Atmospheric pressure loading displacement of SLR stations

Journal of Geodynamics 39 (2005) 247–266

Atmospheric pressure loading displacement of SLR stations

D. Bocka,b,c, R. Noomena,∗, H.-G. Scherneckd

a Department of Earth Observation and Space Systems, Delft University of Technology,Kluyverweg 1, 2629 HS Delft, The Netherlands

b RWTH Aachen, Aachen, Germanyc University of Stuttgart, Stuttgart, Germany

d Chalmers University of Technology, Onsala, Sweden

Received 22 April 2004; received in revised form 15 November 2004; accepted 19 November 2004

Abstract

This paper addresses the local displacement at ground stations of the world-wide Satellite Laser Ranging (SLR)network induced by atmospheric pressure variations. Since currently available modelling options do not satisfythe requirements for the target application (real-time availability, complete coverage of SLR network), a newrepresentation is developed. In a first step, the 3-dimensional displacements are computed from a 6-hourly grid of1◦ × 1◦ global pressure data obtained from the ECMWF, for the period 1997–2002. After having been converted intopressure anomalies, this pressure grid is propagated into horizontal and vertical station displacements using Green’sfunctions and integrating contributions covering the entire globe; oceans are assumed to follow the inverted barometer(IB) approximation. In the next step, a linear regression model is developed for each station that approximates thetime-series of the predicted vertical displacements as well as possible; this regression model relates the verticaldisplacement of a particular station to the local (and instantaneous) pressure anomaly. It is shown that such a simplemodel may represent the actual vertical displacements with an accuracy of better than 1 mm; horizontal displacementsare shown to be negligible. Finally, the regression model is tested on actual SLR data on the satellites LAGEOS-1and LAGEOS-2, covering the period January 2002 until April 2003 (inclusive). Also, two model elements areshown to be potential risk factors: the global pressure field representation (for the convolution method) and the localreference pressure (for the regression method). The inclusion of the atmospheric pressure displacement model gives

∗ Corresponding author. Fax +31 152785322.E-mail address:[email protected] (R. Noomen).

0264-3707/$ – see front matter © 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.jog.2004.11.004

248 D. Bock et al. / Journal of Geodynamics 39 (2005) 247–266

improvements on most of the elements of the computations, although the effects are smaller than expected sincethe nominal effect is absorbed by solved-for satellite parameters.© 2005 Elsevier Ltd. All rights reserved.

Keywords:Atmospheric loading; Space geodesy; SLR

1. Introduction

The accuracy requirements on present-day space geodetic computations are such that a full modellingof a wide variety of dynamical and geometric elements of System Earth is needed. This paper focuses onone of these elements, notably atmospheric pressure loading (APL). Since the global atmospheric massdistribution, and thus, the surface pressure distribution changes with time, the loading of the Earth’s surfaceis not constant. This fluctuation in loading causes a time-dependent displacement of the Earth’s surface,mainly in the vertical direction. The amplitude of such instantaneous displacements is on the centimetrescale, so it cannot be ignored in accurate computations of satellite orbits and derived parameters suchas station coordinates. This holds, in particular, for the Satellite Laser Ranging (SLR) technique, whichis extremely valuable for absolute quantifications of global geophysical phenomena such as geocentreand global scale. This necessitates the modelling of the displacements at SLR ground stations caused byatmospheric pressure loading. Other physical phenomena like non-tidal ocean loading, ground water tableand snow loading may also play a role here, possibly of comparable size, but such effects are beyond thescope of this study and are not further considered. The IERS Conventions 2003(McCarthy and Petit, 2004)mention a number of possibilities for this APL modelling: (i) a convolution method based on the actualloading distribution over the entire surface of the (solid) Earth; (ii) a simplification of such a model; (iii)an empirical model which is based on the actual deformations derived from (space) geodetic observationstaken at individual sites.McCarthy and Petit (2004)report specific advantages and disadvantages foreach of these options. The IERS Conventions 2003 advise to use the convolution method, in particular,the results that are derived by the IERS Sub-Bureau for Loading (SBL). An operational drawback forthis particular model is that the displacement data are available with a delay of at least 24 h only, whichdisqualifies it for semi real-time applications. The same holds for the direct use of the results of otherconvolution studies, such as (Gegout, 2003), which is extended intermittently, causing real-time lags on theorder of months (cf. Section3.3). The IERS Conventions 2003 do offer a more practical representation ofa convolution model, notably a set of regression coefficients for the relation between vertical deformationand local pressure anomaly: the model presented invan Dam et al. (1997)(cf. Section3.3); this isin line with the second possibility mentioned above. A clear drawback of (the implementation of) thisparticular option is that it does not cover the full network of SLR stations currently in operation or active inthe past.

The purpose of the study presented here is to develop and implement a model for atmospheric pressureloading displacement that: (i) can be used for real-time analyses as being performed at Delft Universityof Technology, and (ii) covers the full network of SLR stations. In addition, a model accuracy of 1 mmis aimed at (the SLR observations have an absolute accuracy of several millimetres for the best stations;cf. (ILRS, 2003)).

D. Bock et al. / Journal of Geodynamics 39 (2005) 247–266 249

To do so, the 3-dimensional displacements are computed for the full set of SLR stations which havebeen active during the past decades (i.e. 103 different locations in total). Section2 describes how this isdone by convolving the instantaneous global pressure distribution with a Green’s function and integrat-ing all contributions. The result is a time-series of instantaneous displacements at each SLR location,covering the period January 1997 to December 2002, with a step-size of 6 h. Section3 presents the devel-opment of a more simple representation of such displacements: a linear regression model, where verticaldisplacements are modelled as a function of local pressure and horizontal displacements are ignored.Such a representation is fully in line with the IERS options. This model is tested for internal and externalconsistency. Next, this model is included in orbit computation software and evaluated in terms of quality(Section4). The paper ends with conclusions and recommendations.

