Machine learning-based landslide velocity prediction model: incorporating multi- expression programming and discrete element modeling

doi:10.21203/rs.3.rs-4643461/v1

The estimation of flow parameters for gravitational flows, such as velocity, volume, and runout distance is important for disaster prevention and mitigation. In this study, we have developed a prediction model for the frontal velocity of landslides using multi-expression programming (MEP), and discrete element modeling (DEM) as a function of slope angle, slope length, volume, coefficient of energy transfer, rolling friction and static friction. Moreover, we have also determined the percentage effect of each parameter on the front velocity. The range of the values for these parameters was selected from well-documented historical cases and experimental studies. The physical modeling results indicate that the front velocity was greatly influenced by the variation in slope angle and friction parameters. The developed prediction model was validated by comparing it with various statistical indices, and by performing sensitivity analysis, which validated the experimental observations that slope angle and friction parameters control the frontal velocity by 53% and 25% respectively. Moreover, a second-level validation was carried out by comparing the predicted front velocity with the front velocity of historical rock landslide cases and found to be in good agreement. It is hoped that the proposed model will help disaster mitigation and risk assessment by effectively predicting the front velocity of the imminent slides, and also reduce the computational cost, time, and resources.

Prediction model

front velocity

machine learning

multi-expression programming

discrete element modeling

Human settlement and infrastructure in mountainous regions are constantly exposed to the risk of natural hazards like, rock avalanches, and debris flows. People in these regions are always challenged to find a balance between potential landslide hazards and spatial development, which makes it a complex scientific problem. The estimation of the flow parameters such as landslide velocity, volume, and depth is important for disaster prevention and mitigation (Crozier and Glade, 2012; Fuchs et al., 2008; Guzzetti, 2000). The estimation of these parameters can be carried out using experimental, numerical, and mathematical or analytical solutions. Massive landslides are usually characterized by high speed, large volume, and massive scale of destruction. It is important to understand and estimate their velocity for mitigation measures, design and construction of protective infrastructures, and population settlement (Christen et al., 2010; Kattel et al., 2018; Pudasaini et al., 2008). Moreover, the impact force, which can induce a chain disaster in terms of airblast also strongly related to the front velocity of the landslides (Evans et al., 2009).

The movement of landslides usually depends upon qualitative and quantitative parameters. Qualitative parameters such as the path materials (Hungr and Evans, 2004; Voight and Sousa, 1994), i.e., glacial ice (De Blasio, 2014; Schneider et al., 2010; Sosio et al., 2012), snow (Boultbee et al., 2006; Deline et al., 2011), sediments (Hungr and Evans, 2004; Sassa and Wang, 2007), bedrock (Whittall et al., 2017), etc. influence the mobility, other than path materials phenomena like, fragmentation, frictional heating, and the presence of water also greatly influence the mobility of landslides. There can be numerous quantitative parameters that contribute to the velocity of landslides such as slope/release angle, particle size, particle density, volume, friction coefficients, energy transfer, etc. Over the years scientists have used various parameters to develop models to understand the dynamics of gravitational mass movements.

Previously, scientists have developed models to predict the velocity of landslides, ranging from empirical to semi-empirical to physical-based models. One of the earliest efforts to predict the velocity can be traced back to the Heim (1932) model and formulation of Heim’s ratio H/L, where H is the elevation difference and L is the horizontal distance. Heim’s ratio described the conservation of energy from potential to kinetic and the effect of frictional force. Heim’s ratio can be interpreted, as the ratio of H/l decreases the volume increases and so does the velocity. Similarly, Scheller (1970) empirically plotted tan (α) against the log of volume obtained a curved correlation, and concluded that the velocity of slide increases as their volume increase, whereas the frictional coefficient decreases as the velocity increases either by exponential square or by hyperbola. Based on these assumptions he tried to establish a correlation but at that time the calculations became quite complex and the formula could not be written in his final closed form as there were a lot of parameters involved. So, there was a need to establish a more simplified closed-form relation to predict the velocity and reach of the landslide. So, Scheidegger (1973) extended the work, and based on his empirical observations developed a logarithmic model that relates friction to the volume of landslides, as shown in Eq. 1.

$${log}\left(f\right)=a{log}\left(V\right)+b$$

1

Where f is the friction factor, V is the volume (m³), a is the empirical constant and b is the intercept. It can be interpreted that the friction coefficient has a direct effect on the volume, runout distance, and velocity of the landslide. Later another empirical relation developed by Scheidegger suggests a direct relationship between velocity and geometric parameters such as drop height and runout distance, Eq. 2.

