spatial and temporal correlations of the gps estimation...

Spatial and temporal correlations

of the GPS estimation errors

Borys Stoew and Gunnar Elgered

Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden

OnsalaMarch 16, 2005

This report summarizes the work carried out within Work Package 3000 in the TOUGHProject. It contains the final versions of Delivery D19 on spatial correlation of errors andDelivery D22 on temporal correlation of errors.

TOUGH (Targeting Optimal Use of GPS Humidity Measurements in Meteorology) is aproject supported by the 5th Framework Programme of the European Commission.

Summary

We have studied the spatial correlation structure of the estimation errors when the GPStechnique is used to derive the atmospheric propagation delay in the zenith direction.These error correlations are parameterized using an analytical function. For spatial scalessmaller than 200 km the GPS error correlations become significant; for larger spatialscales, the error correlations are small and slowly decreasing with distance. These resultsare based on a study of the differences between the propagation delays obtained from anumerical weather model, and the corresponding GPS estimates.

The temporal correlations of the propagation delay estimation errors are also exam-ined and the decorrelation times are of the order of 1–2 days. The temporal correlationresults are derived for the Swedish and Finnish GPS sites, using the relation between theestimation errors for the vertical coordinate and the zenith propagation delay.

Contents

1 Introduction 1

2 Measuring and estimating the ZTD 1

2.1 Radiosonde data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 GPS data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Microwave radiometer data . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Extracting ZTD time series from HIRLAM 4

3.1 The non-linear observation operator for the ZWD . . . . . . . . . . . . . . 43.2 The tangent-linear observation operator . . . . . . . . . . . . . . . . . . . 53.3 ZTD Innovation vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Spatial correlations using analytic models 5

5 Spatial correlations based on NWP and GPS data 7

6 Spatial correlations using vertical coordinate residuals 12

7 Temporal correlations of slant delay residuals using ray tracing 13

7.1 Hydrostatic and wet mapping functions . . . . . . . . . . . . . . . . . . . 137.2 Ray tracing using radiosonde ascents . . . . . . . . . . . . . . . . . . . . . 14

8 Temporal correlations of ZTD from GPS position residuals 16

9 Discussion 18

References 19

ii

1 Introduction

In short term weather forecasting, various sources of data describing the atmosphericsystems and processes are used. The measurement techniques providing informationabout the atmospheric parameters vary not only in sampling rate and spatial distribution,but also in accuracy. The potential of using atmospheric delays estimated from GPS datain near real time and the high temporal/spatial resolution of the measurements make theGPS data assimilation worthwhile (Gutman et al., 2004).

Traditionally, the errors in the radiosonde measurements of humidity are assumed tohave no spatial correlation in the numerical weather prediction (NWP) model setup. Spa-tially correlated errors lead to a non-diagonal structure of the error covariance matrices,whose inverting is computationally demanding. The use of a proper error-covariancemodel ensures the positive definiteness of the respective covariance matrices, so that theyhave no eigenvalues near zero and are non-singular. The errors in the estimates of at-mospheric propagation delay, both from GPS processing and from NWP models, exhibitspatial correlation structures which deserve special attention (Berre, 1997; Stoew et al.,

2001).A statistical description of the temporal correlations of errors in the atmospheric delay

estimated from GPS data is needed for the proper operation of an NWP model — in par-ticular, for the weighting of past zenith total delay (ZTD) estimates in an observation biasreduction scheme (Lindskog, 2001; Daley, 1992). The spatial and temporal correlationsof the estimation errors are assessed in the following sections of this report.

Section 2 makes a brief overview of the relations between relevant atmospheric param-eters and the ZTD. We introduce the radiosonde data which we later use to estimatemapping function errors through the method of ray tracing. We also review the estima-tion of the ZTD using GPS and microwave radiometer data. Section 3 describes howthe ZTD time series are obtained from NWP models. Thereafter we focus on the spatial(horizontal) correlation of ZTD errors. In Section 4, we discuss the analytic functionsused to model the horizontal correlations of the GPS estimation errors in the ZTD; weprovide the resulting parameters in Section 5. Using the residuals of the estimated verticalcoordinates, we assess the spatial error correlations in Section 6.

The temporal correlation of ZTD estimation errors is studied in Section 7 where wediscuss two different atmospheric delay mapping functions (MFs). The residual estimatesof the propagation delay, obtained by ray tracing of radiosonde data, are used to estimatethe temporal correlations of the errors contributed by the MFs used in the GPS processing.The mapping function is only one component introducing a temporal correlation of theZTD estimation errors. For example, satellite orbits and clocks also contribute to theeffect. An attempt to estimate the total effect is made in Section 8 which presentstemporal correlations of the ZTD errors based on the assumption that these estimationerrors are strongly correlated with those of the estimated vertical positions of the GPSsites. Section 9 ends the report with a discussion concerning the obtained results.

