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Volume 31 Issue 6
Dec.  2020
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Robert E. Criss, Wenmin Yao, Changdong Li, Huiming Tang. A Predictive, Two-Parameter Model for the Movement of Reservoir Landslides. Journal of Earth Science, 2020, 31(6): 1051-1057. doi: 10.1007/s12583-020-1331-9
Citation: Robert E. Criss, Wenmin Yao, Changdong Li, Huiming Tang. A Predictive, Two-Parameter Model for the Movement of Reservoir Landslides. Journal of Earth Science, 2020, 31(6): 1051-1057. doi: 10.1007/s12583-020-1331-9

A Predictive, Two-Parameter Model for the Movement of Reservoir Landslides

doi: 10.1007/s12583-020-1331-9
More Information
  • Monitoring data show that many landslides in the Three Gorges region,China,undergo step-like displacements in response to the managed,quasi-sinusoidal annual variations in reservoir level. This behavior is consistent with motion initiating when the reservoir water level falls below a critical level that is intrinsic to each landslide,with the subsequent displacement rate of the landslide being proportional to the water depth below that critical level. Most motion terminates when the water level rises back above the critical level,so the annual step size is the time integral of the instantaneous displacement rate. These responses are incorporated into a differential equation that is easily calibrated with monitoring data,allowing prediction of landslide movement from actual or anticipated reservoir level changes. Model successes include (1) initiation and termination of the annual sliding steps at the critical reservoir level,producing a series of steps; (2) prediction of variable step size,year to year; and (3) approximate prediction of the shape and size of each annual step. Annual rainfall correlates poorly with step size,probably because its effect on groundwater levels is dwarfed by the 30 m annual variations in the level of the Three Gorges Reservoir. Viscous landslide behavior is suggested.
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  • Barla, G., Paronuzzi, P., 2013. The 1963 Vajont Landslide:50th Anniversary. Rock Mechanics and Rock Engineering, 46(6):1267-1270. DOI: 10.1007/s00603-013-0483-7
    Bernardie, S., Desramaut, N., Malet, J. P., et al., 2015. Prediction of Changes in Landslide Rates Induced by Rainfall. Landslides, 12(3):481-494. DOI: 10.1007/s10346-014-0495-8
    Corominas, J., Moya, J., Ledesma, A., et al., 2005. Prediction of Ground Displacements and Velocities from Groundwater Level Changes at the Vallcebre Landslide (Eastern Pyrenees, Spain). Landslides, 2(2):83-96. DOI: 10.1007/s10346-005-0049-1
    Criss, R. E., Winston, W. E., 2003. Hydrograph for Small Basins Following Intense Storms. Geophysical Research Letters, 30(6):1314-1318. DOI: 10.1029/2002gl016808
    Criss, R. E., Winston, W. E., 2008. Discharge Predictions of a Rainfall-Driven Theoretical Hydrograph Compared to Common Models and Observed Data. Water Resources Research, 44(10):W10407. DOI: 10.1029/2007wr006415
    Desai, C. S., Samtani, N. C., Vulliet, L., 1995. Constitutive Modeling and Analysis of Creeping Slopes. Journal of Geotechnical Engineering, 121(1):43-56. DOI: 10.1061/(asce)0733-9410(1995)121:1(43)
    Du, J., Yin, K. L., Lacasse, S., 2013. Displacement Prediction in Colluvial Landslides, Three Gorges Reservoir, China. Landslides, 10(2):203-218. DOI: 10.1007/s10346-012-0326-8
    Jian, W. X., Xu, Q., Yang, H. F., et al., 2014. Mechanism and Failure Process of Qianjiangping Landslide in the Three Gorges Reservoir, China. Environmental Earth Sciences, 72(8):2999-3013. DOI: 10.1007/s12665-014-3205-x
    Li, C. D., Fu, Z. Y., Wang, Y., et al., 2019. Susceptibility of Reservoir-Induced Landslides and Strategies for Increasing the Slope Stability in the Three Gorges Reservoir Area:Zigui Basin as an Example. Engineering Geology, 261:105279. DOI: 10.1016/j.enggeo.2019.105279
    Lian, C., Zeng, Z. G., Yao, W., et al., 2014. Extreme Learning Machine for the Displacement Prediction of Landslide under Rainfall and Reservoir Level. Stochastic Environmental Research and Risk Assessment, 28(8):1957-1972. DOI: 10.1007/s00477-014-0875-6
