Pure and Applied Geophysics

, Volume 175, Issue 8, pp 2785–2806 | Cite as

Stable Computation of the Vertical Gradient of Potential Field Data Based on Incorporating the Smoothing Filters

  • Jamaledin Baniamerian
  • Shuang Liu
  • Mahmoud Ahmed Abbas


The vertical gradient is an essential tool in interpretation algorithms. It is also the primary enhancement technique to improve the resolution of measured gravity and magnetic field data, since it has higher sensitivity to changes in physical properties (density or susceptibility) of the subsurface structures than the measured field. If the field derivatives are not directly measured with the gradiometers, they can be calculated from the collected gravity or magnetic data using numerical methods such as those based on fast Fourier transform technique. The gradients behave similar to high-pass filters and enhance the short-wavelength anomalies which may be associated with either small-shallow sources or high-frequency noise content in data, and their numerical computation is susceptible to suffer from amplification of noise. This behaviour can adversely affect the stability of the derivatives in the presence of even a small level of the noise and consequently limit their application to interpretation methods. Adding a smoothing term to the conventional formulation of calculating the vertical gradient in Fourier domain can improve the stability of numerical differentiation of the field. In this paper, we propose a strategy in which the overall efficiency of the classical algorithm in Fourier domain is improved by incorporating two different smoothing filters. For smoothing term, a simple qualitative procedure based on the upward continuation of the field to a higher altitude is introduced to estimate the related parameters which are called regularization parameter and cut-off wavenumber in the corresponding filters. The efficiency of these new approaches is validated by computing the first- and second-order derivatives of noise-corrupted synthetic data sets and then comparing the results with the true ones. The filtered and unfiltered vertical gradients are incorporated into the extended Euler deconvolution to estimate the depth and structural index of a magnetic sphere, hence, quantitatively evaluating the methods. In the real case, the described algorithms are used to enhance a portion of aeromagnetic data acquired in Mackenzie Corridor, Northern Mainland, Canada.


Potential field magnetic data vertical gradient fast Fourier transform enhancement wavenumber extended Euler deconvolution 



We would like to thank the Editor, Hans-Jürgen Götze, and the reviewers Giovanni Florio and Roman Pašteka for their thoughtful comments and criticisms which helped us to improve the original manuscript. The Geological Survey of Canada is appreciated to permit to use the aeromagnetic data set. This research was supported by grants from the NSF of China and Hubei Province (Nos. 41604087, 2016CFB122), the China Postdoctoral Science Foundation (No. 2016M590132), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (CUGL170407, CUG160609).


