# A New Algorithm for the Retrieval of Atmospheric Profiles from GNSS Radio Occultation Data in Moist Air and Comparison to 1DVar Retrievals

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## Abstract

**:**

## 1. Introduction

**x**, given a set of observations

**y**

_{b}and a priori knowledge on a background atmospheric state profile

**x**

_{b}as well as the error covariance matrices of both the observation and background information. The 1DVar can be written as a minimization of the following equation [20]:

**H**[

**x**] denotes a forward operator mapping the state

**x**to the observation space

**y**

_{o}. The matrices

**B**and

**O**are background and observation error covariance matrices, respectively, representing the standard uncertainties and correlations of the background data and the observation (plus forward-modeled) data. Minimizing the cost function J(

**x**) by variation of the state

**x**yields the retrieved state

**x**

_{r}that minimizes the total deviation against background and observational data. The usual selection of

**y**

_{o}in moist-air retrieval by 1DVar is the observed refractivity profile from which temperature, humidity and surface pressure are retrieved as state

**x**

_{r}[17,18,19,20].

**B**and

**O**is critical for the moist air retrieval, since it determines the weights of background and observed data that lead to the formally optimal profiles according to Equation (1). The 1DVar method is successful and retrieved moist profiles have been used in several climate and weather studies and good results were obtained [32,33,34].

## 2. Methodology—The New Moist Air Retrieval Algorithm

#### 2.1. Algorithm Basis

_{1}Rρ

_{d}= c

_{1}(p

_{d}/T

_{d}), where R = 287.06 J kg

^{−1}K

^{−1}is the dry air gas constant [2,35], c

_{1}= 77.60 K hPa

^{−1}is the Smith-Weintraub refractivity formula first constant (“dry term”), and ρ

_{d}, p

_{d}, and T

_{d}are RO-retrieved dry density, dry pressure, and dry temperature, respectively. The profiles p

_{d}(z) and T

_{d}(z) are derived in RO processing by the so-called dry air retrieval step, using the hydrostatic integral and the equation of state (e.g., [2,41]), and are available as input to the moist air retrieval.

_{d}/ρ

_{d}= RT

_{d}, allows formulating the ratio of p

_{d}and T

_{d}in terms of generic refractivity at any altitude level of z:

_{w}is the water vapor volume mixing ratio. The latter relates to pressure p, water vapor partial pressure e, and specific humidity q as:

_{w}= 0.622 is the moist air gas constant ratio, b

_{w}= 1 − a

_{w}= 0.378 is the moist air gas constant ratio complement [35,45].

_{0}is unity or close to unity and f

_{1}is close to (c

_{2}/T)/c

_{1}, where c

_{2}= 3.73 × 10

^{5}K

^{2}hPa

^{−1}represents the Smith-Weintraub refractivity formula second constant (“wet term”). The exact values of f

_{0}and f

_{1}depend on which refractivity formula is used and whether ideal gas behavior is adopted. As the current OPSv5.6 and rOPS baseline, the standard Smith-Weintraub refractivity formula is used, corresponding to f

_{0}= 1 and f

_{1}= (c

_{2}/T)/c

_{1}= c

_{T}/T [K] and c

_{T}= c

_{2}/c

_{1}= 4806.7 K. These values are used later on.

_{0}and f

_{1}will be very small (order 10

^{−3}or smaller) for any more advanced refractivity formulation so that they could be readily added as “epsilon terms” within the step 1a / step 1b iteration algorithm (cf. Section 2.2) if desired. Aparicio and Laroche [50] caution that any use of an advanced refractivity formulation beyond the Smith-Weintraub form should also consistently use a correspondingly advanced equation-of-state formulation accounting for non-ideal gas behavior; an aspect that can as well be accounted for by adding “epsilon terms” in the current algorithm.

