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The gray renewal GM (1,1) landslide prediction model was established by improving the gray model. Based on the established model, the author has made prediction of landslide deformation to the Xiangjiapo landslide and the Lianziya dangerous rock body. The results show that the gray renewal GM (1,1) model can supplement the new information in time and remove the old information which reduces the meaning of the information because of time lapse. Therefore, the model is closer to reality.

Landslide prediction research has started from the 1960s, and has now become a hot topic in landslide research. Due to the complexity of the landslide problem, landslide time prediction is still a worldwide problem. In these recent years, domestic and foreign experts and scholars in the landslide forecast theoretical research and practical work in the geotechnical engineering field have done a lot of work, and achieved remarkable results [

There are many ways to forecast landslides, although the methods have their own unique, but there are corresponding shortcomings and deficiencies. The gray system analysis method has a good applicability for the incomplete or incomplete information. The GM (1,1) model has been widely used in landslide prediction [

Gray system theory is a famous scholar founded by professor Deng J. L. in 1982. It is based on the small sample of “some information is known and some information is unknown”, and the “poor information” uncertainty system is the main research object. “Part” known information generation, development, extraction of valuable information, to achieve the correct understanding of the system and the exact description of the law, and scientific forecasting. Gray system is through the finishing of the original data to seek its changing laws, while the future state of the system to make scientific predictions. The gray prediction model, also known as the GM model, can be used to observe, analyze and long-term predictions of the studied system.

Assuming that the original number as X ( 0 )

X ( 0 ) = ( x ( 0 ) ( 1 ) , x ( 0 ) ( 2 ) , ⋯ , x ( 0 ) ( n ) ) (1)

According to the gray system theory on the original sequence of first-order accumulation (1-AGO) generated, the resulting column is:

X ( 1 ) = ( x ( 1 ) ( 1 ) , x ( 1 ) ( 2 ) , ⋯ , x ( 1 ) ( n ) ) (2)

x ( 1 ) ( k ) can be calculated using the following formula:

x ( 1 ) ( k ) = ∑ i = 1 k x ( 0 ) ( i ) , k = 1 , 2 , ⋯ , n (3)

The differential equation of the albino form of the system prediction model GM (1,1) is expressed as:

d x ( 1 ) d t + a x ( 1 ) = b (4)

Assuming that a ^ = | a , b | T , then identification a can be calculated by the following formula

a ^ = | a , b | T = ( B T B ) − 1 B T Y (5)

where Y = [ x ( 0 ) ( 2 ) x ( 0 ) ( 3 ) ⋮ x ( 0 ) ( n ) ] (6)

B = [ − 1 2 [ x ( 1 ) ( 1 ) + x ( 1 ) ( 2 ) ] 1 − 1 2 [ x ( 1 ) ( 2 ) + x ( 1 ) ( 3 ) ] 1 ⋮ ⋮ − 1 2 [ x ( 1 ) ( n − 1 ) + x ( 1 ) ( n ) ] 1 ] (7)

Then the time response sequence can be expressed as

x ^ ( 1 ) ( k + 1 ) = ( x ( 0 ) ( 1 ) − b a ) e − a k + b a , k = 1 , 2 , ⋯ , n (8)

The simulation value is expressed as

x ^ ( 0 ) ( k + 1 ) = x ^ ( 1 ) ( k + 1 ) − x ^ ( 1 ) ( k ) , k = 1 , 2 , ⋯ , n (9)

In the conventional GM (1,1) model modeling, the past data from the real time t = n is used. However, the development of any gray system, with the passage of time, will continue to have some random disturbance factors into the system, so that the development of the system affected. Therefore, with the conventional GM (1,1) model, the higher accuracy is only a few recent data, farther away from reality, the weaker the prediction. In order to reflect the impact of the future random disturbance on the gray system and improve the prediction accuracy, the GM (1,1) model is grayed out.