2 Measuring and estimating the ZTD

The refractivity in an air parcel is often written

N = k1pd

T+ k2

e

T+ k3

e

T 2, (1)

where the parameters k1, k2 and k3 are empirically determined from laboratory experi-ments, T is the absolute temperature, e represents the water vapour pressure, and pd isthe sum of the partial pressures of the “dry” gases. The refractivity is usually split into

1

a hydrostatic and a non-hydrostatic (wet) part:

N = Nh +Nw . (2)

Accordingly, the ZTD can be presented as a sum of a zenith hydrostatic delay (ZHD) anda zenith wet delay (ZWD):

` = `h + `w . (3)

The ZHD is due to the refractivity associated with the gasses in hydrostatic equilibrium,obeying the hydrostatic equation for an ideal atmosphere:

dp

dz= −ρg . (4)

Above, p is the partial pressure of the gas at an altitude z, ρ is the gas density, and g

is the gravitational acceleration. The ZWD in (3) is mostly due to water vapour (e.g.,Bevis et al., 1992).Davis et al. (1985) proposed an approximate formula for estimating the ZHD at a

given site with latitude θ in degrees and height ho in km above the ellipsoid from pressuremeasurements:

`h = (2.2768± 0.0024)P

f(θ, ho), (5)

where `h is in mm, the ground pressure P is in hPa, and

f(θ, ho) = 1− 0.00266 cos(2θ)− 0.00028ho . (6)

The amount of water vapour expressed as the height of the column equivalent liquidwater is termed integrated precipitable water vapour (IPWV) and is defined as

I =1

ρl

∫ ∞

ho

ρv dz , (7)

where ρl and ρv are the densities of liquid water and water vapour. The IPWV is typicallymeasured in mm. The quotient

Q =`w

I, (8)

where both `w and I are in mm, has been described by Askne and Nordius (1987):

Q = 10−8ρl Rv

(

k′2 +k3

Tm

)

. (9)

Above, Rv = 461.5 J kg−1K−1 is the specific gas constant for water vapour; we assumek′2 ≈ 22 K/hPa and k3 ≈ 3.7·105K2/hPa after Emardson and Derks (2000); the parameterTm is defined as the mean temperature of the water vapour:

Tm =

∫ ∞

ho

ρv dz

∫ ∞

ho

ρv

Tdz

. (10)

Based on radiosonde profiles acquired over the US, Bevis et al. (1992) proposed the linearregression Tm ≈ 70.2 + 0.72To , where To is the surface temperature in Kelvin. Otherrelations for Europe are reported by e.g. Emardson and Derks (2000).

The ZTD time series are dominated by the temporal variation of the ZWD, even thoughthe absolute values of the ZHD are about an order of magnitude larger those of the ZWD.From the viewpoint of numerical weather modelling, the ZTD can easily be calculatedusing integration over the different levels of the atmospheric model. Conversely, the ZTDestimates based on the GPS measurements carry information about atmospheric pressureand humidity. As the current amount of humidity measurements assimilated into NWPmodels is insufficient for many applications, the GPS estimates of the ZTD are likely tobecome valuable for the weather forecasting community.

2

2.1 Radiosonde data

Balloons carrying measurement sensors can be used to acquire vertical profiles of temper-ature, pressure, humidity, and wind. The regular launching of balloon-borne radioson-des (RS) is costly; this limitation has an impact on the amount of the acquired data andon their quality. Recent studies over large regions of the northern hemisphere have shownthat a low RS launch frequency (e.g. only twice a day) may introduce bias of up to 3%to the estimates of average amount of precipitable water (Dai et al., 2002).

The data obtained from the atmospheric soundings are often reduced to profiles con-taining the values at a set of standard pressure levels (surface, 1000, 850, 700, 500,400, 300 hPa), along with the levels representing significant changes in the observed at-mospheric parameters. The radiosondes used now in the Nordic countries are of theVaisala RS-80 type. Recent investigations on the performance of the Vaisala RS-80 rel-ative humidity (RH) sensors suggest an age-related contamination by the packaging ofthe sondes (Niell et al., 2001; Wang et al., 2002). As the sensors typically underestimatethe actual RH values by as much as 5% in very humid conditions, correction algorithmsshould be applied (Lesht, 1999; Wade and Schwartz, 1993).

The vertical profiles of RH, and the air temperature and pressure can be used to derivethe partial pressure of water vapour and subsequently the wet refractivity. The RHexpressed in percent is defined as:

U% =e

esw100 , (11)

where e is the partial pressure of water vapour; esw is the saturation vapour pressureover water surface, and characterizes the conditions for a given volume and temperaturewhen water can be found in equilibrium in both its liquid and gaseous phases. Empir-ical formulas for calculating esw(T ) have been derived by fitting a model to tabulatedmeasurements (e.g. Crane, 1976; List, 1958). Crane (1976) introduced the model

esw(T ) = 6.105 exp

[

25.22

(

T − 273

T

)

− 5.31 ln

(

T

273

)]

(12)

using tabulated values of esw in hPa and T in Kelvin degrees. The model has an uncer-tainty within ±0.4% over the temperature range from 243K to 303K (−30◦C to +30◦C).Note that the partial pressure of water vapour equals the saturation pressure at the dewpoint temperature Td, i.e.

e = esw(Td) . (13)

2.2 GPS data

GPS is a navigation system using several satellite-borne radio transmitters. At present,the constellation consists nominally of 24 active satellites in 6 orbital planes, allowingfor a simultaneous visibility of at least 4 satellites for any receiver on the surface of theearth (Hofmann-Wellenhoff et al., 1997). For the purposes of meteorology (water vapourin the atmosphere), high measurement accuracy is achieved by using state of the art GPSreceivers capable of measuring the carrier phase of the received signal (Blewitt, 1993).