    Liu, Z. B., Shao, J. F., Xu, W. Y., et al., 2014. Comparison on Landslide Nonlinear Displacement Analysis and Prediction with Computational Intelligence Approaches. Landslides, 11(5):889-896. DOI: 10.1007/s10346-013-0443-z
    Mantovani, F., Vita-Finzi, C., 2003. Neotectonics of the Vajont Dam Site. Geomorphology, 54(1/2):33-37. DOI: 10.1016/s0169-555x(03)00053-9
    Sato, H. P., Harp, E. L., 2009. Interpretation of Earthquake-Induced Landslides Triggered by the 12 May 2008, M7.9 Wenchuan Earthquake in the Beichuan Area, Sichuan Province, China Using Satellite Imagery and Google Earth. Landslides, 6(2):153-159. DOI: 10.1007/s10346-009-0147-6
    SEP, 2016. Simplicity: Stanford Encyclopedia of Philosophy. (2016-12-20)[2020-5-29]. https://plato.stanford.edu/entries/simplicity/
    Shihabudheen, K. V., Pillai, G. N., Peethambaran, B., 2017. Prediction of Landslide Displacement with Controlling Factors Using Extreme Learning Adaptive Neuro-Fuzzy Inference System (ELANFIS). Applied Soft Computing, 61:892-904. DOI: 10.1016/j.asoc.2017.09.001
    Song, K., Wang, F. W., Yi, Q. L., et al., 2018. Landslide Deformation Behavior Influenced by Water Level Fluctuations of the Three Gorges Reservoir (China). Engineering Geology, 247:58-68. DOI: 10.1016/j.enggeo.2018.10.020
    Tang, H. M., Li, C. D., Hu, X. L., et al., 2015. Deformation Response of the Huangtupo Landslide to Rainfall and the Changing Levels of the Three Gorges Reservoir. Bulletin of Engineering Geology and the Environment, 74(3):933-942. DOI: 10.1007/s10064-014-0671-z
    Tang, H. M., Wasowski, J., Juang, C. H., 2019. Geohazards in the Three Gorges Reservoir Area, China-Lessons Learned from Decades of Research. Engineering Geology, 261:105267. DOI: 10.1016/j.enggeo.2019.105267
    Tomás, R., Li, Z., Liu, P., et al., 2014. Spatiotemporal Characteristics of the Huangtupo Landslide in the Three Gorges Region (China) Constrained by Radar Interferometry. Geophysical Journal International, 197(1):213-232. DOI: 10.1093/gji/ggu017
    van Asch, T. W. J., Malet, J. P., Bogaard, T. A., 2009. The Effect of Groundwater Fluctuations on the Velocity Pattern of Slow-Moving Landslides. Natural Hazards and Earth System Sciences, 9(3):739-749. DOI: 10.5194/nhess-9-739-2009
    Wu, Q., Tang, H. M., Ma, X. H., et al., 2019. Identification of Movement Characteristics and Causal Factors of the Shuping Landslide Based on Monitored Displacements. Bulletin of Engineering Geology and the Environment, 78(3):2093-2106. DOI: 10.1007/s10064-018-1237-2
    Xia, M., Ren, G. M., Ma, X. L., 2013. Deformation and Mechanism of Landslide Influenced by the Effects of Reservoir Water and Rainfall, Three Gorges, China. Natural Hazards, 68(2):467-482. DOI: 10.1007/s11069-013-0634-x
    Yang, Y. B., Liu, M. G., 2005. The Present Advances and Trends of Landslide Predictions. Soil Engineering and Foundation, 19(2):61-65 (in Chinese with English Abstract) http://en.cnki.com.cn/Article_en/CJFDTOTAL-TGJC20050200I.htm
    Yao, W. M., Li, C. D., Zhan, H. B., et al., 2020. Multiscale Study of Physical and Mechanical Properties of Sandstone in Three Gorges Reservoir Region Subjected to Cyclic Wetting-Drying of Yangtze River Water. Rock Mechanics and Rock Engineering, 53(5):2215-2231. DOI: 10.1007/s00603-019-02037-7
    Yao, W. M., Li, C. D., Zuo, Q. J., et al., 2019. Spatiotemporal Deformation Characteristics and Triggering Factors of Baijiabao Landslide in Three Gorges Reservoir Region, China. Geomorphology, 343:34-47. DOI: 10.1016/j.geomorph.2019.06.024
    Zhao, N. H., Hu, B., Yi, Q. L., et al., 2017. The Coupling Effect of Rainfall and Reservoir Water Level Decline on the Baijiabao Landslide in the Three Gorges Reservoir Area, China. Geofluids, 2017(12):1-12. DOI: 10.1155/2017/3724867
    Zou, Z. X., Tang, H. M., Xiong, C. R., et al., 2017. Kinetic Characteristics of Debris Flows as Exemplified by Field Investigations and Discrete Element Simulation of the Catastrophic Jiweishan Rockslide, China. Geomorphology, 295:1-15. DOI: 10.1016/j.geomorph.2017.06.012
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A Predictive, Two-Parameter Model for the Movement of Reservoir Landslides

