Page 123 - Machine Learning for Subsurface Characterization

Shallow neural networks and classification methods Chapter 3 99

T 2 amplitudes as a function of T 2 times ranging from 0.3 ms to 3000 ms. ϕ N is
formulated as

Z 3000 64
X
ð
ϕ ¼ AT 2 AT 2,i Þ△T 2 (3.D1)
ðÞdT 2 ¼
N
0:3 i¼1
where A(T 2,i ) is the amplitude A of the ith T 2 bin, and △T 2 can be obtained by
knowing the range of T 2 such that

3000 ms 1
64△T 2 ¼ log 10 ¼ 4 ! △T 2 ¼ (3.D2)
0:3ms 16
Now, the Eq. (3.D1) can be written as
64
1 X
ϕ ¼ ∙ AT 2,i Þ (3.D3)
ð
N
16
i¼1
The second parameter T 2,gm is the 64th root of the product of the 64 T 2 bin
amplitudes formulated as
! 1=n
n
Y
T 2,gm ¼ T 2,i (3.D4)
i¼1
where n ¼ 64 in our case and i indicates the ith T 2 bin. The geometric mean
cannot be calculated if any value is equal to 0. Considering the variation in
amplitudes for different T 2 bins, weighted T 2,gm is calculated at each depth that
is expressed as

! 1=Σw i
64
Y
T 2,gm ¼ T w i (3.D5)
2,i
i¼1
A i
where w i ¼ , A i is amplitude of the ith T 2 bin and A T is the sum of all amplitude
A T
P 64
values. Here, w i ¼ 1. The product of weighted T 2 is equivalent to the sum
i¼1
of weighted log(T 2 ). Consequently, T 2 geometric mean can also be expressed as
T 2 logarithmic mean:

ð
ð
log T 2,gm ¼ w 1 log T 2,1 Þ + w 2 log T 2,2 Þ + … + w 64 log T 2,64 Þ (3.D6)
ð
Abbreviations
ANN artificial neural network
AT10 induction resistivity logs at 10-in.
AT90 induction resistivity logs at 90-in.
CMIS committee machine with intelligent systems
CG conjugate gradient