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Bounding Inversion Parameters with Non-linear Transforms
Note
Global bounds on all model parameters can be set in the Mamba2D.m user interface that makes the Resistivity File (.resistivity), or you can hand adjust them in the file by modifying the line:
Global Bounds: 1.0000E-01, 1.0000E+05
where the values given are the lower and upper bounds (given in linear ohm-m). You can also specify which transform method to use in the Resistivity File (.resistivity) by including the line:
Bounds Transform: bandpass
options: exponential or bandpass
Each individual free parameter given in the regions list section at the bottom of the .resistivity file can have its own specific lower and upper bounds specified (and this will override the global bounds set above). See Resistivity File (.resistivity).
The text below gives background theory on how the bounds are implemented
in MARE2DEM. The model update equation (3) does not
place any constraints on the range of values that a parameter can take,
yet often there are geological reasons or ancillary data sets that
suggest the conductivity will be within a certain range of values. When
such inequality constraints are desired, they can be implemented simply
by recasting the inverse problem using a non-linear transformation of
the model parameters so that the objective function and optimization
algorithm remain essentially the same as the unconstrained problem .
The model parameter
where l is the lower bound and u is the upper bound.
The transformed parameter
With the transformed model vector
where
(here the model prejudice term has been omitted for brevity). Thus the
inversion solves for the unbounded parameter
MARE2DEM (like most other EM inversions) already uses such a transform
approach, in this case a one-sided bound by inverting for

Fig. 44 Non-linear transforms used to bound model parameters during
inversion. The bound model parameter function
Both the exponential and band-pass transforms have been implemented in
MARE2DEM so that each model parameter can have its own unique bounds
specified. For some data sets the user will have a priori knowledge
that can guide the use of a narrow range of parameter bounds in certain
localized regions, for example where nearby well logs provided
independent constraints on conductivity. Narrow bounds could also be
prescribed to test hypotheses about the range of permissible
resistivity values that fit a given data set. Yet in most cases the
inversion will be run without any bounds. However, experience has shown
there to be a benefit from applying global bounds on all model
parameters so that extreme values are excluded from the inversion. In
particular, if the line search jumps to very low
Exponential Transform
The exponential transform is
For large positive
The transformed variable
As shown in the figure, the exponential transform results in x being
quite different than the original unbound parameter. This difference
can also be seen in the sensitivity scaling curves. Functionally, this
difference does not affect the ability of the transform to bound the
model parameters, and indeed the exponential transform has been shown to
be useful in practice. However, note that the roughness operator
Bandpass Transform
Consider a flat sensitivity scaling with unit amplitude in the pass-band between l and u and which rapidly decays at values beyond the bounds. The flatness in the pass-band would allow the transformed parameters to be nearly identical to the original parameters within the range of the bounds. Such a function shape can be created by using a bandpass filter response equation for the transform’s sensitivity scaling:
where c is a constant that controls the decay of the scaling past the
bounds. By setting c to be a function of the extent of the bounds, the
shape of the transform shape can be made to be independent of the
specific bounds.
and solving for x yields
The figure about shows that between the bounds, the transformed parameters are identical to the original parameters, as desired, while the sensitivity scaling is flat between the pass-band with steep drop-offs beyond the bounds.