Knowing the source and runout of debris flows can help in planning
strategies aimed at mitigating these hazards. Our research in this paper
focuses on developing a novel approach for optimizing runout models for
regional susceptibility modelling, with a case study in the upper Maipo
River basin in the Andes of Santiago, Chile. We propose a two-stage
optimization approach for automatically selecting parameters for estimating
runout path and distance. This approach optimizes the random-walk and Perla
et al.'s (PCM) two-parameter friction model components of the open-source
Gravitational Process Path (GPP) modelling framework. To validate model
performance, we assess the spatial transferability of the optimized runout
model using spatial cross-validation, including exploring the model's
sensitivity to sample size. We also present diagnostic tools for visualizing uncertainties in parameter selection and model performance. Although there was considerable variation in optimal parameters for individual events, we found our runout modelling approach performed well at regional prediction of potential runout areas. We also found that although a relatively small sample size was sufficient to achieve generally good runout modelling performance, larger samples sizes (i.e.

Knowledge of where debris flows are initiated and how far they travel is crucial for assessing their impact over large regions (Aleotti and Chowdhury, 1999; van Westen et al., 2006). Commonly, debris-flow runout modelling for large areas is performed by first delineating source areas and then applying empirical–statistical or process-based numerical methods for simulating the runout characteristics (Blahut et al., 2010a; Horton et al., 2013; Mergili et al., 2019). There is a wide selection of heuristic, statistical and machine-learning methods suitable for predicting source areas in large regions (Chung and Fabbri, 1999; Carrara et al., 1999; Brenning, 2005; Goetz et al., 2015b; Lombardo et al., 2018). There are also many empirical–statistical and numerical methods available to model runout patterns; see McDougall (2017) for an overview.

Not all runout methods are suitable for application to large areas. Many of the physically based methods require event-specific geotechnical and rheological parameters, such as material composition (e.g. bulk density and source depths) and flow characteristics (e.g. flow discharge rates). These parameters, such as debris-flow volume, can be extremely difficult to obtain for large areas, let alone single unobserved events (Marchi and D'Agostino, 2004; Dong et al., 2009). Consequently, runout modelling at larger scales has been progressing towards applying simplified conceptual models to simulate debris-flow patterns across different environmental conditions. These models combine spreading algorithms to control the runout path with empirical–statistical or numerical friction models to simulate likely runout paths and distances (Guthrie et al., 2008; Horton et al., 2013; Wichmann, 2017; Mergili et al., 2019). Many of the spreading algorithms, including multiple flow direction models (Holmgren, 1994), cellular automata (Guthrie et al., 2008) and random walk (Gamma, 2000; Mergili et al., 2015), simulate runout paths using only topographic data.

Calibration of model parameters continues to be one of the main challenges in runout modelling for single events and over large areas (Hungr, 1995; van Westen et al., 2006; Schraml et al., 2015; McDougall, 2017; Mergili et al., 2019). The objective of model calibration is to determine parameter values that best capture main debris-flow characteristics, such as runout distance, velocity and distribution of deposits (Hungr, 1995; McDougall, 2017). Approaches for model calibration include adjusting parameters based on visual inspection (i.e. trial and error; Hungr, 1995; Mergili et al., 2012), expert knowledge (Horton et al., 2013), posterior analysis (Mergili et al., 2019; Aaron et al., 2019) and optimization algorithms that aim to minimize a cost function, i.e. a quantitative measure of runout model performance. Some measures of performance include estimates of the intersection over union (Galas et al., 2007), area under the receiver-operating characteristic curve (AUROC; Cepeda et al., 2010; Mergili et al., 2015) and depth error (Aaron et al., 2019) of simulated and observed debris flows. Since most of these calibration approaches are for single observed events, they rarely consider how transferable tuned parameter sets are from local to regional applications.