2. Solid Earth deformation

In this section, we consider the elastic yield of the solid Earth to the changing surface pressure inducedby the atmosphere. Farrell, in his further development of the first successful treatment by Longman(Longman, 1962, 1963), solved the point loading problem for an elastic, non-rotating, self-gravitatingand spherically symmetric Earth with a liquid core, by devising Green’s functions that encompass theEarth’s response over all spherical harmonic degrees, with special consideration of the limit at infinitedegree (Farrell, 1972). The essential step comprises the global integration of the load influence, whichis represented by the corresponding loading Green’s function. We employ the Green’s function approachsince it offers advantages when we integrate the load distribution at all spatial scales, from global to gridcells at close distance to an observing site. The advantages are that the global problem is consistentlytreated (no spatial truncation), whereas high-spatial resolution can be obtained at short range. Limitationsof the method are mentioned in the discussion below. The approach has been used in a number of studieswith only slight variations (e.g.van Dam and Wahr, 1987; van Dam and Herring, 1994).

The specific Green’s functions for vertical and horizontal displacement (du and dv, respectively) areemployed (Farrell, 1972):

du = Gudm (1)

dv = Ghdm (2)

where dm is an infinitesimal point mass causing the corresponding infinitesimal displacements. Note thatdv is reckoned positive along the azimuth from the load to the field point. The Green’s functions arecomputed from load Love numbersh′

n andl′n (Farrell, 1972) according to

Gu(D) = G

ga

∞∑n=0

h′nPn(cosD) (3)

Gh(D) = G

ga

∞∑n=0

l′ndPn(cosD)

dD(4)

whereG is the universal gravitational constant,g the mean gravitational acceleration at the surface ofthe Earth,a the mean Earth radius,D the arc distance between the load and the field point, andPn isthe Legendre polynomial. These functions are singular at the origin and decrease roughly as one-over-


distance, so that the displacements are dominated by the local and regional loads. If the one-over-distancefactor is not taken into consideration, the remainder varies within one order of magnitude with the low-amplitudes at distances greater than 10◦.

2.1. Deformation and global frame translation

The reference for the displacements in this study is the centre-of-mass of the solid Earth; translations ofthe solid Earth with respect to the joint centre-of-mass of solid Earth and atmosphere are not considered.The reason is that this study aims at a local correlation of displacement with station pressure variation.The translation terms are excited by degree-1 spherical harmonic components, which have a truly globalbasis. Correlation of the global translation with local air pressure is regarded as of secondary importance.The translation might become visible at the global scale, when the loading-corrected satellite ranges areadjusted to the reference frame. Sometimes in space geodesy loading and rigid translations are treatedsimultaneously in the Love number (Green’s function) approach (like inBlewitt, 2003), and alreadyFarrell had given this problem careful thought (Farrell, 1972). Since displacement and translation canbe separated uniquely if the deformation of the body is known (using a model), translations could beresolved in a two-stage process of global SLR analysis after having eliminated the displacement atindividual stations first. The two-stage concept applies to this paper since a method for displacementreduction will be developed, in particular, based on a regression model.

Description of the displacements needs Love numbers for a wide range of Legendre degrees, whiletranslation employs trivial numbers and only first-order Legendre functions. It is probably the commonsource of the effects that has ledBlewitt (2003)to treat them together. However, it remains to be determinedwhether an inverted barometer (IB) model is as good an approximation for deformation-effective oceanresponse as it is for degree-1 excitation and response. Thus,Blewitt (2003)appears to sacrifice a degreeof freedom of analysis in his approach.

2.2. Limitations

The basis for this formulation is the assumption of a model Earth, consisting of concentric sphericallayers (a radially symmetric, self-gravitating elastic Earth). One problem with the approach addressesradial symmetry, which conflicts with the obvious dichotomy of the Earth’s surface and lithosphere,distinguishing an oceanic and a continental domain. The loading problem may be solved for differentEarth models as to their lithosphere structure, and thus, the loading effects at stations on continentaland oceanic lithosphere can be computed separately. Fortunately, most geodetic stations are located ina continental lithosphere setting. Sites primarily affected are to be expected at the coasts of enclosedbasins, where long-wavelength loads in the ocean are not very well attenuated. At present questionsas to the response of the ocean (see discussion below) appear to have a more important impact onatmospheric loading effects than oceanic lithosphere structure does, so this problem is left here withoutspecial attention.

Another limitation is the assumption of elasticity. Long-term loading is known to create viscous after-effects, with mantle viscosity being the most important parameter. This study concentrates on the day-to-day variability of loading displacements. Present knowledge of mantle viscosity suggests that sucheffects should be neglegible in a time scope of years to decades, including the scope of this study.


2.3. Computation of loading effects

The loading computations are almost identical to the procedures used in the case of ocean tide loading(Scherneck, 1991). The most important ingredients and the adaptations to the case of barometric loadingon land are outlined below. Most notably, assuming a purely elastic response, the atmospheric load isdescribed as a real-valued geographic grid conceived in the time domain, whereas ocean tide loadingpercieves the load as a complex-valued grid of in-phase and out-of-phase components in the frequencydomain.

For practical reasons the global loading mass is represented by a finite number of points on a grid.Here, use is made of a 1◦ × 1◦ grid retrieved from the European Centre for Medium-Range WeatherForecast (ECMWF). The archived quantity is the logarithm of surface pressure in spherical harmonicrepresentation. Conversion into a regular, user-defined grid is carried out by the Centre’s MeteorologicalArchiving and Retrieval System (MARS). In order to keep the resulting displacement within a reasonablenumerical range, a global average pressure of 1013 hPa is subtracted. As this investigation concernstemporal variations only, this uniform pressure reduction has no influence on the temporal aspects of theoutput.

Other authors’ work was based on pressure fields from the National Meteorological Centre (NMC)with a grid resolution of 2.5◦ × 2.5◦ (van Dam et al., 1997; Petrov and Boy, 2004), with oceanic masksregularly exceeding this resolution (e.g. 0.25◦ × 0.25◦ in Petrov and Boy (2004)).