Information about the moisture of the atmosphere can be extracted from the ZTDfields, using a network of GPS receivers. The data processing can be carried out with theGIPSY-OASIS software (Webb and Zumberge, 1993) using the Precise Point Positioning

solution (Zumberge et al., 1997). This processing is now routinely performed at theanalysis centre at the Onsala Space Observatory for a receiver network (Fig. 1) (e.g.,Emardson et al., 1998).

3

10˚E

20˚E

60˚N

65˚N

70˚N

ARJE

BORA

GOTE

HASS

JONK

KARL

KIRU

LEKS

LOVO

MALM

MART

NORR

ONSA OSKA

OSTE

OVER

SKEL

SUNDSVEG

UMEA

VANE

VILH

VISB

JOEN

KEVO

KUUS

METS

OLKI

OULU

ROMU

SODA

TUOR

VAAS

VAST

VIRO

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JOEN

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ONSA OSKA

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OVER

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JOEN

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ONSA OSKA

OSTE

OVER

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UMEA

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JOEN

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METS

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VIRO

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HASS

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ONSA OSKA

OSTE

OVER

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SUNDSVEG

UMEA

VANE

VILH

VISB

JOEN

KEVO

KUUS

METS

OLKI

OULU

ROMU

SODA

TUOR

VAAS

VAST

VIRO

Fig. 1. Locations of the GPS sites in Sweden and Finland.

2.3 Microwave radiometer data

A microwave radiometer is used at the Onsala Space Observatory for independent as-sessment of atmospheric water vapour variability in general and the ZWD in particular.In the following, it is referred to as a Water Vapour Radiometer (WVR). It operates at21.0 and 31.4GHz measuring the sky brightness temperatures with a long-term repeata-bility (order of years) of approximately 1K. This corresponds to an absolute uncertaintyin the ZWD of 7mm; the repeatability of the measured brightness temperatures becomesbetter for shorter time scales. Typically, RMS differences obtained when WVR data arecompared to other independent measurements, such as radiosondes, are of the same order.More details on the performance of the WVR have been documented elsewhere (Elgeredand Jarlemark, 1998).

3 Extracting ZTD time series from HIRLAM

3.1 The non-linear observation operator for the ZWD

The non-linear observation operator H(·) is used to project the background state ontothe observation space. The background IPWV estimate in the current HIRLAM versionis derived from (7):

I ≈1

ρl gs(θ)

N∑

i=1

qi (Pi+1/2 − Pi−1/2) , (14)

where gs(θ) is the gravity acceleration as a function of latitude of the site; qi is thebackground specific humidity at model level i; N denotes the number of model levels. The

4

background pressure Pi±1/2 is evaluated at the model layer boundaries, and approximatesthe partial pressure of the dry gases in the hydrostatic equation; the other parametersare evaluated at the center of each grid cell. The ZWD part of the operator H(·) is basedon (9), and is defined by the numerically calculated IPWV for the model levels:

`w ≈10−8 Rv

gs(θ)

N∑

i=1

qi

(

k′2 +k3

Ti

)

(Pi+1/2 − Pi−1/2) , (15)

where Ti is the background temperature at model level i.

3.2 The tangent-linear observation operator

The tangent-linear form H is the derivative of H(·) with respect to the state x of thenumerical weather model, evaluated at the background xb:

H = ∇x H(x) |x=xb . (16)

The linear form of the observation operator for the IPWV over a single site, based on(14), is:

I ≈1

ρl gs(θ)

[

N∑

i=1

qi (Pi+1/2 − Pi−1/2) +N∑

i=1

qi (Pi+1/2 − Pi−1/2)

]

, (17)

where the tangent-linear pressure and specific humidity are denoted Pi±1/2 and qi, re-spectively, and are the derivatives of the corresponding parameters at the backgroundstate.

3.3 ZTD Innovation vector

ZTD estimates for one year were acquired, starting with 1 May 2000, at a samplingrate of 5 minutes, acquired using 35 receivers on the territory of Sweden and Finland.These time series were subsequently decimated at 6-hour sampling intervals in order tomatch the HIRLAM forecast runs. These forecast runs were performed off-line at theSwedish Meteorological and Hydrological Institute, and covered the same period. Forthe calculation of the high resolution short term forecasts the HIRLAM 3DVAR wasused (Kallen, 1996; Gustafsson et al., 2001; Lindskog et al., 2001).

The differences between the short term ZTD forecast fields and the estimates of ZTDfrom GPS (the innovations) are shown in Fig. 2. Note that the HIRLAM–GPS innovationtime series (`H− `G) appear to have higher variability in the summer. For all the GPSsites (Fig. 1) there is a persistent over estimation of ZTD by the GPS technique of about1 cm. When a rectangular “sliding window” was applied to the differenced time series foreach of the GPS sites, there was no apparent seasonal dependence in the resulting slowlyvarying bias between the HIRLAM and the GPS ZTDs. We do not correct the innovationdata for this time-varying component, and we only remove the mean difference when thehorizontal correlations of the innovations were calculated for each pair of GPS sites.

