Potential Geophysical Fields

CHAPTER 2

Potential Geophysical Fields: Similarity and Difference Chapter Outline 2.1 Gravity Field 5 2.2 Magnetic Field 6 2.3 Temperature Field 7 2.4 Self-Potential Field 7 2.5 Electric Field 7 2.6 Potential Fields Observed at Different Heights: A Common Interpretation Technique 8 References 11 Further Reading 12

2.1 Gravity Field The potential nature of a gravity field, which follows from the law of gravitation, is expressed by differential equations: rot F ¼ 0

(2.1)

F ¼ grad W;

(2.2)

and

where F is the gravity field intensity and W is the gravity potential. The proportionality between the gravity field intensity and the substance density s is determined by the expression: div F ¼ 4pGs;

(2.3)

where G is the gravity constant. As follows from Eqs. (2.3) and (2.2), the gravity potential must satisfy Poisson’s equation (Kaufman, 1992): V2 W ¼ 4pGs;

(2.4)

Here Zv is the vertical magnetic field component at vertical magnetization; J is the magnetization; b is the horizontal semithickness of TB (thin bed); m is the magnetic mass; Geophysical Potential Fields. https://doi.org/10.1016/B978-0-12-811685-2.00002-3 Copyright © 2019 Elsevier Inc. All rights reserved.

5

6 Chapter 2 Table 2.1: Comparison of analytical expressions for natural potential fields.

Magnetic

Gravity

Analytical expression TB Zv ¼ 2J2b

z x2 þ z2

HCC z Dg ¼ 2Gs 2 x þ z2

Self-potential DU ¼ 2

HCC r1 z U0 r0 2 x þ z2 r1 þ r2

Temperature

e

Point source (rod) mz Zv ¼ 3 2 ðx þ z 2 Þ 2 Sphere z Dg ¼ GM ðx2 þ z2 Þ3=2 Sphere 2r1 z U0 R2 DU ¼ 2 2r2 þ r1 ðx þ z 2 Þ3=2 Sphere q m1 R3 Dt ¼ l2 m þ 2 ðx2 þ z2 Þ3=2 =

Field

G is the gravity constant; s is the density; M is the mass of the sphere; r1 is the host medium resistivity; r2 is the anomalous object (HCC [horizontal circular cylinder] or sphere) resistivity; U0 is the potential jump at the source body/host medium interface; r0 is the polarized cylinder radius; R is the sphere radius; t is temperature; q is the heat flow density; l2 is the thermal conductivity of the anomalous object; m is the object-to-medium thermal conductivity ratio; x is the current coordinate; z is the depth of the upper DTB edge (HCC or sphere center) occurrence. whereas points where the mass is zero (s ¼ 0) should satisfy Laplace’s equation: DW ¼

v2 W v2 W v2 W þ 2 þ 2 : vx2 vy vz

(2.5)

It allows to apply in the gravity method advanced interpretation methods developed in magnetic prospecting (Khesin et al., 1996). Let us consider some analytical expressions for typical interpretation models for natural potential fields (Table 2.1). The proportionality of analytical expressions for different fields presented separately in columns 2 and 3 of the table is obvious.

2.2 Magnetic Field The magnetic field (for DTdwhen magnetic susceptibility is below 0.1 SI unit) is a potential field (e.g., Hughes and Pondrom, 1947; Tafeyev and Sokolov, 1981; Khesin et al., 1996) and is expressed as: Ua ¼ grad V; where Ua is the anomalous magnetic field and V represents the magnetic potential. This field satisfies Poisson’s equation.

(2.6)

Potential Geophysical Fields: Similarity and Difference 7

2.3 Temperature Field In solids, heat flows according to Fourier’s law (Tikhonov and Samarsky, 1963): vT ; (2.7) vn where T is the temperature, q is the heat flow density, l is the thermal conductivity, and n is the outward normal to an isothermal surface. q ¼ l grad T ¼ l

If we assume that the heat transfer from the local sources is conductive and the temperature field is stationary, then the temperature distribution in a space without any sources satisfies Laplace’s equation (Tikhonov and Samarsky, 1963). Taking into account similarity of analytical expressions (Table 2.1), we can apply for examination of temperature anomalies the interpretation methods (Eppelbaum et al., 2014) developed in magnetic prospecting (Khesin et al., 1996).

