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A simplification of rigorous atmospheric raytracing based on judicious rectilinear paths for nearsurface GNSS reflectometry
Earth, Planets and Space volume 72, Article number: 91 (2020)
Abstract
Atmospheric delays are known to cause biases in Global Navigation Satellite System Reflectometry (GNSSR) altimetry applications, such as for sealevel monitoring. The main quantity of interest is the reflectionminusdirect or interferometric atmospheric delay. Recently, we have presented a rigorous raytracing procedure to account for linear and angular refraction in conjunction with reflection as observed from nearsurface platforms. Here, we demonstrate the feasibility of simplifying the ray trajectory by imposing a rectilinear wave propagation model. Two variants were assessed, based on the apparent or refracted satellite direction on the one hand and the geometric or vacuum conditions on the other hand. The former was shown to agree with rigorous results in terms of interferometric radio length while the latter agreed in terms of the interferometric vacuum distance. Upon a judicious combination of the best aspects of the two rectilinear cases, we have defined a mixed variant with excellent agreement with rigorous raytracing in terms of interferometric atmospheric delay. We further showed that mapping functions developed for GNSS positioning cannot be reused for GNSSR purposes without adaptations. Otherwise, the total atmospheric delay may be underestimated by up to 50% at low elevation angles. The present work facilitates the adaptation of existing atmospheric raytracing software for GNSSR purposes.
Introduction
Global Navigation Satellite System Reflectometry (GNSSR) (Cardellach et al. 2011; Jin et al. 2014; Zavorotny et al. 2014) has been widely demonstrated for longterm groundbased coastal sea level altimetry (Larson et al. 2013; 2017). Atmospheric refraction is known to cause a propagation delay which produces a bias in GNSSR altimetry, depending on the satellite elevation angle and the reflector height. Almost all assessments of GNSSR against colocated tide gauges ignore a constant offset between the two sensors, except for example SantamaríaGómez et al. (2015); this in part is not only due to the lack of leveling (surveying) across the two locations but also due to the atmospheric bias. Unfortunately, this limitation undermines one of the promoted advantages of GNSSR altimetry, of providing geocentric sea level measurement. Besides a constant offset (average error), systematic atmospheric refraction errors found in sea level retrievals versus satellite elevation angle (Williams and Nievinski 2017) also affect the precision of retrievals when forming a sitewide average sea level across all visible satellites.
Under multipath reception conditions, direct and reflected radio waves are separated by the interferometric propagation delay \( \tau_{i} = \tau_{r}  \tau_{d} \) (Nievinski and Larson 2014). Under the hypothesis of a large flat and horizontal reflector surface in vacuum, and ignoring other effects, the interferometric propagation delay can be expressed as \( \tau_{i} = 2H\sin e \) where \( e \) is the satellite elevation angle and \( H \) is the reflector height, i.e., the vertical distance between the receiver and the reflecting surface. It is the interferometric atmospheric delay which needs to be removed in GNSSR for determining unbiased reflector height.
Atmospheric refraction manifests in both speed retardation and direction bending along the propagating ray (Nilsson et al. 2013). Its linear and angular components are combined, resulting in the atmospheric propagation delay, which affects GNSS observables such as signaltonoise ratio (SNR), pseudorange and carrier phase. In altimetry, the atmospheric delay may be understood as it were causing a mirage effect, in which the reflecting surface appears to be higher than where it actually is.
Several studies have recognized the importance of atmospheric refraction errors in GNSSR altimetric retrievals (Anderson 2000; Treuhaft et al. 2001; Fabra et al. 2012; Semmling et al. 2012; Roussel et al. 2014; SantamaríaGómez and Watson 2017; Williams and Nievinski 2017). To address the issue, some authors have suggested the adoption of mapping functions used in GNSS positioning, developed for lineofsight or direct propagation from satellites (Nafisi et al. 2012) with minimal adaptation for GNSSR applications. For example, Cardellach et al. (2011) states that “…the delays induced by the tropospheric layer above the receiving platform cancel out, and only those due to the bottom layer, between the surface and the receiver, affect the altimetric range…” In line with this concept of a vertically partitioned atmosphere, Zavorotny et al. (2014) state that “Only the effect coming from the troposphere below the receiver needs to be corrected.” A similar assumption is held in Treuhaft et al. (2001), who defined the zenith delay difference (across surface and antenna altitudes) and multiplied it by a direct mapping function.
However, models for direct propagation as used in GNSS positioning may adequately capture only the effect of linear refraction, i.e., that of speed retardation. This is because the angular refraction experienced by incoming rays in the upper atmospheric layer (i.e., in the portion above the antenna) does not necessarily cancel out when forming the interferometric atmospheric delay, even for nearsurface configurations (SantamaríaGómez and Watson 2017). The incoming reflection ray deviates from a straight line as a function of the gradient of refractivity along its entire path; it is thus a cumulative effect, not restricted to the lower portion of the atmosphere, i.e., between the receiver and the reflecting surface.
As the incident ray arrives along the apparent satellite elevation angle, and abiding to Snell’s law, the refracted reflection point will be shifted compared to unrefracted atmospheric conditions. As the baseline or reference condition for comparison is that of propagation in vacuum, angular refraction thus causes an additional atmospheric delay of geometric nature (SantamaríaGómez and Watson 2017).
In Nikolaidou et al. (2020), we have unified the linear and angular components of interferometric atmospheric delay experienced in GNSSR, demonstrating how they can be derived from first principles. We have also explained the twofold effect of ray bending, introducing subcomponents of the atmospheric geometric delay, to express the shifting of the reflection point as well as the deviation of the ray from a straight line. In that work, we have analyzed the bent wave propagation in groundbased GNSSR altimetry applications. We used a rigorous raytracing approach (RI) in which the Eikonal equation was solved for determining the ray trajectory. Results were of high fidelity but somewhat opaque about the refraction effects involved.
Here, we demonstrate a simplified raytracing approach to determine the interferometric atmospheric delay. We show that the largescale atmospheric geometric delay can be well captured by a judicious choice of rectilinear raypaths. We then assess, for varying satellite elevation and receiver height, under what observation conditions smallscale atmospheric geometric delay is negligible. We justify the simplified rectilinear modeling because it was a common approximation in previous GNSS studies, such as in water–vapor GNSS tomography (Rohm and Bosy 2009; Bender et al. 2011). It was also very common in early modeling efforts of atmosphere effects in radio propagation (Hopfield 1969; Saastamoinen 1972).
In Sect. "Interferometric Raytracing", we describe the numerical procedure while in Sect. "Atmospheric Delay Modeling" we lay down a model for the interaction between the various quantities. An alternative formulation is also presented in Sect. "Atmospheric Delay Modeling", based on the atmospheric layer between receiving antenna and reflecting surface. Numerical assessment results are shown and discussed in Sect. "Results and Discussion", while Sect. "Conclusions" concludes the paper with a summary of the main findings.
Interferometric raytracing
Figure 1 depicts the setup involved in a refracted reflection as observed from a nearsurface receiver. The main position vectors refer to an arbitrary ray position, \( \varvec{r} \); transmitting satellite, \( \varvec{r}_{\text{sat}} \); receiving antenna, \( \varvec{r}_{\text{ant}} \); vacuum surface reflection point, \( \varvec{r}_{\text{sfc}} \); and refracted reflection point, \( \varvec{r^{\prime}_{\text{sfc}}} \). Viewing directions are denoted as unit vectors such as \( \Delta {\user2{\hat{r}}}_{\text{sat}} \) and \( \Delta {\user2{\hat{r}}}^{\prime}_{\text{sat}} \) for geometric (vacuum) and apparent (refracted) conditions, respectively.
Rigorous raytracing
A rigorous interferometric raytracing procedure has been developed in a previous study (Nikolaidou et al. 2020), where we gave a detailed description of both the direct and reflection raytracing. The background can be summarized as follows.
We assume a spherical atmosphere, where the spatial gradient of the index of refraction (\( {\mathbf{\nabla }}{n} \)) points to the center of the sphere and the raypath is a plane curve, i.e., there is no outofplane bending:
where \( \varvec{r^{\prime}} = \varvec{r}  \varvec{r}_{o} \) is a ray vector position with respect to the center of the sphere; the sphere is said to osculate the Earth’s ellipsoid, i.e., its center \( \varvec{r}_{o} \) lies along the ellipsoidal normal and has radius equal to the ellipsoidal Gaussian radius of curvature (Nievinski and Santos 2010).
