Applications for space-based GPS technology have extended to the atmospheric field during the last two decades. More recently, numerical weather prediction (NWP) centers started incorporating global positioning system (GPS) radio occultation (RO) soundings into their operational assimilation algorithms, resulting in a significant improvement in weather forecasting skill. The main reasons for such benefits are the unbiased nature of the GPS RO measurements, high accuracy and precision, all-weather capability, and equal accuracy over either land or ocean. Profiles of refractivity or bending angle are typically used, owing to the relative simplicity of their forward operators and the computational economy this implies. Although the NOAA National Centers for Environmental Prediction (NCEP) are using refractivities in their operational configuration, a bending angle forward operator has been developed, tested, and implemented and was scheduled to replace the assimilation of refractivities operationally in May, 2012. Results from the NCEP's bending angle method (NBAM) show improvement over the assimilation of refractivities in all atmospheric fields being evaluated. A detailed description and evaluation of NBAM is presented in this article, as well as the advantages this code offers over the assimilation of refractivities and other existing algorithms that assimilate GPS RO bending angles.
 Global positioning system (GPS) satellites are transmitting signals at two microwave (L-band) frequencies. Use of the dual frequency is motivated by the dispersive nature of the ionosphere. Briefly, the GPS radio occultation (RO) technology consists of placing additional satellites at lower orbits that allow tracking of the GPS signals. From a low-earth-orbit (LEO) satellite standpoint, an occultation occurs when the transmitting GPS satellite sets or rises behind the Earth. During an occultation, the rays connecting the GPS and LEO satellites scan the atmosphere quasivertically, providing information on the thermodynamic state of the atmosphere. The trajectory of the ray path tangent point, the point of closest approach to the Earth, is not strictly vertical during the occultation, mainly because of the different angular velocities between the GPS and the LEO satellites but also because of the existence of horizontal gradients of atmospheric refractivity. While scanning the atmosphere, rays are refracted due to the different density and moisture of the atmospheric layers. The total amount of bending accumulated during the ray path trajectory can be derived from Doppler shift measurements and from knowledge of the position and velocities of the LEO and GPS satellites, along with the assumption of bilateral symmetry around the ray path tangent point [e.g., Melbourne et al., 1994; Kursinski et al., 1997; Rocken et al., 1997]. After removing the contribution from the ionosphere to the bending angle, profiles of refractivity are then derived from soundings of atmospheric bending angle under global spherical symmetry of the Earth atmosphere and with the use of some auxiliary climatology information.
 Soundings of GPS RO observations are being assimilated worldwide at most operational NWP centers [e.g., Healy and Thépaut, 2006; Cucurull and Derber, 2008; Aparicio et al., 2009; Rennie, 2010; Anlauf et al., 2011]. Typically, soundings of retrieved bending angle or refractivity are used, and the data assimilation algorithms neglect the horizontal gradients of atmospheric refractivity. A forward operator for refractivity is technically easier to implement than a forward operator for bending angle. Essentially one has only to interpolate modeled pressure, water vapor, and temperature values from the model grid points to the location of the observation, although the dependence of the geometric height of the model levels on the model variables must be taken into account as well. However, the resulting modeled refractivity would match the retrieved refractivity (assuming perfect model and retrieved refractivities) only if the atmosphere were strictly spherically symmetrical, because only in this case would the GPS RO-retrieved refractivity be a true measurement of the refractivity at that height. A similar statement also applies to bending angles.
 Furthermore, some climatology or auxiliary information is used to retrieve refractivities from bending angle profiles. An additional challenge arises under superrefraction conditions, in which conversion of bending angles to refractivities formally results in a negative bias below the height where superrefraction occurs [Sokolvskiy, 2003]. Bending angles retrieved under the assumption of local bilateral symmetry around the ray path tangent height, rather than global spherical symmetry, do not require climatology information nor suffer from the negative bias in the lower troposphere caused by superrefraction conditions. In addition, measurement errors are vertically less well correlated than in refractivity profiles because there is no use of an Abel transform. Finally, bending angle is retrieved earlier than refractivity in the processing of the GPS RO observations, which makes it more attractive from a data assimilation point of view.