4 Spatial correlations using analytic models

When modelling atmospheric parameters, it is a common practice to assume horizontallyhomogeneous, isotropic random field, locally time-stationary over a given period. Ourapproach to spatial correlation modelling is based on the statistical analysis of turbulentatmosphere (Tatarskii, 1971). We are interested in the spatial structure of the correlation

5

0 50 100 150 200 250 300 350 400−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Days since May 1, 2000

HIR

LAM

−G

PS

inno

vatio

ns in

ZT

D, m

Fig. 2. Time series of the differences between the HIRLAM and GPS zenith total delay estimates (i.e. theinnovations) for the site at Onsala. The corresponding time series of the innovations were produced for all GPSsites, and then used to calculate the correlations of the GPS estimation errors.

coefficient ρ which is the normalized two-dimensional auto-covariance function of a ran-dom field. We use a sampled version (both in time and in space) of this field to calculateρ as a function of the site separation r (e.g., Stoew et al., 2001).

Different authors have proposed various analytical expressions which describe the spa-tial correlation of a random field as a function of distance (Daley, 1991; Tatarskii, 1971).These expressions satisfy the requirement that any proper 2D correlation model shouldyield a non-negative power spectral density (PSD) function. We consider the Gaussian

and the von Karman analytical functions in order to model spatial correlations in polarcoordinates:

%1(r) = a · exp

(

−r2

r21

)

(18)

%2(r) =b

2ν−1Γ(ν)

(

r

r2

)ν

Kν

(

r

r2

)

(19)

where Kν(x) is the modified Bessel function of the second kind with parameter ν > 0; theparameters a and b are scaling factors; r1, r2 are characteristic lengths for the respectivefunction fitted to the data. Any linear combination of (18) and (19) forms a proper 2Dcorrelation function. To describe 2D spatial correlations, a second-order autoregressivefunction is sometimes used in the literature (e.g., Daley, 1991); it is, however, a specialcase of (19).

For a 2D isotropic field, the Fourier transform is equivalent to a zero-order Hankeltransform (Erdelyi et al., 1954). Thus we can write the normalized power spectral densities

6

corresponding to (18) and (19):

g1(κ)

g1(0)= exp

(

−κ2r214

)

(20)

g2(κ)

g2(0)=

1(

1 + κ2r22)ν+1 (21)

where κ corresponds to a set of 2D spatial wave number pairs (k, l), for which κ2 = k2+l2.This approach has its limitations which stem from the ignoring of the curvature of

the earth surface. Hollingsworth and Lonnberg (1986) described the isotropic componentof model-prediction error covariances using finite sums of Bessel functions, taking intoaccount the spherical harmonic expansion of the field. We refrain from using their repre-sentation of spatial correlations, as it yields large variations of the correlation model forbaselines where no data are available (Stoew, 2001).

5 Spatial correlations based on NWP and GPS data

It is possible to separate the contributions from two estimation error fields, assuming thatthe spatial correlation structure is known for one of them. We denote the ZTD true valuesas ì and `j for the locations i and j; the ZTD estimated using the HIRLAM and GPS are`H

i and `G

i , respectively. The differences eH

i = `H

i − ì and eG

i = `G

i − ì represent the timeseries of the estimation errors for the two methods. The covariance of the differenced timeseries (the innovations) carries information about the spatial structure of the eH and eG

fields:

cov[

`H

i − `G

i , `H

j − `G

j

]

= cov[

`H

i − ì + ì − `G

i , `H

j − `j + `j − `G

j

]

= cov[

eH

i − eG

i , eH

j − eG

j

]

= cov[

eH

i , eH

j

]

+ cov[

eG

i , eG

j

]

, (22)

where in the last line we use the fact that there is no correlation between the HIRLAMand the GPS estimation errors in ZTD (no GPS data are assimilated into HIRLAM).An analytical model of the term cov[eH

i , eH

j ], derived using the NMC method (Parrishand Derber, 1992), is used to fit a curve to the remaining term in order to obtain thehorizontal correlations of the GPS errors.

The NMC method was applied at SMHI for three different set-ups of HIRLAM, namelyfor the 44-, 33-, and 22-km horizontal resolution. The resulting models of the HIRLAMprediction error spatial correlations are presented in Fig.3. The spatial correlations de-pend on the chosen HIRLAM resolution, and as the model resolution increases, the corre-lation spatial scales become smaller (i.e., the correlation curves become steeper). In thesame plot we present also the spatial correlations derived using the ensemble assimila-tion method for a 44-km set-up of HIRLAM. The parameter values for the correspondingmodels, %H(r), are summarized in Table 1.

The spatial correlations for the innovations `H− `G are shown in Fig. 4. These correla-tions can be expressed by the normalized pairwise covariances:

%i,j(r) =

cov[

`H

i − `G

i , `H

j − `G

j

]

σi σj, (23)

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0 200 400 600 800 1000 1200 1400 1600 1800−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Baseline length, km

Cor

rela

tion

coef

ficie

nts

Analytic function fit (NMC/44km)Analytic function fit (NMC/33km)Analytic function fit (NMC/22km)Analytic function fit (ENS/44km)

Fig. 3. Analytic functions modelling the horizontal correlations of forecast errors for ZTD calculated for HIRLAM.The solid line presents the model of the forecast error correlations, estimated using the NMC method for a 44-kmresolution NWP model. For the 33-km and the 22-km resolution model, the NMC method reveal even smallercorrelation scales (# and 5, respectively). The results from the use of the ensemble assimilation method appliedto the 44-km resolution version of HIRLAM are also shown (×).

where r is the site separation; σi, σj are the standard deviations of the innovation timeseries at sites i and j, respectively. They are of the order of 1.1–1.3 cm, and decreaseslightly with site latitude and height. Note that because spatially uncorrelated estima-tion/forecast errors contribute to the standard deviation of the innovations, %

i,j(0) ≤ 1 .