In the convolution process, the interval is refined to 0.1◦ × 0.1◦ inside a distance range w.r.t. the stationof interest of 10◦, in order to better approximate the integral in the part of the domain where the Green’sfunction has most of its variation. Normally, there is one such sub-element that contains the station. In thissituation, a semi-analytical approach is taken that integrates the one-over-distance factor of the Green’sfunction, and multiplies the result with the asymptotic value

u0 = 2 limD→0

sinD

2Gu(D)(5)

whereDandGu are arc distance and Green’s function for vertical displacement, respectively, as mentionedbefore. The analytical factor for the integrated near-field loading effect of a rectangular box in flat-Earthapproximation is given by:

f = 2aδβ

[Arsh (cosβ) + cosβ Arsh

(1

cosβ

)](6)

whereδβ is the refined grid interval andβ is the latitude. The Earth’s sphericity is ignored in this factor.The need to take the local effect into account arises only in the vertical component.

Thus, in summary, the pressure anomalyp given at the grid nodesj is resulting in:

uk = +∑ pj

gσj Gu(Dkj) + u0 f

pJ

g(7)

vk = −∑ pj

gσj cos (A) Gh(Dkj) (8)

wk = −∑ pj

gσj sin (A) Gh(Dkj) (9)


whereA is the azimuth, so thatvk andwk specify the east and north displacement, respectively, at pointk.The summation covers all nodes of the loading grid including the refined areas, but except the one nodeJwhere the semi-analytical factor is used. Each node represents a surface areaσj.

2.4. On the ocean response—dynamics

The question what to do with the ocean is a difficult one to answer. This investigation will resort to thesimple IB assumption, i.e. the pressure anomaly is replaced by its oceanic average at all oceanic nodes.

Ideally, the dynamic response of the ocean bottom pressure to changing air pressure at the oceansurface would have to be modelled, or the bottom pressure would have to be measured. Since oceanbottom pressure is not really important for weather forecasting, this parameter is not available from themeteorological services. Since many geodetic stations are near the coast and some are even in pelagicsettings of the open-ocean, the response in coastal waters could be significant.

In this light, the use of the IB concept is to be seen as a crude approximation, employed with thenotion that improvements are needed in the near future. At present, references can be made toWunschand Stammer (1997), who investigated the deep water open-ocean and enclosed basin cases and foundcertain limitations on the IB assumption. The most severe limitations are probably associated with theexcitation of resonant modes, including gravity modes and Rossby waves. The equatorial ocean has beenpointed out by several authors as an area where the ocean appears to respond significantly more sluggishunder the atmospheric load (Gentemann, 1995; Mathers and Woodworth, 2001). For a large part of thedeep ocean, however, the IB assumption appears to be a reasonable first-order approximation.

2.5. On the ocean response—statics

van Dam et al. (1997)have pointed out that the concept of an equilibrium response of the ocean tolarge-scale atmospheric variations should take the gravitational attraction of the load and the actual stateof deformation of the solid Earth into account. Primarily loading the continents, the oceanic responsegenerates a secondary load and subsequently causes deformation. The problem was first studied inWahr(1982). From a multi-year time-seriesvan Dam et al. (1997)found that the ocean surface may be displacedby 75 mm peak-to-peak as an extreme value. The greatest amplitudes were found at high-latitudes in thenorth and south. Crustal loading due to the implied water mass variation may produce 4 mm crustaldisplacement, using a rule of thumb, predominantly near the respective coasts. Looking at the root meansquare (rms) results in the same publication, 95% of the cases seem to have less than 7.5 mm oceanresponse, so that only few cases would exceed a 0.4 mm rms displacement effect.

3. Linear regression model

Having obtained 6-hourly time-series for the 3-dimensional pressure loading displacements of the SLRnetwork as described above, the results can be converted into a model which uses a simple linear relationbetween the modelled (vertical) displacement and the local ECMWF-induced instantaneous atmosphericpressure anomaly w.r.t. a reference value; horizontal displacements will be ignored. This approach isjustified for a number of reasons: (i) for a complete (re-)analysis of SLR data, a time-series starting inJanuary 1997 is not sufficient: effectively, the SLR network started to provide significant information on a


wide variety of characteristics of the Earth in September 1983, with the onset of the so-called MonitoringEarth Rotation and Intercomparison of Techniques (MERIT) campaign; however, the information on theglobal pressure data dating back to the 1980s and early 1990s is restricted in resolution (temporal, spatial)and changes in the gridding have introduced artefacts in the local pressure values at a number of locations(as will be discussed in Section3.2); (ii) as mentioned in Section1, the accuracy target of the modellingof the pressure loading displacements is set at 1 mm; it will be shown that a linear regression model forthe vertical displacements satisfies this requirement (which is confirmed by a number of tests), and thathorizontal displacements are at this same level at worst; (iii) only a regression model is capable to satisfythe real-time requirement of displacement modelling; for convolution methods this is a non-option.

In some areas, the dominating wavelength of the pressure anomalies may be shifted towards long-or short-wavelength during extended periods (e.g. weather patterns changing from fast eastward trav-elling low-pressures to high-pressure blockages on the European continent). During long-wavelengthsituations, the loading response is larger in magnitude than during short-wavelength forcing. We willpay attention to the range of variation of the linear regression coefficient during temporal segments ofthe data.

3.1. Mean position

The following considerations are looking into the future when loading is included in the computationof a next-generation International Terrestrial Reference Frame (ITRF).

In order to arrive at unbiased site coordinates, it must be assured that they are estimated using consistentparameters. If the reduction of atmospheric loading effects is included in the geodetic estimation, onemust therefore make sure to use the same reference pressure in the surveys and campaigns that wewish to compare or combine. In the current modelling, i.e. employing loading regression coefficients,a unique reference pressure needs to be adopted for each station. Although the IERS Conventions atpresent recommend to reduce air pressure loading effects, they do not establish the reference pressure forthe geodetic stations (instead, it is done at the start of the computations when following the convolutionapproach; cf. Eqs.(7)–(9)). The ITRF products that are currently distributed do not involve atmosphericloading corrections. An obvious candidate to quantify such a reference value is a long-term averagepressure value, measured on site. Since the difference between the observed pressure and the referencepressure will determine the displacement, a barometer that is not (regularly re)calibrated against thisreference value will cause a (drifting) bias in the loading correction, and therefore, a (drifting) bias in the(estimated and/or modelled) site coordinates. However, this is a hypothetical situation.

