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. 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 S12. 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

Figure S13. 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

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