Comprehensive Assessment of Tropospheric Effects over a Wide Range of Frequencies Transmitted from LEO Satellites

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Abstract

This paper provides a theoretical examination of tropospheric effects on signals at the L, S, C, X, and Ku bands across four distinct regions characterized by diverse climates. Specifically, we investigate Indonesia, situated near the equator and renowned for its exceptionally high rainfall; Norway, a polar region with markedly dry conditions; Boulder, Colorado, a mid-latitude area with a relatively arid climate; and Maui, Hawaii, a low-latitude locale known for its relatively wet climate. This paper summarizes an analysis of total delay and attenuation. The analysis includes (1) the variation in total delay with elevation angle, (2) effects of attenuation stemming from atmospheric phenomena such as rain, clouds, and gases, and (3) amplitude fading due to scintillation. The modeling shows that the effects increase with increasing frequency and decreasing elevation angle. Further, the assessment suggests that gaseous attenuation can be ignored for our frequencies of interest but that the other tropospheric effects must be accounted for.

Keywords

1 INTRODUCTION

Interest in low Earth orbit (LEO) satellite-based navigation systems is rapidly growing. Despite this growing interest, there remains a gap in research concerning the tropospheric effects on signals transmitted from these satellites—especially at frequencies much higher than those used in current global navigation satellite systems (GNSS) L-band signals. These effects arise from the interaction of signals with the troposphere, involving absorption, scattering, refraction, and reflection. Such interactions lead to range delays, signal attenuation, and scintillation, depending on atmospheric conditions and signal propagation geometry. Because there are only a few LEO satellites transmitting navigation signals and these transmissions are proprietary, we must rely on other signals of opportunity transmitted by communication satellites, beacon satellites, and other scientific research satellites to study tropospheric effects on various signal frequencies at the L, S, C, X, and Ku bands.

Although tropospheric propagation has been extensively studied via active radar systems—particularly in early foundational work (Atlas, 1990; Beckwith et al., 1970; Kostenko et al., 2001) and in synthetic aperture radar applications (Hanssen, 2001; Zebker et al., 1997)—these studies typically involve both signal transmission and reception under fixed or narrowly defined geometries. These works do not examine how tropospheric effects vary systematically with elevation angle, nor do they generalize across diverse environmental conditions. These limitations are significant in the context of LEO satellite applications, where signals sweep through a wide range of elevation angles during each pass, traversing atmospheric paths of varying length and refractive structure.

This paper aims to fill this gap in research by conducting a comprehensive review on signal attenuation and range delay over a wide range of frequencies transmitted by LEO satellites at different elevation angles and under various atmospheric conditions. This knowledge can inform the design and optimization of future LEO satellite systems for navigation, communication, and other applications.

The analysis shows that the tropospheric effects can be large and cannot be ignored, especially at higher frequencies and lower elevation angles. In future work, we plan to validate the theoretical model with real data from the GNSS site deployed on the summit of Haleakala on the island of Maui, Hawaii, by the University of Colorado Boulder Satellite Navigation and Sensing Lab.

2 RANGE DELAY

The troposphere consists of dry gases (N₂, and O₂) and water vapor, which refract radio signals transmitted from satellites. The troposphere is approximately 7 km (poles) to 17 km (equator) thick and is nondispersive for up to 30 GHz (Hobiger & Jakowski, 2017; Brenot et al., 2020). The atmospheric refraction causes a delay that depends upon the actual path of the signal and the refractive index of the gases along the path. This delay is a result of the fact that the larger refractive index n (n > 1) of atmospheric gases, compared with that of free space (n = 1), causes the speed of light (group velocity) in the medium to decrease below its free space value c (Spilker, 1996). The total delay $(T_{r}^{s})$ is determined by integrating the refractive index n along the signal propagation path S, as shown in Equation (1):

$T_{r}^{s} = \int_{s} (n - 1) dS$ 1

The range delay $(T_{r}^{s})$ has two components: the hydrostatic (dry) delay and the wet delay, caused by the dry part of gases and highly varying water vapor in the atmosphere, respectively. These components depend on the local weather conditions, such as the humidity, pressure, and temperature. The dry delay is stable and well-modeled, whereas the wet model is variable and depends on the relative humidity of the atmosphere and the surface temperature of the location.

