Correlation coefficient describes the degree of closeness between two variables, which is generally represented by r. The value of r is between -1 and +1, and the closer the value is to -1 and 1, the closer the relationship is. The correlation coefficient accurately reflects the linear correlation between two variables in a numerical way, to infer the overall situation. Depending on different types of variables, different correlation coefficients are selected for calculation. Correlation coefficients are mainly divided into Pearson simple correlation coefficient, Spearman grade correlation coefficient and Kendallτ correlation coefficient^[10].

Calculating the correlation coefficient of two things can determine the strength of correlation. The higher the correlation coefficient is, the stronger the correlation is. While, correlations between variables are complex and can be influenced by other things, which makes it impossible to accurately reflect the degree of correlation between variables. Generally, the simple correlation coefficient is not pure correlation.

Partial correlation analysis is also called net correlation, which more accurately analyzes correlations between two variables by controlling the effects of other variables. The method of measuring the correlation is to calculate the partial correlation coefficient. Partial correlation analysis can accurately find implied correlations from interferential variables.The process of calculating partial coefficients fos analysis is as follows^{[11, 12]}.

(1) Reflect the strength of the net correlation between two variables.

Calculating the partial correlation coefficient of two variables reflects the net correlation. The larger value of correlation coefficient is, the stronger net correlation is, and vice versa.

When calculating the partial correlation between variable … and y, the linear relation should be controlled. Then, the first-order partial correlation between … and y is defined as:

ry1, y2=ry1-ry2r12(1-ry22)(1-r122)

In the formula, r_y1, r_y2 and r₁₂ respectively represent the correlation coefficient between y and x₁, the correlation coefficient between y and x₂, and the correlation coefficient between x₁ and x₂. The value range of partial correlation coefficient is consistent with that of correlation coefficient.

(2) Put null hypothesis. There is no significant difference between zero and partial correlation of the partial correlation of the two populations.

(3) Construct inspection statistics. In partial correlation analysis, the statistic is T and is defined as

T=rn-q-21-r2

R is the partial correlation coefficient, q is the order, and n is the number of samples.

(4) Calculate the observed value of test statistics and the corresponding probability value p.

(5) If the probability value is less than the given significance level α , then the original hypothesis should be rejected, indicating that two variables are partially correlated. On the contrary, if the probability value p is greater than the given significance level, the original hypothesis should be accepted, indicating that partial correlation coefficients of the two variables are not significant with zero.

The continuous removed method is a common method for analyzing absorption and reflection characteristics of ground objects.The envelope curve is defined as the maximum values of straight line on a spectral curve, and the external angle of the line is greater than 180 degrees. The envelope is equivalent to the shell of spectra.The de-envelope of spectra is the value on the original spectral curve divided by the value on the envelope curve. After spectra are removed, the reflectance of each point of the new spectral curve is ρ (λ ), the absorption depth is, and λ is the wavelength^[13].

Wavelet packet analysis can provide a more precise analysis method for signals, which divides the frequency band into different levels and further decomposes high frequency parts without subdivision of multi-resolution analysis. According to the characteristics of analyzed signals, corresponding frequency bands are adapted to match signal spectra, so as to improve the time-frequency resolution. Fig. 2 is a three-layer wavelet packet decomposition tree diagram.

Fig.2

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	Fig.2 Three-layer wavelet packet decomposition tree

In the figure, S represents the original signal, A represents the low frequency part of signals, and D represents the high frequency part of signals.

S=AAA3+DAA3+ADA3+DDA3+AAD3+DAD3+ADD3+DDD3

In this paper, original spectral curves were decomposed by Three Layers of Wave packet using the Db5 Wavelet Function, and we obtained the high-frequency and low-frequency parts of the spectral curve, which effectively separated the approximate features and details of the spectral curve.

Fig.3

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	Fig.3 Original spectral curve

Fig.4

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	Fig.4 Wavelet packet decomposition node(3, 0)

Fig.5

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	Fig.5 Wavelet packet decomposition node(3, 1)

Fig.6

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	Fig.6 Wavelet packet decomposition node(3, 2)

Fig.7

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	Fig.7 Wavelet packet decomposition node(3, 3)

Fig.8

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	Fig.8 Wavelet packet decomposition node(3, 4)

Fig.9

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	Fig.9 Wavelet packet decomposition node(3, 5)

Fig.10

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	Fig.10 Wavelet packet decomposition node(3, 6)

Fig.11

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	Fig.11 Wavelet packet decomposition node(3, 7)

Fig.12

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	Fig.12 Original spectrum high frequency signal

Fig.13

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	Fig.13 Original spectrum low frequency signal

Taking a spectral curve as an example, the following wavelet packet analysis results are obtained after decomposition and reconstruction of three-layer wavelet packets.

