ORIGINAL PAPER

Structural boundaries delimitation from geomagnetic datausing the continuous wavelet transform. Applicationto Hoggar (Algeria)

Sid-Ali Ouadfeul & Said Eladj & Leila Aliouane

Received: 15 August 2010 /Accepted: 14 December 2010 /Published online: 5 April 2011# Saudi Society for Geosciences 2010

Abstract The main goal of the proposed work is todelineate structural boundaries in a very complex geologyenvironment using the spatial and statistical properties ofthe potential field data. The analysis is performed usingmagnetic anomaly of the total field data over In Ouzzal, anArchaean north–south elongated block belonging to theHoggar (Algeria). This region is geologically and geophys-ically very poorly known except some localized areas. Theintrinsic properties of high-frequency signals and therelated causative sources are explored, thanks to two-dimensional continuous wavelet transform. The obtainedresults, represented by spatial distribution of the maxima ofthe modulus of the wavelet transform at each scale, clearlyshow that the major magnetic singularities of the field maybe related to geological features. Comparison with theEuler’s deconvolution solutions exhibits a very goodcorrelation. Even though where geological structures areknown, our method shows better resolution and accuracy.The proposed multiscale method proves to be morepowerful, easy to use, and versatile where classical methodsof potential field interpretation fail or are very constraining.However, work is still ongoing to try to better and fullycharacterize the causative sources of the potential fields.

Keywords Boundaries . Potential filed data .Wavelettransform . Euler deconvolution

Introduction

During the last decade, wavelet analysis has been appliedin various fields such as in image processing and geo-sciences as an alternative to Fourier analysis (Arneodoet al. 2003; Ouadfeul 2006; Ouadfeul 2007; Ouadfeul2008; Ouadfeul and Aliouane 2010). In geophysics, thisprocessing is a modern analysis of potential fields (gravityand magnetism).

Moreau et al. (1997) have presented the results ofapplication of the continuous wavelet transform (CWT) tolocate and characterize homogonous point sources from thefield they generate in a distant hyperplane.

Sailhac and Gibert (2003) have proposed a newtechnique of identification of sources of potential field withthe continuous wavelet transform; it is based on the two-dimensional (2D) wavelets and multipolar approximations.

Cooper (2006) has published a paper on the interpreta-tion of potential field data using the continuous wavelettransform of their horizontal derivatives. For gravity data,the method uses wavelets based on the integer-orderhorizontal derivatives of the gravity anomaly from a pointsource (the Poisson kernel). For magnetic data, the waveletis based on the integer-order horizontal derivatives of theanalytic signal of the anomaly from a contact or a thinsheet.

Recently, Khattach et al. (2006) have published a paperon the utilization of the maxima of the modulus of thecontinuous wavelet transform on the mapping of contactsfrom potential gravity data; the obtained results show a highrobustness of the CWT on the geological boundarydelimitations from gravity data.

We present in this paper an application of the 2Dcontinuous wavelet transform detailed in Arneodo et al.(2003) on the mapping and analysis of geological contact.

S.-A. Ouadfeul (*) : L. AliouaneGeophysics Department, FSTGAT, USTHB,Algiers, Algeriae-mail: SOuadfeul@ymail.com

S. Eladj : L. AliouaneEarth Physics Laboratory, UMBB,Algiers, Algeria

Arab J Geosci (2012) 5:365–370DOI 10.1007/s12517-010-0273-4

Area of study is situated in the south-west of Algeria. Itrepresents a very important mining field that needs anadvanced exploration tools to get more idea about itspotentialities.

