Inversion and Error Estimation of GPS Radio Occultation Data

Journal of the Meteorological Society of Japan, Vol. 82, No. 1B, pp. 507--531, 2004 507

Inversion and Error Estimation of GPS Radio Occultation Data

Y.-H. KUO, T.-K. WEE, S. SOKOLOVSKIY, C. ROCKEN, W. SCHREINER, D. HUNTand R.A. ANTHES

University Corporation for Atmospheric Research, Boulder, Colorado, USA

(Manuscript received 30 May 2003, in revised form 5 November 2003)

Abstract

In this paper, we describe the GPS radio occultation (RO) inversion process currently used at theUniversity Corporation for Atmospheric Research (UCAR) COSMIC (Constellation Observing System forMeteorology, Ionosphere and Climate) Data Analysis and Archive Center (CDAAC). We then evaluatethe accuracy of RO refractivity soundings of the CHAMP (CHAllenging Minisatellite Payload) and SAC-C (Satellite de Aplicaciones Cientificas-C) missions processed by CDAAC software, using data primarilyfrom the month of December 2001. Our results show that RO soundings have the highest accuracy fromabout 5 km to 25 km. In this region of the atmosphere, the observational errors (which include bothmeasurement and representativeness errors) are generally in the range of 0.3% to 0.5% in refractivity.The observational errors in the tropical lower troposphere increase toward the surface, and reach@3% inthe bottom few kilometers of the atmosphere. The RO observational errors also increase above 25 km,particularly over the higher latitudes of the winter hemisphere. These error estimates are, in general,larger than earlier theoretical predictions. The larger observational errors in the lower tropical tropo-sphere are attributed to the complicated structure of humidity, superrefraction and receiver trackingerrors. The larger errors above 25 km are related to observational noise (mainly, uncalibrated iono-spheric effects) and the use of ancillary data for noise reduction through an optimization procedure.We demonstrate that RO errors above 25 km can be substantially reduced by selecting only low-noiseoccultations.

Our results show that RO soundings have smaller observational errors of refractivity than radio-sondes when compared to analyses and short-term forecasts, even in the tropical lower troposphere. Thisdifference is most likely related to the larger representativeness errors associated with the radiosonde,which provides in situ (point) measurements. The RO observational errors are found to be comparablewith or smaller than 12-hour forecast errors of the NCEP (National Centers for Environmental Predic-tion) Aviation (AVN) model, except in the tropical lower troposphere below 3 km. This suggests that ROobservations will improve global weather analysis and prediction. It is anticipated that with the use ofan advanced signal tracking technique (open-loop tracking) in future missions, such as COSMIC, theaccuracy of RO soundings can be further improved.

1. Introduction

The radio occultation (RO) technique hasplayed an important role in characterizingplanetary atmospheres since the 1960s (Klioreet al. 1964; Fjeldbo and Eshleman 1968; Lindal

et al. 1983; and Lindal 1992). The planetaryexplorations have stimulated the theoreticalstudy of its potential application to the Earth’satmosphere (Phinney and Anderson 1968; Lu-signan et al. 1969). However, until recentlyRO studies of Earth’s atmosphere did not re-ceive serious consideration, primarily becauseof the prohibitive cost of space-borne trans-mitters with stable clocks and the insufficientaccuracy of satellite positioning. With the real-ization that the Global Positioning System(GPS) could be used for occultation observa-

Corresponding author: Ying-Hwa Kuo, Univer-sity Corporation for Atmospheric Research, P.O.Box 3000, Boulder, CO 80307, USA.E-mail: [email protected].( 2004, Meteorological Society of Japan

tions, as suggested by Gurvich and Krasil’ni-kova (1987) and Yunck et al. (1988), applicationof the RO technique to the Earth’s atmospherehas now become a reality (a historical survey ofGPS RO sounding can be found in Yunck et al.2000). To demonstrate the feasibility and accu-racy of active atmospheric limb sounding of theEarth’s atmosphere using the RO technique,UCAR established a program known as GPS/Meteorology (GPS/MET) in May 1993, in col-laboration with the Jet Propulsion Laboratory(JPL) and the University of Arizona. A proof-of-concept satellite, MicroLab-1, carrying an ex-perimental receiver developed by JPL and Al-len Osborne Associates was launched in April1995, which marked the beginning of two yearsof data collection. Comparisons of GPS/METdata with models and other correlative dataindicated that the GPS/MET soundings possessthe equivalent temperature accuracy of@1 K inthe range from the lower troposphere to 40 km(Ware et al. 1996; Kursinski et al. 1996; Rockenet al. 1997) as well as a geopotential height ac-curacy of 10 to 20 m (Leroy 1997). This accu-racy is comparable to that of the radiosondesystem.

Following GPS/MET, two GPS RO missionswere launched in 2000: the German CHAMPand the Argentinean SAC-C. These two satel-lites collectively produce approximately 350 ROsoundings per day (Hajj et al. 2004). BothCHAMP and SAC-C are equipped with an ad-vanced GPS occultation receiver, known as the‘‘Black Jack,’’ developed by JPL. This receiveris capable of tracking L2 GPS signal, modu-lated by Y-code, with fairly reasonable quality(above the moist troposphere), thus producingcontinuous RO observations (in GPS/MET thiswas possible during only limited periods whenthe Y-code was replaced by P-code). CHAMPdata are used for continuous monitoring of theneutral atmosphere and study of the signalsreflected from Earth’s surface (Wickert et al.2001; Beyerle et al. 2002; Wickert et al. 2004).In late 2005, the joint U.S.-Taiwan ROCSAT-3/COSMIC mission will be launched and is ex-pected to collect approximately 3,000 RO sound-ings per day. The COSMIC data will be avail-able in near real-time and can be used todemonstrate the value of RO data for opera-tional numerical weather prediction (NWP).In addition to COSMIC, a number of other

GPS RO missions are also being developedor planned in the next ten years, includingMETOP, EQUARS, ACEþ, and NPOESS.

The raw measurements of RO soundings arephase and amplitude of radio signals trans-mitted by the GPS. Based on these measure-ments and the knowledge of the precise posi-tions and velocities of the GPS and low Earthorbiter (LEO) satellites, which carry GPS re-ceivers, vertical profiles of bending angle andatmospheric refractivity are derived with theuse of the local spherical symmetry assump-tion and Abel inversion (Phinney and Ander-son 1968). Because atmospheric bending anglesand refractivities (which are functions of tem-perature, water vapor, and pressure) are nottraditional meteorological variables forecast bythe models, the use of advanced data assimila-tion techniques, such as three-dimensional orfour-dimensional variational data assimilation(3DVAR/4DVAR) systems, are required to as-similate the RO data into operational NWPsystems (Eyre 1994; Kuo et al. 2000). In orderto assimilate the RO data effectively, one needsto properly account for their measurement char-acteristics and errors. As the inversion of ROdata requires various assumptions, simplifica-tions, and approximations, the data inversiondetails affect the accuracy of the retrieved ROsoundings and they must be taken into accountin error analysis. Descriptions of end-to-endRO data processing, carried out by different re-search groups, can be found in Hocke (1997),Feng and Herman (1999), Steiner et al. (1999)and Hajj et al. (2002, 2004).

In preparation for the COSMIC mission,UCAR’s COSMIC Data Analysis and ArchiveCenter (CDAAC) has developed a GPS RO dataprocessing system. The CDAAC software hasbeen used to process GPS/MET, CHAMP andSAC-C data. In this paper, we provide a briefdescription of the CDAAC data processing pro-cedures. We then evaluate the accuracy of theCHAMP and SAC-C refractivity soundings bycomparing them with global analyses, globalprediction, and available radiosonde observa-tions. The primary objective of this paper is tobetter quantify the observational errors asso-ciated with the RO soundings. This knowledgewill also help provide direction for future im-provement of RO retrievals and data assimila-tion.

Journal of the Meteorological Society of Japan508 Vol. 82, No. 1B

2. Inversions of GPS radio occultationdata

2.1 UCAR CDAAC data processingprocedures

In this section, we briefly summarize theUCAR CDAAC RO retrieval procedure. Figure1 outlines the processing (inverting) of the ROsignals, beginning with the phase and ampli-tude of the radio waves and precise positionsand velocities of the satellites and ending withthe retrieved refractivity profile at the esti-mated ‘‘occultation point.’’ The various steps ofthe data processing can be described as follows:

Step 1: Detection of L1 tracking errors andtruncation of the signal

The detection of L1 tracking errors is basedon the fact that for a large enough distancefrom the Earth’s limb to the receiver, the frac-tional variability of the Doppler frequency shiftof the RO signal is much smaller than the cor-responding variability of the refractivity in theatmosphere. This effect was first noticed andestimated by Sokolovskiy (2001b), and it makesopen-loop tracking of tropospheric RO signalspossible, by accurately modeling their Dopplerfrequency shift prior to an occultation without

a real-time correction during an occultation byuse of feedback. The feedback, in closed-looptracking, is a significant source of errors, in-cluding bias, under conditions of multipathpropagation typical for the moist troposphere(Ao et al. 2003; Beyerle et al. 2003). The smallvariability of the atmospheric Doppler fre-quency shift also allows for detection of closedloop tracking errors, when their magnitude islarge enough, in post-processing. For this pur-pose, at first, the L1 Doppler frequency shift ismodeled with account for positions and veloc-ities of the GPS and LEO and the refractivityclimatology (the algorithm was introduced bySokolovskiy 2001b). The refractivity climatol-ogy is based on CIRAþQ climate model (CIRA-86 with included moisture model below 20 km)developed by Kirchengast et al. (1999). Thenthe modeled Doppler is compared to the ob-served (smoothed) L1 Doppler. If the differenceexceeds a certain threshold (which depends onLEO altitude) then the RO signal is truncated.The truncation is done at earlier time, whenthe difference exceeds some pre-specified frac-tion of the threshold. Figure 2 shows an exam-ple of L1 and L2 excess phase rates for aCHAMP occultation. Imposed is the L1 excess

1) Detection of L1 tracking errors andtruncation of the signal

9) Optimal estimation of thebending angle

Fig. 1. Flow diagram of the UCAR CDAAC RO data processing procedures. Arrows, incoming fromsides, indicate the use of ancillary data (currently CIRAþQ).

Y.-H. KUO et al. 509March 2004

phase rate estimated from GPS and LEOorbit knowledge and CIRAþQ climatology. Thebeginning of an L1 tracking error (A) is de-tected at 117.86 s, and the RO signals at alllater times are discarded.

Step 2: Filtering of raw L1 and L2 DopplerNoise in the raw L1 and L2 Doppler signals

can cause the phase of the RO signals to betaken out of the space restricted by the as-sumption of spherically symmetric refractivity.This, in turn, causes the bending angle, calcu-lated from the Doppler under this assumption(see Step 5), to become a multi-valued functionof the impact parameter. Such a multi-valuedfunction cannot be inverted by the Abel tech-nique (Step 10). Thus, the noise must be fil-tered out prior to the calculation of the bend-ing angles and impact parameters. We use twomethods for the low-pass filtering of phase andsimultaneous calculation of Doppler: (i) cubicspline regression, and (ii) Fourier filtering withGaussian windowing function for the spectrum.Both filters allow analytic differentiation of thephase without finite differencing and also allowthe bandwidth to vary with altitude. To sup-

press the end-effects related to finite durationof a RO signal and discontinuities at ends,the RO signal is subject to extrapolation withsmooth transition to a constant (zero) beyondthe ends prior to Fourier filtering. Currently,we are using a constant bandwidth of 2 Hz,which, on average, provides vertical resolutionconsistent with the size of the Fresnel zone(@1 km) at the mean altitude of the tropo-pause; this is justified by reasonable agreementof Doppler and amplitude inversions (Soko-lovskiy 2000). The complex RO signal, used forradioholographic inversions under conditions ofmultipath propagation in the troposphere, isnot currently subject to any filtering.

Step 3: Estimation of the ‘‘occultation point’’The ‘‘occultation point,’’ i.e. the point on

Earth’s surface to which the retrieved refrac-tivity profile is assigned, is estimated under thetangent point of the ray connecting the GPSand LEO, with account for the ray bending, fora certain height of the ray asymptote. We de-fine the occultation point based on the L1 ex-cess phase ¼ 500 m, which, on average, corre-sponds to 3–4 km height. The bending angle asa function of the height of the ray asymptote isestimated either from CIRAþQ refractivity cli-matology or from strongly smoothed L1 Dop-pler. At this time we use the latter approach.

Fig. 2. Excess phase rate (Doppler) forone of CHAMP RO soundings, for L1(red) and L2 (black) frequencies. Greenline shows L1 Doppler predicted fromGPS and LEO orbits and refractivityclimatology (CIRAþQ). Vertical bluelines show points of the quality degra-dation of L1 (A) and L2 (B,C) signalsbelow critical, as detected by automatedCDAAC software (for details see text).

Fig. 3. L1 bending angle as a functionof the height of ray asymptote derivedfrom Doppler and from a complex signalby different radioholographic methodsfor one of the GPS/MET tropical occul-tations.


Step 4: Transfer of the reference frame to thelocal center of Earth’s curvature

As shown by Syndergaard (1998), the Earth’soblateness can introduce noticeable errors (de-pending on latitude) when solving the inverseproblem under the assumption of spherical sym-metry in the Earth-centered reference frame.In order to minimize that error (by preservingthe assumption of local spherical symmetry)the center of the reference frame is trans-ferred from the Earth’s center to the virtualcenter of sphericity assigned to an occultation.The latter center is defined as the local centerof curvature of the intersection of the Earth’sreference-ellipsoid and the occultation plane(Earth’s center-GPS-LEO) under the estimated‘‘occultation point’’ (see Step 3).

Step 5: Calculation of L1 and L2 bendingangles from the filtered Doppler

At altitudes above the moist troposphere,where multipath propagation is rare, the bend-ing angle as a function of impact parameter ofa ray is calculated from the Doppler frequencyshift. The equation, which relates the Dopplerfrequency shift and the satellite velocities tothe inclination of the phase fronts at GPS andLEO (Kursinski et al. 1997), is solved concur-rently with Snell’s equation (with the assump-tion of the spherical symmetry of refractivity)for the starting and arrival angles of the ray atthe GPS and the LEO. These angles are thenused to calculate both the bending angle andthe impact parameter.

Step 6: Calculation of the bending anglesfrom L1 raw complex signal

Under conditions of multipath propagation,common in the moist troposphere, the calcula-tion of ray arrival angles from Doppler (Step 5)is not applicable. (Its formal application resultsin the bending angle becoming a multi-valuedfunction of the impact parameter.) In the caseof multipath propagation, the L1 bending angleis derived from the raw complex signal (phaseand amplitude) by radioholographic methods,which allow disentangling of multiple tones(rays). CDAAC processing software includesfour radioholographic algorithms: the backpropagation (BP) method (Gorbunov and Gur-vich 2000), the sliding spectral (SS) method(Sokolovskiy 2001a), the canonical transform(CT) method (Gorbunov 2002a), and the full

spectrum inversion (FSI) method (Jensen et al.2003). The SS, CT and FSI methods all providebending angles as a single-valued function ofthe impact parameter under all multipath con-ditions (including superrefraction). The CT andFSI methods, as shown by testing with simu-lated signals, provide better accuracy and reso-lution than the SS method. The FSI is compu-tationally faster than the CT and currently isselected as the main inversion method in thelower troposphere. However, the data in thisstudy were processed by using the CT method.The SS method, besides calculating the bend-ing angles, allows visualization of the spec-tral content of the RO signals. The cutoff ofthe bending angle and refractivity profiles, re-trieved by the SS, CT and FSI methods, is de-termined based on the fading of the amplitudeof the transformed signal. Figure 3 shows acomparison of L1 bending angles derived fromDoppler and from the complex signal by use ofradioholographic methods. The agreement be-tween the different radioholographic methodsis substantially closer than between any ofthem and the Doppler method.

Recently, a heuristic method using the canon-ical transform applied in the sliding window(CTss) was proposed by Beyerle et al. (2004),which seems to be less sensitive to the BlackJack receiver tracking errors in the tropo-sphere. (This comes at the expense of reductionin vertical resolution.) Also, a full spectrum in-version, applied in the sliding window of a spe-cial shape in order to suppress noise (FSIsw),has been proposed by Lohmann et al. (2003).These methods are being considered for testingin CDAAC.

Step 7: Combining (sewing) L1 bending angleprofiles from Steps 5 and 6

Multipath propagation in the moist tropo-sphere results in strong fluctuations and largetracking errors on the L2 signals, which makethem unusable. A typical example is shown inFig. 2. Large and abrupt noise increases on theL2 Doppler (B) or large mean deviation be-tween L1 and L2 Dopplers (C), whichever oc-curs at the higher altitude, are used to deter-mine the altitude Z below which the L2 signalis discarded for the ionospheric calibration (tobe described in Step 8). The geometric optics(calculated from Doppler) L1 bending angle


(Step 5) is replaced by the radioholographic L1bending angle (Step 6).

Step 8: Ionospheric calibration of thebending angle

The ionospheric calibration is performed bya linear combination of the L1 and L2 bendingangles taken at the same impact parameter(Vorob’ev and Krasil’nikova 1994). Below thealtitude where L2 is discarded (Step 7), the ra-dioholographic L1 bending angle is corrected bythe (L1–L2) bending angle extrapolated fromabove.

During some observation periods, CHAMPand SAC-C RO signals are contaminated byspikes in the L1 and L2 Doppler (see Section2.3). The effect of the spikes on calibrated ROphase can be considerably reduced by applyinga modified ionospheric calibration with strongsmoothing of the difference of L1 and L2 excessphases on reference link (reference GPS-LEO)(J. Wickert, personal communication, 2003).While this technique almost completely sup-presses the spikes in the calibrated L1 phase,for some occultations the residual effect of thespikes still remains on the L2 phases. Thenthe modified ionospheric correction, where theregularly smoothed L1 bending angles are cor-rected by strongly smoothed L1–L2 bendingangles, applied for the occulted link, almostcompletely suppresses the effect of spikes inthe ionospheric free bending angle. (A similarapproach had been applied for processing GPS/MET data when L2 was subject to Anti-Spoofing.) However, this reduction of the effectof spikes, which is based on the assumptionthat the ionospheric effect is a smooth enoughfunction of altitude, comes at the expense ofan increase in the uncompensated ionosphericeffect in the calibrated bending angle, which,for some occultations, can be larger than theeffect of the spikes. These spikes are not a fun-damental problem of RO, but are related to thespecific hardware on CHAMP and SAC-C (seealso Section 2.3).

Step 9: Optimal estimation of thebending angle

While the magnitude of the bending angleand its neutral atmosphere-related variations(signal) decrease exponentially with altitude,the magnitude of the noise (after filtering andionospheric calibration) remains about constant

and overshadows the signal above a certainaltitude (which may vary considerably betweendifferent occultations, as discussed below). Toreduce the error propagation from high to lowaltitudes after the Abel inversion (Step 10), theobservational bending angle at high altitudesmust be replaced by a model (first guess) whoseerror is smaller than the observational noise.We use an optimal estimation of the bendingangle profile prior to the Abel inversion, de-scribed by Sokolovskiy and Hunt (1996). A sim-ilar approach has been applied by Gorbunovet al. (1996), Hocke (1997), Steiner et al. (1999),Healy (2001), Gorbunov (2002b), Gobiet et al.(2002). The optimal bending angle vector~aa (i.e.,the profile aðaÞ where a is the impact parame-ter) is found by minimizing the cost function:

Jð~aaÞ ¼ ð~aa�~aaobsÞTBB�1obsð~aa�~aaobsÞ

þ ð~aa�~aaguessÞTBB�1guessð~aa�~aaguessÞ

¼ min ð1Þwhere ~aaobs and ~aaguess are the observational andthe first guess vectors, and BBobs and BBguess arethe error covariance matrices of the observa-tions and the first guess, respectively. The so-lution of the linear equation qJ=q~aa ¼ 0 is:

~aaopt ¼ ðBB�1obs þ BB�1

guessÞ�1ðBB�1

obs~aaobs þ BB�1guess~aaguessÞ

ð2Þ

To simplify the solution, we arbitrarily neglectthe vertical correlation in the observationalnoise, as well as in the error of the first guess(which results in matrices BBobs and BBguess beingdiagonal). Then, the optimal bending angle pro-file is:

aoptðaÞ ¼ wobsðaÞaobsðaÞ þ wguessðaÞaguessðaÞð3Þ

where the weighting functions wobsðaÞ andwguessðaÞ are:

wobsðaÞ ¼s2

guessðaÞs2

guessðaÞ þ s2obsðaÞ

and

wguessðaÞ ¼s2

obsðaÞs2

guessðaÞ þ s2obsðaÞ

ð4Þ

On the one hand, the observational error,sobsðaÞ, which depends mainly on the iono-spheric disturbances, varies significantly be-tween different occultations and is estimated


individually for each occultation in the altituderange of 60–80 km (where the observationalnoise overshadows the signal from the neutralatmosphere). On the other hand, typically, for agiven occultation, the structure and the magni-tude of the observational noise are rather uni-form below E-layer. This allows us to considersobs obtained at 60–80 km as an estimate forlower altitudes, i.e. to assume sobsðaÞ ¼ sobs ¼const for a given occultation. Reliable estima-tion of the error of the first guess, which can berepresented as sguessðaÞ ¼ KðaÞaguessðaÞ is moredifficult. Some studies (e.g., Gorbunov 2002b)estimate K from the deviation aobs � aguess inthe lower stratosphere, where the effect of noiseis not important, and use that K as a constantfor a given occultation. However, rocket sound-ings have shown that the variability of densityincreases with altitude, and at altitudes of 30–60 km (where wobsAwguess and thus the esti-

mate of K is most important for optimization)the variability can be as large as @20%. Cur-rently, we define K ¼ 0:2 for all occultations.Figure 4 shows aobs � aguess, wobs;wguess and theretrieved temperature profiles for two occulta-tions with substantially different noise levels.As seen, the altitude where wobs ¼ wguess is dif-ferent for different occultations, varying fromabout 30 to 60 km, depending on the observa-tional noise sobs.

Optimization (noise reduction) of the bend-ing angles is an underdetermined problem dueto lack of information about the upper strato-sphere, and different approaches discussed inthe literature (see below) suffer from using dif-ferent ad hoc assumptions. Currently, our firstguess aguess is based on CIRA-86. The use ofa refined climatology and upper stratosphericoperational numerical analysis or prediction asthe first guess is under consideration. Other ad

Fig. 4. Optimization and inversion of the bending angles for two GPS/MET occultations: 1995, doy(day of year) 174, 1:56 UTC (upper panels) and 1997, doy 012, 9:58 UTC (lower panels). Left pan-els: deviation of the bending angle from first guess (CIRA-86) (solid line); assumed magnitude ofthe standard error (G20%) of the first guess (dotted line). Middle panels: weighting functions forthe observations (solid line) and for the first guess (dotted line). Right panels: Retrieved temper-atures from the optimized (weighted with the first guess) bending angle (solid line) and for the firstguess (dotted line).


hoc approaches, such as ‘‘scaling’’ of the firstguess caguess and estimation of the magnitude ofsguess on the basis of deviation aobs � aguess ataltitudes where sobs f sguess (Gorbunov 2002b;Gobiet et al. 2002) are being considered for test-ing and validation.

Step 10: Retrieval of refractivity byAbel inversion

We calculate aguessðaÞ and aoptðaÞ in an ex-tended vertical range by exponential extrapola-tion of CIRA-86 refractivity from 120 to 150 km(if an occultation starts at a lower altitude, weset wobs ¼ 0, wguess ¼ 1 above that altitude).Reconstruction of the vertical refractivity pro-file NðzÞ by integration (Abel inversion) of theoptimized bending angle profile aoptðaÞ startsat 150 km. This basically results in retrieval ofthe first guess (CIRA-86) refractivity at highaltitudes, @80 km. At lower altitudes the re-trieved refractivity is gradually affected moreby the observations, according to an increase oftheir weight in soptðaÞ:

Step 11: Retrieval of pressure andtemperature

At altitudes where humidity may be ne-glected (above the troposphere), pressure andtemperature are reconstructed from the re-trieved refractivity (which is then proportionalto density) by integration of the hydrostaticequation. The integration starts at 150 km, bysetting pressure and temperature to zero. This‘‘zero initialization’’ does not affect the re-trieved temperature at @80 km, which is closeto the first guess (CIRA-86). At lower altitudesthe retrieved temperatures are gradually af-fected by observations, according to an increaseof their weight in the retrieved NðzÞ:

2.2 Quality control of inverted RO signalsQuality control (QC) of the inverted RO sig-

nals begins with the raw signals, continues inthe inversion process, and ends at the invertedrefractivity. Some elements of the QC werealready discussed above: truncation of the L1signal (Step 1), truncation of L2 while keep-ing the L1 (Step 7), and estimation of the mag-nitude of residual noise which controls theweighting of the observations and the firstguess at high altitudes (Step 9). Overall, thequality of the retrieved bending angle and re-fractivity profiles is characterized in different

respects by a set of parameters. The most im-portant parameters are listed and their physi-cal meaning is explained in Appendix A. A usercan plot histograms of these parameters for acertain period in order to make a decision aboutusing or discarding the occultations based oncertain values of the parameters, thus, balanc-ing the quality and quantity of the data for aparticular study. An example of the use of a QCparameter for the selection of low-noise occul-tations that provide better retrieval accuracy inthe upper stratosphere (at the expense of re-duction in the amount of the data) is discussedin Section 3.1.

2.3 Raw data quality issuesCurrent missions (CHAMP and SAC-C) are

experimental, and the signal tracking hard-ware and firmware are not yet optimized. In-tensive research and development are ongoingat JPL, and considerable improvements are ex-pected in future RO missions. Currently, bothL1 and L2 signals contain receiver tracking er-rors and, during some periods of the missions,periodic spikes that are related to receiver clockdistribution errors. The 1 s spikes in excessDoppler can be clearly seen in Fig. 2 (beforeMarch 2002, CHAMP RO signals sometimescontained 5 s spikes). The 1 s spikes make itdifficult to use the affected data for study of thestratospheric gravity waves. Even though theeffect of the spikes can be reduced (in a statis-tical sense) by applying strong smoothing ofthe (L1–L2) data (Section 2, Step 8), this addi-tional smoothing can cause an increase of theretrieval errors due to uncalibrated ionosphericeffect. Because of this, the spikes must be elim-inated in the future. The L1 tracking errors willbe eliminated by applying the open loop track-ing in the lower troposphere. The largest re-maining concern is the quality of the L2 signal,which may not be tracked in the open loopmode unless C/A code is available on L2.

Our analysis of SAC-C data shows that thequality of the L2 signal varied significantlyfrom fairly good to rather poor in year 2002.Figure 5 (upper panels) shows the altitudes Z,below which L2 was discarded (see Step 7 ofthe processing), for all processed SAC-C occul-tations during March and August 2002. Thetest for Z was started below the point wherethe L1 excess phase ¼ 30 m, which approxi-


mately corresponds to 16–18 km, depending onlatitude. This starting altitude is traced by asharp upper boundary on the scatter plot in theupper right panel of Fig. 5. It is clearly seenthat L2 signals acquired by the SAC-C receiverin August 2002 are much noisier than the sig-nals acquired in March 2002. In August 2002there are almost no occultations with accept-able L2 quality below 10 km. The fact that theZ values cluster immediately below the start-ing altitude (upper right panel) means that formost occultations, in fact, the L2 signal haspoor quality at higher altitudes. In March 2002,the Z values do not cluster right below the startaltitude, and many occultations have accept-able L2 quality down to 5 km in the tropicsand down to the surface in polar regions. Fig-ure 5 (lower panel) shows Z averaged over eachmonth in 2002 for SAC-C. The fact that the L2signal acquisition was possible with reasonablygood quality during some time periods, as can

be seen in the upper left panel of Fig. 5, in-dicates that the problem can be resolved by im-proving (correcting) receiver firmware.

3. Analysis of CHAMP and SAC-C radiooccultation soundings

3.1 Comparison of radio occultationsoundings with global analyses

In this section, we evaluate the accuracy ofRO data by comparing CHAMP and SAC-C ROsoundings with global analyses, global forecastsand radiosonde soundings. The study is basedon data for the month of December 2001. Theselection of December 2001 was based on theinterest in a case study to examine the impactof GPS RO data on the prediction of an intenseAntarctic storm. This selection was not basedon the relative quality or the quantity of theavailable GPS RO data. Statistically, the qual-ity of the RO signals in the upper stratosphereand the quality of the L2 signal in the tropo-

Fig. 5. Upper panels: altitude Z, below which the L2 signal was discarded, for SAC-C, in March 2002(fairly good L2 quality) and in August 2002 (poor L2 quality). Lower panel shows mean value of Zfor each month of 2002.


sphere for this period are characterized by theuse of two QC parameters in Fig. 6. Upperpanels show the observational error sobs (stan-dard deviation of the bending angle from firstguess based on CIRA-86, estimated between60–80 km), and lower panels show the altitudeZ below which the L2 signal is discarded (fordetails see Section 2.1 and Appendix A) foreach occultation in December 2001. The meanvalue hsobsi (‘‘h i’’ designates average over allthe soundings for the month of December 2001)is 8:1 � 10�6 rad for CHAMP, and 5:3 � 10�6 radfor SAC-C. If we remove noisy soundings (withsobs exceeding 1 � 10�5 rad), the monthly meanof hsobsi becomes 3:8 � 10�6 rad for CHAMP and4:2 � 10�6 rad for SAC-C. Thus, the quality ofCHAMP and SAC-C signals at high altitudesis not substantially different for the period con-sidered, except for somewhat bigger number ofoutliers with large sobs for CHAMP (apparentlyrelated to signal acquisition problems, not toionospheric effects), as can be seen from com-parisons of the upper panels in Fig. 6. Thequality of L2 at low altitudes, as can be seen

from the lower panels in Fig. 6, is differentfor CHAMP (hZi ¼ 8:5 km) and SAC-C (hZi ¼13:8 km). For SAC-C the L2 quality is some-where between those for March and August2002 (upper panels in Fig. 5). For SAC-C thereare almost no occultations, while for CHAMPthere are a fair number of occultations with ac-ceptable L2 quality below 5 km. The fact thatthe CHAMP Z values do not cluster right belowthe start altitude for our L2 QC check (whichwas 13–14 km, corresponding to 40 m L1 ex-cess phase) indicates that most occultationshave reasonable L2 quality at higher altitudes.

Because of differences in vertical resolution,vertical coordinates, and horizontal smearingof RO soundings, a comparison of RO andradiosonde soundings with global analysis andother types of data is not trivial. In thisstudy, we first exclude outliers (suspicious ROsoundings from a statistical point of view) fromthe comparison dataset. We then perform adigital filtering to remove small-scale featureswith a vertical scale of less than 1 km, thusmaking the vertical resolution of the RO data

Fig. 6. Upper panels: estimate of the observational noise of the bending angles at 60–80 km, sobs, forCHAMP and SAC-C occultations in December 2001. Lower panels: altitude Z below which the L2signal was discarded.


compatible with other correlative data (e.g.,global analyses). Finally, we compare the ROand other data on height coordinates at alti-tudes corresponding to constant pressure sur-faces of the global analyses. Approximately6,500 RO soundings passed the quality controlprocedure from the total of @8,500 soundings.The details of data processing and data qualitycontrol procedures are described in AppendixB. Figure 7 shows the mean and the standarddeviations of fractional differences in refrac-tivity between RO (including both CHAMP andSAC-C) and global analyses. The fractional dif-ference in refractivity1 ðDf NijÞ between two re-fractivity values (Ni and Nj) is defined as the

following:

Df Nij ¼Ni � Nj

Nð5Þ

where N is the average between Ni and Nj. Therefractivity values from the global analyses arelocal values interpolated to the time and loca-tion of the corresponding GPS occultation. TheEuropean Centre for Medium Range WeatherForecasts (ECMWF) analysis used here is the

Fig. 7. Comparison of GPS radio occultation soundings with ECMWF and NCEP AVN analyses:(a) for southern hemisphere (30 S to 90 S), (b) for tropics (30 S to 30 N), and (c) for northern hemi-sphere (30 N to 90 N). The upper-right panel shows the mean fractional differences in refractivity,while the upper-left panel shows the standard deviations. The lower-left panel shows the totalnumber of RO soundings used in these calculations as a function of height. The solid lines in thetwo upper panels are comparisons between GPS RO with NCEP AVN analysis, while the dashedlines are comparisons with ECMWF analysis. The mean altitudes corresponding to pressure sur-faces are shown on right side of the upper-right panel.

1 Appendix C describes the relationship betweenfractional errors or differences in refractivity andthe corresponding errors in temperature and wa-ter vapor pressure.


TOGA (Tropical Ocean and Global Atmosphere)analysis at 2.5 degrees horizontal resolutionwith 21 pressure levels. The NCEP AVN anal-ysis is at 1 degree horizontal resolution and 26vertical pressure levels. The comparisons areseparated into three regions: northern hemi-sphere (30 N to 90 N), southern hemisphere(30 S to 90 S), and the tropics (30 S to 30 N).

The comparisons between the RO andECWMF analysis and between RO and AVNanalysis are very similar (Fig. 7). There aresmall biases between the RO soundings andthe two global analyses in the bottom 2 km ofthe troposphere for the middle and higher lat-itudes. Large refractivity biases (negative Nbias relative to global analyses) are found in thetropical lower troposphere, with magnitudesapproaching 4% near the surface. The existenceof the negative N bias was first noticed anddiscussed by Rocken et al. (1997) for GPS/MET,and recently confirmed by modeling receivertracking loops (Ao et al. 2003; Beyerle et al.2003) and superrefraction in PBL (Sokolovskiy

2003). Smaller bias (less than 1%) is found inthe lowest few kilometers in middle and higherlatitudes. The bias is smaller in the winter(northern) hemisphere (@0.5%) than in thesummer (southern) hemisphere (@1.0%). Thisdifference is presumably related to the seasonaldistribution of moisture (more moisture in thesummer than in the winter hemisphere). Itshould be noted that the significant reductionsof the amount of available RO data in the low-er troposphere (see lower-left panel of Fig. 7),particularly over the tropics, may affect theestimate of the bias. Noticeable bias exists inthe upper stratosphere above 25 km betweenRO and the two global analyses. This is likelyrelated to errors in global analyses at these al-titudes as well as to RO observational (mainlyresidual ionospheric) errors and the errors

Fig. 8. (a) shows the mean standard de-viations of fractional differences be-tween RO soundings and the ECMWFglobal analysis for the month of Decem-ber 2001. (b) is the same as (a), exceptthe selection of RO soundings is re-stricted to low measurement noise (seetext for details).

Fig. 9. The standard deviations ofECMWF analysis differences with one-day lag for December 2001: (a) re-fractivity, (b) temperature, and (c)water vapor pressure.


caused by the use of ancillary data for noise re-duction (optimization) in RO processing.

The standard deviations (SD) of fractionalrefractivity differences of RO soundings fromglobal analyses are about 0.5% between500 hPa (@5 km) and 20 hPa (@25 km) forboth hemispheres. Below 500 hPa, the SD in-crease to 1.3% and 2.0% for northern and south-ern hemispheres, respectively. In the tropics,the SD increase from about 0.5% at 300 hPa toabout 3% at 1000 hPa. The SD between RO andglobal analyses also increase above 20 hPa, es-pecially in the northern (winter) hemisphere.

To examine the inversion errors related tothe observational noise and the use of ancillarydata (e.g., CIRA climatology) for the noise re-duction (optimization), we show in Fig. 8a thezonal mean SD of fractional differences in re-fractivity between RO and the ECMWF analy-sis for the month of December 2001. Two re-gions with large SD are found, one in thetropical lower troposphere and the other inthe upper stratosphere in the northern hemi-sphere, from 30 N to 90 N. At 40 km, the SDbetween GPS and ECMWF exceed 5% (which ismuch larger than in the tropical lower tropo-sphere). The upper stratosphere in winter isa region of significant variability. This followsfrom Fig. 9a, which presents the day-to-day re-fractivity variation in ECMWF analysis (e.g.,the SD among one-day lagged ECMWF refrac-tivity analyses) for December 2001. The vari-ability in the summer stratosphere is consider-ably smaller. The large variability means thatthe daily ECMWF analysis is likely to deviateconsiderably from the CIRA climatology (a 40-year climatology) used as the first guess fornoise reduction (optimization). Therefore, if thefirst guess (climatology) is weighted heavily (fora RO sounding with significant noise) in theoptimized bending angle profile (as discussedin Step 9 of Section 2.1), then the resulting re-fractivity profile in the upper stratosphere canbe subject to large errors. The weight of clima-tology is reduced for soundings with low noise.To demonstrate the effect of noise on inver-sions, we restrict the selection of RO soundingsto only those with standard deviation and meanof the observed ionosphere free bending anglefrom the first guess at altitudes of 60–80 km tosobs a 3 � 10�6 rad and hdaia 5 � 10�7 rad, re-spectively. This reduces the sample size from

6,500 to about 1,300 soundings (@20%). Theresult is shown in Fig. 8b. As seen, with the useof only low-noise RO soundings, the large devi-ation between RO and ECMWF over the winterstratosphere is mostly removed, except for thevery high levels (above 37 km). This exampledemonstrates the clear benefit of using QC pa-rameters, generated by the inversion software,in selecting occultations for a particular study.Thus, for climate monitoring and the study ofgravity waves in the upper stratosphere, onlylow-noise occultations should be selected, eventhough this reduces the number of data. Forweather prediction in the troposphere andlower stratosphere, the effects of observationalnoise and the use of ancillary data for noise re-duction are smaller, and a larger number of oc-cultations can be used.

Further examination of the day-to-day vari-ability based on the ECMWF analyses showsthat the variability in refractivity of the upperstratosphere in the winter hemisphere (Fig. 9a)is mainly related to the variability in the tem-perature field (Fig. 9b). The other high vari-ability region in the lower troposphere fromsubtropical to middle latitudes, as mentionedearlier, is mainly related to tropospheric watervapor variability (Fig. 9c). A region of moderatevariability in refractivity near the tropopause(at around 200 hPa near 60 latitudes of bothhemispheres) is related to temperature vari-ability associated with upper tropospheric dis-turbances (Fig. 9b).

Figure 10 presents a comparison between ROsoundings and global analyses over land versusoceans for the northern hemisphere from 25 Nto 65 N. If more than one radiosonde station isavailable within a 500 km radius from a givenRO sounding, the location is classified as landand vice versa. For both ECMWF and AVNanalyses, the results indicate that the SD be-tween RO soundings and the global analysesare noticeably larger over oceans than overland. The accuracy of the RO soundings is ex-pected to be about the same over ocean versusland. This suggests that the accuracy of globalanalyses is lower over oceans (due to the rela-tively smaller number of observations beingassimilated). Rocken et al. (1997) arrived at asimilar conclusion based on comparison of GPS/MET data to global analyses.

Short-range (6 to 12 hours) predictions are


often used as the background fields for a globalanalysis. Short-range predictions provide use-ful data sets for the evaluation of observationalsystems, as the forecasts are less dependenton the observations used in the analysis, theforecast errors are relatively small, and theforecast fields are more dynamically balancedthan the analysis. Figure 11 compares the AVNsix-hour forecast with radiosonde observations(RAOBS) and RO soundings over the tropics(from 30 S to 30 N) and over the northernhemisphere mid-latitudes (30 N to 60 N). In thiscomparison, only RO soundings located overland are used. Both comparisons (RO vs. fore-cast and radiosonde vs. forecast) are made atthe time and location of each observing system.This puts RO in a slightly disadvantageousposition, as additional time interpolation ofthe forecast fields is required for the RO vs.forecast comparison, which is not needed forthe radiosonde vs. forecast comparison. Wenote that for both regions, RO soundings showbetter agreement with the AVN six-hour fore-casts than the radiosonde. It is important tonote that the deviation of radiosondes fromthe AVN forecast is larger than that of the ROobservations even in the tropical lower tropo-sphere, despite the fact that the radiosondeshave smaller bias errors compared to the ROin this region of the atmosphere (figures notshown). The disagreement between radiosondeand forecast might be attributed to the mea-

surement errors of the radiosonde observingsystem such as inaccurate humidity measure-ments in the lower atmosphere (Elliott andGaffen 1991; Schwartz and Doswell 1991; El-liott et al. 1998) and the radiation effect athigher altitudes (McMillin et al. 1992; Luersand Eskridge 1998). In addition, the radio-sonde, which is a point measurement, is likelyto have a larger representativeness error whencompared with a global forecast. As expected,the deviations of the RO and radiosonde datafrom the AVN forecast are larger in the tropicallower troposphere than in the mid-latitudelower troposphere.

3.2 Estimates of GPS RO observational errorand model forecast error

Short-range forecasts can be used to provideestimates of errors of an observing system. Sev-eral methods are available for this based onthe statistics of observation minus forecast dif-ferences. In comparing an observing systemwith a short-range forecast, the difference canbe considered as an apparent or perceived error.This includes the errors of the forecast as wellas the observing system. Under the assumptionthat the observational errors are uncorrelated

Fig. 10. Standard deviations of fractionaldifferences in refractivity between ROsoundings and global analyses from25 N to 65 N. The comparisons arestratified over land versus ocean.

Fig. 11. Standard deviations of frac-tional refractivity differences betweenRO and NCEP AVN six-hour forecast(solid lines), and between radiosondes(RAOBS) and NCEP AVN six-hourforecast (dashed line) for December2001. The mean altitudes correspond-ing to pressure surfaces are shown onthe right side of the figures: (a) for thetropics (30 S to 30 N), and (b) for thenorthern hemisphere (30 N to 60 N).


with the forecast errors, the perceived error canbe separated into two parts:

s2a ¼ s2

b þ s2o ð6Þ

where s2a ; s

2b , and s2

o are variances correspond-ing to apparent error, forecast error, and ob-servational error, respectively. If the forecasterror can be estimated accurately, then the ob-servational error can be estimated by subtract-ing the forecast error variance from the total(perceived) error variance.

a. Estimation of NCEP AVN 12-hourforecast error

Two common methods are used for modelforecast error estimation. The first method,known as the NMC (National MeteorologicalCenter) method (Parrish and Derber 1992) as-sumes that the difference between two fore-casts that are valid at the same time is a goodestimate of the model forecast error. The ad-vantage of this method is that it does not re-quire the use of any observations to estimatemodel error. The weakness is that it tends tounderestimate model error. The model containsonly a limited spectrum of atmospheric circu-lations; it has greatly reduced variability com-pared to the true atmosphere. Moreover, themodel’s implicit or explicit smoothing furtherreduces the model variability. Finally, bias orother model errors correlated in time or be-tween consecutive forecasts tend to cancel inthe NMC method. In this study, we calculatethe differences of 12-hour and 24-hour AVNforecasts for the month of December 2001 (atotal of 62 pairs of forecasts).

Over the region where dense observations areavailable, another method proposed by Hol-lingsworth and Lonnberg (1986) (hereafter re-ferred to as the H-L method) can be used to pro-vide a more realistic estimate of model error.This method assumes that observational errorsare spatially isotropic, uncorrelated with eachother, and uncorrelated with forecast error. Theprinciple of this method is to calculate a histo-gram of ‘‘forecast minus observation’’ covari-ances among all pairs of stations stratifiedagainst the separation distance of each pair,and then perform the least square fit of thecollected correlations to an isotropic correla-tion model. Because any correlated error shouldcome only from forecast fields under the givenassumptions, the forecast error variance can be

obtained from the fitted correlation model atzero separation distance. This method is onlyapplicable to the observational networks thatare dense and large enough to provide infor-mation on diverse scales, preferably over a rel-atively long period of time (usually longer thana month) for a fixed set of stations. Obviously,this requirement is difficult to meet for thepresent study with one month of RO soundingsfrom CHAMP and SAC-C. However, if the fore-cast error is assumed to be uncorrelated withthe observational error, the estimated forecasterror should be the same regardless of the ob-serving system used. Therefore, our strategyis to use radiosonde observations available forthe month of December 2001 to estimate theerror of AVN forecasts and then to evaluatethe observational error of RO soundings fromthe forecast error and the departure statisticsof RO from the same forecast. The resulting ob-servational error of RO is expected to be fairlyreasonable, at least in the vicinity of radiosondestations. We thus use only the RO soundingsover land. The same condition has been appliedfor the NMC method.

The results of both methods are presented inFig. 12. With the NMC method, the fractionalrefractivity error of the AVN 12-hour forecast

Fig. 12. Forecast errors estimated bythe NMC method (dashed line) andthe Hollingsworth and Lonnberg (1986)method (solid lines). The mean alti-tudes corresponding to pressure sur-faces are shown on the right side ofthe figures: (a) for the tropics (30 S to30 N), and (b) for the northern hemi-sphere (30 N to 60 N).


is very small over the tropics, above 300 hPa.The error varies from 0.3% at 300 hPa to 0.4%at 10 hPa. The AVN model error increases from0.3% at 300 hPa to about 2% at 700 hPa. Itthen decreases to about 1.2% near the surface.Over the mid-latitudes (from 30 N to 60 N), theerror varies from about 0.3% at 400 hPa to0.6% at 10 hPa. The maximum error of about1% occurs at 700 hPa.

The forecast error estimated by the H-Lmethod is almost twice as large as those of theNMC method over the tropics above 250 hPa.The two methods produce comparable resultsbelow 500 hPa, except for the bottom 100 hPa.Near the surface, again, the H-L method givesa larger forecast error by about 0.8%. Simi-lar results are obtained over the mid-latitudes(30 N to 60 N), with the H-L method giving amuch larger estimate of forecast error than theNMC method. It is important to recognize thatboth the NMC and the H-L methods have short-comings; there is no perfect way to determinethe true model forecast error. However, in gen-eral, the H-L method is believed to provide amore realistic estimate of the forecast errorthan the NMC method because actual observa-tions are used to estimate the model forecasterror.

b. Estimation of RO observational errorWith the knowledge of model forecast errors,

one can estimate the observational errors bysubtracting the model forecast error variancesfrom the apparent error variances. Figure 13shows the estimates of RO observational errorsbased on the forecast errors obtained by theNMC method and the H-L method. Because ofits larger forecast errors, the H-L method givessmaller observational errors for RO than theNMC method does. Based on the H-L method,the RO observational errors vary from about0.2% at 400 hPa to about 0.5% at 10 hPa overthe tropics. The error increases almost linearlywith height (logarithm of pressure) toward thelower troposphere, from 0.2% at 400 hPa toabout 3% near the surface. The RO observa-tional errors are nearly constant at 0.3% from500 hPa to 30 hPa over the mid-latitudes. Theyincrease from 0.3% at 30 hPa to 1.2% at 10 hPa,and from 0.3% at 500 hPa to about 0.75% nearthe surface. These results show that the ROobservational errors vary with latitude and al-titude; thus, a single error profile is not a goodglobal representation of the RO errors. TheRO soundings have the lowest errors betweenthe upper troposphere (@5 km) and the lowerstratosphere (@25 km).

Fig. 13. RO observational errors (in terms of fractional differences in refractivity) based on the NMCmethod (dashed line) and the Hollingsworth and Lonnberg (1986) method (solid line). The ROmeasurement errors (for all latitudes) as estimated by Kursinski et al. (1997), are shown as dot-dashed lines (adapted from Fig. 13 of Kursinski et al. 1997). Left panel is for the tropics (30 S to30 N), and the right panel is for the midlatitudes northern hemisphere (30 N to 60 N).


The RO errors, as estimated by the NMCmethod, are generally larger than those by theH-L method. For the tropical atmosphere above400 hPa, and for most of the mid-latitudes tro-posphere and lower stratosphere, the observa-tional errors estimated by the NMC methodare about twice as large as those by the H-Lmethod. For the tropical lower troposphere,these two methods give close results. With themore realistic estimates of model errors by theH-L method, the RO errors estimated by thismethod should also be more realistic.

Kursinski et al. (1997) identified a numberof important sources of errors in RO measure-ments, and provided theoretical estimates ofthese errors. These errors, applicable to all lat-itudes, are reproduced in Fig. 13. The predictederrors are generally lower than the errors esti-mated in our study. For example, the maxi-mum error predicted by Kursinski et al. (1997)in the lower troposphere is about 1%, whilethe error in the tropical lower troposphere esti-mated in our study is about 2@3 times larger.This is to be expected, as the theoretical pre-diction does not take into account all the errorsassociated with the RO measurements (e.g.,tracking errors and the effects of superrefrac-tion in the tropical planetary boundary layer).It is important to note that for the upper tro-posphere and lower stratosphere, where RO is

expected to have its best performance, the mea-surement errors predicted by Kursinski et al.(1997) agree fairly well with the errors esti-mated by the H-L method (e.g., generally in therange of 0.2 to 0.3%). This suggests that thedominant error source for this region of the at-mosphere is horizontal inhomogeneity alongthe ray path. We note that the theoreticallypredicted errors are larger than the estimatedobservational errors over the lower tropospherein the middle latitudes (30 to 60 N). In principlethis can be explained by representativenesserrors. Kursinski’s theoretical estimates werebased on comparisons with local values. In ourestimates we compare the RO observations tothe model refractivity, which is a horizontal av-erage, and the difference is likely to be smaller.

Figure 14 compares the model errors withthe observational errors of RO and radiosondes,based on the H-L method. It shows that RO hassmaller errors than the radiosonde. This resultremains unchanged using the NMC method(not shown). This is even true over the tropicallower troposphere, where the RO has the larg-est errors. The larger error associated with theradiosonde, which is a point measurement, islikely related to larger ‘‘representativeness’’ er-ror. This is particularly true for the moisturefield, which varies significantly in time andspace (Fig. 9c). Another important conclusion

Fig. 14. NCEP AVN 12-hour forecast errors (dot-dashed lines) and observational errors of RO (solidlines) and radiosonde (dashed lines) as estimated by the Hollingsworth and Lonnberg (1986)method for the month of December 2001. Left panel is for the tropics (30 S to 30 N), and the rightpanel is for the midlatitudes northern hemisphere (30 N to 60 N).


is that the RO errors are significantly smallerthan the AVN 12-hour forecast errors, exceptfor the tropical lower troposphere. This in-dicates that the RO should contribute posi-tively to global analysis and prediction.

3.3 Cross comparison of RO soundingsOne way to assess the accuracy of the RO

soundings is to compare those that took placein close proximity (Hajj et al. 2004). For themonth of December 2001, we identified ap-proximately 70 pairs of RO soundings fromCHAMP and SAC-C missions that occurredwithin 300 km and two hours of each other.This includes CHAMP-to-CHAMP, SAC-C-to-SAC-C and CHAMP-to-SAC-C pairs. We strat-ify the data into three different regions: thetropics (30 S to 30 N), northern hemisphere(30 N to 90 N), and southern hemisphere (30 Sto 90 S), with each region having approxi-mately 25 pairs of soundings. The number ofpairs available for comparison at a certainaltitude drops significantly below 5 km andis reduced to only 5 in the lowest few kilo-meters. Figure 15a shows the root-mean-square(rms) of fractional differences between theseRO sounding pairs. For middle and high lat-itudes in both hemispheres, the rms are ap-proximately 1% between 5 km and 25 km.Above 25 km, the rms over the northern hemi-sphere increase to about 1.5% at 30 km.

The rms of refractivity differences ðsGÞ amongpairs of neighboring RO soundings (e.g., s2

G ¼ðGi � GjÞ2, where Gi and Gj represent a pair ofRO soundings and overbar denotes the averageover all collocated pairs) includes measurementerrors as well as true atmospheric variability(e.g., s2

T ¼ ðTi � TjÞ2, where Ti and Tj are thetrue refractivity values at the time and loca-tion of the RO soundings). One way to estimatethe atmospheric variability is to repeat thesame calculation using ECMWF analyzed re-fractivities interpolated to the RO soundinglocations. The results are shown in Fig. 15b.We note that the ECMWF to ECMWF compar-isons at the RO sounding sites have compara-ble magnitude of fractional differences as theRO to RO differences from 5 km to about 25 kmfor the northern and southern hemisphere.The rms increase upward from 22 km to about30 km for the northern (winter) hemisphere,while in the southern (summer) hemisphere

they decrease with height. The results over thetropics yield a profile similar to that of theactual RO to RO comparisons, except the rms

Fig. 15. (a) shows the rms of the frac-tional refractivity differences betweenpairs of RO soundings that fall within300 km and two hours of each other.(b) is the same as (a), except for theECMWF to ECMWF comparison at theRO sounding sites. (c) provides an esti-mate of the GPS measurement errorsbased on GPS-to-GPS comparison (seetext for details).


are slightly smaller. The large rms at 15–18 kmrange are related to variability of the tropi-cal tropopause. Above 20 km, the rms in theECMWF simulation do not increase rapidlywith height, unlike the actual RO to RO com-parison. In the tropical lower troposphere, therms of the simulated comparison increase in-versely with height, and reach a value of 3%near the surface.

If we assume that the ECMWF analysis isa good approximation to the real atmosphere,then the true atmospheric variability can be ap-proximated with the values shown in Fig. 15b:s2

T As2E ¼ ðEi � EjÞ2, where Ei and Ej represent

the refractivity of ECMWF analysis interpo-lated to the RO location. Assuming measure-ment errors are uncorrelated with each otherand uncorrelated with true atmospheric vari-ability, the RO measurement error, so can beexpressed as:

2s2o ¼ s2

G � s2T As2

G � s2E: ð7Þ

It is encouraging that the GPS RO measure-ment errors (Fig. 15c), as estimated by thisapproach, are similar to the observational er-rors estimated earlier in this section. From5 km to 25 km, the measurement errors aregenerally in the range of 0.3% to 0.5% in boththe northern and southern hemispheres. Overthe tropics, the errors increase downward andreach a value of 2% at about 3 km. Since thenumber of RO to RO comparisons drops sig-nificantly below 5 km, the results below thisaltitude are not reliable. It is possible that theerror estimates above 5 km may be larger thanthe true measurement errors, as the naturalatmospheric variability is most likely under-estimated by the ECMWF analysis, which hasa horizontal resolution of 2.5 degrees and atemporal resolution of 12 hours.

4. Summary and discussions

In this paper we described the data inver-sion methods applied in the UCAR COSMICData Analysis and Archive Center (CDAAC).We then evaluated the observational errorsof the RO soundings based on the data fromCHAMP and SAC-C missions for the month ofDecember 2001. Our analysis leads to the fol-lowing conclusions:

(i) The CHAMP and SAC-C RO refractivityprofiles are of the highest accuracy from

5 km to 25 km. In this part of the atmo-sphere, there is no superrefraction, track-ing errors are not very common, and theeffect of the residual ionospheric noise isnot significant. In this altitude range theRO refractivity errors are generally in therange of 0.3% to 0.5%. The RO measure-ment errors are comparable to or smallerthan the 12-hour global model forecast er-rors. This is the region where one wouldexpect the RO soundings to have theirmaximum impact on weather analysis andprediction.

(ii) The observational errors estimated fromreal RO soundings are, in general, largerthan the theoretical predictions by Kursin-ski et al. (1997). In the 5-km to 25-kmrange, the estimated errors based on actualRO soundings are only slightly larger thanthe theoretical prediction. However, in thetropical lower troposphere, the estimatedobservational errors (3@4%) are muchlarger than the theoretical predictions of@1%. Larger observational errors are alsofound in the stratosphere above 25 km.These results suggest that the errors in theupper troposphere to lower stratosphere(5 km to 25 km range) are dominatedby along-track horizontal inhomogeneities(Kursinski et al. 1997). Tracking errorsand superrefraction (which are not con-sidered in the theoretical estimates) con-tribute significantly to errors in the tropi-cal lower troposphere. The larger errors inthe upper stratosphere (above 25 km) arerelated to the residual ionospheric noiseand the use of ancillary (climatology) datafor the noise reduction process (optimiza-tion). However, the upper stratosphericerrors can be substantially smaller whenselecting only high quality (low noise) oc-cultations with reduced weight of the firstguess (CIRA-86 climatology) in the opti-mized bending angles and retrieved re-fractivity. This is an important result forthe use of RO soundings in climate mon-itoring.

(iii) Observational errors compared to the anal-yses (which include both the measurementerrors and the representativeness errors)associated with radiosondes are found tobe significantly larger than those of the RO


observations. This is even true for thetropical lower troposphere. This is attrib-uted to measurement errors of the radio-sonde as well as the larger representative-ness errors associated with the radiosonde,which is a point measurement.

(iv) The errors of RO soundings, as estimatedby comparison of independent neighboringRO observations, are, in general, compati-ble with the observational errors estimatedusing the standard methods from 5 km to25 km. This gives support to the estimatesof the RO observational errors based onthe global analyses.

The results presented in this paper indicatethat the quality of RO soundings is very high inthe upper troposphere and lower stratosphere.The measurement (and observational) errors inthis range of the atmosphere are comparablewith the theoretical predictions. However, theerrors in the tropical lower troposphere and thestratosphere are considerably larger than theo-retical predictions. These are the regions wherefurther improvements in signal acquisition andsounding retrieval are most important.

Theoretically, RO remote sensing has a veryhigh potential accuracy, which is fundamen-tally limited by the stability of the transmitterand receiver clocks, positioning accuracy, dif-fractional effects, ionospheric variability, andhorizontal inhomogeneity of refractivity in theatmosphere. The horizontal inhomogeneity ofthe atmosphere, which makes the inverse ROproblem underdetermined, requires either re-duction of the dimension of the problem by us-ing the assumption of local spherical symmetry(which introduces errors) or the inclusion ofancillary information in the inversion (varia-tional assimilation). Superrefraction can be asignificant error source below the sharp top ofa marine boundary layer near 2–3 km altitude(Sokolovskiy 2003). The problem of multipathpropagation, which has been previously consid-ered as the factor restricting accuracy in themoist troposphere, has been successfully re-solved by the development of advanced radio-holographic methods that allow disentanglingof multiple rays. With the use of GPS, the clockstability and the positioning accuracy do notintroduce a dominant error source. However,other features of GPS do not allow the potential

of the RO technique to be realized to its fullestextent. In particular, at GPS frequencies theuncalibrated ionospheric effect related to small-scale plasma irregularities is the dominanterror source at altitudes above @30 km. Thiseffect can be significantly reduced by use ofhigher frequencies. Closed-loop tracking resultsin large errors when tracking multitone sig-nals and is currently one of the dominant errorsources in the moist troposphere. While open-loop tracking (which is free of tracking errors)can be applied for L1 (modulated by C/A code),currently it cannot be applied for L2, which ismodulated by P(Y) code. The planned GPS sig-nal modernization may solve this problem.

Acknowledgements

This work was supported by the NationalScience Foundation, as part of the developmentof the COSMIC Data Analysis and ArchiveCenter (CDAAC) at UCAR under the Coopera-tive Agreement ATM-9732665 and by the Na-tional Aeronautics and Space Administration,through Grant NAG 5-9518 to Ohio State Uni-versity and UCAR. We thank Jay Fein of NSFespecially for his support of GPS research overthe years.

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Appendix A

Some parameters characterizing qualityof radio occultation

sobs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihða� aguessÞ2i

q: standard deviation of

the observed ionosphere free bending anglefrom the first guess at altitudes 60–80 km(where the observational noise overshadowsthe signal from the neutral atmosphere for alloccultations). The dominant source of noiseis the uncalibrated effect of small-scale iono-spheric irregularities and the magnitude sobs

spans a rather large range. This parameter al-lows estimation of the weighting functions forthe observations and for the first guess.

hdai ¼ ha� aguessi: mean deviation of the ob-served ionosphere free bending angle from thefirst guess at altitudes 60–80 km. Normally,for most occultations, hdaif sobs. If hdai@ sobs

and sobs is much larger than its mean value;this can indicate a bias error, and such occulta-tion must be discarded.

Z: altitude below which a low quality of L2signal has been detected and the ionosphericcalibration is performed by extrapolation of(L1–L2) from above. A large value of Z canindicate a low quality of L2 signal and a highprobability of tracking errors at all altitudesbelow Z.

S4: normalized standard deviation of L1 sig-nal amplitude at high altitudes. A large value


of S4 can indicate an ionospheric scintillationand a high probability of tracking errors at allaltitudes.

ða12Þmax: maximum difference of L1 and L2bending angles above Z. A large value can in-dicate extreme ionospheric conditions or track-ing errors.

ððN � NguessÞ/NguessÞmax: maximum fractionaldeviation of the retrieved refractivity from thefirst guess.

�ðdN/dzÞmax: maximum negative retrievedrefractivity gradient. In the future, with elimi-nation of L1 tracking errors by open loop track-ing, this parameter can be used as the indicatorof sharp tops of the moist planetary boundarylayer and, potentially, superrefraction condi-tions (which results in a negative N bias).

Appendix B

Procedures for comparison of GPS ROdata with other types of data

The comparison of RO data with other typesof data is not simple because GPS occultationsare sampled irregularly both in space and time.The RO sounding has much higher verticalresolution than most other correlative data(e.g., global analysis), and the latitude andlongitude of the estimated perigee points of theRO change with height during an occultation(smearing effect). In this appendix, we discussthe data handling procedures for the compari-son of RO data with other types of data.

1) Vertical resolutionRO has much higher vertical resolution than

other satellite data, global analyses, or opera-tional radiosonde data when reported only atstandard levels. Careful attention must be paidto comparisons of RO data with data of lowerresolution. In particular, one should avoid in-terpolation of the lower-resolution data to thelevels of the higher-resolution RO data. A com-parison following such an interpolation wouldproduce small-scale features, which mainly re-flect structures of the higher-resolution ROdata instead of the real differences betweenthe two. Therefore, the down sampling of RO ismore reasonable in such a comparison. How-ever, the down sampling can introduce anothererror—aliasing. By applying a filter, we canmake the resolution of RO compatible with thatof other data without introducing aliasing. Thisis described next.

2) Application of a filterWe first normalize the RO refractivity at

each height level with a corresponding refer-ence value (based on CIRAþQ climatology).We then interpolate the normalized refrac-tivity profile to a dense regular height grid(at 10 m resolution). On this height grid, weremove small-scale features selectively by ap-plying a spatial filter. The filter used in thisstudy is a low-pass filter combined with aDolph-Chebyshev window to prevent high-frequency oscillations (e.g., Gibbs oscillations;Lynch 1997). Since the typical vertical resolu-tion of NWP analyses is @1 km, this is chosenas the cut-off length of the filter. The filter re-moves only the small-scale components, whichhave vertical scales shorter than 1 km, withoutaffecting larger-scale structures. We then in-terpolate the filtered profile back to the originalheight levels. The filtered RO refractivity pro-file, which is devoid of small-scale variations, isused for comparison with other sources of data.

3) Vertical coordinateGPS RO provides refractivity as the function

of height. Consequently, it is more natural tocompare RO refractivity with other types ofdata on height coordinates. Refractivity is afunction of temperature, water vapor pressure,and pressure. The relationship is valid only forheight coordinates and sometimes that tendsto be a source of confusion, especially whenone compares two data sets on pressure coor-dinates. The refractivity difference of the twodata at the same pressure surface is not thesame as a comparison of refractivity on thesame altitude, because the height of the con-stant pressure surface may not be the samefor these two data sets. In this study, we per-form the refractivity comparison on the alti-tudes of the constant pressure surfaces of globalanalyses (models).

4) SmearingThe ray perigee point of a RO sounding

varies with height, while the altitude of aconstant pressure surface for a global analysisvaries horizontally. For a comparison of ROwith global analysis, we need to identify theintersection point between a plane of pressurelevel of the global analysis and the RO sound-ing. Since the RO sounding does not carrypressure information, this intersectional point


is not readily available. In this study, this isdone through an iterative process, by search-ing up or down along the RO sounding whilerepeatedly applying horizontal interpolation tothe analysis. Because the resolution of ROsounding is very high, the intersection pointcan be determined precisely.

5) Removal of outliersAs discussed earlier, RO soundings are sub-

ject to measurement errors and noise. It isfeasible to remove statistically suspicious data(outliers) before comparison with global anal-yses. For this purpose, we first split the entiredata sample (8,500 soundings over one month)into latitude bins with the size of 10�. In eachlatitude bin, we calculate the fractional devia-tion of retrieved RO refractivity from the corre-sponding NWP analyses at each pressure levelof these analyses. At each level, we take aGPS occultation as a sample and treat the re-maining occultations as its mother group. Withthe particular occultation excluded, we calcu-late the mean value and standard deviation ofthe mother group. Assuming that the fractionaldeviations follow a Gaussian distribution, wecan determine whether the sample belongs tothe mother group or not. If a particular RO ob-servation falls outside the 1% significance level,the particular sample is considered an outlierand excluded from the dataset. This procedureis repeated until no additional outliers are de-tected. With the recognition that observationalerrors of RO are vertically correlated, we rejectthe whole sounding if the occultation fails topass the outlier test at any level. We performthe same outlier test using both the NCEPAVN and ECMWF analyses and accept only theoccultations that pass the outlier test for bothglobal analyses. About 6500 occultations outof 8500 original occultations passed the outliertest. The benefit of this approach is that theanalysis errors and RO errors have been im-plicitly considered in the outlier test.

For radiosonde observations, a similar testis performed. However, a loose quality controlfor the parameters of temperature, dew-pointtemperature, and geopotential height is appliedprior to the outlier test for refractivity. Assum-ing the radiosonde observations are verticallyuncorrelated, only the particular outlier at theparticular level that fails the test is removed,

and the data at other levels are kept in the dataset. About 80% of radiosonde observations involume passed the routine quality control pro-cedure and outlier test.

Appendix C

Interpretation of refractivity differences(or errors) in terms of temperature and

water vapor differences (or errors)

Fractional errors (or differences) in refrac-tivity can be related to corresponding errors(or differences) in temperature and/or watervapor pressure for interpretation and compari-son with errors from observing systems thatmeasure temperature and water vapor directly.In general, errors in water vapor pressure con-tribute most to errors in refractivity in moistregions, such as the lower tropical troposphere,while temperature errors contribute most ofthe refractivity error in drier areas such aspolar regions, upper troposphere and strato-sphere. From the equation for the refractivityas a function of temperature and water vaporpressure, one can estimate the order of tem-perature or water vapor pressure error for agiven fractional error in refractivity, assumingthat the other variable is correct. Throughoutmost of the troposphere and stratosphere, a1% error in refractivity corresponds to about a2 K error in temperature (assuming no error inwater vapor). Under the alternate assumptionof no error in temperature, a 1% error in N cor-responds to a maximum error of about 1 mb inwater vapor pressure, and this occurs in theextremely moist tropics near the surface. Theerror in water vapor pressure drops off rapidlywith height; for example, in a standard atmo-sphere the error is approximately 0.35 mb at5 km, 0.15 mb at 10 km, and 0.03 mb at 20 km.(These numbers assume 100% relative humid-ity to estimate the maximum effect.)

In our study we find that errors in ROrefractivity reach 3–4% near the surface inthe moist tropics and average @0.3% to 0.5%over much of the rest of the troposphere andstratosphere. In the tropics, the temperaturethroughout the lower troposphere is fairly uni-form, but the water vapor pressure shows greatvariability (Fig. 9c). A 3% error in refractivitycorresponds to about a 3 mb error in watervapor pressure in the tropics, and it is water


vapor that contributes most to the refractivityerror here. Because the saturation vapor pres-sure is approximately 30 mb, this error pro-duces an error in relative humidity of about10% in the lower tropical troposphere. In the

upper troposphere and lower stratospherewhere water vapor is negligible, most of the re-fractivity error is related to temperature; herea 0.5% error in refractivity corresponds to anerror in temperature of about 1 K.


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Inversion and Error Estimation of GPS Radio Occultation Data