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modelling may be used to produce an approximate match
to the data and inversion used to re
ne the
t.
a reasonable approach provided the form of the objective
function is not overly complex. The problem is that non-
linear inverse problems normally have objective functions
which have extremely complicated forms and, conse-
quently, it can be very hard to
find the overall minima.
An algorithm that goes
2.11.2.1
Inverse modelling methods
The mathematics of geophysical inversion is complex and
beyond our scope. Only the basic principles are described
here to provide the reader with some insight into the
source of possible problems when using inversion tech-
sive description of inversion and its applications to mineral
exploration. Additional examples are presented by Olden-
burg et al.(
1998
).
Inverse modelling requires the interpreter to specify a
starting model (although this may be no more complex
than a half space) which is systematically refined by the
inversion process. The difference between the calculated
and observed responses at each data point is called the
residual. Fundamental to the inversion process is the over-
all match, or the
'
down-hill
'
is likely to be caught in
'
'
'
'
one of many
valleys
or
basins
, which are called local
minima, and has little chance of
the absolutely
lowest point in the terrain, i.e. the global minimum, which
is the ultimate objective.
Figure 2.48
also demonstrates the
reason for non-uniqueness in geophysical modelling, i.e.
more than one model will fit the data (see
Section 2.11.4
).
In
Fig. 2.48d
,
where the depth and physical property
contrast are allowed to vary, the lowest part of the topog-
raphy representing the objective function is not localised;
rather than being a
'
rolling to
'
. Once the
inversion algorithm has found the valley
floor, moving
across the
floor creates little change in the value of the
objective function. The example shows that multiple com-
binations of depth and property contrast values can pro-
duce the same
fit to the observed data, i.e. the result is not
unique.
The terrain analogy is a simpli
cation, since mathemat-
ically the objective function will be de
ned in more than
three variables (dimensions), i.e. it is a hyperspace. Math-
ematically exploring the hyperspace in an ef
cient and
effective manner can be exceptionally dif
cult because
the gradient-based search algorithm can get hopelessly
confused between local and global minima and will not
converge at all. Even when a minima is found it may be
impossible to tell if this is the global minimum.
If local minima are expected then global search methods
are required; these
'
basin
'
, it is instead a
'
valley
'
, between the two responses,
which is mathematically represented by an objective func-
tion that describes the degree of match as a function of the
model parameters. The task for the inversion process is to
adjust the model parameters so as to minimise the object-
ive function, i.e. to minimise the residuals.
There are various mathematical methods for minimising
the residuals, but the most common involves determining
how the residuals vary as each model parameter is altered,
which in turn indicates how the parameters should be
adjusted to minimise the objective function. Normally in
geophysical inversion, it is not possible to directly predict
the optimal value for a particular parameter; the problem is
said to be non-linear. Non-linear inverse problems are
solved using an iterative strategy that progressively alters
the model parameters, calculates the objective function and
then, if necessary, adjusts the parameters again until a
satisfactory match between the observed and calculated
responses is obtained. The inversion algorithm is described
as converging on a solution, i.e. a model that produces a
satisfactory match to the observed data.
Figure 2.48
illustrates how a gradient-based inversion
algorithm, a commonly used strategy, models a set of
gravity observations. The source is spherical, and the par-
ameters that can be varied are its depth, lateral position
and density contrast with its surrounds. A useful analogy is
to visualise the objective function as a terrain, with the
algorithm seeking to
'
degree of
t
'
local features in the func-
tion to seek the overall minima. A well-known global
search algorithm is the Monte Carlo method, which ran-
domly assigns values to model parameters, usually within
de
ned bounds. The match between the computed and
observed responses is then determined. If the match is
acceptable, according to some de
ned criteria, the model
is accepted as a possible solution and becomes one of a
family of solutions. The process is then repeated to
nd
more possible solutions. Monte Carlo methods require a
large number of tests, hundreds to millions depending on
the number and range of model parameters being varied,
and so are computationally demanding. Another disadvan-
tage is that it is still not possible to determine whether all
possible models have been identified, a particular problem
when very different solutions can exist. Global search
'
see through
'
until it reaches the
lowest point in the topography, i.e. the place where the
objective function and the residuals are minimised. This is
'
roll downhill
'
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