It is well known that the 'leveling' process, in which carrier phases are fitting to code observations, is a source of error in the determination slant total electron content (STEC). These leveling errors can be greatly attenuated when using carrier-phase ambiguities obtained from PPP as leveling information. In this post, I show how leveling errors can also impact VTEC and, potentially, DCB estimates.
The standard leveling procedure of fitting carrier-phase to code observations is widely being used by the GNSS community as a simple means of removing the arc-dependency of the geometry-free carrier-phase measurements. Actually, this approach is equivalent to processing uncombined code (P) and phase observations (Phi) to estimate the range (rho), the slant ionospheric delay (I) and the carrier-phase ambiguities (N):
After smoothing, the slant ionospheric delays should be 'leveled' to the code observations. I used the above formulation to highlight that, if it is possible to put a precise constraint on the range parameter, then the precision of the other parameters would be improved. We will get to that later.
First, let us state four additional issues with the standard leveling process:
- Code multipath does not necessarily average out over an arc, especially shorter ones
- It has been shown that there are receiver- and satellite-dependent code biases that can reach decimeter level, see e.g. Hauschild and Montenbruck (2016)
- The procedure is affected by intra-day variations of differential code biases (DCBs)
- It neglects the integer property of the carrier-phase ambiguities
The following figure shows the leveling errors resulting from the standard leveling procedure for station UNB3 on May 29 2016 (with respect to the integer-leveled solution described below). It is clear that leveling errors of a few TECU exist which matches results from previous studies (Ciraolo et al. 2007). In the case of significant intra-day variations of DCBs, leveling errors exceeding 10 TECU can even occur (see below). As one would expect, leveling errors should impact parameters estimated with standard-leveled observations, such as VTEC.
Fig. 1: Leveling errors for station UNB3 when using the standard leveling approach
(each continuous trace is a satellite arc)
An efficient means of reducing leveling errors consists of using the geometric information from a precise point positioning (PPP) solution. To make a link with the above equation, the range parameter could be tightly constrained using the estimated receiver coordinates, receiver clock offset and tropospheric zenith delays from the PPP filter. The next figure shows the leveling errors associated with the PPP approach with float ambiguities.
Fig. 2: Leveling errors for station UNB3 when using PPP with float ambiguities
(each continuous trace is a satellite arc)
The mitigation of leveling errors resulting from including geometric information derived from PPP is obvious. Unfortunately, the four issues enumerated above still impact float ambiguities.
In order to completely eliminate leveling errors, carrier-phase ambiguities in the PPP filter need to be fixed to integer values, a process that I termed “integer leveling” in previous publications. The two figures below show an extreme case of leveling errors with significant intra-day DCB variations for the standard leveling (left) and integer leveling (right) procedures between stations ALGO and ALG3. After correct ambiguity fixing, both leveling errors and intra-day DCB variation effects are removed (notice the scale change in TECU).
Fig.3: Leveling errors between nearby stations ALGO and ALG3 on 17 January 2011
(from Banville and Langley, 2011)
Now what is the impact of leveling errors on VTEC? To measure this, we need to accept the following assumption: VTEC derived from integer-leveled observations should be closer to the true value than other leveling sources. We can then compare the VTEC differences derived from station UNB3 using standard leveling (STD) and using PPP float leveling (PPP), with respect to the integer-leveled VTEC.
Fig.4: VTEC differences ("errors") when derived from standard-leveled observations (STD) or PPP (Float)-leveled observations for station UNB3
For the standard-leveled observations, we see VTEC differences at times exceeding 1 TECU. The concerning part is that the mean VTEC difference is not equal to zero, meaning that the DCB estimate will also differ. It should be noted that VTEC errors could be significantly larger in the presence of intra-day DCB variations.
While there are several error sources involved in creating global ionospheric maps, it seems to me that replacing the leveling procedure with PPP-AR is a simple task that could only be beneficial. Processing a 24-hour data set at a 5-min interval and producing integer-leveled observations takes about 5 seconds per station. Not to mention that DCB estimates can be rigorously aligned from day to day (provided there are no data breaks), but this is a story for another time...
Banville S, Langley RB (2011) Defining the Basis of an Integer-Levelling Procedure for Estimating Slant Total Electron Content. Proceedings of the 24th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2011), Portland, OR, September 2011, pp. 2542-2551.
Ciraolo L, Azpilicueta F, Brunini C, Meza A, Radicella SM (2007) Calibration errors on experimental slant total electron content (TEC) determined with GPS. J Geod 81(2):111-120 doi:10.1007/s00190-006-0093-1
Hauschild A, Montenbruck O (2016) A study on the dependency of GNSS pseudorange biases on correlator spacing. GPS Solutions 20(2):159-171 doi:10.1007/s10291-014-0426-0