  1. Abbas, M. A., & Fedi, M. (2014). Automatic DEXP imaging of potential fields independent of the structural index. Geophysical Journal International, 199, 1625–1632.CrossRefGoogle Scholar
  2. Abbas, M. A., Fedi, M., & Florio, G. (2014). Improving the local wavenumber method by automatic DEXP transformation. Journal of Applied Geophysics, 111, 250–255.CrossRefGoogle Scholar
  3. Baniamerian, J., Oskooi, B., & Fedi, M. (2017). Source imaging of potential fields through a matrix space-domain algorithm. Journal of Applied Geophysics, 136, 51–60.CrossRefGoogle Scholar
  4. Beiki, M. (2010). Analytic signals of gravity gradient tensor and their application to estimate source location. Geophysics, 75, I59–I74.CrossRefGoogle Scholar
  5. Blakely, R. J. (1996). Potential theory in gravity and magnetic applications. Cambridge: Cambridge University Press.Google Scholar
  6. Cella, F., Fedi, M., & Florio, G. (2009). Toward a full multiscale approach to interpret potential fields. Geophysical Prospecting, 57, 543–557.CrossRefGoogle Scholar
  7. Davis, K., Li, Y., & Nabighian, M. (2010). Automatic detection of UXO magnetic anomalies using extended Euler deconvolution. Geophysics, 75, G13–G20.CrossRefGoogle Scholar
  8. Davis, K., Li, Y., & Nabighian, M. N. (2011). Effects of low-pass filtering on the calculated structural index from magnetic data. Geophysics, 76, L23–L28.CrossRefGoogle Scholar
  9. De Lerma, D., Green, C., Cheyney, S., Campbell, S. (2015) Improved high order vertical derivatives of potential field data-extending the ISVD Method. In 77th EAGE Conference and Exhibition 2015.Google Scholar
  10. Evjen, H. (1936). The place of the vertical gradient in gravitational interpretations. Geophysics, 1, 127–136.CrossRefGoogle Scholar
  11. Fedi, M. (2002). Multiscale derivative analysis: A new tool to enhance detection of gravity source boundaries at various scales. Geophysical Research Letters. Scholar
  12. Fedi, M., & Florio, G. (2001). Detection of potential fields source boundaries by enhanced horizontal derivative method. Geophysical Prospecting, 49, 40–58.CrossRefGoogle Scholar
  13. Florio, G., Fedi, M., & Pasteka, R. (2006). On the application of Euler deconvolution to the analytic signal. Geophysics, 71(6), L87–L93.CrossRefGoogle Scholar
  14. Florio, G., Fedi, M., & Pašteka, R. (2014). On the estimation of the structural index from low-pass filtered magnetic data. Geophysics, 79(6), J67–J80.CrossRefGoogle Scholar
  15. Hannigan, P.K., Dixon, J., Morrow, D.W. (2009) Oil and gas resource potential in the Mackenzie corridor, northern mainland, Canada. In Proceedings Canadian Society of Petroleum Geologists Convention, pp. 66–70.Google Scholar
  16. Hood, P. (1965). Gradient measurements in aeromagnetic surveying. Geophysics, 30, 891–902.CrossRefGoogle Scholar
  17. Keating, P., & Pilkington, M. (2004). Euler deconvolution of the analytic signal and its application to magnetic interpretation. Geophysical Prospecting, 52, 165–182.CrossRefGoogle Scholar
  18. Lahti, I., & Karinen, T. (2010). Tilt derivative multiscale edges of magnetic data. The Leading Edge, 29, 24–29.CrossRefGoogle Scholar
  19. Li, Y., Devriese, S. G., Krahenbuhl, R. A., & Davis, K. (2013). Enhancement of magnetic data by stable downward continuation for UXO application. IEEE Transactions on Geoscience and Remote Sensing, 51, 3605–3614.CrossRefGoogle Scholar
  20. Mushayandebvu, M., & Davies, J. (2006). Magnetic gradients in sedimentary basins: examples from the Western Canada Sedimentary Basin. The Leading Edge, 25, 69–73.CrossRefGoogle Scholar
  21. Nabighian, M. N. (1984). Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms: fundamental relations. Geophysics, 49, 780–786.CrossRefGoogle Scholar
  22. Nabighian, M. N., Ander, M. E., Grauch, V. J. S., Hansen, R. O., LaFehr, T. R., Li, Y., et al. (2005a). Historical development of the gravity method in exploration. Geophysics, 70, 63–89.CrossRefGoogle Scholar