2.4 Self-Potential Field SP polarization is brought about by spontaneous appearance of electric double layers on various geological formation contacts (Semenov, 1980). The electric fields E of the electric double layer l caused by natural polarization are defined as the gradient of a scalar potential Pi: ESP ¼ grad Pi :

(2.8)

The potential Pi satisfies Laplace’s equation everywhere outside the layer l (Zhdanov and Keller, 1994).

2.5 Electric Field The electric field E is the electrostatic field (div j ¼ 0, where j is the conduction current density). It is well known that the resistivity field is potential and satisfies the Laplace equation (Kaufman, 1992): E ¼ grad F;

(2.9)

where F is the scalar electric potential. There is a similarity between the analytical expressions applied in magnetic prospecting and those used in the resistivity method (Eppelbaum, 1999a, 2007). Thus, it is possible to apply the advanced methodology developed in magnetic prospecting (e.g., Khesin et al., 1996) to interpret resistivity anomalies.

8 Chapter 2

2.6 Potential Fields Observed at Different Heights: A Common Interpretation Technique In essence, there are only two types of general analytical expressions applicable to the description of these geophysical fields (Alexeyev et al., 1996; Khesin et al., 1996). They are Z ðzs  zÞcos gp þ ðxs  xÞ sin gp U1 ðx; zÞ ¼ P dxs dzs ; (2.10) r2 S h i Z ðzs  zÞ2  ðxs  xÞ2 cos g þ 2ðxs  xÞðzs  zÞ sin g p p dxs dzs ; (2.11) U2 ðx; zÞ ¼ P r4 S 

where gp ¼ 90 4p, 4p is the inclination angle of the polarization vector to the horizon, P is value of this vector (dipole moment of a unit volume; being a scalar in a particular case); S is the cross-section area of the body; P is the polarization vector (dipole moment qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of a unit volume); r ¼ ðxs  xÞ2 þ ðzs  zÞ2 is the distance from the observation point M(x, z) to a certain point of the body P(xs,zs). Therefore, it seems sufficient to illustrate the manipulations taking as examples Eqs. (2.10) and (2.11). The peculiarity of an inclined profile is that the height of the observation point is a linear function of the horizontal distance, namely z ¼ x tan u0 ;

(2.12)

where u0 is the inclination angle of the observation profile (Fig. 2.1). The transformations are carried out in the following sequence. The inclined coordinate system x0 Oz0 is introduced in such a way that   x ¼ x0 cos u0  z0 sin u0 ; (2.13) z ¼ x0 sin u0  z0 cos u0 : The formulas for xs and zs are similar. The Ox0 -axis in this system coincides with the inclined profile, which gives z0 ¼ 0 and results in Eq. (2.12) (Khesin et al., 1996). In the x0 Oz0 system, Eqs. (2.10) and (2.11) are transformed to the following forms:       Z 0 zs cos gp þ u0 þ x0s  x0 sin gp þ u0 U1 ðx; zÞ ¼ P dx0s dz0s ; r 02 S    Z 02  0 _ _ zs  xs  x0 cosg p þ 2 x0s  x0 z0s sing p 0 0 dxs dzs ; U2 ðx; zÞ ¼ P r 04 S 1

=

where r0 ¼ ½ðx0s  xÞ2 þ ðz0s  z0 Þ2 

2

_

and g p ¼ gp þ 2u0 :

(2.14) (2.15)

Potential Geophysical Fields: Similarity and Difference 9 Δ g ,mGal Δ g',mGal 2.0 Δ gf (x,0)

2.0 1.5 1.5

1.0 Δ g(x',0)

1.0 Δ g(x,z) 0.5

0.5

100 -100

0

100

200

300

cf (xof,hf)

500

600

700

x,m

σf

rf

100

400

ωο

100 100

r

200

sf 300

200 200

400 σ 500 c(xo, h)

z',m

300

z,m

s

600 700 800 x' ,m

Figure 2.1 Interpretation of gravity measurements for an inclined profile (Alexeyev et al., 1996).

