Finite-difference modeling of Maxwell viscoelastic media developed from perfectly matched layer

In numerical simulation of wave propagation,both viscoelastic materials and perfectly matched layers(PMLs)attenuate waves.The wave equations for both the viscoelastic model and the PML contain convolution operators.However,convolution operator is intractable in finite-difference time-domain(FDTD)method.A great deal of progress has been made in using time stepping instead of convolution in FDTD.To incorporate PML into viscoelastic media,more memory variables need to be introduced,which increases the code complexity and computation costs.By modifying the nonsplitting PML formulation,I propose a viscoelastic model,which can be used as a viscoelastic material and/or a PML just by adjusting the parameters.The proposed viscoelastic model is essentially equivalent to a Maxwell model.Compared with existing PML methods,the proposed method requires less memory and its implementation in existing finite-difference codes is much easier.The attenuation and phase velocity of P-and S-waves are frequency independent in the viscoelastic model if the related quality factors(Q)are greater than 10.The numerical examples show that the method is stable for materials with high absorption(Q=1),and for heterogeneous media with large contrast of acoustic impedance and large contrast of viscosity.

Non-Uniformity and Generalised Sacks Splitting

We show that there do not exist computable fimetions f_1(e,i).f_2(e,i).g_1(e,i),g_2(e,i)such that for all e,i ∈ω, (1)(W_(f_1)(e,i)-W_(f_2)(e,i))≤T(W_e-W_1): (2)(W_(g_1)(e,i)-W_(g_2)(e,i))≤T(W_e-W_i): (3)(W_w-W_i)≤T(W_(f_1)(e,i)-W_(f_2)(e,i))⊕(W_(g_1)(e,i)-W_(g_2)(e,i)): (4)(W_e-W_i)T(W_(f_1)(e,i)-W_(f_2)(e,i))uuless(W_e-W_i)≤T:and (5)(W_e-W_i)T(E_(g_1)(e,i)-W_(g_2)(e,i))unless(W_w-W_i)≤T. It follows that the splitting theorems of Sacks and Cooper cannot be combined uniformly.