GPR Clutter Modeling Taking into Account Soil Heterogeneity Kazunori Takahashi, Jan Igel, and Holger Preetz Leibniz Institute for Applied Geophysics (LIAG) Hannover, Germany
[email protected] Abstract— In small-scale measurements, a ground-penetrating radar (GPR) is often used with a higher frequency to detect a small object or changes in the ground. GPR becomes more sensitive to the heterogeneity of soil when a higher frequency is used. Soil heterogeneity scatters electromagnetic waves and the scattered waves are observed as unwanted reflections that are often referred to as clutter. Data containing high amplitude clutter are difficult to analyze and interpret because clutter disturbs reflections from objects to be detected. Therefore, modeling GPR clutter is useful to assess the effectiveness of GPR measurements. The authors have developed and demonstrated such a technique with data acquired during an infiltration experiment. In this study, the technique was applied to GPR and time domain reflectometry (TDR) data repeatedly acquired on an outdoor test site for a few months. The modeling results using the TDR data are similar to the clutter power directly extracted from the GPR data. Therefore, the technique works although it requires some modifications for more accurate modeling. Keywords; heterogeneous soil; clutter; geostatistics; modeling; scattering
I.
INTRODUCTION
Ground-penetrating radar (GPR) has been used more and more commonly for small-scale measurements such as in civil engineering and hydrology in these years. Small changes in materials being measured need to be captured with high sensitivity in such applications. Therefore, higher frequencies (typically higher than 1 GHz) are often employed. With a higher frequency, GPR becomes more sensitive also to heterogeneity of the material, resulting in unwanted scattering (commonly referred to as clutter) in the data. Clutter makes the analysis and interpretation of GPR data difficult. Assessing the effectiveness of GPR measurements in a site prior to the actual work by using rapid measurements and analysis may help saving time and costs. The authors developed a simple modeling method of GPR clutter caused by heterogeneous soils and demonstrated its validity with data acquired during an infiltration experiment [1],[2]. In these previous works, a model was constructed using parameters obtained by a geostatistical analysis of the GPR data that were assumed to reflect soil heterogeneity. The model consists of a dielectric sphere and clutter power was calculated as the radar cross-section (RCS) of the sphere that is proportional to the backscattering power. While the model
calculation with the Rayleigh approximation gave significantly different clutter power to clutter actually observed in the GPR data, the modeling with the Mie solution agreed with the experiments. However, it did not fit perfectly and the results indicated some limitations and the necessity of modifications for the wider ranges of applicability as well as for more accurate modeling. The modeling method requires the indications of soil heterogeneity, namely correlation length and variability of the permittivity that can be determined by a geostatistical analysis. In this study, the technique is demonstrated with other data sets that were repeatedly acquired during a few months on an outdoor test site. Time domain reflectometry (TDR) data acquired at the same time of the GPR measurements are available and used to model GPR clutter. Based on the modeling results, the applicability of the method and the characteristics of GPR clutter caused by heterogeneous soil are discussed in this paper. II.
CLUTTER MODELING
A. Determination of Soil Heterogeneity The modeling technique requires input parameters characterizing soil heterogeneity that first need to be determined. The easiest way to directly measure the spatial changes of soil permittivity may be TDR measurements on the ground surface at multiple locations. TDR measures the permittivity (or associated water content) of soil around the probes, and thus the measurement scale depends on the probe configuration. If relatively small probes (e.g., 10 cm length, 2 cm separation) are employed the measurement volume may be similar to that of GPR. Further, TDR typically uses frequencies ranging from 500 MHz to 1 GHz [3] that corresponds to relatively high frequency GPR. Therefore, TDR measurements can be used to characterize soil heterogeneity that is influential on GPR measurements in the similar scale. A semivariogram is a geostatistical analysis tool and it can be used to quantify heterogeneity. For soil permittivity data measured on a 1D profile, the semivariogram γ(h) is calculated as [4]:
This work was supported by the Federal Office of Defense Technology and Procurement, Germany.
