2.1. Phase and Pseudorange Observations
High-precision geodetic measurements with GNSS are performed using the carrier beat phase, the output from a single phase-tracking channel of a GNSS receiver. This observable is the difference between the phase of the carrier wave implicit in the signal received from the satellite, and the phase of a local oscillator within the receiver. The phase can be measured with sufficient precision that the instrumental resolution is a millimeter or less in equivalent path length. The dominant source of error in a phase measurement or series of measurements between a single satellite and ground station is the unpredictable behavior of the time and frequency standards (“clocks”) serving as reference for the transmitter and receiver. Even though the satellites carry atomic frequency standards, the instability of these standards would still limit positioning to the several meter level were it not for the possibility of eliminating their effect through signal differencing.
A second type of GNSS measurement is the pseudo-range, obtained using the codes transmitted by the satellites Pseudo-ranges provide the primary GNSS observation for navigation but are not precise enough to be used directly in geodetic surveys. They are useful, however, for estimating the offsets of receiver clocks, resolving ambiguities, and repairing cycle slips in phase observations.
For a single satellite, differencing the phases (or pseudo-ranges) of signals received simultaneously at each of two ground stations eliminates the effect of bias and instabilities in the satellite clock. This measurement is commonly called the between-stations-difference, or single-difference observable. Forming a second difference, between satellites, eliminates the effect of bias and instabilities in the station blocks.
Since the phase biases of the satellite and receiver oscillators at the initial epoch are eliminated in doubly differenced observations, the doubly differenced range (in phase units) is the measured phase plus an integer number of cycles. (One cycle has a wavelength of 19 cm at GPS L1 and 24 cm at GPS L2 for code-correlating receivers; half these values for squaring-type receivers used prior to the mid-1990s, and slightly different values for the primary frequencies transmitted by Glonass, Beidou, and Galileo. In the manuals we will use the term “L1” to refer to the higher and “L2” the lower of the two L-band frequencies though these may be, e.g. for Galileo E1(L1) and E5.) If the measurement errors, arising from errors in the models for the orbits and propagation medium as well as receiver noise, are small compared to a cycle, there is the possibility of determining the integer values of the biases, thereby obtaining from the initially ambiguous doubly differenced phase an unambiguous measure of doubly differenced range. Resolution of the phase ambiguities improves the uncertainties in relative position measures by about a factor of 15 for 24-hr sessions, 3 for 8-hr sessions and more than 5 for short sessions (see, e.g., Blewitt [1989], Dong and Bock [1989]).
GAMIT incorporates difference-operator algorithms that map the carrier beat phases into singly and doubly differenced phases. These algorithms extract the maximum relative positioning information from the phase data regardless of the number of data outages, and take into account the correlations that are introduced in the differencing process. (See Bock et al. [1986], Schaffrin and Bock [1988] for a detailed discussion of these algorithms.) An alternative, (nearly) mathematically equivalent approach to processing GNSS phase data is to use formally the (one-way) carrier beat phases but estimate at each epoch the phase offset due to the station and satellite clocks. This approach is used by the autcln program in GAMIT to compute one-way phase residuals for editing, display, and estimating atmospheric and ionospheric slant delays.
In order to provide the maximum sensitivity to geometric parameters, the carrier phase must be tracked continuously throughout an observing session. If there is an interruption of the signal, causing a loss of lock in the receiver, the phase will exhibit a discontinuity of an integer number of cycles (“cycle-slip”). This discontinuity may be only a few cycles due to a low signal-to-noise ratio, or it may be thousands of cycles, as can occur when the satellite is obstructed at the receiver site. Initial processing of phase data is often performed using time differences of doubly differenced phase (“triple differences”, or “Doppler” observations) in order to obtain a preliminary estimate of station or orbital parameters in the presence of cycle slips. The GAMIT software uses triple differences in editing but not in parameter estimation. Rather, it allows estimation of extra bias parameters whenever the automatic editor has flagged an epoch as a cycle slip that cannot be repaired. Various algorithms to detect and repair cycle slips are described by Blewitt [1990], and also in Chapter 4 of the GAMIT Reference Manual.
Although phase variations of the satellite and receiver oscillators effectively cancel in doubly differenced observations, errors in the time of the observations, as recorded by the receiver clocks, do not. However, the pseudo-range measurements, together with reasonable a priori knowledge of the station coordinates and satellite position, can be used to determine the offset of the station clock to within a microsecond, adequate to keep errors in the doubly differenced phase observations below 1 mm.