Assessing spatial transferability is essential for testing the assumption a trained model based on a sample of events captures the generic debris-flow characteristics across a region (Fabbri et al., 2003). The distribution of training data and the sample size can have a strong influence on the model calibration and performance of regional models (Heckmann et al., 2014; Petschko et al., 2014; Rudy et al., 2016). For spatially distributed models, spatial transferability can be assessed by exploring model parameter selection and performance under different spatial partitioning scenarios of training and test data (Wenger and Olden, 2012; Brenning, 2012; Schratz et al., 2019; Mergili et al., 2019). Although spatial transferability has been well explored for regional landslide susceptibility models (Brenning, 2005; Lombardo et al., 2014; Petschko et al., 2014; Goetz et al., 2015b; Cama et al., 2017; Knevels et al., 2020), such analysis is not common for regional runout modelling.

In this study, we developed an optimization procedure for process-based models applied for regional simulation of debris-flow runout patterns. The performance evaluation focuses on the spatial transferability and sensitivity to sample size of an optimized regional debris-flow runout model. We achieve this by utilizing the open-source statistical R software to add optimization and evaluation functionality to the open-source Gravitational Process Path (GPP) modelling framework (Wichmann, 2017). Additionally, this study demonstrates the use of spatial cross-validation and visualization techniques to diagnose uncertainties in the prediction of source areas, runout paths and runout distances, including the sensitivity of optimized parameter selection. The aim of this research is to contribute to improving the development of quantitative techniques for runout model calibration and uncertainty estimation (McDougall, 2017). This is especially important in large and inaccessible mountainous areas where various types of mass movements pose unique challenges to the safety of the local population, the integrity of transportation infrastructure and the reliability of drinking water supplies.

Our study area is the upper Maipo River basin (33

High-intensity rainfall (Sepúlveda et al., 2015), rapid snowmelt (Moreiras et al., 2012) and seismic activity (Serey et al., 2019) are the main triggers of debris flows in this region. They occur in steep gullies and talus slopes consisting of gravel, small boulders and a fine sandy–silty matrix. Much of this material is from weathered volcanic and sedimentary rocks of the Abanico and Farellones formations in the western Main Cordillera (Sepúlveda et al., 2006). A typical runout track will cut through previously formed debris-flow channels and alluvial fans, resulting in new erosion and deposition paths (Sepúlveda et al., 2015). Rainfall-triggered runout distances in this area have been observed up to 5.5 km, and the thickness of deposits varies from 1 to 2 m in deposition areas (Sepúlveda et al., 2015).

Runout characteristics of the debris-flow inventory.

Note: IQR is the interquartile range.

Map providing an overview of the debris-flow polygons and source points mapped in the upper Maipo River basin.

Debris-flow polygons and source points were mapped based on
photointerpretation of high-spatial-resolution (0.50 m) satellite imagery
(2000 to 2019) from CNES/Airbus and Maxar Technologies available through
Google Earth Pro software, field observations, reviewed news articles and
the compilation of data collected by public authorities. Each mapped polygon
represents a debris-flow track that includes source, runout and deposition
area. In total, 541 source points and 521 debris-flow polygons were mapped
(Table 1). Manually mapping all debris flows across the upper Maipo
basin is a challenging task due to its large geographical extent and its
high abundance of mass movements. Therefore, a mapping strategy was employed
that divided the basin into 58 sub-drainage basins (5439

Potential debris-flow source areas were spatially predicted using a
generalized additive model (GAM). In general, GAMs demonstrate good
performance for susceptibility modelling compared to other commonly used
physically based and machine-learning techniques (Goetz et al., 2011, 2015b). To improve model generality and avoid overfitting, the GAM
smoothing spline parameters were allowed a maximum five effective degrees of
freedom (Wenger and Olden, 2012; Goetz et al., 2015a). The training and test
data were based on the common

