Integer Ambiguity Resolution

Along with the consideration of ambiguous carrier phase measurements, the ambiguities need be resolved to achieve precise positioning. During the formation of differential measurements, the integer nature of the ambiguities are conserved as much as possible. The Integer Ambiguity Resolution (IAR) is the process of benefitting the integer nature to obtain a fast and reliable centimeter-level navigation solution if it succeeds. But also it can lead to unacceptable errors compromising the position integrity when done wrong. As a consequence, there is always a fine line between keeping float ambiguity values that are reliable or accepting fixed ambiguity values that may be corrupted. This is why IAR should strictly be composed of two processes: (1) integer ambiguity estimation to obtain optimal integer ambiguity estimates and (2) validation process to validate the best integer ambiguity candidate [10], [12].

In the first subsection, the notion of Integer Aperture Estimators (IAE) is devised which enables an overall theoretical analysis of IAR. A brief introduction of several popular integer ambiguity estimation techniques is then provided, including the Least-squares AMBiguity Decorrelation adjustment (LAMBDA) method. At the end, integer ambiguity validation procedures are described to provide a discuss on the reliability of the obtained integer ambiguity solution.

Integer Aperture Estimator

To assess the combination of the fixed ambiguity assessment and validation process and provide an overall theoretical basis for IAR, the idea of Integer Aperture Estimator (IAE) is presented in [12]. The general idea of the IAE is the distinction of three situations:

 success where the correct integer estimation is accepted;

 failure where the false integer estimation is accepted;

 uncertainty where integer estimation is rejected.

The related probability parameters are depicted in following table [48]:

Table 3-1. Relevant parameters of Integer Aperture Estimator

Correct Integers 𝑃𝑠,𝐼𝐿𝑆 Wrong Integers 𝑃𝑓,𝐼𝐿𝑆

Accept 𝑃𝑓𝑖𝑥 = Success 𝑃𝑠 + Failure 𝑃𝑓

+ +

Reject 𝑃𝑓𝑙𝑜𝑎𝑡 = False Alarm 𝑃𝑓𝑎 + Detection 𝑃𝑑

Intuitively, the failure case is the most undesired condition that instead of shrinking the float solution uncertainty level, a severely biased fixed solution might be obtained. Therefore, an acceptable upper threshold for the value 𝑃𝑓 is recommended. An optimal IAE is thus the one which maximizes the probability of success 𝑃𝑠 and on the same time the 𝑃𝑓 does not go beyond the predefined limit [11], [49].

Integer Ambiguity Estimation

Any GNSS navigation system can be parametrized in integers 𝑎 ∈ 𝛧^𝑛and non-integers 𝑏 ∈ 𝑅^m. A generic representation of the observation model is proposed as the following linear(ized) equation:

𝑦 = 𝐴𝑎 + 𝐵𝑏 + 𝜖, 𝑤𝑖𝑡ℎ 𝑎 ∈ 𝛧^𝑛, 𝑏 ∈ 𝑅^𝑚 (3-10)

where

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 𝑎 is the vector of integer parameters (i.e., integer ambiguities),

 𝐴 is the design matrix for integers, and

 𝑏 is the vector of non-integer states to be resolved in 𝑅 (i.e., the baseline increments, receiver clock delay) with 𝐵 the related design matrix, and 𝜖~𝑁(Θ, 𝑄𝑦) is the Gaussian observation noise.

The integer ambiguity estimation is mostly accomplished following three steps [50], [51].

 First, the constraint that 𝑎 ∈ 𝛧^𝑛 is disregarded to obtain the real-valued estimates (𝑎̂, 𝑏̂)^𝑇 from the output of an estimation filter for instance a Kalman Filter (KF) or a LS. The resulting VC-matrix of 𝑎̂ and the so-called float solution 𝑏̂ is thereby

[𝑎̂

𝑏̂]~[𝑄_𝑎̂ 𝑄_𝑎̂𝑏̂

𝑄_𝑏̂𝑎̂ 𝑄_𝑏̂] , 𝑤𝑖𝑡ℎ 𝑎̂ ∈ 𝑅^𝑛, 𝑏̂ ∈ 𝑅^𝑚 (3-11)

 Second, search or calculate the integer ambiguity estimates 𝑎̌ from the float estimates

𝑎̌ = 𝑆(𝑎̂) (3-12)

where 𝑆: 𝑅^𝑛→ 𝛧^𝑛is the integer estimator, a mapping function which directs all float point in R^𝑛 into a specific point in Ζ^𝑛.

 Third, calculate the fixed solution 𝑏̌ by adjusting the float solution in the following way 𝑏̌ = 𝑏̂|𝑎̌=𝑏̂ − 𝑄_𝑏̂𝑎̂𝑄_𝑎̂⁻¹(𝑎̂ − 𝑎̌). _(3-13) Various integer estimators have been proposed since 1980. Among those, the most intuitive and direct integer estimator is the integer rounding estimator [11]. Float ambiguities are rounded to their closest integers. However, the correlation among those ambiguities is then not considered.

A sequential version, the bootstrapped estimator was proposed in [13], [52]. Starting from the first ambiguity term, all following float ambiguities are decorrelated from previous ambiguities before being rounded. Nevertheless, there still lacked a solid theory on the performance evaluation of such integer estimators, until the LAMBDA method and a performance indicator, namely the probability of success fixing 𝑃𝑠, were brought up [14], [15], [53].

