The determination of ionospheric key quantities such as the maximum electron
density of the F2 layer

One of the major tasks in ionospheric research activities concerns the
determination of physically relevant parameters from space geodetic
observations. Exploiting the knowledge about the physical processes improves
the description of the ionospheric behavior in time and space to monitor
ionosphere phenomena and perform space weather studies. One of the most
important parameters in this context is the electron density (

Various global (e.g.,

Complementary, VTEC over the oceans can be derived from radar altimetry (RA)
along the satellite orbits. On most of the current RA missions such as
Jason-2, SARAL, Cyosat and HY-2A, a DORIS (Doppler Orbitography and
Radiopositioning Integrated by Satellite) receiver that is active for orbit
determination purposes can also be considered to derive STEC from the signals
transmitted by DORIS ground beacons

However, empirical models such as NeQuick

In
order to obtain information of the

It stands to reason that, depending on the choice of key parameters, the
question of parameter interdependencies arises and will form the emphasis of
the investigations in this paper. Several studies dealing with the issue of
correlations between ionospheric parameters are purely based on ionosonde
measurements. To mention just a few of these,

The
basic principles of the modeling concept and parameter determination will
therefore be repeated at the beginning of Sect.

The description of the vertical

Model parameters are the maximum electron density

The model approach presented
here is in particular suited for regional applications since localizing
endpoint-interpolating polynomial B-splines

Thus, the target quantities to be determined are the B-spline series
coefficients

Taking the linearized model into account, the estimated set of coefficients
results from an iterative estimation procedure via

Concerning the parameter estimability,

In this paper, correlations between the B-spline series coefficients and also
F2 Chapman key parameters are studied in detail. A reliable separability of
the key parameters is certainly required for obtaining realistic values and
to ensure a safe convergency of the procedure. To discuss this question, the
correlations between key parameters shall be analyzed on the basis of the
correlation matrices

The covariance matrix of the unknown
series coefficients

A stable estimation of B-spline series coefficients applying the adjustment
model presented in Eq. (

Each
simulated profile contains synthetic

Correlation matrix of the coefficients

Correlation matrices of the key parameters

Key parameter correlations contained in

In the first row, Table

As already claimed, those correlations originating from B-splines (appearing
close to the diagonal of the

This output is based on a simulated scenario and purely shows the
correlations between model parameters that are indirectly obtained from IRI07
without input of F3/C measurements. However, the obtained results are in
agreement with other correlation studies for the F2 key parameters that took
ionosonde data into account: for instance, with

Furthermore,

Negative correlations between

In our model, approximate values for

In this simulation,
the preliminary model of Eq. (

According to the simulated scenario described in Sect.

The pre-processed F3/C electron density
profiles kindly provided by the Center of Space and Remote Sensing Research
(CSRSR,

the detection of larger jumps, here

the verification of data availability around the peak region within

the screening of

Based on the retrieval concept
to derive EDPs from ionospheric radio occultation measurements, the position
of each

Finally, a total of 123 F3/C profiles (52 % rejected) for 1 July 2008 and 96 F3/C profiles (38 % rejected) for 1 July 2012 remain above the target region in South America after the data screening. Most of the rejected profiles have been removed because of the high measurement noise, due to incompleteness or because they are affected by a dominant E layer. It thus becomes clear that a decreasing number of rejections can be expected if an E layer model is introduced.

The
distribution of all profiles for 1 July 2008, here independent of the
corresponding measurement epoch, is depicted in
Fig.

Spatial F3/C EDP distribution observed on 1 July 2008 over the study area in South America.

It should be noted that the payload of each F3/C satellite includes two
precise orbit determination (POD) and two occultation (OCC) antennas, all of which can be used for the retrieval of EDPs from radio
occultations, i.e., different profiles from several antenna may be acquired
for the same location and time. Hence, the number of profiles is larger than
the number of orange squares in Fig.

To derive a combined
graphical relation between the localization of the EDPs with their
corresponding measurement epoch, Fig.

F3/C EDP distribution related to spatiotemporal grid cells observed on 1 July 2008 in the South America region.

Again, as also depicted in Fig.

The total number of profiles
measured within each cell spans from 1 to 13 and shows that the B-spline
levels with respect to longitude and latitude have to be chosen rather low in order to
bridge those regions without data. This situation is also visible from
Fig.

To find an adequate level for the time, all profiles belonging to
a specific time segment in Fig.

Temporal data distribution: EDPs are counted within 2 h intervals for 1 July 2008 (light blue) and 1 July 2012 (dark blue).

For instance, there are only three profiles available between 16:00 and
18:00 UT in 2008 and just one between 02:00 and 04:00 UT in 2012. Based on
this distribution, an approximate time sampling of 2.5 h has been chosen,
and, according to Eq. (

In order to process sparse and inhomogeneous data, the adjustment system
introduced with Eq. (

Equation (

The resulting correlation matrices will
still be denoted

The studies on 1 July 2008 include a correlation analysis on the B-spline
series coefficient level (

The correlation matrix of the coefficients

Correlation matrix of the coefficients

It has previously been stated in Sect.

Correlation matrices of the key parameters

As can be seen from Fig. 7, noticeable blue/red-colored patches along the diagonal become
visible, identifying correlations between nearby key parameters. The width of these
patches depends on the resolution of the computed

The corresponding numerical information for

Key parameter correlations contained in

Total minimum and maximum correlations are close to

Discussions about the correlation matrix structure have been neglected in the
context of the simulation (Sect.

Upper triangular structure of the

The numbering of the matrix elements in

1-D representation of endpoint-interpolating polynomial B-splines of
level

With regard to the investigation area covering

A single spline function, e.g., the red-colored spline

Representation of a single B-spline with nonzero interval of

Under consideration that the two interior splines together already cover the
whole region, a negative correlation may appear between them and consequently
also between key parameters located in the interior region. This relation can
clearly be seen from the longitude correlations in Fig.

Additionally, two black bands related to

The appearance of internal correlations
between the variables of a specific key parameter resulting from the B-spline
modeling approach is expected and not surprising. The absence of
interparameter dependencies in this scenario can be explained by studying the
observation and parameter weighting resulting from the VCE. The normal
equations are given by Eq. (

To prove the general validity of the conclusions obtained for the 1 July 2008 scenario during quiet solar conditions, the same procedure has been conducted for 1 July 2012, a day with slightly higher solar activity.

All settings for the studies of 1 July 2012 are maintained as for 2008 –
the configuration for the data screening, B-spline levels and computation
sequence – as explained in Sects.

The results
are comparable to the outcomes of 2008 and have not manifested any
differences. Studies on the coefficient level are therefore neglected here
and only

Correlation matrix of the key parameters

In accordance with Table , a numerical expression of the
correlations is provided in Table . Again, the majority
of correlations are located within the

Key parameter correlations contained in

The investigations performed in this paper are provided as an extension of

The authors would like to thank Lung-Chih Tsai from CSRSR at the NCU, Taiwan, for providing the FORMOSAT-3/COSMIC electron density profiles. This work was carried out as part of the project “Multi-scale model of the ionosphere from the combination of modern space-geodetic satellite techniques”, which is funded by the German Research Foundation (DFG), Bonn, Germany. Topical Editor K. Hosokawa thanks J. F. Conte and M. M. Alizadeh for their help in evaluating this paper.