6.4.2 Dual Frequency Relations

IONOSPHERE-FREE COMBINATION

This is also sometimes known as the "L3" combination. The ionospheric refraction bias can be eliminated by constructing a combined ionosphere-free phase or pseudo-range observable from the L1 and L2 data.

Consider the phase observations on two frequencies. The mathematical model of the undifferenced (or one-way) carrier phase measurement has been given in eqn (6.1-13), and can be simplified (neglecting: (a) the clock error terms, which are eliminated during double-differencing, and (b) the tropospheric and orbit biases, which are significantly reduced for short baselines) for the L1 frequency as:

(6.4-3a)

Similarly for the L2 frequency carrier phase:

(6.4-3b)

Note that n_jⁱ_(L1) is not equal to n_jⁱ_(L2).

In the following equations the subscripts (receiver identifier), superscripts (satellite identifier) and time arguments are dropped. The L1 ambiguity term is simply expressed as n₁, and similarly the L2 ambiguity is denoted by n₂. _(L1) denotes the phase of the L1 signal in cycles, _(L2) refers to the phase of the L2 signal in cycles.

Multiplying each of the above phase observations (in units of cycles) by the signal frequency, and then differencing them gives:

(6.4-4)

where the second term of right hand side of eqn (6.4-4) is equal to zero due to the relation at eqn (6.4-2). Hence the equation degenerates to:

(6.4-5)

In order to combine the L1 and L2 phase observations, which are in units of cycles (of different wavelengths for L1 and L2), they have to be converted to the same units, for example, scaling by the L1 frequency:

(6.4-6)

yields the following corrected L1 phase measurement:

(6.4-7a)

Alternatively, if eqn (6.4-5) is scaled by the L2 frequency, the corrected L2 phase measurement is obtained:

(6.4-7b)

Eqn (6.4-7) has the exact form of the original raw carrier beat phase observation eqn (6.1-13), except that the integer ambiguity term is replaced by the linear combination of the L1 and L2 ambiguities. This is also called the L3 linear combination of the phase data, or simply the L3 ionosphere-free observable. The relationship between the L1, L2 and L3 cycles, measured in units of L1 wavelengths, is:

L3[(L1)cycles] 2.546L1[(L1)cycles] – 1.984L2[(L2)cycles]

(6.4-8a)

and measured in L2 wavelengths is:

L3[(L2)cycles] 1.984L1[(L1)cycles] – 1.54L2[(L2)cycles]

(6.4-8b)

The n₃ ambiguity is related to the L1 and L2 ambiguities as follows:

n3[(L1)cycles] 2.546n1 – 1.984n2	(6.4-9a)
n3[(L2)cycles] 1.984n1 – 1.54n2	(6.4-9b)

Hence n₃ is not an integer combination of L1 and L2 ambiguities! An alternate from with integer coefficients is introduced in the discussion following the Table in section 6.4.6.

It should be emphasised that _(L1) is the measured L1 phase in units of ₁ wavelengths (0.19m), while _(L2) is the measured L2 phase in units of ₂ wavelengths (0.24m). In order to obtain the L3 observable in units of metres, both sides of eqn (6.4-7a) have to be multiplied by ₁, or both sides of eqn (6.4-7b) have to be multiplied by ₂.

The ionosphere-free combination of pseudo-ranges is derived in the following manner. A convenient model of the L1 and L2 pseudo-range observations is (note that the sign of the ionospheric delay is reversed compared to the carrier phase equation):

P(L1) = + dion(L1)	(6.4-10a)
P(L2) = + dion(L2)	(6.4-10b)

Multiplying eqn (6.4-10) by the signal frequency squared, and then differencing gives:

f12.P(L1) - f22.P(L2) = (f12 - f22). + ( f12.dion(L1)- f22.di on(L2))

(6.4-11)

The equivalent pseudo-range L3 combination therefore is (last term in eqn (6.4-11) is zero):

(6.4-12)

Unfortunately the L3 combinations (phase or pseudo-range) have approximately three times the noise of the L1 observations (see Table in section 6.4.6).

Note: all the expressions derived here are just as valid for the double-differenced observables as they are for the one-way observations. For example, eqn (6.4-7) can be also written as:

(6.4-13)

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© Chris Rizos, SNAP-UNSW, 1999