ORIGINAL PAPER

On the depth to anomaly estimation using Karous and Hjeltfilter in VLF-EM data

Mohamed A. Khalil & Fernando M. Santos

Received: 5 February 2013 /Accepted: 10 September 2013 /Published online: 21 September 2013# Saudi Society for Geosciences 2013

Abstract The Karous and Hjelt filter has been long time usedas a qualitative interpretation of VLF-EM data. It is deriveddirectly from the concept of magnetic fields associated withthe current flow in the subsurface and resulted in a 2-D crosssection showing the current density distribution at differentdepths. Practically, as the distance between measuring pointsincreases, the total depth of the 2-D current density distribu-tion section increases. Theoretically, the common guide toestimate the depth of penetration of an electromagnetic waveis the skin depth, which depends on the frequency of theelectromagnetic wave and the conductivity of the host geo-logical material, regardless of the distance interval betweenmeasuring points. Accordingly, the accuracy of the Karousand Hjelt filter regarding depth estimation of the anomaly istested in this study. We proposed a conductive anomaly in adefinite dimension and depth. The response of this conductivebody is calculated as in-phase and out-of-phase synthetic VLFdata via forward modeling. The synthetic VLF data is filteredby the Karous and Hjelt filter at 1, 5, and10 m of intervaldistance between measuring points. The present study showedthat the Karous and Hjelt filter is characterized by a largedegree of accuracy in depth estimation.

Keywords VLF-EM .Karous and Hjelt filter . Depthestimation

Introduction

VLF-EM technique has been widely used in mining explora-tion (Paal 1965; Paterson and Ronka 1971; Frasheri et al.1995; Bayrak 2002) because of the large difference in electri-cal conductivity of the ore body and the host rock. Manystudies are concerned also with underground water explora-tion (Benson et al. 1997; Powers et al. 1999; Sharma andBaranwal 2005; Monteiro Santos et al. 2006; Khalil et al.2009). Some studies tried to utilize VLF-EM in the field ofarcheological prospecting for shallow subsurface targets(Bozzo et al. 1992; Khalil et al. 2010). Others used VLF-EM in cave detection and karst studies (Bosch and Mller2001).

The fundamentals of VLF-EM, in addition to its geologicaland hydrogeological applications, can be found in the litera-ture, e.g., McNeill and Labson (1991). The essential principlesof the VLF-EM method are as follows: a primary low fre-quency electromagnetic field is propagated from many radiotransmitters scattered in different parts of the world, designedfor military communications and navigation. The transmittedfrequency is usually between 15 and 30 kHz. This primaryelectromagnetic field of a radio transmitter (vertical electricdipole) possesses a vertical electric field component (EPz) anda horizontal magnetic field component (HPy), parallel to theground and perpendicular to the propagation direction. At adistance greater than several free wavelengths from the trans-mitter, the primary EM field components can be assumed to behorizontally traveling waves. The primary magnetic fieldcomponent (HPy) penetrates into the ground and induces eddycurrents forming a secondary horizontal electric component(ESx) in buried conductive structures. A secondary magneticfield (HS) is generated which is out of phase with the primarymagnetic field and of smaller amplitude. The intensity of thesecondary magnetic field depends on the conductivity of theground. The interface between the primary and the secondary

M. A. Khalil (*) : F. M. SantosCentro de Geofsica da Universidade de Lisboa-IDL, Universidadede Lisboa, Campo Grande, Ed. C8, 1749-016 Lisbon, Portugale-mail: khalil250@hotmail.com

M. A. KhalilNational Research Institute of Astronomy and Geophysics,Helwan Cairo, Egypt

Arab J Geosci (2014) 7:43554359DOI 10.1007/s12517-013-1110-3

magnetic fields produces a resultant magnetic field which iselliptically polarized. The orientation of this ellipse is randombut is greatly extended along the direction of the primary field.This is referred to as the polarization ellipse. The parametersof interest are (a) the orientation of the minor axis (tilt angle)of the polarization ellipse and (b) the ratio of the minor to themajor axis of the ellipse (the ellipticity). These two parametersare equivalent to the in-phase (real) and the out-of-phase(quadrature) component of the secondary magnetic field, re-spectively. At each measurement point, it is possible to definea scalar tipper B given by HSz=B HSy. The tipper is acomplex quantity originated by the time lag between thehorizontal and vertical components of the magnetic fieldsdue to the electromagnetic induction phenomena. Over a 2-D earth, the tipper varies along the measuring profile showingthe strongest variations in the vicinity of resistivity contrasts.The real and imaginary components of the tipper in the case ofthe VLF-EM method are measured as percentage of the pri-mary field. The best advantage of the VLF-EMmethod is thatno ground contact is needed, which allows a higher speed ofsurvey.