978-1-4577-0333-7/11/,(((
γ (h) =
2 1 N ⎡⎣ z ( xi + h ) − z ( xi ) ⎤⎦ ∑ 2 N i =1
(1)
where h is the lag distance, or separation between two data points, z(xi+h) and z(xi), and N is the number of data pairs with a constant lag distance h from all data points. Often, the semivariance γ(h) increases with the lag distance h up to a certain value and then it becomes constant. The lag distance h and the semivariance γ(h) where γ(h) becomes constant are called range a and sill C, respectively. The range indicates the mean distance at which a data pair does not correlate anymore and thus it is equivalent to correlation length [5]. The sill corresponds to the maximum variance within a data set and thus it is the indication of the variability. The exponential model given as [4]:
γˆ ( h ) = C ⎡⎣1 − exp ( −3h a ) ⎤⎦
(2)
is fitted to the obtained experimental semivariograms in order to determine range a and sill C. B. Model Calculation A simple model is constructed with the model parameters obtained by the semivariogram, i.e., correlation length a and variability C, as well as the mean of measured soil permittivity ϵm. The model considered in this study is shown in Fig. 1. It consists of a dielectric sphere embedded in a homogeneous space. The homogeneous background space is defined as having a permittivity equal to the mean permittivity, ϵ1 = ϵm. The circumference of the dielectric sphere is chosen as the correlation length and thus the diameter d is defined as d = a/π. The permittivity of the sphere ϵ2 is set so that the difference to the ambient medium is equal to the square root of valiability:
Δε = ε 2 − ε1 = C
(3)
With this model, the radar cross-section (RCS) that is proportional to the backscattering power is theoretically calculated, assuming a monostatic configuration and plane wave incidence. There are two ways to calculate RCS for the model: the Mie solution and Rayleigh approximation. The Mie solution is the exact solution of Maxwell’s equations and describes the scattering by an arbitrary sized particle [6]. Therefore, it is valid in all frequency/sphere size ranges including not only the Mie scattering region but also Rayleigh scattering and optical regions. By the Mie solution, the RCS σs of a dielectric sphere is given as [7]:
σs =
1 x2
∑ ( 2n + 1)( −1) ( a n
n
n
− bn )
2
(4)
where x = kr, k is the wave number in the ambient medium and r is the radius of the sphere. In case there is no change in the magnetic permeability between the dielectric sphere and the
ε1
ε2 d = a/π
Figure 1. Model for the calculation of backscattering power that consists of a dielectric shpere with a permittivity ϵ2 and a diameter d embedded in a homogeneous space with a permittivity ϵ1.
ambient medium which is assumed in this study, the coefficients an and bn are given by:
an =
m2 jn ( mx ) ⎡⎣ x jn ( x ) ⎤⎦′ − jn ( x ) ⎡⎣ m x jn ( mx ) ⎤⎦′ m2 jn ( mx ) ⎡⎣ x hn(1) ( x ) ⎤⎦′ − hn(1) ( x ) ⎣⎡ m x jn ( mx ) ⎦⎤′
bn =
(5)
jn ( mx ) ⎣⎡ x jn ( x ) ⎦⎤′ − jn ( x ) ⎣⎡ m x jn ( mx ) ⎦⎤′ jn ( mx ) ⎣⎡ x hn(1) ( x ) ⎦⎤′ − hn(1) ( x ) ⎡⎣ m x jn ( mx ) ⎤⎦′
(6)
where m denotes the refractive index and the function jn(x) and hn(1)(x) are the spherical Bessel function of first kind of order n and the spherical Hankel function of order n, respectively. Primes mean derivatives with respect to the argument. The summation in (4) is theoretically from n = 1 to ∞, but the infinite series can be truncated after nmax = x + 4x1/3 + 2 [7]. The Rayleigh approximation describes scattering by a small particle compared to a wavelength [7], and it is generally valid only in Rayleigh scattering region. In such a case (x≪1), the Mie solution (4)-(6) can be simplified by the Rayleigh approximation as [7]:
σ s = 4 x2
m2 − 1 m2 + 2
(7)
The equation exhibits that the scattered power due to the particle is inversely proportional to the fourth power of the wavelength (or proportional to the fourth power of the sphere size) in Rayleigh scattering region. In this study, both Mie solution and approximation were examined for the comparison.
Rayleigh
surface was mostly covered by grass, which was considered to increase the heterogeneity of soil moisture distribution due to the water consumption by the irregular root system. A GPR system (GSSI SIR-3000) with 1.5 GHz antennas was scanned on the ground one-dimensionally on a 5 m long profile, along with TDR measurements on the same profile. While the GPR scanned the full 5 m length and sampled data every 1 cm, TDR measurements were carried out every 2 cm in the first 1 m. The measurements were repeated seven times from April to June 2010.
III.
DEMONSTRATION OF THE MODELING WITH EXPERIMENTAL DATA In order to demonstrate the modeling technique to see the applicability and limitations, a series of GPR and TDR measurements were carried out.
A. Experiments The GPR and TDR measurements were conducted on an outdoor test site at the Leibniz Institute for Applied Geophysics (LIAG). The texture of the test soil is medium sand with a high humus content. At the times of measurements, the ground
Fig. 2 shows all seven GPR profiles. The travel time was converted to depth using the mean permittivity measured by
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Figure 2. GPR profiles on a 5 m long profile acquired on (a) April 15th, (b) April 21st, (c) April 28th, (d) May 5th, (e) May 26th, (f) June 3rd, and (g) June 17th. Travel time was converted to depth using the mean of permittivity measured by TDR (Fig. 3). Amplitudes of the profiles were normalized throughout the measurement campaign. Only a bandpass filter was applied.
