On the calculation of vertical derivatives of potential fields for downward continuation and related filters

Saulo Pomponet Oliveira; Long Duc Luu; Thu-Hang Thi Nguyen; Kha Van Tran; Luan Thanh Pham

doi:10.15625/2615-9783/24314

Author affiliations

Authors

Saulo Pomponet Oliveira Department of Mathematics and Graduate Program in Geology, Federal University of Paraná, Caixa Postal 19096, Curitiba, PR, Brazil
Long Duc Luu University of Science, Vietnam National University, Hanoi, Vietnam
Thu-Hang Thi Nguyen University of Science, Vietnam National University, Hanoi, Vietnam
Kha Van Tran Institute of Earth Sciences, VAST, Hanoi, Vietnam
Luan Thanh Pham University of Science, Vietnam National University, Hanoi, Vietnam

DOI:

https://doi.org/10.15625/2615-9783/24314

Keywords:

Vertical derivative, potential field data, data enhancement, SW Sub-basin

Abstract

Methods for enhancing and estimating parameters of potential fields from gravimetric and magnetometric surveys typically utilize the vertical derivatives (VDR) of the potential field. These derivatives amplify the high-frequency content of the field, which can be caused by shallow bodies or survey noise. Regularization methods generate approximations of derivatives that must reconcile two objectives: reducing the effect of high-frequency amplification and providing an accurate approximation. Achieving this balance is crucial for methods that require vertical derivatives of successive order, such as Taylor-series implementations of downward continuation and the enhanced horizontal derivative (EHD) filter. This paper evaluates the performance of several vertical-derivative methods for downward continuation and EHD filters using both noise-free and noisy synthetic data. In addition, gravity data over the SW Sub-basin are considered, and the findings are compared with seismic data. Our results show that the β-VDR method provides more accurate and stable derivatives under noisy conditions.

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References

Ai H., Ekinci Y.L., Alvandi A., Deniz Toktay H., Balkaya Ç., Roy A., 2024. Detecting edges of geologic sources from gravity or magnetic anomalies through a novel algorithm based on hyperbolic tangent function. Turk. J. Earth Sci., 33 (6), 684–701. https://doi.org/10.55730/1300-0985.1936.

Ai H., Huang Q., Ekinci Y.L., Alvandi A., Narayan S., 2025. Robust edge detection for structural mapping beneath the Aristarchus Plateau on the Moon using gravity data. Earth and Space Science, 12(8), e2025EA004379. https://doi.org/10.1029/2025EA004379.

Altınoğlu F.F., 2023. Mapping of the Structural Lineaments and Sedimentary Basement Relief Using Gravity Data to Guide Mineral Exploration in the Denizli Basin. Minerals, 13(10), 1276. https://doi.org/10.3390/min13101276.

Alvandi A., Ardestani V.E., Motavalli-Anbaran S.-H., 2024. Novel Detectors Based on the Elliott Function for Mapping Potential Field Data: Application to Aeromagnetic Data from Indiana, United States. Ann. Geoph., 67(6), GP656. https://doi.org/10.4401/ag-9146.

Alvandi A., Ardestani V.E., Motavalli-Anbaran S.-H., 2025. Enhancement of the total horizontal gradient of potential field data using the Modified Gudermannian Function (MGTHG): application to aeromagnetic data from Georgia, USA. Bull. Geophys. Oceanogr., 66(1), 73–94. https://doi.org/10.4430/bgo00479.

Aprina P.U., Santoso D., Alawiyah S., Prasetyo N., Ibrahim K., 2024. Delineating geological structure utilizing integration of remote sensing and gravity data: a study from Halmahera, North Molucca. Indonesia. Vietnam J. Earth Sci., 46(2), 147–168. https://doi.org/10.15625/2615-9783/20010.

Bhattacharyya B.K., 1965. Two-dimensional harmonic analysis as a tool for magnetic interpretation. Geophysics, 30(5), 829–857. https://doi.org/10.1190/1.1439658.

Blakely R.J., 1996. Potential theory in gravity and magnetic applications. Cambridge University Press, Cambridge. ISBN: 0-521-57547-8.

Clarke G.K.C., 1969. Optimum second-derivative and downwardcontinuation filters. Geophysics, 34(3), 424–437. https://doi.org/10.1190/1.1440020.

Cella F., Fedi M., Florio G., 2009. Toward a full multiscale approach to interpret potential fields. Geophys. Prospect., 57(4), 543–557. https://doi.org/10.1111/j.1365-2478.2009.00808.x.

Dean W.C., 1958. Frequency analysis for gravity and magnetic interpretation. Geophysics, 23(1), 97–127. https://doi.org/10.1190/1.1438457.

Eldosouky, A.M., Abd El‑Wahed, M.A., Saada, S.A., Attia, M., 2025. New insights into structural and tectonic evolution of Safaga-Semna shear belt, Eastern Desert, Egypt: advanced integration of aeromagnetic, remote sensing and field studies. Geomech. Geophys. Geo-energ. Geo-resour., 11, 31. https://doi.org/10.1007/s40948-025-00946-2.

