The sensitivity of European Centre for Medium-Range Weather Forecasts (ECMWF) numerical weather prediction analyses to the empirical refractivity coefficients used to assimilate bending angles derived from GPS radio occultation measurements has been investigated. We have compared the Smith and Weintraub (1953) coefficients with the “best average” values proposed by Rüeger (2002). The Rüeger values produce simulated bending angles in the upper troposphere and stratosphere that are larger by ∼0.115%. This produces a cooling in the troposphere by around ∼−0.1 K, which improves the fit to radiosonde geopotential height measurements in the Northern Hemisphere but degrades the fit in the tropics and Southern Hemisphere. The cooling is caused primarily by Rüeger's increase in the “k1” refractivity coefficient, which accounts for the dry air contribution to the total refractivity. It is confirmed that this cooling can be reduced by introducing nonideal gas effects in the hydrostatic integration of the forward model. However, the Rüeger k1 coefficient should also be adjusted to k1 = 77.643 K hPa−1 if it is used in a forward model that includes nonideal gas effects when evaluating the refractivity from the model state. Furthermore, if the nonideal gas effects are introduced in a consistent way, we find that the Rüeger coefficients plus nonideal gas effects produce very similar results to the Smith and Weintraub values, where nonideal gas effects are neglected.
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 One of the key characteristics of GPS radio occultation (GPSRO) measurements is that they can be assimilated into operational numerical weather prediction (NWP) and reanalysis systems without bias correction. This is possible because the assimilated GPSRO quantities, usually either bending angle profiles or refractivity profiles, are derived from precise measurements of a time delay with an atomic clock. In addition, the forward problem is relatively straightforward when compared with the assimilation of satellite radiance measurements, partly because it is not considered to be reliant upon poorly known spectroscopic parameters, or the assumed concentrations of well-mixed gases. In fact, the forward modeling of GPSRO measurements does require empirically derived refractivity coefficients, but to date the level of confidence in these values has been very high. The refractivity coefficients used operationally at European Centre for Medium-Range Weather Forecasts (ECMWF) and the Met Office were derived by Smith and Weintraub  (hereafter referred to as SW53), but their accuracy has been reinforced in more recent work by, for example, Hasegawa and Stokesberry  (hereafter referred to as HS75) and Bevis et al.  (hereafter referred to as BEV94). In particular, there has been a broad consensus in the literature that the k1 coefficient, which accounts for the dry air contribution to the total refractivity (see section 2), is k1 = 77.6 K hPa−1, and this has effectively become the “standard value” used in GPSRO studies. However, Rüeger  (hereafter referred to as RU02) has proposed a new set of “best average” refractivity coefficients, and the new k1 coefficient is 0.115% larger than the standard value. The new coefficients are based on a thorough reappraisal of experimental work from the 1950s to the 1970s.
 In the context of NWP, Rüeger's coefficients were tested for GPSRO data monitoring at Météo-France, but they were not used in assimilation experiments (P. Poli, personal communications, 2009). They are now used operationally at Environment Canada [Aparicio and Deblonde, 2008].
 ECMWF has recently conducted forecast impact experiments comparing SW53 coefficients and the RU02 best average coefficients, and we have found that the RU02 coefficients cool the mean state in the troposphere by ∼−0.1 K. This cooling reduces the biases in the short-range forecast fit to radiosonde temperature and height measurements in the Northern Hemisphere but increases the biases in the tropics and Southern Hemisphere. Experiments show that the cooling is caused primarily by the change in the “k1” refractivity coefficient. Interestingly, the increased bias with respect to radiosonde measurements in the Southern Hemisphere appears to be qualitatively similar to the results presented by J. Aparicio (Environment Canada) at the ECMWF/Global navigation satellite system Receiver for Atmospheric Sounding (GRAS) Satellite Application Facility (SAF) workshop in 2008. Aparicio et al.  use the Rüeger coefficients operationally but also advocate the inclusion of nonideal gas effects in GPSRO operators in order to reduce geopotential height biases against radiosondes. Furthermore, National Centers for Environmental Prediction (NCEP) has also recently reported difficulties with the RU02k1 value, suggesting it is too large [Cucurull, 2010], and they have retained k1 = 77.6 K hPa−1 in their refractivity forward model by adopting the BEV94 coefficients.