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Figure 3. (a) Spatial variations of relative permittivity measured by a time domain reflectometry (TDR) in the first 1 m of the profile with 2 cm spacing and (b) the mean permitivitty of the TDR measuremtns.
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Figure 4. (a) Experimental semivariograms of soil permittivity measured by the TDR and (b) correlation length and variability determined from the semivariograms, which are represented with a and C respectively in (2).
the TDR (shown in Fig. 3) and the amplitudes were normalized throughout all the measurements. Almost the same patterns can be observed in all seven profiles, and only small changes and intensity differences are visible. Fig. 3 shows the spatial change of relative permittivity measured by the TDR at the same time of the GPR measurements and its mean. The associated volumetric water content was given by Topp’s equation [8]. It can be observed that the water content and permittivity changed due to evapotranspiration and some precipitation events.
B. Geostatistical Analysis of the TDR Data Experimental semivariograms were calculated for the TDR data as shown in Fig. 4(a). By semi-automatically fitting the exponential model given by (2) in the least square error manner, the correlation length and variability were obtained as Fig. 4(b). The result showed that the correlation length and variability changed in the ranges of 10-25 cm and 0.5-1.5, respectively, and thus the soil heterogeneity in permittivity seems to have been changing during the experiments. C. Clutter Modeling The determined correlation length and variability as well as the mean permittivity were set into the model as shown in Fig. 1 and clutter power was calculated with this model as described in Section II.B. In this study, both Mie solution and Rayleigh solution were examined and the calculations are shown in Fig. 5. Modeled clutter power with both calculations in this figure was normalized by the mean. D. Validation of the Modeling In order to validate the modeling results, the clutter observed in the GPR data was extracted for the comparison to modeled clutter. The 20 highest amplitudes in a depth section deeper than 7.5 cm that was just below the ground surface reflection were picked from the GPR data shown in Fig. 2. The extracted clutter power and its mean value are plotted with
crosses in Fig. 6. By comparing the results, the modeled clutter with the Mie solution is similar to the mean of the picked clutter power. They do not perfectly fit, but the tendencies (ups and downs) agree. On the other hand, the modeled clutter with the Rayleigh approximation does not really fit to the experiments. Mean squared errors of modeled clutter with the Mie solution and Rayleigh approximation are 0.15 and 0.87, respectively. The results indicate that the power of clutter signals caused by soil heterogeneity can roughly be calculated by the Mie solution and a rather simple model shown in Fig. 1 with the support of soil water content measurements (i.e., TDR measurements) and geostatistics. IV.
DISCUSSION AND CONCLUSIONS
In this study, GPR clutter was modeled from the soil heterogeneity that was determined from TDR data. The modeling results agreed with clutter power observed in the GPR data. The modeling technique could reasonably model the GPR clutter variation caused by change in soil heterogeneity over a few months. However, modeling did not fit perfectly to the experiments. Especially the error for the fourth (on May 5) and sixth (on Jun. 3) data are relatively large. The possible reason can be found in relative correlation length (Fig. 7) and variability (red circles in Fig. 4(b)). These data showed both longer correlation length and large variability. The observation also supports the results of our previous work: the modeling did not fit well to the experiments immediately after irrigation where long correlation lengths and large variability were observed [1],[2]. A possible explanation is that a longer correlation length may indicate the gradual changes of permittivity in contrast to a sharp boundary in the model. Moreover, a large variability may cause more complex wave interactions in the soil. Therefore, the technique needs to be modified for more accurate modeling in these conditions and this should be our further challenges.
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Figure 6. Clutter power ectracted from the GPR data and modeled clutter. The crosses and dash dot line indicate the 20 highest clutter powers in the GPR data and their mean, respectively. Values were normalized by the mean of all clutter power.
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Comparing the modeled (Fig. 5) or measured clutter (Fig. 6) to the determined correlation length and variability of soil permittivity (Fig. 4(b)), a simple relationship between these parameters and clutter power cannot be found. This is because the scattering by heterogeneous soil is in the Mie scattering region as the relative correlation length (Fig. 7) demonstrates and also suggested by [9], and the backscattering power in this region oscillates with correlation length. If the scattering is dominated by Rayleigh scattering, clutter power and these parameters characterizing the soil heterogeneity should have a direct correlation since scattering in Rayleigh scattering region can be expressed by a relatively simple form as given in (7). Therefore, it was demonstrated that GPR clutter caused by soil heterogeneity is dominated by Mie scattering in the case that a relatively higher frequency (i.e., higher than 1 GHz) is employed. It is different to the situation with a lower frequency that is commonly used for large scale measurements. In this case, the scattering due to heterogeneous media is governed by Rayleigh scattering [3].
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Figure 5. Modeled clutter power using the Mie solution (blue dots) and Rayleigh approximation (red circles). Values were normalized by the mean.
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Figure 7. Correlation length relative to wavelength determined from the semivariograms.
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