Calibration of zenith hydrostatic delay model for ... - Wiley Online Library

Radio Science,Volume 35, Number 1, Pages133-140, January-February2000

Calibrationof zenithhydrostaticdelaymodel for local GPS applications Y. Liu,H. BakiIz• andY. Chen Departmentof Land SurveyingandGeo-Informatics, The HongKong PolytechnicUniversity, Hung Hom, Kowloon, Hong Kong

Abstract. Determinationof dry zenithdelay,DZD, is importantfor separating wet zenith delaywith highaccuracyfromGlobalPositioningSystem(GPS) measurements. The existing modelsareusuallyfor the calculationof the zenithhydrostatic delay,ZHD. Becausethe zenithhydrostaticdelayincludesthe contributionof watervaporandis alsonot accurate enough,the DZD, approximately replacedby ZHD, will induce-2 cm errorinto localGPS applications.In this studywe calibratea popularZHD model,namely,the Saastamoinen modelfor local GPS meteorology,usingradiosondedata. We testthe calibratedmodelby comparingthe predicteddelaysfrom the calibratedmodelagainstthe observeddelaysfrom radiosonde data that are not included in the calibration solution.

The calibrated model shows

20 mm improvementoverthe uncalibratedmodelpredictionsoverthe sameperiod. A comparisonof the DZDs calculatedfrom GPS measurements over a monthperiodwith calibratedanduncalibratedZHD modelpredictionsshows15 mm averageimprovement.

1. Introduction

spatialand temporalprecipitablewater vapor content derived from the wet zenith delay from the GPS The tropospheric delayis an importanterrorsource measurements. Hence, separatingthe wet zenith in space geodeticobservations.The conventional delay from the troposphericzenithdelay, which is the approachis to eliminateor reduceits influencevia sumof the wet zenith delay and the dry zenith delay, empiricalmodelsor to estimateits influencethrough DZD, is needed for accurate GPS meteorological unknown parameters.A by-productof this latter applications. Although DZD can be accurately approachis the spatialandtemporalresolutionof the computedfrom radiosondedata, this is a very costly troposphericzenith delay, which is beneficial for approachto implement. Instead, empirical models weatherresearchandglobalclimatemonitoring. that dependon lesscostlysurfacemeteorologicaldata In the caseof Global PositioningSystem,GPS, the are developedfor this purpose. signal is affectedby the troposphericdelay when it DZD dependson the shape of the profile with crossesthe tropospherefrom GPS satelliteto ground height and the dry air/water vapor mix ratio. receivingstation.This delay can be brokeninto two However, there is no specificmodel that meetsthis components: the wet delay, which is causedby the need. Conventionally,the zenith hydrostaticdelay, wet atmosphereor water vapor, and the dry delay, ZHD, is deployedfor calculatingDZD [Tralli et al., which is causedby the dry atmosphere.One of the 1992; Rocken et al., 1995; Bar-Sever and Kroger, basicobjectivesof GPS meteorologyis to acquirethe 1996; Namet al., 1996; Duan et al., 1996]. Accordingto Davis et al. [ 1985]and Spilker[1996], • Now at EarthOrientation Department, U.S. Naval the ZHD is obtainedby integratingthe hydrostatic refractivity Observatory,Washington,D.C.

N h = 22.768p, Copyright2000 by the AmericanGeophysicalUnion.

(1)

where p is total tropospheric density,which includes the dry and wet densities.On the other hand, the DZD is generatedthrough the integrationof dry

Papernumber1999RS900100. 0048-6604/00/1999RS900100511.00 133

134

LIU ET AL.'

CALIBRATION

OF ZENITH

HYDROSTATIC

DELAY

MODEL

reftactivity,whichonlyincludes thedrydensity.This hydrostaticdelay along a vertical path throughthe difference, which may not be very significant totaltroposphereabovea specificstation. sometimes, may introduce error to GPS The Saastamoinen modelfor ZHD is expressed as meteorologicalapplicationsas will be demonstrated in this study. The ZHD canbe calculatedeasilythroughsurface meteorological observations makinguseof empirical modelsthat have been developedfor this need.The most popular ZHD models are the Saastamoinen (SAAS) [Saastamoinen,1973] and Hopfield models [Hopfield,1971].Bothmodelsarein closeagreement and do not produce significant differences in calculatingZHD. In this studywe first calibratethe ZHD model of SAAS usingHong Kong radiosondedata in situ such that it can be used in the computationof DZD for GPS-based meteorologicalapplications.We then comparethe predicteddelaysusingthe calibratedand uncalibrated models with the ones calculated from the radiosonde data. We also use the calibrated model in the GPS solutions to evaluate its

performance. 2. ZHD

Model

P

d• -0.22768-F((p,H)

(2)

F(tp, H) -- 1- 0.0026 cos (2{p)- 0.00028H.

In the aboveexpression,{pis the latitudeof stationin radians,H is the orthometricheightor leveledheight in kilometers, and P is the surface pressure measurement (total) in millibars at the station. The subscriptS denotesthat the ZHD is from the SAAS model.The unit of dj is centimeters. The SAAS model is thereforedependenton the locationof the site and the pressurebut independentof the station temperature.

