With reference to the rockfall risk estimation and the planning of rockfall protection devices, one of the most critical and most discussed problems is the correct definition of the design block by taking into account its return period. In this paper, a methodology for the assessment of the design block linked with its return time is proposed and discussed, following a statistical approach. The procedure is based on the survey of the blocks that were already detached from the slope and had accumulated at the foot of the slope in addition to the available historical data.

Rockfall is one of the most critical slope instabilities
because it can be highly destructive and unpredictable. The analysis of this
phenomenon is very difficult because it is affected by aleatory variability
(irreducible natural variability) and epistemic uncertainty (lack of
knowledge). For these reasons, probabilistic methods are a suitable
approach for modelling rockfall. When risk analysis has to be performed for
forecasting and protection purposes, the size of the involved blocks and the
corresponding return period are the most important variables among the ones
that characterize the phenomenon

Modern design approaches for buildings, for example, aim to guarantee the
structural safety of the building throughout its expected lifetime. In such reliability-based
framework, the buildings have to be robust, i.e. to support forces due to
anthropic and natural hazards without being significantly damaged. Proper
design processes for common natural hazards, such as extreme winds or seisms,
are already present in the building codes

Dealing with natural hazards, one of the common ways to input the external
forces applied to the structures is to establish a link between the magnitude
of the forces and the corresponding return period. A larger return period
implies higher intensity in the force. In a recent work,

The magnitude–frequency relationship is at the basis of the probabilistic
hazard analysis. In seismic analysis, the Gutenberg–Richter law expresses such
relationship.

In designs of engineering works that protect a village or a road from falling rocks (e.g. net fences or embankments), the size of the falling block used in the modelling is currently not linked to its probability of occurrence, i.e. the return period of a block with such volume. The most frequently applied approaches refer to an analysis of the blocks that had already collapsed, integrated with a survey of the slope. The size of the falling design block is chosen from among the already collapsed blocks and the surveyed blocks of the slope.. Adapting the well-known procedures of modern design practice requires that the sizes of the falling blocks have to be related to their probability of occurrence and vice versa.

Examples of volume–frequency laws are proposed in the literature. They are obtained from the analysis of a large number of rockfall events for which each observed event is dated and the volume is estimated. This allows a volume–frequency curve to be drawn, in which each point corresponds to an observation. In general, precise catalogues with a large number of events are rare because the road owners or the territorial administrations started the records of events, which have large return periods, only some tens of years ago. For common uses, e.g. design of protective devices or risk estimation, for which there are no long records of events nor detailed surveys on site, no operative procedures are consolidated and the designer develops the project following his personal experience. In any case, the choice of the “characteristic block volume” (design volume) has to be done by the designer's own engineering judgement. For this reason, it is affected by subjectivity.

With the aim of contributing to overcoming this design problem, this
paper proposes a methodology for estimating the block volume–frequency
relationship that can be used for deriving the size of the falling design
block having a prescribed return period. The procedure, which is described in
detail in Sect.

Statistical analyses of historical data or experimental tests related to a certain natural phenomenon give evidence that it is possible to deduce power laws that link the magnitude of the event to its frequency. These mathematical relationships can be used for predicting type, extent, return time and magnitude of future events.

In the fifties,

The analysis of historical data, which are available in public archives or
catalogues, is therefore extremely important for the study of natural
phenomena. With particular reference to landslides and rockfalls, this
statistical approach has been recently studied and applied by several authors
in many mountain sites. Research has mainly focused on the analysis of the
volume distribution of rockfall events for the sites of Grenoble, Yosemite
Valley, Arly gorges, British Columbia, Hong Kong, Italian Apennines, Aosta
Valley, Christchurch-Canterbury and La Réunion Island

The comparison of the previous studies showed that negative power laws
fit all rockfall recurrence volume distributions well. However, some variability
in the values assigned to the power law coefficients does appear. This has
been mainly attributed to the variability in the sampling procedures of the
landslide volumes. At present, no proper test equipment (which provide, as
for earthquakes, objective and reliable values that are comparable from one
site to another) and standard procedures have been defined for the different
geological and structural settings where rockfalls may occur

Rockfall inventories do not always contain quantitative and detailed
information, and the description of historical events is often characterized
by a low degree of accuracy. For example, in the Yosemite rockfall inventory

Previous considerations, which have to be taken into account in treating
historical data, are related to Yosemite Valley but can be easily referred to
almost all of the historical archives

In addition, the temporal length of the observations can affect the
recurrence volumetric distribution. In particular, a few years time window
underestimates larger collapses. Many authors examined the frequency–size
distributions of both rockfalls and fallen blocks and noted that the
cumulative frequency is linearly related to the magnitude (block volume or
rockfall volume) on a log–log plot. In mathematical terms, the following
power law relationship subsists:

Sketch of a

This formulation implies that (i) larger rockfall events are less frequent
than those characterized by smaller size and (ii) frequency–size
distributions are well fitted by a power law only over a given range of
volumes. The power law exhibits a deviation from the observed distribution
for volumes smaller than a certain value. This discrepancy has been discussed
in the literature. It can be the result of undersampling of the smallest
rockfall events

Regarding power laws that are applied to rockfall volumes, the values of the
parameters of Eq. (

As mentioned, Eq. (

A three-step procedure for deriving a volume–frequency relationship for
blocks with a reduced amount of available data is built up and discussed in
the following. Some aspects of the proposed methodology result from
hydrological approaches in flood-frequency analyses (see

As described in detail in this section, the required data for deriving a
volume–frequency relationship are as follows.