 Given the difficulties with the RU02 coefficients in NWP impact experiments, we have attempted to reconcile the differences between the standard (i.e., SW53/HS75/BEV94) and RU02k1 values. This exercise has shown that there is more uncertainty in the k1 coefficient than is generally recognized by the NWP community. It has highlighted small numerical discrepancies in how the refractivity coefficients are derived from refractivity measurements by different authors, and a misunderstanding over whether the contribution of CO2 is accounted for when deriving the coefficient. A similar analysis is also presented by Cucurull . We have also investigated the connection between the choice of refractivity coefficients and the inclusion of nonideal gas effects in the GPSRO observation operator. In section 2, we introduce and compare the standard values of the refractivity coefficients given by SW53, BEV94, and others, with the new RU02 estimates. The forecast impact experiments examining the sensitivity to the SW53 and RU02 coefficients, and investigating the inclusion of nonideal gas effects are described in section 3. The discussion and conclusions are given in section 4.
 We only give the main points that are considered to be the most relevant for the GPSRO assimilation problem here in order to convey the some of the difficulties with the published literature. We have investigated the inconsistency in the published results but have not assessed the accuracy of individual experiments. A discussion of the refractivity coefficients is also given by Cucurull , but nonideal gas effects have not been addressed in that work.
2.1. Assuming an Ideal Gas
 Neglecting nonideal gas effects, the refractivity N of air can be approximated with a three-term expression
where Pd is the pressure (hPa) of dry air, e is the water vapor partial pressure (hPa), and T is the temperature (K). k1, k2, and k3 are empirically derived coefficients. An alternative form of the refractivity equation is
where P is the total pressure (P = Pd + e) and k2 = k′2 − k1. SW53 (equation (7)) use the alternative form and give the following three-term expression:
but then simplify it, by combining the second and third terms by multiplying the second term by 273/T, to give
Equation (4) is currently used in operations at ECMWF and the Met Office. The (k1 = 77.6) term, which is the primary interest of this report, is derived from an average of three experiments that determined the dielectric constant ε of dry air published in 1951. SW53 quote an uncertainty k1 = 77.607 ± 0.013 K hPa−1, which is less than 0.02%. The SW53k1 has been used extensively within the GPS meteorology community, but this has been justified by more recent studies. For example, both HS75 and BEV94 have performed a statistical analysis of results from 20 experiments from 1951 to 1970. HS75 suggested k1 = 77.600 ± 0.032 K hPa−1 and BEV94 derived k1 = 77.60 ± 0.05 K hPa−1 for their ground-based GPS studies. Furthermore, Kursinski et al.  estimated the uncertainty in the SW53 formula for dry air to be less than 0.06% generally, and less than 0.03% above 500 hPa. Therefore, k1 = 77.6 has essentially become the standard value used in GPS meteorology.
RU02 has performed a similar statistical analysis to HS75 and BEV94 and produced a new set of best average refractivity coefficients, given by a weighted mean of what he considers to be the most reliable experiments. After the removal of data from 11 of the experiments used by HS75 and BEV94 and the addition of one new experiment from 1977, his reanalysis of the experimental data gives
In addition, RU02 has adjusted the k1 value to account for the increase in CO2 concentration from 0.03% to 0.0375%. This modification leads to
The change in k1 as a result of the increase in CO2 is only 0.0042 K hPa−1, which is a 0.005% increase. RU02 has estimated the uncertainty in the new, best average k1 value is less than 0.02% (0.015 K hPa−1).
