GEOPHYSICAL RESEARCH LETTERS, VOL. 28, NO. 9, PAGES 1863-1866, MAY 1, 2001
Identification of sources of potential fields with the continuous wavelet transform: Application to self-potential profiles Dominique Gibert and Marc Pessel G6osciences Rennes- CNRS/INSU, Universit6Rennes1, France Abstract.
We show how the continuouswavelet transform may be used to quickly localize and characterizethe sourcesof self-potential anomalies. The method is applied to synthetic examples and to a self-potential profile crossing a shallow fault
zone.
(1)
Introduction
Self-potentialdata carry uniqueinformation concern- where • istheprimary flow(fluid flow, heatflux,etc.), ing fluid flows underground,geothermaland electro- q is the cross-coupling coefficient,a is the electricalconchemicalphenomena [Corwin,1990,ZhdanovandKeller, ductivity, and •bis the electrical potential. The electri1994]. Despitepotentialdata are acquiredsinceseveral calcurrent•tota• is the sumof the advective current, decades only a few studiesconcerninversion[Fitterman •advec = -q• created bytheprimary flow,andofthe
and Cotwin,1982;Patella,1997;Lapennaet al., 2000],
conductive current •conduc = -••
and most studies use a forward approach and numeri-
Ohm's
law.
If no external
current
resulting from
sources exist
and
cal modeling[Fitterman, 1978, 1979, 1983; Sill, 1983; either the stationary regime approximation or the lessYasukawaet al., 1993; Wurmstichand Morgan, 1994; restrictive quasi-static approximation holds, the total Michel and Zlotnicki, 1998; Adler et al., 1999; Jouniaux
current flowisdivergence-less, •. •tot• - 0,andequa-
et al., 1999]. In the samespirit, recentstudiescon- tion (1) gives: cerning hydrologicalapplicationsfit the data through hydro-dynamical models which ensurethe physicalcoherence of the current sourcesresponsiblefor the ob-
servedself potential [Ishidoand Pritchett, 1999;Revil et al., 1999]. It might be usefulto disposeof a less specialized(i.e. not assumingan hydrologicaloriginfor the sources)method to performa rapid analysisof the data in order to obtain information (e.g. number of sources,depth range, multipolar nature, etc.) concerning the sourcesand which could be used as prior model
For constantelectricalconductivity,equation(2) mW be further simplified,
where• = •/a is the voltagecouplingcoe•cient. The sourceterm of the Poissonequation(3) may be splited
for moresophisticated inversionmethods[e.g. Patella, 1997; Lapennaet al., 2000]. In recentstudies,we pre-
V. (-•)--V•.•-•V.
•,
(4)
where the first right-hand term representssourceslo(almostperpendicwavelet transform which allowsa quick analysisof po- catedwherethe primary flowcrosses tential field data [Moreauet al. 1997, 1999],and we ularly) spatialvariationsof the couplingcoe•cient, and showhowthis methodcan be usedwith self-potent!althe secondright-hand term representsexternal sources data. of primary flow (e.g. well pumping,rainfall, etc.). The secondmathematicalitem of the presentstudy is the continuouswavelettransformof a function •0 which Theoretical Background may be written as the convolutionproduct, The starting equationis the first-order(Born approxw [g, a) (vg ß imation) couplingequation betweenprimary and secsented a theoretical
flamework
based on the continuous
ondaryinducedflows[Sill, 1983], Copyright2001 by the AmericanGeophysical Union.
where g is the analyzing wavelet, and D• is the dilation operator such that
Papernumber2000GL012041. 0094-8276/01/2000GL012041
$05.00
1863
1864
GIBERT AND PESSEL: WAVELET ANALYSIS OF SELF-POTENTIAL
nature (e.g. bipolar) of a singularsourcefrom its potential field anomaly(gravity,magnetic,self-potential).
•J
This
I,IiJlll
0
PROFILES
II
mathematical
flamework
constitutes
a means to
quickly analyze potential field data by representingthe observedfield in terms of homogeneoussources.This is not an inversionmethod but, instead, a pre-processing which may provide prior information useful for a more sophisticatedinversion.
