1. Introduction
Nowadays, highly precise positioning and navigation solutions are obtained by using carrier-phase based positioning techniques such as real-time kinematic (RTK) positioning [
1]. High precision is achieved once the initial phase ambiguities, inherent with carrier-phase observations, are resolved [
2]. However, owing to signal blockage by obstacles and receiver motion, the continuous tracking of GNSS carrier-phase signals between two consecutive measurement epochs might get interrupted. Such a tracking loss, termed as a cycle slip, introduces a bias in carrier-phase measurements. As a consequence of cycle slips (CS), carrier-phase ambiguities need to be resolved again to avail RTK positioning accuracy. Single-epoch ambiguity resolution is very challenging in RTK because of receiver dynamics and the time-sensitive nature of the kinematic solution. Therefore, instead of resolving ambiguities again, it is beneficial if CS are detected and the corresponding measurements are taken care of [
3,
4]. CS detection techniques are able to perform better for dual-frequency receivers due to their ability to eliminate ionospheric effects on measurements. However, single-frequency receivers occupy more than 60% share of the current GNSS receiver market [
5]. Coupled with the demand for higher positioning accuracies for commercial users, the focus has now shifted to using single-frequency receivers for providing accurate and precise navigation solutions for the mass market [
6,
7,
8]. Single-frequency differential positioning is expected to be the primary tool for a plethora of consumer applications in areas as varied as geomatics, precision agriculture, location-based services, internet of things, and mHealth [
9].
For the wide variety of RTK commercial applications, quality control of received data in real-time positioning is an area of concern [
10,
11]. This is due to GNSS data being subjected to measurement errors such as CS, which, if not detected, significantly degrade overall system performance [
12]. Quality control involves the estimation of variances, quality tests of significance on variances, and possibly other checks [
13]. In geodesy, the quality of a navigation solution can be catered to by the least squares (LS) adjustment theory. The LS adjustment theory deals with an optimal combination of unknown measurements, together with the combination of unknown parameters [
14,
15,
16]. Using this theory, it has been established that, given the availability of redundant measurements, measurement biases can be detected by using statistical tests in connection with the adjustment of networks [
17]. For positioning using code-pseudoranges, several receiver autonomous integrity monitoring (RAIM) techniques have been developed for GNSS data quality monitoring based on adjustment theory [
18,
19,
20]. Measurement quality control is practiced by assessing, detecting, and isolating failure situations through a fault detection and exclusion procedure [
18]. By applying RAIM techniques, the quality of the position fix can also be quantified during the design procedure. In this respect, the important aspects to consider are the quality of the position fix result under nominal conditions (precision) and the sensitivity of the position fix to undetected model errors (reliability) [
21]. Although RAIM was traditionally designed for systems utilizing code-phase measurements for positioning, these techniques have recently been applied to CS detection in carrier-phase based positioning schemes [
22]. This CS detection approach can be further extended to incorporate the evaluation of reliability in position fixing.
The concept of reliability was introduced by Baarda [
14] in the context of statistical testing for outlier detection during LS adjustment for the determination of the navigation solution. As per definition, the strength of a system model depends on the level of confidence one has in the outcomes of the statistical tests. This confidence is monitored by the reliability of the fix [
23]. Specifically, reliability refers to the ability of the system to detect measurement outliers (internal reliability) and the effect that undetectable outliers have on the estimated values derived from these measurements (external reliability) [
23,
24]. Internal reliability is defined in terms of minimal detectable bias (MDB) [
23]. These are the biases that may be found with a certain probability in the outlier test. External reliability is defined in terms of the marginally detectable error (MDE) [
11]. It is the influence of undetected bias on the final result of a geodetic adjustment. The statistical testing procedure presented in [
14] was broken down into three parts, namely, detection, identification, and adaptation (DIA) by Teunissen [
25]. It is seen that to determine the reliability of a position fix, the statistical tests in detection and identification steps are interrelated [
14,
23,
25].
For single-frequency receivers, statistical testing during LS adjustment was used for CS detection in references [
22,
26,
27,
28,
29,
30,
31]. All the implemented schemes were able to detect CS for an MDB of one cycle [
9]. However, none of the mentioned schemes have discussed MDE, whereas it is recommended that the reliability measure of a differential position fix should be expressed in terms of external reliability [
11]. Similarly, the relationship between the statistical tests during detection and identification is not considered; as a result, the reliability of the position fix cannot be asserted for any of the CS detection techniques for single-frequency receivers. This paper aims to bridge this void by presenting a detailed procedure to detect CS and determine a reliable position fix for single-frequency RTK positioning. The process is led by the DIA procedure. The chosen level of reliability is achieved by deriving decision thresholds in detection and identification stages through equating their non-centrality parameters determined from their respective probability density function (PDF). The proposed framework is tested on two kinematic datasets, and theoretical values of MDB and MDE are obtained. After CS are introduced in carrier-phase measurements, it is seen that they can be detected and a reliable position fix is obtained given the magnitude of CS is four cycles or more. It is observed that although the theoretical value of MDB is one to two cycles, the actual values are slightly higher. This is mainly attributed to the value of detection and identification thresholds determined from the chosen level of significance in the local test.
The proposed framework provides an in-depth procedure for incorporating the concept of positioning reliability with CS detection for single-frequency receivers. The flow of the paper is as follows.
Section 2 discusses the CS detection algorithm for single-frequency RTK positioning. The single-frequency RTK positioning model developed for this research is introduced, followed by the LS adjustment and CS detection process through the DIA framework. In
Section 3, the concept of positioning reliability is introduced, and the process of determining a reliable position fix by exploring the relationship between decision thresholds in the statistical tests for detection and identification is presented.
