New method for edges detection of magnetic sources using logistic function

The tilt angle of the analytic signal amplitude (TA) is defined as the arctangent of the ratio of the first vertical derivative to the total horizontal derivative of the analytic signal amplitude. It is commonly used as a useful tool to estimate edges of magnetic sources because its value is slightl...

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Дата:	2018
Автори:	Pham, L.T., Oksum, E., Do, T.D., Huy, M.L.
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Мова:	English
Опубліковано:	Інститут геофізики ім. С.I. Субботіна НАН України 2018
Назва видання:	Геофизический журнал
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Цитувати:	New method for edges detection of magnetic sources using logistic function / L.T. Pham, E. Oksum, T.D. Do, M.L. Huy // Геофизический журнал. — 2018. — Т. 40, № 6. — С. 127-135. — Бібліогр.: 11 назв. — англ.

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spelling	irk-123456789-1456532019-01-26T01:23:13Z New method for edges detection of magnetic sources using logistic function Pham, L.T. Oksum, E. Do, T.D. Huy, M.L. The tilt angle of the analytic signal amplitude (TA) is defined as the arctangent of the ratio of the first vertical derivative to the total horizontal derivative of the analytic signal amplitude. It is commonly used as a useful tool to estimate edges of magnetic sources because its value is slightly dependence on the direction of magnetization vector, and it is more effective in estimating the edges of the bodies than the analytic signal amplitude and the standard tilt angle. Based on logistic function (L) that has the same shape with the shape of arctangent function, and the derivatives of the analytic signal amplitude, we introduce some new filters which also can reduce the effect of the magnetization direction. Угол наклона амплитуды аналитического сигнала (TA) определяют как арктангенс отношения первой производной вертикального градиента к суммарной горизонтальной производной амплитуды аналитического сигнала. Определение этого угла обычно используют как полезный метод для оценки граней магнитных источников, поскольку его величина незначительно зависит от направления намагниченности. По аналитической функцией (L), что имеет одинаковую форму с формой функции арктангенс, введены некоторые новые фильтры, которые также могут уменьшить эффект направления намагниченности. Кут нахилу амплітуди аналітичного сигналу (TA) визначають як арктангенс відношення першої похідної вертикального градієнта до сумарної горизонтальної похідної амплітуди аналітичного сигналу. Визначення цього кута зазвичай використовують як корисний метод для оцінювання граней магнітних джерел, оскільки його величина незначно залежить від напрямку намагніченості. За аналітичною функцією (L), що має однакову форму з формою функції арктангенсу, введено деякі нові фільтри, які також можуть зменшити ефект напрямку намагніченості. 2018 Article New method for edges detection of magnetic sources using logistic function / L.T. Pham, E. Oksum, T.D. Do, M.L. Huy // Геофизический журнал. — 2018. — Т. 40, № 6. — С. 127-135. — Бібліогр.: 11 назв. — англ. 0203-3100 DOI: https://doi.org/10.24028/gzh.0203-3100.v40i6.2018.151033 http://dspace.nbuv.gov.ua/handle/123456789/145653 en Геофизический журнал Інститут геофізики ім. С.I. Субботіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The tilt angle of the analytic signal amplitude (TA) is defined as the arctangent of the ratio of the first vertical derivative to the total horizontal derivative of the analytic signal amplitude. It is commonly used as a useful tool to estimate edges of magnetic sources because its value is slightly dependence on the direction of magnetization vector, and it is more effective in estimating the edges of the bodies than the analytic signal amplitude and the standard tilt angle. Based on logistic function (L) that has the same shape with the shape of arctangent function, and the derivatives of the analytic signal amplitude, we introduce some new filters which also can reduce the effect of the magnetization direction.
format	Article
author	Pham, L.T. Oksum, E. Do, T.D. Huy, M.L.
spellingShingle	Pham, L.T. Oksum, E. Do, T.D. Huy, M.L. New method for edges detection of magnetic sources using logistic function Геофизический журнал
author_facet	Pham, L.T. Oksum, E. Do, T.D. Huy, M.L.
author_sort	Pham, L.T.
