Process attribution

Which forcing drives the stress at depth — and with what confidence

Thesis This is step 4 of the uncertainty arc, and its endpoint. Steps 1–3 deliver $\Delta\sigma_{ij}(z,t)$ with a posterior. Attribution asks the community’s real question — which process (thermoelastic, poroelastic, load, damage, tectonic) produced that stress, and how sure are we. The forward physics and the executable module for this step live in the companion coupling-framework project; codameter supplies the measurement covariance $C_d$ and the stress posterior the attribution consumes.

The terms that carry stress in isolation

Each physical process contributes a stress term that can act on its own, so the depth–time stress is a sum of mechanism terms,

\[ \Delta\sigma(z,t) = \Delta\sigma_{\text{thermo}} + \Delta\sigma_{\text{poro}} + \Delta\sigma_{\text{damage}} + \Delta\sigma_{\text{tectonic}} + \dots, \]

where each $\Delta\sigma_k(z,t)$ is predicted by the corresponding forward model in the framework (Berger thermoelastic, Roeloffs poroelastic including partial saturation, Snieder damage/healing, a tectonic term). Attribution is the inverse: given the observed $\Delta\sigma(z,t)$ and its covariance, estimate how much each term contributed and how well it is resolved.

Attribution as optimal fingerprinting

The natural frame is the geophysical analogue of climate detection-and-attribution — optimal fingerprinting scaled by the very measurement covariance $C_d$ this arc builds.

Apportionment. Each forcing $k$ predicts a depth–time stress fingerprint; a generalized-least-squares regression weighted by $C_d$ gives its amplitude, and a variance decomposition gives its share $A_k(z)$ of the stress variability at each depth. The cross-covariances quantify how ambiguous the attribution is when two forcings are collinear (thermoelastic and hydrologic are both seasonal).
Coupling. When forcings couple, the additive decomposition gives way to Sobol variance indices whose interaction term is the coupling contribution — the coupling framework expressed as an attribution component.
Confidence. Sampling the joint posterior — forcing amplitudes, material priors, and the measurement realization — turns each share into a credible interval and, decomposed further, shows whether a weak attribution is data-limited or model-limited — i.e., which additional data would most reduce it.

The deliverable is a depth–source attribution chart: per depth, a stacked bar of $A_k(z)$ with credible-interval whiskers plus an explicit coupling slice (“at 200 m: 60% poroelastic ±15%, 25% thermoelastic ±10%, 10% co-seismic, 5% coupling”).

Best practice versus deviation

Choice	Best practice	Common deviation	Consequence
Forcing separation	Physical diagnostic (seasonal phase lag, branch asymmetry, coupling test)	Attribute by eye	Mis-attribution when forcings are collinear
Regression weighting	GLS weighted by the measurement covariance $C_d$	Unweighted / ordinary least squares	Over-weights low-coherence epochs; wrong shares
Coupling	Sobol indices; report the interaction term explicitly	Assume additive contributions	Hidden coupling mis-assigned to a single forcing
Attribution uncertainty	Credible intervals plus a data- vs model-limited split	A single point attribution	No measure of confidence, nor of which data would most reduce it

Status — lives in the companion framework

Process attribution is the contribution of the dvv-coupling-framework project, which owns the per-mechanism forward models, the fabric diagnostic that fixes the acoustoelastic $\beta_{ij}$, and the uq_attribution module (optimal-fingerprinting shares, Sobol indices, detection, and the data- vs model-limited budget). It consumes the $\Delta\sigma_{ij}(z,t)$ posterior that step 3 delivers. Tracked in framework issues #17 and #20.

References

--- title: "Process attribution" subtitle: "Which forcing drives the stress at depth — and with what confidence" bibliography: references.bib --- ::: {.keyidea} [Thesis]{.k-title} This is **step 4 of the uncertainty arc**, and its endpoint. Steps 1–3 deliver $\Delta\sigma_{ij}(z,t)$ [with a posterior](uncertainty-3-stress.qmd). Attribution asks the community's real question — *which* process (thermoelastic, poroelastic, load, damage, tectonic) produced that stress, and how sure are we. The forward physics and the executable module for this step live in the companion **[coupling-framework project](https://github.com/mdenolle/dvv-coupling-framework)**; codameter supplies the measurement covariance $C_d$ and the stress posterior the attribution consumes. ::: ## The terms that carry stress in isolation Each physical process contributes a stress term that can act on its own, so the depth–time stress is a sum of mechanism terms, $$ \Delta\sigma(z,t) = \Delta\sigma_{\text{thermo}} + \Delta\sigma_{\text{poro}} + \Delta\sigma_{\text{damage}} + \Delta\sigma_{\text{tectonic}} + \dots, $$ where each $\Delta\sigma_k(z,t)$ is predicted by the corresponding forward model in the framework (Berger thermoelastic, Roeloffs poroelastic including partial saturation, Snieder damage/healing, a tectonic term). Attribution is the inverse: given the observed $\Delta\sigma(z,t)$ and its covariance, estimate how much each term contributed and how well it is resolved. ## Attribution as optimal fingerprinting The natural frame is the geophysical analogue of climate detection-and-attribution — **optimal fingerprinting** scaled by the very measurement covariance $C_d$ this arc builds. - **Apportionment.** Each forcing $k$ predicts a depth–time stress fingerprint; a generalized-least-squares regression weighted by $C_d$ gives its amplitude, and a variance decomposition gives its share $A_k(z)$ of the stress variability at each depth. The cross-covariances quantify how ambiguous the attribution is when two forcings are collinear (thermoelastic and hydrologic are both seasonal). - **Coupling.** When forcings couple, the additive decomposition gives way to **Sobol** variance indices whose interaction term *is* the coupling contribution — the coupling framework expressed as an attribution component. - **Confidence.** Sampling the joint posterior — forcing amplitudes, material priors, and the measurement realization — turns each share into a credible interval and, decomposed further, shows whether a weak attribution is **data-limited or model-limited** — i.e., which additional data would most reduce it. The deliverable is a depth–source attribution chart: per depth, a stacked bar of $A_k(z)$ with credible-interval whiskers plus an explicit coupling slice ("at 200 m: 60% poroelastic ±15%, 25% thermoelastic ±10%, 10% co-seismic, 5% coupling"). ## Best practice versus deviation | Choice | Best practice | Common deviation | Consequence | |---|---|---|---| | Forcing separation | Physical diagnostic (seasonal phase lag, branch asymmetry, coupling test) | Attribute by eye | Mis-attribution when forcings are collinear | | Regression weighting | GLS weighted by the measurement covariance $C_d$ | Unweighted / ordinary least squares | Over-weights low-coherence epochs; wrong shares | | Coupling | Sobol indices; report the interaction term explicitly | Assume additive contributions | Hidden coupling mis-assigned to a single forcing | | Attribution uncertainty | Credible intervals plus a data- vs model-limited split | A single point attribution | No measure of confidence, nor of which data would most reduce it | --- ::: {.callout-note} ## Status — lives in the companion framework Process attribution is the contribution of the [dvv-coupling-framework](https://github.com/mdenolle/dvv-coupling-framework) project, which owns the per-mechanism forward models, the fabric diagnostic that fixes the acoustoelastic $\beta_{ij}$, and the `uq_attribution` module (optimal-fingerprinting shares, Sobol indices, detection, and the data- vs model-limited budget). It consumes the $\Delta\sigma_{ij}(z,t)$ posterior that [step 3](uncertainty-3-stress.qmd) delivers. Tracked in framework issues #17 and #20. ::: ### References {.unnumbered} ::: {#refs} :::