It is obvious that the right-hand sides of Eqs. (2.14) and (2.15) correspond to the functions U0 1(x0 , 0), U0 2(x0 , 0) in the inclined system, but for a different inclination angle of the polarization vector. If a body in the initial system was vertically polarized, it turns out to be obliquely polarized (angles u0 or 2u0 respectively) in the inclined system, since the polarization vector does not intersect the Ox0 -axis at a right angle. Eqs. (2.14) and (2.15) can be essentially interpreted in the inclined coordinate system making use of the techniques developed for the horizontal profile, since the changing

10 Chapter 2 heights of observation points are not included there. To interpret in the initial system, we have to continue our manipulations in the following sequence. The entire space with the anomalous object and the polarization vector is turned by the angle u0 and compressed at the compressibility coefficient of (cos u0). In addition, when manipulating with Eq. (2.14), P is multiplied by (sec u0). This done, the inclined profile Ox0 coincides with the horizontal straight line, and the observation points on the profile pass along the vertical into the corresponding points on the horizontal straight line. An anomaly plot, constructed by these points (in horizontal projection) is a standard plot used routinely, although observations are made on an inclined relief. However, after the space rotation and compression the anomalous object occupies a different position with respect to the initial one. Its cross-section is smaller than the initial one, but the outline is similar. After this transformation, Eqs. (2.14) and (2.15) acquire the following forms:     Z zsf cos gp þ u0 þ ðxsf  xÞ sin gp þ u0 dxsf dzsf ; (2.16) U1 ðx; zÞ ¼ U1f ðx; 0Þ ¼ Pf rf2 Sf h i _ _ Z z2  ðxsf  xÞ2 cosg p þ 2ðxsf  xÞzsf sing p sf dxsf dzsf ; (2.17) U2 ðx; zÞ ¼ U2f ðx; 0Þ ¼ P rf4 Sf Here, the subscript “f ” stands for the parameters of a fictitious body. When used with symbols U1 and U2, it denotes that they refer to a fictitious body. The interpretation of the curves U1f and U2f results in obtaining parameters of a fictitious body, which are used to reconstruct those of a real body (denoted by “s” subscript) with the help of the following formulas of transition:

8 zs ¼ zsf þ xsf tan u0 ; > > > > xs ¼ xsf tan u0 þ zsf ; > > > > < S ¼ Ssf sec2 u0 ;

P ¼ Pf cos u0 ðby U1 interpretationÞ

9 > > > > > > > > =

:

> > > > P ¼ Pf ðby U2 interpretationÞ > > > > > > > > g ¼ g  u ðby U interpretationÞ > > 0 1 p pf > : g ¼ g  2u ðby U interpretationÞ > ; p

pf

0

(2.18)

2

Let us consider the analytical expression of the gravity anomaly caused by a certain anomalous body: Z ðzs  zÞ cos gg þ ðxs  xÞ sin gg Dg ¼ 2Gs ds; r2 S where the value gg is an analogue of the value gp in Eq. (2.10).

(2.19)

Potential Geophysical Fields: Similarity and Difference 11 This formula does not differ from Eq. (2.10) by its structure. After the above manipulations, it receives the following form: Z rgf cos ggf þ ðxsf  xÞsin ggf Dgf ¼ 2Gsf dsf ; rf2 S

(2.20)

such that ggf ¼ gg þ u0 :

(2.21)

