David W. Matolak and Ruoyu Sun

University of South Carolina, Columbia, South Carolina

AG Channel Measurement and Modeling Results for

Over-Water and Hilly Terrain Conditions

NASA/CR—2015-218486

April 2015
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David W. Matolak and Ruoyu Sun

University of South Carolina, Columbia, South Carolina

AG Channel Measurement and Modeling Results for

Over-Water and Hilly Terrain Conditions

NASA/CR—2015-218486

April 2015

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared under Grant NNX12AR56G
Available from

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Abstract 

 This report describes work completed over the past year on our project, entitled “Unmanned 

Aircraft Systems (UAS) Research: The AG Channel, Robust Waveforms, and Aeronautical Network 

Simulations.” This project is funded under the NASA project “Unmanned Aircraft Systems (UAS) in 

the National Airspace System (NAS).” In this report we provide the following: an update on project 

progress; a description of the over-freshwater and hilly terrain initial results on path loss, delay 

spread, small-scale fading, and correlations; complete path loss models for the over-water AG 

channels; analysis for obtaining parameter statistics required for development of accurate wideband 

AG channel models; and analysis of an atypical AG channel in which the aircraft flies out of the 

ground site antenna main beam. We have modeled the small-scale fading of these channels with 

Ricean statistics, and have quantified the behavior of the Ricean K-factor. We also provide some 

results for correlations of signal components, both intra-band and inter-band. An updated literature 

review, and a summary that also describes future work, are also included. 

1. Introduction 

 In this report we continue our investigation of the air-ground (AG) channel. Prior reports [1]-

[6] reviewed the existing AG channel literature, outlined the plan for flight tests, discussed the 

channel modeling approach, provided a detailed characterization of our dual-band channel sounder, 

and reported detailed results for measurements and some model results for the over-sea setting. 

 The goals of this project are to quantitatively characterize the AG channel in a number of 

different environments, for contribution to NASA’s work on Unmanned Aircraft Systems (UAS) in 

the National Airspace System (NAS). The different environments pertain primarily to the type of 

environment surrounding the ground site (GS) with which the UAS is communicating. For our 

characterization we have employed measurements in which we have collected data that enables us to 

estimate the channel’s time varying impulse response. This provides a complete description of the 

channel, and from this data, we are developing AG channel models for each setting. 

 The actual measurements collected power delay profiles (PDPs), from which (with associated 

measured phase information), we can estimate the AG channel impulse response (CIR). From these 

PDPs and knowledge of other link parameters, we can also determine propagation path loss. We 

refer the reader to prior reports [1]-[6] for additional background material. 

 The remainder of this report is organized as follows: in Section 2 we briefly summarize 

progress made since the last report, and also summarize our data processing procedures. Section 3 

describes the over-freshwater measurements. In Section 4 we provide new results for the over-water 

AG channels, including completed path loss models for both over-sea and over-freshwater conditions, 

a description of statistical analyses required for obtaining statistical channel characteristics that 

includes estimation of stationarity distance and correlations, some new over-freshwater channel 

results, a description of channel characteristics for an over-harbor setting in which the aircraft flew 

outside the ground site antennas’ main beams, and some initial statistics required for the 

construction of the wideband AG channel models. Section 5 contains new results for the hilly 

terrain/suburban environment, and Section 6 contains an update to the literature review. In Section 

7 we conclude and discuss planned future work. Several appendices provide supporting detail for the 

main body of the report. 

AG Channel Measurement and Modeling Results for 

Over-Water and Hilly Terrain Conditions 

David W. Matolak and Ruoyu Sun 

University of South Carolina 

Columbia, South Carolina 29208 

NASA/CR—2015-218486 1
2. Summary of Project Progress 

Since the last report, we have done the following: 

x processed the vast majority of the flight test data for the over-freshwater flight tests; 

x processed some of the data for the hilly terrain flight tests; 

x processed some of the data for the urban setting flight tests; 

x conducted a detailed statistical analysis of the over-sea data, which will enable us to use the 

techniques we have developed for the remaining data sets. This analysis allows us to estimate 

the AG channel’s stationarity distance, which in turn enables computation of correlations among 

the signals received on the various antennas, computation of small-scale fading statistics, etc.; 

x written several papers to publicize our results [7]-[12]; 

x contributed to a report for the International Civil Aviation Organization (ICAO) working group F 

[13] (another such contribution is imminent); 

x made a presentation to the Radio Technical Commission for Aeronautics (RTCA) Special 

Committee (SC) 228, apprising them of project progress [14]; 

x augmented our update to the literature review on UAS and AG channels; 

x compiled statistics required for development of detailed statistical wideband tapped-delay line 

(TDL) models for the over-sea AG channel. 

The principal investigator (PI) has also been invited to contribute a paper and give an invited talk at 

an upcoming conference [15], with another invited talk to occur in April 2015. We are also currently 

writing a journal paper that will introduce the AG channel characterization project, describe our 

methods and summarize measurements, and provide complete results for the over-water settings. 

Additional journal papers are also planned, and will cover results for the remaining GS 

environments. 

 At this time, we have largely completed our algorithms for processing the measured flight 

test data. As noted in prior reports, we have had to account for several non-idealities in the 

measurements. This includes channel sounder impulse response and noise characteristics, external 

electronic noise from other aircraft electronics, channel sounder sampling clock drift, and a variable 

channel sounder PDP output rate (see Appendix D). A few parameters also remain only partially 

known—primarily the aircraft antenna patterns, which are affected by the airframe itself. Focusing 

first on the relatively simplest over-water settings has allowed us to develop a thorough and reliable 

set of algorithms for extracting the actual AG channel characteristics from the measurement data. 

Figure 1 provides an illustration of the data processing procedure. 

NASA/CR—2015-218486 2
Figure 1. Illustration of data processing procedures. 

 A corresponding table that describes the status of the flight test data processing appears in 

Figure 2. As can be observed, the over 315 million PDPs we have recorded provides—to our 

knowledge—the largest set of data ever compiled for analysis of the AG channel. 

NASA/CR—2015-218486 3
Figure 2. Summary of flight test data processing to date. 

3. Over-Freshwater Measurement Summary 

The over-sea measurements were described in detail in [6]. Prior reports also described the 

channel sounder characteristics in detail, and some detail on portions of the data processing 

algorithms. Here we describe the over-freshwater measurements. 

The over-freshwater flights were conducted on 22 October 2013, over Lake Erie, near 

Cleveland, OH. The ground site was placed near the lake shore line in Edgewater Park, and as with 

the over-sea measurements, both straight and oval-shaped flight tracks (FTs) were flown. Figure 3 

shows an aerial view of the GS location and a view looking east toward downtown Cleveland. 

Figure 3. Views of over-freshwater test area: left, aerial view from south of GS, right, view from aircraft looking eastward. 

 The GS was located at coordinates latitude 41° 29' 33.8" N, longitude 81 44'5.48" W, with 

elevation (AMSL) 177.4 m. The GS antennas were 20 m above ground, oriented 352° from geographic 

north in azimuth, with zero degree elevation angles. The average Lake Erie water level is 174 m 

AMSL. Figure 4 shows a view from the GS looking east. 

Index Location Date Terrain # of Flight 

Tracks

# of PDPs 

(Million)

Raw Data 

Size (GB)

Geometry & 

Analytical

Pairing & 

Rotation

Noise 

Threshold

Persistance 

Check

Cross 

Correlation TDL SD K factor Doppler Shift

Alignment & 

Results

Total Size 

(GB) Backup Date Note

Status N P N N N D N N D N

Update Date

Size (GB) -- -- -- -- -- -- -- -- -- --

Status Y Y Y Y Y D P P D Y

Update Date 9/16/2014 10/19/2013 11/22/2013 11/23/2013 2/23/2014 9/23/2014

Size (GB) 1.53E-03 16.90 3.45 3.53 0.16 -- -- -- -- 3.39

Status Y Y Y Y Y D P P D Y

Update Date 9/15/2014 10/17/2013 10/17/2013 11/4/2013 11/15/2013 9/23/2014

Size (GB) 1.70E-03 33.90 18.50 18.90 0.57 -- -- -- -- 3.28

Status N Y Y Y Y D N N D N

Update Date 12/4/2013 12/5/2013 12/6/2013 12/15/2013

Size (GB) -- 27.30 15.50 15.80 2.49E-01 -- -- -- -- --

Status N Y Y Y Y D N N D N

Update Date 11/20/2013 11/24/2013 11/24/2013 12/13/2013

Size (GB) -- 18.60 8.98 9.18 1.24E-01 -- -- -- -- --

Status N Y Y Y Y D N N D N

Update Date 11/21/2013 11/23/2013 11/24/2013 12/6/2013

Size (GB) -- 13.30 8.32 8.47 1.25E-01 -- -- -- -- --

Status N Y Y Y Y D N N D N

Update Date 11/23/2013 11/23/2013 11/23/2013 11/29/2013

Size (GB) -- 24.90 14.20 14.40 0.24 -- -- -- -- --

Status Y Y Y Y Y D P P D P

Update Date 9/15/2014 9/7/2014 9/7/2014 9/7/2014 2/28/2014

Size (GB) 8.44E-04 11.00 6.65 6.78 0.11 -- -- -- -- --

79 315.95 145.5 4.07E-03 145.90 75.60 77.06 1.58 0.00 0.00 0.00 0.00 6.67 452.31

Y Done

P Partially done or need further checking

N Not start yet

D Developing

33.90 1.64E-03 Size of outputs in GB. Each processing step generates some output files to record some variables, which will be used by following steps.

8 11.2 Cleveland, 

OH 10/22/13 Over lake & Downtown 6 25.32

26.1

10.3

7 Telluride, 

CO 09/12/13 Mountain 12 48.32 22.3

6 Cleveland, 

OH 09/05/13 Suburban 6 28.36

13 52.88 24.0

5 Palmdale, 

CA 06/13/13 Desert & Hilly 13 54.97 18.2

12.53 10.6

2

12

Latrobe, 

PA 04/15/13 Hilly 12

1 Cleveland, 

OH 03/20/13 Suburban 5

26.46 22.8

67.11

Total

3 Oxnard, 

CA 06/11/13 Over sea & Harbor

4 Palmdale, 

CA 06/12/13 Desert & Hilly

9/19/2014

9/19/2014

9/19/2014

9/19/2014

9/19/2014

9/19/2014

9/19/2014

9/19/2014

L-band Rx 2 does not 

have "DATA 

FIELDS".

40.52

76.04

35.74

10.60

50.23

101.25

82.85

55.08

NASA/CR—2015-218486 4
Figure 4. View from GS looking eastward. 

 Over 25 million PDPs were collected over six FTs. Figure 5 shows two example FTs in a 

Google Maps® view, and in ECEF coordinates. The FT2 was a straight flight path toward the GS 

with link distance varying from approximately 2.5-29.35 km. The average altitude difference 

between the GS and aircraft was 566 m (minimum 554 m, maximum 578 m), and the elevation angle 

ranged from 1.4-13q. The oval-shaped (or, U-shaped) FT4 had link distance ranging from 16.34-21.76 

km, average altitude difference 567 m (maximum 578 m, minimum 555 m), with elevation angle 

from 1.7-2.2q. 

 Initial results from the over-freshwater measurements appear in Section 4. Detailed FT 

information and additional results for the other over-freshwater FTs appears in Appendix A. 

Figure 5. Left: Google Maps® view of FT2 (straight) and FT4 (oval); Right: FTs in ECEF coordinates. 

4. Over-Water AG Channel Results 

 In this section we describe in detail the recent analysis we have done in order to develop 

models for the over-water AG channels. We provide detailed path loss results for both the over-sea 

and over-freshwater cases, and some additional preliminary results for the over-freshwater case 

(with added detail in Appendix A). We also provide results for the atypical condition in which the 

aircraft was not within the main beams of the GS antennas, but rather in the “backlobe” region 

(initial results for this case appear in [6]). Although this is not a normal operational condition, it 

NASA/CR—2015-218486 5
certainly can occur, particularly during early phases of UAS GS deployments, and in any “ad hoc” or 

anomalous conditions (e.g., when one GS sector has failed). We also provide example statistics that 

are the next step required for completion of wideband statistical models for the over-sea AG channel. 

4.1 Over-Water Path Loss Models 

As noted in [6], radio propagation in over-water environments has been studied for many 

years for communications, radar, and remote sensing applications [16]. In very open coastal ocean 

and lake settings, obstructions are minimal. We observed no anomalous propagation phenomena 

during our testing. Specifically, we saw no increased attenuation from hydrometeors or atmospheric 

gases—effects of hydrometeors (primarily rain) are well known, and can be estimated using 

established models, with inputs of rain rate and the spatial extent of the rainstorm [17]. Likewise, 

we saw no enhanced propagation via ducting [18], [19]. Since evaporation ducts are essentially 

always present over water, when the altitude difference between GS and aircraft is small (difference 

between transmitter and receiver antenna heights is within tens of meters or less), the effect of these 

ducts should be taken into account in these conditions. The empirical procedures in [20] can be used 

for estimating this effect when required. 

Numerous example path loss results appear in [6], for both C-band and L-band. From these 

results, we have determined that the over-water AG channels are best modeled using the curvedearth two-ray (CE2R) model (see [2] for analysis). For short link distances, the simpler flat-earth 

two-ray model is sufficient. The applicable distance at which one should transition from the flatearth to the CE2R depends upon frequency and antenna heights, in a rather complicated way. The 

criterion of when (at what distance) the two model results differ is also somewhat subjective. 

Nonetheless, one can select some criterion for this and evaluate the transition distance numerically 

given the carrier frequency and antenna heights. For the antenna heights in our over-sea 

measurements, the transitions could be set as approximately 2 km for L-band, and approximately 3 

km for C-band. Note that because these two-ray models have very rapidly “oscillating” loss values for 

these short distances, the very need for modeling the transition from flat to curved earth is 

debatable. In fact, once programmed into a computer for numerical computation, the CE2R is not 

very computationally-intensive, hence one may use it for all values of distance. 

Even though the CE2R model captures the main features of the over-water AG channel, 

measured results will not follow the analytical model perfectly. Aside from any antenna effects or 

equipment imperfections, and additional multipath components (which are sparse in the over-water 

cases) the surface reflection will vary. By this we mean that the sea surface is complex, and 

temporally (and spatially) varying [21]. Hence the phase—and to a lesser degree the amplitude—of 

the specular component of the surface reflection can vary because of the changing geometry of the 

rough sea surface. We do incorporate a simple sea-surface roughness factor into our CE2R model, 

based upon wind speed, to approximate the effect on the specular reflection’s magnitude, but there is 

no easy way to model the dynamic effects upon phase. Thus in addition to the deterministic CE2R, 

we have incorporated random variation to quantify the variation of path loss. Also, since the sea 

surface is smoother at L-band than at C-band, because the electrical length (relative to wavelength 

O) of the sea surface features is smaller at L-band, the CE2R model matches the experimental data 

better at L-band. Finally on this, for freshwater settings, we expect generally smoother surface 

conditions, which would enhance the specular reflection, but this is countered in part by the lower 

conductivity of freshwater compared to that of seawater. Hence the over-freshwater results for path 

loss are comparable to those for the over-sea case. 

The over-sea path loss models for the two bands are composed of two equations (for each 

band’s model), which apply to different ranges of elevation angle (equivalently, different ranges of 

link distance). The models for path loss (PL), in units of dB, are as follows: 

NASA/CR—2015-218486 6
L-band

ܲܮሺ݀ǡ ߠሻ ൌ ൜ܣ଴ǡ௅ǡ௦ ൅ ͳͲ݊௅ǡ௦݈݋݃ሺ݀ሻ ൅ ܺ௅ǡ௦ ൅ ܮߞ௅ǡ௦ǡߠ ൐ ߠ௧ሺ݀ ൐ ݀௧ሻ

௧ሻ݀ ൏ ݀௧ሺߠ ൏ ߠǡκǡ௅ܮߞ ൅ ǡκ௅ܺ଴ ൅ ܮ ൅ ܴʹܧܥ

 (1) 

where threshold elevation angle Tt = 5 degrees, ]= -1 for travel toward the GS and +1 for travel away 

from the GS, LL,s and LL,l

 are small constants, A0,L,s is an intercept value, nL,s is the path loss 

exponent, distance ranges from 1 d d d28 km for fresh water and 2.2 d d d24 km for sea water. The 

variable X is a zero-mean Gaussian random variable, with standard deviation VX dB, and L0 is a 

small offset value. Note that two distinct (both zero-mean) Gaussians pertain to the different 

segments in (1). 

C-band 

ܲܮሺ݀ǡ ߠሻ ൌ ቊܣ଴ǡ஼ǡ௦ ൅ ͳͲ݊஼ǡ௦݈݋݃ሺ݀ሻ ൅ ܺ஼ǡ௦ ൅ ܮߞ஼ǡ௦ǡߠ ൐ ߠ௧ሺ݀ ൐ ݀௧ሻ

ܣ଴ǡκ ൅ ͳͲ݊஼ǡκ݈݋݃ሺ݀ሻ ൅ ܺ஼ǡκ ൅ ܮߞ஼ǡκǡߠ ൏ ߠ௧ሺ݀ ൏ ݀௧ሻ (2) 

The same parameter definitions in the L-band model apply to the C-band model as well. Model 

parameters are provided in Tables 1 and 2. 

Table 1. L-band path loss parameters for eq. (1) (using CE2R). 

Setting A0,L,s

(dB)

nL,s VX,L,s

(dB)

LL,s

(dB)

L0

(dB)

VX,L,l

(dB)

LL,l

(dB)

dt

(km)

Fresh 57.7 1.4 2.8 1.8 1.1 3.2 1.8 6.6 

Sea 50.6 1.5 2.8 1.2 1.0 4.8 1.1 9.1 

Table 2. C-band path loss parameters for eq. (2). 

Setting A0,C,s

(dB)

nC,s VX,C,s

(dB)

LC,s

(dB)

A0,l

(dB)

nC,

l

VX,C,l

(dB)

LC,l

(dB)

dt

(km)

Fresh 53.6 1.8 2.6 2.2 53.7 1.8 2.8 1.5 6.6 

Sea 60.0 1.6 2.3 1.7 63.9 1.5 2.5 0.5 9.1 

We also provide a second model for L-band path loss that uses the log-distance form only (no CE2R), 

for any who may want a simpler (slightly less accurate) model: 

L-band, log-distance 

ܲܮሺ݀ǡ ߠሻ ൌ ቊܣ଴ǡ௅ǡ௦ ൅ ͳͲ݊௅ǡ௦݈݋݃ሺ݀ሻ ൅ ܺ௅ǡ௦ ൅ ܮߞ௅ǡ௦ߠ ൐ ߠ௧ሺ݀ ൐ ݀௧ሻ

ܣ଴ǡκ ൅ ͳͲ݊௅ǡκ݈݋݃ሺ݀ሻ ൅ ܺ௅ǡκ ൅ ܮߞ௅ǡκǡߠ ൏ ߠ௧ሺ݀ ൏ ݀௧ሻ

 (3) 

with parameters provided in Table 3. 

Table 3. L-band path loss parameters for eq. (3) (using log-distance form). 

Setting A0,L,s

(dB)

nL,s VX,L,s

(dB)

LL,s

(dB)

A0,l

(dB)

nL,l VX,L,l

(dB)

LL,l

(dB)

dt

(km)

Fresh 57.7 1.4 2.8 1.8 28.6 2.1 3.9 1.8 6.6 

Sea 50.6 1.5 2.8 1.2 23.2 2.2 4.6 1.0 9.1 

NASA/CR—2015-218486 7
4.2 Stationarity Distance 

Aside from intermittent multipath components and variability of the surface reflection, the 

predominant features of the over-water AG channels are deterministic, and well-approximated by 

the CE2R model. For more detailed channel characterizations that account for wideband effects such 

as multipath, and for effects that are difficult to model deterministically (such as airframe 

shadowing), stochastic approaches are appropriate. Such approaches might also be applied to 

modeling of the water surface reflection. With a channel that is largely deterministic, stochastic 

modeling presents something of a conceptual challenge. Yet it is common for communication 

engineers to employ stochastic models, and their use does increase the realism (accuracy) of the 

resulting model outputs. Thus we have proceeded to develop statistical models (in addition to that 

embodied in the path loss models of (1)-(3)) to fully account for the “random” elements in the 

channel. When employed with the CE2R, this will result in a “mixed” model that has both 

deterministic and stochastic components. 

One of the primary random effects is small-scale fading, attributed mostly to multipath 

propagation. Physically, in the over-water channels, our intermittent multipath is combined with 

two-ray effects, which are primarily the phase variation of the surface reflection. Larger scale effects 

such as airframe shadowing generally vary much more slowly, or equivalently, over larger 

distances—and these are characterized separately. Small-scale fading is characterized by specifying 

the distribution of amplitude (sometimes also phase), after removal of large-scale effects such as 

path loss and shadowing. Small-scale effects must also be considered when evaluating diversity, in 

our case in both the spatial and temporal domains. Thus we have developed procedures for 

estimating small-scale fading parameters. 

In order to estimate statistical parameters for more detailed channel characterization, it is 

necessary to determine the region of space over which the channel statistics can be assumed 

constant1. We quantify this region as the stationarity distance (SD). (Note strictly that the region 

should be a volume of space, but for our AG application, the simpler linear distance should suffice.) 

So-called “rules of thumb” exist for these spatial extents or SDs, but these are based on terrestrial 

propagation in non-line-of-sight (NLOS) conditions. For example, the rule devised in [22], which 

yields a stationarity distance of 20O -40O, has been extensively used for terrestrial cellular radio 

channels. Also underlying the derivation in [22] is the assumption of Rayleigh amplitude statistics. 

Neither the NLOS or Rayleigh conditions apply in our over-water settings, so we should not employ 

this value of SD in the LOS AG channel. 

Before discussing computation of the SD, we briefly describe the channel variation in a tworay case. Consider the (ideal) two-ray model for a straight flight path. Here, the PDP consists of two 

components, the LOS, and the surface reflection. Since absolute delay does not affect the shape of the 

PDP or any statistics of interest, we can fix (shift) the delay of the LOS component to zero. Then for a 

flight away from the GS, the PDP varies as follows: both the LOS and surface reflection amplitudes 

decrease with distance (~1/d), and the relative delay of the surface reflection also decreases. The 

opposite variation occurs for a flight toward the GS. This is illustrated in Figure 6. 

 

1 In general a balance must be struck: it is desirable to have as long a data record as possible in order to obtain reliable 

statistics, yet the data record should not be long enough to span a period (or distance) over which the statistics change.

NASA/CR—2015-218486 8
Figure 6. Magnitude of two-ray CIR (or, PDP) for two different distances: left, at time t1 (distance d1); right, at time t2 >t1

(distance d2>d1). 

Thus in the ideal two-ray case, the PDP is changing continuously. Yet for times (or distances) 

closely separated, the PDP does not change substantially, and in fact changes very slowly as distance 

increases. As noted though, we do have additional intermittent MPCs that change the PDPs, as well 

as the change to the surface reflection from ocean surface dynamics, thus the channel will have a 

quantifiable stationarity distance. 

The stationarity distance can be computed in several ways. For our wideband (50 MHz Cband) measurements, we have chosen to use the method in [23], which essentially computes a 

temporal correlation coefficient for the temporally varying PDP. To begin, the CIR for the ith time 

instant is given by 

݄ሺ߬ǡ ݐ௜ሻ ൌ σ ߙ௞ǡ௜݁௝థೖǡ೔ߜሺ߬ െ ߬௞ǡ௜ሻ ௅೔

௞ୀଵ (4) 

where Dk,i represents the kth MPC amplitude at time instant ti, Ik,i the kth MPC’s phase, and Wk,i the 

kth MPC’s delay. The ith (instantaneous) PDP is then 

ܲሺ߬ǡ ݐ௜ሻ ൌ σ ൫ߙ௞ǡ௜൯

ଶ

ߜሺ߬ െ ߬௞ǡ௜ሻ ௅೔

௞ୀଵ . (5) 

The method for computing the PDP temporal correlation coefficient in [23] begins by averaging N

PDPs of the form of (5). The averaged PDP is denoted Pavg,N(W,ti), 

ܲ௔௩௚ǡேሺ߬ǡ ݐ௜ሻ ൌ ଵ

ே σ ܲሺ߬ǡ ݐ௜ሻ ௜ାேିଵ

௜ (6) 

with the first PDP in the average the ith. The temporal correlation is easily translated to a spatial 

correlation via knowledge of platform velocity. The averaging of the instantaneous PDPs is done to 

smooth somewhat the effects of small scale fading and remove any equipment-related variation, so N

should be large enough to do this, without being so large as to include PDPs that are affected by 

larger scale effects, or PDPs that are—in effect—outside the region of stationarity. Note that this 

selection involves some assumption regarding the very parameter (stationarity distance) we are 

trying to estimate! Hence we rely on engineering judgment, including knowledge of our equipment’s 

variation rate, to select N.

From [23], using our average PDP Pavg,N(W,ti), we compute a temporal correlation coefficient 

c('t,ti) as follows: 

ܿሺοݐǡ ݐ௜ሻ ൌ ׬௉ ೌೡ೒ǡಿሺఛǡ௧೔ሻ௉ೌೡ೒ǡಿሺఛǡ௧೔ାο௧ሻௗఛ

௠௔௫ቄ׬ൣ௉ೌೡ೒ǡಿሺఛǡ௧೔ሻ൧మ

ௗఛǡ׬ൣ௉ೌೡ೒ǡಿሺఛǡ௧೔ାο௧ሻ൧మௗఛቅ. (7)

NASA/CR—2015-218486 9
This metric quantifies how similar the average PDP at time ti is to the average PDP at time ti+'t; 

naturally this is also a function of the starting time ti (or location, or distance di). The denominator of 

(7) simply normalizes c('t,ti) to a maximum value of unity. The coefficient c('t,ti) is also not 

completely equivalent to the usual correlation coefficient definition, since c('t,ti) cannot be negative. 

