Performance assessment of GNSS diffraction models in urban areas

2.1 Huygens-Fresnel principle

The fundamental law for the knife-edge model is the Huygens-Fresnel principle (Hristov, 2000). As shown in Figure 2, the Huygens-Fresnel principle states that each infinitesimal point of a wavefront emits a spherical wavelet. The secondary wavelets interfere with each other and jointly form a new wavefront. When part of the wavefront is obstructed, the receiver can only receive the total effect of those secondary wavelets emitted from the unobstructed part of the wavefront. Hence, the diffraction is the incompleteness of the collective effect from the secondary wavelets on a partially blocked wavefront, noted that the phase difference of each secondary wavelet on a reference point (e.g., the receiver) will introduce constructive or destructive interference in between. Hence, the geometrical relationship between the obstacle and the reference point determines the corresponding total constructive/destructive effect; in other words, the field attenuation.

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FIGURE 2

The demonstration of the Huygens-Fresnel principle for GNSS diffraction

2.2 Evaluation with Fresnel integral

The GNSS diffraction effect for the receiver in Figure 2 can be represented as the superposition of the contribution from all secondary wavelets. Since the satellite is far away from the user compared to the building, the wavefront crossing the building edge and emitting secondary wavelets can be approximated as a plane in Figure 3. By applying the spatial integral on the passing through the wavefront (Orfanidis, 2002), the unobstructed/diffracted field ratio, namely the diffraction coefficient, can be estimated as follows:

1

2

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FIGURE 3

The GNSS diffraction evaluation based on the knife-edge model

where 𝐹(𝜐) denotes the Fresnel integral. Variable 𝜐 is determined by the signal wavelength 𝜆 and the geometrical parameter 𝑏, which relates to the satellite elevation angle, the elevation angle of the building edge and the building height (refer to Figure 3). By knowing the geometrical parameters, the corresponding diffraction coefficient can be derived and further used to evaluate the 𝐶∕𝑁₀ of a diffracted signal.

3.1 Free-space propagation model

The UTD is built up based on the modern geometrical optics (GO), which extends the classical geometrical optics with Maxwell’s equations and the Luneberg-Kline expansion (McNamara et al., 1990). For the high-frequency field, such as GNSS signals, the property variation of the medium or the scatter obstacle is negligible over the scale of the wavelength. Then, the total effect of the field can be asymptotically represented by the local effect of a scatter. In GO, the free-space propagation of high-frequency electric field with spherical wavefront can be expressed by

3

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where 𝐄(𝑠) denotes the electric field with a distance 𝑠 along the direct propagation path from a reference point. The term 𝐄(0) denotes the field on the reference point 𝑠 = 0, consisting of the field amplitude 𝐄_𝑎𝑚𝑝(0) on the polarization direction and the reference phase component 𝑒^{−𝑗𝑘Ψ(0)} with phase function Ψ(0). The term is usually denoted as the spreading factor 𝐴(𝑠), representing the amplitude variation along with propagation. 𝛾₁ and 𝛾₂ are the principal radii of the wavefront curvature. For example, the plane wave has the radii 𝛾₁,𝛾₂ → ∞, reducing the propagation expression as 𝐄(𝑠) = 𝐄(0)𝑒^{−𝑗𝑘𝑠}. The term 𝑒^{−𝑗𝑘𝑠} represents the phase shift with wavenumber 𝑘 during propagation. From this expression, we can evaluate the electric field on a specified location based on the knowledge of that field on a reference point.

3.2 3D Diffraction model

Following Fermat’s principle (Keller, 1962), the local effect of the GNSS diffraction shall be represented by the point achieving the shortest distance between the satellite and the receiver along the line of edge, denoted as the point of diffraction 𝑄 in Figure 4. Consequently, the angle 𝛽^′ between the incident ray and edge is equal to the angle 𝛽 between the diffraction path and edge. Analogous to the reflection effect, the expression of the diffracted field from a perfectly conducting structure follows the GO free-space propagation model, but with the diffraction coefficient (Kouyoumjian & Pathak, 1974) 𝐷_∥,⊥, as follows:

5

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FIGURE 4

The GNSS diffraction evaluation based on UTD (McNamara et al., 1990; Nicolás et al., 2012)

where and denote the two orthogonal components of the diffracted field along the unit vector and on location 𝑅, parallel and perpendicular to the diffraction plane (the plane containing the diffracted ray and the obstacle edge), respectively. Here, the unit vectors of propagation direction, ∥, and ⊥ directions are following the right-hand rule. Similarly, and denote the two orthogonal components of the incidence field on location 𝑄, parallel and perpendicular to the incidence plane (the plane containing the incidence ray and the obstacle edge), respectively. Here, we employ a decomposition coordinate the same as (McNamara et al., 1990), which may have a different definition on ∥ and ⊥ compared to some of the other studies. The decomposition of the incident field is aiming to account for the diffraction effect differently on the ∥ and ⊥ component. The connection between the incident and the diffracted field is described by Equation (5) with the diffraction coefficient 𝐷_∥ and 𝐷_⊥, which are given by (Kouyoumjian & Pathak, 1974) as follows:

