Landslides DOI 10.1007/s10346-020-01347-0 Received: 22 August 2019 Accepted: 2 January 2020 © Springer-Verlag GmbH Germany part of Springer Nature 2020 Cedric Meier I Michel Jaboyedoff I Marc-Henri Derron I Christian Gerber A method to assess the probability of thickness and volume estimates of small and shallow initial landslide ruptures based on surface area Abstract A new inventory of 66 small and shallow landslides within six pilot areas was created based on a high-resolution digital elevation model in the canton of Vaud in Switzerland. The geometrical characteristics of the landslides were recorded (i.e. surface area, maximum thickness and length), and the volumes were estimated. These data permitted the development of a model that provides the probability for a landslide to possess a maximum thickness or volume smaller than a given value based on the landslide horizontal surface area. The results are compared with three existing power-law relationships of surface area–volumes. This new approach constitutes a way to improve the quantification of the uncertainty of volume and maximum depth estimations for small and shallow landslides. Keywords Landslide . Volume . Depth . Failure surface . Probability Introduction The volume of a landslide is an important factor to assess its potential impact on objects at risk because it partly controls the distance of propagation and the affected area and its damages (Corominas, 1996; von Ruette et al. 2016; Jackob, 2005; Jackob et al. 2012). The volumes of landslides are often estimated by multiplying the surface area and the average depth (Guzzetti et al. 2009). The recommendations by Cruden and Varnes (1996) proposed using a half ellipsoid parallel to the slope. There are several other methods to estimate volumes, which are often auto- matic, but this is beyond the scope of this paper because we will analyse the uncertainty of landslide volume based on the surface area–volume relationship, which has been described by many studies. Compilations about such relationships can be found in Guzzetti et al. (2009) and Larsen et al. (2010). Most of these relationships expressed volumes as power-law functions of surface areas. The spread of data is recognised by both Guzzetti et al. (2009) and Larsen et al. (2010), but they did not provide the probability of obtaining a given volume based on the surface area, even though Guzzetti et al. (2009) included a 95% confidence level. For the present study, an inventory of small and shallow landslides was created within six pilot areas in the canton of Vaud (Switzerland). To avoid any misunderstanding, we used the term small landslides in addition to shallow landslides to include landslides that are deeper than 4 m. A recent high- resolution light detection and ranging (LiDAR) digital elevation model (DEM) with 0.5 m raster resolution was used. This in- ventory, made by visual inspection of the DEM and its hillshades, includes the following geometrical characteristics: maximum vertical depth, length, width, horizontal surface and volume estimation of the initial failure surface and not of the deposit. Based on these data, a method is developed to estimate the probability of obtaining the maximum vertical thickness or vol- ume based on the horizontal surface area. This is performed using the principal component analysis (PCA) major axis on the loga- rithmic data values. Then, the distance to this line is used to obtain a log-normal distribution to define quantiles to classify the data. Based on these results, quantiles not exceeding a given vertical depth based on horizontal surface area are constructed. The quantiles for volumes are then calculated assuming half-ellipsoid or elliptic-paraboloid shapes. The proposed method is a first step toward the quantification of landslide volumes in a probabilistic approach, which still needs to incorporate uncertainty when the number of samples is small. Data Six zones were chosen as test areas (Fig. 1). They are distributed in the three main geological units: the Jura unit, the Molasses unit and the Prealps unit. The substratum type was identified for each landslide: it consists of marly and clayish limestone (34%), flysch (32%), marl (8%), sandstone (3%), shale (5%), limestone (1%) and Quaternary deposits. Quaternary superficial formations are mostly talus cones or alluvial fans, glacial formations and deep-seated landslides. A new high-resolution DEM from an airborne LiDAR scan of the canton of Vaud (Switzerland) was used. The raw data pos- sessed a high density of points up to 15 points/m 2 and were rasterised with a grid cell size of 0.5 m over the whole area of the canton (3074 km 2 ) (Source: Geodata Canton of Vaud, Switzerland, 2017—www.geoportail.vd.ch/). Other documents available are a former inventory including more than 610 events from 1889 to 2013 from all kinds of natural hazards and is poorly documented in regard to shallow landslide geometry. Also available are geolog- ical and topographical maps and air photos at 1:25,000 (Swisstopo GA25, Swisstopo National Map, Swisstopo Swissimage; available at: map.geo.admin.ch). The 66 shallow and small landslides registered are located only on hillslopes, “excluding the effect of fluvial undercutting by streams and rivers” (von Ruette et al., 2011) encountered in riverbanks. Methods Data acquisition The first step is the visual identification and mapping of the contours of the shallow landslides directly on the hillshade of the DEM and based on profiles of the topography using the high- resolution DEM. In each test area, individual landslide scars and deposits are contoured manually, including the horizontal length (L rh ) and width (W r ) of the initial surface of the rupture, the Landslides Technical Note