Gravity gradient tensor of a finite line of mass of arbitrary orientation

7/27/2019 Gravity gradient tensor of a finite line of mass of arbitrary orientation

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Gravity gradient tensor of a finite line

of mass of arbitrary orientation

Mark GettingsU.S. Geological Survey, 520 N. Park Ave. Rm 355

Tucson, AZ 85719, USA.([email protected])

1 Abstract

Analytical expressions for the gravitational attraction and gravity gradienttensor are given for a model consisting of a finite length line of mass of ar-bitrary orientation. The expressions are valid for all points outside the lineof mass. Expressions for the tensor magnitude and three standard invariantsof a 3x3 tensor are also given. Fortran code to evaluate the gravitational at-traction, the gravity gradient tensor, tensor magnitude, tensor invarients, andtensor eigenvalues and eigenvectors for an xyz grid of field points is listed in anappendix. For the example presented, the tensor magnitude and second tensorinvariant are essentially mirror images of each other, and the third tensor invari-ant is useful for defining the location of the line of mass. The model is useful for

studying the gravitational effects of approximately linear mass concentrations,tunnels, and off-axis gravitational effects of cylinders.

2 Introduction

There is a continuing interest in gravity gradiometry from several areas ofinterest, especially mineral exploration geophysics and tunnel detection. Con-tinuing efforts in airborne gravity gradiometry in mineral exploration has ledto steady improvements in instrumentation [Lee, 2001; Dransfield et al., 2001;Hinks et al., 2004; DiFrancesco et al., 2008] and analysis techniques [Mickus andHinojosa, 2001; Li, 2001; Heath et al., 2005; Saad, 2006; Murphy, 2007]. In-terest in tunnel and underground void detection [Romaides et al., 2001; Maier,2002] is increasing due to the renewed efforts at border security. Previous modelstudies have focused on a point source [Maier, 2002], a horizontal line of massof fixed or infinite length [Romaides et al., 2001], or using voxels computed fromthe gravitational effects of right rectangular prisms [Montana et al., 1992]. Thevoxel model is very flexible but requires computation of very large numbers ofvoxels for realistic models [Heath et al., 2005]. For some modeling problems, itwould be convenient to have an analytic model of a line of mass of arbitrary

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orientation in order to model effects such as a dipping shaft, the off-axis gravity

field of a cylinder, or any approximately linear density contrast.

In this report, a model for the gravitational attraction and gravity gradi-ent tensor of a finite length line of mass of arbitrary orientation is presentedalong with computer code for routine calculation. Expressions for some invari-ant quantities of the gradient tensor (quantities that are constant regardless ofthe coordinate system) are also given in order to develop exploration schemesindependent of the orientation of the target body. As shown by both Romaideset al. [2001] and Maier [2002], the effects for typical underground tunnels orstructures is quite small and either very low level airborne or ground basedmeasurements are required for detection. Moreover, geologic noise due to localgeologic variations are of the same order or larger in amplitude [Heath et al.,2005; Dransfield et al., 2001; Murphy, 2007]. If a grid of observations is available,a pattern recognition algorithm would be the most reliable form of detection.In the case of profile only data, detection is more difficult, and is the subject ofongoing research.

3 The potential

The line of mass is taken along the x-axis extending from L to L in a right-handed coordinate system (Fig. 1). The body vector r0 is (, 0, 0) (along thex-axis) and the position vector for the observation point in the line of masssystem is r = (x,y,z). The line of mass has density/unit length of and G isthe universal constant of gravitation. The gravitational potential is then [Grantand West, 1965]

U(r ) =

LL

Gd(x )2 + y2 + z2

(1)

and thus

U(r )

G= ln[

L x +

(L x)2 + y2 + z2

L x +

(L + x)2 + y2 + z2] (2)

4 Gravity vector

The gravity vector components were calculated both manually and usingthe differentiation and simplification features of Mathematica 5.1 [Wolfram Re-search, 2004] in order to insure there were no errors.

1

G

U(r )

x=

1(L x)2 + y2 + z2

+1

(L + x)2 + y2 + z2(3)

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Figure 1: Geometry of the finite line of mass in the body-centered system

1

G

U(r )

y=

y

y2 + z2(x(

1(L x)2 + y2 + z2

+1

(L + x)2 + y2 + z2)+

L(1

(L x)

2

+ y2

+ z2

+1

(L + x)

2

+ y2

+ z2

))

(4)

1

G

U(r )

z=

z

y2 + z2(x(

1(L x)2 + y2 + z2

+1

(L + x)2 + y2 + z2)+

L(1

(L x)2 + y2 + z2+

1(L + x)2 + y2 + z2

))

(5)

5 Gravity gradient tensor components

The algebra involved in calculating the tensor components is tedious thoughstraightforward; one component was calculated both manually and using Math-ematica 5.1. The remaining components were calculated with Mathematica 5.1and some final simplification and factoring was done manually.

1

G

2U(r )

xy=

y

((L x)2 + y2 + z2)3/2

y

((L + x)2 + y2 + z2)3/2(6)

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1G

2U(r )xz

= z((L x)2 + y2 + z2)3/2

z((L + x)2 + y2 + z2)3/2

(7)

1

G

2U(r )

yz=

1

(y2 + z2)2(yz(

(L x)(y2 + z2)

((L x)2 + y2 + z2)3/2+

2(L x)(L x)2 + y2 + z2

+

(L + x)(y2 + z2)

((L + x)2 + y2 + z2)3/2+

2(L + x)(L + x)2 + y2 + z2

))

(8)

1

G

2U(r )

x2 =(L x)

((L x)2 + y2 + z2)3/2 (L + x)

((L + x)2 + y2 + z2)3/2 (9)

1

G

2U(r )

y2=

1

(y2 + z2)2(

(L x)y2(y2 + z2)

((L x)2 + y2 + z2)3/2+

(y2 z2)(L x)(L x)2 + y2 + z2

+

y2(y2 + z2)(L + x)

((L + x)2 + y2 + z2)3/2+

(L + x)(y2 z2)(L + x)2 + y2 + z2

)

(10)

1

G

2U(r )

z2

=1

(y2

+ z2

)2

(z2(y2 + z2)(L x)

((L x)2

+ y2

+ z2

)3/2

(L x)(z2 y2)(L x)

2

+ y2

+ z2

+

z2(y2 + z2)(L + x)

((L + x)2 + y2 + z2)3/2+

(L + x)(z2 y2)(L + x)2 + y2 + z2

)

(11)

6 Coordinate transformations

The above expressions were derived in the body-centered system with the lineof mass centered along the x-axis (Fig. 1). To use the formulae in general, weneed to define the coordinate transformation between the field or Earth systemand the body-centered system (Fig. 2). Right-handed coordinate systems wereassumed for this study. A north-east-down (positive x, y, and z directions) was

used for the field system, and the line of mass lies along the x axis centered atthe (x, y, z) body-centered system origin, as above. We have that

x = A(x x0) (12)

where x is the position vector ofP(x,y,z) in the field system, x0 is the positionvector in the field system of the origin of the (x, y, z) system (center of the

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line of mass), andx is the position vector of the point P in the body-centered

system. The angle is the angle between the +x direction and the projection ofthe +x axis on the xy plane, measured counterclockwise from north (xaxis).The angle is the dip of the +x axis relative to the xy plane measured positivecounterclockwise in the xz plane (Fig. 2). Thus a horizontal line of mass wouldhave a of zero and a strike of , and a vertical line of mass would have a of 90. In the vertical case, is not defined and taken as zero. The rotationmatrix A thus defined is independent of the observation point and is unchangedfor all observations of a given source. It is given by

A =

cos cos cos sin sinsin cos 0

sin cos sin sin cos

(13)

and the inverse transformation matrix A1 is given by the transpose

A1 = At (14)

The calculation therefore proceeds with the following steps for each observa-tion (field) point:

1. Transform the field point coordinates P(x,y,z) to the body-centered sys-

tem (x, y, z) withx = A(x x0)

2. Calculate the three gravitation vector componentsg (equations 3-5 above)

and the six gradient tensor components (equations 6-11 above) and formthe gradient tensor g.

3. Transform back to the field system using [Butkov, 1968; Spiegel, 1959]g = Atg and g = AtgA to obtain the desired gravity vector and

gravity gradient tensor.

These formulae are exact outside the body for any geometry symmetric aboutthe x axis, for example a circular tunnel or rod, with the density per unit length appropriately calculated (R2 for a circular cross section of density andradius R, etc.) Example applications are a dipping tunnel and off-axis gravityeffects of a buried vertical cylinder or shaft.

The gravity gradient tensor components all depend on the coordinate systemand the orientation of the line of mass relative to it. It is thus useful to calculatequantities that are invariant with regard to the coordinate system for some stud-

ies. The quantities that are typically calculated include the tensor magnitude,the three tensor invariants, and the eigenvalues and associated eigenvectors ofthe tensor. The tensor magnitude for a tensor Tij is given by the square root ofhalf the tensor double dot product [Dutton, 2002; Anonymous, 2001]

| Tij |=

1

2

i

j

T2ij (15)

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Figure 2: Geometry of the field (right-handed (x,y,z); e.g. north-east-down) andbody-centered coordinate systems (x,y,z). P(x,y,z) is the observation point,

and the blue line shows the finite line of mass. The orientation of the line ofmass is specified by a strike and a dip in the fieldpoint system. The symbol x means parallel to the x-axis.

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The three standard invariants for a 33 tensor are [Wikipedia, 2008; Zhukov

et al., undated] Ig = gxx + gyy + gzz (16)

IIg = gxxgyy + gyygzz + gzzgxx g2

xy g2

yz g2

xz (17)

andIIIg = det (g) (18)

Ig is identically zero for a gravitational field in free space by virtue of Laplacesequation.

In addition, it is useful to compute the eigenvalues and eigenvectors whichcan be used for principal components analysis, trends analysis and other schemesin modeling. A computer code was written to compute the quantities defined

above and is given in the appendix. Eignvalues and eigenvectors were computedusing the open-source subroutine library described by Kopp [2008].

7 Example of a dipping line of mass

As an example, the vertical gravity component and gradient tensor magnitudeand invariants for a line of mass 30m long striking 60 east of north and dipping30 to the northeast are shown in figures 3-6. The model was computed atthe nodes of a 5 m grid extending 100m north-south and 150 m east west withmeasurements made at an altitude of 1m above the ground. The center of theline of mass is at 50 m north and 75 m east, at a depth of 7.5 m. The southwestend of the line of mass just reaches the surface. For this model, the linear

density was 6283 kg/m corresponding to a 1m radius line with a density of 2000kg/m3.

Examination of Fig. 4 and Fig. 5 shows that the tensor magnitude andsecond tensor invariant are essentially mirror images of each other; the tensormagnitude extends over the map to larger distances from the source line thanthe second tensor invariant resulting in a slightly larger map area of the tensormagnitude. The third tensor invariant (Fig.6) contains negative lobes off theends of the mass line that might be helpful in estimating the location of theends of the mass line, particularly the shallow end. This invariant is also morelocal in its positive signature near the mass line than the other two measures(Figs. 4 and 5), with steeper gradients and a more restricted map area about the

source. This sharper anomaly shape of the third tensor invariant could make itmore useful in identifying the mass line location in a noisy environment. Murphy[2007] has shown the utility of two invarients of the horizontal components of thegradient tensor in mineral and hydrocarbon exploration using airborne gravitygradiometer data.

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Figure 3: Gravity effect (z-component of the gravity vector) of a finite line ofmass striking 60 east of north and dipping 30 to NE. Bold black line is theprojection of the line of mass on the surface. Line of mass is at the surface onthe southeast end and the observation plane is 1m above the surface. Contourinterval is 0.0002 mgal.

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Figure 4: Tensor magnitude (equation 15) of a finite line of mass striking 60

east of north and dipping 30 to NE. Bold black line is the projection of theline of mass on the surface. Line of mass is at the surface on the southeast endand the observation plane is 1m above the surface. Contour inerval is 20 Eotvos

units

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Figure 5: The second tensor invariant (equation 17) of a finite line of massstriking 60 east of north and dipping 30 to NE. Bold black line is the projectionof the line of mass on the surface. Line of mass is at the surface on the southeastend and the observation plane is 1m above the surface. Contour interval is 20,000

Eotvos2

.

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Figure 6: Third tensor invariant (equation 18) of a finite line of mass striking60 east of north and dipping 30 to NE. Bold black line is the projection of theline of mass on the surface. Line of mass is at the surface on the southeast endand the observation plane is 1m above the surface. Contour interval is 100,000

Eotvos3

.

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8 Acknowledgments

The author thanks Jeff Phillips and Jeff Wynn of the U.S. Geological Surveyfor helpful reviews of the manuscript. Publication of this report was approvedby the Director, U.S. Geological Survey.

References

Anonymous, Vector and tensor mathematics, [Onlinewww.polymerprocessing.com/notes/root92a.pdf; accessed 6-November-2008], 2001.

Butkov, E., Mathematical Physics, Addison Wesley Publishing Co., 735 p., 1968.

DiFrancesco, D., D. Kaputa, and T. Meyer, Gravity gradiometer systems; ad-vances and challenges, Preview, 133, 3036, 39, 2008.

Dransfield, M. H., A. N. Christensen, M. Rose, P. M. Stone, and P. Diorio, FAL-CON test results from the Bathurst Mining camp, Exploration Geophysics,32(3-4), 243246, 2001.

Dutton, J. A., The Ceaseless Wind: An Introduction to the Theory of Atmo-spheric Motion, Courier Dover Publications, 2002.

Grant, F. S., and G. F. West, Interpretation Theory in Applied Geophysics,McGraw-Hill Book Company, 583 pp., 1965.

Heath, P. J., S. Greenhalgh, and N. G. Direen, Modelling gravity and magnetic

gradient tensor responses for exploration within the regolith, Exploration Geo-physics, 36(4), 357364, 2005.

Hinks, D., S. McIntosh, and R. Lane, A comparison of the FALCON andAir-FTG airborne gravity gradiometer systems at the kokong test block,botswana, in Airborne gravity 2004- Abstracts from the ASEG-PESA air-borne gravity 2004 workshop, edited by R. Lane, pp. 125134, GeoscienceAustralia Record 2004/18, 2004.