In order to use the 3-dimensional displacements obtained from the load convolution directly, theequivalent task remains to establish a mean displacement field for the Earth. Since the displacementswill depend slightly on the method, and the static component can be large (as in areas with considerableaverage topography), the results from the different methods need to be accompanied by a set of averagedisplacements. Alternatively, sufficiently long-joint sections of data must be available.

As an example of the combination,Fig. 1gives the time-series of 6-hourly pressure values for the SLRstation at Beijing, China, covering the full period analysed here. Clearly visible is the yearly variationof the actual pressure, strongly correlated with seasonal weather patterns. It is obvious that the standardsea-level pressure of 1013 mbar is not a particularly good reference here (although Beijing is located atan altitude of 83 m only). The vertical displacement is proportional to the numbers in this plot, of course.The full network of 103 stations is illustrated inFig. 2.


Fig. 1. Time-series of atmospheric pressure at the SLR station at Beijing (7249) during the period 1997–2002.

3.2. Regression coefficient

The regression coefficients, the numerical coupling between the displacement in a particular directionat a certain station and the local pressure anomaly are easily computed by making a least-squares fit of alinear function�r = c × (pinst − pref) through the time-series of displacements as described in Section

Fig. 2. Overview of the global network of SLR stations, and an indication of the relation between vertical displacement andlocal pressure derived from the multi-year time-series, for each individual SLR station.


Fig. 3. Pressure loading regression coefficients derived for the network of SLR stations for individual years (ordinate) versusthe coefficients covering the entire 6-year time-span (abscissa). (Left) original results. (Right) results after elimination of entrieswith suspected pressure anomalies.

2. In this equation,�r, c, pinst and pref represent the displacement (mm), the regression coefficient(mm/mbar) and the instantaneous and reference atmospheric pressure (mbar), respectively.

The 6 years of displacement data are easily converted into a multi-year solution for the regressioncoefficient. Before doing so and discussing the results, it must be noted that it is also possible to deriveregression coefficient values at each station for individual years; each yearly solution is still based on1460 individual data points. This is done to investigate variations in these coefficients, which mightbe introduced by yearly variations in the global weather systems (cf.Fig. 1). The results of the lattercalculation are plotted inFig. 3(left).

For the far majority of stations, a good consistency is obtained for the yearly regression coefficientsolutions. Also, for most of the stations a good agreement between the yearly values and the multi-yearlyvalue is observed.Fig. 3clearly illustrates this; converted into a correlation coefficient, a value of 0.8805can be obtained. However, the plot also suggests that for a number of stations, there is a significant variationin the solution for the regression coefficient, which would support the weather pattern argument againsta simple regression model as mentioned in the previous paragraph. This concerns, in particular, the SLRstations in Albuquerque, Arequipa, Basovizza, Bear Lake, Cerro Tololo, Concepcion, Ensenada, Flagstaff,Goldstone, Maidanak, Medicina, Melengiclick, Monte Generoso, Monte Venda, Mount Hopkins, Riyadh,Santiago, Vernal, Yigilca, Yogzat, Yuma and Zimmerwald.

However, closer analysis shows that the cause for these outliers is a different one. To illustrate this,Fig. 4 (left) shows the vertical displacement for the SLR station Maidanak (1863) as a function of thelocal pressure anomaly (for 8764 points, i.e. 6 years covered by 6-h time steps).

It is clearly visible that a parallel displacement along the pressure axis occurs in some years, in particular,for the vertical direction (which is more susceptible for pressure-induced motions). Clearly, the trend line


Fig. 4. The instantaneous vertical pressure loading displacement at station Maidanak (1863) for the period 1997–2002 as afunction of the local pressure anomaly. The trend line indicates the straight-line fit that can be drawn through the entire dataset.(Left) original dataset and results. (Right) idem, after elimination of the years with an unusual pressure anomaly.

for this displacement, with a regression coefficient of−0.2317 mm/mbar, is not representative for theactual displacement at the station. It turns out that this pressure offset of about 20 mbar is present in asubset of the 6-year time-span, i.e. in 1997 and 1998 only. A closer analysis revealed that this apparentchange in local atmospheric pressure is introduced by changes in the resolution of the orography as beingused by the ECMWF and a simultaneous increase in spectral resolution of the atmospheric fields. Sincethe problem is found in a remote high-altitude location, we suspect that the change of the orographicrepresentation caused the problem. Such changes took place in April 1995 (not relevant here) and April1998. To effectively remedy this problem, 1997 and 1998 have been eliminated from the regressionanalysis (for this station) and the computation is repeated. The result is summarized inFig. 4 (right). Itis clearly visible that the trend line obtained now much better fits the actual displacements (with an rmsresidual of 1.2 mm in the vertical component), and no systematic anomalies are present anymore. Thenew regression coefficient for the vertical displacement now becomes−0.6550 mm/mbar.

The anomalous situation of 1997 and 1998 appears to affect the other SLR stations mentioned abovein a similar way. However, the data for the majority of the stations (80 out of a total of 103) are found tobe fully consistent, also for these 2 years, and the reason for this pressure offset remains unknown. As asolution, the anomalous contributions (which covered some 7% of the total input for these computations,and which are easily identifiable) were simply eliminated. It is emphasized here that the majority of thestations listed before deals with historic stations, which do not contribute to the current SLR network.After this elimination, the calculation of the regression coefficients was repeated, and the new results aresummarized inFig. 3(right). Clearly, the results have a much better consistency: the correlation betweenthe yearly and multi-year regression coefficient now becomes 0.9851. In comparison toFig. 3(left), theanomalous entries in the plot have now shifted to the left, i.e. the multi-yearly regression coefficientsolutions have grown in magnitude.

