7 V o "OI..~'.‘- 00v .‘ .0 o. ~00 QfooofinOo'Oflf-u' v-gcon,. .00 l ‘ ,-V.... .‘ 0.”. (0 IC SECOND — .- ' C a : SOUR MICHIGAN STATE UNIVERSITY MAL “AL .AN0 0 VERTICAL DERIVATIVES - 1971 MICHAEL M. SPURGAT g..-.-- FROM. GRAVITY AND MAGN' ._; Thesis tor-theDegIe'e of M. S‘ . WIDTH 0? " 0-DIMENSIO --w- -o—oo. ~ 9 0-00- 00“ 9-... -o~r. . . vQ. .1; 0 .~. ....'.1...;. 1.... :3. ....3.?..,Is.1...Oz‘fl8.9? 1. . o . O . . . . . . .. . I. , _ . . . ...n I. . . . .r. .1. . 4 r .. u. .n ‘4 . ‘ . o..‘.. ..I ._, . . ....... ..;o . .. . o...;... .2. I o . o'- .o. . .... .. . ... o ,,..o.—-..o. . . . ._.._ . . .J/...... . . .I .f .9 . o. . f. I .c to. .—. _ . .. ,a ... oo,r , . . _. r u. .. . _. .o.’. . _. ....nnv..ov..oia . ‘ .. I . .- JV... . . .. . h.nn.na _ .. .. ,. . .u o... . .o 1.. .o..; .. . . I... ....w;’. .. .I . 4a. I .._ .. A. i . .o n...... . n .. .0. . or ,lua.’ 0 ¢ ._ .... .. o.c _. 1.. ,I. .... .l. .7? ’ . _ .. ct. .f.:..¢¢..o —. . . . c . an... . .5. ..I. n . . o. .19... .. n . . . . 41.. .. '.“ .c. . .. .. o... .....Jo.-o .I . n..a 1.. . .. . . do. . o. . . . . . . I. . ,.v u. . I. .. .o . t .. I. u I a .U C u a .0 ... . a A . . _... . . .. c. u n ‘u D o _ ' u D 0:. . .. . . . . . . o .. u . . . . t .r n _ .. . a . . _ . . . a . .. v .. _ . o . . .. p a r . ‘ . . . I; a ....., . . . .4 .o ..r . s . ..., ... . . . . . ' a . ’. _. , .1 I I. a . . ... . . . . c. o. .. . . . .. I .. ‘ ._ I. n. c u. (I O . _ . ..v I . . . . . .. ... ... o 4- a: r.... I 0‘. I .a.. f. ..c. . , .. ... .. I. . . .0 ...a.. ..v... a . _ . . I I I . . ... I. .. I. .‘o . 3... . .. o .—.r V...:' I . v o . . a. II.) ..{ul.o . . . ,. . .. .. I . P. a . ..v _ . . . I . ...o . :.. .. . . . . . . . I , . I. I A. u o: I o. . . . flv. .. _ .,l 1;? f. _. .«L ._ _ . . I . .. p. ,l'. ’59 .r... “:3 4 . .. n. . 0U... . _ If ’ . OI: . a: . . 1 . I. .o . . .l a . ..4 x . . . .o O . .. . . . ..I. . .. c . _ a . . . _ .._I . ..60... . . . . . h . . .a o . I o . n . I . . . _ n . . I f I. . .. I... . . u r ’ d . I. I I .A. . . .. . . . v . .. I. It . .l ._. J .o}. v. o. ;_ O I. In. . ,n\ . I... . I .0 . . I. I . 5 .-¢. .- I.... I .o o. . a. t . .. . . .o . r . " ‘ o . . . I . _ . . . o . I . .X. . . .. _ .o a I . . I. . I r o . . I I r. . .I . . \ V. o . . . . . . I .o. . oI .—. . . o n I . . 1 . I ’0 o . o O I. .O . I . I lot a . o . . O. I I I. . O I . . I v . . . I .. .4. . . I .I .u . I v a . .\. I .4 . . o . .c . . . . a. I. I v . .. . I I. . . .f o . u.o.I. . JI. . . W. «I . I . Ia._ o . .. f . lI . .I.I .o . . I t . I u. r ’ I I .. . u. I.. . c . o In: iv. 0/. .J ,o I. .‘ ’ g Q . . .I o I ! ... a v .. . .o;.u ... I. , .. ,. . -....olf..o.... 00..Ill.;\.1n..r . I . .o... .\ .¢_. I . . .0... ... .. 77.4.2». ofV r . .. . . . . not. . .: ...OIa-Vl;c . . . .— .vZOQTJ . I . \.. .. ‘g. lefi . . h I I. 0‘1.‘ . .. .. .. .. .. o . ..... .n. ’.1 u .1..vr:£w.‘q ..oI ‘_..I¢ .. .. ...... .. ‘uo ...o. .... o . 0 - 1. LIBRARY Michigan State Univctsity WIDTH OF TWO-DIMENSIONAL ANOMALY SOURCES FROM GRAVITY AND MAGNETIC SECOND VERTICAL DERIVATIVES BY Michael M. Spurgat A theoretical study of second vertical derivatives of gravity and magnetic anomalies shows that the width of the derivatives at zero value is useful in determining the width of two—dimensional tabular anomaly sources. The width of the zero values of the second vertical derivative increases with respect to the source width as the depth and depth extent increases and the dip decreases. Widths of vertical two-dimensional tabular sources can be estimated if the depth and depth extent of the source are known using general families of curves relating true widths to observed widths as determined from the zero values of second vertical derivatives of gravity and vertical magnetic anomaliesv In the gravity case if the width of the vertical tabular source is greater than approximately 4.5 times the depth, the zero values of second vertical derivatives are in error by less than 10 percent for any depth extent in predicting the width of the source. In the case of vertical magnetic anomalies, if the width exceeds the depth to the source by 2.0 or Michael M. Spurgat more, the error in estimating width from second vertical derivatives is less than 10 percent for any depth extent of the source. WIDTH OF TWO-DIMENSIONAL ANOMALY SOURCES FROM GRAVITY AND MAGNETIC SECOND VERTICAL DERIVATIVES BY Michael MfXSpurgat A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Geology 1971 TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . iii LIST OF FIGURES. . . . . . . . . . . . . iv Chapter I. INTRODUCTION . . . . . . . . . . . 1 II. THE SECOND VERTICAL DERIVATIVE METHOD. . . 3 III. THE GRAVITY CASE. . . . . . . . . . 8 IV. RESULTS OF GRAVITY MODELING . . . . . . 13 V. THE MAGNETIC CASE . . . . . . . . . 19 ’VI. RESULTS OF MAGNETIC MODELING. . . . . . 25 VII. A LEAST SQUARES SECOND VERTICAL DERIVATIVE APPROXIMATION METHOD. . . . . . . . 29 VIII. CONCLUSION. . . . . . . . . . . . 40 BIBLIOGRAPHY. . . . . . . . . ‘. . . . . 42 APPENDIX A . . . . . . . . . . . . . . 45 APPENDIX B . . . . . . . . . . . . . . 49 APPENDIX C . . . . . . . . . . . . . . 54 APPENDIX D . . . . . . . . . . . . . . 53 APPENDIX E . . . . . . . . . . . . . . 64 APPENDIX F . . . . . . . .. . . . . . . 73 ii Table 1. LIST OF TABLES Difference between theoretical and approximate second vertical derivatives of gravity of a vertical tabular source using various approximation methods (r=%Zl) . . . . . . . . . . . Difference between theoretical and approximate second vertical derivatives of gravity of a vertical tabular source using various data intervals for the five point method. . . . . . . . Difference between theoretical and approximate second vertical derivatives of magnetics of a vertical tabular source (r=%Z1) . . . . . . .g . . . . Difference between theoretical and approximate second vertical derivatives of magnetics of a vertical tabular source using various data intervals for the five point method . . . . . iii Page 34 36 37 39 Figure 1. LIST OF FIGURES Example of template used in calculating second vertical derivatives from gridded data. (Dots represent data points). . . Definition of parameters used for geological models and second vertical derivative calculations . . . . . . . . . . Cross-section of a two-dimensional geolo— gical model represented by an irregular- shaped polygon.‘ . . . . . . . . . Relationship between W, T, 21' and AX for a vertical tabular source of gravity anomalies . . . . . . . . . . . The effect of dip of tabular Sources on AX of gravitational fields. . . . . . Relationship between W, T, Z , and AX for a vertical tabular source of vertical magnetic intensity anomalies in a vertical magnetic field. . . . . . . The effect of dip of tabular sources on AX of vertical magnetic intensity anomalies . . . . . . . .. . . . Grid for calculating a three dimensional second vertical derivative. . . . . . Approximate second vertical derivative plotted as a function of r2, and shown fitted with a least squares curve of degree two . . . . . . . . . . . iv Page ll l4 17 26 28 31 33 CHAPTER I INTRODUCTION Second vertical derivatives of gravity and magnetic fields have been used extensively for many years to enhance anomalies characterized by high frequency components. Another important prOperty of second vertical derivatives is their relationship to geologic boundaries of anomalous sources. Vacquier and others (1951) have pointed out that the zero curvature or second vertical derivative contour of the total magnetic intensity closely approximates the outline of the vertical prismatic magnetic anomaly sources extending to great depths. Leney (1966) has used this principle in estimating the width of a nearly vertical, tabular magnetic ore body. Rudman and Blakely (1965) have used the second vertical derivative of both gravity and magnetic fields to outline the subsurface configuration of a balsaltic plug intruded into a granitic basement in Indiana. The delineation of the outline of any anomaly source is invaluable in the geophysical mapping of geolOgic units and is useful as a starting point in defining the anomaly source in the indirect method of interpreting gravity and magnetic anomalies. Despite this important property of second vertical