Long GPS coordinate time series Multipath and geometry effects

Surveying and Spatial Science Group, School of Geography andClickHereforFullArticleJOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, B04403, doi:10.1029/2009JB006543, 2010Long GPS coordinate time series: Multipath and geometry effectsMatt A. King1 and Christopher S. Watson2Received 16 April 2009; revised 1 October 2009; accepted 21 October 2009; published 3 April 2010.1 Within analyses of Global Positioning System (GPS) observations, unmodeledsubdaily signals propagate into long period signals via a number of different mechanisms.In this paper, we investigate the effects of time variable satellite geometry andthe propagation of a time constant unmodeled multipath signal. Multipath reflectors atH = 0.1 m, 0.2 m, and 1.5 m below the antenna are modeled, and their effects on GPScoordinate time series are examined. Simulated time series at 20 global IGS sites for2000.02008.0 were derived using the satellite geometry as defined by daily broadcastorbits. We observe the introduction of time variable biases in the time series of up toseveral millimeters. The frequency and magnitude of the signal is dependent on sitelocation and multipath source. When adopting realistic GPS observation geometriesobtained from real data (e.g., including the influence of local obstructions and hardwarespecific tracking), we observe generally larger levels of coordinate variation. In thesecases, we observe spurious signals across the frequency domain, including very highfrequency abrupt changes (offsets) in addition to secular trends. Velocity biases of morethan 0.5 mm/yr are evident at some sites. The propagated signal has noise characteristicsthat fall between flicker and random walk and shows spectral peaks at harmonics ofthe draconitic year for a GPS satellite (351 days). When a perfectly repeating syntheticconstellation is used, the simulations show near negligible time correlated noisehighlighting that subtle variations in the GPS constellation can propagate multipathsignals differently over time, producing significant temporal variations in time series.Citation: King, M. A., and C. S. Watson (2010), Long GPS coordinate time series: Multipath and geometry effects, J. Geophys.Res., 115, B04403, doi:10.1029/2009JB006543.1. Introduction2 Geophysical interpretation of GPS coordinate timeseries most commonly involves the determination of rates ofcrustal motion and amplitudes and phases of periodicmotions caused by seasonal mass loads (for example,atmospheric, hydrological, or oceanic) Dong et al., 2002;van Dam and Wahr, 1998. Each geophysical parameter ofinterest derived from long GPS time series is biased at somelevel by residual systematic error, particular those thatmanifest as long period spurious trends or (quasi) periodicsignals. In the case of GPS, these systematic errors and theirpropagation into GPS time series are not yet all wellunderstood. Recent literature has highlighted that longperiod systematic errors may occur in coordinate time seriesdue to one of two primary mechanisms.3 First, spurious signals may occur directly due tounmodeled long period signals, such as satellite antennamodeling errors that propagate differently as the satelliteconstellation changes Ge et al., 2005. The second origi-nates in the presumption that all subdaily signals are mod-eled at the observation level, either completely within thefunctional model or through partial mitigation from the sto-chastic model (e.g., elevation dependent weighting). How-ever, this is not yet the case and residual subdaily (systematic)errors remain which have been shown to propagate into timeseries e.g., Penna et al., 2007 due to a combination of dif-ferent mechanisms e.g., Stewart et al., 2005.4 One well studied example of how subdaily signalspropagate into longer period signals is the case of an un-modeled subdaily tidal signal. Unmodeled semidiurnal anddiurnal signals at certain tidal periods have been shown topropagate into fortnightly, semiannual and annual periodsPenna and Stewart, 2003; Stewart et al., 2005 with admit-tances exceeding 100% in some cases Penna et al., 2007.The propagated signal was shown to be highly sensitiveto input signal frequency, coordinate component of the un-modeled signal and site location. King et al. 2008 showedthat, even after modeling for solid earth tides and ocean tide1School of Civil Engineering and Geosciences, Newcastle University, loading displacements, substantial signal at subdaily periodsNewcastle upon Tyne, UK.2Environmental Studies, University of Tasmania, Hobart, Tasmania,Australia.Copyright 2010 by the American Geophysical Union.0148 0227/10/2009JB006543remained in GPS coordinate time series and these propagateinto annual and semiannual periods in 24 h solutions withmedian amplitudes of 0.5mm, but reaching several milli-meters at several sites. As indicated, studies of seasonalgeophysical loading phenomena e.g., Blewitt et al., 2001;B04403 1 of 23B04403 KING AND WATSON: MULTIPATH AND GEOMETRY EFFECTS ON GPS B04403Wu et al., 2003 are therefore adversely affected, as are esti-mates of linear velocity, by the presence of such systematicerrors.5 Systematic errors in least squares solutions, such asused in GPS data analysis, propagate according to the leastsquares design matrix. Therefore, the propagation mecha-nism is controlled by the observation geometry (as definedby the receiver location(s), satellite constellation and localobstructions) plus the chosen parameterization of the solu-tion. Parameters typically include (but are not limited to)site coordinates, adjustments to tropospheric zenith delay,atmospheric gradients, phase ambiguity terms (when notfixed to integers) and receiver clocks (depending on the datadifferencing approach adopted). Temporal variations in theobservation geometry or the number or type of parametersestimated will therefore likely produce temporal variationsin the propagated signal.6 It is notable, therefore, that even in the trivial yetunrealistic case of the GPS antenna location and number ofestimated parameters remaining constant with time, the GPSsatellite constellation is constantly evolving as satellites arecommissioned and decommissioned, or removed from thesolution due to satellite eclipse or maneuver. Furthermore,site specific obstructions such as vegetation or man madestructures may change with time, producing a further changein the observation geometry. The consequence is that, evenif an unmodeled signal remains completely constant in time,the way in which it will propagate is likely to change withtime. If these temporal variations are suitably large, then theensuing systematic error will likely bias the GPS time seriessignificantly, resulting in erroneous interpretation of geo-physical signals such as tectonic velocity, glacial isostaticadjustment, vertical motion of tide gauges or seasonalgeophysical loading signals. Furthermore, such errors woulddegrade the GPS contribution to the International TerrestrialReference Frame Altamimi et al., 2007.7 In this paper we test this hypothesis, using one sourceof presently unmodeled subdaily signal: carrier phase mul-tipath, taking “multipath” to mean signal reflections fromplanar surfaces not part of the antenna itself, as adopted byGeorgiadou and Kleusberg 1988. Errors of this type aresimilar in nature to antenna phase center variation mis-modeling and antenna imaging (changes in the antennaphase pattern induced by conducting material in the vicinityof the antenna) Georgiadou and Kleusberg, 1988, in thatthey exhibit an elevation dependency. We do not considerthese additional sources of systematic error here, yet simplynote that the propagation mechanism is likely to be some-what similar. Early studies investigating multipath foundvery small effects on geodetic time series, leading to thethought that multipath effects are mitigated by averagingwhen sufficient observational time spans are used e.g.,Davis et al., 1989; Lau and Cross, 2007a. However, yearsof experience with continuous GPS time series, combinedwith analysis advances leading to improved signal to noiseratio of long time series, have shown that GPS time seriesare highly sensitive to GPS hardware changes (includingreceiver, receiver firmware and antenna), suggesting thatmultipath may be playing an important role; yet the mech-anism for this is not well understood. As a contribution tounderstanding the effect of carrier phase multipath onmultiyear GPS coordinate time series, we perform severaltrials using simulated and real data, using various simulatedcarrier phase multipath signals.8 We commence in section 2 by introducing the adoptedmodel of carrier phase multipath that we use throughout thispaper to perturb a multiyear, simulated GPS coordinate timeseries. The simulation approach is introduced (section 2.2)before detailing the different satellite