Kopp, J., Efficient numerical diagonalization of hermitian 3x3 matrices, Inter-national Journal of Modern Physics C, 19, 523548, 2008.

Lee, J. B., FALCON gravity gradiometer technology, Exploration Geophysics,

32(3-4), 247250, 2001.Li, X., Vertical resolution; gravity versus vertical gravity gradient, The Leading

Edge, 20(8), 901904, 2001.

Maier, M. W., Underground structures and gravity gradiometry, Tech. Rep.ADA400252, AEROSPACE CORP EL SEGUNDO CA RECONNAISSANCESYSTEMS DIV, 2002.

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Mickus, K. L., and J. H. Hinojosa, The complete gravity gradient tensor derived

from the vertical component of gravity; a fourier transform technique, Journalof Applied Geophysics, 46(3), 159174, 2001.

Montana, C. J., K. L. Mickus, and W. J. Peeples, Program to calculate thegravitational field and gravity gradient tensor resulting from a system of rightrectangular prisms, Computers and Geosciences, 18(5), 587602, 1992.

Murphy, C. A., Interpreting FTG gravity data using horizontal tensor compo-nents, EGM 2007 International Workshop - Innovation in EM, Grav and MagMethods: a new perspective for exploration, Capri, Italy, April 2007, 2007.

Romaides, A. J., J. C. Battis, R. W. Sands, A. Zorn, D. O. B. Jr., and D. J.DiFrancesco, A comparison of gravimetric techniques for measuring subsur-face void signals, J. Phys. D: Appl. Phys., 34, 433443, 2001.

Saad, A. H., Understanding gravity gradients; a tutorial, The Leading Edge,25(8), 942949, 2006.

Spiegel, M. R., Vector Analysis and an Introduction to Tensor Analysis,Schaums Outline Series, McGraw-Hill Book Co., 225 p., 1959.

Wikipedia, Invariants of tensors wikipedia, the free encyclopedia, [Online;accessed 20-November-2008], 2008.

Wolfram Research, Mathematica, version 5.1 ed., Wolfram Research, Inc.,Champaign, Illinois, 2004.

Zhukov, L., K. Museth, D. Breen, R. Whitaker, and A. H. Barr, Tensor invari-

ants for modeling and segmentation of diffusion weighted MRI data, depart-ment of Computer Science,California Institute of Technology, undated.

Appendix - Fortran computer code

c Main program module and all associated subroutines

c Calculate gravity vector and gravity gradient tensor

c for a finite line of mass of arbitrary orientation

c MEG/Sep08; Nov 08 add invariants and eigen vals/vecs;

c and output file single ascii lines

c Although this code has been tested and verified, it is not

c guaranteed to perform correctly in all cases, and may containc bugs that have not been identified

c

real*4 a(9),ai(9),x(3),xp(3),x0(3),theta,phi,hl,lamda

real*4 strike, dip, xin(3), g(3), gp(3), gmn(9), gmnp(9)

real*4 Gcons, tmag, tinvar(3), teval(3), tevec(3,3)

real*4 unitsg, unitst

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character qu*80,qr*1,dflt1*1,fnam1*80,df80*80,fmt*80,fnam2*80

character(8) datecharacter(10) time

logical verify

ittin=5

ittout=6

io1 = 10

io2 = 11

verify=.false.

Gcons = 6.674e-11

call date_and_time(DATE=date,TIME=time)

c

c get line of mass parameters, filenames, set up transform matrices

c

write(ittout,200)200 format(Enter line of mass parameters, units m, degrees,,

1 kg/m**3)

qu=Enter x0:

call ttinr4(qu,x0(1),-999.,ittin,ittout,verify)

if(x(1).le.-999.) go to 900

qu=Enter y0:


qu=Enter z0:


qu=Enter strike(deg CL off N, 0-360)[0]:

call ttinr4(qu,strike,0.,ittin,ittout,verify)

theta = 360. - strikequ=Enter dip(deg, + down, -90 to 90)[0]:

call ttinr4(qu,dip,0.,ittin,ittout,verify)

phi = -dip

call matfrm(1,theta,phi,a)

call mattra(a,ai,3,3)

qu=Enter half length of line of mass(m)[10]:

call ttinr4(qu,hl,10.,ittin,ittout,verify)

qu=Enter lamda (density/unit length kg/m)[1000]:

call ttinr4(qu,lamda,1000.,ittin,ittout,verify)

qu=Enter 1 for mks, 2 for mgal,eotvos, 3 for gu,SI units[1]:

call ttini4(qu,iunit, 1, ittin, ittout, verify)

if(iunit.eq.1) then

unitsg=1.0

unitst=1.0

elseif(iunit.eq.2) then

unitsg=1.0e5

unitst=1.0e9

elseif(iunit.eq.3) then

unitsg=1.0e6

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unitst=1.0e9

endifqu=Filename of field point coordinates (


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220 format(body system:,0pf10.1,1p4e11.3)

cc transform g back to fieldpoint system

c

call matmul(ai,gp,g,3,3,1)

c

c gradient tensor transform to fieldpoint system

c use loop 52 to mult g by G*lamda

c

c write(io2,222) (g(k),k=1,3)