 NOAA/NCEP currently assimilates profiles of refractivity in its operational global data assimilation system [Cucurull, 2010]. The idea of assimilating bending angles was first introduced by Eyre , and practical implementations have been developed by several authors [e.g., Zou et al., 1999; Palmer et al., 2000; Liu et al., 2001; Healy and Thépaut, 2006]. Although a preliminary forward operator for bending angle was developed at NCEP in 2006 [Cucurull et al., 2007], its evaluation showed some problems in prediction skill in the Northern Hemisphere extratropics (NH, latitudes above 20°N) [Cucurull et al., 2008]. Recently, this bending angle forward operator has been revisited and significantly improved. On top of the forward operator, which maps model variables to the observation space, the quality control (QC) procedures and observation error characterization have been updated as well. This article describes the improved bending angle forward operator (NBAM) and evaluates its performance against the assimilation of profiles of refractivity in terms of weather forecasting skill. In addition, its sensitivity to several components of the forward operator configuration is analyzed.
 The article is organized as follows. The implemented NBAM is described in section 2. Evaluation of some differences between NBAM and a forward operator for refractivity is presented in section 3 for a single analysis case. Section 4 presents the design of some long-term experiments and discusses the performance of the NBAM configuration in terms of weather forecast skill. Finally, section 5 summarizes the main conclusions.
2 A Local Forward Operator for Bending Angle: NBAM
 The total refractivity (N) of moist air in the neutral atmosphere at microwave wavelengths is given by the following three-term expression [Thayer, 1974]:
where Pd is the pressure of the dry air, Pw is the pressure of the water vapor, and T is the absolute temperature. The k1, k2, and k3 are the atmospheric refractivity constants, and and are the inverse compressibility factors that take into account small departures from the behavior of an ideal gas.
 The NCEP's forward operator for GPS RO observations uses equation (1) to simulate observations of refractivity and approximates the compressibility factors to one, thus ignoring the nonideal behavior of the atmospheric gasses. Instead of using Thayer's original refractive coefficients, these are provided by Bevis et al. . A detailed description of the implementation of this forward operator, as well as the optimization of the QC procedures and observation error structures, has been given by Cucurull . A more accurate expression for the refractivity equation has recently been proposed by Aparicio and Laroche . Despite the good results in terms of weather forecasting skill, the use of refractivity data has some well-known limitations. In particular, it ignores the existence of horizontal gradients of refractivity in the atmosphere, it requires the use of auxiliary data such as a climatological model, and it is affected by negative biases in the lower troposphere under superrefraction conditions. Superrefraction conditions occur when the vertical gradient of atmospheric refractivity is so large that the ray is trapped in a layer and never leaves the atmosphere [Sokolovskiy, 2003]. The residual of the ionospheric compensation will overshadow the bending angle information above a certain altitude (~30 − 60 km, depending on the ionosphere conditions). To overcome this limitation, profiles of bending angle are combined with some auxiliary information (typically, a climatological model) in a statistically optimal way [e.g., Sokolovskiy and Hunt, 1996]. Profiles of refractivity are then derived from these optimized bending angle profiles. As a result, profiles of refractivity are affected by climatology.
 To overcome some of these limitations, the assimilation of bending angle profiles prior to the optimization step instead of refractivity profiles is recommended. On the other hand, because the variability of the atmospheric bending angles as a function of height is larger than that of refractivities, owing to the variability of the vertical gradients of refractivity, their use in data assimilation algorithms is more challenging. Mainly, this is a consequence of the lower vertical resolution of NWP models compared with the GPS RO observations, but, in addition, bending angles in the middle to upper stratosphere can be noisier as a result of the significant residuals of the ionospheric compensation there. Since profiles of refractivity are estimated from optimized profiles of bending angle through an Abel inversion, upper-level refractivities are less noisy.
 Provided the index of refraction n, simulations of bending angle α as a function of the impact parameter a for a spherically symmetric atmosphere can be estimated from the formula of Bouguer [Born and Wolf, 1980, p.123] by evaluating the following integral:
(x = nr),where r is the distance between the center of symmetry and a point on the ray path. The magnitude x is the refractional radius, and it would be the impact parameter for a ray with a tangent point height r. The total refractivity N is defined as N = (n − 1) × 106.
 Note that a singularity exists in equation (2) at the lower limit of the integral. NBAM avoids the numerical singularity in the denominator of the integrand by evaluating the integral in a new grid, s, where
 The integral in equation (2) is then evaluated in an equally spaced grid in s, so the trapezoidal rule can be easily and accurately applied. As is the case for the refractivity operator, the bending angle code accounts for the drift of the tangent point within a profile. Consideration of the obliquity of the GPS RO profiles has been shown to provide better results [Foelsche et al., 2011; Cucurull, 2011; Healy, 2011, personal communication].
