Dilution of precision (navigation)

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dir="ltr"><section class="mf-section-0" id="mf-section-0"> <p><b>Dilution of precision</b> (<b>DOP</b>), or <b>geometric dilution of precision</b> (<b>GDOP</b>), is a term used in <a href="/wiki/Satellite_navigation" title="Satellite navigation">satellite navigation</a> and <a href="/wiki/Geomatics_engineering" class="mw-redirect" title="Geomatics engineering">geomatics engineering</a> to specify the <a href="/wiki/Error_propagation" class="mw-redirect" title="Error propagation">error propagation</a> as a mathematical effect of navigation satellite geometry on positional measurement precision. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Geometric_Dilution_Of_Precision.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Geometric_Dilution_Of_Precision.svg/220px-Geometric_Dilution_Of_Precision.svg.png" decoding="async" width="220" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Geometric_Dilution_Of_Precision.svg/330px-Geometric_Dilution_Of_Precision.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Geometric_Dilution_Of_Precision.svg/440px-Geometric_Dilution_Of_Precision.svg.png 2x" data-file-width="560" data-file-height="450"></a><figcaption>Understanding dilution of precision with an example. In <b>A</b> someone has measured the distance to two landmarks, and plotted their point as the intersection of two circles with the measured radius. In <b>B</b> the measurement has some error bounds, and their true location will lie anywhere in the green area. In <b>C</b> the measurement error is the same, but the error on their position has grown considerably due to the arrangement of the landmarks.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:404px;max-width:404px"><div class="trow"><div class="theader">Bad DOP vs Good DOP</div></div><div class="trow"><div class="tsingle" style="width:402px;max-width:402px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Bad_gdop.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Bad_gdop.png/400px-Bad_gdop.png" decoding="async" width="400" height="358" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Bad_gdop.png/600px-Bad_gdop.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Bad_gdop.png/800px-Bad_gdop.png 2x" data-file-width="858" data-file-height="768"></a></span></div><div class="thumbcaption">Navigation satellites with poor geometry.</div></div></div><div class="trow"><div class="tsingle" style="width:402px;max-width:402px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Good_gdop.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Good_gdop.png/400px-Good_gdop.png" decoding="async" width="400" height="358" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Good_gdop.png/600px-Good_gdop.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Good_gdop.png/800px-Good_gdop.png 2x" data-file-width="858" data-file-height="768"></a></span></div><div class="thumbcaption">Navigation satellites with good geometry.</div></div></div></div></div> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Introduction"><span class="tocnumber">1</span> <span class="toctext">Introduction</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Interpretation"><span class="tocnumber">2</span> <span class="toctext">Interpretation</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Computation"><span class="tocnumber">3</span> <span class="toctext">Computation</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#References"><span class="tocnumber">5</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Further_reading"><span class="tocnumber">6</span> <span class="toctext">Further reading</span></a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Introduction">Introduction</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dilution_of_precision_(navigation)&action=edit&section=1" title="Edit section: Introduction" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>The concept of dilution of precision (DOP) originated with users of the <a href="/wiki/Loran-C" title="Loran-C">Loran-C navigation system</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The idea of geometric DOP is to state how errors in the measurement will affect the final state estimation. This can be defined as:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GDOP} ={\frac {\Delta ({\text{output location}})}{\Delta ({\text{measured data}})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GDOP</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>output location</mtext> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>measured data</mtext> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GDOP} ={\frac {\Delta ({\text{output location}})}{\Delta ({\text{measured data}})}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29d720bd31bef8342fb092021fae9c395dd6949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.111ex; height:6.509ex;" alt="{\displaystyle \operatorname {GDOP} ={\frac {\Delta ({\text{output location}})}{\Delta ({\text{measured data}})}}}"></noscript><span class="lazy-image-placeholder" style="width: 30.111ex;height: 6.509ex;vertical-align: -2.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29d720bd31bef8342fb092021fae9c395dd6949" data-alt="{\displaystyle \operatorname {GDOP} ={\frac {\Delta ({\text{output location}})}{\Delta ({\text{measured data}})}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>Conceptually you can geometrically imagine errors on a measurement resulting in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ({\text{measured data}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>measured data</mtext> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ({\text{measured data}})}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/523baf42dc90b085d3730965d761aad9f26f634c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.424ex; height:2.843ex;" alt="{\displaystyle \Delta ({\text{measured data}})}"></noscript><span class="lazy-image-placeholder" style="width: 18.424ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/523baf42dc90b085d3730965d761aad9f26f634c" data-alt="{\displaystyle \Delta ({\text{measured data}})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> term changing. Ideally small changes in the measured data will not result in large changes in output location. The opposite of this ideal is the situation where the solution is very sensitive to measurement errors. The interpretation of this formula is shown in the figure to the right, showing