intech-new techniques for potential field data interpretation

7/30/2019 InTech-New Techniques for Potential Field Data Interpretation

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New Techniquesfor Potential Field Data Interpretation

Khalid S. EssaCairo University/ Faculty of Science/ Geophysics Department, Giza,

Egypt

1. Introduction

Many of the geological structures in mineral and petroleum exploration can be classifiedinto four categories: spheres, cylinders, dikes and geological contacts. These four simplegeometric forms are convenient approximations to common geological structures oftenencountered in the interpretation of magnetic and gravity data. Two quantitative categoriesare usually adapted for the interpretation of magnetic and gravity anomalies in order todetermine basically the depth, the shape of the buried structure and the other modelparameters. First, two parameters are considered the most important problem in explorationgeophysics. The first category includes 2D and 3D continuous modelling and inversionmethods (Mohan et al., 1982; Nabighian, 1984; Chai & Hinze, 1988; Martin-Atienza &Garcia-Abdeslem 1999; Zhang et al., 2001; Keating & Pilkington, 2004; Numes et al., 2008;

Martins et al., 2010; Silva et al., 2010). The solutions of these methods require magneticsusceptibility and/or remanent magnetization for magnetic interpretation and densityinformation for gravity interpretation a as part of the input, along with the same depthinformation obtained from geological and/or other geophysical data. Thus, the resultingmodel can vary widely depending on these factors because the magnetic and gravity inverseproblem are ill-posed and are, therefore unstable and nonunique (Zhdanov, 2002; Tarantola,2005). Standard techniques for obtaining a stable solution of an ill-posed inverse probleminclude those based on regularization methods (Tikhonov & Arsenin, 1977). The secondcategory is fixed simple geometry methods, in which the spheres, cylinders, dikes, andgeological contacts models estimate the depth and the shape of the buried structures fromresiduals and/or observed data. The models may be shifted from geological reality, but theyare usually sufficient to determine whether the form and the magnitude of both calculatedmagnetic and gravity effects are close enough to those observed to make the geologicalinterpretation reasonable. Several methods have been developed for the second category tointerpret magnetic and gravity data using a fixed simple geometry (Nettleton, 1976; Stanley,1977; Atchuta Rao & Ram Babu, 1980; Prakasa Rao & Subrahmanyan, 1988; Abdelrahman &El-Araby, 1993; Zhang et al., 2000; Salem et al., 2004; Abdelrahman & Essa, 2005; Essa, 2007;Essa, 2011). The drawbacks of these methods are that they are highly subjective, need apriori information about the shape (shape factor) of the anomalous body and use fewcharacteristic points when inverting for the remaining parameters. New algorithms formagnetic and gravity inversion have been developed that successively determines the depth

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New Achievements in Geoscience22

(z) and the other associated model parameters of the buried structure. The inverse problemof the depth estimation from the observed magnetic and gravity data has been transformedinto a nonlinear equation of the form f(z) = 0. This equation is then solved for depth byminimizing an objective functional in the least-squares sense. Using the estimated depth, the

other model parameters are computed from the measured magnetic and gravity data,respectively. The procedures are applied to synthetic data with and without noise formagnetic and gravity data. These procedures are also tested to complicated regional andinterference from neighbouring magnetic rocks. Finally, these techniques are alsosuccessfully applied to real data sets for mineral and petroleum exploration, and it is foundthat the estimated depths and the associated model parameters are in good agreement withthe actual values.

2. Formulation of the problems

2.1 A least-squares minimization approach to depth determination from residualmagnetic anomaly

Following Gay (1963), Prakasa Rao et al. (1986), and Prakasa Rao & Subrahmanyan (1988),the magnetic anomaly expression produced by most geologic structures with center locatedat xi= 0 can be represented by the following equation:

m n n mp2r 2az bx sin cos cx z sin cosi iT(x , z , ) K ,i 1,2,3,...Lqi 2 2x zi

(1)

The geometries are shown in Figure 1. In equation (1), z is the depth, K is the amplitudecoefficient (effective magnetization intensity), is the index parameter (effectivemagnetization inclination), xi is the horizontal coordinate position, and q is the shape factor.Values for a, b, c, m, n, r, p, and q, are given in Table 1.

At the origin (xi =0), equation (1) gives the following relationship:

2 2(0),

(sin ) (cos )

q r

m n

T zK

a

(2)

where T(0) is the anomaly value at the origin (Fig. 2).