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8

9

10

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where 𝐷₁ and 𝐷₂ are related to the diffraction effect when the o-face and the n-face of an obstacle are shadowed, while 𝐷₃ and 𝐷₄ are related to the diffraction effect when there also exist valid reflections from the n-face and the o-face of an obstacle to the location 𝑅. The variable 𝑇(𝑥) is a transition function (McNamara et al., 1990) which can be numerically calculated based on the geometrical parameters in Figure 4 and the wavenumber 𝑘. The term 𝑎^±(𝜙 ± 𝜙^′) = 2cos²{[2𝑛𝜋𝑁^± −(𝜙±𝜙^′)]∕2} is a geometry-related function (Kouyoumjian & Pathak, 1974), where 𝑁^± is the integer most nearly satisfying 2𝑛𝜋𝑁^± −(𝜙±𝜙^′) = ±𝜋 based on the angle 𝜙^′ from the o-face to the incidence plane and 𝜙 from the o-face to the diffraction plane.

For the case of GNSS signals, the diffraction always occurs in a scenario with typical geometrical features, which can simplify the preceding expressions. In practice, the level-of-detail-1 (LOD-1) building model (Groves, 2016) is employed during the diffraction modeling, where the building edges are simplified as straight lines. Hence, the diffracted field wavefront is cylindrically emitted from a line source field along the building edge, resulting in a spreading factor . According to (McNamara et al., 1990), the parameter 𝑛 can be assigned with 1.5 representing a 90-degree diffracting wedge angle, because the LOD-1 building model is always vertically mounted on the ground and has a flat roof. Since the satellite is far away from the building and the receiver (𝑟₁ ≫ 𝑟₂), the incident field can be approximated as plane-wave incidence, making the distance parameter 𝐿 = 𝑟₂sin²𝛽. Since it is hard to precisely describe the building material in the complex urban environment, the structures introducing diffractions are assumed to be perfectly conducting structures for simplification. In most cases, the GNSS receiver suffering diffraction is below the building roof and with a certain distance to the building. It is less likely to enter the zone occurring reflection and diffraction simultaneously for the same building, which reduces the diffraction coefficient as 𝐷_∥,⊥ = 𝐷₁ +𝐷₂.

4.1 Overview

The GNSS diffraction simulation architecture is demonstrated as shown in Figure 5. The ephemeris provides the satellite position 𝐱_𝑆𝑉 for a specific time. The 3D building model is commonly in the LOD-1, which is denoted by matrix 𝐗_𝐵 containing all the corner locations of each building. Based on 𝐱_𝑆𝑉, 𝐗_𝐵, and the receiver location 𝐱_𝑅, the geometrical parameters of the building introducing diffraction can be obtained through the ray-tracing technique. For the diffraction simulation based on the knife-edge model, the diffraction coefficient can be evaluated by the geometry-related parameters 𝑏 and 𝑟₂. For the UTD approach, the location of the diffraction point on the building edge, which dominates the diffraction effect, needs to be obtained beforehand. After searching for the location satisfies the geometrical condition of diffraction by raytracing, the UTD employs the geometrical parameters of the diffraction point to obtain the diffraction coefficient of the field. Then, the modeled open-sky 𝐶∕𝑁₀ and the diffraction coefficient from the knife-edge model or UTD are combined to simulate the 𝐶∕𝑁₀ of the diffracted signal. Moreover, the diffraction effect on GNSS pseudorange can also be simulated after applying UTD. The distances between satellite, receiver, and the diffraction point are estimated during the ray-tracing. The GNSS diffracted pseudorange can be simulated by combining the systematic error 𝜀_𝜌 from the pseudorange error model, which considers the atmospheric delay and satellite clock/orbit bias. During the pseudorange simulation, if the satellite LOS vector is unobstructed, there exists the influence of the direct field, besides the diffracted field on edge. The interference between the direct and the diffracted field on pseudorange needs to be estimated by the corresponding 𝐶∕𝑁₀ and the multipath noise envelope. At this point, both the GNSS 𝐶∕𝑁₀ and pseudorange measurements under diffraction are simulated through the flowchart.

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FIGURE 5

The flowchart of the GNSS diffraction simulation

4.2 𝑪∕𝑵_𝟎 of the diffraction signal

Both the knife-edge model and the UTD are derived based on the linear polarized electric field, whereas the GNSS signal is right-hand circularly polarized (RHCP). The circular polarized field of GNSS can be considered as the superposition of two orthogonal linear polarized fields with a quarter phase shift in between. Similar to GNSS-R (Jia & Pei, 2018), the overall diffraction is evaluated by combining the individual diffraction effect on each decomposed linear polarized field.

For the knife-edge model approach, the 𝐶∕𝑁₀ of the diffracted signal is simulated based on the 2D model in Section 2.2, which simplifies the 3D case with the same cross sections. We first use ray-tracing to search for the building closest to the LOS vector of the satellite, which probably introduces diffraction. The corresponding building height and the elevation angle of the building edge on the satellite’s azimuth angle direction are employed to derive the geometrical parameters 𝑏 and 𝑟₂ based on Algorithm 1. Then, the diffraction coefficient of a linear polarized field related to that building can be estimated by the knife-edge model with Equations (1) and (2) as shown in Figure 5. For the GNSS signal with RHCP, the corresponding diffraction coefficient can be derived as the averaged coefficient of its two orthogonally decomposed linear fields (Stutzman, 1993). Those decomposed fields have the same diffraction coefficient from the knife-edge model, which makes the averaged diffraction coefficient for the RHCP field also the same. Therefore, the 𝐷_{𝑘𝑛𝑖𝑓𝑒−𝑒𝑑𝑔𝑒} from Equations (1) and (2) can be directly employed to indicate the overall diffraction coefficient of the RHCP GNSS signal and used to simulate the 𝐶∕𝑁₀ and pseudorange of the diffracted signal later.