To estimate the line-of-sight (LOS) tropospheric delay $(T_{r}^{s})$ , as shown in Figure 1, We first compute the tropospheric delays in the zenith direction for the dry and wet components. For the present study, the Saastomoinen model (Misra & Enge, 2006) is used to compute the zenith dry delay (ZHD) per Equation (2) and the zenith wet delay (ZWD) per Equation (3). European Centre for Medium-Range Weather Forecasts (ECMWF) data for International GNSS Service stations from the Vienna Mapping Function 3 (VMF3) (5° × 5°) grid (re3data, 2020) are used to obtain the weather parameters required for computing these zenith delays. For ZHD, we have the following:

$Z H D = 0.00227 \times (1 + 0.0026 \cos (2 Φ) + 0.00028 H) \times P_{0}$ 2

FIGURE 1

Relative directions of the LOS and zenith delay θ represents the satellite elevation angle.

where ɸ is the receiver geographic latitude in degrees, H is the altitude of the location in kilometers, and P₀ is the atmospheric pressure in millibars. ZWD is determined as follows:

$Z W D = 0.002277 \times (\frac{1255}{T_{s}} + 0.05) \times e_{0}$ 3

where e₀ is the partial water vapor pressure in millimeters and T_s is the surface temperature in Kelvin (K).

The zenith wet and dry delays are scaled with their corresponding obliquity factors (also referred to as mapping functions) to account for the prolonged propagation path, which is dependent on the satellite elevation (θ) (Misra & Enge, 2006), as shown in Equation (4). VMF3 (Landskron & Böhm, 2018) is used to obtain the obliquity factors. Using m_dry and m_wet to denote the mapping functions for ZHD and ZWD, respectively, we obtain the tropospheric delay along the propagation path through the following formula:

$T_{r}^{s} = (Z H D \times m_{d r y} (θ)) + (Z W D \times m_{w e t} (θ))$ 4

While this study focuses on tropospheric effects—including range delay and attenuation—it does not address ionospheric effects, namely code delay and scintillation. Ionospheric code delay arises from the dispersive nature of the ionized medium and becomes progressively less significant at higher frequencies because of its inverse square dependence on frequency (Morton et al., 2020; ITU-R, 2013a). Moreover, first-order code delay can be estimated and mitigated by using dual-frequency measurements. In contrast, ionospheric scintillation refers to rapid fluctuations in signal amplitude and phase caused by small-scale electron density irregularities. These effects are more pronounced at lower frequencies, where signals experience more frequent fades and greater phase dynamics (Sun et al., 2024). As a result, ionospheric scintillation becomes less prominent at higher frequencies. Similarly, effects due to signal polarization—such as linear, circular, or elliptical polarization—are not included in this analysis. Polarization can influence signal propagation through interactions with atmospheric particles, surface reflections, and receiver antenna characteristics, potentially leading to differential attenuation or phase shifts (Oguchi, 1983; Crane, 1996; ITU-R, 2017a). However, modeling these effects requires detailed knowledge of transmitter and receiver configurations. In this study, polarization is excluded to simplify the analysis and to isolate the fundamental tropospheric contributions to signal degradation. Future work could extend this analysis to jointly model ionospheric and tropospheric effects, while incorporating polarization-dependent interactions and receiver specifications.

3 SIGNAL ATTENUATION

Signal attenuation is caused by different atmospheric conditions such as rain, clouds, gases, and scintillation, as discussed in the ITU-R methods. To compute the total signal attenuation, all of these effects are considered (ITU-R, 2015).

3.1 Attenuation Due to Rain

Signal attenuation due to precipitation is largely dependent on the amount of precipitation at a location. This attenuation is estimated in three steps. First, the rainfall rate RR (mm/h) that is exceeded for a certain percentage (p) of the time for a region is used to determine the specific attenuation (γ) in dB/km by applying a power-law relationship (ITU-R, 2005) given by Equation (5):

$γ_{R} = k \times {(R R)}_{p}^{α}$ 5

The coefficients k and α are determined as functions of frequency f (GHz) using Equations (6) and (7) (ITU-R, 2005):

$k = \frac{[k_{H} + k_{V} + (k_{H} - k_{V}) \cos^{2} θ \cos (2 τ)]}{2}$ 6

$α = \frac{[k_{H} α_{H} + k_{V} α_{V} + (k_{H} α_{H} - k_{V} α_{V}) \cos^{2} θ \cos (2 τ)]}{2 k}$ 7

where θ is the path elevation angle and τ is the polarization tilt angle relative to the horizontal. In this paper, circular polarization is considered, indicating that a polarization tilt angle of 45° is assumed. k_H and α_H denote the coefficients for horizontal polarization whereas k_V and α_V denote the constants for vertical polarization. These constants are functions of frequency and are computed from the methodology presented by ITU-R (2005).

