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GEOPHYSICS, VOL. 72, NO. 1 �JANUARY-FEBRUARY 2007�; P. L1–L12, 7 FIGS., 5 TABLES.10.1190/1.2400010
egrees of homogeneity of potential fields andtructural indices of Euler deconvolution
etar Stavrev1 and Alan Reid2
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ABSTRACT
Homogeneity is a well-known property of the potentialfields of simple point sources used in field inversion. We findthat the analytical expressions of potential fields created bysources of complicated shape and constant or variable densi-ty or magnetization also show this property. This is true if allvariables of length dimension are involved in the test of ho-mogeneity. The coordinates of observation points and thesource coordinates and sizes form an extended set of vari-ables, in relation to which the field expression is homoge-neous. In this case, the principal definition of homogeneityapplied to a potential field can be treated as an operator of aspace transform of similarity. The ratio between the trans-formed and original fields determines the value and sign ofthe degree of homogeneity n. The latter may take on positive,zero, or negative values. The degree of homogeneity dependson the type of field and on the assumed physical parameter ofthe field source, and can be nonunique for a given field ele-ment. We analyze the potential field of one singular point asthe simplest case of homogeneity. Thus, we deduce results forthe structural index, N = −n, in Euler deconvolution. Thestructural index can also be positive, zero, or negative, but ithas a unique value. Analytical considerations, as well as nu-merical tests on the gravity contact model, confirm the pro-posed physical interpretation of N, and lead to an extendedversion of Euler’s differential equation for potential fields.
INTRODUCTION
Euler’s differential equation for homogeneous functions hasound wide application as a theoretical basis for the inversion ofarge magnetic and gravity data sets in terms of simple sourcesThompson, 1982; Reid et al., 1990; Marson and Klingele, 1993;
Manuscript received by the Editor September 26, 2003; revised manuscrip1University of Mining and Geology, St. Iv. Rilski, Department ofApplied G2Reid Geophysics Ltd., 49 Carr Bridge Drive, Leeds LS16 7LB and Unive
-mail: [email protected] Society of Exploration Geophysicists.All rights reserved.
L1
hang et al., 2000; Reid et al., 2003�. Homogeneity can also be es-ablished and put to practical use by the direct application of its orig-nal definition.
Afunction is homogeneous if all of its terms have an equal sum ofxponents. This follows J. Bernoulli’s and Euler’s concept of homo-eneity �see Euler, 1936, p. 93�, which was later expressed by thequivalent form
f�tv1,tv2, . . . ,tv j� = tnf�v1,v2, . . . ,v j� , �1�
here f�v1,v2, . . . ,v j� is a function of a set of variables v =v1,v2, . . . ,v j�, j is an arbitrary number of variables, t is a real num-er, and n is the degree of homogeneity of f�v�, e.g., Courant andohn �1965�. For such functions, if they have a differential at v, theollowing Euler’s partial differential equation �Euler, 1949, p. 157�olds:
v1�f /�v1 + v2�f /�v2 + ¯ + v j�f /�v j = nf�v1,v2, . . . ,v j� .
�2�
onversely, if a function satisfies equation 2, then it is a homoge-eous function of v1, v2, . . . ,v j according to definition 1 �Courant andohn, 1965�.
�The term homogeneous field is also used when portions of a sca-ar or vector field in some part of the space are constant.�
The property of potential fields expressed by equations 1 and 2an be referred to as Euler homogeneity. Thus, Euler’s equation 2 issed as a condition for homogeneity in contrast to the other mean-ngs attributed to the term homogeneity.
The concept of homogeneity finds application in methods forodeling physical phenomena and, in particular, for solving direct
nd inverse problems for potential fields. The analytical expressionssed in these problems may or may not be homogeneous with re-pect to different sets of variables. A given expression is homoge-eous if it satisfies either the defining equation 1 or equation 2 for aertain combination of variables. These conditions were tested on annalytically derived expression of gravity and magnetic field ele-
ed September 7, 2006; published online December 29, 2006.sics, Sofia 1700, Bulgaria. E-mail: [email protected], School of Earth and Environment, Leeds LS2 9JT2, United Kingdom.
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ents for some simple sources �Stavrev, 1997�. In this paper, the ho-ogeneity test is applied to the general integral expressions for the
ravity and magnetic potentials and their spatial derivatives. Weonsider a set of variables having the dimension of length. They areoordinates of the observation points as well as geometrical sourcearameters.
Equation 1 is a much easier way to analyze homogeneity than dif-erential equation 2, which requires a lot of tedious and error-proneifferential calculus. The left-hand side, f�tv�, of equation 1 can beiewed as an operator acting on the set v of all independent variablesith dimension of length, and generates a similarity transform in the
pace. The ratio f�tv�/f�v� determines the sign and value of the de-ree of homogeneity. Indeed, by equation 1,
n = ln�f�tv�/f�v��/ln t �3�
hen t�0 and t�1. The coefficient of similarity t can be taken to belimited number greater than 1, without invalidating the generalityf the above statements.
The multiplication of all variables of length dimension in the left-and side of equation 1 by a given number t is a manipulation of thenalytical expression that affects the observations and the sources asell. The scaling factor t is a coefficient of similarity between theriginal source and the transformed source. The latter copies thehape of the original source, but in another scale and position. Thus,straight line shifts to a straight line, the angle between two lines isreserved, the length measure increases by factor t, the surface mea-ure by t2, and the solid measure by t3. During the above manipula-ion, if we assign the original density distribution to the similaroints of the transformed source, then the ratio between the originaleld and the transformed field will be determined by tn, where n rep-esents the degree of homogeneity according to equations 1 and 3.ere, we assume the mass and magnetic moment, and their line, sur-
ace, and volume densities are limited in value. It is also assumedhat they are continuously �or piecewise-continuously� distributedhysical parameters of limited variations in the analyzed expres-ions of potential fields.
With these conditions in mind, we study the geometrical andhysical meaning of homogeneity shown by the general analyticalxpressions for gravity and magnetic fields. The studies are both an-lytic and numeric.
