Optimization methods for the evaluation of the parameters of a rockfall fractal fragmentation model

Landslides (2019) 16:1385–1396 DOI 10.1007/s10346-019-01182-y Received: 4 January 2019 Accepted: 7 April 2019 Published online: 19 April 2019 © Springer-Verlag GmbH Germany part of Springer Nature 2019 Maddalena Marchelli IValerio De Biagi Optimization methods for the evaluation of the parameters of a rockfall fractal fragmentation model Abstract In rockfall events, the falling blocks impacting the slope can experience fragmentation due to their kinetic energy. This process produces new blocks, smaller than the initial ones, moving along independent trajectories. As a result, the in situ block size distribution (related to the slope face of the source area) and the rockfall block size distribution (related to the deposit) differ. The present paper proposes and compares two optimization procedures for choosing the parameters of an iterative fractal fragmentation model based aimed at describing the rockfall fragmentation process on the base of source and deposit block distributions. To discuss the effectiveness of each approach, the two distributions are considered free of uncertainties. The influence of the number of iterations and optimization approach are discussed in terms of easiness of interpretation of the results. Keywords Fractal model .Fragmentation .Rockfall . Optimization techniques Introduction Rockfall events are one of the most dangerous landslide phenomena, in terms of both loss of lives and damages to structures. They involve the detachment, the fall, and the subsequent bouncing, rolling, or sliding of rock fragments before deposition (Varnes 1978). This kind of events entails the displacement of either a single or several blocks, which can break, impacting with the slope. In this last case, the term fragmental rockfall is applied to the phenomenon (Evans and Hungr 1993). The fragmentation process produces new blocks, smaller than the initial ones, moving along the slope as rigid bodies following independent trajectories (RuizCarulla et al. 2017). At each impact, the number of blocks involved in the phenomenon increases and the total kinetic energy changes (Agliardi and Crosta 2003). Consequently, considering fragmentation in rockfall studies is fundamental in the perspective of performing a rockfall risk analysis and designing risk mitigation solutions. The fragmentation process depends on several parameters: the topography of the slope; the geomechanical properties of impacting boulders (lithology and spacing, persistence, and orientation of the discontinuities); the type of impacted surfaces, such as the presence of vegetation or their lithology; and the consequent involved block volumes and energies (Haug et al. 2016; Zhao et al. 2017). The few numerical possibilities of including the fragmentation in the propagation studies of the rockfall, i.e., the dynamics, often require a used-defined probability distribution of the potentially fragmented blocks (Crosta et al. 2015). Recently, Wang et al. proposed a numerical DEM code for studying the interaction between the boulders and the impacting surfaces (Wang and Tonon 2011). Nevertheless, the knowledge of both number and size of the unstable blocks that can be involved in the rockfall phenomenon, as well as number and size of the fallen blocks, is still a crucial issue. The possibility of creating the in situ block size distribution (IBSD), i.e., the probability distribution of the volumes describing the source area, depends on the type of measurements, the elaboration, and the interpretation of the data surveyed on the unstable rock face. Similarly, the construction of the rockfall block size distribution (RBSD), i.e., the probability distribution of the volumes describing the rockfall deposit, is a challenging task, due to the high number of blocks to be measured and their possible removal before the measurements when interferences exist with the human activities (Ruiz-Carulla et al. 2015). Usually, risk analyses are conducted assuming that the IBSD is equal to the RBSD, neglecting the fragmentation process. This simplifies the calculations but results in a less accurate modeling of the natural phenomenon. The creation of a connection between the IBSD and the RBSD taking into account the fragmentation process would result in more realistic risk analyses. Since the fragmentation process depends on several aspects, an overwhelming model considering the physics of the phenomenon is difficult to achieve. As Hadas (1997) reports, phenomenological models offer an effective alternative to mechanistic models. In the past, this problem was frequently tackled through fractal models (Crosta et al. 2007; Turcotte 1986; Turcotte 1997). The adoption of fractals for the study of complex natural phenomena dates back to the 1980s, with the studies by Mandelbrot (1982). Perfect (1997) proposed a scale-invariant fractal probability model for the fragmentation of brittle materials. The model, based on iterative calculations, has been recently modified by Ruiz-Carulla et