 An earlier version of this forward operator was developed and tested at NCEP in 2006. Although the code performed reasonably well for the Southern Hemisphere extratropics (SH, latitudes below 20S), it had some clear deficiencies in the NH that prevented its implementation in the operational system [Cucurull et al., 2008]. NBAM presents many improvements over the earlier version of the bending angle code. In particular, the two-term refractivity expression [Smith and Weintraub, 1953], which is needed to evaluate equation (2), has been replaced with equation (1), and there is a better fit to the observations in areas of complex topography. The NBAM forward operator passed all the tangent linear and adjoint tests.
 NBAM has the advantage of not requiring the atmospheric refractivity to decay exponentially with height. In fact, if N is formally forced to vary exponentially within a model layer, the NBAM bending angle forward operator can become highly sensitive to perturbations in model variables when the integral is evaluated in the new grid. This is exacerbated by the fact that the model layers are not equally spaced, resulting in spurious features in the gradients of refractivity if the interpolation is not performed carefully. Continuity of the vertical derivative of refractivity is desirable, as it is primarily from the ray integration of this vertical gradient that the bending angle is computed in the forward operator. Our forward operator makes use of a quadratic interpolator that preserves continuity of the refractivity values and their derivatives in both the model vertical grid and the new integration grid and ensures continuity of both the forward model and its derivatives with respect to model variables perturbations. This is accomplished by centering Lagrange polynomials [Abramowitz and Stegun, 1970] of some chosen even degree (such as quadratic) at each grid point, but averaging the proximate pair of them within any grid interval with weights that linearly partition unity across the extent of the grid interval. This forward operator requires only smooth values of refractivity to apply the interpolator method accurately.
 Owing to the simplicity of applying the trapezoidal rule (essentially a simple inner product), the procedure is fast and does not require the relatively costly evaluation of exponential or error functions to achieve adequate accuracy. Provided the intervals of variable s are exactly uniform and the data smooth at the scale of this discretization, the effective numerical accuracy of the evaluation of the whole integral extending between the domain extremes where the data have dropped away to negligible values is extremely good, far better than the merely second order of accuracy one obtains from an otherwise comparable integral either when the discretization intervals are uneven or when the data at the domain extremes remain of appreciable magnitude (this generic result is easily demonstrated numerically in simple idealized simulations). In addition, by not explicitly computing exponential or error functions, we are also avoiding the assumption that these are exactly the functional forms beyond or between the grid points.
 Two additional capabilities have been incorporated in NBAM. First, and based on the work of Aparicio et al. , the forward operator (equation 2) has been modified to allow the use of compressibility factors in the computation of the geopotential heights of the model layers. This step in the forward operator is needed to locate a GPS RO observation within the model vertical grid. When the compressibility factors are considered in the code, the refractive constants k1, k2, and k3 provided by Rüeger  are used instead of the Bevis coefficients. Although Rüeger-derived coefficients are theoretically more accurate, they were found to introduce biases in the model when used without considering departures from ideal gas behavior. The nonideal gas behavior is introduced only in the hypsometric equation, through the geopotential heights, and it is still ignored in the computation of the refractivity in the integrand in equation (2). The capability of including the compressibility factors is available in both bending angle and refractivity forward operators at NCEP.
 Second, we have extended the top of the profiles from 30 to 50 km when assimilating bending angles as a result of improved background error covariances in the NCEP's model. The extension of the GPS RO profile top in NBAM is also motivated by the fact that the refractivity soundings are heavily weighted with climatology in the upper stratosphere, although the bending angles are not. In NBAM, the QC procedures and error characterization have been tuned up to 50 km. In contrast, the assimilation of refractivities cannot reliably extend above 30 km.
 Quality controls and optimized observation errors following Desroziers et al.  were updated for the bending angle assimilation. As an example, the statistics for the differences between bending angle data and model simulations are shown in Figure 1 for one of the COSMIC satellites (FM4) in the SH and for a period of 10 days (2 − 12 February 2011). Results can be seen before (Figure 1a) and after (Figure 1b) the QC. About 97% of the observations within the model vertical grid and below 50 km passed the QC. Figure 1 also shows the differences between observations and model simulations normalized by the observation error standard deviation (Figure 1c). This number ranges between about one and two, which indicates that the observation error has been characterized correctly. Note that observations in the middle to lower troposphere are not down-weighted in the assimilation code.
