# Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation

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## Abstract

**:**

^{−2}(that is, above 60 km). By contrast, the direct induced a 10% error, and the improvement of exp T to exp is limited. Experiment 2 simulated the exact errors of an FM (exp) based on inputs on different vertical resolutions. The inputs are refractivity profiles on model levels of three widely used analyses, including ECMWF 4Dvar analysis, final operational global analysis data (FNL), and ERA5. Results demonstrated that based on exp and log-cubic interpolation, BA on model level of ECMWF 4Dvar has the highest accuracy, whose RD is 0.5% between 0–35 km, 4% between 35–58 km, and 1.8% between 58–80 km. By contrast, the other two analyses have low accuracy. This paper paves the way to better understanding the FM, and simulation errors on model levels of three analyses can be a helpful FM error reference.

## 1. Introduction

#### 1.1. FM Algorithm in Data Assimilation and RO Data Processing

^{−6}N + 1. Hence, the accuracy of BA is strongly affected by n and the FM. This BA is then assimilated into the NWP data assimilation system. Therefore, the FM is critical for RO data assimilation.

#### 1.2. FM Algorithm

## 2. Experiments

#### 2.1. Data, Collocation, and Quality Control

#### 2.1.1. Data

#### 2.1.2. Spatial and Temporal Collocation

#### 2.1.3. Quality Control

#### 2.2. Experiment 1: Algorithm Comparison

`outliers`s. Two statistics were selected for the analysis: the difference of the BA, i.e., (E − T); and the relative difference (RD) of the BA, i.e., ((E − T)/T × 100%). Based on these statistics, we determine how close an FM’s result is to the true value; Another form of RD, which is the difference between two experimental values, i.e., ((E

_{1}− E

_{2})/E

_{2}× 100%), can avoid introducing the bias of true value and provide the difference between the two experimental values. Subsequently, the relatively better FM can be obtained.

_{FMx}− T), is just an estimate showing which algorithm is closer to the truth. By contrast, the difference between any two FMs (E

_{FM1}− E

_{FM2}) is the absolute difference value. Experiment one shows which FM is better, instead of expressing how much error is incurred from the FM. Through experiment 1, although we can determine which FM is better, how much error exactly is incurred from the FM cannot be expressed. Such a question can refer to experiment 2.

#### 2.3. Experiment 2: Evaluation of Errors of FMs on the Fixed Model Level

## 3. Results

#### 3.1. Experiment 1: Algorithm Comparison

#### 3.1.1. Difference Analysis for Direct and Exp

^{−5}is within 5–30 km, and 1 × 10

^{−6}is within 30–60 km. However, the exp is more accurate than the direct because it is closer to the true value except at heights of 12.5–17.5 km, which reflects the errors passed by refractivity. Figure 3c,d shows the difference between the two algorithms. The result of the direct is larger than that of the exp until at a height of 50 km. Figure 3e,f shows the difference due to interpolation. In the height range of 5–30 km, the error induced by log-linear is less than ±1 × 10

^{−6}, and above 30 km, it is less than ±0.5 × 10

^{−7}. Interestingly, log-linear interpolation causes more undulation compared with log-cubic interpolation. The undulation due to interpolation is larger for the exp algorithm. To verify this result, we repeated this experiment using MetOp as the true value, and the same conclusion was drawn (see Figure A4 in Appendix B).

#### 3.1.2. Relative Difference Analysis of Direct and Exp Algorithms

^{−2}above (below) a height of 60 km, the direct algorithm’s result is less (larger) than the true value. (iii) Note that at heights of 18, 48, and 58 km, larger errors are incurred by refractivity rather than the errors of the FM. Unexpectedly, the errors of the exp algorithm at these heights are larger than those of the direct algorithm. This difference can be attributed to the fact that refractivity of the exp algorithm exponentially decays faster than that of the direct algorithm.

#### 3.1.3. Analysis to exp_T

_{ref}(ref is refractivity). This error was then passed to BA’s (E−T)/T. Therefore, what we got in experimentone is the which FM is better, rather than how much error exactly one FM causes. This issue will be tested in experiment two. Since analysis above identified that the exp algorithm with log-cubic interpolation is the most accurate FM, in next experiment we will figure out how much exactly the error of exp algorithm is on three fixed model levels.

_{ref}(where ref denotes refractivity), which are then passed to the (E−T)/T value of the BA. Therefore, experiment 1 demonstrates which FM is better, rather than how much error exactly an FM causes, which will be tested in experiment 2. The above analysis indicated that the exp algorithm with log-cubic interpolation is the most accurate FM; therefore, in experiment 2, we determined the extent of error of the exp algorithm on three fixed model levels.

