Geoscience Reference
In-Depth Information
where Φ i ( t ) is the measured carrier phase expressed in cycles referred to
station i and satellite j at epoch t . The time-independent phase ambiguity
N i is an integer number and, therefore, often called integer ambiguity or
integer unknown or simply ambiguity.
Relative positioning requires simultaneous observations at both the ref-
erence and the unknown point. This means that the observation time tags
for the two points must be the same. Assuming such observations (5-14) at
the two points A and B to satellite j and another satellite k simultaneously
at epoch t , the following measurement equations may be set up:
Φ j A ( t )= 1
λ j A ( t )+ A ( t )+ N A ,
Φ A ( t )= 1
λ A ( t )+ A ( t )+ N A ,
Φ j B ( t )= 1
(5-15)
λ j B ( t )+ B ( t )+ N B ,
Φ B ( t )= 1
λ B ( t )+ B ( t )+ N B .
Introducing the short-hand notations
Φ j AB ( t )=Φ B ( t )
Φ j B ( t )
Φ A ( t )+Φ j A ( t ) ,
j AB ( t )= B ( t )
j B ( t )
A ( t )+ j A ( t ) ,
(5-16)
N j AB
N B
+ N A ,
= N B
N A
we form the double-difference model which is defined as
Φ j AB ( t )= 1
λ j AB ( t )+ N j AB .
(5-17)
Note that the receiver clock biases have canceled; this is the reason why
double-differences are preferably used. This cancellation resulted from the
assumptions of simultaneous observations and equal frequencies of the satel-
lite signals (which is justified for GPS).
Assuming A as reference station with known coordinates, the remain-
ing unknowns of the double-difference model are the desired coordinates
X B ,Y B ,Z B - which are comprised in j B ( t )and B ( t ) - and the ambiguities.
To solve for these unknowns, we need more satellites (to set up additional
double-differences) and also more epochs.
We do not consider linearization, possible redundant measurements, etc.
We just intended to demonstrate the principle. The desired result obtained
from (5-17) is the baseline vector b AB with the components ∆ X AB , Y AB ,
Z AB or, finally, the GPS coordinates X B ,Y B ,Z B
derived from (5-10) via
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