Rockfall Magnitude-Frequency Relationship Based on Multi-Source Data from Monitoring and Inventory

Abstract

Quantitative hazard analysis of rockfalls is a fundamental tool for sustainable risk management, even more so in places where the preservation of natural heritage and people’s safety must find the right balance. The first step consists in determining the magnitude-frequency relationship, which corresponds to the apparently simple question: how big and how often will a rockfall be detached from anywhere in the cliff? However, there is usually only scarce data on past activity from which to derive a quantitative answer. Methods are proposed to optimize the exploitation of multi-source inventories, introducing sampling extent as a main attribute for the analysis. This work explores the maximum possible synergy between data sources as different as traditional inventories of observed events and current remote sensing techniques. Both information sources may converge, providing complementary results in the magnitude-frequency relationship, taking advantage of each strength that overcomes the correspondent weakness. Results allow characterizing rockfall detachment hazardous conditions and reveal many of the underlying conditioning factors, which are analyzed in this paper. High variability of the hazard over time and space has been found, with strong dependencies on influential external factors. Therefore, it will be necessary to give the appropriate reading to the magnitude-frequency scenarios, depending on the application of risk management tools (e.g., hazard zoning, quantitative risk analysis, or actions that bring us closer to its forecast). In this sense, some criteria and proxies for hazard assessment are proposed in the paper.

Full text

Citation: Janeras, M.; Lantada, N.;
Núñez-Andrés, M.A.; Hantz, D.;
Pedraza, O.; Cornejo, R.; Guinau, M.;
García-Sellés, D.; Blanco, L.; Gili, J.A.;
et al. Rockfall Magnitude-Frequency
Relationship Based on Multi-Source
Data from Monitoring and Inventory.
Remote Sens. 2023,15, 1981. https://
doi.org/10.3390/rs15081981
Academic Editors: Giuseppe Casula,
Claudio Vanneschi, Mirko Francioni
and Neil Bar
Received: 5 February 2023
Revised: 27 March 2023
Accepted: 4 April 2023
Published: 9 April 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
remote sensing
Article
Rockfall Magnitude-Frequency Relationship Based on
Multi-Source Data from Monitoring and Inventory
Marc Janeras 1, 2, * , Nieves Lantada 2, M. Amparo Núñez-Andrés2, Didier Hantz 3, Oriol Pedraza 1,
Rocío Cornejo 2, Marta Guinau 4, David García-Sellés4, Laura Blanco 4, Josep A. Gili 2and Joan Palau 1
1Institut Cartogràfic i Geològic de Catalunya (ICGC), 08038 Barcelona, Spain
2Department of Civil and Environmental Engineering, Universitat Politècnica de Catalunya-BarcelonaTech,
08034 Barcelona, Spain
3Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, Univ. Gustave Eiffel, ISTerre,
38000 Grenoble, France
4
Departament de Dinàmica de la Terra i de l’Oceà, GRC RISKNAT, UB-Geomodels, Facultat de Ciències de la
Terra, Universitat de Barcelona (UB), 08028 Barcelona, Spain
*Correspondence: marc.janeras@icgc.cat; Tel.: +34-935671500
Abstract:
Quantitative hazard analysis of rockfalls is a fundamental tool for sustainable risk manage-
ment, even more so in places where the preservation of natural heritage and people’s safety must
find the right balance. The first step consists in determining the magnitude-frequency relationship,
which corresponds to the apparently simple question: how big and how often will a rockfall be
detached from anywhere in the cliff? However, there is usually only scarce data on past activity
from which to derive a quantitative answer. Methods are proposed to optimize the exploitation of
multi-source inventories, introducing sampling extent as a main attribute for the analysis. This work
explores the maximum possible synergy between data sources as different as traditional inventories
of observed events and current remote sensing techniques. Both information sources may converge,
providing complementary results in the magnitude-frequency relationship, taking advantage of
each strength that overcomes the correspondent weakness. Results allow characterizing rockfall
detachment hazardous conditions and reveal many of the underlying conditioning factors, which are
analyzed in this paper. High variability of the hazard over time and space has been found, with strong
dependencies on influential external factors. Therefore, it will be necessary to give the appropriate
reading to the magnitude-frequency scenarios, depending on the application of risk management
tools (e.g., hazard zoning, quantitative risk analysis, or actions that bring us closer to its forecast). In
this sense, some criteria and proxies for hazard assessment are proposed in the paper.
Keywords:
rockfall hazard; detection and observation; TLS monitoring; inventory; magnitude-
frequency; multi-scale; multi-source; spatial-temporal variability; conglomerate; basalt
1. Introduction
This paper is focused on the characterization of rockfall hazard scenarios using the
magnitude-frequency relationship as the first step of any hazard and risk quantitative
analysis. This task corresponds to relating the magnitude of potential rockfall events to
their occurrence probability or frequency, which is based on the rockfall activity in the past.
This basic data can be obtained both with traditional methods of inventory and nowadays
thanks to monitoring with remote sensing techniques. This is the context presented in the
following introductory section.
1.1. Magnitude-Frequency Relationship for Quantitative Assessment
Risk conceptualization and assessment are complex tasks because of the inherent
uncertainties, and rockfall, as a stochastic natural process, is not an exception. Risk in
geohazards is a concept that is often approached by society from an emotional point of
Remote Sens. 2023,15, 1981. https://doi.org/10.3390/rs15081981 https://www.mdpi.com/journal/remotesensing

Remote Sens. 2023,15, 1981 2 of 36
view, either minimizing or maximizing it. If risk management and its communication to
stakeholders are conducted in these terms, it can arbitrarily lead to imprudence or blocking,
both equally unjustified. Instead, quantitative risk analysis (QRA) should provide an
objective tool for risk management [
1
–
6
]. The complete procedure for rockfall QRA can be
summarized in four stages (Figure 1):
1.
Detachment: starting zone characterization to stabilize instability scenarios on the
rock face dealing with rockfall initiation. The questions are: which events of a certain
magnitude should be expected, and what is their probability of occurrence?
2.
Propagation: trajectory analysis to map the resulting hazard distribution. In this step,
it is critical to consider fragmentation, which modulates both the probability and
intensity of impact. Rockfall simulation is required.
3.
Impact: exposure considerations are to be cross checked with the hazard under
several conditions, either from the perspective of individual or collective risk. The
consideration of all possible damage scenarios can be addressed using event trees.
4.
Damage: vulnerability analysis to estimate the intensity of damage for each element
type in statistical terms. Fragility curves help to raise the uncertainty in this last step.
Remote Sens. 2023, 15, x FOR PEER REVIEW 2 of 42
1.1. Magnitude-Frequency Relationship for Quantitative Assessment
Risk conceptualization and assessment are complex tasks because of the inherent
uncertainties, and rockfall, as a stochastic natural process, is not an exception. Risk in
geohazards is a concept that is often approached by society from an emotional point of
view, either minimizing or maximizing it. If risk management and its communication to
stakeholders are conducted in these terms, it can arbitrarily lead to imprudence or
blocking, both equally unjustified. Instead, quantitative risk analysis (QRA) should
provide an objective tool for risk management [1–6]. The complete procedure for rockfall
QRA can be summarized in four stages (Figure 1):
1. Detachment: starting zone characterization to stabilize instability scenarios on the
rock face dealing with rockfall initiation. The questions are: which events of a certain
magnitude should be expected, and what is their probability of occurrence?
2. Propagation: trajectory analysis to map the resulting hazard distribution. In this step,
it is critical to consider fragmentation, which modulates both the probability and
intensity of impact. Rockfall simulation is required.
3. Impact: exposure considerations are to be cross checked with the hazard under
several conditions, either from the perspective of individual or collective risk. The
consideration of all possible damage scenarios can be addressed using event trees.
4. Damage: vulnerability analysis to estimate the intensity of damage for each element
type in statistical terms. Fragility curves help to raise the uncertainty in this last step..
Figure 1. Sketch of the approach considering the contributions of Geomatics to the QRA
(quantitative risk analysis) for rockfalls. Highlighted in the orange box is the part covered by this
article: rockfall detachment and the hazard scenario definition in terms of magnitude-frequency
relationship.
Based on previous research [7], two distinct analyses of rockfall hazards can be
distinguished: diffuse and focused hazard analysis. While the approach to focused
hazards tries to answer the question, “Is this single block unstable?” and “How unstable
is it?”; for diffuse hazards, the question is another one: “How big and how often will a
rockfall be detached from anywhere in the cliff?”. Geomatic remote sensing techniques,
such as LiDAR or photogrammetry, can be used by both approaches to try to answer these
Figure 1.
Sketch of the approach considering the contributions of Geomatics to the QRA (quantitative
risk analysis) for rockfalls. Highlighted in the orange box is the part covered by this article: rockfall
detachment and the hazard scenario definition in terms of magnitude-frequency relationship.
Based on previous research [
7
], two distinct analyses of rockfall hazards can be dis-
tinguished: diffuse and focused hazard analysis. While the approach to focused hazards
tries to answer the question, “Is this single block unstable?” and “How unstable is it?”; for
diffuse hazards, the question is another one: “How big and how often will a rockfall be
detached from anywhere in the cliff?”. Geomatic remote sensing techniques, such as LiDAR
or photogrammetry, can be used by both approaches to try to answer these questions [
8
–
11
].
When applied as monitoring techniques for change detection, rock surface deformation
can be read as instability indications for focused analysis, and rockfall detection provides
activity sampling for the frequency analysis of diffuse hazards. A broader state of the art
on rockfall hazard assessment can be found in [10].

Remote Sens. 2023,15, 1981 3 of 36
This paper aims at diffuse hazard analysis and is part of the Georisk project entitled
“Advances in rockfall quantitative risk analysis (QRA) incorporating developments in
geomatics” (www.georisk.upc.edu, accessed on 5 April 2023). The diagram in Figure 1
shows the scope of the project and the main objective of taking advantage of new geomatic
developments for transferring metrics to risk analysis. At this point, remote sensing allows
us the detection of rockfall activity (location and dating), as well as the measurement of
detached volume or its magnitude [
12
]. Consequently, it provides both data necessary
for magnitude-frequency analysis. Along with rockfall propagation, fragmentation may
frequently occur [
13
]. Then, we can either measure the volume in the compact rock
compartment before detachment or by summing the volume of all the resulting blocks
at each run-out position. In the case of measuring the deposit volume as a whole, it will
be possible to obtain the apparent expanded volume, including the porosity between
fragments, which is necessary to make the appropriate correction. It must be distinguished
between the size distribution of blocks resulting from fragmentation of one or superposed
events [
14
], and the size distribution over time of total detached volume, the latter being
the focus of this article.
Therefore, magnitude does not report the movement but only the dimensions of the
initial trigger. Likewise, the magnitude-frequency relationships are the genuine expression
of the probability of occurrence of rockfall release according to their size. This results
from the properties of the rock mass and escarpment and the geodynamic and climatic
context that causes the triggering actions. In this formulation, subsequent propagation is
not considered, which will depend on the morphology and conditions of the slope located
below the escarpment. Thus, the magnitude-frequency relationship does not fully express
the idea of hazard but only its initial conditions, which should be more precisely called
detachment hazard scenarios. In the rest of the text, we will use the notation McF to denote
the relationship between magnitude and its cumulative frequency. McF has been analyzed
for all landslide types [
3
,
15
–
20
]. According to the recent review of this issue specifically for
rockfalls by [21], our approach is equivalent to the so-called McF methods.
1.2. Rockfall Activity Detection and Registration
In the diffuse risk analysis, rockfall is considered a recurrent phenomenon distributed
over time and space. Thus, it is assumed that its frequency can be derived from past
activity as a representative characterization also for the future, while the underlying
conditions remain invariant. Therefore, the frequency analysis is based on the register of
rockfall activity. Observing an event either directly or retrieved by survey, a documentary
record, a land photographic register, or remote sensing and monitoring are commonly used
methods for acquiring this type of data. Some additional considerations are detailed in
Supplementary Material SM1. Except in the case of direct observation, it is necessary to
do a certain process of extracting information from the available data, and this process
will determine the properties of the inventory obtained. Whatever the data source, a
perfect record of landslide activity is never obtained. In landslides, there is a current trend
to combine the detection capabilities of remote sensing with processing using machine
learning techniques [22,23].
In reporting rockfall activity, each record corresponds to an event, understood as a
single rockfall, either a monolithic block or a disintegrated rock mass. However, they are
physically connected in the same failure mechanism and with a simultaneous or continuous
fall. We consider the following criteria into account in different cases: Despite having the
same trigger (as is sometimes the case during an earthquake), simultaneous rockfalls from
various points are distinct events. Even though it may make sense to consider them as
progressive failure, rockfalls that occur near together but at different times due to various
trigger pulses are considered separate events. For subsequent exploitations of the inventory,
it is desirable to have additional fields of the conditions under which they occur (e.g.,
environmental conditions, identified triggers, linkage in progressive ruptures). However,

Remote Sens. 2023,15, 1981 4 of 36
for the McF analysis that we aim to conduct, three essential data points must be obtained
from each event:
•
Where? The localization of the detachment point. In rockfall from vertical cliffs, xyz
coordinates are needed, as xy coordinates on digital terrain models incur a great deal
of indeterminacy in the elevation. To determine the spatial representativeness, at least
each rockfall is needed to be linked to a cliff sector that will determine the spatial
resolutions of the analysis.
•
When? The accuracy of the dating of events can be highly variable depending on the
data source, from imprecise references in years to detailed dates and times. For the
annual frequency calculation, we simply need the dating by years, but this is not the
case in triggering factor analyses.
•
How big? This is the feature to be analyzed for hazard scenario assessment. It is
expressed in the total volume of rock detached from the cliff as a single rockfall event.
Since the accuracy of these three components can be very variable, it is convenient
to enrich inventories with auxiliary fields describing the accuracy of each variable inde-
pendently, either numerically or qualitatively in typical cases, to reflect the interpretation
process employed until data are registered. It is also important to consider the completeness
of the inventory as a whole and even its heterogeneity according to the data sources used.
1.3. Remote Sensing in Rockfall Problems
A comprehensive review of remote sensing techniques for rockfall research is beyond
the scope of this article, however, it is accessible in [
8
,
10
,
24
–
26
]. Other remote sensing
systems are reported in [
24
,
27
–
31
]. Among them, two technologies are becoming predom-
inant in rockfall analysis due to their versatility: LiDAR and photogrammetry, both of
which provide 3D point clouds of the object’s surface, including the ground [
24
,
27
,
32
]. The
combination of these two techniques on different platforms (terrestrial and airborne, lately
from drones), providing complementary points of view (frontal, oblique, and zenithal),
constitutes a complete remote sensing system for rockfall characterization [
8
,
25
,
28
,
33
]. The
recent irruption of UAVs (unmanned aerial vehicles) in geosciences [
34
,
35
] has greatly facil-
itated access to oblique views, which are particularly useful in the study of rockfalls [
36
,
37
].
For the test sites in this paper (Section 3), we are using LiDAR monitoring data, in particular
from a terrestrial laser scanner (hereafter TLS), which allows a frontal view of vertical cliffs
from opposite slopes or viewpoints at their bottom.
The products that can be derived from rockfall 3D models/clouds cover the successive
analysis steps: morphology and susceptibility of the slope; terrain surface for trajectory
calculation and roughness consideration; rock mass structure and rupture mechanisms;
volume measurement of potentially unstable masses and scars on the wall; volume mea-
surement of deposits and fallen blocks; monitoring for the detection of rockfall activity;
and monitoring for the detection of surface displacements as precursor signs of instabil-
ity [38–43].
1.4. Objectives and Content
In accordance with the scope presented in this introduction, the main objective of
this study is specified as optimizing the exploitation of rockfall inventories for hazard
assessment. We intend to cross check the results of traditional inventories and monitoring
to leverage each other and determine their complementarity. Specific objectives cover other
related issues:
•
Hazard variability over time is due to triggering conditions and other evolving factors.
•Hazard variability at different scales from outcrop to massif.
•Influence of other factors on rockfall detachment conditions and related hazards.
All these investigations aim to better understand factors that influence the preparation
and triggering of rockfalls, with the horizon set in their forecast to enrich risk management
strategies. After this introductory section, the content of the paper is structured as follows:

