# Vehicle Movement Analyses Considering Altitude Based on Modified Digital Elevation Model and Spherical Bilinear Interpolation Model: Evidence from GPS-Equipped Taxi Data in Sanya, Zhengzhou, and Liaoyang

Authors: Jiawei Gui; Qunqi Wu
Journal: Journal of Advanced Transportation
 (2020)

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## Abstract

Aggravating energy shortages and increasing labor costs have become global problems and have garnered special importance in recent years in the transportation sector, especially in taxi markets. Automatic vehicles have a bright future, however, there is an equal amount of skepticism and concern about safety for all the optimism. To unlock the potential of automatic vehicles in intelligent transportation systems, a lot more data and testing are required to promote safety level as far as possible and achieve the organizational aim of reducing accidents to zero tolerance. And it is vital to provide accurate models for vehicle movement analyses. In this study, Modified Digital Elevation (MDE) model and Spherical Bilinear Interpolation (SBI) model were proposed for vehicle movement analyses considering altitude. And the experimental data of 9,990 GPS-enabled taxis in Sanya, Zhengzhou, and Liaoyang were adopted to support comparisons. Measurement results showed that MDE model had over 99% less disparity with direct solution than original model and SBI model could further improve the effects. It indicated that the application of MDE model and SBI model could improve both accuracy and efficiency of vehicle movement analyses and it had a bright future in the field of automatic vehicles. Future directions could be improving models and expanding data.


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## Body

## 1. Introduction

Nowadays, there are plenty of urgent social issues in the transportation research field. For example, aggravating energy shortages and increasing labor costs have become global problems and have garnered special importance in recent years in the transportation sector, especially in taxi markets. With the worsening situation of energy shortages and labor costs, taxi markets are facing increasing opportunities as well as challenges. It is essential that appropriate counter measures must be taken to suppress or reverse or at least alleviate the worsening situation.With the rapid progress of self-driving technologies, there are more and more advanced applications involved in intelligent traffic system. For example, automatic vehicles have a bright future. Dubai Roads and Transportation Authority tested autonomous pods, the world’s first automatic taxi, in 2018. According to the evidence lab of union bank of Switzerland, the number of taxis in New York might drop by two-thirds if automatic vehicles were promoted completely, and the global automatic taxi market may exceed $2 trillion by 2030. Tesla Inc published a full self-driving computer called Autopilot 3.0 and an automatic taxi plan called Robo Taxi. Besides, many companies have already tested their automatic taxis or automatic buses along the road, such as Waymo in USA, Yandex in Russia, Gateway in UK, Easymile in France, Bestmile in Switzerland, Vislab in Italia, ZMP in Japan, KT in South Korea, GAC in China, etc.However, there is an equal amount of skepticism and concern for all the optimism surrounding automatic vehicles. Most crucially, many people doubt the safety of automatic vehicles because they are not controlled by reliable human drivers but computers. On March 19th in 2018, a terrible accident took place in Tempe, Arizona that a 49-year-old woman with a bicycle on the sidewalk was hit to die by an Uber self-driving car at a speed of 65 kilometers per hour. Since then, the willingness of the public to ride in a fully self-driving vehicle fell in a very low level.To unlock the potential of automatic vehicles in intelligent transportation systems, a lot more data and testing are required to promote safety level as far as possible and achieve the organizational aim of reducing accidents to zero tolerance. And it is vital to provide accurate models for vehicle movement analyses. In this study, modified digital elevation model and spherical bilinear interpolation model were proposed for vehicle movement analyses considering altitude. And GPS-enabled taxi data in Sanya, Zhengzhou, and Liaoyang were adopted to support comparisons.The rest of this paper were organized as follows: Section2 reviewed related works; Section 3 introduced formulas, models, data and tools; Section 4 presented a sample and described results in Sanya, Zhengzhou, and Liaoyang; Section 5 made a discussion; Section 6 summarized main conclusions, contributions and proposed future directions.


## 2. Literature Review

In this section, there are several similar problems and some related works till date, including Internet of Vehicles problem (see Section2.1), data analysis for automatic vehicles (see Section 2.2) and taxi service improving problem (see Section 2.3).


### 2.1. Internet of Vehicles Problem

The Internet is a popular network technology all over the world and it has continuous developments in recent years. The evolution of network technology has led to a deployment of various access networks as introduced by Piamrat et al. [1]. Internet of Things (IoT) is a novel paradigm that is rapidly gaining ground in the scenario of modern wireless telecommunications as addressed by Atzori et al. [2]. Miorandi et al. proposed an overview of IoT technologies and services [3]. Xu et al. reviewed classical researches of IoT technologies and major applications in industries [4]. Zanella et al. proposed an urban IoT system and explored the application of the IoT paradigm to smart cities, taking Padova of Italy as an example [5].Intelligent Transportation Systems (ITS) had significant impact on our life as introduced by Wang [6]. Internet of Vehicle (IoV) is one of the revolutions mobilized by IoT as addressed by Kaiwartya et al. [7]. Lu et al. regarded IoV as the next frontier for automotive revolution and the key to the evolution to ITS [8]. Yang et al. proposed an abstract network model of the IoV and presented its applications [9]. However, IoV also posed new challenges to the communication technology [10]. The information security of IoV was a considerable challenge. Singh et al. presented the potential of transforming vehicle communication in terms of efficiency and safety [11]. Chen proposed a trust-based cooperation authentication bit-map routing protocol against insider security threats in wireless ad-hoc networks [12]. Huang et al. proposed a proactive scheme to secure Edge computing-based IoV against RSU hotspot attack [13]. Furthermore, IoV has rapid developments in recent years. Butt et al. presented a scalable Social IoV (SIoV) architecture based on Restful web technology [14]. Chen et al. proposed Cognitive IoV (CIoV) to enhance transportation safety and network security by communication technologies [15].Vehicular Ad-hoc Networks (VANET) comprise communications among vehicles and infrastructures by wireless local network technologies as addressed by Hartenstein and Laberteaux [16]. With the wide spread of Global Positioning System (GPS) and Geographic Information System (GIS), the participants in VANET could acquire much more information than before. Benefit from that, GPS-Equipped cars could not only acquire their real-time locations, but also road directions. Thus, VANET technologies helped improving road safety and providing comfort for passengers [17, 18]. In the field of ITS, IoV and VANET technologies created essential conditions for automatic vehicles.


### 2.2. Data Analysis for Automatic Vehicles

Automatic vehicles are regarded as the future of transportation. It has been a long-lasting dream of robotics researchers and enthusiasts as indicated by Petrovskaya and Thrun [19]. However, safety is the dominant factor in any automatic vehicle control system design as proposed by Shladover et al. [20]. Much more work was required until autonomous vehicles could participate in real-world urban traffic safely and robustly as proposed by Luettel et al. [21]. To ensure the safety of autonomous vehicles, a holistic fleet deployment scheme from interdisciplinary perspectives should be established as proposed by Koopman and Wagner [22]. Wang et al. investigated the acceptance of intelligent driving vehicles in Guangzhou and indicated that consumers focused on not only the developmental prospects but also the technological safety of intelligent driving vehicles [23]. With the rapid progress of IoV and VANET, there is a great deal of technologies involved in data analysis for automatic vehicles, which are described as follows.(i)
Vehicle communication systems. Machine-to-Machine (M2M) communication for the IoT system was considered to be a key technology in future networks as introduced by Chen et al. [24]. Park et al. proposed a transmission strategy for use with a capillary M2M system under wireless personal area networks based on a series of performance analysis for contention adaptation of M2M devices with directional antennas [25, 26]. Zhou et al. proposed a two-stage access control and resource allocation algorithm based on M2M communication and verified it under various simulation scenarios. With the rapid progress of IoV and VANET, M2M communication has been applied to vehicle communication in recent years. The vehicle communication systems include but are not limited to Vehicle-to-Pedestrian (V2P), Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I). Rahimian et al. examined the feasibility that sending traffic warnings to texting pedestrians based on V2P communication when they initiate an unsafe road crossing influences their decisions and actions [27]. Fan et al. proposed a traffic-aware relay selection based on millimeter-wave V2V communication to overcome the Line-of-sight blockage problem [28]. Lyamin et al. proposed a data-mining-based method for V2V communications by random and On-off models [26]. Han et al. proposed an optimal signal control algorithm for signalized intersections using individual vehicle's trajectory data based on V2I communication [30]. Jia et al. verified that autonomous vehicles have significantly improvements on traffic efficiency via V2V and V2I communications [31].(ii)
Machine learning methods. Machine learning were used to identify objects in images, transcribe speech into text, match news items, posts or products with users’ interests, and select relevant results of search as introduced by Lecun et al. [32]. With the rapid progress of IoV and VANET, machine learning methods has been applied to vehicle simulation, emulation and prediction in recent years. Aramrattana et al. proposed a simulation framework for cooperative safety assessment and efficiency evaluation in ITS [33]. Elwekeil et al. proposed a deep learning approach to achieve both reliability and spectrum efficiency of V2V communications [34]. Lv et al. proposed a modified cyber-physical system for automated electric vehicles based on unsupervised machine learning algorithms [35]. Li et al. proposed a proactively load balancing approach for vehicular network traffic control based on V2I communication, convolutional neural networks and deep learning to enable efficient cooperation among mobile edge servers [36].(iii)
Big data technologies. In recent years, a significant change in ITS was that much more data were collected from a variety of sources and can be processed into various forms as proposed by Zhang et al. [37]. And numerous advanced multidisciplinary journals began publishing a special issue of big data, for instance, Nature in 2008 [38] and Science in 2011 [39]. To solve big data, several technologies were proposed, including cloud computing, artificial intelligence and blockchain technology. Gubbi et al. proposed a cloud centric vision for worldwide implementation of IoT [40]. Whaiduzzaman et al. verified that cloud computing was a technologically feasible and economically viable paradigm for vehicles [41]. Al-Fuqaha et al. explored the relationships between IoT and big data analytics [42]. Ning et al. proposed a three-layer model based on fog computing to enable distributed traffic management and verified it by trajectory-based performance analysis [43]. As for applications, more and more researches focus on large scale taxi GPS data [44–46].