We compute (7) for a range of 't, for each value of ti, and select as the stationarity distance (SD) the 

value of distance 'x such that c('t,ti)>0.9, with the translation 'x=v't, with v=velocity. The value 0.9 

is conservative, i.e., with very high probability, the channel is stationary during this time (over this 

distance). For simplicity we apply these computations to our straight flight tracks. 

For the over-sea straight FT1, the aircraft flew straight toward the GS. Average aircraft 

velocity was 92 m/s, and the PDP update rate was approximately 2900 Hz, for all four receivers. 

Each receiver recorded more than 670,000 PDPs in this FT. Link distances ranged from 3 to 24.15 

km in C-band and from 2.25 to 24.15 km in L-band2. Due to periodic variation of our C-band received 

power, caused by a slow relative drift of the sampling clocks in the Tx and Rx [3], the averaging 

distance we selected (corresponding to N in (6)) is 200λ. 

A selected plot of values of c('t,ti) (actually, c('x,di)) versus link distance and ∆x is shown in 

Figure 7. The quasi-periodicity one can observe is still under investigation, but at present we are 

fairly confident that this is also due to the sampling clock drift. The value of 'x in Figure 7 for which 

c('x,di)>0.9 is approximately 6-7 m. To determine an SD we can apply to all data in this 

environment, we collect statistics of c('x,di) over all our data. These statistics appear in Table 4, and 

Figure 8 shows cumulative distribution functions (cdfs) for SD for FT1. The SD follows a lognormal 

distribution fairly well. For computation of all channel statistics, the median value of SD is 

employed: ~250λ in C-band. This is substantially larger than the 20-40O widely used for cellular 

channels, but this is as expected in this LOS channel. We have also found the 250λ value of SD to 

hold for the over-freshwater flight tracks. 

Note that we did not compute an SD for L-band. This is because the PDP in the over-water 

channels has multipath components (including the main surface reflection) that are almost always 

unresolvable, hence the coefficient c('x,ti) would nearly always be one. Based upon the physical 

environment, we expect that the SD for L-band (with physically longer wavelength) will not be 

smaller than the SD for C-band, hence our conservative estimate should apply to the L-band as well. 

Worth pointing out is that our conservative choice for SD (c('x,di)>0.9) is practical in the sense that 

even if the actual SD is larger than our chosen value, computing statistics over intervals shorter 

than the true SD still yields valid statistics. Additional results for stationarity distance appear in 

Appendix B. 

Figure 8. Contour of C-band PDP correlation coefficient c('x,d) vs. link distance d and ∆x for segment of FT1. 

 

2 Different minimum link distances result from slightly different GS antenna patterns for the two bands. 

NASA/CR—2015-218486 10
Table 4. C-band stationarity distance statistics, over-sea FT1. 

Rx1 Rx2

Lognormal μ (m) 2.78 2.66 

σ (m) 0.86 0.79 

Mean 

(m) 23.5 19.0 

(λ’s) 396.6 321.2 

Median (m) 15.2 14.4 

(λ’s) 256.1 242.0 

10th

percentile 

(m) 5.5 5.2 

(λ’s) 92.9 87.3 

Min 

(m) 0.97 0.91 

(‘λs) 16.4 15.4 

Figure 8. SD CDFs for straight flight track (FT1). 

4.3 Correlation Analysis 

With our estimate of SD, we can proceed to estimate statistical channel parameters of 

interest. One of these is the correlation between received signal components. The components in 

question are in general all received signal components, on all four antennas, in both bands. For 

channels with more multipath components (in addition to the surface reflection), correlations among 

the MPCs are also of interest. In our over-water channels, we are primarily interested in the 

correlations of the main (LOS) components, since the vast majority of received power is contained 

within the LOS components. 

In [6] we presented preliminary analysis of the correlations among LOS components for the 

four antennas for the over-sea case. Specifically, for the correlation between the two C-band 

antennas, the amplitude correlation is computed by 

ߩ಴஺భǡ஺಴మ ൌ ாቂቀ஺಴భିఓಲ಴భቁቀ஺಴మିఓಲ಴మቁቃ

ఙಲ಴భఙಲ಴మ

 (8) 

where E denotes expectation, the A’s in the numerator are sample amplitude vectors, and the P’s are 

the means of the A vectors. Specifically, ܣ஼ଵ ൌ ሺܣ஼ଵǡଵǡ ܣ஼ଵǡଶǡǥǡܣ஼ଵǡ௜ǡǥǡܣ஼ଵǡ௡ሻ is the vector of LOS 

10 20 30 40 50 60 70

0

0.2

0.4

0.6

0.8

1

'x (m)

CDF

OxnardCA***06-11-2013***FT1

C-band Rx1

C-band Rx2

NASA/CR—2015-218486 11
amplitude samples for C-band Rx1; analogously, ܣ஼ଶ ൌ ൫ܣ஼ଶǡଵǡ ܣ஼ଶǡଶǡǥǡܣ஼ଶǡ௜ǡǥǡܣ஼ଶǡ௡൯ is the vector of 

Rx2 amplitude samples for its LOS component. Both vectors span only the SD. The V’s are the 

standard deviations of the respective sample vectors. Element ACk,i is the ith amplitude sample (after 

time alignment, from the ith PDP in a 1 ms delay bin) for Rxm (m=1 or 2), and vector length n

corresponds to the SD. 

In order to gain insight into what we might expect for the cross correlation among signal 

components of the AG channel, and to check measurement results, we have developed an analysis for 

the theoretical two-ray channel, specifically the CE2R. We apply our analysis to the over freshwater 

case here. Flight track and measurement details for this case were provided in Section 3, and in this 

analysis we select parameters close to those of the measurements, e.g., we use an altitude difference 

of 566.3 m between the GS antenna and aircraft, approximating the value in the over freshwater 

tests. For the CE2R model, we also require the relative dielectric constant (permittivity) H =81, and 

the conductivity Vfw =0.01 S/m of freshwater (the relative permeability Pr =1) [24]. Also required for 

our over-water CE2R is average wind speed, applied to compute the water surface reflection 

coefficient according to the Miller-Brown surface roughness model [25]. The average wind speed for 

Lake Erie on 22 October 2013 was 11.16 mph [26]. Finally, the atmospheric refractivity is accounted 

for via the modified earth radius approach, i.e., the modified earth radius is given by ka, where 

k=1/[1+(a/n0)dn/dh], with a the earth radius a#6380 km, n0 the index of refraction at the surface 

level, and dn/dh the refractivity gradient with respect to altitude h. Typically k=4/3 is used but one 

can use a more accurate value if n0 and dn/dh are available. We used n0=1.000315, and the common 

exponential form for refractivity gradient (dn/dh)=315(-0.136)exp(-0.136h), with h in km. 

The horizontal distance between aircraft Rx1 and GS is denoted dk1, and for our analysis this 

ranges from 10-5000 m, with an increment of 0.1 m. The link distance, denoted Rk1, ranges from 

566.4-5032.1 m, as computed by the CE2R model equations (15)-(20) in [2]. The link distance for Rx2 

is dk1+∆d, where ∆d is the relative distance between the two intra-band antennas. Two ∆d values of 

0.4 m and 1.84 m are used in this analysis, for illustration (actual maximum antenna separation is 

'd~1.3 m). 

Note that the earth surface reflection is always unresolvable at L-band, and is only 

resolvable within the 20 ns C-band delay resolution for short link distances up to approximately 5 

km (analytically of course, we can still compute for both bands for any values of distance). Thus the 

first five kilometers are of most interest for comparison with measurements. For this analytical 

flight path, the elevation angle ranges from 89.7 to 6.5 degrees. 

We derive the amplitude values required to compute (8) via the CE2R path loss, PLCE2R, used 

in the following link budget equation, 

ܲ௥ ൌ ܲ௧ ൅ ܩ ൅ ஺௉ܩ௅ே஺ ൅ ܩ௧ ൅ ܩ ௥െ ܮ ஼െ ܲܮ஼ாଶோǡ (9) 

where Pt is 40 dBm (10 Watts), GPA is 7 dB and pertains only to C-band, GLNA is 30 dB in C-band and 

15.5 dB in L-band, Gt is 6 dB in C-band and 5 dB in L-band, Gr is 5 dB for both bands, and LC is 7.5 

dB in C-band and 4 dB in L-band. We then obtain the (linear-scale) amplitudes we need for (8) via 

ൌ൅ටͳͲౌ౨

భబିଷ. (10) 

 Figure 9 shows the analytical CE2R path loss difference between the two C-band receivers 

for the two values of antenna separation 'd. As can clearly be seen, the path loss difference takes its 

largest values at small link distances, and the antenna separation 'd does affect the magnitude and 

range of distances where the path loss difference is greater than zero. 

NASA/CR—2015-218486 12
(a) 

(b) 

Figure 9. C-band path loss difference vs. link distance between two antennas, (a) ∆d=0.4; (b) ∆d=1.84. 

The analytical cross correlation vs. distance for the CE two-ray model is shown in Figures 10 

and 11. The correlation is close to one at large values of distance, and varies in an oscillatory fashion 

between one and negative one at short distances. This oscillation is a direct manifestation of the tworay effect. Note also that the correlations depend upon the window length, which takes several 

values in Figures 10 and 11. Detailed behavior also depends somewhat on the distance increment for 

which the figures are plotted, or the graph’s resolution; here this is 0.1 m. Evaluation at smaller 

distance increments did not change the variation beyond that seen in Figure 12B. For our over water 

flight velocities and PDP update rates, the SD of 15 m corresponds to n~144, thus the results in 

Figures 10 and 11 for n=100 correspond most closely to what we should use to compare with 

measurements. As shown in Figure 12 (taken from [6]), correlations computed using measured data 

do exhibit these oscillations, where the negative-valued correlations are seen at values of link 

distance less than approximately 5 km. Analytical L-band results are similar to those for C-band, 

since the only things that change are surface reflection coefficients at the different frequency. Thus, 

our measurement results for correlation are supported by this analysis of the theoretical CE2R 

channel. Interestingly, even at large link distances, the correlation coefficients exhibit periodic 

variation, and this period increases with link distance. The extent of the variation of the correlation 

coefficient decreases with vector length: this is illustrated in Figure 12B. Thus when two-ray effects 

dominate, the intra-band correlation coefficients can be expected to vary significantly over all values 

of link distance, particularly for small window lengths (or assumed small SDs). 

NASA/CR—2015-218486 13
(a) 

(b) 

(c) 

Figure 10. C-band correlation coefficient vs. link distance computed with five vector lengths n=[25, 100, 250, 500, 2000], (a) 

∆d=0.4; (b) ∆d=1.84; (c) vector length fixed at n=250, comparison between U’s for two different ∆d’s. 

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1

-0.5

0

0.5

1

Link Distance (km)

Correlation Coefficient

25

100

250

500

2000

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Link Distance (km)

Correlation Coefficient

25

100

250

500

2000

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Link Distance (km)

Correlation Coefficient

Length 250, deltad 184cm

Length 250, deltad 40 cm

NASA/CR—2015-218486 14
Figure 11. C-band correlation coefficient vs. link distance with five vector lengths n=[25, 100, 250, 500, 2000] (enlarged 

version of Figure 10(a)), ∆d=0.4. 

Figure 12. Measured amplitude correlation coefficient between C-band Rx1 and 2 for over-sea FT1, for three different vector 

lengths, n=[500, 1000, 5000]; (Figure 10(a) from [6]). 

Figure 12B. Analytical CE2R C-band correlation coefficients vs. link distance with antenna separation 2 m, and five vector 

lengths n=[25, 100, 250, 500, 2000]. 

4.4 Preliminary Over-Freshwater Results 

For the over-freshwater flight tests, measurements were described in Section 3, and final 

path loss models were described in Section 4.1. Here we provide some analysis of preliminary 

results, specifically for one flight track: FT2. Results for other FTs appear in Appendix A. Although 

we do have final path loss models ((1)-(3)) for the over-freshwater setting, we term the results here 

“preliminary” because we do not yet have complete wideband models for the tapped-delay line overfreshwater setting, nor have we completed all computations of correlations. We also do not have a 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -1

-0.5

0

0.5

1

Link Distance (km)

Correlation Coefficient

25

100

250

500

2000

0.5 1 1.5 2

x 104

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Link Distance (m)

Correlation Coefficient

OxnardCA***06-11-2013***Track1***C1C2

500

1000

5000

10 20 30 40 50 60 70 80 90 100 -1

-0.5

0

0.5

1

Link Distance (km)

Correlation Coefficient

25

100

250

500

2000

NASA/CR—2015-218486 15
final implementation of the Ricean K-factor that varies with link distance d (although we do have 

the functional forms for K(d)—see subsequent discussion in this section and results in Appendix A). 

Figure 13 shows geometric traces for multiple flight parameters for the over-freshwater FT2. 

This includes the flight path in ECEF coordinates, flight velocity, altitude difference between Tx and 

Rx, azimuth angle from GS to aircraft, elevation angle from GS to aircraft, aircraft heading (relative 

to geographic north), and aircraft pitch and roll angles. The latter two parameters are of use in 

deducing airframe shadowing events. 

(a) (b) 

(c) (d) 

6.84

6.85

6.86

6.87

6.88

x 105

-4.735

-4.73

-4.725

-4.72

-4.715

x 106

4.2

4.205

4.21

4.215

4.22

4.225

4.23

x 106

X

ClevelandOH***10-22-2013***FT2

Y

Z

Tx (Ground)

Rx (Aircraft)

Rx start point

L-band Rx end point

C-band Rx end point

5 10 15 20 25 68

70

72

74

76

78

80

Link Distance (km)

Aircraft Velocity (m/s)

ClevelandOH***10-22-2013***FT2

Whole Track

Rx start point

L-band Rx end point

C-band Rx end point

5 10 15 20 25 545

550

555

560

565

570

Link Distance (km)

Relative height between Tx and Rx (m)

ClevelandOH***10-22-2013***FT2

Whole Track

Rx start point

L-band Rx end point

C-band Rx end point

5 10 15 20 25 0

50

100

150

200

250

300

350

Link Distance (km)

Azimuth Angle (degree)

ClevelandOH***10-22-2013***FT2

Whole Track

Rx start point

C-band Rx end point

Rx end point

NASA/CR—2015-218486 16
(e) (f) 

(g) (h) 

Figure 13. Geometric traces for FT2: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude difference 

between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of aircraft; 

(h) roll angle of aircraft. 

 Path loss vs. distance for all four receivers is shown in Figure 14. As with the over-sea case, 

path loss generally follows the free space path loss line, with variation due to two-ray effects, and 

increased attenuation at short link distances (~ < 5 km) due to aircraft antenna effects. Just as seen 

for the over-sea results, in the freshwater case the CE2R model is also better for L-band than for Cband, due to the larger reflection coefficient and the relatively smoother water surface at L-band. 

Linear fit path loss model lines for this specific FT are also shown in the figure. Note that the path 

loss models reported in Section 4.1 pertain to the overall over-freshwater case, or in other words, the 

path loss models of (1)-(3) are aggregate models, whereas results here, and those reported in 

Appendix A, are “per-FT” models. The aggregate models of (1)-(3) are recommended for general use. 

The “per-FT” models are provided to illustrate some of the variation seen among the various FTs. 

Table 5 provides the “per-FT2” path loss model parameters. 

5 10 15 20 25 0

5

10

15

20

25

30

35

40

Link Distance (km)

Elevation Angle (degree)

ClevelandOH***10-22-2013***FT2

Curved Earth

Flat Earth

ENU

Rx start point

L-band Rx end point

C-band Rx end point

5 10 15 20 25 -200

-150

-100

-50

0

50

100

150

200

Link Distance (km)

True Heading (degree)

ClevelandOH***10-22-2013***FT2

Whole Track

Rx start point

L-band Rx end point

Rx end point

5 10 15 20 25 6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

Link Distance (km)

Pitch (degree)

ClevelandOH***10-22-2013***FT2

Whole Track

Rx start point

L-band Rx end point

C-band Rx end point

5 10 15 20 25 -8

-6

-4

-2

0

2

4

6

8

Link Distance (km)

Roll (degree)

ClevelandOH***10-22-2013***FT2

Whole Track

Rx start point

L-band Rx end point

C-band Rx end point

NASA/CR—2015-218486 17
 

(a) (b) 

(c) (d) 

Figure 14. Measured path loss vs. distance for (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table 5. Path loss model parameters for over-freshwater FT2, straight flight toward GS. 

Receiver

d<dt (θ>θt) d>dt (θ<θt)

Log-distance 

linear PL model

Elevation 

angle range 

(degree)

Distance 

range (km)

Log-distance 

linear PL model CE2R (dB)

Elevation 

angle range 

(degree)

Distance 

range (km)

A0,S

(dB) nS

σXL,S

(dB) θmin,S θmax,S dmin,S dmax,S

A0,L

(dB) nL

σXL,L

(dB) L0 σX2 θmin,L θmax,L dmin,L dmax,L

C-Rx1 47.8 2.0 1.4 5.0 12.7 2.5 6.6 52.1 1.9 2.4 1.0 3.9 1.4 5.0 6.6 29.4

C-Rx2 47.9 2.0 1.1 5.0 12.7 2.5 6.6 51.4 1.9 2.3 0.4 3.8 1.4 5.0 6.6 29.3

L-Rx1 56.1 1.5 2.3 5.0 35.5 1.0 6.6 49.9 1.7 3.3 4.1 1.8 1.4 5.0 6.6 29.4

L-Rx2 64.6 1.2 2.2 5.0 35.5 1.0 6.6 15.4 2.4 3.7 1.2 1.8 1.4 5.0 6.6 29.3

 In Figure 15 we show plots of root-mean square delay spread (RMS-DS) vs. link distance for 

the two C-band receivers. These plots are “post-processing,” as we describe in the following 

paragraph. The RMS-DS is denoted VW and is given by 

ɐத ൌ ටσ ஑ౡ

మதౡ ైషభ మ ౡసబ

σ ஑ౡ ైషభ మ ౡసబ

െ Ɋத

ଶ, (11) 

where as in (4), αk is the kth MPC amplitude out of a total of L MPCs in a PDP, τk is the delay of the 

kth MPC, and μk is the mean energy delay, calculated via 

104 110

115

120

125

130

135

140

145

150

Link Distance (m)

Path Loss (dB)

ClevelandOH***10-22-2013***FT2***C-band Rx1

Measured PL

Free Space PL

Fit [n=2 ,V=1.37]

Fit [n=1.89 ,V=2.44]

2-Ray Curved Earth

104 110

115

120

125

130

135

140

145

150

Link Distance (m)

Path Loss (dB)

ClevelandOH***10-22-2013***FT2***C-band Rx2

Measured PL

Free Space PL

Fit [n=2.01 ,V=1.05]

Fit [n=1.9 ,V=2.32]

2-Ray Curved Earth

103104 80

90

100

110

120

130

140

150

Link Distance (m)

Path Loss (dB)

ClevelandOH***10-22-2013***FT2***L-band Rx1

Measured PL

Free Space PL

Fit [n=1.52 ,V=2.28]

Fit [n=1.68 ,V=3.27]

2-Ray Curved Earth

103104 80

90

100

110

120

130

140

150

Link Distance (m)

Path Loss (dB)

ClevelandOH***10-22-2013***FT2***L-band Rx2

Measured PL

Free Space PL

Fit [n=1.23 ,V=2.16]

Fit [n=2.43 ,V=3.65]

2-Ray Curved Earth

NASA/CR—2015-218486 18
Ɋத ൌ σ ஑ౡ

మதౡ ైషభ

ౡసబ

σ ஑ౡ ైషభ మ ౡసబ

. (12) 

The denominator sum σ Ƚ୩

୐ିଵ ଶ ୩ୀ଴  in (11) and (12) is the total power in the PDP. This power can be 

either the power for the average PDP for a given set of data, or for a single PDP; for the latter case 

the parameters in (11) and (12) are termed the instantaneous RMS-DS and instantaneous mean 

energy delay, respectively. 

(a) (b) 

Figure 15. RMS-DS vs. distance for C-band (a) Rx1; (b) Rx2, over-freshwater FT2. 

As in the over-sea case, we note the presence of multiple “bumps” in the RMS-DS vs. distance 

plots, where the geometry was such that reflections from objects on the lake shore presented 

multipath components (MPCs) that resulted in a temporary increase in RMS-DS. In fact, the original

RMS-DS vs. distance plots contained many more bumps of much larger value, due to strong 

reflections from the large buildings in downtown Cleveland. These large reflections (MPCs) were 

removed via a delay threshold of 1 Ps. 

 A closer look at this is presented in Figure 16 where we show the original RMS-DS vs. 

distance plot before application of the delay threshold. The more numerous and larger RMS-DS 

bumps are clearly evident. Figure 17 shows a decimated sequence of PDPs for this FT both before 

and after thresholding. Note that the two figures have different delay scales, but the distant 

reflections from downtown Cleveland have been removed by thresholding in Fig. 17(b). The original 

(before-thresholding) PDPs will be of use in developing the urban GS setting channel models. 

Figure 16. C-band Rx1 RMS-DS vs. distance for FT2, before thresholding. 

5 10 15 20 25 0

10

20

30

40

50

60

70

80

Link Distance (km)

RMS-DS (ns)

ClevelandOH***10-22-2013***FT2***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

5 10 15 20 25 0

10

20

30

40

50

60

70

Link Distance (km)

RMS-DS (ns)

ClevelandOH***10-22-2013***FT2***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 19
(a) (b) 

Figure 17. Sequence of C-band Rx1 PDPs for FT2, (a) before thresholding, with distant reflections (MPCs from large buildings 

in downtown Cleveland) indicated, and (b) after thresholding. Note two different delay ranges. 

 Figure 18 shows a Google Maps® view in which we have identified the potential short-delay 

“near-shore” reflecting objects along the lake shore. Several large buildings, three harbors with 

watercraft, water barriers, light houses and oil tanks are present. The geometry indicates that these 

obstacles are likely responsible for the MPCs in Figure 17(b). 

Figure 18. Google Maps® view illustrating region of potential near-shore reflectors. 

Figure 19 shows histograms of RMS-DS for FT2, both for the original thresholded data set, 

and a moving-average filtered version of the thresholded instantaneous RMS-DS sequences. The 

filtering naturally reduces the maximum values, but does not change the means. Although the 

distribution may not be strictly “heavy-tailed” (hyperbolic), it clearly indicates that a small 

percentage of PDPs have RMS-DS values well above the mean. Table 6 quantifies the RMS-DS 

statistics. 

NASA/CR—2015-218486 20
Figure 19. Histogram of RMS-DS for FT2 (straight) after thresholding, C-band Rx 1. 

Table 6. Over-freshwater RMS-DS statistics. 

RMS-DS (ns) Mean Max

FT2 

Rx1

Original 

9.7

73.3 

Moving Averaged Over 100 16.2 

Moving Averaged Over 1000 14.0 

Rx2

Original 

9.9

62.2 

Moving Averaged Over 100 17.7 

Moving Averaged Over 1000 15.3 

FT4 

Rx1

Original 

9.9

161.0 

Moving Averaged Over 100 119.3 

Moving Averaged Over 1000 77.3 

Rx2

Original 

10.0

135.8 

Moving Averaged Over 100 116.6 

Moving Averaged Over 1000 96.0 

 In Figure 20 we show intra-band correlation coefficient results vs. link distance for both Cband and L-band. The C-band results do not strongly resemble those described for the analytical 

CE2R. This is in part a manifestation of the rapidly-oscillating two-ray effect (and surface reflection 

variations), but is actually dominated here by equipment stability limitations. Specifically, for some 

FTs the channel sounder exhibited greater variation (sample clock drift) than others, and this causes 

the unusual correlation coefficient results of Figure 20. In the results of Figure 20, we actually used 

a window (vector) length of twice the SD in order to reduce the clock drift effect. The mean value of 

C-band correlation is 0.5, and the mean value of L-band correlation is 0.66. We are currently working 

on an algorithm to remove the sample clock drift effects from the data so that accurate correlation 

results can be obtained for the data from all FTs. As with the over-sea results, inter-band 

correlations are essentially zero. 

0 10 20 30 40 50 60 70 100

102

104

106 ClevelandOH***10-22-2013***FT2***C-band Rx1Original RMS-DS

0 10 20 30 40 50 60 70 100

102

104

106

Density

Moving Averaged over 100 PDPs

NASA/CR—2015-218486 21
(a) (b) 

Figure 20. Measured amplitude correlation coefficient (a) between C-band Rx1 and 2; (b) between L-band Rx1 and 2 for 

FT2. 

 Our final results in this over-freshwater section pertain to small-scale fading, specifically the 

distribution of amplitude fading about the mean (where the mean is given by the path loss models). 

As is common in such LOS channels, we selected the Ricean distribution as the appropriate one to 

quantify this fading. This is fully described via the Ricean K-factor—see Appendix C for a complete 

description of the Ricean density function, and estimation methods for K. Using our stationarity 

distance, we estimated the Ricean K-factor for all four receivers. Plots of K vs. link distance are 

shown in Figure 21. In general, K increases very slightly with distance for FT2, although for other 

FTs, K is more nearly constant, or may decrease slightly (see Appendix A). As with the “per-FT2” 

path loss models, the K-factors here are “per-FT2,” and an aggregate model for K vs. distance will be 

developed in the future (for over-sea as well as over-freshwater). The L-band K is also smaller than 

that for C-band, and in addition shows dependence on two-ray effects; as with path loss, this is due 

to the stronger surface reflection at L-band. Nonetheless, the over-freshwater AG channel has 

relatively large K-factors of ~12 dB for L-band, and ~25 dB for C-band, illustrating the weakness of 

true small-scale fading (aside from the CE2R effects). Table 8 lists the “per-FT2” Ricean K-factor 

statistics. The K-factor estimated via the maximum-likelihood method, KML, is the most accurate. 

