The Effect of Weather Fronts on GPS Measurements
INNOVATION
of how atmospheric refraction maps into a The Effect of GPS positioning error.
ATMOSPHERIC DELAY Weather Fronts As a GPS signal travels from the satellite to a receiver, it passes through the atmosphere, where different layers refract it in various on GPS ways. The first layer it encounters is the ionosphere, which is charged with a high Measurements number of free electrons that refract the sig- nal. The resulting delay depends on the signal frequency (because the ionosphere is a dis- Thierry Gregorius persive medium), which is why we can use Geoffrey Blewitt data from daul-frequency receivers to easily The University of Newcastle upon Tyne estimate and almost entirely eliminate the delay’s magnitude. The ionosphere actually accelerates the carrier phase (with a net phase advance) and slows down the pseudorandom Dr. Blewitt completed his Ph.D. research in noise codes and the navigation message (with high-energy physics in 1985 at the California a net modulation or group delay). Institute of Technology and then spent more The ionosphere’s electron content is tem- than eight years working at the National porally and spatially highly variable. Under Aeronautical and Space Administration’s Jet the influence of solar flares and coronal holes Propulsion Laboratory, where he coauthored and the resulting geomagnetic storms, these the GIPSY OASIS software for high-precision variations may become so rapid and unpre- On the southeast coast of England, not very far GPS applications. In 1994, he returned to the dictable that the higher-order terms of the from where the Battle of Hastings occurred, United Kingdom to join the University of delay not eliminated by the ionosphere-free lies Herstmonceux Castle — a fifteenth- Newcastle upon Tyne, where he is now linear combination of the L1 and L2 data century manor house that was, for many years, professor of space geodesy in the Department could cause a significant bias in the estimated the home of the Royal Greenwich Observatory of Geomatics. A governing board member of station position — the position of the geodetic (RGO). Although the skies above the castle are IGS, he is actively involved in IGS analysis. marker on which the receiver’s antenna rests. generally clearer than those above RGO’s “Innovation” is a regular column featuring Having passed through the ionosphere, the original home in the London borough of discussions about recent advances in GPS signal then undergoes a different kind of Greenwich, the frequently cloudy and rainy technology and its applications as well as the delay in the neutral atmosphere. Termed non- conditions are less than ideal for astronomy. fundamentals of GPS positioning. The column dispersive because it is not frequency-depen- RGO, therefore, built new telescopes on La is coordinated by Richard Langley of the dent and thus cannot be easily eliminated, Palma in the Canary Islands and moved most Department of Geodesy and Geomatics this neutral delay is caused by both the of its administrative and research facilities to Engineering at the University of New stratosphere and troposphere. Because the Cambridge in 1990. Brunswick, who appreciates receiving your bulk of the effect occurs in the troposphere, The same poor conditions dreaded by comments as well as topic suggestions for the geodetic community has taken to mis- astronomers, however, are ideal for studying future columns. To contact him, see the naming the neutral delay as the tropospheric weather fronts in relation to GPS. The grounds “Columnists”section on page 4 of this issue. delay, a convention that we shall follow here. of Herstmonceux Castle (now owned by We can adequately model the tropospheric Canada’s Queen’s University and operated as Weather fronts are an important meteorolog- delay’s dry part (more precisely, the part that an international study center) house an ical phenomenon that not only bring a change is in hydrostatic equilibrium, the bulk of International GPS Service (IGS) station. This in weather but can also significantly disturb which is accounted for by the dry gases) if we site has provided Drs. Thierry Gregorius and GPS observations. Because they occur in the know the surface pressure with high accu- Geoffrey Blewitt with a wealth of data for their troposphere, they leave their mark in the tro- racy, which is information a properly cali- studies of the effects of weather fronts on GPS pospheric propagation delay. brated barometer can provide. The tricky measurements, which they recount in this In the January 1993 issue of GPS World, aspect is the delay’s wet part, caused by month’s column. Fritz K. Brunner and Walter M. Welsch dis- water vapor in the tropo-sphere’s lower lay- Dr. Gregorius studied surveying at the cussed the troposphere’s effect on GPS mea- ers. Similar to the ionospheric electron con- Universities of Karlsruhe (Germany) and New surements. They correctly mentioned that (in tent, the water vapor’s spatial and temporal South Wales (Australia), where he graduated extreme cases) the wet part of the delay can distribution is largely unpredictable and can with a B.E. in geomatic engineering in 1995. vary by more than 3 centimeters per hour undergo rapid variations, especially in the He then went to the United Kingdom to work during the passage of a front. In this article, presence of a weather front. Surface humidity on geophysical and meteorological we will explain in more detail why fronts readings do not usually represent the tropos- applications of GPS at the University of cause such rapid variations in the delay, how phere’s moisture content very well. There- Newcastle upon Tyne. After obtaining his that affects GPS precision, and what we can fore, even with surface meteorological data, Ph.D. this year, he moved to The Netherlands, do to reduce or eliminate the problem. First, it is hard, if not impossible, to properly model where he now works for Shell International. however, we will provide a quick summary or predict the wet delay.