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ALGORITHM 1

Ray-tracing for the knife edge model

For the UTD approach, unlike the knife-edge model, it approximates the diffraction as a local effect on a particular location along the building edge, namely the diffraction point. Hence, the location of the diffraction point needs to be obtained beforehand. Based on the satellite position, receiver position, and the 3D building model, the feasible diffraction point can be located via ray-tracing, searching the points satisfying that the angle between incidence and edge is equal to that between the diffraction path and edge. The detailed procedures of the diffraction point searching and the corresponding geometrical parameters calculation are demonstrated in Algorithm 2. The geometrical parameters of that diffraction point can then be used to estimate the diffraction coefficient related to that building edge for a linear polarized incident field by Equations (5)-(11). However, the preceding UTD is developed for the linear polarized fields, whereas the GNSS signal is normally transmitted using RHCP. Additional conversion is needed for the UTD to model the diffraction effect of the RHCP GNSS signal, noted the conversion between linear polarized and RHCP field here is different from the knife-edge model since the UTD approximates the diffraction as a local effect.

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ALGORITHM 2

Ray-tracing for the UTD approach

In UTD, the GNSS signal experiences a transition on the building edge from the incident field to the diffracted field. As shown in Figure 4, the corresponding RHCP field can be orthogonally decomposed into two linear polarized fields, the soft diffraction component denoted by ∥, and the hard diffraction component denoted by ⊥. The decomposed fields behave differently during this transition according to the geometrical relationship to the edge. Knowing the GNSS field maintains RHCP during diffraction (Maqsood et al., 2013), the dyadic diffraction coefficient for the RHCP field can then be derived based on the dyadic diffraction coefficient for those linear fields (McNamara et al., 1990) and the circular polarization mechanism (Stutzman, 1993):

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where and denote the incident and the diffracted RHCP field, respectively. Variables , , , and are the unit vectors defined in Section 2.2, which are used to describe the transition behavior for each decomposed field during diffraction.

Unlike the knife-edge model naturally includes all the field interferences by the integral, the UTD approximates the phenomenon as a local effect. When the GNSS LOS signal is also unobstructed, the interference between the LOS and the diffracted field needs to be additionally considered. Therefore, by including the spreading factor and phase shift, the overall diffraction coefficient on the receiver location for different conditions are derived as follows:

13

where 𝛿𝑟 is the extra distance of the diffracted path compared to the unobstructed path, 𝑟₂ is the distance between the diffraction point and receiver. Here, the LOS field is simplified with the spreading factor of a plane wave, 𝐴(𝑠) = 1, since the satellite is very far from the receiver.

For the scenario with diffractions from multiple edges, such as the narrow gap between buildings in a dense urban area, the geometrical parameters of the corresponding diffracted field are obtained by repeating Algorithm 2 on those edges. After calculating the diffraction coefficient of each diffracted field, the overall diffraction coefficient is derived by the superposition of diffracted fields from all the related edges, as

14

where 𝐷_{𝑅𝑅,𝑚}, 𝐴_𝑚, and Ψ_𝑚 denote the diffraction coefficient, spreading factor, and phase shift distance from the reference field for the 𝑚^𝑡ℎ edge, respectively. The variable Ψ₀ denotes the phase shift distance between the LOS field and the reference field, which may not be the same in some cases. Then, the overall diffraction coefficient from UTD, 𝐷_𝑈𝑇𝐷, can be used to simulate the 𝐶∕𝑁₀ and pseudorange of the diffracted signal. For most practical cases, only the closest building edge dominating the diffraction effect needs to be considered, which reduces Equation (14) to Equation (13).

After obtaining the overall diffraction coefficient, the incident field strength is also required to simulate the diffracted field strength. However, it is hard to directly obtain the incident field, which relates to atmospheric interference or other effects. A practical approach is to employ an open-sky GNSS elevation-𝐶∕𝑁₀ regression model from long period data (Suzuki & Kubo, 2012). In this study, the model is obtained through the first-order polynomial fitting on the four hours open-sky GNSS elevation angle and 𝐶∕𝑁₀ (in the unit of Hz) data with respect to different satellite orbits. An example of the fitting model for the GPS satellite is shown in Figure 6. Then, the incident field strength of a specific satellite can be described by its unobstructed 𝐶∕𝑁₀ from the model. Since the 𝐶∕𝑁₀ relates to the signal power, the diffraction coefficient expressed in the power ratio is given by

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FIGURE 6

Example of the open-sky GNSS elevation-𝐶∕𝑁₀ regression model for the GPS measurements

where 𝑃^𝑑 and 𝑃^𝑖 denote the diffracted and the incident field power, respectively. The term 𝐷_{𝑘𝑛𝑖𝑓𝑒−𝑒𝑑𝑔𝑒,𝑈𝑇𝐷} is the overall diffraction coefficient from the knife-edge model or UTD. Based on the formulation of signal-to-noise ratio (SNR) (Jia & Pei, 2018) and the conversion between SNR and 𝐶∕𝑁₀ (Joseph, 2010), the 𝐶∕𝑁₀ of a diffracted GNSS signal is derived as follows:

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where 𝐶∕𝑁_{0,𝑜𝑝𝑒𝑛} is the unobstructed 𝐶∕𝑁₀ predicted by the open-sky model based on the satellite elevation angle. Here, the influence of the 𝐶∕𝑁₀ from the diffraction delay is neglected, since this delay is negligible compared to the total traveling distance.

