Combined analysis of gravity and magnetic anomalies using normalized source strength

0 Introduction

Based on the potential field theory, gravity and magnetic field have inherent multi-solution property. The interpretation of potential field data becomes complicated due to the interference effect of neighboring anomalies. Interpreters often identify the correlation of gravity and magnetic data based on their experience. This visual judgment cannot objectively represent the degree of homology of the source.

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Combined analysis of magnetic and gravity anomalies based on Poisson’s relation is carried out, which broke the ice that there were no quantitative analysis criteria for the correlation. Homologous sources are integrated by the linear relationship in Poisson’s relation, and the homology is calculated quantitatively. The deep structure of the lower crust and upper mantle and regional geothermal distribution can be obtained by the combined analysis, which is helpful for the understanding of tectonic and geodynamics of the crust and mantle.

抽象执行的本质是在抽象域将程序变量的抽象表示形式作为程序变量本身带入程序进行计算。首先要对程序的变量和程序中的函数操作进行抽象表示，然后抽象域中执行程序的运行过程。

Garland (1951) applied this combined analysis to identify common gravity and magnetic source, to test whether the magnetization direction is close to the geomagnetic field, and to describe the property of geological bodies using the Poisson’s ratio. Wilson (1970) used this method to separate neighboring sources with different magnetization directions. Liu (1985) first introduced this approach to China. Liu (1992) put all measured points in a Poisson window, calculating the regression analysis of reduction to the pole (RTP) anomaly and the first order vertical derivative of gravity anomaly in Zhalong area of Songliao Basin. Lithology of the underlying rock is identified by calculating the residual density. Zeng et al. (2006) used the slope and intercept to analyze the concealed rock body and estimate the effect of remanence.

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The NSS μ (Wilson, 1985) can be derived from the combination of eigenvalues of the magnetic gradient tensor, which are arranged in non-ascending order as λ1≥λ2≥λ3:

Neglecting the effect of remanent magnetization and self-demagnetization, the effective magnetization is only caused by induced geomagnetic field, then

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1 Theory

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(1)

Where r and r0 are the observed point and source location respectively, q is the geometry factor, C is a constant, n is the structure index. The structure index of the sphere, horizontal cylinder, thin sheet, and contact is 4, 3, 2, and 1 respectively, while the geometry factor is 3m (Am2), 4js(Am), 2jt(A), and 2j(A/m) respectively. m is the magnitude of the magnetic moment, j is the magnitude of the effective magnetization, t is the thickness of the thin sheet model, s is the cross-section area of the cylinder.

Based on the property that NSS has the minimal dependence on the magnetization direction, the gravity field is transformed into the pseudomagnetic field of the direction of geomagnetic field magnetization using Poisson’s relation. The NSS of the pseudomagnetic field and that of the original magnetic field are calculated, and linear regression analysis is carried out. This approach is tested using synthetic model under complex magnetization and real data in Wudalianchi in China.

In a Cartesian coordinate system, with the x-axis pointing to the geographical north, the y-axis to the east, and the z-axis vertically downwards, the magnetic gradient tensor Bij(i,j=x,y,z) of that uniformly magnetized object with uniform density is Formula (2).

(2)

Where Mα(α=x,y,z) is the α-component of the effective magnetization, Uijα(α=x,y,z) is the α-direction derivative of gravity gradient tensor Uij.

Under the homology condition, the gravity potential U and magnetic potential V of an evenly magnetized body with uniform density can be expressed as Formula (1).

The normalized source strength (NSS) is calculated from the eigenvalues of the magnetic gradient tensor (Wynn et al., 1975; Wilson, 1985; Schmidt et al., 2004), which is minimally affected by the direction of source magnetization. Advantages of using NSS to calculate source parameter include its tendency to be centered directly over the source, with high symmetry and good focus, and its insensitivity to magnetization direction. For simple sources, the NSS is a homogeneous function. For complex 3D sources, the NSS is slightly dependent on magnetization direction but less so than the analysis signal amplitude (ASA). Lots of methods have been developed using these advantages to interpret gradient tensor maps (Clark, 2012; Beiki et al., 2012; Pilkington & Beiki, 2013; Beiki et al., 2014; Guo et al., 2014).

