The Analysis on the Applicability of Speed Calculation Methods for Avalanche Events in the G219 Wenquan–Horgos Highway

Abstract

The avalanche speed is an important indicator for measuring the intensity of avalanches, and its measurement method is relatively complex. In practical engineering, empirical formulas based on statistics are usually adopted. However, research on the applicability of existing calculation methods in different regions is still insufficient, and further verification and improvement are urgently needed. Based on the integrated space–air–ground field survey data, this study uses RAMMS::AVALANCHE to conduct dynamic numerical simulations of 78 avalanche events in the Qiet’ akesu Gully of the Wenquan to Horgos transportation corridor in the Western Tianshan Mountains during the winter of 2023–2024, analyses the avalanche movement process, and compares the calculation results of the numerical tests of avalanche movement speed with empirical formulas. The results indicate that the velocities calculated using Formula A and Formula B are generally overestimated, approaching approximately 1.5 times the reference value. The mean absolute percentage error of Formula A (19.46%) is lower than that of Formula B (48.27%). In contrast, Formula C exhibits a significantly lower mean absolute percentage error (8.42%) compared with the other two methods, and its results remain stably around one-half of the reference value. Based on these findings, a comprehensive estimation strategy is proposed: twice the value calculated by Formula C is adopted as the primary reference, while two-thirds of the value from Formula A is taken into consideration, and the larger of the two is selected as the final estimated velocity. This strategy ensures the robustness of the results while effectively avoiding the potential overestimation or underestimation associated with reliance on a single empirical formula. This study provides a scientific basis for highway route selection and the placement of avalanche mitigation measures in high-altitude mountainous areas, and offers technical support for the construction and operational safety of infrastructure along the G219 Wenquan–Horgos transportation corridor.

1. Introduction

Against the backdrop of global warming, natural disasters in the cryosphere occur frequently [1,2,3,4,5,6]. As a kind of mountain natural disaster, avalanches are characterised by strong destructiveness, great impact force, and difficulty in prediction, which seriously threaten human lives and infrastructure in mountainous areas [7,8,9,10,11,12]. To reduce the hazards of avalanches, scholars at home and abroad have formed avalanche engineering prevention and control types such as stabilisation, mitigation, prevention, and blocking through continuous research and engineering practice [13]. However, it is difficult to accurately obtain key dynamic parameters closely related to engineering protection design, such as avalanche movement speed, avalanche run-out distance, and avalanche impact force, which limits the construction and optimisation of efficient avalanche protection systems. Therefore, an in-depth study of avalanche dynamic characteristics has important scientific value for improving the effectiveness of avalanche prevention and control in mountainous areas.

The movement speed of avalanches is a key indicator for measuring avalanche size, and accurately estimating their speed is crucial for ensuring the safety of avalanche protection structures [14,15,16]. To this end, extensive research has been conducted on avalanche speed measurement. YOICHI ITO [17] measured particle velocity and airflow velocity in the powder cloud of artificially released avalanches using a snow particle counter; Emmanuel Thibert [18] calculated avalanche speed based on the moving distance and propagation time of avalanches; L. Rammer [19] measured avalanche speed at a full-scale avalanche test site using pulsed Doppler radar and continuous-wave radar, and verified the results by comparing them with data from photoelectric speed sensors and video measurements; Emma Suriñach [20] estimated the front velocity of avalanches based on seismic wave spectra; Vilajosana [21] measured and analysed the velocity during avalanche flow through scaled snow trough experiments. Although these methods are relatively accurate, they generally have problems such as high cost and susceptibility to weather conditions, visibility, measurement range, as well as the internal structure of avalanches and the reflection characteristics of electromagnetic waves. In addition, these technologies often require complex preprocessing procedures such as image processing and signal processing, making it difficult to popularise them in practical engineering applications outside the field of scientific research. In engineering practice, statistically based empirical formulas are usually used to predict avalanche speeds. These methods are easy to operate and can provide timely estimates of speed magnitudes for preliminary engineering design, risk zoning, and the formulation of emergency plans. However, due to the different factors considered by different calculation methods, their calculation results vary greatly, and their applicability in specific regions still needs further research and verification. The construction of the G219 Highway from Wenquan to Horgos in the Tianshan Mountains of China crosses the complex mountainous area in the western part of the Tianshan Mountains, involving complex alpine canyon areas. Avalanche disasters are densely distributed along the route, with frequent avalanches in winter that are highly destructive, posing a significant threat to the construction and operation safety of the Wenquan to Horgos transportation corridor along G219 [22,23,24,25,26]. Therefore, studying the applicability of avalanche speed calculation methods for this specific region can not only deepen the understanding of the local avalanche disaster-causing mechanism but also provide direct guidance for the precise layout of avalanche prevention projects along the route and the formulation of targeted risk management strategies.

