# Experimental Study of Polarized Radar Scattering from the Tree Canopy at the Ka-Band

Authors: Chiung-Shen Ku; Pao-Chi Chang; Yang-Lang Chang
Journal: International Journal of Antennas and Propagation
 (2019)

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## Abstract

In this study, experimental measurements were carried out to estimate the polarized backscattering coefficient at the Ka-band through deciduous and coniferous trees. By utilizing a ground-based scatterometer, the returned signal was obtained in three polarized states at an incident angle ranging from 30 to 80 degrees. The scattering coefficients were estimated from the radar equation for the distributed target under narrow beam approximation. Radiometric calibration was conducted using a metal sphere, and a set of corner reflectors was used as the calibration target to ensure measurement accuracy. Both frequency and angular response of backscattering coefficients with three polarizations were analyzed for the cases of deciduous (banyan) and coniferous (pine) trees. The results indicated that the dynamic range for HV polarization is higher than that of HH and VV polarizations for both banyan and pine trees. Comparatively, the dynamic range for the pine tree is larger than that of the banyan tree.


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## Body

## 1. Introduction

Microwave propagation, attenuation, and scattering in vegetation are important in the fields of communication [1, 2] and remote sensing [3–7]. Vegetation occupies one-third of the land surface on earth, which plays a key role in the global circulation of carbon and nitrogen and considerably affects the energy and water balance of the biosphere. In electromagnetics, vegetation may be considered as a blockage to obstruct the line of sight between a transmitter and a receiver. In this case, propagating radio waves are forced to go through different paths to the receiver, which leads to signal degradation. The vegetation canopy represents a dynamic and complicated object of investigation in radio wave propagation due to the dimensions of leaves, branches, and trunks being comparable to the wavelengths of radio frequencies.In the last three decades, extensive research by theoretical modeling and numerical simulations on vegetation such as agricultural plants, grasslands, and forests from the VHF to Ku-bands has been reported [3–12]. For example, by solving for the single-scattering solution of the radiative transfer equation, physical and dielectric properties of the canopy have been characterized [11]. The numerical Maxwell model of 3D (NMM3D) [12] comprises randomly distributed dielectric cylinders to simulate the electromagnetic behavior of a vegetation canopy. Additionally, different types of measurement systems provide reliable data from the local to global scale. Ulaby et al. [13] applied a ground-based scatterometer at 1.6 GHz with a 40-degree incident angle to acquire tree attenuation data. Pi-SAR [14] is an airborne radar system that provides 4-look, 2 m resolution, and polarimetric data at the L band, which demonstrates the capability to display the backscattering coefficient as a function of canopy biomass.It is clear that the study of scattering and propagation of the vegetation canopy at the Ka-band is still limited. As the frequency band is pushed higher, such as in the Ka-band, there is existing interest to understand the scattering properties of trees. With the advancement of technology, the effectiveness of the Ka-band for radar applications has been demonstrated. Additionally, several Ka-band satellite missions and instrument concepts have been proposed or are even under development. In this paper, a set of experimental measurements was conducted at the Ka-band using a ground-based scatterometer. A reference measurement was taken in the anechoic chamber, and a conducting sphere and a set of corner reflectors were used as calibration targets to ensure the measurement capabilities of this system. In situ measurements were performed within the categories of deciduous and coniferous trees. The organization of this paper is as follows: Section2 describes the radar equation for distributed targets to invert the backscattering coefficient. The measurement setup and reference calibration are presented in Section 3. In Section 4, the backscattering coefficient as a function of the frequency and incident angle for deciduous and coniferous trees is discussed. Finally, in Section 5, conclusions and future studies are addressed to close the paper.