 It is important to point out that a forward operator for the assimilation of bending angles reverses the procedure of assimilating refractivities. When soundings of refractivity are assimilated, the inversion from bending angles to refractivities takes place at the processing centers. On the other hand, when the assimilation uses profiles of bending angle, the forward conversion of modeled refractivities to bending angles takes place at the NWP centers. However, because equation (2) is used, which is valid only under spherical symmetry of the atmosphere, NBAM and any other forward operator currently used at the operational centers neglects atmospheric horizontal gradients of refractivity. Consequently, NBAM is still affected by errors induced by deviations from spherical symmetry. More sophisticated forward operators that incorporate some information on horizontal gradients is necessary to reduce these errors [see, e.g., Zou et al., 1999; Sokolovskiy et al., 2004; Syndergaard et al., 2004; Healy et al., 2007]. This will be addressed in future work at NCEP.
3 Assimilation of Bending Angle Observations for a Stand-Alone Case
 The fraction of reduction of the initial cost function provided by a group of observations type x during the minimization process in a variational approach can be computed as:
where Jo and Jf are the initial (background) and final (analysis) total observation cost functions, respectively, and Jo_x and Jf_x are the cost function components of the group of observations type x. The contribution in reducing the cost function for the different observation types in a single assimilation case (1 February 2011 at 12UTC) is shown in percentage in Figure 2a for different GPS RO assimilation configurations. In REF, profiles of refractivity up to 30 km are assimilated. BNDT (“T” for truncated) and BNDC (“C” for complete) use the bending angle code, and they differ in the height at which the assimilation of GPS RO data stops: BNDT stops at 30 km, and BNDC extends up to 50 km (i.e., NBAM configuration). Both BNDC and BNDT use the compressibility factors and the Rüeger coefficients, whereas REF does not. The use of the compressibility factors along with Rüeger coefficients in the assimilation of refractivities or bending angles does not modify the results shown in Figure 2. During the minimization, the total cost function is reduced by 42% in BNDC and BNDT and by 44% in REF. As can be seen in Figure 2a, the largest contribution comes from the radiance observations, which account for ~35% of the total reduction of the cost function. The second largest contributor is the wind observations, with a reduction in cost function of ~30%. In all cases, the third contributor is GPS RO. Note that, in this case, the larger reduction among the three GPS RO paradigms is found in BNDC, resulting in a slight decrease of the contribution from the other major observation types in comparison with either REF or BNDT. For this study case, the use of refractivities in REF seems to contribute slightly more to the reduction of the total cost function than the use of bending angles up to 30 km (BNDT). This might result from the greater complexity in assimilating bending angles than refractivities (see section 2). In fact, the partial cost function associated with the GPS RO observation is reduced by ~75% in REF and by only ~55% in BNDT during the minimization. If the assimilation of GPS RO is extended to 50 km in BNDC, the contribution of GPS RO in reducing the total cost function is the greatest (~18%).
 The contribution of each observation type x in reducing the total cost function can be estimated by normalizing equation (4) with the number of observations in x. This is shown in Figure 2b for the different configurations. Since a large number of radiance observations is used (over 2 million), the contribution of each single observation is very small. The larger contribution from ozone observations in the figure is interesting. Close to 17,000 ozone observations are assimilated in the experiments, and the weight associated with each of these observations appears to be quite significant. This indicates that the ozone observations are used very efficiently in the assimilation algorithms and/or that these observations are weighted very strongly in the assimilation algorithms. The second observation type in efficiency is the GPS RO observations. With a larger number of observations than for ozone (~51,000 in REF and BNDT, and ~83,000 in BNDC), they do contribute to the reduction of the cost function more than any other observation type but ozone. Note that, because BNDC is using many more observations than REF and BNDT, the contribution of a single observation is lower. Some correlation exists between the observations within a profile, so using more observations from the same number of profiles does not necessarily mean that these observations will provide significant additional information on the state of the atmosphere. Some of the information might be redundant, so the impact of each observation might decrease. REF is using a smaller number of GPS RO observations, and refractivities are smooth and behave very well in the assimilation system. Consequently, observations of refractivity have a larger individual impact in the analysis.
