https://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&feed=atom&action=historyAnalysis-Forecast Cycle Observation Impacts - Revision history2024-03-28T11:16:50ZRevision history for this page on the wikiMediaWiki 1.36.2https://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5831&oldid=prevRobertson: Text replacement - "W4DPSAS_FCT_SENSITIVITY" to "RBL4DVAR_FCT_SENSITIVITY"2020-07-22T17:24:14Z<p>Text replacement - "W4DPSAS_FCT_SENSITIVITY" to "RBL4DVAR_FCT_SENSITIVITY"</p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Practicalities==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Practicalities==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#The observation impact driver for the forecast cycles is activated using the [[Options#<del style="font-weight: bold; text-decoration: none;">W4DPSAS_FCT_SENSITIVITY</del>|<del style="font-weight: bold; text-decoration: none;">W4DPSAS_FCT_SENSITIVITY</del>]] cpp option.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#The observation impact driver for the forecast cycles is activated using the [[Options#<ins style="font-weight: bold; text-decoration: none;">RBL4DVAR_FCT_SENSITIVITY</ins>|<ins style="font-weight: bold; text-decoration: none;">RBL4DVAR_FCT_SENSITIVITY</ins>]] cpp option.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The default option is 3<sup>rd</sup>-order approximations of the squared forecast error difference <math>\delta e</math>. 1<sup>st</sup>- and 2<sup>nd</sup>-order cases can also be run by changing the parameter <span class="limeGreen">ImpOrd</span> in the routine <span class="red">obs_sen_w4dpsas_forecast.h</span> but this is recommended ''only'' for testing purposes and once you feel confident about how things work.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The default option is 3<sup>rd</sup>-order approximations of the squared forecast error difference <math>\delta e</math>. 1<sup>st</sup>- and 2<sup>nd</sup>-order cases can also be run by changing the parameter <span class="limeGreen">ImpOrd</span> in the routine <span class="red">obs_sen_w4dpsas_forecast.h</span> but this is recommended ''only'' for testing purposes and once you feel confident about how things work.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The length of the analysis cycle is assigned in [[roms.in]] using the parameter [[Variables#ntimes_ana|NTIMES_ANA]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The length of the analysis cycle is assigned in [[roms.in]] using the parameter [[Variables#ntimes_ana|NTIMES_ANA]].</div></td></tr>
</table>Robertsonhttps://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5448&oldid=prevRobertson at 20:49, 17 July 20192019-07-17T20:49:02Z<p></p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>__TOC__</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><!-- Edit Template:Variational_Data_Assimilation_TOC to modify this Table of Contents--></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><div style="float: left;margin: 0 20px 0 0;">{{Variational Data Assimilation TOC}}</div></ins>__TOC__</div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The procedure for computing the observation impacts during the forecast cycle is a little more involved than that for the analysis cycle. However, a separate ROMS driver exists for this. To help illustrate the procedure involved, consider the typical analysis-forecast cycle shown schematically below.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The procedure for computing the observation impacts during the forecast cycle is a little more involved than that for the analysis cycle. However, a separate ROMS driver exists for this. To help illustrate the procedure involved, consider the typical analysis-forecast cycle shown schematically below.</div></td></tr>
</table>Robertsonhttps://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5393&oldid=prevArango: /* Practicalities */2019-07-10T15:00:24Z<p><span dir="auto"><span class="autocomment">Practicalities</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The observation impact driver for the forecast cycles is activated using the [[Options#W4DPSAS_FCT_SENSITIVITY|W4DPSAS_FCT_SENSITIVITY]] cpp option.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The observation impact driver for the forecast cycles is activated using the [[Options#W4DPSAS_FCT_SENSITIVITY|W4DPSAS_FCT_SENSITIVITY]] cpp option.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The default option is 3<sup>rd</sup>-order approximations of the squared forecast error difference <math>\delta e</math>. 