Second, the attenuation level that is exceeded 0.01% of the time is computed using Equation (8) and Equation (9). The 0.01% exceedance level is adopted as a standard reference (ITU-R, 2015), which corresponds to the level of attenuation exceeded for approximately 53 min per year, serving as a benchmark for evaluating worst-case performance in satellite communication links. Because this attenuation level characterizes signal attenuation during statistically rare but heavy rainfall conditions, it is commonly used as a basis for calculating attenuation at other exceedance probabilities through extrapolation. Equation (8) defines the effective path length (km) as the product of the actual path length (L_R) and the correction factor (V_0.01) (ITU-R, 2015). The correction factor is a function of the satellite elevation angle, rain height column, geographic latitude of the location, and altitude of a location from the mean sea level (ITU-R, 2015):

$L_{E} = L_{R} \times V_{0.01}$ 8

The attenuation (in dB) that is exceeded 0.01% of the time (A_0.01) is then determined as the product of the specific attenuation and effective path length given by Equation (8) (ITU-R, 2015):

$A_{0.01} = γ_{R} \times L_{E}$ 9

Third, the estimated attenuation (in dB) exceeded for p percentage of an average year (A_p) is determined from the attenuation to be exceeded for 0.01% for an average year (A_0.01) using Equation (10), where β is a function of the receiver geographic latitude in degrees (ITU-R, 2015):

$A_{P} = A_{0.01} \times {(\frac{p}{0.01})}^{- (0.655 + 0.033 \ln (p) - 0.045 \ln (A_{0.01}) - β (1 - p) sin θ)}$ 10

In this paper, rain rate data are taken from ITU-R digital maps produced using the GPCC Climatology (version 2015) database over land and from the ECMWF Reanalysis (ERA) Interim r-analysis database over water (ITU-R, 2017a). Rain height column data are taken from global maps provided by ITU-R (2013b).

3.2 Attenuation Due to Clouds and Fog

Signal attenuation due to clouds and fog depends on the slant path traversed by the signal. This attenuation (in dB) is a function of the frequency f, integrated cloud liquid water content that is exceeded p percentage of the time, cloud liquid mass absorption coefficient, and elevation angle (ITU-R, 2023).

The predicted slant path cloud attenuation denoted by A_c is given by Equation (11):

$A_{C} (f, p) = \frac{K_{L} (f) \times L (p)}{sin θ}$ 11

where K_L is the cloud liquid mass absorption coefficient in dB/mm. L(p) is the integrated cloud liquid water content at the exceedance probability p, in millimeters, from the surface of the Earth at the desired location. θ is the satellite elevation angle.

The cloud liquid mass absorption coefficient is a function of the complex permittivity of water, which depends on the frequency and temperature. This coefficient is computed from the mathematical model based on Rayleigh scattering provided by ITU-R (2023). Statistics of integrated cloud liquid water content from integral maps provided in the ITU-R (2023) methods are used for this research.

3.3 Attenuation Due to Gases

Gaseous attenuation is divided into two components: attenuation due to dry air and attenuation due to water vapor. Oxygen is the dominant contributor to dry air attenuation, as its absorption lines fall in the frequencies of interest (ITU-R, 2022). The first step in calculating gaseous attenuation is to determine the specific attenuation (in dB/km) due to dry air (γ_o) and water vapor (γ_wv) (ITU-R, 2022; MATLAB, 2024). The dry-air-specific attenuation is a function of dry air pressure, temperature, and the complex refractivity of oxygen. Similarly, the water-vapor-specific attenuation is a function of water vapor partial pressure, temperature, and the complex refractivity of water vapor. In this paper, a line-by-line model (ITU-R, 2022) is used to compute the specific attenuations.