This study focuses on the RAMMS::AVALANCHE model as its core, combining a high-precision Digital Elevation Model (DEM) of the study area, historical avalanche event records, and underlying surface classification data to conduct numerical simulations of avalanche dynamics. Numerical simulations of avalanche dynamics were performed on 78 avalanche events that occurred in the Qiet’ akesu Gully of the Western Tianshan Mountains at the end of the winter of 2023–2024, thereby obtaining the movement processes of these avalanche events. On this basis, three commonly used calculation methods for avalanche speed are selected to conduct a comparative analysis between the avalanche speed simulation results and various theoretical calculation results. It aims to determine the applicability of different calculation methods in the West Tianshan corridor under complex terrain and specific snow conditions. To deepen the understanding of the avalanche movement laws in this area and provide a reference for avalanche engineering prevention, risk management, and disaster prevention and mitigation work.

2. Overview of the Study Area

The study area is located between 80°39′–80°41′ E and 44°28′–44°31′ N, situated in the Qiet’ akesu Gully along the northern slope of the Tianshan Mountains. It is a critical section of the G219 highway, connecting Wenquan and Horgos (Figure 1). The region is characterised by steep terrain with significant topographic variations, predominantly consisting of north–south-oriented U-shaped deep valleys. Additionally, due to the blocking effect of the northern and southern branches of the Tianshan Mountains, moist air from the west is strongly uplifted, causing localised precipitation. This results in abundant rainfall and high humidity, providing a sufficient material base for avalanches. For example, during late winter and early spring, observational data indicate an average snow depth of 56.48 cm and an average relative humidity of 65.94%. The coupling of high-water-content snow layers with steep terrain makes the area a high-risk zone for avalanches, posing a significant threat to the safety of the transportation corridor.

3. Data and Methods

3.1. Data

3.1.1. Digital Elevation Model (DEM)

In this study, a DJI Mavic 3E enterprise-grade UAV was employed for data acquisition, with the maximum flight altitude set to 1500 m, a lateral and forward overlap rate of 70%, and a flight height of 200 m. A 30 m resolution Digital Elevation Model (DEM) was integrated into the mission planning, and the terrain-following flight mode was utilised to ensure consistent ground-relative altitude for all aerial images, thereby improving mapping accuracy. During the flight, positional accuracy was corrected using GNSS (Global Navigation Satellite System) equipment combined with PPK (Post-Processing Kinematic) differential processing, ensuring that the absolute position of each aerial image was aligned within the same control network. After image acquisition, the data were processed using Metashape (version 2.2.0) software to generate a DEM with a spatial resolution of 0.4 m.

3.1.2. Video Monitoring Data

To comprehensively obtain snowpack and avalanche information within the observation area, the research team deployed 15 automatic high-definition avalanche monitoring cameras across the study region in 2023. The monitoring system combined fixed long-range camera positions with timed rotating dome cameras. The preset monitoring viewpoints (Figure 2) were determined based on local topography, geomorphology, and vegetation conditions. Data were recorded every 30 min, allowing real-time capture of key dynamics during avalanche events. In addition, snow stakes were installed around the observation sites to continuously monitor snow depth. These measurements were cross-checked with snow-depth sensor data from nearby meteorological stations to ensure consistency and reliability of the observations.