For practical purposes, it is worth to mention that (1) inorder to optimize the maximum induction effect, the directionof the transmitter should be selected to coincide with the strikeof the conductive geological body, (2) the VLF-EM profileshould be long enough to cover the conductive anomalydesired, and (3) VLF-EM devices (receivers) are so designedthat the anomaly indications in the real component, that is, apositive peak appears ahead of the conductor and a negativepeak appears behind it. This crossover anomaly depends onthe specific direction the receiver operator faces when takingreadings along a traverse. The anomaly profile would be 180out of-phase if the operator faced the opposite direction whentaking readings across the same conductor. The VLF-EM dataused in the present study is essentially a synthetic data. Whena conductive anomaly in a definite dimension and depth isproposed, then the response of this conductive body is calcu-lated as in-phase and out-of-phase synthetic VLF data viaforward modeling. The estimated synthetic VLF data is fil-tered by the Karous and Hjelt filter at 1, 5, and 10m of intervaldistance between measuring points to get the result in the formof 2-D current density distribution section showing the esti-mated depth and location of the anomaly. Accordingly, thecomparison between the real depth and the estimated depth bythe Karous and Hjelt filter using different interval distancesbetween measuring points would be possible.

VLF-EM data processing and interpretation

During the last 50 years, several methods, both analog andnumerical, have been developed by many researchers to in-terpret the VLF-EM data, and attempts have been made to

determine the parameters of heterogeneities. Coney (1977)and Baker and Myers (1979) based their studies principallyon analog models. Fraser (1969) proposed the horizontalgradient filter. Karous and Hjelt (1983) presented the currentdensity cross section. Ogilvy and Lee (1991) evaluated theperformance of the Karous filter. Chouteau et al. (1996) usedMaxwells equations to obtain an autoregressive filter to con-vert the VLF-EMmeasurements into apparent resistivity data.One of the mostly used processing techniques for qualitativeinterpretation is the Karous and Hjelt filter (1983).

Karous and Hjelt (1977, 1983) proposed a method basedon the principle of discrete filtering, and it gives, as a result,such apparent current densities at different depths, whichwould cause a magnetic field equal to the measurements. Theystarted with the BiotSavart law to describe the vertical com-ponent of the magnetic field arising from a subsurface 2-Dcurrent distribution. Karous and Hjelt used linear filter theoryto solve the integral equation for the current distribution,assumed to be located in a thin horizontal sheet of varyingcurrent densities, situated everywhere at a depth equal to thedistance between the measurement stations. By selecting datapoints at progressively greater distances apart, the behavior ofthe current distribution in the assumed sheet, now at progres-sively greater depths, can be inferred. They determined thatthe shortest filter that correctly inverts the field of a singlecurrent line element with an error less than 8 % has the simpleform (Karous and Hjelt 1977, 1983)

Z2

Ia x=2 0:205H2 0:323H11:446H0

1:446H10:323H2 0:205H3Ia x=2 1

where Z is the assumed thickness of the current sheet, Ia isthe current density, x is the distance between the data pointsand also the depth to the current sheet. The values of H2throughH3 are the normalized vertical magnetic field anomalyat each of the six data points. Location of the calculated currentdensity is beneath the center point of the six data points.

This filter provides a pictorial indication of the depth of thevarious current concentrations and hence the spatial disposi-tions of subsurface geological features, such as mineral veins,faults, shear zones, and stratigraphic conductors (Ogilvy andLee 1991).

The finite Karous and Hjelt discrete filter method is a moregeneralized and rigorous form of the Fraser filter (Fraser1969). However, it is derived directly from the concept ofmagnetic fields associated with the current flow in the sub-surface and resulted in a 2-D cross section showing the currentdensity distribution at different depths based on the intervaldistance between stations. As the interval distance betweenstations decreases, the number of calculated data levels in-creases and vice versa, where the vertical offset distance

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between data levels equal to the distance between the mea-surement stations. The present study is a test of the accuracy ofthe Karous and Hjelt filter regarding depth estimation of theanomaly, when VLF data is measured in different intervaldistance between stations.

A common guide to the depth of penetration is known asthe skin depth, which is defined as the depth at which theamplitude of a plane wave has decreased to 1/e or 37 %relative to its initial amplitude (Sheriff 1991).

The depth of the anomaly estimated from the KarousHjeltfilter is completely a numerical solution and depends on theinterval distance between measuring points. Whereas the skindepth depends on the frequency of the electromagnetic wave thatis diffused, frequency of transmitter, and the conductivity of thehost material, environment conductivity (McNeill and Labson1991). Accordingly, the electromagnetic waves of the transmitter,in a specific frequency, should penetrate the ground, in a specificresistivity, till reaching the skin depth, supposing that the groundis isotropic half space andmagnetically nonpolarizable, whateverthe distance interval between the measured stations.

