Structure tensor of potential field tensor data
Potential field gradient tensor data are the secondorder space derivatives of potential field \(f\) (gravitational field or magnetic field) in the three orthogonal directions \(x\), \(y\) and \(z\). The potential field gradient tensor matrix is
$$ F = \left[ {\begin{array}{*{20}c} {f_{xx} } & {f_{xy} } & {f_{xz} } \\ {f_{yx} } & {f_{yy} } & {f_{yz} } \\ {f_{zx} } & {f_{zy} } & {f_{zz} } \\ \end{array} } \right]. $$
(1)
The original structure tensor consists of a Gaussian envelope and horizontal gradient tensor of the potential field data (Sertcelik and Kafadar, 2012). The expression of the original structure tensor matrix is
$$ {\text{M}} = G_{\sigma } * \left[ {\begin{array}{*{20}c} {f_{zx}^{2} } & {f_{zx} f_{zy} } \\ {f_{zx} T_{zy} } & {f_{zy}^{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {G_{\sigma } * f_{zx}^{2} } & {G_{\sigma } * f_{zx} f_{zy} } \\ {G_{\sigma } * f_{zx} f_{zy} } & {G_{\sigma } * f_{zy}^{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {M_{11} } & {M_{12} } \\ {M_{21} } & {M_{22} } \\ \end{array} } \right], $$
(2)
where \(G_{\delta } = \frac{1}{{2\pi \delta^{2} }}e^{{  \frac{1}{2}\left( {\frac{{x^{2} }}{{\delta x^{2} }} + \frac{{y^{2} }}{{\delta y^{2} }}} \right)}}\), \(\delta x\) and \(\delta y\) are the standard deviations of Gaussian envelope in \(x\) and \(y\) directions. The homogeneous characteristic equation for 2D tensor \({M}\) is
$$ \lambda^{2}  \lambda (M_{11} + M_{22} ) + (M_{11} M_{22}  M_{12} M_{21} ) = 0. $$
(3)
The largest eigenvalue of matrix \({M}\) is
$$ \lambda = \frac{1}{2}\left( {M_{11} + M_{22} + \sqrt {\left( {M_{11}  M_{22} } \right)^{2} + 4M_{12} M_{21} } } \right). $$
(4)
Sertcelik and Kafadar (2012) pointed that the largest eigenvalue \(\lambda\) can locate the edges of geological bodies. However, the balancing ability of equalizing the edge signal amplitude of large and small anomalies is weak. The larger standard deviation \(\delta x\) and \(\delta y\) can enhance the balancing ability, but reduce the resolution of the identified edges.
NDC of the largest eigenvalue
The normalized full gradient (NFG) method (Elysseieva and Pasteka 2009; Zeng et al. 2002) is the normalization of the analytic signal modulus at different downward continuation levels. Fedi and Florio (2011) applied the normalization on the analytic signal and on the downward continuation potential field itself, called this generalized method as normalized downward continuation (NDC). The expression of NDC applied to the largest eigenvalue \(\lambda\) of the structure tensor matrix can be expressed as:
$$ N\lambda = \frac{{\lambda \left( {x,y,z} \right)}}{N\left( z \right)}, $$
(5)
where \(\lambda \left( {x,y,z} \right)\) is the edge detector of the structure tensor at point (x, y, z); \(N\left( z \right)\) is the normalization function. Here, arithmetic mean, median and geometric mean are used as the normalization function, shown as:
$$ \left\{ \begin{gathered} N\left( z \right) = \frac{1}{M}\sum\limits_{0}^{M} {\lambda \left( {x,y,z} \right)\begin{array}{*{20}c} {} & {} & {\begin{array}{*{20}c} {{\text{Arithmetic}}} & {{\text{mean}}} \\ \end{array} } \\ \end{array} } \hfill \\ N\left( z \right) = median(\lambda \left( {x,y,z} \right))\begin{array}{*{20}c} {} & {\begin{array}{*{20}c} {{\text{Median}}} & {} \\ \end{array} } \\ \end{array} \hfill \\ N\left( z \right) = \sqrt[M]{{\lambda_{1} \lambda_{2} \cdots \lambda_{M} }}\begin{array}{*{20}c} {} & {} & {{\text{Geometric}}} & {{\text{mean}}} \\ \end{array} \hfill \\ \end{gathered} \right., $$
(6)
where \(M\) is the number of all calculation points.