### 2.3. Taxi Service Improving Problem

Taxi is the main constituent of urban transportation. With the rapid progress of wireless sensor network [47], vehicle markets meet an evolution from intelligent vehicle grid to autonomous, Internet-connected vehicles and vehicular cloud as discussed by Gerla et al. [48]. Automatic taxis have gradually become reality in several advanced regions but not fully promoted yet. Just like every coin has two sides, automatic taxis have advantages and disadvantages. On the one hand, automatic taxis overcome some defects of human drivers, including excessive stress [49], inadequate sleepiness [50], and negative behaviors [51]. On the other hand, automatic taxis need mountains of work to achieve functions and ensure security. Taxi automation is one of the most promising technologies for the future and it is a challenge well worth meeting for taxi companies.Fast and reliable service that can compete with the single occupancy vehicle was one of the demands of transit users as proposed by El-Geneidy et al. [52]. And it has become significant to provide users with a range of security-related and user-oriented vehicular applications as proposed by Ning et al. [53]. Thus, it is necessary for taxi companies improve their service to meet the demand of passengers. Yuan et al. proposed a price equilibrium model for taxi market to improve the service level [54]. Song et al. proposed a planning concept from the perspective of supply and demand economic equilibrium to optimize the transportation markets [55]. Dou et al. proposed a heuristic line piloting method that a taxi deviating from the typical route would raise an alert when malfunction took place or even hijacked by criminals [56]. Tang et al. proposed a customer-search model based on route choice behavior analysis to help taxi drivers find next passengers in urban road networks [57]. Although taxi companies and governments have already made great efforts, there are still plenty of service improving problems to be settled.To unlock the potential of IoV in ITS, in the previous study [58], Gui and Wu measured taxi efficiency based on 2191 GPS-equipped taxi data in Sanya and indicated that the application of Motorcade-Sharing model could not only alleviate urban traffic congestion but also optimize urban taxi markets. However, altitude was not considered in that study which might reduce the measurement accuracy and result in lower safety level when altitude is nonnegligible. Thus, Original Model need improvements.To sum up, some helpful works have already been done. However, few researches took altitude into consideration and proposed accurate models for vehicle movement analyses. Besides, autonomous vehicles need assigning continuous directives in time while it is difficult for complex algorithms to response immediately under the background of big data.How much the altitude will affect the vehicle movement analyses? How to simplify complex algorithms of the Direct Solution for automatic vehicles? And how to handle the situation when some data was missing? In the next section, methods and data are involved to explore these questions.


## 2.1. Internet of Vehicles Problem

The Internet is a popular network technology all over the world and it has continuous developments in recent years. The evolution of network technology has led to a deployment of various access networks as introduced by Piamrat et al. [1]. Internet of Things (IoT) is a novel paradigm that is rapidly gaining ground in the scenario of modern wireless telecommunications as addressed by Atzori et al. [2]. Miorandi et al. proposed an overview of IoT technologies and services [3]. Xu et al. reviewed classical researches of IoT technologies and major applications in industries [4]. Zanella et al. proposed an urban IoT system and explored the application of the IoT paradigm to smart cities, taking Padova of Italy as an example [5].Intelligent Transportation Systems (ITS) had significant impact on our life as introduced by Wang [6]. Internet of Vehicle (IoV) is one of the revolutions mobilized by IoT as addressed by Kaiwartya et al. [7]. Lu et al. regarded IoV as the next frontier for automotive revolution and the key to the evolution to ITS [8]. Yang et al. proposed an abstract network model of the IoV and presented its applications [9]. However, IoV also posed new challenges to the communication technology [10]. The information security of IoV was a considerable challenge. Singh et al. presented the potential of transforming vehicle communication in terms of efficiency and safety [11]. Chen proposed a trust-based cooperation authentication bit-map routing protocol against insider security threats in wireless ad-hoc networks [12]. Huang et al. proposed a proactive scheme to secure Edge computing-based IoV against RSU hotspot attack [13]. Furthermore, IoV has rapid developments in recent years. Butt et al. presented a scalable Social IoV (SIoV) architecture based on Restful web technology [14]. Chen et al. proposed Cognitive IoV (CIoV) to enhance transportation safety and network security by communication technologies [15].Vehicular Ad-hoc Networks (VANET) comprise communications among vehicles and infrastructures by wireless local network technologies as addressed by Hartenstein and Laberteaux [16]. With the wide spread of Global Positioning System (GPS) and Geographic Information System (GIS), the participants in VANET could acquire much more information than before. Benefit from that, GPS-Equipped cars could not only acquire their real-time locations, but also road directions. Thus, VANET technologies helped improving road safety and providing comfort for passengers [17, 18]. In the field of ITS, IoV and VANET technologies created essential conditions for automatic vehicles.


## 2.2. Data Analysis for Automatic Vehicles

Automatic vehicles are regarded as the future of transportation. It has been a long-lasting dream of robotics researchers and enthusiasts as indicated by Petrovskaya and Thrun [19]. However, safety is the dominant factor in any automatic vehicle control system design as proposed by Shladover et al. [20]. Much more work was required until autonomous vehicles could participate in real-world urban traffic safely and robustly as proposed by Luettel et al. [21]. To ensure the safety of autonomous vehicles, a holistic fleet deployment scheme from interdisciplinary perspectives should be established as proposed by Koopman and Wagner [22]. Wang et al. investigated the acceptance of intelligent driving vehicles in Guangzhou and indicated that consumers focused on not only the developmental prospects but also the technological safety of intelligent driving vehicles [23]. With the rapid progress of IoV and VANET, there is a great deal of technologies involved in data analysis for automatic vehicles, which are described as follows.(i)
Vehicle communication systems. Machine-to-Machine (M2M) communication for the IoT system was considered to be a key technology in future networks as introduced by Chen et al. [24]. Park et al. proposed a transmission strategy for use with a capillary M2M system under wireless personal area networks based on a series of performance analysis for contention adaptation of M2M devices with directional antennas [25, 26]. Zhou et al. proposed a two-stage access control and resource allocation algorithm based on M2M communication and verified it under various simulation scenarios. With the rapid progress of IoV and VANET, M2M communication has been applied to vehicle communication in recent years. The vehicle communication systems include but are not limited to Vehicle-to-Pedestrian (V2P), Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I). Rahimian et al. examined the feasibility that sending traffic warnings to texting pedestrians based on V2P communication when they initiate an unsafe road crossing influences their decisions and actions [27]. Fan et al. proposed a traffic-aware relay selection based on millimeter-wave V2V communication to overcome the Line-of-sight blockage problem [28]. Lyamin et al. proposed a data-mining-based method for V2V communications by random and On-off models [26]. Han et al. proposed an optimal signal control algorithm for signalized intersections using individual vehicle's trajectory data based on V2I communication [30]. Jia et al. verified that autonomous vehicles have significantly improvements on traffic efficiency via V2V and V2I communications [31].(ii)
Machine learning methods. Machine learning were used to identify objects in images, transcribe speech into text, match news items, posts or products with users’ interests, and select relevant results of search as introduced by Lecun et al. [32]. With the rapid progress of IoV and VANET, machine learning methods has been applied to vehicle simulation, emulation and prediction in recent years. Aramrattana et al. proposed a simulation framework for cooperative safety assessment and efficiency evaluation in ITS [33]. Elwekeil et al. proposed a deep learning approach to achieve both reliability and spectrum efficiency of V2V communications [34]. Lv et al. proposed a modified cyber-physical system for automated electric vehicles based on unsupervised machine learning algorithms [35]. Li et al. proposed a proactively load balancing approach for vehicular network traffic control based on V2I communication, convolutional neural networks and deep learning to enable efficient cooperation among mobile edge servers [36].(iii)
Big data technologies. In recent years, a significant change in ITS was that much more data were collected from a variety of sources and can be processed into various forms as proposed by Zhang et al. [37]. And numerous advanced multidisciplinary journals began publishing a special issue of big data, for instance, Nature in 2008 [38] and Science in 2011 [39]. To solve big data, several technologies were proposed, including cloud computing, artificial intelligence and blockchain technology. Gubbi et al. proposed a cloud centric vision for worldwide implementation of IoT [40]. Whaiduzzaman et al. verified that cloud computing was a technologically feasible and economically viable paradigm for vehicles [41]. Al-Fuqaha et al. explored the relationships between IoT and big data analytics [42]. Ning et al. proposed a three-layer model based on fog computing to enable distributed traffic management and verified it by trajectory-based performance analysis [43]. As for applications, more and more researches focus on large scale taxi GPS data [44–46].