derivatives little study has been devoted to analysis of the method's applicability and accuracy. In this investi- gation, the relationship of the zero curvature values of gravity and vertical magnetic intensity anomalies to the width of a tabular, two—dimensional, dike-like source is determined analytically over a wide range of source parameters. -The tabular, two—dimensional, dike-like source is considered because of its similarity to many geological features and its relatively well-known anomaly field. The study of the second vertical derivative of the vertical magnetic intensity has been restricted to induced magneti- zation in a vertical geomagnetic field, or the total magnetic intensity in this special case. In order to calculate theoretical values, it was necessary to derive the equations for the second vertical derivative of gravity and vertical magnetic intensity of two-dimensional vertical rectangular prisms. In addition, equations were derived for second vertical derivatives of two-dimensional polygonalrshaped prisms which can be used to approximate the source of any two-dimensional gravity and magnetic feature. Furthermore, a method based on the least squares technique is described for calculating second vertical derivatives from field data derived from two- dimensional sources. CHAPTER II THE SECOND VERTICAL DERIVATIVE METHOD Metnods of approximating the second vertical derivative of gravity and magnetic fields were introduced into the geophysical industry in the early nineteen thirties to aid in isolating economically interesting anomalies. However, the first account of the method was not published until 1949 (Peters, 1949 and Henderson and Zietz, 1949). Subsequently, other approaches to the calculation of second vertical derivatives have been prOposed (e.g.; Elkins, 1951; Rosenbach, 1953; and Henderson, 1960) and their relative merits discussed. Fuller (1967) has calculated the frequency response of several of the more cannon second vertical derivative calculation methods utilizing the Fourier transform. In general, the second vertical derivative is determined from orthogonal second horizontal deriveiives of the anomaly field (A) using LePlace's equation: 2 9.1;... _ (82A + 3.2.2:) (1) 32 5'2 ar‘ where X and Y are orthogonal coordinates on a horizontal surface and Z is the vertical component. The second horizontal derivative can be calculated directly from a contour map of the anomaly field. Most methods utilize a numerical method employing anomaly values at set radial distances (r) from the point of calculation. The second vertical derivative is calculated using an equation of the general form: 0) CD (WC +WC +WC 0 0 1 1 2 2 + ° ' ° + wnCn) (2) Q) N! I ”NI N where CO is the value of the anomaly field at the point of calculation; C C2, . . . Cn are average anomaly values 1! of data points on rings 1, 2, . . , n; W0, W1, W2, . , W n! are weighting factors determined by the particular method either theoretically or empirically; K is a numerical coefficient; and r is the distance represented by the unit grid spacing of the anomaly values (Fig. 1). The choice of weighting factors, number of rings, and the number of data points on each ring are a result of extensive comparative tests on theoretical and field data of various degrees of precision, frequencies, and data intervals. The final product, a second vertical derivative map, is only an approximation to the true second vertical derivative. /_ \// /' /\\\ on \ \ C2 /. FIGURE 1.--Examp1e of template used in calculating second vertical derivatives from gridded data. (Dots repre- sent data points). \ y / The second vertical derivative of gravity and magnetic anomalies has been used primarily to increase the perceptibility of steep-gradient anomalies derived from shallow sources. Romberg (1958) has pointed out that second vertical derivatives decrease two powers faster with depth than the original anomaly field. As a result second vertical derivatives attenuate rapidly as the depth to the source increases, thus increasing the perceptibility of shallow source anomalies. An additional use of second vertical derivatives is to aid in defining the boundaries of the anomaly source. As noted perviously, the zero second vertical derivative contour approaches the outline of the upper surface of the anomaly source. This is illustrated in the principal profile over a two-dimensional anomalous source (Fig. 2). The relationship between the distance between the zero cross-over points of the second vertical derivative profile (AX) and the parameters of tabular magnetic and gravity anomaly sources is the subject of this investigation. \/L_M__l SECOND VERTWCAL DERIVATIVE-2' O Observation Point ea/7 DATUM LEVEL / / / f /1*‘/ f X2 ‘ z. _+_.Xl____il i \\\\V\\V\ GEOLOGICAL MODEL2 \ \ \\\x\\ Xl=Distance from the observation point to the near side of the geological model. X =Distance from the observation point to the far side of the geological model. Zl=Depth to the top of the model. Zz=Depth to the bottom of the model. T =Thickness of the model. a =Dip of the model. AX=Distance between cross-over points. The cross-over point is defined as the point where the second vertical derivative changes sign. FIGURE 2.-—Definition of parameters used for geo- logical models and second vertical derivative calculations. lA11 second vertical derivative values given in the text are in units of 10-10. For all gravitational sources a density of 1.0 gr./cc. was used, and for all magnetic sources a susceptibility of 0.001 emu/cc and induced field of 0.58 oersteds was used. CHAPTER III THE GRAVITY CASE The equation for the gravitational anomaly due to a rectangular—shaped two-dimensional prism of density contrast Ap as given by Heiland (1940) is 2 a as King 2 ”1 1 l (3) X X X X +Zz(tan-l 73 - tan—l—-Z-}-)-Z.,(tan-l 7E - tan 1 §£)] 2 2 * 1 1 where Ag is the calculated gravity anomaly, y is the universal gravitational constant, and the other parameters are as defined in Figure 2. The first vertical derivative of gravity as determined by taking the derivative of equation 3 with respect to Z holding X constant is X X X %%fl = 2yAp[tan l 23 — tan 1 Z- - tan—l-Eg 2 2 1 _ X1 (4) + tan 2.]. 1 Taking the partial derivative of equation 4 with respect to Z we obtain the second vertical derivative of gravity of a two-dimensional rectangular—shaped prism, A2 » A X X XI 0 Eg : ZYApL ,1 + 2 7 — ——$—§ - ‘ l. (5) 322 Kgi'ZZ X2+Zh X2+Z X31“ Z‘.‘ l l 2 2 l 2 2 1 In order to obtain a more general relationship and specifically to calculate the second vertical derivative of dipping, two—dimensional, tabular bodies, we can consider the mathematical expression for the gravity anomaly of a two-dimensional prism of arbitrary cross—section (Grant and West, 1965): } .2 2 I b. ‘ + - . _ 9 K .1 xK+l ZK+1\ [Ag _ ZYAQ )4 I 1,) L-2- ln( I) 2 / + 7-3»! hzl l+aK XKIZK (6) _ k, - i Xv (tan “ "£11."; - "can 1 “3'41 z,.. z" I\_‘1-J 1‘. Where: XK+lnxK a = 7' ‘:*‘ K ‘K+l ZK b _ XKZK+l-XK+IZK K 7 Z ‘Z K+l K Ag is the calculated gravity anomaly, y is the universal gravitational constant, Ap is the density differ- ential between the anomalous source and the country rock, 10 XK (K=1,2,3,4,. . .,N), xK+1 (K=1,2,3,4,. . .,N), 2K (K=l,2,3,4,. . .,N), and ZK+1 (K=l,2,3,4,. . .,N) are horizontal and vertical distances respectively to the intersection points of the straight lines defining the shape of the source (see Fig. 3). The first vertical derivative of gravity as determined by taking the deriva— tive of equation 6 with respect to Z holding X constant is: X2 +Z2 §g3_= ZYAp{[% 1n( K+1 K+1) + a J (7) 2 +1 (XK+1+ ZK+1)ZK Z)ZKl ) Z2 2 K ZK 71+N (X+ (X+ ) (X +Z ) K+1 K+1 W+bomro XK+ZK XK+1+ZK+1 Taking the partial derivative of equation 7 with respect to Z we obtain the second vertical derivative of gravity, ll DATUM LEVEL fr I I I I1T 7 I I I F I I GRAVITY OR MAGNETIC STATION [‘— ZKH GEOLOGIC MODEL FIGURE 3.--Cross-section of a two—dimensional geological model represented by an irregular-shaped polygon. 