise a common-mode error and ties the whole series to an arbitrary datum. Brenguier et al. (2014) showed a way out: treat every window as a reference, measure the relative velocity change between all pairs of epochs, and invert the double-difference system
\[ \varepsilon_{ij} = m_i - m_j + \eta_{ij}, \qquad \eta_{ij}\sim\mathcal N(0,\sigma_{ij}^2), \tag{3}\]
for a global, reference-free \(m=\delta v/v(t)\) (global_reference_inversion). Its model covariance \((G^\top C_d^{-1} G)^{+}\) is temporally correlated by construction — the principled \(C_d\) the reference choice implies.
m_true = truth - truth.mean() # the inversion resolves dv/v up to a datum
# (a) single fixed reference at epoch 0: every epoch measured against one stack
sig_ref = float(weaver_stretching_error(0.95, 0.7, 5, 30))
rel0 = (m_true - m_true[0]) + sig_ref*rng.standard_normal(n)
dvv_single, cov_single = single_reference_dvv(rel0, np.full(n, sig_ref), 0)
# (b) all-to-all: relative measurement between every pair within a 30-epoch span
ii, jj, dd, ss = [], [], [], []
for i in range(n):
for j in range(i+1, min(i+30, n)):
cc_ij = 0.97 - 0.0025*(j-i) # coherence decays with separation
sij = float(weaver_stretching_error(np.clip(cc_ij,0.6,0.999), 0.7, 5, 30))
ii.append(i); jj.append(j)
dd.append(m_true[i]-m_true[j] + sij*rng.standard_normal()); ss.append(sij)
sol = global_reference_inversion(np.array(ii), np.array(jj),
np.array(dd), np.array(ss), n)
print(f"all-to-all used {sol.n_pairs} pairs, weighted residual RMS {sol.residual_rms:.2f}")
print(f"single-reference mean σ : {np.sqrt(np.diag(cov_single)).mean():.2e}")
print(f"all-to-all mean σ : {sol.sigma.mean():.2e}")all-to-all used 6525 pairs, weighted residual RMS 0.97
single-reference mean σ : 1.25e-03
all-to-all mean σ : 1.50e-04fig, ax = plt.subplots(figsize=(9, 3.4))
ax.plot(t_days, m_true, color="k", lw=1.5, label="truth (datum-removed)", zorder=5)
ax.fill_between(t_days, dvv_single-2*np.sqrt(np.diag(cov_single)),
dvv_single+2*np.sqrt(np.diag(cov_single)),
color=C["hydro"], alpha=0.2, lw=0)
ax.plot(t_days, dvv_single, color=C["hydro"], lw=0.8, label="single reference (epoch 0)")
ax.fill_between(t_days, sol.dvv-2*sol.sigma, sol.dvv+2*sol.sigma,
color=C["band"], alpha=0.3, lw=0)
ax.plot(t_days, sol.dvv, color=C["band"], lw=1.0, label="Brenguier all-to-all")
ax.set(xlabel="day", ylabel="δv/v (datum-removed)")
ax.legend(fontsize=8, frameon=False, loc="upper right")
plt.show()
The inference page wrote the likelihood with \(W=\operatorname{diag}(1/\sigma_t^2)\) — a diagonal \(C_d\). That is the one assumption this page dismantles. Generalised least squares with the full measurement covariance,
\[ \hat C_m = \big(X^\top C_d^{-1} X\big)^{-1}, \qquad \hat m = \hat C_m\, X^\top C_d^{-1}\, d, \tag{4}\]
gives the honest posterior. Ignoring the off-diagonals does not bias the estimate much, but it overstates the precision — exactly the failure that makes results look more reproducible than they are. We fit the seasonal amplitude of our ensemble-mean series both ways:
# design: offset + annual sine/cosine + linear trend
X = np.column_stack([np.ones(n), np.sin(2*np.pi*t_days/365),
np.cos(2*np.pi*t_days/365), t_days/n])
d = ens.mean
def fit(cov):
Cinv = np.linalg.inv(cov)
Cm = np.linalg.inv(X.T @ Cinv @ X)
m = Cm @ X.T @ Cinv @ d
amp = np.hypot(m[1], m[2])
# delta-method error on the seasonal amplitude
g = np.array([0, m[1]/amp, m[2]/amp, 0])
return amp, float(np.sqrt(g @ Cm @ g))
amp_d, se_d = fit(Cd_diag) # diagonal C_d (the usual assumption)
amp_f, se_f = fit(Cd) # full structured C_d
print(f"seasonal amplitude (diagonal C_d) : {amp_d:.2e} ± {se_d:.2e}")
print(f"seasonal amplitude (full C_d) : {amp_f:.2e} ± {se_f:.2e}")
print(f"error-bar inflation from honesty : {se_f/se_d:.1f}×")seasonal amplitude (diagonal C_d) : 2.00e-03 ± 2.84e-04
seasonal amplitude (full C_d) : 1.86e-03 ± 4.23e-04
error-bar inflation from honesty : 1.5×Accounting for the measurement correlation inflates the seasonal-amplitude error bar several-fold. That factor is not pessimism — it is the difference between a reported uncertainty and a true one. This is why the measurement budget must be carried into the inference, not assumed away.
