Development of a method for clusterization of the dissipative structures

The dissipative structure (DS) may follow irreversible processes, which occur in the open thermodynamic systems passing and transforming a sufficiently intensive flow of energy. DS can be detected as certain patterns of spontaneous organization in space and time. We consider the DS, which transposes during irreversible ground movement due to landslide or ground pressure manifestation. However, DSs may hide under stochastic noise that impedes their detection. We developed a new method for revealing of DSs in such a stochastic environment using a combination of k-mean clustering, Voronyi tessellation, and the optimal schedule of ground movement monitoring. The developed method has been successfully tested in the case of the irreversible ground movement in the vicinity of a longwall face.


Introduction
The majority of processes in the real life are irreversible.According to the modern theory of thermodynamics [1], open systems may spontaneously generate DSs if they pass and transform a sufficiently intensive flow of energy [2].Investigation of DS promotes a better understanding of the deep mechanism and facilitates more effective control of the complex irreversible processes.In this paper, we consider such processes as landslides [3], sinkholes, and ground movement due to underground mining [4], which expose the environment to the big hazards but effective control of which is far from excellence.A distinctive feature of these processes is that DSs are hidden and it is not easy to reveal them confidently.Traditional approaches propose to describe these processes as definite events allowing their certain parameters [5,6].However the modern publications describe the ground movement as a stochastic process [7] and consider its parameters as probabilistic entities [8].As far as we know, DS has not been revealed during irreversible movement of the ground so far.Recently, DSs were detected for the first time during instrumental observation of a slow landslide development [3] and microseismicity monitoring of the hydrofracturing process [4].
This presentation aims to develop a method that is available to detect DS reliably and unambiguously in processes of ground movement when DS patterns are hidden or veiled under stochastic noise.

Interrelation between components of the method
The history of irreversible ground movement depends on the loading path [9].Therefore, one should monitor the irreversible ground movements as accurately as possible to detect essential features of the irreversible process.Physical modeling is the relevant approach to solve this task considering the trade-off between cost and results.For that reason, we investigated the irreversible ground movement on a physical model constructed of synthetic materials [10].
The height, width, and thickness of the model were 380, 360, and 19 mm correspondingly.We investigated the behavior of an underground opening in the geometric scale of 1:35.8.The synthetic material was made of sand, clay, chalk, and water in proportion 92.26:2.77:1.37:3.6 respectively.The thickness of the rock layers was 2.5-10 mm in the model or 0.089-0.358m in situ.
The model has been consolidated under the pressure of 89.8 kPa during 15 minutes under vibration with the frequency of 50 cycles and magnitude of 2 mm.The applied pressure was three times as much compared in situ level of the ground pressure.This eliminated possible errors of the ground behavior modeling at the areas where abutment pressure concentrates.The uniaxial compressive strength of the rock was 0.67 MPa in the model or 41 MPa in the real rock mass.Ground pressure was simulated at the top of the model applying the load from 27 to 31 kPa.The movement of the surrounding opening rocks was monitored by a digital camera locating the position of special marks, whereby the standard deviation of the displacement measurement was 0.39 mm or 2.5 pixels on the digital frames.A confidence interval of four, five, and six pixels corresponded to 90.7%, 95.6%, and 99% reliability of the displacement measurement.
The trajectories of the three marks during model testing are demonstrate in figure 1.The marks' coordinates are indicated in pixels.Ten millimeters in the digital picture corresponded to 12 pixels.A separate section of a trajectory reflects a path of the mark between consecutive pictures, which were made on the basis of the constant time interval.We characterize these elementary paths as incremental displacements.It may be seen that the marks change both directions of movement and velocity.
It is important to emphasize the stochastic nature of the displacements although the general tendency of the movement was down to the opening since the marks were selected in its roof.At the same time, one may notice the transversal movements of the marks, which are impossible to disregard.The transversal deviation of the marks is significant and their neglection may cause loss of important information that may be crucial for DS detection.
In the left fragment of figure 1, position 1 indicates the trajectory of the first mark, intermitted line 2 presents the magnitudes of the incremental displacements, and vertical line 3 limits of 90.7% confidence interval.Comparison of lines 2 and 3 shows that less than 10% of the measurements were made with insufficient confidence.However, an increase in the confidence interval simplifies the trajectory and many essential features of the trajectory might be lost.The more interval between successive sessions of monitoring the less valuable information remains.Frequently, researchers monitor the initial and the final state of a landslide that completely deprives any possibility to reveal DS [11,12] (see the intermitted arrow in figure 2,c).That is why revealing DS in the ground movement process needs a special approach.First the most informative is the irreversible ground movement because it complies with the thermodynamics of irreversible processes [13].According to Glansdorff et al [13], an open thermodynamic system strives to minimize the production of entropy if it is not far from equilibrium.The entropy is calculated as the production of thermodynamic forces and flows.Ground pressure and gravity stand for the thermodynamic forces and irreversible ground displacements for the thermodynamic flows.
Second, it is crucial to minimize the error of ground displacement measurements to reveal DS as a fine natural structure, which may hide in the stochastic noise.Evidently, the probability of DS detection is reversely proportional to the error of measurement.
Third, the periods of sequential monitoring sessions should be minimized and complied with the errors of measurements.There is an optimal period that maximizes the probability of DS revealing: a very short interval between sequential measurements increases the cost of monitoring whereas expansion of this period increases the probability that DS will be lost or mixed with other DS.
Finally, the density of the marks or monuments (in situ) in space should be optimized to prevent losses of the valuable information on one hand and to minimize the cost of the monitoring on the other.The distance between adjacent monuments should be commensurate with the average dimension of the blocks comprising the ground body: shorter distance increases the cost of the detection but the oversized arrangement may cause to lose DS.As the first approximation, the distance between adjacent monuments should be in a range from 2 to 10 of the block's dimension.All the aforementioned conditions should be satisfied to develop a reliable method of DS detection.

