1. Introduction
Attempts to determine the total ambiguities in phase measurements in GPS observations date back to the late 1970s—i.e., GPS satellites were first placed in Earth orbits in 1978 [
1,
2]. The first research paper to appear on GPS measurement and determination of ambiguities in phase measurements for GPS observations was published by Counselman and Gourevitch [
3]. With the initially quite weak constellation of GPS satellites, it was already proved to achieve precision levels down to centimeters for baselines ca. 10 km long. This was performed during static sessions of 2-3 h [
4]. Even though three decades have elapsed since then, the issue of ambiguity determination is still subject to intense research and development work in numerous research centers all over the world. This has resulted in various methods applicable not only to statistical measurements but also to kinematic measurements, including those performed in real-time. Kim and Langley [
5] provide a general outline on methods of ambiguity resolution developed in the period between 1981 and 1999 where ambiguity resolution methods were divided into three main groups: ambiguity search in the measurement domain, search in the coordinate domain and search in the ambiguity domain. Such divisions still remain valid and correspond well to the current methods. The first group is based on linear combinations of GNSS observations. The linear combinations presented by Melbourne [
6] or Wubbena [
7] were originally invented by Hatch [
8]. This approach uses dual frequency of code and carrier phase measurements to fixed ambiguities for different integer lineal combinations. Many ambiguity resolution strategies resolve firstly the fundamental Melbourne-Wubbena-Hatch combination-the so-called wide-line linear combination. Once the wide-lane ambiguity is resolved, then the L1 and L2 integer ambiguity solutions are easier performed [
9]. The second method is based on the mathematical ambiguity function proposed by Counselman and Gourevitch (1981) and further investigated by Remondi [
10]. Han and Rizos [
11] improved computational efficiency of the ambiguity function algorithms for practical applications. Cellmer et al. [
12] proposed further improvement of the ambiguity function approach. They have developed a new method named the Modified Ambiguity Function Approach (MAFA), in which the dramatically reduced computational load characterizes the computational process. The third group is currently the most popular and is based on estimation with the integer least-squares principle. The least-squares ambiguity search solution is an approximate solution with a description of the volume (e.g., ellipsoid) over which the search is to be conducted [
13]. The start-up procedure was based on the differential code solution for the initial position, and the search was performed over the associated three-sigma region surrounding that position. Based on numerical tests performed by Hatch, a single-epoch solution was possible under the following conditions: (1) dual frequency data are available, (2) distances are limited to a few tens and the ionospheric refraction effect is not too severe, (3) seven or more total satellites are available for processing with good geometry. This was revolutionary research that shows that a precise kinematic surveying or precise navigation is possible when L1 and L2 GPS measurements are available. Frei and Beutler [
14] introduced an efficient algorithm of Fisher’s probability density function to form confidence regions for individual ambiguities between float (real numbers) and fixed (integer) solutions for validation of the final fixed ambiguities. Teunissen improved the idea presented by Hatch by the use of variance covariance matrix of the float solution to define the search area [
15]. The search area is represented by an ellipsoid, with orientation dependent on the variance covariance matrix. The LAMBDA method is the most popular and famous method in GNSS world. The results of the PREFMAR approach can be compared with the LAMBDA solution, because a single epoch solution was presented in detail based on the float solution. All of these groups can be combined and modified to improve efficiency and reliability, especially for the integration of more than one GNSS system. The current research of ambiguity resolution is rather focused on triple-frequencies. The review of triple-frequency ambiguity resolutions was presented in [
16,
17]. Numerical comparisons of multicarrier ambiguity resolution for relative GNSS positioning was investigated by O’Kneefe et al. [
18].
