Introduction

Rainfall-induced shallow landslides are among the most common gravitational mass movements on slopes in steep terrain, contributing to sediment transfer, erosion, and deposition (Benda and Dunne 1997; Bogaard and Greco 2018; Crosta and Frattini 2008; Gariano et al. 2015; Guzzetti et al. 2007; Lazzari et al. 2020; Masi et al. 2021). These types of landslides usually interest the uppermost layer of soil or weathered rock, typically within the depth of a few meters from the surface (Hungr et al. 2014; Phillips et al. 2021), posing a serious threat to people and infrastructure as well as agroforestry production in mountain areas. Lately, the use of vegetation as a Nature-based Solution (NbS) for shallow landslide risk reduction is receiving increased attention in the scientific community (Capobianco et al. 2022; de Jesús Arce-Mojica et al. 2019).

Vegetation can contribute to slope stability through both hydrological and mechanical processes (Fig. 1). Plants can provide mechanical reinforcement through the tensile strength of the root network and the interaction between the roots and the soil and/or bedrock (Li et al. 2023; Pollen and Simon 2005; Schwarz et al. 2010a; Stokes et al. 2009; Waldron 1977; Wu et al. 1979). The hydrological reinforcement effects of plants consist of reducing the amount of water in the soil, i.e., reducing the driving forces in the slope caused by pore water pressures. The water content is reduced through plants’ ability to intercept water on their canopies and by root water uptake through transpiration (Chirico et al. 2013; Lepore et al. 2013; Ng et al. 2016; Wilkinson et al. 2002a). Furthermore, the presence of roots in the vadose zone can increase or decrease the soil water retention capability and the saturated soil permeability, depending on type and age of plants, and vegetation spacing (Ni et al. 2019a, 2019b; Vergani and Graf 2016). Finally, vegetation can affect the stability of a slope by inducing a certain surcharge due to self-weight (Gray and Megahan 1981; Greenway et al. 1987) and by being subjected to wind (Kim et al. 2020).

Fig. 1
figure 1

Overview of principal effects provided by vegetation on a slope. Blue and orange boxes and arrows refer to hydrological and mechanical effects, respectively

Full size image

In order to quantify the performance of vegetation management as a slope stabilizing measure, slope stability models are valuable tools. Recently, a few literature reviews have been published to highlight different aspects of these models. Briefly, Murgia et al. (2022) conducted a thorough review of physically based slope stability models that consider the mechanical effect of roots in the factor of safety (FS) calculation. Their study focused on highlighting the identified models’ scale of application, the dimension of calculation, and which geotechnical and hydrological models the slope stability models are based on. Masi et al. (2021) conducted a similar review of publications published between 2015 and 2020, where in addition to the mechanical effects, also the hydrological effects of vegetation were studied. Bordoloi and Ng (2020) also conducted an extensive review on experimental evidence related to both the influence of mechanical and hydrological traits of vegetation on slope stability, quantified through in situ measurements, laboratory analysis, and modeling. These earlier reviews have contributed to advancing our understandings of slope stability models; however, none go into depth in describing how the governing processes of root-soil interaction in shallow landslide triggering are parametrized in the existing slope stability models.

This study builds on the previous reviews by identifying and analyzing physically based slope stability modes introduced between year 2000 and 2023 that consider both mechanical and hydrological effects of vegetation on slope stability. The novelty of this study is that it aims to provide a comprehensive overview of how these effects are parametrized in the existing models and how many parameters are needed to each model. To this end, a systematic review of the peer-reviewed literature published between January 2000 and June 2023 based on the internationally recognized databases Web of Science, Scopus, and Science Direct is conducted. The analysis is divided in two parts: (1) a qualitative overview of all the identified and selected models and (2) a detailed analysis of how the effects of vegetation on slope stability are parametrized for a selection of the models analyzed in (1). Results of the two analysis approaches employed are shown and discussed. In conclusion, limitations and potential for improvement are discussed, with recommendations about when to use which models, depending on the aim of the study.

Nature-based Solutions for shallow landslide risk reduction

Nature-based Solutions (NbS) are defined by the UNEA (United Nations Environment Programme (UNEP) 2022) as “actions to protect, conserve, restore, sustainably use and manage natural or modified terrestrial, freshwater, coastal and marine ecosystems, which address social, economic and environmental challenges effectively and adaptively, while simultaneously providing human well-being, ecosystem services and resilience and biodiversity benefits.” NbS should not be considered a single approach but an umbrella term for various ecosystem-based approaches (Preti et al. 2022; Seddon et al. 2021). These approaches are considered cost-effective, win-win measures as they are seen as no-regret solutions: on the one hand, they enhance biodiversity conservation, and on the other, they improve livelihood security while reducing risk (Seddon et al. 2021). For shallow landslide risk reduction, protection forests have gained visibility as an NbS from the local to the international level because of their ecological features that assist in the mitigation of this hazard (Dorren and Schwarz 2016; Kalsnes and Capobianco 2019; Moos et al. 2018; Ng et al. 2022; Stokes et al. 2009; Wehrli and Dorren 2013). However, proper forest management is needed in light of the global change impacts that may be threating their protective effect (Moos et al. 2023).

Mechanical and hydrological effects of vegetation on slope stability

The following sections provide an introduction to the commonly used mechanical and hydrological frameworks for modeling the different effects of vegetation on slope stability.

Types of mechanical effects

The most recognized positive effect of vegetation on slope stability is the additional root reinforcement of soil. Plants’ root systems can increase the shear strength of soil through the combined action of fine and large roots (diameter smaller and bigger than 2 mm, respectively (Schwarz et al. 2010b)). Large, woody roots can anchor the superficial soil layers to more stable substrates of soil or bedrock by crossing potential planes of weakness (Stokes et al. 2009). Fine roots can strengthen the bonds with the soil particles, increasing the overall cohesion of the soil-root matrix (Pollen and Simon 2005; Schwarz et al. 2010a; Waldron 1977; Wu et al. 1979). As illustrated in Fig. 1, the root reinforcement can act both on the base of a potential failure plane (often denoted as “basal reinforcement”) and on the lateral sides of the landslide body (often denoted as “lateral reinforcement”).

Surcharge on the slope from vegetation can also influence slope stability. Whether the effect is positive or negative depends on the slope geometry with respect to the mechanical properties of the soil (Gray and Megahan 1981; Greenway et al. 1987). However, this factor is proven generally to be modest (Fan and Lai 2014) and is often neglected in slope stability assessments.

Another mechanical effect of vegetation is the drag force from wind on trees and bushes. The wind load on trees is transmitted as moments and forces through the branches to the trunk and towards the ground and taken up by the root system (Kim et al. 2020). Similar to the vegetation surcharge, this effect is also often neglected in slope stability analysis.

Existing root reinforcement models

The Wu and Waldron model (WWM)

Wu and Waldron (Waldron 1977; Wu et al. 1979) developed a novel approach to quantify roots’ contribution to reinforcement of soil. Their studies recognized that root reinforcement can be expressed as an additional “root cohesion” term, \({c}_{r}\) (kPa), in the Mohr-Coulomb failure criterion. The modified Mohr-Coulomb failure criterion is then (1)

$${\tau }_{b}=c^{\prime}+{c}_{r}+{\sigma }_{n}\text{tan}\varphi$$
(1)

where \({\tau }_{b}\) (kPa) is the basal shear resistance, \(c{\prime}\) (kPa) is the effective soil cohesion, \({\sigma }_{n}\) (kPa) is the normal stress on the failure plane, and \(\varphi\) (°) is the friction angle of the soil.

Waldron (1977) assumed that all roots passing through the shear plane are initially perpendicular to the slip plane and that tensile stress, \({t}_{r}\), is transferred from the soil to the roots as is the case for laterally loaded piles subjected to shearing. The total root cohesion is calculated by summing the tensile forces of individual roots crossing a failure plane (2):

$${c}_{r}=(\text{sin}\varepsilon +\text{cos}\varepsilon \text{tan}\varphi ){t}_{r}\text{RAR}$$
(2)

where \(\varepsilon\) (°) is the shear distortion with respect to the vertical, \({t}_{r}\) (kPa) is average tensile strength of roots per unit area of soil, \(A\) (m2), and \(\text{RAR}=\frac{{A}_{r}}{A}\) is the root area ratio (−), where \({A}_{r}\) (m2) is the total cross section of roots in \(A\).

