12.215 Modern NavigationSummary of Last ClassToday’s ClassBasic atmospheric structureTroposhereRefractivity of airRefractivity of airRefractivity in terms of densityIntegration of RefractivityZenith delayZenith hydrostatic delayZenith wet delayZenith wet delayPW and ZWDMapping functionsContinued fraction mapping functionTruncated versionMapping functionsCoefficients in mapping functionEffects of atmospheric delayParameterization of atmospheric delayExample using NCEP analysis fieldSummary12.215 Modern NavigationThomas Herring11/29/2006 12.215 Modern Naviation L19 2Summary of Last Class• GPS measurements– Basics of pseudorange measurements– Phase measurements (allow millimeter level position with GPS and cm in real-time)– Examine some GPS data.• Positioning modes• Dilution of precision numbers11/29/2006 12.215 Modern Naviation L19 3Today’s Class• Atmospheric propagation.– Basic structure of the atmosphere– Refractive index of atmospheric air– Accounting for atmospheric delays in precise GPS navigation:• Hydrostatic and wet delays– Use of GPS for “weather forecasting”11/29/2006 12.215 Modern Naviation L19 4Basic atmospheric structureTroposphere is where the temperature stops decreasing in the atmosphere. (~10 km altitude)11/29/2006 12.215 Modern Naviation L19 5Troposhere• Lots of examples of web-based documents about the atmosphere: See for example.http://www-das.uwyo.edu/~geerts/cwx/notes/chap01/tropo.html• Tropopause is where temperature stops decreasing. Generally at pressure levels of about 300 mbar but can be as low as 500 mbar.• Sometimes term “tropospheric delay” used but this is only about 70% of delay.• Generally by height of 50-100km all of atmospheric delay accounted for.• Troposphere is where weather system occur and aircraft fly on the tropopause.11/29/2006 12.215 Modern Naviation L19 6Refractivity of air• Air is made up of specific combination of gases, the most important ones being oxygen and nitrogen.• Each gas has its own refractive index that depends on pressure and temperature.• For the main air constituents, the mixing ratio of the constituents is constant and so the refractivity of a packet of air at a specific pressure and temperature can be defined.• The one exception to this is water vapor which has a very variable mixing ratio.• Water vapor refractivity also depends on density/temperature due to dipole component.11/29/2006 12.215 Modern Naviation L19 7Refractivity of air• The refractivity of moist air is given by:• For most constituents, refractivity depends on density (ie., number of air molecules). Water vapor dipole terms depends on temperature as well as density N = k1PdTZd−1Density of dry air1 2 3 + k2PwTZd−1Density of water vapor1 2 3 + k3PwT2Zd−1Dipole compoentof water vapor ρ/T1 2 3 k1= 77.60 ± 0.05 K/mbark2= 70.4 ± 2.2 K/mbark3= (3.730 ± 0.012) ×105 K2/mbar11/29/2006 12.215 Modern Naviation L19 8Refractivity in terms of density• We can write the refractivity in terms of density:•Density ρ is the density of the air parcel including water vapor. R is universal gas constant, Mdand Mware molecular weights. Zwis compressibility (deviation from ideal gas law) See Davis, J. L., T. A. Herring, and I.I. Shapiro, Effects of atmospheric modeling errors on determinations of baseline vectors from VLBI, J. Geophys. Res., 96, 643–650, 1991.N = k1RMdρ+k'2T+k3T2⎛ ⎝ ⎜ ⎞ ⎠ ⎟ PwZw−1k'2= k2− k1Mw/Md= 22.1± 2.2 K/mbar11/29/2006 12.215 Modern Naviation L19 9Integration of Refractivity• To model the atmospheric delay, we express the atmospheric delay as:• Where the atm path is along the curved propagation path; vac is straight vacuum path, z is height for station height Z and m(ε) is a mapping function. (Extended later for non-azimuthally symmetric atmosphere)• The final integral is referred to as the ”zenith delay”D = n(s)ds− dsvac∫atm∫≈ m(ε)(n(z) −1)dz =Z∞∫m(ε) N(z) ×10−6dzZ∞∫11/29/2006 12.215 Modern Naviation L19 10Zenith delay• The zenith delay is determined by the integration of refractivity vertically.• The atmospheric is very close to hydrostatic equilibrium meaning that surface pressure is given by the vertical integration of density. Since the first term in refractivity depends only on density, its vertical integration will depend only on surface pressure. This integral is called the “zenith hydrostatic delay (ZHD)”. (Often referred to as “dry delay” but this is incorrect because has water vapor contribution).11/29/2006 12.215 Modern Naviation L19 11ZHD =10−6k1RMdgm−1Ps≈ 0.00228 m/mbar• The Zenith hydrostatic delay is given by:• Where gmis mean value of gravity in column of air (Davis et al. 1991)gm=9.8062(1-0.00265cos(2φ)-3.1x10-7(0.9Z+7300)) ms-2• Ps is total surface pressure (again water vapor contribution included)• Since Ps is 1013 mbar at mean sea level; typical ZHD =2.3 meters Zenith hydrostatic delay11/29/2006 12.215 Modern Naviation L19 12Zenith wet delay• The water vapor delay (second term in refractivity) is not so easily integrated because of distribution of water vapor with height.• Surface measurements of water vapor pressure (deduced from relative humidity) are not very effective because it can be dry at surface and moist above and visa versa.• Only effective method is to sense the whole column of water vapor. Can be done with water vapor radiometer (WVR) which infers water vapor delay from thermal emission from water vapor molecules and some laser profiling methods (LIDAR). Both methods are very expensive (200K$ per site)11/29/2006 12.215 Modern Naviation L19 13Zenith wet delay• In meteorology, the term “Precipitable water” (PW) is used. This is the integral of water vapor density with height and equals the depth of water if all the water vapor precipitated as rain (amount measured on rain gauge).• If the mean temperature of atmosphere is known, PW can be related to Zenith Wet Delay (ZWD) (See next page)11/29/2006 12.215 Modern Naviation L19 14PW and ZWD• Relationship:• The factor for conversion is ~6.7 mm delay/mm PW• This relationship is the basis of ground based GPS meteorology where GPS data are used to determine water vapor content of atmosphere.• ZWD is usually between 0-30cm.ZWD =10−6RMw(k'2+k3/Tm)PWTm=Pw/Tdz∫Pw/T2dz∫11/29/2006 12.215 Modern Naviation L19 15Mapping functions• Zenith delays discussed so far; how to relate to measurements not