GNGTS 2024 - Atti del 42° Convegno Nazionale

Session 3.3 GNGTS 2024 where is the applied source current density, r = ( x, y, z ) is the spatal locaton, and − is the complex electric feld. In the tme domain, Poisson’s equaton is given as a convoluton between the conductvity and the electric feld as a functon of the tme t (e.g. Kemna, 2000): where is the inverse Laplace transform of and E ( t, r ) = − ∇ u ( t, r ). As mentoned above, during the last two decades, signifcant advancements in induced polarizaton research have taken place, partcularly with respect to spectral IP (SIP) and its increasing applicaton in near-surface investgatons even if surveys are usually modelled by taking into account only the integral chargeability, thus disregarding spectral content and neglectng the efect of the transmited waveform, biasing inversion results. In this context, following Fiandaca et al. (2012, 2013)’s approach, EEMverter has been developed to model IP in electric and electromagnetc (EM) data within the same inversion framework and where the forward response is computed in the frequency domain for all dimensionalites, solving the full version of Poisson’s equaton, and then transformed into the tme domain, thus avoiding the tme-domain approximaton (eq. 3). Here, we will focus only on the galvanic aspects in EEMverter modelling, while the other modelling features of EEMverter, such as EM modelling, tme-lapse and joint inversion of galvanic and EM data are treated in Fiandaca a et al. (2024). From a physical-mathematcal point of view, resistvity and IP forward responses are modelled in the frequency domain for a range of frequencies using the fnite element method. The responses are then transformed into the tme domain for each quadrupole measurement and the transmited current waveform is applied. In 2-D, the FD forward response assumes an isotropic 2-D distributon of the complex conductvity, neglectng electromagnetc inducton. Considering the complex conductvity at a given frequency with a point source at the origin with (zero-phase) current I, the Poisson’s equaton can be defned as follow: where is the Fourier-transformed complex potental, λ is the Fourier transformaton variable for the assumed strike ( y ) directon and δ represents the Dirac delta functon. Once the frequency domain potental is computed, the tme domain computaton is carried out through a cosine/sine transform, solved numerically in terms of Hankel transforms, expressed in terms of Bessel functons of order -1/2 and +1/2, respectvely (Johansen and Sørensen, 1979): (5) j* S E* ( ω , r ) = ∇ u* ( ω , r ) ∇ ∙ j s ( t , r ) = ∇ ⋅ ∞ ∫ 0 σ ( t ′  , r ) E ( t − t ′  , r ) dt ′  (3) σ ( t ) σ* ( ω ) σ* ( x , z , ω ) ω ∂ ∂ x ( σ* δσ * δx ) + ∂ ∂ z ( σ* ∂ ϕ * ∂ z ) − λ 2 ϕ*σ* = − Iδ ( x ) δ ( z ) (4) ϕ* ϕ* 1 π ∞ ∫ 0 f ( ω ) cos sin ( ωt ) dω = r ∞ ∫ 0 f 1 ( λ ) λJ ∓ 1 2 ( λr ) dλ