Later, Wu et al. (1979) demonstrated that the root cohesion is rather insensitive to the distortion angle during shearing and the friction angle of the soil; hence, the root cohesion can be calculated as a constant \(k'\) (varying between 1.1 and 1.3), multiplied by the sum of the tensile stresses of the individual roots, \({t}_{r}\), and the root area ratio, according to the following Eq. (3):

$${c}_{r}=k'{t}_{r}\text{RAR}$$
(3)

One major limitation with the WWM is that it assumes that all roots break simultaneously; hence, it ignores the possibility for roots to slip and the possibility for load distribution between the roots due to progressive failure of the roots. Later studies have shown that this limitation causes the WWM to over-estimate the root reinforcement by up to 215% (Docker and Hubble 2008; Hao et al. 2023). In order to compensate for this over-estimation of root contribution to soil strength, a reduction factor, \(k''\), has been included in Eq. (3), with a value often ranging between 0.3 and 0.8 (Greenwood et al. 2004; Hao et al. 2023; Preti and Schwarz 2006). The most used form of the WWM today can be expressed with Eq. (4):

$$c_r=k'k''t_r\text{RAR}$$
(4)

The fiber bundle model (FBM)

Pollen and Simon (2005) adopted a fiber bundle model (FBM), first proposed by Daniels (1945), to address the problem of quantifying the root reinforcement of soil. The basic principle of the general FBM is that the maximum load withstood by a bundle of fibers is less than the sum of each of the fiber’s individual strength, because the different fibers will mobilize and break under different loading conditions (i.e., the contrary of what is assumed in the WWM) (Pollen and Simon 2005). The FBM proposed by Pollen and Simon (2005) accounts for the progressive rupture of roots, and the redistribution of loads from the ruptured to the remaining roots in the bundle.

The model simulates the process of root breakage by initially loading the bundle until one root breaks. Thereafter, the load this root was holding is redistributed to the other roots in the bundle. If this increment of load applied to the remaining roots causes another root to break, the redistribution process is repeated. This continues until the bundle withstands the applied load (unless all roots are broken, in which case the simulations stop). Subsequently, an additional load increment is applied to the bundle causing a new root to break, and the above-mentioned process restarts. This sequence will continue until all the fibers in the bundle are broken (Pollen and Simon 2005).

The model employs a global load shearing approach, which means that when a root breaks, the stress is redistributed equally to the rest of the bundle (Hidalgo et al. 2001). This assumption is considered adequate for relatively small streambank failures, but is a limitation in the case of larger scale slope failures. Pollen and Simon (2005) emphasize that in the case of slope stability failures, a local load sharing approach would be more suitable, as the redistribution of forces would most likely occur locally in the slope and not globally over the entire failure surface. Following the local shearing approach, the load would be redistributed to intact roots in the vicinity of the broken root (Hidalgo et al. 2001).

The model additionally speculates that all the roots will break and not fail due to slippage from the soil. Also, the elastic properties are assumed to be equal for all roots, and the roots are assumed to be parallel, i.e., root tortuosity is not considered (Pollen and Simon 2005).

As for the WWM, the final output of the model is an additional root cohesion term (\({c}_{r})\) to be included in the Mohr-Coulomb failure criterion (1). The maximum tensile strength of the root bundle can be calculated with Eq. (5):

$${t}_{r}=\text{max}({t}_{r,j}\text{RAR}{N}_{\text{intact}})$$
(5)

where \(j\) [−] represents the weakest root that is still intact upon loading of the root bundle of \({N}_{\text{intact}}\) [−] roots. The value, \({t}_{r}\), obtained from Eq. (5) can then be implemented in Eq. (3) to calculate the additional cohesion provided by the root bundle.

Root bundle model (RBM)

To overcome the limitations embedded in the model proposed by Pollen and Simon (2005) of assuming that all roots have the same elastic properties and that all roots fail due to breakage, Schwarz et al. (2010a) proposed a new version of the FBM, called the root bundle model (RBM). In the RBM, roots with different diameters are assigned different lengths and mechanical properties. Moreover, the model accounts for root-soil interactions by including mechanisms of root breakage and root slippage as a function of soil type, confining pressure and soil moisture. The model is defined as a strain step loading model, where the strain is the controlled parameter (in contrast to the FBM, where the load is incremented to induce strain). The displacements are imposed equally for all the roots in the bundle.

In the RBM, the distribution of roots of different diameters is parameterized using a Weibull probability function (6):

$$p(\varnothing_i;m,k)=\frac m{k^m}\varnothing_i^{m-1}e^{-\left(\frac{\varnothing_i}k\right)^m}$$
(6)

where \(p(\varnothing_i)\) is the probability that a mapped root in a profile belongs to the root diameter class \(\varnothing_i\) (mm), and \(m\) [−] and \(k\) [−] are the shape and scale parameters of the Weibull probability density function, respectively.

The total pullout force of the bundle \({F}_{\text{TOT}}\) (kPa), as a function of displacement, \(\Delta x\) (m), is calculated by summing the maximum pullout force of each individual root, \({F}_{i}\) (kPa), of diameter class \(i\) [−] multiplied by the number of roots in the corresponding diameter class, \(N_{\varnothing_i}\), (7):

$$F_\text{TOT}(\Delta x)=\sum^{i_{max}}_{i=1}F_i(\Delta x)N_{\varnothing_{i}}$$
(7)

An assumption of the model is that the roots bridging a crack will be loaded under tension and that the reinforcing effects of the interaction between neighbouring roots are negligible, as demonstrated in experimental studies conducted by Giadrossich et al. (2013). The final output of the model is a tensional force, which if desired can be considered per unit area and hence providing an additional root cohesion value (\({c}_{r})\), similar to WWM and FBM.

The root bundle model with root-failure Weibull survival function (RBMw)

A further development of the RBM was proposed by Schwarz et al. (2013), referred to as the RBMw. In this model, the strength variability within a root class is accounted for using a Weibull survival function, which is a probability function that captures the failure probability of a complex system beyond a threshold. The probability of a root to survive is a function of a normalized displacement \((\Delta {x}^{*})\) and is given by the Weibull survival function (8):

$$S(\Delta {x}^{*})={e}^{-{\left(\frac{\Delta {x}^{*}}{{\lambda }^{*}}\right)}^{\omega }}$$
(8)

where \(\omega\) is the Weibull exponent (shape factor) and \({\lambda }^{*}\) is the scaling factor.

The tensile force, representing the root reinforcement, of a bundle of roots is obtained by summing the force contribution from each root (7) multiplied by the survival function, \(S\), as shown in Eq. (9).

$$F_\text{TOT}(\Delta x)=\sum^{i_{max}}_{i=1}F_i(\Delta x)N_{\varnothing_{i}}S(\Delta x^\ast)$$
(9)

RBMw is based on the same assumptions of the RBM (i.e., roots bridging a crack will be loaded under tension, negligible interaction forces between neighbouring roots), and the output can be treated as additional root cohesion (\({c}_{r})\), when the tensional force is considered per unit area.

Types of hydrological effects

Precipitation-induced vertical flow of water in slopes is commonly modeled by different variations of Richards’ equation, e.g., expressed with the following one-dimensional Eq. (10):

$$\frac{d{\theta }_{m}}{dt}=\frac{d}{dz}\left[k\left(\psi \right)\left(\frac{d\psi }{dz}+1\right)\right]-{S}_{t}(\psi )$$
(10)

where \({\theta }_{m}\) [−] is the volumetric water content at field moisture content, \(t\) (day) is the elapsed time, \(z\) (m) is depth, \(k\) (m/day) is the soil hydraulic conductivity, \(\psi\) (m) is the matric suction, and \({S}_{t}\) (day−1) is the sink term which represents the volume of water transpired by a plant integrating over the entire root zone for a given time interval (Feddes et al. 1976; Roberts-Self and Tarantino 2023). Hence, the soil moisture distribution is affected by vegetation through root-water uptake driven by transpiration (Fig. 1). Roberts-Self and Tarantino (2023) have recently proposed a change in the Feddes et al. (1976) reduction function to transform it from being an empirical to a physically based function.