1<sup>st</sup>- and 2<sup>nd</sup>-order cases can also be run by changing the parameter <span class="limeGreen">ImpOrd</span> in the routine <span class="red">obs_sen_w4dpsas_forecast.h</span> but this is recommended ''only'' for testing purposes and once you feel confident about how things work.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The default option is 3<sup>rd</sup>-order approximations of the squared forecast error difference <math>\delta e</math>. 1<sup>st</sup>- and 2<sup>nd</sup>-order cases can also be run by changing the parameter <span class="limeGreen">ImpOrd</span> in the routine <span class="red">obs_sen_w4dpsas_forecast.h</span> but this is recommended ''only'' for testing purposes and once you feel confident about how things work.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#The length of the analysis cycle is assigned in [[<del style="font-weight: bold; text-decoration: none;">ocean</del>.in]] using the parameter [[Variables#ntimes_ana|NTIMES_ANA]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#The length of the analysis cycle is assigned in [[<ins style="font-weight: bold; text-decoration: none;">roms</ins>.in]] using the parameter [[Variables#ntimes_ana|NTIMES_ANA]].</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#The length of the <del style="font-weight: bold; text-decoration: none;">foeecast </del>cycle is assigned in [[<del style="font-weight: bold; text-decoration: none;">ocean</del>.in]] using the parameter [[Variables#ntimes_ana|NTIMES_FCT]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#The length of the <ins style="font-weight: bold; text-decoration: none;">forecast </ins>cycle is assigned in [[<ins style="font-weight: bold; text-decoration: none;">roms</ins>.in]] using the parameter [[Variables#ntimes_ana|NTIMES_FCT]].</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#The default configuration of the driver is to use a verifying analysis to compute the forecast errors as in equation (1). Two adjoint model forcing input files are required: one that corresponds to <math>\bold{C}(\bold{x}_f^j-\bold{x}_a^{j+2})</math> in equation (3) and one that corresponds to <math>\bold{C}(\bold{x}_f^{j+1}-\bold{x}_a^{j+2})</math>. These are referred to respectively as [[Variables#FCTnameB|FCTnameB]] and [[Variables#FCTnameA|FCTnameA]] in [[<del style="font-weight: bold; text-decoration: none;">ocean</del>.in]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#The default configuration of the driver is to use a verifying analysis to compute the forecast errors as in equation (1). Two adjoint model forcing input files are required: one that corresponds to <math>\bold{C}(\bold{x}_f^j-\bold{x}_a^{j+2})</math> in equation (3) and one that corresponds to <math>\bold{C}(\bold{x}_f^{j+1}-\bold{x}_a^{j+2})</math>. These are referred to respectively as [[Variables#FCTnameB|FCTnameB]] and [[Variables#FCTnameA|FCTnameA]] in [[<ins style="font-weight: bold; text-decoration: none;">roms</ins>.in]].</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#For forecast error metrics in observation space as in equation (4), it is necessary to also define the [[Options#OBS_SPACE|OBS_SPACE]] cpp option. In this case the input forcing files for the adjoint model have the same structure as an observation file. In the code they are referred to as [[Variables#FOInameA|FOInameA]] and [[Variables#FOInameB|FOInameB]] in [[<del style="font-weight: bold; text-decoration: none;">ocean</del>.in]] corresponding to <math>\bold{C}(\bold{y}_f^{j+1}-\bold{y}^{j+2})</math> and <math>\bold{C}(\bold{y}_f^j-\bold{y}^{j+2})</math> respectively in equation (5). The actual observations assimilated during the analysis cycle prior to the forecast cycle are referred to as <span class="limeGreen">obs_C.nc</span> in the driver.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#For forecast error metrics in observation space as in equation (4), it is necessary to also define the [[Options#OBS_SPACE|OBS_SPACE]] cpp option. In this case the input forcing files for the adjoint model have the same structure as an observation file. In the code they are referred to as [[Variables#FOInameA|FOInameA]] and [[Variables#FOInameB|FOInameB]] in [[<ins style="font-weight: bold; text-decoration: none;">roms</ins>.in]] corresponding to <math>\bold{C}(\bold{y}_f^{j+1}-\bold{y}^{j+2})</math> and <math>\bold{C}(\bold{y}_f^j-\bold{y}^{j+2})</math> respectively in equation (5). The actual observations assimilated during the analysis cycle prior to the forecast cycle are referred to as <span class="limeGreen">obs_C.nc</span> in the driver.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The options [[Options#OBS_IMPACT_SPLIT|OBS_IMPACT_SPLIT]] and [[Options#IMPACT_INNER|IMPACT_INNER]] can also be used.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#The options [[Options#OBS_IMPACT_SPLIT|OBS_IMPACT_SPLIT]] and [[Options#IMPACT_INNER|IMPACT_INNER]] can also be used.</div></td></tr>