Illllllllli,l,ll,11•111llll,ll
JJ
-1.0 •
ß
B•IIj J
Horizontal
•lJJJ
fw
ß
Synthetic Examples
distance
Figure 1. A: Continuouswavelettransformsof a Dirac We now illustrate the method with two synthetic eximpulse(B) of homogeneity a = -1. The wavelettrans- amples. The first one, whose results are shown in the form displaysa cone-likegeometry and the magnitude along any straight line crossingthe cone apex varies
accordingto a powerlaw with exponenta (C).
top part of Figure 3, correspondsto a model of constant
conductivityer- 10-2 S.m-• with two currentsource
points: xs= 90m,zs= 14m,•-?advec= -0-4A.m -3 andx8= 110m,z8= 12m,•-•advec= +0-4A.m -3.
The resultingpotential(labelleda) hasa wavelettransform (labelledb) with a clearcone-likestructurewhose oscillatinglocalizedfunction (i.e. with a compactor apex is located in the (xs, zs) half-spaceby assessing quasi-compactsupport)with a vanishingintegral. The the cone-like geometry of the wavelet transform with useful mathematical property concernsthe geometrical respect to every point in a rectangular domain of the structure of the wavelettransformof homogeneous func- (xs,zs) domain. In practice,the cone-likecoherencyis tions •b0verifying quantified by rescalingthe wavelet transform with re(x), (7) spect to the tested apex and by computingthe slopeof the wavelet transform modulus along a set of straight lines emergingfrom the tested apex. A normalizedhiswith a the homogeneity degree.Equation(5) gives' togramhi with N binsof the slopespj is computedand w = (b, (s) its 5-like behavior is quantified by the followingindex basedon Shannonentropy[Tasset al., 1998] and the whole wavelet transform of the homogeneous function may be extrapolated from the wavelet transform given at a singledilation. The geometricaltransla' lnN ' where the dilation a > 0. The analyzing wavelet is an
p(xs zs) =InN+E//v=i hiInhi
(9)
tion of this property is that the wavelet transform of an homogeneousfunction possesses a cone-like structure The index p varies from 0 for a uniform histogram to convergingat the homogeneitycenter of the analyzed I for a 5-like histogram. For this first example, the
function (Figure 1). Also, the variationof the magni- p(xs, zs) map (labelledc in the top part of Figure 3) tude of the wavelet transform along any straight line indicates a narrow area for the sourceposition with the crossingthe homogeneitycenteris a powerlaw of exponent e with respectto a (Figure 1C) [Alexandrescu et al., 1995,1996;Holschneider, 1995]. The continuationproperty (8) of the wavelettransform has been generalized, and a particular class of wavelets exists such that the wavelet transform of a po-
tential field satisfyingthe Poissonequation(3) may be used to detect, localize and characterizehomogeneous
o
Dilation
singularitiespresentin the causativesourcedistribution
[Moreauet aL, 1997, 1999; Hornby et al., 1999; Sailhac et al., 2000]. The main differencewith respectto the analysisshownin Figure I is that the apex of the cone-like structure
of the wavelet transform
is no more
ß
Log(dilation
located on the a - 0 line, but now at a - -zs below the wavelet half-plane, zs being the depth of the causative Figure 2. A: Continuouswavelettransformthe poten-
tial field (B) causedby a bipolarsourceof homogeneity lationsa with respectto zs [ Moteauet aL, 1997,1999] • = -2. The apex of the coneis located at a negative dilation correspondingto the depth zs of the source. the power-lawvariationof the magnitudeof the wavelet C: The magnitudealong any straight line crossingthe transform is restituted and the homogeneitydegreee cone apex now varies in a complicated manner. D: The of the sourcecan be determined(Figure 2d,f). Hence, power-law behavior is restituted when the dilation axis the wavelet transform givesboth the location and the is properly rescaledwith respect to zs. source(Figure 2A). Applyinga properscalingof the di-
GIBERT
60
AND
DO
PESSEL:
100
1•-0
WAVELET
ANALYSIS
OF SELF-POTENTIAL
PROFILES
1865
The second example has the same current sources as the first one and a conductivity structure with two blocks of different conductivities separated by a vertical fault plane located at z - 108 m. The results are shown in the middle part of Figure 3 and, as expected,
140
4O
the location of the current sources is not so accurate
as in the first examplesincethe constant-conductivity
0
assumption of equation(3) is not valid. This results .•3o
in a horizontal location x s - 104 m of the equivalent c
bipolarsourcebiasedtowardthe lessconductive block,
40
õ0
100
120
140
and zs - 18 m. Despite these perturbations, the homogeneity degree remains unchanged,a - -2.07, still indicating a bipolar current source.
