A note on the method of the intersection point as a modified form of the tri-potential method

(;roc,.u/‘/o~crti~,,~ ~ Elsevier

A NOTE

ON THE

DIFIED

FORM

Publishing

Company,

METHOD

OF THE

Amsterdam

OF THE

~ Printed

INTERSECTION

TRI-POTENTIAL

in The

Netherlands

POINT

AS A MO-

METHOD

P4Ul. EGERSZEGI

(Received

February

20, 1967)

SUMhlARY

Geoelectric sounding curves plotted wit11 four electrodes in an equidistant arrangement by the three-variant method can intersect one another in the case 01 an inhomogeneous half-space. Theoretical curves giving the coordinates of the intersection point for the three-layer case as functions of the parameters of the second layer are plotted. With the theoretical curves the parameters of the ith layer can be determined if: oi_, > pi < pi + , or ili , < pi > p, + ,

The tri-potential method (CARPENTER and HABBERJAM. 1956) is founded on the combined analysis of the apparent resistivity curves measured with the following three electrode electrically

THE

arrangements:

inhomogeneous

xAMNB, /JABNM, yMANB. If sounding half-spaces

is made over

the three curves can be intersecting.

(‘ONDITIONOF INTERSEC'TION

Let the inhomogeneous half-space be represented by a block of homogeneous layers of various resistivities. separated by horizontal planes. In case of equidistant arrangement expressing the potential distribution by the potential functions G(n) as used in MOONEY and WETZEL (1956),an intersection point is obtained when G(2a)-G(N) = G(3a)-G(2a). This condition is fulfilled in a half-space free from horizontal inhomogeneity, when the apparent resistivity curve plotted with the arrangement 2 has an extreme value. In a three-layer case this is equivalent to the curves of type K or H.

P. EGERSZEGI

90 500 MT, 200

100

50

20

78 5

Fig.1. point;

L

theoretical

curves

of the K type

_l,~,

I

point;

Three-layer

for the method

of the interrectiol:

T-o.?.

I

2

Fig.2. Three-layer S = a/e.

5

IO

theoretical

20

curves

5D

of the

100 H type

200

s,IS, 500

for the method

of the intersection

Gwexp/maior~,

5 ( 1967)

89-94

20

?C

3

2

i

0.’

point;

Fig.3. CL.Q.

Three-layer

theoretical

curves

OF the K type

for

the

method

of‘ the

intcrscction

7 point

Fig.4. Three-layer u, I 10.

theoretical

cxr\es

of the Ii

type for the method

of the intersection

YOTE

ON A MODIFIED

THREE-LAYER

TRI-POTENTIAL

THEORETICAL

CURVES

93

METHOD

OF INTERSECTION

POINTS

Let us examine the connection in a three-layer case ofthetype Kor Hbetween the coordinates of the intersection point (pm, a,) and the parameter of the second layer

ipz, h,).

In the case of the type K the product /~,~a,,, is denoted by T,,,, pzh2 by Tz. and [I,/?, by T,:in the case of the type H the product I/p,,, . urn is denoted by S,, 1!PZ . lj2 by S,, and i/p, - h, by S,. Then in the case of the type K Fig. i and 3, and in the case of the type H Fig.2 and 4 show the relationships between the coordinates of the intersection point and the parameters of the second layer. They may be called three-layer theoretical curves of the intersection points.

APPLICATION

OF THE INTERSECTION

POINT

METHOD

Fig.5 shows a set of tri-potential curves plotted zontal inhomogeneity ( ECERSZEGI, 1967).

2

5

on a field freed from hori-

IOa[mJ 20

am Fig.%

Tri-potentiai

CLIIWZS

obtainotl

in the field without

horizontal

inhomogencity.

By evaluating the left hand side section of curve I with the two-layer theoretical curve MOONEY and WETZEL (1956) we get y, = 75fZm, h, = 0,7 m, k, 2 0.6 (p = 4), where p = p2/p1. The coordinates of the intersection point are pm = 223 f.?,m, ff, = 5.3 m, and therefore T, = 52.5 f2m2. T, = 1,182 Qm2, T,jr, = 22.5. The curve is of the type KQ, where p2 2 4p,, p3 z p,, and p4 z 1/5p,. Dispense with the fourth layer; then TJT, = 44, since p3 =: pr (Fig.1). If the third layer is dispensed with, then T,jT,= 51 (Fig.1). On analysing the four-layer theoretical curves of the KQ type we have found that at such high values of T,/T,the solution falls between the two values. Because of this we can take the arithmetical mean of the two values as the real value of TJT, (= 48)from which T, = 2,520Qrn’. A comparison with the two-layer theoretical curve has resulted in ,U = 4. Ptot the corresponding values in Fig.3 on the theoretical curves of the parameters /~~/p, = I and pa/p1 = 3 to assess the values of pz. As the resulting intersectron points are lying far away from the calculated values. the values of pz cannot be

94

P. EGERSZEGI

exactly determined,

yet it is seen, that the estimated

p2/p1 = 4 determined

by the two-layer

theoretical

value is not inconsistent curve.

with

Hence p2 = 300 !2m,

and h, = 8.4 m. It is seen that from the tri-potential intersection point the parameters of the ith value can be determined if /Ii _ 1 > p, < pi + , or pi _ , < pi > P;+~.

REFERENCES

CAKPENT~R, E. W. and HABBERJAM, G. M.,

1956. A tri-potential

method ofresistivity

prospecting.

Geophysics, 2 l(2) : 455-469. EGERSLFGI, P., 1967. The development of the method of the intersection point. Pub/. Tech. Utrir. Heuvy Ind. Miskolc, 28: in press (in Hungarian). MOONEY, H. M. and WETZEL, W. W., 1956. The Potentiuls about u Point Electwde alrd Appcrre~~t Resistivity Curves for (I Two- T/we+ and Forrr-l~.~w Emth. Univ. Minnesota Press, Minneapolis,

Minn.,

146 pp.

Georxplomtion,

5 (1967) 89-94