_{w}, we get the following two mutually equivalent forms:

_{w}= 1/a

_{w}− 1 = 0.608 is the moist air humidity coefficient for virtual temperature [35]. By expressing the moist pressure vertical increment dlnp in terms of the dry pressure increment dlnp

_{d}, and also using Equation (3) to convert q to V

_{w}, we get:

_{d}will be log-linearly discretized over adjacent levels, we can write dlnp = dlnp(z

_{i}) − dlnp(z

_{i}

_{−1}) = ln[p(z

_{i})/p(z

_{i}

_{−1})], where i represents the corresponding level indices. Similarly, we can write dlnp

_{d}= ln[p

_{d}(z

_{i})/p

_{d}(z

_{i}

_{−1})] for the dry pressure increment. Based on these expressions, we can then derive the expression of p at any altitude level z

_{i}as:

_{i}and z

_{i}

_{−1}that leads to matching this change to the fractional moist pressure change. Since T

_{d}is always smaller than T if moisture is non-zero, β is (slightly) smaller than one, expressing that p is changing less than p

_{d}, consistent with the fact that p

_{d}is always larger than p for non-zero moisture [51]. The specific formulation of β, with temperature expressed as mid-layer linear average (arithmetic mean) between the two levels, and water vapor mixing ratio as mid-layer log-linear average (geometric mean), is found helpful for high numerical accuracy at any given level spacing.

_{w}(and hence q via Equation (3)) and p if T is prescribed. We can do this by a simple iteration at any arbitrary altitude level z where a suitably adjacent level has been solved for p before (starting at a “tropospheric top” level with negligible moisture where p

_{d}essentially equals p). If q (and hence V

_{w}) is prescribed, then, for any altitude level, we iterate the pairs of Equations (5) and (11) until T has converged to within a small tolerance dT

_{tol}, and p will be consistent with the converged T.

_{w}has converged to within a small tolerance (dV

_{w}/V

_{w})

_{tol}, and p will then be consistent with the converged V

_{w}. This formulation of the “direct method” of moist air retrieval is highly robust and versatile and applicable to arbitrary non-equidistant vertical grids of any level number (from minimum two levels) and vertical range from a chosen “tropospheric top” level to bottom of profile.

#### 2.2. Algorithm Description

_{b}(z) and its uncertainty u

_{q}

_{b}(z); the observed dry temperature T

_{d}(z) and its uncertainty u

_{T}

_{d}(z); and the observed dry pressure p

_{d}(z) and its uncertainty u

_{p}

_{d}(z). The output profiles include: temperature T

_{q}(z) and its uncertainty u

_{T}

_{q}(z); pressure p

_{q}(z) and its uncertainty u

_{pq}(z), where the subscript q denotes variables retrieved with specific humidity prescribed. Using the prescribed q

_{b}(z), the corresponding water vapor volume mixing ratio V

_{wb}(z) can be calculated using Equation (3). Then, based on Equations (5) and (11), T

_{q}(z) and p

_{q}(z) at altitude level z

_{i}can be expressed as:

_{i}from the initial altitude of our moist retrieval z

_{iniMoist}= 16 km to the bottom level of the profile, iteration over Equations (12) and (13) yields the profiles of T

_{q}and p

_{q}.

_{q}(z) denoted as ${u}_{Tq}^{2}\left(z\right)$ can be obtained from propagating the variance profiles ${u}_{T\mathrm{d}}^{2}\left(z\right)$ and ${u}_{q\mathrm{b}}^{2}\left(z\right)$ based on a linearized version of Equation (12), linearized with some reasonable assumptions (cf. Appendix A):

_{q}

_{2T}= 7727.9 K is the moist air humidity coefficient in temperature error estimation and the square-root of ${u}_{Tq}^{2}\left(z\right)$ is the desired uncertainty profile.

_{b}(z) and its uncertainty u

_{T}

_{b}(z); observed dry temperature T

_{d}(z) and its uncertainty u

_{T}

_{d}(z); and observed dry pressure p

_{d}(z) and its uncertainty u

_{p}

_{d}(z). The output profiles include the specific humidity q

_{T}(z) and its uncertainty u

_{qT}(z) and pressure p

_{T}(z) and its uncertainty u

_{pT}(z). According to Equations (6) and (11), the corresponding water vapor volume mixing ratio profile V

_{wT}(z) and p

_{T}(z) can be expressed as:

_{T}= c

_{2}/c

_{1}= 4806.7 K has been described above. V

_{wT}(z) and p

_{T}(z) can be solved by iterating level by level top-downward from z

_{i}= z

_{iniMoist}to the bottom level. After obtaining V

_{wT}(z), the corresponding specific humidity q

_{T}(z) can be calculated using an inverse version of Equation (3) and its variance ${u}_{qT}^{2}$ can be propagated from the variance profiles of ${u}_{T\mathrm{b}}^{2}\left(z\right)$ and ${u}_{T\mathrm{d}}^{2}\left(z\right)$ based on a linearized version of Equation (16) (cf. Appendix A):

_{qT}(z).

