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Detection of plant water status is important for monitoring plant physiology. Previous studies showed that radio waves are attenuated when passing through vegetation such as trees, and models (both empirical and analytical) were developed. However, for models to be more broadly applicable across a broad range of vegetation types and constructs, basic electrical properties of the vegetation need to be characterised. In our previous work, a model was developed to calculate the RF loss through vegetation with varying water content. In this paper, the model was extended to calculate RF loss through tree canopies with or without an air gap. When the model was compared with the actual RF loss acquired using Eucalyptus
*blakelyi* trees (with and without leaves), there was a systematic offset equivalent to a residual moisture content of 13% that was attributed to bound water. When the model was adjusted for the additional water content, the effective water path (EWP) was found to explain 72% of the variance in the measured RF loss.

Eucalypts are iconic Australian forest trees. The Eucalyptus forest type is by far the most common forest type in Australia covering 101 million hectares, which is 77% of Australia’s total native forest area [

Water stress affects plant growth and development due to reduction in photosynthetic activities [

Radio signals are attenuated when passing through vegetation due to absorption and scattering by the discrete elements such as the branches, stems and leaves [

Radio waves interact strongly with water [

Le Vine and Karam [

The water inside vegetation (leaves and stems) 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 [

Previous studies showed that the RF loss through vegetation is strongly dependent on the water content in vegetation through its dielectric constant. Ulaby and El-Rayes [

In our previous paper, we developed a model to calculate the RF loss through packed Eucalyptus leaves [

In the following model derivation, the radio wave electric field is first related to the intrinsic impedance and complex propagation constant of each medium in the path. Those two parameters are then related to the corresponding complex permittivity of each medium. Permittivity, in turn, is related further to the volume fractions of the substances (especially water) within the media.

We assume a plane wave travels through a set of slabs. In the case of a packed canopy (a group of trees with no space in between), each slab in the model represents one tree. For an open canopy on the other hand, each tree and each space between the trees is represented by a slab in the model (refer

Each tree is assumed to be a homogenous, lossy medium. Also, we assume that the material is non-magnetic.

In

When an incident electromagnetic wave with electric field phasor, E_{TF} is incident at the first interface, it is partially transmitted and partially reflected. The transmitted wave, E_{1F} propagates through first lossy medium with a complex

propagation γ_{1}, intrinsic impedance η_{1} and thickness d_{1} in metres. At the second interface it is again partially transmitted and partially reflected, the transmitted wave again passes through second lossy medium and so on until it transmits through the last interface into the air as E_{RF}. As a result of the reflections there is a reverse-travelling wave also, denoted with subscript R.

In _{1}, γ_{2}, γ_{3} are the complex propagation constants in the 3 different media with thicknesses d_{1}, d_{2}, d_{3}, intrinsic impedances η_{1}, η_{2}, η_{3} and complex permittivity ε_{c}_{1}, ε_{c}_{2}, ε_{c}_{3} respectively.

We consider

The combined effect of cascaded two-port networks is found by multiplying the individual T-parameter matrices, but S-parameters are more intuitive. The S-parameter matrix equation is found first and then converted to a T-parameter equation.

The first two-port network represents the interaction of the radio waves at the left-most interface in _{TF} and E_{1R}, are related to the outgoing phasors, E_{TR} and E_{1F}, by an S-parameter matrix [

[ E T R E 1 F ] = [ Γ 01 t 10 t 01 Γ 10 ] [ E T F E 1 R ] , (1)

where, t_{01} and t_{10} are the transmission coefficients of a forward and reverse travelling waves through the first interface respectively, Γ_{10} and Γ_{01} are the reverse-to-forward and forward-to-reverse reflection coefficients when waves are reflected from the first interface (air-to-first lossy medium interface). The transmission coefficients, t_{01} and t_{10} are complex and these represent both the amplitude change and the radio signal phase shift that occurs when the wave passes through the interfaces. Likewise, the reflection coefficients, Γ_{10} and Γ_{01} are complex and represent the amplitude and phase shift from reflection. These terms are expressed in the following Equations (2)-(5) [

Γ 10 = η 0 − η 1 η 0 + η 1 = ε c 1 − ε c 0 ε c 1 + ε c 0 , (2)

Γ 01 = η 1 − η 0 η 0 + η 1 = − ( ε c 1 − ε c 0 ε c 1 + ε c 0 ) , (3)

t 01 = 2 η 1 η 0 + η 1 = 2 ε c 0 ε c 1 + ε c 0 , (4)

t 10 = 2 η 0 η 0 + η 1 = 2 ε c 1 ε c 1 + ε c 0 , (5)

where, η_{0} and η_{1} are the intrinsic impedances of air and the first lossy medium respectively, and ε_{c}_{0} is the relative permittivity of air (=1).

