132 Seismic Imaging same misfit value. The reduction of the size of the null space can be achieved by introducing log data or a priori information. Memory and CPU requirements FWI is an expensive process, both in terms of memory and CPU requirements. Let’s consider a simple case where the forward and backward wave equations are solved with a standard Finite Difference scheme (Virieux, 1986). We applied the constant density acoustic wave equation (Eq. (5.4)). In the simplest case, one first needs to define a regular grid in time and space; the second-order time derivative and Laplacian operator are approximated in 2D by + − + [ ] ( ) 1 2 1 2 , , / ∆t , + − + [ ] ( ) 1 2 1 2 , , / ∆x and + − + [ ] ( ) 1 2 1 2 , , / ∆z , where Δt, Δx and Δz are the grid increments along the time and space axis. They cannot be chosen arbitrarily and should satisfy the conditions ∆ ∆ x z v f min max = ≤ 1 10 (5.8) ∆ ∆ ∆ t x z vmax ≤ ( ) +( ) 1 1 1 1 2 2 (5.9) The first equation is the dispersion condition: the space discretization should be 10 times smaller than the maximum wavelength v f min max / , obtained for the minimum expected velocity value in the model and for the maximum frequency. This is more restrictive than the Nyquist condition (more than 2 points per wavelength) as this is not a static representation but a dynamic one. If the condition is not satisfied, then numerical dispersion is observed on the signals. Equation (5.9) is related to the stability condition and is a function of the maximum velocity value. If the condition does not hold, the scheme is not stable and does not provide any useful information. Let’s consider an example for a target at 1 km depth, with desired maximum frequencies at 10 Hz and 100 Hz, for typical velocities between 300 m/s to 3000 m/s. Table 5.1 2D discretization in space and time for the same extreme velocity values and for two different maximum frequency values, as well as the storage requirement for a single wave field (right column). Δx (m) Δz (m) Δt (ms) Storage (GB) fmax = 10 Hz vmin = 300 m/s vmax = 3000 m/s 3.0 3.0 0.70 3.2 fmax = 100 Hz vmin = 300 m/s vmax = 3000 m/s 0.3 0.3 0.07 3200
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