# Single-receiver single-channel multi-frequency GNSS integrity: outliers, slips, and ionospheric disturbances

## Abstract

In this contribution the integrity of single- receiver, single-channel, multi-frequency GNSS models is studied. The uniformly most powerful invariant test statistics for spikes and slips are derived and their detection capabilities are described by means of minimal detectable biases (MDBs). Analytical closed-form expressions of the phase-slip, code-outlier and ionospheric-disturbance MDBs are given, thus providing insight into the various factors that contribute to the detection capabilities of the various test statistics. This is also done for the phaseless and codeless cases, as well as for the case of a temporary loss-of-lock on all frequencies. The analytical analysis presented is supported by means of numerical results.

### Keywords

Global navigation satellite systems (GNSS) Single-receiver single-channel model Phase-slips and code-outliers Ionospheric disturbance Uniformly most powerful tests Minimal detectable bias (MDB)## 1 Introduction

Integrity monitoring and quality control can be exercised at different stages of the GNSS data processing chain (Teunissen and Kleusberg 1998; Leick 2004). These stages range from the single-receiver, single-channel case to the multi-receiver/antenna case, sometimes even with additional constraints included. An example of the latter is the quality control of baseline-constrained GNSS attitude models (Giorgi et al. 2012), while geometry-dependent receiver autonomous integrity monitoring (RAIM) is an example of the single-receiver, multi-channel case (Teunissen 1997; Wieser et al. 2004; Feng et al. 2006; Hewitson and Wang 2006).

In the present contribution, we study the single-receiver, single-channel model. It is the weakest model of all, due to the absence of the relative receiver-satellite geometry. Despite its potential weakness, there are several advantages to single-receiver, single-channel data validation. First, since it is the simplest model of all, it can be executed in real-time inside the (stationary or moving) receiver, thus enabling early quality control on the raw data. Second, the geometry-free single-channel approach has the advantage that no satellite positions need to be known per se and thus no complete navigation messages need to be read and used. Additionally, such an approach also makes the method flexible for processing data from any other (future) GNSS or for parallel processing, which could prove relevant when considering a large number of receivers.

We study the carrier phase-slip and code-outlier detection capabilities of the single-receiver, single-channel model. For the integrity monitoring of carrier-phase data, various studies can already be found in the literature. For dual-frequency GPS data, for instance, methods of carrier phase-slip detection are discussed and tested in (Lipp and Gu 1994; Mertikas and Rizos 1997; Blewitt 1998; Teunissen 1998a; Gao and Li 1999; Jonkman and de Jong 2000; Bisnath and Langley 2000; Bisnath et al. 2001; Liu 2010; Miao et al. 2011). More recent studies on triple-frequency carrier phase-slip detection can be found in (Fan et al. 2006; Dai et al. 2009; Wu et al. 2010; De Lacy et al. 2011; Xu and Kou 2011; Fan et al. 2011). Our contribution differs from these previous studies, because of its focus on the detectability of single-receiver, single-channel modeling errors. Next to the phase-slip detection, the detectability of code-outliers is studied as well. Our analysis is analytical, while supported by numerical results. Analytical expressions are derived for the minimal detectable biases (MDBs) of the uniformly most powerful invariant tests (Baarda 1967, 1968; Teunissen 1990a). The MDB is an important diagnostic tool for inferring the strength of model validation. Examples of such studies for geometry-dependent and integrated GNSS models can be found in (Salzmann 1991; Teunissen 1998b; de Jong 2000; de Jong and Teunissen 2000; Hewitson and Wang 2010).