Using equation (2), equation (1) can be rewritten as

2 2 2 2

2 2

( ) (tan )(0)( , , ) ,

( )

pq r r n mi i

i qi

az bx cx zT zT X Z

a x z

(3)

For all shapes (function of q), equation (3) gives the following relationship at xi=N

2 22 2 2 2

2 2( )( ) (0) ( )

(tan ) ,(0)

q q r rn m

p q r

aT N N z T z az bN

cNz T z

(4)

where T (N) is the anomaly value at xi = N (Fig. 2).

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New Techniques for Potential Field Data Interpretation 23

Fig. 1. Various simple geometrical structures diagram.

Table 1. Definition of a, b, c, m, n, p, r, and q values were shown in equation (1). F.H.D. andS.H.D. are the first and the second horizontal derivatives of the magnetic anomaly,respectively.

Substituting equation (4) into equation (3), we obtain the following nonlinear equation in z2 2 2 22 2 2 2 2 2

2 2

(0) ( ) ( )( ) (0) ( )( , ) ,

( )

q r q q r r ri i i

i qi

NT z az bx ax T N N z x T z az bN T x z

aN x z

(5)

The unknown depth z in equation (5) can be obtained by minimizing

2

( ) ( ) ( , ) ,N

i ii

z L x T x z (6)

where L (xi) denotes the observed magnetic anomaly at xi.

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Fig. 2. A typical magnetic anomaly profile over a thin dike. The anomaly value at the originT(0) and the anomaly value T(N) where N is taken to be 1arbitrary unit in this case, theposition of the maximum value (M), and the minimum value (m) are illustrated.

Minimization of (z) in the least - squares sense, i.e., (d/dz) (z)=0 leads to the followingequation:

) ( ) ( , ) ( , ) 0,N

i i ii

f(z L x T x z T x z (7)

where

*( , ) ( ) ( , )i idT x z T x zdz

Equation (7) can be solved for z using the standard methods for solving nonlinear equations.Here, it is solved by an iteration method (Press et al., 1986). The iteration form of equation(7) is given as

( ) ,f jz f z (8)

where zj is the initial depth and zf is the revised depth. zf will be used as zj for next iteration.The iteration stops when |zf - zj |< e, where e is a small predetermined real number close tozero. The source body depth is determined by solving one nonlinear equation in z. Anyinitial guess for z works well because there is only one minimum. The experience with theminimization technique for two or more unknowns is that it always produced good resultsfor synthetic data with or without random noise. In the case of field data, good results mayonly be obtained when using very good initial guesses on the model parameters. Theoptimization problem for the depth parameter is highly nonlinear, increasing the number ofparameters to be solved simultaneously also increases the dimensionality of the energysurface, thereby greatly increasing the probability of the optimization stalling in a localminimum on that surface. Thus common sense dictates that the nonlinear optimization

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should be restricted to as few parameters as is consistent with obtaining useful results. Thisis why we propose a solution for only one unknown, z. Once z is known, the effectivemagnetization inclination, , can be determined from equation (4). Finally, knowing , theeffective magnetization intensity, K, can be determined from equation (2). Then, we measure

the goodness of fit between the observed and the computed magnetic data for each N value.The simplest way to compare two magnetic profiles is to compute the root-mean-square(RMS) of the differences between the observed and the fitted anomalies. The modelparameters which give the least root-mean square error are the best. To this point we haveassumed knowledge of the axes of the magnetic profile so that T(0) can be found. T(0) isdetermined using methods described by Stanley (1977). As illustrated in Figure 2, the lineM-m intersects the anomaly profile at xi = 0. The base line of the anomaly profile lies adistance M-T(0) above the minimum. A semi-automated interpretation scheme based on theabove equations for analyzing field data is illustrated in Figure 3.

Fig. 3. Generalized scheme for semi-automatic depth, index parameter and amplitudecoefficient estimations.

2.1.1 Synthetic example

Synthetic examples of spheres and horizontal cylinders buried at different depths (profilelength = 40 units, K = 100 units, = 60o, sampling interval = 1unit) were interpreted usingthe present method [equations (7), (4), and (2)] to determine depth, index parameter, andamplitude coefficient, respectively. In each case, the starting depth was 5 units. In all cases

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examined, the exact values of z, , and K were obtained. However, in studying the errorresponse of the least-squares method, synthetic examples contaminated with 5% randomerrors were considered. Following the interpretation scheme, values of the most appropriatemodel parameters (z, , K) were computed and percentage of error in model parameters was

plotted against the model depth for comparison (Fig. 4). We verified numerically that thedepth obtained is within 3% for horizontal cylinders and 4% for spheres. The indexparameter obtained is within 5.5% whereas the amplitude coefficient is within 8% (Fig. 4).Good results are obtained by using the present algorithm, particularly for depth estimation,which is a primary concern in magnetic prospecting and other geophysical work.

