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Advances in Transportation Studies an international Journal Section B 34 (2014)

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Safety evaluation: a new operating speed model for two-lane, undivided rural roads

F. G. Praticò1 M. Giunta2

1DIEES Department University Mediterranea of Reggio Calabria

Via Graziella, Feo di Vito 89100 Reggio Calabria email: [email protected]

2DICEAM Department University Mediterranea of Reggio Calabria Via Graziella, Feo di Vito 89100 Reggio Calabria

email: [email protected] subm. 9th January 2014 approv. after rev. 13rd May 2014

Abstract

The main objective of this paper is to examine the influence of road alignment on vehicle operating speed. The authors propose and validate a new operating speed model in which the geometric features of an alignment and the lengths of its elements (curve, tangent) are considered. The results demonstrate that grade, radius, and length affect driver perception and operating speeds and are relevant to safety analysis methods. In contrast, the geometric features of previous and oncoming elements, although essential for an accurate analysis of the single locations and sections, had a negligible statistical influence on operating speed. The paper provides a set of analytical tools for predicting driver behaviour and speed in a given section based on the alignment features (previous and oncoming elements included). Furthermore, a framework for ranking and screening the factors that affect speed profiles is presented. Finally, the model can be used in safety evaluation of horizontal curves and tangents on rural undivided roads. Keywords – driver behaviour, operating speed, road alignment, safety

1. Introduction and objectives of the research

Drivers manage the road environment as it varies in complexity by making adjustments to their behaviour based on a number of parameters (workload, posted limits, perceived risk, road type, traffic density, vehicle types, etc.) [42]. Cafiso and La Cava [9], analysed seven parameters as driving performance indicators. The maximum driving speed differential between two successive elements and between the average section speed and the minimum single element speed were chosen as driving performance indicators since they were not correlated and agreed statistically with the accident history. Threshold values were identified for the identification of those elements characterized by acceptable (good), reasonable (fair), or intolerable (poor) alignment inconsistencies. Cafiso and Cerni [8], proposed a model of driver behaviour in terms of speeds on two-lane rural roads. Averaged horizontal curvature and averaged vertical grade were used. The model was obtained through GPS sampling of the positions of several test drivers who travelled several different roads in both directions and allows the estimation of a continuous speed profile that depends not only on the spot geometry of the section of the road in which the driver is

ISBN 978-88-548-7674-3 – ISSN 1824-5463-14005 – DOI 10.4399/97888548767437 – pag. 67-80


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traveling but also on the horizontal and vertical alignment of the road already travelled and to be travelled. Several studies conducted by Solomon [41], Nilsson [31], Aljanahi et al. [4], Quimby, et al. [39], Elvik, et al. [13], Aarts et al. [2], and Lamm et al. [24] correlated speed with crash rates. Further, it is widely recognised the importance of operating speeds for design consistency and in order to establish a comprehensive safety strategy for rural roads. Operating speed, indicated as V85, is defined as the 85-th percentile of the speed distribution, under free flow conditions, on clean, wet road surfaces. V85 represents the speed scenario in a given road section [16, 20, 32]. Several models of two-lane rural roads have been developed over the past several decades [15, 16, 22, 23, 26, 28]. Operating speeds were found to depend on road alignment [3, 6, 10, 15, 18, 19, 21, 25, 33, 36]. Praticò and Giunta [37], proposed a new model for low volume roads. D’Andrea et al. [12], set out a fuzzy prediction model to identify the characteristics of the road environment that mostly affect operating speeds. Abbas et al. [1], focused on 85th percentile operating speed models at mid-curve, based on approach speed and radius. These studies demonstrate the importance of considering the conditions of alignment preceding a horizontal element as well as the overall alignment of the road to better estimate the driver behaviour and operating speed in a current horizontal element.

In the context of previous studies, this paper focuses on the influence of road alignment on driver behaviour and operating speed (V85). In particular, this study deals with the explanatory potential of the length of an element (curve or tangent).

The objectives and scopes of this paper are to: i) describe the experimental methodology used to collect the speed data along a six- kilometres roadway section in Italy; ii) formalise models for predicting the influence of road alignment on operating speed; iii) deal with the explanatory potential of the length of an element (curve or tangent).