ANALYSIS OF EULER HOMOGENEITYOF GRAVITY FIELD ELEMENTS
otential of the field of a set of point masses
The gravity field of a point mass has the potential V,
V = �m/r , �4�
here � is the gravitational constant, m is the mass, and
r = ��x − x0�2 + �y − y0�2 + �z − z0�2�1/2 �5�
s the distance between an observation point P�x,y,z� and the loca-ion M�x ,y ,z � of the mass �e.g., Blakely, 1995�. In expression 4,
0 0 0he distance r is variable with the dimension of length. It is definedy the components rx = �x − x0�, ry = �y − y0�, and rz = �z − z0� ofhe distance vector r from point M to point P, as well as by the ele-
entary independent variables x, y, z, x0, y0, and z0, in equation 5.he tests of homogeneity for the potential in equation 4 can be im-lemented using equation 1 and taking into account equation 5 to ob-ain
V�tr� = �m/�tr� = t−1V�r� , �6�
V�trx,try,trz� = �m/��t�x − x0��2 + �t�y − y0��2
+ �t�z − z0��2�1/2 = t−1V�rx,ry,rz� , �7�
V�tx,ty,tz,tx0,ty0,tz0� = �m/��tx − tx0�2 + �ty − ty0�2
+ �tz − tz0�2�1/2
= t−1V�x,y,z,x0,y0,z0� . �8�
hese equations show that the coordinates x, y, z, x0, y0, z0 form theull analytical set of independent variables of homogeneity for theravity potential V with a degree of homogeneity n = −1. The minusign implies
V�tx,ty,tz,tx0,ty0,tz0� � V�x,y,z,x0,y0,z0� . �9�
See equation 3 for t�1�. Obviously, the potential at pointP*�tx,ty,tz�, created by a mass m located at point M*�tx0,ty0,tz0�, has
lower value than the original potential at point P�x,y,z�. Physical-y, equation 9 is explained by the larger distance tr�r, although the
ass m remains unchanged. Geometrically, the points P* and M* aremages of similarity of the original points P and M �Figure 1�.
If we accept the condition 0� t�1, then inequality 9 would pro-uce the reverse ��� result. However, the degree of homogeneity re-ains the same, namely n = −1. Thus, the condition t�1 is suffi-
ient for further analysis of the degree n.Aset of point masses creates a gravity field with potential,
V = �Vi = ��mi/rMiP, i = 1,2, . . . , j , �10�
here j is the number of masses, rMiP is the distance between the ob-ervation point P�x,y,z� and the point Mi�x0i,y0i,z0i� locating theass mi. Expression 10 is homogeneous of degree n = −1 with re-
pect to the set of all coordinates, v = �x,y,z,x01,y01,z01 . . . ,x0j,y0j,0j�, because each potential Vi is homogeneous of degree n = −1 byquations 6–8. Thus, all terms in equation 10 satisfy the principal re-uirement of an equal degree of homogeneity. The sign of n showshat V�tv��V�v�. The masses mi, i = 1,2, . . . , j, do not change theiralues in this homogeneity test, i.e., mi
*�tx0i,ty0i,tz0i� = mi�x0i,y0i,0i�.
ield potential of a line mass
Assume that a line mass is distributed continuously at a constantr variable density � along a curved line L. Then, the potential V ofts gravity field can be expressed by the well-known integral forms
V = ���L�
dm/rMP �11�
nd
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Homogeneity of potential fields L3
V = ���L�
�dl/rMP, �12�
here P�x,y,z� is the observation point, M�x0,y0,z0� is a point of theine source L, and rMP is the distance between points P and M accord-ng to equation 5.
The subintegral expression of the integral 11 is a homogeneousunction of the distance rMP, like the potential of a point mass �seequations 4–8�. As a limit of the sum of an infinite number of infini-esimal terms of equal degree of homogeneity n = −1, the first inte-ral has the same degree of homogeneity �see equation 10 for exam-le�. A space transform of similarity shifts the original source L to aransformed source L* with the same mass, because this physical pa-ameter does not change its value in the above test of homogeneity.
The second integral 12 has a subintegral expression containingwo elements of length dimension, rMP and dl. Their ratio, dl/rMP, is a
0 1
1
2
P P *
Mm
M *m
2 3 4 5 6 7 8 9 10
a)35
30
25
20
15
10
5
0
Pot
entia
l (m
Gal
.km
)
x (km)
0 1 2 3 4 5 6 7 8 9 10
b)0
1
2
3
4
5
Dep
th z
(km
)
x (km)
igure 1. Similarity transforms of a point mass and the potential ofts gravity field at coefficient of similarity t = 2. �a� Profiles of theravity potential V �curve 1� from the original mass and the potential* �curve 2� from the similar mass. �b� Original mass m in point�3,2� and the similar mass m* = m in point M*�6,4�. Point P*�0,8�
s the similar image of the original observation point P�0,4�. In point*, the calculated gravity potential V* = 15 mGal.km is half theriginal potential V = 30 mGal.km at point P, thus showing degreef homogeneity n = −1 �V* = 2−1V�.
imensionless quantity, so the degree of homogeneity of the subinte-ral expression is n = 0. Hence, the potential V obtains the same de-ree of homogeneity, n = 0. In this case, the line density ��M� doesot change its value in a space transform of similarity, i.e., ��M*���M�, where M* is the similar image of the original point M of the
ine source L.Evidently, the potential V in expressions 11 and 12 can exhibit dif-
erent degrees of homogeneity. These correspond to various ways inhich the potential can be expressed whenever the mass is distribut-
d continuously. When the mass parameter is assumed, as in expres-ion 11, then n = −1. In terms of the similarity transform, this means*�P*��V�P�, because of the larger distances rM*P and length of L*
t the same mass of L* as the original mass of L. When the line densi-y is used, as in expression 12, then n = 0. In this case, the mass of L*
s greater than the mass of L, as the density �* = � on a longer line L*
see Figure 2�. This compensates for the effect of the increased dis-ances, so that V*�P*� = V�P�. The degree n = 0 reflects this equali-y.
0 1
1 2
P P *M1
M1*
M2
L
L*
l *
l
M2*
2 3 4 5 6 7 8 9 10
a) 100
80
60
40
20
0
Pot
entia
l (m
Gal
.km
)
x (km)
0 1 2 3 4 5 6 7 8 9 10
b)0
1
2
3
4
5
6
7
Dep
th z
(km
)
x (km)
igure 2. Similarity transforms of a vertical rod and the potential ofts gravity field at coefficient of similarity t = 2. �a� Profiles of theotential V �curve 1� from the original rod L and the potential V*
curve 2� from the similar rod L*. �b� Original rod from pointsM1�3,1� to M2�3,3� and the similar rod from points M1
*�6,2� toM2
*�6,6� at the same density, �* = �. A segment l* of L* is the similarmage of the segment l of L. Point P*�0,8� corresponds to the originalbservation point P�0,4�. The calculated potential V*�P*� =6 mGal.km is equal to the original potential V�P�. It means n =�V* = 20V�.
