Modeling of electromagnetic wave reflection from wet soil taken into account of dispersion, heterogeneity and surface roughness

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Introduction

Due to the accelerated growth of technological production processes in the agricultural industry, there is a need to measure soil moisture remotely in real time [1; 2]. Existing methods for determining soil moisture are mainly contact methods and have different labor intensity and error [3]. Synthetic aperture radar data are widely used for remote estimates of soil moisture content [4; 5]. However, such estimates may face difficulties due to various factors such as heterogeneous soil composition, temperature and vegetation cover effects. To improve the accuracy and precision of soil moisture estimates, a new approach based on a mathematical model of moist soil using the concept of artificial metamaterials is proposed [6–8]. This paper considers the adaptation of the metamaterial model to moist soil, where dry soil acts as a container and inclusions act as regions of unknown moisture content. The purpose of constructing such a mathematical model is to analyze the reflection of electromagnetic wave from moist soil, taking into account dispersion and surface roughness. For this purpose, the heterogeneous Maxwell-Garnett model [9], which takes into account the dispersion of the dielectric constant of water in the soil, has been applied. The use of such mathematical models can greatly improve the performance of remote sensing of soil moisture and provide more accurate data for agricultural processes.

1. Heterogeneous mathematical model of complex dielectric permittivity of moist soil with regard to dispersion

Moist soil is considered as a heterogeneous medium consisting of a solid matrix (dry soil) with water filled pores (Fig. 1). The complex dielectric constant (CDC) of dry soil ${\epsilon }_{с}$ can be considered as the permeability of a solid matrix, which is constant for a certain type of soil. The CDC of pure water ${\epsilon }_w,$ however, depends on both the frequency f and the temperature T.

 

Fig. 1. Wet soil as a two-component heterogeneous system

Рис. 1. Влажная почва как двухкомпонентная гетерогенная система

 

The equation for the effective CDC of moist soil based on the heterogeneous Maxwell-Garnett model can be written in the following form:

${\epsilon }_{эфф}\left(f,T,W\right)={\epsilon }_{с}\frac{1+2\alpha \left(W\right){\epsilon }_x\left(f,T\right)}{1-\alpha \left(W\right){\epsilon }_x\left(f,T\right)};$ (1)

${\epsilon }_x\left(f,T\right)=\frac{{\epsilon }_w\left(f,T\right)-{\epsilon }_c}{{\epsilon }_w\left(f,T\right)+2{\epsilon }_c},$

where $\alpha =W{\rho }_{dw}$ is the concentration of moist components in soil; $W$ is the soil moisture; ${\rho }_{dw}$ is the normalized dry soil density.

The pure water CDC is described in general by the following equation:

${\epsilon }_w\left(f,T\right)={{\epsilon }^{\text{'}}}_w\left(f,T\right)-j{{\epsilon }^{\text{'}\text{'}}}_w\left(f,T\right),$ (2)

where ${{\epsilon }^{\text{'}}}_w\left(f,T\right)$ is the real part of the CDC of water; ${{\epsilon }^{\text{'}\text{'}}}_w\left(f,T\right)$ is the imaginary part of the CDC of water; $j$ is the imaginary unit.

Further, for the compactness of writing formulas, we will use simplified designations of the real and imaginary parts of the CDC, assuming their dependence on frequency and temperature: ${{\epsilon }^{\text{'}}}_w,$ ${{\epsilon }^{\text{'}\text{'}}}_w.$

The explicit form of expressions for the real and imaginary parts of the pure water CDC are given in the ITU recommendations [10] and are written in the following form:

${{\epsilon }^{\text{'}}}_w=\frac{{\epsilon }_s-{\epsilon }_1}{1+{\Omega }_1^2}+\frac{{\epsilon }_1-{\epsilon }_{\infty }}{1+{\Omega }_2^2}+{\epsilon }_{\infty };$ (3)

