Integration of root systems into a GISbased slip surface model: computational experiments in a generic hillslope environment
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Abstract
Root systems of trees reinforce the underlying soil in hillslope environments and therefore potentially increase slope stability. So far, the influence of root systems is disregarded in Geographic Information System (GIS) models that calculate slope stability along distinct failure plane. In this study, we analyse the impact of different root system compositions and densities on slope stability conditions computed by a GISbased slip surface model. We apply the 2.5D slip surface model r.slope.stability to 23 root system scenarios imposed on pyramidoidshaped elements of a generic landscape. Shallow, taproot and mixed root systems are approximated by paraboloids and different stand and patch densities are considered. The slope failure probability (P_{f}) is derived for each raster cell of the generic landscape, considering the reinforcement through root cohesion. Average and standard deviation of P_{f} are analysed for each scenario. As expected, the r.slope.stability yields the highest values of P_{f} for the scenario without roots. In contrast, homogeneous stands with taproot or mixed root systems yield the lowest values of P_{f}. P_{f} generally decreases with increasing stand density, whereby stand density appears to exert a more pronounced influence on P_{f} than patch density. For patchy stands, P_{f} increases with a decreasing size of the tested slip surfaces. The patterns yielded by the computational experiments are largely in line with the results of previous studies. This approach provides an innovative and simple strategy to approximate the additional cohesion supplied by root systems and thereby considers various compositions of forest stands in 2.5D slip surface models. Our findings will be useful for developing strategies towards appropriately parameterising root reinforcement in realworld slope stability modelling campaigns.
Keywords
Slip surface models R.slope.stability Root system morphology Shallow landslidesIntroduction
Herein, FoS > 1 depicts stable conditions, whereas FoS ≤ 1 indicates unstable slope conditions.
Slope stability can be computed in one, two or three dimensions (Xie et al. 2003). In general, a 1D slope stability model only considers soil thickness as a linear, metric parameter to compute FoS for an individual pixel (Van Westen et al. 1997). In geotechnics, 2D models are commonly used to assess the mechanical stability along a predefined failure plane of the vertical crosssection of a slope. However, both 1D and 2D models are not able to represent the threedimensionality of slip surfaces. GIS environments are generally defined as 2.5D environments, since only a single z value per GIS raster layer (i.e. a set of discrete z values in the case that more than one layer is used) is assigned to a plane coordinate (x and ytuple) of a raster pixel in a Digital Terrain Model (DTM). This distinguishes 2.5D from 3D environments, in which a location is represented by an isotropic volume element (voxel). Thus, a location with the plane coordinates x and y might contain multiple discrete z values in a 2.5D environment, whereas a continuous representation of the patterns in z direction requires a 3D environment. The expression ‘3D’ or ‘threedimensionality’ is often misleading in literature, due to the discriminative practices of the expression in different communities (e.g. geotechnical engineering and geomorphology) or simply because of inconsistent (or negligent) use. Hence, it is not always discernible whether a 2.5D or 3D was used.
Embedded in a GIS environment (2.5D), the infinite slope model evaluates the stability of a terrain for individual raster cells without taking into account forces that are apparent in adjacent raster cells (e.g. Montgomery and Dietrich 1994; Pack et al. 1998; Baum et al. 2002; Malet et al. 2005). However, this model is not or only conditionally suitable for slip surface lengthtodepth ratios L/D < 16–25 (Griffiths et al. 2011; Milledge et al. 2012), or for soils with discontinuities such as root systems. In such conditions, slip surface models are more appropriate, applying more or less complex limit equilibrium relations to calculate the FoS along predefined or randomly determined failure planes (e.g. Hovland 1979; Hungr 1987, 1988; Hungr et al. 1989; Lam and Fredlund 1993; Xie et al. 2004). Several software applications employing 2.5D or 3D slip surface models are available, such as CLARA (Hungr 1988), TSLOPE3 (Pyke 1991), 3DSLOPE (Lam and Fredlund 1993) and the r.slope.stability (Mergili et al. 2014a). 2.5D and 3D slip surface models have been applied either for model testing (e.g. Hovland 1979; Hungr 1988; Lam and Fredlund 1993; Xie et al. 2003; Mergili et al. 2014a) or for particular case studies (Seed et al. 1990; Chen et al. 2001; Chen et al. 2003; Jia et al. 2012; Mergili et al. 2014b).

