# An Investigation of the Influence on Compacted Snow Hardness by Density, Temperature and Punch Head Velocity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Overview of the Research Area

#### 2.2. Research Equipment

## 3. Orthogonal Test Method

#### 3.1. Factors Affecting the Hardness of Compacted Snow

#### 3.2. Arrangement and Results of Orthogonal Experiment

_{64}(8

^{9}) orthogonal table. The constant penetration velocity of the IMSP punch head, the density, and the temperature of the compacted snow sample are the three factors. The minimum constant penetration velocity of the IMSP punch head is 15.03 mm/s and the maximum is 21.11 mm/s; the minimum density of compacted snow samples is 354 kg/m

^{3}and the maximum is 438 kg/m

^{3}. The minimum temperature value is −38 °C, and the maximum is −3 °C. With equal spacing between each level, these three factors are fixed at eight levels. V represents the cone penetration speed; ρ represents the density of compacted snow samples; and T represents the temperature of compacted snow samples. The design method and experimental factors, in order, are detailed in Table 1.

#### 3.3. Range Analysis Results and Discussions

^{3}. This is similar to the ${\overline{F}}_{\rho}^{7}$ presented in Table 3 of this study (i.e., the average hardness of 402 kg/m

^{3}of snow is 541.64 kPa within the experimental range). This observation suggests that the orthogonal test analysis maintains a certain degree of reliability.

#### 3.4. Variance Analysis

_{i}be

_{i}signifies the effect of the ith-factor on the experimental results, which is also known as the sum of squares of ith-factor’s deviations.

_{i}) is as follows:

_{i}.

_{i}should be evaluated against V

_{e}, calculating FTEST

_{i}(empty column in the orthogonal table serves to show the experimental error variance V

_{e}for this study). The FTEST for the ith-factor is as follows:

_{0}.

_{05}(7, 7), so it can be concluded that the penetration velocity and sample temperature are significant at a 0.05 inspection level. And the FTEST of the sample density is greater than F

_{0}.

_{001}(7, 7), so it can be concluded that, at an inspection level of 0.001, the sample density is statistically significant.

- Sample density has the most significant influence on the hardness of the sample;
- The penetration velocity and sample temperature are significant at an inspection level of 0.05 and cannot be ignored.

## 4. Support Vector Regression (SVR) Algorithm for Model Development

#### 4.1. The Selection of Kernel Function

_{i}is the support coefficient for each Gaussian RBF kernel.

#### 4.2. The Particle Swarm Optimization-Support Vector Regression (PSO-SVR) Parameter Optimization Architecture and Proceeding

- Initialize all particle positions and speeds at random;
- Based on the fitness function of the snow hardness estimate issue, the fitness value of each particle is determined;
- The fitness value of each particle is compared with both Pbest (i.e., the best position visited by this particle so far) and Gbest (i.e., the best position found by all the particles so far). In the event that the fitness value surpasses the current value of Pbest, it is necessary to update Pbest with the new fitness value. In the event that the fitness value surpasses the current value of Gbest, the Gbest should be updated with the new fitness value;
- In order to expand the space for particles and prevent convergence to local optima, it is necessary to reset the penalty factor c of the SVR on each update;
- Update the position and speed of every individual particle until the predetermined maximum number of iterations has been attained. Subsequently, output the optimal parameters. Alternatively, go back to Step 2.

#### 4.3. The Result of PSO-SVR Algorithm

_{min}, y

_{max}] set to [−1, 1]. Equation (8) represents the function expression fitted by the SVR, where ω is the support vector, n is the number of support vectors, k

_{i}is the coefficient corresponding to the i-th support vector Gaussian RBF kernel, and b is the constant term of the function. Through reverse normalization, the ultimate predicted value G is obtained.

^{2}value is 0.901.

^{3}, the density is the most dominant factor that affects compacted snow hardness. Furthermore, it is necessary to guarantee that the snow density is sufficiently high, otherwise it is difficult to obtain compacted snow structures that meet the hardness requirements.