This contribution is organized as follows: In Sect. 2 we formulate the multi-frequency, single-receiver, geometry-free GNSS model. This is done for an arbitrary number of frequencies. An overview of the model’s redundancy for different measurement scenarios gives a first indication of the model’s testability. In Sect. 3 the uniformly most powerful invariant test-statistics for spikes and slips are developed. It is shown how they can be applied to test for code-outliers, phase-slips and ionosphere disturbances. The strength of these test-statistics is described by their corresponding MDBs, for which lower bounds and upper bounds are also given. Due to the relatively simple structure of the geometry-free model, the expression for the MDB can be decomposed into a time-dependent and a time-invariant component. The effect of the time-dependent component is shown in Sect. 3, while the characteristics of the time-independent part are studied in the sections following. The detectability of phase-slips is treated in Sect. 4. An analytical expression for the phase-slip MDB is derived and it is used to assess the single-, dual- and multi-frequency phase-slip detectability for GPS and Galileo. To evaluate the influence of the code data, the analysis is performed for both the case with code data present and without code data present. The latter case is also of interest, for instance, when one wants to avoid the use of multipath corrupted code data. In Sect. 5, an analytical expression for the code-outlier MDB is derived. It is used to study the code-outlier detectability for the single-, dual- and multi-frequency GPS and Galileo case, including the case that phase data are absent. In Sect. 6, the MDB for an ionospheric disturbance is presented and analyzed. Finally, the detectability of temporary loss-of-lock on all phase observables is treated in Sect. 7.

## 2 The multi-frequency, single-receiver geometry-free model

### 2.1 Functional model

*first*frequency. Thus for the \(f_{j}\)-frequency pseudorange observable, its coefficient is given as \(\mu _{j}=f_{1}^{2}/f_{j}^{2}\). The GPS and Galileo frequencies and wavelengths are given in Table 1. The parameters \(b_{\phi _{j}}\) and \(b_{p_{j}}\) are the phase bias and the instrumental code delay, respectively. The phase bias is the sum of the initial phase, the phase ambiguity and the instrumental phase delay.

GPS and Galileo frequencies (\(f\)) and wavelengths (\(\lambda \))

L1 | L2 | L5 | E1 | E5a | E5b | E5 | E6 | |
---|---|---|---|---|---|---|---|---|

\(f\) (MHz) | 1,575.42 | 1,227.60 | 1,176.45 | 1,575.42 | 1,176.45 | 1,207.14 | 1,191.795 | 1,278.75 |

\(\lambda \) (cm) | 19.0 | 24.4 | 25.5 | 19.0 | 25.5 | 24.8 | 25.2 | 23.4 |

*absolute*ionospheric delays, but rather on the

*relative*, time-differenced, ionospheric delays. We therefore have the additional (pseudo) observation equation

### 2.2 Stochastic model

To determine the variance matrix of the time-differenced ionospheric delays, let \(D (\mathcal I )=Q_\mathcal{I \mathcal I }\) be the variance matrix of the *absolute* ionospheric delay vector \(\mathcal I =(\mathcal I (1), \ldots , \mathcal I (k))^{T}\). The variance matrix of the time- differenced ionospheric delay vector \(d I =(D_{k}^{T} \otimes 1) \mathcal I \) is then given as \(D (d \mathcal I )=D_{k}^{T}Q_\mathcal{I \mathcal I }D_{k}\).

It is through the choice of \(Q_\mathcal{I \mathcal I }\) that we can model the time-smoothness of the ionospheric delays. If we assume that the time series of ionospheric delays can be modeled as a *first-order autoregressive* stochastic process, then the covariance between \(\mathcal I (t)\) and \(\mathcal I (s)\) is given as \(\sigma _\mathcal{I }^{2} \beta ^{|t-s|}\), with \(0 \le \beta \le 1\). The two extreme cases are \(\beta =0\) and \(\beta =1\). In the first case, \(Q_\mathcal{I \mathcal I }\) is a scaled unit matrix and \(\mathcal I (t)\) is considered a *white noise* process. In the second case, the variance matrix equals the rank-one matrix \(Q_\mathcal{I \mathcal I }=\sigma _\mathcal{I }^{2}e_{k}e_{k}^{T}\) and \(\mathcal I (t)\) is considered a *random constant*. In the first case we have \(D (d \mathcal I )=\sigma _\mathcal{I }^{2}D_{k}^{T}D_{k}\), while in the second case we have \(D (d \mathcal I )=0\).

### 2.3 Redundancy

*and*code case, and the code-only (phaseless) and phase-only (codeless) cases. When both phase and code data are used, the ionosphere-weighted redundancy equals \((k-1)(2n-1)\). Thus in this case, redundancy exists for any number of frequencies, provided \(k \ge 2\). That at least two epochs of data are needed is of course due to the fact that we are working with time-differenced data. That already single-frequency (\(n=1\)) processing provides redundancy is due to the ionospheric information. Without this information, there would be no redundancy in the single-frequency case, but only in the dual- and multi-frequency cases, provided both phase and code data are used.