Since the resulted depth in the KarousHjelt cross sectionis a function of the distance interval between stations, the

furthest distance can be determined by the following equation(Reynolds 1997):

2= 12 503 f 12 2Equation (2) can be simplified to

500f

r3

where

skin depth, in meter angular frequency=2f , where f stands for frequency in

Hertz, and =3.14 magnetic permeability of free space, 4 107 NA2

conductivity, in Siemens per meter resistivity in ohm per meter

Forward modeling

To verify the reliability and sensitivity of the proposed ap-proach, the following scheme is carried out: (1) proposing an

Fig. 1 The synthetic data (in-phase and out-of-phase) resultedfrom the model and the KarousHjelt filter cross sections at 1 m(a), 5 m (b), and 10 m (c) ofinterval distance betweenmeasuring stations

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initial model of a conductive anomaly in a specific geometry,depth, resistivity, surrounding environmental resistivity, trans-mitter frequency, and distance interval between stations; (2)generating the synthetic tipper data (in-phase and out-of-phase) of this anomaly via forward modeling approach; (3)filtering the synthetic data using the Karous and Hjelt filter indifferent distance intervals between stations (1, 5, and 10 m);and (4) comparing the resulted 2-D current density sectionsregarding the depth of the anomaly.

Model 1

Figure 1 shows an initial model, containing one conductivebody. The dimensions of this body in X and Y directions are1020 m, respectively. The body is located between 30 and50 m in depth. Synthetic data (in-phase and out-of-phase) ofthis body is generated by forward approach (Inv2DVLF)software developed by Monteiro Santos et al. (2006), propos-ing that the resistivity of the body is 10 m, environmentalresistivity is 300 m, transmitter frequency is 21.7 kHz, and

the data is collected every 1 m. According to Eq. (3), the skindepth of this profile considering the mentioned parametersshould be about 60 m. The Karous and Hjelt filter has appliedupon the synthetic VLF data, proposing that the data wasmeasured at 1, 5, and 10 m. No significant difference in thedepth to anomaly has been observed in the resulted 2-Dcurrent intensity sections for 1, 5, and 10m of interval distance(Fig. 1(ac)). The conductive body is located in the same Xand Y positions in the three sections.

Model 2

Figure 2 shows an initial model, containing one conductivebody. The dimensions of this body in X and Y directions are1020 m, respectively. The body is located between 40 and60 m in depth. Synthetic data (in-phase and out-of-phase) ofthis body is generated by forward approach, proposing that theresistivity of the body is 10 m, environmental resistivity is400 m, transmitter frequency is 18 kHz, and the data iscollected every 1 m. The skin depth of this profile considering

Fig. 2 The synthetic data (in-phase and out-of-phase) resultedfrom the model and the KarousHjelt filter cross sections at 1 m(a), 5 m (b), and 10 m (c) ofinterval distance betweenmeasuring stations

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the environmental resistivity and transmitter frequency shouldbe 74.5 m in depth. Figure 2(ac) shows the current densitypseudo-section resulted from the KarousHjelt filter. Theconductive body is located in the same X and Y positionsregardless of the interval distance between measuring points.

Conclusion

The present study is an approach for quantitative application ofthe Karous and Hjelt filter (1983), in particular, the depth esti-mation of the anomaly body. The depth estimated from the filteris completely a numerical solution based on the distance intervalbetween the measured stations. The depth of penetration ofelectromagnetic waves depends on the frequency of the inductedelectromagnetic wave and the conductivity of the host rock.

It begins by (1) proposing an initial model in specificconditions, regarding geometry, depth, resistivity, surroundingenvironmental resistivity, transmitter frequency, and distanceinterval between station; (2) generating the synthetic tipperdata (in-phase and out-of-phase) of this anomaly via forwardmodeling approach; (3) filtering the synthetic data using theKarous and Hjelt filter in different distance intervals betweenstations (1, 5, and 10 m); and (4) comparing the resulted 2-Dcurrent density sections regarding the depth of the anomaly.

The present study shows a large degree of accuracy in thedepth to anomaly estimated by the Karous and Hjelt filterregardless of the distance interval betweenmeasuring stations.

Acknowledgments The corresponding author is indebted to theFundao para a Cincia e a Tecnologia (Portugal) for his support throughthe postdoctoral fellowship (SFRH\BPD\29971/2006). This work waspartly developed in the scope of the scientific cooperation agreementbetween the CGUL and the NRIAG.

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On the depth to anomaly estimation using Karous and Hjelt filter in VLF-EM dataAbstractIntroductionVLF-EM data processing and interpretationForward modelingModel 1Model 2ConclusionReferences