The combination of L1-L2 frequencies of a GPS system is a special one, available in every multi-frequency GNSS receiver and probably the most effective double-frequency combination of GNSS data. A detailed description of mathematical functions used in the PREFMAR approach was investigated earlier for the use of L1-L5 or E1-E5a GPS/GALILEO data [
19], but each double combination of GNSS frequencies must be carefully analyzed to find specific properties for fast and precise ambiguity initialization and reinitialization. Therefore, this paper is devoted to the most important combination in GNSS frequencies—i.e., L1-L2 GPS measurements. This research is focused not only on the initialization of ambiguities but also presents innovative method for instantaneous reinitialization of ambiguities for L1-L2 measurements. Both initialization and reinitialization of ambiguities are realized based on the new mathematical functions:
and
. A mathematical background of the function is based on a strong correlation between float solutions (see
Section 2,
Section 3 and
Section 4). In
Section 5, properties of these functions are discussed in terms of real relative errors of the double difference carrier phase and pseudorange of L1 and L2 GPS measurements. These properties were discovered by the author during analysis of real GPS data. Therefore,
Section 5 presents different templates for ambiguity initialization in terms of magnitude of relative carrier phase and code errors.
Section 6 presents a detailed numerical example for a single epoch were a float GPS L1-L2 solution is available.
Section 7 analyses the efficiency of ambiguity resolution for the 3.6 km GPS baseline. Summary and conclusions of the research are given in
Section 8.
2. Double-Differenced GPS Data Equations and the Correlation of Geometry-Free Ambiguities
Let us make double-difference equations for two satellites, two measurement points and two frequencies: L1 and L2. For a very short baseline, the ionospheric and tropospheric delays have been ignored. Then, for the two frequencies, double-difference (DD) equations for carriers (
) and pseudo-ranges (
) can be written as follows:
where:
—observations of double-difference phase measurements for and frequencies (in cycles); —double-difference geometric range [m]; 0.190293672798365 [m]; 0.244210213424568 [m].
The above equations can be also presented in a matrix notation
, as [
20]:
where the
X vector of unknowns represents the
and
ambiguities, and the
value. The
and
integer values remain invariable in time if a GPS antenna has no interruption in the reception of phase signals.
By examining the
and
frequencies (Equations (1)–(4) separately, we can calculate
and
ambiguities for a single measurement epoch:
If thus, the accuracy of and code measurements would be better than 0.5 of the length of respective waves, then using formulas 6 and 7, we could precisely calculate the and ambiguities by rounding to the nearest integer. Unfortunately, the accuracy of code measurements is far worse, which makes it impossible to calculate and ambiguities in a reliable manner.
To achieve a strong correlation between the
and
values, a common
quantity, which would pertain to code measurements in specific measurement epochs (t), needs to be introduced to the equations above:
where:
or [
21]
Although the value
can be calculated using only a double-difference (DD) observation for a pair of satellites (Equation (11)), the most accurate
is obtained using double differenced geometric distance (Equation (10)) based on code relative or differential GNSS positioning based on Kalman filter and network code DGNSS solutions [
22,
23].
In
Figure 1 that the
and
observations are strongly correlated, as the
and
values are located precisely along a straight line (
Figure 1b). The value of the correlation coefficient between
and
ambiguities for data included in
Figure 1b is equal to 0.999946. The data presented pertain to a vector with a length of 3.6 km and the use of two GPS receivers - Ashtech Z-XII and Topcon Hiper Pro - and based only on two satellites.
3. Ambiguity Regression Line Equations for L1-L2 Measurements
Regression line equation for correlated
ambiguities with reference to
ambiguity in the
system of ambiguities can be written with the following equation:
or
where:
; .
As ambiguities in the form of real numbers (
) lie along a straight line expressed with the Equation (13), the
value can be calculated with the following formula
thus
Based on the Equation (16), it can be seen that the
value can be calculated for a single measurement epoch, without knowing integer
and
values. While the Equation (12) allows the determination of the integer
value if the integer
value is known. If
is known, after transforming the Equation (12), we may calculate the
value:
Finally, for DD observations, the regression line equation in the
, system for a single measurement epoch can be written as:
whereas for n measurement epochs, we have a detailed formula determining the relation between integer
values and calculated (real)
values. Based on this we receive:
or in simplified form
4. Ambiguity Functions for L1-L2 GPS Measurements
Assuming that the
values are equal to zero, for distribution of the
set of ambiguities being integer numbers in the
system, we can write
However, in the majority of cycles, the calculated
value with reference to the integer
value is a real number, thus
Analogically, for the
system, we have:
and
For Equations (22) and (24), we can determine the value of certain
error in the following manner:
or
meeting the relation
, as these are relative errors in phase observations (DD).