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The difference between the two possible degrees of homogeneity,= −1 and n = 0, follows the relationship, � = dm/dl, between the
wo physical parameters. This relationship contains one element ofength dimension. Note that the integral expressions 11 and 12 re-ect the degree of homogeneity without the need to perform the inte-ration. This assumes the above condition requiring a well-behavedass distribution.The general inferences made here are confirmed by the analytical
nd numerical tests on the gravity potential of a line mass model withonstant and variable density described in Appendix A. Appendix Bhows similar basic results for a line mass with a logarithmic 2D po-ential.
ield potential of a surface mass
A surface mass is distributed continuously at constant or variableensity � on a surface S. Its gravitational potential is
V = ���S�
��/r�ds �13a�
r
V = ���S�
dm/r . �13b�
omogeneity analysis can be performed by analogy with the abovease of a line mass. For the first integral 13a, we obtain a degree ofomogeneity of n = −1 + 2 = 1, taking into account the factor r−1
nd a 2D integration area ds. For the second integral expression 13b,he factor r−1 gives a degree of n = −1. These results do not dependn the assumed functional forms of mass distribution along the sur-ace S. The example below illustrates these conclusions.
Consider a potential V of a spherical shell with radius R, constantensity �, and mass m = �4�R2, centered at the point M�x0,y0,z0�,
V = �m/r = ��4�R2/��x − x0�2 + �y − y0�2 + �z − z0�2�1/2,
�14�
here r�R. The degree of homogeneity of the first expression is n−1 with respect to the distance r, or the set of coordinates v�x,y,z,x0,y0,z0�, when m keeps its value. However, the expression
fter the second equality �=� using the full set of geometric vari-bles, including radius R, shows a degree of homogeneity n = 1, astR�2/�tr� = t1R2/r. In this case, V�P*��V�P� because of the nowarger mass m* = �4��tR�2 = t2m.
ield potential of a solid mass
Assume that a solid mass is distributed continuously at a constantr variable density � in a domain D. The resulting potential is
V = ���D�
��/r�d �15a�
r
V = ���D�
dm/r �15b�
e.g., Blakely, 1995, p. 46�. The first integral 15a is homogeneous ofegree n = 2, because the factor r−1 contributes a degree of −1, andhe 3D volume element d contributes a degree of 3, so that we have
= −1 + 3 = 2. The second integral 15b has a degree of homogene-ty n = −1, determined only by the factor r−1.
These results can be illustrated with the example of a solid spheri-al body of radius R, constant density �, and mass m = ��4/3��R3,entered at M�x0,y0,z0�. Outside the sphere, the gravity potential is
V = �m/r = ���4/3��R3/��x − x0i�2 + �y − y0i�2
+ �z − z0i�2�1/2. �16�
his expression is homogeneous of degree n = −1 with respect tohe distance r, or to the coordinates x, y, z, x0, y0, z0. Relative to theomplete set of geometric variables, v = �x,y,z,x0,y0,z0,R�, theightmost expression in equation 16 has a degree of homogeneity n
2. This positive degree shows that V�tv��V�v� because of the in-reased mass m* = ��4/3���tR�3 = t3m of the enlarged sphere. Inhis case, the transformed source has the same density � as the origi-al source, i.e., �* = �.
D potential
In Appendix B we discuss the basic 2D case of a line-mass poten-ial.A2D solid mass with density �, and a 2D surface mass with den-ity �, create fields with potentials
V = 2�����
� ln�1/r�ds ,
V = 2����
�ln�1/r�dl , �17�
and V = 2�����
ln�1/r�d� ,
here � is the cross section of a solid body, is an open or closedine of the cross section of a 2D material surface, and � is � or , re-pectively. The logarithmic function has a degree of homogeneityqual to 0 �Appendix B�. Thus, the first integral yields a degree of ho-ogeneity n = 2, the second n = 1, and the third n = 0. These re-
ults agree with those obtained above for the cases of the 3D poten-ials of a solid, a surface, and a line mass, respectively, when the as-umed physical parameter is the density. When the physical parame-er is the partial mass of segments of equal length along the 2Dource, then n = −1 �Appendix B�. Therefore, the 2D case does notiffer from the 3D case when one accounts for the entire source ge-metry, and not just for the source’s normal cross section. In supportf this conclusion, Figures 3 and 4 depict numerical tests for 2Dodels of thin and thick contacts, respectively.
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Homogeneity of potential fields L5
omogeneity of gravity potential derivatives
If the potential V is homogeneous, then so is any derivative of VBlakely, 1995�. The degree of homogeneity nk of the derivativekV/�x��y �z�, where � + + � = k, decreases by the number kith respect to the degree n0 of the potential V, i.e.,
nk = n0 − k , �18�
here k is the order of the derivative, k = 0,1,2,3, . . . . This relationhows the increasing exponent of the reciprocal distance 1/r as therder k increases. The derivatives of a given order k at different val-es of �, , and �, all have equal degrees of homogeneity. The deriv-tives of V with respect to the geometric variables of the source alsoxhibit regularity of equation 18 �see also equations A-4 in Appen-ix A�.
0 1
1 2
P P *
g *g
M*
M
S
S *
2 3 4 5 6 7 8 9 10
a) 6
5
4
3
2
1
0
g (m
Gal
)
x (km)
0 1 2 3 4 5 6 7 8 9 10
b)0
1
2
3
4
5
Dep
th z
(km
)
x (km)
igure 3. Similarity transforms of a semi-infinite horizontal sheetnd the vertical gravity component g at coefficient of similarity t2. �a� Profiles of the component g �curve 1� from the original sheet
nd the component g* �curve 2� from the similar sheet. �b� Originalheet with edge point M�3,2� and the similar sheet with edge point
*�6,4� at surface density �* = �. Point P*�0,8� is the similar imagef the original observation point P�0,4�. In point P*, the calculatedravity component g* = 4.06 mGal is equal to the original compo-ent g at point P, thus showing degree of homogeneity n = 0 �g*
20g�.
generalized expression for the degree of homogeneity
The results we have obtained so far for the degrees of homogene-ty of the gravity potential and its derivatives can be combined in aommon expression,
ng = p − k − 1, �19�
here k is the order of the gravity field element in terms of the deriv-tive of the potential V, and p is an integer whose value indicates theype of physical parameter in the analytical expression of the field el-ment. Thus, the integer p has value: p = 0 for the mass parameter,
p = 1 for the line density, p = 2 for the surface density, and p = 3 forhe density of a solid mass. For p between 0 and 3, the degree ng var-es between �−k − 1� and �−k + 2� in accordance with equation 19.able 1 shows ng values of the gravity field elements for orders k�3.