(a) (b) 

NASA/CR—2015-218486 22
(c) (d) 

Figure 21. Ricean K factor for over-freshwater FT2 (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table 7. Statistics of over-freshwater Ricean K factor, FT2, Cleveland, OH, 22 October 2013, straight flight toward GS. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 26.3 26.3 13.6 13.6

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 25.1 23.1 25.1 25.3 23.3 25.3 13.0 12.2 12.8 13.2 12.3 13.0

n 0.12 0.09 0.12 0.10 0.07 0.10 0.06 0.11 0.07 0.05 0.09 0.06

σX (dB) 1.5 2.9 1.5 1.4 2.9 1.4 1.3 1.5 1.2 1.3 1.5 1.1

Statistics of K factor (dB)

Max 33.0 32.9 33.0 32.6 32.5 32.6 16.5 16.6 16.5 16.5 16.5 16.6

Min 13.6 11.1 13.9 13.4 11.1 13.9 4.9 -3.5 4.0 4.2 0.0 3.7

Median 27.0 24.5 27.0 26.8 24.4 26.9 13.9 13.7 13.9 13.9 13.7 14.0

μ 27.3 25.0 27.3 27.0 24.8 27.1 14.0 13.9 14.0 14.1 14.0 14.1

σ 1.8 3.0 1.7 1.6 2.9 1.6 1.4 1.8 1.3 1.3 1.7 1.2

4.5 The Over-Harbor “Backlobe” AG Channel

We first described some of these results in [6], which were obtained during an over-sea flight 

from Oxnard, CA in June 2013. Here we provide more detailed analysis of this atypical case, some of 

which will appear in [11]. 

Figure 22 shows the portion of the flight track in question, in both ECEF coordinates (a) and 

in a Google Maps® view (b). Here the aircraft flew over a harbor. For the portion of the flight track 

shown in green (C-band) and brown (L-band) in Figure 22(a), the receivers were not in the GS 

antennas’ main beams, i.e., they were in the GS antennas’ “backlobe” regions. These backlobe areas 

and flight regions at large elevation angles are often neglected by researchers and system engineers, 

and in fact these areas are usually intentionally avoided by communication system designers. Yet in 

cases where the GS does not have an antenna beam directed upward, the aircraft will be out of the 

antenna main beam at large elevation angles. Similarly, there may be situations such as initial 

deployment, or equipment failure for one GS sector, in which signals between the aircraft and the 

backlobes of the GS antenna are of interest. The results here describe the AG channel in such a case. 

NASA/CR—2015-218486 23
Figure 22. Portion of FT2 with ground site at Oxnard, 11 June 2013. 

In the backlobe region, one would expect poor coverage, possibly severe fading and fairly 

large values of delay spread, and these can dramatically degrade the performance of any CPNC 

system operating in the backlobe area. Backlobe channel characteristics may also be used in 

estimating inter-sector interference in sectored antenna schemes. Despite the dependence of these 

results on our specific antenna patterns, the empirical statistical models provide interesting 

practical results. 

Figure 23 illustrates the situation via a simplified diagram. Region (1) is where the aircraft 

is out of the main beam of the GS antenna where the GS transmitter antenna gain Gt is only very 

approximately known, and Region (2) denotes the region where the aircraft is within the half power 

main beam where Gt is almost a constant. The link distance d between GS and aircraft at the 

starting point is 2163 m. The green and brown lines in Fig. 22 are the backlobe areas of the C- and 

L-band antennas, respectively. The link distance at the boundary of the 3 dB main beam is 1857 m 

in C-band and 1175 m in L-band, due to the slightly different main beam patterns. The minimum 

link distance dmin is 1043 m. The blue line in Fig. 22 is the route for the rest of FT2, within the main 

beam for both bands. In Region 1, the earth surface reflections are on the ground. Additional 

potential reflectors include buildings of no higher than three levels, water vehicles in the harbor, the 

harbor itself, ground vehicles, and the ground itself (areas of sand). The GS antennas were elevated 

20 m above the ground, oriented 283q from geographic north with zero elevation angles. In the 

backlobe area, the altitude difference between the aircraft and the GS ranged from 780.6 to 783.8 m. 

The azimuth angle ranged from -40 to 60 degrees relative to geographic north. The elevation angle 

was between 21.2 and 48.5 degrees. Each of the four receivers recorded approximately 70,000 PDPs 

in Region 1. 

Figure 23. Illustration of main beam and backlobe regions. 

The patterns of the directional Tx (GS) antennas are shown in Figure 24, with main beams 

directed to zero azimuth/elevation angles. The maximum gains are 6 dB in C-band and 5 dB in Lband. The locations of the omni-directional monopole Rx antennas (maximum gain 5 dB) are on the 

-2.586

-2.584

-2.582

-2.58

-2.578

x 106

-4.612

-4.61

-4.608

-4.606

x 106

3.5625

3.563

3.5635

3.564

3.5645

3.565

x 106

X

OxnardCA***06-11-2013***FT2

Y

Z

Tx (Ground)

Rx (Aircraft)

Backlobes C-band

Backlobes L-band

Rx start point

Rx end point

NASA/CR—2015-218486 24
bottom of the S3-B fuselage in a rectangular pattern; the front-to-rear and left-to-right distances 

between the Rx antennas are 1.29 and 1.32 m, respectively. 

(a) (b) 

(c) (d) 

(e) (f) 

Figure 24. GS antenna patterns. (a) C-band azimuth plane; (b) C-band elevation plane; (c) C-band 3D pattern in linear scale; 

(d) L-band 3D pattern in linear scale; (e) L-band azimuth plane; (f) L-band elevation plane

We employed the antenna gain patterns in post processing, according to the 

elevation/azimuth angles and the orientation of the GS antennas with respect to the aircraft. (Note 

again that the patterns of the aircraft Rx antennas were measured on a metal ground plate in an 

anechoic chamber, and not while mounted on the aircraft, and the metallic body of the aircraft has a 

significant effect on these patterns. This introduces additional uncertainties into our measurements.) 

Both Gr and Gt have a resolution of one degree. The backlobes are always narrow, and the depths 

and locations of the nulls are sensitive to both angular resolution and the method by which the 

patterns are estimated. Hence Gr and Gt are only very approximately known at large elevation 

angles and in the backlobe regions. 

In Region 1, the aircraft lies in the backlobes of the GS antennas where it is possible that the 

LOS component is heavily attenuated in a null but some other multipath components (MPCs) are in 

0 90 180 270 360 -10

-5

0

5

10

Azimuth Angle (degree)

C-band GS Antenna Gain (dBi)

Azimuth Plane

90 60 30 0 -30 -60 -90 -50

-40

-30

-20

-10

0

10

Elevation Angle (degree)

C-band GS Antenna Gain (dBi)

Elevation Plane

0 90 180 270 360 -15

-10

-5

0

5

10

Azimuth Angle (degree)

L-band GS Antenna Gain (dBi)

Azimuth Plane

90 60 30 0 -30 -60 -90 -20

-15

-10

-5

0

5

10

Elevation Angle (degree)

L-band GS Antenna Gain (dBi)

Elevation Plane

NASA/CR—2015-218486 25
a relatively stronger backlobe. The power of these MPCs can thus be stronger than in the main 

beam. Compared with the over-sea results in the main beam, results in Region 1 show fast fading 

and large delay spread. 

Propagation path loss in dB vs. link distance for the two C-band receivers is shown in Figure 

25. As indicated in Figure 22, the link distance between the aircraft and GS first decreased and then 

increased (hence in Figure 25 two loss values may occur at a single value of distance). The pink circle 

and black square are the Rx starting point and ending point within the backlobe areas, respectively, 

which correspond to the points shown in Figure 22. Compared to the path loss within the main 

beam, for short distances below approximately 2.5 km (Fig. 25(b)), the path loss in the backlobe 

areas is up to 15 dB larger than the expected free space path loss, and has large variation. The 

standard deviation (VPL) of the measured path loss difference from free space path loss is 6.8 dB for 

Rx1 and 7.3 dB for Rx2. In contrast, in Region 2 VPL is less than 2 dB. 

(a) 

(b) 

Figure 25. Path loss vs. link distance for C-band receivers. (a) Back lobe areas only; (b) including backlobe areas and part of

Region 2 up to 5 km. 

The path loss results for L-band are shown in Figure 26. The path loss in the main beam shows 

clear two-ray behavior, but the path loss in the backlobe area does not, and this is likely due to the 

presence of more MPCs and backlobe attenuation of the LOS component. Here the measured path 

loss is up to 20 dB larger than free space path loss. The VPL of the measured L-band path loss with 

respect to free space path loss is 5.8 dB for Rx1 and 6.4 dB for Rx2. In the main beam Region 2 the 

VPL is approximately 2.3 dB. 

1043 1200 1400 1600 1800 2000 2163 90

100

110

120

130

Link Distance (m)

Path Loss (dB)

OxnardCA***06-11-2013***FT2***C-band Rxs

Measured PL Rx1

Measured PL Rx2

Free Space PL

Rx start point

Rx end point

1500 2000 2500 3000 3500 4000 4500 5000 90

100

110

120

130

Link Distance (m)

Path Loss (dB)

OxnardCA***06-11-2013***FT2***C-band Rxs

Measured PL Rx1

Measured PL Rx2

Free Space PL

Rx start point

Rx end point

NASA/CR—2015-218486 26
(a) 

(b) 

Figure 26. Path loss vs. link distance for L-band receivers. (a) Back lobe areas only; (b) including backlobe areas and part in

Region 2 up to 5 km. 

We also extracted amplitude fading data from the path loss measurements, just as we did for 

the Ricean fading fits in the over-water settings. The fading amplitude distributions are shown in 

Figure 27. The Generalized Extreme Value (GEV), lognormal, Weibull and Rayleigh distributions 

are plotted. The probability density function of the GEV distribution is 

݂ሺݔǢ ߤǡ ߪǡ ߦሻ ൌ ଵ

ఙ ቂͳ ൅ ߦ ቀ௫ିఓఙ ቁቃሺିଵȀకሻିଵ ݁݌ݔ൜ െ ቂͳ ൅ ߦ ቀ௫ିఓఙ ቁቃሺିଵȀకሻ

ൠ, (13) 

where ξ is the shape factor, P is a location parameter, and V the scale factor. The parameters of the 

four distributions are listed in Table 8. The Weibull and Rayleigh distributions do not fit the data. 

Due to the complexity of the backlobe patterns combined with multipath, the distributions show two 

or three modes, and often fading is severe, or worse than Rayleigh [27].

1043 1200 1400 1600 1800 2000 2166 80

90

100

110

120

Link Distance (m)

Path Loss (dB)

OxnardCA***06-11-2013***FT2***L-band Rxs

Measured PL Rx1

Measured PL Rx2

Free Space PL

Rx start point

Rx end point

1500 2000 2500 3000 3500 4000 4500 5000 80

90

100

110

120

Link Distance (m)

Path Loss (dB)

OxnardCA***06-11-2013***FT2***L-band Rxs

Measured PL Rx1

Measured PL Rx2

Free Space PL

Rx start point

Rx end point

NASA/CR—2015-218486 27
(a) (b) 

(c) (d) 

Figure 27. Histogram of received signal envelope amplitude for (a) C-band Rx1; (b) C-band Rx 2; (c) L-band Rx1; (d) Lband Rx2. 

Table 8. Distribution parameters for backlobe amplitude fading. 

C-band L-band 

Rx1 Rx2 Rx1 Rx2 

Generalized 

Extreme Value

ξ 0.43 0.59 0.58 0.62 

μ 0.02 0.02 0.02 0.02 

σ 0.02 0.02 0.01 0.01 

Lognormal μ -3.45 -3.45 -3.45 -3.45 

σ 0.78 0.84 0.66 0.74 

Weibull β 1.44 1.38 1.69 1.41 

Rayleigh σ 0.04 0.04 0.03 0.04 

The C-band RMS-DS along with its moving averaged version over 1000 PDPs is shown vs. 

distance in Figure 28. The RMS-DS in the main beam is approximately 10 ns with some “bumps” no 

larger than 50 ns as seen in Figure 28(b).3 The RMS-DS in the backlobe area has mean 35 ns, 

maximum 220 ns for Rx1 and mean of 37 ns, maximum of 391 ns for Rx2, i.e., RMS-DS is 

substantially larger in the backlobe region than in the main beam. 

 

3Due to the present of filters in the channel sounder, the minimum RMS-DS that can be measured (through a back-to-back 

connection) is 10 ns in C-band. 

0.05 0.1 0.15 0

5

10

15

20

25

30

Amplitude

Density

Amplitude

GEV

Lognormal

Weibull

Rayleigh

0.05 0.1 0.15 0

10

20

30

40

Amplitude

Density

Amplitude

GEV

Lognormal

Weibull

Rayleigh

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0

10

20

30

40

Amplitude

Density

Amplitude

GEV

Lognormal

Weibull

Rayleigh

0.05 0.1 0.15 0.2 0

10

20

30

40

Amplitude

Density

Amplitude

GEV

Lognormal

Weibull

Rayleigh

NASA/CR—2015-218486 28
(a) 

(b) 

Figure 28. RMS-DS of C-band receivers vs. link distance, (a) backlobe areas only; (b) including backlobe areas and part in 

region 2 up to 5 km. 

Figure 29(a) is the sequence of C-band PDPs for Rx1 from the starting point to the minimum 

link distance dmin. Figure 29(b) shows the sequence of PDPs from dmin to approximately 2.5 km. In 

both plots, the flight path is along the link distance axis from upper right to lower left. The RMS-DS 

decreases substantially after the half power main beam boundary at approximately 1857 m, as 

indicated by the white line at the base of the plot in Fig. 29(b). Many MPCs are present in the 

backlobe area, most within 1000 ns delay, but a few MPCs have delay up to 1820 ns. The power of 

the strongest (LOS) component increases from the range -40 to -62 dBm up to -31 dBm when the 

aircraft goes from the backlobe area to the main beam. 

1043 1200 1400 1600 1800 2000 2163 0

100

200

300

400

Link Distance (m)

RMS-DS (ns)

OxnardCA***06-11-2013***FT2***C-band Rxs

Measured Rx1

Measured Rx2

Moving Averaged Rx1

Moving Averaged Rx2

Rx start point

Rx end point

1500 2000 2500 3000 3500 4000 4500 5000 0

100

200

300

400

Link Distance (m)

RMS-DS (ns)

OxnardCA***06-11-2013***FT2***C-band Rxs

Measured Rx1

Measured Rx2

Moving Averaged Rx1

Moving Averaged Rx2

Rx start point

Rx end point

NASA/CR—2015-218486 29
(a) 

(b) 

Figure 29. Sequence of PDPs for C-band Rx1 (a) from start point to dmin; (b) from dmin to 2.5 km (inlcuding part of Region 

1 & 2). Flight path in both figures is along link distance axis from upper right toward lower left. 

An example individual PDP is shown in Figure 30; the link distance of this 17150th PDP is 

1287.8 m, which lies in the segment before dmin (Fig. 29(a)). The strongest MPC (likely the LOS) is 

shifted to a delay value of 100 ns. MPCs are present with a maximum excess delay of 540 ns, with 

MPC power from 10 to 30 dB smaller than that of the LOS. The instantaneous RMS-DS of this PDP 

is 61.6 ns. 

NASA/CR—2015-218486 30
Figure 30. Example individual C-band PDP, 17150th PDP at d=1287.8 m. 

In summary, for this “backlobe” AG channel when the aircraft is outside the half power main 

beam of the GS antennas, path loss is up to 15 dB larger than free space path loss in C-band and up 

to 20 dB larger than free space path loss in L-band. Path loss variation is also much larger and more 

rapid than that within the main beam. The received signal amplitude fading is fast and severe, or 

worse than Rayleigh. The C-band RMS-DS is up to 400 ns with mean of 35 ns, whereas the RMS-DS 

in the main beam (for the flight over the Pacific Ocean) is essentially near the measurement lower 

limit of 10 ns with a few values near 50 ns and very rare larger values near 200 ns. The sequence of 

PDPs and example individual PDPs show that the power of the LOS component in the backlobe area 

is 10-30 dB smaller than that in the main beam, and also has rapid variation. The number of MPCs 

and their excess delays (up to ~ 1.8 Ps) within the backlobe are much larger than those in the main 

beam as well. 

4.6 Statistics for Over-Sea Channel Models 

 As reported in [6], we are compiling results that will enable us to develop statistical, tappeddelay (TDL) line models of the wideband over-water AG channels. Although not complete at this 

time, we are close to the final specification of these models4. The basis for the over-water AG channel 

models is the CE2R. This geometry-based model represents the first two components (“taps”) in the 

TDL model, and is the deterministic portion of the complete model. The remaining portion is the 

stochastic part, which is composed of the intermittent MPCs. 

 In [6] we provided example statistics for these intermittent MPCs, termed the 3rd and 4th

rays. Upon a more detailed analysis of the over-sea data, we have concluded that what was termed 

the 4th ray in some results was actually the 3rd ray at a larger value of relative delay. Thus the 

presence of the 4th ray is even more rare than previously reported (~ 0.29% of the time). Hence the 

final over-sea TDL models may not have any 4th ray at all. Since the 4th ray’s power level is very 

small (at least 27 dB down from the LOS component), this simplification will not result in a 

significant loss of accuracy. For purposes of discussion, we repeat the diagram of the (3-ray) TDL 

model in Figure 31. 

 

4 Progress was delayed in part due to a breakdown of our main processing computer on 18 September 2014. This is currently 

under repair, and should be ready to resume use in several weeks. 

0 100 200 300 400 500 600 700

-80

-75

-70

-65

-60

-55

-50

-45

-40

-35

Delay (ns)

Chip Power (dBm)

OxnardCA***06-11-2013***FT2***C-band Rx1

NASA/CR—2015-218486 31
Figure 31. 3-ray TDL for over-sea AG channel. 

 The impulse response of this channel is given by 

݄ሺ߬ǡ ݐሻ ൌ ߙ଴ሺݐሻ݁ି௝థబሺ௧ሻ

ߜሺ߬ሻ ൅ ߙ௦ሺݐሻ݁ି௝థೞሺ௧ሻߜሺ߬െ߬௦ሺݐሻሻ ൅ ݖଷሺݐሻߙଷሺݐሻ݁ି௝థయሺ௧ሻߜሺ߬െ߬௦ሺݐሻ െ ȟ߬ଷሺݐሻሻ (14) 

where again, the D’s denote the MPC amplitudes, the I’s the MPC phases, Ws denotes the relative 

delay of the surface reflection, 'W denotes the additional relative delay of the 3rd ray, and z3 denotes 

the binary on/off (or, “0/1”) random process that controls the frequency of occurrence and duration of 

the intermittent 3rd ray. The LOS component delay is set to a value of zero in Fig. 31 and (14). 

 In addition to the CE2R, to complete the TDL in Fig. 31 requires that we specify the 

statistics of the 3rd ray, specifically its relative delay 'W, its relative amplitude D3, its presence and 

duration (controlled by z3), and its phase (the phase appears to be well-modeled as uniform on 

[0,2S)). Here we present some of the initial statistics. 

 Figure 32 shows statistics for the 3rd ray duration vs. link distance, and the excess delay 

(relative to the LOS component, i.e., 'W=Ws+'W3) for the over-sea straight FT1. Duration Dm is 

provided in units of meters, but this can be translated to an actual time duration via link velocity 

(duration Ds(seconds)=Dm(meters)/velocity(m/s)). 

(a) (b) 

Figure 32. Third-ray statistics vs. distance, (a) duration Dm (m) and (b) excess delay 'W (ns). 

 The results in Figure 32 show that “spatial duration” Dm ranges up to only 4.2 m, 

corresponding to a temporal duration of approximately 5 ms. The 3rd ray duration is greater than 

zero only for specific values of link distance, and most 3rd ray occurrences are at the smaller values of 

link distance—where near-shore objects induce MPCs. Similarly for excess delay, since quantized 

into increments of the sounder’s delay resolution of 20 ns, excess delays take values only in the set 

4 6 8 10 12 14 16 18 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Link Distance (km)

Duration of 3rd Tap (m)

OxnardCA***06-11-2013***FT1***C-band Rx1

4 6 8 10 12 14 16 18 100

120

140

160

180

200

220

240

Link Distance (km)

Excess Delay of 3rd Tap (ns)

OxnardCA***06-11-2013***FT1***C-band Rx1

NASA/CR—2015-218486 32
{100, 120, 140, 200, 220, 240} ns. Receiver 2’s statistics are similar, but not quite identical, and for 

the final set of valid values of duration and excess delay, we will aggregate results from both 

receivers. 

 The dependence on link distance is also evident in a plot of z3(d), shown in Figure 33. Here 

again we see more 3rd ray presence in PDPs at short link distances than at large link distances. The 

results for Rx2 show more 3rd ray presence at larger distances than in Figure 33 for Rx1, but the 

presence of the 3rd ray is still concentrated at the smaller link distances. 

Figure 33. Third-ray “presence” vs. distance: “1” denotes 3rd ray present, “0” denotes 3rd ray absent. 

 In Figure 34 we show a plot of duration Dm vs. excess delay 'W. This can be viewed as 

analogous to a joint distribution of these variables, which illustrates their dependence. From plots 

like that in Figure 34 we can count the instances of each pair (Dm,'W) to construct a histogram that 

represents the actual joint probability mass function of these two variables. One such plot is shown 

in Figure 35. 

Figure 34. Third-ray duration Dm vs. excess delay 'W. 

10 11 12 13 14 15 16 17 18 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Excess Delay (ns)

Tap Duration (m)

OxnardCA***06-11-2013***FT1***C-band Rx1

NASA/CR—2015-218486 33
Figure 35. Third-ray joint distribution for duration Dm and excess delay 'W. 

 The results in Figure 35 can be used along with another histogram, of excess delay vs. 

distance, shown in Figure 36, to obtain the joint probability mass function of duration and excess 

delay vs. distance. 

Figure 36. Third-ray distribution for excess delay 'W as a function of link distance. 

 With these results, one algorithm for generating the wideband 3-ray over-sea channel model 

is as follows: 

1. Run CE2R (vs. distance d); 

2. Given distance d, based upon a model similar to that in Figure 33, draw random variable to 

declare 3rd ray present or absent. Go to Step 3; 

3. If 3rd ray absent, model output=CE2R(d), and return to Step 1; 

If 3rd ray present, draw random variable with distribution similar to that in Figure 36 to 

determine excess delay. Go to Step 4; 

4. With excess delay of Step 3, draw random variable to set duration Dm according to distribution 

similar to that in Figure 35. Return to Step 1 and continue running CE2R (with added 3rd ray) until 

distance d+Dm is reached. For first value of distance d> d+Dm, resume Steps 1, 2, .... 

 Although this algorithm should yield the required dependencies between the variables (based 

upon measurements), it is not the only way to generate the 3-ray TDL. We might instead model the 

“persistence” process z3 via a (statistically non-stationary) discrete Markov chain, with a dependence 

on distance. We are also currently compiling a joint probability mass function that represents the 

third ray’s duration, distance, and excess delay all together (in a sense, combining Figures 35 and 

36)—this will not be representable by diagram, but will instead be in tabular form. Each of these 

alternatives is being explored to determine the procedure that best mimics the actual measurement 

100

150

200

250

1

2

3

4

5

0

20

40

60

80

Excess Delay (ns)

OxnardCA***06-11-2013***FT1***C-band Rx1

Tap Duration (m) Histogram

100

150

200

250

5

10

15

20

0

10

20

30

40

Excess Delay (ns)

OxnardCA***06-11-2013***FT1***C-band Rx1

Distance (km) Histogram

NASA/CR—2015-218486 34
results. As noted, we are aggregating results from both receivers for this. Finally on this wideband 

model algorithm, we have not stated anything about the 3rd ray amplitude D3. From our measured 

data, this value is nearly constant at approximately 25 dB down from the LOS component. We will 

investigate the amplitude variation of the 3rd ray during its “on” periods, but if this variation is 

minimal, it should be sufficient to model D3 as a constant. The remaining part of the wideband model 

would be a model for airframe shadowing—although we did not have any such shadowing in the 

over-sea flights, it was observed in some of the over-freshwater and hilly terrain flights, hence it 

could be incorporated when flight maneuvers (primarily roll angle) can cause such shadowing. 

 Also of interest for CNPC is a narrowband over-water AG channel model. This can be 

constructed as shown in Figure 37, and consists of the CE2R plus a Ricean fader, plus the shadowing 

model. A Ricean fader that has a K-factor varying with distance is easy to construct. The only 

remaining characteristic of the Ricean fader that must be specified is its fading rate. This rate 

should be relatively easy to determine from our results on the Ricean K-factor (see Section 4.4 and 

Appendix A), and once determined, the Ricean fader can be completed by specifying the appropriate 

lowpass filter. 

Figure 37. Narrowband over-sea AG channel model. 

5. Initial Results for the Hilly Terrain AG Channel 

 Flight tests were conducted with the GS in a hilly terrain/suburban environment on 15 April 

2013. These tests were made in southwestern Pennsylvania, near the town of Latrobe. All link 

parameters (transmit powers, antenna gains and GS antenna height, etc.) were identical to those 

used in the over-water flight tests. 

Multiple FTs were flown along both straight and oval-shaped paths, with two orientations 

relative to the predominant ridgeline. Example FTs are shown in Figure 38: FT1 (straight toward GS 

from 17 km - 2.2 km), and FT6 (oval, link distance 16 to 24 km) are shown in a Google Earth® view. 

The average altitude difference between Tx and Rx was 602 m. Elevation angle ranged from 1.6-16q. 

Based upon curved-earth computations, the maximum link distance at which the surface reflection is 

resolvable is 4150 m for C-band, and 615 m for L-band (thus the L-band surface reflection was 

essentially never resolvable since the altitude difference was only 602 m). Figure 39 shows a view of 

the GS near the Latrobe airport, and geometric flight track traces for FT1 are shown in Figure 40. 

Traces for the remaining FTs appear in Appendix A, along with additional hilly terrain FT channel 

measurement results. 

NASA/CR—2015-218486 35
Figure 38. FT1 (brown), FT6 (blue), and ridgeline (double-green), Google Earth®.

Figure 39. GS location, Latrobe, PA: view toward 137q from magnetic north. 