A composite model, based on a linear combination of three von Karman curves (19),

Table 1. Parameters of the composite von Karman correlation models (see text). The values of the parameters νand r2 for the %H(r) component were estimated using the NMC method at a horizontal resolution of 44, 33, and22 km. The ensemble assimilation method was used to derive the parameters in the fourth group in the table.

ParameterModel ai ν r2

%H(r) 0.27 1.05 192.80 NMC, 44-km resolution%G

1 (r) 0.53 3.33 37.02%G

2 (r) 0.07 2.98 685.33


1 (r) 0.40 2.65 38.01%G

2 (r) 0.14 1.01 714.69


1 (r) 0.42 2.21 42.31%G

2 (r) 0.13 1.01 726.27

%H(r) 0.33 1.82 84.93 Ensemble assimilation,%G

1 (r) 0.41 2.68 36.23 44-km resolution%G

2 (r) 0.16 0.80 717.73

8

was fitted to the correlation values:

%(r) = a1 %G

1(r) + a2 %G

2(r) + a3 %H(r) , (24)

where a1, a2, a3 are scaling factors; %H(r) represents the signature of the HIRLAM ZTDforecast error correlations, and is plotted using the dash-dotted line in Fig. 4. The mainimpact of the HIRLAM forecast errors is for baselines r < 1000 km. The other twovon Karman curves, %G

1(r) and %G

2(r), model the contribution of GPS errors to the inno-

vation correlations (the solid line in Fig. 4). The composite model (24) is presented bythe dashed line, and has 7 degrees of freedom: three for each %G

1(r) and %G

2(r), and one

for %H(r). The parameter values for the model are shown in Table 1.The slowly declining spatial correlations for baselines r > 400 km in the same graph

would have to be attributed to large scale spatial correlations in GPS estimation errors.This behavior is in agreement with earlier simulation results (Jarlemark et al., 2001).However, for small separations, the results depend strongly on the correctness of themodel %H(r), and on the availability of ZTD data from nearby GPS sites.

For the short baseline separations, the ZTD error correlations derived using the NMCmethod decline at a slower rate compared to the HIRLAM–GPS difference data. Thisresult indicates that, in reality, the forecast error correlations may have shorter lengthscales than those estimated by Berre (1997). It is further supported by the notion thatthese correlations are calculated for the +6 hour forecasts, utilizing differences betweenforecasts valid at the same time, but generated at different ranges (+24 and +48 hoursrespectively). We speculate that the spatial scales of the forecast errors should decreaseif the forecast range is decreased. These scales also depend on the chosen horizontalresolution of the NWP model, as demonstrated in Figures 4, 5, and 6.

A new method, used by the European Centre for Medium-Range Weather Forecasts,based on ensemble assimilation indicates smaller length scales than those determined bythe NMC technique. The newer ensemble assimilation approach uses sets of randomlyperturbed observations and parameters of the numerical model to derive the error correla-tion structures in the forecasts (Houtekamer et al., 1996). The results from the ensembleassimilation approach are shown in Fig. 7.

9

0 200 400 600 800 1000 1200 1400 1600 1800−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Horizontal correlations of the HIRLAM−GPS ZTD innovations

Site separation, km

Cor

rela

tion

coef

ficie

nts

Innovation correlations Model fit (innovations) Model fit (Hirlam forecasts)Model fit (GPS estimates)

Fig. 4. Horizontal correlations of the HIRLAM–GPS zenith total delay innovations for a one year period (∗).The dash-dotted line represents the contribution of the forecast errors correlations for ZTD, estimated using theNMC method for HIRLAM at a resolution of 44 km. The dashed line shows the best fit to the innovation data of acomposite model of the correlations. The solid line represents the remaining term, corresponding to the correlationsof GPS errors.

0 200 400 600 800 1000 1200 1400 1600 1800−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Site separation, km

Cor

rela

tion

coef

ficie

nts

Innovation correlations Model fit (innovations) Model fit (GPS estimates) Model fit (Hirlam forecasts)

Fig. 5. Horizontal correlations of the ZTD innovations for a one year period (∗). The dash-dotted line representsthe contribution of the forecast errors correlations for ZTD, estimated using the NMC method for HIRLAM at aresolution of 33 km. The dashed line shows the best-fit composite model for the innovation correlations. The solidline represents the remaining term, corresponding to the GPS error correlations.

10

0 200 400 600 800 1000 1200 1400 1600 1800−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Site separation, km

Cor

rela

tion

coef

ficie

nts


Fig. 6. Horizontal correlations of the ZTD innovations for a one year period (∗). The dash-dotted line representsthe contribution of the forecast errors correlations for ZTD, estimated using the NMC method for HIRLAM at aresolution of 22 km. The dashed line shows the best-fit composite model for the innovation correlations. The solidline represents the remaining term, corresponding to the GPS error correlations.

