text:
Journal topic
Nat. Hazards Earth Syst. Sci., 19, 629–653, 2019
https://doi.org/10.5194/nhess-19-629-2019
Nat. Hazards Earth Syst. Sci., 19, 629–653, 2019
https://doi.org/10.5194/nhess-19-629-2019

Research article 25 Mar 2019

Research article | 25 Mar 2019

# Application of the Levenburg–Marquardt back propagation neural network approach for landslide risk assessments

Application of the Levenburg–Marquardt back propagation neural network approach for landslide risk assessments
Junnan Xiong1,3, Ming Sun2, Hao Zhang1, Weiming Cheng3, Yinghui Yang1, Mingyuan Sun1, Yifan Cao1, and Jiyan Wang1 Junnan Xiong et al.
• 1School of Civil Engineering and Architecture, Southwest Petroleum University, Chengdu, 610500, P.R. China
• 2The First Surveying and Mapping Engineering Institute of Sichuan Province, Chengdu, 610100, P.R. China
• 3State Key Laboratory of Resources and Environmental Information System, Institute of Geographic Science and Natural Resources Research, Chinese Academy of Sciences, Beijing, 100101, P.R. China

Correspondence: Junnan Xiong (neu_xjn@163.com) and Hao Zhang (zhanghao412658@163.com)

Abstract

Landslide disasters are one of the main risks involved with the operation of long-distance oil and gas pipelines. Because previously established disaster risk models are too subjective, this paper presents a quantitative model for regional risk assessment through an analysis of the patterns of historical landslide disasters along oil and gas pipelines. Using the Guangyuan section of the Lanzhou–Chengdu–Chongqing (LCC) long-distance multiproduct oil pipeline (82 km) in China as a case study, we successively carried out two independent assessments: a susceptibility assessment and a vulnerability assessment. We used an entropy weight method to establish a system for the vulnerability assessment, whereas a Levenberg–Marquardt back propagation (LM-BP) neural network model was used to conduct the susceptibility assessment. The risk assessment was carried out on the basis of two assessments. The first, the system of the vulnerability assessment, considered the pipeline position and the angle between the pipe and the landslide (pipeline laying environmental factors). We also used an interpolation theory to generate the standard sample matrix of the LM-BP neural network. Accordingly, a landslide susceptibility risk zoning map was obtained based on susceptibility and vulnerability assessment. The results show that about 70 % of the slopes were in high-susceptibility areas with a comparatively high landslide possibility and that the southern section of the oil pipeline in the study area was in danger. These results can be used as a guide for preventing and reducing regional hazards, establishing safe routes for both existing and new pipelines, and safely operating pipelines in the Guangyuan area and other segments of the LCC oil pipeline.

1 Introduction

By the year 2020, the total length of long-distance oil and gas pipelines is expected to exceed 160 000 km in China. This represents a major upsurge in the length of multinational long-distance oil and gas pipelines (Huo et al., 2016). The rapid development of pipelines is associated with significant geological hazards, especially landslides, which increasingly threaten the safe operation of pipelines (Wang et al., 2012; Yun and Kang, 2014; Zheng et al., 2012). Landslide disasters cause great harm to infrastructure and human life. Moreover, the wide impact area of landslides restricts the economic development of landslide-prone areas (Ding et al., 2016; Hong et al., 2015). A devastating landslide can lead to casualties, property loss, environmental damage, and long-term service disruptions caused by massive oil and gas leakages (G. Li et al., 2016; Zheng et al., 2012). Generally, pipeline failure or destruction caused by landslides is much more deleterious than the landslides themselves, which makes it important to research the risk assessment of geological landslide hazards in pipeline areas (Inaudi and Glisic, 2006; Mansour et al., 2011).

Natural disaster risk is comprised of a combination of natural and social attributes (Atta-Ur-Rahman and Shaw, 2015). The United Nations Department of Humanitarian Affairs defines natural disaster risk as a product of susceptibility and vulnerability (Rafiq and Blaschke, 2012; Sari et al., 2017). In recent years, progress in geographic information systems (GIS) and remote sensing (RS) technologies have greatly enhanced our ability to evaluate the potential risks that landslides pose to pipelines (Akgun et al., 2012; Li and Gao, 2015; Sari et al., 2017). The disaster risk assessment model has been widely recognized and applied by experts and scholars all over the world. Landslide risk assessment can take the form of a qualitative (Wu et al., 1996), quantitative (Ho et al., 2000) or semiquantitative assessment (Liu et al., 2015) according to actual demand. Quantitative methods and models that have been proposed for the assessment can be divided into methods of statistical analysis (Sari et al., 2017), mathematical models (Akgun et al., 2012) and machine learning (He and Fu, 2009). However, most of these methods are subjective, such as expert evaluations, analytical hierarchy processes, logistic regressions and fuzzy integration methods, which could affect the accuracy and reasonableness of the evaluation (Fall et al., 2006; Sarkar and Gupta, 2005). This shortcoming can be overcome through the artificial neural network, especially the mature back propagation (BP) neural network that is widely used in function approximation and pattern recognition (Ke and Li, 2014a; Li et al., 2013; Su and Deng, 2003). The evaluation indicator system generally includes disaster characteristics, disaster prevention and pipeline attributes (Li, 2008, 2010). The fault tree analysis, fuzzy comprehensive evaluation and the grey theory are used to evaluate the failure probability of the system through indicator weight and scoring (Shi, 2011; Ye et al., 2013). In previous studies, pipeline vulnerability evaluation indicators only considered the pipeline itself, and the relationship between the pipeline and environment was rarely examined (Feng et al., 2014; Li, 2008; Liu et al., 2015). In this paper, the interaction between landslide hazards and the pipeline itself was considered, which improved the quantitative degree of the evaluation.