title	New method for edges detection of magnetic sources using logistic function
title_short	New method for edges detection of magnetic sources using logistic function
title_full	New method for edges detection of magnetic sources using logistic function
title_fullStr	New method for edges detection of magnetic sources using logistic function
title_full_unstemmed	New method for edges detection of magnetic sources using logistic function
title_sort	new method for edges detection of magnetic sources using logistic function
publisher	Інститут геофізики ім. С.I. Субботіна НАН України
publishDate	2018
url	http://dspace.nbuv.gov.ua/handle/123456789/145653
citation_txt	New method for edges detection of magnetic sources using logistic function / L.T. Pham, E. Oksum, T.D. Do, M.L. Huy // Геофизический журнал. — 2018. — Т. 40, № 6. — С. 127-135. — Бібліогр.: 11 назв. — англ.
series	Геофизический журнал
work_keys_str_mv	AT phamlt newmethodforedgesdetectionofmagneticsourcesusinglogisticfunction AT oksume newmethodforedgesdetectionofmagneticsourcesusinglogisticfunction AT dotd newmethodforedgesdetectionofmagneticsourcesusinglogisticfunction AT huyml newmethodforedgesdetectionofmagneticsourcesusinglogisticfunction
first_indexed	2025-07-10T22:10:06Z
last_indexed	2025-07-10T22:10:06Z
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fulltext	NEW METHOD FOR EDGES DETECTION OF MAGNETIC SOURCES USING LOGISTIC FUNCTION Геофизический журнал № 6, Т. 40, 2018 127 Introduction. Edge detection is an im por- tant field in magnetic interpretation. There are many methods for detecting edges, most of which are based on the vertical or ho ri- zontal derivatives of the field. The most com- monly used filter is the total horizontal de- rivative of the potential field [Cordell, 1979; Cordell, Grauch, 1985]. The greatest advan- tage of this filter is its low sensitivity to the noise in the data because it only requires the first order horizontal derivatives of the field. However, the filter requires a reduction to DOI: 10.24028/gzh.0203-3100.v40i6.2018.151033 New method for edges detection of magnetic sources using logistic function L. T. Pham1, E. Oksum2, T. D. Do1, M. L. Huy3, 2018 1VNU University of Science, Faculty of Physics, Department of Geophysics, Hanoi, Vietnam 2Süleyman Demirel University, Engineering Faculty, Department of Geophyisical Engineering, Isparta, Turkey 3Institute of Geophysics, Vietnam Academy of Science and Technology, Hanoi, Vietnam Received 31 July 2018 Кут нахилу амплітуди аналітичного сигналу (TA) визначають як арктангенс відно- шення першої похідної вертикального градієнта до сумарної горизонтальної похідної амплітуди аналітичного сигналу. Визначення цього кута зазвичай використовують як корисний метод для оцінювання граней магнітних джерел, оскільки його величина незначно залежить від напрямку намагніченості. За аналітичною функцією (L), що має однакову форму з формою функції арктангенсу, введено деякі нові фільтри, які також можуть зменшити ефект напрямку намагніченості. Крім того, ці фільтри ство- рюють амплітудні максимуми над межами джерел і компенсують вплив аномалій від незначних і глибоких джерел. Можливість використання фільтрів продемонстровано на чистій і спотвореній шумами синтетичних 3D магнітних моделях, коли отримані положення граней добре збігаються з реальними межами. Ефективність фільтрів оцінено також порівнянням їх з даними інших методів виявлення положення граней. Показано, що нові фільтри менш чутливі до варіацій глибини розташування джере- ла тіл і що використання модифікованої логістичної функції (Lk) може забезпечити кращі результати, ніж амплітуда аналітичного сигналу (AS), амплітуда аналітичного сигналу кута нахилу (AT), TA- і L-фільтри. Фільтри також застосовують до магнітних даних ділянки на півдні центрального В’єтнаму. Запропоновані фільтри є корисним інструментом для якісної інтерпретації магнітних спостережень. Ключові слова: логістична функція, кут нахилу, амплітуда аналітичного сигналу, визначення граней, інтерпретація магнітних даних. the pole or pseudo-gravity transformation [Cordell, Grauch, 1985] that has serious li- mi tations when used in low latitude are as. Furthermore, the edge detection result is dominated by the response from the shal- lower sources that produce strong ano ma- lies, and hence it cannot display the strong and weak amplitude anomaly edges