Absolute Ionosphere Slant Delays From Ambiguous Carrier Phase Data

Dru A. Smith, *National Geodetic Survey, NOS/NOAA*

# Biography 

Dr. Dru Smith is a research geodesist with NOAA's National Geodetic
Survey (NGS). He received his Ph.D. in geodesy from Ohio State before
joining NGS in 1995. He was the principle investigator for geoid
modeling from 1996 until 2001, at which time he went to work for the
Executive Secretariat for the Interagency GPS Executive Board, working
with GPS policy makers. In 2002 he returned to NGS and has been working
on a method for using the Continuously Operating Reference Station
(CORS) network to model total electron content in the ionosphere. He is
a member of the American Geophysical Union, the International
Association of Geodesy, the Institute of Navigation and serves on the
Board of Directors for the American Association for Geodetic Surveying.

# Abstract 

A new method for computing absolute (unambiguous) levels of Total
Electron Content (TEC) in the ionosphere and subsequently the L1 and L2
phase advances of GPS is presented, dubbed "ICON" (for its primary
purpose of modeling the Ionosphere over CONus). Unlike previous
computational methods, this method relies solely upon dual-frequency,
ambiguous carrier phase data with no reliance on pseudo-range, a-priori
values or other external information. The only requirements for this
method are that mapping functions between sufficiently close slant-views
of the ionosphere are available, and that the GPS data come from a
network of ground stations, geographically separated so as to allow
satellites to be viewed by a variety of stations at overlapping times.
This method can be applied either through independent least squares
adjustments (such as one day of data at a time), or may be applied in an
on-the-fly mode, where new data are adjusted into pre-adjusted values
epoch-by-epoch.

Sensitivity analyses are presented, showing both the advantages and
current limitations of this method. Additionally, comparisons between
ICON and the IGTEC and MAGIC models of the ionosphere are presented and
discussed. Plans to improve the method, as well as its application
toward forecasting of the ionosphere are also included.

# Introduction 

Dual-frequency carrier phase data is currently the most accurate data
for computing precise positions with the Global Positioning System.
However, both integer ambiguities and the phase advance (caused by Total
Electron Content, TEC) along the receiver-satellite path are frequency
dependent. These two unknowns are generally inseparable without either
lengthy observation sessions or else significantly more accurate
pseudo-ranges. As such, it stands to reason that if one were given an
independent estimate of the TEC (and thus the phase advance), that a
faster determination of integer ambiguities (and thus centimeter level
positioning) would be possible \[1,2\]. With this idea in mind, the
National Oceanic and Atmospheric Administration (NOAA), through their
National Geodetic Survey (NGS) embarked upon research to independently
model the ionosphere over the Conterminous USA (CONUS) region.

Many methods already exist for getting around the "ionosphere problem",
from either removing it mathematically, fitting carrier phase data to
pseudo-range data, or computing both ambiguities and ionosphere delay
using lengthy observing sessions. Additionally, in many existing studies
of the GPS related ionosphere (such as \[1\], \[2\]), the presumption is
that data, including the ionosphere, are processed in double difference
mode.

However, in this paper, an entirely new method for unambiguously
computing the absolute level of TEC, from a network of ground stations
and only using ambiguous carrier phase data is being presented. For the
sake of simplicity, this method will be referred to as the "ICON" method
(named for its primary purpose of modeling the [I]{.underline}onosphere
over [CON]{.underline}US). The method presented is fast, and can be kept
updated over time using a sequential least squares adjustment as the
data change (as new satellites rise and old ones set). ICON was first
outlined in \[3\], however the method was in its infancy at that point,
and this paper attempts to clarify and expand upon the both the method's
advantages and limitations.

**ASSUMPTIONS AND DEFINITIONS**

Certain mathematical and physical assumptions are required for ICON to
work. The **first assumption**, and most fundamental is that a network
of GPS receivers exist, geographically spaced so that they view various
satellites from various geographic locations throughout the day. This
has the effect of "sampling" the ionosphere at various locations at a
variety of angles every epoch. (Station spacing is left purposefully
vague, as a variety of spacings would suffice for this method to work).

The **second assumption** as that each receiver receives both the L1 and
L2 carriers for extended periods of time (again, being vague; as long as
loss of lock and cycle slips can be kept to a minimum, ICON will work).
The successive data over time (without loss of lock) for any given
receiver/satellite combination will be referred to as a "track".

The **third assumption** is that the phase advances (due to the
scintillating effects of the ionosphere), in cycles, on L1 and L2
(called I1 and I2, respectively) can be directly related to the Total
Electron Content along the receiver-satellite vector (called TECS) at
any given epoch by the following equations:

$I_{1} = - \frac{\kappa}{f_{1}c}\text{TECS} = (0\text{.}\text{853})\text{TECS}$
(1)

> $I_{2} = - \frac{\kappa}{f_{2}c}\text{TECS} = (1\text{.}\text{095})\text{TECS}$
> (2)

where:

> I~1~,I~2~ = cycles of phase advance in L1 and L2 respectively
>
> c = speed of light \[299,792,458 m / s\]
>
> f~1~, f~2~ = frequency of L1 and L2 respectively (1,575,420,000 and
> 1,227,600,000 cyc / s)
>
> TECS = density of electrons in receiver-satellite direction in TECU
> (10^16^ elec / m^2^)

κ = 40.3 x 10^16^ (m cyc^2^ / TECU / s^2^)

Equations 1 and 2 can be found in \[4\], equation 6.58. The assumptions
which led to these equations are more fully discussed in \[5\]. For the
purposes of this paper it is assumed that the approximations used to
derive these equations are far smaller in magnitude (generally) than
other approximations of ICON.

The **fourth assumption** of ICON is that any two TECS values
"sufficiently close" (defined later) to one another (in space and time)
may be accurately related to one another through some mapping function.
That is, for two different tracks, with TECS samples close in space and
within δi epochs in time, one may write:

$\begin{array}{r}
\text{TECS}(\text{track 1,epoch i}) = \\
f(\text{TECS}(\text{track 2,epoch }(i \pm \delta i)))
\end{array}$ (3)

The actual mapping function used to validate ICON will be discussed in a
later section. It should be pointed out though that any reliable mapping
function could be used in equation 3.

Further assumptions will be discussed as they come up, but these four
are generally all that is required for ICON to work.