  23. Nabighian, M. N., Grauch, V. J. S., Hansen, R. O., LaFehr, T. R., Li, Y., Peirce, J. W., et al. (2005b). The historical development of the magnetic method in exploration. Geophysics, 70, 33–61.CrossRefGoogle Scholar
  24. Nabighian, M. N., & Hansen, R. O. (2001). Unification of Euler and Werner deconvolution in three dimensions via the generalized Hilbert transform. Geophysics, 66, 1805–1810.CrossRefGoogle Scholar
  25. Pašteka, R., Richter, F., Karcol, R., Brazda, K., & Hajach, M. (2009). Regularized derivatives of potential fields and their role in semi-automated interpretation methods. Geophysical Prospecting, 57, 507–516.CrossRefGoogle Scholar
  26. Pedersen, L. B., Bastani, M., & Kamm, J. (2015). Gravity gradient and magnetic terrain effects for airborne applications—a practical fast Fourier transform technique. Geophysics, 80, J19–J26.CrossRefGoogle Scholar
  27. Phillips, J.D. (2000) Locating magnetic contacts: a comparison of the horizontal gradient, analytic signal, and local wavenumber methods. In SEG Technical Program Expanded Abstracts 2000, pp. 402–405. Society of Exploration Geophysicists.Google Scholar
  28. Phillips, J. D., Hansen, R. O., & Blakely, R. J. (2007). The use of curvature in potential-field interpretation. ASEG Extended Abstracts, 2006, 1–7.CrossRefGoogle Scholar
  29. Pilkington, M. (2007). Locating geologic contacts with magnitude transforms of magnetic data. Journal of Applied Geophysics, 63, 80–89.CrossRefGoogle Scholar
  30. Pilkington, M., & Keating, P. (2009). The utility of potential field enhancements for remote predictive mapping. Canadian Journal of Remote Sensing, 35, S1–S11.CrossRefGoogle Scholar
  31. Pilkington, M., & Tschirhart, V. (2017). Practical considerations in the use of edge detectors for geologic mapping using magnetic data. Geophysics, 82, J1–J8.CrossRefGoogle Scholar
  32. Richter, P., & Pašteka, R. (2003). Influence of norms on calculation of regularized derivatives in geophysics. Contributions to Geophysics and Geodesy, 33, 1–16.Google Scholar
  33. Roy, I. G. (2017). An alternative approach in establishing relation between vertical and horizontal gradients of 2D potential field. Geophysical Prospecting. Scholar
  34. Salem, A., & Ravat, D. (2003). A combined analytic signal and Euler method (AN-EUL) for automatic interpretation of magnetic data. Geophysics, 68, 1952–1961.CrossRefGoogle Scholar
  35. Salem, A., Williams, S., Fairhead, D., Smith, R., & Ravat, D. (2007). Interpretation of magnetic data using tilt-angle derivatives. Geophysics, 73(1), L1–L10.CrossRefGoogle Scholar
  36. Thurston, J. B., & Smith, R. S. (1997). Automatic conversion of magnetic data to depth, dip, and susceptibility contrast using the SPI (TM) method. Geophysics, 62, 807–813.CrossRefGoogle Scholar
  37. Tikhonov, A., Glasko, V., Okl, L., & Melikhov, V. (1968). Analytic continuation of a potential in the direction of disturbing masses by the regularization method: Izvestiya. Physics of the Solid Earth, 12, 30–48. (in Russian; English translation: 738–747).Google Scholar
  38. Wang, B., Krebes, E. S., & Ravat, D. (2008). High-precision potential-field and gradient-component transformations and derivative computations using cubic B-splines. Geophysics, 73(5), I35–I42.CrossRefGoogle Scholar
  39. Yin, G., Zhang, Y., Mi, S., Fan, H., & Li, Z. (2016). Calculation of the magnetic gradient tensor from total magnetic anomaly field based on regularized method in frequency domain. Journal of Applied Geophysics, 134, 44–54.CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jamaledin Baniamerian
    • 2
    • 3
  • Shuang Liu
    • 1
  • Mahmoud Ahmed Abbas
    • 3
    • 4
  1. 1.Institute of Geophysics and GeomaticsChina University of Geosciences (Wuhan)WuhanChina
  2. 2.Department of Earth Sciences, College of Sciences and Advanced TechnologiesGraduate University of Advanced TechnologyMahanIran
  3. 3.Department of Earth, Environmental and Resources ScienceUniversity of Naples Federico IINaplesItaly
  4. 4.Geology DepartmentSouth Valley UniversityQenaEgypt

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