We detail a new prediction-oriented procedure aimed at volcanic hazard
assessment based on geophysical mass flow models constrained with
heterogeneous and poorly defined data. Our method relies on an itemized
application of the empirical falsification principle over an arbitrarily wide
envelope of possible input conditions. We thus provide a first step towards a
objective and partially automated experimental design construction. In
particular, instead of fully calibrating model inputs on past observations,
we create and explore more general requirements of consistency, and then we
separately use each piece of empirical data to remove those input values that
are not compatible with it. Hence, partial solutions are defined to the inverse
problem. This has several advantages compared to a traditionally posed
inverse problem: (i) the potentially nonempty inverse images of partial
solutions of multiple possible forward models characterize the solutions to
the inverse problem; (ii) the partial solutions can provide hazard estimates
under weaker constraints, potentially including extreme cases that are
important for hazard analysis; (iii) if multiple models are applicable,
specific performance scores against each piece of empirical information can
be calculated. We apply our procedure to the case study of the Atenquique
volcaniclastic debris flow, which occurred on the flanks of Nevado de Colima
volcano (Mexico), 1955. We adopt and compare three depth-averaged models
currently implemented in the TITAN2D solver, available from

Hazard assessment of geophysical mass flows, such as landslides or
pyroclastic flows, usually relies on the reconstruction of past flows that
occurred in the region of interest using models of physics that have been
successful in hindcasting. The available pieces of data

In a probabilistic framework, for each model

Our approach is characterized by three steps:

For each model

After a preliminary screening, we characterize a specialized input
space,

Our notion of output plausibility is not related
to the model plausibility defined in

Through more detailed testing,

We omit to express the

The philosophy of our method is based on an itemized application of the
empirical falsification principle of Karl Raimund Popper

the intersection space

the partial solutions

each probability

Diagram of input spaces, model functions, and output space (blue), with feasible inputs domain (red), plausible output codomain and specialized inputs (green), and observed data and partial solutions subsets (orange). The question mark emphasizes that the covering of other plausible outputs could be enabled by adding more models if necessary.

Our meta-modeling framework is fully described in Fig.

We remark that the implementation of multiple models is a crucial aspect in
our approach. Typically, the available models are not able to entirely cover

We apply our procedure to the case study of the Atenquique volcaniclastic
debris flow, which occurred on the flanks of Nevado de Colima volcano
(Mexico) in 1955. We adopt and compare the three depth-averaged models
Mohr–Coulomb (MC)

Barranca de Atenquique (Mexico) overview.

The rest of the study is organized as follows. In Sect.

The Colima Volcanic Complex is located in the western portion of the
Trans-Mexican volcanic belt (small box in Fig.

Nevado de Colima (4320 m a.s.l.) occupies the central part of the volcanic
complex, being the most voluminous of the three volcanoes (300–400 km

Figure

On 16 October 1955, at 10:45 LT, the inhabitants of Atenquique were
surprised by the sudden arrival of an 8–9 m high wave carrying mud, boulders
and tree trunks that devastated the buildings in the town and four bridges,
including the railroad bridge. More than 23 people died, and the flood
leveled everything but the tower of the church and the upper part of the
market place that luckily served as shelter for survivors

The main flood probably formed in the Atenquique ravine but was enhanced by
the confluence of flows from its tributaries: Dos Volcanes at 11.2 km, Arroyo
Seco and Los Plátanos, at 22.5 km. The following description of the flow
deposits summarizes

The 1955 debris flow, according to eyewitness accounts and deposit analyses,
emanated from multiple sources throughout the watershed. The existence of
multiple source areas presents a unique challenge when attempting to model
the flow. Eyewitnesses confirm that, after the event, many small landslides
scars were present along the main ravine and its tributaries. It is
hypothesized that these landslides, triggered by rain infiltration, supplied
the bulk of the material

We remark that source number, location, and volume partition will be
preserved in the following analysis. However, we tested whether small
variations affect the character of the simulated flow only proximally. Large
variations, for example additional testing focused on increasing the
weight

Our numerical modeling of the Atenquique flow proceeds by first assuming that
the laws of mass and momentum conservation hold for properly defined system
boundaries. The flow had very small depth compared to its length, and hence
we assume that it is reasonable to integrate through the depth to obtain
simpler and more computationally tractable equations