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</table>Arangohttps://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5392&oldid=prevRobertson at 14:32, 10 July 20192019-07-10T14:32:17Z<p></p>
<a href="https://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5392&oldid=5391">Show changes</a>Robertsonhttps://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5391&oldid=prevRobertson at 06:23, 10 July 20192019-07-10T06:23:58Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In the figure above, each analysis cycle is assumed to be of length <math>\tau</math> and analysis cycle <math>j</math> spans the interval<math>[t_0^j,t_0^j+\tau]</math>. The circulation estimate at time <math>t_0^j+\tau</math> (i.e. the ''end'' of analysis cycle <math>j</math>) is denoted as <math>x_a^j</math> and is the initial condition for the forecast spanning the next analysis interval <math>[t_0^{j+1},t_0^{j+1}+\tau]</math>. In <del style="font-weight: bold; text-decoration: none;">sequel </del>it is assumed that the forecast duration is an integer multiple of <math>\tau</math>, but this is <del style="font-weight: bold; text-decoration: none;">not a necessary constraint</del>. <del style="font-weight: bold; text-decoration: none;">The figure </del>shows the analyses and forecasts that result from three adjacent analysis cycles, namely cycles <math>j</math>, <math>j+1</math> and cycle <math>j+2</math>. The analysis <math>x_a^j</math> at the <del style="font-weight: bold; text-decoration: none;">''</del>end<del style="font-weight: bold; text-decoration: none;">'' </del>of cycle <math>j</math> is used as the initial condition for the forecast <math>x_f^j</math> of duration <math>2\tau</math> that terminates at time <math>t_0^{j+2}+\tau</math>, the end of analysis cycle <math>j+2</math>. Similarly, the analysis <math>x_a^{j+1}</math> at the <del style="font-weight: bold; text-decoration: none;">''</del>end<del style="font-weight: bold; text-decoration: none;">'' </del>of cycle <math>j+1</math> is used as the initial condition for the forecast <math>x_f^{j+1}</math> of duration <math>\tau</math> and also terminates at time <math>t_0^{j+2}+\tau</math>, the end of analysis cycle <math>j+2</math>. After sufficient time has elapsed, a new analysis <math>x_a^{j+2}</math> will be computed at this time. Since <math>x_a^{j+2}</math> represents our best estimate of the ocean circulation at time <math>t_0^{j+2}+\tau</math> it can be used to quantify the veracity of the forecasts <math>x_f^j</math> and <math>x_f^{j+1}</math>. For this reason, <math>x_a^{j+2}</math> is usually referred to as the “verifying analysis.” However, as discussed, <del style="font-weight: bold; text-decoration: none;">later </del>other sources of information can be used to verify the forecasts, such as new or independent observations.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In the figure above, each analysis cycle is assumed to be of length <math>\tau</math> and analysis cycle <math>j</math> spans the interval<math>[t_0^j,t_0^j+\tau]</math>. The circulation estimate at time <math>t_0^j+\tau</math> (i.e. the ''end'' of analysis cycle <math>j</math>) is denoted as <math>x_a^j</math> and is the initial condition for the forecast spanning the next analysis interval <math>[t_0^{j+1},t_0^{j+1}+\tau]</math>. In <ins style="font-weight: bold; text-decoration: none;">the figure </ins>it is assumed<ins style="font-weight: bold; text-decoration: none;">, for convenience only, </ins>that the forecast duration is an integer multiple of <math>\tau</math>, but this <ins style="font-weight: bold; text-decoration: none;">does not have to be the case, and the code </ins>is <ins style="font-weight: bold; text-decoration: none;">set up to handle analysis and forecast cycles that are different lengths</ins>. <ins style="font-weight: bold; text-decoration: none;">Figure 1 </ins>shows the analyses and forecasts that result from three adjacent analysis cycles, namely cycles <math>j</math>, <math>j+1</math> and cycle <math>j+2</math>. The analysis <math>x_a^j</math> at the end of cycle <math>j</math> is used as the initial condition for the forecast <math>x_f^j</math> of duration <math>2\tau</math> that terminates at time <math>t_0^{j+2}+\tau</math>, the end of analysis cycle <math>j+2</math>. Similarly, the analysis <math>x_a^{j+1}</math> at the end of cycle <math>j+1</math> is