Fractal rock slope dynamics anticipating a collapse

Time series of dilatometric measurements of relative displacements on rock cracks on stable and unstable sandstone slopes were analyzed. The inherent dynamics of rock slopes lack any significant nonlinearity. However, the residuals obtained by removing meteorological influences are fat-tailed non-Gaussian fluctuations, with short-range correlations in the case of stable slopes. The fluctuations of unstable slopes exhibit self-affine dynamics of fractional Brownian motions with power-law long-range correlations and are characterized by asymptotic power-law probability distributions with decay coefficients outside the range of stable Lévy distributions.

Figure 1

Time series of dilatometric measurements of relative displacements on rock cracks on stable (a)-(c) and unstable (d), (e) sandstone slopes.

Figure 2

Linearly detrended time series of dilatometric measurements of relative displacements on rock cracks on stable (a) and unstable (b) sandstone slopes. Time series of atmospheric temperature (c), humidity (d), and precipitation (e) in the region.

Figure 3

Testing for nonlinearity in the relationship between atmospheric temperature and the detrended unstable dilatometric time series using mutual information $I\left(X\right(t);Y(t+\tau \left)\right)$ (b), (d) and the check of the surrogate data using linear mutual information $L\left(X\right(t);Y(t+\tau \left)\right)$ (a), (c). The values of mutual information (a,b) from the tested data (solid line), mean (dash-dotted line), and mean $\pm$SD (dashed lines) of a set of 30 realizations of the surrogate data. The statistics, differences in number of standard deviations of the surrogates (c), (d).

Figure 4

Testing for nonlinearity in the relationship between atmospheric temperature and the residuals of the multilinear regression of the detrended unstable dilatometric time series on the meteorological variables, using mutual information $I\mathbf{\left(}X\right(t);Y(t+\tau \left)\mathbf{\right)}$ (b),(d) and the check of the surrogate data using linear mutual information $L\mathbf{\left(}X\right(t);Y(t+\tau \left)\mathbf{\right)}$ (a),(c). See caption of Fig. 3 for the line key.

Figure 5

Testing for nonlinearity in the residuals of the triple linear regression of the detrended unstable dilatometric time series on the meteorological variables, using mutual information $I\mathbf{\left(}X\right(t);X(t+\tau \left)\mathbf{\right)}$ (b),(d) and the check of the surrogate data using linear mutual information $L\mathbf{\left(}X\right(t);X(t+\tau \left)\mathbf{\right)}$ (a),(c). See caption of Fig. 3 for the line key.

Figure 6

The empirical probability $P(\mid x\mid >X)$ of observing amplitudes larger than a given value $X$ (where $x$ is a deviation from the mean value) for the triple regression residuals of an example of a stable (a) and unstable (b) time series of dilatometric measurements. Diamonds and squares illustrate left and right sides of the distribution. The solid line shows the average distribution of ${10}^5$ realizations of a 1024-sample time series randomly drawn from the Gaussian distribution with the same mean and variance as the residuals under study.

Figure 7

Power spectra of regression residuals of an example of stable (a),(b) and of unstable (c),(d) time series of dilatometric measurements. Single (a),(c) and double (b),(d) logarithmic plots.

Figure 8

Power spectra (a),(b) and results of the detrended fluctuation analysis (c)-(e) for the residuals of the single (c) and triple (a),(d) regressions of the linearly detrended time series of dilatometric measurements on an unstable slope; and of the triple regression of the high-pass filtered unstable dilatometric series (b),(e).

Figure 9

Power spectra (a),(b) and results of the detrended fluctuation analysis (c),(d) for the differenced residuals of the single (a),(c) and triple (b),(d) regression of the linearly detrended time series of dilatometric measurements on an unstable slope. Thin curves in (a),(b) and points in (c),(d) are the results of the respective methods; thick solid lines in all figures are fitted robust linear regressions in particular scaling regions.