For a further validation of the results, which are presented inTables 1–3, the multi-yearly regressioncoefficients for vertical displacement have been plotted on a world map (Fig. 2). Here, it is clearly visiblethat the regression coefficients are very consistent, i.e. stations in a particular part of the Earth behave in asimilar fashion. In general, two regimes can be distinguished: continental stations, where a displacementof on average−0.4 mm/mbar is observed, and oceanic stations, where the IB effect compensates for anyatmospheric pressure loading effect. It should be emphasized that the results shown here are of coursemodel values, and not real displacements derived from actual space geodetic data.


Table 1The values for the regression coefficients and the reference pressure as derived from the multi-yearly analysis described in thispaper

Station Site no. Regression coefficient (mm/mbar) pref (mbar)

Albuquerque 7884 −0.5053 814± 0.58Algonquin 7410 −0.4311 987± 0.63American Samoa 7096 0.0034 1008a

Ankara 7589 −0.4518 909± 0.62Arequipa 7403, 7907 −0.2036 761± 0.04Askites 7510 −0.3310 999± 0.41Balkhash 1869 −0.4611 977± 1.67Bar Giyyora 7530 −0.4346 930± 0.27Basovizza 7550 −0.4120 968± 0.51Bear Lake 7046, 7082 −0.3887 806± 0.36Beijing 7249, 7343 −0.4676 1009± 0.42Bermuda Island 7067 −0.0187 1013a

Borowiec 7811 −0.4615 1008± 0.26Cabo San Lucas 7882 −0.0988 995± 0.24Cagliari 7545, 7548 −0.2009 997± 0.46Cerro Tololo 7401 −0.1417 789± 0.12Changchun 7237 −0.4799 987± 0.19Concepcion 7405 −0.2176 1000± 0.30Dionysos 7515, 7940 −0.2322 961± 0.29Diyarbakir 7575 −0.4497 932± 0.38Easter Island 7097 −0.0148 1006± 0.17Ensenada 7883 −0.2694 992± 0.35Evpatoria 1867 −0.3265 1016± 1.31Flagstaff 7891 −0.5043 766a

Grand Turk Island 7068 −0.0266 1013a

Goldstone 7085 −0.3948 914± 0.80Grasse 7835, 7845 −0.3384 875± 0.09Graz 7839 −0.4614 962± 0.10Greenbelt 7101, 7105, 7106, 7125, 7918, 7920 −0.3673 1012± 0.09Hartebeesthoek 7501 −0.5561 866± 0.14Haystack Observatory 7091 −0.3125 1002± 0.94Helwan 7831 −0.4755 996± 0.14Herstmonceux 7840 −0.3645 1014± 0.10Huahine 7121, 7123 0.0237 1008± 0.16Isigaki Sima 7307 −0.1082 1005± 0.45Iwo Jima 7305 −0.0251 1006± 0.85

Stations A-J.a ISA value.

In addition, the variation of the yearly estimates of the vertical regression coefficients (computed as theroot mean square difference w.r.t. the multi-yearly value) can be plotted as a function of location as well:Fig. 5. This plot confirms that the variation in the yearly values is very small: typical scatter values aresmaller than 0.05 mm/mbar (assuming a maximum offset of 20 mbar in the local pressure, this correspondsto an uncertainty of 1 mm, within the accuracy target set for this study). The few stations with a ratherhigh-value for the inconsistency between yearly regression coefficients (there are three such stations, as




Karitsa 7520 −0.3183 949± 0.30Kashima 7335 −0.1233 1011± 0.49Kattavia 7512 −0.2294 1009± 0.22Kazivili 1893 −0.2941 1011± 0.36Koganei 7328 −0.1776 1006± 0.48Komsomolsk-na-Amure 1868 −0.4175 997± 0.38Kootwijk 7833, 8833 −0.3954 1014± 0.65Kunming 7820 −0.7502 798± 0.17Kwajalein Atoll 7092 −0.0451 1009a

La Grande 7411 −0.3578 1012± 1.36Lampedusa 7544 −0.2227 1008± 0.33Maidanak 1863, 1864 −0.6550 730± 0.12Matera 7540, 7541, 7939, 7941 −0.3022 960± 0.09Maui 7120, 7210 0.0081 709± 0.04Mazatlan 7122 −0.2719 1005± 0.07McDonald Observatory 7080, 7086, 7850 −0.4170 800± 0.06Medicina 7546 −0.4183 1012± 0.73Melengiclik 7580 −0.3406 869± 0.37Metsahovi 7805, 7806 −0.4188 1010± 0.69Minami Daito Jima 7304 −0.0623 1020± 0.50Minami Tori Sima 7300 −0.0053 1015± 0.85Miura 7337 −0.1325 1006± 0.67Monte Generoso 7590 −0.3932 847± 0.37Monte Venda 7542 −0.4082 964± 0.71Monument Peak 7110 −0.2991 814± 0.03Mount Hopkins 7888 −0.5011 744a

Mount Stromlo 7849 −0.3449 927± 0.12Natal 7929 −0.1956 1009a

Noto 7543 −0.1832 1003± 0.38Oga 7321 −0.1921 1009± 1.41Okinawa 7301 −0.0915 996± 1.02Orroral Valley 7843, 7943 −0.3386 871± 0.12Otay Mountain 7035, 7062 −0.2256 898± 0.24Owens Valley 7114, 7853 −0.4292 880± 0.38Papeete 7124 0.0083 1003± 0.11Pasadena 7896 −0.1820 962a

Platteville 7112 −0.3840 846± 0.29Potsdam 1181, 7836 −0.4452 1008± 0.22

Stations K-P.a ISA value.

shown inFig. 5) turn out to have a multi-yearly value for this parameter which fits into their surroundingsvery well (cf.Fig. 2); this indicates that the multi-yearly regression coefficients of the SLR ground stationswith a high rms value are not suspect. In conclusion, the multi-yearly regression coefficients derived fromthe time-series of 6-hourly station displacements are considered as very accurate and reliable.




