A Model for RF Loss through Vegetation with Varying Water Content

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1. Introduction

Radio signals are attenuated when passing through vegetation due to absorption and scattering by the discrete elements such as the branches, stems and leaves [1] [2] [3]. The so-called RF loss has been measured for specific frequencies in particular situations (forest, apple orchard & coconut garden for example) and empirical models have been developed from such measurements [2] [4] - [8]. Analytical models have also been developed but the more accurate Radiative Energy Transfer (RET) models depend on experimental measurements for their formulation and validation [9]. For a model to be accurate across a broad range of vegetation however, the relevant electrical characteristics of the vegetation need to be incorporated into the model.

The detection of plant water status is important for monitoring the physiological status of plants, the assessment of drought and fire risk in natural plant communities, and for irrigation scheduling of crops [10] [11]. Although field sampling of single leaves and shoots provides the most accurate assessment of plant water status, such methods are not feasible when estimates are required for large areas of vegetation [12].

Radio waves interact strongly with water [13] and eucalyptus leaves are no exception. While RF loss measurements at any radio frequency would be related to water content, some frequency bands are more suitable than others. Below about 600 MHz, the main RF loss mechanism involves the movement of ions. RF loss then is highly dependent on the medium’s electrical conductivity and hence the concentration of dissolved ions. Such information would be difficult to obtain for different species of vegetation which presents a hurdle for practical estimation of water content. Above about 1 GHz and up to 100 GHz, the main RF loss mechanism is the rotation of water molecules resulting from the interaction between the radio signal electric field and the molecular electric dipole [14] [15]. Frequencies above 1 GHz have an advantage in that electrical conductivity does not play a significant part hence avoiding the need to characterise the highly-variable constituents of electric conductivity in leaves. The higher frequency also offers another advantage; namely directional antennas that can be used to facilitate location-specific measurements are smaller.

Le Vine and Karam [16] calculated the attenuation associated with a vegetation canopy using a discrete scatter model, where the vegetation canopy is presented by a sparse layer of discrete, randomly oriented particles such as leaves, stalks, branches, etc. over a homogeneous ground plane (soil). They found that for frequencies up to 5 GHz the attenuation varies linearly with plant water content but for optical frequencies, the attenuation is relatively independent of both water content and frequency. Nakajima, Ohyama, Juzoji and Ta [17] measured the RF attenuation of individual leaves at 5, 10 and 20 GHz in a waveguide. They also investigated the effect in a living tree by measuring RF attenuation at 10.5 GHz. They asserted that “Microwave attenuation by tree foliage should have a strong link to water content in the leaves”. The dependency of attenuation on water content is through the dielectric constant which is highly dependent on the water content inside the material. Furthermore, moisture on the surface of leaves of the trees also absorbs the RF waves resulting in more attenuation [18] [19] [20] [21].

The water inside leaves/vegetation can be divided into free water and bound water. Free water is the liquid water found in cell lumen and is relatively easy to remove [22]. Bound water is the water molecules that penetrate the cell walls and are chemically bound to cellulose molecules. Bound water cannot always be expelled by heat without damaging the material [23] and the removal of bound water also depends on the temperature and relative humidity of the environment [24].

The aim of this paper is to extend the model of complex permittivity for vegetation developed by Ulaby and El-Rayes [25] with the view to developing a means of estimating water content of vegetation. They developed a Debye-Cole dual-dispersion model for complex relative permittivity (also known as complex dielectric constant) consisting of three parts: a dispersive free-water component, a dispersive bound water component and a nondispersive residual component. Their proposed dielectric model was found to give excellent agreement with data over a wide range of moisture conditions and over the entire 0.2 - 20 GHz range examined in their study. In this paper a model for calculating the radio frequency (RF) signal loss in vegetation is derived. The model in this study uses the complex permittivity for vegetation modelled by Ulaby and El-Rayes. The model is compared against experimental measurements of RF loss for eucalyptus leaves at 2.4 GHz.

2. Model

2.1. Wave Propagation

We assume that a plane wave travels through a lossy homogeneous medium of thickness *d* with complex relative permittivity, *ε** _{v}* as shown in Figure 1. Also, we assume that the material is non-magnetic.