${{\epsilon }^{\text{'}}}_w=\frac{{\Omega }_1\left({\epsilon }_s-{\epsilon }_1\right)}{1+{\Omega }_1^2}+\frac{{\Omega }_2\left({\epsilon }_1-{\epsilon }_{\infty }\right)}{1+{\Omega }_2^2},$ (4)

where ${\Omega }_1=f/f_1;$ ${\Omega }_2=f/f_2;$ ${\epsilon }_s=\mathrm{77,66}+\mathrm{103,3}\beta ;$ ${\epsilon }_1=$$\mathrm{0,0671}{\epsilon }_s;$ ${\epsilon }_{\infty }=\mathrm{3,52}-\mathrm{7,52}\beta ;$ $\beta =$$300/\left(T+\mathrm{273,15}\right)-\mathrm{1,}$ $f_1$ and $f_2$ are the Debye relaxation frequencies, GHz:

$f_1=\mathrm{20,20}-\mathrm{146,4}\beta +316{\beta }^2;f_2=\mathrm{39,8}{f}_1.$ (5)

2. Reflection of a plane electromagnetic wave from the air-soil interface when surface roughness is taken into account

We shall consider the problem of oblique incidence of a plane electromagnetic wave of linear polarization on the air-soil interface taking into account the surface roughness. The geometry of the problem is shown in Fig. 2. The wave falls on the interface at an angle of $\theta .$ Region 1 is a vacuum with permeabilities: ${\epsilon }_1=\mathrm{1,}$ ${\mu }_1=1.$ Moist soil (Region 2) is described by the material parameters and For simplicity, we denote the effective dielectric constant of moist soil as ${\epsilon }_{эфф}.$

 

Fig. 2. Geometry of the problem

Рис. 2. Геометрия задачи

 

To take into account the roughness of the soil surface, we used the model proposed in [11], according to which the reflection coefficients for waves of horizontal $R_e$ and vertical $R_h$ polarization are determined as:

$R_e=\frac{\cos \theta -\sqrt{{\epsilon }_{эфф}-{\sin }^2\left(\theta \right)}}{\cos \theta +\sqrt{{\epsilon }_{эфф}-{\sin }^2\left(\theta \right)}}\Psi \left(h,\theta \right);$ (6)

$R_h=\frac{{\epsilon }_{эфф}\cos \theta -\sqrt{{\epsilon }_{эфф}-{\sin }^2\left(\theta \right)}}{{\epsilon }_{эфф}\cos \theta +\sqrt{{\epsilon }_{эфф}-{\sin }^2\left(\theta \right)}}\Psi \left(h,\theta \right),$ (7)

where $\Psi \left(h,\theta \right)=\exp \left(-\frac{1}{2}h{\cos }^2\theta \right).$

In formulas (6) and (7), $h$ is the roughness parameter, which is defined as follows:

$h=4{\sigma }^2{\left(\frac{2\pi }{\lambda }\right)}^2,$ (8)

where $\lambda$ is the electromagnetic wavelength; $\sigma$ is the standard deviation of roughness on the soil surface.

According to [12], the following is accepted: for a slightly rough surface $\sigma <\mathrm{0,2}$ cm, for a surface with an average roughness of 0,2 cm $\le \sigma \le$ 1 cm, and for a strongly rough surface $\sigma >1$ cm. The values of the electromagnetic wave reflection coefficients calculated by formulas (6) and (7) from the soil were compared with the experimental results given in [13; 14]. There is a steady correspondence between theoretical dependences and experimental values of reflection coefficients of moist and frozen soils at different frequencies, at different soil moisture contents.

3. Calculation results

During the calculations, a model of loose soil ${\rho }_{dw}=\mathrm{1,5}$ (silty loam) with a standard deviation of roughness on the soil surface $\sigma >\mathrm{0,5}$ cm and a temperature $T=20$ $°\text{С.}$ CDC permeability of dry soil ${\epsilon }_c=$$\mathrm{3,556}-j\mathrm{0,361}$ is considered. Figs. 3 and 4 show graphs of calculations of the reflection coefficients of a plane electromagnetic wave of horizontal and vertical polarization depending on the frequency of probing radiation at a fixed value of soil moisture $W=15$ % and the following angles of incidence: $\theta =0°$ is depicted as a solid line, $\theta =30°$ as a dashed line, and $\theta =45°$ as a dotted line. The calculations were performed in the frequency range from 1 GHz to 15 GHz.

From the graph in Fig. 3, it can be seen that the reflection level in the case of horizontal polarization increases with increasing angle of incidence. However, for the case of vertical polarization in the frequency range of interest from 1 to 6 GHz, the reflection level decreases with increasing incidence angle.

