0 Introduction

Edge detection is necessary in the interpretation of potentialfield data, which has been widely used in exploration. The main geological edges are fault lines and the boundaries of geological bodies that are with different density or magnetic nature etc. Many edge detectors have been proposed to detect and enhance the edges. Most of them are based on the horizontal and vertical derivatives of potential field data, such as total horizontal derivatives and analytic signal amplitude (Cordell, 1979; Cordell & Grauch, 1985; Roestet al., 1992). However, these two detectors cannot display the edges of deep anomaliesclearly, which are blurred by the shallow anomalies. In order to equalize the edge amplitude of shallow and deep anomalies, some balanced filters have been proposed (Miller & Singh, 1994; Verduzcoet al., 2004; Wijnset al., 2005; Cooper & Cowan, 2006; Ma & Li, 2012). However, these methods are sensitive to noise and likely to bring additional false edges, if thereal geological bodies contain both positive and negative anomalies.

Structure tensor is one of the image processing techniques thatpresent a local orientation in n-dimensional space (Weickert, 1999a, 1999b). Sertcelik and Kafadar(2012) used the large eigenvalue of 2D structure tensor to extractthe edges and corners of causative bodies.Yuan et al. (2014) made an improvement on it usingthree different normalization methods. Wang and Ma (2003) applied the 3D structure tensor to the identification of local motion information of video object. Jeonget al. (2006) used the 3D structure tensor to identify faults on the seismic images.Zhou et al. (2016) used the directional total horizontal derivatives of 3D structure tensor to delineate the edges of geological bodies, and presented a normalization edge detector with a nonnegative constant in the denominator of the expression. The large constant value can effectively remove the additional false edges, but reduce the ability of balancing deep anomalies. On the contrary, the small constant value can enhance the ability of balancing deep anomaly but reduce the ability of removing the false edges and noise influence. Therefore, this method has some disadvantages, e.g. it is sensitive to the noise, and the selection of constant value is subjective.

常规护理往往只注重患者的个人状况以及生命体征监测，容易忽略患者社会、心理、生理等方面的正常需求，在日常护理工作中有一定的滞后性。相比于常规护理，对患者进行优质护理可以更突显出综合性、全面性和科学性[7] 。护理人员从术前术后各个方面对患者进行护理指导，可以充分满足患者不同方面的需求，充分保证患者多方面对护理的需求。

In order to reduce the noise influence, completely balance the small anomalies, and remove the false edges objectively, the authors redefine the 3D structure tensor with a Gaussian envelop, and propose a new normalized edge detector. We applied the new method to the synthetic model and measured gravity data in Sichuan Basin to testthe performance of the new edge detector.

1 Original 3D structure tensor method

Zhou et al. (2016) have given the 3D structure tensor of potential field data, and developed a new edge detector based on it. The expression of 3D structure tensor is

(1)

They defined the directional total horizontal derivatives of 3D structure tensor in x and y directions as

(2)

(3)

Based on the definitionof THDT, we define a newedge detector THDTσ by matrix Tσ. The expression of THDTσ is

(4)

In order to balance the large and small amplitude anomalies simultaneously, a normalized THDT detector is defined as

(5)

However, this method has some disadvantages. First, it cannot identify the edges accurately. The identified edges are located outside the true edges. Second, the large value of k can decrease the noise influence and remove false edges, but reduce the balance ability, which make the balance incomplete. On the contrary, the small value of k can enhance the balance ability, but increase the noise influence and reduce the ability to remove false edges. Therefore, the selection of k is very important.

2 An improved 3D structure tensor

For the disadvantages of original 3D structure tensor, we redefine the 3D structure tensor method with a Gaussian envelop convolution operator. The expression is

明代新昌地图右上方的“水自金庭来”，既说明古金庭位于新昌境内的东北角，又说明古金庭是一河流的发源地，从图中河流看，为古剡黄泽江上游的一个源头。这刚好与王罕岭位置合。

(6)

σx and σy are the standard deviations of Gaussian envelope in x and y directions, and σ =

Where,

Where, the maximum values of THDx and THDy indicate the N-S and E-W edges, respectively. Combining THDx and THDy, they defined a new edge detector

The maximum value of THDTσ locate the edges of geological bodies. But it cannot balance the small amplitude anomalies. Therefore, we proposed an objective normalization method, defined as

“雪龙号”能以1.5节航速连续冲破1.2米厚的冰层。推冰碾雪时，“雪龙号”那火红的船体在白皑皑的冰雪世界里分外耀眼。行进中，不时有大块大块的冰从冰川上滑落下来，跌入幽蓝的海水中，溅起大大小小的浪花。不一会儿，落下来的冰块又纷纷从海面探出“头”来，有的成了顺水漂流的浮冰，有的成了或大或小的冰山。难道是“雪龙号”的动静太大，惊扰了冰川吗？