 The difference between the RU02 and the standard value (k1 = 77.6) of the k1 coefficient is Δk1 = 0.089 K hPa−1, which is an increase of 0.115%, more than 5 times the uncertainty quoted in RU02. This change in the k1 coefficient is particularly significant for the assimilation of GPSRO data because the use of the RU02 value produces a systematic 0.115% increase in the forward modeled bending angles from the upper troposphere upward [Lewis, 2008], where the measurements are given most weight in the assimilation process. The inconsistency between the k1 value given in RU02 and the values given by HS75 and BEV94 is particularly surprising, since they have essentially performed the same exercise, with the aim of providing a statistically robust estimate of k1, using the available experimental data. Two obvious reasons for the difference are apparent [see also Cucurull, 2010]. First, part of the discrepancy in k1 values can be attributed to a simple numerical inconsistency, whereby it has been assumed that 0°C = 273 K, when deriving k1 from measurements of N, using k1 = NT/P. This approximation has been made by a number of authors, who have tabulated experimental results, including SW53, HS75, and BEV94. As a consequence, the k1 values derived from experiments given in Table 4 of RU02 are systematically larger, by a factor of (273.15/273) or, equivalently, 0.055%, than results from the same experiments given in both Table 1 of HS75 and Table A1 of BEV94. This accounts for almost half of the total difference in the k1 values.
 A second reason for the discrepancy is that the coefficients given in Table 1 of HS75 and Table A1 of BEV94 are for dry, CO2 free air (note that SW53 account for 0.03% CO2 by increasing their raw data by 0.02%). These k1 coefficients should be used in a four term refractivity formula, with a new term that explicitly accounts for the atmospheric CO2 contribution to the refractivity. This fact does not appear to be mentioned by BEV94, where it is used in a three-term expression, but it is noted by HS75 (p 871). The RU02k1 value includes 0.0375% of CO2. Including 0.0375% CO2 increases a CO2 free k1 value by ∼0.02. If we combine this modification with the error associated with misspecifying 0°C, we can increase the HS75/BEV94k1 value from k1 = 77.60 to k1 = 77.66, which is two thirds of the Δk1 = 0.089 value.
 The remaining differences between the k1 values are probably attributable to data selection. RU02 only uses data from 9 of the 20 experiments included by HS75 and BEV94. For example, RU02 disputes the use of refractivity data which has been extrapolated from the visible or infrared, which tends to bias the derived k1 values low. In summary, the k1 coefficient suggested by RU02 appears to be more robust and defendable than the standard value.
2.2. Including Nonideal Gas Effects When Deriving k1 From N
 An additional complication is the inclusion of nonideal gas effects when deriving k1. In general, most authors appear to derive the k1 from a measured N using k1 = NT/P, neglecting nonideal gas effects by assuming the density is given by ρ = P/RT, but in some cases these details are unclear. For example, Smith and Weintraub [1953, p. 1036] state in connection with the data given in their Table 1 that “[t]hese values are also given on a real rather than an ideal gas basis,” but they then use the ideal gas equation when deriving their k1. Similarly, the data given in Table 1 of Boudouris [1963, p. 656] is “after the Van der Vaals correction for real gases for normal temperature and pressure.” However, further details of the nature and size of the correction are not given.
Thayer  introduced a more general, three-term expression for the refractivity,
where Pd is the pressure of dry air and Zd and Zw are the dry air and water vapor compressibilities, respectively. The compressibilities account for nonideal gas effects, such as the finite size of the molecules and mutual attraction, and they are a function of pressure, temperature, and water vapor pressure. In the atmospheric conditions of interest, the compressibility is less than unity, so their inclusion in equation (7) increases the computed value of N. One interesting implication of equation (7) is that if refractivity coefficients are derived neglecting nonideal gas effects, they are not really constant because each coefficient has incorporated the compressibility factor, which is itself a function of the atmospheric state.
Thayer  has included nonideal gas effects when deriving k1 from a measurement of N. In fact, he derives [Thayer, 1974, equation (6)] an estimate of k1 from the same experimental data and refractivity value used by SW53, N = 288.04 ± 0.05, but correctly uses 0°C = 273.15 K. The coefficient is given by k1 = (NTZd)/P, where (1/Zd = 1.000588) is the inverse of the compressibility of dry air for P = 1013.25 hPa and T = 273.15 K. The computed k1 value is
It is interesting to note that the SW53 error associated with assuming 0°C = 273 K is almost completely canceled as a result of the additional error introduced by ignoring nonideal effects; (273.15/273) = 1.000549 and 1/Zd = 1.000588, nwhen P = 1013.25 hPa and T = 273.15 K. This means that the Thayer and SW53k1 values are in very good agreement. Unfortunately these values should differ because they are used in different formulations of the refractivity equation. Kursinski et al. [1997, section 3.9.1, p. 23,447] based their error estimate for the SW53k1 coefficient of less than “0.03% above 500 mbar,” on comparisons with Thayer's expression. However, note that RU02 disputes Thayer's k1 value because, like SW53, it is partly based on an optical measurement which has been extrapolated to radio wavelengths, and this is known to bias the k1 estimate low. Cucurull  has tested the Thayer refractivity coefficients in a refractivity equation which does not include compressibility. This approach is questionable because the Thayer coefficients have been reduced to account for their use in an equation that includes compressibility (e.g., note the 1.000588 factor in equation (6) [Thayer, 1974]).
