It is shown that the deviation of air from an ideal gas has nonnegligible effects when assimilating GPS radio occultation (GPSRO) data in a Numerical Weather Prediction (NWP) system. Therefore an assimilation system that aims to be unbiased to within the threshold of detection should account for this effect. GPSRO data are vertically referenced in terms of mean sea level altitude. Many other data types are vertically referenced in pressure units. The assimilation system may use yet another vertical coordinate. The required transformations between vertical coordinate systems should not induce significant biases. In the context of NWP the threshold of detection for a systematic height bias is on the order of 1–2 m. This study demonstrates that this level of accuracy cannot be obtained unless the deviation of air from an ideal gas, known as compressibility factor, is properly taken into account. With the current volume of GPSRO data an inconsistency between pressure and altitude scales larger than the mentioned threshold can lead to the development of nonnegligible biases in NWP assimilation cycles. Consideration of the compressibility factor realigns the altitude- and pressure-based scales to better than 1 m in the entire troposphere. Impacts are appreciated not only from global averages but from zonal averages as well.
 The vertical coordinate for most meteorological observations used in the context of Numerical Weather Prediction (NWP) has traditionally been pressure or some closely related variable. Indeed, the most accurate vertical location procedure within a meteorological context has been for a long time the barometer reading. The governing equations also take a simpler form or can be numerically better behaved under pressure-like coordinates. It is thus not surprising that pressure-related variables have become a standard vertical coordinate for both data and models. Geopotential altitude, or related variables such as MSL altitude, has traditionally been measured with lower accuracy than pressure.
 Radionavigation technologies, and notably the Global Positioning System [Parkinson and Spilker, 1996], hereafter GPS, allow accurate geometric location in both the vertical and horizontal coordinates. The accuracy obtained from a GPS receiver measuring altitude is at least comparable, and often superior to the accuracy of a barometer measuring pressure. Geometric altitude thus could be expected to progressively replace pressure as the vertical coordinate for at least certain applications. In a simultaneous measurement of pressure and altitude, the largest share of the observation error is now in many cases associated to pressure. This change has already partially taken place for radiosondes equipped with GPS receivers that report measurements as a function of MSL altitude, rather than as a function of pressure.
 At the same time an ensemble of limb-sounding observations including GPS radio occultation (GPSRO) [see Melbourne et al., 1994], and instruments such as GOMOS [Ratier et al., 1999], MLS [Barath et al., 1993], or HALOE [Russell et al., 1993] have appeared, which intrinsically produce measurements as a function of MSL altitude, or some closely related variable. Among them, and particularly accurate in their vertical positioning, are the radio occultation observations [Lee et al., 2000].
 This situation leads to a greater need to bring all observations and model vertical coordinates to a common vertical reference, and particularly to do so without introducing significant biases owing to the aligning process of the different vertical scales. In this paper it is shown that current expectations in meteorological data collection and modeling require that this alignment be unbiased to approximately 1 m in the vertical coordinate. Tolerance for the accuracy of the location of individual observations is less strict, but systematic effects should not exceed the above figure. However, standard procedures for the alignment do not have this level of accuracy. It is shown here how these procedures can be refined to reach the targeted tolerance.
 It is also shown that the equation of state (EOS) of air affects the assimilation of GPSRO data in two ways. The first is the vertical location of the data with respect to the background fields. The EOS affects the relationship between pressure and altitude in the background field. This is a property of the background model and is not particularly related to whether GPSRO data are being assimilated or not. Second, the compressibility can be involved in the relationship between the refractivity and other thermodynamic variables. Although both effects have a nonnegligible size, this paper focuses on the first.
 The possibility that the nonideal behavior of air could lead to a systematic bias in the interpretation of radio occultation data has been studied and quantified previously [Kursinski, 1997; Kursinski et al., 1997]. The authors find relative impacts on the order of 3 × 10−4 and conclude that the effect is smaller than other inaccuracies and can be neglected. However, two types of tolerances should be considered: for random and for systematic errors. Random observation errors tend to disappear in large-scale averaging. Systematic errors, instead, can affect NWP assimilation cumulatively, and small deviations can be evident in data assimilation cycles or after large-scale averaging. Errors in the equation of state are systematic. As shown in section 6, the deviations from ideal gas would be sufficiently small to be neglected if they were random. However, they are above the detection threshold for systematic errors.