Converting the S-parameter matrix to T-parameter matrix [

[ E T R E T F ] = [ 1 t 01 Γ 01 t 01 − Γ 10 t 01 1 t 01 ] [ E 1 R E 1 F ] . (6)

The output electric field phasor, E_{1F} of first two-port network travels through the first lossy medium and comes out of the medium attenuated as E_{2F}. Similarly, E_{2R} enters the lossy medium and exits attenuated as E_{1R}. The S-parameter for this two-port network can be written as

[ E 1 R E 2 F ] = [ 0 e − γ 1 d 1 e − γ 1 d 1 0 ] [ E 1 F E 2 R ] , (7)

where, γ_{1} is the complex propagation constant of the first lossy medium. Complex propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The real part of γ is the attenuation constant, α in Np/m (Nepers per m) and the imaginary part is the phase constant, β in rad/m [

γ 1 = j ω μ 0 ε 0 ε c 1 , (8)

where, ε_{0} and µ_{0} are the permittivity and permeability of vacuum respectively, ω is the angular frequency in rad/sec and ε_{c}_{1} is the complex permittivity of the first lossy medium. Complex permittivity, ε_{c}_{1} is expressed as

ε c 1 = ε ′ c 1 − j ε ″ c 1 , (9)

where, the real part, ε ′ c 1 represents the relative permittivity and the imaginary part, ε ″ c 1 represents the dielectric loss [

Converting the S-parameter matrix to T-parameter matrix, Equation (7) can be written as

[ E 1 R E 1 F ] = [ e − γ 1 d 1 0 0 e γ 1 d 1 ] [ E 2 R E 2 F ] . (10)

The output electric field phasor, E_{2F} of second two-port network transmits through the second interface, travels through the second lossy medium and it continues until it transmits into the air through the last interface, which is the seventh two-port network as per

The electric field phasor on the left-hand side of the cascaded 7 two-port networks can be written as simple multiplication of the T-parameter matrices and the electric field phasor on the right-hand side as

[ E T R E T F ] = [ 1 t 01 Γ 01 t 01 − Γ 10 t 01 1 t 01 ] [ e − γ 1 d 1 0 0 e γ 1 d 1 ] [ 1 t 12 Γ 12 t 12 − Γ 21 t 12 1 t 12 ] [ e − γ 2 d 2 0 0 e γ 2 d 2 ] × [ 1 t 23 Γ 23 t 23 − Γ 32 t 23 1 t 23 ] [ e − γ 3 d 3 0 0 e γ 3 d 3 ] [ 1 t 34 Γ 34 t 34 − Γ 43 t 34 1 t 34 ] [ E R R E R F ] (11)

where the incident wave at the rightmost interface E_{RR} = 0. If there are N number of lossy media then Equation (11) can be written as

[ E T R / E R F E T F / E R F ] = [ 1 t 01 Γ 01 t 01 − Γ 10 t 01 1 t 01 ] [ e − γ 1 d 1 0 0 e γ 1 d 1 ] ⋯ [ 1 t N − 1 , N Γ N − 1 , N t N − 1 , N − Γ N , N − 1 t N − 1 , N 1 t N − 1 , N ] × [ e − γ N d N 0 0 e γ N d N ] [ 1 t N , N + 1 Γ N , N + 1 t N , N + 1 − Γ N + 1 , N t N , N + 1 1 t N , N + 1 ] [ 0 1 ] (12)

Then the total loss for N lossy homogenous slabs in dB is

L s l a b = 20 log 10 | E T F E R F | . (13)

Equation (12) in this paper simplifies to Equation (7) in Peden, et al. [

If the lossy homogenous medium mentioned in Section 2.1 is a tree canopy, then the permittivity ε_{c} in Equation (8) and (9) is the permittivity of a tree canopy. A canopy of a tree consists of leaves, branches/twigs and air, and all these will contribute to the total permittivity of a tree canopy given by,

ε c = v v ε v + v a ε a , (14)

where, ε_{v} and ε_{a} are the permittivity of the vegetation (leaves and branches/twigs) and air in the canopy respectively.v_{v} and v_{a} are the volume fractions of the vegetation and air in the canopy envelope respectively.

