ادامه فروسوی پایدار داده‌های میدان پتانسیل با استفاده از منظم‌سازی تیخونوف برای تخمین عمق توده‌های معدنی

نوع مقاله: مقاله تحقیقی‌ (پژوهشی‌)

نویسنده

دانشکده مهندسی معدن و متالورژی، دانشگاه یزد

چکیده

ادامه فروسوی داده‌های میدان پتانسیل به‌عنوان ابزاری کارآمد در تفسیر و پردازش داده‌های ژئوفیزیکی مورد استفاده قرار می‌گیرد. مشکل استفاده از این روش، ناپایدار بودن نتایج آن است. از لحاظ اصول نظری، گسترش داده‌های میدان پتانسیل (به سوی بالا یا پایین) باید در نواحی بدون حضور توده انجام گیرد. در مورد ادامه فراسو، این اصل مهم رعایت می‌شود، اما در مورد ادامه فروسو گسترش داده‌ها به طرف منبع زیرسطحی باعث نقض اصل نظری و ناپایدار شدن نتایج می‌شود. برای حل این مشکل راهکارهای مختلفی توسط پژوهشگران ارائه شده است. منظم‌سازی تیخونوف یکی از این راهکارها است که در آن بر پایه راه حل کمینه‌سازی، صافی پایین‌گذری در حوزه طیف فوریه طراحی می‌شود که موجب پایدارسازی نتایج ادامه فروسو خواهد شد. در این مقاله انتخاب مقدار بهینه پارامتر منظم‌سازی  با استفاده از تشکیل هنج (نُرم) مقادیر ادامه فروسو انجام می‌گیرد. در این رابطه نقطه کمینه نسبی هنج   به عنوان محلی برای انتخاب پارامتر   در نظر گرفته می‌شود. در این مقاله روش ارائه شده، روی داده‌های میدان پتانسیل مصنوعی در حالت‌های دوبُعدی و سه‌بُعدی به کار برده شده است. نتایج نشان می‌دهند که هنگامی که عمق ادامه فروسو به عمق قرارگیری سطحی‌ترین توده نزدیک می‌شود، نقطه کمینه نسبی هنج   به تدریج از بین می‌رود. لذا از این روش می‌توان به عنوان معیاری برای تخمین عمق سطحی‌ترین توده نیز استفاده کرد. همچنین این روش روی داده‌های مغناطیس چهارگوش زمین‌شناسی استان یزد و داده‌های بی‌هنجاری بوگر معدن سنگ آهن شواز در استان یزد به‌کار برده شده است. در این بررسی از کد برنامه‌نویسی به زبان متلب به نام REGCONT استفاده شده است.

کلیدواژه‌ها


عنوان مقاله [English]

Stable downward continuation of potential field data using Thikhoniv regularization for depth estimation of the mining bodies

نویسنده [English]

  • Kamal Alamdar
چکیده [English]

