## Non Uniqueness

One aspect of geophysical imaging that commonly leads to misconceptions is the concept of non-uniqueness. Non-uniqueness essentially means that in the absence of any other information, there are an infinite number of subsurface property distributions that honor (fit) a set of geophysical measurements. A common misconception is thus that any geophysical image is one of an infinite number of equally likely images. ** This is not correct**. The reason that this is not correct is that as part of the imaging process we **constrain** our model. **Applying constrants allows the selection of a realistic solution based on the available knowledge.** The process of constraining the solution is often referred to as regularization, and the constraints applied are often referred to as regularization constraints, model constraints, or solution constraints (as opposed to data constraints). For example, in an Occam's type inversion, the inversion is formulated to produce the most homogeneous image possible, while maintaining an adequate data fit given the level of noise in the data. This produces a smoothed or blurred representation of the true subsurface, and the only heterogeneity in the model is the heterogeneity that is resolved by the data. Occam's type inversions are often used in the absence of any information about the subsurface, other than what the data provide. When other information is available (e.g. water table location, locations and dimensions of known boundaries, maximum and minimum limits on conductivity, sharp conductivity contrasts etc.) it is often possible to incorporate that information in the form of model constraints in order to improve resolution.

The figure shown on the top right illustrates both the concept of non-uniqueness and model constraints. **Figure A** shows cross sections of a true subsurface conductivity distribution, which consists of two conductive blocks embedded into a homogeneous background. ERT data are produced using the three electrode lines (red dots) on the surface. **Figure B** shows the image produced by a standard Occam's type inversion, which produces a smoothed version of reality, but gives indication of the general location and dimension of the blocks. This image represents the information that the data alone can provide. In **Figure C**, an additional constraint was used stating that the conductivity should be greater than or equal to the true homogeneous background conductivity. This constraint alone significantly improves resolution. In **Figure D**, another constraint was added stating that if the conductivity gradient exceeds a particular value, then a sharp boundary should be produced where that gradient occurs. Finally, in **Figure E**, a 'binning' constraint was added that encourages the conductivity to fall into one of three values (the background value or the value of one of the two blocks). All of the solutions B through E fit the data equally well but provide different resolutions depending on the constraints, illustrating two things. First, the solutions illustrate the general non-uniqueness in the inverse problem. Second, the solutions show how this non uniqueness is addressed by applying constraints. In summary, incorporating information into the inversion through model constraints improves the degree to which the subsurface can be resolved. It should be obvious that choosing and implementing appropriate constraints is an important part of the job of a geophysicist.