Wavelet transform and potential data

The sharp contrasts that show the potential data areassumed to result from discontinuities or interfaces suchas faults, flexures, contrasts intrusive rocks, etc. Forcontacts analysis between geological structures, we usuallyuse the classical methods based on the location of localmaxima of the modulus of the total (Nabighian 1984) or thehorizontal gradient (Blakely and Simpson 1986), or theEuler’s deconvolution (Reid et al. 1990). This techniqueallows, in addition to localization in the horizontal plane ofcontact, an estimate of structures’ depth. The potential field,over a vertical contact, involving the presence of rocks ofdifferent susceptibilities is indicated by a low in side rocksof low susceptibility and a high in side rocks of highsusceptibility. The inflection point is found directly belowthe vertical contact. We can use this characteristic ofgeomagnetic anomalous for localization of abrupt suscep-tibility change. If the contact has a dip, the maxima ofhorizontal gradients move in the direction of dip. Todetermine the dip direction of contacts, we move the mapupward of the potential field at different altitudes. At eachlevel, the maxima of horizontal gradient are located. If thestructures are vertical, all maxima are superposed. However,moving of maxima with upward indicates the direction of thedip. The potential theory lends perfectly to a multiscaleanalysis by wavelet transform (WT).

By choosing an appropriate wavelet, measurement ofgeomagnetic field or its spatial derivatives can be processedas a wavelet transform. Indeed, this analysis unifies variousclassical techniques: it process gradients that have beenmoved upward to a range of altitudes. The expressions ofvarious conventional operations on the potential field arewell-designed in the wavelet domain. The most importantis the equivalence between the concept of scaling and theupward continue. Indeed, the wavelet transform of apotential field F0(x,y) at a certain scale a=Z/Z0 can beobtained from measurements made on the level Z0 by:

1. Upward movement continue the measured filed a levelZ=a×Z0

2. Calculation of the horizontal gradient in the plane (x,y)3. Multiplication by a.

For a multiscale analysis of contacts, it is sufficient tolook for local maxima of the modulus of the CWT fordifferent scales to get an exact information about geologicalboundaries (Khattach et al. 2006).

The 2D wavelet transform modulus maxima lines

Most of the fractal methods used to analyze geophysicalsignals simply rely upon the estimate of the fractaldimension DF which is related to the so-called Hurstexponent H that characterizes statistically the globalroughness of surface. The multifractal formalism accountsfor possible fluctuations of the local regularity of a roughsurface as defined by the Holder (local roughness) exponenth(r) of the function f(r) whose graph defines the roughsurface under study (Arneodo et al. 2003).

f ðr þ lÞ � f ðrÞ � lj jhðrÞ lj j ! 0 ð1ÞThe 2D wavelet transform modulus maxima (WTMM)

method provides a way to estimate the so-called D(h)singularity spectrum defined as the Hausdorff dimension ofthe set of points r where the local roughness exponent h(r)is h(Arneodo et al. 2003). This method consists inperforming wavelet-based multiscale canny edge detection.Let us define two wavelets:

y1ðX ; Y Þ ¼@qðX ; Y Þ

@X; ð2Þ

y2ðX ; Y Þ ¼@qðX ; Y Þ

@Y; ð3Þ

where θ(X,Y) is a 2D smoothing function well localizedaround (X=Y=0). For any function f(r), the wavelettransform defined with respect to y1 and y2 can beexpressed as a vector:

Ty f½ �ðb; aÞ ¼ r Tq f½ �ðb; aÞf g; ð4Þwhere:

Tq f½ �ðb; aÞ ¼ a�2XXd2rqða�1ðr � bÞÞf ðrÞ: ð5ÞIf θ is just a Gaussian qðrÞ ¼ exp �r2ð Þ, then Eq. 4

defines the 2D WT as the gradient of f(r) once smoothed bydilated versions q r

a

� �of this filter. At a given scale a≻0

Calculation of the 2D Continuous WaveletTransform CWT of the image I(X,Y)

Positioning of maxima of themodulus of the 2DC WT

Original 2D potential field map I(X,Y)

Fig. 1 Flow chart of the 2D WTMM

366 Arab J Geosci (2012) 5:365–370

where the WT modulus my f½ � b; að Þis locally maximum inthe direction Ay f½ � b; að Þ of the gradient vector Ty f½ � b; að Þ.When analyzing rough surfaces, these WTMM lie onconnected chains. These WTMM are disposed along

connected curves across scales. The WT skeleton definedby these maxima lines contains a priori all the informationabout the hierarchical organization of the singularities ofthe function f(r). In particular, one can prove that provided

