JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. B8, PAGES 19,455-19,475,AUGUST 10, 2000
Identification of sources of potential fields with the
continuous wavelet transform' Complex wavelets and
application to aeromagnetic profiles in French Guiana
P•sc•l
S•ilh•c
•nd Arm•nd
G•lde•no
Laboratoire de G6omagn6tisme, Institut de Physique du Globe de Paris, Paris, France.
Dominique Gibert and Fr6dariqueMoreau
G6osciencesRennes, Universit6 de Rennes 1, Rennes, France.
Claude Delor
Bureau des RecherchesG•ologiques et Minibres, Orleans, France.
Abstract. A continuouswavelettechniquehas been recentlyintroducedto analyze
potential fields data. First, we summarizethe theory, which primarily consistsof
interpreting potential fields via the properties of the upward continuedderivative
field. Using complexwaveletsto analyzemagneticdata givesan inversescheme
to find the depth and homogeneitydegreeof local homogeneous
sourcesand the
inclination of their magnetizationvector. This is analytically applied on several
local and extended synthetic magnetic sources.The application to other potential
fieldsis also discussed.Then, profilescrossingdikes and faults are extracted from
the recent high-resolutionaeromagneticsurvey of French Guiana and analyzed
using complex one dimensionalwavelets. Maps of estimated depth to sourcesand
their magnetizationinclination and homogeneitydegreeare proposedfor a region
between Cayenne and Kourou.
1.
Introduction
have demonstratedthe general n-dimensionaltheory
for local homogeneous
sources.Moteauet al. [1999]
The high resolutionof recent magnetic(and grav- have analyzed the effects of noise and extent of sources
ity) surveys[Gunn, 1997; Biegertand Millegan, 1998;
on the properties of the wavelet coefficients. We now
Grauchand Millegan,1998]stimulatesthe development
(lof specific interpretation techniqueswhich emphasize presentspecificpropertiesof the one-dimensional
D)
complex
wavelet
coefficients
of
the
total
field
magthe information of interest to the geologistand to the
netic anomaly:Apparentinclinationof magnetization,
geophysicist:Modern magnetometersmeasurethe magin additionto depth, verticalextent, and dip angleof
netic field to .-• 0.01 nT (the Earth's dipolemagnetic
sources
can be estimated. This wavelettechniqueapfield is .-•30,000-60,000 nT worldwide, with an ampliplied
to
magneticstudiesis not only a filteringas used
tude of the anomaly due to the upper crust of hundreds
in
recent
advances
in aeromagnetic
processing
[Fediand
of nanoteslas);
Global PositioningSystems(GPS) can
Quarta,
1998;
Ridsdill-Smith
and
Dentith,
1999]
but acgive the positionfrom a few meters(in aeromagnetic
tuallygivesthe derivatives
(andanalyticsignal)of the
surveys)to a few centimeters(in groundsurveys).
upward continuedanomaly field. This is more like the
We have explored the use of wavelet transforms, as
initially introduced in the analysisof potential fields by continuouswavelet analysisdevelopedfor the location
Moteau[1995]. Shehasdescribedthe techniquefor lo- of singularfeaturesof the sourcedistribution[Hornby
cal and extendedsources,with the emphasison gravity et al., 1999],whichis improvedwhenscalingrelationsof
the waveletcoefficientsare analyzedand when vertical
applications,followedby preliminary resultsfor mag-
netic cases. The principle of this method is to interpret potential fields data via the propertiesof the up-
derivatives are used in addition to the horizontal deriva-
tives. This complementary
development
to existingupward
continuation
techniques
[Paul
et
al., 1966] and
ward continuedderivativefield. Moteau et al. [1997]
to recentadvancesin the interpretationof the gradientsof potentialfields[Pedersen
andRasmussen,
1990;
Copyright 2000 by the American Geophysical Union.
Pilkington,1997;Hsu et al., 1998],providesa theoretiPaper number 2000JB900090.
0148-0227/00/2000JB900090509.00
cal frameworkto enlightenpropertiesof the sourcesvia
their scalingcharacter.
19,455
19,456
SAILHAC
ET AL.' WAVELET
TRANSFORMS
First, we have applied this techniqueto severallocal and extendedsyntheticmagneticsources:Local ele-
OF MAGNETIC
PROFILES
1997,1999],weobtaina classof wavelets
;)v (Figure
1) for whichthe waveletcoefficients
of a potentialfield
mentarymagnetization
(dipole),localmultipolemagne- dueto localhomogeneous
sources
exhibitsimpleproptization, vertical and inclinedsteps,strips, and prisms erties.For the potentialfieldd(x, z = 0) measured
at
of magnetization. Wavelet coefficientshave been calcu- levelz: 0 whichis dueto a localhomogeneous
source
lated for the total magneticanomalyfield at any normal locatedat x = 0 and depth z = z0, the waveletcoeffiapparent inclination.
cientsin the upperhalf planeof positionsand dilations
Then, we have appliedthe techniqueto data profiles (a =-z > 0) obeya double
scaling
lawwithtwoexpofrom French Guiana. A set of 27 profiles each of 80 nent parameters:
km long with flightline direction N30øE has been transformed by wavelets;for the purposeof this study we
W½•lo(.,z=o)(x,a
)have analyzedthree isolatedfeaturesin the CayenneKourou
2.
2.1.
(a)v
•+zo')-•142•lc)(.z:o)
•+zo
•7 (a
a+z0/
'
a+z0
a').
area: two dikes and one fault.
(a)
In (3), x and a are the positionand the dilation, re-
Method
Wavelet
Transform
of Potential
Fields
spectively, for the left-hand side wavelet coefficient;
First, we briefly recall the basictheory [Moveauet x(a•+ zo)/(a + zo) and a• are the positionand the dial., 1997, 1999],which we apply to the caseof a two- lation, respectively,for the right-hand side wavelet codimensionalphysicalspace. This involvesparameters efficient:This definesa set of lines(x, a) whichsatisfy
x/(zo + a) = const. For variousconstantswe obtaina
that are listed in the notation section. We define the
familyof linesthat intersectat the point (0,-z0) inside
continuous
wavelettransformof a functiond0(x C I•)
the lower half plane; therefore the wavelet transform exhibits a cone-likestructure where the top of the coneis
shiftedto the locationof the source:Usingthe modulus
as a convolutionproduct,
1•¾•1½0(b,
a)-f•d__•_x
½(bx)&0(x)
a
maximalines(on whichthe signalto noiseratio is the
best),thisconstitutesa geometrical
procedure
to obtain
a
the locationof a homogeneous
local sourcewith no a pri-
: (va½ ß
ori idea of its homogeneity
degree(seeFigure2). The
where½(x C I•) is the analyzingwavelet.,a • I• + is first exponent7 is the order of the wavelet;)v whichhas
a dilation parameter, and the dilation operatorPa is beenused,the secondexponentfi is associatedwith the
defined by
homogeneity
degreeof the sourcea (• = a- 7 for the
total geomagnetic
field anomaly).Oncethe depthz0 of
.
the anomalyfield has beenobtained,the exponent• is
The sourceis modeled as a homogeneous
function simply obtainedwith the waveletcoefficientsW• along
or(x,z). This means,for instance,whenthe sourceis modulusmaximalinesasthe slopeof log(IWal/aV) verhomogeneous
at x : 0 and z - 0 with homogeneity suslog(a+ z0) (Figure2).
1½(•)
(2)
degreea, that for any positive number A it followsthat
The kernel of the Poisson semigroup which is used
t.o build these special wavelets defines the well-known
By applyinga Fouriermultiplier homogeneous
of de- upwardcontinuationfilter Pa(X) whichtransformsthe
gree7 (equivalent
to a derivativeof order7) anda di- harmonicfield d(., z) from measuredlevelz to the level
lation to the Poissonsemigroupkernel [Moteauet al., z + a [Le Mou•'l, 1970;Bhattachavyya,
1972; Galdeano,
0.2
0.05
0
0.1
-0.05
-0.1
-0.15
-0.2
-0.1
-0.25
•'lx(x)
-0.3
-0.2
0
-5
0
x
5
10
0
-5
0
5
10
x
Figure1. Typicalwavelets
belonging
tothePoisson
semigroup
class:•(x) - (1/7r)[-2x/(x2+
1)2] and•}(x) -(1/7r)[(x2- 1)/(x2+ 1)2].
SAILHAC ET AL.- WAVELET TRANSFORMS OF MAGNETIC PROFILES
19,457
0.6
0.5
0.4
0.3
0.2
0.1
0
0
ß
..
,
i
i
i.
i
i
............
.
i
10
i
i
,
i
!
J
0
-5
-4
-3
-2
-1
0
1
2
3
0
0 2
0 4
Iog(a+z
o)
x
Figure 2. (bottom left) Magnetictotal field T due to a verticaldipoleat x0 - 0 and z0 - 1
analyzedwith verticalwavelet•}' (top left) Modulusmaximalinesconverge
at the locationof
the source,at x:
x0 and a = -z0 (with downwardcontinuationin dotted lines); (right) the
modulusalongthe line at x = x0 followsa scalingrelation with exponent/• = -3 associatedwith
the homogeneitydegreeof the local line sourcec•: -2.
1974; Barahoy, 1975; Gibert and Galdeano,1985]; x
beinga 1-D variable(abscissa
alongthe profile),this is
1
a
Pa(x)
- •ra2+ x•'
(4)
ward vertical derivatives,respectively,of
the harmonic
potentialfield q•(.,z) at levelz.
2.2.
