1. Introduction

The spatial resolution of traditional amplitude-limited radar altimeter observing the sea surface height (SSH) poses a limitation, hindering the observation of mesoscale and submesoscale (meso–submesoscale) ocean dynamic processes (Chen et al. 2019; Fjørtoft et al. 2010; Fu and Ubelmann 2014; Pascual et al. 2006; Peral et al. 2015). The interferometric radar altimeter (IRA) proposed by Rodriguez et al. (1999) utilizes a combination of low incidence angle and short baseline cross-track interferometry, facilitating the effective resolution of this scientific issue through a wide observed swath (Fu and Ferrari 2008; Rodriguez et al. 1999). The existing IRA observation missions contain the Surface Water and Ocean Topography (SWOT) satellite jointly proposed by the National Aeronautics and Space Administration (NASA) and the French National Center for Space Studies (CNES) (Durand et al. 2010; Fu and Ferrari 2008; Fu et al. 2009), as well as the “Guanlan” scientific mission proposed by China National Laboratory of Marine Science and Technology (Bai et al. 2020; Chen et al. 2019). At present, the factors that introduce the SSH errors of the IRA include platform attitude, propagation medium, random noise, and ocean waves. The meso–submesoscale ocean phenomena exhibit variation in SSH only several centimeters, necessitating further investigation into the impact of these SSH errors for IRA observation (Koblinsky et al. 1992; Rodriguez et al. 1999).

From the traditional satellite altimeter to IRA, the propagation medium error is one of the important sources of SSH error, which impacts the SSH measurement accuracy significantly (Chelton et al. 2001; Jeannin et al. 2010; Ubelmann et al. 2014). The SSH error introduced by the charge-neutral atmospheric propagation path proposed in this paper is one of the effects of the propagation medium. The charge-neutral atmosphere is predominantly composed of neutral gas molecules (i.e., O₂, N₂, CO₂, H₂O, et al.). The refractive index of this medium for microwaves is nondispersive approximately, exhibiting variations with temperature, atmospheric pressure, and water vapor pressure (Eineder et al. 2011; Hopfield 1971; Owens 1967). Its value slightly exceeds the vacuum refractive index and increases as height decreases (Valma et al. 2010). According to Snell’s law, the change in refractive index will change the electromagnetic wave propagation speed and direction. Therefore, the impact of signal propagation in the charge-neutral atmosphere encompasses two aspects: (i) path delay and (ii) path bending (Berrada Baby et al. 1988; Hanssen 2001).

Similar to cross-track interferometric SAR (XT-InSAR), IRA also enables three-dimensional surface observation by converting the interferometric phase into a measure of distance change and establishing a correlation with surface elevation information (Bai et al. 2020). Zebker proposed that the uncorrelated atmosphere would induce interferometric phase noise and signal decorrelation during revisited observations of repeat-track InSAR (Zebker et al. 1997). However, the interferometry process of IRA occurs in the same atmospheric environment due to the work mode of single antenna transmits and double antennas receiving signals and the system design of the fixed short baseline. Consequently, the IRA echo is immune to the influence of uncorrelated atmospheric conditions, allowing for the estimation of the interference phase term caused by differential delay through observed geometric relationships.

Unfortunately, the IRA on board estimates the signal propagation pathlength based on the time it takes for the signal to propagate and at the speed of light in a vacuum (Ubelmann et al. 2014). This length is the optical pathlength, which will exceed the geometric pathlength of the signal propagation path and induces a line-of-sight (LOS) delay (Berrada Baby et al. 1988; Goldstein 1995; Hanssen 2001; Onn 2006; Picot et al. 2018; Zebker et al. 1997). The estimated incidence angle based on the optical pathlength undergoes an estimated bias in incidence angle due to the LOS delay, resulting in an offset of the reference phase and introducing errors in sea surface height measurements. In addition, the inhomogeneous spatial distribution of the water vapor will cause the spatial variation of zenith wet delay, which can contaminate the SSH results (Goldstein 1995; Hanssen 2001; Onn 2006; Zebker et al. 1997). Ubelmann (Ubelmann et al. 2014) proposed use of a cross-track radiometer (CTR) to suppress the spatial variation of the zenith wet delay in the swath. In addition, previous studies have suggested that the power-law characteristic of the wet delay spectrum corresponding to the water vapor turbulence for the spatial scale ranging from 0.5 to 1.4 km satisfies the Kolmogorov −8/3 power law (Goldstein 1995; Hanssen 2001; Onn 2006). However, the spatial distribution characteristics of zenith neutral delay in the open ocean, spanning a spatial scale ranging from 10 to 1000 km, warrant further investigation.

The phenomenon of atmospheric refraction leads to a gradual deviation of the signal propagation direction from the initial incidence direction, resulting in path bending (Marini 1972). The estimated imaging incidence angle at the imaging position in the reference plane, based on geometric pathlength, would be slightly smaller than the initial incidence angle of IRA. Hence, a phase-estimated offset still exists in the reference phase estimated through imaging incidence angle. Therefore, the SSH error related to the path bending would be inevitable. For the requirement of IRA altimetry accuracy, the SSH error induced by path bending cannot be ignored.

In this work, the impact of the charge-neutral atmosphere on IRA measurement was analyzed first. The SSH errors resulting from the propagation path in the charge-neutral atmosphere are derived based on the IRA observation geometry. Meanwhile, the derivation results of the above SSH errors are verified by simulation experiments based on the three-dimensional numerical weather model (3D-NWM). The 3D-NWM is constructed using parameters such as pressure, temperature, and specific humidity derived from the single level and pressure levels monthly average data of ERA5 provided by the European Centre for Medium-Range Weather Forecasts (ECMWF). Then, the spatial distribution characteristic of the charge-neutral atmosphere in meso–submesoscale is estimated by performing spectrum analysis on the Jason-3 along-track zenith delay correction data. Finally, the impact of the SSH errors caused by the propagation path in the charge-neutral atmosphere of IRA in observing the meso–submesoscale ocean dynamic phenomena is discussed.

The remainder of this paper is organized as follows. In section 2, the impact of the charge-neutral atmospheric propagation path on the IRA interferogram is evaluated. The impact of three types of the propagation path on SSH measurements is examined in section 3. In section 4, the spatial distribution characteristics of the charge-neutral atmospheric zenith delay is presented. The impact of the above SSH error anomalies caused by the propagation path on the observation performance of IRA is discussed in section 5. Finally, conclusions are given in section 6.

2. The impact of the charge-neutral atmosphere on IRA interferogram

a. Model of propagation path in the charge-neutral atmosphere

The propagation path of the microwave signal in the satellite observation geometry is shown in Fig. 1 without considering the impact of the ionosphere and Earth’s curvature (the default condition in this paper). The geometric pathlength of the signal propagation path (black solid curve) is L. The straight line (black dashed line) length between the radar and the illuminated target is R. The propagation path of the microwave signal in the vacuum (blue dashed line) is the initial propagation direction of the signal propagation. H is the platform height and h_atm is the top-level altitude of the atmosphere. θ_ini represents the initial incidence angle of the signal propagation path. Since the length difference between L and R can usually be neglected (i.e., L ≈ R) (Berrada Baby et al. 1988; Hanssen 2001; Onn 2006; Zebker et al. 1997).

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The satellite radar signal propagation path.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-22-0142.1

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Due to the minimal relative fluctuation of SSH in meso–submesoscale, an approximate estimation of angle θ can be obtained by θ ≈ arccos(H/L). Then, the LOS path delay ΔL^LOS can be calculated by (Hanssen 2001)

$\mathrm{\Delta }{L}^{\text{LOS}}={\displaystyle {\int }_L(n-1)ds}\approx {10}^{-6}{\displaystyle {\int }_{h_0}^{h_{\text{atm}}}\frac{N}{\cos \theta }dh}=-\mathrm{\Delta }{h}_{\text{ZND}}\frac{1}{\cos \theta },$(1)

where the radio refractive index N can be expressed as N ≡ (n − 1) × 10⁻⁶ (Hanssen 2001; Onn 2006), n is the atmospheric refractive index, and ds is the integral element along the actual propagation pathlength L. If considered ds = dh/cos θ, integral from surface altitude h₀ to the top-level altitude h_atm, and ΔL^LOS can be represented by zenith neutral delay (ZND; denoted as Δh_ZND, and default Δh_ZND ≤ 0) at the illuminated position.

The refractive index n is a key parameter in the signal propagation path model. Currently, the vertical distribution of refractive index n established by the 3D-NWM has been widely used in the study of InSAR path delay correction and achieved centimeter-level accuracy improvement (Cao et al. 2021; Cong et al. 2012; Hu and Mallorqui 2018; Shen et al. 2019). Based on the 3D-NWM proposed by Abdalla (Abdalla 2013), the signal propagation path model can be expressed as follows:

$L=\frac{(H-h_{\text{atm}})}{\cos {\theta }_{\text{ini}}}+{\displaystyle \sum _{i=1}^M\frac{h_i}{\cos {\beta }_i}},i=1,2,\dots ,M,$(2)

${\beta }_i=\arcsin \left[\frac{n_0\sin ({\theta }_{\text{ini}})}{n_i}\right],$(3)

$n_i={10}^{-6}\times N_i+1={10}^{-6}\times \left[k_1\frac{P_i}{T_i}+(k_2-k_1)\frac{e_i}{T_i}+k_3\frac{e_i}{T_i^2}\right]+1,$(4)

where i represents the number of atmospheric stratification levels in 3D-NWM, M is the total number of levels, and h_i represents the thickness of each level. Based on the Integrated Forecasting System (IFS) provided by ECMWF, the charge-neutral atmosphere is divided into 137 levels (i.e., M = 137), and the top-level altitude h_atm is about 80 km (Abdalla. 2013). According to Snell’s law, the refraction angle β in each level can be obtained by Eq. (3) to describe the signal propagation direction, where n₀ = 1 is the vacuum refractive index. In Eq. (4), the atmospheric refractive index n in each level can be described by the radio refractive index N in each level depending on the atmospheric parameters such as atmospheric pressure P, temperature T, and water vapor pressure e, where e can be calculated by specific humidity, and k₁ = 77.6 K × hPa⁻¹, k₂ = 71.6 K × hPa⁻¹, and k₃ = 3.75 × 10⁵ K² × hPa⁻¹ are experimental determination constants (Abdalla. 2013; Cong et al. 2012; Onn 2006).