Remote Sens. 2023,15, 1981 5 of 36
In Section 2, the methodology that will be applied is summarized, and the sampling extent
feature is introduced. In Section 3, the case sites used as examples for the developments are
presented with their rockfall activity samples obtained by monitoring and inventory. These
data are analyzed in Section 4separately by each source type to take into consideration the
limitations encountered, apply appropriate treatments, and make the most of the available
data. The results in terms of magnitude-frequency relationships are presented, and in
Section 5the main findings are discussed in view of the various applications to hazard
assessment oriented to rockfall risk management before concluding.
2. Methods
The main definitions and formulations for McF analysis follow in this section.
2.1. Methodological Basis
According to the glossary in use [
3
,
12
], the rockfall magnitude is the volume
V
of the
total detached mass in a single event, and its frequency is expressed in inverse cumulative
terms,
F(V)=F(Volume ≥V)
. This formulation corresponds to the fact that we are not
interested in the probability of any case of a specific size, but of all cases large enough to
cause a certain level of effects. Under uniform and constant conditions of the rock mass
and wall, the frequency increases proportionally to the period and extension of the source
area. Therefore, frequency is normalized to spatial-temporal cumulative frequency,
Fst (V)
,
being expressed in normalized units,
hhm−2·year−1i
, where the source area extension is
computed in
hm2
and the time span in
years
[
7
]. This way,
Fst (V)
describes the cliff activity
in terms of the pursued McF. Multiple rockfall hazard studies in the last 25 years [
19
,
44
–
53
]
have established that the distribution of detached volumes can be adjusted by a power law
such as
Fst (V)=Ast·V−B
, in which two parameters describe the rockfall activity of the
rock cliff:
•Ast
is the normalized or unitary activity, as
Ast =Fst(1)
, being the number of rockfalls
of
V≥
1 m
3
detached from 1 hm
2
of cliff surface in 1 year. It is a significant normalizing
case since in most situations, this size is both frequent and destructive enough to be
considered a reference.
•B
is the uniformity coefficient in the volume distribution. The greater the
B
, the greater
the reduction in probability when going from small to large rockfalls. By definition
of cumulative frequency,
B>
0, and the negative sign of the exponent is already
introduced in the equation.
In Supplementary Material SM2, more details on McF power law are highlighted.
There, we define the center of the hyperbola as the point where the curve
Fst
is orthogonal
to the bisector line of the first quadrant (
F0st =−
1), which is placed on the volume
value
Vc=(Ast·B)1/(B+1)
. By center definition, for volume values
VVc
, small linear
variations in volume correspond to a larger change in frequency, and, inversely, for
VVc
large linear variations in volume have a small influence on the corresponding frequency.
In other words, uncertainty in volume plays a more significant role on the left side than
on the right, which leads to the necessity of measuring properly small volumes using
TLS monitoring as an appropriate tool. The center does not necessarily correspond to the
point of the hyperbola closest to the coordinate origin
(0, 0)
; instead, we call this point the
origin of the hyperbola that is placed at,
V0=A2
st·B1
2(B+1)
, and this minimum distance is
d0=V0√1+1/B.
As established by [
7
], the integration of the area below the hyperbola corresponds to
the erosion rate, which is a proper descriptor of the detachment hazard of the outcrop. This
calculation requires establishing upper and lower limits of plausible volumes linked to
the characteristics of the massif and the slope [
49
]. Without considering these magnitude
thresholds, the hazard cannot be fully concluded, but the center of McF describes the
central part of the hazard and can be used to compare different case sites. For this purpose,

Remote Sens. 2023,15, 1981 6 of 36
two features of the hyperbola can be derived: from its origin (
d0
, already introduced) and
from its center (
pc
, as the product of both coordinates of the center). It can be seen that
pc=Vc·Fst (Vc)=Ast(Ast·B)1−B
1+B
. In this paper, the center,
(Vc;Fc)
, is used to describe McF
by one point instead of a line to easier compare different case sites and to see how rock
mass and environmental conditions influence the hazard potential. As seen in Section 5,
without considering the truncated curve at the maximum expected magnitude,
Vmax
, that
would provide the true meaning of hazards and erosion rates. Under this assumption,
pc
and
d0
are suggested as proxies for the intrinsic hazard of the slope before considering the
geometry of the outcrop.
2.2. Sampling Extent
Rockfall inventories are registers of its activity covering a specific time span
ct
and
source zone area
ca
, which are needed to be assessed for computing
Fst
. Furthermore, they
define the sample contained in the inventory. In this manner, the sampling extent
SE
is
defined as follows:
SE(sampling extent)=covered areahhm2i×covered period [year](1)
Very frequent rockfall activity is well characterized by samples of low
SE
, but not
the infrequent activity or exceptional events in which we are particularly interested in
hazard assessment. Then, a large
SE
is required, which can be provided by a time span
or covered area (Figure 2). The bisector axis represents
SE
as the product of both axes
(spatial and temporal). In this kind of graph, rockfall inventories can be represented to
quickly see their attributes. Samples of equal
SE
are placed on a hyperbolic curve, and
their distance from the bisector axis shows if they are time-over-space enriched samples
or in reverse. On this graph, inventories obtained by the different data sources previously
presented can be compared. Monitoring inventories, such as those from TLS, tend to
be local and short samples, while observational inventories tend to build a sample with
larger coverage in space and time, but then the problem is making sure it is complete.
We propose merging rockfall inventories from different data sources to take advantage of
the completeness of monitoring inventories and the representativeness of observational
and historical inventories. This is because the longer the time span, the more confidence
there is in calculating return periods. As declared in Section 1.4, we attempt to extract as
much detail as possible about the spatial and temporal variability of hazards, such as [
54
].
Thus, we attempt combining multi-source data to compensate for the shortcomings of one
source and another and to cover the widest magnitude range possible, such as [
55
]. The
introduction of
SE
in the analysis allows us to achieve this since it characterizes the samples
when combining them.
SE
happens to be an attribute of the recorded events, in which the
extent of the rockfall has been detected and recorded.
Finally, it should be noted that the
SE
value, which allows obtaining a sufficient
sample to characterize the activity, depends on the degree of the activity itself. In places or
periods—both variables will be analyzed in this paper—of great activity, a small sampling
area and period of time are sufficient to obtain a sufficient sample, whereas, in places or
periods of minimal activity, it is necessary to expand the sampling to achieve a sufficient
level of representativeness. Therefore, it is convenient to put
SE
in relation to the parameter,
Ast
, in such a way that we introduce the unitary sampling extent as
SEu A =SE·Ast
,
which is dimensionless. Note that it is the expected number of rockfalls larger than
1 m
3
to be registered by the sample. In Figure 3, samples of case sites analyzed in this
paper and presented in the next Section 3are shown this way. Every sample can be
rapidly characterized by
SEu A
, because
SEu A <
1 means that there is no sample larger
enough to expect collected events larger than 1 m
3
and to be confident with the
Ast
value.
Alternatively,
Fc
could be also used to normalize
SE
spite less intuitively. In that case,
if
SEuc =SE·Fc
1, the sampling extent covers appropriately the central part of the
hyperbola, providing confidence in the obtained Bvalue.

Remote Sens. 2023,15, 1981 7 of 36
Remote Sens. 2023, 15, x FOR PEER REVIEW 7 of 42
Figure 2. Practical graph for the assessment of sampling extent 𝑆𝐸 according to its spatial and tem-
poral coverage. Monitoring inventories, such as those obtained by TLS, tend to cover a limited cliff
area because of site constraints, and time is equal to the monitoring period. Meanwhile, observa-
tional inventories, including documentary information, can easily cover larger areas for longer pe-
riods of time, but in that case, completeness becomes the main difficulty.
Finally, it should be noted that the 𝑆𝐸 value, which allows obtaining a sufficient
sample to characterize the activity, depends on the degree of the activity itself. In places
or periods—both variables will be analyzed in this paper—of great activity, a small sam-
pling area and period of time are sufficient to obtain a sufficient sample, whereas, in places
or periods of minimal activity, it is necessary to expand the sampling to achieve a suffi-
cient level of representativeness. Therefore, it is convenient to put 𝑆𝐸 in relation to the
parameter, 𝐴, in such a way that we introduce the unitary sampling extent as 𝑆𝐸 =
𝑆𝐸 ∙ 𝐴, which is dimensionless. Note that it is the expected number of rockfalls larger
than 1 m to be registered by the sample. In Figure 3, samples of case sites analyzed in
this paper and presented in the next Section 3 are shown this way. Every sample can be
rapidly characterized by 𝑆𝐸, because 𝑆𝐸 1 means that there is no sample larger
enough to expect collected events larger than 1 m and to be confident with the 𝐴
value. Alternatively, 𝐹 could be also used to normalize 𝑆𝐸 spite less intuitively. In that
case, if 𝑆𝐸 =𝑆𝐸∙𝐹
≫1, the sampling extent covers appropriately the central part of the
hyperbola, providing confidence in the obtained 𝐵 value.
Figure 2.
Practical graph for the assessment of sampling extent
SE
according to its spatial and
temporal coverage. Monitoring inventories, such as those obtained by TLS, tend to cover a limited cliff
area because of site constraints, and time is equal to the monitoring period. Meanwhile, observational
inventories, including documentary information, can easily cover larger areas for longer periods of
time, but in that case, completeness becomes the main difficulty.
Remote Sens. 2023, 15, x FOR PEER REVIEW 8 of 42
Figure 3. Dimensionless expression of sampling extent 𝑆𝐸

=𝑆𝐸∙𝐴

for Montserrat (in orange)
and Castellfollit (in blue). On the left side, samples obtained by TLS monitoring are either global
and disaggregated by regions or levels. On the right side, samples were obtained from observational
inventory, where 𝑆𝐸 is variable with volume according to the analysis presented in Subsection 4.2.
Another key feature of inventories obtained by rockfall monitoring is the sampling
interval, as pointed out by [51,53], which should consider the scope of the monitoring. To
perform trigger analysis, continuous monitoring is required. However, for McF analysis,
discontinuous monitoring is preferred to maximize covered 𝑆𝐸. The surveying period 𝑆𝑃
should be put in relation to the activity level (𝐴) at the site. If surveys are planned with
𝑆𝑃 = 𝑉
𝐴
⁄, it will be expected to mean one single event of 𝑉≥𝑉
 within a unitary
source area of 1 hm
2
. In case sites such as those considered in this paper have 𝐴 =0.1
and 𝐵=0.5, a surveying period of 0.3 years would be appropriate to unequivocally iden-
tify rockfalls larger than 𝑉 =10
m.
Figure 3.
Dimensionless expression of sampling extent
SEu A =SE·Ast
for Montserrat (in orange)
and Castellfollit (in blue). On the left side, samples obtained by TLS monitoring are either global
and disaggregated by regions or levels. On the right side, samples were obtained from observational
inventory, where SE is variable with volume according to the analysis presented in Section 4.2.
Another key feature of inventories obtained by rockfall monitoring is the sampling
interval, as pointed out by [
51
,
53
], which should consider the scope of the monitoring. To
perform trigger analysis, continuous monitoring is required. However, for McF analysis,
discontinuous monitoring is preferred to maximize covered
SE
. The surveying period
SP
should be put in relation to the activity level (
Ast
) at the site. If surveys are planned with
SP =VB
min /Ast
, it will be expected to mean one single event of
V≥Vmin
within a unitary
source area of 1 hm
2
. In case sites such as those considered in this paper have
Ast =
0.1 and
B=
0.5, a surveying period of 0.3 years would be appropriate to unequivocally identify
rockfalls larger than Vmin =10−3m3.

Remote Sens. 2023,15, 1981 8 of 36
3. Test Sites and Data
The test sites in this paper are Montserrat Massif and Castellfollit de la Roca Cliff,
where ICGC is currently conducting monitoring with TLS for rockfall risk mitigation
projects, which must meet the requirements of both safety and the preservation of the
natural heritage. For this reason, research is being carried out in monitoring aimed at rock-
fall forecasts on which to base future sustainable risk management strategies. Monitoring
was performed with two different TLS devices (see Table 1) offering similar performances:
Optech from the beginning in 2007/08 and the Leica P50 from 2018.
Table 1.
Significant features of the TLS devices in use at the case sites according to their technical
specifications. Optech ILRIS-3D has been used since 2007 and Leica P50 since 2018.
Manufacturer Model Maximum
Range
Range
Accuracy
Scan
Rate
Mean
Spacing
Spot
Diameter
(Name) (Name) (m) (mm @100 m) (Hz) (mm @100 m) (mm @100 m)
Optech ILRIS-3D 1500 7 2×10330 29.0
Leica ScanStation P50 570 4 up to 1 ×1068/16/31/63 26.5
The Montserrat Massif and Castellfollit de la Roca are excellent examples of both sorts
of rockfall risks. The Montserrat Massif is an example of a location where boulders fall
down a slope and impact with great force, while Castellfollit de la Roca is an example of a
location where rockfalls cause the escarpment to retreat, which can lead to the collapse of
constructions atop the cliff. In both cases, McF’s analysis helps us define hazard scenarios
for subsequent risk management and communication. As claimed by [
21
], the main
characteristics of the test sites can be found in Supplementary Material SM3, and more
details on rockfall issues follow in the next subsections.
3.1. Conglomerate Massif in Montserrat
Montserrat is an isolated mountain placed 40 km inland from Barcelona, Catalonia,
in the northeastern part of Spain. This massif is formed by thick layers of conglomerate
interleaved by siltstone/sandstone from a Late Eocene fan delta along the southeastern
margin of the Ebro Foreland Basin and adjacent to the Catalan Coastal Ranges [
56
]. At
present, it emerges as inverted relief over the Llobregat River with an overall height
difference of 1000 m (from 200 to 1200 m.a.s.l.). This configuration leads to the characteristic
relief of staggered slopes, where vertical cliffs of conglomerate alternate with steep slopes
of softer ground covered by densely vegetated colluvial deposits.
The mountain is a natural park due to its geological and biological assets and covers
an extension of 35 km
2
. Moreover, at mid-height, there is a thousand-year-old monastery
with a long and intense religious and cultural tradition. The double interest for tourists
and hikers, together with the proximity to the metropolitan region of Barcelona, produces
a great deal of frequentation in this small massif. This exceeds 3 million visitors, who
concentrate especially in the vicinity of the monastery and sanctuary and in other places
throughout the park. This exposure, combined with inherent geohazards, has demonstrated
the risk for the building area and the terrestrial accesses, both roads and a rack railway. This
risk was highlighted in the events of 2007–2008 in Degotalls affecting terrestrial accesses to
monasteries and in 2010 affecting a building on the monastery grounds, which prompted
first efforts to move from reactive to preventive actions [57].
Rock mass outcrops are represented by walls and needles, where rockfalls are gen-
erated. To assess the spatial variability of hazards all over the Montserrat Massif, slope
units were established and called “slope domains.” Their delineation similarly follows the
half-basin to [
58
], which is additionally crossed by the horizontal stratigraphic levels of
main conglomerate units leading to 1070 domains. Since the relief is staggered, rockfall
source areas are properly discretized through these domains. Rockfall source areas are
identified following the approach of geomorphometric analysis of slopes proposed by [
59
].

Remote Sens. 2023,15, 1981 9 of 36
In Montserrat, two morphological units are identified as rockfall source areas: inclined rock
walls
slope ∈[46◦; 70◦]
and vertical walls
slope ∈[70◦; 90◦]
, with
slope
being the raster
derived from the 2.5D terrain elevation model. For the entire Montserrat Mountain and
natural park, the sum of the rockfall source area is 1088 hm
2
. The dominant morphological
unit is
slope ∈[18◦; 46◦]
, which corresponds to steep slopes where rockfalls propagate and
form colluvial deposits.
3.1.1. Observational Inventory
As part of the risk mitigation plan, an inventory was launched in 2014 to record
the activity of geological risks in Montserrat, including rockfalls, shallow landslides, and
torrential flows. It is a local scale inventory fed by different sources of information. On
the one hand, a surveillance task is carried out consisting of periodic field inspections
covering specific areas of interest and also being reactive to rainfall episodes. On the
other hand, another way to obtain information is to search and extract from documents of
heterogeneous nature. Recently, the inventory has been reviewed to improve homogeneity
in the quality of the information. At this stage, three fields have been added to the DB that
qualitatively describe the accuracy of the assessment of the three variables of the event
vector (location, dating, and volume). The sample is formed by a total of 222 rockfalls
that occurred between the years 1546 and 2020, with volumes ranging from 2
×
10
−4
to
2.2 ×103m3
, distributed throughout the massif with greater concentration in areas with
higher human presence (Figure 4). The sample after filtering useless data (8%) is formed
by a total of 205 rockfalls associated with only 97 slope domains. Accordingly, five zones
have been selected for McF analysis, where the highest density of data is achieved because
rockfalls directly affect infrastructures of major interest (Figure 4). They total 236 hm
2
of
rockfall source area or 22% of the total in Montserrat. Despite having recovered an event
from the 16th century, it can be called “young inventory,” which has not been able to go
back much in time since 92% of the records are from the 21st century. In terms of magnitude,
48% have a volume of less than 1 m
3
, 34% are between 1 and 10 m
3
, and the remaining 18%
are greater than 10 m
3
. The inventory in Montserrat is estimated to have a total value of
SE =5156 hm2·year.
3.1.2. TLS Monitoring
TLS monitoring started in May 2007, after a large rockfall in the Degotalls section
of the monastery parking that affected both the road and the railway that has a parallel
track 115 m down the slope. The progressive failure at this point of the wall led to another
rockfall in December 2008, which is the largest event captured by TLS (before and after
scanners were available) until the present. As part of the mitigation plan, other local sites
have been progressively added to the monitoring system [
26
], until there are currently
twelve stations, from which the seven longest series are included in this study (Figure 4).
The most common period between surveys is about 6 months, although it was reduced to
half for two years (2015–2016) looking for seasonal variations of rockfall activity, which
was not clearly observed. The rock wall covered surface is mostly around 1 hm
2
, except for
the panoramic view on the wall above the Monastery area from Fra Garíviewpoint, which
is the triple. Due to the combined effect of coverage in space and time,
SE
varies by up
to an order of magnitude depending on the station. Table 2describes the main attributes
of the sample obtained with TLS in Montserrat. In total, 592 rockfalls have been detected,
which cover a wide range of six orders of magnitude.