222 format(>>>,1p3e11.3)

do 52 jc=1,3

g(jc) = g(jc)*Gcons*lamda*unitsg

52 continue

call matmul(ai,gmnp,gmn,3,3,3)call matmul(gmn,a,gmnp,3,3,3)

do 58 jr=1,9

gmn(jr) = gmnp(jr)*Gcons*lamda*unitst

58 continue

c

c calculate tensor invariants, eigenvalues and eigenvectors

c

call tinvar3x3(gmn, tmag, tinvar, teval, tevec)

c

c output to io2

c

write(io2, 210) xin,g,gmn(1),gmn(5),gmn(9),gmn(4),gmn(7),1 gmn(8),tmag,tinvar,teval,((tevec(i,j),i=1,3),j=1,3)

210 format(1p28e11.3)

go to 500

900 close(io1)

close(io2)

stop

end

subroutine flinmass(xc,hl,gp,gmnp)

c

c Finite line of mass from -L to +L on x axis

c Calculate gravity vector and gradient tensor in body system

c vector xc contains (x,y,z) of field pointin body system

c on return g contains (gx,gy,gz) components of gravitation

c array gmn contains gradient tensor gmn(i), stored by column,

c so gmn(1)=gxx, gmn(3)=gzx, gmn(4)=gxy,gmn(7)=gxz, gmn(9)=gzz

c Units of G*lamda (gravconst, density/unit length)

c MEG/Aug2008

c


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c vectors tevec, store xyz by column - 1st vector is tevec(1,1)

c tevec(2,1),tevec(3,1), 2nd in col 2, 3rd in col3c uses Kopps routines for eigensolution

c arXiv:physics/0610206v3 [physics,comp-ph] 4 Jul 2008

c Efficient numerical diagonalization of hermitian 3 x 3 matrices

c Gettings Nov 2008

c


c guaranteed to perform correctly in all cases, and may contain

c bugs that have not been identified

c

real*4 gmn(9), tmag, tinvar(3), teval(3), tevec(3,3)

real*8 sum, g(3,3), eval(3), evec(3,3)

c

c store tensor in 2D array, calculate tensor magnitude and invariantsc

sum = 0.0

do 50 j=1,3

do 50 i=1,3

k = (j-1)*3+i

g(i,j) = gmn(k)

sum = sum + gmn(k)*gmn(k)

50 continue

tmag = dsqrt(sum * 0.5)

tinvar(1) = g(1,1) + g(2,2) + g(3,3)

tinvar(2) = g(1,1)*g(2,2) + g(2,2)*g(3,3) + g(1,1)*g(3,3) -

1 g(1,2)*g(1,2) - g(2,3)*g(2,3) - g(1,3)*g(1,3)tinvar(3) = gmn(1)*gmn(5)*gmn(9) + gmn(4)*gmn(8)*gmn(3) +

1 gmn(7)*gmn(2)*gmn(6) - gmn(3)*gmn(5)*gmn(7) -

2 gmn(6)*gmn(8)*gmn(1) - gmn(9)*gmn(2)*gmn(4)

c

c calculate eigenvalues and eigenvectors

c

call dsyevh3(g,evec,eval)

do 60 i=1,3

teval(i) = eval(i)

do 60 j=1,3

tevec(i,j) = evec(i,j)

60 continue

return

end

function cosin(theta)

c ** computes cosine of any angle(in degrees) between 0 and 360.




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c

tol=1.0e-8pi=acos(-1.0)

dgr=pi/180.0

if(abs(theta-90.0).le.tol.or.abs(theta-270.).le.tol) go to 50

if(theta.lt.0)theta=360.0+theta

thetar=theta*dgr

itheta=theta/90.0+1.

go to (10,20,30,40),itheta

10 cosin=cos(thetar)

return

20 thetar=pi-thetar

cosin=-cos(thetar)

return

30 thetar=thetar-picosin=-cos(thetar)

return

40 thetar=2*pi-thetar

cosin=cos(thetar)

return

50 cosin=0.0

return

end

function sine(theta)

c ** computes sine of any angle(in degrees) between 0 and 360.


