2-D Niblett-Bostick magnetotelluric inversion - MTNet

J. RODRÍGUEZ et al.
Geologica Acta, 8(1), 15-30 (2010)
DOI: 10.1344/105.000001513
2-D Niblett-Bostick
30
(D.1)
where Bij = 0 or 1, D and U are numbers which depend
on the model precision and amplitude, respectively, and B
numerically equal to unity carries the dimensions of Bij.
Equation (D1) may be viewed as a binary representation of
{s i
g }1
N with D+U+1 bits (e.g., Zhang and Paulson, 1997).
Hence, for the general sequence model we have
+ (a-term-independent-of-the-model). (D.2)
Once again, comparison of (D2) and (12) shows that C
and E have the same form except for the model-independent
term, provided that
(D.3)
(D.4)
and
(D.5)
This way the state of the Bim neuron after dynamics
actualization will be:
(D.6)
The number of neurons in the network is N(D+U+1),
where D and U are large enough to guarantee the amplitude
and precision of the model. It should be noted that in
order to implement the algorithm on a computer, several
optimizations can be made in order to improve both the CPU
computation time and memory resources. The program that
we used to simulate the RHANN was written in C computer
language, in which 32 states of every neuron, namely 0’s and
1’s, can be stored in a 32-bit wide variable (unsigned long int
value). Power of 2 expressions such as 2 (m+n) can be computed
and stored in memory as power2[m+n], we also note that
, the values of the products
can be calculated and stored in a bi-dimensional array, in
order to be used later in the inner loops of the dynamics.
This way, we obtain the term in equation
(D6) by including equation (D4). We can state the term as:
(D.7)
Similarly, in equation (D6), for Iim the three products
within can also be computationally optimized, this way we
will define the values sum1[i][m], sum2[i][m] and sum3[i]
[m] as follows:
(D.8)
(D.9)
(D.10)
therefore, the dynamics of the state of each neuron Bim will
be updated according to:
(D.11)
Furthermore, it should be noted that the simple binary
natural representation of the problem makes it suitable to
be solved by a device which is capable of accumulating
sums of partial products (vector/matrix computation). Such
devices are currently available in the form of an electronic
circuit known as a Digital Signal Processor (DSP) or the
more versatile Field Programmable Gate Array (which
might have embedded DSPs included). Such circuits are
capable of fast multiplication-accumulation processing.
If such a custom digital computer could be realized, it
could reduce the computation times achieved by current
traditional CPU based computers.
7
)}.
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7
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(a−term−independent−of−the−model). (D.2)
7
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s
A
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(a−term−independent−of−the−model). (D.2)
,
im
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B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
1
n
j
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i
for
withT
A
A imjn
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(
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n
0
0
(D.6)
]
kj
ki A
kj
ki A
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)
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1
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[
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2
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,
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2
[
2
2
1
]
][
)
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1
t
im
ki
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k
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A
m
power
m ∑ =
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= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
m σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
1
j
i
matk
U
m
power
m
i
N
j
∑ =
+
= (D.10)
≤
+
+
+
>
+
+
+
;
0
]
][
[
3
]
][
[
2
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[
1
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[
0
0
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[
3
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2
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1
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0
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m
i
sum
m
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sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
(D.11)
,
im
im
B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
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1
n
j
m
i
for
withT
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T imjn
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ki
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m
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k
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=
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−
=
+
=
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.
)
2
(
)
2
(
)
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2
1 )
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1
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1
)
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+
=
=
=
=
∑
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∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
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⎧
≤
+
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+
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= −
=
−
=
=
+
,
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T
if
I
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B
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)
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n
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)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
8
,
im
im
B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑ (D.4)
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
(t+1)
,
im
im
B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑ (D.4)
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
8
,
im
im
B
V =
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
A
A
B
A
I
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
(
1
]
][
[
*
]
[
2
]
][
[
0 j
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
=
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
=
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
=
)
1
( +
t
im
B
⎩
⎨
⎧
+
+
+
+
=
+
]
][
[
2
]
][
[
1
]
][
[
0
0
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
if
B t
im
8
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11
(t+1)
8
,
im
im
B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑ (D.4)
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
7
).
j
a
P
a
j
a
σ
σ ∂
+ (C.20)
+
∂
∂
−
∂
∂
)
|
|
|
1
j
TM
j
TM
a
TM
a
i σ
φ
σ
σ
)}.
|
|
|
|
1
(
|
)
(
j
TE
j
TE
a
TE
a
i
TE
a
i
e
TE
P
σ
φ
σ
σ
σ
φ
φ
∂
∂
−
∂
∂
−
(C.21)
}.