used as the initial condition for the forecast <math>x_f^{j+1}</math> of duration <math>\tau</math> and also terminates at time <math>t_0^{j+2}+\tau</math>, the end of analysis cycle <math>j+2</math>. After sufficient time has elapsed, a new analysis <math>x_a^{j+2}</math> will be computed at this time. Since <math>x_a^{j+2}</math> represents our best estimate of the ocean circulation at time <math>t_0^{j+2}+\tau</math> it can be used to quantify the veracity of the forecasts <math>x_f^j</math> and <math>x_f^{j+1}</math>. For this reason, <math>x_a^{j+2}</math> is usually referred to as the “verifying analysis.” However, as discussed <ins style="font-weight: bold; text-decoration: none;">shortly</ins>, other sources of information can be used to verify the forecasts, such as new or independent observations.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>It should be clear from the figure that the forecast <math>x_f^{j+1}</math> benefits from the observations assimilated into the model during analysis cycle <math>j+1</math> (i.e. during the interval <math>[t_0^{j+1},t_0^{j+1}+\tau]</math>). Therefore, providing that <math>x_f^j</math> and <math>x_f^{j+1}</math> are subject to identical surface forcing and open boundary conditions during the interval <math>[t_0^{j+2},t_0^{j+2}+\tau]</math>, any differences in forecast error must be associated with the observations assimilated into the model during the interval <math>[t_0^{j+1},t_0^{j+1}+\tau]</math>. <del style="font-weight: bold; text-decoration: none;">The </del>impact of <del style="font-weight: bold; text-decoration: none;">each</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>It should be clear from the figure that the forecast <math>x_f^{j+1}</math> benefits from the observations assimilated into the model during analysis cycle <math>j+1</math> (i.e. during the interval <math>[t_0^{j+1},t_0^{j+1}+\tau]</math>). Therefore, providing that <math>x_f^j</math> and <math>x_f^{j+1}</math> are subject to identical surface forcing and open boundary conditions during the interval <math>[t_0^{j+2},t_0^{j+2}+\tau]</math>, any differences in forecast error must be associated with the observations assimilated into the model during the interval <math>[t_0^{j+1},t_0^{j+1}+\tau]</math>.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">observation on </del>the forecast error <del style="font-weight: bold; text-decoration: none;">can </del>be <del style="font-weight: bold; text-decoration: none;">quantified as </del>described <del style="font-weight: bold; text-decoration: none;">next which </del>is <del style="font-weight: bold; text-decoration: none;">based on </del>the <del style="font-weight: bold; text-decoration: none;">work </del>of <del style="font-weight: bold; text-decoration: none;">Langland </del>and <del style="font-weight: bold; text-decoration: none;">Baker </del>(<del style="font-weight: bold; text-decoration: none;">2004</del>), <del style="font-weight: bold; text-decoration: none;">Errico </del>(<del style="font-weight: bold; text-decoration: none;">2007</del>) and <del style="font-weight: bold; text-decoration: none;">Gelaro </del>''<del style="font-weight: bold; text-decoration: none;">et al.</del>'' (<del style="font-weight: bold; text-decoration: none;">2007</del>).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Forecast Error Metrics==</ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">As in the case of the analysis cycle observation impacts described above, the </ins>impact of <ins style="font-weight: bold; text-decoration: none;">the observations during the forecast cycle is computed for a specific metric, in this case a metric of </ins>the forecast error<ins style="font-weight: bold; text-decoration: none;">. The methodology will </ins>be described <ins style="font-weight: bold; text-decoration: none;">first for a standard generic quadratic forecast error metric given by:</ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{| class="eqno"</ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|<math display="block">e=(\bold{x}_f-\bold{x}_t)^T \bold{C}(\bold{x}_f-\bold{x}_t)</math></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|(1)</ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|}</ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">where <math>\bold{x}_f</math> denotes the forecast state-vector, <math>\bold{x}_t</math> denotes the true state-vector, and <math>\bold{C}</math> is a weight matrix. For example, if <math>\bold{C}</math> </ins>is <ins style="font-weight: bold; text-decoration: none;">a diagonal matrix with elements equal to 1 corresponding to all surface temperature grid points, and zero elsewhere, then <math>e</math> would represent the sum of the squared errors in SST. Forecast error metrics of the form (1) are very common in numerical weather prediction and oceanography, so (1) is a good starting point.</ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In the figure there are two forecasts of interest: <math>\bold{x}_f^j</math> initialized at </ins>the <ins style="font-weight: bold; text-decoration: none;">end </ins>of <ins style="font-weight: bold; text-decoration: none;">analysis cycle <math>j</math> at time <math>t_0^j+\tau</math>, </ins>and <ins style="font-weight: bold; text-decoration: none;"><math>\bold{x}_f^{j+1}</math> initialized at the end of analysis cycle <math>j+1</math> at time <math>t_0^{j+1}+\tau</math>. At time <math>t_0^{j+2}+\tau</math> the error in forecast <math>\bold{x}_f^j</math> is given by <math>e_b=(\bold{x}_f^j-\bold{x}_t )^T \bold{C}</ins>(<ins style="font-weight: bold; text-decoration: none;">\bold{x}_f^j-\bold{x}_t </ins>)<ins style="font-weight: bold; text-decoration: none;"></math></ins>, <ins style="font-weight: bold; text-decoration: none;">while the error in <math>x_f^{j+1}</math> is given by <math>e_a=(\bold{x}_f^{j+1}-\bold{x}_t)^T \bold{C}</ins>(<ins style="font-weight: bold; text-decoration: none;">\bold{x}_f^{j+1}-\bold{x}_t</ins>)<ins style="font-weight: bold; text-decoration: none;"></math>. As noted above, if <math>\bold{x}_f^j</math> </ins>and <ins style="font-weight: bold; text-decoration: none;"><math>\bold{x}_f^{j+1}</math> are subject to </ins>''<ins style="font-weight: bold; text-decoration: none;">identical</ins>'' <ins style="font-weight: bold; text-decoration: none;">surface forcing and open boundary conditions during the interval <math>[t_0^{j+2},t_0^{j+2}+\tau]</math>, then the difference in forecast error <math>\delta e=e_a-e_b</math> is due solely to the difference in the forecast initial conditions due to the observations assimilated during analysis cycle <math>j+1</math> spanning the interval <math>[t_0^{j+1},t_0^{j+1}+\tau]</math>. </ins>(<ins style="font-weight: bold; text-decoration: none;">The more realistic case where the two forecasts are subject to different surface forcing fields is addressed below in the “step-by-step procedure” notes</ins>)<ins style="font-weight: bold; text-decoration: none;">. Specifically, if <math>\delta e<0</math> the observations assimilated during cycle <math>j+1</math> lead to an improvement in the forecast skill (i.e. <math>e_a<e_b</math>), while if <math>\delta e>0</math> the observations assimilated during cycle <math>j+1</math> have degraded the forecast (i.e. <math>e_a>e_b</math>). While this convention may seem counter-intuitive, it is the convention used in the numerical weather prediction literature, so it seems prudent to adopt it here</ins>.</div></td></tr>
</table>Robertsonhttps://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5281&oldid=prevRobertson at 14:48, 29 October 20182018-10-29T14:48:09Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:48, 29 October 2018</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">FIGURE HERE</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Image:analysis-forecast_cycle_schematic.png|center|frame|'''Figure:''' A schematic showing the typical configuration of an analysis-forecast cycle. Analysis cycle <math>j</math> spans the interval <math>[t_0^j,t_0^j+\tau]</math>, and is associated with 4D-Var analysis <math>x_a^j</math> and the forecast <math>x_f^j</math>. At the start of each cycle there are three available circulation estimates: two forecasts initialized from the previous two adjacent analysis cycles, and the analysis for the current time. These are illustrated at time <math>t_0^{j+2}+\tau</math> at the start of cycle <math>j+3</math>.]]</ins></div></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">'''Figure:''' A schematic showing the typical configuration of an analysis-forecast cycle. Analysis cycle <math>j</math> spans the interval <math>[t_0^j,t_0^j+\tau]</math>, and is associated with 4D-Var analysis <math>x_a^j</math> and the forecast <math>x_f^j</math>. At the start of each cycle there are three available circulation estimates: two forecasts initialized from the previous two adjacent analysis cycles, and</del></div></td><td colspan="2"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">the analysis for the current time. These are illustrated at time <math>t_0^{j+2}+\tau</math> at the start of cycle <math>j+3</math>.