Based on the theory of the Levenberg–Marquardt back propagation (LM-BP) neural network, a standard sample matrix was developed using interpolation theory, after an analysis of the distribution characteristics of landslides that occurred in the study area was performed and a regional landslide susceptibility assessment was completed. Considering the interaction between landslide disasters and the pipeline itself, a pipeline vulnerability evaluation in the landslide area was realized using the entropy weight method. This paper establishes a risk assessment model and methods for assessing landslide geological susceptibility of oil pipelines by comprehensively utilizing GIS and RS technology, which together improves the quantitative degree of the assessment.

2 Study area

The study area was Guangyuan City in Sichuan province, which was further narrowed to the area from 10515 to 10604 E and 3203 to 3245 N, straddling 19 townships in five counties from south to north (Fig. 1). The Lanzhou–Chengdu–Chongqing (LCC) multiproduct oil pipeline is China's first long-distance pipeline. It begins in Lanzhou City and runs through Shanxi and Sichuan provinces (Hao and Liu, 2008). Our study area covered sloped areas of this range, with 5 km on both sides of the Guangyuan section (82 km) of the oil pipeline. The pipeline within the K558–K642 pipes may be affected by the slope areas. The Guangyuan section, located in northern Sichuan, is a transitional zone from the basin area to the mountain area. It features terrain of moderate and low mountains, crisscrossed networks of ravines and a strong fluvial incision. Altitudes in this area range from 328 to 1505 m. The study area has a subtropical monsoon climate with four distinctive seasons and annual precipitation measuring about 900 to 1000 mm. Moreover, two large unstable faults (the central fault of the Longmen Mountains and Longmen Mountains' Piedmont fault zone) make the area geologically unstable and prone to frequent geological hazards (Li et al., 2012). Guangyuan, through which the pipeline passes, has a high incidence of landslides, which have happened 300 times in the Lizhou and Chaotian districts (Zhang et al., 2011). In this area, landslide geological hazards seriously threaten the safe operation of the LCC oil pipeline.

Figure 1Landslide location map of the study area.

3 Data sources

Landslide susceptibility assessment, pipeline vulnerability assessment and geological hazard risk assessment of the landslide pipeline were made successively. Digital elevation model (DEM) data with 30 m accuracy were sourced from the Geospatial Data Cloud (http://www.gscloud.cn/, last access: January 2017. Precipitation data were downloaded from the dataset of annual surface observation values in China between the years 1981 and 2010, as published by the China Meteorological Administration (http://data.cma.cn/, last access: January 2017). These data were collected from 18 meteorological observatories near and within the study area and interpolated using the kriging method (at a resolution of 30 m ×30 m). Geological maps and landslide data (historical landslides) in the study area were obtained from Sichuan province's geological environmental monitoring station. RS images (GF-1, multispectral 8 m, resolution 2 m) were provided by the Sichuan Remote Sensing Center.

The location of the middle line of the pipeline was detected through the direct connection method (i.e., the transmitter's output line was directly connected to the metal pipeline) using an RD8000 underground pipeline detector. Pipeline midline coordinates were measured using total network real-time kinematic technology, and, simultaneously, the coordinates of the pipe ancillary facilities (including detective poles, mileage pegs and milestones) were acquired. Mileage data obtained through inner pipeline detection were derived from the China Petroleum Pipeline Company.

4 Methods

## 4.1 Assessment unit

Division precision and the scale of the slope unit (i.e., the basic element for a regional landslide susceptibility assessment) were in keeping with the results of the evaluation (Qiu et al., 2015). A total of 315 slope units were divided using hydrologic analysis in ArcGIS (v. 10.4) (Fig. 2a). The irrational unit (a slope unit with an inaccurate boundary) was artificially identified and modified by comparing GF-1 satellite remote sensing images. Boundary correction, fragment combination and fissure filling were used for modification.

This vulnerability study focuses on assessing the vulnerability of transport pipelines to landslides. Considering both previous research and the particulars of the research object, we used a comprehensive segmentation method based on GIS to divide the pipelines in our study. A total of 180 pipes were divided in the study area, of which the longest was about 1.7 km, and the shortest was only about 10 m (Fig. 2b).

Figure 2All slope units (a) and pipeline sections (b) in the study area.

## 4.2 Assessment indicators

Based on selection principles of the indicator system and the formation mechanism of landslide geological hazards, as few indicators as possible were selected to reflect the degree of danger posed by the landslide as accurately as possible (Avalon Cullen et al., 2016; Jaiswal et al., 2010; Ray et al., 2007). The internal factors in these indicators included topography, geological structure, stratigraphic lithology and surface coverage. Similarly, the external factors included mean annual precipitation (MAP) and the coefficient of the variation of annual rainfall (CVAR). The correlations between indicators were analyzed using R (v. 3.3.1), and the results show a significant correlation between MAP and CVAR (R=0.99) and between the normalized difference water index (NDWI) and normalized differential vegetation index (NDVI) (R=0.87). Based on correlation and standard deviation, CVAR and NDWI were eliminated from the original evaluation system for the landslide susceptibility assessment in the pipeline area (Table 1).

Table 1Indicators of landslide susceptibility assessment and pipeline vulnerability assessment.

Generally, the evaluation indicator of pipeline vulnerability as it relates to the relationship between a pipeline and its surrounding environment is rarely considered. The evaluation indicators in this paper were refined to include pipeline parameters and the spatial relationship between a pipeline and a landslide. The pipelines in the study area were based in mountainous areas and have been running for many years. All of these pipelines consisted of high-pressure pipes that were made of steel tubes and had a diameter of 610 mm for conveying oil. In keeping with the theory of the entropy weight method, these indicators (e.g., pressure, materials, diameter and media) were not included in the final evaluation system used to determine pipeline vulnerability.