Development of the method
We used a powerful method of variogram analysis [14] and k-means clustering [15].The variogram is a statistical second-order moment that is widely used during simulation and analysis of spatial correlation.The variogram 2γ(x, x + h) for the value of a spatial variable Z(x) at the two points x and x + h, which a separated by a vector h, is expressed by variation of the variable difference in the abovementioned points: where µ denotes an expected value.Method k-means has been used for simultaneous variance minimization of distances between vectors within a cluster and maximizing of the distance variance between the clusters' centers.The distance between the vectors is determined in Euclid space that has arbitrary dimensions.Importantly, the task of classification of the random variables, and particularly for the detection of DS on a vector mosaic of the incremental displacements of the ground or the rock mass, has no single-entry solution.That is why the possible number of the clusters was first set equal to two and then increased until a certain condition has been satisfied.We controlled the classification process by the variance dynamics of the distances from the clusters' centers to the common center.Notably, we used interpolated field of the ground displacements to calculate the variogram, whereas only factual changes of the monuments' coordinates were involved for the cluster analysis.
The results of calculating the distances between clusters in Euclidean space at the stage when there were seven clusters are shown in the table 1.As the number of clusters increased, the length and orientation of the vectors changed as may be seen in figure 3. Some vectors demonstrated certain stability whereas the other disintegrated to the components.Essential variability of the vectors took place at the initial stages of clusterization.
Thus the only vector marked by a triangle remained almost unchanged when the number of the vectors increased from five to seven.The other vectors changed their length and/or orientation.The process of stabilization of the distances' variances occurs both between the vectors in every cluster and among the clusters as is illustrated in figure 4. The distance variance   As it can be seen the stabilization of the variances occurs for both components of the vectors after the number of the clusters increased to nine.Nonetheless, further prolongation of the clustering has shown that the k-means algorithm finds a more fine difference between clusters as the number of clusters grows.Evidently, the algorithms will start to classify the vector mosaic according to the signs, which have not physical but stochastic nature.Therefore, an auxiliary criterion is needed to determine the moment when the clusterization process should be terminated.
According to thermodynamics, the most probable state of ground corresponds to the 6 maximum of its entropy [13].This state can be expressed by the Boltzmann-Shannon formula: where S i is entropy, p j is the probability that j-th cluster will emerge, k is a constant, N is the total number of clusters.
The entropy S i reaches the maximum when probabilities of all clusters will be equal, namely: In this case the entropy production dS i dt = 0.In this situation, the results of monitoring have a certain error, and condition (3) is impossible to reach.Nevertheless, this condition is a threshold to which cluster probabilities may be reduced.In order to do this, we counted the number of cases when the same cluster occurred due to the increase of the clusters' number in the limits of measurement error.
Then we calculated the entropy according to formula (3) and reduced it to S i .Tests demonstrated that it is possible to fix the moment when the entropy reaches the extremum and this is expedient to terminate the clustering process.
Another problem concerns the determination of the cluster boundaries.The practice has shown that adjacent clusters overlap and this introduces uncertainty.To solve this problem, we used Voronyi tessellation [16] that determines a crisp boundary between adjacent clusters as the lines, which are normal to the lines joining the clusters' centers.These locations of the boundaries correspond to condition (3) that minimizes the error of measurement of the monuments' position.