al. (2017) who proposed an application to a real rockfall case adopting a finite number of iterations. The present paper aims at investigating the influence of the number of iterations on the obtained fragmentation curves and the algorithms for obtaining the more suitable parameters that describe the fractal fragmentation model. The analysis is performed in reference to a real study case in the Northwestern Italian Alps, as detailed. The paper is organized as follows. First, a general overview of block size distributions is presented and fragmentation processes are introduced with particular reference to the adopted fractal fragmentation model. Two optimization approaches are proposed to evaluate the parameters of the model. Briefly, an additional block size distribution derived from the fragmentation model is obtained, and its best fitting with the RBSD is searched. Thanks to the application to a real case study, the two techniques are compared and the obtained results are discussed and commented, also in terms of easiness of the interpretation of the results. Landslides 16 &(2019) 1385 Technical Note In situ and rockfall block size distributions Due to the several uncertainties involved in in situ surveys, the definition of a single-valued representative rockfall block volume is a difficult (say, impossible) task. Accurate measures in the rockfall source area can be related to isolated blocks or to highly fractured cliffs where it is possible to identify discontinuity patterns. During the last years, many studies concentrated on a probabilistic estimate of the rockfall block volume, leading to the so-called IBSD. Various authors discussed the procedure for getting the IBSD. Mainly, the construction of the curve requires two input data: (i) the definition of a fracture network, i.e., the characterization of the discontinuities and joint sets, (ii) the definition of the possible unstable blocks volumes. Varioustechniquesforobtaining the first dataset can be adopted. The International Society for Rock Mechanics proposed a manual procedure based on the geomechanical survey of the rock mass (ISRM 1978). Indirect techniques encompass digital photogrammetry and terrestrial laser scanning, or remote sensing, e.g., LiDAR (Mavrouli et al. 2015; Riquelme et al. 2016). The distribution of the potentially detached blocks volumes in the source area (IBSD) can be estimated through statistical techniques, accounting for the variability of the observed geomechanical properties of the rock mass and the presence of unstable isolated blocks, coupled with geometrical considerations related to the spatial configuration of the discontinuity sets (Elmouttie and Poropat 2012). Nowadays, the fast development of both advanced data acquisition technologies, leading to a high amount of data, and the development of semi-automatic or automatic software for the estimation of the block volumes facilitate the creation of the IBSD curve. As a consequence, it is possible to draw a series of scenarios that are of great value in the field of risk analysis (Marchelli et al. 2019) or for estimating the impact forces for the design of protective structures (De Biagi et al. 2017; Marchelli and De Biagi 2019). Considering the distribution of block volumes at the foot of the cliff, Ruiz-Carulla et al. (2015) observed in many real cases a power law distribution of deposited blocks volumes. They provided a useful method to obtain the RBSD based on scale invariance properties, i.e., local detailed manual measurements in selected uniform parts and normalization over the overall deposit. In particular, the procedure is based on (i) a selective sampling of the small-medium size blocks (subdivided into volume classes) within homogeneous depositional areas (i.e., areas of the deposit with similar average block size) and (ii) a complete measurement of the large blocks on the whole deposit. In addition, a complete survey should be integrated with a rockfall event inventory, if available (Dussauge et al. 2002;2003) in order to get a rockfall volume-frequency law (De Biagi et al. 2017). De Biagi (2017) emphasized the importance of an adequate survey of the volumes located in the deposit at the foot of the cliff, since the quality of the measurements largely modifies the obtained frequency law. Fragmentation process Rockfalls occur in a rock cliff as single blocks or, more frequently, as a rock mass composed by blocks of different sizes and shapes (Perfect 1997). In the latter case, each block follows its trajectory almost independently from the trajectories of the other blocks, i.e., the degree of interaction between the falling elements is low. On the contrary, the interaction between a falling block and the slope originates internal stresses in the boulder that can cause its breaking. The occurrence of the fragmentation process largely depends on many variables, such as motion’s kinematic, material’s properties, the presence of latent discontinuities within the block, and the geological characteristics, in such a way that the rock mass preferentially disaggregates along pre-existing