**CONNECTING RINEX DATA TO TECS**

At any epoch "i", the relationship between the geometric range (^i^d)
and the measured range (^i^r) from a GPS receiver to satellite can be
expressed (for L1 and L2 respectively) as:

$r_{1} = d + c\Delta t + \frac{I_{1}}{\lambda_{1}} + T + m_{1}$ (4)

$r_{2} = d + c\Delta t + \frac{I_{2}}{\lambda_{2}} + T + m_{2}$ (5)

where

> r~1~, r~2~ = Measured range to satellite on L1 and L2 carriers
> respectively
>
> d = Geometric range to satellite
>
> c∆t = Range error due to clock errors
>
> λ~1~, λ~2~ = Wavelengths of L1 and L2
>
> T = Range error due to troposphere
>
> m~1~, m~2~ = Range error due to carrier phase multipath error on L2
> and L2

Now, under our initial assumption, each GPS receiver will be tracking
carrier phase data on L1 and L2. At acquisition of L1 and L2, it will
compare its internal oscillator with the number of L1 and L2 cycles
received and report a more or less nonsensical (i.e. "ambiguous") number
of cycles (called ^i^φ~1~ and ^i^φ~2~ respectively, for epoch i; these
are the values usually found as the L1 and L2 observables in a RINEX
file). Presuming the receiver does not lose lock, it will continue to
generate an internal L1and L2 cycle count, and compare the changes in
cycles of its internal count to those cycles received from the GPS
satellite. The difference between received and internal cycles can be
interpreted as the change in measured range (r) due to satellite motion.
Equivalently it could be considered the change in geometric range (d)
plus the change to all error sources. Thus between two epochs i and j:

$\begin{array}{r}
{\Delta\phi}_{1}\lambda_{1} = {\Delta r}_{1} = \\
\Delta d + c\text{ΔΔ}t + \frac{{\Delta I}_{1}}{\lambda_{1}} + \Delta T + {\Delta m}_{1}
\end{array}$ (6)

$\begin{array}{r}
{\Delta\phi}_{2}\lambda_{2} = {\Delta r}_{2} = \\
\Delta d + c\text{ΔΔ}t + \frac{{\Delta I}_{2}}{\lambda_{2}} + \Delta T + {\Delta m}_{2}
\end{array}$ (7)

where ^i,j^∆φ~1~ and ^i,j^∆φ~2~ are the differences, in cycles, between
the number of L1 or L2 cycles generated inside the receiver with the
number of cycles received from the GPS satellite, over the time period
from epoch i to epoch j. (This is just the difference between any two
RINEX values of L1 or L2 between two epochs). Now, if one subtracts
equation 7 from equation 6 then only frequency dependent terms will
remain:

$\begin{array}{r}
{\Delta\phi}_{1}\lambda_{1} - {\Delta\phi}_{2}\lambda_{2} = \\
(\frac{{\Delta I}_{1}}{\lambda_{1}} - \frac{{\Delta I}_{2}}{\lambda_{2}}) + \text{ΔΔ}m_{1,2}
\end{array}$ (8)

Note that by arriving at equation 8, the geometric range (d) has been
removed, and thus so have the unknown integer ambiguities. Unfortunately
(for the time being) it also removes the absolute TECS value as well,
leaving only TECS gradients.

The assumption will now be made that m~1~ and m~2~ are small, and also
that the difference between m~1~ and m~2~ is small enough to be
neglected \[6,7\]. Then, applying equations 1 and 2 to equation 8 and
solving for the TECS term we end up with the following:

$\begin{array}{r}
\Delta\text{TECS} = \text{TECS} - \text{TECS} = \\
\left( \frac{1}{\kappa} \right)\left( \frac{1}{f_{1}^{2}} - \frac{1}{f_{2}^{2}} \right)^{- 1}\left( {\Delta\phi}_{1} - {\Delta\phi}_{2} \right)
\end{array}$ (9)

This equation shows that if we take the difference (in time) of the
difference (in frequency) of L1 and L2 cycle changes (between internal
and received) we have an unambiguous measure of the difference (in time)
of the electron density as seen along the receiver-satellite path
(TECS). To state it more clearly, if we plot a curve of TECS versus time
for a particular receiver-satellite combination, we know the *shape* of
TECS for that track, but we are *missing a single bias* which would
define the *absolute values* of TECS for that track. This is exemplified
in Figure 1 where five (of an infinite number) of TECS curves have been
drawn for one particular track. Note that all 5 curves have the same
shape, but with an unknown absolute value.

![](media/image1.png){width="3.3541666666666665in"
height="2.5416666666666665in"}

Fig. 1. Five possible versions of TECS for the same track showing
identical shape, but variable absolute values.

One point that bears emphasizing is that, independent of how energetic
the ionosphere is, a GPS receiver tracking L1 and L2 without loss of
lock will successfully yield a ∂TECS/∂t curve that is highly accurate.
This would only break down if either equations 1 or 2 were not reliable,
a situation generally not considered in most literature \[4,5,7\]. As
such[,]{.underline} [the goal of ICON is to retain the full shape of
each TECS track and find the one unknown bias for each track of data,
and thus have a full set of reliable TECS values]{.underline}. That data
may then be put into various forms for distribution, but the raw
information comes from finding the absolute TECS value for each track.
Because it would be desirable to have sub-cycle accuracies (for the
fastest ambiguity resolution), and based on equations 1 and 2, the
accuracy goal of ICON is sub-TECU. It will be shown that, at this time,
the method may reach this goal under certain circumstances.

**MATHEMATIC MODEL**

Assuming we have many tracks of TECS data (from multiple ground
receivers observing multiple GPS satellites), the question arises as to
how to solve for the unknown biases on each track.

Consider any given GPS track of TECS data. For each epoch, pick a [point
of convenience]{.underline} (or "POC"; somewhere that two TECS values
can be reliably mapped into one another). These points can be fixed in
height or varying (and should not be confused with pierce points on a
fixed height shell; points of convenience can vary in height and are
merely "convenient points" on two TECS vectors whose locations can be
compared to one another to see if two TECS vectors are "sufficiently
close" to one another to allow them to be mapped accurately into one
another). If one plots these points geographically (assuming the height
of the points doesn't change significantly from one epoch to the next),
one can see they form a smooth series of points from first acquisition
(satellite rise, for example) to loss of lock (satellite set, for
example). One such track, (using a fixed height of 300 km for
simplicity), is plotted in Figure 2.

![](media/image2.png){width="3.370833333333333in"
height="3.160416666666667in"}

Fig. 2. An example plot of the points of convenience for one track
(formed when station SBT1 tracks GPS SV#3 from rise to set, with points
of convenience at a fixed height of 300 km on July 12 of 2002). Station
SBT1 is shown by the large asterisk.

Consider next any two tracks, where at some epoch, the
satellite-receiver vectors of the two different tracks are "sufficiently
close" to the same POC. We will call such an occurrence a
"[crossover]{.underline}". See Figure 3.

![](media/image3.jpeg){width="3.25in" height="2.4375in"}

Fig. 3. Side view cross-section of the ionosphere showing a crossover
(two different satellite-receiver vectors crossing at a single point of
convenience).

Now as per assumption 4, TECS~1~ must be mappable into TECS~2~. For the
purposes of this paper, the mapping function to be invoked for testing
ICON will rely upon the familiar cos z' mapping between slant (TECS) and
vertical (TECR) views of TEC. See equation 10:

$\text{TECS} = \text{TECR}\text{/cos}z'$ (10)

Because the cos z' mapping is often associated with a "thin shell", ICON
may seem to rely on such a shell, but the only true assumption is that
some suitable mapping function (fulfilling equation 3) can be found to
map "sufficiently close" TECS values into one another. [No direct
requirement that the ionosphere lies on a "shell" is actually called
for!]{.underline} (This contradicts previous statements of \[3\], and
corrects that error.)

Now, notice that (see Figure 3) both TECS~1~ and TECS~2~ may be mapped
into a TECR value at the point of convenience using their respective z'
angles. Note further that, because the two TECS values are "sufficiently
close" to each other (and to the POC), that if the mapping of TECS into
TECR at both tracks is accurate, that they must generate the same value
of TECR. That is, TECR~1~=TECR~2~. Expanding this, the mapping of TECS1
into TECS2 can be written as:

$\text{TECS}1 = \text{cos}z'2\text{TECS}2\text{/cos}z'1$ (11)

thus fulfilling assumption 4. This means that for every crossover that
occurs, we have:

1)  One constraint (TECR~1~=TECR~2~)

2)  Two unknowns (unknown TECS bias for each track)

Standing by itself, one crossover can not solve for two unknowns.
However, if a situation existed where the number of unknowns was equal
to (or exceeded by) the number of crossovers, then the situation would
be solvable.

Such situations can (and do) exist. Consider the case of 3 GPS tracks
(call them 1, 2 and 3), generated by 3 GPS/receiver combinations, and
distributed in such a way that the 3 tracks have 3 crossovers (call them
A, B and C), effectively forming a "triangle" in the ionosphere (see
Figure 4).

![](media/image4.png){width="2.875in" height="2.3854166666666665in"}

Fig. 4. Top view of a "triangle" formed by 3 tracks and 3 crossovers.

As previously stated, we know the shape of the TECS curve (∂TECS/∂t)
along each of the three tracks, but do not know the absolute value. Put
another way, each of the 3 tracks (1, 2 and 3) has 1 unknown bias (b~1~,
b~2~ and b~3~). In fact, one can define b~i~ as the actual value of TECS
on track i at the very first point on that track. If we know that bias,
we could compute the absolute values of TECS (because we know ∆TECS
along the track). The three crossovers at A, B and C provide the
following constraints on the system:

$_{1}^{A}\text{TECR} =_{3}^{A}\text{TECR}( =_{}^{A}\text{TECR})$ (12)

$_{1}^{B}\text{TECR} =_{2}^{B}\text{TECR}( =_{}^{B}\text{TECR})$ (13)

$_{2}^{C}\text{TECR} =_{3}^{C}\text{TECR}( =_{}^{C}\text{TECR})$ (14)

That is, the TECR values at a crossover must be unique, independent of
track. Notice that while all of our actual track information is in the
form of ∆TECS values, the actual constraints on the system are on TECR.
So, in the notation used above, let's write out the conversion from TECR
to TECS and see how the unknown biases fit into the picture:

$\begin{array}{r}
_{1}^{A}\text{TECR} =_{1}^{A}\text{TECS}\text{cos}_{1}^{A}z' = \\
(b_{1} +_{1}^{A}\Delta\text{TECS})\text{cos}_{1}^{A}z'
\end{array}$ (15)

$\begin{array}{r}
_{3}^{A}\text{TECR} =_{3}^{A}\text{TECS}\text{cos}_{3}^{A}z' = \\
(b_{3} +_{3}^{A}\Delta\text{TECS})\text{cos}_{3}^{A}z'
\end{array}$ (16)

$\begin{array}{r}
_{1}^{B}\text{TECR} =_{1}^{B}\text{TECS}\text{cos}_{1}^{B}z' = \\
(b_{1} +_{1}^{B}\Delta\text{TECS})\text{cos}_{1}^{B}z'
\end{array}$ (17)

$\begin{array}{r}
_{2}^{B}\text{TECR} =_{2}^{B}\text{TECS}\text{cos}_{2}^{B}z' = \\
(b_{2} +_{2}^{B}\Delta\text{TECS})\text{cos}_{2}^{B}z'
\end{array}$ (18)

$\begin{array}{r}
_{2}^{C}\text{TECR} =_{2}^{C}\text{TECS}\text{cos}_{2}^{C}z' = \\
(b_{2} +_{2}^{C}\Delta\text{TECS})\text{cos}_{2}^{C}z'
\end{array}$ (19)

$\begin{array}{r}
_{3}^{C}\text{TECR} =_{3}^{C}\text{TECS}\text{cos}_{3}^{C}z' = \\
(b_{3} +_{3}^{C}\Delta\text{TECS})\text{cos}_{3}^{C}z'
\end{array}$ (20)

Substituting equations 15-20 into equations 12-13, and converting to
matrix (Y=A X) form yields:

$\begin{bmatrix}
_{1}^{A}\Delta\text{TECS}\text{cos}_{1}^{A}z' -_{3}^{A}\Delta\text{TECS}\text{cos}_{3}^{A}z' \\
_{1}^{B}\Delta\text{TECS}\text{cos}_{1}^{B}z' -_{2}^{B}\Delta\text{TECS}\text{cos}_{2}^{B}z' \\
_{2}^{C}\Delta\text{TECS}\text{cos}_{2}^{C}z' -_{3}^{C}\Delta\text{TECS}\text{cos}_{3}^{C}z'
\end{bmatrix} = \begin{bmatrix}
 - \text{cos}_{1}^{A}z' & 0 & + \text{cos}_{3}^{A}z' \\
 - \text{cos}_{1}^{B}z' & + \text{cos}_{2}^{B}z' & 0 \\
0 & - \text{cos}_{2}^{C}z' & + \text{cos}_{3}^{C}z'
\end{bmatrix}\begin{bmatrix}
b_{1} \\
b_{2} \\
b_{3}
\end{bmatrix}$ (21)

This system of linear equations has a similarity to those used, for
example, in solving altimetric crossovers or geodetic leveling problems.
In the case of altimetry and geodetic leveling however, the A matrix
contains only +1, -1 and 0 and is thus non-invertible without additional
constraints. In contrast, the cosines in our A matrix allow for an
inversion. In simplest terms, this means that even though we only know
∆TECS (that is, the shape or temporal changes of TECS) on a track and we
only know that TECR must be identical at the crossovers (independent of
track), we can compute one unique absolute value of TECS at the
beginning of each track (b~1~, b~2~ and b~3~). Knowing this value, we
are then able to compute all TECS values for the entire track (and
subsequently all absolute TECR values for the entire track if vertical,
rather than slant, delays are desired.)

Therefore, [within the confines of our opening assumptions, an
unambiguous solution for TECS, based solely on ambiguous carrier phase
data, has been found.]{.underline} It is entirely due to the existence
of a mapping function (in this case, cos z'~2~/cos z'~1~) between TECS
values at crossovers that this is possible. As it turns out, this
mathematical relationship holds for any number of tracks that form a
closed polygon, not just for triangles, since the number of sides
(tracks) of a polygon must equal the number of vertices (crossovers),
and thus would provide a unique and solvable system.

However, while the above triangular example offers a *unique* solution,
it offers no *redundancy*. But if the situation is made just slightly
more complex, say by adding a 4th track, crossing tracks 1 and 2 at
points D and E respectively (forming another triangle, with common
vertex B) we begin to build up redundancy; see Figure 5.

![](media/image5.png){width="2.875in" height="2.4270833333333335in"}

Fig. 5. Adjoining "triangles" showing 4 tracks (unknowns) and 5
crossovers (observations) and thus redundancy

In this case, we now have 4 tracks (with unknowns b~1~, b~2~, b~3~ and
b~4~) but 5 crossovers (constraints), and thus a Least-Squares
Adjustment can be performed on this redundant system. This buildup of
redundancy grows as the number of tracks grows, since each new track can
be expected to cross more and more existing tracks, adding multiple
crossovers (constraints) to the system while only adding 1 unknown bias
for the new track.

Given the fact that a complex network of tracks should yield more
crossovers than tracks, it seems that the ideal application of ICON is
toward a large network of dual-frequency carrier phase GPS receivers,
which are continuously operating and spaced geographically so that their
tracks cross, but offer some wide spatial distribution in the
ionosphere. (Thus our initial assumption that one needs a large network
of receivers, since forming crossovers is the key to this method.)

One important point to consider is that, once a large network of data
has been adjusted, and biases computed, that future tracks of data can
be adjusted into the pre-adjusted network. This application is not the
approach currently taken in production; daily, independent networks are
being computed at NGS under the name "ICON", but nothing prevents an
iterative least squares adjustment from computing biases on an epoch by
epoch basis, as new tracks form new crossovers and enter into the
network adjustment.

**THE CORS NETWORK**

The National Geodetic Survey of NOAA, coordinates a network of
continuously operating reference stations (CORS) that provide GPS
carrier phase and code range measurements in support of 3-dimensional
positioning activities throughout the United States and its territories
(see Figure 6).

![](media/image6.jpeg){width="3.24375in" height="2.5972222222222223in"}

Fig. 6. The CORS Network

The CORS system benefits from a multi-purpose cooperative endeavor
involving many government, academic, commercial and private
organizations. New sites are evaluated for inclusion according to
established criteria. The data are collected at various rates (1, 5, 10
or 30 seconds) and then transmitted (with varying latencies) to NGS for
coordinate computation, data distribution and archival. However, these
data are also perfect for modeling the ionosphere using the ICON
procedure outlined in the previous section.

**TESTING THE MODEL**

As shown in \[3\], extensive testing was done on the ICON method of
modeling TECS values. The initial tests were on small networks, and
proved the mathematical rigor of the method itself. Large scale,
full-day tests were then performed, showing that an entire day of data
could be adjusted (after cleaning cycle slips) in mere seconds.

New tests have since been performed, and are discussed below.

**OHIO STATE UNIVERSITY TESTS**

The Department of Civil and Environmental Engineering and Geodetic
Science at The Ohio State University have been actively testing ICON and
its applicability. In order to facilitate their testing, various nominal
values were chosen (such as crossover limitations, cycle slip detection
and repair methods, location of POCs, etc) and various days of data
provided for analysis. Details of these tests can be found in \[8\],
\[9\], and \[10\], all of which compare and contrast ICON against a
variety of other ionosphere modeling methods. The predominant
conclusions of those tests were positive, but a summary highlights
follows.

According to these OSU studies, ICON appears to very accurately model
the shape of TECS data in double-difference mode, to within a few tenths
of a TECU for most cases, except for the following issues that were
found:

1)  The method used to detect and repair cycle slips at NGS differed
    from that used to generate "truth" values at OSU, and therefore the
    comparison between the two occasionally yields a "step function"
    between "truth" and this method which can be on the order of 1-2 cm
    (\~0.1 TECU).

2)  Occasionally a track has so few crossovers that its bias is poorly
    constrained in the adjustment and causes the double-differenced
    version of the ionosphere to be in error by many TECUs (both
    positively and negatively, allowing or rare (\<0.1%) cases where the
    TECS estimates turn out negative).

Considering the sheer amount of data processed, and the reliance solely
upon crossovers to control this adjustment, the generally good results
found by OSU is a validation of ICON with the caveats that further
controls on data outliers need to be instituted, and a deeper
understanding of the sensitivity of the method to mapping functions
needs to be understood.

**SENSITIVITY ANALYSES**

Although nominal values were found to work for the initial tests of
ICON, a sensitivity analysis was performed at NGS in order to quantify
the stability of the method and its dependence upon such nominal values.

Two major nominal values were tested: location of the POCs and
definition of how close two satellite-receiver vectors need to be to
constitute a crossover.

The points of convenience, in theory, could occur anywhere on the TECS
vectors, so long as a reliable mapping between two "sufficiently close"
TECS values is established at those points. (In fact, the points of
convenience themselves could actually be done away with, and an entirely
generic mapping of one TECS vector to another could be used. Points of
convenience are simply an aid to defining "sufficiently close" TECS
vectors.) Ostensibly, points of convenience could change over time,
though in this study, no allowances are put in ICON for the location of
the points of convenience to be unknowns (variable and known, yes; truly
unknown, no). Some function must be known to describe their locations.

For the sake of simplicity, points of convenience have been defined as
being at a constant height in this study. During the OSU tests, a
nominal value of 300 km was chosen for the heights of the points of
convenience. Sensitivity tests were conducted using constant heights of
250, 300, 350, 400 and 450 km. A specific date (day 298 of 2004) was
chosen as a nominal test day, and TECS estimates were formed using the 5
different definitions of the POCs. The individual values of TECS for
every combination epoch/receiver/satellite were compared for the five
different heights of the POCs. The maximum value and minimum value
computed were differenced for all epoch/receiver/satellite combinations,
and the statistics of those spreads are shown in Table 1.