doi: 10.1007/s12583-020-1331-9

Abstract: Monitoring data show that many landslides in the Three Gorges region,China,undergo step-like displacements in response to the managed,quasi-sinusoidal annual variations in reservoir level. This behavior is consistent with motion initiating when the reservoir water level falls below a critical level that is intrinsic to each landslide,with the subsequent displacement rate of the landslide being proportional to the water depth below that critical level. Most motion terminates when the water level rises back above the critical level,so the annual step size is the time integral of the instantaneous displacement rate. These responses are incorporated into a differential equation that is easily calibrated with monitoring data,allowing prediction of landslide movement from actual or anticipated reservoir level changes. Model successes include (1) initiation and termination of the annual sliding steps at the critical reservoir level,producing a series of steps; (2) prediction of variable step size,year to year; and (3) approximate prediction of the shape and size of each annual step. Annual rainfall correlates poorly with step size,probably because its effect on groundwater levels is dwarfed by the 30 m annual variations in the level of the Three Gorges Reservoir. Viscous landslide behavior is suggested.

Robert E. Criss, Wenmin Yao, Changdong Li, Huiming Tang. A Predictive, Two-Parameter Model for the Movement of Reservoir Landslides. Journal of Earth Science, 2020, 31(6): 1051-1057. doi: 10.1007/s12583-020-1331-9
Citation: Robert E. Criss, Wenmin Yao, Changdong Li, Huiming Tang. A Predictive, Two-Parameter Model for the Movement of Reservoir Landslides. Journal of Earth Science, 2020, 31(6): 1051-1057. doi: 10.1007/s12583-020-1331-9
  • Landslides in reservoir areas exhibit a confounding diversity of behaviors that are well displayed by those near the Three Gorges Reservoir (hereafter, TGR). At one extreme is catastrophic movement during initial reservoir impoundment, as exemplified by the Qianjiangping Landslide of 2003 (e.g., Jian et al., 2014; Fig. 1). The other extreme is steady downslope creep, approximated by the huge Huangtupo Landslide (Tang et al., 2015; Tomás et al., 2014; Fig. 1). Between these behaviors are landslides whose displacement mostly occurs during steep annual steps that may or may not be superimposed on steady creep. Many TGR landslides exhibit this behavior, including the Baijiabao (Yao et al., 2019) and Shuping landslides (Wu et al., 2019), featured here (Fig. 1).

    Figure 1.  (a) Satellite view of the Three Gorges area showing locations of landslides considered here; the inset map of China shows this area of detail as a small red box (satellite map modified from images of CNES/Airbus and Maxar Technologies; map of China after GS(2016)1569). Yangtze River flows eastward toward the giant Three Gorges Dam (lower right). (b), (c) Cross sections through the Baijiabao and Shuping landslides, showing their main sliding zones (red lines) that separate the main sliding masses (green) that respectively lie above Jurassic or Triassic clastic strata that in both cases dip into the hillside. The upper and lower levels of the TGR under current management are shown as blue lines, and the locations of several GPS sites are in orange and red lettering. No vertical exaggeration. Cross sections are simplified after Yao et al. (2019) and Wu et al. (2019).

    Many models have been proposed to explain landslide movements. Hydro-mechanical and viscoelastoplasticity models strive to develop a comprehensive theory of motion by combining failure theory with data on material properties, topographic slopes, rainfall, and groundwater behavior (e.g., Jian et al., 2014; Xia et al., 2013; Corominas et al., 2005; Desai et al., 1995). Empirical models, such as rainfall-driven models, posit proportionalities between driving forces and landslide response, sometimes using convolution methods (e.g., Bernardie et al., 2015; van Asch et al., 2009). However, using such mathematical or numerical approaches the predictive problem is generally under-constrained, because the number of parameters involved greatly exceeds the single quantity being evaluated, which is the landslide movement rate. Besides, several of the mechanical properties are not simple quantities but rather are variables whose values depend on the geotechnical properties, the stress history, or the saturation state, etc. (e.g., Yao et al., 2020), and certain models may only be valid for landslides with very similar features (Yang and Liu, 2005).