The predictor variables of source areas included hillslope angle, elevation,
catchment area, plan curvature and distance to faults. These predictor
variables generally have a high importance for modelling debris-flow
initiation susceptibility as observed in previous works (Blahut et al.,
2010b; Goetz et al., 2015b; Angillieri, 2020). The publicly available ALOS
PALSAR radiometrically terrain corrected (RTC) high-resolution (12.5 m)
digital elevation model (DEM; ASF DAAC, 2011) was used to derive terrain attributes. Before deriving the terrain attributes, mesh denoising was applied to the DEM to mitigate the propagation of artefacts such as
high-frequency noise (Brock et al., 2020) in the prediction of source areas
(Sun et al., 2007; Stevenson et al., 2010). We used the implementation of this algorithm in the System for Automated Geoscientific Analyses geographic information system (SAGA-GIS). After denoising, an algorithm to fill sinks (Planchon and Darboux, 2002) was applied to the DEM, and the terrain attributes were processed. Distance to faults was calculated as the Euclidean distance from the fault lines (Servicio Nacional de Geología y Minería de Chile 2003; scale

The performance of the source-area prediction model was assessed using
repeated

The GPP (Wichmann, 2017) model was used to regionally model runout. The GPP model is an open-source framework in the SAGA-GIS software that provides users with various model components to simulate runout path, distance, velocity and deposition of material of mass movements (e.g. snow avalanches, rockfalls and debris flows). Due to the extent and remoteness of the study area, we focus on modelling the likely spatial patterns of runout. That is, we are not modelling flow velocity and depth.

Runout path was modelled using the random-walk process path component of the
GPP model (Gamma, 2000). It is a common approach for debris-flow runout path
modelling at medium scales (Mergili et al., 2012; Heckmann and Schwanghart,
2013; Mergili et al., 2015). Random-walk models the potential path of runout
by iteratively simulating (via Monte Carlo simulation) the downslope
movement of debris flows originating from source-area grid cells. These
simulations result in a grid with runout frequencies that indicate how many
times a grid cell is traversed: this is a cumulative frequency based on
simulations from all source areas. There are three parameters that need to
be calibrated to obtain a desired runout path: (1) a slope threshold
(

Runout distance was constrained using the two-parameter friction model (PCM;
Perla et al., 1980) component of the GPP model. The PCM model, which is also
a component of the Flow-R model (Horton et al., 2013), has also been used
for modelling debris-flow behaviour at medium scales (Mergili et al., 2012;
Heckmann and Schwanghart, 2013; Mergili et al., 2015). It is a
centre-of-mass model where motion is mainly controlled by (1) the sliding
friction coefficient

For regionally applying the runout model, we needed to determine the combination of model parameters that result in the best match to our debris-flow inventory. Determining optimal parameters was based on two criteria: (1) the ability of the model to capture our observed runout paths and (2) its ability to match the observed runout distances. Therefore, we performed this optimization task using a two-stage approach that first optimizes the random-walk model and then the PCM model parameters.

A random sample of 100 debris-flow tracks and corresponding source points was used for optimizing the runout models. This sample of the inventory was chosen to facilitate quality control and reduce the computational complexity of the optimization. Source areas were determined by buffering each source point by 50 m and masking away the buffered area that exceeds the runout trimline; this ensures the source area is contained within a mapped debris-flow polygon. A sink-filled version of the original DEM was used for the runout modelling.

For each model component, an exhaustive grid search in parameter space was
used to find parameter sets that achieve optimal model performance across
all sampled debris flows. The search ranges were similar to Wichmann's
(2017) suggested parameter limits for debris flows (see Table 2 for
value ranges). We additionally tested the use of a spatially varying sliding
friction coefficient. The value for this spatially varying sliding friction
coefficient

Runout model grid-search optimization setup and results. Optimization performance was assessed using spatial cross-validation (CV).

Note: IQR is the interquartile range;

The AUROC was used as a performance measure for the random-walk model. The
receiver-operating characteristic (ROC) is a plot of the true positive rate
versus the false positive rate. AUROC values range from 0.5 (random
discrimination between classes) to 1.0 (a perfect classifier; Zweig and
Campbell, 1993). Model performance was rated higher if the random-walk model
contained observed debris-flow tracks within its simulated paths. After
optimizing the random-walk model, we fixed these parameters for the PCM
model and optimized the

Runout distance was measured in terms of horizontal length of the debris-flow track. This distance was measured as the length of a minimum area bounding box containing the observed debris-flow track (Fig. 2; Niculiţa, 2016; Taylor et al., 2018). Estimated debris-flow tracks were defined as grid cells with values greater than a median runout frequency (Fig. 2). In this case, the median value represents the most typical simulated debris-flow track. It also provides a conservative estimate of runout distance, which helps mitigate the chance that the optimized model regionally underestimates runout distances.