## 5. Conclusions

- This study uses the orthogonal experiment method, and orthogonal experiments are conducted on 64 different samples of compacted snow. The results of the range analysis indicate that the density of compacted snow samples has the most sensitive impact on hardness. Temperature and penetration velocity have far less sensitive effect on hardness than density.
- According to the variance analysis of the orthogonal experiment, the effect of density on hardness within the range of this experiment is the most significant. Comparatively, the effects of temperature and penetration velocity are limited to the inspection level of 0.05, which cannot be ignored completely. The significance of density is clearly supported by the current findings.
- Employing PSO-SVR analysis, we obtain the continuous function relationship between the IMSP Value and the three factors (i.e., density, temperature, penetration velocity), within the experimental range. This study not only confirms the positive correlation between density and hardness [32], but also discovers the relationship between temperature and hardness. By carefully examining the model, it is found that if the density of compacted snow is low, the hardness of the snow also tends to remain low. In this case, there is no significant positive relationship between snow density and hardness at specific temperatures and punch head speeds. Therefore, when constructing compacted snow roads, it is necessary to avoid the occurrence of weak areas of low-density snow. One of the more significant findings to emerge from this study is that when the temperature approaches the melting point of snow, even with high density, the hardness remains low, which is especially apparent at high penetration velocities. This indicates that the hardness of snow suffers a fundamental decrease at high temperatures. In actual compacted snow road projects, when the temperature approaches 0 °C, even if the density of the compacted snow road is high, maintenance of the piste is still required. In this case, skiing is never permitted because of this sharp decrease in hardness, which could cause serious safety issues.
- The IMSP employs an electric motor to precisely control the penetration speed, and utilizes the sensors to precisely measure the end snow resistance. This device can be used as a penetrating instrument on pistes and lays the groundwork for future research.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Number | Coefficient Term | Super Vector | Number | Coefficient Term | Super Vector | ||||
---|---|---|---|---|---|---|---|---|---|