Complex
Wavele•s
Any of the two real wavelets•
and •
can be used
Two typical real wavelets can then be considered, to determinethe depth and the homogeneitydegreec•of
the "horizontal" ½x, and the "vertical" '½z made by a localdipolesource(Figure 2), but the determination
one horizontalor one vertical (upward) derivativeof of the inclination of this dipole needsthe introduction
P• respectively; they are said to be of order 1. Fol- of complexwavelets• in order for this to be done in
lowing with 3'- 1 derivatives over x gives wavelets a simple way. Using the Hilbert transform 74, which
of order 7: ½•(x) = O•Pl(x) (or in Fourierdomain: changes• into • - -7/[•], we definethe complex
of the hori½J(u)- (i2•'u)•e-2'•l"[);
½7(•)- aj-•aaP•(x)l•=• (or wavelets•,• - ½J+ i7/[½J]ascombinations
zontaland verticalwavelets½• and ½• [Moreau,1995]'
in Fourier
domain'• (u) - (i2•-u)•-•(-2•-]u[)e-2,•l"l).
Then dilating these waveletswith the dilation a trans-
•,• - ½J- i½•. These are actually defined not only
for '7 C N* but for '7 C 5•_ (with the help of frac-
forms •fi•(z) into •(x/a)/a,
which is also the 7th
tional derivatives).They are progressive
waveletsproderivativeof P•(z) multipliedby the scalingfactor a•.
portionalto the Cauchywavelets[Holschneider,
1995],
Thus the convolutionof the harmonicfieldq•(.,z) (meaand their general expressionis
suredat level z) with thesedilated real waveletsgives
the wavelet
coefficients
at scale a which
are also the
ei'•(v+•) iF(7 + 1)
½•(x)- •- (x+i)v+•
'
(6)
derivativesof the upwardcontinuedfield (at levelz + a)
whosedimensionis that of the initial field (in nT for
the geomagnetic
total field anomaly):
where F is the Gamma function. Thus the complex
waveletcoe•cientsof the potentialfield q•(.,z) are
142,:l,(o,z)(X
, a) -- 142•jl,(.,z)(X
, a) - iYV•7l,(.,z)(X
, a).
(7)
whereqS•(.,z) and qS•(.,z)are the horizontaland up-
where F is the Gamma
function.
19,458
SAILHAC
ET AL.: WAVELET
TRANSFORMS
Fu (x=O):-I: - p/6
OF MAGNETIC
PROFILES
FT(x:O):-;I- p/2= -5p/6
6
2
•'--'---'•-'-:----'•..'-•..-•
•-'---'-:•!?'..:_•:ii,;...'•ii:•:ii!!?•,?.--&i•i:•?•?•:..:..
.......
..:.-:::y:.'
..,.....%..
:.
,½.--.------?•....:.-•:.-•:::-•
::-:•,.,'•:....•$•!
.:.....•.•-.....•...•...•iij•::._.•½•i•i•ii:ii!!•:::i•!i•::
..........
.......
.:.....:-::':'
:.::
_• ..................................................
:.:':C..'7•,.'•:•::i:
:.:.:.•
...........................................
ß
-5
ß
0
5
o
1
5
......................
........................
-1
:
.....
............
-2
-5
o
.............
0
5
x
-5
-2
0
5
x
Figure 3. Magneticpotential U and total field T due to a dipole with inclinationI-
•r/6 at
x0 - 0 andz0 - i analyzedwith complexwavelet•b•' Phaseisovalues
(shownin white,every
•r/6) converge
to the source(dottedlines);the phaseisovalue(I>(x- 0) (for x - x0 - 0) gives
the' lIICllIli:t,
"
•:--tlOll I.
For a local dipole of normal inclination I analyzed
with a complex wavelet of order 1, the argument of the
wavelet coefficientsis constant along the vertical to the
source,equal to-I
for the magnetic potential and to
source c•. Thus it relates to the structural
index N used
structurepointingto the source(Figure3).
at al., 1999]. Usingwavelets,the homogeneity
degree
of the sourcec• can be determinedfrom the scalingexponentfi without a priori value. Hereafter, we give the
in Euler deconvolutionwhich is alsobasedupon an homogeneityproperty,that of the field [Thompson,1982;
Raid at al., 1990;Huang,1996].UsingEuler deconvolu-21- •r/2 for the magnetictotal field anomaly(Figure tion, the structural index N must be assumeda priori,
3). Equation(3) showsthat the isovalues
of the phase except when applied in conjunctionwith other proceof the complex wavelet coefficientsalso draw a cone-like dures [Huang,1996; Rayat and Taylor, 1998; Barbosa
2.3.
Comparison With
Classical Techniques
The complex
wavelet
coefficients
W•+•i, -,•) (x,a)
exponent fi of the 1-D wavelet coefficientsas a function
areassociated
with
theupward
continued
analytic
sig-of c•. We alsogivethe homogeneitydegreeof the as-
nal asearly introducedto the inteipretationof geophys- sociatedmagneticpotential-N in differentsituations,
dependingon the type of anomalyfield or potentialq5
ical potentialfields[Nabighian,1972,1974]:
which is analyzed:
O-r
+./)
2
+
0•
+./)
2
V (gravitypotential,or Green's
function),
ß
In the classicaluse of the analytic signal,one consid-
fly - -7 +- + 2- -(7+ N-
]);
ers •he modulus of the analytic signal bu• the inter- g - V'V (gravityfield)or U - -V'V.M (magnetic
popretation of i•s phaseis missing. Recen• improvements tential due to the dipoleM),
are due•o the interpretationof the phase[Smithat al.,
1998]. Within the theory of continuouswavele•trans- /3• - flu - -7 + o•+ 1 - -(7 + N);
forms[Holschnaidar,
1995]•he useof both the modulus
and phase(and bo•h the real and imaginarypar•s) of T- -VU (magneticfield)or O•g (verticalderivative
the upward continued analytic signal a• differen• levels of the gravityfield/,
(complexwavele•coefficients
for differentdilations)is
natural and allows interestingpropertiesregarding the /3• -- -7 + o•-- -(7 + N + 1).
geometryof •he singularities•o be taken into account.
Besides,the exponent/3definedin (3) dependson •he
derivative order 7 and the homogeneiWdegree of the
Classically,one definesN from the potential field but
this givesa valuewhichdependson the analyzeddata,
SAILHAC
ET AL.' WAVELET
TRANSFORMS
OF MAGNETIC
PROFILES
19,459
and this involves a shift of 1 between the gravity and
magneticcases.Instead,we considerits definitiongiven
sin/' (-5•/•cos2F
+ 5T2sin2F);(10)
by Huang[1996];this is morereliableasit givesa value
which dependsonly on the geometryof the source:N is 5T• is equal to the anomaly field reduced to the pole
the oppositeto the homogeneitydegreeof the magnetic (I' = 90ø),and5T2is equalto the anomalyfieldreduced
for the
potential and is also that of the correspondinggravity to the inclination I' = 45ø. Analytic expressions
field.
total field anomaly due to a givenbody are equivalentto
those of south-north profilesby replacinginclinations I
with
apparentinclinationsI' (for a collectionof analytic
3. Synthetic Examples
ST_
(sinI)2
3.1. Total Magnetic Field Anomaly in Profiles
f
expressions,
see Telfordet al. [19901).
In this paper, calculations of the wavelet coefficients
We consider the magnetic total field anomaly producedby a set of elementary•nagnetizationvectorswith
will be doneby taking derivatives(from equations(5),
and (7)). It can be shownthat the modulusof the
declination
real and imaginary parts exhibit extrema controlledby
the derivative order 7 and the mean apparent inclina-
D and inclination
I within a normal field of
declination D'* and inclination I'*. To clarify the analytic expressionsin the caseof profiles,we use two sim-
tion 9:
(F+ I•)/2, whilethe modulusof the complex
plifications. First, when profilesare striking geographic coefficientsexhibit extrema whosegeometryis indepeninclinations(AppendixB). A
north with an angle • and perpendicularto the sources dentof the meanapparent,
special
formulation
for
extended
sourcesand the corre(Figure4), onecanintroducean apparentinclinationI'
and an apparentnormalfieldinclinationI• correspond- spondingcomplex wavelet coefficientsis possibleusing
ing to a geomagnetic
south-northprofile[de Gery and the complex variables method; it is introduced in Appendix C for its potential applicationsin the numerical
Naudy,1957]:
approach to the direct and inverse problems. We have
tan/
tan/,,
calculated analytical expressionsfor sometypical simple
2-D bodies below. For local sourcesat depth z0, exact
tan
I' = cos(D
+p)' tan
I' :
+ . (9)
Second, one can considermagnetization vectors with powerlawsof (a + z0) are shown(a is the dilation,or
declination D and inclination I equal to those of the continuationaltitude);for extendedsources,Taylor exnormal field D'* and I'*, respectively, as if there were pansionsare calculated to analyze the perturbation due
only induced magnetization. In this case,apparent in- to a finite extent.
clinationsare equal(I' = I•).
3.2. Local Elementary Magnetization
Hencethe total magneticfield anomaly5T at (x,z)
As a first synthetic case, let us consider the magis givenby two conjugatedsymmetricaland antisymmetrical functions 5Tt and 5T2 associatedwith second- netic total field anomalyproducedat level z (positive
orderderivativesof the Green'sfunctionV (seedetailed downward)by a localelementarymagnetizationvector
locatedat (x0, z0). For this elementarymagnetization,
expressions
in AppendixA):
Horizontal
Plane
Vertical
Vertical Plane of the Magnetic Meridian
Plane of the Profile
Geomagnetic
Geographic
North North
z'
ISøOho
.