## 2. Formula from Radar Equation for the Backscattering Coefficient

### 2.1. Radar Equation for the Distribution Target

According to the radar equation, the returned power is related to the measurement system, including transmitted power, transmitting and receiving antenna gains, and the propagation path between the antenna and the target. When distributed targets are illuminated by a radar system, the average returned power for a monostatic system is given by [5, 10](1)Pθ0=∬APtG2θ,ϕ;θ0σ°θλ24π3R4dS,where θ0 is the bore sight angle of the transmitting and receiving antenna, R is the distance to the illuminated area, Pt is the transmitted power, σ° is the differential radar cross section, and G is the antenna gain pattern. If the tile of a surface among observation targets can be neglected, the differential unit of area on the radar scattering plane in Figure 1 is given by(2)dS=R2sinθcosθdθdϕ=R2tanθdθdϕ.Figure 1
Geometry of radar scattering.Lettingh be the radar to target distance at normal incidence, then(3)R=hcosθ.Now, equation (1) becomes(4)Pθ0=∫02π∫0π/2Ptλ24π3σ°θG2cos4θh4h2tanθcos2θdθdϕ=Ptλ224π3h2∫02π∫0π/2σ°θG2θ,ϕ;θ0sin2θdθdϕ.


### 2.2. Reference Measurement for Calibration

A reference target can be applied to provide a known radar cross section to obtain the absolute calibration [10]. In this study, a conducting sphere was set as a target to take a reference measurement at normal incidence. The returned power from the sphere is expressed as(5)Prcal=Ptcalλ24π3Rcal4G2θ′,ϕ′,θ0′σcal,where Ptcal is the transmitted power for calibration and Rcal is the distance from the antenna to the sphere. Under Mie scattering, the RCS of the calibration target can be approximated as πrcal2 in the high frequency range, where rcal is the radius of conducting sphere.Equation (4) may be normalized by division as in equation (5), yielding the normalized average power as(6)Pθ0=K∫02π∫0π/2g2θ,ϕ,θ0sin2θσ°θdθdϕ,where K=PrcalPtRcal4/2πPtcalh2rcal2,g2θ,ϕ,θ0=Gθ,ϕ,θ0/Gθ′,ϕ′,θ0′,g20,0,0=1.


### 2.3. Narrow Beam Approximation

If the RCS does not change significantly over the illuminated patch on the target, equation (1) may be simplified by shifting σ0 out from the integral. Additionally, by assuming the constant range R, the scattering coefficient can be expressed as(7)σ°=PrPtcalh2rcal2PrcalPtRcal4β21∬Ag2θ,ϕ,θ0dA,where contours of constant gain on the scattering plane can be approximated as an ellipse (Figure 2):(8)a≈2Rβ1/2cosθ0=2β1/2hcos2θ0,b≈2Rβ1/2=2hβ1/2cosθ0⟹A=π4ab≈πβ1/22h2cos3θ0.Figure 2
Radar scattering plane under narrow beam approximation.In particular, if we defineβ1/2 as the effective antenna beam width, then we may write the expression for the backscattering coefficient as(9)σ°=PrPtcalh2rcal2PrcalPtRcal4β1/22cosθ0.


## 2.1. Radar Equation for the Distribution Target

According to the radar equation, the returned power is related to the measurement system, including transmitted power, transmitting and receiving antenna gains, and the propagation path between the antenna and the target. When distributed targets are illuminated by a radar system, the average returned power for a monostatic system is given by [5, 10](1)Pθ0=∬APtG2θ,ϕ;θ0σ°θλ24π3R4dS,where θ0 is the bore sight angle of the transmitting and receiving antenna, R is the distance to the illuminated area, Pt is the transmitted power, σ° is the differential radar cross section, and G is the antenna gain pattern. If the tile of a surface among observation targets can be neglected, the differential unit of area on the radar scattering plane in Figure 1 is given by(2)dS=R2sinθcosθdθdϕ=R2tanθdθdϕ.Figure 1
Geometry of radar scattering.Lettingh be the radar to target distance at normal incidence, then(3)R=hcosθ.Now, equation (1) becomes(4)Pθ0=∫02π∫0π/2Ptλ24π3σ°θG2cos4θh4h2tanθcos2θdθdϕ=Ptλ224π3h2∫02π∫0π/2σ°θG2θ,ϕ;θ0sin2θdθdϕ.


## 2.2. Reference Measurement for Calibration

A reference target can be applied to provide a known radar cross section to obtain the absolute calibration [10]. In this study, a conducting sphere was set as a target to take a reference measurement at normal incidence. The returned power from the sphere is expressed as(5)Prcal=Ptcalλ24π3Rcal4G2θ′,ϕ′,θ0′σcal,where Ptcal is the transmitted power for calibration and Rcal is the distance from the antenna to the sphere. Under Mie scattering, the RCS of the calibration target can be approximated as πrcal2 in the high frequency range, where rcal is the radius of conducting sphere.Equation (4) may be normalized by division as in equation (5), yielding the normalized average power as(6)Pθ0=K∫02π∫0π/2g2θ,ϕ,θ0sin2θσ°θdθdϕ,where K=PrcalPtRcal4/2πPtcalh2rcal2,g2θ,ϕ,θ0=Gθ,ϕ,θ0/Gθ′,ϕ′,θ0′,g20,0,0=1.


