GEOPHYSICAL RESEARCH LElTERS, YOLo 27, NO 7, PAGES 2661-2664, SEPTEMBER 2000

Obtaining single path phase delays from GPS double differences

Chris Alber. Randolph Ware. Christian Rocken. John Braun

University Corporation for Atmos~ric Research (UCAR). Bwlder. Colorado 80307

Abstract. We describe a method for obtaining single-path phase delays from GPS double differences. Theresulting "zem differences" (ZDs) can be used forremote sensing of atmospheric water vapor. The methodis demonstrated by simulating and observing atmo-spheric delay gmdients. and by comparing ZDs withpointed radiometer observations of integrated watervapor along GPS ray paths. In-situ GPS antenna phasecenter and multi path effect.; are mapped in ZD residualsfor a specific site and network. We conclude that ZDsderived from GPS network data show promise for realtime sensing of water vapor for use in meteorologicalmodeling and forecasting.

rier phase ambiguities. However, since the double dif-ferences include observations along 4 different paths(from 2 observation sites to 2 satellites), they are moredifficult to interpret than single path delays. In thispaper we describe a method for converting double dif-ferences into single path phase delays (ZDs).

Obtaining Zero Differences from DoubleDifferences

Double differencing is widely used to analyze GPS

observations. Let ~IA and ~A be observations of satel-lite-.. I and 2 by receiver A. and +. B and ~B be observa-

tions by receiver B. "r)1ese observations can then be

combined into single differences (SlAB and ~AB).

defined as

AS~~

AB Aand s2 = '2 B

.~; ..

A double difference can be expressed as the differenceof two single differences

B AB AB, '2) = S I - 52

AB Bdd = ("

cfJ <+,

Common mode em>rs in GPS observations cancel indouble differencing. Primarily, this includes errors fromsatellite and receiver clocks. Depending on the equip-ment used and the network size, em>rs from satelliteorbits, atmospheric delay, site coordinates. antennaphase center variations. and multipath effects mayormay not difference out.

To convert double differences to single differences.we write the double differences. dd, as the product of amatrix D and a vector of single differences, s,

Os dd (3)

Introduction

Global Positioning System (GPS) signals are delayedby propagation through the Earth ionosphere and neutralatmosphere. The delay is defined as the excess pn)paga-tion path from the satellite to the receiver compared totravel through a vacuum. Ionospheric delays are dis~r-sive and thus can be corrected using dual frequencyGPS observations. Neutral atmospheric delays can bedivided into hydrostatic (dry) and wet delay terms. Thedry delay, based on the assumption of hydrostatic e(jui-librium, can be computed using surface pressure [Beviset al., 1992] or meteorological model information [Chenand Herring, 1997]. The total delay minus the dry delaygives the wet delay.

Using this approach, precipitable water valX>r (PW)above the GPS antenna can be estimated with betterthan 2 mm accuracy [Duan et al., 1996]. PW is esti-mated aN the average vertical comlX>nent of the inte-grated water vapor along the line-of-sight to each of theobserved GPS satellites. Real time PW measurementsderived from GPS network data are described byRocken et aI., [1997]. More detailed measurements canbe o~ned by estimating the integrated water vapor("slant water", or SW) along the GPS ray paths [Ware etaI., 1997]. Examples of real-time PW and post-pro-cessed SW are available via ~,~t .LJl'ar..:JLJI~I"sr~)r~;lllim~.hlml.

High accuracy GPS applications commonly use dou-ble differences to cancel satellite and receiver clockerrors and to help determine integer values of GPS car-

For an individual ba."eline and n single differences.there are only n-1 linearly independent double differ-ences. and D cannot be inverted. However. if we intro-duce an additional independent constraint on at leaSt oneof the single differences. as shown in Equarion (4). thenD has a well defined inverse. The additional constraintis e~pressed at the upper right as a weighted sum of thesingle differences between sites I and J at one observa-rion epoch. The satellite-dependent weighring for the

sire pair IJ is wi' !.Wjs:J can be estimated using model

parameters from the GPS analysis. or by using a pointedwater vapor radiometer (WVR) with a barometer ormodel for rotal delay esrimation.

Copyright 2000 by the American Geophysical Union

Paper number 20000L0ll 525.0094-8276/00/2000GL011525$05

?Ml

2662 ALBER ET AL.:SINGLE PAm PHASE DELAYS FROM GPS DOUBLE DIFFERENCES

is distributed over a large area (-100 km). this assump-tion is generally valid because the distribution of watervapor at the sites can be considered random and themean zenith delay can be estimated and removed. For asmall network. if all stations observe a satellite throughthe same volume of atmosphere. the delay will havecommon mode elements that cannot be resolved. Onemethod to compute absolute slant delays for a small net-work uses a collocated pointed radiometer and barome-ter (or modeled pressure fields) at one site. Absoluteslant delay calculated from the wet and dry delay mea-surements at the collocated site can be used to leverabsolute slant delays at the other sites. Another methodis to analyze the small network withir1 a larger networkto minimize common mode error.

