GNGTS 2016 - Atti del 35° Convegno Nazionale

GNGTS 2016 S essione 3.3 589 of these methods generally assume that the field is homogeneous, at least locally, that is within a moving window. Homogeneous potential fields f are generated by ideal sources and satisfy the homogeneity equation: (1) or, equivalently, the Euler’s differential homogeneity equation: (2) where n is the homogeneity degree, t >0 and {x 0 , y 0 , z 0 } are the coordinates of the unknown single source, also called one-point source. The homogeneity degree n has integer values, ranging from -3 to 0 in the magnetic case and -2 to 1 in the gravity case. The structural index denoted by N is the opposite of n , as far as the magnetic field or the gravity gradient are considered. Hansen and Suciu (2002) extended the Euler deconvolution to interpret multiple–sources by analyzing each window for multiple sources. A multiple source body generates an inhomogeneous field, whose homogeneity- degree varies with the distance from the source and assumes either integer or fractional values, as shown by Steenland and La Fehr (1968) in early 1960s. Hansen and Suciu (2002) faced the problem by setting, arbitrarily, the structural index to zero, or to 1, for all the points within a window. The authors point out one more drawback: in order to test different numbers of sources for a given data window, it is necessary to form and solve a new least-squares problem at each step. Fedi et al. (2015) demonstrate a different approach called Multi-HOmogeneity Depth Estimation (MHODE), First they showed that homogeneous field may be defined, that can have fractional degree; then they solved for the unknown source coordinates by assuming that at any altitude the inhomogeneous field can be approximated by homogeneous field of either fractional or integer degree. MHODE involves in the inversion of the scaling function, which has the advantage of not depending on any physical quantity so to be inverted directly for source geometry. In the present work, MHODE is used to study the inhomogeneous field generated by complex sources, which are modeled by Talwani’s approach (Talwani et al. , 1959). Inversion for scaling function is performed for the source coordinates using a Very Fast Simulated Annealing (VFSA) algorithm (Ingber, 1989,1993). MHODE Method. The scaling function, τ , of potential fields has a very simple expression along the lines defined by the zeros of the p -order horizontal derivative, of the potential field, called ridges. It is there defined as (Fedi, 2007): (3) In order to consider a generic multiple-source body, here we use the gravity field formula due to Talwani for 2D sources (Talwani et al ., 1959). Any two dimensional body can be defined by a set of q lines, from which the gravity field f is given by (Blakely,1996): (4) where i =1, …, L where γ and ρ are the gravitational constant and density respectively; x q and z q are the coordinates of the polygon vertices; r q and θ q are defined as following: