Detection of Short-Section Ballast Breakdown in Track: A Fractal Analysis Approach with Reduced Window Size

Abstract

Due to increasing demands on the available railway infrastructure, accurate estimates of safety-critical track condition as well as breakdowns of individual track components are crucial. This task can be supported by analyzing track measurement data. Ballast breakdown can be determined by analyzing the longitudinal level using fractal analysis: Commonly, a window with a width of 150 m is dragged over the signal computing an approximation of a fractal dimension of the signal for each position of the window. However, while a large window size can be used to describe the condition of ballast and substructure simultaneously, it fails to precisely localize short-section ballast breakdowns in the track. With the objective of describing and detecting these local effects in the ballast bed, this work analyzes a set of 114 known weak ballast spots. By reducing the width of the sliding window, the position of short-section ballast breakdowns can be reliably depicted. The application of a modified version of fractal analysis allows for a more accurate targeted maintenance on a component-specific basis.

1. Introduction

Continuous monitoring is essential to ensure the safety and proper condition of railway tracks. Knowing the condition of the track is critical, as increasing demands on the rail infrastructure require more efficient maintenance planning and execution [1]. In order to determine the condition of the track and its components, vehicles equipped with measuring tools (measuring cars) routinely inspect the track and assess critical factors such as alignment, cross level, and gauge, in accordance with the European Standard EN 13848-5 [2]. The typical approach to assess the quality of the track is to analyze the longitudinal level as recorded by the track measurement car. This data typically appears as sinusoidal waveforms, reflecting the vertical displacements along the track. The data pertaining to longitudinal level and alignment are subject to filtering across two wavelength (λ) ranges: D1, covering 3 ≤ λ ≤ 25 m, and D2, covering 25 ≤ λ ≤ 70 m. Subsequent to this filtering process, specific quality indicators are calculated. These indicators are dependent on whether the defect under analysis is distributed or isolated. The longitudinal level is used to calculate various quality indicators, most commonly the standard deviation of the longitudinal level [2,3,4,5,6]. By analyzing the standard deviation over time, we can describe the deterioration of the track. This quality indicator is typically used to schedule maintenance works, in particular track tamping [7,8]. Tamping is employed in railway maintenance to restore the alignment and level of the track by lifting the rails and sleepers, compacting the ballast underneath, and ensuring that the track panel is then set in the desired position.

Restoring a defined track geometry through tamping is the most frequently required heavy maintenance and is therefore also cost intensive [6]. The need for tamping arises through irregular track settlements, often being a consequence of a contaminated ballast bed. This contamination leads to a reduction in track elasticity, increases stress within the structure, and ultimately leads to a decline in track geometry quality [9,10,11]. Settlements and variations in ballast bed stiffness have a significant impact on track life and should be corrected as soon as possible [12]. Although tamping can correct track geometry, it may not be the most sustainable method of maintenance, particularly in cases of a poor ballast bed condition [13].

EN 13848-5 emphasizes that the assessment of geometrical track quality should encompass both distributed irregularities and isolated defects [2]. These isolated defects are characterized by their specific wavelength, amplitude, and shape. Typically, the analysis of the longitudinal level, for example through the standard deviation or mean values, focuses primarily on the amplitude of the signal to describe the general condition of the track geometry. In contrast, component-specific analysis aims at assessing the condition of the track by examining changes in wavelength composition within the longitudinal level [14]. In order to identify whether a track geometry defect is caused by the ballast condition, we need to analyze the longitudinal level differently, i.e., based on the wavelengths within the signal. This can be done with power density spectra [15,16] or fractal analyses [17]. The power spectral density approach examines signals based on its wavelength composition. Informally, it describes how the average power of the strength of the signal is distributed over different frequency components [18]. Fractal analysis, on the other hand, assesses the “roughness” of the signal without requiring precisely defined frequency ranges.

In this work we investigate the characteristics of the track’s longitudinal level with the help of fractal analysis. In contrast to other methods for ballast condition assessment, such as ground penetrating radar (GPR) [11,12,19,20] or the analysis of ballast samples, this approach does not necessitate the implementation of additional measures and therefore does not result in any traffic disruptions. Previous work estimates the fractal dimensions of a local patch of signal data and repeats this computation in a sliding window manner. To capture the characteristics of both the ballast and the substructure simultaneously, the size of this window is typically chosen to be 150 m [12,17,21,22]. This makes it possible to assign recorded data to a specific damage pattern. By analyzing the longitudinal level using fractal analysis and estimating fractal dimensions, the root cause of the failure can be narrowed down to different parts of the track structure [21].

