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ISPRS Int. J. Geo-Inf. 2023, 12, 487. https://doi.org/10.3390/ijgi12120487 www.mdpi.com/journal/ijgi
Article
Vertical vs. Horizontal Fractal Dimensions of Roads in Relation
to Relief Characteristics
Klemen Prah
1,
* and Ashton M. Shortridge
2
1
Faculty of Logistics, University of Maribor, 2000 Maribor, Slovenia
2
Department of Geography, Environment, and Spatial Sciences, Michigan State University,
East Lansing, MI 48824, USA; ashton@msu.edu
* Correspondence: klemen.prah@um.si
Abstract: This paper investigated the surface length of roads from both horizontal and vertical per-
spectives using the theory of fractal dimension of surfaces and curves. Three progressive experi-
ments were conducted. The first demonstrated the magnitude of the differences between the planar
road length and the DTM-derived surface road length and assessed its correlation with the DTM-
calculated road slope. The second investigated the road distance complexity through the fractal di-
mension in both planar and vertical dimensions. The third related the vertical with the horizontal
fractal dimension of roads across a range of distinct physiographic regions. The study contributed
theoretically by linking the planimetric complexity to vertical complexity, with clear applications
for advanced transportation studies and network analyses. The core methodology used geographic
information systems (GIS) to integrate a high resolution (1 × 1 m) digital terrain model (DTM) with
a road network layer. A novel concept, the vertical fractal dimension of roads was introduced. Both
the vertical and horizontal fractal dimensions of the roads were calculated using the box-counting
methodology. We conducted an investigation into the relationship between the two fractal dimen-
sions using fourteen study areas within four distinct physiographic regions across Slovenia. We
found that the average slope of a three-dimensional (3D) road was directly related to the length
difference between 3D and two-dimensional (2D) roads. The calculated values for the vertical fractal
dimension in the study areas were only slightly above 1, while the maximum horizontal fractal di-
mension of 1.1837 reflected the more sinuous properties of the road in plan. Variations in the vertical
and horizontal fractal dimensions of the roads varied between the different physiographic regions.
Keywords: geographic information systems; digital elevation model; road slope; road distance;
fractal dimension; box counting
1. Introduction
Roads as built features exemplify the geographic interaction between people and
their environments. Nearly every human lives along a road. Roads connect and convey
people and channel their commerce. The geographical position of any road is the conse-
quence of intentional activity through time. Its endpoints are, or were, shaped by local
and regional economic, cultural, and political factors, while deviations of its route from a
straight line are due to balancing directness with construction and traversal costs [1]. Fur-
ther, the relative importance of those factors and costs has varied substantially over time
and space, so that the geographic position of a modern road is a legacy of past conditions
and actions, as is the fitness of a road network for the efficient movement of people and
goods across a region [2].
Modeling efficient network movement is a key application domain in geographic in-
formation systems for transportation (GIS-T) [3,4]. Such models are used for traversing
networks, determining optimal routes, facility coverage areas, and many other applica-
tions [5]. In these models, the analysis relies on the link distance measurements that are
Citation: Prah, K.; Shortridge, A.M.
Vertical vs. Horizontal Fractal
Dimensions of Roads in Relation to
Relief Characteristics. ISPRS Int. J.
Geo-Inf. 2023, 12, 487.
hps://doi.org/10.3390/ijgi12120487
Academic Editors: Wolfgang Kainz
and Dev Raj Paudyal
Received: 7 September 2023
Revised: 15 November 2023
Accepted: 23 November 2023
Published: 30 November 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Swierland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Aribution (CC BY) license
(hps://creativecommons.org/license
s/by/4.0/).
ISPRS Int. J. Geo-Inf. 2023, 12, 487 2 of 22
calculated from geographic network data. Mismatches between the data-derived link dis-
tances and real-world link distances propagate into the analysis, leading to uncertainty
about all manner of real-world transportation system characteristics. There are many po-
tential sources of error in the distances calculated from road network data. One of partic-
ular interest in this paper arose from the use of a two-dimensional plane as the representa-
tive basis of the network. While 3D network data development methods have been pro-
posed e.g., [6,7], most network data products are strictly planimetric, and analysts inter-
ested in measuring 3D link distances must integrate the vertical dimension themselves.
Several studies have explored approaches to achieve this, and these approaches are sum-
marized in Table 1. While the earlier work focused on the methods for incorporating a
third dimension into network models, more recent studies generally concentrated on par-
ticular transportation applications. The approaches in all these studies depended on the
accurate measurements of 3D distances along network links.
Table 1. Selected research that incorporate the topographical variations of space to model efficient
network movement.
Author(s) Approach
de Smith, 2003 [5]
Algorithm for path finding that takes into account topo-
graphic obstacles to find a route between two points in a
two-dimensional continuous space with topographic varia-
tions.
Dubuc, 2007 [8] Approach to estimate and incorporate the impact of slope
on travel time and on transport cost calculations.
Tavares et al., 2009 [9]
GIS 3D route modeling for waste collection using fuel con-
sumption as a core criterion and considering local road
gradients.
Zhang et al., 2010 [6]
Semi-automated approach for 2D road extraction from or-
thorectified radar imagery and automated elevation assign-
ment from road segments based on an interferometric syn-
thetic aperture radar (IFSAR)-derived digital elevation
model (DEM).
Kaul et al., 2013 [7]
A filtering and lifting framework that augments a 2D spa-
tial network model with the elevation information ex-
tracted from massive aerial laser scan data to produce an
accurate 3D model.
Prah et al., 2018 [10]
Comparison of 2D and 3D GIS models for school vehicle
routing with the travel distance and travel time as core cri-
teria.
Schröder and Cabral, 2019
[11]
GIS 3D route modeling to illustrate the effects of road incli-
nation on fuel consumption and carbon dioxide (CO2)
emissions, and to optimize routes in the domain of road
freight transportation.
The fundamental line length estimation problem has aracted much interest, includ-
ing empirical and theoretical developments [12]. Richardson [13] studied the relationship
between the geographic length and scale and found a tendency for the sampling interval
to be related to the length estimate by a power law. Mandelbrot [14,15] provided a frame-
work for Richardson’s results by introducing the concept of fractal dimension, which can
be defined as an index to characterize fractal paerns or sets by quantifying their com-
plexity as the ratio of the change in detail to the change in scale [16]. In the geographical
sense, the fractal dimension characterizes the complexity of the curves and surfaces [17].