Table 1: Statistics of the deviation (spread) between TECS values
computed for 5 different POC locations

  --------------- ---------- ----------- ----------- ----------- ----------
  Number of TECSs Ave        STD         RMS         Min         Max

  8,204,539       1.98       1.85        2.71        0.00        36.05
  --------------- ---------- ----------- ----------- ----------- ----------

What table 1 shows is that there is a sensitivity to POC location (at
about a 2 TECU level). This sort of sensitivity will enter directly into
the absolute determination of TECS values themselves. This is about an
order of magnitude larger than would be desirable for a high-accuracy
ionosphere model.

A similar sensitivity analysis was performed for 5 different methods of
defining a crossover. Specifically what criteria were used to define if
two POCs were "sufficiently close". These are shown in Table 2 (Note
that crossover type #1 was the nominal choice for data sent to OSU for
testing.)

Table 2: Definitions of 5 different types of crossovers

+----------------+-----------------+-----------------+----------------+
| Crossover      | Max             | Max             | Max            |
|                |                 |                 |                |
| Type           | ∆φ              | ∆λ              | ∆t             |
+----------------+-----------------+-----------------+----------------+
| 1              | 0.1º            | 0.1º            | 60 s           |
+----------------+-----------------+-----------------+----------------+
| 2              | 0.2º            | 0.2º            | 300 s          |
+----------------+-----------------+-----------------+----------------+
| 3              | 0.05º           | 0.05º           | 30 s           |
+----------------+-----------------+-----------------+----------------+
| 4              | 0.15º           | 0.15º           | 150 s          |
+----------------+-----------------+-----------------+----------------+
| 5              | 0.01º           | 0.01º           | 10 s           |
+----------------+-----------------+-----------------+----------------+

Using these 5 different criteria, and now holding the POC definition as
fixed at 300 km height, the same process was applied. The first thing to
point out is that, even with over 400 CORS stations seeing almost 30
satellites over CONUS, there were not enough crossovers for crossover
type 5 to even form an adjustment. As such, while a strict crossover
criteria would mean a more reliable equality of the TECR values at the
POC, the fact remains that tightening that definition too much yields
too little useful information. Of 22,052 total tracks in the whole CORS
network for this day, the number of tracks solved by crossover
adjustment are shown in Table 3.

Table 3: Number of adjusted tracks for day 298 of 2004

  ----------------------------------- -----------------------------------
  Crossover Type                      Number (of 22,052 total) tracks
                                      adjusted

  1                                   12,698

  2                                   14,657

  3                                   9,129

  4                                   13,941

  5                                   0
  ----------------------------------- -----------------------------------

Not unexpectedly, a larger crossover definition would allow more
crossovers and thus more tracks to enter the adjustment. As such,
crossover type 2 has the largest number of tracks adjusted, and track 5
has the least.

So, considering only those tracks solved from crossover types 1 through
4, a sensitivity analysis was done in a similar fashion to that of the
POC definition sensitivity. Table 4 shows the statistics of crossover
definition sensitivity.

Table 4: Statistics of the deviation (spread) between TECS values
computed for 4 different crossover definitions

  --------------- ---------- ----------- ----------- ----------- -----------
  Number of TECSs Ave        STD         RMS         Min         Max

  8,233,223       1.26       1.87        2.25        0.00        26.73
  --------------- ---------- ----------- ----------- ----------- -----------

As can be seen, the definition of crossover is also a point upon which
the final TECS values depend (less so than POC definition, but still
much larger than the ultimate goal of a few tenths of a TECU).

What these sensitivities mean is clear: while the mathematics of ICON
are internally consistent, their reliance upon the simple cos z'~2~/cos
z'~1~ mapping function between TECS values can only yield absolute
accuracies of TECS to one or two TECUs at best, with the potential for
outliers in the tens of TECUs.

Nonetheless, there is light at the end of the tunnel. First, it must be
pointed out that in differential accuracy, the studies of \[3\], \[8\],
\[9\] and \[10\], all point out that this method yields results that are
generally good at the sub-TECU level. But the goal of this method was
not to simply find differential accuracies at that level, but actual
absolute accuracies. As such, more investigation into how best to
perform and constrain the crossover adjustment, over and above the
assumptions of this paper must be investigated.

**COMPARISON WITH IGTEC DATA**

Although ICON has been shown to be mathematically stable and internally
consistent, it also needs validation in comparison with other
independent methods of determining the ionosphere. One such readily
available data set are the IGTEC data files \[11\] compiled daily by the
International GPS Service (IGS). Although the accuracies stated in each
IGTEC file are in the 2-8 TECU range (see the header of any IGTEC file),
these data should provide a basic check on the reliability of the data
produced by ICON.

Data for almost a month (around Dec 2004) were compared, and the results
analyzed. The comparison was done track-by-track by determining an IGTEC
value of TECS (by computing the piercing location at their given shell
height of 450 km, interpolating their TECR value from the IGTEC grid and
dividing by the cos z' at that pierce point) and subtracting the TECS
value from ICON.

The first noticeable issue was that ICON occasionally yields "outliers"
-- tracks where the TECS bias is estimated at an unreasonable value
relative to the surrounding tracks (either being negative in value, or
tens to hundreds of TECUs off of neighboring values). These outliers are
almost always caused by the outlying track's reliance upon a single,
questionable crossover with another track. As such, future work must be
done to remove these outliers from the least squares adjustment process.
After evaluating daily files, and removing the few outlying tracks, the
results were tabulated. The results are shown in Table 5. The statistics
of each day were computed after outlying tracks had been removed from
the comparison. Note that even on the worst day (342) only 35 of 12,306
tracks were identified as outliers (0.28%).

Table 5: Statistics of the difference between ICON and IGTEC TECS values
for part of 2004. A "\*" indicates no IGTEC data was available for that
day. A "!" indicates no solution was available for ICON for that day.

