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Volume 32 Issue 2
Apr.  2021
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Qiuming Cheng. Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes. Journal of Earth Science, 2021, 32(2): 276-278. doi: 10.1007/s12583-021-1454-7
 Citation: Qiuming Cheng. Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes. Journal of Earth Science, 2021, 32(2): 276-278.

# Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes

##### doi: 10.1007/s12583-021-1454-7
• Integral and differentiation are two mathematical operations in modern calculus and analysis which have been commonly applied in many fields of science. Integration and differentiation are associated and linked as inverse operation by the fundamental theorem of calculus. Both integral and differentiation are defined based on the concept of additive Lebesgue measure although various generations have been developed with different forms and notations. Fractals can be considered as geometry with fractal dimension (e.g., non-integer) which no longer possesses Lebesgue additive property. Accordingly, the ordinary integral and differentiation operations are no longer applicable to the fractal geometry with singularity. This paper introduces a recently developed concept of fractal differentiation and integral operations. These operations are expressed using the similar notations of the ordinary operations except the measures are defined in fractal space or measures with fractal dimension. The calculus operations can be used to describe the new concept of fractal density, the density with fractal dimension or density of matter with fractal dimension. The concept and methods are also applied to interpret the Bouguer anomaly over the mid-ocean ridges. The results show that the Bouguer gravity anomaly depicts singularity over the mid-ocean ridges. The development of new calculus operations can significantly improve the accuracy of geodynamic models.
•  Cheng, Q. M., 2016. Fractal Density and Singularity Analysis of Heat Flow Over Ocean Ridges. Scientific Reports, 6(1): 1-10. https://doi.org/10.1007/978-3-319-45901-1_41 Cheng, Q. M., 2018. Mathematical Geosciences: Local Singularity Analysis of Nonlinear Earth Processes and Extreme Geological Events. In: B. S. Daya Sagar, Qiuming Cheng, Frits Agterberg eds., Handbook of Mathematical Geosciences: Fifty Years of IAMG. Springer, 179-208 Dalir, M., Bashour, M., 2010. Applications of Fractional Calculus. Applied Mathematical Sciences, 4(21): 1021-1032 McKenzie, D., 2018. A Geologist Reﬂects on a Long Career. Annual Review of Earth and Planetary Sciences, 46: 1-20. https://doi.org/10.1146/annurev-earth-082517-010111 Parsons, B., Sclater, J. G., 1977. An Analysis of the Variation of Ocean Floor Bathymetry and Heat Flow with Age. Journal of Geophysics Research, 82: 803-827 Schertzer, D., Lovejoy, S., Schmitt, F., et al., 1997. Multifractal Cascade Dynamics and Turbulent Intermittency. Fractals, 5(3): 427-471. https://doi.org/10.1142/s0218348x97000371 Talwani, M., Le Pichon, X., Ewing, M., 1965. Crustal Structure of the Mid-Ocean Ridges: 2. Computed Model from Gravity and Seismic Refraction Data. Journal of Geophysical Research, 70(2): 341-352. https://doi.org/10.1029/jz070i002p00341 Zhao, P. D., 1998. Geological Anomaly Theory and Prediction of Mineral Deposits: Modern Theory and Methods for Mineral Resources Assessments. Geological Publishing House, Beijing (in Chinese)
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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## Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes

##### doi: 10.1007/s12583-021-1454-7
###### Corresponding author:Qiuming Cheng, qiuming.cheng@iugs.org

Abstract: Integral and differentiation are two mathematical operations in modern calculus and analysis which have been commonly applied in many fields of science. Integration and differentiation are associated and linked as inverse operation by the fundamental theorem of calculus. Both integral and differentiation are defined based on the concept of additive Lebesgue measure although various generations have been developed with different forms and notations. Fractals can be considered as geometry with fractal dimension (e.g., non-integer) which no longer possesses Lebesgue additive property. Accordingly, the ordinary integral and differentiation operations are no longer applicable to the fractal geometry with singularity. This paper introduces a recently developed concept of fractal differentiation and integral operations. These operations are expressed using the similar notations of the ordinary operations except the measures are defined in fractal space or measures with fractal dimension. The calculus operations can be used to describe the new concept of fractal density, the density with fractal dimension or density of matter with fractal dimension. The concept and methods are also applied to interpret the Bouguer anomaly over the mid-ocean ridges. The results show that the Bouguer gravity anomaly depicts singularity over the mid-ocean ridges. The development of new calculus operations can significantly improve the accuracy of geodynamic models.

Qiuming Cheng. Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes. Journal of Earth Science, 2021, 32(2): 276-278. doi: 10.1007/s12583-021-1454-7
 Citation: Qiuming Cheng. Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes. Journal of Earth Science, 2021, 32(2): 276-278.
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