This approach, as proposed by , is implemented mainly for convenience in order to provide a simple means to detect the process path along the gradient of gravity. A particle follows the steepest descent of the slope: n=max⁡{(z-zi)/di}, where n is the neighbor of steepest descent, z is the elevation of the currently processed cell, zi is the elevation of neighbor cell i, and di is the horizontal distance to neighbor cell i.

The model result is thus deterministic, with the exception of its behavior (as implemented in the GPP model) when two or more neighbor cells show the same steepest descent or when a flat area is reached. In the first case, one of the neighbor cells is chosen at random. On flat areas a set of potential neighbor cells is determined which is made up of all neighbors with the same elevation as the current cell which have not been traversed yet in the current model iteration. From this set, a process path cell is chosen at random. This introduces a probabilistic component. Further, the terrain could have been modified between two model iterations by sink filling or material deposition.

The Maximum Slope model approach has no special parameters besides those controlling the mode of operation of the GPP model main loop, such as the number of model repetitions or the processing order. The pseudo-random number generator, used to choose a neighbor cell at random under the predescribed conditions, can be initialized either with the current time or a fixed seed value. The latter will always produce the same succession of values for a given seed value and will thus give the same results for consecutive tool runs.

With this approach, the process path is modeled by a variant of the dfwalk model as proposed by . It uses a stochastic way of path finding, which makes it possible to model the lateral spreading of a process by calculating several iterations from the same start position. Besides the parameters controlling the Monte Carlo simulation, such as the number of repetitions, the Random Walk approach has three parameters to calibrate the model in order to mimic the behavior of different geomorphological processes: (i) a threshold parameter defines the terrain slope below which divergent flow is allowed; (ii) this is accompanied by an exponent for divergent flow: below the slope threshold, the parameter controls the degree of divergence; (iii) finally, a persistence factor can be used to preserve the direction of movement by weighting the current flow direction in order to account for inertia, which can be observed for debris flows or wet snow avalanches . Rockfall may be modeled with (almost) no persistence and a higher degree of divergence.

For the currently processed grid cell, a set N of potential flow path cells is determined from all immediate neighbor cells in a 3 by 3 window, which have an equal or lower elevation than the central cell. This is done in several steps. First of all, for each neighbor cell i a slope value γi, based on the slope threshold βthres, is calculated : γi=tan⁡βitan⁡βthres,βi≥0,i∈1,2,…8, where βi is the slope to neighbor cell i. The maximum value γmax⁡=max⁡{γi} is a measure of how close the slope to the steepest neighbor is to the slope threshold. In the case that γmax⁡>1 the set N of potential flow path cells is only made up of the steepest neighbor. Otherwise, the mfdf (multiple flow directions for debris flows; ) criterion is used to decide which neighbor cells are additionally included in N: γi≥γmax⁡a0<γmax⁡≤1,a≥1, where a is the exponent to control the amount of divergent flow (a≥1). If γi is greater than or equal to the mfdf criterion, then the neighbor i is included in N. Thus, the set N is given by : N={i∣γi≥γmax⁡afori∈1,2,…8}if0<γmax⁡≤1andN={i∣γi=γmax⁡fori∈1,2,…8}ifγmax⁡>1.

The slope threshold makes it possible to adjust the model to different topography: in steep sections of the process path, where the terrain slope is near the threshold, only steep neighbors are allowed in addition to the steepest descent. In flat sections, almost all lower neighbor cells are potential flow path cells and the tendency for divergent flow is increased. The degree of divergent flow below the slope threshold can be controlled by the exponent of divergent flow. This sensitivity to the terrain conditions is an important property which is missing in the modeling approaches developed for hydrological processes, which distribute the flow proportionally to the slope to all lower neighbors irrespective of the local topography .