to allow for quantitative statements on the influence of different types and densities of root systems on the slope stability conditions;

to build a basis for developing strategies towards appropriately parameterising root reinforcement in realworld slope stability analyses in a GIS environment.
 i.
the stabilising potential of root systems are diverse according to both root architecture and spatial position along a hillslope;
 ii.
potential slip surface size influences whether a distinct root system type (or a tree stand with a diverse distribution of various root systems) is able to contribute to slope stabilisation.
 iii.
the approximation of root systems by paraboloid solids allows a straightforward parameterisation of the spatial root reinforcement distribution in 2.5D slip surface models.
Next, we explain the generic landscape and root systems used for all the analyses. We then introduce the relevant functionalities of the software r.slope.stability and the computational experiments performed (“Materials and methods”). We present (“Results”) and discuss the results and conclude with the key messages and an outlook (“Discussion”).
Materials and methods
Generic landscape and root system morphology

Three identically shaped octagonal pyramidoids (Fig. 1);

Each pyramidoid has an extent of 2500 m × 2500 m, 2 m × 2 m raster cell grid and is divided into eight equal plots of triangular shape, so that 24 identical plots are available in total. The surface area of each plot is 51.75 ha.

All plots have an equal slope inclination of 25°.

The slope stability analyses are performed only for the lower 75% of the pyramidoid since the convergence of the margins in the upper 25% would not allow a proper analysis of the r.slope.stability results.

Only the mechanical reinforcing effect of roots is considered; hydrological effects of roots are not considered.
Land cover conditions and properties of distinguished plots
Scenario  Plot properties  

ID  Distribution  Root system  S (trees ha^{−1})  P (patches plot^{−1})  
Scenario I  S1.1  No vegetation cover (empty)  
S1.2  No vegetation cover (empty)  
S1.3  Uniform  SRS  50  –  
S1.4  SRS  2000  –  
S1.5  TRS  50  –  
S1.6  TRS  2000  –  
S1.7  MRS  50  –  
S1.8  MRS  2000  –  
Scenario II  S2.1  Separated patches  TRS  150  25–30 
S2.2  TRS  150  35–40  
S2.3  TRS  2000  25–30  
S2.4  TRS  2000  35–40  
S2.5  SRS  2000  35–40  
S2.6  SRS  2000  25–30  
S2.7  SRS  150  35–40  
S2.8  SRS  150  25–30  
Scenario III  S3.1  Uniform patches  SRS and TRS (separated)  150  25–30 
S3.2  150  35–40  
S3.3  2000  25–30  
S3.4  2000  35–40  
S3.5  Mixed patches  MRS  2000  35–40  
S3.6  2000  25–30  
S3.7  150  35–40  
S3.8  150  25–30 
We simulate sparse and dense stand conditions (S = 50, 150 and 2000 trees ha^{−1}, respectively) with different maximum distances between the tree trunks (cf. Schmid and Kazda 2002; Puhe 2003). The overall distribution of trees on each plot are categorised in ‘uniform’ coverage (scenario I), ‘separated patches’ (scenario II), ‘uniform patches’ and ‘mixed patches’ (both scenario III). Each raster cell of a plot (2 m × 2 m) can only accommodate a single tree, which fixes minimum tree distance to 2 m (raster cell length). For a uniform coverage, we assume a full forest cover of the plot and tree distances depending on the simulated stand density condition (S). We create randomly generated values for each raster cell ranging from 0 to 1 and define those raster cells as tree locations, where the random raster value does not exceed the percentile of the tree cells that can occupy the plot. For example, for a uniform tree density of 50 trees ha^{−1}, all raster cells are assigned as tree locations, N_{tree} (N_{tree} ≔ N_{0}), where the random cell value ≤ 50 trees ha^{−1} × 51.75 ha (plot size). For mixed uniform conditions (e.g. S1.7 and S1.8), S halves for each species. Thus, stand density S for both species remains equal, whereas only 50% of trees of each species can occupy a stand. In case of scenario II and III, where forest is simulated to not cover the whole plot, we define patches (P) first that function as tree stands. Therefore, 25–30 or 35–40 random points over the whole plot are selected from which radially ~ 2500 adjacent raster cells define the patch area, where trees can be located. All cells that are defined as tree cells in a uniform coverage and are located within the defined patches are then assigned as tree cells. Raster cells that are located outside the patches are assigned as nontree cells. Herein, the spaces between patches create voids that simulate forest glades in which potential reinforcement from the tree root systems is remarkably reduced or nonexistent. In cases where patches within a single plot shall be separately occupied by SRS and TRS (scenario III, plots 1–4), the number of patches P is again halved for each species, whereas S remains constant. For cases of mixed conditions within the patches (scenario III, plots 5–8), P remains constant but S halves (cf. description above for S1.7 and S1.8).
Constant values for vertical and horizontal distribution of the two considered types of root systems
Root system  z_{max} (m)  r_{max} (m) 

