Positioning Error - Revision history

Christian.Rost: corrected equation 12 and 13

2013-08-16T12:43:02Z

corrected equation 12 and 13

← Older revision		Revision as of 12:43, 16 August 2013
Line 173:		Line 173:

	:<math>		:<math>
	\mbox{2-D rms horizontal error}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\left ( \Delta E_i^2+\Delta N_i^2 \right )^2}}		\mbox{2-D rms horizontal error}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\left ( \Delta E_i^2+\Delta N_i^2 \right )}}
	\qquad \mbox{(12)}		\qquad \mbox{(12)}
	</math>		</math>

	:<math>		:<math>
	\mbox{3-D rms error}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\left ( \Delta E_i^2+\Delta N_i^2 +\Delta U_i^2\right )^2}}		\mbox{3-D rms error}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\left ( \Delta E_i^2+\Delta N_i^2 +\Delta U_i^2\right )}}
	\qquad \mbox{(13)}		\qquad \mbox{(13)}
	</math>		</math>

Jaume.Sanz at 19:53, 4 February 2013

2013-02-04T19:53:34Z

← Older revision		Revision as of 19:53, 4 February 2013
Line 92:		Line 92:


	The Root Mean Square (~~RMS~~) is given by:		The Root Mean Square Error (RMSE) is given by:

	:<math>		:<math>
	\mbox{~~RMS~~}_{_{\mathbf \Delta X}}= \sqrt{trace ({\mathbf P}_{_{\mathbf \Delta X}})}=\sigma \sqrt{trace \left [({\mathbf G}^T\,{\mathbf G})^{-1}\right ]}		\mbox{RMSE}_{_{\mathbf \Delta X}}= \sqrt{trace ({\mathbf P}_{_{\mathbf \Delta X}})}=\sigma \sqrt{trace \left [({\mathbf G}^T\,{\mathbf G})^{-1}\right ]}
	\qquad \mbox{(9)}		\qquad \mbox{(9)}
	</math>		</math>


	what means that, the trace of matrix <math>({\mathbf G}^T\,{\mathbf G})^{-1}</math> is a scale factor on <math>\sigma</math> for the ~~RMS~~<math>_{_{\mathbf \Delta X}}</math>.		what means that, the trace of matrix <math>({\mathbf G}^T\,{\mathbf G})^{-1}</math> is a scale factor on <math>\sigma</math> for the RMSE<math>_{_{\mathbf \Delta X}}</math>.

Carlos.Lopez at 11:39, 23 February 2012

2012-02-23T11:39:50Z

← Older revision		Revision as of 11:39, 23 February 2012
Line 4:		Line 4:
	\|Level=Advanced		\|Level=Advanced
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}

Carlos.Lopez at 14:36, 30 January 2012

2012-01-30T14:36:05Z

← Older revision		Revision as of 14:36, 30 January 2012
Line 51:		Line 51:


	thence, <math>{\mathbf P}_{enu}={\mathbf R}^T\,{\mathbf P}_{xyz}\,{\mathbf R}</math> is given, where <math>{\mathbf P}_{xyz}</math> is the sub-matrix of <math>{\mathbf P}_{_{\mathbf \Delta X}}</math> containing solely the geometric components <ref group="footnotes"> The matrix <math>{\mathbf P}_{enu}</math> can be obtained directly in the ENU coordinates, using the geometry matrix <math>{\mathbf G}</math> computed in the ENU coordinates. In this local system the rows of matrix <math>{\mathbf G}</math> (see [[Code Based Positioning (~~SPP~~)]]) are <math>[\cos el^i \sin az^i, \cos el^i \cos az^i, \sin el^i, 1]</math>, where <math>el^i</math> and <math>az^i</math> are the elevation and azimuth angle of the satellite <math>i</math> observed from the receiver position, (see [[Transformations between ECEF and ENU coordinates]]).</ref>. From this covariance matrix, they can be defined:		thence, <math>{\mathbf P}_{enu}={\mathbf R}^T\,{\mathbf P}_{xyz}\,{\mathbf R}</math> is given, where <math>{\mathbf P}_{xyz}</math> is the sub-matrix of <math>{\mathbf P}_{_{\mathbf \Delta X}}</math> containing solely the geometric components <ref group="footnotes"> The matrix <math>{\mathbf P}_{enu}</math> can be obtained directly in the ENU coordinates, using the geometry matrix <math>{\mathbf G}</math> computed in the ENU coordinates. In this local system the rows of matrix <math>{\mathbf G}</math> (see [[Code Based Positioning (SPS)]]) are <math>[\cos el^i \sin az^i, \cos el^i \cos az^i, \sin el^i, 1]</math>, where <math>el^i</math> and <math>az^i</math> are the elevation and azimuth angle of the satellite <math>i</math> observed from the receiver position, (see [[Transformations between ECEF and ENU coordinates]]).</ref>. From this covariance matrix, they can be defined:

