Residual Error Model to Bound Unmodeled Tropospheric Delays for Terrestrial Navigation Systems at Very Low Elevation Angles

2.1 Tropospheric Delay Decomposition

Tropospheric conditions can be divided into three categories: nominal (𝒯_n), ducting (𝒯_d), and off-nominal (𝒯_o):

$𝒯={𝒯}_n+{𝒯}_d+{𝒯}_0$ 2

Nominal tropospheric errors are described by a tropospheric profile, where the temperature, pressure, and humidity decrease with increasing altitude. Tropospheric errors due to ducting (𝒯_d) arise from atmospheric regions in which the temperature and humidity profiles increase with increasing altitude. Although ducting is not a common phenomenon across all geographical locations, some locations on the western coasts of the Americas, Africa, the Persian Gulf, Australia, and equatorial regions may exhibit ducting probabilities greater than 90% (Narayanan et al., 2019).

For the purposes of this paper, the error due to ducting is considered separately from other off-nominal tropospheric errors. Both the nominal and off-nominal troposphere errors affect the accuracy and integrity of the aircraft position solution and must therefore be characterized and bounded.

The off-nominal component (𝒯_o) stems from atmospheric nonuniformities, such as weather fronts and thunderstorms. The additional error arising from these off-nominal atmospheric effects is not considered in this paper and is left as an item for future work.

2.2 Bounding Unmodeled Tropospheric Delays in a GNSS

The tropospheric delay model currently used in GNSS safety-of-life applications is recommended by the Radio Technical Commission for Aeronautics (RTCA); the associated maximum zenith error has a standard deviation of 0.12 m globally (RTCA SC-159, 2020). As with the delay itself, the residual error for the tropospheric delay equation is given as follows:

$\delta 𝒯=m_h\left(E\right)\cdot \delta Z_h+m_w\left(E\right)\cdot \delta Z_w$ 4

It is typically assumed that the error δ𝒯 predominantly arises from the zenith delay models and not the mapping functions. This assumption is valid for GNSSs, but not for terrestrial ranging. The difference lies in the fact that the most commonly used functional form of the mapping functions in GNSSs is based on continued fractions, such as the Vienna mapping function (Boehm et al., 2006), the Niell mapping function (Niell, 1996), etc., and the parameters of the mapping function are tabulated and regularly updated via least-squares fitting to ray-traced tropospheric delays. With this approach, the GNSS mapping functions are accurate on the order of sub-centimeter level down to elevations of approximately 2° or 3° (Guo & Langley, 2003).

Because the wet and hydrostatic mapping functions are approximately equal for satellite elevation angles typically used in aviation navigation (above a few degrees), the standard deviation of the residual tropospheric error in Equation (4) is further approximated as follows (RTCA SC-159, 2020):

${\sigma }_{tropo}={\sigma }_{TVE}\cdot m\left(E\right)$ 5

where a modified version of the Black and Eisner mapping function is used to project the total zenith delay error (σ_TVE) to the desired elevation angle (RTCA SC-159, 2020).

The GNSS-like approach for deriving a residual tropospheric error is not conservative when applied to terrestrial ranging. Aviation GNSS users typically have access to a relatively high number of satellite measurements; thus, GNSS-based integrity services usually apply an elevation mask of 5°. In contrast, measurements from terrestrial systems are typically used at elevation angles far lower than that used in GNSSs (on the order of 0.2° (Narayanan & Osechas, 2023)), where over-bounding the residual tropospheric delay (σ_tropo) is far more critical than for high-elevation situations faced in GNSSs. At these low elevation angles, tropospheric mismodeling due to error in the mapping functions and the impact of tropospheric error due to ducting become non-negligible and must be considered in the development of a residual error model. Thus, the nominal troposphere correction and its corresponding variance are inadequate to conservatively bound troposphere errors for terrestrial ranging.

