~eoex~~o~u~~on, 17 (1979) 261-268 261 Q Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

THE HARMONIC METHOD OF INVERTrNG A MAGNETIC PROFILE OVER A CONTACT

RONALD GREEN

Department of Geophysics, University of New England, Armidale, N.S. W. 2351 (Australia)

(Received November 21, 19’78; accepted February 1, 1979)

ABSTRACT

Green, R., 1979. The harmonic method of inverting a magnetic profile over a contact. Geoexploration, 1’7: 261-268.

The profile of the magnetic gradient over a buried geological contact has four significant paints: two turning points and two inflexion points. The four points are harmonic. Using this inuariant property of the position along the profile of any three of the four points, a simple graphical method is given which determines: (If the position of tbe contact; (2) the depth to the contact; and (3) the angle of dip of the interface.

INTRODUCTION

It is widely accepted that there is little of a fundamental nature that remains to be discovered about interpreting the form of a magnetic profile over a geo- logical contact. Nev~r~ele~* the question can be looked at again with advan- tage when it can be demonstrated that an interesting result is fo~h~oming from the bringing together of:

(1) New technology enabling magnetic gradients to be measured directly. (2) A restatement of the little known results that a certain four characteris-

tic points on the profile are harnaonic points. (3) The availability of hand-held programmable calculators now enable

simple interpretational procedures to be carried out in the field.

DISCUSSIUN

The Varian VIW 2321 CI portable Caesium gradiometer (1977) measures the horizontal gradient directly with an accuracy greater than 0.1 gamma rn-’ and with a sampling rate faster than eleven samples per second, In practical terms for a ground-borne survey this is equivalent to a continous profile and the accuracy is such that the limit is not in the instrument but in geological noise.

262

The use of the hand-held, pro~ammable calculator for geophysical (seismic ray paths) problems has been pointed out by ~ichaels (1977).

To appreciate the significance of four certain characteristic points on a profile of the magnetic gradient over a geological contact being harmonic points, it is necessary to discuss the matter further before showing how this property can be used as a powerful method of interpretation. It was first pointed out by Roessiger (1943) and again by Jung (1953) that the two turning points and the two inflexion points are harmonic but the significance of this property was not pursued.

The harmonic property of the four points require that if the distance be- tween the four points be: U, u and w, then for every profile the following ratio is always true:

u(v-w) = w(u-v)

If u, u and w are, respectively, 1, 2 and a the points are harmonic. Continuous profiling is essential if the position of the maximum and the

minimum is to be located accurately. Let F(x) by the horizontal gradient of the magnetic field measured directly

with a gradiometer. In the case of a geological contact at a depth h, dipping at an angle, d and

striking at an angle X (Green and Stanley, 1975) where the susceptibility con- trast is k, the analytical expression for the anomaly is given by:

F(x) = -2kTc sin d [sin (8 + #)] /r

where :

F(x) = the horizontal gradient in the magnetic field k = the susceptibility contrast T = the total field

T = 1-cos21 cos2 X, a constant = the inclination of the magnetic field

x = the geological strike (E of N, +ve) d = geological dip of the contact e = x/h, an angle $ = d-2b, an angle b = arc tan (tan I/sin X )

Ll = distance from the measurement point to the contact = depth to the contact

The nomenclature is shown in Fig.1. The anomaly in the profile of the magnetic gradient can be written in the

form:

F = (-K/h) [sin (0 + rj)*cos 81

where K = 2kTcesin (d).

(1)

263

/

k =‘susceptibility contra& _-’ , _,, / ,

F&l. The convention used in the case of a contact.