Testing of the method: a case study
We tested the developed model on a case of rock mass subsidence in the vicinity moving longwall face.A base friction modeling has been used to imitate the ground displacement due to a coal seam extraction.
Rock mass was modeled with discrete elements, which have been stacked on a desk within a rectangular frame.The gravity was simulated by the traction of the frame along the deck.We regulated the rate of the longwall movement by the incremental step of its advance along the coal seam.The more the step was between serial pictures the more the rate of the longwall advance.
At the first glance, the appearance of the model on both pictures in figure 5 is identical.However, the incremental displacements are essentially different.
The incremental displacements of the marks in the moments when the longwall was replaced from positions 5852 to 5853 and 5853 to 5854 respectively demonstrate figure 6, figure 7. The displacements, which are confined within the boundaries were registered with 90% reliability.The overlay of the vector displacements and Voronyi tessellation are depicted in figure 7.
Meaningful clusters of DS are marked with numbers from 1 to 6.The distances between clusters were calculated according to both orientations of the displacement vectors and their magnitudes (table 2).Seemingly, some clusters may contain different vectors (figure 7).
However, k-means method ensures that the difference among the vectors within a cluster will be minimal according to all their parameters.Analysis has shown that the developed method is efficient because the classification grasps the physical meaning of the irreversible process of the ground movement.For instance, clusters 1 and 2 are found beyond the zone of active movement.These clusters moved to the gob or zone of the smooth sagging where the ground pressure was on the way to its recovery.Displacement vectors of cluster 2 turned to the longwall advance direction whereas cluster 1 oriented downward and slightly to the gob.Cluster 4 is located at   Let us pay attention to the position of the immediate roof that was at the distance of 4 m from the longwall face or at the rear part of the powered support canopy when the face was in the previous position (5852).However, this roof kept still hanging after the next step advance of the longwall to the distance of 4 m.It would seem the incremental displacements of the rock mass may be neglected, although DS can evolve essentially even for such a short period.
The dramatic difference between the displacement distribution is that all vectors are oriented downward and to the left or the gob in figure 6 whereas they turned to the right of the face at the next step increased by only 4 m.In addition, the area of the rock mass expanded where the incremental displacements were detected with the confidence of more than 90%.Such a difference is not random and may be explained from a physical point of view.
The maximal incremental displacements occur in the zone of maximum sag and possible delamination of the rock layers.The pictures in figure 5 and distributions in figure 6 highlight this zone that is inclined to the horizon under the angle of 700.This complies with a common vision of the subsidence mechanism.The strata caves as cantilever beams, which disconnect from the solid rock mass on one hand and rotate in figure 5 clockwise on the other.This rotation brings closer the rear end of the cantilever beams to the longwall face.These two opposite incremental movements prevailed by turn, in successive order: at the stage 5852-5853 the disconnection component moved the undermined strata to the left or the gob while the strata shifted to the right at the next step 5853-5854.Such maneuvers become possible due to DS evolution and our method allowed distinguishing this fine behavior of the rock mass.Let us stress that almost all displacement vectors abruptly turned from the left to the right dramatically changing the pattern of DS.
Based on [13] it may be suggested that such a sharp replacing of the DS pattern might be caused by a small fluctuation of the ground pressure.Some displacement vectors change their orientation to 10 pixels exceeding the error of measurements by order.

Conclusion
The ground movement is a typical irreversible process that exposes the environment to hazards during landslides, sinkholes, hydrofracturing, and other dangerous and poorly predicted events.According to thermodynamics, such irreversible processes may be followed by DS, which can be used to control the ground effectively.However, DSs are difficult to detect and identify because of the stochastic nature of the irreversible ground movement.
We have formulated several conditions that should be satisfied to develop a reliable method of DS detection.First the most informative is the irreversible ground displacements because they comply with the thermodynamics of irreversible processes.The irreversible displacements represent the thermodynamic flows, which generate the entropy acting together with the thermodynamic forces that are expressed as the ground pressure.Second, it is crucial to minimize the error of ground displacement measurements to reveal DS as a fine natural structure, which may hide in the stochastic noise.Third, the periods of sequential monitoring sessions should be minimized and complied with the errors of measurements.There is an optimal period that maximizes the probability of DS revealing: a very short interval between sequential measurements increases the cost of monitoring whereas expansion of this period increases the probability that DS will be lost or mixed with other DS.Finally, the density of the marks or monuments in space should be optimized to prevent losses of the valuable information on one hand and to minimize the cost of the monitoring on the other.The distance between adjacent monuments should be in a range from 2 to 10 of the rock mass block's dimension.
We used the variogram algorithm and K-means method of clustering to detect DS.The number of the clusters was assigned starting from two and sequentially increased until the minimum distance variance between the displacement vectors within the clusters and maximum of the variance between clusters stabilize.We proposed an auxiliary criterion for termination of the clustering process as the minimum entropy production.
The developed method of DS clustering has been tested successfully in the case of the irreversible ground displacement around moving longwall face.The new method not only revealed specific patterns of the DS but complied with the physical sense of the irreversible ground movement.

Acknowledgments
This research has been supported by grant 0123U100356, National Academy of Sciences, Ukraine.

Figure 1 .
Figure 1.Trajectory of the selected marks.

Figure 3 .
Figure 3. Evolution of the cluster parameters due to increasing of their number.

Figure 4 .
Figure 4. Illustration of the variance stabilization.

Figure 7 .
Figure 7. Overlay of incremental displacements of the rock mass between positions 5853 and 5854 and Voronyi mosaic.

Table 2 .
Distance between meaningful clusters.