Quincy 7051, 7109, 7886 −0.3512 889± 0.06Richmond 7295 −0.1742 1016± 0.28Riga 1884, 1885, 7560 −0.4746 1021± 0.27Riyad 7832 −0.6590 927± 0.18Roumeli 7517 −0.2011 1004± 0.23San Fernando 7824 −0.2868 1013± 0.16Santiago 7400, 7404 −0.1207 935± 0.35Santiago de Cuba 1953 −0.0433 1016± 0.29Shanghai Observatory 7837 −0.3965 1020± 0.31Simeiz 1873, 7561 −0.2968 978± 0.27Simosato Observatory 7838 −0.1951 1009± 0.14Sofia 7505 −0.4453 902± 0.50Sutherland 7502 −0.3413 834± 0.31Tateyama 7339 −0.1103 1003± 0.42Titi Sima 7844 −0.0265 993± 0.74Tokyo 7308 −0.1727 1007± 0.87Tromso 7602 −0.2452 1000± 0.84Tusima 7302 −0.1687 1017± 0.94Tyosi 7322 −0.0678 1009± 1.31Vernal 7892 −0.3591 830a

Wettzell 7594, 7596, 7597, 7834, 8834 −0.4350 946± 0.11Wuhan 7236 −0.5035 1015± 0.87Xrisokalaria 7525 −0.2059 965± 0.31Yarragadee 7090, 7847 −0.3856 984± 0.06Yigilca 7587 −0.4378 926± 0.27Yozgat 7585 −0.4369 835± 0.21Yuma 7894 −0.2481 986a

Zimmerwald 7810 −0.4502 916± 0.11

Stations Q–Z.a ISA value.

Two other aspects of the generation of this regression model must be mentioned here. First, the linearmodel approximates the observations (i.e. the instantaneous displacement estimates coming out of theGreen’s convolution; cf. Section2) very well: an overall fit of 1.1 mm has been obtained (the maximumvalue is 2.1 mm, for Balkhash (station 1869)). One goal of this study (to derive a simple and easy-to-use model with an accuracy of 1 mm) is clearly met. Second, the horizontal deformations appear to benegligibly small: they have a value of 2–3 mm at most. If a linear regression model were to be developed,it would have to be as a function of the pressure gradient (North–South for latitudinal displacementand East–West for longitudinal displacement). However, this would complicate the modelling effortconsiderably, and considering the small magnitude of this component as addressed in the first part of thisstudy, it is simply ignored altogether. The remainder of this investigation concentrates on the (effects ofthe) regression coefficients for the vertical displacement of the SLR ground stations.


Fig. 5. The scatter of the yearly regression coefficients for the individual SLR stations.

3.3. External validation

In addition to internal consistency checks of the regression coefficients, it is also possible to makecomparisons with information on the same phenomenon, provided by other investigators.

Two such sources are readily available in literature: (i) a time-series on 3-dimensional displacementssimilar to the one derived here, computed by P. Gegout from the University of Strassbourg, France; and(ii) direct values for the regression coefficients as derived by T. van Dam. A brief summary of these resultswill be given here first.

3.3.1. GegoutGegout also provides 3-dimensional displacement data in 6-hourly steps, in this case from January 1992

to September 2002 (Gegout, 2003) (at least, this interval was analyzed here). This dataset covers in total95 different locations of SLR sites, only missing stations that have come on line very recently. Gegoutprovides 2 different time-series of displacements: one in which the offset is computed against the Earth’scentre-of-mass (the so-called CME solution), and a second one w.r.t. the combined centre-of-mass of thesolid Earth and atmosphere (labelled CMEA). To make Gegout’s results comparable to the representationderived in this study, his time-series was converted into regression coefficients first; the procedure for thisis identical to the one described above.

3.3.2. van DamGoing one step further than the previous option, van Dam has converted her solutions for the dis-

placements at ground stations into regression coefficients already (van Dam, 2003). These coefficientswere computed by using Reanalysis Data of the National Center for Environmental Prediction (NCEP)through a time-span of 18 years: January 1980 to December 1997. Her input data was the surface pressuregiven on a 2.5◦ × 2.5◦ global grid in 6-hourly values. The vertical crustal displacement at a particular sitewas modelled by applying the Farrell’s Green’s functions for aGutenberg-Bullen AEarth model; similarto what has been done in this study. van Dam modelled the ocean’s response to atmospheric pressure


Table 4Number of common stations between arbitrary pairs of options for vertical regression coefficients

This study (multi-year) Gegout van Dam

This study (multi-year) 103Gegout 64 95van Dam 34 26 208

Numbers on the main diagonal provide the total number of stations present in a particular model (not necessarily restricted toSLR stations).

anomalies by using an IB function. As usual, the regression coefficients are determined as a linearisationbetween the modelled crustal displacements and the local surface pressure anomaly.

To get a better feel for the consistency of the various solutions and to assess the qualitative andquantitative aspects of the model derived in this study, in particular, a comparison between the differentoptions has been made. This was done by plotting the regression coefficients for common stations for anycombination of two options, and calculating the correlation between these regression coefficients. Thiscomparison is summarized inTables 4 and 5.

Table 4shows that all three models contain a fair amount of stations. However, the solutions by Gegoutand van Dam also contain a large amount of stations equipped with other techniques (like GPS etcetera).This is clearly reflected in the overlaps, where the small number of stations common in this study’snetwork and Gegout’s is caused by the absence of older and very recent stations in the latter set. Thevan Dam solution encompasses the main SLR network but also misses new stations and campaign-stylesites.

Table 5shows that the regression coefficients for vertical displacement as derived here are quiteconsistent with the values derived from Gegout’s data and the values directly available from van Dam’ssolution. On an individual station basis, larger discrepancies may be observed, however. The consistencybetween our multi-yearly and single-yearly solutions (as well as Gegout’s CME and CMEA results)stands out very clearly (although it is realized that each of these combinations have much in common, ofcourse).

In summary, it is concluded that the regression coefficients as derived in this investigation are bestsuited to model the pressure loading displacement at the SLR stations, for a number of reasons: (i) theset comprises the full network of (at the time of writing) 103 locations, whereas the other options arerestricted to subsets; (ii) the current solution shows a good internal consistency, which may not necessarilybe proof of absolute correctness but certainly is a good step in that direction; (iii) the solutions as derived

Table 5Correlations between various solutions for the regression coefficients for vertical pressure loading displacement at individualSLR stations

This study (multi-year) Gegout CME Gegout CMEA van Dam

This study (single-year) 0.9851Gegout CME 0.7762 1.0Gegout CMEA 0.6639 0.9372 1.0van Dam 0.7345 0.6452 0.6125 1.0


here appear to be more consistent with any of the other, external solutions than the latter are amongthemselves.