Complex permittivity, *ε** _{v}* is expressed as

Figure 1. The medium is modelled as a parallel-sided slab of thickness d with air on either side. When a plane radio wave travelling left-to-right meets the medium it is partially transmitted and partially reflected at the first interface. The transmitted wave propagates through the material with a complex propagation constant. At the second interface it is again partially transmitted and partially reflected. The total outgoing wave is a composite of the unreflected “straight through” wave and waves that have been reflected internally 2, 4, …, times.

${\epsilon}_{v}={{\epsilon}^{\prime}}_{v}-j{{\epsilon}^{\u2033}}_{v}$ (1)

where, the real part, ${{\epsilon}^{\prime}}_{v}$ represents the relative permittivity and the imaginary part, ${{\epsilon}^{\u2033}}_{v}$ represents the dielectric loss [23].

When an incident electromagnetic wave with electric field phasor, *E*_{0} passes through a lossy homogeneous medium, the total transmitted field phasor will be a combination of transmitted power across the interfaces and reflections from the interfaces, as shown in Figure 1.

In Figure 1, *T*_{1} is the transmission coefficient of the wave transmitting through air-to-lossy medium interface, *T*_{2} is the transmission coefficient of the wave transmitting through lossy medium-to-air interface, *γ* is the complex propagation constant in the medium and Γ is the reflection coefficient of the wave undergoing multiple internal (partial) reflections inside the slab at the lossy medium-to-air interfaces. These terms are expressed in the following Equations (2)-(5) [26]:

${T}_{1}=\frac{2}{\sqrt{{\epsilon}_{v}}+1}$, (2)

${T}_{2}=\frac{2\sqrt{{\epsilon}_{v}}}{\sqrt{{\epsilon}_{v}}+1}$, (3)

$\Gamma =\frac{\sqrt{{\epsilon}_{v}}-1}{\sqrt{{\epsilon}_{v}}+1}$, (4)

$\gamma =j\omega \sqrt{{\mu}_{0}{\epsilon}_{0}{\epsilon}_{v}}$ (5)

where, *ω* is the angular frequency in rad/sec, *µ*_{0} and *ɛ*_{0} are the permeability and permittivity of air respectively.

The total transmitted complex electric phasor, *E* after passing through a lossy medium slab is given by the “straight through” wave and a series of multiple-reflected waves (refer Figure 1). Depending on the relative phase shifts in the different paths, these contributions to the outgoing wave may reinforce or cancel each other. Summing these terms as phasors accounts for the phase shift

$E={T}_{1}{T}_{2}{\text{e}}^{-\gamma d}{E}_{0}\left(1+{\Gamma}^{2}{\text{e}}^{-2\gamma d}+{\Gamma}^{4}{\text{e}}^{-4\gamma d}+\cdots \right)$. (6)

Using geometric series for |Г| < 1, *E* can be expressed as

$E=\frac{{T}_{1}{T}_{2}{E}_{0}{\text{e}}^{-\gamma d}}{1-{\Gamma}^{2}{\text{e}}^{-2\gamma d}}$. (7)

The total loss for the lossy homogenous slab in dB is

${L}_{slab}=20{\mathrm{log}}_{10}\left|\frac{{E}_{0}}{E}\right|$. (8)

Replacing Equations (2)-(4) & (7) in Equation (8) yields

${L}_{slab}=20{\mathrm{log}}_{10}\left|\left({\left(\sqrt{{\epsilon}_{v}}+1\right)}^{2}-{\left(\sqrt{{\epsilon}_{v}}-1\right)}^{2}\left({\text{e}}^{-2\gamma d}\right)\right)/\left(4\sqrt{{\epsilon}_{v}}{\text{e}}^{-\gamma d}\right)\right|$. (9)

2.2. Complex Permittivity of Vegetation

If the lossy homogenous medium is leaves, then the permittivity *ɛ** _{v}* in Equation (9) is the permittivity of vegetation. Ulaby and El-Rayes [25] developed a dielectric model to calculate the dielectric constant of vegetation. They modelled the dielectric constant of vegetation,

${\epsilon}_{v}={\epsilon}_{r}+{v}_{fw}{\epsilon}_{f}+{v}_{b}{\epsilon}_{b}$ (10)

where, *v _{fw}* is the volume fraction of free water,

Free water may contain dissolved salt and the frequency dependent dielectric constant of bulk saline water is given by the Debye equation [27],