ISPRS Int. J. Geo-Inf. 2023, 12, 487 3 of 22
Since it is meaningless to talk about the length of such curves, we can measure their di-
mension or length complexity, the value of which for a one-dimensional process varies
between one (smooth and straight) and two (very complex and space filling).
The fractal theory indicates that the observed length complexity of linear features
depends on the scale at which these features are measured, i.e., the length complexity will
be greater at a finer scale. The fractal dimension at fine scales may, therefore, provide more
sophisticated estimates for road distance complexity and, consequently, more realistic
transport network analysis results.
The fractal dimension has been used in many studies as an index to describe the com-
plexity of curves and surfaces [17], such as in the study of coastlines and islands [18],
topography [19,20], urban structures [21–23], and point paern analysis [24]. It has been
used for many network studies as well. The fractal dimension also has applications in the
analysis of transportation networks, as summarized in Table 2. Many studies use the box-
counting method to analyze the fractal properties of transportation networks. Other stud-
ies used alternative fractal dimension methods to analyze the fractal properties of
transport networks. For example, Bai et al. [25] used methods based on the Hausdorff
dimension and covering depth, while Mo et al. [26] adapted the Hausdorff method and
Wang et al. [27] used length dimensions and branch dimensions.
Table 2. Existing studies that apply the fractal dimension in transportation network analysis.
Author(s) Application
Lu and Tang, 2004
[28]
Study of the relationship between the mass size of cities in the Dal-
las-Fort Worth, U.S. area and the complexity of their road systems
using a modified box-counting method.
Bai et al., 2010 [25]
Research on the spatial structures and paerns of road networks in
northern and southern Jiangsu Province, China. The methods of
the Hausdorff dimension and the covering depth were used.
Mo et al., 2015 [26]
Analysis of Shenzhen, China’s municipal road network based on
the fractal theory and GIS. The Hausdorff method was used to cal-
culate the road network coverage index, leading to suggestions for
road network optimization.
Lu at al., 2016 [29]
Research on the fractal dimensions of major road networks in large
U.S. metro areas and the associations of these values on the urban
b
uilt environment.
Wang et al., 2017 [27]
Study of the fractal characteristics of urban surface transit and
road networks in Strasbourg, using length dimension and branch
dimension methods.
Abid et al., 2021 [30]
Calculation of five types of fractal dimensions for road networks in
22 districts in the Greater Amman metropolitan area. Each calcu-
lated fractal dimension was regressed to the district’s area urban
form parameters.
Daniel at al., 2021
[31]
Study of the topological parameters of the road network in Tiru-
chirappalli, India based on the graph theory. Relationships be-
tween the fractal dimension and other topological parameters were
explored using geographically weighted regression.
Deng et al., 2023 [32]
Analysis of the road network features in nine urban municipal dis-
tricts in Harbin, China based on five fractal dimension methods,
including box counting. Each fractal dimension result was linearly
regressed with district paern indicators.
ISPRS Int. J. Geo-Inf. 2023, 12, 487 4 of 22
1.1. Motivation
All these studies relating the fractal dimension theory to transport networks (Table
2) treated the network as planar—a graph that can be drawn on a two-dimensional plane
such that no edges cross each other [33]—and without considering the vertical component.
Indeed, we did not find any papers relating the fractal dimension in the horizontal and
vertical dimensions to the network analysis. Since real-world road networks are embed-
ded in three dimensions, important relationships with the vertical fractal dimension may
have been missed, including the relationship of the road distance to the elevation change,
the relative contributions of the horizontal and vertical fractal dimensions to distance, and
the role of spatial scale on the vertical fractal dimension.
The context presented above suggests two research issues. First, the existing plani-
metric network data models and products may benefit by incorporating the third dimen-
sion for measuring the link distance in order to improve the network models and reduce
uncertainty about the real-world transport system characteristics. Second, concepts for the
vertical fractal dimension and formal methods for its measurement must be developed,
along with assessments for the relationships between the horizontal and vertical fractal
dimensions, topography, and road network properties. Addressing these issues would
contribute to a more realistic characterization of transport routes, which could contribute
to a beer assessment and prediction of the travel characteristics, fuel consumption, emis-
sions, and hazards. In addition, it could provide a stronger theoretical basis for the digital
representation of road networks.
Therefore, we pursued the following three goals in this study.
• First, to explicitly demonstrate the magnitude of the length difference between the
planar road length and the surface road length derived from a DTM, in correlation
with the road slope calculated from the DTM.
• Second, to study road distance complexity using the fractal dimension in the planar
and vertical dimensions.
• Third, to investigate the vertical and horizontal fractal dimensions of roads in differ-
ent geographical regions with distinct topographic characteristics.
1.2. Research Originality and Contributions
This research paper provides novel and meaningful contributions, mainly in three
fields. First, it introduced and implemented a new concept, the vertical fractal dimension
of roads, which characterized the vertical complexity of the road distance and was calcu-
lated based on the DTM-derived vertical road profile at different spatial scales. A new
vector-based approach for performing box counting to measure the vertical fractal dimen-
sion of roads was developed. Second, it characterized the relationship between the differ-
ence in the 2D and 3D road length and slope. Third, it identified relationships between
road complexity and the terrain characteristics. Using fine-resolution road and topo-
graphic data, both derived in a consistent manner region-wide, we identified some basic
regularities in the relationship between the vertical and horizontal fractal dimensions and
terrain in a range of physiographic regions. Geographically, we focused on Slovenia due
to its great topographic diversity and ready data availability. Despite its small size, Slove-
nia is very diverse physiographically and is, therefore, suitable for such research. The
country provided an excellent case study for evaluating our three objectives.
1.3. Organization of the Paper
The rest of the paper is organized as follows. Section 2 presents the data and the core
methods. Section 3 comprises the three experiments that pursued the three main research
goals of the study. The experiments are presented in three separate subsections. Section 4
compares this research work with other existing studies and discusses the results and lim-
itations of this work. Section 5 provides conclusions and directions for future research.
ISPRS Int. J. Geo-Inf. 2023, 12, 487 5 of 22
2. Data and Core Method
This study used two datasets covering Slovenia, the first of which was the road da-
taset [34]. It was constructed by a commercial provider through a multi-step process in
which centerline digitization was used to represent the road object as a single line. The
centerline of the road indicated the middle of the roadbed. In the centerline digitization
specification used for this data product, any point along the link could not deviate more
than 3 m perpendicular to the centerline of the road relative to its endpoints. In addition,
the links conformed to the accuracy requirements of +/−5 m for the absolute position and
+/−1 m for the relative position. However, the documentation noted that the accuracy var-
ied by country and source. Unfortunately for the Slovenian region, the accuracy only
reached the specified level for highways; other road types were not so accurate.