In the original data X ( 0 ) = ( x ( 0 ) ( 1 ) , x ( 0 ) ( 2 ) , ⋯ , x ( 0 ) ( n ) ) , the latest information x ( 0 ) ( n + 1 ) is placed and the oldest information x ( 0 ) ( 1 ) , X ( 0 ) = ( x ( 0 ) ( 2 ) , x ( 0 ) ( 3 ) , x ( 0 ) ( n + 1 ) ) is removed. The model established according to the above steps is the gray metabolic GM (1,1) model, and a series of prediction data is obtained at the same time.

The accuracy of the gray prediction model is usually checked by the posterior difference method. The model accuracy is evaluated by the mean square error ratio and the small error probability. The smaller the mean square error ratio and the smaller the probability of small errors, the higher the accuracy of the prediction model. The basic method is as follows: assuming X ( 0 ) for the original sequence, x ^ ( 0 ) for the GM (1,1) model simulation sequence, ε ( 0 ) for the residual series, then

x ¯ = 1 n ∑ k = 1 n x ( 0 ) ( k ) , (10)

S 1 2 = 1 n ∑ k = 1 n ( x ( 0 ) ( k ) − x ¯ ) 2 (11)

Respectively, the mean of X ( 0 ) , variance;

ε ( 0 ) ( k ) = x ( 0 ) ( k ) − x ^ ( 0 ) ( k ) (12)

ε ¯ = 1 n ∑ k = 1 n ( ε ( 0 ) ( k ) − ε ¯ ) 2 (13)

S 2 2 = 1 n ∑ k = 1 n ( ε ( 0 ) ( k ) − ε ¯ ) 2 (14)

C = S 2 S 1 (15)

where ε ¯ is the mean of residual ε ( 0 ) ( k ) , S 2 2 is the residual variance, and C is the mean square deviation. The mean square error ratio C and the small error

probability p are calculated from C = S 2 S 1 and p = P ( | ε ( k ) − ε ¯ | < 0.6745 S 1 ) .

The accuracy of the model is shown in

In order to test the actual prediction effect of the gray metabolic prediction model, Xiangjiapo landslide data were used to predict. The landslide is located in K13+500 - K13+960 of Chongqing-Guizhou expressway. The landslides in the landslide are the quaternary of the quaternary system and the Jurassic lower Pearl Group (J_{1Z}), the spontaneous flow (J_{1-2Z}) and the Triassic upper Xujiahe formation (T_{3XJ}). The landslide occurred mainly in the quaternary landslide accumulation layer ( Q 4 del ) and the quaternary system ( Q 4 col + dl ). The landslide is mainly composed of silty clay clumps and strong weathering sand and mudstone. It belongs to mixed rock and rock landslide. The rock mass structure is loose, the water content is high, unstable and easy to deform.

The highway landslide monitoring information see the literature [

Model 1 is the predicted value using t = 1, 2, 3, 4, 5, and 6 weeks. Model 2 is the predicted value using t = 2, 3, 4, 5, 6 and 7 weeks. Model 3 is the predicted value using t = 3, 4, 5, 6, 7, and 8 weeks. As can be seen from

Model accuracy level | Small probability of error p | Mean square deviation ratio C |
---|---|---|

Level 1 (good) | ≥ 0.95 | ≤ 0.35 |

Level 2 (qualified) | 0.80 ≪ p < 0.95 | 0.35 < C ≤ 0.50 |

Level 3 (barely) | 0.70 ≪ p < 0.80 | 0.50 < C ≤ 0.65 |

Level 4 (unqualified) | < 0.70 | > 0.65 |

Monitoring date | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Measuring point displacement value/mm | 32.300 | 48.600 | 69.600 | 96.370 | 128.700 | 168.200 | 207.870 | 256.400 | 306.800 |

Observed total time/ week | Measured value/mm | Model 1 | Relative error/% | Model 2 | Relative error/% | Model 2 | Relative error/% |
---|---|---|---|---|---|---|---|