The depth-averaged Saint-Venant-type equations that result are as follows:

In this study we adopt the Mohr–Coulomb (MC),
Pouliquen–Forterre (PF), and Voellmy–Salm (VS) models, detailed
in Appendix

Available from

The definition of the input space hierarchy

The input space boundaries

Total volume:

Input space constraints:

The construction of

The flow reaches a minimum elevation

The maximum overspill at the confluence of flows from different sources is

Input space exploration and testing of

Figure

Overview of the specialized experimental design in

In MC we observe overspill issues if

In VS the overspill is observed in the region

Model PF must be treated with greater care because of its higher
dimensionality. We divide its behavior along four different hyperplanes,
corresponding to different values of

Overview of modified specialized experimental designs in PF,
supported over the four different hyperplanes described in Fig.

Initially,

We enhance the sampling procedure by relying on orthogonal arrays

Figure

Figure

Diagram of the steps of our meta-modeling approach. Our statistical summary includes the local analysis of contributing variables.

We devise multiple statistical measures for analyzing the data according to
the specialized LHS design described in the previous section. In general, for
each

In particular, we select five sites and we gather detailed results from our
simulations in those special locations. These are called sites 1–5 (stars
in Fig.

Site 1, section 23, UTM 656690.1

Site 2, section 28, UTM 660380.8

These first two are not described in detail, but the related results are
included in SI4–SI5 in the Supplement. Conversely, we focus our analysis
on the other three points, all placed along the main ravine in proximity to
Atenquique village.

Site 3, section 21, UTM 660258.1

Site 4, section 17, UTM 662453.1

Site 5, section 42, UTM 663539.0

First of all, we report the spatial maps of maximum flow depth,

In our depth-averaged approach the kinetic energy

While the use of dynamic pressure is more common in these applications, we have used kinetic energy as a more stable indicator of potential to damage infrastructure and a quantity with a more stable computation as a conserved scalar across all models.

is defined asThis is equivalent to

Maximum flow height

Maximum kinetic energy

The kinetic energy,

Figure

We further analyze the flow properties at two of the sites selected above, which are located in the distal part of the flow (sites 3 and 4), with very
different results. All the quantities reported are estimated on the element
of the grid, which contains the coordinates of the site. We remark that the
grid is adaptive, and hence the values can be affected by the size and
position of the element. However, the integration over the input space
significantly reduces this effect

Along with the locally analyzed flow height and speed, we calculate the local
contributing variables in the modeling equations: that is, the dominance factors and the expected contributions related to the
force terms in the conservation laws that characterize the models. In
particular,

The higher dimensionality of

Local flow properties at Site 3,

Figure

In summary, MC is characterized by a lower speed and by the dominance of basal friction. The expected contribution of the internal friction is not negligible, meaning the internal shear of the material is important. In PF, the pressure force contribution is significant, and it can even be the dominant force initially. This is related to the steepening of the flow front. An initial short-lasting wave of high speed is observed in either PF or VS, as is particularly evident in the upper bound of the plots. This fast wave is related to the closest source, 3. The uncertainty affecting height and speed is generally higher in VS than in PF, in spite of the higher dimensionality of the second.

Figure

Local flow properties at Site 4, immediately before Atenquique
village. Panels

In VS, the flow reaches the site in [250, 550] s, with a first wave of

In summary, all the models show lower flow height and speed than at the previous site upstream. The flow depth is stable, without decreasing downstream, meaning no formation of a significant deposit. MC is much slower than the other models, and its dynamics is completely dominated by basal friction. An initial, short-lasting wave of high speed is observed in VS. This fast wave is related to source 5 in Arroyo Platános.

In Site 5, reported in SI6 in the Supplement, all the models show a further decrease in flow height and speed compared to the previous site. In MC, the site is not always reached, and in PF some input values inundate the site only at the end of the time domain.

Histograms of local flow height in sites 3–5. Panels

Figures

At Site 3, flow height pdf at

At Site 4, flow height pdf at

At Site 5, MC and PF do not always reach the site in

Histograms of local flow speed in sites 3–5. Panels

Bar plots of data likelihood in sites 3–5. Panel

In Fig.

At Site 3, flow speed pdf at

At Site 4, flow speed pdf at

At Site 5, flow speed pdf at

In Figs.