#### 3.2. Experiment 2: Evaluation of the Error of the FM on the Fixed Model Level

_{BA}− T

_{BA})/T

_{BA}× 100%). In general, the less the interval, the less is the RD. EC4Dvar’s RD is less than the RDs for others because of the higher resolution of its levels. For example, EC4Dvar’s resolution is less than 1 km below 35 km, and its RD is less than ±0.5%. By contrast, the errors for FNL and ERA5 are less than ±1% for heights of 0–30 km owing to their low resolution. Interestingly, ERA5 and FNL show similar vertical resolution. However, the RD for ERA5 below 10 km is mostly less than that for FNL. This is because the Abel integral integrates values from lower levels to higher levels, and ERA5 owns the other six levels (i.e., 875, 825, 775, 225, 175, and 125 hPa), which ensures more accurate interpolation results for ERA5. Figure 6 also shows the comparison of the log-cubic interpolation (a–c) with log-linear interpolation (d–f). The errors of the latter are 1–2% larger than those of the former owing to the lower interpolation accuracy of log-linear.

^{−5}, respectively, while those for FNL and ERA5 frequently oscillate in the range of ±2% to ±4%. Above 35 km, the RD and difference for EC4Dvar are similar to those for FNL and ERA5. The large bias for FNL and ERA5 resulted from two aspects: (i) the low resolution of the level interval; (ii) the rapid change in the atmospheric condition below 10 km. In such conditions, if the vertical grid density of refractivity profiles is insufficient, a larger oscillation of BA profiles will occur. The same comparison as Figure 7 but for log-linear interpolation is shown in Figure A6 in Appendix B.

## 4. Discussion

^{−2}(that is, above a height of 60 km). Above this height, the direct algorithm induced a 10% error. (2) When errors exist in inputs, the exp algorithm tends to amplify the error compared with the direct algorithm. (3) The exp algorithm with log-linear interpolation is often feasible for its computation efficiency. Its errors were 0.2%, −2%, and −20% larger than those of log-cubic for the heights of 5–50, 55–65, and 70–80 km, respectively. (4) The improvement of exp_T algorithm to the exp algorithm is limited. Our results are consistent with results of [35,36].

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Abel Integral Algorithms

#### Appendix A.1. Spherically Symmetric Assumption

_{t}). If the right triangles AOC and BOC are the same, and the right triangles MOB and MAP are similar; then, ∠α is twice the size of ∠θ since they have the shared angle ∠AMP. According to Bouquer’s law, the impact parameter a = nrsin90° [37,38,39]. The direction vector r is the same as the direction of the refractive index n’s gradient. The BA is expressed as

**Figure A1.**Schematic of radio occultation geometry based on the spherically symmetric atmosphere assumption. That is, if triangles of ΔAOC and ΔBOC are symmetric (equal), then α = 2θ. OA(a), OB(a), and OC(rt) on the ray path are the position vectors of a transmitter, receiver, and tangent point, respectively. The ray is over bent for the clarity of the geometry relationship.

`→`∞. The finite upper limit is set as 120 km or 150 km (for Beidou FY3C/D) in a real-world application. The singularity of x = a can be eliminated by different analytical methods, called the Abel integral methods.

#### A.2. Abel Integral

#### A.2.1. Direct Algorithm

#### A.2.2. Exp Algorithm

_{j}= ln(N

_{j}/N

_{j+1})/(x

_{j+1}− x

_{j}), and the value is positive with a minimum of 10

^{−6}. In addition, the term lnn = ln(10

^{−6}N + 1) in Equation (2) is equivalent to 10

^{−6}N based on infinitesimal assumption. Moreover, at the tangent point, we have $\sqrt{{x}^{2}-{a}^{2}}\approx \sqrt{2a\left(x-a\right)}$ when we assume the impact parameter x is equivalent to the impact parameter a because a and x of air in the higher magnitude of order (6000 km) and their difference are less (<80 km).

_{j}) such that the BA is overestimated [35]. In such circumstances, the second line of Equation (7) is written as:

#### A.2.3. Exp_T Algorithm

## Appendix B. Figures

**Figure A2.**Monthly O-B statistics of FY3D, September 2019. The mean relative difference (RD) between ECMWF forecast and FY3D is less than 2%. Copyright © 2022 EUMETSAT.