## 4.3 LM-BP neural network model

A neural network is a nonlinear mathematical structure which is capable of representing complex nonlinear processes that relate the inputs and outputs of any system (Hsu et al., 1995). With its good performance in nonlinear statistical modeling, it is very useful in exploring the hidden relationships between the inputs and the outputs (Wu and Wang, 2016). A BP neural network with many adjustable parameters has a powerful parallel processing mechanism, high flexibility and can incorporate uncertainty information well. The mechanism of landslide evaluation is complex with many uncertainties and incomplete information (Jie et al., 2015). The BP neural network model can calculate the intrinsic rules from the vast amount of complex and fuzzy data in the changing environment and make corresponding inferences. The information about landslides reflected by the data used in the process of susceptibility assessment is mostly qualitative rather than quantitative. Through the analysis of this fuzzy information, accurate assessment results can be obtained. Landslide susceptibility assessment is essentially a study of pattern recognition (Feng et al., 2017). The BP neural network can approximate arbitrary continuous function with arbitrary precision, so it is widely used in nonlinear modeling, pattern recognition and pattern classification (Xiong et al., 2010). Because the BP neural network model is widely used, there are many successful cases as a reference for the number of neurons in each layer, the parameters of network learning and the optimization of algorithms, which can effectively improve the reliability and accuracy of the model (Ke and Li, 2014b).

The LM algorithm, also known as the damped least-squares method, has the advantage of local fast convergence. Its strong global searching ability contributes to the strong extrapolation ability of the trained network. The LM algorithm is a combination of the gradient descent method and Gauss–Newton method. Its iteration process is no longer along a single negative gradient direction, which greatly improves the convergence speed and generalization ability of the network (Li et al., 2016). The BP neural network model, optimized by the LM algorithm, was used to evaluate the regional landslide susceptibility in this study. MATLAB 2014 with the trainlm training function was used to implement the LM-BP neural network. The flowchart of LM-BP neural network algorithm is shown in Fig. 3.

Figure 3Flowchart of the LM-BP neural network algorithm.

Figure 4The frequency distribution of each indicator in the landslide location. Maps (a), (b), (c), (d), (e), (f), (g) and (h) represent the elevation, slope, aspect, height difference, TPC, NDVI, MAP and distance from the fault, respectively.

Data from 106 landslide disasters were collected near the research area. Of these landslides, 23 were within the region of the study area. Most of the landslides located outside the study area were less than 20 km away from the pipeline. Due to comparable environmental conditions, these landslides could still help us identify the relationship between landslides and environment factors. In light of the frequency distribution of each evaluation indicator (Fig. 4), the landslide susceptibility grade corresponding to each interval of the indicators was divided, and then the susceptibility degree monotonicity in each interval was decided. For this study, the landslide susceptibility grade was divided into four levels: low (I), medium (II), high (III) and extremely high (IV). Based on previous research experience and field investigations (Appendix H), the monotonous intervals of different indicators of susceptibility degrees were judged (Appendix A). For instance, there were hardly any landslides, only collapses that occurred in slopes above 60. Besides, the susceptibility degree in the area was monotone decreasing in the slope interval of 60 to 90. Because of the very small sliding force in slopes at 0 to 15, landslides were rare to occur here, even under other extreme conditions (Zhang et al., 2015). On the basis of the classification criteria of the evaluation indicators used to predict landslide susceptibility degree and the functional relationship between the evaluation indicators and landslide probabilities, standard samples (training samples and test samples) were built using a specific mathematical method. When establishing the empty matrix, the sample size of each landslide susceptibility level was set to 200, and the training sample size was 800. According to the order of susceptibility from low to high (Appendix A), the input was constructed by interpolating for each interval. The interval of the susceptibility degree is [0, 1], and the output is obtained by interpolating 800 values equidistantly between the interval of [0, 1] (Appendix B). Using interpolation theory to build samples avoided the excess human influence in the process of building a neural network model by traditional methods. The training samples and test samples were evaluated using similar construction methods but with different sample sizes. Finally, the indicator data were normalized, they were entered into the LM-BP neural network for simulation and 315 slope unit landslide susceptibility values were output.

## 4.4 Vulnerability assessment model for pipelines

The vulnerability evaluation model of pipelines in the landslide area was established using the entropy weight method, which overcame the shortcomings of the traditional weight method that does not consider the different evaluation indicators and the excessive human influence on the process of evaluation (Gao et al., 2017; Pal, 2014). Entropy is a method of measuring the uncertainty of information by using probability theory (Liu and Zhang, 2011). The entropy indicates the extent of difference in an indicator: the more different the data, the greater the role in evaluation (Jia et al., 2007). The extremum difference method difference method was used to normalize each indicator value. The decision information of each index can be expressed by entropy value ei:

$\begin{array}{ll}& {r}_{ij}=\frac{{x}_{ij}-{min}_{j}\mathit{\left\{}{x}_{ij}\mathit{\right\}}}{{max}_{j}\mathit{\left\{}{x}_{ij}\mathit{\right\}}-{min}_{j}\mathit{\left\{}{x}_{ij}\mathit{\right\}}},\\ \text{(1)}& & {r}_{ij}=\frac{{max}_{j}\mathit{\left\{}{x}_{ij}\mathit{\right\}}-{x}_{ij}}{{max}_{j}\mathit{\left\{}{x}_{ij}\mathit{\right\}}-{min}_{j}\mathit{\left\{}{x}_{ij}\mathit{\right\}}},\text{(2)}& & {e}_{i}=\frac{\sum _{j=\mathrm{1}}^{n}p\left({x}_{ij}\right)\mathrm{ln}p\left({x}_{ij}\right)}{\mathrm{ln}\left(n\right)},\text{(3)}& & p\left({x}_{ij}\right)=\frac{{r}_{ij}}{\sum _{j=\mathrm{1}}^{n}{r}_{ij}},\text{(4)}& & {w}_{i}=\frac{\mathrm{1}-{e}_{i}}{m-\sum _{i=\mathrm{1}}^{m}{e}_{i}},\text{(5)}& & {H}_{j}=\sum _{i=\mathrm{1}}^{m}{w}_{i}{r}_{ij},\end{array}$

where n is the number of evaluation objects, and rij represents the ith evaluation indicator values of jth pipe sections. Hj is the evaluation value of the pipeline section's vulnerability; wi is the weight of the evaluation indicator.

Pipeline defect density was obtained from the pipeline internal inspection data, which consisted of both mileage data that needed to be converted into three-dimensional coordinate data and pipeline centerline coordinate data obtained through C# programming. In addition, the main slide direction of the landslide was replaced by the slope direction that was extracted by the DEM. The coordinate azimuth of the pipe section was extracted using the linear vector data of each pipe section, and the angle between the pipeline and the slope was calculated using the mathematical method. The calculation process was solved in the Visual Basic language on ArcGIS using second development functions. Finally, the entropy weight of 5 indicators was calculated by programming in MATLAB 2014. The entropy weight calculation results for pipeline landslide vulnerability assessment are shown in Table 2.

Table 2Entropy weight of evaluation indicator.

5 Results and comparison

## 5.1 Regional landslide susceptibility assessment

The LM-BP neural network was trained and the network was stopped after 182 iterations. An RMSE value of $\mathrm{9.93}×{\mathrm{10}}^{-\mathrm{9}}$ indicated that the goal of precision had been reached. Through the simulation of the network test, none of the absolute error values of test data (20 groups) were found to be greater than 0.02; this result aligned with our expectation of the precision of the landslide susceptibility assessment. The landslide susceptibility grade was divided into four levels by using the equal interval method at intervals of 0.25. The safe (low-susceptibility) section was located in the central part of the study area. The dangerous (high-susceptibility) section was located north and south (Fig. 5). In the study area, most of the exposed rock was dominated by shale, which belongs to the easy-slip rock group.

Figure 5Landslide hazard map of study area.

Average altitude ranged from 450 to 1400 m, and the relative height difference was greater than 80 m, with the slope between 15 and 35. Based on an overlay analysis of historic landslides within the study area, and susceptibility zonation maps, we surmised that the probability of landslides in the study area was extremely high, and that 87 % of the landslides occurred in the medium-, high-, and extremely high-susceptibility areas. Among these landslides, three were located in low-susceptibility areas, which accounted for 13 % of the landslide disaster sites, five occurred in medium-susceptibility areas (accounting for 21.7 % of disaster sites), seven occurred in high-susceptibility areas (accounting for 30.4 % of sites) and eight occurred in extremely high-susceptibility areas (accounting for 34.8 % of sites). The evaluation results were found to accurately reflect the trends and rules of distribution of landslides in the study area. The number and area of slopes in high-susceptibility and extremely high-susceptibility areas accounted for about 70 % of the total (Table 3). The probability of landslide occurrence in the study area was generally high, which was consistent with the fact that the region was landslide-prone.

Table 3Number and area of slopes of four hazard grades.

## 5.2 Vulnerability assessment for oil pipeline in landslide area

The equal interval of 0.25 was used to divide the pipeline vulnerability level into four grades to obtain the pipeline vulnerability zonation of the study area (Fig. 6). The pipeline in the northern part of the study area was given a low vulnerability grade, while the situation in the south of the region is more serious. The number, length and percentage of pipeline segments with different grade vulnerabilities are shown in Table 4. The number and length of pipeline segments in highly vulnerable areas (III) and extremely vulnerable areas (IV) accounted for about 12 % of the total.

Table 4Number and distances of pipeline of four vulnerability grades.

Figure 6Pipeline vulnerability map of study area.

## 5.3 Risk assessment for oil pipeline in landslide area

According to natural disaster risk expressions released by the UN, the definition of risk may be expressed as the product of landslide susceptibility in a pipeline area and pipeline vulnerabilities in the landslide area. Scientific analysis and expression of disaster risk assessment results can simplify complex risk assessments and accelerate findings (Ding and Tian, 2013). There is no unified criterion for disaster evaluation zoning, and the equal interval method is one of the methods to express the results more intuitively (Hu et al., 2011; Jin and Meng, 2011; Wang et al., 2011). The susceptibility and vulnerability degrees were distinguished using the equal interval method, and four risk grades were then automatically generated. Where the comprehensive risk assessment value was within 0 to 0.0625, the corresponding risk grade was Grade I; the corresponding risk grades with the values of 0.0625 to 0.25, 0.25 to 0.5625 and 0.5625 to 1.0 were Grade II, III and IV, respectively. The risk grade of each section of the pipeline within the research area is shown in Fig. 7.

Figure 7Pipeline risk map of study area.

The number of sections with a high-risk grade was 33, which accounted for 18.33 % of all pipeline sections and represented 16.57 % of the total pipeline length of 13 461 km. There were four sections with extremely high-risk grade, which accounted for 2.22 % of all sections and represented 3.31 % of the total pipeline length of 2.538 km. The section number and the length of the pipelines lying in high-risk (III) and extremely high-risk (IV) areas accounted for 20 % of the total pipeline length, and the risk grade of pipelines inside Qingchuan and Jian'ge County was relatively high.