${\epsilon}_{f}={{\epsilon}^{\prime}}_{f}-j{{\epsilon}^{\u2033}}_{f}={\epsilon}_{f\infty}+\frac{\left({\epsilon}_{fs}-{\epsilon}_{f\infty}\right)}{1+j\left(f/{f}_{f0}\right)}-j\frac{\sigma}{2\pi f{\epsilon}_{0}}$ (11)

where, *f* is the operating frequency in Hz, *f _{f}*

${\epsilon}_{f}=4.9+\frac{75}{1+j\left(f/18\right)}-j\frac{\sigma 18}{f}$ (12)

where, *f* is in GHz.

The conductivity *σ *(siemen/metre) may be related to *S* (‰) by,

$\sigma \cong 0.16S-0.0013{S}^{2}$. (13)

For bound water, Ulaby and El-Rayes [25] conducted dielectric measurements on sucrose-water mixture and data was fitted to Cole-Cole dispersion equation. The complex dielectric constant of bound water is given by

${\epsilon}_{b}=2.9+\frac{55}{1+{\left(jf/0.18\right)}^{0.5}}$ (14)

where, *f* is in GHz. Equation (11) includes a loss term associated with the conductivity of the free water and dissolved ions in the medium. In contrast, Equation (14) has no corresponding conductivity term as the water molecules are bound to other substances and do not contribute to the bulk conductivity of the medium.

By inserting Equations (12)-(14) in Equation (10), the dielectric constant of vegetation can be written as

${\epsilon}_{v}={\epsilon}_{r}+{v}_{fw}\left(4.9+\frac{75}{1+j\left(f/18\right)}-j\frac{\sigma 18}{f}\right)+{v}_{b}\left(2.9+\frac{55}{1+{\left(jf/0.18\right)}^{0.5}}\right)$. (15)

The variation of *ε _{r}*,

${\epsilon}_{r}=1.7-0.74{M}_{g}+6.16{M}_{g}^{2}$, (16)

${v}_{fw}={M}_{g}\left(0.55{M}_{g}-0.076\right)$, (17)

${v}_{b}=\left(4.64{M}_{g}^{2}\right)/\left(1+7.36{M}_{g}^{2}\right)$, (18)

where, *M _{g}* is calculated from the weight measurement of leaves before and after drying them inside a vacuum oven as follows

${M}_{g}=1-\left(\frac{\text{weightofdryleaves}}{\text{weightofleaves}\left(\text{differentstagesofdrying}\right)}\right)$. (19)

3. Material and Method

All the experiments were carried out at The University of New England main campus located in Armidale, New South Wales, Australia (latitude 30.4867˚S and longitude 151.6430˚E). Two flat-panel, phased-array directional antennas (ARC Wireless Solutions, USA, PA2419B01, 39.1 cm × 39.1 cm × 4.3 cm) were used, one as a transmitter connected to a transceiver Beacon (Dosec Design, Australia, EnviroNode Beacon) and the other as a receiver connected to a transceiver hub (Dosec Design, Australia, EnviroNode Hub) operated at a frequency of 2.4331 GHz. The antenna had a gain of 19 dBi, front-to-back ratio of >30 dB and 3 dB beamwidth of ±9˚. The antennas were placed outside, facing each other at a separation of 6.10 metres. A constant transmitted power of 100 milliwatts was used. The hub measured and logged the RSSI (received signal strength indicator, dBm) to a removable SD card at 1 minute intervals. The experimental set-up is shown in Figure 2.

Some preliminary measurements were made to verify the experimental set-up. A wooden frame covered, front and back, with 5 mm clear acrylic sheets was used to hold the leaves to be tested. In order to confirm that only the radio waves propagating directly through the wooden frame were received by the receiver (*i.e.* no multipath signals), a metal sheet, impenetrable to radio waves, was temporarily attached to the front of the frame. The position of the frame relative to the transceivers was then optimised such that the RSSI was below the minimum measurable power level for our equipment (−80 dBm).

