A Novel Approach To The Resolution Analysis Of Geophysical Inversion

A novel approach to the resolution analysis of geophysical inversion
Michael S. Zhdanov∗ and Ekaterina Tolstaya, Department of Geology and Geophysics, The University of
Utah
Summary
The analysis of sensitivity and resolution of geophysical
methods is one of the most important problems of
practical geophysics. In this paper we develop a novel
technique for resolution studies, based on theevaluation
of the maximum possible errors in the solution of the inverse problem for the given level of errors in the observed
data. We introduce a new characteristic of geophysical
inversion, a resolution density, which is determined as
the inverse of the upper bounds of the model parameter
errors. The method is demonstrated by the resolution
study of three-dimensional (3-D) electromagnetic(EM)
inversion. The case history includes interpretation of the
helicopter-borne EM data collected by INCO Exploration
in the Voisey’s Bay area of Canada. We believe that this
new technique provides a useful tool for the analysis of
the accuracy and quality of the geophysical inversion.

Introduction
One of the most important problems of practical geophysics is the analysis of sensitivityand the resolution of
geophysical methods. This problem arises in the initial
stage of geophysical investigation when we design the
geophysical survey. The same problem appears at the
final stage when we examine the results of interpretation
of the observed geophysical data. Actually, the question
about sensitivity and resolution of the given geophysical
method is usually the first one askedby geologists
working with geophysical data. In this paper we introduce a new characteristic of the geophysical inversion,
a resolution density, and develop a numerical method
of its analysis for linear inverse problems. Note that
this method can be extended for the nonlinear inverse
problem as well.

Here m is the vector-column of the model parameters of
order Nm , d is the vector-columnof the observed geophysical data of order Nd , and the matrix A is the Nd × Nm
matrix of the linear forward modeling operator. The regularized solution of this inverse problem is given by the
following formulae (Zhdanov, 2002):
2
2
2
2
mα = (A∗ Wd A + αWm )−1 A∗ Wd d + αWm mapr .
(2)
where Wd and Wm are the data and model weighting
matrices, and α is a regularization parameter. Let usapply the variational operator δ to both sides of (2):
2
2
2
δ mα = (A∗ Wd A + αWm )−1 A∗ Wd δ d.

We shall call the matrix
2
2
2
Rα = (A∗ Wd A + αWm )−1 A∗ Wd ,

(4)

an inversion resolution matrix (IRM). The spatial variation of the resolution of the geophysical inversion can be
found by individually analyzing the columns of the IRM,
Rα . Indeed, formula (3) in scalar notation canbe written
as:
Nd

δmi =

Rαij δdj ,
j =1

where Rαij are the scalar components of the Rα . From
the Cauchy inequality, we have:
Nd
2

|δmi | ≤

Nd

|Rαij |
j =1

2

|δdj |2 = ε2 /R2 ,
i

−1

Nd

R2
i

=

(5)

j =1

where

Resolution density distribution

d

2

|Rαij |

2

,

(6)

j =1

The existing techniques available for the appraisal ofgeophysical inverse images are based, primarily, on
the calculation of the model resolution and the model
covariance matrices (Menke, 1989; Tarantola, 1987;
Alumbaugh and Newman, 2000). In many practical
applications, it may be useful, however, to also estimate
the maximum possible errors in the solution of the
inverse problem for the given errors in the observed data.
These upper boundsof the model errors determine the
actual resolution of geophysical method. In this paper,
we introduce a numerical method of solving this problem.
Let us consider a linear matrix equation:
d = Am.

(3)

(1)

and
ε = δd / d ,
is a norm of the relative errors in the data. Note that the
Nd
term j =1 |Rαij |2 represents a sum of the squares of the
scalar components located in the i-th...