Finally, a cell is picked at random from the set N. The probability for each cell probi is given by probi=fi⋅tan⁡βi∑jfj⋅tan⁡βj, where i describes the currently processed neighbor cell, j depicts all neighbor cells in set N, and f is a weighting factor. In the case that the flow direction to neighbor i equals the previous flow direction, f equals the persistence factor p (with p≥1), otherwise f=1. A tendency to move towards the steepest descent is always given as the transition probabilities are weighted by slope. The persistence factor can be used to weight the current flow direction, which results in a higher probability that the neighbor in this direction gets selected. This property can be used to reduce abrupt changes in flow direction. Finally the transition probabilities are scaled to accumulated values between 0 and 1, and the pseudo-random generator is used to select one flow path cell from the set.

In the GPP model, the approach is extended to also handle flat areas. This is done as described for the Maximum Slope approach with the same restriction that a potential successor cell must not have been traversed yet in the current model iteration in order to prevent endless loops.

The result of several model iterations is a raster data set storing the transition frequencies, i.e., how many times a grid cell has been traversed. Figure  shows the effect of different parameter settings for the three calibration parameters slope threshold, exponent for divergent flow and persistence factor (the run-out length was calculated with the Geometric Gradient approach using an angle of 26.5∘; see Sect. , point a). The number of model iterations is set to 1000 in the examples (a) to (j). In Fig. a–e the slope threshold (40∘) and the persistence factor (1.0) are fixed, while the exponent for divergent flow is increased in several steps (1.0, 1.1, 1.2, 1.5 and 2.0). It is obvious that the extent of the process area increases significantly because of the higher degree of lateral spreading.

In Fig. f–j the exponent for divergent flow (1.5) and the persistence factor (1.0) are fixed, while the slope threshold is increased gradually (15∘, 20∘, 30∘, 40∘ and 60∘). It can be seen that the point at which lateral spreading is allowed is moving up the torrential fan, resulting in an increase of the total process area.

Figure k–o show the results of a stepwise increase of the persistence factor (1.0, 1.5, 2.0, 2.5 and 3.0) while the slope threshold (40∘) and the exponent of divergent flow (2.0) are fixed. Here, only a single iteration was calculated from each start cell in order to visualize single trajectories. It is obvious that with higher persistence factors the number of changes in direction along a trajectory is decreasing.

The run-out length of a process is often described by the vertical and horizontal distances covered by a particle from its start to the stopping position: tan⁡α=dv/dh, where α is the angle to the horizontal and dv and dh are the vertical and horizontal offset, respectively. Both offsets can be defined differently, see below. This describes a straight energy line from the start to the stopping position . With a straight energy line, the velocity can be calculated by : vi=2⋅g⋅hv, where vi is the velocity (m s-1) on the currently processed grid cell, g is the acceleration due to gravity (m s-2), and hv is the height difference (m) between the energy line and the current grid cell i. Although the angle α is not constant, it can be observed that it has a characteristic value range for gravitational movements of a specific type. The calibration of the angle α, which can be measured quite easily, is usually done by field observations and mapping. All approaches based on the energy line principle provide the possibility to output raster data sets storing the stopping positions and the maximum velocity encountered in each cell of the process path.

a. Geometric gradient:

The geometric gradient defines the vertical offset as the vertical distance between the release area and the end of the deposit. The horizontal offset is defined as the horizontal distance between these two points. This modeling approach thus requires just the friction angle α as input. The GPP model supports both a global friction angle or a raster data set with friction angles for each start cell. Once the angle between the start cell of the particle and the current position of the particle drops below the friction angle α, the end of the deposit is reached.

The shadow angle can again be provided either as a global value or by a raster data set with shadow angles for each start cell. In order to determine the location of the first impact of a particle on the talus slope, the GPP model implements two different approaches: (i) the user provides a raster data set with impact areas. Once a particle reaches a cell labeled as impact area, the location of this cell is used to measure the shadow angle; (ii) a threshold describing the slope angle above which free fall is assumed is provided. As soon as the angle between the start cell and the current position of the particle drops below the threshold, the location of this cell is used to measure the shadow angle.

