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Knowledge of slip depth and slip rate not only helps quantify the role landslides play in the dynamic equilibrium of hillslope processes (Roering, 2012) but also is valuable for geotechnical sampling operations, properly placing instrumentation, and designing landslide mitigation systems. Constraining subsurface slip, however, can be exceedingly difficult. Contact methods for locating landslide slip surfaces, such as trial pits, boreholes, inclinometers, and geophysical exploration surveys are expensive, time consuming and may require extensive fieldwork in hazardous areas, as well as repeated maintenance when slide deformation destroys instrumentation or access to the subsurface. Therefore, a non-contact method for inferring the depth of slip, and its orientation and magnitude, is desirable (Booth et al., 2013).

Previous work pursued graphic or geometric approaches to infer landslide slip surfaces using ground-surface displacements. Varnes (1978) introduced a graphical method for estimating slip along a circular failure, and a more versatile graphical method (Carter and Bentley, 1985) assumed rigid body motion along a single slip surface. Using this latter method, Baum et al.

(1998) demonstrated that this approach can yield a good approximation of the slip surface depth but this requires prior knowledge of the boundary of the active part of the landslide on the ground surface. The slip depth of a translational landslide has also been estimated by geometrically balancing the area along a cross-section (Bishop, 1999); this approach is similar to that routinely used in structural geology (Woodward et al., 1989). Recently, Booth et al. (2013) presented a method to constrain landslide deformation and thickness by inverting 3D surface change data from repeat stereo imagery, although their approach must be calibrated using thickness measurements.

An alternative, potentially valuable, approach involves inverting surface displacements using elastic dislocation (ED) models (e.g. Hudnut et al., 1996; Brooks and Frazer, 2005) to infer subsurface earthquake slip and fault orientation, as is commonly performed in earthquake geodesy (e.g. Hudnut et al, 1996). Although these models make simplifying assumptions, such as that deformation occurs in an elastic half-space, ED methods are computationally efficient and well-tested; they have been particularly helpful in forming well-posed inverse problems and in constraining first-order characteristics of seismotectonic faulting processes (Brooks and Frazer, 2005). Importatnly, this approach does not require calibration.

Landslides are often modeled with plastic flow rheology (e.g., Savage and Chleborad, 1982;

Iverson, 1986); however, the geometry of faults in slow-moving landslides and earth’s shallow crust can also appear similar (Hobbs et al., 1976). Field observations suggest that, under certain conditions, deformation involved in slip-surfaces underlying slow-moving landslides may be similar to those associated with tectonic faults (Fleming and Johnson, 1989; Gomberg et al., 1995), albeit at different scales as well as temperature and pressure conditions. Some studies (e.g., Fukao, 1995) argue that landslides are often described as dislocations due to gravitational potential energy represented by a single-force in contrast to tectonic earthquakes described as dislocations due to the release of strain energy represented as double-couple forces. The singleforce model is bounded by the sliding surface and therefore the landslide is in the advance stage compared to the dislocation model. Therefore, numerous studies have treated slow-moving landslides as shear dislocations, the equivalent-force system of which is the double-couple. For example, Fleming and Johnson (1989) consider the landslide as a fracture in their conceptual model and Martel (2004) discusses the mechanics of the ED model applied to landslides.

Regardless of the preferred mechanics, the utility of methods to infer subsurface slip character from surface displacements has been limited, in large part, because of the challenges of collecting the necessary data: accurate and spatially complete estimates of surface displacements.

Now, the widespread acquisition of Terrestrial Laser Scanning (TLS) data from active landslides (McCoy et al., 2010; e.g., Aryal et al., 2012) provides opportunities to develop quantitative methods to infer subsurface deformation character from surface displacements. In this paper we test the hypothesis that 3D displacements derived from the particle image velocimetry method applied to the TLS data can be used to quantify the subsurface slip character (orientation and slip magnitude) of slow-moving landslides. To develop the methodology and to illustrate the range of possible solutions, we employ both the purely geometric balanced cross-section (BC) method (Bishop, 1999) and an ED model in an homogeneous elastic half-space (Okada, 1985). We apply the method and ground-truth it with in situ subsurface measurements at the slow-moving Cleveland Corral landslide (CCL) in California's Sierra Nevada range (Reid et al., 2003; Aryal et al., 2012).