The correlation coefficients between density and spectral reflectance, absorption depth and high and low frequency after wavelet packet decomposition are respectively shown in Fig.14— Fig.16.

Fig.14

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	Fig.14 Wavelet packet reconstruction error

Fig.15

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	Fig.15 Correlation coefficient between density and reflectance

Fig.16

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	Fig.16 Correlation coefficient between density and absorption depth

The relationship between density and spectrum is more significant in the range of 0.57~0.85 μ m, and density is negatively correlated with spectral reflectance. Correlation coefficients of igneous rocks are up to -0.41 at 0.63 μ m band, and the density is positively correlated with spectral absorption depth, indicating that densities have effect on spectral absorption in this range. The high frequency of spectral wavelet packet and density have large negative correlation coefficients near 0.57, 0.71, 0.85 μ m bands. The highest correlation coefficient is 0.43, which better confirms the influence of density on spectral absorption in this range.

The Fig.17— Fig.19 show the correlation between susceptibility and spectral absorption depth, spectral high and low frequencies. As we can see from the figure, the susceptibility has negative correlations with spectral reflectance, in which the correlation between susceptibility and spectra of igneous rocks is higher, confirming that contents of susceptibility of igneous rocks are higher.

Fig.17

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	Fig.17 Correlation coefficient between density and high and low frequency

Fig.18

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	Fig.18 Correlation coefficient between susceptibility and reflectance

Fig.19

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	Fig.19 Correlation coefficient between susceptibility and absorption depth

Near 0.53 μ m band, the susceptibility is significantly negatively correlated with absorption depth, and the correlation coefficient is up to -0.36. The susceptibility has a large positive correlation coefficient with spectral high frequency, and the correlation coefficient reaches +0.2, which illustrates that susceptibilities have effect on spectral reflectance in this band.

In the vicinity of the 1.08 μ m band, susceptibilities of igneous rocks are negatively correlated with spectral reflectance, with a correlation coefficient of -0.36. The susceptibilities of igneous rocks are significantly positively correlated with absorption depth with the correlation coefficient +0.29, which is a good indication that the density plays a role in spectral absorption of igneous rocks near this band.

Fig.20— Fig.22 respectively show the correlation between the resistivity and spectral reflectance, absorption depth, high and low spectral frequencies. As can be seen from the figure, the correlation coefficient between resistivity and spectrum is very similar to that between density and spectrum, which also confirms the close relationship between density and resistivity of rocks. The correlation between resistivity and spectrum of sedimentary rocks is more significant, and the variation of correlation coefficient is greater, which exactly confirms that the resistivity of sedimentary rocks is larger.

Fig.20

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	Fig.20 Correlation coefficient between susceptibility and high and low frequency

Fig.21

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	Fig.21 Correlation coefficient between resistivity and reflectance

Fig.22

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	Fig.22 Correlation coefficient between resistivity and absorption depth

In the range of 0.6~0.7 μ m, the resistivity and spectral absorption depth have positive correlation. At 0.62 μ m band, the correlation coefficient is up to 0.47, meaning the negative correlation between resisitivity and spectral reflectance. The resistivity of sedimentary rocks is higher, and the correlation of sedimentary rocks is more significant. The resistivity and high spectral frequency have a large negative correlation coefficient at 0.6 and 0.69 μ m, with the biggest value of -0.25, which is a good illustration that the resistivity has influence on spectral absorption in the range of 0.6~0.7 μ m.

The relationships between permittivity and spectral reflectance, absorption depth, spectral high and low frequency are shown in Fig.23— Fig.25. It can be seen from the figure that the correlations between permittivities and spectra of sedimentary rocks are more significant and changes are more volatile, confirming that the permittivity constant of sedimentary rocks is higher.

Fig.23

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	Fig.23 Correlation coefficient between resistivity and high and low frequency

Fig.24

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	Fig.24 Correlation coefficient between permittivity and reflectance

Fig.25

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	Fig.25 Correlation coefficient between permittivity and absorption depth

Near the 1.85 μ m band, the permittivity and spectral reflectance have negative correlations with the correlation coefficient reaching -0.2. The permittivity has a positive correlation with spectral absorption depth, and the correlation coefficient is up to +0.19. In this band, the permittivity and high frequency have negative correlations with the value of -0.16, which indicates that the permittivity has influence on spectral absorption, in which the influence of sedimentary rocks is more significant.

We respectively analyze the correlation and partial correlation on t densities, susceptibilities, resistivities and permittivities of 590 rocks.The results are shown in Table 2 and Table 3.