During the processing, there are many issues that needed to be overcome. The downward continuation plays the key role in the NDC method. The accuracy of the NDC method is determined by the stability of downward continuation method directly. Therefore, it is necessary to use a stable downward continuation in the calculation process. Many new stable algorithms have been introduced to implement the downward continuation method (Fedi and Florio, 2002; Cooper, 2004; Ma et al., 2013; Zeng et al., 2013, 2014; Zhang et al., 2013; Zhou et al., 2018). Zhou et al. (2018) has compared the errors between the function \(\exp (x)\) and its different approximation functions, including Taylor series, Chebyshev approximation, Pade´ approximation, and Chebyshev–Pade´ approximation. Comparison of results indicate that downward continuation based on Chebyshev–Pade´ approximation can obtain a more precise result. Therefore, in this study, the Chebyshev–Pade´ approximation downward continuation method was chosen for the NDC of the largest eigenvalue of the structure tensor.
In the frequency domain, the downward continuation can be denoted as:
$$ \tilde{f}_{h} \left( {\omega_{x} ,\omega_{y} ,\Delta h} \right) = e^{{\Delta h\omega_{\gamma } }} \tilde{f}\left( {\omega_{x} ,\omega_{y} } \right), $$
(7)
where the downward continuation operator \(e^{{\Delta h\omega_{\gamma } }}\) can be approximated by the (three and two) terms of Chebyshev–Pade´ as follows (Zhou et al. 2018):
$$ e^{{\Delta h\omega_{\gamma } }} = \frac{{0.9196 + 0.5667\Delta h\omega_{\gamma } + 0.1467\left( {\Delta h\omega_{\gamma } } \right)^{2} + 0.01627\left( {\Delta h\omega_{\gamma } } \right)^{3} }}{{0.9194{  }0.3528\left( {\Delta h\omega_{\gamma } } \right){ + }0.0403\left( {\Delta h\omega_{\gamma } } \right)^{2} }}. $$
(8)
Therefore, the expression of the (three and two) terms of Chebyshev–Pade´ approximation downward continuation is
$$ \tilde{f}_{h} \left( {\omega_{x} ,\omega_{y} ,\Delta h} \right) = \frac{{0.9196 + 0.5667\Delta h\omega_{\gamma } + 0.1467\left( {\Delta h\omega_{\gamma } } \right)^{2} + 0.01627\left( {\Delta h\omega_{\gamma } } \right)^{3} }}{{0.9194{  }0.3528\left( {\Delta h\omega_{\gamma } } \right){ + }0.0403\left( {\Delta h\omega_{\gamma } } \right)^{2} }}\tilde{f}\left( {\omega_{x} ,\omega_{y} } \right), $$
(9)
where \(f_{h}\) and \(f\) are the gravity and magnetic data at two observation heights separated by a vertical distance \(\Delta h\), respectively. \(\tilde{f}_{h}\) and \(\tilde{f}\) denotes the Fourier transform of \(f_{h}\) and \(f\), \(\omega_{x}\) and \(\omega_{y}\) are the wavenumbers x and y direction, and \(\omega_{\gamma } = \sqrt {\omega_{x}^{2} + \omega_{y}^{2} }\) is the radial wavenumber. \(\Delta h\) is a positive number for the downward continuation distance and vice versa for the upward continuation. Zhou et al. (2018) have pointed out that Chebyshev–Pade´ approximation can give a more precise approximation in the low frequency and it can suppress the high frequency. During the calculation of downward continuation, the short wavelength (high frequency) components can be suppressed by Chebyshev–Pade´ approximation. Therefore, we can obtain a stability and noise reduced potential field gradient data at different depths by using the Chebyshev–Pade´ downward continuation. The residual noise of the potential field gradient data at different depths can be further removed by the Gaussian envelop when using the NDC of the largest eigenvalue of structure tensor to estimate the geology source.
Therefore, the calculation procedures of the NDC of the largest eigenvalue of structure tensor are:

1.
Calculate the potential field gradient data from the observed potential field data, or obtain from the real measurement.

2.
Set a maximum depth H (the depth that we want to know the maximum range of geologic source) for downward continuation, and divide the underground into n layers (i = 1, 2, 3, …, n).

3.
alculate potential field gradient data at the different depth levels by using the Chebyshev–Pade´ approximation downward continuation method.

4.
Calculate the NDC of the largest eigenvalue of structure tensor of potential field data according to Eqs. 2, 3, 4, 5.