the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 21 Mar 2025
Implementing the Equations of Motion in the Energy Line Principle to Simulate the Runout Zones of Gravitational Natural Hazards
Abstract. The mitigation of the risk from gravitational natural hazards involves a variety of measures in engineering and management. The basis for any measures is the identification of hazard prone areas by considering past events and model simulations. Various models are available for the simulation of gravitational natural hazards but their application is often a trade off between simplicity and accuracy. A physical interpretation of the well established energy line principle allows to formulate the corresponding equations of motion and to introduce a model that is still simple but more accurate. The equations of motion are derived by the application of the Lagrange formalism for a friction block that slides on an inclined plane. Furthermore, a numerical algorithm based on the Euler method is set up to solve the equations of motion on a digital terrain model. This model is applied and compared to the energy line principle based on the equation of energy and to past events in two case studies for rockfall and landslide. The outcomes show that the formulation of the energy line principle with the equations of motion corresponds well to the formulation based on the equation of energy but allows a more differentiated simulation of the runout zone that reproduces better the past events. However, there are artifacts from the numerical solution of the equations of motion that require a deeper theoretical investigation and also uncertainties in the past events that must be worked through in a more elaborated empirical study. Nevertheless, the concept of this approach to enhance the performance of the energy line principle simply by another perspective to the same physical concept is thus proved.
- Preprint
(36486 KB) - Metadata XML
- BibTeX
- EndNote
Status: final response (author comments only)
-
RC1: 'Comment on nhess-2024-226', Stefan Hergarten, 15 Aug 2025
-
AC1: 'Reply on RC1', Elisa Marras, 04 Mar 2026
Dear Stefan Hergarten
Â
Thank you very much for these profound and critical comments. We appreciate your deep understanding of the matter, which encourages us to go deeper in the revised article too. In the submitted version we were a bit biased too and deliberately simplified the physics for also addressing the broader community.
Â
Your bias about this simple frictional rheology is understandable from a theoretical point of view, since it is basics in the field of physics. But also according to our knowledge, this rather simple approach has not been applied yet in the field of natural hazards. A crucial point to highlight better and also to implement into the model, is the difference between the energy line principle and the frictional rheology. This was only briefly mentioned in the submitted paper (line 133), but deserves to be brought into focus. Therefore we will now show more detailed, why the frictional rheology does not exactly match the energy line principle and how the correct Rayleigh function for a kinematic rheology must be defined to achieve the energy dissipation proportional to the trajectory length. Also we agree to publish the code in an online repository with a DOI, which will be cited in the revised manuscript.
Â
We share your concerns about the application of the Lagrange function because by neglecting the mixed velocity terms and the curvature terms, we made two crucial simplifications without reasoning them. The main reason for these simplifications was to avoid cumbersome equations in the method section and because the model is still working well regarding its purpose for preliminary runout analysis on regional scales. But of course this is not a scientific reasoning and we like to show the correct and complete version in the revised version. Whereas we show the complete equations in the mathematical derivation, we still use the simplified equations in the numerical algorithm because we want a code that is not slower than ELINE. The curvature terms require bilinear interpolations and implicit schemes for stable solutions, which would make the code an order of magnitude slower and therefore less suitable for preliminary assessments on regional scale.
Â
(1) This strong focus on the specific situation in Italy in the introduction section is indeed not appropriate for this paper. Therefore, we short this paragraph and focus more on the issues of preliminary runout analysis on regional scale.
Â
(2) Obviously we should differentiate between RAMMS::Rockfall and RAMMS::Debrisflow. Also it is correct and important to mention that RAMMS::Debrisflow can be applied only with Coulomb friction rheology and that RAMMS::Rockfall can be parameterized by predefined terrain classes. Nevertheless, both models requires assumptions about several parameters as well as the volume and the shape, which requires additional information that is usually not available on a regional scale.
Â
(3) Indeed the detour via the Salm model is not needed and there are more appropriate references for the Coulomb model. We will adapt this formulation and cite some work that is more appropriate.
Â
(4) The neglecting of the mixed velocity terms was only done to get a simpler formulation because by considering them, an additional matrix transformation is needed to isolate the acceleration components. Now we present the complete equations without any simplification in a compact tensor notation.
Â
(5) The particle position is defined as a point on the terrain model and therefore the difference quotients should be correct in this form, even though they are only first order. But we will state more clear that we assume a lumped mass model, in which the particle position is a point on the terrain model.
Â
(6) This equation is an empirical correction for the slope angle based on the cell size, which can be used to define the start cells in models such as ELINE and ROCKYFOR3D. In our opinion, it is not needed to write the equation so we will only properly describe it and cite the underlying concept.
Â
(7) This equation for the partition of the total sums of squares is indeed not correctly formulated because it mixes means and sums. Since the principle is pretty elementary, we do not even see the need to write the equation but only to properly describe it and cite the underlying concept.