RU02 does not include nonideal gas effects when deriving k1 from experimental N values, and it is conceded that the introduction of compressibility requires further evaluation. Introducing a compressibility factor of Zd = 1/1.000588 would reduce the RU02 best average value from k1 = 77.689 to k1 = 77.643. RU02 argues that nonideal gas effects should only be included when deriving k1, if the k1 value is going to be used in an expression for N like equation (7) that includes the compressibility terms. Conversely, if k1 is derived ignoring nonideal effects, it should only be used in an expression for N that does not include compressibility (e.g., equation (2)). It is important to emphasize that by construction, the refractivity expression including compressibility, and using the adjusted k1, should produce the same refractivity value at P = 1013.25 hPa and T = 273.15 K as the original RU02k1 in equation (6), in order to be consistent with the refractivity measurements.
3. Forecast Impact Experiments
3.1. Experiments Assuming an Ideal Gas
 There is clearly a degree of uncertainty in the refractivity coefficients that arises as a result of assumptions made by different authors when processing the experimental refractivity data. This has probably not been fully appreciated within the GPSRO community, so it is important to establish whether it has any significant implications for the assimilation of GPSRO data in operational NWP and reanalysis applications. Therefore, we have investigated the analysis and forecast sensitivity to the refractivity coefficients used in the assimilation of bending angle profiles in a series forecast impact experiments. Similar work has been presented recently by Cucurull , who investigated the sensitivity of refractivity profile assimilation to the assumed coefficients in the NCEP assimilation system. The experiments presented here use the CY35R1 version of the ECMWF analysis system and are run in the incremental four-dimensional variational assimilation (4D-Var) configuration, at T159/159 resolution, with 91 levels in the vertical. The experiments cover the period 1 December 2008 to 31 January 2009 and assimilate all the data types that were used operationally during this period. Variational bias correction (VarBC) [Dee, 2005] is used to correct the satellite radiance measurements. The GPSRO measurements from the COSMIC constellation and the GRAS instrument on MetOP-A are assimilated with a 1-D bending angle operator [Healy and Thépaut, 2006]. The control experiment (CTL) uses the current operational implementation of the 1-D bending operator, with the SW53 refractivity coefficients (equation (4)). The Rüeger experiment (RUEG) is identical to the CTL experiment, except for the use of RU02 best available coefficients in a three-term expression (equation (6)), to evaluate the refractivity on the model levels in the 1-D bending angle operator.
Figure 1 shows COSMIC-4, noise normalized observation minus background (o - b) bending angle departures ((o - b)/σo, where σo is the assumed observation error), in the Southern Hemisphere for the RUEG and CTL experiments. Note that the background b is a short-range NWP forecast and the analysis a is after the 4D-Var assimilation. As expected [Lewis, 2008], the bending angles simulated with the Rüeger coefficients are larger in the upper troposphere and stratosphere, so the (o - b) departures are systematically smaller. This negative shift in the departures is evident for all instruments and all geographic regions. Figure 2a shows the zonally averaged mean temperature analysis differences (RUEG minus CTL) averaged from 1 December 2008 to 31 January 2009. The use of the Rüeger coefficients produces a systematic cooling below 300 hPa. This cooling reduces the height of the upper tropospheric/stratospheric model levels, and this partially compensates for the larger forward modeled bending angles. The 300 hPa geopotential differences are shown in Figure 2b, giving an indication of the spatial variation of the cooling. The mean geopotential height, (geopotential/9.80665), of pressure levels above 300 hPa is ∼5 m lower in the RUEG experiment, south of 50°S. We have verified that these changes are primarily caused by the k1 coefficient by running a “mixed” experiment, using the RU02k1 value in SW53 two term formula (equation (4)) giving
The results using (9) for refractivity are almost identical to the RUEG experiment in terms of the temperature and height biases, confirming that the difference in the k1 coefficient is the most significant change.