simul- taneously [Cooper, Cowan, 2008]. In order to make both the shallow and deep sources visible simultaneously, [Miller, Singh, 1994] proposed using tilt angle that based on the L. T. PHAM, E. OKSUM, T. D. DO, M. L. HUY 128 Геофизический журнал № 6, Т. 40, 2018 ratio of vertical derivative to total horizon- tal derivative, [Cooper, Cowan, 2006] used horizontal tilt angle, and [Wijns et al., 2005] used the theta map method. However, all three methods are sensitive to dip and mag- netization effects [Pilkington, Tschirhart, 2017]. Another method which is also based on tilt angle is introduced by [Verduzco et al., 2004], called total horizontal derivative of tilt angle. Although the method is not influenced by dip or magnetization effects, it generates some false edges [Pilkington, Tschirhart, 2017]. Another commonly used method, known as the analytic signal ampli- tude method, is introduced by [Nabighian, 1972] and [Roest et al., 1992]. They showed that, in the 2D case, the shape of the analytic signal amplitude is independent of the direc- tion of the ambient magnetic field and the direction of source magnetization. However, [Li, 2006] pointed that this independent is lost in the 3D case. An enhanced analytic sig- nal is introduced by [Hsu et al., 1996]. They used the higher order derivatives to detect the edges of magnetic sources. Although the method can effectively reduce the interfer- ence due to adjacent geological bodies, the effect of noise increases. Another disadvan- tage of the method is that it cannot display large and small amplitude edges simultane- ously. [Ansari, Alamdar, 2011] suggested the use of the analytic signal amplitude of the tilt angle as a balanced edge detection filter. It is more effective than the analytic signal, but it still performs poorly in detecting all the edges of the body [Cooper, 2014]. G. Cooper proposed modified analytic signal amplitude that based on tilt angle method to balance the different amplitude edges. The advantage of the method is reducing the dependence of the analytic signal amplitude of magnetic anomaly on the direction of magnetization. In this paper, we describe a new edge de- tection filter based on the logistic function. We also improve this by introducing a modi- fied logistic function resulting as an enhance- ment in delineating the geologic contacts. Theory. The analytic signal of magnetic anomaly M is defined in 3D by [Roest et al., 1992] as € € €( , ) M M MAS x y x y i z x y z ∂ ∂ ∂ = + + ∂ ∂ ∂ , (1) where 1i = − and €x , €y and €z are unit vec- tors in x, y and z directions, respectively. From Eq. 1 it follows that the amplitude of the ana- lytic signal is given by: 22 2 ( , ) M M MAS x y x y z ⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ . (2) The ratio of the first vertical derivative and total horizontal derivatives of the analytic sig- nal amplitude is: 22 AS zR AS AS x y ∂ ∂= ⎛ ⎞∂ ∂⎛ ⎞ +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ . (3) G. Cooper used arctangent of R to delin- eate the edges of the magnetic sources, called it the tilt angle of analytic signal amplitude (TA) [Cooper, 2014]. He showed that the TA is theoretically independent of the source mag- netization vector direction for the 2D case, and it can reduce the effect of the magnetiza- tion direction for the 3D case. The TA filter is sensitive to the noise in the data because it uses second order derivatives, and especially the ones that use vertical derivative. To reduce the noise effect, [Cooper, 2014] proposed us- ing the zero order analytical signal amplitude. Although this filter has a much reduced noise sensitivity, the edges of the magnetic sources are not as sharply defined. Here, we introduce a new filter based on logistic function and the ratio of the first vertical derivative and total horizontal derivatives of the AS, which is de- fined as 1 1 RL e−= + . (4) The idea of producing this filter is that logistic function is a mathematical func- tion having a characteristic «S»-shaped curve (or sigmoid curve) that is the same shape with the shape of arctangent function. NEW METHOD FOR EDGES DETECTION OF MAGNETIC SOURCES USING LOGISTIC FUNCTION Геофизический журнал № 6, Т. 40, 2018 129 Therefore, it also can reduce the effect of the magnetization direction, like the TA filter. Similarly to the tilt angle of analytic signal amplitude, using