	:<math>		:<math>
Line 90:		Line 90:


	This expression depends upon two factors: 1) the measurements (prefit-residuals) variance <math>\sigma^2</math>, and 2) the geometry matrix (<math>{\mathbf G}</math>), which is linked only to the receiver-satellite rays geometry, (see [[Code Based Positioning (~~SPP~~)]]).		This expression depends upon two factors: 1) the measurements (prefit-residuals) variance <math>\sigma^2</math>, and 2) the geometry matrix (<math>{\mathbf G}</math>), which is linked only to the receiver-satellite rays geometry, (see [[Code Based Positioning (SPS)]]).

Carlos.Lopez: /* Measured Accuracy */

2012-01-30T09:00:34Z

Measured Accuracy

← Older revision		Revision as of 09:00, 30 January 2012
Line 188:		Line 188:
	:* CEP: Circular Error Probable, as the 50th percentile of horizontal error.		:* CEP: Circular Error Probable, as the 50th percentile of horizontal error.
	:* SEP: Spherical Error Probable, as the 50th percentile of 3-D error.		:* SEP: Spherical Error Probable, as the 50th percentile of 3-D error.
	:* 2drms: calculated as twice the 2-D rms horizontal error given by (13).		:* 2drms: calculated as twice the 2-D rms horizontal error given by (12).

Carlos.Lopez: /* Measured Accuracy */

2012-01-30T09:00:07Z

Measured Accuracy

← Older revision		Revision as of 09:00, 30 January 2012
Line 170:		Line 170:
	:<math>		:<math>
	\mbox{rms vertical error}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\Delta U_i^2}}		\mbox{rms vertical error}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\Delta U_i^2}}
	\qquad \mbox{(12)}		\qquad \mbox{(11)}
	</math>		</math>

	:<math>		:<math>
	\mbox{2-D rms horizontal error}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\left ( \Delta E_i^2+\Delta N_i^2 \right )^2}}		\mbox{2-D rms horizontal error}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\left ( \Delta E_i^2+\Delta N_i^2 \right )^2}}
	\qquad \mbox{(13)}		\qquad \mbox{(12)}
	</math>		</math>

	:<math>		:<math>
	\mbox{3-D rms error}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\left ( \Delta E_i^2+\Delta N_i^2 +\Delta U_i^2\right )^2}}		\mbox{3-D rms error}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\left ( \Delta E_i^2+\Delta N_i^2 +\Delta U_i^2\right )^2}}
	\qquad \mbox{(14)}		\qquad \mbox{(13)}
	</math>		</math>

Line 208:		Line 208:

	A wide discussion on such accuracy equivalences can be found in [Diggelen, 2007] <ref> [Diggelen, 2007] Diggelen, F. v., 2007. GNSS Accuracy: Lies, Damn Lies, and Statistics. GPS World. 18(1), pp. 26-32. </ref>.		A wide discussion on such accuracy equivalences can be found in [Diggelen, 2007] <ref> [Diggelen, 2007] Diggelen, F. v., 2007. GNSS Accuracy: Lies, Damn Lies, and Statistics. GPS World. 18(1), pp. 26-32. </ref>.