_{pT}(z) can be calculated as:

_{q}(z) and its uncertainty u

_{Tq}(z) obtained in step 1a and also on the prescribed background temperature profile T

_{b}(z) and its uncertainty u

_{T}

_{b}(z), the optimally estimated temperature profile T

_{e}(z) can be calculated by combining T

_{q}(z) and T

_{b}(z) based on inverse variance weighting at all altitude levels:

_{T}

_{e}(z).

_{T}(z) and its uncertainty u

_{qT}(z) obtained in step 1b, and also on the prescribed specific humidity profile q

_{b}(z) and its uncertainty u

_{q}

_{b}(z), the optimally estimated specific humidity profile q

_{e}(z) and its variance ${u}_{q\mathrm{e}}^{2}$(z) can be estimated as:

_{q}

_{e}(z).

_{we}(z), pressure p

_{e}(z), density ρ

_{e}(z), and water vapor partial pressure e

_{e}(z) can be calculated quite straightforwardly since the relevant retrieval operators are known. The corresponding uncertainty profiles u

_{V}

_{we}(z), u

_{p}

_{e}(z), u

_{ρ}

_{e}(z), and u

_{e}

_{e}(z), can also be calculated using variance-based uncertainty propagation, given the state retrieval operators.

_{e}(z), the derived water vapor volume mixing ratio V

_{we}(z) can be calculated according to Equation (3):

_{e}(z) can be calculated using the temperature profile T

_{e}(z) and volume mixing ratio profile V

_{we}(z) based on Equation (11):

_{iniMoist}level as the previous pressure retrievals, is effectively based on the hydrostatic equation (Equation (7) or (8)) (in the convenient variant available in the context of this algorithm) and provides a pressure profile hydrostatically consistent with the estimated temperature and humidity profiles. We call this a hydrostatic-equation-based closure scheme for the retrieval of pressure to emphasize that it is improved over the refractivity-equation-based closure scheme used in the OPSv5.6 approach.

_{e}(z) can be computed as:

_{e}

_{e}(z).

_{e}(z) can be derived by using the equation of state in moist air:

_{ρ}

_{e}(z).

#### 2.3. Modeling of Observation and Background Uncertainties

_{T}

_{b}(z), the background specific humidity uncertainty profile u

_{q}

_{b}(z), the observed dry temperature uncertainty profile u

_{T}

_{d}(z), and the observed dry pressure uncertainty profile u

_{p}

_{d}(z). We sequentially describe below how we estimated these uncertainties for this study.

#### 2.3.1. Observation Uncertainty Modeling

_{T}

_{d}and observed dry pressure u

_{p}

_{d}are modeled following the empirically derived error model developed by Scherllin-Pirscher et al. [51]. Currently, both the OPSv5.6 and the dynamic approach use this model to estimate the observation uncertainty. In the future rOPS system, the propagated individual-profile based observation uncertainties (and error correlation matrices) will be used [43].

_{Ttop}is the top altitude of the troposphere domain, z

_{Sbot}is the bottom altitude of the stratosphere domain, s

_{0}is the standard error (uncertainty) in the upper troposphere/lower stratosphere domain, q

_{0}is the best-fit magnitude parameter for the tropospheric model, and p is the associated exponent parameter. The complete parameter settings are summarized in Table 1.

#### 2.3.2. Background Uncertainty Modeling

_{a}; (3) bias of the mean analysis profile b

_{a}; (4) the mean forecast profile of temperature ${\overline{T}}_{\mathrm{f}}$ and specific humidity ${\overline{q}}_{\mathrm{f}}$; (5) the standard deviations of the forecast-minus-analysis difference profiles s

_{f-a}; (6) the number of values in the analysis and the forecast ensemble N

_{a,f}. These basic variables except the bias of the mean analysis profile are calculated based on statistical calculation using a large ensemble of forecast and analysis profiles in each grid cell. The details of how to extract the ensemble of profiles on the grid and how to calculate these profiles were described by Li et al. [52,53].