12 2 X -X K + K 1)] 3—9% = 2YA0{{[( l 2)< BZ l+a K -(x2 +z 2 (XK+Z )Z K+1 K+1 ZK+1—ZK 2 K+1>ZK )+ a 2 +Z2 2 +2 )(XK+1 K+ 2 K [< 2 K K (X 2 )Z _(XK+ +1 WM 2 (XK+Z x K(x2+z K WM 1) 2 1+ZK+1)ZK }+{[( Ki 2 K+1+ (X +Z K 2 K)(X W XK XK+1 LIZ-‘2 ' 2 2 K+1—ZK+1 )JE 2 2 K+1+ZK+1 [(x§+z§)( )(X X2 2 2 ‘{[(XK+ZK)ZK+1-(fK+l 2 2 2(XK+1+ZK+1)ZK]} + a I 2 ZK+1) X -X (—i—§)«——————- 1+aK ZK+1-2K 2 2 + 2 )][(XK+ZK)+(XK+1 ZK+1 2 +Z2 )12 xK+1 K+1 2 2 2 +zK+l)zK][2(xK+zK)z 2XK+1ZK+1 ( _ K 2 2 2 (XK+1+ZK+1) XK+1 K K 2 2 (XK+ZK) 2 +Z2 fl K+1 K+1 )1} BI}. The computer programs for calculating the second vertical derivative of gravity according to equations 5 and 8 are given in Appendices A and B respectively. CHAPTER IV RESULTS OF GRAVITY MODELING The horizontal distance between the zero second vertical derivative values of gravity (AX of Fig. 2) were calculated for a series of two-dimensional, vertical, tabular bodies from equation 5. The direction of the calculated profiles is perpendicular to the Y-axis along which the source is assumed to extend infinitely. A simple interation scheme was used to calculate the horizontal position of the zero value of the second vertical derivative from which AX can be computed. This interation appears as a subroutine to the computer programs in the appendices. The horizontal positions were calculated to a precision of 0.2 percent or less of the minimum distance parameter of the geologic models. The AX values were computed for models varying from 500 to 10,000 units for depth to the top, width and thickness, except for the special case of T/Zl=100 where T was set equal to 100000 units. The results are presented in Figure 4 with AX/Zl plotted against W/Zl where W is the true width of the model and Z is the depth to the 1 top of the source from the observation surface. The values 13 200° my so 20% no 0% 5 I] /I I] if [/7 // / J I’ / % / /I / /’ / / / / 4V/ 4 fl ['1 /1 f/ IF / f/I V / /{ ,// / / I 1/ _[/ / I 7 I? r T ‘I ll /" I // // /{// 3 1% / / _ —— I [I / // / / 91 1 A ,///j / z I / ”lflfié/fl / L“... // // 2 / / .griw/ -II-- ~ 1 Matty/x I , _=::: 7“% ”rm—«Ifl—— I» - - _HL_I.L_ M / // /I‘ I i I I “I / / / 4” I é ' fl / '//‘ I .I 7 I “I / / //Z/ I /// i I I I /// I I . / I | i o 7 _I ____.._ .i- __ -__ . _. __.._.__L.. .. -._..JL.....-__..-_.I O I '6 w 3 4 f) 77 FIGURE 4.-—Re1ationship between W, T, 21, and AX for a vertical tabular source of gravity anomalies. 15 of AX and W are divided by Z to generalize the results. 1 A family of curves is presented for values of T/Zl from 0.1 to 100. Error curves also have been plotted on Figure 4 which show the percent error according to the equation ((AX-W)/W) x 100. A number of very important relationships between the width, depth, and thickness of gravity anomaly sources and AX can be observed from Figure 4. First, for any given values of T, Z and W, AX is always greater than W, 1! except for the specific case of Z =0 where AX equals W. l Secondly, as the ratio of W/Zl decreases, as, for example, when the depth increases while the width remains constant, the ratio of AX/Zl decreases, but the percent error rapidly increases for all values of T/Zl' Thirdly, as the width increases for a constant depth, the ratio of AX/Zl increases and the error decreases for all values of T/Zl° The curves asymptotically approach zero error as W/Zl increases toward large values. Fourthly, with width and depth constant, the error increases directly with the thickness of the source, but at a diminishing rate as thickness increases. The error difference between ratios of T/Zl of 10 and 100 are relatively small. Utilizing the general relationships shown in Figure 4, if AX of a vertical two-dimensional tabular body is determined from a second vertical derivative of gravity map or profile and Z is known from geological or geo- 1 physical data, a range of widths of the causative body can 16 be determined or the true width determined if the thickness of the source is known. For example, if AX is found to be 2700 feet and Z1 and T are estimated to be 1000 and 5000 feet respectively, the width of the body is 2000 feet. As a general statement, if W/Zl is approximately 4.5 or greater, AX will be in error by less than 10 percent, for any ratio of T/Zl' Plotted in Figure 4 is a curve of T/Zl=100. In this case, the thickness of the source can be considered essentially infinite. At great depths the mass of this source has little influence on the observed gravitational field, and therefore little effect on the second vertical derivative and AX. Thus, any value of T can be considered essentially infinite when T/Zl>10. This conclusion can be verified by the following example. Assume the model parameters are T=1000 feet, Zl=500 feet (T/Zl=20), and W=500 feet. Using the T/Zl=10 curve of Figure 4, AX/Zl=2.20, while for the T/Zl=100 curve, AX/Zl=2.23. The difference introduced in calculating AX by this slight variation in AX/Zlis less than 1.5 percent. The actual value of AX/Zl, for T/Zl=20 is 2.22 (not shown in Fig. 4). The effect of dip of tabular bodies on the AX of gravitational fields is illustrated for a few special cases in Figure 5. In this figure the difference in the AX of dipping tabular bodies (AX ) and vertical tabular l bodies (AX2) is plotted against T for specific cases of 17 I000 900 800 700 soo soo - . — I T“ I 400 -- -— I ~» , ~+- *— AXW’AXOO’ ' BOO - Ax‘oo- AX.OO g I000 ' I a“) . I i ‘I__I"I”' / 1 ; ax - AX. -AX2 i . 3, MW.- ax”. 2:000 I00 so ~ - A » ~— —~ 30 - -' - -— « 70 . V‘ Z . " " -- Ax-rs°’Axso°~—W= "000 60 I066“ E.“ AX AX ' I 50 M" 10" l 40 -4:— ' AX AX 30 — — . ~— AX 7" AK». I I000 2 000 3000 4000 5000 6M 7000 00(1) 9WD IOOOO FIGURE 5.--The effect of dip of tabular sources on AX of gravitational fields, 18 anomaly sources dipping at 600 and 750. Only a relatively few data points have been calculated and, therefore, the positions of the curves between data points as indicated by dashed lines may be somewhat in error because of their high curvature. Several conclusions can be reached from this figure. First, AX increases as the dip decreases. Second, the difference in the AX of dipping and vertical tabular bodies increases rapidly as thickness of the bodies increase up to thicknesses of approximately five times the depth to the top of the source. For greater thicknesses the difference in AX remains relatively constant and for one case starts to decrease. Third, the difference in the AX of dipping and vertical tabular bodies increases with increasing depth to the top and decreasing width of the source. CHAPTER V THE MAGNETIC CASE The equation for the vertical magnetic intensity due to a rectangular-shaped two—dimensional prism of susceptibility contrast Ak as given by Cook (1950) is: X X X X Av=2AkH(tan l 7.1.. - tan 1 fl— - tan 1 2—2- + tan-l 72-) (9) ' l ‘2 l 2 where AV is the calculated vertical magnetic intensity, Ak is the susceptibility contrast, H is the Earth's magnetic field, and the other parameters are as defined in Figure 2. The first vertical derivative of the vertical magnetic intensity as determined by taking the derivative of equation 9 with respect to Z holding X constant is: X X X X géy.=2AkH(___£§ _ _§$_§ — —§_g§.+ 2 2). (10) Xl+Zl X1+Z2 X2+Zl X2+Z2 Taking the partial derivative of equation 10 with respect to Z we obtain the second vertical derivative of magnetics of a two-dimensional rectangular-shaped prism, 19 20 BZAV _ 4AkH[ X121 _ X122 _ X251 + X222 1 (11) "72‘ ‘ "'2"2“2‘ '"2"'2'2 '"‘2“2'2 "72“2‘2‘° az ()1+Z;) (Xl+22) (x2+zl) (x2+22) In order to obtain a more general relationship and specifically to calculate the second vertical derivative of dipping, two—dimensional, tabular bodies, we can consider the mathematical expression for the vertical magnetic intensity of a two—dimensional prism of arbitrary cross- section (Grant and West, 1965): N AV = 2AkH Vl-coszlcoszl Z ( l 2) K=1 l+a {[(aK sine + cosB)XK] 2 2 2 [in (1+aK)ZK+l+2aKbKZK+l+bK _ 2 2 '42 k J. (1+a )ZK+2aKbKZK bK ~l (1+aKIZK+l (aKcosB - SinB)XKMian ( bK + aK) 2 _ (1+a )2 —tan l(___B§_;5-+ a )3} (12) K K where: a _ XK+l-XK b _ XKZK+1_XK+1ZK — '***tT‘—r ‘ _ K ZK+1 ZK K ZK+1 ZK B _ ta -1(tanI) 21 where AV is the vertical magnetic intensity, Ak is the susCeptibility contrast between the anomalous source and the country rock, H is the Earth's magnetic field, A is the strike of the body, I is the inclination of the Earth's magnetic field, and X (K=1,2,3,. . N), X (K=l,2,3, . . , K K+1 N), zK+1 (K=1,2,3, . . ,N) are horizontal and vertical distances respectively to the I N), ZK (K=1,2,3,. . I intersection points of the straight lines defining the shape of the source (See Fig. 3). The first vertical derivative of the vertical magnetic intensity as determined by taking the derivative of equation 12 with respect to Z holding X constant is: N a—é—‘i = 2AkHV1—coszlcoszI E e K K=1 [jKVK—iKmKJ - {[(———l ) jKVK gK I+sK (l3) bKuK_CKuKZK+l . 