220.127.116.11 Double Difference Observable
By subtracting two between satellites single differences the double difference observable Φ12ab is formed in metres. From equation 1.27 for two receivers a and b making simultaneous measurements at the same nominal time to satellites and it follows:
Where Φ12α-Φ12b are the single difference carrier phase measurements between receivers α and b to satellites 1 and 2 for an epoch t, observed in cycles. The receiver independent error sources that affect the carrier phase measurements are: a) the ionospheric Ι and tropospheric Τ biases along the signal path, b) multipath effect δm and c) measurement noise ε . The double difference geometric distance is represented by the sumbol ρ.
In the above equation receiver and satellite clock offsets and clock biases cancel, since double differencing is effectively differencing between satellites and between receivers. The single difference ambiguities difference Na12 – Nb12 is commonly parameterised as a new ambiguity parameter Na12 . Another advantage of double differencing is that the Nb12 parameter is an integer because the non-integer terms in the GPS carrier phase observation, due to clock and hardware delays in the transmitter and receiver, are eliminated.
Usually more double difference ambiguities that there are original data can be formed, by differencing parts of other double differences. However, there is no more new information created from the additional equations and such observations should not be processed. The set of double difference equations for a given number of stations and satellites that cannot be formed as a linear combination from the involved equations is the linearly independent set of double differences. For a single baseline the linearly independent set will be s – 1, where s is the number of satellites. A linearly independent set must be used in least squares processing.
The double difference operator D transforms a column vector of measurements l into a column vector of double difference data.
The double difference operator is a rectangular matrix with number of rows equal to the linear independent set of double difference observables and number of columns equal to the number of measurements. The elements of matrix consist of values +1, ‑1 and 0, arranged in a way to produce the independent set of double differences.
Another useful principle of double differencing is that the sum of the double difference ambiguities around a closed loop is zero. This principle is employed by network ambiguity resolution techniques and used in Multiple Rover Network algorithms.