Yuan et al. Earth, Planets and Space (2020) 72:143 https://doi.org/10.1186/s40623-020-01282-3 FULL PAPER Open Access Application of the normalized largest eigenvalue of structure tensor in the interpretation of potential feld tensor data Yuan Yuan1, Xiangyu Zhang2* , Wenna Zhou3, Guochao Wu1 and Weidong Luo2 Abstract Obtaining horizontal edges and the buried depths of geological bodies, using potential feld tensor data directly is an outstanding question. The largest eigenvalue of the structure tensor is one of the commonly used edge detectors for delineating the horizontal edges without depth information of the potential feld tensor data. In this study, we pre- sented a normalized largest eigenvalue of structure tensor method based on the normalized downward continuation (NDC) to invert the source location parameters without any priori information. To improve the stability and accuracy of the NDC calculation, the Chebyshev–Pade´ approximation downward continuation method was introduced to obtain the potential feld data on diferent depth levels. The new approach was tested on various models data with and without noise, which validated that it can simultaneously obtain the horizontal edges and the buried depths of the geological bodies. The satisfactory results demonstrated that the normalized largest eigenvalue of structure ten- sor can describe the locations of geological sources and decrease the noise interference magnifed by the downward continuation. Finally, the method was applied to the gravity data over the Humble salt dome in USA, and the near- bottom magnetic data over the Southwest Indian Ridge. The results show a good correspondence to the results of previous work. Keywords: Potential feld tensor data, Normalized largest eigenvalue, Structure tensor, Downward continuation, Source parameter estimation Introduction the depth of the causative sources by the normalization Te source parameters estimation of potential feld ten- on the analytic signals modulus of the downward con- sor data is an important part for geological interpre- tinuation potential feld, without requiring any geomet- tation, which can provide the horizontal position and ric input parameters or assumptions regarding geological the buried depth of the geological bodies as an a priori properties. information for potential feld inversion (Li and Olden- In recent years, tremendous progresses have been burg 1998). Te normalized full gradient (NFG) method made in the implementation of the NFG technique (Don- proposed by the Russian Geophysical School (Elysseieva durur 2005; Aydin 2007; Oruç and Keskinsezer 2008; Fedi and Pašteka 2009, 2019) can be directly used to obtain and Florio 2011; Pamukcu and Akcig 2011; Aghajani et al. 2011). Besides the potential feld data, the NFG method has also been applied to the electromagnetic data (Don- *Correspondence: [email protected] durur 2005), self-potential data (Sindirgi et al. 2008) and 2 MLR Key Laboratory of Marine Mineral Resources, Guangzhou Marine Geological Survey, Guangzhou 510075, China seismic data (Karsli and Bayrak 2010). Fedi and Florio Full list of author information is available at the end of the article © The Author(s) 2020. 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Earth, Planets and Space (2020) 72:143 Page 2 of 13 2 (2011) applied diferent normalization factors on analyti- − 1 x2 + y G = 1 e 2 δx2 δy2 δx δy cal signals modulus and the downward continued poten- where δ 2πδ2 , and are the stand- tial feld itself, and called this generalized method as ard deviations of Gaussian envelope in x and y directions. normalized downward continuation (NDC). Based on the Te homogeneous characteristic equation for 2D tensor generalized NDC method, some authors have applied it M is on the local wavenumber (Ma et al. 2014), total horizon- 2 − (M + M ) + (M M − M M ) = 0. tal derivatives (Li et al. 2014), directional analytic signals 11 22 11 22 12 21 (Zhou 2015) and directional total horizontal derivatives (3) (Zhou et al. 2017) to obtain the horizontal edges and the Te largest eigenvalue of matrix M is buried depth simultaneously. 1 2 Te structure tensor is one of the image process- = M11 + M22 + (M11 − M22) + 4M12M21 . 