GEOFÍSICA INTERNACIONAL (2012) 5 1 - 2 : 121-127

• • ORIGINAL PAPER

Application of the Wigner-Ville distribution to interpret ground -penetrating radar anomalies

Martha Angélica Elizondo*, Rene E. Chavez, Maria Encarnación Cámara and Andrés Tejero

Received: November 18, 2010; accepted: October 30, 2011; published on line; March 30, 2012

Resumen

Con base en la D is t r ibuc ión de W igne r -V i l l e (WVO) se rea l i zó un aná l i s i s en t i e m p o y frecuencia de datos obtenidos con el Radar de Penetración Terrestre (GPR), basado en el estudio de la descomposición de la señal espectral. Se calcula una correlación entre la señal or iginal y las componentes de t iempo- f recuencia para obtener anomalías estructurales de la información contenida en el radargrama relacionándola con la geología disponible. En pr imer lugar se describe la aplicación de un ejemplo teórico consti tuido por lo que represen ta r ía un túne l ( t ube r í a ) . Se obtuvieron las f i rmas correspondientes en el domin io de l t iempo y en el domin io de la frecuencia. Finalmente se analiza esta metodología en un sido de prueba en la detección de un tambo enterrado donde son conocidas la geometr ía y su profundidad. Este especial sitio fue facil itado por la Universidad Nacional Autónoma de México, en los terrenos del Observator io Magnético de Teoloyucan, Estado de México. Los resul tados obtenidos son bastante alentadores, ya que la WVD es capaz de definir los rasgos morfofógicos relacionados con el tambo y abre la posibilidad de localizar este tipo de estructuras.

Palabras clave: Radar de Penetración Terrestre (GPR), análisis t iempo-frecuencia, distr ibución de Wigner-Vi l le, proceso de señales.

A b s t r a c t

T i m e - f r e q u e n c y a n a l y s i s , based on s igna l spectral analysis decompos i t ion , is per fo rmed by t h e W i g n e r - V i l l e d i s t r i b u t i o n ( W V D ) f o r Ground Penetrating Radar (GPR) data. The cross correlat ion between the original signal and the t ime-frequency components is obtained to get the structural anomalies available in the informat ion provided by the GPR, related to the geology. We describe the application to a theoret ical example represen t ing a tunne l ( p i pe ) . Cor respond ing signatures are obta ined in the t ime- f requency domain. An actual appl icat ion at a test site is presented, where a d r u m of known geometry has been bur ied. This is a special test s i te at the National Autonomous University of Mexico, Magnetic Observatory at Teoloyucan, State of Mexico, Depth and dimensions are well control led, and the results are encouraging, since the WVD is capable to def ine morphological features related to the d r u m and opens the possibil i ty to locate these structures.

Key words: Ground Penetration Radar (GPR); t ime- f requency analysis; Wigner-Vil le d is t r ibut ion; signal processing.

M. A. Elizondo' Posgrado en Ciencias de la Tierra Universidad Nacional Autónoma de México Ciudad Universitaria Delegación Coyoacán, 04510 México D.F., México 'Corresponding author: mass 112@yahoo.com.mx

R. E. Chávez Instituto de Geofísica Universidad Nacional Autónoma de México Ciudad Universitaria Delegación Coyoacán, 04510 México D.F., México

M, E. Cámara Escuela de Ingenieros Industriales Universidad Politécnica de Madrid Madrid, España

A. Tejero Facultad de Ingeniería Universidad Nacional Autónoma de México Ciudad Universitaria Delegación Coyoacán, 04510 México DF., México

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M.A. Elizando, R.E. Chavez, M.E. Cámara and A. Tejero

I n t roduc t ion

GPR (Ground Penetrating Radar) const i tutes an impor tant methodology to reveal subsurface features. Shallow bur ied features such as f ractures, mines, caves and anthropogenic structures (pipes, connect ing lines, etc.) can be successfully detected. Traditionally the processing of GPR data may be done in the Time Domain (TD) or the Frequency Domain (FD) by employing the Fourier t rans form. However, it is not always possible to obtain a reasonable resolution f rom GPR data, due to variat ions in the relat ive dielectric permi t t iv i ty (RDP). Therefore, GPR interpretat ion requires the appl icat ion of TD or FD fi l ters to separate impor tant informat ion f rom noise. Unfortunately, such filtering may be contaminated wi th spurious in format ion. Also, the est imated events or anomalies may be wel l -defined in FD, but not in TD. A t ime- f requency analysis (TF) can reveal the t ime variat ion of t he frequencies contained in a I D GPR signal.

In the TF l inear distr ibut ions as well as in the Wavelet t rans form, the signal is decomposed in the t i m e - frequency domain based on ampl i tude, over a family of functions derived f rom a function base to determine the scale on which depends the resolution (Vera, 2003; Auger e£ ¿tl., 1995) .