Magnetic
Geomagnetic
North
Meridian
x'
Horizontal
I': Apparent Inclination
h Inclination
downward
downward
Meridian
z
Figure4. Apparent
inclination
I' versus
inclination
I: Theunitvector
f (respective
f'*)defining
the directionof the magnetization
M (respective
of the magneticfield F) givesthe apparent
inclinationI' (respectiveI'•) whenprojectedontothe verticalplaneof the profile.
19,460
SAILHAC ET AL.- WAVELET TRANSFORMS OF MAGNETIC
PROFILES
IWol
2
2 .....................
! ...........
:..........
!............
'..
_
-2
-15
-10
-5
0
5
10
-15
-10
x
-5
0
5
10
x
Figure 5. Localelementarymagnetizations
at depth z0 - 1 (and a - -z0 - -1) with apparent
inclinationsI • = 90ø at x = -10 and I •: 29.16ø at x: +5: (bottom) Total magneticfield,
and (left) modulusof real waveletcoefficients
or (right) complexwaveletcoefficients.Isovalues
are shown in white with regular intervals. Modulus maxima lines are shown in black with their
downward
continuation
to the source.
which is concentrated on an infinite horizontal line, the
Green'sfunctionis V(x, z) - ln[(x- x0)2+ (z0- z)2;
so (10) holdswith
inclination
I'= -•/2 + (7 + 2)(•r/4) (modulo•r).
5•(•,•)_ _•[(•(•- •0)
•0)
• -(•0- •)•
• + (•0- •)•]•'
5T2(x,
z)- -4[(x-(xx0)
- xo)
(zo
- z)
• + (•0- •)•]•'
Thus using variables X - x-
Whenthe depthz0is known,modulusIWI andphase
ß alonga modulusmaximaline simplygive the values
of the homogeneitydegree• = -2 and the apparent
When the depthz0 is unknown,adjustinga straight
lineto the plotsof log[)/VV/aV
I versuslog(z0+ a) for a
set of a priori depthsz0 and lookingfor the best least
(Figure6).
(11)squaresfit providesboth z0 and •
x0 and Z - z0- z -
z0+ a andprefactorK - 4(sinI/sin F) 2, onegetsthe
3.3. Local Multipole Magnetization
As a secondsynthetic case, let us considera multi-
pole magnetization
vectorlocatedat (xo,zo) whichis
formallyan obliquederivativeof the local elementary
following wavelet coefficients:
'],'V•ZlST(.,z:0)
(;g
,a)-- --]-'•a
½-i2It(X -JiZ)-3 (12) magnetizationvectorpreviouslyanalyzed.Asymptotic
expansions of the far field due to extended sourcesimWe have computed these wavelet coefficientsfor two ply the sumsof multipoles.For instance,an elementary
orientations of magnetization: in a south-north profile pole plus a dipole mass can be used as a model for an
at the pole (F-90 ø) wherethe anomalyis symmetri- inclinedgravimetricborder[Moreau,1995].Let us call
cal and in the SSW-NNE profilesof the Guiana survey /•1the directionof the obliquederivativecorresponding
(F - 29.16ø).Figure5 showsthesewaveletcoefficients.to the dipole magnetizationvector. Then 0• is the di-
rectionof the secondobliquederivativecorresponding
'at the point (xo,a - -zo) insidethe lowerhalf plane. to the quadrupolemagnetizationvector, and 0• is the
There is only one modulusmaxima line of the complex directionof the nth obliquederivativecorresponding
to the nth-multipolemagnetizationvector. We assume
waveletcoefficients
(with equationx- x0).
Extrema
lines of the real wavelet
coefficients
intersect
x0'
that the structureof this multipolesourceis still infinite
in the horizontaldirectionperpendicularto the profile,
W•JiST(.,z=o)(xo,a
) --
planeof the profile(apparentangles0• areequalto 0).
Wavelet coefficientsfor any derivative order 7 E
read for x-
so that • angles have to be consideredin the vertical
Then for this nth-multipolemagnetization,n additional
2(,
7+1)lß\sin
(sin/
I/)2a'•ei(-2z'+(v+2)•
(Z0
-••)-• )
(in directions
Ojfor 1 _•j _• n) have
(13) obliquederivatives
to be performedin (10), (12), and (13).
SAILHAC ET AL.' WAVELET TRANSFORMS OF MAGNETIC
PROFILES
19,461
20
Best:
13=-2.985
andZo=0.988
._
1
F:10
5
0
0
1
2
3
4
5
-1
-2
o
o15
1
1.5
Iog(a+z
o)
•-4
-•-5
-6
7
-0
i
•
,
3
,
4
5
a priori
depth
zo
Figure 6. From the waveletcoefficients
of Figure 5, (right) computationof the best slope/3
and depth z0 is accomplished
by leastsquareslinear regressions
of •og(IrV•l/a•) versuslog(a+
z0). (left) A set of a priori depths[0.1,5] has beentested;this givesgoodestimatesfor both
homogeneitydegree c• =/• + 3': -1.985 and depth z0 = 0.988.
Thus using variables X - :c- x0 and Z - z0 - z -
,
z0+ a, prefactorIx' - 12(sinI/sin I') •, andangle• =
-2I'-0•,
--
onegetsthe followingwaveletcoefficients
for
-
+
(•4)
the wavelet
coefficients
3• -- 3• o
+
the dipole magnetization sourcein direction 0•'
We have also calculated
tan 1
3• -- 3• o
for
Thus, using variables X - x- xo, Zz = zz- z =
z• + a, and Z• : z•- z : z• + a and prefactor
t•; = 2(sin//sin1')•, onegetsthefollowing
waveletco-
any derivativeorder 7 C N * and any multipoledegree efficients for the vertical step:
n C N, which reads for .r - x0,
c
' a)
¾V½•l•T(.,z=O)(X
- h•ae-i'"
[Z2--iX
•
]4?•OjlaT,,(.,z=O)
(xo,a) --
Zi-iX
'
Figure 7 showsthese wavelet coefficients.The shape
2(•n
t-r/n
t-1)ßI(s¾h-77)
sin/
2a7½
i(-2I'-©'•+(7+n+2)•)
+
'
of modulus
maxima
lines for real wavelets
is not as sim-
ple asfor a localsource(in section3.2). For smalldila-
whereO• - Ey:•0i characterizes
the combination
of tions a, these are not straight but convergeat about.the
directionsin this multipole magnetization.
Again, there is only one modulusmaximum line of the
top of the step source; for dilations a which are large
enough(typically (z0 + a) >> h/2), theseare straight
complexwaveletcoefficients
(with equation
lines and convergeat about the mean depth of the step
The modulus IwI and phase• along this maximum
z0 - (z= + z•)/2, as if the sourcewaslocal.
line simply give the valuesof the homogeneitydegree
The argument and modulus are
a - -(n + 2) and either the apparentinclinationI' =
-(* + O,,)/2 + (7 - a)•/4 (modulo•), or the sumof (I>½•l•T(.,•-0)(x
, a) -- -21' + zr+ t.an-• •x + tan-• x
directionsO,- -•2I' + (7- a)=/2 (modulo2=).
3.4.
Magnetization
Step
As a first exampleof a nonlocalsource,let usconsider
Equations(18) showthat there is only one modulus
the total field anomalygeneratedby a vertical step lo- maximum
line(forwhichO•IYV½2
(x,a)l iszero),defined
catedat x0 anddepths[z•,z=](withheighth- z=- z•)
by X = 0 (verticalto the stepsource,at •' = x0).
of elementary magnetization with normal apparent in-
Thus, when we consider modulus from neither the real
clinationI' (sourceonthe north,for x >_x0). Equation norimaginary
part (IW<l norIrvmI)
ratherfrom
(10) holdswith
thecomplex
itself(]W½}
I), thereisonesingle
19,462
SAILHAC ET AL.: WAVELET TRANSFORMS OF MAGNETIC PROFILES
9
8
7
Stepdipping
60ø
VerbcalStep
6
":-
5
.
'!.:.......
:::::•.:.::':"'
.'•i::•"
-
' :::;.:
:::
:::•:':.:':½..:•-...:•::
.:.. .:•.':
.-
4
--
3
•.5
::'..:..
'•'?•:!:
1
'::i.,.:
::'.'"'::
2
ß -'::-'-'--•<
•:-::;-"
O. 5
1
.
ß
.-.:
,..
............................................
•...:
.,.......
....................
.................
-1
-2
ß
ß
........
'
0.5
'
........
:::::-:::::c
....
;.........
ß
..............
:::::.::::.::::::::::::::::::::::::::::::
...:.:..:..:
_1
.
-40
-38
-36
36
38
40
x
-0.5
0
42
44
x
' '
-80
-60
-40
-20
0
20
40
60
80
IWcl
9
8
7
Vertical
Step
6
Stepdipping
60ø
5
4
3
./...•....:::::::.
2
0.5
I
...
0
......
.--
-1
ß
ß
ß
-2
.
............
.
....
ß
.
-0.5
•....-::c
..........
:....................................
:::::::::::::::::::::::::::::::::::::::::::::::::::
.........
.:.::.:.
......
O.5
....
-44
-4: -•
-3• -3'•
x
3• 3•
40 4• •4-1•
x
-0.5
-80
-60
-40
-20
0
20
40
60
80
Figure 7. Verticalanddippingstepsof magnetization
(at meandepthz0 - 1, with apparent
inclination
I t: 29.16ø):(bottomleft)Totalmagnetic
field,and(topleft)corresponding
modulus
of realor complexwaveletcoefficients
(zoomedat right). Isovalues
areshownshadedwith regular
intervals. Modulus maxima linesare shownin black; their continuationto the sourceis shownin
thelowerhalf-space
(analytically,
abovethe top of the source).
