Although propagation at high altitudes have been studied for aircraft purposes, this is the first ever comprehensive UAV data set of propagation at altitude under 400′ (see prior art discussion below). Propagation modelling for wireless is a vast field, but existing models are formulated to apply to at or near ground level receivers (Fig. 5). Therefore, it should be no surprise that existing models do not apply. Nonetheless, we can discuss the deviation from existing models to elucidate some of the features we have found in wave propagation that should be included in more sophisticated models in the future.

### Flight A

The first flight clearly has some distance to transmitter dependence that is not Friis like, i.e. there is some scatter. Existing models treat scattering off the earth and building as a separate path from transmitter (TX, base station) to RX (airborne receiver). If the Earth is treated as a perfect mirror, it is called the 2-ray or “plane earth” model. (See fig. 2.6 of Ref.^{12}.) If the Earth is treated as a rough scatterer, there is the Sarkar model (based on Sommerfeld 100 years ago)^{10} and empirical models called the Hata-COST231 model^{2,3}. We discuss how they compare to our data below.

### Comparison to ballistic (scatter free) propagation model

The predictions of a ballistic (scatter free) model would be the following:

This assumes the altitude is much less than the distance to the transmitter, so that the horizontal distance d is approximately equal to the total distance from TX to RX (see Fig. 2). In our data, we observe altitude independence, which indicates minimal scatter. However, the path loss dependence on distance is not 20 log d, which we discuss next.

The location of the call tower is not known. Therefore the distance is not known, and the power law is not known. By adjusting the distance, a variety of power laws can be fitted, from 20 to 50 dB/decade of distance, but these are not quantitative since the distance is unknown with enough precision.Therefore, the actual power law is not really known. That being said, there IS a strong distance dependence, and NO altitude dependence in this case, which is opposite the mountainous case (flight B).

### Comparison to the 2 ray model

The 2-ray model assumes ballistic (scatter free) propagation from TX to RX along one path, and a perfect specular reflection from the ground (modeling the Earth as a perfect conductor i.e. mirror) as a second path, and does not take into account any other scattering or multi path, e.g. no diffusion or diffraction is modeled in the 2-ray model (see fig. 2.6 of ref.^{12}).

The predictions of the 2-ray model are given by:

$$begin{aligned} {P_rover P_t} = G_{TX}G_{RX}left( {h_{TX}h_{RX}over d^2}right) ^2 end{aligned}$$

(2)

where (h_{TX}), (h_{RX}) are the heights of the transmit and receive antennas, respectively, and G the gains of the antennas. The assumptions of the 2-ray model are that the height of the receiver (h_{RX}) and transmitter (h_{TX}) are much lower that the distance d. Since the distance is over 1 km and the max height is 100 m this is reasonable. The interesting point about the data for flight A is that the RSSI is independent of the altitude over a large range of altitudes, from 100 to 400 ft. (30–120 m) (100–400 lambda). This is clearly not what is predicted by the 2-ray model, which predicts an (h_{RX}^2) dependence on signal strength.

### Comparison to the Sarkar model

The Sarkar model^{10} predicts a 30 dB/decade decay (but only in the far-field, see below), for altitudes close to the ground. Our data shows a power law decay so is consistent in this sense. There is no comprehensive, analytical prediction for higher altitudes in this model in the far field. However, recent simulation work has addressed the near field, which we discuss in more detail below.

### Comparison to the Hata-COST231 model

The Hata-COST231 model is purely an empirical model. Through many iterations and refinements, it has been codified in a literal industry standard as ITU 2019^{3}. However, the model, being empirical, has limitations, which are of course based on the input data range in the first place! In their case, they did not have drone technology. The model limits (h_{RX}) to 1 to 10 (lambda). We are at well over 100 (lambda). Nonetheless, it is interesting to extrapolate, discuss, and compare.

Hata initial observed a “height gain” with antenna height, but only data up to 10 m in (h_{RX}) was considered. However, there is no empirical prediction above this altitude. (We recently studied this in detail at low altitude, in a sister paper^{13}). The general concept is that, as the receiver is raised, scattering becomes less prevalent. Extrapolation of Hata would give a huge dependence on altitude, which we clearly do not see. Therefore, Hata model does not really apply to flight A, nor does it give predictions even if extrapolated to our case. A similar conclusion can be drawn about the most recent industry standard COST231 which is based also on Hata’s work^{3}.