 GPSRO data have been successfully assimilated at a number of NWP centers [e.g., Healy and Thépaut, 2006; Cucurull et al., 2007; Rennie, 2008; Poli et al., 2008], without noticing the development of systematic biases. As mentioned below, the development of systematic effects here reported are related to an inconsistency between GPSRO and radiosonde data. Since testing consistency is only possible through the use of observation operators and the hydrostatic equation, it is entirely possible to avoid a systematic bias if those two are mutually consistent.
 The nonideal gas behavior is often described trough the compressibility factor Z, a quantity equal to one for an ideal gas. Whereas in the atmosphere, this quantity is indeed very close to unity, it is not sufficiently close to be negligible. This is the object of this work. First, it is shown that properly accounting for the compressibility factor leads to improved alignment of the pressure and altitude coordinates. Next, the impact of the compressibility factor is evidenced in several data assimilation cycles with GPSRO data. Measurably different bias structures are obtained. For the data assimilation system of the Meteorological Service of Canada (MSC), the cycles that take into account the compressibility factor provide better agreement between pressure-registered data such as radiosondes with respect to altitude-registered data such as GPSRO. The impact is particularly noticeable in geopotential height. The improved agreement leads also to forecasts of better accuracy.
2. Radio Occultation Data
 GPS radio occultation [Melbourne et al., 1994] is a satellite limb sounding remote sensing technique. The retrieved measurements have good accuracy and high vertical resolution. Among the different possible postprocessing products that are offered by the data providers, we have chosen the refractivity as a function of MSL altitude. The relative observation error of the refractivity is ∼0.5% in the upper troposphere and lower stratosphere, with a vertical resolution on the order of 500 m [Lee et al., 2000]. The refractivity of the air is a function of thermodynamic variables. There are several expressions available in the literature, which follow in general the functional forms proposed by Smith and Weintraub  and Thayer . Among those, we have adopted the expression of Rueger , who presents a careful review of the available expressions, including the impact in the expression of the nonideal behavior of air, and of the composition of dry air, in particular related to the changing concentration of CO2. The expression proposed is the following:
The refractivity N is expressed as a function of the partial pressures of dry air (Pd) and water vapor (Pw), and of the absolute temperature (T). Units are hPa and K. It is interesting to note that the expressions mentioned differ in their coefficients. The refractivity depends on the density of air and water vapor [Born and Wolf, 1999]. The expression of this dependence as P/T, as in the work of Rueger , implicitly assumes that the ideal gas law holds. The formula presented by Rueger  is empirical, and its coefficients are adjusted to reproduce the real atmosphere, therefore including the effect of compressibility at least for conditions similar to the fitted measurements. However, expressing this functional dependence as P/T can slightly degrade the accuracy of the expression far from its fitting conditions, where the value of the compressibility may differ. Ideally, a new fit should be made following the real gas functional dependence in the expression of the refractivity. Within this work, this would also improve the consistency between the equation of state used in the context of the hydrostatic equation, fully nonideal, and the equation of state implied by the expression of the refractivity, which includes only a partial account of the nonideal effects. The differences between the coefficients in the above expressions are largely the result of the different ways in which the compressibility factor is accounted for [Thayer, 1974; Rueger, 2002] or ignored [Smith and Weintraub, 1953] in the relationship between pressure, density and temperature. Within an NWP system, where the equation of state also appears in many other contexts, the choice of an expression for the refractivity may determine whether the system is internally consistent, or is using different EOS for several related purposes.
 The main purpose of this paper is to study the size of the impact that an error in the EOS can produce. We show this through the details of the vertical alignment between pressure and altitude scales, without modifying the expression of the refractivity (equation (1)). The coherence between radiosondes and GPSRO data requires that this alignment be sufficiently accurate. It also requires that the thermodynamic expression of the refractivity be well calibrated. As mentioned above, the proper calibration of this expression is also related to the compressibility factor [Rueger, 2002].
