SECT_6.4.1

6.4.1 Dual Frequency Relations

INTRODUCTORY REMARKS

As discussed in section 6.2.7, the L-band signals transmitted by the GPS satellites are delayed as they travel through the ionosphere and the troposphere. The residual effects remaining after between-station differencing are generally small for short baselines, and hence are often neglected. For longer interstation distances the ionospheric and tropospheric effects cannot be ignored because:

these unmodelled biases begin to degrade the solution by an appreciable amount (the magnitude of its effect tends to grow in proportion to the baseline length), and
they make the ambiguity resolution process more difficult because they tend to "destroy" the integer nature of the ambiguity parameters.

The ionospheric delay is highly unpredictable, being a function of the latitude of the receiver, the elevation angle to the satellite, the time-of-day of the observations, and the level of solar activity at the time of observation (itself dependent on a number of factors such as the season, the 11 year solar cycle, etc.).

How to handle the ionospheric biases? One way is to use the corrections based on the ionospheric model transmitted by the satellites themselves in the Navigation Message. Although many navigation-style (single frequency) GPS receivers apply these corrections automatically to the pseudo-ranges (section 3.3.4), they are likely to only model 25-50% of the daily variability. Hence, because the ionospheric range bias equivalent is of the order of tens of metres, a significant residual bias remains in the phase observations. Between-station differencing may bring this down to several decimetres in extreme cases, such as for baselines greater than 100km in length, observed in daylight, during a solar cycle maximum.

Another, more appropriate, way of accounting for the ionospheric delay is to observe on both L-band frequencies (section 6.2.7). However, dual-frequency observations offer many possibilities of generating new observables, some with interesting properties and uses. These are discussed below.

LINEAR COMBINATIONS OF DUAL-FREQUENCY OBSERVATIONS

The effect of the ionosphere on GPS observations can be considered in terms of the time delay (_ion), phase change (_ion) or range (or group) delay (d_ion). A first-order approximation for the ionospheric bias is (section 6.2.7):

(6.4-1)

where:

_ion is the ionospheric delay in seconds,

_ion is the ionospheric phase delay in cycles,

d_ion is the ionospheric group delay in metres,

c is the speed of electromagnetic radiation in a vacuum (m/s),

f is the signal frequency (Hz), and

STEC is the Slant Total Electron Content of a column of ionosphere condensed onto a disc, expressed as the number of free electrons per square metre (el/m₂).

The ionosphere causes the integrated carrier phase count to decrease (that is, the apparent phase velocity is greater than the velocity of light!), but causes the pseudo-range to appear longer than the geometric range. In the remainder of these notes only the group delay term d_ion will be used in the pseudo-range and phase observation equations. Note that the time delay is proportional to the inverse of the frequency squared. That is, higher frequencies are less affected by the ionosphere, and hence the ionospheric time delay for L1 observations (1575.42MHz) is less than for L2 observations (1227.60MHz). From eqn (6.4-1) the relationship between the time delays on the two frequencies f1 and f2 due to the ionosphere is:

f₁² . d_ion(L1) = f₂² . d_ion(L2) (6.4-2)

Hence the L2 ionospheric effect is approximately 1.646 times that on L1 (1.646 f₁² / f₂²).

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_ion	is the ionospheric delay in seconds,
_ion	is the ionospheric phase delay in cycles,
d_ion	is the ionospheric group delay in metres,
c	is the speed of electromagnetic radiation in a vacuum (m/s),
f	is the signal frequency (Hz), and
STEC	is the Slant Total Electron Content of a column of ionosphere condensed onto a disc, expressed as the number of free electrons per square metre (el/m₂).