_{a}for temperature is estimated by systematic error modeling according to Li et al. [52,53]. It is applied with no vertical variations but with latitudinal variations. The temperature biases are smallest within the ±40° latitude band, where the values are equal to the basic mean magnitude of s

_{0}(0.5 K). Such values increase with the increase of latitude. Poleward of 60°, s

_{0}are 20% higher than their basic mean magnitude in the summer hemisphere but twice their mean magnitude in the winter hemisphere [51,52]. The bias profile b

_{a}for specific humidity is currently adopted as a relative uncertainty value of 5% of the mean analysis humidity, an educated-guess value.

_{b}(representing both for u

_{T}

_{b}and u

_{q}

_{b}) are estimated as:

_{a}and s

_{f-a}here represent the collocated values obtained from a bi-linear interpolation of their values from the four grid points surrounding the tangent-point location of the given RO event. Preparing for this, u

_{a}at each grid point (denoted for clarity as u

_{a_grid}) is estimated as a combination of the systematic biases and the statistical errors [52,53]:

_{T}

_{b}between 10 km and 16 km needs to be penalized to gradually increase in uncertainty at these high tropospheric altitudes, in order to ensure that the observations always safely take increasing weight towards the stratosphere, we modified u

_{T}

_{b}from 10 km to 16 km and used an intentional uncertainty increase of the form:

_{T}

_{top}is 10 km and H

_{T}

_{b}is the “uncertainty scale height” set to 5 km.

_{q}

_{b}is input in form of relative humidity values into the scheme. That is, we first use the collocated background specific humidity uncertainty divided by the collocated mean forecast humidity profile, ${u}_{q\mathrm{b}}/{\overline{q}}_{\mathrm{b}}$, and then use this relative value to multiply it with the collocated actual background profile in order to obtain the specific humidity uncertainty in absolute values for the algorithm.

_{T}

_{d}increases with decreasing altitude from 0.7 K at 16 km to more than 4 K at the surface. Dry pressure uncertainty stays around 0.2% from 16 km to 10 km and then gradually increases with decreasing altitude to about 1.5% at the surface. As noted above, the observation uncertainties are still used as global static profiles, i.e., used globally in the same way, while in the future rOPS they will be as well used dynamically such as the dynamic background uncertainties discussed next.

_{T}

_{b}is a static global profile, being 2 K at 16 km, 0.6 K at 10 km, and about 1.2 K at the surface. In comparison, the dynamic u

_{T}

_{b}exhibits latitudinal and altitudinal variations. u

_{T}

_{b}in polar regions is larger than that in non-polar regions. It is largest in the southern hemisphere polar regions, with values varying from 1.2 K close to the surface to more than 4 K above 12 km. In non-polar regions, the values gradually decrease from high latitudinal regions to low latitudinal regions. Furthermore, our sensitivity test results (not shown) indicate that the dynamic u

_{T}

_{b}exhibits clear seasonal variations, with largest uncertainty in the polar winter hemisphere.

_{q}

_{b}. It is 10% at the surface, increases to about 40% at 7 km, and then gradually decreases to about 15% at 16 km. The dynamic u

_{q}

_{b}exhibits clear latitudinal variations, with largest values (>40% from 3 km up to 10 km) in tropical regions, decreasing towards the poles.

#### 2.4. Inspection of Intermediate Variables

_{T}

_{b}of the simMetOp event, which is located at higher latitudinal regions, is larger than that of the COSMIC event, with values that decrease from 3 K at 16 km to 1 K at 10 km and further to 0.8 K below. For both events, u

_{Tq}is smaller than u

_{T}

_{b}above 10 km, while below 10 km, u

_{Tq}increases quickly and becomes larger than 6 K at bottom altitude levels. The optimally estimated T

_{e}is bounded between T

_{b}and T

_{q}and properly takes more weight from the profile that has ascribed less uncertainty. The differences between T

_{e}and the corresponding reference profile are generally smaller than the differences of T

_{b}and T

_{q}, indicating the effectiveness of the optimal estimation. Comparing dynamic u

_{T}

_{b}and OPSv5.6 uncertainty u

_{T}

_{dOPSv56}, we can see that u

_{T}

_{b}is of similar magnitude as u

_{T}

_{dOPSv56}, with values larger at high latitudes and smaller at low latitudes.