 Histograms of the incremental refractivity and bending angle (in percentage) for the same analysis case are shown in Figure 3 for observations that passed the QC procedures. Results are shown for one of the COSMIC satellites (FM1) and for the NH. Results have been stratified in three different vertical ranges: 0 − 20 km, 20 − 30 km, and 30 − 50 km. Observations below 20 km appear to be slightly negatively biased against the background field in REF, BNDT, and BNDC, but a more unbiased probability density function is achieved with the use of bending angles in BNDT and BNDC (Figures 3b and 3c, respectively). In REF, slightly heavier tails are found within the 0 − 20 km range than at higher altitudes. More pronounced tails are found in BNDT and BNDC than in REF between 0 and 20 km, and they persist (although they are less pronounced) between 20 and 30 km. Observations between 30 and 50 km in BNDC appear clearly non-Gaussian. This seems to indicate that a more sophisticated treatment, such as a carefully calibrated nonlinear QC scheme [Lorenc and Hammon, 1988; Andersson and Järvinen, 1999] would be advantageous. This seems especially appropriate for the higher-level observations perturbed by ionospheric effects (but, to a lesser extent, for very low-level data too, since they also exhibit distinct non-Gaussianity for other reasons). Even moderately discrepant observations are uncritically discarded, and any residual information that they contain is entirely lost, with a conventional QC scheme. In contrast, nonlinear QC does not entirely nullify a datum's input. Instead, it applies to each datum a down-weighting factor based on that datum's departure from the interpolated assimilation there (note that this definition is implicit, since the assimilation depends on the data). In this way, the nonlinear QC, by never completely and permanently excluding data during the iterative process of solving the assimilation problem, can allow the assimilation to salvage beneficial information even when, by chance, large departures of essentially “good” data from early iterations of the evolving assimilation would have misled a conventional QC scheme into rejecting such data permanently. We plan to develop, test, and report on such a scheme in the near future.
 The assimilation of profiles of bending angle vs. profiles of refractivity also show a different pattern in terms of normalized differences between observations and their model simulation counterpart. These ratios are shown in Figure 4 for refractivity and bending angle as a function of the model vertical level and for the same stand-alone analysis case. In both cases, the number of observations increases with the model vertical level, as the model layers become wider as they move away from the surface. Note that the ratios for the background field are larger for upper levels (levels ~40 − 45, height of ~15 − 20 km) with the use of refractivity, whereas the largest ratios are found at lower levels (levels ~20 − 25, height of ~3 − 5 km) and the uppermost levels if bending angle is used instead. This indicates that the GPS RO information content will project mostly at different vertical ranges of the atmosphere. The fit of the analysis to the GPS RO observations will depend on the specifications of the background error covariance, as well as on all the observations and corresponding errors being assimilated. As expected, the analyses show lower ratios than the background fields. All levels seem to minimize well, regardless of the choice of GPS RO observation.
 In a linear 3D-VAR scheme and following the notation by Ide et al. , the analysis increment δx of the analysis control variables x, as a result of the assimilation of observations type α, yα, can be expressed as follows:
where H is the linearization of the observation operator H, HT is the adjoint of the observation operator, R−1 is the inverse of the observation error covariance, B−1 is the inverse of the background error covariance matrix, xB is the background field, and d = yα − H[xB]. Although equation (5) is the formal solution of the 3D-VAR problem, in practice the analysis is found using iterative methods for minimization, such as the conjugate gradient.
 To investigate the sensitivity of the analysis increments to the assimilation of GPS RO, we have computed the jacobians (i.e., linearization of the observation operator H with respect to perturbations of model variables) for a high- and a low-level observation in REF and BNDC. The vertical structure of the jacobians shows at what model vertical levels the assimilation of a given observation will project its impact, indicating where the contribution of that observation to the analysis increments will be greater. Along with the jacobians, the analysis increment from the assimilation of a given observation will depend on the difference between the observation and the model simulation, the observation error, the B matrix, and the weights of the interpolation of the model variables to the location of the observation.
 Since the values in equation (5) are different whether one choses to assimilate an observation of refractivity or bending angle, one cannot directly compare the absolute magnitude of the jacobians of two different types of observations to infer which observation will produce the greater model increments and where. Furthermore, and for the same reasons, one cannot directly compare the magnitudes of the jacobians between a high-level and a low-level observation. However, by comparing the jacobians between different types of observations, or simply different single observations, one can analyze the differences between the levels at which the assimilation of a given observation will project its information content. Both the refractivity and the bending angle forward operators are sensitive to the temperature, humidity, and pressure model variables. Jacobians for all these three variables are shown in Figure 5 for REF and BNDC. The assimilation of a high- and low-level GPS RO observation is represented in each panel in Figure 5. The impact of the use of the compressibility factors in the jacobians is insignificant.
 As a result of the use of the geopotential heights in the computation of the forward operator (equation 1), the assimilation of a refractivity observation affects the (virtual) temperature at the model vertical levels surrounding the observation plus all levels below the location of the observation. This can be seen in Figure 5a, where the temperature jacobians for a high-level observation (incremental refractivity difference of 0.33%, geometric height of 24.5 km, model level 50) and a low-level observation (incremental refractivity difference of −0.08%, geometric height of 5.1 km, model level 24) are represented. In the low-level observation case, the impact is greater at level 25 because the observation is closer to level 25 than to level 24 and also because the geopotential heights component of the forward operator makes use of the location of the observation within the interface rather than midpoint levels. Since the observation is located at level 25 with respect to the interface levels, additional weight is projected to level 25, which justifies why the jacobian peaks at level 25. On the other hand, the high-level observation is still located at interface level 50. As expected, in both cases, the impact decreases as we move away from the location of the observation.