0 1
1
2
P
P *
g *
g
M1
M2 M1*
M2*
2 3 4 5 6 7 8 9 10
a)10
9
8
7
6
5
4
3
2
1
0
g (m
Gal
)
x (km)
0 1 2 3 4 5 6 7 8 9 10
b)–2
–1
0
1
2
3
4
z (k
m)
x (km)
igure 4. Similarity transforms of a vertical contact and the verticalravity component g at coefficient of similarity t = 2. �a� Profiles ofomponent g �curve 1� from the original contact and the component* �curve 2� from the similar contact. �b� Original contact with edgeoints M1�3,1� and M2�3,2�, and the similar contact with edge points
M1*�6,2� and M2
*�6,4� at density �* = �. Point P*�−2,6� is the similarmage of the original observation point P�−1,3�. In point P*, the cal-ulated component g* = 6.28 mGal is double the original compo-ent g = 3.14 mGal in point P, thus showing degree of homogeneity= 1 �g* = 21g�.
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L6 Stavrev and Reid
The parameter p in equation 19 can be treated in terms of homoge-eity as the negative degree of homogeneity of the density distribu-ion when the mass is the assumed physical parameter, i.e., when m*
t 0m. From the analysis above, �* = t−1�, �* = t−2�, �* = t−3�,here �, �, � are the line, surface, and solid density of the original
ource, and �*, �*, �* are the densities of the respective transformedource. If the density is the assumed physical parameter, then theass parameter undergoes transform, m* = tpm, in the test of homo-
eneity. Thus, the physical parameter distribution can be accepted ashomogeneous function with a specific conditional degree of homo-eneity 0 for the assumed parameter, and p or −p for the alternativearameter. R. O. Hansen �personal communication, 2005� addressedhis problem.
omogeneity of the sum of analytical expressionsf a gravity field element
Consider a field element produced by collections of sources of dif-erent geometry, i.e., by point, line, surface, and solid sources. Theirummary field can be homogeneous if the four potentials have theame degree of homogeneity.According to the above analysis, this isossible when the physical parameter in the appropriate analyticalxpressions is a mass parameter with index p = 0. In this case, theour potentials yield n = −1. The respective common degree n of aeld element of higher order k is ng = −k − 1, which is the minimalegree of homogeneity as indicated by equation 19.
omogeneity as a precondition forotential field transforms
The definition of homogeneity by equation 1 can be treated as aasic expression of a field transform if all observations and geomet-ic variables of length are involved. Equation 8 is an example of suchtransform. Figures 1–5 illustrate model transforms of this type.hey are space transforms of similarity with respect to a given cen-
ral point O, coefficient of similarity t, and degree of homogeneity n.f a data set of gravity �or magnetic� anomaly A�x,y,z� is given, thents direct similarity transform A�tx,ty,tz� is given by
able 1. Degrees of homogeneity ng of gravity field elements.
hysicalarameter in thenalyticalxpressions ofhe field elements
Index, p, of thephysical parameter
0
Poten
uantity of mass�kg�
0 −1
ine density�kg/m�
1 0
urface density�kg/m2�
2 1
olid density�kg/m3�
3 2
A�tx,ty,tz� = tnA�x,y,z� . �20�
he transformed anomaly A�tx,ty,tz� at shifted points P�tx,ty,tz�orresponds to the shifted original source with coefficient of similar-ty t.Assigning n is equivalent to choosing index p �equation 19, andquation 22, below�, where the order k of data A is a known number.hus, a desired transfer of masses �or dipoles� can be implementedsing the field transform by equation 20. This physical aspect of theransform can be used as a basis of inversion techniques for potentialelds. Suitable tools for such inversions are the finite-difference andifferential similarity transforms �Stavrev, 1997�. They are func-ions of the differences between the transformed field and the origi-al field at a set of common points.
ANALYSIS OF EULER HOMOGENEITYOF MAGNETIC FIELD ELEMENTS
The basic expression is the potential U of a dipole at the pointM�x0,y0,z0�,
U = Cm��M�1/r� = − Cm��P�1/r� , �21�
here r is the distance between point M and observation pointP�x,y,z�, � is the magnetic moment, and Cm is a factor depending onhe measuring units �e.g., Blakely, 1995�.
The test for homogeneity with respect to the geometric variableshows that the degree of homogeneity of expression 21 is n = −2,ecause 1/r has a degree of homogeneity �−1�, and the �-operatorontributes �−1�. The magnetic moment �, as a vector physical pa-ameter, is not a variable of homogeneity in this test.
As a consequence, the degree of potential U is n = −2, which is 1ess than the degree n = −1 of the potential V of a point mass �equa-ions 6–8�. The potential of more complicated dipole fixed distribu-ions is calculated by integrating potentials dU = Cmd��M�1/r� ofoint dipoles �equation 21�. Note that the gravity potential is calcu-ated by integrating dV = �dm/r of point masses �equation 15�.ence, for magnetic cases, the degree of homogeneity is always one
ess than the equivalent gravity case because of the �-operator. So,e can use equation 19 to produce an equivalent relation for magnet-
c cases, namely,
Gravity field elements of order k
1 2 3
Strength,components
Gradients,curvatures
Thirdderivatives
−2 −3 −4
−1 −2 −3
0 −1 −2
1 0 −1
tial
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pgt
wDrt�e
coptfon
dim�dtsrB
wfidl
FsfifOtlca�
Homogeneity of potential fields L7
nm = ng − 1 = p − k − 2, �22�
here p is an integer specifying the type of magnetic parameter inhe analytical expression, so that p = 0 for the magnetic moment, p
1 for the line density of dipoles, p = 2 for the surface density of di-oles, and p = 3 for the volume density of dipoles �magnetization�; ks the order of derivative of the potential U. This index p can be treat-d as a specific conditional degree of homogeneity of the magneticarameter distributions, as in the gravity case above.
Table 2 shows the nm values of magnetic field elements of order k3. For the possible values of index p, the degree of homogeneity
aries within the limits �−k − 2� and �−k + 1�. By analogy with theravity case, the minimum degree, �−k − 2�, is the common degreef homogeneity of interfering fields of point, line, surface, and solidipole distributions. Figure 5 shows graphically the numerical testith magnetic anomaly �T�k = 1� of a 2.5D solid source of arbi-
rary polygonal cross section with constant magnetization �p = 3�.he observed degree of homogeneity is nm = 0 as predicted by for-ula 22.
STRUCTURAL INDICES OFEULER DECONVOLUTION
Euler deconvolution �after Reid et al., 1990� is widely used. It as-umes that, at least locally, any source body giving rise to an anoma-ous magnetic or gravity field is simple enough to be represented byne singular point �e.g., point mass or dipole, line source, sheet edge,r thick contact top�. Recently, proposals were made to deal with theeld arising from a source body with two singular points �a thick lay-r, limited in depth contact, dike, rod, etc.� by similarity transformsStavrev, 1997�, and for a multiple-source Euler deconvolutionHansen and Suciu, 2002�.