$${v}_{max}={10}^{{log}\left(H\right)-{log}\left(L\right)+C}$$

2

Whereas V_max is the maximum velocity, H is the vertical drop (m), L is the horizontal runout distance (m), and C is the empirical constant. Another energy-based model which is also attributed to Scheidegger (Eq. 3) suggests that there exists a relationship between maximum velocity as a function of gravitational acceleration ($g$), effective friction($\mu )$, height (H), and length (L).

$$v=\sqrt{2g\left(H-\mu L\right)}$$

3

Later, Hungr et al. (1984) developed a simplified dynamic model by refining a simple energy-based model and incorporating frictional and drag forces in landslide dynamics. The proposed model suggests the velocity of landslide reduces due to internal friction and external forces. Voellmy-Salm developed a semi-empirical model that combines empirical observations with theoretical constructs to cater to varying slopes and friction conditions. Similarly, McDougall and Hungr (2005) developed a model to determine the velocity of a landslide by combining velocity-squared drag and frictional forces. The proposed model emphasizes the role of material heterogeneity and terrain roughness. Iverson (1997), proposed a depth-averaged model by incorporating the depth of sliding mass and momentum conservations. Recently, the field of mass movements has seen an increasing trend of numerical modeling for understanding the physical characteristics and dynamics (Cascini et al., 2014; Cuomo et al., 2016; Gardezi et al., 2023, 2022, 2021; McDougall and Hungr, 2005; Mergili et al., 2020) The models discussed till now use various parameters to predict the velocity of gravitational mass movements, but there is no single model that incorporates all the governing quantitative parameters and predicts the velocity of landslides more accurately. Though the numerical models can predict the velocity more accurately, they are quite time-consuming and require a lot of expertise, and computation power.

Since the experimental and numerical models are time-consuming, laborious, and require expertise to evaluate the landslide dynamics, to overcome this problem there is a need for models that require less time, effort, and expertise such as artificial intelligence (AI) based prediction models for a swift guesstimate of landslide characteristics and dynamics. Previously the prediction models used regression analysis but now the use of artificial intelligence (AI) models is encouraged. Many scientists have used AI-based models such as ANN and other models to predict landslide susceptibility worldwide (Chang et al., 2023; Kainthura and Sharma, 2022; Lv et al., 2022; Merghadi et al., 2020; Wang et al., 2019; Y. Wang et al., 2020; Zhang et al., 2022; Z. Zhao et al., 2022). Despite the usage of AI in susceptibility mapping it is hard to find it in predicting the landslide dynamics.

Other than neural network-based models there has been some latest development in genetic programming (GP) based models, i.e., genetic algorithms (GAs) to identify the optimized solution. Genetic programming (GP) was first introduced by (Cramer, 1985) and was later modified and improved by (Koza, 1994). The most advanced and sophisticated models among the existing genetic programming techniques are multi-expression programming (MEP) and genetic expression programming (GEP). Both MEP and GEP are gene-type programming techniques that produce tree-like models. The models behave like living organisms as they can learn and adapt by modifying their size, shape, and composition (Usama et al., 2023). The MEP adopts a demonstrative method for programming and uses linear chromosomes that can determine the best mutation probability, population size, and evolutionary model. MEP is a cutting-edge methodology and advanced form of GP and can produce precise results even when the complexity of the target is not known.

In this study, we have made an effort to predict the frontal velocity of landslides/rock avalanches using physical and machine learning models. The physical modeling was carried out using Discrete Element Models (DEM) whereas the prediction modeling was carried out using multi-expression programming (MEP). The most important input governing parameters that control the velocity of the landslide in DEM, i.e., the density of rock, slope angle, volume, coefficient of restitution, coefficient of sliding friction, and coefficient of rolling friction, were determined from the literature. The density of the slide was kept constant to an average value obtained from 20 rock avalanche cases. Moreover, we have also determined the percentage effect of input parameters on the velocity of landslides which itself is a novel idea.

Once the input parameters were determined it was important to know the range of the parameters being used. In this study, the range of the input parameters was selected from the previous rock avalanche cases around the world especially from China, Table 1, and experimental studies performed to understand the behavior of landslides (Pudasaini et al., 2008; Qing-zhao et al., 2019; Scheidl et al., 2015).