 The tropospheric cooling produces mixed results when the short-range forecasts are compared with radiosonde temperature and height measurements (Figures 3 and 4). In the Northern Hemisphere, the temperature biases are improved from 850 to 150 hPa by typically a few hundredths of kelvins, and the geopotential height departures are generally better except near 250–300 hPa. The negative temperature bias against radiosondes in the Northern Hemisphere is probably related to the assimilation of aircraft temperature measurements which are biased positive and tend to produce a negative temperature bias against radiosondes. This problem is most significant in the Northern Hemisphere where the number of aircraft temperature measurements is greatest. The results in the tropics and Southern Hemisphere show that the RU02 coefficients tend to increase the positive temperature biases below 500 hPa and geopotential height biases throughout the atmosphere. In the Southern Hemisphere, the bias at 100 hPa increases from 6.1 to 9.3 m and in the tropics it increases from 8.7 to 11.4 m. Note that the results shown in Figures 3 and 4 are for all radiosonde types, but restricting comparisons to just RS92 radiosonde measurements does not change the main results.
 In general, the forecast scores are neutral when the RUEG and CTL experiments are verified against their own analysis. The forecast scores against temperature and geopotential height observations in the tropics and Southern Hemisphere are slightly worse for the RUEG experiment, primarily as a result of increased temperature biases in the troposphere.
3.2. Including Nonideal Gas Effects in a GPSRO Operator
 Nonideal gas effects are normally neglected in NWP, but at the ECMWF/GRAS-SAF workshop, Aparicio et al.  suggested that they needed to be included in the GPSRO observation operators to reduce a forward model bias. Aparicio et al. have implemented the RU02 refractivity coefficients and found that the assimilation of GPSRO refractivity profiles increases the bias of short-range forecasts with respect to radiosonde height measurements. Aparicio et al.'s results are qualitatively similar to those given section 3, but he found that he could reduce the bias by including nonideal gas effects in the integration of the hydrostatic equation used in his refractivity observation operator. Nonideal gas effects have now been introduced into the 1-D bending angle operator and this has been tested in the assimilation system.
 The bending angle observation operator contains essentially three major components. The first is the integration of the hydrostatic equation, to compute the height of the model levels. The second is the evaluation of refractivity on the model levels and the third is the evaluation of the bending angle integral. Nonideal gas effects should be included in both the integration of the hydrostatic equation and the evaluation of the refractivity on the model levels. Note that Aparicio et al.  do not include the impact of nonideal gas effects on the evaluation of the refractivity on the model levels.
where P is the total pressure, ρ is the density, R is the gas constant for dry air, Tv is the virtual temperature, and Z is the compressibility of moist air. For an ideal gas, Z = 1, but for air in the troposphere, typically Z ∼ 0.9995, so the departure is 0.05%. Davis  and Picard et al.  provide a polynomial expansion for Z, which is straightforward to implement in observation operators. If we include the compressibility in the hydrostatic integral, the geopotential height h becomes
where go = 9.80665 m s−2. Given that Z < 1, it is clear that including compressibility reduces the height of model levels. In the bending angle operator, the thickness between the ith and (i + 1)th model levels initially calculated neglecting nonideal gas effects [Simmons and Burridge, 1981], Δh′i,i+1, is scaled by the average of the compressibility at ith and (i + 1)th levels, to give Δhi,i+1 = 0.5 × (Zi + Zi+1) × Δh′i,i+1. In simulations, we have found that the height of model levels near 100 hPa can be reduced by dh ∼ 7 m. Assuming a refractivity scale height of 7 km, ΔN/N = −dh/7 km, this translates into a reduction in the forward modeled refractivity or bending angle value of Δα/α = −0.1%.