Fig. 2 Geographic location of Hoggar (Algeria)

400000 440000 480000 520000 560000 600000

2420000

2440000

2460000

2480000

2500000

2520000

2540000

2560000

2580000

3335033400334503350033550336003365033700337503380033850339003395034000340503410034150342003425034300343503440034450345003455034600

X(m)

Y(m)Fig. 3 Total geomagnetic fieldmap of In Ouzzal

Arab J Geosci (2012) 5:365–370 367

the first Ny, moments of y are zero, then my f½ � � ah r0ð Þ

along a maxima line pointing to the point r0 in the limita ! 0þ, where h r0ð Þ < Ny

� �is the local Hölder exponent

(Arneodo et al. 2003)

The processing algorithm

In this part, we have analyzed the 2D potential field data bythe WTMM to delimitate structural boundaries. The flowchart of this method is detailed in Fig. 1.

Application on geomagnetic data

Geological setting

The lens of the massif of Hoggar that occupies the southernpart of Algeria (Fig. 2) is defined as a wide lapelPrecambrian. It integrates mobile areas caught betweenthe West African craton and East Africa. It extends at1,000 km from east to west and 700 km from north tosouth. It is extending through the “append” branches of Air-Niger at south-east and the Adrar des Iforas-Mali at thesouth-west. Its territory is covered to the north-east andsouth in part by training Paleozoic sedimentary of Tassili.The entire range (Hoggar, Air, and Adrar des Iforas) formsthe Tuareg shield (also known as Tuargi shield). Stablesince the Cambrian, this massif belongs to the chain called“Pan” which belts the West African craton to the east. Thisold chain is probably the result of the evolution of bothintercontinental basins and accretion of microplates involv-

ing the creation of oceans and island arcs. It is surroundedby sediment platform of Paleozoic age.

Multiscale analysis of magnetic data of In Ouzzal

In this section, we have analyzed the geomagnetic datato demonstrate the power of the CWT method toidentify geological contacts. A source code in Clanguage is developed to calculate the 2D continuouswavelet transform and the spatial distribution of itsmaxima at different scales. The area of study is calledIn Ouzzal; it is located in Hoggar. The geomagneticfield data are recorded by a national company andprocessed with a regular grid of 750 m. Figure 3 is a

400000450000

500000550000

600000

8

10

12

2400000

24400002480000

25200002560000

2600000

Y(m) X(m)

a(m

)

Fig. 5 Skeleton of the modulus of the CWT

400000 440000 480000 520000 560000

2420000

2440000

2460000

2480000

2500000

2520000

2540000

2560000

2580000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

7500

8000

8500

9000

9500

X(m)

Y(m)Fig. 4 Modulus of the continu-ous wavelet transform plotted atthe scale a=2.1 km

368 Arab J Geosci (2012) 5:365–370

representation of the data in the plan, and Fig. 4 representsthe modulus of continuous wavelet transform at the scalea=2.1 km. The analyzing wavelet is the Mexican hat(Arneodo et al. 2003).

The next operation consists to calculate maxima of themodulus of the continuous wavelet transform at each scale(scales varied between 2.1 and 136 km). Figure 5 shows thechains of maxima in the space; they are called the skeletonof the modulus of the wavelet transform. At each scale, wemap points of maxima in the plane. The obtained set ofmaxima for all ranged scales will give the geometry ofgeologic contacts (Fig. 6).

Result interpretations

The mapped contacts by the CWT are compared with themagnetic field anomalous map (Fig. 6). One can remark thegood correlation between field defects and contact positionsidentified by the proposed technique. To better ensure ourresults, we have compared it with geological boundariesobtained by Euler’s deconvolution (Fig. 7). We can notethat the CWT method gives not only contacts found by theEuler’s deconvolution but also other contacts which have avery clear anomalous origin in the magnetic field map(Fig. 6).