IJI.z.WjSj

IJ Iddl2 !

IIJ 'I ddl3

IJ IIsl +...+wns..

lJ lJsl -s2

IIJ IJsl -s3

WI w2 w31 -I 0I 0-1

I 0 0

Wn00 (~)

~3

~

»t..IJ IJ

IiI -!in

IJddlnn J

If we model all GPS observation effects and use thepost~fit double difference residuals in Equation (4), then

setting ~WjSrJ equal to zero (the "zero mean" assump-

tion) produces an inverse where the single differencesretain the un-modeled part of the double differences.Included in these single differences are effects such asmulti path or tropospheric inhomogeneities. The con-straint can be improved by downweighting the singledifferences at low angles.

We obtain zero differences using all single differ-ences to a given satellite from all linearly independentsite pairs in the GPS network, Starting with Equation (5)

Dz, = s (5)I I .'

where OJ is a matrix operating on a vector of zero differ-

ences for the i'th satellite, Zj' to produce the vector of

single differences Sj' We obtain a form for OJ includingan additional constraint (upper right) on the zero differ-ences in Equation (6)

Single Path Delay Simulation and Observation

We demonstrate the zero difference method usingsynthetic GPS data including I-mm phase noise and a:t:10 mm delay gradient from east to west at 10 degreeselevation (Figure I). The gradient was applied to one ofeight network sites in Colorado. Kansas and Oklahoma(jiUl:.:i!.\',-~!:;~ \uljj (,Ill! );I;J -.;()1 ~[l' 11111)1).

The zero-mean assumption was violated in the simu-lated data since the gradient delay was added to onlyone site -- data for the other seven sites included noiseonly. Therefore. the retrieved ZD observations fromeach site contained :t:10/8 = :t:1.2 mm error. The rightpanel shows observed ZD variations over a 2-hour timeperiod at Platteville. Colorado. using the same eight sitenetwork. The simulated gradient is similar in magnitude(but opposite in sign) to the observed gradient. BemeseGPS software [Rothacher and Mervart. 1996) was usedto process the simulated and observed data. The ZDmethod wa." used to extract slant delays from theBemese DD residuals.

, Az +A'J

A

'i'W t.Z i

Bz ;.A'1zcrcit; ,"aL,c~ -C

'~.r:if

Sensing Integrated Water Vapor Along GPSRay Paths

The ZD method can be used to sense variations inatmospheric water vapor along GPS ray paths. This is

The weighting for the i'th satellite at the l'th site is

wI' and 1:wIZ: includes single path effects to the i'th

satellite from each site, As before, we can downweightthe zero differences used in the constraint at low eleva-

tion angles. If we assume that 1:w,z: equals zero, wecan obtain zero differences which represent the slantdelay fluctuations about the model used to obtain thedouble and single differences.

In both inversions, any error in the zero-meanassumption is divided equally over all stations in thenetwork. For example, if for a 10 site network a 10 romdelay is present in one ZD, the mean residual will have aI-mm error. The ZDs are relative to the ensemble meanof the network. This implies that large networks can beused with careful modeling to minimize biases. Stationcoordinates should be held fixed to known long-termaverages to avoid correlation with slant delays.

For tropospheric slant delays, the zero mean assump-tion implies that the residual delay in the direction ofone GPS satellite at each epoch, averaged over the entireGPS network, is equal to zero. For a GPS network that

'A wB Wc

1 -I 0

1 0 -I

ALBER ET AI SINGLE PATH PHASE DELA YS FROM GPS DOUBLE DIFFERENCES 2663

ZD residuals for the Dome-Margolin antenna is shownin Figure 3. The ZD phase center variations are relativeto the mean phase center pattern of the ensemble ofantennas. In general. this mean was not zero. However.this map is similar to the phase center pattern deter-mined from anechoic chamber tests [Meertens et 01..J 996] which is also shown. In-situ phase center varia-tions of Dome Margolin antenna~ are currently underdebate within the International GPS Service (IGS) com-munity. Figure 3 shows in-situ antenna phase variationssimilar to anechoic chamber measurements. The in-situresults can be used to correct antenna pha.c;e centereffects in ZD residuals for specific networks and sites.

Day of Year

Figure 2. Comparison of GPS-sensed (ZD method) andWVR-sensed SW above 10 degrees elevation. Samplingintervals are 8 min (WVR) and 30 sec (GPS).

