Seismic velocity changes as Bayesian stress & strain meters

In one sentence codameter turns a noisy \(\delta v/v\) time series into depth-resolved stress with a full posterior, carrying uncertainty from measurement noise and material-property priors through to the interpreted stress rather than reporting a single number.

codameter is a research method and open-source package — the executable companion to Denolle (in prep, JGR Solid Earth). It reads a relative seismic-velocity-change (\(\delta v/v\)) time series plus the environmental forcings that drive it (temperature, precipitation, earthquakes, surface load) and returns:

This site is the narrative companion to the API docs: it shows the reasoning — the theory, the equations, and the life of each figure — not just the function signatures.

Overview

Two ideas motivate the whole project.

1 · Reproducibility, turned into uncertainty quantification. Turning cross-correlations into a \(\delta v/v\) series takes a long sequence of signal-processing choices — estimator, frequency band, coda window, reference, stacking, coherence gating — made ad hoc and seldom reported. Because reasonable analysts choose differently, the same data yields different \(\delta v/v\), and even different stated uncertainties: a reproducibility problem. Rather than crown one pipeline, codameter treats the choice as a nuisance parameter and marginalises it, so the spread across reasonable choices becomes a structured measurement covariance \(C_d\). The diversity of processing choices, usually a weakness, becomes the raw material for an honest, propagatable uncertainty.

2 · Making \(\delta v/v\) interpretable — which stress, and what rheology. A \(\delta v/v\) value is only useful once it is tied to a mechanism. codameter propagates \(C_d\) through depth-sensitivity kernels to a profile of shear-velocity change, then through a petrophysical model to stress, so one can ask which component of the stress explains the observations. Along the way it tests whether the rheology behaves linearly — the forcings superpose and a closed-form posterior holds — or nonlinearly, where drainage and damage couple and the inference must be sampled. The connection is the acoustoelastic bridge relation \(\Delta\sigma_{ij}(z,t)=\beta_{ij}(z)\,\delta V_S/V_S\), which links the stress and strain representations of \(\delta v/v\) and lets a velocity change be read as a named, depth-resolved stress or strain component.

How to read this site

There are two distinct halves, and they are written in two different voices.

① The uncertainty arc in four steps

The theory pages lay out the core contribution: a forward chain that carries uncertainty from the raw measurement to stress at depth and its attribution. Each step consumes the previous step’s covariance.

  • 1 · Measurement — the marginal error in \(\delta v/v\). The error in the measurement itself, marginalised over processing choices (method, window, stacking, gating, and the reference, including the Brenguier (2014) all-to-all scheme). Two pages: first the marginal errors, and how they differ — the within-method floor, the methodological spread, temporal correlation, and reference choice; then turning them into a Bayesian measurement — a posterior \(\delta v/v(t)\) and the structured data covariance \(C_d\).
  • 2 · Depth — \(\delta v/v \to \delta V_S(z)/V_S\). Invert the per-band \(C_d\) through sensitivity kernels for a profile of shear-velocity change \(\delta V_S/V_S(z)\) and, in partially saturated ground, a density change \(\delta\rho/\rho(z)\) that needs a petrophysical model to separate. See propagating to depth.
  • 3 · Stress meter — \(\delta v/v\) as a stress meter. Push the depth posterior through layered acoustoelastic priors to stress and strain at depth, \(\Delta\sigma_{ij}(z,t)=\beta_{ij}(z)\,\delta V_S/V_S\), with a posterior. See stress meter; the companion inference UQ page holds the Bayesian machinery.
  • 4 · Attribution — which forcing drives it. Decompose the stress into the physical terms that can each act in isolation (thermoelastic, poroelastic, damage, tectonic) and quantify each share with uncertainty. See process attribution; the forward physics and the module live in the companion coupling-framework project.

👉 Read them in order: 1 · Measurement → 2 · Depth → 3 · Stress meter → 4 · Attribution.

② The implementation tutorial

A dynamic, runnable walkthrough that follows the scientific arc from completely decoupled forcings — thermoelastic, hydrological, loading, damage, each treated as an independent stress source — to identifying coupled mechanisms where that independence breaks down. Every figure on these pages is generated live from the package.

Step Page Phase
1 Data & site readiness 0–1
2 Decoupled forward models forward
3 Linear superposition & inversion 3–4
4 When decoupling breaks: coupling diagnostics 2
5 Interpretation: stress at depth 6
6 End-to-end Parkfield walkthrough 0–6

The scientific arc in one figure

The whole project runs along one axis: how strongly are the stress sources coupled? On the left, the forcings add linearly and a closed-form Gaussian posterior is exact. On the right, pore pressure, damage, and saturation feed back on each other, the posterior goes non-Gaussian, and we need the coupled MCMC machinery.

Code 
flowchart LR
    A["<b>Decoupled regime</b><br/>thermoelastic ⟂ hydrological ⟂ damage<br/>Eq. 6 linear superposition"]
    B["<b>Coupling diagnostics</b><br/>drainage Péclet number<br/>β<sub>eff</sub>(ω)"]
    C["<b>Coupled regime</b><br/>state-dependent forward operator<br/>Eq. 19 · MCMC posterior"]
    A -- "Pe<sub>d</sub> safe" --> D["Linear-Gaussian<br/>posterior (exact)"]
    A -- "Pe<sub>d</sub> marginal" --> B
    B -- "escalate" --> C
    C --> E["Non-Gaussian<br/>posterior (sampled)"]
    D --> F["Stress at depth<br/>+ full uncertainty"]
    E --> F
    style A fill:#ede7f6,stroke:#5e35b1
    style C fill:#fce4ec,stroke:#c62828
    style F fill:#e8f5e9,stroke:#2e7d32

flowchart LR
    A["<b>Decoupled regime</b><br/>thermoelastic ⟂ hydrological ⟂ damage<br/>Eq. 6 linear superposition"]
    B["<b>Coupling diagnostics</b><br/>drainage Péclet number<br/>β<sub>eff</sub>(ω)"]
    C["<b>Coupled regime</b><br/>state-dependent forward operator<br/>Eq. 19 · MCMC posterior"]
    A -- "Pe<sub>d</sub> safe" --> D["Linear-Gaussian<br/>posterior (exact)"]
    A -- "Pe<sub>d</sub> marginal" --> B
    B -- "escalate" --> C
    C --> E["Non-Gaussian<br/>posterior (sampled)"]
    D --> F["Stress at depth<br/>+ full uncertainty"]
    E --> F
    style A fill:#ede7f6,stroke:#5e35b1
    style C fill:#fce4ec,stroke:#c62828
    style F fill:#e8f5e9,stroke:#2e7d32

Both regimes funnel into the same destination — stress at depth with a quantified posterior — which is exactly what makes the Bayesian framing the unifying idea of the whole package.

Status — v0.1

Phase What it does v0.1
0 Data ingestion + QC ✅
1 Site characterisation; depth–frequency table ✅
2 Coupling diagnostics — Tier 1 (poroelastic) ✅
3 Linear-superposition design matrix (Eq. 6) ✅
4 Linear inversion (WLS, closed-form Gaussian posterior) ✅
5 Anomaly detection (posterior predictive checks) ✅
6 β-bridge stress at depth (uncertainty propagated) ✅
4 Coupled inversion (MCMC) v0.2
6 Water-table inversion v0.2
2 Tiers 2–4 (damage, saturation, thermo-capillary) v0.3 
pixi install
pixi run quarto render quarto/
# open quarto/_site/index.html

Every code cell runs against the installed codameter package, so the figures regenerate from source.

Back to top