Asymptoticbehaviorisobtainedwith a Taylorexpansionfor z0 + a > > hi2 (largedilationsor large average
signal).The phasealongthis maximumline equals½r- depth)
modulusmaximumline whichis straightand vertical(a
well-known property for the maximum of the analytic
ß
2I t.
ah e
z=o)(zo,
) • (•
W½:l•(.,
a)- 2(si--•
•i•½
+ •)(• + •)'
(19)
(20)
SAILHAC ET AL.' WAVELET TRANSFORMS OF MAGNETIC PROFILES
19,463
The first-order term is that of a local source located at
Indeed,moduliof (20) may be first approximatedin
depth z0; its homogeneitydegree,givenby the scal- a form valid for very large dilations:
ing exponent/? = -2, is c• = -1 (that of a step
source whose structural index is known to be N =
in
a)l"'"k q- ln(zoq-a). (21)
-(c• + 1) = 0). The second-order
term is a pertur-
bation(of 100x [(h/2)/(zo+ a)]2 %), thisis another Fitting a straight line to this approximation provides
local sourcelocated at depth z0 but with homogeneity the slope/• = c•- 1 and the log factor k = ln(I•h)
degreec•= -3 (dipoleof magnetization).
Thus(17) to (20)donotonlyprovidethemeandepth
z0 (at the convergence
of modulus
extremalinesof real
waveletcoefficients),
the apparentinclinationI / (from
the phaseof the complexwaveletcoefficients)
and the
(whereI5; is the prefactorusedin (17) whichincludes
informationonI, I' andthe intensityof magnetization).
These then can be used to plot the following function
H(a) which converges
rapidly to a limit whichis the
heighth of the step(seeFigure8):
characteristichomogeneitydegreec• = -1 using the
slope/• in the plot of log(l¾•al/a)versuslog(aq-z0)
(alongmodulus
extremalines),but theyalsoprovidea
way to estimatethe heighth of the stepfromresiduals
. InIwcgl*T(
....ø)(xø'a)l
--k--/•ln(z0
q-a)
in the determination of/•.
Steps
between
z1=0.6
and
z2=1.4
(1solid:
vertical;
dashed:
dipping
60
ø)
......
O.
0.9
0.4
0.8
0.3
-
O.
-•-
-•- 0.7
0.0
0,6
0.1
..
0
1'0
2'0
1
10
100
1
a+z0
a
10
1'0
0
a+z0
2'0
a+zo
Steps
between
z1=0.2
andz2=1.0
(solid:
vertical;
dashed:
dipping
60ø)
0.4
0.7
1
1o
0.6
0.3
0.5
0.2
0.9
ß
0.4
•
0.3
-12
0.8
•' 0.1
,0.1
14
0
10
20
0.1
1
a
10
100
0.6•
1
a+z0
10
0
10
a+z0
20
a+zo
Steps
between
z1=0.7
andz2-1.3
(solid:
vertical;
dashed:
dipping
60ø)
0.35
O,
0.( ;3
0.3
0.( ;2
0.25
0.(
0.2
0.15
•o.o
0
0.0
0.. •
/
0.1
0.05
0
1'0
a
10
a+zo
100
;8
1
lO
a+zo
o
1'0
2'0
a+zo
Figure 8. Scalingrelations
for verticalanddipping(0 = -40 ø) stepsof magnetization
(with
apparentinclinationI' = 29.16ø). (left to right) Modulusof complexwaveletcoefficients
along
x = x0,first-order
scaling(equation
(25)),residual
dueto thesecond-order
term(equation
(31)),
andfunctionH(a) estimating
theheightofthestep(equation
(32)). Correspond
to (top)anomaly
in Figure7 (z0= 1, h = 0.8), (middle)to a lowerdepth(z0= 0.6, h: 0.8), and(bottom)to a
lowerheight(z0 = 1, h = 0.6).
(22)
19,464
SAILHAC ET AL.' WAVELET TRANSFORMSOF MAGNETIC PROFILES
Writing wavelet coefficientsalong modulus maxima
Let us now consideran inclined step with dip angle
•r/2- 0: with a limit betweenthe two points(x•,z•) similarlyto (19) for a verticalstep,thenTaylorexpandand (x2,z2) and middlepoint at xo = (x• + x2)/2 and ing for z0 + a >> hi2 (largedilationsor largeaverage
zo = (z• + z2)/2. Similar algebraicmanipulationsas depth) gives
those usedfor the vertical step allow us to calculate the
total field anomaly and its derivatives.
Using variables X• = x- x•, X2 = x- x2, Z• =
z•-z:
z• +a, and Z2 = z2-z
= zu+a; prefac-
tl ah (zo+•)2
+ (1- tan2
2([sinI
sin/
tor K = 2(sinI/sinI')2; andangle• = -21'+ 0, one
(•o+•)4
.
(27)
getsthe waveletcoefficients
for the inclinedstep (to be Thus, scaling parameters at large dilations are those
comparedwith (17)):
of the verticalstep(equation(20)) exceptfor the relative amplitudes between terms of the Taylor expansion
]42•p}lST(.,z=O)
(x,a) --
Kacos
O½
i• [Z2-iX2
• zxZixx
• ].
whichdependon the dip anglerr/2- 0.
Fitting a straightline as in (21) providesthe slope
fi = a-1 and the log factor k = ln(Kh). Thesecan
(23)
The argument and moduluscan be written, general-
then be used,alongthe extrema lines, to plot a function
H(a) similarto that definedby (22), whichconverges
for 0 • +•r/4 to a limit whichis the heighth of the step
x_•+ tan-•x__z (seeFigure8):
izing equations(18), respectively,
as follows'
(I)•)•lST(.,z=0)(x
-•
• a) -- --2F+ rr+ tan
Zx
Kah
Z2 •
(24)
Solving
for cqIW½}
(x,a)l/Ox- 0 implies
that modulus maxima are defined by solutions of a second-order
polynomialin Z - z- z0 (and fourth-orderin X =
x- x0)' with v- X/(h/2) • 0 (and 0 =/=0). Solutions
2(zo+ a)
l1- tan2
1__
ßInIW½11'•"(
....ø)(xø'a)l
- k-/31n(zo
q-a)
a
The phaseis rr- 2F for large dilations and could be
used to determine
are
-_z0+
(28)
an unknown
inclination
I'.
Note that the searchfor the heightfrom (22) or (28)
½1+
assumesthat one has first determined the dip angle
(25)
Equation(25) tellsusthat for differentdip anglesrr/20, the modulusmaximum line is not a straight line and
its slopedependson the dilation a and morepreciselyon
•r/2- 0: This could be done via the directionof the
modulusmaximalines (seeequation(25)). When the
angle remains unknown, an estimation from equation
(22) insteadof (28) wouldbe usefulanywayasit gives
the correct height within a factor of 2 for any 0 value
(a + zo)/(h/2), •, and 0. It is possibleto makea Taylor
except for values of about 45øor 900 Other possible
expansion of this equation near to the vertical of the
errors are due to the uncerta,inty in x location and
source(for v << 1); this givestwo typesof asymptotic modulus maximum determination, which a,re involved
behavior. For large dilationsa (for a + z0 >> hi2)
in the Taylor expansionalong the modulus maximum
the maximum line is a branch of a hyperbola whose
line. Qua,ntification of this remark is given by manipuasymptote is x - x0. For small dilations the maximum
lations showingtha.t the next term in the expansionof
line is a branch of a polynomial in x - x0 whosefirst
(20)is -(z- x0)2/(z0+ a)4;sothat H(a) maygivean
ordercorresponds
to a stra.ight
linewithslopecot(20)' average
valueof v/h2 -4(x -x0) 2 instead
oftheheight
Ifa+z0
h itself (noisein positioningor continuationalsoleads
to errorsin the waveletcoefficients).
>>h/2
a + zo •
2(h/2)2tanO
3•--
Z0
+ O(z -•r0);
Otherwise
3.5. Magnetization
Strip
Let us now consider the total field anomaly generated by an inclinedstrip with dip angle•r/2- 0: with
a limit betweenthe two points(x•, z•) and (x2,z2) and
This quantifiesa classicalpropertyof potentialfield middlepoint at x0 = (x• + x2)/2 and z0 = (z• + z2)/2.
anomaliesover a dippinghomogeneous
source:At low This strip is formally the horizontal derivativeof a right
altitudethe maincontributionis dueto the upperpart step source.So, as shownby Moteau et al. [1999],its
of the source;while at increasingaltitude the relative wavelet coefficientscan be obtained by a simple horcontributionof deeperparts progressively
increases
and izontal derivativeof (23): The waveletcoefficients
of
the location of the maximum effect moves to the vertical
order 7 for the strip are those of order 7 + 1 for the
passingthroughthe "centerof gravity" of the source. right step divided by a dilation factor a. Note that a
a+ zo
- 0)cot(20) + /2
2
2 2+ 0(. 4)] .
SAILHAC ET AL.: WAVELET TRANSFORMS OF MAGNETIC PROFILES
19,465
calculationbasedupon (10) as donein sections3.1-3.4
where
k - ln(Kh)andf(O)- (211
- tan2el)-« Along
would imply the same results.
Thus using variables X1 = xZ1 : zl-z
= zl + a, and Z2 :
modulusmaximum lines, phasesfor large dilation a con-
prefactorK:
vergeto -21' + 3zr/2, whichis the valuefor an elementary localmagnetization(again,this characterizes
how
4(sinI/sin I')2; angle• = -21'+ 0; and the strip transformsinto a horizontalline for h small).
z2-
z = z2 + a;
the height h = z2 - zl, the wavelet coefficientsread
3.6. Magnetization
}/VO•lST(.,z=o)
(X,a) =
- (Z•-•X•)•
Kacos
0½
i• (Z2_•X2)2
i
Prism
As a final synthetic case, let us consider the total
field anomaly generated by an inclined prism whose
] ' (29)limits are the four points (xx,zl),
(x2,z2), (xa,za),
and
(x4,
z4)
(clockwise,
with
zx
=
z4,
z2 : za, and
The shapeof modulus maximum lines of thesewavelet
xl
x4
=
x2
xa).
