Airborne Gravity 2010 - Geoscience Australia

Airborne Gravity 2010
forms of density variations within a layer: (a) a constant, (b) changes with depth, (c) varying
horizontally, and (d) changes both vertically and horizontally. The most general cases may be defined
by a voxet. In practice, a velocity voxet may be available when gravity and/or gravity gradient data are
used to help interpret a horizon that may be difficult to image using seismic methods (e.g., base salt).
We can convert this velocity voxet into a density voxet using, for example, a simple velocity-density
relationship or a more complex relationship such as a 3D variogram derived from geostatistical
analysis.
A voxet may be divided vertically into a number of laminas (thin sheets). Each lamina is equivalent to a
surface density distribution � on a horizontal plane. The gravity effect g on the top surface of a lamina
can be written in the wavenumber domain as
�g�K�� ��F���K��
F � 2
(1)
where � is the gravitational constant, K the wavenumber vector, and F stands for the Fourier
transform. Applying an upward continuation, we obtain the gravity effect on a horizontal plane at
altitude z:
F
� K z
�g�K�� ��e F���K��
� 2 (2)
Summing up the effects of all laminas produces the total gravity effect. We can apply other operations
in the wavenumber domain to compute the gravity gradient components T XX , T XY , T XZ , T YY , T YZ ,
T ZZ , TUV � �TYY � TXX
� 2 and the total magnetic intensity � T . Here X points to the east, Y to the
north, and Z is vertical down.
For inversion, Bott (1960) proposed a very simple update formula for interpretation of gravity
anomalies over a sedimentary basin. The gravity effect due to an infinite horizontal slab is a constant
independent of the vertical position (i.e., burial depth) and the horizontal location of observation:
g � 2���t
(3)
where � is the bulk density (contrast) of the slab, and t the thickness of the slab. Rearranging
equation (3) produces
g
t � (4)
2���
This equation can be used to compute the depth change of a horizon, point by point. The depth
inversion is accomplished with an iterative process.
More sophisticated algorithms for forward and inverse modeling are needed in practice. For example,
equation (4) cannot be used to update depth when the magnetic anomaly or a gravity gradient
component is inverted. We have developed advanced and efficient algorithms that can invert not only
a field or gradient component individually but also a selection of any number of field and gradient
components simultaneously. Below, I present a joint inversion example.
Example
Our test model consisted of a seawater layer, sediments, a salt body and a basement (Figure 1,
Figure 3a). Each of the depth grids was quite large, having 960 rows and 960 columns. Each cell
represents an area of 25 m by 25 m. The sediments had a complex density variation, defined by a
voxet that each cell is 1500 m by 1500 m by 150 m in size. Figure 1 shows the base of salt and the
density distribution of the initial 3D model. Figure 2a displays three maps and five east-west crosssections
for this starting model. Note that the starting model is the same as the true model except for
the base of salt.
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