_{q}

_{b}and u

_{qT}, we see that u

_{q}

_{b}is larger than u

_{qT}below 8 km for the simMetOp event and below 7 km for COSMIC event. Above 7 km to 8 km, u

_{qT}increases quickly to large values. In the optimal estimation, q

_{e}takes more weight from the profile with smaller uncertainties in the optimal estimation, and its difference against the reference profile is smaller than the one between q

_{b}and q

_{T}, again indicating the effectiveness of the optimization. The OPSv5.6 specific humidity uncertainty u

_{q}

_{bOPSv56}is a static profile globally, starting from near 20% at bottom altitude levels, increasing to about 40% at 7 km and then gradually decreasing to below 20% at 16 km.

## 3. Results—Algorithm Performance Evaluation by Comparison to 1DVar Retrievals

#### 3.1. Insights from Individual Event Profiles

#### 3.2. Statistical Ensemble Results

_{T}

_{e}are larger than the statistically estimated standard deviations, which is especially related to the fact that u

_{T}

_{e}is calculated using u

_{T}

_{d}(cf. Equations (14) and (21)), which is empirically estimated for real rather than simulated data based on the model by Scherllin-Pirscher et al. [51]. That is, for simulated data, u

_{T}

_{d}is overestimated since the quality of dry temperature of our simulated data is better than real observed data. The propagated specific humidity uncertainties are of similar magnitude compared to the statistically estimated uncertainties.

_{T}

_{e}are basically consistent with the statistically estimated uncertainty, which again indicates the reasonableness of this simplified uncertainty propagation.

_{q}

_{e}, is generally smaller below about 10 km than the statistically estimated uncertainties (except in the SHP region, where absolute moisture content is low). This is due to the reason that observed, background, and analysis specific humidity jointly represent more (noisy) variations over the troposphere than the simplified error estimates used here do capture. In other words, part of the “representativeness error” is not captured.

#### 3.3. Simple Observation-to-Background Weighting Ratio Profiles and Comparative Results

_{obw}, of temperature and specific humidity for the dynamic, OPSv5.6 and ROM-SAF approaches (CDAAC-provided moist profile files do not contain uncertainty information, hence these data are not included here). Since both our approaches and the 1DVar approach used by ROM-SAF are not (fully) linear, it is not an easy task to calculate the real r

_{obw}in a comparable manner. Hence we implemented and inspected an approximation as follows.

_{rbu}) is straightforward and consistently possible for all four datasets, we used this ratio to calculate an approximate r

_{obw}. The retrieval-to-background uncertainty ratio is defined and calculated as ${r}_{\mathrm{rbu}}=100\frac{{u}_{\mathrm{ret}}}{{u}_{\mathrm{b}}}$, where u

_{ret}is the uncertainty profile of the retrieved (optimally estimated) profile and u

_{b}is the corresponding background uncertainty profile. Based on the r

_{rbu}profile, we can then estimate r

_{obw}as:

_{obw}results for temperature and specific humidity, shows that the r

_{obw}for temperature from dynamic, OPSv5.6, and ROM-SAF approaches are generally consistent, with dynamic r

_{obw}comparatively largest. This indicates that, by its observational and background uncertainty choices, the dynamic approach uses more observation information in the temperature retrieval. The temperature results of the three approaches reveal clear latitudinal variations, with less observation information used in the tropics (TRO), where humidity is large, and more in polar regions (especially SHP), where humidity is small and RO likewise accurate.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Detailed Numerical-Algorithm Formulations of Steps 1a and 1b

_{b}and its associated uncertainty ${u}_{q\mathrm{b}}$, the observed dry temperature T

_{d}and its uncertainty u

_{T}

_{d}, and the observed dry pressure p

_{d}and its uncertainty u

_{p}

_{d}. As noted in the main text of the paper, ${V}_{\mathrm{wb}}$ can be calculated using Equation (3). Then, based on Equations (12) and (13), the profiles T

_{q}(z) and p

_{q}(z) can be solved by iteration, level by level top-downward from the level below the first level (z

_{iniMoist}= 16 km) to the bottom level, of the (T-β-p)-three-equation system:

_{i}, initial values for the iteration are (k = 0):

_{dthres}= 240 K is a threshold in dry temperature T

_{d}above which it can typically deviate by more than 1 K from actual T;

_{0}= 8 km.

