A new method to integrate different gravity gradient components in reweighted regularized inversion with a minimum support constraint

Qin, Pengbo; Xiang, Min; Liang, Xin; Hou, Zhenlong

doi:10.1093/gji/ggz095

SUMMARY

The gravity gradient tensor has been increasingly used in practical applications. Among them, how to extract information contained in different gravity gradient components is a challenging problem. Gravity gradient joint inversion is one effective method to solve this problem. We integrate different gravity gradient components in a matrix and then apply them in inversion directly. In this paper, we modify the method to get a new gravity gradient joint inversion (NGGJI). The method is based on the reweighted regularized inversion. We choose one component, for example, g_zz, and use the other components to build a weighting matrix. Then we apply the weighting matrix in g_zz inversion. We present the method to construct the weighting matrix based on a single component and multiple components. We analyse the characteristics of different weighting matrices and the noise effects on weighting matrices. We compare the inversion results obtained from the conventional gravity gradient joint inversion (CGGJI) with the inversion results obtained from the NGGJI. We conclude that the NGGJI's requirement for memory storage is lower and the resolution of the NGGJI inversion results is higher. We apply the method to survey data from Vinton Salt Dome, Louisiana, USA. The results have proved to be consistent with known geological information.

Geopotential theory, Gravity anomalies and Earth structure, North America, Inverse theory, Joint inversion

1 INTRODUCTION

Full tensor gradiometry (FTG) technology has been widely used in the geophysical exploration industry. Compared with gravity measurement technology, FTG technology has better noise immunity and is suitable for moving platform measurement. A significant advantage is that it can measure all components of the gravity gradient tensor. Multiple component data sets have been acquired over large areas. The need for processing and interpretation tools is all the more important. One essential aspect is to use the entirety of information contained in multicomponent gradiometry measurements (Martinez et al. 2013). Gravity gradient joint inversion method is one of the commonly used methods (Barnes & Barraud 2012).

Different gravity gradient tensor components enhance different features of the source (Paoletti et al. 2016). The horizontal components (g_xx, g_yy and g_xy) tend to highlight the lateral density boundaries more readily, while the vertical component (g_zz) provides a more intuitive anomaly map (Martinez et al. 2013). Martinez et al. (2013) pointed out that in theory, one component at the observational surface carries all the information. However, in actual applications, the data are measured at discrete observation locations and are affected by noise, so the geological information contained in different gravity gradient components has a difference. When multiple components are integrated into inversion simultaneously, the resolution of the inversion results improve accordingly to some extent (Martinez et al. 2013; Qin et al. 2016). Generally, we use different components to form a matrix, and then apply them in inversion. Li (2001) developed an algorithm for inverting multiple components based on the formalism of generalized inversion. Zhdanov et al. (2004) integrated the differential curvature tensor components and g_xy into inversion and developed a method based on the focusing inversion technique. Uieda & Barbosa (2012) used a systematic search algorithm to solve the inverse problem of gravity gradient data. Martinez et al. (2013) examined multiple density contrast models obtained by inverting different component combinations to understand the information content in different data components. Geng et al. (2014) developed a method based on the cokriging inversion technique. The method can fully integrate multiple gravity gradient data and prior geological information. All these works have explored the advantages of multiple component inversion.

In this paper, we modify the conventional gravity gradient joint inversion (CGGJI) method and present a new gravity gradient joint inversion (NGGJI) method to integrate gravity gradient components into inversion. In the CGGJI method, different components form a matrix, and we apply it in inversion. In the NGGJI method, we choose a component as the inversion data, other components are used to construct the weighting matrix. Since the g_zz component is sufficient to produce geologically reasonable and interpretable results (Martinez et al. 2013) and it is the single tensor component with the largest information value (Pilkington 2012, 2014), we choose g_zz component as the inversion data.

The NGGJI method is based on a regularized function, which is a conventional way of solving an ill-posed inverse problems (Portniaguine & Zhdanov 1999). In many researches, the smoothness constraint is used to get a smooth model (Li & Oldenburg 1996, 1998; Li 2001; Martinez et al. 2013). For a model with a sharp boundary, the minimum gradient support constraint is used (Zhang & Koyama et al. 2012). Last & Kubik (1983) proposed a minimum support constraint to get a model with the minimum area of the anomalous parameters distribution. Zhdanov (2009) further complemented this constraint. In this paper, we adopt this constraint to get a compact density model.

In the NGGJI method, one of the key steps is to construct a weighting matrix based on a single component, or multiple components. The purpose is to extract geological information contained in different components. To avoid the influence of the constraint, we only use the misfit function to achieve this.

To solve the problem, we introduce two different optimization algorithms: the linear conjugate gradient (LCG) algorithm and the non-linear conjugate gradient (NLCG) algorithm. The difference between the LCG and the NLCG is the method to get the step length. In the LCG algorithm, we deduce the expression of the step length; in the NLCG method, we use the line search method to get the step length. When we construct the weighting matrix, the expression of the step length is deduced, and we choose the LCG algorithm. For the regularized inversion with a minimum support constraint, the expression of the step length is difficult to get, and we apply the NLCG algorithm to solve the inverse problem.

In the following, we first introduce the forward formulas of gravity gradient components; next, we discuss the reweighted regularized inverse problem with a minimum support constraint, and introduce the expression of CGGJI method and the NGGJI method. Thirdly, we analyse the feature of weighting matrices which are calculated from different gravity gradient components, and discuss the effect of noise on the weighting matrices. We further analyse the characteristics of the weighting matrix constructed by multiple components. We also make a comparison between the CGGJI method and the NGGJI method. Finally, we apply the NGGJI method to gravity gradient data from Vinton salt dome, southwestern Louisiana, USA (data acquired by Bell Geospace Inc).

2 3-D GRAVITY GRADIENT FORWARD FORMULAS

The formulas are based on a right-handed Cartesian system, where the z-axis points downward. The gravitational potential at an observation point |${{\bf r}} = (x,y,z)$| is defined as:

\begin{eqnarray*} U({{\bf r}}) = G\iiint_D {\frac{{\rho ({{\bf r \prime}})}}{{{{\left\| {{{\bf r \prime}} - {{\bf r}}} \right\|}_2}}}dv \prime}, \end{eqnarray*}

(1)

where |$G$| is Newton's gravitational constant, |$G = 6.67 \times {10^{ - 11}}{{\rm{m}}^{\rm{3}}}{\rm{(kg}}\,\cdot {{\rm{s}}^2})^{-1}$|⁠, |${{\bf r \prime}} = (\xi ,\eta ,\zeta )$| is the point within the domain |$D,\rho ({{\bf r \prime}})$| is the anomalous density at the point |${{\bf r \prime}},{\| {{{\bf r \prime}} - {{\bf r}}} \|_2}$| is the 2-norm of the vector, |${\| {{{\bf r \prime}} - {{\bf r}}} \|_2}{\rm{ = }}{( {{{(x - \xi )}^2} + {{(y - \eta )}^2} + {{(z - \zeta )}^2}} )^{\frac{1}{2}}}$|⁠. The second spatial derivative of |$U({{\bf r}})$| form the gravity gradient tensor |${g_{\alpha \beta }}({{\bf r}})$| (⁠|$\alpha ,\beta = x,y,z$|⁠), defined mathematically by:

\begin{eqnarray*} {g_{\alpha \beta }}({{\bf r}}) & = & \displaystyle\frac{{{\partial ^2}}}{{\partial \alpha \partial \beta }}U({{\bf r}})\nonumber \\ & = &{\rm{G}}\iiint_D {\displaystyle\frac{{\rho ({{\bf r \prime}})}}{{\left\| {{{\bf r \prime}} - {{\bf r}}} \right\|_2^3}}{K_{\alpha \beta }}({{\bf r \prime}} - {{\bf r}})dv \prime}, \end{eqnarray*}

(2)

where the kernels |${K_{\alpha \beta }}$| are equal to:

\begin{eqnarray*} {K_{\alpha \beta }}({{\bf r \prime}} - {{\bf r}}) = \left\{ {\begin{array}{@{}*{1}{c}@{}} {3\displaystyle\frac{{(\alpha \prime - \alpha )(\beta - {\beta ^{ \prime}})}}{{\left\| {{{\bf r \prime}} - {{\bf r}}} \right\|_{2}^{2}}},\alpha \ne \beta }\\ {3\displaystyle\frac{{{{(\alpha \prime - \alpha )}^{2}}}}{{\left\| {{{\bf r \prime}} - {{\bf r}}} \right\|_{2}^{2}}} - 1,\alpha = \beta } \end{array}} \right.,\alpha ,\quad \beta = {\rm{x,y,z}}. \end{eqnarray*}

(3)