 During the period under study, the following missions were providing GPSRO data in near-real time: Challenging Minisatellite Payload for Geophysical Research and Application (CHAMP), Gravity Recovery and Climate Experiment (GRACE), and Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC). The latter mission, consisting of six satellites and recording both rising and setting occultations [Lee et al., 2000], provides more than 90% of the GPSRO data available for assimilation in MSC's NWP system.
3. Vertical Location of GPSRO Data
 Let us review the vertical location procedure for radio occultation data. The measurements are made over certain locations in the atmosphere. These are identified through the known position of the emitter and receiver satellites involved in the radio occultation, whose positions are known within a few centimeters [Lee et al., 2000] in absolute Earth-centered coordinates, normally in the World Geodetic System (WGS 84) Cartesian frame of reference [National Geospatial Intelligence Agency (NIMA), 2004]. The locations of the observations are estimated also within this frame of reference. The Cartesian coordinates are equivalent to a set of ellipsoidal coordinates (altitude over the reference ellipsoid, latitude and longitude).
 For meteorological purposes, the accuracy of the ellipsoidal latitude and longitude is sufficient. This is not the case for the vertical coordinate. The reference surface in NWP models is usually the mean sea level (MSL). This surface is not sufficiently equivalent to the ellipsoid as departures, known as undulations, of several decameters are common, reaching extremes of 100 m [see, e.g., NIMA, 2004]. For this reason, the ellipsoid height is transformed to geoid orthometric height, subtracting the undulations. The geoid is very close to the MSL surface. The difference between the two, known as sea surface topography in the open ocean, is of only a few centimeters, and is associated with ocean currents, local sea temperature, and salinity. This is too small to be of concern for NWP purposes. Thus, in the following, the meteorological MSL surface is considered equivalent to the geoid surface. We will also assume that the geoid model that is used to transform ellipsoid heights to orthometric heights is of sufficient accuracy, which is on the order of a few centimeters. The above ensemble of manipulations, from a point in Earth-centered Cartesian space to its MSL altitude, can be carried out with a formal accuracy on the order of 10 cm. These manipulations are performed by the data providers, who deliver profiles of refractivity as a function of MSL altitude. Although the observation may contain larger errors, the mathematical manipulations mentioned do not introduce errors larger than 10 cm, which are negligible for the present purpose.
 It is then necessary to transform geoid orthometric heights to geopotential altitudes (i.e., altitudes as measured in geopotential energy, rather than in geometric distance). This requires knowledge of the structure of the gravitational acceleration, and can also be estimated from an Earth gravity model. We adopt the WGS 84 specification of the acceleration of gravity for low geopotential altitudes [NIMA, 2004], which is sufficient to deliver centimeter-level geopotential altitudes in the entire atmosphere. The constants are defined by WGS 84 and are reproduced in Table 1. The specification determines the surface acceleration γ0 at latitude ϕ as
The specification also defines the acceleration at finite altitudes z. The following expression is proposed by WGS 84 for low altitudes, which is sufficiently accurate for the atmosphere
The geopotential energy can be calculated [e.g., Holton, 2004] as a line integral of the strength of gravity, from the MSL surface to the height in question
It is common practice in NWP to use pressure, rather than altitude, be it geometric or geopotential, as the vertical coordinate. The pressure levels can be related to geopotential levels using the hydrostatic equation. The World Meteorological Organization  establishes the following procedure for the transformation from pressure to geopotential altitude:
Table 1. Coefficients for the WGS 84 Specification of Gravity
6.69437999014 × 10−3
 1. We have a discretized profile of pressure P, temperature T, and moisture Q. The profile begins at a surface whose geopotential altitude is H0.
 2. Each node of the profile is located at an unknown geometric altitude (MSL) z and at an unknown geopotential altitude H. Again, we use in this paper the WGS 84 estimation for the acceleration of gravity, although other expressions may also be suitable.
 3. The gravity potential Φ with respect to MSL is then expressed in geopotential meters instead of J/kg, dividing by some standard acceleration: H = Φ/γ0. A geopotential meter is the distance between two equipotential surfaces that are separated by one International System (SI) meter when the true acceleration is equal to the standard value. World Meteorological Organization (WMO) establishes = 9.80665 m/s2 as the standard value of the gravitational acceleration. Within meteorological applications, where the altitude is of at most a few tens of kilometers, a geopotential meter is approximately, but not exactly, one SI meter.