joints. Following the impact and the first breaking, the originated fragments can, in turn, break again along the slope. Hypothetically, if no fragmentation process occurs, the resultant RBSD is equal to the initial IBSD. By contrast, if it occurs, the RBSD differs considerably from the initial IBSD (Ruiz-Carulla et al. 2017). The fractal fragmentation model and the analyses herein performed are based under the fundamental hypothesis that IBSD and RBSD, which are site specific, are considered free of epistemic and aleatoric uncertainties. Fractal fragmentation model The model developed by Perfect (1997) describes the fragmentation of a single block (initiator) into smaller elements, and it is based on the hypothesis that the initiator is made of multiple building blocks that are generated following a similar rule (in the present paper, the rule is inserted into inverted commas). The model does not simulate the physics of a real rockfall fragmentation process, but it describes its results in terms of distribution of fragmented elements. Referring to the fragmentation process exemplified in Fig. 1, the fragmentation rule is: “At each level, the block is divided into eight parts.”The symbol Ndenotes the number of part into which the block is divided, with N(∈ℕ). In this example, N= 8. That is, repeating the process three times, j= 3, it can be seen that the initiator is dived into N 3 =8 3 = 512 parts of equal volume. If the initiator has unit volume, at the end of the process, each generated block has a volume equal to 1/512. In the previous example, it is supposed that the single fragment generates smaller blocks of equal size. In reality, this process does not often occur. For example, it might happen that the initiator generates just two of the eight blocks, and so forth. In this case, the fragmentation rule is: “At each level, the block is divided into three part, two of which have volume equal to 1/8.”Figure 2displays the process. At the first step, just two-eighths are generated (blue and gray cubes), each of which being further fragmented. It is said that the probability of failure is p=n/N, where n(∈ℕ) is the number of generated blocks, e.g., n= 2 in the present example. The fragmentation process, repeated three times, generates 15 blocks of various sizes, as shown in Fig. 3. In general, at each ith fragmentation level, the number of remaining parts, having volume equal to Vi¼ 1−p Nii≠j 1 Nii¼j 8 > < > : ð1Þ is equal to ni¼pNðÞ ið2Þ Landslides 16 &(2019)1386 Technical Note Given the number of times the process is repeated, i.e., j, the total number of resulting elements is equal to ∑ i¼j i¼0 pNðÞ ið3Þ By constraints, the variation range of the probability of failure is defined by the inequality: N−1≤p≤1ð4Þ A probability smaller than the lower bound produces no fragments, while for p= 1, the whole block is fragmented in N j parts. The work of Ruiz-Carulla et al. (2017) slightly modifies the fragmentation model just presented by providing a survival rate. Given kinitiators, it is expected that a fraction of them, namely S r , does not undergo any fragmentation process. The authors introduced the possibility to use a volume distribution as initiator (rather than a single block) and to study the resulting volume distribution after the process has occurred. By consequence, if S r = 0, all the blocks will undergo the fragmentation process, while for S r = 1, the initiator and the final distributions coincide. Method Construction of the IBSD The in situ block size distribution was determined on the bases of the results of a manual geomechanical survey adopting a simplified methodology proposed in Philippot (2015). The inputs of the simplified approach are three rock mass discontinuity sets and the orientation of the face, while the output is a set of possible volumes on the cliff face. Each block is considered isolated by the three joint sets: the first plays the role of sliding plane, the second plays the role of tension crack, and the third laterally isolates the block. In the present study, the orientations of each joint set were considered as deterministic quantities (measured onsite). Spacing and persistence of sliding plane and tension crack sets were related to a negative exponential truncated cumulative probability distribution (Hudson and Harrison 1997; Stravropoulou 2014), Fig. 4, expressed as Fig. 1 Simple fragmentation process “At each level, the block is divided into eight parts.