The second dataset was the lidar-derived national digital terrain model (DTM) with
a 1-m horizontal resolution [35]. It was made as part of the Laser Scanning of Slovenia
project with an average density for the last lidar returns of five points per square meter.
The data product, a 1 m × 1 m DTM grid, was built from raw unclassified lidar data. The
grid was based on the iterative approximation of the approximated surface to the actual
terrain. In this process, the points were first divided into a grid of cells of equal size, and
then the height of each cell was determined by its lowest contained point. In the next step,
this point was treated as the control point of the interpolation function to build the DTM.
Thin plate spline interpolation was used for the DTM interpolation [36,37].
This study benefits from the high resolution of the DTM, which enabled the use of
very fine-resolution grids for box counting, which is explained in more detail in Experi-
ment 2. This allowed for the clear identification of relief details and their effects on road
variation.
This research was organized into three general, progressive stages that corresponded
with the paper’s three goals (Figure 1). At the first stage, we demonstrated the magnitude
of the length difference between the planimetric road length and the DTM-derived surface
road length in correlation with the DTM-calculated road slope. The core method of inte-
grating the road network with DTM heights to calculate the 3D surface length of the road,
presented later in this section, is explained and demonstrated in Experiment 1, so both the
core method and Experiment 1 were intertwined during the first stage, as shown in Figure
1. For a geographic scale experiment in this stage, we chose the topographically rugged
area of Gornji Grad. During the second progressive stage of the research, we evaluated
the road distance complexity using the fractal dimension in both the planar and vertical
dimensions. During the third progressive stage of the research, we conducted a survey in
a wide range of physiographic regions to develop the relationship between the vertical
and horizontal fractal dimensions of the roads with respect to the topographic features.
The details of the methods used in each stage are presented in the section of the experi-
ments. All the analyses were performed in Esri’s ArcGIS ArcMap 10.8.1 and its 3D Analyst
extension [38]. The newer ArcGIS Pro application also contained all the necessary tools to
perform these analyses.
ISPRS Int. J. Geo-Inf. 2023, 12, 487 6 of 22
Figure 1. The research design was organized into three general, progressive stages (orange ellipses)
and associated analysis details (blue rectangles).
The road network dataset had to be integrated with the DTM to calculate the surface
road length. To do this, the vector road lines were broken into 5-m planar sections. These
were in turn divided into 1-m planar segments. This one-meter sample distance was cho-
sen to match the raster cell size. The elevations of the start and endpoints of each segment
were estimated using bilinear interpolation from the DTM with a one-meter resolution.
The three-dimensional surface lengths were calculated for the segments, and these were
summed to provide the surface road length of a 5-m road section, while the average slope
was calculated by weighting the slope of each segment by its 3D length. Both the 3D sur-
face length and slope were calculated using the Add Surface Information tool within 3D
Analyst [38]. This methodology is illustrated in Figure 2 and Table 3, where an example
of a five-meter road section [34] was overlain on the one-meter DTM surface and its ele-
vations were interpolated. The vertical profile shown in Figure 2 highlights how the sec-
tion was divided into five segments (s1-5), each with its own surface length and slope val-
ues. The surface length of the whole segment was 5.58 m, and the average slope was 20.32
degrees.
ISPRS Int. J. Geo-Inf. 2023, 12, 487 7 of 22
Figure 2. Vertical profile of a five-meter road section overlain on a one-meter cell size raster DTM
processed using ArcGIS, according to the bilinear interpolation method, to obtain the Z, slope, and
surface length information.
Table 3. Partial results of the surface properties for a five-meter road section to obtain the surface
length and average slope information for the whole section.
Segment Slope (Degrees)
Surface Length
(Meters)
Weighted Slope
(Degrees)
s1 0.29 1.00 0.29
s2 22.05 1.08 23.81
s3 48.24 1.50 72.36
s4 2.86 1.00 2.86
s5 2.30 1.00 2.30
An alternative approach, which ignored the implications of the fractal dimension,
extracted only the start and endpoints of each five-meter section and calculated its surface
length and slope using trigonometry. Applying this approach to the example shown in
Figure 2 resulted in a surface length of 5.05 m and a slope of 7.18 degrees, which was much
lower than the values calculated using our methodology. This comparison highlights the
importance of scale in calculating the surface length.
3. Experiments
In the following section, we present the three aforementioned experiments. The first
experiment demonstrates the magnitude of the length difference between the planar road
length and 3D surface road length, in correlation with the road slope. The second experi-
ment investigates the road distance complexity using the fractal dimension in the planar
and vertical dimensions. The novel methodology of vertical dimension box counting is
demonstrated. The third experiment investigates and evaluates the relationship between
the vertical and plan fractal dimensions in a wide range of topographic conditions.
3.1. Experiment 1: Length Difference between the Planar and Surface Road Lengths, and their
Relationship with the Slope
The entire road network of the topographically rugged municipality of Gornji Grad
in Slovenia was evaluated in order to determine the length difference between the planar
road length and the DTM-derived surface road length, and the relationship with the slope
of the roads calculated by the DTM. The municipality is situated in the Alpine macro-
region and submacro-region of the Alpine high mountains [39]. Gornji Grad is located on
the flanks of large mountains and is characterized by three relief units: a small basin, a
karst and forested plateau that reaches approximately 1130 m above the boom of the
basin, and forested mountains that are slightly higher than the karstic plateau. The mean
slope in the municipality was 23.9 degrees, calculated from a one-meter cell size raster
covering the entire region. The diverse topography of this municipality made it particu-
larly suitable for this experiment.
The road network was split into 63,261 five-meter road sections, the surface length
was calculated according to the previously laid out methodology, and for each section,
the difference between the surface length and planar length was calculated. A polynomial
regression analysis was performed to identify the relationship between the length differ-
ence and the average slope for all the road sections.
The length differences of the road sections increased with their average slope (Figure
3). The polynomial regression analysis with an R2 of 0.983 confirmed a strong and increas-
ingly positive relationship between the variables. It could be noticed how the length dif-
ference, with quite sharp upper and lower bounds, arced upward very smoothly, approx-
imately to an average slope of 45 degrees. Above this slope, however, the model arced
ISPRS Int. J. Geo-Inf. 2023, 12, 487 8 of 22
more intensely and the observations varied more widely. In general, the variance in the
length difference for the road sections with a similar average slope increased with the
increasing average slope. The interquartile range was only 0.01 for the sections with aver-
age slopes less than five degrees and increased by approx. 0.05 for each average slope class
up to 45 degrees. For the higher classes, the growth in the interquartile range was greater.