Then, in the Formula (25), we substitute for
the expression from the Formula (18), thus receiving the following function
=
(in cycles) for the
system.
Analogically, for the
system and the function
=
, where
The above-presented functions are used in the PREFMAR method. Behaviors of the
and
functions for a
value equal to zero have been presented in
Figure 2 and
Figure 3. Additionally, behaviours of the
and
functions have been presented in
Figure 4 and
Figure 5, where that their minima and thus their periodic character (repeatability) can clearly be seen. Based on an analysis of the values of the
functions lying in the (−0.5
; 0.5
) interval, it can be observed that the value of the
function for
and
frequencies of GPS observations repeat precisely every 77
cycles. Additionally, the
function (
Figure 4) has its minima also every 77 cycles—i.e.,
To the contrary, based on an analysis of the values of
, that analogically as for the
function are comprised in the (-0.5
; 0.5
) interval, the values of the
function for
and
frequencies of GPS observations repeat precisely every 60
cycles. Additionally, the
function (
Figure 5) has its minima also every 60 cycles—i.e.,
when analyzing the behavior of the
and
functions, it can be seen that both functions have the same wavelength and frequency:
because
and they are equivalent in the process of determining
and
ambiguities for the value of
5. Ambiguity Search Space in the PREFMAR Method
When analyzing the ambiguity functions of the PREFMAR method, formulated based on correlated
and
observations, certain
(Equations (25) and (26)) values need to be considered as relative errors in L1 and L2 phase observations that will help us to understand interpretations of the
and
functions. Thus, let us write
If errors in DD phase observations for
and
frequencies of GPS observations are equal and have the same value, then:
Thus, if
then the minima of the
and
functions indicate the searched values of
and
ambiguities. The question remains, however, how large the errors in DD phase observations can be for the ambiguities to be determined by the ambiguity function minima, assuming the Equation (37). The answer to this question depends on the smallest value of the functions expressed in the Equations (27) and (28) and, in the case of L1–L2 phase measurements of GPS observations, these are the
and
values equal to
The 38 formula thus determines the smallest unit of the
function in the
system and
in the
system. Thus, if relative errors in DD phase observations are smaller than half the base unit—i.e.,
mm—then the minima of the
and
functions indicate the searched ambiguities on the horizontal axis as, respectively:
using the minimum of the
function and
using the minimum of the
function. If correlated errors are equal to zero, in both of the above cases, approximate values of
and
must be at a distance smaller than 0.5
or 0.5
from the real values (
and
), thus smaller than:
and/or
If, however, the absolute or values are different from zero and larger than 1.6 mm, then ambiguity values will be located in different points than the minima of the and functions, and their selection will strongly depend on the real values and from the approximate and values.