Ulaby and El-Rayes [_{v} as a simple addition of three components: a nondispersive residual component (ε_{r}), free-water component (v_{fw}ε_{f}) and the bulk vegetation-bound water component (v_{b}ε_{b}), expressed as

ε v = ε r + v f w ε f + v b ε b , (15)

where,v_{fw} is the volume fraction of free water, ε_{f} is the dielectric constant of free water, v_{b} is the volume fraction of the bulk vegetation-bound water mixture and ε_{b} is its dielectric constant. Assuming that ε_{r} is a nondispersive residual component is supported by Hasted [^{−1} and 10^{−3}.

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

ε f = ε ′ f − j ε ″ f = ε f ∞ + ε f s − ε f ∞ 1 + j f f f 0 − j σ 2 π f ε 0 , (16)

where, f is the operating frequency in Hz, f_{f}_{0} is the relaxation frequency in Hz, and ε_{fs} and ε_{f}_{∞} are the dimensionless static and high frequency limits of ε ′ f . The salinity, S of a solution is defined as the total mass of salt in grams dissolved in a solution of 1 kg and is expressed as parts per thousand on a weight basis. The salinity for vegetation is taken to be 10‰ [

ε f = 4.9 + 75 1 + j f 18 − j σ 18 f , (17)

where, f is in GHz. The conductivity σ (siemen/metre) may be related to S (‰) by,

σ ≅ 0.16 S − 0.0013 S 2 . (18)

For bound water, Ulaby and El-Rayes [

ε b = 2.9 + 55 1 + ( j f 0.18 ) 0.5 , (19)

where, f is in GHz. Equation (17) includes a loss term associated with the conductivity of the free water and dissolved ions in the medium. In contrast, Equation (19) has no corresponding conductivity term as the water molecules are bound to other substances and do not contribute to bulk conductivity of the medium.

By inserting Equations (17) and (19) in Equation (15), the dielectric constant of a vegetation can be written as

ε v = ε r + v f w ( 4.9 + 75 1 + j f 18 − j σ 18 f ) + v b ( 2.9 + 55 1 + ( j f 0.18 ) 0.5 ) . (20)

The variation of ε_{r}, v_{fw} and v_{b} with gravimetric moisture content, M_{g} were derived by Ulaby and El-Rayes [

ε r = 1.7 − 0.74 M g + 6.16 M g 2 , (21)

v f w = M g ( 0.55 M g − 0.076 ) , (22)

v b = 4.64 M g 2 1 + 7.36 M g 2 , (23)

where, M_{g} is calculated from the weight measurement of tree before and after drying as follows

M g = 1 − ( weightofdrytree weightoftree ( differentstagesofdrying ) ) . (24)

By inserting Equation (20) in Equation (14), the dielectric constant of a tree canopy can be written as

ε c = [ { ε r + v f w ( 4.9 + 75 1 + j f 18 − j σ 18 f ) + v b ( 2.9 + 55 1 + ( j f 0.18 ) 0.5 ) } × v v ] + v a ε a . (25)

All the experiments were conducted in an indoor facility at The University of New England main campus located in Armidale, New South Wales, Australia. 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 facing each other at a separation of 6.15 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

A Eucalyptus blakelyi (also known as Blakely’s red gum) tree about 2.6 m in height (Tree 1) was cut and was mounted on a wooden pallet. The RSSI (dBm) for no obstruction between the transceivers was measured for 4 minutes and then the tree was placed in front of one antenna. The difference between the time-average RSSI with and without the tree in place was converted to a time-averaged RF loss associated with the tree. The sequence of tree and no tree

measurements was repeated three times to provide a measurement average. The RF loss (L) associated with the tree canopy was then calculated using,

L ( dB ) = RSSI ( notree ) − RSSI ( tree ) . (26)

Following the RSSI measurements with and without the tree in place, the tree was left to dry for one hour and the measurement RF loss was repeated.

The process of drying and remeasuring the RSSI was repeated until no further weight loss from drying was achieved (i.e. tree was considered dry). At this end point the mass of the water (m_{w}) in the tree canopy and subsequently each partially-dried tree canopy was retrospectively calculated from the known mass of the tree during drying and the final dry weight of the tree.