 4. To evaluate the ensemble of geometric altitudes z, the hydrostatic equation is used
 5. The integration of the hydrostatic equation requires an equation of state (hereafter EOS), to express the density in terms of the known pressure, temperature, and moisture. The WMO standard cited requires the use of the ideal gas EOS
 The standard establishes that the dry gas constant Rd should be used, and that a new thermodynamic variable called “virtual temperature” Tv must be created, which absorbs the local changes in the gas constant owing to the different molecular weight of dry air and water vapor.
 6. According to the standard, the final equation to be solved is
or in geopotential height terms, this can be expressed as
This procedure has been broadly adopted in both NWP modeling and radiosonde measurement reporting, ensuring coherent results across users, with the exception only of differences in the discretization of a vertical profile.
 In order to assimilate GPSRO data, it is necessary to locate them vertically. This is performed by comparing their geopotential altitude, evaluated from the vertical integral of the gravitational acceleration, equation (4) against the geopotential altitudes of a background NWP field (related to NWP values of pressure, temperature and moisture). It must be stressed here that this involves two different definitions of geopotential altitude. The first stems directly from the knowledge of the equipotential surfaces, and has centimeter-level formal accuracy. The second is a convention to link equipotential surfaces, with profiles of pressure, temperature and moisture. As will be further discussed in section 6, we require that the alignment of both pressure and altitude scales does not introduce a systematic bias larger than 1 m at tropopause levels (i.e., at 10–15 km), as a larger bias would be above the detection threshold. This requires the transformation must be accurate to within 1 part in 10,000, or 0.01%. However, the ideal equation of state, required by the standard WMO algorithm, is not an accurate representation of the thermodynamic behavior of air to within this tolerance in most meteorological conditions, as shown below. In a more general case, we can use a general EOS
where the quantity Z is known as the compressibility factor. This equation actually defines the compressibility factor as the quantity that takes into account any nonideal gas behavior. For a gas like air under meteorological conditions, Z is very close to one, and it is thus often a good approximation to simply assume that Z is equal to one, which would lead equation (10) to simplify to equation (7). However, this approximation is not sufficient to achieve an accuracy of 0.01%, notably under cold conditions where departures on the order of 0.1% can occur. It is thus needed to estimate the value of the compressibility factor, and to include its effect in the hydrostatic equation, which then becomes
 As mentioned in the introduction, this paper focuses on the impact of the compressibility in the vertical location of the GPSRO data. In this sense, we compare the vertical location obtained when the background fields against which the data is compared follow equation (8) between altitude, pressure and virtual temperature, versus the background fields assuming that they follow equation (11).
4. Compressibility Factor
 The targeted accuracy needs to be improved by only a moderate amount. The quantity Z − 1 needs to be known to within an accuracy of ∼10−4. The quantity itself for meteorological conditions is on the order of 10−3, thus a relative accuracy of 10% in the value of Z − 1 suffices.
 A number of expressions can be found in the literature [e.g., Reid et al., 1977; Dymond and Smith, 1980; Hyland and Wexler, 1983; Wagner and Pruss, 2002; Davis, 1992; Picard et al., 2008] to estimate the compressibility factor for pure substances and mixtures, including dry and moist air. Some of them are complex expressions, whereas others are limited in their range of applicability, and would perform very poorly if they were used outside their domain of validity. As the target accuracy in this study is to improve the equation of state of moist air by only a moderate factor with respect to the ideal gas equation, it is not of critical importance to choose the expression of highest accuracy. Since it has to be used within the context of NWP data assimilation, a computationally very intensive task, it is instead more important that the expression be simple, that it can be evaluated with low computational cost, and that it is robust and properly behaved even if used beyond its optimal range of applicability. The expression of Davis , whose validity is confirmed by Picard et al. , is selected here for these reasons. It implicitly treats dry air as a pure substance, and also estimates the cross interactions between air and water. The ensemble of possible meteorological conditions largely exceeds the recommended range of use stated by the authors. This expression is nevertheless well behaved over a broad range of conditions, and comparison against other expressions show that its accuracy is sufficient for the present purpose and in the entire meteorological domain. The expression is
where p is pressure, T is temperature in Kelvin, t is the temperature in Celsius, and x the mole fraction of water vapor in the air. The coefficients in the expression are summarized in Table 2.