Generally, the tensor is expressed in matrix form:

\begin{eqnarray*} {{{\bf g}}_{\alpha \beta }} = \left[ {\begin{array}{@{}*{3}{c}@{}} {{g_{xx}}}&{{g_{xy}}}&{{g_{xz}}}\\ {{g_{yx}}}&{{g_{yy}}}&{{g_{yz}}}\\ {{g_{zx}}}&{{g_{zy}}}&{{g_{zz}}} \end{array}} \right]. \end{eqnarray*}

(4)

Outside of the source domain, |$U({{\bf r}})$| satisfies the Laplace equation |${\nabla ^2}U = {g_{xx}} + {g_{yy}} + {g_{zz}} = 0$|⁠. Moreover, |${{{\bf g}}_{\alpha \beta }}$| is symmetric, the independent components are reduced to five. Here, we give the expressions of |${g_{xx}}$|⁠, |${g_{yy}}$|⁠, |${g_{zz}}$|⁠, |${g_{xy}}$|⁠, |${g_{xz}}$|⁠, |${g_{yz}}$|⁠, where,

\begin{eqnarray*} {g_{xx}} = - G\int\limits_{V}{{\rho \frac{{2{{(x - \xi )}^2} - {{(y - \eta )}^2} - {{(z - \zeta )}^2}}}{{\left\| {{{\bf r \prime}} - {{\bf r}}} \right\|_2^5}}dv}}, \end{eqnarray*}

(5)

\begin{eqnarray*} {g_{yy}} = - G\int\limits_{V}{{\rho \frac{{2{{(y - \eta )}^2} - {{(x - \xi )}^2} - {{(z - \zeta )}^2}}}{{\left\| {{{\bf r \prime}} - {{\bf r}}} \right\|_2^5}}dv}}, \end{eqnarray*}

(6)

\begin{eqnarray*} {g_{zz}} = - G\int\limits_{V}{{\rho \frac{{2{{(z - \varsigma )}^2} - {{(x - \xi )}^2} - {{(y - \eta )}^2}}}{{\left\| {{{\bf r \prime}} - {{\bf r}}} \right\|_2^5}}dv}}, \end{eqnarray*}

(7)

\begin{eqnarray*} {g_{xy}} = 3G\int\limits_{V}{{\rho \frac{{(x - \xi )(y - \eta )}}{{\left\| {{{\bf r \prime}} - {{\bf r}}} \right\|_2^5}}dv}}, \end{eqnarray*}

(8)

\begin{eqnarray*} {g_{xz}} = 3G\int\limits_{V}{{\rho \frac{{(x - \xi )(z - \varsigma )}}{{\left\| {{{\bf r \prime}} - {{\bf r}}} \right\|_2^5}}dv}}, \end{eqnarray*}

(9)

\begin{eqnarray*} {g_{yz}} = 3G\int\limits_{V}{{\rho \frac{{(y - \eta )(z - \varsigma )}}{{\left\| {{{\bf r \prime}} - {{\bf r}}} \right\|_2^5}}dv}}, \end{eqnarray*}

(10)

Generally, in 3-D gravity gradient forward calculation, the subsurface is divided into a finite number of rectangular prisms with associated density. The number of the prisms is |$N = {n_x} \times {n_y} \times {n_z}$|⁠. The |$N \times 1$| vector |${{\bf m}}$| represents the model parameter. The elements |${m_j}$| of |${{\bf m}}$| represent the density in cell |$j$|⁠. The vector |${{\bf d}}$| represents the observed data. If data points coincide with the centre of the horizontal locations of cells, the size of |${{\bf d}}$| is |$M = {n_x} \times {n_y}$|⁠. In practice, gravity measurements cannot be so evenly distributed in the field, |$M$| is the number of observations. |${{\bf A}}$| is the sensitivity matrix, its size is |$M \times N$|⁠. The elements |${a_{ij}}$| of |${{\bf A}}$| represent the effect of unit density in cell |$j$| (⁠|$j = 1, \ldots ,N$|⁠) at data location |$i$| (⁠|$i = 1, \ldots ,M$|⁠). The forward modelling operator for gravity gradient data can be expressed in matrix form as:

\begin{eqnarray*} {{\bf d = Am}}. \end{eqnarray*}

(11)

3 3-D GRAVITY GRADIENT INVERSION PROCEDURE

3.1 Reweighted regularized inversion

Generally, the inverse problem is ill-posed (Tikhonov & Arsenin 1977), that is the solution can be non-unique and unstable. Regularization is used to solve this problem. The method is based on the minimization of the Tikhonov parametric function. The equation is as follows:

\begin{eqnarray*} \phi ({{\bf m}}) = {\phi _d}({{\bf m}}) + \alpha {\phi _s}({{\bf m}}), \end{eqnarray*}

(12)

where |${\phi _d}({{\bf m}})$| is the data misfit function; |${\phi _s}({{\bf m}})$| is the stabilizing function; and |$\alpha $| is the regularization parameter. The expression of |${\phi _d}({{\bf m}})$| is as follows:

\begin{eqnarray*} {\phi _d}({{\bf m}}) = \left\| {{{\bf Am - d}}} \right\|_2^2, \end{eqnarray*}

(13)

where |$\| \bullet \|_2^2$| represents the square of the 2-norm. To produce a unique and stable solution in potential interpretation, particular constraints have to be involved in inversion method (Silva et al. 2001). In addition, Silva et al. demonstrated that the inversion results are valuable only if the mathematical stabilizing constraints are deduced from the geological setting. Thus, we choose different stabilizing functions for different geological models. There are several expressions for stabilizing function (Portniaguine & Zhdanov 1999; Silva et al. 2001). In this paper, we choose the minimum support function to get a compact density model. Last & Kubik (1983) first considered this function. Portniaguine & Zhdanov (1999, 2002) developed this function into a minimum gradient support function. Zhdanov (2009) presented an overview of new advances in the regularization theory related to this function. Its expression is given by:

\begin{eqnarray*} {\phi _s}({{\bf m}}) = \sum\limits_{i = 1}^n {\frac{{m_i^2}}{{m_i^2 + {e^2}}}} , \end{eqnarray*}

(14)

where |${m_i}$| represents the elements in |${{\bf m}}$|⁠. e is a sufficiently small value. It is set to avoid singularities at points that make |${m_i} = 0$|⁠. By substituting eqs (13) and (14) in eq. (12), we get the objective function:

\begin{eqnarray*} \phi ({{\bf m}}) = \left\| {{{\bf Am - d}}} \right\|_2^2 + \alpha \sum\limits_{i = 1}^n {\frac{{m_i^2}}{{m_i^2 + {e^2}}}} . \end{eqnarray*}

(15)

Gravity gradient data, like any static potential field data, has no inherent depth resolution (Li & Oldenburg 1996, 1998; Li 2001). When minimizing eq. (15), structures tend to concentrate near the surface, regardless of the true depth of the causative bodies. This arises because the constructed model is a linear combination of kernels that decay rapidly with distance. The tendency to concentrate density at the surface can be overcome by introducing a depth weighting function to counteract the natural decay of the kernels (Li & Oldenburg 1996, 1998). In this paper, two different types of weighting functions are introduced. One is the depth weighting function, the other one is a weighting matrix. We use |${{\bf W}}$| to represent them, it is a diagonal matrix. |${{{\bf W}}^{{\rm{ - }}1}}$| represent the inverse of |${{\bf W}}$|⁠. We introduce them in eq. (15). The resulting expression can be written as follows:

\begin{eqnarray*} \begin{array}{@{}*{2}{l}@{}} {\phi ({{\bf m}})}&{ = {\phi _d}({{\bf m}}) + \alpha {\phi _s}({{\bf m}})}\\ {}&{{\rm{ = }}\left\| {{{\bf AW}}{{{\bf W}}^{{{\bf - 1}}}}{{\bf m - d}}} \right\|_2^2 + \alpha \sum\limits_{i = 1}^n {\displaystyle\frac{{{{({w_i}{w_i}^{ - 1}{m_i})}^2}}}{{{{({w_i}{w_i}^{ - 1}{m_i})}^2} + {e^2}}}} }\\ {}&{ = \left\| {{{{\bf A}}_w}{{{\bf m}}_w}{{\bf - d}}} \right\|_2^2 + \alpha \sum\limits_{i = 1}^n {\displaystyle\frac{{{{({w_i}{m_{wi}})}^2}}}{{{{({w_i}{m_{wi}})}^2} + {e^2}}}} } \end{array}. \end{eqnarray*}

(16)

where |${{{\bf A}}_w} = {{\bf AW}}$|⁠, |${{{\bf m}}_w} = {{{\bf W}}^{ - 1}}{{\bf m}}$|⁠, |${m_{wi}}$| is an element in |${{{\bf m}}_w}$|⁠. Since |${{\bf W}}{{{\bf W}}^{ - 1}}$| is an identity matrix, the value of eq. (16) is equal to the value of eq. (15). We adopt the NLCG algorithm (Appendix B) to solve the inverse problem. The gradient of the eq. (16) respect to |${{{\bf m}}_w}$| is as follows:

\begin{eqnarray*} \nabla \phi ({{\bf m}}) = {{{\bf A}}_w}^T({{{\bf A}}_w}{{{\bf m}}_w} - {{\bf d}}) + \alpha \sum\limits_{i = 1}^n {\frac{{{e^2}{w_i}^2{m_{wi}}}}{{{{\left[ {{w_i}^2{m_{wi}}^2 + {e^2}} \right]}^2}}}} . \end{eqnarray*}

(17)

eq. (17) is can be rewritten as:

\begin{eqnarray*} \nabla \phi ({{\bf m}}) = {({{\bf AW}})^T}({{\bf Am}} - {{\bf d}}) + \alpha \sum\limits_{i = 1}^n {\frac{{{w_i}{e^2}m}}{{{{\left[ {{m^2} + {e^2}} \right]}^2}}}} . \end{eqnarray*}

(18)

The gradient of the objective function has changed, in such a way that |${{\bf W}}$| takes now part in the inversion process.