Our empirical data include the following:

The deposit thickness, calculated from the envelope of the closest field
sections, is [3.7, 5.5] m at Site 3, [1.7, 3] m at
Site 4, and [1.4, 3.8] m at Site 5

The flow height in Atenquique village, from historical documents and
witnesses, is [8, 9] m at Site 3 and/or Site 4

The peak flow speed following the inundation of the village, based on
the superelevation method, is [4, 6] m s

The estimation of the likelihood of the pieces of observed data is an
essential step towards the definition of partial solutions of the inverse
problem. Besides this, it is also relevant information in the model selection
problem. The likelihood of a data piece,

In summary, model performance is dependent on the selected quantity of interest and on the spatial location. Regarding the deposits, MC performs well at Site 3, while VS does so at Site 5. In the evaluation of the maximum flow depth in the village, both PF and VS can replicate the values at Site 3, and only VS can replicate the values at Site 4. If we focus on the maximum flow speed, at Site 4 both PF and VS perform moderately well, while at Site 5, only VS can provide speed values inside the assumed range.

Example 1 of partial solution inputs in

Example 2 of partial solution inputs in the

Figures

Figure

Example 3 of partial solution inputs in the

The partial solution inputs in

Figure

Flow properties of the VS model, over the input
space

The solution of the partial inverse problems can enable us to select a model,
which nevertheless depends on the required properties and the spatial location.

In Example 1 the partial inverse problem is not well posed.

In Example 2 only in VS can we find solutions

In Example 3 both in PF and VS we find solutions

In Figs.

In Fig.

Flow properties of PF model, over the input
space

In Fig.

In the spatial maps, PF shows slightly lower maximum flow height and
significantly lower energy than VS, especially in the distal part of the
domain. The flow in the tributaries can reach the village, except for the
smallest flows of Arroyo Plátanos in PF, which, however, at

In summary, the statistical analysis of the partial solutions told us the following:

The deposit thickness is the three selected sites is not reproduced by any of the models. In particular, the input choices that fit Site 3 are inconsistent with those that fit the deposit downstream. This advocates the possibility of testing additional models, for example including an entrainment term in the mass conservation equation.

The MC model is not capable of reproducing the required maximum flow height and speed in the village. Its feasible input space does not allow us to reduce the friction further. Even if PF can reproduce the required height and speed when impacting the village, only VS is also capable of maintaining those values in the downstream part of the village.

In particular, models using MC-based rheologies are unlikely to reproduce the properties of the 1955 flow. Instead, the flexible basal friction angle in PF allows for both higher speed and longer runout, consistent with those observed. The higher dimensionality of its parameter space does not significantly increase the uncertainty affecting the outputs. Similarly, the velocity-dependent term in VS is a very robust mechanism for preserving numerical stability, avoiding the spurious results that affect the MC model at equivalently low values of basal friction. Indeed the highest levels of simulated speed are observed with VS.

We remark that the assumed [4, 6] m s

In this study, we have introduced a new prediction-oriented method for hazard
assessment of volcaniclastic debris flows (lahars), based on multiple
geophysical mass flow models. Similar strategies have been applied in
hurricane hazard analysis

We applied our procedure to a case study of the 1955 Atenquique
volcaniclastic debris flow. We adopted and compared three depth-averaged
models based on the Saint Venant equations that are widely used in hazard
assessment, namely Mohr–Coulomb (MC), Pouliquen–Forterre (PF), and
Voellmy–Salm (VS).

We defined a specialized experimental design after assuming the realism of the underlying physics, the numerical simulation is robust in some sense, and the flow dynamics or inundation output is meaningful. This produced a range of output simulations that contain valuable information for hazard assessment.

Indeed, these outputs do not strictly reconstruct past flows, so can provide hazard estimates under constraints weaker than those used therein, potentially including cases of extreme events. Moreover, our designs were not trivial geometrically due to the correlated effects of model inputs. This is a first step towards the development of an objective and partially automated experimental design.

We described the statistics of the outputs and contributing variables by performing a Monte Carlo simulation over the specialized design. We made global maps of the flows and investigated detailed characteristics. This allowed us to calculate the likelihood that different model realizations reasonably represented the 1955 Atenquique flow, given multiple pieces of field data regarding its characteristics. Depending on how it is looked at, the exercise provided useful information in either model selection or data inversion.