In the case of discrete Wavelet t ransform, it corresponds to a dyadic sampling of the t ime-frequency plane in which the funct ion family const i tutes an or thonormal base with a wavelet well localized in t ime and frequency (Auger et a/., 1995; Rosado, 2000; Rivera-Recillas er at., 2005) .

The Wigner-Vil le distr ibut ion is a t ime-frequency bilinear t rans form, which computes the distr ibut ion of signal energy in two variables: time and frequency, as energy density. I t s spectrum depends on t ime , preserving mathemat ical properties such as the marginal and instantaneous frequency. I t employs the analytic signal f rom a real signal and a parametr ic funct ion of constant value (kernel = l ) . T h u s , the interference terms appearing in the Wigner distr ibut ion due to negative frequency components of the real signal can be suppressed, (Ramos, 1997).

The idea is to extract the t ime series profi le that make up the radargram and define a funct ion in TF domain, which shows features related to anomalous bodies or sub-surface structures. I n this case, the Wigner-Vil le Distr ibution (WVD) is applied to def ine these structural objectives (Lopera etal., 2008) . I t Is proposed to obtain the frequency informat ion contained in t ime, for a I D GPR signal processed by the WVD, in order to identify an event such as a strat igraphic change,

a cavity or f racture. First, a I D theoret ical model is presented to show the abil i ty of the WVD, and then a real example is shown f rom a tes t site where the characteristics are known. The test site is located in the facil i t ies of the National Autonomous University of México, Magnetic Observatory at Teoloyucan t own , in the State of Mexico, approximately 60 km NE of Mexico City.

Methodology

The WVD of a discrete signal s ( i ) is given by (Cohen, 1989) :

WVD,(t,,fj) = -Jds(i,+kT)S"(il.-kT)e'i,*f'kT

(1)

I f a signal is sampled at an interval T, the maximum cutoff frequency where the signal is folded in the Fourier spectrum is fe = 1/2T. However, the WVD t ime-frequency spectrum is folded at fc/2. I t means that the original signal must be sampled at an interval T * = T / 2 to avoid aleasing effects. Therefore, the new cutoff frequency can be expressed as fc* = l / 2 T * = l /T=2 fc .

For the theoret ical analysis a scan synthet ic radargram was obtained for a I D strat igraphy land, where the electric f ield at the Earth's surface can be obta ined. Following the method of Weng (1971) , based on the fact that a plane wave enters the Earth's surface and the total field obtained on the surface is the incident f ield plus the ref lected, considering a dissipative dielectric N-layered earth model and a general ized ref lexion coefficient given by Stokes (Bel lman and Wing, 1975) and solved in a recursive form as:

R + R -**W*a-*J 5 _ K M + I + /Cf»l.r>2g

" + l 1 + R R *a i«w*t<kj-4iJ

Later in the frequency domain , the Ricker's wavelet, g iven by (Ricker, 1977) , is used:

,2 L

/ r (3)

Where f = frequency

fc = central f requency of oscil lation

Which is characterized by a slow construct ion and a decline of energy at the extremes of the funct ion, in which the standard of the wavelet is smooth and cont inuous In its l s l and 2nd

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derivatives and can be optimized to meet the center frequency of a GPR antenna, which is designed to match the input impedance at the central feed point and the origin of the wavelet is used to force the electric field vector in the central feed point (Daniels, 2004; Cassidy and Murdie, 2000) Figure 1.

This is done through the Convolution Theorem radagram obtaining the synthetic trace in the frequency domain (Lázaro-Mancilla and Gómez-Trevlño, 1996).

R(f) = E ( f ) S ( f ) (4)

Where

E (f) = the electric field in the frequency domain

S (f) = the Ricker pulse in the frequency domain

Finally, the inverse Fourier transform is computed to go into the time domain and produce a physical meaning to the estimated radagram (Annan and Chua, 1992; Diaz, 2003).

R{t) = F<{E\f)S[r)} (5)

It is expected that reflections produced by each layer represent the double travel time done by the GPR signal, given by:

^-fXJJW^ (6)

The pulse travel time along the layered medium are calculated independently, the speed of propagation of electromagnetic waves is obtained through the formula.

V =

(7)

Where c = speed of light

£ = relative permiiiviiy

h = layer thickness

.0.00

-0 04

Pulse of Ricker Frequency Domain T

166 Pulse of Ricker Time Domain

333 500 6.66

10 Frequency 12

x 10*

6 33 Depth (m)10 0 0

20 40 60 80

Figure 1. Features Ricker pulse used

100 T i rTle 120

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M.A. Elizondo, R.E. Cháveí, M.E. Cámara and A. Tejero

The simple travel t ime of each layer is

(8)

And the depth of the objects known can be est imated fol lowing, Peniche (2008) .

depth = .sample time travel

speed (9)

Finally, we calculated the WVD to obtain the t ime- f requency plane, which is represented in graphical f o rm by his absolute value.

A n a l y s i s a n d r e s u l t s of t h e GPR t races

Table (1) presents the model used to generate the synthetic t race (Figure 2 ) . The synthetic trace model (Figure 2a) is represented in a sampl ing window of 60 ns, tha t would be obtained by an antenna of 270 MHz with 512 samples per record.

