Formal estimation of the random component in global maps of total electron content
Random component of the total electron content (TEC) maps, produced by global navigation satellite system processing centres, was analysed. Helmert transform (HT) and two-dimension singular spectrum analysis (2dSSA) were used. Optimal parameters (in the sense calculation speed versus quality) of the...
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irk-123456789-1199522017-06-11T03:04:36Z Formal estimation of the random component in global maps of total electron content Choliy, V.Ya. Random component of the total electron content (TEC) maps, produced by global navigation satellite system processing centres, was analysed. Helmert transform (HT) and two-dimension singular spectrum analysis (2dSSA) were used. Optimal parameters (in the sense calculation speed versus quality) of the 2dSSA windows were determined along with precision estimations. 2016 Article Formal estimation of the random component in global maps of total electron content / V.Ya. Choliy // Advances in Astronomy and Space Physics. — 2016. — Т. 6., вип. 1. — С. 56-60. — Бібліогр.: 9 назв. — англ. 2227-1481 DOI:10.17721/2227-1481.6.56-60 http://dspace.nbuv.gov.ua/handle/123456789/119952 en Advances in Astronomy and Space Physics Головна астрономічна обсерваторія НАН України |
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Random component of the total electron content (TEC) maps, produced by global navigation satellite system processing centres, was analysed. Helmert transform (HT) and two-dimension singular spectrum analysis (2dSSA) were used. Optimal parameters (in the sense calculation speed versus quality) of the 2dSSA windows were determined along with precision estimations. |
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Choliy, V.Ya. |
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Choliy, V.Ya. Formal estimation of the random component in global maps of total electron content Advances in Astronomy and Space Physics |
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Choliy, V.Ya. |
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Choliy, V.Ya. |
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Formal estimation of the random component in global maps of total electron content |
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Formal estimation of the random component in global maps of total electron content |
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Formal estimation of the random component in global maps of total electron content |
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Formal estimation of the random component in global maps of total electron content |
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Formal estimation of the random component in global maps of total electron content |
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formal estimation of the random component in global maps of total electron content |
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Головна астрономічна обсерваторія НАН України |
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Formal estimation of the random component in global maps of total electron content / V.Ya. Choliy // Advances in Astronomy and Space Physics. — 2016. — Т. 6., вип. 1. — С. 56-60. — Бібліогр.: 9 назв. — англ. |
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Advances in Astronomy and Space Physics |
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AT choliyvya formalestimationoftherandomcomponentinglobalmapsoftotalelectroncontent |
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2025-07-08T16:58:51Z |
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| fulltext |
Formal estimation of the random component
in global maps of total electron content
V.Ya.Choliy∗
Advances in Astronomy and Space Physics, 6, 56-60 (2016) doi: 10.17721/2227-1481.6.56-60
© V.Ya.Choliy, 2016
Taras Shevchenko National University of Kyiv, Glushkova ave., 4, 03127, Kyiv, Ukraine
Random component of the total electron content (TEC) maps, produced by global navigation satellite system
processing centres, was analysed. Helmert transform (HT) and two-dimension singular spectrum analysis (2dSSA)
were used. Optimal parameters (in the sense calculation speed versus quality) of the 2dSSA windows were deter-
mined along with precision estimations.
Key words: ionosphere modelling and forecasting
introduction
Helmert transform (HT) and singular spectrum
analysis (SSA) in one-dimension variant were used by
author and his colleagues in their works [1, 2, 3, 4, 5].
This article closes the series. Beside the TEC
map analysis the article contains the comparison of
Helmert transform (HT) and two-dimensional SSA
(2dSSA). It should be noted that latter of the meth-
ods needs much more calculations than the �rst one.
That is why determination of the optimal values of
2dSSA window parameters as of independent signif-
icance.
All �les containing TEC maps in IONEX format
have been taken from NASA server1 for six successive
solstices and equinoxes: 22/09/2014, 22/12/2014,
21/03/2015, 20/06/2015, 22/09/2015, 22/12/2015.
Very short information about the data sources is
given in Table 1. Di�erent �les contain di�erent
amount of maps calculated for di�erent time mo-
ments inside the day. The only maps equally dis-
tributed during the day with two hour interval were
used. It means 13 maps per day per �le. All the
maps have identical dimensions: 71 rows (latitude
step is 2.5◦) and 72 columns (longitude step is 5◦).