 The adjoint of equation (1) projects the moisture information into the levels surrounding the observation, as can be noted in Figure 5b. The component of the forward operator that involves the geopotential heights is not sensitive to the moisture analysis variables, just to the pressure and virtual temperature. In the case of pressure increments, the information is projected to and below the lowest interface level surrounding the observation. For both observations, the increments peak at the surface pressure. In a hybrid sigma-pressure vertical coordinate system, the pressure increments will all project to the surface pressure.
 The jacobians for the bending angle counterpart are plotted in Figure 5d − f. The different structure of the forward operator with respect to the refractivity results in a clearly different pattern for the temperature, moisture, and pressure jacobians. Increments for temperature extend now to the whole range of the model vertical grid for both the low-level (incremental bending angle difference of −0.98%) and the high-level (incremental bending angle difference of −0.05%) observation. This is a result of the upper limit in the integral in equation (2). In practice, this limit extends to a few levels above the model top. As expected, although increments are nonzero at all levels, these are larger around levels surrounding the observation. Similar structures are found for the pressure jacobian (Figure 5f). In the case of the humidity (Figure 5e), the jacobians peak at the levels surrounding and above the location of the observation within the model vertical grid.
 Profiles of GPS RO observations are assimilated along with many other satellite and conventional observations. As a consequence, the impact of the assimilation of an observation within an NWP model is not limited to the use of the observation per se. The analysis increments (i.e., departures from the background field) with the assimilation of a profile of refractivity (REF) vs. a profile of bending angle (BNDC) along with the other observations routinely assimilated are shown in Figure 6. In both cases, the GPS RO profile is the same (47°N, 70°W); the difference arises only on what type of observation is being used (refractivity or bending angle). Because the GPS RO forward operator is sensitive to temperature, moisture, and pressure, we have selected analysis increments for these three variables. It is clear from Figure 6a that the use of bending angle produces greater temperature increments at the uppermost model levels. This is likely caused by the differences in the top of the profiles being assimilated in REF vs. BNDC. There are no observations of refractivity being assimilated above 30 km. Slightly lower increments are found in BNDC in the lower 20 model levels. Although the moisture increments are lower in REF between model vertical levels 18 and 20 (Figure 6b), the opposite situation is found below level 18. Slight differences between both forward operators are also found for the pressure increments (Figure 6c). In this case, lower increments (<0.4 mb) are found in BNDC below level 35.
 This section has addressed the differences resulted from applying two different forward operators to the assimilation of GPS RO, focusing on a single analysis case. This has helped us to understand the differences between the assimilation of bending angles and refractivities. Next, we investigate the performance of NBAM compared with the assimilation of refractivities in terms of prediction skill by running the assimilation system for an extended time.
4 Performance of NBAM Configuration
4.1 Campaign Design
 We performed two global parallel runs at resolution T382L64 to evaluate the performance of the NBAM configuration in terms of weather forecasting skill. The first experiment (PRREF) uses the forward operator for refractivity (REF), and it is the operational configuration for GPS RO assimilation at NCEP. The test experiment (PRBNDC) replaces the refractivity forward operator with the NBAM operator described in section 2. Both experiments cover from 1 February 2011 to 22 March 2011 and used the operational version of the NCEP's Global Data Assimilation System. GPS RO data from the following missions were assimilated in the experiments: COSMIC 1 − 6, MetOp/GRAS, GRACE-A, C/NOFS, SAC-C, and TerraSAR-X.
4.2 Quality Control
 The processing of the MetOp/GRAS GPS RO data is conducted at two different centers depending on the type of GPS RO-derived product. EUMETSAT processes the bending angles, and the Abel transform inversion to recover soundings of refractivity is performed at GRAS/SAF. Typically, we see a larger number of GRAS bending angle profiles than refractivity profiles. For this study, PRBNDC ingested ~14% more GRAS profiles than PRREF. After passing the QC, both experiments assimilated ~97% of the available observations. This number does not include observations above 30 km in PRREF (50 km in PRBNDC), which are directly rejected in the assimilation system prior to any QC.
