Sect_6.1.1

6.1.1 GPS Observation Equations

CARRIER BEAT PHASE

The GPS satellites transmit on two L-band frequencies: L1 at 1575.42MHz and L2 at 1227.60MHz. The observation equation for the carrier beat phase is developed below, and is valid for measurements made on either the L1 or L2 frequency. The following development is principally taken from GRANT et al (1990). The reader is also referred to TIBERIUS & DE JONGE (1995) for a condensed treatment of the derivation of the GPS carrier phase observation equation.

Clock Phase Error

The phase of an oscillator at time T can be represented by (T). The behaviour of phase with time in general obeys the relation:

(6.1-1)

where f(T) is the time dependent frequency of the oscillator and T₀ is the time at some arbitrary reference epoch. In developing the observation equations for carrier beat phase it is useful to assume that every clock, or oscillator, can be compared directly with a "perfect" oscillator having a known and constant frequency f₀ in the reference time scale (for example, GPS Time). Because the frequency of the "perfect" oscillator is constant at f₀ the relation becomes:

(T) = (T₀) + f₀ . (T – T₀) (6.1-2)

The oscillators in the GPS satellites and receivers are used to generate timed signals such as the P code, the C/A code and the observation time-tags. It is common therefore, to think of them as clocks and to consider the errors caused by the variation of frequency as clock errors. This use of the term "clock error" can cause difficulties when it is necessary to distinguish between the carrier phase error caused by variations in the receiver oscillator frequency, and errors in the observation time-tag which are also affected by these frequency variations. It is preferable to use the terms clock phase error (or clock range error in the case of pseudo-range measurements) for the former and time-tag error for the latter (RIZOS & GRANT, 1990). Although the carrier phase and observation time-tags are generated by the same oscillator they are generally treated as if they were independent in GPS processing. For instance, the time-tags are often corrected according to the clock error estimates obtained from a pseudo-range point position solution. These corrections are not generally accompanied by an equivalent correction to the observed carrier phase. Using the symbol (T) to represent the phase of a real (though imperfect) GPS satellite or receiver oscillator at time T the clock phase error (T) can be defined in terms of the phase error ((T) – (T)):

(6.1-3)

(T) = (T) + f₀.(T) (6.1-4)

Of course, in using eqn (6.1-3) or eqn (6.1-4) account must be taken of both the integral and fractional components of the phase observation. This point is important later when considering how phase is actually measured and modelled.

Signal Transit Time

Consider the transmission of a signal from a satellite i to a receiver j. The time-of-reception of a signal at receiver j is T_j. The time-of-transmission of that signal from satellite i is T_jⁱ. The transit time of the signal from i to j is _jⁱ(T_j) and is defined by:

_jⁱ(T_j) = T_j – T_jⁱ (6.1-5)

The beat phase formed as an observation in a GPS receiver is the difference between the phase of the local receiver oscillator and the phase of the received signal. (This is the sense that is now adopted for the beat observation, earlier instrumentation had this difference reversed -- see, for example, GRANT et al (1990).) The phase of the received signal at the time-of-reception T_j is equal to the phase of the transmitted signal at the time-of-transmission T_jⁱ. Therefore:

_bjⁱ(T_j) = _loj(T_j) – _rjⁱ(T_j)

= _loj(T_j) – ^ti(T_jⁱ) (6.1-6)

where :

_bjⁱ(T_j) is the carrier beat phase for receiver j, satellite i, at reception time T_j,

_rjⁱ(T_j) is the received signal phase from satellite i at receiver j at time T_j,

_loj(T_j) is the local oscillator phase of receiver j at time T_j, and

^ti(T_jⁱ) is the transmitted signal phase from satellite i at transmission time T_jⁱ.

From eqn (6.1-4) the expression relating phase and clock phase error of the local oscillator can be written as:

_loj(T_j) = (T_j) + f₀._rcj(T_j) (6.1-7)

where _rcj(T_j) is the clock phase error of GPS receiver j. Using eqns (6.1-2), (6.1-4) and (6.1-5), an expression relating transmitted phase and satellite clock phase error, at reception time, can be formed:

^ti(T_j) = (T_jⁱ) – f₀.^sci(T_jⁱ)

= (T_j) – f₀.^sci(T_jⁱ) - f₀._jⁱ(T_j) (6.1-8)

where ^sci(T) is the clock phase error attributable to the oscillator of GPS satellite i. Combining eqns (6.1-6), (6.1-7) and (6.1-8), leads to an expression for carrier beat phase:

_bjⁱ(T_j) = f₀.[ _jⁱ(T_j) – ^sci(T_jⁱ) + _rcj(T_j) ] (6.1-9)

The transit time component in eqn (6.1-9) is made up of two parts. The main part is derived from the geometric range _jⁱ between the satellite at the time-of-transmission and the receiver at the time-of-reception. This can be determined from the position vectors of the satellite and the receiver, when expressed in the same reference system. Given the speed of the signals (c, the speed of electromagnetic radiation in a vacuum) it is possible to calculate the time taken for the signal to travel this distance ( _jⁱ ). The value used for c is the speed in a vacuum and hence this is the primary means of defining metric scale in GPS. It is the satellite and receiver position information contained in the geometric range that is important for GPS positioning.