Thus, knowing the true values of
errors, we are able to explicitly calculate the searched
and
ambiguities based on mathematical
and/or
functions, as within the range of
we can only obtain a single candidate for (
) ambiguity that satisfies the equations:
However, in reality, we do not know the true value of
errors. Therefore, searching for unknown
and
ambiguities starts with the smallest values of the
or
functions in a certain, direct neighborhood of float solution (
), with the ambiguities being selected in line with the following relation:
Let us thus assume that the
values for the
function are in the range of ±14.3 mm—i.e., below 4.5 units
. Then, our set of possible ambiguities is represented graphically in
Figure 6 and
Figure 7. Thus, the
function indicates possible solutions for ambiguities with assumed
relative errors, and these are the following values:
For comparison, search areas for
that are in the range of ±27 mm (i.e., up to 8.5 units) have been presented in
Figure 8 and
Figure 9. Then, the
and
functions indicated integer values for
, in which the correct ambiguity can be located, and these are the following items:
If we assume, for example, that the values of relative errors in DD phase observations lie in the
interval, then the
function indicates integer
ambiguities every nine cycles, and these are the following values:
Similarly, if we assume, for example, that the values of relative errors in DD phase observations lie in the
interval, then the
function indicates integer
ambiguities every nine cycles, and these are the following values:
Thus, analysing only the
, system for selected integer ambiguities
, integer
values can be calculated with the use of the 12 and 15 formulas—i.e.,:
Thus, for the
interval and for the
interval, these are [
;
integer sets:
that differ by 9 and 7 cycles for
and
, respectively.
However, if we increase the range of relative errors—i.e.,
—these will be the following proposals for
(
Figure 10) and for
(
Figure 11):
The
Figure 10 and
Figure 11 present templates for possible ambiguities for relative errors in DD phase observations with the values up to
52 mm. This range seems sufficient for determining ambiguities in real-life conditions for short baselines or when using data from virtual reference stations.
6. Numerical Example Using the PREFMAR Method and a Float Solution
As mentioned in the introduction, although there are numerous methods for determining ambiguity, the PREFMAR method can provide sets ambiguities for any pair of satellites without the VC matrix with a float solution. Therefore, the goal of this chapter is to present numerical calculations to indicate the most probable [
,
pairs that can later be used as input data for final ambiguity determination—i.e., validation. However, the PREFMAR allows for (mathematically) unambiguous determination of ambiguities based on known relative errors in DD phase observations—i.e., the
or
values (Equations (25) and (26)). In reality, it is unable to determine these values precisely, but it can be assumed that the search area starts with the
or
values with the smallest absolute values, which seems logical for short baselines. Thus, for calculations, we use real single epoch L1 and L2 of GPS data. For calculations we use values from the books [
1,
24]—i.e., approximate
and
values and elements of the variance-covariance matrix
of a float solution:
Sets of ambiguities can be determined both with the use of the
function and the
function. However, if we decide to use the
function, the search area can be determined as
<
];
—i.e.,
<
;
. Thus, let us present the necessary calculations in the form of
Table 1 where elements of the
columns are determined based on an expression for integer and subsequent
values, in the
and
neighbourhoods, using the following formula:
and the values of the
function are calculated with the formula:
However, if we decide to use the
function, the search area can be determined as
<
];
—i.e.,
<
;
. Detailed calculations are presented in
Table 2.
The most probable solution includes the set of ambiguities:
and
where the functions achieved the smallest absolute values—i.e.,
. Thus, based on two functions, we received the same results (
Figure 12) because
and
Only three sets have been presented in
Figure 12:
;
and
, as these are the proposals with the assumption that relative errors in DD phase observations are smaller than 3.5 cm. It should also be noted that the proposed most likely values [
;
are also consistent with wide lane (
) ambiguities, because
;
and
. Therefore, the indicated sets of ambiguity
with values
and
are the most probable, because they represent neighboring and subsequent values of wide lane ambiguities
in the immediate vicinity of the float solution.
If we assume that relative errors in phase measurements for DD observations are smaller than 6 cm, the number of ambiguities increases,
Figure 13.
Please note, however, the phase errors in DD measurements reaching 6 cm are quite large and can occur in the case of interferences in satellite signals or in the case of longer vectors. In such cases, we strive to increase code measurement accuracy to reduce the search area as far as possible. If, for instance, in the example presented above, we achieve a code measurement accuracy down to ca. 0.5 m, then we only have two ambiguity candidates—
;
—and even with the assumption that relative errors in phase measurements for DD observations—i.e., the values of the
or
functions are below 6.4 cm (see
Table 1 and
Table 2).