Evjen H.M., 1936. The place of the vertical gradient in gravitational interpretations. Geophysics, 1(1), 127–136. https://doi.org/10.1190/1.1437067.

Fedi M., Florio G., 2001. Detection of potential fields source boundaries by enhanced horizontal derivative method. Geophys. Prospect., 49(1), 40–58. https://doi.org/10.1046/j.1365-2478.2001.00235.x.

Fedi M., Florio G., 2002. A stable downward continuation by using the ISVD method. Geophys. J. Int., 151(1), 146–156. https://doi.org/10.1046/j.1365-246X.2002.01767.x.

Fedi M., 2002. Multiscale derivative analysis: A new tool to enhance gravity source boundaries at various scales. Geophys. Res. Lett., 29(2), 16-1-16-4. https://doi.org/10.1029/2001GL013866.

Florio G., Fedi M., Pašteka R., 2006. On the application of Euler deconvolution to the analytic signal. Geophysics, 71(6), L87–L93. https://doi.org/10.1190/1.2360204.

Gang Y., Lin Z., 2018. An improved stable downward continuation of potential fields using a truncated Taylor series and regularized vertical derivatives method. J. Geophys. Eng., 15(5), 2001–2008. https://doi.org/10.1088/1742-2140/aac53a.

Gao H., Zhou D., Qiu Y., 2009. Relationship between formation of Zhongyebei basin and spreading of southwest subbasin, South China Sea (East Sea). J. Earth Sci., 20(1), 66–76. Doi: 10.1007/s12583-009-0007-2.

Henderson R.G., 1960. A comprehensive system of automatic computation in magnetic and gravity interpretation. Geophysics, 25(3), 569–585. https://doi.org/10.1190/1.1438736.

Henderson R.G., Zietz I., 1949. The computation of second vertical derivatives of geomagnetic fields. Geophysics, 14(4), 508–516. https://doi.org/10.1190/1.1437558.

Karcol R., Pašteka R., 2025. Regularized derivatives-Revisited. Geophysics, 90(3), G73–G84. https://doi.org/10.1190/geo2023-0787.1.

Luo X., Liu S., Tang X., 2025. A robust method for calculating the vertical derivative of potential fields based on Hilbert transforms. Sci. Rep., 15(1), 18350. https://doi.org/10.1038/s41598-025-99798-9.

Melo F.F., Barbosa V.C.F., 2020. Reliable Euler deconvolution estimates throughout the vertical derivatives of the total-field anomaly. Comput Geosci, 138, 104436. https://doi.org/10.1016/j.cageo.2020.104436.

Melo J.A., Mendonça C.A., Marangoni Y.R., 2023. Python programs to apply regularized derivatives in the magnetic tilt derivative and gradient intensity data processing: A graphical procedure to choose the regularization parameter. Appl. Comput. Geosci., 19, 100129. https://doi.org/10.1016/j.acags.2023.100129.

Miller H.G., Singh V., 1994. Potential field tilt - a new concept for location of potential field sources. J. Appl. Geophys., 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 polygonal cross-section: its properties and use for automated anomaly interpretation. Geophysics, 37(3), 507–517. https://doi.org/10.1190/1.1440276.

Nabighian M.N., 1984. Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms: Fundamental relations. Geophysics, 49(6), 780–786. https://doi.org/10.1190/1.1441706.

Naidu P., 1966. Extraction of potential field signal from a background of random noise by Strakhov’s method. J. Geophys. Res., 71(24), 5987–5995. https://doi.org/10.1029/JZ071i024p05987.

Narayan S., Sahoo S.D., Pal S.K., Kumar U., Pathak V.K., Majumdar T.J., Chouhan A., 2016. Delineation of structural features over a part of the Bay of Bengal using total and balanced horizontal derivative techniques. Geocarto Int., 32(4), 351–366. https://doi.org/10.1080/10106049.2016.1140823.

Narayan S., Kumar U., Pal S.K., Sahoo S.D., 2021. New insights into the structural and tectonic settings of the Bay of Bengal using high-resolution earth gravity model data. Acta Geophys., 69, 2011–2033. https://doi.org/10.1007/s11600-021-00657-8.

Narayan S., Kumar U., Sahoo S.D., Pal S.K., 2024. Appraisal of lineaments patterns and crustal architectures around the Owen fracture zone, Arabian Sea, using global gravity model data. Acta Geophys., 72, 29–48. https://doi.org/10.1007/s11600-023-01170-w.

Nguyen T.N., Van Kha T., Van Nam B., Nguyen H.T.T., 2020. Sedimentary basement structure of the Southwest Sub-basin of the East Vietnam Sea by 3D direct gravity inversion. Mar. Geophys. Res., 41(1), 7. https://doi.org/10.1007/s11001-020-09406-w.

Öksüm E., Altinoglu F.F., Kafadar Ö., 2025. Structural mapping and depth configuration of the Sinanpaşa and western Afyon-Akşehir grabens (SW Türkiye) using advanced gravity data interpretation methods. J. Mt. Sci., 22, 2191–2210. https://doi.org/10.1007/s11629-025-9533-3.