3.2 CGGJI method

The purpose of the gravity gradient joint inversion method is to reflect the information of underground anomalies contained in different components in the inversion results. In CGGJI method, we integrate different gravity gradient components in a matrix equation (Li 2001; Zhdanov et al. 2004; Martinez et al. 2013; Geng et al. 2014; Qin et al. 2016). The objective function for six components inversion is as follows:

\begin{eqnarray*} {\phi _d} = \left\| {\left[ {\begin{array}{@{}*{1}{c}@{}} {{{{\bf A}}_{xx}}}\\ {{{{\bf A}}_{xy}}}\\ {{{{\bf A}}_{xz}}}\\ {{{{\bf A}}_{yy}}}\\ {{{{\bf A}}_{yz}}}\\ {{{{\bf A}}_{zz}}} \end{array}} \right]{{\bf m}} - \left[ {\begin{array}{@{}*{1}{c}@{}} {{{{\bf d}}_{xx}}}\\ {{{{\bf d}}_{xy}}}\\ {{{{\bf d}}_{xz}}}\\ {{{{\bf d}}_{yy}}}\\ {{{{\bf d}}_{yz}}}\\ {{{{\bf d}}_{zz}}} \end{array}} \right]} \right\|_2^2 + \alpha {\phi _{\rm{s}}}({{\bf m}}). \end{eqnarray*}

(19)

From eq. (19), we find that as the number of components increases, the computation time and the requirement for memory storage increases.

3.3 NGGJI method

Fig. 1 shows the flow of the NGGJI method. Constructing the weighting matrix is a key in the NGGJI method. There are two different kinds of weighting matrices. One (⁠|${{{\bf W}}_2}$|⁠) is constructed from a single component and the depth weighting function |${{{\bf W}}_1}$|⁠, others (⁠|${{{\bf W}}_3}$|⁠, …, |${{{\bf W}}_{i + 1}}$|⁠) are constructed from multiple components. Fig. 1 has been divided into three parts.

Figure 1.

The flow of the new gravity gradient joint inversion (NGGJI) method.

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In part I, we get a weighting matrix based on a single component. The data is |${{{\bf g}}_1}$|⁠, its depth weighting function is |${{{\bf W}}_1}$|⁠. In order to avoid introducing other information, we adopt a depth weighting function based on the sensitivity matrix because the matrix participates in the inversion. The expression of |${{{\bf W}}_1}$| is defined as:

\begin{eqnarray*} {w_j} = {\left( {\sum\limits_{i = 1}^M {a_{ij}^2} } \right)^{\beta /4}},j = 1,...,N, \end{eqnarray*}

(20)

where the elements |${a_{ij}}$| of |${{\bf A}}$| represents the effect of unit density in the cell |$j$|(⁠|$j = 1, \ldots ,N$|⁠) at data location |$i$| (⁠|$i = 1, \ldots ,M$|⁠); |$\beta $| is a constant, the larger the value, the stronger the weight. Generally, the value for |$\beta $| is close to 1 (Mehanee et al. 1998; Portniaguine & Zhdanov 1999, 2002). During the process of the construction of the weighting matrix, only the misfit function is used. The objective function is:

\begin{eqnarray*} {\phi _d}({{\bf m}}) = \left\| {{{{\bf A}}_w}{{{\bf m}}_w} - {{\bf d}}} \right\|_2^2. \end{eqnarray*}

(21)

We adopt the LCG algorithm to solve this problem (Appendix A). The inversion result is |${{\bf m}}$|⁠. Values in |${{\bf m}}$| might be positive, negative, and zero. However, the weight of each cell represents the possibility of the existence of the density contrast, it might be positive or zero, but it cannot be negative. So we take the absolute values of the components of |${{\bf m}}$|⁠. Then we normalize |${{\bf m}}$| to get the weighting matrix |${{{\bf W}}_2}$|⁠. In addition, if the weight is zero, it may cause singularities in the inversion. To avoid this, we assign a sufficiently small value (10⁻⁶ or less) to the zero element in |${{{\bf W}}_2}$|⁠.

Part II shows the method to construct a weighting matrix based on multiple components. After we get |${{{\bf W}}_2}$|⁠, select another component |${{{\bf g}}_2}$|⁠, then apply the method in Appendix A to |${{{\bf W}}_2}$| and |${{{\bf g}}_2}$| to get the inversion result |${{\bf m}}$|⁠. Take the absolute value of the components of |${{\bf m}}$|⁠, and normalize |${{\bf m}}$|⁠. Then assign a sufficiently small value (10⁻⁶ or less) to the elements equal to 0, we get |${{{\bf W}}_3}$|⁠. By analogy, we get the weighting matrix |${{{\bf W}}_{i + 1}}$| based on |${{{\bf W}}_i}$| and |${{{\bf g}}_i}$|⁠.

In part III, we apply the weighting matrix to |${{{\bf g}}_{\alpha \beta }}$| inversion. We build the objective function (eq. 16) based on |${{{\bf g}}_{\alpha \beta }}$| and |${{{\bf W}}_{i + 1}}$|⁠, and apply the NLCG algorithm (Appendix B) to solve the problem.

Generally, we select |${g_{zz}}$| as |${{{\bf g}}_{\alpha \beta }}$|⁠. This is because |${g_{zz}}$| inversion results are sufficient to produce geologically reasonable and interpretable results (Martinez et al. 2013). Pilkington (2012) used singular values to measure the information content of tensor components, he found that the singular value of |${g_{zz}}$| component has the highest magnitude over most of the spectral range. Pilkington (2014) found that of the single tensor components, |${g_{zz}}$| gave the best performance overall. The |${g_{zz}}$| inversion showed the main part of the anomalous body.

4 NUMERICAL MODEL AND DISCUSSION

This section is divided into three parts. In the first part, (1) we make comparisons among different matrices which are constructed from a single component; (2) we evaluate the effect of noise on the weighting matrices and (3) we analyse the characteristic of weighting matrices which are constructed from multiple components. In the second part, we make a comparison between the CGGJI and the NGGJI. In the third part, we further validate the NGGJI method by applying it to the model of the extended dike.

4.1 The characteristics of the weighting matrix

Single prism model with density contrast 1 g cm–3, dimensions 0.8 × 0.8 × 0.8 km3, depth to the top at 0.4 km.

Figure 2.

Single prism model with density contrast 1 g cm^–3, dimensions 0.8 × 0.8 × 0.8 km³, depth to the top at 0.4 km.

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Figure 3.

Contour maps of (a) g_xx, (b) g_xy, (c) g_xz, (d) g_yy, (e) g_yz and (f) g_zz.

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To analyse the characteristic of the weighting matrix, we built weighting matrices by using single-noise free components (Figs 4–6). The distributions of the weighting matrices show consistency with the shape of the model. First, the weighting matrices have a significantly larger value in the range of the anomalous body than other regions, and the largest weight is near the centre of the anomalous body; second, in the x-direction, the highest consistency between the weight distribution and the actual anomaly distribution is the g_xx component, followed by g_xz, g_xy, g_zz and the lowest is g_yy; third, in the y-direction, the highest consistency between the weight distribution and the actual anomaly distribution is the g_yy component, followed by g_yz, g_xy, g_zz and the lowest is g_xx; fourth, in the z-direction, the weighting matrices have a wide range of distribution. These conclusions are important for selecting the appropriate components to participate in the inversion.

Figure 4.

Weighting matrix based on single noise-free component. (a) g_xx, (b) g_xy, (c) g_xz, (d) g_yy, (e) g_yz and (f) g_zz. Depth sections are set to z = 600 m. The ‘+’ indicates the position of the centre of the anomalous body.

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Figure 5.