Our analysis concerned the mean values and uncertainty percentiles of quantities of interest. Moreover, the probabilistic setting allowed us to make inferences regarding the uncertainty affecting the data. We analyzed the contributing variables, which shed light on the different assumptions underlying the three models. In particular, the MC model is generally characterized by a lower speed over its feasible input space, when compared to the other models. The expected contribution of the internal friction is significant, meaning the internal shear of the material is important. In the PF model, the pressure force contribution related to the steepening of the flow front was locally significant and was sometimes even the dominant force. An initial, short-lasting wave of high speed related to the closest of the multiple sources was observed in both PF and VS. The uncertainty in height and speed was generally higher in VS than in PF, in spite of the higher dimensionality of the second.

We constructed partial solutions to the inverse problem, conditioning the specialized experimental design to be consistent with subsets of the observed data. We described the corresponding input sets and investigated their intersection. We found model selection to be inherently linked to the inversion problem. That is, the partial inverse problems enabled us to select models depending on the example characteristics and spatial location.

In particular, when attempting to correctly represent the deposits, MC performed well about 2 km upstream from the village, while VS did so in the village. In the evaluation of the maximum flow or runup depth, both PF and VS replicated the values 2 km upstream from Atenquique, but only VS replicated the values in the village. In terms of maximum flow speed, both PF and VS performed moderately well in the village, but only VS performed well 1 km downstream. These results are consistent with the evolution of flow rheology downstream in the vicinity of the village, from MC above the village to either PF or VS within and downstream from the village. If VS was dominant as the flow propagated downstream, it may reflect an evolution from inertial to macroviscous debris flow behavior near Atenquique, perhaps related to engulfment of the reservoir just upstream from the village.

Data sets are available from references and Supplement.
The fourth release of TITAN2D is available from

Many models based on different assumptions from those adopted in this study
are available in the literature and are either more complex

Based on the long history of studies in soil mechanics

We can summarize the MC rheology assumptions as follows:

basal friction based on a constant friction angle,

internal friction based on a constant friction angle,

Earth pressure coefficient formula depends on the Mohr circle (implicitly depends on the friction angles),

velocity-based curvature effects are included into the equations.

Under the assumption of symmetry of the stress tensor with respect to the

As a result, we can write down the source terms of Eq. (

The scaling properties for granular flows down rough inclined planes led to
the development of the Pouliquen–Forterre rheology (PF), assuming a variable
frictional behavior as a function of Froude number and flow depth

PF rheology assumptions can be summarized as follows:

Basal friction is based on an interpolation of two different friction angles, based on the flow regime and depth.

Internal friction is neglected.

Earth pressure coefficient is equal to one.

Normal stress is modified by a pressure force related to the flow thickness gradient.

Velocity-based curvature effects are included into the equations.

Two critical slope inclination angles are defined as functions of the flow
thickness, namely

An empirical friction law

The functions

In particular,

The depth-averaged Eq. (

The theoretical analysis of dense snow avalanches led to the VS rheology

We can summarize VS rheology assumptions as follows:

Basal friction is based on a constant coefficient, similarly to the MC rheology.

Internal friction is neglected.

Earth pressure coefficient is equal to one.

Additional turbulent friction is based on the local velocity by a quadratic expression.

Velocity-based curvature effects are included into the equations, following an alternative formulation.

The effect of the topographic local curvatures is addressed with terms
containing the local radii of curvature,

Therefore, the final source terms take the following form:

Let

Let

Moreover, we define the random contributions, an additional tool that
we use to compare the different force terms, following a less restrictive
approach than the dominance factors. They are obtained by dividing the force
terms by the dominant force

Let

In particular, for a particular location

The supplement related to this article is available online at:

AB and AP conceived the main conceptual ideas. AB implemented and performed the simulations and the statistical analysis. AB wrote the manuscript. AB, AP, and MIB interpreted the computational results. All authors discussed the results, commented on the manuscript, provided critical feedback, and gave final approval for publication.

The authors declare that they have no conflict of interest.

We would like to acknowledge the support of NSF awards 1339765, 1521855,
1621853, and 1821311, and of project FISR2017, Ministry of Education,
University, and Research (Italy). We would like to thank Byron Rupp for his
fundamental work on the localization and volume constraints of the Atenquique
debris flow

This paper was edited by Giovanni Macedonio and reviewed by two anonymous referees.