In this paper, we illustrate the factors affecting the estimation of source depth in potential fields inversion. We verify the conditions for which it is theoretically and practically possible to invert data derived by non-linear operators (such as Total Gradient or Normalized Source Strength data) (Roest et al., 1992; Wilson, 1985; Pilkington & Beiki, 2013) through a linear algorithm. This approach is indeed often used in literature without investigating its theoretical and practical limits. Total Gradient or Normalized Source Strength data are, along the eigenvalues of the gradient tensor (λ1, λ2, λ3) and the invariants (I1, I2)(Pedersen & Rasmussen, 1990),all deriving from non-linear transformations of the magnetic field. This means that, considering for example the NSS due to two nearby sources, the sum of the two NSS of the field of each source is different from the NSS of the field of the two sources. The lack of linearity implies that the system of equations on which the forward problem is based can’t be built through a kernel matrix whose elements are given by the transformed variable calculated for each cell. Our study was performed using a linear inversion method based on the Generalized Singular Value Decomposition (GSVD) (Hansen, 2010) and through tools such as the Picard Plot and the Depth Resolution Plot (e.g., Paoletti et al., 2014). Our GSVD analysis on Normalized Source Strength data computed from the magnetic field shows even when dealing with well-conditioned systems (with similar number of data and unknowns), the inversion of quantities deriving from non-linear transformation of the magnetic field by a linear algorithm introduces non-negligible errors. The analysis of the Picard Plot (Hansen, 2010) shows indeed that the Picard condition is not fulfilled, so regularization by a truncated GSVD is needed, even with noise-free data. Because of this truncation, the system becomes under-determined, yielding to a deepening in the reconstruction of the sources and, thus, leading to a wrong estimation of their depth. This calls for the need of using a non-linear kernel for inversion of the cited quantities.
Inversion of non-linear quantities of potential fields / Paoletti, V.; Bingham, H. B. B.; AHMED ABBAS AHMED, Mahmoud; Fedi, M.. - (2018). ( 37° Convegno Nazionale Gruppo Nazionale di Geofisica della Terra Solida Bologna 19-21 Novembre, 2018.).
Inversion of non-linear quantities of potential fields
V. Paoletti;AHMED ABBAS AHMED, MAHMOUD;M. Fedi
2018
Abstract
In this paper, we illustrate the factors affecting the estimation of source depth in potential fields inversion. We verify the conditions for which it is theoretically and practically possible to invert data derived by non-linear operators (such as Total Gradient or Normalized Source Strength data) (Roest et al., 1992; Wilson, 1985; Pilkington & Beiki, 2013) through a linear algorithm. This approach is indeed often used in literature without investigating its theoretical and practical limits. Total Gradient or Normalized Source Strength data are, along the eigenvalues of the gradient tensor (λ1, λ2, λ3) and the invariants (I1, I2)(Pedersen & Rasmussen, 1990),all deriving from non-linear transformations of the magnetic field. This means that, considering for example the NSS due to two nearby sources, the sum of the two NSS of the field of each source is different from the NSS of the field of the two sources. The lack of linearity implies that the system of equations on which the forward problem is based can’t be built through a kernel matrix whose elements are given by the transformed variable calculated for each cell. Our study was performed using a linear inversion method based on the Generalized Singular Value Decomposition (GSVD) (Hansen, 2010) and through tools such as the Picard Plot and the Depth Resolution Plot (e.g., Paoletti et al., 2014). Our GSVD analysis on Normalized Source Strength data computed from the magnetic field shows even when dealing with well-conditioned systems (with similar number of data and unknowns), the inversion of quantities deriving from non-linear transformation of the magnetic field by a linear algorithm introduces non-negligible errors. The analysis of the Picard Plot (Hansen, 2010) shows indeed that the Picard condition is not fulfilled, so regularization by a truncated GSVD is needed, even with noise-free data. Because of this truncation, the system becomes under-determined, yielding to a deepening in the reconstruction of the sources and, thus, leading to a wrong estimation of their depth. This calls for the need of using a non-linear kernel for inversion of the cited quantities.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