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THE POSITIONING EFFECT For differential positioning (based on dou- matic-type surveys, however, this is more On top of GPS’s inherent geometric weak- ble differencing), Brunner and Welsch sug- difficult to determine. ness because a receiver cannot track satellites gested the following rule of thumb for the below the horizon, tropospheric delay is the propagation of tropospheric error into the WHAT IS A WEATHER FRONT? main additional ingredient to the heighting GPS estimates of height (for an elevation cut- Although most GPS users know about atmo- error budget. The accuracy of station height off angle of 15 degrees): height difference spheric delay and how it contributes to posi- determinations is less than that of the latitude error equals three times the differential tro- tioning error, they may be unfamiliar with the or longitude by a factor of two or so. pospheric delay error. In other words, an esti- concept of a weather front. Before we High-precision GPS software packages mated total tropospheric delay error of 1 describe how weather fronts affect GPS mea- account for the tropospheric delay by esti- centimeter will propagate into a heighting surements, we will briefly introduce the mating a zenith delay parameter that is linked bias of 3 centimeters. Achieving subcentime- physics of fronts. to arbitrary elevation angles through a map- ter heights, therefore, seems to require mod- Out in Front. A weather front is the boundary ping function. Traditionally, tropospheric eling and estimating the total delay with an between two air masses that display differ- variations over time have been accounted for overall accuracy better than 3 millimeters. ences, especially in temperature, wind direc- by stochastic estimation techniques, ranging One problem with estimating the residual tion, and humidity. Depending on the front’s in sophistication from Kalman filtering and tropospheric delay, though, is the high corre- direction of motion, it is denoted as either equivalent approaches to simply estimating a lation between the height and tropospheric cold or warm. For a stationary observer on new delay bias at regular intervals, for exam- parameters, which increases even further the earth’s surface, cold air replacing warm ple every hour. More recently, to account for with higher elevation cut-off angles. Mathe- air defines a cold front. Conversely, warm air spatial variations of tropospheric refraction, a matically, it is therefore unclear whether succeeding cold air denotes a warm front. number of scientists have attempted addition- observed, short-term changes in height are Often, a faster-moving cold front overtakes a ally to estimate tropospheric gradients in attributable to atmospheric variations or to warm front from behind, eventually resulting north and east directions to allow for any antenna motion. In static mode, one can rea- in a more complex, merged front called an azimuthal variation in the delay (as opposed sonably assume that the stationary antenna occlusion (or occluded front). Such a front to assuming that the delay varies with the (and the ground in which it is anchored) is then slowly dissolves as the differences vertical elevation angle only), apparently stable and, therefore, can attribute any short- between the bordering air masses gradually with some success. term changes to the atmosphere. In kine- disappear.