Transpiration leads to desaturation of the soil which again leads to a lowering of the water table and suction development in the unsaturated zone (Zhu et al. 2018). The transpiration-induced soil suction acts as a reinforcement on slope stability and can be included in the Mohr-Coulomb failure criterion as an additional cohesion term, \({c}_{\psi }\) (kPa), as shown in Eq. (11):

$${\tau }_{b}={c}'+{c}_{r}+{c}_{\psi }+{\sigma }_{b}\text{tan}\varphi ,{ c}_{\psi }={\gamma }_{w}\psi \chi \text{tan}\varphi$$
(11)

where \({\gamma }_{w}\) (kN/m3) is the unit weight of water and the parameter \(\chi\) [−] ranges from 0 and 1 and represents the variation from fully saturated (0) to dry (1) conditions and can be calculated with different approaches (Lepore et al. 2013). One frequently used approach was proposed by Vanapalli et al. (1996), where the parameter is assumed equal to the effective saturation of the soil, calculated following Eq. (12):

$$\chi =\left(\frac{{\theta }_{m}-{\theta }_{r}}{{\theta }_{\text{sat}}-{\theta }_{r}}\right)$$
(12)

where \({\theta }_{m}\), \({\theta }_{r}\), and \({\theta }_{\text{sat}}\) [−] are the field moisture content, residual volumetric water content, and volumetric water content at saturation, respectively.

The matric suction can either be measured, or calculated as a function of soil moisture, e.g., following the Brooks and Corey (1964) approach, expressed by Eq. (13):

$$\psi ={\psi }_{A}{\left(\frac{{\theta }_{m}-{\theta }_{r}}{{\theta }_{\text{sat}}-{\theta }_{r}}\right)}^{-\frac{1}{\lambda }}$$
(13)

where \({\psi }_{A}\) (m) is the air entry pressure and \(\lambda\) [−] is an empirical modeling parameter.

As mentioned above, transpiration by plants can reinforce the soil by inducing suction. Additionally, evaporation from intercepted water on, for example, tree canopies, bushes, and grass, acts to reduce the amount of water in the soil (i.e., reducing the pore water pressure). Several methods for estimating the evapotranspiration exist, but the pioneer in developing a formulation for this purpose was Penman (1948), who developed a model that combines the mass transfer with the energy balance. Later, Monteith (1965) modified Penman’s model, to include physiological characteristics of plants. Hence, the original Penman-Monteith Eq. (14) (Monteith 1965) for estimating actual evapotranspiration, \({E}_{a}\) (mm/day), accounts for the surface resistance, which entails the resistance of water vapor movement through the leaf stomata and soil surface, and for the aerodynamic resistance, which describes the resistance of vertical water vapor diffusion from a surface to the surrounding air:

$${E}_{a}=\frac{{R}_{n}\Delta +\frac{\rho {c}_{p}{\delta }_{e}}{{r}_{a}}}{{\lambda }_{\text{heat}}\left[\Delta +{\gamma }_{\text{psyc}}\left(1+\frac{{r}_{c}}{{r}_{a}}\right)\right]}$$
(14)

where \({R}_{n}\) (W/m3) is the net radiation, \(\Delta\) (Pa/K) is the rate of increase with temperature of saturation vapor pressure of water at air temperature, \(\rho\) (kg/m3) is the density of air, \({c}_{p}\) (J/kgK) is the specific heat of air at constant pressure, \({\delta }_{e}\) (Pa) is the vapor pressure deficit of air, \({\lambda }_{\text{heat}}\) (J/kg) is the latent heat of vaporization per unit mass, \({\gamma }_{\text{psyc}}\) (Pa/K) is the psychrometric constant, and \({r}_{a}\) and \({r}_{c}\) (s/m) are the aerodynamic and canopy resistance to water vapor transport, respectively.

Another process that reduces water infiltration to the soil is the interception of precipitation by tree canopies, bushes, and grass (Fig. 1). The storage capacity of vegetation depends on plant species and age, as well as forest structure (Carlyle-Moses and Gash 2011; Herbst et al. 2008). Also for this process, several models exist, but they are often based on the Rutter et al. (1971) model. The change in stored water on the canopies, \({C}_{w}\) (mm), during a storm can be expressed as Eq. (15) when the amount of stored water is equal to or greater than and Eq. (16) when the amount of stored water is smaller than the storage capacity, \({S}_{c}\) (mm):

$$\frac{d{C}_{w}}{dt}=\left(1-{p}_{t}\right){P}_{r}-{E}_{p,e}-k{e}^{b{C}_{w}},\text{ when }{C}_{w}\ge {S}_{c}$$
(15)
$$\frac{d{C}_{w}}{dt}=\left(1-{p}_{t}\right){P}_{r}-{E}_{p,e}\frac{{C}_{w}}{{S}_{c}}-k{e}^{b{C}_{w}},\text{ when }{C}_{w}<{S}_{c}$$
(16)

where \({p}_{t}\) [−] is the proportion of rainfall passing through the canopy (throughfall), \({P}_{r}\) (mm/day) is the precipitation rate, \({E}_{p,e}\) (mm/day) is the potential evaporation from wet foliage, and \(k\) (mm/day) and \(b\) (m−1) are model parameters.

The presence of roots in the vadose zone can increase or decrease the soil water retention capability, depending on type and age of plants, and vegetation spacing (Ni et al. 2019b). In studies that show an increase in water retention capability, the researchers argue that this is due to blockage of soil pores by roots, while the argument used in studies that present the opposite response, i.e., a decrease in water retention capability, is the crack formation due to, e.g., repeated shrinkage-swelling cycles of the soil and root decay and growth dynamics (Ni et al. 2019a). Decayed roots can also influence slope stability as they create channels in the ground and cause preferential flow paths for water (Fig. 1). This may cause (i) the slopes to drain faster (i.e., reducing pore water pressures) and (ii) more water to infiltrate the soil (i.e., increasing pore water pressures) (Ni et al. 2019b; Shao et al. 2017; Vergani and Graf 2016). It is therefore challenging to evaluate whether an increase in permeability due to macropores in the soil has a positive or negative effect on slope stability. The growth of roots on the other hand can reduce the soil void ratio and hence reduce the hydraulic conductivity of the soil (Ng et al. 2016). Few studies on these particular effects exist in the literature (Ni et al. 2019a, b; Vergani et al. 2017), and the parametrization of these phenomena in the slope stability models analyzed herein is not assessed.

Modeling approaches to slope stability

Approaches for assessing slope stability can be qualitative or quantitative. Qualitative approaches are subjective as they rely on the understanding of geological and geomorphological conditions that govern slope instability (Guzzetti et al. 1999). Using these approaches, landslide susceptibility is portrayed using descriptive (i.e., qualitative) terms. Quantitative methods, on the other hand, produce numerical estimates, i.e., probabilities of occurrence of landslide phenomena in any susceptibility zone (Guzzetti et al. 1999). The quantitative methods can be further categorized into two subgroups: statistical and physically based methods.

Statistical slope stability models use statistical methods to analyze the relationships between landslide occurrence and various triggering factors, such as rainfall intensity and duration, topography, and soil properties. These models assume that the same causes are more likely to produce the same effects (Oliveira et al. 2017). Statistical models can be used to quantify the probability of landslide occurrence in different areas based on historical landslide data and the analysis of relevant factors (Frattini et al. 2009).

Physically based slope stability modeling is a more complex approach that involves the use of mathematical and computational models to simulate the physical processes that control slope stability. These models consider factors such as soil mechanics, hydrological conditions, and the interaction between precipitation and slope materials. Physically based models aim to provide a more detailed and mechanistic understanding of slope stability and can be used to simulate the behavior of slopes under different conditions (Alvioli et al. 2014). They are often used to assess the stability of specific slopes or areas, or to analyze the effects of different factors on slope stability (Meena et al. 2022). Physically based models can be divided into two groups: deterministic and probabilistic models. The deterministic models use finite quantities of input values and yield finite quantities of slope stability (usually the slope’s factor of safety, FS), while the probabilistic models use probability distributions of the input variables and yield probabilities of failure as output values.