Remote Sens. 2023,15, 1981 10 of 36
Remote Sens. 2023, 15, x FOR PEER REVIEW 11 of 42
Figure 4. Map of the Montserrat Massif where the relief and rockfall source areas (𝑠𝑙𝑜𝑝𝑒 ≥ 46°) are
highlighted. Main data sources are shown: location of TLS stations for monitoring, rockfall obser-
vational inventory available, and the areas where inventory analysis is performed.
3.1.2. TLS Monitoring
TLS monitoring started in May 2007, after a large rockfall in the Degotalls section of
the monastery parking that affected both the road and the railway that has a parallel track
115 m down the slope. The progressive failure at this point of the wall led to another rock-
fall in December 2008, which is the largest event captured by TLS (before and after scan-
ners were available) until the present. As part of the mitigation plan, other local sites have
been progressively added to the monitoring system [26], until there are currently twelve
stations, from which the seven longest series are included in this study (Figure 4). The
most common period between surveys is about 6 months, although it was reduced to half
for two years (2015–2016) looking for seasonal variations of rockfall activity, which was
not clearly observed. The rock wall covered surface is mostly around 1 hm2, except for the
panoramic view on the wall above the Monastery area from Fra Garí viewpoint, which is
the triple. Due to the combined effect of coverage in space and time, 𝑆𝐸 varies by up to
an order of magnitude depending on the station. Table 2 describes the main attributes of
the sample obtained with TLS in Montserrat. In total, 592 rockfalls have been detected,
which cover a wide range of six orders of magnitude.
Figure 4.
Map of the Montserrat Massif where the relief and rockfall source areas (
slo pe ≥
46
◦
)
are highlighted. Main data sources are shown: location of TLS stations for monitoring, rockfall
observational inventory available, and the areas where inventory analysis is performed.
Table 2.
Summary of TLS monitoring in Montserrat Massif and obtained rockfall inventory are used
in this paper.
Station First
Survey
Last
Survey
Surveyed
Period Surveys Surface SE Rockfalls Mean
Activity
(Name) (Date) (Date) (Years) (Number) (hm2) (hm2·Year) (Number) (hm−2·Year−1)
Degotalls N 2007-05-11 2020-11-24 13.55 26 1.73 23.44 225 9.60
Degotalls E 2007-05-11 2020-11-24 13.55 26 1.33 18.02 140 7.77
Monastery 2011-02-15 2020-11-24 9.78 25 3.57 34.29 170 4.56
Guilleumes 2016-07-22 2020-11-25 4.35 10 0.83 3.61 11 3.05
Sant Benet 2016-07-22 2020-11-25 4.35 10 1.00 4.35 12 2.76
Collbató2015-07-07 2020-12-01 5.41 7 1.03 5.57 21 3.77
Can Jorba 2016-07-19 2020-12-01 4.37 8 1.29 5.64 13 2.30
min max weighted sum sum sum sum weighted
Total 2007-05-11 2020-12-01 8.86 112 10.78 95.55 592 6.20
3.2. Basaltic Cliff in Castellfollit de la Roca
Castellfollit de la Roca is a village of 940 inhabitants located 40 km inland from Girona,
Catalonia, in the northeastern part of Spain. This small municipality in extension (0.7 km
2
)
is limited to the built-up center of medieval origin, defensively placed on a raised hill.
This relief is caused by a Quaternary basaltic formation, which is the overlap of two lava
flows from around 200,000 years ago that were separated by 20,000 years and a thin layer

Remote Sens. 2023,15, 1981 11 of 36
of paleosol, which makes a weak spot in the whole rock massif. The later erosive fluvial
incisions on its banks (the main river Fluviàat the north flank and a tributary river, Toronell,
at the south) have shaped the elongated cliffs that surround the entire hill [
60
]. This process
has resulted in a basaltic cliff of up to 40 m in height and around 1200 m in perimetral
length, which is currently one of the main geomorphological highlights of the natural park
of the Garrotxa Volcanic Field and has a high value as a natural and cultural landscape.
3.2.1. Observational Inventory
An observational inventory in Castellfollit has been built based on historical data com-
piled by recent documents on the local rockfall risk, whether academic studies or technical
reports. There are some historical clues about the clifftop withdrawal over the centuries, but
so far, no clear data on rockfall events have been found. A low-intensity surveillance task
is carried out in parallel to TLS monitoring, which does not provide significant additional
data. Until 2020, the inventory recorded 19 rockfall events, including large magnitude ones
in 1976 and 2011. The list can be found in Supplementary Material SM3.
3.2.2. TLS Monitoring
In March 2006, TLS monitoring began in Castellfollit de la Roca. Based on the previous
work [
61
], subsequent TLS monitoring has been able to detect both rockfall activity and
precursory displacements. In this way, although the survey cadence is variable, a series
of data from 2008 to 2020 has been obtained and is discussed here. The three operational
TLS stations in Castellfollit (Figure 5) cover a significant extent of the northern side of the
town, corresponding to the main cliff. On the other hand, there is no monitoring available
of the cliff on the south side since it is a densely vegetated ravine that does not allow views
suitable for remote sensing.
Remote Sens. 2023, 15, x FOR PEER REVIEW 13 of 42
Figure 5. Map of the hill formed by basaltic cliffs, where Castellfollit de la Roca village is placed.
The main data sources for this work are highlighted: the relief and rockfall source areas (𝑠𝑙𝑜𝑝𝑒 ≥
46°); the rockfall observational inventory available; TLS stations in monitored sites.
Table 3 describes the main attributes of the sample obtained with TLS in Castellfollit
de la Roca, which totals 𝑆𝐸 = 33.14ℎ𝑚∙𝑦𝑒𝑎𝑟. In total, 298 rockfalls have been detected
since 2008 with sizes between 2×10−4 m3 and 6×100 m3, which cover a limited range of four
orders of magnitude because any large event in the inventory coincided with the scope
covered by monitoring.
Table 3. Summary of TLS monitoring in the Castellfollit de la Roca cliffs and obtained rockfall in-
ventory are used in this paper.
Station First
Survey
Last
Survey
Surveyed
Period Surveys Surface SE Rockfalls Mean
Activity
(name) (date) (date) (years) (number) (hm2) hm2·year) (number) (hm−2·year−1)
River path 2008-01-18 2020-11-23 12.86 8 1.92 24.69 192 7.78
River parking 2011-05-17 2020-11-23 9.53 8 0.24 2.29 93 40.67
Kitchen garden 2008-01-18 2020-11-23 12.86 8 0.48 6.17 13 2.11
min max weighted sum sum sum sum weighted
Total 2008-01-18 2020-11-23 12.55 8 2.64 33.14 298 8.99
4. Data Analysis for McF Results
The data presented in the previous Section 3 are analyzed separately according to
their source: both TLS monitoring (Subsection 4.1) and observational inventory (Subsec-
tion 4.2).
4.1. Monitoring Data Analysis
Characterizing the subsamples with their own sample extent 𝑆𝐸 allows us to group
the data into a single sample that will have the aggregate extent 𝑆𝐸 = ∑𝑆𝐸 , as long as
the areas are not overlaid. Considering the sum of the wall surfaces covered by the differ-
ent TLS locations, we obtain the equivalent sampling period of this integrative sample,
Figure 5.
Map of the hill formed by basaltic cliffs, where Castellfollit de la Roca village is placed. The
main data sources for this work are highlighted: the relief and rockfall source areas (
slo pe ≥
46
◦
); the
rockfall observational inventory available; TLS stations in monitored sites.

Remote Sens. 2023,15, 1981 12 of 36
Table 3describes the main attributes of the sample obtained with TLS in Castellfollit
de la Roca, which totals
SE =
33.14
hm2·year
. In total, 298 rockfalls have been detected
since 2008 with sizes between 2
×
10
−4
m
3
and 6
×
10
0
m
3
, which cover a limited range
of four orders of magnitude because any large event in the inventory coincided with the
scope covered by monitoring.
Table 3.
Summary of TLS monitoring in the Castellfollit de la Roca cliffs and obtained rockfall
inventory are used in this paper.
Station First
Survey
Last
Survey
Surveyed
Period Surveys Surface SE Rockfalls Mean
Activity
(Name) (Date) (Date) (Years) (Number) (hm2) (hm2·Year) (Number) (hm−2·Year−1)
River path 2008-01-18 2020-11-23 12.86 8 1.92 24.69 192 7.78
River
parking 2011-05-17 2020-11-23 9.53 8 0.24 2.29 93 40.67
Kitchen
garden 2008-01-18 2020-11-23 12.86 8 0.48 6.17 13 2.11
min max weighted sum sum sum sum weighted
Total 2008-01-18 2020-11-23 12.55 8 2.64 33.14 298 8.99
4. Data Analysis for McF Results
The data presented in the previous Section 3are analyzed separately according to their
source: both TLS monitoring (Section 4.1) and observational inventory (Section 4.2).
4.1. Monitoring Data Analysis
Characterizing the subsamples with their own sample extent
SEi
allows us to group
the data into a single sample that will have the aggregate extent
SE =∑iSEi
, as long as the
areas are not overlaid. Considering the sum of the wall surfaces covered by the different
TLS locations, we obtain the equivalent sampling period of this integrative sample, which
is a mean value weighted by each surface coverage, as seen in Tables 2and 3. Implicitly
assuming that sectors are compatible, this fusion aims to be representative of a geographic
entity at an upper scale and for a certain period. At the outset, quite similar rockfall activity
can be observed in Castellfollit and Montserrat, with values between 6 and 9 rockfalls per
year and per hm
2
of the wall. However, it is necessary to see the distribution of volumes
to translate these values in terms of hazards. The spatial–temporal cumulative frequency
distribution,
Fst
, of the global set of TLS monitoring in Castellfollit de la Roca is shown
in Figure 6. A very good fit is observed for a potential law in the wide central range of
the sample, but some deviations at the extremes are the subject of analysis in the next
subsections.
The Montserrat and Castellfollit TLS series that are presented in this work have been
processed by the UB RiskNat team on behalf of the ICGC with a methodology and criteria
that have been developed throughout different research works [
61
–
65
]. The tools and
programming environments have been changing, and some criteria have been progressively
adjusted. This evolution could have introduced a certain degree of heterogeneity in the
series, as discussed in more detail in Section 5.2.2. In general, the core methodology, also
called standard by [
66
], is based on the following sequence of steps: (1) co-registration
and alignment of point clouds; (2) comparison and calculation of differences in terms of
distance between models; (3) clustering to identify single events; (4) discrimination of
changes to identify those corresponding to rockfalls; (5) rockfall event volume calculation;
and (6) extraction of the registration list (centroids and volumes).

Remote Sens. 2023,15, 1981 13 of 36
Remote Sens. 2023, 15, x FOR PEER REVIEW 14 of 42
which is a mean value weighted by each surface coverage, as seen in Tables 2 and 3. Im-
plicitly assuming that sectors are compatible, this fusion aims to be representative of a
geographic entity at an upper scale and for a certain period. At the outset, quite similar
rockfall activity can be observed in Castellfollit and Montserrat, with values between 6
and 9 rockfalls per year and per hm
2
of the wall. However, it is necessary to see the distri-
bution of volumes to translate these values in terms of hazards. The spatial–temporal cu-
mulative frequency distribution, 𝐹, of the global set of TLS monitoring in Castellfollit de
la Roca is shown in Figure 6. A very good fit is observed for a potential law in the wide
central range of the sample, but some deviations at the extremes are the subject of analysis
in the next subsections.
Figure 6. Power law fits, 𝐹

(𝑉), for the global sample of rockfall inventory detected by TLS in Cas-
tellfollit de la Roca cliff. The fit improves in the central part, from 0.002 to 3.3 m
3
, involving 246
rockfalls (83% of the total amount of 298).
The Montserrat and Castellfollit TLS series that are presented in this work have been
processed by the UB RiskNat team on behalf of the ICGC with a methodology and criteria
that have been developed throughout different research works [61–65]. The tools and pro-
gramming environments have been changing, and some criteria have been progressively
adjusted. This evolution could have introduced a certain degree of heterogeneity in the
series, as discussed in more detail in Subsection 5.2.2. In general, the core methodology,
also called standard by [66], is based on the following sequence of steps: (1) co-registration
and alignment of point clouds; (2) comparison and calculation of differences in terms of
distance between models; (3) clustering to identify single events; (4) discrimination of
changes to identify those corresponding to rockfalls; (5) rockfall event volume calculation;
and (6) extraction of the registration list (centroids and volumes).
The general trend has been to seek the greatest possible automation in processing,
both to reduce the number of dedicated hours by technicians and to ensure the homoge-
neity of results. In this sense, the last steps have been oriented toward the application of
machine learning techniques [23,67], as described in [68]. Even though the metrics in the
TLS point clouds have high resolution and accuracy, it should be noted that the calcula-
tion of the detached volumes is not univocal. Instead, the value you obtain depends on
the method of integration you employ [69–71]. For the application presented here, it is not
particularly relevant since we work on the volume distribution in orders of magnitude.
4.1.1. Rollover Effect
The roll-over effect at the lower limit of landslide inventories has been widely studied
for all types of landslides and also for rockfall. This relates to the completeness of
Figure 6.
Power law fits,
Fst(V)
, for the global sample of rockfall inventory detected by TLS in
Castellfollit de la Roca cliff. The fit improves in the central part, from 0.002 to 3.3 m
3
, involving
246 rockfalls (83% of the total amount of 298).
The general trend has been to seek the greatest possible automation in processing, both
to reduce the number of dedicated hours by technicians and to ensure the homogeneity of
results. In this sense, the last steps have been oriented toward the application of machine
learning techniques [
23
,
67
], as described in [
68
]. Even though the metrics in the TLS point
clouds have high resolution and accuracy, it should be noted that the calculation of the
detached volumes is not univocal. Instead, the value you obtain depends on the method of
integration you employ [
69
–
71
]. For the application presented here, it is not particularly
relevant since we work on the volume distribution in orders of magnitude.
4.1.1. Rollover Effect
The roll-over effect at the lower limit of landslide inventories has been widely studied
for all types of landslides and also for rockfall. This relates to the completeness of invento-
ries and is commonly excluded from analyzes that are focused on magnitudes capable of
causing significant damage [
72
–
75
]. In rockfalls detected with TLS monitoring, the lower
threshold is small enough to include all the cases of interest. In this sense, we will only make
a theoretical reflection on the conditioning factors in the roll-over effect in our application
cases. It is necessary to consider the possibility that, physically, the massif would have a
lower limit of volume susceptible to detachment conditioned by the failure mechanisms
or the degradation of the rock. In Montserrat, the rock mass is formed by a conglomerate,
according to the mechanism
mec
3 of detachment presented and discussed in Section 5.2.3,
which corresponds to the weathering of the rock matrix and the fall of isolated pebbles.
Detached volumes up to 10
−6
m
3
are physically possible and abundant around 10
−4
m
3
.
Beyond these actual lower limits, a roll-over effect can be caused by undersampling the
smaller volumes. First, the performance of the laser scanner equipment and its on-site
implementation determine an instrumental limitation to capturing small detachments. It
is influenced by the measurement precision, the visual distance, the visibility of the rock
surface, and ultimately the density of points obtained as a result of all the previous factors.
Likewise, post-processing adds some additional restrictions, according to the criteria for
defining clusters of points as detectable entities and the methods of discrimination and
validation of results. As a result, in our case sites, this threshold for roll-over effect can
be placed commonly in the range of 1
·
10
−3≤Vmin ≤
5
·
10
−3m3
. Finally, there may also
be limitations derived from time discontinuities in data capture. It could be the case that
changes detected throughout the period between surveys have actually been different falls,
but due to being adjacent, they are confused into a single major one. This restriction could

Remote Sens. 2023,15, 1981 14 of 36
hide processes of progressive rupture that are very important to risk management [
76
].
These considerations link to those made in Section 2.2 for surveying design.
To identify the threshold at which this roll-over effect is noticeable in the sample, an
empirical method is applied equivalent to the considerations made by other authors [
48
,
77
],
which are simpler than those presented by [
21
]. We use a progressive cut on the sample,
starting from the minimum volume and increasing the sense to find the point where the
potential fitting is stable. If the sample is sufficiently wide in volume range and sufficiently
densely populated, a stretch of stability for the fitting parameters is observed until reaching
the point where the sample is already reduced to the right branch and this stability is lost
again due to the variability of the upper extreme values. Figure 7shows the example of the
Castellfollit merged sample already presented, containing 298 rockfall events detected in
the volume range of 2
×
10
−4
to 6 m
3
. By applying this empirical method, we determine
an upper threshold of the roll-over at 2
×
10
−3
m
3
, in which the value’s stability of the
parameters is reached for
Ast =
0.32 and
B=
0.52, a maximum of the regression coefficient
R2=0.98 as well, and the removal of data is still limited (16%).
Remote Sens. 2023, 15, x FOR PEER REVIEW 16 of 42
Figure 7. Empirical method to determine the central part of the McF relationship and its parameters,
𝐴

and 𝐵. On the left column, filtering the roll-over effect with a progressive cut in the sample
starting from the minimum value of volume and in increasing sense to look for the optimal point
where a stable power law fitting is reached. On the right column, filtering the under-sampling effect
on the largest events with a progressive cut in the sample starting from the maximum value of vol-
ume and in decreasing sense to look for the optimal point where a stable power law fitting is
reached. Example with data from the global sample in the Castellfollit case.
4.1.2. Undersampling of Large Events
This section continues with the same example case of Castellfollit from the previous
subsection, once the lower limit for volume was applied. Considering its sample extent
𝑆𝐸 = 33.14hm
2
·year, the largest volume that would nominally correspond to having been
observed once in the mean (𝐹=1) is that of 𝐹 =𝐹 𝑆𝐸
⁄, which is 𝑉≥93 m
3
. Identically,
according to the available 𝑆𝐸 and the fitted McF, it would be expected to have observed
2 events (𝐹=2) of 𝑉≥25 m
3
, or 3 of 𝑉≥11 m
3
. However, the largest volume detected
by TLS is 6m
3
. The shape of the upper limit in the volume distribution in Figure 6 would
seem to indicate that there is a physical limit that makes events greater than 10 m
3
very
unlikely, following the considerations in [49]. Since 1976, however, a maximum detach-
ment of 1500 m
3
and up to 7 rockfalls of 𝑉≥50 m
3
have been observed, indicating that
this is not the case. This information is available in Supplementary Material SM3. They
are only included in the observational inventory because they were there before the start
of monitoring with TLS or in places outside of TLS coverage, as was the case with the
southern cliff next to the Toronell River in 2011.
On the contrary, we have detected 14 rockfalls in the 1–6 m
3
range until 2020, when
the fitted McF would expect to have only 6 events within this class. Therefore, we con-
clude that in our sample we have an undersampling of the largest volumes, in the sense
that major events have accumulated in similar values of volume instead of differentiating
more in gradation. Another effect could also be present, according to [77], due to the tem-
poral spacing of the monitoring surveys or the criteria in the delimitation of the detached
rock blocks, since these volumes actually represent multiple events interpreted as single
events. In any case, such a distortion in the extreme values leads to a mismatch in the
regression of the potential law. In this upper range of major events, the most significant
fact is their occurrence, not so much their specific volume. In this sense, we can apply a
Figure 7.
Empirical method to determine the central part of the McF relationship and its parameters,
Ast
and
B
. On the left column, filtering the roll-over effect with a progressive cut in the sample
starting from the minimum value of volume and in increasing sense to look for the optimal point
where a stable power law fitting is reached. On the right column, filtering the under-sampling effect
on the largest events with a progressive cut in the sample starting from the maximum value of volume
and in decreasing sense to look for the optimal point where a stable power law fitting is reached.
Example with data from the global sample in the Castellfollit case.
4.1.2. Undersampling of Large Events
This section continues with the same example case of Castellfollit from the previous
subsection, once the lower limit for volume was applied. Considering its sample extent
SE =
33.14 hm
2·
year, the largest volume that would nominally correspond to having been
observed once in the mean (
F=
1) is that of
Fst =F/SE
, which is
V≥
93 m
3
. Identically,
according to the available
SE
and the fitted McF, it would be expected to have observed
2 events (
F=
2) of
V≥
25 m
3
, or 3 of
V≥
11 m
3
. However, the largest volume detected
by TLS is 6m
3
. The shape of the upper limit in the volume distribution in Figure 6would
seem to indicate that there is a physical limit that makes events greater than 10 m
3
very
unlikely, following the considerations in [
49
]. Since 1976, however, a maximum detachment

Remote Sens. 2023,15, 1981 15 of 36
of 1500 m
3
and up to 7 rockfalls of
V≥
50 m
3
have been observed, indicating that this
is not the case. This information is available in Supplementary Material SM3. They are
only included in the observational inventory because they were there before the start of
monitoring with TLS or in places outside of TLS coverage, as was the case with the southern
cliff next to the Toronell River in 2011.
On the contrary, we have detected 14 rockfalls in the 1–6 m
3
range until 2020, when
the fitted McF would expect to have only 6 events within this class. Therefore, we conclude
that in our sample we have an undersampling of the largest volumes, in the sense that
major events have accumulated in similar values of volume instead of differentiating more
in gradation. Another effect could also be present, according to [
77
], due to the temporal
spacing of the monitoring surveys or the criteria in the delimitation of the detached rock
blocks, since these volumes actually represent multiple events interpreted as single events.
In any case, such a distortion in the extreme values leads to a mismatch in the regression
of the potential law. In this upper range of major events, the most significant fact is their
occurrence, not so much their specific volume. In this sense, we can apply a filter to
optimize the adjustment of the potential law. By comparing observed
F(Vi)
with theoretical
Fst (Vi)·SE
, we identify the major events with the largest divergence to be filtered manually.
However, it is not an elimination of their occurrence, as far as the following data maintains
the cumulative frequency.
Alternatively, we can apply the empirical method analogous to that of the preceding
subsection for roll-over. In this case, we are trimming the sample by progressively discard-
ing one record in a decreasing sense from the maximum volume. Using the Castellfollit
example once more, Figure 7demonstrates that the adjustment parameters attain stable
values rapidly, where we select the optimal point with a maximum regression coefficient
R2=
0.994 for
Ast =
0.365 and
B=
0.492. This filter to
V=
3.3 m
3
instead of 6.0 m
3
, which
was the largest detected rockfall, corresponds to collapsing the five largest registers into
the sixth one. This maintains the value F=6 as real cumulative frequency.
4.1.3. Oversampling of Large Events
The variability in extreme values argued in the previous subsection can also occur in
the opposite direction. A clear example of this is the Degotalls north sub-sample. On 2nd
January 2007, a 300 m
3
rockfall on this 180 m high wall caused severe material damage
to the monastery parking lot and the rack railway that runs parallel 115 m further down
the slope. The recovery work included intensive rock clearing and fence installation, as
well as the removal of parking spaces, while the next phases of protection were planned,
including a project to stabilize large, potentially unstable blocks in the wall. In support of
these tasks, the first laser scan of the wall was carried out in May 2007. Before starting this
second phase of work, the progression of the rupture produced a new detachment three
times larger on 28th December 2008, which forced the reconstruction of protective fences
and stabilization, including their extension to cover both walls (N and E). These events
have led the following TLS monitoring series to include a period of exceptional activity
from 2007 to 2009.
Figure 8shows the TLS records of activity on the Degotalls wall (north and east
merged) and their McF distribution for the period 2007–2009. This includes the large
detachment in 2008 and much of the clearing and stabilization work conducted on the
wall. Figure 8also shows the period 2009–2020, with
SE2007−09 =
7.98
hm2·year
and
SE2009−20 =33.48 hm2·year
, respectively. For the total sample extent
SE =
41.47
hm2·year
once the roll-over has been filtered and McF adjusted with
Ast =
0.354
hm−2·year−1
and
B=
0.544, the largest expected detachment would be of volume
V=
140
m3
, which corre-
sponds to
F=Fst ·SE =
1. However, due to the described circumstances, the monitoring
period has coincided with an event of
V1=
786
m3
140
m3
according to the original
cubication, which was subsequently revised upwards (see Section 5.2.2). However, in any
case, the event was much greater than expected. This is not the case for the second sample
value, which corresponds to
F=Fst (V2)·SE =1.6 ∼
2. Equivalently to the treatment in

Remote Sens. 2023,15, 1981 16 of 36
the preceding subsection, the first value is not considered, but its occurrence and successive
cumulative frequency values are. Once the effects of the two extremes, upper and lower,
have been filtered, an optimal adjustment of
Ast =
0.339
hm−2·year−1
and
B=
0.555 with
R2=0.991 results in a range of validity for almost 5 orders of magnitude in volume.
Remote Sens. 2023, 15, x FOR PEER REVIEW 17 of 38
Figure 8. Rockfall activity was detected by TLS in Degotalls's north and east walls in Montserrat.
Two periods are distinguished: 2007–2009 with large events, and works on the walls, and 2009–2020
with clearly lower activity. After roll-over and unexpected extreme values being filtered as ex-
plained in the text, the optimal McF is adjusted for the representative range covered by TLS moni-
toring with the current sample extent.
4.2. Observational Data Analysis
A basic property of the observational inventory is that it samples only if “seen” (a
human view of the phenomenon) or “noticed” (damages and effects). Therefore, the de-
tection capacity depends on the human presence and, by extension, their exposure. Con-
sequently, the first question that arises is about the completeness of the sample that we
can obtain, as already pointed out in Subsection 2.2. This leads to epistemic uncertainty,
as discussed in [78]. This section starts with the question: Have we recorded all the activity
(for all sizes) throughout all the time and covered all the space from which the inventory
is obtained? Additionally, since the answer is negative, we attempt to solve the inverse
question: of what space, of what period, and for what range of volumes is the obtained
sample in the inventory representative?
As seen in Subsection 4.1, the detection of detachment activity with TLS enables a
complete sample within its full extent 𝑆𝐸 to be obtained homogeneously for all volumes
within a central range. 𝑆𝐸 of the sample is an attribute derived from the design of the
monitoring surveys due to the start and end dates and their visual coverage. This fact
allows direct exploitation of the record and the calculation of McF with a unique and con-
stant 𝑆𝐸 for the whole sample. On the contrary, in observational inventories, 𝑆𝐸 is not
previously defined but is a deduction from the sample and data it has been possible to
collect in the past. In this case, we identify that the sample is heterogeneous since 𝑆𝐸(𝑉)
depends on the volume range. The strategy proposed in this case and presented in the
following subsections consists of segmenting homogeneous sub-samples so that they can
be calculated in terms of 𝐹 and then integrating them again into a wider sample. To
accomplish this, if sub-samples are discretized on the same basis as uniform categories of
volumes, the subsequent integration is facilitated. In this work, the half-decimal order of
magnitude is applied to get a uniform distribution on the logarithmic axis. McF undergoes
a slight variation between continuous and discrete samples, as represented in the subsec-
tions below.
4.2.1. Frequency Calculation on Inventories
Spatio–temporal frequency is simply calculated as 𝐹(𝑉)=𝑁(𝑉) 𝑆𝐸
⁄, 𝑁(𝑉) is the
number of events with a volume greater than or equal to 𝑉 and 𝑆𝐸 includes the period
𝑐𝑡 of temporal coverage of the sample and the surface of the source area 𝑐𝑎, according to
Figure 8.
Rockfall activity was detected by TLS in Degotalls’s north and east walls in Montserrat.
Two periods are distinguished: 2007–2009 with large events, and works on the walls, and 2009–2020
with clearly lower activity. After roll-over and unexpected extreme values being filtered as explained
in the text, the optimal McF is adjusted for the representative range covered by TLS monitoring with
the current sample extent.
4.2. Observational Data Analysis
A basic property of the observational inventory is that it samples only if “seen” (a
human view of the phenomenon) or “noticed” (damages and effects). Therefore, the
detection capacity depends on the human presence and, by extension, their exposure.
Consequently, the first question that arises is about the completeness of the sample that
we can obtain, as already pointed out in Section 2.2. This leads to epistemic uncertainty, as
discussed in [
78
]. This section starts with the question: Have we recorded all the activity
(for all sizes) throughout all the time and covered all the space from which the inventory
is obtained? Additionally, since the answer is negative, we attempt to solve the inverse
question: of what space, of what period, and for what range of volumes is the obtained
sample in the inventory representative?
As seen in Section 4.1, the detection of detachment activity with TLS enables a complete
sample within its full extent
SE
to be obtained homogeneously for all volumes within a
central range.
SE
of the sample is an attribute derived from the design of the monitoring
surveys due to the start and end dates and their visual coverage. This fact allows direct
exploitation of the record and the calculation of McF with a unique and constant
SE
for the
whole sample. On the contrary, in observational inventories,
SE
is not previously defined
but is a deduction from the sample and data it has been possible to collect in the past. In
this case, we identify that the sample is heterogeneous since
SE(Vi)
depends on the volume
range. The strategy proposed in this case and presented in the following subsections
consists of segmenting homogeneous sub-samples so that they can be calculated in terms
of
Fst
and then integrating them again into a wider sample. To accomplish this, if sub-
samples are discretized on the same basis as uniform categories of volumes, the subsequent
integration is facilitated. In this work, the half-decimal order of magnitude is applied to get
a uniform distribution on the logarithmic axis. McF undergoes a slight variation between
continuous and discrete samples, as represented in the subsections below.

Remote Sens. 2023,15, 1981 17 of 36
4.2.1. Frequency Calculation on Inventories
Spatio–temporal frequency is simply calculated as
Fst (V)=N(V)/SE
,
N(V)
is the
number of events with a volume greater than or equal to
V
and
SE
includes the period
ct
of temporal coverage of the sample and the surface of the source area
ca
, according to
Section 2.2. This has been conducted with the TLS monitoring data, in which the period
directly obtained by the duration of the campaigns
ct =tn−t0
is considered, with
t0
being the date of the first survey and
tn
being the date of the last available, which we
know precisely by day and even by the hour. If we bypass the spatial normalization
to focus on the temporal frequency, we have that
Ft(V)=N(V)/ct
. In contrast, for
observational inventories, there are two considerations to be made, as shown in Figure 9.
On the one hand, in historical data, the dates are often imprecise, and we treat them only
with the resolution of one year. In this rounding, we consider that the entire calendar
year in which an event is located is covered by the inventory record, given that it has
been covered by the effort to recover data from past activity. In this case, the period is
calculated directly as
ct =last year −f ir st ye ar +
1. In addition, as stated in the previous
paragraphs, in the records of observational inventory there is an indeterminacy in the
time period covered by the sample obtained, especially when the observation has been
indirect (e.g., historical documents, survey). If we consider that the period covered
ct
by the
sample is delimited by the first and last observations, we would incur an overestimation of
this frequency, since intuitively it must correspond to the inverse of their average spacing
ˆ
e=∑N−1
i=1ei,i+1/(N−1)=ct/(N−1)
, where
ei,i+1
is the spacing between each event and
the next one.
Remote Sens. 2023, 15, x FOR PEER REVIEW 19 of 42
Subsection 2.2. This has been conducted with the TLS monitoring data, in which the pe-
riod directly obtained by the duration of the campaigns 𝑐𝑡 = 𝑡−𝑡 is considered, with
𝑡 being the date of the first survey and 𝑡 being the date of the last available, which we
know precisely by day and even by the hour. If we bypass the spatial normalization to
focus on the temporal frequency, we have that 𝐹(𝑉)=𝑁(𝑉) 𝑐𝑡
⁄. In contrast, for observa-
tional inventories, there are two considerations to be made, as shown in Figure 9. On the
one hand, in historical data, the dates are often imprecise, and we treat them only with
the resolution of one year. In this rounding, we consider that the entire calendar year in
which an event is located is covered by the inventory record, given that it has been cov-
ered by the effort to recover data from past activity. In this case, the period is calculated
directly as 𝑐𝑡 = 𝑙𝑎𝑠𝑡 𝑦𝑒𝑎𝑟 − 𝑓𝑖𝑟𝑠𝑡 𝑦𝑒𝑎𝑟 + 1. In addition, as stated in the previous para-
graphs, in the records of observational inventory there is an indeterminacy in the time
period covered by the sample obtained, especially when the observation has been indirect
(e.g., historical documents, survey). If we consider that the period covered 𝑐𝑡 by the sam-
ple is delimited by the first and last observations, we would incur an overestimation of
this frequency, since intuitively it must correspond to the inverse of their average spacing
𝑒 = ∑𝑒,

 (𝑁−1
)=𝑐𝑡 (𝑁−1)
⁄⁄ , where 𝑒, is the spacing between each event and
the next one.
Figure 9. An illustrative example of the frequency calculation considering the covered period is as
follows: In the case of A, the period of TLS monitoring is derived from the surveys and is not influ-
enced by the occurrence of events, and its frequency can be directly calculated from 𝑐𝑡; in the case
of B, if we consider the period fitted by the first and last recorded event, the frequency is over-
estimated and the period must be extended to 𝑐𝑡
∗
; similarly, in the case of C, if we consider the
period between the first record in the past and present time, the period should be extended by half
due to an extreme fit by sampled data.
Consequently, it is appropriate to consider an extended period 𝑐𝑡∗ that corrects this
distortion and becomes truly representative of the events in the sample. This allows the
calculation equivalent to the monitoring data, 𝐹(𝑉)=𝑁(𝑉) 𝑐𝑡∗
⁄ and equivalently,
𝐹(𝑉)=𝑁(𝑉) 𝑆𝐸∗
⁄ in the cases shown in Figure 9. There, we represent an open interval
when both endpoints of the period are defined independently of the sample, as is the case
for TLS. On the other hand, a closed interval corresponds to a period fitted by the first and
last recorded event and then 𝑐𝑡∗=𝑐𝑡∙𝑁 (𝑁−1
)⁄ . Finally, a half-closed interval is used to
represent the most common case in historical inventories, in which the period is set be-
tween the first record in the past and present time and the period should be extended by
the half 𝑐𝑡∗=𝑐𝑡∙𝑁 (𝑁−0.5
)⁄ due to the initial extreme fitted by sampled data.
Since in observational inventories it is common to have few observations, this distor-
tion can be relevant, although it loses weight as the value of 𝑁 grows. This correction of
the extended period 𝑐𝑡∗ will gain influence in the analysis presented in the next subsec-
tion, where different periods are considered for each volume class. For example, in Cas-
tellfollit de la Roca, the historical inventory collected five recent events larger than 10 m
3
in the last 30 years (1995, 2007, 2011, 2015, and 2017) (Figure 10). The closed interval de-
fined by these events is 𝑐𝑡 = 2017 − 1995 + 1 = 23 𝑦𝑒𝑎𝑟𝑠 and the corresponding ex-
tended period 𝑐𝑡∗=𝑐𝑡∙𝑁 (𝑁−1
)⁄ = 28.8 𝑦𝑒𝑎𝑟𝑠. If we consider the half-closed interval
between the first event and the present time at the end of 2020 included in the analysis,
Figure 9.
An illustrative example of the frequency calculation considering the covered period is
as follows: In the case of A, the period of TLS monitoring is derived from the surveys and is not
influenced by the occurrence of events, and its frequency can be directly calculated from
ct
; in
the case of B, if we consider the period fitted by the first and last recorded event, the frequency is
over-estimated and the period must be extended to
ct∗
; similarly, in the case of C, if we consider the
period between the first record in the past and present time, the period should be extended by half
due to an extreme fit by sampled data.
Consequently, it is appropriate to consider an extended period
ct∗
that corrects this
distortion and becomes truly representative of the events in the sample. This allows
the calculation equivalent to the monitoring data,
Ft(V)=N(V)/ct∗
and equivalently,
Fst (V)=N(V)/SE∗
in the cases shown in Figure 9. There, we represent an open interval
when both endpoints of the period are defined independently of the sample, as is the case
for TLS. On the other hand, a closed interval corresponds to a period fitted by the first and
last recorded event and then
ct∗=ct·N/(N−1)
. Finally, a half-closed interval is used
to represent the most common case in historical inventories, in which the period is set
between the first record in the past and present time and the period should be extended by
the half ct∗=ct·N/(N−0.5)due to the initial extreme fitted by sampled data.
Since in observational inventories it is common to have few observations, this dis-
tortion can be relevant, although it loses weight as the value of
N
grows. This correction
of the extended period
ct∗
will gain influence in the analysis presented in the next sub-
section, where different periods are considered for each volume class. For example, in
Castellfollit de la Roca, the historical inventory collected five recent events larger than
10 m
3
in the last 30 years (1995, 2007, 2011, 2015, and 2017) (Figure 10). The closed

