A novel automated procedure for determining steady-state friction conditions in the context of rate- and state- friction analysis

In the framework of empirically-derived rate- and state- friction (RSF) laws, friction constitutive parameters a, b, and D_c  (and further sets of state parameters) are obtained from inverse modelling of laboratory data on the assumption that steady-state conditions are reached following the velocity steps. This method also includes removing any slip-dependent linear trends in friction by linear regression when steady-state conditions are considered to be achieved. The choice of where linear detrending, thereby where to assume the attainment of steady-state friction conditions is therefore key for a correct retrieval of the modelled RSF parameters and their consequent use in modelling of earthquake nucleation. Nonetheless, to date this procedure is still user-dependent and as such, RSF outputs may differ ceteris paribus.

To better elucidate the detrimental consequences of an incorrect assumption of steady-state friction conditions in RSF analysis, in this study synthetic velocity steps were generated with superimposed random Gaussian noise, characterized by increasing characteristic slip distances in the second set of state variables, D_c2,from 0 to 500 µm. In each velocity step, steady-state conditions were assumed starting at progressively larger displacements with respect to the occurrence of the velocity jump. This means that the arbitrarily chosen “steady-state” may or may not correspond to the true steady-state conditions. To retrieve RSF parameters, a slip window of constant size (i.e., 100 µm) was applied from the selected “steady-state” point onwards to remove any linear trend in friction, implying that the remainder of the velocity step beyond the slip window is also at steady-state. During each RSF analysis, the slope calculated from linear regression within the 100 µm long slip window after the velocity steps is systematically compared with the slope computed from linear regression prior to the velocity steps.

Our results show that:

• while a, b₁ and D_c1 are essentially constant regardless of the choices of steady-state and equal to the true values used to generate the synthetic velocity steps, b₂ and D_c2 may significantly differ if D_c2 is commensurate with the whole displacement window that contains the velocity step;
• all modelled RSF parameters coincide with the true ones when the ratio of the slopes before and after the velocity steps approach unity; this observation can be regarded as a proxy for the achievement of the steady-state conditions and becomes increasingly relevant with larger D_c2.

Based on such evidence, we developed a routine that automates the above described work flow, providing a systematic and reproducible technique to determine steady-state friction and thus return the correct RSF parameters. Furthermore, this novel procedure determines the optimal minimum slip window size to remove slip-dependent linear trends in friction and alerts the user when steady-state is not reached within a given step length and hence when D_c2 and b₂ cannot be properly determined with experimental data.