As the aim of this review is to assess how the vegetation effect is parametrized in slope stability models, the choice falls naturally on physically based models as the modeling approach to study. Hence, only models following the physically based approach are included in this review.

Physically based stability models

Physically based models are generally composed of a hydrological and a geotechnical model that are coupled. The hydrological module serves to describe the porewater pressure and water movement within porous materials in space and time, and the geotechnical model serves to quantify the slope stability conditions (Capparelli and Versace 2011).

Physically based slope stability models can be 1-, 2-, or 3-dimensional and be based on different methods such as the limit analysis method, limit equilibrium method, finite element method, or discrete element method. Additionally, the models can allow for the geotechnical and hydrological conditions to be transient in time. Various combinations of analysis spatial scale and method provide the possibility to assess landslide susceptibility with different degrees of detail and accuracy. In general, 1- and 2-dimentional analysis using the simpler methods, such as the limit analysis or limit equilibrium method, requires less computational efforts than 3-dimentional analysis using the more sophisticated finite or discrete element methods.

The limit analysis method (LAM) considers the soil as a perfectly plastic material, which allows for the calculation of an upper and lower bound theorem for assessing a slope’s failure load (Drucker and Prager 1952). These theorems can be helpful as they provide an estimate of the range in which the exact solution can be found (Yu et al. 1998). The upper bound solution is determined by assessing whether the work done by internal stresses when subjected to an external load equals the work done by the same external load in an increment of displacement. If this is the case, the external load is assumed to be greater than the failure load of the slope. However, it is not assured that the external loads are in equilibrium with the internal stresses, and the failure mechanism is not necessarily the actual failure mechanism. The lower bound solution is determined by assessing whether an equilibrium distribution of stress over the whole slope can be found that balances an external loading on the stress boundary, and the stress is lower than the failure criterion of the material. If this is the case, then the external load is lower than the failure load. However, the strain and displacement of the material are not considered; hence, the stress is not necessarily the state of stress at failure (Yu et al. 1998).

The limit equilibrium (LEM) is a widely used method for slope stability assessment. It is based on the principle that a landslide body acts as a rigid block (i.e., it does not deform), and stability is quantified by assessing equilibrium between driving and resisting forces and/or momentum acting on this block. Different versions of the method exist, where the infinite slope method (ISM) and the method of slices (MS) are the most commonly applied. In the infinite slope method (Ewen et al. 2000; Lepore et al. 2013; Simoni et al. 2008), it is assumed that the slope has infinite length and width and that the failure surface is parallel to the slope. Hence, the method is applicable if the failure mechanism is assumed to be planar, and the depth to the failure plane is much less than the total extent of the slope, assumptions that often are valid for shallow landslides. Equilibrium of driving and resisting forces is solved for the slope to quantify its stability. When using the method of slices, on the other hand, the slope is divided into multiple slices and equilibrium of forces and/or moments acting on each slice is analyzed (Bishop 1955; Janbu 1954; Morgenstern and Price 1965). The failure surface is allowed to be of different shapes but must always be defined prior to the analysis. This method is suitable when the topography and/or depth to the failure surface is variable in the longitudinal direction with respect to the slope. Both the ISM and the MS are widely used because of their simplicity and computational efficiency (Chen et al. 2003).

The finite element method (FEM) is considered a powerful alternative to traditional limit analysis and limit equilibrium methods (Griffiths and Lane 1999) as it considers the constitutive relation between stress and strain under loading of the soil body. It is accurate, versatile, and requires less assumptions than LAM and LEM, especially regarding the failure mechanism (Griffiths and Lane 1999). The FEM allows for the analysis of failure surfaces of arbitrary shape, which is of interest as the critical surface of a slope may deviate significantly from a circle or a plane (Morgenstern and Price 1965). The method is based on the principle of continuum mechanics; i.e., it does not allow for opening of voids under deformation (e.g., tensional cracks in a slope) (Abderrahmane and Abdelmadjid 2016).

The discrete element method (DEM) is a discontinuous method that can model the interaction between distinct particles, making it suitable to analyze the behavior of granular materials such as soil (Cundall and Strack 1979). The main advantages of the model are that the need for a constitutive model of the soil is bypassed and that it can be used for modeling of both landslide triggering, propagation, and runout (Abderrahmane and Abdelmadjid 2016).

Review methods

The systematic literature review was conducted following the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) statement (Moher et al. 2009), which consists of a 27-item checklist and a four-phase flow diagram. The phases of the flow diagram consist of identification, screening, eligibility, and inclusion of records (i.e., articles published in peer-reviewed literature). The phases and how they are applied to this research are described below and illustrated in Fig. 2, whereas the full checklist can be found on the PRISMA website (http://www.prisma-statement.org/).

In the identification phase, records are identified via a pre-defined search strategy in selected literature databases. For this study, the databases used were the platforms for peer-reviewed publications Web of Science (WoS), Scopus, and Science Direct (SD). Slightly different Boolean functions were used to specify the search terms in the different platforms (complete search term list in Table 5 in the Appendix). For example, the search in WoS was constructed as

$$\left(Slope\;NEAR/2\;stability\right)\;AND\;model^\ast;\,AND\;xx^\ast$$

where “xx” consisted of the words: root, vegetation, plant, tree, shrub, and grass. For all searches, only research articles were assessed (excluding conference proceedings, grey literature, etc.) in order to assure quality of the identified publications. Duplicates were excluded before moving to the next phase.

In the screening phase, the publications identified from the previous phase are assessed to determine which records are meeting the inclusion criterion, i.e., are relevant to the literature review. For this study, the titles and abstracts of the identified records from the identification phase are screened. The records excluded in this step are those not treating the subjects of slope stability modeling and vegetation or are published before year 2000.

In the eligibility phase, a more thorough assessment of the articles selected after the screening phase is conducted. In this study, the full manuscript of the selected publications was examined in order to assess whether the eligibility criterion was met. Records were excluded in this step if they did not introduce a novel slope stability model including the effect of vegetation and/or was not written in English and/or a full manuscript was not accessible.

In the inclusion phase, records that passed the eligibility-check are analyzed. In this study, the remaining database of records after the eligibility-check phase was integrated with six publications identified through the previously mentioned review articles treating a similar topic as this study (Bordoloi and Ng 2020; Masi et al. 2021; Murgia et al. 2022). To support the first part of this review, which is the qualitative synthesis, a qualitative analysis was conducted for all of the publications in the inclusion phase. This consisted of individually analyzing the type and scale of the vegetation effects included in the models which provided a qualitative synthesis of the state-of-the-art of the investigated topic. Subsequently, for the second part of this review, consisting in the detailed analysis, the parameterization approaches to account of the effects of vegetation in the different models were analyzed and compared. This latter analysis focused on articles published in the first half of 2023 as well as the ten publications with the highest normalized number of citations (i.e., < number of citations > /(2023- < publication year >)). Finally, the application rate of these models was investigated through a supplementary literature search in Scopus.

Results

Collected and selected records

A summary of the process of data collection and selection is illustrated in Fig. 2. The above-mentioned literature searches performed in SD, WoS, and Scopus returned 126, 576, and 869 records (i.e., research articles) respectively. Of these, 525 records were duplicates, leaving a total of 1046 research articles to be screened. After the screening process where the title and abstract of the identified records were analyzed, 940 records were excluded, leaving 106 research articles to be assessed by their full manuscript. This assessment resulted in another 70 of the records to be excluded, leaving a total of 36 relevant records to be included in further analysis. Additionally, another six relevant publications were identified from a non-systematic review. Hence, 42 identified publications, representing 42 unique slope stability models, formed the basis for this literature review (Abdollahi et al. 2023; Arnone et al. 2016; Arnone et al. 2015; Cislaghi et al. 2017; Cohen and Schwarz 2017; Dhakal and Sidle 2003; Emadi-Tafti and Ataie-Ashtiani 2019; Ewen et al. 2000; Gong et al. 2021; Gorsevski et al. 2006; Greenwood 2006; Hess et al. 2017; Huang et al. 2021; Hwang et al. 2015; Ji et al. 2012; Jia et al. 2022; Kim et al. 2013; Kokutse et al. 2006; Kuriakose et al. 2006; Lepore et al. 2013; Li et al. 2022; Mao et al. 2014; Medina et al. 2021; Milledge et al. 2014; Montrasio et al. 2023; Moresi et al. 2020; Ng et al. 2021; Nguyen et al. 2019; Ni et al. 2018; Okada et al. 2023; Raimondi et al. 2023; Rees and Ali 2012; Salvatici et al. 2018; Simoni et al. 2008; Strauch et al. 2018; Switala and Wu 2018; Saadatkhah et al. 2016; Temgoua et al. 2016; Tiwari et al. 2013; van Zadelhoff et al. 2022; Wilkinson et al. 2002a; Wilkinson et al. 2002b).