The
dip
angle
is
rr/20, the height
coefficients(not shown)is similar to that of the step
is
h
=
z2
zx,
and
the
horizontal
length
is
1: xx - x4.
source(Figure 7). With real waveletsthey converge
As
(xl,zl),
(x2,z2)
define
the
right
edge
and
(xa,za),
either to the top of the strip or to its mean depth z0.
(x4,
z4)
define
the
left
edge
(Figure
9),
analytic
expresWhile analytical formulation of modulus maxima of
sionsfor the wavelet coefficientsare given by the differlutionsof a secondorderpolynomialin Z = z-z0 (equa- ence between wavelet coefficientsof two step sources.
complex wavelet coefficientsfor the inclined step are so-
UsingvariablesX01 = x-x1 +1/2, X02: x-x2+l/2,
tions(25) and (26)), the strip caseimpliesa fifth-order
Z1:
polynomial which does not simplify easily. Neverthe-
zx-z:
zx+a, and Z2:
z2-z = z2+a; prefactor
less,graphicalsolutionsfor modulusmaxima (plotted K = 2(sinI/sin i,)2; andangle• = -2I' + 0, onegets
aszerosof x derivativesof complexwaveletmoduli)are the wavelet coefficientsfor the inclined prism (to be
accessible,showingasymptotic behavior similar to that comparedwith equation(23)):
1
obtainedfor the stepsource(Figure 7).
W•iaT(.,•=0)
(x,a)-- Kacos
Oe
i• [z=-i(•o=+Z/2)
In the case of a vertical strip, analytical formulations
are possible: Complex wavelet coefficientson modulus
i
i
1
--Z•-i(Xo•+I/2)Z,-i(Xo,-I/2)
+ Z•-i(Xo•-I/2)]'
maximalines(for x = x0) obey
(33)
For smalllength1 (for I << h) this corresponds
to
}/V•[ST(.,z=o)
(Xo,a) :
the horizontal derivative of the inclined step, so that
we recover the strip. Thus one expects the same kind
of results as those obtained for the strip. As shownon
Figure 9, properties of the wavelet coefficientssuch as
Calling h the height of the strip and z0 its mean
the homogeneitydegree are the same as for the strip.
depth, we obtain the following Taylor expansion for
These are properties correspondingto the asymptotic
z0+ a >> hi2 (largedilationsor largeaveragedepth):
expansionsfor large dilations a. Nevertheless,for small
•2
•2• ,•i(-2I'+3•)
2( sin/•2
a(•,
2--•'1]'-'
"
dilations(for z0+ a •//2),
W,•lar(.,,=0)
(•0,a) =
modulusextremalinesand
isoargumentsexhibit branching;they point toward two
siW)2ah?(--21'+35)
J
4(sin,
[(•o+a)•
1 + 2(•o+a)*
(h/2)2
] ' (al)top edges(stepsources)insteadof onesingleedgeobtained from the strip source.
This issimilarto (20) exceptfor a hctor 2 andfor the
For clarity, exact analytical expressionsof complex
powersof (z0 + a). The sameestimationtechniqueas waveletcoefficientson modulusmaximumlinesare given
that shownfor the verticalstepapplies:Equations(21) onlyfor the verticalprism(at x: x0):
and (22) now apply with •: •-7
= -3. Here, the
first-order term is that of an elementary local magne-
tization(of homogeneity
•:
W½•IST(.,z
•
- 0)(x0,
a)-- 2(si•)
sin
I 2 alh
-2), whichcharacterizes
how the strip transforms into a horizontal line for h
small.
Similarities between the vertical step and strip cases
also apply for both inclined casesin Taylor expansions
(Zl+ z2+ 2a)ei(-2I'+3•)
[(Z1+ a)2+ (•) [(z2+ +
(34)
Calling z0 the mean depth, we obtain the following
alongmodulusmaximalines,so that a functionH(a) Taylor expansionfor z0 + a >> hi2 , zo + a >> 1/2
similarto that definedin (22)and (28), whichconverges(large dilations or large average depth) and
(for almostall dippingangles0) to a limit whichis the h½l - tan20[ • l:
height h, exists:
H(a) • 2(z0+ a)f(O)
a
Iln
Iw½•ar(
....
ø)(•ø'•)1
- k-•ln(z0
+a)=, (32)
WW•lar(.,•=0)(x0
,a)• 4(si•)
sin
I 2 alh
sec 0cos20-(l/2)2
.ei(-2F+35
(zo+a)
•
.
' ) [(zo+a)
I a+2(hi2)
22
19,466
SAILHAC
IWxl:
ET AL.- WAVELET
ThinDipping
Prism
1
TRANSFORMS
OF MAGNETIC
PROFILES
Almost
Isometric
Vertical
Prism Almost
Isometric
Dipping
Prism WidePrism
WithVertical
Edges
:
05
:i
-0.5 .......
'
'
-1.5-1•4-12'2-12'0-11'8-116
-4• -4'• -4•) -3• -3•
X
3'6 3'8 40 z•2 z•4 116 li8 1•0 1:•21•4
X
Almost
Thin Dip )ing Prism
IWøl:
1.5
........ - - X
Isometric
Vertical
Prism
X
Almost IsometricDippingPrism
Wide Prism With Vertical Edges
_.-
:'•iiii!iii•:•;_;
. ::••-
•-.•..-•?½•:•;•.:.-
:
ß--!i
½.-.•
'"..--.--'.!
i-%-':'-:::
....
5-?:-:.?.'.--':i
,:
ß
.
:
'?'!!?ik..-.
":"
.,::
05
..
......
•
0
.
ß
........
ß
:..
--.....-
............
. _...
.
i
.,
..
.-•....•::-
-1.5
.
,..
ß
-0.5
.
-:--
. .:.
-124-122-120-118-11'6 -44 -4'• -4•) -38 -36
x
36
38
40
x
42
44
116
118
120
122
124
x
x
Total Field Anomaly
0.5
-0.5
-120
-80
-40
0
40
80
120
x
Figure 9. Verticalanddipping(0 - -40 ø) prismsof magnetization
(at depthz0 - 1, with
apparentinclinationI • - 29.16ø)
ß (bottom)Total magneticfield and corresponding
modulus
of (top) realor (middle)complex
waveletcoefficients.
Isovalues
areshownshadedwith regular
intervals. Modulus maxima lines are shown in black; their continuation down to the source is
shownfor small dilationsin the lowerhalf-space.
This is similarto (31) exceptfor prefactor1 and addi- H(a) similarto that definedin (32), whichconverges
tionaltermin lU:Thefirstislinkedto thechange
in the (for almostall dip angles•r/2- O) to a limit whichis
sourcedensity(per volumeinsteadof per surface)which the height h of the prism:
is hidden for the calculations; the secondis due to the
,(a) "02(z0+ a)f(O,-•)
changein sourcegeometry. It is difficult to interpret the
prefactorsexcept in casesof well-knowna priori source
density where conventional techniques apply. Nevertheless,it is worth using a multiscale techniquebased
upon the multipole expansion to obtain estimates for wherek - log(2Klh), •-3, and f(O,1/h.) - (21depth and source extensions. There exists a function
0- (t/h)21)-ø-sis a factorwhichdepends
on the
2
InIW*•11*•"(
....ø)(•ø'•)1
- k-/31n(z0
+ a)
(36)
SAILHAC ET AL.' WAVELET
TRANSFORMS
OF MAGNETIC
PROFILES
19,467
f(e,i/h)=(21
l_tan2e_(I/h)2)-o
5 (inclined
prism)
!
!
[
[
[
90
i
I/h=O (inclined strip)
10-•
80
70
_
10-øs
60
50
-,
40
30
20
10
i
3
2
1
f(e)=(211
_tan2e)-o
5
e=O (vertical prism)
4
0
1
i
i
1.5
2
I/h
Figure 10. Multiplicativecorrectionfactor f(O,!/h) for strip and prism to be appliedto the
heightestimatordefinedin equation(26) (undefinedin the dashedcurveneighborhood).
givesa convergence
point at the locationof
ratiobetween
/ andh andequals1/v/• for 0: 0 and coefficients
a localsource(horizontalline). When sourcesare exAlong modulusmaxima line, phasesfor large dilation tended(step,strip, and prism)and haveuniformmag-
1 << h (seeFigure 10).
at the centerof the oba convergeto -2I' + 3•r,/2,asfor the strip. Note that f netization,there is convergence
ject
for
modulus
maxima
lines
at
large dilationswhile
reachesinfinity for 0 = 0 with bodieshaving ahnost the
the
convergence
is
near
the
upper
borders
of the object
same horizontal and vertical extents for which higher-
orderTaylor expansion
is necessary.For d2 = h"•- at smallerdilations(whenthe sourceis not too deep).
12<< (h4 + 14)/(zo+ a)• thisimpliesthefourthpower For extended sourcesthis also dependson the choiceof
of h and l:
the derivativeorder[Moreauet al., 1999].
The modulus maximum line of complex wavelet coefficientsis a vertical through the location of a local
147½}
I•T(.,z:0
) (Wo,
a)
source(suchas a horizontalline), and the modulus
sin
I'/ 2alhei(¾ [(zo+a)
(zo+a)7
4(sinlh
2I'+3•r)
1 3+ 3(hl2)4+(112)
4] ' alongthis maximumline allowsoneto find the depth
(37)
3.7.