## 2.3. Taxi Service Improving Problem

Taxi is the main constituent of urban transportation. With the rapid progress of wireless sensor network [47], vehicle markets meet an evolution from intelligent vehicle grid to autonomous, Internet-connected vehicles and vehicular cloud as discussed by Gerla et al. [48]. Automatic taxis have gradually become reality in several advanced regions but not fully promoted yet. Just like every coin has two sides, automatic taxis have advantages and disadvantages. On the one hand, automatic taxis overcome some defects of human drivers, including excessive stress [49], inadequate sleepiness [50], and negative behaviors [51]. On the other hand, automatic taxis need mountains of work to achieve functions and ensure security. Taxi automation is one of the most promising technologies for the future and it is a challenge well worth meeting for taxi companies.Fast and reliable service that can compete with the single occupancy vehicle was one of the demands of transit users as proposed by El-Geneidy et al. [52]. And it has become significant to provide users with a range of security-related and user-oriented vehicular applications as proposed by Ning et al. [53]. Thus, it is necessary for taxi companies improve their service to meet the demand of passengers. Yuan et al. proposed a price equilibrium model for taxi market to improve the service level [54]. Song et al. proposed a planning concept from the perspective of supply and demand economic equilibrium to optimize the transportation markets [55]. Dou et al. proposed a heuristic line piloting method that a taxi deviating from the typical route would raise an alert when malfunction took place or even hijacked by criminals [56]. Tang et al. proposed a customer-search model based on route choice behavior analysis to help taxi drivers find next passengers in urban road networks [57]. Although taxi companies and governments have already made great efforts, there are still plenty of service improving problems to be settled.To unlock the potential of IoV in ITS, in the previous study [58], Gui and Wu measured taxi efficiency based on 2191 GPS-equipped taxi data in Sanya and indicated that the application of Motorcade-Sharing model could not only alleviate urban traffic congestion but also optimize urban taxi markets. However, altitude was not considered in that study which might reduce the measurement accuracy and result in lower safety level when altitude is nonnegligible. Thus, Original Model need improvements.To sum up, some helpful works have already been done. However, few researches took altitude into consideration and proposed accurate models for vehicle movement analyses. Besides, autonomous vehicles need assigning continuous directives in time while it is difficult for complex algorithms to response immediately under the background of big data.How much the altitude will affect the vehicle movement analyses? How to simplify complex algorithms of the Direct Solution for automatic vehicles? And how to handle the situation when some data was missing? In the next section, methods and data are involved to explore these questions.


## 3. Methods and Materials

In this section, there are several parts of methods and materials. First, an analysis procedure was presented as an overview (see Section3.1). Second, traditional methods were reviewed, including Direct Solution (see Section 3.2) and Original Model (see Section 3.3). Third, modified models were proposed, including MDE model (see Section 3.4) and SBI model (see Section 3.5). At last, materials were described, including data and tools (see Section 3.6).


### 3.1. Analysis Procedure

To solve this problem, a series of analysis procedure was established (see Figure1).Figure 1
Analysis procedure design.In terms of data, GPS data without altitude and digital elevation data were collected at the beginning. GPS data with altitude was calculated on the basis of them afterwards (see Section3.6). And the preciseness of GPS data with altitude was further improved by Spherical Bilinear Interpolation (SBI) Model (see Section 3.5).In terms of model, three models were adopted for analyzing vehicle movements. They are direct solution (see Section3.2), original model (see Section 3.3) and Modified Digital Elevation (MDE) model (see Section 3.4) and they have different functions (see Table 1).Table 1
Function comparison of direct solution, original model, and MDE model.
Data source
Data
Direct solutionD
Original modeld
MDE modeld′
GPS data (without altitude)
Longitude
√
√
√
Latitude
√
√
√
Digital elevation data
Altitude
√
√(i)
Direct solution. It is the traditional approach with large computation requirements because of its square algorithm, which is one of its drawbacks.
(ii)
Original model. It simplifies the square algorithm of direct solution by cosine theorems (see Distance Formula in the previous study [58]). However, it ignores altitude which affects the measurement accuracy of results when altitude is nonnegligible.
(iii)
MDE model. It adopts original model and takes altitude into consideration. In theory, MDE model has more accurate results than original model especially when altitude is nonnegligible. In addition, MDE model adopts SBI model to improve the preciseness of GPS data with altitude.In this study, vehicle movements were analyzed by 3 models above mentioned at the same time and then a series of comparisons were raised on the basis of results. The notations for variables were in the Appendix (see AppendixA).


### 3.2. Direct Solution

Direct solution is the traditional approach and it calculates distances in a direct way (see Figure2).Figure 2
Flow chart of direct solution.There are 4 steps in this section.First, in order to describe the location of vehicles in a mathematical way, the Spherical Coordinate System was established in Formula (1).(1)Pρ,θ,φ,whereρ≥0, 0≤θ≤π, and 0≤φ<2π. The variable ρ represents the radial distance. The variable θ represents the polar angle. The variable φ represents the azimuthal angle. The radial distance can be divided into two parts by Formula (2).(2)ρ=R+h,whereR=6378137.00m, which represents the radius of the earth. The variable h represents the altitude. Based on Formulas (1), (2), the coordinate of vector OP⇀ in Spherical Coordinate System can be represented by Formula (3).(3)OP⇀=R+h,θ,φ.Second, in order to calculate the distance between two locations by Distance Formula, the Rectangular Coordinate System was established in Formula (4).(4)OP⇀=x,y,z.Based on Distance Formula, the distance between pointsP1 and P2 in Rectangular Coordinate System can be calculated by Formula (5).(5)P1P2=Δx2+Δy2+Δz2.Third, spherical coordinates were converted into rectangular coordinates. Based on Formulas (1), (4), the relationship of them can be represented by Formula (6).(6)x=ρsinθcosφ,y=ρsinθsinφ,z=ρcosθ.Based on Formula (6), the spherical coordinate of point P1R+h1,θ1,φ1 was converted into P1R+h1sinθ1cosφ1,R+h1sinθ1sinφ1,R+h1cosθ1 in Rectangular Coordinate System, and the spherical coordinate of point P2R+h2,θ2,φ2 was converted into P2R+h2sinθ2cosφ2,R+h2sinθ2sinφ2,R+h2cosθ2 in Rectangular Coordinate System.Based on Formula (5), the distance between points P1 and P2 under direct solution can be calculated by Formula (7), which was sophisticated to some extent.(7)D=R+h1sinθ1cosφ1−R+h2sinθ2cosφ22+R+h1sinθ1sinφ1−R+h2sinθ2sinφ22+R+h1cosθ1−R+h2cosθ22.Fourth, all the distances were summed up. Supposing that the variableTD represents total movements of vehicles under Direct Solution and it can be calculated by Formula (8).(8)TD=∑i=1I∑t=1TDi,t..


### 3.3. Original Model

Original Model simplifies the square algorithm of Direct Solution by cosine theorems and it calculates distances in a simple way (see Figure3).Figure 3
Flow chart of Original Model.There are 5 steps in this section.First, the Spherical Coordinate System was established in Formula (1).Second, the Rectangular Coordinate System was established in Formula (4).Third, in order to calculate the distance between two locations, the cosine value of the angle between vectorsOP1⇀ and OP2⇀ was calculated by Formula (9).(9)cosOP1⇀,OP2⇀=OP1⇀⋅OP2⇀OP1⇀OP2⇀.Formula (9) can be converted into Formula (10) and the derivation process is in the Appendix part (see Appendix B in the previous study [58]).(10)cosOP1⇀,OP2⇀=sinθ1sinθ2cosφ2−φ1+cosθ1cosθ2.Fourth, considering the equation of arc length calculation, the distance between vectorsOP1⇀ and OP2⇀ was calculated by Formula (11).(11)d=ROP1⇀,OP2⇀.Based on Formula (9), Formula (11) can be converted into Formula (12).(12)d=Rarccossinθ1sinθ2cosφ2−φ1+cosθ1cosθ2.In other words, the distance betweenP1R,θ1,φ1 and P2R,θ2,φ2 can be calculated approximately by the longitudes and latitudes of them based on GPS data, and the radius of the earth under Original Model.Fifth, all the distances were summed up. Supposing that the variableTd represents total movements of vehicles under Original Model and it can be calculated by Formula (13).(13)Td=∑i=1I∑t=1Tdi,t.


### 3.4. Modified Digital Elevation Model

MDE adopts Original Model and SBI model to improve the preciseness of results and it calculates distances in a comprehensive way (see Figure4).Figure 4
Flow chart of Modified Digital Elevation Model.There are 5 steps in this section.First, the Spherical Coordinate System was established in Formula (1).Second, the Rectangular Coordinate System was established in Formula (4).Third, all the unknown altitude values were calculated approximately by Spherical Bilinear Interpolation Model.Fourth, the distance betweenP1 and P2 was calculated approximately by Modified Digital Elevation Model. Based on Formula (12), the distance between points P1R+h1,θ1,φ1 and P2R+h2,θ2,φ2 can be calculated by Formula (14) when ρ=R+h1.(14)dρ=R+h1=R+h1arccossinθ1sinθ2cosφ2−φ1+cosθ1cosθ2.Based on Formula (12), the distance between points P1R+h1,θ1,φ1 and P2R+h2,θ2,φ2 can be calculated by Formula (15) when ρ=R+h2.(15)dρ=R+h2=R+h2arccossinθ1sinθ2cosφ2−φ1+cosθ1cosθ2.Based on Distance Formula, the distance can be calculated by Formula (16).(16)d′=dρ=R+h1+dρ=R+h222+Δh2.Based on Formulas (14), (15), Formula (16) can be converted into Formula (17).(17)d′=R+h1+h22arccossinθ1sinθ2cosφ2−φ1+cosθ1cosθ22+h2−h12.Fifth, all the distances were summed up. Supposing that the variableTd′ represents total movements of vehicles under Modified Digital Elevation Model and it can be calculated by Formula (18).(18)Td′=∑i=1I∑t=1Tdi,t'..