</del></div></td><td colspan="2"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In the figure above, each analysis cycle is assumed to be of length <math>\tau</math> and analysis cycle <math>j</math> spans the interval<math>[t_0^j,t_0^j+\tau]</math>. The circulation estimate at time <math>t_0^j+\tau</math> (i.e. the ''end'' of analysis cycle <math>j</math>) is denoted as <math>x_a^j</math> and is the initial condition for the forecast spanning the next analysis interval <math>[t_0^{j+1},t_0^{j+1}+\tau]</math>. In sequel it is assumed that the forecast duration is an integer multiple of <math>\tau</math>, but this is not a necessary constraint. The figure shows the analyses and forecasts that result from three adjacent analysis cycles, namely cycles <math>j</math>, <math>j+1</math> and cycle <math>j+2</math>. The analysis <math>x_a^j</math> at the ''end'' of cycle <math>j</math> is used as the initial condition for the forecast <math>x_f^j</math> of duration <math>2\tau</math> that terminates at time <math>t_0^{j+2}+\tau</math>, the end of analysis cycle <math>j+2</math>. Similarly, the analysis <math>x_a^{j+1}</math> at the ''end'' of cycle <math>j+1</math> is used as the initial condition for the forecast <math>x_f^{j+1}</math> of duration <math>\tau</math> and also terminates at time <math>t_0^{j+2}+\tau</math>, the end of analysis cycle <math>j+2</math>. After sufficient time has elapsed, a new analysis <math>x_a^{j+2}</math> will be computed at this time. Since <math>x_a^{j+2}</math> represents our best estimate of the ocean circulation at time <math>t_0^{j+2}+\tau</math> it can be used to quantify the veracity of the forecasts <math>x_f^j</math> and <math>x_f^{j+1}</math>. For this reason, <math>x_a^{j+2}</math> is usually referred to as the “verifying analysis.” However, as discussed, later other sources of information can be used to verify the forecasts, such as new or independent observations.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In the figure above, each analysis cycle is assumed to be of length <math>\tau</math> and analysis cycle <math>j</math> spans the interval<math>[t_0^j,t_0^j+\tau]</math>. The circulation estimate at time <math>t_0^j+\tau</math> (i.e. the ''end'' of analysis cycle <math>j</math>) is denoted as <math>x_a^j</math> and is the initial condition for the forecast spanning the next analysis interval <math>[t_0^{j+1},t_0^{j+1}+\tau]</math>. In sequel it is assumed that the forecast duration is an integer multiple of <math>\tau</math>, but this is not a necessary constraint. The figure shows the analyses and forecasts that result from three adjacent analysis cycles, namely cycles <math>j</math>, <math>j+1</math> and cycle <math>j+2</math>. The analysis <math>x_a^j</math> at the ''end'' of cycle <math>j</math> is used as the initial condition for the forecast <math>x_f^j</math> of duration <math>2\tau</math> that terminates at time <math>t_0^{j+2}+\tau</math>, the end of analysis cycle <math>j+2</math>. Similarly, the analysis <math>x_a^{j+1}</math> at the ''end'' of cycle <math>j+1</math> is used as the initial condition for the forecast <math>x_f^{j+1}</math> of duration <math>\tau</math> and also terminates at time <math>t_0^{j+2}+\tau</math>, the end of analysis cycle <math>j+2</math>. After sufficient time has elapsed, a new analysis <math>x_a^{j+2}</math> will be computed at this time. Since <math>x_a^{j+2}</math> represents our best estimate of the ocean circulation at time <math>t_0^{j+2}+\tau</math> it can be used to quantify the veracity of the forecasts <math>x_f^j</math> and <math>x_f^{j+1}</math>. For this reason, <math>x_a^{j+2}</math> is usually referred to as the “verifying analysis.” However, as discussed, later other sources of information can be used to verify the forecasts, such as new or independent observations.</div></td></tr>
</table>Robertsonhttps://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5278&oldid=prevRobertson at 16:47, 26 October 20182018-10-26T16:47:49Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The procedure for computing the observation impacts during the forecast cycle is a little more involved than that for the analysis cycle. However, a separate ROMS driver exists for this. To help illustrate the procedure involved, consider the typical analysis-forecast cycle shown schematically below.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The procedure for computing the observation impacts during the forecast cycle is a little more involved than that for the analysis cycle. However, a separate ROMS driver exists for this. To help illustrate the procedure involved, consider the typical analysis-forecast cycle shown schematically below.