_{i}> z

_{iniMoist}, use the same formulations to assign T

_{q}and p

_{q}as used at z

_{iniMoist}, i.e., the initial-value formulations under iteration condition (1) above.

_{q}, we first derive a linearized version of Equation (12). Using the approximate assumptions of ${V}_{\mathrm{wb}}\approx {q}_{b}/{a}_{w}$, $d{p}_{d}/{p}_{d}\approx d{p}_{q}/{p}_{q}$, $d{T}_{d}/{T}_{d}\approx d{T}_{q}/{T}_{q}$ and $d{T}_{d}/{T}_{d}\wedge d{T}_{q}/{T}_{q}<<d{V}_{wb}/{V}_{wb}$, which are reasonably valid over the moist air retrieval altitude range, the linearized version becomes:

_{T}

_{q}(z).

_{p}

_{d}(z), is obtained via:

_{b}and its uncertainty u

_{T}

_{b}, the observed dry temperature T

_{d}and its uncertainty u

_{T}

_{d}, and the observed dry pressure p

_{d}and its uncertainty u

_{p}

_{d}. Using these input profiles, we can solve for profiles V

_{wT}and p

_{T}based on Equations (16) and (17), based on iterating level by level as for step 1a above. The (V-β-p)-three-equation system in this case is:

_{i}, the initial values for the iteration are (k = 0):

_{0}= 8 km and T

_{dthres}= 240 K.

_{w}/V

_{w})

_{tol}= 0.01% is the convergence tolerance, yielding V

_{wT}(z

_{i}) and p

_{T}(z

_{i}) as converged values. At all higher levels, ${z}_{i}>{z}_{\mathrm{iniMoist}}$, we use the same formulations to assign V

_{wT}and p

_{T}as used at z

_{iniMoist}, i.e., the initial-value formulations under iteration condition (1) above.

_{minE}= 0.001 g/kg, is because we try to prevent unphysical (negative) values in case T

_{b}< T

_{d}occurs, which can happen within errors of T

_{b}and T

_{d}at upper troposphere levels where q is very small (less than about 0.1 g/kg). We note that the error estimation is unaffected by this low-bounding as it does not depend on q itself. Also, the resulting humidity profile after the optimal estimation step is receiving essentially negligible weight at the high tropospheric altitudes from this step 1b profile compared to the background humidity profile.

_{wT}(z) the retrieved specific humidity profile q

_{T}(z) can be computed using the inverse version of Equation (3) in the main text in the form:

_{T}related to T

_{b}and T

_{d}. Using for the purpose the approximate assumptions ${V}_{wT}\approx {q}_{T}/{a}_{w}$ and $d{p}_{\mathrm{d}}/{p}_{\mathrm{d}}\approx d{p}_{T}/{p}_{T}$, which are reasonably valid over the moist air retrieval altitude range of interest, the linearized version reads:

_{T}(z):

_{T}(z) can be propagated in a straightforward manner from the dry pressure uncertainty u

_{p}

_{d}(z) via:

## Appendix B. Bias Correction of Background Profiles and Its Effects

_{f}is co-located ECMWF forecast temperature, $\Delta {\overline{T}}_{\mathrm{f}-\mathrm{a}}$ is the bias-correction term obtained from bi-linear interpolation of $\Delta {\overline{T}}_{\mathrm{f}-\mathrm{a}}$ from the four surrounding grid points, where $\Delta {\overline{T}}_{\mathrm{f}-\mathrm{a}}$ at each grid point is calculated as the difference profile between mean forecast temperature and mean analysis temperature, $\Delta {\overline{T}}_{\mathrm{f}-\mathrm{a}}={\overline{T}}_{\mathrm{f}}-{\overline{T}}_{\mathrm{a}}$. Similarly, the bias-corrected specific humidity profile is calculated as:

**Figure A1.**Differences between RO retrieved profiles and ECMWF co-located analysis profiles obtained from using the bias-corrected background profiles (red, “Bias Corr”) and from using the original background profiles (black, “No Bias Corr”) profiles for three exemplary RO events from simMetOp (upper), CHAMP (middle), and COSMIC (bottom) from 15 July 2008.

**Figure A2.**Systematic differences and standard deviations of moist temperature and specific humidity for simulated MetOp (left), CHAMP (middle), and COSMIC (right) events in the global domain (upper two rows) and TRO regions (bottom two rows). Statistics are shown for the bias-corrected case (Bias Corr) and the no bias corrected case (No Bias Corr).