2. Experiments, analysis, and discussion

2.1. Data collection

An experimental survey was carried out on the road SP21 (two-lane rural road in the Province of Reggio Calabria, southern Italy; traffic volume which usually allows free-flow speeds; grade covers a range of ±10.0%; positive uphill grade in one direction; negative downhill grade in the opposite direction). There are no spiral curves, the general condition of the pavement is good, and the average pavement width is 6.00 m. Speed data were preliminarily collected at six sections (both directions at each section) using a speed laser gun (midpoints of the curves and tangents). Previous investigations demonstrated that the accuracy and precision of this method is optimal when laser gun and car direction are coincident (i.e., at 0 or 180°; only angles in the range 0°± 10° or 180°± 10° were used; accuracy of the measurements within 2 km/h, see Praticò and Giunta, [38]). Time headway was greater than 8 seconds (free flow conditions, see Moses and Mtoi [29]).

This preliminary investigation yielded a standard deviation of 6–9 km/h, and the coefficient of variation ranged from 0.13 to 0.17. Under the hypotheses of normality [5, 27], the minimum sample size required (errors due to sampling lower than 2 km/h) was derived according to [35]:

2

222

22

eUKn (1)

where e=2 km/h, K=1.96 (a confidence level of 95% was chosen), =9.15 km/h (>9, where 9 is the upper limit of the obtained range of standard deviation), and U=1.04 (sample size required for the 85-th percentile speed). As a consequence, a sample size of 125 was chosen.

This sample size was successfully verified ex-post after the real survey.


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Speed data were collected at 74 sites (in both directions). For each site (curve or tangent), at least 125 car speeds were measured for each direction. As for tangents, note that authors assessed that points under investigation were out of acceleration and deceleration lengths. The speeds of buses, motorcycles and trucks were not considered. All measurements were collected under free flow conditions, with the aim of including only drivers who were driving at their personally selected speed [39], in the daytime and under dry pavement conditions.

2.2. Data analysis

Table 1 summarizes the geometric features and operating speeds of the road under investigation. Note that due to the high variability of grades (from -10% to 10%), horizontal radii (from 18 to 148 m) and element lengths (from 8 to 245 m), vehicles can undergo very different conditions when passing from one location to the successive. As a consequence particular attention was paid to assess the main factors affecting V85 variance. In more detail, attention was focused to the improvement of the balance between terms representing the abovementioned “general character” of the alignment and terms specifically related to the single stretch under consideration. Data were analysed by referring to the influence of the main parameters. The conceptual framework was as follows: i) evaluating the effects of the curvature (1/R); ii) evaluating the effects of the grade (g); iii) evaluating the effects of the length (L) of a given alignment element on the operating speed; iv) assessing the relevance of the previous alignment elements on the operating speed; v) evaluating the effects of the oncoming alignment elements on the operating speed; vi) summarizing the effect of each factor; vii) analysing the superposition of effects.

In more detail (points 1 to 3), attention was focused: i) to find subsets of data within which both g and L were approximately constant and to analyse V85 versus 1/R; ii) to find subsets of data within which both 1/R and L were approximately constant and to analyse V85 versus g; iii) to find subsets of data within which both 1/R and g were approximately constant to analyse V85 versus L. After the analyses, data were grouped again to illustrate the overall effect of each of the three variables. Figure 1 illustrates the overall effect of curvature (1/R, m–1, x-axis) on the operating speed (V85, y-axis, km/h). As expected, a monotonically decreasing behaviour was observed. The values of the operating speed when 1/R = 0 illustrate the range of variation of speeds along the tangent.

Tab.1 - Data and statistics of SP 21


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Fig. 1 – curvature vs V85

Fig. 2 – grade vs V85

Fig. 3 – L vs V85

Figure 2 shows the effects of grade (g, x-axis, %) on V85 (y-axis, km/h), whereas Figure 3 shows the correlation between the length of the i-th element of the alignment (L, x-axis, m) and its corresponding operating speed (V85, y-axis, km/h). A negative correlation between grade and V85 was observed. A weak positive correlation between L and V85 was observed.