Illustration of runout-distance optimization of the sliding
friction coefficient

In addition to determining a global optimal parameter set, the best-performing parameters for individual debris-flow events were also explored. We similarly applied a grid search to each event and determined optimal parameter sets based on the performance (AUROC and relative error) of each runout model component.

An example realization of random partitions based on

Based on our random sample of 100 debris-flow tracks, we assessed the transferability of optimized runout model (random-walk and PCM) parameters by performing 5-fold spatial cross-validation with 1000 repetitions (Fig. 3). This approach allows us to explore the sensitivity of grid-search optimized parameter combinations to spatial variation in training and test data. To do this, we observed the frequency of variations in optimized parameter combinations within all cross-validated iterations. Optimal parameter combinations that occurred more frequently were considered to have a higher degree of transferability, thus being considered more reliable for application to the entire study area.

We also assessed if there were any spatial patterns in the optimized
performance for each model component. That is, were there any spatial trends
in model performance that may indicate our model is locally overfitting? We
explored such spatial trends by mapping the distribution of individual
debris-flow runout model performance based on the optimized parameters.
Additionally, we were concerned if the optimized parameters had a stronger
fit to debris flows of a certain magnitude or initiating conditions.
Therefore, the potential to overfit to certain debris-flow characteristics
was assessed by determining Spearman's rank correlation (

We explored how runout model parameter selection, performance and robustness were affected by the number of debris flows used for optimization. Spatial cross-validation was applied to data sets of varying training sample sizes using the random sample of 100 debris flows used for model optimization. To ensure a fair comparison, the size of the test data for each cross-validation iteration was set to 20 debris flows, which is the maximum test sample size when performing 5-fold spatial cross-validation with a sample of 100 debris flows. We tested training samples sizes from 10 to 80. Model performance for each runout modelling component was summarized using the median and interquartile range (IQR) (AUROC and relative error). The optimal parameter sets for a given sample size were determined as the parameter combinations that were most frequent.

For regionally applying the runout model for susceptibility mapping or exposure analysis, a dichotomous classification of the predicted source areas is required to define the grid cells where simulated debris flows initiate. In the case of basing the source areas on a susceptibility model, a suitable threshold of the prediction values needs to be selected. In this study, we determine a suitable prediction threshold to classify source areas by searching for the threshold that results in the best-performing runout model for the entire area. Using the optimized model parameters, we tested runout models based on source areas that were delineated using prediction thresholds from 0.5 to 0.95 with a step of 0.05. The performance of each these models was measured using the AUROC. The AUROC was calculated using a sample of 1000 debris-flow runout locations and 1000 locations outside of the debris-flow polygons. The source-area prediction threshold resulting in the highest AUROC values was selected for regionally computing a debris-flow runout map for our study area.

The methods for runout modelling optimization, validation and visualization of the source-area prediction and runout modelling were implemented using the open-source statistical software R (ver. 3.6.2; R Core Team, 2019) and SAGA GIS (version 6.1; Conrad et al., 2015) with its GPP model tool (Wichmann, 2017). Coupling SAGA GIS with R was done using a combination of the RSAGA (Brenning et al., 2018) and Rsagacmd (Pawley, 2019) packages. The GAM was implemented using the mgcv package (Wood, 2011). General handling of spatial data in R used the sf (Pebesma, 2018), sp (Pebesma and Bivand, 2005), rgeos (Bivand and Rundel, 2019), rgdal (Bivand et al., 2019) and raster (Hijmans, 2020) packages; spatial cross-validation was applied using the sperrorest (Brenning, 2012) and ROCR (Sing et al., 2005) packages. Parallelization of the optimization and validation procedure used foreach (Microsoft and Weston, 2020). Visualization was done using R's ggplot2 (Wickham, 2009) and metR (Campitelli, 2020) packages and ESRI's ArcMap (ver. 10.5).