1 | 32.300 | 32.30000 | 0 | ||||

2 | 48.600 | 51.70845 | 6.39598 | 48.6 | 0 | ||

3 | 69.600 | 69.45098 | 0.21411 | 74.3689 | 6.85182 | 69.6 | 0 |

4 | 96.370 | 93.28144 | 3.20490 | 96.4319 | 0.06426 | 101.6628 | 5.49217 |

5 | 128.700 | 125.28876 | 2.65054 | 125.0405 | 2.84346 | 128.3116 | 0.30181 |

6 | 168.200 | 168.27863 | 0.04675 | 162.1363 | 3.60505 | 161.9458 | 3.71833 |

7 | 207.870 | 226.01947 | 8.73116 | 210.2374 | 1.13889 | 204.3965 | 1.67102 |

8 | 256.400 | 303.57270 | 18.39809 | 272.6087 | 6.32164 | 257.9747 | 0.61416 |

9 | 306.800 | 407.73650 | 32.89977 | 353.4837 | 15.21633 | 325.5973 | 6.12691 |

model | Relative error α | Mean square deviation ratio C | Small probability of error p |
---|---|---|---|

model 1 | 0.09068 | 0.38085 | 1 |

model 2 | 0.05149 | 0.19714 | 1 |

model 3 | 0.02987 | 0.10169 | 1 |

of model 3 is level 1, and the values of C and α are decreasing from model 1 to model 3, indicating that the accuracy of the model is gradually increasing. It can be seen that the gray renewal GM (1,1) model is used to predict, because the system constantly update the modeling data, remove the old data, so that the system’s prediction accuracy has been improved, gradually close to the measured value.

The rock masses is located in the south bank of the Xiling gorge in the Xintan Town, Zigui Town, Zigui County, Hubei Province. It is located at 26 km from the Sandouping dam. The site and the new beach landslide confrontation, its length 700 m, east-west width of 30 - 200 m, nearly 100 m high. According to the monitoring data of the Lianziya hazardous rock masses in the literature [

Model 1 is the predicted value when the observed values are from 1978.12 to 1983.12, the model 2 is the predicted value at 1979.12 to 1984.12, and the model 3 is the predicted value at 1980.12 to 1985.12. As can be seen from

Time (year/ month) | Displacement observations/ mm | model 1 | Relative error/% | model 2 | Relative error/% | model 3 | Relative error/% |
---|---|---|---|---|---|---|---|

1978.12 | 10.32 | 10.32 | 0 | ||||

1979.12 | 26.96 | 29.37167 | 8.94537 | 26.96 | 0 | ||

1980.12 | 34.07 | 32.97365 | 3.21794 | 35.43405 | 4.00365 | 34.07 | 0 |

1981.12 | 38.65 | 37.01735 | 4.22420 | 38.25624 | 1.01879 | 39.87869 | 3.17901 |

1982.12 | 42.98 | 41.55695 | 3.31097 | 41.30320 | 3.90134 | 42.02683 | 2.21771 |

1983.12 | 44.93 | 46.65325 | 3.83542 | 44.59285 | 0.75038 | 44.29068 | 1.42292 |

1984.12 | 47.16 | 52.37454 | 11.05713 | 48.14451 | 2.08759 | 46.67648 | 1.02527 |

1985.12 | 48.38 | 58.79746 | 21.53258 | 51.97904 | 7.43911 | 49.19080 | 1.67590 |

model | Relative error α | Mean square deviation ratio C | Small probability of error p |
---|---|---|---|

model 1 | 0.08018 | 0.33823 | 0.857 |

model 2 | 0.03200 | 0.23306 | 1 |

model 3 | 0.01904 | 0.17360 | 1 |

has high prediction accuracy, but also has strong self-adjustment ability. It can adjust the model parameters with the new data, and reflect the evolution direction of the system in time prediction.

Based on the data of GA point deformation data of Xiangjiapo landslide and Lianziya hazardous rock masses, this paper uses the gray renewal GM (1,1) model method to predict the landslide. The results show that the gray metabolic GM (1,1) model can supplement the new information in time and remove the old information which reduces the meaning of the information because of time lapse. Therefore, it can reflect the current characteristics of the system. The model is closer to reality. The method has high accuracy, the prediction results can be used to guide the construction site, to achieve the dynamic management of landslide management, and application prospects are very broad.

Wang, Z.Y. (2017) Application of Renewal Gray GM (1,1) Model to Prediction of Landslide Deformation. Journal of Geoscience and Environment Protection, 5, 148-154. https://doi.org/10.4236/gep.2017.59011