Geophysics 1998

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GEOPHYSICS, VOL. 63, NO. 6 (NOVEMBER-DECEMBER 1998); P. 1943 –1946, 2 FIGS., 1 TABLE.

Short Note

Fractal behavior and detectibility limitsof geophysical surveys

V. P. Dimri ∗

INTRODUCTION

The detectibility limits of a large-scale geophysical measur-ing network (such as arraysof seismometers for seismic hazardassessment, arraysof airguns andvibratorsin seismic proling,and station spacing in gravity, magnetic, magnetotelluric, andother geophysical surveys for exploration of oil, minerals,and groundwater) depends on the fractal dimension of thenetwork and the anomaly (Lovejoy et al., 1986; Korvin, 1992).The geophysical anomaly resulting from the fractal nature of sources (Turcotte, 1992) [such as length of fault (Okubo andAki, 1987; Robertson et al., 1995), velocity inhomogeneities(Sato, 1988), nonrandom distribution of density (Thorarinssonand Magnusson, 1990) and susceptibility (Pilkington andTodoeschuck, 1993; Pilkington et al., 1994; Maus and Dimri,

1994, 1995, 1996), reectivity (Todoeschuck et al., 1990; Dimri,1992), etc.] cannot be accurately measured unless its fractaldimensiondoes notexceedthe difference of the2-DEuclideanand fractal dimension of the network (Lovejoy et al., 1986;Korvin, 1992).

In this short note, the fractal dimension and detectibilitylimits for homogeneous and inhomogeneous distributions of measuring networks are investigated with the help of a grav-ity eld example. The detectibility limits of an aeromagneticsurvey for fractal magnetization or scaling geology is derived.A numerical example illustrates the detectibility limits of anaeromagnetic survey to resolve the scaling magnetic sourcesas a function of ight height.

FRACTAL DIMENSION OF GRAVITY NETWORK

In traditional Euclidean geometry, a line is onedimensional,a plane two dimensional, and a cuboid three dimensional.Mandelbrot (1982) introduced the concept of fractal geom-etry for ragged edges, broken surfaces, networks, and all kindof irregular shapes. Their dimensions need not be integers. Hecoined the word “fractal” from the Latin word “fractus,” an

Manuscript received by the Editor April 28, 1997; revised manuscript received May 8, 1998.∗National Geophysical Research Institute, Uppal Road, Hyderabad 500 007, India. E-mail: [email protected] 1998 Society of Exploration Geophysicists. All rights reserved.

adjectivefromthe verb “frangere” meaningto break.There arenow many methods to estimate the fractal dimension of irreg-ular surfaces (Mandelbrot, 1982; Turcotte,1992; Korvin, 1992).

I have used as an example the gravity measuring network of theAbitibi Greenstone Belt in eastern Canada(Keating, 1995)to estimate the fractal dimension of the measuring network. Ihave divided the network into two halves, referred to blocks AandB (see Figure 1),sincethereis an unevendistribution of the1442 gravity stations. The well-known box counting technique(Mandelbrot, 1982) has been applied to estimate the fractaldimension of block A as 1.06 and block B as 1.2 (Figure 2). Theinhomogeneous distribution produces a marginally less fractaldimension for block A than the gridded and denser (especiallyat the right bottom corner) block B.

The detectibility limits of geophysical surveys is given byKorvin (1992) as

Ds = E − Dn (1)where Dn and D s are the fractal dimensions of the networkand the source, respectively, and E is the Euclidean dimension(which is 2 here).

So, the lower detectibility limit for blocks A and B are 0.94and 0.8, respectively. Sources with fractal dimensions less thanthese limits cannot be detected.

DETECTIBILITY LIMITS OF AEROMAGNETIC SURVEYS

The problemof lowerdetectibilitylimitsof geophysical mea-suring networks due to inhomogeneous distributioncanbe un-

derstood in terms of locating the sources where the fractal di-mension is not known for most geological settings. Pilkingtonand Todoeschuck (1993), Pilkington et al., (1994), and Mausand Dimri (1994, 1995, 1996) have shown that the spatial vari-ation of geophysical parameters such as density and suscep-tibility exhibit power spectra proportional to f −β , where f isfrequencyand −β is a scaling exponent.Mausand Dimri (1995)derived relations between the scaling exponent of the source

1943



1944 Dimri

and the eld as

γ grav = β dens +1 and γ magn = β susc −1, (2)

where γ grav and γ magn are scaling exponents of gravity andmagnetic eld due to scaling of density β dens and susceptibilitydistribution β susc , respectively.

The obvious advantage of equation (2) is that it enables usto determine an unknown scaling exponent if the other fac-tor is known, which is generally the case with geophysicalstudies.

The spectrum of a 2-D magnetic survey is given by Naidu(1969) and Mishra and Naidu (1974) as

SK (u , v ) =1K

K

k =−1

1 L x L y | X k (u , v )|2, (3)

where S(u , v ) is power spectrum of a 2-D aeromagnetic eld, X (u, v ) is the Fourier transform of the eld, L x and L y arelength dimensions, and u and v are frequency in the x and y

directions, respectively, and given as

u =2π L x

and v =2π L y

. (4)

F IG . 1. Location map of gravity stations, indicated by points. The open circles, crosses, and stars indicate stationswhere the difference between the accuracy and the calculated error is less than 0.5 mGal, between 0.5 and1.0 mGal, and more than 1 mGal, respectively (after Keating, 1995). The accuracy (also called the mean error)is dened as the square root of the sum of the squares of all the errors affecting the gravity anomalies of a givenstation (Keating, 1995).

