Examples of single frequency Cycle-Slip Detectors - Revision history

Jaume.Sanz at 08:03, 22 March 2013

2013-03-22T08:03:18Z

← Older revision		Revision as of 08:03, 22 March 2013
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	::::* Declare cycle-slip when data hole greater than <math>tol_{\Delta t}</math> <ref group="footnotes">For instance, 60 seconds.</ref>.		::::* Declare cycle-slip when data hole greater than <math>tol_{\Delta t}</math> <ref group="footnotes">For instance, 60 seconds.</ref>.
	::::* If no data hole larger than <math>tol_{\Delta t}</math>, thence:		::::* If no data hole larger than <math>tol_{\Delta t}</math>, thence:
	::::* Update an array with the previous <math>N</math> values (after the last ~~cycles~~-~~lip~~) <math>[d(s;k-N),\dots,d(s;k-1)]</math>		::::* Update an array with the previous <math>N</math> values (after the last cycle-slip) <math>[d(s;k-N),\dots,d(s;k-1)]</math>
	::::* Fit a n-degree polynomial <math>P(s;x)</math> to the previous values <ref group="footnotes"> <math>n=2 </math> or <math>3 </math> can be used.</ref>.		::::* Fit a n-degree polynomial <math>P(s;x)</math> to the previous values <ref group="footnotes"> <math>n=2 </math> or <math>3 </math> can be used.</ref>.
	::::* Declare cycle-slip when:		::::* Declare cycle-slip when:

Jaume.Sanz at 07:58, 22 March 2013

2013-03-22T07:58:54Z

← Older revision		Revision as of 07:58, 22 March 2013
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	''Additional comments: Differences in time detector.''		''Additional comments: Differences in time detector.''

	A detector based in the n-order time differences of carrier phase <math>\Phi</math> measurements between consecutive epochs could be considered (see article [[Detector based in carrier phase data: The geometry-free combination]]. Nevertheless, it must be taken into account that ~~now~~ such differences are affected by changes in range and in clocks <ref group="footnotes"> In the case of article [[Detector based in carrier phase data: The geometry-free combination]], the geometry-free combination was used, cancelling all the non dispersive effects (geometric range, clocks...).</ref>. In spite of that, most of the geometric range variation is cancelled from <math>\Delta^3</math>, because it varies as a smooth function. Thence, this detector mainly affected by the receiver clock instabilities, high ionospheric fluctuations (i.e., scintillation...), as well.		A detector based in the n-order time differences of carrier phase <math>\Phi</math> measurements between consecutive epochs could be considered (see article [[Detector based in carrier phase data: The geometry-free combination]]. Nevertheless, it must be taken into account that such differences are affected by changes in range and in clocks <ref group="footnotes"> In the case of article [[Detector based in carrier phase data: The geometry-free combination]], the geometry-free combination was used, cancelling all the non dispersive effects (geometric range, clocks...).</ref>. In spite of that, most of the geometric range variation is cancelled from <math>\Delta^3</math>, because it varies as a smooth function. Thence, this detector mainly affected by the receiver clock instabilities, high ionospheric fluctuations (i.e., scintillation...), as well.

	In the case of the receiver clock effects, they can be removed by considering single differences of measurements between pairs of satellites in view, but it will enlarge the noise. The ionospheric effects will be not completely removed with such differences, but they should not be a problem, except for a highly disturbed scenario.		In the case of the receiver clock effects, they can be removed by considering single differences of measurements between pairs of satellites in view, but it will enlarge the noise. The ionospheric effects will be not completely removed with such differences, but they should not be a problem, except for a highly disturbed scenario.