3.1.3. Meteorological Monitoring Data

A total of five meteorological monitoring stations were installed within the study area. Each station was equipped with a W10 wind speed sensor, a W20 wind direction sensor, a T200B weighing-type rain and snow measurement system, an HMP155 air temperature and humidity sensor, a CS106 atmospheric pressure sensor, and a SnowVUE10 digital snow-depth sensor. These instruments enabled real-time monitoring of key meteorological parameters, including wind speed, wind direction, precipitation, snow depth, atmospheric pressure, air temperature, and relative humidity. Data were recorded at 10 min intervals and stored on local memory cards.

3.1.4. Field Survey Data

Focusing on the G219 Wenquan–Horgos transportation corridor, the research team conducted a systematic field investigation of avalanche events that occurred during the 2023–2024 period. The survey primarily included point-based avalanche distribution mapping, UAV-based avalanche site photogrammetry, snow density and surface moisture observations, and measurements of avalanche deposits (Figure 3). Statistical results show that a total of 78 avalanche events were recorded within the study area during this period. The runout distances of avalanche deposits varied significantly, with the longest reaching approximately 1900 m and the shortest about 70 m.

3.2. Methods

This study establishes a three-tier logical framework of “data foundation-simulation verification-comparative analysis” (Figure 4). Based on field investigations and UAV imagery, a fundamental database of avalanche distribution and characteristics within the study area was developed. Subsequently, the RAMMS::AVALANCHE model was applied to perform dynamic simulations of 78 avalanches in the region, and the reliability of the simulation results was verified using field-measured physical parameters. Subsequently, three widely recognised typical empirical methods commonly used in engineering practice were selected to calculate the corresponding avalanche speed. For ease of presentation, the study designates these three calculation formulas as Formula A (Ma, 1974) [27], Formula B (Xu, 2024) [28], and Formula C (Qiu, 2005) [1]. By systematically comparing the empirical results with the model simulations, the applicability and limitations of each method within the study area were comprehensively evaluated.

3.2.1. RAMMS::AVALANCHE

RAMMS::AVALANCHE (version 1.8.27) is a professional avalanche simulation software developed by the Swiss Federal Institute for Snow and Avalanche Research. It can simulate the flow process of avalanches in complex three-dimensional terrain and predict the path, impact force, flow velocity of avalanches, and the distribution of snow accumulation areas. Moreover, existing studies have proven that this model shows high accuracy in simulating various types of avalanches [29,30]. This model is based on the two-parameter Voellmy–Salm–Gruber friction law, which divides the friction coefficient into the dry Coulomb friction coefficient proportional to the normal stress and the turbulent friction coefficient. The expression for frictional resistance is as follows:

$$S=\mu N+{\displaystyle \frac{\rho g{u}^2}{\xi }},N=\rho gh\cos \theta$$

(1)

In the equations, $\rho$ represents the density; $g$ is the gravitational acceleration; $\theta$ represents the bevel angle; $h$ represents the flow height; $\mathsf{\xi }$ represents the turbulent friction coefficient; $N$ represents normal stress; $u$ represents the dry Coulomb friction coefficient, as a vector $u={\left(u_x,u_y\right)}^{\mathrm{T}}$.

RAMMS::AVALANCHE uses the three-dimensional volume method to numerically simulate the flow, accumulation, and impact processes of avalanches, which mainly includes the following operation steps: first, use drone aerial images to generate high-resolution DEM, and complete coordinate unification and range cropping in ArcMap (version 10.8) to ensure coverage of the formation area, movement area, and accumulation area; then, according to the results of on-site investigations, delineate the location of the avalanche formation area and set the avalanche release volume; Finally, the appropriate dry Coulomb friction coefficient and turbulent friction coefficient are determined by identifying the underlying surface type of the study area [31]. The specific parameter selections are shown in

Table 1. RAMMS::AVALANCHE model parameter values.

Parameter Name	Value	Unit
DEM	0.4	m
Dry Coulomb Friction Coefficient	0.34
Turbulent Friction Coefficient	1250	m/s2
Cohesion	100	pa
Fracture Depth	0.5	m
Snow Density	230	kg·cm−3
Storage Step Size	2	s
Maximum Simulation Duration	300	s

.