logistic function leads to a more balanced response. Both methods are effective in enhancing the edges that produce only low-amplitude analytic signal maxima. Nonetheless, the edges of the shal- low source are clear and refined, whereas the edges of deep source are clear but diffuse. In this study, we resolved this issue of the logis- tic filter by using a modified logistic func- tion, which is defined as 1 k RL k e−= + , (5) Where k is a positive constant less than one. In order to demonstrate the feasibility of L and Lk filters, we choose three other fre- quently used filters to compare the boundary detection results. They are the AS, the AT and the TA. Synthetic Example. The efficiency of the new methods to enhance the detection of edges of the magnetic source bodies is stud- ied by analysis of two synthetic examples. The first example involves a single prism with pa- rameters shown in Table 1. Fig. 1, a shows the magnetic anomaly due to the single prism model whose outlines are shown in black. The results of AS, AT, TA, L, and Lk of the field are shown in Fig. 1, b―f respectively. It can be observed that the AS is only effective in enhancing two of the four edges of the causative body. The AT is more effective than the AS in enhancing all the edges of the causative body. However, the edges detected by the AT are not precise, and it is clearly noisier than the other filters. Both the L and the TA yield similar results, they enhance all the edges, yet their obtained results are diffused to some extent. By comparison among the results in Fig. 1, we can see that Lk can not only de- lineate the edges of the source body clearly and precisely, but also give better resolution of the edges than other filters. The second example involves three prisms models with the same dimensions in size but in increasing depths. Their parameters are shown in Table 2. The outlines in plain view of the prismatic sources are shown by the black lines in all figures. In order to test the stability of the L and Lk filters, we added random noise with amplitude equal to 5 % of the data amplitude to the magnetic anomaly data due to the prisms (Fig. 2). Because the L filter uses second order derivatives which increase the noise influence. Therefore, we need to reduce the noise effect first before we use the filter to detect the edges. Using upward continuation of the magnetic data, the noise effect can be reduced. Fig. 2, b―f display the results of the AS, AT, TA, L and Lk after upward continuation of 1 km, re- spectively. In this case, as can be seen from Fig. 2, b, c, the AS is also only effective in delineating two of the four edges of each causative body, whereas the AT represents a poor view of the edges. Fig. 2, d shows the results calculated by the TA filter and Fig. 2, e is the results from the application of the L filter. It can be observed that the TA and the L filters can enhance all the edges of caus- ative bodies. The edges are clearly enhanced more sharply, compared with AS and AT fil- ters. Fig. 2, f displays the edges detected by the Lk filter. It can be clearly observed that the Lk filter can not only balance anomalies from shallow and deep sources, but also give a higher resolution, and can delineate the edges more clearly and precisely. The ampli- tude of the response from the two causative bodies is similar, although the response from the deeper causative body is rather diffuse. Real Data Example. In order to demon- strate the practical applicability of the sug- gested new filters, we consider their applica- Ta b l e 1. Parameters of the single prism model Center coordinates 31.5 km; 31.5 km Length×Width 30×30 km Inclination 15° Depth of top 2 km Declination 25° Depth of bottom 3.5 km Magnetization 5 A/m Rotation angle 0° L. T. PHAM, E. OKSUM, T. D. DO, M. L. HUY 130 Геофизический журнал № 6, Т. 40, 2018 Fig. 2. Test results of the three prism models P-1-3: a ― synthetic magnetic anomaly of three prisms, b ― AS, c ― AT, d ― TA, e ― L, f ― Lk with k=0.01. Fig. 1. Test results of the single prism model (see Table 1): a ― synthetic magnetic anomaly of the single prism model, b ― AS, c ― AT, d ― TA, e ― L, f ― Lk, with k=0.01. NEW METHOD FOR EDGES DETECTION OF MAGNETIC SOURCES USING LOGISTIC FUNCTION Геофизический журнал № 6, Т. 40, 2018 131 Fig. 3. Location of the study area within (a), the total field