## 2.3. Narrow Beam Approximation

If the RCS does not change significantly over the illuminated patch on the target, equation (1) may be simplified by shifting σ0 out from the integral. Additionally, by assuming the constant range R, the scattering coefficient can be expressed as(7)σ°=PrPtcalh2rcal2PrcalPtRcal4β21∬Ag2θ,ϕ,θ0dA,where contours of constant gain on the scattering plane can be approximated as an ellipse (Figure 2):(8)a≈2Rβ1/2cosθ0=2β1/2hcos2θ0,b≈2Rβ1/2=2hβ1/2cosθ0⟹A=π4ab≈πβ1/22h2cos3θ0.Figure 2
Radar scattering plane under narrow beam approximation.In particular, if we defineβ1/2 as the effective antenna beam width, then we may write the expression for the backscattering coefficient as(9)σ°=PrPtcalh2rcal2PrcalPtRcal4β1/22cosθ0.


## 3. Measurement Setup

Figure3 shows a simplified block diagram of the coherent radar system. A vector network analyzer, N5245A PNA-X, was utilized as a transceiver, and the specification of the vector network analyzer is listed in Table 1. A pair of wideband antennas with 2.9° of 3 dB beam width covering 26.5 GHz to 40 GHz was used as the transmitter and receiver. Additionally, the specifications of the Ka-band antenna are shown in Table 2.Figure 3
A simplified block diagram of the ground-based scatterometer.Table 1
Specifications of a vector network analyzer.
Parameter
Value
Frequency range
10 Hz–50 GHz
IF bandwidth
15 MHz
Output power
+15 dBm
Optimum dynamic range
126 dB system and 129 dB receiver
Noise level
−111 dBm at 10 Hz IF bandwidth
Dynamic accuracy
0.1 dB compression with +15 dBm input power at the receiver
Ports
2 or 4 ports with two built-in sourcesTable 2
Specifications of the Ka-band antenna.
Parameter
Value
Frequency range (GHz)
26.5–40
Diameter of main reflector (mm)
200
Gain dB (typ.)
32.7
Beam width at 3 dB level deg. (Typ.)
2.9
VSWR (typ.)
1.35
Polarization
Single


### 3.1. Reference Measurement

In a real system, under nonideal conditions such as polarization cross-talk errors and channel imbalance between the transmitter and receiver, polarimetric calibration is required. We employed three reference targets with known measured scattering matrices for polarimetric calibration. A trihedral and dihedral corner reflector can offer two equations for copolarization, and a 45° rotated dihedral corner reflector provides two equations in cross-polarization. Based on the relationship between the measured and theoretical scattering matrices, the reference targets can be used to solve the calibration matrix. After the calibration matrix has been determined, the scattering matrices of the reference targets can be calibrated. In addition, the scattering matrices of unknown targets can be calibrated. To evaluate the effectiveness of the calibration technique, the amplitude errorA^ and phase error ϕ^ are calculated according to(10)A^=20log10ScSref,(11)ϕ^=argSc−argSref,where Sc represents the calibrated scattering matrices of point targets and Sref represents the theoretical scattering matrices of point targets.Because the polarimetric calibration is primarily concerned with channel imbalance and cross-talk between the transmitter and receiver, it is preferable to take normalization with respect to a reference channel, for example, HH polarization or HV polarization whichever is applicable. As a result, the amplitude error and phase error are within 0.2 dB and 2.5°, respectively, which is regarded acceptable. A detailed calibration procedure followed is given in [15]. As an illustration, a conducting sphere with a 5 inch radius was set as a calibration target and put 1.6 m away from transmitted and receive antennas. The returned powers from 28 GHz to 36 GHz were acquired at a normal incidence. The setup of the measurement is shown in Figure 4(a), and the HH, VV, and HV polarized returned signals after the polarimetric calibration are plotted in Figure 4(b). Very close values of copolarized HH and VV and much lower values of cross-polarized HV were seen, as expected.Figure 4
(a) Measurement scene; (b) returned signal at HH, HV, and VV polarizations after polarimetric calibration.
(a)
(b)