Taking into account that gg ¼ 0, we obtain ggf ¼ u0. Hence, the anomaly of Dg observed on the inclined relief corresponds to that caused by a fictitious obliquely polarized body observed on the horizontal relief. The latter is affected both by vertical and horizontal gravity components. This is equivalent to the manifestation of “vector properties” of density on the inclined relief. Overall, the effect of the profile inclination is equivalent to an increased effect of oblique polarization, if the vector P and the relief have the same sense of slope, and to a weakened effect, if the relief and the vector P are inclined in opposite senses, up to their mutual neutralization. This fact determines the unified techniques for interpreting obliquely polarized bodies observed on the inclined relief. It facilitates the analysis of the distortions due to the effect of sloping relief, which can be completely attributed to the oblique polarization effect. Thus, Eqs. (2.11) and (2.12) reduce the problem of interpretation of the potential fields observed on the inclined profile to the same problem for the horizontal profile. Eq. (2.13) describes the transition from the parameters obtained while interpreting a fictitious body to those of a real body. The above data substantiate the conversion in the inclined semispace, since a field observed on the complex relief can be expressed using a certain inclined system of coordinates, the reduction being accomplished as in the normal system. The field in the observation points of the relief is converted to an inclined straight line along the normal to the latter. The advantage of such reduction consists in that the inclined plane of the reduction may be selected as close as possible not only to the highest points but also to many points of the relief. As a result, the amplitude decreases appreciably less than when converting to the horizontal level of the highest relief points.

References Alexeyev, V.V., Khesin, B.E., Eppelbaum, L.V., 1996. Geophysical fields observed at different heights: a common interpretation technique. In: Proceed. of the Meeting of Soc. of Explor. Geophys., Jakarta, pp. 104e108.

12 Chapter 2 Eppelbaum, L.V., 1999a. Quantitative interpretation of resistivity anomalies using advanced methods developed in magnetic prospecting. In: Trans. of the XXIV General Assembly of the Europ. Geoph. Soc., Strasbourg, vol. 1, No. 1, p. 166. Eppelbaum, L.V., 2007. Revealing of subterranean karst using modern analysis of potential and quasi-potential fields. In: Proceed. Of the 2007 SAGEEP Conference, 20, Denver, USA, pp. 797e810. Eppelbaum, L., Kutasov, I., Pilchin, A., 2014. Applied Geothermics. Springer. Hughes, D.S., Pondrom, W.S., 1947. Computation of vertical magnetic anomalies from total magnetic field measurements. Transactions e American Geophysical Union 28 (2), 193. Kaufman, A.A., 1992. Geophysical Field Theory and Method. In: Gravitational, Electric and Magnetic Fields, vol. 1. Academic Press, San Diego. Khesin, B.E., Alexeyev, V.V., Eppelbaum, L.V., 1996. Interpretation of Geophysical Fields in Complicated Environments. Kluwer Academic Publishers (Springer. Ser.: Modern Approaches in Geophysics, Boston e Dordrecht e London. Semenov, A.S., 1980. Electric Prospecting by Self-Potential Method. revised and supplemented, 4st ed. Nedra, Leningrad. in Russian. Tafeyev, Y.P., Sokolov, K.P., 1981. Geological Interpretation of Magnetic Anomalies. Nedra, Leningrad in Russian. Tikhonov, A.N., Samarsky, A.A., 1963. Equations of Mathematical Physics. Pergamon Press. Zhdanov, M.S., Keller, G.V., 1994. The Geoelectrical Methods in Geophysical Exploration. Elsevier.

Further Reading Dobrin, M.B., Savit, C.H., 1988. Introduction to Geophysical Prospecting, fourth ed. McGraw-Hill, New York. Eppelbaum, L.V., 2011. Review of environmental and geological microgravity applications and feasibility of their implementation at archaeological sites in Israel. International Journal of Geophysics 1e9. ID 927080. Gupta, H.K. (Ed.), 2011. Encyclopedia of Solid Earth Geophysics. Springer, Dordrecht, The Netherlands. Lowrie, W., 2007. Fundamentals of Geophysics. Cambridge Univ. Press, Cambridge, UK. Parasnis, D.S., 1986. Principles of Applied Geophysics, fourth ed. Chapman & Hall, London. revised and supplemented. Telford, W.M., Geldart, L.R., Sheriff, R.E., 2004. Applied Geophysics, second ed. Cambridge Univ. Press, Cambridge.