3.4. Reference pressure

The previous computation and discussion focused on one aspect of the linear regression model: the valuefor the linear regression coefficient. Before this model can be implemented into the orbit determinationand (geo)physical parameter estimation software, another element of this model needs to be quantified, i.e.the value for the reference pressure. In principle, two options exist for this purpose: (i) the pressure valueas derived from the International Standard Atmosphere (ISA), and (ii) the mean pressure determinedfrom the meteorological pressure data records as registered by each individual SLR ground station.This choice is not difficult to make; however, the ISA pressure may be very easy to calculate given thealtitude of the station, but it clearly does not take into account detailed aspects of local topography orspecific contributions from local (seasonal) weather patterns; therefore, this option has to be discardedimmediately.

To get reliable numbers for the mean pressure derived from the atmospheric registrations on site, it isimportant to consider a long time-span, preferably 20 years or even more. In this way, seasonal and/oryearly effects can be averaged out. In principle, this can be achieved for the current dataset since LAGEOS-1 tracking data is readily available at DEOS for the period September 1983 until current. Here, each passover a ground station is considered as a single entry into the computation (and not each individual rangeobservation, the number of which can differ considerably from pass to pass). The computation mightalso use the data records on LAGEOS-2, which is available since its launch (October 1992), but sincethere is a large overlap in the regular tracking of the two satellites, adding this second one is not expectedto contribute much new information, so the computation was restricted to LAGEOS-1 only. A minorcomplication is that not all stations have been active since 1983, which means that the averaging processmight miss certain long-term variations in pressure patterns.

The reference pressure values were calculated for each station in a straightforward manner. For themajority of stations, several hundreds of passes are available for this purpose. An issue directly relatedto this is the uncertainty of this average pressure value. Assuming a pessimistic possible random error inthe instantaneous pressure readings of 1 mbar, and taking into account the real-variation of the pressuremeasurement on site, the uncertainty of the average pressure value can be computed directly. In thisway a typical outcome for the majority of stations is 0.5 mbar or less; the actual values are shown inTables 1–3. Combined with an average height/pressure regression coefficient of−0.4 mm/mbar, thisuncertainty translates into a minor fraction of a millimetre. One can object that errors in barometerreadings do not show a random character, but will tend to show a more systematic deviation (a bias).In such a case, an assumed 1 mbar error would remain present in the average value, but still result in aminor (average) 0.4 mm effect in the pressure loading displacement (which is within the 1 mm target ofthis study).

As a final check of the outcome of the computation of the average pressure, the results were plottedagainst station height (not depicted here). This plot clearly showed that the results follow a clear patternas expected, with deviations from the trend line of about 5 mbar at most. On average, the offset is below2 mbar, equivalent to better than 1 mm in height.

As for the SLR stations that have not been active in the period 1983 until current, the ISA pressurevalue is taken as the reference (cf.Tables 1–3).


4. Implementation

As mentioned in Section1, the ultimate goal of this investigation is the inclusion of a model for atmo-spheric pressure loading displacement in and hence improvement of the software that is used for satelliteorbit computation and (geophysical) parameter estimation. In this particular case, use is made of theGEODYN-II software version 0104 (Eddy et al., 1990), which has been developed by NASA/GSFC. Thissoftware already has the capability to model pressure loading displacements through a linear regressionmodel. However, it is up to the user to specify the parameter values that are used for any particular stationin an orbit/parameter computation.

To test the current model, SLR data on LAGEOS-1 and LAGEOS-2 taken during a period of about16 months has been processed in a configuration without and with the pressure loading modelling.More specifically, the observations are processed in data arcs of 8 days length, all starting on successiveSaturday mornings 00:00 h and ending on Sunday morning 00:00 h. The combination of arc length andfrequency results in overlaps of exactly 1 day. The analysis period runs from Saturday January 5, 2002,until Sunday May 4, 2003. It is recognized that this test dataset covers a reasonable set of observations,in particular, more than a full year with its possible seasonal pattern in loading effects, but it does notcover all 103 (partly historic) stations for which the current model was generated. Such a test, however,would demand too much computational effort and is expected not to give a different picture of the effectof modelling atmospheric pressure loading. The computation model that was used is summarized inTable 6.

The modelling and analysis results have been checked thoroughly for correctness, and an indicationof the corrections that have been obtained with the modelling of the pressure loading implemented wasalready given inFig. 1. This plot shows the pressure variation taking place at Beijing as a function of time.

Table 6Computation model for the satellite orbit and geophysical parameter estimations

Dynamical modelNASA/GSFC EGM-96 gravity field model, truncated at degree and order 20; C2,1 and S2,1 according to IERS Conven-tions; GM= 398600.4360 km/s;c = 299792.458 km/s;a = 6378.1370 km; 1/f = 298.257; Wahr solid-Earth tides model;NASA/GSFC EGM-96 ocean tides model; direct solar radiation pressure (CR kept fixed at 1.14); 3rd-body attraction fromSun, Moon, Venus, Mars, Jupiter and Saturn (JPL DE-200 ephemerides); empirical accelerations in along-track (constant andonce-per revolution) and cross-track (once-per revolution) directions; no relativistic effects; pole tide applied

Reference frameIERS ITRF2000 for station coordinates; IERS Bulletin, a model fora priori Earth orientation parameters; Lieske model forprecession; Wahr model for nutation; Love solid-Earth tides model; ocean loading applied; pole tide applied

ObservationsLAGEOS-1 and LAGEOS-2 normal points provided through NASA CDDIS and Eurolas EDC; Marini-Murray tropospheric re-fraction correction; centre-of-mass correction 251 mm; no relativistic effects modelled; cut-off elevation 0◦; station-dependentdata weighting