Figure 3. Relationship between the length difference and average slope of the five-meter road sec-
tions in the municipality of Gornji Grad in Slovenia.
Experiment 1 confirmed a strong, non-linear relationship between the slope and sur-
face road length across a wide range of gradients. The differences in the planar and surface
length became substantial—in the order of 20 percent—at slopes of 30 degrees or greater.
Furthermore, this finding sets the stage for further experiments that directly investigate
the role of the fractal dimension in road distance complexity.
3.2. Experiment 2: Road Distance Complexity using the Fractal Dimension
For the second experiment, we developed and described a method for calculating the
fractal dimension in the vertical dimension using GIS tools. As pointed out in the intro-
duction, the vertical fractal dimension of roads presents the vertical complexity of the road
distance and is calculated based on the vertical road profile derived from the DTM. The
concept required analysis at various scales.
1. The ArcGIS Interpolate Shape tool [38] was used to convert a 2D road line into a 3D
road line by interpolating the Z-values from a raster surface of a one-meter cell size.
Similar to the previous experiment, the interpolation method was bilinear, and the
sample distance was equal to the size of the raster cell.
2. A vertical profile of the road section was created using the tool Create Profile Graph
[38], where the horizontal axis (distance) and vertical axis (elevation) had the same
scale, 1:1, in meters. The profile was exported as a table. This table was re-imported
as X–Y data into ArcGIS ArcMap, resulting in a planar vector representation of the
road section profile. This meant that the x-axis was the distance along the road sec-
tion, while the y-axis was the elevation of the section.
3. Box counting, a common approach for calculating the fractal dimension [40,41], was
applied to the profile as follows. A sequence of M square vector meshes was created,
each covering a profile. The initial mesh had a square side length of ƞ1, while each
ISPRS Int. J. Geo-Inf. 2023, 12, 487 9 of 22
subsequent mesh had a finer side length ƞi. For each mesh, the number of squares Ni
that included part of the profile was counted.
The relation between ƞi and the fractal dimension D is as follows.
log(N(ƞi)) = a − D log(ƞi)
Given the M measures, the values for a and D could be estimated by fiing a regres-
sion model between the independent variable log(ƞi) and the dependent variable
log(N(ƞi)), where i = 1, …, M. D was given by the absolute value of the regression line slope
and was a parameter of particular importance in this study. In order to obtain reliable
results and a realistic estimate of D [42], we used a large range of square side lengths,
where each subsequent value was twice the previous one: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512,
1024, 2048, 4096, and 8192. The maximum square side length from the presented set of
values was determined for each study site individually, namely with the first occurrence
of a single-row or single-column mesh in the output. The minimum square side length
was one meter, unless the size and the shape of the line caused the output grid file to be
too large. In this case, the minimum square side length was two meters.
In this experiment, we again focused on the municipality of Gornji Grad, due to its
variable terrain. We selected a road section running from the lowest elevation of the road
in the region at 395 m to the highest elevation of the road at 1444 m. Its planar length was
15.88 km. Using the procedure described above, we calculated the vertical fractal dimen-
sion D1 for this road section. Eleven mesh side lengths ranging from 2 to 2048 m were
developed for the regression model. Figure 4 illustrates the vertical profile of this road
section together with a hundred-meter (planar length) detail of this profile. On a small
scale, the road appeared to maintain a consistent rate of rise almost throughout the profile.
However, it was apparent from the inset that the small scale concealed significant vertical
variation along the entire profile.
Figure 4. Vertical profile of the sample road section. A one-hundred-meter-long detail of the road
illustrated its vertical roughness.
We used a similar procedure to calculate the horizontal fractal dimension of the road
(D2), as we suspected that a relationship between D2 and D1 depended on the surface char-
acteristics. The horizontal fractal dimension referred to the distance complexity in the 2D
projection of the road. Ten mesh side lengths ranging from 16 to 8192 m were developed
for the regression model. Figure 5 illustrates the horizontal variation of this road section
from a top-down map view for four distinct mesh sizes, illustrating how the squares over-
lying the road were identified. It was also visually apparent that this road section was
substantially more variable when viewed as top-down than as a vertical profile. The road
climbed from the lowest point in the valley in the northeast to the highest point on the
ridge in the south. In doing so, it adapted to the surface, so that the path was longer and
sinuous.
ISPRS Int. J. Geo-Inf. 2023, 12, 487 10 of 22
Figure 5. Procedure of box counting to obtain D
2
, the horizontal fractal dimension of a road in the
Gornji Grad region. Ten different square side lengths were calculated, of which four are presented
to illustrate the approach: (a) 1024 m, (b) 256 m, (c) 128 m, and (d) 16 m. The last example (d) presents
in detail a 128-by-128 m area. In all the examples, the squares that intersect with the road are colored
light blue. An elevation map provides context.
We introduced certain differences regarding the minimum and maximum square
side lengths in the box-counting procedure between the vertical and horizontal fractal di-
mensions. Unlike the vertical fractal dimension, where it made sense—due to the very fine
DTM model—to persist down to the smallest square side length of one or two meters, it
did not make sense in the horizontal fractal dimension. Namely, the roads were neces-
sarily smooth features when viewed as top-down at a fine resolution. In addition, the road
network dataset used in this study had shortcomings in its accuracy, as indicated in the
data section. Figure 5d shows that the shortest straight segment of the local road measured
approximately 20 m. For this reason, we used the square side length of 16 m as the mini-
mum value in the box-counting procedure of the road in planar.
By fiing a regression line to all the paired log(ƞ):log(N) values across the different
mesh sizes, D
1
and D
2
were calculated (Figure 6). Both regressions provided an excellent
fit with R
2
values greater than 0.997. The vertical fractal dimension D
1
was 1.0089. Since
this value was only slightly above 1, the road profile visually and in length somewhat
resembled a straight line (Figure 4). The horizontal fractal dimension of this road, also
ISPRS Int. J. Geo-Inf. 2023, 12, 487 11 of 22
shown in Figure 6, was 1.1368. As shown in Figure 5, the road appeared more sinuous in
planar than in profile, leading to a larger horizontal fractal dimension.