Fig. 4. Model parameters error response estimates. Abscissa: model depth. Ordinate: percenterror in model parameters.

2.1.2 Field example

Figure 5 shows a total magnetic anomaly above an olivine diabase dike, Pishabo Lake, Ontario(McGrath & Hood, 1970). The geological cross section of the dike is shown beneath theanomaly. The depth to the outcropping dike (sensor height) is 304 m (Fig. 5). The anomalyprofile was digitized at an interval of 100 m. Equations (7), (4), and (2) were used to determinedepth, index parameter and amplitude coefficient, respectively, using all possible cases on Nvalues. The starting depth used in this field example was 50 m. Then, we computed the root-mean-square of the differences between the observed and the fitted anomalies. The best fitmodel parameters are: z = 306 m, = 38o and K = 1422 nT*Z (Table 2). McGrath & Hood (1970)applied a computer curve-matching method to the same magnetic data employing a least-squares method and obtained a depth of 301 m. Moreover, Figure 5 shows that there is a lackof thinness because the thickness of the dike is only slightly less than the depth. However,because many of the characteristics of the thick dike anomalies are similar to those for thindike (Gay, 1963), our method can be applied not only to magnetic anomalies due to thin dikes,but also to those due to thick dikes to obtain reliable depth estimates.

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Fig. 5. Total magnetic anomaly (above) over an outcropping diabase dike (below), PishaboLake, Ontario, Canada (McGrath & Hood, 1970). The base line and zero crossing shown aredetermined using Stanleys method (1977).

Table 2. Numerical results of the Pishabo field example.

2.2 Three least-squares minimization approach to model parameters estimation frommagnetic anomaly due to thin dikes

From equation (1), the magnetic anomaly expression of a thin dike which extends to infinityin both strike direction and down dip (2-D) is given by:

2 2

sin cos( , , ) , 1,2,3,.....,ii

i

x zT x z zK i N

x z

(9)

where z is the depth to the top of the body, xi is a distance point along x-axis where theobserved anomaly is located, K is the amplitude coefficient, and is the index parameter.

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The values of K and for the anomalies in total, vertical, and horizontal fields are given inTable 3. The index parameter is related to the effective direction of the resultantmagnetization, the dip and strike of the dike.

Table 3. Characteristic index parameter and amplitude coefficient K in vertical (V),horizontal (H), and total (T) magnetic anomalies due to thin dikes where t is thethickness, k is the magnetic susceptibility contrast, d is the dip angle, To and o are thevalues of effective total intensity and effective inclination of magnetic polarization, and isthe strike (after Gay, 1963).

Following Griffin (1949), the moving average residual magnetic anomaly (R) is defined as

(0) ( ),R T T s (10)

where T(0) is the magnetic anomaly value at a given point on the magnetic anomaly map, and

2

0

1( ) ( , )

2T s T s d

(11)

is the average magnetic value at the radial distance s from the point where the magneticvalue is T(0). The geometrical interpretation of equation (11) is that the average value, ( )T s ,is obtained by going around a circle of radius s, having T(0) as its centre, and forming thesum of the products of T(s, ) and d for an infinite number of infinitesimally small ds.The result of the above is then divided by 2. Since an integral form of ( )T s is not know,Griffin (1949) has adopted a simple numerical method to obtain the average magneticanomaly value. The method is to form the arithmetical average of a finite number of pointsabout the circumference of a circle of radius s, or

1 2 3( ) ( ) ( ) ....... ( )( ) .nT s T s T s T s

T sn

(12)

for n points.

Along a profile, the moving average residual magnetic anomaly is then defined as

2 ( ) ( ) ( )( , ) ,2

i i ii

T x T x s T x sR x s

(13)

where s = 1, 2, 3,., M sample spacing units and is defined here as the window length orgraticule spacing.

Applying equation (13) to equation (9), the moving average residual magnetic anomaly dueto a thin dike is obtained as

2 2 2 2 2 2

2 sin 2 cos ( )sin cos ( )sin cos( , , , ) .

2 ( ) ( )i i i

ii i i

x z x s z x s zzKR x z s

x z x s z x s z

(14)

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The specific value R(xi, z, , s) at xi = 0 is given by

2

2 2cos

(0) .Ks

Rs z

(15)

Using equation (15), equation (14) can now be written as

2 2

2 2 2 2 2 2 2

2 tan 2 ( )tan ( )tan( ) (0)( , , , ) .

2 ( ) ( )i i i

ii i i

x z x s z x s zz s z RR x z s

s x z x s z x s z

(16)

Using equation (16), we can obtain the following equations at xi = s

2 2

2 2

( ) 2 3 tan,

(0) 4 iR s z s zs

x sR s z

(17)

And

2 2

2 2

( ) 2 3 tan, .

(0) 4 iR s z s zs

x sR s z

(18)

From equations (17) and (18), we obtain

2 2(4 )tan ,

6F s z

zs

(19)

where

( ) ( ),

(0)R s R s

FR

which is computed directly from observed anomaly. In all cases, the maximum value of s isless than or equal (N-1)/4, where N is the number of the data points along the anomalyprofile whose centre is located at xi = 0. Using equation (19), equation (16) can be written as

( , , ) (0) ( , , ),i iR x z s R W x z s (20)

where

2 2 2 2 2 22 2

3 2 2 2 2

2 2 2

2 2

2 (4 ) 12 ( ) (4 ) 6( )( , , )

12 ( )

( ) (4 ) 6.

( )

i ii

i i

i

i

x F s z z s x s F s z z ss zW x z s

s x z x s z

x s F s z z s

x s z

The unknown depth (z) in equation (20) can be obtained by minimizing

2

1

( ) ( ) (0) ( , , ) ,N

i ii

z L x R W x z s

(21)

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where L(xi) denotes the moving average residual magnetic anomalies at xi. Setting thederivative of (z) to zero with respect to z leads to

*1

( ) ( ) (0) ( , , ) ( , ) 0.N

i i ii

f z L x R W x z s W x z s

(22)

where W*(xi, z, s) is evaluated by analytically differentiating W(xi, z, s) and is given by thefollowing equation

2 2 2 2 2

2 2 2

2 2 2 2 2* 2 2

2 2 2

2 2 2 2 2

2

( )(4 24 ) 2 (2 (4 ) 12 )

( )

(( ) )(2( ) 12 ) 2 (( ) (4 ) 6 )( , , ) ( )

(( ) )

(( ) )(2( ) 12 ) 2 (( ) (4 ) 6 )

(( )

i i i

i

i i ii

i

i i i

i

x z x Fz zs z x F s z z s

x z

x s z x s Fz zs z x s F s z z sW x z s s z

x s z

x s z x s Fz zs z x s F s z z s

x s

2 2

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2

)

2 (4 ) 12 ( ) (4 ) 6 ( ) (4 ) 62

( ) ( ) ( )i i i

i i i

z

x F s z z s x s F s z z s x s F s z z sz

x z x s z x s z

Equation (22) can be solved for z using standard methods for solving nonlinear equations.Here, it is solved by an iterative method (Demidovich & Maron, 1973).

Substituting the computed depth (zc) as a fixed parameter in equation (16), we obtain

2 2

2

( ) (0)( , , ) ( , , ),2

c ci i

z s z RR x s P x ss

(23)

where

2 2 2 2 2 22 tan 2 ( )tan ( )tan

( , , ) .( ) ( )

i c i c i ci

i c i c i c

x z x s z x s zP x s

x z x s z x s z

The unknown index parameter () in equation (23) can be obtained by minimizing

22 2

21

( ) (0)( ) ( ) ( , , ) .

2

Nc c

i ii

z s z RL x P x s

s

(24)

Setting the derivative of () to zero with respect to leads to

2 2*

21

( ) (0)( ) ( ) ( , , ) ( , , ) 0,

2

Nc c

i i ii

z s z RL x P x s P x s

s

(25)

where P*(xi, , s) is obtained by differentiating P(xi, , s) with respect to and is given by thefollowing equation

* 22 2 2 2 2 2

2 ( ) ( )( , , ) .

( ) ( )i i i

ii c i c i c

x x s x sP x s Sec

x z x s z x s z

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Equation (25) can be solved using a simple iterative method (Demodivch & Maron, 1973).Substituting the computed depth (zc) and index parameter (c) in equation (14) as fixedparameters, we obtain

( , ) ( , ),i iR x s KD x s (26)where

2 2 2 2 2 2

2 sin 2 cos ( )sin cos ( )sin cos( , ) .

2 ( ) ( )c i c c c i c c c i c c c

ii c i c i c

z x z x s z x s zD x s

x z x s z x s z

Finally, applying the least-squares method to equation (26), the unknown amplitudecoefficient (Kc) can be determined from

1

2

1

( ) ( , )

.( , )

N

i i

ic N

ii

L x D x s

KD x s

(27)

Theoretically, one value of s is enough to determine the model parameters (z, , K) fromequations (22), (25), and (27), respectively. In practice, more than one value of s is desirablebecause of the presence of noise, interference from neighbouring magnetic rocks, andcomplicated regional in data. Moreover, the advantage of this method over continuousmodelling methods is that it requires neither magnetic susceptibility contrast nor depthinformation, and it can be applied if little or no factual information other than the magneticdata is available. The method may be automated to solve for a number of anomalies along amagnetic profile when the origin of each dike is known from other geophysical and/orgeological data.

2.2.1 Synthetic example

The composite vertical magnetic anomaly in nanoteslas of Figure 6 consisting of thecombined effect of an intermediate structure of interest (thin dike with K = 200 nT, z = 5 km,and = 40o), an interference neighbouring magnetic rocks (horizontal cylinder with K = 40nT, z = 2 km, and = 30o), and a deep-seated structure (sphere with K = 80000 nT, z = 15 km,and = 10o) was computed by the following expression:

2

2 2 2

2 2

2

2

(4 ( 10) )cos30 4( 10)sin 30( ) 40 (( 10) 2 )

( )

sin 40 5cos 40200

5( )

(450 ( 60) )sin10 45( 60)cos1080000

(( 60) 15

i ii

i

i

i

i i

i

x x

T x x

Horizontalcylinder verticalcomponent

x

x

Thindike verticalcomponent

x x

x

52 2.

)( )Sphere verticalcomponent

(28)

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Fig. 6. Noisy composite vertical magnetic anomaly consisting of the combined effect of anintermediate structure (thin dike with K = 200 nT, z = 5 km, and = 40) [anomaly 1], adeep-seated structure (sphere with K = 80000 nT, z = 15 km, and = 10) [anomaly 2], andan interference from neighbouring magnetic rocks (horizontal cylinder with K = 40 nT, z = 2km, and = 30) [anomaly 3].

In Figure 6, anomaly 1 is the anomaly due to the intermediate structure of our interest,

anomaly 2 is the anomaly due to the interference from neighbouring magnetic rocks, andanomaly 3 is the anomaly due to the deep-seated structure. The composite magneticanomaly T(xi) is contaminated with 20% random error using the following equation:

( ) ( ) 1 ( ( ) 0.5) * 0.2 ,rnd i iT x T x RND i (29)

where Trnd(xi) is the contaminated anomaly value at xi, and RND(i) is a pseudo randomnumber whose range is [0, 1]. The interval of the pseudo random number is an openinterval, i.e. it does not include the extremes 0 and 1.

The noisy composite magnetic anomaly (T) is subjected to a separation technique using the

moving-average method. Nine successive moving-average graticule spacings were appliedto each set of input data. Equations (22), (25), and (27) were applied to each of the ninemoving-average residual profiles, yielding depth, index parameter, and amplitudecoefficient solutions. The results and their average values are given in Table 4.

We verified numerically that the depth obtained is within 0.38%. The index parameterobtained is within 0.4%, whereas the error in the amplitude coefficient is within 4.7%. Goodresults are obtained by using the present algorithm for depth, index parameter, andamplitude coefficient, which are a primary concern in magnetic prospecting and othergeophysical work. On the other hand, the Euler deconvolution method (Thompson, 1982)has been applied to the same noisy composite magnetic anomaly to determine the depth to

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the buried dike. The result in this case is shown in Figure 7. The average depth obtained is3.27 km compared with 4.99 km using the least-squares method. This result indicates thatthe present least-squares method is less sensitive to errors in the magnetic anomaly thanEuler deconvolution method.

Table 4. Theoretical results obtained from the noisy composite magnetic anomaly. Thedepth, index parameter, and amplitude coefficient were computed from equations (22), (25),and (27), respectively. The actual dike parameters are: z = 5 km, = 40 and K = 200 nT.

Fig. 7. Euler deconvolution depth estimates from the data of the noisy composite vertical

magnetic anomaly of Figure 6 using a thin dike model. The average depth obtained is 3.27 km.

2.2.2 Field example

Figure 8 shows a total-field magnetic anomaly above a Mesozoic diabase dike (2 m thick)intruded into Palaeozoic sediments of the Parnaiba Basin, Brazil (Silva, 1989; Fig. 10). Thetotal-field data were collected along profile perpendicular to the dike, with samplinginterval of about 2 m. The depth to the outcropping dike (sensor height) is 1.9 m. Thisanomaly profile of 24.64 m was digitised at an interval of 0.385 m.

Table 5. Computed values of depth, index parameter, and amplitude coefficient obtainedfrom a total-field magnetic anomaly above a Mesozoic diabase dike intruded into Palaeozoicsediments from the Parnaiba Basin, Brazil.