c guaranteed to perform correctly in all cases, and may containc bugs that have not been identified

c

tol=1.0e-8

pi=acos(-1.0)

dgr=pi/180.0

if(abs(theta).le.tol.or.abs(theta-180.0).le.tol) go to 50

if(abs(theta-360.0).le.tol) go to 50

if(theta.lt.0)theta=360.0+theta

thetar=theta*dgr

itheta=theta/90.+1

go to (10,20,30,40),itheta

10 sine=sin(thetar)

return

20 thetar=pi-thetar

sine=sin(thetar)

return

30 thetar=thetar-pi

sine=-sin(thetar)

return

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40 thetar=2*pi-thetar

sine=-sin(thetar)return

50 sine=0.0

return

end

subroutine matfrm(k,theta,phi,a)

c**subroutine to set up rotation transformation matrices, using

c**right-handed systems;k=0 for 2-dimension,k=1 for 3-dimensions.

c**matrix stored by column (col1,col2, etc.)




c

dimension a(*)cst=cosin(theta)

snt=sine(theta)

if(k.eq.1)go to 10

a(1)=cst

a(2)=snt

a(3)=0.0

a(4)=-snt

a(5)=cst

a(6)=0.0

a(7)=0.0

a(8)=0.0

a(9)=1.0go to 20

10 csp=cosin(phi)

snp=sine(phi)

a(1)=csp*cst

a(2)= snt

a(3)= cst*snp

a(4)=-csp*snt

a(5)= cst

a(6)=-snt*snp

a(7)=-snp

a(8)=0.0

a(9)= csp

20 return

end

subroutine mattra(a,r,n,m)




c

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23/36


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c Terminal entry of integer*4 variable - F77 version

c qu - Character string prompt que,80 C long.c i4 - variable to be input

c defalt - default value if only given

c itin,itout - read, write LUNs of terminal

c verify - if true, asks for verification of input

c if false no verification

c typical call:

c call ttini4(que,k,16,itin,itout,verify)

c MEG/ Feb 86; SUN conversion Jul 89/ MEG

c




clogical verify

character ans*16, blank*16, qu*80

character que(80)*1, qr*1

integer*4 i4, defalt

blank =

read(unit=qu, fmt=200) que

200 format(80a1)

call lnnobl(que, nq, 80)

300 write(unit=itout, fmt=204)

204 format(1h ,$)

do 40 i = 1, nq

40 write(unit=itout, fmt=201) que(i)201 format(a,$)

read(unit=itin, fmt=101, err=300) ans

101 format(a)

if (ans .eq. blank) goto 910

read(unit=ans, fmt=100) i4

100 format(bn,i16)

if (.not. verify) goto 900

write(unit=itout, fmt=202) i4

202 format(1x,6hValue ,i16,5h ok? ,$)

read(unit=itin, fmt=101) qr

if ((qr .eq. n) .or. (qr .eq. N)) goto 300

900 return

910 i4 = defalt

goto 900

end

subroutine ttini2(qu, i2, defalt, itin,itout, verify)

c Terminal entry of integer*2 variable - F77 version

c qu - Character string prompt que,80 C long.

c i2 - variable to be input

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c itin,itout - read, write LUNs of terminal

c verify - if true, asks for verification of inputc if false no verification

c typical call:

c call ttinaa(que,fnam,16,defalt,itin,itout,verify)

c MEG/ Dec 85; SUN conversion Jul 89/ MEG

c




c

logical verify

character css(80)

character qu*80

character que(80)*1, qr*1c character defalt(*), cs(*) replaced with following:

character*(*) cs,defalt

read(unit=qu, fmt=200) que

200 format(80a1)

call lnnobl(que, nq, 80)

300 write(unit=itout, fmt=204)

204 format( ,$)

do 40 i = 1, nq

40 write(unit=itout, fmt=201) que(i)

201 format(a1,$)

read(unit=itin, fmt=101, err=300) css

101 format(80a1)call lnnobl(css, kcss, 80)

c go to default

if (kcss .eq. 0) goto 910

write(cs,(80a))(css(i),i=1,ncs)

if (.not. verify) goto 900

write(unit=itout, fmt=202) (css(i),i = 1, ncs)

202 format(1x,17hCharacter string /1x,80a:)

write(unit=itout, fmt=203)

203 format(5h ok? ,$)

read(unit=itin, fmt=101) qr

if ((qr .eq. n) .or. (qr .eq. N)) goto 300

900 return

910 cs = defalt

goto 900

end

* ----------------------------------------------------------------------------

* Numerical diagonalization of 3x3 matrcies

* Copyright (C) 2006 Joachim Kopp

* ----------------------------------------------------------------------------

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* This library is free software; you can redistribute it and/or

* modify it under the terms of the GNU Lesser General Public* License as published by the Free Software Foundation; either

* version 2.1 of the License, or (at your option) any later version.

*

* This library is distributed in the hope that it will be useful,

* but WITHOUT ANY WARRANTY; without even the implied warranty of

* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU

* Lesser General Public License for more details.