### Near vs. far field and “breakpoint”

In all of the scatting models above, the predictions implicitly assume that the radiation is in the far field. The far field is traditionally assumed to begin at distances larger than (2l^2/lambda), where *l* is the maximum antenna dimension. However, in the context of scattering off the earth, the situation is not quite so straightforward. Sarkar has shown^{14,15,16} that for distances closer to the transmitter than (R_{breakpoint}), the fields are actually near field:

$$begin{aligned} R_{breakpoint}=4 h_{TX} h_{RX}/lambda end{aligned}$$

(3)

A common feature of all models is that they predict power law like behavior for the distance dependence for distances larger than a “breakpoint”, where the fields are already in the far-field region.

For distances *closer* to the transmitter than the breakpoint, since the fields are in the near-field region, there are no analytical predictions for propagation loss, per se. The reason is that, in the near field, the TX and RX coupling depends on more than just the distance and polar angle between the two. Near-field effects are not possible to predict analytically for general cases, and depend strongly on specific geometries. The exact definition of far-field vs. near field has been extensively discussed in^{17}.

For most uses cases, where the receiver is near the ground, the breakpoint is a few hundred meters, meaning propagation studies and actual deployed systems will be beyond the breakpoint within the cell for a given tower. The case of drones is very different, where the altitude is high. For example, in our measurements here, the breakpoint is roughly 10 km assuming (h_{TX}=1 0) m, since our drone flies up to 100 m. Therefore, our receiver (and by extension the use case of drones in general) is closer to the cell tower than this breakpoint. What that means is that *all* of the propagation models fail to be applicable to our data, and for drones in general. *There is no propagation model for drone-to-ground signal propagation.*

The implications of this are that our data (and drones in general) are not covered by the traditional models and comparison must be made to near-field simulations. We turn next to comparison with existing near-field simulations.

### Comparison to near-field simulations

In the near field, Sarkar has simulated the path loss for the 2-ray model as well as his own model, for a few different specific conditions. What is characteristic of the specific cases in the near field covered (i.e. simulated^{14,15,16} and recently measured^{18}) so far are two characteristics:

Drastic oscillations of signal strength with distance

Very strong, counter-intuitive dependence on transmitter height (e.g. higher transmit gives lower signal strength)

^{19}.

High altitude receivers were not studied, but in one simulation study^{14,15,16}, a 500 m transmitter gave actually *lower* signal strength than a 100 or 10 m transmitter at distances closer than the breakpoint.

Again, although we are in the near field according to this analysis, we do not see strong oscillations in signal strength with distance and we do not see strong altitude dependence. This also contradicts the (very few) simulations in the published literature on near-field propagation models.

### Summary flight A

Our flight A shows some very clear signs that scattering is minimal and that the characteristics are closest to that of ballistic nature. A detailed consideration of all of the existing models (including physics based models as well as purely empirical models) fail to give predictions for this altitude regime because they are all formulated for ground level receivers. In specific cases where trends can be predicted based existing models, the data we measure fails to obey these trends. Thus, not all is solved. There remain some enigmas which are not currently understood.

Therefore, this data we have presented here is significant and new. At this point we can only summarize the qualitative features of the propagation that will be used for future more comprehensive models. In addition, we point out that the data is new and significant, so is the method. In fact, our invention of this method should enable new models for the drone-to-ground propagation case in the future. Any new models will need to be checked and our method is a new, novel, and the best way to do that.

### Flight B

Flight B shows a very strong dependence on altitude, in contrast to flight A.

### Power law

We are unable to extract a meaningful power law. For the 100′, 200′ data there is clearly a change of about 25 dB over 500 m (200′) or 20 dB over 500 m (100′), but the high altitude date interestingly does not show any clear distance dependence. The 300′ is a band switch event as per our interpretation, where the carrier temporarily used a different band. We have since improved our software (http://www.gitlab.com/pjbca/4GUAV) to record the band at each flight but it was not recorded on this flight.

