FDTD Simulation of GPR with a Realistic Multi-Pole Debye Description of Lossy and Dispersive Media M. Loewer 1, 2, J. Igel 1 Leibniz Institute for Applied Geophysics, Stilleweg 2, 30655 Hannover, Germany 2 Leibniz University Hannover, Callinstraße 3, 30167 Hannover, Germany
[email protected],
[email protected]
Abstract— Simulation of electromagnetic wave propagation in lossy and dispersive media requires a realistic description of the electrical and dielectric media parameters. We measured the complex dielectric permittivity of fine sand and silty clay in the GPR frequency range using the coaxial transmission line technique. We inverted the data with singleand multi-pole Debye models and a constant dc conductivity term. FDTD simulations were carried out on the basis of the different Debye pole media descriptions and compared to simulations with constant electromagnetic parameters. We show that for soils with a fine-grained texture, the frequencydependence of the real part of the dielectric permittivity cannot be neglected. Therefore a multi-pole Debye description of the medium must be used for simulations: one Debye pole describing the relaxation of free water in the lower GHz region and two poles describing the bound water and interfacial relaxations in the upper MHz region.
precise description of the frequency-dependent dielectric relaxation mechanisms based on laboratory measurements, together with finite-difference time-domain (FDTD) simulations of the GPR wave propagation. distance [m]
0
time [ns]
1
2.0
3.0
(a)
10
20 0
Index Terms— Debye relaxation, dispersion, soil, complex dielectric permittivity.
1.0
11% clay, 67% sand, θv = 14%,
= 1 mS/m
(b)
10
20 0
time [ns]
One of the first questions, which often come along with a GPR survey, is the question of the soil or subsurface suitability. However, providing an empirical answer to the question can be ambitious. Reference [1] for instance created a useful GPR soil suitability map for the U.S. by defining a GPR component index value that is based on the clay content, the dc electrical conductivity and the calcium carbonate or sulfate content of the soil. However, the map is based on responses from GPR antennas with center frequencies between 100 MHz and 200 MHz and the characteristics of the GPR system itself are not taken into account. Choosing a higher center frequency, as e.g., used for landmine and IED detection can lead to a dominating influence of dielectric relaxation mechanisms [2]. Therefore, a different weighting of the GPR component index with respect to frequency will become necessary. Against this background, we investigate how the soil physical parameters control the frequency-dependence of the effective dielectric permittivity and how this affects GPR wave propagation. In our approach, we combine a
time [ns]
I. INTRODUCTION
27% clay, 23% sand, θv = 33%,
= 10 mS/m
(c)
10
20
42% clay, 12% sand, θv = 44%,
= 10 mS/m
Fig. 1. 900 MHz radar profiles of soils with different amounts of clay and sand, varying volumetric water content θv and dc electrical conductivity σdc. 2 metal plates are buried in 20 cm and 40 cm depth. Static shift, divergence compensation and a bandpass filter have been applied.
II. DIELECTRIC RELAXATION
where ε0 is the permittivity of free space, ε’r,eff and ε’’r,eff are the real and imaginary part of the complex relative effective A. Influence on GPR measurements permittivity ε*r,eff and σdc is the dc electrical conductivity, The strong influence of dielectric relaxations can be with ω the angular frequency (𝜔 = 2𝜋𝑓, where f is the seen in the radargrams in Fig 1. It shows GPR frequency). The data had been measured by means of the measurements of 3 homogenous soils that had been carried coaxial transmission line (CTL) technique in the frequency out with a ground-coupled 900 MHz antenna system. Each range between 1 MHz and 10 GHz [6]. A model consisting soil has different clay and water content and 2 metal plates of 4 Debye poles and a dc electrical conductivity term were with the dimension of 50 cm x 50 cm had been buried in 20 fitted to the data using the Geophysical Inversion and cm and 40 cm depth. The GPR measurements were carried Modelling Library (GIMLi) [7]: out with a ground-coupled 900 MHz system. In soil (c) the N ε σ r , s − ε r ,∞ reflection from the deeper plate can hardly be seen, even so (2) ε *r ,eff (ω ) − ε ∞ = − i dc , the dc electrical conductivity of the soil is comparable to the ωε 0 k =1 1 + iωτ conductivity of soil (b). The latter had been determined by means of geoelectrical resistivity measurements. Therefore, wherein ε r,∞ corresponds to the high-frequency limit of the the attenuation of the signals are caused by the dominating permittivity, ε r,s is the static or low-frequency limit, τk the influence of dielectric relaxations, that in turn are due to the relaxation time and N is the number of Debye poles. In higher water and clay content in soil (c) in comparison to contrast to [6], we used a multi-pole Debye model instead of a single broadband Cole-Cole model to describe the soil (b). relaxations in the MHz range. The multi-pole Debye model B. Fitting with Debye models The variations of the complex effective dielectric measured data fit (a) permittivity with respect to the frequency of the applied EM relax. i relax. ii field are caused by different dielectric relaxation relax. iii mechanisms [3, 4, 5]. If we consider a broad EM spectrum relax. iv iv
∑
TABLE I. CONSTANT ELECTRICAL AND DIELECTRIC PARAMETERS AND
i
iii
SINGLE AND MULTI-DEBYE RELAXATION PARAMETERS THAT WERE USED IN THE FDTD SIMULATIONS. Dielectric Relaxation Parameters const. parameters
Fine sand
ε! = 25.05 τ = 9.2e − 12s σ!" = 0.003 S/m ε! = 5.3
const. parameters
1 Debye relaxation
ε! = 26.07 σ = 0.137 S/m
ε! = 24.07 τ = 8e − 12s σ!" = 0.137 S/m ε! = 2.0
(b) 3Debye relaxations
ε! = 19.99 ε! = 5.46 ε! = 24.07 τ! = 4.83e − 9s τ! = 4.41e − 10s τ! = 8e − 12s σ!" = 0.038 S/m ε! = 2.0
diel. relaxations only
ity tiv uc nd co al ric ct le -e dc
Silty clay
ε! = 25.05 𝝈 = 𝟎. 𝟎𝟎𝟑 𝑺/𝒎
ii
1 Debye relaxation
iv
i
iii ii
with a bandwidth of 100 kHz to 100 GHz and therefore covering by far the entire GPR frequency range, we can Fig. 2. Real (a) and imaginary part (b) of the relative effective complex observe one high frequency dielectric relaxation around 17 dielectric permittivity measured with the CTL technique. The data are fitted GHz and a number of relaxations taking place in the MHz with a 4-pole Debye model (i, ii, iii, iv) plus a dc electrical conductivity frequency ranges and below. Fig. 2 shows the real and term. The inlay shows imaginary permittivity subtracted by the dc electrical conductivity term. imaginary part of the dielectric permittivity for moist silty clay. The data contain all loss and energy storage mechanisms taking place in measured material and are is the most straightforward way to implement the complex described in effective parameters: dielectric permittivity parameter into a FDTD code and no approximations to the Cole-Cole model in time-domain are ⎛ σ ⎞ (1) required in order to solve the Maxwell equations [8, 9]. The ε *r ,eff = ε ' r ,eff −iε ' ' r ,eff = ε ' r −i⎜⎜ ε ' ' r + dc ⎟⎟ , ωε 0 ⎠ intrinsic attenuation α and the phase velocity v were ⎝
calculated from the measured data using:
α=
ω c0
v = c0
ε ' r ,eff ⎛⎜
⎛ ε ' ' r ,eff ⎜ 1 + ⎜⎜ 2 ⎜ ⎝ ε ' r ,eff ⎝
ε ' r ,eff ⎛⎜
⎛ ε ' ' r ,eff ⎜ 1 + ⎜⎜ 2 ⎜ ⎝ ε ' r ,eff ⎝
2 ⎞ ⎞ ⎟ − 1⎟⎟ ⎟ ⎟ ⎠ ⎠
and
(3)
2 ⎞ ⎞ ⎟ + 1⎟⎟ . ⎟ ⎟ ⎠ ⎠
(4)
III. MODELLING AND SIMULATIONS A. Soil sampels and fitting parameters Two soils had been investigated using dielectric spectroscopy by means of the coaxial transmission line technique: fine sand with 97 percent sand content and 1 percent clay and silty clay with 12 percent sand and 42 percent clay. Both soils are nearly saturated with volumetric water content of 38.6 percent in the sand and 40.5 percent in the clay. The measured relative effective dielectric permittivity of the sand with its real and imaginary part is shown in Fig. 5a) and b). The measured dielectric data for the silty clay are shown in Fig. 6a) and b), respectively. Intrinsic attenuation and velocity that had been calculated by (3) and (4) are shown in Fig. 5c), d) and 6c), d). distance [m] 0.0
