IAR Methods
Methods of Integer Ambiguity Resolution
There are many different ways for computing integer ambiguities from the real-valued float ambiguities. The three most popular methods are: (1) integer rounding, (2) integer bootstrapping, and (3) integer least-squares.
![combo](https://gnss.curtin.edu.au/wp-content/uploads/sites/21/2015/09/temp-combo-320x152.jpg)
Integer rounding
This method is the simplest of all. In this case the integer ambiguity solution is obtained from a component-wise nearest integer rounding of the float ambiguity vector. Integer rounding may be used if its probability of correct integer estimation, referred to as success-rate, is sufficiently close to 1. The success-rate depends on the precision of the float solution; the more precise the float solution, the higher the success-rate. The following graph and table show how, for the scalar case, the success-rate of rounding depends on the ambiguity standard deviation. To obtain a success-rate of 99.9%, a standard deviation of about 0.15 cycles is needed.
![formula](https://gnss.curtin.edu.au/wp-content/uploads/sites/21/2015/09/temp-0181-320x180.jpg)
Computing the rounding success-rate is not that easy anymore in the vectorial case. In that case one can resort to simulation or make use of the lower bound.
![formula](https://gnss.curtin.edu.au/wp-content/uploads/sites/21/2015/09/temp-020-320x180.jpg)
For computing the lower bound, only the ambiguity standard deviations are needed.
![adjust](https://gnss.curtin.edu.au/wp-content/uploads/sites/21/2015/09/temp-022-320x180.jpg)
Integer bootstrapping
This method is a combination of integer rounding and sequential conditional least-squares estimation. Before a component of the float ambiguity solution is rounded, it is first adjusted following the integer values of its previous components:
Instead of the standard deviations, one now needs the conditional standard deviations as input for the success-rate computation. These conditional standard deviations are the square-roots of the nonzero entries of the diagonal matrix of the ambiguity variance matrix’ LDU-decomposition.
Bootstrapping is better than rounding, since it can be shown that the bootstrapped success-rate is never smaller than the rounding success-rate (PT 1998).
![formula](https://gnss.curtin.edu.au/wp-content/uploads/sites/21/2015/09/temp-026-320x180.jpg)
Integer least-squares
This method uses all the entries of the ambiguity variance matrix and it is defined as follows,
![formula](https://gnss.curtin.edu.au/wp-content/uploads/sites/21/2015/09/temp-026-320x180.jpg)
This method provides the best integer estimator of all, since it can be shown to have the largest success-rate of all admissible integer estimators.
![formula](https://gnss.curtin.edu.au/wp-content/uploads/sites/21/2015/09/temp-030-320x180.jpg)
Simulation is used to compute the least-squares success-rate. Alternatively, one can make use of the easy-to-compute ADOP-based approximation.
![ADOP](https://gnss.curtin.edu.au/wp-content/uploads/sites/21/2015/09/temp-32-320x204.jpg)
The ADOP (Ambiguity Dilution Of Precision) is defined as the determinant of the ambiguity variance matrix to the power 1/2n. It is a generalized ambiguity precision measure that has the important property of being invariant against ambiguity re-parametrization (PT 1997). The below graph shows the ADOP-based success-rate as function of the ADOP for varying n. It shows that for n ≤ 20, an ADOP of 0.12 cycles gives a success-rate better than 99.9%.
![formula](https://gnss.curtin.edu.au/wp-content/uploads/sites/21/2015/09/temp-034-320x180.jpg)
As an alternative to simulation or approximation, one may use bounds of the least-squares success-rate. The least-squares success-rate is bounded from below by the easy-to-compute bootstrapped success-rate and it is bounded from above as: