**Authors**

Institute of Geophysics, Associate Professor

**Abstract**

Study of the potential fields at different altitudes, constituting a multiscale field is a class of interpretative methods which are used to approximate depth and geometry of sources. Interpretation of potential fields by this class of methods is mainly based on the recognition that the gravity or magnetic fields, generated by ideal sources (point mass, line of mass, sheet and contact) are homogeneous functions satisfying Euler homogeneous equation. In multiscale methods the potential fields have to be known at several altitudes. Because the direct measurement of the field at many altitudes is not often feasible, the upward-continuation algorithm is used to create a multiscale field. Fedi et al (2012) introduced a multiscale method to estimate, depth and the structural index of potential field sources. In this new method, depth to the source of homogeneous fields is determined by a geometric technique. According to the geometric approach, as a consequence of the dilation of potential fields versus the altitude, the maxima of the field modulus at various scales are located along the straight lines that are called ridges. The source depth (singular points of source) can be recovered by simply extrapolating the ridges below the measurement surface and then identifying their intersection point. Simple sources, such as spheres, horizontal cylinders and sills, have singular points corresponding to their center. Dikes, vertical cylinders and contacts have their singular points correspond to the top of the source. Besides, the independent estimate of the structural index is done by the ScalFun method (Fedi and Florio 2006; Florio et al. 2009). The ScalFun method is based on the concept of the scaling function of potential fields which estimates both the structural index and the depth to source either independently or simultaneously. The scaling function is defined as the derivative of the logarithm of a potential field with respect to log(z) where z is altitude.

Finally, the validity of the results is tested by a criterion, called ‘ridge consistency’ criteria. The criterion is based on the principle that the structural index estimations on all the ridges converging towards the same source should be consistent. If there exist some coalescence effects, the gravity or magnetic anomalies measured from high altitudes may not be sufficiently isolated, and the estimated structural index from different ridges will be significantly different. One solution can be testing the field derivatives of any order to lessen the interference effects from nearby sources or regional fields up to obtaining a consistent set of estimates. Discarding low enough levels eliminates the improved high frequency noises produced during the differentiation and improves the results as well. Increasing the resolution with differentiation warrants better depth estimation. As differentiation and upward continuation behave like high pass and low pass filters, respectively, a combined use of them makes the whole procedure a very stable process. Briefly, by the explained multiscale analysis method the interpretation is done in four main steps: 1. Generation of a multiscale data set through the upward continuation algorithm, 2. Estimation of the source position with a geometrical method, 3. Estimation of the structural index for each analyzed ridge by using ScalFun method, 4. validating the results by the ridge consistency criteria.

The Depth from Extreme Points (DEXP) method of Fedi (2007) is the other multiscale method probed in this paper. DEXP approach is based on the explicit scaling of the upward continued field by a power law of the continuation height. The type of power law i.e., its exponent, can be either assumed or determined directly from the field data by the criterion of extreme point position invariance versus derivative order. There is a specific relationship between scaling exponent and source structural index. Moreover, similar to multiscale analysis field derivative of any order can be used. Therefore, in DEXP method, the scaling function is dependent on the structural index, upward continuation height and order of field derivative. Depths to sources are obtained from the position of the extreme points of the DEXP transformed ﬁeld. As the main advantages, these multiscale methods are very fast and stable respect to noises even while applying to high order derivatives.

In order to evaluate the capability of the studied methods, firstly the multiscale analysis and DEXP method are applied to a noise contaminated synthetic dataset due to three thin-magnetic dike. The results obtained by both methods are in a good agreement with the real ones. Finally, the practical utility of these multiscale methods are verified using a real profile extracted from an aeromagnetic data set acquired in Sweden. Also, in the real case the results of the studied methods are consistent.

Finally, the validity of the results is tested by a criterion, called ‘ridge consistency’ criteria. The criterion is based on the principle that the structural index estimations on all the ridges converging towards the same source should be consistent. If there exist some coalescence effects, the gravity or magnetic anomalies measured from high altitudes may not be sufficiently isolated, and the estimated structural index from different ridges will be significantly different. One solution can be testing the field derivatives of any order to lessen the interference effects from nearby sources or regional fields up to obtaining a consistent set of estimates. Discarding low enough levels eliminates the improved high frequency noises produced during the differentiation and improves the results as well. Increasing the resolution with differentiation warrants better depth estimation. As differentiation and upward continuation behave like high pass and low pass filters, respectively, a combined use of them makes the whole procedure a very stable process. Briefly, by the explained multiscale analysis method the interpretation is done in four main steps: 1. Generation of a multiscale data set through the upward continuation algorithm, 2. Estimation of the source position with a geometrical method, 3. Estimation of the structural index for each analyzed ridge by using ScalFun method, 4. validating the results by the ridge consistency criteria.

The Depth from Extreme Points (DEXP) method of Fedi (2007) is the other multiscale method probed in this paper. DEXP approach is based on the explicit scaling of the upward continued field by a power law of the continuation height. The type of power law i.e., its exponent, can be either assumed or determined directly from the field data by the criterion of extreme point position invariance versus derivative order. There is a specific relationship between scaling exponent and source structural index. Moreover, similar to multiscale analysis field derivative of any order can be used. Therefore, in DEXP method, the scaling function is dependent on the structural index, upward continuation height and order of field derivative. Depths to sources are obtained from the position of the extreme points of the DEXP transformed ﬁeld. As the main advantages, these multiscale methods are very fast and stable respect to noises even while applying to high order derivatives.

