IMPÉDANCES ET MATRICES CARACTÉRISTIQUES EN PROSPECTION ÉLECTROMAGNÉTIQUE*

An analogy is established between the propagation of an electromagnetic field in a horizontally stratified conductive medium and that of currents in a linear electric network. This analogy will allow us to know in which case measurements executed at the surface of the ground can provide information about the properties of deeper layers. The situation is the same as when one tries, by means of impedance measurements at the input terminals, to know the properties of electric lines or lumped networks.

A symmetrical linear network establishes between:

Voltages V1 and V2 (respectively at input and output terminals)

Currents I1 and I2 (respectively at input and output terminals), the relationship

is called the “phase constant” and Z the characteristic impedance The table

is called “characteristic matrix”

If one then takes the horizontal components of an electromagnetic field, one can compare the voltage V with the horizontal electric field E (in volts m‐1) and the current I with the horizontal magnetic field H (in Amp m⁻¹)

For plane waves in vacuum, one gets (Schelkunoff)

Z=C (velocity of light)

A layer whose thickness is D has a phase constant

⌈= 2 πD/λ (λ= wavelength)

For plane waves in a conductive, non magnetic medium whose conductivity is δ, one gets

For the system of waves created by a point‐source one expresses the solution, since it is classical, as a sum or integral of different modes. For each of these modes, one can still write a bilinear relationship similar to (1), and calculate Z and ⌈.

A generalization of this result is given.

Boundary conditions that exist at the surfaces of separation are met by writing the continuity of E and H. This condition exactly corresponds to the fact that there one V and one I at the junctions of various sections of lines, or networks.

The characteristic matrix of a cascade of networks‐or here that of a horizontally stratified ground– is the product of the matrixes of each network. If, in addition, one knows the end impedance–here the conductivity λn of the last layer, supposed to be infinite–one can compute the input impedance Re of the whole system. In electromagnetic prospecting, it is Re that conditions the observable field.

Inversely, if measurements were infinitely accurate, the. knowledge of the surface field would give all the thicknesses D and conductivities δ of intermediate layers, in the same way that impedance measurements at the input terminals of a cable would allow to locate any failure. The imperfection of measurements causes the elements located too far away (from the point of view of wave attenuation) to escape detection.