4.3 Diffracted pseudorange

The GNSS pseudorange is a ranging measurement based on the time-of-arrival (TOA) from satellite to receiver. During the UTD simulation, the distance 𝑟₀ from satellite to receiver, 𝑟₁ from satellite to diffraction point, and 𝑟₂ from diffraction point to receiver can be obtained from the raytracing and used to simulate the GNSS ranging measurement on the dominating signal path. However, for the knife-edge model, the diffraction effect is modeled as a cumulating effect via integral. The ray-tracing there is only used to obtain the geometrical parameters of the integral area. Therefore, unlike the UTD approach, the diffraction effect on the pseudorange in the knife-edge model is not dominated by a single signal path, and it needs a complicated integration process that will not be simulated in this study. For the diffracted satellite being blocked by the buildings, namely NLOS diffraction, the signal delay due to diffraction can be evaluated by 𝑟₀, 𝑟₁, and 𝑟₂, denoted as 𝜀_𝑑 = 𝑟₁ +𝑟₂ −𝑟₀. Then, the diffracted pseudorange can be simulated by combing the LOS distance 𝑟₀, diffraction delay 𝜀_𝑑, and the systematic error 𝜀_𝜌, including the atmospheric delay and satellite clock/orbit bias from models (Kaplan & Hegarty, 2017). For the diffracted satellite not being blocked by buildings, namely LOS diffraction, the interference between the diffracted signal and the direct signal needs to be considered in addition. This interference can be modeled by the multipath noise envelope (Liu & Amin, 2009) based on the ranging delay 𝜀_𝑑, the carrier-phase offset 𝛽 = −𝑘𝜀_𝑑, and the 𝐶∕𝑁₀ ratio between the diffracted and the direct signal, denoted as a function 𝜀_𝑚𝑝(𝜀_𝑑,𝐶∕𝑁_{0,𝑜𝑝𝑒𝑛},𝐶∕𝑁_{0,𝑑𝑖𝑓𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛}). In this study, we only consider the strongest two fields for the scenario with multiple fields. Since it is hard to know the time spacing between the early and late correlators for various low-cost receivers, the correlator spacing here is assumed to be 1 chip. Therefore, the diffracted pseudorange for LOS or NLOS condition is simulated by

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Compared to the LOS signal without diffraction, the pseudorange error due to NLOS diffraction and LOS diffraction is 𝜀_𝑑 and 𝜀_𝑚𝑝, respectively. Since the diffraction delay is usually in decimeter level (McGraw et al., 2019), the pseudorange error due to diffraction is generally negligible compared to the user-equivalent-range-error (UERE) 𝜎_{𝑈𝐸𝑅𝐸} ≈ 7𝑚 in practical cases (Kaplan & Hegarty, 2017).

4.4 Diffraction availability

Besides simulating the quantitative value of the 𝐶∕𝑁₀ and pseudorange of the diffracted signal, the availability of these diffracted measurements is also required to be considered. When the satellite elevation angle is descending below the building edge introducing diffraction, the signal is increasingly attenuated, until it is too weak to be received by the receiver. The connection between signal availability and geometrical properties is related to the Fresnel zone of the signal.

From the Huygens-Fresnel principle (Figure 2), each point on the wavefront has a different distance to the receiver, introducing the phase difference between secondary wavelets. Compared to the secondary wavelet on the LOS vector, some of the secondary wavelets are constructive while others are destructive. The secondary wavelets with constructive or destructive interference are respectively located in certain regions, which are the Fresnel zones demonstrated in Figure 7. The boundary of each zone is defined by

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FIGURE 7

The demonstration of Fresnel zone blockage during diffraction

where 𝑁_𝐹 is a positive integer number. 𝑟₀ is the distance between satellite and receiver. The variable 𝑟_𝑎 is the distance between a satellite and a wavefront, and 𝑟_𝑏 is the distance from the secondary wavelet on that wavefront to the receiver. The secondary wavelet located in the 1^st Zone (Figure 7) has an extra travelling distance below half of the wavelength compared to that on the LOS vector, which introduces constructive interference. Similarly, the secondary wavelet in the zones with odd integer 𝑁_𝐹 is constructive with respect to the secondary wavelet on the LOS vector. In contrast, the secondary wavelet lies in the even numbered zones introduces destructive interference. The wave propagation can be regarded as a total effect of secondary wavelets from all these zones with constructive or destructive effects.