A major source of error in single-frequency GNSS measurements is the variable delay ntroduced by the ionosphere. For day-time observations near solar maximum this effect can exceed several parts per million of the baseline length. Fortunately, the ionospheric delay is dispersive and can usually be reduced to a millimeter or less by forming a particular linear combination (LC) of the L1 and L2 phase measurements:
for GPS (see, e.g., Bock et al. [1986] or Dong and Bock [1989]). Forming LC, however, magnifies the effect of other error sources. For baselines less than a few kilometers the ionospheric errors largely cancel, and it is preferable to treat L1 and L2 as two independent observables rather than form the linear combination. The station separation at which the ionospheric errors exceed the phase noise depends on many factors (receiver, antenna, multipath environment, latitude, time of day, sunspot activity) and must be determined empirically by analyzing the data with both observable types.
In examining phase data for cycle slips, it is often useful to plot several combinations of the L1 and L2 residuals. Single-cycle slips in GPS L1 or L2 will appear as jumps of 2.546 or 1.984 cycles, respectively, in LC. Single-cycle slips in both GPS L1 and L2 (a more common occurrence) appear as jumps of 0.562 cycles in LC, which, though smaller, may be more evident than the jumps in L1 and L2 because the ionosphere has been eliminated. If the L2 phase is tracked using codeless techniques, the carrier signal recorded by the receiver is at twice the L2 frequency, leading to half-cycle jumps when it is combined with full-wavelength data. Hence, a jump of a “single” L2 cycle will appear as 0.992 in LC, and simultaneous jumps in (undoubled) L1 and (doubled) L2 will appear as 1.554 cycles in LC. Another useful combination is the difference between L2 and L1 with both expressed in distance units
for GPS, called “LG” because the L2 phase is scaled by the “gear” ratio (\(f_2/f_1 = 60/77 = 1227.6/1575.42\) for GPS). In the LG phase all geometrical and other non-dispersive delays (e.g., the troposphere) cancel, so that we have a direct measure of the ionospheric variations. One-cycle slips in L1 and L2 are difficult to detect in the LG phase in the presence of much ionospheric noise since they are equivalent for GPS to only 0.221 LG cycles.
If precise (P-code) pseudorange is available for both GNSS frequencies, then a “wide-lane” (WL) combination of L1, L2, P1, and P2 can be formed which is free of both ionospheric and geometric effects and is simply the difference in the integer ambiguities for L1 and L2:
The WL observable can be used to fix cycle slips in one-way data [Blewitt, 1990] but should be combined with LG and doubly differenced LC to rule out slips of an equal number of cycles at L1 and L2.
These various combinations of phase and pseudorange observations are used not only in data editing, but also in resolving the phase ambiguities. When the LC observable is used, we determine the L1 and L2 ambiguities by first resolving \(n_2 - n_1\) (“widelane”) and then \(n_1\) (“narrow lane”). If precise and unbiased pseudoranges are available, the widelane ambiguities can be resolved for baselines up to thousands of kilometers under any ionosphere conditions. For measurements prior to 1995, and possibly prior to 2000, inter-channel receiver biases can corrupt the pseudoranges and it is necessary to use the phase observations alone with a constraint on the ionopshere to resolve the widelane ambiguities (see, e.g., Blewitt [1989], Dong and Bock [1989], Feigl et al. [1993]). GAMIT gives you the option of selecting the method to be used, either pseudoranges (LC_AUTCLN) or ionospheric constraints (LC_HELP). When using the pseudorange approach with different receiver types, it is important to use the satellite-dependent differential code biases (DCBs) computed from tracking data by the Center for Orbit Determination in Europe (CODE) [http://www.aiub.unibe.ch/ionosphere.html] and updated monthly at MIT in file dcb.dat. Once the wide-lane ambiguity for a given doubly differenced combination has been resolved, resolving the narrow-lane ambiguity for that combination depends on the level of noise from the receiver, multipathing, and the troposphere, and the accuracy of the models employed for the position and motion of the stations and satellites. It is generally more difficult to resolve these ambiguities for the longest baselines, but for data acquired since 2000 we can usually resolve 80–95% of the ambiguities in a global analysis.
Although GAMIT can process data from single-frequency receivers, ionsopheric errors begin to limit relative positioning accuracy for station separations greater than 100–2000 m, depending on time of day, latitude, and the solar sunspot cycle. So for most applications, the use of two frequencies is required to remove first-order ionsospheric effects. Although some GNSS satellites transmit signals on three of more frequencies, GAMIT is currently limited to using only two. GAMIT is also limited to processing data from only one GNSS at a time, though the GAMIT position estimates from several GNSS tracked during a session can easily be combined in GLOBK.