Behavior of the full period of the
function for input data has been presented in
Figure 14, and below we presented precisely calculated minima of this function— i.e.,
and
based on the formula:
thus:
i.e.,
Generally, for the system, the search area takes the form of a parallelogram with sides identical to those described above. The short sides of this parallelogram are parallel to the vertical lines of the system —i.e., to the axis. The longer side of this parallelogram is slanted with respect to the axis under and angle and the geometrical centre of the search area is located in point , . To precisely determine the search area for the PREFMAR method, it is thus necessary to know the accuracy of code measurements and allowable values of relative errors that can take maximum values of .
In the end, it should be noted that the most probable integer candidates calculated by the PREFMAR—i.e.,
and
are the same as the LAMBDA (Least-squares AMBiguity Decorrelation Adjustment) result [
24].
7. Discussion of the PREFMAR’s Efficiency in Terms of Initialization and Reinitialization Based on Real Positioning Data Using L1-L2 GPS Measurements.
The PREFMAR method uses the functions described above:
and/or
to indicate the most likely sets of ambiguities for L1-L2 GPS measurements. The functions presented relate to relative errors of DD phase observations. Thus, these errors can be both positive or negative. Therefore, function values
or
can more precisely determine the search area in case of reinitialization of ambiguities—i.e., where we have already determined our ambiguities but lost contact with some or all satellites for various reasons. This process of ambiguity reinitialization is more common in practice than the initialization process—i.e., the first determination of sets of ambiguities for a given vector. Thanks to the discovered properties of the developed mathematical functions, used in the PREFMAR method, and using relative errors in the reinitialization process, we are able to reduce the number of combinations in the validation process even several dozen times. For example, let us imagine that in the above example we would reinitialize the ambiguity in time (
), having previously determined ambiguities in time
—i.e., we have at our disposal values of the function
. For short vectors, several kilometers in length, it is sufficient that only the information whether the value of function
is positive or negative. Suppose that
, this means that in the reinitialization process in the example shown above, our most probable sets of ambiguities will be as follows: (
Figure 15)
;
and
, for the function values
. Thus, during the re-initialization process, we obtained a significant reduction in possible sets of ambiguities from 7 pairs to 3 pairs. If we assume that we have four satellites and perform validation with 7 pairs of possible sets of ambiguities for four satellites we obtain 7
3 = 343 combinations. In the proposed reinitialization process, we have only 3
3 = 27 combinations. Therefore, using the described properties of mathematical functions, the PREFMAR method enables immediate reinitialization of ambiguities for single observation epochs for different baseline lengths. Paradoxically, the length of baselines is not a problem in the reinitialization process, as larger relative errors will be more reliable in the process of determining the search area. Therefore, the search area of the PREFMAR method can be presented in several variants, depending on the length of the vector and with regard whether we are dealing with initialization or reinitialization. In the presented calculation example, on the basis of the variance–covariance matrix, we can see that we have quite a large error in code measurements—i.e., about 1.4 m. The current capabilities of GNSS receivers allow quite good code accuracy, especially when using the Kalman filter, and obtaining code accuracy below the wavelength
m is not a problem. This means that for short baselines, it is enough to give four sets of ambiguities, among which the value searched for should be included (
Figure 16).
This means that for the frequency L1-L2, we should try to obtain code accuracy of less than 0.86 m, and then for short vectors we give only four sets of ambiguities—i.e., ; ; ; . This method has been proven to be relevant for relative errors of phase observations below 6 cm; which is absolutely sufficient for baselines of several kilometers or for baselines in relation to virtual reference stations (VRS).
The dependence of the number of ambiguity sets on the accuracy of code measurements below 0.86 m using the PREFMAR method is shown in
Figure 17.
In
Figure 17, we can see that in the case of relative errors in the range
), we have only one set of ambiguity. For relative errors in the range
), we have two sets of ambiguity. However, for the relative errors
we have four sets. Note that whenever we obtain an even number of ambiguity sets, half of the sets are determined by positive values of the function
, and the other half by negative values.