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Fig. 8. A total field magnetic anomaly above a Mesozoic diabase dike (2 m thick) intrudedinto Palaeozoic sediments from the Parnaiba Basin, Brazil (after Silva, 1989).

Fig. 9. Moving average residual magnetic anomalies above a Mesozoic diabase dikeintruded into Palaeozoic sediments from the Parnaiba Basin, Brazil.

Adopting the same technique applied in the example above, we found the results given inFigure 9 and Table 5. The model parameters obtained are: z = 2.22 m, = 53o, and K = -215nT. Silva (1989) applied a method to the same magnetic data employing the M-fittingtechnique and obtained a depth of about 3.5 m. The depth to the top is overestimated byboth methods. This is reasonable because the upper part of the dike was weathered and themagnetite present was oxidized, losing most of its magnetic property (Silva, 1989). This fieldexample shows also that there is a lack of "thinness". However, because many of thecharacteristics of wide dike anomalies are similar to those for thin dikes (Gay, 1963), ourmethod can be applied not only to magnetic anomalies due to thin dikes, but also to thosedue to wide dikes to obtain reliable estimates of model parameters.

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2.3 A window curves method to shape and depth solutions from third moving averageresidual gravity anomalies

Following Abdelrahman & El-Araby (1993), the general gravity anomaly expressionproduced by most geological structures is given by the following equation:

2 2( , , ) ,( )i q

i

Ag x z q

x z

(30)

where q is the shape (shape factor), z is the depth, x is the position coordinate and A isamplitude coefficient related to the radius and density contrast of the buried structures.Examples of the shape factor for the semi-infinite vertical cylinder (3D), horizontal cylinder(2D), and sphere (3D) are 0.5, 1.0, and 1.5, respectively. Also, the shape factor for the finitevertical cylinder is approximately 1.0. The shape factor (q) approaches zero as the structurebecomes a flat slab, and approaches 1.5 as the structure becomes a perfect sphere (pointmass). The moving average (grid) method is an important yet simple technique for the

separation of gravity anomalies into residual and regional components. The basic theory ofthe moving average methods is described by Griffin (1949), Agocs (1951), Abdelrahman &El-Araby (1996) and Abdelrahman et al. (2001).

Let us consider seven observation points (xi-3s, xi-2s, xi - s, xi, xi + s, xi +2s, and xi+3s) alongthe anomaly profile where s=1, 2,, M spacing units and is called the window length. Thefirst moving average regional gravity field Z1(xi, z, q, s) is defined as the average of g(xi-s, z,q) and g(xi+s, z, q), which for the simple shapes mentioned above can be written as

2 2 2 21( , , ) (( ) ) (( ) ) .2q q

i i i

AZ x z s x s z x s z (31)

The first moving average residual gravity anomaly R1(xi, z, q, s) at the point xi is defined as(Abdelrahman & El-Araby, 1996).

2 2 2 2 2 21( , , , ) 2( ) (( ) ) (( ) )2q q q

i i i i

AR x z q s x z x s z x s z

(32)

The second moving average residual gravity anomaly R2(xi, z, q, s) at the point xi is(Abdelrahman et al. 2001)

2 2 2 2 2 22

2 2 2 2

( , , , ) 6( ) 4(( ) ) 4(( ) )4

(( 2 ) ) (( 2 ) )

q q qi i i i

q qi i


x s z x s z

(33)

The third moving average residual gravity anomaly R3(xi, z, q, s) at any point xi is

2 2 2 2 2 23

2 2 2 2

2 2 2 2

( , , , ) 20( ) 15(( ) ) 15(( ) )8

6(( 2 ) ) 6(( 2 ) )

(( 3 ) ) (( 3 ) )

q q qi i i i

q qi i

q qi i


x s z x s z

x s z x s z

(34)

For all shape factors (q), equation (34) gives the following value at xi=0

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32 2 2 2 2 2 2

4 (0)

10 15( ) 6(4 ) (9 )q q q qR

Az s z s z s z

(35)

Using equation (34) and equation (35), we obtain the following normalized equation at xi=s

22 2 2 2 2 2 2 23

2 2 2 2 2 2 23

26( ) 16(4 ) 6(9 ) 15 (16 )( )(0) 20 30( ) 12(4 ) 2(9 )

q q q q q

q q q q

s z s z s z z s zR s

R z s z s z s z

(36)

Equation (36) is independent on the amplitude coefficient (A). Let F= R 3(s)/R3(0) then fromequation (36) we obtain

1215

,* ( , , ) ( , , )

q

zF P s z q W s z q

(37)