### 3.5. Spherical Bilinear Interpolation Model

Spherical Bilinear Interpolation (SBI) Model helps improving the preciseness of GPS data with altitude especially when the data of altitudes is not precise enough or missing (see Figure5).Figure 5
Flow chart of Spherical Bilinear Interpolation Model and its applications.There are 3 steps in this section.First, find out 4 vertexes by positional notation based on the spherical coordinate of pointP. For any point PR+h,θ,φ where θ1′≤θ≤θ2′ and φ1′≤φ≤φ2′, there are 4 points Q1R+h1′,θ1′,φ1′, Q2R+h2′,θ2′,φ1′, Q3R+h3′,θ1′,φ2′, Q4R+h4′,θ2′,φ2′ around it (see Figure 6).Figure 6
Schematic diagram of vertexes.Second, figure out the relationship among the spherical coordinates of 5 pointsP, Q1, Q2, Q3, Q4. Supposing that there are 4 factors α1, α2, α3, α4 satisfying Formula (19).(19)OP⇀=α1OQ1⇀+α2OQ2⇀+α3OQ3⇀+α4OQ4⇀.As a result, the solution of Formula (19) was settled by Formula (20). The derivation process of Formula (20) is in the Appendix (see Appendix B).(20)α1=θ2′−θ×φ2′−φθ2′−θ1′×φ2′−φ1′,α2=θ−θ1′×φ2′−φθ2′−θ1′×φ2′−φ1′,α3=θ2′−θ×φ−φ1′θ2′−θ1′×φ2′−φ1′,α4=θ−θ1′×φ−φ1′θ2′−θ1′×φ2′−φ1′.Based on Formula (20), Formula (19) can be converted into Formula (21).(21)OP⇀=θ2′−θ×φ2′−φθ2′−θ1′×φ2′−φ1′OQ1⇀+θ−θ1′×φ2′−φθ2′−θ1′×φ2′−φ1′OQ2⇀+θ2′−θ×φ−φ1′θ2′−θ1′×φ2′−φ1′OQ3⇀+θ−θ1′×φ−φ1′θ2′−θ1′×φ2′−φ1′OQ4⇀.The checking process of Formula (21) is in the Appendix (see Appendix C). Based on Formula (C.24) in the Appendix C, the altitude value of point PR+h,θ,φ can be calculated by Formula (22).(22)h=θ2′−θ×φ2′−φθ2′−θ1′×φ2′−φ1′×h1′+θ−θ1′×φ2′−φθ2′−θ1′×φ2′−φ1′×h2′+θ2′−θ×φ−φ1′θ2′−θ1′×φ2′−φ1′×h3′+θ−θ1′×φ−φ1′θ2′−θ1′×φ2′−φ1′×h4′.Third, substituteh1′, h2′, h3′, h4′ by the altitude value of points Q1, Q2, Q3, Q4. Finally, the altitude value of point P can be calculated approximately by Spherical Bilinear Interpolation Model.


### 3.6. Data and Tools

(i)
Data Source. The GPS data of taxis in this study was collected from the big data platform, Travel Cloud, which was developed by Ministry of Transport of the People’s Republic of China (see Data Availability). It was provided in part by the transportation department of Liaoning Province, in part by Henan Province department of transportation, in part by Sanya Traffic and Transportation Bureau of Hainan Province. Its major data items include anonymous vehicle ID, longitude, latitude and recording time. And the digital elevation data in this study was collected from Google Map.
(ii)
Data Processing. In terms of GPS data processing, there are 3 steps. First, all the taxi locations were averaged every 60 seconds so as to improve the accuracy of data, that is to say, average locations of those taxis were recorded every 60 seconds entirely. Second, the flaw data was removed so as to ensure the integrity of data. Third, the obtained data was checked to avoid mistakes. As a result, GPS data without altitude was prepared. In terms of digital elevation data processing, there are 4 steps. First, the original digital elevation data was imported by Arcgis. Second, GPS data without altitude in a Microsoft Excel file was converted into a XY table of Arcgis geodatabase by the conversion tool called Excel to Table. Third, the XY table was adopted to create a point feature class on the basis of its coordinate system, World Geodetic System 1984 by the data management tool called Add XY Data. Fourth, the altitude data of the point feature class was extracted by the spatial analyst tool called Extract Values by Points. As a result, GPS data with altitude was prepared (See the file named “Data.xlsx” in the Supplementary Material).
(iii)
Implement Tool. Several kinds of software were adopted in this study as follows. Rivermap X3.1 was applied for extracting the altitude from digital elevation data. Microsoft Excel 2019 was applied for original data processing and result storage. Stata 14.1 was applied for statistic analyzing. Matlab R2018a was applied for programming. Global Mapper 14.1 was applied for map drawing. Arcgis Pro 2.2.0 (trial version) was applied for map drawing and spatial analyzing.


## 3.1. Analysis Procedure

To solve this problem, a series of analysis procedure was established (see Figure1).Figure 1
Analysis procedure design.In terms of data, GPS data without altitude and digital elevation data were collected at the beginning. GPS data with altitude was calculated on the basis of them afterwards (see Section3.6). And the preciseness of GPS data with altitude was further improved by Spherical Bilinear Interpolation (SBI) Model (see Section 3.5).In terms of model, three models were adopted for analyzing vehicle movements. They are direct solution (see Section3.2), original model (see Section 3.3) and Modified Digital Elevation (MDE) model (see Section 3.4) and they have different functions (see Table 1).Table 1
Function comparison of direct solution, original model, and MDE model.
Data source
Data
Direct solutionD
Original modeld
MDE modeld′
GPS data (without altitude)
Longitude
√
√
√
Latitude
√
√
√
Digital elevation data
Altitude
√
√(i)
Direct solution. It is the traditional approach with large computation requirements because of its square algorithm, which is one of its drawbacks.
(ii)
Original model. It simplifies the square algorithm of direct solution by cosine theorems (see Distance Formula in the previous study [58]). However, it ignores altitude which affects the measurement accuracy of results when altitude is nonnegligible.
(iii)
MDE model. It adopts original model and takes altitude into consideration. In theory, MDE model has more accurate results than original model especially when altitude is nonnegligible. In addition, MDE model adopts SBI model to improve the preciseness of GPS data with altitude.In this study, vehicle movements were analyzed by 3 models above mentioned at the same time and then a series of comparisons were raised on the basis of results. The notations for variables were in the Appendix (see AppendixA).


## 3.2. Direct Solution

Direct solution is the traditional approach and it calculates distances in a direct way (see Figure2).Figure 2
Flow chart of direct solution.There are 4 steps in this section.First, in order to describe the location of vehicles in a mathematical way, the Spherical Coordinate System was established in Formula (1).(1)Pρ,θ,φ,whereρ≥0, 0≤θ≤π, and 0≤φ<2π. The variable ρ represents the radial distance. The variable θ represents the polar angle. The variable φ represents the azimuthal angle. The radial distance can be divided into two parts by Formula (2).(2)ρ=R+h,whereR=6378137.00m, which represents the radius of the earth. The variable h represents the altitude. Based on Formulas (1), (2), the coordinate of vector OP⇀ in Spherical Coordinate System can be represented by Formula (3).(3)OP⇀=R+h,θ,φ.Second, in order to calculate the distance between two locations by Distance Formula, the Rectangular Coordinate System was established in Formula (4).(4)OP⇀=x,y,z.Based on Distance Formula, the distance between pointsP1 and P2 in Rectangular Coordinate System can be calculated by Formula (5).(5)P1P2=Δx2+Δy2+Δz2.Third, spherical coordinates were converted into rectangular coordinates. Based on Formulas (1), (4), the relationship of them can be represented by Formula (6).(6)x=ρsinθcosφ,y=ρsinθsinφ,z=ρcosθ.Based on Formula (6), the spherical coordinate of point P1R+h1,θ1,φ1 was converted into P1R+h1sinθ1cosφ1,R+h1sinθ1sinφ1,R+h1cosθ1 in Rectangular Coordinate System, and the spherical coordinate of point P2R+h2,θ2,φ2 was converted into P2R+h2sinθ2cosφ2,R+h2sinθ2sinφ2,R+h2cosθ2 in Rectangular Coordinate System.Based on Formula (5), the distance between points P1 and P2 under direct solution can be calculated by Formula (7), which was sophisticated to some extent.(7)D=R+h1sinθ1cosφ1−R+h2sinθ2cosφ22+R+h1sinθ1sinφ1−R+h2sinθ2sinφ22+R+h1cosθ1−R+h2cosθ22.Fourth, all the distances were summed up. Supposing that the variableTD represents total movements of vehicles under Direct Solution and it can be calculated by Formula (8).(8)TD=∑i=1I∑t=1TDi,t..


## 3.3. Original Model

Original Model simplifies the square algorithm of Direct Solution by cosine theorems and it calculates distances in a simple way (see Figure3).Figure 3
Flow chart of Original Model.There are 5 steps in this section.First, the Spherical Coordinate System was established in Formula (1).Second, the Rectangular Coordinate System was established in Formula (4).Third, in order to calculate the distance between two locations, the cosine value of the angle between vectorsOP1⇀ and OP2⇀ was calculated by Formula (9).(9)cosOP1⇀,OP2⇀=OP1⇀⋅OP2⇀OP1⇀OP2⇀.Formula (9) can be converted into Formula (10) and the derivation process is in the Appendix part (see Appendix B in the previous study [58]).(10)cosOP1⇀,OP2⇀=sinθ1sinθ2cosφ2−φ1+cosθ1cosθ2.Fourth, considering the equation of arc length calculation, the distance between vectorsOP1⇀ and OP2⇀ was calculated by Formula (11).(11)d=ROP1⇀,OP2⇀.Based on Formula (9), Formula (11) can be converted into Formula (12).(12)d=Rarccossinθ1sinθ2cosφ2−φ1+cosθ1cosθ2.In other words, the distance betweenP1R,θ1,φ1 and P2R,θ2,φ2 can be calculated approximately by the longitudes and latitudes of them based on GPS data, and the radius of the earth under Original Model.Fifth, all the distances were summed up. Supposing that the variableTd represents total movements of vehicles under Original Model and it can be calculated by Formula (13).(13)Td=∑i=1I∑t=1Tdi,t.