3. Modeling

This section deals with data modeling. Overall, authors formalised ad hoc equations based on: i) data interpretation and fitting; ii) need to meet expectations based on data interpretation; iii) literature analysis.

3.1. Basics

As is well known, motor vehicle crashes happen more frequently and are more severe on horizontal curves [34].

20

40

60

80

100

0.00 0.02 0.04 0.06

1/R (m-1)V8

5 (k

m/h

)

20

40

60

80

100

-15 -10 -5 0 5 10 15

g (%)

V85

(km

/h)

20

40

60

80

100

20 40 60 80 100

L (m)

V85

(Km

/h)


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Based on a literature analysis [6, 15], the first operating speed prediction model considered in this study was the following:

bR

aVd

85 (2)

where the coefficient a tunes the weight of the main variable, R (radius), d is a scale factor, and a (<0) and d (>0) influence the effects of the horizontal alignment on the operating speed. Low values of a and d yield large effects on V85. The parameter b is the value to which V85 approaches when 1/R tends to zero. As a consequence, this value relates to the so-called desired speed (McLean, 1979). Under these assumptions, the correlation yielded a poor value of R-squared, 0.37 (a=–34; b=53; d=0.47), (see point 4 in Figure 8). Other models in literature correlate V85 with the CCR (curvature change rate) or the DC (degree of curvature) but in this case, considered the absence of spiral curves, it appeared more suitable the use the above model. To obtain information about the grade effect, the following equation was optimized:

gcbV85 (3)

where c is a coefficient, and g (%) indicates the grade [7, 30]. In this case, the percentage of explained variance was 24%, with b=50 and c=–0.48 (see point

3 in Fig. 8). The parameter values were obtained by using the least squares method. Note that by k

(b=50, c = –1.05, K = 0.7).

3.2. Effect of the length of the element

The equations above described do not consider the extension (length or/and travel time) of the element under examination. In contrast, many studies [18, 30, 36, 40] confirmed that long elements influence the operating speed to a greater extent. In more detail, data here gathered suggested that, due to driver risk perception, the higher the length of the tangent the higher the operating speed of drivers. Furthermore, drivers behaviour in high-radius curves resulted similar to the one held in tangents. Due to the above facts, it seems unrealistic to model driver behaviour in terms of the sequence acceleration to the maximum allowed speed - constant speed - deceleration, where the maximum allowed speed depends on legal or/and design instances and not on real, perceived risks. A tool that manages the speeds by controlling the element length is, therefore, needed for both curves and tangents [30]. A framework that accounts for these aspects has been here addressed by introducing the concept of element relevance. In more detail, the element parameter i is here introduced as follows:

ni

i

fL

1

11 10 i . (4)

where Li (m) indicates the length of the i-th element, and f (positive, m) and n (positive, dimensionless) are model parameters to be estimated. Note that L affects the speed by which i approaches 1. High n and low f yield a high speed. Under these conditions, the element parameter,

i, varies over the range above stated. Note that high lengths Li yield a high element parameter value. In particular, if Li tends to infinity, the element factor tends to 1. In contrast, a low Li yields element factor values closer to zero. Note that the main drawback of i is that it doesn’t depend on the element radius.


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In contrast, data gathered, visual observations of driver behaviours on the road under examination and international literature [11] suggest that the effects of acceleration will be higher (and, consequently, will have a higher impact on safety) on tangents or high-radius curves. To this end, let us introduce the effective element coefficient, i, defined by means of a sigmoid factor F(R):

iii RF )( (5)

where:

GRR

RFi

i *exp1

1)( (6)

Combining these expressions (namely, equations 4, 5, 6), the following algorithm may be used to consider the effects of the length of a tangent (or a high-radius curve) on a driver’s behaviour:

e

GRR

fL

bVn *exp1

1

1

1185 (7)

Note that the higher the element length, L, the higher the radius, R, the closer to b+e is the operating speed (maximum V85). In contrast, the lower the element length, L, the lower the radius, R, the closer to b is the operating speed (minimum V85). Furthermore, note that, in practice, low radius-curves (R<R*) aren’t affected by e.