Remote Sens. 2023,15, 1981 18 of 36
interval defined by these events is
ct =
2017
−
1995
+
1
=
23
years
and the correspond-
ing extended period
ct∗=ct·N/(N−1)=
28.8
years
. If we consider the half-closed
interval between the first event and the present time at the end of 2020 included in
the analysis,
ct =
2020
−
1995
+
1
=
26
years
and the corresponding extended period
ct∗=ct·N/(N−0.5)=28.9 years.
Remote Sens. 2023, 15, x FOR PEER REVIEW 20 of 42
𝑐𝑡 = 2020 − 1995 + 1 = 26 𝑦𝑒𝑎𝑟𝑠 and the corresponding extended period 𝑐𝑡∗=𝑐𝑡∙
𝑁(𝑁−0.5
)⁄ = 28.9 𝑦𝑒𝑎𝑟𝑠.
In conclusion, the time–frequency calculation is subject to uncertainty [79], especially
in the range of the largest rockfalls, where the number of records is very low, reaching the
extreme case of a single observation, for which the idea of recurrence loses its meaning
and 𝑐𝑡∗ falls into indeterminacy. Accordingly, when considering 𝑐𝑡∗, the volume range
where a single event is recorded may not be considered sampled enough to be included
for the McF adjustment.
4.2.2. Size Distribution of Registered Events over Time
In the analysis of rockfall inventories, it is revealing to visualize the temporal distri-
bution of the records collected. For example, Figure 10 shows the 19 events of the historical
inventory in Castellfollit de la Roca collected in 2020 when searching for antecedents. On
this distribution, we identify the volume ranges in which we have a sufficiently homoge-
neous sampling: during the 16 years of the period 2005–2020, we have a representative
sample of rockfall with 𝑉≥1 m
; during 26 years from 1995 to 2020 we have a proper
sub-sample of 𝑉≥10 m
; in 45 years of the entire period 1976–2020, we can only charac-
terize the sub-sample of 𝑉 ≥ 100 m. In the bottom-left margin of the graphic, we identify
a data gap in the inventory. In contrast, TLS monitoring detected 15 rockfalls inside the
range of 1 m𝑉10 m
 detached from only half the cliff for 13 years (𝑆𝐸 = 33). It
must be supposed that from 1976 to 2005 there was some 𝑉  10 m rockfall activity, but
this information has not remained as a dated record in the memories of the interviewees
nor in the available documentation. It should be noted that the definition of these sample
limits is quite interpretive, with a margin of variability depending on the analyst.
Figure 10. Time distribution of the rockfall events recorded in the inventory for the Castellfollit de
la Roca cliff. It is clearly seen that there has been a lack of data for lower-magnitude events in the
past. Therefore, a variable covered time 𝑐𝑡 and sample extent 𝑆𝐸 must be considered for the three
volume ranges.
This treatment corresponds to considering a variable value of 𝑐𝑡 and 𝑆𝐸 also in 𝐹
calculation. In Figure 11, several McF adjustments can be observed and compared under
different hypotheses and formats. Firstly, if a uniform 𝑐𝑡 = 2020 − 1976 + 1 = 45 𝑦𝑒𝑎𝑟𝑠
was applied to the whole range of volume values, 𝐴 = 0.097 and 𝐵 = 0.335 would be
obtained. This means that for every 10 rockfalls of 𝑉≥1 m
, one of them has 𝑉≥
1000 m, because this low value of 𝐵 implies that every three orders of magnitude in
volume correspond only to one order of magnitude in frequency. Secondly, if a variable
value of 𝑐𝑡 is considere a function of volume range, according to the considerations of
lack of data in Figure 11, the slope of the regression line increases significantly to values
Figure 10.
Time distribution of the rockfall events recorded in the inventory for the Castellfollit de
la Roca cliff. It is clearly seen that there has been a lack of data for lower-magnitude events in the
past. Therefore, a variable covered time
ct
and sample extent
SE
must be considered for the three
volume ranges.
In conclusion, the time–frequency calculation is subject to uncertainty [
79
], especially
in the range of the largest rockfalls, where the number of records is very low, reaching the
extreme case of a single observation, for which the idea of recurrence loses its meaning and
ct∗
falls into indeterminacy. Accordingly, when considering
ct∗
, the volume range where
a single event is recorded may not be considered sampled enough to be included for the
McF adjustment.
4.2.2. Size Distribution of Registered Events over Time
In the analysis of rockfall inventories, it is revealing to visualize the temporal distribu-
tion of the records collected. For example, Figure 10 shows the 19 events of the historical
inventory in Castellfollit de la Roca collected in 2020 when searching for antecedents. On
this distribution, we identify the volume ranges in which we have a sufficiently homoge-
neous sampling: during the 16 years of the period 2005–2020, we have a representative
sample of rockfall with
V≥
1
m3
; during 26 years from 1995 to 2020 we have a proper
sub-sample of
V≥
10
m3
; in 45 years of the entire period 1976–2020, we can only character-
ize the sub-sample of
V≥
100
m3
. In the bottom-left margin of the graphic, we identify
a data gap in the inventory. In contrast, TLS monitoring detected 15 rockfalls inside the
range of 1
m3<V<
10
m3
detached from only half the cliff for 13 years (
SE =
33). It must
be supposed that from 1976 to 2005 there was some
V<
10
m3
rockfall activity, but this
information has not remained as a dated record in the memories of the interviewees nor in
the available documentation. It should be noted that the definition of these sample limits is
quite interpretive, with a margin of variability depending on the analyst.
This treatment corresponds to considering a variable value of
ct
and
SE
also in
Fst
calculation. In Figure 11, several McF adjustments can be observed and compared under
different hypotheses and formats. Firstly, if a uniform
ct =
2020
−
1976
+
1
=
45
years
was
applied to the whole range of volume values,
Ast =
0.097 and
B=
0.335 would be obtained.
This means that for every 10 rockfalls of
V≥
1
m3
, one of them has
V≥
1000
m3
, because
this low value of
B
implies that every three orders of magnitude in volume correspond
only to one order of magnitude in frequency. Secondly, if a variable value of
ct
is considere

Remote Sens. 2023,15, 1981 19 of 36
a function of volume range, according to the considerations of lack of data in Figure 11,
the slope of the regression line increases significantly to values B∼0.5. Additionally, Ast
arises because
Fst
is maintained for the largest rockfalls and is increased inversely to the
ct
value for successively smaller rockfalls. In this case, a slight difference is obtained between
both considerations of covered time, either direct
ct
or extended
ct∗
. The slope increases a
little bit, being the fixed point in the lowest volumes where the
ct∗≈ct
, as
N
1. Finally,
the result is shown for the case where a discrete scheme is used for the volume. In this case,
groups of data collapse at the beginning of their volume class, for instance: four rockfalls
with a volume of 30
m3≤V<
100
m3
compute for the class
V≥
30
m3
, at a lower value
of volume. With a negative slope, this implies a slight lowering of
Fst
values for infrequent
events and an additional increase in the slope B.
Remote Sens. 2023, 15, x FOR PEER REVIEW 21 of 42
𝐵~0.5. Additionally, 𝐴 arises because 𝐹 is maintained for the largest rockfalls and is
increased inversely to the 𝑐𝑡 value for successively smaller rockfalls. In this case, a slight
difference is obtained between both considerations of covered time, either direct 𝑐𝑡 or
extended 𝑐𝑡∗. The slope increases a little bit, being the fixed point in the lowest volumes
where the 𝑐𝑡∗≈𝑐𝑡, as 𝑁≫1. Finally, the result is shown for the case where a discrete
scheme is used for the volume. In this case, groups of data collapse at the beginning of
their volume class, for instance: four rockfalls with a volume of 30 m≤ 𝑉  100 m
compute for the class 𝑉≥30 m
, at a lower value of volume. With a negative slope, this
implies a slight lowering of 𝐹 values for infrequent events and an additional increase in
the slope 𝐵.
Figure 11. McF relationship obtained from historical inventory in Castellfollit de la Roca Cliff. Four
formats of adjustments are presented: on the one hand, continuous and discrete data series; on the
other hand, considering the direct covered time 𝑐𝑡 or the extended one 𝑐𝑡
∗
, both variable according
to volume ranges as explained in the text. They are compared to the original McF adjustment if a
uniform period 𝑐𝑡 was considered.
4.2.3. Size Distribution of Registered Events over Space
Finally, we address one last issue that affects inventory samples. It is relevant to re-
member the big difference between monitoring with TLS, which makes an exhaustive de-
tection in the starting zone by a repetitive, systematic process, and inventory. This leads
to observations in the arrival zone, often only in a reactive way to events with effects,
which are conditioned by the propagation of falling rocks down the slope. The smaller the
magnitude, the more conditioned the observation is on the fact of producing damage or
effects to the elements at risk. In most inventories, the elements at risk act as data collec-
tors, and their exposure and vulnerability somehow determine their detection capacity.
Therefore, the position of the elements at risk concerning the extent of the rockfall trajec-
tories is very influential in the sampling. When the inventory is built based on photoin-
terpretation, despite not depending on the presence of human elements, it similarly hap-
pens that the detection capacity is conditioned by the rockfall propagation and the imprint
generated on the environment, for example on the forest. This approach is similar to the
idea of an “effective surveyed area,” which was introduced by [73] in inventories for land-
slide susceptibility analyses.
In the previous point, we asked ourselves, “What time span have we effectively con-
trolled with the tasks furnishing the inventory?” Here we ask similarly: what space have
we effectively controlled? Furthermore, equivalently to the answer to the first question
regarding time, we find again a dependence of covered space on magnitude. Focusing on
the case of Montserrat, we can see very clear examples of this idea. For instance, the built
enclosure of the Monastery is surrounded by vertical walls, at the foot of which there are
buildings and occupied spaces. Any event, even one of a small magnitude, can already
Figure 11.
McF relationship obtained from historical inventory in Castellfollit de la Roca Cliff. Four
formats of adjustments are presented: on the one hand, continuous and discrete data series; on the
other hand, considering the direct covered time
ct
or the extended one
ct∗
, both variable according
to volume ranges as explained in the text. They are compared to the original McF adjustment if a
uniform period ct was considered.
4.2.3. Size Distribution of Registered Events over Space
Finally, we address one last issue that affects inventory samples. It is relevant to
remember the big difference between monitoring with TLS, which makes an exhaustive
detection in the starting zone by a repetitive, systematic process, and inventory. This
leads to observations in the arrival zone, often only in a reactive way to events with
effects, which are conditioned by the propagation of falling rocks down the slope. The
smaller the magnitude, the more conditioned the observation is on the fact of producing
damage or effects to the elements at risk. In most inventories, the elements at risk act as
data collectors, and their exposure and vulnerability somehow determine their detection
capacity. Therefore, the position of the elements at risk concerning the extent of the rockfall
trajectories is very influential in the sampling. When the inventory is built based on
photointerpretation, despite not depending on the presence of human elements, it similarly
happens that the detection capacity is conditioned by the rockfall propagation and the
imprint generated on the environment, for example on the forest. This approach is similar
to the idea of an “effective surveyed area”, which was introduced by [
73
] in inventories for
landslide susceptibility analyses.
In the previous point, we asked ourselves, “What time span have we effectively
controlled with the tasks furnishing the inventory?” Here we ask similarly: what space
have we effectively controlled? Furthermore, equivalently to the answer to the first question
regarding time, we find again a dependence of covered space on magnitude. Focusing on
the case of Montserrat, we can see very clear examples of this idea. For instance, the built
enclosure of the Monastery is surrounded by vertical walls, at the foot of which there are
buildings and occupied spaces. Any event, even one of a small magnitude, can already

Remote Sens. 2023,15, 1981 20 of 36
have an impact on human activity and be perceived and recorded in the inventory. On the
other hand, continuing up the slope, there are other more remote outcrops. If rocks fall
from these, even if they are bigger than the previous ones, they are retained in the forest
or the gullies and can go completely unnoticed. This means that the area covered by the
sample, which contributes to the
SE
count, is variable according to the volume range. We
can apply this distinction based on the division of the Montserrat massif into domains
introduced in Section 3.1. In Table 4and Figure 12, it can be seen that the application of this
idea to the Monastery sample can be quantified, although based on expert judgment. Thus,
for example, a rockfall of 1 m
3
is only recorded if it occurs at lower levels, while an event of
100 m3will be recorded anyway due to its visibility and repercussions.
Table 4.
Considered variability of rockfall volume range sufficiently covered by observational
inventory for each domain of source area in Monastery of Montserrat. For each domain and volume
range: Y if considered, N if not. The spatial distribution of the domains can be seen in Figure 12.
Volume Domains 205 206 207 208 209 210 211 220 221 222 223 224
V (m3)Source area
(hm2)1.35 0.41 2.02 2.26 0.23 0.26 2.88 1.67 0.90 0.90 0.45 2.35
0.001 2.5 N N N Y Y N N N N N N N
0.003 3.8 Y N N Y Y N N N N N N N
0.01 4.1 Y N N Y Y Y N N N N N N
0.03 6.1 Y N Y Y Y Y N N N N N N
0.1 6.5 Y Y Y Y Y Y N N N N N N
0.3 9.4 Y Y Y Y Y Y Y N N N N N
1 9.4 Y Y Y Y Y Y Y N N N N N
3 13.3 Y Y Y Y Y Y Y Y Y Y Y N
10 15.7 Y Y Y Y Y Y Y Y Y Y Y Y
30 15.7 Y Y Y Y Y Y Y Y Y Y Y Y
100 15.7 Y Y Y Y Y Y Y Y Y Y Y Y
300 15.7 Y Y Y Y Y Y Y Y Y Y Y Y
1000 15.7 Y Y Y Y Y Y Y Y Y Y Y Y
3000 13.9 Y Y Y Y Y Y Y Y N N Y Y
10,000 11.5 Y Y Y Y N Y Y N N N N Y
This consideration generally implies an increasing value of
SE
with magnitude. How-
ever, at the highest extreme of magnitude, another consideration is also possible: can a
rockfall of enormous magnitude occur at any outcrop of the rocky massif? As several
authors have proposed [
7
,
14
,
49
,
80
], each rocky outcrop has a physical limit to the magni-
tude of events that can occur. This depends on the structure of the massif (mainly joint
persistence) and the intersection with the surface of the escarpment, as well as its extension
and especially its height. In this sense, we can identify some domains where the conditions
are not met to generate failures greater than 3 ×103m3or 1 ×104m3(see Table 4). In the
available inventory, we do not have any cases of this magnitude, so this effect does not
influence the present analysis.

