Several elementary physical models are compatible with step-slip landslide behavior (Table 1). One simple conceptual model, analogous to conventional thinking about earthquake occurrence, features the gradual buildup of strain to the point of sudden brittle failure, producing a sharp displacement step that releases most or all of the accumulated strain; this abrupt release is followed by another static period of gradual strain buildup. This model can produce a series of subequal displacement jumps, separated by long static periods. If strain release is more gradual, a series of asymmetrical jumps with an exponential or diffusive shape can be produced.
Model Displacement rate (dx/dt) Integral (x) Brittle failure Heaviside step function A Exponential k(A–x) A(1–e–kt) Logistic kx(A–x)/A Axi/[xi+(A–xi)e–kt] Diffusive c(k/t)3/2e–k/t A∙Erfc[(k/t)1/2] Simple critical
k(Lc–Lt), for Lt≤Lc A (numerical integration) x. Position; t. time; k. rate or time constant; A. step size; c. constant; Lt. instantaneous water level; Lc. critical water level (constant).
Table 1. Some elementary models for landslide step displacement
Alternatively, a logistic model curve is compatible with steady strain buildup to a point of failure, but with the subsequent displacement rate being instantaneously proportional to the amount of remaining strain. This produces a symmetrical S-shaped curve (Fig. 4).
Figure 4. Graphs of simple functions (Table 1) that produce step-like displacements. The rate constant k was taken as unity for the exponential and logistic functions, but ten times larger for the diffusion function.
Worldwide, many landslides are known to occur immediately after periods of heavy rain. Such rain produces an asymmetrical hydrograph known to exist in both surface streams and shallow groundwater, as seen in springs, and is most realistically modeled by the diffusion hydrograph (Criss and Winston, 2008, 2003). Similar shapes have been argued to resemble the width of monitored landslide cracks, but in the TGR, several studies show that crack widths do not resemble the displacement of the main slide mass.
An extremely simple model is that falling reservoir levels cause the landslide and its included groundwater to progressively lose lateral support. Failure ensues when the reservoir falls to a critical level, with the subsequent displacement rate being proportional to the depth of the reservoir below that critical level, as discussed below.
As discussed above, daily monitoring data for Baijiabao Landslide show that movement in 2018 initiated when the TGR fell below 153 m, and mostly terminated when the reservoir rose nearly back to that level (Fig. 3). The smooth, continuous displacement record (Fig. 3) is incompatible with sudden brittle failure but is compatible with "critical level" models. The correspondence of the 2018 maximum displacement rate with the minimum 2018 TGR level (Fig. 3), and the variability of step size from year to year (Fig. 1), also support critical level models.
The simplest mathematical form for a critical level model assumes that the displacement rate dx/dt is zero when the observed, instantaneous water level Lt is above a constant critical level Lc. However, when Lt is lower than Lc, dx/dt is directly proportional to their difference. Thus
This model is easily calibrated with monitoring data for any given site, which reveal logical choices for Lc on inspection. The rate constant k is a simple scaling factor with units of inverse time, and is easily calibrated for any given site by requiring that the total model displacement over an interval of interest matches the actual total displacement. A simple spreadsheet can then be used to evaluate these equations for any real or hypothetical reservoir level record, for example, by calculating the predicted displacements for each of a series of daily time steps, then conducting a running sum.