| 30 Jul 2026
A physics-based morphometric model to explain the emergence of landslide rupture geometry and to identify hillslope transience
Abstract. Like earthquakes or other rupture processes, landslide tends to universally follow some specific geometrical scaling laws and size distributions. Numerical models have attempted to explain the emergence of these universal laws, putting forward mainly the role of hillslope strength and shape. The main difficulty that models are facing is that the geometry of the surface rupture of a landslide, or of large population of landslides, is not a priori known and must often be guessed using assumptions about the rupture shape, depth, and angle. Here, we develop an analytical solution that defines an optimal rupture depth, below each point of a topography, associated with an optimal rupture angle. This defines a rupture surface that daylights downslope the rupture point. Landslides are then defined as clusters of unstable neighbours sharing the same daylight point. The geometry of the surface rupture of the landslide is then simply defined by the optimal rupture depth at each point belonging to the landslide. Applying this model, coined scaLr (slope curvature analysis for Landslide rupture) to the Central Range of Taiwan, we show it can produce landslide area-volume and area-depth relationships with power-law exponents roughly consistent with observed ones. The distribution of landslide length-to-width ratio is also consistent with the range of observed values and exhibit a distribution similar to an Inverse-Gamma. The distribution of landslide area exhibits a power-law decay for large landslides and no clear rollover is obtained despite a break in slope below a cutoff area. However, the power-law decay tends to be in the lower part of the range of observed values. Amalgamating modelled landslides does however produce more acceptable exponents, which suggests that the difference between modelled and observed landslides might partly results from inherent amalgamation in landslide catalogues. Last, we show that the optimal rupture depth also represents a morphometric index to assess landscape transience along hillslopes. In Taiwan, we tend to observe deeper expected ruptures along streams locally characterized by high river steepness as well on the aggressive side of migrating divides, as identified by the χ index. The optimal rupture depth therefore offers complementary information to fluvial metrics, such as river steepness or the χ index, and to hillslope metrics such as Gilbert’s metrics.
This is an exciting manuscript that tackles the difficult challenge of calculating landslide rupture depths at scale without prior knowledge of landslide geometries using a new morphometric model. The model calculates the relative stability for each pixel within a DEM and identifies a “daylight” point, indicative of the most unstable downslope rupture location. The model then clusters unstable pixels and considered neighbouring pixels with a shared “daylight” point to be a single landslide. The geometric properties (e.g., area, volume, depth) of the modelled landslides are then compared to traditional scaling laws and a case study for Taiwan. The authors find a more realistic fit between modelled and observed scaling relationships when amalgamating several of the individual landslides. The authors argue that this amalgamation is inherent in the manually mapped landslide inventories used for comparison. Finally, the authors argue that the modelled rupture depths can be used as an additional metric to identify landscape transience, with deeper ruptures corresponding to high channel steepness and the asymmetry of migrating divides.
I think this is a compelling and well-written manuscript that presents an exciting advance in our ability to anticipate landslide rupture depth at scale, with clear applications for landscape evolution and hazard. I would recommend that the authors clarify some aspects of their approach prior to publication, since the underlying model and calculations of Hmin, area, and depth are important to assess the comparisons between the modelled output to observed relationships. I also think it would be useful to see an overlay of the landslides triggered in Typhoon Morakot on top of the landslide geometries mapped in Fig. 5 to qualitatively assess the mapped landslide geometries. I appreciate that this is not the primary focus of the model but think it would help to support the approach since most landslide shapes do appear reasonable. Finally, I think an additional figure alongside Fig. 10 could better demonstrate the benefit of this approach for landscape transience. With some clarifications, this manuscript will certainly be of interest to the ESurf community.
General clarifications
Comparison with manual landslide inventory
I appreciate that this approach intends to identify all/most potential failures by reducing the cohesion and therefore does not represent a specific event. However, showing the manually mapped inventory from Typhoon Morakot used to compare the scaling relationships would be very helpful to (1) visualise the importance of amalgamation – the light green landslide in 5b2 looks like a deep-seated landslide that would remain persistent for several years if triggered, it would be interesting to see if it was; (2) demonstrate the proportion of the landscape that did fail in a recent event relative to the proportion that could fail; and (3) explore how the spatial distribution of landslides triggered compares particularly when thinking about landscape transience.
Links to landscape transience
I think the arguments presented by the authors seem reasonable, however I wonder if there is a clearer way to present the complex relationships in Fig. 10 to provide support for the strong conclusion in the abstract (lines 25 to 26).
A simple adjustment to Fig. 10 would be to add labels to highlight key features, such as the central belt of high channel steepness and the higher values of hmin.