0 200 400 600 800 1000 1200 1400 1600 1800−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Site separation, km

Cor

rela

tion

coef

ficie

nts


Fig. 7. Horizontal correlations of the ZTD innovations for a one year period (∗). The dash-dotted line representsthe contribution of the forecast errors correlations for ZTD, estimated using the ensemble assimilation method.The dashed line shows the best fit to the innovation data of a composite model of the correlations. The solid linerepresents the remaining term, corresponding to the correlations of GPS errors.

11

6 Spatial correlations using vertical coordinate residuals

It is well known that there is a strong correlation between the estimation errors in theZTD and the vertical coordinate. In order to assess the results in the previous section itis worthwhile to study the spatial error correlations of the vertical coordinate. Verticalcoordinate residuals have been calculated using GPS data from 1996 to the beginning of2001 from daily solutions (Johansson et al., 2002; Scherneck et al., 2003). The data havebeen collected from the SWEPOS and FinnRef networks (see Fig. 1). We calculated thecross-correlations of the time series for each pair of sites, and the results are shown inFig. 8.

The errors in the estimated station height have common sources with the errors inthe ZTD estimated using GPS. The plot in Fig. 8 supports the simulation results byJarlemark et al. (2001), suggesting a slow decrease of the horizontal correlations of theGPS estimation errors. For the short baselines (∼200 km), the correlation values remainbelow 0.9. This may indicate a random, uncorrelated component in the data, which hasa scaling effect on all correlation coefficient values.

Although the spatial correlations of these residuals are clearly present, the scatter ofthe data points prevents us from providing a reliable fit of an analytical model of the typediscussed in Section 4.

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Site separation, km

Cor

rela

tion

coef

ficie

nt

Horizontal correlations of the vertical coordinate residuals (1996−2001)

Fig. 8. Horizontal correlations of the vertical coordinate residuals for the period 1996–2001, using data fromSWEPOS and FinnRef.

12

7 Temporal correlations of slant delay residuals using ray tracing

7.1 Hydrostatic and wet mapping functions

The slant propagation delay is usually converted to an equivalent zenith delay usingmapping functions (Davis et al., 1985). As an example, the wet-delay MF (WMF) mw(ε)relates the equivalent zenith wet delay to the slant delay `w(ε):

`zw(ε) =`w(ε)

mw(ε), (25)

where ε is the elevation angle.In GPS processing, the ZWD estimate is a weighted average of the equivalent zenith

delays `zw(εi), estimated for the directions towards the satellites visible at a given epoch. Itis somewhat different from the actual value of the delay in the zenith direction, `w(90

◦),due to (a) inhomogeneities of the atmosphere, and (b) imperfections in the mappingfunction. The hydrostatic mapping functions (HMFs) and delays are related in a similarway.

The ray-tracing software described below assumes the atmosphere to be horizontallyisotropic above the RS launch site. Most often, the total slant delays are expressed by

`(ε) = `h mh(ε) + `w mw(ε) , (26)

where `h and `w denote the ZHD and the ZWD, respectively.The GPS estimates utilize mapping functions which at present introduce an rms error

in total delay of less than 6mm at satellite elevation angles larger than 10◦ (Bisnath et al.,

1997; Niell, 1996).The Ifadis (1986) MFs are used for ε > 2◦, and are in the form:

mI(ε) =

1

sin ε+a

sin ε+b

sin ε+ c

, (27)

The Ifadis HMF/WMF utilize surface measurements of the temperature, water vapour-,and total pressure. Time series of the Ifadis HMF values for ε = 10◦ at the Landvettersite are shown in Fig. 9.

The Niell (1996) HMF and WMF are based on statistics covering RS ascents from 26locations and have the general form:

mN(ε) =

1 +a

1 +b

1 + c

sin ε+a

sin ε+b

sin ε+ c

, (28)

where the parameters a, b, and c are obtained using only information about the sitelocation. These MFs are useful for ε > 3◦. For the Niell WMF, these parameters dependonly on site latitude, while the behavior of the HMF is related also to the day of the yearand the station height (see Fig. 9). The differences between the parameterizations of theNiell and the Ifadis mapping functions are summarized in Table 2.

13

0 50 100 150 200 250 300 350 4005.546

5.548

5.55

5.552

5.554

5.556

5.558

5.56

5.562

5.564

Days since Jan 1, 2000

Hyd

rost

atic

MF

val

ue a

t ε

= 1

0°

Niell hydrostatic Ifadis hydrostatic

Fig. 9. Time series of the Niell and Ifadis hydrostatic mapping functions, calculated for ε = 10◦ at Landvetter. Thesystematic differences between these two MFs can reach 0.1% in the winter for this site. The seasonal variationsin the Niell MF are only related to site latitude and the day of the year, while the Ifadis HMF utilizes surfacemeasurements.

7.2 Ray tracing using radiosonde ascents

The ray-tracing software uses vertical profiles of RH, air temperature and pressure tocalculate the wet and hydrostatic propagation delays, and the geometric delays (bending),for a given elevation angle. The results from the ray tracing procedure are then used toproduce the normalized wet/hydrostatic atmosphere thickness (slant delays divided bythe corresponding zenith values) in order to obtain the residuals for the Niell and Ifadismapping functions, for ε = (5◦, 10◦, 15◦).