The evolution of the ray is defined by solving the Eikonal equation (Born and Wolf 1999):
With \( n \) the (scalar) field of index of refraction and \( l \) the incremental raypath arc length. On the one hand, linear refraction will slow down the radio wave (via the atmospheric speed of propagation \( v = c/n \), where \( c \) is the vacuum speed of light). On the other hand, angular refraction will change the direction of propagation, via the gradient of refraction \( {\varvec{\nabla }}n \) (Nievinski and Santos 2010).
To solve Eq. (2), a set of conditions needs to be specified. Often the time is reversed so that raytracing starts at the receiving antenna and ends near the transmitting satellite. A common set of conditions is made of an initial position and an initial direction—the receiver position and the satellite apparent direction; in this case, the final position (the satellite position) is determined as a consequence of the raytracing procedure. Another common choice of boundary conditions is made of initial and final positions (receiver and satellite positions); in this case, the initial or apparent direction follows from raytracing. We start by raytracing the direct ray followed by the reflection. The latter is performed separately for each of the incident and scattered legs (incoming and outgoing segments), which split the whole reflection ray at the specular point. For more details, the reader is referred to Nikolaidou et al. (2020).
Rectilinear raytracing
We simplified the rigorous bent raypath, based on the Eikonal equation, by postulating rectilinear radio propagation:
So, rays are artificially set to coincide with straight line segments, where \( \user2{\check{r}} \) is the initial ray position, \( \user2{\hat{t}} \) the ray tangent direction (constant unit vector), and \( s \) the incremental ray distance.
Under this simplified model, the solution of initial and boundaryvalue problems is no longer necessary, as the ray trajectory is completely known in advance. We denote the infinitesimal straight line distance as \( ds \), e.g., \( \left\ {\user2{r}_{2}  \user2{r}_{1} } \right\ = \int_{\user2{r}}^{{\user2{r}_{2} }} {ds} \). A numerical quadrature is retained, to integrate the propagation delays \( d \) based on refractivity \( N \equiv n  1 \): \( d = \smallint N ds \). Propagation still occurs in an inhomogeneous atmospheric model so the ray is subject to linear refraction and, indirectly, may also be subject to angular refraction, depending on the postulated ray direction, as detailed below.
There are two variants of the rectilinear model. For the rectilinear geometric (RG) model, the direct ray is based on the satellite and antenna position in vacuum:
where the geometric or vacuum satellite relative direction with respect to the antenna is
Continuing with the RG model, the reflection is specified in terms of its incoming and outgoing parts:
where the geometric or vacuum relative surface direction with respect to the antenna is
The second variant of this simplified ray model is the rectilinear apparent (RA) model, for which the direct ray is defined as
The antenna position \( \varvec{r}_{\text{ant}} \) is unchanged and the apparent or refracted satellite relative direction with respect to the antenna \( \Delta {\varvec{\hat{r}}}^{\prime}_{\text{sat}} \) is assumed known; in practice, the latter is obtained from a previous rigorous directpath raytracing (Nikolaidou et al. 2020). The RA reflection is again specified in terms of its incoming and outgoing parts:
where the apparent or refracted relative surface position with respect to the antenna, \( \Delta {\varvec{\hat{r}}}^{\prime}_{\text{sfc}} \) is obtained analogously to \( \Delta {\varvec{\hat{r}}}^{\prime}_{\text{sat}} \). Where necessary, we establish a fictitious apparent satellite position as
lying along a given apparent satellite direction \( \Delta {\varvec{\hat{r}}}^{\prime}_{\text{sat}} \) at a direct distance \(D_{d} = \left\ {\Delta {\user2{\hat{r}}}_{\text{sat}} } \right\ \) which is the same as in vacuum, for convenience. Given these specifications of rectilinear initial and boundary conditions, raytracing proceeds as before.
Atmospheric delay modeling
Now, we describe how to model the interferometric atmospheric delay given the output of the raytracing procedure laid above.
Rigorous delay formulation
Table 1 summarizes the definitions of the intrinsic radio propagation quantities between any two points: vacuum distance: \( D = \left\ {\varvec{r}_{1}  \varvec{r}_{2} } \right\ \); radio length: \(L = \int_{{\varvec{r}}_{1} }^{{\varvec{r}}_{2} } {n\,dl} \); and curve range: \( R = \int_{{\varvec{r}_{1} }}^{{\varvec{r}_{2} }} {1\,dl} \). Table 2 recapitulates the atmospheric delay and its components: total: \( d = L  D = d^{a} + d^{g} \); alongpath: \( d^{a} = L  R = \int_{{\varvec{r}}_{1}} ^{{\varvec{r}}_{2} } {N\,dl} \); and geometric atmospheric delay: \( d^{g} = R  D \). For details, the reader is directed to Nikolaidou et al. (2020).