In the above equations, e is a calibration factor that affects the maximum magnitude of the effect (when L and R tend to infinity, i.e., for long tangents), R* (where R L) distinguishes between low- and high-radius curves, and G is a coefficient (dimension: length; unit of measure: m) that accounts for the gradualness (gradient) of this transition. Higher values of G yield more gradual transitions. As an example, Figure 4 illustrates the variations of V85 as a function of the radius, R, for different values of the parameter L (length of the element), for a threshold value (R*) of 60 m. Figure 5 shows the dependence of V85 on the length, L, of a given element of the alignment. Note that for tangent or high-radius sections, the higher is the length the higher is the speed. On the contrary, for low-radius sections (R<R*), this doesn’t happen. Note that in Figures 4 and 5, b (the constant in equation 7) was not considered. Higher radii with a given L give higher values of V85. Note that the effect of the length of the element with very small radii results negligible.

Fig. 4 – R vs V85 (term b = 0)

0

5

10

0 25 50 75 100

R (m)

V85

(Km

/h) L = 10 m

L = 50 mL = 100 m


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Fig. 5 – L vs V85 (term b = 0; R*=60)

This performance of the algorithm for small radii agrees with driver perception of run-out risks. Indeed, on horizontal curves, because of the limited sight distance and the increased probability of skidding, increased accident rates are observed and the majority of accidents on horizontal curves concern single vehicle run-off accidents and head-on collisions. Nonetheless, it is noted that despite this increased risk perception for small-radius curves, research results show that the number of road accidents still tends to increase when the radii of horizontal curves decrease [14, 17, 43]. Figures 6 and 7 show the corresponding first derivatives (V85', y-axis, 1000/h, where 1h=3600s) as a function of R and L, respectively (x-axes, m). The effect of R on the first derivative (i.e. V85 variations) is always negligible, except for radii close to R*.

Furthermore, the first derivative decreases for increasing L and this fact agrees with the superposition of psychological aspects and car performance. Under these hypotheses, the overdetermined system (174 equations in six variables) was solved using the least squares method. Data fitting yielded an R-square value of 0.37, with b=50, f=6.8 104, n=1.0, R*=95m, G=1, e=3590 (see point 5 in Fig. 8).

Fig. 6 – R vs V85' (1h=3600 s)

Fig. 7 – L vs V85'

0

5

10

0 25 50 75 100

L(m)V8

5 (K

m/h

)

R= 20 mR = 60 mR = 100 m

0.0

0.2

0.4

0.6

0 25 50 75 100

R(m)

V85'

(100

0/h)

L = 10 mL = 50 mL = 100 m

0.0

0.2

0.4

0.6

0 25 50 75 100

L(m)

V85'

(100

0/h) R=60 m

R=100 m


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Fig. 8 – R-Square values corresponding to each factor

It is noted that in this case (in which only the effect of the length of the current element was

considered), R* assumed a value two times higher than the average horizontal radius (i.e., 42m, see Table 1).

3.3. Effect of the previous element

The effect of the (i-1)-th radius on the i-th operating speed was considered using the following equation (where a’ and d’ are calibration coefficients, as in eq. 2):

bRaV di

'1

'85 (8)

In this case, an R-square close to zero was obtained (see point 1 in Fig. 8). Note that, for the considered road, the length of the previous element had a negligible impact on explained variance.

3.4. Effect of the oncoming element

Following data analysis, the effects of the oncoming (i+1)-th alignment element on the i-th operating speed were incorporated according to the following hypotheses: i) If the (i+1)-th element is similar (in curvature) to the i-th element, its additional influence will be negligible; ii) If the (i+1)-th curvature radius is higher than the previous i-th element, its influence will be negligible because the operating speed will be controlled by the current curvature; iii) If the (i+1)-th curvature radius is much lower than that of the previous i-th element, its influence will be significant, and it will affect V85i, especially when the sight distance allows the driver to recognize the effective curvature of the element. Indeed, an important safety effect concerns the presence of a curve (especially for low radii curves) after a tangent and drivers behaviour is affected by the perception of an oncoming risk [18]. In light of the above hypotheses, attention was focused on setting out functional forms able to fit experimental data and observations. Consequently, the influence of future elements was described according to the following equation:

b

RR

hRR

V n

i

iii 111

1

11185 (9)

0 1 2

34 5

0.0

0.2

0.4

0.6

0.8

1.0

cons

t.

previ. e

lem.

oncom. e

lem.

slope

radius

length

R-s

quar

e


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where the coefficients n1 (dimensionless), h (km m/h, where 1h=3600s), and b(km/h) are determined by optimization. h affects the maximum magnitude of an effect, whereas n1 affects the “speed” of the transition, from 0 to 1 of the expression in square brackets containing Ri+1/Ri.

