GNGTS 2013 - Atti del 32° Convegno Nazionale

Km=d (3) where K is the kernel coefficients matrix and d is the measured data vector. This inverse problem is under-determined, as the unknowns are more than data, so it has infinite solutions. The simplest solution is that minimizing the Euclidean norm of the solution m : m = K T ( KK T ) -1 d (4) Cribb (1976) showed, for this solution, this remarkable equation in the frequency domain: (5) where F denotes the Fourier transformation, L is the number of layers, m i is the magnetization intensity vector of the i th layer, h i is its depth, k is the wavevector with components k x , k y and ( k x 2 + k y 2 ) 1 2 , V ( k )=( t . k )( n . k )/| k | 2 , with t and n unit-vectors along the inducing field and magnetization vectors. Therefore the i th layer of the magnetization distribution m i is directly related to the upward continued field of the data d , to a distance equal to the opposite of the layer depth: z =- h i . Anti-transforming the second member of Eq. (5) and assuming, for simplicity, magnetization and inducing field both vertical, we find: (6) showing that and differ only for a numeric constant. Based on Eqs. (5) and (6) we therefore may use the minimum length solution as an effective alternative to common upward continuation techniques. In Fig. 1 we tested the VUC approach in two simple cases: level-to-level and level-to-draped upward continuation of the magnetic anomaly due to a horizontal dipole line located at 30 m depth. We observe that the field obtained from the minimum-length solution well reproduces the computed anomaly at the same altitude. The result has been achieved following these steps: • inverting the magnetic anomaly to obtain the minimum-length solution; • converting the magnetization volume to an upward continued field volume through Eq. (6); • extracting the i th layer corresponding to the desired continuation altitude z =- h i (level- to-level); • or extracting the field corresponding to the desired draped surface (level-to-draped). The most valuable advantage of the VUC method occurs when the field data are on a draped surface (e.g., a topographic surface) and are continued to a constant level. Some synthetic tests (Mastellone et al. , 2013) shows that the VUC result, compared with that produced by using other continuation algorithms, can better avoid topographic effects in the continued data. VUC is helpful also if the profile is severely truncated, and extrapolation is needed before performing upward continuation. In this case, one could obtain very different results, from several extrapolation algorithms. In this case VUC helps to circumvent the border effects by extending the source-volume with additional blocks at its borders. The border effects are better controlled, because of the constraint, inherent in the inversion process, that the predicted data must fit the measured data.. VUC in the gravity case: the role of the vertical derivative. In the gravity case the anomalous density distribution coming from a minimum-length inversion is proportional to the upward continuation of the vertical derivative of the observed data: (7) where G is the gravitational constant, F denotes the Fourier transformation, L is the number of layers, m i is the density vector of the i th layer, h i is its depth, k is the wavevector with components k x , k y and ( k x 2 + k y 2 ) 1 2 . Therefore the i th layer of the density distribution m i is directly 146 GNGTS 2013 S essione 3.2