The interferometric quantities yield as the difference of the corresponding reflection and direct quantities, for example, interferometric vacuum distance: \( D_{i} = D_{r}  D_{d} \); interferometric radio length: \( L_{i} = L_{r}  L_{d} \); and interferometric curve range: \( R_{i} = R_{r}  R_{d} \). The interferometric atmospheric delay follows from two equivalent formulations:
This definition is extended to the interferometric delay components, the alongpath delay: \( d_{i}^{a} = L_{i}  R_{i} \) and the geometric one: \( d_{i}^{g} = R_{i}  D_{i} \). As before, the two parts make up the total delay, i.e., \( d_{i} = d_{i}^{a} + d_{i}^{g} \) (Nikolaidou et al. 2020).
Finally, the atmospheric geometric delay can be further decomposed into the geometricexcess and the geometricshift delays as \( d_{i}^{g} = d_{i}^{g\prime} + d_{i}^{g\prime \prime } \). This is possible with the introduction of a shifted vacuum delay \( D_{i}^{\prime}: \)
where
So, while the ordinary vacuum delay \( D_{i} \) involves the vacuum reflection point, the shifted vacuum delay \( D_{i}^{\prime} \) is obtained freezing the refracted reflection geometry (shifted specular point \( {\varvec{r}}_{\text{sfc}}^{\prime} \) and apparent satellite direction \( \Delta {\varvec{\hat{r}}}^{\prime}_{\text{sfc}} \)) and undressing the atmosphere (i.e., nullifying the refractivity, \( N = 0 \)).
The atmospheric geometricshift delay is a consequence of the application of Snell’s law at the refracted reflection point. It will maintain its magnitude even for small reflector heights, as it is largely formed by the angular refraction taking place above the receiving antenna. In contrast, the atmospheric geometricexcess delay corresponds to the deviation of the ray paths from straightline segments. It may be assumed close to zero for sufficiently small reflector heights, as determined below.
Rectilinear delay formulation
In Table 3 we have adapted the rigorous definitions above (henceforth RI) for the two cases of rectilinear propagation (RG and RA); an overhead bar notation is used for distinction.
The RG vacuum distance equals the ordinary one used in the RI case, \( \bar{D} = D \). In the RA approach, though, it equals the shifted vacuum distance, \( \bar{D}^{\prime} = D^{\prime} \). The RG approach lacks any angular refraction effect and is subject only to linear refraction, albeit on a simplified ray path. The RA approach, on the other hand, includes both types of refraction, although ray bending is accounted in an allornothing manner, only in the incident direction and it is not allowed to vary along the raypath as in the RI case. In both rectilinear cases, the curve range equals the respective vacuum distances, as there is no ray bending:
The rectilinear models may seem overly simplistic, but it turns out a judicious combination proved accurate, as demonstrated by results below. We define a rectilinearmixed (RM) model, denoted with double overhead bars. It utilizes the RG vacuum distance in conjunction with the RA radio length and the RA curve range:
Table 4 summarizes the definitions of the various rectilinear interferometric atmospheric delays and their components.
Where necessary, the atmospheric altimetry correction follows from half the rate of change of delay with respect to the sine of the elevation angle (Nikolaidou et al. 2020):
Results and discussion
Here, we assess results from rectilinear approach against rigorous raytracing. We assess first wave propagation quantities and later the derived atmospheric delays and altimetry corrections.
As atmospheric model source, we employed the COSPAR International Reference Atmosphere 1986 (CIRA86) climatology (Chandra et al. 1990; Fleming et al. 1990); more specifically, file twp.lsn, available for download from https://ccmc.gsfc.nasa.gov/modelweb/atmos/cospar1.html. It provides temperature (0–120 km) and pressure (20–120 km) at 5km intervals; surface pressure is obtained via hydrostatic integration. Temperature and pressure at any other sampling points are obtained via linear and loglinear interpolation, respectively.
In terms of computational cost, for an elevation angle of 5 degrees and a delay convergence tolerance of \( 10^{  6} {\text{m}} \), the processing time decreases by 67%, from 0.45 s in RI to 0.15 s in RM, i.e., RM takes only onethird the time taken by RI.