Section 4 presents the results of the proposed framework. It describes the experimental setup, choice of parameters, and the magnitude of detected CS both theoretically and in practice. The discussion is concluded in
Section 5.
3. Reliable Positioning
To obtain consistent high-precision positioning results with GPS carrier-phase measurements, errors unspecified in the functional or stochastic model should be correctly detected and removed or otherwise handled at the data processing stage [
39]. Reliability refers to the system’s capability to detect such errors and to estimate the effects that they may have on the position. Reliability is measured by stating the size of error that might remain undetected with a specified probability [
11]. Both internal and external reliability are distinguished in this respect. The internal reliability of a GNSS positioning solution is its ability to detect outliers for the chosen level of significance and power of test. External reliability informs of the impact of undetected errors on estimated positions [
40]. A high internal reliability implies that small errors can be detected. High external reliability implies that statistically undetectable outliers have very little effect on the final position [
11]. Reliability is driven by accuracy of observations, adjustment redundancy, and satellite geometry [
16,
23,
35].
To ensure that the model error
is reliably detected, with the same probability by both the overall model test and the
w-test, the B-method of testing is used [
14,
35]. In this method, the
F-test of the detection step and the
w-test of the identification step are linked with each other. Given that
is the non-centrality parameter of the
statistic for GT and
is the non-centrality parameter for the
statistic for LT, the parameter
is given as
The procedure is to make a choice for
and
and calculate
and
from the given relationship. This choice of equal values for the non-centrality parameter
and power
in both tests implies that a certain model error can be found with the same probability by the
F-test and the
w-test. Both tests will, therefore, have the same reliability. Therefore, an adjustment is unreliable if after a GT failure, the LT does not fail because there is an inconsistency between the two tests, i.e.,
is accepted. For a chosen value of
for the LT, the procedure for determining values for
and the corresponding threshold in the GT is given in
Table 1 [
14,
24,
35,
41]. It should be noticed that Step 1 is chosen at the design stage of the system whereas Steps 2 to 5 are conducted on an epoch by epoch basis. The values in Step 3 are derived from the monograms given in [
14].
3.1. Internal Reliability
Internal reliability is defined as the error that can be detected by the generalized likelihood ratio test with a probability of correct detection being
. It is expressed in terms of minimal detectable bias (MDB). By definition, the MDB of an alternative hypothesis is the smallest outlier that can lead to the rejection of a null hypothesis for the given probability level
and
[
37]. Since for the proposed framework, it is assumed that only one observation is corrupted by CS at a single epoch, the following expression can be given for the MDB
as [
16,
42]
where
is the shift in mean for the two hypotheses. The value for
can be determined as [
43]
It is seen that varying
and
directly affects the reliability statement, so whenever an MDB is quoted, it should relate to both
and
[
11]. Unless the data has a very large number of outliers, any level of significance
from 0.1% to 5% is expected to lead to identical results [
11]. On the other hand, since the MDB indicates the magnitude of outliers that can be found with a reasonable certainty, in order for the MDB to be a meaningful figure,
has to be fairly large [
11].
3.2. External Reliability
It is recommended that the reliability measure of a differential position fix should be expressed in terms of external reliability [
11]. External reliability is defined as the influence of undetected bias
on the final results of a geodetic computation or adjustment. It is expressed in terms of a marginally detectable error (MDE) [
11]. The MDE, computed for all observations, is viewed as a measure of the capability of the network to detect blunders with probability
[
16]. A positional MDE is the effect of an undetected observational bias, with a magnitude that corresponds to the size of MDB, on the computed position [
35,
40,
44]. The positional MDE can be determined as [
40]
External reliability is assessed by the largest horizontal positional MDE [
11].
The framework to assess whether a position fix is reliable is represented by the flow chart in
Figure 3. When there are no CS in measurements,
for GT is true, the solution is deemed reliable, and the reliability parameters, i.e., the MDB for all visible satellites and the MDE values, are evaluated. For this scheme, LT is carried out for fault identification only if
of the GT is rejected, and only the observation with the largest value of
is tested and possibly rejected. However, once the GT fails, but no CS are identified in local test, the solution is deemed unreliable and the position fix is computed. The status as to whether the solution is reliable or not, and the case when the former is true, the MDB and MDE values, as well as the position, are displayed at the user front end.
5. Conclusions
A framework for CS detection and determination of a reliable position fix for single-frequency RTK receivers is presented in this paper. The scheme uses DD measurements to detect CS during an LS adjustment. Once detected, CS-contaminated measurements can be eliminated from the adjustment model and position fix, along with reliability parameters MDB and MDE being computed. From the reliability assessment of the proposed scheme on two dynamic datasets, it is seen that MDB depends on the level of significance
chosen in the LT and the number of observed satellites. MDB increases as
decreases. However, the choice of
and
does not affect the MDB significantly. MDB increases as the number of visible satellites decreases. In addition, although theoretically the MDB is one or two cycles for the chosen values of
and
, in practice it is four cycles for the two scenarios. This can be attributed to measurement noise which was ignored while developing the single-frequency RTK model in Equation (5). MDB can be decreased and detection can be improved by lowering the value of
in the LT, which lowers decision thresholds for both the tests. However, it was seen that this causes false flags and several measurements were incorrectly identified as CS. Therefore the recommended values of
and
were retained. Also, it is less likely to have very small cycle slips (e.g., one to two cycles) in the data and it is usually hidden in the higher noise levels in kinematic navigation with low-cost equipment [
49].