Let FM and F, be the value of the maximum and minimum of the profile for values of eM and em, respectively. By differentiating eq. 1 with respect to 0, it is seen that:

0~ = f-45”-#/2); FM = (-K/2h) (sin 45-l) (2)

em = (45”~@/2); F, = (--R/~/Z) (sin @ + 1) (3)

Let Fa and Iii, the value of the points of inflexion of the profile for values of Ba and 6b, respectively. By taking the second derivative of eq. 1 with respect to 0, it is seen that:

B, = -612; F’ = (-K/2/z) sin # (4)

eb = (90”~#/2); Fb = (-K/2/z) sin # (5)

It can be seen from eqs. 2, 3, 4 and 5 that at the inflexion points, the value of the profile is equal to the mean of the maximum value, FM, and the mini- mum value, F,. That is:

Fa=&=(FM +F,)/2

In Fig.3, four straight lines are drawn from the point 0 at angles given by

0,) em9 8,, @, with respect to one another. In Fig.3, the four straight lines diverging from 0, form a pencil of rays. Con- sider a further straight line parallel to Ob and cutting the rays of the pencil at P, Q, R and S. If OQ is unit distance, then:

PQ=QR=OQ--1

and:

RS=m

Because of the ratios, it follows that the points PQRS are harmonic. It therefore follows that any straight line such as the straight line PQRS in Fig.4, which is inclined at an angle, $/2 to Ob, is also harmonic. The points P, Q, R

264

100 T FM

A 0 50

100 4

half-range value = (FM+ F,) 12

-501

Fig. 2. The form of a magnetic gradient profile over a contact FM is the maximum at XM ; FM is the minimum at x,. The inf!exion points coincide with the half-range-values [ = (PM + F, ) / 21 and are located at x, and xb along the traverse. This condition is proved in the text.The points, xa, XM, xb, x, are harmonic, that is

This condition is proved also in the text.

m’ hi

M (1 m a m

Fig. 3. The pencil of rays OM, Oa, Om, Ob is, given, with each ray making an angle of 45” with its neighbour. The straight line PQRS is parallel to Ob. PQ = QR = OQ = 1. Also note that any straight line such as PQRS, which is divided into three sections whose lengths are 1, 1 and -, is divided harmonically.

Fig.4. The pencil of rays (OM, Oa, Om, Ob) is given. Also shown is the straight line such that at:,P: there is a maximum in F(x); Q: a half-range-value point, i.e., (FM + F,,,)/2; R: a minimum in F(x); S: a half-range-value point, i.e., (FM + F, ) / 2. The resulting slope of the line, o/2, allows @(= d-2b) to be read off and knowing the parameter b, the dip, d can be found. Note that the position along the traverse of the half-range-value is the same as the position of the inflexion point.

265

and S correspond to the points on the traverse over a contact where:

P, maximum FM, at xM on the traverse Q, inflexion F,, at X~ R, minimum F,,, , at x, S, inflexion Fb, at Xb

The perpendicular distance, OH is the depth to the top of the contact. The position of xH on the traverse gives the location of the point vertically above the top of the contact.

PRACTICAL CONSIDERATIONS

Because the four points xM , x0, xm, xb are harmonic, the position of only any three need be known to determine the fourth. This is because of the in- variant property of the harmonic-cross-ratio that:

f&M -&I) (xm --xb)l / [(&r&n) (xb-% )I = -1 (6) Consequently, only the maximum FM, and the minimum F,,, and a half-range- value x, or xb need be read off.

If in Fig.4, xM be placed on the ray OM and xm on Om, and distance x, Orxb

to the half-range-value, be placed on either the ray Oa or Ob, then the perpen- dicular distance from 0 to H on the straight line PQRS is the depth to the con- tact. The slope of the straight line, $/2 gives the dip, d of the contact because $ = d-2b as defined previously; hence:

d = f$ + 2 arctan (tan Z/sin X) (7)

So far, a geometrical description of the method of obtaining the angle 4 and the depth, h, has been given, but it is obvious that an analytic method is avail- able (see Fig.4). Let:

U = xa-XM = PQ

and

v=x,-x,=&R

and

W = 3tb-x,,, = RS

Then

tan (#/2) = (u-v) /(u + v) (11)

OR = v/w

and

h = v (cos 412 + sin @/2) (cos #/2) (12)

266

The position of the contact, x0, is given by:

x0 = x, + u (cos Q/2 + sin G/2) - sin e/2

which follows directly from eq. 12.