Â
(8) The volume and the density are mandatory parameters for ELINE, in which they only influence the energy but not the reach. We mentioned them for the reproducibility because they must be defined, but we will describe more clear that they do not matter for the runout analysis that is only based on the frequency grid.
Â
(9) This aspect needs indeed a more careful discussion. But according to the physical considerations, which we will show in more details, the realized tangents of the energy line angles should tend to be smaller than the parameterized Coulomb friction coefficient. Since we now anyway implement the adapted Rayleigh function that guarantees that the energy dissipation is equal to the product of the tangent of the energy line angle and the path length, the discussion must be adapted too.
Â
(10) It can actually happen from several reasons that BLOCKSLIDE reaches locations that are not reached by ELINE. One reason can be the lateral spread in complex terrain, which is in ELINE simply defined by a constant cone width angle independent of the topography but strongly influenced by the topography in BLOCKSLIDE. Another reason can be the variability of the realized energy line angles in BLOCKSLIDE, so that some trajectories can reach locations that are by geometrical definition not reached from ELINE. But we see the need to implement these considerations in the discussion.
Â
(11) The normalization of the areal values is based on the union area. Indeed, this can currently only be guessed from the sums always equal to 100 %. Therefore, we will also describe the definition in the caption of the table.
Â
(12) We agree that this paragraph is indeed too much in a salesman style. Therefore we will be more critical and modest by only mentioning, that this approach allows to improve the energy line principle for preliminary runout analysis on regional scales but also mention that it is still a tread-off between simplicity and accuracy.
Â
In summary, the main changes we suggest are a more profound theory and a published code. We will derive the equations of motion without simplifications in the Lagrange function and also introduce the Rayleigh function not only for the Coulomb friction rheology but also for the kinematic rheology to match exactly the definition of energy line principle regarding the energy dissipation. A revised manuscript with these changes, and also these from the second answer, is ready.
Â
Sincerely Yours
Â
Elisa Marras, Dominik May, Luuk Dorren, Filippo Giadrossich
Citation: https://doi.org/10.5194/nhess-2024-226-AC1
-
AC1: 'Reply on RC1', Elisa Marras, 04 Mar 2026
-
RC2: 'Comment on nhess-2024-226', Anonymous Referee #2, 22 Jan 2026
The manuscript presents a model aimed at simulating the runout zones of gravitational natural hazards. The paper is generally well prepared and falls within the broad scope of topics covered by NHESS. From a scientific standpoint, I agree with most of the comments raised by Referee #1 and, in particular, I encourage the authors to reflect on the added value that could be provided by adopting a more open data and software policy. For these reasons, I recommend major revisions.
In addition to the comments above, I have the following specific remarks:
- Equation formatting. The mathematical notation should be improved. In particular, only round brackets are currently used, whereas the consistent use of round, square, and curly brackets would significantly enhance clarity and readability.
- Lines 190–195. As I understand it, a predefined slope range is adopted to identify unstable source areas (i.e. the potentially triggered mass). This represents a strong simplification. It would be useful to discuss how this assumption affects the results, especially with respect to the ability of the model to reproduce past landslide events. For instance, the authors could consider discussing the potential benefits of coupling runout models with landslide triggering models.
- Lines 239–242. Equation (22) could be reformulated within a ROC analysis framework. This would likely make the formulation more intuitive and easier to interpret for the reader.
Citation: https://doi.org/10.5194/nhess-2024-226-RC2 -
AC2: 'Reply on RC2', Elisa Marras, 04 Mar 2026
Dear Anonymous Referee
Â
Thank you very much for these detailed and critical comments. We appreciate your profound understanding of the matter, which encourages us to critically reconsider the revised article too. Due to your explicit agreement with the comments of the first referee, we refer to the first answer for many points and will only address your specific points here.
Â
We see the added value to publish the code in an online repository with a DOI, which will be cited in the revised manuscript. Indeed, this is more convenient for interested people than just offer to share on request. Furthermore the publishing allows full transparency, whereas the part in the methods can be reduced because the cited code contains all the details of the numerical algorithm.
Â
(1) We agree that there is some room for improvement regarding the equation formatting. In the revised manuscript, only the relevant equations for the derivation of the equations of motion are presented, for which we apply now a more compact tensor notation and an improved formatting.
Â
(2) Your understanding is correct that the start cells are defined by a simple slope threshold criteria and we agree that this is a massive simplification. Since this approach is common for preliminary analyses and the scope of this paper is runout and not failure, we consider this approach as sufficient for this purpose. But we address the limitations of this approach more detailed in the discussion and also the possibility to combine the presented runout model with more sophisticated failure models.