horizontal position (m)
• 2o o flo
IOO
12o
Analysis of the Pont-P•an
4o
The Pont-P6an
is a mineralized
diorite
dike
which has been industrially exploited during the 18th
o
and
c
40 1 O0
120
horizontal position (m) 15
1 oo
15o
800
80
.40
:• •o o
!,0
-
•o 1 O0
t õ0
until
a massive
flood
invaded
the
3 and display a bipolar anomaly located in the vicinity of the fault zone. The dilation range a < 20 of the
•30 (• 50
centuries
(Britanny, France), the area is not industrializedand the geophysicalmeasurements are of goodquality (i.e., good signal-to-noiseratio and repeatability of measurements). The electricalpotential measuredperpendicularly •o the fault is shownin the bottom part of Figure
•-5 50
19th
underground galleries in April 1904. The area has been extensively prospected and the geologicalstructure is well-known, with a linear 4 km-long fault which enables a very good 2D approximation. Although located in the Pont-P6an village 10 krn south of Rennes
•o •o
fault-zone
Data
200
horizontal position (m)
wavelet
transform
was not
considered
because
of
a too low signal-to-noiseratio (seeAlezandrescuet al. [1995}for details concerningthe wavelettransformof noisy signals). However,the cone-likestructureof the
Figure 3. Top: Synthetic model with a constant con- wavelet transform is conspicuousand the location area ductivitya - 10-2 S.m-•. A) potentialcreatedby of the sourceobtained from the p map is rather limited two current source points Zs - 90 m, zs - 14 m, I - -0.4 A.m-3, and • - ll0 m, z, - 12 m, in extend with a clearly marked maximum centered at I - +0.4 A.m-3. B) Continuous wavelettransforms zs = 107 m, zs = 15 m with an homogeneity degree of the potential profile. C) Map of the p measureshow- a = -2.25 very near the one of a pure bipolar source. ing the area where the conedikestructure of the wavelet The shallow depth to the sourceindicatesthat the selftransform is coherent. Middle: Same as above but for potential signalis more likely producedby shallowfluid a - 10-2S.m -• left of x - 108m and a - 2.10-aS.m -• flow in the fault zone rather than by electro-chemical on the right. Bottom- A) Self potential alongan east- phenomena arising in the rnineralized dike whose top west profile perpendicular •o the Pon[-P•an fauk zone. The fauk is approximately located at the x - 100 m has been found muchdeeper(i.e. > 80 rn). This result horizon[a]positionand is inchnedtowardthe left (east- agrees with a recent electrical impedance tornography
ward) wkh an angle of about 80ø. B) Continuous wavelet transform of the data. The small-dilation range a < 20 has been removed becauseof a [oo bad signalto-noiseratio. C) Map of the p measureshowing most likely position of the current sources.
performedalongthe sameprofile [Pesseland Gibert , 2001]whichindicatesthat the electricalsourcesidentified in the present study are localized in a highly conductive zone located above the fault zone and probably associatedwith very altered rocks. Conclusion
best location being Zs = 100 m and Zs = 15 m with In this letter, we show how the continuous wavelet an homogeneitydegreea = -2.05 indicating a bipolar source. The depth to the source is slightly overesti- transform may be used to analyze self-potential data. mated because the two current source points are actu- Straightforwardapplicationto 3D casesis possibleproally separatedby a finite distance6Zs = 20 m. vided potential data acquiredin the (z,•t) plane are
1866
GIBERT AND PESSEL: WAVELET ANALYSIS OF SELF-POTENTIAL PROFILES
available.