depth [m]
-0.0
0.5
1.0
1.5
2.0
GPR
-0.2 -0.4
dielectric permittivity between 50 MHz and 1 GHz remains constant and no low-frequency relaxations can be observed in the sand. In contrast, the complex effective dielectric spectra of the silty clay in Fig. 6 show much stronger frequency dependence over the measured frequency range. We used the same inversion approach and models as for the sand: fitting with constant parameters for dielectric permittivity and electrical conductivity (dashed black line) and fitting with a single-pole Debye relaxation (blue line). Apart from that, we inverted the data using a multi-pole Debye relaxation model consisting of 3 Debye poles and one term that takes the dc-electrical conductivity into account (red line). It can be observed that the latter model approach with the multi-pole Debye model fits the data very well. B. FDTD Simulations For the FDTD simulations we used the software gprMax [11, 12]. In its latest version, different relaxation and dispersion models can be taken into account, like the Debye, Lorentz or Drude model. Furthermore, it enables to add an arbitrary number of relaxation poles to each individual model. We implemented the single- and multi-pole Debye model parameters for the complex dielectric permittivity and electrical conductivity given in Table I to describe the media for the FD simulation. The 2d model is a rectangular box with 2 m width and 0.8 m depth (Fig. 3). Two perfect conductors with 0.5 m length are placed in 20 cm and 40 cm depth, in analogy to the metal plates in our field experiment (Fig.1). We used a small discretization of 2 mm in order to fulfill the Courant-Friedrichs-Lewy condition for numerical stability [14]. Our source is a hertzian dipole that transmits a ricker wavelet with a center frequency of 600 MHz (Fig. 4). 600 MHz ricker wavelet
(a)
(b)
-0.6 -0.8
Fig. 3. Model for the FDTD simulation with two perfect conductors in 20 cm and 40 cm depth.
The measured data have been inverted using different models. The fitting parameters are listed in Table I. The dielectric data of the sand had been fitted with constant values of relative dielectric permittivity and dc electrical conductivity (dashed black line in Fig. 5 a-d) and in Fig. 4. Amplitude and power spectrum of a 600 MHz ricker wavelet comparison, with a single-pole Debye relaxation model that used in the simulation. (blue line in Fig. 5 a-d). It can be observed that the singlepole Debye relaxation with a relaxation frequency of 17.3 C. Simulation results GHz perfectly matches the measured data. This relaxation The result of the simulations for the two soils is shown frequency is typical for the relaxation of free water at room in Fig. 5(M, N, e, f, g, h, i,j) and in Fig. 6(R, S, e, f, g, h, i,j), temperature, as found in [10]. The real part of the effective
(a)
fine sand (1 % clay, 97 % sand)
(b)
θv = 38,6 %
trace m1
0
trace m2
measured data eps., sig. = const. fitted data 1 Debye relax.
(c)
(e)
trace m2 & n2
(f)
eps., sig. = const. 1 Debye relax.
(g)
trace m2 & n2
(h)
trace m1 & n1
(i)
eps., sig. = const. 1 Debye relax.
(j)
trace m1 & n1
5
time [ns]
(d)
10
15
20
600 MHz
M 0.0
0.5
0
trace n1
1.0
1.5
trace n2
time [ns]
5
10
15
20
N 0.0
600 MHz 0.5
1.0
1.5
distance [m]
Fig. 5. Real (a) and imaginary part (b) of the effective dielectric permittivity of fine sand and calculated phase velocity (c) and intrinsic attenuation (d). The data are fitted with constant permittivity and conductivity parameters (dashed black line) and with a single-pole Debye relaxation (blue line, see Table I). The FDTD simulations are based on the two different media descriptions and on the model in Fig. 3 (M and N). The radargrams are compensated for spherical divergence. Time- traces showing the metal plate reflections (e, f, g, h, i) and their individual frequency spectrum (g, j).