Summary and Discussion
and homogeneitydegreeby linear regression
of bilog-
arithmicplots (classicalleast squarescan be used).
Whensources
areextended(step,strip,andprism)and
Analysis of magnetic potential fields anomaliesusing have uniform magnetization, the modulus maxima at
the continuous
wavelet
transform
can be done without
the need to reduce the data to the pole or to the equator. As shown by the above results summarized below,
the interpretation is made via wavelet coe•cients of the
anomaly field itself.
Geometrical analysis of modulus maximum lines of
either the real or imaginary parts of complex wavelet
largedilationsform a verticalline pointingto the center of the source;at smalldilationsthe modulusmaxima
do not form vertical lines, except for a vertical step or
strip(andpossibly
at verticalbordersof a prism)but
rather curvespointingnear to the upper bordersof the
object with an anglelinked to the dip angle.
For these extended sources with finite height h a
19,468
SAILtIAC ET AL.: WAVELET TRANSFORMS
OF MAGNETIC
PROFILES
genericlaw extractedfrom a multipoleexpansiongives sothat (7 + 2)-orderwaveletcoefficients
of 5V havethe
an estimate of the extent of the source. The first-order
sameproperties as */-order wavelet coefficientsof 5T for
expansionis associatedwith linear fits in the bi]ogarith- inclinationF = 7r/2. Similarly,
mic plot correspondingto an equivalent local source;
this depends on three parameters, the depth z0, the
slope• (linkedto the homogeneity
degree),and a third
term k (linkedto the intensity).The second-order
term
W½7+,l•,(.,z
)(x,a)- a
]42•Jl•T(.,z)
(X,a)
sin
I )2ei(-2I'+3¾)
,r •
si-h--F
(40)
implies a transformation of residualsfrom the previous sothat (*/+ 1)-orderwaveletcoefficients
of 5g• havethe
linear fits in bilogarithmic plots and providesan alter- sameproperties as */-order wavelet coefficientsof 5T for
nativemeansof regression
whichis deterministic(equa- inclinationF:
tion (28)). This givesa functionH(a; zo,•,k) which
-•r/4. Also
)&'½Tlsgz(
,•)(x
a)- 14?½215u("z)(X'
a)
should becomea constant.for large a for the best a pri-
(41)
'
'
sin/ i(-I'+•)
ori model(z0,•, k). A correction
factormustbe applied
sinI • ½
to obtain the heightof the source(equations(32) and
of 5g• havethe same
(36) and Figure10). This can alsobe usedasa measure sothat */-orderwaveletcoefficients
of the uncertainty in the estimated location of sources. properties as */-order wavelet coefficientsof 5U for inThe phasealongmodulusmaxima lines at large dila- clinationF = 7r/2.
tions tends to a limiting value which is not dependent
on a possibledip angle of these extendedsourceswith 4. Aeromagnetic Data
finite height:
4.1. Survey of French Guiana and Geological
2•r), (38) Setting
•'½'•I•T
-- --2F+ k•••r (modulo
where k• corresponds
to the nature of the sourceand
The regionof interestfor the applicationof the wavelet
the order '7 of the wavelet; k• = '7 + 2 for a line, a transibrm techniqueis located betweenCayenneand
strip or a prism,and k• - 3'+ 1 for a right step(and Kourou,in FrenchGuiana (Figure 11). To introduce
k• = 7-1 for a left step).Moreprecisely,
k• islinkedto the geologicalsetting, let us first recall that this forms
the homogeneitydegree. In somecasesthis is equal to part of the Guiana Shield, which is made up of an
-/• = 3'- c•, and this correspondsto a multipole mag- Arcbeancomplexand a widely developedPaleoproteronetization whosesum of anglesof successive
derivatives zoic succession
includingmetamorphosedsedimentary
is a multipleof 27r(or ©• = 0 modulo2•r in equation and volcanicformations and granitic and medium- to
(•5)).
high-grademetamorphicterrains and Mezoproterozoic
For other extended sourceshaving a very large ex- formationsat its northernand southernedges[Mil•si
tent (h -->oc), Taylorexpandingfor z0+ a >> h is not et al., 1995; Vanderhaeghe
et al., 1998]. The opening
possible. It is neverthelesspossibleto expand wavelet of the North GuianaTroughbeganwith the formation,
coefficientsfor z• +a > > zl; this showsthat the "depth" in a sinistral strike-slipsetting, of pull-apart basinsin
as previously estimated is the top depth z• instead of
the average z0, and both the homogeneitydegree and
limit phase do not obey the above laws. Let us consider the infinite strip which is a typical model used in
the classicalanalytic signal method. The expressionfor
whichtheUpperDetritalFormation
wasdeposited
(af-
F. These can also be used on other kinds of potential field data. To put them into analytical results that
would be obtained for a magnetic potential anomaly
5U, a vertical gravity field anomaly 5gz, or a gravity
potential anomaly 5V, one can use duality equations
between wavelet coefficientsas shown in Appendix A:
et MiniSres)in 1996[D½loret al., 1997].A total distanceof 135,000
kmhasbeenflownat a lowspeed(•250
ter 2120Ma). Associatedwith morerecentactivity due
to the extensivetectonicsof the early openingof the
Atlantic Ocean(200 Ma), Pertoo-Triassic
magmatism
has emplacedclustersof dolerite dikesstriking NNW
its waveletcoefficients
is givenby (29) whosefirst term andNNE [Deckart,1995;Deckartet al., 1997].This ge(with z2 --> oc) equals0; it is governedby the second- ologicalsetting is similar to that found in West Africa,
order term (with z•) for any dilation a. This implies with largeore deposits.As Guianahasbeengreatlyunthat the homogeneitydegreec• is increasedby I (-1 in- derexploredin comparison,recent exploration interest
steadof-2 for a finite strip), and the phaseis decreased led to a new survey.
by the dip angle(-2F + 0 + 7rinsteadof-21 t + 37r/2).
An airborne survey including radiometricand aeroWe have shown analytical results for the total mag- magneticdata (total field intensity)has been carried
netic field anomaly, as due to a 2-D body of homoge- out by CGG-Gdoterrex(CompagnieGdndralede G•oneous magnetization with normal apparent inclination physique)for
BRGM(BureaudesRecherches
Gdologiques
)- (sini
si_h__77
)2
kin/h) anda lowflightlevel(•120 m clearance).
The
interlinedistancewas0.25,0.5,or 1 km (500m in the
areaof interestto us). UsingdifferentialGPS, the resolution of data after processing
is better than 10 m;
sampling
intervalis •7 m alongprofiles(constanttime
samplingintervalof 0.1 s). In FrenchGuianathe main
, (39)magnetic field had inclination I = 210 and declination
SAILHAC ET AL.' WAVELET
TRANSFORMS
OF MAGNETIC
PROFILES
19,469
N-NE
Mesocenozoic
•
Gabbro
AJluvial
mesozoic
sediments
Fluvialtertiary
formations
and dolerife dikes
/
Gabbro--diodte
Add Plutonisum 'Caraibe'
:>-'--:•
Galibi
granite
.[,-• Monzogranite
Porphyroid
granediodte,
and granRoid
Upper Detrital Formation'Orapu'
':"'::'"'"'":•
Sandstone
andquartzite
':• Mono
andpolygenic
conglomerate
AcidPlutonisum"Guyanais'
.•'1• Granitoides
andMigmatites
Volcano-Sedimentary
Armina
•-•
Black
schist
andmeta-graywacke
Volcano-SedimentaryParamaca
W
Amphiboloschist.
quartzite.
metavolcanic,
metapyroclasfite,
and chlorltoschlst
Basic Plutonisum
Gneiss Substratum "Cayenne"
7.-"..,•
Amphibolite,
leptynite
andmigrnatite
•
MajorFaults
•
Sinistralstrike-slipfault
S-SW
Figure11. Simplified
geological
map.(Theboxedareabetween
Cayenne
andKouroudefines
the areaof interestto us,shownin moredetailon the rightside.)
D = -16.5 ø, and the lines wereflown in a N30øE direction which means an apparent inclination of • 300
whenstrikinganomalystructures
of inducedmagnetization perpendicularly.The referencestationfor diurnal
correctionswas located in the Cayenne area. The con-
fidencein the final magneticdata is better than +3.8
cone-likestructures. In wavelet theory these are typical
conesfor local singularitieslocated at their top: The
dikes and the fault are local singularitiescharacterized
in the wavelet domain by these cone-like structures.
Interpretation by geometricalcontinuationof the extrema
lines down to their
intersections
allows one to
nT (standard
deviationof misfitbetween
crosslines
and model the sources. Figure 13 showsthe depth of homogeneous
sourcesthat have been estimatedby picklinesoverthe wholesurveybeforeleveling).
ing the two dikes and the fault on all profiles;these
4.2. Profiles and Anomalies of Interest
valueshave been correctedfor flight altitude and are
A set of 27 profileseachof 80 km lengthhasbeen almostall positive(a very few have very small values
to hills havingstronglyvaritransformed
usingwavelets
(seecentralprofilein Figure of < 50 m corresponding
12).Wehaveanalyzed
theextrema
linesforthreesin- ablegroundlevel).The fault (rightsideon Figure13)
(200m downto 1 kin)
gularfeatures:
twodikesfromthePertoo-Triassic
and hasa deepermagneticresponse
a sinistralstrike-slipfault of the Upper Detrital Forma- than the two dikes(< 400 m); in the centralprofile,
tion. The geology
is revealed
magnetically
because
of onegetsthe followingdepthsz0 (fromthe SSW to the
the highmagnetization
of gabbroanddioriteof dikes NNE): 370 m (SSWdike),250 m (middledike),and
positions
follow
contrasting
with the low magnetization
of "Caraibe" 930m (thefault). Notethat horizontal
and "Guyanais"
acidplunonic
rocks.The originof the what would have been obtained by classicalmethods
magnetic
anomaly
ofthefaultisin thehighmagnetiza-based on the locations of the extrema of the analytic
tion of the Paramacavolcano-sedimentary
series(Fig- signalor the gradientof the field reducedto the pole.