+------+-----------+--------+-------+--------+--------+-------+-----+
| >    | > \# Tks  | > Ave  | > STD | > RMS  | > Min  | > Max | >   |
|  Day |           |        |       |        |        |       |  \# |
| > of |           |        |       |        |        |       | >   |
| >    |           |        |       |        |        |       |  ou |
| Year |           |        |       |        |        |       | tli |
|      |           |        |       |        |        |       | ers |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 330  | \*        | \*     | \*    | \*     | \*     | \*    | \*  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 331  | \*        | \*     | \*    | \*     | \*     | \*    | \*  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 332  | \*        | \*     | \*    | \*     | \*     | \*    | \*  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 333  | 11,424    | -4.21  | 2.78  | 5.04   | -17.4  | 9.4   | 5   |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 334  | 12,214    | -4.89  | 2.60  | 5.54   | -16.7  | 8.0   | 8   |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 335  | 13,318    | -4.81  | 2.53  | 5.43   | -17.1  | 7.7   | 8   |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 336  | 12,601    | -4.46  | 2.67  | 5.20   | -15.2  | 8.8   | 22  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 337  | 11,482    | -4.85  | 2.70  | 5.55   | -17.1  | 8.6   | 16  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 338  | 10,439    | -4.93  | 2.36  | 5.46   | -15.9  | 6.7   | 10  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 339  | 11,485    | -5.19  | 2.26  | 5.66   | -16.4  | 5.8   | 10  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 340  | 11,913    | -5.76  | 2.69  | 6.36   | -18.2  | 7.6   | 17  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 341  | 12,373    | -4.65  | 3.90  | 6.07   | -21.9  | 14.8  | 31  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 342  | 12,306    | -4.65  | 3.78  | 5.99   | -22.0  | 14.2  | 35  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 343  | 12,779    | -4.58  | 3.22  | 5.60   | -19.5  | 11.4  | 22  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 344  | 13,376    | -4.49  | 3.20  | 5.51   | -19.3  | 11.2  | 10  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 345  | 13,332    | -4.01  | 3.17  | 5.11   | -18.5  | 11.8  | 9   |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 346  | 13,145    | -4.00  | 3.09  | 5.06   | -18.5  | 10.9  | 14  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 347  | 13,212    | -3.74  | 3.10  | 4.85   | -18.5  | 11.4  | 16  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 348  | 13,528    | -4.10  | 2.23  | 4.66   | -14.9  | 7.0   | 17  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 349  | 14,044    | -3.73  | 2.57  | 4.52   | -16.1  | 9.1   | 18  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 350  | 13,952    | -4.15  | 2.67  | 4.99   | -16.9  | 9.6   | 25  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 351  | 13,934    | -4.41  | 2.54  | 5.09   | -15.1  | 8.2   | 32  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 352  | 13,960    | -3.89  | 2.83  | 4.81   | -16.8  | 10.1  | 27  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 353  | 12,808    | -3.47  | 2.55  | 4.31   | -13.5  | 9.1   | 32  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 354  | 12,242    | -3.47  | 2.76  | 4.43   | -14.0  | 10.1  | 20  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 355  | !         | !      | !     | !      | !      | !     | !   |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 356  | 13,742    | -4.35  | 2.96  | 5.26   | -16.5  | 10.2  | 26  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 357  | 13,257    | -4.67  | 3.24  | 5.68   | -20.1  | 11.3  | 10  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 358  | 13,417    | -4.03  | 3.35  | 5.24   | -20.0  | 12.7  | 33  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 359  | 13,357    | -4.10  | 2.97  | 5.06   | -17.0  | 10.7  | 23  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 360  | 13,623    | -4.93  | 2.78  | 5.66   | -17.5  | 8.9   | 26  |
+------+-----------+--------+-------+--------+--------+-------+-----+

A number of clear conclusions can be drawn from Table 5. First, the
number of outliers, relative to the number of adjusted tracks, is quite
small, indicating internal consistency of ICON. Second, it is obvious
that, at least for this month of data, a consistent bias of 3-4 TECUs
exists between ICON and IGTEC. The source of this bias is not clear at
this time, but a brief investigation (see below) was performed to see if
height of points of convenience or method of cross-over determination
was the cause.

Because day 298 of year 2004 had been previously studied for sensitivity
(see previous section), the results of those tests were compared to the
IGTEC data. The results of those comparisons are shown in Tables 6 and
7.

Table 6: Statistics of the difference between IGTEC and ICON TECS values
for various heights of the points of convenience on day 298 of year
2004.

+------+-----------+--------+-------+--------+--------+-------+-----+
| > He | > \# Tks  | > Ave  | > STD | > RMS  | > Min  | > Max | >   |
| ight |           |        |       |        |        |       |  \# |
| > of |           |        |       |        |        |       | >   |
| >    |           |        |       |        |        |       |  ou |
|  POC |           |        |       |        |        |       | tli |
| >    |           |        |       |        |        |       | ers |
| (km) |           |        |       |        |        |       |     |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 250  | 12,041    | -7.30  | 3.47  | 8.90   | -22.8  | 9.3   | 7   |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 300  | 12,698    | -7.49  | 3.13  | 8.12   | -20.6  | 8.0   | 19  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 350  | 12,680    | -7.54  | 2.90  | 8.08   | -18.3  | 5.8   | 8   |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 400  | 12,905    | -7.82  | 2.89  | 8.34   | -22.0  | 5.3   | 7   |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 450  | 13,044    | -7.93  | 2.75  | 8.39   | -20.6  | 5.4   | 14  |
+------+-----------+--------+-------+--------+--------+-------+-----+

Table 7: Statistics of the difference between IGTEC and ICON TECS values
for various methods for selecting crossovers on day 298 of year 2004.
(See Table 2 for definitions of various crossover criteria).

+------+-----------+--------+-------+--------+--------+-------+-----+
| > C  | > \# Tks  | > Ave  | > STD | > RMS  | > Min  | > Max | >   |
| ross |           |        |       |        |        |       |  \# |
| over |           |        |       |        |        |       | >   |
| >    |           |        |       |        |        |       |  ou |
| Crit |           |        |       |        |        |       | tli |
| eria |           |        |       |        |        |       | ers |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 2    | 14,657    | -7.59  | 3.12  | 8.20   | -19.3  | 7.7   | 2   |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 4    | 13,941    | -7.55  | 3.10  | 8.16   | -22.9  | 7.9   | 9   |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 1    | 12,698    | -7.49  | 3.13  | 8.12   | -20.6  | 8.0   | 19  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 3    | 9,129     | -7.73  | 3.91  | 8.67   | -27.1  | 11.6  | 22  |
+------+-----------+--------+-------+--------+--------+-------+-----+
| 5    | 0         | \-     | \-    | \-     | \-     | \-    | \-  |
+------+-----------+--------+-------+--------+--------+-------+-----+

Looking at the results of Tables 6 and 7, one may immediately conclude
that the choice of the height used in the points of convenience has no
significant impact on closing the gap between IGTEC and ICON.
Interestingly, choosing 450 km as the POC height (which matches the
"shell height" listed in IGTEC) makes the bias 0.6 TECU *worse* than the
250 km height *but* it does reduce the standard deviation of errors
around that larger bias. The conclusion is that by matching the POC
height to the shell height of IGTEC the two methods more closely
resemble each other in shape, while maintaining an unresolved bias
between the methods.