respectively. Fig. 5 M shows the simulated radargrams for the fine sand with constant dielectric permittivity and electrical conductivity (see Table I). All radargrams have been compensated for spherical divergence proportional to the square root of travel time. Fig. 4 N shows the simulation (a)
silty clay (42 % clay, 12 % sand)
(b)
θv = 40,5 %
trace r1
0
trace r2
(e)
measured data eps., sig. = const. fit - 1 Debye relax. fit - 3 Debye relax.
trace r2 & s2
result using a single high-frequency Debye relaxation. Beside the B-scans, two traces from the center of the metal plate reflections are plotted (e, f, h, i), together with their individual frequency spectra (g, j). It can be observed that the shape of the reflected waves with constant parameters is (c)
(f)
eps, sig. = const. 1 Debye relax. 3 Debye relax.
(g)
trace r2 & s2
(i)
eps, sig. = const. 1 Debye relax. 3 Debye relax.
(j)
trace r1 & s1
time [ns]
5
10
eps, sig = const. 1 Debye relax. 3 Debye relax.
15
20
R 0.0
600 MHz MHz 600 0.5
1.0
trace s1
0
1.5
trace s2
(h)
trace r1 & s1
5
time [ns]
(d)
10
15
20
S 0.0
600 MHz 0.5
1.0
eps, sig = const. 1 Debye relax. 3 Debye relax.
1.5
distance [m]
Fig. 6. Real (a) and imaginary part (b) of the effective dielectric permittivity of silty clay and calculated phase velocity (c) and intrinsic attenuation (d). The data are fitted with constant permittivity and conductivity parameters (dashed black line) with a single-pole Debye relaxation (blue line) and with a three-term multi-pole Debye relaxation (red line, see Table I). The FDTD simulations are based on the constant and multi-Debye media descriptions and on the model in Fig.3 (R and S). The radargrams are compensated for spherical divergence. Time- traces showing the metal plate reflections (e, f, g, h, i) and their individual frequency spectrum (g, j) Please note the different scaling of the amplitude scales within the subplots.
similar to the simulation with 1 Debye relaxation. The main difference between the traces is the signal amplitude, which is generated by the increasing attenuation above 500 MHz for the Debye fit compared to the zero-attenuation with constant parameters. The rise of the attenuation is caused by the relaxation of the free water molecules [15] and this effect can be only considered in the simulations, when using 1 Debye model in the parameter description of the soil. Fig. 5 R and S shows the simulated radargram for the silty clay. Whereas in Fig 5 R, the radargram was calculated based on constant constitutive parameters, it was calculated with a multi-pole Debye relaxation model in Fig 5 S (relaxation parameters in Table I; the simulated radargram with a single Debye relaxation is not shown). The attenuation is very strong, which is on the one hand caused by dc electrical conductivity and on the other hand, caused by strong dielectric relaxations. The spectra for the complex effective dielectric permittivity (Fig. 5a, b) show that only the multi-pole Debye model, consisting of 3 poles is able to describe the spectra accurately. Dispersion effects can be observed in the traces, which are caused by the frequencydependence of phase velocity. The frequency spectra of the reflected signals (Fig. 5g,j) show that the soil acts as a lowpass filter on the wavelet. This observation is especially pronounced in the multi-pole Debye media, where a frequency shift of the reflected wave center frequency can be observed from 600 MHz to 200 MHz (Fig. 6j). IV. CONCLUSION With respect to the fitting result of the measured dielectric spectra for the fine sand and the silty clay, we can show that a realistic description of the frequencydependence of the effective dielectric permittivity is necessary in order to receive realistic simulation results. This is especially the case for soils with a high content of silt and clay, for which relaxation processes dominate the dielectric properties in the high radar-frequency range. Using simple models with constant properties or a single Debye relaxation describing the free water relaxation, yields simulation results with wrong amplitudes and spectra of received wavelets. Investigating how radar wavelets changes with respect to the complex frequency-dependent media parameters could help to derive petrophysical parameters, like for instance the content of bound water and clay, from GPR measurements in the future.
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[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
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