Suppose gravity or magnetic data are given at a set of observationoints P�x,y,z�. The data contain the anomalous field A with one sin-ular point M�x0,y0,z0�. Then A satisfies equations of type 6–8 andhe corresponding Euler’s differential equations
rdA/dr = nA , �23�
rx�A/�rx + ry�A/�ry + rz�A/�rz = nA , �24�
x�A/�x + y�A/�y + z�A/�z + x0�A/�x0 + y0�A/�y0
+ z0�A/�z0 = nA , �25�
here n �ng or nm� is the degree of homogeneity of A �Tables 1 and 2�.ifferential equation 23 shows directly the field rate along a given
adial axis r originating from the singular point M. Expression 25 ishe most detailed differential equation, where �A/�x0 = −�A/�x,A/�y0 = −�A/�y, �A/�z0 = −�A/�z. These equalities transformquations 24 and 25 into
�x − x0��A/�x + �y − y0��A/�y + �z − z0��A/�z = nA ,
ontaining only the derivatives with respect to the coordinates of thebservation points. This is a case of a simplified homogeneity, com-ared to the more complicated cases considered above and below inhis section. Field inversion for source coordinates using Euler’s dif-erential equation becomes practical because we can use calculatedr measured gradients to obtain derivatives of the source coordi-ates.
Table 3 shows a formal conversion of the general results for theegree n in Tables 1 and 2, to the structural index, N = −n, of simplenterpretation models. Traditionally, index N is given for the total
agnetic anomalies �T, as it was first introduced by Thompson1982�. The other field elements obtain an index equal to N plus or-er k of the �T �or gravity �g� derivatives. In contrast to this prac-ice, the full value of the structural index of each field element ishown in Table 3. Thus, the values of index N can be generated di-ectly, like the degree n in equations 19 and 22, and Tables 1 and 2.y analogy, index N in Table 3 is given by
N = k + s − d , �26�
here s = 1 for a field element created by masses, and s = 2 for aeld element created by dipoles. Here, d = 0 for a point mass, pointipole, and the equivalent spherical sources; d = 1 for line masses,ine dipoles, and the equivalent cylindrical models; d = 2 for a plate
0 1
21
P *
B *
B
P
O 1
1
2
2
2 3 4 5 6 7 8 9 10
a)200
100
0
–100
–200
Ano
mal
ies
(nT
)
x (km)
0 1–1–2 2 3 4 5 6 7 8
b)–2
–1
0
1
2
3
4
z (k
m)
x (km)
igure 5. Similarity transforms of a complex-shaped 2.5D magneticource and anomaly �T along an uneven line of observation at coef-cient of similarity t = 1.5. �a� Profiles of anomaly �T �curve 1�rom the original source and from the similar source �curve 2�. �b�riginal source B �contour 1� and the similar source B* �contour 2� at
he same magnetization vector, J* = J. Point P*�1.5,−3� is the simi-ar image of the original observation point P�1,−2�. In point P*, thealculated magnetic anomaly �T* = 167 nT is equal to the originalnomaly �T in point P, thus showing degree of homogeneity n = 0�T* = 20�T�.
ammdlsvsup
lm
cvoapsoette
s
T
Ppaete
D�
L
S�
SJ
Te
t
Pe
Lmec
Tsc
Cdp
at
L8 Stavrev and Reid
nd the equivalent thin layer, dike, and step; and d = 3 for the contactodel. Index d corresponds to the geometrical dimension of the ele-entary sources. The equivalent sources above have the same indexbecause their sizes �radius or thickness� are coefficients in the ana-
ytical expressions and cannot be determined uniquely in an inver-ion of the field. The deductive result in equation 26 gives the samealues of index N as that known from the particular analysis of pointource models. According to equation 26, the structural index N hasnique values in contrast to the degree of homogeneity n, which de-ends on the type p of the accepted physical parameter.
Table 3 consists of positive, zero, and negative values of N. Theatter in all cases correspond to the theoretical models with indeter-
inate ��� potential and intensity. For them, d − k�1 in the gravity
able 2. Degrees of homogeneity nm of magnetic field element
Magnetic field elem
hysicalarameter in thenalyticalxpressions ofhe fieldlements
Index pof the
physicalparameter
0 1
PotentialStrength,
components
ipole moment�Am2�
0 −2 −3
ine density�Am�
1 −1 −2
urface density�A�
2 0 −1
olid density�A/m�
3 1 0
able 3. Structural index N in Euler’s differential equation folements.
Gravity/magnetic el
Gravity/magneticheoretical models
Geometricaldimensiond of themodel
0 1
PotentialStrength,
components
oint mass/dipolequivalent sphere
0 1/2 2/3
ine ofass/dipole,
quivalentylinder
1 0�/1 1/2
hin semi-infiniteheet, dike, sill,ontact, step
2 −1�/0 0�/1
ontact of infiniteepth extent,yramid
3 −2�/−1� −1�/0
Note: For some model varieties ��� or for all model varieties ��� thre not determinate, and N corresponds to the closest determinate ment.
ase, and d − k�2 in the magnetic case �see formula 26�. The zeroalue of N appears for some theoretical models with indeterminater determinate field elements, where d − k = 1 in the gravity case,nd d − k = 2 in the magnetic case. For example, the vertical com-onent of gravity intensity �k = 1� of a thin, semi-infinite horizontalheet �d = 2� is determinate with index N = 0, although the slopingr vertical sheet does not have determinate vertical intensity. How-ver, all varieties of the same magnetic model have determinate po-ential and intensity �see Table 3�. The most specific in this respect ishe gravity contact model of infinite depth extent that shows the low-r formal index N = −2.
The physical sense of the positive N is well known from Thomp-on’s �1982� functional form f�x,y,z� = G/r N, where r is the dis-
tance in equation 5, and G is not dependenton x, y, z. If N�0, then f shows a naturalattenuation with distance r. If N = 0, thenf = G, which is possible for some modelssuch as the gravity semi-infinite horizontalsheet �small step�. But the case N�0 can-not be explained using Thompson’s form.The negative structural indices in Table 3may gain meaning in terms of the conceptof extended homogeneity. The direct useof equations 1 and 3 is not possible for thispurpose because the theoretical field ele-ments are indeterminate and cannot becompared. However, an infinitesimal ap-proach from a determinate model to the re-spective indeterminate one gives a reason-able answer to the problem. Such a mathe-matical approach corresponds to the reali-ty where measured potential-field ele-ments are fully determinate, so their sourc-es may only approximate the theoreticalindeterminate models. The synthetic testsshown below confirm the appearance ofnegative structural index N for the modelof gravity contact.
