# 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 \(\mathbf{m}\) is bounded as

where *l* is the lower bound and *u* is the upper bound.
The transformed parameter \(\mathbf{x}\) is unbounded:

With the transformed model vector \(\mathbf{x}\), the model update is accomplished with a modified version of (3):

where

(here the model prejudice term has been omitted for brevity). Thus the inversion solves for the unbounded parameter \(\mathbf{x}\) which can have values anywhere from negative to positive infinity. However the forward operations are carried out on the bounded parameter via \({F}(\mathbf{m(x)})\).

MARE2DEM (like most other EM inversions) already uses such a transform approach, in this case a one-sided bound by inverting for \(\log_{10} \rho\) to avoid physically meaningless negative resistivity (and conductivity). MARE2DEM also offers two additional types of bounding transforms that can be used constrain the range of resistivity for any given free parameter: the exponential and bandpass transforms. These are shown in the figure below and the text that follows explains the math behind these transforms.

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 \(\mu\) values, the
Occam model update can produce rough models that have unrealistically
high and low resistivity that usually produce poor misfit. Bounding all
parameters to a plausible range (such as 0.1 to 100,000 ohm-m for marine
models) alleviates this situation. Another consequence is that
unrealistically conductive regions can cause the adaptive mesh
refinement scheme in the forward code to spend excessive effort refining
the mesh to produce accurate responses in these regions; bounding all
the model parameters to a plausible range avoids this computational
time-wasting situation.

## Exponential Transform

The exponential transform is

For large positive \(x, m(x)\) asymptotes to \(u\), while for large negative \(x, m(x)\) asymptotes to \(l\). The sensitivity transform is found by taking the derivative of this expression, yielding:

The transformed variable \(x(m)\) can be found from

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
\(\mathbf{R}\) in eq. (5) operates on the transformed
parameter and not the original parameter; this could have unintended
consequences, for example, where the transformation significantly
increases or decreases the spatial gradient in the transformed model,
effectively resulting in a non-linear roughness operator that
under-penalizes regions where the transform reduces the model gradient
and over-penalizes regions where the gradient is increased. This lead
to the development of the bandpass transform discussed below.

## 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. \(c = 15/(u-l)\) has been found to work well in
practice. Integrating the equation above yields the expression for the bound model parameters:

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.