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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. doi: 10.1007/s12583-021-1454-7

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

doi: 10.1007/s12583-021-1454-7
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  • 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 Accordingly, the ordinary integral and differentiation operations are no longer applicable to the fractal geometry with singularity.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.
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  • 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 doi:  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 doi:  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 doi:  10.1029/JB082i005p00803
    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 doi:  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 doi:  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)
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Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes

doi: 10.1007/s12583-021-1454-7

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. doi: 10.1007/s12583-021-1454-7
  • Classical calculus such as differential equations based on Newton theory have been extensively applied to describe the motion of the plates in plate tectonics including mantle convection, plate subduction, heat flow and ocean floor depth over the mid ocean ridges, as well as basin formation during the whole Wilson circle of plate tectonics (McKenzie, 2018). These models have been commonly used in extrapolations of data to form global distribution of heat flow or temperature of the Earth surface and the estimated temperature of mantle from the surface measurements of heat flow. However, it has been pointed out that these types of models are less successful in fitting energy and mass distributions caused by singular events occurred during the multiplicative cascade plate tectonics and in respond to self-organized criticality (SOC) and phase transition (Cheng, 2018, 2016). There are several potential causes that might limit these models: one of them is due to model resolution and data available for model validation and refining, the second reason can be the model itself which may not consider all necessary factors and parameters (Parsons and Sclater, 1977) and the other cause to be considered here is due to the limitation of the parameters (e.g., density) utilized in these models. The ordinary density used in these models are usually valid for normal regime with regular geometry without singularities. However, for matters with fractal structures or geological anomalies such as the oceanic plate boundaries over the mid-ocean ridges caused by phase transition and exposure of asthenosphere on the surface, ordinary density can no longer be used to characterize the lithosphere (Cheng, 2016). Geological anomalies refer to such geologic features or structures that depart markedly from their surrounding environment with respect to composition, texture, or genes (Zhao, 1998). This paper introduces new concepts and formulations of fractal differentiation and integral operations as well as fractal density. These concepts and methods are used to analyze the Bouguer gravity anomalies over the mid-ocean ridges.

  • Integral and differentiation are two well-known mathematical operations introduced in modern calculus. Both integral and differentiation calculus can be considered as measures defined based on geometry. The former concerns accumulation of quantities, and areas under or between curves. The latter concerns instantaneous rates of change, and the slopes of curves. 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 are geometries with fractal dimension (e.g., non-integer) which no longer possesses Lebesgue additive property (Schertzer et al., 1997). How to define similar calculus operations as integral and differentiation is a challenging task which has not been fully tackled in the literature. Due to the regularity of fractals, the ordinary integral and differentiation operations are no longer applicable to the fractal geometry with singularity (Cheng, 2018). The author of the current has proposed a way to define fractal integral and differentiation which will be introduced below. The fractal integral and fractal differentiation operations defined by the author can be considered as an extension of the ordinary integral and differentiation operations based on fractal measure instead of Lebesgue set. The concept can be found in Cheng (2018) and details are given below,

    where Δf(x) represents the increment of function f(x) over an increment of Δx. The convergence of the limit of form (1) can be defined as α-fractal dimensional derivative of function f(x) (α is a parameter with real value). Similarly, we can define the fractal dimensional integral of function f(x) as follows

    where f(xi) is the average height of function f(x) in the small range of [xi, xix]. If the summation in Eq. (2) converges while Δx→0, then the converged limit can be named the α-fractal dimensional integral of the function f(x). The notations of the fractal differentiation are illustrated in Figure 1. The curve is differentiable everywhere except at x0 where fractal differentiable.

    Figure 1.  Illustration of differentiation of a function. At the location of x0 the ordinary differentiation does not exist.

    It must be reminded that the fractal derivative operation defined in this paper is different from the fractional derivative (fractional order) defined in the literature (Dalir and Bashour, 2010). The fractional derivative operation assumes that the normal integer order derivative exists. The fractal derivative operation defined in Eq. (1) is based on fractal dimension of the measure whereas the fractional derivative operation is based on fractional order of derivative operation defined on normal measure.