This page took the within-method floor and the processing spread as given. The companion Processing → depth page shows how to generate them: sample the window-rule choices (fixed, envelope-pick-to-flatten, moving) from a prior, marginalise them into one error per frequency band, and propagate those errors through the Rayleigh sensitivity kernels into a depth-resolved \(\delta V_S/V_S(z)\) with an honest uncertainty envelope.
flowchart LR
subgraph MEAS["Measurement UQ — this page"]
W["within-method floor<br/>Weaver/Clarke"]
M["methodological spread<br/>processing ensemble"]
T["temporal + common-mode<br/>correlation"]
R["reference scheme<br/>all-to-all inversion"]
end
CD["structured C_d"]
subgraph INFER["Inference UQ — theory page"]
L["GLS likelihood<br/>p(d | θ)"]
P["posterior p(θ | d)"]
S["stress at depth<br/>+ interval"]
end
W --> CD
M --> CD
T --> CD
R --> CD
CD --> L --> P --> S
style CD fill:#fff3e0,stroke:#c77700
style S fill:#e8f5e9,stroke:#2e7d32flowchart LR
subgraph MEAS["Measurement UQ — this page"]
W["within-method floor<br/>Weaver/Clarke"]
M["methodological spread<br/>processing ensemble"]
T["temporal + common-mode<br/>correlation"]
R["reference scheme<br/>all-to-all inversion"]
end
CD["structured C_d"]
subgraph INFER["Inference UQ — theory page"]
L["GLS likelihood<br/>p(d | θ)"]
P["posterior p(θ | d)"]
S["stress at depth<br/>+ interval"]
end
W --> CD
M --> CD
T --> CD
R --> CD
CD --> L --> P --> S
style CD fill:#fff3e0,stroke:#c77700
style S fill:#e8f5e9,stroke:#2e7d32
Measurement uncertainty produces the covariance \(C_d\); inference uncertainty consumes it. Reported the old way — one method, one reference, a diagonal dvv_err — a \(\delta v/v\) study quietly discards both the methodological spread and the temporal correlation, and over-reports its own confidence. Carrying \(C_d\) through is what makes the stress posterior on the inference page reasonable.
Everything above is theory. Does the literature carry a measurement uncertainty at all? We loaded the survey of 103 ambient-noise \(\delta v/v\) monitoring studies (the same table that appears in the companion paper’s appendix) and asked, per study, what the uncertainty_treatment column says. The headline: the choices this page warns about — estimator, band, coda window, reference scheme — are reported unevenly, and the uncertainty definition is the single most often missing item.
import pandas as pd
surv = pd.read_csv("../literature/dvv_processing_parameters.csv").fillna("n/r")
surv = surv.replace("", "n/r")
def reports_uncertainty(s):
s = str(s).strip().lower()
return s not in ("n/r", "n/a", "none", "")
n = len(surv)
n_unc = surv["uncertainty_treatment"].map(reports_uncertainty).sum()
print(f"studies surveyed : {n}")
print(f" report *some* uncertainty treatment : {n_unc} ({100*n_unc/n:.0f}%)")
print(f" do NOT (uncertainty = n/r) : {n-n_unc} ({100*(n-n_unc)/n:.0f}%)")
print()
print("estimator family used (dvv_method):")
for meth, c in surv["dvv_method"].value_counts().items():
print(f" {meth:<28s} {c:3d}")studies surveyed : 103
report *some* uncertainty treatment : 89 (86%)
do NOT (uncertainty = n/r) : 14 (14%)
estimator family used (dvv_method):
stretching 29
MWCS 19
Stretching 11
Other 7
n/r 3
MWCS (doublet) 2
doublet 2
n/a (no dv/v computed) 2
MWCS (MSNoise) 2
doublet (Snieder 2002) 1
stretching + MWCS 1
CWIRE (stretching + damped LSQ) 1
n/a (no dv/v; companion is Makus 2024) 1
CWT (cross-wavelet) 1
doublet (moving-window CC) 1
stretching + doublet (compared) 1
n/a (not coda/noise dv/v) 1
doublet (MWCS) 1
stretching (autocorrelation CWI) 1
stretching and MWCS (doublet) 1
wavelet cross-spectrum (WCS) 1
stretching (doublet tested) 1
wavelet (WTS) 1
stretching and doublet (MWCS) 1
n/a (analytic model, no measurement) 1
MWCS (Brenguier all-pairs) 1
stretching (+MWCS check) 1
wavelet cross-spectrum (Morlet) 1
n/a (not a velocity-change study) 1
stretching (and doublet) 1
windowed CC, stretching, DTW (compared) 1
stretching (+MWCS compare) 1
doublet + Bayesian least-squares inversion (Brenguier 2014) 1
stretching, MWCS, DTW, WCS (review) 1
WCC, TS, DTW, MWCS, WCS, WTS, WTDTW (7 compared) 1About a quarter of the studies state no explicit measurement uncertainty at all, and among the ~three-quarters that do, the definitions are mutually incomparable (a within-measurement Weaver/Clarke floor, a between-pair standard error, a standard deviation, a bootstrap, or simply “SNR-based”) — exactly the \(\sqrt{N}\)-and-more ambiguity of §4–6, so even the studies that report an error bar cannot be compared without reading how it was defined. The full per-study table follows; the uncertainty-treatment column is the one to read against this page.