This model requires surface pressure measurements.Hence the accuracy of pressure directly affects the ZHD. In order to secure millimeter accuracy for the ZHD we need first to discuss and determine

the admissible

error

in the

surfacepressuremeasurements.

If (p andH areknownin (2) at a specificsite,then

and the Influence

the impact of the uncertainty in the pressure of Pressure Error measurementP to the ZHD can be evaluatedusing the variancepropagationlawson P, whichgives The ZHD is usuallycalculatedusingan empirical 0.2277 model that makes use of surface meteorological c•,,•. •c•. (3) measurements. The model is assumed to be accurate F((p, H) enoughto be consideredwithout uncertainty[Tralli In the aboverelationship the coefficientof cre et al., 1992; Bar-Sever and Kroger, 1996]. Several should be maximized for the largestimpact on the studies [Bevis and Businger, 1995; Gendt and ZHD. This condition is achieved only when (p is Beutler, 1995; Namet al., 1996; Dodson et al., 1996,' minimized and H is maximized in (3). Now, if we use Duan et al., 1996] state that the existingempirical (p=0 ø and H=9 km, which are quite extreme for the models are accurate to a few millimeters with the use of accuratesurfacepressuredata.Davis et al. [1985] indicatesthat the uncertaintymust be between 0.5 and 20 mm/1000 mbar. Nonetheless, no studies

confirmtheir accuraciesin detail. Moreover,despite the presenceof numerousmodelsfor calculatingthe ZHD, there are very few studiesabout the use of ZHD for calculatingthe DZD in GPS applications, which is an adopted practice. Therefore, in this section

we

will

first

examine

in

detail

how

Earth, we obtain

c•,,•.- 0.2289c•,.

(4)

Here, the unit of ZHD is in centimeters and the pressureis in millibars. From (4), 0.4 mbar pressure measurementaccuracywill guaranteethat the ZHD precisionwill remain within 1 mm. This result is quite robust with respect to the current barometer standards.As a check, if we consider T=296.16 K,

meteorologicaldata affect the estimationof ZHD, and then comparethe differencebetweenthe ZHD

(p=22.15ø,and H=0.114 km, representingthe mean temperatureand the location of a GPS station in

and the DZD

HongKong,we obtainc•,,.,. - 0.2281c•,.Thisresult

derived from local radiosonde data.

The most popular ZHD model belongs to Saastamoinen [1973]. This model makes use of surface pressure measurement to estimate the

indicatesthat if proper pressuremeasurements are made,we shouldexpectthatthe influenceof pressure error on the ZHD

is < 1 mm in the SAAS model.

LIU ET AL.:

CALIBRATION

OF ZENITH

HYDROSTATIC

DELAY

MODEL

135

The ZHD is always larger than the DZD because the wet density is also included in the hydrostatic refractivity as shown in (1). Water vapor does not conform to the hydrostaticequationbecausewater vapor is subject to myriad effects including condensation and exhibits large variations in concentrationunlike the dry gasesthat are uniformly mixed[Spilker,1996]. Thusthereis no simpleway to removethe water vapor from the ZHD relativeto the surfacewater vapor pressure.To gain insightinto the contributionof the water partialpressurein the zenith hydrostatic delay, we introduce the following independent referencestandardfor comparison.

and height. These observationerrors of pressure, temperature,and height will propagateinto the DZD via (6) and (7). To analyzethe impactof theseobservationerrors, we differentiate(6). Neglecting Z j' on the righthand side, the differential relationshipbetween Na

3. Computing the DZD From Radiosonde

Differentiating(7) and consideringboth (8) and (9) yields

Measurements and Its Uncertainty The dry zenith delay d can be written as [Dodson et al., 1996]

and Pd, T is

dN a 77'6ZJ 1 - 77.6PdZj 1dr. (8) The dry partial pressureis the differenceof total pressure P andwaterpressuree sothat

dPd: dP- de.

(9)

dd,• - (05x10-6)[Z (N;+' "[Nt)(dh,+ - dh, ) ß

d

1

+ +1 + 77.6 Z (h,+, h, )Z•' (.dP,de, alP,+ 1 de, rl Tt+i (5)

d--10 -6INadh,

whereNa is the dry air reftactivityand h is the height

rt

above the site.

