New 3D parallel SGILD modeling and inversion Page: 12 of 20
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Xie et al, LBNL 42252, New Parallel SGILD Modeling and Inversion, September 1, 1998
P(o) e 2 (17)
P(dla)= 1 e 2 Va (18)
By Bayes theorem,
P (od) = P(dlojP(a)dm' (19)
the optimization (16) is equvalent to the following stochastic nonlinear opti-
mization of the random variable a,
I(u- d)II2 (-L(& - ),& -5) +(&.- =m2
V2 + )2 = min, (20)
where P (aid) is the posterior probability, P (a) is the prior probability on
the acoustic velocity, P (dju) is data probability based on the acoustic veloc-
ity model, A is the Laplacian operator, v is the mean value of the random
velocity that will be measured by core analysis or direct observation, & is the
fitting distribution based on v, Ud is the measured data with noises, V.u is
the standard deviation of ud, VUd is normalized deviation, V0 is the standard
deviation of o, and , is normalized deviation. Because there is incomplete
information of V which is measured in a few logging well or on the surface,
we introduce a regularizing parameter a and translate (20) into the following
stochastic nonlinear regularizing optimization,
u - ud) 2 + a (-0 ( - O), Q - Q) + (Q - )2) = min, (21)
the regularizing parameter is relative to the confidence interval of the random
3.3 Gauss-Newton iteration with annealing process
We use the modified annealing Gauss-Newton iterative method to solve the
optimization problem (21). The iteration scheme is as follows,
[sTQ - a0 a=
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Xie, G.; Li, J. & Majer, E. New 3D parallel SGILD modeling and inversion, article, September 1, 1998; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc710929/m1/12/: accessed May 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.