Redundancy for \(k\)-epoch, \(n\)-frequency, iono-weighted and iono-float, single-receiver geometry-free model (6)

Phase and code | Phase-only | Code-only | |
---|---|---|---|

I-weighted | \((k-1)(2n-1)\) | \((k-1)(n-1)\) | \((k-1)(n-1)\) |

I-float | \(2(k-1)(n-1)\) | \((k-1)(n-2)\) | \((k-1)(n-2)\) |

The phase-only and code-only redundancies are the same. In the phase-only and code-only cases we have \((k-1)n\) observations less than in the phase and code case. Hence, this is the number by which the redundancy drops when either the code data or the phase data are left out. Thus in the phase-only or code-only cases, single-frequency testing is impossible even if ionospheric information is provided.

## 3 Testing and reliability

In this section we formulate our alternative hypotheses and present the corresponding test statistics.

### 3.1 Outliers, cycle slips and loss-of-lock

We now formulate our alternative hypotheses for the single-receiver, geometry-free GNSS model. They accommodate model biases such as outliers in the pseudo range data, slips in the carrier phase data and loss-of-lock.

Recall that the undifferenced observational vector of epoch \(t\) is given as \(y(t)=(\phi (t)^{T}, p(t)^{T}, \mathcal I (t))^{T}\). Now assume that a model error has occurred in the data of epoch \(l\) and that this \((2n+1)\)-bias vector can be parametrized as \(Hb\), where \(H\) is a given matrix of order \((2n+1) \times q\) and \(b\) is an unknown vector having \(q\) entries. Then \(Hb\) is the difference between the expectation of \(y(l)\) under the null hypothesis \(\mathcal H _{0}\) and the expectation of \(y(l)\) under the alternative hypothesis \(\mathcal H _{a}\). Thus \(E (y(l)| \mathcal H _{a})=E (y(l)| \mathcal H _{0})+Hb\). Through the choice of matrix \(H\), we can describe the type of model error. For instance, if all the phase data are assumed erroneous, as would be the case after a temporary loss-of-lock on all phase observables, then \(H=(I_{n}, 0, 0)^{T}\) and \(q=n\). But if only the pseudo-range data on frequency \(j\) is corrupted with an outlier, then \(H=(0, \delta _{j}^{T},0)^{T}\) and \(q=1\), where \(\delta _{j}\) is an \(n\)-vector having a \(1\) as its \(j\)th entry and zeros elsewhere.

Apart from describing the model error through matrix \(H\), we also need to specify the time behavior of the model error. Here we consider *spikes* and *slips*. A model error behaves as a spike if it occurs at one and only one epoch. A model error is said to behave as a slip if it persists after occurrence. Examples of spikes are outliers in the pseudo-range data or in the ionospheric delays. Examples of slips are cycle slips in the phase data or momentary loss-of-lock.

If we assume the model error \(H b\) to behave as a spike at epoch \(l\), then \(E (y| \mathcal H _{a})=E (y| \mathcal H _{0})+ (s_{l} \otimes H) b\), where \(s_{l}\) is a \(k\)-vector having a \(1\) as its \(l\)th entry and zeros elsewhere. Would we assume the error to be persistent, however, as would be the case after a loss-of-lock or after a slip, then \(s_{l}\) is a \(k\)-vector having zeros in its first \(l-1\) entries, but \(1\)s in all its remaining entries.

*uniformly most powerful invariant*(UMPI) test is used, see e.g., (Arnold 1981; Koch 1999; Teunissen 2000). It rejects the null hypothesis in favor of the alternative hypothesis, if

### 3.2 Test statistics for spikes and slips

In order to derive the appropriate test statistics, we first determine the least-squares estimator of \(b\) in (9). Here and in the following we assume \(D (d \mathcal I )= \sigma _\mathcal{I }^{2} D_{k}^{T}D_{k}\) and therefore \(D (y(i))=\mathrm{blockdiag}(Q_{\phi }, Q_{p}, \sigma _\mathcal{I }^{2}) \overset{call}{=}Q\). The least-squares estimator of \(b\) and its variance matrix is given in the following theorem:

**Theorem 1**

*Proof*

See Appendix. \(\square \)

The bias estimator and its variance matrix are given the arguments \(l\) and \(k\) to emphasize that it is an estimator of an error occurring at epoch \(l\), based on \(k\) epochs of data.