Fig. 2
figure 2

Flow diagram describing the collection and selection process of records analyzed in the systematic literature review, following the PRISMA approach (Moher et al. 2009)

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Qualitative synthesis of existing models

In the previously described process of collecting and selecting records, a total of 1052 publications were assessed, from which 96% were excluded leaving 4%, or 42 models, as the basis for the analysis presented in this study (Fig. 2). The high percentage of exclusion of records (96%) is due to the use of a broad set of search terms (Appendix, Table 5) and was hence to be expected. The complete overview of the qualitative analysis of the 42 models is available in Table S1 in the supplementary material to this publication.

The number of models presented per year shows a significant positive trend from 2000 until the first half of 2023 (Fig. 3) with 0.64 as the average number of models per year from 2000 to 2011 and 2.75 from 2012 to 2023.

Fig. 3
figure 3

Novel models for slope stability assessment accounting for vegetation effects included in the review. a Number of normalized citations of each publication (i.e., < number of citations > /(2023- < publication year >)). Selected models for a detailed assessment of parametrization of vegetation effect are marked with dark green bars and dark green rectangles. b Models distributed on publication year. In 2023, only records published in the first half year are considered (pale-colored bar). The dotted line shows the linear trend for the investigated time period

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The main author’s country affiliation of the collected 42 studies is spread over 17 countries around the world (Fig. 4), with the four most contributing countries being (ranged from most to least): Italy, USA, UK, and China. Most of the studies are placed in central Europe (20 studies), while 13 and 9 studies are conducted in Asia and North America, respectively. None of the selected publications have their main author affiliations located in the southern hemisphere, nor the Nordic countries or the Arctics.

Fig. 4
figure 4

World map and cake diagram showing affiliation countries of the first author of the 42 publications included in the review. The sizes of the stars in the map are proportional to the number of studies from each country

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The 42 models included in the qualitative analysis can be divided into two groups based on scale of application: 21 of the models are suitable for slope-scale applications, and 21 of the models are suitable for regional-scale applications (Fig. 5). Of the 21 slope-scale models, 20 use a deterministic approach to quantify slope stability; hence, only one model uses a probabilistic approach (Fig. 5, left). For the 21 regional-scale models, 11 of the models use a deterministic approach, and 10 use a probabilistic approach to quantify slope stability (Fig. 5, right).

Fig. 5
figure 5

Central diagram: proportion of models developed for slope- and regional-scale analysis. Left and right diagrams: proportion of the slope-scale and regional-scale models, respectively, that are defined as probabilistic and deterministic models

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For the slope stability calculations, a variety of geotechnical models are employed by the models analyzed in this study (Fig. 6). As many as 32 of the models are based on the limit equilibrium method (LEM), divided between the infinite slope method (LEM–ISM), the method of slices (LEM–MS), and a three-dimensional method (LEM–3D), with a number of 23 (55%), 7 (17%), and 2 (5%) applications, respectively. The second most frequently used model is the finite element model (FEM), with a total of eight applications (19%). Additionally, one (2%) application of the discrete element method (DEM) and one (2%) application of the explicit smoothed particle finite element method (eSPFEM) are observed (consult Jia et al. (2022) for details on eSPFEM).

Fig. 6
figure 6

Proportion of geotechnical models adopted for the slope stability modeling in the analyzed publications

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In order to quantify the mechanical effect of roots, the majority of the models, 22 out of 39 (excluding models which do not take into account mechanical root reinforcement), employ the simple WWM, either as defined by Wu et al. (1979), or including a reduction factor as, e.g., proposed by Greenwood et al. (2004) or Preti and Schwarz (2006) (Fig. 7). More sophisticated root reinforcement models, such as the FBM, RBM or the RBMw, are only applied in one to two of the analyzed models (Fig. 7). Six of the models do not include any root reinforcement model in their formulation, even though they include an additional cohesion term accounting for the root reinforcement in the calculation of the FS. The values of root cohesion are in these studies generally taken from reference values found in literature. The six remaining slope stability models apply less known root reinforcement models (Fig. 7). These latter root reinforcement models are not investigated in this study; however, references to their model descriptions can be found in Table S1 in the supplementary material to this publication.

Fig. 7
figure 7

Proportion of root reinforcement models adopted to account for the mechanical effects of roots in the models. Only models that account for the mechanical effect of root reinforcement are included (i.e., 39 out of 42 models)

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The analyzed models account for the mechanical and hydrological effects of vegetation on slope stability to varying degrees (Fig. 8a). All, but three of the models, account for the mechanical effect(s), while only 27 of the models account for the hydrological effect(s) of vegetation. Slightly less than half of the models account for some extent of both the mechanical and hydrological effects of vegetation. As for the mechanical effects, the most common feature to include is the basal root reinforcement, followed by the surcharge from vegetation (Fig. 8b). Less common is the inclusion of lateral reinforcement and the drag force by wind on trees. The main hydrological effects of vegetation, i.e., accounting for the unsaturated zone, evapotranspiration, and interception loss (intercepted water lost through evaporation or plant absorption), are more evenly distributed between the models (Fig. 8c).

Fig. 8
figure 8

a Type of vegetation effect(s) accounted for in the analyzed models, b type of mechanical effects, and c type of hydrological effects of vegetation accounted for in the models

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The effects of vegetation on slope stability are generally not constant in time and space. Around 40% of the 42 models allow for variation in root reinforcement with depth, whereas around 76% of the models allow for spatial variation in vegetation effect (Fig. 9). Spatial variation is either considered at a local scale (36%), including the effect of a single tree, or at a regional scale, including the effect of land use and land cover (LULC) (40%). Additionally, for some models (19%), the variation of the effects of vegetation on slope stability is allowed to vary with time (Fig. 9).

Fig. 9
figure 9

Temporal and spatial variation of vegetation effects accounted for in the analyzed models

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Detailed analysis of parametrization of vegetation effects in selected models

A detailed study of the parametrization of the effects of vegetation has been carried out for the ten publications with the highest normalized citation number (all above five citations per year), together with the four identified studies published in the first half of 2023 (Fig. 3a). The different approaches to parameterize the effects of vegetation in the 14 selected models are tabulated in Tables 1, 2, 3. Table S3 in the supplementary material available to this publication provides an overview of the parametrization of the factor of safety (FS). Additionally, the results are summarized in this section, divided in two subsections treating slope- and regional-scale models separately. The nomenclature utilized in the equations listed in the tables are provided in Table 4.

Table 1 Synthesis of the 14 slope stability models selected for the detailed study, listed in chronological order. Det. deterministic, Prob. probabilistic, LEM limit equilibrium method, ISM infinite slope method, DEM discrete element method, Basal/lateral basal/lateral root reinforcement, LULC land use and land cover
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Table 2 Parametrization of mechanical root reinforcement in the models
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Table 3 Parametrization of unsaturated zone, evapotranspiration, and interception in the models
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Slope-scale models

Five of the models are developed for slope-scale analysis (Table 1): MD-STAB (Milledge et al. 2014), SOSlope (Cohen and Schwarz 2017), the model proposed by Ni et al. (2018) (herein denoted the “Ni-model”), the model proposed by Okada et al. (2023) (herein denoted the “Okada-model”), and the model proposed by Abdollahi et al. (2023) (herein denoted the “Abdollahi-model”). All off these models employ a deterministic approach for the calculation of FS (Table 1). The MD-STAB model, Ni model, and the Abdollahi model apply the limit equilibrium, infinite slope method (LEM-ISM) as their geotechnical model, while the SOSlope and Okada model apply the discrete element method (DEM) and a three-dimensional limit equilibrium method (LEM-3D), respectively (Table 1). The Ni model and the Abdollahi model employ Richards’ equation for vertical precipitation infiltration (10) whereas the SOSlope model accounts for a combination of slow matrix flow and fast preferential flow, based on the principles of TOPMODEL (Beven and Germann 1982), while the MD-STAB model and the Okada model assume hydrostatic groundwater conditions.