## 3.4. Modified Digital Elevation Model

MDE adopts Original Model and SBI model to improve the preciseness of results and it calculates distances in a comprehensive way (see Figure4).Figure 4
Flow chart of Modified Digital Elevation Model.There are 5 steps in this section.First, the Spherical Coordinate System was established in Formula (1).Second, the Rectangular Coordinate System was established in Formula (4).Third, all the unknown altitude values were calculated approximately by Spherical Bilinear Interpolation Model.Fourth, the distance betweenP1 and P2 was calculated approximately by Modified Digital Elevation Model. Based on Formula (12), the distance between points P1R+h1,θ1,φ1 and P2R+h2,θ2,φ2 can be calculated by Formula (14) when ρ=R+h1.(14)dρ=R+h1=R+h1arccossinθ1sinθ2cosφ2−φ1+cosθ1cosθ2.Based on Formula (12), the distance between points P1R+h1,θ1,φ1 and P2R+h2,θ2,φ2 can be calculated by Formula (15) when ρ=R+h2.(15)dρ=R+h2=R+h2arccossinθ1sinθ2cosφ2−φ1+cosθ1cosθ2.Based on Distance Formula, the distance can be calculated by Formula (16).(16)d′=dρ=R+h1+dρ=R+h222+Δh2.Based on Formulas (14), (15), Formula (16) can be converted into Formula (17).(17)d′=R+h1+h22arccossinθ1sinθ2cosφ2−φ1+cosθ1cosθ22+h2−h12.Fifth, all the distances were summed up. Supposing that the variableTd′ represents total movements of vehicles under Modified Digital Elevation Model and it can be calculated by Formula (18).(18)Td′=∑i=1I∑t=1Tdi,t'..


## 3.5. Spherical Bilinear Interpolation Model

Spherical Bilinear Interpolation (SBI) Model helps improving the preciseness of GPS data with altitude especially when the data of altitudes is not precise enough or missing (see Figure5).Figure 5
Flow chart of Spherical Bilinear Interpolation Model and its applications.There are 3 steps in this section.First, find out 4 vertexes by positional notation based on the spherical coordinate of pointP. For any point PR+h,θ,φ where θ1′≤θ≤θ2′ and φ1′≤φ≤φ2′, there are 4 points Q1R+h1′,θ1′,φ1′, Q2R+h2′,θ2′,φ1′, Q3R+h3′,θ1′,φ2′, Q4R+h4′,θ2′,φ2′ around it (see Figure 6).Figure 6
Schematic diagram of vertexes.Second, figure out the relationship among the spherical coordinates of 5 pointsP, Q1, Q2, Q3, Q4. Supposing that there are 4 factors α1, α2, α3, α4 satisfying Formula (19).(19)OP⇀=α1OQ1⇀+α2OQ2⇀+α3OQ3⇀+α4OQ4⇀.As a result, the solution of Formula (19) was settled by Formula (20). The derivation process of Formula (20) is in the Appendix (see Appendix B).(20)α1=θ2′−θ×φ2′−φθ2′−θ1′×φ2′−φ1′,α2=θ−θ1′×φ2′−φθ2′−θ1′×φ2′−φ1′,α3=θ2′−θ×φ−φ1′θ2′−θ1′×φ2′−φ1′,α4=θ−θ1′×φ−φ1′θ2′−θ1′×φ2′−φ1′.Based on Formula (20), Formula (19) can be converted into Formula (21).(21)OP⇀=θ2′−θ×φ2′−φθ2′−θ1′×φ2′−φ1′OQ1⇀+θ−θ1′×φ2′−φθ2′−θ1′×φ2′−φ1′OQ2⇀+θ2′−θ×φ−φ1′θ2′−θ1′×φ2′−φ1′OQ3⇀+θ−θ1′×φ−φ1′θ2′−θ1′×φ2′−φ1′OQ4⇀.The checking process of Formula (21) is in the Appendix (see Appendix C). Based on Formula (C.24) in the Appendix C, the altitude value of point PR+h,θ,φ can be calculated by Formula (22).(22)h=θ2′−θ×φ2′−φθ2′−θ1′×φ2′−φ1′×h1′+θ−θ1′×φ2′−φθ2′−θ1′×φ2′−φ1′×h2′+θ2′−θ×φ−φ1′θ2′−θ1′×φ2′−φ1′×h3′+θ−θ1′×φ−φ1′θ2′−θ1′×φ2′−φ1′×h4′.Third, substituteh1′, h2′, h3′, h4′ by the altitude value of points Q1, Q2, Q3, Q4. Finally, the altitude value of point P can be calculated approximately by Spherical Bilinear Interpolation Model.


## 3.6. Data and Tools

(i)
Data Source. The GPS data of taxis in this study was collected from the big data platform, Travel Cloud, which was developed by Ministry of Transport of the People’s Republic of China (see Data Availability). It was provided in part by the transportation department of Liaoning Province, in part by Henan Province department of transportation, in part by Sanya Traffic and Transportation Bureau of Hainan Province. Its major data items include anonymous vehicle ID, longitude, latitude and recording time. And the digital elevation data in this study was collected from Google Map.
(ii)
Data Processing. In terms of GPS data processing, there are 3 steps. First, all the taxi locations were averaged every 60 seconds so as to improve the accuracy of data, that is to say, average locations of those taxis were recorded every 60 seconds entirely. Second, the flaw data was removed so as to ensure the integrity of data. Third, the obtained data was checked to avoid mistakes. As a result, GPS data without altitude was prepared. In terms of digital elevation data processing, there are 4 steps. First, the original digital elevation data was imported by Arcgis. Second, GPS data without altitude in a Microsoft Excel file was converted into a XY table of Arcgis geodatabase by the conversion tool called Excel to Table. Third, the XY table was adopted to create a point feature class on the basis of its coordinate system, World Geodetic System 1984 by the data management tool called Add XY Data. Fourth, the altitude data of the point feature class was extracted by the spatial analyst tool called Extract Values by Points. As a result, GPS data with altitude was prepared (See the file named “Data.xlsx” in the Supplementary Material).
(iii)
Implement Tool. Several kinds of software were adopted in this study as follows. Rivermap X3.1 was applied for extracting the altitude from digital elevation data. Microsoft Excel 2019 was applied for original data processing and result storage. Stata 14.1 was applied for statistic analyzing. Matlab R2018a was applied for programming. Global Mapper 14.1 was applied for map drawing. Arcgis Pro 2.2.0 (trial version) was applied for map drawing and spatial analyzing.


## 4. Results

In this section, a sample (see Section4.1) was presented for better description and vehicle movements were analyzed by Direct Solution, Original Model, MDE model at the same time, including Sanya (see Section 4.2), Zhengzhou (see Section 4.3), Liaoyang (see Section 4.4). And the results of them were compared with each other (see Section 4.5).


### 4.1. Sample

In the sample, there was a taxi around Luhuitou Park in Sanya of Hainan Province moving from the left yellow point to the right yellow point (see Figure7). Luhuitou Park was on a hill and the taxi was on the road uphill. It indicated that the altitude of the taxi in the sample would rise from the left yellow point to the right yellow point.Figure 7
Actual position of the sample.In Figure7, Luhuitou Park was surrounded by South of Sea Road, also known as Nanbianhai Road in Chinese. Furthermore, four red points surrounding yellow points were supplementary points drawn for SBI model. Their longitudes and latitudes kept only 3 decimal digits. Actual position in Figure 3 can be converted into Schematic diagram (see Figure 8).Figure 8
Schematic diagram of the sample.In Figure8, there are 2 yellow points P1R+h1,109.4983°,18.2274°, P2R+h2,109.4987°,18.2272° and 4 red points Q1R+152,109.498°,18.227°, Q2R+150.81,109.499°,18.227°, Q3R+139.55,109.498°,18.228°, Q4R+132.45,109.499°,18.228°.According to the original digital elevation data, the altitude value of pointsP1 and P2 were 145m and 147m respectively. The taxi altitude in the sample changed approximately 2m from P1 to P2. Based on SBI model, the altitude value of points P1 and P2 can be further calculated by Formula (17). And the altitude value of points P1 and P2 adjusted by SBI model were 145.00m and 147.30m respectively. Based on Formulas (7), (12), (17), the sample results and comparisons of three methods were figured out (see Table 2). The derivation process of Table 2 is in the Appendix (see Appendix D).Table 2
Sample results and comparisons of three methods. Unit: m.
Method
Variable
V
a
l
u
e
1
V
a
l
u
e
2
V
a
l
u
e
1
−
V
a
l
u
e
2
Direct Solution
D
49.267562
49.280653
0.013091
Original Model
d
49.225885
49.225885
0
MDE model
d
′
49.267623
49.280715
0.013092
(Comparisons)
D
−
d
0.041677
0.054768
0.013091
D
−
d
′
0.000061
0.000062
0.000001In Table2, Value1 adopted original digital elevation data while Value2 adopted the digital elevation data adjusted by SBI model. Under Value1, the vehicle movement P1P2⇀ under Direct Solution was 49.267562m. The deviation between Original Model and Direct Solution was 0.041677m while the deviation between MDE model and Direct Solution was only 0.000061m. Under Value2, the vehicle movement P1P2⇀ under Direct Solution was 49.280653m. The deviation between Original Model and Direct Solution was 0.054768m while the deviation between MDE model and Direct Solution was only 0.000062m. It can be seen that MDE model had much less disparity with Direct Solution than Original Model in the sample results. In addition, even if the original digital elevation data, the altitude value of points P1 and P2 was missing, it could be further calculated by SBI model. By contrast, the deviation between Original Model and Direct Solution was 0.013091m while the deviation between MDE model and Direct Solution was only 0.000001m. It further verified the advantage of MDE model even though some data was missing in the sample results and SBI model was adopted to improve the preciseness of GPS data with altitude.The deviation between Original Model and MDE model may be more obvious when sample size increases.