Figure 6. Vertical and horizontal fractal dimensions for the Gornji Grad road section. The red square
features represent the vertical profile, while the round dark blue features represent the horizontal
profile. The dots indicate the paired log(ƞ):log(N) values that were empirically derived at different
mesh side lengths. The regression model lines are depicted for each set of points, and each regres-
sion equation with parameter estimates and R2 are reported. The vertical and horizontal fractal di-
mensions are given by the absolute value of the line slope in bold.
Figure 6 shows how the eight middle cell side lengths ranging from 16 m (log(ƞ) =
−1.2) to 2048 m (log(ƞ) = −3.3) were common to the box-counting procedure in the vertical
and the horizontal calculation of the fractal dimension. The results only for the log(ƞ) of
the longest cell size length values (8192 m and 4096 m) are shown in the lower left corner,
which were only meaningful in the calculation of the horizontal fractal dimension. The
results for only the log(ƞ) of the shortest cell size length values (8 m, 4 m, and 2 m)) are
shown in the upper right corner, which were meaningful in the calculation of the vertical
fractal dimension.
3.3. Experiment 3: Vertical vs. Horizontal Fractal Dimension of Roads
This experiment expanded on the second one to characterize the relationship be-
tween the vertical and horizontal fractal dimensions of the roads more fully in a wide
range of topographic domains. Fourteen study areas (Figure 7) were selected from all four
Slovenian physiographic macro-regions [39]: Pannonian, Dinaric, Mediterranean, and Al-
pine. Further, the study areas were drawn from four main Slovenian relief units [43]:
plains, low hills, high hills (hereafter “hills”), and mountains.
ISPRS Int. J. Geo-Inf. 2023, 12, 487 12 of 22
Figure 7. Fourteen study areas, represented by numbered dark blue squares, spanning all four Slo-
venian macro-regions, including plains, low hills, high hills, and mountains. Source: Perko 1998,
ARSO, GURS.
The Pannonian macro-region (Figure 7), a densely populated and intensively culti-
vated area, is divided into plains and low hills. Two study areas were selected from the
plains, one along the Mura River (site 1) and another along the Drava River (site 2). Two
low hilly areas, both between the Drava and Mura rivers (sites 3 and 4), were also selected
from this region. Towards the southwest, the Pannonian macro-region transitions to the
Dinaric macro-region. Three plateau-like study areas were selected there. The first was a
low hilly site that was divided into two parts by the Krka River valley (site 5). The next
two areas were in hills, the first of which was more dissected (site 6), while the second
occupied a high plateau (site 7).
The Mediterranean macro-region was divided into flysch low hills and karst plat-
eaus. This region’s first site (site 8) was dissected by a dense network of streams and rivers,
while the second area (site 9) stretched above the Vipava valley. The Alpine macro-region
was located in northern Slovenia. Two study areas were selected from the high mountains
(10 and 11), which were dissected by deep, glacially shaped valleys. On the margins of the
high mountains, the next area (site 12) distinctly featured karstic and forested plateaus,
namely Gornji Grad, known from the previous two experiments. The last two study areas
were Alpine hills and were located in the eastern part of the Alpine macro-region (sites 13
and 14).
A road section was selected that fit within each nine-by-nine km site. Preference was
given to the higher order road categories, i.e., larger roads while excluding motorways or
highways. This was because the laer took less account of relief features and were routed
more directly through tunnels and over viaducts. The intention was to select roads repre-
sentative of these distinct regions that were not subject to substantial landscape engineer-
ing. Once the road sections were selected, their vertical and horizontal fractal dimensions
were calculated using the procedures laid out in Experiment 2.
ISPRS Int. J. Geo-Inf. 2023, 12, 487 13 of 22
The results for all fourteen study areas are presented in Table 4 and in Figure 8, where
the horizontal axis depicts the horizontal fractal dimension D2 and the vertical axis the
vertical fractal dimension D1. Three important characteristics of this relationship were ap-
parent. First, the D2 values were generally larger in magnitude than D1, suggesting that
the magnitude of D1 was limited along the roads. Second, the scaerplot of the observa-
tions showed a positive correlation between D1 and D2. The sites with a low vertical fractal
dimension also tended to have a lower horizontal fractal dimension. A linear regression
analysis was applied to quantify the strength of this relationship between the vertical and
horizontal fractal dimensions. The model’s slope was positive, while the R2 was 0.38 (Fig-
ure 8).
Table 4. Properties of each study site: site #, relief unit, elevation range of the road section, average
slope of the road section, and horizontal and vertical D and R2.
Site # Relief Unit
Elevation
Range of the
Road Section
(Meters)
Average Slope
of the Road
Section
(Degrees)
Vertical Fractal Dimension
of the Road Section
Horizontal Fractal Dimension of
the Road Section
D1 R
2 D2 R
2
1 Plains 11.02 0.58 1.0033 1.0000 0.9913 0.9995
2 Plains 17.19 1.25 0.9990 0.9999 1.0027 0.9999
3 Low hills 140.30 3.77 1.0096 1.0000 1.0748 0.9979
4 Low hills 149.54 3.68 1.0042 1.0000 1.0387 0.9997
5 Low hills 446.14 4.61 1.0093 1.0000 1.0468 0.9993
6 Hills 922.91 12.98 1.0182 0.9999 1.1837 0.9978
7 Hills 178.82 7.62 1.0158 1.0000 1.0880 0.9978
8 Low hills 319.26 3.23 1.0068 1.0000 1.0155 0.9992
9 Low hills 93.93 2.14 1.0040 1.0000 1.0486 0.9974
10 Mountains 970.55 7.38 1.0102 1.0000 1.1774 0.9986
11 Mountains 694.84 5.51 1.0028 1.0000 1.1152 0.9977
12 Hills 1063.39 9.08 1.0089 0.9999 1.1368 0.9976
13 Hills 598.38 9.60 1.0158 0.9999 1.0812 0.9994
14 Hills 1038.71 8.39 1.0042 1.0000 1.0667 0.9991
Figure 8. Vertical and horizontal fractal dimensions of the road sections for all fourteen study sites
(site numbers). The linear regression line and expression quantified the strength of the relationship
between these dimensions.