Table 2. Coefficients From the Davis  Expression for the Compressibility Factor of Moist Air
1.58123 × 10−6 K Pa−1
−2.9331 × 10−8 Pa−1
1.1043 × 10−10 K−1 Pa−1
5.707 × 10−6 K Pa−1
−2.051 × 10−8 Pa−1
1.9898 × 10−4 K Pa−1
−2.376 × 10−6 Pa−1
1.83 × 10−11 K2 Pa−2
−0.765 × 10−8 K2 Pa−2
Figure 1 shows the behavior of the compressibility factor Z for meteorologically relevant pressures and temperatures. An interesting feature is that air is less ideal as it becomes colder or denser. It is also to be noticed that Z − 1 for dry air is negative at lower temperatures, including the entire range found in meteorology, zero at ∼76 C and positive above. The presence of moisture also produces the effect of reducing the compressibility factor. Since cold air can only contain a small amount of moisture before saturation is reached, this effect can be important only for warm air. Thus it can be concluded, a priori, that the deviation of ideal behavior will be most noticeable in two circumstances: in very cold dense air, such as the polar regions, and in very moist air, such as the tropical troposphere.
5. Comparison Against Model and Analysis
 In Figure 2, the observation minus background statistics for GPSRO data are shown. The background field is the short-term (6-h) forecast (hence named F) produced during data assimilation by the global operational system [Gauthier et al., 2007; Laroche et al., 2007; Côté et al., 1998] of the Meteorological Service of Canada (MSC), using the forward operator as described in the work of Aparicio and Deblonde . The forward operator acts over a background profile of temperatures and moistures as a function of pressure. It contains forward operators of refractivity and altitude, from which it interpolates the observation. The forward operator is here modified with respect to Aparicio and Deblonde  only to allow the altitude operator in the GPSRO package to optionally include the compressibility in the hydrostatic equation that links pressure and altitude (equation (11) instead of equation (8)).
 This model has a lid at 10 hPa, or ∼30 km. In Figure 2, the GPSRO data are not assimilated, but are only passively compared. Figure 2 illustrates the global average of the observation minus forecast (O-F) for all available GPSRO data over two months during the boreal winter 2006–2007. Some known biases are noticeable, including (1) a surface-level negative bias, known to be related to the observation [Ao et al., 2003]; (2) a tropopause-level negative bias, related to the fact that most errors, either positive or negative in the height of the tropopause, tend to appear systematically negative in GPSRO space; and (3) a generalized stratosphere positive bias, related to the radiative balance in the stratosphere of MSC's model.
 The surface-level bias could lead to problems if these data were assimilated. However, this can be avoided if low-altitude GPSRO data are discarded, as done at MSC. The other sources of O-F bias mentioned above can also lead to changes in the bias structure if the GPSRO data are assimilated, but this is significant only above the tropopause.
 It can also be seen in Figure 2 that besides the near surface and the tropopause, the entire troposphere presents a small negative bias if the ideal gas EOS is used (i.e., Z = 1). The bias is smaller in magnitude compared to the ones mentioned above, but is present at all heights in the troposphere. However, this bias is not related to any of the causes mentioned. During initial assimilation tests, it was found that geopotential biases developed in the troposphere. This suggests that the assimilated data were already biased with respect to the background. The near-surface data were not being assimilated, so they could not be responsible. Other tests in which GPSRO data were assimilated only within specific layers (midtroposphere, near tropopause, and stratosphere) indicated that the geopotential bias that developed in the troposphere was due to data in the midtroposphere layer, even if their bias was relatively small. The near-tropopause and stratosphere layers of data present substantially larger biases with respect to the background fields. However, their assimilation introduces corrections mostly in the upper levels of the atmosphere, and leaves the midtroposphere relatively untouched.