3.2.2. Avalanche Speed Calculation

The study uses the local maximum avalanche speed Vmax from the calculation results of the RAMMS::AVALANCHE numerical simulation as the benchmark reference, and quantifies the characteristic velocities of avalanche points (3/2 Vmax, Vmax, 2/3 Vmax, 1/2 Vmax) into four-level standards V1–V4. Calculate the avalanche movement speed using three empirical formulas, respectively. Through systematic comparative analysis of the consistency between the calculation results of each method and the four-level speed standards, further judge the accuracy of the calculation methods, thereby evaluating the applicability of the avalanche calculation methods in the Wenquan to Horgos transportation corridor along the G219 line. Finally, form a speed calculation system suitable for avalanche protection projects in transportation corridors, providing a quantitative basis for reducing engineering survey costs and improving the accuracy of avalanche disaster assessment.

• Formula A

Ma Zhenghai et al. [27] regarded an avalanche as a rocket moving from a high-potential energy area to a low-potential energy area along a mountain slope, and there are certain similarities between the two in terms of physical processes. First, the mass of a rocket decreases with time due to the continuous consumption of fuel during its movement, while an avalanche body continuously entrains natural snow along its path through basal erosion during its sliding, resulting in a dynamic increase in its mass. Second, a rocket operates in an air fluid, while an avalanche body moves in two media: snow layers and air. After simplifying the two movement processes, respectively, the avalanche motion equation can be expressed similarly using the rocket motion equation. The derived formula for avalanche velocity is as follows:

$$V=\sqrt{V_o^2+\left[2g\left(\sin \phi -\mu \cos \phi \right)X\right]/\left(2{\displaystyle \frac{K}{\rho h}}+3\right)}$$

(2)

In the equations, $V$ represents the final speed of an avalanche; $V_o$ represents the avalanche initiation speed; $\phi$ represents the slope angle; $h$ represents the thickness of the snow layer removed by an avalanche, taking the average value; $K$ represents the snow layer resistance coefficient; $\rho$ represents the average density of natural snow cover; $u$ represents the dynamic friction coefficient (

Table 2. Reference table for the friction coefficient of Formula A.

Frictional Properties	Value
Dynamic friction coefficient of dry-snow avalanche	0.50
Dynamic friction coefficient of wet-snow avalanche	0.40
Static friction coefficient between granular snow particles	1.28
Static friction coefficient between granular snow and the frozen snow layer	0.77
Static friction coefficient between wet granular snow and turf	0.65

); $X$ represents the avalanche runout distance.

The derivation of this formula provides an in-depth analysis of the entrainment effect of snow during avalanche motion. Throughout the entire avalanche process, the surrounding natural snowpack is progressively incorporated into the moving avalanche mass in the direction of travel. By examining the temporal variation in the avalanche mass, the rate of mass increase is related to factors such as the density, thickness, and width of the entrained snow layers, as well as the travel (runout) distance; these relationships are then combined to establish the avalanche dynamic equation.

2.

Formula B

The Coulomb friction model can provide predictive results for avalanche velocity and impact distance in most cases, and is therefore commonly used in the estimation of macroscopic avalanche dynamics [28]. By considering the presence or absence of basal friction, the upper and lower theoretical limits of the maximum velocity can be estimated. The maximum front velocity of the avalanche changes with the square root of the total vertical drop. The formula for the movement velocity of the avalanche body is expressed as follows:

$$v=\sqrt{2g\left(h-{\displaystyle \frac{Hl}{L}}\right)}$$

(3)

In the equations, $g$ represents gravitational acceleration; $H$ represents the height difference from the avalanche formation area to the accumulation area; $h$ represents the height difference from the avalanche formation area to the calculated position along the avalanche path; $L$ represents the horizontal projection length of the avalanche path from the formation area to the limit boundary of the accumulation zone; $l$ represents the horizontal projection length from the avalanche formation area to the calculation position.

3.