The aim of this work is to use the additional information on the condition of the ballast to optimize track maintenance by detecting short sections of ballast breakdowns. This is based on a data-driven approach analyzing the track condition and determining the root causes of occurring faults. As soon as we know the root cause, we can predict targeted maintenance measures in particular areas, thus improving track quality in the long term. In this way, we can stretch maintenance cycles and ultimately extend the service life of the track [21].

2. Methodology

Fractal analysis is a mathematical approach which examines a signal by analyzing its wavelengths and amplitudes collectively. This technique calculates the fractal dimension quantifying the signal’s “roughness” [23]. Fractal analysis attempts to distinguish and characterize the different wavelength components within a waveform [17]. To estimate fractal dimensions, various methods can be employed, such as the box counting, yardstick, or divider method, depending on the structure to be analyzed [24]. An illustrative example for the calculation of the fractal dimension is the yardstick method [25]. This method generalizes the fractal dimension into the yardstick dimension. Figure 1 depicts a classic fractal, the Koch curve, and illustrates how to determine the fractal dimension, showcasing the procedure’s application [26]:

The blue line in Figure 1a–d, created with the yardstick method, constitutes a simplified representation of the actual shape of the Koch curve. The method establishes a kind of “yardstick” by drawing circles with a fixed radius $r$ at the leftmost point of the curve. The resulting intersection of the circle with the Koch curve is used as the new center point for the next circle with the same radius $r$. When connecting the center points of the generated circles using a straight line, we obtain a piecewise linear approximation of the curve analyzed (shown in blue). Once the entire object has been measured, the number of circles $N\left(r\right)$ is counted and the resulting length $L\left(r\right)$ of the resulting polygon is then given by [26]

$$L\left(r\right)=N\left(r\right)·r$$

(1)

To relate $L\left(r\right)$ with the dimensionality of the curve, we study the behavior of $L\left(r\right)$ as $r\to 0$. The closer $r$ is to $0$, the better the actual shape of the Koch curve is approximated. Plotting the determined values $L\left(r\right)$ and $r$ in a diagram (see Figure 2), we see that the best approximation is a power function. The length $L\left(r\right)$ can therefore also be determined as follows [26]:

$$L\left(r\right)=c·r^a$$

(2)

The value $a$ determined for the Koch curve is $a\approx -0.265$. Following [26], we define $D:=1-a$ and observe that it follows from Equations (1) and (2) that:

$$\begin{gathered} r·N\left(r\right)=c·r^a \\ N\left(r\right)=c·r^{-D} \\ \ln \left(N\left(r\right)\right)=\ln \left(r^{-D}\right)+\ln \left(c\right) \\ \ln \left(N\left(r\right)\right)=\mathrm{b}-\mathrm{D}·\ln \left(r\right) \\ D={\displaystyle \frac{b-\ln \left(N\left(r\right)\right)}{\ln \left(r\right)}} \end{gathered}$$

The fractal dimension can now be determined by considering the following limit value [26]:

$$D=\underset{r\to 0}{\lim }{\displaystyle \frac{b-\mathrm{l}\mathrm{n}\left(N\left(r\right)\right)}{\mathrm{l}\mathrm{n}\left(\mathrm{r}\right)}}$$

(3)

2.1. Fractal Analysis for Track Condition Description

Hansmann and Landgraf propose a fractal analysis for track condition assessment with a window size of 150 m [21]. This method enables a simultaneous evaluation of the condition of the ballast bed and the substructure. The fractal analysis algorithm is an iterative process that can be divided into the following steps, as outlined in [21] and illustrated in Figure 3:

•

Let $f: \mathbb{R}\to \mathbb{R}$ denote the track’s longitudinal level as a function of position

•

Step 1: Given a position, $x\in \mathbb{R}$, we span a window of length $w$ around it i.e., $[x-{\displaystyle \frac{w}{2}},x+{\displaystyle \frac{w}{2}} ]$

•

Step 2: Subdivide the window into subsegments with a length of $l_s$ (originally referred to as λ, but renamed to $l_s$ for clarity)