The above atmospheric parameters of each level in 3D-NWM are given by ERA5 monthly averaged data on levels provided by ECMWF with a spatial resolution of 0.25° × 0.25° (Abdalla 2013). The spatial average atmospheric parameters of each level in the local region of 2° × 2° with the center longitude and latitude of 34°N and 165°E. respectively (Fig. 2a), are selected to estimate the vertical distribution of the monthly average refractive index n in different months. The refractive index n calculated by Eq. (4) from 2011 to 2021 is shown in Fig. 2b. From the results in Fig. 2b, one can find that the value of n in the lower atmosphere fluctuates periodically with the season, while the seasonal variation of n is not significant in the upper atmosphere. The deviation of the signal propagation direction from the initial incidence angle (i.e., θ_ini − β) is shown in Fig. 2c. Here, the refraction angle β is calculated by Eq. (3) based on the monthly average value of n in June 2021. The deviation of the signal propagation direction compared with the initial propagation direction increases with the decrease in altitude. In addition, the θ_ini is greater, resulting in a more pronounced direction of deviation propagation. This phenomenon implies that the impact of path bending would be more pronounced at far range.

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(a) The location of the research region; (b) the seasonal variation of the refractive index; (c) the deflection of signal propagation direction under SWOT system.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-22-0142.1

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b. The charge-neutral atmospheric effect on IRA interferogram

The ocean surface observation geometry of IRA is illustrated in Fig. 3. The baseline (with a length of B) between antenna 1 (master) and antenna 2 (slave) is inclined at an angle α (also known as the baseline angle) for the local tangent. The geometric pathlengths L₁ and L₂ can be evaluated by Eq. (2). However, since the microwave wavelength is affected by the vertical distribution of the refractive index n (Fig. 2b), the optical pathlength between the master–slave antennas and the illuminated target are $L_1+\mathrm{\Delta }{L}_1^{\text{LOS}}$ and $L_2+\mathrm{\Delta }{L}_2^{\text{LOS}}$, respectively. The master–slave images (S₁ and S₂) can be expressed as (Rodriguez and Martin 1992)

$S_1(L_0,x_0)=A{\displaystyle \int dz}{\displaystyle \int dxdy}\exp \left[j\frac{2\pi }{\lambda }(L_1+\mathrm{\Delta }{L}_1^{\text{LOS}})\right]E(x,y,z)\times W(L_1-R_0,x-x_0)+{\epsilon }_1,$(5)

$S_2(L_0+\mathrm{\Delta }{R}_0,x_0)=A{\displaystyle \int dz}{\displaystyle \int dxdy}\exp \left[j\frac{2\pi }{\lambda }(L_2+\mathrm{\Delta }{L}_2^{\text{LOS}})\right]E(x,y,z)\times W(L_1-R_0,\mathrm{\Delta }R,x-x_0)+{\epsilon }_2,$(6)

where x and y denote the position in the azimuth and range direction, z is the position in the direction perpendicular to the x–o–y plane, L₀ is the optical pathlength from the master antenna to the center of the scattering unit, y₀ is the azimuth center position of the scattering unit, and ΔR₀ represents the optical pathlength difference between the master and slave antennas and the center of the scattering unit, A is the coefficient related to the system parameters, E is the electromagnetic scattering field related to the backscattering coefficient, W is the range–azimuth-point target response, and ε₁ and ε₂ are the system thermal noise independent of the target echo signals. Ignoring the effect of thermal noise, the IRA interferogram can be obtained by conjugate multiplication of the master–slave images. The measured interferometric phase φ can be expressed as

$\phi =\arctan \left[\frac{\mathrm{Im}(S_1{S}_2^{*})}{\mathrm{Re}(S_1{S}_2^{*})}\right]=-\frac{2\pi }{\lambda }\times \mathrm{\Delta }{R}_0=-\frac{2\pi }{\lambda }\times (\mathrm{\Delta }L+\delta L_{\text{LOS}})={\phi }_{\text{geo}}+{\phi }_{\text{delay}}.$(7)

Here, ΔL = L₂ − L₁ and $\delta L_{\text{LOS}}=\mathrm{\Delta }{L}_2^{\text{LOS}}-\mathrm{\Delta }{L}_1^{\text{LOS}}$ denotes the geometric pathlength difference and the path delay difference between the master–slave antennas, respectively. Therefore, without considering the additional phase caused by random noise, the interference phase φ considering the impact of the propagation path consists of two parts: geometric path difference phase φ_geo and path delay difference phase φ_delay. In the observation, since the baseline length B of IRA is much smaller than L, the value of ΔR₀ will be approximated by the LOS projection of B on the signal initial propagation direction (Zebker and Goldstein 1986), which can be represented by ΔL ≈ −B sin(θ_ini − α). The geometric path difference phase in Fig. 3 can be approximately expressed as