From the Kronecker product structure of (13) it follows that \(\hat{b}(l,k)\) and its variance matrix can be computed directly from its single-epoch counterparts. We therefore have the following result:

**Corollary 1**

**Spikes**For spike-like biases the \(k\)-vector \(s_{l}\) is a unit vector having a \(1\) as its \(l\)th entry. If we make use of \(P_{D_{k}}= I_{k}-e_{k}(e_{k}^{T}e_{k})^{-1}e_{k}^{T}\), it follows that \(\bar{s}_{l}=\delta _{l}-\frac{1}{k}e_{k}\) and \(\bar{s}_{l}^{T}\bar{s}_{l}=1-\frac{1}{k}\). For spikes, the bias expression (16) therefore simplifies to

**Slips**For slip-like biases the \(k\)-vector \(s_{l}\) is a vector having \(1\)s as its last \(k-l+1\) entries and zeros elsewhere. If we make use of \(P_{D_{k}}= I_{k}-e_{k}(e_{k}^{T}e_{k})^{-1}e_{k}^{T}\), it follows that \(\bar{s}_{l}=s_{l}-\frac{k-l+1}{k}e_{k}\) and \(\bar{s}_{l}^{T}\bar{s}_{l}=\frac{(k-l+1)(l-1)}{k}\). We therefore have

Note that (21) reduces to (18) when \(l=k\). This shows that one cannot discriminate between spikes and slips on the basis of one single epoch. That is, one needs to have a delay (\(k>l\)) to be able to separate spikes from slips.

### 3.3 Minimal detectable biases

Under the alternative hypothesis \(\mathcal H _{a}\), the test statistic \(T_{q}\) is distributed as a noncentral \(\chi ^{2}\)-distribution with \(q\) degrees of freedom, \({T}_{q} \mathop {\sim }\limits ^\mathcal{H _{a}} \chi ^{2}(q,\lambda )\), where \(\lambda = b^{T}Q_{\hat{b}\hat{b}}(l,k)^{-1}b \) is the *noncentrality parameter*. Test (12) is an UMPI-test, meaning that for all \(b\), it maximizes the power within the class of invariant tests. Here, power, denoted as \(\gamma \), is defined as the probability of correctly rejecting \(\mathcal H _{0}\); thus \(\gamma =P [{T}_{q} > \chi ^{2}_{\alpha }(q,0)| \mathcal H _{a}]\).

The power of test (12) depends on the degrees of freedom \(q\) (i.e., the dimension of \(b\)), the level of significance \(\alpha \), and through the noncentrality parameter \(\lambda \), on the bias vector \(b\). Once \(q, \alpha \) and \(b\) are given, the power can be computed.

*quadratic*equation in \(b\):

*reliability*of the null hypothesis \(\mathcal H _{0}\) with respect to the alternative hypothesis \(\mathcal H _{a}\). For \(q=1\), Eq. (23) describes an interval, for \(q=2\) it describes an ellipse and for \(q>2\) it describes an (hyper)ellipsoid. Bias vectors \(b \in R^{q}\) that lie on or outside the ellipsoid (23) can be found with at least probability \(\gamma \).

*Minimal Detectable Bias*(MDB) vector of Baarda’s reliability theory (Baarda 1967, 1968). Its length is the smallest size of bias vector that can be found with probability \(\gamma \) in the direction \(d\) with test (12). By letting \(d\) vary over the unit sphere in \(R^{q}\) one obtains the whole range of MDBs that can be detected with probability \(\gamma \) with test (12). For the one-dimensional case \((q=1)\), we have \(d=\pm 1\) and therefore \(\mathrm{MDB}(l,k)=\sigma _{\hat{b}}(l,k)\sqrt{\lambda _{0}}\).