In addition, in the case of reinitialization in time (t), we use the function values from the time before the disturbance in the reception of the continuity of phase signals—i.e., from time
as follows:
where
represents the time delay—i.e., the initialization time minus the time of the last epoch with fixed ambiguities:
For the DD observation at time (t) during reinitialization, we can, therefore, write the following functions:
where:
,
represent fixed ambiguities in the time
.
The developed algorithms were tested on real L1-L2 observations, for a 10-min session with an interval of 1 s, and a baseline length of 3.6 km. The L1-L2 GPS measurements were performed with the Topcon HiperPro (M) receiver on 13rd August 2010, from 10:30:00 to 10:40:00 UTC, with reference to the OLST reference station. There were five GPS satellites available above the horizon: G09, G12, G15, G17 and G27, for which the following double-difference (DD) observations were created: G27-G09, G27-G12, G27-G15 and G27-G17. For every double-difference carrier phase observation and correctly determined ambiguities, the relative errors
between L1 and L2 double-difference measurements were below 52 mm (see
Figure 18). The ambiguities from the first range
were for 1317 (55%) DD phase observations, in the second range
there were 1039 (43%) DD phase observations. However, for the range
there were only 44 DD phase observations (2%). Detailed data on the number of DD observations depending on the value of the relative errors
are presented in
Table 3.
In the second stage of the numerical tests, in the case of reinitialization at
s, the values of the function
were obtained in the range
for all DD observations (
Figure 19), and, therefore, all (100%) first ambiguity sets at reinitialization determined by the PREFMAR method were the searched unknowns.
The properties of the functions
or
and the stability of the relative errors during the measurements enable to instantaneous ambiguity reinitialization because the search area is the same as for ultra-short baselines. Then, for the float solution, with the use of code measurements P1 and P2 below 0.86 m, and for the relative errors in the range
, we obtain only one set of ambiguities (
Figure 18). Therefore, the PREFMAR allows instantaneous ambiguity reinitialization for L1-L2 GPS measurements, for short as well for longer baselines.
8. Summary and Conclusions
This work presents the new PREFMAR method for determining ambiguities in phase measurements for single measurement epochs, for GPS observations performed at L1 and L2 frequencies. Its efficiency mostly depends on the values of relative errors in DD phase observations. Ambiguity is selected based on a proprietary search function that uses the correlation between precisely determined ambiguity values depending on the frequency of satellite signals. The PREFMAR method allows determining ambiguities for individual measurement epochs without using a variance–covariance matrix with a float solution. For relative errors of
carrier phase measurements
the search area is as if we were using frequencies for a wavelength equal to 1.73 m (see
Figure 17). Thus, two neighboring ambiguities cover an area equivalent to an 18-cycle range
and 14-cycle range
in the
system. This is a remarkable feature compared to the L1-L5 combination.
In the case of relative errors of carrier phase measurements the search area is as if we were using frequencies for a wavelength equal to 0.86 m. Furthermore, for these relative errors, the wide lane ambiguity is unambiguously recalculated to and .
The developed PREFMAR method indicates ambiguity sets (, ) for L1 and L2 frequencies, using the functions or , from which mathematical equations have been derived and described in detail in this work. The identification of the ambiguity sets (, ) can be performed if we have the following input data:
Note that for case III, the PREFMAR method shows ambiguities even for a single DD observation, for any pair of satellites, but the accuracy of the code measurements is of key importance as we use double-difference observations from only two satellites. However, for much more accurate P5 code measurements transmitted at the L5 frequency, the III approach can be effectively used in L1-L5 positioning. The approach II seems to be the most universal, because in the global XYZ solution we can use any configuration of GNSS satellites with the most accurate code measurements and Kalman filter, which significantly increases the accuracy of the solution of the approximate XYZ position, both in static and kinematic positioning. Based on preliminary numerical results, the PREFMAR allows instantaneous ambiguity reinitialization if all satellites lost contact with a GNSS antenna.