(13)

The angle of dip, d, is obtained from eq. 7 where Q/2 is the actual angle the straight line PQRS makes with Ob, as in Fig.4.

If, as in the southern hemisphere, the order of the harmonic points is xm, xb, xM. Then the straight line PQR makes an angle such that:

90” < $12 < 135. This is as to be expected because “b” as defined is negative, and the dip of the contact, d, is obtained from eq. 7.

EXAMPLE

Based on actual field data is the modelled structure and the calculated profile given in Fig.5. The horizontal gradient of the total magnetic field is given in Fig.6. From Fig.6 the position of the maximum xM, and the half-range values, xg and xb, can be found accurately. There is uncertainty about the position of the minimum, x,, but because of the harmonic property of the four points, x,, Xb, x, and xa, the fourth point can always be found from any three. In Fig.6 the half range values, xa and xb, are shown. The actual value

c 4000

Total Magnetic Field

0 100 -I

-4000

Fig.5. Total magnetic field over a contact: depth 10 m; strike 150” E of N; dip 90”.

267

45,x//o half-range value = 40.3

i /

0 (

14,x,

6&X,

\

Horuontal Magnetic Gradient

-50

Fig. 6. Horizontal gradient of the magnetic field over a contact (Fig. 5). The position of the maximum, xM = 54, the half-value-pointsx, and xf, are at 45 and 68, and the minimum can be shown to be x, = 14.

used in the calculations were: (1) depth of contact, h = 10; (2) position of contact, x0 = 51; (3) dip angle, d = 90.

In Table I, the value of xb, xM and xa are as read off the graph (Fig.6) and rounded off to the nearest whole number. In the case of Fig.6, the position of the minimum at xm is poorly defined and its position can be calculated from xb, xM and x, using the harmonic ratio property or it can be read off the

TABLE I

xm = 14; may be read off Fig.6 or calculated from xb, XW, x,

xb = 45; well defined in Fig.6

XiM = 54; well defined in Fig.6

% = 68; well defined in Fig.6 u = xb-x, = 31 u =xj&--xb= 9 W =.%o-x,,f = 14

tan (b/2 = (u-u)I(u + u) OR u/w = 0.55 OR 0.64

o/2 = 28.8” + (90) OR 32.7 + (90) h = 10.7 OR 10.4

X0 = 6.8 + 45 OR 6.7+45 d = tb + 2 arctan (tan I/ sin h)

= 87.6 OR 95.4 I = -62 x = 150

Note. From Fig.6, any three values of xrn, xb, xM and x, are read off. The four points can always be calculated. The characteristic parameters for the contact have been calculated using (1) x,, xb, X&l; and (2) xb, XM, X,.

268

graph (Fig.6). The resulting calculated values for h, x0 and d are as given in Table I. They can be compared with the actual figures listed above and it in- dicates the sensitivity of the result to inaccuracies in the input.

CONCLUSION

The method depends upon the determination of the position of three si~ific~t points - a maximum or a minimum in the gradient, and the two hall-halve-poi~ ts. Provided a fit-sampling gradiometer is used, these points can always be found. From this information the position, depth and dip of the contact can be found.

No information is required about the position of the zero-field. Only points in the vicinity of the contact are required. This minimizes the effect of neighbouring contacts.

The method is simple to apply and it can be used either in a graphical or analytical manner.

REFERENCES

Green, R. and Stanley, J.M., 1975. Application of a Hilbert Transform method to the inter- pretation of surface-vehicle magnetic data. Geophys. Prospect., 23: 18-27.

Jung, K., 1953. Some remarks on the interpretation of gravitational and magnetic anomalies. Geophys. Prospect., 1: 29.

Michaels, P., 1977. Seismic raypath migration with a pocket calculator. Geophysics, 43: 1056-1063.

Roessiger, M., 1943. Magnetisehe Messungen - Taschenbuch der angewandten Geophysik. Akademisehe Verlaggesellshaft, Leipzig, pp. 280-317.

Varian, 1977. Caesium vapor gradiometer manufactured by Varian Associates, 45 River Drive, Georgetown, Ontario.