Semi-automated interpretation methods in applied gravity and magnetic are based on the estimation of source parameters directly from the measured (processed) and/or transformed potential fields data. These methods are built on the introduction of a prior information on the properties of the desired solution, mainly at the mathematical level. In the majority of cases, the priori information is based on the recognition of a predefined source type response in the interpreted data. Transformations of geophysical potential fields (mainly in gravity and magnetic) play an important role in their processing and interpretation. Due to the harmonic property of potential field data derived from Laplace equation, it is possible to realize the operation of analytical continuation of them, upwards and downwards, in the source free domain. In the case, when we continue the data further from the sources, we speak about upward continuation, in the opposite direction (closer to the sources), we deal with downward continuation. This description of the operation is independent of the orientation of the vertical coordinate axis, which can be different depending upon the application (either pointing downwards or upwards). In geophysical data processing, analytical continuation is used in various situations: e.g. to compare airborne and ground geophysical potential surveys data (measured on different height levels). In the interpretation of potential field data, upward continuation is used to enhance the regional components in the original data by suppressing near surface sources manifestation. Conversely, the downward continuation is used to enhance the detection of shallower sources by extracting the local or residual anomalies. Downward continuation is also often used to calculate the depth of the important shallowest sources. A great variety of mathematical treatments of the classical analytical continuation problem, either in space or spectral domain, have been reported. Downward continuation of potential fields is a powerful, but very unstable tool used in the processing and interpretation of geophysical data sets. It has been analytically proved from potential field theory that in downward continuation, we can only continue an interconnected potential field function to the depth of its nearest source (its upper edge). In the very downward continuation process, it is common that even in depths shallower than first source, the noise is amplified. Treatment of the instability problem has been realized by various authors in different ways. The Tikhonov regularization approach is one of the most robust ones. Regularization approach (Tikhonov et al., 1968) gives a straightforward and elegant way to the solution of the problem of achieving stable downward continuation of potential fields. It is based on a low-pass filter, derived in the Fourier spectral domain, by means of a minimization problem solution. We highlight the most important characteristics from its theoretical background and present its realization in the form of a Matlab-based program (REGCONT) written by Pašteka et al (2012). As we have shown in the presented synthetic model studies and practical data transformation, the proposed regularization method gives stable results, which are relatively close to the correct values particularly at shallow continuation depths. In comparison with other approaches to stabilize downward continuation, it shows a relative small dependency on the sampling rate of the data sets to be interpreted. For the selection of the optimum regularization parameter value , the behavior of the constructed norm functions has been used. In the majority of cases, C-norm gave a better developed and necessary local minimum in the function shape, which is connected with the searched optimum value. Positions of local minima for the other Lp-norm functions give in general higher values of , which lead to more smooth solutions. On the other hand, in some cases these norm functions can give a better developed local minimum and so they can be better used for depth estimation purposes (mainly in the case of under-sampled data sets). We demonstrate very good stabilizing properties of this method on several synthetic models and also the real gravity and magnetic datasets respectively form Shavaz Iron ore and Yazd geological quadrangle. In Shavaz Iron ore, the depth to the top of the subsurface body was estimated 60 m which is in agreement with the drilling data. The main output of the proposed solution is the estimation of the depth to the source below the potential field measurement level. In this study, the REGCONT Matlab code was used.

کلیدواژه‌ها [English]

  • Potential field
  • upward continuation
  • Downward continuation
  • Tikhonov regularization
  • regularization parameter
  • norm of function
  • Yazd geological quadrangle
  • Shavaz Iron ore
علمدار، ک.، 1388، تعبیر و تفسیر داده‌های میدان پتانسیل در حوزه فرکانس با کاربرد روی توده‌های معدنی: پایان‌نامه کارشناسی ارشد اکتشاف معدن، دانشگاه یزد.
داده‌های مغناطیس هوایی چهارگوش زمین‌شناسی یزد: سازمان زمین شناسی و اکتشافات معدنی کشور.

