# Optimization and geophysical inverse problems Page: 8 of 37

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Notes

Because o can vary over extremely large ranges in the earth, it is common to write -(z) = em(z)

and let m(z) represent the model parameter. The differential equation then becomes

d2

dz2 E(z, w) + iwpem(z)E(z, w) = 0 , (3)

and the inverse problem is to solve for m(z) at all depths 0 < z < Z.

Problem 3 - Seismic Sounding

The third problem is one version of the type of hyperbolic second-order differential equations

that are common in seismology. Consider an earth in which the seismic velocities and density

vary only as a function of depth z. Then the elastodynamic equations of motion in the frequency-

slowness domain can be put in the form

d

dz v(w, p, z) = w A(p, z) v(w, p, z)+ f(w, p) , (4)

where

w is angular frequency in units of radians s-1,

p is horizontal slowness of the wave in units of s m-

z is depth below the surface of the earth in units of m,

v(w, p, z) is the displacement-stress vector v = [uz, uX, rzz/w, Tzx/W]T

A(p, z) is a matrix depending upon the material properties of the medium,

P (z)+2(z2) A(z)+2 (z) 0

A(p, z) = -p 0 0 1 (5)

-p(z) 0 0 p

S P2'Y(z) - P(z) -Pz +(2 z) 0

where

-y(z) = 4p(z) ,~)+yz

A(z) + 2P(z)

A(z) and lt(z) are elastic constants in units of kg m-1 s-2,

p(z) is the density in units of kg m3,

f(w,p) is the source vector that acts at the depth zs,

0

[0

f(w, p) = .( (0

fz(w, p)/w

f (w, p)/w4

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Barhen, J.; Berryman, J.G.; Borcea, L.; Dennis, J.; de Groot-Hedlin, C.; Gilbert, F. et al. Optimization and geophysical inverse problems, report, October 1, 2000; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc901685/m1/8/: accessed January 21, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.