Table 1

Indicates the rock avalanche (RA) cases from where the range of the parameters was taken.
Rock Avalanche	Density (Kg/m³)	Slope angle	Friction coeff.
Yigong RA	2400	17	0.08	(Yin and Xing, 2012)
Nayong RA	2380	20	0.58	(Luo et al., 2020)
Fuquan RA	2100	23	0.1	(Xing et al., 2016b)
Gualing, Guizhou RA	2350	16	0.1	(Yin et al., 2011)
Zhaotong, Touzhai Valley RA	2675	15	0.12	(Xing et al., 2016a)
Wenjia valley RA	2000	27	0.09	(Zhuang et al., 2019)
Niumen valley RA	2650	18	0.19	(Xing et al., 2015)
Nyexon RA	2600	35	0.18	(Cui et al., 2022b)
Tagarama RA	2860	16	0.17	(Y. F. Wang et al., 2020)
Luanshibao RA	2800	12	-	(Wang et al., 2018)
Sedongpu RA	2000	12	0.27	(C. Zhao et al., 2022)
Tangjia valley RA	2300	18	0.12	(Li et al., 2017)
Ganqiuchi RA	2680	21	-	(Zhou et al., 2020)
Lymek RA	2390	35	0.17	(Shi et al., 2023)
Luchedu RA	2370	16	-	(Cui et al., 2022a)
Ultar RA	2500	35.8	0.25	(Gardezi et al., 2023)
Attabad RA	2100	25	0.13	(Gardezi et al., 2021)
Langtang valley RA	2600	37	0.2	(Zhuang et al., 2023)
Turnoff creek RA	2600	15.5	0.2	(An et al., 2020)

2.1 Model preparation and calibration

Physical modeling was carried out using discrete element modeling (DEM). The geometry was prepared in an ANSY workbench following the physical characteristics of the geometry used by (Qing-zhao et al., 2019) in their experiments to understand the deposition pattern of the sliding materials in mountainous regions, as shown in Fig. 1. The DEM model was calibrated using the same parameters as (Qing-zhao et al., 2019) used in his study and the deposition behavior was observed. The modified model was 3 m long and 1 m wide, as shown in Fig. 1, and composed of three major zones similar to actual landslide zones. A source area having variable dimensions, a runout zone inclined at varying angles of 8° to 48°, and a 0.32m deep V-shaped deposition area, similar to actual valley-shaped deposition areas in mountainous regions. The source is composed of coarse-graded particles, the percentage of various sizes of particles in the source area is shown in Table 2.

The prepared model was calibrated using the parameters proposed by (Qing-zhao et al., 2019) in their experiments. The results indicate that the sliding material entered the deposition valley in 0.8 seconds after the release, which is similar to the experimental outcomes of (Qing-zhao et al., 2019), and also had a similar kind of deposition pattern. The results for model calibration and the parameters used to calibrate are shown in Table 3.

Table 2

Particle size distribution
Particle size (mm)	10	20	30	40	50	60
Percentage (%)	5	13	12	20	40	10

Table 3

Input parameters for DEM to calibrate the model.
Symbol	Value	Description
D	0.005–0.06 m	Particles diameter
ρ_s	2650 kg/m³	Particles density
υ	0.25	Poisson ratio
G	1x10⁸	Shear modulus
e	0.5	Coff. Of restitution
R_s	0.35	Coff. Of rolling friction
R_r	0.01	Coff. Of static friction
g	9.8 m/s²	Gravitational acceleration
Δt	1x10^− 5	Time step

2.2 Discrete element modeling

The understanding of the dynamics of mass movement, especially the estimation of flow velocity is very important for disaster prevention and mitigation. This can be accomplished using various numerical models, in this study we have used 3D-Discrete Element Modeling (DEM) to determine the landslide front velocity. DEM since its first proposed usage by (Cleary and Campbell, 1993) has been popular among the scientific community because of its high efficiency and realistic results. The usage of DEM is not only limited to the field of geomechanics (Chang and Taboada 2009; Li et al. 2012; Liu et al. 2021, 2020; Salciarini et al. 2010; Staron and Hinch 2005; Tang et al. 2009; Yuan et al. 2014; Zhao 2014), it has been used in other filed, i.e., pharmacy, material sciences, and chemical engineering (Kretz et al. 2016; Cui et al. 2017; Chaudry and Wriggers 2018; Yeom et al. 2019). DEM models the solid particles interacting at discrete contact points, and its usage in the analysis of gravitational mass movements has produced useful results, one of the benefits of using DEM is that all the data is available to the user at any time during the test, the motion of particles and interaction forces can be traced at any time step during the test.