 The second step where nonideal effects should be included is the computation of refractivity on the model levels using [Thayer, 1974]
where Zd and Zw are the compressibility of the dry air and water vapor, respectively. Note how the combined effects of these changes partially cancel. The heights of the model levels are reduced, meaning that the refractivity on a given height level is smaller. However, the inclusion of compressibility in the refractivity computation increases the value. In fact, if the compressibility factor was a constant for the entire profile, it would have little impact on the simulated bending angle values, because the hydrostatic and refractivity computation changes should cancel each other out.
 We can split the impact of nonideal gas effects on the assimilation of GPSRO measurements into three parts. The first is how their inclusion changes the hydrostatic integration. The second is the impact of using an expression for refractivity which includes compressibility (equation (12)), and the third is ensuring that the refractivity coefficients have been adjusted appropriately for use with compressibility factors. For nonideal gas effects to be introduced in a consistent way, all three aspects of the problem should be considered. Therefore, the impact of introducing nonideal gas effects has been investigated in three experiments. In the first experiment, “COMP1,” the compressibility has been introduced in the hydrostatic integration, but not in the computation of refractivity on the model levels (which uses the Rüeger coefficients). This experiment essentially reproduces the approach adopted by Aparicio et al.  but applies it to a bending angle operator rather than a refractivity operator. In the second experiment, “COMP2” compressibility is used in both the hydrostatic integration and refractivity computation, where Rüeger's coefficients (equation (6)) are used in equation (12). The third experiment, “COMP3,” is identical to COMP2 except that Rüeger's k1 value has been adjusted downward to k1 = 77.643 = (77.689/1.000588), to account for the fact it is being used in a refractivity formula that includes compressibility factors [Thayer, 1974]. Note that the 1.000588 is the inverse of the compressibility factor at P = 1013.25 hPa and T = 273.15 K. This value is used to ensure that the refractivity expressions with and without compressibility both give the same refractivity value at P = 1013.25 hPa and T = 273.15 K and are therefore consistent with the measurements. Similar adjustments have not been made to the k2 and k3 values, but it has already been noted that these terms do not have a major impact on the temperature and geopotential height biases shown here.
 The inclusion of compressibility reduces the simulated bending angle values by ∼0.1% in the lower/middle stratosphere and improves the agreement with the CTL experiment. The mean temperature analysis differences of the COMP1, COMP2, and COMP3 minus the CTL experiments are shown the in Figure 5. These should be compared with Figure 2, noting that the temperature range has reduced in Figure 5. It is clear that the inclusion of compressibility has in general improved the agreement with the CTL experiment, but the level of improvement varies for each experiment. The COMP1 experiment performs better than COMP2, in terms of reducing the temperature biases relative to the CTL. This is because in the COMP2 experiment the reduction of the simulated bending angles that is a result of including compressibility in the height calculation is partly canceled by the increase in the refractivity values that is a result of including compressibility in the expression for refractivity. However, although COMP1 produces slightly better results because it improves the fit to radiosonde height measurements, it is a partial approach, which is difficult to justify on theoretical grounds. The difficulty with COMP2 is because the RU02 coefficients are being used in an expression that includes compressibility, when they were derived assuming this was not the case. This inconsistency has been removed in COMP3, where the k1 value has been reduced. The implementation tested in COMP3 appears to be the most theoretically consistent, and in fact the COMP3 experiment has a better fit to the radiosonde height measurements in the Southern Hemisphere than the CTL experiment (Figure 6). Overall, these results clearly demonstrate that the introduction of compressibility can have a significant impact on the fit to radiosonde measurements.
4. Discussion and Conclusions
 The results presented above have demonstrated the analysis and forecast sensitivity to the k1 refractivity coefficient used to forward model the bending angle observations. It has been shown that an increase in the k1 of 0.115% suggested in RU02 leads to a systematic increase in the simulated bending angle values in the stratosphere. This results in a cooling in the troposphere which improves radiosonde biases in the Northern Hemisphere but degrades the fit to radiosonde temperature and height measurements in the tropics and Southern Hemisphere. Nevertheless, the evidence in support of the RU02 values appears to be strong. Indeed his detailed analysis has highlighted some problems with the interpretation of the estimates given by HS75 and BEV94, namely (1) assuming 0°C = 273K when deriving k1 from a measured N and (2) the coefficients are valid for dry, carbon dioxide-free air.
