Slip, Differentiate, Observe: State and Parameter Estimation for Rate and State Friction from Noisy Data

TL;DR

Quantifying frictional properties of faults from indirect, noisy observations is challenging due to limited access to shear stress and internal state. The authors develop a nonlinear control framework that combines a Robust Exact Filtering Differentiator (REFD) for noise-robust slip-rate reconstruction with an exponentially convergent adaptive observer to estimate the RSF internal state $\psi$ and the parameters $(a-b)$ and $d_c$. They demonstrate that, in fast-slip regimes with sufficient data, RSF parameters can be recovered with errors on the order of 14–28% and the RSF state is tracked despite noise, while slow-slip periods are typically unobservable. The work provides convergence guarantees, explicit error bounds, and observability/identifiability analysis, offering a pathway to model-based inversions and reduced-order friction parameter estimation in both terrestrial and planetary fault settings.

Abstract

Quantifying frictional properties of interfaces remains a major challenge in both terrestrial and extraterrestrial geomechanics, where available samples, laboratory apparatuses, and geophysical observations are inherently limited. We introduce an analytic and numerical framework, grounded in nonlinear control theory, to infer the emergent frictional behavior of seismic faults. From noisy slip measurements, we first reconstruct the slip rate and frictional response in finite time using a Robust Exact Filtering Differentiator (REFD) that attenuates measurement noise. Building on these reconstructions, we design an exponentially convergent adaptive-gain observer that estimates the internal state variable and the key parameters (a - b) and dc of the rate-and-state friction (RSF) law, widely used in fault mechanics. Numerical experiments show that, in fast slip regimes where data are sufficiently rich, the method recovers RSF parameters with errors on the order of 20% and accurately tracks the RSF state variable despite noise contamination, whereas slowly varying sliding periods lack the observability required for reliable estimation. We also establish observability and identifiability conditions for the extended system, enabling the determination of additional parameters and outlining pathways for more advanced control-theoretic approaches to friction and state identification in fault systems. Although we apply the approach to a reduced spring-slider analogue, it improves on classical RSF calibration methods that depend on laboratory access to shear and normal stress. It also offers convergence guarantees and explicit error bounds, and it can further support model-based inversions that embed RSF in forward simulations.

Figures (6)

• Figure 1: Illustration of a one-degree-of-freedom spring slider model used as an analogue to a fault.
• Figure 2: True and noisy measurements of slip from spring-slider simulation. (a) True slip against time; the red shaded region marks the slow-slip interval, and the inset zooms into the fast-slip event. (b) Noisy measurement of slip during the slow-slip interval, plotted together with the true slip. (c) Noisy measurement of slip during the fast-slip interval, plotted together with the true slip. For clarity, the time axis in (c) and in the inset is plotted as $t-t_{ev}$ where $t_{ev}$ denotes the onset of the fast-slip event.
• Figure 3: Performance of the Robust Exact Filtering Differentiator (REFD) during the slow-slip regime. (a) Noisy slip measurements $y(t)$ (grey), true slip $u(t)$ (black), and REFD slip estimate (red dashed); the inset zooms into a short time interval and highlights the convergence of the estimate to the true signal. (b) True slip rate $v(t)$ (black) and REFD slip rate estimate (red) on a logarithmic scale. (c) True friction coefficient $\mu(t)$ (black) and reconstructed friction $\hat{\mu}(t)$. The subsequent fast-slip episode of the same signal is shown in Figure \ref{['fig4']}.
• Figure 4: Performance of the Robust Exact Filtering Differentiator (REFD) during the fast-slip regime. (a) Noisy slip measurements $y(t)$ (grey), true slip $u(t)$ (black), and REFD slip estimate (red dashed); the inset zooms into a short time interval and highlights the convergence of the estimate to the true signal. (b) True slip rate $v(t)$ (black) and REFD slip rate estimate (red) on a logarithmic scale. (c) True friction coefficient $\mu(t)$ (black) and reconstructed friction $\hat{\mu}(t)$.
• Figure 5: Performance of the observer during the fast-slip regime. (a) Estimation for state $\hat{\psi}(t)$ (red dashed) and true state $\psi(t)$ (black) in time. (b) True $d_c$ (black) and estimated $\hat{d_c}$ against iterations. (c) True $a-b$ (black) and estimated $\hat{a}-\hat{b}$ against iterations.