Table 2

Table 2

Table 2 Partial correlation coefficient between geophysical parameters partial correlation coefficient density susceptibility resistivity permittivity density 1 0.31 0.418 0.06 susceptibility 0.31 1 -0.157 0.148 resistivity 0.418 -0.157 1 -0.038 permittivity 0.06 0.148 -0.038 1

Table 2 Partial correlation coefficient between geophysical parameters

Table 3

Table 3

Table 3 Correlation coefficient between geophysical parameters correlation coefficient density susceptibility resistivity permittivity density 1 -0.033 0.411 0.039 susceptibility -0.033 1 -0.162 0.154 resistivity 0.411 -0.162 1 -0.04 permittivity 0.06 0.154 -0.04 1

Table 3 Correlation coefficient between geophysical parameters

It can be seen from partial correlation coefficients and correlation coefficients that the density has positive correlation with the resistivity, with the correlation coefficient reaching +0.41, which also proves that the relationship between density and spectra is similar to that between resistivity and spectra. In natural state, water contents and salt concentration of aqueous solution are the main factors affecting rocks’ electrical conductivity. The higher the densities are, the smaller the porosities are, the worse conductivities of rocks are, and the higher resistivities values are, which is a good explanation of the positive correlation between density and resistivity.

Fig.26 shows correlation coefficients between densities and each metal elements. As we can see from the figure, correlations between densities and metal elements of rocks are poor. Meanwhile, the density is positively correlated with Ti and Co, with the correlation coefficients of +0.16 and +0.3. The densities are negatively correlated with V with the correlation coefficient of sedimentary rocks reaching -0.39. Correlations between metal elements and densities of rocks are not significant, but the characteristic relationship between them cannot be ignored. Using the change of densities can qualitatively and semi-quantitatively infer contents and changes of geochemical elements.

Fig.26

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	Fig.26 Correlation coefficient between permittivity and high and low frequency

The correlation coefficients between susceptibilities and each metal element are shown in Fig.27. The susceptibility is well correlated with metal elements, where the susceptibility is positively correlated with Fe and Ti, with correlation coefficients of +0.57 and +0.47. On the one hand, that confirms the significant positive correlation between Fe and Ti, as well as the effect of Fe-containing and Ti-containing minerals on magnetism of rocks, such as the vanadium-titanium magnetite deposit in Panzhihua, Sichuan, China. On the other hand, it also shows that the magnetic minerals containing of Fe affect susceptibilities of rocks. The reduction of ferromagnetic minerals in magma splitting also indicates the decrease of susceptibilities of rocks.Therefore, we can use susceptibilities of rocks to study the magma separation and formation process.

Fig.27

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	Fig.27 Correlation coefficient between density and chemical elements

The correlation coefficients between resistivities and each metal element were shown in Fig.28. It can be seen that the relationship between rock resistivity and metal elements is relatively weak, which is due to its close relationship with the characteristics of mineral crystal particles. When conductive mineral contents of rocks are more than 80%, the resistivities will play a significant role. Therefore, the relationship between the content of each metal element and the resistivity is weak.

Fig.28

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	Fig.28 Correlation coefficient between susceptibility and chemical elements

The correlation coefficient between the permittivity and each metal element is shown in Fig.29. The content of metal elements is positively correlated with the permittivity, which shows the increased ability of rocks to separate charges in electric fields with the increase of the content of positively correlated metal elements. And the correlation of sedimentary rocks is more significant. Fe, Bi, Co, Zn have a weak positive correlation with permittivity, and Ti has a significant positive correlation with permittivity with the correlation coefficient reaching +0.31. Meanwhile, there are significant positive correlations between the V, Zn and Bi and permittivities of sedimentary rocks. The correlation coefficient between Bi and permittivitiies of sedimentary rocks reaches +0.47. That well illustrates the relationship between physical properties and chemical element contents of rocks, and there are also great differences in the dependence of physical properties on chemical elements.

Fig.29

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	Fig.29 Correlation coefficient between resistivity and chemical elements

Fig.30 and Fig.31 respectively show correlations between the Fe element and absorption depth, high and low frequency of rocks. Compared with other elements, the correlation between Fe element and spectra is more significant.

There are negative correlations between Fe and absorption depth in the range of 0.4~0.54 μ m, and the correlation of igneous rocks is more significant, which indicates that Fe element make spectra show good reflection characteristics in this range. At 0.4 μ m band, Fe has negative correlations with absorption depth with the correlation coefficient reaching -0.282, and Fe has large positive correlation coefficients with high frequency +0.2, which is a good indication that the 0.4μ m band is the characteristic reflection band of Fe.

Fig.30

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	Fig.30 Correlation coefficient between permittivity and chemical elements

Fig.31

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	Fig.31 Correlation coefficient between Fe and absorption depth

At 0.54 μ m, Fe elements and the high frequencies have large positive correlation coefficient of +0.25, indicating that the Fe element has strong reflection of spectrum near this band. In the range of 1.0~1.15 μ m, there are positive correlation between Fe element and spectral absorption depth of igneous rocks, with the correlation coefficient up to +0.443, indicating that the Fe element makes spectra strongly absorbed in this range.