Third, the magnitudes of both fractal dimensions were strongly affected by the to-
pography of the study sites. The lower left portion featured road sections in plains, while
ISPRS Int. J. Geo-Inf. 2023, 12, 487 14 of 22
the sites in low hills had somewhat higher values in both dimensions. The vertical fractal
dimensions for the low hills sites ranged from 1.0040 (site 9) to 1.0096 (site 3), while their
horizontal fractal dimensions ranged from 1.0155 (site 8) to 1.0748 (site 3). The five hilly
study sites were arranged from the more central part of the graph to its upper right, with
the widest variation in both fractal dimensions of all the topographic classes (for D1, 1.0042
at site 14 to 1.0182 at site 6; for D2, 1.0667 at site 14 to 1.1837 at site 6). In comparison to the
hilly sites, the road sections in the two mountainous sites had relatively low vertical fractal
dimensions (1.0028 and 1.0102). The mountainous study site 11 had a vertical fractal di-
mension smaller than all the hilly and low hilly study sites and even one study site in the
plains (site 1). The horizontal fractal dimension of the mountainous study sites ranged
from 1.152 to 1.174. Evidently, mountainous terrain presented physical constraints that
resulted in roads trading higher horizontal fractal dimensions for lower vertical fractal
dimensions, a tradeoff that was not required in non-mountainous relief units.
The sites falling in the same relief unit category were relatively concentrated, as
shown in Figure 8, which occupied distinct portions of the data space. Taken together, the
vertical and horizontal fractal dimensions, therefore, appeared to capture the manner in
which topography affected the path by which the roads traversed the landscape.
4. Discussion
The comparison of our work to the prior studies presented in Table 2, which applied
the fractal dimension in the transportation network analysis, found both commonalities
and differences across a range of topics (Table 5). Six of these studies used GIS as the soft-
ware environment to perform at least a portion of the research. In addition to our research,
half of the existing studies used the box-counting method. Other fractal dimension meth-
ods have also been applied, as listed in Table 5. Two studies compared five different meth-
ods for calculating the fractal dimension.
Table 5. Comparison of our research with the existing studies presented in Table 2, which applied
the fractal dimension in the transportation network analysis.
Our Research Existing Research from Table 2
Applying
GIS GIS GIS [25,26,29–32]
Fractal di-
mension
method
Novel concept of the vertical
fractal dimension of roads
(vector-based approach for
b
ox-counting method imple-
mented in ArcGIS software)
- Box counting [28,30–32]
- Hausdorff dimension [25,26]
- Covering depth [25]
- Length dimension [27]
- Branch dimension [27]
- Geometric fractal dimension [29]
- Structural fractal dimension [29]
- Comparison between five fractal dimen-
sions (box counting, perimeter area, in-
formation, mass, and ruler dimension)
[30,32]
Definition of
network Non-urban roads
- Urban transportation network [27–30]
- Urban road network [27,30,32]
- Urban surface transit network (public
transportation) [27]
Software
ArcGIS software with the
Add Surface Information tool
and Interpolate Shape tool
- ArcGIS software [25,26,29,31]
- ArcGIS Network Analyst extension [31]
- BENOIT software [30,32]
ISPRS Int. J. Geo-Inf. 2023, 12, 487 15 of 22
Spatial data
coverage
- Vector data layers
(NAVTEQ streets, physio-
graphic regions)
- 1 × 1 m digital terrain
model to derive 3D sur-
face road length
- Vector data layers [25,26,30,32]
- OpenStreetMap vector data [30,32]
- Road network vectorized from raster
image [31]
- Road network rasterized from shapefile
[30,31]
Time situa-
tion Current Ten-year span [25]
Our study was the only one to cover non-urban roads, while other studies strictly
focused on the urban transportation network. Three studies analyzed urban road net-
works, and one dealt with the urban surface public transportation network.
Surprisingly, in addition to our research, four other studies applied the proprietary
software ArcGIS, since there are many other proprietary and open-source GIS software
available. One study also used the ArcGIS Network Analyst extension. Only two studies
highlighted the use of the computer program BENOIT, which is specialized for calculating
the fractal dimension and for which it is necessary to prepare the transportation network
layer as a raster image.
Our fractal dimension calculation was vector-based, and was the only study that used
a DTM, as it was the only one dealing with the vertical dimension. Other research used
variants of the vector and raster transportation network models, with transformations be-
ing necessary in some cases. Only two studies used OpenStreetMap vector data, which
we believe may represent a more up-to-date and accurate database. Only one study com-
pared the fractal dimensions between two time periods at ten years apart.
Experiment 1 demonstrated that the average slope of a 3D road section was directly
related to the length difference between the 3D and 2D for that section. In this experiment,
the length difference arced upward very smoothly to a slope of about 45 degrees. Higher
up, the relationship varied more widely, perhaps in part due to the smaller number of
road sections with such steep gradients. Even in the sections with moderate average
slopes, the length differences varied by half a meter (10% of the total planar length) or
more. These findings indicated that the use of plan models in regions with high gradients
systematically underestimated the actual road distance, with errors propagating to any
subsequent analysis using those distances. If analysts want to characterize roads in such
environments as realistically as possible, their 3D lengths must be taken into account. The
high quality of the regression model suggests that the slope of a road section may be used
to accurately estimate its 3D length.
A limitation of this work was the positional error in the road network data. This error
propagated into the slope information obtained along the network. We obtained a number
of road slope values that we assumed were exaggerated or unreasonable. We mainly were
concerned with values above 20 degrees, although we believe that some of them were also
possible, since Gornji Grad is a very hilly and topographically diverse region. However,
in a number of places that we manually checked, the road line was offset horizontally by
up to 13 m from the road surface presented in the DTM. Instead of following the actual
road alignment, the vector road line crossed steeper terrain near the actual road align-
ment. As a result, the slope of the road line was too high and did not correspond to the
slope at the actual position of the road.
Experiment 2 investigated the fractal dimension directly. We believe this was the first
study to separately calculate the vertical fractal dimension and contrast it to the horizontal
fractal dimension. Furthermore, the use of the vector-based method outlined in this ex-
periment made the results in the vertical direction readily comparable to those of the hor-
izontal. In both cases, the use of box counts at different scales resulted in pronounced
linear relationships, making the regression-based estimation of the fractal dimension reli-
able. The vertical fractal dimension was very close to 1 in this single case study, while the
ISPRS Int. J. Geo-Inf. 2023, 12, 487 16 of 22
horizontal fractal dimension was higher, closer to 1.1. This was understandable from a
visual assessment of the entire vertical profile of the road section shown in Figure 4, which
appeared almost like a straight line. The vertical variability in this road profile only be-
came apparent at the larger spatial scale in the inset of this figure. In contrast, the horizon-
tal variation of the section was visibly larger, as shown in Figure 5. Nevertheless, the hor-
izontal fractal dimension of this road was still small compared to Koch’s snowflake, with
a fractal dimension of 1.2618 [16], indicating that the road was relatively smooth compared
to the much more complex snowflake. In this study, we were able to show that roads are
smooth features when viewed at high resolution, which was not the case for a snowflake
or a coastline.