Returning to Table 5, one can see that the standard deviations range
from 2.30 TECU up to 3.78 TECU around the bias. Considering that IGTEC
lists its formal error estimates around 2-8 TECU, this number is
encouraging. If the source of the bias between IGTEC and this method
were discovered and alleviated, this consistently small standard
deviation would indicate an overall reliability of this method. In order
to further understand how this method compares to IGTEC, a particular
day was chosen (day 334/year 2004) and the differences in TECS values by
epoch were plotted. For the purposes of completeness, outliers were left
in this plot. See Figure 7.

![](media/image7.png){width="3.2444444444444445in"
height="2.4027777777777777in"}

Fig. 7. 7,261,965 differences between TECS (ICON) and TECS (IGTEC) on
day 334, year 2004, plotted by 30 second epoch number (1 to 2881)

Recall from Table 5 that the differences for this day average -4.75 +/-
2.89 TECU, which is clear from Figure 7. There are a few other points of
interest in Figure 7. First, it's clear that there is a slight downward
trend in the data throughout the day, but nothing that could be
considered a significant time dependent error. Second, the spread of
data, while having a standard deviation of 2.89 TECU shows plenty of
differences (visually) from around -11 to +6 TECU. Theoretically, the
cause for this spread can be traced to the difference in resolution of
the models, in addition to errors in the models themselves (including
POC choice and crossover definition sensitivities; see earlier).
Specifically, for any given track of data from ICON, the entire shape of
the TECS is retained at very fine resolution, as opposed to
interpolating off of the IGTEC grid which is at a grid spacing of 2.5º x
5º x 2 hours (and therefore very coarse relative to the amount of
information contained in all the tracks of data.)

To further validate ICON, and compare with the spread of data seen in
Figure 7, one other data set was checked against this method, albeit
briefly.

**COMPARISON WITH MAGIC DATA**

NOAA's Space Environment Center (SEC) in collaboration with NGS,
produced a method for modeling the ionosphere over CONUS, using CORS
phase and pseudo-range data, discrete radial empirical orthonormal
functions and a Kalman filter under the model name "MAGIC". Both MAGIC
and ICON went into prototype daily productions on November 1, 2004 and
are distributed via the NGS web site at
([www.ngs.noaa.gov/ionosphere](http://www.ngs.noaa.gov/ionosphere)).
However, most MAGIC data is currently unavailable so a detailed
comparison between this method and the MAGIC method was not possible;
however, for the day 334 of year 2004 (exemplified in Figure 7), data
from both methods were made available and they were compared to one
another. The results are shown in Figure 8. Again, for purposes of
exemplifying the outliers (3 of 4,778 tracks), they are left in this
plot.

![](media/image8.png){width="3.2465277777777777in"
height="2.436111111111111in"}

Fig. 8. 2,716,181 differences between TECS (ICON) and TECS (MAGIC) on
day 334, year 2004, plotted by 30 second epoch number (1 to 2881)

Because MAGIC only uses a sub-set of all available CORS data, there are
significantly less points in Figure 8 than Figure 7. Statistically, for
this day, the differences average +1.00 +/- 2.91 TECU (after outlier
removal). Note that the standard deviation about the average is about
the same as that when compared to IGTEC data, while the bias has
switched to +1.00. Because MAGIC was set up specifically to model the
ionosphere over CONUS at CORS stations (like ICON), it is gratifying to
see a small bias in this comparison. However, similar to IGTEC, the
MAGIC model is not specifically built around the retention of the exact
shape of TECS curves the way ICON is. As such, differences in final
resolution of track data as well as model errors lead to the \~3 TECU
standard deviation of the differences. However, MAGIC's method prevents
the sort of outliers seen in this method by constraining the ionosphere
through use of the orthonormal functions and fitting to the
International Reference Ionosphere, making it likely that some
combination of MAGIC and ICON will be tested at NGS in future research.

**DATA PRODUCTION**

While the daily production of ICON yields TECS values at all CORS
stations over a day, these values are difficult to interpret in a
network sense without a visual aid. As such, animated GIFs are produced
daily as part of ICON's production. These GIFs have been found
invaluable for identifying outliers on tracks, as well as observing the
differences between calm and stormy ionosphere days.

For example, Figures 9 through 12 show 4 different views of the vertical
TECR values (implied by taking ICON's TECS values and mapping them to
TECR at 300 km) on a grid for day 311 of 2004 at GPS times 0h, 6h, 12h
and 18h (being local times over CONUS 4-7pm, 10pm-1am, 4-7am and
10am-1pm).

Two things should be noted from Figures 9-12. First, one can
occasionally see an outlying track causing a local "bump" in the TECR
values (such as in the northwest of Figure 9). But also, this particular
day is fairly "normal" in that most maps of the TECR look about like
these figures. However, on the very next day (312, Nov 8), an ionosphere
storm began (and continued to affect the ionosphere for about 4 days).
Plotting hours 0, 6, 12 and 18 for day 312 in Figures 13-16, it is
interesting how similar they are during the night, but in mid day
(Figure 16), the effect of the ionosphere storm is highly pronounced.

On the positive side, these results show that ionosphere storms can be
detected through the ICON method. Unfortunately, the intensity of this
particular storm was so high that the accuracy of the mapping functions
used in ICON failed in their day-to-day accuracy, and no successful
least squares solution could be solved for the days of highest intensity
(Nov 10-12). This further points out the need to investigate more
accurate mapping functions for ICON to remain stable in all conditions.

![](media/image9.png){width="3.245138888888889in" height="1.91875in"}

Figure 9: ICON based TECR map for GPS time 0h on day 311 (Nov 7) 2004.
(Local EST: Nov 6, 7:00 p.m.)

![](media/image10.png){width="3.245833333333333in"
height="1.9361111111111111in"}

Figure 10: ICON based TECR map for GPS time 6h on day 311 (Nov 7) 2004.
(Local EST: Nov 7, 1:00 a.m.)

![](media/image11.png){width="3.2465277777777777in"
height="1.9097222222222223in"}

Figure 11: ICON based TECR map for GPS time 12h on day 311 (Nov 7) 2004.
(Local EST: Nov 7, 7:00 a.m.)

![](media/image12.png){width="3.248611111111111in"
height="1.886111111111111in"}

Figure 12: ICON based TECR map for GPS time 18h on day 311 (Nov 7) 2004.
(Local EST: Nov 7, 1:00 p.m.)