(a) (b) 

NASA/CR—2015-218486 36
(c) (d) 

(e) (f) 

(g) (h) 

Figure 40. Geometric traces for FT1, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of 

aircraft; (h) roll angle of aircraft. 

 Propagation path loss results are shown in Figure 41. As with the over-water results, path 

loss generally follows the free space path loss lines, with larger attenuation at small link distances 

from aircraft antenna effects. The two-ray model does not fit the measured data as well as in the 

over-water cases, but this is expected because of the lower reflectivity of dry ground as compared to 

that of water, and also to the generally rougher surface. Figure 41 also shows linear least-squares fit 

lines to the measured path loss data (two segments, for two ranges of elevation angle, or equivalently 

two ranges of link distance). The “per-FT” path loss model parameters for these linear fits appear in 

Table 9. 

NASA/CR—2015-218486 37
(a) (b) 

(c) (d) 

Figure 41. Measured path loss vs. distance for FT1, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table 9. Path loss model parameters, FT1, Latrobe, PA, 15 April 2013, straight flight toward GS. 

d<dt (θ>θt) d>dt (θ<θt)

Log-distance linear 

PL model

Elevation 

angle range 

(degrees)

Distance 

range (km)

Log-distance linear 

PL model CE2R (dB)

Elevation 

angle range 

(degree)

Distance 

range (km)

A0,S

(dB) nS

σXL,S

(dB) θmin,S θmax,S dmin,S dmax,S

A0,L

(dB) nL

σXL,L

(dB) L0 σX2 θmin,L θmax,L dmin,L dmax,L

C-Rx1 5.6 1.7 1.7 5.0 15.5 2.3 6.9 27.4 1.4 2.0 -0.4 4.3 2.1 5.0 6.9 17.5

C-Rx2 -11.8 2.0 1.3 5.0 15.5 2.3 6.9 24.5 1.5 2.3 -0.7 4.4 2.1 5.0 6.9 17.4

L-Rx1 -25.1 2.0 2.9 5.0 27.9 1.3 6.9 -37.6 2.2 3.7 1.4 6.1 2.1 5.0 6.9 17.5

L-Rx2 29.5 1.2 2.9 5.0 27.9 1.3 6.9 -63.2 2.5 3.8 1.8 6.2 2.1 5.0 6.9 17.5

 Figure 42 shows plots of RMS-DS vs. link distance for this FT. As seen in the overfreshwater results where near-shore obstacles induced multipath components and caused temporary 

increases in RMS-DS, similar RMS-DS “bumps” can be observed for this hilly terrain environment. 

RMS-DS values of larger than 500 ns are evident. Figure 43 shows sequences of PDPs where we 

have identified the MPCs for two of the RMS-DS bumps, with the detailed PDPs for the bump at link 

distance 5.8 km shown in Fig. 43(b). 

 In this environment, using maps and geometry, we have identified the likely reflectors that 

cause the MPCs as large buildings in/near the town of Latrobe. Some moderate reflections may also 

arise from portions of the hillsides [28]. Statistics for RMS-DS for two FTs are provided in Table 10. 

NASA/CR—2015-218486 38
(a) (b) 

Figure 42. RMS-DS vs. distance for FT1, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

(a) (b) 

Fig. 43. Sequence of PDPs for C-band Rx1 FT1, hilly terrain: (a) entire FT1; (b) portion of FT1 near 5.8 km. 

Table 10. RMS-DS statistics for hilly terrain, two FTs. 

RMS Delay Spread Mean (ns) Median (ns) Max (ns) 

Standard 

Deviation

(ns) 

FT1

Original 

12.9 

10.9 267.2 9.8 

Moving Averaged over 100 PDPs 11.1 87.0 6.6 

Moving Averaged over 1000 PDPs 11.2 57.0 5.7 

FT6

Original 

35.7 

10.9 995.7 55.7 

Moving Averaged over 100 PDPs 18.6 371.8 42.9 

Moving Averaged over 1000 PDPs 19.3 349.2 40.4 

 Plots of the correlation coefficient vs. link distance are not presented here—the data is still 

being processed to compute those. The statistics of correlations for FT1 are shown in Table 11. 

Table 11. Statistics of spatial correlation, FT1, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 101 101 101 101 101 101

Mean 0.33 0.62 0.03 -0.01 -0.01 -0.02

Median 0.66 0.92 0.07 -0.02 -0.02 -0.05

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.71 0.54 0.78 0.79 0.79 0.79

4 6 8 10 12 14 16 0

100

200

300

400

500

600

700

800

900

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT1***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

4 6 8 10 12 14 16 0

100

200

300

400

500

600

700

800

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT1***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 39
 Our final results for the hilly terrain AG channel are the Ricean K-factors, shown in Figure 

44. Very similar to the results for over-water, the Ricean K-factors increase very slightly with link 

distance, and are approximately 12 dB for L-band, and 25 dB for C-band. Two-ray behavior is again 

evident for the L-band results as well. The (“per-FT”) K-factor statistics are listed in Table 12. 

(a) (b) 

(c) (d) 

Figure 44. Ricean K factor for FT1, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table 12. Statistics of Ricean K factor, FT1, Latrobe, PA, 15 April 2013, straight flight toward GS. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 26.5 26.6 13.5 13.5

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 24.8 24.6 24.8 26.0 25.8 26.0 13.5 11.9 13.2 13.2 11.6 12.9

n 0.24 0.09 0.24 0.13 -0.04 0.13 0.02 0.12 0.03 0.05 0.15 0.06

σX (dB) 1.5 2.5 1.5 1.4 2.7 1.4 1.0 1.3 1.0 0.9 1.2 1.0

Dmax(km) 2.2 2.2 2.2 2.2 2.2 2.2 1.2 1.2 1.2 1.2 1.2 1.2

DMin(km) 17.5 17.5 17.5 17.4 17.4 17.4 17.5 17.5 17.5 17.5 17.5 17.5

Statistics of K factor 

(dB)

Max 32.4 32.1 32.3 31.8 31.8 31.8 18.5 17.0 18.4 18.7 17.1 18.7

Min 12.5 2.6 12.0 7.9 0.0 6.9 8.2 2.4 8.6 9.2 0.9 9.4

Median 27.1 25.4 27.1 27.2 25.4 27.2 13.7 13.0 13.4 13.7 12.9 13.4

μ 27.2 25.7 27.2 27.4 25.9 27.4 13.5 13.0 13.3 13.5 13.0 13.2

σ 1.8 2.5 1.8 1.5 2.7 1.5 1.0 1.4 1.1 0.9 1.4 1.0

NASA/CR—2015-218486 40
6. Literature Review Update 

 This section provides an update to the ongoing project literature review. With the growing 

popularity of UAS, it is not hard to find material. In fact, articles related to UAS appear in the 

popular press daily. For example, the American Institute of Aeronautics and Astronautics (AIAA) 

provides regular daily email news articles, which often address various aspects of UAS, including 

policies, experiments, applications, etc. One recent article noted that the popular social media 

company Facebook is looking to expand internet access via solar powered UAS [29]. The same issue 

of the AIAA email had an article that stated UAS will be used for aerial photography at future major 

sporting events. Another recent online article that received worldwide publicity appeared on the 

CNN website [30], and described how a tourist crashed a UAS into the largest hot spring in 

Yellowstone National Park. An airline magazine [31] predicted “personal UAS” in the future. Since 

much of the reporting in the popular press tends toward non-technical issues, henceforth we will 

limit review of such articles to only those that have sufficient technical content, and direct bearing 

on the UAS in the NAS project. 

 The establishment of six FAA UAS test sites in the United States is also generating publicity 

[32], as is the public’s reaction to UAS in general [33]. Reputable popular media are also 

contributing, often identifying clear challenges to UAS safety and security [34]. 

 In terms of more technical literature, this too is expanding, with publications on networking, 

multiple antenna techniques, and some on the air-ground wireless channel as well. Reference [35] is 

a good review article on the challenges in designing airborne tactical networks. These challenges 

include an expectation of significantly increasing data rates in the future, latency for long-range 

links and networks, mobility-induced network topology changes, scarcity of spectrum, and “body 

blockage” or airframe shadowing. Interference resilient waveforms (e.g., spread spectrum) were cited 

as critical for these tactical networks. 

 The authors of [36] describe the L-band Digital Aeronautical Communication System-1 

(LDACS1). This article does not compare LDACS1 with competing technologies (as reported in [6], 

several publications have already done so, with, for example, LDACS1 being superior to LDACS2), 

Rather, the article provides an overview of the system, and how it has been designed to effectively 

coexist with existing ground-based distance measuring equipment (DME) transmitters. 

 In [37], the authors describe how UAS can be used in rescue services after natural disasters. 

The article provides an overview, and some description of image processing to locate victims. 

Reference [38] also describes the use of UAS in military settings, with a custom spread-spectrum 

waveform based upon the 3rd-generation cellular standard WCDMA. The authors briefly discuss 

required antenna characteristics for their Ku-band system, and show a few example results from a 

field trial. 

 Reference [39] provides a summary of UAS network topologies and recommends several 

protocols for their effective operation. Little technical detail is provided. The authors of [40] consider 

the use of UAS for serving as LTE base stations. Simulation results are provided for throughput and 

delay, but unfortunately the channel employed in the simulations is a terrestrial cellular-like 

channel, so results are of questionable validity. 

 The effects of interference between aircraft (UAS as well as piloted) and terrestrial systems 

are addressed in [41]. The authors employ simplified path loss models, plus models for fuselage 

attenuation, to estimate interference densities and the consequent effects on system capacity. 

Although not an exceptionally clear treatment of the problem, the analysis does illustrate the 

potential degradations to both airborne and cellular systems if spectrum management is not 

enforced. 

 In [42], the authors describe experiments conducted with very small, or “micro” UAS. 

Interesting results obtained with WiFi (IEEE 802.11b) radios address the effects of different antenna 

types on the radio throughput. Their main conclusion is that allowing the WiFi radios to employ 

their normal adaptation (for physical layer parameters, e.g., modulation and coding), does not yield 

the largest throughput when the UAS are moving. This is simply because the WiFi system was not 

NASA/CR—2015-218486 41
designed with mobility in mind, i.e., the adaptation is not fast enough. The authors also identify 

shadowing as a significant degrading effect that should be characterized. 

 Reference [43] describes experiments with UAS in a 200 nautical mile link. Two different 

radios were used: one a Ku-band radio with a steerable beam antenna, and the other an L-band 

frequency hopping radio. Both radios had data rates above 1 Mbps. The authors characterize link 

availability with the two radios for both air-air and air-ground links. Availabilities ranged from as 

high as 99.9% to as low as 63%, depending on link and flight patterns. Statistics on latency and 

throughput were also provided. 

 In [44] the authors analyze the maximum achievable capacity in airborne multipleinput/multiple-output (MIMO) air-air links. The channel model is a pure free space channel, and the 

analysis is geometry-based, with the assumption of equal aircraft altitudes. No shadowing or 

multipath propagation was considered, and one main conclusion is that capacity decreases with link 

distance (as SNR decreases). This paper was an extension of similar work in [45]. 

 The author of [46] conducts a comparison of the spectral efficiency of several types of systems 

and levels of signal processing complexity in interference limited environments. One system is an 

air-ground system with 4 by 4 MIMO links. The comparison is via simulation, and the AG channel 

was modeled as a free space channel without any shadowing or multipath propagation. Results were 

in terms of spectral efficiency as a function of antennas and cell radius. 

 A slightly older treatment of MIMO in the AG channel appears in [47]. This paper analyzes 

MIMO capacity as a function of distance (coming to conclusions similar to those in [44], but with a 

less rigorous analysis). Another older paper on AG MIMO is [48], in which the authors also assume 

the same Ricean channel as in [47]. Their main conclusion is that MIMO capacity can approach that 

of the uncorrelated Rayleigh channel as antenna separation increases, in general agreement with 

results in [44]. 

 In [49] the authors address the effects of rain on a satellite to ground channel in both L- and 

C-bands. The most interesting results in this paper show that during heavy rain events, C-band 

attenuation can increase by up to 4 dB (in general agreement with predictions available via ITU 

recommendations), and that the differential delay between L-band and C-band signals can reach 

approximately 68 mm (only 22 ps) during these rain events. 

 Reference [50] describes measurements made with IEEE 802.11a radios in AG links. The 

authors characterized throughput versus link distance, with some results resembling two-ray 

channel variations. The most substantial paper addressing AG channel characteristics we have 

recently found is [51]. The authors configured an OFDM system to measure channel impulse 

responses at 2.3 GHz over the island of Oahu, Hawaii. Multiple flight tracks were flown at altitudes 

500-1500 m. A large number of multipath components (up to 20) were seen in some flights over 

populated urban areas, and some with extremely large values of excess delay, up to 30 Ps. No data 

on path loss or delay spreads were reported. 

7. Summary, Conclusions, and Future Work 

 In this report we provided additional measurement and modeling results on the AG channel. 

Over-water, hilly/suburban, and over-harbor backlobe environments were addressed. We also 

described procedures for estimating statistical channel characteristics, namely, our procedure for 

estimation of the stationarity distance, over which other channel parameters can be estimated, and 

procedures for estimating small-scale fading (Ricean K-factors) and inter-antenna correlations. We 

also described our evaluation of the inter-antenna correlations for an analytical (two-ray) channel, 

for comparison with measurements. Some initial statistics compiled for development of the wideband 

over-sea channel models were also provided, and the literature review was updated. 

 The AG channel path loss is best described by the curved-earth two-ray (CE2R) model, plus a 

Gaussian random variable to account for deviations from this geometric model. The CE2R model 

pertains best to the over-water channels, and is slightly less accurate for the hilly terrain setting. 

Complete path loss models for the over-water channels were provided. 

NASA/CR—2015-218486 42
 In terms of channel dispersion, quantified by RMS-DS VW, our results show the expected 

relationship VW,urban > VW,hilly >VW,over-water, and because of the greater number of large obstacles in the 

freshwater Lake Erie environment than in the ocean area near Oxnard, CA, we also have the 

relationship VW,over-freshwater >VW,over-sea for our results. The RMS-DS values were quantified for all FTs. 

 Intra-band correlations should be large in the LOS environments for straight FTs, with rapid 

variation from -1 to +1 at shorter link distances, and slower variation at longer link distances, in 

general agreement with measurements. Some the difference between measured and analytical 

correlations emanates from channel sounder instability, i.e., the varying sample clock drift between 

Tx and Rx. This does not affect absolute power values significantly, hence it does not affect path loss 

or RMS-DS, and it has little effect on Ricean K-factors. Inter-band correlations are near zero, as 

expected. 

Fairly large Ricean K-factors were found for both bands, with slight variation with distance 

in some FTs. As with path loss, the L-band K-factors (smaller than C-band) are more strongly 

affected by the stronger surface reflection, in both over-water and hilly terrain settings. Stationarity 

distances computed in the three environments, and for some urban GS data, are all similar at 

approximately 250 C-band wavelengths, or ~ 15 m. 

 Future work includes the following: 

1. continuation of data processing for the remaining GS environments: desert, urban, and 

mountainous terrain; 

2. completion of computations of inter-antenna correlations after development of a method to remove 

effects of channel sounder sample clock drift for selected FTs; 

3. development of aggregate Ricean K-factor vs. distance models for over-water and hilly terrains; 

4. quantification of Ricean fading rates for completion of Ricean fading models; 

5. development of hilly terrain path loss models; 

6. completion of wideband tapped-delay line statistical models for the over-sea setting; 

7. validation of Ricean fading and wideband tapped-delay line simulation models with 

measurements. 

For tasks 3, 5, and 6, model development for the remaining GS environments will also be done, 

following the model development for the settings specified in these tasks. 

 

NASA/CR—2015-218486 43
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Mag., vol. 52, no. 7, pp. 141-149, July 2014. 

[43] B. N. Cheng, R. Charland, P. Christensen, L. Veytser, J. Wheeler, “Evaluation of Multihop 

Airborne IP Backbone with Heterogeneous Radio Technologies,” IEEE Trans. Mobile Computing, vol. 

13, no. 2, pp. 299-310, February 2014. 

[44] W. Su, J. D. Matyjas, M. J. Gans, S. Batalama, “Maximum Achievable Capacity in Airborne 

MIMO Communications with Arbitrary Alignments of Linear Transceiver Antenna Arrays,” IEEE 

Trans. Wireless Comm., vol. 12, no. 11, pp. 5584-5593, November 2013. 

NASA/CR—2015-218486 45
[45] M. J. Gans, “Aircraft Free-Space MIMO Communications,” Proc. Asilomar Conf. on Signals, 

Systems, & Computers, Pacific Grove, CA, November 2009. 

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Proc. DASC, Seattle, WA, 16-20 October 2011. 

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Systems,” Siena, Italy, 7-10 September 2009. 

[49] M. Schonhuber, S. Junker, I. Bartunkova, K. Plimon, R. O. Perez, “Synchronized Measurements 

of L- and C-band Tropospheric Propagation Parameters,” Proc. EuCAP 2014, The Hague, 

Netherlands, 6-11 April 2014. 

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Propagation Characteristics for Unmanned Aircraft Systems,” Proc. EuCAP 2014, The Hague, 

Netherlands, 6-11 April 2014. 

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[54] C. Tepedelenlioglu, A. Abdi, G. B. Giannakis, "The Ricean K factor: estimation and performance 

analysis," IEEE Trans. Wireless Comm., vol. 2, no. 4, pp. 799-810, July 2003. 

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NASA/CR—2015-218486 46
Appendix A: Results from Additional Flight Tracks 

This appendix contains path loss, delay spread, spatial correlation and Ricean K-factor 

results for the over-freshwater and hilly terrain settings, for the remaining flight tracks not 

described in the main body of the report. New K-factor results, and some new correlation results for 

the over-sea FTs, are also provided. Most of the results are presented without discussion because of 

their similarity to those for FTs reported in the report’s main body (or in that of [6]), but some 

atypical results are briefly described. The analysis of all results here has not been completed, and 

this will be done in the near future. Also, complete results for all FTs in this Appendix are not yet 

available: for examples, for the hilly terrain FT10, much of the data may be invalid, since the 

sounder PDP update rate was unusually small, and significant terrain shadowing was present. Note 

also that all tabulated results in this appendix, for path loss parameters, correlations, and K-factors, 

are “per-FT. For data in this appendix that we are confident is invalid, we have highlighted table 

entries or figure captions in grey. 

In Tables A1 and A2 we list the basic information of the over-freshwater and hilly terrain 

FTs including shape, range of distances, velocity and number of PDPs. We also provide an expanded 

description of the over-sea FTs in Table A3. Note that the over-freshwater FT2 in Table A1 was 

already described in detail in the main body of the report; the FT details are shown here in Table A1 

for completeness. 

Table A1. Descriptions of FTs for over freshwater FTs, Cleveland, OH, 22 October 2013. 

Band Rx 

Index # PDPs

Time 

duration

∆t (s)

Average 

velocity 

(m/s)

PDP 

update 

rate (Hz)

dmin (km) dmax (km)

FT2, flew 

straight 

toward GS

C-band

Rx1 1080854 367 73.1 2942 2.51 29.35

Rx2 1077204 367 73.1 2933 2.51 29.34

L-band

Rx1 1109829 390 65.3 2843 0.97 29.35

Rx2 1136858 390 65.3 2912 0.97 29.35

FT3, flew 

straight 

away from 

GS

C-band

Rx1 1039849 353 76.9 2948 1.21 28.09

Rx2 912903 353 76.9 2588 1.21 28.09

L-band

Rx1 974564 360 76.8 2708 0.76 28.09

Rx2 995626 360 76.9 2766 0.76 28.09

FT4, Oval 

FT

C-band

Rx1 1691453 586 76.6 2884 16.31 21.76

Rx2 1632930 586 76.5 2785 16.31 21.76

L-band

Rx1 1513727 586 76.3 2581 16.31 21.76

Rx2 1599337 586 76.4 2727 16.31 21.76

 

NASA/CR—2015-218486 47
Table A2. Descriptions of FTs in hilly terrain, Latrobe, PA, 15 April 2013. 

Band Rx 

Index # PDPs

Time 

duration

∆t (s)

Average velocity 

(m/s)

PDP update 

rate (Hz)

dmin

(km)

dmax

(km)

FT1, flew straight toward 

GS

C-band

Rx1 443224 205 74.6 2157 2.27 17.46

Rx2 411707 205 74.6 2005 2.27 17.45

L-band

Rx1 455082 222 74.6 2054 1.27 17.46

Rx2 478889 221 74.5 2162 1.27 17.45

FT2, flew straight away 

from GS

C-band

Rx1 352034 170 91.9 2072 1.39 16.91

Rx2 309421 170 91.8 1820 1.39 16.91

L-band

Rx1 385010 179 92.1 2153 0.79 16.91

Rx2 403085 179 92.0 2254 0.79 16.91

FT3, flew straight toward 

GS

C-band

Rx1 425798 195 74.3 2184 2.25 16.60

Rx2 396254 195 74.3 2033 2.25 16.60

L-band

Rx1 467080 214 74.2 2187 1.03 16.60

Rx2 446435 213 74.2 2091 1.03 16.60

FT4, flew straight away 

from GS

C-band

Rx1 338902 164 90.0 2069 1.28 15.80

Rx2 301524 164 89.9 1841 1.28 15.81

L-band

Rx1 337449 171 89.9 1979 0.79 15.80

Rx2 313380 171 89.8 1837 0.79 15.80

FT5, flew straight away 

from GS

C-band

Rx1 286055 272 90.0 1052 17.01 41.49

Rx2 181805 272 90.1 669 17.02 41.49

L-band

Rx1 493141 272 90.0 1814 17.00 41.49

Rx2 453757 272 90.0 1670 17.01 41.49

FT6, Oval FT

C-band

Rx1 1131539 731 83.2 1549 15.60 24.00

Rx2 894030 730 81.5 1224 15.60 24.00

L-band

Rx1 1244883 731 82.4 1704 15.60 24.00

Rx2 1285370 731 82.6 1760 15.60 24.00

FT7, Oval FT

C-band

Rx1 974889 672 81.2 1452 15.83 23.76

Rx2 814172 672 79.4 1212 15.83 23.76

L-band

Rx1 1219353 672 81.5 1815 15.83 23.76

Rx2 1257997 672 81.4 1873 15.83 23.76

FT8, flew straight toward 

GS

C-band

Rx1 293118 165 90.5 1772 2.50 17.52

Rx2 279700 121 90.4 2319 2.50 13.33

L-band

Rx1 219470 180 91.3 1219 1.24 17.53

Rx2 223819 180 91.2 1244 1.24 17.51

FT9, flew straight away 

from GS

C-band

Rx1 357710 200 75.9 1792 1.98 17.02

Rx2 323670 200 76.0 1621 1.98 17.02

L-band

Rx1 259612 212 76.8 1225 1.22 17.02

Rx2 263313 212 76.8 1242 1.22 17.02

FT10, Oval FT

C-band

Rx1 23545 196 85.8 120 19.19 23.95

Rx2 14600 196 87.2 75 19.20 23.95

L-band Rx1 39826 196 86.9 203 19.19 23.95

NASA/CR—2015-218486 48
Rx2 46449 196 86.7 237 19.19 23.95

FT11, Oval FT

C-band

Rx1 616485 641 85.9 962 13.28 19.84

Rx2 514186 641 86.6 803 13.28 19.84

L-band

Rx1 690892 641 85.5 1078 13.28 19.84

Rx2 786794 641 85.4 1228 13.28 19.84

FT12, Oval FT

C-band

Rx1 668296 522 90.2 1281 12.34 19.58

Rx2 561444 521 90.7 1077 12.34 19.58

L-band

Rx1 798047 522 89.0 1530 12.34 19.58

Rx2 861001 522 88.5 1651 12.34 19.58

FT13, flew straight 

toward GS

C-band

Rx1 288092 157 94.8 1836 2.50 17.16

Rx2 271241 157 94.8 1730 2.50 17.16

L-band

Rx1 260000 161 95.0 1619 2.24 17.17

Rx2 250000 156 95.0 1598 2.57 17.16

Table A3. Descriptions of FTs for over sea FTs, Oxnard, CA, 11 June 2013. 