    Models based on computational intelligence technology, e.g., artificial neural network (ANN), support vector machines (SVMs) and extreme learning machine (ELM) have also been widely studied (Shihabudheen et al., 2017; Lian et al., 2014). Advanced computational intelligence models show good nonlinear fitting ability for landslide displacement prediction. However, the mathematical functions or neural networks used in these predictive models do not have clear physical meanings (Liu et al., 2014), and they generally require a large amount of data to use and verify. In addition, the predictive performance greatly depends on the choice and determination of the associated parameters, as well as on the quality of the displacement data and other information, such as rainfall amounts and groundwater levels (Du et al., 2013).

    We propose an alternate approach that is grounded in the principle of parsimony, which posits that a simple theory that captures the essential behavior of a phenomenon is best (SEP, 2016). This venerable philosophical notion was accepted by Newton and Galileo, and is especially important for purposes such as emergency hazard response, where rapid evaluation of a problem is essential. Accordingly, we sought to develop a predictive model for the displacement rate of reservoir landslides that is easily calibrated and utilizes an absolute minimum of fitting parameters and assumptions. Our resultant model, which proposes that landslide movement is initiated when the reservoir level falls below a critical value, compactly explains many first-order behaviors of several TGR landslides, and provides a useful benchmark for judging the effectiveness of more complex predictive models.

  • The TGR area of the Yangtze River basin hosts thousands of landslides, owing to its steep terrane and abundance of weak, Triassic and Jurassic strata (e.g., Li et al., 2019). These strata are slide prone because they commonly contain abundant, steeply dipping shale interbeds. Since the impoundment of the TGR began in 2003, regional seismicity has increased and many landslides have been reactivated, partly because the cyclic changes in reservoir level affect body forces as well as rock strength (Tang et al., 2019).

    This study focuses on the Baijiabao and Shuping landslides, which are two giant colluvial reservoir landslides whose toes are now continuously submerged beneath the TGR (Fig. 1). Significant movements of both masses have been observed since TGR impoundment, but in this paper we primarily focus on data acquired since 2009, because water levels have been consistently managed to annually vary between ~145 and 175 m since that time (see below). Continuous monitoring of surface movements and rainfall has been performed on these two landslides for purposes of geological hazards prevention and scientific research. Reservoir level data are available at https://cj.msa.gov.cn/xxgk/xxgkml/aqxx/swgg/.

    The Baijiabao Landslide is situated in Guizhou, Zigui County, Hubei Province of China, on the right bank of Xiangxi River (30°58′59.9″N, 110°45′33.4″E), only 2.5 km upstream of its confluence with the Yangtze River (Fig. 1a). This landslide is 550 m long, 400 m wide, and about 40 m thick, so its estimated volume is nearly ten million m3. The subjacent bedrock is quartz sandstone and argillaceous siltstone of the Early Jurassic Xiangxi Formation (J1x), which dips into the hillside at an angle of 30° to 40°. Four GPS monitoring sites provide position data at ~3 week intervals starting October 2006, and three additional, automatic GPS monitoring sites provide daily data since October 2017. Zhao et al. (2017) and Yao et al. (2019) provided additional details about the geologic setting and available observations.

    The Shuping Landslide is situated in Shazhenxi, Zigui County, Hubei Province of China, on the right bank of the Yangtze River (30°59′37″N, 110°37′0″E) (Fig. 1a). This landslide is ~800 m×700 m and about 50 m thick, so the volume of the sliding mass is approximately 2.7×107 m3. The Shuping Landslide lies above interbedded mudstone and siltstone of the Triassic Badong Formation (T2b), that dip from 9° to 38° into the hillside. Eight GPS monitoring sites provide surface displacement data since July 2003; monitoring data at ~3 week intervals from October 2006 to October 2013 were used in this study. Song et al. (2018) and Wu et al. (2019) provided many additional details about this site.

