The non unique solution (and instability) of geophysical inversion may have different reasons. Most of cases it is related to the mathematically ill-posed problem definition which characterize almost all geophysical inversion problems, but eg. the discretization itself may also resulted non uniqueness. To avoid the inversion instability, some kind of regularization (Tikhonov type) is needed which ensure smoother and stable results. In most cases additional model dependent function is added to the functional (to be minimized). In this additional function may contain the quadratic form of model parameters (or derivative of parameters). If a priori information are available it may also built in the regularization function. Sometimes the reason of non uniqueness is the parameter correlation when the available a priori information may reduce the uncertainty.

Geophysical data tend to be inherently non-unique. Things like the internal density field can vary continuously, and you are always restricted to a finite amount of data. It is also common to have data types that are simply not sensitive to certain variations of internal quantities. A very simple example is that the mass and volume of a body tells you the mean internal density, potentially very accurately, but nothing about where density variations might occur. Similarly, external gravity data are inherently non unique, providing information about the horizontal structure of internal density variations, but not the depth of those variations.

The Backus–Gilbert method was invented to deal with such non-unique geophysical inversion problems. Backus–Gilbert seeks the most stable solutions under sampling noise, and, unlike the Tikhonov method, does not impose a smoothness constraint on the solution. (I believe that Backus & Gilbert were interested in the sensitivity of seismic and free oscillation data to internal discontinuities such as the core-mantle boundary, for which a smoothness constraint is not that appropriate.)

The non-uniqueness in the inverse problem can not be eliminated but can be reduced .

To reduce the non-uniqueness is necessary:

a) In the space of the data ( measurements, observables )

- increase the completeness of the data space (spatial and time sampling of observables)

- decrease the measurement error of the data space

b) In the parameters space (model parameters)

- set the parameters as appropriate (number of model parameters and their spatial spacing and physical attributes)

- starting with a correct assumptions on the model (starting models)

- have a priori information

In general we should use a theory that connects in a consistent way the data space to the space of model parameters (direct problem and inverse problem) with the observed data ( data-driven inversion , direct inversion methods or a priori information).

However, we must not confuse the reduction of non- uniqueness with the accuracy of the response of a model (precision). In the space of models exist an infinite number (or a finite sub set) of precise models whose responses fit a criterion of residual reduction (a function of calculated and observed data) or other statistical indicators of precision. This aspect could be improved (stability of solution and its precision) by the appropriate choice of the regularization approach but not reduce the non-uniqueness.

On the other hand , in the parameters space there are few models compatible with the data and with the stringent assumptions underlying the model (model appraisal).

For example, in the seismic inversion from first arrivals (seismic refraction tomography), we can obtain from travel time trend if we are in the presence of a model velocity characterized by genuine discontinuities, or high gradient discontinuities or pure gradient (which correspond respectively to critical refracted wave, or continuous refracted wave or diving wave). In this case, we must adopt an approach to the direct-inverse (starting model) and direct problem (model response) consistent with the physical processes and the hypothesis deductible from the data.

i.e. if travel times indicate critical refracted arrivals (stright lines) we should start with a model including genuine interfaces and we should use for the response of the model (calculated data) a modelling including critical raypaths.

Only the evaluation of hypotheses on the model reconstruction which is compatibile with the data (data-driven model appraisal) in general, and really, reduces the non-uniqueness.

Potential geohysical methods interpretations are solutions to inverse problems theoretically admitting an infinity of mathematical solutions. However the physical solution ( which is the real situation) is unique. There are techniques to approach the reasonable plausible interpretation often requiring more data, calibrations, use of complimentary integrated geophysical methods. Probably one of the best modern free references can be found in my teacher’s late Prof. Albert Tarantola (RIP). Following web page.

http://www.ipgp.fr/~tarantola/