The following consequences apply to low radii (Ri+1) elements:

hR

bViRi 1

185lim (10)

The road and data set under investigation did not yield a satisfactory correlation under these conditions (see point 2 in Fig. 8). Note that the interaction among the different curvatures of the alignment was considered in three main steps: previous (i-1), current (i), oncoming (i+1). In more detail, differences, and synergetic influences were addressed by using three independent coefficients (a, a’, h, respectively), which operated on three different radii (Ri-1, Ri, Ri+1), as well on one combined interaction (Ri+1, vs. Ri).

3.5. Influence of each factor

Figure 8 summarizes the analysis conducted to obtain information about the influence of each of the following factors on the i-th V85: 1) Ri-1; 2) Ri+1; 3) g; 4) Ri; 5) Li. The trivial case of V85=b (point 0 in Fig. 8) is reported only for comparison. It should be noted that the length alone may contribute appreciably to the percentage of V85 variance.

Furthermore, the grade itself explains around 24% of the variance. This fact highlights the importance of considering grades when analysing operating speeds and safety in mountainous terrain.

3.6. Superposition effect and results of fitting model

Finally, the model was validated by considering four progressive models (see Tables 2 and 3 and Figures 9 and 10) that combined each contribution. Note that all the models are valid for both tangents and curves.

Table 2 lists the models considered. For the sake of completeness all the models were reported (even in the case supplementary parts of the equation didn’t pursue any significant optimization of data fitting).

In each case, data were examined using an algorithm that differed from the previous in only one component. R-squared values ranged from 0.37 to 0.69.

Fig. 9 – V85 Scatter plot (independent variable: Radius)

20

40

60

80

100

20 40 60 80 100Observed V85 [km/h]

Pred

icte

d V8

5 [K

m/h

]


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Fig. 10 – V85 Scatter plot (independent variables: current and previous radii, grade, length)

Tab.2 - Models considered

Tab.3 - Contributions of factors to V85

(Additional) Factor Contribution to V85 Variance Total contribution %

Present Radius 0.37

68% Grade 0.18 Element coefficient 0.13

Past Previous element 0.01 1% Future Oncoming element 0.00 0%

Unexplained variance 0.31 31%

The significance of each correlation (p-values) is summarized in the last column of Table 2, and represents the probability of rejecting the null hypothesis (variables are not correlated) when the null hypothesis is actually true (Type I error, or “false positive determination”). Small p-values indicate the significant results. All correlations were significant (at a 1% level of significance). Figures 9 and 10 plot the correlation between true and predicted V85 values. The equality line indicates a perfect correlation. It is noted that gathered data were divided into two sets. The first one was used for model calibration and the second one was used to derive the predicted V85 values. The results indicate that (see also Table 3): i) For a simple model that does not account for the influence of grade (c=0), agreement between the model and experimental data was not good.

20

40

60

80

100

20 40 60 80 100Observed V85 [km/h]

Pred

icte

d V8

5 [K

m/h

]


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The curvature (1/R) of the single element in this case did not sufficiently describe the factors contributing to operating speed. A threshold was observed in Figure 9, corresponding to the value of b (53 km/h). This value is close to the average speed. This fact demonstrates the assumption that b is related to the desired speed; ii) The correlation between model and experimental data improved appreciably when the influence of grade was considered. An appreciable range of grades was considered in the data (mountainous context); iii) The introduction of the element coefficient, i, as a correction factor, tangibly and substantially “tuned” the effects of length on operating speeds. The estimated and actual speeds yielded a correlation that agreed closely with the equality line. The R-squared values increased appreciably (see Tables 2 and 3 and Figures 9 and 10); iv) Consideration of the curvature of the previous element (eq. 11) slightly increased the coefficient of determination (see Table 3). The consideration of the curvature of the (i+1)-th element did not substantially change the correlation between the experimental data and the model with respect to the case in which only the previous element was considered.

