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Extra resources for Compressibility, Internal Pressure and Atomic Magnitudes

Example text

Nˆ j (x) −∞ j = c¯is (x,t) + ∂V d τ ∑ ψ i jkl (x,t − τ )∂l uks (x, τ ) kl dS(x ) T 0 dt Ccis cis (x,t; x ,t )† uis (x , T − t ). 34). Requiring this variational derivative to be zero, we obtain the boundary condition for the adjoint wave equation, ∑ nˆ j (x) j ∞ −∞ d τ ∑ ψ i jkl (x, T − t − τ )∂l † uks (x, τ ) = 0. 26). Requiring this variational derivative to be zero, we obtain fsi (x,t) = 0 fsi (x,t) + V dV (x ) T 0 dt C fsi fsi (x,t; x ,t )† uis (x , T − t ). 26). Requiring this variational derivative to be zero and applying the divergence theorem, we obtain Msi j (x,t) = 0 Msi j (x,t) − T dV (x ) V dt CMi j Mi j (x,t; x ,t )∂ j † uis (x , T − t ).

32) can also be transferred to the adjoint wave field by applying Green’s identity. 32) can be expressed as Ns ∑∑ s=1 i = V Ns ∑∑ s=1 i ∂V j V 0 dS(x) − ∑ nˆ j (x) + T dV (x) dV (x) ∞ −∞ T 0 ∞ dt ∑ ∂ j −∞ j T 0 dt d τ ∑ ψ i jkl (x,t − τ )∂l δ uks (x, τ ) † uis (x, T − t) kl ∞ − t) ∑ nˆ j (x) † i us (x, T −∞ j d τ ∑ ψ i jkl (x,t − τ )∂l δ uks (x, τ ) kl d τ ∑ ψ i jkl (x, T − t − τ )∂l † uks (x, τ ) δ uis (x,t) kl dt ∑ ∂ j j ∞ −∞ d τ ∑ ψ i jkl (x, T − t − τ )∂l † uks (x, τ ) δ uis (x,t). 34), we obtain the variational derivative of Y with respect to us (x,t) as follows: δY = ρ (x)† u¨is (x, T − t) − ∑ ∂ j 2δ uis (x,t) j ∞ −∞ d τ ∑ ψ i jkl (x, T − t − τ )∂l † uks (x, τ ) kl i i (t)Wdsrn + ∑ δ (x − xr )Jsrn i d i dsrn .

27). Collecting terms containing δ ψ i jkl (x,t) and applying the divergence theorem, we obtain ψ i jkl (x,t) = 0 ψ i jkl (x,t) − Ns T ∑ s=1 0 V T dV (x ) 0 dt Cψ i jkl ψ i jkl (x,t; x ,t ) d τ ∂ j † uis (x , T − t − τ )∂l uks (x , τ ) . 47) By now, we have completed our derivation of the Euler-Lagrange equations for the full-wave seismic data assimilation problem. , the structure and source parameters that maximize the posterior probability for one iteration can be used as the prior estimates for the next iteration.

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Compressibility, Internal Pressure and Atomic Magnitudes by Richards Th. W.


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