Ideally, the time correlations of these residuals should represent the temporal behaviorof the GPS slant delay estimation errors. The results for the Niell HMF/WMF are shownin Fig. 10, spanning the year 2000. Note that the temporal correlations of the Niell WMFresiduals appear uncorrelated on scales larger than one day, while the HMF residuals arecorrelated on scales of several days.

The results for the Ifadis HMF/WMF are shown in Fig. 11. Note the slowly-varyingbias in the HMF residuals (the HMF underestimates the ray-trace results in the winter);this bias causes the correlations of the Ifadis HMF residual to be rather high on monthlyscales. Furthermore, there is an apparent diurnal signature in these correlations, whichshould be attributed to the differences in the day-time and night-time variations of thetotal pressure and temperature in the vertical direction.

The values of the temporal correlations for these structure function residuals changeinsignificantly, only by about 1% for the different elevation angles.

Table 2. Summary of the properties of the Niell and the Ifadis mapping functions.

Hydrostatic MF Wet MF

Niell Depends on height, lati-tude and day of the year.

Depends only on stationlatitude.

Ifadis Parameterized using surface temperature, pressureand humidity.

14

0 50 100 150 200 250 300 350 400−0.02

−0.01

0

0.01

0.02

Sla

nt d

elay

res

idua

ls, m


Niell wet Niell hydrostatic

0 5 10 15 20 25 30−0.2

0

0.2

0.4

0.6

0.8

1

Cor

rela

tion

coef

ficie

nt

Time lag τ, days

a)

b)

Fig. 10. Residuals of the Niell HMF, calculated for ε = 10◦ at Landvetter (upper graph). The rms variations ofthe HMF residuals are about 5 times larger than those of the Niell WMF. The temporal correlations of the Niellmapping function residuals are shown in the lower graph. The WMF residuals (#) are practically uncorrelated onscales larger than one day, while the Niell HMF residuals (×) are correlated on scales of several days.

0 50 100 150 200 250 300 350 400−0.02

−0.01

0

0.01

0.02

Sla

nt d

elay

res

idua

ls, m


Ifadis wet Ifadis hydrostatic

0 5 10 15 20 25 30−0.2

0

0.2

0.4

0.6

0.8

1

Cor

rela

tion

coef

ficie

nt

Time lag τ, days

a)

b)

Fig. 11. Residuals of the Ifadis HMF, calculated for ε = 10◦ at Landvetter (upper graph). The rms variations ofthe HMF residuals are about 6 times larger than those of the Ifadis WMF; there is also an apparent systematicbias in the HMF for this site. The temporal correlations of the Ifadis MF residuals are shown in the lower graph.The Ifadis WMF residuals (#) seem to be uncorrelated for time scales larger than one day. The Ifadis HMFresiduals (×) are correlated on scales of several days and exhibit a diurnal pattern.

15

8 Temporal correlations of ZTD from GPS position residuals

We calculated an estimate of the normalized auto-covariance function for a 1-year longtime series of differences between ZWDs, estimated using a WVR and GPS data at Onsala.The results are shown in Fig. 12. These correlation values decrease to e−1 at a time lag ofabout 1 day. The temporal structure of the WVR errors is unknown; therefore, the GPSerror temporal correlations cannot be deduced from the GPS–WVR differences withoutreservations.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1ONSA

Calendar year

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Cor

rela

tion

coef

ficie

nt

Time lag τ, days

e−1

Fig. 12. The temporal correlation values for the differences between GPS and WVR data at Onsala, for the year2001. The time correlation of these differences decreases to about 1/e at a time lag τ ≈ 1 day.

Another way to analyze the temporal correlations of the errors in GPS-estimatedZWD is to study the differences between time series of integrated water vapour derivedradiosonde- and GPS measurements. Although the errors in the RS data can be as-sumed temporally uncorrelated, the variance of the RS errors actually makes the time-correlations of the differences appear smaller (see Fig. 13). Furthermore, there are fewcolocated RS and GPS sites, and using similar differenced time series from nearby siteswould introduce a component due to the atmosphere itself.

0 50 100 150 200 250 300 350 400−10

−5

0

5

10IWV Differences between RS and GPS


IWV

diff

eren

ce, m

m

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

e−1

Time lag τ, days

Cor

r. c

oeffi

cien

t

Fig. 13. The temporal correlation values for the differences between GPS and RS data at Sundsvall, for the year1997. The time correlation of these differences decreases to about 1/e at a time lag τ ≈ 1/2 day, partly due to thevariance of the random measurement errors in the RS data.

The temporal behavior of the errors in GPS-estimated ZWD/ZTD can be describedindirectly, using the residual of the vertical coordinate of a permanent GPS site, as thesetwo parameters have common error sources. Their relation further depends on the dataprocessing strategy, the used zenith delay MFs, the antenna phase center modelling, andthe modelling of the ocean/atmosphere loading effects.