*

* You should have received a copy of the GNU Lesser General Public

* License along with this library; if not, write to the Free Software

* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA

* ----------------------------------------------------------------------------

* ----------------------------------------------------------------------------

SUBROUTINE DSYEVH3(A, Q, W)

* ----------------------------------------------------------------------------

* Calculates the eigenvalues and normalized eigenvectors of a symmetric 3x3

* matrix A using Cardanos method for the eigenvalues and an analytical

* method based on vector cross products for the eigenvectors. However,

* if conditions are such that a large error in the results is to be

* expected, the routine falls back to using the slower, but more

* accurate QL algorithm. Only the diagonal and upper triangular parts of A need

* to contain meaningful values. Access to A is read-only.

* ----------------------------------------------------------------------------* Parameters:

* A: The symmetric input matrix

* Q: Storage buffer for eigenvectors

* W: Storage buffer for eigenvalues

* ----------------------------------------------------------------------------

* Dependencies:

* DSYEVC3(), DSYTRD3(), DSYEVQ3()

* ----------------------------------------------------------------------------

* .. Arguments ..

DOUBLE PRECISION A(3,3)

DOUBLE PRECISION Q(3,3)

DOUBLE PRECISION W(3)

* .. Parameters ..

DOUBLE PRECISION EPS

PARAMETER ( EPS = 2.2204460492503131D-16 )

* .. Local Variables ..

DOUBLE PRECISION NORM, N1, N2

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* Calculate second eigenvector by the formula

* v[1] = (A - lambda[1]).e1 x (A - lambda[1]).e2Q(1, 2) = Q(1, 2) + A(1, 3) * W(2)

Q(2, 2) = Q(2, 2) + A(2, 3) * W(2)

Q(3, 2) = (A(1,1) - W(2)) * (A(2,2) - W(2)) - Q(3, 2)

NORM = Q(1, 2)**2 + Q(2, 2)**2 + Q(3, 2)**2

IF (NORM .LE. ERROR) THEN

CALL DSYEVQ3(A, Q, W)

RETURN

ELSE

NORM = SQRT(1.0D0 / NORM)

DO 40, J = 1, 3

Q(J, 2) = Q(J, 2) * NORM

40 CONTINUE

END IF

* Calculate third eigenvector according to

* v[2] = v[0] x v[1]

80 Q(1, 3) = Q(2, 1) * Q(3, 2) - Q(3, 1) * Q(2, 2)

Q(2, 3) = Q(3, 1) * Q(1, 2) - Q(1, 1) * Q(3, 2)

Q(3, 3) = Q(1, 1) * Q(2, 2) - Q(2, 1) * Q(1, 2)

END SUBROUTINE

* End of subroutine DSYEVH3

* ----------------------------------------------------------------------------

* Numerical diagonalization of 3x3 matrcies* Copyright (C) 2006 Joachim Kopp

* ----------------------------------------------------------------------------


* modify it under the terms of the GNU Lesser General Public

* License as published by the Free Software Foundation; either


*





*




* ----------------------------------------------------------------------------

* ----------------------------------------------------------------------------

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SUBROUTINE DSYEVC3(A, W)

* ----------------------------------------------------------------------------* Calculates the eigenvalues of a symmetric 3x3 matrix A using Cardanos

* analytical algorithm.

* Only the diagonal and upper triangular parts of A are accessed. The access

* is read-only.

* ----------------------------------------------------------------------------

* Parameters:


* W: Storage buffer for eigenvalues

* ----------------------------------------------------------------------------

* .. Arguments ..



* .. Parameters ..

DOUBLE PRECISION SQRT3

PARAMETER ( SQRT3 = 1.73205080756887729352744634151D0 )


DOUBLE PRECISION M, C1, C0

DOUBLE PRECISION DE, DD, EE, FF

DOUBLE PRECISION P, SQRTP, Q, C, S, PHI

* Determine coefficients of characteristic poynomial. We write

* | A D F |

* A = | D* B E |* | F* E* C |

DE = A(1,2) * A(2,3)

DD = A(1,2)**2

EE = A(2,3)**2

FF = A(1,3)**2

M = A(1,1) + A(2,2) + A(3,3)

C1 = ( A(1,1)*A(2,2) + A(1,1)*A(3,3) + A(2,2)*A(3,3) )

$ - (DD + EE + FF)

C0 = A(3,3)*DD + A(1,1)*EE + A(2,2)*FF - A(1,1)*A(2,2)*A(3,3)

$ - 2.0D0 * A(1,3)*DE

P = M**2 - 3.0D0 * C1

Q = M*(P - (3.0D0/2.0D0)*C1) - (27.0D0/2.0D0)*C0

SQRTP = SQRT(ABS(P))

PHI = 27.0D0 * ( 0.25D0 * C1**2 * (P - C1)

$ + C0 * (Q + (27.0D0/4.0D0)*C0) )

