Geophysical methods are based on the study of different physical fields being propagated through the earth’s interior. The most important geophysical fields are gravity, magnetic, electromagnetic, and seismic wave fields. The observed values of these fields depend, first, on the physical properties of rocks. The conventional approach to geophysical data analysis consists of constructing different geological models and comparing the theoretical geophysical data computed for these models with the observed data.
Forward problem
Numerical modeling of geophysical data for given model parameters is usually called a forward problem. The forward problem solution makes it possible to predict geophysical data for specific geological structures.
Inverse problem
Forward problem :
estimates of model parameters → quantitative model → predictions of data
Inverse problem :
observations of data → quantitative model → estimates of model parameters
Formulation of Forward and Inverse Problems
The definition of general forward and inverse problems can be described schematically by:
Forward problem:
model {model parameters m} → data d.
In studying the geophysical methods, we should also take into account that the field can be
generated by some source. So we have
generated by some source. So we have
model {model parameters m, sources “s”} → data d:
d = As(m),
where As is the forward problem operator depending on a source “s.”
d = As(m),
where As is the forward problem operator depending on a source “s.”
Inverse problem:
data d → model {model parameters m}
{data d, sources “s”} → model {model parameters m}:
m = As^-1 (d)
or {data d} → model and sources {model parameters m, sources “s”}:
(m, s) = A^-1(d),
where As^-1 and A^-1 are inverse problem operators.
(m, s) = A^-1(d),
where As^-1 and A^-1 are inverse problem operators.
Key terms in Inversion
Data and Model parameters
The observations of the world will consist of a tabulation of
measurements, or data. The questions we want to answer will be stated in terms of the numerical values (and statistics) of specific (but not necessarily directly measurable) properties of
the world. These properties will be called model parameters.
Sparse and Dense matrix
A matrix is said to be a sparse matrix or sparse array. if most of the elements of the matrix are zero. By contrast, if most of the elements are nonzero, then the matrix is considered dense. The number of zero-valued elements divided by the total number of elements is called the sparsity of the matrix which is also equal to 1 minus the density of the matrix. So a matrix will be sparse when its sparsity is greater than 0.5.