Jaume.Sanz at 07:51, 22 March 2013

2013-03-22T07:51:58Z

← Older revision		Revision as of 07:51, 22 March 2013
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	::::* Declare cycle-slip when data hole greater than <math>tol_{\Delta t}</math> <ref group="footnotes">For instance, 60 seconds.</ref>.		::::* Declare cycle-slip when data hole greater than <math>tol_{\Delta t}</math> <ref group="footnotes">For instance, 60 seconds.</ref>.
	::::* If no data hole larger than <math>tol_{\Delta t}</math>, thence:		::::* If no data hole larger than <math>tol_{\Delta t}</math>, thence:
	::::* Update an array with the ~~last~~ <math>N</math> ~~continuous epochs~~ <math>[d(s;k-N),\dots,d(s;k-1)]</math>		::::* Update an array with the previous <math>N</math> values (after the last cycles-lip) <math>[d(s;k-N),\dots,d(s;k-1)]</math>
	::::* Fit a n-degree polynomial <math>P(s;x)</math> to the previous values <ref group="footnotes"> <math>n=2 </math> or <math>3 </math> can be used.</ref>.		::::* Fit a n-degree polynomial <math>P(s;x)</math> to the previous values <ref group="footnotes"> <math>n=2 </math> or <math>3 </math> can be used.</ref>.
	::::* Declare cycle-slip when:		::::* Declare cycle-slip when:
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	'''''Algorithm description: Example 2'''''		'''''Algorithm description: Example 2'''''

	The detection is based on computing the mean and sigma values of the code pseudorange and carrier phase (<math>\Phi-R</math>) differences over a sliding window of <math>N</math> samples (e.g., <math>N=60</math> with 1Hz data). A cycle-slip is declared when a measurement differs from the mean bias value over a predefined threshold.		The detection is based on computing the mean and sigma values of the code pseudorange and carrier phase (<math>\Phi-R</math>) differences over a sliding window of <math>N</math> samples (e.g., <math>N=100</math> with 1Hz data). A cycle-slip is declared when a measurement differs from the mean bias value over a predefined threshold.

	'': Input data:'' Code pseudorange (<math>R</math>) and carrier phase (<math>\Phi</math>) measurements.		'': Input data:'' Code pseudorange (<math>R</math>) and carrier phase (<math>\Phi</math>) measurements.
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	:::::<math> d(s;k)=\Phi(s;k)-R(s;k) \qquad \mbox{(2)} </math>		:::::<math> d(s;k)=\Phi(s;k)-R(s;k) \qquad \mbox{(2)} </math>
	:::: That is: <math>[d(s;k-N),\dots,d(s;k-1)]</math>		:::: That is: <math>[d(s;k-N),\dots,d(s;k-1)]</math>
	::::* Compute the mean and sigma discrepancy over the previous <math>N</math> ~~continuous~~ epochs <math>[k-N, \dots , k-1]</math>:		::::* Compute the mean and sigma discrepancy over the previous <math>N</math> epochs (after the last cycle-slip) <math>[k-N, \dots , k-1]</math>:
	::::<math>		::::<math>
	\begin{array}{l}		\begin{array}{l}
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	\end{array} \qquad \mbox{(3)}		\end{array} \qquad \mbox{(3)}
	</math>		</math>
	::::* Compare the difference at the epoch <math>k</math> with the mean value of differences computed over the previous <math>N</math> samples window. If the value is over a <math>threshod=n_T*S_d</math> (e.g., <math>n_T=5</math>), declare cycle-slip <ref group="footnotes"> The mean bias between code and phase computed from the "previous <math>N</math> samples" is taken as the prediction of the actual difference at epoch <math>k</math> (i.e., <math>\hat{d}(s;k)\equiv m_d(s;k-1)</math>) with confidence <math>n_T\,S_d(s;k-1)</math>. </ref>.		::::* Compare the difference at the epoch <math>k</math> with the mean value of differences computed over the previous <math>N</math> samples window. If the value is over a <math>threshod=n_T*S_d</math> (e.g., <math>n_T=6</math>), declare cycle-slip <ref group="footnotes"> The mean bias between code and phase computed from the "previous <math>N</math> samples" is taken as the prediction of the actual difference at epoch <math>k</math> (i.e., <math>\hat{d}(s;k)\equiv m_d(s;k-1)</math>) with confidence <math>n_T\,S_d(s;k-1)</math>. </ref>.
	::::That is:		::::That is:
	:::: If <math>\|d(s;k)-m_d(s;k-1)\| > n_T \,S_d(s;k-1)</math>,		:::: If <math>\|d(s;k)-m_d(s;k-1)\| > n_T \,S_d(s;k-1)</math>,
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	:* To avoid ~~noisier or~~ unrealistic estimations of sigma during the first iterations of the filter, the following weighted average with an initial value <math>S^2_0</math> can be used (see in figure 1 an example of implementation performance):		:* To avoid unrealistic estimations of sigma during the first iterations of the filter, the following weighted average with an initial value <math>S^2_0</math> can be used (see in figure 1 an example of implementation performance):
	:::<math>		:::<math>
	S_d^2(s;k) = \frac{k-1}{k}S_d^2(s;k)+\frac{1}{k}S^2_0		S_d^2(s;k) = \frac{k-1}{k}S_d^2(s;k)+\frac{1}{k}S^2_0
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	:where, <math>a=k</math> when <math>k<N</math> and <math>a=N</math> when <math>k \geq N</math>.		:where, <math>a=k</math> when <math>k<N</math> and <math>a=N</math> when <math>k \geq N</math>.

	:This equation allow computation of a sequential estimate of the mean value, but this filter has infinite memory, propagating forward the divergence of the ionospheric refraction <ref group="footnotes"> On the contrary, the equations (3) provide and estimate based in a (sliding) window, with a finite number of points.</ref>. Nevertheless, such an accumulated effect, although biasing the ambiguity estimate, should not ~~be an issue, since~~ the cycle-slips ~~can be detected by looking~~ for jumps ~~in "<math> d-m_d</math>"~~.		:This equation allow computation of a sequential estimate of the mean value, but this filter has infinite memory, propagating forward the divergence of the ionospheric refraction <ref group="footnotes"> On the contrary, the equations (3) provide and estimate based in a (sliding) window, with a finite number of points.</ref>. Nevertheless, such an accumulated effect, although biasing the ambiguity estimate, should not affect the cycle-slips detection, because it varies smoothly and the detector looks for large jumps.

	::[[File: Single Freq Cycle Slip De Fig 1.png\|none\|thumb\|640px\|'''''Figure 1:''''' Fault cycle-slip detection on GPS <math>L_1</math> signal: A multipath drift, after a quite stable period (i.e., with small sigma), produces a fault detection. The Measured and Predicted differences are compared with the confidence threshold (computed with the sliding window algorithm, using the equation (3) combined with (4).]]		::[[File: Single Freq Cycle Slip De Fig 1.png\|none\|thumb\|640px\|'''''Figure 1:''''' Fault cycle-slip detection on GPS <math>L_1</math> signal: A multipath drift, after a quite stable period (i.e., with small sigma), produces a fault detection. The Measured and Predicted differences are compared with the confidence threshold (computed with the sliding window algorithm, using the equation (3) combined with (4).]]
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	''Additional comments: Differences in time detector.''		''Additional comments: Differences in time detector.''

	A detector based in the n-order time differences of carrier phase <math>\Phi</math> measurements between consecutive epochs could be considered (see article [[Detector based in carrier phase data: The geometry-free combination]]. Nevertheless, it must be taken into account that now such differences are affected by ~~the change of geometry (geometric~~ range and clocks) <ref group="footnotes"> In the case of article [[Detector based in carrier phase data: The geometry-free combination]], the geometry-free combination was used, cancelling all the non dispersive effects (geometric range, clocks...).</ref>. In spite of that, most of the geometric range variation is cancelled from <math>\Delta^3</math>, because it varies as a smooth function. Thence, this detector mainly affected by the receiver clock instabilities, high ionospheric fluctuations (i.e., scintillation...), as well.		A detector based in the n-order time differences of carrier phase <math>\Phi</math> measurements between consecutive epochs could be considered (see article [[Detector based in carrier phase data: The geometry-free combination]]. Nevertheless, it must be taken into account that now such differences are affected by changes in range and in clocks <ref group="footnotes"> In the case of article [[Detector based in carrier phase data: The geometry-free combination]], the geometry-free combination was used, cancelling all the non dispersive effects (geometric range, clocks...).</ref>. In spite of that, most of the geometric range variation is cancelled from <math>\Delta^3</math>, because it varies as a smooth function. Thence, this detector mainly affected by the receiver clock instabilities, high ionospheric fluctuations (i.e., scintillation...), as well.

	In the case of the receiver clock effects, they can be removed by considering single differences of measurements between pairs of satellites in view, but it will enlarge the noise. The ionospheric effects will be not completely removed with such differences, but they should not be a problem, except for a highly disturbed scenario.		In the case of the receiver clock effects, they can be removed by considering single differences of measurements between pairs of satellites in view, but it will enlarge the noise. The ionospheric effects will be not completely removed with such differences, but they should not be a problem, except for a highly disturbed scenario.

Jaume.Sanz at 08:27, 3 February 2013

2013-02-03T08:27:14Z

← Older revision		Revision as of 08:27, 3 February 2013
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	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}
	The examples of single frequency detectors presented next are based only on data measurements of a single receiver and does not use any geometrical delay model. They are simple algorithms, suitable to run in real-time, but with worse performance that the two frequency detectors of previous sections. Other algorithms using geometrical modelling or single or double differences between satellites, or satellites and receivers, improve the detection threshold and the reliability.		The examples of single frequency detectors presented next are based only on data measurements of a single receiver and does not use any geometrical delay model. They are simple algorithms, suitable to run in real-time, but with worse performance that the two frequency detectors of previous sections. Other algorithms using geometrical modelling or single or double differences between satellites, or satellites and receivers, can improve the detection threshold and the reliability.


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	'''Algorithm description: Example 1'''		'''Algorithm description: Example 1'''

	The detection is based on fitting a <math>n</math>-degree polynomial over a sliding window of <math>N</math> samples (e.g., <math>N=200</math> with 1Hz data). The residuals (predicted-observed) ~~for two consecutive epochs~~ are compared to identify the carrier phase cycle-slips <ref group="footnotes">This approach is inspired in the algorithms developed by [Blewitt, 1990] for the geometry-free combination of two frequency signals. Notice that, a similar scheme is applied here for the code and phase combination of a single frequency signal.</ref>~~. The predicted value from this polynomial is compared with the observed value to detect the cycle-slip~~. As the geometry-free combination is affected by the ionospheric refraction, a sampling rate dependent threshold is considered.		The detection is based on fitting a <math>n</math>-degree polynomial over a sliding window of <math>N</math> samples (e.g., <math>N=200</math> with 1Hz data). The residuals (predicted-observed) are compared to identify the carrier phase cycle-slips <ref group="footnotes">This approach is inspired in the algorithms developed by [Blewitt, 1990] for the geometry-free combination of two frequency signals. Notice that, a similar scheme is applied here for the code and phase combination of a single frequency signal.</ref>. As the geometry-free combination is affected by the ionospheric refraction, a sampling rate dependent threshold can also be considered.

	:'' Input data:'' Code minus phase combination of a single frequency data.		:'' Input data:'' Code minus phase combination of a single frequency data.
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	::::* If no data hole larger than <math>tol_{\Delta t}</math>, thence:		::::* If no data hole larger than <math>tol_{\Delta t}</math>, thence:
	::::* Update an array with the last <math>N</math> continuous epochs <math>[d(s;k-N),\dots,d(s;k-1)]</math>		::::* Update an array with the last <math>N</math> continuous epochs <math>[d(s;k-N),\dots,d(s;k-1)]</math>
	::::* Fit a n-degree polynomial <math>P(s;x)</math> to the previous values <ref group="footnotes"> ~~The empirical value of~~ <math>n=~~\min[(N~~/~~100+1),6]~~</math> ~~for the polynomial suggested in [Blewitt, 1990]~~can be used ~~as well~~.</ref>.		::::* Fit a n-degree polynomial <math>P(s;x)</math> to the previous values <ref group="footnotes"> <math>n=2 </math> or <math>3 </math> can be used.</ref>.
	::::* Declare cycle-slip when:		::::* Declare cycle-slip when:
	:::::<math>\begin{array}{l}		:::::<math>\begin{array}{l}
	\left \|d(s;k) -P(s;k~~) \right) - \left (d(s;k-1) -P(s;k-1~~) \right \|> Threshold		\left \|d(s;k) -P(s;k) \right \|> Threshold
	\end{array}		\end{array}
	</math>		</math>
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	:* This detector is affected by the pseudorange noise and multipath, as well as the divergence of the ionosphere. Thence, higher sampling rates ~~(i.e., <math>\Delta t \leq 1\,sec</math>)~~ improve the detection performance, but still the shortest jumps can escape to this detector. On the other hand, a minimum number of samples is needed for the filter initialisation, in order to assure a reliable value of <math>S_d</math> for the detection threshold.		:* This detector is affected by the pseudorange noise and multipath, as well as the divergence of the ionosphere. Thence, higher sampling rates improve the detection performance, but still the shortest jumps can escape to this detector. On the other hand, a minimum number of samples is needed for the filter initialisation, in order to assure a reliable value of <math>S_d</math> for the detection threshold.


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	\begin{array}{l}		\begin{array}{l}
	m_d(s;k)= \displaystyle \frac{a-1}{a} m_d(s;k-1) + \frac{1}{a} d(s;k)\\		m_d(s;k)= \displaystyle \frac{a-1}{a} m_d(s;k-1) + \frac{1}{a} d(s;k)\\
	~~\\[-0.3cm]~~
	~~m_{d^2}(s;k)= \displaystyle \frac{a-1}{a} m_{d^2}(s;k-1) + \frac{1}{a} d^2(s;k)\\~~
	\end{array} \qquad \mbox{(5)}		\end{array} \qquad \mbox{(5)}
	</math>		</math>
	:where, <math>a=k</math> when <math>k<N</math> and <math>a=N</math> when <math>k \geq N</math>.		:where, <math>a=k</math> when <math>k<N</math> and <math>a=N</math> when <math>k \geq N</math>.

	:~~These equations~~ allow ~~computing~~ a sequential ~~estimation~~ of the mean ~~and sigma values~~, but this filter has infinite memory, propagating forward the divergence of the ionospheric refraction <ref group="footnotes"> On the contrary, the equations (3) provide and estimate based in a (sliding) window, with a finite number of points.</ref>. Nevertheless, such accumulated effect, although biasing the ambiguity estimate, should not ~~affect to~~ the cycle-~~slip estimation, because it varies smoothly and the detector looks~~ for jumps.		:This equation allow computation of a sequential estimate of the mean value, but this filter has infinite memory, propagating forward the divergence of the ionospheric refraction <ref group="footnotes"> On the contrary, the equations (3) provide and estimate based in a (sliding) window, with a finite number of points.</ref>. Nevertheless, such an accumulated effect, although biasing the ambiguity estimate, should not be an issue, since the cycle-slips can be detected by looking for jumps in "<math> d-m_d</math>".

	::[[File: Single Freq Cycle Slip De Fig 1.png\|none\|thumb\|640px\|'''''Figure 1:''''' Fault cycle-slip detection on GPS <math>L_1</math> signal: A multipath drift, after a quite stable period (i.e., with small sigma), produces a fault detection. The Measured and Predicted differences are compared with the confidence threshold (computed with the sliding window algorithm, using the equation (3) combined with (4).]]		::[[File: Single Freq Cycle Slip De Fig 1.png\|none\|thumb\|640px\|'''''Figure 1:''''' Fault cycle-slip detection on GPS <math>L_1</math> signal: A multipath drift, after a quite stable period (i.e., with small sigma), produces a fault detection. The Measured and Predicted differences are compared with the confidence threshold (computed with the sliding window algorithm, using the equation (3) combined with (4).]]
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	''Additional comments: Differences in time detector.''		''Additional comments: Differences in time detector.''

	A detector based in the n-order time differences of carrier phase <math>\Phi</math> measurements between consecutive epochs could be considered (see article [[Detector based in carrier phase data: The geometry-free combination]]. Nevertheless, it must be taken into account that now such differences are affected by the change of geometry (geometric range and clocks) <ref group="footnotes"> In the case of article [[Detector based in carrier phase data: The geometry-free combination]], the geometry-free combination was used, cancelling all the non dispersive effects (geometric range, clocks...).</ref>. In spite of that, most of the geometric range variation is cancelled from <math>\Delta^3</math>, because it varies as a smooth function, ~~being~~ this detector mainly affected by the receiver clock instabilities, high ionospheric fluctuations (i.e., scintillation...), as well.		A detector based in the n-order time differences of carrier phase <math>\Phi</math> measurements between consecutive epochs could be considered (see article [[Detector based in carrier phase data: The geometry-free combination]]. Nevertheless, it must be taken into account that now such differences are affected by the change of geometry (geometric range and clocks) <ref group="footnotes"> In the case of article [[Detector based in carrier phase data: The geometry-free combination]], the geometry-free combination was used, cancelling all the non dispersive effects (geometric range, clocks...).</ref>. In spite of that, most of the geometric range variation is cancelled from <math>\Delta^3</math>, because it varies as a smooth function. Thence, this detector mainly affected by the receiver clock instabilities, high ionospheric fluctuations (i.e., scintillation...), as well.

	In the case of the receiver clock effects, they can be removed by considering single differences of measurements between pairs of satellites in view, but it will enlarge the noise. The ionospheric effects will be not completely removed with such differences, but they should not be a problem, except for a highly disturbed scenario.		In the case of the receiver clock effects, they can be removed by considering single differences of measurements between pairs of satellites in view, but it will enlarge the noise. The ionospheric effects will be not completely removed with such differences, but they should not be a problem, except for a highly disturbed scenario.

Jaume.Sanz at 12:13, 13 January 2013

2013-01-13T12:13:24Z

← Older revision		Revision as of 12:13, 13 January 2013
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	::::* Declare cycle-slip when data hole greater than <math>tol_{\Delta t}</math> <ref group="footnotes">For instance, 60 seconds.</ref>.		::::* Declare cycle-slip when data hole greater than <math>tol_{\Delta t}</math> <ref group="footnotes">For instance, 60 seconds.</ref>.
	::::* If no data hole larger than <math>tol_{\Delta t}</math>, thence:		::::* If no data hole larger than <math>tol_{\Delta t}</math>, thence:
	::::* Update an array with the last <math>N</math> ~~values~~ <math>[d(s;k-N),\dots,d(s;k-1)]</math>		::::* Update an array with the last <math>N</math> continuous epochs <math>[d(s;k-N),\dots,d(s;k-1)]</math>
	::::* Fit a n-degree polynomial <math>P(s;x)</math> to the previous values <ref group="footnotes"> The empirical value of <math>n=\min[(N/100+1),6]</math> for the polynomial suggested in [Blewitt, 1990]can be used as well.</ref>.		::::* Fit a n-degree polynomial <math>P(s;x)</math> to the previous values <ref group="footnotes"> The empirical value of <math>n=\min[(N/100+1),6]</math> for the polynomial suggested in [Blewitt, 1990]can be used as well.</ref>.
	::::* Declare cycle-slip when:		::::* Declare cycle-slip when:
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	\end{array}		\end{array}
	</math>		</math>
			:::::* Reset algorithm after cycle slip.
	::: End		::: End
	:: End		:: End
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	:::::<math> d(s;k)=\Phi(s;k)-R(s;k) \qquad \mbox{(2)} </math>		:::::<math> d(s;k)=\Phi(s;k)-R(s;k) \qquad \mbox{(2)} </math>
	:::: That is: <math>[d(s;k-N),\dots,d(s;k-1)]</math>		:::: That is: <math>[d(s;k-N),\dots,d(s;k-1)]</math>
	::::* Compute the mean and sigma discrepancy over the previous <math>N</math> ~~samples~~ <math>[k-N, \dots , k-1]</math>:		::::* Compute the mean and sigma discrepancy over the previous <math>N</math> continuous epochs <math>[k-N, \dots , k-1]</math>:
	::::<math>		::::<math>
	\begin{array}{l}		\begin{array}{l}
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	:::: If <math>\|d(s;k)-m_d(s;k-1)\| > n_T \,S_d(s;k-1)</math>,		:::: If <math>\|d(s;k)-m_d(s;k-1)\| > n_T \,S_d(s;k-1)</math>,
	:::: Thence, cycle-slip.		:::: Thence, cycle-slip.
			::::* Reset algorithm after cycle slip.
	::: End		::: End
	:: End		:: End

Jaume.Sanz at 09:36, 13 January 2013

2013-01-13T09:36:00Z

← Older revision		Revision as of 09:36, 13 January 2013
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	::::* Declare cycle-slip when:		::::* Declare cycle-slip when:
	:::::<math>\begin{array}{l}		:::::<math>\begin{array}{l}
	\left(d(s;k) -P(s;k) \right) - \left (d(s;k-1) -P(s;k-1) \right ) > ~~k\, \mbox{cycles}~~		\left \|d(s;k) -P(s;k) \right) - \left (d(s;k-1) -P(s;k-1) \right \|> Threshold
	\end{array}		\end{array}
	</math>		</math>

Carlos.Lopez at 11:28, 23 February 2012

2012-02-23T11:28:36Z

← Older revision		Revision as of 11:28, 23 February 2012
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	\|Level=Advanced		\|Level=Advanced
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}

Jaume.Sanz at 17:31, 1 February 2012

2012-02-01T17:31:39Z

← Older revision		Revision as of 17:31, 1 February 2012
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	==References==		==References==
	[Blewitt, 1990] Blewitt, G., 1990. An automatic editing Algorithms for GPS data. Geophysical Research Letters. 17(3), pp. 199{202.		[Blewitt, 1990] Blewitt, G., 1990. An automatic editing Algorithms for GPS data. Geophysical Research Letters. 17(3), pp. 199-202.

	[[Category:Fundamentals]]		[[Category:Fundamentals]]
	[[Category:GNSS Measurements and Data Preprocessing]]		[[Category:GNSS Measurements and Data Preprocessing]]

Jaume.Sanz at 17:28, 1 February 2012

2012-02-01T17:28:18Z

← Older revision		Revision as of 17:28, 1 February 2012
Line 104:		Line 104:
	:where, <math>a=k</math> when <math>k<N</math> and <math>a=N</math> when <math>k \geq N</math>.		:where, <math>a=k</math> when <math>k<N</math> and <math>a=N</math> when <math>k \geq N</math>.

	:These equations allow computing a sequential estimation of the mean and sigma values, but this filter has infinite memory, propagating forward the divergence of the ionospheric refraction <ref group="footnotes"> On the contrary, the equations (3) provide and estimate based in a (sliding) window, with a finite number of points.</ref>. Nevertheless, such accumulated effect, although biasing the ambiguity estimate, should not affect to the cycle-slip estimation, because it ~~vary~~ smoothly and the detector looks for jumps.		:These equations allow computing a sequential estimation of the mean and sigma values, but this filter has infinite memory, propagating forward the divergence of the ionospheric refraction <ref group="footnotes"> On the contrary, the equations (3) provide and estimate based in a (sliding) window, with a finite number of points.</ref>. Nevertheless, such accumulated effect, although biasing the ambiguity estimate, should not affect to the cycle-slip estimation, because it varies smoothly and the detector looks for jumps.

	::[[File: Single Freq Cycle Slip De Fig 1.png\|none\|thumb\|640px\|'''''Figure 1:''''' Fault cycle-slip detection on GPS <math>L_1</math> signal: A multipath drift, after a quite stable period (i.e., with small sigma), produces a fault detection. The Measured and Predicted differences are compared with the confidence threshold (computed with the sliding window algorithm, using the equation (3) combined with (4).]]		::[[File: Single Freq Cycle Slip De Fig 1.png\|none\|thumb\|640px\|'''''Figure 1:''''' Fault cycle-slip detection on GPS <math>L_1</math> signal: A multipath drift, after a quite stable period (i.e., with small sigma), produces a fault detection. The Measured and Predicted differences are compared with the confidence threshold (computed with the sliding window algorithm, using the equation (3) combined with (4).]]

Jaume.Sanz at 17:26, 1 February 2012

2012-02-01T17:26:22Z

← Older revision		Revision as of 17:26, 1 February 2012
Line 104:		Line 104:
	:where, <math>a=k</math> when <math>k<N</math> and <math>a=N</math> when <math>k \geq N</math>.		:where, <math>a=k</math> when <math>k<N</math> and <math>a=N</math> when <math>k \geq N</math>.

	:These equations allow ~~to compute~~ a sequential estimation of the mean and sigma values, but this filter has infinite memory, propagating forward the divergence of the ionospheric refraction <ref group="footnotes"> On the contrary, the equations (3) provide and estimate based in a (sliding) window, with a finite number of points.</ref>. Nevertheless, such accumulated effect, although biasing the ambiguity estimate, should not affect to the cycle-slip estimation, because it vary smoothly and the detector looks for jumps.		:These equations allow computing a sequential estimation of the mean and sigma values, but this filter has infinite memory, propagating forward the divergence of the ionospheric refraction <ref group="footnotes"> On the contrary, the equations (3) provide and estimate based in a (sliding) window, with a finite number of points.</ref>. Nevertheless, such accumulated effect, although biasing the ambiguity estimate, should not affect to the cycle-slip estimation, because it vary smoothly and the detector looks for jumps.

	::[[File: Single Freq Cycle Slip De Fig 1.png\|none\|thumb\|640px\|'''''Figure 1:''''' Fault cycle-slip detection on GPS <math>L_1</math> signal: A multipath drift, after a quite stable period (i.e., with small sigma), produces a fault detection. The Measured and Predicted differences are compared with the confidence threshold (computed with the sliding window algorithm, using the equation (3) combined with (4).]]		::[[File: Single Freq Cycle Slip De Fig 1.png\|none\|thumb\|640px\|'''''Figure 1:''''' Fault cycle-slip detection on GPS <math>L_1</math> signal: A multipath drift, after a quite stable period (i.e., with small sigma), produces a fault detection. The Measured and Predicted differences are compared with the confidence threshold (computed with the sliding window algorithm, using the equation (3) combined with (4).]]