•

Step 3: For different values of $l_s$, calculate the polygonal length $L_p$

$$L_p=\sum _{k=0}^{N-1}\sqrt{{l_s}^2+{\left(f\right(x-{\displaystyle \frac{w}{2}}+k·l_s)-\mathrm{f}(x+{\displaystyle \frac{w}{2}}+(k+1)·l_s\left)\right)}^2}$$

(4)

•

Step 4: Plot the polygonal length $L_p$ and the corresponding subsegment length $l_s$ in a log-log diagram (Richardson plot [17,21])

•

Step 5: Apply a linear regression in the Richardson plot and use its slope to approximate dimension of $f$ in the window

Steps 1: The fractal analysis calculation commences with reading the longitudinal level measurement, which is then checked for existing error values. Subsequently, a window with a length of 150 m is spanned, covering 75 m in both directions, with the track position under consideration situated in the center (Figure 4). With a sampling rate of the measurement car of 25 cm, this yields a data segment of 601 individual measurements.

The signal analyzed in this study represents vertical track geometry irregularities with wavelengths ranging from 3 to 70 m. These irregularities primarily arise due to differential settlements in the ballast bed and the substructure [27]. Using fractal analysis, the track geometry irregularities can be assigned to different root causes (see Figure 5) [21,28,29]. In the fractal analysis, Hansmann and Landgraf use a window size of 150 meters to cover at least two full waves of the longest wavelength that may occur within the analyzed signal (i.e., $>2·70 \mathrm{m}$).

Step 2: The choice of subsegment length $l_s$ depends on the window size and is determined using a predefined divider matrix which defines the number of subsegments $\left(N\right)$. The divider matrix specifies the number of subsegments that the window size should be divided into. The subsegment lengths $l_s$ are determined based on this specification. (Figure 6).

If the starting and ending points of the subsegment do not coincide with a measurement point of the signal, we use linear interpolation to determine the values at the intersection points between the existing measurement values. To determine the length of the resulting polygon $L_p$, we calculate the lengths of the subsegments’ straight lines $l_c$ using the Pythagorean theorem, which are then summed up [28].

$$\begin{gathered} L_p=\sum l_c=\sum _{i=1}^N\sqrt{{\left(Y_{i-1}-Y_i\right)}^2+{\left({\displaystyle \frac{W}{N}}\right)}^2} \\ N \dots \mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{g}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s} \\ W \dots \mathrm{W}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{w} \mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e} \mathrm{i}\mathrm{n} \mathrm{m} \end{gathered}$$

(5)

Step 4: The fractal value is determined by plotting the subsegment length $l_s$ on the x-axis and the length of the respective polygon course $L_p$ on the y-axis in a double logarithmic plot (Richardson plot, see Figure 7). As the length of the subsegments is decreasing, the resulting length of the polygon increases. Consequently, the established polygon gradually converges towards the original measurement signal.

Step 5: The fractal value is approximated using the divider matrix. In the Richardson plot, three fractal areas can be distinguished based on the subsegment length $l_s$. The short-wave range ($l_s\in [0.25, 0.75]$) refers to the rails and the sleeper–ballast interface, the medium-wave range ($l_s\in [0.75, 5]$) to the ballast, and the long-wave range ($l_s>5$) to the substructure. Linear regression is utilized in each range to approximate the fractal value. Hansmann and Landgraf fix the first six points from the left in the Richardson plot for the short and medium ranges and the first three points from the right for the long wavelength range. For each subsegment range, the linear regression is calculated using the selected data points before additional data points from the respective range are gradually included. This is done until the highest coefficient of determination R² is found. The slope of the final regression line indicates the extent of the fractal value and provides information about the cause of the error.

The methodology used to determine the fractal value follows a modified variant of the yardstick method. In this method, the data segment under consideration is divided into constant subsegment lengths $l_s$ ($l_{\mathrm{s}}$ is denoted as $\mathsf{\lambda }$ in the following equation in accordance with the original sources) rather than drawing circles in the signal (see Figure 1). It is assumed that the points generated in the Richardson plot can best be approximated by a power function [32]. Therefore, the following applies:

$$L\left(\mathsf{\lambda }\right)=c·{\mathsf{\lambda }}^a$$

(6)