1 b K K bKuK-cKuKZK } ( 2 )1 bK where: C = xK—XK+1 K Z Z K+1‘ K 22 eK = aK SinB + cosB gK = aK cosB - sinB ' = u Z2 +2a b Z +b2 1K K K+1 K K K+1 K . _ 2 2 3K - uKZK+2aKbKZK+bK mK = 2(uKZK+aKbK+aKcKZK+chK) u Z _ K K+1 2 K u Z _ K K 2 tK — l+( b + aK) K _ 2 uK — (1+aK) VK = 2(uKAK+l+aKbK+aKCKZK+l+bKCK)' Taking the partial derivative of equation 13 with respect to Z we obtain the second vertical derivative of the vertical magnetic intensity, BZAV VI 2 2 N ——7— = 2AkH 1-cos Acos I Kgl {8K I[jKVK(JKnK+mKVK-1KHK+mKVKI3+ (l4) 2 (lKjK) [(jKVK_iKmK)OnKVK+anK)]J}+ 23 {gK 35:K—leKJ} - rK{£(I%§—> KJK K bK K—CK KZK+1 1 bK K K K K 1 K (bKuK'CKuKZK+1)] -( 1 ) b2 L 1+sK K 2 (b (CKuK)_(CKuK)+(bKuK_CKuKZK+l) b4 I K (ZbKCK))J — [( l )(bKuK_CKuKZK>1 1+wK b2 K b2(c u )—(c u )+ +[( l )( K, K K K K l+t ~ 4 K bK (bKuK-CKuKZk)(2bKCK) )1} where: 2 = U. nK 2E K+2aKCK+CK] 24 + + = aKbK uKZK+1 uK pK b K rK = -a1 SinB + cosB w = aKbK+uKZK+uK K b ' K The computer programs for calculating the second vertical derivative of the vertical magnetic intensity according to equations 11 and 14 are given in Appendices C and D respectively. An inclination of 900 was used for the models, therefore, the vertical magnetic intensity is equal to the total magnetic intensity. CHAPTER VI RESULTS OF MAGNETIC MODELING The horizontal distance between the zero second vertical derivative of vcrt cal magnetic intensity (AX of Fig. 2) was calculated for the same series of two-dimensional, vertical tabular bodies that was consid- ered in the gravity case. Vertical magnetic polarization of the anomaly sources was assumed. The results of these calculations are presented in Figure 6 in the same manner as the results of the gravity calculations are shown in Figure 4. The single exception is that the T/lelO curve is not plotted for the magnetic case because it is nearly the same as the T/Zl=5 curve. Figure 6 may be used in magnetic interpretation as Figure 4 can be used in gravity interpretation and i1 general the conclusions reached from Figure 4 are applic- able to the magnetic case considered here. However, the measured horizontal distance between the zero second vertical derivatives of vertical magnetic intensity is more nearly equivalent to the true width of the anomaly source than AX in the gravity case. As a general state- ment, if W/Zl is 2.0 or greater, AX will be in error by 25 26 . a. 1. “I L _ a a. 2. a n _ _ _ r M _ _m m. _ w 2 . I r i . III IIIITII It I IIIIII LI c i _ . i 1 . . m w i N _ u _ . m m w _ i . 1 U // . 6 .fi H M i _ . . - . M/ /L _ M m a u /: MI: . IL A L i - - II a . / _I // _ w “1 fl 4.. 4 . I 2/ /l 1 _ i , I _ r 1 _ .II - IV. Int/I- I -I I: I- .- - :- III I . an / / . _ . w/ x/ /L H _ / . . . . // W n / .7/ i . u M 14-: yi,.:.o I .. I- - - I: . ./ . _ . A //_ n // // i i. m // f x i r mfl: I - I: - II. I . II :5: I -I I / - I/i /// i /.//N / _ / . H/ .II- -_I/:/- IIiI V. I :/.7_/ ~ 1 / L / fl _ /,i _/ i m _ /.../ m / h ~ . / _ / .II I III: I I -- Iiyl.I--II4I ml ‘ L . _ // _ nvall _ . [I 2.. ll/l . _ _ Ti I -IIL/LI.-NI4I IIILIIII L m _ w /+I//H fl h H _ .I/lr _ IIII+II I L F I, 17 _ I i L h . . . II LII-II I L i 5 .5 X A 1... and AX for Z1, a vertical tabular source of vertical magnetic intensity FIGURE 6.--Relationship between W, T, anomalies in a vertical magnetic field. 27 less than 10 percent for any value of T/Zl' Also the effect of varying T is less in the magnetic case than in gravity. For all practical purposes in this regard, the source can be assumed to have an infinite thickness if T/Zl>5. As in gravity case, the effect of dip of the anomaly source on AX was studied for a few special cases in magnetics. The results of this study are shown in Figure 7 in an equivalent manner to the gravity case (Fig. 5). Only a relatively few data points have been calculated and, therefore, as in the gravity case, the position of the curves between data points as indicated by dashed lines, may be somewhat in error because of their high curvature. The significant conclusions reached from a study of Figure 7 are the following. First, AX increases with decreasing dip for any given set of body dimensions. Second, the difference in AX of dipping and vertical tabular sources increases as the depth increases and the width decreases all other dimensions being equal. Third, the difference in AX of dipping and vertical tabular bodies increases rapidly as the thickness (T) of the source increases. The difference in AX reaches a maximum and then gradually decreases. The value of T at which the maximum occurs increases as the depth to the top of the source increases. 28 IOOO 900 000 700 600 400 300 200 AXI-AXz :00 so so 70 60 50 4O 20 IOOO 2 000 3000 4 000 5000 m 7WD 0000 9” IOOOO FIGURE 7.--The effect of dip of tabular sources on AX of vertical magnetic intensity anomalies. CHAPTER VII A LEAST SQUARES SECOND VERTICAL DERIVATIVE APPROXIMATION METHOD The relationships previously determined in this study are based on theoretical second vertical derivatives. However, in actual practice second vertical derivatives calculated from observed anomalies are only approximations to the true or theoretical second vertical derivatives. Therefore, conclusions based on the results previously presented will be in error by an amount determined by the accuracy of the method used to calculate the approximate second vertical derivative. Elkins (1951) and Swartz (1954) have outlined a second vertical derivative procedure which is applicable to two-dimensional gravity and magnetic anomalies. This method, which is based on LaPlace's equation, utilizes an extra- polation of the curve of “r2" versus the average anomaly at a distance r" from the calculation point to the zero value of r. The curve can be approximated by the method of least squares to determine the second vertical derivative. The theory is developed in the following manner assuming that the field data collected from a gravity or magnetic survey obeys LaPlace's equation. 29 30 LaPlace's equation is, ) (15) 2 BY where A is the observed anomaly. Equation 15 can be simpli- fied to 82 2 A 8 . _ = - (16) Z2 M 2 0) if the anomaly is two—dimensional in the Y direction, i.e., 82A/8Y2 is equal to zero. Referring to Figure 8 where c is the anomaly value at the point of calculation of the second vertical derivative, and bi and b2 are the anomaly values along the X direction at a distance "r" from c, the second horizontal derivative parallel to the X axis is: (17) ) substituting —82A/322 for 8“A/8X2 the approximate relation— ship in equation 17 becomes exact as the distance r approaches zero, and thus: a A _ LIM 2C-bl'b3 ,18) “Z: _ r+0 r“ . \ G Utilizing a series of values at different distances ("r") from c, a least-squares curve method can bw readily 31 b2 b3 C bl b4 FIGURE 8.--Grid for calculating a three dimensional second vertical derivative. 32 Y 22A 3 22 / f2——>- FIGURE 9.--Approximate second vertical derivative plotted as a function of r2, and shown fitted with a least squares curve of degree twe. applied to calculate 32A/322 using equation 18. The procedure consists of calculating the approximate second vertical derivative values from profile data at various values of r. An example is illustrated in Figure 9 where azA/BZ2 is plotted against the appropriate "r2" values. Next a polynomial equation of degree N is fitted to the data points using the least squares method. After solving for the coefficients of the equation, "r" is set equal to zero and the resulting equation is solved for BZA/BZ2 according to equation 18. The data points represent a smooth function, therefore, a second degree equation (N=2) was chosen for the analysis. A listing of the computer programs used for the gravity and magnetic cases are given in Appendices E and F respectively. A theoretical gravity anomaly was calculated for a vertical tabular source and used to test the least squares method of finding second vertical derivatives. The theo- retical second vertical derivative was calculated for the vertical tabular source using the same set of parameters (Zl=lOOO feet, W=1000 feet, and Z2=1OO,OOO feet). The theoretical gravity and theoretical second vertical derivative of gravity are presented in Table l. A varying number of data points were used in calculating the approximate second vertical derivative. The "five point method" implies that the center point and two adjacent points, two on either side, were used in the calculation. Columns 4 through 7 show the difference between the approxi- mate and theoretical second vertical derivative fur various "point methods". 