Multifractal Analysis of Geological Data Using a Moving Window Dynamical Approach
Abstract
1. Introduction
2. Methodology
2.1. Dynamical Approach Method
2.2. Moving Windows-Based Strategy
- First phase: Evaluate , which is the average fractal dimension associated with the data within the window , centered at . Thus, is related to (see Figure 1).
- Second phase: Define a new set of windows , centered at . This new set of windows is obtained by shifting all windows from the previous set by positions.
- Third phase: Evaluate , which is the average fractal dimension associated with the data within the window , centered at . Thus, is related to .
- Fourth phase: Repeat the second and third phases for subsequent steps, until the step is reached. Hence, a vector with fractal dimensions , respectively, associated with the vector is obtained.
- •
- The number of windows W. There must be a balance between computational efficiency and the accuracy of the vector . A larger W will comprise more data points, potentially improving the reliability of the fractal dimension estimates for . However, it will also increase the computational load. Thus, selecting W involves a trade-off: minimizing computational costs without compromising the accuracy of the estimates for .
- •
- The window size. The window size should be selected based on the desired resolution and the intrinsic characteristics of the data. Larger windows capture broader trends in fractal dimensions but may smooth out local details. Smaller windows are more sensitive to local variations but may be more susceptible to noise and imprecision inherent in the calculation of fractal dimension with few data points.
- •
- The step size . This parameter dictates the amount by which the windows are shifted along the dataset at each step. Typically, is less than or equal to the length of the smallest window to ensure overlapping, which provides smoother transitions between segments. The choice of is a trade-off between available computational resources and the desired resolution for locating points that indicate a change in the fractal behavior of the signal.
3. Validation
- •
- Number of windows. After several preliminary numerical tests, it was verified that six windows provided a good precision for the evaluation of fractal dimensions without significantly increasing the computational costs.
- •
- Window size. Two analyses were conducted to evaluate the methodology using windows with smaller lengths and larger lengths. For smaller windows, six windows were used with lengths of 401, 501, 601, 701, 801, and 901 data points. For larger windows, sizes of 1001, 1101, 1201, 1301, 1401, and 1501 data points were adopted.
- •
- The step size. To ensure a robust description of the multifractality, the number of steps was adjusted according to the domain size. By using , it resulted in 155 components for smaller windows and 149 for larger ones, providing sufficient detail for the vector .
3.1. Case 1
3.2. Case 2
3.3. Case 3
3.4. Case 4
4. Multifractality of Geological Data
- Number of windows. This number was kept at 6 (six).
- Window size. Smaller windows were used (401, 501, 601, 701, 801, and 901 data points) to focus on local variations in fractal behavior.
- The step size. A step size of was used to provide sufficient detail for the fractal dimension vector .
4.1. Evaluation of Well #1
4.2. Evaluation of Well #2
4.3. Evaluation of Well #3
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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401 Points | 501 Points | 601 Points | 701 Points | 801 Points | 901 Points | Set of Small Windows | |
---|---|---|---|---|---|---|---|
Mean | 1.5022 | 1.5033 | 1.5025 | 1.5025 | 1.5018 | 1.5009 | 1.5022 |
Standard deviation | 0.0129 | 0.0117 | 0.0106 | 0.0086 | 0.0085 | 0.0081 | 0.0063 |
1001 Points | 1101 Points | 1201 Points | 1301 Points | 1401 Points | 1501 Points | Set of Larger Windows | |
---|---|---|---|---|---|---|---|
Mean | 1.5006 | 1.5006 | 1.5002 | 1.5011 | 1.5001 | 1.5007 | 1.5005 |
Standard deviation | 0.0063 | 0.0057 | 0.0056 | 0.0043 | 0.0043 | 0.0046 | 0.0028 |
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Silva, G.; Miranda, F.P.d.; Michelon, M.; Ovídio, A.; Venturelli, F.; Moraes, L.; Ferreira, J.; Parêdes, J.; Cury, A.; Barbosa, F. Multifractal Analysis of Geological Data Using a Moving Window Dynamical Approach. Fractal Fract. 2025, 9, 319. https://doi.org/10.3390/fractalfract9050319
Silva G, Miranda FPd, Michelon M, Ovídio A, Venturelli F, Moraes L, Ferreira J, Parêdes J, Cury A, Barbosa F. Multifractal Analysis of Geological Data Using a Moving Window Dynamical Approach. Fractal and Fractional. 2025; 9(5):319. https://doi.org/10.3390/fractalfract9050319
Chicago/Turabian StyleSilva, Gil, Fernando Pellon de Miranda, Mateus Michelon, Ana Ovídio, Felipe Venturelli, Letícia Moraes, João Ferreira, João Parêdes, Alexandre Cury, and Flávio Barbosa. 2025. "Multifractal Analysis of Geological Data Using a Moving Window Dynamical Approach" Fractal and Fractional 9, no. 5: 319. https://doi.org/10.3390/fractalfract9050319
APA StyleSilva, G., Miranda, F. P. d., Michelon, M., Ovídio, A., Venturelli, F., Moraes, L., Ferreira, J., Parêdes, J., Cury, A., & Barbosa, F. (2025). Multifractal Analysis of Geological Data Using a Moving Window Dynamical Approach. Fractal and Fractional, 9(5), 319. https://doi.org/10.3390/fractalfract9050319