The one-parameter friction model has been developed to simulate rockfall and is based upon concepts introduced by , which have been extended by various authors . The GPP model implements several of these approaches, more details can be found in and . The one-parameter friction model calculates the velocity of the currently processed grid cell according to the velocity on the previous cell of the process path, the slope and a friction parameter. Once the velocity becomes zero, the end of the deposit is reached. Once a block is detached from the rock face, it is falling in free air: vi=2⋅g⋅hf, where vi is the velocity (m s-1) on the currently processed grid cell, g is the acceleration due to gravity (m s-2), and hf is the height difference (m) between the start cell and the current grid cell i. The impact on the talus slope occurs, similar to the shadow angle model, if (a) a particle reaches a cell labeled as impact area or (b) the angle between the start cell and the current position of the particle drops below the free fall threshold. The decrease of velocity on the talus slope due to energy loss on the first impact can be calculated in two different ways: i.

energy reduction :

vi=2⋅g⋅hf⋅K,

where K is the amount of unspent energy (K≤1, i.e., for an energy reduction of 75 % K is 0.25);

Approach (i) requires the user to specify the amount of energy reduction as calibration parameter. Approach (ii) usually results in larger run-out distances. The strong dependence of approach (ii) on the slope of the impact cell complicates the model calibration . Approach (i) is used as the default in the GPP model. After the impact, two different modes of motion can be modeled : i.

sliding:

vi=v(i-1)2+2⋅g⋅(h-μs⋅D),

where v(i-1) is the velocity (m s-1) on the previous grid cell of the process path, h is the height difference (m) between adjacent grid cells, D is the horizontal difference (m) between adjacent grid cells, and μs is the sliding friction coefficient (–).

Once the velocity on a grid cell becomes zero, the end of the deposit is reached. The model calibration usually requires only two parameters: the amount of energy loss on impact (%) and, depending on the chosen mode of motion, either the sliding or the rolling friction coefficient (–). The friction coefficient can be provided as a global value or spatially distributed by providing a raster data set with friction values. Impact on the talus slope can be modeled either by providing an input raster data set with impact areas or by using a slope threshold (see Sect. , point c). Besides the possibility to output a raster data set storing the stopping positions, a raster data set with the maximum velocity encountered in each cell of the process path can be output.

The PCM model is a two-parameter friction model originally developed to calculate the run-out distance of avalanches. It is based on the model of . The model has also been applied to debris flows by various authors . It is a center-of-mass model and it is assumed that the motion is mainly governed by a sliding friction coefficient μ and a mass-to-drag ratio M/D. In steeper parts of the process path, the velocity is mainly influenced by M/D, whereas the velocity in the run-out area is dominated by μ. The velocity on the currently processed grid cell depends on the velocity of the previous cell, the slope, the slope length and the two friction coefficients: vi=αi⋅M/Di⋅1-eβi+v(i-1)2⋅eβi and αi=gsin⁡θi-μicos⁡θi,βi=-2LiM/Di, where vi is the velocity (m s-1) on the currently processed grid cell, g is the acceleration due to gravity (m s-2), θ is the local slope (∘), L is the slope length between adjacent grid cells (m), μ is the sliding friction coefficient (–), and M/D is the mass-to-drag ratio (m). assume the following velocity correction for v(i-1) before vi is calculated in the case of a concave transition in slope direction: v(i-1)*=v(i-1)cos⁡θ(i-1)-θiifθ(i-1)≥θiv(i-1)ifθ(i-1)<θi.

The correction is based on the conservation of linear momentum and has a higher magnitude in the event of abrupt transitions. The accurate stopping position on a grid cell may be calculated by the following: s=M/Di2ln⁡1-vi-12αiM/Di, where s is the length (m) of the process path segment on the grid cell. In the GGP model, s is not calculated and the process stops as soon as the square root in Eq. () becomes undefined. Thus the raster cell size determines the precision of the stopping position, which is a reasonable compromise for a grid-based model.

proposed incorporating the velocity correction (Eq. ) directly into the velocity calculation (Eq. ): vi=αi⋅M/Di⋅1-eβi+v(i-1)2⋅eβi⋅cos⁡Δθi and Δθi=θ(i-1)-θiifθ(i-1)>θi0ifθ(i-1)≤θi.