V | ρ | T | V | ρ | T | ||||

1 | −2.773 | −1.000 | 0.714 | 1.000 | 30 | −8.594 | −0.141 | 0.429 | −0.143 |

2 | 0.615 | −1.000 | 1.000 | 0.714 | 31 | 8.594 | 0.141 | 0.714 | −0.143 |

3 | −3.694 | −1.000 | −1.000 | 0.429 | 32 | −5.231 | 0.141 | 1.000 | −0.429 |

4 | 8.594 | −1.000 | −0.714 | 0.143 | 33 | 0.681 | 0.141 | −1.000 | −0.714 |

5 | −0.921 | −1.000 | −0.429 | −0.143 | 34 | 8.594 | 0.141 | −0.714 | −1.000 |

6 | −4.251 | −1.000 | −0.143 | −0.429 | 35 | −8.594 | 0.141 | −0.429 | 1.000 |

7 | −8.594 | −1.000 | 0.143 | −0.714 | 36 | −0.577 | 0.141 | −0.143 | 0.714 |

8 | 0.134 | −1.000 | 0.429 | −1.000 | 37 | −8.594 | 0.141 | 0.143 | 0.429 |

9 | −8.594 | −0.714 | 0.714 | 0.714 | 38 | −8.594 | 0.141 | 0.429 | 0.143 |

10 | 5.390 | −0.714 | 1.000 | 1.000 | 39 | 8.594 | 0.428 | 0.714 | −0.429 |

11 | −8.594 | −0.714 | −0.714 | 0.429 | 40 | −8.165 | 0.428 | 1.000 | −0.143 |

12 | −7.315 | −0.714 | −0.429 | −0.429 | 41 | −8.147 | 0.428 | −1.000 | −1.000 |

13 | 8.594 | −0.714 | −0.143 | −0.143 | 42 | 8.594 | 0.428 | −0.714 | −0.714 |

14 | 8.594 | −0.714 | 0.143 | −1.000 | 43 | −8.594 | 0.428 | −0.429 | 0.714 |

15 | 8.594 | −0.714 | 0.429 | −0.714 | 44 | 8.594 | 0.428 | −0.143 | 1.000 |

16 | 8.594 | −0.428 | 0.714 | 0.429 | 45 | 8.594 | 0.428 | 0.143 | 0.143 |

17 | 4.388 | −0.428 | 1.000 | 0.143 | 46 | 8.594 | 0.428 | 0.429 | 0.429 |

18 | −4.978 | −0.428 | −1.000 | 1.000 | 47 | −5.572 | 0.763 | 0.714 | −0.714 |

19 | 8.594 | −0.428 | −0.714 | 0.714 | 48 | 6.969 | 0.763 | 1.000 | −1.000 |

20 | 8.594 | −0.428 | −0.429 | −0.714 | 49 | −3.816 | 0.763 | −1.000 | −0.143 |

21 | −2.825 | −0.428 | −0.143 | −1.000 | 50 | −4.590 | 0.763 | −0.714 | −0.429 |

22 | 8.594 | −0.428 | 0.143 | −0.143 | 51 | 8.594 | 0.763 | −0.429 | 0.429 |

23 | −8.594 | −0.428 | 0.429 | −0.429 | 52 | 1.699 | 0.763 | −0.143 | 0.143 |

24 | −8.594 | −0.141 | 0.714 | 0.143 | 53 | −1.500 | 0.763 | 0.143 | 1.000 |

25 | 3.708 | −0.141 | 1.000 | 0.429 | 54 | −5.748 | 0.763 | 0.429 | 0.714 |

26 | 8.594 | −0.141 | −0.714 | 1.000 | 55 | −6.869 | 1.000 | 0.714 | −1.000 |

27 | −8.594 | −0.141 | −0.429 | −1.000 | 56 | 6.076 | 1.000 | 1.000 | −0.714 |

28 | −8.594 | −0.141 | −0.143 | −0.714 | 57 | 8.594 | 1.000 | −1.000 | −0.429 |

29 | 4.343 | −0.141 | 0.143 | −0.429 | 58 | −8.594 | 1.000 | −0.714 | −0.143 |

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**Figure 1.**Schematic diagram of experimental geographical location. The red box is Bin County urban district.

**Figure 3.**Snow crystal after the first stage of metamorphism: (

**a**) Single snow crystal; (

**b**) Multiple snow crystals.

**Figure 4.**Compacted snow sample: (

**a**) Control the density of the snow sample with a high-precision electronic scale; (

**b**) Preparation of snow samples by layering compaction.

**Figure 5.**FTEST of various factors and Fα(7, 7) comparison results. A higher FTEST than Fα(7, 7) indicates that this factor is significant, at least at an inspection level of α.

**Figure 7.**Comparison of true value and prediction value. Red line is the true IMSP Value, while blue line is the prediction value by SVR model. This comparison demonstrates that the trained SVR model is extremely close to the actual values, showing the accuracy performance of this SVR model.

**Figure 8.**SVR model of relationship between the IMSP Value, temperature, and density with different penetration velocities.

Level Number | Factors | ||
---|---|---|---|

V (mm/s) | ρ (kg/m^{3}) | T (°C) | |

1 | 15.03 | 426 | −3 |

2 | 15.90 | 438 | −8 |

3 | 16.77 | 354 | −13 |

4 | 17.64 | 366 | −18 |

5 | 18.50 | 378 | −23 |

6 | 19.37 | 390 | −28 |

7 | 20.39 | 402 | −33 |

8 | 21.11 | 414 | −38 |

Test Number | Factor Level | Empty Column | IMSP Value (kPa) | Test Number | Factor Level | Empty Column | IMSP Value (kPa) | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

