GNGTS 2018 - 37° Convegno Nazionale

672 GNGTS 2018 S essione 3.2 differences in confining pressure, etc). Geophysical methods can help in the reconstruction of a representative hydrologic model of the subsoil since they investigate a much greater volume of the subsurface, are able to image the complexity and heterogeneity of the subsurface itself and provide a continuous model of it (Doetsch et al. , 2010). Neverthless, they leave some ambiguities in the reconstruction of the subsoil features, because of the nonuniqueness which is inherent of the inversion process (Doetsch et al. , 2010; Moorkamp, 2017). The joint inversion of two or several geophysical methods, defined as the approach in which different types of data are inverted “within a single algorithm, with a single objective function...” (Moorkamp, 2017) can improve the reconstruction of the subsoil. The reason for this improvement is straightforward: since the various geophysical methods are sensitive to different physical properties, their integration can bring much more confidence in the estimated models. Joint inversion can be conducted using an explicit petrophysical relationship, even if it depends on many physical parameters that vary in the space and cannot be known precisely, or imposing structural similarity between models. Specifically, Gallardo and Meju (2004) developed a structural approach in which the vector cross-product of the gradients of two different models is forced to be zero at each iteration of the inversion, implying similar directions of the gradient vectors (Doetsch et al. , 2010). This method has been widely used and slightly modified by various authors (Linde et al. , 2006; Demirci et al. , 2017) and it is considered one of the most robust methods in the joint inversion of near surface geophysical field data (Linde et Doetsch, 2016). In this work, we present the joint inversion of Electrical Resistivity Tomography (ERT) and Seismic Refraction Tomography (SRT). The choice of the two geophysical methods is due to their high resolution for the characterization of the shallow subsurface, that is important from an engineering and environmental point of view. Joint inversion algorithm Before the description of the joint inversion, we present the inversion methods of individual data set. First, the investigated area is divided in cells, using an uniform regular grid. The forward modelings, f( m ) , for the examined geophysical methods are both nonlinear and consequently approximated with numerical methods. For the electrical method, the Poisson’s equation, that describes the electric field behavior, can be approximated through the finite elements method (Rücker, 2011), while for the seismic method, the eikonal equation can be approximated through the finite-difference scheme of Sethian (1999) (implemented in the pyGIMLi package (Rücker et al. , 2017)). The inverse problem can be solved as an optimization problem, in which an objective function is formulated (Gunther, 2004): (1) where: d is the vector of field data, whose noise is taken into account in matrix D ; f( m ) is the vector of predicted or synthetic data; C is the constraint matrix; m is the model vector; m 0 the reference model vector and λ the regularization parameter that weights the regularization term and is a trade-off parameter between the two terms (Gunther, 2004). Furthermore, Φ d is called data misfit, while Φ m the regularization term. The cross-gradients function, developed and used in the joint inversion of two geophysical methods by Gallardo and Meju (2003, 2004) is defined as the cross product of the gradient vectors of two models, m 1 and m 2 , which in our work are m ERT and m SRT respectively: (2) Adding the cross-gradients function to the objective function, we obtain: (3)