(7)

当用户通过在线社交应用发布消息时，应用会调用标注模块。消息的内容经过标注模块的处理，标注过的内容将用于评估隐私敏感度。

The surface potential ψsλin GSGCDMT-SON MOSFET can be calculated by solving a three dimensional Poisson’s equation [21] given by

动态心电图是对心律失常予以检查的常用仪器，近年来，置入永久起搏器的患者人数逐渐增加，动态心电图已经成为置入起搏器患者的重要随访检查项目之一，以便明确患者的起搏器功能情况[1]。本文样本资料是本医院予以诊断和治疗的18例置入起搏器患者，研究动态心电图检测用在起搏器间歇性感知功能字符异常的临床诊断价值。现报告如下。

In order to test the application ability, we construct a more complex model which contains both posi-tive and negative anomalies. The synthetic gravity anomaly is shown in Fig.3a. The edge results detected by the methods proposed above are shown in Fig.3b-h. It shows that THDTcan outline the edges properly, without any additional false edge information. But, the edge of deep anomalyis not clear (Fig.3b). When k=0, traditional normalized method NTcan effectively balance the deep anomaly, but introduce some additional false edges between the positive and negative anomalies, (Fig.3c). Increasing the value of k can effectivelyremove the false edges, but reduce the balance ability (Fig.3d-e). Also, methods TDX and Theta map bring out some false edges (Fig.3f-g). Besides, the identified edges are located outside the real edges. However, thenew improved method TDR_THDTσcan balance the edges of large and small anomalies clearly and precisely, without creating any false edges(Fig.3h).

(8)

The maximum value of TDR_THDTσ locate the edges of geological bodies.

In order to validate the feasibility of TDR_THDTσ, the results with two well-known methods TDX (Cooper & Cowan, 2006) and Theta map (Wijns et al., 2005) are compared.

3 Synthetic model experiments

In this section, we construct a gravity model which contains two identical prisms at top depth of 10m and 20m respectively. Both prisms are 40 m thick, with residual density of 0.2 g/cm3. Fig.1a displays the synthetic gravity anomaly. Fig.1b-f show the edges identified by edge detectors THDT, NT(with k=0), TDX, Theta map and TDR_THDTσ, where the black lines indicate the true horizontal position of the two prisms. We can see that THDT cannot display the edges of deep anomaliesclearly. Although NT, TDX and Theta map can well balance the edges of shallow and deep anomalies, the identified edges are located outside the real edges. Compared with the edge results detected by the traditional methods, our improved method TDR_THDTσ can not only balance different amplitude edges, but also delineate the edges accurately (Fig.1f). Fig.2 shows the edge results of NT with different values of k. It shows that with the increasing of k, the balance ability has been reduced, indicating that traditional normalization method NT has some disadvantages.

TDR_THDTσ=

In order to test the stability of TDR_THDTσ, we add Gaussian noise with standard deviation equal to 10% of maximum amplitude of the synthetic gravity anomaly of model 2. Fig.4a shows the noisy gravity anomaly. Fig.4b-h displays the edge results identified by edge detectors THDT, NT (with different values of k), TDX, Theta map and TDR_THDTσ. It is indicated that with the increasing of k of NT, the ability of removing noise influence and balancing small anomaly will decrease (Fig.4c-e). Fig.4f-g shows the edge results of traditional methods TDX and Theta map, which are contaminated by the noise. However, our new method TDR_THDTσ can effectively reduce the noise influence and outline the edges clearly and precisely. This is because Gaussian envelopwas introduced in the improved 3D structure tensor, which can filter the noise influence.

4 Application to real case

The method is appliedto the processing of real gravity data from Sichuan Basin, Southwest China.

(a) Synthetic gravity anomaly; (b) THDT edge result; (c) NT edge result; (d) TDX edge result; (e) Theta map edge result; (f) TDR_THDTσ edge result. Fig.1 Edge results of synthetic gravity model 1

(a)k=0; (b)k=0.1; (c)k=0.5; (d)k=1 Fig.2 Edge result of NT with different k values

(a) Synthetic gravity anomaly; (b) THDT edge result; (c) NT edge result, withk=0; (d) NT edge result, with k=0.01; (e) NT edge result, with k=0.05; (f) TDX edge result; (g) Theta edge result; (h) TDR_THDTσ edge result. Fig.3 Edge results of synthetic gravity model 2

(a) Synthetic gravity anomaly; (b) THDT edge result; (c) NT edge result, with k=0; (d) NT edge result, with k=0.01; (e) NT edge result, with k=0.05; (f) TDX edge result; g/Theta edge result; (h) TDR_THDTσ edge result. Fig.4 Edge results of synthetic gravity model 2 with Gaussian noise

(a) gravity anomaly of Sichuan Basin; (b) THDT edge result; (c) NTedge result, with k=0; (d) NT edge result, with k=0.01; (e) NTedge result, with k=0.05; (f) TDX edge result; (g) Theta edge result; (h) TDR_THDTσ edge result. Fig.5 Edge results of synthetic gravity model 2

Fig.5 shows the geology map of the Sichuan Basin (Li et al., 2013).