**Figure A3.**Level interval of FNL, ERA5, and EC4Dvar. Interval decreases with increasing height. (

**a**) 0–12 km, (

**b**) 12–40 km, (

**c**) 40–80km.

^{−5}in the range of 5–30 km and 1 × 10

^{−6}in the range of 30–60 km, as shown in Figure A4a,b and Figure 3a,b. The results of FY3D have less error because it has passed 4QC. The experiment based on MetOp used the same collocation method as experiment 1. But we did not use 4QC. Secondly, they have the same conclusion on which algorithm is more accurate, that if the exp. Thirdly, the log-linear interpolation also causes more undulation to (${\mathrm{E}}_{\mathrm{dir}}-{\mathrm{E}}_{\mathrm{exp}}$) than which is caused by log-cubic interpolation. Moreover, the error induced by interpolation for exp is larger than that for direct. Therefore, the results of experiment one are reliable.

^{−2}above (below) the height of 60 km, the direct algorithm’s result is less (larger) than the true value.

**Figure A4.**Mean difference from MetOp and experimental value calculated from direct and exp. (

**a**,

**b**) Difference (E − T) of BA based on the log-cubic interpolation; (

**c**,

**d**) difference of BA between exp and direct based on the log-linear and log-cubic interpolation; (

**e**,

**f**) difference of BA between log-linear and log-cubic interpolation based on exp and direct. (

**a**,

**c**,

**e**) 10–30 km; (

**b**,

**d**,

**f**) 30–60 km; Results from MetOp are the same as the results using FY3D.

**Figure A5.**Mean refractivity in 0–40 km, 40–60 km, and 60–80 km, and their order of magnitude are 1 × 10

^{−0}, 1 × 10

^{−1}, 1 × 10

^{−2}, respectively. (

**a**) 0–40 km, (

**b**) 40–60 km, (

**c**) 60–80km.

^{−5}from 0.5% and $\pm $0.5× 10

^{−5}in Figure 7. Between 35 to 50 km, those values are the same, less than 5%. In terms of FNL and ERA5, more RD values reach $\pm $4% below 35 km, and more of those are over $\pm $5% above 35 km. There is more undulation for results based on linear interpolation.

**Figure A6.**The same as Figure 7 but in log-linear interpolation, the difference (

**a**–

**c**) and RD (

**d**–

**f**) of FM based on the exp algorithm and log-linear interpolation. The BA RD of EC4Dvar below 35 km is less than ±1%, whereas it of FNL and ERA5 below 35 km is less than ±5%. Above 35 km, the RD EC4Dvar is less than ±5%, while it of FNL and ERA5 is up to 10%.

## Appendix C. 4QC

**Figure A7.**Spatial (

**upper**) and temporal (

**lower**) distribution of 386 samples after collocation and 4QC. In general, these samples are evenly distributed in longitude and latitude, and are evenly distributed in different hours on each day in September.

**Figure A8.**Mean value of refractivity profiles’ relative difference (RD), $\left(\mathrm{E}-\mathrm{T}\right)/\mathrm{T}$, between FY3D (True value, T) and EC4Dvar (Experimental value, E). There are large differences at the heights of 18 (0.3%), 48 (1.3%), 58 (3.1%), 78 (10%) km. (

**a**–

**c**) shows the mean value of it.

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**Figure 3.**Difference between the BA of FY3D (T) and EC4Dvar BA (E) using the direct and exp Abel integral algorithms, together with log-linear and log-cubic interpolation methods. (

**a**,

**b**) (E

_{exp/dir}− T) using the log-cubic interpolation; (

**c**,

**d**) difference of the Abel integral algorithms (E

_{dir}− E

_{exp}) based on the log-linear and log-cubic interpolation, respectively; (

**e**,

**f**) difference of interpolation methods (E

_{lin}− E) based on exp and direct Abel integral algorithms, respectively.

**Figure 4.**RD of the BAs of FY3D (T) and EC4Dvar (E) based on the direct and exp Abel integral algorithms, with log-linear and log-cubic interpolation methods. (

**a**–

**c**) (E − T)/T, where E can be calculated by either the exp or direct algorithm, with both interpolation methods; (

**d**–

**f**) (E

_{1}− E

_{2})/E

_{2}, where 1, 2 indicate two different FMs.

**Figure 6.**Effects of various model level intervals (km) on the FM error. The x-axis represents the RD, ((E − T)/T × 100%), of the BA between true value (T) and three simulated analyses (E): (

**a**,

**d**) 137 levels of EC4Dvar; (

**b**,

**e**) 31 levels of FNL; (

**c**,

**f**) 37 levels of ERA5. The FM is the exp algorithm, together with (

**a**–

**c**) log-cubic or (

**d**–

**f**) log-linear.

**Figure 7.**Under various model levels, the difference (

**a**–

**c**) and RD (

**d**–

**f**) of FM based on the exp algorithm and log-cubic interpolation. The RD for EC4Dvar below 35 km is less than $\pm $0.5%, whereas RDs for FNL and ERA5 below 35 km are less than$\pm $ 5%. Above 35 km, the RD for EC4Dvar is less than $\pm $5%, while RDs for FNL and ERA5 are up to 10%.