(3)

I and D are the inclination and declination of the geomagnetic field respectively.

The homologous gravity and magnetic anomalies may display irrelevant results in the linear regression calculation due to the effect of the remanence. If ignoring the effect of remanent magnetization and using the direction of geomagnetic field as the direction of total magnetization, the RTP anomaly would shift and not illustrate the real shape of target geological body.

(4)

Importantly, the NSS of simple source may be generalized as (Beiki et al., 2012)

(5)

Where M is the effective magnetization vector, ρ is the density contrast, G is the gravitation constant.

The pseudomagnetic field can be transformed from gravity data using Equation (2), assuming there exists only induced magnetization. The NSS of pseudomagnetic field μP and the NSS of observed magnetic field μB have a linear relationship.

(6)

Where B is the slope indicating the Poisson’s ratio, A is the intercept representing the influence of neighboring source and remanence. Correlation analysis of μP and μB is carried out by a moving window.

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(7)

Where R is the correlation coefficients, cov(μB, μP)=E[(μB-E(μB))·(μP-E(μP))] is the covariance between μB and μP, E is the mathematical expectation, D is variance. R is used to calculate the homology degree. In the homological gravity and magnetic source condition, a high R indicates a reliable Poisson’s ratio of the rock.

2 Synthetic data test

The composite model under complex magnetization is established to illustrate the validity of this method. The inclination and declination of geomagnetic field are 10° and 50° respectively. The parameters of six spheres are shown in Table 1. The observation grid is 201×201, with 10m spacing. Among them, sphere 5 and 6 are none-homology gravity and magnetic anomalies. Five of the six models contain remanence, with different degrees of oblique magnetization. Fig.1a-b shows the theoretical gravity anomaly and magnetic anomaly. It can be seen that the magnetic anomaly is complicated due to the effect of oblique magnetization. Fig.1c-d shows the first-order vertical derivative of gravity anomaly and the RTP anomaly, which is calculated using the direction of geomagnetic field as that of magnetization of models. The anomaly is closer to the actual location of the source after the above treatment. However, RTP anomaly has a northeast trend indicating the influence of remanent magnetization. Fig.1e-f are the normalized source strength calculated from the pseudomagnetic anomalies using the gravity anomaly magnetized according to the direction of the geomagnetic field and the normalized source strength calculated from the actual magnetic anomalies. Compared with the results in Fig.1c-d, results of the normalized source strength have a better illustration of the source location.

Table 1 Model parameter

ModelCentraldepth/mRadius/mMagnetizationdirectionInclination/°Declination/°Magnetization/A·m-1Residualdensity/g·cm-31200150105020．12200100204040．133001505012050．541501003016040．251501000．1615050-601301

(a) Gravity anomalyof the combined model; (b) TMI magnetic anomaly of the combined model; (c) the first-order vertical derivative of gravity anomaly; (d) RTP anomaly using the direction of geomagnetic field; (e) the NSS result of the pseudomagnetic field; (f) and that of the original magnetic field. Fig.1 Synthetic model anomalies

Combined analysis of the first-order vertical derivative of gravity anomaly and RTP magnetic anomaly, and combined analysis using NSS calculated from pseudomagnetic and original magnetic anomaly are carried out. The sliding window selection is 200 m. Fig.2a shows the correlation coefficients distribution calculated using the first-order vertical derivative of gravity anomaly and RTP magnetic anomaly. Due to the existence of remanent magnetization, the distribution of correlation coefficients is disordered.