## 5.4 Analysis of risk assessment results

Large or huge landslides were common in areas that we categorized as extremely high risk, which we defined as those that were geologically evolving or had experienced obvious deformations within the last 2 years with cracks that are still visible. These pipelines were subject to dangers at any time, as the pipelines within the areas prone to landslides were found to contain many defects or extensive damage. These areas also posed considerable threats; for example, pipeline ruptures or breaks could lead to leakages or serious deformations that cause transportation failure. Because these are unacceptable events, risk prevention and control measures must be taken as soon as possible. Pipelines with extremely high risk were mainly distributed in the following areas: (1) Xiasi village in Xiasi County (pipe no. K628–K630) and (2) Shiweng village–Maliu village of Xiasi County (pipe no. K635–K637). This section lay in the south of the research area, with an altitude of 500 to 750 m. Here, the slope conditions affected the distribution of groundwater pore pressure and the physical and mechanical characteristics of the rock and soil in three areas: vegetation cover, evaporation and slope erosion. Ultimately, these three factors affected slope stability (Luo and Tan, 2011). Vertical and horizontal ravines were also seen in this section, with a relative height difference greater than 100 m and slopes between 15 to 35. Slope degrees with obvious changes had a great influence on slope stability (Chang and Kim, 2004; Hu et al., 2015). The exposed rocks in this area were mainly shale and belonged to the sliding-prone rock group. Rock type and interlayer structure were found to be important internal indicators that a landslide could occur (Guzzetti et al., 1996; Xiang et al., 2010; Xin et al., 2009). The distance between the fault and the pipeline in the section was about 2 km with a NDVI of about 0.75 and MAP of about 970 mm. Faulted zones and nearby rock and earth masses that were destroyed in a geologic event reduced the integrity of a slope, and the faults and important groundwater channels could also cause deformation and damage to a slope (Liu, 2009). The pipelines in these areas exhibited many defects. Most pipelines passed through the slope in an inclined or horizontal way, an attribute that typically increased the risk of a landslide occurring.

In high-risk areas, small or moderate landslides commonly occurred in areas that we have categorized as high risk. They were in the process of deformation, or had obvious deformation recently (within 2 years), such as cracks, subsidence, or tympanites on the landslide, and, in some cases, even shear. The pipelines in these areas had defects and were buried at a shallow depth. If a landslide occurred in this pipeline area, it could cause pipe suspension, floating and damage. It could also contribute to a small to moderate leakage of the multiple petroleum products, such as gasoline, diesel and kerosene. However, damaged pipes can be welded or repaired. Monitoring is critical in high-risk areas. In our study, the pipeline high-risk area was defined as the following areas: (1) Xiasi town–Xiasi village–Shiweng village (pipe no. K622–K633) and (2) Xiasi town–Maliu village–Jinzishan Xiangdasang village (pipe no. K635–K642). This area was located south of the pipe, which was buried in the study area. The altitude of the study area was between 450 and 800 m, the relative elevation difference was over 100 m and the slope was between 15 and 40. Most of the outcrops in this area were quartz sandstone, which belongs to the easy-sliding rock group. The pipes in this area were about 2.5 km away from faults. The NDVI was about 0.6 to 0.8, and MAP was about 970 nm. Pipes showed many defects, most of them either crossing the slope or lying in the center of slope. All of the above factors provided sufficient conditions for the formation of landslide.

In the medium-risk areas, only small landslides were found to occur, and we observed no sign of deformation. But through the analysis of geological structure, topography and landform, we found the area to demonstrate a tendency for developing landslides. The pipes in this risk area exhibited almost no faults and were buried deep beneath the ground. However, under bad conditions, the landslides in these areas could also affect the pipes' safety, causing the pipes to become exposed or deformed. These areas need simple monitoring. For our study, medium-risk areas were defined as follows: (1) Sanlong village of Dongxihe township–Panlong town–Dongsheng village (pipe no. K559–K593). (2) Panlong town–Qinlao village–Wu'ai village (pipe no. K595–K597). (3) Baolun town–Laolin'gou village–Xiasi town–Youyu village (pipe no. K599–K630).

In the low-risk areas, landslides did not occur under ordinary conditions, but they could occur if a strong earthquake hit or if the area experienced continuous or heavy rain. The pipes in low-risk areas showed no defects and were buried very deep. They were also located far away from areas affected by landslides. Therefore, landslides in these areas caused no obvious damage to the pipes and few threatened the safety of pipes. However, regular inspection is necessary to ensure that the pipes continue to operate safely. The low-risk areas were defined as follows: (1) Panlong town–Dongsheng village–Qinlao village (pipe no. K591–K597) and (2) Baolun town–Xiaojia village–Baolun town–Laolin'gou village (pipe no. K599–K608).

Through comprehensive analysis of each risk level area, we compiled a list of pipeline landslide risks (Table 6). This list describes each landslide risk level in four respects: pipeline risk, landslide susceptibility, pipeline vulnerability and risk control measures.

Table 5Number and distances of pipeline of four risk grades.

Table 6Description of pipeline risk level.

The main purpose of this study was to provide managers and planners a comprehensive assessment of landslide risk in areas containing pipelines. The results offer information on the possibility of failure of slopes. The landslide susceptibility maps could help planners reorganize and plan future pipeline construction. Pipeline vulnerability maps could assist engineers in pipeline maintenance operations. Based on this final risk map, managers and engineers can then make decisions and formulate prescriptions that will have highly predictable results for safely transporting petroleum products, relocating settlements and significantly reducing the risk of any adverse effects.