A precise formula for the dry air refractivity by Thayer[ 1974] gives

Na=77.6 Pa Z-I T

t+l

Assumingthat the measurements are uncorrelated and applyingthe variancepropagationlaw to (10) gives

d

(J'd R-- (J'h + B(o•,+cye)+

0.52• -4)•T2], t Z•'=1+Pd[(57.90 x10-8)(1 +•---) --(9.4611 x10 whereA, B, and C are definedby

,

(11)

(6)

- (0.s x +'+ d) where Pd is the partial pressureof dry air in mbar, T 1 1 is the temperaturein Kelvin, and t is the temperature in degreesCelsius. The dry partial pressureand temperaturealongthe p2 p2 verticalpathshouldbe knownto calculatethe dry air Z dt dt+ I refractivity along the vertical path, which can be •+1 provided by the radiosonde measurements.This (12) informationcanthenbe usedto convertthe upperdry air refractivity into the dry delay in the vertical The empiricalvaluesfor A, B, and C are given in direction using the following approximation Table 1. They are basedon the 13-monthradiosonde

- (3.0 088 x

(h,+, -

+ )

C-(3.01088x10 -9) (]'t,+,-ht)2(r7--[T4).

expression to (5):

datacollected inHongKongt ]•ft+l

Nt+I

dR_lo-OZ(h,+,_h,)..a + 2 d' i

(7)

The empiricalvalues for A, B, and C are given in Table 1. They are based on the 13-months Radiosondedatacollectedin Hong Kong.

where, the subscriptsand superscriptsi and i + 1 Also, the height observationerror o-h is better denote the top and the bottom of each layer, than 1 rn in GPS wind finding systememployedin respectively,for heightand dry air refractivity. The Hong Kong. Pressureand temperaturemeasurement DZD is functionof dry partial pressure,temperature, errors are cye=0.5 mbar, and cy•=0.2 K (the RS80

136

LIU ET AL.'

CALIBRATION

OF ZENITH

DELAY

MODEL

0.8

Table 1. The EmpiricalValues for A, B, C, andthe

Corresponding (XdR

0.0

Maximum

Minimum

Mean

A

1.38x10 -•

1.06x10 -6

1.24x10 -6

B

1.55x10 -6

0.77x10 -6

1.52x10 -6

C

3.13x10 -•

2.51x10 -6

2.95x10 -6

2.3 mm

1.5 mm

2.3 mm

oaR

HYDROSTATIC

.

-1.6 -2.4 ..n. -3 2 96 6

96.8

Uncalibrated

97.0

97.2

97.4

97.6

07.8

technical specificationsare available from Vaisala Date in year Company: http://www.vaisala.corn.). The water Figure 2. Differences between radiosondeand model vapor pressure e can be calculated through before and after the calibration. temperatureand relative humidity by an empirical formula [MassachusettsInstitute of Technology (MIT), 1999] 4. Difference Between DZD and ZHD

e = RH ß[6.11x 1075t/(t+2373) ]

(13)

whereRH is relativehumidityand t is temperaturein degreesCelsius.Similarly, it is easyto showthat the accuracyC•eof water vaporpressureshouldbe better than 1.5 mbar considering that the maximum temperatureis 40 øC in Hong Kong andthe errorof relative humidity is-2% from radiosonde(from Vaisala Company "the RS80 technical specifications"). Substitutingthesevaluesand thosein Table 1 into (11) and expressing the resultingunit of millimeters,

the corresponding {Xd, is givenin the lastrow of Table 1. Theseresultsshowthat the impactof the errorsof the observedmeteorologicalparametersas the DZD

from the radiosonde data is-2

mm.

Becausethe radiosondedata are locally collected and accurate,we use them as a standardto compare the DZD to the ZHD computedfrom the empirical SAAS model. Figure 1 showsthe ZHDs from the SAAS

model

and the DZDs

calculated

from

the

radiosondedatafor the periodof September1, 1996, to September30, 1997, that were collectedtwice daily by the Hong Kong Observatoryin King's Park radiosondestation.The offset is clearly revealed in Figure 2, which is constructedusingthe differences between DZD computedfrom radiosondedata and ZHD from the SAAS models(we reserveone month of radiosondedata for model validation). Statistics for the differences

data and ZHD

between DZD

from the radiosonde

from the SAAS model show a 17 mm

averageoffset with a standarddeviationof 3.7 mm. The maximum 235

-e- DZDfromRadiosonde 233

231

ß •, 229

the radiosonde

227

225

223

96.6

91•.8

9•.0

9•.2

9•4

9•6

and minimum

differences

are-8.7

mm

and-25.4 mm, respectively. This systematicoffset doesnot, however,fully explainall the variationsthat are presentin the differencesin Figure2. Figure 3 and Table 2 give the correlationbetween the differencesand the pressureand temperaturedata used in calculating the model zenith delays. Theoretically,the differencebetweenthe DZD from

97.8

Date in year

Figure 1. DZDs calculatedfrom the radiosondedata and ZHDs from the SAAS model from September1, 1996, to September30, 1997.

and the ZHD

from the SAAS

model

reveals the contributionof water vapor to ZHD. When the temperatureis increasing,such as in summer,water vapor contentof the troposphereis larger, and its contributionto the ZHD is larger. Consequently,the absolutedifferencebetweenDZD and ZHD is larger while temperatureis increasing. This is why the temperatureis negativelycorrelated with the above differences.Water vapor pressure,

LIU ET AL.'

36

CALIBRATION

'•o. Temperature(L)

32 28

,,,

';•'-Dry air pressure(R) .

"

"

ß•

•:-• ....') 423

OF ZENITH HYDROSTATIC

o /'/'"'

e. _e ß o '4-•m ,, • ;., •',:•,,','