Geoscience Reference
In-Depth Information
methods that retain and use information about the object-
ive function when searching for the global minima are
usually more ef cient than Monte Carlo methods. Algo-
rithms of this type include simulated annealing and genetic
algorithms; see Smith et al.( 1992 ) .
example Pratt and Witherly ( 2003 ) and the modelling
example described in Inverse modelling in Section 3.11.2 .
In practice, there are often limited data available to con-
strain the nature and distribution of the geological units in
the model. Starting models are invariably built from a
combination of geological observations and inferences.
Petrophysical data, if available, may be incorporated.
A relatively small number of constraints can make a big
difference to the result (Farquharson et al., 2008 ; Fullagar
and Pears, 2007 ) .
Constrained inversion
Inversion can be directed towards a plausible solution by
including known or inferred information about the area
being modelled into the inversion process. This is known
as constrained inversion and the information helps direct
the inversion algorithm to that part of the objective
function hyperspace where the global minimum exists.
Provided the information is accurate, the result will usually
be a more useful solution than that from an unconstrained
inversion (see Section 3.11.2 ). One way to constrain the
inversion is to set bounds, or limits, on the values of
selected parameters (with or without probability compon-
ents). For example, the variation of physical property may
be constrained; non-negativity being an obvious constraint
for a property such as density, and one that greatly
improves the likelihood of obtaining a geologically realistic
result from the inversion. The possible locations of parts of
the model may also be constrained: forcing them to below
the ground being an obvious constraint; or to honour a
drilling intersection. Particularly important in many
instances are distance weighting parameters which counter
the tendency of inversion algorithms to place regions with
anomalous physical properties close to the receiver, which
equates to close to the surface for airborne and ground
surveys. Other common constraints include geometric
controls, for example the source should be of minimal
possible volume; it should be elongated in a particular
direction (useful in layered sequences); it should not con-
tain interior holes; and adjacent parts of the model should
be similar, i.e. a smoothing criterion. Smooth inversion
tends to de
Joint inversion
The reliability of a solution may be improved by modelling
two or more types of data simultaneously in a process
known as joint inversion, which is becoming increasingly
common. These data have properties containing common
or complementary information about the subsurface. The
incremental results for one data type guide the changes
made to the model during the inversion of the other data
type, and vice versa. Early forms of joint inversion tended
to assume a correlation between, say, the density and
magnetism in equivalent parts of the model. This assump-
tion is rarely justi ed, and more recent work has concen-
trated on correlating regions where the physical properties
are changing rather than an explicit correlation of their
magnitudes (Gallardo, 2007 ) .
A joint inversion model may be more accurate as it is the
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to two disparate data types; and it has the add-
itional advantage of reducing the time and effort needed to
analyse two different datasets independently.
A disadvantage is that more computational effort is
required and that the results critically depend on the
assumed relationship between the physical properties
being inverted for. If the assumption is good then the
results will probably be more reliable. If it is not, then the
results could be less reliable than individual inversions of
each dataset. When jointly inverting different types of
geophysical data, it is important to account for the differ-
ent physics of the methods. For example, and as shown in
Fig. 3.65 , magnetic data are more in uenced by the shallow
subsurface than are gravity data. A joint inversion of these
two data types will be mostly in uenced by the gravity data
in the deeper parts of the model, so in this sense the
inversion is not
best t
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boundar-
ies and with properties that are less accurately resolved,
rather than as compact sharp-boundary zones whose prop-
erties are more accurately determined, but with greater
probability of uncertainty in their locations and shapes.
The inversion process can also be restricted to adjust one
or a few of the possible model parameters whilst fixing
others, e.g. to invert for dip, in other words to nd the
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ne zones with gradational or
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fuzzy
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'
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joint
.
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dip if the source is assumed to be a sheet.
It is preferable to direct the inversion towards more
likely solutions through the use of an approximate, but
appropriate, forward solution as a starting model; see for
best- tting
Tomography
An important, but specialised, type of inverse modelling
is based on tomography (Dyer and Fawcett, 1994 ;
 
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