### 3.2. Outdoor Measurement Sites

Ka-band radar measurements were conducted on banyan tree and pine tree canopies, and the measurement scene and two types of trees are shown in Figure5. A receiver was placed either under the tree canopy or outside it, and for additional assurance, a NLOS with the transmitter was placed on top of a 20 m pole above the ground. The major characteristics including the basal diameter, canopy diameter, and canopy height are illustrated in Table 3. Random samples were collected by moving the receiver. The slant range between the transmitter and the receiver was measured by laser distance in meters. Note that the wind-induced Doppler shift can be ignored due to wide bandwidth used in the receiver. To obtain a backscattering coefficient for trees, the measured system was mounted on the top of a 15 m height platform from which the radar had an unobstructed view of selected target trees. The banyan and pine trees were substantially larger than the footprint of the 2.9° of antenna beam width. Target trees were selected where adjacent (ground and tree) scatterers could be conveniently rejected by range gating in the time domain. For each measurement, the ensemble average of two thousand samples was taken to estimate the mean returned power. Then, by utilizing the radar equation, the backscattering coefficient as a function of frequency and incident angle can be calculated.Figure 5
Radar measurement of the tree canopy: (a) photo of the pole-mounted transmitter; (b) a banyan tree; (c) a pine tree.
(a)
(b)
(c)Table 3
Geometric parameters of the banyan and pine trees.
Parameter
Banyan tree (m)
Pine tree (m)
Basal diameter
0.8 ± 0.1
0.6 ± 0.1
Canopy height
12 ± 0.2
13 ± 0.2
Canopy diameter
17 ± 0.2
12 ± 0.2


## 3.1. Reference Measurement

In a real system, under nonideal conditions such as polarization cross-talk errors and channel imbalance between the transmitter and receiver, polarimetric calibration is required. We employed three reference targets with known measured scattering matrices for polarimetric calibration. A trihedral and dihedral corner reflector can offer two equations for copolarization, and a 45° rotated dihedral corner reflector provides two equations in cross-polarization. Based on the relationship between the measured and theoretical scattering matrices, the reference targets can be used to solve the calibration matrix. After the calibration matrix has been determined, the scattering matrices of the reference targets can be calibrated. In addition, the scattering matrices of unknown targets can be calibrated. To evaluate the effectiveness of the calibration technique, the amplitude errorA^ and phase error ϕ^ are calculated according to(10)A^=20log10ScSref,(11)ϕ^=argSc−argSref,where Sc represents the calibrated scattering matrices of point targets and Sref represents the theoretical scattering matrices of point targets.Because the polarimetric calibration is primarily concerned with channel imbalance and cross-talk between the transmitter and receiver, it is preferable to take normalization with respect to a reference channel, for example, HH polarization or HV polarization whichever is applicable. As a result, the amplitude error and phase error are within 0.2 dB and 2.5°, respectively, which is regarded acceptable. A detailed calibration procedure followed is given in [15]. As an illustration, a conducting sphere with a 5 inch radius was set as a calibration target and put 1.6 m away from transmitted and receive antennas. The returned powers from 28 GHz to 36 GHz were acquired at a normal incidence. The setup of the measurement is shown in Figure 4(a), and the HH, VV, and HV polarized returned signals after the polarimetric calibration are plotted in Figure 4(b). Very close values of copolarized HH and VV and much lower values of cross-polarized HV were seen, as expected.Figure 4
(a) Measurement scene; (b) returned signal at HH, HV, and VV polarizations after polarimetric calibration.
(a)
(b)