2.2.1 Hertz-Mindlin no-slip contact model

There are numerous contact models in the DEM that can be used based on the user requirements. Here in this study, we have used the Hertz-Mindlin (no-slip) contact model, which is a non-linear elastic model and is frequently used because of its accuracy and efficient calculations. This model has been used in the source area in the present study. The two components of force, i.e., a normal component and a tangential component used in the Hertz-Mindlin contact model are shown in. Figure 2, using two spring-dashpot responses. The first response presents the normal contact among geometry and particles, and the second response indicates rolling friction interaction. The normal force indicated by Eq. 4 is a function of normal overlap, whereas the tangential force appendages the Coulomb frictional model.

$${F}_{n}=\frac{4}{3}{E}^{*}\sqrt{{R}^{*}}{\delta }_{n}^{\frac{3}{2}}$$

4

Whereas, ${\delta }_{\text{n}}$ indicates the normal overlap, ${R}^{*}$ is the equivalent radius, and ${E}^{*}$ indicates equivalent to Young's modulus.

In this study, EDEM developed by DEM Solutions has been used to estimate the front velocity of landslides by varying the geometric parameters, i.e., slope angle, slope length, volume, and other input parameters of DEM, i.e., coefficients of restitution, rolling friction, and sliding friction. A schematic of the DEM model in Fig. 3 indicates the DEM geometry, particles generated in the source area, and then the calculated front velocity of the particles. The volume variation was achieved by varying the height in the z-direction, this was done to determine the effect of the ratio of the source area height to its length h_o/l_o. Following the procedures, a data set of 150 mass movements, as shown in Table 4, was analyzed with various combinations, and the results for frontal velocity were obtained.

3.1 Multi-expression programming

Multi multi-expression programming (MEP) model for predicting the front velocity of landslides/rock avalanches was developed by using numerical records as shown in Table 4. A database of 150 landslides was prepared by varying the parameters obtained from previous rock avalanche cases and experimental studies. The collected data was numerically modeled in EDEM to determine the front velocity of landslides against each data point. MEP initiates by generating a random population of chromosomes, and it continues and subsequent steps are repeated until a terminal condition is achieved, targeting to generate an optimal expression from the data having input and output pairs over a certain number of generations as shown in Fig. 4 following are the steps involved in the MEP process.

1. Following a binary tournament process a pair of parents is chosen and then it is put through to recombination through a consistent crossover probability.
2. The recombination generates two offspring.
3. The offspring generated in step 2, undergo mutation, and if subsequent offspring perform better than the least fitting individuals in the current population, the better offspring replace the foulest.

The illustrations adopted by MEP are approximately similar to machine codes obtained using C + + and Pascal compilers. The MEP chromosomes are composed of multiple genes combined using mathematical operators such as addition (+), subtraction (-), multiplications (x), and division (/), and these genes create expression trees (ETs) (Cheng et al., 2020). There are several parameters that govern the overall performance/behavior of MEP such as code length, sub-population size and number, crossover probability, and other sets of various functions. The population size tells us about the number of programs, whereas an increase in sub-population number and size can result in complex computations and increased calculation time. Moreover, the code length greatly influences the length of the predicted model.