 A realistic compressibility factor in the hydrostatic equation introduces some noticeable changes, albeit small, in the a priori bias structure. This is also shown in Figure 2. The two cases differ in bias by about 0.0005 in terms of (O-F)/F or 1 part in 2000. Although small, this is larger than the tolerance stated above. If these data were assimilated, a change in the bias structure of the resulting fields should indeed be expected. In terms of geopotential altitude, a systematic error of 1 part in 2000 means a bias of 8 m at tropopause altitudes. In terms of temperature, this means a bias of about 0.1–0.2 K. It is important to note that random errors, measured by the standard deviation, are larger than these by over an order of magnitude. The compressibility factor, however, does not affect these data randomly but systematically.
6. Data Assimilation Cycles
 In order to check the impact of the compressibility factor in the equation of state, a total of six global data assimilation cycles were compared in two groups of three. The two groups represent boreal winter (20 December 2006 to 29 January 2007) and boreal summer (15 June to 17 July 2006). There is abundant GPSRO data covering the first period. However, the amount of GPSRO data in the second was significantly smaller. As shown below, the tendencies are consistent in all cases, although the size of the differences between cycles is smaller in the boreal summer group, as was to be expected. A more complete analysis of the impact of the assimilation of GPSRO data is beyond the scope of this paper and will be presented elsewhere.
 The three cycles in each group include a reference where GPSRO data is not assimilated, but that otherwise includes all data that is operationally assimilated at MSC. This data set includes radiosondes, surface stations, aircraft data, several sources of radiance data (AMSU, AIRS, and SSMI), atmospheric motion vectors, and marine surface winds. All radiance data contains a dynamic bias correction, where the past 15 days of data is used to correct the bias of each radiance channel. The dynamic bias correction uses the background fields as references, which are themselves anchored to data assumed to be free of bias. In these tests, radiosondes and GPSRO data are assumed to be free of bias.
 The reference cycle shows that the system reproduces the geopotential heights of the radiosondes with a bias not larger than 1–2 m in the entire troposphere, which can be seen in global and regional averages. The model stratosphere, as already mentioned, presents a systematic cold bias. Since the behavior of the system does not reproduce the radiosonde data with sufficient accuracy before introducing GPSRO data, we do not draw conclusions from the behavior of the model's stratosphere after the introduction of GPSRO data.
 The two other cycles in each group add GPSRO data to the assimilation data set. The GPSRO data include all available refractivity profiles from near-real-time missions (CHAMP, GRACE, and COSMIC). For a large fraction of the periods under study, and especially the boreal winter case, the volume of GPSRO data is largely dominated by COSMIC. The initial period of the boreal summer group includes only CHAMP and GRACE data. COSMIC was still under its commissioning phase and the use of data collected prior to 14 July 2006 is not recommended (C. Rocken, private communication, 2006). These two cycles in which GPSRO data is assimilated differ only in the equation of state that is used for the vertical location of GPSRO data. Namely, in one of the cycles, the EOS is ideal (Z = 1), whereas in the other, the compressibility factor Z is a more realistic estimation [Davis, 1992].
Figure 3 presents the global verification of the 6-h forecasts against radiosondes for the three assimilation experiments for the boreal winter season 2006–2007. The verification set consists of a selected subset of 374 radiosonde sites, reporting twice daily (i.e., at 0000 and 1200 UTC). The verification discards the first 7 days as spin up, and averages the remaining 34 days, totaling about 20,000 radiosonde profiles. Statistics were produced for winds, geopotential height, temperature and dew point depression at a number of levels. The levels of significance of the interexperiment difference were also evaluated for both the bias and the standard deviation.
 The results show that the control experiment (GPSRO data not assimilated) did indeed present an acceptable geopotential bias structure in the entire troposphere. The system forecasts the tropospheric geopotentials with a bias not larger than 1–2 m. The cycle in which GPSRO data are assimilated with ideal gas EOS (i.e., Z = 1) develops a noticeable geopotential bias. This can also be seen in Figure 3 as a small but systematic temperature bias. Finally, the cycle with realistic Z shows also an acceptable geopotential bias structure, of quality comparable to the control experiment. The statistical significance of these differences in the bias structure between any two pairs of experiments is larger than 99.5% in the entire troposphere, and particularly in the upper troposphere (over 99.9%). The two experiments that assimilate GPSRO are in most of the other respects better than the control, showing that the assimilation of these data is otherwise beneficial. We must mention in particular the reduced standard deviation of temperature and geopotential height (significance over 95% in the mid and upper troposphere, over 99% in the stratosphere). Beyond the better bias structure, the GPSRO experiment with realistic Z also presents a slight reduction of the standard deviation (STD), with respect to the ideal EOS experiment.