IntegrationCowell 11th-order predictor-corrector; step-size 100 s

Estimated parametersSatellite state-vectors at epoch; satellite empirical accelerations at 4-day intervals; epoch station coordinates (weak solutionsin ITRF2000 only); Earth orientation parameters at 1-day intervals


Table 7Statistical summary of the satellite orbit and geophysical parameter estimation without (center) and with (right) modelling ofatmospheric pressure loading (APL)

Criterion Without APL With APL

Weighted rms-of-fit (mm) 14.74 14.72

rms range bias (mm)LAGEOS-1 18.89 18.78LAGEOS-2 18.35 18.32

rms time bias (�s)LAGEOS-1 11.80 11.80LAGEOS-2 13.26 13.26

rms along-track orbit difference (mm)LAGEOS-1 34.96 34.53LAGEOS-2 37.53 38.72

Repeatability station height (mm)Maidanak (1863) 14.15 14.66Beijing (7249) 22.71 22.27Arequipa (7403) 11.47 11.51Concepcion (7405) 8.41 8.16Chania (7830) 6.64 6.51Riyadh (7832) 6.69 6.41Potsdam (7841) 5.98 7.00Matera (7941) 6.12 5.52

It is clearly visible that min-max variations of over 40 mbar may occur, with a specific yearly pattern.When combined with the regression coefficient of−0.4676 mm/mbar obtained for this station, verticaldisplacements of over 15 mm are the outcome. The proper modelling in the software of these correctionshas been verified.

To assess whether the quality of the overall modelling improves or not, a number of different criteriaare inspected: (i) the weighted rms-of-fit of the SLR observations to the model; (ii) the apparent rangeand time biases of the passes; (iii) the difference between overlapping satellite ephemerides solutions;and (iv) the stability of other parameters. The results of the orbit/parameter estimations, in terms of thecriteria mentioned above, are summarized inTable 7.

Table 7shows that an improvement has been obtained when the atmospheric pressure loading effect ismodelled, but the magnitude of this improvement is rather small. First, the overall rms-of-fit reduces atsub-mm level. A similar conclusion can be drawn for the apparent range biases which can be interpretedas a combination of an error or uncertainty in the radial component of the satellite orbit at the time ofpassage of a ground station and in the vertical station position. There appears to be no noticeable effectin the apparent time bias (which can be related to along-track satellite position or uncertainty duringstation overflight, or horizontal station position mismodelling); the latter result basically confirms thatthe pressure loading effect mainly works in vertical direction, and horizontal components are hardlyaffected if at all. A small change in the consistency of the orbital overlaps is observed: for LAGEOS-1,an improvement of about 0.5 mm is found, whereas for LAGEOS-2 a slight worsening of results canbe seen.Table 7gives orbit overlap statistics for the along-track direction only, since this is typically


the most sensitive satellite position component, in which modelling effects can be expected to show upfirst. The orbit overlap statistics inTable 7show that the orbital solutions are consistent at the level of3–4 cm (overlaps in other directions, not presented here, are up to a factor 5 smaller), and that the absoluteorbital accuracies in direction of flight are in the order of about 6 cm (which can be derived from the timebias statistics). The most “convincing” evidence of the value of including pressure loading displacementsin the computation model comes from the consistency of the height components of the solutions forstation coordinates (only stations of choice have been solved for): here, typical reductions in the order of0.1–0.5 mm are found (although degradations are also observed).

The overall improvement of the results may be disappointing at first sight, but it must be realized thatthe presence or absence of a pressure loading model is to a large part compensated for by the parametersthat are solved for in the computations (cf.Table 6). In particular, the orbital parameters (state-vector,empirical acceleration) will do their best to reduce the fit of the observations to the model as much aspossible, and hence may obscure a large part of the nominal pressure loading effects. It is also importantto note that the loading effect may show very significant variations over a year, whereas the variationwithin an arc length as used here (8 days) will be relatively small (cf.Fig. 1). In conclusion, the relativelyminor improvement in terms of fit, bias, overlap and consistency statistics should not come as a surprise,but should also not be taken as a reason not to model atmospheric pressure loading. The results suggestthat there are probably other elements of the computation model where more can be gained in terms ofabsolute fit and/or parameter solution consistency.

5. Conclusions and recommendations

To summarize this investigation, we have been able to derive a time-series of instantaneous displace-ments induced by atmospheric pressure loading for the network of (effectively) 103 SLR stations, coveringthe period 1997–2002 and with a temporal resolution of 6 h. These displacements are computed in 3 di-mensions, and are based on an integration of pressure anomalies throughout the globe.

The time-series of 6-hourly displacements has been converted into a regression model that approximatesthe vertical displacement due to atmospheric pressure anomalies with an accuracy of better than 1 mm.It has been shown that the displacements in horizontal directions are typically 2–3 mm at maximum.Assuming that the target accuracy of orbit computations and (geo)physical parameter estimations usingabsolute SLR observations is 1 mm, it is justified to conclude that the 1-dimensional regression modelas derived here satisfies these requirements and provides a simple tool to satisfy modelling accuracyrequirements for arbitrary epochs.

Although the nominal effect of pressure loading displacement may range well over 15 mm, its modellingturns out to have positive, yet minor consequences for the total fit of the observations to the computationmodel, orbit uncertainties and other statistical parameters. Successive solutions for the height of SLRstations show the effect of the addition of such a model most clearly.

An advantage of the regression model pursued in this study is that it provides for a stable and accuratelong-term model for pressure loading displacement; the alternative, i.e. a more detailed convolutionmethod covering the entire Earth, proves very susceptible to various aspects (grid size, orography, etc.)of the (computation of the) global pressure field, whereas real-time applications become impossible bydefinition.


As a recommendation, it is advised that the geodetic stations calibrate their barometer at regular intervalswith an aimed-for accuracy of 1 mbar or better, and maintain continuity of this series and the pressuremeasurement location.

Acknowledgements

The authors thank the International Laser Ranging Service (ILRS) for providing the SLR measurementswhich are processed in this investigation. We sincerely thank E. Pavlis (JCET) and an anonymous reviewerfor providing valuable input to improve the contents of this paper.

References

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Atmospheric pressure loading displacement of SLR stations