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Circle 36 Circle 37 May 1998 GPS WORLD 53 For some applications it is convenient to the passage, these properties usually do not causing a kink in the isobars. Such a trough, model a front as a two-dimensional boundary change much. however, may also be nonfrontal so that surface. In reality, however, it is a relatively This scenario is slightly different for a fronts cannot simply be inferred by such thin (40Ð200 kilometers thick), three-dimen- cold front, and any changes at the passage are anomalies in the pressure field. sional sheet of air that separates the two con- often more sudden than for a warm front. In The frequency of weather fronts largely trasting air masses. A front is therefore often advance of the cold front, the temperature depends on a site’s latitude or climate. Fronts referred to as a frontal zone. Because of the changes little, although humidity and wind usually divide polar and tropical air and are clash of different wind patterns, tempera- speed both increase, and pressure falls therefore mainly found in midlatitudes. In tures, and humidities, this zone is subject to slowly. During the passage, temperature falls other words, fronts are most frequent in tem- rather strong turbulence and cloud formation, rapidly, the wind veers, humidity stays high, perate, humid climates. Great Britain, for which then results in precipitation. Near the and pressure may undergo a sudden jump. example, is crossed by weather fronts about ground, the evaporation of this rain can After the passage, wind and temperature may once every two to three days. Under stormy increase the air’s moisture content up to satu- continue to vary but humidity rapidly falls as conditions, as many as three or four fronts ration level (100 percent relative humidity). the sky clears, with pressure rising slowly. can pass in 24 hours. On a satellite image, we can identify fronts On a weather chart, the surface front is often by the long, narrow bands of cloud that recognizable as a trough of low pressure, THE DELAY EFFECT accompany them. The rapid variations weather fronts can cause Sample Fronts. Figure 1 shows a warm and a in the tropospheric delay is well illustrated by cold front moving across the British Isles. -30° ° Figure 3. The figure shows the passage of 70° -20° -10 0 70° Typically, such a system’s velocity ranges fronts over an International GPS Service from 30 to 50 kilometers per hour, with the (IGS) site at Herstmonceux Castle, in south- cold front often moving slightly faster than ern England, during a 14-day period. Their the warm one. The main difference between passage usually stands out as a peak in the warm and cold fronts is their inclination, as LOW delay time series. It therefore is no surprise 60° 60° can be seen from the schematic diagram in that such rapid variations can introduce a Figure 2. Warm fronts have a very gentle form of systematic error in the estimated slope, generally not more than 0.5Ð1 degree, GPS station position (especially in height). and incline toward their direction of move- Delay Estimation Models. To assess the effect of ment because of surface friction and the rela- ° weather fronts on tropospheric delay, we first 50° 50 tively low density of warm air. Cold fronts, HERS estimate the tropospheric zenith delay with on the other hand, incline backward because GPS (using the random walk model in the kilometers their dense, heavy air subsides and slides 0 500 least-squares estimation engine, which -30° 0° underneath the lighter warm air. Near the -20° -10° allows the tropospheric delay parameter to ground, surface friction causes the cold cold front warm front occluded front vary over time —— this model characterizes frontal zone to bulge forward. Cold fronts the random component of tropospheric generally have a steeper slope than warm Figure 1. A typical frontal system behavior, which is not much different, math- fronts (about 1.5Ð2 degrees). moving across the British Isles (fictitious ematically, from the steps of a drunken Because of its gentle slope, the cross- example inspired by U.K. Meteorological sailor). We then compare these estimates section of a front can span many hundreds of Office data). The Herstmonceux GPS with two models based on surface meteoro- kilometers. In practice, the cold, warm, and site is marked with a red star. logical observations. We stress that the frontal air layers are stacked on top of each other almost horizontally. For a warm front, upper-air clouds associated with the frontal direction of motion zone can herald the front’s arrival at the tropopause ground surface 12 hours or more in advance. Note that weather charts always mark the surface fronts, which can lie several hundred Cold Frontal Zone kilometers behind (warm front) or ahead RAIN & CLOUD (cold front) of the frontal zone at upper lev- COLD WARM COLD AIR dry AIR AIR els. dry
For an observer on the ground, the U.K. ~ 10 km
Meteorological Office suggests the following Warm Frontal Zone telltale signs as the most reliable way of rec- R A I N & C L O U D 0.5 - 1¼ ognizing a front’s passage. In advance of a ~ 1 km warm front, temperature rises steadily, the ground surface wind speed increases, pressure falls, and rel- ative humidity rises because of precipitation. ~500 km ~100 km ~100 km ~800 km During the passage, temperature rises more intensely, the wind changes direction, pres- Figure 2. Idealized cross-section of a system containing two fronts. Note that the sure stops falling, and the air’s moisture con- vertical scale is greatly exaggerated. The dimensions stated (in kilometers) are only tent may increase up to saturation level. After rough estimates and can vary considerably from front to front.