Band 

Index

Rx 

Index # PDPs

Time

duration

∆t (s)

Average velocity 

(m/s)

PDP update 

rate (Hz)

dmin

(km)

dmax

(km)

FT1, flew straight 

towards the GS

C-band

Rx1 693717 235 94.8 2954 2.50 24.15

Rx2 690832 235 94.8 2943 2.50 24.15

L-band

Rx1 683450 240 94.8 2850 2.22 24.15

Rx2 679470 240 94.8 2834 2.22 24.15

FT2, flew straight away 

from the GS

C-band

Rx1 1360107 460 94.9 2955 1.88 45.24

Rx2 1333364 460 94.9 2896 1.88 45.25

L-band

Rx1 1338635 470 94.8 2847 1.18 45.24

Rx2 1361656 470 94.8 2896 1.18 45.25

FT3, flew straight 

towards the GS

C-band

Rx1 976201 330 91.8 2954 18.81 48.99

Rx2 976374 330 91.8 2955 18.81 48.98

L-band

Rx1 950131 330 91.8 2875 18.82 48.99

Rx2 958074 330 91.8 2899 18.81 48.99

FT4, Oval FT

C-band

Rx1 1926266 686 88.4 2809 19.52 26.79

Rx2 1841367 686 88.5 2686 19.52 26.79

L-band

Rx1 1999086 686 88.4 2915 19.52 26.79

Rx2 2013971 686 88.4 2937 19.52 26.79

FT5, Oval FT

C-band

Rx1 1834243 645 89.0 2844 19.33 26.76

Rx2 1662261 645 89.1 2577 19.33 26.76

L-band

Rx1 1859686 645 89.0 2883 19.33 26.76

Rx2 1878599 645 89.0 2912 19.33 26.76

FT6, Oval FT

C-band

Rx1 1810296 645 89.8 2807 18.91 27.33

Rx2 1664753 645 90.0 2582 18.91 27.33

L-band

Rx1 1863404 645 89.8 2889 18.91 27.33

Rx2 1886979 645 89.8 2926 18.91 27.33

FT7, Oval FT C-band Rx1 1827301 645 90.0 2832 19.32 27.75

NASA/CR—2015-218486 49
Rx2 1609671 645 90.2 2495 19.32 27.75

L-band

Rx1 1798433 645 90.1 2788 19.32 27.75

Rx2 1838679 645 90.0 2850 19.32 27.75

FT8, flew straight 

towards the GS

C-band

Rx1 795032 268 87.8 2966 2.50 25.76

Rx2 785290 268 87.8 2930 2.50 25.75

L-band

Rx1 818696 284 87.8 2885 1.29 25.76

Rx2 825874 284 87.8 2911 1.29 25.75

FT9, flew straight away 

from the GS

C-band

Rx1 1528210 520 86.2 2937 1.79 46.04

Rx2 1470922 520 86.1 2827 1.79 46.05

L-band

Rx1 1549847 531 86.1 2916 1.04 46.04

Rx2 1556759 532 86.1 2929 1.04 46.05

FT10, flew straight 

towards the GS

C-band

Rx1 876770 305 90.1 2871 19.15 45.18

Rx2 873236 305 90.1 2860 19.15 45.18

L-band

Rx1 886269 305 90.1 2902 19.15 45.18

Rx2 893365 305 90.1 2926 19.15 45.18

FT11, Oval FT

C-band

Rx1 1565001 706 79.4 2216 19.28 26.56

Rx2 1374307 706 79.8 1947 19.28 26.56

L-band

Rx1 1357075 706 79.6 1922 19.28 26.56

Rx2 1422747 706 79.6 2015 19.28 26.56

FT12, Oval FT

C-band

Rx1 1656305 755 79.8 2195 19.27 27.09

Rx2 1461223 755 79.9 1936 19.27 27.09

L-band

Rx1 1505207 755 79.9 1995 19.27 27.09

Rx2 1550215 755 79.9 2054 19.27 27.09

A.1 Over Fresh Water Flight Track 3 

(a) (b) 

6.85

6.86

6.87

6.88

x 105

-4.74

-4.73

-4.72

-4.71

x 106

4.2

4.205

4.21

4.215

4.22

4.225

4.23

x 106

X

ClevelandOH***10-22-2013***FT3

Y

Z

Tx (Ground)

Rx (Aircraft)

L-band Rx start point

Rx end point

C-band Rx start point

5 10 15 20 25 72

74

76

78

80

82

84

Link Distance (km)

Aircraft Velocity (m/s)

ClevelandOH***10-22-2013***FT3

Whole Track

L-band Rx start point

Rx end point

C-band Rx start point

NASA/CR—2015-218486 50
(c) (d) 

(e) (f) 

(g) (h) 

Figure A1. Geometric traces for FT3: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude difference 

between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of aircraft; 

(h) roll angle of aircraft. 

5 10 15 20 25 545

550

555

560

565

570

575

Link Distance (km)

Relative height between Tx and Rx (m)

ClevelandOH***10-22-2013***FT3

Whole Track

L-band Rx start point

C-band Rx start point

Rx end point

5 10 15 20 25 0

50

100

150

200

250

300

350

Link Distance (km)

Azimuth Angle (degree)

ClevelandOH***10-22-2013***FT3

Whole Track

L-band Rx start point

C-band Rx start point

Rx end point

5 10 15 20 25 0

5

10

15

20

25

30

35

40

45

50

Link Distance (km)

Elevation Angle (degree)

ClevelandOH***10-22-2013***FT3

Curved Earth

Flat Earth

ENU

L-band Rx start point

C-band Rx start point

Rx end point

5 10 15 20 25 -12

-10

-8

-6

-4

-2

0

2

4

Link Distance (km)

True Heading (degree)

ClevelandOH***10-22-2013***FT3

Whole Track

L-band Rx start point

C-band Rx start point

Rx end point

5 10 15 20 25 4

5

6

7

8

9

10

11

Link Distance (km)

Pitch (degree)

ClevelandOH***10-22-2013***FT3

Whole Track

L-band Rx start point

C-band Rx start point

Rx end point

5 10 15 20 25 -14

-12

-10

-8

-6

-4

-2

0

2

4

6

Link Distance (km)

Roll (degree)

ClevelandOH***10-22-2013***FT3

Whole Track

L-band Rx start point

C-band Rx start point

Rx end point

NASA/CR—2015-218486 51
 

(a) (b) 

(c) (d) 

Figure A2. Measured path loss vs. distance for (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A4. Path Loss Model, FT3, Cleveland, OH, 22 October 2013, straight flight away from GS. 

d<dt (θ>θt) d>dt (θ<θt)

Log-distance 

linear PL model

Elevation 

angle range 

(degree)

Distance 

range (km)

Log-distance 

linear PL model

CE2R 

(dB)

Elevation 

angle range 

(degree)

Distance 

range (km)

A0,S

(dB) nS

σXL,S

(dB) θmin,S θmax,S dmin,S dmax,S

A0,L

(dB) nL

σXL,L

(dB) L0 σX2

θmin

,L

θmax,L dmin,L dmax,L

C-Rx1 71.0 1.2 1.5 5.0 27.5 1.2 6.6 51.3 1.8 1.7 -3.5 4.1 1.4 5.0 6.6 28.1

C-Rx2 73.8 1.2 1.4 5.0 27.5 1.2 6.6 50.8 1.9 2.1 0.0 3.9 1.4 5.0 6.6 28.1

L-Rx1 50.2 1.6 2.0 5.0 48.5 0.8 6.6 37.3 1.9 3.2 -0.6 3.0 1.4 5.0 6.6 28.1

L-Rx2 61.9 1.2 1.7 5.0 48.5 0.8 6.6 24.3 2.2 3.0 -0.8 2.9 1.4 5.0 6.6 28.1

 Note that some of the RMS-DS results in Fig. A.3 show values below our minimum value of 

10 ns (near zero). This may be due to a very weak signal at large distances, in which case only the 

LOS component was above our threshold. It may also be attributable to variation in the channel 

sounder. In any case, such small values are inconsequential. 

104 100

105

110

115

120

125

130

135

140

145

150

Link Distance (m)

Path Loss (dB)

ClevelandOH***10-22-2013***FT3***C-band Rx1

Measured PL

Free Space PL

Fit [n=1.24 ,V=1.5]

Fit [n=1.8 ,V=1.74]

2-Ray Curved Earth

104 100

105

110

115

120

125

130

135

140

145

150

Link Distance (m)

Path Loss (dB)

ClevelandOH***10-22-2013***FT3***C-band Rx2

Measured PL

Free Space PL

Fit [n=1.18 ,V=1.43]

Fit [n=1.9 ,V=2.11]

2-Ray Curved Earth

103104 80

90

100

110

120

130

140

150

Link Distance (m)

Path Loss (dB)

ClevelandOH***10-22-2013***FT3***L-band Rx1

Measured PL

Free Space PL

Fit [n=1.57 ,V=1.96]

Fit [n=1.87 ,V=3.15]

2-Ray Curved Earth

103104 80

90

100

110

120

130

140

150

Link Distance (m)

Path Loss (dB)

ClevelandOH***10-22-2013***FT3***L-band Rx2

Measured PL

Free Space PL

Fit [n=1.21 ,V=1.75]

Fit [n=2.18 ,V=3.03]

2-Ray Curved Earth

NASA/CR—2015-218486 52
(a) (b) 

Figure A3. RMS-DS vs. distance for C-band (a) Rx1; (b) Rx2, FT3. 

(a) (b) 

Figure A4. Measured amplitude correlation coefficient (a) between C-band Rx1 and 2; (b) between L-band Rx1 and 2 for 

FT3. 

Table A5. Statistics of Spatial Correlation, FT3, Cleveland, OH, 22 October 2013. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 98 98 98 98 98 98

Mean 0.62 0.76 0.03 0.01 0.02 0.01

Median 0.94 0.96 0.00 0.01 0.00 0.00

Max 1.00 1.00 1.00 0.78 1.00 1.00

Min -1.00 -0.97 -1.00 -0.78 -1.00 -1.00

Standard deviation 0.55 0.39 0.77 0.16 0.77 0.77

5 10 15 20 25 0

5

10

15

20

25

30

35

40

45

Link Distance (km)

RMS-DS (ns)

ClevelandOH***10-22-2013***FT3***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

5 10 15 20 25 0

5

10

15

20

25

30

35

40

45

Link Distance (km)

RMS-DS (ns)

ClevelandOH***10-22-2013***FT3***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 53
(a) (b) 

(c) (d) 

Figure A5. Ricean K factor for FT3 (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A6. Statistics of Ricean K factor, FT3, Cleveland, OH, 22 October 2013, straight flight away from GS 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 26.4 25.7 11.3 11.8

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 24.9 22.5 24.9 26.2 23.7 26.2 11.3 10.6 11.1 12.1 11.6 12.1

n 0.16 0.13 0.16 0.03 0.01 0.03 0.01 0.07 0.04 -0.01 0.05 0.02

σX (dB) 1.6 3.0 1.6 2.0 3.3 2.0 1.0 1.2 1.0 0.9 1.1 0.9

Statistics of K factor (dB)

Max 32.1 32.1 32.1 32.5 32.0 32.5 14.9 -17.5 14.6 15.1 15.0 15.0

Min 12.4 10.0 12.8 14.6 10.3 14.8 5.0 0.0 3.5 5.6 -1.9 4.3

Median 27.2 24.4 27.2 26.5 23.7 26.5 11.4 11.6 11.7 11.9 12.2 12.4

μ 27.7 24.7 27.7 26.8 24.3 26.8 11.4 11.7 11.8 11.9 12.4 12.4

σ 2.0 3.2 2.0 2.0 3.3 2.0 1.0 1.3 1.0 1.0 1.2 0.9

NASA/CR—2015-218486 54
A.2 Over Fresh Water Flight Track 4 

(a) (b) 

(c) (d) 

(e) (f) 

6.7

6.8

6.9

7

x 105

-4.735

-4.73

-4.725

-4.72

x 106

4.2

4.205

4.21

4.215

4.22

x 106

X

ClevelandOH***10-22-2013***FT4

Y

Z

Tx (Ground)

Rx (Aircraft)

Rx start point

Rx end point

17 18 19 20 21 60

65

70

75

80

85

90

Link Distance (km)

Aircraft Velocity (m/s)

ClevelandOH***10-22-2013***FT4

Whole Track

Rx start point

Rx end point

17 18 19 20 21 545

550

555

560

565

570

575

Link Distance (km)

Relative height between Tx and Rx (m)

ClevelandOH***10-22-2013***FT4

Whole Track

Rx start point

Rx end point

17 18 19 20 21 0

50

100

150

200

250

300

350

Link Distance (km)

Azimuth Angle (degree)

ClevelandOH***10-22-2013***FT4

Whole Track

Rx start point

Rx end point

17 18 19 20 21 1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

Link Distance (km)

Elevation Angle (degree)

ClevelandOH***10-22-2013***FT4

Curved Earth

Flat Earth

ENU

Rx start point

Rx end point

17 18 19 20 21 -100

-50

0

50

100

150

Link Distance (km)

True Heading (degree)

ClevelandOH***10-22-2013***FT4

Whole Track

Rx start point

Rx end point

NASA/CR—2015-218486 55
(g) (h) 

Figure A6. Geometric traces for FT4: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude difference 

between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of aircraft; 

(h) roll angle of aircraft. 

(a) (b) 

Figure A7. Measured path loss vs. distance for (a) C-band Rx1 & 2; (b) L-band Rx1 & 2. 

(a) (b) 

Figure A8. RMS-DS vs. distance for C-band (a) Rx1; (b) Rx2, FT4. 

17 18 19 20 21 3

4

5

6

7

8

9

10

11

12

Link Distance (km)

Pitch (degree)

ClevelandOH***10-22-2013***FT4

Whole Track

Rx start point

Rx end point

17 18 19 20 21 -30

-25

-20

-15

-10

-5

0

5

10

Link Distance (km)

Roll (degree)

ClevelandOH***10-22-2013***FT4

Whole Track

Rx start point

Rx end point

16.4 17 18 19 20 21 21.7

125

130

135

140

145

150

Link Distance (m)

Average Path Loss (dB)

ClevelandOH***10-22-2013***FT4***C-band Rxs

C-band Rx1

C-band Rx2

Free Space PL

CE2R

16.4 17 18 19 20 21 21.7

110

115

120

125

130

135

140

Link Distance (m)

Average Path Loss (dB)

ClevelandOH***10-22-2013***FT4***L-band Rxs

L-band Rx1

L-band Rx2

Free Space PL

CE2R

17 18 19 20 21 0

20

40

60

80

100

120

140

160

180

Link Distance (km)

RMS-DS (ns)

ClevelandOH***10-22-2013***FT4***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

17 18 19 20 21 0

20

40

60

80

100

120

140

Link Distance (km)

RMS-DS (ns)

ClevelandOH***10-22-2013***FT4***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 56
(a) (b) 

Figure A9. Measured amplitude correlation coefficient (a) between C-band Rx1 and 2; (b) between L-band Rx1 and 2 for 

FT4. 

Table A7. Statistics of Spatial Correlation, FT4, Cleveland, OH, 22 October 2013. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 98 98 98 98 98 98

Mean 0.42 0.59 0.01 0.00 0.02 0.00

Median 0.85 0.91 0.03 0.00 0.06 0.00

Max 1.00 1.00 1.00 0.58 1.00 0.61

Min -1.00 -1.00 -1.00 -0.58 -1.00 -0.73

Standard deviation 0.72 0.59 0.80 0.13 0.80 0.13

The small K-factors at large distances in Figure A.15 are likely due to wing (fuselage) 

shadowing since the roll angle was near -30 degrees, thus significantly attenuating the LOS 

component. The L-band path loss is also larger than expected as a result of this shadowing. The Cband K-factor is smaller than the values in the straight FTs, whereas the C-band path loss is larger 

than in the straight FTs. Interestingly, the shadowing effect appears to be less severe in C-band, and 

this is unexpected—we expect shadowing to more strongly attenuate the C-band signal. This is a 

topic for future investigation of shadowing. 

(a) (b) 

NASA/CR—2015-218486 57
(c) (d) 

Figure A10. Ricean K factor for FT4 (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A8. Statistics of Ricean K factor, FT4, Cleveland, OH, 22 October 2013, oval-shaped FT. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 14.7 14.5 12.5 12.5

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 29.0 27.9 28.9 30.3 28.9 30.4 24.8 30.3 29.6 28.3 28.8 27.8

n -0.68 -0.73 -0.67 -0.75 -0.82 -0.75 -0.63 -0.95 -0.89 -0.83 -0.86 -0.78

σX (dB) 3.4 4.0 3.4 3.5 4.4 3.5 1.6 2.8 2.7 3.6 2.6 2.6

Statistics of K factor (dB)

Max 28.1 27.8 28.1 26.5 25.2 26.5 15.5 -24.0 -23.2 15.6 -22.0 -24.5

Min 5.5 0.0 6.4 4.0 0.0 4.0 1.1 0.0 0.0 -89.7 0.0 0.4

Median 16.4 14.4 16.4 16.5 13.8 16.4 13.1 12.8 13.1 12.9 12.8 13.2

μ 16.5 14.7 16.5 16.7 14.2 16.7 13.4 13.5 13.8 13.2 13.3 13.6

σ 3.6 4.1 3.5 3.7 4.6 3.7 1.8 3.1 3.0 3.8 2.9 2.9

A.3 Hilly Terrain Flight Track 2 

(a) (b) 

NASA/CR—2015-218486 58
(c) (d) 

(e) (f) 

(g) (h) 

Figure A11. Geometric traces for FT2, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of 

aircraft; (h) roll angle of aircraft. 

NASA/CR—2015-218486 59
(a) (b) 

(c) (d) 

Figure A12. Measured path loss vs. distance for FT2, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band 

Rx2. 

Table A9. Path loss model parameters, FT2, Latrobe, PA, 15 April 2013, flight straight away from GS. 

d<dt (θ>θt) d>dt (θ<θt)

Log-distance 

linear PL model

Elevation 

angle range 

(degree)

Distance 

range (km)

Log-distance 

linear PL model

CE2R 

(dB)

Elevation 

angle range 

(degree)

Distance 

range (km)

A0,S

(dB) nS

σXL,S

(dB) θmin,S θmax,S dmin,S dmax,S

A0,L

(dB) nL

σXL,L

(dB) L0 σX2 θmin,L θmax,L dmin,L dmax,L

C-Rx1 27.3 1.3 1.1 5.0 25.5 1.4 6.9 -0.4 1.8 2.2 -

4.1 4.1 2.2 5.0 6.9 16.9

C-Rx2 24.6 1.4 1.9 5.0 25.5 1.4 6.9 -6.8 1.9 2.3 -

3.3 4.3 2.2 5.0 6.9 16.9

L-Rx1 9.0 1.4 2.8 5.0 47.5 0.8 6.9 -

104.4 3.1 3.6 -0.7 5.9 2.2 5.0 6.9 16.9

L-Rx2 12.9 1.4 2.8 5.0 47.5 0.8 6.9 -32.5 2.0 4.0 -

1.7 6.1 2.2 5.0 6.9 16.9

NASA/CR—2015-218486 60
(a) (b) 

Figure A13. RMS-DS vs. distance for FT2, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

Table A10. Statistics of spatial correlation, FT2, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 82 82 82 82 82 82

Mean 0.38 0.70 0.01 0.01 0.01 0.05

Median 0.82 0.96 0.04 0.02 0.02 0.19

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.73 0.51 0.80 0.80 0.80 0.82

(a) (b) 

2 4 6 8 10 12 14 16 0

200

400

600

800

1000

1200

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT2***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

2 4 6 8 10 12 14 16 0

200

400

600

800

1000

1200

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT2***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 61
(c) (d) 

Figure A14. Ricean K factor for FT2, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A11. Statistics of Ricean K factor, FT2, Latrobe, PA, 15 April 2013, flight straight away from GS. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 29.1 28.9 12.4 12.5

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 29.3 28.8 29.3 29.8 28.7 29.8 11.3 9.8 10.8 11.7 10.2 11.2

n 0.12 -0.21 0.11 0.06 -0.21 0.06 0.15 0.24 0.17 0.11 0.20 0.13

σX (dB) 2.0 4.0 2.0 2.1 4.1 2.0 1.2 1.4 1.3 1.1 1.2 1.2

Dmax(km) 1.4 1.4 1.4 1.4 1.4 1.4 0.8 0.8 0.8 0.8 0.8 0.8

DMin(km) 16.9 16.9 16.9 16.9 16.9 16.9 16.9 16.9 16.9 16.9 16.9 16.9

Statistics of K factor (dB)

Max 36.9 36.0 36.9 40.2 34.8 40.2 18.4 18.2 18.3 19.3 18.5 19.3

Min 12.8 7.9 13.6 12.9 7.6 13.4 8.2 -0.1 7.2 7.0 0.9 7.8

Median 30.3 26.9 30.3 30.3 26.8 30.3 12.6 11.8 12.3 12.7 11.9 12.3

μ 30.6 27.9 30.5 30.5 27.9 30.5 12.3 11.7 11.9 12.5 11.9 12.1

σ 2.1 4.1 2.0 2.1 4.2 2.1 1.4 1.8 1.5 1.2 1.5 1.3

A.4 Hilly Terrain Flight Track 3 

(a) (b) 

NASA/CR—2015-218486 62
(c) (d) 

(e) (f) 

(g) (h) 

Figure A15. Geometric traces for FT3, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of 

aircraft; (h) roll angle of aircraft. 

NASA/CR—2015-218486 63
(a) (b) 

(c) (d) 

Figure A16. Measured path loss vs. distance for FT1, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band 

Rx2. 

Table A12. Path loss model parameters, FT3, Latrobe, PA, 15 April 2013, flight straight toward GS. 

d<dt (θ>θt) d>dt (θ<θt)

Log-distance 

linear PL model

Elevation 

angle range 

(degree)

Distance 

range (km)

Log-distance 

linear PL model

CE2R 

(dB)

Elevation 

angle range 

(degree)

Distance 

range (km)

A0,S

(dB) nS

σXL,S

(dB) θmin,S θmax,S dmin,S dmax,S

A0,L

(dB) nL

σXL,L

(dB) L0 σX2 θmin,L θmax,L dmin,L dmax,L

C-Rx1 7.1 1.7 1.9 5.0 15.5 2.3 7.0 43.9 1.2 2.1 0.2 4.3 2.2 5.0 7.0 16.6

C-Rx2 22.4 1.5 1.2 5.0 15.5 2.3 7.0 25.5 1.4 2.2 -

0.1 4.1 2.2 5.0 7.0 16.6

L-Rx1 23.0 1.3 3.2 5.0 35.5 1.0 7.0 -73.7 2.7 4.1 -

0.1 6.7 2.2 5.0 7.0 16.6

L-Rx2 17.2 1.3 2.9 5.0 35.5 1.0 7.0 -54.3 2.4 4.0 1.1 6.7 2.2 5.0 7.0 16.6

NASA/CR—2015-218486 64
(a) (b) 

Figure A17. RMS-DS vs. distance for FT3, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

Table A13. Statistics of spatial correlation, FT3, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 101 101 101 101 101 101

Mean 0.38 0.61 0.00 0.00 0.00 0.01

Median 0.90 0.93 0.00 0.03 0.03 0.04

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.78 0.54 0.84 0.83 0.83 0.82

(a) (b) 

4 6 8 10 12 14 16 0

100

200

300

400

500

600

700

800

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT3***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

4 6 8 10 12 14 16 0

50

100

150

200

250

300

350

400

450

500

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT3***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 65
(c) (d) 

Figure A18. Ricean K factor for FT3, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A14. Statistics of Ricean K factor, FT3, Latrobe, PA, 15 April 2013, flight straight toward GS. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 26.9 26.7 13.3 14.1

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 25.5 24.6 25.5 25.6 25.1 25.6 12.7 11.2 12.3 13.4 11.9 13.1

n 0.19 0.08 0.19 0.16 0.02 0.16 0.09 0.18 0.10 0.10 0.20 0.11

σX (dB) 1.4 2.5 1.4 1.4 2.4 1.3 1.1 1.3 1.2 0.9 1.2 1.0

Dmax(km) 2.2 2.2 2.2 2.2 2.2 2.2 1.1 1.1 1.1 1.1 1.1 1.1

DMin(km) 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6

Statistics of K factor (dB)

Max 32.2 31.6 32.2 32.4 32.3 32.3 18.1 17.1 18.0 18.9 17.2 18.9

Min 13.7 9.8 14.0 14.5 10.1 14.7 6.5 2.0 5.2 6.5 1.9 5.8

Median 27.3 25.3 27.3 27.0 25.3 27.1 13.5 12.8 13.2 14.3 13.6 14.0

μ 27.5 25.7 27.5 27.2 25.6 27.2 13.3 12.9 13.1 14.2 13.8 14.0

σ 1.6 2.5 1.6 1.5 2.4 1.5 1.2 1.5 1.3 1.0 1.5 1.1

A.56 Hilly Terrain Flight Track 4 

(a) (b) 

NASA/CR—2015-218486 66
(c) (d) 

(e) (f) 

(g) (h) 

Figure A19. Geometric traces for FT4, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of 

aircraft; (h) roll angle of aircraft. 

NASA/CR—2015-218486 67
(a) (b) 

(c) (d) 

Figure A20. Measured path loss vs. distance for FT4, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band 

Rx2. 

Table A15. Path Loss Model, FT4, Latrobe, PA, 15 April 2013, flight straight away from GS. 

d<dt (θ>θt) d>dt (θ<θt)

Log-distance 

linear PL model

Elevation 

angle range 

(degree)

Distance 

range (km)

Log-distance 

linear PL model

CE2R 

(dB)

Elevation 

angle range 

(degree)

Distance 

range (km)

A0,S

(dB) nS

σXL,S

(dB) θmin,S θmax,S dmin,S dmax,S

A0,L

(dB) nL

σXL,L

(dB) L0 σX2 θmin,L θmax,L dmin,L dmax,L

C-Rx1 50.8 1.0 0.7 5.0 27.5 1.3 6.9 37.4 1.2 2.1 -

3.9 3.9 2.3 5.0 6.9 15.8

C-Rx2 45.6 1.1 1.6 5.0 27.5 1.3 6.9 19.7 1.5 2.2 -

2.8 4.0 2.3 5.0 6.9 15.8

L-Rx1 6.9 1.5 2.9 5.0 48.6 0.8 6.9 -30.9 2.0 3.9 -

1.1 6.1 2.3 5.0 6.9 15.8

L-Rx2 14.3 1.4 2.9 5.0 48.6 0.8 6.9 -38.1 2.1 4.0 -

1.7 6.2 2.3 5.0 6.9 15.8

NASA/CR—2015-218486 68
(a) (b) 

Figure A21. RMS-DS vs. distance for FT4, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

Table A16. Statistics of spatial correlation, FT4, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 83 83 83 83 83 83

Mean 0.44 0.70 -0.06 -0.06 -0.06 -0.04

Median 0.83 0.96 -0.18 -0.19 -0.19 -0.11

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.68 0.49 0.78 0.77 0.77 0.80

(a) (b) 

2 4 6 8 10 12 14 0

100

200

300

400

500

600

700

800

900

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT4***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

2 4 6 8 10 12 14 0

100

200

300

400

500

600

700

800

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT4***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 69
(c) (d) 

Figure A22. Ricean K factor for FT4, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A17. Statistics of Ricean K factor, FT4, Latrobe, PA, 15 April 2013, flight straight toward GS. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 28.9 29.1 12.4 13.3

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 27.5 27.9 27.5 28.1 28.0 28.1 11.6 10.0 11.1 12.5 10.9 12.1

n 0.27 -0.12 0.27 0.23 -0.17 0.23 0.14 0.23 0.15 0.13 0.22 0.14

σX (dB) 2.0 3.5 1.9 1.8 3.7 1.8 1.1 1.4 1.2 1.0 1.4 1.0

Dmax(km) 1.3 1.3 1.3 1.3 1.3 1.3 0.8 0.8 0.8 0.8 0.8 0.8

DMin(km) 15.8 15.8 15.8 15.8 15.8 15.8 15.8 15.8 15.8 15.8 15.8 15.8

Statistics of K factor (dB)

Max 35.8 34.3 35.8 35.6 33.7 35.6 18.4 17.5 18.4 20.5 17.7 20.4

Min 13.9 11.8 14.2 13.9 10.6 14.4 7.4 0.3 6.4 7.5 0.2 6.3

Median 29.9 26.8 29.9 30.0 26.6 30.0 12.6 11.8 12.3 13.5 12.6 13.2

μ 30.3 27.3 30.3 30.3 27.2 30.3 12.4 11.6 12.0 13.4 12.8 13.1

σ 2.3 3.5 2.2 2.1 3.8 2.0 1.3 1.7 1.4 1.1 1.7 1.2

A.6 Hilly Terrain Flight Track 5 

(a) (b) 

NASA/CR—2015-218486 70
(c) (d) 

(e) (f) 

(g) (h) 

Figure A23. Geometric traces for FT5, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of 

aircraft; (h) roll angle of aircraft. 

