8.1.3 GPS Baseline Processing

PROCESSING OF DIFFERENCED DATA

There are a number of comments that can be made with regards to the various differenced phase data solutions:

• Triple- and double-differenced data solutions have different strengths and weaknesses, and are therefore offered as a "package" of processing options by the GPS instrument manufacturer. All three processing strategies are generally used (although it may not always be obvious from a study of the output results file).

   

• Triple-, double- (ambiguity-free) and double-differenced (ambiguity-fixed) solutions represent a hierarchy of processing strategies that are generally applied in sequence: first the triple-differenced solution, through to the double-differenced solutions, with the ultimate aim to obtain an ambiguity-fixed solution (see Figure below).

   

• The sequence of processing strategies: triple-difference through to the double-difference ambiguity-fixed solution, is from a robust, but relatively imprecise, algorithm to one that is capable of high precision and reliability only if the following (incomplete) list of conditions are met:
  • the data is free of cycle slips,
  • the observation session length is long enough (generally 30 to 60 minutes),
  • Favourable receiver-satellite geometry (tracking as many satellites as are visible, with low PDOP),
  • the baseline length is relatively short (<20km) for single frequency observations.

   

• The "optimum" baseline result is dependent on the quality and reliability of the double-difference solution. If an ambiguity-fixed solution is not possible (that is, it is not reliable), the double-difference ambiguity-free solution may be accepted as the "best". Under certain conditions, the triple-difference solution is the one preferred over the relatively weak double-difference solutions.

   

• GPS manufacturers provide guidelines for the selection of the "optimum" solution, on the basis of the length of the baseline. For example, it may be recommended that the "optimum" solution for baselines is: (a) if shorter than 15km it is the double-difference ambiguity-fixed solution (assuming ambiguity resolution was possible), (b) if baseline lengths greater than 15km and less than 50km it is the double-difference ambiguity-free solution unless session length is greater than 30mins, and (c) for baselines greater than 50km it is the triple-difference solution. (Note: recommendations vary from manufacturer to manufacturer and "age" of the hardware/software.)

   

• Baseline reductions are invariably performed using the broadcast ephemerides. The main reasons are that it is instantly available (via the Navigation Message), and it is of sufficient accuracy. Investigations have shown that the accuracy of the broadcast ephemerides is of the order of ten metres, often much higher, hence contributing less than 1-2ppm to the baseline error. (Note: precise IGS ephemerides are now available with very little delay over the Internet.)

   

GPS phase data processing sequence.

Triple-Differenced Phase Solution

The following are some characteristics of triple-difference solutions:

• The functional model for the solution is eqn (7.2-5), containing only coordinate parameters (the ambiguity and clock phase errors were eliminated during differencing).
• Triple-difference solutions are "robust", being relatively immune to the effect of cycle slips in the data, which have the characteristics of "spikes" in the data (see Table in section 6.3.7 and Figure in section 7.3.5).
• This low susceptibility to data that is not free from cycle slips is due to the correlations in the differenced data not being taken into account (assume a diagonal Weight Matrix P).
• The algorithm used to construct the triple-differenced observables is ideally suited for detecting and repairing cycle slips in the double-differenced data. Hence these solutions are generally carried out as part of the overall data cleaning (pre-)process (section 7.3.1). An automatic procedure would be based on scanning the residuals of the triple-difference solution for those close to an integer value of one or more cycles.
• Relatively simple algorithm that can easily handle a changing satellite constellation.
• The triple-difference solution provides good apriori values for the baseline components.
• Under extreme circumstances the triple-difference solution may be the only one that is reliable.

The triple-differences solution algorithm : Difference epoch data between-satellites, form double-differences. Difference double-differences between epochs at some sample rate (for example, every 5th observation epoch), form triple-differences. Assume all triple-difference observations are independent when forming Weight Matrix (no correlations taken into account), define P matrix. Form Observation Equations, construct the A matrix. Accumulate Normal Equations, scaled by the Weight Matrix ATPA. At end of data set, invert Normal Matrix and obtain geodetic parameter solution, = (ATPA)-1.ATP . Update parameters. Optionally scan triple-difference residuals for cycle slips in double-difference observables.