${\phi }_{\text{geo}}=\frac{2\pi }{\lambda }\times \mathrm{\Delta }L\approx -\frac{2\pi }{\lambda }B\sin ({\theta }_{\text{ini}}-\alpha ).$(8)

Combined with Eq. (1), the path delay difference phase can be expressed as follows:

${\phi }_{\text{delay}}=-\frac{2\pi }{\lambda }\delta L_{\text{LOS}}=\frac{2\pi }{\lambda }\mathrm{\Delta }{h}_{\text{ZND}}\left(\frac{L_1}{H}-\frac{L_2}{H+B\sin \alpha }\right).$(9)

The SSH measurement of IRA is based on the relationship between the phase information of the interferogram and the geometry. In the above analysis, we derive the composition of the IRA interference phase considering the impact of the charge-neutral atmospheric propagation path. The SSH errors caused by the charge-neutral atmospheric propagation path are analyzed by combining the SSH measurement process of IRA in the next section.

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IRA observation geometry in the charge-neutral atmosphere.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-22-0142.1

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3. Impact of charge-neutral atmosphere on SSH measurement

b. The swath transfer function for SSH errors caused by the propagation path

We believe that the charge-neutral atmospheric propagation path has two aspects to the SSH measurement process. On the one hand, as discussed in section 2, the path delay difference corresponding to the master–slave antennas will introduce additional phase terms in the interferometric phase and cause the SSH error during the measurement process. On the other hand, the path delay and path bending also can cause the deviation of the estimated reference phase, and the introduced flat phase shift can be reflected in the elevation phase and cause SSH errors. In the following analysis, we assume that the observed surface is a plane with zero elevation, and ignore the effects of platform attitude, random noise, and coregistration bias.

1. The SSH error caused by the difference delay: This SSH error is caused by φ_delay reflected in Δφ_h. As shown in Fig. 4, LOS delay can be estimated by ZND approximation according to Eq. (1). Due to the spatial separation, there is a certain difference in imaging incidence angle for the master–slave antennas at the corresponding imaging position P. So, there is a difference delay between the echoes of master–slave antennas, which is described in detail in section 2.

   

   

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   Geometry relationship of the difference delay.

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According to Eq. (7), the φ_delay is contained in φ during the IRA observation process, so it cannot be corrected by φ_p. Substitute Eq. (9) into Eq. (12), and this SSH error further called the differential delay error (DDE) can be expressed as follows:

$\mathrm{\Delta }{h}_{\text{DDE}}=-\frac{{\lambda }_{0}L\sin \theta }{2\pi B\cos (\theta -\alpha )}{\phi }_{\text{delay}}=-\mathrm{\Delta }{h}_{{\rm Z}\text{ND}}\tan \theta \frac{\cos \theta \sin (\theta -\alpha )+\sin \alpha }{\text{cos}\theta \cos (\theta -\alpha )}=-\mathrm{\Delta }{h}_{{\rm Z}\text{ND}}{\tan }^2\theta .$(13)

• ii. The SSH errors caused by path delay and path bending: The geometry relationships of SSH errors caused by path delay and path bending are shown in Fig. 5. In the zero-Doppler plane, P denotes the imaging position in the reference plane determined by L and H. The SSH errors caused by path delay and path bending are Δh_PDE and Δh_PBE, respectively.

  

  

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  Imaging geometry of IRA SSH error caused by the path delay and path bending in the charge-neutral atmosphere: (a) path delay error, and (b) path bending error.

  Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-22-0142.1

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The Δh_PDE in Fig. 5a is further called the path delay error (PDE), which is caused by the incidence angle estimation deviation Δθ_PDE introduced by ΔL^LOS. As shown in Fig. 5a, if we replace the actual geometrical pathlength L with the optical pathlength L + ΔL^LOS during IRA imaging, the imaging position P would shift to P₁ in the IRA interferogram. The imaging incidence angle θ_PDE (i.e., θ_PDE = θ + Δθ_PDE) estimated based on the optical pathlength L + ΔL^LOS would be significantly larger than the actual imaging incidence angle θ estimated by true geometric pathlength L. The ΔL^LOS will cause a flat Earth phase shift Δφ_f according to Eqs. (10) and (11). This phase change will be reflected in the elevation phase after removing the reference phase at position P₁ and result in PDE. Using Eqs. (10) and (11), the SSH error Δh_PDE can be written as

$\mathrm{\Delta }{h}_{\text{PDE}}=-L\sin \theta \cdot \mathrm{\Delta }{\theta }_{\text{PDE}}=-\mathrm{\Delta }{L}^{\text{LOS}}\cos \theta \approx \mathrm{\Delta }{h}_{\text{ZND}},$(14)

with

$\mathrm{\Delta }{\theta }_{\text{PDE}}=\frac{\mathrm{\Delta }{L}^{\text{LOS}}}{L\text{tan}\theta }.$(15)

