An Application of Time-Domain Multiscale Waveform Tomography to Marine Data

We present an application of time-domain waveform tomography to a marine dataset. The data are lowpass filtered using a Wiener filter, and the inversion is carried out using a multiscale method with a dynamic early-arrival muting window. The adjoint method is used in the inversion for an efficient computation of the gradient directions. A staggered-grid, explicit finitedifference method with 4-order accuracy in space and 2-order accuracy in time is utilized for forward modeling and the adjoint calculation. Our method is applied to a marine seismic dataset. The initial velocity model for waveform tomography is obtained by using traveltime tomography with dynamic smoothing filters. The velocity tomogram obtained from waveform tomography has a higher resolution and a better accuracy than the initial model. The tomogram accuracy is verified by comparing common image gathers. INTRODUCTION Waveform tomography can theoretically provide an accurate and highly resolved estimate of the subsurface velocity structure. However, in practice, the problem is highly nonlinear, and a waveform tomography method will tend to converge to a local minimum if the starting model is not in the vicinity of the global minimum (Gauthier et al., 1986). Therefore, a good initial velocity model is required by waveform tomography to partially overcome the local minima problem. To further mitigate the nonlinearity, a multiscale approach can be utilized in either the frequency domain (Sirgue and Pratt, 2004; Brenders and Pratt, 2007) or the time domain (Bunks et al., 1995; Boonyasiriwat et al., 2008). A multiscale approach can be effectively applied to solve the local minima problem, and is computationally efficient. The nonlinearity of waveform tomography depends on the frequency content of seismic data. The misfit function at low frequencies is more linear than at high frequencies. Therefore, the inversion process that sequentially proceeds from low to high frequencies has a better chance to reach the global minimum compared to using highfrequency, raw data (Sirgue and Pratt, 2004; Boonyasiriwat et al., 2008). At low frequencies, coarser grids can be used for computing numerical solutions of the wave equation than at high frequencies resulting to a computational efficiency. In this paper, we present a multiscale method for timedomain waveform tomography proposed by Boonyasiriwat et al. (2008) and apply it to a marine dataset. The data are low-pass filtered into 2 frequency bands to allow the inversion to proceed from low to high frequencies. The inversion is carried out using a multiscale method and a dynamic early-arrival muting window. The inverted tomogram is more accurate than the initial model, and it further enhances the migration image. ACOUSTIC WAVEFORM TOMOGRAPHY In this section, we briefly review the theory of time-domain waveform tomography. The constant-density acoustic wave equation is used as our forward model, given by 1 c(r) ∂p(r, t|rs) ∂t −∇p(r, t|rs) = s(r, t|rs), (1) where p(r, t|rs) is the pressure field at position r at time t from a source at rs, c(r) is the velocity model, and s(t) is the source function. In term of the Green’s function G(r, t|r, 0) associated with equation 1, the solution, p(r, t|rs), can be written as