Weighting matrix based on single noise-free component. (a) g_xx, (b) g_xy, (c) g_xz, (d) g_yy, (e) g_yz and (f) g_zz. Vertical sections show y = 1500 m. The ‘+’ indicates the position of the centre of the anomalous body.

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Figure 6.

Weighting matrix based on single noise free component. (a) g_xx, (b) g_xy, (c) g_xz, (d) g_yy, (e) g_yz and (f) g_zz. Vertical sections show x = 1500 m. The ‘+’ indicates the position of the centre of the anomalous body.

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In practice, the gravity gradient data contains noise. Here, noise was added to the data to evaluate the noise effect on the weighting matrix. Li (2001) added Gaussian noise of 1 E. Zhdanov et al. (2004) added 3 per cent Gaussian noise. Oliveira & Barbosa (2013) added Gaussian noise of 3 E. Geng et al. (2014) added 5 per cent noise. We added 10 per cent of the maximum value pseudorandom Gaussian noise to the data. The noise level is higher than that in the predecessors' work. Generally, the lower the noise level, the less effect on the result. Based on this, we only evaluate the 10 per cent Gaussian noise's effect on the weighting matrix.

We built weighting matrices based on a single noise-containing component (Figs 7–9). We compare Figs 7–9 with Figs 4–8. The outline of the higher weight in Figs 4 and 7 is basically consistent with each other. The distribution of the weighting matrix in the depth sections changes slightly (Figs 4 and 7). For vertical sections (Figs 5, 6, 8 and 9), we get the same conclusion. Thus, the distribution of the noise-free data's weighting matrix and the distribution of the noise-contaminated data's weighting matrix are similar, the noise has a minor effect on the weighting matrix with this method.

Figure 7.

Weighting matrix based on single noise-containing component. (a) g_xx, (b) g_xy, (c) g_xz, (d) g_yy, (e) g_yzand (f) g_zz. Depth sections were set to z = 600 m. The ‘+’ indicates the position of the centre of the anomalous body.

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Figure 8.

Weighting matrix based on single noise-containing component. (a) g_xx, (b) g_xy, (c) g_xz, (d) g_yy, (e) g_yz and (f) g_zz. Vertical sections show y = 1500 m. The ‘+’ indicates the position of the centre of the anomalous body.

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Figure 9.

Weighting matrix based on single noise-containing component. (a) g_xx, (b) g_xy, (c) g_xz, (d) g_y_y, (e) g_yz and (f) g_zz. Vertical sections show x = 1500 m. The ‘+’ indicates the position of the centre of the anomalous body.

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Next, we built weighting matrices by using multiple components (Fig. 10). The data here contains noise. Figs 10(a)–(c) show the weighting matrix based on g_xx and g_yy components. It is compared with the weighting matrix based on g_xx (Figs 7a–9a) and the weighting matrix based on g_yy (Figs 7b–9b). The weighting matrix's distribution in spatial converge to the centre of the anomalous body, and it is consistent with the distribution of the model. Figs 10(d)–(f), (g)–(i) and (j)–(l) show the weighting matrix based on three components (g_xx|g_xy|g_yy), four components (g_xx|g_xy|g_yy|g_yz), five components (g_xx|g_xy|g_yy|g_yz|g_xz), respectively. We found that as the number of the components increase, the range of the high-weight distribution in the weighting matrix gradually converges to the centre of the anomalous body. This indicates that the inversion results tend to be more focused with more components added to the inversion. The reason is that different components affect each other, the common information is enhanced and the different information is attenuated. Thus, when we apply this method to gravity gradient inversion, appropriate component combinations must be chosen to get satisfactory results.

Figure 10.

Weighting matrix based on multiple noise-containing components. (a–c) Based on g_xx| g_yy, (d–f) based on g_xx|g_yy|g_zz, (g–i) based on g_xx|g_yy|g_xy|g_yz,(j–l) based on g_xx|g_yy|g_xy |g_xz|g_yz. Depth sections are at z = 600 m (left-hand column), vertical sections are at y = 1500 m (centre column) and x = 1500 m (right-hand column).

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4.2 Comparison between different methods

The parameters of the survey area are unchanged. Fig. 11 shows multiple models with different property. Two cubic bodies, each 0.4 |$ \times $| 0.4 |$ \times $| 0.4 km³ in dimension, with density contrast of 1 g cm^–3 over background, and centres locate at (600, 600, 500) m and (1400, 600, 500) m. Three prisms are used, one is 1.2 |$ \times $| 0.3 |$ \times $| 0.4 km³ in dimension, and another is 0.3 |$ \times $| 1.2 |$ \times $| 0.4 km³ in dimension, the third one is 0.3 |$ \times $| 0.3 |$ \times $| 0.5 km³ in dimension, with density contrast of 0.8 g cm^–3 over background, and centres locate at (1500, 2200, 600) m, (2500, 2100, 600) m and (1400, 1500, 450) m. The synthetic observed data were contaminated by 10 per cent of the maximum pseudorandom Gaussian noise with zero means (Fig. 12). Figs 13(a)–(e) show one depth section and four vertical sections of the synthetic model. We apply the CGGJI method to the g_zz component, the inversion results are shown in Figs 13(f)–(j).

Figure 11.

Perspective view of the synthetic model. Two cubic models have a density of 1 g cm^–3, three prism models have a density of 0.8 g cm^–3. The models are buried at different depth.

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Figure 12.

Observed gravity gradient data produced by synthetic model. The data were contaminated by 10 per cent of the maximum pseudorandom Gaussian noise with zero means.

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Figure 13.

True and recovered model for the multiple body example, including: (a, b, c, d, e) true model, (f, g, h, I, j), g_zz inversion.

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4.2.1 Two components inversion

From Section 4.1, the weighting matrix based on g_xx shows a better resolution in the x-direction, the weighting matrix based on g_yy shows a better resolution in the y-direction, the distribution of the weighting matrix based on g_xy is close to the true model. We applied the CGGJI method and the NGGJI method to three different component combinations: g_xx|g_zz, g_yy|g_zz, g_xy|g_zz. The inversion results are shown in Figs 14–16.

Figure 14.

Recovered model for the multiple body example. The inversion method was applied to g_xx|g_zz. (a–e) CGGJI; (f–j) NGGJI.

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Figure 15.

Recovered model for the multiple body example. The inversion method was applied to g_yy|g_zz. (a–e) CGGJI; (f–j) NGGJI.

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Figure 16.

Recovered model for the multiple body example. The inversion method was applied to g_xy|g_zz. (a–e) CGGJI; (f–j) NGGJI.

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First, we compare the recovered models (Figs 14a–e to 16a–e) obtained by the CGGJI method with g_zz inversion result (Figs 13f–j). The recovered models (Figs 14a–e to 16a–e) have a higher value at the centre of the anomalous bodies than the g_zz inversion result. They have a clear outline, but the boundaries of the anomalous bodies are less clear than that in the g_zz inversion result. It means that in the CGGJI method, different components affect each other, although the overall resolution was improved, the local resolution is reduced. Additionally, we make comparisons among the three groups of figures (Figs 14a–e to 16a–e), the difference is marginal. We make a comparison between Figs 14(d) and 15(d), we find that the model in the centre in Fig. 15d has a higher value. There is also a minor difference in Figs 15(c) and 16(c). The model in the centre in Fig. 15(d) is closer to the true model. From the inversion results, of the two components inversion with CGGJI, the improvement of the inversion result is marginal. Thus, we conclude that the characteristic information of the components participating in the inversion is not well reflected in the inversion results.

Secondly, we compare the recovered model (Figs 14f–j to 16f–j) obtained from the NGGJI method with g_zz inversion result (Figs 13f–j). The outline of the recovered models is more clear, the maximum value of the models are larger. The recovered models show a better consistency with the true model (Figs 13a–e). The resolution of the recovered models in Figs 14f–j to 16f–j have been improved significantly.

Third, we compare the recovered model obtained from the NGGJI method (Figs 14f–j to 16f–j) with the recovered model obtained from the CGGJI method (Figs 14a–e to 16a–e). The results with NGGJI show a clearer outline, and the maximum value of the density is close to the true density. In addition, the results with NGGJI are more consistent with the true model. Thus, we conclude the results with NGGJI have a higher resolution than the results with CGGJI.

What's more, g_xx|g_zz inversion results (Figs 14f–j) converges to the centre of the recovered model in x-direction, it shows a better resolution in this direction. Similarily, g_yy|g_zz inversion results (Figs 15f-j) shows a better resolution in the y-direction, and g_xy|g_zz inversion result (Figs 16f–j) is more consistent with the actual model. Based on the results in Section 4.1, different components show different characteristic information. They are well reflected in the inversion results. For two components inversion, the NGGJI method provides a better way to extract and utilize the information contained in data.