We note here that our results were, to some extent, due to the spatial scale of our road
measurements. A more detailed data scale, either in the planar or for the vertical, could
theoretically return a higher variability. However, it was important to consider the limits
of fine-scale variability related to the passage of vehicles along a road, namely (1) substan-
tial vertical changes at scales finer than the circumference of automobile wheels (about
two meters); (2) a vertical variation greater than a vehicle’s ground clearance at scales finer
than the vehicle’s length; and (3) planar sinuosity greater than a vehicle’s turn radius.
These limits on scale variation, with respect to vehicles, meant that roads below the 3–5 m
scale must be relatively smooth. Large-scale, one-meter DTMs such those used in this
study, therefore, appeared to be sufficient to reliably estimate the fractal dimension at
spatial scales relevant to transportation applications.
A related finding of this work was that the spatial scale of analysis was different for
the horizontal and vertical fractal dimensions of the roads. The advantage of using a DTM
with a high resolution 1 × 1 m was fully expressed only in the vertical dimension. This was
not the case with the horizontal fractal dimension, since the roads were planimetrically
smooth features at resolutions below a few meters; a smoothness which was further com-
pounded by the generalization of our road data. The box-counting calculation with a
square side length of less than 16 m was, therefore, not meaningful on this road dataset.
A reasonable minimum square side length would be much higher for expressways and
highways due to the higher speeds and smoothness these road categories require.
Experiment 3 extended the analysis by investigating the variation of the vertical and
horizontal fractal dimensions across the different geographical regions and relief types. In
general, the positions of the study sites shown in Figure 8 were arranged along the diag-
onal from the lower left to the upper right corner, reflecting a sequence of plains, low hills,
hills, and mountains. In terms of the horizontal fractal dimension, the mountain sites were
most similar to the hill sites, but generally had a lower vertical fractal dimension. We spec-
ulate below why this might be.
Figure 9 shows a top-down view on 6-by-6 km excerpts of the four study areas (plain,
low hills, hills, and mountainous terrain) presented in Figure 7 and their road sections
overlaid on a shaded relief from the DTM used in the study. The differences in the surface
variations are visually apparent. Site 1 on the plain (Figures 9a and 7) looked nearly flat.
Geologically, its former valleys were filled with gravel and sand by the Mura River at the
end of the past glacial period. It is a traditional agricultural landscape dominated by reg-
ularly shaped fields. As a result, the road was not very winding, and its deviations were
more due to the two clustered selements it serves than to topographic obstacles. The
vertical profile of the same road section was a straight horizontal line (Figure 10a).
ISPRS Int. J. Geo-Inf. 2023, 12, 487 17 of 22
Figure 9. 6-by-6 km excerpts from selected study sites with roads (red lines): (a) plain (site 1); (b)
low hills (site 3); (c) hills (site 6); and (d) mountains (site 10). The shaded relief was generated from
the DTM used in the study. The streams and rivers (blue lines) provide context. Source: ARSO,
GURS, NAVTEQ.
ISPRS Int. J. Geo-Inf. 2023, 12, 487 18 of 22
Figure 10. Vertical profiles of the road sections presented in Figure 9: (a) plains of study area 1; (b)
low hills of study area 3; (c) hills of study area 6; and (d) mountains of study area 10. In each case,
the lowest grid line represents the lowest elevation of the road section (also labeled on the axis). For
beer representation, the ratio between the horizontal and vertical axes was 1:2.
In the case of site 3, a low hills site (Figures 9b and 10b), the terrain consisted of low
ridges running from the northwest to southeast separated by straight-running valleys. The
selements in this agricultural landscape were dispersed, and the homes were arranged
in long rows along the ridges. The studied road ran along one ridge and crossed a small
valley in the upper part of the site (Figure 9b). In this planar view, the road adapted to the
topography and was more winding than that on the plains. This resulted in a higher value
for the horizontal fractal dimension. Due to the characteristics of the terrain, the vertical
profile was visually rougher than the plains site (Figure 10b). On the right side of this
graph, we can see a noticeable depression that represents the crossing of the valley.
The hilly terrain of site 6 looked quite different from the previous two (Figure 9c). Its
location was at the transition between the Alpine and the Dinaric regions, and the area
represents a landscape of undulating plateaus, into which rivers and streams have cut
deep valleys and ravines. Individual rounded peaks rise above river valleys and plateaus.
There was lile flat terrain except on the higher plateaus. The landscape was sparsely pop-
ulated [44]. The selected road looked very winding from the top-down view (Figure 9c)
as it crossed this terrain. The lower left road crossed a relatively flat plateau, then in the
middle descended into and crossed a valley that ran along a tectonic fault in a northwest
to southeast direction. The vertical profile of the road was the roughest of all four cases
(Figure 10c). On the left, the course of the road across the flat plateau was clearly visible.
ISPRS Int. J. Geo-Inf. 2023, 12, 487 19 of 22
It then crossed the valley in the center and rose to the right. Due to the pronounced vari-
ation of both the horizontal and vertical profiles, the relatively high values for both fractal
dimensions were not surprising.
The mountain area of site 10 (Figure 9d) was characterized by exceptional pictur-
esqueness and landscape diversity. The surface was relatively young, as evidenced by
deeply incised valleys. Above them rose narrow mountain ranges and ridges with pointed
peaks. The slopes were steep and the valley sides acted like walls. Selements were sparse
and uneven, and an important economic activity is tourism [44]. The road generally ran
along the boom of the alpine valley in the lower half of the site (Figure 9d). In the central
part, it began to climb intensively towards the high pass. First, it climbed in a serpentine
manner along the southern slope of the mountain, then crossed the ridge and, with a few
switchbacks, climbed up the gentler opposite slope towards the pass. The serpentine part
of the road was the most variable part of any road in this illustration (Figure 9), and indeed
this road almost had the highest horizontal fractal dimension (Table 4 and Figure 8). The
vertical profile of the road could be divided into two main parts (Figure 10d). The left side
slopes more gently but generally rose slightly as the road ran along the valley floor. The
right side climbed steadily at an almost constant rate, even in the serpentine area. Com-
pared to the hilly vertical profile (Figure 10c), the mountainous one was less rough, result-
ing in a lower vertical fractal dimension. It was apparent that road builders in mountain-
ous areas with major vertical obstacles may trade directness in the planar for reductions
in the gradient. The fractal consequence of this was a larger horizontal fractal dimension
and a relatively smaller vertical fractal dimension compared to that seen on roads in the
hills.