The zenith dry air or oxygen attenuation (ZA_o) and water vapor attenuation (ZA_wv) are obtained (in dB) by multiplying the specific attenuation (dB/km) by the path length (R) (in km) given by Equation (12) and Equation (13) (ITU-R, 2022). The slant or LOS attenuations (dB) for both oxygen and water vapor are obtained using the cosecant law (ITU-R, 2022) shown in Equation (14) and Equation (15):

$Z A_{o} = γ_{o} \times R$ 12

$Z A_{w v} = γ_{w v} \times R$ 13

$A_{o} = \frac{Z A_{o}}{\sin θ}$ 14

$A_{w v} = \frac{Z A_{w v}}{\sin θ}$ 15

where θ is the elevation angle.

The reference standard atmosphere parameters provided by ITU-R (2017b) are used to compute the gaseous attenuation. Dry air has a water vapor density of zero (ITU-R, 2017b), and a path length of 1 km is used (ITU-R, 2022).

3.4 Attenuation Due to Scintillation

Rapid fluctuations in the refractive index of the neutral atmospheric medium along the propagation path of a signal lead to tropospheric scintillation, which can reduce the strength of a signal, causing signal amplitude fading. While the term “scintillation” is often associated with ionospheric effects in GNSS literature, in this paper, we refer to tropospheric scintillation as defined by ITU-R (2015), which specifically models small-scale refractive turbulence in the lower atmosphere.

In this paper, the ITU-R model presented by ITU-R (2015) is used to compute scintillation fading. This prediction method is based on temperature and relative humidity and reflects the specific climatic conditions at the receiver location. The model requires the wet term of the surface refractivity N_wet, the height of the turbulent layer, and the satellite elevation angle as inputs. The elevation angle and the height of the turbulent layer are used to compute the effective path length. Then, the standard deviation of the signal amplitude for the applicable period and propagation path, denoted by σ, is calculated as a function of the frequency, effective path length, satellite elevation angle, and N_wet (ITU-R, 2015). The fade depth that is exceeded p percentage of the time, A(p), is defined as the product of the standard deviation of the signal amplitude and the time percentage factor a(p), as shown in Equation (15). a(p) is determined by p as given in Equation (16):

$A (p) = a (p) \times σ$ 16

$a (p) = - 0.061 {({log}_{10} p)}^{3} + 0.072 {({log}_{10} p)}^{2} - 1.71 ({log}_{10} p) + 3.0$ 17

The wet component refractivity maps for this study are taken from the ECMWF ERA data (ITU-R, 2017a). The height of the turbulent layer is assumed to be 1 km (ITU-R, 2015).

4 RESULTS

The range delay and signal attenuation are computed at different elevation angles for four different regions: Cibinong, Indonesia; Maui, Hawaii; Ny-Ålesund, Norway; and Boulder, Colorado. Indonesia is a typical tropical region near the equator and is known to receive some of the highest rainfall on Earth. Norway is a polar region with a dry climate. Maui, Hawaii, is a relatively wet low-latitude region, and Boulder, Colorado, is a relatively dry mid-latitude region. The signal frequencies considered in this paper are 1.5 GHz for the L band, 3 GHz for the S band, 6 GHz for the C band, 10 GHz for the X band, and 15 GHz for the Ku band. The tropospheric effects are highlighted for elevation angles of 5° and 30°. In multipath-benign environments, elevations lower than 5° can suffer from multipath effects for our frequencies of interest (ITU-R, 2015; Morton et al., 2019; Collett et al., 2020). Thus, we have not considered those elevation angles in this paper.

Figure 2 shows the tropospheric delay variation with elevation angle for the four regions mentioned above. Because tropospheric delay is nondispersive up to 30 GHz (Spilker, 1996; ITU-R, 2022), the results shown here are frequency-independent and apply across the microwave spectrum in this range. Thus, tropospheric delay varies only with elevation angle and the atmospheric conditions at a location. This paper uses the VMF3 mapping functions from Landskron and Böhm (2018) and the associated 5° × 5° grid data of ZHD and ZWD from re3data (2020) for the weather parameters and tropospheric delay estimation. In Figure 2, the delay is shown to be the highest for Indonesia and the lowest for Boulder, as expected. The range delay increases for lower elevation angles, with the delay value reaching up to 37 m for Indonesia at an elevation angle of 3°. The delay values for the four regions at 5° and 30° elevation angles are listed in the table inserted in the figure. The delay was also computed across different seasons, but the seasonal variations were found to be negligible.