4.2.2 Multiple component inversions

We apply the CGGJI method and the NGGJI method to different component combinations (g_xx|g_yy|g_zz, g_xx|g_yy|g_xy|g_zz, g_xx|g_yy|g_xy|g_xz|g_zz, g_xx|g_yy|g_xy|g_yz|g_zz, g_xx|g_yy|g_xy|g_xz|g_yz|g_zz). For three component combinations, we chooseg_xx|g_yy|g_zz. Because g_xx and g_yy show better consistency in x-direction and y-direction, respectively. Then we add other components in inversion gradually. Figs 17–21 show the inversion results.

Figure 17.

Recovered model for the multiple body example. The inversion method was applied to g_xx|g_yy|g_zz. (a–e) CGGJI; (f–j) NGGJI.

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Figure 18.

Recovered model for the multiple body example. The inversion method was applied to g_xx|g_xy|g_yy|g_zz. (a–e) CGGJI; (f–j) NGGJI.

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Figure 19.

Recovered model for the multiple body example. The inversion method was applied to g_xx|g_xy|g_yy|g_xz|g_zz. (a–e) CGGJI; (f–j) NGGJI.

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Figure 20.

Recovered model for the multiple body example. The inversion method was applied to g_xx|g_xy|g_yy|g_yz|g_zz. (a–e) CGGJI; (f–j) NGGJI.

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Figure 21.

Recovered model for the multiple body example. The inversion method was applied to g_xx|g_xy|g_yy|g_xz|g_yz|g_zz. (a–e) CGGJI; (f–j) NGGJI.

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First, we analyse the inversion results obtained by the CGGJI methods. Figs 17(a)–(e) show the g_xx|g_yy|g_zz inversion results, we compare Figs 17(c)–(d) with Figs 14(c)–(d) and 15(c)–(d), the centre of the anomaly is more clear, the resolution of the inversion result is improved. Figs 18(a)–(e) show the g_xx|g_yy|g_xy|g_zz inversion results, compared with Figs 17(a)–(e), the improvement is minor. Figs 19(a)–(e) and 20(a)–(e) show the g_xx|g_yy|g_xy|g_xz|g_zz inversion results and the g_xx|g_yy|g_xy|g_yz|g_zz inversion results. The results were similar to Figs 18(a)–(e). Further, we give the six components inversion results in Fig. 21, we make a comparison among Figs 19(d)–20(d), the maximum value of the anomaly's centre is higher in Fig. 21(d). Compared to the five components inversion result, the six components inversion result is improved.

Secondly, we analyse the inversion results obtained by the NGGJI method. Figs 17(f)–(j) shows the g_xx|g_yy|g_zz inversion results. The recovered model converges to the centre of the anomaly. The maximum value of the density is close to the true model's density contrast. According to Scetion 4.1.2, the weighting matrix based on g_xx has higher weights at the centre of the anomalous body, it is convergent in x direction. The weighting matrix based on g_yy also has higher weights at the centre of the anomalous body, it is convergent in y direction. When we build a weighting matrix based on g_xx and g_yy, the information contained in different components overlaps, the common information is enhanced, and the different information is weakened, the weighting matrix converges to the centre of the anomalous dody. It gives more weights to the centre of the model. This affects the inversion result. Thus, the recovered model converges to the centre of the anomalous bodies.

Figs 18(f)–(j) showed the g_xx|g_yy|g_xy|g_zz inversion results. The information contained in g_xy shows a good resolution in both the x-direction and the y-direction. This improves the resolution of the boundary. Figs 19(f)–(j) and 20(f)–(j) show the g_xx|g_yy|g_xy|g_xz|g_zz inversion results and the g_xx|g_yy|g_xy|g_yz|g_zz inversion results respectively. Compared with Figs 18(f)–(j), the inversion results in Figs 19(f)–(j) and 20(f)–(j) converges to the centre of the anomalous body. Figs 21(f)–(j) shows the inversion result for the six components. Compared with Figs 19(f)–(j) and 20(f)–(j), the centre of the anomalous bodies were better outlined by the six components inversion result.

In conclusion, when we integrate multiple components (equal to or more than 3) in inversion, different components affected each other. For multiple components inversion, the recovered model of g_xx|g_yy|g_xy|g_zz inversion was most consistent with the actual models. Additionally, six components inversion better outlines the centre of the anomalous bodies.

4.2.3 Comparison of the degree of data fit

The degree of data fit is used to measure the difference between the standard deviation of the residuals (SDR), which is obtained from the difference between the observed and predicted data, and the standard deviation of Gaussian noise (SDG). It provides a quantitive method to evaluate the reasonableness of the inversion result. If the SDR is close to the SDG, the degree of data fit is desirable.

Table 1 shows the SDG and the SDR obtained by CGGJI method. We first calculate the difference between the SDG and the SDR of different inversions, then take its absolute value. The results are shown in Fig. 22(a). We use different colours to represent different inversions. From Fig. 22(a), we find that for two components inversion, the degree of data fit for different components changes rapidly. This is because the component which participates in the inversion has an over-fit phenomenon. The SDR of this component is far less than its SDG (Table 1). We also find that for three components inversion, four components inversion, five components inversion, and six components inversion, the line approximate to the x-axis, and the difference between the SDG and SDR of these inversions is less than that of two components inversion. So, the degree of data fit is better with more components in inversion.

Figure 22.

Comparison of data fit of different inversions, (a) CGGJI, (b) NGGJI. The black line represents the g_zz inversion, the purple line represents two components inversion, the green line represents three components inversion, the blue line represents four components inversion, the yellow line represents five components inversion, the red line represents six components inversion.

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Table 1.

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Standard deviations of Gaussian noise and the estimated standard deviation of the residuals between observed and predicted data of different inversions by CGGJI method. The unit is E.

	g_xx	g_xy	g_xz	g_yy	g_yz	g_zz
Std	3.87	1.45	3.42	3.45	3.42	6.13
g_zz	4.34	1.63	3.73	3.94	3.74	6.90
g_xx\|g_zz	3.21	1.65	3.67	4.36	3.82	4.08
g_yy\|g_zz	4.66	1.65	3.76	2.67	3.63	4.28
g_xy\|g_zz	4.55	1.25	3.65	4.14	3.67	3.88
g_xx\|g_yy\|g_zz	4.32	1.60	3.58	3.91	3.58	6.94
g_xx\|g_yy\|g_xy\|g_zz	4.21	1.55	3.58	3.82	3.55	6.82
g_xx\|g_yy\|g_xy\|g_xz\|g_zz	3.53	1.36	3.34	2.96	3.54	5.35
g_xx\|g_yy\|g_xy\|g_yz\|g_zz	4.20	1.54	3.58	3.81	3.53	6.79
g_xx\|g_yy\|g_xy\|g_xz\|g_yz\|g_zz	4.11	1.52	3.53	3.76	3.52	6.69

	g_xx	g_xy	g_xz	g_yy	g_yz	g_zz
Std	3.87	1.45	3.42	3.45	3.42	6.13
g_zz	4.34	1.63	3.73	3.94	3.74	6.90
g_xx\|g_zz	3.21	1.65	3.67	4.36	3.82	4.08
g_yy\|g_zz	4.66	1.65	3.76	2.67	3.63	4.28
g_xy\|g_zz	4.55	1.25	3.65	4.14	3.67	3.88
g_xx\|g_yy\|g_zz	4.32	1.60	3.58	3.91	3.58	6.94
g_xx\|g_yy\|g_xy\|g_zz	4.21	1.55	3.58	3.82	3.55	6.82
g_xx\|g_yy\|g_xy\|g_xz\|g_zz	3.53	1.36	3.34	2.96	3.54	5.35
g_xx\|g_yy\|g_xy\|g_yz\|g_zz	4.20	1.54	3.58	3.81	3.53	6.79
g_xx\|g_yy\|g_xy\|g_xz\|g_yz\|g_zz	4.11	1.52	3.53	3.76	3.52	6.69

Table 1.