## 3.2. Outdoor Measurement Sites

Ka-band radar measurements were conducted on banyan tree and pine tree canopies, and the measurement scene and two types of trees are shown in Figure5. A receiver was placed either under the tree canopy or outside it, and for additional assurance, a NLOS with the transmitter was placed on top of a 20 m pole above the ground. The major characteristics including the basal diameter, canopy diameter, and canopy height are illustrated in Table 3. Random samples were collected by moving the receiver. The slant range between the transmitter and the receiver was measured by laser distance in meters. Note that the wind-induced Doppler shift can be ignored due to wide bandwidth used in the receiver. To obtain a backscattering coefficient for trees, the measured system was mounted on the top of a 15 m height platform from which the radar had an unobstructed view of selected target trees. The banyan and pine trees were substantially larger than the footprint of the 2.9° of antenna beam width. Target trees were selected where adjacent (ground and tree) scatterers could be conveniently rejected by range gating in the time domain. For each measurement, the ensemble average of two thousand samples was taken to estimate the mean returned power. Then, by utilizing the radar equation, the backscattering coefficient as a function of frequency and incident angle can be calculated.Figure 5
Radar measurement of the tree canopy: (a) photo of the pole-mounted transmitter; (b) a banyan tree; (c) a pine tree.
(a)
(b)
(c)Table 3
Geometric parameters of the banyan and pine trees.
Parameter
Banyan tree (m)
Pine tree (m)
Basal diameter
0.8 ± 0.1
0.6 ± 0.1
Canopy height
12 ± 0.2
13 ± 0.2
Canopy diameter
17 ± 0.2
12 ± 0.2


## 4. Results and Discussion

The backscattering coefficients of the banyan tree for HH, HV, and HH polarizations from 30 to 80 degrees of incidence are shown in Figure6. The dynamic ranges of the backscattering coefficients for HH, HV, and VV were approximately 8.7 dB, 7 dB, and 4.7 dB, respectively. The dynamic ranges of HV polarizations returned close to the levels of VV and HH polarizations for the banyan tree. Among the three polarizations, the HH polarization maintains stronger angular dependence. In general, for higher frequencies, backscattering is marginally stronger for all three polarizations, and the smaller the incidence, the stronger the backscattering. The frequency responses for HV and VV polarizations show bumpy fluctuations in terms of incidence variations, perhaps implicitly indicating higher radar clutter.Figure 6
Frequency responses of the backscattering coefficient for the banyan tree in (a) HH, (b) HV, and (c) VV polarizations in a 30 to 80 degree range of incidence.
(a)
(b)
(c)Figure7 displays the backscattering coefficient of the pine tree for VV, HV, and HH polarizations at an incident angle from 30 to 80 degrees, with a frequency from 28 GHz to 36 GHz. The dynamic ranges of the backscattering coefficients of VV, HV, and HH polarizations were approximately 4.7 dB, 8.7 dB, and 5.5 dB, respectively. The dynamic range of the HV polarized returns was clearly higher than the VV and HH polarizations for the pine tree.Figure 7
Frequency responses of the backscattering coefficient for the pine tree in (a) HH, (b) HV, and (c) VV polarizations.
(a)
(b)
(c)Comparing the backscattering coefficients for banyan and pine trees, due to the leaf stems of the pine trees constituting a larger portion of the beam area, the HV polarization backscattering is larger for the pine tree than for the banyan tree. Table4 indicates the dynamic ranges of pine trees and banyan trees from 28 GHz to 36 GHz for VV, HV, and HH polarizations at different angles of incidence.Table 4
The dynamic range of the pine tree and banyan tree from 28 GHz to 36 GHz for VV, HV, and HH polarizations.
Incident angle (degree)
Polarization
Dynamic range
Pine
Banyan
30
HH
3.5
5.9
HV
6.2
3.8
VV
2
2.4
40
HH
3.1
7.3
HV
5.6
5.3
VV
2.1
3.4
50
HH
2.8
5.9
HV
5.4
5
VV
1.8
4.8
60
HH
3.8
6.7
HV
6.1
5.2
VV
2.9
6.4
70
HH
4.3
4.9
HV
6.7
4.1
VV
1.4
6
80
HH
4.7
4.3
HV
6.8
3.4
VV
1.6
6.7Figure8 plots the dynamic range of the backscattering from trees. It shows that the dynamic range in the HV polarized state is higher than in HH and VV states for the pine tree. However, for the banyan tree, the dynamic range of cross-polarization is close to the level of the copolarization.Figure 8
The dynamic ranges of angular response in (a) pine trees and (b) banyan trees.
(a)
(b)Figures9 and 10 present the angular response of the backscattering coefficient for the banyan tree and pine tree. The results show that the coefficients of VV and HH polarization are at the same level, but the value of HH is slightly higher than that of VV. The HV backscattering is weaker than that of HH and VV.Figure 9
The angular responses of the backscattering coefficient for the banyan tree in (a) 28 GHz, (b) 29 GHz, (c) 30 GHz, (d) 31 GHz, (e) 32 GHz, (f) 33 GHz, (g) 34 GHz, (h) 35 GHz, and (i) 36 GHz.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 10
The angular responses of the backscattering coefficient for the pine tree in (a) 28 GHz, (b) 29 GHz, (c) 30 GHz, (d) 31 GHz, (e) 32 GHz, (f) 33 GHz, (g) 34 GHz, (h) 35 GHz, and (i) 36 GHz.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)The angular patterns of backscattering coefficients in the HH, HV, and VV polarizations are shown in Figure11 for the banyan tree. The results illustrate the dynamic range of backscattering coefficients in HH polarization varies from 4.3 dB to 7.3 dB, the dynamic range of backscattering coefficients changes from 3.4 dB to 5.3 dB for HV polarization, and the dynamic range of backscattering coefficients in VV polarization changes from 2.4 dB to 6.7 dB.Figure 11
Angular patterns of measured backscattering coefficients of the banyan tree in (a) HH, (b) HV, and (c) VV polarizations from 28 GHz to 36 GHz.
(a)
(b)
(c)Similarly, Figure12 presents the angular pattern of the backscattering coefficients of the pine tree in three polarized states. The results indicate that the dynamic range changes from 2.8 dB to 4.7 dB in HH polarization, the dynamic range changes from 1.6 dB to 2.9 dB for VV polarization, and the dynamic range for HV polarization varies from 5.4 dB to 6.8 dB, increasing almost 3 dB.Figure 12
Angular patterns of measured backscattering coefficients of the pine tree in (a) HH, (b) HV, and (c) VV polarizations from 28 GHz to 36 GHz.
(a)
(b)
(c)The detailed values of the dynamic range for the angular response from banyan and pine trees are listed in the Table5.Table 5
Dynamic range of measured backscattering coefficients of pine and banyan trees in HH, HV, and VV polarizations from 30 to 80 degrees.
Frequency (GHz)
Dynamic range
Pine
Banyan
VV
HV
HH
VV
HV
HH
28
5.5
10.2
7.7
4.3
5.5
6.7
29
2.9
7.8
6.3
4.2
7.4
8
30
3.5
10
5.4
7.4
7.5
8.8
31
5.6
10.4
4.7
5
8.1
7.5
32
4.3
8.9
3.4
2.7
6.7
9.7
33
3.4
8.1
6.5
4.3
5.8
10.2
34
3.4
6.5
6.6
4.4
7.5
9
35
4
9.7
3.5
4.8
7.1
9.8
36
5
7.3
5.4
5.6
7.4
8.7