In order to measure any potential effects of the wooden frame itself, each of the empty frames was positioned between the transceivers and the RF signal loss, relative to no frame in place, was measured. Irrespective of frame size, the RF

Figure 2. Experimental set-up. Two flat-panel antennas were mounted facing each other 6.10 m apart. A frame containing the leaves being tested was mounted on a tripod immediately in front of one antenna. Solar panels for electrical power are also visible. The RSSI (dBm) was recorded to a removable SD card inside the transceiver Hub every minute.

loss through the empty frame was confirmed to be below the measurement resolution of 0.1 dB, which indicated that any loss associated with the frame of leaves would be solely due to the presence of the contained material inside.

A wooden frame (inner dimensions 600 mm × 560 mm × 42 mm) was completely filled with freshly-plucked, turgid *Eucalyptus laevopinea* (silver top stringybark) leaves such that there was no visible gap between the frame and leaves when the frame is flipped over. The filled frame was then weighed. The RSSI (dBm) for no obstruction between the transceivers was measured for 3 minutes and then the frame filled with leaves was placed on the stand and signal strength was again measured for 3 minutes. The difference between the time-average RSSI with and without frame in place was converted to a time-averaged RF loss associated with the leaves. The sequence of frame and no frame measurements was repeated three times to provide a measurement average. The RF loss (*L*) associated with the sample of leaves was then calculated using,

$L\left(\text{dB}\right)=\text{RSSI}\left(\text{noframe}\right)-\text{RSSI}\left(\text{frame}+\text{leaves}\right)$. (16)

Following the RSSI measurements with and without the frame in place, the leaves were removed from the frame and oven-dried in a vacuum oven at a temperature of 60˚C and 80 kPa (60 cmHg) vacuum gauge pressure for one hour. As condensation and the temperature of the transmitting material effects radio wave transmission, the hot leaves were then spread on a table surface to both cool down and allow residual water vapour to dissipate rather than condense. The frame was then refilled with the cooled leaves and weighed. The frame containing the now partially-dried leaves was then placed on the stand at the experimental site and the process of measuring the RSSI was repeated.

The process of partial drying and remeasuring the RSSI was repeated until no further weight loss from drying was achieved. At this point, the desiccated leaves were yellow/brown in colour and brittle.

At this end point the mass of the water (m_{w}) in the freshly plucked and each partially-dried, leaf sample was retrospectively calculated from the known mass of the leaf samples and the final dry weight of the desiccated leaves.

The measurement sequence was repeated for frame thicknesses of 63 mm, 105 mm, 147 mm and 195 mm. Any RF loss measurements exceeding 30 dB, as occurred in the 147 mm and 195 mm frame thicknesses, exceeded the reliable measurement range of the equipment and were excluded from subsequent analyses.

The radio wave passes through vegetation thickness, *d _{v}* containing a distributed mass, m

${m}_{w}={\rho}_{w}\times \text{EWP}\times x\times y$, (21)

where,*ρ _{w} *is the density of pure water 1000 kg/m

$\text{EWP}=\left(\frac{{m}_{w}\times {d}_{v}}{{\rho}_{w}\times V}\right)\times 1000$ (22)

where, *V* is the volume of the frame in m^{3}.

4. Result and Discussion

The RF loss through vegetation was analysed at 2.4 GHz and for 5 different vegetation thicknesses (42 mm, 63 mm, 105 mm, 147 mm and 195 mm). For each thickness, the RF loss in dB is maximum when the leaves are wet and the RF loss monotonically decreases with the reduction of EWP associated with drying (Figure 3). The calculated value of the RF loss versus EWP (Equation (9)) is also given in the graphs of Figure 3 (red curves).

All graphs exhibit similar gradients (Figure 3(f)) and in all cases the measured RF loss is generally higher than modelled loss irrespective of vegetation thickness. The consistent offset between the measured and modelled values, we believe, is attributable to not adequately accounting for the unbound water in estimating EWP using Equation (21). The method we used to dry the leaves may not have removed the water completely, especially the bound water [23] [24]. When dried to constant weight by heating, vegetation is in an equilibrium state with the drying air [24]. Moreover, the oven dried leaves may then re-absorb water from the air in the period of time between drying and testing. For example, the moisture content of dried and ground alfalfa could be up to 20% with the normal range of humidity for the time period of testing [24] (Figure 4).