Where

2 2 2 2 2 2 2( , , ) 20 30( ) 12(4 ) 2(9 ) ,q q q qP s z q z s z s z s z

and

2 2 2 2 2 2 2 2( , , ) 16(4 ) 6(9 ) (16 ) 26( ) .q q q qW s z q s z s z s z s z

Equation (37) can be used not only to determine the depth, but also to simultaneouslyestimate the shape of the buried structure as will be illustrated by theoretical and field

examples.2.3.1 Synthetic example

The composite gravity anomaly in mGals of Figure 10 consisting of the combined effect of ashallow structure (horizontal cylinder with 3 km depth) and a deep-seated structure(dipping fault with depth to upper faulted slab = 15 km, depth to lower faulted slab = 20km, and dip angle of fault plane = 50 degrees) was computed by the above expression:

1 12 2

5 5200( ) 100(tan ( cot(50 )) tan ( cot(50 ))),

15 20( 3 )i i

ii

x xg x

x

(38)

(Horizontal cylinder + dipping fault)

The composite gravity field (g) was subjected to our third moving average technique. Thethird moving average residual value at point xi is computed from the input gravity datag(xi) using the following equation:

320 ( ) 15 ( ) 15 ( ) 6 ( 2 ) 6 ( 2 ) ( 3 ) ( 3 )

( ) .8

i i i i i i ii

g x g x s g x s g x s g x s g x s g x sR x

(39)

Five successive third moving average windows (s= 2, 3, 4, 5, and 6 km) were applied toinput data. Each of the resulting moving average profiles was analyzed by equation (37). For

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Fig. 12. Family of window curves of z as a function of q for s= 2, 3, 4, 5, and 6 km as obtainedfrom gravity anomaly (g) using the second moving average method. Estimates of q and zare, respectively, 0.75, and 2.7 km.

Fig. 13. Family of window curves of z as a function of q for s= 2, 3, 4, 5, and 6 km as obtained

from gravity anomaly (g) using the present third moving average method. Estimates of qand z are, respectively, 1.03, and 3.05 km.

In the case of using first moving average technique, the window curves intersect at anarrow region where 0.55>q>0.05 and 2.5 km>z> 1.8 km (Fig. 11). The central point of thisregion occurs at the location q= 0.26 and z= 2.1 km. On the other hand, in the case of usingsecond moving average technique, the curves intersect each other nearly at a point whereq= 0.75 and z= 2.7 km (Fig. 12). Finally, in the case of using the third moving averagemethod (present method), the curves intersect each other at the correct locations q= 1.03and z= 3.05 km (Fig. 13). From these results (Figs. 11-13), it is evident that the first and thesecond moving average methods give erroneous results for the shape and depth solutions

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whereas the model parameters obtained from the third moving average method are inexcellent agreement with the parameters of the shallow structure given in model[equation (38)]. The conclusion is inescapable that the third moving average techniquegives the best result. Moreover, random errors were added to the composite gravity

anomaly g(xi) to produce noisy anomaly. In this case, we choose a white random noisewith amplitude being 5% of the maximum amplitude variation along the entire profile, sothat the noise will be equally along the profile. The noisy anomaly is subjected to aseparation technique using only the third moving average method. Five successive thirdmoving average windows were applied to the noisy input data. Adapting the sameinterpretation procedure, the results are shown in Figure 14.

Fig. 14. Family of window curves of z as a function of q for s= 2, 3, 4, 5, 6, and 7 km as obtainedfrom gravity anomaly (g) after adding 5% random errors to the data using the present thirdmoving average method. Estimates of q and z are, respectively, 1.09, and 2.98 km.

Figure 14 shows the window curves intersect at approximately q= 1.09 and z= 2.98 km. Theresults (Fig. 14) are generally in very good agreement with the shape factor and depthparameters shown in [equation (38)]. This indicates that our method will produce reliableshape and depth estimates when applied to noisy gravity data.

2.3.2 Field example

A Bouguer gravity anomaly profile along A of the gravity map of the Humble salt dome,near Houston (Nettleton, 1962, Fig. 22) is shown in Figure 15. The gravity profile has beendigitized at an interval of 0.26 km. The Bouguer gravity anomalies thus obtained havebeen subjected to a separation technique using the third moving average method. Filterswere applied in four successive windows (s= 2.60, 2.86, 3.12, and 3.38 km). In this way,four third moving average residual anomaly profiles were obtained (Fig. 16). The sameprocedure described for the synthetic examples was used to estimate the shape and thedepth of the salt dome. The results are plotted in Figure 17. Figure 17 shows that thewindow curves intersect at a point where the depth is about 4.95 km and at q=1.41. This

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suggests that the shape of the salt dome resembles a sphere or, practically, a three-dimensional source with a hemi- spherical roof and root. This result is generally in goodagreement with the dome form estimated from drilling top and contact (Nettleton, 1976;Fig. 8-16).