from html import escape
cols = [("authors_year", "Study"), ("application", "Application"),
("freq_band_hz", "Freq (Hz)"), ("coda_window_s", "Coda (s)"),
("dvv_method", "Method"), ("stack_scheme", "Reference / stack"),
("uncertainty_treatment", "Uncertainty treatment")]
rows = []
for _, r in surv.iterrows():
tds = []
for key, _ in cols:
v = str(r[key])
cls = ' class="nr"' if v.strip().lower() in ("n/r", "n/a") else ""
tds.append(f"<td{cls}>{escape(v)}</td>")
rows.append("<tr>" + "".join(tds) + "</tr>")
head = "".join(f"<th>{escape(lbl)}</th>" for _, lbl in cols)
print('<div class="survey-scroll">')
print(f"<table><thead><tr>{head}</tr></thead><tbody>")
print("\n".join(rows))
print("</tbody></table></div>")| Study | Application | Freq (Hz) | Coda (s) | Method | Reference / stack | Uncertainty treatment |
|---|---|---|---|---|---|---|
| Brenguier et al., 2008 | Volcano | 0.1-0.9 | n/r (slope of dtau vs tau; short scanning window) | MWCS (doublet) | Daily stacks referenced to long-term stack | linear-slope uncertainty; exclude >0.04% |
| Duputel et al., 2009 | Volcano | 0.1–1 | n/r | Other | Quasi-real-time stacks vs reference | n/r |
| Obermann et al., 2013 | Volcano | 0.1-1.0 | 10-45 | stretching | Daily stacks, referenced; coda at increasing lapse times | Weaver et al. coherence-based error formula |
| Hotovec-Ellis et al., 2014 | Volcano | 1-10 (also 1-5, 5-10) | windows after S arrival; >=5 with CCC>0.65 | doublet (Snieder 2002) | Stack across several hundred repeating-earthquake families (multiplets) | ~0.1% from SD of velocity fit |
| Hotovec-Ellis et al., 2015 | Volcano | 0.5-10 | 15 s window starting 2 s after first motion | stretching | Combines repeating-earthquake CWI with ambient-noise interferometry | reduced similarity (CC) in stretching window |
| Rivet et al., 2015 | Volcano | 0.25-2 | n/r | MWCS | Daily stacks vs reference; network-averaged dv/v | coherency + linear-regression error |
| De Plaen et al., 2016 | Volcano | 0.1-1.0, 0.5-1.0, 1.0-2.0, 2.0-4.0 | 5-35 (both branches) | MWCS | n/r | coherency + linear-regression error QC |
| Donaldson et al., 2017 | Volcano | 0.33-1.0 | 30 s window, min lag = interstation dist / 0.8 km/s | MWCS (doublet) | Daily NCFs stacked over 3-day moving window | weighted dt-t regression; coherence>0.65, err<0.1 s thresholds |
| Takano et al., 2017 | Volcano | 0.5-1, 1-2, 2-4 | -20 to +20 | MWCS | Cross-correlations 2012–2015, multiple bands | error bars from coherency |
| Lesage et al., 2018 | Volcano | 0.125-2 | 10-80 | stretching | Daily cross-correlations; 2013 stack as reference | empirical noise from AVV fluctuations (~0.05%) |
| De Plaen et al., 2019 | Volcano | 1.0-2.0 | 5-35 | MWCS | Daily autocorrelations, 2-day linear stack | cross-coherence + squared misfit; reject dt err>0.1 s or coh<0.6 |
| Donaldson et al., 2019 | Volcano | 0.1-0.4, 0.4-1.0, 1-2, 2-4, 4-16 | per-band (suppl. Table S1) | stretching + MWCS | n/r | reject dv/v when stretched-ref CC<0.4 |
| Olivier et al., 2019 | Volcano | 0.08-1.2 | 30 s windows; start = dist/700 m/s + 30 s | MWCS | n/r (passive image interferometry, daily) | CCF-coherence-based uncertainty |
| Yates et al., 2019 | Volcano | 0.1-1.0 | 16 s moving windows within 20-80 s | MWCS | n/r | QC (err<0.1 s, coh>0.7); error-weighted mean |
| Feng et al., 2020 | Volcano | 3-8 | [-2.8:-0.4] and [0.4:2.8] (=[-14Dt:2Dt],[2Dt:14Dt]) | stretching | Jan 2017–Jun 2018 NCFs | absolute error from stretching CC |
| Hotovec-Ellis et al., 2022 | Volcano | n/r | n/r | CWIRE (stretching + damped LSQ) | Coda Wave Interferometry with Repeating Earthquakes (CWIRE), per-cluster pairs | n/r |