Could the authors calculate the average or total depth of material hmin “draining” into each channel segment and plot the longitudinal profile or a scatter plot for the relationship between hmin and steepness. The asymmetric relationship χ is more challenging to represent but perhaps a ratio could be used to show the large differences either side of an actively migrating divide. The relationship for χ would also benefit from annotations on Fig. 10.
In Fig. 10, a lot of the landscape appears to have an assigned hmin – what proportion of the landscape does the model expect to fail? This could be added as another panel to Fig. 9.
Minor Comments
Unsure about the term emergence in the title.
Introduction
Lines 34-35: This sentence could be clearer - how are smaller landslides are affected by power-law scaling?
Line 38: what is meant by geometrical scaling here?
Lines 42 – 43: This comment feels important for framing the manuscript but could be clearer.
Line 48: “From these models it has been suggested…”? I would expect each of the numerical points to be referenced.
Lines 74: define what is meant by geometry (e.g., area)
Line 75: expand on how it is not prescribed by modelling assumptions.
line 80: new model is established
line 80: space between show and the
lines 80 to 82: sentence could be clearer.
Methods
The derivation of equations (1) to (6) and those in Appendix A seem reasonable to me, though hopefully another reviewer can critique the approach here more thoroughly.
I have primarily commented on where descriptions of the approach could be clearer.
Line 95: It would be helpful here to summarise the approach of Alvioli et al. 2014 and explicitly state the objectives of this approach.
Line 98: It would be helpful to clarify use of the term “daylight”
Lines 120 to 125: It may be helpful to explain that every point in a DEM is considered to be a possible rupture point earlier in the text – if this is the case.
Line 129 to 130: I found this sentence to be confusing.
Figure 1 – I really like this schematic. Apologies if this is a misunderstanding, but should rupture lines then also be drawn for points j, k and l to point m to produce a landslide?
Figure 2: This figure is very effective. Would these two landslides be combined based on the amalgamation approach here? If so, it would be interesting to add this to the caption. Is there a specific reason why only one rupture point is shown in panel A? Another interesting panel would be the Fmin scores for each cell.
Line 173: “emergence of landslide” a bit confusing.
Line 175: circle
Section 2.4 to 2.5: It would be helpful to add an additional section that clarifies how landslide area, volume and depth are calculated (e.g., is depth maximum or mean Hmin?, is the daylight point (900 m2) considered in the area calculation?) since these are important to understand the scaling relationships proposed.
Line 199: “To test the model abilities, we apply it to a DEM…”
Results
Figure 4: For the Fmin values over 1, does this mean a landslide will not happen within certain parts of the hillslope at this cohesion (100 kPa) or friction angle (40 degrees)? And that the cells would not be mapped as a landslide? I think this is covered in the text (such as lines 221) but wonder if it could be made clearer in the figure too – a dotted or dashed line for non-landslides for example?
Line 236: I appreciate that you are not modelling an individual event, but could you compare these landslide areal densities of 9% to the expected densities from different events? They seem high – which is not a reflection of performance necessarily but would highlight how the model looks for all potential unstable slopes.
Figure 5: Add to the caption that the numbers correspond to Fig. 3
Line 255 to 257: I’m a bit confused by this sentence; would it be possible to explain how the amalgamation affects the volume? This step may be worth clarifying in the methods.
Line 299: What is considered a large landslide here?
Figure 8: I wondered if the pdf units should be m-1 and m-2 respectively? If it’s not too messy, it could be interesting to add the underlying landslide area data to the plots as a histogram showing count for example on a secondary axis to see how the populations compare.
Discussion
Figure 9: Add more points to the colour bar of 9C.
Line 317: double check sentence, maybe since instead of if?
Section 4.1:
The assumptions and explanations of these seem reasonable to me but hopefully another reviewer can provide greater feedback.
I found the use of first, second, third to separate points twice in the same paragraph a bit confusing. Could these be separate paragraphs?
I think a simple overlay of the landslides triggered in Typhoon Morakot from the author’s paper in 2020 would be insightful and provide a first-order assessment for the limitation stated in lines 341 to 343.
Section 4.2
Additional points that it may be beneficial to discuss are
Section 4.4
I wonder if the authors can find a way to display these complex relationships more clearly, see above comments. This could include a series of longitudinal profiles with the asymmetry between hmin on the aggressor and victim hillslopes shown?
The relationships described here also rely on accurate lateral landslide predictions and so I think showing the manual inventory from a previous event will be helpful.
Could you overlay results/show the area with the landslides mapped by Chen et al. 2023?
It is great that the authors have made the code available.