The theoretical framework established by
Holschneider, M., Wavelets: An Analysis Tool, 423 pp., Moteau et al. [1997, 1999] may formally be applied Clarendon, Oxford, England, 1995. only if the electrical conductivity is a constant. How- Hornby, P., F. Boschetti, and F.G. Horowitz, Analysis of potential field data in the wavelet domain, Geophys.J. Int.,
ever, the synthetic example correspondingto the inho-
137, 175-196, 1999.
mogeneousconductivitymodel (middle part of Figure Ishido, T., and J.W. Pritchett, Numerical simulation of elec3) showsthat the theory may still be applied with an trokinetic potentials associatedwith subsurfacefluid flow, J. acceptable accuracy level. The wavelet analysis provides both
an estimate
of the location
and of the na-
ture (dipole,etc.) of the currentsourceresponsible of a given self-potentialsignal. This may constitutea prior information useful to initiate a more sophisticatedinversion strategy like the one recently proposedby Patella
[1997].It mustbe kept in mind that the sources identified by the wavelet method proposedin the presentpa-
Geophys.Res., 10•, 15247-15259, 1999. Jouniaux, L., J.-P. Pozzi, J. Berthier, and P. Mass•, Detection of fluid flow variations at the Nankai Trough by electric and magnetic measurementsin boreholesor at the seafloor, J. Geophys.Res., 10d, 29293-29309, 1999. Lapenna, V., D. Patella, and S. Piscitelli, Tomographicanalysis of self-potential data in a seismicarea of SouthernItaly, Annali di Geofisica,d3, 361-373, 2000. Michel, S., and J. Zlotnicki, Self-potential and magnetic sur-
veying of La Fournaisevolcano(Rb•unionisland): Correla-
per are homogeneous (e.g. bipolar) and constitutethe tion with faulting, fluid circulation, and eruption, J. Geosimplest sourcemodel obtained from the information phys. Res., 103, 17845-17857, 1998.
contained in the potential data alone. More physically Moreau, F., D. Gibert, M. Holschneider, and G. Saracco, realistic sources could subsequentlybe derived by re- Wavelet analysis of potential fields, Inverse Probl., 13, 165178, 1997.
placingthe localizedhomogeneous sourcesby extended Moreau, F., D. Gibert, M. Holschneider, and G. Saracco, onesproducingthe samemultipolarpotentialfields(e.g. Identification of sourcesof potential fields with the continua sphereis equivalentto a monopolarsource). How- ous wavelet transform: Basic theory, J. Geophys.Res., 10•, ever, this may be achievedonly at the expanseof ad- 5003-5013, 1999. ditional information as shown in the paper by $ailhac Patella, D., Introduction to ground surfaceself-potential tomography, Geophys.Prospecting,d5, 653-681, 1997. and Marquis[2000]whichuseshydrological constraints Pessel, M., and D. Gibert, Multiscale electrical impedance (e.g. continuityequation)togetherwith potentialfield tomography, J. Geophys.Res., 106, in press,2001. data.
Acknowledgments. We thank GeorgesRuelloux and Aude Chambodut for kind assistanceduring the field op-
Revil, A., H. Schwaeger,L.M. Cathles III, and P.D. Manhardt, Streaming potential in porous media 2. Theory and application to geothermal systems,J. Geophys. Res., 10•, 20033-20048, 1999.
erations, and the Gal•ne association(http://pont-pean. le- Sailhac, P., and G. Marquis, Analytic potentials for the forvillage.corn) for both historicaland geologicalinformations ward and inversemodeling of SP anomaliescausedby subconcerningthe Pont-P•an mining industry. Andr• Revil surface fluid flow, Geophys. Res. Left., in press, 2001. made a very constructivereview. This work is financially Sailhac, P., A. Galdeano, D. Gibert, F. Moreau, and C. Desupported by the CNRS and ANDRA through the GdR lor, Identification of sourcesof potential fields with the conFORPRO (Researchaction number99.II) and corresponds tinuous wavelet transform: Complex wavelets and applicato the GdR FORPRO contributionnumber2000/34A. This tion to aeromagneticprofilesin French Guiana, J. Geophys. study is a part of the Electrical Tomography Project of the
Res., 105, 19455-19475, 2000.
CNRS
Sill, W.R., Self-potentialmodelingfrom primary flows, Geo-
ACI
Eau et Environnement.
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D. Gibert, G•osciencesRennes, Universit• Rennes 1, Bat.
15 Campus de Beaulieu, 35042 Rennes cedex, FRANCE. (e.mail:
[email protected]) M.
Pessel, CEREGE,
Europ6le M•diterran•en
de
l'Arbois, BP 80, 13545Aix-en-Provencecedex04, FRANCE. (e.maih pessel@cerege'fr)
(ReceivedJuly 12, 2000; revisedJanuary 22, 2001; acceptedJanuary 30, 2001.)