Formula C

This formula is based on the relationship between the forces acting on the avalanche and its velocity. The avalanche velocity depends on the gravitational driving force of the snow parallel to the slope and the following resistances: basal dynamic friction, which decreases with increasing velocity; viscous shear force within the moving snow, which is proportional to the velocity; basal disturbance resistance, which is proportional to the square of the velocity; and the resistance generated by the additional snow incorporated into the avalanche after snow fracture, which is independent of velocity [1]. Considering that the equation based on the above complex relationships cannot yield practical solutions, the simplified expression is as follows:

$$V_T=\sqrt{\zeta R\sin \beta }$$

(4)

In the equations, $\zeta$ represents the disturbance friction coefficient, encompassing all factors proportional to the square of the velocity (

Table 3. Recommended Friction Coefficient Values.

Topographic Features	Value (m/s2)
Flat, hard snow with a uniform slope angle, free of trees and visible rocks	1200~1600
Treeless, commonly open slopes	750
Open mountain slopes with shrubs and rocks	500
Typical valley	400~600
Valleys with rocks and undulating, winding topography	300
Forest	150

); $R$ represents the hydraulic radius, which is equal to the height of the flowing snow on a clear slope; $\beta$ represents the slope angle.

4. Results

4.1. RAMMS::AVALANCHE Reconstruction

Based on the aerial images taken by drones, a large area of approximately 30 km in length and 10 km in width along the G219 Line Wenquan to Horgos Transportation Corridor was divided into nine regions according to the distribution of avalanches, which were named model 1 to model 9, respectively (Figure 5). Based on this, RAMMS::AVALANCHE was used to simulate 78 avalanche events that occurred in 2023–2024, obtaining the avalanche movement process (Figure 6) and its maximum speed (Figure 7).

To verify the reliability of the RAMMS::AVALANCHE simulation, this study selected a typical avalanche event with complete observation records as the verification object and compared the simulation results with field survey data (Figure 8). This avalanche event has detailed process images and field measurement data, providing a reliable basis for model verification. The simulation results show that the entire avalanche process lasted approximately 146 s: after the avalanche initiated, it accelerated rapidly along the slope, with the flow velocity reaching a peak at about 40 s, then gradually sliding into the valley bottom and expanding. After 80 s, the avalanche entered a stable phase and began to accumulate, with the flow velocity significantly decreasing and the erosion and carrying effects gradually weakening; finally, a stable accumulation body was formed between approximately 120–146 s, with a maximum thickness of about 2.5 m. By comparing the simulated flow trajectories and accumulation ranges with the on-site measured results, it is further concluded that their coincidence degree reaches 95%, meeting the expected similarity standards. This indicates that the RAMMS::AVALANCHE model has a high level of reproducibility and reliability in simulating the avalanche motion process.

The results of on-site image comparison (Figure 9) further indicate that the simulated final accumulation range is generally consistent with the measured avalanche coverage area, but there are local deviations. The deviations mainly stem from the blocking of part of the snow flow by large vehicles parked on the road on the day of the avalanche, resulting in local thickening and flow direction disturbance of the accumulation body on the windward side of the vehicles. However, such details were not fully reflected in the macroscopic calculations of the model. In general, the RAMMS::AVALANCHE model shows high accuracy and applicability on the macroscopic scale in terms of avalanche movement range, accumulation location, and morphology.

4.2. Avalanche Speed Calculation

4.2.1. Formula A

Based on Formula (2), the maximum avalanche velocities within the study area were calculated (Figure 10a). The results indicate that the computed velocities are generally overestimated and tend to be closer to the V1 reference values, which is 2/3 of the maximum avalanche velocity calculated by the RAMMS::AVALANCHE model. Under different velocity standards, the theoretical results of Formula A exhibit substantial fluctuating errors when compared with the actual observations. Statistical analysis (

Table 4. Error between the results of Formula A and the simulation results.