Propagation quantities
We start by illustrating in Fig. 2 the discrepancy in interferometric vacuum distance, \( D_{i} \). The rectilineargeometric (RG) result, \( \bar{D}_{i} \), is in near absolute agreement with that of RI. In contrast, rectilinear apparent (RA) result, \( \bar{D}^{\prime}_{i} \), falls short of \( D_{i} \) by an amount which is a consequence of angular refraction (ray bending angle, \( \delta e = e^{\prime}  e \)). At zenith, all interferometric vacuum distances agree to \( 2H \). Their discrepancy increases at low elevation angles, reaching 6.5 cm at 5° elevation angle for a 10m reflector height.
Next, Fig. 3 shows the discrepancy in interferometric curve range, \( R_{i} \), among the various approaches. Contrary to the previous comparison, here the RG curve range, \( \bar{R}_{i} \), has a large discrepancy with respect to RI. In this comparison, it is RA that best matches RI, \( \bar{R}^{\prime}_{i} \approx R_{i} \), as both are subject to angular refraction on the raypath. So, although the fictitious refracted satellite is very far from the actual satellite position, it is more representative for the calculation of the interferometric curve range. The agreement between RA and RI is not exact because rectilinear propagation neglects pathdependent incremental ray bending, accounting only for the total ray bending. The RA curve range degenerates to the respective (modified) vacuum distance, \( \bar{R}^{\prime}_{i} = \bar{D}^{\prime}_{i} \).
Last, the discrepancy in interferometric radio length \( L_{i} \) follows a similar pattern than that of the interferometric curve range (Fig. 3), with RI agreeing better with RA, \( \bar{L}^{\prime}_{i} \), than with RG, \( \bar{L}_{i} \). This characteristic will be further analyzed below, in terms of the atmospheric delay.
In summary, RI vacuum distance is best approximated by RG while RI curve range and RI radio length are best approximated by RA. With this we justify the rectilinearmixed (RM) approach (17), which borrows the best of each rectilinear models: RG \( \left( {\bar{\bar{D}}}_{i} = {\bar{D}}_{i} \right) \) and RA \( \left( {\bar{\bar{L}}}_{i} = {\bar{L}}^{\prime}_{i} {\text{ and }} {\bar{\bar{R}}}_{i} = {\bar{R}}^{\prime}_{i} \right). \)
Total atmospheric delay and altimetry correction
In this section, we shall assess rectilinear results in terms of atmospheric delay and the resulting atmospheric altimetry correction. Figure 4 shows the discrepancy, \( \bar{\bar{d}}_{i}  d_{i} \), in total interferometric atmospheric delay, between RM (\( \bar{\bar{d}}_{i} = \bar{\bar{L}}_{i}  \bar{\bar{D}}_{i} \)) and RI (\( d_{i} = L_{i}  D_{i} \)) approaches. For a 10m reflector height, the agreement is excellent, having a maximum submm discrepancy near the horizon. Further, it demonstrates that rigorous results (RI) can be approximated well by a judicious rectilinear propagation scheme (RM). It is remarkable that the effect of ray bending can be accurately represented by a straight line at the appropriate direction in the interferometric case.
The discrepancy in the resulting interferometric atmospheric altimetry correction (Fig. 5) follows a similar pattern than in the previous result, but scaled approximately by a factor of ten. The maximum discrepancy in atmospheric altimetry is found near the horizon, amounting to 0.4 cm for a 10m reflector height. Both figures illustrate the proportional increase in the discrepancy with reflector height. For a reflector height of 20 m, the RM–RI agreement is better than 1 cm in altimetry correction for any elevation above 5 degrees; so the 20m antenna height may be adopted as a threshold of validity for the assumption of nearsurface conditions for the rectilinear model.
Atmospheric delay components
Figure 6 compares interferometric geometric atmospheric delay, across RI \( \left( {d_{i}^{g} = R_{i}  D_{i} } \right) \) and RM \( \left( \bar{\bar{d}}_{i}^{g} = \bar{\bar{R}}_{i}  \bar{\bar{D}}_{i} \right) \) approaches. The discrepancy \( \bar{\bar{d}}_{i}^{g}  d_{i}^{g} \) converges to zero at zenith, where elevation bending is null, and it grows to at most 1 mm toward the horizon for a 10m reflector height. Figure 4 demonstrates that the bulk of angular refraction is well captured by the atmospheric geometricshift delay, the difference between RA and RG interferometric vacuum distances, \( \bar{\bar{d}}_{i}^{g \prime \prime} = \bar{\bar{D}}_{i}^{\prime}  {\bar{D}}_{i} \). The present RM–RI discrepancy in geometric delay is dominated by the atmospheric geometricexcess delay, which also equals the RA–RI discrepancy in curve ranges: \( {\bar{\bar{d}}_{i}^{g}}  {d_{i}^{g}} = {\bar{\bar{d}}_{i}^{g \prime}}  {d_{i}^{g \prime}} = \bar{R}^{\prime}_{i}  R_{i} \). Thus, it follows from the incremental elevation bending present in rigorous raytracing but absent in the rectilinear approaches.
