Approximation of depth and structural index of magnetic sources using multiscale analysis and DEXP methods

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.