2.2 Methods for Inferring Subsurface Slip 2.2.1 Balanced cross-section (BC) The BC method (Bishop, 1999) assumes conservation of area and plane strain deformation (displacement is only in the downslope direction). Accordingly, this method assumes that the loss of area in the head-scarp (‘depletion zone’) of a landslide is caused by the downslope

movement of the slide. Then, slip depth, D for a profile line can be written (Figure 2. 1a):

–  –  –

where R is the mean of the displacement along a profile line, and A is the area of depletion at the landslide headscarp (Figure 2. 1a). We estimate R applying the PIV method to TLS data, and A by integrating changes in elevations along the profile line. Both A and R have associated errors δA and δR respectively. Assuming a normal Gaussian error distribution and using the general

equation of error propagation, uncertainty in the estimated depth is:

–  –  –

2.2.2 Dislocation in an Elastic Half-Space Geodetically measured coseismic surface displacements are commonly used to infer fault parameters at depth assuming that deformation is due to a displacement dislocation embedded in an elastic half-space. Okada (1985) presents a complete set of compact closed analytical expressions for surface deformation due to inclined shear and tensile faults in an elastic halfspace and the detailed derivation and description of dislocation modeling is also presented in Segall (2010). Briefly, Okada’s formalism estimates a surface displacement vector field due to a dislocation discontinuity at depth in an elastic half-space characterized by Poisson’s ratio. A set of parameters further defines the dislocation’s geometry and slip vector (Figure 2. 1b). Although the elastic dislocation (ED) model is inherently valid for small and elastically recoverable displacement, it has been successfully applied to model co-seismic deformation and faults with larger strains (e.g., Healy et al., 2004). We acknowledge that landslide deformation is nonrecoverable but we justify the use of the ED model here for the following reasons. First, as described below, we apply the model only when surface displacements ( ~ 30 cm) and strains (~0.12) are relatively small. In cases such as these, the first order linear term in the strain tensor is much larger than the higher order non-linear terms and the small strain assumption can be a fair approximation. Second, in the interest of developing and exploring the application of a new general methodology for landslides rather than delivering the best characterization of a particular site, we are willing to trade the imperfect ED mechanical assumptions for its computational efficiency and ease of implementation in an inverse approach.

We implement a multidimensional grid search in parameter space to obtain marginal probability distributions for eight nonlinear parameters (X and Y location, depth, length, width, strike, dip and slip magnitude) that characterize slip on a rectangular displacement discontinuity. In order to render the elastic dislocation problem computable via a grid search, we place prior constraints (e.g. length of the slip patch is approximately equal to the length of the landslide) on all parameters. We calculate an approximation-minimum norm solution in the least square sense for each parameter combination and estimate the marginal probability distribution (mpd) of the

model parameter space (P) following Menke (1989):

Figure 2. 1.

Sketch of the two models used to infer landslide subsurface slip geometry. Gray arrows on right side of each sketch show displacement-depth profile inferred by the model. a) Longitudinal slice for the balanced cross-section (BC) method. b) Slide geometry relative to the ground surface for the elastic dislocation (ED) model. Red line is the projection of a rectangular slip surface.

–  –  –

where data (d) and model parameters (m) are related by function (G).

2.3 The Cleveland Corral landslide CCL is a large (~ 450 m long and 20-70 m wide) earth-slide located along U.S. Highway 50 in the Sierra Nevada Mountains of California (Figure 2. 2). Shallow seismic exploration indicates that typically 5-10 m of sliding material overlies the schist bedrock (Reid et al., 2003), although the active sliding mass is thinner in some locations. Since monitoring began in the late 1990s, the landslide has moved only in years when precipitation exceeds the mean annual precipitation;

a neighboring slide with similar characteristics failed catastrophically in 1997 (Reid et al., 2003).

Measured surface displacements at the CCL vary in time and space ranging from millimeters to several meters per year (Reid et al., 2003; Aryal et al., 2012).

