_{1}

Typically, dual-frequency geodetic grade GNSS receivers are utilized for positioning applications that require high accuracy. Single-frequency high grade receivers can be used to minimize the expenses of such dual-frequency receivers. However, user has to consider the resultant positioning accuracy. Since the evolution of low-cost single-frequency (LCSF) receivers is typically cheaper than single-frequency high grade receivers, it is possible to obtain comparable positioning accuracy if the corresponding observables are accurately modelled. In this paper, two LCSF GPS receivers are used to form short baseline. Raw GPS measurements are recorded for several consecutive days. The collected data are used to develop the stochastic model of GPS observables from such receivers. Different functions are tested to determine the best fitting model which is found to be 3 parameters exponential decay function. The new developed model is used to process different data sets and the results are compared against the traditional model. Both results from the newly developed and the traditional models are compared with the reference solution obtained from dual-frequency receiver. It is shown that the newly developed model improves the root - mean-square of the estimated horizontal coordinates by about 10% and improves the root-mean-square of the up component by about 39%.

Typically, differential GPS based on carrier phase observables is the first alternative for users seeking centimeter level accuracy [

Unlike dual-frequency receivers, however, ionosphere delay represents a major challenge for single-frequency receivers. There are two main techniques to account for ionosphere delay in case of single-frequency receivers [

The second technique to account for ionosphere delay is to form ionosphere-free linear combination using both code and carrier phase observations on L1 from the single-frequency receiver. This technique is based on the Group and Phase Ionosphere Calibration (GRAPHIC) [

In addition to the low-cost, high sensitivity single-frequency receiver are able to acquire signals with low decibel watt (dBW) [

In this paper, performance of LCSF receivers is improved by developing its unique stochastic model that fits its observables. Two u-blox NEO-7P LCSF GPS receivers are used to form short baseline. Both pseudorange and carrier phase measurements are recorded at sampling rate of 1 Hz. The collected data are used to determine the stochastic characteristics hence improving the positioning performance of such receivers.

The mathematical models for undifferenced pseudorange and carrier phase measurements can be written as follows [

where,

The final solution of least-squares of positioning model (Equations (1) and (2)) does not depend only on the mathematical formulation of the unknowns, but also depend on the statistical representation of the observations and unknowns. The observations stochastic properties are reflected in the weight matrix of observations, which includes the corresponding relative and absolute accuracies. The GPS signal’s power can be used as a measure of the signal quality. For example, the signal-to-noise ratio and carrier-to- noise power density ratio can be used as a measure of the GPS signal power and used to weight different signals [

Stochastic properties of GPS receiver’s signals can be determined through the calibration process. Typically, receiver noise can be examined using zero baseline [

Alternatively, the GPS system noise can be tested by differencing pseudorange and the carrier phase measurements [

Equation (4) can be differenced between the two receivers forming between receiver single difference, which sufficiently cancels out the ionosphere delay. The remaining terms include the integer ambiguity, hardware delay, system noise and multipath effect. Since the multipath effect is repeated each sidereal day, it can be sufficiently removed by differencing over two consecutive days. The integer ambiguity number is constant as long as the receiver tracks the satellite and hardware delay is stable over several days. As such, both can be removed from the combination by subtracting the first value of the time series. At this stage, we have only the differenced system noise in the time series. The differenced combination can be divided into bins based on the satellite elevation angle, and the best fitted mathematical function for observations standard deviation can be determined.

To investigate the stochastic properties of LCSF receiver, two u-box New-7P GPS receivers are used to form short baseline. The short baseline is fixed on the roof top of Faculty of Maritime Studies (FMS) building beside a base station established using Topcon GR3 GNSS receiver (used as a reference). GPS data are collected at sampling frequency of 1 Hz from both single-frequency and dual-frequency receiver. The collected data are used to examine the noise level of single-frequency receivers. The developed model is used to process new session of GPS data collected using the same single-frequency receivers.

The single frequency data collected for two consecutive days is used to compute code- carrier observable (Equation (4)) for both receivers. The new observable from the two receivers is used to form between-receiver single-difference combination to remove the ionosphere delay. However, the resultant combination is affected by the multipath effect. Since the multipath effect is repeatable every sidereal day, subtracting such combination from two consecutive days will successfully remove the multipath effect. It should be noted here that the sidereal day is 23 hours 56 minutes and 4 seconds which is less than the solar day by 3 minutes and 56 seconds (i.e., 256 Seconds). Hence, to remove the multipath effect, the second day series should be shifted by 236 seconds.

It is clear from

apply the time shift for the second day then subtract the two days to eliminate the multipath effect. The next step is to classify the resultant noise for each satellite according to elevation angle. The elevation angles are divided into bins of 5 degrees each. The mean value of the noise and the corresponding standard deviation is then calculated. The last step to develop the stochastic model is to perform fitting to determine the best mathematical function that fits the data. Different techniques are applied to determine the best model that fits the data including 2 parameters exponential decay function, 3 parameters exponential decay function, 4 parameters exponential decay function, 3 parameters rational equation, 2 parameters rational equation, 2 parameters hyperbolic decay function, and 3 parameters hyperbolic decay function.

^{°} and 05^{°} elevation angles, respectively. These results makes sense for single-frequency receiver if Trimble R7 GNSS receiver’s model has corresponding values of 0.2 m and 0.7 m at elevation angles of 90^{°} and 05^{°}, respectively [

Model | Mathematical formula | Model parameters | R^{2}/standard error |
---|---|---|---|

2 parameters exponential decay | 0.9771/0.0482 | ||

3 parameters exponential decay | 0.9878/0.0344 | ||

4 parameters exponential decay | 0.9869/0.0356 | ||

3 parameters rational equation | 0.9851/0.0380 | ||

2 parameters rational equation | 0.9767/0.0475 | ||

2 parameters hyperbolic decay | 0.9767/0.0475 | ||

3 parameters hyperbolic decay | 0.9851/0.0380 |

Where, el is the satellite elevation angle, and

cluding ocean loading, Earth tides, carrier phase windup, relativity, and sagnac effect are accounted for using existing models [

where

In this paper, we investigated the stochastic properties of low-cost-single-frequency receivers. The main objective is to develop stochastic model to improve the positioning performance of such receivers. Raw GPS measurements are collected using two low-cost single-frequency GPS receivers fixed to form short baseline. A third dual-frequency receiver is used as a reference to evaluate the performance of low-cost single-frequency receivers. Between receivers, single difference is formed using the code-carrier combination from both receivers and the ambiguity term and hardware delay constants are removed from each satellite pass. The resultant combination is differenced over two consecutive days to eliminate the multipath effect. The final noise is used to develop the relationship between the satellite elevation angle and signal standard deviation. Different functions are tested to determine the best model that fits such relationship. It is found that the 3 parameters exponential decay function is the best fit model. The developed model is used to process different data sets. Both solutions from the traditional (sine function) and the developed models are compared with the reference solution. It is shown that the developed model can improve the accuracy of the estimated coordinates by about 10% and 39% for the horizontal and up components,

Error Parameter | Total Error (m) | % Improvement | |
---|---|---|---|

Traditional Model | Developed Model | ||

2.220 | 2.000 | ||

5.410 | 4.880 | 9.9% | |

1.000 | 0.614 | ||

1.960 | 1.203 | 39.0% |

respectively. These results can be considered significant to improve the performance of low-cost single-frequency receivers.

Elsobeiey, M.E. (2016) Stochastic Analysis of Low-Cost Single- Frequency GPS Receivers. Positioning, 7, 91- 100. http://dx.doi.org/10.4236/pos.2016.73009