GNGTS 2016 - Atti del 35° Convegno Nazionale

GNGTS 2016 S essione 3.3 605 Adjoint method. To obtain the gradient of the misfit function, one can compute the partial derivatives by using a finite difference approximation. For example, the partial derivative , related to a model ĉ i,j , can be approximated by with a small ∈ >0. To do this, the numerical solution of the wave equation for two different velocity models must be evaluated. Clearly, this method is impracticable when the number of unknowns is large and the solution of the wave equation is computationally expensive. Another way to obtain the gradient of Χ( ĉ i,j ) is to use the adjoint state method (Plessix, 2006; Fichner, 2010). By this method the gradient is obtained by the approximation of the following integral where p is the regular solution of the wave equation and p * T is the solution of the adjoint equation subject to the terminal conditions p ( x → i,j , T ) = ṗ ( x → i,j , T ) = 0, ∀ x → ∈ D , and where g * T is the adjoint source, that is Thus, if x → i,j is a receiver node, the adjoint source is the difference between the observed and the predicted data. Since the adjoint equation is of the same type of wave equation, it can be solve using the same numerical method examined before, but backward in time, to satisfy the terminal conditions. Computational aspects of the adjoint method. The adjoint state method is a powerful tool to reduce the computational time of the gradient, however some practical precautions are necessary. First at all, to approximate the time integral, both p and p * T must be know simultaneously for each time step. Unfortunately, the regular p solution propagates forward in time, while the adjoint p * T solution propagates backward in time. To solve this inconvenient, first the regular solution is solved forward in time and the adjoint source g * T is computed. Then both the regular p and adjoint p * T solutions are solved backward in time. By this way, both p and p * T are known simultaneously. This procedure needs to solve the wave equation three times: the first to compute the regular solution, forward in time, the second and the third to compute the solution backward in time. Another important aspect is the presence of absorbing boundaries. Due to their attenuation, it is not possible to reconstruct the regular solution p , backward in time from its final value. In fact, the regular solution must be multiplied at each time step, by the inverse of the Gaussian tapering factor, that is bigger than 1, causing numerical instability. To prevent this risk, the regular solution inside the absorbing layers must be stored theoretically each time step. Test on the Marmousi model. We consider the Marmousi velocity model (Bourgeois et al. , 1991) (Fig. 1a), and a seismic acquisition formed by two different seismic shots, situated on the top left top and right of the model respectively, characterized by a Ricker wavelet with a peak frequency of 6 Hz, and a maximum frequency of 18 Hz. The recording spread is composed by 190 receivers for each shot, equally spaced by 48 m, with a depth of 24 m and an offset of 300 m for the first receiver. A time sampling of dt =0.002 s, a recording time of 6 s, a space sampling of dx =24 m. The space grid is thus formed by 46948 grid nodes.