An Anisotropic Model Extension to Develop New Railroad Routes
MASTERS THESIS
Francois De Wet Rousseau
Submitted in partial fulfilment of the requirements for the degree of Master of Science in Geographical Information Systems (UNIGIS)
Centre for GeoInformatics (ZGIS)–Salzburg University Sub-Saharan Africa Study Centre
November 2013
Abstract
The purpose of this research thesis is to propose a model that can produce routes for new railroads based on data from an underlying raster elevation model. The result of many
conventional least-cost path raster-based techniques produces routes that have a strong zigzag line characteristic, and need additional vertical and horizontal smoothing before transport system routes can be deduced. Adjacent search techniques on conventional raster models that are wider produce better routes.
The act of planning and identifying routes for new railroads has specific requirements regarding maximum slope and turn radius, as required by the trains that will use the railroad.
The length, travelling speed and weight of the train determine these requirements. The model proposed in this thesis uses the wider search function in a specific direction, and identifies possible routes subject to constraint parameters on maximum slope and minimum turn radius from an underlying raster. Results are compared numerically, statistically and with profile graphs.
Table of Contents
1. Introduction ...1
2. Literature Review ...6
3. Methodology...16
3.1 Study Area ...16
3.2 Model ...18
4. Results and Discussion ...39
4.1 Parameters for tests ...39
4.2 Further analysis ...41
4.3 Comparison of indicators for results...51
5. Conclusion ...60
5.1 Anisotropic Slopes ...60
5.2 Horizontal Curve Constraints ...61
6. Future Work...64
7. Bibliography ...65
List of Figures
Figure 1 - Force on the Curve of a Track...2
Figure 2 - Direction of travel ...4
Figure 3 - Thesis Overview...5
Figure 4 - Flat surface distances ...8
Figure 5 - Geometric Representation of a Digital Elevation Model (Yu, Lee, & Munro- Stasiuk, 2003) ...9
Figure 6– “The allowed directions of movement from a focal cell associated with different raster-based neighbourhood patterns” (Antikainen, 2013)...11
Figure 7– “Three types of neighbouring patterns in raster data formats” (Yu, Lee, & Munro- Stasiuk, 2003) ...12
Figure 8– “Full anisotropic least cost path model” (Moghaddam & Delavar, 2007)...13
Figure 9 - Orientation Map ...17
Figure 10 - Hill Shade and Elevation of the Study Area ...17
Figure 11 - Aerial Photograph of the Study Area (Microsoft Corporation, 2013) ...18
Figure 12 - Point Identification Overview ...20
Figure 13 - Point Identification Detail ...20
Figure 14 - Model Overview...23
Figure 15 - Data tables ...25
Figure 16 - Input screen for parameters ...28
Figure 17 - Calculation of points ...29
Figure 18 - Calculation of points–Detail ...30
Figure 19 - Angle for Arc Length Calculation...33
Figure 20 - Radian Calculation for Cost Direction ...34
Figure 22 - Backtrack sample ...37
Figure 23 - Input for First Analysis ...40
Figure 24 - Map PI-87 and 88...41
Figure 25 - Map PI-89...42
Figure 26 - Map PI-91...43
Figure 27 - Map PI-92...43
Figure 28 - Map PI-97...44
Figure 29 - Map PI-100...45
Figure 30 - Map PI-106...45
Figure 31 - Map PI-110 and PI-114 ...46
Figure 32 - Map PI-118...47
Figure 33 - Map PI-121...48
Figure 34 - Map PI-122...48
Figure 35 - Map PI-122 (Points)...49
Figure 36 - Map PI-121 and 122 intersecting ...50
Figure 37 - Map PI-121 and 122 Combination...50
Figure 38 - PI-97 Line 0 Histogram...53
Figure 39 - PI-97 Line 1 Histogram...54
Figure 40 - PI-97 Line 2 Histogram...54
Figure 41 - PI-97 Line 3 Histogram...55
Figure 42 - PI-106 Histogram ...56
Figure 43 - PI-121 and 122 Histogram ...56
Figure 44 - Profile PI_97 Line 0 ...57
Figure 45 - Profile PI_97 Line 1 ...57
Figure 47 - Profile PI_97 Line 3 ...58
Figure 48 - Profile PI_106 ...58
Figure 49- Profile PI_121 and 122 vertices ...59
Figure 50- Profile PI_121 and 122 line...59
Figure 51 - Map slope evaluation ...61
Figure 52 - Map circle radius...62
List of Tables
Table 1- Tier Cost Addition...21 Table 2 Analyses Results ...52
List of Equations
Equation 1–Diagonal Distance Calculation ... 8
Equation 2–Direct Connections (Yu, Lee & Munro-Stasiuk, 2003) ... 9
Equation 3–Diagonal Connections (Yu, Lee & Munro-Stasiuk, 2003)... 9
Equation 4–Theorem of Pythagoras... 28
Equation 5–Calculating Angles ... 29
Equation 6–Longitude Left ... 30
Equation 7–Latitude Left ... 30
Equation 8–Longitude Centre ... 30
Equation 9–Latitude Centre ... 31
Equation 10–Longitude Right... 31
Equation 11–Latitude Right ... 31
Equation 12–Inverse Distance Weighting (De Smith, Goodchild & Longley, 2013) .. 32
Equation 13–Theorem of Pythagoras... 32
Equation 14–Arc Length (Arc Length, 2013)... 33
Equation 15–Difference in Radians ... 34
Equation 16–Slope Difficulty Function ... 35
Disclaimer
The results in this thesis are based onthe author’sown research at the UNIGIS Sub-Saharan Africa Study Centre, Centre for GeoInformatics (ZGIS) of the Salzburg University.
Signed:
Francois De Wet Rousseau Johannesburg, November 2013
Acknowledgements
The author would like to thank the following persons:
Mr Dirk Engelbrecht, Rail Engineer at Transnet SOC Ltd for the background information and explanations supplied during the interview. Your inputs contextualised the problem statement and helped to connect the model to what is happening in practise.
Mr Henk Bester, MD Rail at Hatch for explaining in an interview the current methodologies and important parameters, in railroad route planning and selection, currently practised in South Africa.
1. Introduction
The purpose of this thesis is to attempt to address the problem of selecting a route for a new railway line based on a raster grid with elevation values of the study area.
Finding the easiest route over mountainous and uneven terrain is a problem that has been experienced by man since the earliest times. Following footpaths created by wild animals over mountainous terrain provided a good solution for travellers on foot (Berry, n.d.), since animals seem to have a build-in navigation system that allows them to find an easy route. Animals and humans on foot have the ability to traverse fairly steep gradients and turn very sharply, allowing them to travel on a zigzag route up steep mountainsides. In the 19th and early 20th Centuries, railroad planners in South Africa used horses on a free reign to help identify the best routes to build railroads through uneven terrain
(Engelbrecht, 2013).