For the GNSS signal with a short wavelength, the constructive or destructive zones are narrowly spread in the space and rapidly canceling each other. The total effect can be represented by the first several zones (Hristov, 2000). Then, the field attenuation under diffraction is depending on the blockage of the first several zones, or more directly, the first zone with the highest constructive effect (Bradbury, 2008). As shown in Figure 7, the signal is less likely to be available for the receiver when most of the first zone is being blocked. A rule of thumb to determine the signal availability is by a threshold of 60% blockage (Bradbury, 2008). By checking the blockage ratio of the first Fresnel zone, the signal availability for the receiver can be predicted during the simulation. However, this prediction is only appropriate for the geodetic receiver. For the low-cost receiver with higher sensitivity, the measurement may still be available even though the 1^st Fresnel zone is entirely blocked.

5.1 Experiment setup

To evaluate the performance of the simulated GNSS 𝐶∕𝑁₀ after diffraction using the knife-edge model and the UTD, two static experiments are conducted in the urban canyons in Hong Kong, as shown in Figure 8. The first experiment is located with one-side blockage by a single building. Here, the receptions of the blocked satellites are only due to diffractions since no valid reflection exists from other buildings (means no long- and middle-range reflections exist). The second experiment is located in a deep urban canyon surrounded by multiple buildings, where reflections and diffractions may occur simultaneously. A consumer-grade GNSS receiver ublox EVK-M8T with a patched antenna is used to record measurements on both sites. The GNSS antenna is placed on a flat surface for both scenarios, as shown in Figure 8, to mitigate the ground reflections. The receiver’s true location is obtained based on the landmarks on Google Earth, which the accuracy is within 1-2 meters based on our experience. The 3D building model for simulation is also extracted from Google Earth. From the skyplots of Figure 8, the 𝐶∕𝑁₀ of the satellites moving across the building edge in both two tests are having significant attenuations, which are very likely due to diffractions. In the following sections, the measurements of these satellites are used to validate the diffraction simulation performance.

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FIGURE 8

The surrounding environment and the diffracted satellite 𝐶∕𝑁₀ in the skyplot during the validation experiments. The 𝐶∕𝑁₀ is given by a low-cost commercial GNSS receiver

5.2 Validation on the one-side building scenario

For the diffracted signal from satellite G27 in the one-side building scenario, the comparison of 𝐶∕𝑁₀ between real received measurement and the simulation from the knife-edge model or UTD are shown in Figure 9. The blockage ratio of the first Fresnel zone is also shown correspondingly, which is commonly used to evaluate the signal strength attenuation. The signal gets higher attenuation when the blockage ratio of the first Fresnel zone is increased. Here, the LOS vector of the satellite is nearly perpendicular to the building edge, which is ideal for the knife-edge model. The knife-edge model and UTD yield similar simulation results for the 𝐶∕𝑁₀, which agree with the received measurements. In line with the prediction, the measurements 𝐶∕𝑁₀ starts decreasing when the building edges enter the first Fresnel zone of the signal. Therefore, the measurement has already been attenuated before its LOS vector is blocked by the building; in other words, the LOS measurement close to the building edge could already be affected by diffraction. However, even though the first Fresnel zone has been completely blocked, the measurement is still available with very low 𝐶∕𝑁₀. It is probably because the sensitivity of a low-cost receiver is much higher than that of the geodetic receiver (Bradbury, 2008, Walker & Kubik, 1996). A similar phenomenon is also observed in (Icking et al., 2020). Hence, for a low-cost receiver, the diffraction may still occur even when the satellite elevation angle is much lower than the building (see Section 5.6 for the details).

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FIGURE 9

The diffraction simulation result for satellite G27 in the one-side building scenario, including: (a) comparison of the 𝐶∕𝑁₀ from the knife-edge model or UTD simulation and real measurements; (b) the blockage ratio of the first Fresnel zone; (c) the elevation angles of the satellite and the elevation angles of the building edge in the azimuth direction of the satellite; and (d) the pseudorange error from simulated diffraction and real measurement

The pseudorange error from simulated diffraction is compared with that from the real measurement for G27 in Figure 9 (d). The pseudorange error from measurement is estimated using the double difference (DD) approach based on the measurement and the actual location of the receiver and a reference station (Xu et al., 2019). During the diffraction period between purple dash lines, the simulated pseudorange error is in decimeter level and negligible compared to 𝜎_{𝑈𝐸𝑅𝐸}. Meanwhile, the DD-estimated pseudorange error from the real measurement is around 4 meters, which is the same as the period before the first Fresnel zone blockage starts. Noted that the DD approach is employed to eliminate the receiver/satellite clock errors during the pseudorange error estimation through a differential process with the measurement from a selected master satellite (usually with the highest elevation angle). However, the multipath error from the master satellite will be transferred to the pseudorange error estimation of the target satellite, although it is usually very small for the master satellite with a high elevation angle. Therefore, the 4 meters error here could be due to the multipath error from the master satellite, which probably overestimates the pseudorange error of G27. During this period, no significant error due to the diffraction delay is observed. Noted that the fluctuation of the pseudorange error from the measurement increases after the first Fresnel zone has been entirely blocked. It is possibly due to the severe signal attenuation when the constructive effect from the first Fresnel zone is lost. The resultant low 𝐶∕𝑁₀ will further introduce large pseudorange noise in the tracking loop (Kaplan & Hegarty, 2017). The relationship between the measurement 𝐶∕𝑁₀ and the pseudorange noise is demonstrated in Appendix B with a numerical example. In the later period, there occur a few epochs with enormous pseudorange errors in the measurement, which may be introduced by unknown interferences or phenomena related to weak signals.