In the range of 1.7~1.8 μ m, Fe elements have significant positive correlations with spectral absorption depth of igneous and sedimentary rocks, and the difference of correlation coefficients between the two is not distinct. Fe element content is negatively correlated with the high frequency of spectra, which indicates Fe has effects on absorption of spectra in this range.

The correlations between absorption depth, high and low frequency of rocks and Ti element are shown in in Fig.32 and Fig.33. The correlation between Ti element and absorption depth is similar to that between Fe and absorption depth, which is a good evidence of a strong positive correlation between Fe and Ti elements.

Fig.32

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	Fig.32 Correlation coefficient between Fe and high and low frequency

Fig.33

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	Fig.33 Correlation coefficient between Ti and absorption depth

Fig.34

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	Fig.34 Correlation coefficient between Ti and high and low frequency

In the range of 0.4~0.55 μ m, Ti elements have negative correlations with spectral absorption depth, and correlations of igneous rocks are more significant than those of sedimentary rocks, which proves that element contents of Ti of igneous rocks are higher than those of sedimentary rocks. At 0.55 μ m, Ti is negatively correlated with absorption depth and positively correlated with high frequency of spectrum with a correlation coefficient of +0.32, indicating that Ti has effects on spectral reflection near this band.

In the range of 0.6~0.65 μ m, there are positive correlations between Ti elements and absorption depth of igneous rocks, and negative correlations between that of sedimentary rocks, which shows that Ti makes spectra of igneous rocks have good absorption characteristics and makes spectra of sedimentary rocks have good reflection. At 0.6 μ m, Ti has negative correlation with high frequency, and the correlation coefficient is up to -0.31. At 0.64 μ m, Ti has positive correlation with high frequency, and the correlation coefficient is up to +0.27. That is a good illustration of characteristic bands of Ti near 0.6 and 0.64 μ m bands.

The correlations between V elements and absorption depth, high and low frequencies of rocks are shown in Figs.34 and 35. On the whole, the correlations between V and absorption depth are weak. While the correlations are more prominent and different when rocks are classified into igneous and sedimentary rocks.

In the range of 0.44~0.49 μ m, the V element has positive correlations with spectral absorption depth of igneous rocks. Near 0.5 μ m band, there are positive correlation coefficients between V element and absorption depth of igneous rocks, which is up to +0.275, indicating that the V element has effects on spectral absorption of igneous rocks in this band. Near 0.58 μ m band, positive correlation coefficients between V elements and spectral absorption depth of sedimentary rocks are up to +0.21, indicating that V element has influence on spectral absorption of sedimentary rocks near this band.

In the range of 0.75~0.95 μ m, there are positive correlation coefficients between V element and spectral absorption depth of igneous rocks and negative correlation coefficients between that of sedimentary rocks, indicating that V element has effects on spectral absorption of igneous rocks and spectral reflection of sedimentary rocks.

In the vicinity of 1.94 μ m, the V element of igneous rocks is positively correlated with spectral absorption depth, and the correlation coefficient is up to +0.34. Therefore, there may be characteristic absorption bands of V element of igneous rocks near this band.

Fig.36 and Fig.37 respectively show correlations between Mn element and spectral absorption depth, high and low frequencies of rocks. It can be seen from the figure that correlation coefficients between spectra and Mn elements are similar to those between spectra and Fe and Ti elements, which can also indicates that Mn, Fe and Ti are positively correlated.

Fig.35

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	Fig.35 Correlation coefficient between V and absorption depth

Fig.36

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	Fig.36 Correlation coefficient between V and high and low frequency

Fig.37

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	Fig.37 Correlation coefficient between Mn and absorption depth

Fig.38

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	Fig.38 Correlation coefficient between Mn and high and low frequency

In the range of 0.4~0.48 μ m, Mn element has negative correlations with spectral absorption depth, indicating that Mn element has influence on spectral reflection in this wavelength range. The Mn element has positive correlations with high frequencies of spectra at 0.41 μ m band, indicating there are characteristic reflection bands of Mn near this band.

Near 1.93 μ m band, Mn element has negative correlation with spectral absorption depth. The correlations of sedimentary rocks are more significant with the correlation coefficient reaching -0.16. And Mn is positively correlated with high frequencies of spectra, and the correlation coefficient is +0.13. In the vicinity of the 2.22 μ m band, Mn element has negative correlation coefficients with absorption depth with the value reaching -0.2, and positive correlation coefficients with high frequency with the value reaching +0.11. That all well indicate that it is likely to exist feature reflection bands of Mn elements near 1.93 and 2.22 μ m bands.