FIGURE 2

Tropospheric delay (m) for the four regions as a function of elevation angle (°) The delays at 5° and 30° elevation angles are marked by blue dashed lines and are also listed in the table.

Figure 3 shows the attenuation due to precipitation (dB) at 5° and 30° elevation angles. The rain column heights considered are 4-5 km for the four regions. We observe that a rainfall rate of 145 mm/h, occurring 0.01% of the time in Indonesia, results in Ku band signal attenuation of 115 dB for a 5° elevation and 37 dB for a 30° elevation. Similarly, in Maui, Hawaii, where the rainfall rate is 63 mm/h for the same duration, the Ku band attenuation is 65 dB for a 5° elevation and 15 dB for a 30° elevation. The other regions have lower attenuations, as their rainfall rates are lower.

FIGURE 3

Attenuation due to rain (dB) as a function of the rain rate (RR; mm/h) for (a) 5° and (b) 30° elevation angles

The x axis represents the rain rate on a log scale, and the y axis represents attenuation on a log scale. Attenuations for the four regions for 0.01% and 1% of the time at different frequencies are marked in black dashed lines. Each region is represented by a unique color shown in the legend. Attenuations at different rain rates for each frequency are represented by a unique shape shown in the legend, with the color of that region.

The trends shown in Figure 3 are consistent with those reported in early empirical studies compiled by Schanda (1976), as presented by Ulaby et al. (1981). These attenuation values, expressed as specific attenuation (dB/km), were derived from earlier data sets (Haroules & Brown, 1969; Beckwith et al., 1970) based on horizontal path assumptions and fixed rain rates. Schanda (1976) noted that these curves were compiled from multiple sources with differing assumptions and should be used with caution, particularly because of variations in drop size distribution models. The empirical fits used in those curves—combining linear and logarithmic trends—are primarily applicable for frequencies above 2.8 GHz. In contrast, our study includes lower frequencies such as the L band and follows the ITU-R (2005) model, which calculates specific attenuation via a frequency-dependent power-law relationship with region-specific rain rate statistics exceeded for a given percentage of time. The total attenuation is then computed using slant path corrections from ITU-R (2015), as discussed in Section 3.1, which account for satellite elevation angle and rain height. Whereas the polarization used in the figure presented by Schanda (1976) is not specified, our study assumes circular polarization (45° tilt). The drop size distribution is not explicitly modeled in either approach, but the k and α coefficients used in the ITU-R model are empirically derived and implicitly account for average drop size distribution effects across a wide range of observational conditions. Despite differences in assumptions and methodology, both approaches show consistent trends in the dependence of rain attenuation on frequency and rain rate. Similar calculations and assumptions have also been discussed by Long and Ulaby (2015), further supporting the validity of the rain attenuation trends presented in this study.

Figure 4 shows the attenuation due to cloud/fog (dB) at 5° and 30° elevation angles. The results show that Indonesia, which has the highest liquid water content of 0.88 mm among the four regions, has the largest attenuation of 1.75 dB. This value occurs for a signal at 5° elevation in the Ku band when there is a 0.01% probability of cloud cover. The second-largest attenuation is observed for Maui, Hawaii, with an attenuation of 1.35 dB at a 5° elevation for the same probability of cloud cover and the same signal band. The attenuation values are much lower for a 50% probability of cloud/fog, with the highest value of 0.35 dB observed for Indonesia for the same frequency band and satellite elevation angle. Attenuation becomes negligible at the L and S bands. The attenuation values for a 30° elevation angle are much lower than those at 5° elevation and can be assumed to be negligible, as the highest value is only 0.32 dB, corresponding to a 0.01% probability of cloud cover for Indonesia at the Ku band.

