Seismic acoustic problem

Consider the integral operator for the anomalous (scattered) field pa(z, zs,ω) which is generated by the impulsive source -δ(z-zs), located at the point zs:

p^a (z^′,z_s,ω)=ω^2 ∫_0^(+∞)▒〖α(z)/(V_n^2 (z) ) p^n (z,z_s,ω) 〗 p^n (z,z′,ω)dz

  This is a linear integral equation for α(z) in terms of the observed data. It has been shown that in the simplest case when Vn(z)=C=const the normal field can be expressed in the form:

p^n (z,ζ,ω)=-C/2iω exp[±iω/C (z-ζ)]

where we use the sign + for z>ζ and the sign − for z<ζ.

  Suppose that we have observed the anomalous (scattered) field pa(0, zs,ω) at the surface of the earth (z′=0) at some frequencies ω1, ω2, ω3, …., ωN.

  Let us assume that the source of the field is located on the surface, zs=0. We assume that we measure voltage in seismometer, which is proportional to the time derivative of the field. So, voltages W(ωj) can be expressed as

W(ω_j )=iω_j  k/4 ∫_(z_1)^(z_2)▒〖α(z) e^(((2iω_j z)/C) ) dz〗

where k is a conversion constant. Here k=1.

According to the basic principles of the regularization method we have to find a quasi-solution of the inverse problem as the model mλ minimizing the parametric functional,

Pλ(mλ,d) = min

The parameter of regularization λ is determined from the misfit condition:

Amλd

where δ is some a priori estimation of the level of noise of the data:

‖δd  = δ

  Construct the algorithm of the solution of the 1-D seismic inverse problem with respect to α(z), using the regularized least square method.

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