54 GPS WORLD May 1998 humidity recorded on the ground is not usu- hypothesis with data acquired under such model assumes that the troposphere has a ally representative of the water vapor levels conditions. certain vertical profile with average physical found in the troposphere above. Some days, The first model simply uses a standard properties. however, do present a high correlation atmosphere to extrapolate temperature, The second model, the front model, between surface humidity and the wet zenith humidity, and pressure from the ground accounts for fronts and works out separate delay, such that we can test the weather front throughout the troposphere. That is, this delays for each air mass using individual
2.50 2.44 Cold Warm surface front Warm Cold surface front Cold Cold 2.42 2.45 Warm Warm Cold 2.40 Cold 2.40 Occluded Occluded 2.38 2.35 2.36
2.30 otal tropospheric zenith delay (m) T 2.34 GPS estimates Front model: r.m.s. = 6.1 mm Estimated total tropospheric zenith delay (m) Standard model: r.m.s. = 11.6 mm 2.25 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2.32 0 12 24 36 48 60 72 Day of November 1996 Time past 00:00 on 28 November 1996 (hours)
Figure 3. Tropospheric zenith delay at Herstmonceux, southern Figure 4. GPS-estimated and model-predicted total zenith England, during a fortnight in November 1996. The passages of path delay at Herstmonceux, 28 to 30 November 1996 the surface fronts are marked as warm, cold, or occluded.
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temperature and humidity gradients based on laps with the true estimates almost perfectly. passage (Figure 5). The dry delay actually typical values published in the literature for What’s more, the estimated frontal parame- decreases because it is directly proportional warm and cold air masses. To compensate for ters confirm the typical values for fronts cited to atmospheric pressure, and the two fronts in the lack of upper-air data, which is mostly in the meteorological literature, and the com- this example are linked to a field of low pres- obtainable only from expensive weather bal- puted passage times of the surface fronts per- sure (a frontal depression). loons (radiosondes), the front model esti- fectly agree with those derived from official mates the front’s geometry and velocity to weather charts. This preliminary example, FRONTS AND GPS PRECISION obtain the best possible fit with the GPS esti- for which the correlation between surface Even though a well-tuned random walk mates. This way, we actually kill two flies humidity and wet zenith delay is very high model should in theory be able to cope with with one whack. Not only do we generate a (0.92), indicates that GPS could indeed have rapid variations in delay, removing days prediction of the tropospheric delay caused the power to resolve the geometry (especially affected by fronts from a long time series of by fronts, but we also find out whether GPS gradient-related parameters, such as the incli- GPS heights still improves the repeatability could actually have the data strength to nation) and passage times of weather fronts of the vertical station component. Repeata- explicitly solve for parameters related to with relatively high confidence. Such a capa- bility is a measure of internal precision that is weather fronts. bility naturally has potential benefits not just often used to describe the reliability of mea- Testing the Models. Figure 4 shows the curve for GPS positioning, but for meteorology and surements recorded at permanent GPS sites. of the total tropospheric zenith delay over climate research as well. It is essentially a standard deviation that also three days at Herstmonceux while a warm During the warm front’s approach, the takes day-to-day formal errors into account and a cold front passed by. If we assume that total tropospheric zenith delay increased by and is computed as 1 2 the GPS estimates of tropospheric delay ade- about 8 centimeters in 11 hours. After the r2 Σ i 2 quately represent the troposphere’s true state cold front had passed on the ground, the ei repeatability = n i (which has been suggested by many studies), delay dropped back down by about the same n ± 1 Σ 1 e2 we can see that the model using a standard amount in 7 hours. When we plot the delays’ i i atmosphere is relatively inaccurate under dry and wet portions separately, it becomes where n is the number of data points (days), ri frontal influence, especially during the warm clear that this rise and fall is caused by the is the daily residual (derived from linear front’s approach. wet part of the delay (for example, by water regression of the time series plot) and ei is the The front model, on the other hand, over- vapor, clouds, and rainfall) during the front’s daily formal error (standard deviation).