With $b≔\ln \left(c\right)$ it follows that

$$\begin{gathered} \ln \left(L\left(\mathsf{\lambda }\right)\right)=\mathrm{b}+\mathrm{a}·\ln \left(\mathsf{\lambda }\right) \\ a={\displaystyle \frac{\ln \left(L\left(\mathsf{\lambda }\right)\right)-b}{\ln \left(\mathsf{\lambda }\right)}} \end{gathered}$$

It can be shown that the fractal dimension can be approximated with $D≔1-a$ [26,32]

$$D=1-{\displaystyle \frac{\ln \left(L\left(\mathsf{\lambda }\right)\right)-b}{\ln \left(\mathsf{\lambda }\right)}}$$

As with the yardstick method, the following limit value must now be determined:

$$\underset{\mathsf{\lambda }\to 0}{\lim }1-{\displaystyle \frac{\ln \left(L\left(\mathsf{\lambda }\right)\right)-b}{\ln \left(\mathsf{\lambda }\right)}}=1-\underset{\mathsf{\lambda }\to 0}{\lim }{\displaystyle \frac{\ln \left(L\left(\mathsf{\lambda }\right)\right)}{\ln \left(\mathsf{\lambda }\right)}}$$

The fractal analysis algorithm presented here uses linear regression in the Richardson plot to estimate the size

$$k\approx \underset{\mathsf{\lambda }\to 0}{\lim }{\displaystyle \frac{\ln \left(L\left(\mathsf{\lambda }\right)\right)}{\ln \left(\mathsf{\lambda }\right)}}$$

(7)

Accordingly, the fractal dimension of the analyzed signal corresponds to $D \approx 1-k$. Instead of 1 $-k$, we use $k·{-10}^7$ to reproduce existing experiments from the literature [28,29] and to establish comparisons with them and with the standard deviation of the longitudinal level. Notably, since $k$ takes on very small values, it is advantageous to avoid subtracting $k$ from 1 to prevent computational errors arising from excessive decimal places.

2.2. Modified Fractal Analysis with REDuced Window Size (FRED)

In this work, the algorithm for fractal analysis was modified to introduce a reduced window size in order to improve the detection and localization of ballast bed problems. This change is aimed at detecting ballast bed problems that have predominantly localized effects, which typically affect areas much shorter than 150 m (the window size chosen by Hansmann and Landgraf). Within the longitudinal level, the condition of the ballast bed is mainly represented in the wavelength range D1 (wavelengths between 3 and 25 m). Consequently, we investigated the effectiveness of eight different window sizes, ranging from 6.25 to 50 m, at identifying local ballast bed problems. A reduction in the length of the data segment allows for more precise localization of individual errors in the signal, as their impact is limited to a smaller area. However, this approach has the consequence that the condition of the substructure can no longer be evaluated simultaneously, as this requires the investigation of longer wavelengths with larger windows. In addition, it is important to validate that the condition of the ballast bed can be accurately characterized by the fractal analysis with a reduced window size. For the purpose of validation, we analyzed track sections that had undergone short ballast bed renewal and studied the behavior of the modified fractal analysis on these sections.

For FRED, instead of using a predefined Divider Matrix like in Hansmann and Landgraf’s fractal analysis, we adjust the window sizes and determine the number and length of subsegments based on the maximum possible number of subsegments that can fit within the specified maximum and minimum lengths. Additionally, it is noteworthy that the linear regression in the Richardson plot is calculated using the RANSAC algorithm [33], eliminating the need for the manual selection of boundaries for the regression. The application of RANSAC significantly reduces the computation time of the algorithm.

In order to evaluate the performance of the modified fractal analysis and the different window sizes, we employ the F₁ score as a key metric. The F₁ score is a statistical measure that balances precision (the accuracy of positive predictions) and recall (the model’s ability to identify all actual positives). The F₁ score, which ranges from 0 to 1, is the harmonic mean of precision and recall and allows us to compare the effectiveness of different models in datasets with potentially unbalanced class distributions. By using the F₁ score, the study ensures a balanced evaluation of the models, taking into account both false positives and negatives, which is important for an accurate model evaluation [34].

The investigated window sizes and computation parameters are listed in

Table 1. Comparison of window sizes under investigation.

	FRED								Hansmann/Landgraf
Input	Longitudinal Level (3–25 m)								Longitudinal Level (3–70 m)
Window size W [m]	50	25	20	15	12.5	10	8	6.25	150
$l_s$, min [m]	s								0.75
$l_{s,}$max [m]	50	25	20	15	12.5	10	8	6.25	30
Interval [m]	1 m								2.5 m
Number of subsegments	66	33	26	20	16	13	10	8	73

. These comprise eight versions of the modified fractal algorithm (FRED) as well as the original fractal algorithm (“Hansmann/Landgraf”).