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Standard deviations of Gaussian noise and the estimated standard deviation of the residuals between observed and predicted data of different inversions by CGGJI method. The unit is E.

	g_xx	g_xy	g_xz	g_yy	g_yz	g_zz
Std	3.87	1.45	3.42	3.45	3.42	6.13
g_zz	4.34	1.63	3.73	3.94	3.74	6.90
g_xx\|g_zz	3.21	1.65	3.67	4.36	3.82	4.08
g_yy\|g_zz	4.66	1.65	3.76	2.67	3.63	4.28
g_xy\|g_zz	4.55	1.25	3.65	4.14	3.67	3.88
g_xx\|g_yy\|g_zz	4.32	1.60	3.58	3.91	3.58	6.94
g_xx\|g_yy\|g_xy\|g_zz	4.21	1.55	3.58	3.82	3.55	6.82
g_xx\|g_yy\|g_xy\|g_xz\|g_zz	3.53	1.36	3.34	2.96	3.54	5.35
g_xx\|g_yy\|g_xy\|g_yz\|g_zz	4.20	1.54	3.58	3.81	3.53	6.79
g_xx\|g_yy\|g_xy\|g_xz\|g_yz\|g_zz	4.11	1.52	3.53	3.76	3.52	6.69

	g_xx	g_xy	g_xz	g_yy	g_yz	g_zz
Std	3.87	1.45	3.42	3.45	3.42	6.13
g_zz	4.34	1.63	3.73	3.94	3.74	6.90
g_xx\|g_zz	3.21	1.65	3.67	4.36	3.82	4.08
g_yy\|g_zz	4.66	1.65	3.76	2.67	3.63	4.28
g_xy\|g_zz	4.55	1.25	3.65	4.14	3.67	3.88
g_xx\|g_yy\|g_zz	4.32	1.60	3.58	3.91	3.58	6.94
g_xx\|g_yy\|g_xy\|g_zz	4.21	1.55	3.58	3.82	3.55	6.82
g_xx\|g_yy\|g_xy\|g_xz\|g_zz	3.53	1.36	3.34	2.96	3.54	5.35
g_xx\|g_yy\|g_xy\|g_yz\|g_zz	4.20	1.54	3.58	3.81	3.53	6.79
g_xx\|g_yy\|g_xy\|g_xz\|g_yz\|g_zz	4.11	1.52	3.53	3.76	3.52	6.69

Table 2 shows the SDG and SDR by NGGJI method. We get Fig. 22(b) based on Table 2 and find that the degree of data fit of different inversions are close to each other except the six components inversion. This is because, when we apply the NGGJI method to six components inversion, the inversion results converge to the centre of the anomalous body, this affects the degree of data fit.

Table 2.

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Standard deviations of Gaussian noise and the estimated standard deviation of the residuals between observed and predicted data of different inversions by NGGJI method. The unit is E.

	g_xx	g_xy	g_xz	g_yy	g_yz	g_zz
Std	3.87	1.45	3.42	3.45	3.42	6.13
g_zz	4.34	1.63	3.73	3.94	3.74	6.90
g_xx\|g_zz	4.74	1.73	4.01	4.16	4.02	5.01
g_yy\|g_zz	4.60	1.72	3.89	4.10	3.93	4.85
g_xy\|g_zz	4.58	1.76	4.11	4.29	4.24	5.54
g_xx\|g_yy\|g_zz	4.53	1.67	4.01	3.85	3.81	5.42
g_xx\|g_yy\|g_xy\|g_zz	4.43	1.65	4.06	3.74	3.86	5.97
g_xx\|g_yy\|g_xy\|g_xz\|g_zz	4.37	1.84	4.39	4.02	4.60	6.90
g_xx\|g_yy\|g_xy\|g_yz\|g_zz	4.47	1.76	4.39	3.86	4.24	6.54
g_xx\|g_yy\|g_xy\|g_xz\|g_yz\|g_zz	5.45	2.73	6.47	6.33	6.59	9.02

	g_xx	g_xy	g_xz	g_yy	g_yz	g_zz
Std	3.87	1.45	3.42	3.45	3.42	6.13
g_zz	4.34	1.63	3.73	3.94	3.74	6.90
g_xx\|g_zz	4.74	1.73	4.01	4.16	4.02	5.01
g_yy\|g_zz	4.60	1.72	3.89	4.10	3.93	4.85
g_xy\|g_zz	4.58	1.76	4.11	4.29	4.24	5.54
g_xx\|g_yy\|g_zz	4.53	1.67	4.01	3.85	3.81	5.42
g_xx\|g_yy\|g_xy\|g_zz	4.43	1.65	4.06	3.74	3.86	5.97
g_xx\|g_yy\|g_xy\|g_xz\|g_zz	4.37	1.84	4.39	4.02	4.60	6.90
g_xx\|g_yy\|g_xy\|g_yz\|g_zz	4.47	1.76	4.39	3.86	4.24	6.54
g_xx\|g_yy\|g_xy\|g_xz\|g_yz\|g_zz	5.45	2.73	6.47	6.33	6.59	9.02

Table 2.

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Standard deviations of Gaussian noise and the estimated standard deviation of the residuals between observed and predicted data of different inversions by NGGJI method. The unit is E.

	g_xx	g_xy	g_xz	g_yy	g_yz	g_zz
Std	3.87	1.45	3.42	3.45	3.42	6.13
g_zz	4.34	1.63	3.73	3.94	3.74	6.90
g_xx\|g_zz	4.74	1.73	4.01	4.16	4.02	5.01
g_yy\|g_zz	4.60	1.72	3.89	4.10	3.93	4.85
g_xy\|g_zz	4.58	1.76	4.11	4.29	4.24	5.54
g_xx\|g_yy\|g_zz	4.53	1.67	4.01	3.85	3.81	5.42
g_xx\|g_yy\|g_xy\|g_zz	4.43	1.65	4.06	3.74	3.86	5.97
g_xx\|g_yy\|g_xy\|g_xz\|g_zz	4.37	1.84	4.39	4.02	4.60	6.90
g_xx\|g_yy\|g_xy\|g_yz\|g_zz	4.47	1.76	4.39	3.86	4.24	6.54
g_xx\|g_yy\|g_xy\|g_xz\|g_yz\|g_zz	5.45	2.73	6.47	6.33	6.59	9.02

	g_xx	g_xy	g_xz	g_yy	g_yz	g_zz
Std	3.87	1.45	3.42	3.45	3.42	6.13
g_zz	4.34	1.63	3.73	3.94	3.74	6.90
g_xx\|g_zz	4.74	1.73	4.01	4.16	4.02	5.01
g_yy\|g_zz	4.60	1.72	3.89	4.10	3.93	4.85
g_xy\|g_zz	4.58	1.76	4.11	4.29	4.24	5.54
g_xx\|g_yy\|g_zz	4.53	1.67	4.01	3.85	3.81	5.42
g_xx\|g_yy\|g_xy\|g_zz	4.43	1.65	4.06	3.74	3.86	5.97
g_xx\|g_yy\|g_xy\|g_xz\|g_zz	4.37	1.84	4.39	4.02	4.60	6.90
g_xx\|g_yy\|g_xy\|g_yz\|g_zz	4.47	1.76	4.39	3.86	4.24	6.54
g_xx\|g_yy\|g_xy\|g_xz\|g_yz\|g_zz	5.45	2.73	6.47	6.33	6.59	9.02

Further, we make a comparison of the degree of data fit between the NGGJI method and CGGJI method. The results are shown in Fig. 23. We find that the degree of data fit are close to each other for two components inversion, three components inversion, and four components inversion. When we add more components in the inversion, the degree of data fit of the CGGJI was better than that of the NGGJI.

Figure 23.

Comparison of data fit of different inversions, the solid line represented the CGGJI method, the dashed line represents the NGGJI method: (a) g_zz, (b) g_xx|g_zz, (c) g_yy|g_zz, (d) g_xy|g_zz, (e) g_xx|g_yy|g_zz, (f) g_xx|g_yy|g_xy|g_zz, (g) g_xx|g_yy|g_xy|g_xz|g_zz, (h) g_xx|g_yy|g_xy|g_yz|g_zz and (i) g_xx|g_yy|g_xy|g_xz|g_yz|g_zz.

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4.2.4 Discussion

The results of Section 4.2 show that in two components inversions, the recovered models with the NGGJI have a higher resolution than the recovered model with the CGGJI. Compared to the g_zz inversion result, they show the characteristic information contained in the relevant component. The NGGGJI provides a better way to extract and utilize the characteristic information contained in the data. In addition, in multiple components inversions, with more components in inversion, the resolution of the recovered models with the CGGJI are improved, and the recovered model with the NGGJI tend to converge to the centre of the anomalous bodies. The g_xx|g_yy|g_xy|g_zz inversion was the most consistent with the actual models. Through the comparison of the degree of data fit, in two components inversion, three components inversion, four components inversion, and five components inversion, the NGGJI is better than the CGGJI. For six components inversion, the CGGJI is better than the NGGJI. The reason for this is that, with more components in NGGJI, the weighting matrix tend to converge to the centre of the anomalous bodies, this affects the inversion result, and the degree of data fit.

4.3 Application to the model of the extended dike

To further validate the method, we applied the method to the model of the extended dike. The parameters of the survey area are unchanged. There are two extended dikes (Fig. 24). Both buried at a depth of z = 400 m. The red one has a density of 1 g cm^–3. It extends 1000 m in the z-direction. The orange one has a density of 0.8 g cm^–3. It extends 500 m in the z-direction. We added 10 per cent of the maximum value pseudorandom Gaussian noise with zero mean to the data (Fig. 25).

Figure 24.

Two extended dike models. The red one have a density of 1 g cm^–3, the orange one have a density of 0.8 g cm^–3. The models are buried at z = 400 m.