We surveyed the CCL with an Optech Ilris-3D scanner in January, May and June, 2010 while the toe portion of the slide was active. TLS surveys were conducted from a high elevation vantage point across the valley from the slide (ranging 500 – 700 m). In each scan, point-cloud spotspacing varied from 6-12 cm. The first scan was georeferenced to a 0.5 m DEM in a UTM coordinate system (NAD83) derived from aerial photographs acquired in 2007. We aligned subsequent scans to the georeferenced scan, masking out the data points from the potentially moving toe area (Aryal et al., 2012). In addition to the TLS scans, we installed vertical copper shear rods at two locations in the toe of the CCL during May 2010 when this part of the slide was active (Figure 2. 2). When sub-surface slip occured, the rods were severed by the slip surface, and subsequent measurement of the rod sections remaining in the slide provided highly accurate slip depths.

2.4 3D Displacement Field We use the Particle Image Velocimetry (PIV) method adapted to point cloud data (Aryal et al.,

2012) to estimate horizontal displacement fields. To obtain vertical displacement, we translate the ground surface according to the PIV-estimated 2D horizontal displacement and then difference the elevations (see appendix).

Figure 2. 2.

Displacement fields for an active part of the Cleveland Corral landslide (red box in the inset), obtained for two time periods using repeat TLS scans and PIV. Black vectors represent horizontal displacements and background color represents the estimated vertical displacement. White vectors are from tracking identifiable features in the TLS data. Landslide surface features were mapped in 2010;

slide boundaries in the inset are from Reid et al. (2003). Origin of the UTM coordinates shifted to (724000, 4295000) in order to avoid large numbers in the axes. (a) January-May 2010 time period. (b) May-June time period. The cumulative displacement from these two displacement fields (max. 1±0.14

m) is consistent with the independently estimated January-June displacement field (see Figure A2 in the appendix).

We estimate the 3D displacement field for three time period pairs: January-May (Figure 2. 2a), May- June (Figure 2. 2b.), and January-June (Figure A2). The plan view displacement field for the active toe of the slide is elongated in a ~20 m wide and ~55 m long pattern. The horizontal displacement vectors agree well with the GPS measurements and displacements from tracking of identifiable features in the TLS data (Aryal et al., 2012; Figure 2. 2). Overall, both the JanuaryMay and May-June displacement fields record about the same displacement magnitude (maximum of ~0.5 m horizontal and ~0.3 m vertical). However, the May-June displacement vectors trend slightly SW compared to the January-May vectors. The maximum estimated displacement over the five-month time period (January-June) is 1.±0.1 m (Figure A2); this is consistent with the sum of the January-May and May-June displacements. The relatively larger noise in the May-June displacement field compared to the January-May field (Figure 2. 2) is likely due to early summer vegetation growth.

2.5 Subsurface Inference Results 2.5.1 Balanced Cross-section From these 3D displacement fields, we estimate the slip depth of the CCL for all three acquisition time pairs along 22 transects at 1m spacing using the BC method. The resultant crosssectional slip surface profile is asymmetric (Figure 2. 3): it drops quickly to a maximum depth of

-6±0.7 m near the western lateral margin of the active area and then decreases with fluctuations towards the eastern lateral margin. Errors in the estimated depth could be due to violation of the model assumptions or errors in the TLS data (e.g., vegetation and alignment errors). Mostly south facing displacement vectors indicate that the out-of-plane component of the displacement is very small for the Jan-May displacement (generally 10% of mean displacement) although it is slightly larger for the May-Jun displacement (Figure 2. 2). Because we use TLS-derived DEMs to estimate the loss of the area in the zone of depletion, trees and bushes may add errors in the DEMs, which then can propagate to the depth estimates. Therefore, in our analysis, we discard any DEM nodes with greater than 2- mean standard deviation of the TLS elevations as returns from trees and bushes are characterized by scattered heights. Figure 2. 3 demonstrates that the BC method depth estimates agree quite well with the shear rod measurements. The estimated slip depth at both shear rod locations agrees with the observed depths (2.37 m and 3.18

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