Table 3 shows both well-known and lit-tle-known �but useful� regularities. Thestructural index N has equal value for allfield elements of equal order k caused bysources of equal index d. Such elementsare the magnetic field components X,Y, Z, H, magnitude T, and anomalies�T�k = 1�, and all members of the gravityor magnetic gradient tensors �k = 2�, alsoHilbert transforms �the latter shown byNabighian and Hansen, 2001�. In all cases,N is higher by 1 for a magnetic model thanN for the same gravity model.
Synthetic tests for negativeindices N
Consider the gravity model of a verticalcontact �Figure 4b�, creating vertical grav-ity attraction g defined by the analyticalexpression
f order k
3
ntsThird
derivatives
−5
−4
−3
−2
ity/magnetic field
s of order k
3
ents,ors
Thirdderivatives
4/5
3/4
2/3
1 1/2
etical field elementsf a considerable ex-
s.
ents o
2
Gradie
−4
−3
−2
−1
r grav
ement
2
Graditens
3/4
2/3
1/2
0�/
e theorodel o
wzoon→r
v
wztt
Bm
w−sMe
ot
wc
Aszs
T
Ap
viNlcstb7Nt−f
FgaCt�fn
Homogeneity of potential fields L9
g = �����z2 − z1� + 2�z2 − z�tan−1��x − x0�/�z2 − z��
− 2�z1 − z�tan−1��x − x0�/�z1 − z�� + �x − x0�
�ln���x − x0�2 + �z2 − z�2�/��x − x0�2 + �z1 − z�2��� ,
�27�
here x, z are the coordinates of observation points, and x0, z1 and x0,
2 are the coordinates of the upper M1 and the lower M2 edge pointsf the contact �e.g., Telford et al., 1990�. The degree of homogeneityf anomaly g with respect to all geometric variables �x, z, x0, z1, z2� is= 1 according to the definition by equation 1 �see Table 1�. If z2
� �infinite thickness�, then g→� �indeterminate value�. Table 3epresents this case by negative N = −1.
Differential equation 2 for anomaly g with respect to all geometricariables is
x�g/�x + z�g/�z + x0�g/�x0 + z1�g/�z1 + z2�g/�z2 = ng ,
�28�
here �g/�x0 = −�g/�x, and �g/�z1 + �g/�z2 = −�g/�z. When
2�z1 and �x��z2, the terms z1�g/�z1 and z2�g/�z2, which containhe unknown derivatives �g/�z1 and �g/�z2, can be approximated byhe equations
z1�g/�z1 = − z1�g/�z − ����z1 + 2���z1/z2��x − x0��;
�29�
z2�g/�z2 = ����z2 + 2���x − x0�� . �30�
y a substitution of these expressions in equation 28 and after simpleanipulations, Euler’s equation takes the form
x0�g/�x + z1�g/�z = N2g + x�g/�x + z�g/�z
+ �����z2 − z1� + 2���x − x0�� ,
�31�
here N2 is an approximation to the negative structural index N =n = −1. The bracketed term � .� must be taken into account when
olving the inverse problem of determining the upper edge point
1�x0,z1�. This term contains unknown variables but can be includ-d by using its linear character.
If one accepts the contact model of considerable depth extent as ane-point source with singular point M1�x0,z1�, then Euler’s equa-ion 2 should be written as
x0�g/�x + z1�g/�z = N1g + x�g/�x + z�g/�z , �32�
here N1 is the supposed structural index. The value of this indexan be estimated analytically from equation 32 by the expression
N1 = ��x0 − x��g/�x + �z1 − z��g/�z�/g . �33�
simple test at �x0 − x� = 0 when �g/�z = 0 and g = ����z2 − z1�hows N1 = 0. But this is the structural index of a semi-infinite hori-ontal sheet model �Table 3�. Hence, equation 32 does not corre-pond to the contact model of considerable depth extent.
A similar test can be carried out for the index N2 in equation 31.aking into account equations 31–33,
N2 = N1 − ����z2 − z1�/g − 2���x − x0�/g . �34�
t �x0 − x� = 0 when g = ����z2 − z1�, we obtain N2 = −1 as isredicted in Table 3 for the solid contact model.
Equations 31 and 32 correspond to the models of very thick andery thin contact structures, respectively. If the real structure has anntermediate relative thickness w = �z2 − z1�/�z1 − z�, then N1 and
2 values in these equations deviate from the expected ones. Calcu-ated results from equations 33 and 34 along model profiles g aboveontacts are shown in Figures 6a and 7a, respectively. At a givenmall thickness w = 0.02, index N1 maintains expected values closeo 0. For a thick contact with w = 4, index N1 changes significantlyetween −0.125 and 0.5 along the tested profile. Index N2 �Figurea� shows similar properties. At a considerable thickness, w = 19,2 is close to the expected negative index −1. At an intermediate
hickness, w = 4, index N2 undergoes considerable changes between1.2 and −0.1. These changes indicate a deviation of the real source
rom the accepted model.
0 1–1–2–3–4–5
3
2
1
1
2
3
2 3 4 5
a) 0.5
0.25
0.0
–0.25
–0.5
0.0
1.0
2.0
3.0
N1
x/h1
0 1–1–2–3–4–5 2 3 4 5x/h1
b)
Est
imat
ed d
epth
h/h
1
igure 6. Structural index in Euler’s differential equation for theravity anomaly g, created by a vertical contact model, considereds a one-point source with the upper edge at depth h1 = �z1 − z�. �a�alculated index N1 at a relative thickness w/h1 of the contact struc-
ure: �1� w/h1 = 0.02 �thin sheet�, �2� w/h1 = 1.0, �3� w/h1 = 4.0thick contact�. �b� Estimated relative depth, h/h1, from Euler’s dif-erential equation at a prescribed index N1 = 0 for a relative thick-ess of the contact w/h , as in the case �a� above.
1
3tvttdp
tovTsdei
ehagriicagdtafiii
mctctt
pi
actfiw
hzs
wdstf
FgaCw=ea
L10 Stavrev and Reid
Figures 6b and 7b show calculated depths from equations 31 and2 at prescribed indices N1 = 0 and N2 = −1, respectively. For an in-ermediate thickness of the contact model, the estimated depth mayary significantly along the interpreted profile. Using equation 32,he estimated depth tends to the average depth �z1 + z2�/2 with dis-ance from the point above the edge �Figure 6b�. For the negative in-ex N2 = −1, equation 31 gives better results near the above-edgeoint �Figure 7b�.