Fig.5a shows the gravity data for Sichuan Basin, collected from the Bouguer gravity anomaly, on a scale of 1∶1 000 000, provided by the National Administration of Surveying, Mapping, and Geo-information, China. We interpolate the data with a grid interval of 500 m. Fig.5b-h shows the edge results of different edge detectors. We can see that original normalized 3D structure tensor cannot delineate theanomaly completely. When k=0, NT has the same edge results with TDX and Theta map. THDT and NT (k=0.01, k=0.05) cannot identify the boundaries of the deep and shallow anomaly at the same time, while the NT (k=0), TDX and Theta can be balanced deep shallow exception, but it doesn’t get more details of the border,and the boundaries that they get are diverging.Compared with these results, the new proposed method TDR_THDTσ can highlight the edges more clearly than other methods and reveal more details.

5 Conclusion

This study proved that the original 3D gravity structure tensor method has certain disadvantages, e.g. it is sensitive to noise, the selection of the value of k is subjective, and the large value of k will reduce the balance ability. Therefore, we have made an improvement to the original 3D structure tensor method to detect the edges of potential anomaly clearly. This method has been tested on synthetic model data and real measured data. In summary, our improved method can effectively avoid these disadvantages, which can completely balance the edges of large and small anomalies, and avoid bringing false edges when the model and real geological anomalies contain both positive and negative anomalies, while the traditional methods NT, TDX and Theta map may produce false edges. Moreover, with the introducing of Gaussian envelop, the new method can effectively filter the noise influence.

References

Cooper G R J, Cowan D R. 2006. Enhancing potential field data using filters based on the local phase. Computers & Geosciences, 32:1585-1591.

Cordell L. 1979. Gravimetric expression of graben faulting in Santa Fe Country and the Espanola Basin, New Mexico// New Mexico Geol. Soc. Guidebook.30th Field Conf., 59-64.

Cordell L, Grauch V J S. 1985. Mapping basement magnetization zones from aeromagnetic data in the San Juan Basin,New Mexico//Hinzc W J. (ed.) The Utility of regional gravity and magnetic anomaly. [s. l.]: Society of Exploration Geophysics, 181-197.

Jeong W K, Whitaker R, Dobin M. 2006. Interactive 3D seismic fault detection on the Graphics Hardware//Proceedings of the International Workshop on Volume Graphics, 111-118.

Li L L, Ma G Q,Du X J. 2013. Edge detection in potential field data by enhanced mathematical morphology filter. Pure and Applied Geophysics, 170(4):645-653.

Ma G Q, Li L L. 2012.Edge detection in potential fields with the normalized total horizontal derivatives.Computer & Geosciences, 41: 83-87.

Miller H G, Singh V. 1994. Potential field tilt: a new concept for location of potential field sources. Journal of Applied Geophysics, 32: 213-217.

Roest W R, Verhoef J. Pilkington M. 1992. Magnetic interpretation using the 3-D analytic signal.Geophysics, 57: 116-125.

Sertcelik I, Kafadar O. 2012. Application of edge detection to potential field data using eigenvalue analysis of structure tensor.Journal of Applied Geophysics, 84:127-136.

Verduzco B, Fairhead J D. 2004. New insights into magnetic derivatives for structural mapping.The Leading Edge, 23: 116-119.

Wang H Y, Ma K K. 2003. Automatic video object segmentation via 3D structure tensor//International Conference on Image Processing, 1: 153-156.

Weickert J. 1999a. Coherence-enhancing diffusion of color images. Image and Vision Computing, 17(3/4): 201-212.

Weickert J. 1999b. Coherence-enhancing diffusion filtering.International Journal of Computer Vision, 31(2/3):111-127.

Wijns C, Perez C, Kowalczyk P. 2005. Theta map: edge detection in magnetic data. Geophysics, 70(4): L39-L43.

Yuan Y, Huang D N, Yu Q L, et al. 2014. Edge detection of potential field data with improved structure tensor methods.Journal of Applied Geophysics, 108(9): 35-42.

Zhou S, Huang D N, Jiao J. 2016. Total horizontal derivatives of potential field three-dimensional structure tensor and their application to detect source edges.ActaGeodaetica et Geophysica, 52(3) :1-13.