The second part of the transit time term accounts for the extra time taken for the signal to travel through the earth's atmosphere. This is caused by a change in speed but can also be modelled as a time delay, a phase delay or as an increase in range. This can be incorporated as a phase correction term _atmos. (This may be further broken down into components for the different parts of the atmospheric delay; _ion for the ionosphere, _trop for the troposphere -- see section 6.2.1.) Hence:

f₀._jⁱ(T_j) = ( f₀ / c )._jⁱ(T_j) + _atmos (6.1-10)

Combining eqns (6.1-9) and (6.1-10) a model for carrier beat phase in units of cycles can be developed:

_bjⁱ(T_j) = ( f₀ / c )._jⁱ(T_j) + f₀.[ _rcj(T_j) – ^sci(T_jⁱ) ] + _atmos (6.1-11a)

and in metric units:

_bjⁱ(T_j) = _jⁱ(T_j) + c. [ _rcj(T_j) - ^sci(T_jⁱ) + ._atmos (6.1-11b)

where is the wavelength of the carrier wave ( = c / f₀ ). The range term and its relationship to the two clock errors (receiver and satellite clocks) is illustrated in Figure 1 below (note: the notation used is: dT = _rcj, dt = ^sci, and p = _bjⁱ).

Figure 1. Geometric range and clock errors.

Measured Carrier Beat Phase

The measured carrier beat phase differs from the above modelled carrier beat phase in a number of important respects. Firstly, the carrier beat phase measurements will have random noise (and signal interference such as multipath) associated with the measurement process. This is contained in an additional term _noise.

Secondly, the definition of clock phase error in eqn (6.1-4) depends on the integral and fractional part of the carrier phase. When the carrier beat phase is measured in the receiver the measurement is ambiguous with regards to the number of integer cycles. This integer is set in a "counter" to some arbitrary value when the satellite signal is first acquired (for example, zero). There is therefore an unknown integer number of cycles n_jⁱ difference between the measured carrier beat phase and the model beat phase in eqn (6.1-11). This will be unique to a particular satellite-receiver pair. Furthermore, it will be a constant for as long as the receiver continues to track and count the integer number of cycles from the time the satellite signal is first acquired (see Figure 2). At any epoch other than the initial measurement epoch, the instrument measures the fractional phase _fjⁱ(T_j) and, in addition, takes a reading C_R(T_j) on the counter. This combined fractional and integer phase observation is therefore referred to as integrated carrier beat phase, and n_jⁱ is the cycle ambiguity.

Thirdly, if the receiver at any time fails to track the signal correctly, and the ambiguity changes between observation epochs, then a cycle slip S(T_j) has occurred.

Figure 2. Integrated carrier beat phase measurement.

The model for the measured carrier beat phase therefore includes these three additional terms:

(6.1-12)

This is the basic observation equation for integrated carrier beat phase. The dependence of _atmos and _noise on the receiver, satellite or epoch, are not explicitly shown, however these terms will vary slightly for every observation. The change in observed carrier beat phase with time is therefore equal to the change in any one of the following quantities:

the satellite-receiver geometric range,
the satellite-receiver clock phase error difference,
the number of cycle slips,
the tropospheric delay,
the ionospheric delay, or
the measurement noise and signal disturbance.

Mathematical Model for Integrated Carrier Beat Phase

Eqn (6.1-12) is an example of a physical model of the integrated carrier beat phase observation. In order to develop an appropriate parameter model for carrier phase data processing, a mathematical model needs to be developed. The mathematical model incorporates only those terms of the physical observation model that will be parameterised for the adjustment, but without explicitly defining the functional models for these terms. Eqn (6.1-12) can therefore be written in the standard form:

_bjⁱ(T_j) = ( f₀ / c )._jⁱ(T_j) + f₀.[_rcj(T_j) – ^sci(T_jⁱ)] + n_jⁱ + _atmos (6.1-13a)

or in terms of metric units:

_bjⁱ(T_j) = _jⁱ(T_j) + c.[_rcj(T_j) – ^sci(T_jⁱ)] + .n_jⁱ + ._atmos (6.1-13b)

Note that the fractional and integer part of the observation are now combined into a single term ⁱ(T_j), and that the noise and cycle slip terms have been dropped. (Cycle slips are assumed to be independently determined during a "pre-processing" step, and hence are absent from the phase data as modelled above -- see section 7.3.)

Eqn (6.1-13) is valid for either L1 or L2 carrier phase observations, however, there are several frequency dependent biases: the ionospheric delay and the ambiguity term (section 6.2). All other terms are identical.

The mathematical model for pseudo-range, or "code-phase" as it is sometimes referred to, is also eqn (6.1-13), but without the cycle ambiguity term. However the sign of the ionospheric delay is reversed (section 6.4.1).

The observation model for Doppler data can be derived from eqn (6.1-13) by taking the time derivative of all the quantities. In that case, the constant ambiguity term is eliminated and other biases such as those due to the atmosphere can also be considered as being relatively insignificant. The rate of change of clock error, however, may still be significant enough to warrant additional parameterisation in the observation equation.

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_bjⁱ(T_j)	= _loj(T_j) – _rjⁱ(T_j)
	= _loj(T_j) – ^ti(T_jⁱ)	(6.1-6)

_bjⁱ(T_j)	is the carrier beat phase for receiver j, satellite i, at reception time T_j,
_rjⁱ(T_j)	is the received signal phase from satellite i at receiver j at time T_j,
_loj(T_j)	is the local oscillator phase of receiver j at time T_j, and
^ti(T_jⁱ)	is the transmitted signal phase from satellite i at transmission time T_jⁱ.

^ti(T_j)	= (T_jⁱ) – f₀.^sci(T_jⁱ)
	= (T_j) – f₀.^sci(T_jⁱ) - f₀._jⁱ(T_j)	(6.1-8)