The mechanical root reinforcement effect is parametrized by the WWM (Wu et al. 1979) in the Ni model and in the Okada model and by the RBMw (Schwarz et al. 2013) in the SOSlope model. In the MD-STAB model, the mechanical effect is parametrized by the Dunne/Benda-Dunne model (Benda and Dunne 1997; Dunne 1991), which allows a reference cohesion value to decrease with depth of the failure plane. Finally, in the Abdullahi model, the analytical fiber bundle model (AFBM) (Cohen et al. 2011) is used, which is an analytical solution to the FBM. Details about the parametrization of the mechanical effect of vegetation are found in Tables 1 and 2. For all the models, except the SOSlope model, the mechanical root reinforcement is included in the FS calculation as an additional root-cohesion term (Table S3). In the SOSlope model on the other hand, root reinforcement is accounted for as lateral bond forces between the discrete elements in the model when calculating FS. Additionally, the Okada model includes the effect of vegetation weight in the FS formulation (Table 1, Table S3).

The hydrological effects of vegetation are not included in the MD-STAB model or the Okada model (Tables 1 and 3). The SOSlope model includes the effect of suction in the unsaturated zone by using a dual-porosity model, but not the effects of evapotranspiration and interception loss (Table 3). The Ni model and the Abdollahi model include the effect of the unsaturated zone by including a suction-induced cohesion term in the FS calculation (Table 3, Table S3) and evapotranspiration by introducing a sink term in Richards’ Eq. (10) (Table 3). The effect of interception loss is not included in the latter models (Tables 1 and 3).

Regional-scale models

The remaining nine models are developed mainly for regional scale applications (Table 1). Four of these models use a deterministic approach for the calculations, namely, SHETRAN (Ewen et al. 2000), tRIBS-VEGGIE-Landslide (Lepore et al. 2013), SPRIn-SL (Raimondi et al. 2023), and G-SLIP (Montrasio et al. 2023), and five use a probabilistic approach, namely, GEOtop-FS (Simoni et al. 2008), the tRIBS-VEGGIE-Landslide model modified by Arnone et al. (2016) (herein denoted tRIBS-VEGGIE-Landslide-Arnone), Landslide-Probability (Strauch et al. 2018), the HIRESS model modified by Salvatici et al. (2018) (herein denoted HIRESS-Salvatici), and the FSLAM (Medina et al. 2021). All the regional-scale models employ the limit equilibrium, infinite slope method (LEM-ISM) in their geotechnical model (Table 1). The SHETRAN model, GEOtop-FS model, both versions of the tRIBS-VEGGIE-Landslide model, and the HIRESS-Salvatici model use a version of Richards’ Eq. (10) for vertical precipitation infiltration (Table 1). The Landslide-Probability model uses the VIC hydrological model (Liang et al. 1994), the FSLAM uses the Montgomery and Dietrich (1994) model for lateral flow and the SCS-CN model (USDA 1986) for vertical flow, the SPRIn-SL uses the TOPOG model (O’Loughlin 1972) and Green and Ampt (1911) model as hydrological models, and the G-SLIP model uses a formulation proposed by Montrasio (2000) for modeling vertical infiltration of precipitation (Table 1).

The original tRIBS-VEGGIE-Landslide model does not include the mechanical root reinforcement. This model was modified by Arnone et al. (2016) and now allows a probabilistic modeling approach of the mechanical root reinforcement by employing the RBMw (Schwarz et al. 2013) (Tables 1 and 2). The mechanical root reinforcement effect is parametrized by the WWM (Wu et al. 1979) in the HIRESS-Salvatici model, the SPRIn-SL model, and the G-SLIP model (Tables 1 and 2). The remaining models (SHETRAN, GEOtop-FS, Landslide-Probability, and FSLAM) do not have a specified model for root reinforcement, but rely instead on, e.g., reference values from literature for including root cohesion in the FS calculation (Tables 1 and 2, Table S3).

Hydrological effects of vegetation are included in all the regional-scale models. Both versions of the tRIBS-VEGGIE-Landslide model, the HIRESS-Salvatici model, and the G-SLIP model include the unsaturated zone by adding a suction-induced cohesion term in the FS-calculation (Table 3, Table S3). The SHETRAN model, GEOtop-FS model, both the versions of the tRIBS-VEGGIE-Landslide model, and the Landslide-Probability model include evapotranspiration and interception loss following energy-mass-balance approaches such as the Penman-Monteith equation (Monteith 1965) and the Rutter equation (Rutter et al. 1971) (Table 3). In the FSLAM, the SPRIn-SL model, and the G-SLIP model, the effects of evapotranspiration and interception loss are indirectly accounted for by allowing a reduced amount of the precipitation to infiltrate the ground (Table 3). The effects of evapotranspiration and interception loss are not accounted for in the HIRESS-Salvatici model (Table 3).

Validation of models by case-study application

For all the 14 analyzed models, a validation through a case-study application is presented together with the model in the original publication. However, an additional search in Scopus, the database that yielded the most records in the collection phase in this study (Fig. 2), indicated a low application rate of all the 14 models to other case studies than the ones presented in the model-description publications. The SHETRAN model (Ewen et al. 2000) is reported in six other publications, with the tRIBS-VEGGIE-Landslide model (Lepore et al. 2013) and the FSLAM (Medina et al. 2021) registered in one and four applications, respectively. The countries of the case-study applications included Spain, Italy, Northern Ireland, India, China, Turkey, Puerto Rico, and Ecuador. For the remaining 11 models, no additional applications were documented in the peer-reviewed literature available through Scopus.

Limitations of the models

In Table 1, an effort has been made to point out some limitations to each of the models analyzed in this detailed study. However, these limitations should not be considered to be absolute; i.e., they might not be considered limitations as long as the purpose of future applications of the models are in line with the developer’s idea for the model. For example, the SOSlope model (Cohen and Schwarz 2017) was developed primarily to quantify the mechanical reinforcement effect of roots, clearly stating the choice of leaving out the different hydrological effects of vegetation. The fact that the model does not include the hydrological effects can therefore only be viewed as a limitation if the purpose of future applications of the model includes the study of the hydrological effects of vegetation. Other noted limitations might be of more universal importance, as, for example, the deterministic approach to regional-scale modeling adopted by tRIBS-VEGGIE-Landslide (Lepore et al. 2013). In fact, this point was later modified to a probabilistic approach in the newer version of the model, tRIBS-VEGGIE-Landslide-Arnone developed by Arnone et al. (2016).

Discussion

By analyzing the countries of affiliation of the main author of the identified studies (Fig. 4), it is found a clear relation between the number of studies and how susceptible to landslide that country is. For example, Italy, USA, and China are among the five countries with the highest economical losses due to landslides from 1930 to 2018 (CRED 2018). The same countries where on the top five list of number of studies analyzed in this review. Similar results were found by de Jesús Arce-Mojica et al. (2019). From these observations, one can conclude that the focus on protection forests for landslide mitigation has gained highest focus in landslide-prone countries.

Different models for different applications

The steady increase in the development of slope stability models that include the effect of vegetation since year 2000 (Fig. 3) reflects the research community’s increased attention towards NbS for hydromechanical risk reduction as an alternative for grey solutions. This is in agreement with the findings of Sudmeier-Rieux et al. (2021). An equal number of models developed for slope- and regional-scale applications were identified (Fig. 5), indicating that that the two fields of application are, on average since year 2000, receiving similar attention in the scientific community. The focus of the most recently proposed models can be divided in three groups based on topic as discussed in the following.