2.3 Modeling Tropospheric Delay in Terrestrial Navigation

Tropospheric delay modeling for terrestrial navigation with mapping functions utilizes a model for zenith delays (Z_t), which is then projected to the slant delay (𝒯) at the desired vacuum elevation angle (E). In prior work (Narayanan & Osechas, 2023), a tropospheric mapping function specific to terrestrial ranging, called the polynomial mapping function (PMF), was developed based on a refractivity profile representative of the atmosphere up to the tropopause. This mapping function developed for terrestrial ranging is a total mapping function that projects the total zenith delay (Z_t) into the slant direction. In general, this is represented as follows (Narayanan & Osechas, 2023):

$𝒯\left(E\right)=Z_t\cdot m\left(E\right)$ 6

where m(E) represents the mapping function evaluated at elevation angle E. The total zenith delay (Z_t) is given as follows:

$Z_t=Z_h+Z_w+Z_d$ 7

where Z_d is the additional zenith delay error due to ducting. Thus, the tropospheric delay estimated via the PMF accounts for the nominal delay (𝒯_n) modeled by the mapping function and the additional error due to ducting (𝒯_d).

Note that the mapping function m(E) described in Equation (6) is a total mapping function that uses a single mapping function to project the total zenith delay (Equation (7)) into the slant direction, yielding the total tropospheric delay. In contrast, the approach described in Equation (3) is the established methodology for modeling the tropospheric delay for GNSS applications, wherein the tropospheric delay is the sum of the contributions due to hydrostatic and non-hydrostatic components only, yielding the nominal tropospheric delay.

The mapping functions used in terrestrial ranging, such as the PMF, may introduce uncertainty into the measurements. Recent studies (Narayanan & Osechas, 2023; Narayanan et al., 2017) have validated the existing tropospheric mapping functions for terrestrial navigation using flight test data, showing that, at elevation angles below 0.5°, the error can increase from subcentimeter level to 40–50 cm (Narayanan & Osechas, 2023) at an aircraft range of approximately 250 km. This trend clearly illustrates the need to consider the impact of mapping functions on the tropospheric residual error model.

Anomalous tropospheric conditions, such as ducting, result in worst-case ranging errors on the order of magnitude of the ranging accuracy of terrestrial navigation systems (Narayanan et al., 2019). Recent work acknowledged the need to quantify and define a means for bounding potential errors due to tropospheric ducting for terrestrial ranging. Pressure, temperature, and humidity typically decrease with increasing altitude, but in the presence of ducting, the temperature increases with increasing altitude. Consequently, the refractivity profile deviates from the nominal behavior, resulting in additional tropospheric delay. At elevation angles close to the local horizon, terrestrial ranging signals traverse a significant distance over the first few kilometers of the atmosphere, where ducting is more likely to occur (Von Engeln & Teixeira, 2004). Thus, the relative impact of ducting-induced ranging errors is non-negligible in terrestrial ranging and must be accounted for within the residual error model.

4.1 Computational Setup

The computational setup estimates the residual tropospheric delay error by comparing the reference values from ray tracing with those derived from zenith delay models and mapping functions. Figure 2 presents a flowchart indicating the various steps involved in estimating the residual error contributions from zenith delay models and the PMF to model the tropospheric delays for terrestrial ranging.

The ray-traced delays, regarded as “reference” values, are used in validating the accuracy of the zenith delay models and the PMF to model tropospheric delays. Ray tracing utilizes the ground station coordinates, the vacuum elevation angle, and the azimuth of observation between the ground station and aircraft to yield reference values of the tropospheric delay (𝒯_r), mapping function (m_r), and zenith delays (Z_r), which are later used to separately estimate the residual error contributions due to mapping functions and zenith delay models. The numerical data underlying the ray tracing stems from the ERA-5 ECMWF re-analysis NWM data. These meteorological data include the geopotential, pressure (P), temperature (T), and specific humidity (q) on 137 model levels (at predefined heights), at a horizontal resolution of 0.125° × 0.125° in latitude and longitude, sampled every 1 h.

The model-level data require further processing of the meteorological information to determine the atmospheric parameters on individual model levels. In NWMs, and often in meteorology, the height is not represented as geometric height or altitude, but in terms of geopotential height. Before ray-tracing calculations can be performed, the vertical coordinate system of the meteorological data from the NWM should be transformed into an ellipsoidal-height-based reference system that is used in geodetic data processing. In this work, the ray-tracing tool uses a piecewise-linear approach. A more detailed description of the ray-tracing approach is given in the work by Narayanan (2023).

