7.2.2 Basic Modelling Concepts

GPS OBSERVATION MODELS

Dealing with Biases: To obtain a GPS solution, strategies have to been developed to overcome the problem of systematic biases. There are a number of options: They can be estimated as explicit parameters. Those biases linearly correlated across different datasets can be eliminated (or significantly reduced) by data differencing. They can be directly measured. They can be assumed known from models. They can be ignored. The most commonly used option is the second. Observation DIFFERENCING is closely identified with the GPS surveying technique, requiring two or more GPS receivers to track the same set of satellites -- hence appropriate field and data analysis procedures must be used.

GPS measurements, both pseudo-range and carrier phase, are affected by biases and errors (section 6.1.1). Different levels of GPS accuracy are associated with a different partitioning of "biases" and "errors". At one extreme, in the case of GPS pseudo-range point positioning, all effects with the exception of the receiver and satellite clock biases are treated as errors. At the other extreme, GPS baseline determination to accuracies of 1 part in 10⁸ requires that almost all measurement biases are explicitly accounted for in any solution. In the case of GPS surveying there is a compromise between fully accounting for all biases and keeping the computational burden down by not over-parameterising the observation model.

First the commonly used observation models in GPS surveying are described, and the reasoning behind the particular parameterisation that is adopted will be discussed. The bias categories are defined as follows (section 6.2.1):

Satellite dependent:

• Ephemeris uncertainties
• Satellite clock uncertainties
• Selective Availability effects

Receiver dependent:

• Receiver clock uncertainties
• Reference station coordinate uncertainties

Receiver-Satellite (or observation) dependent:

• Ionospheric delay
• Tropospheric delay
• Carrier phase ambiguity

It is assumed that the data has been "cleaned" (and hence, in the case of carrier phase data, there are no cycle slips present), and that all other effects are indistinguishable from data "noise", which the Least Squares adjustment must accommodate.

Accounting for Biases in GPS Surveying

The first point that needs to be made is that by adopting the observation strategy of simultaneous tracking of the same set of satellites from a number of receivers, the most powerful technique of bias minimisation can be used. Observation differencing, or more generally the principle of relative positioning, takes advantage of the correlated nature of GPS biases.

In the case of GPS phase data reductions:

• Between-satellite and between-receiver differencing effectively eliminates all clock biases due to oscillator errors, while significantly reducing most of the other biases (with the exception of the phase ambiguity, which requires special treatment).
• The ambiguity term in the phase observation model can be eliminated by forming between-epoch (or triple) differences (section 6.3.6).
• The ambiguity term in the phase observation model can also be accounted for by estimating it in a preliminary "ambiguity-free" solution as a real-valued quantity, and then resolve it to its most likely integer in a subsequent "ambiguity-fixed" solution.
• Although all non-clock biases may be "significantly reduced" as a result of data differencing, some comments are worth making:
  • With the implementation of Selective Availability, the orbit errors may become larger and baseline accuracies may suffer. Although there is no evidence at present that the orbits are degraded, if this were the case surveyors are likely to request the "precise ephemerides" (for example, from the International GPS Service).
  • Errors in the coordinates assigned for the fixed station in a baseline, or network, reduction are not likely to have a significant effect on GPS surveying accuracies attained. The influence is similar to orbit errors, hence if seeking better than 5ppm relative positioning accuracy, coordinates derived from the averaging of pseudo-range solutions are suitable.
  • The ionospheric refraction is ignored in single frequency observations, but can be largely eliminated in dual-frequency observations for longer baselines (see section 6.4). For baselines <30km in length, even when dual-frequency data is available, analysts usually only use the L1 observations.
  • The tropospheric refraction is more problematic. In general it is ignored, though some software packages apply refraction "corrections" to the data using some atmospheric model, either with or without the aid of observed met parameters (BRUNNER & WELSCH, 1993).
    

