GNSS integer ambiguity posterior probability calculation with controllable accuracy

Abstract

Integer ambiguity resolution (IAR) is one of the key techniques in GNSS high precise positioning. However, an overlooked incorrect integer ambiguity solution may cause severe biases in the positioning results. The optimal integer aperture estimator (IAE) has the largest possible success rate given a certain fail rate. An alternative approach that take advantage of ambiguity integer nature to minimize the solution’s mean squared error (MSE) is known as the best integer equivariant (BIE) estimator. Both of which are associated with the posterior probability of the GNSS integer ambiguity. It is therefore of great significance to calculate posterior probability precisely and efficiently. Due to the occurrence of infinite sums, practical calculation approaches approximate the exact value by neglecting sufficiently small terms in the sum. As a result, they can only produce posterior probability calculation result, information about the result’s accuracy cannot be produced. In this contribution, the value of the posterior probability is bounded from below and from above by dividing the infinite sum into two parts: the major finite part and the minor infinite part. They are calculated partly by enumeration and partly by algebraical bounding. The obtained upper and lower bounds are rigorous and in closed form, so that can be conveniently used. Based on both of the bounds, a method of posterior probability calculation with controllable accuracy is proposed. It not only produces posterior probability calculation result, but also calculation error, which is always smaller than the user-defined acceptable error. Numerical experiments have verified that the proposed approach has advantages on both controllable calculation accuracy and adjustable computational workload.