magnetic anomaly of the study area (b). L. T. PHAM, E. OKSUM, T. D. DO, M. L. HUY 132 Геофизический журнал № 6, Т. 40, 2018 Fig. 4. The total field magnetic anomaly at upward continuation level of 1 km (a), b ― AS, c ― AT, d ― TA, e ― L, f ― Lk, with k=0.01. tion to field magnetic data from an area in south-central Vietnam. The study area lies between longitude 107.8°E and 109.3°E and latitude 12°N and 13.5°N, covering an area of approximately 27000 km2 (Fig. 3, a). The total field magnetic anomaly data (Fig. 3, b) are NEW METHOD FOR EDGES DETECTION OF MAGNETIC SOURCES USING LOGISTIC FUNCTION Геофизический журнал № 6, Т. 40, 2018 133 compiled by Geological Survey of Japan and Coordinating Committee for coastal and off- shore geoscience programs in East and South- east Asia [Geological …, 1996]. The magnetic anomaly values vary from −280 to +50 nT with many positive and negative anomalies that have E-W trends. As a pre-process of the data, upward continuation of the total field magnetic anomaly was performed to reduce noise effect (Fig. 4, a). The upward continu- ation height used is 1 km. The upward con- tinuation produced results that are smoother and less sensitive to random noise than the original anomaly data, but will not change the primary shapes. Fig. 4, b shows the AS of the magnetic data in Fig. 4, a. We can see that the AS fil- ter performs poor, and it is dominated by the high amplitude anomalies. Fig. 4, c shows the AT of the magnetic data. As expected (and discussed in the above sections), it is clearly more noisy than the results of using other filters, and it gives insufficient results to ac- curately determine the boundary of the magnet- ic sources. Fig. 4, d and e show the TA and L, re- spectively. By comparing the results, we can see that the TA and FS filters provided similar results, and they are effective in bringing out the details of the small amplitude anomalies. Fig. 4. f dis- plays the results of the Lk filter. It can be observed from this figure that the Lk filter cannot only bal- ance the large and small amplitude anoma- lies, but provides the best resolution of the magnetic boundaries in the study area. Conclusions. We have presented two new edge detection filters that are based on logis- tic function and the ratio of the first vertical derivative and total horizontal derivatives of the analytic signal amplitude. As in the tilt angle of analytic signal, the L and Lk filters can be applied directly to the magnetic data. The disadvantage of the filters is that they are sensitive to noise. Using upward continu- ation of magnetic data can help reduce the effects of noise, and increase the coherency of the solutions. The filters have been dem- onstrated on two synthetics and real magnet- ic data. The results showed that both the L and Lk filters can balance the large and small amplitude edges, and can bring out more de- tails than the AS and AT filters. The results also showed that the Lk filter give a higher resolution, compared with other edge detec- tion filters. Ta b l e 2. Parameters of the three prism model Prism ID P-1 P-2 P-3 Center coordinates 12 km; 31.5 km 31.5 km; 31.5 km 51 km; 31.5 km Inclination 12° 18° 20° Declination 25° 26° 24° Magnetization 5 A/m 5 A/m 5 A/m Length × Width 45×10 km 45×10 km 45×10 km Depth of the top 1 km 2 km 3 km Depth of the bottom 2 km 3 km 4 km Rotation angle 0° 0° 0° New method for edges detection of magnetic sources using logistic function L. T. Pham1, E. Oksum2, T. D. Do1, M. L. Huy3, 2018 The tilt angle of the analytic signal amplitude (TA) is defined as the arctangent of the ratio of the first vertical derivative to the total horizontal derivative of the analytic signal amplitude. It is commonly used as a useful tool to estimate edges of magnetic sources L. T. PHAM, E. OKSUM, T. D. DO, M. L. HUY 134 Геофизический журнал № 6, Т. 40, 2018 Ansari, A. H., & Alamdar, K. (2011). A new edge detection method based on the analytic sig- nal of tilt angle (ASTA) for magnetic and gravity anomalies. Iranian Journal of Science and Technology, 35(2), 81―88. doi: 10.22099/ ijsts.2011.2131. Cooper, G. R. J. (2014). Reducing the dependence of the analytic signal amplitude of aeromagnet- ic data on the source vector direction. Geophys- ics, 79(4), J55―J60. https://doi.org/10.1190/ geo2013-0319.1. Cooper, G. R. J., & Cowan, D. R. (2008). Edge