The second column of Table 1 contains six sym-
bols. Each of them is to be used for the single date
from the just speci�ed date list. Their meanings are:
(+) the maps of the �le were used for comparison,
(−) the maps were rejected during steps of compari-
son, (0) there was no �le for the date. Each data
processing centre uses di�erent methods to build
TEC map: di�erent amount of satellites, stations,
smoothing, combination, forecasting or just present
raw data. None of those di�erences were taken into
account during the analysis. We just used the maps
as they were provided. That is why in Table 1 we
have presented the data from �le headers only with
minimum comments. It should be taken in mind that
combined solutions Igrg and Igsg for di�erent dates
were created with di�erent composing source maps.
Table 1: Main properties of the TEC maps.
Name Key O�ce Comment
C1pg ++++++ AIUB 1 day forecast
C2pg ++++++ 2 days forecast
Codg ++++++
Corg ++++++ raw
E1pg � � � � � 0 ESA/ESOC 1 day forecast
E2pg 0 � � � � 0 1 day forecast (?)
Ehrg ++++++
Emrg 000 � � + NRCan/CGS
Esag ++++++ ESA/ESOC
Esrg ++++++
Igrg ++++++ GRL/UWM combined
Igsg ++++++ combined
Jplg ++++++ JPL-GNISD
Jprg ++++++
U2pg � � � � � 0 gAGE/UPS 2 day forecast
Uhrg ++++++
Upcg ++++++ UPC-IonSAT raw
Uprg ++++++ raw
Uqrg ++++++
All maps for the single day were analysed sepa-
rately. Each element of the map represents the point
on the surface with the radius-vector is TEC value:
r = TEC(cosλ cosφ, sinλ cosφ, sinφ),
and λ, φ are its coordinates (longitude and latitude).
No pre-selection criteria were applied to the map
list before analysis.
∗Choliy.Vasyl@gmail.com
1ftp://cddis.gsfc.nasa.gov/pub/gps/products/ionex/
56
Advances in Astronomy and Space Physics V.Ya.Choliy
Table 2: Random error estimation from HT (in 0.1 TECU).
name 22/09/2014 22/12/2014 21/03/2015 20/06/2015 22/09/2015 22/12/2015
C1pg 4.40 1.50 7.41 2.64 2.88 1.25 2.11 0.42 1.73 0.81 3.23 1.48
C2pg 4.10 0.70 7.43 2.12 3.13 0.92 2.12 0.41 1.90 0.20 3.08 0.66
Codg 1.86 0.69 3.15 1.37 2.10 0.81 0.65 0.09 0.77 0.27 0.89 0.29
Corg 1.41 0.38 4.32 1.18 1.77 0.56 0.78 0.16 0.79 0.27 1.59 0.56
Ehrg 1.46 0.78 2.61 1.27 1.69 0.72 1.05 0.31 0.66 0.23 1.24 0.63
Emrg � � � � � � � � � � 3.83 1.09
Esag 1.91 0.54 3.64 1.33 1.94 0.80 0.96 0.26 0.82 0.18 1.53 0.51
Esrg 2.07 0.71 3.64 1.28 2.23 0.65 1.10 0.27 0.96 0.23 1.72 0.48
Igrg 0.53 0.16 0.43 0.15 0.79 0.34 0.19 0.03 0.26 0.04 0.28 0.10
Igsg 0.85 0.38 1.13 0.43 0.96 0.42 0.35 0.07 0.35 0.16 0.54 0.21
Jplg 1.54 0.52 2.69 0.83 1.74 0.43 0.56 0.09 0.76 0.29 1.08 0.24
Jprg 1.54 0.53 2.49 0.77 1.91 0.71 0.62 0.11 0.78 0.30 1.02 0.25
Uhrg 3.34 1.16 4.88 1.79 3.05 0.82 0.90 0.11 1.36 0.20 1.91 0.63
Upcg 1.79 1.05 2.52 1.70 1.51 0.91 0.80 0.23 0.63 0.35 0.84 0.58
Uprg 1.79 1.05 2.52 1.70 1.51 0.91 0.80 0.23 0.63 0.35 0.84 0.58
Uqrg 3.34 1.16 4.88 1.79 3.05 0.82 0.90 0.11 1.36 0.20 1.91 0.63
Fig. 1: First step of TEC map analysis for 20/06/2015. Fig. 2: Next step of TEC map analysis for 20/06/2015.
helmert transform
However for HT such a preliminary analysis is
necessary, but however it may be done with the help
of HT itself. Let us show it with three steps of the
analysis for 20/06/2015, where the analysis is the
most illustrative. All �gures (Figs. 1�3) have TEC
(in units of 0.1 TECU) along Y axis, but in di�erent
scales. Abscissas are the map sequential number, or,
in other words, the time from 0hUTC of the date in
two-hour intervals. Di�erent lines on the same �gure
are for di�erent �les, say, di�erent processing centres.