Â
(3) We were considering the use a ROC framework for the analysis, but the missing definition of the true negative and a probability threshold in this context prevents such an approach. Therefore, we keep the simple analysis based on the confusion matrix but remove the equation because it could be misleading for this simple approach.
Â
In summary, the main changes we suggest, beside of these in the first answer, are the more detailed discussion about the limitation of the slope threshold criteria. The modeling of the failure zone is indeed crucial, although the analysis in this work is based on relative comparisons of the runout zone that should not be affected from these initial conditions. But for an absolute comparison, this is of course crucial. A revised manuscript with these changes, and also these from the first answer, is ready.
Â
Sincerely Yours
Â
Elisa Marras, Dominik May, Luuk Dorren, Filippo Giadrossich
Citation: https://doi.org/10.5194/nhess-2024-226-AC2
Viewed
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 752 | 157 | 34 | 943 | 46 | 53 |
- HTML: 752
- PDF: 157
- XML: 34
- Total: 943
- BibTeX: 46
- EndNote: 53
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
The manuscript presents the formulation and the implementation of a lumped-mass model for rapid mass movements. The model assumes Coulomb friction with a given coefficient of friction. The main physical approximation is that the normal force is only the normal component of gravity (so the static component), while the dynamic component arising from the curvature of the surface is neglected (centrifugal force). As an advance over the basic energy-line principle, the approach presented here predicts not only which points of the topography can be reached energetically, but also dynamically.
Maybe I am a bit biased because I have used basically the same approach (with and without dynamic effects on friction) as a homework assignment in a (admittedly quite advanced) M.Sc. class on mass movements. To my knowledge, the 2-D approach presented here has not been published in a paper so far. However, I would never thought of publishing it because I always considered it a students' exercise. In turn, it might make sense to publish it if it was available as an easy-to-use tool, either as a lightweight standalone software or as a GIS plugin. However, rather not in the old-fashioned mode "Data will be made available on request."
Before going into detail, I am not convinced about the application of the Lagrange formalism (Eqs. 8 and 9) here. I think that dL/dx should also contain the x- and y- derivatives of dh/dx and dh/dy. Then we get the Hessian matrix of h (d^2h/dx^2, d^2h/dxdy, d^2h/dy^2) into th equations and everything becomes a bit messy. To illustrate that it cannot be as simple as in Eqs. 10 and 11, let us assume g = 0, so a particle moving along the surface without driving forces and friction. Eqs. 10 and 11 would predict d^2x/dt^2 = d^2y/dt^2 = 0 then, which means that the horizontal velocity components remain constant. This would, however, violate energy conservation on a curved surface. The problem might be not crucial for velocity-dependent friction (e.g. Hergarten and Robl, 2015, doi 10.5194/nhess-15-671-2015), but I think it is critical for Coulomb friction. Fixing it with the help of the Hessian matrix would cost much of the simplicity achieved by considering only x(t) and y(t) explicitly.
In contrast to this central point, the following points are mainly related to the presentation of the theory and the results.
(1) The introduction focuses a bit too much on the situation in Italy and tends to overrate the achievements of this study.
(2) If we use models such as RAMMS (or one to the numerous similar models), we could also use only Coulomb friction. Then the coefficient of friction would be the only parameter, similar to the approach proposed here. Why is it challenging to parameterize and costly to execute?
(3) Line 62: The reference Salm (1993) does not fit well here. This paper mainly explains the "turbulent" friction term. There are numerous considerations of lumped-mass models in the literature (mainly 1-D) with different frcition laws.
(4) Neglecting the mixed terms in (dz/dt)^2 (Eq. 5): As far as I can see, this approximation could cause a factor of up to 2 in (dz/dt)^2, which is not very small.
(5) How are the difference quotients treated when the particle position is not a point of the terrain model?
(6) Eq. 20 is not clear to me. Why does it make sense to use a relation from ROCKYFOR3D, which uses a different physical basis?
(7) Eq. 21 does not make sense to me. Should there be mean values instead of the sums?
(8) Lines 230-31: Where do volume and density occur?
(9) Lines 294-97: This needs much more discussion. The results should theoretically agree perfectly if the movement follows the steepest descent (talweg line). According to this argument, it is not surprising that the numerically obtained angles are bigger than atan(mu). However, the missing term in the Lagrange formalism may also play a part here.
(10) Table 4: BLOCKSLIDE without ELINE (which should physically not occur) seems to be quite large for the Como example.
(11) Table 5: Takes some time to understand how the data are normalized.
(12) Lines 431-438: I have no doubt that the approach is somewhat better than the original energy-line principle, but this paragraph overrates its potential in my opinion.
Overall, I feel that this study might be publishable if the problem with the curvature is fixed and the software becomes publicly available, either as a simple standalone tool or as a GIS component. In any case, however, it would need a quite extensive revision.
Best regards,
Stefan Hergarten
Â
Â
Â