Figure 7 shows the discrepancy, \( \bar{\bar{d}}_{i}^{a}  d_{i}^{a} \), in interferometric alongpath atmospheric delay, across RI \( \left( {d_{i}^{a} = L_{i}  R_{i} } \right) \) and RM \( \left( {\bar{\bar{d}}_{i}^{a} = \bar{\bar{L}}_{i}  \bar{\bar{R}}_{i} } \right) \) cases. The agreement is even better (at \( 50 \, \mu {\text{m}} \) level), with discrepancy values more randomly distributed, resembling numerical noise (likely caused by interpolation in the CIRA atmospheric model).
Slant factors
For a better comparison to the standard approach reported in the literature, based on mapping functions, we provide an analysis based on slant factors, \( f = d/d^{z} \), defined as the ratio between slant delay and zenith delay at a particular elevation angle. Slant factors computed from direct raytracing are the input data for developing mapping function models, such as the Global Mapping Function (GMF) (Boehm et al. 2006), after fitting to a particular functional expression valid over a given space–time domain (Urquhart et al. 2012).
The direct slant factor is defined as \( f_{d} = d_{d} /d_{d}^{z} \), where the direct zenith delay is that at the antenna: \( d_{d}^{z} = d_{\text{ant}}^{z} \). The interferometric slant factor, \( f_{i} = d_{i} /d_{i}^{z} \), uses the total interferometric zenith delay, \( d_{i}^{z} = 2\left( {d_{ant}^{z}  d_{sfc}^{z} } \right) \), which is twice the zenith delay difference across antenna and surface. The slant factors for interferometric components \( f_{i} = f_{i}^{a} + f_{i}^{g} \) are computed similarly, as \( f_{i}^{a} = d_{i}^{a} /d_{i}^{z} \) and \( f_{i}^{g} = d_{i}^{g} /d_{i}^{z} \) for alongpath and geometric terms, respectively.
Figure 8 shows the slant factors defined above. They all follow the exponential decay of delay with elevation angle. However, at the lowest elevation angle (5 degrees) the interferometric slant factor measures twice the direct one (20.47 m/m vs. 10.29 m/m). At zenith, where angular refraction is null, they both converge to unity. At low elevation angle, though, using a direct mapping function will underestimate the interferometric delay systematically with decreasing elevation angle by up to 50%.
In relation to the components, the interferometric alongpath slant factor resembles the direct slant factor, \( f_{i}^{a} \approx f_{d} \). They are both related to the thickness of atmospheric layers, albeit different ones: respectively, the inner and outer ones, below and above the antenna. Furthermore, the direct slant factor \( f_{d} = f_{d}^{a} + f_{d}^{g} \) also involves a weighting of the layer slant distances by refractivity in the integrand of \( \smallint Ndl \), as well as a minor contribution from the direct atmospheric geometric delay, \( f_{d}^{g} \). The interferometric geometric slant factor \( f_{i}^{g} \) approaches the alongpath one \( f_{i}^{a} \) at low elevations, e.g., 9.97 m/m vs. 10.51 m/m at 5° elevation angle, but converges to zero at zenith due to the absence of bending.
Finally, it should be emphasized that mapping functions developed for GNSS positioning, such as the GMF, are supposed to agree only with the direct slant factors, as in fact it does the agreement with our results is within 2% at 5° (not shown). However, correcting for the atmospheric interferometric delay in GNSSR using GMF or a similar mapping functions will introduce an exponentially increasing bias with elevation angle. The remaining geometricshift atmospheric delay, which is a result of Snell’s law on the refracted specular point, cannot be captured using only the direct or LOS propagation effects, and would require a model for the angular refraction.
