Seite - 1166 - in Book of Full Papers - Symposium Hydro Engineering

3. MATHEMATICAL MODEL 3.1. FORWARD MODEL The elastic wave equation is solved numerically for the displacements u, stresses τ, density and Lamé parameters (λ and μ) using the finite difference method. The wave equation is discretized in time and space with each of the parameters placed on a staggered grid [20, 21]. The 2D elastic wave equation employed for the forward modeling is the stress-displacement formulation (τ- u) in Eq. [1]. The Vp and Vs velocities can be calculated from Eq. [2]. 𝜌 (𝜕 ^2 𝒖 _𝑖 )/(𝜕 𝑡 ^2 ) − (𝜕 𝜏 _𝑖 𝑗 )/(𝜕 𝑥 _𝑗 ) = 𝑓 _𝑖 𝜏 𝑖 𝑗 − 𝐶 𝑖 𝑗 𝑘 𝑙 𝜖 𝑖 𝑗 = 𝑇 𝑖 𝑗 [1] 𝜖 _𝑘 𝑙 = (1/2 (𝜕 𝒖 _𝑖 ))/(𝜕 𝑥 _𝑗 ) + ( 1/2 (𝜕 𝒖 _𝑗 ))/(𝜕 𝑥 _𝑖 )) With 𝒖 𝒕 =𝟎 = 𝒖 𝟎 𝑜 𝑛 Ω (𝜕 𝒖 _(𝒕 = 𝟎 ))/𝜕 𝑡 = ¯𝒗 _𝟎 𝑜 𝑛 Ω 𝒖 = 𝒖 1𝑜 𝑛 Γ1 (𝜕 𝐶 _𝑖 𝑗 𝑘 𝑙 𝜖 _𝑘 𝑙 )/𝜕 𝑥 = ¯(𝑡 ) 𝑜 𝑛 Γ_2 𝑉 _𝑝 = (√((𝜆 +2𝜇 )))/𝜌 ; 𝑉 _𝑠 = √𝜇 /𝜌 [2] where: - 𝑢 𝑖 is the particle displacement - 𝜏 𝑖 𝑗 = 𝜆 𝜃 𝛿 𝑖 𝑗 + 2 𝜇 𝜖 𝑖 𝑗 are stress tensor components - 𝐶 𝑖 𝑗 𝑘 𝑙 = 𝛿 𝑖 𝑗 𝛿 𝑘 𝑙 𝜆 + (𝛿 𝑖 𝑘 𝛿 𝑗 𝑙 + 𝛿 𝑖 𝑙 𝛿 𝑗 𝑘 )𝜇 is the stiffness tensor - 𝜖 𝑖 𝑗 is the strain tensor - ρ is the density of the medium - 𝑓 𝑖 ,𝑇 𝑖 𝑗 are volume and surface source terms - 𝜆 and 𝜇 are the Lamés’ parameters - 𝑉 𝑝 and 𝑉 𝑠 are the primary and secondary wave velocity According to [22, 23], the stability of the staggered grid code is guaranteed when 𝜆 , 𝜇 and ρ are averaged harmonically and arithmetically. The accuracy of the finite difference (FD) operator used to approximate the spatial derivatives is dependent on the order of the FD operator (i.e. truncation error of the Taylor series expansion). Reflection of the wavefield at the boundaries are prevented by using a PML to damp out the wavefield at the boundaries. Furthermore, it is necessary to satisfy spatial and temporal sampling conditions [24] for the wavefield to prevent numerical artifacts and instabilities during the finite difference simulation. 1166