The overall performance of the source-area prediction based on the GAM was good with a spatially cross-validated median AUROC of 0.80 and an IQR of 0.001. We found the source-area prediction map was also geomorphologically plausible. Locations most likely to be source areas were within steep terrain associated with channels, gullies and scree slopes. Shallow-flat terrain and areas along ridgelines were modelled as least likely to be source areas (Fig. 4). This geomorphological knowledge was also expressed in the plots of the GAM spline transformations (Fig. 5). Relatively steep terrain, slightly concave plan curvature and areas near faults were modelled as more likely being source areas.

Map of the debris-flow source-area prediction based on a GAM.

Transformation of predictor variables in the generalized additive
model, where the

The parameter optimization produced runout models with a good spatially
cross-validated performance. The optimal parameters for the runout-path
model were a slope threshold of 40

By visualizing the runout-distance optimization results across grid search
space, we can observe model performance and sensitivity to different
parameter combinations. In this case, we observed only slight model
performance differences for

Density contour plots of parameter optimization of sliding
friction coefficient

Exploring the optimal combination of parameter values using spatial
cross-validation provides insights into performance reliability of the
optimized model. Given different spatial combinations of testing and
training data, we found that the optimal combinations of parameters were
associated with high performance values (

Model performance and frequency of optimal parameters for the
runout path

The best threshold for delineating source areas from the GAM prediction for
runout modelling was 0.7, which results in runout affecting 22 % of the
study area. This threshold had the peak AUROC value of 0.83. The performance
of the runout model drastically decreased with thresholds

Performance of debris-flow runout model for different source-area
thresholds

Map of optimized debris-flow runout model based on global parameters and source-area threshold of 0.70.

We observed no clear spatial pattern of individual debris-flow performance
of the runout model components (Fig. 10), which is evidence that
there was no local overfitting to a particular region of the study area. The
distribution of individual AUROC performance of the runout path model was
generally high for most of the region, showing that optimization of the path
model performs well across the study area. One debris flow was not captured
within the runout path model (

Maps of the performance of the runout path

Histogram of performance of the runout path

The runout-distance model had generally more spatial heterogeneity in
performance (Fig. 10), but again no clear spatial patterns in the
distribution of relative errors were observed. We investigated the
debris flows that did not perform well for modelling runout distance
(relative error

Overall, the regionally optimized runout modelled distances fit well with
our mapped observations (Fig. 11). It tended to slightly
overestimate debris-flow travel lengths with a median runout-distance error
of 16.6 m (Fig. 11). This error is just slightly over a single
grid cell size of the DEM. The highest runout-distance errors were due to
misclassified debris flows, as previously mentioned. The optimization of the
runout model avoided overfitting to debris-flow tracks of a certain
magnitude and general terrain conditions. That is, we did not observe a
strong correlation between runout-distance performance to length of observed
debris flow (

The optimal parameters for individual debris flows were also computed for
general comparison to the regional model. We found that the optimized runout
model parameters were highly variable for individual debris flows
(Fig. 12). The relative errors were low (

Performance and frequency of runout path

Most individual events optimized runout paths with parameter sets leading to
high lateral spreading. The optimal-path parameters for most of the
individual events had a 40

There was no clear spatial pattern in optimal

Map of runout-distance model optimal parameters determined for individual debris flows.

Runout model performance and variation tends to improve slightly when using larger sample sizes (Fig. 14). Large sample sizes also resulted in more consistently selected model parameters across spatial-cross-validation iterations (Fig. 15). The larger spread across grid search space of the relative frequency of optimal parameter combinations for smaller sample sizes illustrates that we may be less likely to find the best model parameters that generalize well for a large region (Fig. 15). In general, as we increased sample size, we reduced sampling variability and narrowed the number of optimal combinations of parameters in grid search space, meaning we became more confident that the selected parameters would transfer better to estimate runout distance in adjacent areas.

Comparison of runout path

Relative frequencies of optimal parameter combinations for different training sample sizes computed using spatial cross-validation.