## Appendix C. Vertical Correlations of Observations and Background Errors

**Figure A3.**Correlation matrices (left), exemplary correlation functions at three exemplary altitude levels of 11 km, 7 km, and 3 km (middle), and estimated correlation length for correlation functions (right) for the observed dry temperature uncertainty (first row), observed dry pressure uncertainty (second row), background temperature uncertainty (third row), and background specific humidity uncertainty (fourth row). The correlation matrices are shown for 15th July 2008 only, and the correlation functions and correlation lengths are shown for 5th, 15th, and 25th of July 2008.

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**Figure 1.**Schematic illustration of the algorithmic steps of the new moist air retrieval algorithm, for description see Section 2.

**Figure 2.**Mean profiles and uncertainties of the four input variables, i.e., observed dry temperature (first row), observed dry pressure (second row), background temperature (third row), and background specific humidity (fourth row) on 15 July 2008. For the input observed profiles (first row and second row), which the OPSv5.6 and the dynamic approach share the same uncertainties, the left, middle, and right panels show for the observed mean profile, uncertainty profile as a function of altitude, and the uncertainty profile as a function of altitude and latitude, respectively. For the background profiles (third and fourth row), the left, middle, and right panels show for the background mean profile as a function of altitude, global mean static uncertainty profile of OPSv5.6 approach as a function of altitude, and dynamic background uncertainty as a function of altitude and latitude, respectively.

**Figure 3.**Illustration of input, intermediate, and result variables relevant towards the estimation of optimal temperature (top row), specific humidity (middle row), and pressure (bottom row) of an exemplary simMetOp event (identified on top), for the dynamic approach. In the top row, the left panel shows temperature profiles for background temperature (blue), temperature calculated with specific humidity prescribed (green), and the optimally estimated temperature (red). The middle panel shows the estimated uncertainty profiles for the three profiles shown left and for the observed dry temperature (black) as well as the uncertainty of the background temperature from the OPSv5.6 approach (dashed blue). The right panel shows the differences between the temperature profiles shown left and the reference profiles, where the references profiles are the ECMWF co-located analysis profiles. In the three panels of the middle row, the same type of variables is shown as in the upper row, but for specific humidity q; thus the intermediate variable here is specific humidity with temperature prescribed (subscript “T”) and there is no dedicated input uncertainty profile in the middle panel (such as u

_{Td}in the upper row). Similarly, the bottom row shows the corresponding variables for pressure, whereby here the intermediate pressures from both humidity prescribed (subscript “q”) and temperature prescribed (subscript “T”) are shown together with the optimally estimated pressure (subscript “e”), and the middle panel also illustrates the input uncertainty profile of the dry pressure p

_{d}.

**Figure 4.**Illustration of input, intermediate, and result variables relevant towards the estimation of optimal temperature (top row), specific humidity (middle row), and pressure (bottom row) of an exemplary COSMIC event (identified on top), for the dynamic approach. Figure format and style are the same as for Figure 3; see that caption for explanation.

**Figure 5.**Difference profiles between RO-retrieved temperature (left column), specific humidity (middle column), and pressure profiles (right column) and their corresponding ECMWF co-located analysis profiles, for three exemplary events (identified on top of each row) from simMetOp (top row), CHAMP (middle row), and COSMIC (bottom row), respectively. The results for the dynamic (red), OPSv5.6 (black), CDAAC (green), and ROM-SAF (blue) approaches are shown.

**Figure 6.**Number of RO profiles from simMetOp (left), CHAMP (middle), and COSMIC (right) as function of altitude for the global domain (top row) and five latitudinal bands (bottom row), including TRO (tropics; 20°S to 20°N), NHP (northern hemisphere polar; 60°N to 90°N), SHP (southern hemisphere polar; 60°S to 90°S), NHSM (northern hemisphere subtropics and mid-latitudes, 20°N to 60°N), and SHSM (southern hemisphere subtropics and mid-latitudes, 20°S to 60°S), on 15 July 2008 for simMetOp and COSMIC and on 14-16 July 2008 for CHAMP. The red, black, green, and blue colors denote the dynamic, OPSv5.6, CDAAC, and ROM-SAF approaches, respectively, with the dynamic one plotted last (hence shadowing other colors above the lower to middle troposphere) and the profiles for different latitude bands denoted by distinct symbols (see legend).