The difference between the PMF (m) and the reference mapping function (m_r) determined from ray tracing illustrates the error due to the mapping function (δm) as a result of propagation of the ranging signal under nominal atmospheric conditions:

$\delta m\left(E\right)=m_r\left(E\right)-m\left(E\right)$ 21

As shown in Figure 2, this error is then projected to the desired elevation angle via the total zenith delay (Z_t), yielding a residual error due to the mapping function:

$\delta {𝒯}_m\left(E\right)=\delta m\left(E\right)\cdot Z_t$ 22

Note that because ray tracing utilizes NWMs, the derived tropospheric delay values account for the additional propagation delay, if any, due to non-nominal atmospheric behavior. Thus, when estimating the total zenith delay, we consider the zenith delay error arising from models that model the nominal zenith delay (hydrostatic: Z_h and non-hydrostatic: Z_w) and ducting-induced delay (Z_d).

The residual zenith delay (δZ_t) is the difference between the zenith delay predicted by models such as the Saastamoinen model (for hydrostatic) (Saastamoinen, 1972) and GPT2w model (for wet) (Böhm et al., 2015) and the zenith delay error due to tropospheric ducting events (Z_d) with the reference zenith delay from ray tracing (Z_r). The difference between the zenith delay computed from ray tracing and that from these models yields an estimate of the residual error due to the zenith delay models:

$\begin{array}{ccc}\delta Z_t& =& Z_r-(Z_h+Z_w+Z_d)\\ & =& Z_r-Z_t\end{array}$ 23

As shown in Figure 2, this error is then projected to the desired elevation angle via the ray-traced mapping function (m_r), yielding a residual error due to the zenith delay:

$\delta {𝒯}_Z\left(E\right)=m_r\left(E\right)\cdot \delta Z_t$ 24

The zenith delay Z_t is determined as follows:

1. Under the operational context for terrestrial navigation (as shown in Figure 1), we must estimate the zenith delay between the ground station and the aircraft.

2. Given the location of the ground station and aircraft, the meteorological parameters, such as pressure, temperature, and humidity, required to compute the zenith delays are used within the Saastamoinen and GPT2w models to estimate the zenith delay values: Z_g and Z_a for the ground station and aircraft, respectively. Here, Z_g and Z_a each include the hydrostatic (Z_h) and wet (Z_w) zenith delay components estimated using the Saastamoinen and GPT2w models, respectively. An important aspect to note here is that the zenith delay values determined from these models yield the delay from the respective user location up to the mesosphere (~80 km).

3. The zenith delay (Z_t) between the ground station and aircraft is then given as follows:

$Z_t=Z_g-Z_a+Z_d$ 25

The tropospheric ducting-induced zenith delay error (Z_d) and the associated variance $\left({\sigma }_d^2\right)$ can be estimated either from a statistical analysis of 10 years (2008-2017) of re-analysis numerical weather data for ducting-induced zenith delay error or from the uncaptured zenith delay error due to ducting from a duct monitor. In a recent study (Narayanan et al., 2022), a global ducting climatology was created based on 10 years of ECMWF re-analysis NWM data. In this study, the seasonal and diurnal behavior of various tropospheric duct parameters, including the additional zenith delay error introduced because of ducting, was investigated. It was observed that ducting can result in worst-case zenith delay errors on the order of ~20 cm. In the work by Narayanan et al. (2022), a duct monitor was developed using ECMWF forecast NWM data, and it was shown that ducting-induced zenith delay errors can be estimated with an accuracy of ~3 cm. Thus, following the current approach of tropospheric residual error modeling for GNSSs, a global constant value of ~20 cm can be used, or alternatively, the tropospheric error due to ducting can be estimated via the duct monitor.

Note that the re-analysis NWM is the most accurate information available on the state of the atmosphere and inherently captures the effects of ducting, weather fronts, and thunderstorms. Subsequently, the ray-traced reference delays generated from these NWM data already account for these off-nominal atmospheric effects at the resolution of the NWM grid. Thus, the residual errors δm and δZ that represent the error due to the mapping function and zenith delay models, respectively, do account for these off-nominal effects to a certain extent. However, because of the limited resolution of the NWM, certain atmospheric effects that occur on smaller scales may go uncaptured. These uncaptured error components must be accounted for in the off-nominal tropospheric delay (𝒯_o) error term.