Assessing the spatial transferability of runout models is essential when extending their use from single or local events to regionally modelling runout across unknown space. Overall, this study demonstrates that our novel optimization approach performed well at regionally modelling the spatial distribution of runout path and distances across the upper Maipo River valley basin. A key component of the success of our modelling approach was its ability to generalize. The transferability of a regional runout model can be affected by the generalization ability of both the source-area prediction and the optimized process-based runout model.

In the development of our source-area prediction model, we aimed to produce a simplified empirical model that would result in good performance and transferability. Wenger and Olden (2012) recommended to control the flexibility of the GAM smoothing parameters to avoid overfitting that would lead to poorer model transferability. Therefore, similarly to previous landslide susceptibility studies using the GAM (Goetz et al., 2011, 2015a; Bordoni et al., 2020), we limited the degrees of freedom for smoothing-spline fitting.

A detailed model based on a large set of predictors can impede its ability to transfer to other locations (Tuanmu et al., 2011). As we and others have demonstrated, good predictive performance of susceptibility models of debris flows can be achieved with a relatively small set of predictors, which are primarily DEM-derived terrain attributes (Blahut et al., 2010b; Heckmann et al., 2014; Goetz et al., 2015b). Additionally, by fitting our models with multi-temporal data, we may be more likely to achieve better transferability in time (Tuanmu et al., 2011; Knevels et al., 2020). Event-specific inventories may not be large enough for regional optimization and risk the potential of overfitting source conditions to spatially varying conditions of that event (e.g. precipitation and snowmelt patterns). A smaller sample may also lead to not capturing the range of terrain conditions across the study area required for robust empirical modelling of source-area locations (Petschko et al., 2014; Rudy et al., 2016).

The interpretability of the GAM allows us to explore modelled behaviour. For
our study, the GAM did well at representing the general geomorphic
characteristics of source areas. Some of the relationships between
predictors, such as elevation, and debris-flow activity can be complex. In
the upper Maipo River basin, elevation can be a proxy for vegetation, snow
cover duration, terrain ruggedness, permafrost and glacial bodies, and
geology. It is therefore difficult to discern any direct relationships
between elevation and likelihood of being debris source areas. However, we
suspect that lower elevations were predicted to be less prone to be source
areas due to increased vegetation cover and less rugged terrain. The
decrease observed at the highest elevations may relate to permafrost and
glacial bodies holding potentially mobilized sediment (e.g. Sattler et al.,
2011). A decrease in predicted likelihood of source areas occurring at high
slope angles (

The best-performing regional random-walk parameters allowed for maximum
lateral spreading of the runout path given the range of parameters for
optimization. Individual events tended to also optimize for high lateral
spreading but not as strongly as the regional model. We believe this high
lateral spreading may be due to the location of the observed debris flows
relative to simulated paths and the quality of the DEM. A large proportion
of the observed debris-flow tracks were located at the fringe of the most
frequent simulated paths. Thus, a higher slope threshold and exponent of
divergence are required to capture these fringe debris flows. Additionally,
the surface of DEMs with resolutions greater than 20 m can be too general to
capture minor gullies that may have high flow accumulation (Blahut et al., 2010b).
The 12.5 m resolution ALOS DEM used in this study is derived from
downsampled Shuttle Radar Topography Mission (SRTM) data and would likely contain some of the topographic
generalizations of the original DEM (

By optimizing the runout-distance model using the median relative error as a metric, we managed to reduce the impact of possible outliers in our training and test data. We additionally reduced our chances of overfitting the regional model to larger debris-flow events, which was crucial for a model to maintain a generalization that makes it transferable across large areas.

Plotting the individual optimized models and exploring correlations between terrain attributes and optimal parameters allowed us to see if there were any broad trends in the parameter selection. In our study, we observed high variability in optimal parameters of the PCM model. While applying a trial-and-error approach, Mergili et al. (2012) also observed different optimal parameter combinations for individual events when modelling runout for a couple of debris flows just north of our study area. It seems apparent that determining runout parameter values for regional modelling without an optimization procedure would be difficult. It is also no surprise that the errors of the individually optimized debris flows were low (Fig. 12). Whether meaningful or not, the optimization approach should be able to match the runout distance with very low error. Poorly individually optimized events could be attributed to locally poor DEM quality (Horton et al., 2013) and mapping uncertainties (Ardizzone et al., 2002).