  • The motion of many landslides in the TGR area is dominated by rather sharp annual steps, followed by long intervals of little or no motion. GPS measurements have been made on ~monthly intervals for many TGR landslides for a decade or more, revealing many essential features of this motion. First, while different parts of the same landslide exhibit similar patterns of motion, some parts move consistently faster than others (Fig. 2). Second, for landslides exhibiting step-slip behavior, most motion occurs in a period of about a month or less. For example, GPS data for the Baijiabao Landslide during 2009 to 2019 show that 50% to 90% of the annual motion typically occurs in a period of ~35 days (Fig. 2). Third, monitored crack widths generally do not closely resemble each other nor do they resemble the motion of the main slide mass (e.g., Yao et al., 2019). Fourth, at any given site, the size of the annual step varies year to year (Fig. 2).

    Figure 2.  Cumulative displacements of the Baijiabao Landslide (sites ZG323– ZG326, green lines) and Shuping Landslide (sites ZG85–ZG90, red lines) compared to the level of the Three Gorges Reservoir (blue line, right scale) for the period 2007 to 2019. Data from Yao et al (2019) and Wu et al. (2019).

    Given the sharpness of these displacement steps, the monitoring interval of the data in Fig. 2 is too infrequent to reveal several other essential details. Fortunately, daily GPS monitoring was recently initiated on a few TGR landslides, providing sufficient data density to resolve additional aspects of the step-slip behavior. Such data for Baijiabao Landslide (Yao et al., 2019) show that 85% of the 2018 displacement occurred in a 30 day period, corresponding to the interval when the reservoir level remained below 153 m (Fig. 3). The cumulative displacement has a smooth pattern bracketed by slow rates at the beginning and the end of the step. Finally, the maximum displacement rate occurred at the center of this pattern, with the infection point occurring on June 10, 2018, when the reservoir first fell to within a few cm of its lowest level for 2018 (Fig. 3). This correspondence supports the conclusion of Yao et al. (2019) that Baijiabao Landslide responds to the water level of the reservoir on a time frame of only about 1 day.

    Figure 3.  Daily monitoring data showing the cumulative displacements of the Baijiabao Landslide (sites AG1, AG2, AG3, left scale, from Yao et al., 2019) compared to the level of the Three Gorges Reservoir (blue line, right scale) for the period Oct., 2018 to Nov., 2019. Most displacement occurred when the reservoir was below 153 m (horizontal line), with the maximum displacement rate occurring about June 10, 2018 when the reservoir first fell to a very low level (thin vertical line).

  • 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(Ax) A(1–ekt)
    Logistic kx(Ax)/A Axi/[xi+(Axi)ekt]
    Diffusive c(k/t)3/2ek/t A∙Erfc[(k/t)1/2]
    Simple critical
    level
    k(LcLt), for LtLc 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.

  • Figure 5 compares the predictions of the simple critical model (Eqs. 1a, 1b) with the monitoring data acquired every ~3 weeks that is available for two sites in the Baijiabao and Shuping landslides. The modeling interval was chosen to begin on January 1, 2009, about when the modern management style of the TGR began. Critical levels were taken as 153 m for Baijiabao and as 163 m for Shuping. Correspondence between prediction and observation is good, given the simplicity of the model. Successes include (1) initiation and termination of the annual sliding steps at the critical reservoir level, producing a series of steps; (2) prediction of variable step size, year to year; and (3) approximate prediction of the shape and size of each annual step (Fig. 5).

    Figure 5.  Cumulative displacements of Baijiabao site ZG324 and Shuping site ZG85 with the predictions of the simple critical model (Eq. 1, red lines). Critical reservoir levels were taken as 153 m for the Bajiabao Landslide and 163 m for Shuping Landslide. The models are "unbiased", in that a constant scaling factor (k value) was chosen for each site so that the total model displacement over the entire plotted time interval matches the data.

  • Available daily data for the Baijiabao Landslide permit close comparison of the actual shape of the 2018 step and that predicted by the critical level model (Fig. 6). The model is the same as before (Eq. 1), again adopting a value of 153 m for Lc, except the scaling factors for the predicted curves were chosen to match the actual size of the 2018 steps at sites AG2 and AG3.

    Figure 6.  Daily position data for Baijiabao sites AG1, AG2 and AG3 (dots), compared to the unbiased critical level model (red lines) that are identical except for two different scaling factors.

