12

A Second Derivative Method of Interpretation of Gravity Anomalies of Anticlines

By B. S. R. RAO, I. V. RADHAKRISHNA MURTHY and Y. V. SUBBA RAO i)

Summary - A simple method for the interpretation of second derivative gravity anomalies of an anticline is presented. The method utilises only three readily measurable distances, from the origin to the points of half maximum second derivative anomaly, zero anomaly and minimum second derivative anomaly. Charts for the computation of the various parameters are constructed.

Introduction

Anticlines, synclines, dipping faults and other structures commonly occur in oil prospecting. But no systematic methods of interpreting the gravity and magnetic anomalies of anticlines and synclines are available in Geophysical literature, though

rigorous methods of analysis of anomalies of simple models are available. The con- struction of any method of interpretation of these bodies is difficult due to the lengthy expressions involved and too many quantities to be interpreted.

The interpretation of the gravity anomalies of symmetrical anticlines and syn- clines seems to be apparently easy, as the comparison of the observed anomalies with the theoretical ones is possible on log-log papers. But such a method is seriously handi- capped by the number of parameters to be interpreted and as the anomaly variation with all the parameters cannot be included in a single chart. Thus the problem re- quires 'a priori ' knowledge of approximate values of at least some of the parameters

to enable the selection of a proper chart f rom a given number of charts. The method proposed here enables one to arrive at the approximate values of the

various parameters, with whose knowledge accurate values can be obtained by actual matching of the field profile with the theoretical ones. The method at present deals with only the calculations of the second derivative anomalies of anticlines. Results of the other calculations will be published in due course.

Fundamental equations and character&tic distances

The method to be developed in this paper requires only three distances on the second derivative profile of the gravity anomaly. The distances are measured from the

i) Geophysics Department, Andhra University, Waltair, India.

Interpretation of Second Derivative Gravity Anomalies 13

origin and each to the point of half maximum anomaly, point of zero anomaly and

the point of minimum anomaly. They are respectively denoted by x l / 2 , x o and xm. The interpretation by second derivative profile rather than the gravity profile itself

has the advantage that the three characteristic distances occur only within a fraction of the total width of the model. Hence interpretation can be carried out even when the gravity observations are available only over a part of the body.

The second derivative anomaly of an anticline can be shown to be

62g 2 y a sin2 c~[(D - d) 2 + co 2] [(2 D + d ) x 2 - d ( D 2 + 0)2)]

az ~ = (x 2 + D 2) [D e -l- (X -}- 09) 2] [D 2 + (x - 0))2] ( I )

so that the points of zero anomaly are defined by the equation

~o ~ - d(D~ + 0)~) (2 ) 2 D + d

where all the parameters are as defined in figure 1.

�9 tO 'b , Figure 1

The points of half maximum anomaly are defined by the real roots of the equation

x 6 + A x 4 + B X 2 + C = 0 (3)

where A = dZ + 2 D2 - 2 0) 2

B = (D 2 + 0)2)~ + 4 Od(0) ~ + D 2 + Da) C = - d 2 ( O z + 0 ) 2 ) 2 .

The distances of the points of the minimum anomaly are obtained by solving the equation

X 6 -{- a x 4 --it- b x 2 + c = 0 (4)

14 B. S. R. Rao, I. V. Radhakrishna Murthy and Y. V. Subba Rao

r /

it /

f ......... f ........... J .......... ,-

/

/ , /

(Pageoph,

t'xl

t

where

and

I0

8-0

6.0

5-0

4-0

3.0

2 D 2 -a t- d 2 - - 2 0.) 2 3 ,)

a = 2 - ~ x ~

b = - (2 D ~ - 2 ~o 2 + d ~) x~

( Xo ~ + d 2) o)2)~ = - 2 (D~ + - ~~ d ~ ( D ~ - ~

where x o is as defined in equation (2).

Z.O

,s i f !

I. 0 t 0"8

0

Vol. 85, 1971/II) Interpretation of Second Derivative Gravity Anomalies 15

n c u r v e s indicate ~es of D

s

i i I [0 2 ; 31o 40 50 6o

Figure 3

16 B.S.R. Rao, I. V. Radhakrishna Murthy and Y. V. Subba Rao (Pageoph,

bOO

0.9(

0,8(

0 ' 7 0

0 - 6 0

0.50

T - I N

x

N u m b e r ~ o n C u r v e s i n d i c a t e v a [ u e ~ o s D

i 1 I0 2 0

I I i ! 3 0 4 0 50 6 0

Figure 4

The three characteristic distances namely Xo, the distance of the point of zero anomaly from the origin, xm, the distance of the point of minimum anomaly from the origin and xl/2, the distance of the point of half maximum anomaly from the origin, are thus calculated from equations 2, 3 and 4 for various values of Did and the dip angle ~. The characteristic distances are schematically represented in Figs. 3 to 5. In Fig. 2, the ratio of x~/2/Xo is plotted against the ratio xm/x o.