A catalogue of events, i.e. events with quantitative rockfall volume estimates observed in the
representative area, is denoted as

A list of measured volumes that may have fallen down at any time is denoted as

Obviously, both the catalogue and the list must be related to the same area
of the slope, i.e. its foot. All the blocks in catalogue

The first step of the analysis consists of choosing “relevant” events
within catalogue

The second step of the analysis consists of the choice of two probabilistic
models. One should be able to describe the temporal occurrences of the events
of catalogue

Knowing the annual mean number of blocks bigger than

The third step of the analysis consists of the estimation of the parameters
of the statistical laws by means of the measured rockfall data contained in

The catalogue of the event

Since the recording of the events is related to in situ observations after
the occurrence, events involving small rock blocks are not always recorded.
Therefore, there is the possibility that catalogue

A reduced catalogue, which is mathematically described by Eq. (

Sketch of the catalogues of events

Under the hypothesis of independence between the observations, the rockfall
phenomenon is considered to be a completely random process for which any
realization consists of a set of isolated stochastically independent points
in time

The probabilistic model of the volume distribution at the foot of the slope
is determined using the records contained in list

In the present framework, substituting

An estimate of the parameter

The estimates of the scale and shape parameters,

With the aim of better explaining the proposed methodology, it was applied to
two areas affected by rockfalls. Both Buisson and Becco dell'Aquila are
located in Aosta Valley, north-western Italian Alps, as shown in
Fig.

Map of the two test site locations in the north-western Italian
Alps. The Buisson site is shown with a red bullet in

The Buisson site (UTM: 392267, 5077165, 32,

A detailed survey in the deposition area was performed: 60 blocks with volume
ranging from 0.02 to 308 m

Volumes of the surveyed blocks in the deposition area of the Buisson
site. The blocks are divided into two classes, depending on their size
(smaller or larger than

The reduced list,

Input and results of the analyses performed at the Buisson site. The estimates of the parameters of the distribution are reported in the bottom rows.

Figure

Volume–annual frequency of occurrence plot related to Buisson site.

Becco dell'Aquila site (UTM: 341345, 5074157, 32,

The historical catalogue of the Regional Geological Service of Aosta Valley
reports three events in this site since 1998 (April 1998, April 2001,
May 2012). The size of the fallen blocks is always larger than 5 m

The number of events considered in the analysis is equal to

Referring to the distribution of the volumes, the reduced list,

List of the grouped volumes of the surveyed blocks on the slope of
Becco dell'Aquila site. All the blocks belong to list

Input and results of the analyses performed on the Becco dell'Aquila site. The estimates of the parameters of the distribution are reported in the bottom rows (the standard deviations are detailed into brackets).

Based on the previously discussed data it was possible to obtain the
volume–annual frequency of occurrence that is reported in
Fig.

The definition of the relationship between the volumes that can stop on a slope and their return period is a parameter of paramount importance for a correct design procedure. The proposed methodology allows a volume–frequency law to be computed, which can be used in engineering calculations. Two different probabilistic models are considered: one for the Poisson's point process related to the occurrences of the events, the other for the fallen-block volume distributions (the GPD, which is independent of the year of rockfall occurrence). In order to make these considerations and use these probabilistic models, hypotheses are necessary.

The two probabilistic models are merged considering the hypothesis that the
annual frequency of a rock block having a volume equal to the threshold
volume is the parameter

Volume–annual frequency of occurrence plot related to Becco dell'Aquila site.

Bootstrap statistical parameters of the estimates of the parameters of the generalized Pareto distribution related to the two example sites.

The events described by Poisson's probabilistic models need to be
independent. In other words, no causality links have to subsist. Under this
hypothesis, the process is random

The GPD has been chosen for fitting the values of list

Pareto family distributions are very similar to power law distributions except for
the fact that the former are bounded distributions. The bound is represented
by the location parameter

GPD differs from the classical Pareto model for the introduction of a location parameter,
which does not affect the slope of the right part of the plot, being governed
by the exponent

GPD is suitable for extreme value analysis.

The degree of precision of the estimates of the parameters of a generalized
Pareto distribution is determined through a bootstrap analysis

From the set of the estimates of the parameters, the bootstrap mean, variance
and median and the values of 90 and 95 % confidence bounds are determined
(Table

Histogram of the volumes having 1000 years return period fitted by a log-normal law (red dashed line). The empirical distribution function is plotted in the box: the squares bound the 90 % confidence interval.

In addition, for each bootstrap replication, once parameters

It results that the width of the 90 % confidence interval increases as much as the return period increases. This implies a spread of the value of the volumes of the blocks. Detailed and long records of the events as well as a proper survey of the volumes of the blocks would permit an increase in the quality of the volume–frequency law and, as a consequence, reduce the statistical errors in the procedure.

Results of the bootstrap analysis on the records on Buisson and
Becco dell'Aquila sites. The continuous lines are plotted from
Eq. (

The proposed method allows the relationship between the return period and the
volume of the blocks to be defined. This is a key aspect in land management
and planning, design of protection devices

The historical data can be accessed at

The authors declare that they have no conflict of interest.

The authors acknowledge the two anonymous referees for their observations and comments of the manuscript. F. Laio is particularly acknowledged for having generously shared his experience and for his valuable comments. This research was supported by Regione Autonoma Valle d'Aosta under the framework of the project “Realizzazione di scenari di rischio per crolli di roccia”. Edited by: T. Glade Reviewed by: two anonymous referees