34 TABLE l.--Difference between theoretical and approximate second vertical derivatives of gravity of a vertical tabular source using various approximation methods (r = 1/2 21). (1) (2) (3) <4) (5) (6) (7)—— 1 12.112 - 1.570 0.000 +0.000 +0.000 +0.000 2 12.523 — 1.886 0.000 +0.000 +0.000 $0.001 3 12.977 - 2.297 0.000 +0.000 +0.000 +0.002 4 13.485 — 2.837 0.000 +0.001 +0.002 +3.008 5 14.059 - 3.536 +0.002 +0.002 +0.007 40.022 6 14.716 - 4.381 —0.011 +0.003 +0.011 -0.005 7 15.474 - 5.113 -0.038 -0.002 —0.053 -0.203 8 16.348 - 4.382 -0.043 —0.060 -0.159 0.177 9 17.313 + 2.689 -0.316 -0.099 —0.382 -) 971 10 18.188 +21.880 —0.005 -0.054 —0.l44 -0.l72 11 18.567 +35.011 :0.055 +0.194 +0.671 :LLEE4 Absolute mean error = 0.043 0.038 0.130 0.254 Column Column Column Column Column Column Column (1) (2) (4) (5) (6) (7) Station number (interval = 500 feet). Observed gravity anomaly (mgals) over a vertical tabular source. Station 11 is over the center of the source. Theoretical second vertical derivative of gravity in mgals/cm2 x 10‘10. Difference between the theoretical and approxi- mate second vertical derivative using the five point method. Same as (4) for the seven point method. Same as (4) for the nine point method. Same as (4) for the eleven point method. TABLE 2.--D 3‘- ifference between theoretical and approximate second vertical derivatives of gravity of a vertical tabular using various data intervals for the five point method. (l) (2) (3) (4) (5) (6) 1 10 11 12.112 12.523 12.977 13.485 14.059 14.716 15.474 16.348 17.313 18.188 18.567 - 1.570 +0.000 +0.000 - 1.886 +0.000 +0.000 +0.001 - 2.297 +0.000 +0.000 — 2.837 +0.001 +0.000 —U.013 - 3.536 +0.001 +0.002 - 4.381 —0.001 -0.011 -U.239 - 5.113 -0.002 -0.038 - 4.382 -0.004 —0.043 +0 486 + 2.689 —0.021 —0.316 +21.880 +0.005 -0.005 +0.238 +35.011 +0.041 +0.055 Source Parameters: Z =1000 units; =100,000 units. 22 Column (1) Column (2) Column (3) Column (4) Column (5) Column (6) 1 W=1000 units; Station number (interval = 500 feet). Observed gravity anomaly tabular source. of the source. (mgals) over a vcrtical Station 11 is over the center Theoretical second vertical derivative of gravity in mgals/cm2 x 10710. Difference between the theoretical and approxi- mate second vertical derivative for r=l/4 Z1. Same as (4) for r=1/2 Z 1' Same as (4) for r=Zl. 36 The average mean error was calculated to aid in evaluating the various "point methods". In the example given the "Seven point method” gave the best overall results. However, in other cases depending on (1) the station spacing, (2) the noise within the data, and (3) the frequency spectra of the anomalies; one of che other methods may be more suitable. Comparative tests on the set of data being studied would be necessary in most cases to determine which method gave the best results. This could be easily programmed as a simple input parameter. A test was also made to determine the effect of the data interval on the accuracy in calculating the seconfi vertical derivative of gravity of the same model used in evaluating the various "point methods". Various data intervals ("r") were chosen as a function of Z1. A value of r equal to 1/4 Zl gave the best results for the theoretical case (see Table 2). However, field data may contain high frequency noise, therefore, the closest data interval chosen would not necessarily give the best results. A compromise between theoretical accuracy and the influence of the noise within the data would have to be made in choosing the optimum data interval. The same procedure and anomaly source used to evaluate the second vertical derivative of gravity was applied to the magnetic case assuming vertical magnetic polarization. Table 3 for magnetics is equivalent to Table 1 for gravity. The same general conclusions can be drawn for TABLE 3.--D 3f ifference between theoretical and approximate second vertical derivatives of magnetics of a vertical tabular source (r=1/2 Z1). (l) (2) (3) (4) (5) (6) (7) 1 3.35 - 10.824 + 0.008 - 0.001 - 0.004 — 0.017 2 4.36 - 16.095 + 0.016 - 0.002 - 0.011 — 0.038 3 5.76 - 24.900 + 0.035 — 0.055 — 0.021 — 0.058 4 7.75 — 40.354 + 0.079 - 0.007 - 0.011 — 0.177 5 10.70 — 68.990 + 0.158 + 0.021 + 0.326 + 2.094 6 15.30 - 124.859 + 0.073 + 0.337 + 2.179 + 4.399 7 22.91 - 235.855 — 2.267 + 1.127 — 0.669 - 3.156 8 36.16 - 424.515 -11.032 - 7.167 -22.936 - 35.5-5 9 59.06 — 444.462 +26.095 + 7.070 +32.650 + 76.145 10 89.95 + 624.256 -10.871 - 6.852 -20.?95 « 30.023 11 106.41 +1598.162 +48.049 +16.406 +64.540 +129.997 Absolute mean error = 8.97 4.20 13.11 25.60 Column (1) Station number (interval = 500 feet). Column (2) Observed vertical magnetic intensify {ginmmsh over a vertical tabular source in a vertical magnetic field. Station 11 is over the center of the source. Column (3) Theoretical second vertical derivative of magnetics in gammas/cm2 x 10‘10. Column (4) Difference between the theoretical and approxi- mate second vertical derivative for the five point method. Column (5) Same as (4) for the seven point method. Column (6) Same as (4) for the nine point method. Column (7) Same as (4) for the elven point method. 38 TABLE 4.--Difference between theoretical and approximate second vertical derivatives of magnetics of a vertical tabular source using various data intervals for the five point method. (1) (3) (4) 1 3.35 - 10.824 +0.000 + 0.008 2 4.36 - 16.095 +0.000 + 0.016 t .313 3 5.76 - 24.900 +0.000 + 0.033 4 7.75 — 40.354 +0.004 + 0.079 + 1.381 5 10.70 — 68.990 +0.008 + 0.158 6 15.30 - 124.859 +0.006 + 0.073 - 9.002 7 22.91 — 235.855 -0.107 - 2.267 8 36.16 - 424.515 -0.836 —11.032 ~25.199 9 59.06 - 444.462 +1.480 +26.095 10 89.95 + 624.256 -0.802 -10.871 -32.426 11 106.41 +1598.162 +2.769 +48.049 Source Parameters: Z =1000 units; W=1000 units; ant Z2=100,000 un ts. Column (1) Station number (interva1=500 feet). Column (2) Observed vertical magnetic intensity (gammas) over Column Column Column Column (3) field. source. magnetics in gammas/cm2 x 10‘10. (4) a vertical tabular source in a vertical magnetic Station 11 is over the center of the Theoretical second vertical derivative of Difference between the theoretical and approxi— mate second vertical derivative for r-1/4 Z]. (5) (6) Same as (4) for r=Z Same as (4) for r=1/2 Z 1' 1° 39 magnetics except that the magnitude of error is greater for magnetics. This can be explained since the equivalent magnetic anomaly is larger in magnitude and narrower in width, thus containing sharper gradients. Table 4 for magnetics is equivalent to Table 2 for gravity. The data interval r=l/4 Zl gave the best results as in the equivalent gravity case. Observed magnetic data in general contains higher frequency components than gravity data from the same saurces. Thus, the selection of the best "point method" is more critical for magnetics due to greater difficulty in separat- ing noise from signal. CHAPTER VIII CONCLUSION The distance between zero values of the secord vertical derivative of gravity and magnetic anomalies is useful as a guide to the width of two-dimensional tabular sources. The accuracy of the zero value in defining the width decreases as the true width decreases, the depth extent increases, and the dip decreases for a given depth to the source. The effecr of these variations is to increase the width as defined by the.zero second vertical derivative values over the true width. The error is greater for gravity than for vertical magnetic intensity. General family of curves relating true widths of vertical two-dimensional tabular