Double-Differenced Phase Solution (Ambiguity-Free)

The following are some characteristics of double-differenced phase (ambiguity-free) solutions:

• The functional model for the solution is eqn (7.2-4), containing both coordinate parameters and ambiguity parameters (the exact form depending upon the ambiguity parameter model used -- section 7.2.1).
• Double-difference solutions are vulnerable to cycle slips in the (double-differenced) data.
• The solution can be quite sensitive to a number of internal software factors such as:
  • between-satellite differencing strategy (see below),
  • data rejection criteria,
  • whether correlations are taken into account during differencing (see below),
  • whether the observation time-tags are in the GPS Time system (section 6.3.8 and section 7.3.3).
• The solution is also sensitive to such external factors as:
  • length of observation session,
  • receiver-satellite geometry (including the number of simultaneously tracked satellites),
  • residual biases in the double-differenced data due to such things as atmospheric disturbances, multipath, etc.,
  • the length of the baseline.
• Only the independent epoch double-differences are constructed: (S-1) double-differences per baseline per epoch, where S is the number of satellites tracked.
• The algorithm used to construct the independent double-differenced observables must take into account the situations such as the rising and setting of a satellite during an observation session (and the appropriate definition of the ambiguity parameters in such a case).

The double-difference solution algorithm: Difference epoch data between-satellites, form double-differences. Apply data reductions, such as a troposphere bias model. Construct Weight Matrix (depending on whether correlations are to be taken into account), define the P matrix. Form Observation Equations --> construct the A matrix. Accumulate Normal Equations, scaled by the Weight Matrix ATPA. At end of session, invert Normal Matrix and obtain geodetic and ambiguity parameter solution, = (ATPA)-1.ATP . Update parameters. Decide (a) iterate solution, or (b) iterate solution only after ambiguity resolution attempted.

Double-Differenced Phase Solution (Ambiguity-Fixed)

The following are some characteristics of double-differenced phase (ambiguity-fixed) solutions:

• The functional model for the solution is eqn (7.2-4), containing coordinate parameters and any unresolved ambiguity parameters (or none if all ambiguities have been resolved to their integer values). As ambiguities are resolved the (integer) value of the ambiguity becomes part of the apriori known information, and this has the effect of converting ambiguous phase observations into unambiguous range observations.
• Such a double-difference solution is comparatively strong (there are less parameters to estimate!), but is reliable only if the correct integer values of the ambiguities have been identified.
• The solution can be quite sensitive to the strategy used to resolve the ambiguities, for example:
  • whether all ambiguities are to be resolved as a set, or only a subset,
  • the resolution criteria used for decision making,
  • the search strategy used for integer values.
• The ambiguity resolution process is also sensitive to such external factors as:
  • length of observation session,
  • receiver-satellite geometry,
  • residual biases in the double-differences due to such things as atmospheric disturbances, multipath, etc.,
  • whether satellites rise or set during the session,
  • the length of the baseline.

The ambiguity-fixed solution algorithm: Difference epoch data between-satellites, form double-differences as before but without ambiguities as solve-for parameters. Apply data reductions, such as a troposphere bias model. Construct Weight Matrix (depending on whether correlations are to be taken into account), define the P matrix. Form Observation Equations, construct the A matrix. Accumulate Normal Equations, scaled by the Weight Matrix ATPA. At end of session, invert Normal Matrix and obtain geodetic parameter solution, = (ATPA)-1.ATP . Update parameters. This process can be iterated to resolve other ambiguities until (a) all have been resolved (and "fixed" to integers), or (b) no more can be reliably resolved.

Once ambiguities have been resolved, the ambiguous phase measurements are converted to precise range observations. As in conventional GPS navigation, single epoch positioning is now possible and hence "carrier-range" observations are ideal for kinematic positioning applications.

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