### 4.2. Sanya

Sanya is a city in Hainan Province where the longitude was in the range of [108.928°, 109.807°] while the latitude was in the range of [18.144°, 18.625°]. It located in the southern China and its altitudes had differences. (see Figure9).Figure 9
Elevation map of Sanya.In Figure9, colors reflected the elevation. The altitudes of blue area were low. The altitudes of green area were medium. The altitudes of yellow area were high.In the origin data of Sanya, the locations of 2,506 taxis were recorded every 15 seconds from 9:00 a.m. to 9:59 a.m. on Nov. 15th in 2016, adding up to 766,042 records (see Figure 10).Figure 10
Taxi positions of Sanya.In Figure10, taxi positions of Sanya located minutely at a fixed monitor where the longitude was in the range of [109.0°, 109.8°] while the latitude was in the range of [18.1°, 18.5°]. After data processing, there were 2,191 taxis and 131,460 records left in the experimental data. And 107,427 movements of 2,096 taxis were extracted as a result. Based on Formulas (7), (12), (17), the Sanya results and comparisons of three methods were figured out (see Table 3).Table 3
Sanya results and comparisons of three methods. Unit: m.
Method
Variable
V
a
l
u
e
1
V
a
l
u
e
2
V
a
l
u
e
1
−
V
a
l
u
e
2
Direct Solution
T
D
34,065,354.87
34,064,690.71
664.16
Original Model
T
d
34,058,703.99
34,058,703.99
0
MDE model
T
d
′
34,065,343.29
34,064,679.28
664.01
(Comparisons)
T
D
−
T
d
6,650.88
5,986.72
664.16
T
D
−
T
d
′
11.58
11.43
0.15In Table3, Value1 adopted original digital elevation data while Value2 adopted the digital elevation data adjusted by SBI model. Under Value1, total vehicle movements under Direct Solution was 34,065,354.87m. The deviation between Original Model and Direct Solution was 6,650.88m while the deviation between MDE model and Direct Solution was only 11.58m. Under Value2, total vehicle movements under Direct Solution was 34,064,690.71m. The deviation between Original Model and Direct Solution was 5,986.72m while the deviation between MDE model and Direct Solution was only 11.43m. It can be seen that the deviation between Original Model and MDE model in Sanya was much more obvious than sample. The results in Sanya verified that MDE model had much less disparity with Direct Solution than Original Model. In addition, even if some original digital elevation data of actual positions in Sanya were missing, it could be further calculated by SBI model. By contrast, the deviation between Original Model and Direct Solution was 664.16m while the deviation between MDE model and Direct Solution was only 0.15m. It further verified the advantage of MDE model even though some data was missing in Sanya and SBI model was adopted to improve the preciseness of GPS data with altitude.


### 4.3. Zhengzhou

Zhengzhou is a city in Henan Province where the longitude was in the range of [112.714°, 114.206°] while the latitude was in the range of [34.262°, 34.985°]. It located in the midland China and its altitudes had differences. (see Figure11).Figure 11
Elevation map of Zhengzhou.In Figure11, colors reflected the elevation. The altitudes of blue area were low. The altitudes of green area were medium. The altitudes of yellow area were high.In the origin data of Zhengzhou, the locations of 9,703 taxis were recorded every 15 seconds from 14:50 p.m. to 15:38 p.m. on Nov. 15th in 2016, adding up to 1,048,575 records (see Figure 12).Figure 12
Taxi positions of Zhengzhou.In Figure12, taxi positions of Zhengzhou located minutely at a fixed monitor where the longitude was in the range of [112.8°, 114.5°] while the latitude was in the range of [34.2°, 35.0°]. After data processing, there were 5,774 taxis and 282,926 records left in the experimental data. And 229,370 movements of 5,597 taxis were extracted as a result. Based on Formulas (7), (12), (17), the Zhengzhou results and comparisons of three methods were figured out (see Table 4).Table 4
Zhengzhou results and comparisons of three methods. Unit: m.
Method
Variable
V
a
l
u
e
1
V
a
l
u
e
2
V
a
l
u
e
1
−
V
a
l
u
e
2
Direct Solution
T
D
94,767,484.51
94,767,008.47
476.04
Original Model
T
d
94,764,096.35
94,764,096.35
0
MDE model
T
d
′
94,767,481.20
94,767,005.17
476.03
(Comparisons)
T
D
−
T
d
3,388.16
2,912.12
476.04
T
D
−
T
d
′
3.31
3.30
0.01In Table4, Value1 adopted original digital elevation data while Value2 adopted the digital elevation data adjusted by SBI model. Under Value1, total vehicle movements under Direct Solution was 94,767,484.51 m. The deviation between Original Model and Direct Solution was 3,388.16 m while the deviation between MDE model and Direct Solution was only 3.31m. Under Value2, total vehicle movements under Direct Solution was 94,767,008.47 m. The deviation between Original Model and Direct Solution was 2,912.12 m while the deviation between MDE model and Direct Solution was only 3.30m. It can be seen that the deviation between Original Model and MDE model in Zhengzhou was much more obvious than sample. The results in Zhengzhou verified that MDE model had much less disparity with Direct Solution than Original Model. In addition, even if some original digital elevation data of actual positions in Zhengzhou were missing, it could be further calculated by SBI model. By contrast, the deviation between Original Model and Direct Solution was 476.04m while the deviation between MDE model and Direct Solution was only 0.01m. It further verified the advantage of MDE model even though some data was missing in Zhengzhou and SBI model was adopted to improve the preciseness of GPS data with altitude.


### 4.4. Liaoyang

Liaoyang is a city in Liaoning Province where the longitude was in the range of [112.588°, 123.684°] while the latitude was in the range of [40.710°, 41.615°]. It located in the northern China and its altitudes had differences. (see Figure13).Figure 13
Elevation map of Liaoyang.In Figure13, colors reflected the elevation. The altitudes of blue area were low. The altitudes of green area were medium. The altitudes of yellow area were high.In the origin data of Liaoyang, the locations of 2,237 taxis were recorded every 30 seconds from 9:59 a.m. to 10:58 a.m. on Aug. 8th in 2016, adding up to 268,440 records (see Figure 14).Figure 14
Taxi positions of Liaoyang.In Figure14, taxi positions of Liaoyang located minutely at a fixed monitor where the longitude was in the range of [122.7°, 123.6°] while the latitude was in the range of [40.8°, 41.6°]. After data processing, there were 2,025 taxis and 121,440 records left in the experimental data. And 87,879 movements of 1,810 taxis were extracted as a result. Based on Formulas (7), (12), (17), the Liaoyang results and comparisons of three methods were figured out (see Table 5).Table 5
Liaoyang results and comparisons of three methods. Unit: m.
Method
Variable
V
a
l
u
e
1
V
a
l
u
e
2
V
a
l
u
e
1
−
V
a
l
u
e
2
Direct Solution
T
D
29,796,201.87
29,796,059.67
142.20
Original Model
T
d
29,795,356.19
29,795,356.19
0
MDE model
T
d
′
29,796,201.11
29,796,058.91
142.20
(Comparisons)
T
D
−
T
d
845.68
703.48
142.20
T
D
−
T
d
′
0.76
0.76
0.00In Table5, Value1 adopted original digital elevation data while Value2 adopted the digital elevation data adjusted by SBI model. Under Value1, total vehicle movements under Direct Solution was 29,796,201.87 m. The deviation between Original Model and Direct Solution was 845.68m while the deviation between MDE model and Direct Solution was only 0.76m. Under Value2, total vehicle movements under Direct Solution was 29,796,059.67 m. The deviation between Original Model and Direct Solution was 703.48m while the deviation between MDE model and Direct Solution was only 0.76m. It can be seen that the deviation between Original Model and MDE model in Liaoyang was much more obvious than sample. The results in Liaoyang verified that MDE model had much less disparity with Direct Solution than Original Model. In addition, even if some original digital elevation data of actual positions in Liaoyang were missing, it could be further calculated by SBI model. By contrast, the deviation between Original Model and Direct Solution was 142.20m while the deviation between MDE model and Direct Solution was less than 0.01m. It further verified the advantage of MDE model even though some data was missing in Liaoyang and SBI model was adopted to improve the preciseness of GPS data with altitude.


### 4.5. Comparisons

Based on Tables3–5, the Comprehensive results and comparisons of three methods were figured out (see Table 6).Table 6
Comprehensive results and comparisons of three methods. Unit: m.
Variable
Data
Sanya
Zhengzhou
Liaoyang
Total
T
D
−
T
d
V
a
l
u
e
1
6,650.88
3,388.16
845.68
10,884.72
V
a
l
u
e
2
5,986.72
2,912.12
703.48
9,602.32
T
D
−
T
d
′
V
a
l
u
e
1
11.58
3.31
0.76
15.65
V
a
l
u
e
2
11.43
3.30
0.76
15.49
T
D
−
T
d
′
T
D
−
T
d
V
a
l
u
e
1
0.17%
0.10%
0.09%
0.14%
V
a
l
u
e
2
0.19%
0.11%
0.11%
0.16%In Table6, Value1 adopted original digital elevation data while Value2 adopted the digital elevation data adjusted by SBI model. Under Value1, total deviation between Original Model and Direct Solution was 10,884.72 m while total deviation between MDE model and Direct Solution was only 15.65m. The latter is 0.14% that of the former. Under Value2, total deviation between Original Model and Direct Solution was 9,602.32 m while total deviation between MDE model and Direct Solution was only 15.49m. The latter is 0.16% that of the former. Comprehensive results verified that MDE model had over 99% less disparity with Direct Solution than Original Model because MDE model took altitude into consideration but Original Model did not. In other words, MDE model was much more accurate than Original Model.