	==Notes==		==Notes==

Carlos.Lopez: /* Predicted Accuracy: Dilution of Precision */

2012-01-30T08:59:43Z

Predicted Accuracy: Dilution of Precision

← Older revision		Revision as of 08:59, 30 January 2012
Line 84:		Line 84:

	the covariance matrix of <math>{\mathbf{\Delta X}}</math> estimate is given by (see [[Parameters adjustment]])		the covariance matrix of <math>{\mathbf{\Delta X}}</math> estimate is given by (see [[Parameters adjustment]])
	~~:<math>~~

	:<math>		:<math>
	{\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1} \qquad \mbox{(8)}		{\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1} \qquad \mbox{(8)}
Line 165:		Line 163:

	where, basically, DOP represents an approximate ratio factor between the precision in the measurements (<math>\sigma_0</math>) and in positioning. This ratio is computed only from the satellites-receiver geometry.		where, basically, DOP represents an approximate ratio factor between the precision in the measurements (<math>\sigma_0</math>) and in positioning. This ratio is computed only from the satellites-receiver geometry.


	== Measured Accuracy ==		== Measured Accuracy ==

Carlos.Lopez at 08:59, 30 January 2012

2012-01-30T08:59:12Z

← Older revision		Revision as of 08:59, 30 January 2012
Line 85:		Line 85:
	the covariance matrix of <math>{\mathbf{\Delta X}}</math> estimate is given by (see [[Parameters adjustment]])		the covariance matrix of <math>{\mathbf{\Delta X}}</math> estimate is given by (see [[Parameters adjustment]])
	:<math>		:<math>
	~~{\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1}\;;\;\;\;~~
	~~\mbox{MSE}_{_{\mathbf \Delta X}}=n\,\sigma^2~~
	~~\qquad \mbox{(8)}</math>~~

	~~:as:~~

	:<math>		:<math>
	{\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1} \qquad \mbox{(9)}		{\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1} \qquad \mbox{(8)}
	</math>		</math>

Line 104:		Line 99:
	:<math>		:<math>
	\mbox{RMS}_{_{\mathbf \Delta X}}= \sqrt{trace ({\mathbf P}_{_{\mathbf \Delta X}})}=\sigma \sqrt{trace \left [({\mathbf G}^T\,{\mathbf G})^{-1}\right ]}		\mbox{RMS}_{_{\mathbf \Delta X}}= \sqrt{trace ({\mathbf P}_{_{\mathbf \Delta X}})}=\sigma \sqrt{trace \left [({\mathbf G}^T\,{\mathbf G})^{-1}\right ]}
	\qquad \mbox{(10)}		\qquad \mbox{(9)}
	</math>		</math>

Line 124:		Line 119:
	q_{xt}&q_{yt}&q_{zt}&q_{tt} \\		q_{xt}&q_{yt}&q_{zt}&q_{tt} \\
	\end{array}		\end{array}
	\right] \qquad \mbox{(11)}		\right] \qquad \mbox{(10)}
	</math>		</math>

Jaume.Sanz at 14:26, 23 January 2012

2012-01-23T14:26:47Z

← Older revision		Revision as of 14:26, 23 January 2012
Line 89:		Line 89:
	\qquad \mbox{(8)}</math>		\qquad \mbox{(8)}</math>

			:as:

			:<math>
			{\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1} \qquad \mbox{(9)}
			</math>

Jaume.Sanz at 14:24, 23 January 2012

2012-01-23T14:24:28Z

← Older revision		Revision as of 14:24, 23 January 2012
Line 90:		Line 90:


	~~:as:~~

	~~:<math>~~
	~~{\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1} \qquad \mbox{(9)}~~
	~~</math>~~