The Δh_PBE in Fig. 6b further called the path bending error (PBE) is introduced by the variation of n with altitude in the charge-neutral atmosphere. Because of the atmospheric refraction effect, the radar beam with the initial incidence angle θ_ini illuminates the target at position P instead of the target at position P₂. In the IRA interferogram, the incidence angle θ estimated by the geometric pathlength L of the actual propagation path would be slightly smaller than θ_ini, and resulted in an incidence angle offset Δθ_PBE (i.e., Δθ_PBE = θ_ini − θ). The reference phase evaluated based on θ would be underestimated, which induces the flat Earth phase shift Δφ_f after removing the reference phase at position P. The elevation SSH error Δh_PBE is introduced inevitably and can be expressed as

$\mathrm{\Delta }{h}_{\text{PBE}}=L\sin {\theta }_{\text{ini}}\cdot \mathrm{\Delta }{\theta }_{\text{PBE}}=\mathrm{\Delta }{L}^{\text{PBE}}\cos {\theta }_{\text{ini}},$(16)

with

$\mathrm{\Delta }{\theta }_{\text{PBE}}=\frac{\mathrm{\Delta }{L}^{\text{PBE}}}{L\text{tan}{\theta }_{\text{ini}}}.$(17)

Here, ΔL^PBE represents the difference between the geometric pathlength L of the actual propagation path and the geometric pathlength L_ini of the propagation path with the incidence angle θ_ini in the vacuum (i.e., ΔL^PBE = L_ini − L). Using Eqs. (2) and (3), it is obtained by the imaging geometry in Fig. 5b and expressed as

$\begin{array}{l}\mathrm{\Delta }{L}^{\text{PBE}}=L_{\text{ini}}-\frac{(H-h_{\text{atm}})}{\cos {\theta }_{\text{ini}}}-{\displaystyle \sum _{i=1}^M\frac{h_i}{\cos {\beta }_i}}\\ =\frac{h_{\text{atm}}}{\cos {\theta }_{\text{ini}}}-{\displaystyle \sum _{i=1}^M\frac{h_i{n}_i}{\sqrt{n_i^2-{\sin }^2}{\theta }_{\text{ini}}}},\end{array}$(18)

with

$\cos {\theta }_{\text{ini}}=\frac{\cos \theta }{1+\mathrm{\Delta }{L}^{\text{PBE}}/R}.$(19)

Substitute Eq. (18) into Eq. (16), we can find that the PBE is not only related to the refractive index n in each level but also affected by the initial incidence angle θ_ini. The relationships between θ_ini and θ can be expressed by Eq. (19). Then, the Δh_PBE using the refractive index n from 2011 to 2021 in Fig. 2b are presented in Table 1.

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Simulation verification of SSH errors caused by charge-neutral atmospheric propagation path: The black solid line is the theoretical result of the overall propagation path error estimated by Eq. (21), where the theoretical values of DDE and PBE [calculated by Eqs. (13) and (20)] are represented by the red solid line and the blue dashed line shown in the insert, respectively. Referring to Table 2, the overall propagation path error simulation results of SWOT (orange square), GL in the Ku band (blue circle), and GL in the Ka band (yellow circle) are obtained, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-22-0142.1

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Table 1.

Impact of the charge-neutral atmosphere with different refractive index vertical distribution (Fig. 2b) on PBE for different initial incidence angles.

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The mean values of Δh_PBE increase significantly with the initial incidence angle θ_ini in Table 1. Although the charge-neutral atmospheric refractive index n exhibited seasonal variation (in Fig. 2b), the values of the root-mean-square (RMS) of Δh_PBE for different θ_ini are all less than centimeter. This phenomenon means that the PBE for IRA is not sensitive to the changes of n in different atmosphere environments. Therefore, PBE can be considered as a systematic error only related to θ (by radians), and the fitting function $\mathrm{\Delta }{h}_{\text{PBE}}^{\text{fit}}$ (by meters) is established as

$\mathrm{\Delta }{h}_{\text{PBE}}^{\text{fit}}=2.1876\times {\theta }^4-0.2534\times {\theta }^3+2.4843\times {\theta }^2-0.0027\times \theta +5.21\times {10}^{-5}.$(20)

Note that the application of Eq. (20) requires that the platform height H is higher than the top-level altitude h_atm of the charge-neutral atmosphere (i.e., H > h_atm) and that the imaging incidence angle θ is nonnegative and should range from 0.6° to 6°.

In the above analysis, the SSH error in Eq. (12) introduced by the system parameter (such as L and θ) estimated error caused by the path delay is ignored, which is not the focus of this paper. Then, the total SSH error of the charge-neutral atmospheric propagation path on the IRA altimetry (i.e., Δh_PE) can be expressed as

$\mathrm{\Delta }{h}_{\text{PE}}=\mathrm{\Delta }{h}_{\text{DDE}}+\mathrm{\Delta }{h}_{\text{PDE}}+\mathrm{\Delta }{h}_{\text{PBE}}^{\text{fit}}.$(21)

Based on the signal propagation path model in the charge-neutral atmosphere, the master–slave images of IRA are established by Eqs. (5) and (6). Referring to the system parameters of IRA designed by SWOT and the “Guanlan” mission (abbreviated GL), the simulation experiment system parameters are shown in Table 2 (Chen et al. 2019; Fu et al. 2009).

Table 2.

The parameters in the simulation experiment.

