5.5 The half space response
A transmitter of time varying magnetic field will induce emf in the
ground itself and induced currents will flow in proportion to the conductivity
of the ground. The general physics of the ground response is the same as that
for an isolated conductor but the currents are no longer confined within a
finite target.
For sources on the surface of a (layered) half space the various
components of the magnetic and electric fields at a distance r away can also
be plotted as a function of a half space induction number  = 0 r2,
where, now r is the spacing between T and R. Plots of the amplitude
response vs. the induction number for six configurations of T-R are given in
Figure 5.6. The field is plotted as the ratio of the total field (primary plus
secondary) to the primary, where the primary means the free space value of
that component at the receiver position. In the case of the minimum-coupled
system, IV, the total field is normalized by the maximum-coupled field at
the same point.
All the maximally coupled systems begin at a field ratio of 1.0 at low
frequencies - there is no secondary field. For the horizontal loop - loop
system the response grows slightly with induction number and then falls to
zero. The induced currents in the ground act by Lenz’s law to oppose the
primary field, in effect excluding it from the ground so that in the high
frequency limit the field is entirely horizontal and the vertical component is
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zero. The same effect causes the horizontal field from the coaxial vertical
loop system, II, to double in the high induction number limit.
Finally the response for a horizontal loop-loop system, this time
plotted as an Argand diagram in the secondary field as a percent of the freespace primary field, is presented in Figure 5.7. The response of a thin layer
of thickness s replacing the half-space is also shown on this plot (here the
induction number is s0 r). The thin layer is useful for estimating the
response of an overburden layer.
Typical transient responses for a horizontal loop transmitter over a
uniform half-space of resistivity 100 Ohm-m are shown in Figure 5.8a, b, c.
The form of these transients follows from a brief explanation of the physics
of the response. At the instant the DC current is terminated in the transmitter
loop the magnetic field through the ground begins to collapse. The Faraday
currents that immediately form try to maintain the field and so they are
initially a mirror image of the transmitter current immediately beneath the
wire of the transmitter loop. Because they are not confined these image
currents diffuse downwards and outwards from the transmitter. The
secondary magnetic field from these currents also decays because the
currents move farther and farther away with time and because they lose
energy to ohmic heating in the ground.
At the center of the transmitter loop the vertical field simply decays to
zero from its initial value, Figure 5.8a. At “late” time the field decays as
t 3 2 and the time derivative as t 5 2 . The transient response of the vertical
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field 100 m from a vertical magnetic dipole (a small horizontal loop) is
shown in Figure 5.8b. The fields go through zero and reverse in sign as the
diffusing current passes beneath the receiver. The late-time decays for the
field and the time derivative of the field are t 3 2 and t 5 2 respectively. The
transient response of the horizontal component of the field is shown in
Figure 5.8c and it is seen that the decay rates are higher than for the vertical
component, t 2 and t 3 for the field and time derivative of the field
respectively. There is no horizontal field transient at the center of a
horizontal loop on a homogeneous layered half-space.
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