Seismic acoustic problem

Consider the integral operator for the anomalous (scattered) field p^{a}(z, z_{s},ω) which is generated by the impulsive source -δ(z-z_{s}), located at the point z_{s}:

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 V_{n}(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 p^{a}(0, z_{s},ω) at the surface of the earth (z′=0) at some frequencies ω_{1}, ω_{2}, ω_{3}, …., ω_{N}.

Let us assume that the source of the field is located on the surface, z_{s}=0. We assume that we measure voltage in seismometer, which is proportional to the time derivative of the field. So, voltages W(ω_{j}) 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 **m**_{λ} minimizing the parametric functional,

P^{λ}(**m**_{λ},**d**) = min

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

‖**Am**_{λ}–**d****‖**=δ

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

‖δ**d****‖**** ** = 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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