The two-parameter PCM model has a uniqueness problem (Perla et al., 1980).
Possibly infinitely many pairs of

Although we obtained a unique regional model solution, runout-distance
relative errors were only slightly higher than the best performer for
combinations of

In theory, it is possible that the minimum area-bounding box could
contribute to parameter insensitivity. Abrupt changes in flow perpendicular
to the initial flow direction, such as a flow meeting a channel, may only
slightly increase the length of the bounding box for several iterations of
decreasing

As with any optimization problem, using a suitable cost function is critical to ensure the model parameters are optimized to solve a very specific problem. It can be difficult to define a single metric that simultaneously measures both performance of path and distance simulations. As shown in this study, it may also not be necessary. The modular framework of the GPP model provides the ability to optimize two distinct runout components, the runout path including lateral spreading and the runout distance. In our study, we used the random-walk and PCM components of the GPP model to simulate spatial extent of runout. By using the two-stage approach to optimization, where we first optimize the runout path model and then plug in those values to optimize the runout-distance model component, we can considerably reduce computational complexity, while ensuring that our final model result explicitly optimizes for runout distance – a key characteristic for spatially predicting areas potentially impacted by debris-flow runout. In our case, we only needed to solve two separate problems with three (random-walk) and two (PCM) unknown parameters, as opposed to solving simultaneously for five unknowns. We used an exhaustive grid search to optimize the runout model components because it allowed us to visualize and assess model performance across parameter space. However, a drawback of this optimization method is that it can be computationally slow to explore all candidate parameter combinations. If speed is a requirement for regional runout analysis, then a faster method like random search (Bergstra and Bengio, 2012) may be preferred (Schratz et al., 2019).

In terms of model improvement, analysis of the spatial distribution of optimal parameters may lead to better parameter optimization based on terrain or geological characteristics. The spatially varying values used in this study were based on modelling for alpine regions (Gamma, 2000) and do not account for the potential variability in sliding conditions between different catchments (Guthrie, 2002). This challenge may be overcome by using regional modelling strategies already applied for landslide initiation susceptibility modelling, where the study area is divided into geologically similar regions, runout model optimization is performed for each region and then combined (e.g. Petschko et al., 2014).

Modelling the spatial pattern of debris-flow runout in large, mainly remote
areas, with sparse data, requires model calibration and validation methods
that ensure spatial transferability. In this study, we demonstrated that the
combination of the statistical prediction (GAM) of source areas and our
regional optimization of the GPP runout model (random walk and PCM)
performed well at generalizing runout patterns across the upper Maipo River
basin. In addition to high model performance, the transparency and
interpretability of the GAM provided further confidence in the prediction of
source areas by illustrating regionally geomorphically plausible modelled
behaviour. The optimized runout model parameters sets were consistently
similar within grid search space when assessing transferability using
spatial cross-validation. We believe this strong transferability of our
runout model was due to the hillslope gradient of the deposition area being
one of the major controls of runout distance in the PCM model. The
regionally optimized runout model also resulted in geomorphically plausible
results, with best-performing

To complement this paper, we developed the runoptGPP R package for optimizing mass movement runout models using the random-walk and PCM model components of the GPP tool in SAGA-GIS. It is available for download from Zenodo:

The data are available for download from Zenodo:

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was funded by CETAQUA Chile on behalf of Aguas Andinas from the project titled “Mass movement processes in the upper Maipo basin: Modelling the susceptibility and probable sediment transfers”, which was done in collaboration between the Friedrich Schiller University Jena and the Institute of Geography at the Pontificia Universidad Católica de Chile. We acknowledge support by the German Research Foundation and the Open Access Publication Fund of the Thueringer Universitaets- und Landesbibliothek Jena project no. 433052568.

This paper was edited by Margreth Keiler and reviewed by two anonymous referees.