Undifferenced Carrier Phase Observable

Superscripts refer to the satellite, subscripts refer to ground station

: Carrier phase observable in cycles refers to the carrier phase observable from SV p to Station k.

: Carrier frequency

: Speed of light

: The topocentric range is the range from SV p to Station k.

: Receiver clock error as a function of time

: SV clock error as a function of time

: The integer ambiguity from SV p to Station k

: Ionospheric advance is the Ionospheric advance from SV p to Station k in cycles

: Tropospheric delay is the tropospheric delay from SV p to Station k

: Receiver hardware delays in cycles as a function of time

: Multipath in cycles as a function of time

: Satellite hardware delays in cycles as a function of time

: Measurement noise in cycles

is the actual phase observable recorded in the receiver. The terms to the right of the equal sign model various components that make up the observable. is the initial integer count of the number of cycles from SV p to Station k. This is also referred to as the integer ambiguity. Unlike the other modeled terms to the right of the equal sign, it is not a function of time, as long as the receiver maintains lock on the SV signal this number will not change.

When a receiver locks onto a signal from the GPS satellite, it continuously monitors the satellite transmission. At predetermined epochs, the receiver records the data at that epoch. The frequency with which the receiver records data is the data sampling rate. The data sampling rate is frequently incorrectly described as "epochs". For example, it is often, "Data was collected at 30 second epochs." The correct terminology is, "Data was collected with a data sampling rate of 30 seconds." An epoch is a particular instant in time. The time between epochs is an interval.

The single difference observable, double difference observable and the triple difference observable are all linear combinations of the undifferenced observable.

Single Difference Observable

A single difference is the difference between two undifferenced observables for the same satellite at the same epoch.

is the single difference between SV p and Stations k and m at epoch t.

The satellite clock errors and satellite hardware delays cancel.

Double Difference Observable

A double difference observable is the difference between two single difference observables for the same pair of ground stations with different satellites at the same epoch.

is the double difference observable between SV p and q and Stations k and m at epoch t.

Now the receiver clock errors and hardware delays cancel.

For short baselines, the double difference ionospheric term and the double difference tropospheric term will trend towards zero.

The resulting double difference observable is:

The double difference multipath term and the double difference measurement noise term are unmodeled so the final double difference observable is:

Triple Difference Observable

A triple difference observable is the difference between two double difference observables for successive epochs.

is the triple difference between SV p and q and Stations k and m at epoch and epoch .

If the receiver retains lock between epochs, the double difference ambiguity remains the same for each epoch. This being the case the double difference ambiguity cancels in the triple difference observable. If the receiver loses lock between sampling epochs this is no longer true. In that case, the triple difference solution that includes the interval during which satellite lock was lost will show up as an outlier. For this reason the triple difference solution can be used to spot during post processing.