## 4.1. Sample

In the sample, there was a taxi around Luhuitou Park in Sanya of Hainan Province moving from the left yellow point to the right yellow point (see Figure7). Luhuitou Park was on a hill and the taxi was on the road uphill. It indicated that the altitude of the taxi in the sample would rise from the left yellow point to the right yellow point.Figure 7
Actual position of the sample.In Figure7, Luhuitou Park was surrounded by South of Sea Road, also known as Nanbianhai Road in Chinese. Furthermore, four red points surrounding yellow points were supplementary points drawn for SBI model. Their longitudes and latitudes kept only 3 decimal digits. Actual position in Figure 3 can be converted into Schematic diagram (see Figure 8).Figure 8
Schematic diagram of the sample.In Figure8, there are 2 yellow points P1R+h1,109.4983°,18.2274°, P2R+h2,109.4987°,18.2272° and 4 red points Q1R+152,109.498°,18.227°, Q2R+150.81,109.499°,18.227°, Q3R+139.55,109.498°,18.228°, Q4R+132.45,109.499°,18.228°.According to the original digital elevation data, the altitude value of pointsP1 and P2 were 145m and 147m respectively. The taxi altitude in the sample changed approximately 2m from P1 to P2. Based on SBI model, the altitude value of points P1 and P2 can be further calculated by Formula (17). And the altitude value of points P1 and P2 adjusted by SBI model were 145.00m and 147.30m respectively. Based on Formulas (7), (12), (17), the sample results and comparisons of three methods were figured out (see Table 2). The derivation process of Table 2 is in the Appendix (see Appendix D).Table 2
Sample results and comparisons of three methods. Unit: m.
Method
Variable
V
a
l
u
e
1
V
a
l
u
e
2
V
a
l
u
e
1
−
V
a
l
u
e
2
Direct Solution
D
49.267562
49.280653
0.013091
Original Model
d
49.225885
49.225885
0
MDE model
d
′
49.267623
49.280715
0.013092
(Comparisons)
D
−
d
0.041677
0.054768
0.013091
D
−
d
′
0.000061
0.000062
0.000001In Table2, Value1 adopted original digital elevation data while Value2 adopted the digital elevation data adjusted by SBI model. Under Value1, the vehicle movement P1P2⇀ under Direct Solution was 49.267562m. The deviation between Original Model and Direct Solution was 0.041677m while the deviation between MDE model and Direct Solution was only 0.000061m. Under Value2, the vehicle movement P1P2⇀ under Direct Solution was 49.280653m. The deviation between Original Model and Direct Solution was 0.054768m while the deviation between MDE model and Direct Solution was only 0.000062m. It can be seen that MDE model had much less disparity with Direct Solution than Original Model in the sample results. In addition, even if the original digital elevation data, the altitude value of points P1 and P2 was missing, it could be further calculated by SBI model. By contrast, the deviation between Original Model and Direct Solution was 0.013091m while the deviation between MDE model and Direct Solution was only 0.000001m. It further verified the advantage of MDE model even though some data was missing in the sample results and SBI model was adopted to improve the preciseness of GPS data with altitude.The deviation between Original Model and MDE model may be more obvious when sample size increases.


## 4.2. Sanya

Sanya is a city in Hainan Province where the longitude was in the range of [108.928°, 109.807°] while the latitude was in the range of [18.144°, 18.625°]. It located in the southern China and its altitudes had differences. (see Figure9).Figure 9
Elevation map of Sanya.In Figure9, colors reflected the elevation. The altitudes of blue area were low. The altitudes of green area were medium. The altitudes of yellow area were high.In the origin data of Sanya, the locations of 2,506 taxis were recorded every 15 seconds from 9:00 a.m. to 9:59 a.m. on Nov. 15th in 2016, adding up to 766,042 records (see Figure 10).Figure 10
Taxi positions of Sanya.In Figure10, taxi positions of Sanya located minutely at a fixed monitor where the longitude was in the range of [109.0°, 109.8°] while the latitude was in the range of [18.1°, 18.5°]. After data processing, there were 2,191 taxis and 131,460 records left in the experimental data. And 107,427 movements of 2,096 taxis were extracted as a result. Based on Formulas (7), (12), (17), the Sanya results and comparisons of three methods were figured out (see Table 3).Table 3
Sanya results and comparisons of three methods. Unit: m.
Method
Variable
V
a
l
u
e
1
V
a
l
u
e
2
V
a
l
u
e
1
−
V
a
l
u
e
2
Direct Solution
T
D
34,065,354.87
34,064,690.71
664.16
Original Model
T
d
34,058,703.99
34,058,703.99
0
MDE model
T
d
′
34,065,343.29
34,064,679.28
664.01
(Comparisons)
T
D
−
T
d
6,650.88
5,986.72
664.16
T
D
−
T
d
′
11.58
11.43
0.15In Table3, Value1 adopted original digital elevation data while Value2 adopted the digital elevation data adjusted by SBI model. Under Value1, total vehicle movements under Direct Solution was 34,065,354.87m. The deviation between Original Model and Direct Solution was 6,650.88m while the deviation between MDE model and Direct Solution was only 11.58m. Under Value2, total vehicle movements under Direct Solution was 34,064,690.71m. The deviation between Original Model and Direct Solution was 5,986.72m while the deviation between MDE model and Direct Solution was only 11.43m. It can be seen that the deviation between Original Model and MDE model in Sanya was much more obvious than sample. The results in Sanya verified that MDE model had much less disparity with Direct Solution than Original Model. In addition, even if some original digital elevation data of actual positions in Sanya were missing, it could be further calculated by SBI model. By contrast, the deviation between Original Model and Direct Solution was 664.16m while the deviation between MDE model and Direct Solution was only 0.15m. It further verified the advantage of MDE model even though some data was missing in Sanya and SBI model was adopted to improve the preciseness of GPS data with altitude.


## 4.3. Zhengzhou

Zhengzhou is a city in Henan Province where the longitude was in the range of [112.714°, 114.206°] while the latitude was in the range of [34.262°, 34.985°]. It located in the midland China and its altitudes had differences. (see Figure11).Figure 11
Elevation map of Zhengzhou.In Figure11, colors reflected the elevation. The altitudes of blue area were low. The altitudes of green area were medium. The altitudes of yellow area were high.In the origin data of Zhengzhou, the locations of 9,703 taxis were recorded every 15 seconds from 14:50 p.m. to 15:38 p.m. on Nov. 15th in 2016, adding up to 1,048,575 records (see Figure 12).Figure 12
Taxi positions of Zhengzhou.In Figure12, taxi positions of Zhengzhou located minutely at a fixed monitor where the longitude was in the range of [112.8°, 114.5°] while the latitude was in the range of [34.2°, 35.0°]. After data processing, there were 5,774 taxis and 282,926 records left in the experimental data. And 229,370 movements of 5,597 taxis were extracted as a result. Based on Formulas (7), (12), (17), the Zhengzhou results and comparisons of three methods were figured out (see Table 4).Table 4
Zhengzhou results and comparisons of three methods. Unit: m.
Method
Variable
V
a
l
u
e
1
V
a
l
u
e
2
V
a
l
u
e
1
−
V
a
l
u
e
2
Direct Solution
T
D
94,767,484.51
94,767,008.47
476.04
Original Model
T
d
94,764,096.35
94,764,096.35
0
MDE model
T
d
′
94,767,481.20
94,767,005.17
476.03
(Comparisons)
T
D
−
T
d
3,388.16
2,912.12
476.04
T
D
−
T
d
′
3.31
3.30
0.01In Table4, Value1 adopted original digital elevation data while Value2 adopted the digital elevation data adjusted by SBI model. Under Value1, total vehicle movements under Direct Solution was 94,767,484.51 m. The deviation between Original Model and Direct Solution was 3,388.16 m while the deviation between MDE model and Direct Solution was only 3.31m. Under Value2, total vehicle movements under Direct Solution was 94,767,008.47 m. The deviation between Original Model and Direct Solution was 2,912.12 m while the deviation between MDE model and Direct Solution was only 3.30m. It can be seen that the deviation between Original Model and MDE model in Zhengzhou was much more obvious than sample. The results in Zhengzhou verified that MDE model had much less disparity with Direct Solution than Original Model. In addition, even if some original digital elevation data of actual positions in Zhengzhou were missing, it could be further calculated by SBI model. By contrast, the deviation between Original Model and Direct Solution was 476.04m while the deviation between MDE model and Direct Solution was only 0.01m. It further verified the advantage of MDE model even though some data was missing in Zhengzhou and SBI model was adopted to improve the preciseness of GPS data with altitude.


## 4.4. Liaoyang

Liaoyang is a city in Liaoning Province where the longitude was in the range of [112.588°, 123.684°] while the latitude was in the range of [40.710°, 41.615°]. It located in the northern China and its altitudes had differences. (see Figure13).Figure 13
Elevation map of Liaoyang.In Figure13, colors reflected the elevation. The altitudes of blue area were low. The altitudes of green area were medium. The altitudes of yellow area were high.In the origin data of Liaoyang, the locations of 2,237 taxis were recorded every 30 seconds from 9:59 a.m. to 10:58 a.m. on Aug. 8th in 2016, adding up to 268,440 records (see Figure 14).Figure 14
Taxi positions of Liaoyang.In Figure14, taxi positions of Liaoyang located minutely at a fixed monitor where the longitude was in the range of [122.7°, 123.6°] while the latitude was in the range of [40.8°, 41.6°]. After data processing, there were 2,025 taxis and 121,440 records left in the experimental data. And 87,879 movements of 1,810 taxis were extracted as a result. Based on Formulas (7), (12), (17), the Liaoyang results and comparisons of three methods were figured out (see Table 5).Table 5
Liaoyang results and comparisons of three methods. Unit: m.
Method
Variable
V
a
l
u
e
1
V
a
l
u
e
2
V
a
l
u
e
1
−
V
a
l
u
e
2
Direct Solution
T
D
29,796,201.87
29,796,059.67
142.20
Original Model
T
d
29,795,356.19
29,795,356.19
0
MDE model
T
d
′
29,796,201.11
29,796,058.91
142.20
(Comparisons)
T
D
−
T
d
845.68
703.48
142.20
T
D
−
T
d
′
0.76
0.76
0.00In Table5, Value1 adopted original digital elevation data while Value2 adopted the digital elevation data adjusted by SBI model. Under Value1, total vehicle movements under Direct Solution was 29,796,201.87 m. The deviation between Original Model and Direct Solution was 845.68m while the deviation between MDE model and Direct Solution was only 0.76m. Under Value2, total vehicle movements under Direct Solution was 29,796,059.67 m. The deviation between Original Model and Direct Solution was 703.48m while the deviation between MDE model and Direct Solution was only 0.76m. It can be seen that the deviation between Original Model and MDE model in Liaoyang was much more obvious than sample. The results in Liaoyang verified that MDE model had much less disparity with Direct Solution than Original Model. In addition, even if some original digital elevation data of actual positions in Liaoyang were missing, it could be further calculated by SBI model. By contrast, the deviation between Original Model and Direct Solution was 142.20m while the deviation between MDE model and Direct Solution was less than 0.01m. It further verified the advantage of MDE model even though some data was missing in Liaoyang and SBI model was adopted to improve the preciseness of GPS data with altitude.