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56 GPS WORLD May 1998 Circle 40 2.50 22.0 All days Cold surface front Days without fronts Warm surface front 2.45 19.0 Days with fronts
2.40 Total delay 16.0
2.35 Wet delay 13.0
2.30 Vertical repeatability (mm) Dry delay 10.0
Total tropospheric zenith delay (m) 2.25 7.0 0 2 4 6 8 10 12 14 16 18 20 Random walk process noise in mm/(hours)1/2 2.20 0 12 24 36 48 60 72 Time past 00:00 hours on 28 November 1996 (hours) Figure 7. Repeatability as a function of random walk process noise and the passage of weather fronts at Herstmonceux. Figure 5. Total, dry, and wet delays as fronts pass Days affected by weather fronts are more sensitive to the choice of process noise level. The number of days processed for each time 10 series is 155 for ‘all days’, 57 for ‘days with fronts’ and 98 for ‘days without fronts.’ The values shown for zero 5 process noise have not actually been derived with a random walk model. In this case, the tropospheric delay was 0 estimated as a constant parameter over successive 24-hour periods (see text). -5 Height variation (cm) motion derived from the height time series. Whereas the repeatability is only a measure -10 of internal precision, this vertical “velocity” 1996.45 1996.55 1996.65 1996.75 1996.85 1996.95 1997.05 actually gives an indication as to the global Time (years) accuracy of the estimated station height. In our example, the vertical velocity magnitude Figure 6. Height variability time series of Herstmonceux, southern England. The does indeed become smaller (from Ð3.1 to repeatability is 8 millimeters (derived with the best processing strategy). Ð0.6 millimeters per year) and agrees with what is expected from geophysical consider- Improving Repeatability. For example, the limeters when using this strategy. In this ations when removing days affected by fronts repeatability of a six-month series of daily case, the effect of weather fronts is much from the time series. (Southern England is height estimates at Herstmonceux (Figure 6), more detrimental compared with when thought to subside by less than 1 millimeter where fronts are frequent and the GPS employing the more flexible and powerful per year because of postglacial rebound in receiver tracks data 24 hours a day, improves random walk model. By eliminating the days Scotland.) This indicates that the front-free by as much as a millimeter (from about 8 to 7 affected by fronts, the repeatability improves solution represents the true situation more millimeters), depending on the level of by 3 millimeters, and the difference between correctly, although the six-month timespan process noise applied in the random walk frontal and nonfrontal days is statistically of this example is too short to provide con- model. When we compare the time series of significant by a large margin (14 versus 21 clusive results in this matter. frontal and nonfrontal days separately, the millimeters). If we knew the exact beginning and end repeatability of the series containing only the This proves that GPS observations times of the periods during which weather days affected by fronts is considerably worse. acquired during periods of frontal influence fronts had a noticeable effect on the esti- Depending on the level of process noise, the can be systematically biased, as shown in mated tropospheric delay, then their effect on discrepancy reaches almost 2 millimeters. Figure 7. One should not use such data, there- the station height repeatability could be Some commercial software packages fore, for high-accuracy GPS applications if worked out more accurately and would, with- assume that the tropospheric delay remains the processing software is of the “black box” out a doubt, be higher still than the figures we constant and estimate it as such in the least- type and less sophisticated than the high-pre- present here. Classifying a whole day as squares process. This is obviously less than cision packages available from various frontal even if only a few hours were under ideal, particularly under variable conditions research institutions. the influence of a front certainly weakens the in the troposphere. In the Herstmonceux Vertical Velocity. The effect of weather fronts results. Without extensive upper-air data, example, the vertical repeatability is 17 mil- on GPS is confirmed by the vertical site however, no objective measure for determin-
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ing those periods of influence exists, and if data from the MicroLab-1 prototype satellite bly resulting in higher positioning accuracy. we tried to guess their times by inspecting the for several years. This so-called GPS/Met The GPS/Met system’s only limitation is plots of estimated tropospheric delay, we system aims to combine space-based GPS that atmospheric profiles are still difficult to would introduce a human, subjective bias. measurements with currently available mete- determine for the troposphere’s bottom layer Thus, one has to remember that weather orological data and thus provide global cov- because of problems related to signal propa- fronts can have a detrimental effect on GPS erage of detailed temperature and humidity gation, including multipath and topographic that is at least as bad as we suggest in this profiles throughout the atmosphere. The obstructions. Unfortunately, most water article. observed gradients could then be fed back vapor is found in that particular region. One The Horizontal Factor. The effect on the hori- into a user’s GPS processing software, possi- could alleviate this problem by incorporating zontal station component is similar to what we found for height, only much smaller in magnitude. Our experiments with the Herst- monceux site have shown that the degrada- tion of the horizontal repeatability is as much as 80 percent smaller than that of the vertical and therefore largely negligible.