NASA/CR—2015-218486 71
(a) (b) 

(c) (d) 

Figure A24. Measured path loss vs. distance for FT5, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band 

Rx2. 

Table A18. Path loss model parameters, FT5, Latrobe, PA, 15 April 2013, flight straight away from GS. 

d>dt (θ<θt)

Log-distance linear PL model CE2R (dB) Elevation angle range (degree) Distance range (km)

A0,L (dB) nL σXL,L (dB) L0 σX2 θmin,L θmax,L dmin,L dmax,L

C-Rx1 23.1 1.4 1.6 -4.6 5.7 1.2 2.2 17.0 41.5

C-Rx2 37.7 1.3 1.7 -3.6 5.5 1.2 2.2 17.0 41.5

L-Rx1 -48.9 2.3 2.4 0.6 4.4 1.2 2.2 17.0 41.5

L-Rx2 -55.1 2.4 2.2 -0.1 4.3 1.2 2.2 17.0 41.5

NASA/CR—2015-218486 72
(a) (b) 

Figure A25. RMS-DS vs. distance for FT5, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

Table A19. Statistics of Spatial Correlation, FT5, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 83 83 83 83 83 83

Mean 0.27 0.81 0.27 0.25 0.25 0.15

Median 0.67 0.97 0.64 0.63 0.63 0.40

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -0.88 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.76 0.35 0.75 0.76 0.76 0.77

(a) (b) 

20 25 30 35 40 0

500

1000

1500

2000

2500

3000

3500

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT5***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

20 25 30 35 40 0

500

1000

1500

2000

2500

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT5***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 73
(c) (d) 

Figure A26. Ricean K factor for FT5, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A20. Statistics of Ricean K factor, FT5, Latrobe, PA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 29.6 29.1 15.0 15.5

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 31.5 31.4 31.5 31.6 31.5 31.6 12.3 11.9 11.9 12.7 12.3 12.4

n -0.05 -0.08 -0.05 -0.08 -0.11 -0.08 0.10 0.11 0.11 0.10 0.11 0.11

σX (dB) 1.9 2.6 1.9 1.8 2.7 1.8 0.9 1.0 1.0 0.6 0.6 0.6

Dmax(km) 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0

DMin(km) 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5

Statistics of K factor (dB)

Max 37.5 37.2 37.6 38.6 38.1 38.6 18.3 18.5 18.4 18.3 18.5 18.4

Min 13.8 11.2 14.0 17.0 13.7 17.8 11.8 8.6 8.9 11.6 11.1 11.1

Median 30.2 29.5 30.2 29.7 28.8 29.7 15.2 15.0 15.0 15.6 15.5 15.5

μ 30.5 30.1 30.5 29.8 29.2 29.8 15.2 15.0 15.0 15.6 15.5 15.5

σ 1.9 2.6 1.9 1.9 2.8 1.9 1.1 1.2 1.3 0.9 1.0 1.0

A.7 Hilly Terrain Flight Track 6 

(a) (b) 

NASA/CR—2015-218486 74
(c) (d) 

(e) (f) 

(g) (h) 

Figure A27. Geometric traces for FT6, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of AC; (g) pitch angle of AC; 

(h) roll angle of AC. 

As in the over-freshwater FT4, the hilly terrain FT6 likely also has airframe shadowing due 

to the large values of roll angle for link distances from ~19-22.5 km. Increased path loss and reduced 

K-factors result. Similar comments apply to hilly terrain FT7. 

NASA/CR—2015-218486 75
(a) (b) 

Figure A28. Measured path loss vs. distance for FT6, Latrobe, PA (a) C-band Rx1 & 2; (b) L-band Rx1 & 2. 

(a) (b) 

Figure A29. RMS-DS vs. distance for FT6, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

Table A21. Statistics of spatial correlation, FT6, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 90 90 90 90 90 90

Mean 0.15 0.61 0.03 0.02 0.02 0.04

Median 0.39 0.95 0.10 0.05 0.05 0.12

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.76 0.60 0.77 0.77 0.77 0.79

16 17 18 19 20 21 22 23 24 0

500

1000

1500

2000

2500

3000

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT6***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

16 17 18 19 20 21 22 23 24 0

200

400

600

800

1000

1200

1400

1600

1800

2000

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT6***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 76
(a) (b) 

(c) (d) 

Figure A30. Ricean K factor for FT6, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A22. Statistics of Ricean K factor, FT6, Latrobe, PA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 25.1 25.6 15.1 15.3

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 16.6 17.8 16.5 17.9 18.1 17.9 17.2 17.4 18.1 16.7 17.0 17.9

n 0.52 0.39 0.52 0.46 0.38 0.47 -0.10 -0.13 -0.15 -0.06 -0.10 -0.13

σX (dB) 2.6 3.1 2.6 2.2 2.8 2.2 1.6 1.7 1.8 1.6 1.8 2.0

Dmax(km) 15.6 15.6 15.6 15.6 15.6 15.6 15.6 15.6 15.6 15.6 15.6 15.6

DMin(km) 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0

Statistics of K factor (dB)

Max 34.1 34.1 34.1 34.5 33.3 34.5 33.4 30.4 33.4 33.9 34.0 33.8

Min 8.4 2.8 7.8 13.3 8.8 13.2 -33.2 0.0 0.0 -35.6 0.0 0.0

Median 26.3 25.1 26.3 26.4 25.1 26.4 15.4 15.0 15.3 15.6 15.2 15.5

μ 26.4 25.2 26.4 26.4 25.2 26.5 15.4 15.2 15.4 15.6 15.4 15.6

σ 2.9 3.2 2.9 2.4 2.9 2.4 1.6 1.7 1.8 1.6 1.8 2.0

NASA/CR—2015-218486 77
A.8 Hilly Terrain Flight Track 7 

(a) (b) 

(c) (d) 

(e) (f) 

NASA/CR—2015-218486 78
(g) (h) 

Figure A31. Geometric traces for FT7, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of AC; (g) pitch angle of AC; 

(h) roll angle of AC. 

(a) (b) 

Figure A32. Measured path loss vs. distance for FT7, Latrobe, PA (a) C-band Rx1 & 2; (b) L-band Rx1 & 2. 

(a) (b) 

Figure A33. RMS-DS vs. distance for FT7, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

16 17 18 19 20 21 22 23 0

500

1000

1500

2000

2500

3000

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT7***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

16 17 18 19 20 21 22 23 0

200

400

600

800

1000

1200

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT7***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 79
Table A23. Statistics of spatial correlation, FT5, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 92 92 92 92 92 92

Mean 0.16 0.64 0.06 0.05 0.05 0.04

Median 0.37 0.95 0.18 0.12 0.12 0.12

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.75 0.57 0.78 0.77 0.77 0.78

(a) (b) 

(c) (d) 

Figure A34. Ricean K factor for FT7, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

NASA/CR—2015-218486 80
Table A24. Statistics of Ricean K factor, FT7, Latrobe, PA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 25.0 14.2 15.3 15.3

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 18.9 20.2 18.9 18.5 19.2 18.6 15.7 15.9 16.1 15.6 15.8 16.3

n 0.37 0.25 0.37 0.39 0.30 0.39 -0.01 -0.04 -0.04 0.00 -0.03 -0.04

σX (dB) 2.3 2.6 2.3 2.4 2.6 2.2 1.0 1.4 1.4 1.5 1.6 1.7

Dmax(km) 15.8 15.8 15.8 15.8 15.8 15.8 15.8 15.8 15.8 15.8 15.8 15.8

DMin(km) 23.8 23.8 23.8 23.8 23.8 23.8 23.8 23.8 23.8 23.8 23.8 23.8

Statistics of K factor (dB)

Max 33.5 33.3 33.5 33.9 34.1 33.9 30.1 29.5 30.1 30.2 29.6 30.1

Min 11.0 0.5 11.2 -39.8 5.9 11.8 -32.6 0.0 0.0 -74.0 0.0 0.0

Median 25.9 24.9 25.9 25.8 24.8 25.8 15.4 15.1 15.3 15.5 15.3 15.5

μ 25.9 25.0 25.9 25.6 24.7 25.6 15.5 15.3 15.5 15.6 15.5 15.6

σ 2.5 2.7 2.5 2.6 2.7 2.4 1.0 1.4 1.4 1.5 1.6 1.7

A.9 Hilly Terrain Flight Track 8 

(a) (b) 

(c) (d) 

NASA/CR—2015-218486 81
(e) (f) 

(g) (h) 

Figure A35. Geometric traces for FT8, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of 

AC; (h) roll angle of aircraft. 

(a) (b) 

NASA/CR—2015-218486 82
(c) (d) 

Figure A36. Measured path loss vs. distance for FT8, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band 

Rx2. 

In FTs 8, 9, and 13, the aircraft flew over a mountain ridge, and the LOS component was 

blocked. Mountain ridge shadowing can clearly be seen in the path loss figures. The C-band path loss 

reached its “ceiling” of approximately 145 dB (the limit of our sounder). Hence the path loss models 

for these FTs for the larger distance segments are NOT valid, but are shown for completeness. The 

results for RMS-DS and K-factors at the largest distances are also likely invalid.

Table A25. Path loss model parameters, FT8, Latrobe, PA, 15 April 2013, flight straight toward GS. 

d<dt (θ>θt) d>dt (θ<θt)

Log-distance linear 

PL model

Elevation 

angle range 

(degree)

Distance 

range (km)

Log-distance linear 

PL model

CE2R 

(dB)

Elevation 

angle range 

(degree)

Distance 

range (km)

A0,S

(dB) nS

σXL,S

(dB) θmin,S θmax,S dmin,S dmax,S

A0,L

(dB) nL

σXL,L

(dB) L0 σX2 θmin,L θmax,L dmin,L dmax,L

C-Rx1 -6.9 1.9 1.6 5.0 18.3 2.5 9.2 -

284.8 5.9 2.4 2.2 4.4 2.8 5.0 9.2 17.5

C-Rx2 12.5 1.6 1.3 5.0 18.3 2.5 9.2 -

113.8 3.4 1.3 0.9 3.4 3.5 5.0 9.2 13.3

L-Rx1 -23.0 2.0 2.7 5.0 39.6 1.2 9.2 -

391.4 7.2 3.2 7.0 6.1 2.8 5.0 9.2 17.5

L-Rx2 30.1 1.2 2.3 5.0 39.6 1.2 9.2 -

613.9 10.3 3.8 3.5 8.0 2.8 5.0 9.2 17.5

(a) (b) 

Figure A37. RMS-DS vs. distance for FT8, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

4 6 8 10 12 14 16 0

500

1000

1500

2000

2500

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT8***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

4 6 8 10 12 0

50

100

150

200

250

300

350

400

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT8***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 83
Table A26. Statistics of spatial correlation, FT8, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 82 82 82 82 82 82

Mean -0.03 0.53 0.02 -0.01 -0.01 0.02

Median -0.06 0.83 0.03 -0.04 -0.04 0.05

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -0.99 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.72 0.57 0.76 0.76 0.76 0.73

(a) (b) 

(c) (d) 

Figure A38. Ricean K factor for FT8, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

NASA/CR—2015-218486 84
Table A27. Statistics of Ricean K factor, FT8, Latrobe, PA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 29.3 28.5 13.0 13.2

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 30.4 27.9 30.4 29.6 27.5 29.6 11.9 10.2 11.7 11.7 9.9 11.5

n -0.04 0.12 -0.04 0.01 0.14 0.01 0.14 0.26 0.13 0.18 0.31 0.17

σX (dB) 1.9 2.6 1.9 2.0 2.5 2.0 1.2 1.7 1.3 0.9 1.5 1.0

Dmax(km) 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

DMin(km) 17.5 17.5 17.5 13.3 13.3 13.3 17.5 17.5 17.5 17.5 17.5 17.5

Statistics of K factor (dB)

Max 35.3 34.5 35.3 34.5 34.6 34.5 18.8 18.5 18.6 17.7 17.7 18.0

Min 14.1 10.7 14.3 10.7 3.7 10.4 4.0 0.0 2.6 4.8 0.0 5.4

Median 30.0 28.9 30.0 29.7 28.7 29.7 13.2 12.6 12.9 13.3 12.7 13.0

μ 30.2 29.3 30.1 29.8 29.0 29.8 13.1 12.7 12.8 13.1 12.7 12.8

σ 1.9 2.6 1.9 2.0 2.5 2.0 1.3 1.9 1.3 1.1 1.8 1.2

A.10 Hilly Terrain Flight Track 9

(a) (b) 

(c) (d) 

NASA/CR—2015-218486 85
(e) (f) 

(g) (h) 

Figure A39. Geometric traces for FT9, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of 

aircraft; (h) roll angle of aircraft. 

(a) (b) 

NASA/CR—2015-218486 86
(c) (d) 

Figure A40. Measured path loss vs. distance for FT9, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band 

Rx2. 

Table A28. Path loss model parameters, FT9, Latrobe, PA, 15 April 2013, flight straight away from GS. 

d<dt (θ>θt) d>dt (θ<θt)

Log-distance 

linear PL model

Elevation 

angle range 

(degree)

Distance 

range (km)

Log-distance linear 

PL model

CE2R 

(dB)

Elevation 

angle range 

(degree)

Distance 

range (km)

A0,S

(dB) nS

σXL,S

(dB) θmin,S θmax,S dmin,S dmax,S

A0,L

(dB) nL

σXL,L

(dB) L0 σX2 θmin,L θmax,L dmin,L dmax,L

C-Rx1 0.1 1.8 1.0 5.0 23.5 2.0 9.3 -76.5 2.9 2.1 -

1.4 3.3 2.8 5.0 9.3 17.0

C-Rx2 6.5 1.7 1.7 5.0 23.5 2.0 9.3 -

189.1 4.5 1.8 -

1.2 3.3 2.8 5.0 9.3 17.0

L-Rx1 13.8 1.4 2.6 5.0 40.3 1.2 9.3 -

554.2 9.5 2.9 2.4 6.3 2.8 5.0 9.3 17.0

L-Rx2 1.5 1.5 2.5 5.0 40.3 1.2 9.3 -

539.6 9.3 3.8 1.5 6.8 2.8 5.0 9.3 17.0

(a) (b) 

Figure A41. RMS-DS vs. distance for FT9, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

2 4 6 8 10 12 14 16 0

100

200

300

400

500

600

700

800

900

1000

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT9***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

2 4 6 8 10 12 14 16 0

100

200

300

400

500

600

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT9***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 87
Table A29. Statistics of spatial correlation, FT9, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 98 98 98 98 98 98

Mean 0.44 0.65 0.04 0.05 0.05 0.08

Median 0.82 0.89 0.09 0.11 0.11 0.25

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.68 0.49 0.73 0.73 0.73 0.76

(a) (b) 

(c) (d) 

Figure A42. Ricean K factor for FT9, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

NASA/CR—2015-218486 88
Table A30. Statistics of Ricean K factor, FT9, Latrobe, PA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 27.1 27.3 12.5 13.1

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 25.6 24.2 25.5 26.2 24.7 26.2 11.4 9.8 10.9 11.8 10.1 11.3

n 0.23 0.31 0.24 0.19 0.26 0.19 0.15 0.25 0.16 0.16 0.28 0.18

σX (dB) 1.5 2.0 1.5 1.6 2.3 1.6 1.2 1.4 1.3 1.1 1.3 1.1

Dmax(km) 3.0 3.0 3.0 3.0 3.0 3.0 2.8 2.8 2.8 2.8 2.8 2.8

DMin(km) 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0

Statistics of K factor (dB)

Max 33.1 33.0 33.1 34.6 34.7 34.6 17.5 19.5 19.1 18.1 18.4 18.5

Min 14.1 10.6 14.3 15.6 9.5 15.7 6.5 -0.4 6.7 7.5 -5.7 7.2

Median 27.5 26.8 27.5 27.8 26.9 27.8 12.7 12.1 12.3 13.3 12.7 13.0

μ 27.5 27.0 27.5 27.9 27.2 27.9 12.5 11.9 12.1 13.0 12.5 12.6

σ 1.7 2.2 1.7 1.7 2.4 1.7 1.3 1.7 1.4 1.2 1.7 1.3

A.11 Hilly Terrain Flight Track 10

(a) (b) 

(c) (d) 

NASA/CR—2015-218486 89
(e) (f) 

(g) (h) 

Figure A43. Geometric traces for FT10, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of 

aircraft; (h) roll angle of aircraft. 

(a) (b) 

Figure A44. Measured path loss vs. distance for FT10, Latrobe, PA (a) C-band Rx1 & 2; (b) L-band Rx1 & 2. 

As seen from Figure A.44, the C-band attenuation also hit its “ceiling” in FT10. Thus the Cband path loss and RMS-DS results for this FT are likely invalid. Because of the smaller number of 

PDPs and much slower PDP update rate from this FT, the K-factors as well are likely inaccurate. 

NASA/CR—2015-218486 90
(a) (b) 

Figure A45. RMS-DS vs. distance for FT10, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

Table A31. Statistics of spatial correlation, FT10, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 87 87 87 87 87 87

Mean 0.31 0.61 0.35 0.19 0.19 0.26

Median 0.34 0.93 0.73 0.43 0.43 0.61

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -0.82 -0.99 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.53 0.57 0.72 0.74 0.74 0.74

(a) (b) 

19.5 20 20.5 21 21.5 22 22.5 23 23.5 0

500

1000

1500

2000

2500

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT10***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

19.5 20 20.5 21 21.5 22 22.5 23 23.5 0

2

4

6

8

10

12

14

16

18

20

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT10***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 91
(c) (d) 

Figure A46. Ricean K factor for FT10, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A32. Statistics of Ricean K factor, FT10, Latrobe, PA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 28.1 27.7 13.1 13.4

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 51.9 50.2 51.8 28.7 26.4 28.6 7.4 7.7 7.9 2.5 2.7 2.8

n -1.10 -1.03 -1.10 -0.02 0.08 -0.01 0.38 0.34 0.34 0.63 0.60 0.60

σX (dB) 2.3 2.5 2.3 2.1 2.2 2.1 5.1 5.6 5.6 5.7 5.6 5.6

Dmax(km) 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.2

DMin(km) 23.9 23.9 23.9 23.9 23.9 23.9 23.9 23.9 23.9 23.9 23.9 23.9

Statistics of K factor (dB)

Max 39.0 38.4 39.0 37.7 37.2 37.7 26.8 26.8 26.8 26.6 26.7 26.5

Min 20.2 16.2 21.4 20.6 16.8 20.6 -33.4 0.0 0.0 -61.8 0.0 0.0

Median 29.0 28.8 29.0 28.3 28.1 28.3 15.5 14.8 15.0 15.8 15.2 15.3

μ 28.7 28.5 28.7 28.2 28.1 28.2 16.3 16.1 16.2 16.4 16.3 16.3

σ 2.8 2.9 2.8 2.1 2.2 2.1 5.1 5.6 5.6 5.8 5.7 5.7

A.12 Hilly Terrain Flight Track 11

(a) (b) 

NASA/CR—2015-218486 92
(c) (d) 

(e) (f) 

(g) (h) 

Figure A47. Geometric traces for FT11, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of AC; (g) pitch angle of AC; 

(h) roll angle of AC. 

NASA/CR—2015-218486 93
(a) (b) 

Figure A48. Measured path loss vs. distance for FT11, Latrobe, PA (a) C-band Rx1 & 2; (b) L-band Rx1 & 2. 

(a) (b) 

Figure A49. RMS-DS vs. distance for FT11, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

Table A33. Statistics of spatial correlation, FT11, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 87 87 87 87 87 87

Mean 0.12 0.63 0.03 0.03 0.03 0.12

Median 0.31 0.95 0.08 0.09 0.09 0.37

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.76 0.60 0.79 0.78 0.78 0.79

14 15 16 17 18 19 0

500

1000

1500

2000

2500

3000

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT11***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

14 15 16 17 18 19 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT11***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 94
(a) (b) 

(c) (d) 

Figure A50. Ricean K factor for FT11, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A34. Statistics of Ricean K factor, FT11, Latrobe, PA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 25.0 25.1 14.9 14.8

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 13.9 14.3 13.8 16.1 17.8 16.1 15.6 16.5 16.3 16.3 17.8 17.5

n 0.80 0.69 0.80 0.65 0.47 0.65 -0.03 -0.11 -0.08 -0.08 -0.20 -0.17

σX (dB) 2.9 3.2 2.9 2.7 3.1 2.6 1.2 1.8 1.8 1.3 2.1 2.1

Dmax(km) 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3

DMin(km) 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8

Statistics of K factor (dB)

Max 36.6 36.7 36.6 35.5 35.2 35.5 21.8 22.1 22.1 21.5 -24.4 21.8

Min 11.0 9.2 11.4 10.2 8.1 10.0 5.1 0.0 0.0 4.0 0.0 0.0

Median 26.3 25.1 26.3 26.2 25.1 26.2 15.2 14.8 15.0 15.0 14.6 14.8

μ 26.3 25.0 26.3 26.2 25.1 26.2 15.4 15.2 15.3 15.2 15.0 15.1

σ 3.2 3.3 3.1 2.9 3.2 2.8 1.2 1.8 1.8 1.3 2.1 2.1

NASA/CR—2015-218486 95
A.13 Hilly Terrain Flight Track 12

(a) (b) 

(c) (d) 

(e) (f) 

NASA/CR—2015-218486 96
(g) (h) 

Figure A51. Geometric traces for FT12, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of AC; (g) pitch angle of AC; 

(h) roll angle of AC. 

(a) (b) 

Figure A52. Measured path loss vs. distance for FT12, Latrobe, PA (a) C-band Rx1 & 2; (b) L-band Rx1 & 2. 

(a) (b) 

Figure A53. RMS-DS vs. distance for FT12, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

13 14 15 16 17 18 19 0

500

1000

1500

2000

2500

3000

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT12***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

13 14 15 16 17 18 19 0

500

1000

1500

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT12***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 97
Table A35. Statistics of spatial correlation, FT12, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 83 83 83 83 83 83

Mean 0.15 0.66 0.03 0.04 0.04 0.08

Median 0.38 0.95 0.10 0.11 0.11 0.23

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.76 0.55 0.79 0.79 0.79 0.78

(a) (b) 

(c) (d) 

Figure A54. Ricean K factor for FT12, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

NASA/CR—2015-218486 98
Table A36. Statistics of Ricean K factor, FT12, Latrobe, PA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 25.1 25.2 14.8 14.7

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 17.6 18.0 17.5 18.2 19.2 18.2 16.6 17.2 17.6 17.1 17.7 18.2

n 0.60 0.47 0.60 0.55 0.40 0.56 -0.10 -0.16 -0.18 -0.13 -0.20 -0.22

σX (dB) 2.6 3.0 2.6 2.6 3.0 2.6 1.8 2.1 2.2 2.1 2.2 2.4

Dmax(km) 12.3 12.3 12.3 12.3 12.3 12.3 12.3 12.3 12.3 12.3 12.3 12.3

DMin(km) 19.6 19.6 19.6 19.6 19.6 19.6 19.6 19.6 19.6 19.6 19.6 19.6

Statistics of K factor (dB)

Max 34.8 33.4 34.8 34.5 34.3 34.5 34.9 32.7 34.9 33.7 34.0 33.7

Min 12.5 7.4 12.4 11.8 8.8 11.4 -30.7 0.0 0.0 3.8 0.0 0.0

Median 26.3 24.9 26.3 26.4 25.1 26.4 15.2 14.8 15.0 15.1 14.7 15.0

μ 26.6 25.0 26.6 26.5 25.1 26.5 15.1 14.9 15.0 15.0 14.9 15.0

σ 2.9 3.1 2.9 2.9 3.1 2.9 1.8 2.1 2.3 2.1 2.3 2.5

A.14 Hilly Terrain Flight Track 13

(a) (b) 

(c) (d) 

NASA/CR—2015-218486 99
(e) (f) 

(g) (h) 

Figure A55. Geometric traces for FT13, Latrobe, PA: (a) flight track in ECEF coordinates; (b) aircraft velocity; (c) altitude 

difference between aircraft and ground station; (d) azimuth angle; (e) elevation angle; (f) heading of aircraft; (g) pitch angle of 

aircraft; (h) roll angle of aircraft. 

As noted for hilly terrain FTs 8 and 9, the path loss for hilly terrain FT13 is likely not valid 

for large values of distance, due to terrain shadowing. RMS-DS and K-factor results for the largedistance section of FT13 are also likely inaccurate. 

(a) (b) 

NASA/CR—2015-218486 100
(c) (d) 

Figure A56. Measured path loss vs. distance for FT13, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band 

Rx2. 