The measurement sequence was repeated for another tree with leaves (Tree 2) and a third, bare tree without leaves (Bare tree) as shown in

The tree canopy was considered as an ellipsoid (refer

V = π 6 × A × B × C , (27)

where, A, B and C are the lengths of the principal axes and these lengths were measured using a measuring tape for each tree. A, B and C of the trees used for the experiments are listed in

Eucalyptus Tree | A (m) | B (m) | C (m) | |
---|---|---|---|---|

1 | Tree 1 with leaves | 1.30 | 1.70 | 2.10 |

2 | Tree 2 with leaves | 1.55 | 1.80 | 2.15 |

3 | Bare tree | 1.36 | 1.40 | 2.10 |

The volume fraction of vegetation, v_{v} is a summation of volume fraction of the leaves in the canopy, v_{L} and the volume fraction of the woody part (branches) in the canopy, v_{wood} are given by,

v v = v L + v w o o d , (28)

v L = V L / V , (29)

v w o o d = V w o o d / V , (30)

where, V_{L} and V_{wood} are the volume of leaves and woody part of the canopy respectively. Then the volume fraction of air, v_{a} is given by

v a = 1 − v v . (31)

The volume of leaves and woody part of the canopy are calculated using their mass and density as shown below

V L = m L / ρ L , (32)

V w o o d = m w o o d / ρ w o o d , (33)

where m_{L} and m_{wood} are the weight of the leaves and woody part in the canopy respectively. The leaves were taken off from the (third) tree and weighed, and the tree was weighed separately in order to yield the values for m_{L} and m_{wood}. The densities of the leaves,ρ_{L} = 876 kg/m^{3} and woody parts, ρ_{wood} = 1110 kg/m^{3} were determined from the measurements of weight and volume of fresh leaves and fresh woody parts respectively. The volume was measured using a displacement method in water. An assumption was made here that the leaves do not shrink when the tree canopy dries out and the volume remains the same throughout the measurement period.

The radio wave passes through a tree with vegetation thickness, d containing a distributed mass, m_{w} of water (kg), we introduce the effective water path (EWP) in mm expressed as

EWP = ( m w × d ρ w × V ) × 1000 , (34)

where, V is the volume of the tree canopy envelope in m^{3} (refer Equation (27)), ρ_{w} is the density of pure water (1000 kg/m^{3}) and d in our experiment is equal to the dimension A mentioned in

For N number of trees, EWP is a summation of EWP of each tree as follows

EWP = EWP 1 + EWP 2 + EWP 3 + ⋯ + EWP N . (35)

The measured 2.4 GHz RF loss through the tree canopies for Tree 1, Tree 2, Bare tree (after leaves were removed by hand) and Trees 1 and 2 in series are depicted in

values of the RF loss versus EWP (Equation (13)) is also depicted in the graphs of

The volume fractions were calculated using the equations mentioned in Section 3.2. The tree canopy consisted of 0.6% vegetation (v_{v}) and the remainder is air (v_{a}). The vegetation (0.6%) subdivides to 0.2% leaves (v_{L}) and 0.4% woody parts (v_{wood}) of the tree canopy. For the bare tree, v_{L} = 0%, v_{wood} = 0.4% and the remainder is air. An assumption here was made that these volume fractions remain unchanged throughout the experimental period.

The measured RF loss is generally higher than modelled in all the cases shown in

Quantifying the bound water in leaves, on the other hand, is difficult although it can be estimated using a calorimetric methodology [

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. [

An empirical approach available in this work is to identify the value of bound water that would elevate the modelled data values in

Nevertheless, and with the new adjustment in the dry weight, the offset between the modelled and measured data collapses (refer

A plane wave model, including an estimation of the bound water content of tree canopies, was developed to calculate the RF loss through eucalyptus tree canopy as a function of EWP 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 tree when the latter is expressed as EWP in mm. When the model was adjusted for additional water content of 13%, EWP was found to explain 66% and 90% of the variance in the observed RF loss for single tree canopies with leaves and single tree without leaves respectively. It was also found to explain 75% of the variance when two trees with leaves were positioned in series.

The model developed in this research is compared against eucalyptus leaves and trees of some species. To generalize this model for wide range of tree types, the model needs to be compared against experiments acquired using other types of trees and other species of Eucalyptus. Verification of the model could also be done by using other parts of the tree as the lossy medium.

The first author acknowledges receipt of a Tuition Fee-Wavier Scholarship from the University of New England. One of us (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 Antony McKinnon from UNE for their help in setting up the experiment, and Prof. Jeremy Bruhl from UNE for helping us identify the eucalyptus species used for the experiment.

The authors declare no conflicts of interest regarding the publication of this paper.

Peden, S., Bradbury, R.C., Lamb, D.W. and Hedley, M. (2021) RF Loss Model for Tree Canopies with Varying Water Content. Journal of Electromagnetic Analysis and Applications, 13, 83-101. https://doi.org/10.4236/jemaa.2021.136006