    The fidelity of the shape of the critical level model curve to the data is only fair. Of the models in Table 1, the diffusion model provides the most realistic curve shape. Specifically, the function: A∙Erfc [(k/t)0.5], where k is taken as 1 or 3 days, and t is the time in days since attainment of the critical level (here, 151 m), provides smooth curves that closely match actual displacements when values for A are scaled to match the observed 2018 displacements. Regarding this "superior" match, note that (1) an additional fitting parameter is required; (2) unlike Eq. 1, the Erfc function does not predict interannual step size (cf. Fig. 5). Doing so would require even more assumptions and fitting parameters.

  • While rather small, the deviations between actual step size for the Bajiabao and Shuping landslides and those predicted by the simple critical level model suggest that factors other than reservoir level play a secondary role. Rainfall amount is a logical candidate, but available data for these two landslides provide no support. For example, annual rainfall in the TGR regionf was well above average during 2010 and 2017, yet during those years the step size was small yet over-predicted by the critical level model (Fig. 5), contrary to the expectation if neglect of rainfall was the cause. Conversely, annual rainfall in the TGR region was well below average during 2012, yet during that year the step size was underpredicted by the critical level model, likewise contrary to expectation if rainfall neglect was responsible. Finally, note that Song et al. (2018) and Yao et al. (2019) also concluded that rainfall amounts had little correlation with the behavior of Baijiabao Landslide. Many other factors could affect step size, such as construction activities or engineering efforts to achieve landslide stabilization.

  • Landslides, mudflows, rock falls and other downslope mass movements are commonly driven by catastrophic events and processes. Worldwide, landslides have accompanied earthquakes and initial reservoir impoundments (e.g., Barla and Paronuzzi, 2013; Sato and Harp, 2009; Mantovani and Vita-Finzi, 2003). Heavy rain is an even more common cause of sudden downslope movements, particularly when intense storms follow volcanic ash eruptions or forest fires. Infiltrating rain can also cause progressive deterioration of rock strength until the point of sudden failure (Yao et al., 2020; Zou et al., 2017). All such events are catastrophic, and so are best likened to transient, one-time failures. In contrast, the limited, steady or repeated downslope movements considered here are more akin to creep.

    Given the well-established connections between rainfall and landslides, the poor correlation between the annual rainfall and the annual displacements of the landslides considered here was unexpected. The likely explanation is that the effects of interannual rainfall variations on groundwater levels are much smaller than the 30 m annual change of TGR water levels under the current management protocol. In particular, average annual rainfall in the steep, vegetated TGR area is only about 1 m, and > 80% of this rain is lost to evapotranspiration and surface runoff. Interannual variations of the small amount of rain that actually infiltrates the ground are therefore very small compared to the 30 m annual variations in reservoir level, even though the effect of percolating rain is magnified by the inverse porosity of the substrate.

    The shape of the displacement curve, well known only for 2018 at Baijiabao, is best matched by the diffusion model, although this match requires additional fitting parameters. The Erfc function (Table 1) represents the diffusive response to a sharp event, so unlike the critical level model, the step size that would be attained each year would represent the strain accumulated before some critical value is reached, not after as proposed by the Eq. 1 model. This difference is crucial and central to the behavior of these landslides, as it effectively represents the difference between a catastrophic trigger, with a somewhat delayed response, and the instantaneous response of flowing, weak material to instantaneous stress. More daily monitoring data are needed to resolve this crucial point.

    In short, the first-order behavior of the slip-step landslides in the TGR area appears to be controlled by the magnitude of annual variations in water levels, with a critical water level playing a key role, and with landslide response to level changes being rapid. The critical level model explains many first-order features of this complex behavior with an absolute minimum of free parameters. This elementary model should serve as a useful benchmark for the evaluation of more complex models. Moreover, the deviations between critical model predictions and actual behavior might help identify factors that work in concert with reservoir level to govern the behavior of reservoir landslides.

  • A critical level model can explain the first-order behavior of landslides in the Three Gorges Reservoir area that feature step- like annual displacements. Most annual motion of the Baijiabao and Shuping landslides occurs within ~35 day intervals, that are followed by long periods of little or no motion. These landslides respond rapidly to variations in reservoir level, suggesting viscous flow behavior. The effect of annual variations in rainfall is much smaller than the effect of the 30 m annual variations in the level of the Three Gorges Reservoir. Given the above, we propose a parsimonious model where the instantaneous displacement rate is directly proportional to the depth of reservoir water below a critical level, intrinsic to each landslide, where sliding begins. This model has an absolute minimum of free parameters, is trivial to compute and calibrate, and can serve as a baseline for further studies of landslide dynamics.

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