Future research could explore detailed comparison of different methods and recommend one or more optimal approaches. Moreover, This study shows that landslide risk assessments can be performed with a minimal number of relatively easy to obtain datasets. We advocate establishing a database with assessment parameters similar to the one described by this study to construct dynamic landslide risk assessment models.

6 Conclusions

The faults inherent to traditional landslide risk assessments include excessive human influence, failure of pipeline vulnerability assessments to consider the interaction between landslide disaster and pipeline ontology, and the low quantification degree of risk assessment results.

Taking the Guangyuan section (82 km) of the LCC oil and gas pipeline as an example, we used GIS and RS technology to establish a regional landslide susceptibility assessment model based on the LM-BP neural network. We determined that there were 112 and 108 slopes in high-susceptibility and extremely high-susceptibility areas that accounted for 33.18 % and 40.46 % of the total area of the study area, respectively. Then, we established the model of pipeline vulnerability evaluation based on the entropy weight method by combining the pipeline body and the environmental information. The number and length of pipe segments in the highly vulnerable (III) and extremely vulnerable areas (IV) accounted for about 12 % of the total. Finally, based on the susceptibility assessment and the vulnerability assessment, we completed the risk assessment and risk division of the oil pipeline, thus forming a geological disaster risk assessment model and a method for oil pipeline and landslide risk assessment. The risk assessment results demonstrated that the number and length of high-susceptibility and extremely high-susceptibility pipeline segments represented 20 % of the total. Similarly, the pipeline risk within Qingchuan and Jian'ge counties was relatively high. Our pipeline landslide risk assessment has laid a foundation for the future study of pipeline safety management and pipeline failure consequence loss assessment.

Data availability
Data availability.

DEM data can be downloaded from the Geospatial Data Cloud (http://www.gscloud.cn/, last access: October 2016). Precipitation data (dataset of annual surface observation values in China between the years 1981 and 2010) are provided by China Meteorological Administration (http://data.cma.cn/, last access: January 2017). Geological maps, landslide data and pipeline mileage data can be requested by email from the author at neu_xjn@163.com.

Appendix A

Table A1Classification of landslide susceptibility grade corresponding to different intervals.

Appendix B

Table B1Standard training sample matrix and standard test sample matrix.

Appendix C

Table C1Test error of LM-BP neural network.

Appendix D

Table D1Coordinates of the centerline and ancillary facilities of the pipeline. Secrecy regulations regarding geographical coordinate data in the People's Republic of China stipulate that the first three digits of each location's coordinates remain confidential. In this table, this has been represented by ellipses.

Appendix E

Table E1Internal detection data of each pipeline.

Appendix F: Core code of pipeline defect point coordinate calculating program

Appendix G

Table G1Pipeline landslide risk assessment results.

Appendix H: Field environment of study area

Figure H1Vegetation distribution in a watershed in the study area.

Figure H2Vegetation environment of a pipeline section in the study area.

Figure H3Outcropping of rock strata in the study area.

Author contributions
Author contributions.

JX conceived and designed the experiments, MS performed the experiments and analyzed the data, HZ wrote the paper and WC helped to prepare the paper. YY helped edit language and MS, YC and JW helped draw the maps. All authors read and approved the final paper.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement
Special issue statement.

Acknowledgements
Acknowledgements.

The study has been funded by the Strategic Priority Research Program of Chinese Academy of Sciences (XDA20030302), IWHR (China Institute of Water Resources and Hydropower Research) National Mountain Flood Disaster Investigation Project (SHZH-IWHR-57), Scientific and Technological Innovation Team Project of Southwest Petroleum University (2017CXTD09) and the Open Topic of Digital Fujian Institute of Large Data for Natural Disaster Monitoring (NDMBD2018003).