Only some of the maps were marked, as these �gures
(Figs.1�3) outlines the pre-selection process and the
�gures are only for illustration.
Graph in Fig. 1 is essential for the case of patchy
data. At this step the most shifted maps (E1pg,
E2pg) and the map U2pg (declared by its authors as
a preliminary one) were removed from the analysis.
It gave much better but still not quite good picture
with the only shifted map (Fig. 2). It is Emrg map.
Fig. 3 was left for further analysis despite of two
clearly distinct graphs for C2pg and C1pg (two upper-
most lines from top). These are AUIB forecasts for
two and one days. They demonstrate a little worse
precision as compared with another ones, but still are
quite good for forecasts. Another two (bottommost)
are combines map Igrg and Igsg. All other maps'
precision lies around and lower than ∼ 0.1TECU.
This situation is repeatable for another dates
where some of the maps must be removed due to
their shifts and uncompensated errors, not analysed
here.
Comparison results in TECU units are given in
Table 2. Each cell contains daily averaged estima-
tion of the random precision of the TEC map and
its standard deviation.
57
Advances in Astronomy and Space Physics V.Ya.Choliy
Fig. 3: Final step of TEC map analysis for 20/06/2015.
two dimension SSA
There are some speci�c features of 2dSSA. Let
us shortly outline them here. The algorithm of one
dimension SSA clearly explained in original works
[6, 7], was used by author for the analysis and fore-
casting time series of pole coordinates. 2dSSA and
principal component analysis are widely used in the
analysis of photo and video, e. g. [8]. Applying
2dSSA to the precision analysis of ionospheric maps,
most likely, is attempted here for the �rst time.
At this stage, the TEC map is black and white
image, where brightness of its pixels is de�ned by
TEC value. In general the SSA algorithm does not
undergo any changes and consists of the steps:
� building of the trajectory matrix X;
� singular value decomposition of the matrix S =
X ·XT , say, in the sum of the principal compo-
nents;
� grouping of the eigenvalues and eigenvectors of
the matrix S to subdivide principal components
into deterministic and random ones;
� restoration of the random (noisy) principal
components to determine their standard devia-
tion;
� restoration of the deterministic part of the map
for its analysis (for example, for forecasting).
There are some features of the 2dSSA making dif-
ference in some of its steps from 1d variant. In 1d
the SSA window is applying to the time series and
the only window parameter is its length L. In 2d the
window is rectangular and has width w and height
h. The image itself has size W ×H. The SSA win-
dow is scanning the image from upper left to bottom
right corners like the beam in television tube and in
every of its position it generates new column of the
trajectory matrix:
⌜I1,1 I1,2 I1,3
⌝ I1,4 . . .
I2,1 I2,2 I2,3 I2,4 . . .
⌞I3,1 I3,2 I3,3 ⌟ I3,4 . . .
I4,1 I4,2 I4,3 I4,4 . . .
. . . . . . . . . . . . . . .
→
→
I1,1
⌜I1,2 I1,3 I1,4
⌝ . . .
I2,1 I2,2 I2,3 I2,4 . . .
I3,1 ⌞I3,2 I3,3 I3,4 ⌟ . . .
I4,1 I4,2 I4,3 I4,4 . . .
. . . . . . . . . . . . . . .
→ · · · →
→
. . . ⌜I1,w−3 I1,w−2 I1,w−1
⌝ I1,w
. . . I2,w−3 I2,w−2 I2,w−1 I2,w
. . . ⌞I3,w−3 I3,w−2 I3,w−1 ⌟ I3,w
. . . I4,w−3 I4,w−2 I4,w−1 I4,w
. . . . . . . . . . . . . . .