Fig. 15. Observed gravity profile of the Humble salt dome, near Houston, TX, USA (afterNettleton 1962)

Fig. 16. Third moving average residual gravity anomalies on line A of the Humble saltdome, for s= 2.60, 2. 86, 3.12 and 3.38 km.

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Fig. 17. Family of window curves of z as a function of q for s= 2.60, 2. 86, 3.12 and 3.38 km asobtained from the Humble gravity anomaly profile

3. Conclusion

Three different numerical algorithms to depth and other associated model parametersdetermination from magnetic and gravity data have been presented. In the first method, aleast-squares minimization approach has been developed to determine the depth to a buriedstructure from magnetic data. The problem of determining the depth of a buried structure

from the magnetic anomaly has been transformed into the problem of finding a solution of anonlinear equation. The method presented is very simple to execute. The advantages of thepresent method over previous techniques, which use only a few points, distances,standardized curves, and nomograms are: (1) all observed values cal be used, (2) the methodis automatic, and (3) the method is not sensitive to errors in the magnetic anomaly. Finally,the advantage of the proposed method over previous least-squares techniques is that anyinitial guess for the depth parameter works well. In the second method, the problem ofdetermining the depth, index parameter, and amplitude coefficient of the buried thin dikefrom a magnetic anomaly profile has been transformed into the problem of solving twoindividual nonlinear equations and a linear equation, respectively. A scheme forinterpreting the magnetic data to obtain the model parameters based on the least-squaresmethod is developed. The method provides two advantages over previous least-squarestechniques: (1) each model parameter is computed from all observed data, and (2) themethod is less sensitive to errors in the magnetic anomaly. The method involves using a thindike model convolved with the same moving average filter as applied to the observed data.As a result, the method can be applied not only to residuals, but also to measured magneticdata having noise, interference from neighboring magnetic rocks, and complicated regional.Synthetic and field studies demonstrate the efficiency of the present technique. Finally, theproblem of determining the shape and depth from gravity data of long or short profilelength can be solved using the window curves methods for simple anomalies. The windowcurves methods are very simple to execute and work well even when the data contains

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noise. The methods involve using simple models convolved with the same moving averagefilter as applied to the observed gravity data. As a result, the methods can be applied notonly to residuals, but also to measured gravity data. Given their relative strength, thepresent techniques complement the existing methods of shape and depth determination

from moving average residual gravity anomaly profiles and overcome some of theirshortcomings. The present iterative methods converge to a depth solution for any shapefactor value and when applying any window length. The methods are developed to deriveshaperelated parameter and depth and remove the regional trend from gravity data. Thesetwo parameters might be used to gain geological insight concerning the subsurface.However, the advantages of the third moving average method over the first and the secondmoving average techniques are: (1) the method removes adequately the regional field due todeep-seated structure from gravity data and (2) the method can be applied to large griddeddata. Theoretical and field examples have illustrated the validity of the algorithmspresented.

4. Acknowledgment

The author thanks Prof. Aleksander Lazinica, CEO, Prof. Hwee-San Lim, the Book editorand Mrs. Ana Skalamera, Publishing Process Manager for their excellent suggestions andcollaborations. He would like to thank Prof. Sharf El Din Mahmoud, Chairman and Prof. El-Sayed Abdelrahman, Geophysics Department, Faulty of Science, Cairo University for theircontinuous support and encouragement.

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Abdelrahman, E. M., El-Araby, T. M., El-Araby, H. M., & Abo-Ezz, E. R. (2001). A newmethod for shape and depth determinations from gravity data. Geophysics, Vol. 66,No. 6, pp. 1774-1780, ISSN 0016-8033.

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Chai, Y., & Hinze, W. J. (1988). Gravity inversion of an interface above which the densitycontrast varies exponentially with depth. Geophysics, Vol. 53, No. 6, pp. 837845,ISSN 0016-8033.

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New Achievements in Geoscience

Edited by Dr. Hwee-San Lim

ISBN 978-953-51-0263-2

Hard cover, 212 pages

Publisher InTech

Published online 23, March, 2012

Published in print edition March, 2012

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New Achievements in Geoscience is a comprehensive, up-to-date resource for academic researchers in

geophysics, environmental science, earth science, natural resource managements and their related support

fields. This book attempts to highlight issues dealing with geophysical and earth sciences. It describes the

research carried out by world-class scientists in the fields of geoscience. The content of the book includes

selected chapters covering seismic interpretation, potential field data interpretation and also several chapters

on earth science.

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