Currently, a large portion of freight train volumes in Southern Africa are made up of mineral ore transported from mines to harbours for export to industrialised countries.
Exports of coal and iron ore, in fact, made up 59.7% of freight volumes transported by Transnet SOC Ltd during 2012 in South Africa (Transnet SOC Ltd, 2012). According to an article published under the auspices of the World Bank (Foster, Butterfield, Chen, &
Pushak, 2008), hydroelectric power plants and railroads will make up a large portion of themoney spent in terms of China’s role asan infrastructure financier in Africa. Railroad route planning, therefore, will be an important activity in infrastructure investment.
Trains cannot traverse steep gradients and have specific limitations in the radius of curves that they can negotiate. The type of train (freight or passenger), speed, track width, length etc, all play a role in what route is most suitable for a train. Trains, especially loaded
Therefore, the effort required to ascend a slope is not the most important consideration since more locomotives can be added for steeper gradients. The ability to stop a fully loaded train in an emergency during a descent is the most important consideration when determining the maximum gradient that can be traversed (Engelbrecht, 2013).
To increase productivity and reduce costs, the length of the trains employed to transport coal and ore is substantiallymore than the length of ‘normal’trains. A typical freight train transporting general goods will usually have 40 wagons, but when transporting coal, this number may be increased to as much as 200 wagons. This is five times the norm and calls for very specific designs of the horizontal curves of the track to be considered when building a new railroad. Curves with a small radius (sharp turns) place high physical stresses on the railroad track; with a 200-wagon train, the locomotive may have already cleared a curve (and started the next curve) while some wagons have not even entered the first curve. This means that the pulling force provided by the locomotives at the front is in a different direction than what is required by the wagons towards the back. Thus the pulling force must be transferred by leveraging the railroad track, exerting a lateral force on the track that create stress and friction (as shown in Figure 1).
This problem may be alleviated by placing locomotives in the middle of the train and not only in the front (distributed power), and also by specifying large radius curves when designing the track (Engelbrecht, 2013 and Bester, 2013).
Distance calculations and cost surfaces form an integral part of the technique and solution provided by Geographic Information Sciences (GIS) (De Smith, Goodchild, & Longley, Geospatial Analysis, 2013).
In practise civil and rail engineers use commercial software such as Quantm™ (Bester, 2013). Algorithms developed and used by commercial software publishers are well- guarded trade secrets with significant commercial value. Trimble, the owners of the Quantm™software package, sells access to their algorithms with a cost that is dependent on the complexity of the route (Route Planner, 2010). Yu, Lee & Munro-Stasiuk (2003) concluded that in most of the current GIS software products, distance and anisotropic costs have not been fully considered for the functions of finding least-cost paths.
A Digital Elevation Model (DEM) of a geographic area to be traversed provides a useful base to extract slope and aspect for the determination of least cost paths. The slope value is calculated as the change in elevation in a direction where the value is the greatest (De Smith, Goodchild, & Longley, Geospatial Analysis, 2013). Aspect defines the direction in which slope is greatest. This implies that if the direction of traversing a point in a DEM is diagonal to the aspect (as shown by the blue line in Figure 2) the change in elevation will be smaller (easier to traverse) than represented by the calculated slope value.
Figure 2 - Direction of travel
Surface cost calculations that take direction of travel into consideration (anisotropic) provide more options to consider for route determination than isotropic models that only consider slope and not direction. Directional differences are an important consideration for determining optimal paths (Yu, Lee, & Munro-Stasiuk, 2003). Anisotropic models can produce zigzag results which are not acceptable for railroad routes that command smooth changes in horizontal direction.
From the research provided herein, it can be concluded that new railroad developments will be an important construction activity over the next decade or two in Southern Africa.
GIS techniques, and specifically anisotropic surface analysis for route planning, could play an important part in helping to select the best suitable route.
Research question:
This thesis aims to contribute a model that will consider anisotropic slope and horizontal curve constraints concurrently, while developing possible routes for new railroad
development.
The objective is to find a route for a new railroad that conforms to the maximum slope parameter and minimum turn radius parameter.
An overview of the chapters of this thesis is shown in Figure 3. The model depicts the steps to be followed and how these steps support and build on one another.
Figure 3 - Thesis Overview
2. Literature Review
2.1 GIS Surface Modelling and Route Planning
Jankowski (1995) concluded that the role of GIS Modelling for decision support systems is first and foremost, the search of viable alternatives, then for the evaluation of alternatives according to priority weights, and finally for the visualization of results.
The surface of a geographical area can be represented by raster or grid structures.
Raster structures consist of a grid made up of a rectangular array of identical square cells (De Smith, Goodchild, & Longley, Geospatial Analysis, 2013). The size of the cells may differ; the smaller the area represented by each cell, the finer the
resolution. Each cell may also have multiple attributes, such as height values or reflective values from remote sensing sources. Although the cell represents a square area, the value is assigned to a point in the middle of the cell. This value is usually an average for the area of the cell.
The finer the resolution or smaller the areas represented by each cell, the more accurate the representation of reality. The larger the cell, the more generalised the information represented over the area. To increase (double) the resolution from a cell size of 100 metres (m) to 50m requires four times the storage capacity, since four cells are needed to represent the area where one is needed with the courser resolution.
According to De Smith et al (2013) surfaces represented as a grid have some disadvantages:
Large storage and processing requirements;
Details are lost in variation within a cell, for example, a cell resolution of 90m x 90m may have very large variations in elevation especially on steep slopes.
Increasing the resolution to 25m x 25m may alleviate this, but this is dependent on the availability of data or interpolation methods; and
Use of small cell neighbourhoods may result in bias for surface computations, however, this issue can be alleviated by using larger neighbourhoods for calculations.
Although raster grids have disadvantages, they are very useful for data manipulation and processing.
Van Bemmelen, Quak, Van Hekken & Van Oosterom (1993) concluded that raster- based algorithms produce better results than vector-based algorithms. At the time of their study however, computation performance of the models was a major drawback due to the size of raster datasets. Since 1993, major strides have been made in
computing performance and affordability, opening up possibilities for more elaborate raster-processing algorithms.
To determine the least cost path (best path) from one point to another, raster cost distance calculations are used. These costs are not necessarily financial; they may also be described as the friction or resistance experienced when moving through the cell. The costs differentiate the effort of travel through different portions in a study area. This approach allows for the identification of a path that requires the lowest
and each cell is assigned a value based on the criteria of what it‘costs’to traverse the cell. Accumulated cost surfaces are used to identify the least cost path, while costs are added up along alternative paths to identify the path with the lowest accumulated cost, that is, the least cost path (De Smith, Goodchild, & Longley, Geospatial
Analysis, 2013).