</div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">FIGURE HERE</ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Figure:''' A schematic showing the typical configuration of an analysis-forecast cycle. Analysis cycle <math>j</math> spans the interval <math>[t_0^j,t_0^j+\tau]</math>, and is associated with 4D-Var analysis <math>x_a^j</math> and the forecast <math>x_f^j</math>. At the start of each cycle there are three available circulation estimates: two forecasts initialized from the previous two adjacent analysis cycles, and</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Figure:''' A schematic showing the typical configuration of an analysis-forecast cycle. Analysis cycle <math>j</math> spans the interval <math>[t_0^j,t_0^j+\tau]</math>, and is associated with 4D-Var analysis <math>x_a^j</math> and the forecast <math>x_f^j</math>. At the start of each cycle there are three available circulation estimates: two forecasts initialized from the previous two adjacent analysis cycles, and</div></td></tr>
</table>Robertsonhttps://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5276&oldid=prevRobertson at 16:29, 26 October 20182018-10-26T16:29:28Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:29, 26 October 2018</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="title">Analysis-Forecast Cycle <del style="font-weight: bold; text-decoration: none;">Oberservations </del>Impacts</div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="title">Analysis-Forecast Cycle <ins style="font-weight: bold; text-decoration: none;">Observation </ins>Impacts</div></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The procedure for computing the observation impacts during the forecast cycle is a little more involved than that for the analysis cycle. However, a separate ROMS driver exists for this. To help illustrate the procedure involved, consider the typical analysis-forecast cycle shown schematically below.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The procedure for computing the observation impacts during the forecast cycle is a little more involved than that for the analysis cycle. However, a separate ROMS driver exists for this. To help illustrate the procedure involved, consider the typical analysis-forecast cycle shown schematically below.</div></td></tr>
</table>Robertsonhttps://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5274&oldid=prevRobertson at 16:23, 26 October 20182018-10-26T16:23:58Z<p></p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="title">Analysis-<del style="font-weight: bold; text-decoration: none;">Forecast_Cycle_Oberservations </del>Impacts</div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="title">Analysis-<ins style="font-weight: bold; text-decoration: none;">Forecast Cycle Oberservations </ins>Impacts</div></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The procedure for computing the observation impacts during the forecast cycle is a little more involved than that for the analysis cycle. However, a separate ROMS driver exists for this. To help illustrate the procedure involved, consider the typical analysis-forecast cycle shown schematically below.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The procedure for computing the observation impacts during the forecast cycle is a little more involved than that for the analysis cycle. However, a separate ROMS driver exists for this. To help illustrate the procedure involved, consider the typical analysis-forecast cycle shown schematically below.</div></td></tr>
</table>Robertsonhttps://www.myroms.org/wiki/index.php?title=Analysis-Forecast_Cycle_Observation_Impacts&diff=5273&oldid=prevRobertson at 16:23, 26 October 20182018-10-26T16:23:48Z<p></p>
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<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><div class="title">Analysis-Forecast_Cycle_Oberservations Impacts</div></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The procedure for computing the observation impacts during the forecast cycle is a little more involved than that for the analysis cycle. However, a separate ROMS driver exists for this. To help illustrate the procedure involved, consider the typical analysis-forecast cycle shown schematically below.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The procedure for computing the observation impacts during the forecast cycle is a little more involved than that for the analysis cycle. However, a separate ROMS driver exists for this. To help illustrate the procedure involved, consider the typical analysis-forecast cycle shown schematically below.</div></td></tr>
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</table>Robertson