In order to evaluate the capability of the studied methods, firstly the multiscale analysis and DEXP method are applied to a noise contaminated synthetic dataset due to three thin-magnetic dike. The results obtained by both methods are in a good agreement with the real ones. Finally, the practical utility of these multiscale methods are verified using a real profile extracted from an aeromagnetic data set acquired in Sweden. Also, in the real case the results of the studied methods are consistent.

**Keywords**

- depth estimation
- Field derivative
- magnetic sources
- multiscale analysis
- structural index
- Upward continuation

**Main Subjects**

Blakely, R. J. and Hassanzadeh, S., 1981, Estimation of depth to magnetic source using maximum entropy power spectra with application to the Peru- Chile Trench in Nazca Plate; Crustal formation and Andean Convergence, 667-682, Geological Society of America Memoir 154, Boulder, CO.

Fedi, M. and Rapolla, A., 1997, Space-frequency analysis and reduction of potential field ambiguity, Annali Di Geofisica, XL(5), 1189-1200.

Fedi, M. and Florio, G., 2006, SCALFUN: 3D analysis of potential field scaling function to determine independently or simultaneously structural index and depth to source, SEG Expanded Abstract,25, 963-967.

Fedi, M., 2007, DEXP: a fast method to determine the depth and the structural index of potential fields sources, Geophysics,72(1), I1-I11.

Fedi, M. and Pilkington, M., 2012, Understanding imaging methods for potential field data, Geophysics, 77(1), G13-G24, doi: 10.1190/ geo2011-0078.1.

Fedi, M., Florio, G., and Cascone, L., 2012, Multiscale analysis of potential fields by a ridge consistency criterion: the reconstruction of the Bishop basement, Geophys. J. Int., 188, 103-114.

Florio, G., Fedi, M. and Rapolla, A., 2009, Interpretation of regional aeromagnetic data by multiscale methods: the case of Southern Apennines (Italy), Geophys. Prospect*, *57, 479-489.

Gological survey of Sweden, 2005, Airborne geomagnetic and geological maps of Sweden.

Hartman, R. R., Teskey, D. J. and Friedberg, J. L., 1971,A system forrapid digital aeromagnetic interpretation, Geophysics, 36, 891-918.

Li, X., 2006, Understanding 3D analytic signal amplitude, Geophysics,71, L13-L16.

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, 507-517.

Nabighian, M. N., 1974, Additional comments on the analytic signal oftwo-dimensional magnetic bodies with polygonal cross section,Geophysics, 39, 85-92.

Nabighian, M. N.,1984, Toward a three-dimensional automatic interpretation of potential ﬁeld data via generalized Hilbert transforms: Fundamental relations, Geophysics, 49, 780-786.

Peters, L. J., 1949, The direct approach to magnetic interpretation and itspractical application, Geophysics 14, 290-320.

Phillips, J. D., 1979, ADEPT: A program to estimate depth to magnetic basement from sampled magnetic profiles, Open-File Report, 79-367, U.S. Geological Survey.

Rajagopalan, S. and Milligan, P., 1994, Image enhancement of aeromagnetic data using automatic gain control, Exploration Geophysics, 25, 173-178.

Ravat, D., Pignatelli, A., Nicolosi, I. and Chiappini, M., 2007, A study of spectral methods of estimating the depth to the bottom of magnetic sources from near-surface magnetic anomaly data, Geophysical Journal International, 169, 421-434.

Reid, A. B., Allsop, J. M., Granser, H., Millett, A. J. and Somerton, I. W., 1990, Magnetic interpretation in three dimensions using Euler deconvolution, Geophysics, 55, 80-91,

Salem, A. and Ravat, D., 2003, A combined analytic signal and Euler method (AN-EUL) for automatic interpretation of magnetic data,Geophysics, 68, 1952-1961.

Salem, A., Williams, S., Samson, E., Fairhead, D., Ravat, D. and Blakely, R. J., 2010, Sedimentary basins reconnaissance using the magnetic Tilt-Depth method, Exploration Geophysics, 41, 198-209.

Smith, R. S. and Salem, A., 2005, Imaging the depth, structure, and susceptibility from magnetic data, The advanced source parameter imaging method, Geophysics, 70(4), L31-38.

Skillbrei, J. R., 1993, The straight-slope method for basement depthdetermination revisited, Geophysics 58, 593-595.

Spector, A. and Grant, F. S., 1970, Statistical model for interpreting aeromagnetic data, Geophysics, 35(2), 293-302.

Stavrev, P. and Reid, A. B., 2007, Degrees of homogeneity of potential ﬁeldsand structural indices of Euler deconvolution, Geophysics, 72(1), L1-L2.

Stavrev, P. and Reid, A. B., 2010, Euler deconvolution of gravity anomalies from thick contact/fault structures with extended negative structural index.

Thompson, D. T., 1982, EULDPH - A new technique for making computer-assisted depth estimates from magnetic data, Geophysics, 47, 31-37.

Thurston, J. B. and Smith, R. S., 1997, Automatic conversion of magnetic data to depth, dip, susceptibility contrast using the SPI method, Geophysics, 62, 807-813.

Werner, S., 1953, Interpretation of magnetic anomalies at sheet-likebodies, Sveriges Geologiska Undersok., Arsbok, 43(6), series C, no.508.