Experimental results of Reid et al. �2003� on model and real gravi-y data find an explanation based on the above analysis. The authorsbtain returned structural indices �SI� between −0.05 and 0.8, meanalue of 0.5, and standard deviation of ±0.18, for a contact model.he gravity data from Penola Trough, Otway Basin in Australia,how SI values between −0.5 and 0.6, mean value of 0.11, and stan-ard deviation of ±0.23. These results are consistent with the expect-d variations in the sign and amplitudes of index N1 �Figure 6a� forntermediate contact models in equation 33.
0–1–2–3–4–5
3
3
2
1
12
3
21 3 4 5
a) 0.0
–0.5
–1.0
–1.5
–2.0
0.0
1.0
2.0
3.0
N2
x/h1
0–1–2–3–4–5 21 3 4 5x/h1
b)
Est
imat
ed d
epth
h/h
1
igure 7. Structural index in Euler’s differential equation for theravity anomaly g created by a vertical contact model considered astwo-point source with the upper edge at depth h1 = �z1 − z�. �a�alculated index N2 at a relative thickness w/h1 of the contact: �1�/h1 = 19.0 �considerable thickness�, �2� w/h1 = 9.0, �3� w/h1
4.0. �b� Estimated relative depth h/h1 from Euler’s differentialquation at a prescribed index N2 = −1 for a relative thickness w/h1,s in the case �a� above.
CONCLUSIONS
We have shown that the analytical expressions of potential fieldlements caused by arbitrarily distributed sources satisfy the tests ofomogeneity with respect to the set of all independent length vari-bles. The homogeneity analysis using the initial definition of homo-eneity is easier than that using Euler’s differential equation. A di-ect use of the initial definition assumes a space similarity transformnvolving both the observations and the sources. In terms of the sim-larity transform, the value and sign of the degree of homogeneityan be explained by the ratio between the transformed field elementnd the original field element at similar observation points. The de-ree of homogeneity depends on the order of the field element as aerivative of the gravity or magnetic potential and on the assumedype of the physical parameter in the analytical expression. The latterssumption leads to different degrees of homogeneity of a giveneld element. The existing regularities of the degrees of homogene-
ty can be synthesized in common formulas and shown numericallyn table form for the gravity and the magnetic field elements.
The proposed extended treatment of potential-field homogeneityakes it possible to deduce results concerning theoretical and practi-
al problems in Euler deconvolution. The structural indices reflecthe regularities of the degrees of homogeneity. It is shown analyti-ally and numerically that a negative structural index is possible forhe gravity contact interpretation models of considerable depth ex-ent.An appropriate Euler differential equation is composed.
The space similarity transform of a potential field element ex-resses the property of its homogeneity. This can be used to createnversion techniques for potential fields exploiting such transforms.
ACKNOWLEDGMENTS
We are grateful to John Peirce and Yonghe Sun for their consider-ble patience. We also thank Martin Mushayandebvu for his usefulomments. We are particularly indebted to Richard Hansen for assis-ance well beyond the duties of reviewer in helping us to present dif-cult ideas more clearly, and to Sven Treitel for his “swatting” work,hich has considerably improved the readability.
APPENDIX A
HOMOGENEITY OF THE GRAVITYPOTENTIAL OF A ROD WITH CONSTANT
OR VARIABLE DENSITY
Consider a vertical line source �rod� L of constant line density �,orizontal coordinates x0, y0, vertical coordinates z0 between z1 and2, and length l = z2 − z1. From equation 12, the potential V at the ob-ervation point P�x,y,z� is
V = ���L�
��/r�dz0 = �� ln��z2 − z + r2�/�z1 − z + r1�� ,
�A-1�
here ri = ��x − x0�2 + �y − y0�2 + �z − zi�2�1/2, i = 1,2, are theistances from point P to the upper and lower points of the rod, re-pectively. The degree of homogeneity of expression A-1 in relationo the coordinates and distances is n = 0. The same result for n can beound from Euler’s differential equation,
as
dot
wbA=sif
warsfpseie
=A
Etct
b
wedpFsfi
t
Tv
Homogeneity of potential fields L11
x�V/�x + y�V/�y + z�V/�z + x0�V/�x0 + y0�V/�y0
+ z1�V/�z1 + z2�V/�z2 = nV = 0, �A-2�
fter substituting the terms in the left-hand side of A-2 using expres-ions A-4 in the case of constant density �.
Homogeneity is also observed when the line mass has variableensity. Indeed, if ��z0� = �1 + ���/l��z0 − z1� changes continu-usly from �1 at the upper point to �2 = �1 + �� at the lower point ofhe rod, then
V = ���L�
���1 + ���/l��z0 − z1��/r�dz0
= ��1ln��z2 − z + r2�/�z1 − z + r1��
− ����/l���z1 − z�ln��z2 − z + r2�/�z1 − z + r1��
+ r2 − r1� = Vc + V�, �A-3�
here Vc corresponds to the case A-1, and V� is the potential causedy the density changes. The degree of homogeneity of expression-3 with respect to all geometric variables x, y, z, x0, y0, z1, z2, is n0. The set of geometric variables is the same as in expression A-1,
o Euler’s differential equation will consist of the same members asn equation A-2. The derivatives of potential V = Vc + V� yield theollowing expressions
�V/�x = � �x − x0��1E + � �x − x0����/l�
���z − z1�E + 1/r2 − 1/r1� ,
�V/�y = � �y − y0��1E + � �y − y0����/l�
���z − z1�E + 1/r2 − 1/r1� ,
�V/�z = ��1�1/r1 − 1/r2� + � ���/l��ln�Q2/Q1� − l/r2�� ,
�V/�x0 = − �V/�x, �V/�y0 = − �V/�y ,
�V/�z1 = − ��1/r1 + � ���/l�
��ln�Q2/Q1��z − z1�/l + �r2 − r1�/l�� ,
�V/�z2 = ��1/r2 + � ���/l��− ln�Q2/Q1��z − z1�/l − �r2
− r1�/l + l/r2� , �A-4�
able A-1. The terms of Euler’s differential equation A-2 andariation of density �.
Coordinates ofobservations
�km�Potential
�mGal.km�Term
x y z V �x − x0��V/�x �
0.0 0.0 −0.1 0.368 −0.072
1.0 2.0 −0.3 0.693 −0.120
2.0 4.0 −0.2 2.561 0.000
3.0 6.0 0.1 0.725 −0.137
4.0 8.0 0.0 0.369 −0.072
Note: Rod parameters are x = 2.0 km, y = 4.0 km, z = 0.1 km,
0 0 1 2here Q1 = r1 + z1 − z, Q2 = r2 + z2 − z, E = 1/�Q2r2� − 1/�Q1r1�,nd � = z2 − z1.An analytical proof of Euler’s equation A-2 with de-ivatives A-4 requires a complex manipulation of their long expres-ions. Alternately, we can prove the result using rod models. Resultsrom such calculations are shown in Table A-1. The observationoints lie along the epicentral line on an uneven surface. As can beeen, the sum � of the left-hand side terms in Euler’s equation A-2 isqual to 0 for all observation points. This shows degree n = 0, whichs the same as obtained above from a direct homogeneity analysis ofquation A-3.