For satellite B10 in the one-side building scenario, diffracted by an oblique building edge (not perpendicular to the LOS vector), the simulation results are shown in Figure 10. In this scenario, the elevation angle of the building edge on the satellite’s azimuth angle is rapidly increased during the satellite movement. Consequently, the conventional knife-edge model also shows a rapid drop in the 𝐶∕𝑁₀ simulation, which is inconsistent with the real measurements. Noted that the fluctuation of the simulated 𝐶∕𝑁₀ from the knife-edge model is significantly changed at the epoch 1385. This is because the geometric parameters 𝑟₂ and 𝑏 for the knife-edge model are under the transition from Edge-1-related to Edge-2-related (as shown in Figure 11), further influencing the switching frequency of the constructive/destruction interferences. On the other hand, by considering the three-dimensional (3D) angle in diffraction, the signal attenuation by UTD is slower, which is more similar to the real measurements. Notice that the simulated 𝐶∕𝑁₀ by UTD has a bias compared with the measurement. It is because the open-sky model may have a constant error when predicting the 𝐶∕𝑁₀ of the unobstructed signal. Analogous to the adjustment for unexpected interferences in (Nicolás et al., 2012), after manually adjusting the UTD simulation with -4 dB-Hz, the UTD (model corrected) simulation result is consistent with the real measurement. Moreover, the UTD can simulate the 𝐶∕𝑁₀ fluctuation of the real measurement before attenuation, which is described as the interference between the LOS and the diffracted field. After the first Fresnel zone is entirely blocked, the diffracted signal remains available for a certain period. Therefore, the reception of those severely attenuated signals from diffraction may be more often than expected, which relates to the configuration of the receiver and antenna. The criteria of an unavailable signal based on the complete blockage of the first Fresnel zone is valid for geodetic setups, but not appropriate for low-cost setups. Similar to the result of satellite G27, the pseudorange error from simulated diffraction is only a few decimeters. Meanwhile, the DD-estimated pseudorange error from the measurement is dominated by the noises from the master satellite or receiver tracking loop and has no obvious error from diffraction delay. The pseudorange error analysis from G27 and B10 verify the assumption that the pseudorange error due to the diffraction delay is negligible compared to the 𝜎_{𝑈𝐸𝑅𝐸} for most practical cases. The diffraction influence on the pseudorange error is mainly the increment of tracking loop noise due to the signal attenuation.

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FIGURE 10

The diffraction simulation results for satellite B10 in one-side building scenario, including: (a) comparison of the 𝐶∕𝑁₀ between simulations and measurements, the UTD (model corrected) denotes the simulation with a -4 dB-Hz correction on the open-sky 𝐶∕𝑁₀ model; (b) the blockage ratio of the first Fresnel zone; (c) the values of the geometric-related parameter 𝑟₂ and 𝑏 during the knife-edge model simulation; (d) the elevation angles of satellite and building edge; and (e) the pseudorange error from simulated diffraction and real measurement

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FIGURE 11

The demonstration of the involving building edges in the knife-edge model simulation during the transition on epoch 1385

5.3 Validation on surrounded by buildings scenario

For the satellite G13 in the surrounded by buildings scenario, its diffraction simulation results from the knife-edge model and the UTD that are compared with real measurements in Figure 12. Before the first Fresnel zone is entirely blocked, the simulated 𝐶∕𝑁₀ has the same trend compared with the real measurement, but with a certain bias. This bias is, again, caused by the inaccurate open-sky model. However, after the first Fresnel zone is entirely blocked, the signal is not only still available, but also with a high 𝐶∕𝑁₀ value. During this period, the corresponding pseudorange errors raise to over 50 meters, which is apparently larger than the typical pseudorange error within a dozen meters from diffraction delay or tracking loop noise. By using a popular ray-tracing technique for reflections (Hsu et al., 2016), a valid reflection path with nearly 50 meters delay is found in the later period of the experiment. Although the ray-tracing found no valid reflection around epoch 3,500-5,300, the measurement still has a typical reflection-related feature that the pseudorange contains an enormous positive delay. Therefore, as another frequently occurring effect in the urban area, the reflection interference needs to be considered and combined with the diffraction interference in order to achieve a good GNSS 𝐶∕𝑁₀ and pseudorange simulation in dense urban scenarios.

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FIGURE 12

The diffraction simulation results for satellite G13 in surrounded by buildings scenario, including: (a) comparison of the 𝐶∕𝑁₀ between simulations and measurements; (b) the blockage ratio of the first Fresnel zone; and (c) the pseudorange error from measurement, diffraction simulation and the reflection effect simulated by the ray-tracing

For the satellite G05 in the surrounded by buildings scenario, it experiences a transition involving three building edges, as shown in Figure 13. Therefore, the UTD simulation is based on the superposition of the diffracted fields from three involving building edges and the unobstructed field when the satellite is LOS. The diffraction simulation and real measurements are demonstrated in Figure 14, including the 𝐶∕𝑁₀ simulation result of each individual diffracted field during UTD. Similar to the result of G13, the ray-tracing reflection simulation is also employed to find valid reflected signals during the experiment. The corresponding reflection delay is shown in Figure 14 (d). Although the simulation from the knife-edge model correctly reflects the rise of 𝐶∕𝑁₀ when the satellite is LOS between the building gap, the change is too rapid since it only considers the building elevation angles. On the other hand, the UTD approach considers all three adjacent edges related to the diffraction in the manner of the superposition of multiple diffracted fields. The UTD simulation result is smoother than the knife-edge model and more consistent with the real measurement.