SRS  1.5  7.5 
TRS  3.5  5.0 
The tool r.slope.stability
To take into account the uncertainty of the input parameters, the r.slope.stability includes the option to consider a userdefined space of c′ and φ′ values instead of fixed values (Mergili et al. 2014a). Following a probability density function (PDF), FoS is computed for a large number of parameter combinations within the given space. Assuming a uniform PDF, the slope failure probability P_{f} for a given slip surface essentially corresponds to the fraction of parameter combinations yielding FoS < 1. Consequently, P_{f} can assume values in the range 0–1.
The value of FoS or P_{f} for a given raster cell results from the overlay of a large number of slip surfaces: the lowest value of FoS or the highest value of P_{f} out of all intersecting slip surfaces is considered most relevant and therefore applied as the final result. We refer to Mergili et al. (2014a,b) for the technical and mathematical details of the r.slope.stability.
Computational experiments
Spatial dimensions of the slip surfaces in each computational experiment
E  Spatial dimensions (m)  

L  W  D  
1  30  20  2 
2  120  80  2 
3  30–120*  20–80*  2 
Values of geotechnical parameters used in the r.slope.stability
c′_{s} (N m^{−2})  c′_{r} (N m^{−2})  γ (N)  φ′ (°)  

Min  1500  8000  15,000  15 
Max  3000  16,000  15,000  45 
Results
Slip surface size
Scenario I: uniform tree distributions
Scenario II: separated patches
Whilst E1 and E2 yield comparable M of P_{f} for S = 150 and P = 25–30, we note that for TRS, increasing the patch density has a comparatively larger effect on the average P_{f} with E2 (ΔP_{f} = − 0.30) than with E1 (ΔP_{f} = − 0.20): With the higher patch densities, large slip surfaces cannot any more squeeze between the patches (the same principle as observed for E1 with stand density). In general, P_{f} is lower for E2 than for E1 with TRS. In contrast, the average values of P_{f} are similar or higher with E2 than with E1 when assuming SRS. The reason for this phenomenon consists in the fact that the slip surface bottom strongly interacts with TRS, whilst only the edges (< 1.5 m) interact with SRS, so that the influence of SRS is much smaller.
Similarly to scenario I, SD of P_{f} is larger for E1 than for E2 due to a reduced amount of smoothing. However, due to the additional variation in the spatial distribution of the root systems, SD is generally higher than in scenario I.
E3 generally yields higher M of P_{f} than E1 and E2. This phenomenon is particularly pronounced for S2.7: Whilst E1 leads to a distinctive spatial pattern with several raster cells of low P_{f}, E2 results in a smoothed pattern of intermediate values of P_{f} without notable peaks. E3 represents a combination of both, imposing the peaks (small slip surfaces) upon the smoothed background (large slip surfaces). As in scenario I, the M of P_{f} are highest for E3 compared to E1 and E2. SD of P_{f} is intermediate between the values yielded for E1 and E2.
Scenario III: mixed patches
Discussion
Impact of the slip surface size
Root systems are only able to contribute to reinforcement when the potential slip surface intersects but not entirely contains the paraboloid, since (Greenwood et al. 2004; Cammeraat et al. 2005; Van Beek et al. 2007). In the context of the present work, this concerns SRS, whereas TRS are always able to add root cohesion to the slope because r_{max}(TRS) > D. We construe that the reinforcement potential of TRS decreases with slip surface size for low stand densities due to the decreasing share of the potential slip surface coinciding with the root systems. The paraboloids seem to influence stability more effectively when they are rather located at the edges of the slip surface. This appears to be plausible since the intersected area of paraboloid and slip surface boundary of both SRS and TRS is larger when the paraboloid is located at the edge. However, since the angle that determines the curvature of the SRS paraboloid is more pointed than the one of TRS paraboloid, the added cohesion of SRS is larger at the slip surface edge compared to TRS.
This finding is in accordance with the issue highlighted in the research of Danjon et al. (2008), who stated that trees with many thick vertical roots fix a soil package better when located in the middle of a slope. In contrast, the contribution to slope reinforcement of root systems with oblique roots is rather higher, when located at the edge of a potential slip surface. Therefore, we assert that the r.slope.stability can reconstruct these relations, which appear consistent with the findings of Danjon et al. (2008).
Effect of speciesrelated root system type and stand density