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The simulation verification results of the above theory are shown in Fig. 6. The results indicate that the overall SSH error caused by the charge-neutral atmospheric propagation path in theory described as Eq. (21) is consistent with the simulation results obtained using SWOT and GL system parameters. Meanwhile, the theoretical results of PBE and DDE show significant consistency, both of which increase significantly with the increase of incident angle. Given the distinct genesis of PBE and DDE in the previous analysis, we believe that despite their similarities, these two phenomena are fundamentally different. The simulation results also verify this point, only a single PBE or DDE cannot meet the final SSH error value in Fig. 6. Under the ZND conditions given in Table 2, both PBE and DDE caused about 1.2 cm SSH errors for SWOT at the far range (θ = 3.9°), while these errors caused about 2.2 cm in the Ku-band (θ = 5.5°) and 2.8 cm in the Ka-band (θ = 6°) SSH errors for GL, respectively. Therefore, GL will be more significantly affected by PBE and DDE than SWOT. Additionally, Eq. (14) indicates that PDE is only related to ZND, so PDE is a constant in the experiment. Nevertheless, ZND exhibits certain anomalies in practice. According to Eqs. (13) and (14), this will lead to changes in PDE and DDE and affect the observation performance of IRA, which will be discussed in the following sections.

4. Spatial characteristics of charge-neutral atmospheric zenith delay

The spatial characteristics of the charge-neutral atmospheric zenith delay in the research region of Fig. 2a are analyzed based on the vertical distribution of the refractive index n shown in Fig. 2b. According to the Dalton pressure decomposing law, the refractive index n can be decomposed into two parts corresponding to the hydrostatic and wet atmospheres, respectively (Cao et al. 2021; Cong et al. 2012; Onn 2006). Meanwhile, ZND also can be decomposed into the zenith hydrostatic delay (ZHD) and zenith wet delay (ZWD), respectively.

The variation of the zenith delay from 2011 to 2021 at the research region in Fig. 2a is shown in Fig. 7. We can find that the value of ZND (the black line in Fig. 7) is mainly determined by ZHD (the red line in Fig. 7). The variation trend of ZND with month is highly correlated with ZWD and changes periodically with the season due to the variation of water vapor in different seasons (Xu et al. 2011; Yu et al. 2018). Compared to ZWD and ZHD, the variation of ZHD is much smaller than ZWD with month, but ZHD has a much larger value of delay than ZWD. Therefore, the ZHD has a significant contribution to the PDE, while ZWD can lead to a variation of PDE with different conditions of the average water vaper.

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The variation of the zenith delay from 2011 to 2021.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-22-0142.1

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The spatial distribution characteristics of charge-neutral atmospheric zenith delay in the IRA observation swath are more concerned. The inhomogeneous spatial distribution of the atmosphere will cause the anomaly of ZND in the horizontal direction (Esteban-Fernandez 2017; Ubelmann et al. 2014), which is mainly induced by the spatial variation of water vapor (Ubelmann et al. 2014; Zebker et al. 1997). Jason-3 satellite altimeter GDR data provide along-track correction data for ZHD and ZWD. The ZHD correction data of Jason-3 satellite altimeter GDR data are provided by ECMWF, and the uncertainty was about 0.7 cm (Picot et al. 2018). The ZWD correction data are estimated by the nadir-looking radiometer [Advanced Microwave Radiometer (AMR)], whose uncertainty is about 1.2 cm (Esteban-Fernandez 2017; Picot et al. 2018). Therefore, the obtained ZND based on GDR data from 2018 to 2021 can describe the spatial variation of the charge-neutral atmosphere and can estimate the along-track one-dimensional (1D) ZND spectrum of Jason-3 satellite.

In Fig. 8, the zenith delay correction data in the open ocean area are selected, and the data are resampled at a 5 km spatial grid. Meanwhile, a Gaussian high-pass filter with a spatial window of 1500 km is used to remove the large-scale signal of the along-track charge-neutral atmospheric zenith delay (i.e., ZND, ZWD, and ZHD) in each pass. The spatial average spectrums of the charge-neutral atmospheric zenith delay (i.e., ZND average spectrum, ZWD average spectrum, and ZND average spectrum) for all passes at the spatial scale ranging from 10 to 1000 km are presented in Fig. 8. Here, the spatial frequency is κ = 1/λ_c, and λ_c is the along-track spatial scale (Xu and Fu 2011, 2012). The power-law reference lines (the gray dashed lines in Fig. 8) with power law −5/3 and −8/3 in Kolmogorov turbulence theory are also presented in Fig. 8.