The distance of the route can be calculated using the size of the cells in the raster.
Spatial distance is the distance between the centres of two cells.This distance varies when direction and slope change (Yu, Lee, & Munro-Stasiuk, 2003).
Moving on a flat surface from point O to point P2, as depicted in Figure 4, the distance is equal to the resolution - µ (cell size) of the raster. When moving from point O to P1, the distance (D) may be calculated as shown in Equation 1, which is based on the theorem of Pythagoras (Weisstein):
D = 2μ2
Equation 1 - Diagonal distance calculation
Figure 4 - Flat surface distances
Yu et al (2003) created equations to be used with spatial distance calculations where points do not have the same elevation. The following equations are based on Figure 5.To calculate distance from point O’ to point P2’, Yu et al proposed Equation 2:
D( , ) = μ + (H − H )^2
Equation 2–Direct Connections (Yu, Lee, & Munro-Stasiuk, 2003)
The same equation mayalso be used for P4’, P5’ and P7’. To calculate distance from point O’ to point P1’, Yu et al proposedEquation 3:
D( , ) = 2μ + (H − H )^2
Equation 3 - Diagonal Connections (Yu, Lee, & Munro-Stasiuk, 2003)
This equation can then also be used for the other diagonal connections; P3’, P6’and P8’.
Figure 5 - Geometric Representation of a Digital Elevation Model (Yu, Lee, & Munro-Stasiuk, 2003)
2.2 Isotropic Models
Djenaliev (2007) considered slope in his model for multicriteria decision making for railroad planning. Isotropic slope (without direction) was considered and the author assigned cost value of one to ten, where one is the lowest cost and ten the highest. In this,, a value of 1 000 was applied to values of slopes greater than 35%, which are not traversable according to parameters supplied. A scale value was assigned of ten to one where ten was the most suitable slope, and one the least suitable slope. These values were then used in a model considering factors such as ground cover, geology, settlements, etc. The model generated a cost surface used for path analysis.
Antikainen (2013) argued that raster-based path finding could be improved by using larger connectivity patterns between raster cells, and by placing nodes on the side of the cell rather than in the middle. Four neighbourhood patterns were proposed by Antikainen (2013) to increase cell connections to 48 (as shown in Figure 6).
Figure 6– “The allowed directions of movement from a focal cell associated with different raster-based neighbourhood patterns”(Antikainen, 2013)
“The results confirmed the expectations of the use of large search neighbourhoods producing better paths.”(Antikainen, 2013) The improvement of Border Connectivity versus Centre Connectivity (traditional method) was dependent on the type of rasters tested on. Random rasters and rasters with uniform regions benefited from the Border Connectivity method while rasters with smooth variations worked better with the Centre Connectivity. Computation time was approximately 30% longer with the 9 x 9 neighbourhood than with the 3 x 3 neighbourhood. Antikainen (2013) further
concluded that the Border Connectivty model performed poorly with large datasets.
2.3 Anisotropic Models
Dijkstra’s(1959) algorithm is used in many least-cost path applications. Cell centres in a raster grid act as nodes and the connections between cells act as links between the nodes. Yu et al (2003) cites three patterns for connecting cells (as shown in Figure 7):
(a) Rook’s pattern which connects four neighbouring cells (cardinal directions);
(b) Queen’s pattern which connects eight adjacent cells (cardinal and diagonal directions); and
(c) Knight’s pattern which connects 16 cells. The Knight’s pattern not only looks at adjacent cells but includes a second tier of cells beyond the adjacent cells.
The patterns are named after the pieces in the game of chess as they mimic the movement characteristics of chess pieces.
Figure 7– “Three types of neighbouring patterns in raster data formats”
(Yu, Lee, & Munro-Stasiuk, 2003)
Yu et al (2003) argue that many algorithms are not successful in complex terrain applications because they only consider adjacent cells to determine where the path should go. The costs in different directions are important for road planning therefore anisotropic surfaces should be considered when planning a route.
Results show that paths obtained using the Knight’spattern has lower total costs than paths obtained usingthe Queen’s pattern (Yu, Lee, & Munro-Stasiuk, 2003).
Therefore it seems that using algorithms that look further than just immediately neighbouring cells can result in more optimal paths. Yu et al (2003) further concluded that the patterns result in a zigzag path that may be averted by applying a directional constraint, although this is not demonstrated in the cited paper.
Moghaddam & Delavar (2007) developed a model based on values to move from one cell to the next in a specific direction.“…the model generated accumulative cost surface and direction surface that assigns number to each cell showing the direction of movement to calculate accumulated cost surface.”The differentated directional values are shown in Figure 8. The model follows the miniumum cost in each cell from the end point to the start point to find the best route.
Figure 8– “Full anisotropic least cost path model”
(Moghaddam & Delavar, 2007)
2.4 Anisotropic Models with Horizontal Curves
The model proposed by De Smith (2005) makes provision to define maximum gradient and horizontal curvature constraints. The model first selects a route according to the gradient constraints, and then smoothes the route horizontally to eliminate sharp turns that fall outside of the specified parameters. The horizontal movement of the route results in the gradient constraints not being adhered to. The route is then smoothed vertically through proposed cut-and-fill operations, to bring it in line with the gradient constraints.
De Smith (2005) proposed a larger search area neighbourhood be used for smaller allowed gradients. For a steep gradient, a small search area can be used, while for a slope of 1:5 a 7 x 7 cell search neighbourhood will work. For a flatter gradient of 1:10, the search area must be increased to a 13 x 13 cell neighbourhood.
Computing time increases with the increase in the size of the neighbourhood
searched; De Smith (2003) found that a 9 x 9 search neighbourhood used more than double the computing time than that of a 5 x 5 neighbourhood. The unreachable areas in the study area declined to almost nothing when using the 9 x 9 instead of the 5 x 5 neighbourhood (De Smith, 2005).
2.5 Conclusion
Many of the papers and sources cited suggest that larger search areas result in better least cost path calculations. However, one major drawback of the increased search area is the additional computational resources required to do the analysis. Another shortcoming is that the analysis results in zigzag routes. De Smith (2005) does address the curvature issue by smoothing it afterwards with a separate procedure.
More papers and sources were considered during the literature review but were not used.
3. Methodology 3.1 Study Area
An orientation map is shown in Figure 9. The area that will be evaluated starts on the Highveld of Mpumalanga at an elevation of 1 700m and runs down the escarpment to the border of Swaziland ending at an elevation of 1 476m. Although the railway line will run further than this, 25m elevation data is not available for the rest of the area to be traversed. This study will concentrate on the area where 25m data is available to test the model and theory. Figure 10, the Hill Shade Analysis and a colour ramp based on elevation are combined to create this image. Figure 11 shows the land cover of the study area. The Euclidean distance is approximately 34km, made up of rolling farmlands. There is farming activity, as well as roads and other obstacles that need to be taken into consideration for the final route. This model will only evaluate the elevation data to select routes. Other criteria and costs may be added to the routes proposed for before a final decision is made. The large dam in the middle of the image is marked as an area that cannot be traversed.
