Fractal analysis of horizontal velocity deformation time series data: an example from open pit mine in Ghana
DOI:
https://doi.org/10.55779/ng61581Keywords:
fractal analysis, time series, mine slope deformation, horizontal velocity, Hurst exponentAbstract
Geodetic deformation data exhibit complexity and roughness in self-similarity. This study applied fractal geometry to analyse and describe the fractal behaviour of geodetic deformation time series datasets to comprehend their long-range dependence. To achieve this, pit wall stability monitoring time series data consisting of horizontal velocity (HV) deformations from an Open Pit Mine in Ghana were analysed using six different methods of estimating Hurst exponent. The methods applied include the range rescale (R.S), Higuchi, aggregated variance (AV), absolute moments (AM), modified periodogram (MP), and residuals of regression (RR). The Hurst estimation methods were applied to five monitoring prisms established on the berms of the pit walls, consisting of 521 HV deformation data points, to examine and determine the fractal behaviour. An empirical comparison and performance evaluation of the results showed that the Hurst estimation methods applied can explain the fractal characteristics of the HV deformations of the pit wall with satisfactory results. However, for the data type utilised, the study identified AM, AV and Higuchi methods as the most reliable, offering guidance for slope stability analyst. The findings further revealed that the HV deformation in the pit was a persistent type, thus proving that fractal geometry can be used to describe the nature of the deformation dataset and that such a dataset is also susceptible to long range dependence.
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References
Abdelrahman MM, Szabó NP (2024). Integrated workflow incorporating the Hurst exponent and interval inversion for evaluating groundwater formations. Hydrogeology Journal 32(2):487-507. https://doi.org/10.1007/s10040-023-02752-0
Alipour MH, Rezakhani AT, Shamsai A (2016). Seasonal fractal-scaling of floods in two US water resources regions. Journal of Hydrology 540:232-239. https://doi.org/10.1016/j.jhydrol.2016.06.016
Barani S, Mascandola C, Riccomagno E, Spallarossa D, Albarello D, Ferretti G, Scafidi D, Augliera P, Massa M (2018). Long-range dependence in earthquake-moment release and implications for earthquake occurrence probability. Scientific Reports 8(1):5326. https://doi.org/10.1038/s41598-018-23709-4
Bui Q, Ślepaczuk R (2022). Applying Hurst Exponent in pair trading strategies on Nasdaq 100 index. Physica A: Statistical Mechanics and its Applications 592:126784. https://doi.org/10.1016/j.physa.2021.126784
Chen C (2018). Hurst parameter estimate (https://www.mathworks.com/matlabcentral/fileexchange/19148-hurst-parameter-estimate), MATLAB Central File Exchange. Retrieved 2025 November 1.
Dlask M, Kukal J (2019). Hurst exponent estimation from short time series. Signal, Image and Video Processing 13:263-269. https://doi.org/10.1007/s11760-018-1353-2
Fuss FK, Weizman Y, Tan AM (2021). The non-linear relationship between randomness and scaling properties such as fractal dimensions and Hurst exponent in distributed signals. Communications in Nonlinear Science and Numerical Simulation 96:105683. https://doi.org/10.1016/j.cnsns.2020.105683
Gómez-Águila A, Trinidad-Segovia, JE, Sánchez-Granero MA (2022). Improvement in Hurst exponent estimation and its application to financial markets. Financial Innovation 8(1):86. https://doi.org/10.1186/s40854-022-00394-x
Hai L, Lv Y, Tan S, Feng L (2023). Study on the influences of the fractal dimension of the root system and slope degree on the slope stability. Scientific Reports 13(1):10282. https://doi.org/10.1038/s41598-023-37561-8
Hao Y, Huang B, Sulowicz M (2024). A practical prediction model for surface deformation of open-pit mine slopes based on artificial intelligence. Elektronika ir Elektrotechnika 30(3):46-53. https://doi.org/10.5755/j02.eie.36642
Hayat U, Barkat A, Ali A, Rehman K, Sifat S, Iqbal T (2019). Fractal analysis of shallow and intermediate-depth seismicity of Hindu Kush. Chaos, Solitons & Fractals 128:71-82. https://doi.org/10.1016/j.chaos.2019.07.029
Hurst HE (1951). Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116(1):770-799. https://doi.org/10.1061/TACEAT.0006518
Kane IL, Usman D (2013). Long memory analysis of daily average temperature time series. International Journal of Environmental Research and Earth Science 1(5):030-032.