Table 4

Input parameters for experimental and machine learning modeling
S. No.	Slope angle(α)	Slope length(L)	Volume (V)m³	Restitution coeff. (e)	Rolling friction coeff. (µ_r)	Static friction coeff. (µ_s)	Modeled Velocity (v) m/s
1	8°	1.4	0.025	0.3	0.1	0.15	0.9
2	8°	1.8	0.05	0.38	0.18	0.23	0.92
3	8°	2.2	0.075	0.46	0.26	0.31	0.87
4	8°	2.6	0.1	0.54	0.34	0.39	0.65
5	8°	3	0.125	0.62	0.42	0.47	0.11
6	18°	1.4	0.025	0.3	0.1	0.15	2.03
7	18°	1.8	0.05	0.38	0.18	0.23	1.95
8	18°	2.2	0.075	0.46	0.26	0.31	1.86
9	18°	2.6	0.1	0.54	0.34	0.39	1.75
10	18°	3	0.125	0.62	0.42	0.47	1.6
11	28°	1.4	0.025	0.3	0.1	0.15	2.5
12	28°	1.8	0.05	0.38	0.18	0.23	2.85
13	28°	2.2	0.075	0.46	0.26	0.31	2.99
14	28°	2.6	0.1	0.54	0.34	0.39	3.05
15	28°	3	0.125	0.62	0.42	0.47	3.15
16	38°	1.4	0.025	0.3	0.1	0.15	3.27
17	38°	1.8	0.05	0.38	0.18	0.23	3.5
18	38°	2.2	0.075	0.46	0.26	0.31	3.7
19	38°	2.6	0.1	0.54	0.34	0.39	4.4
20	38°	3	0.125	0.62	0.42	0.47	4.5
21	48°	1.4	0.025	0.3	0.1	0.15	3.37
22	48°	1.8	0.05	0.38	0.18	0.23	4.15
23	48°	2.2	0.075	0.46	0.26	0.31	4.6
24	48°	2.6	0.1	0.54	0.34	0.39	4.75
25	48°	3	0.125	0.62	0.42	0.47	4.89
.	.	.	.	.	.	.	.
.	.	.	.	.	.	.	.
.	.	.	.	.	.	.	.
146	8°	1.4	0.025	0.3	0.1	0.47	0.73
147	18°	1.8	0.05	0.38	0.18	0.47	1.29
148	28°	2.2	0.075	0.46	0.26	0.47	2.4
149	38°	2.6	0.1	0.54	0.34	0.47	3.9
150	48°	3	0.125	0.62	0.42	0.47	4.89

3.2 Development of MEP model

The basic purpose was to develop an explicit model for front velocity prediction of landslides, it is expected that the predicted model will be a function of six variables, i.e., slope angle, slope length, volume, coefficient of restitution, coefficient of rolling friction and coefficient of static friction, and can be described as follows,

$$v=f\left(\alpha ,L, V, e,{\mu }_{r},{\mu }_{s}\right)$$

The relation between the parameters was assessed using Pearson’s correlation. It is important to analyze the multicollinearity and interdependency among the parameters, as they can muddle the interpretation of the predicted model. The summarized mathematical expression for MEP is achieved by extracting C + + code and expressing it according to optimized hyperparameter settings as shown in Table 5. The execution of the MEP code in MATLAB, gave the prediction model for the front velocity of the landslides as shown in Eq. 5, and sub-equations 6,7,8, and 9.

Table 5

Parametric settings for MEP algorithm
Sr.No.	Parameters	Values
1	Number of sub-populations	100
2	Sub-population size	150
3	Crossover probability	0.9
4	Code length	70
5	Tournament size	10
6	Mutation probability	0.03
7	Number of generations	550
8	Crossover type	Uniform
9	Error measure	Mean absolute error
10	Problem type	Regression
11	Function set	+, -, ⅹ, /, ^, √, Exp, log10, Tan, Tan^− 1
12	Terminal set	Problem Input
13	Operators	0.5
14	Simplified	Yes
15	Variables	0.5
16	Random seed	0
17	Constants	0

$$v={y}_{1}+{y}_{2}-{y}_{3}+{y}_{4}$$

5

Whereas,

$${y}_{1}=\frac{{tan}\left(V\right)}{{u}_{r}}+\frac{{tan}\left(V\right)}{{u}_{s}}$$

6

$${y}_{2}={\left({\left({u}_{s}^{{\left(\frac{{log}\left({\left(\alpha \right)}^{3}\right)}{{log}{\left(10\right)}^{3}}\right)}^{{e}^{2}}}\right)}^{L+\frac{{log}\left(\frac{{log}\left({\alpha }^{3}\right)}{{log}{\left(10\right)}^{3}}\right)}{{log}\left(10\right)}}\right)}^{\frac{{tan}\left(V\right)}{{u}_{r}}+\frac{{tan}\left(V\right)}{{u}_{s}}+\frac{{log}\left({\alpha }^{3}\right)}{{log}{\left(10\right)}^{3}}}$$

7

$${y}_{3}={\left({u}_{s}^{{\left({log}\frac{({\alpha )}^{3}}{{log}{\left(10\right)}^{3}}\right)}^{e}}\cdot {tan}\left({exp}\left({tan}\left(V\right)\right)\right)\right)}^{L-{log}\frac{{\left(\alpha \right)}^{2}}{2.\text{t}\text{a}\text{n}\left(\text{exp}\left(\text{tan}\left(V\right)\right)\right) }}$$

8

$${y}_{4}={log}\frac{{\left(\alpha \right)}^{3}}{3-\frac{{log}\left({log}\frac{{\left(\alpha \right)}^{3}}{{log}{\left(10\right)}^{3}}\right)\left({u}_{s}-{tan}\left(\frac{V}{{u}_{\gamma }}\right)\right)}{{\left({log}\frac{{\left(\alpha \right)}^{3}}{{log}{\left(10\right)}^{3}}\right)}^{{e}^{2}}}}$$

9

Whereas $v$ is the velocity (m/s), $\alpha$ is the slope angle in degrees respectively, L is the length of the sliding path (m), $V$ is the volume (m³), whereas, e, ${\mu }_{r}$ and ${\mu }_{s}$ is the coefficient of energy transfer, rolling, and sliding friction respectively.