^A single fragment is sketched at each fragmentation level Fig. 2 Sketch of the fragmentation process with N=8,n= 2, and p= 2/8, j= 3. The location of the blue and gray cubes is arbitrary (for sake of clearness) Landslides 16 &(2019) 1387 FxðÞ¼ 0x<xm e−xm=λ−e−x=λ e−xm=λ−e−xM=λxm≤x≤xM 1x>xM 8 > < > : ð5Þ where x m and x M are the minimum and maximum onsite measured values of the parameter, say the spacing of a particular joint set. λis the shape factor of the distribution, which is equal to the average value observed onsite. A reference rock mass domain is subjected to a hypothetical fracturing process in which the characteristics of each discontinuity (spacing and persistence) are obtained through a Monte Carlo generation. The spacing of the lateral plane is supposed to be known (the average value was adopted) and its persistence to be large enough to isolate each block generated by the sliding plane and the tension crack sets. The connectivity between the generated fractures allowed the identification of the volumes isolated by the discontinuity sets. Throughout all the study, the blocks larger than 8 × 10 −4 m 3 were classified into the volume classes reported in Table 1. It is said that a block belongs to a class if its volume is smaller than the threshold volume reported in the second column of Table 1. For example, a class I block has volume larger than 0.008 m 3 and smaller (or equal) to 0.022 m 3 . Construction of the RBSD The rockfall block size distribution was obtained from field measurements related to block volumes following the two steps procedure proposed in Ruiz-Carulla et al. (2015), slightly modified for the selective sampling, as reported in the following. The deposit can be subdivided into homogenous areas identified through visual field observation and orthophoto interpretation in such a way that the average block size is similar (Fig. 5). Details on such operation are reported in Ruiz-Carulla et al. (2015). The block size distributions of each single area are extrapolated from the corresponding subarea. The RBSD is obtained by summing these distributions and adding the large blocks. Taking advantage from the observed scale invariance property in block volume distribution in rockfall debris, the authors proposed a modified approach for performing the selective sampling, as described in the following. For each homogeneous area, a squared shape subarea of size 12 m × 12 m, into which four plots (6 m × 6 m) are drawn (Fig. 5), was identified. The blocks smaller than 10 m 3 were grouped into the volume classes reported in Table 1. Within A1 and A2 areas, all blocks belonging to volume classes 0 to V are counted, while in B1 and B2, only those larger than 0.512 m 3 (i.e., belonging at least to class V). Similarly to the IBSD, all the blocks larger than 8 × 10 −4 m 3 were considered in the sampling. For extending the results to the sampling subarea, the total number of blocks belonging to class V was obtained by summing Fig. 3 Resulting blocks with N=8,n=2,p= 2/8, and j=3. Fig. 4 Probability density, i.e., f, (in black) and cumulative probability, i.e., F=∫ fdx, (in red) functions for a negative exponential truncated probability distribution. The area subtended by the black curve equals to 1 Table 1 Volume classes adopted in the present study Volume class Volume (m 3 ) Class 0 0.008 Class I 0.022 Class II 0.064 Class III 0.181 Class IV 0.512 Class V 1.448 Class VI 2.5 Class VII 5 Class VIII 10 Landslides 16 &(2019)1388 Technical Note the single values on A1, A2, B1, and B2. For smaller blocks, i.e., classes 0 to IV, the counted blocks from A1 and A2 were summed and doubled. For larger blocks, i.e., classes VI to VIII, the counted blocks from B1 and B2 were summed and doubled. For extending the results to the homogeneous area, and then to the whole deposit, the mathematical procedure described in Ruiz-Carulla et al. (2015) was adopted. Estimation of the parameters of the fragmentation model The fractal fragmentation model previously illustrated was applied to the in situ blocks size distribution in order to generate a fragmented block size distribution (named FBSD). In particular, for given values of S r ,N,andn, i.e., a set of parameters π= (S r ;N;n), a single volume (initiator) belonging to a given volume class produces a set of blocks of various sizes, determined by Eq. (1), and quantities, defined by Eq. (2), depending on the number of iterations j. The results for a single initiator are multiplied times the quantity of blocks of the given volume class. The procedure was repeated for all the volume classes considered in the study. The resulting fractal fragmented blocks larger than 8 × 10 −4 m 3 were divided into the volume classes reported in Table 1and the FBSD created. It was considered that the best estimate of the parameters of the fragmentation model (i.e., π) leads to have the FBSD as closest as possible to the RBSD. The obtained set πis site specific since it is generated from IBSD and RBSD, which are site-specific distributions. To estimate the parameters of the fragmentation model, two different optimization techniques were formulated, as detailed in the following. Optimization by best interpolation The optimization by best interpolation method takes advantage of the fact that Pareto Type I distribution, which is a probability distribution originally used in economy science that well describes extreme events (Arnold 2015) and well characterizes the rockfall block size distribution (De Biagi 2017). By consequence, the logarithm of the volumes against the logarithm of the complementary cumulative probabilities is aligned along a decreasing straight line. The complementary cumulative probability function Pis the complement