A couple of months ago, researchers from Wuhan University published a paper entitled “A review on the inter-frequency biases of GLONASS carrier-phase data,” aiming at clarifying distinctions between inter-frequency phase biases (IFPBs) and differential code-phase biases (DCPBs). While the paper offers good descriptions and a detailed analysis, I thought I might share a few additional insights.

GLONASS inter-frequency phase biases have always been a nuisance to GLONASS ambiguity resolution for RTK processing. Wanninger (2012) demonstrated that the IFPBs of all geodetic receivers can be modeled using a linear function, and that receivers of the same type generally have similar values of IFPBs. The IFPBs determined empirically in this study have been adopted by many as a priori values to constrain the IFPB parameter in the positioning filter.

Sleewaegen et al. (2012) have then explained that inter-frequency phase biases are only apparent biases and actually originate from a misalignment between the carrier-phase and code observables in the receiver, hereafter referred to as the differential code-phase bias (DCPB). Even though this problem could also occur for the GPS observables, it mainly affects GLONASS observables due to FDMA since a common timing offset would create different ambiguity biases in units of cycles.

Using the DCPB concept, Geng et al. (2016) argue that the linear model suggested by Wanninger is in fact theoretically incorrect, although the practical impact is negligible. The proof provided by the authors use double-differenced ambiguities, and I would like to offer my own derivation based on undifferenced observations, which should allow gaining additional insights on this issue.

Let us start with the basic carrier-phase observation (L) on frequency i and satellite j:

where rho is the range, dT is the receiver (code) clock, c is the speed of light, lambda is the wavelength of the carrier of channel k, and N is the carrier-phase ambiguity. It is assumed here without loss of generality that other error sources were properly accounted for.

If we do not account for the DCPB, the estimated ambiguities will absorb this quantity and will be:

or, equivalently:

where f is the frequency and delta f is the frequency separation between channels. Since the second term on the right-hand side of the previous equation is common to all satellites, forming between-satellite differences will cancel this quantity. Keeping this in mind, let us define:

If we substitute this quantity in the first equation, we get:

The quantity between brackets is satellite independent and, therefore, **the rigorous partial derivative for the inter-frequency phase biases should consider both the wavelength and
the frequency channel number**. The approximation used by Wanninger, determined empirically, effectively assumes that the wavelengths for all satellites are equal which allows to use
solely the frequency channel number as a partial derivative.

With this formulation, it then becomes easy to compute the error in the IFPB model used by Wanninger:

which will most likely be at the sub-mm level.

In summary, if you are really concerned with having a rigorous implementation, I suggest modifying the partial derivative for the GLONASS IFPB to include the wavelength, although it will not have any impact on your positioning results… and if you do that, don’t forget to scale your a priori values for the IFPBs accordingly!

**References**

Geng J, Zhao Q, Shi C, Liu J (2016) A review on the inter-frequency biases of GLONASS carrier-phase data. J Geod doi:10.1007/s00190-016-0967-9

Sleewaegen JM, Simsky A, de Wilde W, Boon F, Willems T (2012) Demystifying GLONASS inter-frequency carrier phase biases. InsideGNSS 7(3):57–61

Wanninger L (2012) Carrier-phase inter-frequency biases of GLONASS receivers. J Geod 86(2):139–148. doi:10.1007/s00190-011-0502-y

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2#1STAVROS MELACHROINOS(Tuesday, 28 February 2017 23:46)But if we do account for the DCPB in the first equation above (phase expressed in length) and we form single differencing between a satellite j and satellite m wouldn't that eliminate the DCPBs? However if we express the phase in cycles. the DCPB become linearly dependent on the frequency channel. Doesn't this come down on how we simply process the phase (in length or cycles) ?

#2Simon Banville(Wednesday, 01 March 2017 20:08)@Stavros Forming single differences will effectively cancel the term DPCB in the first equation (this is equivalent to using a phase clock as opposed to the code clock in the model above). However, if you have n satellites, you will be left with n-1 single-differenced observations and n ambiguities (because of FDMA, the ambiguity of the reference satellite can't be merged with the ambiguities of other satellites). Now, how do you remove the singularity? The most popular option is to provide a value for the ambiguity of the reference satellite using (phase - code) which will re-introduce the DCPB...

Processing carrier-phase observations in units of cycles would indeed make the equations linearly dependent on the frequency channel number, although the partial derivatives for the position and receiver clock would need to consider the wavelengths.

In summary, changing units will not make anything disappear, unfortunately! I hope this helps!