Oliveira S.P., Pham L.T., 2022. A stable finite difference method based on upward continuation to evaluate vertical derivatives of potential field data. Pure Appl. Geophys., 179(12), 4555–4566. https://doi.org/10.1007/s00024-022-03164-z.

Oliveira S.P., Pham L.T., Pašteka R., 2024. Regularization of vertical derivatives of potential field data using Morozov’s discrepancy principle. Geophys. Prospect., 72(8), 2880–2892. https://doi.org/10.1111/1365-2478.13534.

Pašteka R., Richter F.P., Karcol R., Brazda K., Hajach M., 2009. Regularized derivatives of potential fields and their role in semi-automated interpretation methods. Geophys. Prospect., 57(4), 507–516. https://doi.org/10.1111/j.1365-2478.2008.00780.x.

Peters L.J., 1949. The direct approach to magnetic interpretation and its practical application. Geophysics, 14(3), 290–320. https://doi.org/10.1190/1.1437537.

Pham L.T., 2025a. A generalized β-VDR method for computing high-order vertical derivatives: application to downward continuation. Geophys. J. Int., 243(2), ggaf334. https://doi.org/10.1093/gji/ggaf334.

Pham L.T., 2025b. A robust method for computing vertical derivatives of potential fields using upward continuation data. Geophysics, 91(1), G11–G22. https://doi.org/10.1190/GEO-2025-0274.

Pham L.T., Oliveira S.P., Le-Huy M., Nguyen D.V., Nguyen-Dang T.Q., Do T.D., Tran K.V., Nguyen H.-D.-T., Ngo T.-N.-T., Pham H.Q., 2024. Reliable Euler deconvolution solutions of gravity data throughout the β-VDR and THGED methods: application to mineral exploration and geological structural mapping. Vietnam J. Earth Sci., 46(3), 432–448. https://doi.org/10.15625/2615-9783/21009.

Pham L.T., Oliveira S.P., Luu L.D., Do L.T., 2025. Enhancing potential fields using stable downward continuation and boundary filters: application to the Central Highlands, Vietnam. Vietnam J. Earth Sci., 47(2), 236–251. https://doi.org/10.15625/2615-9783/22702.

Reid A.B., Allsop J.M., Granser H., Millett A.J., Somerton I.W., 1990. Magnetic interpretation in three dimensions using Euler deconvolution. Geophysics, 55(1), 80–91. https://doi.org/10.1190/1.1442774.

Reilly W.I., 1969. Interpolation, smoothing, and differentiation of gravity anomalies. New Zealand J. Geol. Geophys., 12(4), 609–627. https://doi.org/10.1080/00288306.1969.10431100.

Roy I.G., 2013. On computing gradients of potential field data in the space domain. J. Geophys. Eng., 10(3), 035007. https://doi.org/10.1088/1742-2132/10/3/035007.

Saada S.A., Eleraki M., Mansour A., Eldosouky A.M., 2025. Insights on the structural framework of the Egyptian Eastern Desert derived from edge detectors of gravity data. Interpretation, 13(1), T71–T85. https://doi.org/10.1190/INT-2024-0023.1.

Sahoo S., Narayan S., Pal S.K., 2022a. Fractal analysis of lineaments using CryoSat-2 and Jason-1 satellite-derived gravity data: Evidence of a uniform tectonic activity over the middle part of the Central Indian Ridge. Phys. Chem. Earth, 128, 103237. https://doi.org/10.1016/j.pce.2022.103237.

Sahoo S., Narayan S., Pal S.K., 2022b. Appraisal of gravity-based lineaments around Central Indian Ridge (CIR) in different geological periods: Evidence of frequent ridge jumps in the southern block of CIR. J. Asian Earth Sci., 239, 105393. https://doi.org/10.1016/j.jseaes.2022.105393.

Sandwell D.T., Müller R.D., Smith W.H.F., Garcia E.S.M., Francis R., 2014. New global marine gravity model from CryoSat-2 and Jason-1 reveals buried tectonic structure. Science, 346(6205), 65–67. https://doi.org/10.1126/science.1258213.

Tatchum C.N., Tabod C.T., Koumetio F., Manguelle-Dicoum E., 2011. A gravity model study for differentiating vertical and dipping geological contacts with application to a Bouguer gravity anomaly over the Foumban shear zone, Cameroon. Geophysica, 47(1–2), 43–55.

Thompson D.T., 1982. EULDPH: A new technique for making computer- assisted depth estimates from magnetic data. Geophysics, 47(1), 31–37. https://doi.org/10.1190/1.1441278.

Tran K.V., Nguyen T.N., 2020. A novel method for computing the vertical gradients of the potential field: application to downward continuation. Geophys. J. Int., 220(2), 1316–1329. https://doi.org/10.1093/gji/ggz524.

Zhang H., Ravat D., Hu X., 2013. An improved and stable downward continuation of potential field data: The truncated Taylor series iterative downward continuation method. Geophysics, 78(5), J75–J86. https://doi.org/10.1190/geo2012-0463.1.