**Figure 7.**Systematic differences (SysDiff) and standard deviations (StDev) of retrieved temperature (left column), specific humidity (middle column), and pressure (right column), relative to ECMWF co-located analysis profiles as reference, of the ensemble of simMetOp events on 15 July 2008. Statistics for both the dynamic (red) and OPSv5.6 (black) approach are shown for four representative regions (top to bottom: Global, TRO, NHP, SHP). The propagated uncertainties of retrieved profiles from the dynamic approach (UncertDyn; red-dashed) are shown as well.

**Figure 8.**Systematic differences (SysDiff) and standard deviations (StDev) of retrieved temperature (left column), specific humidity (middle column), and pressure (right column), relative to ECMWF co-located analysis profiles as reference, of the ensemble of CHAMP events on 14-16 July 2008. Statistics for the dynamic (red), OPSv5.6 (black), CDAAC (green), and ROM-SAF (blue) approach are shown for four representative regions (top to bottom: Global, TRO, NHP, SHP). The propagated uncertainties of retrieved profiles from the dynamic approach (UncertDyn; red-dashed) are shown as well.

**Figure 9.**Systematic differences (SysDiff) and standard deviations (StDev) of retrieved temperature (left column), specific humidity (middle column), and pressure (right column), relative to ECMWF co-located analysis profiles as reference, of the ensemble of COSMIC events on 15 July 2008. Statistics for the dynamic (red), OPSv5.6 (black), CDAAC (green), and ROM-SAF (blue) approach are shown for four representative regions (top to bottom: Global, TRO, NHP, SHP). The propagated uncertainties of retrieved profiles from the dynamic approach (UncertDyn; red-dashed) are shown as well.

**Figure 10.**Observation-to-background weighting ratio profiles for temperature (six panels in upper two rows) and specific humidity (six panels in lower two rows), for the COSMIC data ensemble of 15 July 2008, are shown for the global ensemble (Global) and the five latitudinal bands TRO, SHSM, NHSM, SHP, and NHP (identified in the panel titles). The results for the dynamic, OPSv5.6 and ROM-SAF approaches are all shown.

**Table 1.**Parameter settings for the observational uncertainty model for dry temperature and dry pressure.

${\mathit{z}}_{\mathbf{Ttop}}$ | ${\mathit{z}}_{\mathbf{Sbot}}$ | ${\mathit{s}}_{0}$ | ${\mathit{q}}_{0}$ | p | |
---|---|---|---|---|---|

${u}_{T\mathrm{d}}$ | 10.0 km | 20.0 km | 0.7 K | 3 K km^{p} | 0.5 |

${u}_{p\mathrm{d}}$ | 10.0 km | 17.0 km | 0.15% | 0.7 %km^{p} | 0.5 |

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## Share and Cite

**MDPI and ACS Style**

Li, Y.; Kirchengast, G.; Scherllin-Pirscher, B.; Schwaerz, M.; Nielsen, J.K.; Ho, S.-p.; Yuan, Y.-b.
A New Algorithm for the Retrieval of Atmospheric Profiles from GNSS Radio Occultation Data in Moist Air and Comparison to 1DVar Retrievals. *Remote Sens.* **2019**, *11*, 2729.
https://doi.org/10.3390/rs11232729

**AMA Style**

Li Y, Kirchengast G, Scherllin-Pirscher B, Schwaerz M, Nielsen JK, Ho S-p, Yuan Y-b.
A New Algorithm for the Retrieval of Atmospheric Profiles from GNSS Radio Occultation Data in Moist Air and Comparison to 1DVar Retrievals. *Remote Sensing*. 2019; 11(23):2729.
https://doi.org/10.3390/rs11232729

**Chicago/Turabian Style**

Li, Ying, Gottfried Kirchengast, Barbara Scherllin-Pirscher, Marc Schwaerz, Johannes K. Nielsen, Shu-peng Ho, and Yun-bin Yuan.
2019. "A New Algorithm for the Retrieval of Atmospheric Profiles from GNSS Radio Occultation Data in Moist Air and Comparison to 1DVar Retrievals" *Remote Sensing* 11, no. 23: 2729.
https://doi.org/10.3390/rs11232729