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Figure 25.

Observed gravity gradient data produced by the extended dike models. The data were contaminated by 10 per cent of the maximum pseudorandom Gaussian noise with zero means.

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Figs 26(a) and (b) show the depth section and the vertical section of the true model. Figs 26(c) and (d) show the g_zz inversion results obtained by the CGGJI. The extended directions of the recovered models are consistent with the true models. The boundaries of the models are not clear. Because the models extend in the y-direction, so we first apply the NGGJI to g_yy|g_zz (Figs 26e and f). The extended directions of the recovered models (Figs 26e and f) are also consistent with that of the true models. We compare Figs 26(e) and (f) with Figs 26(c) and (d), the g_yy|g_zz inversion results (Figs 26e and f) have a higher value, and the boundaries of the recovered models are more clear.

Figure 26.

True and recovered model for the extended dike model, including: (a, b) true model, (c, d) g_zz inversion and (e, f) g_yy|g_zz inversion by NGGJI.

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Next, we apply the NGGJI to different component combinations (g_xx|g_yy|g_zz, g_xx|g_yy|g_xy|g_zz, g_xx|g_yy|g_xy|g_xz|g_yz|g_zz). The inversion results are shown in Fig. 27. First, we make a comparison between g_yy|g_zz (Figs 26e and f) and g_xx|g_yy|g_zz (Figs 27a and b). The inversion results in Figs 27(a) and (b) are more focused to the centre than that in Figs 26(e) and (f). They are more consistent with the true model. The g_xx|g_yy|g_xy|g_zz inversion results are shown in Figs 27(c) and (d). Compare with Figs 27(a) and (b), the recovered models in Figs 27(c) and (d) are more focused to the centre, and boundaries of the models are more consistent with that of the true models. The resolution of the recovered models is improved. For six components inversion (Figs 27e and f), the recovered models are more focused to the centre of the extended dikes than the four components inversion results (Figs 27c and d), the extended directions are consistent with the true models, and the centre of the models are better outlined. From the above results, the NGGJI method provides a new way to extract and utilize the characteristic information contained in the data. We got reasonable results by applying it to the model of the extended dike. With more components in NGGJI, the inversion results tend to converge to the centre of the anomalous bodies.

Figure 27.

Recovered model for the extended dike model by NGGJI, including: (a and b) g_xx|g_yy|g_zz, (c and d) g_xx|g_yy|g_xy|g_zz and (e and f) g_xx|g_yy|g_xy|g_xz|g_yz|g_zz.

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5 APPLICATION TO REAL DATA

5.1 Vinton salt dome

In this part, we select the gravity gradient data measured in the Vinton salt dome to verify the NGGJI method. The Vinton salt dome is located in southwestern Louisiana, USA. Coker et al. (2007) pointed that the Vinton salt dome is a characteristic salt dome. It has a core of massive salt and a well-defined cap-rock. The data was taken by Bell Geospace Inc in 2008 (Fig. 28). It was measured at a mean altitude of 80 m. For this study, the survey area is 3.9|$ \times $|3.45 km². We divided the subsurface into |${n_x} \times {n_y} \times {n_z} = 26 \times 23 \times 20$| prisms. The dimension of the prism is |$150 \times 150 \times 50$||${{\rm{m}}^3}$|⁠. The density to remove the terrain effect is 2.2 g cm^–3 (Oliveira & Barbosa 2013). Here, we first apply the inversion method to g_zz component, and then to different component combinations (g_xx|g_zz; g_yy|g_zz; g_xx|g_yy|g_zz; g_xx|g_yy|g_xy|g_zz; g_xx|g_xy|g_xz|g_yy|g_yz|g_zz).

Figure 28.

The real data of Vinton salt dome.

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5.2 Single-component inversion

We first perform the inversion calculation of the g_zz component. We make a comparison between the predicted g_zz data (Fig. 29a) and the observed g_zz data. The range of the predicted data is [–32.02, 59.53] E, its mean value is –4.73 E. The range of the observed data is [–32.00, 64.66] E, its mean value is –3.76 E. The difference between the observed data and the predicted data of g_zz ranges from –0.75 E to 6.03 E. Its standard derivation is 1.88 E. The difference map (Fig. 29b) shows a clearly positive anomaly, but its rather small compared to the observed data. The results indicate that the major signals in the observed data have been well reproduced (Figs 28f and 29a).

Figure 29.

(a) The predicted g_zz data of g_zz inversion, (b) the difference map between the observed g_zz data and the predicted g_zz data.

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The recovered model shows that the cap-rock extends 1750 m in the east–west direction (Fig. 30b), 1500 m in the north–south direction (Fig. 30c). The high-density part extends from 200 to 600 m in the depth direction (Fig. 30a). Ennen (2012) showed that the caprock extends 1520 m in the east–west direction, 1280 m in the north–south direction. Its top was located 160 m below the surface (Ennen & Hall 2011). Although there are differences, the g_zz inversion produces geologically reasonable and interpretable results.

Figure 30.

Recovered models from the inversion of different component combinations: (a–c) g_zz, (d–f) g_xx|g_zz, (g–i) g_yy|g_zz. Depth sections are at z = 400 m (left-hand column), vertical slices are at Northing = 1350 m (centre column) and Easting = 2250 m (right-hand column).

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5.3 Multiple component inversion

Figs 30(d)–(f) show the g_xx|g_zz inversion results. The cap-rock extends 1050 m in the east–west direction, 1200 m in the south–north direction, 200–800 m in the z-direction. Figs 30(g)–(i) show the g_yy|g_zz inversion results. The caprock extends 1350 m in the east–west direction, 1050 m in the south–north direction, 200–800 m in the z-direction. Although the results are different from the predecessors’ conclusions (1520 m in the east-west direction, 1280 m in the south–north direction, Ennen et al. 2011; Ennen 2012), we find that the boundary of the g_xx|g_zz inversion results tend to be more convergent in the x-direction, and the g_yy|g_zz inversion results tend to be more convergent in the y-direction. As it is shown in Section 4.1.2, the weighting matrix based on g_xx is convergent in x direction, the weighting matrix based on g_yy is convergent iny direction. The inversion result is affected by the weighting matrix.

Then, we apply the NGGJI method to g_xx|g_yy|g_zz inversion (Figs 31a–c). The inversion results tend to concentrate to the centre of the anomaly body. The cap-rock extends 200–600 m in the z-direction, and the high-density part focuses at 300–450 m. We also apply the NGGJI method to g_xx|g_yy|g_xy|g_zz (Figs 31d–f) and g_xx|g_yy|g_xy|g_xz|g_yz|g_zz (Figs 32g–i). The g_xx|g_yy|g_xy|g_zz inversion (Figs 31d–f) shows that the cap-rock extends 1350 m in the east–west direction, about 1050 m in the south–north direction. The high-density part extends 350–550 m in the z-direction. The g_xx|g_yy|g_xy|g_xz|g_yz|g_zz inversion results (Figs 31g–i) show a more focused recovered model than others.

Figure 31.

Recovered models from the inversion of different component combinations: (a–c) g_xx|g_yy|g_zz, (d–f) g_xx|g_yy|g_xy|g_zz, (g–i) g_xx|g_yy|g_xy|g_xz|g_yz|g_zz. Depth sections are at z = 400 m (left-hand column), vertical slices are at Northing = 1350 m (centre column) and Easting = 2250 m (right-hand column).

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Figure 32.

Volume-rendered image of the 3-D density-contrast model constructed by g_xx|g_xy|g_yy|g_zz inversion. The density contrast is greater or equal to 0.15 g cm^–3.

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In Section 4.1.4, with more components in weighting matrix, the range of the high-weight distribution tends to converge to the centre of the anomalous body. In Section 4.2.3, six components inversion better outline the centre of the anomalous bodies. The results in this section are consistent with these results. Finally, we choose g_xx|g_yy|g_xy|g_zz inversion for interpretation.

The 3D density-contrast model constructed by g_xx|g_xy|g_yy|g_zz inversion is shown in Fig. 32. Eti (2004) pointed that the gradients in the southern and eastern sides of the dome are from 36° to 56°, and the gradients in the northerm and western sides are from 16° to 20°. This means that the southern and eastern sides of the dome are steeper than the northern and western sides. This is consistent with the model in Fig. 32. However, previous works have placed the cap-rock at 160–360 m (Ennen et al. 2011), 160–460 m (Oliveira & Barbosa 2013), 200–650 m (Geng et al. 2014), 200–550 m (Qin et al. 2016), these results are different from our results (350–550 m). To explain this, we analysed the g_xx|g_yy|g_zz inversion results in Figs 31(a)–(c). The cap-rock extends 200–600 m in the z-direction. With the NGGJI method, when we add g_xy in inversion, the results tend to converge to the centre of the anomalous body.