Thismethodgivesrealisticdepthestimates
with regard
ure 11).
to the geological
natureof the sources.Other computa-
4.3. Depth: From the Real Part
tionsusingtheEulerdeconvolution
method(notshown)
have
given
lower
depth
estimates,
with
negative
values
Wefirst computedthe realwaveletcoefficients
of each
when
corrected
for
flight
altitude
[Sailhac
et
al.,
1997].
profiles(using½•). Figure12 shows
typicalfeatures
of the region,obtainedon the centralprofile. Above This traducesthe problemof noiseeffectin Euler de-
the dikes and the fault sources,the wavelet coefficients convolutionwhich is automaticallysolvedby upward
plottedasa functionof altitudeexhibittracksforming continuation in the wavelet method.
19,470
SAILHAC ET AL- WAVELET TRANSFORMS
OF MAGNETIC
PROFILES
420
o
350 -•
.__o
'
;.
140
70
.•-
-100 •
½•o6d ......
6•o6d ......
&dobd ......
•o66 ......
•8obd ......
Position from Sea coast (in m)
•o66
' ' '
•i•ure t2. Typical SSW-•E
profile of the survey: Total m•netic field &nom&ly&nd its
w&velettr&nsformcoe•cJents(m•xJm• •re Jndark •nd mJnJm•in clearshadins).
30000
nT
2500O
95
65
45
3O
15
0
-20
-40
-65
2O00O
15000
10000
5000
0
100000
12OOOO
I
0
m
-2oo
-4oo
.AA
-600
-800
- 1000
-1200
!
!
100000
lOOO
5oo
45o
4oo
35o
3oo
25o
2oo
15o
lOO
75
5o
25
o
12oooo
Position (in m)
Figure 13. Imagingsourcesfrom the useof 1-D real wavelets'Eachprofileanomalyprovides
onesourcelocation.For the set of profilesin the area this mapshorizontalpositionsand depths
(top)asseenfromabovesuperimposed
on the anomalies
in greyscaleand(bottom)its vertical
crosssectionprojection.For the centralprofileonegetsthe followingdepthsz0 (fromthe SSW
to the NNE): 370m (SSWdike),250m (middledike),and930m (thefault).
SAILHAC ET AL.' WAVELET TRANSFORMS OF MAGNETIC PROFILES
19,471
ComplexCoef. alongModulusMaximaLines
Modulus Maxima Lines
1400
150
ModulusMaximaLines(ComplexWavelet,order1)
100
,
1200
ooo
•
S
80C
40i.-i"'
'".•.'•-•.-_•.:.•..4•.•>•.,,•..,:•
-•.•'•-•*.
-' '
"
i
.
• 30i
ß
50
.:.
.....
•• •0i
....
•
600
';
:':
.....
/-
.. .... ".
-..
..
.... _
400
'' .
,0
i-."?."'•:
"•/?
•0•
200
I•'
105
1•0
115
120
relative coordinates(in km)
25
130
•
-_.'.
••.
'..?•
"::'•'"'"
'. ß' ".:.ff -100
:?" -:•,,, dilation
(ainm)
relative
coordinates
X(inkm)._.+•
125 0
de[tees
Phase vs. Scale
Modulus vs. Scale
LogLogplot1orthe seallingol WaveletCoellicients
Modulus(Complex,order
Plotfor the scallingof WaveletCoefficientsPhase (Complex,order 1)
10ø
150
• 10
4
½-10-2
0
ß•
50
__.•
10-3
1 O0
1-50
10-4
02
103
dilation+ apparent depth (a+zo' in m)
104
0
200
400
600
800
1000
dilation(a in m)
1200
1400
Figure14. Characterization
ofthesources
from
theuseof1-Dcomplex
wavelets:
(top)The
modulus
maxima
linesandthephase
along
these
linesobey(bottom)
scaling
relations
characterizing
thetwodikes
(dotted
anddashed)
andthefault(plain);
(bottom
left)homogeneity
degree
isgiven
bytheplotof]og(IWal/a)
ve•u•]og(a
+ z0),and(bottom
right)
inclination
isgiven
by
the limit of Arg(Wa) for largea.
rural indexis N =-0.1) for the middledike. The SSW
dike and the fault havean homogeneitydegreewhichis
Usingdepthestimatedfrom the geometrical
proce- in therangeof that of a prismor a strip[Moreau,1995],
dure on modulus maxima lines of the real wavelet coeffi- nearto that of a horizontalpipe(a = -2.0). This correanomalies
asexpected(seccients(section
4.3),wehaveestimated
thehomogeneityspondswith magnetization
dikeof highmagnetization
degrees
fromthe modulus
maximalinesof thecomplex tion 4.2): a gabbro-dioritic
4.4. Homogeneity Degree: From the Modulus
coefficientscalculated on the central profile of Figure
12.
and a fault whosemagneticsignatureis the Paramaca
volcano-sedimentary
series. "fault": this can also be
As shownon Figure 14, modulusmaxima lines are the signatureof an edgeof a sill-likesourcecreatedby
abatting againstthe
both vertical for the two dikes and display an angle the Paramacavolcano-sedimentary
coastal
sedimentary
rocks.
0 •_ 50ø for the fault, this gives a rough estimate
The middle dike is seenwith a surprisinghomogeneof its angle(dippingNNE). Usingthe depthsz0 esity
degreewhichis nearto that of a step(a = -1.0)
timated from real wavelets,the linear regressionof
rather
than a prism. A first interpretationof this value
log([W•(a)I/a) versus
log(a+ z0)implies
a homogeneity
is
that
there is also, associatedwith this dike, a condegree
ofa = -1.9 (theslopeis/3= -2.9 andthestructrast
in
the magnetizationof the background.Indeed,
tural indexis N = 0.9) for the SSWdikeandthe fault
of amphiandof a = -0.9 (the slopeis/3: -1.9 andthe struc- a contactwith the CayenneSeries(composed
19,472
SAILHAC ET AL.: WAVELET TRANSFORMS OF MAGNETIC PROFILES
bolites,leptynitesandmigmatites)hasbeenrecognized to the interpretation of a gravity profile crossingthe
(seeFigure 11); they constitutea surfaceof migrated Himalayas[Marteletet al., 2000].
The applicationto high-resolutionaeromagneticproferromagneticelements. This remark agreeswith the
files
of French Guiana has shown the utility of the
shapeof the observedanomaly(seeFigure12) as comtechnique
for the interpretation of magneticstructures.
paredto the syntheticone (appearingon Figure7). A
secondinterpretation is that this is a strip-like sourceas There is no need to reduce to the pole, and the phase
expectedbut with an infiniteheightextent (suchthat of complexwaveletcoefficientsindicatesthe inclination
the first term in equation(34) tendsto zero and the of the magnetization. This is helpful, especiallyin arsecondterm impliesthe observedexponent/3 = -2). easwheregeophysicalprospectingis the only accessto
In this case,the significanceof estimated depths is not information on depthsof structures,as we have shown
the middleof the objectbut for the top of the dike (as for dikes and a fault in Guiana.
Further possibilitieswith other waveletsof this kind
in the classicalanalyticsignalinterpretationmethod).
could be exploredfor reductionto the pole in the first
4.5.
Inclination:
From
the
Phase
The plots of the phaseof the complexcoefficientsver-
instance. This wavelet techniquecombinesdifferent directions of derivative
of the field.
Also note that this
could be usedfor interpretation of surveyswhere differsus scale exhibit a different behavior for the dikes and
the "fault" (seeFigure14). A constantphaseof •--120ø ent componentsof the field have been simultaneously
is observedfor the two vertical dikes. A moving value recorded: The wavelet analysisof the magnetic potential U is equivalentto the analysisof the horizontaland
from 150 ø to -120 ø is observed for the inclined "fault"
These differencesare due to the dependenceof the phase vertical componentsH and Z that are upward continued to different levels.
on the dip angleof the sourcethat is significantfor small
This article is the secondof a series;forthcomingones
scalesand disappearsfor large scaleswhen the source
will
relate to statisticsof the sources,automation of the
has a finite extent. In this case,the limit for large scales
technique,
considerationsof large-scaleasymptoticbecan be usedto obtainthe inclination(equation(43)).
havior,
and
applicationof 2-D waveletsto aeromagnetic
The value of-120 ø gives an apparent inclination of
about 15ø (modulo180ø) for the verticaldike and the
maps.
fault whosehomogeneitydegreeis ct = -2.0. This implies that values of inclination I and declination D of
Appendix A: Wavelet Coefficients
the source are similar to the inclination I,• and decli-
Versus
nation D,• of the normalfield (for whichI'=30 ø in the
casethat the profileis perpendicularto the source).