to one of the cumulative probability functions and it is a decreasing function (Fig. 6a). In this sense, the RBSD can be written in terms of its complementary cumulative probability function as: log  P¼m⋆logVþq⋆ð6Þ where m ⋆ < 0 and q ⋆ ≤0. The optimization technique consisted in computing the linear interpolation of the distribution resulting from the fractal fragmentation model (FBSD) for a given set of parameters π, i.e., log  Pπ¼mπlogVþqπð7Þ and determining the norm of the residuals R π ≥0 for estimating the goodness of the interpolation (R π →0forperfect interpolation). The optimization process consisted in minimizing the following objective terms: jmπ−m⋆jObj1 jqπ−q⋆jObj2 RπObj3 8 < : ð8Þ The multi-objective optimization leads to multiple solutions, i.e., a set of optimal parameters, lying on the so-called Pareto front (Michalewicz and Fogel 2013). Optimization by least error The second optimization method took advantage of the fact that the FBSD and the RBSD are evaluated for the same volumes (i.e., the volume classes). In this sense, for a given set of parameters π,it is possible to evaluate for each ith volume class an “error”; measured as ei¼jlog  Pπ;i−log  Pijð9Þ as illustrated in Fig. 6b. The optimization considered in minimizing the following expression: ∑ i e2 ið10Þ Considering that for class 0 the following equalities subsist  P0¼ Pπ;0¼1;ð11Þ the summation ranges to all the classes for which fallen blocks were classed. The minimization leads to a single set of optimal parameters. Goodness of the optimization The goodness of the optimization models was evaluated as a function of the residuals, i.e., the difference between the logarithms of the FBSD and the RBSD. Adopting the same notation as previously, the expression of the goodness parameter Eis E ∑i nc log Pπ;i−log  Pi  2 nc ð12Þ where n c is the total number of classes belonging both to FBSD and RBSD. In the present case, n c =6sincetheRBSD ranges between class 0 and class VI. For E→0, the goodness of the obtained solution increases. The adopted approach differs from the classical Komorogov-Smirnov technique (see Clausetetal.2009 for the details) to account for the following issues. Considering the logarithm of the probability instead of its real value serves for highlighting potential inaccuracies for the larger volume classes, where P≈0. Furthermore, averaging the residuals instead of taking their maximum gives the same importance to all volume classes. Landslides 16 &(2019) 1389 Test site The fragmentation model previously described was adopted for the analysis of a real rockfall fragmentation process. The test site is located in Saint-Christophe municipality (45.7696 N, 7.3555 E), Valle d’Aosta, Italy, in the Northwestern Italian Alps at an altitude of about 1530 m a.s.l. (Manzotti et al. 2017). The site consists in a moderately steep fractured cliff (dip angle equal to 52°) at the base of which a deposit area extends. The size of the test site is limited, allowing the possibility to perform accurate measurements in both the source area and on the deposit. Figure 7illustrates an overall view from the bottom of the deposit. From a geological point of view, the rock composing the cliff belongs to the undifferentiated polymetamorphic complex of Mont-Mary and it is paragneiss with porphyroblasts (Polino et al. 2015). Moderate fracturing is observed with the presence of regular blocks with an estimated GSI in the range of 50–60 and a friction angle of the rock mass approximatively equal to 30° (Hoek Fig. 5 Sketch of a deposit of a rockfall prone area. One or more homogeneous areas are determined. In each area, a sampling subarea is identified. Each subarea is divided into four equally sized areas into which the sampling is performed Fig. 6 Sketch of the optimization techniques for the estimation of the parameters of the fragmentation model. The x-axis relates to the logarithm (base 10) of the volume, the y-axis relates to the logarithm (base 10) of the complementary cumulative probability. aOptimization by best interpolation. bOptimization by least error Fig. 7 Overall view from the base of the deposit area Landslides 16 &(2019)1390 Technical Note and Bray 1981). The onsite activities consisted in a traditional geomechanical survey according to the information given in ISRM (1978). Table 2reports the geomechanical data of the front and of the four detected joint sets. The stereonet of the observed joint sets was plot (Figure 8, DIPS 7.0 RocScience software) and the Markland test conducted under the hypothesis of a resisting friction angle equal to 30° revealed that a planar sliding can occur along plane B. No wedge failure can occur since the intersection of two discontinuities does not match the sliding condition (Hoek and Bray 1981). Toppling failure cannot develop since the dip is reduced. A detailed survey on the deposit was performed in order to obtain the fallen block size distribution. The procedure presented by Ruiz-Carulla et al. (2015) and modified by one of the authors was adopted, as described in the following. The in situ block size distribution