Figure 9 - Orientation Map
Figure 10 - Hill Shade and Elevation of the Study Area
Figure 11 - Aerial Photograph of the Study Area (Microsoft Corporation, 2013)
3.2 Model
3.2.1 Overview of the model
The model attempts to find the best route within parameters supplied for maximum gradient and minimum radius for horizontal curvature. The model starts with an origin location, defined in terms of latitude and longitude, direction of travel and a target or end location.
The model first needs to identify points to evaluate for the next point in the proposed path solution. Figure 12 shows an overview of the process that is explained while Figure 13 contains a close-up view of the area. These two figures’details can be referred to for the following explanation:
The figure contains a grid with a cell size of 25m x 25m, representing the cells of the raster. The dots in the grid mark the point where the elevation value is measured in the cell centre.
The current position is marked by point CP in blue on the figure. Direction of travel for point CP is indicated by the green line from D to A. The two orange circles form a tangent with the green line at point CP with a radius of 500m, specified by the parameter of minimum curve for the proposed route. This implies that the next point for the route can be chosen between the two orange circles in the direction of A.
Moving approximately six cells towards A, the circles have diverted the width of one cell from the green line, presenting a choice between three cells as the next possible point. The points are marked in red and identified as Tier 1-Left (1L), Tier 1-Centre (1C) and Tier 1-Right (1R).
In the literature review, it was established that a wider neighbourhood search
provides better results with least cost path analysis (Antikainen, 2013) and (Yu, Lee,
& Munro-Stasiuk, 2003). Therefore a second tier of points have been identified for evaluation. The processes described above are iterated using points 1L, 1C and 1R as origin points to select the next three cells.
The blue line through point 1L indicates the direction of travel at 1L; another circle is added (pink circle on right) as a tangent to point 1L. From this three second tier points, 2LL, 2LC and 2LR, can be identified.
The two blue circles are added to form a tangent to the green line at point 1C. From that point, points 2CL, 2CC and 2CR are identified.
The pink circle on the left is added to form a tangent on the blue line towards C at point 1R. From there points 2RL, 2RC, and 2RR are identified. The resulting spacing between points on the second tier is approximately one cell width.
Figure 12 - Point Identification Overview Figure 12 - Point Identification Overview Figure 12 - Point Identification Overview
Because the identified points do not intersect with the centre point of cells where elevation values exist, interpolation using the Inverse Distance Weighted (IDW) method is used to calculate a height value from the surrounding known elevation values. The IDW method has been chosen because the elevation data is represented by a dense set of points spaced 25m apart (Childs, 2004).
Costs are calculated for moving from Point CP to the individual first tier points (1L, 1C and 1R) as shown in Figure 13. Costs are also calculated moving from the first tier points to the associated second tier points. The first and second tier costs are added together to choose the best next point. The following scenario is possible where the lowest cost tier move is not necessarily the best way to move to the second tier. Table 1 shows that although the first tier cost for moving through point 1L is higher, the total cost moving to point 2LR is the lowest.
Table 1- Tier Cost Addition
Movement Tier 1 Cost Tier 2 Cost Total
CP-1L-2LL 35 25 60
CP-1L-2LC 35 20 55
CP-1L-2LR 35 15 50
CP-1C-2CL 30 25 55
CP-1C-2CC 30 33 63
CP-1C-2CC 30 32 62
CP-1R-2RL 28 35 63
CP-1R-2RC 28 36 64
CP-1R-2RR 28 40 68
Points 1L and 2LR are added to the chosen path and the current position changed to point 2 LR.
The process described is then iterated until the target or a dead-end is encountered. A dead-end is reached when the edge of the raster model with elevation data is met, or forward movement within the gradient constraint is not possible. The model then returns to the previous point and marks the movement it retreated from as a dead- end. The options with the lowest cost that is not marked as dead-ends, are then explored for further points.
If the returned to position is the same as the start point, no solutions are possible without relaxing the parameter constraints.A schematic overview of the model is shown in Figure 14.
Figure 14 - Model Overview
The components of the model shown in Figure 14 are discussed in paragraph 3.2.5 to paragraph 3.2.9.
3.2.2 Parameters
The selection parameters used for the study are based on information gathered during the researcher’sinterview with Engelbrecht (2013). The parameters for railroad route selection for freight trains of over 200 wagons in length transporting coal from mines
in Mpumalanga to the Richards Bay Coal Terminal at the KwaZulu-Natal coast in South Africa include:
The minimum radius of horizontal curves is 500m to accommodate the longer than normal train (200 wagons rather than 40); and
Vertical gradients are expressed in ratios, for example the maximum allowable ratio for this train is 1:80. That means that a 1m change in elevation is allowed for every 80m travelled. A weighted average of not more that 1:100 is required, and 1:150 is best. The gradient is an important safety requirement to make sure it is possible to stop the train safely in case of emergency. A fully loaded train of this length must be able to dissipate a lot of energy in an emergency stop situation. The steeper the slope, the more potential energy is added to the equation and puts more stresses on the equipment and track, potentially leading to failures and/or increased stopping distances.
Different parameters can be specified for other study areas and applications based on engineer’s design specification.
3.2.3 Data and Data Structures
The Digital Elevation Model (DEM) data used was obtained from the National Geospatial Information (NGI) section that is part of the Department of Rural Development and Land Reform in South Africa. The data has a resolution of 25m supplied in American Standard Code for Information Interchange (ASCII) format.
Three tables are used in the model to store decision making information, weighting criteria and the elevation data of the study area. Figure 15 shows the three tables used and some of the fields. The‘Inputs’table is used to store the parameters and input
different scenarios and inputs. The dbo_Calc table is an SQL Server table where the calculation parameters and results for each point evaluated are stored. The
dbo_dembox table contains the elevation data for each point in the study area imported from the ASCII files supplied by the NGI. This data is used to interpolate and calculate elevation values for points being evaluated.
Performance of the model has been improved by moving the two large data tables to an SQL Server environment and creating indexes of the relevant search columns.
Figure 15 - Data tables
The routines implementing the logic of the model are written in Visual Basic for Applications and run in a Microsoft Access environment.
3.2.4 Initializing and Weighting
The model is initialised with the following parameters:
3.2.4.1 Constraints
Maximum Slope Ration and Minimum Turn Radius are design constraints as specified by the design engineer for the current train design.