Conclusions
A simplification of the rigorous interferometric raytracing approach (RI) was carried out, imposing a rectilinear ray propagation model for GNSS reflectometry (GNSSR) applications. Two initial variants were developed, considering the apparent (refracted) and the geometric (vacuum) satellite directions. The rectilineargeometric (RG) agreed with the rigorous (RI) for the vacuum distance, while the rectilinearapparent (RA) agreed with RI for the radio length. Both RG and RA had poor performance in terms of atmospheric delays, though.
Upon combination of the best matching aspect of the two above, RG and RA, we defined a third variant, the rectilinearmixed (RM) model. It demonstrated excellent agreement in the interferometric atmospheric delay, both in total value and in all components (alongpath and geometric). GNSSR altimetry corrections can, therefore, be predicted by performing a single rigorous raytracing in the direct or lineofsight direction to determine the ray bending, followed by two rectilinear raytracings in the direct and reflection directions.
The rectilinear models demonstrated for the interferometric atmospheric delay allow for faster and more efficient raytracing, as the reflection threepoint boundaryvalue problem (satellite–surface–antenna) can be replaced for an easier twopoint problem. Thus, existing raytracing software can be adapted more easily for groundbased GNSSR applications. The simplifications demonstrated here also pave the way for the future development of more convenient closedform expressions.
Another key demonstration is that mapping functions developed for GNSS positioning, or even a direct raytracing procedure, cannot be reused for GNSSR purposes without adaptations. The interferometric atmospheric delay is induced by the atmosphere both above and below the receiver, roughly corresponding to its angular refraction and linear refraction components. At low elevation angles, where the interferometric delay components are similar, directonly mapping functions will underestimate the total delay by nearly half. In the current study, however, we showed how the interferometric delay and its components can be deduced with a directonly raytracing procedure by employing a judicious combination of two simpler rectilinear models for the raypath.
Availability of data and materials
No external datasets were used for this study. The study uses simulation results.
Abbreviations
 GNSSR:

Global Navigation Satellite System Reflectometry
 RA:

Rectilinear apparent
 RG:

Rectilinear geometric
 RM:

Rectilinearmixed
 RI:

Rigorous raytracing approach
References
Anderson KD (2000) Determination of water level and tides using interferometric observations of GPS signals. J Atmos Ocean Technol 17:1118–1127. https://doi.org/10.1175/15200426(2000)017%3c1118:DOWLAT%3e2.0.CO;2
Bender M, Dick G, Ge M et al (2011) Development of a GNSS water vapour tomography system using algebraic reconstruction techniques. Adv Sp Res 47:1704–1720. https://doi.org/10.1016/J.ASR.2010.05.034
Boehm J, Niell A, Tregoning P, Schuh H (2006) Global Mapping Function (GMF): a new empirical mapping function based on numerical weather model data. Geophys Res Lett 33:L07304. https://doi.org/10.1029/2005GL025546
Born M, Wolf E (1999) Principles of optics Electromagnetic theory of propagation, interference and diffraction of light, 7th expand. Cambridge University Press
Cardellach E, Fabra F, NoguésCorreig O et al (2011) GNSSR groundbased and airborne campaigns for ocean, land, ice, and snow techniques: application to the GOLDRTR data sets. Radio Sci. https://doi.org/10.1029/2011RS004683
Chandra S, Fleming EL, Schoeberl MR, Barnett JJ (1990) Monthly mean global climatology of temperature, wind, geopotential height and pressure for 0–120 km. Adv Space Res 10(6):3–12. https://doi.org/10.1016/02731177(90)90230W
Fabra F, Cardellach E, Rius A et al (2012) Phase altimetry with dual polarization GNSSR over sea ice. IEEE Trans Geosci Remote Sens 50:2112–2121. https://doi.org/10.1109/TGRS.2011.2172797
Fleming EL, Chandra S, Barnett JJ, Corney M (1990) Zonal mean temperature, pressure, zonal wind and geopotential height as functions of latitude. Adv Space Res 10(12):11–59. https://doi.org/10.1016/02731177(90)90386E
Hopfield HS (1969) Twoquartic tropospheric refractivity profile for correcting satellite data. J Geophys Res. 74(18):4487–4499