Baarda, in his work on the strength analysis of general purpose networks, applied his general MDB-form to data snooping, thus obtaining the scalar boundary values (in Dutch: ‘grenswaarden’). Applications of the vectorial form can be found, for example, in Van Mierlo (1980, 1981) and Kok (1982a) for deformation analysis, in Foerstner (1983) for photogrammetric linear trend testing and in Teunissen (1986) for testing digitized maps. For recursive testing, with application of testing time and types of error, the first innovation-based vectorial MDB was given in Teunissen (1990b). Application of the generalized eigenvalue problem to the vectorial form to obtain MDB-bounds can be found in e.g., (Teunissen 2000; Knight et al. 2010).

Earlier (cf. 12) it was assumed that the variance matrix \(Q_{\hat{b}\hat{b}}\) in the teststatistic \(T_{q}\) is known. In case, however, the variance factor \(\sigma ^{2}\) in \(Q_{\hat{b}\hat{b}}=\sigma ^{2}G_{\hat{b}\hat{b}}\) is unknown, then the Chi-square distributed teststatistic needs to be replaced by the \(F\)-distributed teststatistic \(T_{q,df}^{\prime }=\hat{b}^{T}G_{\hat{b}\hat{b}}^{-1}\hat{b}/(\hat{\sigma }^{2}q)\), in which \(\hat{\sigma }^{2}\) is the unbiased estimator of the variance factor under the alternative hypothesis and \(df\) is its degrees of freedom. This teststatistic is distributed under \(\mathcal H _{a}\) as \(T_{q,df}^{\prime } \sim F(q, df, \lambda )\), with the same noncentrality parameter as that of \(T_{q}\) (Kok 1982b; Koch 1999). Therefore also in case \(\sigma ^{2}\) is estimated, will the same MDB expression be found as given in (24). Hence, similarly as with the other statistical parameters, the effect on the MDB, for the two cases \(\sigma ^{2}\) known versus \(\sigma ^{2}\) estimated, is only felt through the scaling factor \(\lambda _{0}\). In this contribution we work with \(\lambda _{0}\) computed from the Chi-square distribution.

In the following, we present the MDBs for the two-epoch case (\(k=2, l=2\)). The corresponding MDB-values for arbitrary epochs can then be obtained by using the multiplication factor \(f(l,k)\) of (25).

## 4 Detectability of carrier phase slips

In this section we present and analyze the phase-slip MDBs. This is done for the single-, dual- and multi-frequency case. We also analyze the phase-slip MDBs in case code data are absent.

### 4.1 Minimal detectable carrier phase slips

In the following, MDB values have been computed numerically using the complete set of zenith-referenced code and phase variances according to Table 2, and analytical MDB expressions have been derived for which the phase and code variance matrices have been simplified to scaled unit-matrices, i.e., \(Q_{\phi \phi }=\sigma _{\phi }^{2}I_{n}\) and \(Q_{pp}=\sigma _{p}^{2}I_{n}\). For the standard deviation of the scaled unit-matrices we have used the mean value of the standard deviations of the available signals (except when explicitly stated otherwise). The MDB values from both the numerical computations and the analytical expressions are presented graphically. The differences between the two are shown to be small, especially when the used signals have comparable precision, which indicates that the analytical expressions indeed give a proper representation of the single-receiver, single-channel detectability (in the MDB graphs below, the dashed curves correspond to the analytical expressions, while the full curves correspond to the MDBs computed from the actual variance matrices). For all MDBs which pertain to an error on a single frequency, we have chosen to compute the MDB for an error on the first frequency of the available frequencies. For GPS this is the L1 frequency and for Galileo the E1 frequency when available.

The analytical expression for the multi-frequency phase-slip MDB is given in the following theorem:

**Theorem 2**

*Proof*

see Appendix. \(\square \)

Note that the slip MDB is scaled with \(\sigma _{\phi }\), the small standard deviation of the carrier phase observable. Thus if the bracketed ratio of (28) is not too large, small carrier phase slips will be detectable. The bracketed term depends on \(\lambda _{0}\) and on \(n^{*}\). The scalar \(n^{*}\) is dependent on \(\epsilon , \mu _{i} (i=1, \ldots , n)\) and \(\sigma _{\phi }^{2}/\sigma _{d\mathcal I }^{2}\). Hence, it depends on the measurement precision, on the number of frequencies and their spacings, and on the smoothness of the ionosphere. The MDB gets smaller if \(\epsilon \) gets larger, i.e., if more precise code data are used. The MDB also gets smaller if \(n\) gets larger, i.e., if more frequencies are used. Finally, note that the MDB-dependence on the frequency diversity (i.e., on \(\mu _{i}, i=1, \ldots , n\)) is driven to a large part by the smoothness of the ionosphere. This dependence is absent for \(\sigma _{d\mathcal I }=0\) and it gets more pronounced the larger \(\sigma _{d\mathcal I }\) gets.

We now analyze the slip MDB for the GPS and Galileo single-, dual- and triple-frequency cases.

#### 4.1.1 Single-frequency case

#### 4.1.2 Dual- and multi-frequency case

### 4.2 Cycle slip detection without code data

Since code data are generally much less precise than carrier phase data, one may wonder whether code data are really needed for carrier phase slip detection. This is also of interest for those applications where the code data are corrupted by multipath. In this section we therefore investigate what happens when \(\sigma _{p} \rightarrow \infty \). The corresponding MDBs can be obtained from Theorem 1 by taking the limit \(\lim _{\sigma _{p} \rightarrow \infty } \mathrm{MDB}_{\phi _{j}}\). We have the following result.

**Lemma 1**

*Proof*

Expression (32) follows from taking the limit of (29). Expression (33) follows from inverting (32) and rearranging terms. \(\square \)

The above two expressions clearly show the effect of frequency diversity. The MDB gets smaller, the larger the frequency diversity \(\sum _{i=1}^{n}(\mu _{i}-\bar{\mu })^{2}\) is. And within a set of \(n>1\) MDBs, the smallest \(\mathrm{MDB_{\phi _{j}}}\) is the one for which \(\mu _{j}\) is closest to the average \(\bar{\mu }\).

#### 4.2.1 Single-frequency case

In the single-frequency case we have \(n=1\) and thus no frequency diversity, i.e., \(\sum _{i=1}^{n}(\mu _{i}-\bar{\mu })^{2}=0\). This shows that \(n^{*}=1\) for \(n=1\) (c.f. 32) and that therefore \(\mathrm{MDB}_{\phi _{j}}=\infty \). Hence, phase-slip detection without code data is impossible for the single-frequency case (see also the redundancy Table 3).

#### 4.2.2 Dual-frequency case

When we compare Fig. 5 with Fig. 4, we note that the phase-slip MDB values are not too different for sufficiently small \(\sigma _{d\mathcal I }\), but that their differences increase for larger \(\sigma _{d\mathcal I }\). Thus the presence of the code data are particularly needed when the ionospheric delays are not smooth enough. Codeless dual-frequency phase-data is sufficient to detect phase-slips otherwise.

#### 4.2.3 Multi-frequency case

When we compare the codeless results of Fig. 6 with the triple- and quadruple-frequency results of Fig. 4, no big differences can be seen. Hence, the impact of the code data on the phase-slip MDBs becomes less pronounced if more than two frequencies are used. The only noteworthy difference between the results of the two figures is the performance of E1–E5a–E5b. This difference in performance, compared with the other triple-frequency results, is due to the small frequency separation of E5a and E5b.

## 5 Detectability of code outliers

In this section we present and analyze the code-outlier MDBs. This is done for the single-, dual- and multi-frequency case. We also analyze the code-outlier MDBs in case phase data are absent.

### 5.1 Minimal detectable code outliers

The analytical expression for the multi-frequency code-outlier MDB is given in the following theorem:

**Theorem 3**

*Proof*

As the phase observables and code observables play a dual role in the two-epoch model (4), the code-outlier MDB can be found from the expression of the phase-slip MDB (28) through an interchange of the phase and code variance. \(\square \)

Although (39) and (40) have the same structure as (28) and (29), respectively, there are marked differences between these expressions. First note that (39) is scaled by the standard deviation of code and not by that of phase as in (28). Second, the frequency-dependent term between brackets in (40) is multiplied with the very small phase-code variance ratio \(\epsilon \), while this is not the case with the bracketed term of (29). The consequences of these differences are that the dual- and multi-frequency code-outlier MDBs are generally larger than those of the phase-slip MDBs and that they are less sensitive to the frequencies. The exception occurs in the single-frequency case.

