Geoscience Reference
In-Depth Information
i. e.
S
Y
in (4.37) is obtained as an estimation of the covariance matrix using the
known formulas:
M
M
1
M
−1
1
M
(
y
(
m
i
−
y
i
)(
y
(
m
)
=
=
y
(
m
)
(
S
Y
)
ij
−
y
j
),
y
i
,
j
i
=
=
m
1
m
1
=
=
i
1,...,
N
,
j
1,...,
N
.
Then the analogous estimations are inferred for matrix
S
X
with (4.38). On
the one hand, if just random observational errors are implied, then all
M
measurements will relate to one real magnitude of the measured value. But
on the other hand the elements of matrix
S
Y
could be treated more widely,
as characteristics of variations of the vector
Y
components caused not by
the random errors only but by any changes of the measured value. In this
case, (4.38) is the estimation of the variations of parameters
X
by the known
variations of values
Y
Consequence 3.
Consider the simplest case of the relations similar to (4.36)
- the calculation of the mean value over all components of vector
Y
i. e.
x
=
N
i
=
1
y
i
(here
K
1
=
=
|
N
for
all numbers
i
and the following is derived from (4.38) for the SD of value
x
:
1, so value
X
is specified as a scalar). Then
a
ki
1
N
N
1
N
=
s
(
x
)
(
S
Y
)
ij
.
(4.40)
=
=
i
1
j
1
For the non-correlated observational errors in sum (4.40) only the diagonal
terms of the matrix remain and it transforms to the well-known
errors sum-
mation rule
:
N
1
N
=
s
(
x
)
(
S
Y
)
ii
.
(4.41)
=
i
1
SD of the mean value decreases with the increasing of the quantity of the av-
eraged values as
√
N
(for the equally accurate measurements
s
(
x
)
|
√
N
),
as per (4.41). As not only the uncertainties of the direct measurements could
be implied under
S
Y
, the properties of (4.40) and (4.41) are often used dur-
ing the interpretation of inverse problem solutions of atmospheric optics. For
example, after solving the inverse problem the passage from the optical char-
acteristics of thin layers to the optical characteristics of rather thick layers or
of the whole atmospheric column essentially diminishes the uncertainty of
the obtained results (Romanov et al. 1989). Note also that we have used the
relations similar to (4.41) in Sect. 2.1 while deriving the expressions for the
irradiances dispersion (2.17) in the Monte-Carlo method.
Consequence 4.
Analyzing (4.41) it is necessary to mention one other obsta-
cle. It is written for the real numbers, but any presentation of the observational
=
s
(
y
)
Search WWH ::
Custom Search