The “Hansmann/Landgraf” model, which has been validated and is applied in various countries, serves as a basis for comparison with the modified fractal algorithm. By default, the fractal algorithm inspects the longitudinal level of the left rail for wavelengths between 3 and 70 m. For the modified version, aimed specifically at analyzing local ballast bed issues, the analysis is limited to wavelengths between 3 and 25 m and employs window sizes between 6.25 and 50 m. The minimum subsegment length $l_s$ remains constant at 0.75 m, in accordance with Hansmann and Landgraf. The maximum length ($l_{s,}$_max) varies in dependence with the window size selected. The computation interval, i.e., the interval at which the fractal value is estimated, has been reduced from 2.5 m to 1 m. The efficiency of the new implementation allows narrower intervals, so the choice of window size requires a trade-off between the level of detail in the analysis and the associated computational cost.

3. Results

Before evaluating the best window size for the identification of short-section ballast breakdowns, we have to verify that the modified fractal analysis is indeed able to assesses the condition of the ballast bed.

3.1. Investigation of Changes in Ballast Bed Condition Using FRED

To investigate whether the ballast bed condition can be accurately characterized with the modified fractal analysis, the established algorithm with varying window size (Table 1) was applied at 19 track sections that had undergone ballast bed renewal over the last years. For each window, the fractal values were calculated. The underlying hypothesis is that prior to ballast bed renewal, the condition of the ballast in the specified areas is very poor and improves significantly after renewal. To substantiate this assumption, the dates of ballast bed renewal were determined, allowing for three key comparisons:

•

PR—Comparison between the last two measurement runs prior to ballast renewal.

•

BA—Comparison between the measurement runs before and after ballast renewal.

•

PO—Comparison between the last two measurement runs after ballast bed renewal (post-renewal).

To facilitate the comparison of different track sections with varying section lengths, the average fractal values over this length are calculated. These average values are used to describe the changes in fractal value over time. The average fractal value at time $t_i$ is subtracted from the average fractal value at time $t_{i+1}$. The resulting differences represent the increase in fractal values in the respective period. Before the comparison, it is essential to identify the specific measurement runs between which the ballast bed renewal was carried out. In addition, this analysis aims at evaluating the functionality of the modified fractal analysis and to demonstrate the differences and advantages of using the RANSAC algorithm within this adapted fractal analysis framework.

3.1.1. PR Comparison

In the two measurement runs immediately prior to ballast bed renewal, it is expected that the magnitude of the fractal values tends to increase as the proximity to the renewal event decreases. An increase in the fractal value indicates a deterioration in the condition of the ballast bed. After calculating the difference between consecutive fractal measurements, a negative difference is expected due to the increase in fractal values. Figure 8 illustrates the differences between fractal values of the two last measurement runs prior to the ballast bed renewal.

The y-axis in Figure 8 lists the different window sizes in meters, starting from 6.25 m and going up to 50 m. The box plots are colored light blue and turquoise and differ in the regression method used to estimate the fractal values. The turquoise box plots use the RANSAC regression method, while the light blue box plots use the conventional fractal analysis method (fitted linear regression). Figure 8 demonstrates that both the RANSAC and linear regression methods exhibit similar behavior across all window sizes examined, with the median of all window sizes falling between a difference in fractal values of −0.1 and −0.3.

The PR scenario in Figure 8 shows the expected deterioration of the ballast quality condition over time. The medians of all window sizes assume negative differences; however, the third quartile value for all window sizes is in the positive range. The outliers observed in both the RANSAC and linear regression methods indicate an improvement in ballast quality over time (positive fractal difference). These apparent improvements are probably due to average consideration of the fractal values or could indicate track sections where spot maintenance has been performed, improving ballast quality locally despite the need for ballast bed renewal.

3.1.2. BA Comparison

Considering the measurement runs before and after the ballast bed renewal, it is assumed that the differences in fractal values will show positive values, as an improvement in ballast condition is expected. These considerations are illustrated in Figure 9.

