Supporting Information

Seismic Velocity Changes as Stress and Strain Meters: A Unified Framework for Environmental, Tectonic, and Volcanic Monitoring

These supporting figures use synthetic \(\delta v/v\) with physically realistic shapes to illustrate the framework’s forward predictions and diagnostics.

Figure S1. Annual temperature diffusion into the subsurface for three thermal diffusivities (0.15, 0.6, 2.0 mm\(^2\)/s), showing depth profiles at different phases of the annual cycle. Red dashed lines mark the thermal skin depth \(1/\gamma\)

Figure S2. Berger (1975) thermoelastic stress solution. Left: shallow view dominated by term 1 (direct thermal stress, decaying with skin depth \(1/\gamma\)). Right: deep view dominated by term 2 (equilibrium response, decaying with horizontal wavenumber \(1/k\))

Figure S3. Thermoelastic \(\delta v/v\) sensitivity analysis. Panels show the depth profile for varying nonlinear response strength \(\partial(\rho v^2)/\partial\sigma_c\), surface \(\delta v/v\) versus temperature amplitude, sensitivity to Poisson’s ratio, and thermal skin depth versus diffusivity

Figure S4. Synthetic thermoelastic \(\delta v/v\) time series, including surface temperature with Fourier fit, \(\delta v/v\) for different temperature sensitivities, and thermal diffusion time delay

Figure S5. Poroelastic pore-pressure response to surface loading following Roeloffs (1988), including depth profiles after loading, hydraulic diffusivity dependence, undrained versus drained components, and Skempton coefficient sensitivity

Figure S6. Groundwater-level model following Okubo et al. (2024), including synthetic precipitation, \(\Delta\)GWL response for different hydrological memory timescales, and resulting hydrological \(\delta v/v\)

Figure S7. Detailed regime diagram showing which physical process dominates \(\delta v/v\) as a function of measurement frequency and depth sensitivity, including capillary/vadose-zone, hydrological, thermoelastic, and tectonic domains

Figure S8. Murnaghan (1937) equation-of-state diagnostics, including pressure-volume relations and velocity versus confining pressure for different nonlinear response strengths

Figure S9. Detailed nonlinear-elasticity diagnostics from \(\delta v/v\)–strain crossplots, including tidal/thermal strain, linear and nonlinear velocity response, and curvature from higher-order nonlinearity

Figure S10. Logarithmic healing models, including sensitivity to \(\tau_{\max}\) and \(\tau_{\min}\), a Parkfield-like synthetic time series, and the \(\sim 1/t\) healing-rate decay

Figure S11. Illustrative (synthetic) depth–source stress attribution, showing the shape of the §7.6 output. (a) Stacked per-mechanism shares \(A_k(z)\) of the depth–time stress \(\Delta\sigma(z,t)\) across sensitivity depths, with 68% credible-interval whiskers; (b) the coupling slice \(S_{Tk}-S_k\) (Sobol total-minus-first-order index) per mechanism. Values are hand-set exemplars; the data-driven version is produced by the companion codameter pipeline from the Eq. 7b stress posterior

Figure S12. Detailed homogeneous-half-space validity tests, including sensitivity kernels, peak sensitivity depth versus frequency, and relative error from velocity contrasts within the sensitivity kernel

Figure S13. Validity of the linear acoustoelastic approximation \(\delta v/v = \beta\epsilon_{kk}\) as a function of strain magnitude, including tidal, thermoelastic, and coseismic/strong-motion regimes

Table S1. Comprehensive parameter overview table summarizing typical ranges, controlling physical effects, and validity limits for all key model parameters (\(\kappa_T\), \(c\), \(\nu\), \(\nu_u\), \(B\), \(\beta\), \(\mu'\), \(\alpha\), \(\phi\), \(\epsilon_c\), \(\tau_{\min}\), \(\tau_{\max}\))