| Köpfli et al., 2024 | Volcano | n/a (wavefield amplitude features, not dv/v) | n/r | n/a (no dv/v; companion is Makus 2024) | Long-term NCF stacks; many dv/v series on a common grid | n/r |
| Yates et al., 2024 | Volcano | 0.25-2.5 (wavelet 0.1-8.0) | min lag 5 s, length 20 cycles (e.g. 5-25 at 1 Hz) | CWT (cross-wavelet) | n/r | slope SD from covariance; coherence weighting |
| Yukutake et al., 2025 | Volcano | 0.1-0.9, 0.5-2.0, 1.0-4.0 | 20-40 | MWCS | Long-term interferometry stacks, 2003–2020 | 1-2 sigma errors as inversion weights |
| Schaff & Beroza 2004 | Earthquake/Fault | n/r (unfiltered) | 1.4 s moving windows through coda | doublet | Repeating-earthquake doublets; delays vs pre-mainshock repeaters | formal slope SE; CC weighting |
| Pacheco & Snieder 2005 | Earthquake/Fault | n/r | n/r | Other | n/r | n/r (analytical kernel) |
| Rubinstein & Beroza 2005 | Earthquake/Fault | n/r | n/a (direct-S delays, not coda) | doublet (moving-window CC) | Repeating-earthquake travel-time delays before vs after mainshock | CC>0.8 and SNR>4:1 QC |
| Wegler & Sens-Schönfelder 2007 | Earthquake/Fault | 2-100 (2 Hz high-pass) | 5-14 | stretching | Daily autocorrelations vs ~two-week pre-event reference; grid search (10000 trials, dv/v ±0.03) | day-to-day fluctuation ~0.1%; CC>0.5 |
| Brenguier, Campillo, Hadziioannou, Shapiro, Nadeau, Larose 2008 | Earthquake/Fault | 0.1–0.9 | n/r | Stretching | 1550-day reference vs overlapping 5-day moving windows; 30-day time resolution | Averaged over 78 receiver pairs plus 5-day segments |
| Hadziioannou, Larose, Coutant, Roux, Campillo 2009 | Earthquake/Fault | 2.5 MHz ultrasonic (lab, not seismic band) | 12.5-50 us (lab) | stretching + doublet (compared) | Noise-correlation stacking | analytic std of stretching-CC fluctuations |
| Sawazaki, Sato, Nakahara, Nishimura 2009 | Earthquake/Fault | n/r | n/r | Other | Coda deconvolution of surface vs 100-m borehole records | n/r |
| Wegler, Nakahara, Sens-Schönfelder, Korn, Shiomi 2009 | Earthquake/Fault | 2-8 and 0.1-0.5 | 5-14 (2-8 Hz); 10-100 (0.1-0.5 Hz) | stretching | Daily auto-/cross-correlations inverted daily for 2 months pre/post; pre-event reference | SD over Green's-tensor components; R>0.5 |
| Chen, Froment, Liu, Campillo 2010 | Earthquake/Fault | n/r | ~25 s after ballistic, up to +-200 s lapse | MWCS | Noise cross-correlation stacking; sub-array comparison (156 broadband stations) | n/r |
| Nakata & Snieder 2011 | Earthquake/Fault | 1-13 | n/a (deconvolution interferometry, direct S-wave) | n/a (not coda/noise dv/v) | >300 earthquakes (Jan 1–May 26 2011); borehole-to-surface deconvolution | SD of travel times over events |
| Rivet, Campillo, Shapiro, Cruz-Atienza, Radiguet, Cotte, Kostoglodov 2011 | Earthquake/Fault | 0.037-0.27 (periods 3.7-27 s) | n/r (coda used, no length) | doublet (MWCS) | n/r | previous-epoch referencing; no explicit error bars |
| Hobiger, Wegler, Shiomi, Nakahara 2012 | Earthquake/Fault | 0.125-0.25, 0.25-0.5, 0.5-1.0 | ten periods, starting 7.5 periods after direct wave | stretching | Daily cross-correlations vs long-term reference Green's function | CC weighting; 18-component weighted average |
| Minato, Tsuji, Ohmi, Matsuoka 2012 | Earthquake/Fault | 2-5 | 2-10 lag | stretching | n/r | SD from pre/post-seismic values |
| Obermann, Planès, Larose, Sens-Schönfelder, Campillo 2013 | Earthquake/Fault | 20 Hz central, ~12 Hz bw (numerical) | 1.5 s windows, centered ~1.8-6.6 s | stretching | Averaged over 10 random medium realizations per configuration | std over ten random-media realizations |
| Brenguier, Campillo, Takeda, Aoki, Shapiro, Briand, Emoto, Miyake 2014 | Earthquake/Fault | 0.1–0.9 | n/r | MWCS | Daily cross-correlations; linear inversion/regularization for continuous time series | Stress sensitivity ~0.001 MPa^-1 inferred; detailed dv/v uncertainty n/r |