constellation config-urations adopted to assess the different characteristics of themultipath propagation (section 2.3). First, in order to showthe influence of time variability on the propagation, we startwith a theoretical constellation that has a fixed orbit repeattime, a constant number of satellites and no obstructionsabove the elevation mask at each site (section 2.3.1).Second, we adopt the same clear horizon but with a realistictime variable satellite constellation taken from the broadcastorbits, (section 2.3.2). Finally, we detail the most realisticconstellation that includes site specific time variablechanges to the observation tracking as observed in real data(for example hardware changes and physical obstructions onthe horizon, section 2.3.3).9 In section 3 we compare and discuss the simulatedtime series generated using the three constellation config-urations, in addition to investigating the effect of changesto the adopted functional and stochastic models withinthe evolving constellation scenario. In section 4, we pro-vide a comparison against time series computed using realdata in a PPP approach with GIPSY 5 software Webb andZumberge, 1995, and in section 5 we present two possiblemitigation strategies that involve novel weighting strategiesof the input observations in order to minimize the time andgeometry dependent propagation of the multipath signal.2. Simulations2.1. Signal Multipath10 The space surrounding an antenna may be subdividedinto three regions including the reactive near field in theregion nearest the antenna, the radiating near field and thefar field out to infinity e.g., Balanis, 2005. The boundariesof these regions are not sharply defined although criteriahave been developed in order to delineate them in practice.Given an antenna with maximum dimension D, and signalwith wavelength l, and for D l Balanis, 2005 definesthe first and second boundaries occurring at distancesr D2 2D2R1 0:62 and R2 from the antenna surface, respectively. For a GPS choke ringantenna with D = 0.38 m, lL1 = 0.19 m and lL2 = 0.24 m,then R1L1 = 0.03 m, R1L2 = 0.02 m, R2L1 = 1.52 m andR2L2 = 1.20 m. In this paper we consider multipath sourcessolely within the radiating near field, or distances in the range0.03 m to 1.21.5 m. For an examination of the effect ofreactive near field sources on GPS time series, see Dilssneret al. 2008, and for the effect of phase center modelingerrors on site velocities, see Steigenberger et al. 2009.11 To date there is no widely accepted model for multi-path, partly due to the complexity of real world GPS antennaenvironments. One relatively simple model shown to approx-imate observed multipath has been described by Elosegui2 of 232 L “;B04403 KING AND WATSON: MULTIPATH AND GEOMETRY EFFECTS ON GPS B04403Figure 1. Carrier phase bias due to reflectors using two different multipath models. (left) The Eloseguiet al. 1995 model with two different attenuation values, a, at H = 0.1 m (gray), H = 0.2 m (blue), andH = 1.5 m (magenta), sampled every 1. (right) The HMM for the same values of H and with twodifferent surface reflectivity terms, S, sampled every 1. Np is the refractive index of the reflective medium.The dotted black line (bottom right) is for S = 1.0 for H = 0.16 m.et al. 1995, based on the earlier work of Georgiadou andKleusberg 1988 and Young et al. 1985. This model isbased on the assumption that multipath is caused solely bya horizontal reflector at some height, H, below the GPSantenna phase center causing an attenuation, a, of the signalvoltage amplitude. So, for a satellite with elevation angle, “,phase bias d L (in units of meters) of the L1 or L2 phase dueto multipath may be modeled as Elosegui et al., 1995approximately uniformly scaling the bias as seen by com-paring Figure 1 (left).13 Despite this model providing a useful approximation,it is based on geometric ray optics which is not appropriatein the radiating near field. In addition, values of d LC willbe modified by the antenna gain pattern. To improve on this,we adopt a model as developed by T. A. Herring (personalcommunication, 2009, hereafter denoted HMM) that extendsL 0BBtan 11 Hsin 4cos 4sin “Hsin “1CCA 1the Elosegui et al. 1995 model through the use of Fresnelequations relating to electric field amplitudes, and attemptingto take into account antenna gain properties:!1 a sin 4 H sin “L tan 32 gd a cos 4 H sin “where l is the carrier phase wavelength for the L1 or L2carrier phase signal. Since most geodetic GPS positioningis performed using the ionosphere free linear combination(LC) of the raw carrier phase observables (L1 and L2), theLC phase delay can be computed aswith the antenna gain pattern, consisting of the direct (gd) andreflected (gr) gain, modeled using a simple modified dipolemodel expressed as a function of rate of change (G) of theantenna gain with signal zenith angle, such thatgd cosz=GLC ; H 2:54571:5457L“; ; H;L“; ; H;L1L22gr cos90=G1 sin“12 We show in Figure 1 (left) d LC for a range of heightsabove the reflector (H = 0.1, 0.2, and 1.5 m) and for twodifferent attenuation values (a = 0.05 and 0.1). As noted byElosegui et al. 1995, the effect of changing the heightabove the reflector is to change the rate at which d LC varieswith elevation (increased rate of variation with increasingheight). Within a reasonable range of attenuation values(0.01 to 0.1), a change in attenuation has the effect ofalso, a = SgrRa is the amplitude of the reflected signal for agiven surface roughness (S), with2 q 36n1 cos z n22 n1 sin z27Ra 4 q 5n1 cos z n22 n1 sin z2being the Fresnel equation for an electric field perpendicularto the plane of incidence.3 of 23B04403 KING AND WATSON: MULTIPATH AND GEOMETRY EFFECTS ON GPS B04403Figure 2. Site locations from the IGS network used in this study.14 This depends on refractive indices n1 and n2, withn1 = 1 for air and n2 appropriate for the reflective medium.In practice this can be derived from the square root oftabulated dielectric constants, and for concrete this is typi-cally taken as 4 (n2 = 2).15 Here d LC may be formed analogously toequation (2).16 For the purposes of this paper we adopt values whichapproximate the amplitude of the signal in GPS phaseresiduals (T.A. Herring, personal communication, 2009),namely S = 0.3, G = 1.1 and n2 = 2, although these are notdefinitive, and indeed some sites may require a differentvalue of S, as we shall demonstrate. We show in Figure 1(right) d LC for S = 0.3 and 0.5 based on HMM. The ampli-tude of the modeled signals scales approximately linearlywith variations in S. Increasing either G to 1.2 or n2 to 100result in increased model signal amplitude of about 50%.Only small changes in frequency or phase occur. Two dif-ferences to the model of Elosegui et al. 1995 can beidentified. First, the signal in HMM is substantially reducedat high elevations, which is in general agreement with thepattern seen in GPS carrier phase residuals. Second, bothamplitude and frequency of the HMM is proportional to H,whereas the Elosegui et al. 1995 model does not show thesame sensitivity of amplitude to variations in H (compareFigure 1, left versus right).2.2. Simulation Approach17 Our GPS simulator is based on the undifferenced GPSobservable, and is conceptually similar to that of Santerre1991. Observation level biases, such as d LC, are enteredvia the “observed minus computed” term (b) of the batch leastsquares adjustment:parameters is reflected in the estimated values x. W is theinverse of the observation variance covariance matrix.19 Initial station coordinates are taken from the ITRF2005coordinate set and satellite positions are taken from one ofthree orbit configurations (section 2.3). Satellite orbits andclocks are assumed known and fixed. Unless otherwisespecified below, estimated parameters are the correctionsto initial station coordinates, tropospheric zenith delays,receiver clocks (normally every epoch) and real valuedambiguity terms (normally one per satellite pass). In thesesimulations, nonzero values of the estimated parametersrepresent parameter bias. Correct integer ambiguity fixingmay be simulated by simply removing these terms from thedesign matrix Santerre, 1991.20 This simulator has previously been used by Kinget al. 2003 to study propagation of tidal signals in sub-daily coordinate estimates; the observed biases in outputtime series were accurately reproduced in the simulator. Wehave also verified the simulator by reproducing the propa-gated periodic biases observed in the GIPSY solutions ofPenna et al. 2007 to within very small errors. We thereforeassume the simulator is capable of reproducing the effectsof systematic errors on real GPS solutions (that use anundifferenced observation strategy), but with the advantageof controlling all systematic error sources.21 For the tests described here, we used a typical 24 h“observation” session and estimated adjustments to topo-centric sta