Jin S, Cardellach E, Xie F (2014) GNSS remote sensing theory, methods and applications. Springer, Netherlands
Larson KM, Ray RD, Nievinski FG, Freymueller JT (2013) The accidental tide gauge: a GPS reflection case study from Kachemak Bay, Alaska. IEEE Geosci Remote Sens Lett. https://doi.org/10.1109/LGRS.2012.2236075
Larson KM, Ray RD, Williams SDP et al (2017) A 10year comparison of water levels measured with a geodetic GPS receiver versus a conventional tide gauge. J Atmos Ocean Technol 34:295–307. https://doi.org/10.1175/JTECHD160101.1
Nafisi V, Urquhart L, Santos MC et al (2012) Comparison of raytracing packages for troposphere delays. IEEE Trans Geosci Remote Sens 50:469–481. https://doi.org/10.1109/TGRS.2011.2160952
Nievinski FG, Larson KM (2014) Forward modeling of GPS multipath for nearsurface reflectometry and positioning applications. GPS Solut 18:309–322. https://doi.org/10.1007/s102910130331y
Nievinski FG, Santos MC (2010) Raytracing options to mitigate the neutral atmosphere delay in GPS. Geomatica 64:191–207
Nikolaidou T, Santos CM, Williams DPS, GeremiaNievinski F (2020) Raytracing atmospheric delays in groundbased GNSS reflectometry. J Geod. https://doi.org/10.1007/s00190020013908
Nilsson T, Böhm J, Wijaya DD et al (2013) Path delays in the neutral atmosphere. Springer, Berlin, pp 73–136
Rohm W, Bosy J (2009) Local tomography troposphere model over mountains area. Atmos Res 93:777–783. https://doi.org/10.1016/J.ATMOSRES.2009.03.013
Roussel N, Frappart F, Ramillien G et al (2014) Simulations of direct and reflected wave trajectories for groundbased GNSSR experiments. Geosci Model Dev. https://doi.org/10.5194/gmd722612014
Saastamoinen J (1972) Atmospheric correction for the troposphere and stratosphere in radio ranging satellites. American Geophysical Union (AGU), Washington, D. C, pp 247–251
SantamaríaGómez A, Watson C (2017) Remote leveling of tide gauges using GNSS reflectometry: case study at Spring Bay, Australia. GPS Solut 21:451–459. https://doi.org/10.1007/s102910160537x
SantamaríaGómez A, Watson C, Gravelle M et al (2015) Levelling colocated GNSS and tide gauge stations using GNSS reflectometry. J Geod 89:241–258. https://doi.org/10.1007/s001900140784y
Semmling AM, Schmidt T, Wickert J et al (2012) On the retrieval of the specular reflection in GNSS carrier observations for ocean altimetry. Radio Sci. https://doi.org/10.1029/2012RS005007
Treuhaft RN, Lowe ST, Zuffada C, Chao Y (2001) 2cm GPS altimetry over Crater Lake. Geophys Res Lett 28:4343–4346. https://doi.org/10.1029/2001GL013815
Urquhart L, Nievinski FG, Santos MC (2012) Raytraced slant factors for mitigating the tropospheric delay at the observation level. J Geod 86:149–160. https://doi.org/10.1007/s001900110503x
Williams SDP, Nievinski FG (2017) Tropospheric delays in groundbased GNSS multipath reflectometry—experimental evidence from coastal sites. J Geophys Res Solid Earth 122:2310–2327. https://doi.org/10.1002/2016JB013612
Zavorotny VU, Gleason S, Cardellach E, Camps A (2014) Tutorial on remote sensing using GNSS bistatic radar of opportunity. IEEE Geosci Remote Sens Mag 2:8–45. https://doi.org/10.1109/MGRS.2014.2374220
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FGN acknowledges funding from CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico; 457530/20146, 433099/20186) and Fapergs (Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul; 26228.414.42497.26062017). TN acknowledges funding from Mitacs (Grant No. IT11988).
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FGN and TN conceived the idea. FGN and TN developed the theory and performed the computations. MCS and SDPW verified the analytical methods. SDPW provided experimental data. FGN and TN wrote the manuscript. MCS contributed to the final version of it. All authors provided critical feedback and helped shape the research. All authors read and approved the final manuscript.
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Nikolaidou, T., Santos, M., Williams, S.D.P. et al. A simplification of rigorous atmospheric raytracing based on judicious rectilinear paths for nearsurface GNSS reflectometry. Earth Planets Space 72, 91 (2020). https://doi.org/10.1186/s40623020012061
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Keywords
 GPS
 GNSS
 GNSSR
 Reflectometry
 Atmospheric delay
 Atmospheric refraction
 Raytracing
 Radio wave propagation