Edited by: Albert J. Kettner
Reviewed by: two anonymous referees

References

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text: Given the coordinates of the vertices of a polygon in the coordinate plane, the side length of the polygon can be calculated using the distance formula: where d is the distance between the two points with coordinates (x 1, y 1) and (x 2, y 2). Determine height/width of rectangle in perspective, Find third vertex of a right triangle given two vertices and one side. If you have three vertices, then you have half of the rectangle: a right triangle. $C^2=A^2+B^2-2AB \cdot\cos(c)$ getting a different answer than creating a third triangle with the distance formula? Formula of rectangle perimeter in terms of diagonal and rectangle side: 4. Why does gpg's secret and public key have the same keyid? The sum of the squares two diagonals is equal to the sum of the squares of the sides: 8. In other words opposite sides of a quadrilateral are equal in length then the quadrilateral is called a rectangle. A saddle rectangle has 4 nonplanar vertices, alternated from vertices of a cuboid, with a unique minimal surface interior defined as a linear combination of the four vertices, creating a saddle surface. Unexpected result when subtracting in a loop, Trilogy in the 80’s about space travel to another world. 9. The measure of the distance is the perimeter of the figure or body. For a rectangle, that means adding four sides: P = a + b + c + d Where P is the perimeter and a , b , c , and d are the lengths of the four sides. For each point $(x,y)$ we can rotate it to a new point $(x', y')$ through an angle $\theta$ by this transformation: $$x' = x \cos \theta - y \sin \theta$$ 1 0. This Site Might Help You. $$y' = x \sin \theta + y \cos \theta$$. This web site owner is mathematician Dovzhyk Mykhailo. Formula of rectangle diagonal in terms of sine of the acute angle between the diagonals and the area of a rectangle. In an amplifier, does the gain knob boost or attenuate the input signal? 1. Some examples of the quadrilaterals are square, rectangle, rhombus, trapezium, and parallelogram. To be noted, a parallelogram also has its opposite sides equal and parallel to each other but the angles are not equal to 90 degrees. A rectangle is a 2d shape which has four sides and four vertices. (ii) The rectangle can be divided into two right triangles. Remember that' the formula to calculate the distance between two points is equal to Find the length side AB. Formula of rectangle diagonal in terms of cosine of the angle that adjacent to the diagonal and the adjacent side of the angle: 8. Perimeter of Rectangle. Start with a triangle. Determine he coordinates of the fourth vertex. Area of a Rectangle, A = a × b, where a and b are the length and the breadth of the rectang… Formula of rectangle area in terms of diagonal and rectangle side: 4. The three vertices are: A = (0, 3), O = (0, 0), B = (5, 0) We know that, the diagonals of a rectangle are of equal length, Length of the diagonal AB = Distance between the points A and B. All four angles of the rectangle is right: 5. The widthis the distance between B and C (or A,D). The diagonal formula in mathematics is used to calculate the diagonals of a polygon including rectangles, square, and more similar shapes. Diagonals are a line that is drawn inside the rectangle connecting opposite corners or vertices and hence the diagonals of a rectangle are congruent. The lengths of the sides will be the same as for the unrotated rectangle. Formula of rectangle diagonal in terms of sine of the angle that adjacent to the diagonal and the opposite side of the angle: 7. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 3. Since a rectangle has four sides, it has four angles. Formula of rectangle circumscribed radius in terms of sine of the acute angle between the diagonals and the area of a rectangle: 1. In rectangle with the different size of sides never enter the incircle. Formula of rectangle diagonal in terms of rectangle sides (by the Pythagorean theorem):: 2. Edited it. The user cross-multiplies corresponding coordinates to find the area encompassing the polygon, and subtracts it from the surrounding polygon … 2. Anonymous. (See Distance between Two Points)So in the figure above: 1. Rectangles differ only ratio of long side to short but four angles is right, that is 90 degrees. All angles of a rectangle are equal. coordinates of all the rest of the rectangle's vertices. Diagonal of a rectangle is the diameter of the circumcircle. The length of a rectangle is the longest side whereas the width is the shortest side. Formula of rectangle sides in terms of perimeter and another rectangle side: 4. A diagonal of a rectangle in half divides each other: 10. Formula of rectangle diagonal in terms of perimeter and rectangle side: 4. In case of a rectangle, the opposite sides of a rectangle are equal and so, the perimeter will be twice the width of the rectangle plus twice the length of the rectangle and it is denoted by the alphabet “p”. Formulas Of Rectangle. Formula of rectangle perimeter in terms of area and rectangle side: 3. A four-sided polygon having two dimensions i.e. we have. The diameter of the circumcircle is equal to the diagonal of the rectangle. This is a rectangle with one vertex at the origin, and sides of length $a$ and $b$ which are parallel to the $x$ and $y$ axes. The midpoint of BD will also be the same. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Formula of rectangle diagonal in terms of radius of the escribed circle (excircle): 5. Do you mean a function that gives all points inside it, or just the perimeter? The opposite sides of the rectangle are equal and parallel to each other. It has 4 right angles. How to rewrite mathematics constructively? You know the angle that this is to make with the $x$ axis (let's say $\beta$). You can do this with any rectangle you want. Formula of rectangle circumscribed radius in terms of sine of the angle that adjacent to the diagonal and the opposite side of the angle: 7. So, to get the answer from your inputs, you have some coordinates you're inputting. Let us derive the formula for its perimeter … Formulas and properties, Ellipse. 6 years ago. An opposite sides of the rectangle are parallel: 3. coordinates of the vertex closest to the X axis and the origin. Let A and B be two points in the plane of the paper as shown in fig. Also find its area. Now, you can position the other points of the rectangle (unrotated) and rotate them through an angle $+\beta$ to get your final coordinates. Mov file size very small compared to pngs. Find the value of y and also the fourth vertex. Welcome to OnlineMSchool. Or the area of the rectangle? Are creature environmental effects a bubble or column? (i) In a rectangle the length of opposite sides will be equal. Formula of rectangle sides in terms of diagonal and angle. It has 4 corners (vertices). First you need to determine which point is at the right angle. The heightof the rectangle is the distance between A and B (or C,D). These unique features make Virtual Nerd a viable alternative to private tutoring. Formula of rectangle circumscribed radius in terms of area and rectangle side: 4. When we go round a closed figure or body, along its boundary, for once, we cover a distance. Formula of angle between the diagonal and rectangle side in terms of diagonal and rectangle side: 2. Formula for calculating rectangle vertices [duplicate], Find Point At Edge of Rectangle, Given Unrelated Angle. Again use midpoint formula! they are equal: 2. A way is to apply Pythagoras theorem: find the two vertices which are the farthest apart. What is the difference between a chess puzzle and a chess problem? Sorry, forgot to add what the outputs were. Formula of rectangle perimeter in terms of diagonal and rectangle side: P = 2 ( a + √ d2 - a2) = 2 ( b + √ d2 - b2) 4. Formula of rectangle circumscribed radius in terms of cosine of the angle that adjacent to the diagonal and the adjacent side of the angle: 8. Formulas, characterizations and properties of a cone. If you have the side lengths, the angle with the axes, and the coordinates of the corners I don't know what you are looking for. Three vertices of rectangle A B C D are A - 5 , 1 , B - 2 , - 3 , and C 6 , y . A(1, 2),B(5, 0) substitute in the formula. The dimensions of the rectangle are found by calculating the distance between various corner points.Recall that we can find the distance between any two points if we know their coordinates. I have a programming problem and I've forgotten some of my math rectangle formulas. Plot the rectangle to better understand the problem. How much did J. Robert Oppenheimer get paid while overseeing the Manhattan Project? length and breadth is called a rectangle. Rectangle can be a parallelogram, rhombus or square in which all the angles right. What do you mean by "formula for a rectangle"? The formula for the area is: What output are you trying to generate? Formula of rectangle perimeter in terms of radius of the escribed circle (excircle) and rectangle side: 5. Formula of rectangle area in terms of perimeter and rectangle side: 3. Formula of rectangle area in terms of radius of the escribed circle (excircle) and rectangle side: 6. It is an equiangular rectangle with four right angles which is 90 degrees. This example shows 4 blue edges of the rectangle, and two green diagonals, all being diagonal of the cuboid rectangular faces. For example, your students already know and understand the formula for finding the area of a rectangle, A = l x w, where A represents the area, l the length, and w the width of the rectangle. Formula of rectangle circumscribed radius in terms of perimeter and rectangle side: 3. Formula of rectangle circumscribed radius in terms of rectangle sides: 2. The length of a diagonalsis the distance between opposite corners, say B and D (or A,C since the diagonals are congrue… 1. How can I defeat a Minecraft zombie that picked up my weapon and armor? $\text{Diagonal of Rectangle Formula} = \sqrt{(a^{2} + b^{2})}$ Where, a is the length of the rectangle. They will see the relationship between this formula and the formula for finding the area of a triangle from examples like the following. 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# tikz intersection fails in combination with the library calc

