where g
m
¼ 2K
i ,
j
K
i ¹ 1 ,
j
Dy
2
K
i ,
j
þ K
i ¹ 1 ,
j
m . 1 ¼
m 0 7a
c
m
¼ 2K
i ,
j
K
i ,
j ¹ 1
Dx
2
K
i ,
j
þ K
i ,
j ¹ 1
i ,
j Ó G
¼ i
, j
[ G 7b
b
m
¼ c
m
7c d
m
¼ g
m
7d a
m
¼ ¹ b
m
þ c
m ¹ 1
þ d
m
þ g
m ¹ n
x
7e
f
m
¼ ¹
w
i ,
j
¹ h
i ,
j
i ,
j [ G
¹ w
i ,
j
i ,
j Ó G
7f These equations can be written in a matrix form as,
A{h} ¼ {f } 8
where A is a band matrix with dimension n
A ¼ a
1
b
1
… d
1
c
1
a
2
] :
] d
n ¹ n
x
g
1
: ]
] b
n ¹ 1
o g
n ¹ n
x
… c
n ¹ 1
a
n
2 6
6 6
6 6
6 6
6 6
6 6
4 3
7 7
7 7
7 7
7 7
7 7
7 5
9
The above system of linear equations is often solved by iterative methods, including successive over-relaxation
14
or conjugate gradient methods.
9
3 PERTURBED SOLUTIONS
In this section, we analyze the perturbed solutions for changes in the matrix A and the vector {f} . We will first
consider the change in A caused by changes in hydraulic conductivity, keeping the vector {f} fixed. We will then
consider the change in vector {f} due to changes in the sinksource term, holding matrix A constant. Finally, we
will address the case where both matrix A and vector {f} change simultaneously. Since we deal with systems of linear
equations, the solutions are additive when both A and {f} change.
3.1 Change in matrix A
To decompose the matrix we denote by {e
i
} the column vector whose i-th element is one and zeros otherwise and
let E
ij
¼ {e
i
}{e
j
}
T
denote the n by n matrix whose only non-zero element is at location i,j with value one. The
matrix A can now be decomposed into either column or row vectors,
A ¼ X
n i ¼ 1
X
n j ¼ 1
a
i ,
j
E
i ,
j
¼ X
n i ¼ 1
{e
i
} X
n j ¼ 1
a
i ,
j
{e
j
}
T
¼ X
n j ¼ 1
X
n i ¼ 1
a
i ,
j
{e
i
} {e
j
}
T
ð 10Þ
From the discussion in the previous section, one can see that when the K value is changed at one location in the grid,
say, location i,j, only a small number of entries in matrix A will be changed. Let A9 ¼ A þ DA be the changed
matrix and {h9} ¼ {h} þ {Dh} be the changed heads. From Noble and Daniel,
17
we know that if A is an n by n nonsingular matrix, C is an n by p matrix and R is an p by
n p , n matrix, then the disturbed matrix A9 ¼ A þ CR is also nonsingular and its inverse is given by
A9
¹ 1
¼ A
¹ 1
¹ A
¹ 1
CZ
¹ 1
RA
¹ 1
11 where Z ¼ I þ RA
¹ 1
C is also assumed to be a nonsingular matrix.
At first glance, eqn 11 appears to be unrelated to our problem. However, suppose that we have A{h} ¼ {f} and
A9{h9} ¼ {f}, and multiply eqn 11 by {f} on both sides, we find that the new solution {h9} can be obtained from
the old one {h} as
{h9} ¼ {h} ¹ A
¹ 1
CZ
¹ 1
R{h} 12
If p is much smaller than n, the size of Z will be much smaller than the size of matrix A, and computing inverse of
matrix Z will be much easier and faster than computing the inverse of the entire matrix A9. For example, if the value a
ij
in A is changed from a
ij
to a9
ij
¼ a
ij
þ d , then from eqn 10
the corresponding changes are DA ¼ dE
ij
, C ¼ {e
i
}, R ¼ d{e
j
}
T
, and Z reduces to a scalar, Z ¼ 1 þ d{e
j T
}{b}. Thus, the disturbed solution {h9} would be
{h9} ¼ {h} ¹ d
1 þ d{e
T j
}{b} {b}{e
T j
}{h} 13
where b is the solution of A{b} ¼ {e
i
}. In this example, to obtain the perturbed solution the
system of linear equations has to be solved twice, and thus there is no advantage of using the proposed method.
However, if the problem requires changing each entry in the matrix A, then multiple perturbed solutions are obtained
readily by updating eqn 13.
When the K value changes at a specified node in the grid, several entries in the matrix A will be affected.
Suppose that the K value at i,j is changed. After some algebraic manipulation, it can be shown that the perturbed
502 P. P. Wang, C. Zheng
matrix becomes A9 ¼ A þ DA where the perturbed portion DA is given by
Recall m ¼ j þ i ¹ 1n
x
. All the 13 non-zero entries are listed in the matrix. The specific changes can be calculated
by using eqn 7. The matrix DA can be written as the product of two smaller matrices DA ¼ CR where matrices
C and R are, respectively,
C ¼ e
m ¹ n
x
e
m ¹ 1
e
m
e
m þ 1
e
m þ n
x
15
R ¼ DA
m ¹ n
x
DA
m ¹ 1
DA
m
DA
m þ 1
DA
m þ n
x
2 6
6 6
6 6
6 6
6 4
3 7
7 7
7 7
7 7
7 5
16
with DA
i
defined as the i-th row of matrix DA. The matrix Z ¼
I þ RA
¹ 1
C appearing in eqn 11 has dimension of 5 by 5 with 13 non-zero entries located at
Z ¼ z
11
z
13
z
22
z
23
z
31
z
32
z
33
z
34
z
35
z
43
z
44
z
53
z
55
2 6
6 6
6 6
6 6
6 4
3 7
7 7
7 7
7 7
7 5
17
Inversion of Z is simple and straightforward. After the inverse of Z is updated, the inverse of matrix A and
the new solution {h9} can be obtained readily from eqns 11 and 12, respectively.
3.2 Change in vector {f}
