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 Accordingly, the ordinary integral and differentiation operations are no longer applicable to the fractal geometry with singularity.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.
Similar content being viewed by others
References Cited
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
Acknowledgments
The research has been jointly supported by the National Key Technology R & D Program of China (No. 2016YFC0600501) and the State Key Program of the National Natural Science of China (No. 41430320). This paper is dedicated to celebrating the 90th birthday of Professor Pengda Zhao, China University of Geosciences. The final publication is available at Springer via https://doi.org/10.1007/s12583-021-1454-7.
Rights and permissions
About this article
Cite this article
Cheng, Q. Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes. J. Earth Sci. 32, 276–278 (2021). https://doi.org/10.1007/s12583-021-1454-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12583-021-1454-7