In the second example I am trying to work the the library intersection. It works fine as long as I don't use the calc library.

The first example shows my intention. I am using ($2*(w1')$) which produces the expected result:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections,calc}

\begin{document}
\begin{tikzpicture}[x=3cm,y=3cm]
\coordinate (O) at (0,0);
\coordinate (F') at (-0.8,-0.1);
\coordinate (F'') at (0.8,0);
\draw[thick](O) ++(F') ++ (F'');
\draw[gray,name path=circle around F'] (F') circle (1.2);
\draw[gray,name path=circle around F''] (F'') circle (1.2);
\draw[name intersections={of=circle around F' and circle around F'',by={w1,w1'}}]   (intersection-1) node[dot,draw]{}
-- (intersection-2)node[dot,draw]{};
\draw[red,thick] (w1) -- (w1');
\draw[blue,thick] (w1) -- ($2*(w1')$);
\end{tikzpicture}
\end{document}

In the picture you can see that the blue line is expanded.

In the second (not minimal) example it fails:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections,calc}

\begin{document}
\tikzset{dot/.style={circle,minimum size=3pt,inner sep=0pt,outer sep=0pt,thick,fill,}}
\begin{tikzpicture}[x=3cm,y=3cm]
\coordinate (B) at (-1,0);\node[dot] at (B) {};\node[left] at (B) {$B$};
\coordinate (D) at (0,1);\node[dot] at (D) {};\node[anchor=south east] at (D) {$D$};
\coordinate (F) at (0,-.25);\node[dot] at (F) {};\node[anchor=west] at (F) {$F$};

\draw[thick,name path=line FB] (F) -- (B);
\draw[thick,name path=line FD] (F) -- (D);
\draw[gray,name path=circle around F] (F) circle (0.8);

\path[name intersections={of=circle around F and line FB,by=F'}];
\path[name intersections={of=circle around F and line FD,by=F''}];

\draw[gray,name path=circle around F'] (F') circle (0.6);
\draw[gray,name path=circle around F''] (F'') circle (0.6);
\draw[name intersections={of=circle around F' and circle around F'',by={w1,w1'}}]
(w1) node[dot]{}
-- (w1')node[dot]{};
\draw[red] (w1') -- ($2*(w1)$);
\end{tikzpicture}
\end{document}

The red line misses the intersection. I don't know why.

What is the problem?

-
I can get the result with \draw[red] (w1') -- ($(w1')!2!(w1)$);. So probably calc library is assuming something different for the omitted coordinate in ($coord1!2!coord2$) type of input. – percusse Mar 2 '12 at 16:50
@percusse: Thanks for your comment. This input method I didn't try. – Marco Daniel Mar 2 '12 at 16:56
If TikZ is deep magic, the calc library is the deeper magic from beyond the dawn of time. – Loop Space Mar 2 '12 at 16:57
@AndrewStacey: Sounds really obscure ;-). Thanks for your answer. – Marco Daniel Mar 2 '12 at 17:02
Incidentally, I didn't know that this sort of syntax was allowed in TikZ calc so I've learnt something too. Good question. – Loop Space Mar 2 '12 at 17:04

The coordinate ($2*(w1)$) means "take the coordinate represented by (w1) and double it". Traditionally, we scale vectors from the origin. If you draw blobs at the origin, (w1) and ($2*(w1)$) then you will see that they line up on a straight line.

What you want is "take the vector from (w1') to (w1), double it, and anchor it back at (w1') again". You can do this with ($(w1') + 2*(w1) - 2*(w1')$) (parentheses seem to confuse it a little) or ($(w1')!2!(w1)$).

-
Experimenting shows that when I try to use parentheses then it thinks they are specifying coordinates so can't be used for mathematical grouping. – Loop Space Mar 2 '12 at 17:00
For grouping you can use {} An example is shown in the manual ;-) – Marco Daniel Mar 2 '12 at 17:03
$(w1') + 2*{(w1') - (w1)}$ makes my machine hang. Where's the example? – Loop Space Mar 2 '12 at 17:05
I meant this example \fill [black] (${3*(4-3)}*(1,0.5)$) circle (2pt); page 120. – Marco Daniel Mar 2 '12 at 17:10