4.1 Prediction performance of the developed model

This section describes the robustness of the developed MEP model by comparing it with certain statistical indices, i.e., root mean square error (RMSE), mean absolute error (MAE), Nash–Sutcliffe efficiency (NSE), correlation coefficient (Cr), performance index (PI), represented by Eq. (10–14) (Alavi et al., 2010; Khan et al., 2022). The indices were calculated using the equations. The predictive model was developed by dividing the entire data set into training and testing subsets. The random distribution was facilitated by MEP itself. Following the general criteria, 70% of the data, i.e., 105 data points were taken as training data, whereas 30% of data, i.e., 45 data points were considered for model validation/testing.

$$RMSE=\frac{{\sum }_{i=1}^{n}{\left({e}_{i}-{p}_{i}\right)}^{2}}{n}$$

10

$$MAE=\frac{{\sum }_{i=1}^{n}\left|{e}_{i}-{p}_{i}\right|}{n}$$

11

$$NSE=1-\frac{{\sum }_{i=1}^{n}{\left({e}_{i}-{p}_{i}\right)}^{2}}{{\sum }_{i=1}^{n}{\left({e}_{i}-{\stackrel{-}{e}}_{i}\right)}^{2}}$$

12

$${R}^{2}={\left(\frac{{\sum }_{i=1}^{n}\left({e}_{i}-{\stackrel{-}{e}}_{i}\right)\left({p}_{i}-{\stackrel{-}{p}}_{i}\right)}{{\sum }_{i=1}^{n}{\left({e}_{i}-{e}_{i}\right)}^{2}{\sum }_{i=1}^{n}{\left({P}_{i}-{\stackrel{-}{P}}_{i}\right)}^{2}}\right)}^{2}$$

13

$$PI=\frac{RRMSE}{1+R}$$

14

Whereas, ${\stackrel{-}{e}}_{i}$ and ${\stackrel{-}{p}}_{i}$ are the average values for the modeled and predicted results, and e_i and p_i are i_th values of experimental and predicted results. Whereas, n indicates the number of samples. It is important to consider the error indices while evaluating the predictive efficiency of a non-linear model, higher values of RSQ and NSE show the robustness and accuracy of the model, whereas lower values of MAE and PI indicate a strong correlation between actual and predicted value. Moreover, the near-zero value of RMSE plays a vital role in developing a precise prediction model. Table 6 represents the values of performance indicators for the MEP model for the training and validation data sets. The indices values describe the exactness in estimating the frontal velocity of landslides/rock avalanches. Furthermore, the higher values of Cr for training and validation data sets indicate a strong correlation, i.e., 0.98 for training and 0.99 for validation.

Whereas the PI values for training and testing are 0.032 and 0.0325 respectively. According to the standards, a Cr value greater than 0.8 is satisfactory, and a PI value less than 0.2 indicates the robustness of the model. Moreover, The values of RMSE and MAE for the validation dataset are approximately 6% and 3%, more than the training dataset, but still in the range of less than 10%, thus indicating how efficiently the MEP model has learned the non-linear relation among input and output parameters with comparatively low error values, and higher generalization ability. The indices mentioned in the Table 6 tell us that our predicted model is highly precise and accurate, and can be used to predict the front velocity of landslides. A comparison of experimental and predicted results for the front velocity of the landslide against input parameters is shown in Fig. 5.