For example, one can remark that in the range intervallimited by 520,000≤X≤560,000 and 2,400,000≤Y≤2,480,000, where X and Y indicate the Cartesian coordi-nates, the Euler’s deconvolution is not able to predict anygeological contacts. However, the proposed method hasgiven many boundaries that have a big correlation with themagnetic anomaly field.

In conclusion, comparison of the obtained contacts withEuler’s deconvolution boundaries and even the magneticfield shows that the CWT is a very robust tool in detectingcontacts from potential field.

Conclusion

The results of this study show that the multiscale analysisbased on the wavelet transform is very effective forgeological contacts analysis from maps of geomagnetic

400000 440000 480000 520000 560000 600000

2420000

2440000

2460000

2480000

2500000

2520000

2540000

2560000

2580000

X(m)

Y(m)

Obtained contacts by the CWT

Euler's deconvolution boundaries

Fig. 7 Geological boundaries identified by the CWT compared withcontacts obtained by Euler’s deconvolution technique

400000 440000 480000 520000 560000 600000

2420000

2440000

2460000

2480000

2500000

2520000

2540000

2560000

2580000

3335033400334503350033550336003365033700337503380033850339003395034000340503410034150342003425034300343503440034450345003455034600

X(m)

Y(m)Fig. 6 Geological boundariesidentified by the multiscaleanalysis compared withthe measured geomagneticanomalous

Arab J Geosci (2012) 5:365–370 369

anomalies. Indeed, this kind of analysis of magnetic mapscan enhance conventional structural studies. The identifiedgeological structures have a great interest in hydrogeolog-ical exploration, mining, and in the study of natural hazards(earthquakes and landslides).

References

Arneodo A, Decoster N, Kestener P, Roux SG (2003) A wavelet-basedmethod for multi- fractal image analysis: from theoretical conceptsto experimental applications. Adv Imaging Electron Phys 126:1–92

Blakely RJ, Simpson RW (1986) Approximating edges of sourcebodies from magnetic or gravity anomalies. Geophysics51:1494–1498

Cooper GRJ (2006) Interpreting potential field data using continuouswavelet transforms of their horizontal derivatives. ComputGeosci 32(7):984–992

Khattach D, Mraoui H, Sbibih D, Chennouf T (2006) Analyse multi-échelle par ondelettes des contacts géologiques :application à lacarte gravimétrique du Maroc nord-oriental. C R Geosci338:521–526

Moreau F, Gibert D, Holschneider M, Saracco G (1997) Waveletanalysis of potential fields. Inverse Problems 13:165–178

Nabighian MN (1984) Toward a three-dimensional automatic inter-pretation of potential field data via generalized Hilbert trans-forms: fundamental relations. Geophysics 49:957–966

Ouadfeul S (2006) Automatic lithofacies segmentation using thewavelet transform modulus maxima lines (WTMM) combinedwith the detrended fluctuation analysis (DFA). In: 17th interna-tional geophysical congress and exhibition, Ankara, Turkey, 14–17 November 2006 (expanded abstract)

Ouadfeul S (2007) Very fines layers delimitation using the wavelettransform modulus maxima lines WTMM combined with theDWT. In: Society of Exploration of Geophysicists summerresearch workshop, Antalya, Turkey, 13–18 May 2007(expanded abstract)

Ouadfeul S (2008) Reservoir characterization using the continuouswavelet transfrom combined with the Self Organizing map(SOM) neural network. In: SPE ECMOR XI, Bergen, Norway,8–11 September 2008 (expanded abstract)

Ouadfeul S, Aliouane L (2010) Multiscale analysis of GPR data usingthe continuous wavelet transform. Presented in GPR 2010, IEEEXplore Compiliance, doi:10.1109/ICGPR.2010.5550177

Reid AB, Allsop JM, Granser H, Millett AJ, Somerton IW (1990)Magnetic interpretation in three dimensions using Euler decon-volution. Geophysics 55:80–91

Sailhac P, Gibert D (2003) Identification of sources of potential fieldswith the continuous wavelet transform: two-dimensional wave-lets and multipolar approximations. J Geophys Res 108:noB5

370 Arab J Geosci (2012) 5:365–370