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Along-track charge-neutral atmospheric zenith delay spectrum estimated based on each pass Jason-3 GDR delay correction data from 2018 to 2022, where the solid black line represents the ZND average spectrum, the pink dashed line represents the ZHD average spectrum, and the cyan dashed line represents the ZWD average spectrum.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-22-0142.1

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5. The impact of the charge-neutral atmospheric propagation error on IRA performance

To evaluate the impact of the propagation path on the performance of IRA for observing the meso–submesoscale ocean phenomena, the two-dimensional (2D) spatial distribution of ZND anomaly within the observed area of IRA should be given first. The ZND 2D random anomaly in the swath of IRA can be simulated by the Monte Carlo method and expressed as

$\mathrm{\Delta }{h}_{\text{ZND}}(x,y)=\frac{1}{L_x{L}_y}{\displaystyle \sum _{p=-M/2}^{P/2-1}{\displaystyle \sum _{q=-N/2}^{Q/2-1}{F}_{\text{ZND}}(k_{x,p},k_{y,q})\times \exp [j(k_{x,p}x+k_{y,q}y)]}},$(23)

where L_x and L_y represent the distance along the azimuth and the range directions of the IRA image, respectively; k_x,p and k_y,q represent the corresponding spatial wavenumber components in the corresponding direction. F_ZND(k_x,p, k_y,q) denotes the Fourier transform coefficient of the ZND anomaly and it is expressed as

$\begin{array}{l}{F}_{\text{ZND}}(k_{x,p},k_{y,q})=2\pi {[L_x{L}_y{E}_{\text{ZND}}^{2\mathrm{D}}(k_{x,p},k_{y,q})]}^{1/2}\\ \times \{\begin{array}{cc}\frac{[N_1(0,1)+j{N}_2(0,1)]}{\sqrt{2}},& p\ne 0,P/2\text{and}q\ne 0,Q/2,\\ N_1(0,1),& p=0,P/2\text{and}q=0,Q/2.\end{array}\end{array}$(24)

Here, N₁(0, 1) and N₂(0, 1) represents the random number that conforms to the normal distribution. In Eq. (24), the change of ZND anomaly with time is ignored, because it is a slow process compared with the imaging time of IRA. Meanwhile, to ensure the obtained value of Δh_ZND is real, F_ZND(k_x,p, k_y,q) is necessary to satisfy the conditions of complex conjugate symmetry. The researchers (Esteban-Fernandez 2017; Ubelmann et al. 2014) have highlighted that it is possible to approximate the atmosphere as isotropic within the spatial scale in the IRA observed swath. Then, $E_{\text{ZND}}^{2\mathrm{D}}(k_{x,p},k_{y,q})$ is the isotropic 2D wavenumber spectrum of ZND in Eq. (24), and the relationship between $E_{\text{ZND}}^{2\mathrm{D}}(k_x,k_y)$ and $S_{{}_{\text{ZND}}}^{1\text{D-single-sided}}(\kappa )$ is given by (Esteban-Fernandez 2017; Ubelmann et al. 2014)

$E_{\text{ZND}}^{2\mathrm{D}}(k_x,k_y)=\frac{1}{4{\pi }^{2}k}{S}_{{}_{\text{ZND}}}^{1\text{D-single-sided}}(\kappa ),$(25)

with

$\kappa =\frac{1}{2\pi }\sqrt{k_x^2+k_y^2}.$(26)

Through the above method, we simulate the 2D ZND anomaly. The 2D spatial distribution of PDE anomaly (PDEA) and DDE anomaly (DDEA) can be obtained by substituting it into Eqs. (14) and (13), respectively, and be presented alongside 2D PBE in Fig. 9. The spatial resolution in Fig. 9 is 1 km × 1 km, and the other parameters are referred to the KaRIn designed by SWOT shown in Table 2.

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The 2D SSH error induced by (a) PDE anomaly; (b) PBE and (c) DDE anomaly.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-22-0142.1

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According to the SWOT system parameters, the maximum fluctuation height of the SSH error caused by the simulated PDEA (Fig. 9a) is about 2.2 cm, while the SSH error caused by DDEA (Fig. 9c) is much smaller than a centimeter. As discussed in section 3, the PBE (Fig. 9b) induces a systematic error resembling a ramp in the single-side swath of SWOT, resulting in an approximate 1.2 cm SSH error at far ranges. Therefore, PDEA and PBE are numerically close to the SSH anomaly of meso–submesoscale, while the impact of DDEA has little contribution to the IRA altimetry error at this scale.

As shown in Fig. 10, the different meso–submesoscale SSHs (the first column in Figs. 10a–d) are utilized to exhibit the impact of PDEA, PBE, and DDEA on the observation performance of IRA more directly. Here, the ocean model HYCOM is used to simulate the actual meso–submesoscale SSHs and the spatial resolution is upsampled to 1 km × 1 km. For comparisons, the SSH superimposed with the 2D SSH errors caused by the propagation path in Fig. 10 are presented. The mean and RMS values of the original SSHs and SSH results considering the impact of propagation path error in the SWOT observation swath shown in Fig. 10 are presented in Table 3.

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Simulation results of the meso–submesoscale SSH anomaly with the impact of charge-neutral atmospheric propagation: (a) the SSH anomaly with different values of RMS, and the results after superimposing (b) PDE anomaly, (c) PBE, and (d) DDE anomaly (given in Fig. 9b).

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-22-0142.1

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Table 3.

The impact of propagation path error on various mean and RMS values of SSHs. All values are in cm.