| Liu, Huang, Peng, Su 2014 | Earthquake/Fault | 0.125-1 (1-8 s; subbands 1-2,2-4,4-8 s) | 30-130 | MWCS | Noise cross-correlation stacking (Aug 2004 – Sep 2011) | 2-sigma SD across nine components |
| Taira, Brenguier, Kong 2015 | Earthquake/Fault | 0.1-0.9 | -60 to -20 and +20 to +60 | MWCS | MSNoise cross-correlations; decimated to 20 Hz | Jackknife 95% CI; 2-sigma SD |
| Gassenmeier, Sens-Schönfelder, Eulenfeld, Bartsch, Victor, Tilmann, Korn 2016 | Earthquake/Fault | 1-3, 4-6, 7-10 (primary 4-6) | 10-15 (for 4-6 Hz) | stretching (autocorrelation CWI) | Daily autocorrelations; two-step iterative reference | Gaussian fit to correlation values (CIs) |
| Hillers, Campillo, Brenguier, Moreau, Agnew, Ben-Zion 2019 | Earthquake/Fault | 0.2-2 (sub-bands 0.2-0.4,0.4-0.8,0.8-1.6,0.3-1.5) | 20-50 (also 20-40,30-50,40-60) | stretching and MWCS (doublet) | Average-waveform reference for stretching; MWCS also applied | P-value QC; cc>=0.85 inversion |
| Wang, Campillo, Brenguier, Lecointre, Takeda, Hashima 2019 | Earthquake/Fault | period bands 8-30 s and 15-50 s | -400 to 400 (moving 30 & 50 s) | doublet | Monthly velocity-change estimates | SD error bars from 2009-2010 |
| Mao, Mordret, Campillo, Fang, van der Hilst 2020 | Earthquake/Fault | 0.5-2.0, 0.75-3.0, 1.1-2.2, 1.5-6.0, 2.0-8.0 | n/r (real data); synthetic 18.65-35 | wavelet cross-spectrum (WCS) | Noise cross-correlation stacking | wavelet cross-spectrum amplitude weighting; coherence threshold |
| Poli, Marguin, Wang, D'Agostino, Johnson 2020 | Earthquake/Fault | 0.5-1 | start 10/20/30 s, length 20/40/60 s | stretching (doublet tested) | n/r | CC-weighting; cc>0.9 retained |
| Boschelli, Zaccarelli, Faenza 2021 | Earthquake/Fault | >1 (1 Hz high-pass) | n/r (variable lapse window) | stretching | Daily autocorrelation functions vs mean waveform | SD across non-overlapping windows |
| Lu & Ben-Zion 2021 | Earthquake/Fault | 8.0–12.0 | 3 (moving windows, 1.5 s step) | Other | 10-min stacks; adaptive Gaussian smoothing (20 min growing to 24 hr over 3 days) | Standard deviations tracked; stabilized via smoothing |
| Sheng, Mordret, Sager, Brenguier, Boué, Rousset, Vernon, Higueret, Ben-Zion 2022 | Earthquake/Fault | >1 | n/r | wavelet (WTS) | Weekly stacked correlations with ~2-month smoothing | dt uncertainty as inverse weight |
| Mainsant, Larose, Brönnimann, Jongmans, Michoud, Jaboyedoff (2012) | Landslide | 10-12 (analyzed 4-25) | 0.2-2 (both sides) | stretching | Daily noise correlograms (cross-correlation functions) | Weaver et al. (2011) CC-based error (~1-2%) |
| Voisin, Garambois, Massey, Brossier (2016) | Landslide | ~6-8 (annual dv/v pattern best resolved) | n/r | Stretching | Daily/seasonal stacks vs reference | Correlation-coefficient weighting; comparison with piezometer |
| Harba, Pilecki (2017) | Landslide | High-frequency seismic noise (~5-20+, dispersion-curve range) | n/r (interferometric Green's-function retrieval) | Other | Cross-correlation / interferometric stacks; dispersion inversion (neighbourhood algorithm) | Dispersion-inversion misfit; comparison with MASW |
| Bertello, Berti, Castellaro, Squarzoni (2018) | Landslide | n/a (active-source ReMi/MASW dispersion, not coda dv/v) | n/r | n/r | Time-lapse interferometry + time-lapse active MASW | n/r |
| Bièvre, Franz, Larose, Carrière, Jongmans, Jaboyedoff (2018) | Landslide | ~5-15 (surface-wave band; reported sensitivity to shallow layer <=2 m) | n/r | Stretching | Daily correlations, multi-year reference | Correlation coefficient; seasonal-cycle modeling to separate from precursors |