en- hancement of potential-field data using nor- malized statistics. Geophysics, 73(3), H1―H4. https://doi.org/10.1190/1.2837309. Cooper, G. R. J., & Cowan, D. R. (2006). Enhanc- ing Potential Field Data Using Filters Based on the Local Phase. Computers & Geosciences, 32(10), 1585―1591. https://doi.org/10.1016/j. cageo.2006.02.016. Cordell, L. (1979). Gravimetric Expression of graben faulting in Santa Fe Country and the Espanola Basin, New Mexico. In R. V. Inger- soll (Ed.), Guidebook to Santa Fe Country (pp. 59―64). New Mexico Geological Society, So- corro. Cordell, L., & Grauch, V. J. S. (1985). Mapping References Basement Magnetization Zones from Aero- magnetic Data in the San Juan Basin, New Mexico. In The Utility of Regional Gravity and Magnetic Anomaly Maps (pp. 181―197). Soci- ety of Exploration Geophysicists, Tulsa. Geological Survey of Japan and Coordinating Committee for Coastal and Offshore Geosci- ence Programs in East and Southeast Asia (CCOP). (1996). Magnetic anomaly map of East Asia 1:4 000 000. CD-ROM. Hsu, S. K., Coppense, D., & Shyu, C. T. (1996). High- resolution detection of geologic boundaries from potential field anomalies: An enhanced analytic signal technique. Geophysics, 61(2), 1947―1957. https://doi.org/10.1190/1.1443966. Li, X., (2006). Understanding 3D analytic sig- nal amplitude. Geophysics, 71(2), L13―L16. https://doi.org/10.1190/1.2184367. Miller, H. G., & Sing, V. (1994). Potential field tilt a new concept for location of potential field sources. Journal of Applied Geophysics, 32(2- 3), 213―217. https://doi.org/10.1016/0926- 9851(94)90022-1. Nabighian, M. N. (1972). The analytic signal of two-dimensional magnetic bodies with po ly- gonal cross-section: Its properties and use of because its value is slightly dependence on the direction of magnetization vector, and it is more effective in estimating the edges of the bodies than the analytic signal amplitude and the standard tilt angle. Based on logistic function (L) that has the same shape with the shape of arctangent function, and the derivatives of the analytic signal amplitude, we introduce some new filters which also can reduce the effect of the magnetization direc- tion. Other notable features of these filters are that they produce amplitude maxima over the edges of sources and that they balance anomalies from shallow and deep sources. The feasibility of the proposed filters is demonstrated on noise-free and noisy synthetic mag- netic data from two 3D models where the obtained results coincide well with the actual edges. The effectiveness of the filters is also evaluated by comparing it with other edge detection methods. The results also show that our filters are less sensitive to variations in the depth of the source bodies and that a modified logistic function (Lk) can achieve better edge detection results than the analytic signal amplitude (AS), the analytic signal amplitude of the tilt angle (AT), the TA and L filters. The filters are also applied to real magnetic data from an area in south-central Vietnam, and the results demonstrate that the proposed filters is a useful tool for the qualitative interpretation of magnetic data. Key words: logistic function, tilt angle, analytic signal amplitude, edge detection, interpretation of magnetic data. NEW METHOD FOR EDGES DETECTION OF MAGNETIC SOURCES USING LOGISTIC FUNCTION Геофизический журнал № 6, Т. 40, 2018 135 automated anomaly interpretation. Geo physics, 37(3), 507―517. https://doi.org/10.1190/ 1.1440276. Pilkington, M., & Tschirhart, V. (2017). Practical considerations in the use of edge detectors for geologic mapping using magnetic data. Geo- physics, 82(3), J1―J8. https://doi.org/10.1190/ geo2016-0364.1. Roest, W. R., Verhoef, J., & Pilkington, M. (1992). Magnetic interpretation using the 3-D analytic signal. Geophysics, 57(1), 116―125. https://doi. org/10.1190/1.1443174. Verduzco, B., Fairhead, J. D., Green, C. M., & Mac- Ken zie, C. (2004). New insights to magne tic de rivatives for structural mapping. The Lea- ding Edge, 23(2), 116―119. https://doi.org/ 10.1190/1.1651454. Wijns, C, Perez, C., & Kowalczyk, P. (2005). The ta map: Edge detection in magnetic data. Geo- physics, 70(4), L39―L43. https://doi.org/10.1190/ 1.1988184.

New method for edges detection of magnetic sources using logistic function

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