Quantifying the bound water in leaves, on the other hand, is difficult although it can be estimated using a calorimetric methodology [29] [30] [31]. Note this methodology refers to the notion of unbound and bound water as being, respectively, “freezable” and “unfreezable”. Assuming they are related, we were unable

(a)(b)(c)(d)(e)(f)

Figure 3. (a)-(e) Plots of measured and modelled (Equation (9)) RF loss (dB) as functions of EWP (mm) for packed *Eucalyptus laevopinea* leaves of varying thickness (d = 42 mm, 63 mm, 105 mm, 147 mm and 195 mm), (f) Combined plot of measured RF loss (dB) as a function of EWP for all the five thicknesses. The average RMSE (f) is ±5.6 dB.

Figure 4. Equilibrium Moisture Content (EMC, % dry basis) versus Relative Humidity (%) for ground alfalfa at 25˚C. “Ads” and “Des” stand for adsorption and desorption, respectively [24].

to discern this value for eucalyptus leaves using available literature. However, Whitman [32] provides an insight into at least the possible orders of magnitude of this value on the basis of his work on a range of Prairie grasses in the U.S. during the summer season. Sagebrush, for example, has a bound water content in its leaves ranging from 10% - 30% (dry-weight basis) with other grass species exhibiting similar ranges and sometimes higher. In this earlier work, however, the bound water content is measured from freshly-sourced leaves which were not subjected to further desiccation. Here the values would be influenced by external factors such as soil moisture content etc. [32].

An empirical approach available in this work is to identify the value of bound water that would elevate the modelled data values in Figure 3 to the measured values, effectively considering the actual water content of our leaves to be higher (by this additional, bound contribution). We identified this value by finding the minimum total variance between measured and modelled values. The residual water from 1% to 15% in 0.1% increments was evaluated. The best fit between the modelled and the measured values is achieved by assuming that the vegetation contained an additional 6.5% of water content when dried to constant weight. Residual water of this order of magnitude is plausible when compared against measurements of other leaf types [32]. Of course, what remains unclear is whether or not the “unfreezable” and “freezable” components of water identified by Whitman and others [29] [30] [31] is accessible through the leaf desiccation process in this work (or not) and whether the bound component is a contributor to the RF losses observed in this work.

Nevertheless, and with the new adjustment in the dry weight, the offset between the modelled and measured data collapses, reducing the average RMSE from ±5.6 dB to ±2.2 dB (Figure 5(f)). It is notable that the modelled and measured data retain similar gradients; in other words, there appears to be no systematic deviation in the relationship between the amended model and measured values. Despite this encouraging outcome, and in light of the comments above, further work will be undertaken to confirm the value of the bound water components of our leaf samples.

(a)(b)(c)(d)(e)(f)

Figure 5. (a)-(e) Plots of measured and revised-modelled (Equation (9) with additional 6.5% leaf water content) RF loss as functions of EWP for packed *Eucalyptus laevopinea* leaves of varying thickness (d = 42 mm, 63 mm, 105 mm, 147 mm and 195 mm); (f) Combined plot of measured RF loss (dB) as a function of EWP (with additional 6.5% leaf water content) for all the five thicknesses. The average RMSE (f) is ±2.2 dB.

5. Conclusions

A plane wave model, including an estimation of the water content of leaves, was developed to calculate the RF loss through packed eucalyptus leaves as a function of the effective water path of the assemblages at a frequency of 2.4 GHz. There was a positive non-linear relationship between RF loss in dB and the water content of the leaves when the latter is expressed as effective water path (EWP) in mm. The modelled values versus actual measurements yielded an average RMSE of ±2.2 dB.

Further work is being undertaken by us to relate this mathematical model of radio propagation to empirical measurements of water content and RF loss through whole trees with the view to assessing plant water status for monitoring plant physiology.

Acknowledgements

The first author acknowledges receipt of a Tuition Fee-Wavier Scholarship from the University of New England. DWL acknowledges the support of Food Agility CRC Ltd, funded under the Commonwealth Government CRC Program. The CRC Program supports industry-led collaborations between industry, researchers, and the community. All authors gratefully acknowledge the contribution of Derek Schneider and Patrick Littlefield from UNE for their help in setting up the experiment, and Professor Jeremy Bruhl from UNE for helping us identify the eucalyptus species used for the experiment.

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