FIGURE 4

Attenuation due to cloud/fog (dB) as a function of the liquid water content for the four regions at (a) 5° and (b) 30° elevation angles, with 0.01% and 50% probability of cloud/ fog marked in black

Figure 5 shows the gaseous attenuation (dB) as a function of frequency (GHz) at a 5° elevation angle. The results are shown for an atmospheric pressure of 101.3 kPa and a water vapor density of 19.6 g/m³ at 27.3°C. This scenario corresponds to Indonesia, as per the methodology presented by ITU-R (2017a), to assess a low-latitude annual reference atmosphere. Because Indonesia has the highest water vapor content in the atmosphere among our four regions, these atmospheric parameters correspond to the worst-case scenario in our paper. Figure 5 shows that the attenuation due to water vapor is much higher than that for oxygen. The attenuation increases with frequency, but the highest value is only 0.8 dB, which corresponds to the Ku band. This finding implies that the attenuation due to both dry air and water vapor is negligible at all locations and all satellite elevation angles for our frequencies of interest.

FIGURE 5

Gaseous attenuation (dB) due to water vapor and oxygen as a function of frequency at a 5° elevation angle for Indonesia

The y axis represents attenuation on a log scale. Attenuations for the frequencies of interest are marked by black dashed lines.

Figure 6 shows amplitude fading due to tropospheric scintillation at 5° and 30° elevation angles. The fading increases with increasing frequency and decreasing elevation angle. For very small percentages of time, the fading due to scintillation at very low elevation angles is large, with a fading depth of 12 dB seen for Indonesia for 0.01% of the time.

$Amplitude scintillation (dB) at (a) 5° and (b) 30° elevation angles as a function of the wet term of the surface refractivity Attenuations for the four regions for 0.01% and 50% of the time at different frequencies are marked by black dashed lines. Each region is represented by a unique color, as shown in the legend. Attenuations at different rain rates for each frequency are represented by a unique shape shown in the legend, with the color of that region.$

FIGURE 6

Amplitude scintillation (dB) at (a) 5° and (b) 30° elevation angles as a function of the wet term of the surface refractivity

Attenuations for the four regions for 0.01% and 50% of the time at different frequencies are marked by black dashed lines. Each region is represented by a unique color, as shown in the legend. Attenuations at different rain rates for each frequency are represented by a unique shape shown in the legend, with the color of that region.

Overall, we observe that the range delay can be large at some locations, especially for the relatively wet regions of Indonesia and Maui, Hawaii. This trend is more prominent for lower elevation angles. Signal attenuation due to rain, cloud, and scintillation fading increases with increasing frequency and decreasing elevation angle, and the effect is stronger for tropical regions such as Indonesia and Maui, Hawaii. In future work, we will verify this analysis with real data from Maui, Hawaii.

5 CONCLUSION

In this work, a comprehensive study of various tropospheric effects of range delay and signal attenuation on signals was performed for a wide range of carrier frequencies. The range delay includes dry and wet components, which were predicted via established models to determine the total delay. The total delay increases for lower elevation angles and displays regional variation, with Indonesia showing the highest delay, followed by Maui, Hawaii. Signal attenuation due to rain, clouds, gases, and scintillation was estimated using ITU-R methods. Attenuation is negligible for the Global Positioning System (GPS) L-band frequencies across all regions but becomes significant for higher frequencies. Indonesia demonstrates the most pronounced attenuation, followed by Maui, Hawaii. Additionally, assessments of gaseous attenuation indicate an insignificant effect at the frequencies under consideration, even at low elevation angles.

Among the sources of attenuation discussed in this paper, attenuation due to scintillation is the highest for GPS frequencies. At a 5° elevation in all regions considered, greater than 1 dB of amplitude fading due to scintillation is observed for 0.01% of the time. Scintillation-induced attenuation can exceed 10 dB at the Ku band in Maui and at the Ku and X bands in Indonesia for 0.01% of the time. In Indonesia, attenuation at the Ku and X bands due to rain can also reach several decibels for 0.01% of the time at 5° elevation. In future work, we plan to confirm our results with real data collected at the summit of Haleakala in Maui, Hawaii.

HOW TO CITE THIS ARTICLE

Sonth, N., Morton, J., & Scott, L. (2025). Comprehensive assessment of tropospheric effects over a wide range of frequencies transmitted from LEO satellites. NAVIGATION, 72(4). https://doi.org/10.33012/navi.725

ACKNOWLEDGMENTS

This work was sponsored by Air Force Research Laboratory (AFRL) award #282109-874X.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Abstract

1 INTRODUCTION

2 RANGE DELAY