Several additional limitations and future directions of this study are worth noting.
First, we strongly recommend using the most accurate possible road dataset. The posi-
tional error in the road data can affect various calculations, such as the elevation difference
and slope along the road, horizontal road distance, and indirectly, the length difference
between planar and surface road lengths.
Second, although there are only four macro-regions in Slovenia, there are many sub-
macro-regions and mesoregions, which differ significantly from each other in terms of
their geological and topographical character. Experiment 3, therefore, by using 14 study
areas, i.e., two to five study areas for each macro-region, comprehensively evaluated the
diversity of the influences of the Slovenian landscape on its roads, but only to a certain
extent. Since the preparation and analysis of the data for each study area was time con-
suming, an important next step would be the automation of the important phases of the
analysis, such as the selection of the individual sites, the selection of a road section within
a site, and the calculation of both fractal dimensions.
Third, we encountered a limitation with the ArcGIS ArcMap software, which allowed
for a maximum size of 2 GB for the fishnet output file. For large areas, this cap limited the
reduction in the cell sides to the desired small sizes. Fourth, it would be interesting to
consider both fractal dimensions of the roads in relation to more specific relief features,
such as the relief coefficient or surface undulation [45]. Fifth, it would be interesting to
consider the entire road network within each study area instead of one typical road sec-
tion. This would, of course, require additional computer resources and time.
Finally, the estimates of the fractal dimension of the roads, both vertical and horizon-
tal, were quite sensitive for several reasons. 1) Especially in the case of the vertical dimen-
sion, the roads may have had small degrees of curvature. 2) The size of the study area and
the resulting road coverage affected the estimates. 3) The chosen square side lengths for
the box-counting process were known to affect these estimates as well. This concern
prompted us to define a clear, uniform, and sophisticated methodology that could account
for topographic diversity.
We conclude this section by considering a few potential applications for the vertical
fractal dimension of roads. First, the method is clearly useful for improving the estimates
of the actual travel distances, as widely used planimetric techniques are not capable of
ISPRS Int. J. Geo-Inf. 2023, 12, 487 20 of 22
detecting vertical distances, resulting in a persistent underestimation of route distances.
Length underestimation is likely higher on high local relief non-vehicle routes, such as
trails, and in other contexts like terrain profiles, where the vertical fractal dimension may
also be useful. Second, the vertical fractal dimension provides insight into the tradeoffs
involved in road placement, slope, and roadbed engineering in different types of terrain.
Finally, its relationship to the slope and curvature deserves more study, as it may be a
useful new terrain “derivative” for a range of landscape analyses.
5. Conclusions
In this article, we addressed a series of questions about road length and its relation-
ship to the horizontal and vertical components. We conceptualized the road length com-
plexity in an innovative way, in connection with fractal theory. We introduced the term
vertical fractal dimension of roads and developed a new vector-based approach for per-
forming box counting to measure the vertical fractal dimension of roads.
The important relationships with the vertical fractal dimension may have been
missed in former studies, including the relationship of the road distance to elevation
change, the relative contributions of the horizontal and vertical fractal dimensions to dis-
tance, and the role of spatial scale on the vertical fractal dimension. Therefore, concepts
for the vertical fractal dimension and formal methods to measure it had to be developed,
along with assessments for the relationships between both the horizontal and fractal di-
mensions, topography, and road network properties.
We expect the horizontal fractal behavior of roads to persist down to approximately
16 m for minor roads. With 16 m length sides, we generally met the scale at which vehicles
interface with them. For coastlines, which were one of the first objects of fractal analysis
e.g., [14], “the closer you look the more you see” applies to finer and finer detail until the
definition of coastline breaks down. However, this was not the case for roads, which are
necessarily smooth features when viewed at a fine resolution. We expect the fractal be-
havior of roads to persist to some limiting fine scale, which depends on the category of
the road. This finest scale might be the shortest on local roads, where the shortest straight
road sections in our road dataset were no longer than about 13 m. On major roads, where
individual straight road sections can reach up to approx. 120 m long, we would observe a
fractal behavior ending at these length scales. Fractal persistence at fine scales would be
the lowest on highways, where rather long smooth road sections can reach a maximum
length of approx. 1500 m. Nevertheless, future research would be very valuable to inves-
tigate the fractal behavior of roads of different categories, e.g., local roads and main roads,
both in the horizontal and vertical dimensions. In our experience, future work should use
the most up-to-date and accurate road database, for example OpenStreetMap.
A study of the relationship between the vertical and horizontal fractal dimensions of
roads was conducted according to the type of relief. The use of topographic regions in this
study suggested that the results could be expected to apply to similar topographic regions
elsewhere. However, certain caveats presented above are necessary.
In any case, this work and its results represent a useful contribution, both theoreti-
cally for the separation of the planar and vertical fractal dimensions and its calculation, as
well as for its practical value in the field of transport modeling and planning. These meth-
ods can allow analysts to beer estimate road distance and complexity, and to link road
undulations to fuel consumption, tire wear, and emissions. The implications for more ef-
ficient and environmentally sustainable transport systems should be of interest to plan-
ners, policy makers, and the public.
Author Contributions: Conceptualization, Ashton M. Shortridge and Klemen Prah; methodology,
Klemen Prah and Ashton M. Shortridge; validation, Klemen Prah and Ashton M. Shortridge; formal
analysis, Klemen Prah; investigation, Klemen Prah and Ashton M. Shortridge; resources, Klemen
Prah and Ashton M. Shortridge; data curation, Klemen Prah; writing—original draft preparation,
Klemen Prah and Ashton M. Shortridge; writing—review and editing, Klemen Prah and Ashton M.
ISPRS Int. J. Geo-Inf. 2023, 12, 487 21 of 22
Shortridge; visualization, Klemen Prah. All authors have read and agreed to the published version
of the manuscript.
Funding: This research received no external funding.
Data Availability Statement: The data presented in this study are available upon request from the
corresponding authors. The data are not publicly available due to privacy.
Acknowledgments: The research was possible due to the cooperation between the Faculty of Logis-
tics of the University of Maribor and the Department of Geography, Environment and Spatial Sci-
ences of Michigan State University. The establishment of cooperation dates back to 2018 and began
between the Maribor Faculty of Logistics and MSU RS&GIS.
Conflicts of Interest: The authors declare no conflict of interest.
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