# The Analysis of Experimental Deployment of IGLUNA 2019 Trans-Ice Longwave System
## 5 Transmission Performance Comparison of Propagation Models
### Proposed Propagation Models
For the region adjacent to LF, the VLW/VLF (very long wave/very low frequency), the following formula was proposed in [28] (approximate; for planetary waveguide propagation):  
\[E\left[\frac{mV}{m}\right]=\frac{245\sqrt{P_{\Sigma}G}}{\sqrt{\left(a+\frac{ Z_{eff.}}{2}\right)}Z_{eff.}\sin\alpha} \tag{8}\]  
with \(Z_{eff. }\) calculated using Equation (5). As the performed experiments used a great advantage of the transmitter's elevation above the sea level to achieve a large distance, a formula incorporating this mechanism could be taken into consideration--a formula by [PERSON] [28] (for the UHF range):  
\[E\left[\frac{mV}{m}\right]=\frac{2.18\sqrt{P_{\Sigma}G}h_{1}h_{2}}{r^{2}\lambda} \tag{9}\]  
where \(h_{1}\) and \(h_{2}\) are the heights of transmitting and receiving antenna positions [m], here 3868 and 27 m, respectively. This formula is to provide best results for \(h_{1}h_{2}\leq r\)[m]\(\cdot\lambda/18\) (in this case: \(h_{1}h_{2}=104,\)436 m\({}^{2}<r\lambda/18\approx 128\times 10^{6}\) m\({}^{2}\). The last formula (the 'diffractive' formula) allows to calculate the E field strength slightly farther than in the visual distance r\({}_{\rm H}\) (best for r\({}_{\rm H}<8.24\)(\(h_{1}^{0.5}+\)\(h_{2}^{0.5}\))) and is approximately described as [28]:  
\[E\left[\frac{mV}{m}\right]=\ \frac{4.25\sqrt{P_{\Sigma}G}h_{1}h_{2}r_{H}^{n_{ t}-2}}{r^{n_{t}}} \tag{10}\]  
where \(n_{t}\) is the indicator of attenuation of the radio wave behind the horizon-defined for LW as close to 2.5 [28]. Equations (4)-(10) have been plotted altogether for a comparison of their results (Figures 14 and 15). The diffractive formula gave results the closest to the ones achieved by P1A emission on 266 kHz, yet the course of this function remains far from expected values; [PERSON]. [PERSON]'s formula gave too low values. The ground wave formula by the UER (best fitting for the frequencies used in the experiment), the Austin-Cohen formula and the VLW formula (long-range low frequency communication) appear best for adaptation to approximately describe the propagation of A3E-SC and P1A/CW (continuous wave) emissions, respectively, with the given constraints:  
* 140-150 times higher elevation of the transmitting antenna in relation to the receiving antenna;
* upper part of the LF frequency band;
* horizontal quarter-wavelength singular loop transmitting antenna; and
* antenna positioned in a diamagnetic structure. The re-formulation of the Austin-Cohen formula (Equation (5)) requires its division by 10 to describe the E field strength at A3E-SC at large distances; if the VLW is multiplied by 3.33, it could be used to describe the E field strength for P1A at large distances. The re-formulation of Equation (2) requires the development of a coefficient which includes the height square root sum, mentioned above in the description of Equation (8). Assuming that this coefficient would take the following approximate form (for its parameter values given above, it would put approximately equal the results of Equation (2) to the experiment's results):  
\[n_{e}(h_{1},h_{2},\ r)=52.4\cdot 10^{-4}\frac{\sqrt{h_{1}}+\sqrt{h_{w}}}{r} \tag{11}\]  
Equation (4) would be reformed into (Equation (5) remains unchanged):  
\[E^{\prime}\left[\frac{mV}{m}\right]=1.573\cdot 10^{8}\sqrt{P_{\Sigma}G}\frac{ \left(\sqrt{h_{1}}+\sqrt{h_{2}}\right)}{\left(r^{2}+4Z_{eff.}^{2}\right)^{ \frac{3}{2}}} \tag{12}\]  
The accuracy and utility of these formulas for determining the \(E\) field strength distribution for this transmission case are to be verified in the next planned experiments--as the effects of the comparison and the reformulations are based on the signal data at maximum distance with few data on shorter distances (this verification would give an insight on the \(E\) field strengths on shorter distances--which formula in terms of its course is best applicable for the achieved results).