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)
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