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FIGURE 13

The skyplot of satellite G05 with a transition involving three different building edges in surrounded by buildings scenario

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FIGURE 14

The diffraction simulation results for the satellite G05 in surrounded by buildings scenario, including: (a) comparison of the 𝐶∕𝑁₀ between measurements and the simulations from knife-edge model and UTD combining multiple diffracted fields; (b) The individual simulation result of each involving diffraction field; (c) the blockage ratio of the first Fresnel zone; (d) the pseudorange error from measurement, diffraction simulation, and the reflection effect simulated by the ray-tracing

From the 𝐶∕𝑁₀ of each diffracted field in Figure 14 (b), the overall field strength is dominated by the diffraction on edge1 when the satellite is blocked by the edge1-corresponded building. Then, the satellite is moving away from edge1 and becomes LOS for the receiver, which increases the 𝐶∕𝑁₀ by considering the unobstructed field. As the satellite moving to the center of the building gap, the first Fresnel zone is almost without blockage, resulting in a high 𝐶∕𝑁₀ close to an open-sky signal. Consistently, the 𝐶∕𝑁₀ from the real measurement is also increased to the level of the open-sky model during this period. Besides diffractions, the measurement could also be degraded by the reflected signal, such as the enormous pseudorange error at around epoch 1,000, which is probably introduced by the reflected signal traced in Figure 15 (a). After that, the total field is dominated by the diffraction from edge2 when it approaches edge2. The total field decays after the blockage from edge2. Finally, the satellite is moving closer to edge3 rather than edge2, switching the dominance of the diffraction to edge3. As a result, the overall 𝐶∕𝑁₀ simulation has a flatter decay in the final stage. However, the 𝐶∕𝑁₀ plot of each diffracted field only demonstrates the amplitude of the signal, whereas each signal also has a different phase. The phase differences between different signals will introduce constructive or destructive interferences, leading to a fluctuation in the overall 𝐶∕𝑁₀ simulation result in Figure 14 (a). As the results in the later period shows, such fluctuation becomes more significant when each signal has a similar amplitude. Noted that the 𝐶∕𝑁₀ from the real measurement in the later stage is higher than the simulation. It is probably due to the reception of the signals reflected from other buildings, since an enormous pseudorange error is observed at the same period with a magnitude similar to the delay of the reflected signal found by ray-tracing in Figure 15 (b). In brief summary, the UTD can appropriately simulate the diffraction of the GNSS signal and the corresponding 𝐶∕𝑁₀ attenuation and even be applicable during the transition involving multiple building edges.

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FIGURE 15

The ray-tracing simulation result of satellite G05 at epoch 1,000 and 3,900 during the experiment

5.4 Building model accuracy

Besides testing the consistency between the UTD simulation and real measurements, the influence of building model accuracy on the UTD is also worthy of analysis. Among all the geometrical parameters used in the UTD, the elevation angle of the building edge has the highest impact on the diffraction result since it controls the major parameter 𝜙 in Equations (7)-(10). The impact of the building model accuracy is evaluated based on the UTD simulation of satellite G08 in the one-side building scenario. The building introducing diffraction effect is around 34 meters away from the receiver with 60 meters height, corresponding to around the 60 degrees building elevation angle. Here, we manually assign ±2^◦ error on 𝜙 in the building model when evaluating the influence of the building model accuracy for the UTD, which is equivalent to around a 5 meters building height error or around a 3 meters building distance error. As Figure 16 shows, the diffraction simulation results from UTD(±2^◦) with ±2^◦ error on the elevation angle of the building edge are compared with UTD(original), the original simulation without manually assigned error, and the real measurement. Compared with UTD(original), UTD(+2^◦) occurs diffraction attenuation earlier, due to a larger elevation angle of the building edge and earlier blockage of the first Fresnel zone. On the contrary, UTD(−2^◦) with a lower edge elevation angle will have a delayed attenuation from diffraction. Moreover, the building accuracy has a greater influence on the first Fresnel zone blockage ratio when the satellite is adjacent to the edge (the satellite elevation angle is close to the building elevation angle). In other words, the diffraction simulation is more sensitive to the error on the building edge when the satellite is closer to the edge. The building model error in height direction could also introduce error on the elevation angle of the edge, but related to the distance between building and receiver. For example, 1m height error only introduces 0.28^◦ error on elevation angle for the 25m height building 200m away, but introduces 1.12^◦ error on elevation angle when the building is 25m away. Therefore, the diffraction simulation is more sensitive to the dimensional model error from a nearby building. In general, the building model error will introduce an error on the simulation result as an overall shift on time series, due to the advance or delay of the first Fresnel zone blockage. This characteristic is also applicable to the knife-edge model approach.