was generated following the methodology previously reported. For the analysis, the joint sets A1, B, and C were considered. A set of isolated in situ volumes was obtained repeating the process 10 3 times. The set of generated volumes was subdivided into the volume classes reported in Table 1. The complementary cumulative distribution is reported in blue in Fig. 9. The rockfall block size distribution was obtained following the methodology previously reported. In the present case, only one homogenous area was identified, i.e., only one subarea was considered. The obtained complementary cumulative distribution is displayed in orange in Fig. 9. A power law trend, i.e., a linear interpolation on a log-log plot, emerges, confirming the scale invariance addressed in the literature (Dussauge et al. 2003). A rough comparison between IBSD and RBSD highlights that the two Table 2 Geomechanical properties of the front and the measured discontinuity sets Set Dip/dip direct Spacing (m) mean (min; max) Persistence (m) mean (min; max) A1 40°/274° 1.05 (0.37; 1.91) 0.55 (0.3; 0.8) A2 15°/274° 1.0 1.33 B 50°/215° 1.0 (0.32; 0.68) 2.68 (0.8; 9.7) C 76°/291° 1.3 2.98 Front 52°/215° Fig. 8 Stereonet of the measured discontinuity sets, made with the DIPS 7.0 RocScience software Landslides 16 &(2019) 1391 distributions differ. In the RBSD, 50% of the blocks belong to class 0, while in the IBSD, 50% of the blocks are smaller than class IV, showing that fragmentation process occurs during rockfall. Results and discussion The proposed optimization strategies were implemented on MATLAB and solved through genetic algorithms solvers (Deb 2001) using the IBSD and RBSD as inputs. Starting from an arbitrary vector (S r ;N;n), the values of the three objective functions reported in Eq. (8) or the result of Eq. (10) are computed. The genetic algorithm is an iterative procedure that generates a set of input vectors (named as parents) and identifies among them those (named as child) that better minimize the objective functions. At each iteration, namely, generation, the same procedure is repeated using as input modified values of the child parameters of the previous iteration (Michalewicz and Fogel 2013). The fragmentation model considers a different number of iterations, i.e., values of jranging from 1 to 4, as discussed in the following. A unique best solution cannot be identified in an optimization problem involving more than one objective function. On the 0.001 0.01 0.1 1 10 Volume (m3) 10-4 10-3 10-2 10-1 100 Complementary cumulative probability RBSD IBSD Fig. 9 In situ block size distribution (in blue) and rockfall block size distribution (in red) for the test site 0.001 0.01 0.1 1 10 Volume (m3) 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Complementary cumulative probability BI: Obj1 BI: Obj2 BI: Obj3 LE RBSD IBSD Fig. 10 Fragmented block size distributions for the best interpolation optimization (BI) and the least error optimization (LE) methods. The three BI curves represent the solutions which minimize each objective of Eq. (8), as described in the text. The IBSD and the RBSD curves are reported Landslides 16 &(2019)1392 Technical Note contrary, a set of solutions defining the tradeoff between competing objectives can be determined. In other words, a set of nondominated points, i.e., solutions for which none of the objective functions can be improved in value without degrading some of the other objective values, composing the Pareto front are found (Deb 2001). In this sense, the best interpolation (BI) optimization strategy, which is based on three objective functions, gives a set of values of πrather than a unique value. The number of best Table 3 Adopted parameters of the fractal fragmentation model for the curves displayed in Fig. 11. From columns (7) to (9), the values of the objective functions related to BI optimization method are reported. Column (10) reports the value of the objective function of LE optimization method. Column (11) reports the goodness parameter jS r Nn b p Obj1 Obj2 Obj3 ∑e2 iE Best interpolation optimization 1 0.236 79 61 4.291 0.772 0.7232 1.2695 1.5256 –0.5302 2 0.214 52 10 3.733 0.192 0.2103 0.2035 1.6634 –0.1035 3 0.277 49 11 3.659 0.224 0.2902 0.1712 1.6319 –0.1161 4 0.188 46 13 3.583 0.283 0.2520 0.3303 1.6276 –0.0793 Least error optimization 1 1.3×10 −7 91 18 4.498 0.198 –––0.4673 0.0668 2 0.026 30 7 3.107 0.233 –––0.4385 0.0626 3 0.026 35 7 3.271 0.200 –––0.4385 0.0626 4 0.026 31 7 3.141 0.226 –––0.4385 0.0626 0.001 0.01 0.1 1 10 Volume (m3) 10-6 10-4 10-2 100 Compl. cumul. probability j=1 0.001 0.01 0.1 1 10 Volume (m3) 10-5 10-4 10-3 10-2 10-1 100 Compl. cumul. probability j=2 0.001 0.01 0.1 1 10 Volume (m3) 10-5 10-4 10-3 10-2 10-1 100 Compl. cumul. probability j=3 0.001 0.01 0.1 1 10 Volume (m3) 10-5 10-4 10-3 10-2 10-1 100 Compl. cumul. probability j=4 BI LE RBSD IBSD Fig. 11 Fragmented block size distributions for the best interpolation optimization (BI) and the least error optimization (LE) methods for jranging from 1 to 4. The IBSD and the RBSD curves are reported Landslides 16 &(2019) 1393