[1] Accelerating displacements preceding some catastrophic landslides have been found to display a finite time singularity of the velocity v $1/(t c À t) [Voight, 1988a[Voight, , 1988b. Here we provide a physical basis for this phenomenological law based on a slider block model using a state-and velocity-dependent friction law established in the laboratory. This physical model accounts for and generalizes Voight's observation: Depending on the ratio B/A of two parameters of the rate and state friction law and on the initial frictional state of the sliding surfaces characterized by a reduced parameter X i , four possible regimes are found. Two regimes can account for an acceleration of the displacement. For B/A > 1 (velocity weakening) and X i < 1 the slider block exhibits an unstable acceleration leading to a finite time singularity of the displacement and of the velocity v $ 1/(t c À t), thus rationalizing Voight's empirical law. An acceleration of the displacement can also be reproduced in the velocity-strengthening regime for B/A < 1 and X i > 1. In this case, the acceleration of the displacement evolves toward a stable sliding with a constant sliding velocity. The two other cases (B/A < 1 and X i < 1 and B/A > 1 and X i > 1) give a deceleration of the displacement. We use the slider block friction model to analyze quantitatively the displacement and velocity data preceding two landslides, Vaiont and La Clapière. The Vaiont landslide was the catastrophic culmination of an accelerated slope velocity. La Clapière landslide was characterized by a peak of slope acceleration that followed decades of ongoing accelerating displacements succeeded by a restabilization. Our inversion of the slider block model in these data sets shows good fits and suggests a classification of the Vaiont landslide as belonging to the unstable velocity-weakening sliding regime and La Clapière landslide as belonging to the stable velocity-strengthening regime.

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents the passage from one approximant to another as the motion realized by a dynamical system with the property of group self-similarity. The derived approximants, because of their form, are named the self-similar factor approximants. These complement the obtained earlier self-similar exponential approximants and self-similar root approximants. The specific feature of the self-similar factor approximants is that their control functions, providing convergence of the computational algorithm, are completely defined from the accuracy-through-order conditions. These approximants contain the Padé approximants as a particular case, and in some limit they can be reduced to the self-similar exponential approximants previously introduced by two of us. It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions which include a variety of transcendental functions. For other functions, not pertaining to this exactly reproducible class, the factor approximants provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Padé approximants. This is illustrated by a number of examples showing the generality and accuracy of the factor approximants even when conventional techniques meet serious difficulties.

We consider the method of self-similar renormalization for calculating critical