→ · · · →
→
. . . . . . . . . . . . . . .
. . . Ih−3,w−3 Ih−3,w−2 Ih−3,w−1 Ih−3,w
. . . Ih−2,w−3
⌜Ih−2,w−2 Ih−2,w−1 Ih−2,w
⌝
. . . Ih−1,w−3 Ih−1,w−2 Ih−1,w−1 Ih−1,w
. . . Ih,w−3 ⌞Ih,w−2 Ih,w−1 Ih,w ⌟
,
which leads to the trajectory matrix like this:
X =
I1,1 I1,2 . . . I1,w−2 . . . Ih−2,w−2
I1,2 I1,3 . . . I1,w−1 . . . Ih−2,w−1
I1,3 I1,4 . . . I1,w . . . Ih−2,w
I2,1 I2,2 . . . I2,w−2 . . . Ih−1,w−2
I2,2 I2,3 . . . I2,w−1 . . . Ih−1,w−1
I2,3 I2,4 . . . I2,w . . . Ih−1,w
I3,1 I3,2 . . . I3,w−2 . . . Ih,w−2
I3,2 I3,3 . . . I3,w−1 . . . Ih,w−1
I3,3 I3,4 . . . I3,w . . . Ih,w
. (1)
Thereby the trajectory matrix owns the block-
hankel structure. It has (H − h + 1) × h blocks of
(W −w+1)×w size. It should be noted when build-
ing the transform:
(image) ↔ (trajectory matrix).
Let us index the block with uppercase letters:
I = 0, . . . , H − h, J = 0, . . . , h− 1,
and pixels inside the block with the lowercase ones:
i = 0, . . . ,W − w, j = 0, . . . , w − 1.
Now we can explain block-hankel structure via block
and inner block indices. Any point within the im-
age with the coordinates (R,C), where R is a row
58
Advances in Astronomy and Space Physics V.Ya.Choliy
number (from top to bottom), C is a column num-
ber (from left to right), will participate in the blocks
of the trajectory matrix where:
I + J = R,
moreover, in the upper leftmost point of any block
there is the point with C = 0, in the upper rightmost
the one with C = W −w, and in lower rightmost the
point with C = W − 1. It is easily recognizable
from (1). Using that information one can select the
starting point for hankelization and then follow the
anti-diagonal, skip the unnecessary blocks, do the
averaging.
In short words the 2dSSA hankelization is the av-
eraging of the values for:
� any blocks where I + J = R,
� along anti-diagonal, starting in every appropri-
ate blocks at the point:
0 ≤ C < w − 1 : (C, 0),
w ≤ C < W : (W − w,W − 1− C).
Transformation from (R,C) to (i, j) + (I, J) is
used on the �rst step of 2dSSA, but backward al-
gorithm � on the two last steps, during principal
component restoration. We left implementation is-
sues beside the article scope.
Any other steps of the analysis does not diverge
from the 1d variant.
map analysis with 2dSSA
To determine the applicability of 2dSSA algo-
rithms to the postulated task we �rst run some nu-
merical experiment. For this purpose the maps from
22/09/2014 were analysed by 2dSSA with di�erent
windows, starting from 6× 5 window (6 longitudinal
points by 5 latitudinal ones), and ending with the
36× 35 window. The size of the window varied with
step 6 for longitude and 5 for latitude. The smallest
window led to eigenvalue spectrum of 30 points, the
greatest one (it is nearly the quarter of the whole
map) to the spectrum of 1260 values.
Typical overview of the eigenvalue spectra (dec-
imal logarithm of the eigenvalue against its sequen-
tial number after descending sort) is shown in Fig. 4.
Three regions at the spectra are clearly recognizable.
Each of them can be approximated with linear func-
tion of di�erent slopes. In our opinion, there can be
an interpretation of these region. Region 3 of the
spectra is mostly due to computing rounding errors.
Whole bunch of region 3 principal components can
explain a very tiny (10−6 ∼ 10−8) portion of the
image standard deviation. Region 1 is for the main
(deterministic) part of the map. Then the region 2
is for random portion of the TEC map.
Some additional arguments for this point of view
can be found in Fig. 5, where the dependence of the
total standard deviation of the �rstm principal com-
ponents for di�erent 6 × n windows are presented.