The center of sphere 2 and 3 are partially irrelevant due to the effect of remanence. It cannot display the homologous gravity and magnetic anomaly under oblique magnetization condition. Fig.2b shows good response in both the homologous and non-homologous areas. It can be seen that the new combined analysis method can effectively identify homologous gravity and magnetic anomalies under remanent magnetization. For the slope in the analysis in Fig.2c, the Poisson’s ratio cannot be estimated due to the low correlation coefficient. The slope of the new combined analysis (Fig.2d) shows there are three kinds of sources in the region. Sphere 1 and 2 have the same Poisson’s ratio, while Sphere 3 has half of that ratio, and Sphere 4 has the highest ratio, indicating a more reliable result than Fig.2c. All the ratio distribution is in accordance with the original model parameters. Overall, the proposed method can distinguish the homologous gravity and magnetic source and estimate the Poisson’s ratio under strong remanent.

3 Test with real data

The research area is located in the Wudalianchi volcano group. The fault structure of the study area is well developed, and the basalt is widely distributed with large amount of remanent magnetization. Local gravity anomaly is shown in Fig.3a. The distribution of gravity anomaly shows a general trend of high in the northeast and low in the southwest. The Laoheishan region is in a low gravity anomaly area, and there may be mass loss caused by magma eruption. There is a surplus on the western side of the Weishan area which is caused by the high-density rock mass. The local magnetic anomalies in the region (Fig.3b) indicate that the distribution of geomagnetic anomalies is mainly controlled by the shallow magnetic body. There are obvious positive and negative anomalies traps in the region, and the distribution is very complicated.

(a) correlation coefficient of the traditional method; (b) correlation coefficient of the proposed method; (c) slope of the traditional method; (d) slope of the proposed method. Fig.2 Correlation coefficients and slopes for two combined analysis methods

Fig.3 Local gravity anomaly (a) and local magnetic anomaly (b) in Wudalianchi

Fig.4 Correlation coefficient (a) and slope (b) of the proposed combined analysis method in Wudalianchi

The method proposed in this article is used to analyze the distribution of homologous gravity and magnetic source; the correlation coefficients are illustrated in Fig.4a. In the Laoheishan area, the correlation coefficients are close to 1, which means there exists homologous gravity and magnetic source. Volcanic rocks in the Laoheishan region mainly occurred in the fractures as filling, confirming that the low gravity may be caused by the mass loss related to the magma eruption. The correlation coefficient of gravity and magnetic anomalies in the Weishan area showed a ring-shape distribution, with the ring fissures developed in this area. The fast imaging results of Xu (2016) showed that in the eastern part of the Weishan area, from the center to the east, it displays obvious low-high-low-high-low extreme point arrangement. Remanent magnetization dominates the magnetism of volcanic rocks with formation age of 0.25--0.75Ma in this area. The negative magnetic anomaly in the Weishan area is not the result of reverse magnetization, but may be due to the magnetic geologic bodies with different occurrences, which is consistent with the results indicated by the correlation coefficients. For the Poisson’s ratio in Fig.4b, Laoheishan area has a higher Poisson’s ratio that is in accordance with the mass loss caused by the magma eruption and distribution of basalt with lower density and higher magnetization. For the high Poisson’s ratio in the northwest part of Laoheishan area and the southern part of Weishan area, it may be caused by the distribution of Indosinian granite, which is the same as Xu’s conclusion (2016).

4 Conclusion

To solve the calculated irrelevant correlation coefficient of a homologous gravity and magnetic source under strong remanent in the traditional way, we proposed a new combined analysis using NSS. The NSS is minimally affected by the magnetization so it can be used to demolish the oblique magnetization caused by the remanence. This method is easy to perform and we utilize this property in Poission’s formula. Compared with traditional combined analysis method, the synthetic model test and the application in real magnetic data show that the proposed method can identify the homologous source and accurately estimate the Poission’s ration under strong remanent, and therfore provide reference for the subsequent interpretation.

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