PHI = (1.0D0/3.0D0) * ATAN2(SQRT(ABS(PHI)), Q)

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P(I) = P(I) - K * U(I)

30 CONTINUE

D(1) = A(1,1)

D(2) = A(2,2) - 2.0D0 * P(2) * U(2)

D(3) = A(3,3) - 2.0D0 * P(3) * U(3)

* Store inverse Householder transformation in Q

* --- This loop can be omitted if only the eigenvalues are desired ---

DO 40, J = 2, N

F = OMEGA * U(J)

D O 4 1 I = 2 , N

Q(I,J) = Q(I,J) - F * U(I)

41 CONTINUE

40 CONTINUE

* Calculated updated A(2, 3) and store it in E(2)

E(2) = A(2, 3) - P(2) * U(3) - U(2) * P(3)

ELSE

D O 5 0 I = 1 , N

D(I) = A(I, I)

50 CONTINUE

E(2) = A(2, 3)

END IF

END SUBROUTINE

* End of subroutine DSYTRD3

* ----------------------------------------------------------------------------

* Numerical diagonalization of 3x3 matrcies

* Copyright (C) 2006 Joachim Kopp

* ----------------------------------------------------------------------------


* modify it under the terms of the GNU Lesser General Public

* License as published by the Free Software Foundation; either


*





*




* ----------------------------------------------------------------------------

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* ----------------------------------------------------------------------------

SUBROUTINE DSYEVQ3(A, Q, W)

* ----------------------------------------------------------------------------

* Calculates the eigenvalues and normalized eigenvectors of a symmetric 3x3

* matrix A using the QL algorithm with implicit shifts, preceded by a

* Householder reduction to real tridiagonal form.

* The function accesses only the diagonal and upper triangular parts of

* A. The access is read-only.

* ----------------------------------------------------------------------------

* Parameters:


* Q: Storage buffer for eigenvectors

* W: Storage buffer for eigenvalues* ----------------------------------------------------------------------------

* Dependencies:

* DSYTRD3()

* ----------------------------------------------------------------------------

* .. Arguments ..


DOUBLE PRECISION Q(3,3)


* .. Parameters ..

INTEGER N

PARAMETER ( N = 3 )


DOUBLE PRECISION E(3)

DOUBLE PRECISION G, R, P, F, B, S, C, T

INTEGER NITER

INTEGER L, M, I, J, K

* .. External Functions ..

EXTERNAL DSYTRD3

* Transform A to real tridiagonal form by the Householder method

CALL DSYTRD3(A, Q, W, E)

* Calculate eigensystem of the remaining real symmetric tridiagonal

* matrix with the QL method

*

* Loop over all off-diagonal elements

DO 10 L = 1, N-1

NITER = 0

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* Iteration loopDO 11 I = 1, 50

* Check for convergence and exit iteration loop if off-diagonal

* element E(L) is zero

DO 20 M = L, N-1

G = ABS(W(M)) + ABS(W(M+1))

IF (ABS(E(M)) + G .EQ. G) THEN

GO TO 30

END IF

20 CONTINUE

30 IF (M .EQ. L) THEN

GO TO 10

END IF

NITER = NITER + 1

IF (NITER >= 30) THEN

PRINT *, DSYEVQ3: No convergence.

RETURN

END IF

* Calculate G = D(M) - K

G = (W(L+1) - W(L)) / (2.0D0 * E(L))

R = SQRT(1.0D0 + G**2)

IF (G .GE. 0.0D0) THEN

G = W(M) - W(L) + E(L)/(G + R)

ELSEG = W(M) - W(L) + E(L)/(G - R)

END IF

S = 1.0D0

C = 1.0D0

P = 0.0D0

D O 4 0 J = M - 1 , L , - 1

F = S * E(J)

B = C * E(J)

IF (ABS(F) .GT. ABS(G)) THEN

C = G / F

R = SQRT(1.0D0 + C**2)

E(J+1) = F * R

S = 1.0D0 / R

C = C * S

ELSE

S = F / G

R = SQRT(1.0D0 + S**2)

E(J+1) = G * R

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C = 1.0D0 / R

S = S * CEND IF

G = W(J+1) - P

R = (W(J) - G) * S + 2.0D0 * C * B

P = S * R

W(J+1) = G + P

G = C * R - B

* Form eigenvectors

* --- This loop can be omitted if only the eigenvalues are desired ---

D O 5 0 K = 1 , N

T = Q(K, J+1)

Q(K, J+1) = S * Q(K, J) + C * TQ(K, J) = C * Q(K, J) - S * T

50 CONTINUE

40 CONTINUE

W(L) = W(L) - P

E(L) = G

E(M) = 0.0D0

11 CONTINUE

10 CONTINUE

END SUBROUTINE

* End of subroutine DSYEVQ3

36

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Gravity gradient tensor of a finite line of mass of arbitrary orientation