For the vertical dike whose homogeneity degree is
c• = -1.0, one obtains the same value for the apparent
inclination by assumingthis is a dike of infinite extent
Green
Function
Here we show usetiff relations between wavelet coeffi-
cientsof a potential field due to a 2-D body of constant
sourcedensityand the Green'sfunctionV (whichcorre-
spondsto its gravitypotential). Factorscorresponding
whileoneobtainsa valueof ,-,-30ø (modulo180ø) for a to the nature of the field (massor magnetizationinfinite step. However it is unlikely that the Cayenne Se- tensity)havebeenomittedfor the relationsto be genries has a reversedremanent magnetization. Therefore eral (theseare givenby Telfordet al. [1990]or Blakely
(5), and (7), complexwavelet
this vertical dike is probably of large extent with a top [1996]).Fromequations
at ,-,250 m depth and a magnetization with apparent
inclinationof ,-,15ø (modulo180ø).
coefficientsof 1/ are given by
WcZlv(.,z)(Z,a)
=
5. Conclusion and Perspectives
The wavelet technique is aimed at analyzing both
the geometry and locations of the sourcesof potential
fields via relations of derivatives of the upward continued field. This can be applied to any potential fields
and components. A geometricalinterpretation of the
scaling over the cone-like structure of the real wavelet
coefficientsinduced by the sourcecan be used for modeling. An automatic depth to sourcedetermination is
basedupon adjustmentsof a scalinglaw for the mod-
'
(A1)
Now let us introducea complexgravity field anomaly
5go= 5gx+iSgz, whosereal part is givenby the horizontal gravityfield anomaly5gx: OV/Ox and whoseimaginary part is given by the vertical gravity field anomaly
5gz= OV/Oz (wherethe z axisis downwardoriented).
With this definition, 5gois actually a complex function
of the complex variable ( : x + ia, which is defined
ulusof complexwaveletcoefficients
(of
vers•,s from5g, with the Hilberttransform:5gc= (1+ i7-l)Sg•.
a + z0). This canbe appliedfor somenonlocalsources Hence complex wavelet coefficientsof V are
such as prismatic bodies where residualsof a simple
power scalinglaw form an estimateof the extent of
5g•(x,z + a). (A2)
the source;this has been recently succesfullyapplied
SAILHAC ET AL.' WAVELET TRANSFORMS OF MAGNETIC PROFILES
Similary, the complexwaveletcoefficientsof 5#z are
_
5a(,z+
(^3)
19,473
of W½•+•lsT(.,z
). Algebraicmanipulations
lead to the
equationR•(x, a): tan(2•) (respectivelyR•(x, a):
-cot(2•)) for the definitionof the modulusmaxima
lines(x, a)of W½•l,T(.,z)
(respectively
W•715T(.,z)
).
Thus their geometry is controlled by the derivative
Then we consider the magnetic potential anomaly
order •/ and the mean apparentinclination•:
(F +
producedby a set of elementarymagnetizationvectors
with declination
D and inclination
I within
a normal
field of declination D• and inclination I•. Along profiles perpendicular to the sourcesand making an
I•)/2. Nevertheless,
whenconsidering
the modulusof
the complexwaveletscoefficients,it is obvious(from
equation(A7)) that the derivativeorder•/, but not •,
controls the geometry of modulus maxima lines.
gle • with geographicnorth (Figure 4), a simplification followsby introducingan apparentinclination
corresponding
to a geomagnetic
south-northprofile[do Appendix C' Complex Variables
Gory and Naudy, 1957]. Thus the magneticpotential Method and Numerical Applications
anomaly 5U is given by an oblique derivative of V in
A general formulation for the direct calculation of
direction F'
' a)_a
I' e-iI'(•O
z+a).
W½•l•U(.,z)(X
•sin
sinI
)•5g•(x,
the complex wavelet coefficientdue to a 2-D body of
any shape can be obtained with the help of complex
variables w = X + iZ, where X = x-x0 and Z = z0+a
are the horizontal and vertical coordinates, respectively,
Eventually, we consider the magnetic total field from the sourceat coordinatesx0 and z0 (as usedin
anomaly.Similarto the apparentinclination(magneti- section3). This method has been previouslyapplied
zationdirection),onecanintroducean apparentnormal to the calculation of derivatives of any order from the
inclinationI• (normalfield direction).Thus the total verticalgravitycomponent[Kwok,1989].
magneticfield anomaly5T at (x, z) is givenby two conThis allowsus to write the coefficientsat (x,a) of
jugatedsymmetricalandantisymmetrical
functions5T• the complexwavelets½} for the total magneticfield
and 5T2. Let us introduce the mean apparent inclina- anomalydue to the 2-D body of contour(x0, z0) C S:
tion • - (F + I•)/2 (suchthat in the specialcaseof
normal apparentinclination,one gets • - I • - I•); ¾•2•p
}lST(.,z: o)(x, a) -this now reads
5T -
sin I
sin I•
[-ST• cos2• + 5T2 sin2•],
sin I • sin I•
4(sini•]
sin
I a
(A5)
((x-•o)+i(zo+a)) (el)
For½• andanyderivative
order•/C IR.+ thisbecomes
where 5T• and 5T2 are defined by second-orderderivatives of the Green's function
, z)-
•
¾•2•p:lST(.,z=o)
(X,a) =
V'
, z)- -••(•,
z) -
z),
si•
z).
((•-- •-o)+i(zo+•))•+• ß
(C•)
As shownby t•wok[1989],because
the integrandis
analytic,the doubleintegralabovecan be simplified
Complexwaveletcoefficientsof 5T are
W•Zl,r(.,•)(x,a)-
usingthecomplex
variable
w: (X-xo)+i(zo+a). The
complexformof Green'stheoremgivesintegrations
over
-
a•SinlsinI•-i2•(O)
sinI • sinI•
• •+1 '
'
theclosed
contour
C in thecomplex
plane(x0,-iz0):
2Re
Appendix B' Modulus Maxima Lines
of Real
Wavelet
Coefficients
real
wavelet
coefficients
and
its relation
to the
Hence,equation(C2) reads
mean apparentinclination• (equation(A5)). Let us
¾•2•pJlST(.,z:o)
(x, a) --
considerthe following characteristicratio function:
R'•(x,a)-
O•+•ST•(x'-a)/
Ox•+
•
Ox•+
•'
/
w•+•'
2Ira
1•((•-•o)+i(•o+•))•+•
• • -- - Re
/cw•+2'
• dw(C3)
We look for the geometry of modulus maxima lines
from
((•-•o)+i(•o+•))
•+•-
(B1)
si-h-/r
Y
w•
+2ß (C4)
F(,-I2)(sin/]2aq
½i(-2I'+3•)/c
• dw
Also, the real and imaginary parts give the real wavelet
Now,if welookfor the modulusmaximaof W½•isT(.,z
) coefficients,
W• (x, a) and-W•7 (x, a), respectively.
(maximaof a profilewhile dilation is kept constant), For numerical applicationson closed n-sidedpolygthe question is reduced to searching for the zero-set onal 2-D bodies, one may considerthe case of integer
19,474
SAILHAG ET AL.' WAVELET TRANSFORMS OF MAGNETIC PROFILES
Acknowledgments.
This paper benefited from comderivativeorder'7 C I•l*. To the coordinates(Zj, Zj)
of eachvertex j C {1, ..., n}, one associates
the series ments made by Associate Editor Kathy Whaler and two
reviewers Alan Reid and Dhananjay Ravat. We also had nuof complexnumberswj = (x - xj) + i(zj + a) (and merousdiscussionswith our colleaguesMatthias Holschneiw•+•: w•). This is usedto transformequation(C4) der, Guillaume Martelet, and Ginette Saracco. This is IPGP
into
contribution
number
NS 1680.
-('7 + 1)a'•
' a)
W,Zi,T(.,z=0)(x
--(3'
--1)(3'
--2)(SinI'
si,•z
)2½i(-2I'+3•)
,• References
'Z e-2i•(w3+•-w3)
-[e-i('-2)•(w3+•)
]' pp., Gebruder Borntraeger, Stuttgart, Germany, 1975.
Iw+l
Baranov, W., Potential Fields and Their Transformations in
Applied Geophysics, Geoexplor. Monogr. Ser., vol. 6, 121
(c5)
where(I)(w)- Arg(w) is the argumentof w.
Notation
function
f(x)
of a real variable
x.
Fourier transform of f(z), equal to
f_+•dxf(x)e-i•',•'
Hilbert
transform
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dilation
operator
(Pal(x):
.x)
(1/a)f(x/a)).
Poissonkernel(upwardcontinuation
+1),
to
dilated Poissonkernel (upwardcontinuation filter, at level +a), equal
to PaP(X).
i
square root of -1.
partial differential operator with
spect to x.
½J(x) real wavelet(of order•) whosefirst
deriwtive
is in x.
real wavelet(of order•) whosefirst
deriwtive
is in z.
complexwavelet(of orderif).
wavelet(of order•) dilated by dilation
]/V•pVl½o
(x, a)
a.
waveletcoe•cient of •0 (by convolu-
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onW½•l•(.,•)(x, a)
volutionof •(x,) with •(x), for
xe•).
•½•l•(.,•)(x,a) argument(or phase)of the wavelet
coe•cientWZ•,(.,•)(x, a).
r(x)
Gamma
function
of the real wriable
x
conjugateof the complexwriable w.
inclination
and
declination
of the
magnetization vector.
In, Dn
inclination
and
field.
declination
of the
normal
apparentinclinationof the magnetization
vector.
apparent inclination of the normal
field.
mean apparent inclination, equal to
+
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A.
Galdeano
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P.
Sailhat,
Labora-
toirede G•omagn•tisme(UMR 7577du CNRS), Institut de
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France.(galdeano@ipgp.jussieu.fr;
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D. Gibert andF. Moreau,Gaosciences
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(ReceivedMarch 17, 1999; revisedJanuary3, 2000;
acceptedMarch 15, 2000.)