GNGTS 2016 - Atti del 35° Convegno Nazionale

606 GNGTS 2016 S essione 3.3 A second velocity model obtained by a mean along the horizontal direction of the Marmousi is considered (Fig. 1b). We then compute the gradient of the L 2 -norm difference between the observed data (solution of the wave equation with Marmousi model) and the predicted data (solution of the wave equation with the 1D model in Fig. 1b). Fig. 1c shows the computation of gradient using a second order finite difference approximation of the spatial derivatives, with ∈ =0.01. Fig. 1d shows the computation using the adjoint state method. To compute the first approximation, it is necessary a huge number of forward modellings, while for the adjoint method it is required to solve the forward modelling only three times, as said before. To prevent numerical instability during the backward propagation of the regular solution, we store the part of solution inside the absorbing region. As can be observed both methods produce very similar solutions, but the computational time of the adjoint method is considerably reduced. To show Fig. 1 – a) Marmousi velocity model, b) the model obtained by a mean along the horizontal direction of the Marmousi, c) gradient using second order finite difference approximation of spatial derivatives, and d) using adjoint method. To improve the visualization, both the gradients are multiplied by depth and we do not consider the water layer. Fig. 2 – a) Marmousi velocity model with a different velocity layer of 4.5 km/s instead of 3.2 km/s, and b) the gradient computed using the adjoint method. To improve the visualization, the gradient is multiplied by depth and we do not consider the water layer.