3.2.4.2 Start Parameters
The start point is defined in terms of meters referenced in the Lo31 coordinate system. This system is based on the Gauss Conform Coordinate System. The central meridian of projection is 31˚ East and‘Eastings’(x) coordinates are measured from this longitude. The‘Southings’(y) coordinate is measured from the equator (Parker, 2012). The‘Start Direction’is expressed in radians and is used to identify candidate points on the first iteration of the model that falls in the turn radius constraint.
3.2.4.3 End Parameters
The‘End Point’will be the target point to move towards when calculating the route.
Although this is defined as a single point with an x and y coordinate, a‘target box’is calculated around the point as a reference to determine whether the target has been reached during the iterations of the model. The size of the box defined in the model is 400m x 400m with the end point right in the middle. This is to‘catch’a point where the steps in the model are between 300m and 320m. An‘End Direction’
measured in radians is used when the model is reversed to calculate a solution from the end point to the start point.
3.2.4.4 Weights
Three types of weights for each of the cost factors can be applied to the model, namely: the Cost Distance Factor, Cost Direction Factor and Cost Slope Factor, which are used as multiplication weights in the mathematical calculation of costs to identify the best candidate point to move to next for the model. As the factor is increased, the weight or importance of the cost component is increased, for instance by increasing the Cost Direction Factor, more emphasis is placed on points that will take the route in the direction of the target point.
Figure 16 shows the screen where parameters are specified. The‘ID’column is an autonumber assigned to the set of parameters stored in the database. The route generated is also identified by this number. In this thesis each analysis is identified by this number as the Parameter ID (PI). Next follow the parameters for the start position, the end position, cell size of the raster, maximum-allowed slope, minimum turn radius and then the weights for cost calculations. On the right part of the screen the option can be specified to generate the route in reverse. The‘Generate Routes’
button starts this process and previously generated routes can be deleted. The last four fields show the result of the analysis, and the number of iterations taken to generate the route. The start and finish times are also recorded to evaluate and measure performance of the model.
Figure 16 - Input screen for parameters
3.2.5 Identifying candidate points
The first step in moving forward is to calculate the location of possible points as illustrated in Figure 13. The current position and direction of travel is taken into account to calculate the candidate points to be evaluated.
Figure 17 includes the first quadrant of a circle with a radius of 500m and shows a wide view of Figure 18 used in the explanations below. From the sketch shown in Figure 18, it is determined that the distance of c from the current point (CP) to the centre point on the first tier (1C) is 156.12m, the distance of u from 1C to 1L was set at 25m (raster cell size) and the distance of l from CP to 1L has been calculated using the theorem of Pythagoras as shown in Equation 4:
= +
Equation 4 - Theorem of Pythagoras
Equation 4 resulted in a value of 158.109m. From this,the angle of α wascalculated in Equation 5 using the arctangent function.
= ( )
Equation 5 - Calculating Angles
The arctangent function as used in Equation 5, works in the first quadrant (0-90˚) of a circle. The model’s routines test the quadrant into which the angle falls, and adjust the calculation accordingly for the other three quadrants. The resulting valuefor α is 9.0977˚.
Figure 17 - Calculation of points
Figure 18 - Calculation of points–Detail
The following formulas are then used to calculate new first tier candidate points:
Left first tier (1L) x = l ∗ cos( β + α) + x
Equation 6 - Longitude Left
y = l ∗ sin( β + α) + y
Equation 7 - Latitude Left
Centre first tier (1C) x = c ∗ cos( β) + x
Equation 8 - Longitude Centre
y = c ∗ sin( β) + y
Equation 9 - Latitude Centre
Right first tier (1R) x = l ∗ cos( β − α) + x
Equation 10 - Longitude Right
y = l ∗ sin( β − α) + y
Equation 11 - Latitude Right
Where xl, xc and xr are the new longitude coordinates for the new candidate points;
yl, yc and yr are the new latitude coordinates for the new candidate points;lis the length from point CP to 1L as calculated above; c is the length from point CP to 1C;
β is the current direction of travel (CP);α is the change in direction as calculated above; xj is the current longitude (CP); yj is the current latitude (CP).
The same calculations are then repeated from point 1L, 1C and 1R as current positions (CP) to identify the second tier points.
3.2.6 Calculating the elevation values of candidate points
Looking at Figure 13, it is clear that possible points may not coincide with the centre points of cells as represented by the DEM. Once the location is calculated, the height value of the point is calculated by using Inverse Distance Weighting (IDW) method of interpolation.
According to De Smith et al (2013) IDW works well where the surface is represented by a dense number of points. The DEM used in this model consists of a large number
The closest four surrounding points are identified and used in the IDW formula as shown in Equation 12. In this formula, the following values are represented: zj is the value to be calculated for point j; dij is the distance between the interpolation point I and the calculated point j; zi is the value of the interpolation point;α is the distance power applied (2 in this model); and kj is an adjustment value to make sure the weights add up to 1 (De Smith, Goodchild, & Longley, Geospatial Analysis, 2013).
z = k 1 d∝z
Equation 12–Inverse Distance Weighting (De Smith, Goodchild, & Longley, Geospatial Analysis, 2013)
3.2.7 Calculating costs
The total cost to move to a candidate points is calculated by adding the distance, direction and slope costs.
3.2.7.1 Distance costs
The distance costs are based on the number of metres travelled. The distance (d) from point CP to 1C as shown Figure 18, is a straight line therefore the length of line c is used with the change in elevation (Δe) between points CP and 1C. Using
Equation 13, the following calculation results in the distance:
= + ∆
Equation 13 - Theorem of Pythagoras
Travelling from CP to L1 will not happen in a straight line but in the form of an arc or curve as identified by p in Figure 18. The distance (length of the arc) is calculated
with the formula shown in Equation 14, where R is the radius of the circle, and C the angle as shown in Figure 19.
= 2 (360) Equation 14 - Arc Length (Arc Length, 2013)
The final distance is the calculated as d = p + ∆e to take the change in elevation into consideration.
Figure 19 - Angle for Arc Length Calculation
The Distance Cost = Distance (d) * Distance Cost Factor.
3.2.7.2 Direction costs
To ensure the model chooses points that will lead to the target point, directional costs are considered during the selection process.The model calculates direction to the target point from the current position (CP) in radians. The difference is then calculated between the direction of the candidate point considered and the target point in radians.
Consider Figure 20. If the target direction is represented by line v and the direction to a candidate point is u, the radian value ofangle αis0.322 and that of angle β is 5.962
with a difference of 5.64. The value that will be more practical is actually the angle of γ.
To accommodate radian values that might have a large difference but are in reality quite close, the model calculates the difference using Equation 15:
q = |α − β|
= | − | − 2
Equation 15 - Difference in radians
The smallest value of q and r is then multiplied with the Cost Direction Factor supplied as a weighting input parameter. This results in the Direction Cost.