4.2 Validation Setup

An effective residual error model should over-bound the tropospheric delay error irrespective of the distance propagated by the ranging signal through the atmosphere for any given elevation angle above the local horizon or the azimuth of observation. Thus, we validate the residual error model over a uniformly sampled elevation angle–aircraft range space with the following combination of parameters:

• Elevation angle ranging from 0.2° to 90°

• Aircraft range from 1 km to 250 km

• Azimuth of measurement between the ground station and aircraft ranging from 0° to 360°

• Aircraft altitude varying from ~30 m to ~12 km

Figure 3 shows the range of elevation angles and aircraft ranges for which we evaluate the residual error model. At elevation angles above 20°, aircraft ranges greater than 30 km would cause the aircraft altitude to exceed 40,000 feet, which is the maximum cruising altitude for most civil aviation. As shown in Figure 3, this case results in a relatively increasing number of two-dimensional (2D) elevation angle–aircraft range bins for elevation angles close to the local horizon at the aircraft range above 30 km. Each of these 2D bins contains 10,000 samples varying in elevation angle and aircraft range. This approach provides a sufficiently sampled data set that can be used to derive a residual error model for tropospheric delays for terrestrial navigation.

• Download figure
• Open in new tab
• Download PowerPoint

Figure 3

Considering a maximum aircraft altitude of 12 km, ranging signals from terrestrial nav-aids at elevation angles close to the local horizon can lead to aircraft ranges on the order of 200 km, in contrast to higher elevation angles.

Although the residual error model can be tailored to specific locations, it is advantageous to derive the simplest models possible, as they must be implemented by an airborne user. One of the advantages of the RTCA MOPS recommendation is its ultimate simplicity, as it provides just a single global value to model the zenith delay error.

To maintain the simplicity of the model to the extent possible while providing a less conservative yet safe model, we derive three types of residual tropospheric error models:

• A model that provides a single standard deviation value representing the maximum residual tropospheric delay error irrespective of elevation angle or aircraft range

• A model that considers variations in residual tropospheric delay error with respect to aircraft range while neglecting the variation in residual tropospheric error across different elevation angles

• A model that takes into account both the elevation angle and the aircraft range as inputs to provide an estimate of the standard deviation, considering the impact of elevation angle and distance propagated by the terrestrial ranging signal on the tropospheric delay error

Note that the three options described above are validated via the computational setup shown in Figure 2. Figure 2 presents the general flow used to compute the residual tropospheric delay error. The three options indicate how we analyze these tropospheric delay residuals by considering both elevation angle and aircraft range, considering aircraft range only, or neglecting their interdependence in order to derive appropriate over-bounds.

5.1 Modeling the Error Contribution from the Tropospheric Mapping Function

The residual error due to the mapping function has a mean of zero and can be over-bounded by a Gaussian over-bound. Figure 4 shows a histogram of the residual error contribution when the zenith delay is scaled with the mapping function. Because the mapping function is simply a scale factor, it is a dimensionless quantity. Thus, as described in Equation (20), the error in the mapping function $\left({\sigma }_m^2\right)$ projected on the slant range using the reference zenith delay (Z_t) as $Z_t^2\cdot {\sigma }_m^2$ . is Figure 4 shows that this error can be on the order of ±40 cm. Note that this residual error covers all of the elevation angles and aircraft ranges used in this analysis. The relatively large tails of the distribution shown in Figure 4 primarily correspond to elevation angles close to the horizon coupled with aircraft ranges above 200 km.

• Download figure
• Open in new tab
• Download PowerPoint

Figure 4

The residual tropospheric delay error (r_n) due to the PMF across all locations, elevation angles, aircraft altitudes, and aircraft distances has a mean of zero and can be conservatively modeled with a zero-mean Gaussian over-bound.