Method of interpretation

All the parameters of the anticline can be calculated with the aid of Figs. 2 to 5. The steps to be followed are simply as follows:

(1) Obtain the second derivative profile from the observed gravity profile. The second derivative calculation can easily be made by a simple formula (1)

aZg

Oz ~ = 2 g(0) - 2 ~(1)

where g(0) is the observed gravity at the point of observation and ~(1) is the average of the two gravity anomalies, on either side of the point.

(2) From the second derivative profile, thus calculated, obtain the distances Xo, Xm and xl/2-

(3) Find out the ratios xm/xo and xl/2/Xo. Pick out the two curves in Fig. 2,

Vol. 85, 1971/I1) Interpretation of Second Derivative Gravity Anomalies 17

~ ~ .... 5 "

Figure 5

corresponding to xm/xo and X1/2/)C O. The point of intersection of these curves will

define ~ and Did. (4) With the interpreted values of c~ and D/d, obtain the ratio xo/d f rom Fig. 3,

which on comparison with Xo obtained in step 2, yields d. Similarly use Figs. 4 and 5 to obtain two more values for d. The three values of d thus obtained must be close to

each other. (5) Knowing Did from step 3 and d from 4, D can be obtained. Thus the three parameters c~, D and d completely define the anticline. The density

contrast a can be calculated from the maximum gravity anomaly G, by the formula

G=4vtr{ (D-dc~ - ds in~c~176176

2 P A G E O P H 85 (1971/11)

18 B.S .R. Rao, I. V. Radhakrishna Murthy and Y. V. Subba Rao (Pageoph,

Some limiting rules for depth and dimensions of the body

The fol lowing t h u m b rules are derived f rom the analysis o f the var ious distances

calculated. They will be often useful in readi ly determining the approx ima te dimen-

sions of the body.

(1) The ha l f width of the second derivat ive anomaly o f an anticl ine is approx i -

mate ly equal to the depth d to its top for small angles of the dip of the flanks. The

dep th thus calculated is always an under-es t imated value. W h e n the dip of the flanks

is less than 20 ~ and the ra t io D/d is not less than 3, the error in the ca lcula t ion o f d by

this rule will not be more than 10~o. When the angle is less than 10 ~ the error will be

as low as 3 to 59/0.

(2) The dis tance of the po in t of 1 m a x i m u m anomaly f rom the origin is approx i -

mate ly equal to a/3 t imes the depth to the top. The da ta concerning the + m a x i m u m

points is not however presented in this paper . This dep th rule is val id for more num-

ber o f cases than the ha l f width rule above.

(3) The dis tance f rom the origin to the min imum anomaly is approx ima te ly equal

to the ha l f width co of the base of the anticline. This rule, jus t l ike the ha l f width rule,

give excellent results for small angles o f the dip of the flanks and large rat ios of D/d.

Errors in equating an anticlinal structure to a horizontal cylinder

Antic l inal models are often approx ima ted as hor izon ta l cyl indrical bodies and the

anomal ies are accordingly interpreted. But it is present ly found out that such approxi -

ma t ion yields most inaccurate results. The errors ob ta ined in the mean depth calcula-

t ions of anticl ines o f 60 ~ dip, in app rox ima t ing them to o rd ina ry hor izonta l cyl inders

are b rough t out in Table 1. I t can be clearly seen f rom this table that such approx ima-

t ion does no t p rovide rel iable results. The depths es t imated are only accurate for

small ra t ios o f D/d, i.e. when the body can be app rox ima ted to a line mass.

Table I

Did Mean depth of anticline in units ofd

Depth calculated by the formulae of a cylinder

Depth ~ 3.059 x1/2 Depth = 1.732 x0

Depth in % Depth in units of d error units of d error

1.5 1.33 1.41 +6.0 1.32 --0.8 2.0 1.67 1.57 -- 6.0 1.61 -- 3.6 2.5 2.00 1.76 -- 12.0 1.87 -- 6.5 3.0 2.33 1.92 --17.6 2.10 --9.9 4.0 3.00 2.17 -- 27.7 2.52 -- 16.0 6.0 4.33 2.39 -- 44.8 3.20 -- 26.1 8.0 5.67 2.54 -- 55.2 3.77 -- 33.5

10.0 7.00 2.64 -- 62.3 4.26 -- 39.1

Vol. 85, 1971/II) Interpretation of Second Derivative Gravity Anomalies 19

As can be easily seen, the errors in such an app rox ima t ion will be m i n i m u m for

anticl ines o f dip o f 60 ~ . Hence the errors for those with dip other than 60 ~ will be more

serious. The anomal ies o f an anticl ine canno t therefore be in terpre ted on the basis o f

the dep th ru les / in te rpre ta t ion techniques o f a hor izon ta l cylinder.

REFERENCE

[1] B. S. R. RAO, I. V. RADHAKRISHNA MURTHY, and S. JEEVANANDA REDDY, A Simple FormulaJor the Second Derivative Method of Interpretation, Pure and Applied Geophysics (in press).

(Received June 10th 1970)