The whole rod mass is m = m1 + �m, where m1 = �1l and �m��l/2. By substituting �1 = m1/l and �� = 2�m/l in equation-3,
V = � �m1/l�ln�Q2/Q1� + 2� ��m/I2�
���z1 − z�ln�Q2/Q1� + r2 − r1� . �A-5�
xpression A-5 shows degree of homogeneity n = −1 with respecto the set of geometric variables z, z1, r1, r2, and l. This result is identi-al to that obtained from the analysis of integral expression 11, whenhe physical parameter is the mass.
Table A-2 shows results from a numerical test with the same rodut with a variable density according to the sine function
��z0� = �1�1 + sin�2�k�z2 − z0�/�z2 − z1� + ��� ,
�A-6�
here k is the frequency, and � is the phase. Euler’s differentialquation A-2 is satisfied for the sinusoidal density of equation A-6 ategree of homogeneity n = 0. A more complicated continuous oriecewise-continuous density distribution can be represented by aourier series of sinusoidal components. Thus, for an arbitrary den-ity distribution �see Introduction�, Euler homogeneity of a createdeld element is valid.
APPENDIX B
HOMOGENEITY OF THE LOGARITHMIC 2DPOTENTIAL OF A LINE SOURCE
The logarithmic potential �e.g., Blakely, 1995� introduced in theheory of 2D fields is represented here by the equivalent expression
sum � for the potential V of a vertical rod with a linear
he Euler’s differential equation�mGal.km�
Sum�mGal.km�
�V/�y z�V/�z z1�V/�z1 z2�V/�z2 �
87 −0.001 −0.037 0.397 0.000
80 −0.030 −0.074 0.704 0.000
00 −0.933 −0.607 1.540 0.000
50 0.006 −0.075 0.756 0.000
90 0.000 −0.037 0.399 0.000
1 km, � = 0.3 Mt/km, �� = −0.1 Mt/km.
their
s of t
y − y0�
−0.2
−0.4
0.0
−0.5
−0.2
z = 1.
1
wartte
Ttrw
�
�
vc
B
C
E
—H
M
N
R
R
S
T
T
Z
Tv
z2 = 1.
L12 Stavrev and Reid
V = − 2�� ln�r/r0� , �B-1�
here r0 is a standard unit of the distance r between the line sourcend the observation point. This expression uses the well-known Fou-ier rule stating that in physical equations the transcendental func-ions may only have a dimensionless argument. It is easy to checkhat the derivatives of V do not contain the measurement unit r0. Forxample, from expression B-1,
�V/�r = − 2���r0/r��1/r0� = − 2��/r . �B-2�
he degree of homogeneity of potential of equation B-1 is equal tohe degree of the logarithmic function with respect to r and r0. Theesult is n = 0 as predicted by the general integral expression 12hen the physical parameter is density �.
The density � = �m/�y, where �m is the mass of a unit segmenty along the line source. In expression B-1, if � is substituted form/�y, then the degree of homogeneity in relation to all geometricariables will change to n = −1. This result corresponds with theonclusion drawn for expression 11 for a mass source.
able A-2. The terms of Euler’s differential equation A-2 andariation A-6 of density �.
Coordinates ofobservations
�km�Potential
�mGal.km�Term
x y z V �x − x0��V/�x �
0.0 0.0 −0.1 0.723 −0.141
1.0 2.0 −0.3 1.354 −0.232
2.0 4.0 −0.2 4.670 0.000
3.0 6.0 0.1 1.422 −0.268
4.0 8.0 0.0 0.725 −0.142
Note: Rod parameters are x0 = 2.0 km, y0 = 4.0 km, z1 = 0.1 km,
REFERENCES
lakely, R. J., 1995, Potential theory in gravity and magnetic applications:Cambridge University Press.
ourant, R., and F. John, 1965, Introduction to calculus and analysis: WileyInterscience.
uler, L., 1936, Introduction to the analysis of infinitesimal: ONTI �Russiantranslation�.—–, 1949, Differential calculus: Gostechizdat �Russian translation�.
ansen, R. O., and L. Suciu, 2002, Multiple-source Euler deconvolution:Geophysics, 67, 525–535.arson, I., and E. E. Klingele, 1993, Advantages of using the vertical gradi-ent of gravity for 3-D interpretation: Geophysics, 58, 1588–1595.
abighian, M. N., and R. O. Hansen, 2001, Unification of Euler and Wernerdeconvolution in three dimensions via the generalized Hilbert transform:Geophysics, 66, 1805–1810.
eid, A. B., J. M. Allsop, H. Granser, A. J. Millet, and I. W. Somerton, 1990,Magnetic interpretation in three dimensions using Euler deconvolution:Geophysics, 55, 80–91.
eid, A. B., D. FitzGerald, and P. McInerny, 2003, Euler deconvolution ofgravity data: 73rd Annual International Meeting, SEG, Expanded Ab-stracts, 580–583.
tavrev, P., 1997, Euler deconvolution using differential similarity transfor-mations of gravity or magnetic anomalies: Geophysical Prospecting, 45,207–246.
elford, W. M., L. P. Geldart, and R. E. Sheriff, 1990, Applied Geophysics,2nd ed.: Cambridge University Press.
hompson, D. T., 1982, EULDPH — A new technique for making computerassisted depth estimates from magnetic data: Geophysics, 47, 31–37.
hang, Ch., M. F. Mushayandebvu, A. B. Reid, J. R. Fairhead, and M. E.Odegard, 2000, Euler deconvolution of gravity tensor gradient data: Geo-physics, 65, 512–520.
sum � for the potential V of a vertical rod with a sinusoidal
e Euler’s differential equation�mGal.km�
Sum�mGal.km�
�V/�y z�V/�z z1�V/�z1 z2�V/�z2 �
63 −0.002 −0.073 0.779 0.000
30 −0.061 −0.145 1.368 0.000
00 −1.547 −1.006 2.553 0.000
73 0.013 −0.147 1.475 0.000
68 0.000 −0.073 0.783 0.000
1 km, �1 = 0.3 Mt/km, k = 0.5, � = 0.
their
s of th
y − y0�
−0.5
−0.9
0.0
−1.0
−0.5
