Otto Koefoed:

# The Application of the Kernel Function in Interpreting Geoelectrical Resistivity Measurements

Ed.: Geza Kunetz; S. Saxov

1968. XII, 111 pages, 47 figures, 9 plates, 17x24cm, 600 g
Language: English

(Geoexploration Monographs, Number 2)

ISBN 978-3-443-13002-2, bound, price: 19.00 €

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# Synopsis top ↑

In the present book new methods are discussed, by which the distribution of resistivity layers in the subsurface can be determined directly from the results of resistivity measurements. The contribution of the present author is restricted to the derivation of a practical procedure of determining the kernel function from the resistivity measurements.
The book has been written on two different levels at the same time. Readers, who wish to make themselves familiar with the practical interpretation technique without fully understanding the theoretical background of it, may omit the chapters 1 to 5 and 7 to 8. The remaining chapters form a closed unit, which can be understood with only elementary knowledge of algebra and of differential calculus. To understand the whole book, a good knowledge of differential and of integral calculus is required. No advanced knowledge is needed, however, of special subjects, in particular of the theory of determinants and of the theory of BESSEL functions. Both these subjects are essential to the interpretation of resistivity measurements, but the properties of determinants and of BESSEL functions are treated in the present book itself, to the extent that is required for understanding the interpretation methods of resistivitv measurements.

Chapter 1
The fundamental relation between the electrieal potential at the
surface and the distribution of resistivity layers in the subsurface.
1.1 Statement of the problem 1
1.2 Simple solutions of the equation of LAPLACE 2
1.3 The general solution of the equation of LAPLACE with cylindrical symmetry 4
1.4 Adaptation of the solution to the boundary conditions 5
Chapter 2
The apparent resistivity and its relation to the kernel function.
2.1 General considerations 9
2.2 The apparent resistivity for a SCHLUMBERGER electrode configuration 9
2.3 The explicit expression for the kernel function 11
2.4. The apparent resistivity for a WENNER electrode eonfiguration 12
2.5 The general shape of the kernel curve 13
2.6 The effect of errors in the apparent resistivity upon the kernel function 14
Chapter 3
Determination of the kernel function by the decomposition method.
3.1 The principle of the decomposition method 16
3.2 Conditions imposed on the partial apparent resistivity functions 17
3.3 Classification of partial apparent resistivity functions 18
3.3.1 Exponential functions approximating to zero at zero distance and at
infinity 18
3.3.2 Exponential functions approximating to a finite value at zero distance 18
3.3.3 Algebraic functions approximating to zero both at zero and at infinity 15
3.3.4 Algebraic functions approximating to a finite value at infinity 19
3.4. Descrition of standard curves 19
Chapter 4
Derivation of the partial kernel functions.
4.1.Exponential functions approaching to zero at zero distance and at infinity 21
4.2 Exponential functions approximating to a finite value at zero distance 22
4.3 Irrational algebraic functions 23
4.4 The modified BESSEL function and the modified STRUVE function 25
4.5 Derivation of the kernel functions corresponding to the rational algebraic
functions 26
4.6 General remarks concerning the partial kernel functions 27
Chapter 5
Partial apparent resistivity functions for the WENNER configuration.
5.1 Methods of derivation of the WENNER partial apparent resistivity functions 28
5.2 Relation between the WENNER apparent resistivity curve and the
SCHLUMBERGER apparent resistivity curve 29
5.3 Determination of the expressions for the WENNER partial apparent resistivity functions 30
5.3.1 Exponential functions 30
5.3.2 Irrational algebraic functions 31
5.3.3 Rational algebraic functions 32
Chapter 6
Practical applications of determination of the raised kernel function.
6.1 Introduction 33
6.2 Description of the interpretation method 34
6.3 First example 38.
6.5 Third example 40
6.6 Fourth example 42
6.7 Fifth example 45
6.8 Sixth example 46
6.9 Seventh example 49
6.10 Eighth example
Chapter 7
Introduction on determinants.
7.1 Definitions 54
7.2 Expansion of a determinant after the elements of a row or of a column 55
7.3 Some properties of determinants 56
7.4 Expansion of a determinant after two rows (or columns) 57
7.5 Solution of a system of linear equations 59
Chapter 8
The relation between the kernel function and the depths and resistivity con-
trasts of the boundary planes.
8.1 The expression of the kernel function in terms of the parameters of the
boundary planes 60
8.2 The modified kernel function 62
8.3 Asymptotic behaviour of the modified kernel function 62
8.4 Reduction of the modified kernel function to a lower boundary plane 64
8.5 Possible disadvantage of the modified kernel function 66
8.6 Reduction of the raised kernel function to a lower boundary plane 67
8.7 Zero points of three-layer kernel functions 68
Chapter 9
First method of determining the layer distribution from the raised kernel
function.
9.1 Outline of the method 69
9.2 First example 71
9.3 Second example 73
9.4. Third example 75
9.5 Fourth example 77
9.6 Fifth example 79
Chapter 10
Second method of determining the layer distribution from the raised kernel
function.
10.1 Limitations of the method discussed in chapter 9 82
10.2 Outline of the second method 83
10.3 First example 84
10.4 Second example 86
10.5 Third example 88
10.6 Fourth example 89
10.7 Fifth example 90
Chapter 11
Some special procedures.
11.1 The effect of a surface layer of high resistivity 93
11.2 The zero point of a three la er modified kernel curve 95
Chapter 12
The inverse problem: prediction of the detectability of subsurface layers.
12.1 Statement of the problem 96
12.2 The relation between a change in an apparent resistivity curve and that
in the raised kernel curve 96
12.3 The mathematical expressions for the raised kernel functions 98
12.4 Application of the method to a three layer case 98
12.5 Application of the method to a four layer case 101
Appendix
The mathematical expressions for the partial apparent resistivity functions and
the partial kernel functions 104
References 108
Index 109