There was no sense to present lines for every win-
dow, as the �gure become too crowded. We gave
only some of them. For another windows the general
view of the graph is quite similar. For low n the map
deviations was not accounted in full. Greater n bet-
ter represents the standard deviation of the map and
asymptotically n → w × h it tends to the standard
deviation of the whole map. Position of the break-
ing point on the graph in Fig. 5 is like the A point in
Fig. 4.
That is why we interpret the A point as a border
between the deterministic and random parts of the
TEC map. Doing precision analysis one can limit
itself to the region 2, as with the region 3 principal
components only very little part of the total stan-
dard deviation may be explained. It was proven by
direct calculations.
Additional analysis with linear approximations of
the spectra and counting of the principal components
led us to the conclusion that in the region 1 there is
a small number of the �rst principal components.
Sometimes it is the only one of them. Empirical rule
for amount of the principal components in region 1 is
given in Table 3. There are window sizes along verti-
cal and horizontal table limits. There are data lacks
in the Table as it was too much time consumed to
calculate them. In Table 4 there are standard devia-
tions of the region 2 principal components for Igrg
map calculated according to Table 3 recommenda-
tions. Despite the variability, the results are similar.
Total standard deviations of the region 2 princi-
pal components for all TECmaps of 20/06/2016 were
calculated. After that the random error estimations
as a part of the standard deviation in its full value
were calculated and presented in Table 5. We used
the greatest possible window with the smallest pos-
sible amount of principal components in region 1 �
namely one. This is 10×12 window. The coincidence
of these values (given in Table 5) with the data in
Table 2 is poor. The correlation between them never
exceeds 0.5. It means that 2dSSA can be used for the
task, but the results stay uncertain. The only item
we should point on now is that 2dSSA estimations
are much more smooth which is very essential for the
results based on the same raw data and nearly the
same processing strategy.
The �classical� variant of the subdivision of prin-
cipal components by the analysis of correlation func-
tions shows consistency with the proposed method-
ology.
conclusion
Singular spectrum analysis method may be used
with some success for determination of the formal
precision of the TEC maps if left behind the scene
the fact of the calculation time. In real applications
59
Advances in Astronomy and Space Physics V.Ya.Choliy
Table 3: Recommended number of region 1 principal
components.
6 12 18 24 30 36
5 1 1 2 2 2 2
10 1 1 2 2 2 2
15 1 2 3 2 2 3
20 2 2 3 3 3 -
25 2 3 4 - - -
30 3 4 4 - - -
35 3 4 - - - -
Table 4: Standard deviations estimated for region 2 for
20/06/2016, Igrg map.
6 12 18 24 30 36
5 0.1155 0.4580 0.1136 0.2120 0.3458 0.3596
10 0.3366 0.5797 0.2650 0.3265 0.4167 0.4550
15 0.5772 0.4523 0.1919 0.6244 0.6704 0.6976
20 0.2665 0.5321 0.2611 0.3374 0.4175 -
25 0.5679 0.4650 0.6160 - - -
30 0.3013 0.4091 0.5564 - - -
35 0.3044 0.3780 - - - -
Table 5: Random error for TEC map from 2dSSA (in 0.1 TECU).
Name C1pg C2pg Codg Corg Ehrg Esag Esrg Igrg Igsg Jplg Jprg Uhrg Upcg Uprg Uqrg
precision 0.71 0.55 0.86 0.65 0.67 0.62 0.59 0.58 0.78 0.72 0.66 0.78 0.62 0.62 0.78
Fig. 4: General view of the eigenvalue spectra for di�er-
ent assorted windows.
Fig. 5: Standard deviation via principal component
amount.
the necessary time is huge as compared with HT. By
the way, if one has the only �eld to be analysed, SSA
may be the only possibility, since other methods need
intercomparison of the data sets (in the case of HT
there are three sets necessary).
Comparison of our results with [9] shows that our
estimations are lesser than ones from [9]. Unfortu-
nately we cannot estimate the level of random errors
introduced by processing methods, but they may be
di�erent and in�uence the estimations.
There is another di�erence in HT and 2dSSA,
which may cause the di�erence in the results. HT
builds the systematic di�erence model according to
some prede�ned formulae. It is a�ne transform in-
cluding rotation, shift and possible inclination. In
2dSSA the systematic di�erence is build from the
scratch for each of the data sets and may be di�erent
for di�erent maps. It needs additional investigation.
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