Statistical Indicators	V1 Standard	V2 Standard	V3 Standard	V4 Standard
Mean absolute error	6.41	8.92	9.66	9.94
Percentage error (%)	19.46	37.04	61.47	81.36

) shows that the percentage error of this formula ranges from a minimum of 19.46% to a maximum of 81.63%, indicating considerable uncertainty in its estimations. In particular, Avalanches 3-1, 5-5, and 7-13 display exceptionally large deviations (Figure 10b). The overall overestimation may stem from the formula’s use of avalanche runout distance without accounting for path curvature, channel constriction and expansion, or lateral divergence. By simplifying a complex three-dimensional path into a single scalar distance—and by using average values for snow density and snow-layer thickness—the formula fails to capture the incremental mass changes along the avalanche path that dilute momentum during motion, ultimately leading to an overestimation of avalanche velocity.

4.2.2. Formula B

Based on Formula (3), the maximum avalanche velocities within the study area were calculated (Figure 11a). The results show that the computed velocities are generally overestimated, with considerable deviation from the reference values (Figure 11b). Statistical analysis (

Table 5. Error between the results of Formula B and the simulation results.

Statistical Indicators	V1 Standard	V2 Standard	V3 Standard	V4 Standard
Mean absolute error	14.36	14.96	15.45	15.82
Percentage error (%)	48.27	78.94	120.24	161.13

) indicates that the maximum mean percentage error reaches 161.13%. This substantial discrepancy is likely since the formula accounts only for the conversion of gravitational potential energy using geometric elevation difference and horizontal distance, while neglecting factors such as frictional resistance and air drag. In essence, it assumes that the avalanche descends along a perfectly smooth slope and ignores terrain-induced energy dissipation. As a result, the computed velocities approach idealised theoretical values and deviate substantially from the actual dynamic behaviour of real avalanche flows.

4.2.3. Formula C

Based on Formula (4), the maximum avalanche velocities within the study area were calculated (Figure 12a). This formula demonstrates significantly better stability compared to Formula A and Formula B, with smaller errors (Figure 12b). Statistical analysis (

Table 6. Error between the results of Formula C and the simulation results.

Statistical Indicators	V1 Standard	V2 Standard	V3 Standard	V4 Standard
Mean absolute error	5.25	3.47	2.27	2.16
Percentage error (%)	25.38	14.52	11.3	8.42

) reveals that the maximum mean percentage error is 25.38%, with a minimum of only 8.42%. The computed results are primarily close to V4, which is stabilised at approximately half of the maximum avalanche velocity calculated by the RAMMS::AVALANCHE model. Therefore, in practical engineering calculations, the velocity can be first doubled and then combined with a safety factor to estimate the peak avalanche velocity.

5. Discussion

This study evaluated the accuracy of avalanche velocity estimation by comparing the results obtained from Formula A, Formula B, Formula C, and the RAMMS::AVALANCHE numerical simulations in the study area (Figure 13). The results indicate that although empirical formulas can, to some extent, reflect the actual dynamic processes of avalanches under certain conditions, significant discrepancies still exist when compared with the RAMMS::AVALANCHE simulation results. These errors mainly originate from differences in theoretical assumptions, energy dissipation mechanisms, and terrain sensitivity. Formula B oversimplifies the effects of friction and flow morphology, neglecting the influence of underlying surface conditions and snow type along the avalanche path, which results in large deviations in complex terrain and renders it unsuitable for application in the present study area. Although Formula A incorporates certain dynamic adjustments, it is still based on idealised assumptions, leading to an overestimation of avalanche velocities for large-scale events. In contrast, Formula C integrates constraints from both energy conservation and resistance dissipation, allowing it to better approximate real avalanche processes, particularly in capturing frontal velocity and path evolution characteristics. However, as demonstrated by the study of B. Sovilla [32], the omission of snow density in this formula also leads to an underestimation of the calculated velocities.