In the GPP model Eq. () is implemented. The model has to be calibrated by the friction parameters μ and M/D. In order to overcome the problem of mathematical redundancy – various combinations of the two parameters can result in the same run-out distance – the parameter M/D is usually taken to be constant along the process path. It is only calibrated once in order to obtain realistic maximum velocity ranges for a given process. Both friction parameters can be provided either as a global value or spatially distributed by a raster data set. In the GPP model implementation it is also required to provide an initial velocity (m s-1) in order to avoid the process already stopping on the first grid cell along the process path. As with the one-parameter friction model, it is possible to output raster data sets storing the stopping positions and the maximum velocities.

The sink-filling approach is immediately activated once a raster data set with material heights per start cell is provided as input. As soon as a sink is detected, the particle stops and material is deposited. The deposition approach attempts to preserve a downward slope if procurable, thus avoiding the creation of new sinks and making it possible to overcome the obstacle in subsequent model iterations.

The sink-filling approach is based on with slight modifications: (i) the overflow cell and the depth of the sink are determined; (ii) if the depth of the sink cannot be filled with the material available for the current model iteration, all material available is deposited and the computation stops; (iii) the sink is filled up to the height which is preserving a user-specified minimum slope to the overflow cell; (iv) in order to avoid the creation of another sink, material is deposited on the process path above the sink; therefore it is tested if the material left is enough to fill up the process path above the sink while preserving the minimum slope; in the case that the available material is not enough to preserve this slope, the angle is continuously decreased until a minimal downward slope can be preserved. In the case that material is left, it is made available for the subsequent iterations. did not use a user-specified minimum slope to preserve, but determined the average slope along the process path above the sink for the last 50 m. In performance tests of the GPP model this turned out to be too dependent on the local slope conditions, often resulting in large angles and thus using too much material which is then missing to fill the sink upwards.

This approach simply deposits material on the grid cell of the modeled stopping position. The amount of material deposited on this cell is controlled by the Initial Deposition on Stop parameter, which describes the percentage of the available material which is deposited at the stopping position. The rest of the material is used to fill up the process path above the stopping position. The angle used to do this while preserving a downward slope is determined in a way that all material left in this iteration is used.

The approach makes it possible to adjust the deposition behavior to different geomorphological processes: simulating a rock fall event, the Initial Deposition on Stop parameter can be set to 100 %, resembling the deposition of single rocks. With debris flows or snow avalanches, it can be set lower in order to achieve a more lobe-like deposition pattern. Nevertheless, the approach is not intended to realistically simulate the deposition pattern. But it can be used for scenario modeling, forcing the process path into different directions in subsequent model iterations.

The On Stop deposition approach can be extended by slope- and/or velocity-based components, which can be used to force the deposition of material along the process path. Such components have been proposed by and are used in a modified way in the GPP model. Again, this approach is most useful for scenario modeling in order to simulate debris jamming or channel plugging. It is also useful if a high-resolution DTM with great detail is used. The deposition starts once the slope or the velocity drops below a specific threshold. At a slope or velocity of zero, the Maximum Deposition along Path parameter controls the percentage of material (available in this model iteration) that is deposited. At the threshold the material deposition is zero, which results in a linear relation.

The slope- and velocity-based approaches can be used separately or in combination. In the latter case, a deposition height is calculated with both approaches and the lower deposition height is applied. This reduces artefacts resulting from the usage of a single threshold. For example, on flat areas, no material is deposited as long as the velocity is still high.

The slope- and velocity-based approaches have a further parameter, the Minimum Path Length, which describes the distance along the process path that must be exceeded before deposition sets in. This is required to simulate the behavior of a volume (and not single particles) and to prevent the deposition of material shortly after the process has initiated or even within the release area itself. It is also useful to have more control on the position along the process path where deposition should set in, especially in the case of cascades with alternating steeper and gently dipping slope profile sections.