The 3-D TTI medium can be characterized as eight parameters (5 independent elastic moduli in constitutive coordinate system, density, tilt as well as azimuth angle) at each spatial point. One of the key issues in implementing FWI for parameter characterizing TTI media is efﬁcient gradient calculation of objective function with respect to each model parameter. To calculate the gradient of each independent parameter involves the synthetic and adjoint data, as well as the derivatives of elastic moduli with respect to the independent parameters of the model. In this paper, the synthetic and adjoint waveﬁelds are simulated with a staggered-grid ﬁnite-difference algorithm in anisotropic media. The derivatives of the elastic modulus tensor for TTI media are also analyzed in this paper. Numerical examples of the gradients calculation are thus illustrated in a three-layer TTI model. One of the issues addressed in the discussion is that the synthetic data at each time step should be stored on the disk so as to perform cross-correlation with the adjoint waveﬁelds to generate the gradient for FWI in time domian. The huge dataset storage during synthetic waveﬁeld simulation and loading when calculating the gradient is highly memory cost and time consuming. The use of random boundary layer allows us to compute both the adjoint and synthetic waveﬁeld simultaneously without the necessity of storing total synthetic data. In this paper, cubic grains are implemented as random boundary layers. The randomized elastic moduli instead of velocities are thus added in elastic wave equations in TTI medium. The synthetic waveforms with random boundary layers are ﬁnally illustrated.
View full article as PDF (2.36 Mb)