112.215 Modern NavigationThomas Herring ([email protected]),http://geoweb.mit.edu/~tah/12.21511/05/2006 12.215 Modern Naviation LSextant 2Todayʼs class• Analysis of Sextant measurements• Homework was broken into a number of small steps:– Determining the maximum observed angle to thesun and time this maximum occurred– Obtaining the mean index error– Computing maximum elevation to the sun– Computing the atmospheric bending correction– Computing the latitude– Computing the longitude211/05/2006 12.215 Modern Naviation LSextant 3Simpler parts of calculation• Mean of index error: Simply the sum of the valuesdivided by the number of values• Also we can compute a standard deviation about themean (also called a root-mean-square (RMS) scatter).This gives is an indication of how well we can makemeasurements with the sextant. The standarddeviation of our measurements was 0.9ʼ• We use this today and in later lectures we will showhow to use this to allow us to estimate the uncertaintyof our final latitude and longitude determination.11/05/2006 12.215 Modern Naviation LSextant 4Atmospheric refraction• We can use the simple formula given in class or wecan look up the values in the Nautical Almanac.• The formula result is slightly greater than 1ʼ sincetan(ε) ~ 1• Using the almanac we can explore how much thisvalue will vary due to atmospheric conditions.• (For latitude determination, atmospheric refractionbecomes a bigger problem the closer we get to thepole where the meridian crossing elevation angle willbe much smaller. It will also be a bigger problem inmid-winter than in mid-summer).311/05/2006 12.215 Modern Naviation LSextant 5Geometry of measurement• Sphericaltrigonometry thatwe can solve (weinterpret on themeridian and soeasy)EquatorPath followed by SunVertical at MITGHAΔGHA90-δsunZdθ11/05/2006 12.215 Modern Naviation LSextant 6Spherical Trigonometry• Based on the figure, we can write the solution for thezenith distance to the sun:• If we assume we know our latitude and longitude thenwe can compute the expected variations in the zenithdistance to the Sun• In addition, since we measured 2*(elevation tosun+refraction)+ index error , we can include this inwhat is called a “forward model”€ cos Zd = cosθcos(90 −δ) + sinθsin(90 −δ)cos(ΔGHA)411/05/2006 12.215 Modern Naviation LSextant 7Results of forward model• GPS latitude 42.36; longitude -71.0890• Declination of Sun at MIT meridian crossing -12.2 deg• Interpolating the Almanac GHA, UT meridian crossing16.470 hrs (-4 hrs to EST)• The forward model can be computed and compared tomeasurements.11/05/2006 12.215 Modern Naviation LSextant 8Forward Model CalculationBlue:quadraticRed: ForwardModel511/05/2006 12.215 Modern Naviation LSextant 9Comparison• Agreement looks good but when totals are displacedthe results can be be deceptive in that the details cannot been seen.• Normal to look at the difference between theobservations and the model• On the quadratic fit residuals we show “error bars”based on the index measurements. These arecomputed from sqrt(Sum(residuals^2)/(number-1)).Also called Root-mean-square (RMS) scatter• In class we will vary the parameters of the model tosee there effect on the fit to the data.11/05/2006 12.215 Modern Naviation LSextant 10Residuals (Quadratic and Model)Black Stars:Residual tomodelRed circles:residuals toquadratic fitRMSFit: 5.7’611/05/2006 12.215 Modern Naviation LSextant 11Some neglected effects• Refraction and index error not included in forwardmodel but these can be easily added into Matlabcode.• Motion of Sun during measurements was accountedfor during the run• Later we will use the forward model to obtain rigorousestimate of latitude and longitude.11/05/2006 12.215 Modern Naviation LSextant 12Summary:• Today we explored the latitude and longitude problemin more detail looking at the actual data collected withthe sextant.• Introduced the notion of a forward model forcomparing with data and varying the parameters ofthe model to better match the observations.• Differences between observations and models can bequantified with an estimated standard deviation orRMS scatter.• These issues are returned to when we addressstatistics and
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