Comparisons in terms of computational efficiency and performance between different integer ambiguity estimation techniques are presented in [15], [50], [53], [54]. As a result, the LAMBDA methodology has been acknowledged as the optimal integer estimator in terms of maximizing 𝑃𝑠. A brief introduction of LAMBDA is herein given. For more detailed information, refer to [55], [56].

The LAMBDA is fundamentally an integer least-squares (ILS) estimator as it strives to minimize the following quadratic objective function

𝑎∈𝛧𝑚𝑖𝑛^𝑛,𝑏̂∈𝑅^𝑚‖𝑦 − 𝐴𝑎 − 𝐵𝑏‖_𝑄²_𝑦 = 𝑚𝑖𝑛

𝑎∈𝛧^𝑛,𝑏̂∈𝑅^𝑚 (‖𝑒̂‖_𝑄²_𝑦+ ‖𝑎 − 𝑎̂‖_𝑄²_𝑎̂+ ‖𝑏 − 𝑏̂|𝑎‖_𝑄

𝑏̂|𝑎

2 ) (3-14) In the right side of the previous equation is the orthogonal decomposition of the objective function into three parts. 𝑒̂ is the measurement residual adhering to the float estimates (𝑎̂, 𝑏̂)^𝑇. 𝑏̂|𝑎 is the conditional least-squares baseline vector conditioned on 𝑎 ∈ Ζ^𝑛, having 𝑄𝑏̂|𝑎 as the VC-matrix. The first part ‖𝑒̂‖𝑄2_𝑦

is irrelevant to the integer estimate 𝑎̌, and the third part can be null by taking the fixed baseline solution as 𝑏̌ = 𝑏̂|𝑎̌. Consequently, the mapping function under the least-squares idea is fundamentally to derive the ILS estimates 𝑎̌ by resolving

𝑎∈𝛧𝑚𝑖𝑛^𝑛‖𝑎 − 𝑎̂‖_𝑄²_𝑎̂ = (𝑎 − 𝑎̂)^𝑇𝑄_𝑎̂⁻¹(𝑎 − 𝑎̂) (3-15)

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In case of a diagonal 𝑄𝑎̂, meaning that integer ambiguities are independent from each other, the search for ILS estimates 𝑎̌ would be quite simple. However, for GNSS applications, all ambiguities are strongly correlated. Hence, a reduction stage by a Z-transformation is primarily settled to simplify the next discrete searching stage. A modified version was brought up in [55] accounting for effectively reducing computational burdens.

The probability of having the final ILS estimates 𝑎̌ equal to the real integer values is quantified by 𝑃𝑠,𝐼𝐿𝑆. The whole procedure of ILS search makes it impossible to derive a compact theoretical formula of 𝑃𝑠,𝐼𝐿𝑆. However, a lower bound is provided by the IB estimator [13], [15]

𝑃_{𝑠,𝐼𝐿𝑆}≥ 𝑃_{𝑠,𝐼𝐵}= ∏ (2𝛷 ( 1

 𝝈𝒂̂(𝒊|𝑰,𝒊|𝑰) is the standard deviation of the i-th ambiguity obtained through a conditioning on the previous 𝑰 = 𝟏, … , (𝒊 − 𝟏) ambiguities, and

Theoretically, a parameter estimation should not be considered complete without an appropriate process to validate the solution [11]. Practically, the original intention of fixing the integer parameters in GNSS applications is to decrease the positioning uncertainty on the basis of float solution. A wrong fixing of integer ambiguities may lead to a much severer consequence (i.e., a remarkable positioning bias leading to integrity issues). Therefore, the integer ambiguity validation process is never an omissible part for ensuring a reliable GNSS precise positioning.

The most popular validation methods are the Ratio-test (RT), the F-ratio test (FT) and the Difference test (DT). Comparison between them have been the topic of many publications [48], [57]–[59].

Those three validation methods are all striving to make a reliable discrimination between the best (𝑎̌) and the second best (𝑎̌2) integer candidates of ambiguities, but with different considerations. For its implementation simplicity and well-functioning in practical tests, the detailed implementations of the RT are described herein as an example:

Accept 𝑎̌ if ^(𝑎̌_{(𝑎̌−𝑎̂)}^{2−𝑎̂)𝑇}𝑇^𝑄𝑄^𝑎_𝑎^̂−1_̂^−1(𝑎̌(𝑎̌−𝑎̂)^2−𝑎̂)≥ 𝑐 (3-18) where 𝑐 is a pre-defined threshold or the critical value that the squared norm of ambiguity residuals of the best and second best candidates should overpass to validate the integer estimation.

Regarding the determination of the critical value, a rigorous theoretical analysis of the Eq(3-18) should be done to derive the value. However, it is theoretically too complex and the computation burden is heavy [59]–[61]. Therefore, an empirical fixed value is generally taken (e.g., 𝑐 = 3 as in [7], [62]), namely a Fixed Threshold RT (FT-RT). However, it should be pointed out that it is not the correctness of integer candidates but rather the closeness to the float value that is tested with RT [62], [63]. The popular indicator of IAR performance, the 𝑃𝑓 can not be controlled with the FT-RT and the confidence put on the final solution should therefore be relatively low.

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