Digital Signal Processing Reference
In-Depth Information
6.1.4
1D-Var (Variational Method)
Another approach to obtain optimal estimation of the water vapor, temperature and
pressure is through the variational methods. These methods combine the occultation
measurements (e.g., refractivity) with the a-priori (or background) atmospheric
information in a statistically optimal way (e.g., Zou et al.
1995
; Healy and Eyre
2000
; Kursinski et al.
2000a
). For example, the optimal solution to the state vectors
x
(e.g.,
T
,
P
w
and
P
) can be found by adjusting the state vector elements in a way that is
consistent with the estimated background errors, to produce simulated measurement
values that fit the observations to within their expected observational errors (Healy
and Eyre
2000
). When assuming Gaussian error distribution, the optimal solution of
the state vectors can be achieved by minimizing a cost function
J
(
x
) given by,
2
x
x
b
T
B
1
x
x
b
C
1
1
2
.y
o
y.x//
T
.O
C
F /
1
.y
o
J.x/
D
y.x//
(6.9)
where
x
is the state vectors (e.g.,
T
,
P
w
and
P
),
y
is the measurement vector (e.g.,
RO refractivity
N
),
y
(
x
) is the forward model to map the state vectors into the
measurement space (e.g., Eq. (
6.1
) for refractivity), the superscript
b
and
o
denote
the background (or a-priori) and measurement information,
O
and
F
represents
the error covariance of the measurement and the forward model, respectively. The
superscripts
T
and
1
denote matrix transpose and inverse. Similarly bending angle
˛ can be the measurement vectors instead of refractivity, however, the forward
model becomes much more complicated and computation cost also becomes much
higher. Nerveless, the measurement errors covariance of bending is relatively
simpler as compared with refractivity, as the RO refractivity generally includes
vertically correlated errors as a result of the Abel integration.
The variational method demonstrates that the retrieval results are less sensitive to
the errors in the a-priori information than the simpler direct retrieval method (e.g.,
Eqs.
6.6
,
6.7
and
6.8
). However, the errors of the geophysical parameters derived
from the variational method could become more challenging to interpret as the
errors will consist of the model background errors (generally less understood) with
the RO measurement errors.
6.2
Characteristics of GNSS RO Observations
The limb sounding geometry of the GNSS RO technique leads to high vertical
resolution measurements but with relatively coarse horizontal resolution and is a
highly complementary to the conventional nadir-viewing infrared and microwave
satellite sounders. The RO L-band microwave signals are not sensitive to aerosol,
cloud and precipitation. Such all-weather sounding capability makes the high
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