## 4.5. Comparisons

Based on Tables3–5, the Comprehensive results and comparisons of three methods were figured out (see Table 6).Table 6
Comprehensive results and comparisons of three methods. Unit: m.
Variable
Data
Sanya
Zhengzhou
Liaoyang
Total
T
D
−
T
d
V
a
l
u
e
1
6,650.88
3,388.16
845.68
10,884.72
V
a
l
u
e
2
5,986.72
2,912.12
703.48
9,602.32
T
D
−
T
d
′
V
a
l
u
e
1
11.58
3.31
0.76
15.65
V
a
l
u
e
2
11.43
3.30
0.76
15.49
T
D
−
T
d
′
T
D
−
T
d
V
a
l
u
e
1
0.17%
0.10%
0.09%
0.14%
V
a
l
u
e
2
0.19%
0.11%
0.11%
0.16%In Table6, Value1 adopted original digital elevation data while Value2 adopted the digital elevation data adjusted by SBI model. Under Value1, total deviation between Original Model and Direct Solution was 10,884.72 m while total deviation between MDE model and Direct Solution was only 15.65m. The latter is 0.14% that of the former. Under Value2, total deviation between Original Model and Direct Solution was 9,602.32 m while total deviation between MDE model and Direct Solution was only 15.49m. The latter is 0.16% that of the former. Comprehensive results verified that MDE model had over 99% less disparity with Direct Solution than Original Model because MDE model took altitude into consideration but Original Model did not. In other words, MDE model was much more accurate than Original Model.


## 5. Discussion

In this section, several operational details over the course of this research were presented for fellow colleagues and follow-up studies (see Section5.1). In addition, the authors discussed the application prospects in the field of automatic vehicles (see Section 5.2).


### 5.1. Operational Details

Over the course of this research, there were several operational details that had not been contemplated within, including software use and data processing, which are described as follows.(i)
During processing original data by Microsoft Excel 2019, we found that Excel had a processing limit that its maximum capacity was 1,048,576 rows. If original data exceeds this limit, we had to divide them into several pieces before processing. Otherwise, there were other software, Snapde for instance, suiting for the big data more than 1,048,576 rows. In this research, the biggest original data was 1,048,575 records in Zhengzhou (see Section4.3). Therefore, we adopted Excel as a result.(ii)
During spatial analyzing by Arcgis Pro (trial version), we found that the function of the spatial analyst tool called Extract Values by Points was available in v2.2.0 but unavailable in v2.3.0. Some errors may exist in v2.3.0. Therefore, we adopted v2.2.0 as a result.(iii)
In SBI model, the first step was to find out 4 vertexes by positional notation based on the spherical coordinate of pointP. We tried both 3 decimal digits and 4 decimal digits and found that 3 decimal digits of longitudes and latitudes were sufficient for vertexes.(iv)
In this research, we defined thatR=6378137.00m, which represents the radius of the earth (see Section 3.2). However, the radius of the earth has differences in different places.


### 5.2. Application Prospects

In the field of automatic vehicles, vehicle security is always under the spotlight. And elevation changes have a significant effect on the security of automatic vehicles because it is difficult for them to respond as human beings when traffic changes. Most crucially, the specific calculations used to vehicle movement analyses must be not only accurate but also fast. Otherwise, those automatic vehicles will be in danger, especially when they are running at high speed. Thus, appropriate counter measures could be taken to suppress or reverse or at least alleviate its limitations. The application prospects had not been contemplated within former sections, including MDE model and SBI model, which are described as follows.(i)
MDE model. In last section, results verified that altitude was a key element of vehicle movement analyses, and it indicated that MDE model was much more accurate than Original Model in terms of vehicle movement analysis (see Section4.2–4.5). It is technically feasible to send alerts to the automatic vehicles on the road uphill or downhill in advance so that they can adjust their brake systems in time. Compared with Direct Solution, MDE model reduced the computational costs by avoiding square algorithms. Compared with Original Model, MDE model raised the computational accuracy by taking altitude into consideration. Thus, analyzing vehicle movements by MDE model could be faster and more accurate and it has a bright future in the field of automatic vehicles.(ii)
SBI model. In last section, results indicated that SBI model could improve the preciseness of GPS data with altitude especially when some data was missing in terms of vehicle movement Analysis (see Section4.2–4.4). It is technically feasible for automatic vehicles to establish an offline GIS database and evaluate real-time GIS information when complete data is difficult to acquire, in the boondocks for instance, so that they can make vehicle movement analyses continuously. SBI model can be viewed as a supplementary for MDE model. Thus, SBI model is necessary in the field of vehicle movement analyses and it also has a bright future in the field of automatic vehicles.To sum up, the authors discussed that MDE model and SBI model have great practical prospects and they are suitable and useful for autonomous vehicles, especially for automatic taxis.


## 5.1. Operational Details

Over the course of this research, there were several operational details that had not been contemplated within, including software use and data processing, which are described as follows.(i)
During processing original data by Microsoft Excel 2019, we found that Excel had a processing limit that its maximum capacity was 1,048,576 rows. If original data exceeds this limit, we had to divide them into several pieces before processing. Otherwise, there were other software, Snapde for instance, suiting for the big data more than 1,048,576 rows. In this research, the biggest original data was 1,048,575 records in Zhengzhou (see Section4.3). Therefore, we adopted Excel as a result.(ii)
During spatial analyzing by Arcgis Pro (trial version), we found that the function of the spatial analyst tool called Extract Values by Points was available in v2.2.0 but unavailable in v2.3.0. Some errors may exist in v2.3.0. Therefore, we adopted v2.2.0 as a result.(iii)
In SBI model, the first step was to find out 4 vertexes by positional notation based on the spherical coordinate of pointP. We tried both 3 decimal digits and 4 decimal digits and found that 3 decimal digits of longitudes and latitudes were sufficient for vertexes.(iv)
In this research, we defined thatR=6378137.00m, which represents the radius of the earth (see Section 3.2). However, the radius of the earth has differences in different places.


## 5.2. Application Prospects

In the field of automatic vehicles, vehicle security is always under the spotlight. And elevation changes have a significant effect on the security of automatic vehicles because it is difficult for them to respond as human beings when traffic changes. Most crucially, the specific calculations used to vehicle movement analyses must be not only accurate but also fast. Otherwise, those automatic vehicles will be in danger, especially when they are running at high speed. Thus, appropriate counter measures could be taken to suppress or reverse or at least alleviate its limitations. The application prospects had not been contemplated within former sections, including MDE model and SBI model, which are described as follows.(i)
MDE model. In last section, results verified that altitude was a key element of vehicle movement analyses, and it indicated that MDE model was much more accurate than Original Model in terms of vehicle movement analysis (see Section4.2–4.5). It is technically feasible to send alerts to the automatic vehicles on the road uphill or downhill in advance so that they can adjust their brake systems in time. Compared with Direct Solution, MDE model reduced the computational costs by avoiding square algorithms. Compared with Original Model, MDE model raised the computational accuracy by taking altitude into consideration. Thus, analyzing vehicle movements by MDE model could be faster and more accurate and it has a bright future in the field of automatic vehicles.(ii)
SBI model. In last section, results indicated that SBI model could improve the preciseness of GPS data with altitude especially when some data was missing in terms of vehicle movement Analysis (see Section4.2–4.4). It is technically feasible for automatic vehicles to establish an offline GIS database and evaluate real-time GIS information when complete data is difficult to acquire, in the boondocks for instance, so that they can make vehicle movement analyses continuously. SBI model can be viewed as a supplementary for MDE model. Thus, SBI model is necessary in the field of vehicle movement analyses and it also has a bright future in the field of automatic vehicles.To sum up, the authors discussed that MDE model and SBI model have great practical prospects and they are suitable and useful for autonomous vehicles, especially for automatic taxis.


## 6. Conclusions

In this study, the authors proposed Modified Digital Elevation (MDE) Model, Spherical Bilinear Interpolation (SBI) Model and adopted the experimental data of 9,990 GPS-enabled taxis in Sanya, Zhengzhou and Liaoyang to make movement comparisons by three methods, including Direct Solution, Original Model and MDE model. MDE model was for improving the accuracy of vehicle movement analyses. SBI model was for further improving the preciseness of GPS data with altitude. Given the arguments above mentioned, the major findings in this article include several contents, which are described as follows.(i)
The application of MDE model could improve the accuracy of vehicle movement analyses and it resulted in over99% less disparity with Direct Solution than Original Model because it took altitude into consideration (see Section 4.2–4.5).(ii)
The application of SBI model could improve the preciseness of GPS data with altitude especially when some data was missing (see Section4.2–4.4).(iii)
The application of MDE model and SBI model could improve effects according to statistic results (see Section5.2) and it has a bright future in the field of automatic vehicles.The contributions of this study could be as follows:(i)
MDE model was built to calculate vehicle movements by digital elevation data based on mathematical equations (see Section3.4).(ii)
SBI model was proposed and applied to improve the preciseness of GPS data with altitude of collaborative vehicles (see Section3.5).(iii)
A sample was presented for better description (see Section4.1).(iv)
It was verified that MDE model had higher accuracy than the original (see Section4.2–4.5).(v)
The Derivation Process and Checking Process of SBI model was proposed (see AppendixC, D).Future directions for research can be in two ways. The one is improving MDE model and SBI model. The other is expanding the data size.



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*Source: 6301703-2020-01-20.xml*