The first group includes models that try to reproduce, as realistically as possible, processes governing the effects of vegetation on slope stability. For example, the SOSlope model (Cohen and Schwarz 2017) is a strain-step discrete element model that reproduces the self-organized redistribution of forces on a slope during rainfall-triggered shallow landslides. In the model, the effects of tree size, spacing, weak zones, maximum root-size diameter, and different root strength configurations on slope stability can be investigated. Another model belonging to this group is the tRIBS-VEGGIE-Landslide-Arnone model (Arnone et al. 2016), including a branching topology model based on Leonardo’s rule (Richter 1970), which can provide an estimation of the amount of roots and the distribution of diameters with depth. Additionally, the model also accounts for the hydrological effects of vegetation, through accounting for the unsaturated zone and calculating the interception loss and evapotranspiration with an energy-mass-balance approach. Both models employ the RBMw (Schwarz et al. 2013) for root reinforcement quantification. The models in this group are particularly suitable for applications to case studies where the user has a good base of data for the characterization of the site, or when a detailed sensitivity analysis for investigating the effect of certain parameters is desired.

The second group of models is developed for regional susceptibility assessments with the aim to be computationally inexpensive and/or be user friendly, appealing to practitioners for their simplicity. These models generally employ simpler methods to account for the effect of vegetation on slope stability than the ones belonging to the first group. Examples of models belonging to this group are the FSLAM (Medina et al. 2021) and the SPRIn-SL model (Raimondi et al. 2023). The FSLAM is specifically developed to assess the probability of failure for large areas with a high-resolution topography in a short computational time, by combining a simplified hydrological model and an infinite slope model. The root reinforcement effect is accounted for by including a basal root cohesion term in the FS-formulation, but it is up to the user to estimate this input parameter (i.e., no root reinforcement model is included in the slope stability model). The SPRIn-SL model on the other hand aims to be a user-friendly framework for shallow-landslide susceptibility mapping and can be run directly from the QGIS processing toolbox supported by a user-friendly geographical interface. The mechanical effect of vegetation is included using the WWM (Wu et al. 1979) and accounted for in the FS-calculation as a basal root cohesion term. In both FSLAM and SPRIn-SL, the hydrological effect of vegetation is included indirectly by reducing the groundwater recharge by precipitation as a function of LULC. These models are particularly useful when the purpose is to obtain a quick overview of an area’s susceptibility to shallow landslide, without performing extensive field investigations.

The third group consists of models that are developed for specific sites or specific problems. An example of a site-specific model is the Okada model (Okada et al. 2023), developed specifically to study the effects on slope stability of the root-decay of the Japanese cedar trees. In the model, a shallow landslide is modeled as a sliding block, simplified as a three-prism model. The mechanical reinforcement is included using the WWM (Wu et al. 1979), and hydrological effects of vegetation are not included. An example of a problem-specific model is the Abdollahi model (Abdollahi et al. 2023), which is developed specifically to investigate the effect of wildfires on the stability of vegetated slopes. The model combines a hydromechanical infiltration model and an infinite slope stability model to simulate temporal changes in the depth profiles of soil water content, pressure head, and the resulting FS of vegetated slopes. Mechanical root reinforcement is accounted for using the analytical fiber bundle model (AFBM) (Cohen et al. 2011), which is a simplified version of the RBM proposed by Schwarz et al. (2010a), excluding root failure due to slippage, effect of effective soil pressure, and water content. The hydrological effect of vegetation is included by considering soil suction-induced cohesion in the FS formulation, and by including a transpiration-induced sink term in the calculation of evapotranspiration. Even though these models are stated by the authors to be site- or problem-specific, their applicability can easily be extended to other sites or other cases, as their formulations are physically based.

Vegetation effects typically included in the slope stability models

This study indicates that the mechanical reinforcement effects provided by roots are more often included in slope stability models than the hydrological effect of vegetation in the form of evapotranspiration, interception loss, or suction-induced cohesion in the unsaturated zone (Fig. 8). However, previous studies have found that the hydrological effects of vegetation can be at least, if not more, important than the mechanical effect of root reinforcement on slope stability (Arnone et al. 2016; Boldrin et al. 2017, 2021; Capobianco et al. 2021; Chirico et al. 2013), especially during dry summer months in temperate climates. Hence, to be able to realistically model the effects of vegetation on slope stability, it is important to include both the mechanical and hydrological effects of vegetation in the modeling scheme.

In the slope stability models assessed in detail (14 models), the approaches for including the hydrological effects of vegetation through evapotranspiration and interception loss can be divided in two main groups (Table 3): a sophisticated energy-mass-balance approach and a simplified conceptual approach. The models using the sophisticated energy-mass-balance approach (Arnone et al. 2016; Ewen et al. 2000; Lepore et al. 2013; Simoni et al. 2008; Strauch et al. 2018) are typically using variants of the Penman-Monteith equation (Monteith 1965) for the estimation of the evapotranspiration and variations of the Rutter model (Rutter et al. 1971) for the estimation of interception loss. The models using a simpler approach (Abdollahi et al. 2023; Medina et al. 2021; Montrasio et al. 2023; Ni et al. 2018; Raimondi et al. 2023) are typically using a reduction factor or a sink term in Richards’ equation for estimation of infiltration rate to account for the loss off available water to the soil due to evapotranspiration and interception loss. As mentioned, some models do not include the hydrological effects of vegetation at all (Cohen and Schwarz 2017; Milledge et al. 2014; Okada et al. 2023; Salvatici et al. 2018). A simple way to improve these models could be to include the hydrological effects using a simplified approach as described above.

The mechanical reinforcement is, as seen in Fig. 8 and Table S1, included mostly in the form of basal root cohesion, and only six of the 42 identified models include the effect of lateral reinforcement (Cislaghi et al. 2017; Cohen and Schwarz 2017; Hess et al. 2017; Milledge et al. 2014; Okada et al. 2023; van Zadelhoff et al. 2022). Despite the clear prevalence of including basal root reinforcement in stability models, studies have shown that the slip surface in shallow landslides often is found to be deeper than the depth of the roots; hence, roots do not necessarily provide basal root reinforcement (Bischetti et al. 2005; Tron et al. 2014). The validity of the current practice of modeling mechanical root reinforcement solely considering basal root cohesion can therefore be questioned. As a minimum, when slope stability modeling is to be performed for a specific site, it is of upmost importance to investigate the relative depth of the roots and typical slip surfaces to ensure the validity of the chosen slope stability model.

Of the in-depth analyzed articles in this study, only the MD-STAB model (Milledge et al. 2014), the SOSlope model (Cohen and Schwarz 2017), and the Okada-model (Okada et al. 2023) include the lateral effect of root reinforcement (Tables 2 and 5). The MD-STAB model includes the lateral reinforcement considering a depth-averaged cohesion applied to cross-slope vertical boundaries and to the failure surfaces of the upslope and downslope wedges of the landslide body. The Okada model also considers a depth-averaged lateral cohesion, but in this model, the principle of infinite slope is assumed in the longitudinal direction; hence, no upslope and downslope wedges are considered. In the SOSlope model, the lateral root reinforcement is accounted for as bond forces between the discrete elements in the model, both in the transversal and longitudinal directions with respect to the slope. Whether or not it is necessary to include the reinforcement contribution of the upslope and downslope wedges, as done in the MD-STAB model and the SOSlope model, must be viewed in the light of the characteristics of the studied landslide. In the case where the length of the landslide exceeds greatly the width, the reinforcement contribution by the upslope and downslope wedges might be of negligible importance (Milledge et al. 2012).

Vegetation shows generally large spatial variation; hence, it is important to include such variation when performing slope stability modeling, especially at the regional scale (Sidle 1992). In the qualitative analysis of the 42 models, regional spatial variation of vegetation reinforcement (LULC-dependent) is implemented in 17 models, and the possibility to account for single-tree contribution to vegetation reinforcement is present in 15 models (Fig. 9). According to previous studies (Roering et al. 2003; van Zadelhoff et al. 2022; Vergani et al. 2014), single-tree based modeling has a higher accuracy in predicting shallow landslides than when the variation of vegetation effects are limited to large scale variation. Among the selected models for the described parametrization analysis (14 models), only the SOSlope model (Cohen and Schwarz 2017) includes the possibility to account for single-tree effects, and it does so by the following three assumptions: (1) linear allometric relation between trunk size and root density, (2) power-law decay of root density with distance from the tree trunk, and (3) logarithmic decrease in root density with root-diameter size.