6 CONCLUSION

We have proposed a new gravity gradient joint inversion (NGGJI) method in this paper. We choose one component, for example, g_zz, to perform in the inversion directly and used other components to construct a weighting matrix. We analysed the characteristic of the weighting matrix based on a single component and concluded that: (1) the largest weight is near the centre of the anomalous body; (2) in the x-direction, the highest consistency between the weight distribution and the actual anomaly distribution was the g_xx component; (3) in the y-direction, the highest consistency between the weight distribution and the actual anomaly distribution was the g_yy component; (4) in the z-direction, the weighting matrices have a wide range of distribution and (5) we add a high-level noise to gravity gradient data, its effect on the weighting matrices is minor. We also analysed the weighting matrices based on multiple components and found that as the number of the components increase, the range of the high-weight's distribution in the weighting matrix gradually converges to the centre of the anomalous body. The reason is that different components affect each other, the common information is enhanced and the different information is attenuated. Thus, when we apply this method to gravity gradient inversion, we must choose appropriate component combinations to get satisfactory results.

We also introduced the conventional gravity gradient joint inversion (CGGJI) method and made a comparison between the NGGJI method and the CGGJI method. We found that: (1) for two components inversion, the NGGJI method significantly improves the resolution of the inversion results. It provides a better way to extract and utilize the information contained in data; (2) for multiple components inversion, g_xx|g_yy|g_xy|g_zz is the most consistent with the actual models; (3) six components inversion results which are obtained by NGGJI method better outline the centre of the anomalous bodies and (4) the degree of data fit of NGGJI was worse than CGGJI. We further validated the NGGJI method by applying it to the model of the extended dike. The inversion results are reasonable.

Finally, we applied our method to FTG data from the Vinton salt dome. The results further prove the conclusions in the synthetic part. We chose g_xx|g_xy|g_yy|g_zz inversion as the final results. The results are different from the predecessors’ conslusions. To further confirm this estimate, more prior geological information about the Vinton salt dome must be introduced. For now, we can accept this recovered model as a possible geometry of the cap-rock.

ACKNOWLEDGEMENTS

The authors thank Dr Qin Xuwen and Dr Lu Jingan for their support to this work. Dr Geng Meixia, Dr Zhang Dailei, and Dr Zhang Chong for interesting discussion and constructive feedback. They also thank Jimmy Gorimar for helping to polish the paper. Associate editor Fern Storey, reviewer Jerome Verdun, and two other reviewers helped to clarify many parts of the paper. They also thank Bell Geospace Inc. for permission to use the real data from Vinton salt some. This work was supported by Project 2017YFC0307601 supported by National Key R&D Plan.

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APPENDIX A: THE LINEAR CONJUGATE GRADIENT METHOD

The linear conjugate gradient (LCG) method is a commonly used optimization algorithm. We adopt this method to construct the weighting matrix. The expression of the objective function was defined as

\begin{eqnarray*} {\phi _d}({{\bf m}}) = \left\| {{{{\bf A}}_w}{{{\bf m}}_w} - {{\bf d}}} \right\|_2^2. \end{eqnarray*}

(A1)

The expression of |$\nabla {\phi _d}({{\bf m}})$| was

\begin{eqnarray*} \nabla {\phi _d}({{\bf m}}) = {{\rm{(}}{{{\bf A}}_w})^T}({{{\bf A}}_w}{{{\bf m}}_w} - {{\bf d}}). \end{eqnarray*}

(A2)

The key step of the LCG method is the calculation of the step length. After the |$i{\rm{th}}$| iteration, we get |${{{\bf m}}_{w(i)}}$|⁠, the search direction is |${{\bf l}}$|⁠, assume the step length is |$k$|⁠, then we get |${{{\bf m}}_{w(i + 1)}} = {{{\bf m}}_{w(i)}} + k{{\bf l}}$|⁠. To calculate the step length, we require that

\begin{eqnarray*} \phi ({{{\bf m}}_{w(i + 1)}}) = \phi ({{{\bf m}}_{w(i)}} + k{{\bf l}}) = {\rm{min}}. \end{eqnarray*}

(A3)

The expression is regarded as a function about |$k$|⁠, then we get

\begin{eqnarray*} \nabla \phi (k) = {{\rm{(}}{{{\bf A}}_w}{{\bf l}})^T}({{{\bf A}}_w}{{\bf l}}k + {{{\bf A}}_w}{{{\bf m}}_w} - {{\bf d}}) = 0. \end{eqnarray*}

(A4)

From above expressions, we get

\begin{eqnarray*} k = \frac{{{\rm{ - (}}{{{\bf A}}_w}{{\bf l}}{)^T}({{{\bf A}}_w}{{{\bf m}}_w} - {{\bf d}})}}{{{{{\rm{(}}{{{\bf A}}_w}{{\bf l}})}^T}{\rm{(}}{{{\bf A}}_w}{{\bf l}})}}. \end{eqnarray*}

(A5)

The flow of the LCG method is referred to Pilkington (1997) and Zhdanov (2009).

APPENDIX B: NON-LINEAR CONJUGATE GRADIENT METHOD

The difference between the NLCG method and the LCG method is the method to calculate the step length. The NLCG method adopt the line search method to calculate the step length. The expression of the objective function is defined as

\begin{eqnarray*} \phi = {({{{\bf A}}_w}{{{\bf m}}_w} - {{\bf d}},{{{\bf A}}_w}{{{\bf m}}_w} - {{\bf d}})_D} + \alpha \sum\limits_{i = 1}^n {\frac{{{{({w_i}{m_{wi}})}^2}}}{{{{({w_i}{m_{wi}})}^2} + {e^2}}}}. \end{eqnarray*}

(B1)

Substitute |${{{\bf m}}_{w(i + 1)}} = {{{\bf m}}_{w(i)}} + k{{\bf l}}$| into the eq. (B1), the expression of |$k$|is non-unique. Thus, we adopt the NLCG method to solve this problem. The method is based on the Wolfe condition, it was defined as

\begin{eqnarray*} \left\{ {\begin{array}{@{}*{1}{c}@{}} {\phi \left( {{{{\bf m}}_{w(i)}} + k{{\bf l}}} \right) - \phi ({{{\bf m}}_{w(i)}}) \le {c_1}k\nabla \varphi {{({{{\bf m}}_{w(i)}})}^T}{{\bf l}}}\\ {\nabla \varphi {{\left( {{{{\bf m}}_{w(i)}} + k{{\bf l}}} \right)}^T}{{\bf l}} \ge {c_2}\nabla \varphi {{({{{\bf m}}_{w(i)}})}^T}{{\bf l}}}, \end{array}} \right. \end{eqnarray*}

(B2)

where |$0 < {c_1} < {c_2} < 1$| and usual values are |${c_1} = {10^{ - 4}}$| and |${c_2} = 0.9$|⁠.

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Article Contents

A new method to integrate different gravity gradient components in reweighted regularized inversion with a minimum support constraint

SUMMARY

1 INTRODUCTION

2 3-D GRAVITY GRADIENT FORWARD FORMULAS

3 3-D GRAVITY GRADIENT INVERSION PROCEDURE

3.1 Reweighted regularized inversion

3.2 CGGJI method

3.3 NGGJI method

4 NUMERICAL MODEL AND DISCUSSION

4.1 The characteristics of the weighting matrix

4.2 Comparison between different methods

4.2.1 Two components inversion

4.2.2 Multiple component inversions

4.2.3 Comparison of the degree of data fit

4.2.4 Discussion

4.3 Application to the model of the extended dike

5 APPLICATION TO REAL DATA

5.1 Vinton salt dome

5.2 Single-component inversion

5.3 Multiple component inversion

6 CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX A: THE LINEAR CONJUGATE GRADIENT METHOD

APPENDIX B: NON-LINEAR CONJUGATE GRADIENT METHOD

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Article Contents

A new method to integrate different gravity gradient components in reweighted regularized inversion with a minimum support constraint

SUMMARY

1 INTRODUCTION

2 3-D GRAVITY GRADIENT FORWARD FORMULAS

3 3-D GRAVITY GRADIENT INVERSION PROCEDURE

3.1 Reweighted regularized inversion

3.2 CGGJI method

3.3 NGGJI method

4 NUMERICAL MODEL AND DISCUSSION

4.1 The characteristics of the weighting matrix

4.2 Comparison between different methods

4.2.1 Two components inversion

4.2.2 Multiple component inversions

4.2.3 Comparison of the degree of data fit

4.2.4 Discussion

4.3 Application to the model of the extended dike

5 APPLICATION TO REAL DATA

5.1 Vinton salt dome

5.2 Single-component inversion

5.3 Multiple component inversion

6 CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX A: THE LINEAR CONJUGATE GRADIENT METHOD

APPENDIX B: NON-LINEAR CONJUGATE GRADIENT METHOD

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Email alerts

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