REMEDIES AND POSSIBILITIES We have estimated that, depending on the weather front’s intensity and the estimation strategy used, a front can degrade the heights in a regional network by as much as several millimeters. This effect is worst if the GPS data-reduction software is unable to solve for the tropospheric delay in a time-varying manner, which is the case for many commer- cial packages. Several ways to overcome the problem exist. The most obvious is to simply not per- form a survey when a front is crossing the area. Also, a front may not affect solutions from a small GPS network, because most errors will cancel out in differential mode. However, weather fronts are likely to 1/2 Page Island degrade the station heights, if the baselines are longer than 10 kilometers or so. There- Ad Goes Here fore, before conducting a GPS campaign, one should check the weather forecast for any fronts. If height precision is important, wait, Keyline does not print perhaps, until the fronts have passed. Ideally, the troposphere should be moni- tored by launching radiosondes or deploying page 59 water vapor radiometers during the survey. One would then feed the results into GPS processing software capable of making opti- mal use of such information. Both these methods, however, are very expensive, and radiometers lose their reliability during peri- ods of rainfall (which, unfortunately, usually accompanies the passage of weather fronts). Alternatively, estimating frontal parameters in addition to the tropospheric delay could possibly improve the station height. Supplementing with Satellites. In the future, cheaper satellite-based systems could replace or complement radiosondes. A constellation of low-orbiting microsatellites with onboard GPS receivers presents one potential solu- tion. To investigate such a possibility, the University Corporation for Atmospheric Research and others have been analyzing
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data from permanent ground networks, such variability on that day. Again, by using an duce tropospheric disturbances similar to as IGS stations, into the GPS/Met analysis. appropriate threshold, we can improve the those caused by fronts. Fixing the Time Series. For permanently operat- vertical coordinate repeatability of most per- ing GPS stations, the most straightforward manent GPS stations. CONCLUSION remedy is to remove the periods affected by We found that the gradients method Overall, users need only worry about weather fronts from the time series. If users don’t achieves the highest front detection rate, fronts in midlatitudes and if the GPS network want to rely on daily weather information, whereas the curve length index usually pro- employed is reasonably large (spanning, say, they could use an algorithm that detects duces the greatest improvements in repeata- more than 10 kilometers). The problem is fronts or other major tropospheric distur- bility. The index using fractal dimension most severe for permanent GPS sites because bances from the GPS data alone. We have tends to pick out the most variable days while fronts introduce a systematic bias in station developed and tested algorithms to provide maximizing the amount of data being coordinate time series. In small, local net- indicators or indices of such tropospheric dis- retained in the time series. However, the fact works, all stations will probably be affected turbances. We found they successfully that none of the methods achieve a 100 per- in more or less the same way, such that any improved the long-term repeatability of most cent front detection rate indicates that other errors would largely cancel out when form- mid-latitude and tropical station coordinates. forces are also at work that occasionally pro- ing double differences. Mainly the station These indices work on a very simple prin- height is affected; the bias in the horizontal ciple: We first quantify the troposphere’s component seems to be largely negligible. variability for each day using an objective Further Reading To recap, the best remedy to prevent measure and then eliminate those days that For an introduction to meteorology, see unnecessary height errors is to avoid survey- suffer from tropospheric variability higher Ⅲ Atmosphere, Weather and Climate, by ing during the passage of weather fronts. If R.G. Barry and R.J. Chorley, 7th Edition. than a certain, empirical threshold. This cut- Published by Routledge, London and New this is impossible and the highest level of off must be carefully tuned to avoid rejecting York, 1998. accuracy is required, the tropospheric delay too much data from the time series. To quan- Ⅲ Essentials of Meteorology, by D.H. parameter should be estimated in a stochastic tify the troposphere’s behavior, we first plot McIntosh and A.S. Thom. Published by way, allowing it to vary over time within the estimates of total tropospheric zenith Wykeham, London, 1969. realistic bounds. Ideally, one would launch delay for each day (one data point every 15 Ⅲ Handbook of Aviation Meteorology, radiosondes at regular intervals during the by The Meteorological Office, 3rd Edition. minutes, as done for Figures 3 through 5). Published by The Stationery Office GPS survey and implement an explicit Along this curve, we can then simply count (formerly by Her Majesty’s Stationary weather front model in the tropospheric delay the number of gradients greater than, for Office), London, 1994. estimation. Because this scenario is highly example, 1 centimeter per hour. If any day Ⅲ unlikely, all that can be done right now is to has more than a certain number of steep gra-
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