 It is clear that the HS75 and BEV94 studies should not be cited in support of the SW53 value. Despite these deficiencies, Cucurull  has recently adopted the BEV94 values for operational use at NCEP, based on the pragmatic approach that they produce the better forecast scores. This possibly illustrates that the performance within an NWP system is only relevant to that particular system, and this approach is probably not well suited for an objective assessment of the most accurate coefficients.
 One limitation with the RU02 best average values are that they neglect nonideal gas effects, and it is noted that this simplification requires further evaluation. Neglecting nonideal gas effects when deriving the refractivity coefficients produces coefficients that are weakly dependent on the atmospheric state. Furthermore, in the context of GPSRO assimilation, compressibility should also be included in the integration of the hydrostatic equation; Aparicio et al.  have recently demonstrated that including nonideal gas effects in a GPSRO refractivity forward model that uses the RU02 coefficients improves the short-range forecast biases with respect to radiosonde geopotential height measurements in the Southern Hemisphere. We have essentially reproduced these results with a bending angle forward model but have also shown that the RU02 coefficients with nonideal gas effects give similar results to the control experiment, using SW53 and neglecting nonideal gas effects. We obtain the most consistent results in the Southern Hemisphere when compressibility is included in both the hydrostatic integration and the evaluation of refractivity, and with the RU02 value reduced to k1 = 77.643, to account for its use in an expression that includes compressibility. Therefore, it appears we may be obtaining the right results for the wrong reasons when using the SW53 or BEV94 values operationally. However, this good fortune should not detract from the fundamental problem, which is the larger than expected uncertainty in the k1 refractivity coefficient. This study reinforces the need for new measurements, because it is not possible to determine the best coefficients from the assimilation results. On the contrary new, accurate measurements of the coefficients should be used in the GPSRO forward models, and then we need to investigate how the assimilation system reacts to them. Nearly all of the experimental measurements date from the 1950s and 1960s, and RU02 considers the best available measurements for deriving k1 were performed by Newell and Baird . Bevis et al. [1994, p. 382] acknowledged this problem and stated that “we would like to see some contemporary determinations of the refractivity coefficients,” and Cucurull  has recently expressed a similar view. In addition, some clarification of how and when nonideal gas effects are included is required because this is unclear in some of the older literature.
 All GPSRO forward modeling and retrieval techniques are reliant on accurate knowledge of the refractivity coefficients, and it appears that previous estimates of their accuracy have been too optimistic. It is now clear that the systematic error in the forward modeled bending angle values may be of order ∼0.1%, as a result of the uncertainty in the refractivity coefficients. As noted earlier, one of the strengths of GPSRO measurements is that they can be assimilated without bias correction, and as a consequence they anchor the bias correction of satellite radiances in operational NWP and reanalysis (ERA-Interim) systems. In addition, it has also been suggested that GPSRO measurements be used in climate monitoring and climate model testing activities [Leroy et al., 2006]. Clearly, uncertainty in the refractivity coefficients will also have some impact on these applications.
 In conclusion, we have examined the sensitivity to the assumed refractivity coefficients and confirmed the results of Aparicio et al. , showing that nonideal gas effects in GPSRO bending angle operator have a demonstrable impact on NWP analyses. New measurements of the refractivity coefficients at radio frequencies, with measurement uncertainties sufficiently small to resolve differences between current estimates, should be encouraged. These would have important applications in operational NWP, reanalyses and climate studies and would strengthen claims that GPSRO is one of few measurement types to offer (metrological) traceability.
 Sean Healy is funded by the EUMETSAT GRAS SAF. Mike Rennie is thanked for many useful discussions and for commenting on an early draft of the paper. J. M. Rüeger is thanked for providing a copy of Rüeger . Josep Aparicio is thanked for providing a draft copy of Aparicio et al. , and Lidia Cucurull is thanked for providing a draft copy of Cucurull .