Abedi, M., Gholami, A., and Norouzi, G. H., 2013, A stable downward continuation of airborne magnetic data: A case study for mineral prospectivity mapping in Central Iran: Computers & Geosciences, 52, 269–280.
Abedi, M., Gholami, A., and Norouzi, G. H., 2014, A new stable downward continuation of airborne magnetic data based on Wavelet deconvolution: Near Surface Geophysics, 12(6), 751–762.
Bullard, E. C., and Cooper, R. I., 1948, Determination of masses necessary to produce a given gravitational field: Proc. Roy. Soc., A, 194, 332–347.
Baranov, W., 1975, Potential Fields and their Transformations in Applied Geophysics: Gebrüder Borntraeger, Berlin Stuttgart, 151 pp.
Berezkin, V. M., 1988, Method of the Total Gradient in Geophysical Prospecting: Nedra Moscow, 189 pp [in Russian].
Berezkin, V. M., and Buketov, A. P., 1965, Application of the harmonic analysis for the interpretation of gravity data: Applied Geophysics, 46, 161–166.
Berdichevski, M. N., and Dmitriev, V. I., 2002, Magnetollurics in the Context of the Theory of Ill-posed Problems: Society of Exploration Geophysicists, Tulsa, 230 pp.
Blakely, R. J., 1996, Potential Theory in Gravity and Magnetic Applications: Cambridge University Press, 464 pp.
Braun, J., Thieulot, C., Fullsack, P.,et al., 2008, DOUAR: A new three-dimensional creeping flow numerical model for the solution of geological problems: Physics of the Earth and Planetary Interiors,171, 76–91.
Cooper, G., 2004, The stable downward continuation of potential field data: Exploration Geophysics, 35, 260–265.
Cai, J., Grafarend, E. W., and Schaffrin, B., 2004, The A-optimal regularization parameter in uniform Tykhonov–Phillips regularization: Alpha weighted BLE: In F. Sansó (ed.): 5th Hotine– Marussi Symposium on Math. Geodesy, IAG Symposia, 127, Springer Verlag, Berlin–Heidelberg, pp. 309–324.
Evjen, H. M., 1936, The place of the vertical gradient in gravitational interpretations: Geophysics, 1, 127–136.
Fedi, M., and Florio, G., 2002, A stable downward continuation by using the ISVD method: Geophys. J. Int., 151, 146–156.
Fedi, M., and Florio, G., 2011, Normalized downward continuation of potential fields within the quasi-harmonic region: Geophysical Prospecting, 59, 1087–1100.
Glasko, V. B., Litvinenko, O. K., and Melikhov V. R., 1970, Possibilities of regularizing algorithms for continuation of potential functions close to source masses: Applied Geophysics, 60, 142–157.
Hughes, D. S., 1942, The analytic basis for gravity interpretation: Geophysics, 7, 169–178.
Hansen, P. C., 2007, Regularization Tools Version 4.0 for Matlab 7.3.: Numerical Algorithms, 46, 189–194.
Ku, C., Telford, W., and Lim, S., 1971, The use of linear filtering in gravity problems: Geophysics, 36, 1174–1203.
Lee, S. K., Kim, H. J., Song, Y., and Lee, C., 2009, MT2DinvMatlab, A program in Matlab and Fortran for two dimensional magnetotelluric inversion: Computer and Geoscience, 35, 1722–1735.
Mudretsova, E. A., and Veselov, K. A., Ed., 1990, Gravimetry: Nedra, Moscow, 607 pp. [in Russian with English expanded abstract].
Nabighian, M. N., 1974, Additional comments on the analytic signal of two dimensional magnetic bodies with polygonal cross-section: Geophysics, 39, 85–92.
Pašteka, R., Richter, F. P., Karcol, R., Brazda, K., and Hajach, M., 2009, Regularized derivatives of potential fields and their role in semi-automated interpretation methods: Geophysical Prospecting, 57(4), 507–516.
Pašteka, R., Karcol, R., Pašiaková, M., Pánisová, J., Kuširák D., and Béreš J., 2011, Depth estimation of microgravity anomalies sources by means of regularized downward continuation and Euler deconvolution: Extended abstracts from the 73nd EAGE Conference and Exhibition, Vienna, P051, 4 pp.
Pašteka, R., Richter, F. P., Karcol, R., Brazda, K., and Hajach, M., 2012, Stable downward continuation of potential field data using Thikhonov regularization: Geophysical Prospecting, 57(5), 517–526.
Parker, R. L., 1977, Understanding Inverse Theory: Annual Review of Earth and Planetary Sciences, 5, 35–64.
Peters, L.J, 1949, The direct approach to magnetic interpretation and its practical application: Geophysics, 14, 290–320.
Pawlowski, R. S., 1995, Preferential continuation for potential-field anomaly enhancement: Geophysics, 60, 390–398.
Roy, K. K., 2008, Potential Theory in Applied Geophysics: Springer–Verlag, Berlin, Heidelberg, 651 pp.
Rodi, W. L., and Mackie, R. L., 2001, Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion: Geophysics, 66, 174–187.
Sasaki, Y., 1989. Two-dimensional joint inversion of magnetotelluric and dipole–dipole resistivity data: Geophysics, 54, 254–262.
Trompat, H., Boschetti F., and Hornby P., 2003, Improved downward continuation of potential field data: Exploration Geophysics, 34(4), 249–256.
Tikhonov, A. N., Glasko, V. B., Litvinenko, O. K., and Melikhov, V. R., 1968, Analytic continuation of a potential in the direction of disturbing masses by the regularization method: Izv., Earth Physics, 12, 30–48 [in Russian; English translation: 738–747].
Tikhonov, A. N., and Glasko, V. B., 1965. Application of the regularization method to nonlinear problems: Applied Geophysics, 5(3), 463–473.
Tikhonov, A. N., and Arsenin, B. J., 1977, Solutions of Ill-posed Problems: John Wiley & Sons. New York.
Tsuboi, C., and Fuchida, T., 1937, Relations between the gravity values and corresponding subterranean mass distribution: Earth Research Institute of Tokyo Imperial University Bulletin, 15, 639–649.
Zhdanov, M. S., 2002, Geophysical Inverse Theory and Regularization Problems: Elsevier, 609 pp.
Xu, S., Yang, J., Yang, C., Xiao, P., Chen, S., and Guo, Z., 2007, The iteration method for downward continuation of a potential field from a horizontal plane: Geophysical Prospecting, 55, 883–889.