Geology Reference
In-Depth Information
2
2
;
EZx h m
{[
(
)
]
EZx m
[
( )
]
2
EZ x
[
(
.
h
)
m
]
EZ x
[
(
h
)
m
]}
0
;
2(0) 2() 0
C
C h
;
C
(0)
$
C h
()
Hence:
$ $ ; $
CCh
(0)
( )
C Ch
(0)
( )
C
(0)
Question 2.2
Let
Z
(
x
) be a stationary random function. Show that the variogram that
2
is equal to
()
h
1
2
[ (
Z x
h
)
Z x
()]
has the following properties:
() ()0
(0)
0
h
h
Answer
(0)
1
2
EZ x
[ ()
Z x
()] 0
(i)
(
h
) is the expectation of a square, hence it has to be positive.
2
()
h
1
2
[(
Z x
h
) ( ]
Z x
Let
y
=
x
-
h
(, )
xh
(, )
yh
, because it is intrinsic
2
() (,
h
yh
)
1
2
EZy
[() (
Zy h
]
()
h
;
() ()
h
h
Question 2.3
Let
Z
(
x
) be a stationary random function. Show that
()
Answer
hC
(0)
Ch
()
2
1
2
EZx h m
{[
(
)
]
[
Zx m
( )
]}
(
h
)=
2
2
1
2
{[ (
EZ x
h
)
m
]
EZ x
[ ()
m
]
=
2[(
EZx h
(
)
mZx
)(
( )
m
)]}
1
2
[2
C
(0)
2
C h
( )]
=
(
h
)= (0)
CCh
( )
Question 2.4
Let
Z
*(
x
) be a stationary random function.
n
)
*
where
i
are constants. Show that
Zx
()
Zx
( )
i
i
i
1
))
*
VZ x
(
( ))
Cov (
Zx Zx
(
),
(
))
i
j
i
j
i
j
