Fractal-Based Wavelet Filter for Separating Geophysical or Geochemical Anomalies from Background
- 216 Downloads
The widely used wavelet filtering technique holds potential to approach anomaly–background separation in geophysical and geochemical data processing. Wavelet statistics provide crucial information on such filtering methods. In general, conventional (Gaussian-type) statistical modeling is insufficient to adequately describe the heavily tailed and sharply peaked (at zero) distribution of the wavelet coefficients of irregular geo-anomaly patterns. This paper demonstrates that the cumulative (frequency) number of the wavelet coefficient yields a power-law scaling relationship with the coefficient based on wavelet transform of a fractal/singular measure. This wavelet coefficient–cumulative number power-law model is proven to be more flexible and appropriate than the Gaussian model for characterizing the scaling nature of the coefficient distribution. Accordingly, a fractal-based filtering technique is developed based on the wavelet statistical model to decompose mixed patterns into components based on the distinct self-similarities identified in the wavelet domain. The decomposition scheme of the fractal-based wavelet filtering method considers not only the coefficient frequency distribution but also the fractal spectrum of singularities and the self-similarity of real-world features. Finally, a synthetic data test and real applications from two metallogenic provinces of China are used to validate the proposed fractal filtering method for anomaly–background separation and identification of geophysical or geochemical anomalies related to mineralization and other geological features.
KeywordsWavelet transform Multifractal modeling Fractal filtering Anomaly–background separation
This paper is based on the Ph.D. dissertation of the first author. Thanks are due to Eric Grunsky, Frits Agterberg, Jian Wang, and two anonymous reviewers for their comments that improved this manuscript. This research was jointly supported by the National Key Research and Development Program of China (no. 2016YFC0600508), National Natural Science Foundation of China (nos. 41702355, 416723328, and 41572315), and Fundamental Research Funds for Central Universities (CUG170635) and Most Special Fund from the State Key Laboratory of Geological Processes and Mineral Resources (MSFGPMR09), China University of Geosciences (Wuhan).
- Abry P, Jaffard S, Wendt H (2012) Irregularities and scaling in signal and image processing: multifractal analysis. In: Benoit Mandelbrot: a life in many dimensions. Yale University, pp 31–116Google Scholar
- Agterberg F (2014) Geomathematics: theoretical foundations, applications and future developments. Quantitative geology and geostatistics, vol 18. Springer, BerlinGoogle Scholar
- Chen G (2016) Identifying weak but complex geophysical and geochemical anomalies caused by buried ore bodies using fractal and wavelet methods. Doctor dissertation, China University of GeosciencesGoogle Scholar
- Chen G, Cheng Q (2017) Fractal density modeling of crustal heterogeneity from the KTB deep hole. J Geophys Res Solid Earth 122:1919–1933Google Scholar
- Choi HK, Baraniuk R (1998) Multiscale texture segmentation using wavelet-domain hidden Markov models. In: Conference record of the Asilomar conference, pp 1692–1697Google Scholar
- de Mulder EF, Cheng Q, Agterberg F, Goncalves M (2016) New and game-changing developments in geochemical exploration. Episodes 39:70–71Google Scholar
- Evertszy CJ, Mandelbrot BB (1992) Multifractal measures. In: Peitgen H-O, Jurgens H, Saupe D (eds) Chaos and fractals. Springer, New YorkGoogle Scholar
- Jansen M (2001) Noise reduction by wavelet thresholding. Lecture notes in statistics, vol 161Google Scholar
- Mallat S (1999) A wavelet tour of signal processing. Academic Press, San DiegoGoogle Scholar
- Mandelbrot B (1983) The fractal geometry of nature. W.H Freeman and Company, New YorkGoogle Scholar
- Sornette D (2004) Critical phenomena in natural sciences: chaos, fractals selforganization and disorder: concepts and tools. Springer, New YorkGoogle Scholar