Time series of the vertical coordinate residuals, discussed earlier, are shown in Fig. 14,together with the temporal correlations for the Onsala GPS site. The decrease in thecorrelations of these residuals is consistently slow for all studied SWEPOS GPS sites; thedecorrelation time is about 1–2 days, and is summarized in Table 3. The variances ofthe individual time series were found to be less than 100mm2. Apparently, the northern-

16

1996 1997 1998 1999 2000 2001 2002−0.06

−0.04

−0.02

0

0.02

0.04

Ver

tical

coo

rdin

ate

resi

dual

s, m

Calendar year

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Cor

rela

tion

coef

ficie

nt

Time lag τ, days

e−1 e−1

a)

b)

Fig. 14. Time series of the vertical coordinate residuals for Onsala (upper graph), and the respective temporalautocorrelation values (lower graph). The time correlation typically decreases to 1/e at a time lag τ ≈ 2 days.

most sites have slightly longer decorrelation times and slightly higher variances. This canfurther be attributed to snow and ice, accumulating on the antenna radomes during thewinter (Jaldehag et al., 1996). Lastly, the data are generated at different rates (dailycoordinate estimates vs. 5-minute intervals for the ZTDs). These considerations implythat the presented vertical residual correlations should be used as an approximation ofthe ZWD/ZTD temporal error correlations cautiously, as the former over-estimate thelatter.

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Cor

rela

tion

coef

ficie

nt

Time lag τ, days

e−1 e−1 e−1 e−1 e−1 e−1

Fig. 15. Temporal correlation values for the vertical coordinate residuals from 2001 at Onsala (#). The timecorrelations of the GPS–WVR differences for 2001 are plotted for comparison (dashed line), and show consistentcorrelation values.

Although the linear trends, annual and sub-annual oscillations (up to 3 cycles per year),and air-pressure loading have been removed from these time series (Scherneck et al.,

2003), there are unmodelled components on time scales larger than one month, causingthe correlation estimates to remain rather high for time lags τ > 10 days for some GPSsites, including the one at Onsala. In particular, the years 1996 and 2000 appear to haveshort-term trends (see Fig. 14). Figure 15 presents the temporal correlations obtained

17

Table 3. Decorrelation of the residuals of the vertical coordinates for the SWEPOS sites. The data sets span theyears 1996–2001. The temporal correlation function is shown for time lag τ = 2 days. The GPS sites are sorted byincreasing latitude values.

Site Latitude Correlation %(τ) Variance[◦N] at τ = 2 days [mm2]

HASS 56.09 0.28 71OSKA 57.06 0.30 72ONSA 57.40 0.33 70VISB 57.65 0.34 72BORA 57.72 0.32 60JONK 57.74 0.28 63NORR 58.59 0.30 64VANE 58.69 0.30 62LOVO 59.34 0.33 67KARL 59.44 0.32 64MART 60.59 0.36 69LEKS 60.72 0.35 76SVEG 62.02 0.35 68SUND 62.23 0.32 69OSTE 63.44 0.39 80UMEA 63.58 0.34 69VILH 64.70 0.38 75SKEL 64.88 0.39 80OVER 66.31 0.33 72ARJE 66.32 0.36 85KIRU 67.88 0.40 100

from a subset without an apparent trend in the vertical residual data for Onsala (the year2001), together with a plot of the correlations of GPS–WVR differences from Fig. 12. Wealso performed a study of smaller, seasonal subsets of these data and found no seasonaldependence of the time correlation coefficients.

9 Discussion

The estimation of the atmospheric delay using GPS is associated with errors whose behav-ior is assumed to be random. The spatial structure of the error correlations in the ZTDis derived most reliably using innovation time series. The spatial scales of the horizontalcorrelations are of the order of several hundred kilometres, and the proposed analyticalmodels are described in Table 1 of Section 5. An implementation of the von Karmanfunction for the purpose of a 3DVAR data assimilation should be simple.

Some contribution to the ZTD estimation errors can be systematic, due to mod-elling (imperfect mapping functions). The temporal correlations of the correspondingerrors have time-scales of days, and this is comparable to the time-scales of the atmo-spheric processes.

Furthermore, temporally correlated errors can be introduced into the ZTD estimatesby the GPS data processing strategy itself. In particular, a Kalman filter based dataprocessing software assumes the ZTD to be a random walk process with chosen parame-ters, which reflect the average behavior of the atmosphere. However, a rapid change inthe atmospheric state cannot be modelled well enough, resulting in short-term system-atic differences. Such systematics have time scales of less than an hour, and are shorterthan the typical range of the numerical forecasts of today. These systematics will becomeimportant in the future 4D data assimilation, when GPS data will be used at the pre-cise acquisition epochs. A typical solution to the problem of time-correlated estimationerrors is “thinning” of the Kalman filter estimates, i.e. using results that are sufficientlyseparated in time (Brown and Hwang, 1997).

The comparisons to WVR data in Section 8 suggest that the contribution of the atmo-sphere related systematic errors is small. The assumption that the daily vertical coordi-

18

nate residuals and the ZTD estimation errors have a similar time-correlation structure issupported by the WVR-GPS comparison for the site at Onsala. This indicates that long-term, unmodelled trends in the satellite orbits and tidal motions of the sites propagateinto the ZTD estimation errors, with a variance of less than 100mm2.

Acknowledgements. The work on the spatial correlations of the zenith delay estimation errors was carried out inclose collaboration with Nils Gustafsson. We are grateful to Alan Rogers, Jim Davis, Tom Herring, and ArthurNiell for providing us with the ray-tracing software.

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