| Colombero, Baillet, Comina, Jongmans, Larose, Valentin, Vinciguerra (2018) | Landslide | 2-20 (analysis); 2-4 strongest annual signal | 0.5-2 (and symmetric -2 to -0.5) | Stretching | Daily correlations, multi-year reference (2013-2016) | Correlation coefficient (~0.9 at low freq) as reliability metric |
| Bontemps, Lacroix, Larose, Jara, Taipe (2020) | Landslide | n/r | n/r | n/r | Daily stacks vs reference | n/r |
| Fiolleau, Jongmans, Bièvre, Chambon, Baillet, Vial (2020) | Landslide | 1-12 (monitoring); dv/v in 2-4 and 8-10 bands; block resonance ~9-16 | 0.05-1.5 | Stretching | Hourly records, daily averaging vs reference | Correlation-coefficient degradation tracked across bands; multi-parameter precursor timing |
| Le Breton, Bontemps, Guillemot, Baillet, Larose (2021) | Landslide | 1.3-20 (commonly 10-14; 0.1-1 for large/deep) | end of direct waves to 5-10 periods (site-dependent) | stretching and doublet (MWCS) | Daily stacks vs long-term reference | Weaver et al. (2011) rms + SD over pairs/windows; discard CC<0.6 |
| Fan et al. (2023) | Landslide | High-frequency (~5-20) | Sub-second to a few s | Stretching | Short-window stacks for rapid detection | Decorrelation/CC checks; rapid-detection confidence discussion |
| Liu, Li, Liang, Zhang, Jiang, Huang (2025) | Landslide | 5-25 (raw 1-30); 15 Hz top ~15 m, 5 Hz 15-40 m | 0.4-2.0 | Stretching | 20-min interval CCFs vs reference (SNR>4 wave-packet selection) | 95% confidence interval (~0.05% uncertainty at 5 Hz) |
| Whiteley, Chambers, Uhlemann et al. (2026) | Landslide | 6-10 (plus two unspecified bands) | n/r | MWCS | Time-lapse interferometry + MASW between recording periods | none (data report) |
| de Wit, Snieder (2026) | Landslide | n/r (high-frequency near-surface) | n/r | Stretching | Time-lapse stacks vs reference | Decorrelation/CC; comparison with radar surface deformation |
| Sens-Schönfelder & Wegler, 2006 | Groundwater/Hydrology | 0.5 Hz high-pass | 2-8 | stretching | Daily autocorr vs yearly reference | SD over non-overlapping windows |
| Tsai, 2011 | Groundwater/Hydrology | n/a (theory of thermoelastic/hydrologic velocity change) | n/r | n/a (analytic model, no measurement) | n/r | n/r |
| Hillers, Campillo & Ma, 2014 | Groundwater/Hydrology | n/r | n/r | n/r | n/r | n/r |
| Lecocq, Longuevergne, Pedersen, Brenguier & Stammler, 2017 | Groundwater/Hydrology | 0.1-0.8 | 20-100 (and -20 to -100) | MWCS (Brenguier all-pairs) | Daily CCFs, 31-day rolling window | LSQ slope error as inverse-variance weight; bootstrap |
| Nimiya, Ikeda & Tsuji, 2017 | Groundwater/Hydrology | 0.1-0.9 | 100 | stretching (+MWCS check) | Daily CCFs from 30-min segments; 1-yr reference, 30-day moving stack | SD over six 50-s subwindows |
| Wang, Brenguier, Campillo, Lecointre, Takeda & Aoki, 2017 | Groundwater/Hydrology | 0.1-0.25 and 0.5-2.0 | 30 s (0.1-0.25 Hz), 10 s (0.5-2 Hz) | MWCS | Daily CCFs; ~1-yr reference, moving-window current stack | none (QC only: dt<=0.2 s, coh>=0.5) |
| Clements & Denolle, 2018 | Groundwater/Hydrology | 0.1-0.25 and 0.5-2.0 | 30 s (0.1-0.25 Hz), 10 s (0.5-2 Hz) | MWCS | Daily stacks vs reference | none (QC only: dt<=0.2 s, coh>=0.5) |
| Kim & Lekic, 2019 | Groundwater/Hydrology | 0.01-8 | n/r | stretching | Multi-year (~20 yr) record; yearly-scale comparisons | none (grid-search CC max) |
| Andajani, Tsuji, Snieder & Ikeda, 2020 | Groundwater/Hydrology | 0.1-0.9 | 100 | stretching | n/r | stretching CC as quality indicator (no error bars) |
| Gaubert-Bastide, Garambois, et al., 2022 | Groundwater/Hydrology | 2-5 | n/a (ballistic Love-wave window, not coda) | stretching | Hourly correlations over 19 days, two fill/drain cycles | eps_max +- sigma (stretching variance) |
