GNGTS 2016 - Atti del 35° Convegno Nazionale

604 GNGTS 2016 S essione 3.3 wave equation. However due to the high computational cost of 3D modeling, in this work we consider only the 2D acoustic wave equation: where t ∈ [ 0, T ] is the recording time, x → ∈ D ( x , z ) ⊂ R 2 is a bi-dimensional space domain, p is the acoustic pressure of the wave, x → 0 is the location of the source, s ( t ) is the seismic wavelet, and c is the acoustic wave speed. As initial condition we consider p ( x → ,0) = ṗ ( x → ,0) = 0, ∀ x → ∈ D . The solution of the wave equation is obtained using the following explicit finite difference scheme, The time domain sampling is dt . The space domain D is sampled with a uniform step dx along the horizontal and vertical direction, obtaining a regular grid D i,j formed by nx * nz grid nodes, with i =1,…, nx and j =1,…, nz and with x → i,j =( x i ,z j ). Besides ∆ˆ is an optimized finite difference operator used to approximate the Laplacian ∆ with a small dispersion error. Because of numerical stability, it is necessary to limit the time sampling on the basis of the relation: where λ is the Courant number (Courant et al. , 1967). Moreover, the limitation of the computational domain to only a part of the true physical domain introduces artificial reflecting boundaries. To suppress these undesired reflections we use the Gassian taper method (Cerjan et al. , 1985). The main idea is to introduce a thin narrow strip along the artificial boundary and to multiply the solution, at each time step, by a Gaussian taper factor G . This term is 1 along the boundaries between the computational domain and the narrow strip, and decreases until 0.92 along the outside boundaries of the narrow strip. Misfit function and its derivative. Put in mathematical terms, the aim of FWI is to find an optimal Earth model c – , that minimizes a misfit functional: where C is the set of all possible geological models, p 0 are the observed data and p ( c ) are the predicted data. A classical functional used in the contest of FWI is the L 2 -norm difference between the observed and the synthetic seismograms (Tarantola, 1986; Igel et al. , 1996), where nr is the number of receivers and { x → r } represents their positions. SinceΧis generally a complicated non-linear functional of themodel c , iterativeminimization algorithms are used to obtain the solution c – . Starting from a plausible initial model c 0 , iterative minimization updates the current model c k , to a new model c k +1 , c k +1 = c k + γ k h k where Χ( c k +1 ) < Χ( c k ) and h k and γ k > 0 are the descend direction and the step length respectively. A descend direction is given by h k = – A * ∇ c Χ( c k ), where A is a positive definite matrix (Nocedal and Wright, 2006), and ∇ c Χ( c k ) is the gradient at the k-th iteration. Because of the space discretization of the wave equation, also the models c ∈ C are discretized on the grid nodes with components c i,j . Discretized model are indicated by ĉ i,j . Thus, FWI problem becomes an optimization problem with number of unknowns nx * nz , that is the number of nodes of the modelling grid.