bodies to observed widths as determined from the zero values of second vertical derivatives of gravity and vertical magnetic intensity can be used to estimate true widths providing the depth and depth extent of the source are known. In any event, in the case of gravity, if the width of the vertical tabular source is greater than approximately 4.5 times the depth, the zero values of second vertical derivatives are in error by less than 10 percent for any depth extent in predicting the width 40 41 of the source. In the case of magnetics, if the width exceeds the depth to the source two-fold or more, the error in estimating the width from second vertical deriva- tives is less than 10 percent for any depth extent of the source. The least squares method of approximating the second vertical derivatives of two-dimensional gravity and magnetic anomalies is a viable method of approximating the true second vertical derivative. BIBLI OGPAPHV 42 BIBLIOGRAPHY Cook, Kenneth L., 1950, Quantitative Interpretation of Vertical Magnetic Anomalies Over Veins: Ceophys cs, V. 15, p. 667-686. Elkins, T. A., 1951, The Second Derivative Method «f Gra"itv Interpretation: Geophysics, V. 16, p. Xv—gu- Fuller, B. D., 1967, Two~Dimensional Frequency ~na.ysi3 :id Design Of Grid Operators in Mining Geophysics: 33G, V. 2, p. 658—708. Grant, F. S., and West, G. F., 1965, Interpretation Theory in Applied Geophysics: New York, McGraw—Hill Book Co., Inc. Heiland, C. A., 1940, Geophysical Exploration: New York, Prentice Hall, Inc. Henderson, R. G., 1960, A Comprehensive System of A Luxa?ic Computation in Magnetic and Gravity Intu;p_ita;iun: Geophysics, V. 25, p. 569-585. Henderson, R. G., and Zietz, I., 1949a, The Corputntion «E Second Vertical Derivatives of Geomagnetic Fields: Geophysics, V. 14, p. 508-516. Leney, G. W., 1966, Field Studies in Iron Ore quphysics- in S.E.G., Mining Geophysics, case Histoxi*s, V. I, p. 391-417. Peters, L. J., 1949, The Direct Approach to Magnetic Inter— pretation and Its Practical Application: Geophysics, V. 14, p. 290-320. Romberg, F. E., 1958, Key Variables of Gravity: Geopnychs, V. 23, p. 684-700. Rosenbach, Otto, 1953, A Contribution to the Computation oi the "Second Derivative" from Gravity Data: Geophysics, V. 18, p. 894-912. Rudman, A. J., and Blakely, R. F., 1965, A Geophysical Study of a Basement Anomaly in Indiana: Geophysics, V. 30, p. 740-761. 43 Swartz, Charles A., 1954, Some Geometrical Properties of Residual Maps: Geophysics, V. 19, p. 46—70. Vacquier, V., Steenland, N. C., Henderson, R. G., and Zietz, I., 1951, Interpretation of Aeromagnetic Maps: Geol. Soc. Am. Mem. 47, p. 1-151. APPENDIX A COMPUTER PROGRAM FOR CALCULATING THE THEORETICAL SECOND VERTICAL DERIVATIVE OF GRAVITY OVER A VERTICAL TABULAR MODEL 45 §§G§§§§§'§!§*§*&Ifi§#§§*Q§*§§§ii§§§§i§**ii§i§§I’liD!§§§I*§fi§ PRBGRAM FBR CALCULATING YHE THEBRETICAL SECBND VERTICAL ,DERIVATIVE BF GRAVITY BF A VERTICAL TABULAR,SBURCE PRBGRAHMED IN FBRTRAN FBR A XDS SIGMA 5 (1/72) §&§§§§§§**§§*§!§G*§l**§§§l§§*l**§***§*§*ll‘ii§lilfiifiliilii . elitiefikfiiiififiifiiifi*i**§*i*§§*******§*iiiiiliibi!hiaeiiiiii DESCRIPTIBN 0? INPUT DATA: CARD 1 (15) LTBTITBTAL NUMBER 6F PREFILES .ALARD 2 .(15) JYRTITGTAL NUMBER 9F SBUQCES PER PRBFILE CARD 3 (2F10o2.215) DELXISTATIBN SPACING FBax-CBBRDINATE BF THE FIRST STATIBN M-TBYAL NUMBER 6: STATIaMs PER PRBFILE -.mNnNUMBER 3F CBBRDINATES PLUS aNEHFBR EACH SauRCE LLCARDS 4ov>N (2:10.3> X(I)§X-CBBRDINATES BF THE SBURCE Z(I)92-CBBRDINATES BF THE SBURCE CAQDS 4-.>N ARE REPEATED JTGT NUMBER 8F TIMES _._WCARDS 2 THRBUGH 4-9>N ARE REPEATED LTBT NUMBER 6F TIMES **i!**§*l§§***§§§*u*§l§§§§§lfiéfiiaiiil*ifilfifii§ulga;§§§**I*§§ 47 rqwmam SECDEPIV<500>.FX.M:Xt50’:Z(SO):N READ (5:5) LTBT 5 FBRMAT (15) DB 49 LSIJLTOT nFAD (5.7) JTBT 7 FBRMAT (15) READ (5:9) DEanFfllMtV 9 rfiQMAT (2F1002:215) fig “7 J'IaJTBI WRITE (6111) 11 rQQMAT ('1') WRITE (6113) L 11 IQQWAT t//.50X.'RESULTS Fen anFILE NUMBER '.15.///) DB 15 1'11” 91'! 15 FX(I)'FB'DELX*DELX*RI WRITE (6117) J 17 quMAT (50X,'C68RDINATES F69 BBDY NUMBER 'IIS://) WRITE (6119) 19 FBRMAT (50X:'X-CBBRDIVATE'16X:'Z-CBBRDINAYE';//) DB 25 1810” READ (5:21) X(I):Z(I) 21 FBQMAT (2F10o8) WRITE (6:23) X(I):Z(I) 23 FBRMAT (“6X02F1503) P5 CBVTINUE 2182(1) 22'Z(3) IE (21) 28:27:28 27 218001 23 D“ 30 1'10” X1!FX(11'X(2) X?IFX(I)-X(1) CALC3'(X2/(Zl**8*x2**2)1-(X1/(ZI**2*X1**2)) (ALC“8(X1/(ZE**?*X1**2))-(X9/(ZE**2*X2**2)) SECDEPIV(I)I4o377E-06*(CALC3+CALC“> 30 CBVTINUE WRITE (6:35) 35 FBRMAT (///.30x.'I'.8x.'rx¢rz':8X:'SECBND DERIVATIVE GRAVITY ‘30 Y IHICK PIKE CASE)':/) D9 39 1'13” WRITE (6:37) IIFX(I)ISECDEQIV(I) 37 ERRMAT (27X1141713o31?2XoEi396) 39 CPVTIMUE (ALL XIVTERCT (XINTFRCP) WRITE (6:41) 41 FBRMAY (//,50X:'THE CQBSS-HVFQ PBINT IS LBCATED Af':/) WRITE (6:43) XIVTERCP . #3 FBRMAT (60x,'x-';F1o-3x 47 CBNTINUE J £59 CGVTIMUE STBP F‘JD ‘ SUBRBUTINE XINTERCT (XINTERCP) CBNMB“ DERIV<5001pFX(SOO),NUMRFP,X(50),Z(50),N XXI-xtl) . XX23X(?) 21'Z(1) 2237(1) FXI'F¥(1) IF (DERIV(1)OLEoO) ISETnO IF (DERIV(1)OGT00) ISETu1 IF (21) 1015110 21"01 FXi‘FX1+10 XlsEX1-XX2 Y?IFX1-XX1 XXI'X(1) YXE'X(2) CALC3=(X2/(Z1**2+¥2*aa)).(X1/(21*«2+x1*«2)) CALC4'(X1/(ZE'*2*X1**8))-(X2/(Z2e*2*X2§§2)) DERIVI'903775'06*(CALC3+CALC4) IF (ISEToEQ.OoANDoDERIV1.LT.01 GB T5 10 IF (IqEToEQoIOANDoDERIVI.GT.O) GB TB 10 XIVTEPCPIFXl RETURN END APPENDIX B COMPUTER PROGRAM FOR CALCULATING THE THEORETICAL SECOND VERTICAL DERIVATIVE OF GRAVITY OVER A IRREGULAR-SHAPED POLYGON 49 50 a}il§l§§§§§§.*§§l§*§i§§i§¥§§§*l**lill‘§*§*.I**'§§§*§DG**§§§ DQHGRAM FnR CALCULATIVG THF THEORETICAL SECBND VERTICAL pEQIVATIVE 3F GRAVITY BF A PRISM.SHAPED SBURCE anGRAMwED 1N FeRTRAN Fan A xDS SIGMA 5 (1/72) g!i§§§§§§§ilfi**§§ill§&*§*§*§*§**l§*§G§*DO*I§******§§§§III *§G§§§*§I*§§.'*§§§§if§§§§i§§*****§fil§§§**§*li***§§l*§§*iilfi DESCRIPTIBN 5F INPUT DATA: CARD 1 (IS) LTnT-YHTAL NUMBER 8F PPBFILES CARD 2 (15) JTBT'TBTAL MUMRER 9F SBUQCES PEP PRBFILE CARD 2 (2F1oo2.215> MsTBTAL NUMBER HF STATIBNS PER PRBFILE DELX:STATIBN SPACIVG Fesx.CBBRDINATE BE THE FIRST STATIBN N-MUMBER BF CBBRDINATES DLUS 8N6 F89 EACH SBURCE CARDS 4-->N (EPIC-3) X(I)uX-CBBRDINATES er THE SBURCE z(r)sZ.CBBRDINATES 9: THE SBUPCE CAQDS a..>N ARE PEPEATED JTBT NUMBER er TIVES CAQDS 2 THROUGH u-.>N APE REPEATED LTBT NUMBER BF TIMES i!!!§i*§i§¥§§*§§l¥iii§§§i§fli*I§§§§§*§#§§§§§§i**§§lii§§**§** 1 1 1 3 15 17 19 2 2 1 3 ?5 2 2 7 9 51 CRMMBM SECDERIV(SOO):FX(SOO)aX(SO):Z(50);xINTERCP(50):M:NpLL IMPLICIT DBUBLE PRECISIBN (A-Hlecl) READ (5:5) LIST FBRMAT (15) DH #9 L'IILTBT READ (5:7) JTBT rRRMAT (15) READ (519) DELX3F31M1N FRRMAT (2710.2:218) DB 47 JsladTBT WRITE (6:11) FBRMAT ('1') wRITE (6113) L FBRMAT (I/gSOX:'RESULTS FBR PRBFILE NUMBER ':IS.///) DB 15 1'11“ R!!! rx(I)=F5-0ELX+DELX*RI WRITE (6117) J FPRMAT (SOXp'CBQRDINATES Fe? RBDY NUMBER 1:15;//) WRITE (6:19) reRMAT (50X1'X'CBBRDINATE';6X1'Z-CBBRDINATE'o//) DB 25 IEIIN READ (5:21) X(I):Z(I) FBRMAT (EPIC-2) WRITE (6:23) X(I):Z(I) FBRMAT (46X12F1503) CSVTIMUE NNstl DB 33 1'1:~1 SECDER2¥O D9 31 K'ltVN x1=X(K)-FX(1) x2-X(K+1)-FX(1) ZZl!Z(K) Zi-Z(K) ZZEIZ(K+1) ZEaZ e3-(x1552+Z1**2)*(X2*«2+22**2) RusA1*((X1/(X1**2+Zl**2)1-(Y2/ 97-A1*((X1/(X1‘*2+21**2))-(X2/(X2“*?+ZE**2>)) 92c<1/<1+A1»v2))«< 52 R9IBBi/(1+A1#»2) 910-(xiuu3+z1lvg)»(x3;;3+22ua2) R11'(X1*'P*Z1**2)-(XPI*2+ZEM*2) 813"(X1**3*21**2)*Z2)-((XP**2+Z2**2)*21) p1331(x1..?+21¢r2)§2u22)+((x2**2+22**2)*2*21) e14=((X1'I2+ZII§2)§(X2#42+Z?I*2))*'2 .. _ 915311*(((2*X2#z2)/((x2§u2+22»*2)d¢2))-((2'x1#21)/((x1#i2+z19.2) 1&02))) (28(((ES/961+R7)*981+(B9§((((910*811)'(B12*813)1/814)+915)) SECDER1!C1+C2 SECDERE=SECDE91*SFCDER2 31 CHNTINUE SECDERIV(I)I4-377E-06*SECDER2 33 CBVTINUE WRITE (6:35) 35 EDQMAT (///o3OX:'I'18XI'FX(11'18X1'SECBND DERIVATIVE GRAVITY (ED 3 1ENERAL CASE)’:/) DB 39 181:M WRITE (6:37) IpEX(I):SECDERIV(I) 37 FBRMAT (27X:I“IF1303122X:E13o6) 39 CBNTIMUE CALL XIVTERCT WRITE (6:41) #1 FBRMAT (I/150X2'THE CRBSS-BVE3 pBINTS ARE LBCATED AT':/) 99 “5 I‘1tLL WRITE (6:43) XIVTERCP(I1 h? FHQMAT (60X)'X"IF1003) #5 CBNTIVUE #7 CBNTINUE us CBNTINUE STBP END SUBRGHTINE XINTERCT , QBMMBN DERIV(50011FX(50011X(50):Z(50)IXINTERCP(50)1NUMQER1NILL IMPLICIT DBUBLE PRECISIRN (A.H;8.Z) LL'O MNINol JJ'1 5 IF (DERIV(JJ)) 717115 7 DB 13 IIIpNUMBER 1r (NUMBER-JJ) 35:35.9 9 JJvJJ+1 IF (DERIV(JJ)’ 13:11:11 11 EXI'FX(JJ'11 KK-l LL'LL+1 GB TB 23 CBNTINUE 09 21 1-1.MUMBER 1F (NUMBER-JJ) 35:35:17 HH U'luJ 17 19 21 23 25 27 29 31 33 34 35 53 JJ'JJ+1 1F (DER1V(JJ)) 19:19:21 rXiIFX(JJ-1) KKIO LL-LL+1 GB TB 23 CRVTIMUE EXI'FX1+10 sscozoeso DB 29 K!1:NN XIIX(K)OFX1 XEIX(K+1)-FX1 221'Z(K) 2182(K) ZZB=Z(K+1) Z2-Z(K+1) 1F (Z71-ZZE) 27:25:27 ZZ2IZZE+oOOOOi RRIt(X1*ZE~X2*Zl1/(ZZE-ZZ1) A1'(X2-X1)/(222-221) 91-(1/(1+A1**2))*((X1vX2)/(722~Z21)) R28((X1**2‘Z1**2)*22)-((x2§:2+22**214211 n3-tx1::2+z1:*2):(x2::2+22::2) g4:A1:((X1/(x1::2+z1:.2)1.1x2/(X2l*2+22**2))) €1881‘((82/83)+R4) g5-((x1**2+21**2)*22)-((x2:*2+224*2)*21) as:(x1»-2+Z1**21'(X2442+22*‘2> 97-A1*((x1/(X1*~2+Zle:2))-(x2/(X2'*2+Z2442))) 888(1/(1*41**2))i((X1-X?)