Table A37. Path loss model parameters, FT13, Latrobe, PA, 15 April 2013, flight straight toward GS. 

d<dt (θ>θt) d>dt (θ<θt)

Log-distance 

linear PL model

Elevation 

angle range 

(degree)

Distance 

range (km)

Log-distance linear 

PL model

CE2R 

(dB)

Elevation 

angle range 

(degree)

Distance 

range (km)

A0,S

(dB) nS

σXL,S

(dB) θmin,S θmax,S dmin,S dmax,S

A0,L

(dB) nL

σXL,L

(dB) L0 σX2 θmin,L θmax,L dmin,L dmax,L

C-Rx1 -

82.9 3.1 1.5 5.0 18.6 2.5 9.2 -

180.7 4.4 2.5 2.5 4.0 2.8 5.0 9.2 17.2

C-Rx2 -

42.4 2.5 1.1 5.0 18.6 2.5 9.2 -

186.6 4.5 2.6 2.5 4.0 2.8 5.0 9.2 17.2

L-Rx1 -8.2 1.7 2.5 5.0 20.8 2.2 9.2 -

529.8 9.1 3.9 3.5 7.2 2.8 5.0 9.2 17.2

L-Rx2 3.6 1.6 2.5 5.0 18.1 2.6 9.2 -

589.0 10.0 3.5 3.9 7.5 2.8 5.0 9.2 17.2

(a) (b) 

Figure A57. RMS-DS vs. distance for FT13, Latrobe, PA of C-band (a) Rx1; (b) Rx2. 

4 6 8 10 12 14 16 0

500

1000

1500

2000

2500

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT13***C-band Rx1

Measured

Moving Averaged 100

Moving Averaged 1000

4 6 8 10 12 14 16 0

200

400

600

800

1000

1200

1400

Link Distance (km)

RMS-DS (ns)

LatrobePA***04-15-2013***FT13***C-band Rx2

Measured

Moving Averaged 100

Moving Averaged 1000

NASA/CR—2015-218486 101
Table A38. Statistics of spatial correlation, FT13, Latrobe, PA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 79 79 79 79 79 79

Mean -0.01 0.55 0.01 -0.03 -0.03 -0.03

Median -0.06 0.87 0.03 -0.07 -0.07 -0.07

Max 1.00 1.00 1.00 1.00 1.00 1.00

Min -1.00 -0.99 -1.00 -1.00 -1.00 -1.00

Standard deviation 0.74 0.57 0.74 0.74 0.74 0.72

(a) (b) 

(c) (d) 

Figure A58. Ricean K factor for FT13, Latrobe, PA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

NASA/CR—2015-218486 102
Table A39. Statistics of Ricean K factor, FT13, Latrobe, PA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 29.4 29.3 13.0 13.2

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 28.8 26.3 28.8 29.5 27.7 29.5 12.0 10.2 11.7 12.2 10.3 11.9

n 0.16 0.30 0.16 0.05 0.14 0.05 0.12 0.26 0.13 0.12 0.27 0.13

σX (dB) 1.7 2.4 1.7 1.6 2.1 1.6 1.1 1.5 1.2 1.0 1.5 1.1

Dmax(km) 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

DMin(km) 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.2

Statistics of K factor (dB)

Max 33.9 34.0 33.9 33.6 33.5 33.6 18.4 19.9 19.8 17.1 17.1 17.0

Min 13.9 11.2 14.3 14.4 11.5 14.8 7.1 -0.5 7.3 7.2 -5.8 7.0

Median 30.1 28.9 30.1 29.9 28.9 29.9 13.2 12.6 12.9 13.4 12.8 13.1

μ 30.5 29.5 30.5 30.2 29.4 30.1 12.9 12.5 12.7 13.2 12.8 12.9

σ 1.8 2.6 1.8 1.6 2.2 1.6 1.2 1.7 1.2 1.1 1.7 1.2

A.15 Over Sea Flight Track 1 

The inter- and intra-band spatial correlations reported previously for the over-sea FTs in [6] 

were based on arbitrarily-selected values of window length. Since we have now determined the 

appropriate stationarity distance, we provide updated results for spatial correlation for the over sea 

setting, computed over the stationarity distance. The Ricean K-factors for the over sea setting are 

also included in this section. 

Table A40. Statistics of spatial correlation, FT1, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 79 79 79 79 79 79

Mean 0.60 0.37 0.00 0.00 0.00 0.00

Median 0.98 0.80 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.63 0.64 0.64 0.72

Min -1.00 -0.97 -0.75 -0.77 -0.77 -0.70

Standard deviation 0.64 0.69 0.14 0.14 0.14 0.14

(a) (b) 

NASA/CR—2015-218486 103
(c) (d) 

Figure A59. Ricean K factor for FT1, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A41. Statistics of Ricean K factor, FT1, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 30.4 29.9 12.8 12.8

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 29.1 25.1 29.0 29.0 25.6 29.0 12.0 11.4 11.8 12.1 11.5 11.9

n 0.17 0.22 0.17 0.12 0.15 0.12 0.07 0.10 0.08 0.07 0.10 0.08

σX (dB) 1.7 3.4 1.7 1.5 3.2 1.5 1.0 1.0 1.0 0.9 0.9 0.9

Dmax(km) 3.0 3.0 3.0 3.0 3.0 3.0 2.2 2.2 2.2 2.2 2.2 2.2

DMin(km) 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.1

Statistics of K factor (dB)

Max 35.5 34.4 35.4 34.0 33.6 34.0 20.2 19.9 20.4 19.7 19.6 19.9

Min 13.0 11.3 13.4 13.4 11.3 13.8 10.3 7.4 10.0 10.6 4.2 10.4

Median 31.3 28.0 31.3 30.6 27.6 30.6 12.9 12.7 12.8 12.9 12.7 12.8

μ 31.7 28.7 31.6 30.9 28.4 30.9 12.7 12.5 12.6 12.8 12.5 12.6

σ 2.0 3.6 2.0 1.7 3.3 1.6 1.1 1.2 1.1 1.0 1.1 1.0

A.16 Over Sea Flight Track 2

Table A42. Statistics of spatial correlation, FT2, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 79 79 79 79 79 79

Mean 0.72 0.78 0.00 0.00 0.00 0.00

Median 0.96 0.97 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.76 0.71 0.71 0.78

Min -1.00 -1.00 -0.73 -0.78 -0.78 -0.86

Standard deviation 0.48 0.41 0.13 0.14 0.14 0.14

NASA/CR—2015-218486 104
(a) (b) 

(c) (d) 

Figure A60. Ricean K factor for FT2, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A43. Statistics of Ricean K factor, FT2, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 29.6 29.0 12.6 12.8

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 31.4 26.5 31.4 31.3 26.3 31.3 11.1 10.4 10.7 11.6 11.0 11.3

n -0.03 0.08 -0.03 -0.05 0.06 -0.05 0.07 0.09 0.08 0.06 0.08 0.07

σX (dB) 2.0 3.3 2.0 2.1 3.4 2.1 0.9 1.1 1.0 0.9 1.1 1.1

Dmax(km) 1.8 1.8 1.8 1.8 1.8 1.8 1.2 1.2 1.2 1.2 1.2 1.2

DMin(km) 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2

Statistics of K factor (dB)

Max 34.7 34.5 34.7 35.6 34.8 35.6 17.5 17.5 17.9 17.2 -17.0 17.8

Min 11.1 10.6 11.8 11.4 10.7 12.1 8.0 1.3 1.7 3.0 0.0 3.7

Median 30.6 28.3 30.6 30.1 27.8 30.1 12.8 12.5 12.6 13.0 12.8 12.9

μ 31.0 29.1 31.0 30.4 28.6 30.4 12.7 12.6 12.6 12.9 12.8 12.8

σ 2.0 3.4 2.0 2.2 3.4 2.2 1.3 1.6 1.5 1.2 1.5 1.4

NASA/CR—2015-218486 105
A.17 Over Sea Flight Track 3

Table A44. Statistics of spatial correlation, FT3, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 82 82 82 82 82 82

Mean 0.72 0.44 0.00 0.00 0.00 0.00

Median 0.97 0.85 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.38 0.52 0.52 0.46

Min -1.00 -0.95 -0.40 -0.39 -0.39 -0.68

Standard deviation 0.52 0.65 0.10 0.10 0.10 0.10

(a) (b) 

(c) (d) 

Figure A61. Ricean K factor for FT3, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

NASA/CR—2015-218486 106
Table A45. Statistics of Ricean K factor, FT3, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 29.9 29.9 13.9 14.0

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 32.6 29.6 32.6 31.6 28.8 31.6 12.2 12.0 12.0 12.9 12.8 12.8

n -0.05 0.00 -0.05 -0.03 0.02 -0.03 0.05 0.06 0.06 0.03 0.04 0.04

σX (dB) 2.0 2.7 2.0 1.8 2.6 1.8 0.8 0.9 0.9 0.8 0.8 0.8

Dmax(km) 18.8 18.8 18.8 18.8 18.8 18.8 18.8 18.8 18.8 18.8 18.8 18.8

DMin(km) 49.0 49.0 49.0 49.0 49.0 49.0 49.0 49.0 49.0 49.0 49.0 49.0

Statistics of K factor (dB)

Max 35.0 34.8 34.9 34.4 34.3 34.4 18.3 18.4 18.7 18.9 19.0 19.2

Min 12.8 11.3 13.3 13.0 11.6 13.4 10.2 7.1 7.8 11.2 10.9 11.0

Median 30.9 29.5 30.9 30.7 29.4 30.7 14.1 14.1 14.2 14.0 14.1 14.1

μ 31.4 30.2 31.4 31.0 30.0 31.0 14.0 14.1 14.1 14.0 14.0 14.0

σ 2.1 2.7 2.0 1.8 2.6 1.8 1.0 1.0 1.0 0.8 0.9 0.9

A.18 Over Sea Flight Track 4

Table A46. Statistics of spatial correlation, FT4, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 85 85 85 85 85 85

Mean 0.30 0.68 0.00 0.00 0.00 0.00

Median 0.69 0.96 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.87 0.85 0.85 0.83

Min -1.00 -1.00 -0.86 -0.87 -0.87 -0.83

Standard deviation 0.75 0.56 0.13 0.13 0.13 0.13

(a) (b) 

NASA/CR—2015-218486 107
(c) 

Figure A62. Ricean K factor for FT4, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1. 

Table A47. Statistics of Ricean K factor, FT4, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 26.2 26.4 15.2 14.0

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 31.3 29.9 31.3 33.1 31.0 33.1 19.8 19.2 20.1 19.0 19.9

n -0.17 -0.17 -0.17 -0.25 -0.22 -0.25 -0.19 -0.17 -0.20 -0.16 -0.19

σX (dB) 2.7 3.1 2.7 2.5 2.9 2.5 1.8 1.6 1.9 1.7 2.0

Dmax(km) 19.5 19.5 19.5 19.5 19.5 19.5 19.5 19.5 19.5 19.5 19.5 19.5

DMin(km) 26.8 26.8 26.8 26.8 26.8 26.8 26.8 26.8 26.8 26.8 26.8 26.8

Statistics of K factor (dB)

Max 33.7 33.3 33.7 34.0 33.8 33.9 31.6 30.8 31.6 30.4 31.2

Min 13.6 6.8 13.8 13.6 11.3 13.9 -51.3 0.0 0.0 0.0 0.0

Median 27.4 25.9 27.4 27.4 25.9 27.4 15.5 15.3 15.4 15.4 15.5

μ 27.9 26.2 27.9 27.7 26.2 27.7 15.4 15.4 15.4 15.5 15.6

σ 2.7 3.1 2.7 2.5 3.0 2.5 1.9 1.7 2.0 1.8 2.1

A.19 Over Sea Flight Track 5

Table A48. Statistics of spatial correlation, FT5, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 84 84 84 84 84 84

Mean 0.25 0.55 0.00 0.00 0.00 0.00

Median 0.62 0.87 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.74 0.78 0.78 0.80

Min -1.00 -1.00 -0.81 -0.83 -0.83 -0.77

Standard deviation 0.77 0.60 0.12 0.12 0.12 0.12

NASA/CR—2015-218486 108
(a) (b) 

(c) (d) 

Figure A63. Ricean K factor for FT5, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A49. Statistics of Ricean K factor, FT5, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 26.2 26.2 15.2 15.2

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 30.5 28.9 30.5 31.1 29.3 31.1 14.3 14.4 14.5 14.0 14.1 14.2

n -0.14 -0.14 -0.14 -0.17 -0.16 -0.17 0.05 0.04 0.04 0.06 0.05 0.05

σX (dB) 2.6 3.1 2.6 2.5 3.0 2.5 1.2 1.2 1.3 1.2 1.2 1.3

Dmax(km) 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3

DMin(km) 26.8 26.8 26.8 26.8 26.8 26.8 26.8 26.8 26.8 26.8 26.8 26.8

Statistics of K factor (dB)

Max 36.3 35.5 36.3 35.1 34.7 35.0 32.3 31.5 32.3 32.3 32.3 32.3

Min 13.2 5.8 13.6 14.1 8.9 14.4 6.1 0.0 4.9 5.2 0.1 0.0

Median 27.2 25.6 27.3 27.2 25.7 27.2 15.3 15.3 15.3 15.4 15.3 15.4

μ 27.6 25.9 27.6 27.5 26.0 27.5 15.5 15.4 15.5 15.5 15.5 15.5

σ 2.6 3.1 2.6 2.6 3.0 2.6 1.2 1.2 1.3 1.2 1.2 1.3

NASA/CR—2015-218486 109
A.20 Over Sea Flight Track 6

Table A50. Statistics of spatial correlation, FT6, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 83 83 83 83 83 83

Mean 0.11 0.54 0.00 0.00 0.00 0.00

Median 0.35 0.87 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.74 0.78 0.78 0.66

Min -1.00 -1.00 -0.75 -0.84 -0.84 -0.70

Standard deviation 0.80 0.60 0.12 0.12 0.12 0.12

(a) (b) 

(c) (d) 

Figure A64. Ricean K factor for FT6, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

NASA/CR—2015-218486 110
Table A51. Statistics of Ricean K factor, FT6, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 26.1 26.0 14.9 14.7

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 29.3 28.9 29.2 30.6 29.3 30.6 16.0 16.1 16.2 17.6 17.0 17.2

n -0.08 -0.13 -0.08 -0.16 -0.16 -0.15 -0.04 -0.05 -0.05 -0.11 -0.08 -0.09

σX (dB) 2.8 3.1 2.8 2.5 2.9 2.5 1.3 1.3 1.3 2.4 1.6 1.7

Dmax(km) 18.9 18.9 18.9 18.9 18.9 18.9 18.9 18.9 18.9 18.9 18.9 18.9

DMin(km) 27.3 27.3 27.3 27.3 27.3 27.3 27.3 27.3 27.3 27.3 27.3 27.3

Statistics of K factor (dB)

Max 35.3 34.2 35.3 35.7 35.6 35.7 23.1 22.8 23.5 31.8 32.4 32.4

Min 12.5 7.5 12.9 13.7 9.9 14.0 -38.6 0.0 0.0 -40.8 0.0 0.0

Median 27.4 25.8 27.4 27.0 25.6 27.0 15.0 15.0 15.0 15.1 15.1 15.2

μ 27.8 26.1 27.8 27.3 25.9 27.3 15.1 15.1 15.2 15.3 15.3 15.4

σ 2.8 3.1 2.8 2.6 2.9 2.6 1.3 1.3 1.3 2.5 1.6 1.7

A.21 Over Sea Flight Track 7

Table A52. Statistics of spatial correlation, FT7, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 83 83 83 83 83 83

Mean 0.07 0.52 0.00 0.00 0.00 0.00

Median 0.22 0.84 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.58 0.68 0.68 0.55

Min -1.00 -1.00 -0.64 -0.61 -0.61 -0.60

Standard deviation 0.80 0.60 0.12 0.12 0.12 0.12

(a) (b) 

NASA/CR—2015-218486 111
(c) (d) 

Figure A65. Ricean K factor for FT7, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A53. Statistics of Ricean K factor, FT7, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 26.3 26.1 14.6 14.6

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 29.9 28.7 29.9 32.7 31.7 32.8 14.3 14.8 14.9 13.8 14.3 14.4

n -0.11 -0.13 -0.11 -0.24 -0.27 -0.25 0.02 0.00 -0.01 0.04 0.02 0.01

σX (dB) 2.4 3.0 2.4 2.5 2.9 2.4 0.8 1.1 1.1 0.9 1.3 1.3

Dmax(km) 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3

DMin(km) 27.8 27.8 27.8 27.8 27.8 27.8 27.8 27.8 27.8 27.8 27.8 27.8

Statistics of K factor (dB)

Max 34.3 35.6 35.4 34.0 34.2 34.4 19.7 -18.7 -22.1 19.8 21.2 22.0

Min 13.7 8.0 13.9 15.8 10.0 14.9 -8.9 0.0 0.1 -2.5 0.0 0.0

Median 27.2 25.6 27.2 27.0 25.5 27.1 14.7 14.7 14.7 14.7 14.7 14.8

μ 27.5 25.9 27.5 27.4 26.0 27.4 14.7 14.8 14.8 14.8 14.9 14.9

σ 2.4 3.1 2.4 2.5 3.0 2.5 0.8 1.1 1.1 0.9 1.3 1.3

A.22 Over Sea Flight Track 8

Table A54. Statistics of spatial correlation, FT8, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 85 85 85 85 85 85

Mean 0.57 0.51 0.00 0.00 0.00 0.00

Median 0.96 0.88 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.90 0.88 0.88 0.83

Min -1.00 -1.00 -0.86 -0.91 -0.91 -0.76

Standard deviation 0.66 0.61 0.14 0.14 0.14 0.13

NASA/CR—2015-218486 112
(a) (b) 

(c) (d) 

Figure A66. Ricean K factor for FT8, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A55. Statistics of Ricean K factor, FT8, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 29.1 29.0 12.9 12.9

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 28.2 24.9 28.2 28.1 25.0 28.1 12.3 11.4 12.0 12.5 11.7 12.2

n 0.12 0.18 0.12 0.11 0.17 0.11 0.05 0.10 0.07 0.03 0.08 0.05

σX (dB) 1.7 3.0 1.6 1.5 2.9 1.4 0.9 1.1 1.0 0.8 1.0 0.9

Dmax(km) 2.9 2.9 2.9 2.9 2.9 2.9 1.3 1.3 1.3 1.3 1.3 1.3

DMin(km) 25.8 25.8 25.8 25.8 25.8 25.8 25.8 25.8 25.8 25.8 25.8 25.8

Statistics of K factor (dB)

Max 35.1 34.3 35.4 33.7 34.5 34.5 16.4 17.2 17.2 15.9 16.9 16.9

Min 12.4 10.6 12.7 12.5 10.5 12.9 9.3 2.4 9.1 8.4 3.6 7.8

Median 30.0 27.4 30.0 29.7 27.3 29.7 13.0 12.8 12.9 13.0 12.7 12.9

μ 30.2 28.0 30.2 30.0 27.9 30.0 12.9 12.7 12.8 12.9 12.6 12.7

σ 1.8 3.2 1.8 1.6 3.1 1.6 1.0 1.3 1.1 0.8 1.2 1.0

NASA/CR—2015-218486 113
A.23 Over Sea Flight Track 9

Table A56. Statistics of spatial correlation, FT9, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 87 87 87 87 87 87

Mean 0.75 0.86 0.00 0.00 0.00 0.00

Median 0.98 0.98 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.78 0.82 0.82 0.77

Min -1.00 -0.98 -0.82 -0.81 -0.81 -0.83

Standard deviation 0.45 0.29 0.11 0.11 0.11 0.12

(a) (b) 

(c) (d) 

Figure A67. Ricean K factor for FT9, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

NASA/CR—2015-218486 114
Table A57. Statistics of Ricean K factor, FT9, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 28.6 28.0 12.9 13.2

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 28.4 25.4 28.4 28.7 25.7 28.7 11.8 11.2 11.5 12.3 11.6 11.9

n 0.04 0.10 0.05 0.00 0.06 0.00 0.05 0.07 0.06 0.05 0.06 0.05

σX (dB) 1.9 2.8 1.9 2.0 2.9 2.0 1.0 1.1 1.1 0.9 1.1 1.0

Dmax(km) 1.8 1.8 1.8 1.8 1.8 1.8 1.0 1.0 1.0 1.0 1.0 1.0

DMin(km) 46.0 46.0 46.0 46.0 46.0 46.0 46.0 46.0 46.0 46.0 46.0 46.0

Statistics of K factor (dB)

Max 34.1 34.5 34.5 33.6 33.9 34.0 19.3 20.8 21.4 20.7 21.3 22.1

Min 13.1 10.3 13.1 13.5 10.2 13.8 9.7 2.9 8.9 10.6 4.7 8.9

Median 29.4 27.7 29.4 28.7 27.0 28.7 13.1 12.8 12.9 13.4 13.1 13.2

μ 29.9 28.4 29.9 29.1 27.5 29.1 13.1 12.9 12.9 13.2 13.0 13.0

σ 2.0 3.0 2.0 2.0 3.0 2.0 1.2 1.4 1.3 1.1 1.3 1.3

A.24 Over Sea Flight Track 10

Table A58. Statistics of spatial correlation, FT10, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 83 83 83 83 83 83

Mean 0.57 0.52 0.00 0.00 0.00 0.00

Median 0.92 0.91 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.56 0.48 0.48 0.55

Min -1.00 -1.00 -0.45 -0.51 -0.51 -0.45

Standard deviation 0.62 0.62 0.11 0.11 0.11 0.11

(a) (b) 

NASA/CR—2015-218486 115
(c) (d) 

Figure A68. Ricean K factor for FT10, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A59. Statistics of Ricean K factor, FT10, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 28.8 28.9 14.2 9.7

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 35.2 31.6 35.1 34.7 31.7 34.7 12.1 11.8 11.9 12.9 12.5 12.6

n -0.16 -0.10 -0.16 -0.15 -0.10 -0.15 0.07 0.08 0.08 0.04 0.05 0.05

σX (dB) 2.2 3.2 2.2 2.0 3.1 2.0 0.9 1.0 1.0 1.7 1.0 1.1

Dmax(km) 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1

DMin(km) 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2

Statistics of K factor (dB)

Max 34.8 36.4 36.5 34.5 34.9 35.0 18.0 17.1 18.5 17.4 17.0 18.4

Min 12.2 10.3 12.1 12.4 10.7 12.7 10.4 9.8 9.8 -31.4 -8.8 -0.3

Median 30.0 28.5 30.1 29.9 28.4 29.9 14.3 14.4 14.4 14.1 14.1 14.1

μ 30.6 29.1 30.6 30.5 29.0 30.5 14.4 14.5 14.5 14.1 14.1 14.1

σ 2.5 3.3 2.5 2.3 3.2 2.3 1.1 1.1 1.1 1.7 1.1 1.2

A.25 Over Sea Flight Track 11

Table A60. Statistics of spatial correlation, FT11, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 94 94 94 94 94 94

Mean 0.25 0.55 0.00 0.00 0.00 0.00

Median 0.64 0.89 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.79 0.78 0.78 0.81

Min -1.00 -1.00 -0.70 -0.71 -0.71 -0.75

Standard deviation 0.78 0.60 0.13 0.13 0.13 0.13

NASA/CR—2015-218486 116
(a) (b) 

(c) (d) 

Figure A69. Ricean K factor for FT11, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

Table A61. Statistics of Ricean K factor, FT11, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 25.8 26.3 14.6 14.6

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 23.5 24.7 23.5 28.7 29.5 28.8 16.1 16.5 16.6 17.0 17.3 17.5

n 0.14 0.03 0.14 -0.07 -0.16 -0.07 -0.06 -0.09 -0.09 -0.09 -0.12 -0.12

σX (dB) 2.6 2.8 2.6 2.4 2.8 2.4 1.6 2.2 2.2 1.9 2.1 2.1

Dmax(km) 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3

DMin(km) 26.6 26.6 26.6 26.6 26.6 26.6 26.6 26.6 26.6 26.6 26.6 26.6

Statistics of K factor (dB)

Max 33.5 33.9 33.9 33.3 34.3 34.2 19.9 -19.5 20.1 19.1 -22.8 -20.6

Min 10.9 5.9 10.6 16.7 7.0 16.1 -36.7 0.0 0.0 -39.5 0.0 0.0

Median 26.7 25.5 26.7 27.1 25.9 27.1 14.9 14.6 14.6 14.9 14.6 14.7

μ 27.0 25.6 27.0 27.3 26.2 27.4 14.9 14.8 14.9 15.0 14.9 15.0

σ 2.6 2.8 2.6 2.4 2.8 2.4 1.6 2.2 2.2 1.9 2.1 2.2

NASA/CR—2015-218486 117
A.26 Over Sea Flight Track 12

Table A62. Statistics of spatial correlation, FT12, Oxnard, CA. 

C1C2 L1L2 C1L1 C1L2 C2L1 C2L2

Window Length 94 94 94 94 94 94

Mean 0.22 0.54 0.00 0.00 0.00 0.00

Median 0.55 0.86 0.00 0.00 0.00 0.00

Max 1.00 1.00 0.73 0.72 0.72 0.84

Min -1.00 -1.00 -0.78 -0.77 -0.77 -0.70

Standard deviation 0.78 0.59 0.13 0.13 0.13 0.13

(a) (b) 

(c) (d) 

Figure A70. Ricean K factor for FT12, Oxnard, CA (a) C-band Rx1; (b) C-band 2; (c) L-band Rx1; (d) L-band Rx2. 

NASA/CR—2015-218486 118
Table A63. Statistics of Ricean K factor, FT12, Oxnard, CA. 