Mapping Multipath Effects in ZD Residuals

Residual phase variations that repeat with the period-icity of the GPS orbits (sidereal time) can be mappedwith the ZD method. Assuming that site coordinates.carrier phase ambiguities, GPS orbits, in-situ antennaphase center variations. and mean atmospheric delay areall accurately modeled, the W residuals are dominatedby the un-modeled atmospheric slant delay and ground-scattered multipath. Un-modeled atmospheric delay var-ies on a daily ba£is. but the multipath tends to repeat insidereal time. Detailed multi path maps can be computedby "stacking" daily GPS residuals. Daily residuals. a 21-day stacked multi path map. and a corrected set of resid-uals are shown for a single frequency site [Braun et al..19991. Multipath effects can be corrected by subtractingthe stacked multipath map from the re~idual£. Thereduction of multi path error, particularly at low eleva-tion angles, is evident in the COlTected plot. Residualvariations are reduced from 3.8 to 2.5 mm rms over allangles. Stacking of residuals allows for the separation ofmultipath and unmodeled atmospheric slant delay.

Conclusions

The ZD method obtains single path phase delaysfrom double differences. The resulting residuals can be

demonstrated by comparing 8-min pointed radiometerand 30-sec ZD !lensed SW in Figure 2. Once again. weuse GPS data from the eight site network. The residualsare total SW minus 30-min PW mapped to the elevationangle. Good agreement (0.9 mm rms) is seen betweenthe two measurements. Braun et al. [2000] made a simi-lar comparison using three days of data and found 0.7mm rms agreement above 20 degrees elevation. For thesame data set. Ware et aI. [1997] found 1.3 mm rmsagreement between double difference pointed radiome-ter and GPS residuals. This corresponds to 0.6 mm rms(RSS) agreement for each of the 4 ZDs in the doubledifference residuals. The zero mean assumption is theonly additional em>r introduced in deriving ZDs fromthe double differences. We therefore conclude that theem>r resulting from this assumption is relatively smallfor this case.

It is evident from Figure 2 that the ZD SW noise leveldecreases at high elevation angles near plot center. Sim-ilarly. Braun et al. [2000] found that the ZD SW noiselevel is 0.2 mm rms near zenith and 1.4 mm rms near 10degrees elevation. We attribute the large oscillations inthe GPS residual at low angles (both ends of the plot) toground-scattered multi path.

ZD sensed SW data show pumise for meteorologicalapplications. MacDonald and Xie [2000] simulated theuse of SW data from a GPS network with 4O-km spac-ing. They found that high resolution 3D humidity fieldscould be determined from SW data. They concluded thatassimilation of SW data into weather models shouldprovide significant improvements in forecasting. A vari-ety of meteorological (and other) applications for singlepath GPS residuals are described by Ware et al [2000].

Mapping Antenna Effects in ZD Residuals

We apply the W method to data from four GPSreceivers with their antenna... located on the comers of a10m square at Table Mountain. Colorado. Three of theantennas were designed for minimal phase center varia-tions. The fourth antenna was an Allen Osborne andA~..ociates Dome Mar2olin choke ring. A map of the

elevation angle (degrees)

Figure 3. Dome-Margolin phase center and multipathmap in ZD residuals (blue) compared with anechoicchamber measurements (red). We attribute themodulation in the ZD residuals at low elevations toground-scattered multipath.

ALBER ET AL.: SINGLE PATH PHASE DELAYS FROM GPS DO~DIFFERENCES2664

References

0bseIv~multipath

map

Coo'eCted

Figure 4. ZD corrections at Lamont. Oklahoma.obtained by subtracting the multi path map from theobserved 24-hr ZDs (sky plots similar to Figure I).

used to sense integrated water vapor along GPS raypaths. The ZD method assumes a "zero mean" assump-tion to derive single and zero differences from doubledifferences. During a three day test. we showed that theerror in the ZD residuals resulting from this assumptionis relatively small. We also showed that errors resultingfrom antenna phase center and ground-scattered multi-path effects in ZD residuals can be mapped and cor-rected for specific networks and sites. Overall, the ZDmethod shows promise for real time sensing of atmo-spheric water vapor in meteorological applications.

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Acknowledgments: This work. was supported by the Officeof Naval Research, Dr. Scott Sandgathe, Code 322MM, and bythe Department of Energy Atmospheric Radiation Measure-ment (ARM) Batelle grant 354I06-AQ5. GPS network datawere orovided by NOAA's Forecast Systems Laboratory.

(Received February 24. 200); revised May 13.200);accepted June 19. 200)