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measurement point. We denote the derivative of the X
component of
a)
x, and
the derivatives of the same component in the Y and Z
directions are
P
(P
X
) in the X direction as
P
X
/
Horizontal gradient of response
∂
∂
z, respectively; and
similarly for the Y and Z components. They form a tensor
and are displayed and manipulated in matrix form:
P
X
/
y and
P
X
/
∂
∂
∂
∂
0
0
@
1
A
∂
P
X
∂
∂
P
X
∂
∂
P
X
∂
Response
x
y
z
0
@
1
A
ð
P
XX
P
XY
P
XZ
P
YX
P
YY
P
YZ
P
ZX
∂
P
Y
∂
∂
P
Y
∂
∂
P
Y
∂
or
2
:
1
Þ
x
y
z
0
P
ZY
P
ZZ
∂
P
Z
∂
∂
P
Z
∂
∂
P
Z
∂
x
y
z
Source
Several components of the tensor are related as follows: P
XY
¼
P
ZY
, so it is not necessary to
measure all of them. This means that less complex sensors
are needed and measurements can be made more quickly.
The full-gradient tensor of nine components, i.e. the gra-
dients in the three components in all three directions, pro-
vides diagnostic information about the nature of the source
of a geophysical anomaly. Tensor measurements aremade in
airborne gravity surveying (see
Section 3.3.2
)
but are other-
wise comparatively rare in other geophysical surveys at
present. It seems likely that they will become more common
in the future because of the extra information they provide.
P
YX
, P
XZ
¼
P
ZX
and P
YZ
¼
b)
Horizontal gradient of response
0
Response
0
Source
2.3
The nature of geophysical responses
Figure 2.3
Horizontal gradient data across (a) a localised source,
and (b) a contact. Note how the gradient response is localised near
the source edges.
As described in
Section 1.1
and shown schematically in
contrasts, so changes in the local geology can produce
changes in the geophysical response of the subsurface.
When the measured property of a target zone is greater
than that of the host rocks, the contrast is positive; when
lower, it is negative. Typically the changes are localised,
arising perhaps from a body of mineralisation or a contact
of some kind. These deviations from background values are
called anomalies. The simplest form of anomaly is an
increase or decrease of the measured parameter as the survey
traverses the source of the anomaly. Often, though, peaks in
the anomaly are offset from their source and/or may be more
complex in form; for example, the response from magnetic
sources may comprise both an increase and an adjacent
decrease in response, forming a dipole anomaly.
Although the underlying physics of each geophysical
method is different, some important aspects of the measured
responses are the same.
Figure 2.4
shows some general
Gradient measurements have the advantage of not being
affected by temporal changes in the parameter being meas-
ured; the changes affect both sensors in the same way so any
difference in the parameter at each sensor is maintained.
Gradient data are very sensitive to the
of sources.
They comprise variations that are more spatially localised
than non-gradient data and so have an inherently greater
spatial resolution (
Fig. 2.3
). The main disadvantage of gra-
dient measurements is that they are very sensitive to vari-
ations in the orientation of the sensor. Also, long-
wavelength variations in the parameter, which produce very
small gradients, are often not large enough to be detected.
The derivatives in the three perpendicular directions of
each of the three components of a vector parameter (
'
edges
'
P
)
(
Fig. 2.2b
) completely describe the parameter at
the
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