C-band L-band

Rx1 Rx2 Rx1 Rx2

Moving Window Length 250λ 15 m

Overall K factor (dB) 25.9 26.4 14.9 14.9

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K factor

A(dB) 28.5 28.4 28.5 29.9 29.5 29.9 16.8 17.1 17.2 17.4 18.0 18.1

n -0.06 -0.11 -0.06 -0.12 -0.15 -0.11 -0.08 -0.09 -0.10 -0.10 -0.13 -0.14

σX (dB) 2.9 3.0 2.9 2.5 2.9 2.5 1.1 1.2 1.2 1.1 1.3 1.3

Dmax(km) 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3

DMin(km) 27.1 27.1 27.1 27.1 27.1 27.1 27.1 27.1 27.1 27.1 27.1 27.1

Statistics of K factor (dB)

Max 34.2 35.5 35.6 34.6 35.1 35.2 18.8 19.5 19.6 21.5 21.9 21.9

Min 11.1 8.1 10.8 13.4 6.7 12.9 -30.1 0.0 0.0 -28.7 0.0 0.0

Median 27.0 25.8 27.0 27.3 26.1 27.3 15.0 14.9 14.9 15.0 14.9 14.9

μ 27.3 25.9 27.3 27.5 26.3 27.5 15.1 15.0 15.0 15.1 15.0 15.1

σ 2.9 3.1 2.9 2.5 2.9 2.5 1.1 1.2 1.2 1.1 1.4 1.4

 

NASA/CR—2015-218486 119
Appendix B: Additional Stationarity Distance Statistics 

 Here we provide some additional statistics on (C-band) stationarity distance, for two overfreshwater FTs. These are termed “urban” statistics since we did include the strong MPCs at large 

delays from buildings in downtown Cleveland. Results are comparable to those found in the over-sea 

case, i.e., an SD of 250OC (~15m) is sufficient. 

Table B1. C-band stationarity distance statistics, urban FT5. 

Rx1 Rx2 

Lognormal μ (m) 2.70 2.64 

σ (m) 0.74 0.77 

Mean (m) 19.15 18.41 

(λ’s) 323.00 310.55 

Median (m) 15.22 14.40 

(λ’s) 256.71 242.93 

10th

percentile 

(m) 5.51 5.04 

(λ’s) 92.87 84.96 

Min (m) 0.84 0.91 

(λ’s) 14.23 15.29 

Figure B1. SD CDFs for straight flight track (FT5). 

Table B2. C-band stationarity distance statistics, urban FT6. 

Rx1 Rx2

Lognormal μ (m) 2.67 2.67 

σ (m) 0.66 0.67 

Mean 

(m) 17.66 17.83 

(λ’s) 297.86 300.74 

Median (m) 14.45 14.54 

(λ’s) 243.78 245.21 

10th

percentile 

(m) 6.08 6.02 

(λ’s) 102.47 101.59 

Min 

(m) 0.82 0.86 

(‘λs) 13.91 14.50 

10 20 30 40 50 60 70

0

0.2

0.4

0.6

0.8

1

'x (m)

CDF

ClevelandOH***10-22-2013***FT5***C-band

C-band Rx1

C-band Rx2

NASA/CR—2015-218486 120
Figure B2. SD CDFs for straight flight track (FT6). 

10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

'x (m)

CDF

ClevelandOH***10-22-2013***FT6***C-band

C-band Rx1

C-band Rx2

NASA/CR—2015-218486 121
Appendix C: Ricean K-Factor Estimation 

The envelope probability density function (pdf) of the Ricean distribution is given by [52] 

݌௥ሺݎሻ ൌ ௥

ఙమ ሺെ ௥మା௦మ

ଶఙమ ሻܫ଴ሺ

௥௦

ఙమሻ (C.1) 

where I0(·) is the modified Bessel function of the first kind and zero order, s is the mean of the 

dominant (often, LOS) component, and 2V2 is the power in the remaining (Rayleigh distributed) 

components. Often the pdf is normalized to have a mean-square value of unity. In this case it is 

given by 

݌ሺݎǡܭሻ ൌ ʹݎሺͳ ൅ ͳͲ௄Ȁଵ଴ሻሾെݎଶ൫ͳ ൅ ͳͲ௄Ȁଵ଴൯ െ ͳͲ௄Ȁଵ଴ሿܫ ଴ቀʹݎͳͲ௄Ȁଶ଴ሻඥͳ ൅ ͳͲ௄Ȁଵ଴ቁ (C.1a) 

The K factor in decibels is defined as 

ܭሺ݀ܤሻ ൌ ͳͲ݈݋݃ଵ଴ሺ ௦మ

ଶఙమሻ (C.2) 

If K=0, the Ricean degenerates into the Rayleigh distribution; if K>>1 the PDF looks like a Gaussian 

density with mean value s. Three K factor computation methods are considered. 

C.1 Method I: Maximum Likelihood (ML) Fitting of Empirical Data 

For this method we collect histograms of the measured data, and employ a maximumlikelihood (ML) fitting procedure. The steps employed to compute the empirical K factor are as 

follows: 

1. Collect data (PDPs); the link distance d of each individual PDP is known. Find the narrow 

band received power1 Pr in dBm. 

2. Estimate local moving average value of Pr with a sliding window 

ܲ஺௩௚ሺ݀ሻ ൌ σ ௉ೝሺௗ೔ሻ

ௐ

௜ାௐȀଶ

௜ିௐȀଶ (C.3) 

where the window length W is determined by the stationarity distance (SD), the PDP update 

rate RPDP and aircraft velocity V as 

ܹ ൌ ܵܦ ή ܴ௉஽௉Ȁܸ (C.4) 

We then remove the local moving average from the empirical data to create the residuals ܲோ௦ௗ 

ܲோ௦ௗሺ݀ሻ ൌ ܲ௥ሺ݀ሻ െ ܲ஺௩௚ሺ݀ሻ (C.5) 

Alternative 1: We can also use a linear fit to Pr (~path loss model) to estimate the local 

average instead of the sliding window, but the linear fit does not fit the empirical data as 

well in some segments of data. 

 

1 Our power is the sum of the power in all chip samples. Note that the power recorded by the sounder directly can not be used 

to compute the K factor since the values are quantized in steps of approximately 0.25 dB; this would significantly degrade the 

estimation of the amplitude distribution. 

NASA/CR—2015-218486 122
Alternative 2: Another alternative to remove the large scale path loss is to subtract the free 

space path loss. However, since the window length is only a few meters, the variation of free 

space path loss is very small (it is almost a constant). This method is not recommended for 

removing the local average although it can still be used for the computation of the K factor 

for the complete FT. 

3. Convert ܲோ௦ௗ from decibel scale to linear scale. 

4. Find amplitude ܣோ௦ௗ by taking the (positive) square root of ܲோ௦ௗ. 

5. Find the Ricean fit of ܣோ௦ௗሺ݀௜ǣ ݀௜ାௐᇲ ሻ via the Matlab® function “fitdist(ܣோ௦ௗ, 'rician')”, where 

W’ is the length of another sliding (or stepped) window; W’=W is 250λ in C-band and 15 m in 

L-band. The function “fitdist” is a command within the Matlab® distribution fitting tool, in 

Matlab’s statistics toolbox. Detailed descriptions of W (stationarity distance) appear in 

Section 4.2 and Appendix B. The “fitdist” function returns the maximum-likelihood values of 

s and σ, from which KML can be calculated by (2). 

Processing steps 1-4 are also used in the other K-factor estimation methods. Step 5 can be 

replaced by two other moment-based methods, as we describe next. 

C.2 Method II: Second Order Moment 

The authors of [53] proposed a second-order moment-based K-factor computation method for 

the fixed wireless channel (Tx & Rx motionless). The sliding window length in (C.3) is only a few 

meters, which is much smaller than the link distance. The signal amplitude within this window can 

be treated as a fixed value, hence the moment method in [53] can be applied for the AG channel. 

Assume r(d) is the received signal, V is a fixed complex value and v(d) is a zero mean complex 

random variable with standard deviation of σ. 

ݎሺ݀ሻ ൌ ܸ ൅ ݒሺ݀ሻ (C.6) 

The second moment Gv is the RMS fluctuation of received power Pr about its moving average PAvg: 

ܩ௩ሺ݀௜ሻ ൌ ට ଵ

ௐ σ ൣܲ௥ሺ݀௜ሻ െ ܲ஺௩௚ሺ݀௜ሻ൧ ௜ାௐȀଶ ଶ

௜ିௐȀଶ (C.7) 

From this, |V|2 and σ2 can be obtained as 

ȁܸȁ

ଶ ൌ ටܲ஺௩௚ଶ െ ܩ௩

ଶ

 (C.8) 

ߪଶ ൌ ܲ஺௩௚ െ ටܲ஺௩௚ଶ െ ܩ௩

ଶ (C.9) 

Finally, the K-factor in linear scale is obtained as follows: 

ܭଶ ൌ ȁܸȁ

ଶȀߪଶ. (C.10) 

C.2 Method III: Fourth Order Moment 

In [54], the authors provided a widely used K-factor computation method based on fourth 

moments. The Rician K factor in linear scale is given by, 

NASA/CR—2015-218486 123
ܭଶǡସ ൌ ିଶఓమ

మାఓరିఓమටଶఓమ

మିఓర

ఓమ

మିఓర

, (C.11) 

where μ2 and μ4 are the second and fourth moments, respectively, 

ߤଶ ൌ σ ஺ೃೞ೏మሺௗ೔ሻ

ௐ

௜ାௐȀଶ

௜ିௐȀଶ (C.12) 

ߤସ ൌ σ ஺ೃೞ೏రሺௗ೔ሻ

ௐ

௜ାௐȀଶ

௜ିௐȀଶ . (C.13) 

C.4 Example K-factor Results 

The signal amplitude within a window follows the Ricean distribution well (it is actually 

nearly Gaussian since K>>1). Some example results for both bands are shown in Figures C.1-C.6. 

Figure C.1. Path loss (left) and amplitude histogram and fit (right) for 1000th stepped window, C-band Rx2. 

Figure C.2. Path loss (left) and amplitude histogram and fit (right) for 2000th stepped window, C-band Rx2. 

2.3621 2.3622 2.3622 2.3623

x 104

2.8

3

3.2

3.4

3.6

3.8 x 10-8

Link Distance (m)

Path Loss (dB)

OxnardCA***06-11-2013***FT1***C-band Rx2

Empirical Pr

Filtered Pr

4.1 4.15 4.2 4.25 4.3 4.35 4.4

x 10-4

0

5

10

15

20

Amplitude

Histogram

OxnardCA***06-11-2013***FT1***C-band Rx2

Empirical Data

Rician K=38dB

2.2736 2.2737 2.2738 2.2738

x 104

6.5

7

7.5

8

8.5 x 10-8

Link Distance (m)

Path Loss (dB)

OxnardCA***06-11-2013***FT1***C-band Rx2

Empirical Pr

Filtered Pr

4 4.1 4.2 4.3 4.4 4.5 4.6

x 10-4

0

5

10

15

20

25

Amplitude

Histogram

OxnardCA***06-11-2013***FT1***C-band Rx2

Empirical Data

Rician K=34dB

NASA/CR—2015-218486 124
Figure C.3. Path loss (left) and amplitude histogram and fit (right) for 5000th stepped window, C-band Rx2. 

Figure C.4. Path loss (left) and amplitude histogram and fit (right) for 500th stepped window, L-band Rx2. 

Figure C.5. Path loss (left) and amplitude histogram and fit (right) for 1000th stepped window, L-band Rx2. 

2.1899 2.19 2.19 2.19 2.19012.19022.19022.1902

x 104

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7 x 10-8

Link Distance (m)

Path Loss (dB)

OxnardCA***06-11-2013***FT1***C-band Rx2

Empirical Pr

Filtered Pr

4.15 4.2 4.25 4.3

x 10-4

0

5

10

15

20

25

Amplitude

Histogram

OxnardCA***06-11-2013***FT1***C-band Rx2

Empirical Data

Rician K=43dB

1.599 1.59911.59921.59931.59941.59951.59961.5997

x 104

0

0.5

1

1.5

2

2.5

3 x 10-9

Link Distance (m)

Path Loss (dB)

OxnardCA***06-11-2013***FT1***L-band Rx2

Empirical Pr

Filtered Pr

5.9 6 6.1 6.2 6.3 6.4

x 10-4

0

20

40

60

80

100

Amplitude

Histogram

OxnardCA***06-11-2013***FT1***L-band Rx2

Empirical Data

Rician K=37dB

8195 8200 8205 8210 8215

0.5

1

1.5

2

2.5

3

3.5

4

4.5 x 10-8

Link Distance (m)

Path Loss (dB)

OxnardCA***06-11-2013***FT1***L-band Rx2

Empirical Pr

Filtered Pr

5 6 7

x 10-4

0

20

40

60

80

100

Amplitude

Histogram

OxnardCA***06-11-2013***FT1***L-band Rx2

Empirical Data

Rician K=25dB

NASA/CR—2015-218486 125
Figure C.6. Path loss (left) and amplitude histogram and fit (right) for 1415th stepped window, L-band Rx2. 

The K-factor results for the three methods for the over sea FT1 are shown in Figures C.7-C.10. 

The statistics and linear fit parameters are listed in Table C.1. Method I is the most accurate, but 

the slowest for processing, and on rare occasions, the Ricean fit may not converge. Computing K2 by 

Method II yields results that are always smaller than the results of the other two methods. Note that 

K2 was developed for a fixed Tx & Rx, so it may not be best for the AG channel, even though Method 

II is the fastest computationally. Method III has been widely used in recent years, but the term 

ଶߤʹ

ଶ െ ߤସ in (C.4) may on occasion be negative, making K2,4 a complex number with a very small 

imaginary part. For these rare results we used only the real part for our estimates. 

(a) (b) 

2262 2264 2266 2268 2270

0

0.2

0.4

0.6

0.8

1

1.2

1.4 x 10-7

Link Distance (m)

Path Loss (dB)

OxnardCA***06-11-2013***FT1***L-band Rx2

Empirical Pr

Filtered Pr

4 5 6 7 8 9 10

x 10-4

0

10

20

30

40

50

60

70

80

Amplitude

Histogram

OxnardCA***06-11-2013***FT1***L-band Rx2

Empirical Data

Rician K=17dB

NASA/CR—2015-218486 126
(c) (d) 

Figure C.7. K factor for C-band Rx1, over sea FT1. (a) Method I: ML fit & Method III; (b) Method II: moments; (c) PDF of 

Method I; (d) CDF of Method I. 

(a) (b) 

(c) (d) 

Figure C.8. K factor for C-band Rx2, over sea FT1. (a) Method I: ML fit & Method III; (b) Method II: moments; (c) PDF of 

Method I; (d) CDF of Method I. 

15 20 25 30 35

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

K factor (dB)

PDF

OxnardCA***06-11-2013***FT1***C-band Rx1

KML

Gaussian Fit

15 20 25 30 35

0

0.2

0.4

0.6

0.8

1

K factor (dB)

CDF

OxnardCA***06-11-2013***FT1***C-band Rx1

KML

Gaussian Fit

15 20 25 30

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

K factor (dB)

PDF

OxnardCA***06-11-2013***FT1***C-band Rx2

KML

Gaussian Fit

15 20 25 30

0

0.2

0.4

0.6

0.8

1

K factor (dB)

CDF

OxnardCA***06-11-2013***FT1***C-band Rx2

KML

Gaussian Fit

NASA/CR—2015-218486 127
(a) (b) 

(c) (d) 

Figure C.9. K factor for L-band Rx1, over sea FT1. (a) Method I: ML fit & Method III; (b) Method II: moments; (c) PDF of 

Method I; (d) CDF of Method I. 

(a) (b) 

12 14 16 18 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

K factor (dB)

PDF

OxnardCA***06-11-2013***FT1***L-band Rx1

KML

Gaussian Fit

12 14 16 18 20

0

0.2

0.4

0.6

0.8

1

K factor (dB)

CDF

OxnardCA***06-11-2013***FT1***L-band Rx1

KML

Gaussian Fit

NASA/CR—2015-218486 128
(c) (d) 

Figure C.10. K factor for L-band Rx2, over sea FT1. (a) Method I: ML fit & Method III; (b) Method II: moments; (c) PDF of 

Method I; (d) CDF of Method I. 

Table C.1. Statistics of Ricean K factors, over-sea FT1, 11 June 2013, Oxnard, CA, with aircraft flown straight toward GS. 

C-band L-band 

Rx1 Rx2 Rx1 Rx2 

Moving Window Length 250λ 15 m 

Overall K factor (dB) 

30.4 

29.9 12.8 12.8 

Methods KML K2 K2,4 KML K2 K2,4 KML K2 K2,4 KML K2 K2,4

Linear fit of K

factor 

A(dB) 29.1 25.1 29.0 29.0 25.6 29.0 12.0 11.4 11.8 12.1 11.5 11.9 

n 0.17 0.22 0.17 0.12 0.15 0.12 0.07 0.10 0.08 0.07 0.10 0.08 

σX (dB) 1.7 3.4 1.7 1.5 3.2 1.5 1.0 1.0 1.0 0.9 0.9 0.9 

Dmax(km) 3.0 3.0 3.0 3.0 3.0 3.0 2.2 2.2 2.2 2.2 2.2 2.2 

DMin(km) 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.1 

Statistics of K

factor (dB) 

Max 35.5 34.4 35.4 34.0 33.6 34.0 20.2 19.9 20.4 19.7 19.6 19.9 

Min 13.0 11.3 13.4 13.4 11.3 13.8 10.3 7.4 10.0 10.6 4.2 10.4 

Median 31.3 28.0 31.3 30.6 27.6 30.6 12.9 12.7 12.8 12.9 12.7 12.8 

μ 31.7 28.7 31.6 30.9 28.4 30.9 12.7 12.5 12.6 12.8 12.5 12.6 

σ 2.0 3.6 2.0 1.7 3.3 1.6 1.1 1.2 1.1 1.0 1.1 1.0 

C.5 Discussion 

The K factor for both bands is on average more than 12 dB. Although from some results (e.g., 

Fig. C8, C.10) it appears to increase with link distance, this is not true for all data or all flight 

tracks: in some cases K stays nearly constant, and in some it decreases. Since all increases or 

decreases are small, it is reasonable to model the mean K-factor as constant with link distance. The 

AG channel K factor is larger than that of terrestrial (e.g., V2V and cellular) environments. This is 

as expected because the LOS component is strong and the environment provides only a few, often 

intermittent, MPCs. Ricean K-factors for the AG channel were also estimated in [55]. In this work, 

the authors computed K for the VHF band over land. Values of K ranged from 2-20 dB, with a mean 

value of 16 dB; no other statistics were reported. From Figures C.1-C.10 and Table C.1 it is also clear 

that the C-band K-factors are considerably larger than those in L-band, and this is most likely due to 

the stronger surface-ray cancellation that occurs in L-band. 

12 14 16 18

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

K factor (dB)

PDF

OxnardCA***06-11-2013***FT1***L-band Rx2

KML

Gaussian Fit

12 14 16 18

0

0.2

0.4

0.6

0.8

1

K factor (dB)

CDF

OxnardCA***06-11-2013***FT1***L-band Rx2

KML

Gaussian Fit

NASA/CR—2015-218486 129
Although some reflecting objects may be present near the GS, the reflected rays and the sea 

surface reflection(s), as seen at the Rx antenna, arrive from nearly the same direction as that of the 

LOS component. A close inspection of the plots of K vs. distance also that the K-factor roughly 

follows a two-ray pattern. Figure C.11(a) shows total received power via measurements, and Figure 

C.11(b) shows the analytical value of received power, from the CE2R model. Figure C.12 shows the 

corresponding fading values from the measurements and analysis, with respect to the means. As can 

be seen, when the total received power (Fig. C.11) “fades” to a null, the “fading about the mean” (Fig. 

C.12) decreases, and a smaller variation corresponds to a larger K-factor. The absolute values of K

are much larger for the analytical CE2R computation, which does not account for any noise, antenna 

effects, equipment variation, etc. 

(a) (b) 

Figure C11. Received power vs. link distance for (a) empirical data; (b) analytical CE2R. 

(a) (b) 

Figure C.12. Power fade, with respect to mean, vs. link distance, for (a) empirical data; (b) analytical CE2R. 

5 10 15 20

-70

-65

-60

-55

-50

-45

-40

-35

Link Distance (km)

Total Received Power (dBm)

OxnardCA***06-11-2013***FT1***L-band Rx1

5 10 15 20

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Link Distance (km)

Received Power Fade (dB)

OxnardCA***06-11-2013***FT1***L-band Rx1

NASA/CR—2015-218486 130
(a) (b) 

Figure C.13. Rician K factor vs. link distance for (a) empirical data; (b) analytical CE2R. 

 

5 10 15 20

20

40

60

80

100

120

140

160

180

OxnardCA***06-11-2013***FT1***L-band Rx1

Distance (km)

Analytical K factor (dB)

NASA/CR—2015-218486 131
Appendix D: PDP Update Rate 

The rate at which power delay profiles (PDPs) are recorded by the sounder is not a constant. 

This is due to the fact that the actual data recording rate and data transfer rate inside the sounder is 

slower than the rate at which correlation measurements are made. The sounder’s PDP length is 2046 

samples (two samples per the 1023 chips of the underlying m-sequences). The actual time duration of 

a 2046-sample PDP is 204.6 microseconds in L-band, and 20.46 microseconds in C-band, from the 

200 ns and 20 ns chip durations for these bands, respectively. The aircraft’s flight recorder gathers 

GPS information, from which a UNIX timestamp is generated. The UNIX timestamp is recorded for 

each PDP, and the accuracy is on the order of 10-7 seconds. If ti is the UNIX timestamp of the ith

PDP, the recording duration for one PDP ∆ti is ti - ti-1. The minimum recording duration ∆tmin is 

333.31 microseconds (reciprocal of 3002 Hz), which is much larger than the time duration of any 

single PDP (for either band). 

Since a 25 (or 30) dB threshold is used in the air-ground channel measurements, only a few 

samples above the threshold are recorded when the channel contains a line-of-sight (LOS) 

component and very few multipath components. If the LOS is blocked, and/or many multipath 

components are present, more samples must be recorded, hence the recording time for such a PDP 

increases. We have found that the practical PDP recording duration can only be integer multiples of 

∆tmin, i.e., 

οݐ ௜ൌ ܰοݐ௡௜௠,) D.1) 

where N is a positive integer. The PDP update rate K is the reciprocal of the PDP recording time 

duration ∆t:

ܭ ௜ൌ ଵ

ο௧೔

ൌ

ە

ۖ

۔

ۖ

ۓ

͵ͲͲʹݖܪܰ ൌ ͳ

ͳͷͲͳݖܪܰ ൌ ʹ

ͳͲͲͲݖܪܰ ൌ ͵

͹ͷͲݖܪܰ ൌ Ͷ

͸ͲͲݖܪܰ ൌ ͷ

ǥ ǥ

 (D.2) 

Since the PDP update rate is the reciprocal of the recording time, the average update rate ܭഥ

cannot is 

ܭഥ ൌ ெ

σ భ

಼೔

ಾ

೔సభ

 (D.3) 

where M is the number of PDPs. 

The PDP update rate vs. link distance for the four sounder receivers is shown in Figure 1. 

These results are based on flight track (FT) 2, with flight straight away from the ground site (GS), 

taken over Lake Erie on 22 October 2013. The histogram of PDP update rate for the C-band Rx1 is 

shown in Figure D.2, and statistics of the PDP update rate are provided in Table D.1. The average 

update rate is approximately 2.9 kHz, which is close to the maximum of 3 kHz. The percentage at 

the maximum value is more than 97%. Note that FT2 is one of the “cleanest” FTs, which always has 

a clear LOS, hence the number of samples in each PDP is relatively small. For FTs (or segments of 

FTs) in which the LOS is blocked, e.g., FTs in Telluride where the LOS was blocked by mountain 

ridges, or FTs with a large number of multipath components, the number of samples per PDP is 

expected to be much larger, and the PDP update rate is consequently expected to be much smaller. 

NASA/CR—2015-218486 132
(a) (b) 

(c) (d) 

Figure D.1. PDP update rate for FT2 (straight away from GS) over Lake Erie, 22 October 2013. (a) C-band Rx1; (b) C-band 

Rx2; (c) L-band Rx1; (d) L-band Rx2. 

Figure D.2. Histogram of PDP update rate for Figure D.1(a). 

5 10 15 20 25 0

500

1000

1500

2000

2500

3000

3500

Link Distance (km)

PDP Update Rate (Hz)

ClevelandOH***10-22-2013***FT2***C-band Rx1

Averaged PDP Update Rate is2864 (Hz)

5 10 15 20 25 0

500

1000

1500

2000

2500

3000

3500

Link Distance (km)

PDP Update Rate (Hz)

ClevelandOH***10-22-2013***FT2***C-band Rx2

Averaged PDP Update Rate is2849 (Hz)

5 10 15 20 25 0

500

1000

1500

2000

2500

3000

3500

Link Distance (km)

PDP Update Rate (Hz)

ClevelandOH***10-22-2013***FT2***L-band Rx1

Averaged PDP Update Rate is2839 (Hz)

5 10 15 20 25 0

500

1000

1500

2000

2500

3000

3500

Link Distance (km)

PDP Update Rate (Hz)

ClevelandOH***10-22-2013***FT2***L-band Rx2

Averaged PDP Update Rate is2907 (Hz)

0 500 1000 1500 2000 2500 3000 3500 0

2

4

6

8

10

12 x 105

PDP Update Rate (Hz)

Density

ClevelandOH***10-22-2013***FT2***C-band Rx1

NASA/CR—2015-218486 133
Table D.1. Statistics of PDP update rate for over-freshwater flight, FT2, Lake Erie, 22 October 2013 (see also eq. (D.2)). 

Band|Receiver 

# of 

PDPs

M 

Time 

duration 

∆t (s) 

Average 

update 

rate 

K (Hz) 

% with 

N=1

% with 

N=2

% with 

N=3

% with 

N=4

% with 

N=5

C

Rx1 1,149,310 401.31 2864 97.88% 0.88% 0.68% 0.39% 0.09% 

Rx2 1,143,111 401.26 2849 97.56% 1.01% 0.84% 0.43% 0.08% 

L

Rx1 1,139,534 401.32 2839 97.49% 1.27% 0.59% 0.34% 0.11% 

Rx2 1,166,408 401.29 2907 98.55% 0.81% 0.34% 0.16% 0.04% 

NASA/CR—2015-218486 134