)
(
TE
i
I
TE
P φ
φ −
(C.22)
,
,...,
2
,
1
, N
i = (D.1)
−
+ )
(
)
(
)
(
]
t
jn
t
im
kj
ki
n
m
B
B
A
A
+
+
+
=
=
∑
∑
)
(
]
)
(
1
1
1
]
)
)
2
(
)
2
(
t
im
kj
ki
U
m
N
j
M
k
a
k
ki
m
B
B
A
A
s
A
−term−independent−of−the−model). (D.2)
8
,
im
im
B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑ (D.4)
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
,
im
im
B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑ (D.4)
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
8
),
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑ (D.4)
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
],
2
[
1
withT
A
A
T imj
kj
ki
k
imjn
−
=
=
∑
)
2
(
)
2
(
2
1
1
)
(
2
1
A
B
A
I ki
m
M
k
t
im
ki
m
M
k
im
=
=
∑
∑
+
−
= σ
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
[
2
]
][
[
0
U
D
n
m
power
m
i
sum ∑ −
=
+
−
=
im
I
*
2
[
2
2
1
]
][
[
1 power
m
i
sum −
=
[
2
]
][
[
2 m
power
m
i
sum =
[
2
]
][
[
3 U
m
power
m
i
sum +
=
)
1
( +
t
im
B
⎩
⎨
⎧
+
+
=
+
]
][
[
0
0
]
][
[
0
1
)
1
(
sum
m
i
sum
if
sum
m
i
sum
if
B t
im
8
,
im
im
B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑ (D.4)
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
8
,
im
im
B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑ (D.4)
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
8
,
im
im
B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑ (D.4)
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
,
im
im
B
V = (D.3)
),
,
(
)
,
(
,
0
],
2
[
1
n
j
m
i
for
withT
A
A
T imjn
kj
ki
n
m
M
k
imjn
=
=
−
=
+
=
∑ (D.4)
.
)
2
(
)
2
(
)
2
(
2
1 )
(
1
1
1
)
(
2
1
B
A
A
A
B
A
I kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
im
+
=
=
=
=
∑
∑
∑
∑
+
+
−
= σ (D.5)
)
1
( +
t
im
B
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≤
+
>
+
=
∑∑
∑
∑
= −
=
−
=
=
+
,
I
B
T
if
I
B
T
if
B
im
)
t
(
jn
N
j
imjn
U
D
n
im
)
t
(
jn
imjn
U
D
n
N
j
)
t
(
im
0
0
0
1
1
1
1
(D.6)
]
2
[
1
kj
ki
n
m
M
k
imjn
A
A
T
+
=
∑
−
=
kj
ki
M
k
A
A
j
i
matk ∑ =
= 1
]
][
[
im
t
jn
imjn
U
D
n
N
j
I
B
T +
∑
∑ −
=
=
)
(
1
)
(
1
]
][
[
*
]
[
2
]
][
[
0
t
jm
N
j
U
D
n
B
j
i
matk
n
m
power
m
i
sum ∑
∑
=
−
=
+
−
= (D.7)
im
I
,
)
(
*
]
*
2
[
2
2
1
]
][
[
1
)
(
2
1
t
im
ki
M
k
B
A
m
power
m
i
sum ∑ =
−
= (D.8)
,
)
(
*
]
[
2
]
][
[
2
1
a
k
ki
M
k
A
m
power
m
i
sum σ
= ∑ =
(D.9)
],
][
[
*
]
[
2
]
][
[
3
1
j
i
matk
U
m
power
m
i
sum
N
j
∑ =
+
= (D.10)
)
1
( +
t
im
B
⎩
⎨
⎧
≤
+
+
+
>
+
+
+
=
+
;
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
0
0
]
][
[
3
]
][
[
2
]
][
[
1
]
][
[
0
1
)
1
(
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
m
i
sum
m
i
sum
m
i
sum
m
i
sum
if
B t
im (D.11)
.
.