Irregular GIS Curve Fitting based High Speed Railway Earthquake Influence Range Calculation Model

In this paper, to guarantee that the train can take measures to reduce the damage caused by the earthquake, it propose an irregular GIS curve fitt ing based high-speed railway earthquake influence range calculation model. Firstly, this model eliminates the abnormal points, calculates feature points and finds demarcation points of the highspeed railway GIS curve to get the processed point collection in Mercator coordinate. Secondly, though usin g the processed point collection, this model applies least square polynomial segmentation fitt ing method to implement complex high-speed GIS curve fitt ing. Thirdly, calculate the earthquake influence rang on high-seed railway line, according to the scope of the earthquake equation and the high-speed railway GIS curve fitt ed equation. Finally, the paper selects the Beijing So uth to Dezhou East high-speed railway section which is part of Beijing-Shanghai line as a case study, which proves that the model can calculate the earthquake influence scope on the railway line offering decision support for train operation to ensure safety.


Introduction
By the end of 2016, China's total mileage of h igh-speed railway was more than 20,000 kilo meters [1,2], accounted for over 60% of the world's total mileage of high speed railway, which ranked first in the world. China is located in the earthquake area which is vulnerable to earthquake disasters [3]. When the earthquake occurs, within its influence scope, high-speed railway line will vib rate and distort as the ground, leading to the trains jump ing up and down, wh ich will cause a devastating disaster [4,5].
At present, the discretizat ion of high-speed railway lines' GIS curve mainly through the direct method and the indirect method for the data acquisition [6]. Both ways use a finite number of coordinate points to characterize the railway line but without a mathematical equation to represent. The main process of curve discretizat ion of h igh-speed railway lines is easy to produce abnormal points and lose feature points, which are caused by the system error o r random select ion of high-speed railway GIS curve points, leading to the coordinate points deviate fro m the correct position gravely. How to eliminate abnormal points, calculate feature points and on this basis to achieve the high-speed GIS curve fitt ing is an important problem urgent to tackle [7,8].
The high-speed railway lines are mainly co mposed of line segment and circular arc. In view of co mmon ly used methods of irregular curve fitting [9,10] and comb ined with the characteristics of the high-speed railway lines, this paper puts forward an irregular GIS curve fitting based high-speed railway earthquake influence range calculation model. The model main ly includes two parts . The first part is to use the least square polynomial fitting method [11] to get the mathematical model o f GIS curve including eliminate abnormal points, calculate feature points, find demarcation points. The second part is based on the mathematical model in co mbination with influence scope of earthquake equation to get the earthquake influence range of the high-speed railway line. Finally, this paper uses Beijing South to Dezhou East high speed railway section which is part of the Beijing-Shanghai high-speed railway line as a case to implement the proposed model. The R-square and Adjusted R-square are all above 0.99, shows the excellent goodness of fit by the least square polynomial fitting method proposed above. Moreover though simulate the earthquake happens and the curve fitted, this paper validates the correctness of the high-speed railway earthquake influence range calculation model.

Irregular high-speed lines GIS curve fitting
Irregular GIS curve fitting of high-speed railway line is to select the appropriate type of fitting method to the coordinate points. Curve fitting model consists of linear and nonlinear fitting and the polynomial fitting belongs to the linear fitting. Exponential function and trigonometric function belong to nonlinear fitting. In base of weather the fitting method can generate parameter equation or not, curve fitt ing, can also be divided into the parameter fitting and nonparametric fitting. The parameter fitt ing is mainly based on the least square fitting. Nonparametric fitting mainly use interpolation method such as cubic spline interpolation which can pass all data points and smoothness is better, but it is difficu lt to get the parameter equation. High-speed railway line is mainly made up of line segments and the circular arc with the large rad ius. Meanwhile the polynomial fitt ing can adapt to the characteristics of high-speed railway line. Therefore, in o rder to get parameters equation model, and guarantee the good fitting effect, this article adopts the method of least square polynomial fitting.
For the discretization of high-speed railway lines GIS curve, the discretization coordinate points may have abnormal points and redundancy points, which serious influence the goodness of curve fitting. Thus the first step of curve fitting is to eliminate abnorma l points and reduce the redundancy points. To the complex h igh-speed lines curve also need to extract the boundary point. This chapter will introduce each part of the process in detail.

Eliminate abnormal points
Abnormal points refer to the points of GIS curve deviates fro m the position gravely. The abnormal point elimination is according to the distance ratio as shown in  to calculate the distance is shown in (1)-(3).
Generally l value of GIS curve is small, but the l value of abnormal point will increase significantly.

Calculate feature points
Feature points are generally recognized as the major points of the curve, which can describe the curve effectively. The method of selecting feature points mainly consists of angle test method and a polygon approximation method. This paper adopts an improved feature extract ion method of vector curves, which is more simp le and accurate than Douglas -Peucker algorith m [12]. Suppose the points of the high-speed railway GIS line The slope of Set the threshold be a small value, if the The selection of feature points is mainly evaluated by the data compression and fidelity, which is measured by compression ratio and integral square error co mmonly. The compression ratio is calculated as (7).

Find demarcation points
Demarcat ion points represent the key position points of the curve, such as turning point. The demarcation points are mainly to select the largest slope point among the feature points.

The least square polynomial fitting
The least square polynomial fitting is mainly according to the given set of data points in the space of polynomial:

T T C A A A Y
Above all, the equation of high-speed railway line under the least square polynomial method is obtained.

High speed railway earthquake influence range calculation model
At the time o f the earthquake happens, it produces P wave and the S wave, wh ich is also known as the longitudinal wave and transverse wave. The P wave's vibration direct ion is parallel to the direction of earthquake and its transmission speed is very fast.
However the S wave's vibration direction is vertical to the direction of earthquake and its transmission speed is very slow. P wave is more destructive than the S wave.
High-speed railway earthquake monitoring and early warning system is based on the P wave and S wave velocity difference to realize earthquake early warning. The principle of h igh-speed railway earthquake monitoring and early warning is shown in Fig 3.

Case study
To validate the proposed model, this paper selects Beijing South to Dezhou East high-speed railway section which is part of the Beijing-Shanghai h igh-speed railway line as a test case. At the same time, the railway line is through the earthquake zone as shown in

The least square polynomial fitting
For the processed Mercator coordinate points set of Beijing South-Dezhou East high-speed railway GIS curve, adopt the least square polynomial fitting, which cannot obtain good fitting effect. The fitted figure is shown in Fig 10 (a) Fig 10 (b-d) and the fitted polynomial equation is (20).The R-square is nealy 1and SSE is very sall,which are shown in Table 1.  (20)

Earthquake influence range calculation
This paper simu lates a earthquake that happens with the magnitude of 5.9 and the epicentral distance of 63 KM.
By using the proposed model, the in fluence range of the earthquake on the high-speed railway line was obtained and as shown in Fig 11.

Conclusion
The model proposed by this paper still need to be improved to reduce the earthquake damage to the highspeed railway. The future work of this paper is to simp lify the equation of the high-speed railway line and improve the efficiency of the calcu lation speed to quickly get the influence scope of earthquake on the high-speed railway line. Also, trying to use the big data and cloud computing technology to improve the efficiency of earthquake early warning computing is also urgent to study.

Figure 11
Influence range of the earthquake on the Beijing-Shanghai East high-speed railway gis curve