Figure 4 shows the verification results of the same experiments but for the Antarctic region only. It can be seen that the (GPSRO, Z = 1) experiment develops a particularly large bias over this region. Figure 5 shows similar results but for the tropics, where a bias that is larger than the global average also appears.
Figure 6 shows a similar ensemble of experiments, but for the boreal summer season of 2006, averaged over the globe. The experiment is shorter here: the average includes 25 days after the initial 7 days of spin up. The results obtained are qualitatively similar to the winter season, even if the volume of available GPSRO data is smaller, which in this case is largely dominated by CHAMP and GRACE data. The polar and tropical regions (not shown) present results similar to the boreal winter ones in Figures 4 and 5.
 Some of the results, and notably the boreal summer global average, but also others not shown, present a small negative O-F geopotential bias in the control experiment. The positive O-F bias shift introduced after GPSRO assimilation with ideal EOS could have been interpreted in these cases as a beneficial correction. However, the more global picture emerges that the GPSRO data with ideal EOS introduces a systematic shift, not only when it would be desirable, but also when the control already presents a positive O-F bias.
 The deviation of air from ideal behavior is shown to be nonnegligible for the purpose of NWP assimilation of GPSRO data. Given the level of agreement between model and radiosondes without GPSRO data, with tropospheric geopotential biases not larger than 1–2 m, a new data type should not introduce a bias larger than this amount. This implies that statistical disagreements between GPSRO and radiosondes on the order of 1 part in 10,000 can be detected, although it is an order of magnitude smaller than random errors present in both GPSRO and radiosonde data.
 The impact of the compressibility factor is above this threshold, and is therefore detectable. A system intended to be well calibrated to within the threshold of detection should account for it. As shown in section 6, the difference between the ideal and nonideal EOS can be clearly seen, with a high level of statistical significance (>99%). This impact of the compressibility factor is larger in either cold or wet conditions. The results over polar and tropical regions are thus particularly prone to show a bias owing to the EOS. The introduction of a realistic estimation of the compressibility factor during the vertical location of GPSRO data largely eliminates the observation bias in the entire troposphere, and subsequently the development of a geopotential bias during the assimilation cycles. The observed difference is coincident in size, sign, and thermal behavior, with the expected effect of the compressibility factor.
 The geopotential bias that is noticed after the inclusion of GPSRO data reveals an inconsistency between the assimilated data from GPSRO and the background fields, which were calibrated against radiosonde data. Since the compressibility factor appears also in the expression that relates the refractivity with other thermodynamic variables, we conclude that, if the EOS is not applied uniformly, an inconsistency between background profiles, GPSRO data and radiosondes can reach a detectable level.
 The assimilation of GPSRO data is otherwise beneficial. Generalized improvements in the standard deviation of nearly all other magnitudes are seen worldwide, in both winter and summer, with or without the introduction of the compressibility. The adjustments to the equation of state of air, even if used only for the improvement of the vertical location of GPSRO data, allow benefiting of the assimilation of those data without incurring the development of geopotential height biases, nor their associated temperature biases.
 It is shown that a systematic misalignment of the vertical scales larger than 0.01% is, for data like GPSRO, sufficiently large to lead to the development of measurable biases in NWP data assimilation cycles. The deviation of air from ideal gas behavior is, in the troposphere, larger than 0.01% by up to an order of magnitude, and thus should not be ignored during NWP assimilation. A corollary of this assertion is that the internal consistency of radio occultation data is also shown to be sufficiently high as to be able to identify, albeit after large-scale averaging, systematic biases as small as 0.01%.
 The authors wish to thank the teams at GFZ and UCAR for making available the radio occultation data from missions CHAMP and GRACE (GFZ) and from COSMIC (UCAR), respectively. We also thank the two anonymous referees for their comments.