Remote Sens. 2023,15, 1981 21 of 36
1
Figure 12.
Detailed map from Figure 4for the Monastery area in Montserrat. The relief is highlighted
by applying a hillshade to the slope map. Two kinds of rockfall source areas are distinguished:
inclined walls (
slo pe ≥
46
◦
) and vertical walls (
slo pe ≥
70
◦
). Rockfall domains with influence on the
Monastery built-up area are highlighted and identified with the code used in Table 4, and rockfalls
from the observational inventory are included. The TLS station of Fra Garíis placed at the opposite
slope and covers 75% of the rockfall source area of domains 207 to 210, where the scanner is focused.
If we proceed with the case already presented by the Monastery of Montserrat, once
this correction is applied to the McF adjustment, its influence on the parameters is evident.
Here, the inventory collects 20 rockfalls with a total extension of 15.7 hm
2
of rock walls,
two of which are ancient and considered outside the continuity of registration. Both are
well dated: one in 1546 has been obtained by documentary source, and the other by survey,
and thanks to registration in a memorial, we know that it occurred on 19 May 1927. Since
we lack information on the rockfalls themselves, we only know how they influence the
surrounding landscape. This makes it difficult to estimate the volume, which is assumed to
be equal to or greater than 300 m
3
and clearly greater than 10 m
3
respectively. Thus, we are
considered to have only a sample suitable for McF analysis for lower volumes (Figure 13).
If a uniform
SE
is applied to the entire sample resulting from 18 rockfalls in 21 years,
the slope of the McF curve is very low (
B=
0.20), which is equivalent to saying that the
probability of occurrence is quite similar for any size. Considering the correction described
in the previous section, which makes
SE
variable over time and leads us to discard another
set of data outside the confidence margin in the sampling, parameter B already takes a
value greater than double (
B=
0.44). Moreover, if we consider the spatial variability of
SE
that is presented in this section, according to the values in Table 4, the slope (
B=
0.64)
is already assimilated to other values in Montserrat for other sectors, although it is still

Remote Sens. 2023,15, 1981 22 of 36
significantly lower than what TLS monitoring indicates (
B=
0.88) for the local Monastery
station placed at Fra Garíviewpoint.
Remote Sens. 2023, 15, x FOR PEER REVIEW 24 of 42
Figure 13. Process to obtain the McF relationship for the Monastery sector in Montserrat based on
inventory. On the left side: time distribution of recorded rockfalls (left) according to their volume,
with date excluded from the analysis. On the right side: a progressive improvement of the analysis
from considering a uniform 𝑆𝐸 for all volume ranges within 21 years, considering its time depend-
ence in the graph on the left side, and adding the space variability of 𝑆𝐸 explained in the text.
Results are compared to those obtained by TLS monitoring.
Given the results, we conclude that the inventory achieves sample completeness for
𝑉≥1 m
, but smaller rockfalls, it is sure to have missing records due to a lack of system-
aticity in data capture. This makes the adjustment fall short compared to the TLS moni-
toring for the range of 1×10
−3
to 1×10
−1
m
3
, which covers a fairly similar time span. It should
be noted that there are five major rockfalls in the inventory dated within the monitored
period by TLS. However, they have not occurred in the space covered by the TLS surveys
(3.57 hm
2
), which is 38% of the 9.4 hm
2
that we consider covered by inventory at this range
of volumes. Consequently, we see a high complementarity of inventory with monitoring,
as we will detail in subsection 5.1. In this case, it allows us to have confidence in the ex-
trapolation of the TLS McF adjustment up to around 𝑉 = 10 m. Therefore, it is assumed
that 𝑆𝐸 of TLS monitoring is large enough in time and space to reach a representativeness
similar to the inventory, thanks to its completeness. The treatment discussed in this sec-
tion has a significant impact on the McF adjustment for the rack railway industry. This is
due to the fact that the position of the rack railway relative to the cliffs and rockfall runout
varies along its route. More details on this sample can be found in Supplementary Material
SM5.
In this work, the corrections have been applied to covered time and space depending
on the volume in a coupled way, that is, for a certain magnitude range, a limitation of the
time covered by the inventory is applied everywhere and a space restriction in sampling
applies at all time spans. A possible extension and generalization of the method would
consist of decoupling both variables and being able to determine the spatial–temporal
coverage of the inventory independently. For each range of volumes in each domain, con-
sider the period of time covered by the inventory. Even the temporal coverage could not
be unique until the present but could be discontinuous when the available inventory is
very heterogeneous as a result of work in disjointed stages.
5. McF Discussion
Both case sites (Montserrat Massif and Castellfollit de la Roca) and both data sources
(TLS monitoring and observational inventory) were analyzed using the methods de-
scribed in Section 4. Results for Montserrat are collected in Tables 5 and 6, and for Cas-
tellfollit in Table 7. In the present section, results are discussed, focusing on different
points of interest in each subsection.
Table 5. McF parameters were obtained for the Montserrat Massif case site at different scales based
on TLS monitoring.
Figure 13.
Process to obtain the McF relationship for the Monastery sector in Montserrat based on
inventory. On the left side: time distribution of recorded rockfalls (
left
) according to their volume,
with date excluded from the analysis. On the right side: a progressive improvement of the analysis
from considering a uniform
SE
for all volume ranges within 21 years, considering its time dependence
in the graph on the left side, and adding the space variability of
SE
explained in the text. Results are
compared to those obtained by TLS monitoring.
Given the results, we conclude that the inventory achieves sample completeness
for
V≥
1
m3
, but smaller rockfalls, it is sure to have missing records due to a lack of
systematicity in data capture. This makes the adjustment fall short compared to the TLS
monitoring for the range of 1
×
10
−3
to 1
×
10
−1
m
3
, which covers a fairly similar time
span. It should be noted that there are five major rockfalls in the inventory dated within
the monitored period by TLS. However, they have not occurred in the space covered by the
TLS surveys (3.57 hm
2
), which is 38% of the 9.4 hm
2
that we consider covered by inventory
at this range of volumes. Consequently, we see a high complementarity of inventory with
monitoring, as we will detail in Section 5.1. In this case, it allows us to have confidence in the
extrapolation of the TLS McF adjustment up to around
V=
10
m3
. Therefore, it is assumed
that
SE
of TLS monitoring is large enough in time and space to reach a representativeness
similar to the inventory, thanks to its completeness. The treatment discussed in this section
has a significant impact on the McF adjustment for the rack railway industry. This is due to
the fact that the position of the rack railway relative to the cliffs and rockfall runout varies
along its route. More details on this sample can be found in Supplementary Material SM5.
In this work, the corrections have been applied to covered time and space depending
on the volume in a coupled way, that is, for a certain magnitude range, a limitation of the
time covered by the inventory is applied everywhere and a space restriction in sampling
applies at all time spans. A possible extension and generalization of the method would
consist of decoupling both variables and being able to determine the spatial–temporal
coverage of the inventory independently. For each range of volumes in each domain,
consider the period of time covered by the inventory. Even the temporal coverage could
not be unique until the present but could be discontinuous when the available inventory is
very heterogeneous as a result of work in disjointed stages.
5. McF Discussion
Both case sites (Montserrat Massif and Castellfollit de la Roca) and both data sources
(TLS monitoring and observational inventory) were analyzed using the methods described
in Section 4. Results for Montserrat are collected in Tables 5and 6, and for Castellfollit in
Table 7. In the present section, results are discussed, focusing on different points of interest
in each subsection.

Remote Sens. 2023,15, 1981 23 of 36
Table 5.
McF parameters were obtained for the Montserrat Massif case site at different scales based
on TLS monitoring.
Sample Vmin Vmax Ast BR2SE·Ast Vcd0
(Name) (m3) (m3)(hm−2·Year−1) (–) (–) (–) (m3)(–)
TLS Station
Degotalls N 3.0 ×10−35.0 ×1010.593 0.478 0.993 13.90 0.426 0.547
Degotalls E 3.0 ×10−32.5 ×10−10.061 0.846 0.982 1.11 0.201 0.210
Monastery 5.2 ×10−34.8 ×10−10.041 0.879 0.983 1.44 0.171 0.177
Guilleumes 2.0 ×10−32.2 ×10−10.189 0.461 0.962 0.68 0.188 0.245
Sant Benet 1.6 ×10−34.8 ×10−10.132 0.454 0.965 0.57 0.144 0.189
Collbatóroad 4.0 ×10−41.7 ×10−10.091 0.483 0.977 0.51 0.121 0.155
Can Jorba 1.5 ×10−31.0 ×10−10.057 0.570 0.952 0.32 0.113 0.135
Region
Monastery 5.2 ×10−34.8 ×10−10.041 0.879 0.983 1.44 0.171 0.177
Parking 3.0 ×10−35.0 ×1010.339 0.555 0.991 14.04 0.341 0.413
Railway 1.6 ×10−34.8 ×10−10.113 0.510 0.982 0.90 0.151 0.189
Collbató1.0 ×10−31.1 ×10−10.070 0.518 0.978 0.78 0.112 0.140
Massif
Montserrat 3.0 ×10−35.0 ×10+1 0.163 0.621 0.991 15.57 0.243 0.282
Table 6.
Results of the observational inventory in Montserrat. McF obtained for the five considered
regions and the integration into a global McF at massif scale. The analysis adopted is distinguished
between the upper part of optimal fitting for each region and the lower part for the calculation of the
global McF merging the partial samples of the inventory.
Analysis Region Monastery Parking Rack Railway CollbatóCaves Northern Road
Continuous analysis
with extended period
ct*and
Vmin ≥0.03 m3
Ast 0.023 0.119 0.029 0.139 0.060
B0.635 0.579 0.781 0.593 0.796
R20.966 0.960 0.990 0.803 0.948
Discrete analysis with direct period ct to merge all partial samples of the inventory into a global one
Volume
limit (V≥)∑N∑SE N SE N SE N SE N SE N SE
m3#
hm
2·
year
#
hm
2·
year
#
hm
2·
year
#
hm
2·
year
#
hm
2·
year
#
hm
2·
year
0.01 71 264 8 41 18 55 6 61 31 44 8 64
0.03 61 289 7 61 15 55 6 61 26 44 7 68
0.1 48 391 5 65 13 55 3 118 19 44 8 110
0.3 37 459 5 94 7 55 4 152 14 48 7 111
1 40 1080 7 197 9 118 10 518 10 58 4 188
3 23 1420 4 280 6 182 5 570 6 153 2 235
10 14 2921 0 329 6 348 3 907 0 213 5 1125
30 7 3413 0 329 4 348 2 907 0 213 1 1616
100 8 4995 0 329 3 348 1 900 0 196 4 3222
300 5 4978 0 329 2 348 1 897 0 196 2 3208
1000 2 4978 0 329 0 348 1 897 0 196 1 3208
Ast 0.026 0.029 0.061 0.014 0.118 0.015
Global B0.603 0.422 0.436 0.446 0.461 0.580
R20.983 0.971 0.978 0.958 0.861 0.950
5.1. Spatial Variability in McF
We have obtained the McF relationship that can be considered best adjusted to each
available sample at different scales: outcrop covered by a TLS station, sector by a grouping
of nearby TLS stations, or by areas covered by homogeneous inventory, and the massif as
a whole.

Remote Sens. 2023,15, 1981 24 of 36
Table 7.
McF parameters were obtained for the case site of Castellfollit de la Roca Cliff based on
TLS monitoring.
Sample Minimum
Volume
Maximum
Volume Ast BR2SE·Ast Vcd0
Name m3m3hm−2·Year−1– – – m3–
TLS-Station
River path 2.1 ×10−33.7 ×1000.403 0.447 0.981 9.94 0.306 0.404
River parking 2.0 ×10−33.8 ×1000.826 0.591 0.994 1.89 0.637 0.752
Kitchen garden 7.1 ×10−31.1 ×1000.382 0.363 0.911 2.36 0.235 0.340
Level
Upper lava flow 2.1 ×10−33.7 ×1000.373 0.443 0.990 11.51 0.287 0.381
Lower lava flow 2.0 ×10−33.8 ×1000.826 0.591 0.994 1.89 0.637 0.752
Cliff
Castellfollit 2.0 ×10−33.7 ×1000.389 0.479 0.994 12.90 0.321 0.412
5.1.1. Multi-Scale Comparison
In the results from TLS in Montserrat (Table 5), we observe a large variation in McF
laws comparing all the stations:
Ast
ranges from 0.04 to 0.6, and
B
ranges from 0.45 to 0.88.
When we group the samples into regions, disparities are reduced, and each behavior can be
attributed to different conditions over the mountain: upper/lower levels and north/south
flanks. These regions are: Monastery (single station) is an upper-south slope; Parking
(merging Degotalls north and east) is an upper-north slope; Railway (merging Guilleumes
and Sant Benet) is a lower-north slope; and Collbató(merging Collbatóroad and Can Jorba)
is a lower-south slope. These regions are equivalent to those obtained by the inventory
analysis (Table 6). There are significant differences between McF provided by TLS and
inventory (Figure 14), as will be discussed below. When regions are combined into a single
McF for the massif as a whole, both sets of results are the same. Then, we obtain the
coincident value for
B=
0.62 and the value for
Ast =
0.16 from TLS is more confident than
the derived from inventory, which is seven times lower due to the incompleteness of the
sample that is not achieved to offset. In any case, Montserrat is a rocky mountain with
many conglomerate walls of varying slope, aspect, and height, so any McF result for the
massif can only be intended as an approach to the average trend, for which it should be
expected to have large spatial dispersion.
Remote Sens. 2023, 15, x FOR PEER REVIEW 26 of 42
TLS-Station
River path 2.1×10
−3
3.7×10
0
0.403 0.447 0.981 9.94 0.306 0.404
River parking 2.0×10
−3
3.8×10
0
0.826 0.591 0.994 1.89 0.637 0.752
Kitchen garden 7.1×10
−3
1.1×10
0
0.382 0.363 0.911 2.36 0.235 0.340
Level
Upper lava flow 2.1×10
−3
3.7×10
0
0.373 0.443 0.990 11.51 0.287 0.381
Lower lava flow 2.0×10
−3
3.8×10
0
0.826 0.591 0.994 1.89 0.637 0.752
Cliff
Castellfollit 2.0×10
−3
3.7×10
0
0.389 0.479 0.994 12.90 0.321 0.412
5.1. Spatial Variability in McF
We have obtained the McF relationship that can be considered best adjusted to each
available sample at different scales: outcrop covered by a TLS station, sector by a grouping
of nearby TLS stations, or by areas covered by homogeneous inventory, and the massif as
a whole.
5.1.1. Multi-Scale Comparison
In the results from TLS in Montserrat (Table 5), we observe a large variation in McF
laws comparing all the stations: 𝐴 ranges from 0.04 to 0.6, and 𝐵 ranges from 0.45 to
0.88. When we group the samples into regions, disparities are reduced, and each behavior
can be attributed to different conditions over the mountain: upper/lower levels and
north/south flanks. These regions are: Monastery (single station) is an upper-south slope;
Parking (merging Degotalls north and east) is an upper-north slope; Railway (merging
Guilleumes and Sant Benet) is a lower-north slope; and Collbató (merging Collbató road
and Can Jorba) is a lower-south slope. These regions are equivalent to those obtained by
the inventory analysis (Table 6). There are significant differences between McF provided
by TLS and inventory (Figure 14), as will be discussed below. When regions are combined
into a single McF for the massif as a whole, both sets of results are the same. Then, we
obtain the coincident value for 𝐵=0.62 and the value for 𝐴 =0.16 from TLS is more
confident than the derived from inventory, which is seven times lower due to the incom-
pleteness of the sample that is not achieved to offset. In any case, Montserrat is a rocky
mountain with many conglomerate walls of varying slope, aspect, and height, so any McF
result for the massif can only be intended as an approach to the average trend, for which
it should be expected to have large spatial dispersion.
Figure 14. Parameters 𝐴

and 𝐵 of the McF relationship were obtained from TLS monitoring and
observational inventory in the Montserrat massif and Castellfollit de la Roca cliff at different scales,
highlighting global laws. For Montserrat, results by region are shown and paired for both source
Figure 14.
Parameters
Ast
and
B
of the McF relationship were obtained from TLS monitoring and
observational inventory in the Montserrat massif and Castellfollit de la Roca cliff at different scales,
highlighting global laws. For Montserrat, results by region are shown and paired for both source types.
In Castellfollit, two levels of the cliff are distinguished thanks to TLS data. On the right side graph,
McF parameters of both sites are compared to previous results at other sites in France and Yosemite,
USA [7,81], where results coming from remote sensing (RS) and inventory (IN) are distinguished.

Remote Sens. 2023,15, 1981 25 of 36
The results of the inventory analysis are presented in Table 6. McF parameters were
obtained for each sub-sample of an inventory region according to the methodologies
presented in Section 4.2. Optimal fitting was found through the analysis that applies the
extended period
ct∗
to the continuous sample and consider the variable value of
SE(V)
with magnitude according to the time and domain of each registered event. In contrast to
homogeneous samples of TLS with constant value
SE
, where the merge can be conducted
directly by mixing all data and adding
SE
, in inventory samples, it is proposed to discretize
the samples on magnitude classes to be able to apply the equivalent procedure. In that
case, direct period
ct
is used to calculate frequencies because we expect the event count to
increase in the merged sample. It is worth noting that regions where any major rockfall was
observed also contribute to adding
SE
for the largest events. In other words, the absence of
recorded events of large magnitude in a sample of inventory is also data to be considered
for a proper calculation of their frequency when merged with other regions where this kind
of event was observed. This point agrees with considerations made for TLS monitoring in
Degotall’s north wall that include an exceptionally large event (see Section 4.1). To improve
the frequency calculation of the largest events, a sub-sample without correspondence to the
TLS monitoring station (northern road in Table 6) has been added to the sample. This way
we collected two rockfalls with V≥1000 m3detected over SE =4978 hm2·year.
In contrast, for Castellfollit de la Roca (Table 7), differences between TLS and inventory
are reduced (see Figure 14). In this site, we consider a single region because the extension
of the basaltic cliff is reduced, but two analyses of spatial variability were attempted. It
could be distinguished between north and south flanks corresponding to each fluvial valley
using inventory data, although samples are very poor (9 and 10 events each). McF differs
from the north flank (
Ast ≈
0.15 and
B≈
0.33) to the south flank (
Ast ≈
0.4 and
B≈
0.6),
which agrees with the observation that larger rockfalls seem more frequent at the higher
north cliff, while rockfalls of
V≈
50
m3
would have a similar frequency in both flanks.
TLS stations cover different parts of the cliff: the station in river parking is focused on the
lower lava flow forming the cliff (as explained in Section 3.2), and the other two stations
are focused on the upper lava flow. These results (shown in Table 7and Figure 14) indicate
that the lower level has the higher activity of lower magnitude than the upper level, which
agrees with the observation of upward erosion that configures the overhang profile of the
cliff and leads to larger failures in the central-upper part. However, the sample attributed
to the lower level has the smallest
SE
value, it watches to the end extreme of the cliff, and
it covers a scar from the large event that occurred in 2011, so it may reflect other factors
modulating rockfall activity. In Supplementary Material SM4, it is highlighted that maps
or location figures of the inventoried activity are not necessarily indicative of the hazard
distribution within the wall but must be interpreted according to the cycles of activity that
may manifest.
Similarly to both lava flow levels at the cliff in Castellfollit, at Degotalls Wall in
Montserrat, slight differences in rockfall activity are seen according to stratigraphic levels
in the same outcrop. The rock mass shows smaller values of joint spacing in the lower half
of the wall compared to the upper half, and correspondingly higher activity is detected in
the lower part. However, these results are affected by the artefacts analyzed in Section 5.2.2,
preventing them from being conclusive. Nevertheless, this variability has been observed at
other sites [50] and is fully consistent with the model and methods.
5.1.2. Multi-Source Comparison
As seen in Figure 14, McF results derived from TLS monitoring and observational
inventory differ. In Castellfollit de la Roca, this difference is small, and both McF can be
attempted to be merged into a fusion sample (see Figure 15) to get validity on the maximum
range of magnitudes and be confident on the hazard scenarios at any point of the McF
curve It has already been demonstrated [
81
] that there is a similar degree of coherence
in Yosemite (USA) between rockfall inventory and remote sensing detection (combined
TLS and SfM applied to aerial and terrestrial images), thanks to the high-quality inventory

