Geoscience Reference
In-Depth Information
absolute value of 52,000 nT. In terms of relative values, the
base station is assigned a value of zero. In large surveys
there may be a master base station from which a series of
subsidiary base stations are established. This facilitates
surveying by reducing the distance that needs to be trav-
elled to the nearest base station. Note that the accuracy of
the absolute value of a parameter obtained by relative
measurement from a base station is dependent on the
accuracy of the absolute value at the base station and the
accuracy of the relative measurement itself.
a)
Vertical
gradient
Horizontal
gradient
Gradient
distance
Gradient
distance
Sensors
b)
Z
P
ZZ
2.2.2
Scalars and vectors
P
ZY
P
ZX
P
YZ
Physical quantities are classified into two classes. Those
that have magnitude only are known as scalar quantities or
simply scalars. Some examples include mass, time, density
and speed. Scalar quantities are described by multiples of
their unit of measure. For example, the mass of a body is
described by the unit of kilogram and a particular mass is
described by the number of kilograms. Scalar quantities are
manipulated by applying the rules of ordinary algebra, i.e.
addition, subtraction, multiplication and division. For
example, the sum of two masses is simply the addition of
the individual masses.
Some physical quantities have both magnitude and dir-
ection and are known as vector quantities or simply vectors.
Some examples are velocity, acceleration and magnetism.
They are described by multiples of their unit of measure
and by a statement of their direction. For example, to
describe the magnetism of a bar magnet requires a state-
ment of how strong the magnet is (magnitude) and its
orientation (direction). The graphical presentation and
algebraic manipulation of vectors are described in online
Measuring vector parameters in geophysics implies that
the sensor must be aligned in a particular direction. Often
components of the vector are measured. Measurements in
perpendicular horizontal directions are designated as the X
and Y directions, which may correspond with east and
north; or with directions de
ned in some other reference
frame, for example, relative to the survey traverse along
which measurements are taken. Usually the X direction is
parallel to the traverse. Measurements in the vertical are
designated as Z, although either up or down may be taken
as the positive direction depending upon accepted stand-
ards for that particular measurement. We denote the com-
ponents of a vector parameter (
P
YY
P
YX
Y
P
Z
P
Y
P
XZ
P
XY
P
X
X
P
XX
Figure 2.2
Gradient measurements. (a) Vertical and horizontal
gradiometers. (b) The three perpendicular gradients of each of the
three perpendicular components of a vector parameter P forming the
gradient tensor of P, shown using tensor notation; see text for details.
2.2.3
Gradients
Sometimes it is useful to measure the variation in the
amplitude of a physical parameter (
) over a small distance
at each location. The difference in the measurements from
two sensors separated by a fixed distance and oriented in a
particular direction is known as the spatial gradient of the
parameter. It is specified as units/distance in the measure-
ment direction, and so it is a vector quantity. As the
measurement distance decreases, the gradient converges
to the exact value of the derivative of the parameter, as
would be obtained from calculus applied to a function
describing the parameter
P
field. For the three perpendicular
directions X, Y and Z, we refer to the gradient in the X
direction as the X-derivative and, using the notation of
calculus, denote it as
∂
P
/
x. Similarly, we denote the
∂
Y-derivative as
z.
Gradients may be measured directly using a gradi-
ometer, which comprises two sensors positioned a short
distance apart (
Fig. 2.2a
). Alternatively, it is usually
possible to compute gradients, commonly referred to as
derivatives, directly from the non-gradient survey meas-
urements of the
∂
P
/
y and the Z-derivative as
∂
P
/
∂
∂
P
) in these directions as P
X
,
field (see Gradients and curvature in
P
Y
and P
Z
, respectively.
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