Similar to Scenario PR, Figure 9 compares the fractal differences for all window sizes considered, with the respective regression methods highlighted in color. Since the differences in fractal values are calculated before and after ballast renewal, Figure 9 shows consistently positive fractal differences, reflecting an improvement in the ballast condition. The median of the calculated differences per window size, determined using RANSAC, is between 1 and 2 and decreases for smaller window sizes. The fractal differences calculated using linear regression reveal median differences between 2 and 3 and decrease with larger window sizes. Therefore, it can be assumed that all window sizes are, in principle, suitable for describing the ballast condition. It is also noteworthy that the fractal differences calculated using RANSAC are consistently smaller at the median, interquartile range, and overall range than those calculated using linear regression. This is because RANSAC detects outliers in the Richardson plot and excludes them from the calculation of the fractal value.

3.1.3. PO Comparison

In the course of a ballast bed renewal, new ballast is installed, which can be assumed to exhibit an optimal ballast condition. Consequently, the condition of the ballast bed is not expected to remain largely unchanged over the subsequent measurement runs. This implies that the fractal values will remain approximately equal for the two measurement runs performed immediately after the ballast bed renewal, and after, the calculated difference between the successive fractal values will be small. Figure 10 displays the performance of the average fractal values after the ballast bed renewal is carried out.

Figure 10 depicts the fractal differences from the two measurement runs conducted immediately after ballast bed. For both regression models and over all window sizes, the median is close to zero. The differences in fractal values between RANSAC and the fitted linear regression are minimal in this instance. This indicates that the quality of the ballast bed changes very little after the ballast renewal, suggesting an effective renewal process.

3.2. Determination of the Optimal Window Size

In order to determine the ideal window size for the modified fractal analysis to detect and locate short-section issues in the ballast bed, we evaluate the performance of three distinct model types, each of which is calculated for the eight defined window sizes (see settings for “FRED” in Table 1). The models under consideration are (1) the standard deviation of the longitudinal level, (2) the fractal value, calculated with the modified fractal algorithm, and (3) the combination of the fractal values with their temporal change. With three distinct model types and eight different window sizes, this results in a total of 24 models (3 model types for each of the 8 window sizes). To verify the occurrence of short-section ballast breakdowns, we selected 48 locations with known ballast bed problems. After defining thresholds for each of the 24 models, we performed an analysis to determine which combination of model and window size was most effective in verifying these isolated ballast bed failures. However, in addition to the 48 already known ballast bed impurities, the results of the computations revealed even more locations with similar behavior in the fractal values. We checked these additional locations against trackside images taken by the track measurement car. As a result, an additional 66 locations exhibiting visually observable ballast bed could be included in the analysis. Therefore, we could ultimately use a total of 114 local ballast bed issues to determine the optimal threshold value of each model with respect to detecting ballast bed contaminations.

Through rigorous experimentation, an accurate delineation of ballast contamination within a localized section with a spatial accuracy of ±7.5 m was achieved. Weak spots were identified in regions where defined thresholds were exceeded. These thresholds were carefully calibrated to optimize the F₁ score within the validation dataset corresponding to each specified window size. Figure 11 shows a section of the track where ballast contamination has a predetermined fractal value threshold of 25 over a window size of 12.5 m. Using this threshold, all three known localized ballast bed problems within this section can be detected. The location highlighted in orange highlights a ballast bed contamination that has been confirmed by a track manager. In addition, positions highlighted in yellow indicate weaknesses within the ballast structure, as identified by trackside imaging data.

3.3. Comparison Between the Different Model Types

As the fractal analysis is considered an enhancement to the classical method of track geometry analysis (i.e., analyzing the standard deviation) [29], we investigated whether the newly calculated quality values (fractal values) provide additional information. For this purpose, we applied all three model types—the standard deviation, the fractal value, and the fractal value including their temporal change—at the aforementioned sections with known local weak spots in the ballast bed. All three model types were calculated with the eight defined window sizes of 6.25, 8, 10, 12.5, 15, 20, 25, and 50 m. For each model type, we set a threshold for the standard deviation that aims to maximize the F₁ score. The standard deviation for each window size is calculated with the goal of maximizing the F₁ score in detecting known ballast bed problems. To determine an optimal threshold for identifying the 114 documented ballast bed issues, we divided the dataset into a training set and a validation set. Specifically, 28 issues were used to determine the threshold, while the remaining 86 issues were used to validate the performance of the threshold. The resulting thresholds and the corresponding model performance per window size are summarized below.