two possible scenarios with acceptable and poor GDOP. </p><p>With the wide adoption of <a href="/wiki/Satellite_navigation" title="Satellite navigation">satellite navigation</a> systems, the term has come into much wider usage. Neglecting ionospheric <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and tropospheric<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> effects, the signal from navigation satellites has a fixed precision. Therefore, the relative satellite-receiver geometry plays a major role in determining the precision of estimated positions and times. Due to the relative geometry of any given satellite to a receiver, the precision in the <a href="/wiki/Pseudorange" title="Pseudorange">pseudorange</a> of the satellite translates to a corresponding component in each of the four dimensions of position measured by the receiver (i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></noscript><span class="lazy-image-placeholder" style="width: 1.088ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" data-alt="{\displaystyle z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></noscript><span class="lazy-image-placeholder" style="width: 0.84ex;height: 2.009ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" data-alt="{\displaystyle t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>). The precision of multiple satellites in view of a receiver combine according to the relative position of the satellites to determine the level of precision in each dimension of the receiver measurement. When visible navigation satellites are close together in the sky, the geometry is said to be weak and the DOP value is high; when far apart, the geometry is strong and the DOP value is low. Consider two overlapping rings, or <a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">annuli</a>, of different centres. If they overlap at right angles, the greatest extent of the overlap is much smaller than if they overlap in near parallel. Thus a low DOP value represents a better positional precision due to the wider angular separation between the satellites used to calculate a unit's position. Other factors that can increase the effective DOP are obstructions such as nearby mountains or buildings. </p><p>DOP can be expressed as a number of separate measurements: </p> <dl><dt>HDOP</dt> <dd>Horizontal dilution of precision</dd> <dt>VDOP</dt> <dd>Vertical dilution of precision</dd> <dt>PDOP</dt> <dd>Position (3D) dilution of precision</dd> <dt>TDOP</dt> <dd>Time dilution of precision</dd> <dt>GDOP</dt> <dd>Geometric dilution of precision</dd></dl> <p>These values follow mathematically from the positions of the usable satellites. Signal receivers allow the display of these positions (<i>skyplot</i>) as well as the DOP values. </p><p>The term can also be applied to other location systems that employ several geographical spaced sites. It can occur in electronic-counter-counter-measures (<a href="/wiki/Electronic_warfare" title="Electronic warfare">electronic warfare</a>) when computing the location of enemy emitters (<a href="/wiki/Radar_jammer" class="mw-redirect" title="Radar jammer">radar jammers</a> and radio communications devices). Using such an <a href="/wiki/Interferometry" title="Interferometry">interferometry</a> technique can provide certain geometric layout where there are degrees of freedom that cannot be accounted for due to inadequate configurations. </p><p>The effect of geometry of the satellites on position error is called geometric dilution of precision (GDOP) and it is roughly interpreted as ratio of position error to the range error. Imagine that a <a href="/wiki/Square_pyramid" title="Square pyramid">square pyramid</a> is formed by lines joining four satellites with the receiver at the tip of the pyramid. The larger the volume of the pyramid, the better (lower) the value of GDOP; the smaller its volume, the worse (higher) the value of GDOP will be. Similarly, the greater the number of satellites, the better the value of GDOP. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Interpretation">Interpretation</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dilution_of_precision_(navigation)&action=edit&section=2" title="Edit section: Interpretation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <table class="wikitable"> <tbody><tr> <th>DOP Value </th> <th>Rating<sup id="cite_ref-isik_5-0" class="reference"><a href="#cite_note-isik-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </th> <th>Description </th></tr> <tr> <td>< 1 </td> <td>Ideal </td> <td>Highest possible confidence level to be used for applications demanding the highest possible precision at all times. </td></tr> <tr> <td>1–2 </td> <td>Excellent </td> <td>At this confidence level, positional measurements are considered accurate enough to meet all but the most sensitive applications. </td></tr> <tr> <td>2–5 </td> <td>Good </td> <td>Represents a level that marks the minimum appropriate for making accurate decisions. Positional measurements could be used to make reliable in-route navigation suggestions to the user. </td></tr> <tr> <td>5–10 </td> <td>Moderate </td> <td>Positional measurements could be used for calculations, but the fix quality could still be improved. A more open view of the sky is recommended. </td></tr> <tr> <td>10–20 </td> <td>Fair </td> <td>Represents a low confidence level. Positional measurements should be discarded or used only to indicate a very rough estimate of the current location. </td></tr> <tr> <td>> 20 </td> <td>Poor </td> <td>At this level, measurements should be discarded. </td></tr></tbody></table> <p>The DOP factors are functions of the diagonal elements of the <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> of the parameters, expressed either in a global or a local geodetic frame. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Computation">Computation</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dilution_of_precision_(navigation)&action=edit&section=3" title="Edit section: Computation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>As a first step in computing DOP,<sup id="cite_ref-isik_5-1" class="reference"><a href="#cite_note-isik-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> consider the unit vectors from the receiver to satellite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></noscript><span class="lazy-image-placeholder" style="width: 0.802ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" data-alt="{\displaystyle i}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\left({\frac {x_{i}-x}{R_{i}}},{\frac {y_{i}-y}{R_{i}}},{\frac {z_{i}-z}{R_{i}}}\right),&R_{i}&={\sqrt {(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd></mtd> <mtd> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>x</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>y</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>z</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\left({\frac {x_{i}-x}{R_{i}}},{\frac {y_{i}-y}{R_{i}}},{\frac {z_{i}-z}{R_{i}}}\right),&R_{i}&={\sqrt {(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4210c7f9690f471d89777fa698729a6b86421b78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:72.773ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&\left({\frac {x_{i}-x}{R_{i}}},{\frac {y_{i}-y}{R_{i}}},{\frac {z_{i}-z}{R_{i}}}\right),&R_{i}&={\sqrt {(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}}\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 72.773ex;height: 6.176ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4210c7f9690f471d89777fa698729a6b86421b78" data-alt="{\displaystyle {\begin{aligned}&\left({\frac {x_{i}-x}{R_{i}}},{\frac {y_{i}-y}{R_{i}}},{\frac {z_{i}-z}{R_{i}}}\right),&R_{i}&={\sqrt {(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"></noscript><span class="lazy-image-placeholder" style="width: 5.641ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" data-alt="{\displaystyle x,y,z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> denote the position of the receiver and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i},y_{i},z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i},y_{i},z_{i}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/870a7fc353eb64581a93a7196101340892b273b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.017ex; height:2.009ex;" alt="{\displaystyle x_{i},y_{i},z_{i}}"></noscript><span class="lazy-image-placeholder" style="width: 8.017ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/870a7fc353eb64581a93a7196101340892b273b5" data-alt="{\displaystyle x_{i},y_{i},z_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> denote the position of satellite i. Formulate the matrix, A, which (for 4 pseudorange measurement residual equations) is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}{\frac {x_{1}-x}{R_{1}}}&{\frac {y_{1}-y}{R_{1}}}&{\frac {z_{1}-z}{R_{1}}}&1\\{\frac {x_{2}-x}{R_{2}}}&{\frac {y_{2}-y}{R_{2}}}&{\frac {z_{2}-z}{R_{2}}}&1\\{\frac {x_{3}-x}{R_{3}}}&{\frac {y_{3}-y}{R_{3}}}&{\frac {z_{3}-z}{R_{3}}}&1\\{\frac {x_{4}-x}{R_{4}}}&{\frac {y_{4}-y}{R_{4}}}&{\frac {z_{4}-z}{R_{4}}}&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>x</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>y</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>z</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>x</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>y</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>z</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mi>x</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mi>y</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mi>z</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mi>x</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mi>y</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mi>z</mi> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}{\frac {x_{1}-x}{R_{1}}}&{\frac {y_{1}-y}{R_{1}}}&{\frac {z_{1}-z}{R_{1}}}&1\\{\frac {x_{2}-x}{R_{2}}}&{\frac {y_{2}-y}{R_{2}}}&{\frac {z_{2}-z}{R_{2}}}&1\\{\frac {x_{3}-x}{R_{3}}}&{\frac {y_{3}-y}{R_{3}}}&{\frac {z_{3}-z}{R_{3}}}&1\\{\frac {x_{4}-x}{R_{4}}}&{\frac {y_{4}-y}{R_{4}}}&{\frac {z_{4}-z}{R_{4}}}&1\end{bmatrix}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a0d52ae008e3c3699038b0e7e8390656c9c212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.505ex; width:30.7ex; height:18.176ex;" alt="{\displaystyle A={\begin{bmatrix}{\frac {x_{1}-x}{R_{1}}}&{\frac {y_{1}-y}{R_{1}}}&{\frac {z_{1}-z}{R_{1}}}&1\\{\frac {x_{2}-x}{R_{2}}}&{\frac {y_{2}-y}{R_{2}}}&{\frac {z_{2}-z}{R_{2}}}&1\\{\frac {x_{3}-x}{R_{3}}}&{\frac {y_{3}-y}{R_{3}}}&{\frac {z_{3}-z}{R_{3}}}&1\\{\frac {x_{4}-x}{R_{4}}}&{\frac {y_{4}-y}{R_{4}}}&{\frac {z_{4}-z}{R_{4}}}&1\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 30.7ex;height: 18.176ex;vertical-align: -8.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a0d52ae008e3c3699038b0e7e8390656c9c212" data-alt="{\displaystyle A={\begin{bmatrix}{\frac {x_{1}-x}{R_{1}}}&{\frac {y_{1}-y}{R_{1}}}&{\frac {z_{1}-z}{R_{1}}}&1\\{\frac {x_{2}-x}{R_{2}}}&{\frac {y_{2}-y}{R_{2}}}&{\frac {z_{2}-z}{R_{2}}}&1\\{\frac {x_{3}-x}{R_{3}}}&{\frac {y_{3}-y}{R_{3}}}&{\frac {z_{3}-z}{R_{3}}}&1\\{\frac {x_{4}-x}{R_{4}}}&{\frac {y_{4}-y}{R_{4}}}&{\frac {z_{4}-z}{R_{4}}}&1\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>The first three elements of each row of <i>A</i> are the components of a unit vector from the receiver to the indicated satellite. The last element of each row refers to the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a> of pseudorange w.r.t. receiver's clock bias. Formulate the matrix, <i>Q</i>, as the <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> resulting from the <a href="/wiki/Least-squares_normal_matrix" class="mw-redirect" title="Least-squares normal matrix">least-squares normal matrix</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\left(A^{\mathsf {T}}A\right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\left(A^{\mathsf {T}}A\right)^{-1}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da87ec5ab24f2bd985d9d66f1b670f7235392acc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.237ex; height:3.843ex;" alt="{\displaystyle Q=\left(A^{\mathsf {T}}A\right)^{-1}}"></noscript><span class="lazy-image-placeholder" style="width: 14.237ex;height: 3.843ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da87ec5ab24f2bd985d9d66f1b670f7235392acc" data-alt="{\displaystyle Q=\left(A^{\mathsf {T}}A\right)^{-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>In general: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\left(J_{\mathsf {x}}^{\mathsf {T}}\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}J_{x}\right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\left(J_{\mathsf {x}}^{\mathsf {T}}\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}J_{x}\right)^{-1}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d25afe1c4e6e15a80375f48a301797a347c747fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.861ex; height:5.176ex;" alt="{\displaystyle Q=\left(J_{\mathsf {x}}^{\mathsf {T}}\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}J_{x}\right)^{-1}}"></noscript><span class="lazy-image-placeholder" style="width: 27.861ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d25afe1c4e6e15a80375f48a301797a347c747fa" data-alt="{\displaystyle Q=\left(J_{\mathsf {x}}^{\mathsf {T}}\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}J_{x}\right)^{-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{x}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/573c4eac7701ac45330a1677abe1c3a7fc054ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.463ex; height:2.509ex;" alt="{\displaystyle J_{x}}"></noscript><span class="lazy-image-placeholder" style="width: 2.463ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/573c4eac7701ac45330a1677abe1c3a7fc054ae1" data-alt="{\displaystyle J_{x}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the Jacobian of the sensor measurement residual equations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}\left({\underline {x}},{\underline {d}}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>x</mi> <mo>_</mo> </munder> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>d</mi> <mo>_</mo> </munder> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}\left({\underline {x}},{\underline {d}}\right)=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d1b690361e73f23f66d058ff091cd1938f106f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:11.979ex; height:3.343ex;" alt="{\displaystyle f_{i}\left({\underline {x}},{\underline {d}}\right)=0}"></noscript><span class="lazy-image-placeholder" style="width: 11.979ex;height: 3.343ex;vertical-align: -0.583ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d1b690361e73f23f66d058ff091cd1938f106f" data-alt="{\displaystyle f_{i}\left({\underline {x}},{\underline {d}}\right)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, with respect to the unknowns, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>x</mi> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {x}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14a33c3e6e935d7bbafe48a5166f035cc37c4e75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.537ex; margin-bottom: -0.801ex; width:1.332ex; height:2.676ex;" alt="{\displaystyle {\underline {x}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.332ex;height: 2.676ex;vertical-align: -0.537ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14a33c3e6e935d7bbafe48a5166f035cc37c4e75" data-alt="{\displaystyle {\underline {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{d}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d409f8f94210ec359721cf70ebc383905a7bad24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.509ex;" alt="{\displaystyle J_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 2.382ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d409f8f94210ec359721cf70ebc383905a7bad24" data-alt="{\displaystyle J_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the Jacobian of the sensor measurement residual equations with respect to the measured quantities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>d</mi> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {d}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deefe3d520fef2afcf038759c33883f93c3f2eeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.534ex; margin-left: -0.005ex; margin-bottom: -0.804ex; width:1.222ex; height:3.176ex;" alt="{\displaystyle {\underline {d}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.222ex;height: 3.176ex;vertical-align: -0.534ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deefe3d520fef2afcf038759c33883f93c3f2eeb" data-alt="{\displaystyle {\underline {d}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{d}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d638b409d293db1874a1503bd0f87d20b3d57ed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.754ex; height:2.509ex;" alt="{\displaystyle C_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 2.754ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d638b409d293db1874a1503bd0f87d20b3d57ed7" data-alt="{\displaystyle C_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the correlation matrix for noise in the measured quantities. </p><p>For the preceding case of 4 range measurement residual equations: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {x}}=(x,y,z,\tau )^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>x</mi> <mo>_</mo> </munder> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {x}}=(x,y,z,\tau )^{\mathsf {T}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55519d2150cb584ec05ee12c060c57b902edea35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:15.468ex; height:3.676ex;" alt="{\displaystyle {\underline {x}}=(x,y,z,\tau )^{\mathsf {T}}}"></noscript><span class="lazy-image-placeholder" style="width: 15.468ex;height: 3.676ex;vertical-align: -0.583ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55519d2150cb584ec05ee12c060c57b902edea35" data-alt="{\displaystyle {\underline {x}}=(x,y,z,\tau )^{\mathsf {T}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {d}}=\left(\tau _{1},\tau _{2},\tau _{3},\tau _{4}\right)^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>d</mi> <mo>_</mo> </munder> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {d}}=\left(\tau _{1},\tau _{2},\tau _{3},\tau _{4}\right)^{\mathsf {T}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c556a36624fb9967dc1d7afca7896f15e7c30d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-left: -0.005ex; margin-bottom: -0.755ex; width:18.864ex; height:3.843ex;" alt="{\displaystyle {\underline {d}}=\left(\tau _{1},\tau _{2},\tau _{3},\tau _{4}\right)^{\mathsf {T}}}"></noscript><span class="lazy-image-placeholder" style="width: 18.864ex;height: 3.843ex;vertical-align: -0.583ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c556a36624fb9967dc1d7afca7896f15e7c30d" data-alt="{\displaystyle {\underline {d}}=\left(\tau _{1},\tau _{2},\tau _{3},\tau _{4}\right)^{\mathsf {T}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =ct}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mi>c</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =ct}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2476a7ab60af6fadfba49ed5ae89c1a992d8c6c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.147ex; height:2.009ex;" alt="{\displaystyle \tau =ct}"></noscript><span class="lazy-image-placeholder" style="width: 6.147ex;height: 2.009ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2476a7ab60af6fadfba49ed5ae89c1a992d8c6c7" data-alt="{\displaystyle \tau =ct}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{i}=ct_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>c</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{i}=ct_{i}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/204a2701af634925a14377b392bd5183e0f2eeb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.56ex; height:2.343ex;" alt="{\displaystyle \tau _{i}=ct_{i}}"></noscript><span class="lazy-image-placeholder" style="width: 7.56ex;height: 2.343ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/204a2701af634925a14377b392bd5183e0f2eeb4" data-alt="{\displaystyle \tau _{i}=ct_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i}=|\tau _{i}-\tau |={\sqrt {(\tau _{i}-\tau )^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i}=|\tau _{i}-\tau |={\sqrt {(\tau _{i}-\tau )^{2}}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88c14ea07d3c980cbd307ebdde0a5e70ea4088cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:26.958ex; height:4.843ex;" alt="{\displaystyle R_{i}=|\tau _{i}-\tau |={\sqrt {(\tau _{i}-\tau )^{2}}}}"></noscript><span class="lazy-image-placeholder" style="width: 26.958ex;height: 4.843ex;vertical-align: -1.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88c14ea07d3c980cbd307ebdde0a5e70ea4088cc" data-alt="{\displaystyle R_{i}=|\tau _{i}-\tau |={\sqrt {(\tau _{i}-\tau )^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}\left({\underline {x}},{\underline {d}}\right)={\sqrt {(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}}-{\sqrt {(\tau _{i}-\tau )^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>x</mi> <mo>_</mo> </munder> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>d</mi> <mo>_</mo> </munder> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}\left({\underline {x}},{\underline {d}}\right)={\sqrt {(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}}-{\sqrt {(\tau _{i}-\tau )^{2}}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0ec0a327143f8fdce2ff45a3d8d55a0aeefca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:59.341ex; height:4.843ex;" alt="{\displaystyle f_{i}\left({\underline {x}},{\underline {d}}\right)={\sqrt {(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}}-{\sqrt {(\tau _{i}-\tau )^{2}}}}"></noscript><span class="lazy-image-placeholder" style="width: 59.341ex;height: 4.843ex;vertical-align: -1.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0ec0a327143f8fdce2ff45a3d8d55a0aeefca0" data-alt="{\displaystyle f_{i}\left({\underline {x}},{\underline {d}}\right)={\sqrt {(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}}-{\sqrt {(\tau _{i}-\tau )^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{x}=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{x}=A}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c0c0b34ac2cae1956269934b7d6c5bc6688507d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.304ex; height:2.509ex;" alt="{\displaystyle J_{x}=A}"></noscript><span class="lazy-image-placeholder" style="width: 7.304ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c0c0b34ac2cae1956269934b7d6c5bc6688507d" data-alt="{\displaystyle J_{x}=A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{d}=-I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{d}=-I}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4287996c4c1531dd8a2bb4c3006647c278f7ce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.461ex; height:2.509ex;" alt="{\displaystyle J_{d}=-I}"></noscript><span class="lazy-image-placeholder" style="width: 8.461ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4287996c4c1531dd8a2bb4c3006647c278f7ce7" data-alt="{\displaystyle J_{d}=-I}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and the measurement noises for the different <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{i}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814ca8b33360ac3b7db8e9435271b5654175c853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.816ex; height:2.009ex;" alt="{\displaystyle \tau _{i}}"></noscript><span class="lazy-image-placeholder" style="width: 1.816ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814ca8b33360ac3b7db8e9435271b5654175c853" data-alt="{\displaystyle \tau _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> have been assumed to be independent which makes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{d}=I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{d}=I}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c361093b866100ba1657207d0a79753826b3d0db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.024ex; height:2.509ex;" alt="{\displaystyle C_{d}=I}"></noscript><span class="lazy-image-placeholder" style="width: 7.024ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c361093b866100ba1657207d0a79753826b3d0db" data-alt="{\displaystyle C_{d}=I}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </p><p>This formula for Q arises from applying best linear unbiased estimation to a linearized version of the sensor measurement residual equations about the current solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\underline {x}}=-Q*\left(J_{x}^{\mathsf {T}}\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}f\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>x</mi> <mo>_</mo> </munder> </mrow> <mo>=</mo> <mo>−</mo> <mi>Q</mi> <mo>∗</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\underline {x}}=-Q*\left(J_{x}^{\mathsf {T}}\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}f\right)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4516b448c26b2cbdb9aea2708df9522d8cf213f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.615ex; height:4.843ex;" alt="{\displaystyle \Delta {\underline {x}}=-Q*\left(J_{x}^{\mathsf {T}}\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}f\right)}"></noscript><span class="lazy-image-placeholder" style="width: 31.615ex;height: 4.843ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4516b448c26b2cbdb9aea2708df9522d8cf213f" data-alt="{\displaystyle \Delta {\underline {x}}=-Q*\left(J_{x}^{\mathsf {T}}\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}f\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, except in the case of B.L.U.E. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{d}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d638b409d293db1874a1503bd0f87d20b3d57ed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.754ex; height:2.509ex;" alt="{\displaystyle C_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 2.754ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d638b409d293db1874a1503bd0f87d20b3d57ed7" data-alt="{\displaystyle C_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is a noise covariance matrix rather than the noise correlation matrix used in DOP, and the reason DOP makes this substitution is to obtain a <b>relative</b> error. When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{d}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d638b409d293db1874a1503bd0f87d20b3d57ed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.754ex; height:2.509ex;" alt="{\displaystyle C_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 2.754ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d638b409d293db1874a1503bd0f87d20b3d57ed7" data-alt="{\displaystyle C_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is a noise covariance matrix, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></noscript><span class="lazy-image-placeholder" style="width: 1.838ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" data-alt="{\displaystyle Q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is an estimate of the matrix of covariance of noise in the unknowns due to the noise in the measured quantities. It is the estimate obtained by the <i>first-order second moment</i> (F.O.S.M.) uncertainty quantification technique which was state of the art in the 1980s. In order for the F.O.S.M. theory to be strictly applicable, either the input noise distributions need to be Gaussian or the measurement noise standard deviations need to be small relative to rate of change in the output near the solution. In this context, the second criteria is typically the one that is satisfied. </p><p>This (i.e. for the 4 time of arrival/range measurement residual equations) computation is in accordance with [6] where the weighting matrix, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e9f295aeef664fc64ee48aa232843fa75715bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.32ex; height:3.676ex;" alt="{\displaystyle P=\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}}"></noscript><span class="lazy-image-placeholder" style="width: 17.32ex;height: 3.676ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e9f295aeef664fc64ee48aa232843fa75715bd" data-alt="{\displaystyle P=\left(J_{d}C_{d}J_{d}^{\mathsf {T}}\right)^{-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> happens to simplify down to the identity matrix. </p><p>Note that P only simplifies down to the identity matrix because all the sensor measurement residual equations are time of arrival (pseudo range) equations. In other cases, for example when trying to locate someone broadcasting on an <a href="/wiki/International_distress_frequency" title="International distress frequency">international distress frequency</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> would <b>not</b> simplify down to the identity matrix and in that case there would be a "frequency DOP" or FDOP component either in addition to or in place of the TDOP component. (Regarding "in place of the TDOP component": Since the clocks on the legacy <a href="/wiki/International_Cospas-Sarsat_Programme" title="International Cospas-Sarsat Programme">International Cospas-Sarsat Programme</a> LEO satellites are much less accurate than GPS clocks, discarding their time measurements would actually increase the geolocation solution accuracy.) </p><p>The elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></noscript><span class="lazy-image-placeholder" style="width: 1.838ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" data-alt="{\displaystyle Q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> are designated as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q={\begin{bmatrix}\sigma _{x}^{2}&\sigma _{xy}&\sigma _{xz}&\sigma _{xt}\\\sigma _{xy}&\sigma _{y}^{2}&\sigma _{yz}&\sigma _{yt}\\\sigma _{xz}&\sigma _{yz}&\sigma _{z}^{2}&\sigma _{zt}\\\sigma _{xt}&\sigma _{yt}&\sigma _{zt}&\sigma _{t}^{2}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>t</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>t</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>t</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>t</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>t</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>t</mi> </mrow> </msub> </mtd> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\begin{bmatrix}\sigma _{x}^{2}&\sigma _{xy}&\sigma _{xz}&\sigma _{xt}\\\sigma _{xy}&\sigma _{y}^{2}&\sigma _{yz}&\sigma _{yt}\\\sigma _{xz}&\sigma _{yz}&\sigma _{z}^{2}&\sigma _{zt}\\\sigma _{xt}&\sigma _{yt}&\sigma _{zt}&\sigma _{t}^{2}\end{bmatrix}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a75571e7be5746f3797a3afa531d1ab5539c7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; margin-top: -0.196ex; width:28.753ex; height:13.509ex;" alt="{\displaystyle Q={\begin{bmatrix}\sigma _{x}^{2}&\sigma _{xy}&\sigma _{xz}&\sigma _{xt}\\\sigma _{xy}&\sigma _{y}^{2}&\sigma _{yz}&\sigma _{yt}\\\sigma _{xz}&\sigma _{yz}&\sigma _{z}^{2}&\sigma _{zt}\\\sigma _{xt}&\sigma _{yt}&\sigma _{zt}&\sigma _{t}^{2}\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 28.753ex;height: 13.509ex;vertical-align: -6.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a75571e7be5746f3797a3afa531d1ab5539c7b9" data-alt="{\displaystyle Q={\begin{bmatrix}\sigma _{x}^{2}&\sigma _{xy}&\sigma _{xz}&\sigma _{xt}\\\sigma _{xy}&\sigma _{y}^{2}&\sigma _{yz}&\sigma _{yt}\\\sigma _{xz}&\sigma _{yz}&\sigma _{z}^{2}&\sigma _{zt}\\\sigma _{xt}&\sigma _{yt}&\sigma _{zt}&\sigma _{t}^{2}\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>PDOP, TDOP, and GDOP are given by:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {PDOP} &={\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+\sigma _{z}^{2}}}\\\operatorname {TDOP} &={\sqrt {\sigma _{t}^{2}}}\\\operatorname {GDOP} &={\sqrt {\operatorname {PDOP} ^{2}+\operatorname {TDOP} ^{2}}}\\&={\sqrt {\operatorname {tr} Q}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>PDOP</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>TDOP</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>GDOP</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>PDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>TDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>tr</mi> <mo>⁡</mo> <mi>Q</mi> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {PDOP} &={\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+\sigma _{z}^{2}}}\\\operatorname {TDOP} &={\sqrt {\sigma _{t}^{2}}}\\\operatorname {GDOP} &={\sqrt {\operatorname {PDOP} ^{2}+\operatorname {TDOP} ^{2}}}\\&={\sqrt {\operatorname {tr} Q}}\\\end{aligned}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66db071cfbc3c8f5895fb4bb21a954dda7db81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.417ex; margin-bottom: -0.254ex; width:31.45ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {PDOP} &={\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+\sigma _{z}^{2}}}\\\operatorname {TDOP} &={\sqrt {\sigma _{t}^{2}}}\\\operatorname {GDOP} &={\sqrt {\operatorname {PDOP} ^{2}+\operatorname {TDOP} ^{2}}}\\&={\sqrt {\operatorname {tr} Q}}\\\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 31.45ex;height: 18.509ex;vertical-align: -8.417ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66db071cfbc3c8f5895fb4bb21a954dda7db81" data-alt="{\displaystyle {\begin{aligned}\operatorname {PDOP} &={\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+\sigma _{z}^{2}}}\\\operatorname {TDOP} &={\sqrt {\sigma _{t}^{2}}}\\\operatorname {GDOP} &={\sqrt {\operatorname {PDOP} ^{2}+\operatorname {TDOP} ^{2}}}\\&={\sqrt {\operatorname {tr} Q}}\\\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>Notice GDOP is the square root of the <a href="/wiki/Matrix_trace" class="mw-redirect" title="Matrix trace">trace</a> of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></noscript><span class="lazy-image-placeholder" style="width: 1.838ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" data-alt="{\displaystyle Q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> matrix. </p><p>The horizontal and vertical dilution of precision, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {HDOP} &={\sqrt {\sigma _{n}^{2}+\sigma _{e}^{2}}}\\\operatorname {VDOP} &={\sqrt {\sigma _{u}^{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>HDOP</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>VDOP</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {HDOP} &={\sqrt {\sigma _{n}^{2}+\sigma _{e}^{2}}}\\\operatorname {VDOP} &={\sqrt {\sigma _{u}^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd6342d6459cd187ae37ebdac289310cfec8bb55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.854ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {HDOP} &={\sqrt {\sigma _{n}^{2}+\sigma _{e}^{2}}}\\\operatorname {VDOP} &={\sqrt {\sigma _{u}^{2}}}\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 20.854ex;height: 7.176ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd6342d6459cd187ae37ebdac289310cfec8bb55" data-alt="{\displaystyle {\begin{aligned}\operatorname {HDOP} &={\sqrt {\sigma _{n}^{2}+\sigma _{e}^{2}}}\\\operatorname {VDOP} &={\sqrt {\sigma _{u}^{2}}}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</dd></dl> <p>are both dependent on the coordinate system used. To correspond to the <a href="/wiki/Local_tangent_plane_coordinates" title="Local tangent plane coordinates">local east-north-up</a> coordinate system, </p> <pre>EDOP^2 x x x x NDOP^2 x x x x VDOP^2 x x x x TDOP^2 </pre> <p>and the derived dilutions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {GDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}+\operatorname {VDOP} ^{2}+\operatorname {TDOP} ^{2}}}\\\operatorname {HDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}}}\\\operatorname {PDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}+\operatorname {VDOP} ^{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>GDOP</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>EDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>NDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>VDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>TDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>HDOP</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>EDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>NDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>PDOP</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>EDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>NDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>VDOP</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {GDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}+\operatorname {VDOP} ^{2}+\operatorname {TDOP} ^{2}}}\\\operatorname {HDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}}}\\\operatorname {PDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}+\operatorname {VDOP} ^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ecbfcffedd35bfa3eaadb1b34c64c9806f9e4ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:52.542ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {GDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}+\operatorname {VDOP} ^{2}+\operatorname {TDOP} ^{2}}}\\\operatorname {HDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}}}\\\operatorname {PDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}+\operatorname {VDOP} ^{2}}}\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 52.542ex;height: 14.843ex;vertical-align: -6.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ecbfcffedd35bfa3eaadb1b34c64c9806f9e4ad" data-alt="{\displaystyle {\begin{aligned}\operatorname {GDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}+\operatorname {VDOP} ^{2}+\operatorname {TDOP} ^{2}}}\\\operatorname {HDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}}}\\\operatorname {PDOP} &={\sqrt {\operatorname {EDOP} ^{2}+\operatorname {NDOP} ^{2}+\operatorname {VDOP} ^{2}}}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dilution_of_precision_(navigation)&action=edit&section=4" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <ul><li><a href="/wiki/Circular_error_probable" title="Circular error probable">Circular error probable</a></li> <li><a href="/wiki/GNSS_positioning_calculation" class="mw-redirect" title="GNSS positioning calculation">GNSS positioning calculation</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dilution_of_precision_(navigation)&action=edit&section=5" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRichard_B._Langley1999" class="citation web cs1">Richard B. Langley (May 1999). <a rel="nofollow" class="external text" href="http://gauss.gge.unb.ca/papers.pdf/gpsworld.may99.pdf">"Dilution of Precision"</a> <span class="cs1-format">(PDF)</span>. <i>GPS World</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111004205627/http://gauss.gge.unb.ca/papers.pdf/gpsworld.may99.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2011-10-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-10-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=GPS+World&rft.atitle=Dilution+of+Precision&rft.date=1999-05&rft.au=Richard+B.