The results indicate that TRS contribute more to stability than SRS for all observed slope sections, expressed by a lower P_{f} throughout all scenarios (see Figs. 7, 8 and 9). TRS and MRS generally show lower values of P_{f} than SRS, indicating that paraboloids representing deeprooting have a positive effect on slope stability. This is a consequence of the spatial interaction of the ellipsoidshaped slip surfaces and the paraboloid root systems, where SRS do not penetrate the potential slip surface where D ≥ 1.5 m. The spatial extent of the root system itself—and thus its ability to penetrate a potential slip surface—strongly depends on the soil depth and inclination (Nilaweera and Nutalaya 1999). Kusakabe (1984) gave indications on the stabilising potential of different root systems in respect to specific slope conditions. In this regard, the relation between soil thickness (that strongly determines the depth of a potential slip surface) and the root system morphology should be under particular consideration in further studies.
The findings of scenario I have shown that stand density strongly influences P_{f} (see Figs. 7a and 8). This is in line with field observations and tree mapping of Roering et al. (2003) in a landslide prone area on the Oregon Coast range, which showed the effect of thinning (e.g. due to fires or timber harvesting) on slope failure occurrences. The varying extent of root systems of different tree species in landslide scarp adjacencies has thus a considerable effect on slope stability. The results of scenario I reproduce the patterns reported by Roering et al. (2003), since P_{f} is strongly associated with the planar extent of the root systems. The effects on slope stability caused by thinning and tree removal that lead to a decrease of stand density (and thus to a decreased spatial extent of root systems that can contribute to stability) were also highlighted by Genet et al. (2008).
Separated vs. mixed stands
The results of scenario III show that mixed stands have a beneficial effect on slope reinforcement when P is lower (Fig. 10). We assume that a consistent portioning of SRS and TRS considerably reduces P_{f}. In this regard, P, the distance between the patches and the size of the patches themselves might be the determining factors whether separated mixed stands (SRS/TRS) or totally mixed stands (MRS) can reduce slope failure probability. Thus, MRS appear to be preferable when P is low because then, the chance is higher that a root system occupies a position where it can deploy its maximal reinforcement potential (e.g. SRS close to the landslide margins and TRS in the middle of the landslide). Results presented in Fig. 11 indicate that completely mixed stands tend to decrease P_{f} when stand density is sparse, compared to plots where species are separated (SRS/TRS). Moreover, the findings suggest that in general, a higher amount of trees within a plot contribute more to slope stability and thus to a reduction of P_{f}. This is observable in scenario III (cf. Figs. 6 and 11); when the patch density is higher, overall, more trees occupy the whole plot, and voids between patches are smaller. This indicates that particularly a higher patch density leads to a reduction of P_{f} in environments with voids (forest glades), because more trees are able to reinforce the slope and number/size of voids are reduced (compare plots AEI and CGK with plots BFJ and DHL in Fig. 11). Herein, particularly plots with mixed stands (MRS) tend to be more stable. We suggest that the stand mixture (MRS) leads to this remarkable decrease of P_{f}. This is because SRS and TRS trees are then located at positions where they might be able to deploy a maximum of their positiondependent stabilising potential (e.g. SRS at ellipsoid edges, TRS in the middle of ellipsoids), compared to stand compositions where SRS and TRS are separated. Genet et al. (2010) reported that stand diversity does not primarily affect slope stability, however, rather the tree position on the slope and the architecture of the respective root system that crosses the potential slip surface (also suggested by results of Danjon et al. 2008). The model outputs indicate that the r.slope.stability is able to reproduce the positive effect of favourable positions of the different root systems, considering their distinct root system morphologies.
Implications for model parameterisation
It was reported by many former studies that soil characteristics highly influence the rooting behaviour of plants and thus the development of the root system (e.g. Greenway 1987; Bischetti et al. 2009; Ghestem et al. 2011; Stokes et al. 2014). However, we did not consider any spatial changes of the approximated root systems due to changing geotechnical soil parameters. We reason that a connection between the distribution of the root system architecture and apparent soil properties appears to be challenging for implementation in 2.5D slip surface models and applications in realworld case studies. Further investigations could tackle this issue and consider the variability of root systems in different soil environments and incorporate hydrological impacts of root systems on the soil, e.g. by root water uptake (Zhu and Zhang 2015). Based on the results of this study, we highlight that the spatial distribution of cohesion values within the approximated root system should be favoured compared to uniform cohesion values. For example, Bischetti et al. (2005), Bischetti et al. (2009), Thomas and PollenBankhead (2010), Schwarz et al. (2010) and Schwarz et al. (2012) addressed the spatial distribution of roots, root tensile strength and thus dispersal of rootsoil cohesion values. However, the spatial diversity