### Comparison to Hata, Sarka, 2 ray model

All three models predict an altitude dependence, which is consistent with our data, and this is also consistent with our interpretation that the flight is in the diffusive regime. However, none of those models take mountainous terrain into account, therefore our interpretation is an extension of the concepts of diffuse , multi path scattering to mountainous terrain. The 10 dB gain per 100′ of altitude (100 lambda) is empirical and we await more quantitative modeling based on terrain to explain our data.

### Flight C

Flight B shows little dependence on altitude OR position, in contrast to flight A. However, it shows strong wave like (interference) properties. This seems to be most consistent with the near field characteristics discussed above.

### Single scattering vs. diffusion

The two extremes of wave propagation regimes [ballistic and diffuse (Fig. 1)] are illustrated through two different sets of experimental data (flights A, B) in this work. However, the actual experiments are likely a combination of both. The main conclusion of this paper is that both can contribute to the propagation characteristics and both regimes can be realized in the physical world, not just in abstract mathematical manipulations, and that the regime that dominates depends on the local scattering environment.

There is one more in-between case this is important to consider, which is the result of interference of the direct wave with those scattered at most once (or a small number of times) off of a smooth ground. This is indeed one of the most well-known models in wireless signal propagation theory, and is called the “two ray” model or the “plane earth” model. In the two-ray model, the direct path (ray) from transmitter to receiver interferes with the ray reflected specularly off the surface of the earth, which is model as a perfect conductor. The earth is assumed to be smooth in this model^{12}. In this model, the propagation loss is predicted to be 40 dB/decade, which is not what we observe. Therefore, the experimental data we present is not consistent with a model which is the result of interference of the direct wave with those scattered at most once (or a small number of times). For this reason, we find the diffusive interpretation more appropriate. An extensive discussion of scattering off of multiple sites with many lines (or rays) from transmitter to receiver has been developed by Sarkar^{10}. It therefore seems that the description as diffusive is the most appropriate interpretation of the experimental data, although at present there is no diffusion based model that can predict or even postdict our experimental data. We would like to again emphasize that this work is primarily a data driven exercise. While the data contradict existing models, there is no model at present that explains all the data we present herein. In our opinion, models are fine as exercises in thought. However, experimental data is what makes up the real world, so models must be subservient to data, and not the other way around. This is especially true in engineering fields such as wireless communications, where system performance and economic impact are ultimately beholden to real system performance data and physical reality, not mathematical models thereof.

### Scattering vs. absorption

In addition to scattering, electromagnetic fields also experience absorption. However, one must be careful when one uses the word absorption. One meaning is the complete loss of the signal upon encountering an obstacle. However, this assumes the obstacle blocking the path reflects the wave and the wave never makes it to the receiver. This is in fact not what is assumed in the Hata-COST321 and subsequent models. Those models explicitly allow for propagation from transmitter to the receiver even if there is no direct line of site. In that case, there is propagation but only after scattering off of many scattering sites (buildings, trees, etc). So this definition of absorption is equivalent to the Hata-COST321 model and its subsequent refinements. The other definition of absorption is where energy is lost due to resistive currents in scattering centers, like a black body absorbing rather then reflecting light. (We recently demonstrated this regime by perfectly impedance matching a material to free space to get of order unity absorption of RF^{20}, but this is not normally the case in the physical world we live in.) Because this is not typically the case in the RF regime, it is unlikely to be a significant contributor to signal loss, and has not been modeled in the industry standard Hata-COST321 and refinement models, and is not considered as a significant effect here.

### Summary of discussion

None of the existing models were developed for drone-to-ground communications, and therefore this data (and method) is a fundamentally new area of research that we are opening up. We have shown this through detailed analysis of each model, and each flight.

Although the models are not developed yet, we have shown three significant aspects that appear in physical reality: Ballistic wave propagation, diffusive wave propagation, and strong interference of only a few paths. This lays the experimental and intellectual foundation for the field of drone-to-ground propagation models and our method can be used to verify it. Drone-to-drone communication is also new frontier that this work enables.