Table 2. F₁ scores for models based on the standard deviation.

Standard Deviation
Window Size [meters]	SigmaH Threshold	F1 Score (Training Set)	F1 Score (Validation Set)
6.25	4.5	0.8571	0.6538
8	5.25	0.8514	0.6818
10	5.25	0.8378	0.6667
12.5	5	0.8322	0.6667
15	4.75	0.8322	0.6977
20	4	0.8101	0.7111
25	4.25	0.8108	0.6818
50	3.5	0.7568	0.6383

represents the results for the standard deviation model.

The values presented in Table 2 indicate that the model with a window size of 20 m and a threshold value of 4 achieves an F₁ score of 0.71 (validation), representing the highest F₁ score among all the models evaluated.

Table 3. F₁ score for models based on the FRED-algorithm.

Fractal Values
Window Size [meters]	Fractal Value Threshold	F1 Score (Training Set)	F1 Score (Validation Set)
6.25	33	0.8645	0.7547
8	32	0.8742	0.7692
10	25	0.8442	0.7843
12.5	25	0.8533	0.7917
15	25	0.8400	0.7660
20	20	0.8400	0.7451
25	17	0.8289	0.7925
50	13	0.7516	0.7917

lists the calculated threshold values and the associated F₁ scores for the second model, which is the modified fractal analysis (FRED). The fractal values were calculated using RANSAC.

The results listed in Table 3 reveal that the fractal analysis variant with a window size of 25 m is most effective (validation set) in detecting short-section ballast breakdown with an accuracy of ±7.5 m. The threshold value of the fractal value was determined to be 17, and the F₁ score achieved at the validation set was 0.79. In the validation set, none of the window sizes achieved an F₁ score over 0.80 when applying the respective threshold value.

While analyzing the available data, we observed that the fractal values at the locations of existing single failures in the ballast bed showed a significant temporal increase. Consequently, the approach we used in this study was not only to detect ballast problems based on the fractal value at a single point in time, but also to consider the temporal variation of the fractal value. We achieved this by calculating the differences between the fractal values of consecutive measurement runs, summing the absolute values of these differences, and dividing the sum by the number of measurement series. With this approach, the model type requires two threshold values to achieve the optimal F₁ score, one for the current fractal value and one for the temporal change.

Table 4. F₁ score for the model combining fractal values and their temporal changes.

Fractal Values + Temporal Changes
Window Size [meters]	Fractal Value Threshold	Temporal Changes Threshold	F1 Score (Training Set)	F1 Score (Validation Set)
6.25	33	0	0.8645	0.7547
8	32	0	0.8742	0.7692
10	5	2.5	0.8519	0.7170
12.5	25	1.5	0.8591	0.7917
15	7	1.75	0.8690	0.7170
20	18	2	0.8552	0.6939
25	16	1.75	0.8378	0.7234
50	6	0.75	0.7889	0.7119

presents the calculated threshold values and the corresponding F₁ score for each window size.

Table 4 reveals that six out of the eight models examined the temporal change of fractal values in order to maximize the F₁ score of the training set. The models with a window size of 6.25 and 8 m did not react on this temporal change, as there is no improvement of the F₁ score (compared to the F₁ score of the training set in Table 3). The F₁ scores determined using the validation set show that all models achieve an F₁ score < 0.80. We obtained the highest F₁ score of 0.79 with a window size of 12.5 m, a threshold for the fractal value of 25, and a threshold for the sum of the difference set at 1.5.

4. Discussion

Before interpreting and comparing the performance of each model, it is important to note that the validation set will generally perform slightly worse than the training set due to overfitting. During training, the model learns specific patterns within the training data, which may include noise or irrelevant details. This can result in a model that is overly tuned to the training data, reducing its ability to generalize to new, unseen data. As a result, when applied to the validation set, the model’s accuracy decreases, resulting in lower performance metrics because it has not effectively generalized beyond the training data. It should also be noted that the effectiveness of the models, as measured by the F₁ score, indicates their ability to identify recognized ballast defects, with an F₁ score of approximately 0.8 indicating potential for improvement. The F₁ score could be significantly improved by considering that additional, unidentified ballast defects may be present in the analyzed track sections. These unclassified defects may already be present and could affect the performance of the model if detected and included in the evaluation.

Table 5. Model performance of the analyzed models.