## 5. Conclusions

Backscattering measurements at the Ka-band for deciduous and coniferous trees were taken via vegetation monitoring. A coherent ground-based scatterometer was utilized to record the backscattering signal within HH, VV, and HV polarizations from 28 GHz to 36 GHz at an angle of incidence from 30 to 80 degrees. A reference measurement was taken using a metal sphere at normal incidence in an anechoic chamber to ensure the measurement capabilities of this system were satisfactory. Both the frequency and angle corresponding to the backscattering coefficients were analyzed and discussed.At the Ka-band, the wavelength (1.1 cm∼0.7 cm) is much smaller than most leaves and branches. For the banyan tree, the leaf stems constitute a very small percentage of the visible surface and can be reasonably ignored. In the pine tree, the leaf stems constitute a large percentage of the visible surface, and contributions from multiple scattering reflections are larger than those in banyan trees. For example, at a 30 GHz angle of incidence, the pine tree backscattering is −13.8 dB for VV, −18.7 dB for HV, and −12.9 dB for VV, which are slightly larger than those of the banyan tree at −14.7 dB for VV, −19.2 dB for HV, and −13.2 dB for VV. The factors that affect the properties of backscattering include frequency, polarization, incidence angle, and canopy parameters. The results indicate that cross-polarized returns generally present higher dynamic ranges over the frequency and angular dependencies. By applying characteristics and statistics, the results can be used to identify the category of tree so that the effective propagation constant can be determined.



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