+Langley&rft_id=http%3A%2F%2Fgauss.gge.unb.ca%2Fpapers.pdf%2Fgpsworld.may99.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADilution+of+precision+%28navigation%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDudekJenkin2000" class="citation book cs1"><a href="/wiki/Gregory_Dudek" title="Gregory Dudek">Dudek, Gregory</a>; Jenkin, Michael (2000). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/computationalpri0000dude"><i>Computational Principles of Mobile Robotics</i></a></span>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-56876-5" title="Special:BookSources/0-521-56876-5"><bdi>0-521-56876-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Principles+of+Mobile+Robotics&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=0-521-56876-5&rft.aulast=Dudek&rft.aufirst=Gregory&rft.au=Jenkin%2C+Michael&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcomputationalpri0000dude&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADilution+of+precision+%28navigation%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul_Kintner,_Cornell_UniversityTodd_HumphreysUniversity_of_Texas-AustinJoanna_Hinks2009" class="citation web cs1">Paul Kintner, Cornell University; Todd Humphreys; University of Texas-Austin; Joanna Hinks; Cornell University (July–August 2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111106011219/http://www.insidegnss.com/node/1579">"GNSS and Ionospheric Scintillation: How to Survive the Next Solar Maximum"</a>. <i><a href="/wiki/Inside_GNSS" title="Inside GNSS">Inside GNSS</a></i>. Archived from <a rel="nofollow" class="external text" href="https://www.insidegnss.com/node/1579">the original</a> on 2011-11-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-10-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Inside+GNSS&rft.atitle=GNSS+and+Ionospheric+Scintillation%3A+How+to+Survive+the+Next+Solar+Maximum&rft.date=2009-07%2F2009-08&rft.au=Paul+Kintner%2C+Cornell+University&rft.au=Todd+Humphreys&rft.au=University+of+Texas-Austin&rft.au=Joanna+Hinks&rft.au=Cornell+University&rft_id=http%3A%2F%2Fwww.insidegnss.com%2Fnode%2F1579&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADilution+of+precision+%28navigation%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.trimble.com/gps_tutorial/howgps-error.aspx">"GPS errors (Trimble tutorial)"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160307124113/http://www.trimble.com/gps_tutorial/howgps-error.aspx">Archived</a> from the original on 2016-03-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=GPS+errors+%28Trimble+tutorial%29&rft_id=http%3A%2F%2Fwww.trimble.com%2Fgps_tutorial%2Fhowgps-error.aspx&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADilution+of+precision+%28navigation%29" class="Z3988"></span></span> </li> <li id="cite_note-isik-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-isik_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-isik_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIsikHongPetruninTsourdos2020" class="citation journal cs1">Isik, Oguz Kagan; Hong, Juhyeon; Petrunin, Ivan; Tsourdos, Antonios (25 August 2020). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Frobotics9030066">"Integrity Analysis for GPS-Based Navigation of UAVs in Urban Environment"</a>. <i>Robotics</i>. <b>9</b> (3): 66. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Frobotics9030066">10.3390/robotics9030066</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Robotics&rft.atitle=Integrity+Analysis+for+GPS-Based+Navigation+of+UAVs+in+Urban+Environment&rft.volume=9&rft.issue=3&rft.pages=66&rft.date=2020-08-25&rft_id=info%3Adoi%2F10.3390%2Frobotics9030066&rft.aulast=Isik&rft.aufirst=Oguz+Kagan&rft.au=Hong%2C+Juhyeon&rft.au=Petrunin%2C+Ivan&rft.au=Tsourdos%2C+Antonios&rft_id=https%3A%2F%2Fdoi.org%2F10.3390%252Frobotics9030066&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADilution+of+precision+%28navigation%29" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141122153439/http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap1/149.htm">Section 1.4.9 of <i>Principles of Satellite Positioning</i></a>.</span> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dilution_of_precision_(navigation)&action=edit&section=6" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul><li><a rel="nofollow" class="external text" href="http://gauss.gge.unb.ca/papers.pdf/gpsworld.may99.pdf">DOP Factors</a></li> <li><a rel="nofollow" class="external text" href="http://www.colorado.edu/geography/gcraft/notes/gps/gif/gdop.gif">manually calculating GDOP</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140222000521/http://www.crowtracker.com/hdop-gps-position-errors/">HDOP AND GPS HORIZONTAL POSITION ERRORS</a></li></ul> <ul><li>Article on DOP and Trimble's program: <a rel="nofollow" class="external text" href="http://freegeographytools.com/2007/determining-local-gps-satellite-geometry-effects-on-position-accuracy">Determining Local GPS Satellite Geometry Effects On Position Accuracy</a>.</li> <li>Notes & <a href="/wiki/GIF" title="GIF">GIF</a> image on manually calculating GDOP: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050823013233/http://www.colorado.edu/geography/gcraft/notes/gps/gps.html#Gdop">Geographer's Craft</a></li> <li>GPS Errors & Estimating Your Receiver's Accuracy: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160310132600/http://edu-observatory.org/gps/gps_accuracy.html">Sam Wormley's GPS Accuracy Web Page</a></li> <li>GPS Accuracy, Errors & Precision: <a rel="nofollow" class="external text" href="http://www.radio-electronics.com/info/satellite/gps/accuracy-errors-precision.php">Radio-Electronics.com</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140222164642/http://www.radio-electronics.com/info/satellite/gps/accuracy-errors-precision.php">Archived</a> 2014-02-22 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul>    </section></div> <noscript><img 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dilution of precision" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Diluizione_della_precisione" title="Diluizione della precisione – Italian" lang="it" hreflang="it" data-title="Diluizione della precisione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/DOP" title="DOP – Hebrew" lang="he" hreflang="he" data-title="DOP" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dilution_of_precision" title="Dilution of precision – Polish" lang="pl" hreflang="pl" 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Dilution of precision (navigation) - Wikipedia