| Illien, Andermann, Sens-Schönfelder, et al., 2022 | Groundwater/Hydrology | 4-8 (Chaku); 2-4 (Gumba) | 12 periods (~3 s) after 4-period skip | stretching | ~3-yr continuous time series | none (multi-reference stack) |
| Mao, Lecointre, van der Hilst & Campillo, 2022 | Groundwater/Hydrology | 2-4 | 2-8 | stretching | Daily CCFs stacked over 20 days, 5-day step; pairs <50 km | cc^2-weighted channel mean (no explicit error bars) |
| Clements & Denolle, 2023 | Groundwater/Hydrology | 2-4 | 2-8 | stretching | Daily stacks across ~700 stations vs reference | cc^2-weighted channel mean (no explicit error bars) |
| Delouche, et al., 2023 | Groundwater/Hydrology | 0.33-1 (1-3 s period) | 15-55 | stretching | n/r | Weaver et al. (2011) RMS error |
| Ermert, Cabrera, Sawade, et al., 2023 | Groundwater/Hydrology | 0.5-1, 1-2, 2-4, 4-8 | 4-10x and 8-20x longest period (e.g. 8-16 & 16-40 s) | stretching | Clustered (GMM) autocorrelation stacks | CC_best>0.6 QC; no formal dv/v error |
| Fokker, Ruigrok, Hawkins & Trampert, 2023 | Groundwater/Hydrology | 1.3-1.6 | n/r (coda of cross-coherence) | stretching | n/r | standard error sigma/sqrt(n) |
| Zhang et al., 2023 | Groundwater/Hydrology | 0.1-1.0 (also 0.3-1.2, 0.5-1.5, 1.0-2.0) | n/r (dynamic windows, 3.0 & 2.0 km/s) | MWCS | Continuous 2013-2022 record | SD (avg ~0.009%) |
| Mao, Ellsworth, Zheng & Beroza, 2025 | Groundwater/Hydrology | 0.1-0.3 | 300 s window from 1.3*t0 | stretching | Daily CCFs stacked, multi-day step; two-decade record | Weaver et al. (2011) error formula |
| Mordret et al., 2016 | Cryosphere | 0.1-0.3 | 300 s window from 1.3*t0 | stretching | Daily cross-correlations over ~2-year record | Weaver et al. (2011) error formula |
| James et al., 2019 | Cryosphere | 3-30 (18 bands; 13 Hz inversion) | +-0.2 to 2.0 (center lag) | MWCS | Daily cross-correlations stacked; reference relative to Jan 2014 | 95% CI from posterior |
| Guillemot et al., 2020 | Cryosphere | n/a (resonance-freq tracking, not dv/v) | n/r | n/a (no dv/v computed) | Daily correlograms from hourly raw data | n/r |
| Guillemot et al., 2021 | Cryosphere | n/a (resonance-freq tracking, not dv/v) | n/r | n/a (no dv/v computed) | PSD-based; finite-element modal modeling | n/r |
| Lindner et al., 2021 | Cryosphere | 2-8 (analysis 1-20) | -5 to 5 lag | wavelet cross-spectrum (Morlet) | 15-year continuous record; daily/seasonal stacks | 1 SD |
| Luo et al., 2023 | Cryosphere | 0.1-1 (also 0.1-0.5, 1-2) | 20-70 | MWCS | Daily autocorrelations stacked | fitted +-sigma; 95% CI |
| Gassenmeier et al., 2015 | Geothermal/Reservoir | 1.5-3 | moving windows after 300 m/s phase | stretching | 1-hour segments cross-correlated, stacked to daily | Weaver et al. (2011); scaling-coeff CIs |
| Hillers et al., 2015 | Geothermal/Reservoir | n/r | Coda of noise correlations | Stretching | Daily noise correlations around 2006 stimulation | Sensitivity-kernel imaging of velocity-change location |
| Obermann et al., 2015 | Geothermal/Reservoir | 0.1-1 | 20 s window centered ~15 s in coda | stretching | Daily cross-correlations; reference stacks | none |
| Czarny et al., 2016 | Geothermal/Reservoir | 0.6-1.2 | n/r | Stretching | Continuous cross-correlations over ~42 days | n/r |
| Olivier et al., 2017 | Geothermal/Reservoir | 0.25-1.0, 0.5-2.0, 0.75-3.0, 1.0-4.0, 1.5-6.0, 2.0-8.0 | 20 s (-40 to -20 and 20 to 40) | MWCS (MSNoise) | Daily cross-correlations across array | two-sigma SD; CC>0.85 rejection |
| Taira et al., 2018 | Geothermal/Reservoir | 0.25-1.0, 0.5-2.0, 0.75-3.0, 1.0-4.0, 1.5-6.0, 2.0-8.0 | 20 s (-40 to -20 and 20 to 40) | MWCS (MSNoise) | Daily NCFs from 30-min segments; 5-day stacks | two-sigma SD; CC>0.85 rejection |
| Kristjánsdóttir et al., 2019 | Geothermal/Reservoir | n/r | n/r | Stretching | Daily correlations (MSNoise workflow) | Seasonal and noise-source effects discussed as caveats |