Following Mishra and Naidu (1974), the radial spectrum for a2-D survey is

R( f ) =1

2π 2π

0 f x( f cos θ, f sin θ ) d θ, (5)

F IG . 2. The log-log plot between thesizeofbox(L) andnumberof gravity stations (N). The slope of the line gives the fractaldimension for block A (solid line) and block B (dashed line).



Fractal Behavior of Geophysical Surveys 1945

where u = f cos θ, v = f sin θ , and

f = u 2 +v2. (6)

For the scaling elds, equation (6) becomes

f = [ u 2 +v2]−γ

= (2π/ L x)2 +(2π/ L y)2 −γ . (7)

Letusassumethatthegridhasequalspacingof L x = L y = L .Then, equation (7) becomes

f =[2π √ 2]−γ

L−γ . (8)

Several authors (e.g., Connard et al., 1983; Blakely, 1995),while analysing the spectra of magnetic data due to a nitesource bounded by depths d 1 and d 2 (d 2 > d 1), found that themaximum peak has a frequency f max given by

f max ≥loge d 2

−loge d 1

d 2 −d 1 . (9)

Connard et al. (1983) showed that f max should be at least twicethe fundamental frequency in order to resolve a peak in thespectrum. Though f in equation (5) is a variable, f max in equa-tion (9) is a xed number as in equation (8). Hence, equations(8) and (9) can be combined to get

L−γ ≥[2π √ 2]−γ (2)( d 2 −d 1)

loge d 2 −loge d 1(10)

or

L

≥2π √ 2 2(d 2 −d 1)

loge d 2 − loge d 1

−1/γ

. (11)

A similarexpression fora 1-D survey lengthcanbe obtainedas

L ≥2π2(d 2 −d 1)

loge d 2 −loge d 1

−1/γ

. (12)

For −γ =1, equation (12) reduces to

L ≥4π (d 2 −d 1)

loge d 2 −loge d 1, (13)

as obtained by Blakely (1995) for white noise distribution.

Table 1. Survey and grid lengths of aeromagnetic surveys conducted at 1 km in order to resolve 4-km and 50-km depths fordifferent scaling exponents.

Source scaling exponent, Field scaling exponent, Survey length, Grid length,Survey number Depth (km) −β −γ L(km) L(km)

1 4 2 1 27 382 4 3 2 13 183 4 4 3 10 144 50 2 1 157 2225 50 3 2 31 456 50 4 3 18 26

Equations (11) and(12) show that thegrid lengths area func-tion of the scaling exponent. A numerical example illustratesthe application of estimating the optimum value of grid andsurvey length.

NUMERICAL EXAMPLE

Consider d 1 =1km, d 2 =4km.Letusndtheoptimumvalue

of grid length for different scaling exponents. The eld scalingexponent can be known by estimating the power spectrum of the eld. For this numerical example, let us consider setting

−β =2 to 4 or −γ =1 to 3, which have been normally reportedfor different aeromagnetic sources and their elds (Maus andDimri, 1996; Maus et al., 1997). The survey and grid length incase of 1-D and 2-D aeromagnetic measurements are shownin Table 1. The table also shows the results for d 1 = 1 km andd 2 =50 km with varying scaling exponents. Both examples arerepresentative of basin and regional structure, and the bottomin the latter example maybe a petrological or a temperatureboundary (Maus et al., 1997).

From Table 1, we see that the survey length variesfrom10 to27 km in order to resolve the magnetic source at 4-km depth

from theaeromagneticsurvey conducted1 km above the topof magnetic source. For thesame depthandresolution, a grid areafrom 14 ×14 km to 38 ×38 km is optimum. Similarly, surveylengths intherangeof 18to 157 kmandgrid areas ontheorderof 26 to 222 km2 are required to resolve a depth of 50 km. Asexpected,the gridlength for2-D is more than thesurveylengthfor 1-D. Also, the greater the depth to be resolved, the largeris the required survey length or grid area.

Another interesting nding of the numerical example is thatthe length of grid survey decreases with increased scaling ex-ponents. It means that the detectibility limits depend on thevalue of the scaling exponents. We know that higher scalingexponents show stronger afnity to or more dependency onvariables. The scaling exponent may represent the proportion

of long and short wavelength variation of signals (Maus andDimri, 1995). Thus, the value of the eld scaling exponent of the sources may be a guide to the optimum height of an aero-magnetic survey depending on the depth of the source to beresolved.

CONCLUSION

A dense gridded distribution of a gravity measuringnetworkproduces a higher fractal dimension than an uneven distribu-tion. The detectibility limits of an aeromagnetic survey dependonthe scalingexponents, whichcan bedetermineda priorifromthe power spectrum of the eld and the sources under investi-gation or elsewhere from the adjoining area (Maus and Dimri,



1946 Dimri

1996). The numerical example illustrates that an aeromageticsurvey conducted 1 km above the top of magnetic sources witha scaling exponent of 2 to 4 (Maus et al., 1997) must have agrid survey area of 14 to 38 km 2 and 26 to 222 km 2 in order toresolve depths of 4 km and 50 km, respectively.

ACKNOWLEDGMENTS

The comments of reviewers have improved the manuscript.For this, I thank Dr. Kees Wapenaar. This work is a part of theproject on fractals in geophysics funded by the Department of Science and Technology, Government of India.Thanksare dueto Dr.Ajai Manglikand Mr. Abhey Ramforuseful discussionsandotherhelp.IamgratefultoDr.H.K.Gupta,DirectoroftheNGRI, for constant encouragement and permission to publishthis work.

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Dimri, V. P., 1992, Deconvolution and inverse theory: Application togeophysical problems: Elsevier Science Publishers.Keating,P., 1995, Errorestimation andoptimizationof gravitysurveys:

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Geophysics 1998