 

Fig. 3. Dependences of the absolute values of the reflection coefficients of an electromagnetic wave of horizontal polarization on frequency at different angles of incidence

Рис. 3. Зависимости модулей коэффициентов отражения электромагнитной волны горизонтальной поляризации от частоты при различных углах падения

 

Fig. 4. Dependences of the modules of the reflection coefficients of an electromagnetic wave of vertical polarization on frequency at different angles of incidence

Рис. 4. Зависимости модулей коэффициентов отражения электромагнитной волны вертикальной поляризации от частоты при различных углах падения

 

Figs. 5 and 6 are plots of calculations of the reflection coefficient moduli of a plane electromagnetic wave of horizontal and vertical polarization as a function of soil moisture at a fixed value of the angle of incidence $\theta =30°$ and the following frequencies: 1 GHz is depicted as a solid line, 5 GHz as a dashed line, 10 GHz as a dotted line.

Calculations were carried out in the range of soil moisture variation up to 50 %. From the graphs presented in Figs. 5, 6, it can be seen that with increasing soil moisture the level of reflection smoothly increases.

 

Fig. 5. Dependences of the modules of the reflection coefficients of an electromagnetic wave of horizontal polarization on soil moisture at various frequencies

Рис. 5. Зависимости модулей коэффициентов отражения электромагнитной волны горизонтальной поляризации от влажности почвы на различных частотах

 

Fig. 6. Dependences of the modules of the reflection coefficients of an electromagnetic wave of vertical polarization on soil moisture at various frequencies

Рис. 6. Зависимости модулей коэффициентов отражения электромагнитной волны вертикальной поляризации от влажности почвы на различных частотах

 

Figs. 7 and 8 are plots of calculations of the reflection coefficient moduli of a plane electromagnetic wave of horizontal and vertical polarization as a function of the angle of incidence at a fixed value of soil moisture $W=30$ % and the following frequencies: 1 GHz is depicted as a solid line, 5 GHz as a dashed line, 10 GHz as a dotted line.

 

Fig. 7. Dependences of the absolute values of the reflection coefficients of an electromagnetic wave of horizontal polarization on the angle of incidence at various frequencies

Рис. 7. Зависимости модулей коэффициентов отражения электромагнитной волны горизонтальной поляризации от угла падения на различных частотах

 

Fig. 8. Dependences of the modules of the reflection coefficients of an electromagnetic wave of vertical polarization on the angle of incidence at various frequencies

Рис. 8. Зависимости модулей коэффициентов отражения электромагнитной волны вертикальной поляризации от угла падения на различных частотах

 

From the graphs presented in Figs. 7, 8, it can be seen that in the case of horizontal polarization with increasing angle of incidence there is an increase in the level of reflection of the electromagnetic wave, and in the case of vertical polarization the Brewster phenomenon is clearly manifested at an angle of incidence of 74°.

Conclusion

The results of calculations of electromagnetic wave reflection coefficients from moist soil obtained in this work are valuable information for various fields of science and industry. They can be used to determine the optimal plant watering regime, to control water drainage systems, and to develop systems for automated control of soil moisture in greenhouse farming. In addition, these data can be useful for ecologists when studying the effects of soil moisture on vegetation and animal life, and for geologists when studying soil composition and structure. The results of the calculations can also be used for remote sensing of the earth’s surface by Unmanned Aerial Vehicles (UAVs). This opens up new opportunities for monitoring studies of soils and land moisture, as well as for assessing the state of ecosystems.

About the authors

Dmitry N. Panin

Povolzhskiy State University of Telecommunications and Informatics

Author for correspondence.
Email: d.panin@psuti.ru
ORCID iD: 0000-0003-0598-8591
SPIN-code: 9999-0844
ResearcherId: AAT-1882-2020

Candidate of Physical and Mathematical Sciences, head of the Department of Theoretical Foundations of Radio Engineering and Communication

Russian Federation, 23, L. Tolstoy Street, Samara, 443010

Oleg V. Osipov

Povolzhskiy State University of Telecommunications and Informatics

Email: o.osipov@psuti.ru
ORCID iD: 0000-0002-2125-9228
SPIN-code: 2741-3794
ResearcherId: B-7134-2018

Doctor of Physical and Mathematical Sciences, head of the Department of Higher Mathematics

Russian Federation, 23, L. Tolstoy Street, Samara, 443010

References

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