Because pseudo-range data is of the order of a hundred to a thousand times "noisier" than phase data, there is not the same imperative to eliminate (or minimise) biases, as in the case of carrier phase data. In the case of pseudo-range data reductions:

• Explicit data differencing is not generally carried out.
• The satellite clock error is assumed known from the broadcast Navigation Message (section 3.3.2).
• The satellite orbit is assumed known, and any error in the ephemeris (together with any unmodelled satellite clock error) has a significant effect on point positioning -- exactly the impact intended of Selective Availability.
• The receiver clock error must be estimated directly from the data.
• By differencing the point positions obtained at two (or more) GPS receivers, the effects of orbit and satellite clock errors are reduced. Hence the relative position of the receivers derived from pseudo-range solutions is an order of magnitude more accurate than absolute position.
• Generally the ionospheric refraction effect is accounted for by using the broadcast ionospheric model within the Navigation Message (section 3.3.4).

Explicit expressions for the mathematical models for the following observations types which have a role in GPS surveying can now be developed:

pseudo-range data

one-way carrier phase

double-differenced phase

triple-differenced phase

Pseudo-Range Model

The basis for the model is the parametric equation:

(7.2-1)

where P_jⁱ is the pseudo-range measurement in metric units, _jⁱ is the geometric range, c is the speed of electromagnetic radiation (=299799458m/s), Tj is the time-of-reception and ercj is the receiver clock error (in seconds). Note that the satellite clock error is not included in the above observation equation. Hence there are four unknown parameters per receiver per epoch. A minimum of four observations would permit all the parameters to be estimated on an epoch-by-epoch basis. This of course is the basis of the standard GPS "navigation solution".

Pseudo-range observations, in the context of GPS data post-processing, are used in a slightly different manner than they would in the navigation mode:

• Advantage is taken of the fact that the receiver does not move during an entire observation session. Hence there are many pseudo-range observations available to estimate the three coordinate component parameters. The coordinate solution provides apriori values for the subsequent phase reduction step.
• Because the receiver clock error is the means by which receiver clock time (and hence the observation time-tags) can be related to GPS Time, a polynomial model of the clock error may suffice for this purpose:

(7.2-2)

One-Way Phase Model

In length units, eqn (6.1-13) becomes, for either L1 or L2 phase data:

(7.2-3)

where ^sci is the satellite clock error, T_jⁱ is the time of signal transmission, n_jⁱ is the ambiguity in the phase measurement, and is the wavelength of the carrier wave used to scale cycles into length units (19cm for L1 and 24cm for L2). All other terms have been defined previously. Note the absence of atmospheric delay biases -- it is assumed that after data differencing the residual biases are so small that they can be considered as data "noise".

Note that n_jⁱ is an integer, though .n_jⁱ is not. Eqn (7.2-3) contains three clock biases that must be either explicitly estimated or eliminated by data differencing: n_jⁱ, _rcj and ^sci.

Consequences of data differencing: Differencing increases the measurement noise level. Differencing introduces correlations between the resulting "observables". Differencing decreases the number of parameters in the observation model. Differencing decreases the number of "observables" that need to be processed. Differencing possibilities: between-receivers --> operator between-satellites --> operator between-epochs --> operator between-observation types --> P1, P2, C, 1, 2

Double-Differenced Phase Model

In length units, eqn (6.3-7) can be written as:

(7.2-4)

where t is the time-tag of the observable constructed from four one-way phase observations which have been made within 30 millisecond of each other (section 6.3). Note that there are now only two classes of parameters: coordinates and ambiguities. There are a number of alternative ambiguity modelling options (see discussion later in this section). In the event that the ambiguities are resolved, eqn (7.2-4) can be rewritten as a double-differenced range equation in which the ambiguity terms are eliminated from the parameter set. A solution using this "reduced" phase observable is known as an "ambiguity-fixed" solution.

Triple-Differenced Phase Model

In length units, eqn (6.3-8) is rewritten as:

(7.2-5)

where a and b designate the two epochs involved. In general the epochs are consecutive, but this need not be so (for example, the between-epoch differences can be constructed always with respect to the first epoch in the session).

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