Constraints

Prior information about the real medium can be introduced in the inversion in two ways: with the starting model and by imposing bounds on the variations of the parameters being estimated. In this particular algorithm, these two forms of prior information are essential for a reliable estimation of the vector $\mbox{\boldmath$m$}$.

If the starting model is ``far'' from the ``true'' answer the algorithm may not converge at all, or if it does, it may produce a wrong answer. This may happen with any nonlinear inversion. However, it can be shown that if the starting model is ``close'' to the true answer, a linearized solution of the nonlinear system of equations (5) converges quadratically (Gill et al., 1981). Therefore, any information introduced in the initial model (especially about dips) will help to algorithm to converge to a reliable estimate of the real medium.

Since the interfaces are allowed to change position from one iteration to the next, the algorithm has to make sure that each new model does not contain crossing interfaces in the area of interest. If two interfaces cross after adding $r \Delta \mbox{\boldmath$m$}$ to the given model, the step length r is reduced until those interfaces do not cross after the correction. Once the model has been updated and there are no crossing interfaces, the algorithm checks whether thin layers have been created or not and if so, those layers are eliminated from the inversion, reducing by a multiple of 5 the number of model parameters (five parameters per each eliminated layer). This is done by comparing the thickness of all layers with a predetermined minimum layer thickness, as long as the corresponding boundaries are parallel (also within a predetermined range of angles). If the predetermined minimum thickness is too small, the solution may contain layers with artificially high (or low) velocities and if the minimum is too big, spatial resolution in the model may be lost.

Whenever information is available about the position and dip of certain layers of the medium (from well logs for example) it should also be used to constrain the corresponding interfaces in the model.