It is now well-understood that a misalignment between carrier and code measurements within a GNSS receiver causes an apparent inter-frequency bias in GLONASS carrier-phase observations. Receiver-dependent calibration values are typically used to reduce this error source and a constrained parameter can be added to the filter to account for residual effects. A few years ago, I proposed an alternative which consists of using two reference satellites with adjacent frequency channel numbers to explicitly estimate inter-frequency phase biases. In this post, I share a few additional insights on this approach which were not included in the original paper.
Let me briefly recall the fundamentals of the approach described by Banville et al. (2013): by fixing the between-station ambiguities of two reference satellites (let's label them 1 and 2) to arbitrary integer values and by estimating an unconstrained inter-frequency phase-bias parameter, it can be shown that the double-differenced ambiguities of all other satellites will take the following form:
where N represents a (double-differenced) ambiguity, k stands for the (integer) frequency channel number of a given GLONASS satellite and n identifies another (non-reference) satellite. When | k1 - k2 | = 1, it can be deduced that the ambiguity parameter of any satellite n will be an integer. The identifier (R) is simply used to refer to GLONASS satellites.
This approach is often characterized as “not practical” since two satellites with adjacent frequency channel numbers are not always visible simultaneously. Here are a few additional details that I should perhaps have included in the manuscript:
1. You don't actually need two satellites with adjacent frequency channel numbers
It should be clear that multiplying both sides of the above equation by (k1 - k2) will not affect the integer properties of the equation. However, when using satellites with non-adjacent frequency channel numbers, multiplying the left-hand side by this quantity will effectively reduce the wavelength by (k1 - k2). In a multi-GNSS solution, reducing the wavelength of GLONASS ambiguities by 2 for instance may not preclude successful ambiguity resolution.
2. Once ambiguities are fixed, any satellite can become the reference
Let's suppose that you have achieved successful ambiguity resolution using two reference satellites with adjacent frequency channel numbers and that one of them is affected by a cycle slip or is not being tracked anymore. What happens? You can simply pick any other ambiguity-resolved satellite as the other reference satellite, regardless of its channel number (without reducing the wavelengths).
3. The two reference satellites don't need to be observed simultaneously
Theoretically, as long as there are no discontinuity in the data (i.e., cycle slips on all satellites), any two satellites in the whole observation session can be selected as reference satellites. Hence, for a data set of a couple of hours processed in batch mode, it is almost certain that you will have observed two satellites with adjacent frequency channels.
4. The additional inter-frequency bias parameter can be modeled as a constant
My paper explained that the ADOP of this approach was larger than constraining the inter-frequency bias parameter by using a priori calibration values. It is true that, for single-epoch ambiguity resolution, this can impact performance. However, since the unconstrained inter-frequency phase-bias parameter can be modeled as a constant, it should not be a limitation after successful ambiguity resolution.
This being said, I entirely agree that using calibration values is simpler than defining two reference satellites. But I think that the proposed method can still be quite useful in short-baseline RTK, especially when considering the emergence of many low-cost receivers for which calibration values are not available or can be complex to maintain.