Table 6

Performance index values for MEP-based velocity prediction model

Performance parameters

Training data set

Validation data set

RSQ

0.9798

0.9876

RMSE

0.1920

0.2039

MAE

0.1409

0.1453

RSE

0.0105

0.0109

RRMSE

0.0636

0.0648

P. index

0.0320

0.0325

NSE

0.9894

0.9890

C_r

0.9898

0.9937

Previously scientists have also used the slope of the regression line as a performance indicator for AI models, thus representing a correlation between experimental and predicted results. Figure 6 represents the slope of the regression line for our velocity model. It can be observed that the slope values for training and validation data are quite close to the ideal fit (1:1). The regression line having values of 0.97 and 0.95 for training and validation indicates a strong correlation for the developed model. The accuracy of the developed model can be further validated using residual error values which are then obtained from the difference between experimental and model-estimated velocity values (Alavi et al., 2013). The minimum and maximum, i.e., negative/positive error values obtained for the velocity model are − 0.59 m/s and 0.39 m/s, as shown in Fig. 7 Moreover, most of the error values are placed along the x-axis, highlighting the low error frequency, and accuracy of the developed model.

4.2 Validation of the developed model

The validation of a developed AI model is a vital facet in predictive modeling, as the model can perform better during the training stage, but it may perform poorly for a new/unseen dataset. So, the developed model was validated using an unused dataset, as mentioned previously 30% data was used for validation. However, the model was further validated by performing the sensitivity and parametric analysis and by comparing the predictive results with actual landslide cases.

4.2.1 Sensitivity and Parametric Analysis

Sensitivity and parametric analysis play a vital role in determining the robustness of the proposed model. The sensitivity analysis (SA) of the proposed model on the entire dataset determines how sensitive is our developed model to any change in the input parameter values. Since SA is being carried out to investigate the effect of input parameters on the front velocity of landslides, for an independent variable Y_i the SA can be conducted using equations 15 and 16. That means one of the parameters was changed between its extreme values and others were kept constant to their average values and the output was determined in the form of Y_i, and then a similar process was done for the next and all other parameters.

$${R}_{k}={f}_{max}\left({Y}_{k}\right)-{f}_{min}\left({Y}_{k}\right)$$

15

Relative Importance $SA \left(\%\right)=\frac{{R}_{k}}{{\sum }_{n}^{j=1}{R}_{j}}\times 10$ (16)

Whereas, ${f}_{max}\left({Y}_{k}\right)$ and ${f}_{min}\left({Y}_{k}\right)$ are the minimum and maximum values of the predicted results grounded on the kth domain of the input parameters in the above equation. Figure 8 represents the results of the sensitivity analysis of the developed prediction model for the front velocity of the landslides. The figure indicates that the velocity is greatly influenced by the slope angle (α) of the landslide and has an effect of almost 53%. The length of the sliding path (L) contributes 8% to the velocity, volume (v) contributes 11.5% and the coefficients of restitution, rolling friction (µ_r), and static friction (µ_s) contribute 2.8%, 10.8%, and 13.9% respectively. That model tells us that the slope angle and friction factors contribute approximately 78% to the velocity of the avalanche. It can be summarized as the effect of α > µ_s > V > µ_r > L > e on the front velocity of landslides.

The results of the parametric analysis for the input parameters (slope angle, slope length, volume, coefficient of restitution, coefficient of rolling friction, and coefficient of static friction) used in this study to determine the front velocity of landslides are displayed in Fig. 9 It can be observed that the frontal velocity increases linearly with the increase in slope angle, slope length, and energy transfer, Fig. 9 (a, c, and d) whereas the relation between volume and velocity is second-degree polynomial Fig. 9 (f). Overall, Fig. 9 (a, b, c, and d) indicates that as the values of the input parameters increase, the front velocity increases. Whereas, the front velocity decreases with the increase in the value of Coeff. of static friction, and rolling friction. The results of the parametric analysis are in agreement with the various statistical velocity and runout prediction models (Heim, 1932; Scheidegger, 1973; Scheller, 1970), indicating that the velocity has a direct relation with slope angle, length, height, and volume, and inverse relation with friction coefficients.

4.2.2 Comparison based on actual cases

In the last stage, the predictive model was validated by comparing it with the results of the actual rock slide cases. Here, we have compared the front velocity results of the two historical cases, i.e., Ultar rock avalanche, Pakistan and Yiziyan rock topple, and Fuquan landslide, Guizhou, China (Bilal et al., 2021; Gardezi et al., 2023; Zhang et al., 2024) As, these cases have used discrete element modeling for analysis and have already well-established parameters, so it is easy to compare the predictive modeling results to evaluate the robustness of the proposed model. The original results from all the cases are shown in Fig. 10.

The figure indicates that the maximum front velocity for the Ultar rock avalanche ranges between 72 to 90 m/s whereas for the Yiziyan rock topple the velocity ranges between 20 to 25 m/s. A comparative analysis of the predicted and observed results is presented in Table 7.