**Figure 8.**BA difference ($\mathrm{E}-\mathrm{T}$, in gray) and RD ($\mathrm{E}-\mathrm{T})/\mathrm{T}$, in red) for EC4Dvar, FNL, ERA5, and FY3D based on the exp algorithm and log-cubic interpolation (

**a**–

**c**) and log-linear interpolation (

**d**–

**f**). (

**g**–

**i**) The difference between the RD of log-cubic interpolation and that of log-linear interpolation.

**Table 1.**Relative difference, (E−T)/T × 100%, between the FY3D BA (T) and the forward-modeled BA (E), in which E can be the exp and direct algorithm, and each algorithm has two interpolation methods: log linear (denoted by lin) and cubic spline (no label).

H (km) | exp (%) | exp_lin(%) | direct(%) | direct_lin(%) | exp_T(%) | exp_T_lin(%) |
---|---|---|---|---|---|---|

8–15 | (−0.3, 0.5) | exp + 0.3 | exp ± 0.002 | exp_lin ± 0.002 | ||

15–20 | (0.5, 1) | exp + 0.3 | ||||

20–45 | (−0.3, 0.5) | exp + 0.2 | ||||

45–55 | (−0.5, 1.5) | exp + 0.1 | ||||

55–60 | (0, 5) | (0, 4) | (0, 3.8) | |||

60–70 | (−1, 2.5) | (−4, 2.3) | (−10, 1.8) | |||

70–80 | (−1, 5) | (−20, −5) | (−80, −10) | direct + 4% |

Statistics | MSL Height (km) | Order of Magnitude | EC 4Dvar (cubic) | EC 4Dvar (lin) | Msl Height (km) | FNL(31) (cubic) | ERA5(37) (cubic) |
---|---|---|---|---|---|---|---|

Relative difference (RD) | 0–35 | % | $\pm $0.5% | $\pm $1% | 0–30 | $\pm $2.5% | 3% |

35–58 | $\pm $4% | $\pm $4% | 30–40 | $\pm $5% | 5% | ||

58–80 | $\pm $1.8% | $\pm $2% | 40–50 | $\pm $15% | 10% | ||

Difference | 0–10 | 1 × 10^{−5} | $\pm $4 × 10^{−5} | $\pm $6 × 10^{−5} | 0–11 | $\pm $4 × 10^{−4} | $\pm $2 × 10^{−4} |

10–35 | 1 × 10^{−6} | $\pm $2 × 10^{−6} | $\pm $8 × 10^{−6} | 11–22 | $\pm $4 × 10^{−5} | $\pm $5 × 10^{−5} | |

35–50 | 1 × 10^{−6} | $\pm $3 × 10^{−6} | $\pm $4 × 10^{−6} | 22–40 | $\pm $2 × 10^{−5} | $\pm $2 × 10^{−5} | |

50–60 | 1 × 10^{−7} | $\pm $2 × 10^{−7} | $\pm $4 × 10^{−7} | 40–46 | $\pm $2 × 10^{−6} | $\pm $2 × 10^{−6} | |

60–80 | 1 × 10^{−7} | $\pm $4 × 10^{−8} | $\pm $6 × 10^{−8} | 46–50 | $\pm $6 × 10^{−6} | $\pm $6 × 10^{−6} |

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## Share and Cite

**MDPI and ACS Style**

Deng, N.; Bai, W.; Sun, Y.; Du, Q.; Xia, J.; Wang, X.; Liu, C.; Cai, Y.; Meng, X.; Yin, C.;
et al. Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation. *Remote Sens.* **2022**, *14*, 1081.
https://doi.org/10.3390/rs14051081

**AMA Style**

Deng N, Bai W, Sun Y, Du Q, Xia J, Wang X, Liu C, Cai Y, Meng X, Yin C,
et al. Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation. *Remote Sensing*. 2022; 14(5):1081.
https://doi.org/10.3390/rs14051081

**Chicago/Turabian Style**

Deng, Nan, Weihua Bai, Yueqiang Sun, Qifei Du, Junming Xia, Xianyi Wang, Congliang Liu, Yuerong Cai, Xiangguang Meng, Cong Yin,
and et al. 2022. "Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation" *Remote Sensing* 14, no. 5: 1081.
https://doi.org/10.3390/rs14051081