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The SSH anomaly overlaid with the PDEA (the second column in Fig. 10) indicates that the observed results of IRA are contaminated by the PDEA, which could potentially impact IRA’s performance to observe meso–submesoscale oceanic phenomena, particularly when actual SSH anomaly have low RMS values (such as Figs. 10a,b). Observing Table 3, PDEA also makes some changes in the RMS of SSH results but has little effect on its mean. The impact of PBE (the third column in Fig. 10) causes a ramp phenomenon along the range direction of SSH results as shown in Fig. 9b, which would also distort the observed oceanic phenomenon. Meanwhile, the value of PBE increases significantly at the far range, which will raise the mean sea level (MSL) in the observation swaths. Therefore, it is important to eliminate the impact of PBE on IRA. As shown in Fig. 10 and Table 3, the impact of DDEA (the fourth column in Fig. 10) on the observation of ocean phenomena can be ignored, as well as the values of mean and RMS under this influence are almost consistent with the original SSH. As expected, the presence of these SSH errors significantly complicates the observed SSH results by the IRA.

Currently, the research on the atmospheric propagation error of IRA mainly focuses on the SSH error induced by PDE. According to the research of Ubelmann, the spatial variation of PDE caused by water vapor in the swath can be restrained based on the atmosphere zenith delay data acquired by a cross-track radiometer (CTR), and the residual error of PDE is reduced to below 1 cm at spatial wavelengths ranging from 10 to 500 km (Ubelmann et al. 2014). Combined with Eqs. (13) and (14), the impact of DDEA is also far less than a centimeter after path delay correction. However, the impact of PBE on the SSH measurement has not attracted enough attention. Even if the PDE in the observation area is completely rectified, the meso–submesoscale ocean phenomena can still be influenced by PBE, thereby posing a challenge to the reliability of oceanic phenomena observed by IRA.

6. Summary and conclusions

In this paper, the impact of the charge-neutral atmospheric propagation path (i.e., differential delay, path delay, and path bending) on the performance of IRA for observing the meso–submesoscale oceanic phenomena has been analyzed. We focused on the SSH errors caused by the charge-neutral atmospheric propagation path and the impacts of each SSH error on IRA observation performance.

Combined with the vertical distribution of refractive index simulated by 3D-NWM and the SSH measurement principle of IRA, we identify three types of SSH errors resulting from the propagation path: the SSH error (DDE) about additional interference phase term caused by differential delay, and the SSH errors (PDE and PBE) due to flat phase shift caused by path delay and path bending. According to the derived results, DDE is proportional to the zenith neutral delay (ZND) and increases significantly with the increase of the imaging incident angle. PDE is only related to ZND; the PBE will increase significantly with the increase of the imaging incident angle. Meanwhile, we are surprised to find that PBE seems not sensitive to changes in the atmosphere, which guides us to fit the change of PBE with the imaging incident angle.

The above theoretical derivation results are simulated and verified under the system parameters of SWOT and GL, exhibiting a consistency. From the results, the variation of DDE and PBE with the imaging incident angle is very close, which causes significant SSH error at the far range for IRA. The maximum incident angle of SWOT is less than 4°, and the resulting SSH errors caused by PBE and DDE are approximately 1.2 cm. However, for GL in the far range, the corresponding SSH errors can reach up to 2.8 cm. Consequently, the impact of PBE and DDE on GL is expected to be more pronounced compared to SWOT.

Through the spectral analysis of Jason-3 along-track zenith delay correction data from 2018 to 2022, this paper proposes a 1D spectrum fitting model that statistically describes the spatial distribution characteristics of ZND in the range of scale from 10 to 1000 km. The 2D random field of ZND anomaly is simulated by the Monte Carlo method under the isotropic assumption. The PDE anomaly (PDEA) and DDE anomaly (DDEA) can be obtained from the theory of SSH errors caused by the propagation path with the SWOT system parameters Then, the meso–submesoscale SSHs obtained from the SSH model data provided by HYCOM are superimposed with PDEA, PBE, and DDEA to discuss the influence of charge-neutral atmospheric propagation path on the observation performance of IRA.

According to the results, PDEA and PBE have a major impact on the observation performance of IRA, while the impact of DDEA can be ignored. On one hand, PDEA introduces a random variation of SSH errors of about 2 cm in the swath of IRA. This will contaminate the IRA observation results of the meso–submesoscale ocean phenomena and change the RMS of SSH results. On the other hand, PBE presents a raised ramp phenomenon along the range direction, which can distort the SSH result observed by IRA. In addition, PBE also raises the MSL of the observed region by several centimeters. Therefore, the effect of PBE on the performance of the IRA cannot be neglected.

In summary, the SSH errors about charge-neutral atmospheric propagation will reduce the reliability of the SSH result by IRA. The reasonable correction methods for DDE, PDE, and PBE errors still need to be researched in depth. Meanwhile, the influence of PDEA and PBE on the observation performance of IRA cannot be ignored. The ramp phenomenon caused by PBE along the range is like an SSH system error, yet it has not garnered significant attention.

Data availability statement.

The ERA5 monthly averaged data on pressure levels used in the 3D-NWM can be downloaded from https://cds.climate.copernicus.eu/. The charge-neutral atmospheric zenith delay along-track correction data for the Jason-3 satellite altimeter can be obtained from ftp://ftp.oceans.ncei.noaa.gov/. The HYCOM global SSHs model data are available at https://hycom.org/dataserver/.