Figure 20 - Radian Calculation for Cost Direction
3.2.7.3 Slope costs
The input parameter for slope for this model specifies that of a slope ration between 1:100 and 1:150, the latter is preferable with a maximum ratio of 1:80 (0.716 degrees). The cost increase in slope increase is not well represented by a linear calculation; a slope of 2° is not twice as difficult to traverse, but exponentially more
Equation 16 shows the function used to calculate the difficulty of slope, where x is the slope in degrees.
( ) =
Equation 16 - Slope Difficulty Function
This is the result of the function is shown in Figure 21.
Figure 21 - Slope Difficulty Function
The result of Equation 16 is then multiplied with the cost slope factor to calculate the final slope cost. Slopes beyond the maximum slope are not considered as a possible candidate point and are marked as‘too steep’.
3.2.8 Moving forward
After all the costs have been calculated for each candidate point, the nine second tier points are compared to find the point with the lowest total cost. The model then moves to this point and becomes the current position (CP). The calculation results and decision are written to the database. The next iteration of the model is then
3.2.9 Backtrack
If the slope of all the second tier candidate points exceeds the maximum allowable slope or the calculated points fall outside the bounds of the search area, no forward motion is possible. The parent point of the current position is retrieved from the database as previously stored and the point where no forward motion was possible is marked as a‘dead end’. This point is then excluded from the comparison of
candidate points when the next best point is selected. If no forward motion is possible from the previously retrieved parent record, that is, all candidate points are marked as dead ends, the grandparent record is retrieved to re-evaluate candidate points in that record. In this way, the model steps back every time a dead end is reached to find alternative routes around the dead end.
Figure 22 shows a sample of an analysis where a dead end has been encountered.
The green points are selected points on the route; the red point indicates an area where no forward movement is possible (dead end). The route generated comes in from the top of the figure and the model will have attempted to go where the red dots are indicated first, but have found that no forward movement was possible due to the steep slope ahead. The model will then have backtracked two steps and found a route bypassing the steep slope to continue to the bottom right.
Figure 22 - Backtrack sample
3.2.10 Iterations
Depending on the parameters and the topography of the terrain, the possibility exists that the model cannot find a solution. To stop the model from going in an endless loop, the maximum number of iterations has been limited. After 1 000 iterations in the model, there will be a return to the start point and an increment of the start direction to force a new direction of search that may yield a result. If no solution has been found after exploring the different directions, the model stops and returns a‘No Solution Possible’result. Different costing weights and/or the relaxing of maximum slope or radius parameters can be attempted to find a solution. Each iteration of the model considers three first-tier and nine second-tier candidate points.
3.2.11 Opposite direction
Analysing the same set of parameters in the opposite direction usually results in a different solution to the required linking of the start and end points. Even if a
opposite directions. The reason for this is that the local topography in the first few hundred metres plays an important part in selecting the route that the model will take.
This phenomenon can further be used by breaking the route up into sections and having many different start points from which to initiate the model. This technique can also be used to refine certain sections of the route.
3.2.12 Comparing routes
To compare the routes the following indicators have been calculated for each route generated:
Maximum slope;
Average slope;
Length of the route; and
Total absolute change in elevation.
4. Results and Discussion 4.1 Parameters for tests
Numerous sets of parameters were tested on the model and each set was assigned a Parameter Identification (PI) number. The analyses discussed in this thesis indicate where the results were different from the previous and new insights are gained. The result is that not all PI numbers discussed follow each other in strict chronological order.
To establish the correct weighting parameters for analysis, the following variables were used as a starting point:
Cost distance factor: the distance between the current position and a
candidate second tier point is approximately 300m to 325m depending on the direction and the slope. Therefore a weighting factor of 1 was assigned to the cost distance factor for the first analysis;
Cost Direction Factor: the value of cost direction is based on the difference between the direction of the target and the direction of the candidate points in radians. The possible value is between 0 and 3.14 (0° and 180°). To put this on an equal footing with the Cost Distance Calculation the first multiplication factor of 300. A radian difference of 1 will then have approximately the same directional cost as the distance cost. This allows the directional cost to have an influence on the selection process; and
Slope cost factor: the slope cost is calculated according to the function shown in Equation 16. A slope of 1° will return a value of 1. For the first analysis this has also been multiplied by a factor of 300 to have an influence on the overall cost. Figure 23 shows the input screen for the first analysis with the parameters
Figure 23 - Input for First Analysis
This resulted in the points generated as shown in Figure 24. Two analyses are represented: PI-87 is started in the top left corner of the map at the start point; and PI-88 is started in the reverse direction at the end point. The red dots represent a dead end where no forward motion is possible and the grey dots, points that were selected as part of a route but could not be completed within the 1 000 iteration limit of the model. From the map it is evident that the model explored the high ground around the start point, but could not find a way down the slopes that adhered to the limitations. The analyses around the end point suggest more uneven ground with very few points that meet the slope parameter as specified.
Figure 24 - Map PI-87 and 88
Given the results obtained in the first two analyses, it is clear that adjustment of the input parameters is needed in order to find a route between the two points. Either the weighting parameters should be adjusted or the slope parameters should be relaxed.
If a solution is not possible with the given slope ratio of 1:80 (0.716°), the model must find the route with the best possible slope as a solution instead.
To refine and experiment with the weighting of the parameters, the Maximum Slope Ratio is relaxed for the next analyses. Once the weighting parameter values that produce good results are identified, the slope constraint will be tightened again to try and find the best possible solution in terms of maximum slope.
4.2 Further analysis
For the next two analyses, the slope constraint ratio is relaxed to 1:25 (2.29°). All the weighting parameters for the slope and direction weight are left on the value of 300 and the distance weight on 1. The resulting maps from the start point are shown in Figure 25 for PI-89.
None of the routes is able to reach the end point. The route intersecting the dam from the left is deflected and a route around the dam to the north must be found, but once the dam is cleared, the route does not turn in the direction of the end point but continues North along the slope with the lowest cost. It only turns South again a few kilometres later and backtracks over itself. Considering some of the other routes, it is clear that these also lack direction and go around in circles.
Figure 25 - Map PI-89
The same parameters run in the opposite direction are shown in Figure 26 for analysis PI-91 where the points are converted to lines. The same result as found in Figure 25 is achieved with many unnecessary loops and backtracking. The green line however, has managed to find a solution and reached the target.
Figure 26 - Map PI-91
The Cost Direction Factor is increased to 600 for the analysis PI-92 in Figure 27 to try and correct the problem in the two previous analyses that lacked direction. Two of the routes generated (blue and grey) managed to reach the end point within 1 000 iterations of the models. Increasing the cost distance factor did improve the routes generated but a route such as the purple route went in the wrong direction.
