Geoscience Reference
In-Depth Information
at (x 0 , y 0 , z 0 ), in terms of the first-order derivatives (
f
=∂
x
Moving
window
etc.) of the field in the following form:
Sample #
12345
x 0 Þ
f
f
f
y 0 Þ
z 0 Þ
a)
ð
x + ð
y + ð
z ¼
ð
Þð
:
Þ
x
y
z
N
B
f
3
28
g
z
1st vertical
derivative
which includes a background (regional) component (B).
Note that for magnetic data, information about the direc-
tion of the magnetism is not required, so remanent mag-
netism does not present a problem.
The structural index (N) accounts for the rate of
decrease in the amplitude of the response with distance
from the source (see Section 3.10.1.1 ) . This affects the
measured gradients and depends on the source geometry.
For the case of a spherical source, N is equal to 2 for gravity
data and 3 for magnetics. Indices have been derived for a
variety of source types, and they fall in the range 0 to 3 (for
0 the equation has to be modi
X
b)
g
x
1st horizontal
derivative
X
ed slightly).
The source position (x 0 , y 0 , z 0 ) and the background
c)
eld
are obtained by solving the Euler equation ( Eq. (3.28) ). If N
is too low the depth estimate (z 0 ) will be too shallow, and if
N is too high, the depth will be overestimated. The hori-
zontal coordinates are much less affected. An effective
strategy is to work with all values of N between 0 and 3,
in increments of, say, 0.5. This will account for the geology
not being properly represented by any one of the idealised
model shapes, and also it has been shown that for more
realistic models N varies with depth and location.
The derivatives/gradients are usually calculated but, as
discussed in Gradients and curvature in Section 2.7.4.4 ,
gradients are susceptible to noise, especially as their order
increases, so the quality of the results will be affected
accordingly. Note that in magnetic data the gradient in
the across-line can be poorly constrained owing to spatial
asymmetry in the sampling, and is potentially a major
cause of error in Euler deconvolution.
Figure 3.67 shows the implementation for pro
Gravity g
B
X
x 1 x 2 x 3 x 4 x 5
x 0
d)
X
Solution (current window)
z 0
Source
Z
Figure 3.67 Schematic illustration of 2D Euler deconvolution using a
window of five data points applied to gravity data. See text for details.
unknowns is set up for each location in the window,
forming a set of n simultaneous equations of the form:
Þ
g
g
Þ
ð
x i
x 0
x i + ð
z i
z 0
z i ¼
N
ð
B
g i Þ
ð
3
:
29
Þ
le gravity
(g) data. In this case the across-line (Y) component of the
field is assumed to be symmetric about the pro le and a 2D
result is obtained (x 0 , y 0 ). A window of prede ned length
(n) is progressively moved along the gravity pro le and the
pro les of its vertical and horizontal derivatives, and the
background field and source coordinates are obtained for
each measurement location. Where there are three
unknowns (x 0 , z 0 and B), the window must span a min-
imum of three points; in practice at least double this
number are used, which allows the reliability of the result
to be estimated. For simplicity, Fig. 3.67 shows a five-point
window (n
where i identi
es the data points in the window i
¼
1ton.
The equations are solved for the three unknowns.
The window is then moved to the next point along the
pro le and the process repeated. Each window position
provides an estimate of source location. For 3D implemen-
tation, the window is moved across the gridded data and a
3D (x 0 , y 0 , z 0 ) solution for the source position is obtained.
Window size in either case represents a trade-off between
resolution and reliability; increasing the size reduces the
former and increases the latter. The window size must be
large enough to include signi cant variations in the eld
being analysed, and is usually set to the wavelength of the
anomalies of interest. It can be used as a crude means of
¼
5, centred at x 0 ). An expression for the
 
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