 QC for the lower troposphere used to reject a significant number of observations, because of limitations in the quality of the data at these low altitudes. However, the quality of the observations in the lower troposphere was greatly improved in recent years with the open-loop technology [Sokolovskiy, 2001], and a large number of soundings now extend down to the surface. For example, ~80% of all COSMIC soundings in the tropics (TR, latitude between 20°S and 20°N) penetrate to within 2 km of the Earth's surface and this number increases to 100% of the profiles in the polar regions. As a result of an increase in quality, the QC was updated accordingly, and NCEP has been assimilating a significant percentage of the observations in the lower troposphere, both in the operational suite and in the NBAM configuration for the missions that have open-loop capability (COSMIC 1 − 6, SAC-C, and TerraSAR-X). This can be seen in Figure 7, in which the number of counts and the percentage of observations being assimilated are shown for both PRREF and PRBNDC experiments. Results correspond to one of the COSMIC satellites (FM4). The data have been distributed in 1-km bins.
 The number of counts in PRREF (Figure 7a) is pretty stable down to 5 km, where it starts to slightly decrease. About 72% of the soundings reach down to the Earth's surface in the NH, 66% in the SH, and 50% in the TR. From the observations received at NCEP, the percentage that passed the QC is close to 100% along the whole vertical range of the atmosphere above 5 km. Below 5 km, the percentage of observations assimilated starts to decrease due to stricter QC. At the surface, we assimilate 68% of the data in the NH, 40% in the SH, and 24% in the TR. A detailed discussion on the GPS RO QC within the NCEP's analysis code for the different latitude ranges of the atmosphere is given by Cucurull .
 The bending angle counterpart is plotted in Figure 7b. In this case, the counts and the percentage of observations being used are shown as a function of the impact height above the ellipsoid (i.e., difference between the impact parameter of the ray and the local radius of curvature). The counts increase below 25 km as a results of (a) the nonlinear relationship between the geometric height and the impact parameter (in the processing of the COSMIC profiles, the geometric height for refractivities is computed first and then the impact height is derived) and (b) the fact that the bins are 1-km fixed bins. The decrease in counts in the lower troposphere corresponds to the decrease in counts in refractivities (Figure 7a). The percentage of observations being used is close to 100% between 9 and 30 km, with a decrease above and below this height range due to the stricter QC. Lower numbers of tropical observations are being assimilated between 16 and 19 km. When the impact height reaches its minimum value, the percentage of data assimilated is ~79% in NH, 55% in SH, and 47% in TR. Note that, by the definition of the impact height, this value is always positive.
4.3 Differences in Analyses
 The assimilation of high-level GPS RO data (>30 km) with the earlier version of the bending angle forward operator showed nonphysical analysis temperature increments [Cucurull et al., 2008]. These larger increments were already evident after just a few assimilation cycles and were noticeable at the higher model levels. This problem has been fixed in NBAM owing to upgrades in the background error covariance and improvements in the assimilation algorithms, as shown in Figure 8. Campaign-average temperature analysis differences between PRBNDC and PRREF are shown at 850 and 10 mb. Differences between the two experiments are reasonable at both levels (<0.4 K at 850 mb and <1.8 K at 10 mb). Verification against radiosondes also shows small differences between the assimilation of refractivities and bending angles.
 The assimilation of bending angles with NBAM results in a slightly warmer tropopause around latitudes 30°N and 30°S by ~0.5 K (Figure 9a). As a result, lower tropopause geopotential heights are found in these latitude zones (Figure 9b). It is interesting that the different forward operators and type of observation to be assimilated can reflect changes in the tropopause characteristics. The differences in surface pressure between PRREF and PRBND are small (< 0.5 mb), and the differences in relative humidity are negligible.
 It is interesting to look at the contribution of the high-level observations to the assimilation of bending angles in NBAM. This is shown in Figure 10, where the net effect of assimilating bending angle observations between 30 and 50 km is quantified. Temperature differences between the assimilation of bending angles stopping at 50 and 30 km are depicted for the 850- and 10-mb pressure levels, respectively. As expected, Figure 10b resembles Figure 8b, indicating that the principal differences between PRBNDC and PRREF at the higher levels are caused by the assimilation of high-level observations in PRBNDC. However, because the assimilation of higher level bending angles affects the lower model levels, differences still exist at lower levels between the two bending angle configurations (Figure 10a).
4.4 Dynamic Forecast Skill
 The assimilation of bending angles with NBAM results in a slight improvement in weather forecasting skill over the assimilation of refractivities. Anomaly correlation scores for the 500-mb geopotential heights are shown in Figure 11 for the NH and SH. The improvement in anomaly correlation in NBAM is 0.1% in NH and 1.3% in SH. Results in the SH are statistically significant at the 95% confidence level. A slight increase in anomaly correlation is also found at the other pressure levels.