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FIGURE 16

The influence of building model inaccuracy on the UTD simulation results for satellite G08 in one-side building scenario, including: (a) comparison of 𝐶∕𝑁₀ between measurements and simulations, where UTD(±2^◦) denote the simulation with ±2^◦ errors on the elevation angle of the building edge and UTD(original) denotes the simulation without any assigned error on building edge; (b) the blockage ratio of the first Fresnel zone; and (c) the elevation angles of the satellite and the elevation angles of the building edges in the azimuth direction of the satellite

5.5 Comparison between the knife-edge and the UTD models

The computational load of these two methods during the simulations on different satellites is demonstrated in Figure 17. For the diffraction simulation of satellite G27, B10, G08, and G13 only involving a single building edge, both methods averagely cost around 0.001 seconds for one epoch simulation, while the UTD costs a slightly larger computation. By dividing the total computation load into the ray-tracing process and the attenuation model process, it is clear that most of the computations are spent on employing the ray-tracing to obtain geometrical parameters for each model. The UTD requires much more geometrical parameters than the knife-edge model, resulting in a higher computation for its ray-tracing process. Moreover, during the simulation of G05 in a complex environment, the UTD considers the combination of three different diffracted fields. Hence, the UTD needs to apply ray-tracing on three building edges, making the computation load three times higher than the knife-edge model. Besides the computational load, the UTD has better accuracy than the knife-edge model on the 𝐶∕𝑁₀ simulation of the diffracted signal, since it considers the addition of geometrical details in 3D. The root-mean-square (RMS) error of the 𝐶∕𝑁₀ simulation result compared to the real measurement for each satellite is shown in Table 1. Besides the similar results from G08 and G13, the UTD has less simulation error than the knife-edge model, noted that the simulation is based on the low-cost receiver with higher noise than the geodetic receiver, resulting in a small error even in the open-sky scenario. Both methods have over 8 dB-Hz 𝐶∕𝑁₀ modeling RMS error in the urban scenario, possibly due to the reflection interferences after the first Fresnel zone has been fully blocked. For the UTD approach before the full blockage of the first Fresnel zone, the standard deviation of the 𝐶∕𝑁₀ modeling errors for G13 and G05 in the urban area are 2.5 meters and 3.4 meters, which still show good consistency with the real measurements. Finally, the UTD simplifies the diffraction as a local effect. This idea can be employed to describe the pseudorange error caused by diffraction as a delay introduced by a specific signal traveling through the diffraction point. On the other hand, the knife-edge model based on the integral is too complicated to simulate the pseudorange error. The characteristics of the knife-edge model and the UTD on urban GNSS diffraction simulation are summarized in Table 2.

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FIGURE 17

The computational load of diffraction simulation based on the knife-edge model and the UTD for different scenarios. The total computation load of each method is divided into two parts: the ray-tracing estimation process (unshaded area) and the attenuation modeling process (shaded area)

5.6 Diffraction effect verification by the correlator output

Besides evaluating the diffraction modeling performance with the collected measurements, it is also necessary to verify whether those diffracted measurements are legitimate. For example, even when a satellite becomes unavailable to the receiver, the corresponding measurements may still be generated from the tracking loops until the lock detector concludes its disappearance. Such illegitimate measurements due to the delay of the disappearance conclusion could result in a degradation similar to that of a diffraction effect. A feasible way to verify the legitimacy of the available measurements during diffraction is to identify the navigation bit stream from the correlator outputs corresponding to those measurements. Therefore, an additional experiment is conducted in the one-side building scenario to simultaneously collect the receiver independent exchange format (RINEX) data and the correlator outputs data during the diffraction effect. Since the ublox EVK-M8T receiver does not support the correlator outputs, we employ another low-cost GNSS receiver, a smartphone, to collect the correlator outputs simultaneously at the same location.

The experimental results are shown in Figure 18, including the simulation results, the available measurements from the ublox EVK-M8T, and the prompt in-phase and quadrature phase plot from the smartphone corresponding to four labeled 1-minute periods. Consistent with the preceding analysis, the UTD achieves adequate simulation performance on the diffraction effect. During periods 1 and 2, the first Fresnel zone of the satellite signal is not fully blocked. The corresponding I-Q plot shows a clear ring, which verifies the existence of the navigation message. Note that the smartphone here employs the frequency locked loop (FLL) in the tracking loop, which makes the navigation message form a ring due to the energy spreading on Q. For the noise without the navigation message, the corresponding I-Q plot will be concentrated in the center. During period 3, with a fully blocked first Fresnel zone, the available 𝐶∕𝑁₀ measurements are under the degradation consistent with the UTD simulation. Meanwhile, the existence of the navigation message can be observed by the ring-like I-Q plot from the correlator outputs data at the same location, which verifies the legitimacy of those available measurements. Even during the later period 4, the corresponding I-Q plot is a bit sparse in the center, which could indicate that the navigation message may still be there. In summary, verified by the correlator outputs indirectly, the UTD can adequately model the diffraction degradation in the legitimate measurements from a low-cost receiver.

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FIGURE 18

The diffraction simulation result for satellite G16 in the one-side building scenario, including: (a) comparison of the 𝐶∕𝑁₀ from the UTD simulation and real measurements; (b) the blockage ratio of the first Fresnel zone; (c) the elevation angles of the satellite and the elevation angles of the building edge in the azimuth direction of the satellite; (d) the pseudorange error from simulated diffraction and real measurement; (e) the prompt I-Q plots of the correlator output from another low-cost receiver corresponding to four labeled 1-min periods