4. Examples of pratical application of the model

The proposed model allows to accurately predict the operating speed for both short and long tangents. Indeed, short tangents are often present between two curves, and the operating speeds on such very short elements strongly depend on the previous and successive elements. Other instances relate to long tangents characterized by operating speed profiles that differ substantially from those typical of previous and oncoming curves as well as from the profiles of design speeds. Potential safety adjustments can be made. For example, the maximum allowable length of a tangent can be determined, and this length depends on the maximum allowable difference in speeds on adjacent elements. Under the hypotheses of Lamm et al., 1998, by comparing the design and operating speeds in a tangent (1st Lamm criterion (middle of the tangent, see equation 7), it results:

cgefLbaLtgfRgVV

n

tad

1

1185 (11)

and

cgefLbVVV

n

dd

1

1185 (12)

where Vd is the design speed, ga is the gravitational acceleration, ft is the side friction factor, tg is the superelevation rate, R is the radius of the curve that precedes the tangent, a indicates the longitudinal acceleration (0.8m/s2 in Italian standards), and L is the length of the tangent. To this end, note that the plot of the 85th percentile speed versus inferred design speed can present two different domains: i) a first domain in which V85 exceeds Vd (lower speeds); ii) a second domain in which the opposite occurs [30]. From the comparison between the i-th and the (i-1)-th operating speed (independent tangent vs. successive curve) (2nd Lamm criterion) it results:

(13)

ef

LR

ae

fL

e

GRR

fLR

aVn

i

idRRn

iinii

di

1

1

*1

11

1

11)

*exp(1

1

1

1185


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Note that: a) equations (11–13) permit to derive the maximum allowable length (L) in order to obtain a speed difference lower than a given limit (for example, 20 km/h) [23]; b) the term containing e and the term that depends on Ri-1 are positive (a<0) in equation 16. It follows that small values of the (i-1)-th radius and large tangent lengths originate large absolute differences and small safety potentials for a given independent tangent (a threshold of 20 km/h is usually assumed in comparing successive operating speeds). To this end, for a given speed threshold and curvature, equation (16) can provide maximum allowable length of the tangent. Therefore, the above model provides a tool for analysing the safety potential in a curve or tangent. 5. Conclusions

The following conclusions may be drawn from an analysis of the data under investigation: - grade, horizontal curvature, and length of an element within the road alignment under

consideration are the main factors that influence speed behaviour. Specifically, grade greatly influences operating speed, and the synergistic effects of grade and length may potentially affect the safety of rural roads in mountainous terrain;

- the lengths of elements in curves or tangents are relevant in terms of driver perception and behaviour, especially when comparing short and long tangents (or/and high-radius curves);

- by introducing element and logistic factors, driver speed choice and as a consequence the relationship V85 vs. 1/R were tuned according to the element length. The model provided a reasonable, albeit simplified, representation of the speed profiles on horizontal alignments consisting of long tangents and/or long high radius curves;

- safety analysis techniques and the modelling of driver performance can be improved through the use of the above algorithms and concepts. Speeds can be accurately predicted for both short and long tangents. Indeed, short tangents are often present between two curves, and the operating speeds on such very short elements strongly depend on the previous and successive elements. Other instances relate to long tangents characterized by operating speed profiles that differ substantially from those typical of previous and oncoming curves as well as from the profiles of design speeds. Potential safety adjustments can be made. For example, the maximum allowable length of a tangent can be determined, and this length depends on the maximum allowable difference in speeds on adjacent elements;

- under the above boundary conditions (lengths and radii can vary over the range 0–algorithms described here provide a basis for developing a general framework that may be applied to other cases as well;

- although previous and oncoming geometric features may be essential for the accurate analysis in a given section, from a statistical standpoint, the influence of the geometric features of the current element prevails over the influence of both past and oncoming elements.

Authors are aware that further research is needed on the above issues and a wider sample for practical application will be required (alignment, terrain, users).

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