  • Density is a fundamental physical parameter or variable involved in most geodynamics models for simulation and prediction in the earth sciences. Since the principle of density was discovered by the Greek scientist Archimedes approximately 2000 years ago, density has become a fundamental property of mass or energy as a well-known physical concept with a variety of applications. The volumetric density of a material is defined as its mass per unit volume which can be expressed as the first-order derivative of the mass over volume which is independent of scale v,

    The preceding density exists only if the limit converges when the volume becomes infinitesimal. It has been demonstrated that the limit in Eq. (3) does not always converge for complex objects with fractal properties (Cheng, 2016). A new fractal density can be defined as its mass per unit "fractal set" which can be expressed as fractal derivative,

    The value of can be considered as generalized density because the ordinary density defined in Eq. (3) becomes a special case of Eq. (4) when α=3. Combining Eqs. (3), (4) gives the association of the ordinary and fractal densities,

    where ∝ stands for proportional of two variables. The relation (5) indicates that the ordinary density, expressed as first-order derivative of mass over volume, becomes scale dependent and divergent with decrease of the volumetric scale, thus no longer valid for characterizing matters with fractal structure.

  • Mid-ocean ridges occur along divergent plate boundaries, where tectonic plates spread apart, and new ocean crusts form from volcanism under water. Uneven speeds of spreading segments of ridges adjusted by transform faults form zig-zag type of ridges and the speeds of spreading affects the shape of a ridge. Slower spreading plates result in steep and irregular topography while faster spreading plates produce wider and gentle slopes. The Mid-Atlantic Ridge and the East Pacific Rise are the two well-known global scale ridges. The Mid-Atlantic Ridge spreads slowly at a rate of 2 to 5 centimeters per year and forming a rift valley and irregular margins. Several studies indicated that the low Bouguer gravity anomaly over oceanic rifts could reflect that the mass density of lithospheric over these areas are of significant variability due to the change of rock porosity, fracturing, serpentinization of mantle peridotite, crustal thickening, temperature increase by asthenosphere and pressure reduction caused by extensional force at the divergent boundaries (Cheng, 2016). The mid-ocean ridges depict complex property which has been characterized by singularity of Bouguer gravity anomaly. The Bouguer anomaly data used in Talwani et al. (1965) are reexamined here. Figure 2a shows the Bouguer gravity anomalies across the north mid-Atlantic ridge. Figure 2b shows the relation between the Bouguer anomalies and the distance from the mid-Atlantic ridge. The curves fitted by least squares methods indicate that the relations can be described by power-law functions, y=154.83x0.079, with an exponent (singularity value) 0.079 and coefficient of determination, R2=0.945. The derivative of the above power-law is y'=12.39x-0.921 which approaches infinity when distance x approaches to zero. This means the first order derivative of the function does not converge around the ridges. Using Eq. (1), the fractal derivative of the function at the ridges is y'0.921=12.39. This example shows that the singularity of the low Bouguer gravity anomaly caused by the irregular structure of mid-Atlantic ridges can be characterized by fractal differentiation at 0.921-dimensional space. This example may imply that the lithosphere density over the ridges with strong singularity must be treated as fractal density. Further discussions about the fractal density of lithosphere over the mid-ocean ridges and its implication will be published elsewhere.

    Figure 2.  Bouguer gravity anomalies across the north mid-Atlantic ridge (a) and plot showing the relation between the Bouguer anomalies and the distance from the mid-Atlantic ridge (b). The data were from Talwani et al. (1965). The red curve is fitted by least squares method with a power-law function. Dots represent the observed values.

  • New fractal differentiation and integral operations introduced in the current paper can be utilized to describe the nonlinear properties of fractal geometry. Particularly fractal differentiation calculus is useful for defining fractal density as fractal derivative of mass over fractal geometry. Bouguer anomalies cross the mid-Atlantic ridge were used for validation and demonstration purposes. The singularity found in the Bouguer anomalies cross the mid-Atlantic ridge indicates that the lithosphere density over the mid-Atlantic ridge depicts singularity around the ridges which must be treated as fractal density rather than the ordinary density. The singularity should be considered in the thermal models for prediction of heat flow and depth of ocean floor over the mid-ocean ridges. The fractal calculus concepts and operations proposed in the current are appliable to characterize general geological anomalies.

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