/(ZZE-Z21)) 99-881/(1+A1**2) 910-(Y1*'2+21‘*2)'(X2*‘2+Z?**21 811-(X1"2*21'*2)-(X8**2+Z2'*2) a12=<(x1'*2+21**2)*22)-((x2~:2+22**2)*21) Bl3l((x1§§2+21§.2)§2*22)§((x2§§2+225§2)§2I21) B143((X1*I2+Zl**21*(X2442+2?r*2))**2 915-A1M(((24X2*ZE)/((x2442+224*2)**2))-((2*x1MZ11/((X1**2*21'*?) 1442))) C2'(((RS/861+B71i981+(99a((((Q104811)'(B194313))/91“)+q15)) SECDERI-C1*C2 SECDERE'SECDERI+SEC0ER2 CBVTINUE DER1V1=40377E-06¢SECDER2 1F (KK) 31:33:31 IF (DERIV1) 23:34:34 IF (DERIVI) 34:34:23 XIVTERCP(LL)IFX1 08 TB 5 CBVTIMUE RETURN END APPENDIX C COMPUTER PROGRAM FOR CALCULATING THE THEORETICAL SECOND VERTICAL DERIVATIVE OF MAGNETICS OVER A VERTICAL TABULAR MODEL 54 55 §§§I§¥§§G§l§*§§§§§*§i§i*§*§*§***§*l§§i§§§**§l§*§¥l§§****§*§ DRBGRAM Fan CALCULATING THE THEBRETICAL SgCeND VERTICAL DERIVATIVE 8F MAGNETICS BF A VERTICAL TABULAR Seuch pRssRAMMED IN FBRTRA) ERR A XDS SIGMA 5 11/72) fii§§§§l§§4§§§R§§§§i¥§I!§§§§§§§i§§**’§*§§§§i§§*'*§§§*§l*§§§ *Riilii§§§§§§§§§§§§§§l§§§iii§§i§§l§i§I§§**§***.************ DFSCQIPTIQV 6: INPUT DATA: CARD 1 (IS) LTBT-TBTAL NUMBER 9F PREFILES CARD 2 (IS) JTRTsTRTAL NUMBER BF seURCES PER RRBFTLE CARD 3 (2F10-2:215:3F10:2) DELX=STATIBN SPACIVG FB-XoCBBRDINATE BF THF FIRST sTATIBN MnTBTAL NUMBER er STATIBNS PER PRBFILE N-NUMBER 6F CBBRDIHATES PLUS BNE FBR EACH SBuRCE Bus-SUSCEPTI9ILITY CBNTRAST (EMU/CC) ZINDUCED.MAGNITUDE OF THE EARTH'8 REGIBNAL MAGNETIC FIELD (BERSTEDS) CARDS 4-->N (2F10.3) X(I)¢X-CBBRDINATES BF THE SBURCE Z(I):Z-CBBRDINATES BF THE SBURCE CARDS 4-->M ARE REPEATED qTBT NUMBER BF TIMES CARDS ? THRBUGH 4-->N ARE REPEATED LTBT NUMBER BF TIMES 4! l§§1I§¥IiHIS§§1I§I§§§IHI{I‘Ilefii*l§§‘*§.**§**fifi}§*§***§.****§§* 56 CBMMRV SECDERIV (500),FX(500):M:X(SD):Z(50),SUS,IVCL, IZIVDUCED:N:VERTMAG REAL INCL READ (5:5) LIST 5 FBRMAT (Is) 08 70 L91:LTBT RFAD (5:7) JTBT 7 raRMAT (IS) READ (5:9) 3ELX.F9.M:V:SUS:ZIMDUCFD:INCL 9 FBRHAT (2F1002:315:3F10:2) D810 131:1“) 10 rX(I)=F3-DELX+DELX*I DH 60 J=1:JTBT WRITE (6:20) 20 FPRMAT ('1') wRITE (6:22) L P? EBRMAT (///:50X:'RESULTS FOR PRBFILE NUMBER':15:///) HRIIE (6:24) SUS:ZIVDUCED:INCL u 24 FBQHAT (/:'SUSCEPIIRILITYI':F7o4:IOX:'INDUCED MAGNETIZATIBN": 1F10:2:1X:'BERSTEDS':1OX:'INCLINATION BE THE MAGNETIC FIELD-'1 1F502:1X:'DEGREESI://) WRITE (6:30) J 39 EBRMAT (50X:'CBBRDINATES Fa: Reov NUMBER':IS://) WRITE (6:32) 32 reQMAT (50X,)X-CBQRDINATEV'6XI'ZDCBQPDINATE'l//) DB 38 I'1:N READ (5:34) XII):Z(I) 34 FBRMAT (EFIOOE) wRITE (6:36) X(I):Z(I) 36 FBRMAT (46X:2F15:3) 38 CRNTIMUE RADANGLE'INCL*-31745329 VgRTMAGszINDUCED48IMIRADAAGLE) 21:2(1) 2282(3) IF (21) 45:40:45 40 21-001 45 DB 50 I!1:M x1-FX(I)'X(2) X2'FX(I)'X(1) CALC1=(X1*ZB)/(X1‘*2+22*42)**2 CALC2s(X1*Z1)/(X1**2*21**2)**2 CALC3'(X2*21)/(X2**2+21*4£)e§2 CALC4'(X2*22)/(X2**2+22*‘2)**2 . SECDERIV(I)'SUSRVERTVAGR4:0356E02*(CALCI-CALCBTCALCB-CALC4) CBNTINUE . WRITE (6:52) EBRMAT I///:30X:'I':8x:'FX(I)':8x:'SEC6ND DERIVATIVE MAGNETICS (23 1 THICK DIKE CASE)':/) DR. 5‘4 1311” UT 0 UI 1U 56 b7 60 f‘ 'v’ 15 20 57 WRITE (6:53) I:FX(II:SECDEPIV(I) EBQMAT (27X: 141F1303122X1E1306) CRNITVUE CALL YINTERCT (XIVTERCP) WRITE (6:56) FDQHAT (//:50X:'THE CRBSS-BVEQ POINT IS LBCATED AT':/) WRITE (6:57) XINTERCP FQQMAT (60X:'X8'171003) CDNTINUE CQMIINUE STOP END SUBRBWTINE XINTERCT (XINTERCP) CRMMBV sEcDERIV (500):FX(500):M:XISD):Z(50):SUS:IVCL: IZIVDUFED:N:VERTWAG XX18X(1) XX28X(2) 2182(1) 2282(3) FX18FX(1) 1F (SFCDERIV(1)oLEoO) ISETIO IF (SECDERIV(1)oGToO) ISET-l IE (21) 20:15:20 21-:01 rx1=Fx1+1. xlIFXI-XXE XE-FXI-XXI CALC18(X1'ZEI/(X1**2+22§§2)4§2 CALC28(X1421I/(X1842+z1**2)::2 CALC38(X28Z1I/(X2882+Z1442)v:2 CALC48(X2422)/(X2:§2+28**2)l§? . DERIVI-SUS*VERYMAG*4o0356602~(CALCI-CALC2+CALC3.CALc4T IF IISET8FQ000ANDODEPIV1oLToC) GB TB 20 1F (ISET:EQ.1:ANDoDERIV1:GT:c) GB TB 20 XIVTERcP-FX1 RETURN END APPENDIX D COMPUTER PROGRAM FOR CALCULATING THE THEORETICAL SECOND VERTICAL DERIVATIVE OF MAGNETICS OVER A IRREGULAR-SHAPED POLYGON 58 59 gIiléiifi‘fiffliifiiiliilliii*fl¥§**§**§§§****§‘§§§I§**§*§§*§**§ DRBGRAM ERR CALCULATING THE THFBRETICAL SECBND VERTICAL DERIVATIVE 3F MAGNETICS 8F A PRISM-SHAPED SBURCE oRQGRAMwEn 1N FRRTRAM ERR A XDS SIGMA 5 (1/72I a;i54&&§§§§§#§§*§*i*¥n44ifi*&*§*§o§*§&*ii*u§oi§!§§&*§§§§*§i §§G*§I*&‘#‘liiii§§*§§{bidifiifi§i§§***lfifilifl*lii§*§iifififi§§*§§ DESCRIPTIRN 6F INPUT DATA: CARD 1 (IS) LTRT-TRTAL NUMBER 8F DRRFILES CARD 2 (15) . JTBTsTRTAL NUMRER 8F SBURCES PER PREFILE CARD 3 (2F10:2:215:3F10.2) DELX:STATIBN SPACING Fnsx-CBBRDINATE RE THE FIRST STATISN McTBTAL UUMBER BF STATIPMS PER PRDFILE N-MUMBER eF ceeRDIUATEs PLUS BNE FBR EACH SBURCE sus-SURCEPTIBILITY CBNTRAST (EMU/CC) _ ZIMDUCFD-MAGNITUDE BE THE EARTH'S REGIBNAL MAGNETIC FIELD (BERSTEDS) CARDS 4.->M (2F10.3) 'X(I)-X.CRBRDINATES BF THE sauna; Z(I)-Z.CRBRDINATES BF THE SBURCE cAonS 4-->N ARE REPEATED JTBT NUMBER 0F TIMES CARDS 2 THRBUGH 4-->M ADE REPEATED LTBT NUMBER BF TIMES ***§§#D§‘§§§**§i‘*§§¥«Iii!!!“{v{*‘R‘il§vl§§filN (2F10o3) X(I)=X.COORDINATES BE THE SBURCE 7(I)tZ-CBBRDINATES BF THE SBURCE CARDS 4-->N ARE REPEATED JTBT NUMBER BF TIMES CARDS 2 THQBUGH 4-.>N ARE REPEATED LTBT NUMBER 6F TIMES fi§§§§*§I**I§**I*§&*&§I§&§lilfififllGIl§§*§*§§*****§§Ii§§l§§§§* 66 mmaa . ' . 12/08/66 PROGRAM OGDZ_. _ _ A 1 COMMON/1/GRAVI500).FXISoo).M.APPRDER(500).x(50);i(50).N.DELx.F0. 1RH0,THEODER<50n).NUMBODY.MM,NN COMMON/ZI DUMMYIZOIII MM=1 $ NNPZ READ3.LTOT 13 FORMATIISI. DO 20 J81.LTOT ‘PRINT 7 A 7 FORMAT (1H1) PRINT 4,J , A _*,q 4 FORMAT (ll/50X.tRESULTS FOR PROFILE NUMBERt.15///) . CALL APPROX" PRINT 11 imw 11 FOPMAT(;H;[ PRINT 12 12 FORMAT (5X.*I0.10x,wSTATIONt,1nX,¢GRAy1TY ANOMALYP.10X.*APPROXIMAT 1E SECOND DERIVATIVFt.1OX.oTHEORETICAL SECOND DERIVATIVEPII) .7 DO 13 I?1#MH. .... V 13 THEOUER(I)ITHEODERII’filEio PRINT 15.(IaFXII).GRAY(I>1APPRDEPIIIoTHEODERXI’alilaMI 15 FORMAT (16.8X.F10.3.9X.F10.3.23x.E13.6.26X.E13.6) 7 PUNCH 510.(GRAV(I).APPRDERII).THEQDERIIQ;Is;1H)Ed" 51o FORMAT (6613.6) go CONTINUE END .... fl..- ~.-. .. "S.-- .v . ..--... _.4 a -... .--. - -H- n"..- .u--u.—. » ,,._.,H, u— <- vw—-- “fiqm .- --‘—~. -.... . u‘mw-o ”.M—M'w a... -.._.. . , . h “-mofl~-w _..—-...—. l- .a— .u..I .,u 7v - --._._, ,._ 67 .*m4 _. ._,.- ~.——m..— ._.. >m-‘“l-A‘-1'V—.-A-—u- ‘— . -o-.. .. E- Mn 7.. I. _ NLSB ' 12/08/66 SURROUTINE APPROX . .. . COMMONIIIGRAVI5OD).DUMMY1CSOO}.M.APPRDER(5003oDUHMY2(101)oDELXo 1DUMHYSISOS) COMMON/ZIDUMMY4(931).VAL1(30>.VAthso).DUMMY5(1OZO) CALL GRAVZO PRINT 5 5 FORMAT (1H1) "MAX5N4§WW”WW‘ DO 15 IF31MAX RAD 80 DO 1" K31.? RADBRADODELX VAL1¢KI=IRAD*SO.483t*2 ” vAE2 PUNI;U 505.220 230, 23n 220 TRETE-ATANFIZEFE/EXXXI 3.1415927flm” GO T0300 _ 230 THETRBATANF(ZEEE/EXXX)*3,1415927 GO TO 300 240 IF(ZEEE)250.260.27O 250 THET93'1.5707963 _ _M GO TO 300 260 THETPBO. 0 GO TO 300 270 THETR31.57O7963 GO TO 300 280 THFTRBATANF(ZEEE/EXXX) 390 TFTLPLXQQQ11§fi9213O911_- 3002 EXXBFXXX 40? '4on 3003 3005 ‘420 13 ”"TFTCHECK33?OQ310i350‘- -.———-—---..————r 7 . - .. ,T , , I ZEE=ZEEE R=PR ' THETAzTHETR IF(I-1)205,200.205 1:2 GO TO 205 CHECK=FXXt7EEEezEEtEXXX DELZ=000 OMEGA=THFTAufHETB IF(0MEGA)3201.3202.3202 IFIOMEGAE3.141592733302330p3N0 IF(OMEGAES.1415927)340:330.330 OTHETaOMEGA ' N" GO TO 370 IFIOMEGA)350.360a3OO DTHETaOMFGAo6.2831853 GO TO 370 OTHET=OMEGAo6.?831353 *AAA2.OTHET IFIRRIS71.371.403 C=O.n GO TO 402 IFIRIS71.371.401 CCC:RR/R CCILOGFICCC) _ C=.5¢(7EEE-ZEE)*CC DELZ:AO(B+C) _ SOELZ:SDEL7+DELZ’ IFIIcN)3003.SOOSa3OO5 I=I+1 GO To 3002 SUV =.OO4066032*RHOtSDELZ GRAV(K)=GRAV(K)¢SUM CONTINUE CONTINUE END .,A-_.——.—_.—4 —__--..- ...... 12/08/66 M5.3B 10 16 14 15 ..—- .‘i (1.....— .o.-.s.-- an.