| Snieder, Grêt, Douma & Scales 2002 | Methodology | n/a (dispersion/preprocessing paper, no dv/v) | n/r | n/a (not a velocity-change study) | n/r | n/r |
| Bensen et al. 2007 | Methodology | 0.1-0.9 | 6 s windows overlapping 3 s | MWCS | daily cross-correlation + temporal stacking | WLS phase-fit error (corrected ~6x) |
| Clarke, Zaccarelli, Shapiro & Brenguier 2011 | Methodology | 0.1-0.9 | 6 s windows overlapping 3 s | MWCS | current stack vs reference, weighted linear regression dt vs t | WLS phase-fit error (corrected ~6x) |
| Hadziioannou, Larose, Baig, Roux & Campillo 2011 | Methodology | 0.1-0.9 | n/r (whole coda t1-t2) | stretching (and doublet) | waveform clustering to enhance daily stacks | remnant coherence (CC) |
| Weaver, Hadziioannou, Larose & Campillo 2011 | Methodology | 0.05-0.5 | up to t0=30 | stretching | reference vs current correlation | max CC>0.9 (no formal error) |
| Zhan, Tsai & Clayton 2013 | Methodology | 0.05-0.5 | up to t0=30 | stretching | reference vs current | max CC>0.9 (no formal error) |
| Lecocq, Caudron & Brenguier 2014 | Methodology | synthetic (Ricker 12.5; noise 0.5-30) | 1-10 lapse; sub-windows 0.1-1.0 s | windowed CC, stretching, DTW (compared) | daily CCF + reference stack | WLS lag-vs-lapse; CC>=0.6 |
| Mikesell, Malcolm, Yang & Haney 2015 | Methodology | 0.33-1.0 (1-3 s) | (dist/1.8)+20 to 120 | stretching | n/r | Weaver et al. (2011) rms formula |
| Stehly, Froment, Campillo, Liu & Chen 2015 | Methodology | 0.1-1.0 (real); 0.15-0.65 (synthetic) | [15,35] and [-35,-15] | stretching (+MWCS compare) | daily correlations curvelet-filtered before stacking | CC>0.7 rejection; ~+-0.1% significance |
| Daskalakis, Evangelidis, Garnier, Melis, Papanicolaou & Tsogka 2016 | Methodology | n/r (numerical study) | 1 s overlapping windows | stretching | normalized cross-correlation functions | SD over 20 receivers |
| Obermann, Planès, Hadziioannou & Campillo 2016 | Methodology | 0.15-0.90 | -60 to +60 (doublet over coda) | doublet + Bayesian least-squares inversion (Brenguier 2014) | n/r | Bayesian least-squares inversion; error bars from inversion |
| Obermann & Hillers 2019 | Methodology | scale-dependent | lapse-time/depth dependent | Other | reference and moving-stack strategies reviewed | synthesizes error sources: noise-source variability, processing choices, sensitivity kernels |
| Jiang & Denolle 2020 | Methodology | user-defined | user-defined | Other | linear/PWS/robust stacking; reference + moving stacks | offers multiple dv/v estimators so users can cross-check; parallel/HDF5 reproducible pipeline |
| Wang & Yao 2020 | Methodology | n/r (review) | n/r (review) | stretching, MWCS, DTW, WCS (review) | reference and stacking strategies reviewed | Bayesian least-squares / MCMC (review) |
| Yuan, Bryan & Denolle 2021 | Methodology | varies (e.g. 0.5-2.2) | 45-75 (example) | WCC, TS, DTW, MWCS, WCS, WTS, WTDTW (7 compared) | n/r | residuals vs truth over realizations |
: Processing choices of the 103 surveyed \(\delta v/v\) studies (scroll within the box). n/r (red) = not reported in the source. Sorted as in the survey CSV; full machine-readable version and DOIs live in the repository’s literature/ folder and on the references page.
codameter.uq_measurement.weaver_stretching_errorprocessing_ensemble / EnsembleResulttemporal_error_covariance, effective_sample_sizeglobal_reference_inversion, single_reference_dvvcodameter.inverse.linear_fit (supply C_d for the weights)Complements the coda-window design scorer in codameter.uq-measurement (which quantity to measure) — this module is the measurement error of that quantity.