Remote Sens. 2023,15, 1981 26 of 36
available [
82
]. In Montserrat,
Ast
obtained by inventory is six times lower than the value
from TLS. We conclude that the inventory lacks quality and is insufficiently systematic to
overcome the high variability present in Montserrat. However, similarity in both values for
B
gives us confidence to extend McF resulting from TLS two orders of magnitude up to
major observed rockfalls (V<104m3).
Remote Sens. 2023, 15, x FOR PEER REVIEW 28 of 42
As seen in Figure 14, McF results derived from TLS monitoring and observational
inventory differ. In Castellfollit de la Roca, this difference is small, and both McF can be
attempted to be merged into a fusion sample (see Figure 15) to get validity on the maxi-
mum range of magnitudes and be confident on the hazard scenarios at any point of the
McF curve It has already been demonstrated [81] that there is a similar degree of coherence
in Yosemite (USA) between rockfall inventory and remote sensing detection (combined
TLS and SfM applied to aerial and terrestrial images), thanks to the high-quality inventory
available [82]. In Montserrat, 𝐴 obtained by inventory is six times lower than the value
from TLS. We conclude that the inventory lacks quality and is insufficiently systematic to
overcome the high variability present in Montserrat. However, similarity in both values
for 𝐵 gives us confidence to extend McF resulting from TLS two orders of magnitude up
to major observed rockfalls (𝑉10
 m).
Figure 15. McF comparison between data sources: TLS monitoring and observational inventory for
Castellfollit de la Roca (left) and Montserrat Massif (right). Best McF fittings obtained at cliff and
massif scales are presented.
We conclude a large margin of complementarity between both data sources, which
could still be expanded with others of those presented in Subsection 1.2. On the one hand,
TLS provides systematic detection that is homogeneous for all magnitudes, leading to a
complete sample. Attention must be paid to 𝑆𝐸 value because it tends to have very local
and short monitoring that limits its representativeness. Finally, new studies can only be
planned from the present to the future, forcing them to wait a certain amount of time for
a sufficient sample to be built. On the other hand, inventory allows going back to the past
as long as there are preservation sources of activity data (e.g., written, oral, graphic). It
can also cover a large sample size in both space and time, but there is no longer homoge-
neity with magnitude, and its completeness must always be questioned.
5.1.3. Multi-Site Comparison
In Figure 14, McF values for Montserrat and Castellfollit are compared to other ref-
erence sites in France and Yosemite, USA [7,81], and agreement is observed. According to
these previous references and present results, the following conclusions are drawn. Pa-
rameter 𝐵 is related to rock mass structure, and its value increases as the interlocking of
the rock pieces decreases, which is related to the joint’s pattern and strength. The alterna-
tion of soft and hard layers could produce an additional increase in 𝐵. Parameter 𝐴 is
related to the geodynamical context and the intensity and constancy of erosive and trig-
gering actions. For instance, it is expected highest to the lowest value of 𝐴 when moving
from coastal cliffs or deglaciated valleys to riverside slopes or mountains in a temperate
climate. Thus, it is suggested to distinguish degrees of rockfall activity according to 𝐴
(see Figure 14). The scale step factor used a one and a half decimal order of magnitude to
keep similarity to the frequency classes adopted by the Austrian norm ONR-24810 [83] to
define nominal blocks for rockfall fence design.
Figure 15.
McF comparison between data sources: TLS monitoring and observational inventory for
Castellfollit de la Roca (
left
) and Montserrat Massif (
right
). Best McF fittings obtained at cliff and
massif scales are presented.
We conclude a large margin of complementarity between both data sources, which
could still be expanded with others of those presented in Section 1.2. On the one hand,
TLS provides systematic detection that is homogeneous for all magnitudes, leading to a
complete sample. Attention must be paid to
SE
value because it tends to have very local
and short monitoring that limits its representativeness. Finally, new studies can only be
planned from the present to the future, forcing them to wait a certain amount of time for a
sufficient sample to be built. On the other hand, inventory allows going back to the past as
long as there are preservation sources of activity data (e.g., written, oral, graphic). It can
also cover a large sample size in both space and time, but there is no longer homogeneity
with magnitude, and its completeness must always be questioned.
5.1.3. Multi-Site Comparison
In Figure 14, McF values for Montserrat and Castellfollit are compared to other refer-
ence sites in France and Yosemite, USA [
7
,
81
], and agreement is observed. According to
these previous references and present results, the following conclusions are drawn. Param-
eter
B
is related to rock mass structure, and its value increases as the interlocking of the
rock pieces decreases, which is related to the joint’s pattern and strength. The alternation of
soft and hard layers could produce an additional increase in
B
. Parameter
Ast
is related to
the geodynamical context and the intensity and constancy of erosive and triggering actions.
For instance, it is expected highest to the lowest value of
Ast
when moving from coastal
cliffs or deglaciated valleys to riverside slopes or mountains in a temperate climate. Thus,
it is suggested to distinguish degrees of rockfall activity according to
Ast
(see Figure 14).
The scale step factor used a one and a half decimal order of magnitude to keep similarity
to the frequency classes adopted by the Austrian norm ONR-24810 [
83
] to define nominal
blocks for rockfall fence design.
This comparison of different sites is difficult to read in terms of hazard, due to the
dispersion in the plane of both parameters
Ast
and
B
. As stated, it is quite clear that an
increase in the value of normalized activity
Ast
is directly related to a greater hazard, but
the influence of parameter
B
in terms of hazard is less clear. When the value of
B
differs
between two cases, it is difficult to directly compare McF curves. For this purpose, it
would be useful to know the center of the hyperbola (Vc;Fc)defined in Section 2.1, which
represents McF in a single point (see Figure 16). On this plane,
pc
and
d0
are suggested
as proxy descriptors of intrinsic hazard, which means, all conditions of the rock mass

Remote Sens. 2023,15, 1981 27 of 36
(e.g., structure, geotechnical properties) and environmental context (e.g., climatic, seismic)
influence the hazard before the slope geomorphology itself. Features of the outcrop (e.g.,
height, slope, aspect) will modulate this intrinsic hazard to the complete idea of rockfall
detachment hazard when the area below the McF curve is integrated with the range of
expected magnitudes [Vmin ;Vmax ]according to [7].
Remote Sens. 2023, 15, x FOR PEER REVIEW 29 of 42
This comparison of different sites is difficult to read in terms of hazard, due to the
dispersion in the plane of both parameters 𝐴 and 𝐵. As stated, it is quite clear that an
increase in the value of normalized activity 𝐴 is directly related to a greater hazard, but
the influence of parameter 𝐵 in terms of hazard is less clear. When the value of 𝐵 differs
between two cases, it is difficult to directly compare McF curves. For this purpose, it
would be useful to know the center of the hyperbola (𝑉;𝐹
) defined in subsection 2.1,
which represents McF in a single point (see Figure 16). On this plane, 𝑝 and 𝑑 are sug-
gested as proxy descriptors of intrinsic hazard, which means, all conditions of the rock
mass (e.g., structure, geotechnical properties) and environmental context (e.g., climatic,
seismic) influence the hazard before the slope geomorphology itself. Features of the out-
crop (e.g., height, slope, aspect) will modulate this intrinsic hazard to the complete idea
of rockfall detachment hazard when the area below the McF curve is integrated with the
range of expected magnitudes 󰇟𝑉;𝑉
󰇠 according to [7].
Figure 16. McF alternatively represented by the centre of the hyperbola (𝑉

;𝐹

) as defined in sub-
section 2.1. Gray lines show the correspondence with 𝐴

and 𝐵 parameters. As explained in the
text, on the traffic light color scale 𝑝

and 𝑑

are suggested as proxies for intrinsic hazards.
5.2. Temporal Variability in McF
A direct reading of the rockfall activity as the total number of events per year or the
corresponding rockfall total volume rate 󰇟myear
⁄󰇠 is not sufficient to assess a possible
evolution of the hazard. Obviously, in one period there may be more activity of a lower
magnitude compared to another of the same duration in which a single major event oc-
curs, and the reading in terms of hazards may be different depending on the magnitude-
frequency balance.
5.2.1. McF Relationship over Time
Taking the longest series of Montserrat (Degotalls and Monastery), an analysis of the
variability of McF has been completed over the years. Sub-samples have been taken for
periods ranging from one to two years, depending on the availability of surveys and the
data obtained. Figure 17 shows the variability of the parameters 𝐴 and 𝐵 in the adjust-
ment for each period in the raw data, avoiding filtering extreme effects explained in sub-
section 4.1 to keep the entire sample. In the walls of Degotalls (N and E treated together
to have wider sub-samples), the initial period corresponding to the great rockfall and sub-
sequent works on the rock face stands out with a much higher activity than the rest, as
already seen grouped in Figure 8. In this detail, we see that the value of 𝐴 in the critical
period 2007–2009 has no similarity with the rest, which have an average about eight times
Figure 16.
McF alternatively represented by the centre of the hyperbola
(Vc;Fc)
as defined in
Section 2.1
. Gray lines show the correspondence with
Ast
and
B
parameters. As explained in
the text, on the traffic light color scale pcand d0are suggested as proxies for intrinsic hazards.
5.2. Temporal Variability in McF
A direct reading of the rockfall activity as the total number of events per year or the
corresponding rockfall total volume rate
m3/year
is not sufficient to assess a possible
evolution of the hazard. Obviously, in one period there may be more activity of a lower
magnitude compared to another of the same duration in which a single major event
occurs, and the reading in terms of hazards may be different depending on the magnitude-
frequency balance.
5.2.1. McF Relationship over Time
Taking the longest series of Montserrat (Degotalls and Monastery), an analysis of the
variability of McF has been completed over the years. Sub-samples have been taken for
periods ranging from one to two years, depending on the availability of surveys and the
data obtained. Figure 17 shows the variability of the parameters
Ast
and
B
in the adjustment
for each period in the raw data, avoiding filtering extreme effects explained in Section 4.1
to keep the entire sample. In the walls of Degotalls (N and E treated together to have
wider sub-samples), the initial period corresponding to the great rockfall and subsequent
works on the rock face stands out with a much higher activity than the rest, as already
seen grouped in Figure 8. In this detail, we see that the value of
Ast
in the critical period
2007–2009 has no similarity with the rest, which have an average about eight times lower,
and this comparison factor raises to 58 between the maximum (
Ast =
1.6 in 2007–2009) and
minimum values (
Ast =
0.028 in 2019). Some inverse effect is also observed between the
profiles of
Ast
and
B
, which is related to the maximum volume detected in each period. In
a period when a noticeable event occurs, the value of
B
decreases and that of
Ast
increases
compared to the general trend. We also observe a global trend of an exponential decrease
in activity (number of rockfalls) that can be related to the protective works conducted and
perhaps also to the evolution in analysis methodology, criteria, and tools, as reviewed
in Section 5.2.2 and Supplementary Material SM5. Figure 17 includes the same analysis

Remote Sens. 2023,15, 1981 28 of 36
for Monastery rockfalls detected by TLS, where the same effect is also present but with
less variability.
Remote Sens. 2023, 15, x FOR PEER REVIEW 30 of 42
lower, and this comparison factor raises to 58 between the maximum (𝐴 =1.6 in 2007–
2009) and minimum values (𝐴 = 0.028 in 2019). Some inverse effect is also observed be-
tween the profiles of 𝐴 and 𝐵, which is related to the maximum volume detected in
each period. In a period when a noticeable event occurs, the value of 𝐵 decreases and that
of 𝐴 increases compared to the general trend. We also observe a global trend of an ex-
ponential decrease in activity (number of rockfalls) that can be related to the protective
works conducted and perhaps also to the evolution in analysis methodology, criteria, and
tools, as reviewed in Subsubsection 5.2.2 and Supplementary Material SM5. Figure 17 in-
cludes the same analysis for Monastery rockfalls detected by TLS, where the same effect
is also present but with less variability.
It must be concluded that rockfall activity and hazards vary over time on this annual
or multi-year scale. The available series are not long enough to relate this to large cycles
of triggering factors, and the surveys are not frequent enough to search for seasonal vari-
ability. Continuous monitoring could provide us with this information about triggers as
become successively more common [84], or alternatively adapting survey cadence to trig-
gering factors occurrence such as conducted by [85]. Finally, in both cases, we can see the
wide variability of the McF adjustments obtained, a fact that highlights the importance of
extending SE sufficiently to be able to obtain representative results of the representative
hazards of the place in a timeless way, not conditioned by transitory circumstances.
An important point in this analysis is the representativeness of each sub-sample. In
Figure 17, 𝑆𝐸 of each period is compared with a minimum value 𝑆𝐸 =1 𝐹

⁄ that is
equivalent to the condition 𝑆𝐸 =𝑆𝐸∙𝐹
=1 as introduced in subsection 2.2. Except for
the first surveying period in Degotalls (2007–2009), where 𝑆𝐸 ≫1, the rest of the peri-
ods in Degotalls and Monastery are at the limit of 𝑆𝐸~1, so the adjusted values for 𝐴
and 𝐵 must be taken with caution. The number of data points is also shown, but it is less
indicative of the achieved representativeness.
Figure 17. McF variability over time for Degotalls Wall (left column) and Monastery Wall (right
column) in Montserrat. Parameters 𝐴

and 𝐵 fitted for every sub-sampling period (top) and
Figure 17.
McF variability over time for Degotalls Wall (left column) and Monastery Wall (right
column) in Montserrat. Parameters
Ast
and
B
fitted for every sub-sampling period (
top
) and global
fitting in dashed lines; description of the sub-samples in terms of number of recorded events and
sample extent
SE
compared to a minimum value,
SEmi n
for representativeness (
center
); McF curves
for short sub-samples in blue showing the volume range covered, and global sample in orange in raw
data without any filters applied to the extremes (bottom).
It must be concluded that rockfall activity and hazards vary over time on this annual
or multi-year scale. The available series are not long enough to relate this to large cycles
of triggering factors, and the surveys are not frequent enough to search for seasonal
variability. Continuous monitoring could provide us with this information about triggers
as become successively more common [
84
], or alternatively adapting survey cadence to
triggering factors occurrence such as conducted by [
85
]. Finally, in both cases, we can see
the wide variability of the McF adjustments obtained, a fact that highlights the importance
of extending SE sufficiently to be able to obtain representative results of the representative
hazards of the place in a timeless way, not conditioned by transitory circumstances.
An important point in this analysis is the representativeness of each sub-sample. In
Figure 17,
SE
of each period is compared with a minimum value
SEmin =1/Fc
that is
equivalent to the condition
SEuc =SE·Fc=
1 as introduced in Section 2.2. Except for the
first surveying period in Degotalls (2007–2009), where
SEuc 
1, the rest of the periods
in Degotalls and Monastery are at the limit of
SEuc ∼
1, so the adjusted values for
Ast
and
B
must be taken with caution. The number of data points is also shown, but it is less
indicative of the achieved representativeness.
Since rockfalls are not continuous movements but, on the contrary, isolated events of
very short duration compared to the long inactive periods between falls, the measurement

Remote Sens. 2023,15, 1981 29 of 36
of the activity is highly dependent on the period of observation, as was seen in the previous
section. This variability may reflect many underlying factors as follows [85,86]:
•
Variability of activity is due to natural cycles of the different triggering agents (e.g.,
rainfall regime, thermal regime, seismic activity, the evolution of the rock massif,
and degradation of its resistant properties depending on environmental physical and
chemical agents).
•
Modification of the activity due to the work carried out. The clearing of blocks of
precarious stability concentrates in a short period a large part of the activity that would
have happened more evenly over a longer period. Stabilizing potentially unstable
masses makes the future activity less likely in a wide range of sizes. Therefore,
McF could provide a quantitative method for assessing the effectiveness of hazard
mitigation measures, as explored in Supplementary Material SM5.
In complex cases, such as the wall of Degotall’s north, it is surely a mixture of all
these factors. This brings great uncertainty to the definition of hazardous scenarios. In
this example of extreme variability, although we take the joint sample of north and east
walls and McF adjustments for the 2007–2009 or 2009–2020 periods from Figure 8, we
get, on the one hand, an event similar to that of 2008 (
V=
900
m3
), which corresponds
to an occurrence frequency of
Fst =
0.001 to
Fst =
0.1. On the other hand, a detachment
hazard scenario with a return period,
T=
100
years
, for a unitary wall extension of 1
hm2
(
Fst =
0.01) is equivalent to a rockfall of
V=
32
m3
to
V=
2.3
·
10
5m3
, practically two
orders of magnitude larger than any record in the inventory. This uncertainty would
invalidate any further calculation of hazard. Therefore, which is the McF to be taken as a
reference? It will surely depend on the time scale of the analysis.
5.2.2. McF Sensitivity to Detection Algorithms
Throughout the sixteen years of TLS applications to rockfall detection at these case sites,
methods and tools applied for data processing have been evolving. It can be distinguished
four major stages that have given rise to different series of data, particularly in Montserrat,
the walls of Degotalls being the most common test area:
•
S1: Classic or standard processing with Polyworks and DBscan tools, initiated by [
61
]
and developed by [62], with data from an Optech TLS device, from 2007 to 2017.
•
S2: Classic or standard processing with M3C2 and DBscan tools developed by [
63
]
and applied by [64], with data from an Optech TLS device, from 2007 to 2017.
•
S3: Machine learning processing with PCM and CC tools developed by [
23
,
67
] and
applied to data from an Optech TLS device, from 2007 to 2020.
•
S4: Machine learning processing with PCM and CC tools developed by [
23
,
67
] and
applied to data from a Leica TLS device, from 2018 to 2020.
Monitoring series in Degotalls’ north wall obtained by Optech’s ILRIS-3D TLS device
from 2007 to 2017 were initially processed by [
62
], reviewed by [
64
], and finally computed
by [
23
] using the developed workflow PCM-CC that uses machine learning for the cluster
classification of rockfall candidates. This comparison can be seen in Figure 18. Apparently,
the distributions are quite similar, but there is significant variability in the parameters of
the McF fit. Changes in criteria are seen in the interpretation of false positives, especially
for small volumes. It is assumed that during the first years, there could be a tendency to
preventively classify as rockfall the changes with this appearance, even if there was no
reliable information to prove it. In 2018, we introduced imagery monitoring in parallel to
TLS using the GigaPan device [
69
] to obtain high-definition images that have been helpful
in the process of interpreting the detected changes. At the other end of the graph, for the
largest volumes, there are also differences in the interpretation of the blocks successively
detached in the period of high activity without intermediate monitoring from 2007 to 2009,
which represents significant differences.