	Fractal Values		Standard Deviation (Reference)		Fractal Values + Temporal Changes
Window Size [meters]	THR	F1	THR	F1	THR Frac.	THR Temp.	F1
6.25	33	0.7547	4.5	0.6538	33	0	0.7547
8	32	0.7692	5.25	0.6818	32	0	0.7692
10	25	0.7843	5.25	0.6667	5	2.5	0.7170
12.5	25	0.7917	5	0.6667	25	1.5	0.7917
15	25	0.7660	4.75	0.6977	7	1.75	0.7170
20	20	0.7451	4	0.7111	18	2	0.6939
25	17	0.7925	4.25	0.6818	16	1.75	0.7234
50	13	0.7917	3.5	0.6383	6	0.75	0.7119

provides an overview of the calculated model performances with regard to the detection of locally occurring weak spots in the ballast bed. It presents the F₁ scores of all analyzed calculation variants, comprising the three model types—standard deviation, fractal value, fractal value plus temporal change—and the eight window sizes. The window sizes that performed best in the fractal value model were 12.5 m and 25 m. The model with a window size of 25 m achieved the highest F₁ score of 0.7925, making it the best model for this criterion. The model with a window size of 12.5 m also showed good model performance with an F₁ score of 0.7917. This indicates that both window sizes are very effective in detecting degradation in the ballast bed based on the fractal value alone, with a window size of 25 m having a slight advantage in terms of the raw F₁ score.

However, when temporal changes are considered in addition to fractal values, the 12.5 m model not only retains its high F₁ score, but also proves that it is possible to account for temporal dynamics without compromising accuracy. Furthermore, the models using fractal values systematically achieved higher F₁ scores than the models based on standard deviation. This substantiates the assumption that fractal values are a more reliable measure for identifying ballast bed problems and provide a valuable benefit for infrastructure managers.

Figure 12 compares different track quality indices: the standard deviation calculated with a sliding window of 100 m, the original fractal analysis by Hansmann and Landgraf using a window size of 150 m, and a modified version called “FRED” with a window size of 12.5 m.

When analyzing the standard deviation shown in Figure 12, two significant anomalies appear along the track. However, when considering the fractal value, only one of these two positions is associated with issues originating from the ballast bed. The corresponding fractal value estimated with a sliding window of 150 m does not clearly indicate whether the ballast bed contamination extends over nearly 100 m or is confined to a smaller area. In this context, FRED demonstrates its advantage due to its reduced sliding window size. The position detected using FRED correlates with a localized weak point in the ballast bed. An image from the track recording car confirms a localized weak point in the ballast bed within the analyzed track section. While the fractal analysis with a 150 m window size can detect such weak points, it lacks precise localization capabilities and fails to reflect the severity of the issue. This is because the magnitude of the fractal value is mitigated by the track condition adjacent to the weak spot. In contrast, the fractal analysis with a window size of 12.5 m not only identifies problem areas but also locates them precisely.

5. Conclusions

The application of the modified fractal analysis for track condition assessment has proven to be a promising approach. The results presented in this paper confirm that a reduced window size in the fractal model improves the accuracy of the localization and characterization of weak points in the ballast bed compared to the conventional fractal analysis approach of Hansmann and Landgraf. However, the fractal analysis with 150 m is still used for a general condition assessment of the ballast bed and the substructure on a large scale. The utilization of the modified fractal analysis should instead be regarded as a supplementary analysis method. By reducing the window size to 12.5 m, new applications of fractal analysis have become possible, including the assessment of ballast bed conditions in switches and crossings. Previously, this was not possible due to the dependence on a larger window size of 150 m. This advance therefore supports cause-based maintenance and assessment practices in these critical areas of railway systems. In conclusion, this approach represents a step towards a more data-driven and predictive maintenance strategy in the railway system.

In order to reinforce the findings of this study, further investigations into the occurrence of local ballast contamination are essential. Therefore, future work should focus on analyzing specific track locations where single failures occur due to various factors, such as differences in stiffness or failures due to poor subgrade conditions. This analysis is necessary to ensure that FRED accurately captures the condition of the ballast bed without being influenced by other issues. By reducing the window size to 12.5 m, new applications of fractal analysis have become possible, including the assessment of ballast bed conditions in switches and crossings. Further development of the methodology has the potential not only to increase the reliability of the rail network, but also to reduce maintenance costs and extend the service life of the railway infrastructure.