An important application of slope stability modeling accounting for the effect of vegetation is for land use planning and in developing forest management strategies. For these applications, it is particularly important to quantify changes in reinforcement effect provided by vegetation over time, as this will provide a good overview of how plants can contribute positively (or negatively) to slope stability within their life span. This aspect is becoming more and more important as a future climate is expected to be wilder, warmer, wetter, and including more intense precipitation events and longer periods of droughts, processes that will stress the plants’ life cycles and hence their contribution to slope stability (Li et al. 2023). For instance, an increase in temperature might cause a reduction of the water retention capacity of the soil due to root shrinkage, thermal expansion of liquid phase, and reduction in forces of capillarity and adsorption; i.e., the suction-induced cohesion effect will decrease (Wang et al. 2023). Furthermore, a more variable climate including frequent alternation between wetting and drying periods might cause the mechanical reinforcement effects of roots to deteriorate (Ma et al. 2021). Slope stability models accounting for vegetation effects could be used to forecast landslide susceptibility due to climate change impacts on forests. Despite this, only eight of the 42 models include the possibility to account for transient vegetation effects (mechanical and/or hydrological). The lack of attention to time dependency of vegetation effects is in agreement with what was found by Moos et al. (2023) in their review on the effects of climate change on mountain forests protective capabilities against natural hazards.

One of the models that do account for the transient effect of vegetation is the tRIBS-VEGGIE-Landslide (both versions) (Arnone et al. 2016; Lepore et al. 2013), where the evapotranspiration is time dependent as a function of the leaf area index (LAI). Other models such as the Okada model (Okada et al. 2023) or the Abdollahi model (Abdollahi et al. 2023) aim to model the change of slope stability in time due to the evolution of reinforcement effects of the vegetation. However, none of the models incorporate a function to automatically account for changes in these parameters. Instead, the parameters are changed manually between simulations, and the results then compared.

Simplicity versus complexity

As seen, the degree of complexity of both the geotechnical and hydrological models, and models for considering mechanical root reinforcement, varies between the different existing models (Table S1, Tables 2–3). The vast majority of the models analyzed in the qualitative study are relatively simple models that employ a type of limit equilibrium method as their geotechnical model (77%, Fig. 6), and the WWM (Wu et al. 1979), or no model at all, for mechanical root reinforcement (71%, Fig. 7). The motivation for choosing these types of simplified modeling schemes over the more complex modeling techniques is likely due to the limited field and laboratory data commonly available, as well as time limitations. For example, to run simulations with a limit equilibrium approach (LEM) is generally more time efficient than to run simulations with a finite or discrete element approach (FEM or DEM, respectively). Additionally, input values for LEM simulations are limited to strength parameters of the soil and roots, slope geometry, and pore water pressure distribution, while the FEM and DEM simulations require additional knowledge about the stress-strain relationship of the modeled root-permeated material. Another element is that in order to employ the simple WWM (Wu et al. 1979), one would need the tensional strength of the individual roots (measured or derived from root diameter) and a record of the number of roots of different diameters crossing an assumed failure plane. For the more complex RBMw (Schwarz et al. 2010a, 2013) on the other hand, one would need additional information, such as root diameter distribution along the length of the roots and number of branching points, and root length, hence a more comprehensive field campaign to collect the necessary data is required.

As mentioned above, several of the slope stability models do not include a specified mechanical root reinforcement model in their modeling scheme (Fig. 7). However, this does not prevent the user to pre-process available information on roots by the means of a selected root reinforcement model when applied to a specific case study. The same principle is valid for models that only include a reduction factor applied to precipitation to account for evapotranspiration and/or interception loss (Abdollahi et al. 2023; Medina et al. 2021; Montrasio et al. 2023; Ni et al. 2018; Raimondi et al. 2023). This reduction factor can, if desired, be estimated through a pre-analysis using a sophisticated modeling scheme similar to the techniques incorporated in models that include evapotranspiration and interception loss estimates directly (Arnone et al. 2016; Ewen et al. 2000; Lepore et al. 2013; Simoni et al. 2008; Strauch et al. 2018).

Further research

One of the main interesting insights from this literature review is that most of the models are found to seldomly be reutilized by other researchers than their developers. This implies that the models are generally not validated for a broad spectrum of environmental conditions. For example, in addition to there being no models developed by researchers from the global south or in colder regions such as the Nordic countries (Fig. 4), no application of any of the analyzed models herein is reported in these areas. Even though these regions are not topping the statistics for landslide damage (CRED 2018), they are still experiencing landslides today, and with climate change, they will most likely face larger landslide problems in the future (Tang et al. 2018). A future research potential could therefore be to validate selected models to other conditions than what they were originally designed for and with this increase the focus towards NbS and slope stability modeling in new regions as well as facilitate for these areas to be more prepared for a future climate. Site-specific empirical data are needed to validate the existing models for new environmental conditions, as it cannot be assumed that the same tree species has equal reinforcing capabilities in different environments. An unfortunate example where the significance of environment-specific data has been neglected is in the management of snow avalanches in Norway. The Norwegian guidelines for determining the protective effects of forests from snow avalanches are heavily based on studies from the Swiss Alps, despite there being significant differences between the region’s climatic and geologic conditions (Breien and Høydal 2013). Similar examples are assumed to exist for the case of shallow landslides.

As seen in this study, a large number of slope stability models which include different effects of vegetation exist today, and as discussed above, few of these models are being used by multiple researchers. To bring forward the research on this topic, it would be beneficial if more researchers were to continue working on and further develop existing models, instead of using their time and efforts in creating new slope stability models. A key factor for success in this matter is that the developers behind slope stability models provide open access to their models by sharing their code.

The transient effect of vegetation on slope stability is poorly incorporated in the existing slope stability models. Future research potential lies in the interdisciplinary collaboration between geoscientists and ecologists to ensure that plant-climate interactions are properly implemented in the models. An interesting research direction would be to couple a Dynamic Global Vegetation Model (DGVM) with a slope stability model in order to capture the effects of vegetation composition and cover through time and under different climate scenarios.

Finally, as mentioned by way of introduction, the effect of vegetation on soil water retention capability and hydraulic conductivity is poorly studied and understood (Ni et al. 2019a, b; Vergani et al. 2017). Furthermore, the effects of vegetation on a soil’s hydraulic characteristics might also be altered with climate change, though, e.g., frequent wildfires, heavy and intense rainfall, and long dry spells. Future research efforts should be directed towards both understanding the physical processes involved in the hydraulic alteration of soil by vegetation and to implement these effects in slope stability models.

Conclusions

This study presents the results of a systematic literature review of peer-reviewed publications, published between January 2000 and June 2023, which propose novel, physically based, slope stability models that include the effects of vegetation. The aim of the study was to provide a comprehensive overview of how the hydrological and mechanical effects of vegetation are parametrized in existing, physically based slope stability models.

Table 4 Nomenclature utilized within this study
Full size table

The study shows that the research activity is concentrated in countries experiencing most landslides. The increase in publications per year reflects the increasing interest of NbS as shallow landslide mitigation measures in the research community. Existing slope stability models include the effects of vegetation with various degrees of complexity with regard to how accurately they attempt to mimic the physical processes present in nature.

It is difficult to provide recommendations on which of the analyzed models are better suited to assess slope stability considering the effects of vegetation, as the examined models are developed to meet different needs. When an assessment of landslide susceptibility is to be conducted for an area, it is important to decide in advance what is the purpose and scale of the assessment and what is the desired outcome of the analysis. This review provides valuable guidance for researchers and practitioners in their choice of appropriate slope stability model for their studies.

However, not all cases can be modeled by the currently available slope stability models that include the effects of vegetation. This study has illustrated the need for further validation of the existing models, especially extended to mountainous landscapes in the global south and in colder regions such as the Nordic countries. Efforts to close this research gap would expand the focus of NbS for landslides to new regions, and increase preparedness for a higher landslide frequency in a future climate. Moreover, studies of time dependency in vegetation reinforcement capabilities are lacking, an aspect which is especially important in the light of climate change. Finally, more research is needed to better understand the effects of vegetation on soil water retention capability and hydraulic conductivity. Further research by the authors will therefore be directed towards the validation of selected models for colder regions such as the Nordic countries, with special emphasis on studying the variation with time of the vegetation reinforcement capabilities and the effect of vegetation on a soil’s hydraulic characteristics.