In addition, commonly used empirical formulas exhibit systematic biases in estimating the velocities of different types of avalanches (e.g., dry-snow versus wet-snow avalanches). Gauer [33] pointed out that conventional models tend to underestimate the frontal velocity of high-speed dry-snow avalanches, whereas Takeuchi [34] demonstrated that wet-snow avalanches generally propagate at lower velocities and that empirical formulas have limited capability to distinguish between these two types. In contrast, Takeuchi [35], based on a large-scale dry slab avalanche in the Myoko Makunosawa valley, Japan, reported that the velocity of dry-snow avalanches can reach 40–60 m/s, which far exceeds the estimation range of some traditional empirical formulas. Furthermore, chute experiments conducted by Upadhyay [36] indicated that the frontal velocity of wet-snow avalanches is generally lower than 12 m/s, with tail velocities even lower (3–5 m/s). These studies collectively demonstrate that, due to the higher water content and enhanced frictional effects, wet-snow avalanches exhibit significantly lower velocities than dry-snow avalanches. This further highlights the limitations of empirical formulas in adequately resolving the velocity differences between dry- and wet-snow avalanches, thereby explaining the relatively large deviations observed in the empirical results of this study. Specifically, the velocities calculated using Formula A and Formula B are generally higher than those obtained from the RAMMS simulations, with large errors, and tend to approach approximately 3/2 of the simulated values. In contrast, Formula C yields systematically lower results but remains relatively stable at about one-half of the simulated values. Under large-scale avalanche conditions and in complex terrain, such errors in empirical formulas may lead to either overestimation or underestimation of avalanche velocities, thereby affecting the effectiveness of avalanche mitigation measures. Based on these considerations, a combined estimation strategy is proposed, in which twice the value of Formula C is adopted as the primary reference, while two-thirds of the value from Formula A is considered as a supplementary reference. The larger of the two is finally selected as the representative estimate of avalanche velocity.

In summary, each of the three empirical formulas has its own strengths and limitations. Formula A and Formula B simplify several key physical processes and are therefore suitable for rapid estimation or application in relatively simple terrain conditions. In contrast, Formula C shows more robust performance in dynamic and complex terrain and under variable snow conditions. Nevertheless, all three formulas exhibit a certain degree of deviation in their results. For complex terrain and diverse snow conditions, numerical simulation methods provide more accurate and reliable predictions of avalanche velocity. Future research should focus on improving the applicability of empirical formulas so that they can better adapt to changing terrain and climatic conditions. At the same time, integrating the advantages of numerical simulations with empirical formulas will offer more precise support for avalanche prevention and mitigation engineering.

6. Conclusions

Based on field survey data from the Qiet’ akesu Gully along the G219 Wenquan–Horgos transportation corridor, this study simulated and reconstructed 78 avalanche events and calculated avalanche velocities using multiple methods. By comparing the computed results with the standard reference value, only analyse the applicability of the three calculation methods in this region. The following main conclusions were drawn:

• There are significant differences in the estimation accuracy of the various velocity calculation methods. Formula B exhibits the largest error, with the average absolute error ranging from 14.36 to 15.82 and the maximum percentage error reaching 161.13%. Formula A follows, with an error range of 6.41 to 9.94 and a maximum percentage error of 81.63%. Formula C has the smallest error, with the average absolute error ranging from 2.16 to 5.25, and the average percentage error is only 8.42%.
• Under complex terrain conditions, the three empirical formulas adopted in this study demonstrate considerable uncertainty, with their calculated results showing relatively large deviations from those produced by the RAMMS::AVALANCHE numerical model in the study area. To improve the reliability of velocity estimation, a comprehensive estimation strategy is proposed: twice the value calculated by Formula C is taken as the primary reference, while two-thirds of the value calculated by Formula A is considered as a supplementary reference. The larger of the two is finally selected as the representative avalanche velocity. This approach ensures the robustness of the results while effectively avoiding the potential overestimation or underestimation associated with reliance on a single empirical formula.
• Future research should further refine the limitations of existing empirical formulas and explore their integration with the RAMMS::AVALANCHE numerical simulation framework. Empirical formulas may serve as useful references during route selection and preliminary engineering design, whereas numerical simulations can provide detailed reconstructions of avalanche velocities under complex terrain conditions and diverse snow types by comprehensively accounting for snow physical properties and topographic influences. Integrating these two approaches is expected to substantially enhance the reliability of avalanche velocity predictions, thereby providing stronger technical support for disaster prevention engineering design and risk assessment.