Over-bounding the error due to the mapping function with a zero-mean Gaussian distribution results in a non-negligible variance. This finding emphasizes the need to consider this error within the residual error modeling for terrestrial ranging. Normalizing the residual error via its standard deviation enables us to determine an over-bounding Gaussian distribution that provides a constant value for the residual tropospheric error sigma (σ_tropo); this value can be used by an airborne user irrespective of their elevation angle or range from the ground station. The normalized residual (r_n) is given as follows:

$r_n=\frac{\delta {𝒯}_m-\overline{\delta {𝒯}_m}}{\sigma \left(\delta {𝒯}_m\right)}$ 26

where δ𝒯_m = δm ∙ Z_t is the residual error component due to the mapping function described by the first term in Equation (22). Figure 5 shows the cumulative distribution function (CDF) and 1-CDF of the mapping function residual error, along with the unit-variance Gaussian distribution for reference and the over-bounding Gaussian distribution N(0, σ) with a standard deviation (σ) of 11.7 cm. Although this approach is advantageous from a safety viewpoint, it has a negative effect on the availability and continuity of the positioning service.

• Download figure
• Open in new tab
• Download PowerPoint

Figure 5

The CDF and 1-CDF of the normalized residual tropospheric delay error (r_n) due to the mapping function (blue: with σ = 3.16 cm)

The residual error can be over-bounded with a zero-mean Gaussian distribution (green) with σ = 11.7 cm. The heavy tails of the residual error (blue) are due to its dependence on aircraft distance and elevation angle.

5.2 Modeling the Error Due to Zenith Delay Models

The zenith delay error has a mean of zero and can be over-bounded with a zero-mean Gaussian distribution. Figure 6 shows the CDF and 1-CDF of the zenith delay model residual error $\left({\sigma }_Z^2\right)$, along with the unit-variance Gaussian distribution for reference and the over-bounding Gaussian distribution N(0, σ) with a standard deviation (σ) of 6.7 cm, which is smaller by a factor of two than the current RTCA-adopted value of 12 cm. This result clearly illustrates that using the zenith delay model residual error standard deviation of 12 cm is overly conservative for terrestrial navigation applications.

• Download figure
• Open in new tab
• Download PowerPoint

Figure 6

The CDF and 1-CDF of the normalized residual zenith delay error due to the zenith delay models used in this study (blue: with σ = 4.67 cm)

The residual error can be over-bounded with a zero-mean Gaussian distribution (green) with σ = 6.4 cm.

5.3 Dependence of Residual Tropospheric Delay on Aircraft Distance

Using a constant value to model the variance of the residual tropospheric delay irrespective of the aircraft distance results in an excessively conservative over-bound. Because the mapping function is dependent on the elevation angle, we consider the effects of different aircraft ranges on the residual error of the mapping function (δ𝒯_m). For a given elevation angle, the propagation distance of the ranging signal through the atmosphere depends on the distance between the ground station and the aircraft. The tropospheric delay varies on the order of approximately 10 cm/km with respect to aircraft range. Thus, considering a constant standard deviation of the residual tropospheric error irrespective of aircraft range leads to an over-compensation of the residual error for smaller aircraft ranges in comparison to relatively larger aircraft ranges.

Normalizing the residuals of the mapping function (δ𝒯_m) with aircraft distance leads to a less conservative over-bound in contrast to using a constant value for the standard deviation, as shown in Figure 5. Assuming that the tropospheric delay varies approximately linearly with aircraft range (d), the normalized residual error due to the mapping function is given as follows:

$r_n=\frac{\delta {𝒯}_m}{d}$ 27

Figure 7 shows the CDF and 1-CDF of the mapping function residual error (δ𝒯_m) normalized with a linear function of aircraft range (d), along with the unit-variance Gaussian distribution for reference and the over-bounding Gaussian distribution N(0, σ) with a standard deviation (σ) of 0.75 mm/km. For example, for an aircraft range of 10 km, a comparison of Figures 5 and 7 clearly illustrates that, irrespective of the elevation angle, normalizing the residual error by aircraft distance reduces the residual error model standard deviation by an order of magnitude, from approximately 11.7 cm to 0.75 mm/km. The results shown in Figure 7 are not frequency-dependent for navigation signals in the L-band, as the neutral atmosphere is a non-dispersive medium with respect to radio waves up to frequencies of 15 GHz.