#### 5.1.1 Single-frequency case

In the single-frequency case, for \(k=2\), the code-outlier MDB is identical to that of the phase-slip MDB. This follows if we interchange the role of \(\sigma _{p}^{2}\) and \(\sigma _{\phi }^{2}\) in (30). For \(k>2\), however, the MDBs differ of course. In case of a slip the multiplication factor is \(\frac{1}{\sqrt{2}} \sqrt{\frac{1}{k-l+1}+\frac{1}{l-1}}\), while for an code-outlier it is \(\frac{1}{\sqrt{2}} \sqrt{1+\frac{1}{k-1}}\).

#### 5.1.2 Dual- and multi-frequency case

### 5.2 Code outlier detection without phase data

#### 5.2.1 Single-frequency case

Just like single-frequency codeless phase-slip detection is impossible, so is single-frequency code-outlier detection without phase data. This follows directly from Table 1, which shows that redundancy is absent, when phase data is absent in case \(n=1\). It also follows from (42), which shows that \(m^{*}=1\) if \(n=1\).

#### 5.2.2 Dual-frequency case

#### 5.2.3 Multi-frequency case

## 6 Detectability of ionospheric disturbances

The analytical expression for the multi-frequency ionospheric disturbance MDB is given in the following theorem:

**Theorem 4**

*Proof*

Let \(\hat{d \mathcal I }\), with variance (45), be the LS estimator of \(d\mathcal I \) based on the two-epoch model under \(\mathcal H _{0}\) (cf. 9) using phase and code data only. Then it follows from the structure of the model under \(\mathcal H _{a}\) that the LS bias estimator of the ionospheric disturbance is given as the difference \(\hat{b}_{d \mathcal I }=d\mathcal I -\hat{d \mathcal I }\). Therefore, its variance is given by the sum \(\sigma _{d\mathcal I }^{2}+\sigma _{\hat{d\mathcal I }}^{2}\), from which the result follows. \(\square \)

## 7 Detectability of phase loss-of-lock

Phase loss-of-lock is defined as the simultaneous occurrence of an unknown multivariate slip in all \(n\) carrier phase observables. To study its detectability, we first determine the variance matrix of the multivariate slip under \(\mathcal H _{a}\) and then provide bounds on the norm of the phase loss-of-lock MDB-vector.

### 7.1 The variance matrix of the multivariate slip

In the presence of a phase loss-of-lock, the design matrix of the alternative hypothesis (9) takes for \(k=l=2\) the form \([G, H]\), with \(H=[I_{n},0,0]^{T}\). From the structure of \([G,H]\), it follows that the carrier phase vector \(\phi (t,s)\) will not contribute to the estimation of the parameters \(\rho ^{*}(t,s)\) and \(\mathcal I (t,s)\) under \(\mathcal H _{a}\). These parameters are therefore solely determined by the code observables and a priori ionospheric information. As a consequence, the two-epoch bias estimator is given as the difference \(\hat{b}=\phi (t,s)-\hat{\phi }(t,s)\), where \({\hat{\phi }}(t,s)=e_{n}{\hat{\rho }}^{*}(t,s)- \mu {\hat{\mathcal{I }}} (t,s)\) is the least-squares phase estimator based solely on the code observables and a priori ionospheric information. Solving for \(\hat{\rho }^{*}(t,s)\) and \({\hat{\mathcal{I }}}(t,s)\), followed by applying the variance propagation law to \(\hat{b}=\phi (t,s)-e_{n}\hat{\rho }^{*}(t,s)+ \mu {\hat{\mathcal{I }}}(t,s)\) gives then the variance matrix of the multivariate slip. The result is given in the following Lemma:

**Lemma 2**

Note that the variance matrix is a sum of three matrices, the entries of which may differ substantially in size. The first matrix is governed by the precision of the phase observables and will therefore have small entries. The second matrix is governed by the precision of the code observables and will therefore have generally much larger entries than the first matrix in the sum. The third matrix depends, next to the precision of the code observables, also on \(\mu \) and \(\sigma _{d\mathcal I }^{2}\). Its entries will become smaller if \(\sigma _{d{\hat{\mathcal{I }}}}^{2}\) gets smaller. This happens for smaller \(\sigma _{d\mathcal I }^{2}\) (smoother ionospheric delays) and/or larger \(\mu ^{T}Q_{pp}^{-1}P_{e_{n}}^{\perp }\mu \) (better code precision and/or larger frequency diversity). Thus if \(\sigma _{d\mathcal I }^{2}=\infty \), frequency diversity is needed (i.e., \(\mu ^{T}Q_{pp}^{-1}P_{e_{n}}^{\perp }\mu \ne 0\)) so as to avoid the entries of the third matrix in (48) to become infinite.