Figure 27 - Map PI-92
For the analysis PI-97 shown in Figure 28 the cost direction factor is increased to 1 800 and the cost slope factor is increased to 600. This analysis resulted in four of the seven lines reaching the target within 1 000 iterations.
Figure 28 - Map PI-97
For the next analysis, the maximum allowed slope ration was reduced to 1:40, resulting in a slope of 1.43°. To compensate for the reduction in the maximum allowed slope ratio, the maximum number of iterations per route for the model were increased from 1 000 to 5 000 for all subsequent analyses.
Figure 29 shows the analysis PI-100 with a cost slope factor of 600 and a cost
direction factor of 1200. Although the green line managed to reach the end point, it is obvious that many twists and turns were taken due to the relatively high value of the slope cost factor and the relatively low value of the cost direction factor compared to previous analyses.
Figure 29 - Map PI-100
The analysis PI-106 shown in Figure 30 returned the best analysis for the slope factor of 1:40. The parameters used for this route were 2 400 for the cost direction factor and a 450 for the cost slope factor. This analysis is run in reverse order, starting at the end point and ending at the start point.
Figure 30 - Map PI-106
As the value for the maximum slope ratio decreased, a higher value in the cost direction factor was needed to ensure the end point was reached in the most efficient way.
For the next analyses, the maximum slope ratio has been reduced to 1:50. Figure 31 shows the analyses using a cost direction weight of 2 400 and a cost slope weight of 450. The analysis, PI-110, is run from the start point and is represented by the lines.
PI-114 is run in reverse order and is represented by the dots. The analysis from the endpoint could not find a solution to get closer to the target point (‘Start Point’) in this instance. The lines representing the analyses from the normal start point indicated that with the decrease in the maximum slope ratio, a higher value was needed for the cost direction factor; many loops are evident that do not bring the analyses closer to the target point.
Figure 31 - Map PI-110 and PI-114
For the next analysis the value for the cost direction factor has been doubled from 2 400 to 4 800 to force the model closer to the end point. Figure 32 shows the lines for
the resulting analyses. The result is not an improvement on the results of the previous analyses and the end point is not reached.
Figure 32 - Map PI-118
For analyses shown in Figure 33 and Figure 34, the cost direction factor is increased to 9 600. Although none of the lines reach their target, they do overlap in certain instances. Figure 33 shows that the green and orange lines were the most successful, but were not able to find the valley to the East of the dam needed to get down to the end point with this stringent maximum slope ratio of 1:50. Considering the line generated in the reverse direction represented by the grey line in Figure 34 for analysis PI-122, it is clear that this analysis is forced to find a way out of the valley and is guided by the valley walls. Once the flatter area is reached, it takes the shortest possible route towards the target, in this case the‘Start Point’. The slopes near the start point approaching from the South cannot be traversed with the set maximum slope ratio, hence no solution is reached. The search for a traversable route is shown in Figure 35 and it is clear that the model cannot find a route up.
Figure 33 - Map PI-121
Figure 34 - Map PI-122
Figure 35 - Map PI-122 (Points)
Figure 36, shows where the two lines meet. This area is just northeast of the dam.
The pink dots represent the points identified during the analysis from the start point, while the blue dots are generated in reverse from the end point. The red circle
indicates the point where the line came the closest in terms of distance and elevation.
The elevation of the blue dot is 1 564.59m and that of the pink dot 1 564.09m. The two points are less than 20m apart. The direction of travel is close to 180°; this makes it an ideal point to combine the two lines up to this point and find a route with the parameters supplied.
Figure 36 - Map PI-121 and 122 intersecting
The combination of the analyses for PI-121 and PI-122 is shown in Figure 37.
Figure 37 - Map PI-121 and 122 Combination
4.3 Comparison of indicators for results
To evaluate the effectiveness of the model and to compare different routes the following indicators are compared:
4.3.1 Numerical
Table 2 shows the summarised results for some of the analyses discussed in the preceding paragraphs. The table shows the numerical results for the four lines that reached the target. The columns of the table show the PI analyses number, the line number, the colour of the line as displayed on the relevant maps, the allowed rise over distance (Maximum Slope Ratio) and the calculated maximum slope in degrees.
The input parameters for the Cost Slope Factor, Cost Distance Factor and Cost Direction Factor are also shown. The next column shows whether the calculation is done in the normal direction or in reverse.
The final four columns show the results from the calculation; distance is the total distance of the route in metres. The next column shows the total absolute change in elevation (up or down), while the next column shows the maximum slope and the last column the average slope of the line.
Figure 28 shows the results for analysis PI-97 as discussed. Four lines reached the target point and are shown in the first four rows of Table 2. The input parameters are the same for the four lines; the only difference was the starting direction of the analyses from the start point. Comparing the results shown in the table, the following is observed; distances vary between 48.8km (green line) and 65.3km (white line).
The green line has the lowest change in absolute elevation, and all four lines adhered to the maximum slope as defined with the input parameters. The average slope of the
the best choice since it is the shortest route, has the lowest absolute change in elevation and compares favourably in terms of the average slope with other lines.
Table 2 also shows the results for analysis PI 106 on the fifth line. This analysis is also shown visually in Figure 30 and has a more stringent Maximum Slope Ratio of 1:40 as an input parameter. This analysis returned the shortest route of all those compared in the table, a lower absolute change in elevation than any of the lines of PI-97 and a maximum slope within the parameters supplied with a resulting lower average slope.
The results for the combination line of PI-121 and 122 are shown on the sixth line of Table 2. Here two analyses are selectively merged in order to arrive at a solution.
The total distance of this analysis is longer than PI106 but shorter than any of PI-97.
The absolute change in elevation of 519.96m is the lowest of all the analysis. The maximum slope is within limits as specified in the input parameter and the average slope is also lower.
