Fractal-Based Wavelet Filter for Separating Geophysical or Geochemical Anomalies from Background

Abstract

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.

Appendix A: Derivation of Fractal Model of Wavelet Coefficients

Suppose that a measure follows a multifractal distribution with continuous dimension (or singularity) spectrum $f(\alpha )$. The wavelet transform of such a measure also generates wavelet coefficients with multifractal properties or multiple singularities. As explained in Mallat and Hwang (1992), the small coefficients generally have low singularity strength, being due to the details or noise, whereas the large coefficients show strong singularity, as they are due to significant features. Therefore, two special groups of wavelet coefficients can be considered according to their degree of singularity: (a) h relatively close to the maximum value $h_{\max }$, and (b) h relatively close to the minimum value $h_{\mathrm{min}}$. Note that the following derivation of the W–N fractal model follows the idea for demonstrating the C–A fractal model; much more detail is available in Cheng et al. (1994).

According to Eq. (11), the cumulative number of subsets having singularity $h_{\max }^{-} \le h\le h_{\max }$ can be expressed as

$$\begin{aligned} N(\ge h_{\max }^{-} )=\int \limits _{h_{\max }^- }^{h_{\max }} {Ca^{-D(h)}} \mathrm{d}h, \end{aligned}$$

(A-1)

where C is a constant. Neglecting higher-order terms in the Taylor’s expansion, it approximately follows that

$$\begin{aligned} N(\ge h_{\max }^- )\approx C\left[ a^{-D(h_{\mathrm{max}} )}-a^{-D(h_{\max }^- )}\right] \big /\left[ {D}'(h_{\max })\ln (a)\right] . \end{aligned}$$

(A-2)

Then, use of the Lagrange mean value theorem gives

$$\begin{aligned} N(\ge h_{\max }^- )\approx & {} Ca^{-D(h_{\max })}(h_{\max } -h_{\max }^- ) \nonumber \\\approx & {} Ca^{-D(h_{\max })}[\ln (W_{\max } /C_1 )-\ln (W/C_1 )], \end{aligned}$$

(A-3)

where $C_{1}$ is a constant satisfying the relation $W_{\max } =C_1 a^{h_{\max } -1}$. Since each coefficient has an individual singularity h, thus $N(\ge h_{\mathrm{max}}^- )=N(\ge \lambda _1 )$, where $\lambda _1$ is a certain W-value. Using the approximation ln$(1+x)=x$ if $\left| x \right| \ll 1$, then log transformation of both sides of Eq. (A-3) gives

$$\begin{aligned} \ln N(\ge W)=\ln [Ch_{\max } a^{-D(h_{\max } )}]-\ln (W/C_1 )/\ln (W_{\max } /C_1 ). \end{aligned}$$

(A-4)

Therefore, the frequency number of the wavelet coefficients having smaller singularity obeys a fractal power-law relation of

$$\begin{aligned} N(\ge W)\propto W^{\alpha _2 }, \end{aligned}$$

(A-5)

where the scaling exponent is $\alpha _2 =1/\ln (W_{\mathrm{max}} /C_1 )=1/[h_{\max } \ln (a)]$, being related to the maximum Hurst regularity.

Similarly, an approximate relationship between the total number of subsets $h_{\mathrm{min}}\le h<h_{\min }^+$ and the wavelet coefficient W can be written as

$$\begin{aligned} N(<h_{\min }^+ )=C\int \limits _{h_{\min }}^{h_{\min }^+ } {a^{-D(h)}} \mathrm{d}h. \end{aligned}$$

(A-6)

Therefore,

$$\begin{aligned} N(\ge W)\approx N(T)-Ca^{-D(h_{\mathrm{min}} )}(h_{\min }^{+} -h_{\min } ), \end{aligned}$$

(A-7)

where N(T) denotes the total number of all coefficients. It follows that

$$\begin{aligned} \ln N(\ge W)=\ln N(T)-Ca^{-D(h_{\mathrm{min}} )}/N(T)[h_{\min } {+}\ln (W/C_1 )/\ln (1/a)]. \end{aligned}$$

(A-8)

Consequently, the frequency number of wavelet coefficients with larger singularity obeys a fractal power-law relation of

$$\begin{aligned} N(\ge W)\propto W^{\alpha _1 }, \end{aligned}$$

(A-9)

where $\alpha _1 =Ca^{-D(h_{\mathrm{min}} )}/[N(T)\ln (1/a)]$, being related to the minimum Hurst exponent.