We now discuss the detectability of the phase loss-of-lock for \(n \ge 2\). The single-frequency case \(n=1\) is already treated (cf. 30), since phase loss-of-lock is then equivalent to having a phase-slip.

### 7.2 Bounding the MDB vector

The situation changes drastically, however, if \(d\) is taken to lie in \(\mathrm{span}\{e_{n}, \mu \}\). In that case the large code variance dependent second and third term of (48) will contribute as well. The largest possible value that the length of the MDB vector can take corresponds with \(d\) being the eigenvector of \(Q_{\hat{b}\hat{b}}\) with largest eigenvalue. The corresponding bounds for the ionosphere-fixed and ionosphere-float case are given in the following Lemma:

**Lemma 3**

*Proof*

see Appendix. \(\square \)

The ionosphere-fixed upper bound corresponds with a slip in the \(e_{n}\)-direction, while the ionosphere-float upper bound corresponds with a slip in the direction \(e_{n}+\frac{\sqrt{n}}{||\mu ||}\mu \). Thus phase loss-of-lock is most difficult to detect if it results in a slip with such direction. For example, for the ionosphere-fixed, dual- and triple-frequency (\(q=2, 3\)) cases, the length of the MDB-vector will then be about six to seven times the code standard deviation.

## 8 Summary and conclusions

In this contribution we presented an analytical and numerical study of the integrity of the multi-frequency single-receiver, single-channel GNSS model. The UMPI test statistics for spikes and slips are derived and their detection capabilities are described by means of the concept of minimal detectable biases (MDBs). Analytical closed-form expressions of the phase-slip and code outlier MDBs have been given, thus providing insight into the various factors that contribute to the detection capabilities of the various test statistics. This was also done for the phaseless and codeless cases, as well as for the case of loss-of-lock.

The MDBs were evaluated numerically for the several GPS and Galileo frequencies. From these analyses it can be concluded that detectability of dual- and triple-frequency phase-slips works well for \(k=l=2\). Single-frequency phase-slip detectability, however, is problematic, thus requiring more epochs of data.

From the codeless phase-slip MDBs it follows that detectability is not possible in the single-frequency case, but that it is possible for the dual- and triple-frequency cases. In the dual-frequency case the codeless phase-slip MDBs are less then 10 cm if \(\sigma _{d \mathcal I } \le 3\) cm, while for the triple-frequency case this is already true for \(\sigma _{d \mathcal I } \le 1\) m. In the triple-frequency case, the phase-slip MDBs get even as small as a few centimeters if \(\sigma _{d \mathcal I } \le 3\) cm. These codeless results are important as it shows that in the presence of code multipath, one can do away with the code data and then still have integrity for phase slips.

The code outlier MDBs are, except for the single-frequency case, all relatively insensitive to the smoothness of the ionosphere. The effect of the frequencies is also hardly present in the multi-frequency code-outlier MDBs. Their value is predominantly determined by the precision of the code measurements. From the phaseless code-outlier MDBs it follows that, except for the single-frequency case, code outlier detection is still possible. The multi-frequency phaseless code-outlier MDBs all lie around 1 m for \(\sigma _{d \mathcal I } \le 10\) cm. They increase rapidly, however, for larger values of \(\sigma _{d \mathcal I }\).

## Notes

### Acknowledgments

The first author is the recipient of an Australian Research Council (ARC) Federation Fellowship (project number FF0883188). Part of this work was done in the framework of the project ‘New Carrier-Phase Processing Strategies for Next Generation GNSS Positioning’ of the Cooperative Research Centre for Spatial Information (CRC-SI2). The work of the second author was partly supported by the EU Marie Curie program in the framework of the FP7-PEOPLE-IAPP-2008 (SIGMA) project. All this support is gratefully acknowledged.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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