Geoscience Reference
In-Depth Information
If the errors
ε
P
and
ε
Q
are uncorrelated, then the error covariance
σ
PQ
=0.
From (9-48) we have generally
σ
PQ
=
M
∆
g
P
−
α
Qk
∆
g
k
=
M
∆
g
P
∆
g
Q
− α
Pi
∆
g
Q
∆
g
i
− α
Qk
∆
g
P
∆
g
k
+
α
Pi
α
Pk
∆
g
i
∆
g
k
(9-56)
α
Pi
∆
g
i
∆
g
Q
−
and finally
σ
PQ
=
C
PQ
− α
Pi
C
Qi
− α
Qi
C
Pi
+
α
Pi
α
Qk
C
ik
.
(9-57)
The notations are self-explanatory; for instance,
C
PQ
=
C
(
PQ
).
The error covariance function
The values of the error covariance
σ
PQ
, for different positions of the points
P
and
Q
, form a continuous function of the coordinates of
P
and
Q
.This
function is called the
error covariance function
,orbriefly,the
error function
,
and is denoted by
σ
(
x
P
,y
P
,x
Q
,y
Q
). If
P
and
Q
are different, then we simply
have
σ
(
x
P
,y
P
,x
Q
,y
Q
)=
σ
PQ
;
(9-58)
if
P
and
Q
coincide, then (9-57) reduces to (9-53) so that
σ
(
x
P
,y
P
,x
P
,y
P
)=
m
2
(9-59)
P
is the square of the standard prediction error at
P
.
Thus the error covariances
σ
PQ
may be considered as special values of
the error covariance function, just as the covariances
C
PQ
of the gravity
anomalies may be considered as special values of the covariance function
C
(
s
). To repeat, the error function is the covariance function of the prediction
errors, defined as
,
(9-60)
whereas
C
(
s
) is the covariance function of the gravity anomalies, defined as
M
{
ε
P
ε
Q
}
M
{
∆
g
P
∆
g
Q
}
.
(9-61)
The term “covariance function” in the narrower sense will be reserved for
C
(
s
) - in contrast to least-squares adjustment, where “covariances” auto-
matically mean error covariances. Covariances are “isotropic”, which means
independent of directions; the error covariances are nonisotropic.
From (9-53) and (9-57)
the error function can be expressed in terms of
the covariance function
; we may write more explicitly
σ
(
x
P
,y
P
,x
Q
,y
Q
)=
C
(
PQ
)
−
α
Pi
C
(
Qi
)
−
α
Qi
C
(
Pi
)+
α
Pi
α
Qk
C
(
ik
)
.
(9-62)