2 ing techniques and presents a local orientation in an (4) n-dimensional space (Weickert 1999a, b). Te largest Sertcelik and Kafadar (2012) pointed that the largest eigenvalues of the structure tensor of the potential feld data have been used to delineate the edges of the geologi- eigenvalue can locate the edges of geological bodies. cal bodies (Sertcelik and Kafadar 2012; Oruç et al. 2013; However, the balancing ability of equalizing the edge sig- Yuan et al. 2014). In contrast to traditional derivative- nal amplitude of large and small anomalies is weak. Te δx δy based edge detection methods, the structure tensor has larger standard deviation and can enhance the bal- a property of which can reduce noise in the data while ancing ability, but reduce the resolution of the identifed enhancing discontinuity boundaries, due to a Gaussian edges. envelop within it. Based on the advantages of the largest NDC of the largest eigenvalue eigenvalue of the structure tensor, in this paper, we apply the NDC on the largest eigenvalue of the structure tensor Te normalized full gradient (NFG) method (Elysseieva of the downward continued feld to estimate the depth and Pasteka 2009; Zeng et al. 2002) is the normalization and the horizontal edges. In order to increase the stabil- of the analytic signal modulus at diferent downward con- ity of the NDC process, the Chebyshev–Pade´ approxi- tinuation levels. Fedi and Florio (2011) applied the nor- mation downward continuation method developed by malization on the analytic signal and on the downward Zhou et al. (2018) was brought to compute each down- continuation potential feld itself, called this generalized ward continuation level feld. Te synthetic models with method as normalized downward continuation (NDC). Te expression of NDC applied to the largest eigenvalue and without noise and the real measured potential feld data are utilized to validate the efectiveness of this new of the structure tensor matrix can be expressed as: method. x, y, z N = , (5) NDC of largest eigenvalue of structure tensor N(z) Structure tensor of potential feld tensor data where x, y, z is the edge detector of the structure ten- Potential feld gradient tensor data are the second-order sor at point (x, y, z); N(z) is the normalization function. f space derivatives of potential feld (gravitational feld Here, arithmetic mean, median and geometric mean are x y or magnetic feld) in the three orthogonal directions , used as the normalization function, shown as: and z . Te potential feld gradient tensor matrix is 1 M fxx fxy fxz N(z) = x, y, z Arithmetic mean M F = fyx fyy fyz . � (1) 0 � � , fzx fzy fzz N(z) = median( x, y, z ) Median = M ···� � Te original structure tensor consists of a Gaussian N(z) 1 2 M Geometric mean envelope and horizontal gradient tensor of the potential � (6) feld data (Sertcelik and Kafadar, 2012). Te expression of where M is the number of all calculation points. the original structure tensor matrix is 2 2 fzx fzxfzy Gσ ∗ fzx Gσ ∗ fzxfzy M11 M12 M = Gσ ∗ 2 = 2 = , (2) fzxTzy fzy Gσ ∗ fzxfzy Gσ ∗ fzy M21 M22 Yuan et al. Earth, Planets and Space (2020) 72:143 Page 3 of 13 During the processing, there are many issues that the low frequency and it can suppress the high frequency. needed to be overcome. Te downward continuation During the calculation of downward continuation, the plays the key role in the NDC method. Te accuracy of short wavelength (high frequency) components can be the NDC method is determined by the stability of down- suppressed by Chebyshev–Pade´ approximation. Tere- ward continuation method directly. Terefore, it is nec- fore, we can obtain a stability and noise reduced potential essary to use a stable downward continuation in the feld gradient data at diferent depths by using the Cheby- calculation process. Many new stable algorithms have shev–Pade´ downward continuation. Te residual noise been introduced to implement the downward continu- of the potential feld gradient data at diferent depths can ation method (Fedi and Florio, 2002; Cooper, 2004; Ma be further removed by the Gaussian envelop when using et al., 2013; Zeng et al., 2013, 2014; Zhang et al., 2013; the NDC of the largest eigenvalue of structure tensor to Zhou et al., 2018). Zhou et al. (2018) has compared the estimate the geology source. errors between the function exp(x) and its diferent Terefore, the calculation procedures of the NDC of approximation functions, including Taylor series, Che- the largest eigenvalue of structure tensor are: byshev approximation, Pade´ approximation, and Che- byshev–Pade´ approximation. Comparison of results 1. Calculate the potential feld gradient data from the indicate that downward continuation based on Cheby- observed potential feld data, or obtain from the real shev–Pade´ approximation can obtain a more precise measurement. result. Terefore, in this study, the Chebyshev–Pade´ 2.