First we calculated the electric f ield, and the synthet ic t race. Then , the t ime-f requency representat ion for this trace is obta ined. However, small features such as a layer change can be diminished by t he direct wave and its ref lect ion surface, so the direct wave and its reflection can be removed by applying a f i l ter (a simple average moving autoregressive to el imínate the unwanted echoes). The final result displays only impor tant reflection components (Lopera, 2007) .

The position of high amplitudes indicate events as at the t imes t, = 18 and t , = 39 ns. I t is interesting to note that the ampli tudes are compacted into the centre of the plane in a frequency range f rom 100 to 300 MHz. These represent the energy spectrum of the signal, which has a max imum value of 270 MHz ( f igure 2) .

Finally, a practical example is presented. A GPR profi le at the test si te, obtainid using a mono-stat ic antenna, whose offset is zero, in which no lateral effects have a signif icant influence in g iv ing a I D representat ion; in the exper imental site a cylindrical drum is buried at

0 83 Synthetic Trace

166 2 50 3 33 4.16 5 00 Depth (ml

60

0.01

•0.01 10 20 30 40 50 Time

Wigner-Vile Distribution Plane Time-Frequency

30 40 50

Plane Time-Frequency

10-83 ^ 30 5 o « •J-J SO, „ Time

• 4,16 500

2 Frequency x 10*

Depth (m) Figure 2. The synthetic trace, the time-frequency plane, the spectrum energy of the signal,

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Table 1 . Data used to generate the synthet ic trace.

Layer Material Depth (m) Magnetic Electric Electric Permeability ( h / m ) Conductivity(s/m) Permitivity ( f / m )

1 2 3

Sandy soil Air Sandy soil

4 3 -

uO MO MO

0.001 0 0.001

SE 0

8 E „

depth of 1.0 m, wi th the fol lowing dimensions: diameter of 0.60 m. An area of 1.20 x 8.00 m was scanned. GPR data were analyzed fol lowing the methodology already described, employing the WVD, Each trace presents a similar behavior (Figure 3), providing the possibility t o locate w i th bet ter resolution the posit ion at depth of the d r u m .

We have taken a trace of the GPR profi le. Then, the t ime-frequency representat ion for this trace is obta ined; the posit ion of high ampl i tudes can indicate the location where the cylindrical d rum was bur ied; this event is indicated at the t ime t j = 18 ns. These plots show us di f ferent events and are represented in a t ime- f requency plot as high energy values. I t is interest ing to note tha t the ampl i tudes are compacted into the centre of the plane in a frequency range f rom 100 to 300 MHz. These represent the energy spect rum of t he signal, which has a m a x i m u m value of 270 MHz (Figure 4 ) .

Conclusions

WVD t ime-f requency representat ion may provide another view of the GPR signal tha t complements

what other methods may achieve. We provide a more accurate representat ion of t he evolut ion of a non-stat ionary signal , which enables a more complete analysis of the signal providing classif ication, model ing and pattern identi f ication parameters.

Non-stat ionary propert ies of the GPR signal al lowed the application of WVD based analysis. A t ime- f requency representat ion was obta ined, which allowed visualization of the frequency range contr ibut ing to the signal in the t ime scale studied. I t al lows a better visuaiization of the posit ion of high ampl i tude that we can indicate the location of events of interest.

The spectral components represented in the t ime- f requency plane and dep th , can be correlated with the characteristic events of the temporal signals or indirect ly allow the characterization of the informat ion contained in the GPR signal in te rms of depths.

The study object ives f rom wich the were to allow characterizat ion of the informat ion contained in t he GPR signal, applying the Wigner-Vil le t ransform to non-stat ionary signals,

Ana I iced t race

me ta l barre l

Figure 3. Diagram showing the location of the buried drum, analyzed GPR transect is also shown.

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M.A. Elizondo.R.E. Chávez, M.E. Cámara and A. Tejero

x10* Synthetic Trace

10 20 30 40

x i o * Wigner-Ville Distribution

50 GO 70 aoTimeso

Plane Time-Frequency

20 30 40 50 60 Plane Time- Frequency 3D

70 80 Time

40 60

TimeTimeTime 80 Time Frequency

|

Figure 4. Results obtained at the test site. Noise-free signal is observed at the top. The t-w plane follows, note the effect at t=18 ns and w=270 MHz. The spectrum beJow depicts clearly the effect of the drum. Finally, the spectrum energy of the signal

is depicted.

to visuaiize the evolut ion over t ime of the signal in a frequency range, to help in the classif ication, model l ing and identif ication of patterns wi th their local geological correlat ion.

One of the goals of developed GPR methodologies was to recognize with adequate resolution structures lying at depths of 5.00 m at the most. New numerical approximations should be applied in the future to deal wi th noise and to enhance useful GPR signals to detect anomalies in the subsoil.

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