Remote Sens. 2023,15, 1981 30 of 36
Remote Sens. 2023, 15, x FOR PEER REVIEW 32 of 42
TLS using the GigaPan device [69] to obtain high-definition images that have been helpful
in the process of interpreting the detected changes. At the other end of the graph, for the
largest volumes, there are also differences in the interpretation of the blocks successively
detached in the period of high activity without intermediate monitoring from 2007 to 2009,
which represents significant differences.
Figure 18. Examples of changes introduced by different computation methods and tools when pro-
cessing TLS data for rockfall detection. On the left graph, three computing stages (S1 to S3 presented
in the text) were applied to TLS monitoring with an Optech ILRIS 3D device in Degotall's north wall
from 2007 to 2017. On the right graph, it can be seen the comparison between the results of rockfall
detection using machine learning PCM-CC applied to monitor data obtained by Optech ILRIS-3D
and Leica P50 during the 2018–2020 period for the mixed sample of Degotalls' north and east walls.
Another example of comparison shown in Figure 18 corresponds to the period 2018–
2020 when parallel monitoring was conducted using both different TLS devices presented
in Table 1. Differences in data capture (e.g., point cloud density, laser footprint) lead to
different interpretations despite applying the same analysis procedure (in this case, the
machine learning workflow PCM-CC) [23]. Thus, different data capture systems and pro-
cessing methods introduce variability in results at different stages of the treatment of the
point clouds [87], including the uncertainty in the volume calculation of point clouds [71].
This factor has a higher influence in small samples, such as the presented example.
5.2.3. Detachment Mechanisms Overlap
In the McF distribution for the global Montserrat Massif by TLS, a slight ripple shape
is observed (Figure 19). Instead of resulting from the previously analyzed specificities in
the activity of sites and periods of the sub-samples that make it up, it could also reflect
another intrinsic heterogeneity of rockfalls in this massif. As a follow-up to the description
of the massif in Section 3.1, three mechanisms of rockfall can be distinguished, according
to the structural element that fails and the factors preparing the detachment. They go be-
yond the classic differentiation (fall-slide-topple) already revealed by [88] that reflects the
kinematics of the failure and the first movement when detaching, which have all been
observed in Montserrat. Those are related to a range of sizes, but all three classes are over-
layed:
• Mechanism 1 (𝑚𝑒𝑐1) corresponds to the most considered rockfall type, that is, rock
blocks delimited by mechanical discontinuities in the rock mass. For the conglomer-
ates in Montserrat, these are fracture sets and stratigraphic layers, both of high per-
sistence, which delimit prismatic blocks of a wide range of volumes depending on
the joint spacing at the specific point, from less than one cubic meter to several thou-
sands of cubic meters (mainly from 10
−1
to 10
4
m
3
). The basic properties of the discon-
tinuities can be found in Supplementary Material SM3.
Figure 18.
Examples of changes introduced by different computation methods and tools when
processing TLS data for rockfall detection. On the left graph, three computing stages (S1 to S3
presented in the text) were applied to TLS monitoring with an Optech ILRIS 3D device in Degotall’s
north wall from 2007 to 2017. On the right graph, it can be seen the comparison between the results
of rockfall detection using machine learning PCM-CC applied to monitor data obtained by Optech
ILRIS-3D and Leica P50 during the 2018–2020 period for the mixed sample of Degotalls’ north and
east walls.
Another example of comparison shown in Figure 18 corresponds to the period
2018–2020
when parallel monitoring was conducted using both different TLS devices
presented in Table 1. Differences in data capture (e.g., point cloud density, laser footprint)
lead to different interpretations despite applying the same analysis procedure (in this case,
the machine learning workflow PCM-CC) [
23
]. Thus, different data capture systems and
processing methods introduce variability in results at different stages of the treatment of the
point clouds [
87
], including the uncertainty in the volume calculation of point clouds [
71
].
This factor has a higher influence in small samples, such as the presented example.
5.2.3. Detachment Mechanisms Overlap
In the McF distribution for the global Montserrat Massif by TLS, a slight ripple shape
is observed (Figure 19). Instead of resulting from the previously analyzed specificities in the
activity of sites and periods of the sub-samples that make it up, it could also reflect another
intrinsic heterogeneity of rockfalls in this massif. As a follow-up to the description of the
massif in Section 3.1, three mechanisms of rockfall can be distinguished, according to the
structural element that fails and the factors preparing the detachment. They go beyond the
classic differentiation (fall-slide-topple) already revealed by [
88
] that reflects the kinematics
of the failure and the first movement when detaching, which have all been observed in
Montserrat. Those are related to a range of sizes, but all three classes are overlayed:
Remote Sens. 2023, 15, x FOR PEER REVIEW 33 of 42
• Mechanism 2 (𝑚𝑒𝑐2) corresponds to weathering flakes produced by thermal exfolia-
tion, forming curved plates or slabs of intermediate volume from 10
−3
to 10
2
m
3
, alt-
hough the most common are in the range of few to several dm
3
.
• Mechanism 3 (𝑚𝑒𝑐3) corresponds to pebble detachment from the conglomerate due
to the matrix weathering in contrast to the resistance of the pebbles. Additionally,
masses of aggregates can fall together, especially if small and local fractures are pre-
sent. These rockfalls are of irregular shape and generally of small volume, ranging
from 10
−6
to less than 10
−1
m
3
.
Since each of these mechanisms responds to different factors of rock mass resistance
and destabilizing actions, they could each follow their own McF law over the range of
magnitudes they have, which partially overlap. Figure 19 suggests a possible theoretical
distinction between the three mechanisms with their parameters (𝐴, 𝐵 and range of vol-
umes) corresponding to the characteristics observed on the field, which would give rise
to a unified McF distribution undulating similarly to that obtained by monitoring.
In this line of work, [23] has begun to apply the distinction of these mechanisms in
cluster classification with machine learning techniques. Preliminarily, it has been seen that
with the training samples used for the model, a satisfactory classification of 𝑚𝑒𝑐2 rock-
falls has been achieved, while lower effectiveness is observed in the identification of the
𝑚𝑒𝑐1 mechanism. This is probably due to the central position of 𝑚𝑒𝑐2 within the sample
in contrast to the low number of records of other mechanisms involved in the training
phase. Future developments will attempt to achieve a more efficient classification of rock-
fall clusters, in which the mechanism is also distinguished, and thus be able to confirm if
different McF applies and even adapt specific volume calculation formulations to the dif-
ferent morphologies.
In Castellfollit, the toppling mechanism of basaltic columns has been clearly identi-
fied, which has led to the detection of precursor movements thanks to its large displace-
ments from [60] to [29] with TLS and photogrammetry. However, lava flows have heter-
ogeneous levels, and there could be other rupture mechanisms as suggested by [60], such
as the 2011 large rupture at the fractured base, that may differ slightly in the McF pattern.
It will be necessary to deepen it in future studies.
Figure 19. Influence of rockfall detaching mechanisms identified in Montserrat Massif (left) on the
resulting McF distribution if three different power laws are assumed (right).
6. Conclusions
The magnitude-cumulated frequency relationship (McF) is the first step in any rock-
fall hazard analysis, providing the hazard scenarios to be considered in the source area. It
is found that McF adjustments are very sensitive to many factors related to the available
data (e.g., nature of the data source, quality of the recording, extent and completeness of
the sample). The statistical treatment was performed to calculate temporal frequencies.
These factors have been analyzed at two different sites, the Montserrat conglomeratic
Figure 19.
Influence of rockfall detaching mechanisms identified in Montserrat Massif (
left
) on the
resulting McF distribution if three different power laws are assumed (right).

Remote Sens. 2023,15, 1981 31 of 36
•
Mechanism 1 (
mec
1) corresponds to the most considered rockfall type, that is, rock
blocks delimited by mechanical discontinuities in the rock mass. For the conglomerates
in Montserrat, these are fracture sets and stratigraphic layers, both of high persistence,
which delimit prismatic blocks of a wide range of volumes depending on the joint
spacing at the specific point, from less than one cubic meter to several thousands of
cubic meters (mainly from 10
−1
to 10
4
m
3
). The basic properties of the discontinuities
can be found in Supplementary Material SM3.
•
Mechanism 2 (
mec
2) corresponds to weathering flakes produced by thermal exfoliation,
forming curved plates or slabs of intermediate volume from 10
−3
to 10
2
m
3
, although
the most common are in the range of few to several dm3.
•
Mechanism 3 (
mec
3) corresponds to pebble detachment from the conglomerate due to
the matrix weathering in contrast to the resistance of the pebbles. Additionally, masses
of aggregates can fall together, especially if small and local fractures are present. These
rockfalls are of irregular shape and generally of small volume, ranging from 10
−6
to
less than 10−1m3.
Since each of these mechanisms responds to different factors of rock mass resistance
and destabilizing actions, they could each follow their own McF law over the range of
magnitudes they have, which partially overlap. Figure 19 suggests a possible theoretical
distinction between the three mechanisms with their parameters (
Ast
,
B
and range of
volumes) corresponding to the characteristics observed on the field, which would give rise
to a unified McF distribution undulating similarly to that obtained by monitoring.
In this line of work, [
23
] has begun to apply the distinction of these mechanisms in
cluster classification with machine learning techniques. Preliminarily, it has been seen
that with the training samples used for the model, a satisfactory classification of
mec
2
rockfalls has been achieved, while lower effectiveness is observed in the identification
of the
mec
1 mechanism. This is probably due to the central position of
mec
2 within the
sample in contrast to the low number of records of other mechanisms involved in the
training phase. Future developments will attempt to achieve a more efficient classification
of rockfall clusters, in which the mechanism is also distinguished, and thus be able to
confirm if different McF applies and even adapt specific volume calculation formulations
to the different morphologies.
In Castellfollit, the toppling mechanism of basaltic columns has been clearly identified,
which has led to the detection of precursor movements thanks to its large displacements
from [
60
] to [
29
] with TLS and photogrammetry. However, lava flows have heterogeneous
levels, and there could be other rupture mechanisms as suggested by [
60
], such as the 2011
large rupture at the fractured base, that may differ slightly in the McF pattern. It will be
necessary to deepen it in future studies.
6. Conclusions
The magnitude-cumulated frequency relationship (McF) is the first step in any rockfall
hazard analysis, providing the hazard scenarios to be considered in the source area. It is
found that McF adjustments are very sensitive to many factors related to the available data
(e.g., nature of the data source, quality of the recording, extent and completeness of the
sample). The statistical treatment was performed to calculate temporal frequencies. These
factors have been analyzed at two different sites, the Montserrat conglomeratic massif and
the Castellfollit de la Roca basaltic cliff, where rockfall risk management is an ongoing
challenge. Consequently, it has been revealed that it is necessary to carefully identify the
attributes of the activity record to know the actual validity of McF over space, time, and
magnitude range and what controls its applicability. Uncertainty in both parameters
Ast
and Bin McF laws leads to large indeterminacy when choosing return periods for further
risk assessment. We propose to work with the attribute of sampling extent
SE
defined in
Section 2.2 and various ways of expressing the results of McF for better communication
of hazards.

Remote Sens. 2023,15, 1981 32 of 36
In Section 4, a methodological technique for optimizing the utilization of rockfall
activity data from remote sensing and traditional inventory is described. On the one
hand, monitoring series provide homogeneous samples, which are very useful for the
characterization of the central part of the McF curve once the roll-over effect and distortions
in extreme values have been filtered, following the indications provided in Section 4.1.
Observational inventories, on the other hand, contain sample heterogeneities throughout
both space and time since not all magnitude ranges are identified similarly. To fit the
available data to the correct ranges of time, space, and magnitude, a processing strategy
has been proposed in Section 4.2: inventory is decomposed into homogeneous sub-samples
where the frequency calculation is feasible, which are integrated later into a representative
joint sample. There may be dispersion between the results coming from monitoring
detection and observational inventory, but at the same time, they can complement each
other due to the completeness achieved by the first data source and the sampling extent
that can be provided by the second. If a wider range of magnitudes is covered, the McF law
will allow a wider application in quantitative risk analysis.
At the scale of a mountain massif (i.e., Montserrat), there can be great variability in McF
laws, which must be considered to limit the scale of application of each one. Conversely, it
is easier to achieve consistency at the scale of local homogeneous cliffs (i.e., Castellfollit de
la Roca). In any case, clear variability over time is also detected at various temporal scales,
making a careful review of the space-time coverage of the available samples necessary.
Additionally, variations in McF adjustment have been found when changes in data capture
or processing are introduced. To create a more homogeneous series, it is advisable to apply
consistent approaches and methods or, if this is not possible, to account for this variability.
Despite the limited data available, it seems that the McF curves reflect the heterogeneity
in changes of scale when we face different failure mechanisms according to the rock mass
properties and destabilizing agents. Results from the case sites have been compared to other
McF laws in the literature. It is proposed a qualitative scale of rockfall activity according
to
Ast
parameter that can be mainly correlated with how geodynamic agents act on rock
mass. Additionally, two parameters derived from McF,
pc
and
d0
are proposed as proxies
for intrinsic hazards. With this term, we refer to the detachment regime conditioned by
rock mass properties and environmental conditions before considering the local outcrop
disposal on the slope. This last step leads to the maximum volume likelihood and the full
picture of detachment hazards related to the retreat rate. However, this is beyond the scope
of this paper.
Our findings suggest that every massif has slopes in different conditions, and each
rock outcrop has its activity cycles, which, when characterized, open the door to new
strategies for risk management, closer to forecast. Although a temporal prediction of the
phenomenon of extensive’ coverage in diffuse risk has not yet been achieved, it is hoped to
be able to focus interest and mitigation efforts throughout space–time thanks to monitoring.
Likewise, the effectiveness of stabilization works can be quantified by a change in behavior
in McF, and, in this challenge, remote sensing techniques such as laser scanning and digital
photogrammetry are definitely very useful.
Supplementary Materials:
The following supporting information can be downloaded at: https:
//www.mdpi.com/article/10.3390/rs15081981/s1, File SM1: Data sources on rockfall activity; File
SM2: Power law for McF; File SM3: Test site features; File SM4: Rockfall activity mapping; File SM5:
Hazard mitigation effectiveness.
Author Contributions:
Conceptualization, M.J. and J.A.G.; methodology, M.J. and D.H.; validation,
M.J., N.L., M.A.N.-A. and D.H.; formal analysis, M.J., N.L., M.A.N.-A., D.H. and R.C.; investigation,
M.J. and R.C.; resources, O.P., M.G., D.G.-S. and L.B.; data curation, M.J., O.P., R.C., M.G., D.G.-S. and
L.B.; writing—original draft preparation, M.J.; writing—review and editing, all authors; visualization,
M.J. and O.P.; supervision, J.A.G. and J.P.; project administration, N.L. and M.A.N.-A.; funding
acquisition, N.L., M.A.N.-A. and J.P. All authors have read and agreed to the published version of
the manuscript.

Remote Sens. 2023,15, 1981 33 of 36
Funding:
This research was funded by project Georisk, “Advances in rockfall quantitative risk analy-
sis (QRA) incorporating developments in geomatics (GeoRisk)”, grant number PID2019-103974RB-I00,
funded by MCIN/AEI/10.13039/501100011033, the Ministerio de Ciencia e Innovación and the Agen-
cia Estatal de Investigación of Spain. The APC was covered by Remote Sensing.
Data Availability Statement:
More data and details can be found in the Supplementary Materials.
There are available online for viewers of the rockfalls detected by TLS in both case sites: Montserrat
and Castellfollit de la Roca. Finally, for people interested in more details, please, feel free to contact
the corresponding author.
Acknowledgments:
This work was performed in the framework of the risk mitigation program
started in 2014 and covers all tasks focused on risk management, including studies, protective
works, and monitoring. It is funded by the Catalan Government, promoted by the Patronat de la
Muntanya de Montserrat (PMM, the public entity that manages the site as a natural and cultural
heritage), and executed by the ICGC. Initial data from Castellfollit de la Roca were acquired with
the support of the Spanish Ministry of Science and Education (predoctoral grant 2004-1852) and
funded by the Natural Park of the Garrotxa Volcanic Field (PNZVG). The authors and members of
RiskNat want to acknowledge the support from PROMONTEC (CGL2017-84720-R AEI/FEDER, UE)
funded by the Spanish MINECO and the following projects: MEC CGL2006-06596 (DALMASA),
TopoIberia CSD2006-0004/Consolider-Ingenio2010, MEC CGL2010-18609 (NUTESA), and the grant
TED2021-130602B-I00 funded by MCIN/AEI/ 10.13039/501100011033 and by the “European Union
NextGenerationEU/PRTR.”
Conflicts of Interest: The authors declare no conflict of interest.
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