Table 7

Comparing the observed front velocity and predicted front velocity
Event	Observed velocity	Predicted velocity
Ultar rock avalanche	80 m/s	73 m/s
Fuquan landslide	31.7 m/s	34.6 m/s
Yiziyan rock topple	15–23 m/s	17 m/s

Though the predicted results are quite close to the observed results, which indicates the robustness of the developed model, a slight difference can be observed in both values, this can be due to the fact that in developing the prediction model we have used an average density based on the historical cases obtained from Table 1, but usually the density is slightly different in every case, moreover the difference can also be attributed to the particle size, though we have used a range of particle size as mentioned in Table 2, there can be a difference in the distribution of the particle size.

In this experimental-cum-modeling study, the effect of control variables, such as slope angle, volume, slope length, energy transfer, and coefficient of static and rolling friction has been determined on the front velocity of landslides/rock avalanches, moreover, the percentage effect of every single parameter on the front velocity has also been determined. The results obtained from discrete element modeling were utilized to develop an MEP-based prediction model for frontal velocity. The study aimed to enhance prediction efficiency and reduce the cost, time, and resources involved in determining the velocity of landslides. Based on the study following conclusions are drawn.

Among the numerous parameters that control the velocity of landslides, the selected six parameters and their range of values efficiently produced the results during the modeling and prediction stage.

To develop an optimal MEP model, 12 trials were performed by varying (a) subpopulation numbers, (b) population size, (c) coed length, (d) tournament size, (e) and generations, and their performance was evaluated using numerous performance indices. The optimal MEP model for front velocity was achieved at 100 subpopulations, 150 subpopulation size, 70 code length, 10 tournament size, and 550 number of generations, this combination produced a 0.98 R-value.

The developed regression equation can be directly used for predicting the front velocity of landslides. The equation has been developed from fairly high-accuracy models, and evaluated using R, RMSE, MAE, RSE, P-index, NSE, and C_r (0.98, 0.19, 0.14, 0.01, 0.032, 0.98, 0.98) for the training dataset of frontal velocity.

The developed model was analyzed by performing sensitivity and parametric analysis as second-level validation. The parametric results indicate a change in frontal velocity with respect to changes in input parameters, thus confirming to literature that the parameters, i.e., the slope angle, volume, slope length, and energy transfer have a direct relation to the frontal velocity, whereas coefficients of rolling and static friction have an inverse relation.

The sensitivity analysis produced a percentage contribution of each parameter to the front velocity, which had never been determined before. The results indicate that the percentage contribution of slope angle, slope length, volume, energy transfer, rolling friction, and static friction is 52.9%, 8%, 11.5%, 2.8%, 10.8%, and 13.85% respectively.

A third-level validation was performed by comparing the results of historic rock avalanches to the ones obtained from the prediction model. It was revealed that the prediction model produced excellent results considerably close to the actual velocities.

The current study was performed by considering the six most suitable parameters that generally govern the velocity of gravitational mass movements, it is suggested that in the future other parameters may be added, i.e. width of sliding path and geometric features, and their effect may be determined on the frontal velocity. Moreover, the current model is based on an average density value of 2500 kg/m³, it is suggested that in the future the effect of material density may also be determined on the frontal velocity.

Funding

This work was supported by the National Key R&D Program of China (Grant No. 2023YFC3007001), the National Natural Science Foundation of China (grant no. 42202312), and the Fundamental Research Funds for the Central Universities.

Author Contribution

H.G. collected data, did the simulations, and developed a machine learning-based prediction model.H.G. and X.L. wrote the manuscript. X.L. and Y.H. reviewed the modeling results and helped in revising the manuscript.

Data Availability

The data can be accessed by emailing at [email protected]

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No competing interests reported.

Machine learning-based landslide velocity prediction model: incorporating multi- expression programming and discrete element modeling

Status:

Version 1

Abstract

Figures

1. Introduction

2. Physical modeling

2.1 Model preparation and calibration

2.2 Discrete element modeling

2.2.1 Hertz-Mindlin no-slip contact model

3. Landslide velocity prediction model

3.1 Multi-expression programming

3.2 Development of MEP model

4. Results and discussions

4.1 Prediction performance of the developed model

4.2 Validation of the developed model

4.2.1 Sensitivity and Parametric Analysis

4.2.2 Comparison based on actual cases

5. Conclusions and discussion

Declarations

Author Contribution

Data Availability

References

Additional Declarations

Status:

Version 1