Geoscience Reference
In-Depth Information
the physical concepts of changes the direction the EM field propagates as a function of time into
the mathematical language by the use of derivatives and vectors. All of the fundamental equations
in Newtonian and Maxwellian physics describe the physical phenomenon occurring at a particu-
lar point in space, and at a particular instant in time. The fundamental question that needs to be
answered is: “What is the reaction to a propagating EM wave that is acting on a particular point at
a particular instant in time?” These changes in the EM fields are described by a partial differential
equation, which mathematically can be expressed in a general way as
f(x ,y,z,t)
, where
f
is the func-
tional expression,
x ,y,z
represent the spatial changes in Cartesian coordinates, and
t
is time. The dif-
ferential equation is needed because fields (force and electric) change with time, and the derivatives
are the mathematical way to express changes in time and space. It should be noted that geophysical
forward models for all geophysical measurements that follow the laws Newton and Maxwell are
related to, in the form of, or can be derived from the wave equation.
6.5.2.3
Inverse Modeling
Geophysical data are generally very simple measurements made at a single point below, on, or
above the surface of the earth. These relatively simple measurements are generally used to infer
some pretty complicated events, or physical property distributions in the subsurface. Inverse model-
ing involves the process of manipulating the parameters of a theoretical model until the values com-
puted from the mathematical model match the field measurements. The process of inverse modeling
for some geophysical methods (e.g., gravity, magnetic, resistivity, thermal) is often called curve
matching, because the process involves matching, or fitting, the curve computed from mathematical
modeling with the curve of values from field measurements. Figure 6.16 illustrates field data along
a profile and the computed response from a hypothetical model. The generalized procedure that
this used in interpretation (cut-and-try, analog curve match, or inversion) is shown in Figure 6.16c.
The parameters of the model are adjusted in an iterative manner until the computed model response
matches the field measurements.
Curve matching (or in the case of wave propagation, trace matching) can be achieved using
several different approaches, including cut-and-try, analog overlays and templates, and automated
inversion. The cut-and-try technique is a simple process that involves generating individual forward
models, comparing the model to the field data visually, and iteratively changing the model param-
eters until the values for the theoretical model are close to the field measurement values. The analog
overlay technique involves looking through a catalog of theoretical curves until the interpreter finds
a curve that is very close to the field measurement curve. The analog overlay method was the only
method available to interpreters prior to the widespread use of digital computers in the early 1960s.
These are approximation techniques because of the simplifying assumptions made in arriving at a
solution. Automated, or computer, inversion has become so standard in geophysics that it is nearly
a specialization in itself. Most inverse modeling has been developed over the years for potential
field and resistivity methods, and the following discussion reflects this fact. More recently, inverse
models have been developed for EM methods.
Least-squares inversion is a numerical way to find the physical properties that generate a forward
model response that most closely approximates the field measurements. Least-squares inversion is
an automated way to implement the curve-matching flowchart shown in Figure 6.16c. Least-squares
inversion can be applied to any data and model where changing the parameters in the mathematical
equation changes the result of the equation in a linear fashion. Therefore, least-squares inversion
can be applied to resistivity, gravity, and EM data by “linearizing” the model. An excellent explana-
tion and summary of inverse modeling is provided at the UBC Geophysical Inversion Facility Web
site (www.eos.ubc.ca/research/ubcgif/).
Inversion by the process of least-squares is a numerical problem of adjusting the physical prop-
erties of the model until the model curve matches (or comes close to matching) the curve traced