 Improvement in forecasting skill is also found for other variables. For example, a decrease in the root-mean-squared errors is found for the 3-day tropical winds (Figure 12). Similar results are found for other latitude ranges and pressure levels. In general, a slight overall improvement is found for all fields and pressure levels when the assimilation of refractivities is replaced with bending angles. Verification against radiosondes also shows positive impact with the use of bending angles. A slight positive impact over the assimilation of refractivities was also found at the U.K. Met Office by using a bending angle forward operator based on the error function and formally assuming an exponential decay of refractivity with the model vertical layers [Rennie, 2010].
 The effects of the compressibility factors in NBAM are much smaller than the differences between PRBNDC and PRREF. The use of observations above 30 km in NBAM results in an improvement in low- and high-level winds for the different latitude ranges. However, results using bending angles only up to 30 km already show better skill scores than PRREF. In summary, the benefits of assimilating bending angles over refractivities already exist regardless of using compressibility factors or extending the top of the profiles. However, results are best when the complete NBAM configuration is used.
 A new methodology to implement a forward operator for the assimilation of bending angles under the assumption of spherical symmetry of the atmosphere has been developed and tested at NCEP. The NBAM method makes use of bending angles rather than refractivities and thus avoids the need for auxiliary meteorological data in the retrieval of the observations. In addition, the negative bias present in the lower troposphere in the refractivity profiles under superrefraction conditions does not apply to profiles of bending angles. Although the use of bending angles is more challenging from an assimilation standpoint, mainly because of vertical water vapor gradients and larger residual errors from the ionospheric correction, its choice is preferred. Furthermore, being an earlier product than refractivities in the GPS RO processing steps, bending angles instead of refractivities are more robust for changes in the processing of the data outside NWP centers.
 This article describes the NBAM algorithms and evaluates its performance against a refractivity forward operator. A detailed analysis has been performed for a single analysis case first, followed by a full parallel testing. Overall, a slight improvement in weather forecasting skill over the assimilation of refractivities has been found for all levels and model variables evaluated in this study. Because the use of bending angles is preferred over refractivities and results are superior from an NWP point of view, NCEP switched to the use of NBAM operationally in May 2012. Although NBAM is still affected by the assumption of spherical symmetry of the atmosphere, it overcomes some of the limitations present in other existing bending angle forward operators.
 GPS RO observations, either bending angle or refractivities, are still affected by receiver noise as well as noise caused by turbulence or convection and ionospheric noise associated with the L1 GPS signal. Better technology should further address these issues. The horizontal gradients of refractivity, caused mainly by water vapor, still require some improvement in the forward operators. The deviation from spherical symmetry of the atmosphere being taken into account is expected to improve the use of the GPS RO observations, particularly for the lower troposphere.
 Finally, our examination of histograms of GPS RO observation innovations stratified by altitude indicates a distinct and systematic heavy-tailed non-Gaussianity, both at near-surface levels and, more decidedly, at the very highest altitudes. These features suggest that the use of a carefully calibrated nonlinear QC scheme might enhance the utility of such measurements in the variational assimilation. This task should be approached with caution; in the general case, the inclusion of carelessly introduced nonlinearities within a variational assimilation can easily inadvertently lead to the formation of multiple minima of the cost function, incurring consequent undesirable numerical complications impeding the iterative search for the desired global minimum (or an adequate approximation to it). However, as elaborated by Purser , systematic ways do exist to formulate plausible heavy-tailed measurement error distributions for nonlinear QC that specifically avoid these difficulties (essentially by maintaining cost function convexity). A future article will report on the performance of a scheme of this kind adapted to the particular statistical characteristics of GPS RO bending angle data.
 L. Cucurull was funded by the Joint Center for Satellite Data Assimilation (JCSDA) Science Development and Implementation (JSDI) program through the FY11 NOAA's Internal and Directed Research Funding Opportunity proposal entitled “GPSRO Support for JCSDA.” The authors thank Drs. Louis Uccellini, Bill Lapenta, and John Ward for providing computer resources. The authors also acknowledge UCAR, NSPO, the U.S. Air Force, EUMETSAT, GRAS SAF, and GFZ for providing the different GPS RO profiles assimilated in the experiments. L. Cucurull thanks Dr. Sokolovskiy for helpful discussions. Finally, we are grateful to the three anonymous reviewers, who helped to improve the manuscript.