Table 2 Analyses Results
PI LineID Line Colour
Allowed Rise
Over
Distance Max Slope Cost Slope Factor
Cost Distance
Factor Cost Direction
Factor Reverse
Direction Start Direction Distance Abs Elev
Change Max Of
Slope Avg Abs
Slope
97 0 Grey 1 25 2.290610 600 1 1800 No 5.986 56737.96 1007.24 2.25 1.014420
97 1 Orange 1 25 2.290610 600 1 1800 No 0.656 54858.63 875.27 2.28 0.926919
97 2 White 1 25 2.290610 600 1 1800 No 1.608 65350.67 1129.58 2.27 0.999515
97 3 Green 1 25 2.290610 600 1 1800 No 2.561 48857.54 802.54 2.24 0.949870
106 0 Grey 1 40 1.432096 450 1 2400 Yes 2.844 45634.53 611.35 1.40 0.772917
121-122 0 and 1 Red 1 50 1.145763 450 1 9600 Both 5.986 and 3.797 48746.62 519.96 1.12 0.669220
4.3.2 Histogram
A histogram analysis for each generated route represents the grouped absolute slope (up or down) value on the x-axis and the number of points that fall in that value band on the y-axis. The shape of the histogram gives an indication of the characteristics of
Figure 38, 39, 40 and Figure 41 show the four histograms for the four lines generated by analysis PI97. As established in Table 2, the route generated by Line 3 produced the best results followed by the route for Line 1, represented by the green and orange lines in Figure 28. The histogram for Line 3 shows a high value in the left-most column and then a tapering off to the right. This indicates a large grouping in points on the lower slope values and fewer points as the slope gets steeper, which is more favourable for rail route selection than the one for Line 0 where high value columns are visible towards the right of the histogram. This histogram for Line 1 also tapers off from left to right, while the histogram for Line 1 is more weighted to the right.
The value of the 3rd Quartile in the statistics box on each of the figures indicates a cut-off point for 75% of the values. 75% of the values have a slope lower than this figure. (Yau, 2013). Line 3 has the lowest value in this comparison.
Figure 38 - PI-97 Line 0 Histogram
Figure 39 - PI-97 Line 1 Histogram
Figure 40 - PI-97 Line 2 Histogram
Figure 41 - PI-97 Line 3 Histogram
The histograms shown in Figure 42 and Figure 43 are more weighted towards the right of the graph when compared with the histograms for analysis PI-97. The first reason for this is the more stringent Maximum Slope Ratio parameter that causes the x-axis of the histogram graph to stretch over a smaller range of values. The second reason is that lower values for Slope Cost Factor and higher values for Cost
Direction Factor are specified in the parameters. This causes a slope within the range of slopes to be lower than the maximum allowed slope ratio, and thus has less effect on the point evaluation and selection process.
Figure 42 - PI-106 Histogram
The median of 0.73 for the slope values as shown in Figure 43 is close the maximum slope of 0.716° (ration 1:80) specified by the rail design for which no solution could be found as shown in Figure 24. This means that over 50% of the points will need artificial smoothing by cut and fill operations during construction to meet
specifications.
4.3.3 Profile
The profiles for the different lines for analysis PI-97 are shown from Figure 44 to Figure 47. The profiles help to visualise the figures presented in Table 2. Considering Figure 44, the difficulty (hump) around the 43km mark is not present in all the other lines, and therefore is probably not the optimal solution. The profile of Line 1 shown in Figure 45 seems much better than that of Line 0, but the first 15km do have a lot of ups and downs. Figure 46 is relatively uneven all the way.
Looking at the profile for Line 3 shown in Figure 47, it is evident that the profile is much smoother than the three discussed. The conclusion reached when the table was considered is confirmed by the profile; Line 3 is the best route generated by analysis PI-97.
Figure 44 - Profile PI_97 Line 0
Figure 45 - Profile PI_97 Line 1
PI_97 Line 0
Vertices
55 000 50 000
45 000 40 000
35 000 30 000
25 000 20 000
15 000 10 000
5 000 0
1 700 1 650 1 600 1 550 1 500 1 450
PI_97 Line 1
Vertices
50 000 45 000
40 000 35 000
30 000 25 000
20 000 15 000
10 000 5 000
0 1 700 1 650 1 600 1 550 1 500 1 450
Figure 46 - Profile PI_97 Line 2
Figure 47 - Profile PI_97 Line 3
Figure 48 shows the profile for the route generated with analysis PI-106. The progress is fairly even from the start and has just one significant hump and valley before the end point of the graph.
Figure 48 - Profile PI_106
As expected, the profile as shown in Figure 49, for the combination of analysis PI- 121 and 122, is the smoothest of all the analyses with no major humps or valleys
PI_97 Line 2
Vertices
60 000 55 000 50 000 45 000 40 000 35 000 30 000 25 000 20 000 15 000 10 000 5 000 0
1 700 1 650 1 600 1 550 1 500
1 450
PI_97 Line 3
Vertices
45 000 40 000
35 000 30 000
25 000 20 000
15 000 10 000
5 000 0
1 700 1 650 1 600 1 550 1 500 1 450
PI_106
Vertices
40 000 35 000
30 000 25 000
20 000 15 000
10 000 5 000
0 1 700 1 650 1 600 1 550 1 500 1 450
vertices of the lines generated, which in turn are based on the points identified by the model. The points identified are approximately 300m apart and have a smoothing effect on the topology. By comparing Figure 49 and Figure 50, the effect is visualised. The profile shown in Figure 50 is based on the line and not only the vertices. In the first 12km of the route it is evident that a lot of smoothing has taken place that will have to be compensated with cut and fill operations during
construction.
Figure 49- Profile PI_121 and 122 vertices
Figure 50- Profile PI_121 and 122 line
PI_121 and 122
Vertices
44 000 42 000 40 000 38 000 36 000 34 000 32 000 30 000 28 000 26 000 24 000 22 000 20 000 18 000 16 000 14 000 12 000 10 000 8 000 6 000 4 000 2 000 0 1 700 1 650 1 600 1 550 1 500 1 450
PI_121 and 122
Line
44 000 42 000 40 000 38 000 36 000 34 000 32 000 30 000 28 000 26 000 24 000 22 000 20 000 18 000 16 000 14 000 12 000 10 000 8 000 6 000 4 000 2 000 0 1 700 1 650 1 600 1 550 1 500 1 450
5. Conclusion
The research question posed in the beginning of this thesis stated:
“This thesis aims to contribute a model that will consider anisotropic slope and
horizontal curve constraints concurrently while developing possible routes for a new rail road development. The objective will be to find a route for a new railroad that conforms to the maximum slope parameter and a minimum turn radius parameter.”
5.1 Anisotropic Slopes
Applying limits and weights to slopes and calculating slopes based on direction has satisfied the question in terms of considering anisotropic surfaces as shown in the results in Table 2.
Figure 51 shows an extract from a route selection process, with a red dot representing a dead end, and green dots indicating part of the selected route. This route travels from East to West. From the map it is evident how the model evaluated different valleys to find a route and then backtracked once the slopes became too steep.
Considering the histogram and statistics shown in Figure 43, it is evident that the constraint for the Maximum Slope Parameter is adhered to for the route generated by the analysis.
Figure 51 - Map slope evaluation
5.2 Horizontal Curve Constraints
Being able to accept horizontal curve constraints and applying these constraints during route generation has satisfied the second criterion of the research question to adhere to the Minimum Turn Radius parameter. The model has implemented this requirement successfully as shown by Figure 52 that includes a circle with a radius of 500m superimposed over the points selected for a route as it was approaching the end point from the North.
