Geoscience Reference
In-Depth Information
results has a discrete character in reality, i. e. it corresponds finally to inte-
gers. The discreteness becomes apparent in an uncertainty of the process of
the instrument reading. Hence, real dispersion
s
(
x
) could not be diminished
infinitely, even if
N
[indeed the length value measured by the ruler
with the millimeter scale evidently can't be obtained with the accuracy 1
→∞
µ
m
even after a million measurements, although it does follow from (4.41)]. Re-
gretfully, not enough attention is granted to the question of influence of the
measurement discreteness on the result processing in the literature. The topic
by Otnes and Enochson (1978) could be mentioned as an exception. However,
this phenomenon is well known in practice of computer calculations where the
word length is finite too. It leads to an accumulation of computer uncertain-
ties of calculations, and special algorithms are to be used for diminishing this
influence even during the simplest calculation of the arithmetic mean value (!)
(Otnes and Enochson 1978). As per this brief analysis, the discreteness causes
the underestimation of the real uncertainties of the averaged values.
Consequence 5.
In addition to the considered averaging, the interpolation,
numerical differentiation, and integration are the often-met operations similar
to (4.36). Actually, they are all reduced to certain linear transformations of
value
y
i
and could be easily written in the matrix form (4.36). Thus, (4.38)
isasolutionoftheproblemofuncertaintyfindingduringtheoperationsof
interpolation, numerical differentiation, and integration of the results. Note
that in the general case the mentioned uncertainties will correlate even if the
initial observational uncertainties are independent.
Consequence 6.
Matrix
S
X
does not depend on vector
A
0
in (4.36). Assuming
=
A
0
AY
0
,where
Y
0
is the certain vector consisting of the constants, (4.38)
turns out valid not for the initial vector only but for any
Y
+
Y
0
vector, i. e.
thecovarianceerrormatrixofparametersvector
X
does not depend on the
addition of any constant to observation vector
Y
.
Consequence 7.
Consider nonlinear dependence
X
=
A
(
Y
). It could be re-
duced to the above-described linear relationship (4.36) using linearization, i. e.
expanding
A
(
Y
)intoTaylorseriesaroundaconcretevalueof
Y
and accounting
only for the linear terms as shown in the previous section. Then the elements
of matrix
A
will be partial derivatives
a
ki
=
∂
|∂
y
i
,allconstantterms
as per consequence 6 will not influence the uncertainty estimations and the
same formula as (4.38) will be obtained. For example, the uncertainties of the
surface albedo have been calculated in this way with the covariance matrix of
the irradiance uncertainties obtained at the second stage of the processing of
the sounding results in Sect. 3.3. The uncertainties of the retrieved parameters,
while solving the inverse problem in the case of the overcast sky have been
calculated in this way, as will be considered in Chap. 6. Note, that relation (4.38)
is
an approximate estimation
of the parameters of uncertainty in the nonlinear
case because for exact estimation all terms of Taylor series are to be accounted.
The accuracy of this estimation is higher if the observational uncertainties (i. e.
the matrix
S
X
elements are less).
Return to the inverse problem solution and to begin with again consider the
case of the linear relationship of observational results
Y
and desired parame-
ters
X
(4.9):
Y
(
A
(
Y
))
k
=
G
0
+
GX
. Let the observational errors obey the law of normal
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