of cohesion values within root systems remains to be a vague physical input which is hard to parameterise. In terms of 2.5D slip surface modelling, the consideration of the spatial distribution of root cohesion should be considered in further studies. Moreover, the complexity that arises, when estimating root systems properly (and considering soil characteristics or external environmental influences), identifies the general challenge of an accurate implementation of root properties in physically based models. Many deterministic approaches use the FoS to state whether a hillslope becomes unstable according to distinct physical parameters. Mergili et al. (2014a) and De Lima Neves Seefelder et al. (2016) emphasised that FoS—derived with a fixed set of geotechnical parameters—might fail to capture the details of a landscape. The wide range of root cohesion forces associated to different root systems would rather promote misinterpretations of the FoS due to the uncertainties associated with the dissimilarities of root distribution. Therefore, we suggest to use P_{f} rather than FoS as slope stability indicator particularly in those cases where root properties are used as input. Regarding a practical application of our approach, we highlight that studies applied on realworld conditions require groundtruth data about the geotechnical parameters, information about landslide dynamics (e.g. provided by a multitemporal landslide inventory) and knowledge about forest stand conditions to avoid the induction of uncertainties.
The findings presented shall be employed as a basis to better parameterise root system morphology in realworld slope stability modelling. However, finding strategies to transform the results to realworld conditions remains a challenge:
Compared to traditional catchmentscale slope stability modelling, a finer spatial resolution if the GIS raster cells is needed to appropriately capture the root morphology. Whilst this would not be a problem with the infinite slope stability model, computationally, more complex approaches such as employed by the r.slope.stability could run into difficulties with the computational systems usually available, particularly for larger areas. Our study is performed on a 2 m × 2 m raster grid with a relatively manageable extent. However, it should be considered to elaborate in more detail on the ‘optimum’ cell size for realworld case studies that allows a reliable representation of the root system morphology whilst ensuring a justifiable computational time.
Alternatively, we propose the derivation of a set of rules to implement the findings gained on the generic topography in realworld case studies. However, we note that the results obtained in the present study build on particular parameter assumptions. Even though we postulate that the general patterns obtained have a broader range of validity, more computational experiments with a broad range of conditions of root system morphology, slope, water status etc. have to be performed. We further note that, in the present study, all nontree roots are disregarded.
The findings reveal that the implementation of even simplified approximated root systems in a 2.5D slip surface model yield highly nonlinear effects on the model output. Further studies that apply root system approximations in 2.5D slip surface models should therefore particularly focus on an accurate representation of sensitive parameters such as stand density, stand arrangement (mixed or separated) and tree species.
Conclusions
The added cohesion from root systems and its implementation in 2.5D or 3D slip surface models has not been performed so far. To explore the potential of root system morphology implementation in a 2.5D slip surface model and its impact on the model performance, we used a set of idealised root systems as input for the computational tool r.slope.stability. The model was tested for 23 plots depicting different types and configurations of root systems, summarised in three scenarios (uniform tree distribution, separated patches and mixed conditions). Moreover, the scenarios were tested for three different sizes of potential shallow slip surfaces. We computed P_{f} to determine the added soil reinforcement by the assumed root systems. Our results show that the differently approximated root system morphologies exert distinct effects on the slope stability, where stand density, approximate position on the simulated slip surface and the ability of the root system to cross the potential slip surface are the driving factors for the model output. These findings are in accordance with the results of studies that revealed similar findings in field. Further work will focus on the implementation of more sophisticated ways of root system approximation, considering the spatial distribution of root cohesion values within the root system. In this regard, a better representation of the biomechanical interactions of the plantsoilcontinuum is expected. We will further attempt to employ the insights gained for better parameterising realworld slope stability modelling campaigns.
Notes
Acknowledgements
Open access funding provided by University of Vienna.
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