• Download figure
• Open in new tab
• Download PowerPoint

Figure 7

Normalizing the residual tropospheric error (δ𝒯_m) by aircraft distance (d), as shown in Equation (27), significantly reduces the constant σ_T (shown in Figure 5, valid for all elevation angles and aircraft distances) of 11.7 cm compared with using a value of σ_T that is adapted based on the aircraft range across all elevation angles.

5.4 Modeling Residual Error Model Variance as a Function of Elevation Angle and Aircraft Range

Inflating the residual error model variance to over-bound all of the residuals (as shown in Figures 5 and 7) leads to very conservative bounding at the core of the distribution. As shown in Figure 7, normalizing the residual error variance with respect to aircraft range enables us to consider a standard deviation that varies according to the distance of the aircraft from the ground station. However, the impact of tropospheric delay on terrestrial ranging is more prominent at elevation angles close to the horizon in comparison to ray paths at higher elevation angles. Thus, it is useful to consider an expression for the residual error model variance as a function of elevation angle and aircraft range. This approach allows for a less conservative model to be used, which would greatly benefit the integrity and availability of navigation using terrestrial nav-aids.

In an operational context, the aircraft user is only aware of its elevation angle and range with respect to the ground station. Thus, to determine the residual error due to tropospheric delay, the aircraft would require a function for the residual error model standard deviation that is dependent on elevation angle and aircraft range. As described in Section 4.2, considering the mapping function residual error values (δ𝒯_m) within each of the 2D bins, we compute the corresponding value for the standard deviation to analyze its variability across the elevation angles and aircraft ranges considered within our simulated data. For elevation angles close to the horizon, Figure 8(a) shows that the standard deviation of the residual error due to the mapping function increases from a few centimeters to approximately 40 cm, highlighting the significance of modeling the residual error as a function of both elevation angle and aircraft range.

• Download figure
• Open in new tab
• Download PowerPoint

Figure 8

(a) Heatmap of the standard deviation of residual error values as a function of elevation angle and aircraft slant range; (b) standard deviation of the residual error, showing a nonlinear relationship with respect to aircraft slant range at elevation angles below 10° Different colors indicate different elevation angles.

The tropospheric delay does not follow a linear relationship with respect to aircraft range at elevation angles close to the local horizon. Figure 7 shows that using a linear function for the aircraft range results in a reduced residual error model variance. However, an analysis of the variability of the standard deviation values shown in Figure 8(b) indicates that the assumption of a linear dependence of aircraft range on tropospheric delay is only valid at elevation angles above 20°. At elevation angles close to the local horizon, we find that the residual error is well described by a quadratic function of aircraft range.

A residual error model representing an exponential variability with respect to elevation angle and a quadratic variability with respect to aircraft range leads to an unmodeled error on the order of 1–3 cm for the PMF-based model (Narayanan & Osechas, 2023). Figure 9(a) shows a wire-frame plot (blue) generated using the five-parameter model shown in Table 2, to estimate the residual error model standard deviation values shown in Figure 8 as a function of aircraft range and elevation angle. The corresponding residual difference between the model and the geometric bending delay is shown in Figure 9(b). Considering a terrestrial nav-aid with accuracy on the order of 5–6 m (e.g., L-Band Digital Aeronautical Communication Systems (Osechas et al., 2019) or long-term evolution-based nav-aids (Shamaei et al., 2018)), the residual error plot shown in Figure 9(b) illustrates that the five-parameter model can be effectively used to estimate the residual error due to the mapping function (δ𝒯_m) with sufficient accuracy on the order of 2–3 cm in most cases.

• Download figure
• Open in new tab
• Download PowerPoint

Figure 9

(a) Standard deviation values of the residual error due to the mapping function estimated by the model described in Table 2 (shown by the blue wire-frame) accurately align with the standard deviation values (shown as red dots) across all elevation angles and aircraft ranges. (b) For elevation angles below 0.5° and aircraft ranges above 100 km, the unmodeled component of the residual error model shown in Table 2 is smaller by an order of magnitude, on the order of ±2 cm, in comparison to the standard deviation values of 40 cm shown in (a).