V | ρ | T | e | V | ρ | T | e | ||||

1 | 1 | 1 | 1 | 1 | 717.88 | 33 | 5 | 1 | 5 | 2 | 869.88 |

2 | 1 | 2 | 2 | 2 | 1030.80 | 34 | 5 | 2 | 6 | 1 | 888.56 |

3 | 1 | 3 | 3 | 3 | 187.04 | 35 | 5 | 3 | 7 | 4 | 457.92 |

4 | 1 | 4 | 4 | 4 | 309.32 | 36 | 5 | 4 | 8 | 3 | 630.40 |

5 | 1 | 5 | 5 | 5 | 325.68 | 37 | 5 | 5 | 1 | 6 | 255.24 |

6 | 1 | 6 | 6 | 6 | 371.40 | 38 | 5 | 6 | 2 | 5 | 441.52 |

7 | 1 | 7 | 7 | 7 | 436.44 | 39 | 5 | 7 | 3 | 8 | 469.88 |

8 | 1 | 8 | 8 | 8 | 791.60 | 40 | 5 | 8 | 4 | 7 | 519.20 |

9 | 2 | 1 | 2 | 3 | 696.52 | 41 | 6 | 1 | 6 | 4 | 820.04 |

10 | 2 | 2 | 1 | 4 | 1075.16 | 42 | 6 | 2 | 5 | 3 | 799.80 |

11 | 2 | 3 | 4 | 1 | 326.76 | 43 | 6 | 3 | 8 | 2 | 363.64 |

12 | 2 | 4 | 3 | 2 | 295.84 | 44 | 6 | 4 | 7 | 1 | 468.60 |

13 | 2 | 5 | 6 | 7 | 405.80 | 45 | 6 | 5 | 2 | 8 | 415.92 |

14 | 2 | 6 | 5 | 8 | 574.20 | 46 | 6 | 6 | 1 | 7 | 525.88 |

15 | 2 | 7 | 8 | 5 | 651.32 | 47 | 6 | 7 | 4 | 6 | 619.52 |

16 | 2 | 8 | 7 | 6 | 799.40 | 48 | 6 | 8 | 3 | 5 | 562.60 |

17 | 3 | 1 | 3 | 6 | 938.96 | 49 | 7 | 1 | 7 | 5 | 891.88 |

18 | 3 | 2 | 4 | 5 | 1000.28 | 50 | 7 | 2 | 8 | 6 | 1150.84 |

19 | 3 | 3 | 1 | 8 | 231.96 | 51 | 7 | 3 | 5 | 7 | 365.88 |

20 | 3 | 4 | 2 | 7 | 376.12 | 52 | 7 | 4 | 6 | 8 | 420.48 |

21 | 3 | 5 | 7 | 2 | 463.60 | 53 | 7 | 5 | 3 | 1 | 634.56 |

22 | 3 | 6 | 8 | 1 | 466.40 | 54 | 7 | 6 | 4 | 2 | 567.28 |

23 | 3 | 7 | 5 | 4 | 588.44 | 55 | 7 | 7 | 1 | 3 | 402.20 |

24 | 3 | 8 | 6 | 3 | 667.64 | 56 | 7 | 8 | 2 | 4 | 466.52 |

25 | 4 | 1 | 4 | 8 | 426.20 | 57 | 8 | 1 | 8 | 7 | 909.32 |

26 | 4 | 2 | 3 | 7 | 925.00 | 58 | 8 | 2 | 7 | 8 | 1116.48 |

27 | 4 | 3 | 2 | 6 | 311.40 | 59 | 8 | 3 | 6 | 5 | 390.84 |

28 | 4 | 4 | 1 | 5 | 388.48 | 60 | 8 | 4 | 5 | 6 | 384.04 |

29 | 4 | 5 | 8 | 4 | 379.04 | 61 | 8 | 5 | 4 | 3 | 646.20 |

30 | 4 | 6 | 7 | 3 | 344.92 | 62 | 8 | 6 | 3 | 4 | 397.24 |

31 | 4 | 7 | 6 | 2 | 592.48 | 63 | 8 | 7 | 2 | 1 | 572.80 |

32 | 4 | 8 | 5 | 1 | 613.24 | 64 | 8 | 8 | 1 | 2 | 612.36 |

Factors | ${\overline{\mathit{F}}}_{\mathit{i}}^{1}$ | ${\overline{\mathit{F}}}_{\mathit{i}}^{2}$ | ${\overline{\mathit{F}}}_{\mathit{i}}^{3}$ | ${\overline{\mathit{F}}}_{\mathit{i}}^{4}$ | ${\overline{\mathit{F}}}_{\mathit{i}}^{5}$ | ${\overline{\mathit{F}}}_{\mathit{i}}^{6}$ | ${\overline{\mathit{F}}}_{\mathit{i}}^{7}$ | ${\overline{\mathit{F}}}_{\mathit{i}}^{8}$ | $\mathbf{\Delta}{\mathit{F}}_{\mathbf{i}}$ |
---|---|---|---|---|---|---|---|---|---|

V | 521.27 | 603.13 | 591.68 | 497.60 | 566.57 | 572.01 | 612.46 | 628.66 | 131.07 |

ρ | 783.84 | 998.36 | 329.43 | 409.16 | 440.75 | 461.11 | 541.64 | 629.07 | 668.94 |

T | 526.14 | 538.95 | 551.39 | 551.85 | 565.14 | 569.66 | 622.41 | 667.83 | 141.69 |

Sources | SS_{i} | df_{i} | V_{i} | FTEST_{i} |
---|---|---|---|---|

V | 114,441.90 | 7 | 16,348.84 | 4.44 |

ρ | 2,765,486.06 | 7 | 395,069.44 | 107.35 |

T | 126,117.76 | 7 | 18,016.82 | 4.90 |

e | 25,760.88 | 7 | 3680.13 | —— |

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## Share and Cite

**MDPI and ACS Style**

Zhao, Q.; Li, Z.; Lu, P.; Wang, Q.; Wei, J.; Hu, S.; Yang, H.
An Investigation of the Influence on Compacted Snow Hardness by Density, Temperature and Punch Head Velocity. *Water* **2023**, *15*, 2897.
https://doi.org/10.3390/w15162897

**AMA Style**

Zhao Q, Li Z, Lu P, Wang Q, Wei J, Hu S, Yang H.
An Investigation of the Influence on Compacted Snow Hardness by Density, Temperature and Punch Head Velocity. *Water*. 2023; 15(16):2897.
https://doi.org/10.3390/w15162897

**Chicago/Turabian Style**

Zhao, Qiuming, Zhijun Li, Peng Lu, Qingkai Wang, Jie Wei, Shengbo Hu, and Haorong Yang.
2023. "An Investigation of the Influence on Compacted Snow Hardness by Density, Temperature and Punch Head Velocity" *Water* 15, no. 16: 2897.
https://doi.org/10.3390/w15162897