Karelina EA, Stepanov AA, Smirnov PI, Podgornuy AV, Guly VV (2023). Detection of abnormal traffic flow with the Hurst parameter for traffic management. Intelligent Technologies and Electronic Devices in Vehicle and Road Transport Complex (TIRVED), Moscow, Russian Federation pp 1-5. https://doi.org/10.1109/TIRVED58506.2023.10332752
Khan A, Hussain S, Bakhet A, Anwer A, Raza SM, Ali S, Zakarya M (2025). Climate time series variability analysis of Islamabad Capital Territory using fractal dimension and Hurst exponent methods. Journal of Atmospheric and Solar-Terrestrial Physics 267:106406. https://doi.org/10.1016/j.jastp.2024.106406
Kon D, Shu J, Han L, Ngoie S (2025). Quantifying failure surface roughness and fractal characteristics in open-pit mining: the case of Mashamba West Mine. Open Journal of Civil Engineering 15(1):91-112. https://doi.org/10.4236/ojce.2025.151006
Krampah F, Amegbey N, Ndur S, Ziggah YY, Hopke PK (2021). Fractal analysis and interpretation of temporal patterns of TSP and PM10 mass concentration over Tarkwa, Ghana. Earth Systems and Environment 5(3):635-654. https://doi.org/10.1007/s41748-021-00237-2
Lei Q, Sornette D (2023). A stochastic dynamical model of slope creep and failure. Geophysical Research Letters 50(11):e2022GL102587. https://doi.org/10.1029/2022GL102587
Mandelbrot BB, Wallis JR (1969) Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence. Water Resources Research 5(5):967-988. https://doi.org/10.1029/WR005i005p00967
Nazarychev SA, Zagretdinov AR, Ziganshin SG, Vankov YV (2019). Classification of time series using the Hurst exponent. Journal of Physics: Conference Series 1328:012056. https://doi.org/10.1088/1742-6596/1328/1/012056
Parsian A (2022). Flood hazard mapping using fractal analysis in a semiarid environment. Global Journal of Environmental Science and Technology 10(3):1-2.
Raubitzek S, Corpaci L, Hofer R, Mallinger K (2023). Scaling exponents of time series data: a machine learning approach. Entropy 25(12):1671. https://doi.org/10.3390/e25121671
Rea W, Oxley L, Reale M, Brown J (2009). Estimators for long range dependence: An empirical study. arXiv preprint arXiv:0901.0762. https://doi.org/10.1142/S0218348X95000692
Rosu IA, Grillakis M, Papadopoulos A, Agop M, Voulgarakis A (2024). Fractal and spectral analysis of recent wildfire scars in Greece. Fire Technology 60(1):167-192. https://doi.org/10.1007/s10694-023-01497-2
Sun S, Su Z, Zhang Y, Tian B, Guo P (2015). Prediction of slope deformation time series based on Quasi-Newton. Proceedings of the 2015 International Conference on Architectural, Civil and Hydraulics Engineering. Series: Advances in Engineering Research, pp 299-304. https://doi.org/10.2991/icache-15.2015.58
Takahashi T, Nakanishi A, Kodaira S, Kaneda Y (2023). Estimation of the Hurst exponents of irregularly sampled subsurface fault geometries by the lifting scheme. Geophysical Journal International 235(2):1102-1116. https://doi.org/10.1093/gji/ggad275
Wu H, Dong Y, Shi W, Clarke KC, Miao Z, Zhang J, Chen X (2015). An improved fractal prediction model for forecasting mine slope deformation using GM (1, 1). Structural Health Monitoring 14(5):502-512. https://doi.org/10.1177/1475921715599050
Xu WY, Meng QX, Wang RB, Zhang JC (2016). A study on the fractal characteristics of displacement time-series during the evolution of landslides. Geomatics, Natural Hazards and Risk 7(5):1631-1644. https://doi.org/10.1080/19475705.2015.1081633
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