![](media/image13.png){width="3.245138888888889in"
height="1.867361111111111in"}

Figure 13: ICON based TECR map for GPS time 0h on day 312 (Nov 8) 2004.
(Local EST: Nov 7, 7:00 p.m.)

![](media/image14.png){width="3.247916666666667in"
height="1.8416666666666666in"}

Figure 14: ICON based TECR map for GPS time 6h on day 312 (Nov 8) 2004.
(Local EST: Nov 8, 1:00 a.m.)

![](media/image15.png){width="3.247916666666667in"
height="1.8645833333333333in"}

Figure 15: ICON based TECR map for GPS time 12h on day 312 (Nov 8) 2004.
(Local EST: Nov 8, 7:00 a.m.)

![](media/image16.png){width="3.245833333333333in"
height="1.8854166666666667in"}

Figure 16: ICON based TECR map for GPS time 18h on day 312 (Nov 8) 2004.
(Local EST: Nov 8, 1:00 p.m.)

**CONCLUSIONS AND FUTURE WORK**

A new method for computing absolute TEC values along GPS
receiver-satellite vectors using only ambiguous dual-frequency carrier
phase data has been presented. This method, dubbed "ICON", relies upon
the idea that two different skew views of the ionosphere (TECS) that are
near one another (a crossover) can both be successfully mapped into one
another with reliability and a network of such crossovers provides a
unique solution in absolute TECS space. Significant study has been done
on ICON, showing that in double-difference applications, this method is
reliable to sub-TECU levels. But ICON's strong reliance upon the simple
cos z'~2~/cos z'~1~ mapping function means that even minor, relatively
small changes to certain nominal values, such as where and how
crossovers are defined, introduces uncertainty of the absolute TECS
values at the +/- 2 TECU level. This is about one order of magnitude
greater than the target for using this method to get instantaneous
ambiguity resolution.

Comparisons between ICON and other data (IGTEC and MAGIC) show that ICON
is able to match those data at the level +/- 3 TECUs off of a bias, but
the errors in all three methods don't allow for a strong conclusion
about which is nearest to the true absolute TECS values.

Thus ICON is seen as a reasonable first step toward modeling the
absolute ionosphere, though recognizing that it remains in its infancy
with much work yet to be done. To encourage further testing, this method
is being applied in a prototype production mode on a daily basis and
distributed via the NGS web site at
([www.ngs.noaa.gov/ionosphere](http://www.ngs.noaa.gov/ionosphere)). The
data is being produced and released for free to the public, while
research into improving the method is ongoing.

Future work on this model will include ways to make the least squares
adjustment more robust and less sensitive to the choices made for
nominal values. These methods include introducing more physically
realistic constraints on the mapping of TECS values into one another at
crossovers, identification and removal of outlying tracks, and possibly
introducing constraints on the overall network. Additionally, direct,
in-situ measurements of the ionosphere need to be obtained and compared
directly to ICON to help quantify its ability to accurately model true
absolute ionosphere delays.

## ACKNOWLEDGMENTS

The author wishes to acknowledge those who have provided input to this
process along the way. To Dr. Gerry Mader for providing the initial
encouragement to seek out a way to model the ionosphere from the CORS
network itself, and to Dr. Gerry Mader and Dr. Dennis Milbert for a
continuing series of technical discussions throughout this entire
research project. Also thanks to Dr. Doug Robertson for helping with
questions and data issues on MAGIC. Thanks also go Professor Dorota
Grejner-Brzezinska, Pavel Wielgosz and Israel Kashani at The Ohio State
University for providing such excellent independent analysis of this
method.

## REFERENCES

\[1\] Odjik, D., 2000: Weighting ionospheric corrections to improve fast
GPS positioning over medium distances, Proceedings of the Institute of
Navigation 2000 meeting

\[2\] Teunissen, P.J.G. and D. Odjik, 2003: Rank-defect integer
estimation and phase-only modernized GPS ambiguity resolution, Journal
of Geodesy, v. 76, pp. 523-535.

\[3\] Number range CHAPTERSmith, D.A. 2004: Computing unambiguous TEC
and ionospheric delays using only carrier phase data from NOAA's CORS
network. *Proceedings of IEEE PLANS 2004*, Monterey, California, April
26-29, 2004, pp. 527-537.

\[4\] Hofmann-Wellenhof, B., H. Lightenegger and J. Collins, 1993:
*Global Positioning System, Theory and Practice*, 2^nd^ edition,
Springer-Verlag, Wien, New York.

\[5\] Hartmann, G.K. and R. Leitinger, 1984: Range Errors Due to
Ionospheric and Trophspheric Effects for Signal Frequencies above 100
MHz, *Bulletin Geodesique*, No. 58, pp. 109-136.

\[6\] Wanninger, L. and M. Manja, 2001: Carrier-phase multipath
calibration of GPS reference stations, NAVIGATION, v. 48, n. 2, pp.
113-124.

\[7\] Langley, R.B., 1996: Propogation of the GPS Signals, Lecture Notes
in Earth Sciences, v. 60 (Kleusberg and Teunissen Eds.)

\[8\] Grejner-Brzezinska, D, P. Wielgosz, I. Kashani, 2004a: Accuracy
analysis of various NGS ionosphere estimation models, *Contractual
Report to NOAA/NGS*

\[9\] Grejner-Brzezinska, D, P. Wielgosz, I. Kashani, 2004b: Analysis of
the ambiguity resolution, error modeling techniques and support
infrastructure for nationwide three-frequency real-time kinematic (RTK)
GPS positioning, Part 1(Comparative analysis of various ionosphere
estimation models), *NOAA Technical Report for grant NA04NOS40000067*,
National Geodetic Survey, Silver Spring, MD

\[10\] Grejner-Brzezinska, D., P. Wielgosz, I. Kashani, D. Smith, P.
Spencer, D. Robertson and G. Mader, 2004: The analysis of the effects of
different network-based ionosphere estimation models on the rover
positioning accuracy, *Proceedings of GNSS 2004*, Sydney.

\[11\] Hernández-Pajares, M., 2004: IGS Ionosphere WG Status Report:
Performance of IGS Ionosphere TEC Maps (Position Paper), *Presented at
the IGS Technical meeting, Bern, Switzerland, March 2004*.

\[12\] Spencer, P.S.J., D. Robertson and G. Mader, 2004: Ionospheric
data assimilation methods for geodetic applications, *Proceedings of
IEEE PLANS 2004*, Monterey, California, April 26-29, 2004, pp. 510-517.