— I.,..E,—H. “CONTINUE 70 12/08/66 SUBROUTINE CALCDER , h _w,. COMMON/I/GRAVISon).FXISOOI.N.OUN~Y1I50n).xc50).ZIso).N,OELx.FO.RHO 1aTHEODERISOO).DUMMY2(3) COMMON/Z/ DUMMYSIZOil) NN‘Ni-l DO 15 1:1.M SEC DERZ=0 DO 14 K=1,NN X1‘XIK)0FXII) $ X2=XIK*1)-EXII) $ Zl=ZZl=£(K) S ZZ=ZZZ=Z(K*1) IFI21-72)16.10:16 72:22+.01 RBlzIX1*ZZD- X2*ZZl)/I22-21) A1=Ix2- X1)/I22 21) ~_ BitIi/I1*A1**2))‘(IX1~X2)/IZ? 21)) RZBIIX1**2+21**2)*22)-IIX2**?*ZZ*f2)*Z1)_ 838(X1tt2¢71tt?)t(x2it2tZZtt2) 84=A1tI(Xi/(Xltt2¢71**2))-IX?/Ix2t*2f123*2))) C1381tII92/BS)*B4) 85=IIX1**2¢21**2)*22)-(IX2**?¢Z_2**?)*ZS_) 96=IX1**2+71**7)*IX2**2¢22.*?) B7=A A1tIIX1/IX1tt2+71*t2))'IX?/Ix2t*2*l?**2))) 988(1/I1¢A1**2))*IIX1-X2)/IZ?'21)) 99=891II1+A1**2) A 810=IX1P¢2¢21-*2)*IX2**2¢Z?**2) 811:Ix1**2¢21**2)-IX2**2*Z?**2) 8122IIX1**7+21**2)*22)9 ((X?t*?¢Z2'*2‘*71) 8133IIX1PP?+Zi*t2)*2*Z?)+IIX?*02*Z?*rd)*°*71) “.1 R14:((x1tt7¢21¢t2)t(X2**2*22**?))**2 B15: A1*(I(?*X2*ZZ)/I(X?**2+Z?**2)**2))'((2*X1*Z1)/((X10‘2*21**2)** 12))) OZRII(85/862f87)*8fl)*IB9*(I(I810f311)PI812RB1311181410915)) ' SECDER1=C1¢CZ ISECDER2=SECDER1+SEODER2 CONTINUE SECDERTV =4,377E=06tSECDFR?*RHC THEODERII):THEODERIII+SECUERTV END 71 .. ..'._——-——__.__..~ I~———-—.—---_~'~ —— 1 ‘—w—&o7I 109 73 73 DET=-DFT DO 12 L=1IN T=AIIROH,LI AIIRnN,L)=AIIcoL.LI 1? AIICOL LI=T T= EIIHONI $ RIIRONI=RIICOLI $ BIICOLI=T an“; .7...— -“———50 PIVOTITIEAIICOLIICOL) s no 205 L=1.N 169 INDEXII.1):IRON S INOEXII. ?)=ICOL nET= OETIPIVOTIII__$V_AIIC0L.ICOLI=1.0 ?05 A‘ICOLpL)=A(ICQLIL,/PIVOT(T) B‘ICOL)BB(TC0LT/PIV0T(I) ‘DD 135 LI=1,N IF(LT'TCOL) 211135321 21 T=A(L111C0L) A(LI,ICOL)=000 DO 89 L311“ _. _--—o-..--—- -r’.._ 89 A(LII LIzAILl. L)'A(ICOLIL).T RILII=R_I_LI>-BIICOLIoT, 135 CONTINUE ,22210013 I=1_oN L: N—I¢1 19'JRON=INDEXIL 1) -999LFINDEXIEI2) DO 549 K=1.N I_FI1NDEXIL,1I-INDEYIL 2))19.3 19 - --— .. T=AIK,JRCNIL S AIKIJROHI=AIKIJCOLI 3 AIKIJCOLI=T 549 CONTINUE 1”} CONTINUED 81 CONTINuE END .1. 59 4560.7. {‘”' TEETION"STARTEO AT 9224 .31 APPENDIX F COMPUTER PROGRAM FOR CALCULATING THE SECOND VERTICAL DERIVATIVE OF THEORETICAL MAGNETICS USING THE LEAST SQUARES METHOD 73 74 anfi§§ulii§§§lfii§§§§§§uiggiuu§§§§§§§§§§§§§§§§§§I*§§*g§.§*§*q DRBGRAM Fen CALCULATING THF APPROXIMATE SECOND VERTICAL DERIVATIVE 5p TqERQETICAL MAGNETICS USING THE LEAST SQUARES METHRD DRRGRAMMED IN FSRTQAN FeR A cDC 3600 COMPuTeR (1/67I pNNNNs#**§§§§§§*§vu§§§aNidgbR*#§&&§§¢§N§§§§*§gug*.‘.‘§g.... giI&§.Q§§IGG**§&I§§§*§§¢51¢»*****l**§**§IN§§§§*§i§§§§§*§§*§ OFSCRIPTIRN 8F INPUT DATA: CARD 1 (IS) LTRTaTRTAL NUMRER RF pRpFTLES CARD P (?IIO!2F100?) JTRTaTRTAL NUMBER 5F SBUQCES PER PPRFTLE M-TBTAL NUMBER OF STATIBNS PER PROFILE oELx.STATIRN SPACING rasx.CRRROINATE er THE FIRST STATIBN CARD 3 (110:4F10-2I N-NUNBFR BF COORDINATES PLUS BNE FGR EACH SBURCE SUR-SUGCEPTIRILITY CGNTPAST (EMU/CC) STRIKE-STRIKE OF THE SOURCE (POSITIVE CLOCKWISE (DFGQEES) FRQM THE MAGNETIC NORTH PBLE) _ DIDsINCLINATIBN RF THE EARTH'S REGIONAL MAGNETIC FIELD IPRSITTVE DBNNWARD (DEGREES), ZINDUCEDxMAGNITUDF BF THE EARTH'S REGIHNAL MAGNETIC FIELD (DERSTEDSI INCL -INCLINATIBN RF THE EARTH'S PFGIBmAL MAGNFYIC FIELD (PRSITTVE DSNNNAQD (DEGREESI) CARDS A-->N (2:10.?) XIII-X-CBBQDINATES OF THE SOURCE ZIIIsZ-CGHQDINATES OF THE SQURCE CADDS a-->N ARE REPEATEH dTBT NUMRED 5p TIWES g!libiiifli§*'§§§§§&*§G§§¥§§§§‘&§**‘fi*liil*li‘*§iD§§§§‘§§*‘ _' I 1““ u I I W” TN5.38 11 1? 13 15 51h an 75 01/06/67 DROGPAH TGDZ CONMnN/1/2MAGIBOFI,FX(500I.M.APPHDEHISOUI.XISOI.ZISUI.N.DELX.FO. 1suR,THFoIEQIRQQ),NHMBOHY.MH.NN.SIRIKE,DIPIZINDUCED COWMHN/Zl DUMMY(?OIII MMz2 R NN:3 HEAD¢,LTIT FORM£T(I5) . D0 2F J:1,LTUT PRINT 7 FORMAT (1H1) PRINT 4,5 FORMAT (///5DX.0HESJLTS FOR PRCFILE NUMBFR*.IS///) CALL APPROX PRINT 11 F04M£T(1I1) PRINT 12 FOKMAT (EX,*It’lnX,tSTATIDN*910XD'MAGNFTIC ANOMALY*.10X.9APPROYIMA 1TE SFCONF DERIVATIVE*.]OX.*THECRtTICAL SECOND DERIVATIVEil/I D0 1! =1,M THcozERII)=THEODFR(IIt1810 PRINT 15,I1,er1),zNAG(II.APPRDEN(II.THE0DER(II.1:1.MI FQQMAT IT6,ax,r1h,3,9X.F10.3.23X-E13.6.26X.E13.6) pux‘lch 51f, (ZMA[;( I ) 'APP'QDER‘ [)1THtODER(I)D [=11M, FOTMAT (6513.6) CONTINUE FNJ 76 Ni}? 01/06/67 SUfiRHUTIPE ADDHOX COquN/1/7yAQ(50fi),DUMHY1(SOU),M:APPQD;R(500)IUUMMYZ(101)’DELX' IDUMMV3(SFE) COWMFN/Q/DUMWY4(951):VAL1(5J).VAL2(30).DUMMY5(107O) CALL MAGFD DRINY 5 R FO4M£T (1H1) MAK=¥-3 00 15 I=¢.“AV HA3 :0 HO 1“ K=1,i QA3=FAH+!ELX VAL1(K):(QAU.3fi,AH).¢2 VAL2(K3=(2¢ZMA3(I)~ZMAG(I-K)-ZVAU(I+K)3/VAL1(K) 1n VALZiK)=xAL2(K)*1E10 CALL LQTSQDOL (DERIV) APPRIEP(I):DFHIV 1% COMM NLE FNU ”15.38 1? 16 2n Sun 503 14 1% 1o 1n 3‘ 32 4e 49 77 01/06/67 QUnHFUTIhE MAG?D COWMHN/1/2MARI50J),Fxcfififl).A.APPHDER(5fiU).Y(50).Z(5U),N,DELX:FO, 19US,THEO{EQ(SUO).JTOT,UUMMY1(2).bTRIKE.DIP,ZINDUCED C0“MFM/2/ DUMMY2(2011) HEAD 19.JTOT,M.UFLX.FO FORMAT (?I1U.2F1fi.2) HO 6 1:1.“ 7 APPRVFR(I)=0.G FX(I)=FO-DFLX+DEL¥*I DO 13 J:19JTOT READ 16,“,SU§,9TWIKE,D[P,ZINFJCEU $ FLE=0.0 F0%MIT (110,4F13.2) READ 7n.(X(I).7(I).I=1.N) FO4MAT(2F10.3) pUVCL 50p,M,M,FD,UELX,3US,STHYKE:DIP:ZYNUUCED FORMAT (215.6F10.4) PUVCV 505.(X(I),7(I).I=1.N> FO*MAT (10F8.n) PRINT H,J FQQMLT(5nx,.COORUINATE§ FOR BOFY NUMBE9*.IS//) DRINT 9 F0&MLT(50X.tX-CDORDIKATE*.6X,vZ-UOORDINATEa/I) DRINY1.4.(XH),Z(I):I=.1.N) FORMAT (49X.F15.3.7X,F15.3) PRINT 505U93_.7INDUCEDpDIP FO4MAT (. QUSCEPTIRILIfY=t.F7.4.10X.*IMDUCFD MAGNETIZATION=t.F10.2 .10x.rINCLINATION OF THE MAGNETIU FIELH=*.F5.2.1X.«DEGREEst/l) PRINY 15.8TRIKE F0%MAT (. STQIKE OF THE Rouv=Nt,r5.2,«:.//) IF(STRIKE)1U.1° 9T91“E=,P0001 STQI“E=STRIKFt.01745329 $ DIP=UIP*.O174S329 CALL CALCDER RETA=ATAFF(TANF(PIP)/SINF(STPIKE)) NN=N~1 S HRH:SQRTF(1-((CUSF(STHIKt))**?*(COSV(UIP))**2)) D0 15 133:” 7IJTFN?=“ no 14 K=3,NN X1=X(K)-FX(I) $ Y2=X(K+1)-FX(I) $ 7Zl=71=Z(K) $ Z72=ZZ=Z(K¢1) 722=7ZZ+,01 A1=(x2-X1)/(7Z?-721) Q1=(>1t22-X?*11)/<7Z2-221) IF(B])38.36 R1:.“01 _ H2=1+Alfit2 $ AA=1/h2 RB=A1tSINF(BFTb)+COSF(RETA) DD=AJtCOSF(BETA)-SIVF(BETA) FF=<(1+A1tt2)«72*t?)+(2*A1*Bfl*22)+81**? GG=((1+A1**2)*Zl**?)+(2*51*81*21)+81*t? HH=S£RTF :?&5*SUStZINDUCED*BQF*ZINTEN2 1R CONTINUE 13 COVTINHE END 05.38 17 1A 21 29 14 1: V 79 01/06/67 guapqu1~E CAlflflFH COV‘NH N/l/ZMA“(H[]F),F’(‘)UO),M'ULIMMY1(Snn)9X(5n)12(50)3NpDELXDFOISUS lgTHE‘HER(500).“U*MYZ(3),STHI“E,DIP,ZIN“UCED CoflM'N/Z/ HUMNV3(?UII) FETA=ATA‘F(TAVF(FIP)/SIVF(bTPIKE)) "QN=N".1 g HLquzsrigTF(l'((CUSF(FTHIKE))**?‘(COSF(DIP),**2’) DO 1“ 1:1)M DE&IV?=0 DO 14. K:1,t\|f\: x1=X-¥?*ZJ)/(72?-221) IF (F1)2?,?1 R1=.l1 H2=1+A1tfi2 ¢ AA:1/H2 RB=A1¢SI\F(EETA)+COSFtBETA) PD=A3rCOFF(BETA)-SINF(BETA) FF=(IJ+A1**2)ri2**?)+(2*A1*81*22’+Bl**? GG=(<1+A1t*2)*71**?)+(2*A1*81+21)+91**9 H1=(\1-x$)/(7Z?-721) 9N:((H9t72)/P1)+A1 $ SP:((h?*Zl)/Bl)+A1 HH=(7*72‘H?)*(2*‘1*33)+(2*A1*Z2*H1)+(2*91*H1) SI=(?’71*H7)+(7*A1*HJ)+(?*A1*ZI*H1)+(?*81*H1) qJ=2*("?+(?*M*H1)+HJ**2) SL:1+SN*¢? 1 SN:1+SP*Q2 m0=(LG*FF)*(((RG*SJ)+(HHtSI))-((*F*SJ)*(SI«HH))) PR=((GCtrH)'(FF*SI))*((GG.HH)+(FP*SI)) q3:.‘>*H8v(WU-PR)/(GG*FF)M~2 UUH=~1-Z?*H1 W VVV=Hl-71*H1 HU=1ISL*(-?*H1*H2tUdU)/Blt*3 ww=1/$M*(-?*H1*H7tVVV)/d1**3 Vv=((H?*LHH)/FJ**2)*(-2*SN*(F2*UUU)/Bitr?)/SL**2 Xx=((H?*VVV)/BJ**2)*(-?*SP*(H?*VVV)/Plti?)/SM**2 YY=DIt((LU+VV)-(WN+XX)) UEQIV1=AIt(SS-YY) DENI\?=HEPIV1+DEP1v2 CONTINVE THEO¥ED(I)=2.1527F2*SU9*ZINDPCFD*BRBtDCRIV9 CONTINUE END 80 N5.KB 01/06/67 QURRfUTIPE LQTSUPOLchERIV) CO4MFN/1/DHNMV1(9306).WQN.UU“PY2(3) COMMrN/Z/A(5n,30),R(30),NP1,Y{60),Y(30).C(3U.3fl).DUMMY3(120) D0 3f =1.M 3P C(I.1)=1.P MPI='+1 D0 35 J=?,MP1 D0 3: 1319‘,“ 3‘3 C(I,~1)=C(I,.J-1>*HI) PO 4“ 1:1,MP] no 4v J:1,MP1 A+C(F,I)*C(K.J) DO 43 1:9,NP1 H(I):H.O no 4? K=fi.~ 48 L2(I)=R(I)+C(V,E)*V(r<) CALL MATPIY PRINT 60r,w,w 60h FORMAT (. MUMRPR 0F GIVEN DATA PUINTS:.,12,1nX,oDEGHEE POLYNCMIAL: 1*.123 DO 5“ 1:1,MP1 II=I-1 5n DRINT Hof,11,«c1) _ Ron FOQMIT (Ex,1?.* TEGQFE COEFFICIENT=*,E13.6) DEF(I\-:L‘4(°f) FNU 81H TN5.38 ._ _ _rwv ‘ffi L 7 01/06/67 SUHRfUTINE MATRIX coMMrN/1/'fiumfivl(211d)’" COMMnM/2/A(3n,50).99301,N ,FUNMY2(9603.IPVOT(30),INDEX(30.2)o 1PIV0T(3D) EQUIVALEPCF(lHOW,JQOL).(ICULJJCOL) DFT=J.n DD 17 J:*,N 17 IPVOT(J)=O no 135 1:1,N % T=0 DO 9 J=1.N _ IF(I»V0T(J)31)13.9.ISH_U* 13 no 23 K=3.N ’F(1“VOT‘K)'1’43’23:94mwmu _ 41 IF(AFSF(T)-AHSF(A:A(IC0L,L) 19 A(IC(7LpL):T H T:H(IRON) s H(IRO~)=BL1§QL}_ S' B=T 10o INDEX(I,3):IRHN $ INDEX(I,?)=IUOL PIV0T(I)=A(ICJL,IC0L) “$W,DET=DE|*PIVOT(I) $ A=R(LI)rW(ICfiL)~T 13R COwTINUE 22? DO 3 I=1.N L:\'-I+1 ._ IF $ A(K.JR0N)=A __ FNU _W “11-59165'10171 1 "Ma‘. r.~ ‘ ‘- .. 4,.._._......---- ‘7 ECHJTInN STAPTED AT n3?2 ~56