570 17 Inversion of Potential Field Data
the initial model with available known geology of the area. Difference in the
choice of forward model will generate different answers. It is interesting to
note that the interpreter often over parameterises the model to make i t an
underdetermined problem and get interpretable encouraging results. Even if
the initial forwar d mo del is totally a wrong choice and do es not have any
closeness with the loca l geology, on e may get both convergence and stability
of an inverse problem and get an answer. This is a note of caution. The
convergence i.e., the reductions of the discrepancy between the field data
and the model data in successive iteration does not guarantee any acceptable
result. Interpreter should have some knowledge about the local geology of the
area to get a meaningful results.
A class of inverse problems which use differential op erators are generally
made linear by truncating higher order terms of the Taylor’s series expa nsion
of non li n ear problems. This approximation may b e too severe for a certain
class of strongly nonlinear prob lems. This linearisation, coupled with limited
resolving power of the potential problems, inadequate and inaccurate data
generate non uniqueness. Interpreter dependent factors are the (i) nature of
smoothness used in data processing (ii) choice of a particular approach for
inversion (iii) differe nt way of discretisation to generate data. (iv) use of dif-
ferent softwares.
Even with all these problems of non uniqueness if we have good quality
adequate data, one can get a cluster of models in the parameter space or M-
space near the actual answer. If several earth models obtained using different
approaches of inversion have some common feature then earth must have that
property (Bachus a nd Gilbert 1967). Thus inverse theory survives with con-
siderable success while imaging extremely complicated, inhomogeneous an d
anisotropic earth even in this high level of non uniqueness.
17.2 Wellposed and Illposed Problems
The concept of well posed and ill posed problems were introduced by
J. Hadamard (1902, 1932) in an attempt to classify the different types of
differential equations along with their boundary conditions.
A solution is said to be well posed for solution of the equation
Az =u
n × mm× 1n× 1
(17.1)
where the initial choice of model parameter z is in the model space M and u is
in the data space D. These spaces are M and N dimensional abstract spaces.
These spaces are metric spaces, Euclidean space, pre-Hilbert space, Hilbert
space, normed inner product space etc(Sect. 17.4).
A problem of determining a solution z in the space M from the initial data
u in the D space is well posed o n these spaces (D and M) if the following
condition are satisfied: